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\hypertarget{classLocalCommunicator}{}\section{Local\+Communicator Class Reference}
\label{classLocalCommunicator}\index{Local\+Communicator@{Local\+Communicator}}
{\ttfamily \#include $<$local\+\_\+communicator.\+hpp$>$}
Inheritance diagram for Local\+Communicator\+:\begin{figure}[H]
\begin{center}
\leavevmode
\includegraphics[height=2.000000cm]{classLocalCommunicator}
\end{center}
\end{figure}
\subsection*{Public Member Functions}
\begin{DoxyCompactItemize}
\item
\hyperlink{classLocalCommunicator_a477260797392154cc43815852ced969c}{Local\+Communicator} (\hyperlink{structcubiumServerSocket__t}{cubium\+Server\+Socket\+\_\+t} $\ast$sock, std\+::shared\+\_\+ptr$<$ \hyperlink{classRoutingTable}{Routing\+Table}$<$ \hyperlink{structcubiumServerSocket__t}{cubium\+Server\+Socket\+\_\+t} $>$$>$ \hyperlink{classLocalCommunicator_ac28392070111396dabbadbbcb81a7ef7}{routing\+Table}, \hyperlink{structLogicalAddress}{Logical\+Address} la)
\item
\hyperlink{classLocalCommunicator_ac5845882ec445ca5f3bedbaf666073c0}{Local\+Communicator} (\hyperlink{structcubiumClientSocket__t}{cubium\+Client\+Socket\+\_\+t} $\ast$sock, std\+::shared\+\_\+ptr$<$ \hyperlink{classRoutingTable}{Routing\+Table}$<$ \hyperlink{structcubiumServerSocket__t}{cubium\+Server\+Socket\+\_\+t} $>$$>$ \hyperlink{classLocalCommunicator_ac28392070111396dabbadbbcb81a7ef7}{routing\+Table}, \hyperlink{structLogicalAddress}{Logical\+Address} la)
\item
\hyperlink{classLocalCommunicator_a22dbee26502e3ac2bdf46b9f094d881d}{Local\+Communicator} (\hyperlink{structcubiumServerSocket__t}{cubium\+Server\+Socket\+\_\+t} $\ast$sock, \hyperlink{structLogicalAddress}{Logical\+Address} la)
\item
virtual bool \hyperlink{classLocalCommunicator_afb8fc8a82069dac7bf08f736fa76c4c8}{send\+Msg} (\hyperlink{structSpaMessage}{Spa\+Message} $\ast$msg, ssize\+\_\+t len)
\item
virtual void \hyperlink{classLocalCommunicator_a0d6816af83fd55a79990f0b8caa4e164}{handle\+Failure} (std\+::string)
\item
virtual bool \hyperlink{classLocalCommunicator_a6175c2c35727bb081c5f518e1750dd86}{server\+Send} (\hyperlink{structSpaMessage}{Spa\+Message} $\ast$message, ssize\+\_\+t)
\item
virtual bool \hyperlink{classLocalCommunicator_afca7f865eda5dca2fc4c3bf5ca697ab8}{client\+Send} (\hyperlink{structSpaMessage}{Spa\+Message} $\ast$message, ssize\+\_\+t)
\item
void \hyperlink{classLocalCommunicator_a4dc9ea76a7d6d1d363ec8b69c1d5d586}{client\+Connect} (\hyperlink{structSpaMessage}{Spa\+Message} $\ast$, size\+\_\+t, std\+::function$<$ void(\hyperlink{structcubiumClientSocket__t}{cubium\+Client\+Socket\+\_\+t} $\ast$)$>$)
\item
void \hyperlink{classLocalCommunicator_a8ed10b0a1f9be12597a2e095b7b8fb3e}{init\+Sub\+Dialogue} (\hyperlink{structSpaMessage}{Spa\+Message} $\ast$message, size\+\_\+t len, std\+::function$<$ void(\hyperlink{structcubiumClientSocket__t}{cubium\+Client\+Socket\+\_\+t} $\ast$)$>$ callback)
\item
void \hyperlink{classLocalCommunicator_a03deedab1d3f79fe328a21aa2d5c6bfb}{client\+Listen} (std\+::function$<$ void(\hyperlink{structcubiumClientSocket__t}{cubium\+Client\+Socket\+\_\+t} $\ast$)$>$)
\item
virtual void \hyperlink{classLocalCommunicator_a320a09f88a1eb840e517753b603cd08b}{listen} (std\+::function$<$ void(\hyperlink{structcubiumServerSocket__t}{cubium\+Server\+Socket\+\_\+t} $\ast$)$>$)
\item
virtual void \hyperlink{classLocalCommunicator_a33c7148ee35ec38773b7b4c8f32c643c}{listen} (std\+::function$<$ void(\hyperlink{structcubiumClientSocket__t}{cubium\+Client\+Socket\+\_\+t} $\ast$)$>$)
\item
void \hyperlink{classLocalCommunicator_a3321b932c8c555587b8b3200bc4dfcdb}{set\+Server\+Sock} (\hyperlink{structcubiumServerSocket__t}{cubium\+Server\+Socket\+\_\+t} $\ast$s)
\item
void \hyperlink{classLocalCommunicator_a96569a4dc0b07439ab880385428a53f7}{print\+Table} ()
\end{DoxyCompactItemize}
\subsection*{Protected Attributes}
\begin{DoxyCompactItemize}
\item
\hyperlink{structcubiumServerSocket__t}{cubium\+Server\+Socket\+\_\+t} $\ast$ \hyperlink{classLocalCommunicator_a4c0d8806e53030c1af6124cd55a34335}{server\+Sock}
\item
\hyperlink{structcubiumClientSocket__t}{cubium\+Client\+Socket\+\_\+t} $\ast$ \hyperlink{classLocalCommunicator_af52bd4819c3dc433bd27d3dbefbaac70}{client\+Sock}
\item
std\+::shared\+\_\+ptr$<$ \hyperlink{classRoutingTable}{Routing\+Table}$<$ \hyperlink{structcubiumServerSocket__t}{cubium\+Server\+Socket\+\_\+t} $>$ $>$ \hyperlink{classLocalCommunicator_ac28392070111396dabbadbbcb81a7ef7}{routing\+Table}
\end{DoxyCompactItemize}
\subsection*{Additional Inherited Members}
\subsection{Constructor \& Destructor Documentation}
\mbox{\Hypertarget{classLocalCommunicator_a477260797392154cc43815852ced969c}\label{classLocalCommunicator_a477260797392154cc43815852ced969c}}
\index{Local\+Communicator@{Local\+Communicator}!Local\+Communicator@{Local\+Communicator}}
\index{Local\+Communicator@{Local\+Communicator}!Local\+Communicator@{Local\+Communicator}}
\subsubsection{\texorpdfstring{Local\+Communicator()}{LocalCommunicator()}\hspace{0.1cm}{\footnotesize\ttfamily [1/3]}}
{\footnotesize\ttfamily Local\+Communicator\+::\+Local\+Communicator (\begin{DoxyParamCaption}\item[{\hyperlink{structcubiumServerSocket__t}{cubium\+Server\+Socket\+\_\+t} $\ast$}]{sock, }\item[{std\+::shared\+\_\+ptr$<$ \hyperlink{classRoutingTable}{Routing\+Table}$<$ \hyperlink{structcubiumServerSocket__t}{cubium\+Server\+Socket\+\_\+t} $>$$>$}]{routing\+Table, }\item[{\hyperlink{structLogicalAddress}{Logical\+Address}}]{la }\end{DoxyParamCaption})\hspace{0.3cm}{\ttfamily [inline]}}
\mbox{\Hypertarget{classLocalCommunicator_ac5845882ec445ca5f3bedbaf666073c0}\label{classLocalCommunicator_ac5845882ec445ca5f3bedbaf666073c0}}
\index{Local\+Communicator@{Local\+Communicator}!Local\+Communicator@{Local\+Communicator}}
\index{Local\+Communicator@{Local\+Communicator}!Local\+Communicator@{Local\+Communicator}}
\subsubsection{\texorpdfstring{Local\+Communicator()}{LocalCommunicator()}\hspace{0.1cm}{\footnotesize\ttfamily [2/3]}}
{\footnotesize\ttfamily Local\+Communicator\+::\+Local\+Communicator (\begin{DoxyParamCaption}\item[{\hyperlink{structcubiumClientSocket__t}{cubium\+Client\+Socket\+\_\+t} $\ast$}]{sock, }\item[{std\+::shared\+\_\+ptr$<$ \hyperlink{classRoutingTable}{Routing\+Table}$<$ \hyperlink{structcubiumServerSocket__t}{cubium\+Server\+Socket\+\_\+t} $>$$>$}]{routing\+Table, }\item[{\hyperlink{structLogicalAddress}{Logical\+Address}}]{la }\end{DoxyParamCaption})\hspace{0.3cm}{\ttfamily [inline]}}
\mbox{\Hypertarget{classLocalCommunicator_a22dbee26502e3ac2bdf46b9f094d881d}\label{classLocalCommunicator_a22dbee26502e3ac2bdf46b9f094d881d}}
\index{Local\+Communicator@{Local\+Communicator}!Local\+Communicator@{Local\+Communicator}}
\index{Local\+Communicator@{Local\+Communicator}!Local\+Communicator@{Local\+Communicator}}
\subsubsection{\texorpdfstring{Local\+Communicator()}{LocalCommunicator()}\hspace{0.1cm}{\footnotesize\ttfamily [3/3]}}
{\footnotesize\ttfamily Local\+Communicator\+::\+Local\+Communicator (\begin{DoxyParamCaption}\item[{\hyperlink{structcubiumServerSocket__t}{cubium\+Server\+Socket\+\_\+t} $\ast$}]{sock, }\item[{\hyperlink{structLogicalAddress}{Logical\+Address}}]{la }\end{DoxyParamCaption})\hspace{0.3cm}{\ttfamily [inline]}}
\subsection{Member Function Documentation}
\mbox{\Hypertarget{classLocalCommunicator_a4dc9ea76a7d6d1d363ec8b69c1d5d586}\label{classLocalCommunicator_a4dc9ea76a7d6d1d363ec8b69c1d5d586}}
\index{Local\+Communicator@{Local\+Communicator}!client\+Connect@{client\+Connect}}
\index{client\+Connect@{client\+Connect}!Local\+Communicator@{Local\+Communicator}}
\subsubsection{\texorpdfstring{client\+Connect()}{clientConnect()}}
{\footnotesize\ttfamily void Local\+Communicator\+::client\+Connect (\begin{DoxyParamCaption}\item[{\hyperlink{structSpaMessage}{Spa\+Message} $\ast$}]{message, }\item[{size\+\_\+t}]{len, }\item[{std\+::function$<$ void(\hyperlink{structcubiumClientSocket__t}{cubium\+Client\+Socket\+\_\+t} $\ast$)$>$}]{callback }\end{DoxyParamCaption})}
\mbox{\Hypertarget{classLocalCommunicator_a03deedab1d3f79fe328a21aa2d5c6bfb}\label{classLocalCommunicator_a03deedab1d3f79fe328a21aa2d5c6bfb}}
\index{Local\+Communicator@{Local\+Communicator}!client\+Listen@{client\+Listen}}
\index{client\+Listen@{client\+Listen}!Local\+Communicator@{Local\+Communicator}}
\subsubsection{\texorpdfstring{client\+Listen()}{clientListen()}}
{\footnotesize\ttfamily void Local\+Communicator\+::client\+Listen (\begin{DoxyParamCaption}\item[{std\+::function$<$ void(\hyperlink{structcubiumClientSocket__t}{cubium\+Client\+Socket\+\_\+t} $\ast$)$>$}]{func }\end{DoxyParamCaption})}
\mbox{\Hypertarget{classLocalCommunicator_afca7f865eda5dca2fc4c3bf5ca697ab8}\label{classLocalCommunicator_afca7f865eda5dca2fc4c3bf5ca697ab8}}
\index{Local\+Communicator@{Local\+Communicator}!client\+Send@{client\+Send}}
\index{client\+Send@{client\+Send}!Local\+Communicator@{Local\+Communicator}}
\subsubsection{\texorpdfstring{client\+Send()}{clientSend()}}
{\footnotesize\ttfamily bool Local\+Communicator\+::client\+Send (\begin{DoxyParamCaption}\item[{\hyperlink{structSpaMessage}{Spa\+Message} $\ast$}]{message, }\item[{ssize\+\_\+t}]{len }\end{DoxyParamCaption})\hspace{0.3cm}{\ttfamily [virtual]}}
\mbox{\Hypertarget{classLocalCommunicator_a0d6816af83fd55a79990f0b8caa4e164}\label{classLocalCommunicator_a0d6816af83fd55a79990f0b8caa4e164}}
\index{Local\+Communicator@{Local\+Communicator}!handle\+Failure@{handle\+Failure}}
\index{handle\+Failure@{handle\+Failure}!Local\+Communicator@{Local\+Communicator}}
\subsubsection{\texorpdfstring{handle\+Failure()}{handleFailure()}}
{\footnotesize\ttfamily void Local\+Communicator\+::handle\+Failure (\begin{DoxyParamCaption}\item[{std\+::string}]{message }\end{DoxyParamCaption})\hspace{0.3cm}{\ttfamily [virtual]}}
\mbox{\Hypertarget{classLocalCommunicator_a8ed10b0a1f9be12597a2e095b7b8fb3e}\label{classLocalCommunicator_a8ed10b0a1f9be12597a2e095b7b8fb3e}}
\index{Local\+Communicator@{Local\+Communicator}!init\+Sub\+Dialogue@{init\+Sub\+Dialogue}}
\index{init\+Sub\+Dialogue@{init\+Sub\+Dialogue}!Local\+Communicator@{Local\+Communicator}}
\subsubsection{\texorpdfstring{init\+Sub\+Dialogue()}{initSubDialogue()}}
{\footnotesize\ttfamily void Local\+Communicator\+::init\+Sub\+Dialogue (\begin{DoxyParamCaption}\item[{\hyperlink{structSpaMessage}{Spa\+Message} $\ast$}]{message, }\item[{size\+\_\+t}]{len, }\item[{std\+::function$<$ void(\hyperlink{structcubiumClientSocket__t}{cubium\+Client\+Socket\+\_\+t} $\ast$)$>$}]{callback }\end{DoxyParamCaption})}
\mbox{\Hypertarget{classLocalCommunicator_a320a09f88a1eb840e517753b603cd08b}\label{classLocalCommunicator_a320a09f88a1eb840e517753b603cd08b}}
\index{Local\+Communicator@{Local\+Communicator}!listen@{listen}}
\index{listen@{listen}!Local\+Communicator@{Local\+Communicator}}
\subsubsection{\texorpdfstring{listen()}{listen()}\hspace{0.1cm}{\footnotesize\ttfamily [1/2]}}
{\footnotesize\ttfamily void Local\+Communicator\+::listen (\begin{DoxyParamCaption}\item[{std\+::function$<$ void(\hyperlink{structcubiumServerSocket__t}{cubium\+Server\+Socket\+\_\+t} $\ast$)$>$}]{message\+Handler }\end{DoxyParamCaption})\hspace{0.3cm}{\ttfamily [virtual]}}
Reimplemented from \hyperlink{classPhysicalCommunicator_a4886f4453c1f830ceaef56bc8602423f}{Physical\+Communicator}.
\mbox{\Hypertarget{classLocalCommunicator_a33c7148ee35ec38773b7b4c8f32c643c}\label{classLocalCommunicator_a33c7148ee35ec38773b7b4c8f32c643c}}
\index{Local\+Communicator@{Local\+Communicator}!listen@{listen}}
\index{listen@{listen}!Local\+Communicator@{Local\+Communicator}}
\subsubsection{\texorpdfstring{listen()}{listen()}\hspace{0.1cm}{\footnotesize\ttfamily [2/2]}}
{\footnotesize\ttfamily void Local\+Communicator\+::listen (\begin{DoxyParamCaption}\item[{std\+::function$<$ void(\hyperlink{structcubiumClientSocket__t}{cubium\+Client\+Socket\+\_\+t} $\ast$)$>$}]{message\+Handler }\end{DoxyParamCaption})\hspace{0.3cm}{\ttfamily [virtual]}}
\mbox{\Hypertarget{classLocalCommunicator_a96569a4dc0b07439ab880385428a53f7}\label{classLocalCommunicator_a96569a4dc0b07439ab880385428a53f7}}
\index{Local\+Communicator@{Local\+Communicator}!print\+Table@{print\+Table}}
\index{print\+Table@{print\+Table}!Local\+Communicator@{Local\+Communicator}}
\subsubsection{\texorpdfstring{print\+Table()}{printTable()}}
{\footnotesize\ttfamily void Local\+Communicator\+::print\+Table (\begin{DoxyParamCaption}{ }\end{DoxyParamCaption})\hspace{0.3cm}{\ttfamily [inline]}}
\mbox{\Hypertarget{classLocalCommunicator_afb8fc8a82069dac7bf08f736fa76c4c8}\label{classLocalCommunicator_afb8fc8a82069dac7bf08f736fa76c4c8}}
\index{Local\+Communicator@{Local\+Communicator}!send\+Msg@{send\+Msg}}
\index{send\+Msg@{send\+Msg}!Local\+Communicator@{Local\+Communicator}}
\subsubsection{\texorpdfstring{send\+Msg()}{sendMsg()}}
{\footnotesize\ttfamily bool Local\+Communicator\+::send\+Msg (\begin{DoxyParamCaption}\item[{\hyperlink{structSpaMessage}{Spa\+Message} $\ast$}]{msg, }\item[{ssize\+\_\+t}]{len }\end{DoxyParamCaption})\hspace{0.3cm}{\ttfamily [virtual]}}
Reimplemented from \hyperlink{classPhysicalCommunicator_a9fc5595b693f9908a20d0e64a6579bb5}{Physical\+Communicator}.
\mbox{\Hypertarget{classLocalCommunicator_a6175c2c35727bb081c5f518e1750dd86}\label{classLocalCommunicator_a6175c2c35727bb081c5f518e1750dd86}}
\index{Local\+Communicator@{Local\+Communicator}!server\+Send@{server\+Send}}
\index{server\+Send@{server\+Send}!Local\+Communicator@{Local\+Communicator}}
\subsubsection{\texorpdfstring{server\+Send()}{serverSend()}}
{\footnotesize\ttfamily bool Local\+Communicator\+::server\+Send (\begin{DoxyParamCaption}\item[{\hyperlink{structSpaMessage}{Spa\+Message} $\ast$}]{message, }\item[{ssize\+\_\+t}]{len }\end{DoxyParamCaption})\hspace{0.3cm}{\ttfamily [virtual]}}
\mbox{\Hypertarget{classLocalCommunicator_a3321b932c8c555587b8b3200bc4dfcdb}\label{classLocalCommunicator_a3321b932c8c555587b8b3200bc4dfcdb}}
\index{Local\+Communicator@{Local\+Communicator}!set\+Server\+Sock@{set\+Server\+Sock}}
\index{set\+Server\+Sock@{set\+Server\+Sock}!Local\+Communicator@{Local\+Communicator}}
\subsubsection{\texorpdfstring{set\+Server\+Sock()}{setServerSock()}}
{\footnotesize\ttfamily void Local\+Communicator\+::set\+Server\+Sock (\begin{DoxyParamCaption}\item[{\hyperlink{structcubiumServerSocket__t}{cubium\+Server\+Socket\+\_\+t} $\ast$}]{s }\end{DoxyParamCaption})\hspace{0.3cm}{\ttfamily [inline]}}
\subsection{Member Data Documentation}
\mbox{\Hypertarget{classLocalCommunicator_af52bd4819c3dc433bd27d3dbefbaac70}\label{classLocalCommunicator_af52bd4819c3dc433bd27d3dbefbaac70}}
\index{Local\+Communicator@{Local\+Communicator}!client\+Sock@{client\+Sock}}
\index{client\+Sock@{client\+Sock}!Local\+Communicator@{Local\+Communicator}}
\subsubsection{\texorpdfstring{client\+Sock}{clientSock}}
{\footnotesize\ttfamily \hyperlink{structcubiumClientSocket__t}{cubium\+Client\+Socket\+\_\+t}$\ast$ Local\+Communicator\+::client\+Sock\hspace{0.3cm}{\ttfamily [protected]}}
\mbox{\Hypertarget{classLocalCommunicator_ac28392070111396dabbadbbcb81a7ef7}\label{classLocalCommunicator_ac28392070111396dabbadbbcb81a7ef7}}
\index{Local\+Communicator@{Local\+Communicator}!routing\+Table@{routing\+Table}}
\index{routing\+Table@{routing\+Table}!Local\+Communicator@{Local\+Communicator}}
\subsubsection{\texorpdfstring{routing\+Table}{routingTable}}
{\footnotesize\ttfamily std\+::shared\+\_\+ptr$<$\hyperlink{classRoutingTable}{Routing\+Table}$<$\hyperlink{structcubiumServerSocket__t}{cubium\+Server\+Socket\+\_\+t}$>$ $>$ Local\+Communicator\+::routing\+Table\hspace{0.3cm}{\ttfamily [protected]}}
\mbox{\Hypertarget{classLocalCommunicator_a4c0d8806e53030c1af6124cd55a34335}\label{classLocalCommunicator_a4c0d8806e53030c1af6124cd55a34335}}
\index{Local\+Communicator@{Local\+Communicator}!server\+Sock@{server\+Sock}}
\index{server\+Sock@{server\+Sock}!Local\+Communicator@{Local\+Communicator}}
\subsubsection{\texorpdfstring{server\+Sock}{serverSock}}
{\footnotesize\ttfamily \hyperlink{structcubiumServerSocket__t}{cubium\+Server\+Socket\+\_\+t}$\ast$ Local\+Communicator\+::server\+Sock\hspace{0.3cm}{\ttfamily [protected]}}
The documentation for this class was generated from the following files\+:\begin{DoxyCompactItemize}
\item
lib/\hyperlink{local__communicator_8hpp}{local\+\_\+communicator.\+hpp}\item
lib/\hyperlink{local__communicator_8cpp}{local\+\_\+communicator.\+cpp}\end{DoxyCompactItemize}
|
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| 2,963
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← Sun Through the Eyes of a Runner.
Arizona Through the Eyes of a Runner.
As I stare out at the red Earth surrounding me, I take a deep, dry breath. I look heavenwards towards the light blue sky. The beauty in this place is too majestic, too beautiful, too spiritual.
My eyes can't take in all of this at once. As I close my thin lids, I feel the sun, I feel the breeze, I feel his spirit. How great is our God?
Deeper than the valleys below me, grander than the mountains before me, and more than my feeble mind can ever imagine.
This Earth, this creation. We are here. We are now.
My trip to Arizona was far too short, crazily packed, and wonderfully enjoyed!
I was blessed to be able to vacation for 3 days with my father, my little brother, and our family friend, Chris. We always have an adventure when we go on vacation.
This one included lots of rock climbing, lots of mountain scaling, jumping 20 feet into a freezing cold river, seeing the Grand Canyon, hiking the Grand Canyon, flying a helicopter over the Grand Canyon, catching up with family, and many laughs, and even more smiles!
|
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"redpajama_set_name": "RedPajamaC4"
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| 4,661
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{"url":"https:\/\/www.gradesaver.com\/textbooks\/science\/physics\/CLONE-afaf42be-9820-4186-8d76-e738423175bc\/chapter-7-exercises-and-problems-page-130\/14","text":"## Essential University Physics: Volume 1 (4th Edition)\n\nWe can find the potential energy as follows: $U=\\frac{1}{2}Kx^2$ We plug in the known values to obtain: $U=\\frac{1}{2}(320)(0.18)^2$ $U=5.2J$","date":"2021-03-06 23:35:39","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\": 0.9441081285476685, \"perplexity\": 626.6943191955153}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178375529.62\/warc\/CC-MAIN-20210306223236-20210307013236-00410.warc.gz\"}"}
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\section{Introduction}
As applications for robotics and automation increase, so will the need for robot platforms which are able to perform well in not just one, but a variety of tasks. This versatility usually comes at the cost of simplicity - leading to more versatile, yet complex robots with many Degrees of Freedom (DoF). Often the design of these complex robots mimics the design of biological systems such as humanoid robots \cite{collins2001three}, \cite{hubicki2016atrias}. These high DoF robots are technically capable of performing the complex dynamic behaviors such as those we see in nature, however their control is very difficult due to the many system states and inputs which are coupled in nonlinear ways.
Optimal control ideas have proven useful in modeling high DoF biological systems and in reproducing life-like motion with robots \cite{Li2004}, however the computational complexity of many optimal control algorithms often prohibits their real-time use on systems with many DoF. Because of this, there is a significant amount of active research in how to solve optimal control problems associated with high DoF control quickly enough to perform real-time control.
In this work we propose a method of control trajectory parameterization which can be used to extend the tractability of Model Predictive Control (MPC - a form of optimal control) to systems with more DoF or to lengthen the look-ahead horizon of MPC. We demonstrate that the proposed method drastically decreases MPC solve times while having little effect on performance and potentially improving robustness. We detail and compare two methods used for solving the modified MPC problem. We call these methods parameterized convex solver MPC and evolutionary MPC (EMPC). While parameterized convex solver MPC uses a fast convex solver like many implementations of traditional MPC, EMPC uses the parallel computation available in graphics processing units (GPUs) in order to solve the optimization.
Specifically the contributions of this paper are:
\begin{itemize}
\item Presentation of experimental results demonstrating the effects of control trajectory parameterization on MPC performance and robustness for robot control.
\item The development of a parameterized version of MPC which can be solved using a fast convex solver
\item The development of a very flexible parallelized and parameterized version of MPC based on an evolutionary optimization algorithm - Evolutionary MPC (EMPC).
\item Comparisons of EMPC, parameterized convex MPC, and traditional MPC both in simulation and on real robot hardware.
\end{itemize}
The remainder of this paper is organized as follows: Section \ref{related_work} highlights related work done in the fields of high DoF control and MPC parameterization and parallelization. Section \ref{parameterized_mpc} explains how control trajectories are parameterized for use with MPC, as well as the effects of this parameterization on MPC performance and robustness. Section \ref{parameterized_convex_mpc} explains two implementations of parameterized MPC which can be solved using a convex solver as well as experiments designed to test the solve times of each. Section \ref{empc} explains the implementation of a parameterized and GPU parallelized variant of MPC (EMPC). Section \ref{comparing_mpc_methods} contains simulation and hardware experiments which highlight the advantages and drawbacks of the proposed methods as well as comparisons to traditional MPC methods. Section \ref{conclusion} summarizes our findings and proposes future work.
\section{Related Work}\label{related_work}
\subsection{Optimal Control of High DoF Systems}
A well studied and intuitive method of performing optimal control for a high DoF robot is to take a hierarchical approach. Using the Operational Space Formulation \cite{Khatib1987},
one may prioritize tasks to be completed and ensure that lower priority tasks are only executed in the null space of higher priority task control. In \cite{Sentis2006} as well as \cite{Dietrich2012} this is used to control a humanoid robot, while in \cite{Hutter2014} it is used as part of a locomotion scheme for a quadruped. In \cite{Escande2014} a similar control hierarchy is achieved through quadratic programming which also allows for control of a humanoid.
Another way to perform optimal control in a hierarchical way is the approach taken in \cite{Kuindersma2016} where a high level optimization is done to find footsteps for the humanoid robot Atlas, a lower level optimization is done to plan joint trajectories given simplified dynamics \cite{Dai2014}, and an even lower level LQR controller is used to track those joint trajectories. This stacking of optimizations allows simplifications which make each layer tractable while still finding optimal or near-optimal solutions for the full high DoF problem.
Alternatively, methods such as iLQR/DDP \cite{Li2004} \cite{Tassa2014} work by forward simulating nonlinear dynamics given an input trajectory, and then using derivative information about the cost and the dynamics to calculate an improvement to the trajectory. These methods have been shown to work on high DoF systems with nonlinear dynamics (such as humanoid robots) \cite{Koenemann2015}, however there is still a high computational cost associated with forward simulating nonlinear dynamics and calculating its derivatives. The need to calculate these dynamics and derivatives quickly has even driven the development of software specially designed to do these at speeds which allow for MPC (MUJOCO) \cite{Todorov2012}.
The literature on walking robots is rich with examples of high DoF controllers, many based on optimal control ideas. In \cite{Pratt2001} Virtual Model Control is presented as a method of simplifying the high DoF system into a lower DoF system through feedback. The lower DoF system can then be controlled using standard optimal control techniques. Building on the Zero Moment Point (ZMP) preview control approach to walking \cite{Kajita2004}, several approaches use MPC as a method to control the ZMP toward some desired trajectory or point \cite{Krause2012}, \cite{Wieber2009}. In the Springer Handbook of Robotics \cite{springer_handbook} they claim that the motion generation schemes which power most of the great humanoids use MPC in one form or another.
Advances in direct optimization methods for MPC have come from advances in convex optimization solvers such as OSQP \cite{osqp} and CVXGEN \cite{cvxgen}, as well as researchers exploiting known structures in optimal control problems \cite{Kuindersma2014efficient}. Fast QP solvers have grown to handle larger and larger problems and have also decreased the time taken to find solutions, allowing us to perform MPC for larger and more complex systems at higher rates or with longer horizons. Fast direct solvers usually assume linear dynamics, however there are solvers which admit nonlinear dynamics at the cost of more computation time. There has even been research done in order to allow direct solvers to handle contact dynamics \cite{Posa2013}.
There are also methods to decrease the problem size and complexity to allow convex solvers not only to solve at real-time speeds, but to avoid infeasible problems which will fail to solve at all \cite{Rossiter2010}. In \cite{jon_terry} a modeling method is developed which decouples portions of the system, allowing each portion to be controlled independently using separate MPC controllers. In \cite{Ling2012} and \cite{Ling2011} a similar idea is used, assuming that each input acts independently and therefore can be optimized separately. These controllers exhibit better disturbance rejection because they do something good soon, rather than something better later.
Some high DoF robots have been successfully controlled using different forms of Reinforcement Learning in order to mimic the behavior of optimal controllers. Guided policy search \cite{Levine2013} for example, trains a Neural Network using examples from a nonlinear trajectory optimization (DDP). Because the training does not need to happen in real-time, this allows guided policy search to learn to approximate a nonlinear optimal controller. Although the optimal controller may not be able to solve at real-time rates, the execution times for machine learned models are typically very fast, allowing for very fast nonlinear control. This method has been used for humanoid walking, complex contact-rich tasks \cite{Levine2015contact}, and even learning visuomotor policies \cite{Levine2015}, which are all high DoF tasks.
The pattern that we see in the literature is that in order to solve the MPC problem at fast enough rates for real-time control, some simplifying assumptions or approximations must be made. In this work we present a novel assumption which allows for MPC solutions at higher rates and with longer horizons. The unique assumption made in this work is that a time-varying control trajectory may be represented as a linear function of relatively few equally spaced points. We believe that this novel approach may be combined with others in the literature in order to decrease solve times or extend MPC to more high DoF systems. We present two methods in this work which use this assumption to solve the MPC problem at high rates. The effects of this assumption on MPC performance and robustness are discussed in Sections \ref{parameterized_mpc_performance_results} and \ref{parameterized_mpc_robustness_results}.
\subsection{Parallelized and Parameterized MPC}
Recently, parameterization methods have begun to gain attention a way to reduce the complexity of MPC. In \cite{Muehlebach2019} both inputs and states are parameterized, taking advantage of known properties. In \cite{Khan2013} orthogonal basis polynomials are explored as a form of parameterization for MPC, while in \cite{Lengagne2013} B splines are used to represent states and inputs in an MPC optimization.
Dynamic Movement Primitives (DMPs) represent motions or behaviors using stable nonlinear attractor functions \cite{schaal2006dynamic} instead of time-based trajectories. This idea has been used to model the complex motions seen in humans \cite{kulic2012incremental} and to compile libraries of ``skills" which can be used to generate movements in robots \cite{konidaris2012robot}. A major advantage of DMPs is that they represent closed loop behaviors which are more robust to disturbances. This makes them well suited to real world tasks in unstructured environments such as furniture assembly \cite{niekum2015learning}.
In \cite{mukadam2018continuous} continuous time trajectories are modeled as Gaussian processes. This allows the entire trajectories to be represented using a small number of states and enables very fast interpolation needed for fast planning. This is similar to the idea of parameterization found in \cite{paraschos2013probabilistic} where the idea of a probabilistic motion primitive is introduced.
One popular MPC method involves solving the iLQR trajectory optimization problem rapidly and using the input and/or feedback policy from the first time step in an MPC scheme. Recently, work has been done to parallelize the computations needed for iLQR. iLQR can be thought of as a single-shooting method for solving the initial value problem defined by the dynamics constraint, followed by a backwards policy update using Ricatti recursions. In \cite{Giftthaler2017} the authors use a multiple-shooting method to forward simulate sections of the dynamic trajectory in parallel, followed by a backwards policy and state trajectory update. This allows for faster solution times and is more robust to poor initial guesses. This has been shown to successfully control a quadruped \cite{Neunert2017}.
In very early work in parallelized MPC, the authors of \cite{Rogers2013} and \cite{Ilg2011} used sampling-based optimization methods on a GPU to find trajectories for parafoils and projectiles which were robust to wind disturbances. More recently, the authors of \cite{Williams2016} and \cite{Williams2017} used a parallelized policy improvement method seeded with a learned policy to run MPC on a GPU at real-time rates. Our EMPC method is most similar to this method, however our method of parameterizing the input space makes EMPC tractable without the need for a prior policy to improve upon. The reduction in the search space afforded by the control trajectory parameterization means that a global search is possible using a heuristic global optimization method (the evolutionary algorithm). This is in contrast to gradient-based or policy improvement methods, which start at an initial point in the search space and descend to a local minima.
\section{Effects of Control Trajectory Parameterization on MPC}
\label{parameterized_mpc}
\subsection{Brief Review of Model Predictive Control}
Given a linear or linearized system, we can describe the system dynamics in state variable form as
\begin{equation}
\mathbf{\dot{x} = Ax + Bu + w}
\end{equation}
where $\mathbf{x}$ is the vector of states, $\mathbf{u}$ is the vector of system inputs, and $\mathbf{w}$ is a vector of constant disturbances. If there are $n$ states and $m$ inputs, then the matrices $\mathbf{A}$ and $\mathbf{B}$ are $n$x$n$ and $n$x$m$ respectively, while $\mathbf{w}$ is an $n$x1 vector.
Using any discretization method (Euler, semi-implicit Euler, matrix exponential, etc.) we can create a discretized state space model:
\begin{equation}
\label{eq:first_order_integration}
\mathbf{x_{k+1} = A_dx_k + B_du_k + w_d}.
\end{equation}
The above equation can be used to forward simulate the states of our system, given initial conditions and inputs. In traditional MPC these discretized dynamic equations will become the constraints of our optimization. In an MPC solver looking forward over a horizon of $T$ time steps, the optimization may be formulated as:
\begin{equation}
\label{eq:mpc_problem}
\begin{split}
J &= \sum_{k=0}^{T-1} \bigg[(\mathbf{x_{goal}-x_{k})}^T Q (\mathbf{x_{goal}-x_{k})} \\
&+ (\mathbf{u_{goal}-u_k)}^T R (\mathbf{u_{goal}-u_k)}\bigg]\\
&s.t. \\
&\mathbf{x_{k+1} = A_dx_k + B_du_k + w_d}\quad \forall \quad k=0,...,T\\
&\mathbf{x_{min} \leq x_k \leq x_{max}} \quad \forall \quad k=0,...,T \\
&\mathbf{u_{min} \leq u_k \leq u_{max}} \quad \forall \quad k=0,...,T \\
\end{split}
\end{equation}
where $J$ is the objective function value, $\mathbf{x_{goal}}$ and $\mathbf{u_{goal}}$ are the goal states and inputs respectively, and $\mathbf{x_{min}}$, $\mathbf{x_{max}}$, $\mathbf{u_{min}}$, $\mathbf{u_{max}}$, are the state and input bounds. For all of the experiments in this work $\mathbf{u_{goal}}$ is defined as the zero vector, meaning the cost is quadratic on the input and weighted by the matrix $R$.
MPC solves the above optimization for the entire horizon of length $T$, however only the first input ($\mathbf{u_0}$) is applied to the system. After applying this input, the optimization is solved again using new state and model information. This process is repeated with MPC only ever applying the first input, but solving over an entire horizon of value $T$.
Measures of control performance such as rise time, settling time, and percent overshoot can be affected by altering the weighting matrices $Q$ and $R$, as well as the horizon length $T$. These quantitative performance measures however are not explicitly incorporated into the objective function and therefore solutions which are technically optimal may have higher rise times, settling times, and overshoot. In order to make fair comparisons between MPC solutions, when evaluating the performance of MPC for the rest of this paper, we choose to evaluate the same objective function used when solving the MPC problem, but instead using the actual inputs and resulting states on the real system over a defined amount of time. The actual cost evaluated over $H$ time steps may be stated as
\begin{equation}
\label{eq:actual_cost}
Actual\ Cost = \sum_{i=0}^{H} J(\mathbf{x_{i}},\mathbf{u_{i}})
\end{equation}
where $\mathbf{x_{i}}$ and $\mathbf{u_{i}}$ represent the values of the state and input at time step $i$ and $J(\cdot, \cdot)$ represents the objective function. Throughout the rest of this paper we will refer to the cost calculated using actual states and inputs applied to the system as ``actual cost.''
\subsection{Method of MPC Parameterization}
\label{mpc_parameterization_method}
Traditionally, the optimization defined in Equation \ref{eq:mpc_problem} is solved by finding separate values of $\mathbf{u_k}$ for each discrete time step $k$. This amounts to an optimization over $T*m$ variables, or an optimization in a $T*m$ dimensional space. This optimization becomes difficult to solve fast enough for real-time control over long horizons (large $T$) or for systems with many inputs (large $m$). Reducing the dimension of this search space is the goal of control trajectory parameterization.
By parameterization of a control trajectory, we mean a method to represent the inputs of a time varying control trajectory using fewer than $T*m$ parameters. This is similar to the idea of curve fitting, where many discrete data points are represented by a smaller set of numbers such as coefficients of a polynomial expression, or the Fourier Transform which represents a signal using a small set of coefficients of trigonometric functions.
Instead of choosing to represent control trajectories using coefficients of polynomial or trigonometric functions, we choose to represent them using piece wise linear functions which cross through equally spaced ``knot" points as seen in Figure \ref{fig:one_parameterized_line}. This parameterization is fairly intuitive, allows for simple bounding of the control trajectory, and also preserves the convexity of the optimization problem. We do not intend to claim that this method of parameterizing MPC is the best form of parameterizing the input trajectory. However, experimental results in Section \ref{parameterized_mpc_performance_results} demonstrate that performance comparable to un-parameterized MPC can be achieved by using a sufficient number of knot points.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/one_parameterized_line.png}
\caption{An example of a control trajectory over T time steps parameterized by three ``knot" points.}
\label{fig:one_parameterized_line}
\end{figure}
In the case of a convex cost function with affine constraints, it is possible to show that there exists one optimum which is both the local and global optimum \cite{boyd2004convex}. This means that linear MPC, when implemented with a perfect model on a linear system with a sufficiently long horizon, will execute the optimal control at each time step and will achieve the optimal actual cost. While it is possible for parameterized MPC to also achieve the optimal actual cost, it can never achieve a lower actual cost. Again, this is only true for linear MPC executed on a linear system with a sufficiently long horizon.
\subsection{Parameterized MPC Performance - Experimental Setup}
\label{parameterized_mpc_performance_setup}
In order to evaluate the performance of MPC using a parameterized control trajectory, we conduct experiments on three simulated systems of varying complexity: a pendulum unaffected by gravity (linear), a pendulum in the presence of gravity (nonlinear), and a Puma 560 robot (nonlinear).
\subsubsection{Pendulum}
The continuous time dynamics of an inverted pendulum are
\begin{equation}\label{eq:inverted_pendulum}
m l^2 \ddot{q} + b \dot{q} + m g l sin(q) = \tau_{motor}
\end{equation}
where $m$ is the mass at the end of the link, $l$ is the length of a massless link, $b$ is a viscous damping coefficient, and $g$ is the acceleration of gravity. For experiments without gravity we let $g=0$.
\subsubsection{Puma 560}
We assume the robot is comprised of rigid links and pin joints, so that the dynamic equations take the canonical form
\begin{equation}\label{eq:puma_dynamics}
M(q)\Ddot{q} + C(q,\dot{q}) + b\dot{q} + \tau_{grav} = \tau
\end{equation}
where $q$ is the vector of generalized coordinates, $M(q)$ is a configuration dependent inertia matrix, $C(q,\dot{q})$ represents torques produced by centrifugal and Coriolis forces, $b$ is a viscous damping coefficient, $\tau_{grav}$ are the torques applied by gravity on the robot and $\tau$ are applied torques from the motors.
In order to isolate the effect of parameterization on the MPC problem, initial experiments were performed in simulation using MATLAB's fmincon function to perform the optimization for MPC. MPC is used to control the simulated system for one second from an initial position at rest, to a goal position at rest. To avoid any bias from a particular part of the robot's workspace, we ran several trials with initial and goal positions sampled from a random uniform distribution and we report statistics over all trials.
The metric that we use to evalute the effect of parameterization on MPC performance is the actual cost calculated by evaluating the cost function over the one second simulation using actual states and inputs. However because trials are run for different initial and different goal states, we would expect a large amount of variation in costs even using the same MPC controller. To eliminate this variation due to different initial and goal positions, we normalize the actual cost of each trial by dividing it by the actual cost achieved by traditional MPC with a horizon equal to the actual simulation time (one second). We would expect this ratio to be greater than one for most cases because traditional MPC with a long horizon should find an optimal or near-optimal solution compared to our parameterized version. A ratio of less than one indicates that the performance is better than traditional MPC with a long horizon. For reference, the actual cost ratio described above, evaluated over $H$ time steps can be expressed as
\begin{equation}
\label{eq:actual_cost_ratio}
Actual\ Cost\ Ratio = \frac{\sum_{i=0}^{H} J(\mathbf{x_{param,i}},\mathbf{u_{param,i}})}{\sum_{i=0}^{H} J(\mathbf{x_{trad,i}},\mathbf{u_{trad,i}})}
\end{equation}
where $\mathbf{x_{param,i}}$ and $\mathbf{u_{param,i}}$ represent the values of the state and input at time step $i$ using a parameterized MPC, while $\mathbf{x_{trad,i}}$ and $\mathbf{u_{trad,i}}$ represent the values of the state and input at time step $i$ using traditional MPC.
For each of the robots, we use the setup described above and vary the number of parameters (knot points) used in parameterized MPC with a set horizon (50 time steps) in order to control the robot from the initial position to the goal position. We simulate running MPC at a rate of 100 Hz with a model that has been discretized at a time step of .01 s.
Using our piecewise linear parameterization we can use anywhere between one and $Tm$ parameters where $m$ is the number of inputs. In fact if we use $Tm$ parameters, then the problem and solution are identical to those in traditional MPC. When using $Tm$ parameters, the distance between points is exactly the same as the discretization time step and each knot point becomes the input applied over the discrete time interval, just as in traditional MPC.
By varying the number of parameters between one and $Tm$, we are able to clearly see the effect that using fewer parameters has on MPC performance. These results are shown in Figure \ref{fig:num_parameters_vs_log_cost}.
For comparison, we also ran experiments varying the horizon length for traditional MPC. This provides a useful context for reasoning about performance because horizon length is commonly used as a tuning parameter. Horizon length can also be shortened until the MPC problem can be solved fast enough for real-time control, often at the expense of performance. The results of the experiments varying horizon length are found in Figure \ref{fig:horizon_vs_log_cost}.
\subsection{Parameterized MPC Performance - Experimental Results}
\label{parameterized_mpc_performance_results}
The results of the first experiment can be seen in the box plot in Figure \ref{fig:num_parameters_vs_log_cost}. All box plots in this work follow the following convention: the box contains data between the first and third quartile, the median is represented by a marker within the box, and whiskers contain all data not considered outliers (more than 1.5 times the interquartile range away from the box).
The effect on cost of decreasing the number of parameters is very slight until the number of parameters reaches a threshold between three and four points. This suggests that the important features of the optimal control trajectory are able to be represented fairly well with four or more points using our simple parameterization. The cost is of course the highest for the parameterization with one point. This corresponds to the optimization picking one input which will be applied over the entire horizon.
It can be seen that the actual cost ratio never goes below one for the linear pendulum case. This is the expected result because the long horizon traditional MPC has a perfect model which is valid over the entire horizon and so has found the actual optimal control solution. As we increase the number of parameters it can be seen that parameterized MPC quickly converges to the same solution.
The median cost ratio for the nonlinear pendulum is seen to be lower than that of the linear pendulum for the lower order parameterizations and the cost ratio is also seen to dip below one occasionally.
Traditional linear MPC is not expected to find the absolute optimal solution for a nonlinear system because the linearized model that is used for optimization is only valid for a small region surrounding the linearization point. The fact that the cost ratio is generally lower for the nonlinear pendulum than the linear pendulum does not necessarily indicate that parameterized MPC does better with nonlinear systems, but rather that it is more comparable to traditional MPC for nonlinear systems. This is likely because traditional linear MPC does not perform as well for nonlinear systems as it does for linear systems. This effect is further demonstrated with the results on the Puma 560 robot, which is an even more nonlinear system.
The results found by varying the horizon length as seen in the box plot in Figure \ref{fig:horizon_vs_log_cost} are not surprising. As expected, a shorter horizon length encourages more greedy behavior, which leads to a less optimal solution.
The purpose of this experiment however, is to provide a context when reasoning about MPC performance. When we compare the scale of Figures \ref{fig:num_parameters_vs_log_cost} and \ref{fig:horizon_vs_log_cost} we observe that the effect of decreasing horizon is far more dramatic than that of decreasing the number of parameter knot points.
This is an important result which should influence the design of MPC controllers. It indicates that instead of shortening the horizon length of traditional MPC until solve times allow for real-time control, we should often instead parameterize the control space to reduce solve times. The effect of parameterizing the control space is often smaller than the effect of decreasing horizon length, especially if the horizon length is not long to start with.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{figures/num_pts_vs_actual_cost_ratio_with_puma_zoomed.png}
\caption{Ratio of actual cost using parameterized MPC with a given number of parameters to traditional MPC for different robot platforms.}
\label{fig:num_parameters_vs_log_cost}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/horizon_vs_actual_cost_ratio_zoomed.png}
\caption{Ratio of actual cost using traditional MPC with a given horizon to traditional MPC with a horizon of 50 for different robot platforms.}
\label{fig:horizon_vs_log_cost}
\end{figure}
\subsection{Parameterized MPC Robustness - Experimental Setup}\label{parameterized_mpc_robustness_setup}
While specific MPC methods have been developed aimed at improving robustness \cite{bemporad1999robust} \cite{ostafew2016robust}, traditional MPC itself has been shown to be robust to modeling error and disturbances. The experiments carried out in this section seek to determine the effect of parameterization on the inherent robustness of MPC.
In order to experimentally test the robustness of our controllers to modeling error we intentionally introduce error into the model used for MPC. We then use MPC with an incorrect model to control the simulated system for which we know the model perfectly. Error is introduced in the form of an ``error multiplier" which is multiplied by certain model parameters. When the error multiplier is equal to one there is no modeling error, while values less than or greater than one correspond to underestimates or overestimates of model parameters respectively. For experiments using the inverted pendulum, the error multiplier was applied to both the mass and length of the pendulum. For experiments using the Puma robot, the error multiplier was applied to the entire inertia matrix.
We first quantify the effect of modeling error on MPC performance using the ``actual cost" metric defined in Equation \ref{eq:actual_cost}. In order to compare the sensitivity of several controllers to modeling error, we normalize the actual cost incurred by each controller by dividing by the actual cost incurred by a controller without modeling error. The normalized cost can be stated as
\begin{equation}
\label{eq:actual_cost_ratio}
Normalized\ Cost = \frac{Actual\ cost_{with\ error}}
{Actual\ cost_{without\ error}}.
\end{equation}
This metric shows the sensitivity of MPC performance to modeling error, however we are also interested in finding the sensitivity of MPC \textit{stability} to modeling error. In general, it is difficult to prove the stability of an MPC controller without the use of a local stabilizing controller and terminal constraints as well as costs \cite{mayne2000constrained}, \cite{friedbaum2018model}. These proofs also often operate under the assumption of accurate modeling, which is not the case in these experiments. It is generally the case however that conservative controllers tend to be more stable than aggressive controllers. We choose to quantify the ``conservative-ness" of a controller by measuring the rise time and percent overshoot attained using that controller. Aggressive controllers produce small rise times and large percent overshoot, while conservative controllers produce larger rise times and smaller percent overshoot.
\subsection{Parameterized MPC Robustness - Experimental Results}\label{parameterized_mpc_robustness_results}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/normalized_cost_inverted_pendulum.png}
\caption{How traditional MPC compares to parameterized MPC when an error multiplier is applied for an inverted pendulum. Due to exponential increase with error multiplier values of .5 and .6, these values are omitted from this figure to emphasize how a system is affected closer to an error multiplier of 1 (+ or - 10-30\% modeling error for a system).}
\label{fig:normalized_cost_inverted_pendulum_param_v_vanilla}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/normalized_cost_puma.png}
\caption{How traditional MPC compares to parameterized MPC when an error multiplier is applied for the Puma 560 robot.}
\label{fig:normalized_cost_puma560}
\end{figure}
Comparisons of parameterized MPC to traditional MPC performance sensitivity can be seen in Figures \ref{fig:normalized_cost_inverted_pendulum_param_v_vanilla} and \ref{fig:normalized_cost_puma560}. The results from Figure \ref{fig:normalized_cost_inverted_pendulum_param_v_vanilla} reveal that the sensitivity of MPC performance using parameterized MPC with more than two parameters is similar to that of traditional MPC. While parameterized MPC performance using two knot points is less sensitive to overestimating inertial parameters, it is more sensitive to underestimating them. As more knot points are used in parameterized MPC, the sensitivity approaches that of traditional MPC. This is the expected result because the parameterized MPC is more closely approximating traditional MPC. We do not believe that these results conclusively show that either parameterized or traditional MPC performance is less sensitive to modeling error than the other. In other words, parameterizing MPC does not increase or decrease the sensitivity of MPC performance to modeling error.
Figure \ref{fig:normalized_cost_puma560} serves to confirm that the results found in Figure \ref{fig:normalized_cost_inverted_pendulum_param_v_vanilla} hold for a more complex system (the Puma robot). It also exposes an interesting trend in MPC sensitivity which is true for both parameterized and traditional MPC. While MPC performance degrades very quickly for the case of underestimated inertia, it degrades relatively slowly in the case of overestimated inertia. This information should inform the modeling of dynamic systems for use with MPC. Specifically, mass and length estimates should err on the side of overestimating inertia when used for MPC because underestimating inertia has a strong negative impact on MPC performance.
The results of quantifying percent overshoot and rise time in the inverted pendulum experiments can be seen in Tables \ref{tab:percent_overshoot} and \ref{tab:rise_time}. In both tables it can be seen that when inertia is overestimated (error multiplier greater than one), the MPC controllers act more conservatively and reduce or eliminate overshoot while increasing rise times. Underestimating the inertia (error multiplier less than one) has the opposite effect, leading to greater overshoot and shorter rise times for the case of traditional MPC and parameterized MPC with 8 knot points. This result agrees with the results seen in Figures \ref{fig:normalized_cost_puma560} and \ref{fig:normalized_cost_inverted_pendulum_param_v_vanilla}. Interestingly, parameterized MPC using two or four knot points does not strictly decrease rise time when underestimating inertia.
There is a clear trend which indicates that by parameterizing with fewer knot points, rise time is increased and percent overshoot is decreased. This trend seems consistent for all modeling errors included in these experiments. In other words, using fewer knot points in parameterized MPC results in more conservative control. This result makes sense intuitively because within a prediction horizon, a parameterized MPC controller cannot change inputs instantaneously. Within the prediction horizon inputs must change linearly and can only change direction at knot points. This makes very aggressive ``bang-bang" maneuvers impossible within the MPC prediction horizon. Thus by parameterizing the control trajectory of MPC we have enforced a certain amount of ``conservative-ness".
Because conservative controllers are generally more stable than aggressive controllers when modeling error or disturbances are introduced, the inherent conservative nature of parameterized MPC favors stability over aggressiveness. This also helps to explain the results found in Section \ref{parameterized_mpc_performance_results} where parameterized MPC was found to perform slightly worse than traditional MPC in most cases, especially with few knot points. The decrease in performance can at least partially be attributed to the fact that the parameterized MPC controller was more conservative and therefore incurred higher cost.
As a partial summary, in Section \ref{parameterized_mpc_performance_results} it is shown that the performance difference between parameterized and traditional MPC is relatively small for a sufficient number of knot points. Results from Section \ref{parameterized_mpc_robustness_results} demonstrate that this difference in performance is at least partially because parameterized MPC favors more conservative control. These results together suggest that the performance of parameterized MPC can be very close to that of traditional MPC and, especially when a model is not known very well, will favor more conservative control actions leading to more stable control.
\begin{table}
\begin{center}
\begin{tabular}{|m{3em}|| m{1cm} | m{1cm} | m{1cm} | m{1cm}|}
\hline
\multicolumn{5}{|c|}{Percent Overshoot for Parameterized Systems} \\
\hline
Error Multiplier & 2 Knot Points & 4 Knot Points & 8 Knot Points & Traditi- onal MPC \\ [0.5ex]
\hline\hline
.5 & 37.7 & 40.2 & 40.3 & 43.5\\
\hline
.6 & 26.5 & 29.3 & 29.3 & 31.1 \\
\hline
.7 & 18.6 & 19.8 & 20.8 & 22 \\
\hline
.8 & 12.1 & 12.3 & 13.8 & 14.6 \\
\hline
.9 & 6.88 & 6.78 & 8.13 & 8.64 \\
\hline
1 & 3.05 & 2.96 & 3.89 & 4.17 \\
\hline
1.1 & .772 & .732 & 1.18 & 1.3 \\
\hline
1.2 & .0252 & .0171 & .0729 & .0919 \\
\hline
1.3 & 0 & 0 & 0 & 0 \\
\hline
1.4 & 0 & 0 & 0 & 0 \\
\hline
1.5 & 0 & 0 & 0 & 0 \\[1ex]
\hline
\end{tabular}
\caption{\label{tab:percent_overshoot}Data on the percent overshoot for an inverted pendulum, comparing traditional MPC simulation to different parameterized MPC simulations.}
\end{center}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{|m{3em}|| m{1cm} | m{1cm} | m{1cm} | m{1cm}|}
\hline
\multicolumn{5}{|c|}{Rise Time for Parameterized Systems} \\
\hline
Error Multiplier & 2 Knot Points & 4 Knot Points & 8 Knot Points & Traditi- onal MPC \\ [0.5ex]
\hline\hline
.5 & .85 & .41 & .3 & .28 \\
\hline
.6 & .65 & .36 & .3 & .29 \\
\hline
.7 & .56 & .35 & .31 & .3 \\
\hline
.8 & .54 & .36 & .32 & .31 \\
\hline
.9 & .54 & .38 & .35 & .34 \\
\hline
1 & .59 & .43 & .39 & .38 \\
\hline
1.1 & .67 & .5 & .45 & .44 \\
\hline
1.2 & .78 & .6 & .55 & .54 \\
\hline
1.3 & .89 & .71 & .67 & .66 \\
\hline
1.4 & .96 & .81 & .77 & .76 \\
\hline
1.5 & 1.02 & .88 & .85 & .84 \\[1ex]
\hline
\end{tabular}
\caption{\label{tab:rise_time}Data on the rise time for an inverted pendulum, comparing traditional MPC simulation to different parameterized MPC simulations.}
\end{center}
\end{table}
\section{Parameterized Convex Solver MPC}
\label{parameterized_convex_mpc}
While MPC has been used in many domains such as the chemical process industry \cite{binder2001introduction}, the focus of this work is on the application of MPC to robotics. In order to use MPC in real-time for robots, it is necessary to solve the optimization problem very quickly. For this reason, fast convex solvers have become the method of choice for use with MPC in robotics.
In this section we detail a method for implementing a parameterized version of MPC for use with a convex solver. In addition to outlining the method of implementation, we also provide experimental results demonstrating the effects of this parameterization on MPC solve times. Because the parameterization is the same as in Section \ref{mpc_parameterization_method}, the effects on performance and robustness remain the same as in Sections \ref{parameterized_mpc_performance_results} and \ref{parameterized_mpc_robustness_results}.
Formulating the MPC problem to fit within a convex solver framework is somewhat restrictive, since the optimization must be convex with only linear equality or inequality constraints. However, there are still a couple of ways to formulate the problem. Most convex solvers admit problems of the form
\begin{equation}\label{eq:convex_solver_form}
\begin{split}
minimize& \ \ \ z^T P z + 2q^T z \\
s.t.& \ \ \ lb <= Az <= ub.
\end{split}
\end{equation}
The objective function we wish to minimize is stated in Equation \ref{eq:mpc_problem}. We present two equivalent formulations of the optimization problem. In order to differentiate the methods, we refer to them based on the size of the optimization matrices $P$ and $q$ which result. The two methods will be referred to in this work as the large matrix formulation and the small matrix formulation.
\subsection{Large Matrix Formulation}
Perhaps the most common way to formulate the optimization for MPC is to choose $z = [x_0^T, x_1^T, ..., x_{T}^T, u_0^T, u_1^T, ..., u_{T-1}^T, 1]^T$ which allows us to rewrite Equation \ref{eq:mpc_problem} as
\begin{equation}
J = (z - z_{goal})^T\begin{bmatrix}
Q_{big} & 0 & 0\\
0 & R_{big} & 0\\
0 & 0 & 0
\end{bmatrix}(z - z_{goal})
\end{equation}
where $Q_{big} = I_{nT}\otimes Q$, $R_{big} = I_{mT}\otimes R$, and $\otimes$ represents the Kronecker product of two matrices. The zeros in the above matrix are included because of the value 1 included in the $z$ vector. This value is necessary to include the constant disturbance term $w$ as will be shown hereafter. Choosing $P$ as the block matrix in the above equation, we can further simplify this as
\begin{equation}
\begin{split}
J &= (z - z_{goal})^T P (z - z_{goal}) \\
&= z^T P z - 2z_{goal}^T P z + z_{goal}^T P z_{goal}.
\end{split}
\end{equation}
We now recognize that the last term in this expression is a constant offset and has no effect on the solution of the optimization. Eliminating the constant term we may write
\begin{equation}
\begin{split}
J &= z^T P z - 2z_{goal}^T P z \\
&= z^T P z + 2q^T z
\end{split}
\end{equation}
where $q^T = - z_{goal}^T P$ in order to fit our objective function into the form of Equation \ref{eq:convex_solver_form}.
In order to enforce the dynamics constraints which are necessary for MPC by solving an optimization of the form in Equation \ref{eq:convex_solver_form}, we must define a matrix $A$, as well as lower bounds $lb$ and upper bounds $ub$ on $z$. Note that when $lb=ub$, the constraint becomes an equality constraint. Given the discrete linear states space matrices $A_d$, $B_d$, $w_d$ and the initial state $x_0$, a common way to incorporate the constraints given in Equation \ref{eq:mpc_problem} into the form of Equation \ref{eq:convex_solver_form} is to choose
\begin{equation}
\label{eq:large_matrix_constraints}
A = \begin{bmatrix}
-I & 0 & \dots & 0 & 0 & \dots & 0 & 0\\
A_d & -I & & 0 & B_d & & 0 & w_d\\
& \ddots & \ddots & \vdots & & \ddots &
\vdots & \vdots\\
0 & \dots & A_d & -I & 0 & \dots & B_d & w_d
\end{bmatrix}
\end{equation}
and
\begin{equation}
lb = ub = \begin{bmatrix}
-x_0 \\
0 \\
\vdots \\
1
\end{bmatrix}
\end{equation}
Constraints on the inputs may be enforced by appending these constraints the bottom of the $A$, $lb$, and $ub$ matrices. Assuming the only other constraints are bounds on inputs, this optimization formulation exactly represents the minimization outlined in Equation \ref{eq:mpc_problem} with an optimization over $n(T+1) + Tm + 1$ variables with $n(T+1) + mT + 1$ constraints. Because the resulting $P$, $q$, and $A$ matrices are large, sparse matrices, we refer to this formulation of the optimization as the large matrix formulation of MPC.
\subsection{Small Matrix Formulation}
Another common way to formulate the optimization for MPC involves writing the vector of states over the time horizon as a function of only the inputs and initial condition. Because we have assumed linear dynamics, the vector of states over the time horizon can be written as
\begin{align}
\begin{bmatrix}
x_1 \\
x_2 \\
\vdots \\
x_{T}
\end{bmatrix} \nonumber
&=
\begin{bmatrix}
A_d x_0 + B_d u_0 + w_d\\
A_d x_1 + B_d u_1 + w_d\\
\vdots \\
A_d x_{T-1} + B_d u_{T-1} + w_d
\end{bmatrix} \\ \nonumber
&=
\begin{bmatrix}
A_d x_0 + B_d u_0 + w_d\\
A_d^2 x_0 + A_d B_d u_0 + A_d w_d + B_d u_1 + w_d\\
\vdots \\
A_d ^{T} x_0 + \sum_{i=0}^{T-1}\big[A_d^i(B_d u_{T-1-i} + w_d)\big]
\end{bmatrix} \\ \nonumber
&=
\begin{bmatrix}
B_d & 0 & 0 & \dots\\
A_d B_d & B_d & 0 &\dots\\
& \vdots \\
A_d^{T-1} B_d & A_d^{T-2} B_d &\dots & B_d
\end{bmatrix}
\begin{bmatrix}
u_0 \\
u_1 \\
\vdots \\
u_{T-1}
\end{bmatrix} \\
&\hspace{50pt} +\begin{bmatrix}
A_d x_0 + w_d\\
A_d^2 x_0 + A_d w_d + w_d\\
\vdots \\
A_d^{T} x_0 + \sum_{i=0}^{T-1} A_d^i w_d
\end{bmatrix} \\
&= S z + v
\label{eq:small_matrix_sz_v}
\end{align}
where $z=[u_0^T, u_1^T, ..., u_{T-1}^{T}]^T$.
By rewriting the states at each time step as a linear function of our inputs, we can now rewrite the cost function as
\begin{equation}
\begin{split}
J =& (Sz+v - x_{goal})^T Q_{big} (Sz+v - x_{goal}) \\
&+ (z-u_{goal})^T R_{big} (z-u_{goal}) \\
=& z^T(S^T Q_{big} S + R_{big}) z \\
&+ 2 z^T (S^T Q_{big} v - S^T Q_{big} x_{goal} - R_{big} u_{goal}) \\
&+ 2 v^T Q_{big} v - 2v^T Q_{big} x_{goal} + x_{goal} Q_{big} x_{goal} \\
&- u_{goal}^T R_{big} u_{goal}
\end{split}
\end{equation}
Realizing again that the last four terms which do not contain the optimization design variable $z$ are a constant offset and do not affect the optimization result, we can simplify further to
\begin{equation}
\begin{split}
J =& z^T(S^T Q_{big} S + R_{big}) z \\
&+ 2 z^T (S^T Q_{big} v - S^T Q_{big} x_{goal} - R_{big} u_{goal}) \\
=& z^T P z + 2z^T q \\
\end{split}
\end{equation}
where $P = S^T Q_{big} S + R_{big}$ and $q = (S^T Q_{big} v - S^T Q_{big} x_{goal} - R_{big} u_{goal})$. Note that in this formulation, the dynamics constraints are implicitly enforced in the cost function and it is in the form of Equation \ref{eq:convex_solver_form} needed for use with a convex solver.
Assuming the only other constraints are bounds on inputs, this optimization formulation exactly represents the minimization outlined in Equation \ref{eq:mpc_problem} with an optimization over $mT$ variables with $mT$ constraints. Because the resulting $P$, $q$, and $A$ matrices are smaller than in the other formulation presented, we refer to this formulation of the optimization as the small matrix formulation of MPC.
\subsection{Parameterization}
Because the parameterization introduced in Section \ref{mpc_parameterization_method} is linear, it can also be incorporated into the convex optimization methods described above.
Given $p$ knot points and a horizon of $T$ time steps, and assuming that one point is placed at the beginning and one at the end of the trajectory, we can define the distance between knot points in units of time steps ($\Delta T$) as
\begin{equation}
\Delta T = \frac{T-1}{p-1}.
\end{equation}
At any time step $k$, the input $u$ can be written as a linear combination of at most two knot points $U_{idx1}$ and $U_{idx2}$:
\begin{equation}
u = (1-c) U_{idx1} + (c) U_{idx2}.
\end{equation}
where
\begin{equation}
\begin{split}
idx1 &= floor(\frac{k}{\Delta T}) \\
idx2 &= idx1 + 1 \\
c &= \frac{k - (idx1) \Delta T}{\Delta T}.
\end{split}
\end{equation}
The dynamics can then be written as
\begin{equation}
x_{k+1} = A_d x_k + (1-c) B_d U_{idx1} + (c)B_d U_{idx2} + w_d.
\end{equation}
In order to parameterize both the large matrix formulation or the small matrix formulation of MPC, we include the knot points ($U$) in the vector of design variables ($z$) instead of the inputs at each time step ($u$). Other matrices in the optimization must also be modified. Although the modifications to these matrices are not complex, a general form for any horizon and number of parameters quickly becomes difficult to understand. Instead, we provide simple examples for the specific case of a horizon of $T=5$ time steps using $p=3$ knot points.
\subsection{Parameterized Large Matrix Formulation}
For the large matrix formulation of MPC, in addition to the vector of design variables ($z$), the constraint matrix $A$ must be modified. Instead of a matrix of the form described in Equation \ref{eq:large_matrix_constraints}, for the case of $T=5$ and $p=3$ we obtain the matrix
\begin{equation}
\begin{split}
&A = \\
&\begin{bmatrix}
-I & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
A_d & -I & 0 & 0 & 0 & 0 & B_d & 0 & 0 & w_d\\
0 & A_d & -I & 0 & 0 & 0 & \frac{1}{2}B_d & \frac{1}{2}B_d & 0 & w_d\\
0 & 0 & A_d & -I & 0 & 0 & 0 & B_d & 0 & w_d\\
0 & 0 & 0 & A_d & -I & 0 & 0 & \frac{1}{2}B_d & \frac{1}{2}B_d & w_d\\
0 & 0 & 0 & 0 & A_d & -I & 0 & 0 & B_d & w_d
\end{bmatrix}.
\end{split}
\end{equation}
The only change to this matrix is in the portion containing the matrix $B_d$ and involves the linear combination of at most two knot points for each input.
Assuming the only constraints are bounds on inputs, the parameterized large matrix formulation of MPC requires optimization over $n(T+1) + mp + 1$ variables with $n(T+1) + mp + 1$ constraints. This represents a reduction in the number of optimization design variables and constraints by $m(T+1-p)$.
\subsection{Parameterized Small Matrix Formulation}
For the small matrix formulation, in addition to the vector of design variables ($z$), the matrix $S$ in Equation \ref{eq:small_matrix_sz_v} must be modified which in turn affects the matrix $P$. Note that the matrix $v$ in Equation \ref{eq:small_matrix_sz_v} does not change at all. For the parameterized $S$ matrix in the case of $T=5$ and $p=3$ we obtain
\begin{equation}
\begin{split}
&S = \\
&\begin{bmatrix}
B_d & 0 & 0\\
(A_d + \frac{1}{2}I)B_d & \frac{1}{2}B_d & 0\\
(A_d^2 + \frac{1}{2}A_d)B_d & (\frac{1}{2}A_d + I)B_d & 0\\
(A_d^3 + \frac{1}{2}A_d^2) B_d & (\frac{1}{2}A_d^2 + A_d + \frac{1}{2}I) B_d & \frac{1}{2}B_d \\
(A_d^4 + \frac{1}{2}A_d^3) B_d & (\frac{1}{2}A_d^3 + A_d^2 + \frac{1}{2}A_d) B_d & (\frac{1}{2}A_d + I) B_d \\
\end{bmatrix}
\end{split}
\end{equation}
where $I$ denotes the identity matrix.
A simple way to calculate this matrix is to start from the top and work downwards. Each progressive row is simply the previous row pre-multiplied by $A_d$, then summed with $B_d$ in the correct proportions in each column.
The parameterized small matrix formulation of MPC requires optimization over $mp$ variables with $mp$ constraints. The difference in the number of design variables and constraints in this case is also $m(T+1 - p)$. Although this reduction in the number of variables and constraints is identical to the reduction for the large matrix formulation, because the small matrix formulation started with fewer variables and constraints it represents a proportionally larger reduction in the optimization size.
\subsection{Solve Times of Parameterized Convex Solver MPC - Experimental Setup}
\label{convex_solver_experimental_setup}
Because we are interested in how our methods scale to high DoF robots, we choose to perform experiments with simulated robots with between one and thirteen links. We refer to these robots as N link robots. Each link is assumed to be the same mass (1 kg) and length (.25m) and each robot's workspace is assumed to be in the plane perpendicular to the gravity vector. The dynamic equations take the canonical form
\begin{equation}\label{eq:threelink}
M(q)\Ddot{q} + C(q,\dot{q}) + b\dot{q} = \tau
\end{equation}
where $q$ is the vector of generalized coordinates, $M(q)$ is a configuration dependent inertia matrix, $C(q,\dot{q})$ represents torques produced by centrifugal and Coriolis forces, $b$ is a viscous damping coefficient (.01), and $\tau$ are applied torques from the motors.
The main advantages of MPC are that it easily incorporates constraints and performs an optimization over a future time horizon, leading to less ``greedy" behavior. In order to test our methods in the most useful scenarios we therefore choose a damping coefficient $b$ such that the robots are all underdamped and enforce a torque constraint of two Nm at each joint. This poses a more interesting and challenging problem than that of an over-damped robot or one with infinite available torque.
Experiments are performed to quantify the effects of different MPC formulations, as well as parameterization, on MPC solve times.
We use a state of the art convex solver (OSQP) \cite{osqp} to perform the optimization. We record both the run time reported by the solver and the total MPC solve time, which includes the time needed to calculate the matrices $P$, $q$, $A$, $lb$, and $ub$. We report these solve times as the optimization solve time and MPC solve time respectively. Because the performance of parameterized methods was already explored in Section \ref{parameterized_mpc_performance_setup}, we do not attempt to quantify performance in these experiments.
The first experiment is performed by choosing random initial and goal states for a robot, calculating the dynamics matrices for the robot based on initial conditions, and then using each MPC solver to solve for the next input to apply. Because we are only interested in solve times, the input is never actually applied to any system, but the process of sampling random initial and goal conditions is repeated 100 times. This process is carried out for robots with one to thirteen links. A horizon of 50 time steps is used for every solve, and the parameterized methods used five parameters. Median optimization solve times and MPC solve times for each MPC formulation and each number of links can be seen in Figures \ref{fig:horizon_50_median_solve_times} and \ref{fig:horizon_50_median_total_times} respectively.
The second experiment carried out is exactly the same as the first, but with a horizon of 100 instead of 50. The parameterized methods still use five parameters. Median optimization solve times and MPC solve times for each MPC formulation and horizon are reported in Figures \ref{fig:horizon_100_median_solve_times} and \ref{fig:horizon_100_median_total_times} respectively.
\subsection{Solve Times of Parameterized Convex Solver MPC - Experimental Results}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/horizon_50_solve_times.png}
\caption{Median optimization solve times over hundreds of trials for a horizon of 50 time steps.}
\label{fig:horizon_50_median_solve_times}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/horizon_100_solve_times.png}
\caption{Median optimization solve times over hundreds of trials for a horizon of 100 time steps.}
\label{fig:horizon_100_median_solve_times}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/horizon_50_total_times.png}
\caption{Median MPC solve times over hundreds of trials for a horizon of 50 time steps.}
\label{fig:horizon_50_median_total_times}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/horizon_100_total_times.png}
\caption{Median MPC solve times over hundreds of trials for a horizon of 100 time steps.}
\label{fig:horizon_100_median_total_times}
\end{figure}
Looking first at just the optimization solve times in Figures \ref{fig:horizon_50_median_solve_times} and \ref{fig:horizon_100_median_solve_times} we note the expected result that the solve times for robots with more links are higher than those with fewer links. As can be seen in both figures, the small matrix formulation of MPC solves very quickly for small problem sizes, but seems to scale poorly to larger problem sizes. Perhaps surprisingly, even though the number of variables and constraints is larger for the large matrix formulation, it solves faster than the small matrix formulation for large problems.
This counter-intuitive result may be explained by the optimization method used. Many fast convex solvers depend on a factorization of the $P$, $A$, and $q$ matrices from Equation \ref{eq:convex_solver_form} to solve a system of linear equations. Very efficient solution methods exist for performing these linear system solves for large, sparse matrices. By decreasing the search space in the small matrix formulation we have exchanged large sparse matrices for small dense matrices.
A similar phenomenon is observed with the parameterization of the large matrix formulation of MPC. We see that for small problem sizes, parameterization of the large matrix formulation has little effect on the solve time. In fact if anything, shrinking the search space of the large matrix formulation seems to have the effect of increasing the solve time. Again, we believe this is related to the linear system solution of a sparse matrix that we make more dense through parameterization.
The parameterized small matrix formulation of MPC consistently has the lowest optimization solve times of any MPC formulation. Although the matrices used in the parameterized small matrix formulation are very dense, they are also very small. Although it is difficult to tell because of the scale of the plots, the parameterized small matrix MPC optimization solve times in Figures \ref{fig:horizon_50_median_solve_times} and \ref{fig:horizon_100_median_solve_times} are near identical, despite the length of the horizon doubling.
When we remember that for the small matrix formulation, the optimization problem size is not a function of horizon length, but only number of parameters, this result makes sense. If the number of parameters is held constant, the horizon can be made arbitrarily large with no effect whatsoever on solve time. The same is not true however, for the time taken to calculate the matrices $P$ and $q$.
By looking at Figures \ref{fig:horizon_50_median_total_times} and \ref{fig:horizon_100_median_total_times} we can see the total MPC solve times. These solve times include the optimization solve time plus the time taken to calculate the matrices $P$, $Q$, $A$, $lb$, and $ub$. We recognize that these results may vary based on specific methods and code implementations for calculating these matrices, however we submit that the trends presented represent trends dictated by the size and complexity of the matrices. For the large matrix formulations, most of this extra time is taken to form the large constraint matrix $A$ which contains the dynamics constraints.
For the small matrix formulations, there is greater time spent calculating the matrices $P$ and $q$.
For both the horizon of 50 and the horizon of 100, we can see that the overall effect of parameterization was a reduction in the MPC solve time. This is true even for the large matrix formulation, where the optimization solve time is not decreased by parameterization. This is likely because the matrices used in the parameterized version are much smaller and take less time to form before optimization.
Apart from the trend that parameterization decreases MPC solve time for either formulation of MPC, we can see that parameterization has a greater effect on the MPC solve time of the small matrix formulation, especially for long horizons and complex systems. We see that for robots with many links, the small matrix formulation has the longest MPC solve times, while the parameterized version has the shortest MPC solve times. We also note that the parameterized small matrix formulation seems the least sensitive to the number of links. This is evidence that this method scales better to high DoF systems than the others
These results combined with those found in Section \ref{parameterized_mpc_performance_results}, strongly suggest that when performing MPC for high DoF systems, instead of decreasing horizon length to satisfy demands on solve time, we should instead reformulate and parameterize the input trajectory to decrease solve times. We have established that this method leads to faster solve times with little to no loss in performance. In fact, these results suggest that most traditional MPC controllers can be run at a higher rate through parameterization.
While the results found for this section are found using one specific solver, we expect that similar results would be found with other fast solvers which utilize factorization of sparse matrices for finding the solutions to a system of linear equations.
\section{Evolutionary MPC}
\label{empc}
Up to this point we have discussed the parameterization of MPC, its effects on MPC performance and robustness, and how to implement parameterized MPC for use with a convex solver. We have established that MPC solve times can be decreased drastically by using the small matrix formulation with parameterization.
We now turn our attention to another method of implementing parameterized MPC which is parallelizable, scalable, and can admit much more flexible cost and dynamics constraints. Because this method utilizes an evolutionary algorithm to perform optimization, we refer to this method as evolutionary MPC (EMPC)\footnote{We are grateful to Dr. Greg Stewart from The University of British Columbia for suggesting the random sampling of the configuration/state space.} and we implement it on a GPU. This is similar to the work in \cite{hyatt2017real} where preliminary work with EMPC applied it to a seven DoF compliant robot arm. The current work builds on \cite{hyatt2017real} by relaxing heuristic assumptions about the knot points, using a linear parameterization, and simplifying the evolutionary algorithm.
GPUs are designed to launch thousands of threads which all perform the same function on different data. In the CUDA programming interface these threads are assigned unique indices and are organized in structures called blocks as seen in Figure \ref{fig:cuda_heirarchy}. This hierarchy is important to understand because only threads within the same block can be made to run synchronously and may share common memory. These considerations are important when efficiently simulating and evaluating costs in parallel on a GPU.
\begin{figure}[hbt]
\centering
\includegraphics[width=1.0\linewidth]{figures/gpu_hierarchy.png}
\caption{Graphical explanation of CUDA programming hierarchy as implemented in this paper. One simulation is launched in each block and numSims blocks are launched simultaneously. After all blocks have finished, each block/simulation has calculated a cost associated with an input trajectory.}
\label{fig:cuda_heirarchy}
\end{figure}
At a high level, EMPC utilizes the parallel computing capability of GPUs by evaluating the fitness of hundreds or thousands of input trajectories simultaneously. The best of these trajectories are then mated and mutated to produce a new generation of trajectories which should theoretically be better than the last. As in typical MPC, only the first input of the best trajectory is applied to the actual system. Unlike typical MPC however, EMPC does not wait for any convergence criteria before applying the next input to the system. Instead, the best input is applied to the system after each generation. Despite applying what are almost certainly sub-optimal inputs, because EMPC is continuously improving its solution as it progresses towards the goal, EMPC performs similarly to MPC using a state of the art convex solver, while allowing for shorter solve times and longer horizons (see Section \ref{simulation_experiments}).
Because evolutionary algorithms are gradient-free and non-local optimizations, the type of parameterization used is very flexible. For the sake of generality, for this section we will let $\mathbf{U}$ represent the parameters which define all system inputs over a trajectory, and $\mathbf{U}(i)$ represent the parameters which represent the $i$th input over the trajectory. We will let $get\_u\_from\_U(\mathbf{U}(i),t)$ represent the function which maps from a set of parameters and a time to the actual input applied to the system at that time.
The EMPC algorithm is highly parallelizable, especially because of the assumption of linear dynamics. In order to more fully exploit the inherent parallelism of the problem using the CUDA programming interface for GPUs, we organize our calculations as seen in Figure \ref{fig:cuda_heirarchy}. Each simulation is carried out within one block and each thread within that block is assigned one system state. The threads with $threadIdx < m$ are assigned one input each as well. This assumes that the number of states is equal to or greater than the number of inputs.
The computations for one thread are outlined in Algorithm \ref{alg:EMPC} lines 3-17 for reference, and are discussed more in detail below. At the beginning of each generation, in the case of a cold start, the parameters $\mathbf{U}$ are selected from a random uniform distribution. In the case of a warm start, these parameters are produced by mating two sets of randomly chosen ``parent" parameters. This mating is accomplished by looping through each parameter in the trajectory and giving the child 50\% probability of inheriting each parameter from each parent. This process of crossover allows the best trajectories to be combined with the hope of finding new trajectories which are the best parts of both parents.
In order to encourage exploration and avoid the tendency of populations to stagnate in local optima, we introduce changes in the input population with mutations. After mating is finished, each input has a 50\% probability of being mutated. If mutated, additive noise sampled from a mean-zero normal distribution with standard deviation $\sigma_{noise}$ is added to the parameters.
The number of parents ($numParents$), the number of simulations $numSims$, and the exploration noise $\sigma_{noise}$ are tune-able parameters of the algorithm. If $threadIdx >= m$, then the thread waits for the first $m$ threads to reach line 11 of Algorithm 1 before continuing.
\begin{algorithm}
\caption{Evolutionary Model Predictive Control}\label{alg:EMPC}
\begin{algorithmic}[1]
\For{each Simulation (Parallel block)}
\For{$i = 0$ to $n$ (Parallel thread)}
\If{Cold Start}
\State $\mathbf{U}(i) \sim \mathcal{U}(\mathbf{u_{min}}(i),\mathbf{u_{max}}(i))$
\ElsIf{Warm Start}
\State $P1 \sim \mathcal{U}(0,numParents)$
\State $P2 \sim \mathcal{U}(0,numParents)$
\State $Noise \sim \mathcal{N}(0,\sigma_{noise})$
\State $\mathbf{U}(i) = mate(\mathbf{U_{P1}}(i),\mathbf{U_{P2}}(i)) + Noise$
\EndIf
\For{$t = 0$ to ${T}$}
\State $\mathbf{u}(i) = get\_u\_from\_U(\mathbf{U}(i),t)$
\State Synchronize $\mathbf{x}$ and $\mathbf{u}$ across block
\State $\mathbf{x}(i) = A(i,:)\cdot \mathbf{x} + B(i,:) \cdot \mathbf{u} + \mathbf{w}$
\State $J_i += stage\_cost(\mathbf{x}(i),\mathbf{u}(i),\mathbf{x_{goal}},\mathbf{u_{goal}})$
\EndFor
\State $J_i += terminal\_cost(\mathbf{x}(i),\mathbf{u}(i),\mathbf{x_{goal}},\mathbf{u_{goal}})$
\EndFor
\State $J = \sum_{i=0}^{n} J_i$
\EndFor
\State $\mathbf{U_0, U_1, ..., U_{numParents}} = \mathbf{U}$'s with lowest J's
\For{$i = 0$ to $m$}
\State $\mathbf{u^*}(i) = get\_u\_from\_U(\mathbf{U_{best\_parent}}(i),0)$
\EndFor
\end{algorithmic}
\end{algorithm}
\subsection{Simulation and Cost Calculation}
The simulation of an input trajectory over a horizon is done sequentially using Equation \ref{eq:first_order_integration}. Because of the coupling between states, the calculation of each element of $\mathbf{x_{i+1}}$ depends on the entire vector of states $\mathbf{x_i}$ as well as the entire vector of inputs $\mathbf{u_i}$. Because threads in the same block can access the same shared memory and can be made to synchronize at certain points, each thread can calculate its own state in the vector $\mathbf{x_{i+1}}$ using the states and inputs saved in shared memory. Then after all states have performed this calculation, the states and inputs in shared memory can be updated again.
Some of EMPC's greatest strengths come from this approach to simulation. Although the state space dynamics are linear, because we calculate each state and input at each timestep, we can include some nonlinear effects such as
hysteresis, contact, and impacts. These types of dynamic effects can simply be placed in if/else statements during simulation. Another benefit of this type of simulation is that while convex solvers may find a problem infeasible and return no solution at all, EMPC can be made to always return the best possible solution, and the constraints describing the dynamics of the system are satisfied implicitly as part of the simulation.
At each time step in this forward simulation, each thread can evaluate the stage cost associated with its state as well as a terminal cost at the final time step. The total cost is updated at each time step, then is summed with the rest of the costs from the other threads and is saved to device memory as the simulation cost. Because we are performing a gradient-free optimization, this allows almost unlimited flexibility in the type of cost function used. Cost functions need not be convex or even continuous, so they may contain if/else statements, piecewise functions, and other functions.
Once all simulations are finished, we are left with $numSims$ $\mathbf{U}$ arrays, each with an associated cost. Parent selection simply consists of taking the $numParents$ $\mathbf{U}$ arrays with lowest associated costs and preserving them for the next generation. In order to find $u^*$ (the optimal $u$) for use with MPC, we take the $\mathbf{U}$ array with lowest associated cost, and find the $u(i)$ defined by $get\_u\_from\_U(\mathbf{U_{best\_parent}}(i),t=0)$ for each input $i$. This corresponds to the inputs applied at the first time step given the $\mathbf{U}$ array from the lowest cost simulation.
\section{Comparing Evolutionary, Parameterized Convex Solver, and Traditional MPC}
\label{comparing_mpc_methods}
\subsection{Simulation Experiments - Setup}
\label{simulation_experiments}
In this section experiments are performed in order to compare the solve times and performance of EMPC to traditional and parameterized gradient-based MPC. We are especially interested in how these different solution methods scale to high DoF problems. We again use OSQP as the gradient-based solver in MPC. We perform experiments for this section using the same simulated N link robots described in Section \ref{convex_solver_experimental_setup}.
To test the performance of each MPC controller, we simulate using the controllers to perform control at 100 Hz for ten seconds. Performance is measured using the same ``actual cost" metric as in Section \ref{parameterized_mpc_performance_setup}, where ``actual cost" is calculated by evaluating the cost function used in MPC over the ten second simulation. The same exact cost function and weightings are used for both MPC methods and control is simulated to be done at 100 Hz, even when solve times exceed .01s. The parameterized versions of MPC are using only three knot points. The horizon used for MPC is 100 time steps with each step representing .01s. Each MPC method is used to control from an initial position at rest, to a goal position at rest. To avoid any bias from a particular part of the robot's workspace, we again ran several trials with initial and goal positions sampled from a uniform random distribution and report statistics over all trials.
Solve times are recorded for each MPC solve in every trial. The optimization solve time recorded for MPC utilizing OSQP corresponds to the ``run time" reported by the OSQP solver. This includes time taken to update the sparse matrices within the solver and solve the optimization. Likewise, the optimization solve time recorded for EMPC includes the time taken to copy needed data to the GPU, perform one or more iterations of the genetic algorithm, and copy needed information back from the GPU. Time taken to calculate the dynamics matrices was not included in either solve time. The OSQP MPC solver also required that the dynamics matrices be formatted into large constraint and cost matrices which are then fed into the solver, while for EMPC this step is not necessary. The time taken to construct these matrices plus the optimization solve time is reported as the MPC solve time.
In order to examine the scalability of both traditional MPC and EMPC, we perform experiments on robots with one to thirteen links. We report both performance and solve time statistics as a means of measuring scalability. Because EMPC is parallelized for use on a GPU, we also ran experiments on two different GPUs (an NVIDIA GeForce GTX 750 Ti and an NVIDIA GeForce GTX Titan X). This comparison is interesting because the Titan X contains many more parallelized processing cores than the 750 Ti (3072 vs 640). These experiments help demonstrate what effects GPU improvement may have on parallelized MPC methods.
We also perform experiments using a variant of EMPC which performs three iterations of the evolutionary algorithm before returning a value for use with MPC. This should allow the evolutionary algorithm to find a better solution at the cost of longer solve times.
An example of the joint angle response of a six link robot for these trials is demonstrated in Figure \ref{fig:sample_trajectory}.
\subsection{Simulation Experiments - Results}
\begin{figure*}[hbt]
\centering
\includegraphics[width=1.0\textwidth]{figures/sample_inputs.png}
\caption{Sample joint angle response for a six link planar robot controlled by three different Model Predictive Controllers. Note that the parameterized methods (blue and green) are more conservative and produce less overshoot.}
\label{fig:sample_trajectory}
\end{figure*}
A direct comparison of performance between parameterized convex MPC and EMPC, using traditional MPC as a baseline, can be seen in Figure \ref{fig:lempc_vs_osqp_performance}. We report the ratio of actual cost using each controller to the actual cost using traditional MPC. A ratio of one (the dashed line) corresponds to the same level of performance as traditional MPC. Consistent with the findings in Section \ref{parameterized_mpc_performance_results}, the performance of MPC with parameterization is generally not as good as that of traditional MPC for simple systems. The increased cost for any robot except the one link however is generally pretty small, accruing less than 20\% more cost when using the convex solver. For higher DoF systems however, the convex parameterized MPC accrues lower costs than traditional MPC. This is likely because of the conservative nature of parameterized MPC as outlined in Section \ref{parameterized_mpc_robustness_results}. A clear example of the conservative nature of the parameterized MPC approach is shown in Figure \ref{fig:sample_trajectory} for a six link robot. It can be seen that the un-parameterized MPC method has a faster rise time, but more overshoot.
The performance of EMPC is slightly worse than that of the convex solver, but again, the difference is not very large. It is interesting to note that by performing more iterations or ``generations" for the evolutionary algorithm, the EMPC cost begins to approach that of the parameterized convex solver. This essentially represents the classical trade-off between speed and quality of solution.
Figure \ref{fig:lempc_vs_osqp_solve_times} represents the relationship between the number of robot links and the optimization solve times for all of the MPC controllers. We see the same trend in solve times for the gradient-based solvers as in Figures \ref{fig:horizon_100_median_solve_times} and \ref{fig:horizon_50_median_solve_times}, however the median solve times in these experiments are much lower. That is because in these experiments the solvers are allowed to warm start themselves with a previous solution, which was often near the optimum once the robot reached a steady state equilibrium. The fact that each of our simulations lasts ten seconds means that a great deal of the solve times recorded are from this steady state regime.
For low DoF systems, EMPC optimization solve times are greater than both gradient-based solvers. However we see in Figure \ref{fig:lempc_vs_osqp_solve_times} that one iteration EMPC using either of the GPUs we presented has faster optimization solve times than traditional MPC for ten or more links. By increasing the number of iterations in EMPC by a factor of three we seem to have also increased the solve time by about the same factor.
It is also interesting to note the improvement in solve time by changing only the GPU. It seems that by using the larger GPU, the solve time is decreased by about three ms, but this three ms decrease does not seem affected by the number of links. It is not unreasonable to assume that by using a more powerful GPU (such as the Titan V) solve times would be further reduced. Reductions in speed could also be achieved by using multiple GPUs. This scalability to high DoF problems through parallel computation is one of the main benefits of the EMPC approach.
Figure \ref{fig:lempc_vs_osqp_mpc_solve_times} shows the MPC solve times for each MPC controller for a varying number of robot links. Again, consistent with the results in Section \ref{parameterized_mpc_performance_results}, we find that traditional MPC scales poorly with increasing robot complexity. We also confirm the result that the parameterized gradient-based MPC increases the tractability of MPC. The most impressive feature of these results however is how EMPC scales with increasing DoF of the robot. Because EMPC does not require the formation of large matrices for definition of the cost function or constraints, the MPC solve time is identical to that of the optimization solve time. This demonstrates the fact that the EMPC method scales very gracefully to higher DoF systems.
We see from these results an expected trade-off between tractability and the quality of the solution. Both methods presented in this work are more tractable than traditional MPC and provide high quality solutions. For high DoF systems, one iteration EMPC has the fastest solution time by far, however it also has incurs the highest cost. By increasing the number of ``generations" within an EMPC solve the cost can be decreased, but with slightly higher solve times. Parameterized convex solver MPC provides the lowest cost solutions (even lower than traditional MPC for the high DoF cases), but has a higher solve time for high DoF cases.
Because of the tractability and performance trade-off which exists between parameterized convex solver MPC and EMPC, it is difficult to say that one will be more tractable or provide better performance than the other in all cases. However Figure \ref{fig:lempc_vs_osqp_mpc_solve_times} of MPC solve times and Figure \ref{fig:lempc_vs_osqp_performance} of performance provide convincing evidence that control trajectory parameterization can make MPC far more tractable with very little effect on performance. This is an important result which may enable MPC for many robotic systems for which it was not previously possible.
\begin{figure}[hbt]
\centering
\includegraphics[width=1.0\linewidth]{figures/actual_cost_ratios_n_links.png}
\caption{Ratio of actual cost using EMPC and a parameterized gradient-based solver to traditional MPC. The dashed line at one denotes performance equal to traditional MPC, below one is better, and above is worse.}
\label{fig:lempc_vs_osqp_performance}
\end{figure}
\begin{figure*}[hbt]
\centering
\includegraphics[width=1.0\textwidth]{figures/optimization_solve_times_n_links.png}
\caption{Comparison of optimization solve times for increasingly complex robots}
\label{fig:lempc_vs_osqp_solve_times}
\end{figure*}
\begin{figure*}[hbt]
\centering
\includegraphics[width=1.0\textwidth]{figures/mpc_solve_times_n_link.png}
\caption{Comparison of MPC solve times for increasingly complex robots}
\label{fig:lempc_vs_osqp_mpc_solve_times}
\end{figure*}
\subsection{Hardware Experiments - Setup}
\label{hardware_experiments_setup}
\begin{figure}[hbt]
\centering
\includegraphics[width=\linewidth]{figures/baxter.png}
\caption{Seven DoF Baxter robot used for the hardware experiments}
\label{fig:baxter}
\end{figure}
In order to demonstrate that both EMPC and parameterized gradient-based MPC are viable approaches to performing high DoF MPC for real systems, we implement them both on hardware. Specifically, we implement both for position control of one seven DoF arm of the Baxter robot seen in Figure \ref{fig:baxter}. A PD controller on position was run at 500 Hz and tuned with very low gains ($K_p$, $K_d$) in order to produce an under-damped system which is more safe to operate near humans or delicate equipment, but harder to control for smooth motion. The inputs selected by MPC are set points in joint space ($q_{des}$) for the PD controller which then applies torques directly to the robot.
We model the kinematics and dynamics of the robot using parameters provided by the manufacturer for lengths and masses. The robot automatically applies torques to oppose those of gravity, so we choose to model the robot as if there are no gravity torques. The full dynamics of the arm are
\begin{equation}\label{eq:blow_molded}
M(q)\Ddot{q} + C(q,\dot{q}) = K_{p}(q_{des}-q) - K_{d}\dot{q}
\end{equation}
where $q$ is the vector of joint angles, $M(q)$ and $C(q,\dot{q})$ are the inertia matrix and Coriolis and centrifugal terms respectively. $K_p$ and $K_d$ are the PD control gains and $q_{des}$ is a vector of inputs to the system.
In the hardware experiment we send several step commands which involve moving all of the joints at once. The goal of using MPC for a system such as this is to use model information to move all of the joints quickly and simultaneosly while reducing the overshoot and oscillation which are characteristic of under-damped systems. Each MPC controller is using a cost function of the form found in Equation \ref{eq:mpc_problem}, however the relative weightings on state and input error are tuned individually for each controller. Each controller used a horizon of 100 time steps and a discretization of .01 seconds, leading to a look-ahead time of one second.
MPC was run on one computer equipped with an intel E5-1603 CPU and Nvidia Titan X GPU while impedance control was run on a separate computer on the same local network.
\subsection{Hardware Experiments - Results}
As can be seen in Figure \ref{fig:baxter_jangles}, both MPC controllers are able to control the robot to the commanded joint positions with little to no overshoot or oscillation. In order to demonstrate the natural underdamped-ness of the system, the step response is also included in Figure \ref{fig:baxter_jangles}. While overshoot and oscillation can also be mitigated using command smoothing techniques, MPC allows for faster rise times than these methods afford.
The performance of the three control methods is quantified using the Integral Time Absolute Error (ITAE). This is defined as
\begin{equation}
ITEA = \int_{t_0}^{t_1} (\tau-t_0)|q_{cmd}-q(t)| d\tau.
\end{equation}
This integral must be performed for each joint and each step input, resulting in 21 values for our experiment with seven joints and three step inputs. For easier comparison, we report the mean and median of these these 21 values for each trajectory in Table \ref{tab:ITAE}. As can be seen in the table, Parameterized OSQPMPC and EMPC reduce the median ITAE by 31\% and 42\% respectively.
The accuracy of the model leads to impressive MPC dynamic performance, however there is slight steady state error. This is due to the fact that the gravity compensation is not perfect. This could be decreased with an integrator or by using higher gains for the low level PD controller, however increasing the PD gains would lead to a stiffer system which is less safe in human environments.
\begin{table}[h!]
\begin{center}
\caption{Integral Time Absolute Error statistics of joint trajectories for hardware experiment}
\label{tab:ITAE}
\begin{tabular}{l|c|c}
\textbf{Controller} & \textbf{Mean ITAE} & \textbf{Median ITAE} \\
\hline
Step Input & 1.61 & 1.30\\
\hline
Parameterized OSQPMPC & 1.29 & .90\\
\hline
EMPC & 1.17 & .75
\end{tabular}
\end{center}
\end{table}
While both controllers were able to successfully control this robot, they were not able to run at the same rate. Figure \ref{fig:baxter_solve_times} shows a histogram of MPC solve times for both EMPC and the parameterized gradient-based MPC. As can be seen from the figure, EMPC is able to solve consistently faster than the gradient-based method and also has far less variation in solve time. These hardware results demonstrate that by parallelizing the MPC problem, and offloading much of the computation to the GPU, MPC solve times can be greatly reduced without great loss of actual performance. This also means that the EMPC solver likely could have been run with a longer horizon if desired while still solving faster than the parameterized gradient-based method.
We should also note that this comparison does not include a traditional formulation of MPC because traditional MPC could not solve fast enough for real-time control given a horizon of 100 time steps. While EMPC enabled very fast MPC solves compared to parameterized gradient-based MPC, the fact that the gradient-based MPC is able to be run in real-time at 100 Hz is only due to the parameterization of the input trajectory that we present in this paper.
\begin{figure*}[hbt]
\centering
\includegraphics[width=1.0\textwidth]{figures/baxter_lempc_jangles.png}
\caption{Joint angle response for the seven joints of the robot during the hardware experiment.}
\label{fig:baxter_jangles}
\end{figure*}
\begin{figure}[hbt]
\centering
\includegraphics[width=1.0\linewidth]{figures/baxter_lempc_solve_times.png}
\caption{Hardware experiment solve times for the parameterized gradient-based MPC and EMPC}
\label{fig:baxter_solve_times}
\end{figure}
\section{Conclusion}\label{conclusion}
In this work we have shown for the presented form of input parameterization in the application of robot manipulators that input parameterization:
\begin{itemize}
\item Produces more conservative trajectories, slightly favoring robustness over aggressive behavior
\item Can be implemented using traditional convex solvers for very fast solve times
\item Enables easily parallelized global optimization methods by drastically decreasing the optimization search space for high-DoF long horizon MPC problems
\end{itemize}
While we have shown these results only for robot manipulators, we expect that the methods presented in this paper may be generalized to other robotic systems.
The input-parameterized convex solver MPC presented in this paper was shown to drastically decrease solve times for high-DoF systems while maintaining long horizons. Many MPC controllers for robotics applications currently use this type of convex solver. We expect that most, if not all, may benefit from the input parameterization presented in this paper. The MPC controllers could either be made to run at higher rates using the same horizon length, or at the same rate using a longer horizon.
We have developed a parallelized form of MPC (EMPC) which can be run using a GPU and have shown that the solutions found using this method are also similar to that of traditional MPC. EMPC is shown to have faster solve times than both traditional MPC and the parameterized convex solver MPC for high DoF systems. Furthermore, experiments performed in this work demonstrate that EMPC solve times can be decreased by using a higher performance GPU. This indicates that with the development of better GPUs, parallelized MPC methods could be run at faster rates, with longer horizons, or could find higher quality solutions.
Another advantage of the EMPC approach which was not explored in detail in this work is the ability to change the cost function or dynamics into a form not previously admitted by MPC solvers. Because the optimization is gradient-free, it is possible to include sharp discontinuities in the cost if desired. Because system states are simulated over the horizon, the cost function could even be a direct function of rise time, settling time and overshoot. Furthermore, the assumption of linear dynamics could be relaxed and nonlinear models could be simulated on the GPU. Although this would lead to many local minima instead of one global minimum, it would more accurately represent a real system and because EMPC employs a global gradient-free optimization, a global minimum could be found.
A current weakness of the EMPC approach, and any sampling-based optimization used for MPC, is that it would be very difficult to prove stability using this controller. In practice we find that because we are able to sample such a large number of trajectories, we are able to find ``good" trajectories fairly quickly. Also, by making the exploration noise ($\sigma_{noise})$ a function of distance to a goal state, we find that once near the goal, EMPC keeps the system near the goal.
It seems that, if desired, statistical likelihood of stability using a sampling-based MPC could be calculated, however a guarantee seems impossible.
In this work we have only explored one form of parameterization (piecewise linear functions). This parameterization is linear and so was able to be fit into the form required by gradient-based solvers, however many different parameterizations are possible using the EMPC approach. Future work may include exploring parameterizations such as combinations of learned basis functions or other parameterizations of open-loop trajectories.
If a parameterization is chosen for a feedback gain matrix it may even be possible to use a parallelized MPC solver such as EMPC to search the space of feedback controllers or policies directly. This could potentially lead to an MPC solver which returns a feedback controller which can be run at a high rate to minimize a given cost function over a finite horizon. While this idea is similar to iLQR or DDP, the EMPC based approach enables the global gradient-free search of a nonlinear space which may find better solutions than the gradient-based search used in iLQR and DDP.
Future work may also include other parallelizable optimization methods apart from the evolutionary strategy used in this work and implementations to other robotic platforms. The results presented in this work provide evidence that through the use of parameterization and parallelization, MPC may be applied to many high DoF systems for which it was not previously tractable.
\section*{Acknowledgements}
This work was partially funded by NSF Emerging Frontiers in Research and Innovation grant \# 1935312.
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 7,902
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\section{Introduction}
Electro-thermal behavior is a key ingredient for understanding charge carrier transport phenomena in semiconductor devices including two-dimensional (2D) materials, hetero Junctions, and strong correlated systems \cite{Aninkevicius.V,Matulionis.A,Levi.A,Atar.F,Pumarol.M,Lan.Y,Guo.Z,Lee.S,Zhang.P,Li.Z,Lee.H,Lee.Y,Shao.W,Konstantinova.T,Wu.K}. In small devices on nanoscales, hot electron generation and the resulting characteristic interaction with the host crystal lattice (or phonons) complicates the electro-thermal analysis and limits the device performance \cite{Tisdale.W,Ashalley.E}. Whereas knowing the detailed local profile of the electron effective temperature, $T_{e}$, separately from that of the lattice temperature, $T_{L}$, in the presence of current is prerequisite for understanding the transport characteristics on nanoscales \cite{Dubi.Y,Hartmann.M,Mahajan.R,Brongersma.M,Peng.S},$T_{e}$ has been experimentally hardly accessible \cite{Reparaz.J,Xu.Y,Zhang.X,Nonnenmacher.M,Harzheim.A,Yalon.E,Cui.L,Shi.L,Mecklenburg.M} until quite recently \cite{Weng.Q,Komiyama.S}. It follows that the study of electro-thermal properties has so far been restricted only to the simulation methods such as those of Monte Carlo (MC) simulation based on the Boltzmann transport equations, hydrodynamic equations or molecular dynamics \cite{SINHA.S,Vasileska.D,Fan.A,Garcia.S,Wang.Y}. Unfortunately, however, MC simulations comprise involved calculation procedures, which are not necessarily convenient to gain intuitive understanding of the electro-thermal transport phenomena of given devices. On the other hand, the nonequilibrium condition cannot be incorporated in commercially available semiconductor device simulators.
Here, we propose a simplified electro-thermal model for layered conductors on the basis of the assumption that the electron- and the lattice-subsystems are, respectively, in quasi-equilibrium states characterized by the effective electron temperature $T_{e}$ and the lattice temperature $T_{L}$. The model is applied to a U-shaped layered conductor, where electric field is concentrated at the inner corner of the U-shaped bend. In a semicodncuctor device, simulating GaAs, remarkable hot electron distribution ($T_{e}\gg T_{L}$) is found to develop at the corner, forming a sharp hot spot with $T_{e}$ reaching $\sim2000K$. Differently, in metal devices, simulating NiCr, hot electron effects are found to be absent ($T_{e} \approx T_{L}$), whereas a hot spot profile is visible. These findings are consistent with recent experimental results reported on metals \cite{Weng.QC} and semiconductors \cite{Weng.Q}, indicating the validity of the present model for simulating the electro-thermal behavior of layered conductors in nonequilibrium conditions.
\section{Simulation model}
Figure 1 describes the model conductor considered in this study. A layered conductor with the electric conductivity $\sigma_{e}$ is deposited on an insulating substrate, which is anchored by the heat sink at 300K. The lateral shape of the conductor is arbitrary, so that the electric field $\mathbf{E}$, the current density $\mathbf{j}$, the electron temperature $T_{e}$ and the lattice temperature $T_{L}$ in the conductor are variables to be consistently derived as functions of the lateral position $\mathbf{r}$ for a given conductor with a given bias voltage. In the conductor electrons gain energy from $\mathbf{E}$ through $P=\mathbf{j}(\mathbf{r})\cdot \mathbf{E}(\mathbf{r})=\sigma_{e}E^{2}$ and the energy gained from the field is, in turn, released to the lattice via electron phonon interaction, characterized by the electron-phonon energy relaxation time $\tau_{e-ph}$. The excess energy (or heat) of electrons is transferred, as well, within the electron system through the electron thermal conduction $-\kappa_{e}\bigtriangledown T_{e}$ with $\kappa_{e}$ the electron thermal conductivity. The heat is transferred similarly within the lattice system through lattice thermal conduction $-\kappa_{L}\bigtriangledown T_{L}$ with $\kappa_{L}$ the lattice thermal conductivity. The heat is eventually transferred to the substrate $(T_{L}-T_{LS})/h_{I}$ with $h_{I}$ being the interface thermal resistance and $T_{LS}(\mathbf{r})$ the local lattice temperature of the substrate on its top surface, and finally absorbed by the heat sink. Heat is transferred as well through electrical leads connected to the conductor, as represented by the arrows marked with $\kappa_{e}$ and $\kappa_{L}$ in fig.1 (c), which is taken into account in the model through an appropriate boundary condition as mentioned below for fig.2 (a).
\begin{figure}
\includegraphics[width=1.0\linewidth]{fig1.eps}
\caption{Model for electro-thermal analysis. (a) Physical structure of the sample. (b) Simulated structure, in which the electron- and the lattice-systems of the conductor are separated to thermally contacted two layers. (c) Diagram of energy/heat flow.}
\label{fig.1}
\end{figure}
As schematically shown in figs.1 (b) and (c), our model represents the energy transfer from the electron system to the lattice system in the conductor by the interface heat transfer between the electron sublayer at $T_{e}$ to the lattice sublayer at $T_{L}$. The energy flux released from the electron system to the lattice system through the electron phonon interaction is given by
$$P_{e-ph}=(T_{e}-T_{L})C_{e}/\tau_{e-ph}\eqno{(1)}$$
with $C_{e}$ the electron specific heat per unit area, so that the effective interface thermal resistance $h_{e-ph}$ is
\
$$h_{e-ph}=\tau_{e-ph}/C_{e}.\eqno{(2)}$$
\
The specific heat is approximated by
\
$$C_{e}=C_{ec}=(3/2)N_{2D}k_{B},\eqno{(3)}$$
\
for a classical electron system ($k_{B}T_{e}\gg \varepsilon_{F}$) and by
\
$$C_{e}=C_{eF}=\{(3/2)k_{B}T_{e}/\varepsilon_{F}\}C_{ec}\eqno{(4)}$$
\
for an electron system with the Fermi energy $\varepsilon_{F}$ much higher than the thermal energy ($k_{B}T_{e}\ll \varepsilon_{F}$). Here, $k_{B}$ is the Boltzmann constant, and $N_{2D}$ is the 2D electron density.
\section{Simulated structure}
\begin{figure}
\includegraphics[width=1.0\linewidth]{fig2.eps}
\caption{(a) Simulated sample structure, where the electron- and the lattice-systems of a U-shaped conducting channel of a thickness $t=35nm$ are separately represented, respectively, by Layers A and B. The conducting channels with a width of $w=1\mu m$ $(r=1\mu m)$ and a length of $l=4\mu m$ extend from the two $5\times7\mu m^{2}$ contact pads. The radius of curvature of the inner corner of the U-shape is 15nm, making a gap of $S=30nm$ between the two channels. Bias voltage $V_{b}$ is defined as the voltage difference between the end faces of Layer A (marked by yellow). (b) and (c) Simulated electric field distribution and temperature distribution in the electron system (Layer A) of n-GaAs.}
\label{fig.2}
\end{figure}
As schematically illustrated in fig.2 (a), we consider a 35nm thick conductor layer shaped into a $1\mu m$-wide U-shaped channel with the radius of inner curvature of the U-shape is 15nm and the gap between the channels is $S=30nm$. For the simulation, the electron- and the lattice-systems of the conducting channel are separately represented by Layers A and B, where Layer B is placed on the $10\mu m$-thick substrate (Layer C). The boundary condition of temperature is given by assuming $T=300K$ on the bottom face Layer C and on the end faces of Layers A and B as marked by the orange lines in fig.2 (a). As to the bias condition, a constant voltage is assumed on each end face of the conducting channel (Layer A), and a bias voltage $V_{b}$ is assumed to give the voltage difference between the two end faces.
Two different conductors are considered. One is a doped n-GaAs channel and the other is a NiCr channel, similar to those studied, respectively, in Refs.32 and 39. Substrates are assumed to be lattice-matched GaAs/AlGaAs for n-GaAs sample \cite{Weng.Q} and single crystal Si covered with a thin $SiO_{2}$ layer for NiCr sample \cite{Weng.QC}. The electron density in n-GaAs and NiCr samples are, respectively, $N_{3D}=3.3\times10^{24}/m^{3}$ and $1.0\times10^{30}/m^{3}$; in terms of the sheet electron density, $N_{2D}=1.1\times10^{17}/m^{2}$ and $3.5\times10^{22}/m^{2}$. The specific heat is taken to be $C_{e}=2.3\times10^{-6}Ws/(Km^{2})$ and $1.1\times10^{-2}Ws/(Km^{2})$, respectively assuming Eqs. (3) and (4) for n-GaAs and NiCr samples. In n-GaAs sample, the interface thermal resistance ($h_{I}$) is negligibly small because the n-GaAs conducting layer is epitaxially grown on the lattice matched substrate. The electron-phonon energy relaxation time is assumed to be $\tau_{e-ph}=1ps$ and 3ps, respectively for n-GaAs \cite{Weng.Q} and NiCr \cite{Weng.QC}. Parameter values used are summarized in Table 1.
\begin{table*}[htbp]
\caption{Parameters used in the simulation.}
\label{tab.1}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Quantity & $\sigma_{e}$ & $\kappa_{e}$ & $\kappa_{L}$ & $h_{e-ph}$ & $h_{I}$ & $\kappa_{S}$\\\hline
Unit & $S/m$ & $W/(m\cdot K)$ & $W/(m\cdot K)$ & $Km^{2}/W$ & $Km^{2}/W$ & $W/(m\cdot K)$\\\hline
n-GaAs & $8.8\times 10^{4}$ & 0.1 & 50 & $4.3\times10^{-7}$ & 0 & 50\\\hline
NiCr & $2.89\times 10^{5}$ & 15 & 1 & $2.7\times10^{-10}$ & $3\times10^{-8}$ & 150\\\hline
\end{tabular}
\end{center}
\end{table*}
\section{Results and discussions}
Joule heating caused by electric field and thermal conduction generated by temperature gradient or difference are consistently treated by using a commercial multiphysics software (COMSOL), where the bias voltage is taken to be $V_{b}=4.5V$ in all the calculations described below. It is a common feature of both n-GaAs and NiCr samples that the electric field is concentrated around the U-shaped inner corner as exemplified by the result for n-GaAs sample: Electric field is nearly uniform and $E=2\sim3kV/cm$ in a region away from the U-shaped corner, but rapidly increases to reach about $E=10kV/cm$ in the vicinity of the U-shaped inner corner. As a consequence of this $E$-field enhancement, remarkable nonuniform hot-electron distribution is found to be generated at the inner corner of the n-GaAs sample as shown in fig.2 (c). While the trend of the $E$-field enhancement is substantially the same in the NiCr sample, resulting temperature distribution in the electron- and the lattice-systems is largely different as described in detail below.
\begin{figure}
\includegraphics[width=1.0\linewidth]{fig3.eps}
\caption{(a) and (b) Temperature distributions of electrons and the lattice in the NiCr sample. (c) and (d) Temperature distributions of electrons and the lattice in the n-GaAs sample. White lines indicate the borders of conducting layers.}
\label{fig.3}
\end{figure}
Figures 3 (a), (b) and figures 4 (a)-(d) display the distributions of $T_{e}$ (Layer A) and $T_{L}$ (Layers B) for the NiCr sample. The profile of $T_{e}$ is similar to that of $T_{L}$, and both exhibit spatially varying heating in accord with the $E$-field enhancement peaked at the U-shaped inner corner. The highest temperature at the hot spot is about $150^{\circ}$C above the heat sink (300K). The amplitude of the temperature rise at the hot spot ($\bigtriangleup T_{e}\approx150^{\circ}$C) as well as the quasi-equilibrium feature between the electron- and the lattice-systems ($T_{e}\approx T_{L}$) substantially reproduce the experimental findings reported in Ref.39.
\begin{figure}
\includegraphics[width=1\linewidth]{fig4.eps}
\caption{One-dimensional plots of $T_{e}$ and $T_{L}$, along the black dashed arrows shown in the inset, for the NiCr sample (a)-(d) and for the n-GaAs sample (e)-(h).}
\label{fig.4}
\end{figure}
The feature of the hot spot formation is largely different in the n-GaAs sample as shown in figs.3 (c)(d) and figs.4 (e)-(h). The electron temperature $T_{e}$ is much higher than the lattice temperature $T_{L}$, indicating nonequilibrium hot electron generation, and it assumes a very sharp prominent peak reaching as high a value as $T_{e}\sim 2000K$. On the other hand, the highest value of $T_{L}$ ($<330K$) is at most only $\sim30^{\circ}$C above the temperature of the heat sink (300K). In addition, the hot-spot feature is practically missing as evident in figs.4 (f) and (h). The generation of remarkable hot electron distribution is consistent with the experimental finding reported on n-GaAs constriction devices \cite{Weng.Q}.
The large difference in the electro-thermal properties noted between the n-GaAs sample and the NiCr sample in this study is suggested to be generally inherent to semiconductors and metals. When energy flux $P$ is fed to the electron system in a steady state, the electrons are heated above the lattice temperature by
\
$$T_{e}-T_{L}=(\tau_{e-ph}/C_{e})P=h_{e-ph}P\eqno{(5)}$$
\
if temperature gradient is ignored. For a given $P$, the rise of $T_{e}$ is proportional to $\tau_{e-ph}$, and $1/C_{e}$. In general, $\tau_{e-ph}$ is not largely different between semiconductors and metals, but the heat capacity $C_{e}$ is by orders of magnitude smaller in semiconductors because the electron density is far lower. It follows that the electron system is readily driven away from the equilibrium with lattice in semiconductors. In terms of our model ($h_{e-ph}$), thermal contact between the electron- and the lattice-systems are weak in semiconductors so that they are readily driven out of equilibrium. We mention that a high mobility of electrons is often ascribed to be the cause of hot electron generation in semiconductors. The present study makes this assumption questionable; namely, a high mobility implies a high electrical conductivity (and a large $P$), but the electrical conductivity is usually higher in metals and does not explain why semiconductor is more feasible for hot electron generation.
In this study simulation calculation assumed linear transport. Namely the electrical conductivity and the thermal conductivities are assumed to be constants. In metals, nonlinear effects may not be significant since the $T_{e}$ rise is not too large. In the doped n-GaAs sample at room temperature (as in this work), nonlinear effects may not be serious up to $E\approx10kV$ \cite{Weng.Q}, so that the findings in the present study are supposed to be valid. In the higher $E$ region above $10kV/cm$, however, the electron mobility will be reduced due to the transfer of electrons to upper (X and/or L) valleys. Even in such a higher-$E$ region, our model will provide a useful guideline at the starting point.
\section{Summary}
We demonstrate that in layered conductors nonequilibrium nature between the electrons and the lattice can be explicitly pursued by separating the electron- and the lattice-subsystems into two physical layers that exchange heat at the interface. Highly nonequilibrium distribution of electrons from that of the lattice is found in a doped n-GaAs sample. In a NiCr sample with a similar configuration, the electron- and the lattice-systems are in quasi-equilibrium. Remarkable difference of the electro-thermal properties of semiconductors and metals is suggested to arise from the difference in the electron specific heat. This work provides a simple and convenient method for modeling layered conductors in a nonequilibrium condition, and will give useful hints for deeper understanding of the nonequilibrium properties of electrical conductors
\acknowledgments
We acknowledge funding support from National Key Research Program of China under grant No. 2016YFA0302000, National Natural Science Foundation of China under grant Nos. 11674070/11427807/11634012, and Shanghai Science and Technology Committee under grant Nos.18JC1420402, 18JC1410300, 16JC1400400. S.K. acknowledges support by the Chinese Academy of Sciences Visiting Professorships for Senior International Scientists.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,334
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\section{Introduction}
The improvement of the sensitivity is
a key objective for sensing technologies.
Recent developments have enabled us to perform high precision sensing in variety of areas such as in life science \cite{le2013optical,taylor2016quantum},
investigation of semiconductor devices \cite{iwasaki2017direct}, and study of the condensed matter physics
\cite{casola2018probing}.
In particular, the detection of a weak magnetic field with the use of quantum technologies has been attracting great attentions.
Quantum metrology using qubits \cite{CavesInterfero1981,giovannetti2004quantum,QuantumMetrology2006,RevQSens2017RevMod} is
an essential
technique to improve the estimation precision.
Typical sensing scheme consists of the three procedures: preparing the probe qubits in a specific quantum state, exposing the state to the target magnetic field, and performing a measurement for the readout.
It is well known that the standard quantum limit (SQL) bounds the sensitivities when we prepare separable states for the probe qubits \cite{giovannetti2004quantum}.
On the other hand,
by exploiting quantum properties of entanglement among the probe qubits, the bound can be relaxed to the Heisenberg limit (HL) \cite{BollingerHL1996,leibfried2004toward,QuantumMetrology2006}.
The challenge of the entanglement-enhanced sensing is to develop a practical method for preparing useful entangled states between the probe qubits.
For the entanglement-enhanced sensing of a magnetic field, we need both
the strong coupling with the target magnetic field and controllability of the qubits.
The former is essential to achieve a better sensitivity, while the latter is crucial for the creation of the entangled states.
However,
as the coupling with the target magnetic field is increased, decoherence rate of the
qubit due to the unavoidable coupling with the environment typically increases, resulting in poor controllability of the qubit.
This means that it is difficult to
accurately control the qubits for the magnetic field sensing.
Because of this difficulty, although a great number of methods requiring high controllability
have been proposed,
practical entanglement-enhanced sensing is still challenging in this area.
Typical entangled state for quantum metrology is the Greenberger-Horne-Zeilinger (GHZ) state \cite{greenberger1990bell,MerminGHZ1990}.
The GHZ state can be created by using a sequence of gate operations \cite{GHZin4NVneumann2008,jones2009magnetic,GHZ3preparationDicarlo2010,neeley2010generation,GHZbarends2014superconducting,GHZin4NV2015APL,GHZ18VerifyingWei2020}.
However, entangling gate operations such as
controlled-NOT (CNOT)
gates require an accurate control of the interaction between
qubits.
Such a requirement of the high controllability could be a bottleneck for the practical entanglement enhanced sensing.
In the present paper, we propose an entanglement-enhanced sensing protocol to measure a magnetic field with an always-on nearest-neighbor interaction.
Our scheme does not require either entangling gate operations or switching on/off the interaction among qubits.
More specifically, we consider the one-dimensional spin chain with a nearest-neighbor ferromagnetic Ising interaction accompanied by the homogeneous transverse magnetic field for control.
In our protocol, performing a single-qubit measurement on one of the edges of the chain at equilibrium induces unitary dynamics due to the intrinsic Hamiltonian of the system, and this
generates an entangled state suitable for quantum metrology.
When we expose our probe qubits to the target magnetic field, the interaction is still on; we just need to turn on/off global magnetic fields.
Furthermore, the readout for the estimation of the target magnetic field can be implemented with a single qubit measurement on the edge of the chain.
Our protocol does not require
complicated operations such as
turning on/off the interaction,
which is in stark contrast to the conventional scheme with GHZ states that requires precise control of the interaction.
This paper is organized as follows.
In Sec.~\ref{sec:setting}, we review concepts of the Ramsey-type quantum sensing protocol and the transverse field Ising chain.
In Sec.~\ref{sec:protocol}, we illustrate our protocol in an analytical way and compare it with the conventional protocol.
In Sec.~\ref{sec:numRES} we then numerically calculate the uncertainty to estimate the target magnetic field in our protocol without the presence of noise.
In Sec.~\ref{sec:NMnoise}, we consider the effect of dephasing, and numerically investigate the performance of our protocol with dephasing.
We summarize our paper in Sec.~\ref{sec:disc}.
\section{Setting}\label{sec:setting}
\subsection{Quantum sensing with separable states}
We here briefly review the Ramsey-type-quantum sensing protocol with $L$ separable qubits to probe a magnetic field along the $z$ axis \cite{RevQSens2017RevMod}.
Throughout the paper we take $\hbar=1$.
First, prepare qubits in a separable state $\bigotimes_{n=1}^{L} \ket{+}_n$, where $\ket{+}_n$ denotes the eigenstate of ${\hat \sigma}_{n}^x$ with the eigenvalue $+1$.
Here, $\hat{\sigma}^{\nu}_n$, $\nu\in\{x,y,z\}$ denotes Pauli spin operator on site $n$.
Second, let the state interact with the static and homogeneous target magnetic field.
The Hamiltonian to describe the interaction with the target magnetic field is
\begin{align}
{\hat H}_{\omega}&=\frac\omega2 \sum_{n=1}^{L}{\hat \sigma}^{z}_{n},\label{eq:Homega}
\end{align}
where $\omega$ denotes a frequency shift due to the target magnetic field, and $n$ denotes the sites of the probe qubits.
Third, perform a projective measurement ${\hat P}_{n,\mu}^{+}=({\hat \sigma}^{\mu}_{n}+1)/2$ ($\mu\in\{x,y\}$) with $\mu =y$ on each qubit.
Finally, repeat these three steps, and we estimate the frequency shift $\omega$ due to the target field
from the distribution of the outcomes.
The uncertainty of the estimated $\omega$ for each qubit is obtained as
\begin{align}
\delta \omega =\frac{\sqrt{P(1-P)}}{\left|\frac{\partial P}{\partial \omega}\right|\sqrt{M}},
\label{eq:errorOmegaDef}
\end{align}
where $P$ denotes a probability of inducing the projection of ${\hat P}_{1,y}^{+}$, and $M$ denotes the experimental repetition number \cite{RevQSens2017RevMod,takeuchi2019quantum}.
This probability is calculated as $P=(1 + \sin{\omega T_{\rm int}})/2$,
where $T_{\rm int}$ denotes the duration time for which we expose the probe to the target field.
For a total available time $T_{\rm all}$, the number $M$ is calculated as $M=T_{\rm all}/T_{\rm sensing}$, where $T_{\rm sensing}$ denotes a combined time of the three procedures of a sensing protocol, i.e.,
$T_{\rm sensing}:= T_{\rm reset} + T_{\rm prep} +T_{\rm int}+ T_{\rm read}$.
Here, $T_{\rm reset}$ denotes the duration time for initializing the probe system, $T_{\rm prep}$ denotes the time that is required for creating a metrologically useful state from this initial state, and $T_{\rm read}$ denotes the time to readout the phase information acquired in the quantum state during
the exposure to the magnetic field.
Since the state is separable and consists of $L$ individual qubits, the uncertainty $\delta \omega$ scales as $L^{-1/2}$ because of the central limit theorem.
We assume that the interaction time accounts for a large fraction of the sensing time, i.e., $T_{\rm sensing}\simeq T_{\rm int}$.
Actually, we obtain $\delta \omega=\left({L T_{\rm all}T_{\rm int}}\right)^{-1/2}$, which shows the SQL.
The scaling of $\delta \omega$ can be improved to $L^{-1}$ if we appropriately exploit an entanglement among the qubits,
as we describe below.
\subsection{Quantum sensing with the GHZ state}\label{sec:RamseyGHZ}
Next, we illustrate quantum sensing protocol with the entangled state \cite{HuelgaMarkov1997,RevQSens2017RevMod}.
For the sake of the notation, we define a CNOT gate between qubits on site $n$ and site $n+1$ as
${\rm CNOT}_{n,n+1}:=\left(1-\hat{\sigma}_{n}^{z}+ (1+\hat{\sigma}_{n}^{z})\hat{\sigma}_{n+1}^{x} \right)/2$.
A typical protocol to create the GHZ state by gate operations is summarized as follows.
(i) prepare an $L$-qubit state
$\bigotimes_{n=1}^{L}\ket{0}_{n}$,
where $\ket{0}_n$ ($\ket{1}_n$) denotes the eigenstate of ${\hat \sigma}_{n}^{z}$ with the eigenvalue $-1$ ($+1$).
(ii)
implement a Hadamard gate on the first qubit and perform a sequence of CNOT gates between adjacent qubits
in order to create the GHZ state $\ket{\psi}$, i.e.,
\begin{align}
\ket{\psi}=\frac1{\sqrt{2}}\left(\bigotimes_{n=1}^{L}\ket{0}_{n}+\bigotimes_{n=1}^{L}\ket{1}_{n}\right)
\end{align}
Here, the gates of
${\rm CNOT}_{1,2}$, ${\rm CNOT}_{2,3}$, $\cdots$, and ${\rm CNOT}_{L-1,L}$ are performed in sequence.
(iii)
expose the state to the target magnetic field (\ref{eq:Homega}) for a time $T_{\rm int}$, and obtain the state with a phase shift, i.e.,
\begin{align}
\ket{\psi(T_{\rm int})}=\frac{{\rm e}^{i \omega L T_{\rm int}/2}}{\sqrt{2}}\left(\bigotimes_{n=1}^{L}\ket{0}_{n} + {\rm e}^{-i \omega L T_{\rm int}}\bigotimes_{n=1}^{L}\ket{1}_{n}\right).\label{eq:psiTintConv}
\end{align}
(iv)
implement a sequence of CNOT gates again on the qubits, and obtain a disentangled state
\begin{align}
\ket{\psi'(T_{\rm int})}=\frac{{\rm e}^{i L\omega T_{\rm int}/2}}{\sqrt{2}}\left(\ket{0}_{1} + {\rm e}^{-i L \omega T_{\rm int}}\ket{1}_{1}\right)\otimes\bigotimes_{n=2}^{L}\ket{0}_{n}.
\end{align}
Here, the CNOT gates are performed in a reverse order compared with the case in the step (ii).
More specifically, the gates of
${\rm CNOT}_{L-1,L}$, ${\rm CNOT}_{L-2,L-1}$, $\cdots$, and ${\rm CNOT}_{1,2}$ are performed in sequence.
(v)
measure the first qubit in the ${\hat \sigma}_{1}^{y}$ basis and obtain an outcome of either $+1$ or $-1$.
The combination of the steps (iv) and (v) effectively measures the probability of projecting $\ket{\psi(T_{\rm int})}$ in Eq.~(\ref{eq:psiTintConv}) to $\left(\bigotimes_{n=1}^{L}\ket{0}_{n} + i\bigotimes_{n=1}^{L}\ket{1}_{n}\right)/\sqrt{2}$.
By repeating these steps, we obtain a distribution of the outcomes, and we can estimate the value of $\omega$.
The probability of obtaining an outcome of $+1$ in the ${\hat \sigma}_{1}^{y}$ basis is then calculated as
\begin{align}
P &= \bra{\psi'(T_{\rm int})} {\hat P}_{1,y}^{+} \ket{\psi'(T_{\rm int})} \nonumber\\
&= \frac12 + \frac12\sin{L\omega T_{\rm int}}.
\label{eq:ghzghz}
\end{align}
Using Eq.~(\ref{eq:errorOmegaDef}), the uncertainty of $\omega$ is now obtained as
$\delta \omega = L^{-1}\left(T_{\rm int}T_{\rm all}\right)^{-1/2}$.
This is the HL, which is $L^{-1/2}$ times smaller than the SQL.
The probability $P$ is linear in $\omega$ for $L\omega T_{\rm int} \ll 1$, which is suitable for measuring a weak magnetic field $\omega\ll1$.
Let us estimate the required time for the GHZ state generation
when we use gate operations.
To implement a gate of ${\rm CNOT}_{n,n+1}$
to qubits
in a system with nearest-neighbor Ising interactions and magnetic fields,
we can use
Hadamard gates and a {\rm CZ} ({\rm CPHASE}) gate.
These correspond to the unitary dynamics induced by Hamiltonians of
$\hat{H}_{\rm H}^{(n)}:=h_{\rm H}(\hat{\sigma}^x_{n}+\hat{\sigma}^z_{n})/\sqrt2$ and $\hat{H}_{\rm CZ}^{(n,n+1)}:=(J_{\rm CZ}/4)\left( 1-\hat{\sigma}_{n}^{z} + (1+\hat{\sigma}_{n}^{z})\hat{\sigma}_{n+1}^{z} \right)$, where $J_{\rm CZ}$ denotes the interaction strength and $h_{\rm H}$ denotes the strength of the magnetic field.
More specifically, the CNOT gate can be described as
\begin{align}
{\rm CNOT}_{n,n+1}=&\exp\left(i\frac{\pi}{2h_{\rm H}}\hat{H}_{\rm H}^{(n+1)}\right)\nonumber\exp\left(i\frac{\pi}{J_{\rm CZ}}\hat{H}_{\rm CZ}^{(n,n+1)}\right)\\
&\cross
\exp\left(i\frac{\pi}{2h_{\rm H}}\hat{H}_{\rm H}^{(n+1)}\right).\label{eq:cnot}
\end{align}
Hence
the necessary time is $\pi/h_{\rm H}+\pi/J_{\rm CZ}$.
We show a schematic picture of the conventional protocol in this system
in Fig.~\ref{fig:compare}
\begin{figure}
\centering
\includegraphics[width=0.92\columnwidth]{fignewADD.png}
\caption{
Schematic picture of required operations in the conventional protocol (i)--(v).
Initial state $\ket{\phi_0}$ in the picture denotes $\ket{\phi_0}:=\otimes_{n=1}^{L}\ket{0}_{n}$.
We assume that the system has nearest-neighbor Ising interactions and
gate operations consist of a sequence of CNOT gates, which can be expressed as in Eq.~(\ref{eq:cnot}).
}
\label{fig:compare}
\end{figure}
\subsection{Transverse field Ising chain}
We now introduce the transverse field Ising chain \cite{LIEB1961,PFEUTY1970}.
The Hamiltonian is described as follows:
\begin{align}
{\hat H}_{\rm TFI}&={\hat H}_{\rm Ising} + {\hat H}_{\rm x}, \label{eq:Htfi}\\
{\hat H}_{\rm Ising}&= - \frac{J}4\sum_{n=1}^{L-1}{\hat \sigma}^{z}_{n}{\hat \sigma}^{z}_{n+1},\label{eq:Hising}\\
{\hat H}_{\rm x}&=\frac{h_{x}}2\sum_{n=1}^{L}{\hat \sigma}^{x}_{n}, \label{eq:Htransverse}
\end{align}
where $J>0$ denotes the strength of the ferromagnetic interaction and $h_{x}$ denotes the magnitude of the transverse magnetic field.
Without loss of generality, we assume $h_{x}>0$.
The model exhibits a quantum phase transition at $h_{x}/J=1/2$ at the zero temperature in the thermodynamic limit and shows ferromagnetic order in the $z$ direction for $h_{x}/J<1/2$.
For a finite $L$ and $h_{x}/J<1/2$, the system has two almost degenerate ground states with an exponentially small energy difference.
More specifically, the ground state and the first excited state can be approximated as
$\left(\bigotimes_{n=1}^{L}\ket{0}_{n}\pm\bigotimes_{n=1}^{L}\ket{1}_{n}\right)/\sqrt2$.
with the energy difference which is exponentially small in $L$ \cite{kitaev2001unpaired}.
The excited states are separated from them by a finite energy gap $\sim J/2-h_{x}$.
Throughout of this paper,
we assume that we use thermal equilibrium states as initial states unless specifically mentioned.
For a finite system at equilibrium with an inverse temperature $\beta$, the thermal equilibrium state ${\hat \rho}_{\beta} := {\rm e}^{-\beta {\hat H}_{\rm TFI}} / {\rm Tr }[{\rm e}^{-\beta {\hat H}_{\rm TFI}} ]$ can be well approximated by the mixed state
\begin{align}
{\hat \rho}_{\beta}\simeq{\hat \rho}_{\rm mix}:=&\frac12\left(\bigotimes_{n=1}^{L}\ket{0}_{n}\right)\left(\bigotimes_{n=1}^{L}\bra{0}_{n}\right)
\nonumber\\
&+
\frac12\left(\bigotimes_{n=1}^{L}\ket{1}_{n}\right)\left(\bigotimes_{n=1}^{L}\bra{1}_{n}\right)
\label{eq:RhoMix}
\end{align}
for $h_{x}/J\ll 1/2$ and $1/\beta \ll J/2-h_{x}$.
More specifically, we should decrease $1/\beta$ as we increase $L$ because the probability of having the ground states in $\hat{\rho}_{\beta}$ becomes extremely small for a large $L$.
In Sec.~\ref{sec:protocol},
we will assume that the temperature is sufficiently low in our protocol so that the condition of
${\hat \rho}_{\beta} \sim {\hat \rho}_{\rm mix}$ should approximately hold.
To illustrate how low the temperature should be for satisfying the condition ${\hat \rho}_{\beta} \sim {\hat \rho}_{\rm mix}$,
we calculate a fidelity
$F=F({\hat \rho}_{\beta},{\hat \rho}_{\rm mix})={\rm Tr}\left[\left({\hat \rho}_{\beta}^{1/2}{\hat \rho}_{\rm mix}{\hat \rho}_{\beta}^{1/2}\right)^{1/2}\right]$,
and obtain $F=87\%$ for $\beta=10$, $h_{x}=0.1$ and $L=12$.
We can prepare the thermal equilibrium state just by using the energy relaxation process from the environment, and so a precise control is not required.
\subsection{Quantum domino dynamics }\label{sec:DominoDynamics}
\begin{figure*}
\centering
\includegraphics[width=1.93\columnwidth]{fig1new.png}
\caption{Quantum domino dynamics induced by the Hamiltonian (\ref{eq:Hsecular}) with the initial state $\ket{1}_{1}\otimes\bigotimes_{n=2}^{L}\ket{0}_{n}$.
Black bars show the magnetization at each site $n$ with time steps of $t=0,t_{\rm ideal}^{*}/2,t_{\rm ideal}^{*},3t_{\rm ideal}^{*}/2$, and $2t_{\rm ideal}^{*}$.
We choose $L=20$ and $h_{x}t_{\rm ideal}^{*}=1.05L$ here.
}
\label{fig:domino}
\end{figure*}
We review a concept of ``quantum domino'' dynamics in the transverse field Ising chain, which was theoretically discussed in Ref.~\cite{SpinAmpLee2005,SpinAmpFurman2006PRB,SpinAmpBalachandran2009cat,AmpRapiRobaClose2011} and demonstrated in Ref.~\cite{SpinAmpLee2007NJP}.
It is observed when we prepare
a state $\ket{1}_{1}\otimes\bigotimes_{n=2}^{L}\ket{0}_{n}$ as the initial state and let the state evolve according to the Hamiltonian (\ref{eq:Htfi}) with a weak transverse magnetic field; the qubit-flip on the first site propagates, and this induces a sequence of flipping across the system up to the ($L-1$)-th qubit.
In short, quantum domino dynamics can approximately realize the following transformation when we appropriately tune the evolution time:
\begin{align}
{\hat U} \left(\ket{1}_{1}\otimes\bigotimes_{n=2}^{L}\ket{0}_{n}\right) \simeq \left(\bigotimes_{n=1}^{L-1}\ket{1}_{n}\right)\otimes\ket{0}_{L}\label{eq:IdealAmp1},\\
{\hat U} \left(\ket{0}_{1}\otimes\bigotimes_{n=2}^{L}\ket{1}_{n}\right) \simeq \left(\bigotimes_{n=1}^{L-1}\ket{0}_{n}\right)\otimes\ket{1}_{L}\label{eq:IdealAmp2},
\end{align}
where ${\hat U}$ denotes the unitary dynamics due to the Hamiltonian (\ref{eq:Htfi}).
On the other hand, if all qubits are initialized in the same direction, for instance
$\bigotimes_{n=1}^{L}\ket{0}_{n}$
, the system stays almost in the same state i.e.,
${\hat U}\bigotimes_{n=1}^{L}\ket{0}_{n} \simeq \bigotimes_{n=1}^{L}\ket{0}_{n}$
Therefore, we can approximately generate the GHZ state of $(L-1)$ qubits if we induce the quantum domino dynamics with an initial state of $\ket{+}_{1}\otimes\bigotimes_{n=2}^{L}\ket{0}_{n}$ \cite{SpinAmpBalachandran2009cat}.
Importantly,
the quantum domino dynamics can also occur in the opposite direction, i.e.,
\begin{align}
{\hat U} \left[\left(\bigotimes_{n=1}^{L-1}\ket{1}_{n}\right)\otimes\ket{0}_{L}\right] \simeq \ket{1}_{1}\otimes\bigotimes_{n=2}^{L}\ket{0}_{n}\label{eq:IdealAmp1REVERSE},\\
{\hat U} \left[\left(\bigotimes_{n=1}^{L-1}\ket{0}_{n}\right)\otimes\ket{1}_{L}\right] \simeq \ket{0}_{1}\otimes\bigotimes_{n=2}^{L}\ket{1}_{n}\label{eq:IdealAmp2REVERSE}.
\end{align}
These equations now imply that the entangled state ${\hat U}\left( \ket{+}_{1}\otimes\bigotimes_{n=2}^{L}\ket{0}_{n}\right)$ can go back to the initial state approximately by applying ${\hat U}$ again.
We will refer to the dynamics (\ref{eq:IdealAmp1REVERSE}) and (\ref{eq:IdealAmp2REVERSE}) as well as (\ref{eq:IdealAmp1}) and (\ref{eq:IdealAmp2}) as
ideal domino dynamics.
The emergence of the quantum domino dynamics in the transverse field Ising chain
can be explained as follows.
Regarding ${\hat H}_{\rm x}$ as a perturbative term, the Hamiltonian in the interaction picture is described as ${\rm e}^{i{\hat H}_{\rm Ising}t}{\hat H}_{\rm x}{\rm e}^{-i{\hat H}_{\rm Ising}t}$.
By using a secular approximation to ignore oscillating terms with a high frequency of $J$ \cite{SpinAmpLee2005}, we obtain the following Hamiltonian
\begin{align}
{\hat H}_{\rm secular}=\frac{h_{x}}4 \sum_{n=2}^{L-1} {\hat \sigma}_{n}^{x}\left( 1- {\hat \sigma}_{n-1}^{z}{\hat \sigma}_{n+1}^{z}\right)\label{eq:Hsecular}
\end{align}
as the effective Hamiltonian in the interaction picture.
This Hamiltonian shows that the qubit flip on the site $n$ by the operator ${\hat \sigma}_{n}^{x}$ occurs only when its two adjacent qubits are in the opposite direction.
Suppose that the qubits at the sites $n=1,2,\cdots, k$ are
aligned up
while the other qubits are
aligned down.
In this case, only the $k$-th and $(k+1)$-th qubits could flip while the other qubits remain in the original state.
Therefore, a system which is initialized in the state $\ket{1}_{1}\otimes\bigotimes_{n=2}^{L}\ket{0}_{n}$ exhibits a sequence of qubit flip from the second to the $(L-1)$-th qubits.
We show in Fig.~\ref{fig:domino} quantum domino dynamics according to the Hamiltonian (\ref{eq:Hsecular}). Starting from the state $\ket{1}_{1}\otimes\bigotimes_{n=2}^{L}\ket{0}_{n}$, the qubit flip propagates forwardly until a time
$t=t^{*}_{\rm ideal}$, and then this propagates back for $t^{*}_{\rm ideal}<t<2t^{*}_{\rm ideal}$
, where $t_{\rm ideal}^{*}$ denotes the optimal time to maximize the total magnetization density of the time-evolved state according to the Hamiltonian (\ref{eq:Hsecular}).
Strictly speaking, there is still a small difference between the unitary dynamics induced by the Hamiltonian (\ref{eq:Hsecular}) and the ideal quantum domino dynamics (\ref{eq:IdealAmp1})--(\ref{eq:IdealAmp2REVERSE}).
As the flipping propagates further, the difference between the ideal domino dynamics and the dynamics by the Hamiltonian (\ref{eq:Hsecular}) becomes larger as shown in Fig.~\ref{fig:domino}.
In the ideal domino dynamics,
the total magnetization density, i.e., $M_{z}/L:=(1/2L)\sum_{n=1}^L \langle{\hat \sigma}_{n}^{z}\rangle$, would be $M_{z}/L=1/2-1/L$ for the right hand side of Eq.~(\ref{eq:IdealAmp1}).
On the other hand, for the real dynamics, it is not trivial whether the maximum total magnetization density converges to a finite value as we increase the system size $L$.
Fortunately, it has been found that, when we prepare an initial state $\ket{1}_{1}\otimes\bigotimes_{n=2}^{L}\ket{0}_{n}$
and let this state evolve by the Hamiltonian (\ref{eq:Hsecular}) for a certain time,
we can obtain a finite magnetization density $M_{z}/L\sim0.37$ for a large $L$ \cite{SpinAmpLee2005}.
The optimal time $t^{*}_{\rm ideal}$ is also numerically estimated as $t^{*}_{\rm ideal}\sim 1.06L/h_{x}$ in Ref.~\cite{SpinAmpLee2005}.
We will estimate the appropriate time of the duration time for the original Hamiltonian (\ref{eq:Htfi}) with a finite $L$ and $h_{x}$ in Sec.~\ref{sec:numRES}.
\section{Our quantum sensing protocol with always-on interaction }\label{sec:protocol}
\subsection{Description of our sensing protocol}\label{sec:DescOURS}
Here, we present our sensing protocol with an always-on interaction between the nearest-neighbor qubits in the probe chain.
In the following protocol, the only necessary operations are to initialize the system, to perform projective measurements on the first qubit, and to turn on/off global magnetic fields.
Our protocol can be summarized as follows (see also Fig.~\ref{fig:scheme}):
(i)' prepare a thermal equilibrium state of the Hamiltonian (\ref{eq:Htfi}) with $h_{x}/J<1/2$;
(ii)' perform a projective measurement on the first qubit along the $x$ direction at $t=0$, and then let the system evolve according to the same Hamiltonian (\ref{eq:Htfi}) until $t=t^{*}$;
(iii)' turn off the transverse magnetic field in Eq.~(\ref{eq:Htfi})
and instead let the system interact with the target magnetic field (\ref{eq:Homega}) for a time $T_{\rm int}$;
(iv)' let the system evolve according to (\ref{eq:Htfi}) again for the time $t^{*}$;
(v)' perform a projective measurement on the first qubit in the ${\hat \sigma}_{1}^{y}$ basis.
By repeating these steps, we obtain the probability distribution of the outcomes.
For our protocol, $T_{\rm reset},T_{\rm prep}$, and $T_{\rm read}$ are expressed as
$T_{\rm reset}=T_{1,\rm init}$, $T_{\rm prep}=t^{*} + t_{\rm measure}$, and $T_{\rm read}=t^{*} + t_{\rm measure}$, where $t_{\rm measure}$ denotes the time required for the projective measurements, and $T_{1,\rm init}$ denotes the relaxation time of the system to thermalize, i.e., the time for the step (i)'.
In the present paper, we assume that $T_{1,\rm init}$ and $t_{\rm measure}$ are much shorter than $t^{*}$ and
$T_{\rm int}$.
In general, the interaction time $T_{\rm int}$ needs to be comparable with $T_{\rm{all}}$ ($T_2$) to maximize the sensitivity without (with) noise (, where $T_{2}$ denotes the dephasing time).
For a noiseless case, we can set $T_{\rm{int}}$ to be much longer than the other time scales.
On the other hand, when there is dephasing, we need more careful consideration.
For a long lived qubit, $T_{2}$ can be
much longer than $t^*$ and $t_{\rm readout}$.
However, in most of the solid-state systems, the natural energy relaxation time, which we denote $T_{1,\rm relax}$, becomes longer than $T_2$ \cite{T1AmsussCavity2011,Bylander2011noise,T1ProbstRareEarth2013,yan2016flux,T1AngereBIstable2017,T1BudoyoBottleSCI2018,ShuntedFluxQubitabdurakhimov2019}.
For example, nitrogen vacancies in diamond have an energy relaxation time of $T_1\simeq 45$ seconds \cite{T1AmsussCavity2011}, while the dephasing time is around $T_2\simeq 2$ ms \cite{herbschleb2019ultra}.
Fortunately, there are experimental techniques that temporarily decrease the energy relaxation time \cite{reed2010fast,bienfait2016controlling, ResetSCQ2018,TunableRefrig2020}.
We call such an artificial and short energy relaxation time $T_{1,\rm init}$.
To reset or thermalize the system,
we assume that such resetting techniques are available
\begin{figure}
\centering
\includegraphics[width=0.97\columnwidth]{fig2.png}
\caption{Schematic picture of our protocol.
The upper figure represents a prescription of our scheme. The middle figure shows how we apply global magnetic fields.
The lower figure shows the procedure (i)'--(v)' and the duration time for each step.
During the step (iii)', we turn off the transverse magnetic field and expose the probe to the target magnetic field.
We assume that the interaction time $T_{\rm {int }}$ is much longer than the other times involved.
}
\label{fig:scheme}
\end{figure}
The key idea of the protocol is the use of the quantum domino dynamics.
Although the state is mainly described by the Schr\"{o}dinger picture in this manuscript, we use the interaction picture ${\hat \rho}_{\rm I}(t) = {\rm e}^{itH_{\rm Ising}} {\hat \rho}_{\rm S}(t) {\rm e}^{-itH_{\rm Ising}}
$ in this paragraph to avoid cumbersome expressions, i.e., the state ${\hat \rho}_{\rm S}$ in the Schr\"{o}dinger picture is obtained after considering a time evolution according to the Hamiltonian ${\hat H}_{\rm Ising}$
(see also Sec.~\ref{sec:DominoDynamics}).
The state after the measurement in the step (ii)' is approximated as
$\ket{+}_{1}\prescript{}{1}{\bra{+}}\otimes\left(
\bigotimes_{n=2}^{L}\ket{0}_{n}\prescript{}{n}{\bra{0}}+\bigotimes_{n=2}^{L}\ket{1}_{n}\prescript{}{n}{\bra{1}}\right)/2$
from Eq.~(\ref{eq:RhoMix}) for the case in which the measurement outcome is $+1$, and this state evolves into
\begin{align}
{\hat \rho}_{\rm I} (t^{*})=\frac12
\left(\bigotimes_{n=1}^{L-1}\ket{0}_{n}+\bigotimes_{n=1}^{L-1}\ket{1}_{n}\right)
\left(\bigotimes_{n=1}^{L-1}\prescript{}{n}{\bra{0}}+\bigotimes_{n=1}^{L-1}\prescript{}{n}{\bra{1}}\right)\nonumber\\
\otimes\frac12 \left(\ket{0}_{L}\prescript{}{L}{\bra{0}}+\ket{1}_{L}\prescript{}{L}{\bra{1}}\right)
\end{align}
under the ideal domino dynamics (\ref{eq:IdealAmp1}) and (\ref{eq:IdealAmp2}),
which is approximately induced by the Hamiltonian (\ref{eq:Htfi}) in the step (ii)'.
After obtaining a phase shift at the third step (iii)', i.e.,
\begin{align}
{\hat \rho}_{\rm I}(t^{*}+T_{\rm int})=&
\frac12\left(\bigotimes_{n=1}^{L-1}\ket{0}_{n}+{\rm e}^{-i \omega (L-1) T_{\rm int}}\bigotimes_{n=1}^{L-1}\ket{1}_{n}\right)
\nonumber\\
&\times
\left(\bigotimes_{n=1}^{L-1}\prescript{}{n}{\bra{0}}+{\rm e}^{i \omega (L-1) T_{\rm int}}\bigotimes_{n=1}^{L-1}\prescript{}{n}{\bra{1}}\right)
\nonumber\\
&\otimes\frac12 \left(\ket{0}_{L}\prescript{}{L}{\bra{0}}+\ket{1}_{L}\prescript{}{L}{\bra{1}}\right),\label{eq:rhoTintConv}
\end{align}
the state evolves into
\begin{align}
{\hat \rho}_{\rm I}(2t^{*}+T_{\rm int})=
\frac12\left(\ket{0}_{1}+{\rm e}^{-i \omega (L-1) T_{\rm int}}\ket{1}_{1}\right)
\left(\prescript{}{1}{\bra{0}}\right.
\nonumber\\
+
\left.{\rm e}^{i \omega (L-1) T_{\rm int}}\prescript{}{1}{\bra{1}}\right)
\otimes\frac12 \left(\bigotimes_{n=2}^{L}\ket{0}_{n}\prescript{}{n}{\bra{0}}+\bigotimes_{n=2}^{L}\ket{1}_{1}\prescript{}{1}{\bra{1}}\right),
\end{align}
in the step (iv)', which replaces the disentangling procedure (iv) in the conventional scheme in Sec.~\ref{sec:RamseyGHZ} by the time evolution with the Hamiltonian (\ref{eq:Htfi}).
In this case, the Hamiltonian (\ref{eq:Htfi}) approximately induces the ideal domino dynamics (\ref{eq:IdealAmp1REVERSE}) and (\ref{eq:IdealAmp2REVERSE}).
Here,
the combination of the steps (iv)' and (v)' effectively measures the probability of projecting ($L-1$)-qubit state of ${\hat \rho}_{\rm I}(t^*+T_{\rm int})$ in Eq.~(\ref{eq:rhoTintConv}) to $\left(\bigotimes_{n=1}^{L-1}\ket{0}_{n} + i\bigotimes_{n=1}^{L-1}\ket{1}_{n}\right)/\sqrt{2}$.
The probability $P$ of obtaining $+1$ as the measurement outcome in the step (v)' is written as
\begin{align}
P={\rm Tr}[{\hat U}_{\rm protocol} {\hat \rho}_{0} {\hat U}_{\rm protocol}^{\dagger}
{\hat P}_{1,y}^{+}],\label{eq:DefPinProtocol}
\end{align}
where
\begin{align}
{\hat \rho}_{0} :=& {{\hat P}_{1,x}^{+} {\hat \rho}_{\beta} {\hat P}_{1,x}^{+}} / { {\rm Tr}[ {\hat P}_{1,x}^{+} {\hat \rho}_{\beta} ] },\\
{\hat U}_{\rm protocol} :=& {\rm e}^{ - i {\hat H}_{\rm TFI}t^{*}} {\rm e}^{ - i ({\hat H}_{\rm Ising} + {\hat H}_{\omega})T_{\rm int}} {\rm e}^{ - i {\hat H}_{\rm TFI}t^{*}}.\label{eq:DefUinProtocol}
\end{align}
Hereafter, we assume that the measurement outcome in the step (i)' is $+1$ without loss of generality.
For the case in which the outcome in the step (i)' is $-1$, we exchange the measurement basis in the step (v)' from ${\hat \sigma}_{1}^{y}$ to $-{\hat \sigma}_{1}^{y}$.
Let us derive a sensitivity in our scheme by using some approximations.
In Sec.~\ref{sec:numRES}, we will numerically calculate the sensitivity without approximations by directly calculating Eq.~(\ref{eq:DefPinProtocol}).
Assuming the validity of the approximation (\ref{eq:RhoMix}) and the ideal domino dynamics (\ref{eq:IdealAmp1})--(\ref{eq:IdealAmp2REVERSE}), we can estimate the probability (\ref{eq:DefPinProtocol}) as
\begin{align}
P\simeq\frac12+\frac12\cos{\left[J\left(t^{*} + \frac{T_{\rm int}}2 \right)\right]}\sin{[(L-1)\omega T_{\rm int}]}\label{eq:approxPinOurProtocol}.
\end{align}
The oscillating part $\cos{\left[J\left(t^{*} + {T_{\rm int}}/2 \right)\right]}$ in Eq.~(\ref{eq:approxPinOurProtocol}), which did not appear in Eq.~(\ref{eq:ghzghz}), represents the effect of the presence of the Ising interaction.
By tuning $t^{*}$
, the probability becomes the same as that with the GHZ state composed of $(L-1)$ qubits; see Eq.~(\ref{eq:ghzghz}).
Therefore, we can achieve the HL in this case, similar to the case in Sec.\ref{sec:RamseyGHZ}.
We emphasize here that, even though Eq.~(\ref{eq:approxPinOurProtocol}) is an approximation, the effect of the presence of the Ising interaction at the third step (iii)' can always be canceled out by setting $T_{\rm int}=m 4\pi/J$, where $m$ denotes a natural number.
When the probe qubits interact with the target field,
they can be affected by an additional Hamiltonian (such as residual interactions between qubits).
The effect of such an additional Hamiltonian
has been discussed in some studies.
It was shown in Ref.~\cite{UnitNoisePasquale2013}
that,
when one prepares an optimal state for sensing the target field in the presence of additional Hamiltonian, this term cannot enhance the sensitivity anymore compared to the case where there is no such terms.
Reference \cite{TFISkotiniotis2015} considered estimation of the target longitudinal magnetic field in the transverse Ising chain, where the interaction is of XX type, and showed that
the sensitivity can still achieve the Heisenberg limited scaling
if an appropriate GHZ-type state is used.
On the other hand, in our case,
the sensitivities with and without residual interactions are the same.
Here, we take an
advantage of the fact that the Ising interaction commutes with the target magnetic field and we can cancel out the additional phase shift by tuning the interaction time.
We hence can obtain the HL if we could prepare and disentangle the GHZ state perfectly.
Equation (\ref{eq:approxPinOurProtocol}) also shows that the probability approaches to $1/2$ as $\omega$ goes to $0$.
Although we derived Eq.~(\ref{eq:approxPinOurProtocol}) with several approximations,
we can derive this from a more general setup as follows.
The Hamiltonian ${\hat H}_{\rm TFI}$ and the measurement ${\hat P}_{1,x}^{+}$, as well as the initial state ${\hat \rho}_{\beta}$ commutes with the parity symmetry ${\hat U}_{x}:=\Pi_{n=1}^{L}{\hat \sigma}_{n}^{x}$, while ${\hat \sigma}_{1}^{y}$ in ${\hat P}_{1,y}^{+}$ anti-commutes with ${\hat U}_{x}$.
From these relations and Eqs.~(\ref{eq:DefPinProtocol})--(\ref{eq:DefUinProtocol}),
the expectation value of ${\hat \sigma}_{1}^{y}=2{\hat P}_{1,y}^{+}-1$ always vanishes for $\omega T_{\rm int}=0$.
This shows that the probability distribution of the measurement outcome for the case of the vanishing $\omega$ always takes the same value.
\subsection{Comparison with the conventional protocol}\label{sec:Comperison}
Here we summarize the difference between our protocol and the conventional one introduced
in Sec.~\ref{sec:RamseyGHZ}.
First, there is a difference in state preparation.
We use
the time evolution according to the time-independent Hamiltonian Eq.~(\ref{eq:Htfi}) in preparing a metrologically useful state.
Importantly, since we use the natural dynamics induced by the Hamiltonian for these processes, our protocol does not require any temporal control over the individual Ising interactions between qubits.
This is
in stark contrast to the conventional protocol that uses gate operations for the entanglement generation, which typically requires turning on/off the interaction.
Second, the way to readout the state is different.
We use the Hamiltonian dynamics to transform the entangled probe state into an almost separable state so that we could extract the information of the target magnetic field from the single qubit measurement. On the other hand, in the conventional approach, a combination
of gate operations and projective measurements are required.
Finally, we compare the time required for our scheme
which uses
the quantum domino dynamics and that for the conventional scheme
which uses
the gate operations.
When a system has Ising interactions with strength
of
$J_{\rm CZ}$, an operation time for implementing one CNOT gate is
$({\pi}/{J_{\rm CZ}}+2\tau _{\rm{H}})$
from our estimation in Sec.~\ref{sec:RamseyGHZ} where $\tau _{\rm{H}}$ denotes a necessary gate time to implement the Hadamard gate.
On the other hand, it takes $\sim 1.06/h_x$ for flipping single qubit on average in the domino dynamics.
In Sec.~\ref{sec:numRES} and \ref{sec:NMnoise} we demonstrate that our protocol beats the SQL
by a constant factor, where we set $J=1$ and $h_x=0.1$.
This shows that even if we ignore the operation time for the Hadamard gates, the preparation time
for the case of using the quantum domino dynamics
is only around three times longer
than
that for
the case of using a sequence of CNOT gates,
under the assumption that the Ising interaction strength is the same, i.e., $J_{\rm CZ}=J$.
As long as the coherence time is long,
it is more advantageous to use quantum domino dynamics than gate type operations.
Therefore, our protocol can be a practical way to realize entanglement-enhanced sensing in a qubit system with fixed Ising interaction.
\section{Numerical results about the sensitivity without environmental noise}\label{sec:numRES}
We now present numerical results to show the performance of our protocol without noise.
We calculate the uncertainty (\ref{eq:errorOmegaDef}) using Eqs.~(\ref{eq:DefPinProtocol})--(\ref{eq:DefUinProtocol}).
We here take the interaction strength and the transverse magnetic field to be $J=1$ and $h_{x}=0.1$.
For each size $L$, we numerically find the optimal duration time $t^{*}_{\rm opt}$ in order to obtain the smallest uncertainty $\delta \omega$.
In Fig.~\ref{fig:tstar}, we show the size dependence of $t^{*}_{\rm opt}$, which takes the value around $t^{*}_{\rm opt}\sim1.06L/h_x\simeq t^{*}_{\rm ideal}$ as we mentioned in Sec.~\ref{sec:DominoDynamics}.
Throughout the paper, we use these values of $t^{*}_{\rm opt}$ as $t^{*}$ when we plot $\delta \omega$ and $T_{\rm int}$ for each $L$.
We have numerically checked that $t^{*}_{\rm opt}$ does not depend on $\beta$ in the parameter sets which we use in the present paper.
In order to take into account the effect of the preparation time on the uncertainty, we take $T_{\rm sensing} = T_{\rm int} + 2t^{*}_{\rm opt}$ (and ignore the other times involved for simplicity) although $T_{\rm{int}}$ is much longer than $t^{*}_{\rm opt}$ in the following calculations.
In Fig.~\ref{fig:Oscilation}, we observe an oscillation in $\sim Jt^{*}$ with the probability (\ref{eq:DefPinProtocol}) as
we have discussed in Eq.~(\ref{eq:approxPinOurProtocol}).
The oscillation frequency is almost the same as $J=1$, which is consistent with our approximate analytical expression (\ref{eq:approxPinOurProtocol}).
The optimal time $t^{*}_{\rm opt}$ which provides the smallest uncertainty $\delta \omega$ corresponds to the minimal point of the oscillation.
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{fig3.pdf}
\caption{The optimal duration time $t^{*}_{\rm opt}$ which we use in the step (ii)' and the step (iv)' when we numerically simulate our protocol.
The blue points show $t^{*}$ at which $\delta \omega$ can be minimized for $h_{x}=0.1$ and $\beta=10$. The broken line shows the function $t^{*}_{\rm opt}=1.06 L /h_{x}$.
All parameters are normalized by $J=1$.
}
\label{fig:tstar}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.89\columnwidth]{fig4new.pdf}
\caption{The oscillation of $\delta \omega$ in the duration time $t^{*}$.
Three blue points show the minimum of $T_{\rm all}^{1/2}\delta \omega$ at $t^{*}_{\rm opt}$ for each size $L$ with $L=4$, $7$, and $10$.
The dotted line and the broken line show the uncertainty of the SQL for $L$ qubits and the HL for $(L-1)$ qubits, respectively.
The parameters are, $h_{x}=0.1$,
$\beta=10$,
$\omega=10^{-6}$, and $T_{\rm int}=500\pi$.
All parameters are normalized by $J=1$.
}
\label{fig:Oscilation}
\end{figure}
Figure~\ref{fig:deltaU} shows the uncertainty against the number of the qubits
with different initial states.
These results demonstrate that our protocol achieves the high precision sensing beyond the SQL by a constant factor.
However, when we increase $L$ with a fixed $\beta$, the uncertainty starts to saturate, as a tendency of which can be observed in the plot for $\beta=5$ in Fig.~\ref{fig:deltaU}.
This is due mainly to the breakdown of the approximation (\ref{eq:RhoMix}), which requires $\beta$ to be large.
We will discuss this point again in Sec.~\ref{sec:disc}.
In the conventional quantum domino dynamics the initial state is assumed to be pure, namely
$\bigotimes_{n=1}^{L}\ket{0}_{n}$.
For comparison, we calculate the uncertainty when the initial state is $\bigotimes_{n=1}^{L}\ket{0}_{n}$.
Interestingly, the uncertainty with this pure initial state is almost the same as that with the thermal equilibrium states ${\hat \rho}_{\beta}$ with $\beta=10$ and $20$, as shown in Fig.~\ref{fig:deltaU}.
Therefore, the use of the thermal states does not necessarily degrade the sensitivity compared with the case of using a pure state.
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{fig5new.pdf}
\caption{Size dependence of the uncertainty $\delta\omega$ in our protocol.
The dotted line and the broken line show the SQL for $L$-qubits and the HL for $(L-1)$ qubits, respectively.
The other symbols in the figure show $\delta\omega$ for the cases in which the initial states are ${\hat \rho}_{\beta}$ with $\beta=5,10$, and $20$, and $\bigotimes_{n=1}^{L}\ket{0}_{n}$, respectively.
The parameters are $h_{x}=0.1$,
$\omega=10^{-6}$, and $T_{\rm int}=500\pi$.
All parameters are normalized by $J=1$.
}
\label{fig:deltaU}
\end{figure}
\section{Sensing under time-inhomogeneous dephasing}\label{sec:NMnoise}
Next, we include the effect of dephasing during the interaction with the target magnetic field, and also show that our protocol beats the SQL by a constant factor even in this case.
For the Ramsey protocol which uses the GHZ state as the probe state,
it has been found that improved sensitivity with the scaling $\delta\omega=\mathcal{O}(L^{-3/4})$ can be achieved with an interaction time of $T_{\rm {int }}=\mathcal{O}(L^{-1/2})$
when the noise is time-inhomogeneous dephasing \cite{Matsuzaki2011NMmetro,ChinNM2012PRL,ChinNM2012PRL,Zeno2015,tanaka2015proposed,Matsuzaki2018PRL,DepGrecka2018,DepBinHo2020}, which is also referred to as Non-Markovian dephasing \cite{Matsuzaki2011NMmetro,ChinNM2012PRL,ChinNM2012PRL,Matsuzaki2018PRL,TATSUTA2019cat}.
This scaling is called the Zeno limit.
On the other hand, no improvement of scaling over the SQL is realized in the presence of Markovian noise \cite{HuelgaMarkov1997,Shaji2007}.
Time-inhomogeneous dephasing can be observed when the correlation time $\tau_c$ of the environment is longer than a coherence time of the qubits.
It is known that solid-state systems that have a strong coupling with magnetic fields such as a superconducting flux qubit \cite{YoshiharaNMFluxQubit2006,Bylander2011noise,kakuyanagiscFluxQubit2007NM}, a spin qubit in a quantum dot \cite{kawakami2014electrical,kawakami2016gate}, and an NV center in diamond \cite{GHZin4NVneumann2008,GHZin4NV2015APL,MazeNV2008,StanwixNV2010} are typically subject to such time-inhomogeneous dephasing, and the correlation time of these systems is much longer than the coherence time in these systems.
In this section, we consider the effect of time-inhomogeneous dephasing acting on each qubit independently.
We assume that the dephasing time $T_{2}$ and the relaxation time $T_{1,\rm relax}$ of the qubits satisfy $T_{1,\rm init},\,t_{\rm measure},\,t^{*}\ll T_{2}\ll T_{1,\rm relax}\ll\tau_{c}$.
This implies the following three: first, necessary condition of our noise model $T_{2} \ll\tau_{c}$ is satisfied; second, the relaxation time of the probe qubits $T_{1,\rm relax}$ is much longer than $T_{2}$ during the exposure; third, the total sensing time $T_{\rm sensing}$ is well approximated by $T_{\rm int}$.
For most of the solid-state systems, $T_{1,\rm relax}$ is much longer than $T_2$ especially at a low temperature \cite{T1AmsussCavity2011,Bylander2011noise,T1ProbstRareEarth2013,yan2016flux,T1AngereBIstable2017,T1BudoyoBottleSCI2018,ShuntedFluxQubitabdurakhimov2019}.
Therefore we assume that the effect of the energy relaxation is negligible compared to that of the dephasing during the exposure of the probe qubits to the target magnetic field.
We specifically consider the following master equation of the system during the step (iii)';
\begin{align}
\frac{d}{dt}{\hat \rho}(t)=-i[{\hat H}_{\rm Ising}+{\hat H}_{\omega},{\hat \rho}(t)] - \frac{t}{2{T_{2}}^{2}}\sum_{n=1}^{L} [{\hat \sigma}_{n}^{z},[{\hat \sigma}_{n}^{z},{\hat \rho}(t)]].
\end{align}
This kind of model has been used to describe noise in many solid-state systems \cite{Paladino2002SCQubitPRL,YoshiharaNMFluxQubit2006,kakuyanagiscFluxQubit2007NM,MazeNV2008,de2010universal,StanwixNV2010,NMsoltionMatsuzaki2010,Bylander2011noise,Paladino2014SCQubitRevMod,Matsuzaki2018PRL}.
Solving the above equation provides us with the following solution:
\begin{align}
{\hat \rho}(T_{\rm int})&=\varepsilon_{1}(\varepsilon_{2}(\cdots\varepsilon_{L}({\hat \rho}_{I}(0))\cdots)),\label{eq:channel}\\
\varepsilon_{n}({\hat \rho})&:=\frac{1+{\rm e}^{-(T_{\rm int}/T_{2})^{2}}}2 {\hat \rho} + \frac{1-{\rm e}^{-(T_{\rm int}/T_{2})^{2}}}2 {\hat \sigma}_{n}^{z}{\hat \rho}{\hat \sigma}_{n}^{z},\label{eq:channelDel}\\
{\hat \rho}_{I}(0)&:={\rm e}^{-i({\hat H}_{\rm Ising}+{\hat H}_{\omega})T_{\rm int}}{\hat \rho}(0){\rm e}^{i({\hat H}_{\rm Ising}+{\hat H}_{\omega})T_{\rm int}}.\label{eq:channelInt}
\end{align}
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{fig6new.pdf}
\caption{The optimal interaction time $T_{\rm int}$ in the presence of noise. We numerically obtained $T_{\rm int}$ at which the minimum uncertainty is achieved in our protocol under the time-inhomogeneous dephasing.
The broken line shows the function $T_{\rm int}=(1/2)T_{2}(L-1)^{-1/2}$ at which the minimum uncertainty is achieved when the probe is in the GHZ state of $(L-1)$ qubits
\cite{Matsuzaki2011NMmetro,ChinNM2012PRL,TATSUTA2019cat}.
The parameters are $h_{x}=0.1$,
$\beta=10$,
$\omega=10^{-6}$, and
$T_{2}=10^{4}$.
All parameters are normalized by $J=1$.
}
\label{fig:Tint4}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{fig7new.pdf}
\caption{Size dependence of the uncertainty $\delta\omega$ in our protocol under the time-inhomogeneous dephasing.
The dotted line shows the SQL for $L$-qubits, i.e.,
$T_{\rm all}^{1/2}\delta\omega={\sqrt{2} \exp (1 / 4)}(LT_{2})^{-1/2}$
, and the broken line shows the Zeno limit for $(L-1)$ qubits, i.e.,
$T_{\rm all}^{1/2}\delta\omega={\sqrt{2} \exp (1 / 4)}(L-1)^{-3 / 4} T_{2}^{-1/2}$
\cite{HuelgaMarkov1997,Matsuzaki2011NMmetro,ChinNM2012PRL,TATSUTA2019cat}.
The other symbols in the figure show $\delta\omega$ for the cases in which the initial states are ${\hat \rho}_{\beta}$ with $\beta=5,10$, and $20$, and
$\bigotimes_{n=1}^{L}\ket{0}_{n}$, respectively.
The parameters are $h_{x}=0.1$,
$\omega=10^{-6}$, and $T_{2}=10^{4}$.
All parameters are normalized by $J=1$.
The interaction time $T_{\rm int}$ is chosen so that $\delta\omega$ achieves the smallest value (see Fig.~\ref{fig:Tint4}).
}
\label{fig:deltaNM}
\end{figure}
We numerically calculate $\delta\omega$ using Eqs.~(\ref{eq:channel})--(\ref{eq:channelInt}).
As in the noiseless case we take $T_{\rm sensing} = T_{\rm int} + 2t^{*}_{\rm opt}$ in the calculation.
In contrast to the case in Sec.~\ref{sec:numRES},
where the probability (\ref{eq:DefPinProtocol}) is the function of $\omega T_{\rm int}$,
the slope $|dP/d\omega|$ depends nontrivially on $T_{\rm int}$ in the presence of noise \cite{Matsuzaki2011NMmetro,ChinNM2012PRL,TATSUTA2019cat}.
We thereby numerically tune the interaction time $T_{\rm int}$,
so that the uncertainty (\ref{eq:errorOmegaDef}) takes a minimum value.
The size dependence of the interaction time $T_{\rm int}$ is shown in Fig.~\ref{fig:Tint4}.
This size dependence is consistent with the previous results using the GHZ state for sensing under the effect of time-inhomogeneous dephasing \cite{HuelgaMarkov1997,Matsuzaki2011NMmetro,ChinNM2012PRL,TATSUTA2019cat}.
We stress here that the interaction time $T_{\rm{int}}$ is much longer than the duration time $t^{*}_{\rm opt}$ with the parameter sets we choose in Figs.~\ref{fig:Tint4} and \ref{fig:deltaNM}.
Figure~\ref{fig:deltaNM} shows the uncertainty $\delta\omega$ in the presence of the time-inhomogeneous dephasing at the step (iii)' in our protocol
with three values of $\beta$.
They demonstrate that our protocol beats the SQL by a constant factor except when the temperature of initial state is $\beta=5$.
However, we find that the improvement of $\delta \omega$ in our scheme over the SQL of the conventional scheme becomes smaller compared with the case
without dephasing (see Fig.~\ref{fig:deltaU}).
\section{Discussion}\label{sec:disc}
\begin{figure}
\centering
\includegraphics[width=0.7\columnwidth]{fig8.png}
\caption{For a large number $N\,(\gg L^*)$ of qubits, improved sensitivity by a constant factor can be maintained by separating the qubits into the chains of length $L^*$.
In this case, the probe consists of $N/L^*$ copies of the transverse field Ising chain.
}
\label{fig:chainstar}
\end{figure}
Finally,
let us discuss the metrological advantage of our protocol in a large system.
As we have seen in Fig.~\ref{fig:deltaU}, the uncertainty begins to saturate as we increase $L$ with a fixed $\beta$, and eventually our protocol may give no advantage over the conventional protocol with separable $L$-qubit states.
However, as long as there is at least one length $L^{*}$ at which our protocol with the initial state ${\hat \rho}_{\beta}$ beats the SQL, one can make use of our protocol to obtain an improved sensitivity by taking the length of the chain as $L^{*}$.
For an $N$ qubit probe, using $N/L^{*}$ copies of the transverse field Ising chain of length $L^{*}$ allows us to obtain the sensitivity which is $\delta\omega_{\rm SQL}(L^{*})/\delta\omega(L^{*},\beta)$ times better than the SQL (see Fig.~\ref{fig:chainstar}), where $\delta\omega_{\rm SQL}(L)$ denotes the SQL with $L$ qubits and $\delta\omega(L,\beta)$ denotes the uncertainty in our protocol with $L$ qubit equilibrium state ${\hat \rho}_{\beta}$.
This constant-factor improvement can be maximized by tuning the length of each chain
under the restriction that $\delta\omega(L,\beta) < \delta\omega_{\rm SQL}(L)$.
A similar technique was discussed in Ref.~\cite{DooleyNemoto2016}.
Summarizing the above, the uncertainty in our protocol can beat the SQL as we show in Figs.~\ref{fig:deltaU} and \ref{fig:deltaNM}, as long as the the following assumptions (a) and (b) in addition to the validity of the secular approximation (\ref{eq:Hsecular}) are valid.
(a) the temperature of the initial state $1/\beta$ is small enough for a fixed chain length $L$ so that the approximation Eq.~(11) becomes good.
(b) the decoherence times $T_1$ and $T_2$ are long enough compared to $T_{1,\rm init}$, $t_{\rm measure}$, and $t^*_{\rm opt}$ for a fixed chain length $L$ so that $T_{\rm int}$ dominates the sensing time $T_{\rm sensing}$.
(When the noise is present, $T_{\rm int}\simeq (1/2)T_{2}(L-1)^{-1/2}$ is needed for achieving the minimum uncertainty \cite{HuelgaMarkov1997,Shaji2007}, see Fig.~\ref{fig:Tint4}, and
hence $T_2$ should also be much longer than $2(L-1)^{1/2}$.)
We note that if we keep increasing the length of the chain of qubits while keeping $\beta$, $T_1$ and $T_2$ fixed, the sensitivity in our protocol will be eventually degraded
with the increase of $L$.
However, if we increase the number of chains as the number of available qubits grows, while keeping the length of the chains fixed but large enough, we can achieve scaling of the SQL with an improved constant.
Concluding, we have proposed a way for qubit-based magnetic field sensing with always-on
nearest-neighbor interaction.
Our protocol consists of three operations: initialization of the system in a thermal equilibrium state,
switching on/off global magnetic fields,
and projective measurements on a single qubit at the start and the end of the protocol.
Specifically, we approximately create the GHZ state from the equilibrium of the transverse field Ising chain by inducing the quantum domino dynamics.
We have numerically shown that our protocol beats the SQL by a constant factor even in the presence of time-inhomogeneous dephasing.
Since neither an accurate control of the qubits such as the entangling gate operations nor long-range interaction between qubits is required in the whole process, our protocol may provide an experimentally feasible way to realize entanglement enhanced sensors.
\begin{acknowledgments}
We are grateful to Naomichi Hatano and Hideaki Hakoshima for useful discussions.
This work was supported by Leading Initiative for Excellent Young Researchers MEXT Japan and JST presto (Grant No. JPMJPR1919) Japan.
This work was also supported by CREST (JPMJCR1774), JST.
MT is supported by JSPS fellowship (JSPS KAKENHI Grant No. 20J01757).
\end{acknowledgments}
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{"url":"http:\/\/www.theinfolist.com\/php\/SummaryGet.php?FindGo=Geographic_information_system","text":"TheInfoList\n\nA geographic information system (GIS) is a conceptualized framework that provides the ability to capture and analyze spatial and geographic data. GIS applications (or GIS apps) are computer-based tools that allow the user to create interactive queries (user-created searches), store and edit spatial and non-spatial data, analyze spatial information output, and visually share the results of these operations by presenting them as maps. Geographic information science (or, GIScience)\u2014the scientific study of geographic concepts, applications, and systems\u2014is commonly initialized as GIS, as well. Geographic information systems are utilized in multiple technologies, processes, techniques and methods. They are attached to various operations and numerous applications, that relate to: engineering, planning, management, transport\/logistics, insurance, telecommunications, and business. For this reason, GIS and location intelligence applications are at the foundation of location-enabled services, that rely on geographic analysis and visualization. GIS provides the capability to relate previously unrelated information, through the use of location as the \"key index variable\". Locations and extents that are found in the Earth's spacetime, are able to be recorded through the date and time of occurrence, along with x, y, and z coordinates; representing, longitude (''x''), latitude (''y''), and elevation (''z''). All Earth-based, spatial\u2013temporal, location and extent references, should be relatable to one another, and ultimately, to a \"real\" physical location or extent. This key characteristic of GIS, has begun to open new avenues of scientific inquiry and studies.\n\nHistory and development\n\nTechniques and technology\n\nModern GIS technologies use digital information, for which various digitized data creation methods are used. The most common method of data creation is digitization, where a hard copy map or survey plan is transferred into a digital medium through the use of a CAD program, and geo-referencing capabilities. With the wide availability of ortho-rectified imagery (from satellites, aircraft, Helikites and UAVs), heads-up digitizing is becoming the main avenue through which geographic data is extracted. Heads-up digitizing involves the tracing of geographic data directly on top of the aerial imagery instead of by the traditional method of tracing the geographic form on a separate digitizing tablet (heads-down digitizing). Heads-down digitizing, or manual digitizing, uses a special magnetic pen, or stylus, that feeds information into a computer to create an identical, digital map. Some tablets use a mouse-like tool, called a puck, instead of a stylus. The puck has a small window with cross-hairs which allows for greater precision and pinpointing map features. Though heads-up digitizing is more commonly used, heads-down digitizing is still useful for digitizing maps of poor quality. Geoprocessing is a GIS operation used to manipulate spatial data. A typical geoprocessing operation takes an input dataset, performs an operation on that dataset, and returns the result of the operation as an output dataset. Common geoprocessing operations include geographic feature overlay, feature selection and analysis, topology processing, raster processing, and data conversion. Geoprocessing allows for definition, management, and analysis of information used to form decisions.\n\nRelating information from different sources\n\nGIS uses spatio-temporal (space-time) location as the key index variable for all other information. Just as a relational database containing text or numbers can relate many different tables using common key index variables, GIS can relate otherwise unrelated information by using location as the key index variable. The key is the location and\/or extent in space-time. Any variable that can be located spatially, and increasingly also temporally, can be referenced using a GIS. Locations or extents in Earth space\u2013time may be recorded as dates\/times of occurrence, and x, y, and z coordinates representing, longitude, latitude, and elevation, respectively. These GIS coordinates may represent other quantified systems of temporo-spatial reference (for example, film frame number, stream gage station, highway mile-marker, surveyor benchmark, building address, street intersection, entrance gate, water depth sounding, POS or CAD drawing origin\/units). Units applied to recorded temporal-spatial data can vary widely (even when using exactly the same data, see map projections), but all Earth-based spatial\u2013temporal location and extent references should, ideally, be relatable to one another and ultimately to a \"real\" physical location or extent in space\u2013time. Related by accurate spatial information, an incredible variety of real-world and projected past or future data can be analyzed, interpreted and represented. This key characteristic of GIS has begun to open new avenues of scientific inquiry into behaviors and patterns of real-world information that previously had not been systematically correlated.\n\nGIS uncertainties\n\nGIS accuracy depends upon source data, and how it is encoded to be data referenced. Land surveyors have been able to provide a high level of positional accuracy utilizing the GPS-derived positions. High-resolution digital terrain and aerial imagery, powerful computers and Web technology are changing the quality, utility, and expectations of GIS to serve society on a grand scale, but nevertheless there are other source data that affect overall GIS accuracy like paper maps, though these may be of limited use in achieving the desired accuracy. In developing a digital topographic database for a GIS, topographical maps are the main source, and aerial photography and satellite imagery are extra sources for collecting data and identifying attributes which can be mapped in layers over a location facsimile of scale. The scale of a map and geographical rendering area representation type, or map projection, are very important aspects since the information content depends mainly on the scale set and resulting locatability of the map's representations. In order to digitize a map, the map has to be checked within theoretical dimensions, then scanned into a raster format, and resulting raster data has to be given a theoretical dimension by a rubber sheeting\/warping technology process known as georeferencing. A quantitative analysis of maps brings accuracy issues into focus. The electronic and other equipment used to make measurements for GIS is far more precise than the machines of conventional map analysis. All geographical data are inherently inaccurate, and these inaccuracies will propagate through GIS\u00a0operations in ways that are difficult to predict.\n\nData representation\n\nGIS data represents real objects (such as roads, land use, elevation, trees, waterways, etc.) with digital data determining the mix. Real objects can be divided into two abstractions: discrete objects (e.g., a house) and continuous fields (such as rainfall amount, or elevations). Traditionally, there are two broad methods used to store data in a GIS for both kinds of abstractions mapping references: raster images and vector. Points, lines, and polygons represent vector data of mapped location attribute references. A new hybrid method of storing data is that of identifying point clouds, which combine three-dimensional points with RGB information at each point, returning a \"3D color image\". GIS thematic maps then are becoming more and more realistically visually descriptive of what they set out to show or determine. For a list of popular GIS file formats, such as shapefiles, see .\n\nData capture\n\nData capture\u2014entering information into the system\u2014consumes much of the time of GIS\u00a0practitioners. There are a variety of methods used to enter data into a GIS where it is stored in a digital format. Existing data printed on paper or PET film maps can be digitized or scanned to produce digital data. A digitizer produces vector data as an operator traces points, lines, and polygon boundaries from a map. Scanning a map results in raster data that could be further processed to produce vector data. Survey data can be directly entered into a GIS from digital data collection systems on survey instruments using a technique called coordinate geometry (COGO). Positions from a global navigation satellite system (GNSS) like Global Positioning System can also be collected and then imported into a GIS. A current trend in data collection gives users the ability to utilize field computers with the ability to edit live data using wireless connections or disconnected editing sessions. This has been enhanced by the availability of low-cost mapping-grade GPS units with decimeter accuracy in real time. This eliminates the need to post process, import, and update the data in the office after fieldwork has been collected. This includes the ability to incorporate positions collected using a laser rangefinder. New technologies also allow users to create maps as well as analysis directly in the field, making projects more efficient and mapping more accurate. Remotely sensed data also plays an important role in data collection and consist of sensors attached to a platform. Sensors include cameras, digital scanners and lidar, while platforms usually consist of aircraft and satellites. In England in the mid 1990s, hybrid kite\/balloons called helikites first pioneered the use of compact airborne digital cameras as airborne geo-information systems. Aircraft measurement software, accurate to 0.4\u00a0mm was used to link the photographs and measure the ground. Helikites are inexpensive and gather more accurate data than aircraft. Helikites can be used over roads, railways and towns where unmanned aerial vehicles (UAVs) are banned. Recently aerial data collection has become more accessible with miniature UAVs and drones. For example, the Aeryon Scout was used to map a 50-acre\u00a0area with a ground sample distance of in only 12\u00a0minutes. The majority of digital data currently comes from photo interpretation of aerial photographs. Soft-copy workstations are used to digitize features directly from stereo pairs of digital photographs. These systems allow data to be captured in two and three dimensions, with elevations measured directly from a stereo pair using principles of photogrammetry. Analog aerial photos must be scanned before being entered into a soft-copy system, for high-quality digital cameras this step is skipped. Satellite remote sensing provides another important source of spatial data. Here satellites use different sensor packages to passively measure the reflectance from parts of the electromagnetic spectrum or radio waves that were sent out from an active sensor such as radar. Remote sensing collects raster data that can be further processed using different bands to identify objects and classes of interest, such as land cover. Web mining is a novel method of collecting spatial data. Researchers build a web crawler application to aggregate required spatial data from the web. For example, the exact geo-location or the neighborhood of apartments can be collected from online real estate listing websites. When data is captured, the user should consider if the data should be captured with either a relative accuracy or absolute accuracy, since this could not only influence how information will be interpreted but also the cost of data capture. After entering data into a GIS, the data usually requires editing, to remove errors, or further processing. For vector data it must be made \"topologically correct\" before it can be used for some advanced analysis. For example, in a road network, lines must connect with nodes at an intersection. Errors such as undershoots and overshoots must also be removed. For scanned maps, blemishes on the source map may need to be removed from the resulting raster. For example, a fleck of dirt might connect two lines that should not be connected.\n\nRaster-to-vector translation\n\nData restructuring can be performed by a GIS to convert data into different formats. For example, a GIS may be used to convert a satellite image map to a vector structure by generating lines around all cells with the same classification, while determining the cell spatial relationships, such as adjacency or inclusion. More advanced data processing can occur with image processing, a technique developed in the late\u00a01960s by NASA and the private sector to provide contrast enhancement, false color rendering and a variety of other techniques including use of two dimensional Fourier transforms. Since digital data is collected and stored in various ways, the two data sources may not be entirely compatible. So a GIS must be able to convert geographic data from one structure to another. In so doing, the implicit assumptions behind different ontologies and classifications require analysis. Object ontologies have gained increasing prominence as a consequence of object-oriented programming and sustained work by Barry Smith and co-workers.\n\nProjections, coordinate systems, and registration\n\nThe earth can be represented by various models, each of which may provide a different set of coordinates (e.g., latitude, longitude, elevation) for any given point on the Earth's surface. The simplest model is to assume the earth is a perfect sphere. As more measurements of the earth have accumulated, the models of the earth have become more sophisticated and more accurate. In fact, there are models called datums that apply to different areas of the earth to provide increased accuracy, like North American Datum of 1983 for U.S. measurements, and the World Geodetic System for worldwide measurements. The latitude and longitude on a map made against a local datum may not be the same as one obtained from a GPS receiver. Converting coordinates from one datum to another requires a datum transformation such as a Helmert transformation, although in certain situations a simple translation may be sufficient. In popular GIS software, data projected in latitude\/longitude is often represented as a Geographic coordinate system. For example, data in latitude\/longitude if the datum is the 'North American Datum of 1983' is denoted by 'GCS North American 1983'.\n\nSpatial analysis with GIS\n\nGIS spatial analysis is a rapidly changing field, and GIS packages are increasingly including analytical tools as standard built-in facilities, as optional toolsets, as add-ins or 'analysts'. In many instances these are provided by the original software suppliers (commercial vendors or collaborative non commercial development teams), while in other cases facilities have been developed and are provided by third parties. Furthermore, many products offer software development kits (SDKs), programming languages and language support, scripting facilities and\/or special interfaces for developing one's own analytical tools or variants. The increased availability has created a new dimension to business intelligence termed \"spatial intelligence\" which, when openly delivered via intranet, democratizes access to geographic and social network data. Geospatial intelligence, based on GIS spatial analysis, has also become a key element for security. GIS as a whole can be described as conversion to a vectorial representation or to any other digitisation process.\n\nSlope and aspect\n\nSlope can be defined as the steepness or gradient of a unit of terrain, usually measured as an angle in degrees or as a percentage. Aspect can be defined as the direction in which a unit of terrain faces. Aspect is usually expressed in degrees from north. Slope, aspect, and surface curvature in terrain analysis are all derived from neighborhood operations using elevation values of a cell's adjacent neighbours. Slope is a function of resolution, and the spatial resolution used to calculate slope and aspect should always be specified. Various authors have compared techniques for calculating slope and aspect. The following method can be used to derive slope and aspect:\nThe elevation at a point or unit of terrain will have perpendicular tangents (slope) passing through the point, in an east-west and north-south direction. These two tangents give two components, \u2202z\/\u2202x and \u2202z\/\u2202y, which then be used to determine the overall direction of slope, and the aspect of the slope. The gradient is defined as a vector quantity with components equal to the partial derivatives of the surface in the x and y directions. The calculation of the overall 3\u00d73 grid slope ''S'' and aspect ''A'' for methods that determine east-west and north-south component use the following formulas respectively: : $\\tan S = \\sqrt$ : $\\tan A = \\frac$ Zhou and Liu describe another formula for calculating aspect, as follows: : $A = 270^\\circ + \\arctan \\left\\left( \\frac \\right\\right) - 90^\\circ\\cdot \\frac$\n\nData analysis\n\nIt is difficult to relate wetlands maps to rainfall amounts recorded at different points such as airports, television stations, and schools. A GIS, however, can be used to depict two- and three-dimensional characteristics of the Earth's surface, subsurface, and atmosphere from information points. For example, a GIS can quickly generate a map with isopleth or contour lines that indicate differing amounts of rainfall. Such a map can be thought of as a rainfall contour map. Many sophisticated methods can estimate the characteristics of surfaces from a limited number of point measurements. A two-dimensional contour map created from the surface modeling of rainfall point measurements may be overlaid and analyzed with any other map in a GIS covering the same area. This GIS derived map can then provide additional information - such as the viability of water power potential as a renewable energy source. Similarly, GIS can be used to compare other renewable energy resources to find the best geographic potential for a region. Additionally, from a series of three-dimensional points, or digital elevation model, isopleth lines representing elevation contours can be generated, along with slope analysis, shaded relief, and other elevation products. Watersheds can be easily defined for any given reach, by computing all of the areas contiguous and uphill from any given point of interest. Similarly, an expected thalweg of where surface water would want to travel in intermittent and permanent streams can be computed from elevation data in the GIS.\n\nTopological modeling\n\nA GIS can recognize and analyze the spatial relationships that exist within digitally stored spatial data. These topological relationships allow complex spatial modelling and analysis to be performed. Topological relationships between geometric entities traditionally include adjacency (what adjoins what), containment (what encloses what), and proximity (how close something is to something else).\n\nGeometric networks\n\nGeometric networks are linear networks of objects that can be used to represent interconnected features, and to perform special spatial analysis on them. A geometric network is composed of edges, which are connected at junction points, similar to graphs in mathematics and computer science. Just like graphs, networks can have weight and flow assigned to its edges, which can be used to represent various interconnected features more accurately. Geometric networks are often used to model road networks and public utility networks, such as electric, gas, and water networks. Network modeling is also commonly employed in transportation planning, hydrology modeling, and infrastructure modeling.\n\nHydrological modeling\n\nGIS hydrological models can provide a spatial element that other hydrological models lack, with the analysis of variables such as slope, aspect and watershed or catchment area. Terrain analysis is fundamental to hydrology, since water always flows down a slope. As basic terrain analysis of a digital elevation model (DEM) involves calculation of slope and aspect, DEMs are very useful for hydrological analysis. Slope and aspect can then be used to determine direction of surface runoff, and hence flow accumulation for the formation of streams, rivers and lakes. Areas of divergent flow can also give a clear indication of the boundaries of a catchment. Once a flow direction and accumulation matrix has been created, queries can be performed that show contributing or dispersal areas at a certain point. More detail can be added to the model, such as terrain roughness, vegetation types and soil types, which can influence infiltration and evapotranspiration rates, and hence influencing surface flow. One of the main uses of hydrological modeling is in environmental contamination research. Other applications of hydrological modeling include groundwater and surface water mapping, as well as flood risk maps.\n\nCartographic modeling\n\nDana Tomlin probably coined the term \"cartographic modeling\" in his PhD dissertation (1983); he later used it in the title of his book, ''Geographic Information Systems and Cartographic Modeling'' (1990). Cartographic modeling refers to a process where several thematic layers of the same area are produced, processed, and analyzed. Tomlin used raster layers, but the overlay method (see below) can be used more generally. Operations on map layers can be combined into algorithms, and eventually into simulation or optimization models.\n\nMap overlay\n\nThe combination of several spatial datasets (points, lines, or polygons) creates a new output vector dataset, visually similar to stacking several maps of the same region. These overlays are similar to mathematical Venn diagram overlays. A union overlay combines the geographic features and attribute tables of both inputs into a single new output. An intersect overlay defines the area where both inputs overlap and retains a set of attribute fields for each. A symmetric difference overlay defines an output area that includes the total area of both inputs except for the overlapping area. Data extraction is a GIS process similar to vector overlay, though it can be used in either vector or raster data analysis. Rather than combining the properties and features of both datasets, data extraction involves using a \"clip\" or \"mask\" to extract the features of one data set that fall within the spatial extent of another dataset. In raster data analysis, the overlay of datasets is accomplished through a process known as \"local operation on multiple rasters\" or \"map algebra\", through a function that combines the values of each raster's matrix. This function may weigh some inputs more than others through use of an \"index model\" that reflects the influence of various factors upon a geographic phenomenon.\n\nGeostatistics\n\nGeostatistics is a branch of statistics that deals with field data, spatial data with a continuous index. It provides methods to model spatial correlation, and predict values at arbitrary locations (interpolation). When phenomena are measured, the observation methods dictate the accuracy of any subsequent analysis. Due to the nature of the data (e.g. traffic patterns in an urban environment; weather patterns over the Pacific Ocean), a constant or dynamic degree of precision is always lost in the measurement. This loss of precision is determined from the scale and distribution of the data collection. To determine the statistical relevance of the analysis, an average is determined so that points (gradients) outside of any immediate measurement can be included to determine their predicted behavior. This is due to the limitations of the applied statistic and data collection methods, and interpolation is required to predict the behavior of particles, points, and locations that are not directly measurable. Interpolation is the process by which a surface is created, usually a raster dataset, through the input of data collected at a number of sample points. There are several forms of interpolation, each which treats the data differently, depending on the properties of the data set. In comparing interpolation methods, the first consideration should be whether or not the source data will change (exact or approximate). Next is whether the method is subjective, a human interpretation, or objective. Then there is the nature of transitions between points: are they abrupt or gradual. Finally, there is whether a method is global (it uses the entire data set to form the model), or local where an algorithm is repeated for a small section of terrain. Interpolation is a justified measurement because of a spatial autocorrelation principle that recognizes that data collected at any position will have a great similarity to, or influence of those locations within its immediate vicinity. Digital elevation models, triangulated irregular networks, edge-finding algorithms, Thiessen polygons, Fourier analysis, (weighted) moving averages, inverse distance weighting, kriging, spline, and trend surface analysis are all mathematical methods to produce interpolative data.\n\nReverse geocoding\n\nReverse geocoding is the process of returning an estimated street address number as it relates to a given coordinate. For example, a user can click on a road centerline theme (thus providing a coordinate) and have information returned that reflects the estimated house number. This house number is interpolated from a range assigned to that road segment. If the user clicks at the midpoint of a segment that starts with address\u00a01 and ends with\u00a0100, the returned value will be somewhere near 50. Note that reverse geocoding does not return actual addresses, only estimates of what should be there based on the predetermined range.\n\nMulti-criteria decision analysis\n\nCoupled with GIS, multi-criteria decision analysis methods support decision-makers in analysing a set of alternative spatial solutions, such as the most likely ecological habitat for restoration, against multiple criteria, such as vegetation cover or roads. MCDA uses decision rules to aggregate the criteria, which allows the alternative solutions to be ranked or prioritised. GIS MCDA may reduce costs and time involved in identifying potential restoration sites.\n\nData output and cartography\n\nCartography is the design and production of maps, or visual representations of spatial data. The vast majority of modern cartography is done with the help of computers, usually using GIS but production of quality cartography is also achieved by importing layers into a design program to refine it. Most GIS\u00a0software gives the user substantial control over the appearance of the data. Cartographic work serves two major functions: First, it produces graphics on the screen or on paper that convey the results of analysis to the people who make decisions about resources. Wall maps and other graphics can be generated, allowing the viewer to visualize and thereby understand the results of analyses or simulations of potential events. Web Map Servers facilitate distribution of generated maps through web browsers using various implementations of web-based application programming interfaces (AJAX, Java, Flash, etc.). Second, other database information can be generated for further analysis or use. An example would be a list of all addresses within one\u00a0mile (1.6\u00a0km) of a toxic spill.\n\nGraphic display techniques\n\nTraditional maps are abstractions of the real world, a sampling of important elements portrayed on a sheet of paper with symbols to represent physical objects. People who use maps must interpret these symbols. Topographic maps show the shape of land surface with contour lines or with shaded relief. Today, graphic display techniques such as shading based on altitude in a GIS can make relationships among map elements visible, heightening one's ability to extract and analyze information. For example, two types of data were combined in a GIS to produce a perspective view of a portion of San\u00a0Mateo County, California. *The digital elevation model, consisting of surface elevations recorded on a 30-meter horizontal grid, shows high elevations as white and low elevation as black. *The accompanying Landsat Thematic Mapper image shows a false-color infrared image looking down at the same area in 30-meter\u00a0pixels, or picture elements, for the same coordinate points, pixel\u00a0by\u00a0pixel, as the elevation information. A GIS was used to register and combine the two images to render the three-dimensional perspective view looking down the San Andreas Fault, using the Thematic Mapper image pixels, but shaded using the elevation of the landforms. The GIS\u00a0display depends on the viewing point of the observer and time of day of the display, to properly render the shadows created by the sun's rays at that latitude, longitude, and time of day. An archeochrome is a new way of displaying spatial data. It is a thematic on a 3D\u00a0map that is applied to a specific building or a part of a building. It is suited to the visual display of heat-loss data.\n\nSpatial ETL\n\nSpatial ETL tools provide the data processing functionality of traditional extract, transform, load\u00a0(ETL) software, but with a primary focus on the ability to manage spatial data. They provide GIS\u00a0users with the ability to translate data between different standards and proprietary formats, whilst geometrically transforming the data en route. These tools can come in the form of add-ins to existing wider-purpose software such as spreadsheets.\n\nGIS data mining\n\nGIS or spatial data mining is the application of data mining methods to spatial data. Data mining, which is the partially automated search for hidden patterns in large databases, offers great potential benefits for applied GIS-based\u00a0decision\u00a0making. Typical applications include environmental monitoring. A characteristic of such applications is that spatial correlation between data measurements require the use of specialized algorithms for more efficient data analysis.\n\nApplications\n\nOpen Geospatial Consortium standards\n\nThe Open Geospatial Consortium (OGC) is an international industry consortium of 384\u00a0companies, government agencies, universities, and individuals participating in a consensus process to develop publicly available geoprocessing specifications. Open interfaces and protocols defined by OpenGIS\u00a0Specifications support interoperable solutions that \"geo-enable\" the Web, wireless and location-based services, and mainstream\u00a0IT, and empower technology developers to make complex spatial information and services accessible and useful with all kinds of applications. Open Geospatial Consortium protocols include Web Map Service, and Web Feature Service. GIS\u00a0products are broken down by the\u00a0OGC into two categories, based on how completely and accurately the software follows the OGC\u00a0specifications. ''Compliant Products'' are software products that comply to OGC's\u00a0OpenGIS\u00a0Specifications. When a product has been tested and certified as compliant through the OGC\u00a0Testing Program, the product is automatically registered as \"compliant\" on this site. ''Implementing Products'' are software products that implement OpenGIS\u00a0Specifications but have not yet passed a compliance test. Compliance tests are not available for all specifications. Developers can register their products as implementing draft or approved specifications, though OGC\u00a0reserves the right to review and verify each entry.\n\nWeb mapping\n\nIn recent years there has been a proliferation of free-to-use and easily accessible mapping software such as the proprietary web applications Google\u00a0Maps and Bing\u00a0Maps, as well as the free and open-source alternative OpenStreetMap. These services give the public access to huge amounts of geographic data; perceived by many users to be as trustworthy and usable as professional information. Some of them, like Google Maps and OpenLayers, expose an application programming interface (API) that enable users to create custom applications. These toolkits commonly offer street maps, aerial\/satellite imagery, geocoding, searches, and routing functionality. Web mapping has also uncovered the potential of crowdsourcing geodata in projects like OpenStreetMap, which is a collaborative project to create a free editable map of the world. These mashup projects have been proven to provide a high level of value and benefit to end users outside that possible through traditional geographic information.\n\nThe condition of the Earth's surface, atmosphere, and subsurface can be examined by feeding satellite data into a GIS. GIS\u00a0technology gives researchers the ability to examine the variations in Earth processes over days, months, and years. As an example, the changes in vegetation vigor through a growing season can be animated to determine when drought was most extensive in a particular region. The resulting graphic represents a rough measure of plant health. Working with two variables over time would then allow researchers to detect regional differences in the lag between a decline in rainfall and its effect on vegetation. GIS\u00a0technology and the availability of digital data on regional and global scales enable such analyses. The satellite sensor output used to generate a vegetation graphic is produced for example by the advanced very-high-resolution radiometer (AVHRR). This sensor system detects the amounts of energy reflected from the Earth's surface across various bands of the spectrum for surface areas of about 1 square kilometer. The satellite sensor produces images of a particular location on the Earth twice a day. AVHRR and more recently the moderate-resolution imaging spectroradiometer (MODIS) are only two of many sensor systems used for Earth surface analysis. In addition to the integration of time in environmental studies, GIS is also being explored for its ability to track and model the progress of humans throughout their daily routines. A concrete example of progress in this area is the recent release of time-specific population data by the U.S.\u00a0Census. In this data set, the populations of cities are shown for daytime and evening hours highlighting the pattern of concentration and dispersion generated by North American commuting patterns. The manipulation and generation of data required to produce this data would not have been possible without\u00a0GIS. Using models to project the data held by a GIS\u00a0forward in time have enabled planners to test policy decisions using spatial decision support systems.\n\nSemantics\n\nTools and technologies emerging from the World Wide Web Consortium's Semantic Web are proving useful for data integration problems in information systems. Correspondingly, such technologies have been proposed as a means to facilitate interoperability and data reuse among GIS\u00a0applications. and also to enable new analysis mechanisms. Ontologies are a key component of this semantic approach as they allow a formal, machine-readable specification of the concepts and relationships in a given domain. This in turn allows a GIS to focus on the intended meaning of data rather than its syntax or structure. For example, reasoning that a land cover type classified as ''deciduous needleleaf trees'' in one dataset is a specialization or subset of land cover type ''forest'' in another more roughly classified dataset can help a GIS automatically merge the two datasets under the more general land cover classification. Tentative ontologies have been developed in areas related to GIS\u00a0applications, for example the hydrology ontology developed by the Ordnance Survey in the United Kingdom and the SWEET ontologies developed by NASA's Jet Propulsion Laboratory. Also, simpler ontologies and semantic metadata standards are being proposed by the W3C Geo Incubator Group to represent geospatial data on the web. GeoSPARQL is a standard developed by the Ordnance Survey, United States Geological Survey, Natural Resources Canada, Australia's Commonwealth Scientific and Industrial Research Organisation and others to support ontology creation and reasoning using well-understood OGC literals (GML, WKT), topological relationships (Simple Features, RCC8, DE-9IM), RDF and the SPARQL database query protocols. Recent research results in this area can be seen in the International Conference on Geospatial Semantics and the Terra Cognita \u2013 Directions to the Geospatial Semantic Web workshop at the International Semantic Web Conference.\n\nImplications of GIS in society\n\nWith the popularization of GIS in decision making, scholars have begun to scrutinize the social and political implications of GIS. GIS can also be misused to distort reality for individual and political gain. It has been argued that the production, distribution, utilization, and representation of geographic information are largely related with the social context and has the potential to increase citizen trust in government. Other related topics include discussion on copyright, privacy, and censorship. A more optimistic social approach to GIS\u00a0adoption is to use it as a tool for public participation.\n\nIn education\n\nAt the end of the 20th century, GIS began to be recognized as tools that could be used in the classroom. The benefits of GIS in education seem focused on developing spatial thinking, but there is not enough bibliography or statistical data to show the concrete scope of the use of GIS in education around the world, although the expansion has been faster in those countries where the curriculum mentions them. GIS seem to provide many advantages in teaching geography because they allow for analyses based on real geographic data and also help raise many research questions from teachers and students in classrooms, as well as they contribute to improvement in learning by developing spatial and geographical thinking and, in many cases, student motivation.\n\nIn local government\n\nGIS is proven as an organization-wide, enterprise and enduring technology that continues to change how local government operates. Government agencies have adopted GIS technology as a method to better manage the following areas of government organization: * Economic Development departments use interactive GIS mapping tools, aggregated with other data (demographics, labor force, business, industry, talent) along with a database of available commercial sites and buildings in order to attract investment and support existing business. Businesses making location decisions can use the tools to choose communities and sites that best match their criteria for success\nGIS Planning\ns the industry's leading vendor of GIS data web tools for economic development and investment attraction. A service from th\nFinancial Times\nGIS Planning''\nZoomProspector Enterprise\nan\nIntelligence Components\nsoftware are in use around the world. This includes 30 US statewide economic development organizations, the majority of the top 100 metro areas in North America and a number of investment attraction agencies in Europe and Latin America. * Public Safety operations such as Emergency Operations Centers, Fire Prevention, Police and Sheriff mobile technology and dispatch, and mapping weather risks. * Parks and Recreation departments and their functions in asset inventory, land conservation, land management, and cemetery management. * Public Works and Utilities, tracking water and stormwater drainage, electrical assets, engineering projects, and public transportation assets and trends. * Fiber Network Management for interdepartmental network assets * School analytical and demographic data, asset management, and improvement\/expansion planning * Public Administration for election data, property records, and zoning\/management. The Open Data initiative is pushing local government to take advantage of technology such as GIS technology, as it encompasses the requirements to fit the Open Data\/Open Government model of transparency. With Open Data, local government organizations can implement Citizen Engagement applications and online portals, allowing citizens to see land information, report potholes and signage issues, view and sort parks by assets, view real-time crime rates and utility repairs, and much more. The push for open data within government organizations is driving the growth in local government GIS technology spending, and database management.\n\n* AM\/FM\/GIS * ArcGIS * At-location mapping * Automotive navigation system * Cadastral map * Collaborative mapping * Comparison of GIS software * Counter-mapping * CyberGIS * Digital geologic mapping * Distributed GIS * Geographic information systems in China * Geographic information systems in geospatial intelligence * Geomatics * GIS and aquatic science * GIS and public health * GISCorps * GIS Day * GIS in archaeology * GRASS GIS * GvSIG * Historical GIS * Integrated Geo Systems * List of GIS data sources * List of GIS software * Map database management * Participatory GIS * QGIS * SAGA GIS * TerrSet * Traditional knowledge GIS * Virtual globe\n\nReferences","date":"2021-04-22 20:14:31","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 3, \"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\": 0.2929033935070038, \"perplexity\": 2190.457175713318}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618039604430.92\/warc\/CC-MAIN-20210422191215-20210422221215-00341.warc.gz\"}"}
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{"url":"https:\/\/www.assignguru.com\/multiple-choice\/l\/computer-science-engineering\/information-systems-and-engineering-economics\/information-systems-and-engineering-economics-set-5\/","text":"### Information Systems And Engineering Economics Set 5\n\nThis set of Information systems and engineering economics Multiple Choice Questions & Answers (MCQs) focuses on Information Systems And Engineering Economics Set 5\n\nQ1 | is a document to convey theAadhaar number to a resident.\n\u2022 b) cidr\n\u2022 c) uid\nQ2 | Record date of birth of the resident, indicating day, month and \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0inthe relevant ?eld.\n\u2022 a) initial\n\u2022 b) surname\n\u2022 c) year\n\u2022 d) name\nQ3 | \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0has to be recorded by the EnrolmentAgency as declared by the enrollee in the box provided by recording Male, Female or Transgender.\n\u2022 a) date of birth\n\u2022 b) gender\n\u2022 d) fingerprint\nQ4 | Which of the following is used to retrievepre-enrolment data?\n\u2022 a) pre-enrolment id\n\u2022 c) resident\\s name\nQ5 | The factors of time and \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0are the de?ning aspects of anyengineering economic decisions\n\u2022 uncertainty\n\u2022 certainty\nQ6 | Economic decisions differ in a fundamental way from the types ofdecisions typically encountered in engineering design.\n\u2022 TRUE\n\u2022 FALSE\nQ7 | ENGINEERING ECONOMICS INVOLVES\n\u2022 formulating\n\u2022 estimating\n\u2022 evaluating economicoutcomes\n\u2022 all\nQ8 | The factors of time and uncertainty are the de?ning aspects of anyengineering economic decisions\n\u2022 TRUE\n\u2022 FALSE\nQ9 | An instant dollar is worth more than a distant dollar\n\u2022 TRUE\n\u2022 FALSE\nQ10 | engineering economic decision refers to all investment decisionsrelating to engineering projects.\n\u2022 TRUE\n\u2022 FALSE\nQ11 | engineering economic decision is the evaluation of costs and benefassociated with making a capital investment.\n\u2022 TRUE\n\u2022 FALSE\nQ12 | Engineering economics is needed for many kinds of decision making\n\u2022 TRUE\n\u2022 FALSE\nQ13 | The factors of \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0and uncertainty are the de?ning aspects ofany engineering economic decisions\n\u2022 time\n\u2022 investment\nQ14 | Additional risk is not taken without an expected additional return ofsuitable magnitude\n\u2022 TRUE\n\u2022 FALSE\nQ15 | Money has a time value because it can earn more money over time(earning power).\n\u2022 TRUE\n\u2022 FALSE\nQ16 | F dollars at the end of period N is equal to a single sum P dollars now,if your earning power is measured in terms of interest rate i.\n\u2022 TRUE\n\u2022 FALSE\nQ17 | Initial amount of money in transactions involving debt or investmentsis called the principal (P).\n\u2022 TRUE\n\u2022 FALSE\nQ18 | engineering economic decision is the evaluation of costs and benefassociated with making a capital \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0.\n\u2022 expenditure\n\u2022 investment.\nQ19 | Initial amount of money in transactions involving debt or investmentsis called\n\u2022 interest\n\u2022 principal\nQ20 | How many years would it take an investment to double at 10% annualinterest?\n\u2022 7.27 years\n\u2022 8 years\n\u2022 9 years\nQ21 | Marginal revenue must exceed marginal cost, in order to carry out apro?table increase of operations\n\u2022 TRUE\n\u2022 FALSE\nQ22 | A plan for receipts or disbursements (An) that yields a particular cash ?ow pattern over a speci?ed length of time is called monthly equalpayment\n\u2022 TRUE\n\u2022 FALSE\nQ23 | At 8% interest, what is the equivalent worth of $2,042 after 5 yearsfrom now? \u2022 5000 \u2022 4000 \u2022 2000 \u2022 3000 Q24 | If you had$2,000 now and invested it at 10%, how much would it beworth in 8 years?\n\u2022 4200\n\u2022 4287\n\u2022 5000\nQ25 | Money has a time value because its purchasing power changes overtime (in?ation).\n\u2022 TRUE\n\u2022 FALSE","date":"2023-02-06 12:05:20","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.2019694745540619, \"perplexity\": 13841.532791139163}, \"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\/1674764500339.37\/warc\/CC-MAIN-20230206113934-20230206143934-00455.warc.gz\"}"}
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The Gospel According to André (2017)
PG-13 | 1h 33min | Documentary | 25 May 2018 (USA)
From the segregated American South to the fashion capitals of the world, operatic fashion editor André Leon Talley's life and career are on full display, in a poignant portrait that ... See full summary »
André Leon Talley, Bethann Hardison, Marc Jacobs | See full cast & crew »
Pacific Northwest Pictures Taps eOne's Mark Slone As President
01 February 2019 | Deadline
Pacific Northwest Pictures names Mark Slone president, expands national strategy
01 February 2019 | ScreenDaily
166 Documentary Features Submitted for Oscars
08 November 2018 | Variety
8 Indie Picks for May
Tribeca 2018: Special Screenings
TIFF 2017: TIFF Docs
updated 26 Aug 2017
HuffPost - All the movies that are cool for the summer (2018)
Liz's List
Not That There's Anything Wrong With That
2. Fashion Documentaries
Search for "The Gospel According to André" on Amazon.com
Title: The Gospel According to André (2017)
Follows the creation of The Metropolitan Museum of Art's most attended fashion exhibition in history, "China: Through The Looking Glass," an exploration of Chinese-inspired Western fashions by Costume Institute curator Andrew Bolton.
Director: Andrew Rossi
Stars: Anna Wintour, Andrew Bolton, John Galliano
Westwood: Punk, Icon, Activist (2018)
The first film to encompass the remarkable story of one of the true icons of our time, as she fights to maintain her brand's integrity, her principles - and her legacy.
Director: Lorna Tucker
Stars: Pamela Anderson, Christina Hendricks, Kate Moss
McQueen (2018)
Documentary | Biography
The life and career of fashion designer Lee Alexander McQueen: from his start as a tailor, to launching and overseeing his eponymous line and his untimely death.
Directors: Ian Bonhôte, Peter Ettedgui
Stars: Bernard Arnault, Joseph Bennett, Detmar Blow
Always at The Carlyle (2018)
The iconic Carlyle hotel has been an international destination for a particular jet-set as well as a favorite haunt of the most discernible New Yorkers.
Director: Matthew Miele
Stars: Woody Allen, Herb Alpert, Wes Anderson
Manolo: the Boy Who Made Shoes for Lizards (2017)
Documentary | Biography | Drama
An in-depth portrait of MANOLO BLAHNIK, self-confessed 'cobbler' and the man regarded by most influential fashion figures as 'the best shoe-maker of the 20th and 21st centuries. A film for ... See full summary »
Director: Michael Roberts
Stars: Manolo Blahnik, David Bailey, Mary Beard
Studio 54 (2018)
Documentary | History
The story of the notorious 1970s New York City nightclub.
Director: Matt Tyrnauer
Stars: Ian Schrager, Steve Rubell, Donald Rubell
Kevyn Aucoin: Beauty & the Beast in Me (2017)
Never-before-seen footage, archival videos and celebrity interviews provide insight into the life, career and legacy of makeup artist Kevyn Aucoin.
Director: Lori Kaye
Stars: Isidore Aucoin, Berta Camal, Naomi Campbell
Scotty and the Secret History of Hollywood (2017)
A portrait of unsung Hollywood legend Scotty Bowers, whose bestselling memoir chronicled his decades spent as sexual procurer to the stars.
Stars: Scotty Bowers, Lois Bowers, William Mann
A look at the life and work of the influential fashion editor of Harpers Bazaar, Diana Vreeland.
Directors: Lisa Immordino Vreeland, Bent-Jorgen Perlmutt, and 1 more credit »
Stars: Diana Vreeland, Richard Avedon, David Bailey
Halston (2019)
Prodigiously talented, Halston reigned over fashion in the 1970s and became a household name. But everything changed in the Wall Street era. With his empire under threat, Halston took the biggest gamble of his life.
Director: Frédéric Tcheng
Stars: Liza Minnelli, Marisa Berenson, Joel Schumacher
RBG (2018)
The exceptional life and career of U.S. Supreme Court Justice Ruth Bader Ginsburg, who has developed a breathtaking legal legacy while becoming an unexpected pop culture icon.
Directors: Julie Cohen, Betsy West
Stars: Ruth Bader Ginsburg, Ann Kittner, Harryette Helsel
The September Issue (2009)
A documentary chronicling Vogue editor-in-chief Anna Wintour's preparations for the 2007 fall-fashion issue.
Director: R.J. Cutler
Stars: Anna Wintour, Thakoon Panichgul, André Leon Talley
André Leon Talley ... Himself
Bethann Hardison ... Herself (archive footage)
Marc Jacobs ... Himself
Anna Wintour ... Herself
Tom Ford ... Himself
Eboni Marshall Turman ... Herself
Bruce Weaver ... Himself
Alexis Thomas ... Herself
Anne Bibby ... Herself
Michelle Obama ... Herself (archive footage)
Tamron Hall ... Herself
Wanda Garrett ... Herself
Julia Child ... Herself (archive footage)
Martin Luther King ... Himself (archive footage)
Barbra Streisand ... Herself (archive footage)
From the segregated American South to the fashion capitals of the world, operatic fashion editor André Leon Talley's life and career are on full display, in a poignant portrait that includes appearances by Anna Wintour, Marc Jacobs, Tom Ford, Bethann Hardison, Valentino, and Manolo Blahnik.
A Film About Andre Leon Talley
Rated PG-13 for some thematic and suggestive content | See all certifications »
25 May 2018 (USA) See more »
The Gospel According to André See more »
$47,802, 27 May 2018
a big man with big talent
8 June 2018 | by ferguson-6 – See all my reviews
Greetings again from the darkness. Being an expert, or even a genius, in one's chosen field doesn't necessarily translate to celebrity or a life in the public eye. Few of us can name the best structural engineer or the best commercial airline pilot, yet we regularly drive over bridges and book flights to our vacation spots. However some professions lend themselves to a bit of fame ... and that's either a burden or an opportunity depending on perspective.
Director Kate Novack (writer of PAGE ONE: INSIDE THE NEW YORK TIMES, 2011) focuses her camera on one who seized the opportunity. Andre Leon Talley was raised in the Jim Crow South of North Carolina and rose up to become a literal giant in the fashion industry. It's mostly a biopic of a fascinating, larger than life figure, but also a quasi-history of the fashion industry since the 1970's. Andre crossed paths with all of the greats, and many of them are interviewed here: Marc Jacobs, Anna Wintour, Tom Ford, Valentino, Fran Leibowitz, Manolo Blahnik, and Isabella Rossellini - along with her pigs, a chicken and a turkey. We learn that he worked for Andy Warhol, was mentored by Diana Vreeland, and worked alongside Anna Wintour (teaching her as much as he learned).
"Fashion is fleeting, style remains." So Andre tells us as the film begins. He knows the difference between the two, and understands that beauty comes in many forms. Certainly the first, and often the only black man on the front row of runways in Paris and New York, Andre has lived quite the life. Director Novack's film is at its best when Andre is front and center. He commands attention with his size, his clothes, his voice, his charisma, and mostly his talent. Claiming his eye developed watching the Sunday fashions at the black church of his youth, we also learn young Andre preferred shopping to attending a ballgame with his taxi-driving father.
Thin until age 40, Andre now describes himself as a manatee. The racism he faced within the industry is vivid as he recalls being called "Queen Kong". Sometimes criticized for not taking a more active and vocal stance against racism, Andre simply proclaims that he was too busy with his career ... his same reason for having 'no love life'. The emotional moments of his recollections fade quickly in the segments where he discusses capes, and later veils. His expertise is on full display.
Looming over much of the film is the backdrop of the 2016 Presidential election. It's often distracting, but does lead to one of the more powerful moments. This verbose, grandiose couture figure is stunned and mostly at a loss for words as Donald Trump takes his oath. For most of the film and for most of his life, Andre has talked the talk and walked the walk - and continued talking while he walked. As one of style and influence, he has plenty to say and there's a reason for us to listen.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,846
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{"url":"https:\/\/github.com\/patrickt\/recschemes\/blob\/master\/src\/Part4.lhs","text":"# patrickt\/recschemes\n\nSwitch branches\/tags\nNothing to show\nFetching contributors\u2026\nCannot retrieve contributors at this time\n1097 lines (921 sloc) 46.1 KB\n \\long\\def\\ignore{} Though we've only just begun our dive into \\emph{Bananas, Lenses, Envelopes, and Barbed Wire}, the next natural step in understanding recursion schemes brings us outside its purview. We must turn our attention to a paper written seven years later--- \\href{http:\/\/cs.ioc.ee\/~tarmo\/papers\/inf99.pdf}{\\emph{Primitive(Co)Recursion and Course-of-Value (Co)Iteration, Categorically}}, by Tarmo Uustalu and Varmo Vene. \\emph{Primitive (Co)Recursion} explores and formalizes the definition of apomorphisms (introduced first by Meijer et. al, and which we discussed, briefly, in the \\href{http:\/\/blog.sumtypeofway.com\/recursion-schemes-part-iii-folds-in-context\/}{previous installment}) and describes two new recursion schemes, the \\emph{histomorphism} and the \\emph{futumorphism}. \\emph{Primitive (Co)Recursion} is a wonderful and illuminating paper, but it is dense in its concepts for those unfamiliar with category theory, and uses the semi-scrutable bracket syntax introduced by \\emph{Bananas}. But there's no need for alarm if category theory isn't your cup of tea: Haskell allows us, once again, to express elegantly the new recursion schemes defined in \\emph{Primitive (Co)Recursion}. Guided by Uustalu and Vene's work, we'll derive these two new recursion schemes and explore their ways in which they simplify complicated folds and unfolds. Though these new morphisms are, definition-wise, simple variations on paramorphisms and apomorphisms, in practice they provide surprising power and clarity, as Uustalu and Vene assert: \\begin{quote} {[}We{]} argue that even these schemes are helpful for a declaratively thinking programmer and program reasoner who loves languages of programming and program reasoning where programs and proofs of properties of programs are easy to write and read. \\end{quote} That sure sounds like us. Let's get going. This article is literate Haskell; you can find the source code \\href{https:\/\/github.com\/patrickt\/recschemes\/blob\/master\/src\/Part4.lhs}{here}. % Now we take care of the Haskell stuff. \\ignore{ \\begin{code} {-# LANGUAGE StandaloneDeriving #-} {-# LANGUAGE DeriveFunctor #-} {-# LANGUAGE OverloadedStrings #-} module Part4 ( Attr (..) , Cent , CoAttr (..) , CVAlgebra , CVCoalgebra , Nat (..) , coins , change , compress , futu , histo ) where import Part1 (Term (..)) import Part2 import Part3 (RAlgebra, RCoalgebra) import Prelude hiding (lookup) import Control.Arrow hiding (left, right) import Data.List hiding (lookup) import qualified System.Random as Random import Text.PrettyPrint.Boxes \\end{code} } \\subsubsection{A Brief Recap}\\label{a-brief-recap} In our first entry, we defined \\texttt{Term}, the fixed-point of a Haskell \\texttt{Functor}, with an \\texttt{In} constructor that wraps one level of a structure and an \\texttt{out} destructor to perform the corresponding unwrap\\footnote{ Bob Harper, in \\emph{Practical Foundations for Programming Languages}, refers to \\texttt{In} and \\texttt{out} as rolling'' and unrolling'' operations. This is a useful visual metaphor: the progression \\texttt{f (f (Term f)) -> f (Term f) -> Term f} indeed looks like a flat surface being rolled up, and its opposite \\texttt{Term f -> f (Term f) -> f (f (Term f))} looks like the process of unrolling.}. \\begin{verbatim} newtype Term f = In { out :: f (Term f) } \\end{verbatim} Given an algebra --- a folding function that collapses a Functor \\texttt{f} containing \\texttt{a}'s into a single \\texttt{a}--- \\begin{verbatim} type Algebra f a = f a -> a \\end{verbatim} we use the catamorphism \\texttt{cata} to apply a leaf-to-root\\footnote{ Rob Rix \\href{https:\/\/twitter.com\/rob_rix\/status\/793430628637274112}{points out} that, though catamorphisms are often described as bottom-up'', this term is ambiguous: catamorphisms' recursion occurs top-down, but the folded value is constructed bottom-up. I had never noticed this ambiguity before. (The words of Carroll come to mind: \\,When I use a word,' Humpty Dumpty said, in rather a scornful tone, it means just what I choose it to mean --- neither more nor less.'\\,'')} fold over any recursively-defined data structure. \\texttt{cata} travels to the most deeply-nested point in the data structure by \\texttt{fmap}ing itself, recursively, into the next level of the stucture. When \\texttt{fmap\\ cata\\ x} returns an unchanged \\texttt{x}, we cease recursing (because we have hit the most-deeply-nested point); we can then begin constructing the return value by passing each node to the algebra, leaf-to-root, until all the recursive invocations have finished. \\begin{verbatim} cata :: (Functor f) => Algebra f a -> a -> Term f cata f = out >>> fmap (cata f) >>> f \\end{verbatim} But the catamorphism has its limits: as it is applied to each level of the structure, it can only examine the current carrier value from which it is building. Given the F-algebra \\texttt{f\\ a\\ -\\textgreater{}\\ a}, each of the structure's children---the \\texttt{a} values contained in the \\texttt{f} container---has already been transformed, thus losing information about the original structure. To remedy this, we introduced \\texttt{para}, the paramorphism, and an R-algebra to carry the original structure with the accumulator: \\begin{verbatim} type RAlgebra f a = f (Term f, a) -> a para :: Functor f => RAlgebra f a -> Term f -> a para f = out >>> fmap (id &&& para f) >>> f \\end{verbatim} \\subsubsection{Running a Course with Histomorphisms}\\label{running-a-course-with-histomorphisms} Paramorphisms allow us, at each stage of the fold, to view the original structure of the examined node before the fold began. Though this is more powerful than the catamorphism, in many cases it does not go far enough: many useful functions are defined not just in terms of the original argument to the function, but in terms of previous computed values. The classic\\footnote{Unfortunately, in this context I think classic'' can be read as hackneyed and unhelpful''. I dislike using \\texttt{fib()} to teach recursion schemes, as the resulting implementations are both more complicated than a straightforward implementation and in no way indicative of the power that recursion schemes bring to the table. Throughout this series, I've done my damnedest to pick interesting, beautiful examples, lest the reader end up with the gravely mistaken takeaway that recursion schemes aren't useful for any real-world purpose.} example is the Fibonacci function, the general case of which is defined in terms of two previous invocations: \\begin{code} fib :: Int -> Int fib 0 = 0 fib 1 = 1 fib n = fib (n-1) + fib (n-2) \\end{code} We could express this function using a catamorphism---though one of the carrier values (\\texttt{fib\\ (n-1)}) would be preserved, as the accumulator of our fold, we would need another explicit recursive call to \\texttt{cata} to determine the historical value of \\texttt{fib\\ (n-2)}. This is a bummer, both in terms of efficiency---we're recalculating values we've already calculated---and in terms of beauty: a function so fundamental as \\texttt{fib} deserves a better implementation, especially given the expressive power of recursion schemes. The imperative programmers among us will have a solution to this inefficiency: iterate!'', they will yell, or perhaps they will clamor introduce a cache!'' in a great and terrible voice. And it's true: we could compute \\texttt{fib} with a for-loop or by memoizing the recursive call. But the former approach entails mutable state---a very big can of worms to open for such a simple problem---and the latter leaves us \\href{https:\/\/twitter.com\/importantshock\/status\/241173326846898176}{with two problems}. Uustalu and Vene's histomorphism provides a way out: we will \\emph{preserve the history} of the values our fold computes, so that further recursive calls to compute past values become unnecessary. This style of recursion is called \\emph{course-of-value recursion}, since we record the \\emph{values} evaluated as our fold function \\emph{courses} through the structure. Rather than operate on an \\texttt{f\\ a}, a data structure in the process of being folded, we'll operate on a more sophisticated structure, so that the argument to our fold function contains the history of all applications of the fold itself. Instead of a just a carrier value \\texttt{a}, our \\texttt{f} will contain a carrier value and a recursive, unrollable record of past invocations, to wit: \\begin{code} data Attr f a = Attr { attribute :: a , hole :: f (Attr f a) } \\end{code} We'll call this \\texttt{Attr}, since it's an attributed' version of \\texttt{Term}. An \\texttt{Attr\\ f\\ a} contains an \\texttt{a}---a carrier value, storing the in-progress value of the fold---as well as a fixed-point value (analogous to \\texttt{Term}) at each level of recursion. Thanks to the fixed-point \\texttt{hole} within the \\texttt{f}, further \\texttt{Attr} items are preserved, each of which contains the shape of the folded functor \\texttt{f}. And within the \\texttt{f} there lie further \\texttt{Attr} values, each of which contains a carrier value yielded by \\emph{their} application in their \\texttt{attribute} slot. And those \\texttt{Attr} values in turn contain further \\texttt{hole}s, which contain the historical records pertaining to \\emph{their} childrens' history, and so on and so forth until the bottom of the data structure has been reached. As such, the entire history of the fold is accessible to us: the \\texttt{holes} preserve the shape of the data structure (which was lost during \\texttt{cata}), and the \\texttt{attribute} holds the record of applying the fold to each entity in said data structure. We have a word for preserving a record of the past, of course---\\emph{history}\\footnote{A word with a rich pedigree---most directly from the Greek \u1f31\u03c3\u03c4\u03bf\u03c1\u03af\u03b1', meaning \\emph{a narration of what has been learned}, which in turn descended from \u1f31\u03c3\u03c4\u03bf\u03c1\u03ad\u03c9', \\emph{to learn through research}, and in turn from \u1f35\u03c3\u03c4\u03c9\u03c1', meaning \\emph{the one who knows} or \\emph{the expert}--- a term commensurate with the first histories being passed from person to person orally. And the Greek root \u1f31\u03c3\u03c4\u03bf', according to the OED, can be translated as web': a suitable metaphor for the structural web of values that the \\texttt{Attr} type generates and preserves.}. A fold operation that uses \\texttt{Attr} to provide both an accumulator and a record of prior invocations is known as a \\emph{histomorphism}---a shape-changing (\\emph{morpho}) fold with access to its history (\\emph{histo}). Let's define the histomorphism. It will, like its cousins \\texttt{cata} and \\texttt{para}, use an algebra for its fold function. But unlike the F-algebra of \\texttt{cata} or the R-algebra of \\texttt{para}, we'll be using an algebra that operates on an \\texttt{Attr\\ f\\ a}, yielding an \\texttt{a} out of it. We call this a course-of-value algebra, abbreviated to a \\emph{CV-algebra}, and define a type alias for it, so we end up with a more comprehensible type signature in the histomorphism: \\begin{code} type CVAlgebra f a = f (Attr f a) -> a \\end{code} That is, a CV-algebra maps from a container \\texttt{f} containing children of type \\texttt{Attr\\ f\\ a} (which in turn contain \\texttt{f\\ (Attr\\ f\\ a)} children, as far down as is needed in the nested structure), to a final result type \\texttt{a}. The shape of the folded structure and the history of its applications are all contained in its \\texttt{Attr} values: all you have to do is unroll the \\texttt{hole} value to go back one level in history and use \\texttt{attribute} to examine the stored value. Our \\texttt{histo} function will be similar to \\texttt{cata} and \\texttt{para} at its heart. We start by unpacking the \\texttt{Term}---the initial argument must be a \\texttt{Term} rather than an \\texttt{Attr}, since as we haven't started the fold yet we have no value to fill in for \\texttt{attribute}. We will then recurse, with \\texttt{fmap}, into the thus-revealed structure until we hit its root. We then use the CV-algebra to build the value, starting at the root and continuing upwards to the topmost leaf. These steps are analogous to how we defined \\texttt{cata} and \\texttt{para}, so let's start defining it: \\begin{verbatim} histo :: Functor f => CVAlgebra f a -> Term f -> a histo h = out >>> fmap worker >>> h \\end{verbatim} But what type should the worker have? Well, we can ask GHC, thanks to one of its most useful features\\footnote{A feature taken wholesale, we must note, from dependently-typed languages like Agda and Idris.}---type holes. By prepending an underscore to the use of \\texttt{worker}, we can allow the program compilation to continue as far as is possible---however, when the compilation process has finished, GHC will remind us where we used a type hole, and inform us of the type signature it inferred for \\texttt{\\_worker}. (As a full-time Haskell programmer, I use this feature nearly every day.) \\begin{verbatim} histo :: Functor f => CVAlgebra f a -> Term f -> a histo h = out >>> fmap _worker >>> h \\end{verbatim} Running this code in GHC yields the following type-hole message: \\begin{verbatim} \/Users\/patrick\/src\/morphisms\/src\/Main.hs:14:24: error: \u2022 Found hole: \u2018_worker\u2019 with type :: Term f -> Attr f a \\end{verbatim} Okay, that makes sense! We're operating on \\texttt{Term\\ f} values (lifted into this context by the \\texttt{fmap} within \\texttt{histo}), and we need to yield an \\texttt{Attr\\ f\\ a}, so that the outside \\texttt{Term\\ f} can be transformed into an \\texttt{f\\ (Attr\\ f\\ a)} and then passed into the CV-algebra. An \\texttt{Attr\\ f\\ a}, as defined above, contains two values: a plain \\texttt{a} type, and a recursive \\texttt{f\\ (Attr\\ f\\ a)} hole. Given a \\texttt{Term\\ f} and our ability to invoke both \\texttt{histo} and \\texttt{worker} recursively, we can build the \\texttt{Attr\\ f\\ a} we need. Let's start by defining the skeleton of \\texttt{worker}: given a \\texttt{Term\\ f}, called \\texttt{t}, it constructs an \\texttt{Attr}, containing two fields. \\begin{verbatim} worker t = Attr _ _ \\end{verbatim} The first field, the \\texttt{a}, is yielded by recursing with \\texttt{histo} on the provided \\texttt{Term}---easy enough. This is just like the catamorphism---indeed, a catamorphism is a histomorphism that ignores the provided history. \\begin{verbatim} worker t = Attr (histo h term) _ \\end{verbatim} The second field's construction is more clever: we unwrap \\texttt{term} with the \\texttt{out} function, which gives us an \\texttt{f\\ (Term\\ f)} out of a \\texttt{Term\\ f}. Since we don't know exactly what type \\texttt{f} is yet, we can't extract the contained \\texttt{Term\\ f}---but we can operate on it, with \\texttt{fmap}, provided by the \\texttt{Functor} constraint. So, to go from an \\texttt{f\\ (Term\\ f)} to an \\texttt{f\\ (Attr\\ f\\ a)}, we need a function of type \\texttt{Term\\ f\\ -\\textgreater{}\\ Attr\\ f\\ a}\\ldots{} hang on, that's just \\texttt{worker} itself! \\begin{verbatim} worker t = Attr (histo h term) (fmap worker (out t)) \\end{verbatim} This is the heart of \\texttt{histo}s elegance: it's 'doubly recursive', in that its \\texttt{worker} function invokes both \\texttt{histo} and \\texttt{worker} itself. Now we have a \\texttt{histo} function that passes the typechecker: \\begin{verbatim} histo :: Functor f => CVAlgebra f a -> Term f -> a histo h = out >>> fmap worker >>> h where worker t = Attr (histo h t) (fmap worker (out t)) \\end{verbatim} However, this function does not share its subcomputations properly: each iteration of \\texttt{worker} recomputes, rather than reuses, all the nested \\texttt{hole} values within the constructed \\texttt{Attr}. We can fix this by promoting \\texttt{worker} to operate on \\texttt{Attr} values; by recursing with \\texttt{fmap worker}, placing the input and output of the CV-algebra in a tuple with \\texttt{&&&}, and then unpacking the tuple into an \\texttt{Attr}, we ensure that all the constructed \\texttt{Attr} values share their subcomputations. \\begin{code} histo :: Functor f => CVAlgebra f a -> Term f -> a histo h = worker >>> attribute where worker = out >>> fmap worker >>> (h &&& id) >>> mkAttr mkAttr (a, b) = Attr a b \\end{code} But what does this function \\emph{mean}? We've filled in all these type holes, and we have a working \\texttt{histo} function, but why does it work? Why does this preserve the history? The answer lies in \\texttt{worker}, in the \\texttt{id} function that captures and preserves the \\texttt{Attr} the worker function is operating on. If we omitted that expression, we would have a function equivalent to \\texttt{cata}---one that throws all its intermediate variables away while computing the result of a fold. But our worker function ensures that the result computed at each stage is not lost: as we flow, root-to-leaf, upwards through the data structure, we construct a new \\texttt{Attr} value, which in turn contains the previous result, which itself preserves the result before that, and so on. Each step yields an up-to-date snapshot of what we have computed in the past. By \\emph{not throwing out intermediate results}, and pairing these intermediate results with the values used to calculate them, we automatically generate \\emph{and update} a cache for our fold. Now, I may have used \\texttt{fib} as an example of a course-of-value recursive function, but I won't provide an example of using \\texttt{histo} to calculate the nth Fibonacci number (though it's a good exercise). Let's solve a toy problem that's slightly more interesting, one that histomorphisms make clear and pure, and one whose solution can be generalized to all other problems of its ilk. \\subsection{C-C-C-Changes}\\label{c-c-c-changes} The \\href{https:\/\/en.wikipedia.org\/wiki\/Change-making_problem}{change-making problem} is simple: given a monetary amount \\texttt{N}, and a set of denominations (penny, nickel, dime, \\&c.), how many ways can you make change for \\texttt{N}? While it's possible to write a na\u00efve recursive solution for this problem, it becomes intolerably slow for large values of \\texttt{N}: each computation for \\texttt{N} entails computing the values for \\texttt{N\\ -\\ 1}, and \\texttt{N\\ -\\ 2}, and \\texttt{N\\ -\\ 3}, and so forth: if we don't store these intermediate amounts in a cache, we will waste our precious time on this earth. And, though this era may be grim as all hell, slow algorithms are no way to pass the time. We'll start by setting up a list of standard denominations. Feel free to adjust this based on the denominational amounts of your country of residence. \\begin{code} type Cent = Int coins :: [Cent] coins = [50, 25, 10, 5, 1] \\end{code} So our fundamental procedure is a function \\texttt{change}, that takes a cent amount and returns a count of how many ways we can make change for said cent amount: \\begin{verbatim} change :: Cent -> Int \\end{verbatim} It is here where we hit our first serious roadblock. I asserted earlier that the change-making problem, and all the other \\href{https:\/\/en.wikipedia.org\/wiki\/Knapsack_problem}{knapsack problems} of its ilk, are soluble with a histomorphism---a cached fold over some sort of data structure. But here we're dealing with\\ldots{} natural-number values. There are no lists, no vectors, no rose trees---nothing mappable (that is to say, nothing with a \\texttt{Functor} instance) and therefore nothing to fold over. What are we supposed to do? All is not lost: we can fold over the natural numbers, just as we would fold over a list. We just have to define the integers in an unconventional, but simple, way: every natural number is either zero, or 1 + the previous. We'll call this formulation of the natural numbers \\texttt{Nat}--- the zero value will be \\texttt{Zero}\\footnote{Natch.}, and the notion of the subsequent number \\texttt{Next}. Put another way, we need to encode \\href{https:\/\/en.wikipedia.org\/wiki\/Peano_axioms}{Peano numerals} in Haskell\\footnote{Keen-eyed readers will note that this data type is isomorphic to the \\texttt{Maybe} type provided by the Prelude. We could've just used that, but I wanted to make the numeric nature of this structure as clear as possible.}. \\begin{code} data Nat a = Zero | Next a deriving Functor \\end{code} We use \\texttt{Term} to parameterize \\texttt{Nat} in terms of itself---that is to say, given \\texttt{Term}, we can stuff a \\texttt{Nat} into it so as to represent an arbitrarily-nested hierarchy of contained \\texttt{Nat}s, and thus represent all the natural numbers: \\begin{verbatim} one, two, three :: Term Nat one = In (Next (In Zero)) two = In (Next one) three = In (Next two) \\end{verbatim} For convenience's sake, we'll define functions that convert from standard \\texttt{Int} values to foldable \\texttt{Term\\ Nat}s, and vice versa. Again, these do not look particularly efficient, but please give me the benefit of the doubt. \\begin{code} -- Convert from a natural number to its foldable equivalent, and vice versa. expand :: Int -> Term Nat expand 0 = In Zero expand n = In (Next (expand (n - 1))) compress :: Nat (Attr Nat a) -> Int compress Zero = 0 compress (Next (Attr _ x)) = 1 + compress x \\end{code} While this is, at a glance, obviously less-efficient than using integers, it's not as bad as it seems. We only have three operations: increment, converting from zero, and converting to zero. Restricting our operations to these---rather than writing our own code for addition or subtraction, both of which are linear-time over the Peano numerals---means that operations on our \\texttt{Term\\ Nat} types are almost the same as hardware-time costs, barring GHC-specific operations. As such, the expressivity we yield with our foldable numbers is well worth the very slight costs. Given an amount (\\texttt{amt}), we solve the change-making problem by converting that amount to a \\texttt{Term\\ Nat} with \\texttt{expand}, then invoking \\texttt{histo} on it with a provided CV-algebra---let's call it \\texttt{go}. We'll define it in a where-clause below. \\begin{code} change :: Cent -> Int change amt = histo go (expand amt) where \\end{code} Since we're operating on foldable natural values (\\texttt{Nat}) and ultimately yielding an integral result (the number of ways it is possible to make change for a given \\texttt{Nat}), we know that our CV-algebra will have as its carrier functor \\texttt{Nat} and its result type \\texttt{Int}. \\begin{code} -- equivalent to Nat (Attr Nat Int) -> Int go :: CVAlgebra Nat Int \\end{code} Because \\texttt{histo} applies its algebra from leaf-to-root, it starts at the deepest nested position in the \\texttt{Term\\ Nat}---that is to say, \\texttt{Zero}. We know that there's only one way to make change for zero coins---by giving zero coins back---so we encode our base case by explicitly matching on a Zero and returning 1. \\begin{code} go Zero = 1 \\end{code} Now comes the interesting part---we have to match on \\texttt{Next}. Contained in that \\texttt{Next} value will be an \\texttt{Attr\\ Nat\\ Int} (which we'll refer to as \\texttt{attr}), containing the value yielded from applying \\texttt{go} to the previous \\texttt{Nat}ural number. Since we'll need to feed this function into \\texttt{compress} to perform actual numeric operations on it (since we did not write the requisite boilerplate to make \\texttt{Nat} an instance of the \\texttt{Num} typeclass\\footnote{There is no reason why we couldn't do this---I just chose to omit it for the sake of brevity.}), we'll use an @-pattern to capture it under the name \\texttt{curr}. \\begin{code} go curr@(Next attr) = let \\end{code} Because we need to find out what numeric amounts (from \\texttt{coins}) are valid change-components for \\texttt{curr}, we have to get an \\texttt{Int} out of \\texttt{curr}. We'll call this value \\texttt{given}, since it's our given amount. \\begin{code} given = compress curr \\end{code} Now we have to look at each value of the \\texttt{coins} list. Any values greater than \\texttt{given} are right out: you can't use a quarter to make change for a dime, obviously. \\begin{code} validCoins = filter (<= given) coins \\end{code} Now we subtract the \\texttt{given} amount from each element of \\texttt{validCoins}. This list represents, for each coin in \\texttt{validCoins}, how much change we have remaining after using that coin to make change for \\texttt{given}---if \\texttt{given} were equal to 10, the list would be \\texttt{{[}9,\\ 5,\\ 0{]}}. \\begin{code} remaining = map (given -) validCoins \\end{code} Now we partition this \\texttt{remaining} list into two sublists: the items equal to zero and those that are not. We don't need to consult the lookup table for the items that are zero, obviously, but we need to do so for the others. \\begin{code} (zeroes, toProcess) = partition (== 0) remaining \\end{code} Given each number in \\texttt{toProcess}, we have to consider how many ways we could make change out of that number---but, since we know that that we've already calculated that result, because it's by definition less than \\texttt{given}! So all we have to do is look up the cached result in our \\texttt{attr}. (We'll implement the \\texttt{lookup} function later on---it is two lines of code.) We'll add all these cached results together with \\texttt{sum}. \\begin{code} results = sum (map (lookup attr) toProcess) \\end{code} Then all that's left to do is add \\texttt{zeroCount} and \\texttt{others} together. \\begin{code} in length zeroes + results \\end{code} Let's take a look at what we've written so far. \\begin{verbatim} change :: Cent -> Int change amt = histo go (expand amt) where go :: Nat (Attr Nat Int) -> Int go Zero = 1 go curr@(Next attr) = let given = compress curr validCoins = filter (<= given) coins remaining = map (given -) validCoins (zeroes, toProcess) = partition (== 0) remaining results = sum (map (lookup attr) toProcess) in length zeroes + results \\end{verbatim} Wow. This is pretty incredible. Not only do we have a simple, pure, concise, and performant solution to the change-making problem, but the caching is \\emph{implicit}: we don't have to update the cache ourselves, because \\texttt{histo} does it for us. We've stripped away the artifacts required to solve this problem efficiently and zeroed in on the essence of the problem. This is remarkable. I told you I would show you how to look up the cached values, and indeed I will do so now. An \\texttt{Attr\\ Nat\\ a} is essentially a nonempty list: if we could pluck the most-final \\texttt{Attr\\ Nat\\ a} after \\texttt{change} has finished executing, we would see the value of \\texttt{change\\ 0} stored inside the first \\texttt{attribute} value, the value of \\texttt{change\\ 1} stored inside the \\texttt{attribute} within the first attribute's \\texttt{hole}, and the value for \\texttt{change\\ 2} inside that further \\texttt{hole}. So, given an index parameter \\texttt{n}, we return the \\texttt{attribute} if \\texttt{n} is 0, and we recurse inside the \\texttt{hole} if not, with \\texttt{n\\ -\\ 1}. \\begin{code} lookup :: Attr Nat a -> Int -> a lookup cache 0 = attribute cache lookup cache n = lookup inner (n - 1) where (Next inner) = hole cache \\end{code} \\subsection{A Shape-Shifting Cache}\\label{a-shape-shifting-cache} Something crucial to note is that the fixed-point accumulator---the \\texttt{f\\ (Attr\\ f\\ a)} parameter to our CV-algebra---\\emph{changes shape} based on the functor \\texttt{f} contained therein. Given an inductive functor \\texttt{Nat} that defines the natural numbers, \\texttt{Nat\\ (Attr\\ Nat\\ a)} is isomorphic to \\texttt{{[}{]}}, the ordinary linked list: a \\texttt{Zero} is the empty list, and a \\texttt{Next} that contains a value (stored in \\texttt{Attr}'s \\texttt{attribute} field) and a pointer to the next element of the list (stored in the \\texttt{hole\\ ::\\ Nat\\ (Attr\\ Nat\\ a))} field in the given \\texttt{Attr}). This is why our implementation of \\texttt{lookup} is isomorphic to an implementation of \\texttt{!!} over \\texttt{{[}{]}}---because they're the same thing. But what if we use a different \\texttt{Functor} inside an \\texttt{Attr}? Well, then the shape of the resulting \\texttt{Attr} changes. If we provide the list type---\\texttt{{[}{]}}---we yield \\texttt{Attr\\ {[}{]}\\ a}, which is isomorphic to a rose tree---in Haskell terms, a \\texttt{Tree\\ a}. If we use \\texttt{Either\\ b}, then \\texttt{Attr\\ (Either\\ b)\\ a} is a nonempty list of computational steps, terminating in some \\texttt{b} value. \\texttt{Attr} is more than an attributed \\texttt{Term}''---it is an \\emph{adaptive cache} for a fold over \\emph{any type of data structure}. And that is truly wild. \\subsection{Obsoleting Old Definitions}\\label{obsoleting-old-definitions} As with \\texttt{para}, the increased power of \\texttt{histo} allows us to express \\texttt{cata} with new vocabulary. Every F-algebra can be converted into a CV-algebra---all that's needed is to ignore the \\texttt{hole} values in the contained Functor \\texttt{f}. We do this by mapping \\texttt{attribute} over the functor before passing it to the F-algebra, throwing away the history contained in \\texttt{hole}. \\begin{code} cata :: Functor f => Algebra f a -> Term f -> a cata f = histo (fmap attribute >>> f) \\end{code} Similarly, we can express \\texttt{para} with \\texttt{histo}, except instead of just fmapping with \\texttt{attribute} we need to do a little syntactic juggling to convert an \\texttt{f\\ (Attr\\ f\\ a)} into an \\texttt{f\\ (Term\\ f,\\ a)}. (Such juggling is why papers tend to use banana-bracket notation: implementing this in an actual programming language often requires syntactic noise such as this.) \\begin{code} para :: Functor f => RAlgebra f a -> Term f -> a para f = histo (fmap worker >>> f) where worker (Attr a h) = (In (fmap (worker >>> fst) h), a) \\end{code} \\subsection{Controlling the Future with Futumorphisms}\\label{controlling-the-future-with-futumorphisms} Throughout this series, we can derive unfolds from a corresponding fold by reversing the arrows''---viz., finding the function dual to the fold in question. And the same holds true for histomorphisms---the dual is very powerful. But, to find the dual of \\texttt{histo}, we must first find the dual of \\texttt{Attr}. Whereas our \\texttt{Attr} structure held both an \\texttt{a} and a recursive \\texttt{f\\ (Attr\\ f\\ a)} structure, its dual---\\texttt{CoAttr}---holds \\emph{either} an \\texttt{a} value---we'll call that \\texttt{Automatic}---or a recursive \\texttt{f\\ (CoAttr\\ f\\ a)} value, which we'll call \\texttt{Manual}. (Put another way, since \\texttt{Attr} was a product type, its dual is a sum type.) The definition follows: \\begin{code} data CoAttr f a = Automatic a | Manual (f (CoAttr f a)) \\end{code} And the dual of a CV-algebra is a CV-coalgebra: \\begin{code} type CVCoalgebra f a = a -> f (CoAttr f a) \\end{code} So why call these \\texttt{Automatic} and \\texttt{Manual}? It's simple---returning a \\texttt{Manual} value from our CV-coalgebra means that we specify manually how the unfold should proceed at this level, which allows us to unfold more than one level at a time into the future. By contrast, returning a \\texttt{Automatic} value tells the unfold to continue automatically at this level. This is why we call them \\emph{futu}morphisms---our CV-coalgebra allows us to determine the \\emph{future} of the unfold. (The term futumorphism' is etymologically dubious, since the futu-' prefix is Latin and the -morpho' suffix is Greek, but there are many other examples of such dubious words: television', automobile', and monolingual', to name but a few.) Like its predecessor unfolds \\texttt{ana} and \\texttt{apo}, the futumorphism will take a coalgebra, a seed value \\texttt{a}, and produce a term \\texttt{f}: \\begin{verbatim} futu :: Functor f => CVCoalgebra f a -> a -> Term f \\end{verbatim} We derived the anamorphism and apomorphism by reversing the arrows in the definitions of \\texttt{cata} and \\texttt{para}. The same technique applies here---\\texttt{\\textgreater{}\\textgreater{}\\textgreater{}} becomes \\texttt{\\textless{}\\textless{}\\textless{}}, and \\texttt{In} becomes \\texttt{out}. And as previously, we use a type hole to derive the needed signature of the helper function. \\begin{verbatim} futu :: Functor f => CVCoalgebra f a -> a -> Term f futu f = In <<< fmap _worker <<< f \\end{verbatim} \\begin{verbatim} \/Users\/patrick\/src\/morphisms\/src\/Main.hs:28:32: error: \u2022 Found hole: \u2018_worker\u2019 with type :: CoAttr f a -> Term f \\end{verbatim} This also makes sense! The worker function we used in \\texttt{histo} was of type \\texttt{Term\\ f\\ -\\textgreater{}\\ Attr\\ f\\ a}---by reversing the arrows in this worker and changing \\texttt{Attr} to \\texttt{CoAttr}, we've derived the function we need to define \\texttt{futu}. And its definition is straightforward: \\begin{code} futu :: Functor f => CVCoalgebra f a -> a -> Term f futu f = In <<< fmap worker <<< f where worker (Automatic a) = futu f a -- continue through this level worker (Manual g) = In (fmap worker g) -- omit folding this level, -- delegating to the worker -- to perform any needed -- unfolds later on. \\end{code} When we encounter a plain \\texttt{Continue} value, we continue recursing into it, perpetuating the unfold operation. When we encounter a \\texttt{Stop} value, we run one more iteration on the top layer of the in-progress fold (transforming its children from \\texttt{Coattr\\ f\\ a} values into \\texttt{Term\\ f} values by recursively invoking \\texttt{worker}), then wrap the whole item up with an \\texttt{In} constructor and return a final value. The product of this nested invocation of \\texttt{worker} is then similarly passed to the \\texttt{In} constructor to wrap it up in a fixpoint, then returned as the final output value of \\texttt{futu}. What differentiates this from \\texttt{apo}---which, if you recall, used an \\texttt{Either} type to determine whether or not to continue the unfold---is that we can specify, \\emph{in each field of the functor f}, whether we want to continue the unfold or not. \\texttt{apo} gave us a binary switch---either stop the unfold with a \\texttt{Left} or keep going with a \\texttt{Right}. \\texttt{futu}, by contrast, lets us build out as many layers at a time as we desire, giving us the freedom to manually specify the shape of the structure or relegate its shape to future invocations of the unfold. This is an interesting way to encode unfolds! A CV-coalgebra that always returns a \\texttt{Continue} value will loop infinitely, such as the unfold that generates all natural numbers. This means that we can tell, visually, whether our unfold is infinite or terminating. But Patrick,'' you might say, this looks like a cellular automaton.'' And you would be right---CV-coalgebras describe tree automata. And in turn, coalgebras describe finite-state automata, and R-coalgebras describe stream automata. We'll use this fact to define an example CV-coalgebra, one that grows\\footnote{which brings an amusing literalism to the term seed value'} random plant life. \\subsubsection{Horticulture with Futumorphisms}\\label{horticulture-with-futumorphisms} Let's start by defining the various parts of a plant. \\begin{code} data Plant a = Root a -- every plant starts here | Stalk a -- and continues upwards | Fork a a a -- but can trifurcate at any moment | Bloom -- eventually terminating in a flower deriving (Show, Functor) \\end{code} Let's define a few rules for how a plant is generated. (These should, as I mentioned above, remind us of the rules for tree automata.) \\begin{verbatim} 1. Plants begin at the ground. 2. Every plant has a maximum height of 10. 3. Plants choose randomly whether to fork, grow, or bloom. 4. Every fork will contain one immediate bloom and two further stems. \\end{verbatim} Rather than using integers to decide what action to take, which can get obscure very quickly, let's define another sum type, one that determines the next step in the growth of the plant. \\begin{code} data Action = Flower -- stop growing now | Upwards -- grow up with a Stalk | Branch -- grow up with a Fork \\end{code} Because we need to keep track of the total height and a random number generator to provide randomness, we'll unfold using a data type containing an \\texttt{Int} to track the height and a \\texttt{StdGen} generator from \\texttt{System.Random}. \\begin{code} data Seed = Seed { height :: Int , rng :: Random.StdGen } \\end{code} We'll define a function \\texttt{grow} that takes a seed and returns both an randomly-chosen action and two new seeds. We'll generate an action by choosing a random number from 1 to 5: if it's 1 then we'll choose to \\texttt{Flower}, if it's 2 we'll choose to \\texttt{Branch}, and otherwise we'll choose to grow \\texttt{Upwards}. (Feel free to change these values around and see the difference in the generated plants.) The \\texttt{Int} determining the height of the plant is incremented every time \\texttt{grow} is called. \\begin{code} grow :: Seed -> (Action, Seed, Seed) grow seed@(Seed h rand) = (choose choice, left { height = h + 1}, right { height = h + 1}) where (choice, _) = Random.randomR (1 :: Int, 5) rand (leftR, rightR) = Random.split rand left = Seed h leftR right = Seed h rightR choose 1 = Flower choose 2 = Branch choose _ = Upwards \\end{code} And now we'll define a CV-coalgebra, one that takes a \\texttt{Seed} and returns a \\texttt{Plant} containing a \\texttt{CoAttr} value. \\begin{code} sow :: CVCoalgebra Plant Seed \\end{code} The definition falls out rather quickly. We'll start by growing a new seed, then examining the current height of the plant: And now we'll define a CV-coalgebra, one that takes a \\texttt{Seed} and returns a \\texttt{Plant} containing a \\texttt{CoAttr} value. \\begin{verbatim} sow :: CVCoalgebra Plant Seed \\end{verbatim} The definition falls out rather quickly. We'll start by growing a new seed, then examining the current height of the plant: \\begin{verbatim} sow seed = let (action, next) = grow seed in case (height seed) of \\end{verbatim} Since we'll start with a height value of 0, we'll begin by generating a root (rule 1). Because we want to immediately continue onwards with the unfold, we pass a \\texttt{Continue} into this \\texttt{Root}, giving it the subsequent seed (so that we get a new RNG value). \\begin{verbatim} 0 -> Root (Continue next) \\end{verbatim} Rule 2 means that we must cap the height of the plant at 10. So let's do that: \\begin{verbatim} 10 -> Bloom \\end{verbatim} Otherwise, the height is immaterial. We must consult the \\texttt{action} variable to know what to do next. \\begin{verbatim} _ -> case action of \\end{verbatim} If the action is to \\texttt{Flower}, then we again return a \\texttt{Bloom}. \\begin{verbatim} Flower -> Bloom \\end{verbatim} If it's to grow \\texttt{Upwards}, then we return a \\texttt{Stalk}, with a contained \\texttt{Continue} value to continue our fold at the top of that \\texttt{Stalk}: \\begin{verbatim} Upwards -> Stalk (Continue next) \\end{verbatim} And now we handle the \\texttt{Branch} case. Our rules dictate that one of the branches will stop immediately, and the other two will continue, after a given length of \\texttt{Stalk}. So we return a \\texttt{Fork} with one \\texttt{Stop} and two \\texttt{Continues}. \\begin{verbatim} Branch -> Fork -- grow a stalk then continue the fold (Stop (Stalk (Continue next))) -- halt immediately (Stop Bloom) -- again, grow a stalk and continue (Stop (Stalk (Continue next))) \\end{verbatim} Note how, even though we specify the construction of a \\texttt{Stalk} in the first and third slots, we allow the fold to \\texttt{Continue} afterwards. This is the power of the futumorphism: we can choose the future of our folds, layer by layer. This is not possible with an anamorphism or apomorphism. Here's our full \\texttt{sow} function, rewritten slightly to use one \\texttt{case} statement: \\begin{code} sow seed = let (action, left, right) = grow seed in case (action, height seed) of (_, 0) -> Root (Automatic left) (_, 10) -> Bloom (Flower, _) -> Bloom (Upwards, _) -> Stalk (Automatic right) (Branch, _) -> Fork (Manual (Stalk (Automatic left))) (Manual Bloom) (Manual (Stalk (Automatic right))) \\end{code} \\ignore{ \\begin{code} -- I can't find the original implementation I had of this function. I will -- do it more properly later. render :: Algebra Plant Box render Bloom = \"8\" render (Root a) = vcat center1 [\"X\", a] \\end{code} } This is pretty remarkable. We've encoded a complex set of rules, one that involves both nondeterminism and strict layout requirements, into one CV-coalgebra, and it took just eleven lines of code. No mutable state is involved, no manual accumulation is required---the entire representation of this automaton can be reduced to one pure function. Now, in our \\texttt{main} function, we can grab an RNG from the global state, and call \\texttt{futu} to generate a \\texttt{Term\\ Plant}. \\begin{verbatim} main :: IO () main = do rnd <- newStdGen let ourPlant :: Term Plant ourPlant = futu sow (Seed 0 rnd) \\end{verbatim} Using a rendering function (which I have omitted for brevity's sake, though you can be assured that it is implemented using \\texttt{cata} rather than explicit recursion), we can draw a picture of the plant we've just generated, with little flowers. \\begin{verbatim} \u2698 | \u2698 \u2698 \u2698 |\u2698| | | \u2514\u2500\u2518 | | | | | \u2698 | \u2698 | | | \u2514\u2500\u2500\u2500\u2500\u2500\u2518 | \u2698 | | \u2514\u2500\u2500\u2500\u2500\u2500\u2500\u2518 | \u2698 | \u2514\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2518 | _ \\end{verbatim} Admittedly, the vaguaries of \\href{https:\/\/en.wikipedia.org\/wiki\/Code_page_437}{code page 437} leave us with a somewhat unaesthetic result---but a nicer representation of \\texttt{Plant}, perhaps using \\href{https:\/\/hackage.haskell.org\/package\/gloss}{gloss} or \\href{https:\/\/hackage.haskell.org\/package\/Rasterific}{Rasterific}, is left as an exercise for the reader. One final detail: just as we can use an apomorphism to express an anamorphism, we can express anamorphisms and apomorphisms with futumorphisms: \\begin{code} ana :: (Functor f) => Coalgebra f a -> a -> Term f ana f = futu (fmap Automatic <<< f) apo :: Functor f => RCoalgebra f a -> a -> Term f apo f = futu (fmap (either termToCoattr Automatic) <<< f) where termToCoattr = Manual <<< fmap termToCoattr <<< out \\end{code} \\subsubsection{My God, It's Full of Comonads}\\label{my-god-its-full-of-comonads} Now we know what histomorphisms and futumorphisms are. Histomorphisms are folds that allow us to query any previous result we've computed, and futumorphisms are unfolds that allow us to determine the future course of the unfold, multiple levels at a time. But, as is so often the case with recursion schemes, these definitions touch on something deeper and more fundamental. Here's the kicker: our above \\texttt{CoAttr} definition is equivalent to the \\texttt{Free} monad, and \\texttt{Attr} (being dual to \\texttt{CoAttr}) is the \\texttt{Cofree} comonad. We usually represent \\texttt{Free}, aka \\texttt{CoAttr}, as two constructors, one for pure values and one for effectful, impure values: \\begin{verbatim} data Free f a = Pure a | Impure (f (Free f a)) \\end{verbatim} And we usually represent the cofree comonad with an infix constructor, since the cofree comonad is at its heart a glorified tuple: \\begin{verbatim} data Cofree f a = a :< (f (Cofree f a)) \\end{verbatim} The various packages in the Haskell ecosystem implement \\texttt{cata} and \\texttt{para} in much the same way, but the same is not true of \\texttt{histo} and \\texttt{futu}. Edward Kmett's \\href{https:\/\/hackage.haskell.org\/package\/recursion-schemes}{recursion-schemes} package uses these definitions of \\texttt{Free} and \\texttt{Cofree} (from the \\href{https:\/\/hackage.haskell.org\/package\/free}{free} package). \\href{https:\/\/hackage.haskell.org\/package\/fixplate}{\\texttt{fixplate}} uses a different definition of \\texttt{Attr}: rather than being a data type in and of itself, it is defined as a \\texttt{Term} over a more-general \\texttt{Ann} type. \\href{https:\/\/hackage.haskell.org\/package\/compdata}{\\texttt{compdata}}'s is slightly more complicated, as it leverages other typeclasses \\texttt{compdata} provides to define attributes on nodes, but is at its heart the same thing. Each is equivalent. The free monad, and its cofree comonad dual, lie at the heart of some of the most fascinating constructions in functional programming. I have neither the space nor the qualifications to provide a meaningful explanation of them, but I can enthusiastically recommend \\href{https:\/\/twitter.com\/GabrielG439}{Gabriel Gonzales}'s blog post on \\href{http:\/\/www.haskellforall.com\/2012\/06\/you-could-have-invented-free-monads.html}{free monads}, \\href{https:\/\/twitter.com\/sigfpe}{Dan Piponi}'s post on the \\href{http:\/\/blog.sigfpe.com\/2014\/05\/cofree-meets-free.html}{cofree comonad}, and (of course) Oleg Kiselyov's \\href{http:\/\/okmij.org\/ftp\/Computation\/free-monad.html}{groundbreaking work} on the free and freer monads. But I think the fact that, as we explore as fundamental a construct as recursion, we encounter another similarly fundamental concept of the free monad, provide an argument for the beauty and unity of the category-theoretical approach to functional programming that is far more compelling than any I could ever make myself. I'd like to thank Rob Rix, who was essential to this work's completion, and Colin Barrett, who has been an invaluable resource on the many occasions when I find myself stuck. I'd also like to thank Manuel Chakaravarty, who has done this entire series a great favor in checking it for accuracy, and Jeanine Adkisson, who found some outrageous bugs in the provided futumorphism. Greg Pfiel, Scott Vokes, and Josh Bohde also provided valuable feedback on drafts of this post. Mark Needham, Ian Griffiths, and Bryan Grounds found important bugs in the first published version of this post; I owe them a debt of gratitude. Next time, we'll explore one of the most compelling reasons to use recursion schemes---the laws that they follow---and after that, we'll discuss the constructs derived from combining unfolds with folds: the hylomorphism and the chronomorphism.","date":"2018-10-24 05:35:23","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\": 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\": 0.676184356212616, \"perplexity\": 4442.037522145412}, \"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-2018-43\/segments\/1539583519859.77\/warc\/CC-MAIN-20181024042701-20181024064201-00503.warc.gz\"}"}
| null | null |
layout: post
title: "EVENT: Fly Fishing Techniques for Steelhead"
modified:
categories: [fishing]
tags: []
published: True
comments: True
ads: False
excerpt: I will be presenting on fly fishing techniques, gear, and knots on Jan 15th 2019
image:
feature: north-shore-steelhead-8.jpg
teaser: AFF.jpg
thumb: AFF.jpg
date: 2019-01-07T05:38:35-05:00
---
I will be presenting on fly fishing techniques, gear, and knots used to catch Lake Superior Steelhead during the next <a href="http://arrowheadflyfishers.com/" target="_blank">Arrowhead Fly Fishers</a> meeting on Jan 15th, at 7 p.m. at Clyde Iron Works.
When I first started steelheading it was crazy to think that you could tie on a fly or colorful ball of yarn the size of your fingernail, toss it into a muddy fast moving river and somehow catch a steelhead. It took me an entire season struggling to get my first fish on the fly, and looking back at my flies and the technique I was using, it was no wonder I didn't catch any. Skip the learning curve and learn new techniques and rigging tips to help you catch steelhead on the fly!
<ul>
<li>When: Tuesday, January 15, 2019 at 7 p.m.</li>
<li>Where: <a href="https://goo.gl/maps/3qkYXKDwyUz" target="_blank" title="Get Directions">Clyde Iron Works</a></li>
<li>Cost: FREE (AFF membership appreciated)</li>
</ul>
<a class="btn btn-social facebook" href="https://www.facebook.com/events/331346617470683/" target="_blank">RSVP on Facebook</a>
<figure>
<a href="https://www.facebook.com/events/331346617470683/" target="_blank"><img title="RSVP on Facebook" src="/images/AFF.jpg"></a>
</figure>
**Techniques that will be covered include:**
- Nymphing
- Chuck-n-Duck
- Drop Shot
- Slinky Rig
- In-line
- Euro Nymphing
- Indicator Nymphing
- Horizontal
- Vertical
- Streamer Fishing
- Swinging Streamers
- Stripping Streamers
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La loi d'Ohm est une loi physique empirique qui lie l'intensité du courant électrique traversant un dipôle électrique à la tension à ses bornes. Cette loi permet de déterminer la valeur d'une résistance. La loi d'Ohm a été ainsi nommée en référence au physicien allemand Georg Simon Ohm qui la publie en 1827, dans son œuvre .
Énoncé de la loi d'Ohm
On note :
la tension aux bornes de la résistance ;
l'intensité du courant qui circule à travers la résistance ;
la valeur de la résistance.
La loi d'Ohm établit que (en convention récepteur) :
Un conducteur ohmique est un dipôle vérifiant la loi d'Ohm.
Unités
Dans la loi d'Ohm, la tension est exprimée en volts (), la résistance en ohms () et l'intensité en ampères ().
Interprétation de la loi d'Ohm
La loi d'Ohm indique que la tension aux bornes d'une résistance est proportionnelle à l'intensité du courant qui la traverse. Ce coefficient de proportionnalité est la valeur de la résistance.
La valeur de la résistance est une constante et ne varie donc pas lorsque l'on modifie la tension ou l'intensité.
Utilisation de la loi
Selon son expression et les grandeurs connues, la loi d'Ohm permet d'obtenir différentes grandeurs :
sous la forme U = R × I, elle permet de calculer la tension lorsque la résistance et l'intensité sont connues ;
sous la forme I = U / R, elle permet de calculer l'intensité lorsque la tension et la résistance sont connues ;
sous la forme R = U / I, elle permet de calculer la résistance lorsque la tension et l'intensité sont connues.
Caractéristique d'un conducteur ohmique
Lorsqu'on trace la caractéristique d'un conducteur ohmique (c'est-à-dire le graphique de la tension en fonction de l'intensité), on obtient une droite passant par l'origine. La pente de cette droite est la valeur de la résistance.
Point de vue macroscopique
En courant continu et en régime établi
La différence de potentiel ou tension (en volts) aux bornes d'un résistor de résistance (en ohms) est proportionnelle à l'intensité du courant électrique (en ampères) qui la traverse, ou la d'un dipôle est égale au quotient de sa par l' du courant :
avec et orientées en sens opposés (dipôle en convention récepteur).
N.B. : si et sont orientées dans le même sens (dipôle en convention générateur), la loi devient alors :
.
On peut en déduire :
ou .
Cette loi porte le nom de Georg Ohm qui a travaillé sur le comportement des conducteurs métalliques. Elle s'applique de manière satisfaisante aux conducteurs métalliques thermostatés. Lorsque la température varie, la valeur de la résistance varie également de manière plus ou moins simple, ce qui impose d'introduire des termes correctifs. Par convention, on conserve la loi et on introduit les termes correctifs dans la valeur de la résistance du conducteur.
En courant alternatif
La loi précédente se généralise au cas des courants sinusoïdaux en utilisant les notations complexes. On note et respectivement la tension et le courant complexes. La loi d'Ohm s'écrit alors :
où est l'impédance complexe du dipôle considéré, qui peut être constitué de dipôles linéaires (résistances, condensateurs et inductances).
Dans un circuit RLC série
Par application de la loi des mailles,
Avec :
R la résistance du circuit, en ohms (Ω) ;
L l'inductance de la bobine, en henrys (H) ;
C la capacité électrique du condensateur, en farads (F).
Point de vue local (mésoscopique)
Énoncé de la loi d'Ohm locale
D'un point de vue local, c'est-à-dire mésoscopique, la loi (locale) d'Ohm s'énonce en disant que la mobilité des porteurs de charge est indépendante de .
À noter que la loi d'Ohm doit respecter certaines conditions :
l'homogénéité et l'isotropie du milieu ;
la grandeur considérée ne doit pas varier trop rapidement dans le temps.
Si on note la mobilité des porteurs de charge, leur vitesse s'écrit alors (la direction du mouvement dépend du signe des porteurs) ; la densité de courant associée à une densité de porteurs vaut quant à elle :
,
où est la charge électrique du porteur (en valeur absolue).
On note la conductivité électrique du matériau (pour un seul type de porteur).
On a alors la loi locale d'Ohm pour un seul type de porteur :
.
Si on a plusieurs types de porteurs, comme les électrons et les trous dans un semi-conducteur ou des ions différents dans un électrolyte, la densité de courant devient :
,
avec ,
donc .
On a alors la conductivité totale :
.
Voir aussi Loi de Nernst-Einstein.
Rapport avec la loi d'Ohm macroscopique : définition de la résistance
Considérons une portion de conducteur d'un point A à un point B et de section droite S, on a alors la différence de potentiel qui vaut :
et l'intensité :
.
Multiplions par une constante la différence de potentiel , alors les conditions aux limites sont inchangées ainsi que les lignes de champ de , et l'expression est multipliée par la même constante. Par conséquent le rapport est indépendant de cette constante, c'est une « constante » (il dépend quand même de divers paramètres tels la température) appelée résistance électrique et notée . Elle se calcule comme suit :
Cette formule permet de calculer la résistance de diverses géométries de matériaux (filiforme, cylindrique, sphérique).
Notes et références
Notes
Références
Voir aussi
Articles connexes
Liens externes
Théorie électrique
Électrodynamique
Ohm
|
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| 2,771
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Copyright © 2019 by Laura Anne Lapp
All rights reserved. No part of this book may be reproduced in any manner without the express written consent of the publisher, except in the case of brief excerpts in critical reviews or articles. All inquiries should be addressed to Good Books, 307 West 36th Street, 11th Floor, New York, NY 10018.
Good Books books may be purchased in bulk at special discounts for sales promotion, corporate gifts, fund-raising, or educational purposes. Special editions can also be created to specifications. For details, contact the Special Sales Department, Good Books, 307 West 36th Street, 11th Floor, New York, NY 10018 or info@skyhorsepublishing.com.
Good Books is an imprint of Skyhorse Publishing, Inc.®, a Delaware corporation.
Visit our website at www.goodbooks.com.
10 9 8 7 6 5 4 3 2 1
Library of Congress Cataloging-in-Publication Data is available on file.
Cover design by Peter Donahue
Cover photo by iStockphoto
Print ISBN: 978-1-68099-456-8
Ebook ISBN: 978-1-68099-459-9
Printed in China
TABLE OF CONTENTS
Getting Started
Fruits
Applesauce
Blueberry Pie Filling
Cherry Pie Filling
Apple Pie Filling
Fruit Pudding (or Mush)
Tomatoes
Stewed Tomatoes
Hot Sauce
Tomato Soup
Chili Base
(Basic) Tomato Juice
Seasoned Tomato Juice
Homemade V-8 Juice
Pizza Sauce (Variation 1)
Pizza Sauce (Variation 2)
Pizza Sauce (Variation 3)
Ketchup (Variation 1)
Ketchup (Variation 2)
Salsa (Variation 1)
Salsa (Variation 2)
Pickles
Sweet Kosher Dill Pickles
Bread-and-Butter Pickles
Mustard Pickles
Sweet Dill Pickles
Banana Pickles
Refrigerator Pickles
Pickled Vegetables
Crisp Dilly Green Beans
Pickled Beets
Harvard Beets
Relishes
Green Tomato Relish
Onion Relish
Pepper Relish
Pickle Relish
Zucchini Relish
Jams and Jellies
Strawberry Jam
Grape Jelly
Raspberry Jelly
Apple Jelly
Peach Jelly
Elderberry Jelly
Mint Jelly
Hot Pepper Jam
Old-Fashioned Jams and Jellies
Hot Pepper Jam (without Pectin)
Blackberry Jelly
Rhubarb Jam
Sweet Spreads
Pear Butter
Pear Honey
Easy Apple Butter
Soups
Cream of Celery Soup
Ham and Bean Soup
Chunky Beef Soup
Chicken Noodle Soup (to Can)
Vegetable Soup
Chili Soup
Conversion Charts
Index
GETTING STARTED
Growing up Amish, the eldest daughter in a family of seven, canning and preserving were always a part of family life, a part of summertime that went hand in hand with gardening. Now, as a mother of four growing boys with a large garden, I still find canning and preserving a part of my summer tradition. I love gardening, and preserving the beauty of my garden is just another part of that tradition.
Canning and preserving can be hard work, but with the right equipment and some time spent in the kitchen, you will reap the benefits of your labor all winter long. There's such a wonderful feeling of satisfaction placing your gleaming jars on shelves or cupboards and knowing that it was your efforts that put them there, and for me it's knowing exactly what's in those jars. No artificial ingredients at all, just fresh food from my garden. I still buy fruit from local orchards (as we don't have fruit trees), but that's okay, as I still know where the food's coming from.
There are some items you will need before you begin, but most are fairly inexpensive, and you can reuse them year after year. I recommend buying new jars for your first season of canning. That way, you won't have to worry about chips around the outer edges or cracks anywhere in the jars. If the outer edge of a jar is nicked or chipped, the jar won't obtain a tight seal.
Jars are available in quart, pint, or half-pint sizes. You can also buy even smaller jars, what I call jelly jars, which hold approximately ½ cup. I can all my pickles, applesauce, peaches, pears, beets, and tomato juice in quart-sized jars. My jellies, sauces, salsas, and relishes go in pint-sized jars, but we are a family of six and eat lots at every meal. You can use any size jar that works well for your family.
New canning jars (or mason jars, as they are usually called) come complete with lids and rings, also called screw bands. Lids are flat and have a rubber seal on the inside rim that conforms to the top of the glass jar and seals tightly in the canning process, preserving your food. Lids can be bought separately and are typically used only once. Screw bands do not have an infinite lifespan like jars; they get bent and sometimes rust—even now—and they need to be replaced as they wear out.
The recipes in this book are all processed using a boiling-water-bath method. You can also use a pressure canner, but I don't have one myself and have never used one, so I didn't feel comfortable using that method in my book, and water-bath canning has always worked well for me. You will also need a canner or hot-water-bath canner. A canner is basically a large stockpot, so if you have a large pot, you don't necessarily have to buy a canner. I often use my stainless-steel twelve-quart kettle as a canner.
A jar lifter is a handy tool that you will use often while canning to take your jars out of boiling water, so it is a necessity.
A canning funnel is not necessary but recommended, especially when canning sauces or fruits. I like stainless-steel funnels as they don't stain or melt.
A canning rack is also recommended to keep your jars off the bottom of the pot away from direct heat. Most canners come complete with the rack, but if you're using a pot, you can use a cake cooling rack or can tie screw bands together to make your own rack. In a pinch, I sometimes use a clean dishcloth on the bottom of the pot for my canning rack. It keeps the jars from clattering as they boil and also keeps them from breaking.
It's important to heat your jars before you fill them, not to sterilize them (as they will sterilize during the canning process) but to reduce the stress of temperature change. The best way to heat them is in hot water. Some people recommend heating them in your canner, by placing them in water until they reach a simmer (180°F, almost boiling). You can also use your dishwasher, and I've found that washing my jars in hot soapy water and leaving them on the counter to dry is enough to keep them hot, but my kitchen is always extremely warm in the summertime because I have no AC.
If you're nervous about your jars not sealing while in the canner, you can also heat the lids the same way you heat the jars, but placing the lids on warm jars and then putting them in a canner of boiling water has always worked for me. The screw bands never need to be heated, as that's the part that you fasten securely before putting the jars into the canner.
When filling the jars, be sure to follow each recipe's instructions about headspace, or the space between the top of the jar and the top of the food. Sticky foods like canned fruit or applesauce can cause quite a mess if the jars are too full. Most recipes require ½ inch of headspace, although soft jams or fruit juices require a little less.
After the jars are filled with food, be sure to wipe the surface, especially the outer rim, clean of all food residue or particles. Not doing this step correctly can prevent a tight seal from forming during the canning process.
When all the jars are filled and the screw bands are fastened just fingertip tight (do not use a tool or device to tighten them), place your jars in the canner and adjust the water level so that all the jars are covered, and process at a full rolling boil according to the recipe's instructions. When the jars are done, remove them carefully using the jar lifter. Place them on your kitchen counter or sturdy surface, and don't move them for 24 hours. At this time, it's not necessary to dry the jars or remove the screw bands. After 24 hours, you can remove the bands, check for a tight seal, and wipe down the jars.
Any jars that haven't sealed can be kept in the refrigerator and enjoyed within a few days. Food safety concerns related to this would seem to be dependent on what was being canned; one wouldn't want to eat, for instance, chicken soup if it had been sitting at room temperature for at least twenty-four hours.
Store your jars in a cool, dark space such as a pantry, cupboard, or basement. Basement storage is ideal for canned foods as the temperature is often cooler and more consistent.
Home-canned food is generally considered safe to eat for one year. I have kept food for up to two years and it's been fine, although the taste is not quite as good. It is important to label all canned food with dates to facilitate using them in a timely fashion. If you notice any discoloration or mold growing in your jar, obviously that food is not safe to eat anymore. Any food that is opened and not consumed right away must be refrigerated. After the vacuum seal is released, the food can spoil quickly just like fresh food.
I hope you enjoy this book and all the fruits of your labor.
A note on yields:
I've found that lots of canning recipes yields are only approximate. It's hard to say exactly how many quarts, pints, etc. you'll get from one recipe, so as you're canning please remember that the yield won't always be just what the recipe says.
A note on the recipes:
The recipes in this book are all recipes that I or members of my extended family have used or still use every summer when preserving the bounty of our gardens. I hope you enjoy them and return to them year after year, just as I do.
Fruits
Preserving Fruit
One of the easiest ways to preserve fresh fruit is using a syrup. The fruit is peeled (usually), then sliced or halved and put into jars. The type of syrup you make depends on your tastes. Sugar helps fruit keep its flavor and texture, so traditionally heavy syrups with lots of sugar are used, but it's really not necessary to use all the sugar. You can even use unsweetened fruit juice or water, but please note that if you do use water to can fruit, the fruit will not keep its shape and color or flavor and may look a little discolored and have a bland taste.
A heavy syrup is made up of equal parts water and sugar, so it's easy to adjust to taste. To make a syrup for fruit, measure water and sugar to your taste into a large pot. Bring to a boil, stirring until sugar is dissolved. Turn heat to low and keep syrup warm until ready to use. The yield will be approximately 1 to 2 cups of syrup per quart of fruit.
Canning Fruit
To can fruit, start with ripe fruit that has no spots or blemishes. Any fresh fruit that's ready to eat raw is ready to can. The procedure for most fruit is the same. Wash fruit well; peel if using peaches or pears, and remove stems if using cherries or berries.
Pack the fruit into clean jars allowing plenty of headspace. My own rule of thumb is never packing past the first "ring" from the top of the jar (that's what I tell my boys when they help me). Pour sugar syrup over fruit, again leaving headspace of at least ½ inch. Canned fruit can boil over, and that makes quite a sticky mess, so follow a basic hot-water-bath canning procedure to process. Most fruit should be canned for 20 minutes.
APPLESAUCE
Applesauce is the main fruit I can, usually three bushels every year. Amish people traditionally serve applesauce at all meals except breakfast. It is served as a side to almost everything, even soup. My own family eats applesauce with their pizza. My husband grew up eating it that way, and now our boys prefer that, too. I just smile and watch them dip their pizza in their applesauce. Ginger Gold or similar "sweet" apples need less sugar than more tart ones such as the Smokehouse variety, although in recent years I've done Smokehouse apples and they make delicious applesauce! You can also use a combination of apples to suit your tastes.
Yield: 1 bushel of apples for approx. 25 quarts
1. Wash apples well and remove stems. If using a Victorio strainer, there's no need to core or peel, but if you want completely "clear" applesauce with no specks, it's better to core and peel the apples.
2. Cover the bottom of a large pot with water about 2 inches deep.
3. Pour prepared apples (cut into fourths) into pot and bring to a boil.
4. Boil until soft and fluffy.
5. Very carefully (apples will be hot) press apples through Victorio strainer/food mill. The resulting puree should have a nice smooth consistency.
6. After apples are pureed, add sugar to taste and stir well.
7. Ladle applesauce into jars, leaving plenty of headspace (at least ½ inch or more).
8. Process in hot-water bath for 20 minutes.
BLUEBERRY PIE FILLING
Canned pie filling is a great product to have on hand in the winter. It's so easy to open a jar for desserts or pies.
Yield: approx. 9–10 pints
3 quarts fresh blueberries or other berry of choice (washed and picked over)
1 quart + 1 cup cold water, divided
3 cups sugar
1½ cups clear jel (or therm-flo)
Combine berries, 1 quart water, and sugar.
1. Bring blueberries, water, and sugar to a boil over medium heat.
2. Mix clear jel (or therm-flo) with 1 cup cold water.
3. Stirring constantly, add to berry mixture.
4. Boil until thickened.
5. Ladle into clean jars.
6. Process in hot-water bath for 20 minutes.
CHERRY PIE FILLING
Yield: approx. 9–10 pints
7 cups + 1½ cups cold water, divided
5 cups sugar
3 quarts pitted sour cherries
1¾ cups clear jel (or therm-flo)
1. Bring 7 cups water, sugar, and cherries to a boil over medium heat.
2. In a separate bowl combine 1½ cups cold water and therm-flo.
3. Add to cherry mixture, stirring well, until cherries are thickened.
4. Ladle into jars.
5. Process in hot-water bath for 20 minutes.
APPLE PIE FILLING
Yield: approx. 8–9 pints
6 cups + 1 cup cold water, divided
4 cups sugar
1 tablespoon cinnamon
1¾ cups clear jel (or therm-flo)
6 quarts fresh apples, peeled and sliced
1. Bring 6 cups water, sugar, and cinnamon to boiling.
2. In a separate bowl combine 1 cup cold water and therm-flo.
3. Slowly add clear jel (therm-flo) mixture to boiling sugar mixture, stirring constantly.
4. Pour mixture over apples, and stir to combine.
5. Ladle into clean jars.
6. Process in hot-water bath for 20 minutes.
FRUIT PUDDING (OR MUSH)
Fruit pudding is a very simple Amish dessert that is great on its own or poured over ice cream. It's simply fruit juice of your choice, thickened with therm-flo. You can also add fresh fruit before serving.
Yield: approx. 9–10 pints
3 quarts fruit juice of your choice (grape or raspberry are especially delicious)
3 quarts water + extra for therm-flo
4 cups sugar
2½ cups therm-flo
1. Bring fruit juice, 3 quarts water, and sugar to a boil.
2. In a separate bowl mix therm-flo with enough water to make a smooth paste.
3. Add therm-flo to juice mixture, stirring constantly.
4. Boil until thickened.
5. Ladle into clean jars.
6. Process in hot-water bath for 15 minutes.
Tomatoes
A note on lemon juice in tomato products:
Food safety organizations recommend adding 1 tablespoon bottled lemon juice per pint of tomato product and 2 tablespoons per quart. I never use lemon juice in my own tomato products and have never had a problem, but if you prefer to do so you can always add it in as an added precaution.
A note on therm-flo:
Therm-flo is a modified food starch that works well in canning recipes as a thickener. It is usually mixed with water to make a paste, then added to tomato products to thicken them. If you don't like the idea of adding starch to your product, tomato paste is another option to thicken home-canned tomato products. Therm-flo is always available at Amish or Mennonite bulk food/grocery stores, and I'm sure it's available online, too.
STEWED TOMATOES
Yield: This recipe can be adapted to any size jar.
tomatoes, enough to fill your jar of choice
1 heaping tablespoon chopped onion
1 heaping tablespoon chopped green peppers
1 heaping tablespoon chopped celery
1 tablespoon sugar
1 teaspoon salt
1.Peel and quarter tomatoes.
2.Fill jars with tomatoes.
3. To each quart add all remaining ingredients.
4. Process in hot-water bath for 1 hour.
HOT SAUCE
Yield: approx. 2–4 pints
1 pound tomatoes
1 pound hot peppers of your choice
1 pound onions
1 teaspoon salt
½ cup white vinegar
1 teaspoon garlic salt
1. Wash and cut up tomatoes, hot peppers, and onions.
2. Mix in all remaining ingredients and bring to a boil over medium heat.
3. Cook until soft.
4. Press through Victorio strainer or tomato press.
5. Ladle into jars.
6. Process in hot-water bath for 30 minutes.
TOMATO SOUP
This recipe makes a condensed soup. To serve, add an equal amount of milk (or to taste). This is an old family recipe that my mother always made. It's mildly flavored and delicious paired with grilled cheese.
Yield: approx. 8–10 quarts
14 quarts tomatoes, sliced (not peeled)
3 quarts + 2 quarts water, divided
7 medium onions
14 stalks celery
14 sprigs parsley
2 cups flour
3 cups brown sugar
½ cup salt
1 cup butter, melted
1. Cook tomatoes in 3 quarts water until soft.
2. Cook onions, celery, and parsley with 2 quarts water until soft. Vegetables need to be soft enough to pass through a Victorio strainer or food mill.
3. Strain cooked vegetables and pour juice into large stockpot.
4. Whisk flour, brown sugar, and salt together.
5. Add melted butter and enough cold water to make a nice thick slurry about the consistency of cake batter.
6. Stirring constantly, whisk flour mixture into tomato juice mixture.
7. Bring to a boil and boil for 1 minute.
8. Ladle into jars and process for 1 hour in hot-water bath.
CHILI BASE
Yield: approx. 20 pints
12 quarts tomatoes, peeled
12 onions, chopped
2 bunches celery, chopped
1 cup salt
3 cups vinegar
7 cups sugar
3 peppers or more to taste, chopped
1 tablespoon black pepper or more to taste
1 tablespoon cinnamon or more to taste
1 tablespoon allspice or more to taste
1 tablespoon ginger or more to taste
1. Mix tomatoes, onions, celery, and salt.
2. Let set at room temperature for 8 hours or overnight.
3. Add all remaining ingredients.
4. Bring to a boil.
5. Ladle into jars.
6. Process in hot-water bath for 30 minutes.
Note: This is a sweeter chili base, so feel free to adjust seasonings to your own tastes.
(BASIC) TOMATO JUICE
This recipe can easily be adapted to suit your taste or needs. See page 26 for a more seasoned variation of this basic recipe.
Yield: approx. 3 pounds tomatoes for 1 quart juice
1. Heat tomatoes until soft enough to pass through a food mill or Victorio strainer.
2. Bring tomato juice to a boil and add seasonings of your choice.
3. Ladle into jars.
4. Process in boiling-water bath for 15 minutes.
SEASONED TOMATO JUICE
Yield: approx. 4–6 quarts
½ bushel tomatoes
5–6 bell peppers
3 large onions
7 stalks celery (leaves and all)
¼ cup salt
1. Cut up vegetables.
2. Bring to a boil.
3. Cook until soft enough to pass through Victorio strainer or food mill.
4. Bring juice to boiling and add salt.
5. Ladle into jars.
6. Process in hot-water bath for 15 minutes.
HOMEMADE V-8 JUICE
Yield: 6 quarts
6 quarts tomato juice
¾ cup sugar
2 teaspoons onion salt or more to taste
2 teaspoons garlic salt or more to taste
2 teaspoons celery salt or more to taste
1 teaspoon salt or more to taste
½ teaspoon pepper or more to taste
1. Bring all ingredients to a boil.
2. Ladle into jars.
3. Process in hot-water bath for 30 minutes.
PIZZA SAUCE (VARIATION 1)
I make this recipe every summer. I use it for other things as well as pizza, such as spaghetti or lasagna. For more pizza sauce recipes see pages 30 and .
Yield: approx. 6–7 pints
9 pounds fresh tomatoes (or enough to make 13 cups tomato puree)
½ cup lemon juice, fresh or bottled, optional
2 teaspoons dried oregano or more to taste
1 teaspoon ground black pepper or more to taste
1 teaspoon salt or more to taste
1 teaspoon garlic powder (or minced garlic to taste)
1. Wash tomatoes well and remove stems.
2. Add approximately 1 inch of water to a large stockpot.
3. Quarter tomatoes and bring to a boil.
4. Boil for 3 to 5 minutes until tomatoes are slightly soft.
5. Pass softened tomatoes through a Victorio strainer, food mill, or tomato press.
6. Pour tomato sauce into a large stockpot and add seasonings.
7. Boil, stirring occasionally until the sauce is thin
8. Ladle into jars and process in hot-water bath for 30 minutes.
PIZZA SAUCE (VARIATION 2)
Yield: approx. 10 pints
2½ gallons diced tomatoes
8–10 medium onions
4 green peppers
2 cups vegetable oil
½ gallon tomato paste + more to taste
1 cup sugar
2 tablespoons oregano
2 teaspoons pepper
3 tablespoons pizza seasoning
1 tablespoon Italian seasoning
2 tablespoons dried basil
1 teaspoon garlic powder
½ cup salt (or salt to taste)
Therm-flo, optional
1. In a large stockpot, bring tomatoes, onions, and peppers to a boil.
2. Boil for 20 minutes.
3. Pass through tomato press or Victorio strainer.
4. Pour puree into large stockpot and add remaining ingredients.
5. Boil for 1 hour.
6. If sauce needs to be thickened, add therm-flo mixed with water or another ½ gallon of tomato paste.
7. Ladle into jars and process in hot-water bath for 30 minutes.
PIZZA SAUCE (VARIATION 3)
Yield: approx. 6–7 pints
6 quarts tomato juice or puree from fresh tomatoes
1 tablespoon garlic
1 tablespoon oregano
1½ teaspoons black pepper
2 tablespoon salt
2 cups sugar
1 package Mrs. Wages Pizza Sauce Tomato Mix
1 cup therm-flo
1. Mix all ingredients, except therm-flo, in a large stockpot.
2. Boil for 30 minutes.
3. Mix therm-flo with enough water to make a paste.
4. Stirring constantly, add therm-flo mixture to tomato mixture.
5. Boil until thickened.
6. Ladle into jars and process for 30 minutes in hot-water bath.
KETCHUP (VARIATION 1)
Homemade ketchup has a completely different flavor than store bought. The consistency is usually thinner and it tastes much sweeter. See page 34 for a different take on this ketchup recipe.
Yield: approx. 8–10 pints
4 quarts tomato juice
2¼ cups sugar
1 teaspoon pepper or more to taste
¾ teaspoon allspice or more to taste
1½ teaspoons onion powder or more to taste
¾ teaspoon cinnamon or more to taste
¾ teaspoon cloves or more to taste
1 tablespoon salt
2 cups vinegar
1½ cups therm-flo
1. Combine all ingredients, except therm-flo, in a large stockpot.
2. Bring to a boil and boil for 15 minutes.
3. Add enough water to therm-flo to make a paste.
4. Stirring constantly, add therm-flo mixture to tomato mixture.
5. Boil until thickened.
6. Ladle into jars and process 30 minutes in hot-water bath.
KETCHUP (VARIATION 2)
Yield: approx. 6–7 pints
4 quarts tomato juice
¾ cup brown sugar
¾ cup white sugar
2 teaspoons dry mustard or more to taste
1 teaspoon ginger or more to taste
1 teaspoon celery seed or more to taste
1 teaspoon pepper or more to taste
1½ teaspoons salt or more to taste
½ teaspoon cayenne pepper or more to taste
¾ cup vinegar
¾ cup therm-flo
1. Mix all ingredients, except therm-flo, in a large stockpot.
2. Bring to a boil and boil for 1 hour.
3. Mix therm-flo with enough water to make a paste.
4. Stirring constantly, add to tomato mixture.
5. Boil until thickened.
6. Ladle into jars and process for 30 minutes in hot-water bath.
SALSA (VARIATION 1)
Yield: approx. 12–15 pints
4–5 quarts tomatoes, peeled, chopped
4 large onions, chopped
1½ cup jalapeños (approx. 6 peppers), chopped
6 bell peppers, chopped
2 cups white vinegar
1½ tablespoons chili powder
½ teaspoon alum
½ cup pickling salt
1 tablespoon garlic powder
1 teaspoon onion powder
4½ teaspoons pepper
⅓ cup sugar
1 (12-oz.) can tomato paste or ½–1 cup therm-flo, optional
1. Combine all ingredients, except tomato paste or therm-flo.
2. Bring to a boil in a large stockpot.
3. Simmer for 30 minutes.
4. For a thicker salsa, add tomato paste or therm-flo.
5. Ladle into jars and process for 20 minutes in hot-water bath.
SALSA (VARIATION 2)
Yield: approx. 7–8 pints
14 cups peeled and chopped tomatoes
3 cups chopped onions
½ cup chopped jalapeño peppers or more to taste
2½ cups chopped bell peppers
¼ cup salt
4 teaspoons chili powder
4 teaspoons cumin
1 cup tomato sauce
3 tablespoons brown sugar
6 tablespoons therm-flo
⅔ cup vinegar
1. Combine tomatoes, onions, and peppers.
2. Add seasonings and mix well.
3. Combine tomato sauce, brown sugar, and therm-flo.
4. Stir well and pour over vegetables.
5. Add vinegar and mix thoroughly.
6. Boil for 20 minutes or until thickened.
7. Ladle into jars and process for 20 minutes in hot-water bath.
Pickles
Pickles are one of the easiest and most forgiving things to can. Pickles start with fresh cucumbers that are sliced any way you prefer them: thin, thick, ridged, or straight. If you grow your own, you can pick them when they're tiny and can little gherkin-sized pickles. Most pickle recipes ask for a hot brine to be poured over sliced cucumbers and then processed immediately, so I usually mix the brine and start heating that while I wash and slice the cucumbers. I try to have the brine as hot as possible, not quite boiling, before I pour it into the jars, and also try to have the water in my canner at a full rolling boil before I put the jars into it. Use a jar lifter and be very cautious—I have burned my forearm often!
Many pickle recipes list Mrs. Wages Kosher Dill Pickles mix as an ingredient. This seasoning mix is easily found at many stores that sell canning supplies. I have even found it at Walmart. If you have an Amish or Mennonite bulk-food-style grocery in your area, I'm sure they would have it, too.
Pickles taste best if you wait 2–4 weeks before opening them. I like to refrigerate mine prior to serving, although you don't have to.
Most of the recipes in this section describe different brines, using the same basic instructions for each type of pickle. For most pickle recipes you will need approximately 2 gallons of sliced cucumbers. If you have leftover brine, that's okay. You can always refrigerate it for the next time.
SWEET KOSHER DILL PICKLES
These are my family's personal favorite. I make jar after jar of them as long as my cucumbers keep producing in the garden. We eat them with everything, but especially sandwiches. Grilled ham and cheese on homemade bread with pickles tucked inside tastes amazing!
Yield: approx. 9 quarts
16 cups water
4½ cups sugar
4 cups white vinegar
2 teaspoons salt
1½ packages Mrs. Wages Kosher Dill Pickles mix
2 gallons cucumbers, sliced
1. Mix water, sugar, white vinegar, salt, and Mrs. Wages mix in a large stockpot.
2. Heat on medium until sugar and seasoning mix are dissolved and mixture is just beginning to boil.
3. As the brine is heating, wash and slice the cucumbers.
4. Fill clean, prepared jars with cucumbers, making sure they are full, but not packed in too tightly. (I fill them and tap them on the countertop firmly a few times to settle the cucumbers.) Allow 1 inch of headspace, leaving room for brine.
5. Before pouring the brine over the cucumbers, make sure your canner is full of boiling water, ready to process the pickles.
6. Using a funnel, carefully pour the hot brine into the filled jars, leaving ½ inch of headspace.
7. Wipe the jars and rims, making sure to remove any sticky remnants of brine.
8. Place the lids on clean jars and close them with the jars' accompanying screw bands.
9. Using a jar lifter, place the jars into the canner, making sure they are covered with water.
10. Bring water to a rolling boil and process pickles for 5 minutes.
11. Remove from canner and let set on a sturdy surface for 24 hours. After 24 hours, your jars are ready to be stored. (I remove the screw bands before storing, as I always need them for my next canning project.)
BREAD-AND-BUTTER PICKLES
These pickles are sweeter than the kosher dill type, but the added onions go well with sandwiches.
Yield: approx. 5 quarts
1 gallon thinly sliced cucumbers
2 cups thinly sliced onions
¼ cup salt
4 cups sugar
2 cups white vinegar
1½ cups water
1 teaspoon turmeric
1 teaspoon celery seed
1. Combine cucumbers, onions, and salt.
2. Soak for 3 hours, then drain well.
3. Heat sugar, vinegar, water, turmeric, and celery seed over medium heat until just boiling.
4. Fill clean, prepared jars with cucumber-onion mixture, making sure they are full, but not packed in too tightly. (I fill them and tap them on the countertop firmly a few times to settle the cucumbers.) Allow 1 inch of headspace, leaving room for brine.
5. Before pouring the brine over the cucumbers, make sure your canner is full of boiling water, ready to process the pickles.
6. Using a funnel, carefully pour the hot brine into the filled jars, leaving ½ inch of headspace.
7. Wipe the jars and rims, making sure to remove any sticky remnants of brine.
8. Place the lids on clean jars and close them with the jars' accompanying screw bands.
9. Using a jar lifter, place the jars into the canner, making sure they are covered with water.
10. Bring water to a rolling boil and process pickles for 5 minutes.
11. Remove from canner and let set on a sturdy surface for 24 hours. After 24 hours, your jars are ready to be stored. (I remove the screw bands before storing, as I always need them for my next canning project.)
MUSTARD PICKLES
These pickles are a nice addition to a vegetable or cheese tray.
Yield: approx. 4 quarts
1½ cups white vinegar
3 cups water
1 tablespoon dry mustard
2 cups sugar
1 teaspoon pickling spice
1 teaspoon salt
1 gallon small, whole cucumbers
1. Mix white vinegar, water, dry mustard, sugar, pickling spice, and salt in a large stockpot.
2. Heat on medium until sugar and seasonings are dissolved and mixture is just beginning to boil.
3. As the brine is heating, wash and slice the cucumbers.
4. Fill clean, prepared jars with cucumbers, making sure they are full, but not packed in too tightly. (I fill them and tap them on the countertop firmly a few times to settle the cucumbers.) Allow 1 inch of headspace, leaving room for brine.
5. Before pouring the brine over the cucumbers, make sure your canner is full of boiling water, ready to process the pickles.
6. Using a funnel, carefully pour the hot brine into the filled jars, leaving ½ inch of headspace.
7. Wipe the jars and rims, making sure to remove any sticky remnants of brine.
8. Place the lids on clean jars and close them with the jars' accompanying screw bands.
9. Using a jar lifter, place the jars into the canner, making sure they are covered with water.
10. Bring water to a rolling boil and process pickles for 5 minutes.
11. Remove from canner and let set on a sturdy surface for 24 hours. After 24 hours, your jars are ready to be stored. (I remove the screw bands before storing, as I always need them for my next canning project.)
SWEET DILL PICKLES
Sweet dill pickles, without the kosher dill mix, taste more like a sweet pickle.
Yield: approx. 4 quarts
1 teaspoon dill seed or 1 head fresh dill
1 onion, sliced
1 clove garlic or ½ teaspoon garlic powder
2 cups water
3 cups sugar
2 cups vinegar
2 tablespoons salt
1 gallon cucumbers
1. Heat water, sugar, vinegar, and salt in a large stockpot until sugar is dissolved and mixture is just beginning to boil.
2. As the brine is heating, wash and slice the cucumbers.
3. Fill clean, prepared jars with cucumbers, making sure they are full, but not packed in too tightly. (I fill them and tap them on the countertop firmly a few times to settle the cucumbers.) To each jar allow 1½ inches of headspace for 1 teaspoon or 1 head fresh dill, 1 onion, and 1 garlic clove or ½ teaspoon garlic powder.
4. Before pouring the brine over the cucumbers, make sure your canner is full of boiling water, ready to process the pickles.
5. Using a funnel, carefully pour the hot brine into the filled jars, leaving ½ inch of headspace.
6. Wipe the jars and rims, making sure to remove any sticky remnants of brine.
7. Place the lids on clean jars and close them with the jars' accompanying screw bands.
8. Using a jar lifter, place the jars into the canner, making sure they are covered with water.
9. Bring water to a rolling boil and process pickles for 5 minutes.
10. Remove from canner and let set on a sturdy surface for 24 hours. After 24 hours, your jars are ready to be stored. (I remove the screw bands before storing, as I always need them for my next canning project.)
BANANA PICKLES
Banana pickles are traditional Amish pickles that are often served for lunch following church services. They are cut from large cucumbers and sliced into spears.
Yield: approx. 6 quarts
2 cups vinegar
2 cups water
6 cups sugar
2 teaspoons salt
2 teaspoons celery seed
2 teaspoons turmeric
2 teaspoons mustard seed
1 gallon large pared, deseeded cucumber spears
1. Mix vinegar, water, sugar, salt, celery seed, turmeric, and mustard seed in a large stockpot.
2. Heat on medium until sugar and seasonings are dissolved and mixture is just beginning to boil.
3. As the brine is heating, prepare the cucumbers.
4. Fill clean, prepared jars with cucumbers, making sure they are full, but not packed in too tightly. (I fill them and tap them on the countertop firmly a few times to settle the cucumbers.) Allow 1 inch of headspace, leaving room for brine.
5. Before pouring the brine over the cucumbers, make sure your canner is full of boiling water, ready to process the pickles.
6. Using a funnel, carefully pour the hot brine into the filled jars, leaving ½ inch of headspace.
7. Wipe the jars and rims, making sure to remove any sticky remnants of brine.
8. Place the lids on clean jars and close them with the jars' accompanying screw bands.
9. Using a jar lifter, place the jars into the canner, making sure they are covered with water.
10. Bring water to a rolling boil and process pickles for 5 minutes.
11. Remove from canner and let set on a sturdy surface for 24 hours. After 24 hours, your jars are ready to be stored. (I remove the screw bands before storing, as I always need them for my next canning project.)
REFRIGERATOR PICKLES
These pickles are not actually canned. They are sliced, have brine poured over them, and are refrigerated. They can be stored in the fridge for up to 30 days and are a great way to use a few leftover cucumbers.
Yield: approx. 2 quarts
7–9 cups sliced cucumbers
1 cup sliced green peppers
1 cup thinly sliced onion
2 tablespoons salt
1 tablespoon celery seed
1 cup white vinegar
2 cups sugar
1. Slice cucumbers into a large bowl.
2. Add the peppers and onions.
3. Sprinkle salt and celery seed over vegetables.
4. Mix vinegar and sugar until sugar is dissolved.
5. Pour mixture over vegetables and stir well.
6. Refrigerate and enjoy for up to 30 days.
Pickled Vegetables
CRISP DILLY GREEN BEANS
Dilly beans are similar to pickles but with a different texture. They are a tasty side to barbeques and especially delicious when served ice-cold on a warm summer day.
Yield: approx. 4 pints
2 pounds small, tender green beans
1 teaspoon cayenne pepper
4 cloves garlic
4 large heads dill
2 cups water
¼ cup salt
1 pint white vinegar
1. Destem green beans and pack into jars, allowing 1½ inches of headspace.
2. To each jar add ¼ teaspoon cayenne pepper, 1 clove garlic, and 1 head dill.
3. Heat water, salt, and vinegar until boiling.
4. Pour brine over beans, allowing ½ inch of headspace.
5. Wipe the jars and rims, making sure to remove any sticky remnants of brine.
6. Place the lids on clean jars and close them with the jars' accompanying screw bands.
7. Using a jar lifter, place the jars into the canner, making sure they are covered with water.
8. Bring water to a rolling boil and process green beans for 5 minutes.
9. Remove from canner and let set on a sturdy surface for 24 hours. After 24 hours, your jars are ready to be stored. (I remove the screw bands before storing, as I always need them for my next canning project.)
PICKLED BEETS
Pickled beets is another traditional Amish dish that is served at lunch following church services. These beets are sweet but also sour.
Yield: approx. 10 pints
10 pounds fresh, raw beets
2 cups vinegar
2 cups water or liquid from cooking beets (beet juice)
2 cups sugar
1 teaspoon salt
1. Wash the beets thoroughly.
2. Cut off the leafy tops, but leave the tops of the stems intact. You will remove the tops later. Do not remove the root end, either.
3. Cover the beets with water in a large stockpot and boil until fork tender.
4. Drain cooking liquid, reserving 2 cups for the brine.
5. Cool beets until they aren't too hot to touch.
6. Cut off the tops and root ends and peel beets using your fingers to slide the peels off. Try not to cut into the sides of the beets.
7. Slice or cut beets into chunks and fill jars, leaving 1 inch of headspace.
8. Heat reserved cooking liquid with all remaining ingredients until just boiling.
9. Before pouring the brine over the beets, make sure your canner is full of boiling water, ready to process the beets.
10. Using a funnel, carefully pour the hot brine into the filled jars, leaving ½ inch of headspace.
11. Wipe the jars and rims, making sure to remove any sticky remnants of brine.
12. Place the lids on clean jars and close them with the jars' accompanying screw bands.
13. Using a jar lifter, place the jars into the canner, making sure they are covered with water.
14. Bring water to a rolling boil and process beets for 20 minutes.
15. Remove from canner and let set on a sturdy surface for 24 hours. After 24 hours, your jars are ready to be stored. (I remove the screw bands before storing, as I always need them for my next canning project.)
HARVARD BEETS
You can serve this type of beet as a quick vegetable or side dish to almost any meal.
Yield: approx. 4 pints
1½ cups sugar
2½ tablespoons cornstarch
2½ teaspoons salt
½ cup vinegar
¾ cup water
8 cups cooked beets, sliced or diced
1. Mix sugar, cornstarch, and salt.
2. Add in vinegar and water.
3. Mix well.
4. Stirring often, heat brine over medium heat until boiling.
5. Remove brine from heat and pour over beets.
6. Ladle into jars.
7. Wipe the jars and rims, making sure to remove any sticky remnants of brine.
8. Place the lids on clean jars and close them with the jars' accompanying screw bands.
9. Using a jar lifter, place the jars into the canner, making sure they are covered with water.
10. Bring water to a rolling boil and process beets for 10 minutes.
11. Remove from canner and let set on a sturdy surface for 24 hours. After 24 hours, your jars are ready to be stored. (I remove the screw bands before storing, as I always need them for my next canning project.)
Relishes
Relishes are good condiments to have in stock in your basement or pantry. They taste great with burgers and hot dogs, of course, but also with grilled meats or sandwiches. Relishes take a bit more time than pickles, and the process is a little different, but still perfectly doable. I enjoy making relishes, even if it takes a lot of chopping!
GREEN TOMATO RELISH
This is a great recipe to use up any extra green tomatoes at the end of summer.
Yield: approx. 12 pints
24 large green tomatoes
6 red and/or green bell peppers
12 large onions
3 tablespoons celery seed
3 tablespoons mustard seed
1 tablespoon salt
5 cups white sugar
2 cups cider vinegar
1. Grind or finely chop tomatoes, peppers, and onions.
2. In a mesh sieve or colander, let drain for 1 hour.
3. Combine chopped vegetables with all remaining ingredients in a large stockpot.
4. Boil for 20 minutes.
5. Ladle into jars.
6. Process in hot-water bath for 30 minutes.
ONION RELISH
Yield: approx. 3 pints
14 medium onions
6 medium green peppers
3 small hot peppers, optional
4 cups white vinegar
3 cups sugar
2 tablespoons salt
1. Chop or grind onions and peppers.
2. In a large pot, heat vinegar, sugar, and salt.
3. Bring to a boil and add in vegetables.
4. Simmer for 15 minutes.
5. Ladle into hot jars.
6. Process for 10 minutes in a hot-water bath.
PEPPER RELISH
Yield: approx. 3 pints
2 dozen large peppers
15 medium onions
½ cup salt
Boiling water (enough to cover vegetables)
3 cups vinegar
3 cups sugar
1 teaspoon mustard seed
1 teaspoon celery seed
1 teaspoon salt
1. Chop or grind peppers and onions.
2. Sprinkle with salt and cover with boiling water.
3. Soak vegetables in boiling water for 10 minutes.
4. Drain well.
5. Mix vinegar, sugar, mustard seed, celery seed, and salt.
6. Pour over vegetables.
7. Heat until boiling and boil for 15 minutes.
8. Ladle into hot jars and process for 10 minutes.
PICKLE RELISH
Yield: approx. 15 pints
4 quarts cucumbers, chopped or ground
¼ cup salt + 1 teaspoon salt, divided
1 quart onions, chopped
1 pint peppers, chopped
2 teaspoons mustard seed
2 teaspoons celery seed
1 teaspoon turmeric
4 cups sugar
2 cups vinegar
1. Chop or grind cucumbers.
2. Sprinkle with ¼ cup salt and let soak for 1 hour.
3. Drain well and add in remaining vegetables.
4. Mix 1 tablespoon salt, mustard seed, celery seed, turmeric, sugar, and vinegar.
5. Pour over vegetable mixture.
6. Bring to a boil over medium heat.
7. Simmer for 20 minutes.
8. Ladle into hot jars and process for 10 minutes.
ZUCCHINI RELISH
This relish takes a bit more time as the vegetables need to soak longer. This recipe calls for soaking overnight, but I have made this relish often, sometimes only soaking for two to three hours, and it turned out fine! This is my family's favorite relish.
Yield approx. 15 pints
12 cups ground or chopped zucchini
4 cups ground or chopped onion
1 large pepper, ground or chopped
5 tablespoons salt
2½ cups vinegar
6 cups sugar
1 tablespoon dry mustard
½ teaspoon pepper
¾ tablespoon cornstarch
¾ teaspoon turmeric
1½ teaspoons celery seed
1. Mix chopped vegetables with salt.
2. Let soak overnight.
3. Drain well.
4. Mix remaining ingredients and pour over vegetables.
5. Bring to a boil over medium heat.
6. Simmer for 30 minutes.
7. Ladle into hot jars and process for 10 minutes.
Jams and Jellies
Jams, jellies, and sweet spreads are a great way to preserve fresh summer fruit. I always make strawberry jam in the spring, as that's our family favorite. I have tried others as well, but strawberry remains the favorite. The one ingredient that you'll definitely have to buy is fruit pectin. Pectin comes in powdered or liquid form and is easy to use. It's just important to remember: Do not try to change the amount of sugar in a recipe, because the jelly will not gel correctly. I've tried that with disastrous results! (We had strawberry sauce that year instead of jam.) There are different versions of most recipes, some are made with liquid pectin, some with powdered pectin, and some are made the old-fashioned way with no added pectin at all. I always use the powdered pectin as that seems to work well. I prefer using Sure-Jell brand fruit pectin, but that's just my preference. I'm sure other kinds work just as well.
A note on freezing versus canning:
Most jam or jelly recipes can be both frozen or canned. The only real difference you'll notice is the color of your product. When you freeze jelly it retains the bright color of fresh fruit. If you can it, the heat of the canner will change the color to a darker shade. They both taste great, so the choice is yours, depending on what color you prefer your jams and jellies.
A note on jam versus jelly:
The only difference in jam or jelly is that jam has pieces of fruit in it and jelly is juice of fruit, so it will be completely smooth with fewer seeds, such as strawberry seeds when making strawberry jelly.
STRAWBERRY JAM
This is my favorite recipe, passed to me from my friend Karen. Since I started using this strawberry jam recipe, it's the only jam we like. We eat it on toast and also paired with sharp cheddar cheese. Even the boys love cheese and strawberry jam! I freeze my jam because I love the bright red color, but you don't have to. It still tastes great canned.
Yield 4–5 pints
2 quarts fresh strawberries
1 box regular Sure-Jell fruit pectin
7 cups sugar
1. Remove stems of strawberries and wash and drain berries well.
2. Crush berries to the right consistency for your taste.
3. Mix crushed berries and Sure-Jell together in a large pot.
4. Stirring constantly, heat on high until mixture comes to a rolling boil.
5. Add in sugar.
6. Cook until mixture returns to a rolling boil.
7. Boil on high for 1 minute.
8. Pour into prepared jars or freezer containers.
9. Let jam set at room temperature for 24 hours before putting in freezer.
10. If canning, wipe rims of jars and add lids.
11. Process in a hot-water bath for 10 minutes.
12. Let cool for 24 hours before storing.
GRAPE JELLY
You can do the first four steps of this recipe the evening before actually making the jelly.
Yield: approx. 9 (8-oz.) jelly jars
5 cups fresh grape juice from 5 pounds fresh Concord grapes
1 (1.75-oz.) package fruit pectin
6 cups sugar
1. To make fresh grape juice, wash the grapes well and remove from stems.
2. Heat in a large saucepan with just enough water to prevent scorching.
3. Line a stainless-steel colander with several layers of cheesecloth and set over a bowl.
4. Let drip undisturbed for 2 hours minimum or overnight.
5. In a large saucepan, whisk fruit juice and pectin until pectin is dissolved.
6. Stirring frequently, bring to a boil.
7. Add sugar all at once and bring to a boil.
8. Boil on high for 1 minute.
9. Pour into prepared jars.
10. Wipe rims and add lids and screw bands.
11. Process in hot-water bath for 10 minutes.
12. Let set at room temperature for 24 hours before storing.
RASPBERRY JELLY
You can do the first four steps of this recipe the evening before actually making the jelly.
Yield: approx. 6 (8-oz.) jelly jars
4 pounds raspberries
4 tablespoons lemon juice
1 package powdered fruit pectin
5½ cups white sugar
1. To make fresh raspberry juice, wash the raspberries well.
2. Heat in a large saucepan with just enough water to prevent scorching.
3. Line a stainless-steel colander with several layers of cheesecloth and set over a bowl.
4. Let drip undisturbed for 2 hours minimum or overnight.
5. In a large saucepan, add lemon juice to fruit juice, then whisk in pectin until pectin is dissolved.
6. Stirring frequently, bring to a boil.
7. Add sugar all at once and bring to a boil.
8. Boil on high for 1 minute.
9. Pour into prepared jars.
10. Wipe rims and add lids and screw bands.
11. Process in hot-water bath for 10 minutes.
12. Let set at room temperature for 24 hours before storing.
APPLE JELLY
You can do the first five steps of this recipe the evening before actually making the jelly.
Yield: approx. 6 (8-oz.) jelly jars
5 pounds apples
1 package powdered fruit pectin
5 cups white sugar
1. To make fresh apple juice, wash the apples well and quarter them, removing stem and blossom ends.
2. Heat in a large saucepan with enough water to cover all the fruit.
3. Stirring frequently, boil until apples are softened, approximately 30 minutes.
4. Line a stainless-steel colander with several layers of cheesecloth and set over a bowl.
5. Let drip undisturbed for 2 hours minimum or overnight.
6. In a large saucepan, whisk fruit juice and pectin until pectin is dissolved.
7. Stirring frequently, bring to a boil.
8. Add sugar all at once and bring to a boil.
9. Boil on high for 1 minute.
10. Pour into prepared jars.
11. Wipe rims and add lids and screw bands.
12. Process in hot-water bath for 10 minutes.
13. Let set at room temperature for 24 hours before storing.
PEACH JELLY
You can do the first six steps of this recipe the evening before actually making the jelly.
Yield: approx. 5 (8-oz.) jelly jars
3 pounds peaches
1½ cup lemon juice
1 package powdered fruit pectin
5 cups white sugar
1. To make fruit juice, wash and quarter the peaches. The pits don't have to be removed, but it makes the boiling process easier if they are.
2. Add 1½ cup lemon juice for each pound of peaches.
3. Stirring frequently, bring to a boil.
4. Boil gently for 20 minutes.
5. Line a stainless-steel colander with several layers of cheesecloth and set over a bowl.
6. Let drip undisturbed for 2 hours minimum or overnight.
7. In a large saucepan, whisk fruit juice and pectin until pectin is dissolved.
8. Stirring frequently, bring to a boil.
9. Add sugar all at once and bring to a boil.
10. Boil on high for 1 minute.
11. Pour into prepared jars.
12. Wipe rings and add lids and screw bands.
13. Process in hot-water bath for 10 minutes.
14. Let sit at room temperature for 24 hours before storing.
ELDERBERRY JELLY
You can do the first four steps of this recipe the evening before actually making the jelly.
Yield: approx. 5 (8-oz.) jelly jars
3 pounds elderberries
4 tablespoons lemon juice
1 package powdered fruit pectin
4½ cups white sugar
1. To make fresh elderberry juice, wash the elderberries well and remove from stems.
2. Heat in a large saucepan with just enough water to prevent scorching.
3. Line a stainless-steel colander with several layers of cheesecloth and set over a bowl.
4. Let drip undisturbed for 2 hours minimum or overnight.
5. In a large saucepan, whisk lemon juice and pectin until pectin is dissolved.
6. Stirring frequently, bring to a boil.
7. Add sugar all at once and bring to a boil.
8. Boil on high for 1 minute.
9. Pour into prepared jars.
10. Wipe rims and add lids and screw bands.
11. Process in hot-water bath for 10 minutes.
12. Let set at room temperature for 24 hours before storing.
MINT JELLY
Yield: approx. 5–6 (8-oz.) jelly jars
3 cups water
1½ cups fresh spearmint or peppermint leaves
2 tablespoons white vinegar
⅓ cup powdered pectin
½ teaspoon butter
Green food coloring, optional
4 cups sugar
Mint sprigs, for garnish
1. Bring water to a boil.
2. Add mint leaves and turn off heat.
3. Steep for 15 minutes.
4. Strain tea.
5. Add vinegar, pectin, butter, and food coloring (if using).
6. Stirring constantly, mix well and bring to a boil.
7. Add sugar and return to boil.
8. Boil on high for 1 minute.
9. Remove from heat and skim off foam.
10. Pour into jars with a mint sprig to garnish each jar.
11. Wipe rims and add lids and screw bands.
12. Process in hot-water bath for 10 minutes.
13. Let set at room temperature for 24 hours before storing.
HOT PEPPER JAM
Yield: approx. 8 (8-oz.) jelly jars
10 large jalapeño peppers
3 medium red bell peppers
2 medium green bell peppers
1 cup cider vinegar
1 package powdered fruit pectin
5 cups white sugar
1. Chop all peppers and measure them out to 4 cups total.
2. In a saucepan, add vinegar and pectin.
3. Stir well and bring to a boil over medium heat.
4. Add sugar all at once.
5. Return to a boil.
6. Boil on high for 1 minute.
7. Pour into prepared jars.
8. Wipe rims and add lids and screw bands.
9. Process in hot-water bath for 10 minutes.
10. Let set at room temperature for 24 hours before storing.
Old-Fashioned Jams and Jellies
HOT PEPPER JAM (WITHOUT PECTIN)
Yield: approx. 8 (8-oz.) jelly jars
14 cups chopped hot or sweet peppers
¼ cup salt
6 cups sugar
4 cups vinegar
1. Sprinkle chopped peppers with salt.
2. Soak for 3 hours.
3. Do not drain peppers.
4. In a saucepan, add sugar and vinegar.
5. Stirring occasionally, bring to a boil and boil for 45 minutes.
6. Pour into prepared jars.
7. Wipe rims and add lids and screw bands.
8. Process in hot-water bath for 10 minutes.
9. Let set at room temperature for 24 hours before storing.
BLACKBERRY JELLY
Variations of this jelly can be made using different fruits. Just remember to measure the fruit juice after it has drained for at least 2 hours and then add equal amounts of sugar.
Yield: approx. 6 (8-oz.) jelly jars
2 quarts blackberries
3 cups water
White sugar
1. Wash and drain blackberries.
2. Add water and cook blackberries until soft.
3. In a cheesecloth-lined colander, drain berries.
4. Let drip at least 2 hours or overnight.
5. Measure fruit juice and add sugar equal to the amount of juice.
6. Stirring frequently, boil on high until jelly thickens.
7. Pour into prepared jars.
8. Wipe rims and add lids and screw bands.
9. Process in hot-water bath for 10 minutes.
10. Let set at room temperature for 24 hours before storing.
RHUBARB JAM
Yield: approx. 6 (8-oz.) jelly jars
3 pounds rhubarb stalks
½ cup water
4 cups white sugar
2 oranges, zest and juice
1. Cut rhubarb into ½-inch pieces.
2. Place in saucepan with water, sugar, and orange zest and juice.
3. Bring to a boil.
4. Stirring constantly, simmer for 30 minutes.
5. Pour into prepared jars.
6. Wipe rims and add lids and screw bands.
7. Process in hot-water bath for 10 minutes.
8. Let set at room temperature for 24 hours before storing.
Sweet Spreads
PEAR BUTTER
Yield: 8–9 (8-oz.) jelly jars
3 quarts mashed pears
1 quart sugar
1. Peel ripe pears and cook until soft.
2. Drain and mash well.
3. Stir mashed pears and sugar together.
4. Stirring frequently, bring to a boil.
5. Boil on high for 1 minute.
6. Pour into prepared jars.
7. Wipe rims and add lids and screw bands.
8. Process in hot-water bath for 10 minutes.
9. Let set at room temperature for 24 hours before storing.
PEAR HONEY
Yield: approx. 12 (8-oz.) jelly jars
12–14 medium ripe pears, peeled and cored
8 cups sugar
1 (20-oz.) can crushed pineapple
3 tablespoons lemon juice
1. Puree pears in blender or food processor.
2. Stir remaining ingredients into pears.
3. Bring to a boil in a large saucepan.
4. Reduce heat and boil gently, stirring occasionally for 50 to 60 minutes or until thickened.
5. Pour into prepared jars.
6. Wipe rims and add lids and screw bands.
7. Process in hot-water bath for 10 minutes.
8. Let set at room temperature for 24 hours before storing.
EASY APPLE BUTTER
Yield: approx. 5–6 pints
2 quarts applesauce
2 cups brown sugar
1 cup white vinegar
1 tablespoon cinnamon
½ teaspoon cloves
1. Stir all ingredients together, mixing well.
2. Bring to a boil over medium heat.
3. Simmer for 2 hours.
4. Pour into jars and process for 15 minutes.
Soups
Having an array of canned soup in your basement is a great idea. There's nothing easier than grabbing a can of soup for a quick meal.
A note on canning soup:
Most food safety organizations recommend using a pressure canner for soup. My family processes canned soup for 3 hours to make sure it maintains a high temperature all the way through, but if you prefer to use a pressure canner the time is 1 hour at 10 pounds pressure.
CREAM OF CELERY SOUP
This is a condensed-style soup that can be used in recipes as is or diluted with milk or water. You can also substitute mushrooms for the celery and have cream of mushroom soup instead.
Yield: approx. 9–10 pints
3 bunches celery, finely chopped
3 quarts chicken broth
1 cup butter
½ cup onion, chopped
1½ cup flour
1 tablespoon salt
1. Combine celery and broth in a large stockpot.
2. In a separate pan, melt butter and sauté onions until nice and brown.
3. Whisk flour and salt into butter mixture.
4. Stir butter mixture into celery mixture and boil until thickened.
5. Ladle into clean jars, allowing 1½ inch of headspace.
6. Process in hot-water bath for 3 hours.
HAM AND BEAN SOUP
Yield: approx. 12 quarts
1 pound butter
1 large onion, chopped
3 quarts diced cooked ham
2 gallons great northern beans or other beans of your choice
3 cups ketchup
2½ cups brown sugar
Salt, pepper, or other seasonings of your choice
1. Melt butter and sauté onion until soft.
2. Add ham and cook for 15 minutes.
3. Pour beans, ketchup, brown sugar, and seasonings into a 12-quart stockpot.
4. Add in ham mixture and enough water to fill stockpot.
5. Stir well and adjust seasonings to taste.
6. Ladle into clean jars.
7. Process in hot-water bath for 3 hours.
CHUNKY BEEF SOUP
Yield: approx. 10–12 quarts
2½ quarts water
2 cups flour
¼ cup beef base or beef bouillon
1 can beef broth
1 quart tomato juice
½ cup brown sugar
1½ tablespoons salt
1 quart diced carrots
1 quart diced potatoes
1 quart peas
1 bunch celery
1 onion, diced
2 tablespoons butter
2 pounds hamburger
1. Whisk water, flour, beef base or bouillon, beef broth, tomato juice, brown sugar, and salt, mixing well.
2. Stirring often, bring to a boil until mixture is thickened.
3. Allow to cool.
4. Sauté onion in butter with hamburger.
5. Combine all ingredients and stir well.
6. Ladle into clean jars, allowing at least 1½ inch of headspace.
7. Process in hot-water bath for 3 hours.
CHICKEN NOODLE SOUP (TO CAN)
Yield: 1 quart
1 cup uncooked egg noodles or 2–3 tablespoons uncooked rice
⅓ cup chopped celery
¼ cup chopped or grated carrots
⅓ cup diced cooked chicken
1 teaspoon salt
1 teaspoon dried onion flakes
Chicken broth
1. Add all ingredients to a quart-sized jar.
2. Fill jar with chicken broth, leaving 1-inch headspace.
3. Process in hot-water bath for 3 hours.
VEGETABLE SOUP
Yield: approx. 16 quarts
1 quart green beans
1 quart sliced or diced carrots
1 quart diced potatoes
1 quart peas
1 quart diced onions
1 quart corn
1 quart lima beans
3 large peppers, chopped
3 pounds hamburger, optional
2 tablespoons butter
7 quarts tomato juice
2 tablespoons beef bouillon
¾ cup brown sugar
2 tablespoons chili powder
3 cup alphabet or other small pasta
1. Precook all vegetables and cool before mixing soup.
2. Sauté peppers and hamburger (if using) in butter.
3. Mix vegetables, browned peppers and hamburger (if using) and all remaining ingredients together.
4. Adjust seasonings to taste.
5. Ladle into clean jars.
6. Process in hot-water bath for 3 hours.
Note: Vegetables for this soup can be switched according to personal tastes.
CHILI SOUP
Chili soup is an Amish version of chili that is not as thick or meaty as a typical chili.
Yield: approx. 16 quarts
6 onions
6 pounds hamburger meat
2 tablespoons butter
12 cups kidney beans
9 quarts tomato juice
3 pints pizza sauce or plain tomato sauce
3 teaspoons paprika or more to taste
3 tablespoons chili powder or more to taste
9 tablespoons brown sugar
3 tablespoons Worcestershire sauce
1 tablespoon salt or more to taste
1. Sauté onions and hamburger in butter.
2. Add all remaining ingredients and mix well.
3. Bring to a boil.
4. Ladle into clean jars.
5. Process in hot-water bath for 3 hours.
CONVERSION CHARTS
METRIC AND IMPERIAL CONVERSIONS
(These conversions are rounded for convenience)
OVEN TEMPERATURES
INDEX
A
allspice
Chili Base,
Ketchup,
alum
Salsa,
Apple Butter,
Apple Jelly,
Apple Pie Filling,
Applesauce,
Easy Apple Butter,
B
Banana Pickles,
basil
Pizza Sauce,
beans
great northern
Ham and Bean Soup,
kidney
Chili Soup,
beef
ground
Chili Soup,
Chunky Beef Soup,
Vegetable Soup,
beets
Harvard Beets,
Pickled Beets,
bell peppers
Hot Pepper Jam, ,
Pepper Relish,
Pickle Relish,
Salsa, ,
Seasoned Tomato Juice,
Vegetable Soup,
Blackberry Jelly,
Blueberry Pie Filling,
Bread-and-Butter Pickles,
butters
Easy Apple Butter,
Pear Butter,
C
carrots
Chicken Noodle Soup,
Chunky Beef Soup,
Vegetable Soup,
cayenne
Crisp Dilly Green Beans,
Ketchup,
celery
Chicken Noodle Soup,
Chili Base,
Chunky Beef Soup,
Cream of Celery Soup,
Seasoned Tomato Juice,
Stewed Tomatoes,
Tomato Soup,
celery salt
Homemade V-8 Juice,
celery seed
Banana Pickles,
Bread-and-Butter Pickles,
Green Tomato Relish,
Ketchup,
Pepper Relish,
Pickle Relish,
Refrigerator Pickles,
Zucchini Relish,
Cherry Pie Filling,
chicken
Chicken Noodle Soup,
chicken broth
Chicken Noodle Soup,
Cream of Celery Soup,
Chicken Noodle Soup,
Chili Base,
chili powder
Chili Soup,
Salsa, ,
Vegetable Soup,
Chili Soup,
Chunky Beef Soup,
cinnamon
Apple Pie Filling,
Chili Base,
Easy Apple Butter,
Ketchup,
cloves
Easy Apple Butter,
Ketchup,
corn
Vegetable Soup,
Crisp Dilly Green Beans,
cucumbers
Banana Pickles,
Bread-and-Butter Pickles,
Mustard Pickles,
Pickle Relish,
Refrigerator Pickles,
Sweet Kosher Dill Pickles,
cumin
Salsa,
D
dill
Crisp Dilly Green Beans,
E
Easy Apple Butter,
Elderberry Jelly,
F
Fruit Pudding,
G
garlic
Crisp Dilly Green Beans,
Pizza Sauce, ,
garlic powder
Pizza Sauce, ,
Salsa,
garlic salt
Homemade V-8 Juice,
Hot Sauce,
ginger
Chili Base,
Ketchup,
Grape Jelly,
grape juice
Fruit Pudding,
Grape Jelly,
green beans
Crisp Dilly Green Beans,
Vegetable Soup,
green pepper
Green Tomato Relish,
Hot Pepper Jam,
Onion Relish,
Pizza Sauce,
Refrigerator Pickles,
Stewed Tomatoes,
Green Tomato Relish,
H
ham
Ham and Bean Soup,
Ham and Bean Soup,
Harvard Beets,
Homemade V-8 Juice,
Hot Pepper Jam, ,
hot peppers
Hot Sauce,
Hot Sauce,
J
jalapeño
Hot Pepper Jam,
Salsa, ,
jams
Hot Pepper Jam, ,
Rhubarb Jam,
Strawberry Jam,
jellies
Apple Jelly,
Blackberry Jelly,
Elderberry Jelly,
Grape Jelly,
Mint Jelly,
Peach Jelly,
Raspberry Jelly,
K
Ketchup, ,
Ham and Bean Soup,
L
lima beans
Vegetable Soup,
M
Mint Jelly,
mustard
Ketchup,
Zucchini Relish,
Mustard Pickles,
mustard seed
Banana Pickles,
Green Tomato Relish,
Pepper Relish,
N
noodles
Chicken Noodle Soup,
Vegetable Soup,
O
onion
Bread-and-Butter Pickles,
Chili Base,
Cream of Celery Soup,
Green Tomato Relish,
Ham and Bean Soup,
Hot Sauce,
Pepper Relish,
Pickle Relish,
Pizza Sauce,
Refrigerator Pickles,
Salsa, ,
Seasoned Tomato Juice,
Stewed Tomatoes,
Tomato Soup,
Vegetable Soup,
Zucchini Relish,
onion flakes
Chicken Noodle Soup,
onion powder
Ketchup,
Salsa,
Onion Relish,
onion salt
Homemade V-8 Juice,
oranges
Rhubarb Jam,
oregano
Pizza Sauce, , ,
P
paprika
Chili Soup,
parsley
Tomato Soup,
Peach Jelly,
Pear Butter,
Pear Honey,
peas
Chunky Beef Soup,
Vegetable Soup,
pectin
Apple Jelly,
Elderberry Jelly,
Grape Jelly,
Hot Pepper Jam,
Mint Jelly,
Peach Jelly,
Raspberry Jelly,
Strawberry Jam,
peppermint
Mint Jelly,
Pepper Relish,
Pickled Beets,
Pickle Relish,
pickles
Bread-and-Butter Pickles,
Mustard Pickles,
Refrigerator Pickles,
Sweet Kosher Dill Pickles,
pie filling
apple,
blueberry,
cherry,
pineapple
Pear Honey,
Pizza Sauce, , ,
potatoes
Chunky Beef Soup,
Vegetable Soup,
R
Raspberry Jelly,
raspberry juice
Fruit Pudding,
Refrigerator Pickles,
relishes
Green Tomato Relish,
Onion Relish,
Pepper Relish,
Pickle Relish,
Zucchini Relish,
Rhubarb Jam,
S
Salsa, ,
Seasoned Tomato Juice,
soups
Chicken Noodle Soup,
Chili Soup,
Chunky Beef Soup,
Cream of Celery Soup,
Ham and Bean Soup,
Vegetable Soup,
spearmint
Mint Jelly,
Stewed Tomatoes,
Strawberry Jam,
Sweet Kosher Dill Pickles,
sweet peppers
Hot Pepper Jam,
T
therm-flo,
Apple Pie Filling,
Blueberry Pie Filling,
Cherry Pie Filling,
Fruit Pudding,
Ketchup, ,
Pizza Sauce, ,
Salsa, ,
tomatoes,
Chili Base,
green
Green Tomato Relish,
Hot Sauce,
Pizza Sauce, ,
Salsa, ,
Stewed Tomatoes,
Tomato Soup,
Tomato Juice,
Chili Soup,
Chunky Beef Soup,
Homemade V-8 Juice,
Ketchup, ,
Pizza Sauce,
Seasoned,
Vegetable Soup,
tomato paste
Pizza Sauce,
Salsa,
tomato sauce
Chili Soup,
Tomato Soup,
turmeric
Banana Pickles,
Bread-and-Butter Pickles,
Pickle Relish,
Zucchini Relish,
V
Vegetable Soup,
vinegar
Banana Pickles,
Chili Base,
cider
Green Tomato Relish,
Hot Pepper Jam,
Harvard Beets,
Hot Pepper Jam,
Ketchup, ,
Pepper Relish,
Pickled Beets,
Pickle Relish,
Salsa,
white
Bread-and-Butter Pickles,
Crisp Dilly Green Beans,
Easy Apple Butter,
Hot Sauce,
Mint Jelly,
Mustard Pickles,
Onion Relish,
Refrigerator Pickles,
Salsa,
Sweet Kosher Dill Pickles,
Zucchini Relish,
W
Worcestershire sauce
Chili Soup,
Z
Zucchini Relish,
|
{
"redpajama_set_name": "RedPajamaBook"
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| 410
|
100,000 Pounds of Marine Debris Removed from Papahānaumokuākea
SWIPE LEFT OR RIGHT
Approximately 100,000 pounds of marine debris was removed from sites within the Papahānaumokuākea Marine National Monument.
The mountain of marine debris was loaded aboard 12 shipping containers and was transported to Honolulu from Midway Atoll aboard the charter vessel Kahana.
The debris was collected from the reefs and beaches of Midway and Kure Atolls over the last six years and stored on the tarmac at Midway until it could be removed and shipped to Honolulu.
Now, through the Nets to Energy Program, it will be processed by Schnitzer Steel Corporation and transported to the City and County of Honolulu's H-POWER plant (a Covanta Energy Corporation facility), where it is being incinerated to produce electricity.
"The success of this project is linked to effective inter-agency coordination, communication and action," said Jason Misaki, Oʻahu Wildlife Manager, DLNR's Division of Forestry and Wildlife. "Marine debris in the Monument affects all partners, making joint efforts like this one extremely instrumental to our continued protection of resources."
Monument representatives say marine debris is a potentially lethal entanglement and ingestion hazard for wildlife—including threatened Hawaiian green sea turtles and endangered Hawaiian monk seals.
Plastic debris and fishing line are often ingested by wildlife such as seabirds and can lead to their starvation and death, according to state officials.
Authorities say that at Midway and Kure Atolls, plastics, derelict fishing gear and other marine debris is present in nests along the beach and often consumed by albatross chicks.
It is estimated that more than five tons of plastics end up on Midway Atoll each year because adult albatrosses feed their chicks plastics they forage in the ocean, mistaken for food. Plastics break down into smaller pieces but never go away, affecting all levels of the ocean ecosystem.
"Marine debris is not something you can clean up just once; it takes a sustained effort over time," said US Fish and Wildlife Service Superintendent Matt Brown. "By working with the state of Hawaiʻi, Office of Hawaiian Affairs and NOAA, we can accomplish more than any one agency on its own to clean up marine debris and educate the public to prevent it from entering the ecosystem."
Each year the USFWS, the State of Hawaiʻi, and the National Oceanic and Atmospheric Administration staff and volunteers collect tons of marine debris from the atolls and throughout the Monument.
"With the high rate of marine debris accumulation in the Northwestern Hawaiian Islands, removing debris is imperative to ensure the health of this valuable habitat and the species that call it home. We are happy to have the opportunity to work with partners on this important initiative," said Mark Manuel, NOAA Marine Debris Program Pacific Islands Regional Coordinator.
This marine debris removal effort represents a collaborative partnership between the USFWS, State of Hawai'i DLNR's Division of Forestry and Wildlife, NOAA's Pacific Islands Fisheries Science Center, NOAA's Marine Debris Program, NOAA's Office of National Marine Sanctuaries and Papahānaumokuākea Marine National Monument. Special thanks go to Nets to Energy Program, Schnitzer Steel Co. and Covanta Honolulu/H-POWER for their support of this project.
Agencies Remove Estimated 100,000 Pounds of Marine Debris from
Papahānaumokuākea. PC: NOAA IRC/NOS/ONMS/PMNM
82,600 Pounds of Debris and Trash Removed From… November 15, 2020
47.2 Tons of Marine Debris Removed from… April 22, 2021
46,000 Pounds of Marine Debris Removed from… January 14, 2021
Over 5,000 Pounds of Debris Removed From Lānaʻi Shores February 9, 2020
Hawaiian Names Given to Five Species Living Within… December 3, 2020
2,100 Pounds of Cocaine Seized in Eastern Pacific August 7, 2019
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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Hello! Tonight I have a few pretty things to share with you.
The first creation is one of my DT pieces made for Shimmerz this month.
It's a canvas I made for a friend. Her precious little niece was born and I got to go all out girly and pretty on this one!!!!
I always like to add some crystals or enamel dots to my creations. I try to match the colours as close as possible so they almost blend into the Shimmerz background.
I don't think you can quite tell by my photographs but I added a lot of sparkle to this creation. It's sooooo pretty! I tone my flower edges down with a little Gesso. This is when I will sprinkle on the Enamelz in Sparklicious so that in dries into the wet Gesso.
Oh wow your canvas is stunning, really love your background details, and the flowers. Your cards are amazing too, very pretty colors, really love these gorgeous creations!
Wow - your creations are stunning! I love them! You are so talented. I am going to have to visit you more often - I feel totally inspired after looking are your work.
Oh wow, these sure are pretty! Your projects are all absolutely stunning!
|
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"redpajama_set_name": "RedPajamaC4"
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| 6,580
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{"url":"http:\/\/read.somethingorotherwhatever.com\/entry\/CryptographicProtocolsWithEverydayObjects","text":"# Cryptographic Protocols with Everyday Objects\n\n\u2022 Published in 2011\nIn the collections\nMost security protocols appearing in the literature make use of cryptographic primitives that assume that the participants have access to some sort of computational device. However, there are times when there is need for a security mechanism to evaluate some result without leaking sensitive information, but computational devices are unavailable. We discuss here various protocols for solving cryptographic problems using everyday objects: coins, dice, cards, and envelopes.\n\n### BibTeX entry\n\n@article{CryptographicProtocolsWithEverydayObjects,\ntitle = {Cryptographic Protocols with Everyday Objects},\nauthor = {James Heather and Steve Schneider and Vanessa Teague},\nurl = {http:\/\/www.computing.surrey.ac.uk\/personal\/st\/S.Schneider\/papers\/2011\/cryptoforma11a.pdf},\nurldate = {2016-11-17},\nabstract = {Most security protocols appearing in the literature make use of cryptographic primitives that assume that the participants have access\nto some sort of computational device.\n\nHowever, there are times when there is need for a security mechanism\nto evaluate some result without leaking sensitive information, but computational devices are unavailable. We discuss here various protocols for\nsolving cryptographic problems using everyday objects: coins, dice, cards, and envelopes.},\ncomment = {},\nyear = 2011,\ncollections = {Easily explained,Protocols and strategies,Things to make and do}\n}","date":"2018-08-20 10:35:00","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5818977952003479, \"perplexity\": 3892.798533810017}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-34\/segments\/1534221216333.66\/warc\/CC-MAIN-20180820101554-20180820121554-00396.warc.gz\"}"}
| null | null |
Q: Get Pandas DataFrame first column This question is odd, since I know HOW to do something, but I dont know WHY I cant do it another way.
Suppose simple data frame:
import pandasas pd
a = pd.DataFrame([[0,1], [2,3]])
I can slice this data frame very easily, first column is a[[0]], second is a[[1]]. Simple isnt it?
Now, lets have more complex data frame. This is part of my code:
var_vec = [i for i in range(100)]
num_of_sites = 100
row_names = ["_".join(["loc", str(i)]) for i in
range(1,num_of_sites + 1)]
frame = pd.DataFrame(var_vec, columns = ["Variable"], index = row_names)
spec_ab = [i**3 for i in range(100)]
frame[1] = spec_ab
Data frame frame is also pandas DataFrame, such as a. I canget second column very easily as frame[[1]]. But when I try frame[[0]] I get an error:
Traceback (most recent call last):
File "<ipython-input-55-0c56ffb47d0d>", line 1, in <module>
frame[[0]]
File "C:\Users\Robert\Desktop\Záloha\WinPython-64bit-3.5.2.2\python- 3.5.2.amd64\lib\site-packages\pandas\core\frame.py", line 1991, in __getitem__
return self._getitem_array(key)
File "C:\Users\Robert\Desktop\Záloha\WinPython-64bit-3.5.2.2\python- 3.5.2.amd64\lib\site-packages\pandas\core\frame.py", line 2035, in _getitem_array
indexer = self.ix._convert_to_indexer(key, axis=1)
File "C:\Users\Robert\Desktop\Záloha\WinPython-64bit-3.5.2.2\python- 3.5.2.amd64\lib\site-packages\pandas\core\indexing.py", line 1184, in _convert_to_indexer
indexer = labels._convert_list_indexer(objarr, kind=self.name)
File "C:\Users\Robert\Desktop\Záloha\WinPython-64bit-3.5.2.2\python- 3.5.2.amd64\lib\site-packages\pandas\indexes\base.py", line 1112, in _convert_list_indexer
return maybe_convert_indices(indexer, len(self))
File "C:\Users\Robert\Desktop\Záloha\WinPython-64bit-3.5.2.2\python- 3.5.2.amd64\lib\site-packages\pandas\core\indexing.py", line 1856, in maybe_convert_indices
raise IndexError("indices are out-of-bounds")
IndexError: indices are out-of-bounds
I can still use frame.iloc[:,0] but problem is that I dont understand why I cant use simple slicing by [[]]? I use winpython spyder 3 if that helps.
A: using your code:
import pandas as pd
var_vec = [i for i in range(100)]
num_of_sites = 100
row_names = ["_".join(["loc", str(i)]) for i in
range(1,num_of_sites + 1)]
frame = pd.DataFrame(var_vec, columns = ["Variable"], index = row_names)
spec_ab = [i**3 for i in range(100)]
frame[1] = spec_ab
if you ask to print out the 'frame' you get:
Variable 1
loc_1 0 0
loc_2 1 1
loc_3 2 8
loc_4 3 27
loc_5 4 64
loc_6 5 125
......
So the cause of your problem becomes obvious, you have no column called '0'.
At line one you specify a lista called var_vec.
At line 4 you make a dataframe out of that list, but you specify the index values and the column name (which is usually good practice).
The numerical column name, '0', '1',.. as in the first example, only takes place when you dont specify the column name, its not a column position indexer.
If you want to access columns by their position, you can:
df[df.columns[0]]
what happens than, is you get the list of columns of the df, and you choose the term '0' and pass it to the df as a reference.
hope that helps you understand
edit:
another way (better) would be:
df.iloc[:,0]
where ":" stands for all rows. (also indexed by number from 0 to range of rows)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 683
|
\section{Introduction}
In this paper we explore the utility of using coherent elastic neutrino-nucleus scattering in dark matter experiments for detecting the neutrinos produced in the collapse of supermassive stars to black holes. Two issues are at the heart of why detecting these neutrinos is problematic: (1) Unlike in conventional massive star core collapse, the neutrinos generated in the collapse of a supermassive star are relatively lower energy, reflecting their thermal origin, and making them hard to detect; and (2) As yet there is no direct observational evidence for the existence of supermassive stars. However, new exploration of this subject is called for, first because the mystery of the existence of supermassive black holes at high redshift continues to deepen, and second because experimental techniques have dramatically improved, leading to the first detection of coherent elastic neutrino-nucleus scattering in 2017~\cite{Akimov:2017ade}.
Supermassive black holes (SMBHs) with masses $ \sim10^6 - 10^9 M_{\odot}$ are thought to be ubiquitous in the centers of galaxies~\cite{vanderMarel:1997hr}, and serve as the central engines powering quasars and Active Galactic Nuclei (AGN)~\cite{Rees:1984}. The existence of $10^9 M_\odot$ SMBHs at redshifts as high as $z \sim 7$ presents an intriguing astrophysical problem, as both Eddington-limited accretion and successive mergers are challenged in growing $\mathcal{O}(M_\odot)$ black holes to these masses on the relevant timescales~\cite{Mortlock:2011,Venemans:2013npa,Wu:2015,Banados:2017unc}.
Many proposals have been put forth to explain the origin and formation mechanism of SMBHs~(e.g.~\cite{Kroupa:2020}, see Ref.~\cite{Volonteri:2010} for review).
Of the standard astrophysical pathways to SMBH formation~\cite{Begelman:1978},
several go through an intermediate supermassive star (SMS) with mass $M \gtrsim 3 \times 10^4 M_{\odot}$. Large black holes would be the likely result of the collapse of such SMSs. In turn, these black holes would act as seeds~\cite{Woods:2018lty}. Through accretion or mergers, these could grow into SMBHs. There is no compelling argument for the existence of such SMSs, and no direct observation of them. However, they could plausibly arise either from a primordial gas cloud or as a consequence of the evolution of a dense star cluster. All we can say for certain is that such a configuration, should it arise, will collapse via the Feynman-Chandrasekhar general relativistic instability once it is primarily supported against gravitation by components moving at or near the speed of light, photons in the case of SMSs, and stars in the dense star cluster case. SMS collapse to a black hole will be accompanied by a prodigious neutrino burst, with luminosities capable of exceeding conventional core-collapse supernova by several orders of magnitude~\cite{Shi:1998nd}.
The physics accompanying the collapse of SMSs has been extensively studied in a variety of environmental conditions, including accretion and rotation~(e.g.~\cite{Fuller:1986,Baumgarte:1999nh,Saijo:2002qt,Umeda:2016smj,Haemmerl:2017,Haemmerl:2018,Nagele:2020xqq}).~These events are expected to produce an array of experimental signatures (e.g.~\cite{Shapiro:1979,Fuller:1986,Sun:2017}), including the generation of gravitational waves~\cite{Shibata:2016vzw,Uchida:2017qwn,Li:2018}, gamma-ray bursts~\cite{Fuller:1997em,Sun:2017}, and neutrinos~\cite{Shi:1998nd,Shi:1998jx,Linke:2001mq}.
Detection of neutrinos from SMS explosions would provide invaluable information regarding SMBH seed formation. In contrast to standard core-collapse supernovae, SMS neutrinos would be produced with an energy spectrum generated by the annihilation of thermal $e^\pm$ pairs, and that is similar among the various emitted neutrino species. Note, however, that the $\nu_e$ and $\bar\nu_e$ fluxes will be larger than those of the $\mu$ and $\tau$ flavor species~\cite{Shi:1998nd} because of the charged current annihilation channel available for production of electron flavor neutrinos. The neutrinos produced via thermal $e^\pm$-pair annihilation could be detected either directly from the collapse of individual relatively nearby objects, or via the diffuse background produced from the cumulative history of SMS collapses (see e.g.~\cite{Shi:1998jx,Shi:1998nd}). In the latter case, the spectrum may suffer significant redshift, causing the entirety of the spectrum to become buried under the large neutrino fluxes generated by the Sun, reactors, nuclear processes in the Earth, etc. The possibility of detecting neutrinos from SMS explosions through inverse beta decay (IBD) $\overline{\nu}_e + p \rightarrow n + e^+$ has been previously considered~\cite{Shi:1998nd,Shi:1998jx}, both with conventional neutrino telescopes, such as Cerenkov-based Super-Kamiokande~\cite{FUKUDA:2003,Abe:2013gga}, and with IceCube~\cite{Ahrens:2002dv}.
Coherent elastic neutrino-nucleus scattering (CE$\nu$NS) could provide a new way to search for the low energy neutrinos of a SMS collapse-generated neutrino burst. In contrast to IBD, CE$\nu$NS is unconstrained by the IBD kinematic threshold on neutrino energy. Moreover, CE$\nu$NS will have sensitivity to all six $(\nu_{e}, \overline{\nu}_e, \nu_{\mu}, \overline{\nu}_{\mu}, \nu_{\tau}, \overline{\nu}_{\tau})$ neutrino flavors. CE$\nu$NS has been recently directly observed~\cite{Akimov:2017ade}, and has been considered in a range of studies related to neutrino physics, including
sterile neutrinos (e.g.~\cite{Pospelov:2011ha,Billard:2014yka}),
non-standard neutrino interactions (e.g.~\cite{Harnik:2012ni,Dutta:2017nht,Flores:2020lji}),
solar neutrinos (e.g.~\cite{Billard:2014yka,Schumann:2015cpa,Franco:2015pha,Gelmini:2018ogy}),
geoneutrinos~\cite{Gelmini:2018gqa},
neutrinos from dark matter (DM) annihilation and decays~\cite{Cherry:2015oca,Cui:2017ytb,McKeen:2018pbb}, as well as supernova~\cite{Chakraborty:2013zua,XMASS:2016cmy,Lang:2016zhv,Kozynets:2018dfo,Khaitan:2018wnf} and pre-supernova neutrinos~\cite{Raj:2019wpy}.
Large scale direct detection experiments, whose primary target is dark matter (DM) observation, are themselves effective neutrino telescopes and can explore complementary parameter space compared to that of conventional neutrino experiments. In particular, such experiments have very low keV-scale thresholds, potentially providing sensitivity to a complementary part of the neutrino spectrum. Furthermore, with heavy nuclei as detector targets, these experiments are particularly well suited for signal detection via CE$\nu$NS, whose cross-section scales approximately as neutron number squared.
In this study we explore the detection capabilities of large scale direct DM detection experiments via CE$\nu$NS of neutrinos produced from SMS collapse. We examine both the signal arising from the collapse of individual objects, as well as the diffuse signal generated by the cumulative collapse rate throughout their history. The former of these could be detectable from collapsing stars in nearby galaxies.
This work is organized as follows. In \Sec{sec:sms} we describe neutrino production from the collapse of supermassive stars. \Sec{sec:dd} presents an overview of large direct detection experiments and their sensitivity to coherent neutrino scattering. The sensitivity of these experiments to an individual explosion of a supermassive star and the diffuse background is presented in \Sec{sec:signal}. In \Sec{sec:darkmatter} we elaborate on the extent to which supermassive star collapses may contribute to the background of dark matter searches. We conclude in \Sec{sec:con}.
\section{Neutrinos from Supermassive Star Collapse}\label{sec:sms}
\subsection{Neutrino burst}\label{sec:sms}
Supermassive stars with masses $M \gtrsim 3 \times 10^4 \, M_\odot$ are expected to directly collapse into a black hole as a result of the Feynman-Chandrasekhar instability, unless centrifugal forces from rapid rotation or magnetic fields are sufficiently strong~\cite{Fuller:1986}. During the collapse, only a fraction of the initial star, the homologous core (HC) comprising $M_5^{\rm HC}/M_5^{\rm init} \simeq 0.1$ of the initial mass, with $M_5$ being stellar mass in units of $10^5 M_{\odot}$, plunges through the event horizon, resulting in prompt black hole formation. Most of the HC binding energy will be trapped within the BH, but a small fraction could be emitted in the form of neutrinos and (an even smaller fraction) in gravitational waves.
Neutrino emission from SMS collapse has been analyzed in Ref.~\cite{Shi:1998jx,Shi:1998nd}, whose discussion we follow. The entropy-per-baryon in SMS is large, corresponding to low density and modestly high temperature, with electromagnetic equilibrium consequently implying a large electron-positron ($e^\pm$) pair density. Neutrino pairs are produced by $e^+e^-$ annihilation in the in-falling HC, with most of the neutrino luminosity being generated as the radius of the star nears trapped surface formation, its Schwarzschild radius. Unlike core collapse supernovae, the in-falling material is transparent to emitted neutrinos. Consequently, the luminosity, spectrum, and time profile are well-defined quantities\footnote{Numerical hydrodynamic simulations of Ref.~\cite{Linke:2001mq} show emission suppressed by up to two orders compared to the analytic results of Ref.~\cite{Shi:1998nd}. These differences stem from differing treatments of the in-fall and collapse timescales, pressure,
and the adiabat of collapse, and are exacerbated by the $T^9$ dependence of the neutrino emissivity. Significant uncertainties remain. We employ the results of Ref.~\cite{Shi:1998nd} throughout this study as an example (for comparison of models see Fig. 3 of Ref.~\cite{Li:2018})}. In particular, the total neutrino luminosity is expected to be a sizable fraction of the HC gravitational binding energy $E_s \simeq 10^{59} M_5^{\rm HC}$~erg. Reference \cite{Li:2018} showed that the SMS HC mass range that gives an optimal fraction of the rest mass radiated as neutrinos is $5\times{10}^4\,{M_\odot} < M^{\rm HC} < 5\times{10}^5\,{M_\odot}$. Other factors that determine the ultimate neutrino fluence from collapse of these objects include the time profile of the collapse, dictated by a number of features\footnote{The in-fall time can increase in the presence of rotation or strong magnetic fields.} of the HC. Roughly, this collapse time scale will be $t_s \simeq M_5^{\rm HC}$~s. The neutrino energy spectra and fluxes are determined mostly by the evolution of the density and temperature distributions near the Schwarzschild radius (i.e. a rapid rise as the mass in-falls, followed by a rapid fall as material is absorbed by the black hole).
\begin{figure}[t]
\begin{center}
\includegraphics[width=1\columnwidth]{Nu_flux_SNOLAB}
\caption{Neutrino flux from a single SMS explosion (black) and the diffuse background generated from the cosmological evolution of SMS collapse (DSMSB, blue). The band detailing the burst signal assumes a SMS with mass $10^{5}M_{\odot}$ collapsing at 100 kpc (solid), at 1 Mpc (dashed), at 1 Mpc with enhancement due to non-negligible rotation or magnetic field (dash-dotted) and a SMS with mass $3\times10^{4} \,M_{\odot}$ exploding at a distance of 1 Mpc (dotted). The band detailing the diffuse background signal corresponds to the $R^{\rm flat}$ (upper) and $R^{\rm Pop3}$ (lower) models. Shown for comparison are the solar ($^8$B,$^{17}$F,$^{15}$O,$^{13}$N,$^7$Be,hep,pp,pep), geo ($^{238}$U,$^{232}$Th,$^{40}$K), reactor, atmospheric neutrino and diffuse supernovae neutrino background (DSNB) fluxes -- reactor and geoneutrino fluxes have been computed assuming the experiment is located at SNOLAB~\cite{Gelmini:2018ogy}. }
\label{fig:Nu_flux}
\end{center}
\end{figure}
Considering the peak emission occurring near the Schwarzschild radius, the resulting neutrino luminosity from pair-production during SMS star collapse can be estimated as~\cite{Shi:1998nd}
\begin{equation}
L_\nu \simeq 3 \times 10^{57} \, (M_5^{\rm HC})^{-1.5} \, {\rm erg \, cm^{-3} \, s^{-1}}~.
\end{equation}
The associated neutrino spectrum can be well-fit by
\begin{equation}
f_\nu(E) \simeq \left(\frac{1.2 \, {\rm MeV}}{ \sqrt{M_5^{\rm HC}}}\right)^{-3} \, \frac{1}{F_2(2)} \frac{E^2}{e^{(E \sqrt{M_5^{\rm HC}} / 1.2 \, {\rm MeV}) - 2}+1}~,
\end{equation}
where
\begin{equation}
F_k(\eta_\nu) = \int_0^\infty \, \frac{x^k \, dx}{e^{x-\eta_\nu} +1} \,
\end{equation}
and the average neutrino energy is $\left<E_\nu \right> \simeq 4 \, (M_5^{\rm HC})^{-1/2}$ MeV.
The presence of a strong magnetic field or rapid rotation will delay the SMS collapse. Neutrinos produced near the Schwarzschild radius will then have a higher chance of escaping before the core moves through the event horizon. This results in the possibility of an increase in the emitted neutrino fluence by up to an order of magnitude, and an increase in neutrino energies by a factor of two compared to the case of a non-rotating and non-magnetized collapse scenario~\cite{Shi:1998nd}. The partition of energy among the neutrino species remains the same, however.
Despite the enormous neutrino luminosities from SMS collapse, the detection of this signal on large cosmological scales is unlikely~\cite{Shi:1998nd}. However, detection prospects are favorable if the redshift of SMS collapse event is $z \lesssim 0.2$ (i.e. $\sim 1$~Gpc distance). Since there exist many quasars and AGN at these redshifts, the collapse rate could be sufficiently high so as to be within reach of detection.
\subsection{Diffuse neutrino background}\label{sec:sms}
An isotropic background of redshifted neutrinos will be generated by the cosmological history of SMS collapse. Given the complete ignorance of the formation and collapse rate of such large stars, we adopt a phenomenological perspective in which we motivate a variety of different redshift-dependent collapse rates, and investigate the detection prospects for each.
The flux of diffuse neutrinos from SMSs can be computed from the neutrino emission spectrum and the collapse rate $R_{SMS}(z)$ via
\begin{equation}\label{eq:DNB}
\frac{d\phi}{ dE_{\nu}}(E_{\nu}) = \int \, dz \, \frac{R_{SMS}(z)}{H(z)} \, f_{\nu}( E_{\nu} (1+z)) \, ,
\end{equation}
where we adopt cosmological parameters consistent with the latest \textit{Planck-2018} measurements~\cite{Akrami:2018vks}.
In what follows, we adopt five different parameterizations of the collapse rate in order to obtain a rough estimation of uncertainty in the flux and spectral shape. The models assume
\begin{enumerate}
\item The collapse rate of SMSs traces the quasar formation rate. If we assume the typical quasar lifetime (which is much shorter than the Hubble time) is redshift-independent, then we can assume that the formation rate directly follows the quasar number density. We take this rate to be consistent with the results of~Ref.~\cite{shaver1996decrease,veron1998eso}, and call this model $R^Q$.
\item In order to asses the impact of additional redshift-dependent factors not directly included in the quasar formation rate, we consider two models in which $R^Q$ is re-scaled by a factor of $(1+z)^\alpha$. In order to understand extreme variations in this factor, we adopt $\alpha = \pm 3$, and denote each model by $R^{\pm 3}$.
\item As will be shown, $R^Q$ decreases dramatically at redshifts $z \sim 2$. Should SMSs be the origin of SMBHs, the collapse rate must extend to much larger redshifts. To account for this, we adopt a model which is consistent with the quasar formation rate at $z \leq 1.5$, and is flat at $1.5 \leq z \lesssim 20$, the upper cut-off taken to be roughly consistent with the onset of star formation. We call this model $R^{\rm flat}$.
\item Finally, we adopt a model in which SMSs are assumed to form predominately in metal-free environments at high redshifts. It has been suggested that low-metallicity environments could allow for the rapid cooling and formation of such objects, implying a preferential formation rate peaking near $z \sim 15$. We model this using a Gaussian distribution centered at $z = 15$ with a width $\Delta z = 1$. We call this model $R^{Pop3}$, as it would suggest these stars are among those first produced in the Universe (i.e. Pop-III stars, or perhaps stars produced by tidal disruption in dense star clusters).
\end{enumerate}
The aforementioned models are assumed to characterize only the redshift dependence of the collapse rate. In order to determine normalization of the rate, we assume that less than $10\%$ of the baryons have resided in SMSs. That is, we define the baryon density in SMSs to be
\begin{equation}
\rho_{SMS}= \int dt\, M \, \frac{R_{SMS}(z)}{(1+z)^{3}} \, .
\end{equation}
If these black holes do indeed serve as the seeds for supermassive black holes at the center of galaxies, it would be reasonable to estimate that approximately one SMS exists per galaxy, or equivalently $\rho_{SMS} \sim (M / 10^{10} M_\odot) \rho_b$. We use this to normalize the SMS collapse rate, and show the resultant histories, and the subsequent neutrino fluxes as observed here at Earth, in \Fig{fig:collapse}.
\begin{figure*}[tb]
\centering
\begin{subfigure}
\centering
\includegraphics[width=0.45\linewidth,trim={0 0 0 0},clip]{collapse_rate.pdf}
\end{subfigure}
\begin{subfigure}
\centering
\includegraphics[width=0.45\linewidth,trim={0 0 0 0},clip]{DSMSB_All.pdf}
\end{subfigure}
\caption{Historical SMS collapse rate [Left] and the resulting diffuse neutrino flux at Earth [Right] for each of the models considered in \Sec{sec:sms}. }
\label{fig:collapse}
\end{figure*}
\section{Large Direct Detection Experiments} \label{sec:dd}
\subsection{Experimental configurations}
\begin{table}[t]
\centering
\begin{tabular}{c|c|c|l}
\hline
\hline
Target & Mass & Threshold & Reference \\
& (tons) & (keV)& \\
\hline
Ar & 300 & 0.6 & ARGO~\cite{zuzel2017darkside,Aalseth:2017fik} \\
Xe & 50 & 0.7 & DARWIN~\cite{Aalbers:2016jon,Aprile:2016wwo} \\
Pb & 2.4 & 1.0 & RES-NOVA~\cite{Pattavina:2020cqc} \\
\hline
\hline
\end{tabular}
\caption{Considered detector configurations.}
\label{tab:exp_conf}
\end{table}
In this study, we consider detector configurations consistent with the proposed specifications of the upcoming direct DM detection experiments DARWIN~\cite{Aalbers:2016jon,Aprile:2016wwo}, using xenon (Xe) as a target material, and ARGO~\cite{zuzel2017darkside,Aalseth:2017fik} using argon (Ar) as a target material. These experiments are able to achieve considerable fiducial volume while also taking advantage of a keV-level energy threshold. In addition, we also consider\footnote{We note that low-background xenon and argon detectors have been in development for many years and the scalability of these setups has been established. The feasibility of Pb-based detector on a competitive scale is still to be demonstrated.} a configuration based on lead (Pb), following the recently proposed RES-NOVA~\cite{Pattavina:2020cqc} experiment for detection of core-collapse SN neutrinos via CE$\nu$NS. An overview of these configurations is listed in \Tab{tab:exp_conf}.
We assume the experiments are located at SNOLAB
(Sudbury, Canada), which is likely to host a number of next-generation direct detection experiments.
We stress, however, that this assumption does not strongly affect our conclusions.
The depth of this lab (6010 m.w.e.) ensures that backgrounds due to cosmogenic muons are highly suppressed.
Throughout this work, we will optimistically consider that experiments have perfect detection efficiency and energy resolution, and we adopt detection thresholds consistent with the targeted low-energy searches of each experiment. Furthermore, when considering neutrino coherent interactions with the nuclei, the expected background is assumed to arise exclusively from other neutrino sources\footnote{This assumption is in principle not fully realistic as, e.g., the ionization signal ``S2-only'' analyses of argon and xenon~\cite{Aprile:2019xxb} have unavoidable electronic backgrounds (but allow for lower signal thresholds). However, since the SMS burst signal occurs over a period of $\sim \mathcal{O}(1 {\rm s})$, time correlations should easily allow one to differentiate this signal from background.}. This assumption allows us to treat all analyses on an equal footing, and provide general results independent of specific configurations that could change in the future.
\subsection{Scattering rates}
Given a neutrino flux $\phi_{\nu} (E_{\nu})$ the resulting differential event rate per unit time and detector mass as a function of the recoil energy $E_R$, per unit time and mass $m_I$ of a target nuclide $I$ in a detector is given by
\begin{equation} \label{eq:nu_diff_rate}
\dfrac{d R_{\nu}^I}{d E_R} = \dfrac{C_I}{m_I} \int_{E_{\nu}^{\rm min}} \phi_{\nu} (E_{\nu}) \dfrac{d \sigma^I (E_{\nu}, E_R)}{d E_R} d E_{\nu}~,
\end{equation}
where $d \sigma^I (E_{\nu}, E_R) / d E_R$ is the coherent neutrino-nucleus scattering differential cross-section and $C_I$ is the fraction of nuclide $I$ in the material. In case several nuclides are present, individual contributions are summed.
For a target mass $m_I$ at rest, the minimum neutrino energy required to produce a recoil of energy $E_R$ is
\begin{equation}
E_{\nu}^{\rm min} = \sqrt{\dfrac{m E_{R}}{2}}~.
\end{equation}
The maximum recoil energy due to a collision with a neutrino of energy $E_{\nu}$ is
\begin{equation}
E_R^{\rm max} = \dfrac{2 E_{\nu}^2}{m + 2 E_{\nu}}~.
\end{equation}
\begin{figure*}[tb]
\centering
\begin{subfigure}
\centering
\includegraphics[width=0.45\linewidth,trim={0 0 0 0},clip]{SMS_evts.pdf}
\end{subfigure}
\begin{subfigure}
\centering
\includegraphics[width=0.45\linewidth,trim={0 0 0 0},clip]{SMS_evts_tonn_vs_recoil.pdf}
\end{subfigure}
\caption{[Left] Number of events detected from the burst of a SMS with mass $M = 10^{5} M_{\odot}$ for each experimental configuration as a function of distance. The shaded regions indicate the potential enhancement in the signal that may arise if the SMS has a non-negligible rotation or magnetic field. Shown for comparison are distance markers denoting the location of the galactic center, the edge of the Milky Way, and Andromeda. [Right] Number of events per tonne of detector mass for each target element as a function of threshold energy $E_{th}$, computed assuming $M = 10^5 \, M_\odot$ and $d = 1 $ [Mpc]. The adopted experimental thresholds are shown with vertical lines.}
\label{fig:burstevents}
\end{figure*}
\subsection{CE$\nu$NS}
The Standard Model coherent-scattering neutrino-nucleus cross-section is given by~\cite{Freedman:1977xn}
\begin{equation}\label{eq:nucxsec}
\dfrac{d \sigma^I (E_{\nu}, E_R)}{d E_R} = \dfrac{G_f^2 m_I}{4 \pi} Q_w^2 \left(1 - \dfrac{m_I E_R}{2 E_{\nu}^2}\right) F_{I}^2 (E_R)~,
\end{equation}
where $m_I$ is the target nuclide mass, $G_f$ is Fermi coupling constant, $F_{I}(E_R)$ is the form factor, which we take to be the Helm form factor \cite{Helm:1956zz}, $Q_w = [(1 - 4 \sin^2 \theta_{\rm W}) Z_I-N_I]$ is the weak nuclear charge, $N_I$ is the number of neutrons, $Z_I$ is the number of protons, and $\theta_{\rm W}$ is the Weinberg angle. Since $\sin^2 \theta_{\rm W} = 0.223$ \cite{Patrignani:2016xqp}, the coherent neutrino-nucleus scattering cross-section follows an approximate $N_I^2$ scaling.
\section{Supermassive Star Neutrino Signal Detection} \label{sec:signal}
In Fig.~\ref{fig:Nu_flux} we depict the expected neutrino flux for SMS collapse at a distance of 0.1-1 Mpc and with varying HC mass and inclusion of rotation/magnetic fields. In Fig.~\ref{fig:burstevents} we illustrate the expected number of events from the collapse of a SMS as a function of explosion distance. In this case, we illustrate the enhancement effect (shaded band) that may arise should the SMS star rotate or have strong magnetic fields. Fig.~\ref{fig:burstevents} shows the event rate normalized by the fiducial volume as a function of the detection threshold, highlighting that lead and xenon will benefit particularly from lowering the detection threshold. Note that the adopted thresholds are shown with the colored vertical lines.
In Fig.~\ref{fig:diffuse} we illustrate the event rate produced in a xenon-based experiment by the diffuse SMS neutrino background. Various background neutrino sources are shown for comparison. We expect no more than one event will be detected using the experimental configurations listed in \Tab{tab:exp_conf}, implying that it will be a difficult task to disentangle the diffuse background from the other neutrino sources. Nevertheless, DSMSB will contribute to the irreducible background in the searches for dark matter. For neutrino sources with a well-defined spectrum and flux, this irreducible background may be partially circumvented via background subtraction techniques; this is not the case, however, for the diffuse neutrino flux from SMSs. We now turn our attention toward addressing the potential difficultly that could arise from such a background in the search for dark matter.
\begin{figure}[tb]
\centering
\includegraphics[width=1\columnwidth]{Nu_Recoils_Xenon.pdf}
\caption{Detection of diffuse supermassive star neutrino background (DSMSB), assuming a xenon target and experiment located at SNOLAB. DSMSB signal is shown for $R^Q$ model (solid) and the $R^{\rm flat}$ model (dashed), with shaded region extending down to the signal detection obtained for the $R^{\rm Pop3}$ model. Contributions for solar (hep), geo ($^{238}$U,$^{232}$Th) and reactor neutrinos are displayed (see Ref. \cite{Gelmini:2018ogy} for details). }
\label{fig:diffuse}
\end{figure}
\section{Dark Matter and the Diffuse Neutrino Background} \label{sec:darkmatter}
Individual collapses of super-massive stars are unlikely to obstruct the search for dark matter, as they will typically generate multiple nuclear recoils within a time window of $t \lesssim \mathcal{O}({\rm few})$ seconds. The diffuse background on the other hand has no strong time correlation, and in analogy to the effect of the diffuse supernova background (DSNB), this will necessarily contribute to the irreducible background in the direct detection searches for DM. We stress, however, that this will likely be a sub-dominant effect to other backgrounds.
The extent to which CE$\nu$NS inhibits DM searches has been discussed extensively in the literature within the context of an irreducible neutrino background constituting a ``neutrino floor'' (e.g.~\cite{Billard:2013qya,Monroe:2007xp,Gelmini:2018ogy,OHare:2020lva}). This question is often posed in the following manner: What exposure is required in order for an experiment to identify a particular DM candidate (with a well-defined mass and scattering cross section) at the statistical confidence level of $X \sigma$ (where $X$ is often taken to be $3$)? For a particular model of DM, and for a fixed experimental exposure, this defines a ``discovery floor''. The extent to which this discovery floor scales with exposure is critically dependent upon the level of degeneracy between the recoil spectrum of DM interactions and neutrinos.
The limitations on the DM discovery potential could prove rather difficult to quantify, because the associated SMS diffuse neutrino energy spectrum is determined by the assumed SMS collapse rate. In turn, this is a completely unknown function of redshift. As discussed previously, it is reasonable to conjecture that the SMS redshift-dependent collapse rate could be strongly related to the quasar and AGN formation rate. However, this need not be the case and possible deviations from such scaling can lead to significant differences in the shape of the resulting scattering rate within the experiments. Furthermore, the normalization of the SMS collapse rate contains only an upper limit, which we can estimate by ensuring no more than $\sim 10\%$ of the baryons have resided in SMS.
In order to demonstrate the impact of the diffuse SMS background, we plot in \Fig{fig:diffuse} a comparison of the nuclear recoil event rate produced by the diffuse SMS background and that from solar, geo, and reactor neutrinos. We show both the $R^Q$ model (blue, solid) and the $R^{flat}$ model (blue, dashed), and we shade down to the event rate produced by the $R^{Pop3}$ model (not shown). While the rate never exceeds those coming from known neutrino sources, it does become sizeable at low energies. In \Fig{fig:bestfit}, assuming for illustration the idealized scenario that other backgrounds can be subtracted or suppressed and SMS collapses pose the dominant background, we fit the event rate arising from the $R^Q$ model assuming dark matter interacts with nuclei through a spin-independent contact interaction (SI), a electric dipole (ED), a magnetic dipole (MD), or a pseudo-scalar contact interaction (PS) (see Ref.~\cite{Gelmini:2018gqa} for the details of each interaction). In~\Tab{Tab:bf} we display the approximate DM masses and cross-sections that would be recovered if the diffuse SMS background was mistakenly interpreted in the context of DM. We observe that the diffuse SMS background could further hamper the search for DM candidates with masses $m \lesssim 5$ GeV. Both \Fig{fig:bestfit} and the fit performed in \Tab{Tab:bf} assume that the other low energy neutrino backgrounds (e.g.~solar, reactor, geoneutrinos) can be effectively suppressed, which might be a challenging experimental task. Given both experimental and theoretical uncertainties (related \emph{e.g.} to the experimental response or the difficulties in computing the spectrum of other neutrino sources), the diffuse SMS background is unlikely to be the dominant inhibitor to the DM searches in the low energy regime.
If SMS neutrino indeed eventually becomes a sizable effect in the context of DM searches, there are a number of possible ways in which this could be circumvented. First, experiments that run for multiple years could search for the annual modulation of the scattering rate, induced by the time-variation between the motion of Earth and DM rest frame. Uncertainties in the DM-nucleon interaction~\cite{DelNobile:2015tza,DelNobile:2015rmp} and the astrophysical distribution of DM~\cite{DelNobile:2015nua}, however, can significantly complicate the amplitude and phase of the annual modulation. In the absence of backgrounds, using the annual modulation to differentiate between DM models using only one detector typically requires $N_{\rm evts} \gg \mathcal{O}(10^3)$ events (neglecting astrophysical uncertainties) ~\cite{Witte:2016ydc}. With backgrounds, this number is likely orders of magnitude higher. Consequently, such a technique will not prove easy. A better understanding of halo uncertainties~\cite{OHare:2018trr,Buch:2019aiw,OHare:2019qxc} or the use of novel analysis methods~\cite{Gondolo:2017jro,Gelmini:2017aqe} may improve the situation. Alternatively, directional detection could allow one to efficiently remove isotropic backgrounds, leaving only the dark matter scattering rate~\cite{Mayet:2016zxu,OHare:2017rag,Vahsen:2020pzb}.
\begin{figure}
\begin{center}
\includegraphics[width=1\columnwidth]{best_fit_recoils}
\caption{ Comparison of event rates for recoil spectra induced in a Xenon target from a DSMSB with the recoil spectra arising in various DM interaction models (see text). The best fit values for the DM parameters are given in table ~\ref{Tab:bf}. }
\label{fig:bestfit}
\end{center}
\end{figure}
\begin{table}[t]
\centering
\begin{tabular}{c|c|c}
\hline
\hline
Model & Mass (GeV) & $\sigma$ (cm$^{2}$) \\
\hline
SI & 4.34 & 1.02 $\times$ $10^{-50}$ \\
ED & 4.15 & 4.41 $\times 10^{-48} $ \\
MD & 3.79 & 2.18 $\times 10^{-42} $ \\
PS & 3.79 & 1.42 $\times 10^{-41} $ \\
\hline
\hline
\end{tabular}
\caption{ Best-fit mass and cross section for dark matter scattering with nuclei via a spin-independent (SI), electric dipole (ED), magnetic dipole (MD), or pseudoscalar (PS) interaction. Fits are to the $R^Q$ model of SMS collapse rate.}
\label{Tab:bf}
\end{table}
\section{Conclusions} \label{sec:con}
Supermassive stars with mass $M \gtrsim 3\times 10^4 M_\odot$ are expected to directly collapse to black holes via the Feynman-Chandrasekhar instability. While no such stars have yet been directly observed, supermassive black holes at redshifts as high as $z \sim 7$ suggest at the least that initial seed black holes with large masses might be required. This also serves as a rationale for exploring the consequences of the existence of progenitor stars in this mass range at redshifts as early as $z \sim 15$. Should these objects exist, their collapse can yield a broad array of observable signatures, including gamma-rays, gravitational waves, and neutrinos. In this paper we have analyzed the extent to which neutrinos emitted from the collapse of such objects could be detected via coherent neutrino scattering, focusing on massive direct dark matter experiments.
We have demonstrated that large scale underground experiments built for the purpose of detecting dark matter might be capable of identifying the collapse of individual supermassive stars in nearby galaxies, such as in Andromeda. A diffuse and isotropic neutrino background will also be produced from the cumulative historical collapse of such objects. We have analyzed a variety of potential redshift-dependent collapse rates that may arise, e.g., if the SMS collapse rate follows the AGN formation rate, or if SMSs are preferentially formed in metal-free environments, as would occur at higher redshifts (e.g. near $z \sim 15$). While we have focused on comparison of signal with other neutrino flux sources, future work on non-neutrino background suppression is essential for signal discrimination.
While the existence of SMSs has not been definitively established, such objects provide a simple and plausible explanation of the origin of the supermassive black holes observed to reside at the centers of galaxies, or a least the seed black holes needed to build them up by redshift $z\sim 7$. The only way to truly reveal the existence of these objects is to observe them. The neutrino flux produced from the collapse of SMSs offers a particularly intriguing channel in which to test their existence, as the neutrino energy spectra are non-thermal and easily distinguishable from other sources. Current direct dark matter experiments are already designed in a manner that is ideal for the search of such neutrino flux, with near-future experiments capable of probing the collapse of such objects on extra-galactic scales.
\acknowledgments
The work of S.J.W. was supported by a Juan de la Cierva Formacion fellowship, and is part of a project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 864035 – UnDark). The work of V.M. was supported by CONICYT PFCHA/DOCTORADO BECAS CHILE/2018 - 72180000. The work of V.T. was supported by the U.S. Department of Energy (DOE) Grant No. DE-SC0009937. V.T. was also supported by the World Premier International Research Center Initiative (WPI), MEXT, Japan. G.M.F. and V.T. would like to thank Kavli IPMU, U. of Tokyo for hospitality where this work was initiated. G.M.F. acknowledges NSF Grant No. PHY-1914242 at UCSD and the NSF N3AS Physics Frontier Center, NSF Grant No. PHY-2020275, and the Heising-Simons Foundation (2017-228).
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Don't waste tomorrow, drink responsibly. It is an offence to supply alcohol to a person under the age of 18 years. Penalties apply.
Enjoy great Giancarlo Coffee, superb milkshakes and great light dining.
Serving up an exciting array of dishes 7 days a week, including a regular changing special board and dish of the month.
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Robert Pettena
Bio - CV
The Stones are My Ideas of Imagination
Curated by Mike Watson, Museo del Marmo di Carrara
An exhibition called "The Stones are my Ideas of Imagination" opened on the 15th December in the Carrara Marble Museum. It was the result of a residency and workshop curated by Mike Watson for the Database Project, organised by Ars Gratia Artis with the support of the Carrara Town Council.
Mike Watson, resident curator of the Nomas Foundation in Rome, invited the artists Graham Hudson, Robert Pettena and Andrew Rutt to work in Carrara between September and November.
During this period Pettena devised a new way of organising a workshop. The artist chose a section of the general public to work with, but then he ran the workshop horizontally. He didn't instruct the public but rather used the result of the public's interaction to structure his work.
The title of the exhibition: The Stones are My Ideas of Imagination, is a reference to William Blake. The workshop material became the basis for the new works displayed in the exhibition. Pettena went to the Germinal and Fiaschi anarchist clubs in the centre of Carrara to talk to the miners who work in the marble quarries of Collonata, Torano and Miseglia, in order to explore the concept of anarchy.
© Robert Pettena 2019
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{"url":"https:\/\/math.stackexchange.com\/questions\/2527021\/how-to-figure-out-if-there-is-an-actual-horizontal-tangent-without-a-graph","text":"# How to figure out if there is an actual horizontal tangent without a graph\n\nThere is this practice problem that asks to determine the points at which the graph of $y^4=y^2-x^2$ has a horizontal tangent.\n\nSo I did implicit differentiation to find that\n\n$$\\displaystyle\\frac{dy}{dx} = \\frac{-x}{2y^3-y}$$\n\nTo find the horizontal tangent, I set $\\frac{dy}{dx}=0$ and solved for $x$:\n\n\\begin{align} \\displaystyle\\frac{dy}{dx} &= 0 \\\\ \\frac{-x}{2y^3-y} &= 0 \\\\ -x &= 0 \\\\ x &= 0 \\end{align}\n\nThen I substituted $x=0$ into the equation of the curve:\n\n\\begin{align} y^4&=y^2-(0)^2 \\\\ 0 &= y^4 - y^2\\\\ 0 &= y^2(y+1)(y-1) \\\\ y&=-1,\\,0,\\,1 \\end{align}\n\nI concluded that the points $(0,0)$, $(0,-1)$, and $(0,1)$ were the points with a horizontal tangent.\n\nHowever, when I graphed this using Desmos, it turns out that the point at $(0,0)$ did not look like it has horizontal tangent.\n\nGraph of y^4=y^2-x^2\n\nHow would I have been able to figure this out without graphing it?\n\n\u2022 At $y = 0$ derivative is not defined! Simply look at your expression for derivative for that. \u2013\u00a0SJ. Nov 19 '17 at 4:25\n\u2022 Ah! did not even consider that. \u2013\u00a0user502227 Nov 19 '17 at 5:26\n\nOne of the answers is $(0,0)$. However, at this point, $\\frac{dy}{dx}$ attains $\\frac{0}{0}$ form. So, I think you should not consider the point $(0,0)$.\nIf $y^4=y^2-x^2$, then diffing implicitly, $4y^3y' =2yy'-2x$ or $2y^3y' =yy'-x$.\nIf $y' = 0$, then $x = 0$.\nPutting this in, $y^4 = y^2$ so the possible values are $y = 0, \\pm 1$.\nAt $x=y=0$, suppose $y' = c$. For small $x$ and $y$, $y \\approx cx$ so $c^4x^4 \\approx c^2x^2-x^2$ or, dividing by $x^2$, $c^4 x^2 \\approx c^2-1$. Since $c^4x^2$ is small, $c^2-1$ must also be small so $c^2 \\approx 1$ so $c \\approx \\pm 1$.\nThis means that, at $(0, 0)$, $y' = \\pm 1$.","date":"2020-03-31 08:13:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 2, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 1.0000073909759521, \"perplexity\": 194.2141480219702}, \"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-2020-16\/segments\/1585370500331.13\/warc\/CC-MAIN-20200331053639-20200331083639-00201.warc.gz\"}"}
| null | null |
аль-Малик аль-Музаффар Шамс ад-Дин Юсуф ибн Умар (1222, Мекка — 1295, Таиз) — султан средневекового южно-аравийского государства Расулидов в 1250—1295 годах.
Происхождение и приход к власти
Шамс ад-Дин Юсуф происходил из одного из тюрко-огузских кланов, находившихся на службе у айюбидских султанов Йемена. Он родился в 1222 году. Его отец Нур ад-Дин Умар ибн Али ибн Расул занимал высокое положения при последнем султане йеменской ветви Айюбидов аль-Малике аль-Масуде Юсуфе (ум. 1229), исполняя обязанности его заместителя (наиба). Оставленный наместником султана аль-Масуда Юсуфа, который собрал всё ценное и покинул Йемен, отправившись в Сирию, Нур ад-Дин Умар в 632 году Хиджры (1234/1235 год) провозгласил себя новым султаном под именем аль-Малика аль-Мансура, получив официальное признание аббасидского халифа в качестве независимого суверена Йемена.
Примечания
Литература
Музаффар
Правители Азии XIII века
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\section{Introduction}
\setcounter{equation}{0}
The issue of color confinement in quantum chromodynamics [QCD], widely
accepted as the theory of the strong interaction, is a longstanding problem
both conceptually and quantitatively. One necessary prerequisite for color
confinement is the conservation of color charge: a `leaky' system clearly
cannot be confining. Within the framework of functional field-theoretic
methods in nonabelian theories (i.e., QCD), expressed in terms of local
Green's functions, the Slavnov--Taylor identities
\cite{Taylor:1971ff,Slavnov:1972fg} are the consequence of this charge
conservation.
The Slavnov--Taylor identities come in various guises and have been applied
to many problems, in particular to the nonperturbative study of the Dyson--Schwinger
equations. For example, in quantum electrodynamics [QED] where the
Slavnov--Taylor identities reduce to the Ward identity, their study led to
the Ball--Chiu vertex \cite{Ball:1980ay} with its subsequent improvement,
the Curtis--Pennington vertex \cite{Curtis:1990zs}, which have been used to
study dynamical mass generation (see for example
Refs.~\cite{Pennington:1998cj,Alkofer:2000wg}). In axial gauge Yang--Mills
theory, the Slavnov--Taylor identities \cite{Kummer:1974ze} have been used
to study the gluon propagator Dyson--Schwinger equation, with the inference that the
propagator diverges as $1/q^4$ in the infrared (one potential signal for
confinement) \cite{Baker:1980gf}. This result is however, not without
ambiguity \cite{West:1982gg}. In Landau gauge Yang--Mills theory, the
Mandelstam approximation \cite{Mandelstam:1979xd} (see also
Ref.~\cite{BarGadda:1979cz}), where ghost contributions to the three-gluon
vertex Slavnov--Taylor identity are neglected, one can also obtain an
infrared enhanced gluon propagator. The above examples can be colloquially
referred to as Schwinger--Dyson studies (these are early studies and a quirk
of history has subsequently resulted in the reordering of the names) and
they share two common features: the utilization of the Slavnov--Taylor
identities and either require no, or assume no ghost contributions.
In attempting to further the Mandelstam approximation to Landau gauge
Yang--Mills theory by including the ghost contributions, a startling new
phenomenon emerged --- infrared ghost dominance \cite{von Smekal:1997vx}
(see also \cite{Alkofer:2000wg,Fischer:2006ub} for contemporary reviews).
The starting point was an approximation to the Slavnov--Taylor identity for
the ghost-gluon vertex. This approximation was shown to be inconsistent
with perturbation theory \cite{Watson:1999ha}, but this turned out to be
unimportant: using general arguments one can show that the tree-level part
of the ghost-gluon vertex is the important ingredient
\cite{Watson:2001yv}. Indeed, there are indications that the dressing of
the ghost-gluon vertex function is only slight
\cite{Schleifenbaum:2004id}. The results of this infrared ghost dominance
can be summarized as follows: the gluon propagator is infrared suppressed,
whereas the ghost propagator is infrared enhanced. In the context of
confinement, the results imply positivity violation (formalized in the
Oehme--Zimmermann superconvergence relations \cite{Oehme:1979ai}) and are
in agreement with the Kugo--Ojima \cite{Kugo:1979gm} and Gribov--Zwanziger
\cite{Gribov:1977wm,Zwanziger:1995cv,Zwanziger:1998ez} confinement
scenarios. These results have been contested (see for example
Ref.~\cite{Boucaud:2008ky}) by appealing to the Slavnov--Taylor identity
for the three-gluon vertex, applying the further assumption that certain
dressing functions that compose the ghost-gluon vertex kernel remain
infrared finite and with the conclusion that the ghost propagator is not
infrared enhanced. The existence of such additional solutions has been
verified in Ref.~\cite{Fischer:2008uz} (this type of solution was
incidentally also observed in Ref.~\cite{Epple:2007ut} within the canonical
approach to Coulomb gauge Yang--Mills theory \cite{Feuchter:2004mk}): it was
concluded however that the Slavnov--Taylor identities alone cannot
discriminate between the two types of solution and that one needs explicit
additional input (the authors arguing that the infrared enhanced ghost
solution is preferred).
From the above (admittedly historically incomplete discussion of) works,
several lessons emerge. Firstly, the Slavnov--Taylor identities play a
central role in nonperturbative Dyson--Schwinger studies since they provide information
about the higher $n$-point functions that enter the Dyson--Schwinger equations. Second,
in all the above studies, the Slavnov--Taylor identities relate the
$4$-vector contraction of a vertex to some combination of inverse
propagators (proper two-point functions). This contraction means that only
part of the vertex is constrained by the identity and it is not necessarily
that part which enters the Dyson--Schwinger equations. One either assumes that the
`transverse' vertex (i.e., that part not constrained by the Slavnov--Taylor
identity) can be neglected or considers more sophisticated input to complete
the closure of the system such that the Dyson--Schwinger equations can be solved (e.g.,
the Curtis--Pennington vertex uses multiplicative renormalizability
constraints \cite{Curtis:1990zs}). Third, even if the Slavnov--Taylor
identities are applied to close the Dyson--Schwinger equations, the solution (and in
particular the infrared behavior) may not be uniquely specified and requires
additional input, as demonstrated in Ref.~\cite{Fischer:2008uz}. These
latter two points are emphatically not intended as a criticism of the Dyson--Schwinger
approach, merely as reminders of some of the problems encountered
(confinement is after all, not entirely trivial) and in this paper, we will
see what Coulomb gauge has to say about the subject.
The importance of Coulomb gauge in studying nonperturbative QCD was
recognized early on, as was the inherent difficulty in technical calculation
for such noncovariant gauges \cite{Abers:1973qs}. The significance of
Coulomb gauge is based in the observation that in this gauge, the system
reduces naturally to physical degrees of freedom (explicitly demonstrated
in Ref.~\cite{Zwanziger:1998ez}). Given that the Slavnov--Taylor identities
are the expression of charge conservation as applied to Green's functions,
there exists a clear motivation to derive them in Coulomb gauge. In
addition, there has recently been much technical progress in the Coulomb
gauge functional formalism that provides the background to the study of the
Dyson--Schwinger equations: the derivation of the Dyson--Schwinger equations themselves
\cite{Watson:2006yq,Watson:2007vc}, their one-loop perturbative analysis
\cite{Watson:2007mz,Watson:2007vc,Popovici:2008ty}, the resolution of more
formal aspects of the (incomplete) gauge-fixing and the existence of a
conserved, vanishing total charge (and the absence of the infamous Coulomb
gauge energy divergences) \cite{Watson:2007fm,Reinhardt:2008pr}. With the
Slavnov--Taylor identities, most of the components required for at least an
initial study of the Dyson--Schwinger equations are in place.
In this paper, we thus derive the Slavnov--Taylor identities of Coulomb
gauge Yang--Mils theory in the second order functional formalism and discuss
some of their immediate consequences. In Section~2, the basic formulation
of the functional formalism and the Gauss--BRST invariance is presented.
The Slavnov--Taylor identities for the two-point functions are derived in
detail in Section~3. In Section~4, the identities for the vertex functions
are derived and it is shown how these identities form closed sets such that
the temporal Green's functions may be obtained as their solution. Section~5
is concerned with the possibility of extracting unambiguous information
about Green's functions from the identities in the infrared. The paper
concludes with a discussion of various aspects of the identities. For the
convenience of the reader, selected relevant results from earlier works are
listed in Appendix~\ref{app:decomp}. Lengthy configuration space
expressions, important to the derivation of the Slavnov--Taylor identities
but not to the narrative of the paper are relegated to
Appendix~\ref{app:eqlist}.
\section{Functional formalism and Gauss--BRST invariance}
\setcounter{equation}{0}
Let us begin by considering Yang--Mills theory in the functional formalism.
We will use the notation and conventions established in
\cite{Watson:2006yq,Watson:2007vc}. We work in Minkowski space with metric
$g_{\mu\nu}=\mbox{diag}(1,-\vec{1})$. Roman subscripts ($i,j,\ldots$)
denote spatial indices and superscripts ($a,b,\ldots$) denote color
indices. We will often write configuration space coordinates ($x,y,\ldots$)
as subscripts where no confusion arises.
The Yang--Mills action is defined as
\begin{equation}
{\cal S}_{YM}=\int\dx{x}\left[-\frac{1}{4}F_{\mu\nu}^aF^{a\mu\nu}\right]
\end{equation}
where the (antisymmetric) field strength tensor $F$ is given in terms of the
gauge field $A_{\mu}^a$:
\begin{equation}
F_{\mu\nu}^a
=\partial_{\mu}A_{\nu}^a-\partial_{\nu}A_{\mu}^a+gf^{abc}A_{\mu}^bA_{\nu}^c.
\end{equation}
In the above, the $f^{abc}$ are the structure constants of the $SU(N_c)$
group whose generators obey $\left[T^a,T^b\right]=\imath f^{abc}T^c$. The
Yang-Mills action is invariant under a local $SU(N_c)$ gauge transform
characterized by the parameter $\th_x^a$:
\begin{equation}
U_x=\exp{\left\{-\imath\th_x^aT^a\right\}}
\end{equation}
such that for infinitesimal $\th_x^a$, the gauge field transforms as
\begin{equation}
A_{\mu}^a\rightarrow A_{\mu}^{\prime a}
=A_{\mu}^a-\frac{1}{g}\hat{D}_{\mu}^{ab}\th^b
\end{equation}
with the covariant derivative in the adjoint color representation given by
\begin{equation}
\hat{D}_{\mu}^{ac}=\delta^{ac}\partial_{\mu}+gf^{abc}A_{\mu}^b.
\end{equation}
In terms of the temporal and spatial components, the above transform reads
(we rewrite the temporal component $A_0$ as $\sigma$)
\begin{eqnarray}
\sigma^a\rightarrow\sigma^{\prime a}
&=&\sigma^a-\frac{1}{g}\partial_{0}\th^a-f^{abc}\sigma^b\th^c,\nonumber\\
\vec{A}^a\rightarrow\vec{A}^{\prime a}
&=&\vec{A}^a+\frac{1}{g}\div\th^a-f^{abc}\vec{A}^b\th^c.
\end{eqnarray}
Consider the functional integral
\begin{equation}
Z=\int{\cal D}\Phi\exp{\left\{\imath{\cal S}_{YM}\right\}}
\end{equation}
where ${\cal D}\Phi$ denotes the functional integration measure for the
collection of all fields. Since the action is invariant under gauge
transformations, $Z$ is divergent by virtue of the integration over the
gauge group. To overcome this problem we use the Faddeev-Popov technique
and introduce a gauge-fixing term along with an associated ghost term
\cite{IZ}. Using a Lagrange multiplier field, $\lambda^a$, to implement the
gauge-fixing, in Coulomb gauge ($\s{\div}{\vec{A}}^a=0$) we can then write
\begin{equation}
Z=\int{\cal D}\Phi\exp{\left\{\imath{\cal S}_{YM}+\imath{\cal S}_{FP}\right\}},\;\;\;\;
{\cal S}_{FP}=\int d^4x\left[-\lambda^a\s{\vec{\nabla}}{\vec{A}^a}
-\ov{c}^a\s{\vec{\nabla}}{\vec{D}^{ab}}c^b\right]
\label{eq:zgf}
\end{equation}
where $\ov{c}^a$ and $c^b$ are the Grassmann-valued ghost fields. The new
term in the action, ${\cal S}_{FP}$, is invariant under the
Gauss--Becchi--Rouet--Stora--Tyutin [Gauss-BRST] transform
\cite{Zwanziger:1998ez} whereby the infinitesimal, spacetime dependent
gauge parameter $\th_x^a$ is factorized into two Grassmann-valued
components: $\th_x^a=c_x^a\delta\lambda_t$, where $\delta\lambda_t$ is the
\emph{time-dependent} infinitesimal variation (not to be confused with the
colored Lagrange multiplier field $\lambda^a$). The Gauss-BRST transform is
peculiar to Coulomb gauge --- the time-dependent variation is allowed
simply because the gauge-fixing does not involve any explicit
time-derivatives. The variations of the new fields read:
\begin{equation}
\delta\ov{c}_x^a=\frac{1}{g}\lambda_x^a\delta\lambda_t,\;\;\;\;
\delta c_x^a=-\frac{1}{2} f^{abc}c_x^bc_x^c\delta\lambda_t,\;\;\;\;
\delta\lambda_x^a=0.
\end{equation}
By including a source term, the generating functional is given by
\begin{equation}
Z[J]=
\int{\cal D}\Phi\exp{\left\{\imath{\cal S}_{YM}+\imath{\cal S}_{FP}+\imath{\cal S}_s\right\}}
\end{equation}
where
\begin{equation}
{\cal S}_s=\int d^4x\left[\rho^a\sigma^a+\s{\vec{J}^a}{\vec{A}^a}+\ov{c}^a\eta^a
+\ov{\eta}^ac^a+\xi^a\lambda^a\right].
\end{equation}
Regarding the Gauss-BRST transform as a change of integration variables
under which the generating functional is invariant, noting that the
associated Jacobian factor is trivial \cite{Watson:2006yq} and only the
source term varies, we deduce that
\begin{eqnarray}
0&=&\left.\int{\cal D}\Phi\frac{\delta}{\delta\left[\imath\delta\lambda_t\right]}
\exp{\left\{\imath{\cal S}_{YM}+\imath{\cal S}_{FP}
+\imath{\cal S}_s+\imath\delta{\cal S}_s\right\}}\right|_{\delta\lambda_t=0}
\nonumber\\
&=&\int{\cal D}\Phi\exp{\left\{\imath{\cal S}_{YM}+\imath{\cal S}_{FP}+\imath{\cal S}_s\right\}}
\int d^4x\delta(t-x_0)\times
\nonumber\\&&
\left[-\frac{1}{g}(\partial_x^0\rho_x^a)c_x^a+f^{abc}\rho_x^a\sigma_x^bc_x^c
-\frac{1}{g}J_{ix}^a\nabla_{ix}c_x^a+f^{abc}J_{ix}^aA_{ix}^bc_x^c
+\frac{1}{g}\lambda_x^a\eta_x^a
+\frac{1}{2} f^{abc}\ov{\eta}_x^ac_x^bc_x^c\right].
\label{eq:wtid0}
\end{eqnarray}
This equation is the starting point for deriving the Slavnov--Taylor identities. Notice
the $\delta(t-x_0)$ constraint, which arises because of the time-dependent
variation $\delta\lambda_t$ and is characteristic to the Gauss-BRST transform.
It leads eventually to a nontrivial energy injection into the Slavnov--Taylor identities
which is not present in the covariant gauge case.
It is pertinent at this stage to discuss some nontrivial points associated
with the above. Coulomb gauge is in fact not a complete gauge. Even after
adding the gauge-fixing terms, the functional integral, \eq{eq:zgf}, still
contains zero-modes generated by purely temporal gauge transforms (and for
that matter, global transforms too). Explicitly separating these temporal
zero-modes within the Faddeev--Popov procedure, one can formally show
(within the first order formalism, but since this is connected to the
second order formalism used here via identities, the same conclusions
apply) that the total color charge of the system is constrained to be
conserved and vanishing \cite{Reinhardt:2008pr}. The above expression,
\eq{eq:wtid0}, is the dynamical statement of color charge conservation ---
it shows how the external sources must be arranged in accordance with the
gauge symmetry of the underlying theory. In the two differing contexts,
the incompleteness of the gauge plays a central role: the formal isolation
of the zero-modes gives the total charge constraint and the time-dependent
variation ($\delta\lambda_t$) will give (later) the extra temporal scale. We will
discuss this connection at the end.
Note also that the presence of the temporal zero-modes precludes deriving a
similar expression to \eq{eq:wtid0} above by considering a purely
time dependent (spatially independent) full gauge transform, i.e.,
$\th(t,\vec{x})\rightarrow\th(t)$, since the functional integration is
ill-defined. In such a case, the identity gives a resultant $\Gamma_{\sigma\si}$
Green's function (see later for details of the definition) with a dressing
function that vanishes at zero momentum and this would hold even at
tree-level which clearly contradicts the perturbative behavior for which
the dressing function is unity (see Ref.~\cite{Watson:2007vc} or
Appendix~\ref{app:decomp}). The reason for this is that when considering
the gauge-variant Green's functions, the integration over the gauge group
associated with the zero-modes averages such quantities to zero. For the
Gauss-BRST transform considered here, the spatial dependence of the
ghost-field ensures that this problem is not encountered (since the
gauge-fixing part of the action can be rewritten in terms of spatial
derivatives of the ghost-field, they can always be implicitly defined as
having no spatially constant component).
So far, the generating functional, $Z[J]$, generates all Green's functions,
connected and disconnected. The generating functional of connected Green's
functions is $W[J]$ where
\begin{equation}
Z[J]=e^{W[J]}.
\end{equation}
We define the classical fields to be
\begin{equation}
\Phi_\alpha=\frac{1}{Z}\int{\cal D}\Phi\,\Phi_\alpha\exp{\imath{\cal S}}
=\frac{1}{Z}\frac{\delta Z}{\delta\imath J_\alpha}
\end{equation}
(we use the same notation for both the classical and quantum fields since no
confusion will arise). The generating functional of proper Green's
functions is the effective action, $\Gamma$, which is a functional of the
classical fields and is defined through a Legendre transform of $W$:
\begin{equation}
\Gamma[\Phi]=W[J]-\imath J_\alpha\Phi_\alpha.
\label{eq:legtran}
\end{equation}
In the above, we use a compact notation for the sources and fields: a
generic field is denoted $\Phi_\alpha$, with source $J_\alpha$ and with the
index $\alpha$ standing for all attributes of the field in question (including
its type); further, summation over all discrete indices and integration over
all continuous arguments is implicitly understood. We introduce a bracket
notation for derivatives of $W$ with respect to sources and of $\Gamma$ with
respect to classical fields (no confusion arises since the two sets of
derivatives are never mixed):
\begin{equation}
\ev{\imath J_\alpha}=\frac{\delta W}{\delta\imath J_\alpha},\;\;\;\;
\ev{\imath\Phi_\alpha}=\frac{\delta\Gamma}{\delta\imath\Phi_\alpha}.
\end{equation}
We can now rewrite \eq{eq:wtid0} as
\begin{eqnarray}
0&=&\int\dx{x}\delta(t-x_0)\left\{
\frac{1}{g}\left(\partial_x^0\ev{\imath\sigma_x^a}\right)c_x^a-f^{abc}
\ev{\imath\sigma_x^a}\sigma_x^bc_x^c+\frac{1}{g}
\ev{\imath A_{ix}^a}\nabla_{ix}c_x^a-f^{abc}
\ev{\imath A_{ix}^a}A_{ix}^bc_x^c-\frac{1}{g}\lambda_x^a\ev{\imath\ov{c}_x^a}
\right.\nonumber\\&&\left.
+\frac{1}{2}f^{abc}\ev{\imath c_x^a}c_x^bc_x^c-f^{abc}\ev{\imath\sigma_x^a}
\ev{\imath\rho_x^b\imath\ov{\eta}_x^c}-f^{abc}\ev{\imath A_{ix}^a}
\ev{\imath J_{ix}^b\imath\ov{\eta}_x^c}+\frac{1}{2}f^{abc}\ev{\imath c_x^a}
\ev{\imath\ov{\eta}_x^b\imath\ov{\eta}_x^c}
\right\}.
\label{eq:wtid1}
\end{eqnarray}
Knowing the functional form of the ghost equation of motion
\cite{Watson:2007vc},
\begin{equation}
\ev{\imath\ov{c}_x^a}=-\nabla_x^2c_x^a
+gf^{abc}\nabla_{ix}
\left[\ev{\imath J_{ix}^b\imath\ov{\eta}_x^c}+A_{ix}^bc_x^c\right],
\label{eq:gheom0}
\end{equation}
allows us to write
\begin{eqnarray}
\lefteqn{
\int\dx{x}\delta(t-x_0)\frac{1}{g}\ev{\imath A_{ix}^a}\nabla_{ix}c_x^a=
\int\dx{x}\delta(t-x_0)\frac{1}{g}\left[\frac{\nabla_{ix}}{(-\nabla_x^2)}
\ev{\imath A_{ix}^a}\right]\nabla_{x}^2c_x^a}
&&\nonumber\\
&=&\int\dx{x}\delta(t-x_0)\left\{-\frac{1}{g}
\left[\frac{\nabla_{ix}}{(-\nabla_x^2)}\ev{\imath A_{ix}^a}\right]
\ev{\imath\ov{c}_x^a}-f^{abc}\ev{\imath A_{ix}^a}
\frac{\nabla_{ix}\nabla_{jx}}{(-\nabla_x^2)}
\left[\ev{\imath J_{jx}^b\imath\ov{\eta}_x^c}+A_{jx}^bc_x^c\right]\right\}.
\end{eqnarray}
Inserting the above into \eq{eq:wtid1} then gives
\begin{eqnarray}
0&=&\int\dx{x}\delta(t-x_0)\left\{
\frac{1}{g}\left(\partial_x^0\ev{\imath\sigma_x^a}\right)c_x^a-f^{abc}
\ev{\imath\sigma_x^a}\sigma_x^bc_x^c
-\frac{1}{g}\left[\frac{\nabla_{ix}}{(-\nabla_x^2)}
\ev{\imath A_{ix}^a}\right]\ev{\imath\ov{c}_x^a}
\right.\nonumber\\&&\left.
-f^{abc}\ev{\imath A_{ix}^a}t_{ij}(\vec{x})A_{jx}^bc_x^c
-\frac{1}{g}\lambda_x^a\ev{\imath\ov{c}_x^a}+\frac{1}{2}f^{abc}
\ev{\imath c_x^a}c_x^bc_x^c-f^{abc}\ev{\imath\sigma_x^a}
\ev{\imath\rho_x^b\imath\ov{\eta}_x^c}
\right.\nonumber\\&&\left.
-f^{abc}\ev{\imath A_{ix}^a}t_{ij}(\vec{x})
\ev{\imath J_{jx}^b\imath\ov{\eta}_x^c}+\frac{1}{2}f^{abc}
\ev{\imath c_x^a}\ev{\imath\ov{\eta}_x^b\imath\ov{\eta}_x^c}
\right\}
\label{eq:wtid2}
\end{eqnarray}
where $t_{ij}(\vec{x})=\delta_{ij}+\nabla_{ix}\nabla_{jx}/(-\nabla_x^2)$ is
the transverse projector in configuration space.
Before considering the full content of the Slavnov--Taylor identities arising from
\eq{eq:wtid2}, let us show that functional derivatives with respect to the
Lagrange multiplier field, $\lambda_x^a$, play no further role. The field
equation of motion for $\lambda_x^a$, valid in the presence of sources reads
\cite{Watson:2007vc}
\begin{equation}
\ev{\imath\lambda_x^a}=-\nabla_{ix}A_{ix}^a
\end{equation}
such that the only non-zero functional derivative of $\ev{\imath\lambda_x^a}$ is
\begin{equation}
\ev{\imath A_{jy}^b\imath\lambda_x^a}=\imath\delta^{ba}\nabla_{jx}\delta(y-x).
\label{eq:la0}
\end{equation}
All other functional derivatives of $\ev{\imath\lambda_x^a}$ vanish, even in
the presence of sources. Taking the functional derivative of \eq{eq:wtid2}
with respect to $\imath\lambda_z^d$ gives
\begin{eqnarray}
0&=&\int\dx{x}\delta(t-x_0)\left\{
\frac{\imath}{g}\delta(z-x)\ev{\imath\ov{c}_x^d}
-\frac{1}{g}\left[\frac{\nabla_{ix}}{(-\nabla_x^2)}
\ev{\imath\lambda_z^d\imath A_{ix}^a}\right]\ev{\imath\ov{c}_x^a}
\right.\nonumber\\&&\left.
-f^{abc}\ev{\imath\lambda_z^d\imath A_{ix}^a}t_{ij}(\vec{x})A_{jx}^bc_x^c
-f^{abc}\ev{\imath\lambda_z^d\imath A_{ix}^a}t_{ij}(\vec{x})
\ev{\imath J_{jx}^b\imath\ov{\eta}_x^c}
\right.\nonumber\\&&\left.
-f^{abc}\ev{\imath\sigma_x^a}\frac{\delta}{\delta\imath\lambda_z^d}
\ev{\imath\rho_x^b\imath\ov{\eta}_x^c}
-f^{abc}\ev{\imath A_{ix}^a}t_{ij}(\vec{x})\frac{\delta}{\delta\imath\lambda_z^d}
\ev{\imath J_{jx}^b\imath\ov{\eta}_x^c}
+\frac{1}{2}f^{abc}\ev{\imath c_x^a}\frac{\delta}{\delta\imath\lambda_z^d}
\ev{\imath\ov{\eta}_x^b\imath\ov{\eta}_x^c}
\right\}.
\end{eqnarray}
Now, taking partial functional derivatives of the Legendre transform,
\eq{eq:legtran}, one can show that
\begin{eqnarray}
\frac{\delta}{\delta\imath\lambda_z^d}\ev{\imath J_\alpha\imath J_\beta}&\sim&
\ev{\imath J_\alpha\imath J_\kappa}\ev{\imath\Phi_\kappa\imath\lambda_z^d\imath\Phi_\gamma}
\ev{\imath J_{\gamma}\imath J_\beta}\nonumber\\
&=&0
\end{eqnarray}
since all three-point proper functions involving functional derivatives
with respect to $\imath\lambda_z^d$ vanish. Further using the result
\eq{eq:la0} we arrive at
\begin{equation}
0=\int\dx{x}\delta(t-x_0)\left\{
\frac{\imath}{g}
\left[\delta(z-x)-\frac{\nabla_{ix}\nabla_{iz}}{(-\nabla_x^2)}\delta(x-z)\right]
\ev{\imath\ov{c}_x^d}
-\imath f^{dbc}\left[\nabla_{iz}\delta(x-z)\right]t_{ij}(\vec{x})
\left[A_{jx}^bc_x^c+\ev{\imath J_{jx}^b\imath\ov{\eta}_x^c}\right]
\right\}.
\end{equation}
Knowing that $\nabla_{iz}\delta(x-z)=-\nabla_{ix}\delta(x-z)$ and using
integration by parts on the last term, the above expression is a trivial
identity. Thus, even in the presence of sources, functional derivatives
of \eq{eq:wtid2} with respect to $\imath\lambda_z^d$ give rise to an identity
from which no further information can be obtained and we can set the
classical field $\lambda_x^a=0$ (except for within partial derivatives used in
conjunction with the Legendre transform) and ignore the Lagrange multiplier
field from now on.
Equation (\ref{eq:wtid2}) is Grassmann valued and to proceed, we must first
take the functional derivative with respect to $\imath c_z^d$. Since we
will be taking further derivatives, fields/sources must be retained (with
the exception of $\lambda$ as discussed previously). The subsequent Slavnov--Taylor
identities are thus functional derivatives of
\begin{eqnarray}
0&=&\int\dx{x}\delta(t-x_0)\left\{
-\frac{\imath}{g}\left[\partial_x^0\ev{\imath c_z^d\imath\sigma_x^a}\right]
\imath c_x^a
-\frac{\imath}{g}\left[\partial_x^0\ev{\imath\sigma_x^d}\right]\delta(z-x)
\right.\nonumber\\&&
-\frac{1}{g}\left[\frac{\nabla_{ix}}{(-\nabla_x^2)}
\ev{\imath c_z^d\imath A_{ix}^a}\right]\ev{\imath\ov{c}_x^a}
+\frac{1}{g}\left[\frac{\nabla_{ix}}{(-\nabla_x^2)}
\ev{\imath A_{ix}^a}\right]\ev{\imath\ov{c}_x^a\imath c_z^d}
\nonumber\\&&
-f^{abc}\ev{\imath c_z^d\imath\sigma_x^a}
\left[\ev{\imath\rho_x^b\imath\ov{\eta}_x^c}-\imath\sigma_x^b\imath c_x^c\right]
-f^{abc}\ev{\imath\sigma_x^a}
\left[\frac{\delta}{\delta\imath c_z^d}\ev{\imath\rho_x^b\imath\ov{\eta}_x^c}
-\imath\sigma_x^b\delta^{dc}\delta(z-x)\right]
\nonumber\\&&
-f^{abc}\ev{\imath c_z^d\imath A_{ix}^a}t_{ij}(\vec{x})
\left[\ev{\imath J_{jx}^b\imath\ov{\eta}_x^c}
-\imath A_{jx}^b\imath c_x^c\right]
-f^{abc}\ev{\imath A_{ix}^a}t_{ij}(\vec{x})
\left[\frac{\delta}{\delta\imath c_z^d}\ev{\imath J_{jx}^b\imath\ov{\eta}_x^c}
-\imath A_{jx}^b\delta^{dc}\delta(z-x)\right]
\nonumber\\&&\left.
+\frac{1}{2}f^{abc}\ev{\imath c_z^d\imath c_x^a}
\left[\ev{\imath\ov{\eta}_x^b\imath\ov{\eta}_x^c}
-\imath c_x^b\imath c_x^c\right]
-\frac{1}{2}f^{abc}\ev{\imath c_x^a}
\left[\frac{\delta}{\delta\imath c_z^d}\ev{\imath\ov{\eta}_x^b\imath\ov{\eta}_x^c}
-2\delta^{db}\delta(z-x)\imath c_x^c\right]
\right\}.
\label{eq:wtid3}
\end{eqnarray}
\section{Slavnov--Taylor identities: two-point functions}
\setcounter{equation}{0}
The first of the Slavnov--Taylor identities is generated by taking the functional
derivative of \eq{eq:wtid3} with respect to $\imath\sigma_w^e$ and setting
sources to zero. The resulting equation reads
\begin{eqnarray}
0&=&\int\dx{x}\delta(t-x_0)\left\{
-\frac{\imath}{g}\left[\partial_x^0\ev{\imath\sigma_w^e\imath\sigma_x^d}\right]\delta(z-x)
+\frac{1}{g}\left[\frac{\nabla_{ix}}{(-\nabla_x^2)}
\ev{\imath\sigma_w^e\imath A_{ix}^a}\right]\ev{\imath\ov{c}_x^a\imath c_z^d}
\right.\nonumber\\&&\left.
-f^{abc}\ev{\imath\sigma_w^e\imath\sigma_x^a}\left.\frac{\delta}{\delta\imath c_z^d}
\ev{\imath\rho_x^b\imath\ov{\eta}_x^c}\right|_{J=0}
-f^{abc}\ev{\imath\sigma_w^e\imath A_{ix}^a}t_{ij}(\vec{x})
\left.\frac{\delta}{\delta\imath c_z^d}
\ev{\imath J_{jx}^b\imath\ov{\eta}_x^c}\right|_{J=0}
\right\}.
\label{eq:xstid0}
\end{eqnarray}
This equation is best expressed in momentum space. We define the momentum
space two-point Green's functions via their respective Fourier transforms
\begin{equation}
\ev{\imath\sigma_w^a\imath\sigma_x^b}=\int\dk{k}e^{-\imath k\cdot(w-x)}
\Gamma_{\sigma\si}^{ab}(k_0,\vec{k}),\label{eq:ss0}
\end{equation}
(similarly for $\Gamma_{AA}$ etc. and also for the propagators $W_{AA}$ etc.)
where the arguments of the momentum space functions reflect their
noncovariant nature, we impose translational invariance in the usual way
and $\dk{k}=d^4k/(2\pi)^4$. We also define
\begin{eqnarray}
f^{abc}\left.\frac{\delta}{\delta\imath c_z^d}
\ev{\imath\rho_x^b\imath\ov{\eta}_x^c}\right|_{J=0}&=&
\int\dk{k}e^{-\imath k\cdot(x-z)}
\tilde{\Sigma}_{\sigma;\ov{c}c}^{ad}(k_0,\vec{k}),\label{eq:tscc0}\\
f^{abc}\left.\frac{\delta}{\delta\imath c_z^d}
\ev{\imath J_{jx}^b\imath\ov{\eta}_x^c}\right|_{J=0}&=&
\int\dk{k}e^{-\imath k\cdot(x-z)}
\tilde{\Sigma}_{A;\ov{c}cj}^{ad}(k_0,\vec{k}).\label{eq:tacc0}
\end{eqnarray}
The latter two definitions and notations may at first sight appear somewhat
artificial. In fact, as will be shortly justified, their exact form in
terms of standard Green's functions is irrelevant since neither term will
contribute. However, in the remainder of the paper, similar (but
increasingly complicated) objects will occur so some explanation is in
order. The Dyson--Schwinger equations are derived as functional derivatives of the
field equations of motion and this gives rise to the familiar loop
expressions. For example (taken from Ref.~\cite{Watson:2007vc}),
\begin{equation}
\frac{\delta}{\delta\imath A_{jw}^f}\int d^4yd^4z\,\Gamma_{\ov{c}cAi}^{(0)bca}(y,z,x)
\ev{\imath\ov{\eta}_z^c\imath\eta_y^b}\;\longrightarrow\;
\int\dk{\omega}\Gamma_{\ov{c}cAi}^{(0)bca}(\omega-k,-\omega,k)W_{\ov{c}c}^{cd}(\omega)
\Gamma_{\ov{c}cAj}^{def}(\omega,k-\omega,-k)W_{\ov{c}c}^{eb}(\omega-k)
\end{equation}
gives the ghost loop term of the Dyson--Schwinger equation for $\Gamma_{AA}$ ($W_{\ov{c}c}$
is the ghost propagator, $\Gamma_{\ov{c}cA}$ is the spatial ghost-gluon vertex
and $\Gamma_{\ov{c}cA}^{(0)}$ is the tree-level counterpart). Importantly,
all occurrences of objects such as $\ev{\imath\ov{\eta}_z^c\imath\eta_y^b}$
in the field equations of motion have an associated tree-level vertex --
these terms originate directly from the interaction terms of the original
action such that the functional derivatives have a clear meaning as loops.
However, in the case of the (nonabelian) Slavnov--Taylor identities one has
a different structure which arises from the Gauss-BRST transform (or indeed
generally from the BRS transform) and is exemplified in \eq{eq:wtid3}. In
this case, one has objects such as $\ev{\imath\rho_x^b\imath\ov{\eta}_x^c}$
which must be functionally differentiated, but which are separated from any
interaction type of factor. However, even without the interaction term,
these functional derivatives do still have a partial meaning as loop
expressions; it is simply that the tree-level vertex is missing (although
the color factor and momentum conservation are present). We will denote
such irregular expressions with a tilde (as in $\tilde{\Sigma}$ above or
$\tilde{\Gamma}$ later) and since these terms are important to the arguments
presented in this study, we will write out their explicit forms where
necessary. As alluded to above, this type of pseudo-loop expression is
common to the Slavnov--Taylor identities of nonabelian theories and in
linear covariant gauges, there is a familiar example, referred to as the
ghost-gluon scattering-like kernel which appears in the identity for the
three-gluon vertex \cite{Marciano:1977su}.
Returning to \eq{eq:xstid0}, we can now write
\begin{eqnarray}
0&=\int\dk{q_0}\dk{k}e^{-\imath q_0(t-z_0)-\imath k\cdot(w-z)}&\left\{
\frac{k_0}{g}\Gamma_{\sigma\si}^{ed}(k_0,\vec{k})
-\frac{\imath}{g}\frac{k_i}{\vec{k}^2}\Gamma_{\sigma Ai}^{ea}(k_0,\vec{k})
\Gamma_{\ov{c}c}^{ad}(q_0+k_0,\vec{k})
\right.\nonumber\\&&\left.
-\Gamma_{\sigma\si}^{ea}(k_0,\vec{k})
\tilde{\Sigma}_{\sigma;\ov{c}c}^{ad}(q_0+k_0,\vec{k})
-\Gamma_{\sigma Ai}^{ea}(k_0,\vec{k})t_{ij}(\vec{k})
\tilde{\Sigma}_{A;\ov{c}cj}^{ad}(k_0+q_0,\vec{k})
\right\}.
\end{eqnarray}
In the above, notice how the $\delta(t-x_0)$ factor characterizing the
Gauss-BRST transform leads to the energy insertion $q_0$ which breaks the
energy flow through the various Green's functions and in particular how
this insertion affects only the ghost functions. Since $t$, $w$ and $z$
are arbitrary, we can write down the momentum space Slavnov--Taylor identity:
\begin{equation}
\frac{k_0}{g}\Gamma_{\sigma\si}^{ed}(k_0,\vec{k})=
\frac{\imath}{g}\frac{k_i}{\vec{k}^2}\Gamma_{\sigma Ai}^{ea}(k_0,\vec{k})
\Gamma_{\ov{c}c}^{ad}(q_0+k_0,\vec{k})
+\Gamma_{\sigma\si}^{ea}(k_0,\vec{k})
\tilde{\Sigma}_{\sigma;\ov{c}c}^{ad}(q_0+k_0,\vec{k})
+\Gamma_{\sigma Ai}^{ea}(k_0,\vec{k})t_{ij}(\vec{k})
\tilde{\Sigma}_{A;\ov{c}cj}^{ad}(k_0+q_0,\vec{k}).
\label{eq:mstid0}
\end{equation}
Now consider the definition of $\tilde{\Sigma}_{Aj;\ov{c}c}$, given by
\eq{eq:tacc0}: under a parity transform, the vector source $J_{jx}^b$
changes sign \cite{Watson:2006yq} and we see that
\begin{equation}
\tilde{\Sigma}_{A;\ov{c}cj}^{ad}(k_0,-\vec{k})=
-\tilde{\Sigma}_{A;\ov{c}cj}^{ad}(k_0,\vec{k})
\end{equation}
from which we can infer that
\begin{equation}
t_{ij}(\vec{k})\tilde{\Sigma}_{A;\ov{c}cj}^{ad}(k_0+q_0,\vec{k})
\sim t_{ij}(\vec{k})k_j=0
\label{eq:tilde0}
\end{equation}
such that the last term of \eq{eq:mstid0} vanishes. Because the energy
scale, $q_0$, is arbitrary, we can make a further translation
$q_0\rightarrow-q_0-2k_0$ and \eq{eq:mstid0} becomes
\begin{equation}
\frac{k_0}{g}\Gamma_{\sigma\si}^{ed}(k_0,\vec{k})=
\frac{\imath}{g}\frac{k_i}{\vec{k}^2}\Gamma_{\sigma Ai}^{ea}(k_0,\vec{k})
\Gamma_{\ov{c}c}^{ad}(-q_0-k_0,\vec{k})
+\Gamma_{\sigma\si}^{ea}(k_0,\vec{k})
\tilde{\Sigma}_{\sigma;\ov{c}c}^{ad}(-q_0-k_0,\vec{k}).
\label{eq:mstid1}
\end{equation}
Knowing that the $\sigma$-field and $\rho$-source change sign under
time-reversal, whereas the ghost-antighost pair is invariant
\cite{Watson:2006yq}, we have that
\begin{eqnarray}
\tilde{\Sigma}_{\sigma;\ov{c}c}^{ad}(-q_0-k_0,\vec{k})&=&
-\tilde{\Sigma}_{\sigma;\ov{c}c}^{ad}(q_0+k_0,\vec{k}),\nonumber\\
\Gamma_{\ov{c}c}^{ad}(-q_0-k_0,\vec{k})&=&\Gamma_{\ov{c}c}^{ad}(q_0+k_0,\vec{k}),
\end{eqnarray}
such that
\begin{equation}
\frac{k_0}{g}\Gamma_{\sigma\si}^{ed}(k_0,\vec{k})=
\frac{\imath}{g}\frac{k_i}{\vec{k}^2}\Gamma_{\sigma Ai}^{ea}(k_0,\vec{k})
\Gamma_{\ov{c}c}^{ad}(q_0+k_0,\vec{k})
-\Gamma_{\sigma\si}^{ea}(k_0,\vec{k})\tilde{\Sigma}_{\sigma;\ov{c}c}^{ad}(q_0+k_0,\vec{k}).
\label{eq:mstid2}
\end{equation}
Comparing this with the original \eq{eq:mstid0} above, we find that
\begin{equation}
\tilde{\Sigma}_{\sigma;\ov{c}c}^{ad}(q_0+k_0,\vec{k})=0.
\label{eq:tilde1}
\end{equation}
The Slavnov--Taylor identity now reads
\begin{equation}
\frac{k_0}{g}\Gamma_{\sigma\si}^{ed}(k_0,\vec{k})=
\frac{\imath}{g}\frac{k_i}{\vec{k}^2}\Gamma_{\sigma Ai}^{ea}(k_0,\vec{k})
\Gamma_{\ov{c}c}^{ad}(q_0+k_0,\vec{k}).
\end{equation}
However, since $q_0$ is arbitrary, we can further state that the two-point
ghost proper function $\Gamma_{\ov{c}c}$ is independent of energy, i.e.,
\begin{equation}
\Gamma_{\ov{c}c}^{ad}(q_0+k_0,\vec{k})\rightarrow\Gamma_{\ov{c}c}^{ad}(\vec{k}).
\end{equation}
This was known to all orders in perturbation theory
\cite{Watson:2006yq,Watson:2007vc}, but it is reassuring that the result is
confirmed nonperturbatively. We now have the final form of the first of
the Slavnov--Taylor identities:
\begin{equation}
\frac{k_0}{g}\Gamma_{\sigma\si}^{ed}(k_0,\vec{k})=
\frac{\imath}{g}\frac{k_i}{\vec{k}^2}\Gamma_{\sigma Ai}^{ea}(k_0,\vec{k})
\Gamma_{\ov{c}c}^{ad}(\vec{k}).
\label{eq:stid0}
\end{equation}
Repeating the above analysis but starting by functionally differentiating
\eq{eq:wtid3} with respect to $\imath A_{kw}^e$ and setting sources to zero
leads to the similar Slavnov--Taylor identity
\begin{equation}
\frac{k_0}{g}\Gamma_{A\sigma k}^{ed}(k_0,\vec{k})=
\frac{\imath}{g}\frac{k_i}{\vec{k}^2}\Gamma_{AAki}^{ea}(k_0,\vec{k})
\Gamma_{\ov{c}c}^{ad}(\vec{k})
\label{eq:stid1}.
\end{equation}
Now, since the energy, $k_0$, is a scalar quantity we see immediately from
Eqs.~(\ref{eq:stid0}) and (\ref{eq:stid1}) that the two-point proper
Green's functions involving functional derivatives with respect to the
temporal gluon field, $\sigma$, (these will be referred to as the temporal
Green's functions) can be unambiguously expressed in terms of the ghost
and (longitudinal) spatial gluon two-point proper Green's functions. (We
will discuss the identities, Eqs.~(\ref{eq:stid0}) and (\ref{eq:stid1}),
and their general kinematical decompositions further in
Section~\ref{sec:five}.) In a sense, up to the level of the two-point
functions the field $\sigma$ has been eliminated from the system or
equivalently has been integrated out of the functional form of the
action. The two identities, Eqs.~(\ref{eq:stid0}) and (\ref{eq:stid1}),
have been derived previously from the standard BRS invariance, using
perturbative arguments to eliminate the $\tilde{\Sigma}$ terms and were
verified to one-loop order in perturbation theory \cite{Watson:2007vc}.
The above derivation is however slightly superior since it makes reference
only to symmetry arguments and does not rely on the implicit assumption
that the all-orders resummation of perturbation theory is equivalent to the
nonperturbative theory. Together, the two identities are the Coulomb gauge
equivalent of the standard covariant gauge result that the longitudinal part
of the gluon polarization is not dressed
\cite{Taylor:1971ff,Slavnov:1972fg}. This also highlights an important
difference between Landau gauge and Coulomb gauge: in the former, the gluon
polarization is transverse to the four-momentum; in the latter, the gluon
polarization ($\Gamma_{AA}$ in our notation) is explicitly \emph{not}
transverse --- instead, the spatially longitudinal, ghost and temporal
two-point proper functions are all related. This connection is reminiscent
of the quartet mechanism in the Kugo--Ojima confinement scenario
\cite{Kugo:1979gm}. This is hardly surprising since the Kugo--Ojima
confinement scenario is based on the identification of a well-defined
conserved color charge and the Slavnov--Taylor identities here are the
dynamical expression of charge conservation.
\section{Slavnov--Taylor identities: vertex functions}
\setcounter{equation}{0}
In this section, we study further functional derivatives of \eq{eq:wtid3}
and show that the resultant Slavnov--Taylor identities eventually form a
closed set from which the temporal Green's functions (i.e., those with at
least one external $\sigma$-leg) can be unambiguously derived. This closed set
turns out to be somewhat extended and so, for clarity of presentation, we
separate the various classes of functional derivatives according to how many
pairs of ghost/antighost functional derivatives are present. By explicitly
separating the Grassmann-valued fields, we may then modify our notation for
the generic field $\Phi_\alpha$ or source $J_\alpha$ to be restricted to only the
$\vec{A}$ or $\sigma$ fields/sources and their associated attributes which
allows us to compactify the formalism.
\subsection{No further ghost derivatives}
Let us begin by functionally differentiating \eq{eq:wtid3} twice with
respect to $\imath\Phi_{\lambda(k)w}^e$ and $\imath\Phi_{\tau(l)v}^f$ where as
mentioned above, the indices $\lambda(k)$ and $\tau(l)$ refer here to either the
$A$-field with its associated spatial index ($k$ or $l$) or to the
$\sigma$-field with no associated spatial index. Setting sources to zero, the
resulting equation reads:
\begin{eqnarray}
0&=&\int\dx{x}\delta(t-x_0)\left\{
-\frac{\imath}{g}\left[\partial_x^0
\ev{\imath\Phi_{\tau(l)v}^f\imath\Phi_{\lambda(k)w}^e\imath\sigma_x^d}\right]\delta(z-x)
+\frac{1}{g}\left[\frac{\nabla_{ix}}{(-\nabla_x^2)}
\ev{\imath\Phi_{\tau(l)v}^f\imath\Phi_{\lambda(k)w}^e\imath A_{ix}^a}\right]
\ev{\imath\ov{c}_x^a\imath c_z^d}
\right.\nonumber\\&&
+\frac{1}{g}\left[\frac{\nabla_{ix}}{(-\nabla_x^2)}
\ev{\imath\Phi_{\lambda(k)w}^e\imath A_{ix}^a}\right]
\ev{\imath\ov{c}_x^a\imath c_z^d\imath\Phi_{\tau(l)v}^f}
+\frac{1}{g}\left[\frac{\nabla_{ix}}{(-\nabla_x^2)}
\ev{\imath\Phi_{\tau(l)v}^f\imath A_{ix}^a}\right]
\ev{\imath\ov{c}_x^a\imath c_z^d\imath\Phi_{\lambda(k)w}^e}
\nonumber\\&&
-\ev{\imath\Phi_{\lambda(k)w}^e\imath\sigma_x^a}
\left[f^{abc}\left.\frac{\delta^2}{\delta\imath\Phi_{\tau(l)v}^f\delta\imath c_z^d}
\ev{\imath\rho_x^b\imath\ov{\eta}_x^c}\right|_{J=0}
-f^{afd}\delta_{\sigma\tau}\delta(v-x)\delta(z-x)\right]
\nonumber\\&&
-\ev{\imath\Phi_{\tau(l)v}^f\imath\sigma_x^a}
\left[f^{abc}\left.\frac{\delta^2}{\delta\imath\Phi_{\lambda(k)w}^f\delta\imath c_z^d}
\ev{\imath\rho_x^b\imath\ov{\eta}_x^c}\right|_{J=0}
-f^{aed}\delta_{\sigma\lambda}\delta(w-x)\delta(z-x)\right]
\nonumber\\&&
-\ev{\imath\Phi_{\lambda(k)w}^e\imath A_{ix}^a}t_{ij}(\vec{x})
\left[f^{abc}\left.\frac{\delta^2}{\delta\imath\Phi_{\tau(l)v}^f\delta\imath c_z^d}
\ev{\imath J_{jx}^b\imath\ov{\eta}_x^c}\right|_{J=0}
-f^{afd}\delta_{jl}\delta_{A\tau}\delta(v-x)\delta(z-x)\right]
\nonumber\\&&\left.
-\ev{\imath\Phi_{\tau(l)v}^f\imath A_{ix}^a}t_{ij}(\vec{x})
\left[f^{abc}\left.\frac{\delta^2}{\delta\imath\Phi_{\lambda(k)w}^e\delta\imath c_z^d}
\ev{\imath J_{jx}^b\imath\ov{\eta}_x^c}\right|_{J=0}
-f^{aed}\delta_{jk}\delta_{A\lambda}\delta(w-x)\delta(z-x)\right]
\right\}.
\end{eqnarray}
In the above, we have made use of the results Eqs.~(\ref{eq:tilde0}) and
(\ref{eq:tilde1}) to eliminate such terms. Because the indices $\lambda$ and
$\tau$ refer to field types that commute, the above equation actually
represents three separate equations. Let us define
\begin{eqnarray}
\tilde{\Gamma}_{\sigma;\ov{c}c\tau(l)}^{adf}(x,z,v)&=&
gf^{abc}\left.\frac{\delta^2}{\delta\imath\Phi_{\tau(l)v}^f\delta\imath c_z^d}
\ev{\imath\rho_x^b\imath\ov{\eta}_x^c}\right|_{J=0},
\nonumber\\
\tilde{\Gamma}_{A;\ov{c}c\tau j(l)}^{adf}(x,z,v)&=&
gf^{abc}\left.\frac{\delta^2}{\delta\imath\Phi_{\tau(l)v}^f\delta\imath c_z^d}
\ev{\imath J_{jx}^b\imath\ov{\eta}_x^c}\right|_{J=0},
\label{eq:tildedef}
\end{eqnarray}
(these expressions will be explained later). We further notice that by
taking two functional derivatives of \eq{eq:gheom0} and setting sources to
zero then we have
\begin{equation}
\ev{\imath\ov{c}_x^a\imath c_z^d\imath\Phi_{\tau(l)v}^f}=
-gf^{adf}\delta_{A\tau}\delta_{jl}\nabla_{jx}\delta(v-x)\delta(z-x)
-\nabla_{jx}\tilde{\Gamma}_{A;\ov{c}c\tau j(l)}^{adf}(x,z,v).
\label{eq:ghgldse0}
\end{equation}
This equation is of course the precursor to the Dyson--Schwinger equation for the
temporal and spatial ghost-gluon vertices. Our identities can thus be
written
\begin{eqnarray}
0&=&\int\dx{x}\delta(t-x_0)\left\{
-\imath\left[\partial_x^0
\ev{\imath\Phi_{\tau(l)v}^f\imath\Phi_{\lambda(k)w}^e\imath\sigma_x^d}\right]\delta(z-x)
+\left[\frac{\nabla_{ix}}{(-\nabla_x^2)}
\ev{\imath\Phi_{\tau(l)v}^f\imath\Phi_{\lambda(k)w}^e\imath A_{ix}^a}\right]
\ev{\imath\ov{c}_x^a\imath c_z^d}
\right.\nonumber\\&&
-\ev{\imath\Phi_{\lambda(k)w}^e\imath A_{ix}^a}
\left[\tilde{\Gamma}_{A;\ov{c}c\tau i(l)}^{adf}(x,z,v)
+gf^{adf}\delta_{il}\delta_{A\tau}\delta(v-x)\delta(z-x)\right]
\nonumber\\&&
-\ev{\imath\Phi_{\tau(l)v}^f\imath A_{ix}^a}
\left[\tilde{\Gamma}_{A;\ov{c}c\lambda i(k)}^{ade}(x,z,w)
+gf^{ade}\delta_{ik}\delta_{A\lambda}\delta(w-x)\delta(z-x)\right]
\nonumber\\&&
-\ev{\imath\Phi_{\lambda(k)w}^e\imath\sigma_x^a}
\left[\tilde{\Gamma}_{\sigma;\ov{c}c\tau(l)}^{adf}(x,z,v)
+gf^{adf}\delta_{\sigma\tau}\delta(v-x)\delta(z-x)\right]
\nonumber\\&&\left.
-\ev{\imath\Phi_{\tau(l)v}^f\imath\sigma_x^a}
\left[\tilde{\Gamma}_{\sigma;\ov{c}c\lambda(k)}^{ade}(x,z,w)
+gf^{ade}\delta_{\sigma\lambda}\delta(w-x)\delta(z-x)\right]
\right\}.
\label{eq:inter0}
\end{eqnarray}
Let us now introduce our convention for the Fourier transform of the
three-point functions (similarly for higher $n$-point functions):
\begin{equation}
f(x,y,z)=\int\dk{k_1}\dk{k_2}\dk{k_3}(2\pi)^4\delta(k_1+k_2+k_3)
e^{-\imath k_1\cdot x-\imath k_2\cdot y-\imath k_3\cdot z}f(k_1,k_2,k_3).
\end{equation}
All momenta of the three-point functions are defined as incoming and for
brevity we do not split the temporal and spatial sets of arguments.
Returning to \eq{eq:inter0}, after some manipulation we arrive at the
three Slavnov--Taylor identities ($k_1+k_2+k_3=0$):
\begin{eqnarray}
k_3^0\Gamma_{\tau\lambda\sigma(lk)}^{fed}(k_1,k_2,k_3)
&=&
\imath\frac{k_{3i}}{\vec{k}_3^2}\Gamma_{\tau\lambda A(lk)i}^{fea}(k_1,k_2,k_3)
\Gamma_{\ov{c}c}^{ad}(-\vec{k}_3)
\nonumber\\&&
-\Gamma_{\lambda A(k)i}^{ea}(k_2)
\left[\tilde{\Gamma}_{A;\ov{c}c\tau i(l)}^{adf}(k_2+q_0,k_3-q_0,k_1)
+gf^{adf}\delta_{il}\delta_{A\tau}\right]
\nonumber\\&&
-\Gamma_{\tau A(l)i}^{fa}(k_1)
\left[\tilde{\Gamma}_{A;\ov{c}c\lambda i(k)}^{ade}(k_1+q_0,k_3-q_0,k_2)
+gf^{ade}\delta_{ik}\delta_{A\lambda}\right]
\nonumber\\&&
-\Gamma_{\lambda\sigma(k)}^{ea}(k_2)
\left[\tilde{\Gamma}_{\sigma;\ov{c}c\tau(l)}^{adf}(k_2+q_0,k_3-q_0,k_1)
+gf^{adf}\delta_{\sigma\tau}\right]
\nonumber\\&&
-\Gamma_{\tau\sigma(l)}^{fa}(k_1)
\left[\tilde{\Gamma}_{\sigma;\ov{c}c\lambda(k)}^{ade}(k_1+q_0,k_3-q_0,k_2)
+gf^{ade}\delta_{\sigma\lambda}\right].
\label{eq:stid2}
\end{eqnarray}
These identities can easily be verified at tree-level, using the Feynman
rules presented in Ref.~\cite{Watson:2007vc} (and for completeness presented
in Appendix~\ref{app:decomp}). They are the Coulomb gauge analogue of the
familiar Slavnov--Taylor identity for the three-gluon vertex
\cite{Marciano:1977su}. As in the previous section, the energy $q_0$ is
injected into the ghost line of the various factors but one cannot cancel
terms using symmetry properties anymore (as was the case for the
$\tilde{\Sigma}$ kernels) because of the presence of other energy arguments
in the equations. Neglecting the $\tilde{\Gamma}$ terms, we can see that each
vertex involving a functional derivative with respect to the temporal
$\sigma$-field is fully determined (again because $k_3^0$ is a scalar quantity)
given the corresponding (spatially contracted) vertex involving the
derivative with respect to the spatial $A$-field.
Let us now discuss the form of the $\tilde{\Gamma}$ factors. From the
definition, \eq{eq:tildedef}, and taking functional derivatives in standard
fashion, we have
\begin{eqnarray}
\tilde{\Gamma}_{\sigma;\ov{c}c\tau}^{adf}(x,z,v)&=&
gf^{abc}\left.\frac{\delta^2}{\delta\imath\Phi_{\tau v}^f\delta\imath c_z^d}
\ev{\imath\rho_x^b\imath\ov{\eta}_x^c}\right|_{J=0}\nonumber\\
&=&gf^{abc}\left\{
\ev{\imath\rho_x^b\imath J_\varepsilon}
\ev{\imath\Phi_\varepsilon\imath\Phi_{\tau v}^f\imath\Phi_\lambda}
\ev{\imath J_\lambda\imath J_\kappa}\ev{\imath\ov{\eta}_x^c\imath\eta_u^g}
\ev{\imath\ov{c}_u^g\imath c_z^d\imath\Phi_\kappa}
\right.\nonumber\\&&
-\ev{\imath\rho_x^b\imath J_\kappa}\ev{\imath\ov{\eta}_x^c\imath\eta_u^g}
\ev{\ov{c}_u^g\imath c_z^d\imath\Phi_\kappa\imath\Phi_{\tau(l)v}^f}
\nonumber\\&&\left.
+\ev{\imath\rho_x^b\imath J_\kappa}\ev{\imath\ov{\eta}_x^c\imath\eta_u^g}
\ev{\imath\ov{c}_u^g\imath c_s^h\imath\Phi_{\tau v}^f}
\ev{\imath\ov{\eta}_s^h\eta_r^i}\ev{\imath\ov{c}_r^i\imath c_z^d\imath\Phi_\kappa}
\right\}
\label{eq:rnd1}
\end{eqnarray}
where we recall that having explicitly extracted the Grassmann-valued
fields, the repeated indices refer here to only the $\sigma$ or $A$-fields.
The coordinates $u$, $s$ and $r$ are implicitly integrated over. We further
omit the possible spatial index ($l$) associated when $\tau$ refers to the
$\vec{A}$-field for notational convenience. A similar expression exists
for $\tilde{\Gamma}_{A;\ov{c}c\tau}$. In momentum space, we have ($p_1+p_2+p_3=0$)
\begin{eqnarray}
\tilde{\Gamma}_{\sigma;\ov{c}c\tau}^{adf}(p_1,p_2,p_3)&=&
gf^{abc}\int\dk{k}W_{\sigma\varepsilon}^{bg}(k)\Gamma_{\varepsilon\tau\mu}^{gfh}(k,p_3,-k-p_3)
W_{\mu\kappa}^{hi}(k+p_3)W_{\ov{c}c}^{cj}(p_1-k)
\Gamma_{\ov{c}c\kappa}^{jdi}(p_1-k,p_2,k+p_3)
\nonumber\\&&
-gf^{abc}\int\dk{k}W_{\sigma\kappa}^{bg}(k)W_{\ov{c}c}^{ch}(p_1-k)
\Gamma_{\ov{c}c\kappa\tau}^{hdgf}(p_1-k,p_2,k,p_3)
\nonumber\\&&
+gf^{abc}\int\dk{k}W_{\sigma\kappa}^{bg}(k)W_{\ov{c}c}^{ch}(p_1-k)
\Gamma_{\ov{c}c\tau}^{hif}(p_1-k,p_2+k,p_3)W_{\ov{c}c}^{ij}(-p_2-k)
\Gamma_{\ov{c}c\kappa}^{jdg}(-p_2-k,p_2,k),
\nonumber\\&&
\label{eq:ghgldse1}
\end{eqnarray}
(we make the internal color indices explicit) again with similar expressions
for the other $\tilde{\Gamma}$ factors. Given that there is the implicit
summation over $\varepsilon$, $\mu$ and $\kappa$ (referring to the field types $\sigma$
and $A$), what this means is that the $\tilde{\Gamma}$ considered so far are
in general dependent not only on the non-ghost temporal three-point
functions that are explicit in \eq{eq:stid2} (thus showing that
\eq{eq:stid2} actually forms a set of nonlinear integral equations) but on
the further temporal Green's functions $\Gamma_{\ov{c}c\sigma}$,
$\Gamma_{\ov{c}c\sigma A}$ and $\Gamma_{\ov{c}c\sigma\si}$ (and of course all the other
two-point functions, the purely spatial $A$ and the ghost three- and
four-point functions).
\subsection{One pair of further ghost derivatives}
Let us now consider the Slavnov--Taylor identities that arise if we functionally
differentiate \eq{eq:wtid3} with respect to at least $\imath\ov{c}_v^f$ and
$\imath c_w^e$. If we restrict ourselves to considering no further
ghost/antighost functional derivatives, then we are able to set the
corresponding fields/sources to zero whilst maintaining the rest without
confusion. The resulting expression reads:
\begin{eqnarray}
0&=&\int\dx{x}\delta(t-x_0)\left\{
\frac{\imath}{g}\left[\partial_x^0
\ev{\imath\ov{c}_v^f\imath c_z^d\imath\sigma_x^e}\right]\delta(w-x)
-\frac{\imath}{g}\left[\partial_x^0
\ev{\imath\ov{c}_v^f\imath c_w^e\imath\sigma_x^d}\right]\delta(z-x)
\right.\nonumber\\&&
-\frac{1}{g}\left[\frac{\nabla_{ix}}{(-\nabla_x^2)}
\ev{\imath\ov{c}_v^f\imath c_z^d\imath A_{ix}^a}\right]
\ev{\imath\ov{c}_x^a\imath c_w^e}
+\frac{1}{g}\left[\frac{\nabla_{ix}}{(-\nabla_x^2)}
\ev{\imath\ov{c}_v^f\imath c_w^e\imath A_{ix}^a}\right]
\ev{\imath\ov{c}_x^a\imath c_z^d}
\nonumber\\&&
+\frac{1}{g}\left[\frac{\nabla_{ix}}{(-\nabla_x^2)}
\ev{\imath A_{ix}^a}\right]
\ev{\imath\ov{c}_v^f\imath c_w^e\imath\ov{c}_x^a\imath c_z^d}
\nonumber\\&&
+f^{abc}\ev{\imath\ov{c}_v^f\imath c_z^d\imath\sigma_x^a}
\left[\frac{\delta}{\delta\imath c_w^e}
\ev{\imath\rho_x^b\imath\ov{\eta}_x^c}-\imath\sigma_x^b\delta^{ec}\delta(w-x)\right]
-f^{abc}\ev{\imath\ov{c}_v^f\imath c_w^e\imath\sigma_x^a}
\left[\frac{\delta}{\delta\imath c_z^d}
\ev{\imath\rho_x^b\imath\ov{\eta}_x^c}-\imath\sigma_x^b\delta^{dc}\delta(z-x)\right]
\nonumber\\&&
-f^{abc}\ev{\imath\sigma_x^a}
\frac{\delta^3}{\delta\imath\ov{c}_v^f\delta\imath c_w^e\delta\imath c_z^d}
\ev{\imath\rho_x^b\imath\ov{\eta}_x^c}
+f^{abc}\ev{\imath\ov{c}_v^f\imath c_z^d\imath A_{ix}^a}t_{ij}(\vec{x})
\left[\frac{\delta}{\delta\imath c_w^e}
\ev{\imath J_{jx}^b\imath\ov{\eta}_x^c}-\imath A_{jx}^b\delta^{ec}\delta(w-x)\right]
\nonumber\\&&
-f^{abc}\ev{\imath\ov{c}_v^f\imath c_w^e\imath A_{ix}^a}t_{ij}(\vec{x})
\left[\frac{\delta}{\delta\imath c_z^d}
\ev{\imath J_{jx}^b\imath\ov{\eta}_x^c}-\imath A_{jx}^b\delta^{dc}\delta(z-x)\right]
-f^{abc}\ev{\imath A_{ix}^a}t_{ij}(\vec{x})
\frac{\delta^3}{\delta\imath\ov{c}_v^f\delta\imath c_w^e\delta\imath c_z^d}
\ev{\imath J_{jx}^b\imath\ov{\eta}_x^c}
\nonumber\\&&\left.
+\frac{1}{2}f^{abc}\ev{\ov{c}_v^f\imath c_x^a}
\left[\frac{\delta^2}{\delta\imath c_w^e\delta\imath c_z^d}
\ev{\imath\ov{\eta}_x^b\imath\ov{\eta}_x^c}
-2\delta^{db}\delta^{ec}\delta(z-x)\delta(w-x)\right]
\right\}_{\ov{\eta}=\eta=0}.
\label{eq:wtid4}
\end{eqnarray}
To derive the identity for the ghost-gluon vertex, we further set the
remaining sources to zero (we will return to \eq{eq:wtid4} later in this
subsection to derive more identities) and again using the results given by
Eqs.~(\ref{eq:tilde0}) and (\ref{eq:tilde1}) we get
\begin{eqnarray}
0&=&\int\dx{x}\delta(t-x_0)\left\{
\frac{\imath}{g}
\left[\partial_x^0\ev{\imath\ov{c}_v^f\imath c_z^d\imath\sigma_x^e}\right]\delta(w-x)
-\frac{\imath}{g}
\left[\partial_x^0\ev{\imath\ov{c}_v^f\imath c_w^e\imath\sigma_x^d}\right]\delta(z-x)
\right.\nonumber\\&&
-\frac{1}{g}\left[\frac{\nabla_{ix}}{(-\nabla_x^2)}
\ev{\imath\ov{c}_v^f\imath c_z^d\imath A_{ix}^a}\right]
\ev{\imath\ov{c}_x^a\imath c_w^e}
+\frac{1}{g}\left[\frac{\nabla_{ix}}{(-\nabla_x^2)}
\ev{\imath\ov{c}_v^f\imath c_w^e\imath A_{ix}^a}\right]
\ev{\imath\ov{c}_x^a\imath c_z^d}
\nonumber\\&&\left.
+\frac{1}{2g}\ev{\imath\ov{c}_v^f\imath c_x^a}
\tilde{\Gamma}_{\ov{c};\ov{c}cc}^{ade}(x,z,w)
-f^{ade}\ev{\imath\ov{c}_v^f\imath c_x^a}\delta(z-x)\delta(w-x)
\right\}.
\end{eqnarray}
In the above, we have defined
\begin{eqnarray}
\tilde{\Gamma}_{\ov{c};\ov{c}cc}^{ade}(x,z,w)&=&
gf^{abc}\left.\frac{\delta^2}{\delta\imath c_w^e\delta\imath c_z^d}
\ev{\imath\ov{\eta}_x^b\imath\ov{\eta}_x^c}\right|_{J=0}
\nonumber\\
&=&gf^{abc}\ev{\imath\ov{\eta}_x^b\imath\eta_\nu}
\ev{\imath\ov{\eta}_x^c\imath\eta_\mu}
\left[2\ev{\imath J_\kappa\imath J_\varepsilon}
\ev{\imath\ov{c}_\nu\imath c_z^d\imath\Phi_\kappa}
\ev{\imath\ov{c}_\mu\imath c_w^e\imath\Phi_\varepsilon}
-\ev{\imath\ov{c}_\nu\imath c_w^e\imath c_z^d\imath\ov{c}_\mu}\right]
\end{eqnarray}
(internal indices $\kappa$ and $\varepsilon$ referring only to $A$ or $\sigma$-fields as
in the previous subsection). Considering now the Fourier transform and
using the same conventions as before, we arrive at the Slavnov--Taylor identity for the
temporal ghost-gluon vertex
\begin{eqnarray}
0&=&k_3^0\Gamma_{\ov{c}c\sigma}^{fde}(k_1,k_2,k_3)
-\frac{\imath k_{3i}}{\vec{k}_3^2}\Gamma_{\ov{c}cAi}^{fda}(k_1,k_2,k_3)
\Gamma_{\ov{c}c}^{ae}(-\vec{k}_3)
\nonumber\\&&
-(k_2^0+q_0)\Gamma_{\ov{c}c\sigma}^{fed}(k_1,k_3-q_0,k_2+q_0)
+\frac{\imath k_{2i}}{\vec{k}_2^2}\Gamma_{\ov{c}cAi}^{fea}(k_1,k_3-q_0,k_2+q_0)
\Gamma_{\ov{c}c}^{ad}(-\vec{k}_2)
\nonumber\\&&
+\frac{1}{2}\Gamma_{\ov{c}c}^{fa}(\vec{k}_1)
\tilde{\Gamma}_{\ov{c};\ov{c}cc}^{ade}(k_1+q_0,k_2,k_3-q_0)
-gf^{ade}\Gamma_{\ov{c}c}^{fa}(\vec{k}_1)
\label{eq:stid3}
\end{eqnarray}
with
\begin{eqnarray}
\tilde{\Gamma}_{\ov{c};\ov{c}cc}^{ade}(p_1,p_2,p_3)&=
gf^{abc}\int\dk{k}W_{\ov{c}c}^{bf}(p_1-k)W_{\ov{c}c}^{cg}(k)&
\left\{2\Gamma_{\ov{c}c\kappa}^{fdh}(p_1-k,p_2,p_3+k)
\Gamma_{\ov{c}c\varepsilon}^{gei}(k,p_3,-p_3-k)W_{\kappa\varepsilon}^{hi}(-p_3-k)
\right.\nonumber\\&&\left.
-\Gamma_{\ov{c}cc\ov{c}}^{fedg}(p_1-k,p_3,p_2,k)
\right\}.
\end{eqnarray}
Recall that we are free to choose the energy injection, $q_0$, at will. If
we set $k_2^0+q_0=0$ in \eq{eq:stid3}, then (given that for such general
spacelike momentum configurations, the Green's functions can have no
singularities) the $\Gamma_{\ov{c}c\sigma}$ term of the middle line drops out and
we have an unambiguous nonlinear integral equation for $\Gamma_{\ov{c}c\sigma}$.
Crucially, as far as the three-point functions involving the temporal
$\sigma$-field are concerned, the equation forms a closed expression without
approximation. One can immediately verify this identity at tree-level using
the Feynman rules of Ref.~\cite{Watson:2007vc} (and summarized in
Appendix~\ref{app:decomp}). A similar Slavnov--Taylor identity for the
ghost-gluon vertex in Landau gauge does in fact exist
\cite{von Smekal:1997vx,Alkofer:2000wg}.
Returning to \eq{eq:wtid4}, we take the further functional derivative with
respect to $\imath\Phi_{\lambda(k)u}^g$ (again the field type $\lambda$ refers only
to either the spatial $A$-field with index $k$ or to the temporal
$\sigma$-field) and set sources to zero. Making use of the results
Eqs.~(\ref{eq:tilde0}) and (\ref{eq:tilde1}) to eliminate such terms and if
we further use the appropriate functional derivatives of \eq{eq:gheom0} as
before, then we have \eq{eq:inter1} (such lengthy configuration space
expressions are relegated to Appendix~\ref{app:eqlist}).
Equation~(\ref{eq:inter1}) involves three new kernels:
$\tilde{\Gamma}_{\sigma;\ov{c}cc\ov{c}}$ and $\tilde{\Gamma}_{\ov{c};\ov{c}cc\lambda}$ are
given by Eqs.~(\ref{eq:rnd2}) and (\ref{eq:rnd3}), respectively; the third
kernel, $\tilde{\Gamma}_{A;\ov{c}cc\ov{c}}$, has a similar expression to
\eq{eq:rnd2}. There is only one possible new Green's function involving
derivatives with respect to the $\sigma$-field and this is
$\Gamma_{\ov{c}c\ov{c}c\sigma}$. Taking the Fourier transform of \eq{eq:inter1}
leads to the following Slavnov--Taylor identity in momentum space ($\sum k_a=0$):
\begin{eqnarray}
0&=&k_4^0\Gamma_{\ov{c}c\lambda\sigma(k)}^{fdge}(k_1,k_2,k_3,k_4)
-\frac{\imath k_{4i}}{\vec{k}_4^2}
\Gamma_{\ov{c}c\lambda A(k)i}^{fdga}(k_1,k_2,k_3,k_4)\Gamma_{\ov{c}c}^{ae}(-\vec{k}_4)
\nonumber\\&&
-(k_2^0+q_0)\Gamma_{\ov{c}c\lambda\sigma(k)}^{fegd}(k_1,k_4-q_0,k_3,k_2+q_0)
+\frac{\imath k_{2i}}{\vec{k}_2^2}
\Gamma_{\ov{c}c\lambda A(k)i}^{fega}(k_1,k_4-q_0,k_3,k_2+q_0)
\Gamma_{\ov{c}c}^{ad}(-\vec{k}_2)
\nonumber\\&&
+\Gamma_{\ov{c}c\sigma}^{fda}(k_1,k_2,k_3+k_4)
\left[\tilde{\Gamma}_{\sigma;\ov{c}c\lambda(k)}^{aeg}(q_0-k_3-k_4,k_4-q_0,k_3)
-gf^{age}\delta_{\sigma\lambda}\right]
\nonumber\\&&
+\Gamma_{\ov{c}cAi}^{fda}(k_1,k_2,k_3+k_4)
\left[\tilde{\Gamma}_{A;\ov{c}c\lambda i(k)}^{aeg}(q_0-k_3-k_4,k_4-q_0,k_3)
-gf^{age}\delta_{ki}\delta_{A\lambda}\right]
\nonumber\\&&
-\Gamma_{\ov{c}c\sigma}^{fea}(k_1,k_4-q_0,k_2+k_3+q_0)
\left[\tilde{\Gamma}_{\sigma;\ov{c}c\lambda(k)}^{adg}(-k_2-k_3,k_2,k_3)
-gf^{agd}\delta_{\sigma\lambda}\right]
\nonumber\\&&
-\Gamma_{\ov{c}cAi}^{fea}(k_1,k_4-q_0,k_2+k_3+q_0)
\left[\tilde{\Gamma}_{A;\ov{c}c\lambda i(k)}^{adg}(-k_2-k_3,k_2,k_3)
-gf^{agd}\delta_{ki}\delta_{A\lambda}\right]
\nonumber\\&&
-\Gamma_{\lambda\sigma(k)}^{ga}(k_3)
\tilde{\Gamma}_{\sigma;\ov{c}cc\ov{c}}^{adef}(k_3+q_0,k_2,k_4-q_0,k_1)
-\Gamma_{\lambda A(k)i}^{ga}(k_3)
\tilde{\Gamma}_{A;\ov{c}cc\ov{c}i}^{adef}(k_3+q_0,k_2,k_4-q_0,k_1)
\nonumber\\&&
+\frac{1}{2}\Gamma_{\ov{c}c\lambda(k)}^{fag}(k_1,k_2+k_4,k_3)
\left[\tilde{\Gamma}_{\ov{c};\ov{c}cc}^{ade}(q_0-k_2-k_4,k_2,k_4-q_0)
-2gf^{ade}\right]
\nonumber\\&&
+\frac{1}{2}\Gamma_{\ov{c}c}^{fa}(\vec{k}_1)
\tilde{\Gamma}_{\ov{c};\ov{c}cc\lambda(k)}^{adeg}(k_1+q_0,k_2,k_4-q_0,k_3),
\label{eq:stid4}
\end{eqnarray}
with the kernels (because of the proliferation of color indices, we resort
to Greek superscripts)
\begin{eqnarray}
\lefteqn{
\tilde{\Gamma}_{\sigma;\ov{c}cc\ov{c}}^{adef}(p_1,p_2,p_3,p_4)=
gf^{abc}\int\dk{k}W_{\sigma\nu}^{b\nu}(k)W_{\ov{c}c}^{c\gamma}(p_1-k)\times}
\nonumber\\&&
\left\{
\Gamma_{\ov{c}c\nu}^{f\mu\nu}(p_4,-p_4-k,k)W_{\ov{c}c}^{\mu\varepsilon}(p_4+k)
\left[\Gamma_{\ov{c}c\alpha}^{\varepsilon d\alpha}(p_4+k,p_2,-k-p_2-p_4)
W_{\alpha\kappa}^{\alpha\kappa}(k+p_2+p_4)\Gamma_{\ov{c}c\kappa}^{\gamma e\kappa}(p_1-k,p_3,k+p_2+p_4)
\right.\right.\nonumber\\&&\left.
-\Gamma_{\ov{c}c\alpha}^{\varepsilon e\alpha}(p_4+k,p_3,p_1+p_2-k)
W_{\alpha\kappa}^{\alpha\kappa}(k-p_1-p_2)
\Gamma_{\ov{c}c\kappa}^{\gamma d\kappa}(p_1-k,p_2,k-p_1-p_2)\right]
\nonumber\\&&
+\Gamma_{\ov{c}c\varepsilon}^{f\mu\varepsilon}(p_4,k+p_2,p_1+p_3-k)
W_{\ov{c}c}^{\mu\alpha}(-p_2-k)
\Gamma_{\ov{c}c\nu}^{\alpha d\nu}(-p_2-k,p_2,k)W_{\varepsilon\kappa}^{\varepsilon\kappa}(k+p_2+p_4)
\Gamma_{\ov{c}c\kappa}^{\gamma e\kappa}(p_1-k,p_3,k+p_2+p_4)
\nonumber\\&&
-\Gamma_{\ov{c}c\varepsilon}^{f\mu\varepsilon}(p_4,k+p_3,p_1+p_2-k)W_{\ov{c}c}^{\mu\alpha}(-p_3-k)
\Gamma_{\ov{c}c\nu}^{\alpha e\nu}(-p_3-k,p_3,k)W_{\varepsilon\kappa}^{\varepsilon\kappa}(k+p_3+p_4)
\Gamma_{\ov{c}c\kappa}^{\gamma d\kappa}(p_1-k,p_2,k+p_3+p_4)
\nonumber\\&&
+\Gamma_{\ov{c}c\alpha\nu}^{fe\alpha\nu}(p_4,p_3,-k-p_3-p_4,k)
W_{\alpha\kappa}^{\alpha\kappa}(k+p_3+p_4)\Gamma_{\ov{c}c\kappa}^{\gamma d\kappa}(p_1-k,p_2,k+p_3+p_4)
\nonumber\\&&
-\Gamma_{\ov{c}c\alpha\nu}^{fd\alpha\nu}(p_4,p_2,-k-p_2-p_4,k)
W_{\alpha\kappa}^{\alpha\kappa}(k+p_2+p_4)\Gamma_{\ov{c}c\kappa}^{\gamma e\kappa}(p_1-k,p_3,k+p_2+p_4)
\nonumber\\&&
+\Gamma_{\ov{c}c\ov{c}c}^{f\kappa\gamma d}(p_4,k+p_3,p_1-k,p_2)
W_{\ov{c}c}^{\kappa\alpha}(-p_3-k)\Gamma_{\ov{c}c\nu}^{\alpha e\nu}(-p_3-k,p_3,k)
\nonumber\\&&
-\Gamma_{\ov{c}c\ov{c}c}^{f\kappa\gamma e}(p_4,k+p_2,p_1-k,p_3)
W_{\ov{c}c}^{\kappa\alpha}(-p_2-k)\Gamma_{\ov{c}c\nu}^{\alpha d\nu}(-p_2-k,p_2,k)
\nonumber\\&&\left.
+\Gamma_{\ov{c}c\nu}^{f\mu\nu}(p_4,-p_4-k,k)W_{\ov{c}c}^{\mu\kappa}(p_4+k)
\Gamma_{\ov{c}c\ov{c}c}^{\kappa e\gamma d}(p_4+k,p_3,p_1-k,p_2)
-\Gamma_{\ov{c}c\ov{c}c\nu}^{fe\gamma d\nu}(p_4,p_3,p_1-k,p_2,k)
\right\},
\\
\lefteqn{
\tilde{\Gamma}_{\ov{c};\ov{c}cc\sigma}^{adeg}(p_1,p_2,p_3,p_4)=
gf^{abc}\int\dk{k}W_{\ov{c}c}^{b\nu}(k)W_{\ov{c}c}^{c\gamma}(p_1-k)
\times}
\nonumber\\&&
\left\{
2\Gamma_{\ov{c}c\sigma}^{\nu\mu g}(k,-p_4-k,p_4)W_{\ov{c}c}^{\mu\varepsilon}(p_4+k)
\Gamma_{\ov{c}c\alpha}^{\varepsilon e\alpha}(p_4+k,p_3,p_1+p_2-k)
W_{\alpha\kappa}^{\alpha\kappa}(k-p_1-p_2)\Gamma_{\ov{c}c\kappa}^{\gamma d\kappa}(p_1-k,p_2,k-p_1-p_2)
\right.\nonumber\\&&
-2\Gamma_{\ov{c}c\sigma}^{\nu\mu g}(k,-p_4-k,p_4)
W_{\ov{c}c}^{\mu\varepsilon}(p_4+k)
\Gamma_{\ov{c}c\alpha}^{\varepsilon d\alpha}(p_4+k,p_2,p_1+p_3-k)
W_{\alpha\kappa}^{\alpha\kappa}(k-p_1-p_3)\Gamma_{\ov{c}c\kappa}^{\gamma e\kappa}(p_1-k,p_3,k-p_1-p_3)
\nonumber\\&&
-2\Gamma_{\ov{c}c\varepsilon\sigma}^{\nu e\varepsilon g}(k,p_3,-p_3-p_4-k,p_4)
\Gamma_{\ov{c}c\kappa}^{\gamma d\kappa}(p_1-k,p_2,k+p_3+p_4)W_{\varepsilon\kappa}^{\varepsilon\kappa}(k+p_3+p_4)
\nonumber\\&&
+2\Gamma_{\ov{c}c\varepsilon\sigma}^{\nu d\varepsilon g}(k,p_2,-p_2-p_4-k,p_4)
\Gamma_{\ov{c}c\kappa}^{\gamma e\kappa}(p_1-k,p_3,k+p_2+p_4)W_{\varepsilon\kappa}^{\varepsilon\kappa}(k+p_2+p_4)
\nonumber\\&&
+2\Gamma_{\ov{c}c\varepsilon}^{\nu e\varepsilon}(k,p_3,-p_3-k)W_{\varepsilon\mu}^{\varepsilon\mu}(p_3+k)
\Gamma_{\mu\sigma\alpha}^{\mu g\alpha}(p_3+k,p_4,p_1+p_2-k)
W_{\alpha\kappa}^{\alpha\kappa}(k-p_1-p_2)\Gamma_{\ov{c}c\kappa}^{\gamma d\kappa}(p_1-k,p_2,k-p_1-p_2)
\nonumber\\&&\left.
-2\Gamma_{\ov{c}c\sigma}^{\nu\varepsilon g}(k,-p_4-k,p_4)W_{\ov{c}c}^{\varepsilon\kappa}(p_4+k)
\Gamma_{\ov{c}c\ov{c}c}^{\kappa e\gamma d}(p_4+k,p_3,p_1-k,p_2)
+\Gamma_{\ov{c}c\ov{c}c\sigma}^{\nu e\gamma dg}(k,p_3,p_1-k,p_2,p_4)
\right\}.
\end{eqnarray}
(A similar expression exists for the kernel
$\tilde{\Gamma}_{A;\ov{c}cc\ov{c}}$.) As before, since we may choose $q_0$ at
will and the energy ($k_4^0$) is scalar, we now have two more expressions
that are able to give $\Gamma_{\ov{c}c\sigma A}$ and $\Gamma_{\ov{c}c\sigma\si}$ in terms
of all the other previously considered Green's functions involving the
$\sigma$-field except one: $\Gamma_{\ov{c}c\ov{c}c\sigma}$. Also as before, one can
immediately verify this identity at tree-level using the Feynman rules of
Ref.~\cite{Watson:2007vc} (and repeated in Appendix~\ref{app:decomp}).
\subsection{Two pairs of further ghost derivatives}
In order to close the system of Slavnov--Taylor identities, we must find an equation for
$\Gamma_{\ov{c}c\ov{c}c\sigma}$. This equation arises by functionally
differentiating \eq{eq:wtid3} with respect to $\imath c_w^e$,
$\imath\ov{c}_v^f$, $\imath c_u^g$ and $\imath\ov{c}_r^h$ and then setting
sources to zero. Using the results Eqs.~(\ref{eq:tilde0}) and
(\ref{eq:tilde1}) and the appropriate functional derivatives of
\eq{eq:gheom0} we obtain \eq{eq:inter2} (the new kernel will be discussed
below). This expression is cyclic symmetric in the three ghost derivatives
$\imath c_z^d$, $\imath c_w^e$ and $\imath c_u^g$ and is antisymmetric in
the two derivatives $\imath\ov{c}_v^f$ and $\imath\ov{c}_r^h$. In momentum
space, the identity reads:
\begin{eqnarray}
0&=&k_5^0\Gamma_{\ov{c}c\ov{c}c\sigma}^{hgfde}(k_1,k_2,k_3,k_4,k_5)
-\frac{\imath k_{5i}}{\vec{k}_5^2}
\Gamma_{\ov{c}c\ov{c}cAi}^{hgfda}(k_1,k_2,k_3,k_4,k_5)
\Gamma_{\ov{c}c}^{ae}(-\vec{k}_5)
\nonumber\\&&
+(k_2^0+q_0)\Gamma_{\ov{c}c\ov{c}c\sigma}^{hdfeg}(k_1,k_4,k_3,k_5-q_0,k_2+q_0)
-\frac{\imath k_{2i}}{\vec{k}_2^2}
\Gamma_{\ov{c}c\ov{c}cAi}^{hdfea}(k_1,k_4,k_3,k_5-q_0,k_2+q_0)
\Gamma_{\ov{c}c}^{ag}(-\vec{k}_2)
\nonumber\\&&
+(k_4^0+q_0)\Gamma_{\ov{c}c\ov{c}c\sigma}^{hefgd}(k_1,k_5-q_0,k_3,k_2,k_4+q_0)
-\frac{\imath k_{4i}}{\vec{k}_4^2}
\Gamma_{\ov{c}c\ov{c}cAi}^{hefga}(k_1,k_5-q_0,k_3,k_2,k_4+q_0)
\Gamma_{\ov{c}c}^{ad}(-\vec{k}_4)
\nonumber\\&&
+\Gamma_{\ov{c}c\sigma}^{fda}(k_3,k_4,-k_3-k_4)
\tilde{\Gamma}_{\sigma;\ov{c}cc\ov{c}}^{aegh}(q_0+k_3+k_4,k_5-q_0,k_2,k_1)
\nonumber\\&&
+\Gamma_{\ov{c}c\sigma}^{fea}(k_3,k_5-q_0,q_0-k_3-k_5)
\tilde{\Gamma}_{\sigma;\ov{c}cc\ov{c}}^{agdh}(k_3+k_5,k_2,k_4,k_1)
\nonumber\\&&
+\Gamma_{\ov{c}c\sigma}^{fga}(k_3,k_2,-k_2-k_3)
\tilde{\Gamma}_{\sigma;\ov{c}cc\ov{c}}^{adeh}(q_0+k_2+k_3,k_4,k_5-q_0,k_1)
\nonumber\\&&
+\Gamma_{\ov{c}cAi}^{fda}(k_3,k_4,-k_3-k_4)
\tilde{\Gamma}_{A;\ov{c}cc\ov{c}i}^{aegh}(q_0+k_3+k_4,k_5-q_0,k_2,k_1)
\nonumber\\&&
+\Gamma_{\ov{c}cAi}^{fea}(k_3,k_5-q_0,q_0-k_3-k_5)
\tilde{\Gamma}_{A;\ov{c}cc\ov{c}i}^{agdh}(k_3+k_5,k_2,k_4,k_1)
\nonumber\\&&
+\Gamma_{\ov{c}cAi}^{fga}(k_3,k_2,-k_2-k_3)
\tilde{\Gamma}_{A;\ov{c}cc\ov{c}i}^{adeh}(q_0+k_2+k_3,k_4,k_5-q_0,k_1)
\nonumber\\&&
-\Gamma_{\ov{c}c\sigma}^{hda}(k_1,k_4,-k_1-k_4)
\tilde{\Gamma}_{\sigma;\ov{c}cc\ov{c}}^{aegf}(q_0+k_1+k_4,k_5-q_0,k_2,k_3)
\nonumber\\&&
-\Gamma_{\ov{c}c\sigma}^{hea}(k_1,k_5-q_0,q_0-k_1-k_5)
\tilde{\Gamma}_{\sigma;\ov{c}cc\ov{c}}^{agdf}(k_1+k_5,k_2,k_4,k_3)
\nonumber\\&&
-\Gamma_{\ov{c}c\sigma}^{hga}(k_1,k_2,-k_1-k_2)
\tilde{\Gamma}_{\sigma;\ov{c}cc\ov{c}}^{adef}(q_0+k_1+k_2,k_4,k_5-q_0,k_3)
\nonumber\\&&
-\Gamma_{\ov{c}cAi}^{hda}(k_1,k_4,-k_1-k_4)
\tilde{\Gamma}_{A;\ov{c}cc\ov{c}i}^{aegf}(q_0+k_1+k_4,k_5-q_0,k_2,k_3)
\nonumber\\&&
-\Gamma_{\ov{c}cAi}^{hea}(k_1,k_5-q_0,q_0-k_1-k_5)
\tilde{\Gamma}_{A;\ov{c}cc\ov{c}i}^{agdf}(k_1+k_5,k_2,k_4,k_3)
\nonumber\\&&
-\Gamma_{\ov{c}cAi}^{hga}(k_1,k_2,-k_1-k_2)
\tilde{\Gamma}_{A;\ov{c}cc\ov{c}i}^{adef}(q_0+k_1+k_2,k_4,k_5-q_0,k_3)
\nonumber\\&&
+\frac{1}{2}\Gamma_{\ov{c}c\ov{c}c}^{hdfa}(k_1,k_4,k_3,k_2+k_5)
\left[\tilde{\Gamma}_{\ov{c};\ov{c}cc}^{aeg}(q_0-k_2-k_5,k_5-q_0,k_2)
-2gf^{aeg}\right]
\nonumber\\&&
+\frac{1}{2}\Gamma_{\ov{c}c\ov{c}c}^{hefa}(k_1,k_5-q_0,k_3,q_0+k_2+k_4)
\left[\tilde{\Gamma}_{\ov{c};\ov{c}cc}^{agd}(-k_2-k_4,k_2,k_4)
-2gf^{agd}\right]
\nonumber\\&&
+\frac{1}{2}\Gamma_{\ov{c}c\ov{c}c}^{hgfa}(k_1,k_2,k_3,k_4+k_5)
\left[\tilde{\Gamma}_{\ov{c};\ov{c}cc}^{ade}(q_0-k_4-k_5,k_4,k_5-q_0)
-2gf^{ade}\right]
\nonumber\\&&
+\frac{1}{2}\Gamma_{\ov{c}c}^{fa}(\vec{k}_3)
\tilde{\Gamma}_{\ov{c};\ov{c}ccc\ov{c}}^{adegh}(q_0+k_3,k_4,k_5-q_0,k_2,k_1)
-\frac{1}{2}\Gamma_{\ov{c}c}^{ha}(\vec{k}_1)
\tilde{\Gamma}_{\ov{c};\ov{c}ccc\ov{c}}^{adegf}(q_0+k_1,k_4,k_5-q_0,k_2,k_3).
\label{eq:stid5}
\end{eqnarray}
Aside from the Green's function that we wish to calculate
($\Gamma_{\ov{c}c\ov{c}c\sigma}$), there is only one further unknown kernel:
$\tilde{\Gamma}_{\ov{c};\ov{c}ccc\ov{c}}$, given in \eq{eq:rnd4}. Importantly,
this kernel introduces no new Green's functions involving functional
derivatives with respect to the $\sigma$-field and in momentum space, it reads:
\begin{eqnarray}
\lefteqn{
\tilde{\Gamma}_{\ov{c};\ov{c}ccc\ov{c}}^{adegf}(p_1,p_2,p_3,p_4,p_5)=
gf^{abc}\int\dk{k}W_{\ov{c}c}^{b\mu}(k)W_{\ov{c}c}^{c\nu}(p_1-k)
\left\{
-\frac{1}{3}\Gamma_{\ov{c}c\ov{c}c\ov{c}c}^{\mu gfe\nu d}(k,p_4,p_5,p_3,p_1-k,p_2)
\right.}\nonumber\\&&
+2\Gamma_{\ov{c}c\ov{c}c}^{f\kappa\nu d}(p_5,k+p_3+p_4,p_1-k,p_2)
W_{\ov{c}c}^{\kappa\varepsilon}(-k-p_3-p_4)
\Gamma_{\ov{c}c\ov{c}c}^{\mu g\varepsilon e}(k,p_4,-k-p_3-p_4,p_3)
\nonumber\\&&
+2\Gamma_{\ov{c}c\ov{c}c\beta}^{fd\nu g\beta}(p_5,p_2,p_1-k,p_4,k+p_3)
W_{\alpha\beta}^{\alpha\beta}(k+p_3)\Gamma_{\ov{c}c\alpha}^{\mu e\alpha}(k,p_3,-k-p_3)
\nonumber\\&&
+2\Gamma_{\ov{c}c\gamma\beta}^{fd\gamma\beta}(p_5,p_2,p_1+p_3-k,k+p_4)
\Gamma_{\ov{c}c\alpha}^{\mu g\alpha}(k,p_4,-k-p_4)W_{\alpha\beta}^{\alpha\beta}(k+p_4)
W_{\gamma\delta}^{\gamma\delta}(k-p_1-p_3)\Gamma_{\ov{c}c\delta}^{\nu e\delta}(p_1-k,p_3,k-p_1-p_3)
\nonumber\\&&
+2\Gamma_{\ov{c}c\beta}^{f\kappa\beta}(p_5,-k-p_2-p_5,k+p_2)
\Gamma_{\ov{c}c\alpha}^{\mu d\alpha}(k,p_2,-k-p_2)
\Gamma_{\ov{c}c\delta}^{\nu e\delta}(p_1-k,p_3,k-p_1-p_3)
\Gamma_{\ov{c}c\gamma}^{\varepsilon g\gamma}(k+p_2+p_5,p_4,p_1+p_3-k)
\nonumber\\&&
\times W_{\alpha\beta}^{\alpha\beta}(k+p_2)W_{\gamma\delta}^{\gamma\delta}(k-p_1-p_3)
W_{\ov{c}c}^{\kappa\varepsilon}(k+p_2+p_5)
\nonumber\\&&
+2\Gamma_{\ov{c}c\gamma}^{f\kappa\gamma}(p_5,k+p_2+p_4,p_1+p_3-k)
\Gamma_{\ov{c}c\alpha}^{\mu d\alpha}(k,p_2,-k-p_2)
\Gamma_{\ov{c}c\delta}^{\nu e\delta}(p_1-k,p_3,k-p_1-p_3)
\Gamma_{\ov{c}c\beta}^{\varepsilon g\beta}(-k-p_2-p_4,p_4,k+p_2)
\nonumber\\&&
\times W_{\alpha\beta}^{\alpha\beta}(k+p_2)W_{\gamma\delta}^{\gamma\delta}(k-p_1-p_3)
W_{\ov{c}c}^{\kappa\varepsilon}(-k-p_2-p_4)
\nonumber\\&&
-2\Gamma_{\ov{c}c\beta}^{f\kappa\beta}(p_5,-k-p_3-p_5,k+p_3)
\Gamma_{\ov{c}c\ov{c}c}^{\varepsilon d\nu g}(k+p_3+p_5,p_2,p_1-k,p_4)
\Gamma_{\ov{c}c\alpha}^{\mu e\alpha}(k,p_3,-k-p_3)
\nonumber\\&&
\times W_{\alpha\beta}^{\alpha\beta}(k+p_3)W_{\ov{c}c}^{\kappa\varepsilon}(k+p_3+p_5)
\nonumber\\&&
+2\Gamma_{\ov{c}c\ov{c}c}^{f\kappa\nu d}(p_5,k+p_3+p_4,p_1-k,p_2)
\Gamma_{\ov{c}c\alpha}^{\mu e\alpha}(k,p_3,-k-p_3)
\Gamma_{\ov{c}c\beta}^{\varepsilon g\beta}(-k-p_3-p_4,p_4,k+p_3)
\nonumber\\&&
\times W_{\alpha\beta}^{\alpha\beta}(k+p_3)W_{\ov{c}c}^{\kappa\varepsilon}(-k-p_3-p_4)
\nonumber\\&&
-2\Gamma_{\ov{c}c\ov{c}c}^{f\kappa\nu d}(p_5,k+p_3+p_4,p_1-k,p_2)
\Gamma_{\ov{c}c\alpha}^{\mu g\alpha}(k,p_4,-k-p_4)
\Gamma_{\ov{c}c\beta}^{\varepsilon e\beta}(-k-p_3-p_4,p_3,k+p_4)
\nonumber\\&&\left.
\times W_{\alpha\beta}^{\alpha\beta}(k+p_4)W_{\ov{c}c}^{\kappa\varepsilon}(-k-p_3-p_4)
+\mbox{c.p. ($c^d(p_2),c^e(p_3),c^g(p_4)$)}\right\}
\end{eqnarray}
where we utilize the cyclic symmetry (cyclic symmetric terms denoted by
$\mbox{c.p.}$ and this includes the first factor
$\Gamma_{\ov{c}c\ov{c}c\ov{c}c}$ which is the origin of the factor $1/3$).
Because of the cyclic symmetry and the fact that $q_0$ is arbitrary,
\eq{eq:stid5} can be solved for $\Gamma_{\ov{c}c\ov{c}c\sigma}$ and we have finally
managed to close the set of Slavnov--Taylor identities. The identity,
\eq{eq:stid5}, trivially has no tree-level form.
We have thus shown that just as for the temporal two-point functions, all
temporal Green's functions considered so far can be deduced (at least in
principle) from their non-temporal counterparts as solutions to the
Slavnov--Taylor identities. The temporal $\sigma$-field has been effectively
eliminated from the system, or integrated out of functional form of the
action. Again, as for the two-point functions we see that the
Slavnov--Taylor identities relate temporal, (spatially) longitudinal and
ghost Green's functions in a manner reminiscent of the Kugo--Ojima quartet
mechanism \cite{Kugo:1979gm}.
\subsection{Further Slavnov--Taylor identities}
It has been shown so far that there exists a closed set of Slavnov--Taylor
identities that includes the vertex (three-point proper) Green's
functions. However, one may also consider four-point functions (e.g.,
$\Gamma_{AAA\sigma}$). Because of their extended nature, we do not attempt to
derive these identities completely; rather, we shall merely sketch their
form in order to justify that the equations close just as before. We begin
as previously with \eq{eq:wtid3} and again, the sets of derivatives are
distinguished by how many pairs of ghost/antighost functional derivatives
are taken. We will highlight only those temporal Green's functions or
kernels that have not previously appeared and leave those quantities that
have already been derived as implicit. The sequence is as follows:
\begin{enumerate}
\item
Taking functional derivatives of \eq{eq:wtid3} with respect to
$\imath\Phi_\kappa$, $\imath\Phi_\tau$ and $\imath\Phi_\lambda$ (as before, the
ghost derivatives will be made explicit and the field types here refer only
to either the $\vec{A}$ or $\sigma$-fields), one clearly obtains an identity
for $\Gamma_{\kappa\tau\lambda\sigma}$ in terms of $\Gamma_{\kappa\tau\lambda A}$ and a new kernel
$\tilde{\Gamma}_{\sigma;\ov{c}c\lambda\tau}$. This kernel, following (as a further
functional derivative of) \eq{eq:rnd1} introduces the new quantity:
$\Gamma_{\ov{c}c\kappa\tau\lambda}$. Starting with $\Gamma_{AAAA}$, we can then
sequentially build up to $\Gamma_{\sigma\si\sigma\si}$, given all the
$\Gamma_{\ov{c}c\kappa\tau\lambda}$.
\item
Next we take one pair of ghost functional derivatives (i.e., functionally
differentiate \eq{eq:wtid3} with respect to $\imath\ov{c}$ and $\imath c$)
and then derivatives with respect to $\imath\Phi_\tau$ and
$\imath\Phi_\lambda$. This gives us equations for the $\Gamma_{\ov{c}c\tau\lambda\sigma}$
in terms of the $\Gamma_{\ov{c}c\tau\lambda A}$ and we have two new kernels:
$\tilde{\Gamma}_{\sigma;\ov{c}cc\ov{c}\lambda}$ and
$\tilde{\Gamma}_{\ov{c};\ov{c}cc\lambda\tau}$. These kernels follow from
Eqs.~(\ref{eq:rnd2}) and (\ref{eq:rnd3}), respectively, and both introduce
the new function: $\Gamma_{\ov{c}c\ov{c}c\tau\lambda}$.
\item
With two pairs of ghost functional derivatives of \eq{eq:wtid3}, plus one
further derivative with respect to $\imath\Phi_\lambda$ we generate equations
for $\Gamma_{\ov{c}c\ov{c}c\lambda\sigma}$ in terms of $\Gamma_{\ov{c}c\ov{c}c\lambda A}$,
again with a new kernel: $\tilde{\Gamma}_{\ov{c};\ov{c}ccc\ov{c}\lambda}$. This
kernel follows from \eq{eq:rnd4} and introduces a final Green's function:
$\Gamma_{\ov{c}c\ov{c}c\ov{c}c\lambda}$.
\item
We finally take three pairs of ghost functional derivatives to get an
equation for $\Gamma_{\ov{c}c\ov{c}c\ov{c}c\sigma}$ in terms of
$\Gamma_{\ov{c}c\ov{c}c\ov{c}cA}$. However, this equation does not involve any
further kernels and the set of equations terminates.
\end{enumerate}
Now, all these equations have the same characteristics as in the previous
subsections: the energy is a scalar quantity and there is the energy
injection scale $q_0$, such that one has a set of unambiguous (albeit
nonlinear and extremely long) equations from which the temporal Green's
functions may be derived, given the set of spatial and ghost Green's
functions as external input.
We conjecture that in principle, the closed sets of Slavnov--Taylor
identities in Coulomb gauge may be extended to include all higher
$n$-point Green's functions and that they may be solved to give exact
expressions for all the temporal Green's functions. As may be appreciated
though, how to prove such a general statement is not clear. One may
formulate some ideas based on the following observations: Firstly, the
scalar nature of the energy and the energy injection scale ($q_0$) will
always be present in the Slavnov--Taylor identities of Coulomb gauge
courtesy of the particular Gauss-BRST invariance of the theory (this
certainly improves on the situation in linear covariant gauges where one
has contractions of tensors such that transverse parts cannot be directly
deduced). Second, because ghost derivatives must come in pairs, each
further pair reduces the number of possible non-ghost internal lines within
the kernels -- the external ghost legs must be connected by an internal
ghost propagator or a vertex with these two ghost legs. This restricts the
number of possible internal temporal Green's functions such that eventually
the set of identities closes. One can see the emergence of a characteristic
pattern for the ghost functions necessary to form the kernels in the steps
above: i.e., $(1.)~\Gamma_{\ov{c}c\kappa\tau\lambda}\rightarrow(2.)
~\Gamma_{\ov{c}c\ov{c}c\tau\lambda}\rightarrow(3.)
~\Gamma_{\ov{c}c\ov{c}c\ov{c}c\lambda}\rightarrow(4.)~0$.
\section{\label{sec:five}Simplifications of the identities}
\setcounter{equation}{0}
Given that the Slavnov--Taylor identities derived in previous sections
relate the various types of Green's functions in an extended manner, it is
pertinent to ask whether these relationships reduce in specific
circumstances such that unambiguous statements can be made about the
behavior of individual Green's functions, in particular the two-point
functions. The motivation is clear: such information (particularly in the
infrared region) may be useful for determining how the confinement mechanism
manifests itself (asymptotic freedom already being perturbatively
established in the ultraviolet region of Coulomb gauge \cite{Watson:2007vc})
and also provides for input in other nonperturbative studies. We argue here
that unfortunately such information cannot be unambiguously extracted.
In the noncovariant Coulomb gauge, we must first sort out the temporal
(energy) and spatial (momentum) scales. The most obvious simplification of
the Slavnov--Taylor identities concerns purely spacelike momenta where we
know that if the postulate of Euclidicity is to hold (i.e., that the Wick
rotation is valid), the Green's functions presumably can have no
singularities. One may in principle also consider timelike configurations,
but this is certainly beyond the scope of the present analysis. Thus, we
must first set all energy scales to zero. We define the infrared region as
the limit as one or more of the momenta vanishes and we approach the
light-cone, where singularities (or nontrivial zeroes) may appear.
Let us begin by considering the Slavnov--Taylor identities for the
two-point functions: Eqs.~(\ref{eq:stid0}) and (\ref{eq:stid1}). Using the
general decompositions of Ref.~\cite{Watson:2007vc} (presented also in
Appendix~\ref{app:decomp}) in terms of the (scalar) dressing functions, we
get for general kinematical configurations:
\begin{eqnarray}
-\imath k_0\vec{k}^2\Gamma_{\sigma\si}(k_0^2,\vec{k}^2)&=&
-\imath k_0\vec{k}^2\Gamma_{A\sigma}(k_0^2,\vec{k}^2)\Gamma_{\ov{c}c}(\vec{k}^2),
\nonumber\\
\imath k_0^2k_k\Gamma_{A\sigma}(k_0^2,\vec{k}^2)&=&
\imath k_0^2k_k\ov{\Gamma}_{AA}(k_0^2,\vec{k}^2)\Gamma_{\ov{c}c}(\vec{k}^2).
\end{eqnarray}
Since both equations have common kinematical prefactors, we can then write
\begin{equation}
\Gamma_{\sigma\si}(k_0^2,\vec{k}^2)
=\Gamma_{A\sigma}(k_0^2,\vec{k}^2)\Gamma_{\ov{c}c}(\vec{k}^2)
=\ov{\Gamma}_{AA}(k_0^2,\vec{k}^2)\left[\Gamma_{\ov{c}c}(\vec{k}^2)\right]^2.
\label{eq:2ptstid}
\end{equation}
This holds for \emph{any} kinematical configuration and we conclude that
the Slavnov--Taylor identities for the two-point functions alone can give
no information about the value of the Green's functions, only the
relationship between them.
Slightly less trivial is the Slavnov--Taylor identity for the ghost-gluon
vertex, \eq{eq:stid3}. Concentrating on spacelike momenta as discussed
above, we see that neither of the $\Gamma_{\ov{c}c\sigma}$ vertices will contribute
when the energy is set to zero. To study the equation, let us firstly
decompose the ghost-gluon vertex into tree-level and dressed parts as
follows:
\begin{equation}
\Gamma_{\ov{c}cAi}^{abc}(p_1,p_2,p_3)=-\imath gf^{abc}p_{1i}
-\imath gf^{abc} p_{1j}\tilde{\Gamma}_{A;\ov{c}cAji}(p_1,p_2,p_3).
\label{eq:ghgldse}
\end{equation}
Let us explain this decomposition. Since we are dealing with a three-point
function, the color factor ($f^{abc}$) and coupling ($g$) will always be
common and can be extracted. The appearance of $\tilde{\Gamma}$ stems from the
momentum space form of \eq{eq:ghgldse0} and in fact, given the definition
\eq{eq:ghgldse1}, the above equation, \eq{eq:ghgldse}, is the Dyson--Schwinger equation
for the spatial ghost-gluon vertex. As will be seen shortly, the
contraction of $p_{1j}$ with $\tilde{\Gamma}_{A;\ov{c}cAji}$ is important.
Lastly, for the $\tilde{\Gamma}_{\ov{c};\ov{c}cc}$ kernel, we can only extract
the color and coupling factors and we write
\begin{equation}
\tilde{\Gamma}_{\ov{c};\ov{c}cc}^{abc}=gf^{abc}\tilde{\Gamma}_{\ov{c};\ov{c}cc}
\end{equation}
At zero energy, \eq{eq:stid3} in terms of dressing functions is thus
\begin{eqnarray}
0&=&-\s{\vec{k}_1}{\vec{k}_3}\Gamma_{\ov{c}c}(\vec{k}_3^2)
-\s{\vec{k}_1}{\vec{k}_2}\Gamma_{\ov{c}c}(\vec{k}_2^2)
-\vec{k}_1^2\Gamma_{\ov{c}c}(\vec{k}_1^2)
-k_{1j}k_{3i}\tilde{\Gamma}_{A;\ov{c}cAji}(\vec{k}_1,\vec{k}_2,\vec{k}_3)
\Gamma_{\ov{c}c}(\vec{k}_3^2)
-k_{1j}k_{2i}\tilde{\Gamma}_{A;\ov{c}cAji}(\vec{k}_1,\vec{k}_3,\vec{k}_2)
\Gamma_{\ov{c}c}(\vec{k}_2^2)\nonumber\\
&&+\frac{1}{2}\vec{k}_1^2\Gamma_{\ov{c}c}(\vec{k}_1^2)
\tilde{\Gamma}_{\ov{c};\ov{c}cc}(\vec{k}_1,\vec{k}_2,\vec{k}_3).
\label{eq:lim0}
\end{eqnarray}
This equation is directly analogous to the Landau gauge Slavnov--Taylor
identity for the ghost-gluon vertex which was studied under the truncation
$\tilde{\Gamma}_{\ov{c};\ov{c}cc}=0$ \cite{von Smekal:1997vx}. Quite generally,
it states that the kernels ($\tilde{\Gamma}_{A;\ov{c}cAji}$ and
$\tilde{\Gamma}_{\ov{c};\ov{c}cc}$) and the two-point ghost dressing function
$\Gamma_{\ov{c}c}$ are nontrivially related.
Let us now discuss the above identity, \eq{eq:lim0}, in the infrared to
assess whether it simplifies further. Because of the symmetry, there are
two limits of interest: $k_1\rightarrow0$ and $k_2\rightarrow0$. For the
infrared limit $k_1\rightarrow0$, we see that the entire equation has an
overall factor $\sim|k_1|$ (in fact, one can show that there is the overall
factor $|k_1|^2$). Therefore, in the limit, no single Green's function is
isolated from which to determine a value for this kinematical
configuration. Further, the Green's functions or their combination may even
be singular. For the limit $k_2\rightarrow0$, we recall that in Coulomb
gauge, ghost vertex dressing functions vanish as the ``in-ghost" momentum
vanishes \cite{Watson:2006yq}, i.e.,
\begin{eqnarray}
\tilde{\Gamma}_{A;\ov{c}cAji}(k_1,k_2,k_3)&
\stackrel{\vec{k}_2\rightarrow0}{\propto}&|\vec{k}_2|,\nonumber\\
\tilde{\Gamma}_{\ov{c};\ov{c}cc}(k_1,k_2,k_3)&
\stackrel{\vec{k}_2\rightarrow0}{\propto}&|\vec{k}_2|.
\label{eq:p2zero}
\end{eqnarray}
This is, of course, exactly the same as in Landau gauge
\cite{Taylor:1971ff}. Using the momentum conservation, $k_3=-k_1-k_2$, and
we see that \eq{eq:lim0} has the overall factor $|k_2|$ as
$k_2\rightarrow0$. This leads to the conclusion that no information about
an individual Green's function may be extracted in this limit either. We
conclude that on its own, the Slavnov--Taylor identity for the ghost-gluon
vertex, \eq{eq:stid3}, does not yield unambiguous information about the
Green's functions without further knowledge.
For the Slavnov--Taylor identities given by \eq{eq:stid2}, there is one
special case where simplification occurs and this again results in a
situation exactly analogous to Landau gauge. Considering the $\Gamma_{AA\sigma}$
identity (i.e., setting the external indices $\lambda$ and $\tau$ to be
referring to $\vec{A}$-fields), one has explicitly
\begin{eqnarray}
k_3^0\Gamma_{AA\sigma lk}^{fed}(k_1,k_2,k_3)
&=&
\imath\frac{k_{3i}}{\vec{k}_3^2}\Gamma_{3Alki}^{fea}(k_1,k_2,k_3)
\Gamma_{\ov{c}c}^{ad}(-\vec{k}_3)
\nonumber\\&&
-\Gamma_{AAki}^{ea}(k_2)
\left[\tilde{\Gamma}_{A;\ov{c}cAil}^{adf}(k_2+q_0,k_3-q_0,k_1)
+gf^{adf}\delta_{il}\right]
\nonumber\\&&
-\Gamma_{AAli}^{fa}(k_1)
\left[\tilde{\Gamma}_{A;\ov{c}cAik}^{ade}(k_1+q_0,k_3-q_0,k_2)
+gf^{ade}\delta_{ik}\right]
\nonumber\\&&
-\Gamma_{A\sigma k}^{ea}(k_2)\tilde{\Gamma}_{\sigma;\ov{c}cAl}^{adf}(k_2+q_0,k_3-q_0,k_1)
\nonumber\\&&
-\Gamma_{A\sigma l}^{fa}(k_1)\tilde{\Gamma}_{\sigma;\ov{c}cAk}^{ade}(k_1+q_0,k_3-q_0,k_2).
\end{eqnarray}
Further setting all energy scales to zero and using the general
decompositions ($\Gamma_{3A}^{abc}=-\imath gf^{abc}\Gamma_{3A}$, for the rest, see
above or Appendix~\ref{app:decomp}) one has
\begin{eqnarray}
k_{3i}\Gamma_{3Alki}(\vec{k}_1,\vec{k}_2,\vec{k}_3)\Gamma_{\ov{c}c}(\vec{k}_3^2)
&=&\vec{k}_2^2t_{kl}(\vec{k}_2)\Gamma_{AA}(\vec{k}_2^2)
+\vec{k}_2^2t_{ki}(\vec{k}_2)\Gamma_{AA}(\vec{k}_2^2)
\tilde{\Gamma}_{A;\ov{c}cAil}(\vec{k}_2,\vec{k}_3,\vec{k}_1)
\nonumber\\
&&-\vec{k}_1^2t_{lk}(\vec{k}_1)\Gamma_{AA}(\vec{k}_1^2)
-\vec{k}_1^2t_{li}(\vec{k}_1)\Gamma_{AA}(\vec{k}_1^2)
\tilde{\Gamma}_{A;\ov{c}cAik}(\vec{k}_1,\vec{k}_3,\vec{k}_2).
\end{eqnarray}
Taking the contraction of the above equation with $k_{2k}$, one has thus
\begin{equation}
k_{2k}k_{3i}\Gamma_{3Alki}(\vec{k}_1,\vec{k}_2,\vec{k}_3)=
-\vec{k}_1^2\frac{\Gamma_{AA}(\vec{k}_1^2)}{\Gamma_{\ov{c}c}(\vec{k}_3^2)}
t_{lj}(\vec{k}_1)
\left[k_{2j}
+k_{2k}\tilde{\Gamma}_{A;\ov{c}cAjk}(\vec{k}_1,\vec{k}_3,\vec{k}_2)\right].
\end{equation}
Now, the overall Bose--symmetry of the three-gluon vertex means that after
extracting the color factor ($f^{abc}$), the dressing function $\Gamma_{3A}$ is
antisymmetric under exchange of any two legs and so, by interchanging
$k_{2k}\leftrightarrow k_{3i}$ one can eliminate the three-gluon vertex to
obtain an expression involving only the two-point ghost dressing function
($\Gamma_{\ov{c}c}$) and the kernel $\tilde{\Gamma}_{A;\ov{c}cAjk}$ which, after
canceling out the overall factors reads:
\begin{equation}
0=t_{lj}(\vec{k}_1)\left\{k_{2j}\Gamma_{\ov{c}c}(\vec{k}_2^2)
+k_{3j}\Gamma_{\ov{c}c}(\vec{k}_3^2)+\Gamma_{\ov{c}c}(\vec{k}_2^2)k_{2k}
\tilde{\Gamma}_{A;\ov{c}cAjk}(\vec{k}_1,\vec{k}_3,\vec{k}_2)
+\Gamma_{\ov{c}c}(\vec{k}_3^2)k_{3k}
\tilde{\Gamma}_{A;\ov{c}cAjk}(\vec{k}_1,\vec{k}_2,\vec{k}_3)\right\}.
\label{eq:tripleA}
\end{equation}
In Landau gauge, this identity is well-known \cite{Kim:1979ep,Ball:1980ax}
and was considered in Ref.~\cite{Boucaud:2008ky} under reasonable
assumptions to be indicative of an infrared finite ghost dressing function
($\Gamma_{\ov{c}c}$). In the infrared limit $k_2\rightarrow0$ (equivalently
for $k_3\rightarrow0$), we see that all terms have the prefactor $|k_2|$
since $k_{3j}t_{lj}(\vec{k}_1)=-k_{2j}t_{lj}(\vec{k}_1)$ and for the kernel
we have the general infrared result \eq{eq:p2zero}. Without further
assumption, the following (conservative) statement is true of
\eq{eq:tripleA}: if the \emph{particular contraction} above of the
ghost-gluon vertex kernel ($\tilde{\Gamma}_{A;\ov{c}cAjk}$) is vanishing, then
the ghost dressing function is constant. However, notice that the
`particular contraction' above is not the same as that appearing in either
the Slavnov--Taylor identity for the ghost-gluon vertex, \eq{eq:lim0} or
the Dyson--Schwinger equations from which $\Gamma_{\ov{c}c}$ can be obtained. Thus we
conclude, as previously, that the above component, \eq{eq:tripleA}, of the
Slavnov--Taylor identity, \eq{eq:stid2} does not yield unambiguous
information about particular Green's functions without further input.
The special symmetric contraction case (above) of the three-gluon vertex
Slavnov--Taylor identity aside, in order to say anything about the two-point
functions from the identities \eq{eq:stid2}, one must know something about
at least one of the vertices $\Gamma_{\tau\lambda\sigma}$ or $\Gamma_{\tau\lambda A}$. Such
information is not available from general considerations and so, one cannot
make any simple statement. Further, the rest of the Slavnov--Taylor
identities, equations~(\ref{eq:stid4}) and (\ref{eq:stid5}), clearly involve
too many higher $n$-point functions to have any hope of simplification.
From the above discussion, it seems clear that no information about the
behavior of the two-point functions (or vertices) can be obtained from the
Slavnov--Taylor identities alone. As is obvious from the simple
Slavnov--Taylor identities for the two-point functions summarized by
\eq{eq:2ptstid}, one can only determine the relationship between the
various Green's functions and this must be true for any kinematical
configuration. Our only unambiguous `outside' knowledge about the Green's
functions was the peculiar infrared behavior of the ghost vertices, the
absence of singularities for spacelike momenta and the Bose-symmetry of the
three-gluon vertex (which is why the ghost-gluon vertex identity,
\eq{eq:stid3}, and the identity \eq{eq:tripleA} were of particular
interest), but even then, the dimensionality or complexity of the
Slavnov--Taylor identities denied concrete conclusions.
\section{Summary, discussion and conclusions}
\setcounter{equation}{0}
In this paper, the Slavnov--Taylor identities for Coulomb gauge Yang--Mills
theory within the second order formalism have been derived. The starting
point was the Gauss-BRST invariance of the action \cite{Zwanziger:1998ez},
characterized by the \emph{time-dependent} BRS variation and peculiar to
Coulomb gauge. It was found that for the two-point and vertex Green's
functions, the Slavnov--Taylor identities form closed sets from which the
temporal Green's functions can be unambiguously derived given the relevant
spatial gluon and ghost Green's functions as input. The extension of this
to higher order Green's functions was also discussed. Special cases of the
identities were studied and it was seen that there is no simplification such
that information about a particular Green's function could be extracted.
It is worthwhile at this stage to discuss the Coulomb gauge Slavnov--Taylor
identities in their wider context since the connection between several
different themes becomes apparent. The first of these themes centers around
Gauss' law. In classical electrodynamics, Gauss' law allows one to
determine the temporal component of the gauge field (i.e., $\sigma$) directly
from the physical charge distribution. In the canonical (Hamiltonian-based)
formulation of Yang--Mills theory, Gauss' law is applied as an operator
identity to define the physical state space and can be explicitly resolved
in Coulomb gauge, yielding the so-called Coulomb term which comprises the
confining properties of the theory \cite{Feuchter:2004mk}. In the
functional formalism, Gauss' law appears after the elimination of the
temporal ($\sigma$) field from the action, either directly as in the case of
the first order formalism \cite{Reinhardt:2008pr,Zwanziger:1998ez} or as
here as the elimination from the effective action via the Slavnov--Taylor
identities. The resolution of Gauss' law and its connection to the physical
state space is certainly less apparent in the functional formalism since
Green's functions are not directly related to physical observables but this
is compensated for by the observation that the cancellation of the
gauge-dependent degrees of freedom (i.e., temporal, longitudinal and ghost)
is manifest --- in the first order formalism one can reduce the functional
integral to transverse spatial gluon degrees of freedom and the
Slavnov--Taylor identities here express this explicitly for the Green's
functions in local fashion. However, the connection between the physical
state space and the Green's functions is understood conceptually within
the framework of the Kugo--Ojima confinement scenario \cite{Kugo:1979gm}:
by postulating a well-defined BRS charge and physical state space, the
cancellation of the gauge-dependent degrees of freedom followed and in
Landau gauge led to the prediction that the ghost propagator is infrared
enhanced. In the Coulomb gauge functional formalism, we know explicitly
that there exists at least a total charge that is conserved and vanishing
\cite{Reinhardt:2008pr} which partly confirms the Kugo--Ojima postulate.
This total charge arises from considering the temporal zero-modes inherent
to Coulomb gauge and leads to the formal demonstration of the cancellation
of the gauge dependent degrees of freedom. In this study, we have shown
that the Slavnov--Taylor identities stemming from the temporally nontrivial
Gauss-BRST transform supply this cancellation in local fashion. In this
respect, the temporal features of the Coulomb gauge functional formalism can
be seen to supply a link between the physical charge and states to the
Green's functions of the underlying theory.
The temporally nontrivial nature of Coulomb gauge is manifested in two
ways. On the one hand, resolving the temporal zero-modes leads (as
mentioned above) to the vanishing and conserved total charge
\cite{Reinhardt:2008pr}; on the other hand, the extra temporal degree of
freedom in the Gauss-BRST transform leads to the closure of the
Slavnov--Taylor identities. The temporal zero-modes of the Faddeev--Popov
operator lead us to consider the connection to the Gribov--Zwanziger picture
of confinement \cite{Gribov:1977wm,Zwanziger:1995cv,Zwanziger:1998ez}. In
this scenario, it is recognized that the zero-modes (which are induced by
incomplete gauge-fixing) should be separated from the functional integration
and via stochastic quantization and entropy arguments, the authors were
able to show that the resulting spatial transverse gluon propagator would
be suppressed in the infrared (and thus drops out from the physical
spectrum) whereas the temporal propagator provided for a long-range
confining force. Just as with the Kugo--Ojima scenario, the ghost
propagator in Landau gauge would also be infrared enhanced. In the Coulomb
gauge functional formalism insofar as the temporal zero-modes are
concerned, one can see the parallels: their resolution leads to a total
charge and the cancellation of gauge degrees of freedom, providing an
explicit demonstration of selected features of both the Gribov--Zwanziger
and Kugo--Ojima confinement scenarios. However, one part is evidently
missing --- a prediction for the infrared behavior of the ghost (or
something similar). Resolving the temporal zero-modes in Coulomb gauge
restricted the functional integral to field configurations such that the
total charge is conserved and vanishing, whereas the same temporal feature
gave rise to Slavnov--Taylor identities that form closed sets but
explicitly no information about a particular Green's function. This
underlies the quite general feature of functional techniques, namely, that
in order to talk about Green's functions one must take functional
derivatives and in this respect, the Slavnov--Taylor identities (and for
that matter, the Dyson--Schwinger equations too) represent functional differential
equations. Their solution is known only up to some `constant' of
integration (for explicit examples of this, see
Refs.~\cite{Epple:2007ut,Reinhardt:2008ij,Fischer:2008uz} and references
therein). In general this is not obvious since one necessarily has an
infinite tower of equations to consider, but in the case of the Coulomb
gauge Slavnov--Taylor identities derived here, the closure allows us to see
clearly that this is precisely the case --- functional techniques lead to
\emph{relations} between Green's functions.
Pragmatically, there are two natural avenues which to explore further.
The first is to search for the `missing' prediction about the value of a
specific Green's function (most likely the infrared behavior of the ghost)
in order to complete the connection between the temporal aspects of Coulomb
gauge and the two confinement scenarios: Kugo--Ojima and
Gribov--Zwanziger. The second direction to take is to use the
Slavnov--Taylor identities here in order to construct a charge-conserving
truncation scheme from which to solve the Dyson--Schwinger equations of Coulomb gauge
Yang--Mills theory, allowing for the eventuality that the solution may only
be determined up to some external `boundary condition' in analogy to the
solution of standard differential equations. Both these directions are
being pursued.
On a final note, one further consideration for the Slavnov--Taylor
identities derived in this paper is their verification to one-loop order
in perturbation theory (the tree-level forms are trivial). This has in
fact been done. However, as can be appreciated from the lengthy
expressions, such a technical exercise is not suitable for inclusion in the
present paper. The verification of the one-loop identities involves three
components: the one-loop expansion of the vertex Dyson--Schwinger equations (and the
kernels presented in the text), the use of the inherent translational
invariance of the loop integrals and identities for the color factors.
Importantly, none of the loop integrals need be explicitly evaluated.
\begin{acknowledgments}
This work has been supported by the Deutsche Forschungsgemeinschaft (DFG)
under contracts no. DFG-Re856/6-2,3.
\end{acknowledgments}
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\section{Introduction}
During last years, one witnesses the growth of interest to the
the study of compound, or more generally marked Poisson processes
and application of these to different problems of
probability theory, stochastic
analysis and mathematical physics, see e.g.\ \cite{Kallenberg,
Kingman}. The Gamma process is an important example of a compound
Poisson process with a non-finite measure
of the values of jumps of the process in the L\'evy formula.
In this paper, we derive a white noise calculus for the Gamma
process, studying in detail the so-called Gamma field
operators. Our interest in the Gamma analysis was inspired,
in particular, by the papers by Vershik {\it et al.}\
\cite{Vershik,Vershik1,Vershik2}, where the Gamma measure was used in construction of
a representation of a group of flows. On the other hand, we improve
the results of the paper \cite{Silva}, in which the Fock type
structure of the Gamma space was discovered.
Let us shortly describe our results.
Let $\sigma$
be a non-atomic Radon measure on ${\Bbb R}^d$ and suppose here,
for simplicity, that
$\operatorname{supp} \sigma={\Bbb R}^d$.
In Sect.~1, we introduce the so-called
extended Fock space ${\cal F}_{\mathrm \Ext}({\cal H})$ over ${\cal H}=L^2({\Bbb R}^d,d\sigma)$.
This space is defined as a
completion of the set ${\cal F}_{\mathrm fin}(S)$ with respect to (w.r.t.)\
a scalar product $(\cdot,\cdot)_{{\cal F}_{\mathrm \Ext}({\cal H})}$, which is defined by using
a simple combinatorial rule. Here, ${\cal F}_{\mathrm fin}(S):=\bigoplus_{n=0}
^\infty S_{\Bbb C}^{\hat \otimes n}$ is the topological direct sum of the complexified
symmetric tensor powers of the Schwartz test space $S$. Thus,
${\cal F}_{\mathrm fin}(S)$ consists of finite sequences $f=(f^{(0)}, f^{(1)},
\dots,f^{(n)},0,0,\dots)$, where $f^{(j)}$ is a smooth symmetric rapidly
decreasing function of $j$ variables. We show also the way how to include
the usual Fock space over ${\cal H}$, denoted by ${\cal F}({\cal H})$, into ${\cal F}_{\mathrm \Ext}({\cal H})$,
so that ${\cal F}({\cal H})$ becomes a proper subspace of ${\cal F}_{\mathrm \Ext}({\cal H})$.
Recall now that the Gamma white noise measure $\mu_{\mathrm G}$ with intensity
measure $\sigma$ is defined on $S'$---the dual of $S$ w.r.t.\
the zero space ${\cal H}$---by its Laplace transform
\begin{equation}\label{2.4a}
\ell_{\mathrm G}(\varphi)=
\int_{S'}\exp[\langle\omega,\varphi\rangle]\,d\mu_{\mathrm G}(\omega)=\exp\bigg[-\int_{{\Bbb R}^d}\log(1-\varphi(x))\,d\sigma(x)\bigg],
\qquad 1>\varphi\in S.
\end{equation}
In Sect.~2, we introduce a family of commuting, essentially selfadjoint operators
$(a(\xi))_{\xi\in S}$ in ${\cal F}_{\mathrm \Ext}({\cal H})$ with domain ${\cal F}_{\mathrm fin}(S)$
by the formula
\begin{equation}\label{str}
a(\xi)=a^+(\xi)+2a^0(\xi)+a^-(\xi)+\int_{{\Bbb R}^d}\xi\,d\sigma\cdot\operatorname{id},
\end{equation}
where $a^+(\xi)$ is a usual creation operator:
$$a^+(\xi)\varphi^{\otimes n}=\xi\hat\otimes\varphi^ {\otimes n},$$
$a^0(\xi)$ is a usual neutral operator:
$$a^0(\xi)\varphi^{\otimes n}=n(\xi\varphi)\hat\otimes\varphi^{\otimes(n-1)},$$
and $a^-(\xi)$ is an annihilation operator acting as follows:
$$a^-(\xi)\varphi^{\otimes n}=n\langle\xi,\varphi\rangle\varphi^{\otimes(n-1)}
+n(n-1)(\xi\varphi^2)\hat\otimes\varphi^{\otimes(n-2)}$$
(notice that the first addend corresponds to the usual annihilation operator,
while the second addend is an annihilation of a new type).
By using the spectral approach to commutative Jacobi fields in the Fock space
\cite{Bee, BeLi,Ly, Benew}, we construct the Fourier transform in generalized joint
eigenvectors of the family $(a^\sim(\xi))_{\xi\in S}$, where $a^\sim(\xi)$
is the closure of $a(\xi)$. This transform, denoted by $I$, is a unitary between
${\cal F}_{\mathrm \Ext}({\cal H})$ and the Gamma space $L^2(S',d\mu_{\mathrm G}):=(L^2_{\mathrm G})$,
which was already
constructed in \cite{Silva}. The image of $a^\sim(\xi)$ under $I$ is the operator
of multiplication by the monomial $\langle\cdot,\xi\rangle$. Thus, $a^\sim(\xi)$'s
are actually Gamma field operators.
In Sect.~3, we study the ``chaos decomposition'' of the Gamma space which
naturally appears from its Fock type structure. We show also that the restriction
$I\restriction{\cal F}({\cal H})$ of the unitary $I$ to the Fock space as a
subspace of ${\cal F}_{\mathrm \Ext}({\cal H})$ is exactly the standard isomorphism between
${\cal F}({\cal H})$ and the part of $(L^2_{\mathrm G})$ which
appears as a result of
the chaos
expansion in multiple stochastic integrals w.r.t.\ the compensated Gamma process.
Thus, our approach gives a natural way how, in the Gamma case, to overcome
the main difficulty in compound Poisson analysis connected with the fact that
these processes do not possess the chaotic representation property.
In Sect.~4, we introduce and study spaces of test and generalized functions of Gamma
white noise. This allows us, in particular, to introduce the coordinate operators
$\omega(x)\cdot$, $x\in{\Bbb R}^d$, which, as follows from
\eqref{str}, have the form
$$\omega(x)\cdot=\partial^\dag_x+2\partial^\dag_x\partial_x+1+\partial_x+\partial^\dag_x\partial_x\partial_x,$$
where $\partial^\dag_x$ and $\partial_x$ are the (images of the) creation and annihilation
operators at $x$.
Finally, in Sects.~5 and 6, we derive explicit formulas for the action of the
operators $a^+(\xi)$, $a^0(\xi)$, and $a^-(\xi)$. Some formulas appearing here
are nothing but infinite-dimensional analogs of the formulas obtained in the
one-dimensional case by Meixner in his classical work \cite{Meixner}.
\section{Extended Fock space and its rigging}
Let $\sigma$ be a Borel, regular, non-atomic, $\sigma$-finite
measure on
${\Bbb R}^d$, $d\in {\Bbb N}$. First, we
recall the construction from \cite{Hir} of a rigging
of the real
$L^2$-space
$L^2({\Bbb R}^d,d\sigma)={\cal H}$ by spaces of test and generalized functions of
Schwartz type. (Notice that, since $\sigma$ is not necessarily the
Lebesgue measure,
we need some additional consideration.)
Let $(e_j)_{j=0}^\infty$ be the system of Hermite functions on ${\Bbb R}$. For each
$p\ge1$, define ${\cal S}_p({\Bbb R})$ to be the real Hilbert space spanned by the
orthonormal basis $(e_j(2j+2)^{-p})_{j=0}^\infty$, and let ${\cal S}_p
({\Bbb R}^d)={\cal S}_p({\Bbb R})^{\otimes d}$.
Considered as a subspace of $L^2({\Bbb R}^d,dx)$, every space ${\cal S}_p({\Bbb R}^d)$
coincides with the domain of the operator $(H^{\otimes d})^p$, where $H^{\otimes d}$ is
the harmonic oscillator in $L^2({\Bbb R}^d,dx)$: $H^{\otimes d}=-\sum_{i=1}^d\big(\frac
d{dx_i}\big)^2+\sum_{i=1}^d x_i^2+1$.
As well known, ${\cal S}({\Bbb R}^d)=\projlim_{
p\to\infty}{\cal S}_p({\Bbb R}^d)$ is the Schwartz space of rapidly decreasing functions
on ${\Bbb R}^d$, ${\cal S}_1({\Bbb R}^d)$
consists of continuous functions, and
$$ {\Bbb R}^d\ni x\mapsto \delta_x\in {\cal S}_{-1}({\Bbb R}^d)$$
is a continuous mapping, where ${\cal S}_{-p}({\Bbb R}^d)$ denotes the dual of
${\cal S}_p({\Bbb R}^d)$.
We suppose the existence of $\epsilon\ge0$ such that
the space ${\cal S}_{1+\epsilon}({\Bbb R}^d)$ is continuously embedded into
${\cal H}$; for example, the following condition holds:
$$\int_{{\Bbb R}^d}\|\delta_x\|^2_{{\cal S}_{-1-\epsilon}({\Bbb R}^d)}\,d\sigma(x)
<\infty.$$
Let $O_p\colon S_p({\Bbb R}^d)\hookrightarrow{\cal H}$ be an embedding operator.
Since ${\cal S}({\Bbb R}^d)$ is a nuclear space, we can suppose without loss of
generality that the operator
$O_{1+\epsilon}$ is of Hilbert--Schmidt type
(in case of the Lebesgue measure, $d\sigma(x)=dx$, we can take
$\epsilon=0$).
Note that, because of the regularity and $\sigma$-finiteness of the
measure $\sigma$, ${\cal S}({\Bbb R}^d)$ is a dense subset of $L^2({\Bbb R}^d,\sigma)$.
Define now $S_p$ to be the Hilbert factor space ${\cal S}_{p+\epsilon}/\ker
O_{p+\epsilon}$. By \cite{BeKo}, Ch.~5, Sect.~5, subsec.~1,
$\big\{{\cal H},\, S_p\mid p\ge1\big\}$ is a sequence of compatible Hilbert spaces.
Thus, we
obtain the rigging
\begin{equation}\label{1.1}
S'=\operatornamewithlimits{ind\,lim}_{p\to\infty} S_{-p}\supset
L^2({\Bbb R}^d,d\sigma)={\cal H}\supset\projlim_{p\to\infty} S_p=S,
\end{equation}
where $S_{-p}$, resp.\ $S'$ is the dual of $S_p$, resp.\ $S$
w.r.t.\ the zero space ${\cal H}$.
We stress that, for arbitrary $p\ge p'$, the space $S_p$ is continuously
embedded into $S_{p'}$ and $|\cdot|_{p}\ge|\cdot|_{p'}$, where $|\cdot|_p$
denotes the $S_p$ norm.
Notice that, in fact, the spaces $S_p$ and $S$ are completely
determined by the support of $\sigma$, i.e., $S$
is the Schwartz test space on $\operatorname{supp}\sigma$.
Now we wish to define an $n$-particle extended Fock space over ${\cal H}$,
${\cal F}_{\mathrm \Ext}^{(n)}({\cal H})$, $n\in {\Bbb N}$. Under a loop $\kappa$
connecting points $x_1,\dots,x_m$, $m\ge2$, we understand a class of ordered
sets $(x_{\pi(1)},\dots,x_{\pi(m)})$, where $\pi$ is a permutation of
$\{1,\dots,n\}$, which coincide up to a cyclic permutation. For example,
$(x_1,x_2,x_3)$ and $(x_3,x_1,x_2)$ define the same loop, while
$(x_1,x_2,x_3)$ and $(x_2,x_1,x_3)$ define different loops. We put
$|\kappa|=m$.
We will interpret also a set $\{x\}$ as a ``one-point'' loop $\kappa$, i.e., a loop that
comes out of $x$, $|\kappa|=1$.
Let $\alpha_n=\{\kappa_1,\dots,\kappa_{|\alpha_n|}\}$ be a collection of loops
$\kappa_j$ that connect points from the set $\{x_1,\dots,x_n\}$ so that every
point $x_i\in\{x_1,\dots,x_n\}$ goes into one loop $\kappa_j=\kappa_{j(i)}$
from $\alpha_n$. Here, $|\alpha_n|$ denotes the number of the loops in
$\alpha_n$, evidently $n=\sum_{j=1}^{|\alpha_n|}|\kappa_j|$.
Let $A_n$ stand for the set of all possible collections of loops $\alpha_n$
over the points $\{x_1,\dots,x_n\}$. Every $\alpha_n\in A_n$ generates the
following continuous mapping
\begin{multline}\label{1.2}
S^{\hat\otimes n}_{\Bbb C}\ni f^{(n)}=f^{(n)}(x_1,\dots,x_n)\mapsto\\
\mapsto f^{(n)}_{\alpha_n}(
\underbrace{x_1,\dots,x_1}_{\text{$|\kappa_1|$ times}} ,
\underbrace{x_2,\dots,x_2}_{\text{$|\kappa_2|$ times}} ,\dots,
\underbrace{x_{|\alpha_n|},
\dots,x_{|\alpha_n|}}_{\text{$|\kappa_{|\alpha_n|}|$ times}} )\in
S_{\Bbb C}^{\otimes|\alpha_n|},
\end{multline}
where the lower index ${\Bbb C}$ denotes complexification of a real space and
the symbol $\hat\otimes$ stands for the symmetric
tensor power. Indeed, for any $p\ge1$, the diagonalization operator $\frak D
$ given by
$$f^{(2)}=f^{(2)}(x_1,x_2)\mapsto {\frak D}f^{(2)}=\big({\frak
D}f^{(2)}\big)(x)=f^{(2)}(x,x)$$
acts continuously from $S_{p,{\Bbb C}}^{\hat\otimes 2}$ into $S_{p,{\Bbb C}}$,
\begin{equation}\label{1.3}
|{\frak D}f^{(2)}|_p\le C_p|f^{(2)}|_p,\end{equation} where
$|\cdot|_p$ denotes also the norm of each space $S_{p,{\Bbb C}}^{\otimes
n}$, $n\in {\Bbb N}$, which yields
\begin{equation}\label{1.4}
|f^{(n)}_{\alpha_n}|_p\le C_p^{(|\kappa_1|-1)+(|\kappa_2|-1)+\cdots+
(|\kappa_{|\alpha_n|}|-1)}|f^{(n)}|_p,\end{equation}
giving the continuity of the mapping \eqref{1.2}.
Thus, we define a scalar product on $S_{{\Bbb C}}^{\hat\otimes n}$ by
\begin{equation}\label{1.5}
( f^{(n)},g^{(n)})_{{\cal F}_{\mathrm \Ext}^{(n)}({\cal H})}=\sum_{\alpha_n\in A_n}\int
_{{\Bbb R}^{d|\alpha_n|}}\big(\overline{ f^{(n)}}g^{(n)}\big)_{\alpha_n}\,
d\sigma^{|\alpha_n|},
\end{equation}
where $\overline{f^{(n)}}$ is the complex conjugate of $f^{(n)}$. Let
${\cal F}_{\mathrm \Ext}^{(n)}({\cal H})$ be
the closure of $S_{{\Bbb C}}^{\hat \otimes n}$ in the norm generated by \eqref{1.5}.
\vspace{2mm}
\noindent {\it Remark\/} 1. It is easy to see that the number of the summands in
the series \eqref{1.5} is exactly $n!$. Let us prove this by induction. For
$n=1$, this is trivial. Let the statement hold for $n$. At the step
$(n+1)$ we
add a point $x_{n+1}$ to the set $
\{x_1,\dots,x_n\}$. If $x_{n+1}$ goes into a ``one-point'' loop, then by the
supposition there are exactly $n!$ variants. If $x_{n+1}$ goes into a loop that
connect this point with other points from $\{x_1,\dots,x_n\}$, then to each
loop connecting $i$
points from $\{x_1,\dots,x_n\}$ we can add the point $x_{n+1}$ in $i$ different
ways. That gives $n\,n!$ variants. Thus, in total we have exactly $(n+1)!$
variants.\vspace{2mm}
The extended Fock space $
{\cal F}_{\mathrm \Ext}({\cal H})$
over ${\cal H}$ is defined as a weighted direct sum of the spaces ${\cal F}_{\mathrm \Ext}^{(n)}({\cal H})$:
\begin{equation}\label{1.5a}
{\cal F}_{\mathrm \Ext}({\cal H})=\bigoplus_{n=0}^\infty {\cal F}_{\mathrm \Ext}^{(n)}({\cal H})\,n!,\end{equation}
where ${\cal F}_{\mathrm \Ext}^{(0)}({\cal H})={\Bbb C}$ and $0!=1$. I.e., ${\cal F}_{\mathrm \Ext}({\cal H})$ consists of
sequences $f=(f^{(0)}, f^{(1)},$ $f^{(2)},\dots)$
such that $f^{(n)}\in {\cal F}_{\mathrm \Ext}^{(n)}({\cal H})$ and
$$ \|f\|_{{\cal F}_{\mathrm \Ext}({\cal H})}^2=\sum_{n=0}^\infty\|f^{(n)}\|^2_{{\cal F}_{\mathrm \Ext}^{(n)}({\cal H})}
n!<\infty.$$
Throughout the paper we will identify $f^{(n)}\in {\cal F}_{\mathrm \Ext}^{(n)}({\cal H})$ with
the vector
$$(0,\dots,0,f^{(n)},0,0\dots)\in{\cal F}_{\mathrm \Ext}({\cal H}).$$
In the following section, we will need
a nuclear space that is topologically, i.e., densely and continuously, embedded
into ${\cal F}_{\mathrm \Ext}({\cal H})$, and its dual space. Thus, we take
\begin{equation}
\label{a}
{\cal F}_{\mathrm \Ext}({\cal H})\supset {\cal F}_{{\mathrm fin}}(S),\qquad {\cal F}_{{\mathrm fin}}^*(S)\supset{\cal F}_{{\mathrm fin}}(S).
\end{equation}
Here, ${\cal F}_{{\mathrm fin}}(S)$ is the topological direct sum of the spaces
$S_{{\Bbb C}}^{\hat \otimes n}$, i.e., ${\cal F}_{{\mathrm fin}}(S)$ consists of all finite sequences
$f=(f^{(0)},f^{(1)},\dots,f^{(m)},0,0,\dots)$ such that $f^{(n)}\in S_{\Bbb C}^{\hat \otimes
n}$ and the convergence in ${\cal F}_{{\mathrm fin}}(S)$ means the uniform finiteness and the
coordinate-wise convergence. That ${\cal F}_{{\mathrm fin}}(S)$ is dense in ${\cal F}_{\mathrm \Ext}({\cal H})$
follows directly from the definition of ${\cal F}_{\mathrm \Ext}({\cal H})$. \eqref{1.4}, \eqref{1.5},
and
Remark~1 give
\begin{align}
\|f^{(n)}\|^2_{{\cal F}_{\mathrm \Ext}^{(n)}({\cal H})}&\le n!\max_{\alpha_n\in A_n}\|f^{(n)}_{\alpha_n}
\|^2_{{\cal H}_{\Bbb C}^{\hat \otimes |\alpha_n|}}\notag\\
&\le n!K_1^{2n}\max_{\alpha_n\in A_n}|f^{(n)}_{|\alpha_n|}|_1^2\le
n!\,K_1^{2n}C_1^{2(n-1)}|f^{(n)}|_1^2, \label{1.6}\end{align}
where $K_1$ is the norm of the inclusion operator $S_1\hookrightarrow{\cal H}$ (we suppose that
the constants $K_1,\,C_1\ge1$). \eqref{1.6} implies the continuity of the
embedding ${\cal F}_{{\mathrm fin}}(S)\hookrightarrow{\cal F}_{\mathrm \Ext}({\cal H})$.
The space ${\cal F}_{{\mathrm fin}}^*(S)$ in \eqref{a} is the dual of ${\cal F}_{{\mathrm fin}}(S)$.
It will be convenient for us to take it as the dual of ${\cal F}_{{\mathrm fin}}(S)$ w.r.t.\ the
zero space ${\cal F}({\cal H})$, the usual Fock space over ${\cal H}$:
$${\cal F}({\cal H})=\bigoplus_{n=0}^\infty {\cal H}_{\Bbb C}^{\hat \otimes n}\,n!.$$
So, the second inclusion in \eqref{a} is part of the nuclear triple (cf.\ e.g.\
\cite{BeKo})
\begin{equation}\label{standard}
{\cal F}_{{\mathrm fin}}^*(S)\supset{\cal F}({\cal H})\supset{\cal F}_{{\mathrm fin}}(S).\end{equation}
The space ${\cal F}_{{\mathrm fin}}^*(S)$ consists of infinite sequences
$F=(F^{(0)},F^{(1)},F^{(2)},\dots)$, where $F^{(n)}
\in S_{\Bbb C}^{\prime\,\hat \otimes n}$ and the dualization with $f\in{\cal F}_{{\mathrm fin}}(S)$ is given by
$$ \langle\!\langle F,f\rangle\!\rangle=\sum_{n=0}^\infty \langle \overline{F^{(n)}},f^{(n)}
\rangle\,n!,$$
where $\langle\cdot,\cdot\rangle$ denotes the dualization generated
by the scalar product in ${\cal H}^{\hat \otimes n}$, which is supposed to be
linear in both dots.
Finally, we note that the usual Fock space ${\cal F}({\cal H})$
can be included into ${\cal F}_{\mathrm \Ext}({\cal H})$. To this end, we construct a subset of $S_{\Bbb C}^{\hat \otimes
n}$
of the form
$${\cal R}^{(n)}=\operatorname{l.s.}\big\{\,
\varphi_1\hat \otimes\cdots\hat \otimes\varphi_n\mid\varphi_i\in
S_{\Bbb C},\
\sigma (\operatorname{supp} \varphi_i\cap\,\operatorname{supp}\varphi_j)=0\ \forall i,j=1,\dots,n,\, i\ne j\},$$
where $\operatorname{l.s.}$ denotes the linear span.
It follows from \eqref{1.5} that, for arbitrary $f^{(n)},g^{(n)}\in{\cal
R}^{(n)}$,
\begin{equation}\label{1.7}
( f^{(n)},g^{(n)})_{{\cal F}_{\mathrm \Ext}^{(n)}({\cal H})}=( f^{(n)},g^{(n)})_{{\cal H}_{\Bbb C}^{\hat \otimes n}}.\end{equation}
Let $\overline{{\cal R}^{(n)}}$
be the closure of ${\cal R}^{(n)}$ in the ${\cal F}_{\mathrm \Ext}^{(n)}({\cal H})$ norm. Evidently,
$\overline{{\cal R}^{(n)}}$ is a subspace of ${\cal F}_{\mathrm \Ext}^{(n)}({\cal H})$. On the other
hand, the set ${\cal R}^{(n)}$ is dense in ${\cal H}_{\Bbb C}^{\hat \otimes n}$. Therefore, in view
of \eqref{1.7}, we can identify $\overline{{\cal R}^{(n)}}$ with ${\cal H}_{\Bbb C}^{\hat \otimes
n}$.
Then, the subspace $\bigoplus_{n=0}^\infty \overline{{\cal R}^{(n)}}\,n!$ of ${\cal F}_{\mathrm \Ext}({\cal H})$
is identified with ${\cal F}({\cal H})$.\vspace{2mm}
\noindent {\it Remark\/} 2. Let us explain how the extended Fock space happens to
be greater than the usual Fock space. Take, for example, $n=2$. Then, the space
${\cal H}_{\Bbb C}^{\ho2}=\hat L^2({\Bbb R}^{2d},d\sigma^2)$ is the
complex space of quadratic integrable symmetric functions of two variables from ${\Bbb R}^d$. Of
course, any function $f^{(2)}$ from ${\cal H}_{\Bbb C}^{\hat \otimes 2}$ is defined on ${\Bbb R}^{2d}$ up
to a set of zero $\sigma^2$ measure. The diagonal $\{(x,x)\mid x\in {\Bbb R}^d\}$ in
${\Bbb R}^{2d}$ has $\sigma^2$ measure zero, while $\sigma$ is non-atomic. On the
other hand, every function $f^{(2)}\in{\cal F}_{\mathrm \Ext}^{(2)}({\cal H})$ must also be defined on
the diagonal up to a set of zero $\sigma$ measure, if we identify this diagonal
with ${\Bbb R}^d$. Thus, any function $f^{(2)}\in{\cal H}_{\Bbb C}^{\hat \otimes 2}$ is included into
${\cal F}_{\mathrm \Ext}^{(2)}({\cal H})$ if we put additionally that $f^{(2)}$ is equal to zero on the
diagonal. An analogous situation takes place in all the other $n$-particle
spaces with $n\ge3$.\vspace{2mm}
\noindent{\it Remark\/} 3. Let us stress that we have constructed two
different types of inclusion: the first one ${\cal F}_{{\mathrm fin}}(S)\subset{\cal F}({\cal H})$ in
\eqref{standard} and ${\cal F}({\cal H})$ as a subspace of ${\cal F}_{\mathrm \Ext}({\cal H})$.
\section{Gamma field operators\\ and the Fourier transform}
In this section, we will define a family of Gamma field operators and construct
a Fourier transform in generalized joint eigenvectors of this family.
For each $\xi\in S$, let $a^+(\xi)$ be the standard creation operator defined
on ${\cal F}_{{\mathrm fin}}(S)$:
\begin{equation}\label{2.0}
a^+(\xi)f^{(n)}=\xi\hat \otimes f^{(n)},\qquad f^{(n)}\in S^{\hat \otimes n}_{\Bbb C},\ n\in
{\Bbb N}_0=\{0,1,2,\dots\}.
\end{equation}
Because of the estimate
\begin{equation}\label{2.1}
|\xi\hat \otimes f^{(n)}|_p\le|\xi|_p|f^{(n)}|_p,\end{equation}
the operator $a^+(\xi)$ acts continuously on ${\cal F}_{{\mathrm fin}}(S)$.
Let us calculate the adjoint operator of $a^+(\xi)$ in ${\cal F}_{\mathrm \Ext}({\cal H})$, which will
be denoted by $a^-(\xi)$. By \eqref{1.5}, \eqref{1.5a}, and \eqref{2.0}, we have for arbitrary
$\xi,\varphi,\psi\in S$,
\begin{gather*}
( a^+(\xi)\varphi^{\otimes
n},\psi^{\otimes(n+1)})_{{\cal F}_{\mathrm \Ext}({\cal H})}=(n+1)!\,(\xi\hat \otimes\varphi^{\otimes
n},\psi^{\otimes(n+1)}
)_{{\cal F}_{\mathrm \Ext}^{(n+1)}({\cal H})}\\
=(n+1)!\,\sum_{\alpha_{n+1}\in
A_{n+1}}\int_{{\Bbb R}^{d|\alpha_{n+1}|}}\big((\xi\psi)\hat \otimes(\varphi\psi)^{\otimes
n}\big)_{\alpha_{n+1}}
\,d\sigma^{|\alpha_{n+1}|}
\\
=(n+1)!\,\bigg[\langle\xi,\psi\rangle\sum_{\alpha_n\in
A_n}\int_{{\Bbb R}^{d|\alpha_n|}}\big((\varphi\psi)^{\otimes
n}\big)_{\alpha_n}\,d\sigma^{|\alpha_n|}
\\
\text{}
+n\sum_{\alpha_n\in
A_n}\int_{{\Bbb R}^{d|\alpha_n|}}\big((\xi\varphi\psi^2)\hat \otimes(\varphi\psi)^{\otimes(n-1)}\big)
_{\alpha_n}\,d\sigma^{|\alpha_n|}\bigg]\\
=\big(\varphi^{\otimes n},(n+1)\langle\xi,\psi\rangle\psi^{\otimes
n}+(n+1)n(\xi\psi^2)\hat \otimes\psi
^{\otimes(n-1)}\big)_{{\cal F}_{\mathrm \Ext}({\cal H})},
\end{gather*}
where we used just as in Remark~1 the observation that a new point can be added
to a loop connecting $i$ points in $i$ different ways. Thus,
$$ a^-(\xi)=a_1^-(\xi)+a_2^-(\xi),$$
where $a_1^-(\xi)$ is the standard annihilation operator:
$$a_1^-(\xi)\varphi^{\otimes n}=n\langle\xi,\varphi\rangle\varphi^{\otimes (n-1)}$$
and $a_2^-(\xi)$ given by
$$a_2^-(\xi)\varphi^{\otimes n}=n(n-1)(\xi\varphi^2)\hat \otimes\varphi^{\otimes(n-2)}$$
is a ``Gamma annihilation'' operator, which appears because of the nonstandard
scalar product in our Fock space. Due to \eqref{1.3}, $a^-(\xi)$ acts
continuously on ${\cal F}_{{\mathrm fin}}(S)$ and
\begin{align}
|a_1^-(\xi)f^{(n)}|_p&\le n|\xi|_{-p}|f^{(n)}|_p,\notag\\
|a_2^-(\xi)f^{(n)}|_p&\le n(n-1)C_p^2|\xi|_p|f^{(n)}|_p,\label{2.2}
\end{align}
Finally, we define on ${\cal F}_{{\mathrm fin}}(S)$ the neutral operator $a^0(\xi)$, $\xi\in S$,
as
the differential second quantization of the operator of multiplication by
$\xi$:
$$a^0(\xi)\varphi^{\otimes n}=n(\xi\varphi)\hat \otimes\varphi^{\otimes(n-1)},\qquad \varphi\in S.$$
Again, $a^0(\xi)$ acts continuously on ${\cal F}_{{\mathrm fin}}(S)$ and
\begin{equation}\label{2.3}
|a^0(\xi)f^{(n)}|_p\le nC_p|\xi|_p|f^{(n)}|_p,\qquad f^{(n)}\in S_{\Bbb C}^{\hat \otimes
n}.\end{equation}
Thus, we are in position to define, for each $\xi\in S$, the Gamma field
operator $a(\xi)$ on ${\cal F}_{{\mathrm fin}}(S)$:
\begin{equation}\label{2.4}
a(\xi)=a^+(\xi)+2a^0(\xi)+\langle\xi\rangle
\operatorname{id}+a^-(\xi),\end{equation}
where $\langle\xi\rangle=\int_{{\Bbb R}^d}\xi(x)\,d\sigma(x)$ and $\operatorname{id}$ denotes the identity
operator. Each $a(\xi)$ with domain ${\cal F}_{{\mathrm fin}}(S)$ is a Hermitian operator in
${\cal F}_{\mathrm \Ext}({\cal H})$.
\begin{lem}
The operators $a(\xi)$\rom, $\xi\in S$\rom, with domain ${\cal F}_{{\mathrm fin}}(S)$ are
essentially selfadjoint in ${\cal F}_{\mathrm \Ext}({\cal H})$ and their closures
$a^\sim(\xi)$ constitute a family of commuting selfadjoint operators\rom, where
the commutation is understood in the sense of the resolutions of the
identity\rom.
\end{lem}
\noindent {\it Proof}. Let us show
that every $f^{(n)}\in S_{\Bbb C}^{\hat \otimes n}$, $n\in {\Bbb N}_0$, is an analytical vector of
each $a(\xi)$, i.e., the series
$$\sum_{m=0}^\infty \frac{\|a(\xi)^mf^{(n)}\|_{{\cal F}_{\mathrm \Ext}({\cal H})}}{m!}\,|z|^m,\qquad
z\in{\Bbb C},$$
has a positive radius of convergence.
To this end, we define, for each $p\ge1$ and $k\in{\Bbb N}_0$, a Hilbert space
\begin{equation}\label{space}{\cal F}_{1,k}(S_p)=\bigoplus_{n=0}^\infty S_{p,{\Bbb C}}^{\hat \otimes n}\,(n!)^2
2^{nk},\end{equation}
i.e., for $f=(f^{(0)},f^{(1)},f^{(2)},\dots)\in {\cal F}_{1,k}(S_p)$,
$$\|f\|^2_{{\cal F}_{1,k}(S_p)}=\sum_{n=0}^\infty
|f^{(n)}|^2_p\,(n!)^2 2^{nk}.$$
Because of \eqref{1.6}, we have that topologically
${\cal F}_{1,k_0}(S_1)\subset {\cal F}_{\mathrm \Ext}({\cal H})$ if $2^{k_0/2}\ge K_1C_1$
and
$$\|\cdot\|_{{\cal F}_{1,k_0}(S_1)}\ge\|\cdot\|_{{\cal F}_{\mathrm \Ext}({\cal H})}.$$
Taking to notice that each $a(\xi)^mf^{(n)}$ belongs to ${\cal F}_{{\mathrm fin}}(S)$, it suffices
to prove that the series
$$\sum_{m=0}^\infty\frac{\|a(\xi)^mf^{(n)}\|_{{\cal F}_{1,k_0}(S_1)}}{m!}\,|z|^m$$
has a positive radius of convergence. We have
\begin{align*}
a(\xi)^m&=(a^+(\xi)+2a^0(\xi)+\langle\xi\rangle\operatorname{id}+a^-_1(\xi)+a_2^-(\xi))^m\\
&=\sum_{j=1}^{5^m}Y_{j,m}(\xi)\end{align*}
where every operator $Y_{j,m}(\xi)$ is a composition of $m$ operators every one
of which is one of the operators
$a^+(\xi)$, $2a^0(\xi)$, $\langle\xi\rangle\operatorname{id}$, $a^-_1(\xi)$, $a_2^-(\xi)$. From
\eqref{2.1}--\eqref{2.3}, we deduce that
\begin{align*}
\|a^+(\xi)f^{(n)}\|_{{\cal F}_{1,k_0}(S_1)}&\le(n+1)2^{k_0/2}|\xi|_1\|f^{(n)}\|_{{\cal F}_{1,k_0}(S_1)},\\
\|2a^0(\xi)f^{(n)}\|_{{\cal F}_{1,k_0}(S_1)}&\le2nC_1|\xi|_1\|f^{(n)}\|_{{\cal F}_{1,k_0}(S_1)},\\
\|a_1^-(\xi)f^{(n)}\|_{{\cal F}_{1,k_0}(S_1)}&\le 2^{-k_0/2}|\xi|_{-1}\|f^{(n)}\|_{{\cal F}_{1,k_0}(S_1)},\\
\|a_2^-(\xi)f^{(n)}\|_{{\cal F}_{1,k_0}(S_1)}&\le
(n-1)C_1^2 2^{-k_0/2}|\xi|_1\|f^{(n)}\|_{{\cal F}_{1,k_0}(S_1)}
.\end{align*}
Hence,
$$\|a(\xi)^mf^{(n)}\|_{{\cal F}_{1,k_0}(S_1)}\le
5^m(2C_1^2 2^{k_0/2})^m\frac{(n+m)!}{n!}\,\max\{|\xi|_1,\langle\xi\rangle\}
\|f^{(n)}\|_{{\cal F}_{1,k_0}(S_1)}.$$
Thus, it remains to note that the series
$$\sum_{m=0}^\infty\frac{(10 C_1^2 2^{k_0/2})^m(n+m)!}{m!}\,|z|^m $$
converges as $|z|<(10 C_1^2 2^{k_0/2})^{-1}$.
By using, e.g., \cite{BeKo}, Ch.~5, Th.~1.7, we conclude that the operators
$a(\xi) $ are essentially selfadjoint on ${\cal F}_{{\mathrm fin}}(S)$.
Let us show that the operators $a(\xi)$ commute on ${\cal F}_{{\mathrm fin}}(S)$.
Any operators $a^+(\xi_1)$ and $a^+(\xi_2)$ evidently
commute. Hence, their adjoints $a^-(\xi_1)$ and $a^-(\xi_2)$ also commute. Next,
the operators of second quantization $a^0(\xi_1)$ and $a^0(\xi_2)$ commute,
since
do the operators of multiplication by $\xi_1$ and $\xi_2$. Next, we have to
show that
\begin{equation}\label{star}
a^+(\xi_1)a^0(\xi_2)+a^0(\xi_1)a^+(\xi_2)=a^+(\xi_2)a^0(\xi_1)+a^0(\xi_2)a^+(\xi_1),
\end{equation}
which evidently yields
$$
a^-(\xi_1)a^0(\xi_2)+a^0(\xi_1)a^-(\xi_2)=a^-(\xi_2)a^0(\xi_1)+a^0(\xi_2)a^-(\xi_1).$$
But \eqref{star} can be easily verified on the vectors of the form $\varphi^{\otimes
n}$, $\varphi\in S$, if one uses the equality
\begin{align*}
a^0(\xi)\varphi_1\hat \otimes\varphi_2\hat \otimes\cdots\hat \otimes\varphi_n&=(\xi\varphi_1)\hat \otimes\varphi_2\hat \otimes\cdots\hat \otimes\varphi_n\\
&\quad+\varphi_1\hat \otimes(\xi\varphi_2)\hat \otimes\cdots\hat \otimes\varphi_n+\cdots+\varphi_1\hat \otimes\cdots\hat \otimes\varphi_{n-1}\hat \otimes(\xi\varphi_n),
\end{align*}
which gives, in particular,
\begin{equation}\label{starstar}
a^0(\xi)\varphi_1\hat \otimes\varphi_2^{\otimes(n-1)}
=(\xi\varphi_1)\hat \otimes\varphi_2^{\otimes(n-1)}+(n-1)\varphi_1\hat \otimes(\xi\varphi_2)\hat \otimes\varphi_2^{\otimes(n-2)}.
\end{equation}
Finally, we have to show that
$$a^+(\xi_1)a^-(\xi_2)+a^-(\xi_1)a^+(\xi_2)=a^+(\xi_2)a^-(\xi_1)+a^-(\xi_2)a^+(\xi_1).$$
It is not hard to see that
$$ a_2^-(\xi)\varphi_1\hat \otimes\varphi_2^{\otimes n}=2n(\xi\varphi_1\varphi_2)\hat \otimes\varphi_2^{\otimes(n-1)}+n
(n-1)\varphi_1\hat \otimes(\xi\varphi_2^2)\hat \otimes\varphi_2^{\otimes (n-2)}.$$
Then,
\begin{align*}
&(a^+(\xi_1)a^-(\xi_2)+a^-(\xi_1)a^+(\xi_2))\varphi^{\otimes n}\\
&\qquad
=n\langle\xi_2,\varphi\rangle\xi_1\hat \otimes\varphi^{\otimes(n-1)}+n(n-1)\xi_1\hat \otimes(\xi_2\varphi^2)\hat \otimes\varphi^{\otimes(n-2)}
+\langle\xi_1,\xi_2\rangle\varphi^{\otimes
n}\\
&\qquad\quad+n\langle\xi_1,\varphi\rangle\xi_2\hat \otimes\varphi^{\otimes(n-1)}
+
2n(\xi_1\xi_2\varphi)\hat \otimes\varphi^{\otimes(n-1)}+n(n-1)\xi_2\hat \otimes(\xi_1\varphi^2)\hat \otimes\varphi^{\otimes
(n-2)},
\end{align*}
which is symmetric in $\xi_1$ and $\xi_2$.
Thus, arbitrary $a(\xi_1)$ and $a(\xi_2)$ commute.
Since the operators $a^\sim(\xi)$ are essentially selfadjoint on ${\cal F}_{{\mathrm fin}}(S)$, the
set
$$(a^\sim(\xi)-z\operatorname{id}){\cal F}_{{\mathrm fin}}(S),\qquad z\in{\Bbb C},\ \Im z\ne 0,$$
is dense in ${\cal F}_{\mathrm \Ext}({\cal H})$. In addition
$$(a^\sim(\xi)-z\operatorname{id}){\cal F}_{{\mathrm fin}}(S)\subset{\cal F}_{{\mathrm fin}}(S).$$
Therefore, for arbitrary $\xi_1,\xi_2\in S$, the operators $a^\sim(\xi_1)$,
$a^\sim(\xi_2)$, and
$$a^\sim(\xi_1)\restriction(a^\sim(\xi_2)-z\operatorname{id}){\cal F}_{{\mathrm fin}}(S)$$
have a total
set of analytical vectors. Thus, by \cite{BeKo}, Ch.~5, Th.~1.15, the operators
commute in the sense of the resolutions of the identity. \quad $\Box$
\begin{th} There exists a unitary isomorphism
$$ I\colon {\cal F}_{\mathrm \Ext}({\cal H})\to L^2(S',{\cal B}(S'), d\mu_{\mathrm G})=(L^2_{\mathrm G})$$
between the extended Fock space ${\cal F}_{\mathrm \Ext}({\cal H})$ and the complex $L^2$-space over $S'$
with the Borel measure $\mu_{\mathrm G}$ of Gamma white noise\rom, whose Laplace transform
is given by \eqref{2.4a}\rom.
This isomorphism is defined on the dense set ${\cal F}_{{\mathrm fin}}(S)$ by the formula
\begin{equation}\label{2.5}
{\cal F}_{{\mathrm fin}}(S)\ni f=(f^{(n)})_{n=0}^\infty\mapsto If=(If)(\omega)=\sum_{n=0}^\infty
\langle\wom n,f^{(n)}\rangle\end{equation}
\rom(the series in \eqref{2.5} is actually finite\rom)\rom, where $\wom n\in
S^{'\,\hat \otimes n}$ is the $n$-th Gamma-Wick power of $\omega\in S'$ given by the recurrence
formula
\begin{equation}\label{2.6}\begin{gathered}
\wom{(n+1)}=\wom{(n+1)}(x_1,\dots,x_{n+1})=\big(\wom{n}(x_1,\dots,x_n)\omega(x_{n+1})\big)^\sim\\
\text{}-n\big(
\wom {(n-1)}(x_1,\dots,x_{n-1})\delta(x_{n+1}-x_n)
\big)^\sim\\
\text{}-n(n-1)\big(
\wom{(n-1)}(x_1,\dots,x_{n-1})\delta(x_{n}-x_{n-1})\delta(x_{n+1}-x_n)
\big)^\sim\\
\text{}-2n\big(
\wom n(x_1,\dots,x_n)\delta(x_{n+1}-x_n)
\big)^\sim
-\big(
\wom n(x_1,\dots,x_n)1(x_{n+1})
\big)^\sim,
\\
\wom0=1,\ \wom1=\omega-1,
\end{gathered}\end{equation}
where $(\cdot)^\sim$ denotes the symmetrization of a function.
The image of each
operator $a^\sim(\xi)$\rom, $\xi\in S$\rom, under $I$ is the operator of
multiplication by the monomial $\langle\cdot,\xi\rangle$ in $(L^2_{\mathrm G})$\rom:
\begin{equation}\label{2.7}
Ia^\sim(\xi)I^{-1}=\langle\cdot,\xi\rangle\cdot\, .
\end{equation}
\end{th}
\noindent {\it Remark\/} 4. Let us explain the term ``Gamma noise,''
introduced in \cite{Silva}.
Let $\Delta$
be a bounded Borel set in ${\Bbb R}^d$ and let $\chi_\Delta$ denote the indicator of
$\Delta$. Then, one can define the random variable (r.v.)\
$X_\Delta=X_\Delta(\omega)=\langle\omega, \chi_\Delta\rangle$ as an $(L^2_{\mathrm G})$-limit of a
sequence of r.v.'s $\langle\omega,\xi_j\rangle$ such that $\xi_j\to\chi_\Delta$ in
$L^2({\Bbb R}^d,d\sigma)$ as $j\to\infty$. Then, the Laplace transform
$$\int_{S'}\exp[\lambda X_\Delta(\omega)]\,d\mu_{\mathrm G}(\omega)=\exp\big[
-\sigma(\Delta)\log(1-\lambda)
\big],\qquad \lambda<1,$$
coincides with the Laplace transform of a r.v.\ having Gamma distribution
with parameter $\sigma(\Delta)$, i.e.,
the density of the distribution of $X_\Delta$ is
\begin{equation}\label{gamma}
p_\Delta(s)=\frac{s^{\sigma(\Delta)-1}e^{-s}}{\Gamma(\sigma(\Delta))},\qquad
s>0.\end{equation}
In case $d=1$ and $d\sigma(x)=dx$, the process $(X_t)_{t\ge0}$, where
$X_t=X_{[0,t]}$ and $X_0=0$, is known as a Gamma process, see, e.g., \cite{Ta}, Sect.~19.
Thus, the triple $(S',{\cal B}(S'),\mu_{\mathrm G})$ is a direct representation of the
generalized stochastic process $(\dot X_t)_{t\ge0}$, see \cite{GeVi}. So, the term ``Gamma
noise'' is natural for $\mu_{\mathrm G}$.
Let us also note that the Gamma white noise is a special case of
a compound
Poisson white noise, see e.g.\ \cite{GGV75,GeVi,LyReShch,Silva},
because
the Laplace transform \eqref{2.4a} can
also be presented in the form
\begin{equation}\label{CP}
\int_{S'}\exp[\langle\omega,\varphi\rangle]\,d\mu_{\mathrm G}(\omega)
=\exp\bigg[\int_0^\infty\!\!\int_{{\Bbb R}^d}(e^{s\varphi(x)}-1)\,d\sigma(x)\,\frac{e^{-s}}{s}\,
ds\bigg].
\end{equation}
This yields, in particular, that the measure $\mu_{\mathrm G}$ is concentrated on the
following subset of $S'$, e.g., \cite{Kal83,KMM78}. Let $\Gamma$
denote the marked configuration space over
$\operatorname{supp}\sigma$ with marks in $]0,\infty[$ that is defined as follows:
$\Gamma$ consists of sets $\gamma\subset \operatorname{supp}\sigma\times]0,\infty[$
such that, for arbitrary disjoint $(x,s_x),(y,s_y)\in\gamma$, we have $x\ne y$, and
for any $a>0$ and $K\subset{\Bbb R}^d$ compact, $\gamma\cap(K\times[a,\infty[)$ is a
finite set.
Then, the measure $\mu_{\mathrm G}$ is
concentrated on the set of those $\omega\in S'$ which can be presented in the
form
\begin{equation}\label{configuration}
\omega=\sum_{(x,s_x)\in\gamma}s_x\delta_x\end{equation}
with $\gamma\in\Gamma$.\vspace{2mm}
\noindent {\it Proof of Theorem\/} 1. Just as in the case of Gaussian and
Poisson measures, see \cite{Bee,BeLi,Ly}, we deduce, by using the projection spectral theorem
\cite{BeKo}, Ch.~3, Th.~2.7, the existence of a unitary isomorphism $I$ between
${\cal F}_{\mathrm \Ext}({\cal H})$ and an $L^2$-space $L^2(S',{\cal B}(S'),d\mu)$ for which the
formulas \eqref{2.5}--\eqref{2.7} hold. Here, $\mu$ is a Borel probability
measure on $S'$, which is the spectral measure of the family
$(a^\sim(\xi))_{\xi\in S}$. Indeed, the major step in the proof of this fact is the
following lemma.
\begin{lem}
\rom{1)} The family $(a^\sim(\xi))_{\xi\in S}$ is connected with
\eqref{a} in a standard way\rom, i\rom.e\rom{.,} for each $\xi\in S$\rom, $a(\xi)$
is a linear continuous operator on ${\cal F}_{{\mathrm fin}}(S)$\rom.
\rom{2)} For an arbitrary fixed $f\in{\cal F}_{{\mathrm fin}}(S)$\rom, the mapping
$$ S\ni\xi\mapsto a(\xi)f\in{\cal F}_{{\mathrm fin}}(S)$$
is linear and continuous\rom.
\rom{3)} The vacuum $\Omega=(1,0,0,\dots)\in{\cal F}_{{\mathrm fin}}(S)$ is a strong cyclic vector
of the family $(a^\sim(\xi))_{\xi\in S}$\rom, i\rom.e\rom{.,} the linear span of
the set
$$\{\Omega\}\cup\{\,a(\xi_1)\cdots a(\xi_n)\Omega\mid \xi_i\in S,\ i=1,\dots,n,\
n\in {\Bbb N}\,\}$$
is dense in ${\cal F}_{{\mathrm fin}}(S)$\rom.
\end{lem}
The item 1 of Lemma 2 we have already proved, while the item~2 follows from the
definition of the operators $a(\xi)$ and the estimates \eqref{2.1}--\eqref{2.3}.
The item~3 is implied by the fact that we have the standard creation operators,
which allows us just to repeat the proof of the corresponding fact in the
Gaussian and Poisson cases.
Thus, having Lemma~2, we apply the projection spectral theorem, which gives us
the existence of a spectral measure $\mu$ of the family $(a^\sim(\xi))_{\xi\in S}$ as
a probability measure on $S'$. For $\mu$-almost every $\omega\in S'$, there exists
a generalized joint eigenvector
$$P(\omega)=(P^{(n)}(\omega))_{n=0}^\infty\in{\cal F}_{{\mathrm fin}}^*(S)$$
satisfying
\begin{equation}\label{2.9}
\langle\!\langle P(\omega),a(\xi)f\rangle\!\rangle=\langle\omega,\xi\rangle\langle\!\langle P(\omega),f\rangle\!\rangle,
\qquad f\in {\cal F}_{{\mathrm fin}}(S).\end{equation}
Then, the operator $I$ defined by
\begin{gather}
If=(If)(\omega)=\langle\!\langle P(\omega),f\rangle\!\rangle=\sum_{n=0}^\infty\langle
P^{(n)}(\omega),f^{(n)}\rangle\,n!,\label{2.10}\\
f=(f^{(n)})_{n=0}^\infty\in{\cal F}_{{\mathrm fin}}(S),\notag\end{gather}
can be extended by continuity to a unitary operator between ${\cal F}_{\mathrm \Ext}({\cal H})$ and\linebreak
$L^2(S',{\cal B}(S'),d\mu)$, under which any operator $a^\sim(\xi)$ goes over
into the operator of multiplication by $\langle\omega,\xi\rangle$ (see \eqref{2.9} and
\eqref{2.10}). Denoting $\wom{n}=P^{(n)}(\omega)\,n!$, we present the unitary $I$ in
the form \eqref{2.5}. The recurrence formula \eqref{2.6} can now be derived from
the equality
\begin{gather*}
\langle\omega,\xi\rangle\langle\wom{n},\xi^{\otimes n}\rangle=\langle\wom{(n+1)},\xi^{\otimes(n+1)}\rangle
+\langle\wom n,2n(\xi^2)\hat \otimes\xi^{\otimes(n-1)}+\langle\xi\rangle\xi^{\otimes n}\rangle\\
\text{}+\langle\wom{(n-1)},n\langle\xi^2\rangle\xi^{\otimes(n-1)}+n(n-1)(\xi^3)\hat \otimes\xi^{\otimes(n-2)}\rangle.
\end{gather*}
As easily seen, $\wom{n}$ is well defined as an element of $S^{\prime\,\hat \otimes n}$
for {\it each\/} $\omega\in S'$, not only for $\mu$-almost all $\omega\in S'$.
Thus, we need only to prove that $\mu$ is, in fact, the measure of
Gamma white noise.
Let ${\cal B}_{\mathrm b}({\Bbb R}^d)$ denote the set of all bounded Borel
sets in ${\Bbb R}^d$, let $\Delta\in{\cal B}_{\mathrm b}({\Bbb R}^d)$, and let
$\chi_\Delta$ denote the indicator of $\Delta$. Denote by $a^\sim(\chi_\Delta)$
the operator in ${\cal F}_{\mathrm \Ext}({\cal H})$ whose image under the unitary $I$ is the operator of
multiplication by the function
$\langle\omega,\chi_\Delta\rangle=\langle\wom1,\chi_\Delta\rangle+\sigma(\Delta)
=(I\chi_\Delta)(\omega)+\sigma(\Delta)$.
Each of the vectors $\chi_\Delta^{\otimes n}$, $n\ge0$,
$\chi_\Delta^{\otimes 0}:=\Omega$,
belongs to ${\cal F}_{\mathrm \Ext}({\cal H})$, and let ${\cal K}_\Delta$ be the subspace of ${\cal F}_{\mathrm \Ext}({\cal H})$
spanned by these vectors. A limiting procedure shows that
\begin{equation}\label{2.12}a^\sim(\chi_\Delta)\chi_\Delta^{\otimes
n}=\chi_\Delta^{\otimes(n+1)}+
(2n+\sigma(\Delta))\chi_\Delta^{\otimes
n}+n(n-1+\sigma(\Delta))\chi_\Delta^{\otimes(n-1)}.\end{equation}
Therefore, ${\cal K}_\Delta$ is an invariant subspace for the operator
$a^\sim(\chi_\Delta)$. Let $a_\Delta$ stand for the restriction of $a^\sim(\chi_\Delta)$ to the linear span of the vectors
$\chi_\Delta^{\otimes n}$. Then, $a_\Delta$ is a densely defined, Hermitian operator
in ${\cal K}_\Delta$.
Let $c_{n,\Delta}=\|\chi_\Delta^{\otimes n}\|_{{\cal F}_{\mathrm \Ext}({\cal H})}$, then the vectors
$(e_\Delta^{(n)})_{n=0}^\infty$, with
$e_\Delta^{(n)}=c_{n,\Delta}^{-1}\chi_\Delta^{\otimes n}$, form an orthonormal basis
(ONB) in ${\cal K}_\Delta$.
\eqref{2.12} yields
\begin{equation}
a_\Delta e^{(n)}_\Delta=\frac{c_{n+1,\Delta}}{c_{n,\Delta}}\,e_\Delta^{(n+1)}+
(2n+\sigma(\Delta))e_\Delta^{(n)}
+n(n-1+\sigma(\Delta))\,\frac{c_{n-1,\Delta}}{c_{n,\Delta}}\,e_\Delta^{(n-1)}.\label{2.13}
\end{equation}
Since $a_\Delta$ is Hermitian, the matrix of this operator in the ONB
$(e^{(n)}_\Delta)_{n=0}^\infty$ must be symmetric, which together with the
formula \eqref{2.13} implies that this matrix is a Jacobi one, see \cite{Be}, i.e.,
\begin{equation}\label{2.14}
a_\Delta e_\Delta^{(n)}=\alpha_{n+1,\Delta}e_\Delta^{(n+1)}+
\beta_{n,\Delta}e_\Delta^{(n)}+\alpha_{n,\Delta}
e_\Delta^{(n-1)}.
\end{equation}
Comparing \eqref{2.13} and \eqref{2.14} gives the equality
$$\frac{c_{n,\Delta}}{c_{n-1,\Delta}}=n(n-1+\sigma(\Delta))\,\frac{c_{n-1,\Delta}}{c_{n,\Delta}},$$
from where
\begin{equation}\label{2.15}
\frac{c_{n,\Delta}}{c_{n-1,\Delta}}=\sqrt{n(n-1+\sigma(\Delta))}.\end{equation}
Substituting \eqref{2.15} into \eqref{2.13}, we derive that the coefficients in
\eqref{2.14} equal
\begin{equation}\label{2.16}
\alpha_{n,\Delta}=\sqrt{n(n-1+\sigma(\Delta))},\qquad \beta_{n,\Delta}=2n+\sigma(\Delta).
\end{equation}
By using e.g.\ \cite{Be}, Ch.~7, Th.~1.3, we conclude that the Jacobi matrix
defined by the
coefficients \eqref{2.16} has a unique spectral measure. This means that there
exists a unique probability measure $\mu_\Delta$ on ${\Bbb R}$ such that the system of
polynomials $(P_{n,\Delta})_{n=0}^\infty$ defined by the recurrence formula
\begin{equation}\label{2.17}
P_{n+1,\Delta}(s)=\frac{1}{\alpha_{n+1,\Delta}}\big((s-\beta_{n,\Delta})P_{n,\Delta}(s)
-\alpha_{n,\Delta}P_{n-1,\Delta}(s)
\big)\end{equation}
forms an ONB in $L^2({\Bbb R},d\mu_\Delta)$, and under the unitary $I_\Delta\colon
{\cal K}_\Delta\to L^2({\Bbb R},d\mu_\Delta)$ given by
$I_\Delta e_\Delta^{(n)}=P_{n,\Delta}$ the
operator $a_\Delta^\sim$---the closure in ${\cal K}_\Delta$ of the
essentially selfadjoint operator $a_\Delta$---goes
over into the operator of multiplication by
$s$. Moreover, the form of the coefficients \eqref{2.16} gives (see, e.g., \cite{Chihara}) that the measure
$\mu_\Delta$ is exactly the Gamma distribution having density \eqref{gamma}.
On the other hand, since $Ia^\sim(\chi_\Delta)I^{-1}$ is the operator of
multiplication by $\langle\omega,\chi_\Delta\rangle$, \eqref{2.14} and \eqref{2.17} imply that
$$(Ie_\Delta^{(n)})(\omega)=P_{n,\Delta}(\langle\omega,\chi_\Delta\rangle).$$
Therefore, $\mu_\Delta$ coincides with the distribution of the r.v.\
$\langle\omega,\chi_\Delta\rangle.$
Next, for arbitrary disjoint sets $\Delta_1,\Delta_2\in{\cal B}
_{\mathrm b}({\Bbb R}^d)$,
the spaces ${\cal K}_{\Delta_1}\ominus \widehat \Omega$ and ${\cal
K}_{\Delta_2}\ominus\widehat\Omega$
are orthogonal in
${\cal F}_{\mathrm \Ext}({\cal H})$. Here, $\widehat \Omega$ denotes the vacuum space spanned by
$\Omega$, and $\ominus$ denotes the orthogonal difference. Therefore,
the r.v.'s $\langle\omega,\chi_{\Delta_1}\rangle$ and
$\langle\omega,\chi_{\Delta_2}\rangle$ are independent.
Thus, $\mu$ is indeed the measure of Gamma white noise.\quad $\Box$\vspace{2mm}
\noindent {\it Remark\/} 5. Note that the polynomial $P_{n,\Delta}(s)$ is up to
a multiplier $(-1)^n$ the normalized Laguerre polynomial with parameter
$\sigma(\Delta)-1$.
Therefore,
$$\langle\wom{n},f^{(n)}\rangle=(-1)^n\langle L_n^\sigma(\omega),f^{(n)}\rangle,$$
where $L_n^\sigma(\omega)$ is the kernel of the generalized Laguerre polynomial
introduced in \cite{Silva}. This yields, in particular,
that the Gamma-Wick exponential defined by
\begin{equation}
\label{exponential}
\wexp{\omega}{\varphi}=\sum_{n=0}^\infty\frac1{n!}\langle\wom{n},\varphi^{\otimes n}\rangle,\qquad \varphi\in
S_{\Bbb C},\end{equation}
has
the following
representation,
for $\varphi\in S$ such that $\varphi>-1$,
\begin{equation}\label{silva}
\wexp{\omega}{\varphi}=\exp\Big[\langle\omega,\frac{\varphi}{\varphi+1}\rangle-\langle\log(1+\varphi)\rangle\Big].\end{equation}
\vspace{2mm}
\noindent {\it Remark\/} 6. Following the tradition of quantum probability,
e.g., \cite{Mey}, the family of the operators $(a^\sim(\chi_\Delta))_{\Delta\in\Lambda}$
can be called a quantum Gamma process in the extended Fock space ${\cal F}_{\mathrm \Ext}({\cal H})$.
\section{Chaos decomposition of the Gamma space}
In this section, we will discuss some properties of the unitary $I$ and the
space of Gamma white noise $(L^2_{\mathrm G})$.
Denote by ${\cal P}(S')$ the set of continuous polynomials on $S'$, i.e., functions
on $S'$ of the form
\begin{equation*}
\phi(\omega)=\sum_{i=0}^n\langle\omega^{\otimes i},f^{(i)}\rangle,\qquad f^{(i)}\in S_{\Bbb C}^{\hat \otimes i},\
\omega^{\otimes 0}=1,\ i\in{\Bbb N}_0.\end{equation*}
The greatest number $n$ for which $f^{(n)}\ne0$ is called the power of a
polynomial. Denote also by ${\cal P}_n(S')$ the set of continuous polynomials of
power $\le n$.
By using the isomorphism
\begin{equation}\label{3.1}
{\cal F}_{{\mathrm fin}}(S)\ni f=(f^{(n)})_{n=0}^\infty\mapsto\sum_{n=0}^\infty\langle\omega^{\otimes
n},f^{(n)}\rangle\in{\cal P}(S'),
\end{equation}
one induces from ${\cal F}_{{\mathrm fin}}(S)$ a topology on ${\cal P}(S')$, which makes it a nuclear
space.
\begin{prop} We have
$$I({\cal F}_{{\mathrm fin}}(S))={\cal P}(S'),$$
and the topology on ${\cal P}(S')$ induced from ${\cal F}_{{\mathrm fin}}(S)$ by $I$ coincides with that
induced by \eqref{3.1}\rom.\end{prop}
\noindent {\it Proof}. Using the recurrence relation
\eqref{2.6}, one obtains by induction the inclusion $I({\cal F}_{{\mathrm fin}}(S))\subset{\cal P}(S')$,
and moreover, the equality
\begin{equation}\label{3.2}
\langle\wom{n},f^{(n)}\rangle=\langle\omega^{\otimes n},f^{(n)}\rangle+p_{n-1}(\omega),\end{equation}
where $p_{n-1}(\omega)\in{\cal P}_{n-1}(S')$. Using \eqref{3.2}, one obtains again by
induction the inverse inclusion ${\cal P}(S')\subset{\cal F}_{{\mathrm fin}}(S)$. The statement about
the topology on ${\cal P}(S')$ follows directly from the above
consideration. \quad$\Box$\vspace{2mm}
For every $n\in{\Bbb N}_0$, put $(L^2_{{\mathrm G},n})=I({\cal F}_{\mathrm \Ext}^{(n)}({\cal H}))$, so that
we obtain the following decomposition of the space $(L^2_{\mathrm G})$:
\begin{equation}\label{3.3}
(L^2_{\mathrm G})=\bigoplus_{n=0}^\infty(L^2_{{\mathrm G},n}).\end{equation}
\begin{prop}For each $n\in{\Bbb N}$\rom, we have
\begin{equation}\label{gamman}
(L^2_{{\mathrm G},n})={\cal P}_{{\mathrm G},n}(S')\ominus{\cal P}_{{\mathrm G},n-1}(S'),\end{equation}
where ${\cal P}_{{\mathrm G},n}(S')$ denotes the closure of ${\cal P}_n(S')$ in the $(L^2_{\mathrm G})$
norm\rom, and $\ominus$ stands for the orthogonal difference in $(L^2_{\mathrm G})$.\end{prop}
\noindent {\it Proof}. As appears from the proof of Proposition~1,
$$I\Big(\bigoplus_{i=0}^n S_{\Bbb C}^{\hat \otimes i}\Big)={\cal P}_n(S'),$$
and hence
$$I\Big(\bigoplus_{i=0}^n{\cal F}_{\mathrm \Ext}^{(n)}({\cal H})\Big)=\bigoplus_{i=0}^n(L^2_{{\mathrm
G},n})={\cal P}_{{\mathrm G},n}(S'),$$
from where \eqref{gamman} follows.\quad$\Box$
\begin{prop}
Let $P_{{\mathrm G},n}$ stand for the orthogonal projection of $(L^2_{\mathrm G})$ onto the
subspace $(L_{{\mathrm G},n}^2)$\rom. Then
\begin{equation}\label{3.4}
P_{{\mathrm G},n}(\langle\omega^{\otimes
n},f^{(n)}\rangle)=\langle\wom{n},f^{(n)}\rangle,\qquad f^{(n)}\in S_{\Bbb C}^{\hat \otimes n}.\end{equation}\end{prop}
\noindent{\it Proof}. \eqref{3.4} is derived from \eqref{3.2}
and Proposition~2
by
applying to the left and right hand sides of the
latter formula the operator $P_{{\mathrm G},n}$ and taking to notice that
$\langle\wom{n},f^{(n)}\rangle\in(L^2_{{\mathrm G},n})$.\quad$\Box$\vspace{2mm}
Propositions 2 and 3 make it possible to interpret \eqref{3.3} as a kind of chaos
decomposition of $(L^2_{\mathrm G})$. However, one should be careful with this term.
Indeed, since ${\Bbb E}[X_\Delta]=\sigma(\Delta)$ and ${\Bbb E}[(X_\Delta-\sigma(\Delta))^2]=\sigma(\Delta)$,
the compensated Gamma process
$$\tilde X_\Delta=\tilde X_\Delta(\omega)=
X_\Delta(\omega)-\sigma(\Delta)=\langle\omega,\chi_\Delta\rangle-\sigma(\Delta)
=\langle\wom1,\chi_\Delta
\rangle
,\qquad \Delta\in{\cal B}_{\mathrm b}({\Bbb R}^d),$$
is a normal martingale, and so one can introduce multiple stochastic integrals
w.r.t.\ $\tilde X_\Delta$. But it follows from the general result of \cite{Der}
that this process does not possess the chaotic representation property, so that
\eqref{3.3} cannot be a chaos decomposition of $(L^2_{\mathrm G})$ in the sense of multiple
stochastic integrals. To be more precise, we present the following propopsition.
\begin{prop}
The chaos in $(L^2_{\mathrm G})$ generated by multiple stochastic integrals
w\rom.r\rom.t\rom.\ the compensated Gamma process $\tilde X_\Delta$
coincides with the image under the unitary $I$ of the usual Fock space ${\cal
F}({\cal H})$ as a subspace of ${\cal F}_{\mathrm \Ext}({\cal H})$\rom, and moreover
$$ \int_{{\Bbb R}^{dn}}f^{(n)}(x_1,\dots,x_n)\,d\tilde X_{x_1}\cdots d\tilde
X_{x_n}=If^{(n)}$$
for an arbitrary $f^{(n)}\in{\cal H}_{\Bbb C}^{\hat \otimes n}$\rom, where ${\cal H}_{\Bbb C}^{\hat \otimes n}$ is
considered as a subspace of ${\cal F}_{\mathrm \Ext}^{(n)}({\cal H})$\rom.\end{prop}
\noindent{Proof}. We remind that the $n$-fold stochastic integral w.r.t.\ $\tilde
X_\Delta$ are defined as follows (e.g., \cite{Mey,Der}). First, one takes
arbitrary disjoint sets $\Delta_1,\dots,\Delta_n\in\Lambda$ and sets
\begin{equation}\label{3.5}
\int_{{\Bbb R}^{dn}}\big(
\chi_{\Delta_1}(x_1)\cdots\chi_{\Delta_n}(x_n)
\big)^\sim
d\tilde X_{x_1}\cdots d\tilde X_{x_n}
=\tilde X_{\Delta_1}\cdots \tilde X_{\Delta_n}.
\end{equation}
Then, one extends the equality \eqref{3.5} by linearity and continuity in
the ${\cal H}_{\Bbb C}^{\hat \otimes n}$ norm. But Theorem~1 yields that
\begin{gather*}
\tilde X_{\Delta_1}(\omega)\cdots\tilde
X_{\Delta_n}(\omega)=\langle\wom1,\chi_{\Delta_1}\rangle\cdots\langle\wom{1},\chi_{\Delta_n}\rangle\\
=I\big(
(a(\chi_{\Delta_1})-\sigma(\Delta_1)\operatorname{id})\cdots(a(\chi_{\Delta_n})-\sigma(\Delta_n)\operatorname{id})\Omega
\big)\\
=I(a^+(\chi_{\Delta_1})\cdots a^+(\chi_{\Delta_n})\Omega)
=I(\chi_{\Delta_1}\hat \otimes\cdots\hat \otimes\chi_{\Delta_n})
,\end{gather*}
which together with our consideration at the end of Section~1 gives the statement.\quad$\Box$
\section{Spaces of test and generalized functions\\
and coordinate operators on
them}
In this section, we will construct spaces of test and generalized functions of
Gamma white noise and consider some their properties.
The simplest ones can be obtained by applying $I$ to \eqref{a}.
Thus, taking to notice Proposition~1, we get
$$(L^2_{\mathrm G})\supset{\cal P}(S'),\qquad {\cal P}(S')^*\supset{\cal P}(S').$$
The dual ${\cal P}(S')^*$ of the space of continuous polynomials ${\cal P}(S')$ consists
of generalized functions, which we present in the form
$$\Phi=\Phi(\omega)=\sum_{n=0}^\infty\langle\wom n,F^{(n)}\rangle,$$
where $F=(F^{(n)})_{n=0}^\infty\in{\cal F}_{{\mathrm fin}}^*(S)$ and the dualization of $\Phi$ with
$\phi\in{\cal P}(S')$, $\phi(\omega)=\sum_{n=0}^\infty \langle\wom n,f^{(n)}\rangle$, is
given by
$$\langle\!\langle\Phi,\phi\rangle\!\rangle =\sum_{n=0}^\infty\langle \overline{F^{(n)}},f^{(n)}\rangle n!.$$
However, the test space ${\cal P}(S')$ is too small. This is why we will consider
the following nuclear space and its dual: (cf.~\cite{KoLeS,Koetal,KSWY,GAS})
\begin{gather}
{\cal F}_{\mathrm \Ext}({\cal H})
\supset\projlim_{p,k\to\infty}{\cal F}_{1,k}(S_p)={\cal F}_1(S_1),\notag\\
{\cal F}_{-1}(S')=\operatornamewithlimits{ind\,lim}_{p,k\to\infty}{\cal F}_{-1,-k}(S_{-p})\supset
\projlim_{p,k\to\infty}{\cal F}_{1,k}(S_p)={\cal F}_1(S_1),
\label{4.1}\end{gather}
where the spaces ${\cal F}_{1,k}(S_p)$, $p\ge1$, $k\ge k_0$, are defined by \eqref{space}, ${\cal
F}_{-1,-k}(S_{-p})$ are their respective duals (w.r.t.\ ${\cal F}({\cal H})$):
$${\cal F}_{-1,-k}(S_{-p})=\bigoplus_{n=0}^\infty S_{-p,{\Bbb C}}^{\hat \otimes n} 2^{-nk}.$$
The application of $I$
to \eqref{4.1} gives the following spaces of test and generalized functions:
\begin{gather*}
(L^2_{\mathrm G})\supset\projlim_{p,k\to\infty}(S_{\mathrm
G})_{p,k}^1=(S_{\mathrm G})^1,\\
(S_{\mathrm G})^{-1}=\operatornamewithlimits{ind\,lim}_{p,k\to\infty}(S_{\mathrm G})^{-1}_{-p,-k}\supset
\projlim_{p,k\to\infty}(S_{\mathrm G})^1_{p,k}=(S_{\mathrm G})^1.
\end{gather*}
On the space $(S_{\mathrm G})^{-1}$, we introduce an $\S$-transform in a standard way
(cf.~\cite{KoLeS,Koetal,KSWY,GAS}): If $\Phi\in(S_{\mathrm G})^{-1}$, then by the definition of $(S_{\mathrm G})^{-1}$ there are
$p\ge1$ and $k\ge k_0$ such that $\Phi\in(S_{\mathrm G})_{-p,-k}^{-1}$, and we put
\begin{equation}\label{4.2}
\S[\Phi](\theta)=\ll\overline{\Phi},\wexp{\cdot}{\theta}\ra\!\ra,\qquad \theta\in
S_{\Bbb C},\ |\theta|_p<2^{-k/2},\end{equation}
where $\overline{\Phi}$ is the complex conjugate of $\Phi$ and $\wexp{\cdot}{\theta}$
is defined by \eqref{exponential}.
The condition on the norm of $\xi$ in \eqref{4.2} implies that $\wexp{\cdot}\theta\in(S_{\mathrm G})_{p,k}^1
$, so that the dualization in \eqref{4.2} is well-defined. From the definition
of the $\S$-transform, we have for each
$$\Phi=\Phi(\omega)=\sum_{n=0}^\infty\langle\wom{n},F^{(n)}\rangle,$$
that
$$\S[\Phi](\theta)=\sum_{n=0}^\infty\langle F^{(n)},\theta^{\otimes
n}\rangle.$$
Notice that if $\theta\in S_{\Bbb C}'$, we can still define the Wick exponential
$\wexp{\cdot}{\theta}$ by the same formula \eqref{exponential}, considering it
as a generalized function from $(S_{\mathrm G})^{-1}$. Hence, the $\S$-transform of a test
function $\phi(\omega)=\sum_{n=0}^\infty\langle \wom{n},f^{(n)}\rangle\in(S_{\mathrm G})^1$ can be
extended on $S_{\Bbb C}'$ by setting
\[\S[\phi](\theta)=\ll \overline{\phi},\wexp{\cdot}{\theta}\ra\!\ra=\sum_{n=0}^\infty
\langle f^{(n)},\theta^{\otimes n}\rangle,\qquad \theta\in S_{\Bbb C}'.\]
The following two theorems, which are due to \cite{KoLeS}, give the
description of the spaces $(S_{\mathrm G})^1$ and $(S_{\mathrm G})^{-1}$ in terms of the $\S$-transform.
\begin{th} For any $\Phi\in(S_{\mathrm G})^{-1}$\rom, $\S[\Phi]$ is a function holomorphic
at zero\rom, i\rom.e\rom{.,} $\S[\Phi]\in\operatorname{Hol}_0(S_\C)$\rom. And conversely\rom, for any
$F\in\operatorname{Hol}_0(S_\C)$\rom, there exists a unique $\Phi\in(S_{\mathrm G})^{-1}$ such that
$\S[\Phi]=F$ \rom(i\rom.e\rom{.,} there is a neighborhood of zero\rom, $U$\rom,
such that $\S[\Phi](\theta)=F(\theta)$ for all $\theta\in U$\rom{).}
\end{th}
Denote by $\Emin{S_{\Bbb C}'}$ the set of entire functions on $S_{\Bbb C}'$ of the first
order of growth and of minimal type, i.e., a function $u$ entire on $ S_{\Bbb C}'$
belongs to $\Emin{S_{\Bbb C}'}$ iff
\[ \forall p>0\ \forall \epsilon>0\ \exists C>0:\quad |u(\theta)|\le
Ce^{\epsilon|\theta|_{-p}}, \qquad \theta\in S_{-p,{\Bbb C}}.\]
The set $\Emin{S_{\Bbb C}'}$ can be topologized by the following family of norms
\begin{equation}\label{complexnorm}
\pmb{\pmb |} u\pmb{\pmb |}_{1,p,k}=\sup_{\theta\in S_{-p,{\Bbb C}}}\Big\{|u(\theta)|\exp\big[-\frac
1k\,|\theta|_{-p}\big]\Big\},\qquad p,k\in{\Bbb N},\end{equation}
so that $\Emin{S_{\Bbb C}'}$ becomes a countably normed space.
\begin{th}
For any $\phi\in(S_{\mathrm G})^{1}$\rom, $\S[\phi]\in\Emin{S_{\Bbb C}'}$\rom. And
conversely\rom, for any $f\in\Emin{S_{\Bbb C}'}$\rom, there exits a unique $\phi\in
(S_{\mathrm G})^1$ such that $\S[\phi]=f$\rom. Moreover\rom, the $\S$-transform is a
homeomorphism between the topological spaces $(S_{\mathrm G})^1$ and $\Emin{\S_{\Bbb C}'}.$
\end{th}
Denote now by $\Emin{S'}$ the set of the restrictions to $S'$
of functions from $\Emin{S_{\Bbb C}'}$.
The topology on $\Emin{S'}$ is induced by that of $\Emin{S'_{\Bbb C}}$.
The next theorem gives the inner description of the test space $(S_{\mathrm G})^1$.
\begin{th}
We have
\begin{equation}\label{4.3} (S_{\mathrm G})^1=\Emin{S'},\end{equation}
where \eqref{4.3} is understood as an equality of topological
spaces\rom.\end{th}
\noindent{\it Proof}.
This theorem is, in fact, a direct corollary of results of \cite{GAS} (see also
\cite{LyUs}),
since the Gamma-Wick monomials
$\langle\wom{n},f^{(n)}\rangle$
form a generalized Appell system in the sense of \cite{GAS} with the
transformation function $\alpha(\varphi)=\dfrac{\varphi}{\varphi+1}$
(cf.~\cite{Silva}).
However, for the further exposition, we need to state some details of the theory
of (generalized) Appell polynomials.
Let $\mu$ be a probability measure on $(S',{\cal B}(S'))$ satisfying the following
two conditions:\vspace{1.7mm}
\noindent{\bf Condition 1.} The Laplace transform $\ell_\mu(\varphi)$
of the measure $\mu$ can be extended to an analytic function in a neighborhood
of zero in $S_{\Bbb C}'$. , \vspace{1.7mm}
\noindent{\bf Condition 2.} If $\phi\in{\cal P}(S')$ and $\phi=0$ $\mu$-almost
everywhere, then $\phi\equiv 0$.\vspace{1.7mm}
Put
$$e_\mu(\varphi;\omega)=\frac{e^{\langle\omega,\varphi\rangle}}{\ell_\mu(\varphi)}.$$
Due to Condition 1 and the fact that $\ell_\mu(0)=1$, for every fixed $\omega\in
S'$
$e_\mu(\varphi;\omega)$ is
an analytic function of $\varphi$ in a neighborhood of zero.
Define now $P^{(0)}_\mu(\omega)=1$ and $P_\mu^{(n)}(\omega)\in S^{\prime\,\hat \otimes n}$,
$n\in {\Bbb N}$, by
\begin{equation}\label{zyp}
\langle P_\mu^{(n)}(\omega),\varphi^{\otimes
n}\rangle =\frac{d^n}{dt^n}\Big|_{t=0}e_\mu(t\varphi;\omega),\end{equation}
so that
$$e_\mu(\varphi;\omega)=\sum_{n=0}^\infty\frac{1}{n!}\,\langle
P^{(n)}_\mu(\omega),\varphi^{\otimes n}\rangle.$$
By using the kernel theorem, one shows that this definition is correct, i.e.,
that \eqref{zyp} really determines an element of $S^{\prime\,\hat \otimes n}$. A function
of $\omega$ of the form $\langle P_\mu^{(n)}(\omega),f^{(n)}\rangle$, where $f^{(n)}\in S_{\Bbb C}^{\hat \otimes
n}$, is called an Appell polynomial. These polynomials have the following very important property:
let $\upsilon\in S'$ and let $D_\upsilon$ denote the G\^ateaux derivative in
direction $\upsilon$:
$$D_\upsilon \phi(\omega)
=\frac{d}{dt}\Big|_{t=0}\,\phi(\omega+t\upsilon)
=\lim_{t\to0}\frac{\phi(\omega+t\upsilon)-\phi(\omega)}{t},$$
then
$$D_\upsilon \langle P^{(n)}_\mu(\omega),\varphi^{\otimes n}\rangle=n\langle\upsilon,\varphi\rangle\langle
P^{(n-1)}_\mu(\omega), \varphi^{\otimes(n-1)}\rangle,$$
i.e., $D_\upsilon$ acts as an annihilation operator w.r.t.\ the Appell
polynomials. Particularly, for the gradient
\begin{equation}\label{gradient}
\nabla_x=D_{\delta_x},\qquad x\in{\Bbb R}^d,\end{equation}
we have
$$\nabla_x\langle P_\mu^{(n)}(\omega),\varphi^{\otimes n}\rangle=n\varphi(x)\langle
P^{(n-1)}_\mu(\omega),\varphi^{\otimes(n-1)}\rangle.$$
and so
$$\nabla_x e_\mu(\varphi;\omega)=\varphi(x)e_\mu(\varphi;\omega).$$
Now, for each $p\ge1$ and $k\in{\Bbb N}_0$, one introduces a norm $\pmb{\boldsymbol{\|}}\cdot\pmb{\boldsymbol{\|}}_{1,p,k}$ on ${\cal P}(S')$
as follows: for
$$\phi(\omega)=\sum_{n=0}^k\langle P_\mu^{(n)}(\omega),f^{(n)}\rangle,$$
we have
$$\pmb{\boldsymbol{\|}}\phi\pmb{\boldsymbol{\|}}_{1,p,k}^2=\sum_{n=0}^\infty |f^{(n)}|_p^2(n!)^2 2^{nk},$$
and let $[S_\mu]_{p,k}^1$
denote the closure of ${\cal P}(S')$ in this norm. Put
$$[S_\mu]^1=\projlim_{p,k\to\infty}[S_\mu]_{p,k}^1.$$
Thus, one has \cite{KSWY} that there exist $p_1\ge1$ and $k_1\in{\Bbb N}_0$ such that $[S_\mu]^1_{p_1,k_1}$
is topologically embedded into $L^2(S',d\mu)=(L^2_\mu)$, and moreover, one has
the topological equality
$$[S_\mu]^1={\cal E}^1_{\mathrm min}(S').$$
Next, we proceed to consider a generalized Appell system. Let $\alpha$ be an
arbitrary function that maps a neighborhood of zero in $S_{\Bbb C}$ into itself, which
is supposed to be holomorphic, invertible and to satisfy $\alpha(0)=0$.
Let
$$e_{\mu,\alpha}(\varphi;\omega)=e_\mu(\alpha(\varphi);\omega)=\frac{e^{\langle
\omega,\alpha(\varphi)}\rangle}{\ell_\mu(\alpha(\varphi))}.$$
Then, analogously to the above, one defines generalized Appell polynomials\linebreak $\langle
P^{(n)}_{\mu,\alpha}(\omega),f^{(n)}\rangle$ in such a way that
$$e_{\mu,\alpha}(\varphi;\omega)=\sum_{n=0}^\infty \frac1{n!}\,\langle
P^{(n)}_{\mu,\alpha}(\omega),\varphi^{\otimes n}\rangle.$$
The $\alpha$-gradient $\nabla_x^\alpha$ is defined now by
$$\nabla_x^\alpha \langle P^{(n)}_{\mu,\alpha}(\omega),\varphi^{\otimes n}\rangle=n\varphi(x)\langle
P^{(n-1)}
_{\mu,\alpha}
(\omega),\varphi^{\otimes(n-1)}\rangle,$$
so that
$$\nabla_x^\alpha e_{\mu,\alpha}(\varphi;\omega)=\varphi(x)e_{\mu,\alpha}(\varphi;\omega).$$
Using the polynomials $\langle P_{\mu,\alpha}^{(n)}(\omega),f^{(n)}\rangle$, one defines
just as above the spaces $[S_\mu]_{p,k,\alpha}^1$, and again \cite{GAS} there
exist $p_2\ge 1$ and $k_2\in{\Bbb N}_0$ such that $[S_\mu]^1_{p_2,k_2,\alpha}$ is topologically embedded
into $(L^2_\mu)$ and
$$[S_\mu]_\alpha
^1=\projlim_{p,k\to\infty}[S_\mu]^1_{p,k,\alpha}={\cal E}^1_{\mathrm
min}(S').$$
Let us come back to the case of the Gamma measure $\mu=\mu_{\mathrm G}$. Evidently the
Laplace transform $\ell_{\mathrm G}(\varphi)$ of $\mu_{\mathrm G}$ satisfies Condition~1 (see
\eqref{2.4a}). Proposition~1 implies that Condition~2 is also satisfied for
$\mu_{\mathrm G}$. Thus, we can put
\begin{equation}\label{sp}e_{\mathrm G}(\varphi;\omega)=\frac{e^{\langle\omega,\varphi}\rangle}{\ell_{\mathrm
G}(\varphi)}=\exp\big[ \langle\omega,\varphi\rangle+\langle\log(1-\varphi)\rangle\big].\end{equation}
By setting
\begin{equation}\label{spp} \alpha(\varphi)=\frac{\varphi}{\varphi+1},\end{equation}
which satisfies the conditions on $\alpha$, we get
\begin{equation}\label{neznaju}
e_{{\mathrm
G},\alpha}(\varphi;\omega)=\exp\Big[\big\langle\omega,\frac{\varphi}{\varphi+1}\big\rangle-\langle\log(1+\varphi)\rangle\Big].\end{equation}
Comparing the last formula with \eqref{silva}, we conclude that
$$\langle\wom{n},f^{(n)}\rangle=\langle P^{(n)}_{{\mathrm G},\alpha}(\omega),f^{(n)}\rangle,$$
which yields the theorem.
\quad $\blacksquare$\vspace{2mm}
On the space $(S_{\mathrm G})^{-1}$, one can introduce a Wick product $\diamondsuit$ as follows
(cf.~\cite{KoLeS,KSWY,GAS}):
for arbitrary $\Phi,\Psi\in(S_{\mathrm G})^{-1}$, $\Phi\,\diamondsuit\,\Psi$ is an element of $(S_{\mathrm G})^{-1}$
such that
\begin{equation}\label{4.4}
\S[\Phi\,\diamondsuit\,\Psi](\theta)=\S[\Phi](\theta)\,\S[\Psi](\theta)\end{equation}
for $\theta\in S_{\Bbb C}$ from some neighborhood of zero. Since $\operatorname{Hol}_0(S_\C)$
is an algebra under pointwise multiplication of functions, this definition is
correct. \eqref{4.4} implies that
\[\langle\wom{n},F^{(n)}\rangle\,\diamondsuit\,\langle\wom{m},G^{(m)}\rangle=\langle\wom{(n+m)},F^{(n)}\hat \otimes
G^{(m)}\rangle.\]
Notice that the Wick product of two test functions from $(S_{\mathrm G})^1$
belongs again to $(S_{\mathrm G})^1$.
\vspace{2mm}
\noindent{\it Remark\/} 7. Because of \eqref{sp}--\eqref{neznaju}, the above introduced Wick
product on $(S_{\mathrm G})^{-1}$ coincides with the Wick product on this space defined in
the framework of biorthogonal analysis \cite{KSWY}, i.e., by using the
function $e_{\mathrm G}(\cdot\,;\theta)$ instead of $\wexp{\cdot}{\theta}$.\vspace{2mm}
Let us consider a pair of simplest examples of generalized functions from $(S_{\mathrm G})^{-1}$.
The first one is the Gamma white noise:
\[
\omega(x)={:}\,\omega\,{:}_{\mathrm
G}(x)+1=\langle\wom{1},\delta_x\rangle+1\in(S_{\mathrm G})_{-1,-k_0}^{-1},\qquad x\in\operatorname{supp}\sigma,
\]
the function $\omega(x)-1={:}\,\omega\,{:}_{\mathrm G}(x)$ can be thought of as the
compensated Gamma white noise.
Next, by taking the Wick product of ${:}\,\omega\,{:}_{\mathrm G}(x_i)$, $i=1,\dots,n$,
we obtain a Gamma white noise monomial as
\begin{gather*}
{:}\,\omega\,{:}_{\mathrm G}(x_1)\,\diamondsuit\,
{:}\,\omega\,{:}_{\mathrm G}(x_2)\,
\diamondsuit\cdots\diamondsuit\,
{:}\,\omega\,{:}_{\mathrm G}(x_n)=\langle\wom{n},\delta_{x_1}\hat \otimes\cdots\hat \otimes\delta_{x_n}\rangle\\
={:}\,\omega^{\otimes n}\,{:}_{\mathrm G}(x_1,\dots,x_n)\in(S_{\mathrm G})_{-1,-k_0}^{-1}.
\end{gather*}
The delta function of Gamma white noise, $\tilde\delta_{\upsilon}$, where
$\upsilon\in S'$, is defined by
\[
\ll\tilde\delta_\upsilon,\phi\ra\!\ra=\phi(\upsilon),\qquad
\phi\in(S_{\mathrm G})^{1}.\]
Evidently, $\tilde\delta_\upsilon$ belongs to $(S_{\mathrm G})^{-1}$ and has
the representation:
\[\tilde \delta_\upsilon=\tilde\delta_\upsilon(\omega)=\sum_{n=0}^\infty \langle\wom{n},\frac1{n!}\,\wup{n}
\rangle.\]
Now, we wish to introduce operators of coordinate multiplication,
$\omega(x){\cdot}$, acting from $(S_{\mathrm G})^1$ into $(S_{\mathrm G})^{-1}$.
To this end, we define first linear operators $\partial_x\colon(S_{\mathrm G})^1\to(S_{\mathrm G})^1$ and
$\partial^\dag_x\colon (S_{\mathrm G})^{-1}\to(S_{\mathrm G})^{-1}$ for each $x\in\operatorname{supp}\sigma$ by
\begin{align*}
\partial_x\langle\wom{n},f^{(n)}\rangle&=n\langle\wom{(n-1)},f^{(n)}(x,\cdot)\rangle,\\
\partial^\dag_x\langle\wom{n},F^{(n)}\rangle&=\langle\wom{(n+1)},\delta_x\hat \otimes F^{(n)}\rangle.
\end{align*}
It easy to show that these operators are continuous.
Let us preserve the same notations $a^+(\xi)$, $a^0(\xi)$, $a_1^-(\xi)$, and
$a_2^-(\xi)$ for the images of the corresponding operators under the unitary
$I$. Then, the operators $a^+(\xi)$ and $a_1^-(\xi)$ have the integral
representation (cf.\ \cite{Hidaetal}):
\begin{align}
a_1^-(\xi)&=\int_{\operatorname{supp}\sigma}d\sigma(x)\,\xi(x)\partial_x,\notag\\
a^+(\xi)&=\int_{\operatorname{supp}\sigma}\,d\sigma(x)\,\xi(x)\partial^\dag_x.\label{4.5}\end{align}
Integrals of such type are understood usually in the sense that one applies
pointwisely the integrand operator to a test function, then dualizes the result
with another test function, and finally integrates the obtained function of $x$
w.r.t.\ the measure $\sigma$.
As well known, the neutral operator $a^0(\xi)$ has the representation
\[a^0(\xi)=\int_{\operatorname{supp}\sigma}d\sigma(x)\, \xi(x)\,\partial^\dag_x\partial_x.\]
A new point appearing in Gamma analysis is, of course, the second annihilation
operator, $a_2^-(\xi)$, which has now the representation
\begin{equation}\label{4.7}
a_2^-(\xi)=\int_{\operatorname{supp}\sigma}d\sigma(x)\,\xi(x)\,\partial^\dag_x\partial_x\partial_x .\end{equation}
The formula \eqref{4.7} can be verified in a standard way.
Thus, we define the operator $\omega(x){\cdot}$ by
\begin{equation}
\omega(x){\cdot}=\partial^\dag_x+2\partial^\dag_x\partial_x+1+\partial_x+\partial^\dag_x\partial_x\partial_x,\label{multiplication}\end{equation}
which acts continuously from $(S_{\mathrm G})^1$ into $(S_{\mathrm G})^{-1}$.
Evidently,
\[
\langle\omega,\xi\rangle{\cdot}=
\int_{\operatorname{supp}\sigma}d\sigma(x)\,\xi(x)\omega(x){\cdot}
\, .\]
By analogy with \cite{Huang} and \cite{LyReShch}, the family of operators
$(\omega(x){\cdot})_{x\in\operatorname{supp}\sigma}$ can be also called a Gamma field, or a
quantum Gamma white noise process.
As will be shown now, the action of all the above operators can be easily
represented in terms of the $\S$-transform. Indeed,
let $\phi\in(S_{\mathrm G})^1$ and let $U(\theta)=\S[\phi](\theta)$.
The formula for $\partial^\dag_x$, resp.\ $a^+(\xi)$ is
well known (e.g., \cite{HKPS}):
\begin{alignat}{2}
\S[\partial^\dag_x\phi](\theta)&=\theta(x)U(\theta),&&\qquad \theta\in S_{\Bbb C},\notag\\
\S[a^+(\xi)\phi](\theta)&=\langle\xi,\theta\rangle U(\theta),&&\qquad \theta\in S_{\Bbb C}'
\notag\end{alignat}
(of course, the formula for $\partial^\dag_x$ holds for $\Phi\in (S_{\mathrm G})^{-1}$ with $\theta$
from a neiborhood of zero in $S_{\Bbb C}$).
Next, analogously to \cite{ItKu}, we have that
\begin{alignat}{2}
\S[(\partial_x+2\partial^\dag_x\partial_x)\phi](\theta)&=D_{\delta_x(1+2\theta)}U(\theta),&&\qquad
\theta\in S_{\Bbb C},\label{eins}\\
\S[(a_1^-(\xi)+2a^0(\xi))\phi](\theta)&=D_{\xi(1+2\theta)}U(\theta),&&\qquad
\theta\in S_{\Bbb C}',\label{zwei}\end{alignat}
where $D_\upsilon$, $\upsilon\in S_{\Bbb C}'$, denotes also the G\^ateaux derivatives
in direction $\upsilon$ of a function defined on $S_{\Bbb C}'$. Indeed, let $\phi$ be
of the form $\phi(\omega)=\langle\wom{n},\varphi^{\otimes n}\rangle$, then
$U[\phi](\theta)=\langle\varphi^{\otimes n},\theta^{\otimes n}\rangle$, and
\begin{gather*}
\frac d{dt}\Big|_{t=0}\langle\varphi^{\otimes n},(\theta+t\delta_x(1+2\theta))^{\otimes n}\rangle
=n\varphi(x)(1+2\theta(x))\langle \varphi^{\otimes(n-1)},\theta^{\otimes(n-1)}\rangle\\
=\S[n\varphi(x)\langle\wom{(n-1)},\varphi^{\otimes(n-1)}\rangle+2n\varphi(x)\langle\wom
n,\delta_x\hat \otimes\varphi^{\otimes(n-1)}\rangle](\theta)\\
=\S[(\partial_x+2\partial^\dag_x\partial_x)\langle\wom n,\varphi^{\otimes n}\rangle](\theta),\qquad \theta\in S_{\Bbb C}.
\end{gather*}
Hence, the formula \eqref{eins} will be proved if we show that, if
$\phi_m\to\phi$ in $(S_{\mathrm G})^1$ as $m\to\infty$ and $U_m(\theta)=\S[\phi_m](\theta)$,
then for any fixed $\theta\in S_{\Bbb C}$,
\begin{equation}\label{drei}
\lim_{m\to\infty}D_{\delta_x(1+2\theta)}U_m(\theta)=D_{\delta_x(1+2\theta)}U(\theta).\end{equation}
To this end, we will use the following lemma:
\begin{lem}
For each $\upsilon\in S_{\Bbb C}'$\rom, $D_\upsilon$ defines a linear continuous
operator on $\Emin{S_{\Bbb C}'}.$\end{lem}
\noindent {\it Proof}. Every function $u\in\Emin{S_{\Bbb C}'}$ can be
represented in the form (see e.g.\ \cite{KoLeS})
\[ u(\theta)=\sum_{n=0}^\infty \langle f^{(n)},\theta^{\otimes n}\rangle,\qquad f^{(n)}\in
S_{\Bbb C}^{\hat \otimes n}.\]
For each $p\ge1$ and $k\in{\Bbb N}_0$, define the norm
\begin{equation}\label{drym}
\,{|}\!{|}\!{|}\, u \,{|}\!{|}\!{|}\, _{1,p,k}^2=\sum_{n=0}^\infty |f^{(n)}|_p^2(n!)^2 2^{nk},\end{equation}
and let ${\cal E}^1_{p,k}(S'_{\Bbb C})$ be the closure of $\Emin{S_{\Bbb C}'}$ in this norm.
Theorem~3 (see also \cite{KoLeS}) implies that
\[\Emin{S_{\Bbb C}'}=\projlim_{p,k\to\infty}{\cal E}^1_{p,k}(S_{\Bbb C}'),\]
i.e., that the sequence of norms \eqref{drym} is equivalent to
\eqref{complexnorm}
Let $\upsilon\in S_{-p',{\Bbb C}}$ with $p'\ge1$. Since
\[ D_\upsilon \langle\varphi^{\otimes n},\theta^{\otimes n}
\rangle=n\langle\varphi,\upsilon\rangle\langle\varphi^{\otimes(n-1)},\theta^{\otimes(n-1)}\rangle,\]
the norm of $D_\upsilon$ on each $({\cal E})_{p,k}^1$, $p\ge p'$, $k\in{\Bbb N}_0$,
is not greater than $|\upsilon|_{-p}2^{-k/2}$, and therefore $D_\upsilon$ acts
continuously on $\Emin{S_{\Bbb C}'}$.\qquad $\blacksquare$\vspace{2mm}
Now, from Theorem~3 we conclude that the convergence $\phi_m\to\phi$ in $(S_{\mathrm G})^1$
implies the convergence $U_m\to U$ in $\Emin{S_{\Bbb C}'}$, and hence by Lemma~3
$D_\upsilon U_m\to D_\upsilon U$ in $\Emin{S_{\Bbb C}'}$ for any fixed $\upsilon\in
S_{\Bbb C}'$. Therefore, by \eqref{complexnorm} $D_\upsilon U_m(\theta)\to
D_\upsilon U(\theta)$ for any fixed $\theta\in S_{\Bbb C}'$, which evidently implies
\eqref{drei} because $\delta_x(1+2\theta)\in S_{\Bbb C}'$ for any $\theta \in S_{\Bbb C}$.
The formula \eqref{zwei} can be proved absolutely analogously to
\eqref{eins} if we take into account that $\xi(1+2\theta)\in S'_{\Bbb C}$ for an
arbitrary $\theta\in S_{\Bbb C}'$.
Again analogously to \eqref{eins}, we have
\[ S[\partial^\dag\partial_x\partial_x\phi](\theta)=\theta(x)\nabla_x^2 U(\theta),\qquad \theta\in
S_{\Bbb C},\]
where $\nabla_x$ is defined by \eqref{gradient}. Then
\begin{equation}\label{vier}
\S[a_2^-(\xi)\phi](\theta)=\langle\xi(x)\nabla_x^2 U(\theta),\theta(x)\rangle,\qquad \theta\in
S_{\Bbb C}',\end{equation}
where $x$ denotes the variable in which the dualization is carried out.
This formula is evidently true for $\phi(\omega)=\langle\wom n,\varphi^{\otimes n}\rangle$: then
$U(\theta)=\langle\varphi^{\otimes n},\theta^{\otimes n}\rangle$ and
\begin{align*}
\langle \xi(x)\nabla_x^2\langle\varphi^{\otimes n},\theta^{\otimes n}\rangle,\theta(x)\rangle
& =\langle
\xi(x)n(n-1)\varphi(x)^2\langle\varphi^{\otimes(n-2)},\theta^{\otimes(n-2)}\rangle,\theta(x)\rangle\\
& =n(n-1)\langle\varphi^{\otimes(n-2)},\theta^{\otimes(n-2)}\rangle\langle\xi\varphi^2,\theta\rangle\\
& =n(n-1)\langle \varphi^{\otimes(n-2)}\hat \otimes(\xi\varphi^2),\theta^{\otimes(n-1)}\rangle.
\end{align*}
Hence, it suffices to show that, if $U_m\to U$ in $\Emin{S_{\Bbb C}'}$, then for any
fixed $\theta\in S_{\Bbb C}'$ $\nabla_x^2U_m(\theta)\to\nabla_x^2U(\theta)$ in each
$S_{p,{\Bbb C}}$, $p\ge1$, as a function of $x$.
Representing
\begin{equation}\label{fuenf}
U_m(\theta)=\sum_{n=0}^\infty \langle f_m^{(n)},\theta^{\otimes n}\rangle\to
U(\theta)=\sum_{n=0}^\infty\langle f^{(n)},\theta^{\otimes n}\rangle\quad\text{in
}\Emin{S_{\Bbb C}'},\end{equation}
we get
\begin{align*}
\nabla_x^2 U_m(\theta)&=\sum_{n=0}^\infty \nabla_x^2\langle f_m^{(n)},\theta^{\otimes
n}\rangle\\
&=\sum_{n=2}^\infty n(n-1)\langle f^{(n)}_m(x,x,\cdot),\theta^{\otimes(n-2)}\rangle.
\end{align*}
\eqref{fuenf} yields that $f_m^{(n)}\to f^{(n)}$ in $S_{\Bbb C}^{\hat \otimes n}$, and
therefore
\[ \langle f_m^{(n)}(x,x,\cdot),\theta^{\otimes(n-2)}\rangle\to\langle
f^{(n)}(x,x,\cdot),\theta^{\otimes(n-2)}\rangle\quad\text{in }S_{\Bbb C}.\]
Moreover, upon \eqref{1.3} we have, for an arbitrary $\upsilon\in S_{-p,{\Bbb C}}$,
\begin{align*}
&\big| \langle \upsilon(x),\langle f_m^{(n)}(x,x,\cdot),\theta^{\otimes(n-2)}\rangle\ra
\big|\\
&\qquad =\big|\langle
f_m^{(n)}(x_1,x_1,x_2,\dots,x_{n-1}),\upsilon(x_1)\theta^{\otimes(n-2)}(x_2,\dots,x_{n-1})\rangle\big|\\
&\qquad\le |\upsilon|_{-p}|\theta|_{-p}^{n-2}C_p|f_m^{(n)}|_p,
\end{align*}
whence taking to notice that $S_{p,{\Bbb C}}$ can be thought of as the dual of
$S_{-p,{\Bbb C}}$, we get
\[
\big|\langle f_m^{(n)}(x,x,\cdot),\theta^{\otimes (n-2)}\rangle\big|_p\le
C_p|\theta|_{-p}^{n-2}|f_m^{(n)}|_p.\]
Estimating
\begin{align*}
&\sum_{n=2}^\infty n(n-1)\big|\langle
f_m^{(n)}(x,x,\cdot),\theta^{\otimes(n-2)}\rangle\big|_p\\
&\qquad \le C_p\sum_{n=2}^\infty n(n-1)|\theta|_{-p}^{n-2}|f_m^{(n)}|_p\\
&\qquad \le C_p\sum_{n=2}^\infty \big((n-2)!\big)^{-1}|\theta|_{-p}^{n-2}|f_m^{(n)}|_pn!\\
&\qquad \le C_p\Big(\sum_{n=0}^\infty (n!)^{-2}|\theta|_{-p}^{2n}\Big)^{1/2} \,{|}\!{|}\!{|}\,
U_m \,{|}\!{|}\!{|}\, _{1,p,0},
\end{align*}
we obtain the desired statement.
Thus, we have proved the following theorem.
\begin{th}
Let $\phi\in(S_{\mathrm G})^1$\rom, then the action of the operators
$\omega(x){\cdot}$\rom, $x\in\operatorname{supp}\sigma$\rom, and $\langle\omega,\xi\rangle\cdot$\rom, $\xi\in
S$\rom, can be represented in terms of the
$\S$-transform as follows\rom:
\begin{alignat*}{2}
\S[\omega(x){\cdot}\,\phi](\theta)&=(\theta(x)+1)U(\theta)+D_{\delta_x(1+2\theta)}U(\theta)
+\theta(x)\nabla_x^2 U(\theta),&\qquad \theta\in S_{\Bbb C},
\\
\S[\langle\omega,\xi\rangle\cdot\phi](\theta)&=\langle\xi,\theta+1\rangle
U(\theta)+D_{\xi(1+2\theta)}U(\theta)+
\langle\xi(x)\nabla_x^2U(\theta),\theta(x)\rangle,&\qquad
\theta\in S_{\Bbb C}',\end{alignat*}
where $U(\theta)=\S[\phi](\theta).$
\end{th}
\section{Standard annihilation operator\\ on Gamma space}
In this section, we will study the standard annihilation operators $\partial_x$ and
$a_1^-(\xi)$.
First, we note that
\[ \sup_{x\in\operatorname{supp}\sigma}|\delta_x|_{-p}=\|\delta\|_{p,\infty}<\infty.\]
Therefore, for each $x\in\operatorname{supp}\sigma$, $\partial_x$ can be extended to a continuous
operator on $(S_{\mathrm G})^1_{p,k}$, $p\ge1$, $k\ge k_0$, with norm $\le \|\delta\|_{p,\infty} 2^{-k/2}$.
It follows from the proof of Theorem~4 that $\partial_x$ is nothing but the $\alpha$-gradient
with $\alpha(\varphi)=\frac{\varphi}{\varphi+1}$, and on the total set in
$(S_{\mathrm G})_{p,k}^1$ consisting of the Wick exponentials $\wexp{\cdot}{\varphi}$
with $|\varphi|_p<2^{-k/2}$ we have
\[
\partial_x\wexp{\omega}{\varphi}=\nabla_x^\alpha\wexp{\omega}{\varphi}=\varphi(x)\wexp{\omega}{\varphi}.\]
Note that
\[ \alpha^{-1}(\varphi)=\frac{\varphi}{1-\varphi}=\sum_{n=1}^\infty \varphi^n,\]
and hence in virtue of \eqref{1.3}
\[ |\alpha^{-1}(\varphi)|_p\le\frac{|\varphi|_p}{1-C_p|\varphi|_p}\quad\text{if
}|\varphi|_p<C_p.\]
Choosing $\varphi\in S$ such that
\[ |\varphi|_p<\frac{2^{-k/2}}{1+C_p 2^{-k/2}},\]
we get that $|\alpha^{-1}(\varphi)|_p<2^{-k/2}$,
and then upon \eqref{sp}--\eqref{neznaju}
\begin{align*}
\partial_x e_{\mathrm G}(\varphi;\omega)&=\partial_x\wexp{\omega}{\alpha^{-1}(\varphi)}\\
&=\frac{\varphi(x)}{1-\varphi(x)}\, e_{\mathrm G}(\varphi;\omega)\\
&=\sum_{n=1}^\infty \varphi(x)^n \, e_{\mathrm G}(\varphi;\omega).\end{align*}
Since
\[ \alpha(\varphi)=\frac{\varphi}{\varphi+1}=\sum_{n=1}^\infty
(-1)^{n+1}\varphi^n,\]
the functions $e_{\mathrm G}(\varphi;\omega)$ also constitute a total set in
$(S_{\mathrm G})^1_{p,k}$, and therefore, at least formally, we can write down
\begin{equation}\label{5.1}
\partial_x =\alpha^{-1}(\nabla_x)
=\frac{\nabla_x}{1-\nabla_x}
=\sum_{n=1}^\infty \nabla_x^n.\end{equation}
Let ${\cal E}^{1}_{p,k}(S')$ denote the Hilbert space constructed in the same
way as ${\cal E}^1_{p,k}(S_{\Bbb C}')$
(see the proof of Lemma~3)
but only starting from $\Emin{S'}$. Choosing an arbitrary $\tilde p\ge1$ and
$\tilde k\in {\Bbb N}_0$ such that $2^{\tilde k/2}>\|\delta\|_{\tilde p,\infty}$, we
get that each $\nabla_x$ is a continuous operator on ${\cal E}^1_{\tilde
p,\tilde k}(S')$ with norm less than one. Hence, the series $\sum_{n=1}^\infty
\nabla_x^n$ converges in operator norm. Let now $p\ge1$ and $k\ge k_0$ be
chosen so that the space $(S_{\mathrm G})_{p,k}^1$ is topologically embedded into ${\cal
E}^1_{\tilde p, \tilde k}(S')$. Then, the above functions $e_{\mathrm
G}(\varphi;\omega)$ constitute also a total set in ${\cal E}^1_{\tilde p,\tilde
k}(S')$, which implies the equality \eqref{5.1} on each space ${\cal
E}^1_{\tilde p,\tilde k}(S')$, and therefore on $\Emin{S'}$.
In the same way, one can derive the inverse representation of \eqref{5.1}
\begin{equation}\label{5.2}
\nabla_x =\alpha(\partial_x)=\frac{\partial_x}{\partial_x+1}=\sum_{n=1}^\infty
(-1)^{n+1}\partial_x^n.\end{equation}
As a corollary of \eqref{5.1} or \eqref{5.2}, we have the commutation of
arbitrary $\nabla_{x_1}$ and $\partial_{x_2}$ on $(S_{\mathrm G})^1$.
Now, we will show that analogously to the one-dimensional case (see e.g.\
\cite{OGF}) the operators $\partial_x$ and $a_1^-(\xi)$ have a representation as an integral
w.r.t.\ a difference operator.
\begin{th}
For an arbitrary $\phi\in (S_{\mathrm G})^1$, we have
\begin{align}
(\partial_x\phi)(\omega)&=\int_0^\infty\big(\phi(\omega+s\delta_x)-\phi(\omega)\big)e^{-s}\,ds\notag\\
&=\int_0^\infty \phi(\omega+s\delta_x)e^{-s}\,ds-\phi(\omega),\qquad
x\in\operatorname{supp}\sigma,\label{5.1a}\\
(a_1^-(\xi)\phi)(\omega)&=\int_{\operatorname{supp}\sigma}\int_0^\infty\xi(x)\big(\phi(\omega+s\delta_x)-\phi(\omega)\big)
e^{-s}\,ds\,d\sigma(x),\qquad \xi\in S.\label{5.2a}
\end{align}
\end{th}
\noindent {\it Proof}.
Let $\theta\in S_{-p,{\Bbb C}}$, then for any $u\in\Emin{S_{\Bbb C}'}$,
\begin{align*}
\Big|\int_0^\infty
u(\theta+s\delta_x)e^{-s}\,ds\Big|&\le\pmb{\pmb |} u\pmb{\pmb |}_{1,p,k}\int_0^\infty\exp
\big[\tfrac1k|\theta+s\delta_x|_{-p} \big] e^{-s}\,ds
\\
&\le\pmb{\pmb |} u\pmb{\pmb |}_{1,p,k}\exp\big[\tfrac1k|\theta|_{-p}\big]\int_0^\infty
\exp\big[ -s\big(1-\tfrac1k
\|\delta|_{p,\infty}
\big)\big]\,ds\\
&=\pmb{\pmb |} u\pmb{\pmb |}_{1,p,k}
\exp\big[\tfrac1k|\theta|_{-p}\big]\,\frac1{1-\frac
1k\|\delta\|_{p,\infty}},\qquad k>\|\delta\|_{p,\infty},
\end{align*}
where the norm $\pmb{\pmb |}\cdot\pmb{\pmb |}_{1,p,k}$ is defined by \eqref{complexnorm}.
Hence
\[\pmb{\pmb{\Big|}}\int_0^\infty u(\cdot+s\delta_x)e^{-s}\,ds\,\pmb{\pmb{\Big|}}_{1,p,k}\le
\pmb{\pmb |} u\pmb{\pmb |}_{1,p,k}\,
\frac{1}{1-\frac 1k\|\delta\|_{p,\infty}}.\]
Therefore, by Theorem~4 the operator $A_x$ defined by the right hand side of
\eqref{5.1a} determines a linear continuous operator on $(S_{\mathrm G})^1$, i.e., for
arbitrary $p\ge 1$ and $k\ge k_0$, there are $p'\ge p$ and $k'\ge k$ such that
$A_x$ acts continuously from $(S_{\mathrm G})_{p',k'}^1$
into $(S_{\mathrm G})_{p,k}^1$. On the other hand, $\partial_x$ acts also continuously on
$(S_{\mathrm G})^1$, and in particular on each $(S_{\mathrm G})^1_{p',k'}$. Hence, it suffices to prove
the equality \eqref{5.1a} on a total set in $(S_{\mathrm G})_{p',k'}^1$. As such a set we
take the functions $\wexp{\cdot}{\varphi}$ with $|\varphi|_{p'}<2^{-k'/2}$. Then,
\[ \partial_x\wexp{\omega}{\varphi}=\varphi(x)\wexp{\omega}{\varphi},\]
and from \eqref{silva} we derive that
\begin{align}
\int_0^\infty\wexp{\omega+s\delta_x}{\varphi}\,e^{-s}\,ds&=
\int_0^\infty \exp\big[
\langle\omega+s\delta_x,\frac\varphi{\varphi+1}\rangle-\langle\log(1+\varphi)\rangle
\Big]
e^{-s}\,ds\notag\\
&=\exp\Big[
\langle\omega,\frac{\varphi}{\varphi+1}\rangle-\langle\log(1+\varphi)\rangle
\Big]
\int_0^\infty\exp\Big[
\frac{s\varphi(x)}{\varphi(x)+1}-s
\Big]\,ds\notag\\
&=\wexp{\omega}{\varphi}\int_0^\infty\exp\Big[
\frac{-s}{\varphi(x)+1}
\Big]\,ds\notag\\
&=\wexp\omega\varphi(\varphi(x)+1),
\label{syr}\end{align}
which proves \eqref{5.1a}. The formula \eqref{5.2a} can be proved absolutely
analogously.
\qquad$\blacksquare$
\vspace{2mm}
\noindent{\it Remark\/} 9. Let $\mu_{{\mathrm CP}}$ be a measure of compound Poisson (CP)
white noise on $S'$ with L\'evy measure $\sigma\nu$, i.e., the Laplace transform
of $\mu_{{\mathrm CP}}$ is given by
\[\int_{S'}\exp[\langle\omega,\varphi\rangle]\,d\mu_{{\mathrm CP}}(\omega)=\exp\Big[\int_{{\Bbb R}^{d+1}}(e^{s\varphi(x)}-1)\,d\sigma(x)\,d\nu(s)\Big].\]
Then, one can study CP analysis by using an isomorphism between the usual Fock
space over $L^2({\Bbb R}^{d+1},d\sigma\,d\nu)$ and the $L^2$-space of CP white noise
$(L^2_{\mathrm CP})=L^2(S',{\cal B}(S'),d\mu_{{\mathrm CP}})$
\cite{LyReShch,Silva,compound}.
The CP-field operators $a(\xi)$, $\xi\in S$, have in the Fock space the
following representation:
\[a(\xi)=a^+(\xi\otimes\operatorname{id})+a^0(\xi\otimes\operatorname{id})+\int_{{\Bbb R}^d}\xi(x)\,d\sigma(x)\int_{\Bbb R}
s\,d\nu(s)\operatorname{id}+a^-(\xi\otimes\operatorname{id}),\]
where $\xi\otimes\operatorname{id}=(\xi\otimes\operatorname{id})(x,s)=\xi(x)s$ and $a^+$, $a^0$, $a^-$ are the
standard creation, neutral, and annihilation operators, respectively.
(The image of) the annihilation operator $a^-(\xi\otimes\operatorname{id})$ acts in the following
way in the space $(L^2_{\mathrm CP})$:
\begin{equation}\label{an}
\big(
a^-(\xi\otimes \operatorname{id})\phi
\big)(\omega)=\int_{{\Bbb R}^{d+1}}\xi(x)s(\phi(\omega+s\delta_x)-\phi(\omega))\,d\sigma(x)\,d\nu(s).\end{equation}
In case of the Gamma measure, $d\nu(s)=\frac{e^{-s}}{s}\,ds$ (see \eqref{CP}),
and therefore the formula \eqref{an} takes the form
\[ \big(
a^-(\xi\otimes\operatorname{id})\phi
\big)(\omega)=\int_{{\Bbb R}^{d+1}}\xi(x)\big(\phi(\omega+s\delta_x)-\phi(\omega)\big)e^{-s}\,d\sigma(x)\,ds.\]
Hence, $a^-(\xi\otimes\operatorname{id})$ is just the operator $a_1^-(\xi)$ under consideration in
this paper.
As a consequence of this, we can write down the explicit action of the adjoint
operator of $a_1^-(\xi)$ in $(L^2_{\mathrm G})$, denoted by $a_1^+(\xi)$, which corresponds to the
operator $a^+(\xi\otimes\operatorname{id})$, see \cite{NuVi,Silva}.
Thus, for an arbitrary $\omega\in S'$ of the form \eqref{configuration}
\[
a_1^+(\xi)\phi(\omega)=\langle\omega(x),\xi(x)\phi(\omega-s_x\delta_x)\rangle-\langle\xi\rangle\phi(\omega),
\qquad \phi\in\operatorname{Dom} a^+(\xi).\]
\section{Creation, neutral, and Gamma annihilation \\ operators on the Gamma space}
In this section, we will obtain the explicit formulas for the operators
$a^+(\xi)$, $a^0(\xi)$, and $a^-_2(\xi)$ acting on $(L^2_{\mathrm G})$ by using the formula
\eqref{multiplication}, which expresses the operator of coordinate multiplication
via $\partial_x$ and $\partial^\dag_x$ and Theorem~6.
Let us fix arbitrary $\tilde p\ge1$ and $\tilde k\ge k_0$, then each $\partial_x$ can
be extended to a continuous operator on $(S_{\mathrm G})^1_{\tilde p,\tilde k}$ and $\partial^\dag_x$
and $\omega(x)\cdot$ to continuous operators from $(S_{\mathrm G})_{\tilde p,\tilde k}^1$ into
$(S_{\mathrm G})_{-\tilde p,-\tilde k}^{-1}$. Choose now $p\ge \tilde p$ and $k\in{\Bbb N}_0$ so
that the space ${\cal E}_{p,k}^1(S')$ is topologically embedded into
$(S_{\mathrm G})^1_{\tilde p,\tilde k}$.
The restrictions of $\partial_x$, $\partial^\dag_x$, and
$\omega(x)\cdot$ to ${\cal E}_{p,k}^1(S')$ are continuous operators from this space
into $(S_{\mathrm G})_{\tilde p,\tilde k}^1$ and $(S_{\mathrm G})_{-\tilde p,-\tilde k}^{-1}$,
respectively. We rewrite the formula \eqref{multiplication} as follows:
\begin{equation}\label{nov}\omega(x)\cdot=\partial^\dag_x(\partial_x+1)^2+(\partial_x+1).\end{equation}
Due to the (proof of) Theorem~6, we have for an arbitrary $\phi\in{\cal
E}_{p,k}^1(S')$:
\[(\partial_x+1)\phi(\omega)=\int_0^\infty\phi(\omega+s\delta_x)e^{-s}\,ds,\]
and hence on the total set in ${\cal E}_{p,k}^1(S')$ consisting of the functions
$e^{\langle\omega,\varphi\rangle}$ with $|\varphi|_p< 2^{-k/2}$ we have analogously to
\eqref{syr}:
\[(\partial_x+1)e^{\langle \omega,\varphi\rangle}=(1-\varphi(x))^{-1}e^{\langle\omega,\varphi\rangle}.\]
Therefore, by \eqref{nov},
\[ \omega(x)\cdot e^{\langle\omega,\varphi\rangle}=(1-\varphi(x))^{-2}\partial^\dag_x
e^{\langle\omega,\varphi\rangle}+(1-\varphi(x))^{-1}e^{\langle\omega,\varphi\rangle},\]
which yields
\begin{align*}
\partial_x^+ e^{\langle\omega,\varphi\rangle}&=\omega(x)\cdot (\varphi(x)-1)^2
e^{\langle\omega,\varphi\rangle}+(\varphi(x)-1)e^{\langle\omega,\varphi\rangle}\\
&=\omega(x)\cdot(\nabla_x-1)^2 e^{\langle\omega,\varphi\rangle}+(\nabla_x-1)e^{\langle\omega,\varphi\rangle}.
\end{align*}
Since $\nabla_x$ is a continuous operator on ${\cal E}_{p,k}^1(S')$, this
immediately implies the central lemma of this section:
\begin{lem} We have on $(S_{\mathrm G})^1$\rom:
\[\partial^\dag_x=\omega(x)\cdot(\nabla_x-1)^2+(\nabla_x-1),\]
\end{lem}
Now, by using \eqref{4.5} and Lemma 4, we can calculate, at least formally, the
action of $a^+(\xi)$:
\begin{align*}
a^+(\xi)&=\int_{\operatorname{supp}\sigma}d\sigma(x)\,\xi(x)\big(\omega(x)\cdot(\nabla_x-1)^2+(\nabla_x-1)\big)\\
&=\langle\omega(x),\xi(x)(\nabla_x-1)^2\rangle+\langle\xi(x),\nabla_x-1\rangle.\end{align*}
\begin{th} For any $\phi\in(S_{\mathrm G})^1$\rom,
\[
a^+(\xi)\phi(\omega)=\langle\omega(x),\xi(x)(\nabla_x-1)^2\phi(\omega)\rangle+(D_\xi-\langle\xi\rangle)\phi(\omega),\qquad
\omega\in S',\]
where $x$ denotes the variable in which the dualization is carried out\rom.
\end{th}
\noindent{\it Proof}. Analogously to the above, we will consider $a^+(\xi)$ as a
continuous operator from ${\cal E}_{p,k}^1(S')$ into $(S_{\mathrm G})_{\tilde p,\tilde
k}^1$. Take again $\phi(\omega)=e^{\la\om,\ttt\ra}$ with $|\varphi|_p<2^{-k/2}$. Then, using
\eqref{4.5} and Lemma~4, we have for an arbitrary $\psi\in(S_{\mathrm G})^1$:
\begin{align*}
&\ll a^+(\xi)e^{\la\om,\ttt\ra} ,\psi(\omega)\ra\!\ra\\
&\qquad
=\int_{\operatorname{supp}\sigma}d\sigma(x)\,\xi(x)\ll\big(\omega(x)\cdot(\nabla_x-1)^2+(\nabla_x-1)\big)e^{\la\om,\ttt\ra},\psi(\omega)\ra\!\ra\\
&\qquad= \int_{\operatorname{supp}\sigma}d\sigma(x)\,\xi(x)(\varphi(x)-1)^2\ll\omega(x)\cdot
e^{\la\om,\ttt\ra},\psi(\omega)\ra\!\ra\\
&\qquad\quad+\int_{\operatorname{supp}\sigma}d\sigma(x)\,\xi(x)(\varphi(x)-1)\ll
e^{\la\om,\ttt\ra},\psi(\omega)\ra\!\ra\\
&\qquad=\ll \langle\omega,\xi(\varphi-1)^2\ranglee^{\la\om,\ttt\ra},\psi(\omega)\ra\!\ra+\langle\xi,\varphi-1\rangle\ll
e^{\la\om,\ttt\ra},\psi(\omega)\ra\!\ra\\
&\qquad=\ll
\langle\omega(x),\xi(x)(\varphi(x)-1)^2e^{\la\om,\ttt\ra}\rangle+\langle\xi,\varphi-1\ranglee^{\la\om,\ttt\ra},\psi(\omega)\ra\!\ra.
\end{align*}
Therefore,
\begin{equation}\label{6.1}
a^+(\xi)e^{\la\om,\ttt\ra}=\langle\omega(x),\xi(x)(\varphi(x)-1)^2e^{\la\om,\ttt\ra}\rangle+\langle\xi,\varphi-1\ranglee^{\la\om,\ttt\ra},\end{equation}
where the equality in understood as that in $(S_{\mathrm G})^{-1}$. But the right hand side
of \eqref{6.1} considered as a function of $\omega$ belongs to ${\cal
E}_{p,k}^1(S')$, which follows from the representation
\[\langle\omega(x),\xi(x)(\varphi(x)-1)^2e^{\la\om,\ttt\ra}\rangle=\sum_{n=1}^\infty\langle\omega^{\otimes n},\frac
1{(n-1)!}\,\varphi^{\otimes (n-1)}\hat \otimes(\xi(\varphi-1)^2)\rangle.\]
Therefore, \eqref{6.1} holds for each $\omega\in S_{-p}$.
It remains only to note that $D_\xi$ is a continuous operator on ${\cal
E}_{p,k}^1(S')$ and for any fixed $\omega\in S_{-p}$
$(\nabla_x-1)^2\phi_m(\omega)\to(\nabla_x-1)^2\phi(\omega)$ in $S_p$ as a function
of $x$ if $\phi_m\to\phi$ in ${\cal E}_{p,k}^1(S')$, the latter being proved
in the same way as the formula \eqref{vier}.\qquad$\blacksquare$
We proceed to consider the neutral operator $a^0(\xi)$ on the Gamma space.
\begin{lem}We have on $(S_{\mathrm G})^1$\rom:
\[\partial^\dag_x\partial_x=\omega(x)\cdot\nabla_x(1-\nabla_x)-\nabla_x,\qquad x\in\operatorname{supp}\sigma.\]
\end{lem}
\noindent {\it Proof}. Using Lemma 4 and its proof, we get
\begin{gather*}
\partial^\dag_x\partial_xe^{\la\om,\ttt\ra}=\big(\omega(x)\cdot(\nabla_x-1)^2+(\nabla_x-1)\big)\partial_xe^{\la\om,\ttt\ra}\\
=\big(\omega(x)\cdot(\nabla_x-1)^2+(\nabla_x-1)\big)\frac{\varphi(x)}{1-\varphi(x)}\,e^{\la\om,\ttt\ra}\\
=\bigg(\omega(x)\cdot\frac{(\varphi(x)-1)^2\varphi(x)}{1-\varphi(x)}+\frac{(\varphi(x)-1)\varphi(x)}{1-\varphi(x)}\bigg)e^{\la\om,\ttt\ra}\\
=\big(\omega(x)\cdot(1-\varphi(x))\varphi(x)-\varphi(x)\big)e^{\la\om,\ttt\ra}\\
=\big(\omega(x)\cdot(1-\nabla_x)\nabla_x-\nabla_x\big)e^{\la\om,\ttt\ra}.
\end{gather*}
Again, due to the continuity of $\nabla_x$ on ${\cal E}^1_{p,k}(S')$, we obtain
the lemma.\qquad $\blacksquare$
Since formally
\begin{align*}
a^0(\xi)&=\int_{\operatorname{supp}\sigma}d\sigma(x)\,\partial^\dag_x\partial_x\\
&=\int_{\operatorname{supp}\sigma}d\sigma(x)\,\xi(x)(\omega(x)\cdot\nabla_x(1-\nabla_x)-\nabla_x)\\
&=\langle\omega(x),\xi(x)\nabla_x(1-\nabla_x)\rangle+\langle\xi(x),\nabla_x\rangle,\end{align*}
we come to the following theorem, whose proof is analogous
to that of Theorem~7.
\begin{th}For any $\phi\in(S_{\mathrm G})^1$\rom,
\[
a^0(\xi)\phi(\omega)=\langle\omega(x),\xi(x)\nabla_x(1-\nabla_x)\phi(\omega)\rangle-D_\xi\phi(\omega),\qquad
\omega\in S'.\]\end{th}
Finally, we will shortly consider the Gamma annihilation operator. Analogously to
Lemma~5, we get
\begin{lem}We have on $(S_{\mathrm G})^1$
\begin{align*}
\partial^\dag_x\partial_x^2&=\omega(x)\cdot \nabla_x^2-\partial_x\nabla_x\\
&=\omega(x)\cdot\nabla_x^2-\nabla_x\partial_x.\end{align*}
\end{lem}
\begin{th} For any $\phi\in(S_{\mathrm G})^1$
\begin{align}
a_2^-(\xi)\phi(\omega)&=\langle\omega(x),\xi(x)\nabla_x^2\phi(\omega)\rangle+D_\xi\phi(\omega)\notag\\
&\quad-\int_{\operatorname{supp}\sigma}\int_0^\infty \xi(x)\phi(\omega+s\delta_x)e^{-s}\,ds\,d\sigma(x)-\langle\xi\rangle\phi(\omega)
\label{6.2}\\
&=\langle\omega(x),\xi(x)\nabla_x^2\phi(\omega)\rangle+D_\xi\phi(\omega)\notag\\
&\quad-\int_{\operatorname{supp}\sigma}\int_0^\infty\xi(x)\nabla_x\phi(\omega+s\delta_x)e^{-s}\,ds\,d\sigma(x).\label{6.3}
\end{align}\end{th}
\noindent{\it Proof}. The formula \eqref{6.2} follows directly from the equality
\[ a_2^-(\xi)=\langle\omega,\xi\rangle\cdot-a^+(\xi)-2a^0(\xi)-\langle\xi\rangle\operatorname{id}-a_1^-(\xi)\]
and from Theorems~6--8. The equivalent formula \eqref{6.3} is obtained from
Lemma~6 and Theorem~6 just as above. The only new point here is to prove that
\begin{equation}
\int_{\operatorname{supp}\sigma}d\sigma(x)\,\xi(x)\partial_x\nabla_x\phi_m(\omega)\to\int_{\operatorname{supp}\sigma}d\sigma(x)\,\xi(x)
\partial_x\nabla_x\phi(\omega),\qquad \omega\in S',\label{6.4}\end{equation}
if $\phi_m\to\phi$ in $(S_{\mathrm G})^1$ as $m\to\infty$. Bur for each $x\in\operatorname{supp}\sigma$
the norm of $\nabla_x$ on the space ${\cal E}^1_{p,k}(S')$
does not exceed $\|\delta\|_{p,\infty}2^{-k/2}<1$ under an appropriate choice of
$k$, so that each $\nabla_x\partial_x=\sum_{n=2}^\infty\nabla_x^n$ (see \eqref{5.1})
is a continuous operator on ${\cal E}_{p,k}^1(S')$ whose norm is bounded by a
constant uniformly in $x$. Therefore, since $\xi\in S\subset
L^1(\operatorname{supp}\sigma,d\sigma)$, $\int_{\operatorname{supp}\sigma}d\sigma(x)\,\xi(x)\nabla_x\partial_x$
is a continuous operator on ${\cal E}_{p,k}^1$, which implies \eqref{6.4}.\qquad
$\blacksquare$
\begin{center}\bf ACKNOWLEDGMENTS\end{center}
We would like to thank J. L. Silva for useful
discussions.
The authors were partially supported by the SFB 256, Bonn University.
Yu.K. acknowledges partial financial support through the INTAS-Project
Nr.~97-0378.
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\section{Introduction}
Understanding and classifying topological states of matter are central issues in today's condensed matter physics~\cite{RevModPhys.82.3045,RevModPhys.83.1057,RevModPhys.89.041004}.
Among a number of such systems, topological states in gapped free-fermion systems are the most well-understood example.
There, the keen interplay between symmetry and topology gives rise to various non trivial states,
and the elegant mathematical tools such as group theory, Clifford algebra, and K-theory allow us to classify topological insulators (TIs) and superconductors (TSCs)
under the internal~\cite{
PhysRevB.78.195125,doi:10.1063/1.3149495,1367-2630-12-6-065010}, space-group~\cite{Slager2012,PhysRevB.88.125129,Po2017}
and magnetic space-group~\cite{Watanabeeaat8685} symmetries.
However, this is not the end of the story.
Very recently, a new class of topological insulators has been found, which is called
a higher-order topological insulator (HOTI)~\cite{Hayashi2018, PhysRevB.95.165443, PhysRevB.96.245115, Benalcazar61, PhysRevLett.119.246402, Schindlereaat0346,PhysRevLett.120.026801,PhysRevB.97.241405, Xu2017}.
In this newly-introduced concept, the ``order'' means the dimensionality of the boundary state.
Namely, in conventional TIs in $d$-dimensions, the gapless boundary states emerge at the $(d-1)$-dimensional boundary
~\cite{PhysRevLett.71.3697,PhysRevB.48.11851}, and in that sense, conventional TIs are the first-order TIs.
Similarly, in $n$th order TIs ($2 \leq n \leq d $),
gapless boundary states emerge at the $(d-n)$-dimensional boundary.
In many cases, the emergence of such gapless states is protected by the spacial symmetries
such as mirror symmetries and rotational symmetries.
\begin{figure}[b]
\begin{center}
\includegraphics[width=\linewidth]{./Fig1.pdf}
\caption{
A kagome flake with (a) a rhombus geometry and (d) a triangle geometry.
In each geometry, corners are denoted by green circles.
The energy spectrum of the rhombus as a function of $t_1/t_2$
for (b) a rhombus geometry and (e) a triangle geometry.
The energy spectrum with the disorder $W=1.1$ for (c) a rhombus geometry and (f) a triangle geometry.
}
\label{fig:1}
\end{center}
\end{figure}
To reveal the physical properties of HOTIs, studying the effects of disorders is important.
Indeed, for the first-order TIs protected by the time-reversal symmetry,
the most essential property is the prohibition of back-scatterings of the helical edge/surface states.
As a results, the phase diagrams of the disordered TIs can be obtained by investigating the conductance~\cite{PhysRevB.79.045321,1367-2630-12-6-065008,PhysRevB.85.155138,PhysRevLett.110.236803},
and their validity is evidenced by the direct calculation of topological indices~\cite{PhysRevB.31.3372,PhysRevB.83.195119,doi:10.1063/1.4942494,doi:10.1063/1.5026964,doi:10.7566/JPSJ.86.123710}.
For HOTIs, on the other hand,
the definition of the disordered HOTIs,
relying on either physical quantities or topological indices,
is still unclear.
Nevertheless, we expect that
the boundary states of the HOTI are localized even in the presence of weak disorders,
and that the existence of the boundary states may serve as an indicator of the HOTI phase in the disordered systems.
To examine this scenario, it is desirable to employ a method that can distinguish the disordered HOTI from other phases
in a systematic manner.
To this end, in this paper, we investigate the robustness of the HOTI against disorders by using a machine learning method.
The machine learning method, especially the neural network,
is widely used in the various fields in physics,
including quantum many-body systems~\cite{Carleo602,PhysRevB.96.205152,PhysRevB.97.075114, PhysRevB.97.134109,
Huembeli2018},
Monte Carlo simulations~\cite{PhysRevB.97.205140},
high-energy physics~\cite{WHITESON20091203,Baldi2014} and astrophysics~\cite{doi:10.1093/pasj/psw096}.
As for the topological states, the first order TIs and TSCs
have been successfully classified in the presence of disorders,
and the resulting phase diagrams reproduce those obtained by the other methods.
~\cite{doi:10.7566/JPSJ.85.123706, doi:10.7566/JPSJ.86.044708,PhysRevB.97.205110}.
This motivates us to apply this method to disordered HOTIs.
In the present work, we employ a tight-binding Hamiltonian on a breathing kagome lattice~\cite{0295-5075-95-2-20003,PhysRevB.97.241405,PhysRevLett.120.026801, Xu2017}.
The Hamiltonian, as we will explain later, is known to possess the second-order TI phases
with zero-dimensional corner state
protected by symmetries~\cite{PhysRevB.97.241405,PhysRevLett.120.026801, Xu2017}.
For this system, we first perform supervised learning in the clean limit, where
the phase diagram has already been known~\cite{PhysRevB.97.241405,PhysRevLett.120.026801, Xu2017}.
Once the supervised ``model'' for classification is obtained,
we then introduce on-site disorders and classify the phases by using that model.
We obtain phase diagrams for the disordered breathing kagome model
in both rhombus and triangle geometries.
For both of them, we find that the HOTI phase survives as far as the energy gap does not collapse,
due to the robustness of the corner states against the disorders.
The rest of this paper is organized as follows.
In Sec. \ref{sec:model}, we introduce our model and review the phase diagrams in the clean limit.
The symmetries of Hamiltonian and the topological invariants proposed in the previous works are also summarized.
In Sec. \ref{sec:method}, we explain our method to identify the phases by using the supervised learning.
In Sec. \ref{sec:result}, we present our main results, namely, the phase diagrams in the presence of disorders.
In Sec. \ref{sec:summary}, we present a summary and discussions.
The comparison with the obtained results of the inverse participation ratio is discussed in Appendix.
\section{Model \label{sec:model}}
We consider a tight binding Hamiltonian for spinless fermions on a breathing kagome lattice:
\begin{equation}
\label{eq:H_0}
\mathcal{H}_0 = -\sum_{i,j} t_{ij} c_i^\dagger c_j
\end{equation}
where $c_i^\dagger$ and $c_i$ are, respectively, the creation and annihilation operators of an electron on
a site $i$, and
$t_{ij} = t_1 (t_2)$ if the bond between $i$ and $j$
belongs to the nearest-neighbor bond and
it lives on the upward
(downward) triangles [Fig. \ref{fig:1}(a) and \ref{fig:1}(d)].
In the following, we take $t_2$ to be positive.
The band structure of the bulk is obtained by diagonalizing $ \mathcal{H}_0$
in the Fourier space.
There exists a flat band with energy $t_1 + t_2$.
The other two bands are gapped when $|t_1| \neq |t_2|$;
the gapless linear dispersion appears at $K$ and $K'$ for $t_2 =t_1$,
and at $\Gamma$ point for $t_2 = -t_1 $
\subsection{Phase diagrams in the clean limit for rhombus and triangular geometries}
To study HOTI phases, we have to consider finite system with an open boundary condition.
So far, it is known that there are two choices of global geometries, namely, rhombus and triangular geometries
[see Fig. \ref{fig:1}(a) and \ref{fig:1}(d), respectively].
In both of these geometries, the model has HOTI phases with zero-dimensional
localized states, i.e. corner states~\cite{PhysRevLett.120.026801, Xu2017,PhysRevB.97.241405}.
This can be understood by considering the limit of $|t_1| = 0$ or $|t_2| = 0 $.
For instance, if $|t_1| = 0 $, the \lq \lq trimers" are formed on all downward triangles,
and the \lq \lq dimers" at all edges which do not belong to the downward triangles.
For a rhombus geometry, only the site at $\Gamma_1$ corner is isolated and serves as a zero-energy mode,
and it survives even for finite $t_1$, as long as $|t_1| < |t_2|$ is satisfied.
Similarly, for $|t_1| > |t_2|$, the corner state appears at $\Gamma_2$ corner.
Indeed, in Ref. \onlinecite{PhysRevB.97.241405},
it was shown that the corner zero modes can be explicitly constructed when $|t_1| \neq |t_2|$.
Note that the other two corners, $\Gamma_3$ and $\Gamma_4$, do not possess corner states at any parameters.
For a triangular geometry, on the other hand, all three corners belong to upward triangles, so the corner states exist
only when $ -1 \leq t_1 /t_2 \leq 1/2 $,
and those corner states have three-fold degeneracy~\cite{PhysRevLett.120.026801}.
The difference of corner states with different geometry is also crucial when defining the phases.
In a rhombus geometry, the corner state exists for every parameter, but it matters whether the corner state are in-gap state or not.
So, we classify the phases in such a way that we fix the electron number as $N = \nu_{\rm R} = M^2$, where we consider the system contains
$M \times M$ rhombuses ($3M^2 - 2M$ sites), and see what the highest occupied state is.
With $t_1/t_2$ as a tuning parameter, there are four phases~\cite{Xu2017} [Fig. \ref{fig:1}(b)].
For $t_1/t_2 > 2$, the system is in a HOTI phase, in which a zero-energy state localized at
the $\Gamma_2$ corner exists in a band gap.
We referred to this state as ``HOTI 2" named after the position of the corner state.
For $1/2 < t_1/t_2 < 2$, the system is in a metallic phase, which has zero-energy bulk or edge states,
and the corner state is buried into those states.
For $-1 < t_1/t_2 < 1/2$, there is another HOTI phase, in which a zero-energy state localized at
the $\Gamma_1$ corner and it is in the band gap.
We label this state ``HOTI 1".
For $t_1/t_2 < -1$, the system is in a ``trivial'' phase.
In this phase, there is
a zero-energy state localized at the $\Gamma_2$ corner in the gap, but
it is not the highest occupied state.
Rather, one of the degenerated flat modes is the highest occupied state and it masks the corner state.
In contrast to a rhombus geometry, there are only three phases for a triangle geometry
since there is only one HOTI phase for $ -1 \leq t_1 /t_2 \leq 1/2 $~\cite{PhysRevLett.120.026801}.
In this phase,
the corner states appear at $(\nu_{\rm T}-1)$-th, $\nu_{\rm T}$-th, and $(\nu_{\rm T}+1)$-th states, with $\nu_{\rm T} = \frac{M(M+3)}{2}$,
for the system consists of $\frac{M(M-1)}{2}$ upward triangles [i.e. $\frac{3 M(M-1)}{2}$ sites].
We therefore fix the electron number as $N =\nu_{\rm T}$ so that the highest occupied state is the corner state in this region.
The other two phases are trivial ($ t_1 /t_2 \leq-1$) and metallic ($ t_1 /t_2 \geq 1/2 $) phases,
which are essentially the same as those for the rhombus geometry.
\subsection{Symmetries and topological invariants}
So far, it was revealed that zero-energy corner state in the HOTI phase is protected by the symmetries of the Hamiltonian.
However, the protecting symmetries and corresponding topological invariants vary according to the geometries of the system.
For the triangle geometry, the gapless corner state is associated with
the distance of the ``Wannier center'' from the origin, $P_3$, which
is quantized as $1/2$ for the HOTI phase and $0$ for the non-topological phases
\cite{PhysRevLett.120.026801}.
The quantization of $P_3$ originates from the combination of the mirror and three-fold rotational ($C_3$) symmetries.
For the rhombus geometry, the topological invariant can be obtained from the trajectory of the eigenvalues of the ``Wannier Hamiltonian''
during the adiabatic deformation of the Hamiltonian~\cite{Xu2017}.
In this case, the emergence of the gapless corner state is associated with the composite symmetry operation of the three-fold rotation and the complex conjugation.
Importantly, for both of these two,
the spacial symmetries of the Hamiltonian plays a crucial role, while the random impurity potential breaks these symmetries.
Hence, the topological invariants discussed above are not quantized in the disordered systems.
Also, the corner state, if exists, is no longer pinned at the zero-energy [see Figs.~\ref{fig:1}(c) and \ref{fig:1}(f)].
In such a situation, the definition of the HOTI phase itself is more or less subtle, but
it may be reasonable to adopt the existence of the corner states as a working definition of the (disordered) HOTI phase.
This is the reason why we employ a machine learning method, by which we can systematically judge whether the corner states exist as an in-gap state.
The alternative approach to define the HOTI phase is to use the topological invariant which is well-defined even in the absence of the translational symmetry.
In this respect, the $\mathbb{Z}_3$ Berry phase~\cite{0295-5075-95-2-20003}, which is defined for the local gauge twist of the Hamiltonian, will be a candidate.
The relation between the $\mathbb{Z}_3$ Berry phase and the HOTI phases will be discussed elsewhere~\footnote{H. Araki, T. Mizoguchi, and Y. Hatsugai, in preparation.}.
\subsection{Effect of disorders}
To study the effects of disorders, we introduce a on-site random potential, $\mathcal{H}_R$,
where
\begin{eqnarray}
\label{eq:H_R}
\mathcal{H}_R = \sum_{i} w_i c_i^\dagger c_i.
\end{eqnarray}
Here $w_i$ are randomness with a uniform distribution in $[-W/2, W/2)$ with fixed $W$.
The total Hamiltonian we consider is $\mathcal{H} = \mathcal{H}_0 + \mathcal{H}_R$.
\section{Identification of phases by machine learning \label{sec:method}}
We classify the phases of the disordered system
by using a machine learning technique.
Here a neural-network-based algorithm is implemented by
using the open-source library PyTorch~\cite{pytorch}.
\begin{figure*}[htb]
\begin{minipage}[t]{0.9\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{arc.pdf}
\caption{The architecture of the neural network. The first two layers are two-dimensional
convolutional layers and the rests are linear layers.
We use some max pooling layers and
batch normalization. The size of the layers are shown in the figure.}
\label{fig:arc}
\end{minipage}
\end{figure*}
\subsection{Architecture of the neural network}
A schematic picture of the architecture of the neural network used in this study is shown in Fig. \ref{fig:arc}.
We use the convolutional neural network with five layers, in which the first two layers are
two dimensional convolutional layers and the rests are linear layers.
All the activation functions are the ReLU functions $f(x)=\max(0, x)$.
After the first convolutional layer, we use the max pooling layer.
After the second convolutional layer,
we use the max pooling layer and the batch normalization.
The max pooling layer takes the max value in $2\times 2$ sectors.
Next layer is a linear layer. We use the batch normalization for it.
Then the neural network outputs the probabilities of phases.
The number of output is four for rhombus geometry and three for triangle geometry,
corresponding the number of phases.
\subsection{Input and output data}
Our input data is the electron density of the highest occupied single-particle state.
This choice is similar to the previous works on the first order TIs~\cite{doi:10.7566/JPSJ.85.123706,PhysRevB.97.205110}.
For a rhombus geometry, the input data is
$|\phi_{i}^{\nu_{\rm R} }|^2$ defined as $c_{\nu_{\rm R}} = \sum_{i} \phi_{i}^{\nu_{\rm R}}c_{i}$,
with $c_{\nu_{\rm R}}$ being the annihilation operator of the state $\nu_{\rm R}$.
For a triangle geometry, the input data is $( |\phi_{i}^{\nu_{\rm T}-1 }|^2 +|\phi_{i}^{\nu_{\rm T} }|^2 + |\phi_{i}^{\nu_{\rm T}+1 }|^2) /3$,
which is the average electron density among three corner states.
This particular choice is convenient since it preserves the three-fold rotational symmetry inherent in this geometry.
We remark that we employ the present choice of input data because it is suitable for detecting the corner state in the HOTI phase.
There are other possible choices of the input data, such as the all of the occupied states~\cite{PhysRevB.97.134109}
and topological numbers~\cite{PhysRevB.98.085402}.
\begin{figure}[htb]
\begin{minipage}[t]{0.5\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{rhombus_alignment.pdf}
\caption{The modification of the kagome lattice to the square lattice
for a rhombus geometry (upper) and a triangle geometry (lower).
We add empty sites denoted by gray circles. }
\label{fig:alignment}
\end{minipage}
\end{figure}
To make the input data as two-dimensional square-shaped images, we transform the geometry from a kagome lattice to a square lattice with vacancy sites
(Fig. \ref{fig:alignment}).
For a rhombus geometry, this is achieved by adding redundant sites (denoted by gray circles)
at the center of hexagonal plaquettes, and deform
equilateral triangles into right triangles.
Then we make input data from a single-particle wave function, in which probabilities on the added sites
are always zero.
Through this procedure, the size of image data is increased from about $3N$ to $4N$, where $N$ is the
number of unit cells.
Although the computational cost is increased, we do not lose any information during this process.
Similarly, for a triangle geometry, we add the redundant sites at the center of the plaquettes.
In addition, we also need to add a large downward triangle composed of the redundant sites.
Then, in the resulting image, more than half of the sites are always empty.
The output data are in the form of four- (three-) dimensional vectors for a rhombus (triangular) geometry whose
$n$th component is interpreted as a probability that the state belongs to $n$th phase.
\subsection{Supervised learning}
To classify the phases, we first perform a supervised learning to create a model.
To do this, we first prepare $5000$ images with randomly-chosen values of $t_1/t_2$ in the clean limit.
Each image is tied with the label of the phases which is represented by
a four- (three-) dimensional vector for a rhombus (triangular) geometry.
For a vector corresponding to the $n$-th phase,
only $n$-th component is unity and all the others are zero,
namely, we employ a one-hot representation of the phases.
Then the model is trained such that the deviation from the answer and the output (or a cost function) is minimized.
This can be achieved by updating the parameters in the model on the basis of a gradient method.
Note that
In the present system, the obtained model can reproduce the data over 99\% of accuracy.
Then, using the trained model, we identify the phases of the disordered system.
To do this, we input the wave functions of the disordered system to the model, and see the output vectors which predict the probability for each phase.
For a given parameters, $t_1/t_2$ and $W$,
we prepare $30$ samples and investigate the phase diagram in two different methods of averaging these samples.
The first method is to take an average for the input data.
Namely, the input is chosen as the averaged wave functions.
As is pointed out in Ref. \onlinecite{PhysRevB.97.205110}, this procedure is necessary to restore the translational symmetry that is broken by disorders.
The second method is to take an average for the output data.
Namely, we input the single-shot wave functions for $30$ samples and take an average over the obtained output vectors.
These two methods complement each other.
To be more specific, the former method is advantageous to distinguish the corner states
from the randomly-localized states for large $W$, because the randomly-localized states have uniform distributions after taking the average.
However, this method can not distinguish the trivial, metallic and randomly-localized states,
because all of these states have uniform distributions.
On the other hand, the latter method can distinguish them, because the difference between them is clear in the single-shot wave functions.
To clarify the difference between the averaged and the single-shot wave functions, we show
the averaged wave functions at $W=1$ for several parameters in Fig.~\ref{fig:states}(a) (for a rhombus) and Fig.~\ref{fig:states_triangle} (a) (for a triangle),
and the single-shot wave functions at $W=1$ for the same parameters in Fig.~\ref{fig:states}(b) (for a rhombus) and Fig.~\ref{fig:states_triangle}(b) (for a triangle).
Clearly, the restoration of the translational symmetry due to the averaging occurs at, for example, $t_1/t_2 = -3.0$, where the flat-band states are dominant.
For other phases, the corner states exist in both the averaged and the single-shot wave functions, which serve as an indicator of the HOTI phase for the neural networks, as we will see later.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.98\linewidth]{wavefunctions_rhombus.pdf}
\caption{
Wave functions on a rhombus geometry.
(a) The averaged wave functions of the highest occupied state
with $W=1$.
The average is taken over 30 samples.
(b)
The single-shot wave functions of the highest occupied state
with $W=1$.}
\label{fig:states}
\end{center}
\end{figure*}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.98\linewidth]{wavefunctions_triangle.pdf}
\caption{
Wave functions on a triangle geometry.
(a) The averaged wave functions of the highest occupied state
with $W=1$.
The average is taken over 30 samples.
(b)
The single-shot wave functions of the highest occupied state
with $W=1$.
The average over three corner states is also taken (see the main text).
}
\label{fig:states_triangle}
\end{center}
\end{figure*}
\begin{figure*}[t]
\begin{center}
\begin{tabular}{c}
\begin{minipage}{0.5\hsize}
\begin{flushleft}
{\Large (a) }
\end{flushleft}
\begin{center}
\includegraphics[width=0.9\linewidth]{phase_diagram1.pdf}
\end{center}
\end{minipage}
\begin{minipage}{0.5\hsize}
\begin{flushleft}
{\Large (b) }
\end{flushleft}
\begin{center}
\includegraphics[width=0.9\linewidth]{phasediagram_singleshot.pdf}
\end{center}
\end{minipage}
\end{tabular}
\caption{
The phase diagram of the breathing kagome lattice with disorders for the rhombus geometry.
(a) The phase diagrams is plotted for the averaged wave functions.
The blue, right-blue, yellow and orange areas
denote trivial, HOTI 1, HOTI 2 and metallic phases, respectively.
Green dashed lines denote the energy gap between the corner states and the bulk/edge states.
(b) The phase diagram is plotted by the averaged probabilities for 30 single-shot wave functions.
}
\label{fig:phase_diagram}
\end{center}
\end{figure*}
\begin{figure*}[t]
\begin{center}
\begin{tabular}{c}
\begin{minipage}{0.5\hsize}
\begin{flushleft}
{\Large (a) }
\end{flushleft}
\begin{center}
\includegraphics[clip, width=0.98\linewidth]{phasediagram_triangle_1.pdf}
\end{center}
\end{minipage}
\begin{minipage}{0.5\hsize}
\begin{flushleft}
{\Large (b) }
\end{flushleft}
\begin{center}
\includegraphics[clip, width=0.98\linewidth]{phasediagram_triangle_singleshot_1.pdf}
\end{center}
\end{minipage}
\end{tabular}
\caption{
The phase diagram of the breathing kagome lattice with disorders for the triangle geometry.
(a) The phase diagrams is plotted for the averaged wave functions.
The red, blue and green areas
denote trivial, HOTI, and metallic phases, respectively.
(b) The phase diagram is plotted by the averaged probabilities for 30 single-shot wave functions.
}
\label{fig:phase_diagram_triangle}
\end{center}
\end{figure*}
\section{Results \label{sec:result} }
\subsection{Rhombus geometry}
We first consider the rhombus geometry with the system size $M=10$.
The phase diagram for the averaged wave functions is shown in Fig. \ref{fig:phase_diagram}(a), and
that for the averaged wave functions is shown in Fig. \ref{fig:phase_diagram}(b).
The colors represent the phases: Blue, right-blue, yellow, and orange regions, respectively, correspond to
trivial, HOTI 1, metallic, and HOTI 2 phases.
First, let us focus on the HOTI phases.
We see in both of two phase diagrams that the HOTI phases survive up to the critical strength of the disorders.
The HOTI 1 phase turns into the trivial phase for $W > W^{(1)}_c$,
while HOTI 2 phase into the gapless phase for $W > W^{(2)}_c$,
indicating that the level crossing between corner states and the states nearby zero-energy
occurs in these regimes.
If it is the case, one can estimate $W^{(1,2)}_c$ with respect to the gap
between corner states and nearby bulk/edge states.
To be specific, the formulas for $W_c^{(1)}$ are $2 t_1/t_2 + 2 $ for $-1 \leq t_1/t_2 \leq 0 $,
and $ -2 t_1/t_2 + 2 $ for $ 0 \leq t_1/t_2 \leq 1/2 $;
similarly, those for $W_c^{(2)}$ are $ 2 t_1/t_2 -2$ for $t_1/t_2 > 2 $.
These estimated values are denoted by green lines, which turn out to be good approximations.
Notice that this behavior is reminiscent of the Su-Schrieffer-Heeger model in the presence of on-site disorders,
where the zero-dimensional edge states become unstable when the disorder strength is comparable with bulk band gap~\cite{Perez-Gonzalez2018,Munoz2018}.
Looking at the other phases, the trivial phase is the most fragile against disorders.
Indeed, we see in Fig. \ref{fig:phase_diagram}(a) that there is a mosaic-like region,
which means that the phase is ``unknown'' for the trained neural network.
To be more specific, in this region, the confidence is less than 98\% for all four phases.
We expect that this phase is likely to be an Anderson-localized (AL) phase, because of the following reason:
in this regime, the highest occupied state belongs to a flat band, which has a massive degeneracy.
Then, the disorders lift the degeneracy and pick one of the localized state.
Qualitative argument on this localization transition by using the inverse participation ratio (IPR) is presented in Appendix.
We remark that the previous study predict the existence of the critical phase between the trivial and the Anderson-localized phase
~\cite{PhysRevB.82.104209},
but the transition occurs at very small disorders ($W < 10^{-2}$),
which is much smaller than our minimum strength of the disorder ($W = 0.2$).
Also, the re-entrance to the metallic phase predicted in Refs. \onlinecite{PhysRevLett.96.126401,doi:10.1143/JPSJ.76.024709} occurs at much larger $W$ than the maximum value we study.
\subsection{Triangle geometry}
Next, we show the results for the triangle geometry.
The phase diagrams for $M=16$ are shown in Fig. \ref{fig:phase_diagram_triangle}.
Here, the output in the form of three-dimensional vectors is mapped onto the RGB component of the color map;
red, blue, and green regions respectively, correspond to the trivial, HOTI, and metallic phases.
We see that the HOTI phase
vanishes above a critical strength of the disorder potential, which is again roughly estimated by the energy gap.
This behavior is similar to the rhombus geometry, so
one may speculate that the HOTI phase is stable against the disorders as far as the corner states survive.
We also see that there is a metallic phase above the trivial phase for the result of the averaged wave function [Fig. \ref{fig:phase_diagram_triangle}(a)].
This is the artifact of the averaging of the wave functions, as we described in the previous section. Indeed,
we clearly see the mosaic-like pattern in the result for the single-shot wave function [Fig. \ref{fig:phase_diagram_triangle}(b)],
clearly indicating the existence of the AL phase: this is consistent with the observation from the wave function (Fig. \ref{fig:states_triangle}).
Further, we also observe that the Anderson localization occurs above the metallic phase,
and that its critical value is smaller compared with the case of the rhombus geometry.
\section{Summary and discussions \label{sec:summary}}
To summarize, we have investigated the phase diagram of the disordered breathing kagome model by using a machine learning method.
We use the wave function of the highest occupied state as input data.
By doing so, the neural network can distinguish the HOTI phase from other phases by the existence of the corner states.
Our results reveal that the HOTI phase is robust against the disorders as far as the disorder strength does not exceed the energy gap.
Besides the HOTI phase, the AL state and the trivial/metallic states can be successfully distinguished by combining
the results of averaged wave functions and of the single-shot wave functions.
It is important to justify our results of the machine learning method by using conventional approaches to study the disordered models.
As an example of such conventional approaches, we investigate the IPR for the present model with the rhombus geometry, which provides a consistent result
with that of the machine learning (see. Appendix).
Testing various other approaches, such as the transfer matrix method~\cite{PhysRevLett.110.236803,doi:10.7566/JPSJ.85.123706, doi:10.7566/JPSJ.86.044708,PhysRevB.97.205110,MacKinnon1983},
will be an interesting future problem.
Finally, let us remark on the related works.
Recently, the robustness of the corner states against disorders was investigated
in the electric circuits~\cite{Imhof2018, PhysRevB.98.201402}.
There, it was found that the corner-mode-induced resonance peak of the impedance survives even though the capacitances and inductances are disordered slightly.
Although disorders of this kind in the electric circuits induce both on-site and bond disorders in the language of the tight-binding model~\cite{Lee2018},
these results on the robustness of the corner states are consistent with our present results.
\acknowledgments
The authors would like to thank Nobuyuki Yoshioka for drawing our attention to
application of machine learning techniques to topological phases.
They also thank Yutaka Akagi and Hosho Katsura for the useful comments.
This work is partly supported by JSPS KAKENHI Grants No. JP17H06138 and JP16K13845.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,794
|
// Copyright 2016 Proyectos y Sistemas de Mantenimiento SL (eProsima).
//
// 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.
/**
* @file LatencyTestTypes.h
*
*/
#include <stdint.h>
#include <vector>
#ifndef LATENCYTESTTYPES_H_
#define LATENCYTESTTYPES_H_
#include "fastrtps/fastrtps_all.h"
#define MAX_TYPE_SIZE 10 * 1024 * 1024
class LatencyDataSizes
{
public:
LatencyDataSizes()
{
sample_sizes_ = {16, 1024, 64512, 1048576};
}
inline std::vector<uint32_t>& sample_sizes()
{
return sample_sizes_;
}
private:
std::vector<uint32_t> sample_sizes_;
};
/*
* Loanable DataTypes should be flat (only used basic types and arrays).
* Consequently the compiler cannot generate this type allocation code because the array member size is unknown at build
* time. The LatencyDataType must allocate a suitable buffer for these objects based on the array member size given at
* runtime. A beforehand knowledge of this type alignment in needed to calculate the right buffer size. The default
* alignment turn out to be 4 in both msvc and gcc for x86 and x64 architectures. The alignas specifier is used to match
* this default behaviour in other platforms.
* This type does not define a comparison operator because the actual data size referenced is unknown. Use the
* comparison method provided in LatencyDataType.
* */
struct alignas (4) LatencyType
{
// identifies the sample sent
uint32_t seqnum = 0;
// extra time devoted on bouncing in nanoseconds
uint32_t bounce = 0;
// actual payload
uint8_t data[1];
// this struct overhead
static const size_t overhead;
};
class LatencyDataType : public eprosima::fastdds::dds::TopicDataType
{
// Buffer size for size management
size_t buffer_size_;
public:
LatencyDataType()
: buffer_size_(MAX_TYPE_SIZE - LatencyType::overhead)
{
setName("LatencyType");
m_typeSize = MAX_TYPE_SIZE;
m_isGetKeyDefined = false;
}
LatencyDataType(
const size_t& size)
: buffer_size_(size)
{
setName("LatencyType");
m_typeSize = sizeof(decltype(LatencyType::seqnum)) +
sizeof(decltype(LatencyType::bounce)) +
((size + 3) & ~3) +
eprosima::fastrtps::rtps::SerializedPayload_t::representation_header_size;
m_isGetKeyDefined = false;
}
~LatencyDataType()
{
}
bool serialize(
void* data,
eprosima::fastrtps::rtps::SerializedPayload_t* payload) override;
bool deserialize(
eprosima::fastrtps::rtps::SerializedPayload_t* payload,
void* data) override;
std::function<uint32_t()> getSerializedSizeProvider(
void* data) override;
void* createData() override;
void deleteData(
void* data) override;
bool getKey(
void* /*data*/,
eprosima::fastrtps::rtps::InstanceHandle_t* /*ihandle*/,
bool force_md5 = false) override
{
(void)force_md5;
return false;
}
bool compare_data(
const LatencyType& lt1,
const LatencyType& lt2) const;
void copy_data(
const LatencyType& src,
LatencyType& dst) const;
bool is_bounded() const override
{
// All plain types are bounded
return is_plain();
}
bool is_plain() const override
{
// It is plain because the type has a fixed size
return true;
}
// Name
static const std::string type_name_;
};
enum TESTCOMMAND : uint32_t
{
DEFAULT,
READY,
BEGIN,
STOP,
END,
STOP_ERROR
};
typedef struct TestCommandType
{
TESTCOMMAND m_command;
TestCommandType()
{
m_command = DEFAULT;
}
TestCommandType(
TESTCOMMAND com)
: m_command(com)
{
}
}TestCommandType;
class TestCommandDataType : public eprosima::fastdds::dds::TopicDataType
{
public:
TestCommandDataType()
{
setName("TestCommandType");
m_typeSize = 4;
m_isGetKeyDefined = false;
}
~TestCommandDataType()
{
}
bool serialize(
void* data,
eprosima::fastrtps::rtps::SerializedPayload_t* payload) override;
bool deserialize(
eprosima::fastrtps::rtps::SerializedPayload_t* payload,
void* data) override;
std::function<uint32_t()> getSerializedSizeProvider(
void* data) override;
void* createData() override;
void deleteData(
void* data) override;
bool getKey(
void* /*data*/,
eprosima::fastrtps::rtps::InstanceHandle_t* /*ihandle*/,
bool force_md5 = false) override
{
(void)force_md5;
return false;
}
};
#endif /* LATENCYTESTTYPES_H_ */
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,043
|
\section{Introduction}
\label{Intro}
The central goal in stochastic multi-echelon inventory optimization (SMEIO) is maintaining inventory levels by regulating the order quantities to optimize a cost function. The cost function usually consists of a shortage (penalty) cost plus holding costs, so that there is a tradeoff between ordering too much vs.~too little compared to the stochastic demand, each of which incurs its own cost. In this study, we focus on identifying order-up-to levels (OULs) for multiple pairs of nodes in a complex supply chain network (SCN) under a finite decision horizon. Customer demands are stochastic with a known distribution; shipment lead-times are known constants; and decision makers have global information about the system states.
To this end, we use the deep neural network (DNN) paradigm \cite{lecun2015deep} as a means for decision-making. We model general SCNs (we use the terms "mixed" and "general" SCN interchangeably) in which several DNNs can regulate separate inventory levels for different parts of the network jointly, and by interacting with other parts of the SCN environment, yet our model only suggests a order-up-to level for each decision.
Early approaches using machine learning (ML) to optimize inventory in SCNs primarily used finite Markov decision processes (MDP), dating to about two decades ago. \citet{giannoccaro2002inventory} introduce a simulation-based RL to optimize inventory decisions in a three-node serial SCN. Several modified Q-learning \cite{watkins1992q} algorithms have been presented, with Q-tables containing possible joint state--action spaces for SMEIO \cite{chaharsooghi2008reinforcement}.
Recently, \citet{oroojlooyjadid2017deep} explore the well-known beer game inventory problem (a four-node serial SCN) via the deep Q-learning framework \cite{mnih2013playing}, which integrates the DNN concept into a Q-learning algorithm. Previous ML studies are mainly focused on cost comparisons between their proposed approach and some known optimal or heuristic policies, and no further inventory policy behaviors have been reported \cite{chaharsooghi2008reinforcement,oroojlooyjadid2017deep}. Moreover, previous studies restrict the SMEIO settings to fit their assumptions. For instance, the action spaces (decisions made by ML approaches) are discretized in advance to make the problem tractable.
In this study, not only do we consider general complex SCNs with multiple decisions to be made at the same time by DNN agents at several echelons of the SCN, but we directly suggest OULs as the DNN's output. In other words, the proposed framework is capable of suggesting interpretable inventory actions. Moreover, there is no restriction on the order quantities (action space) and/or inventory levels (state space) of the proposed method. (For example, they are not discretized.)
As our first contribution, we model the SCNs considering pair-wise (edge) decision makers. Then, we propose a method that finds OULs for complex MEIO systems under a finite horizon. In order to demonstrate that our method is effective, we compare it against classical inventory optimization (IO) models and algorithms. However, these algorithms assume an infinite horizon. Therefore, to make the best comparison possible, we train our model in a finite-horizon setting that approaches steady-state sufficiently well that it is a reasonable approximation for the infinite-horizon setting. We do this by (a) setting the initial conditions carefully, (b) setting the decision horizon, and possibly the warm-up interval, carefully. For instance, we initialize the inventory levels to be lead-time demands. If our method is reasonably close to optimal for these classical IO settings, then we have confidence that it will also be close to optimal for settings that cannot currently be solved by classical IO models. The true value of our method is its ability to solve complex MEIO systems settings.
The rest of the paper is as follows: We briefly review the literature in Section \ref{Literature}, divided into two separate parts, one on SMEIO background and one on ML in the SMEIO framework. Then we explain our model in Section \ref{Model}. We describe our numerical experiments in Section \ref{Experiments} and we conclude the paper in Section \ref{Conclusion}. A python package containing an implementation of the framework is available at the paper's \href{https://github.com/mamadpierre/DNN-SMEIO}{repository}.
\section{ML Background in SMEIO}
\label{Literature}
ML approaches for stochastic inventory optimization have been studied for a long time. Global supply chain management has been studied in \cite{pontrandolfo2002global} via the semi-Markov average reward technique (SMART). \citet{stockheim2003reinforcement} train RL agents to explore an optimal job acceptance strategy in a decentralized SCN. Many early works modify different versions of the Q-learning algorithm. Competitive supply chains are studied by \cite{van2007q}, where SCN nodes make their decisions separately and independently in an interconnected system.
\citet{oroojlooyjadid2017deep} study a four-node serial system via deep Q-learning \cite{mnih2013playing}. They study cases in which (non-ML) nodes follow a base-stock policy, as well as cases in which those nodes display irrational behavior and diverge from their expected inventory policy. \citet{zhao2010application} investigate a multi-agent RL framework model to solve SMEIO considering multiple echelons and multiple commodities. \citet{chaharsooghi2008reinforcement} analyze supply chain ordering management with a focus on the beer game and suggest a reinforcement learning ordering mechanism. Recently, \citet{gijsbrechts2019can} extensively investigate the use of deep RL for three otherwise intractable inventory problems---dual sourcing, lost sales, and multi-echelon optimization problems.
Most previous studies consider discrete state and/or action spaces. A countable action set is a necessity for Q-learning convergence \cite{watkins1992q}; therefore, papers using Q-learning discretize the agent's possible action values. For instance, \citet{giannoccaro2002inventory} explain that in SMEIO, inventory position (IP) has no bound, which implies an infinite-size MDP. Then, they discretize the IP values and associate an integer number to an actual IP interval. New advances that integrate DNN into the RL framework open further opportunities to explore complex SCNs. There exist very few studies in which the policies are approximated by deep neural networks \cite{oroojlooyjadid2017deep}, but even these studies do not report the optimal base-stock levels or other inventory policy parameters. In other words, the ML framework aims to minimize the SCN cost function, but the appropriate interpretation of the solution into a near-optimal policy remains unexplored. We aim to provide a framework to present clear base-stock levels for general SCNs. An overview of SMEIO approaches is provided in Section \textcolor{blue}{1} of the supplementary.
\section {SMEIO Model and Environment}
\label{Model}
\subsection{SMEIO Model}
We consider a multi-echelon supply chain network with a general topology. The network must be connected and may not contain directed cycles; otherwise, any topology is allowed, including assembly nodes (nodes with more than one predecessor) and/or distribution nodes (nodes with more than one successor). We use $\mathcal{G}=(\mathcal{N}, \mathcal{E})$ to denote the SCN graph, in which $\mathcal{N}$ is the set of all nodes, and $\mathcal{E} \subseteq \{ (i,j): i,j \in \mathcal{N} \}$ is the set of all edges. We consider periodic review, with a finite horizon consisting of $T$ periods.
Demand at each customer-facing (leaf) node is stochastic and may have any probability distribution, discrete or continuous, so long as the demand is drawn iid from that distribution. Different leaf nodes may have different demand distributions. (Our example networks use normally distributed demands, except when stated otherwise.) Figure \ref{mixedSCN} illustrates an example of an SCN, which we will return to later in the paper.
\begin{figure}
\centering
\includestandalone[width = 8cm]{mixedSCN}
\caption{Mixed SCN.}
\label{mixedSCN}
\end{figure}
Each node in the network accepts raw materials from its supply node(s), processes them into finished goods, and ships the finished goods to its customer node(s). A given node's supply and customer nodes may be other nodes in the network (these are called {\em internal} supplier or customer nodes) or {\em external} suppliers or customers. We assume that a node cannot have {\em both} an internal {\em and} an external supplier, nor can it have both an internal and an external customer, but both of these assumptions can be relaxed through suitable use of dummy nodes. Let $\mathcal{U}_j$ be the set of immediate upstream nodes (i.e., predecessor nodes) from node $j$, and $\mathcal{D}_j$ be the set of immediate downstream nodes (successor nodes) from node $j$, for $j\in \mathcal{N}$. For modeling convenience, if a node has an external supplier or customer node, these are treated as dummy nodes in the network and are assumed to be contained in $\mathcal{U}_j$ and $\mathcal{D}_j$, respectively. Dummy supplier nodes are assumed to have infinite capacity. (The dummy supplier node for node 1 and the dummy customer nodes for nodes 4 and 5 are indicated in Figure~\ref{mixedSCN} with dashed lines.)
Our framework allows for two different kinds of assembly nodes. An ``assembly-{\em and}'' node requires one unit of raw material from {\em each} of its predecessors---it is an ``and'' relationship. (It is straightforward to modify our approach to handle cases in which different numbers of units are required from different predecessors.) An ``assembly-{\em or}'' node requires one unit of raw material from {\em any} of its predecessors---it is an ``or'' relationship.
Each node has a finished-goods inventory that contains items that have been processed but not yet shipped to a customer. It also has one or more raw-material inventories, one for each supplier (including the external supplier, if any). When a node receives raw material items from its supplier(s), those items are placed into the raw material inventories. The node immediately processes as much raw material inventory as possible to produce finished goods. If node $j$ is an assembly-{\em and} node, then it processes
\begin{equation} R_{jt} = \min_{i\in \mathcal{U}_j} \{IL^r_{jit}\} \label{eq:Rjt_and} \end{equation}
items into finished goods in period $t$, and if it is an assembly-{\em or} node, then it processes
\begin{equation} R_{jt} = \sum_{i\in \mathcal{U}_j} \{IL^r_{jit}\} \label{eq:Rjt_or} \end{equation}
items, where $IL^r_{jit}$ is the number of units of the raw material from node $i$ that are in inventory at node $j$ in period $t$. $R_{jt}$ units are immediately added to node $j$'s finished goods inventory. For an assembly-{\em and} node, $R_{jt}$ units are subtracted from each of node $j$'s raw material inventories. For an assembly-{\em or} node or a non-assembly node (i.e., a node with at most one predecessor), all units are removed from the raw material inventory.
The shipment lead time for orders placed by node $j$ from its predecessor $i\in \mathcal{U}_j$ is denoted $L_{ij}$.
Shipment lead-times are deterministic, but due to possible upstream shortages, downstream nodes might experience stochastic lead-time. The processing time to convert raw materials to finished goods at a given node is assumed to be zero, though this assumption can be relaxed by adding dummy nodes whose shipment lead times equal the processing times.
If a node has insufficient inventory to meet its demands in a given period, the available inventory is allocated to customer nodes in proportion to the size of their orders in that period, and the remaining demands are backordered. Backorders are modeled as negative finished-goods inventory, as is common in the literature. If the inventory level is $IL$, then the number of items on hand is $IL^+$ and the number of backorders is $IL^-$, where $x^+ = \max\{x,0\}$; and $x^- = \max\{-x,0\}$. Backorders may occur at any node, whether it has internal or external customers. However, since a node never processes more items than it has raw material inventory for, raw material inventories are always non-negative.
Holding and shortage costs may be arbitrary linear or nonlinear functions of the on-hand inventory (including in-transit inventory) and backorders, respectively, at the end of a period.
In particular, the {\em state variables} at a given node $j\in \mathcal{N}$, evaluated at the end of period $t$, are as follows:
\begin{itemize}
\item $IL_{jt}$ = the inventory level of finished goods
\item $BO_{jkt}$ = the backorders at node $j$ that are owed to customer node $k$ ($k\in \mathcal{D}_j$); note that $\sum_{k\in \mathcal{D}_k} BO_{jkt} = IL^-_{jt}$
\item $IL^r_{jit}$ = the inventory level of raw material $i$ at node $j$ ($i\in \mathcal{U}_j$)
\item $IT_{jkt}$ = the inventory in transit (being shipped) from node $j$ to node $k\in \mathcal{D}_j$
\end{itemize}
The {\em cost functions} at node $j$ are as follows:
\begin{itemize}
\item $h_{ij}(\cdot)$ = the holding cost function for items from node $i\in \mathcal{U}_j$ that are held in raw material inventory at $j$ or as a component of node $j$'s finished goods inventory or of in-transit inventory from node $j$ to node $k\in\mathcal{D}_j$
\item $p_{jk}(\cdot)$ = the stockout cost function for backorders at node $j$ that are owed to node $k\in \mathcal{D}_j$
\end{itemize}
Then the total cost incurred in period $t$ is given by
\begin{equation} \label{eq:ct}
c_t = \sum_{j\in \mathcal{N}} \sum_{i\in \mathcal{U}_j} h_{ij}\left(IL^r_{jit} + IL^+_{jt} + \sum_{k\in \mathcal{D}_j} IT_{jkt}\right)
+ \sum_{j\in \mathcal{N}} \sum_{k\in \mathcal{D}_j} p_{jk}\left(BO_{jkt}\right)
\end{equation}
where $h_{ij}(\cdot)$ and $p_{jk}(\cdot)$ are general (possibly non-linear or non-convex) functions. In this calculation, an item from node $i$ that is shipped to $j$ is counted in the holding cost function $h_{ij}(\cdot)$ when it is in raw-material inventory at node $j$, in finished-product inventory at node $j$, and in transit from node $j$ to its customer(s). This approach for calculating holding costs is somewhat non-standard, but is meant to provide more flexibility, and many common settings are special cases. For example, in a distribution system (each node has at most one predecessor) in which node $j$ has a holding cost charged on finished-goods inventory and inventory in transit to its successors, we can simply set $h_{ij}(x) = h_jx$ for all $(i,j)\in \mathcal{E}$. Moreover, we note that it is straightforward to modify our approach for cost functions with other functional forms. It is also worth mentioning that if one wants to consider an added holding cost value for finished items, one could model this by adding a dummy raw material.
At the end of the planning horizon (after period $T$ ends), any remaining inventory or backorders at node $j$ are reimbursed or charged according to a {\em salvage function} $v_j(x)$. That is, at the end of the horizon, the system incurs a cost of
\begin{equation} \sum_{j\in \mathcal{N}} v_j(IL_{jT}), \label{eq:salvage} \end{equation}
where $v_j(x)$ may be positive (indicating a cost), negative (indicating a revenue), or zero, for either positive or negative values of $x$. Salvage values and costs are a common mechanism in inventory models to avoid end-of-horizon effects such as excess inventory buildup or selloffs near the end of the horizon.
To facilitate traversing through the SCN graph, we number the nodes with integer values. We assume that the nodes are numbered $1,\ldots,N$ in ascending order by their total shipment lead-times from the infinite source. That is, if node $i$ has a longer total lead-time than $j$ as calculated from the infinite source, then $i > j$. Ties are broken arbitrarily, and duplicate node labels are not allowed.
\subsection{State Variables and Sequence of Events}
\label{sec:soe}
Each node $j$ follows a base-stock policy to place orders from its predecessors, and it may use a different order-up-to level (or base-stock level) for each predecessor. In particular, we use $OUL_{ji}$ to represent the order-up-to level used by node $j\in \mathcal{N}$ when it places orders from predecessor $i\in\mathcal{U}_j$. The $OUL_{ji}$ values may be chosen by our DNN agent or by some other mechanism.
The sequence of events at each node $j$ in each time period $t$ is as follows:
\begin{enumerate}
\item The demand $D_{jkt}$ is observed from each $k\in \mathcal{D}_j$. If $k$ is an internal customer, then $D_{jkt}$ is the order quantity placed by node $k$, and if $k$ is an external customer, then $D_{jkt}$ is an exogenous random variable.
\item For each predecessor $i\in \mathcal{U}_j$, node $j$ orders $OUL_{ji} - IP_{jit}$ units from predecessor node $i$, where
\begin{equation} IP_{jit} = IL_{j,i,t-1} - \sum_{k\in \mathcal{D}_j} D_{jkt} + IT_{i,j,t-1} + BO_{i,j,t-1} \label{eq:IP_def} \end{equation}
is the inventory position of item-$i$ materials at node $j$ immediately before the order is placed. Note that it includes only the raw-material inventory of item $i$ at node $j$, and not the finished goods inventory at node $j$.
\item For each predecessor $i\in \mathcal{U}_j$, node $j$ receives all items that were shipped from node $i$ $L_{ij}$ time periods ago. (There are $S_{i,j,t-L_{ij}}$ such units.) Raw material and in-transit inventories are updated as
\begin{align*}
IL^r_{jit} & = IL^r_{j,i,t-1} + S_{i,j,t-L_{ij}} \\
IT_{ijt} & = IT_{i,j,t-1} - S_{i,j,t-L_{ij}}.
\end{align*}
\item Node $j$ processes $R_{jt}$ units, where $R_{jt}$ is given by \eqref{eq:Rjt_and} or \eqref{eq:Rjt_or} depending on whether node $j$ is an assembly-{\em and} node or an assembly-{\em or} node. (If $j$ is not an assembly node, then the two equations are equivalent.) The raw-material inventory levels are further updated as
\begin{equation} IL^r_{jit} = IL^r_{jit} - R_{jt} \label{eq:ending_ILr_and} \end{equation}
if node $j$ is an assembly-{\em and} node, and as
\begin{equation} IL^r_{jit} = 0 \label{eq:ending_ILr_or} \end{equation}
otherwise. The finished goods inventory is updated as:
\[ IL_{jt} = IL_{j,t-1} + R_{jt}. \]
\item For each successor $k\in \mathcal{D}_j$, node $j$ ships $S_{jkt}$ units to node $k$. If $IL_{jt} \ge \sum_{k\in \mathcal{D}_j} (D_{jkt} + BO_{j,k,t-1})$, then node $j$ has sufficient inventory to meet all of its backorders and new demands; the shipment quantity and new backorder level are given by
\begin{align}
S_{jkt} & = D_{jkt} + BO_{j,k,t-1} \label{eq:S_enough} \\
BO_{jkt} & = 0 \label{eq:ending_BO_enough}
\end{align}
If, instead, $IL_{jt} < \sum_{k\in \mathcal{D}_j} (D_{jkt} + BO_{j,k,t-1})$, then available inventory is allocated proportionally according to the current demands:
\begin{align}
S_{jkt} & = (D_{jkt} + BO_{j,k,t-1}) \frac{D_{jkt}}{\sum_{l\in \mathcal{D}_j} D_{jlt}} \label{eq:S_not_enough} \\
BO_{jkt} & = BO_{j,k,t-1} - S_{jkt}. \label{eq:ending_BO_not_enough}
\end{align}
In either case, the finished-goods and in-transit inventory levels are updated as
\begin{align}
IL_{jt} & = IL_{jt} - \sum_{k\in \mathcal{D}_j} D_{jkt} \label{eq:ending_IL} \\
IT_{jkt} & = IT_{jkt} + S_{jkt}. \label{eq:ending_IT}
\end{align}
(Note that only new demands, not old backorders, are subtracted from $IL_{jt}$ since old backorders are already counted as negative inventory in $IL_{jt}$. Note also that demands are subtracted whether or not they are actually shipped out, since the inventory level decreases in either case, either by a reduction in on-hand inventory or an increase in backorders.)
\item Holding and stockout costs are assessed according to \eqref{eq:ct}.
The ending raw-material inventory levels are given by \eqref{eq:ending_ILr_and} or \eqref{eq:ending_ILr_or}; the ending finished-goods inventory level is given by \eqref{eq:ending_IL}; the ending backorders are given by \eqref{eq:ending_BO_enough} or \eqref{eq:ending_BO_not_enough}; and the ending in-transit inventories are given by \eqref{eq:ending_IT}.
\end{enumerate}
The sequence described above is the sequence of events for each node. However, the events are split into two phases: In the first phase, the nodes follow events 1--3 in order from downstream to upstream, and in the second phase, the nodes follow events 4--6 in order from upstream to downstream. That is, the downstream-most nodes place their orders to their predecessors, their predecessors place their orders, etc.; then the upstream-most nodes ship units to their successors, who ship units to their successors, etc.
Note also that the multi-period newsvendor problem (in which there is a single node, which can hold inventory and backorders from one period to the next) can be modeled using the framework above by setting the lead time to 1. (The newsvendor problem is often described as having zero lead time, but it also uses a different sequence of events, in which we observe the demand after we place the order. Setting the lead time to 1 converts our sequence of events to this one.)
The notation is summarized in Section \textcolor{blue}{2} of supplementary material.
\subsection{Interaction between Agents and Environment}
\label{sec:interaction}
\begin{figure}
\hspace*{-1cm}
\centering
\includestandalone[width = 8cm]{frameworkDNN}
\caption{DNN-SMEIO framework.}
\label{framework}
\end{figure}
Each node $j$ must choose the order-up-to level (OUL) it uses when placing order from each of its predecessor nodes $i\in \mathcal{U}_j$; we refer to the OULs as being chosen ``for the edge $(i,j)$.'' Decisions for each edge $(i,j)$ may be made by a separate DNN agent, or using some exogenous mechanism. For ease of exposition, we will assume that {\em all} edges are governed by a DNN agent, but it is straightforward to adapt our method if some edges have non-DNN decision makers. This structure implies that there can be up to $|\mathcal{E}|$ independent DNN decision makers.
The agents are trained by interacting with an {\em environment} consisting of a simulation of the SCN described above. In particular, the agent makes decisions for multiple {\em episodes}, each of which consists of $T$ time periods. At the beginning of each episode $k$, the DNN chooses order-up-to levels $OUL_{ji}$ for all $(i,j)\in \mathcal{E}$ and sends these levels to the environment. The environment simulates the SCN to calculate the cost of the current OULs, given by
\begin{equation}\label{eq:ck}
\mathcal{C}_k = \sum_{t=1}^{T} c_t,
\end{equation}
where $c_t$ is as given by equation \eqref{eq:ct}. The DNN weights are then updated, new OULs are chosen, and a new episode begins. When the DNNs are trained, the output $OUL_{ji}$ converges to a single OUL. In practice, the weight-update procedure happens considering mini-batches of episodes together for computational purposes. The inputs to the DNN are largely irrelevant and can be set in any number of ways. This is because the DNN is optimizing an objective function, rather than trying to determine labels for a given input. In our numerical experiments, we use the inventory positions as the input, but the DNN could just as easily be given a vector of 1s as its input.
Figure \ref{framework} shows the proposed framework, in which a DNN is responsible for deciding the quantities of items ordered by node $j$ from node $i$. We use fully connected DNNs, with several possible hidden layers for each decision maker.
Moreover, a batch normalization procedure is considered after every network layer, which considerably stabilizes the learning procedure.
We allow the agents to be optimized by individual DNNs or by a single DNN. That is, networks corresponding to different agents can be completely separate, or they can share their first few layers.
The loss function used to train the DNN(s) is given by the per-episode costs $\mathcal{C}_k$ given in \eqref{eq:ck}. (A similar approach was used by \cite{oroojlooyjadid2020applying}. This is in contrast to the more common approach in which the DNN loss function measures the distance between an estimated and actual value.) To optimize the weights of the DNN network(s), we use the adaptive moment estimation (Adam) optimizer \cite{kingma2014adam} with fixed learning rate. Each environment consists of many hyperparameters that might need to be tuned. Details on framework structure, optimization algorithm is provided in Section \textcolor{blue}{3} of supplementary material.
\section{Numerical Experiments}
\label{Experiments}
We report the results of our experiments on single-node, serial, assembly and mixed SCNs. At first, we investigate the effectiveness of the model against infinite-horizon classical IO models to check whether our proposed method can produce results that are close to those produced by established methods for classical IO problems. We utilize Spearmint Bayesian optimization, and in particular, the Gaussian process expected improvement (GPEI) method for hyperparameter tuning of our framework \cite{swersky2013multi} (Refer to Section \textcolor{blue}{3} of supplementary material).
\subsection{Single-Node SCN}
\begin{table}[]
\caption{OUL and cost comparison for single-node inventory instances}
\label{tab1}
\setlength\tabcolsep{1pt}
\centering
\begin{tabular}{ccccccccccccccccccc}
& & & & \multicolumn{7}{c}{$L=0$} & & \multicolumn{7}{c}{$L=1$} \\
& & demand & & \multicolumn{3}{c}{analytical} & & \multicolumn{3}{c}{DNN} & & \multicolumn{3}{c}{analytical} & & \multicolumn{3}{c}{DNN} \\ \cline{5-11} \cline{13-19}
case \# & & distribution & & OUL & & cost & & OUL & & cost & & OUL & & cost & & OUL & & cost \\
\hline
1 & & $\mathcal{N}(10,1)$ & & $ 0 $ & & $ 0 $ & & $ 0 $ & & $ 0 $ & & $10.67 $ & & $12.71$ & & $10.68$ & & $12.71$ \\
2 & & $\mathcal{N}(10,2)$ & & $ 0 $ & & $ 0 $ & & $ 0 $ & & $ 0 $ & & $11.35$ & & $25.42$ & & $11.50$ & & $25.47$ \\
3 & & $\mathcal{N}(50,1)$ & & $ 0 $ & & $ 0 $ & & $ 0 $ & & $ 0 $ & & $50.67$ & & $12.71$ & & $50.58$ & & $12.75$ \\
4 & & $\mathcal{N}(50,5)$ & & $ 0 $ & & $ 0 $ & & $ 0 $ & & $ 0 $ & & $ 53.37$ & & $63.56$ & & $ 53.30$ & & $63.59$ \\
5 & & $\mathcal{N}(100,1)$ & & $ 0 $ & & $ 0 $ & & $ 0 $ & & $ 0 $ & & $100.67$ & & $12.71$ & & $100.77$ & & $12.75$ \\
6 & & $\mathcal{N}(100,5)$ & & $ 0 $ & & $ 0 $ & & $ 0 $ & & $ 0 $ & & $103.37$ & & $63.56$ & & $103.28$ & & $63.58$ \\
7 & & $\mathcal{N}(100,10)$ & & $ 0 $ & & $ 0 $ & & $ 0 $ & & $ 0 $ & & $106.74$ & & $127.11$ & & $106.79$ & & $127.12$ \\ \hline
\end{tabular}
\end{table}
In this section, we consider a simple SCN consisting of a single node. In particular, the node has holding and stockout costs of $h=10$ and $p=30$, respectively. The demand per period is normally distributed; we consider various means and standard deviations. We set the holding cost $h=10$ and shortage cost $p=30$ for all the cases. This means critical ratio equals $0.75$. We consider a short time horizon of $T=2$ periods, with no salvage value ($v(x) = 0$ for all $x$). We consider two settings for the lead time, one in which $L=1$ and one in which $L=0$. In the $L=0$ case, the decision maker sees the demand and then places an order; there is no stochasticity. The optimal action is simply to order the realized demand value, and the optimal cost is 0. This is not a typical setting, but we examine it to evaluate the learning process of the DNN-SMEIO framework in a very simple case.
The $L=1$ case is mathematically equivalent to the newsvendor problem. (Recall that in the classical newsvendor problem, the order is received before the demand is observed, whereas the opposite is true in our sequence of events. Therefore, our model is equivalent to the newsvendor model if we set $L=1$.)
Table \ref{tab1} reports the 7 instances we tested. The table shows the demand distribution, the optimal OUL and average cost per period, and the OUL and average cost given by the DNN, for both the $L=0$ and $L=1$ cases. The optimal OULs were found using an analytical approach. For the DNN solutions, we calculate the cost using the base-stock policy simulation.
One can see that the results are very close; the DNN finds near-optimal OULs for these instances. The largest relative error between DNN-SMEIO approach and the analytical solutions are 1.32\% and 0.31\% for the OUL and cost values, respectively.
Figure \ref{fig:newsvendorConvergenceCurve} illustrates the loss function and OUL convergence curves for instances 1 and 7 from Table \ref{tab1} and for both lead-time settings. We train the DNN-SMEIO for 50000 training episodes. The x-axes contain 500 points. After each 100 episodes of training, a new set of episodes is conducted for testing. The cost and OULs are calculated, which corresponds to a single point in the respective figures. One can stop the training much sooner than 50000 episodes. For instance, case 1 reaches its best result after only 1500 training episodes. We let the process continue only to demonstrate the method's stability. The black dotted lines for the OULs are the true optimal values. However, for the loss function, we simulate the process identically to the one that DNN-SMEIO is trained over (i.e., we consider the same initialization and episode horizon).
\begin{figure*}
\hspace*{-1.5cm}
\scalebox{0.4}{\input{newsvendorConvergenceCurve2.pgf}}
\caption{Loss function and OUL convergence curves for single-node inventory instances case 1 and 7.}
\label{fig:newsvendorConvergenceCurve}
\end{figure*}
\subsection{Serial SCN}
In this section, we consider instances of serial SCNs. First, we discuss the comparison structure through a three-echelon serial system, which is illustrated in Figure \ref{serialSCN} and used as an example by \citet{fosct2e}. The network has 3 nodes and therefore 3 separate decisions to be made. There is a single external customer with normally distributed demand $\mathcal{N}(5,1)$. The (local) holding costs increase as one moves downstream, and there is a shortage cost only at the furthest downstream node. \citet{fosct2e} report the optimal OULs for this system under an infinite-horizon, continuous-review environment, based on the Clark--Scarf recursive method \cite{clark1960optimal,ChenZheng94}. However, as mentioned previously, our environment is a finite-horizon, periodic-review environment. Although the two environments are not strictly comparable, our time horizon (we use $T=10$) is long enough so that the system approximately reaches steady state and can therefore be compared to the infinite-horizon case.
Moreover, after the agents report their OULs, we use a simple base-stock policy simulation to find the cost. Nevertheless, because of the inherent differences between our assumptions (finite horizon, periodic review, cost evaluation by simulation) and those of the Clark-Scarf method (infinite horizon, continuous review, analytical cost evaluation), we also consider a third approach for optimizing the OULs: derivative-free optimization (DFO). We use the Trust Region DFO (DFO-TR) method \cite{bandeira2012computation}, which is a model-based DFO method and is well-known to use as few function evaluations as possible. DFO can be used to explore the best possible OULs, so we use them as an additional benchmark against which to compare the DNN-SMEIO method.
\begin{figure}
\centering
\includestandalone[width = 7.5cm]{serialSCN}
\caption{Three-echelon serial SCN.}
\label{serialSCN}
\end{figure}
The methods see the optimal OULs neither at the initialization nor during the training process. The problem settings are reported in Table \ref{tab31}, and the analytical, DNN-SMEIO and DFO results are presented in Table \ref{tab32}. As one can see, the DNN results are quite close to the ones suggested by the analytical approach. For instance, the difference between the best cost for the case discussed above (case 3) achieved by the proposed framework ($47.90$) and the cost obtained by simulation using the exact OULs ($47.65$) is less than 1\%. DFO gives considerably worse results for cases 7 and 8 but performs as well as the other approaches for the rest of the cases.
The convergence curves for the loss function and the OULs of serial SCN case 3 are provided in Figure \ref{fig:THreeEchelonSMEIO}. Training and testing procedures are similar to those discussed in the previous section. One can see that the DNN-SMEIO method is stable to a great extent.
\begin{table*}[]
\caption{Serial SCNs settings}
\label{tab31}
\setlength\tabcolsep{2pt}
\centering
\begin{tabular}{ccccccccccc}
case && \# echelons & & demands & & holding cost & & shortage cost & & lead-time \\
&& & & & & (per item) & & (per item) & & \\
\hline
1&& 2 & & $\mathcal{N}(3,0.5)$ & & $\left(5,8.2\right)$ & & $\left(0,25.5\right)$ & & $\left(1,1\right)$ \\
2&& 2 & & $\mathcal{N}(6,1.5)$ & & $\left(1.9,4.1\right)$ & & $\left(0,11.3\right)$ & & $\left(2,1\right)$ \\
3&& 3 & & $\mathcal{N}(5,1)$ & & $\left(2,4,7\right)$ & & $\left(0,0,37.12\right)$ & & $\left(2,1,1\right)$ \\
4&& 3 & & $\mathcal{N}(50,3)$ & & $\left(5,10,25\right)$ & & $\left(0,0,50\right)$ & & $\left(2,1,1\right)$ \\
5&& 3 & & $\mathcal{N}(100,5)$ & & $\left(25,25,50\right)$ & & $\left(0,0,100\right)$ & & $\left(1,2,2\right)$ \\
6&& 3 & & $\mathcal{N}(100,10)$ & & $\left(10,20,30\right)$ & & $\left(0,0,100\right)$ & & $\left(1,1,1\right)$ \\
7&& 4 & & $\mathcal{N}(3,0.4)$ & & $\left(4,5.75,7.90,10.8\right)$ & & $\left(0,0,0,35.5\right)$ & & $\left(1,1,1,1\right)$ \\
8&& 4 & & $\mathcal{N}(5,1.2)$ & & $\left(5,5,5,10\right)$ & & $\left(0,0,0,30\right)$ & & $\left(1,1,1,1\right)$ \\
9&& 5 & & $\mathcal{N}(80,4)$ & & $\left(10,20,30,40,50\right)$ & & $\left(0,0,0,0,200\right)$ & & $\left(1,1,1,1,1\right)$ \\
10&& 5 & & $\mathcal{N}(25,2)$ & & $\left(5,10,25,50,50\right)$ & & $\left(0,0,0,0,150\right)$ & & $\left(2,1,1,1,1\right)$ \\
\hline
\end{tabular}
\end{table*}
\begin{table*}[]
\caption{OUL and cost comparison for serial SCNs}
\label{tab32}
\setlength\tabcolsep{2pt}
\hspace*{-1.5cm}
\small
\begin{tabular}{ccccccccccccc}
& & \multicolumn{3}{c}{Analytical} & & \multicolumn{3}{c}{DNN} && \multicolumn{3}{c}{DFO} \\
\cline{3-5} \cline{7-9} \cline{11-13}
case & & OULs & & cost & & OULs & & cost & & OULs & & cost \\
\hline
1 & & $(2.91, 3.64)$ & & $22.21$ & & $(2.91, 3.72)$ & & $22.34$ && $(1.22, 5.10)$ && $22.55$\\
2 & & $(12.58,7.60)$ & & $23.07$ & & $(12.58,7.65)$ & & $23.17$ && $(12.05, 7.58) $&& $23.20$\\
3 & & $(10.69,5.53,6.49) $ & & $47.65$ & & $(10.08,5.39,6.64)$ & & $47.90$ && $(10.54, 5.35, 6.57)$ && $50.01$\\
4 & & $(101.45,51.40,52.7040) $ & & $879.88$ & & $(99.29,51.03,52.71)$ & & $885.63$ && $(97.02, 53.59, 53.02)$ && $885.49$ \\
5 & & $(71.026, 228.29, 207.04) $ & & $10568.23$ & & $(87.71,204.63,208.90)$ & & $10625.01$ && $(79.33, 211.20, 208.51)$ && $10695.88$ \\
6 & & $(99.53,102.58,114.05) $ & & $3630.14$ & & $(93.36,103.42,114.26)$ & & $3651.63$ && $(95.83, 100.87, 117.90)$&& $3638.18$\\
7 & & $(2.78,3.13,3.19,3.60) $ & & $63.39$ & & $(2.78, 3.13, 3.19, 3.74)$ & & $63.84$ && $(-12.03, -8.86, -3.93, 1.90)$ && $592.51$\\
8 & & $(-3.80,9.80,9.80,6.35) $ & & $101.48$ & & $(1.48,6.12,7.00,6.46)$ & & $104.04$ && $(-4.96, -3.69, -1.90, 0.02)$&& $674.62$ \\
9 & & $(80.15,80.15,81.17,81.68,86.99) $ & & $8559.85$ & & $(76.83,78.02,79.60,81.62,87.40)$ & & $8678.38$ && $(80.49, 77.62, 80.04, 77.45, 92.18)$ && $8585.50$\\
10 & & $(51.57, 26.30, 25.05, 20.25, 33.01)$ & & $2500.79 $ & & $(48.40, 25.65, 24.02, 22.90, 30.12)$ & & $2581.41 $ && $(49.44, 25.51, 23.04, 23.82, 31.17)$ && $2527.1$ \\
\hline
\end{tabular}
\end{table*}
\begin{figure*}
\hspace*{-1.25cm}
\scalebox{0.40}{\input{ThreeStageResult2.pgf}}
\caption{Total cost and OUL decisions convergence curves for serial SCN case 3}
\label{fig:THreeEchelonSMEIO}
\end{figure*}
\subsection{Assembly SCNs}
We consider two separate assembly structures, which are illustrated in Figure \ref{AssemblySCN}. Each structure contains three echelons. There are 10 and 11 OUL decisions to be made for assembly SCNs 1 and 2, respectively. For each of these structures, we consider 5 separate cases having different holding and shortage costs, lead-times and demand distributions. We use DFO, coordinate descent (CD), and enumeration as heuristic benchmarks to compare our method against. For CD and enumeration, each candidate solution is evaluated by simulating the system for 3 trials, each consisting of 200 periods; one solution is considered superior to another if the mean of the 3 total costs (one per trial) for that solution is smaller than that of the other. Lower and upper bounds for both CD and enumeration are set equal to $0.75D$ and $2D$, respectively, where $D$ is the mean lead-time demand observed by the node. For enumeration, the resulting range was discretized into 10 equal intervals. Moreover, we restricted the OULs to be equal at all nodes within a given echelon, which is optimal given the symmetries in the SCNs, and which therefore reduces the search space. Note that we did not make the same restriction for the DNN method, meaning that we are providing an advantage to the benchmark methods. The complete specifications of the parameters of the assembly system instances, as well as the OULs suggested by each method, are presented in Tables \textcolor{blue}{3} and \textcolor{blue}{4} of the supplement.
For all solutions produced by all methods, we evaluated the cost by simulating the system for 10 trials, each consisting of 10,000 periods. The mean of the 10 total costs (one per trial) is reported in the Table \ref{tabAssembly}. We observe that the costs of the solutions suggested by DNN are very close to those from CD and enumeration, even though the search space was restricted as described above for CD and enumeration but not for DNN.
DFO is inferior compared to the other approaches. For four of the cases, the DFO OULs did not converge to reasonable values, resulting in a very large cost. There are at least three possible reasons for this. First, we may have exceeded the number of variables that can be properly handled by DFO. Second, DFO methods are generally local methods, and DFO may become trapped in local minimum. Third, the performance of DFO heavily depends on the starting point, and we may have started from OULs that are far from the optimal ones, although we tried to alleviate its effect by starting with lead-time demands as initial inventory levels.
\begin{table}[]
\caption{Cost comparisons for assembly SCNs}
\label{tabAssembly}
\setlength\tabcolsep{1pt}
\centering
\begin{tabular}{cccccc}
& Case &DNN-SMIO & CD & enumeration & DFO\\
\hline
assembly1 & 1 & 40.55 & 40.27 & 40.34 &233.45 \\\cline{1-1}
&2 & 103.77 & 101.59 & 101.47 &482.63 \\
&3 & 163.15 & 161.30 & 161.13 &441.43 \\
&4 & 37.49 & 35.97 & 35.98 &139.77 \\
&5 & 29.04 & 27.53 & 27.45 &36.03 \\ \hline
average & & 74.80 & 73.33 & 73.27 &324.32 \\
\hline
& & & & & \\
assembly2 & 1 & 93.94 & 90.40 & 90.54 &116.38 \\\cline{1-1}
&2 & 23.00 & 22.43 & 22.48 &25.75 \\
&3& 86.61 & 82.67 & 82.32 &90.71 \\
&4& 34.62 & 34.04 & 34.17 &42.35 \\
&5& 30.98 & 28.19 & 27.96 &62.76 \\ \hline
average & & 53.83 & 51.55 & 51.49 & 67.59
\end{tabular}
\end{table}
\begin{figure*}
\centering
\begin{minipage}{.35\textwidth}
\includestandalone[width = 5cm]{Assembly1SCN}
\end{minipage}
\begin{minipage}{.15\textwidth}
\includestandalone[width = 4cm]{Assembly2SCN}
\end{minipage}
\caption{Assembly SCN structures. Left figure: assembly SCN (1) and the right figure assembly SCN (2).}
\label{AssemblySCN}
\end{figure*}
\subsection{Mixed SCN}
From an inventory optimization perspective, there are no known analytical solutions for mixed SCNs, other than computationally intensive, enumeration-based approaches. Therefore, providing a stable numerical approach is greatly desirable. In this section, we demonstrate the performance of our method in finding OULs for multiple nodes of a mixed SCN simultaneously. We consider the SCN illustrated in Figure \ref{mixedSCN}. The network has 3 echelons, 5 nodes, 7 edges and two customers having independent stochastic normal distributions $\mathcal{N}(5,1)$ as their demands. The third echelon nodes, $4$ and $5$, are ``assembly-{\em and}'' nodes (see the definition in Section \ref{Model}.), and nodes 2 and 3 are distribution nodes, making the SCN a mixed one. We consider a time horizon of $T=10$. All 7 OULs are required to be optimized simultaneously. Table \ref{tab4} shows the parameters used in the study. Figure \ref{fig:mixedSCNResult} illustrates the convergence curves of the DNN approach. The black dotted lines are the best results, corresponding to the minimum costs achieved by DNN.
We allow the algorithm to restart the learning procedure from scratch using the best OULs found previously as the new initial inventory levels until there is no extra improvement in the objective value. We do this for two reasons. First, we do not know whether the objective functions of mixed SCN inventory problems are convex or nonconvex. In case we are minimizing a nonconvex objective, we do not wish the optimization algorithm to be stuck in a local minimum. Second, we have no idea how far away the initial inventory levels are from the optimal OULs. Considering that we are dealing with finite-horizon ($T=10$) episodes, this can significantly slow the method's learning procedure. Hence, we use the OULs found the previous time as the initial inventory levels of the next learning procedure. We set stopping criteria so that the procedure terminates when there is less than a 1\% change in the objective value.
One can observe the following based on Figure \ref{fig:mixedSCNResult}:
\begin{itemize}
\item The jumps in the subplots are due to the restart procedure discussed previously. The importance of better initialization can be seen in the total cost spike at the restart. Although the OUL decisions are random and not necessarily close to their optimal values, at the restart, the total cost is almost half of the cost at the beginning of the procedure.
\item The farther downstream the node is, the noisier the OULs are.
\item Similar to the previous examples, the agents farther upstream need fewer episodes to be optimized compared with those downstream.
\end{itemize}
\begin{table*}[]
\caption{Mixed SCN parameters}
\label{tab4}
\setlength\tabcolsep{2pt}
\centering
\begin{tabular}{ccccccccccccc}
Echelon & & Edge & & Holding cost & & Shortage Cost & & Shipment lead-time & & Order lead-time & & Initialization \\
& & & & (per item) & & (per item) & & & & & & \\
\cline{1-1} \cline{3-3} \cline{5-5} \cline{7-7} \cline{9-9} \cline{11-11} \cline{13-13}
1 & & (0,1) & & $2$ & & $4$ & & $2$ & & $0$ & & $40$ \\
2 & & (1,2) & & $4$ & & $12$ & & $1$ & & $0$ & & $10$ \\
2 & & (1,3) & & $4$ & & $12$ & & $1$ & & $0$ & & $10$ \\
3 & & (2,4) & & $7$ & & $37.12$ & & $1$ & & $0$ & & $5$ \\
3 & & (2,5) & & $7$ & & $37.12$ & & $1$ & & $0$ & & $5$ \\
3 & & (3,4) & & $7$ & & $37.12$ & & $1$ & & $0$ & & $5$ \\
3 & & (3,5) & & $7$ & & $37.12$ & & $1$ & & $0$ & & $5$ \\ \hline
\end{tabular}
\end{table*}
\begin{figure*}
\hspace*{-1.25cm}
\scalebox{0.40}{\input{mixedSCNResultFinal2.pgf}}
\caption{Total cost and OUL decisions for mixed SMEIO.}
\label{fig:mixedSCNResult}
\end{figure*}
Because there are no reliable algorithms for general mixed systems in the literature, we use a randomized search as a benchmark to compare with the DNN-SMEIO framework. We consider 100 separate randomly generated solutions and for each run we average over 2000 episodes. (Note that this is more than 20 times the number of environment interactions that the DNN approach requires; recall that DNN-SMEIO reaches its best results in at most $10^4$ episodes.)
We select candidate solutions randomly by setting each node's OUL equal to the mean demand of the node plus the absolute value of a random, zero-mean, normal random variate. We use random variates with larger standard deviations for nodes with larger demand means to allow for a search space that is likely to contain good solutions. Table \ref{tab5} shows the parameters for the random variates, as well as a comparison between the 5 best solutions found using this method and the solution found by the DNN approach. As one can see, the DNN performs better than all of the randomly generated solutions.
\begin{table*}[]
\caption{Top 5 randomized solutions and DNN results comparison for mixed SCN}
\label{tab5}
\setlength\tabcolsep{2pt}
\centering
\begin{tabular}{cccccccccccccccccc}
& & & & & & & \multicolumn{11}{c}{OULs} \\
\cline{8-18}
& Echelon & & Edge & & Random search parameters & & \multicolumn{9}{c}{5 best randomly generated solutions} & & DNN \\ \cline{2-2} \cline{4-4} \cline{6-6} \cline{8-16} \cline{18-18}
& 1 & & (0,1) & & $40 + | \mathcal{N}(0,4)|$ & & 40.00 & & 43.58 & & 41.22 & & 45.50 & & 41.19 & & 42.87 \\
& 2 & & (1,2) & & $10 + | \mathcal{N}(0,2)|$ & & 13.54 & & 12.84 & & 10.57 & & 12.49 & & 13.07 & & 11.65 \\
& 2 & & (1,3) & & $10 + | \mathcal{N}(0,2)|$ & & 13.72 & & 12.02 & & 11.57 & & 10.62 & & 13.21 & & 11.58 \\
& 3 & & (2,4) & & $5 + | \mathcal{N}(0,2)|$ & & 6.31 & & 7.58 & & 5.08 & & 6.99 & & 8.03 & & 6.73 \\
& 3 & & (2,5) & & $5 + | \mathcal{N}(0,2)|$ & & 6.50 & & 5.85 & & 5.77 & & 5.36 & & 9.07 & & 6.73 \\
& 3 & & (3,4) & & $5 + | \mathcal{N}(0,2)|$ & & 5.84 & & 7.79 & & 7.54 & & 6.36 & & 5.14 & & 6.99 \\
& 3 & & (3,5) & & $5 + | \mathcal{N}(0,2)|$ & & 5.16 & & 5.16 & & 9.45 & & 5.39 & & 8.00 & & 6.41 \\
\hline
& Total Cost & & & & & & 215.05 & & 214.72 & & 214.45 & & 212.97 & & 211.90 & & 208.80 \\
\end{tabular}
\end{table*}
We compare the performance of the proposed method with two additional alternatives: derivative-free optimization (DFO) \cite{conn2009introduction} and Spearmint Bayesian optimization \cite{swersky2013multi}. Each DFO or Spearmint step can be reduced to three parts: first, a suggestion of OULs; second, an independent simulation run to obtain the objective value for that suggestion; third, a suggestion of a new set of OULs based on an optimization algorithm. We consider 2000 episodes to allow each simulation run to converge. We test two cases, one in which there is no explicit bound on the number of function evaluations and one in which we restrict them to make sure the number of interactions between agents and the environment stays equal for the DFO, Spearmint, and DNN approaches. For this instance, 25 function evaluations for DFO or Spearmint were conducted, each of which consists of 2000 episodes, so we allowed DNN to use 50,000 episodes to reach its result. The only stopping criteria for the cases without an upper bound on the number of function evaluations are either having 100 algorithm steps without any improvement or having the last 10 improvements be less than 0.5\% of the cost function value. In addition, DFO-TR requires an initial guess to start the algorithm. We use the lead-time demand means for this purpose. Spearmint, however, requires an interval for each decision variable. We consider lead-time demand means as the lower bounds and an acceptable range (at least twice the lead-time demand standard deviation) to cover possible solutions. Both algorithms might suffer from a dependence on these initial conditions.
Table \ref{tab5-2} shows the comparison results. DFO and Spearmint find marginally better values in terms of objective function value for the cases without any upper bound on the number of function evaluations. However, restricting them to have the same number of interactions with the environment as the DNN has results in inferior performance of these alternatives compared to the DNN. Furthermore, because the optimal OULs for this mixed SCN structure happen to be close to the lead-time demand means, this biases the experiment in favor of DFO and Spearmint, which are given the lead-time demand means (or a small interval containing them) as initial values. In the next section, however, we investigate a more realistic case study and explore the comparison further.
\begin{table*}[]
\caption{Comparison results between DFO, Spearmint and DNN approaches for Mixed SCN}
\label{tab5-2}
\hspace*{-1.5cm}
\setlength\tabcolsep{1.5pt}
\centering
\begin{tabular}{cccccccccccccccccc}
& & & & & \multicolumn{11}{c}{Alternatives} & & \multirow{3}{*}{DNN} \\ \cline{6-16}
& & & & & \multicolumn{3}{c}{Initial Value Choosing} & & \multicolumn{3}{c}{Results with 25 function evaluations} & & \multicolumn{3}{c}{Best results without restrictions} & & \\ \cline{6-8} \cline{10-12} \cline{14-16}
\multirow{8}{*}{OUL decisions} & Echelon & & Edge & & DFO & & Spearmint & & DFO & & Spearmint & & DFO & & Spearmint & & \\ \cline{2-2} \cline{4-4} \cline{6-6} \cline{8-8} \cline{10-10} \cline{12-12} \cline{14-14} \cline{16-16}
& 1 & & (0,1) & & $40$ & & $[40,48]$ & & 47.69 & & 41.45 & & 43.73 & & 43.80 & & 42.87 \\
& 2 & & (1,2) & & $10$ & & $[10,14]$ & & 12.45 & & 12.34 & & 11.46 & & 11.45 & & 11.65 \\
& 2 & & (1,3) & & $10$ & & $[10,14]$ & & 12.62 & & 11.76 & & 11.46 & & 11.49 & & 11.58 \\
& 3 & & (2,4) & & $5$ & & $[5,9]$ & & 5.51 & & 5.39 & & 5.77 & & 5.80 & & 6.73 \\
& 3 & & (2,5) & & $5$ & & $[5,9]$ & & 5.58 & & 5.54 & & 5.77 & & 5.77 & & 6.73 \\
& 3 & & (3,4) & & $5$ & & $[5,9]$ & & 5.53 & & 7.02 & & 5.77 & & 5.78 & & 6.99 \\
& 3 & & (3,5) & & $5$ & & $[5,9]$ & & 5.40 & & 5.63 & & 5.78 & & 5.78 & & 6.41 \\
Total Cost & & & & & & & & & 215.21 & & 214.66 & & 206.35 & & 206.36 & & 208.80 \\ \hline
\end{tabular}
\end{table*}
\subsection{Complex SCN}
In this section we introduce a case study of a general-structured SCN with realistic settings such as nonlinear holding and penalty costs and salvage values. For this comparison, We consider trust-region derivative-free optimization (DFO-TR) \cite{bandeira2012computation,pirhooshyaran2020feature}, GPEI Bayesian along with simple yet powerful random search techniques as alternative methods. We consider the complex SCN shown in Figure \ref{fig:caseStudy}. The SCN contains 7 nodes and 13 edges. Nodes 5, 6 and 7 are ``assembly-{\em and}'' nodes. That is, to produce one item at any of these nodes, one item from each of its predecessors is required. In this case study, holding costs are considered to be piecewise linear instead of linear. When the number of items in inventory increases beyond a certain threshold, the cost per item decreases. On the other hand, shortage penalty costs are considered to be nonlinear, and the cost per item increases when the shortage quantity is greater. There are three leaf nodes that see customer demands. One follows an independent normal distribution $\mathcal{N}(5,1)$, the second follows a discrete uniform distribution $\mathcal{U}\{1,2,\dots,5\}$, and the third follows a two-sided truncated Poisson distribution $\mathcal{TP}(\lambda =3;6;10)$, where 3 is the distribution parameter and 6 and 10 are the beginning and the end of the possible values, respectively. We consider $T=10$ as the episode horizon length.
We assume that the three leaf nodes have different salvage functions that represent the cost or reward incurred based on the inventory level at the end of the horizon. Nodes 5 and 6 have linear rewards, with different slopes; for example, these might model situations in which excess inventory can be sold for a per-unit cost at the end of the horizon. However, for node 7, we assume a nonlinear salvage reward policy to clear the remaining inventory. Up to a threshold, the price per item is considered to be high because there are only few items left. The price per item then reduces drastically to reach the second threshold and thereafter it is fixed.
Table \ref{tab6} provides further details about this problem instance. In the table, $(x<\text{threshold},f_1(x),f_2(x))$ means that when $x < \text{threshold}$, the function $f_1(x)$ is used, and afterwards $f_2(x)$ is used. The initial inventories are set equal to the demand mean. For instance, $\mathbb{E}\left( \mathcal{TP}(\lambda =3,6,10) \right) = 7.58$ is considered as the initialization values for the $(2\leftarrow7),(3\leftarrow7)$ and $(4\leftarrow7)$ decisions.
\begin{figure*}
\centering
\includestandalone[width = 10cm]{mixedSCNCaseStudy}
\caption{Mixed SCN with customer demands and salvage policies.}
\label{fig:caseStudy}
\end{figure*}
\begin{table*}[]
\caption{Case study mixed SCN parameters$^\ast$}
\label{tab6}
\setlength\tabcolsep{2pt}
\hspace*{0.25cm}
\begin{small}
\begin{tabular}{ccccccccccc}
Edge & & Holding cost & & Shortage Cost & & Shipment lead-time & & Salvage reward & & Initialization \\
& & (for $x$ units on hand) & & (for $x$ backorders) & & & & (for $x$ units on hand) & & \\
\cline{1-1} \cline{3-3} \cline{5-5} \cline{7-7} \cline{9-9} \cline{11-11}
(0,1) & & $2x$ & & $4x$ & & $2$ & & --- & & $45.24$ \\
(1,2) & & $(x<3,4x,3x)$ & & $(x<3,12x,4x^2)$ & & $1$ & & --- & & $15.08$ \\
(1,3) & & $(x<3,4x,3x)$ & & $(x<3,12x,4x^2)$ & & $1$ & & --- & & $15.08$ \\
(1,4) & & $(x<3,4x,3x)$ & & $(x<3,12x,4x^2)$ & & $1$ & & --- & & $15.08$ \\
(2,5) & & $(x<3,7x,6x)$ & & $(x<3,36x,12x^2)$ & & $1$ & & $1.25x$ & & $5.00$ \\
(2,6) & & $(x<3,7x,6x)$ & & $(x<3,36x,12x^2)$ & & $1$ & & $1.5x$ & & $2.50$ \\
(2,7) & & $(x<3,7x,6x)$ & & $(x<3,36x,12x^2)$ & & $1$ & & $(x<2,15-0.5x,max(-3.5x^2 +14x,3))$ & & $7.58$ \\
(3,5) & & $(x<3,7x,6x)$ & & $(x<3,36x,12x^2)$ & & $1$ & & $1.25x$ & & $5.00$ \\
(3,6) & & $(x<3,7x,6x)$ & & $(x<3,36x,12x^2)$ & & $1$ & & $1.5x$ & & $2.50$ \\
(3,7) & & $(x<3,7x,6x)$ & & $(x<3,36x,12x^2)$ & & $1$ & & $(x<2,15-0.5x,max(-3.5x^2 +14x,3))$ & & $7.58$ \\
(4,5) & & $(x<3,7x,6x)$ & & $(x<3,36x,12x^2)$ & & $1$ & & $1.25x$ & & $5.00$ \\
(4,6) & & $(x<3,7x,6x)$ & & $(x<3,36x,12x^2)$ & & $1$ & & $1.5x$ & & $2.50$ \\
(4,7) & & $(x<3,7x,6x)$ & & $(x<3,36x,12x^2)$ & & $1$ & & $(x<2,15-0.5x,max(-3.5x^2 +14x,3))$ & & $7.58$ \\ \hline
\end{tabular}
$^\ast$ {\footnotesize All of the costs listed below equal 0 if $x < 0$.}
\end{small}
\end{table*}
First, we compare the DNN-SMEIO method with the randomized approach. We consider 400 runs of 5000 episodes each. Table \ref{tab7} shows the results of this comparison. The intervals of possible OULs used for the randomized approach contain the best solution found by DNN-SMEIO. One can see the clear advantage of the proposed method over the randomized approach.
\begin{table*}[]
\caption{Top 5 randomized solutions and DNN results comparison for mixed SCN}
\label{tab7}
\setlength\tabcolsep{2pt}
\centering
\begin{small}
\begin{tabular}{cccccccccccccccccc}
& & & & & & & \multicolumn{11}{c}{OULs} \\
\cline{8-18}
& Echelon & & Edge & & Random search parameters & & \multicolumn{9}{c}{5 best randomly generated solutions} & & DNN \\ \cline{2-2} \cline{4-4} \cline{6-6} \cline{8-16} \cline{18-18}
& 1 & & (0,1) & & $45.24 + |\mathcal{N}(0,50)|$ & & 102.73 & & 102.56 & & 111.44 & & 100.40 & & 100.40 & & 101.44 \\
& 2 & & (1,2) & & $15.08 + | \mathcal{N}(0,5)|$ & & 19.50 & & 17.96 & & 22.71 & & 22.75 & & 15.91 & & 18.75 \\
& 2 & & (1,3) & & $15.08 + | \mathcal{N}(0,5)|$ & & 14.25 & & 14.51 & & 18.72 & & 18.81 & & 19.84 & & 20.82 \\
& 2 & & (1,4) & & $15.08 + | \mathcal{N}(0,5)|$ & & 15.51 & & 15.10 & & 20.05 & & 20.51 & & 21.85 & & 21.48 \\
& 3 & & (2,5) & & $5 + | \mathcal{N}(0,5)|$ & & 7.52 & & 6.91 & & 6.06 & & 5.33 & & 6.26 & & 7.29 \\
& 3 & & (2,6) & & $2.5 + | \mathcal{N}(0,5)|$ & & 7.75 & & 10.38 & & 5.51 & & 8.95 & & 7.16 & & 6.95 \\
& 3 & & (2,7) & & $7.58 + | \mathcal{N}(0,5)|$ & & 9.38 & & 8.83 & & 12.52 & & 12.29 & & 9.84 & & 10.28 \\
& 3 & & (3,5) & & $5 + | \mathcal{N}(0,5)|$ & & 8.03 & & 7.87 & & 9.65 & & 9.39 & & 6.47 & & 7.21 \\
& 3 & & (3,6) & & $2.5 + | \mathcal{N}(0,5)|$ & & 6.99 & & 6.55 & & 4.08 & & 5.19 & & 5.73 & & 6.06 \\
& 3 & & (3,7) & & $7.58 + | \mathcal{N}(0,5)|$ & & 11.87 & & 11.14 & & 8.17 & & 8.52 & & 9.17 & & 9.48 \\
& 3 & & (4,5) & & $5 + | \mathcal{N}(0,5)|$ & & 7.04 & & 7.23 & & 9.11 & & 5.16 & & 5.02 & & 6.11 \\
& 3 & & (4,6) & & $2.5 + | \mathcal{N}(0,5)|$ & & 8.51 & & 6.70 & & 8.85 & & 5.54 & & 5.15 & & 6.5 \\
& 3 & & (4,7) & & $7.58 + | \mathcal{N}(0,5)|$ & & 10.10 & & 9.84 & & 10.48 & & 8.25 & & 7.71 & & 9.38 \\ \hline
& Total Cost & & & & & & 534.06 & & 526.75 & & 522.18 & & 519.68 & & 514.69 & & 478.61 \\
\end{tabular}
\end{small}
\end{table*}
We further investigate a comparison between DFO, Spearmint, and the proposed DNN approach. We do not put an upper limit on the number of interactions with the environment for the DFO and Spearmint approaches. (This biases the experiment in favor of alternative approaches.) Table \ref{tab8} tabulates the results. One can see the clear advantage of the proposed method over the alternatives. Both DFO and Spearmint fail to find results that are nearly as good as the randomized method. We also emphasize the fact that choosing a suitable interval for the Spearmint method requires expert knowledge, trial and error or multiple runs of the algorithm. For instance, based on the results reported in Table \ref{tab7}, we already knew that OULs greater than 100 should be considered for the $0\leftarrow 1$ decision. Consequently, we assumed a large interval of $[45, 150]$ for this decision variable, but without this prior knowledge, one might miss important regions for the decision variable. We refer the reader to see the learning curves and loss behavior plus their discussions in Section \textcolor{blue}{5} of supplement.
\begin{table*}[]
\caption{Comparison results between DFO, Spearmint and DNN approaches for case study SCN}
\label{tab8}
\setlength\tabcolsep{2pt}
\centering
\begin{small}
\begin{tabular}{cccccccccccccc}
& & & & & \multicolumn{7}{c}{Alternatives} & & \multirow{3}{*}{DNN} \\ \cline{6-12}
& & & & & \multicolumn{3}{c}{Initializing} & & \multicolumn{3}{c}{results} & & \\ \cline{6-8} \cline{10-12}
\multirow{8}{*}{OUL decisions} & Echelon & & Edge & & DFO & & Spearmint & & DFO & & Spearmint & & \\ \cline{1-1}\cline{2-2} \cline{4-4} \cline{6-6} \cline{8-8} \cline{10-10} \cline{12-12}
& 1 & & (0,1) & & $45.24$ & & $[45, 150]$ & & 99.15 & & 93.86 & & 101.44 \\
& 2 & & (1,2) & & $15.08$ & & $[15, 45]$ & & 15.94 & & 29.93 & & 18.75 \\
& 2 & & (1,3) & & $15.08$ & & $[15, 45]$ & & 16.01 & & 24.40 & & 20.82 \\
& 2 & & (1,4) & & $15.08$ & & $[15, 45]$ & & 15.99 & & 26.01 & & 21.48 \\
& 3 & & (2,5) & & $5$ & & $[2.5, 15]$ & & 5.20 & & 5.08 & & 7.29 \\
& 3 & & (2,6) & & $2.5$ & & $[2.5, 15]$ & & 3.13 & & 3.24 & & 6.95 \\
& 3 & & (2,7) & & $7.58$ & & $[2.5, 15]$ & & 7.62 & & 8.02 & & 10.28 \\
& 3 & & (3,5) & & $5$ & & $[2.5, 15]$ & & 5.41 & & 5.21 & & 7.21 \\
& 3 & & (3,6) & & $2.5$ & & $[2.5, 15]$ & & 2.84 & & 3.41 & & 6.06 \\
& 3 & & (3,7) & & $7.58$ & & $[2.5, 15]$ & & 7.52 & & 8.22 & & 9.48 \\
& 3 & & (4,5) & & $5$ & & $[2.5, 15]$ & & 5.44 & & 5.17 & & 6.11 \\
& 3 & & (4,6) & & $2.5$ & & $[2.5, 15]$ & & 2.77 & & 3.37 & & 6.5 \\
& 3 & & (4,7) & & $7.58$ & & $[2.5, 15]$ & & 7.52 & & 8.37 & & 9.38 \\
Total Cost & & & & & & & & & 644.41 & & 618.44 & & 478.61 \\ \hline
\end{tabular}
\end{small}
\end{table*}
\begin{table}[]
\caption{Cost comparisons for complex mixed SCN instances}
\label{TabComplex}
\setlength\tabcolsep{3pt}
\centering
\begin{small}
\begin{tabular}{ccc}
Instance & DNN-SMIO & DFO \\ \hline
1 & 380.95 & 402.41 \\
2 & 419.13 & 442.42 \\
3 & 407.83 & 408.27 \\
4 & 379.31 & 400.04 \\
5 & 478.61 & 644.41 \\ \hline
average & 426.43 & 446.32 \\
\end{tabular}
\end{small}
\end{table}
We further conduct a cost comparison between the DFO and DNN approaches for 4 new instances based on the complex SCN structure shown in Figure \ref{fig:caseStudy}. Table \ref{TabComplex} shows the cost comparison results for the new instances plus the one already discussed earlier on the structure. We focused on different salvage costs and demand distributions for this comparison. We refer the reader to Table \textcolor{blue}{5} of the supplement for the structure details and suggested OULs. As can be seen, the DNN approach outperforms DFO for all five cases.
\section{Conclusion}
\label{Conclusion}
This research studies simultaneous decision-making for stochastic multi-echelon inventory optimization with arbitrary SCN topologies, demand distributions, and cost structures, using deep neural networks as decision makers, considering a finite-horizon. We introduce pairwise modeling of SMEIOs and associate a DNN to each edge in need of decision making. The DNNs constantly interact with their environment (the supply chain network) and aim to learn the OULs minimizing the total network cost. We assume that the demand distribution as well as the inventory levels are known to all agents. Our research is one of the first works considering deep neural networks as joint decision makers in an SMEIO framework that can suggest clear and interpretable OULs as an output.
The findings indicate the effectiveness of the method both in terms of its accuracy compared to analytical exact solutions and versus alternatives such as DFO and Spearmint Bayesian optimization, as well as in terms of the computational expense (interactions with environment) compared to enumeration methods. The convergence curves shown for the single-node newsvendor, serial, and mixed systems validate the stability of the framework. For mixed supply networks with more advanced cost schemes, findings suggest that upstream echelons hold more items compared to their demand means than downstream echelons. For example, the OUL found by DNN-SMEIO for the first echelon $(0\leftarrow 1)$ in the complex SCN shown in Figure \ref{fig:caseStudy} is more than twice the demand it sees.
This study can be extended to cases in which the demands are auto-correlated and/or the decision makers/DNNs have partial information about the structure of the SCN. Another future study can be conducted to investigate closed-loop supply chain networks. In addition, considering time dependent OULs would be a desirable improvement to this method.
\section{Stochastic multi-echelon inventory optimization}
Research on SMEIO can be classified according to several dimensions. The number of echelon layers as well as their topological structure are essential factors. One can categorize SMEIO into single-, two-, and multi-echelon networks with serial, divergent, convergent, or general structures \cite{fosct2e}. Furthermore, inventory review can be continuous or periodic, meaning that the system's inventory levels evolve smoothly over time or update once in each decision period. Demands can be discrete (mainly Poisson \cite{topan2017heuristics}, compound Poisson \cite{johansson2020controlling}, and Erlang \cite{bitton2019joint}), continuous (mainly normal \cite{cobb2016lead}) or general stochastic \cite{sapra2017dual}.
Solution approach is another dimension along which to classify SMEIO. Simulation--optimization \cite{xu2019simulation}, computational and numerical \cite{oroojlooyjadid2017deep}, approximate \cite{topan2017heuristics}, and exact (theoretical) methods \cite{sapra2017dual} are among the most common solution approaches. In most approaches, we impose a particular form of the inventory policy (e.g., a base-stock policy), and an algorithm is used to obtain a near-optimal set of parameters for that policy to optimize the objective function \cite{de2018typology}. This approach is especially prevalent for systems for which the optimal policy structure is unknown, a category which includes distribution systems \cite{federgruen1984approximations} as well as general networks. However, this approach has some limitations. The methods are dependent on the problem settings, and adjustments in the chosen policy or system structure require the algorithm to be initiated once again. We aim to reduce these limitations by using trainable DNN algorithms in which one can transfer the knowledge (learned weights of agents' networks) from a SCN to a modified one.
Most SMEIO models fall either into the category of {\em stochastic service models} (SSM) or of {\em guaranteed service models} (GSM). SSM models allow for the possibility of stockouts at various nodes of the SMEIO system, which then result in stochastic lead times to their downstream nodes. All of the models we have discussed so far are SSM models, as is the approach we propose. In contrast, GSM models impose an upper bound on the demand, which allows all nodes in the system to guarantee delivery within a fixed number of time periods. This guarantee leads to greater tractability (and therefore greater adoption in practice), even for general network topologies, but also requires some strong assumptions, the demand bound chief among them. To the best of our knowledge, no GSM models in the literature can accommodate general holding and stockout cost structures, as our approach can. Finally, we note that the inventory dynamics in SSM and GSM models operate essentially in the same way; the difference lies in how the base-stock levels are optimized, rather than in how the systems are managed. For further discussion of SSM and GSM models, see \cite{GrWi03chapter,fosct2e}.
\section{Summary of Notation}
The notation is summarized in Table~\ref{table:notation} of Supplementary.
\section{Framework Structure, Optimization Algorithm and Hyperparameter Tuning}
\begin{table}
\caption{Notation summary.}
\label{table:notation}
\centerline{
\begin{tabular}{lll}
\hline
\multicolumn{3}{l}{Inputs} \\
& $T$ & Length of the planning horizon \\
& $\mathcal{N}, \mathcal{E}$ & Sets of nodes and edges in SCN \\
& $\mathcal{U}_j$, $\mathcal{D}_j$ & Sets of nodes upstream from and downstream from (respectively) node $j$ \\
& $h_{ij}(\cdot)$ & Holding cost function for items from node $i$ in raw material, finished goods, or in-transit inventory at node $j$ \\
& $p_{ij}(\cdot)$ & Stockout cost function for stockouts at node $i$ that are owed to node $j$ \\
& $L_{jk}$ & Lead time for items shipped from node $j$ to node $k$ \\
\multicolumn{3}{l}{State Variables} \\
& $D_{jkt}$ & Demand placed by node $k$ to node $j$ in period $t$ \\
& $S_{jkt}$ & Number of units shipped from node $j$ to node $k$ in period $t$ \\
& $IL_{jt}$ & Finished-goods inventory level at node $j$ at the end of period $t$ \\
& $IL^r_{jit}$ & Raw-material inventory level at node $j$ for items from node $i$ at the end of period $t$ \\
& $BO_{jkt}$ & Backorders at node $j$ owed to node $k$ at the end of period $t$ \\
& $IT_{jkt}$ & Inventory in transit from node $j$ to node $k$ at the end of period $t$ \\
& $IP_{jit}$ & Inventory position of item-$i$ materials at node $j$ in period $t$ immediately before order is placed \\
\multicolumn{3}{l}{Decision Variables} \\
& $OUL_{ji}$ & Order-up-to level (base-stock level) used by node $j$ when placing orders from node $i$ \\
\hline
\end{tabular}
}
\end{table}
\begin{figure}[]
\centering
\includestandalone[width = 14cm]{GPEI}
\caption{Tuning DNN-SMEIO via GPEI Bayesian optimization}
\label{Hyper}
\end{figure}
In this study, we utilize Spearmint Bayesian optimization, and in particular, the Gaussian process expected improvement (GPEI) method \cite{swersky2013multi}. Figure \ref{Hyper} shows a sketch of GPEI Spearmint. The number of shared layers (if any) between separate networks, the number of hidden layers for each network, the learning rate, the activation function type, the number of nodes in each layer and the size of the mini-batches are considered as the tunable hyperparameters.
Table \ref{tab0} shows the values we considered for the parameters to be tuned. In most of our experiments, the best settings are to use totally separated DNNs (Multi-agent structure = 0) with Softplus activation functions, learning rate 0.01, and 4 hidden layers. The results appear to be quite quite insensitive to the mini-batch size.
\begin{table}
\hspace*{-1cm}
\caption{Hyperparameter values for DNN-SMEIO framework.}
\label{tab0}
\vspace{1em}
\begin{tabular}{clll}
& parameter & possible values & notes \\
\hline
1 & multi-agent structure & \{0,1\}& 0: totally separated, 1: sharing first layers \\
2 & \# hidden layers & \{1,2,3,4,5\} & total number of layers is this number plus 2 \\
3& \# shared layers & \{1,2,3\} & only used if multi-agent structure = 1 \\
4& activation function & softplus, leakyReLU, ReLU &\\
5& \# nodes in each layer & \{8,12,16,24,36,48\} & \\
6& mini-batch size & \{1,2,5,10\} & \\
7& learning rate & \{0.0005,0.001,0.005,0.01,0.02\} &
\end{tabular}
\end{table}
Because the episodes have a finite horizon, the initial values of the state variables can have a significant impact on the training process. Of course, in practice, one would set the initial values to the actual values of the inventory levels at the start of the horizon. For training, however, in each episode we chose to initialize the raw-material inventories to 0 and the finished-goods inventory levels to the lead-time demand. That is, we set $IL_{j0} = \mu_jL_j$ and $IL^r_{ji0} = 0$ for all $j\in \mathcal{N}$ and $i\in \mathcal{U}_j$. The system is therefore ``primed'' with enough inventory to meet the upcoming expected demand.
\section{Details of Assembly SCN Experiments}
The complete specifications of the parameters of the assembly system instances in Section \textcolor{blue}{4.3} of the main article, as well as the OULs suggested by each method, are presented in Tables \ref{tabassem1} and \ref{tabassem2} of the supplement.
\begin{table*}[]
\caption{Assembly(1) SCN cases details}
\label{tabassem1}
\setlength\tabcolsep{4pt}
\centering
\begin{scriptsize}
\begin{tabular}{llllllllll}
Instance 1 & & & & & & & & & \\
& & & & & & \multicolumn{4}{c}{OUL} \\
Node & Predecessor & H & S & LT & Demand & DNNSMIO & CD & Enumeration & DFO \\
1 & -- & 0.25 & 0 & 2 & -- & 26.91 & 25.77 & 26.72 & 6.08 \\
2 & -- & 0.25 & 0 & 2 & -- & 26.8 & 25.77 & 26.72 & 27.32 \\
3 & -- & 0.25 & 0 & 2 & -- & 26.86 & 25.77 & 26.72 & 21.77 \\
4 & -- & 0.25 & 0 & 2 & -- & 26.85 & 25.77 & 26.72 & 26.89 \\
5 & 1 & 0.8 & 0 & 1 & -- & 13.55 & 13.87 & 13.36 & 12.20 \\
5 & 2 & 0.8 & 0 & 1 & -- & 13.57 & 13.87 & 13.36 & 14.75 \\
6 & 3 & 0.8 & 0 & 1 & -- & 13.58 & 13.87 & 13.36 & 12.85 \\
6 & 4 & 0.8 & 0 & 1 & -- & 13.57 & 13.87 & 13.36 & 15.36 \\
7 & 5 & 1.9 & 10 & 1 & N(13,(1.2)\textasciicircum{}2) & 14.64 & 15.08 & 15.16 & 14.41 \\
7 & 6 & 1.9 & 10 & 1 & N(13,(1.2)\textasciicircum{}2) & 14.64 & 15.08 & 15.16 & 14.46 \\
& & & & \multicolumn{2}{l}{Cost} & 40.55 & 40.26 & 40.34 & 233.45 \\
& & & & & & & & & \\
Instance 2 & & & & & & & & & \\
& & & & & & & & & \\
& & & & & & & & & \\
& & & & & & \multicolumn{4}{c}{OUL} \\
Node & Predecessor & H & S & LT & Demand & DNNSMIO & CD & Enumeration & DFO \\
1 & -- & 2 & 0 & 2 & -- & 10.08 & 7.50 & 7.5 & 10.19 \\
2 & -- & 2 & 0 & 2 & -- & 10.08 & 7.50 & 7.5 & 11.83 \\
3 & -- & 2 & 0 & 2 & -- & 10.13 & 7.50 & 7.5 & 1.00 \\
4 & -- & 2 & 0 & 2 & -- & 10.12 & 7.50 & 7.5 & 11.15 \\
5 & 1 & 4 & 0 & 1 & -- & 5.42 & 6.04 & 3.75 & 4.63 \\
5 & 2 & 4 & 0 & 1 & -- & 5.42 & 6.04 & 3.75 & 4.67 \\
6 & 3 & 4 & 0 & 1 & -- & 5.38 & 6.04 & 3.75 & 1.46 \\
6 & 4 & 4 & 0 & 1 & -- & 5.33 & 6.04 & 3.75 & 4.62 \\
7 & 5 & 7 & 37.12 & 1 & N(5,(1)\textasciicircum{}2) & 6.54 & 8.58 & 10.69 & 6.81 \\
7 & 6 & 7 & 37.12 & 1 & N(5,(1)\textasciicircum{}2) & 6.59 & 8.58 & 10.69 & 6.99 \\
& & & & \multicolumn{2}{l}{Cost} & 103.77 & 101.59 & 101.47 & 482.63 \\
& & & & & & & & & \\
Instance 3 & & & & & & & & & \\
& & & & & & & & & \\
& & & & & & \multicolumn{4}{c}{OUL} \\
Node & Predecessor & H & S & LT & Demand & DNNSMIO & CD & Enumeration & DFO \\
1 & -- & 0.4 & 0 & 2 & -- & 40.98 & 36.67 & 35.55 & 39.66 \\
2 & -- & 0.4 & 0 & 2 & -- & 40.99 & 36.67 & 35.55 & 22.78 \\
3 & -- & 0.4 & 0 & 2 & -- & 41.03 & 36.67 & 35.55 & 12.47 \\
4 & -- & 0.4 & 0 & 2 & -- & 41.07 & 36.67 & 35.55 & 46.63 \\
5 & 1 & 0.9 & 0 & 2 & -- & 42.83 & 45.19 & 41.11 & 39.61 \\
5 & 2 & 0.9 & 0 & 2 & -- & 43.29 & 45.19 & 41.11 & 42.00 \\
6 & 3 & 0.9 & 0 & 2 & -- & 43.26 & 45.19 & 41.11 & 42.41 \\
6 & 4 & 0.9 & 0 & 2 & -- & 42.17 & 45.19 & 41.11 & 40.04 \\
7 & 5 & 2.1 & 15 & 2 & N(20,(3)\textasciicircum{}2) & 46.19 & 47.85 & 52.22 & 47.02 \\
7 & 6 & 2.1 & 15 & 2 & N(20,(3)\textasciicircum{}2) & 46.14 & 47.85 & 52.22 & 48.82 \\
& & & & \multicolumn{2}{l}{Cost} & 163.15 & 161.30 & 161.13 & 441.43 \\
& & & & & & & & & \\
Instance 4 & & & & & & & & & \\
& & & & & & & & & \\
& & & & & & \multicolumn{4}{c}{OUL} \\
Node & Predecessor & H & S & LT & Demand & DNNSMIO & CD & Enumeration & DFO \\
1 & -- & 0.3 & 0 & 2 & -- & 10.12 & 9.21 & 10.27 & 1.26 \\
2 & -- & 0.3 & 0 & 2 & -- & 10.19 & 9.21 & 10.2 & 7.59 \\
3 & -- & 0.3 & 0 & 2 & -- & 10.73 & 9.21 & 10.27 & 9.89 \\
4 & -- & 0.3 & 0 & 2 & -- & 10.21 & 9.21 & 10.27 & 1.60 \\
5 & 1 & 0.8 & 0 & 2 & -- & 12.74 & 11.55 & 10.27 & 8.06 \\
5 & 2 & 0.8 & 0 & 2 & -- & 12.38 & 11.55 & 10.27 & 7.26 \\
6 & 3 & 0.8 & 0 & 2 & -- & 12.55 & 11.55 & 10.27 & 5.27 \\
6 & 4 & 0.8 & 0 & 2 & -- & 12.20 & 11.55 & 10.27 & 7.94 \\
7 & 5 & 2 & 15 & 2 & N(5,(1)\textasciicircum{}2) & 12.25 & 12.62 & 13.05 & 15.52 \\
7 & 6 & 2 & 15 & 2 & N(5,(1)\textasciicircum{}2) & 12.25 & 12.62 & 13.05 & 13.81 \\
& & & & \multicolumn{2}{l}{Cost} & 37.49 & 35.97 & 35.98 & 139.77 \\
& & & & & & & & & \\
Instance 5 & & & & & & & & & \\
& & & & & & \multicolumn{4}{c}{OUL} \\
Node & Predecessor & H & S & LT & Demand & DNNSMIO & CD & Enumeration & DFO \\
1 & -- & 0.5 & 0 & 1 & -- & 5.01 & 4.38 & 3.75 & 8.79 \\
2 & -- & 0.5 & 0 & 1 & -- & 5.02 & 4.38 & 3.75 & 7.35 \\
3 & -- & 0.5 & 0 & 1 & -- & 5.11 & 4.38 & 3.75 & 8.91 \\
4 & -- & 0.5 & 0 & 1 & -- & 5.06 & 4.38 & 3.75 & 5.23 \\
5 & 1 & 1 & 0 & 1 & -- & 7.04 & 6.23 & 6.52 & 3.07 \\
5 & 2 & 1 & 0 & 1 & -- & 7.07 & 6.23 & 6.52 & 5.71 \\
6 & 3 & 1 & 0 & 1 & -- & 6.98 & 6.23 & 6.52 & 3.08 \\
6 & 4 & 1 & 0 & 1 & -- & 6.91 & 6.23 & 6.52 & 5.74 \\
7 & 5 & 3 & 16 & 1 & N(5,(1)\textasciicircum{}2) & 6.33 & 6.49 & 6.52 & 5.75 \\
7 & 6 & 3 & 16 & 1 & N(5,(1)\textasciicircum{}2) & 6.23 & 6.49 & 6.52 & 6.38 \\
& & & & \multicolumn{2}{l}{Cost} & 29.04 & 27.53 & 27.45 & 36.03 \\
& & & & & & & & &
\end{tabular}
\end{scriptsize}
\end{table*}
\begin{table*}[]
\caption{Assembly(2) SCN cases details}
\label{tabassem2}
\setlength\tabcolsep{4pt}
\centering
\begin{scriptsize}
\begin{tabular}{llllllllll}
Instance 1 & & & & & & & & & \\
& & & & & & \multicolumn{4}{c}{Suggested OUL} \\
Node & Predecessor & H & S & LT & Demand & DNNSMIO & CD & Enumeration & DFO \\
1 & -- & 2 & 0 & 1 & -- & 5.23 & 3.97 & 3.75 & 6.76 \\
2 & -- & 2 & 0 & 1 & -- & 5.01 & 3.97 & 3.75 & 6.53 \\
3 & -- & 2 & 0 & 1 & -- & 5.22 & 3.97 & 3.75 & 6.80 \\
4 & -- & 2 & 0 & 1 & -- & 4.96 & 3.97 & 3.75 & 6.80 \\
5 & -- & 2 & 0 & 1 & -- & 4.92 & 3.97 & 3.75 & 5.93 \\
6 & 4 & 4 & 0 & 1 & -- & 5.9 & 5.49 & 5.83 & 0.80 \\
6 & 5 & 4 & 0 & 1 & -- & 6.31 & 5.49 & 5.83 & 5.92 \\
7 & 1 & 7 & 40 & 1 & N(5,(1)\textasciicircum{}2) & 5.89 & 7.77 & 7.91 & 6.64 \\
7 & 2 & 7 & 40 & 1 & N(5,(1)\textasciicircum{}2) & 6.97 & 7.77 & 7.91 & 5.75 \\
7 & 3 & 7 & 40 & 1 & N(5,(1)\textasciicircum{}2) & 5.82 & 7.77 & 7.91 & 6.10 \\
7 & 6 & 7 & 40 & \multicolumn{2}{l}{1} & 6.33 & 7.77 & 7.91 & 6.27 \\
& & & & & Cost & 93.9432 & 90.4087 & 90.5432 & 116.38 \\
& & & & & & & & & \\
Instance 2 & & & & & & & & & \\
& & & & & & \multicolumn{4}{c}{Suggested OUL} \\
Node & Predecessor & H & S & LT & Demand & DNNSMIO & CD & Enumeration & DFO \\
1 & -- & 0.3 & 0 & 1 & -- & 9.93 & 8.98 & 7.5 & 10.89 \\
2 & -- & 0.3 & 0 & 1 & -- & 9.89 & 8.98 & 7.5 & 10.49 \\
3 & -- & 0.3 & 0 & 1 & -- & 9.98 & 8.98 & 7.5 & 11.01 \\
4 & -- & 0.3 & 0 & 1 & -- & 9.86 & 8.98 & 7.5 & 11.24 \\
5 & -- & 0.3 & 0 & 1 & -- & 9.87 & 8.98 & 7.5 & 10.79 \\
6 & 4 & 0.5 & 0 & 1 & -- & 10.99 & 10.53 & 10.27 & 5.59 \\
6 & 5 & 0.5 & 0 & 1 & -- & 11.39 & 10.53 & 10.27 & 10.47 \\
7 & 1 & 0.9 & 3.5 & 1 & N(10,(1)\textasciicircum{}2) & 12.47 & 12.54 & 14.44 & 10.88 \\
7 & 2 & 0.9 & 3.5 & 1 & N(10,(1)\textasciicircum{}2) & 12.31 & 12.54 & 14.44 & 11.25 \\
7 & 3 & 0.9 & 3.5 & 1 & N(10,(1)\textasciicircum{}2) & 12.5 & 12.54 & 14.44 & 11.12 \\
7 & 6 & 0.9 & 3.5 & 1 & N(10,(1)\textasciicircum{}2) & 11.85 & 12.54 & 14.44 & 11.28 \\
& & & & & Cost & 23.00 & 22.43 & 22.48 & 25.75 \\
& & & & & & & & & \\
Instance 3 & & & & \multicolumn{2}{l}{} & & & & \\
& & & & & & \multicolumn{4}{c}{Suggested OUL} \\
Node & Predecessor & H & S & LT & Demand & DNNSMIO & CD & Enumeration & DFO \\
1 & -- & 0.5 & 0 & 1 & -- & 10.7 & 11.1868 & 10.27 & 11.57 \\
2 & -- & 0.5 & 0 & 1 & -- & 10.43 & 11.1868 & 10.27 & 19.27 \\
3 & -- & 0.5 & 0 & 1 & -- & 10.71 & 11.1868 & 10.27 & 11.59 \\
4 & -- & 0.5 & 0 & 1 & -- & 10.27 & 11.1868 & 10.27 & 13.77 \\
5 & -- & 0.5 & 0 & 1 & -- & 10.24 & 11.1868 & 10.27 & 13.63 \\
6 & 4 & 3 & 0 & 1 & -- & 11.06 & 11.1868 & 10.27 & 5.71 \\
6 & 5 & 3 & 0 & 1 & -- & 11.07 & 11.1868 & 10.27 & 11.56 \\
7 & 1 & 6 & 25 & 1 & N(10,(2)\textasciicircum{}2) & 11.54 & 12.4788 & 13.05 & 11.81 \\
7 & 2 & 6 & 25 & 1 & N(10,(2)\textasciicircum{}2) & 11.94 & 12.4788 & 13.05 & 11.78 \\
7 & 3 & 6 & 25 & 1 & N(10,(2)\textasciicircum{}2) & 10.87 & 12.4788 & 13.05 & 11.77 \\
7 & 6 & 6 & 25 & 1 & N(10,(2)\textasciicircum{}2) & 11.62 & 12.4788 & 13.05 & 11.90 \\
& & & & & Cost & 86.61 & 82.67 & 82.32 & 90.71 \\
Instance 4 & & & & & & & & & \\
& & & & & & \multicolumn{4}{c}{Suggested OUL} \\
Node & Predecessor & H & S & \multicolumn{2}{l}{LT} & DNNSMIO & CD & Enumeration & DFO \\
1 & -- & 0.6 & 0 & 1 & -- & 7.26 & 6.2706 & 5.25 & 7.47 \\
2 & -- & 0.6 & 0 & 1 & -- & 7.16 & 6.2706 & 5.25 & 7.95 \\
3 & -- & 0.6 & 0 & 1 & -- & 7.1 & 6.2706 & 5.25 & 6.10 \\
4 & -- & 0.6 & 0 & 1 & -- & 6.93 & 6.2706 & 5.25 & 5.57 \\
5 & -- & 0.6 & 0 & 1 & -- & 6.88 & 6.2706 & 5.25 & 7.52 \\
6 & 4 & 1.1 & 0 & 1 & -- & 7.72 & 7.53 & 7.19 & 10.76 \\
6 & 5 & 1.1 & 0 & 1 & -- & 8.02 & 7.53 & 7.19 & 5.42 \\
7 & 1 & 2.5 & 5.4 & 1 & N(7,(1)\textasciicircum{}2) & 8.22 & 8.72 & 10.11 & 7.60 \\
7 & 2 & 2.5 & 5.4 & 1 & N(7,(1)\textasciicircum{}2) & 7.87 & 8.72 & 10.11 & 7.87 \\
7 & 3 & 2.5 & 5.4 & 1 & N(7,(1)\textasciicircum{}2) & 7.73 & 8.72 & 10.11 & 8.18 \\
7 & 6 & 2.5 & 5.4 & 1 & N(7,(1)\textasciicircum{}2) & 7.43 & 8.72 & 10.11 & 9.30 \\
& & & & & Cost & 34.62 & 34.04 & 34.17 & 42.35 \\
& & & & & & & & & \\
Instance 5 & & & & & & & & & \\
& & & & & & \multicolumn{4}{c}{Suggested OUL} \\
Node & Predecessor & H & S & LT & Demand & DNNSMIO & CD & Enumeration & DFO \\
1 & -- & 0.25 & 0 & \multicolumn{2}{l}{1} & 12.11 & 10.55 & 8.25 & 14.88 \\
2 & -- & 0.25 & 0 & 1 & -- & 11.97 & 10.55 & 8.25 & 14.55 \\
3 & -- & 0.25 & 0 & 1 & -- & 11.9 & 10.55 & 8.25 & 14.48 \\
4 & -- & 0.25 & 0 & 1 & -- & 11.5 & 10.55 & 8.25 & 4.95 \\
5 & -- & 0.25 & 0 & 1 & -- & 11.59 & 10.55 & 8.25 & 14.22 \\
6 & 4 & 0.59 & 0 & 1 & -- & 13.52 & 12.62 & 12.83 & 2.07 \\
6 & 5 & 0.59 & 0 & 1 & -- & 13.59 & 12.62 & 12.83 & 13.95 \\
7 & 1 & 0.88 & 49.7 & 1 & N(11,(2)\textasciicircum{}2) & 14.86 & 18.44 & 20.47 & 15.89 \\
7 & 2 & 0.88 & 49.7 & 1 & N(11,(2)\textasciicircum{}2) & 14.49 & 18.44 & 20.47 & 15.05 \\
7 & 3 & 0.88 & 49.7 & 1 & N(11,(2)\textasciicircum{}2) & 14.51 & 18.44 & 20.47 & 14.55 \\
7 & 6 & 0.88 & 49.7 & 1 & N(11,(2)\textasciicircum{}2) & 14.29 & 18.44 & 20.47 & 17.67 \\
& & & & & Cost & 30.98 & 28.19 & 27.96 & 62.76
\end{tabular}
\end{scriptsize}
\end{table*}
\begin{figure*}
\hspace*{-2.5cm}
\scalebox{0.45}{\input{CaseStudyResults2.pgf}}
\caption{Total cost and OUL decisions for the case study SCN.}
\label{fig:CaseStudyResult}
\end{figure*}
\section{Complex SCN Learning Curve and Extra Discussion}
Figure \ref{fig:CaseStudyResult} illustrates the learning process and the optimal values for all 13 OULs to be optimized. The jumps in the subplots are related to the restarting procedure discussed in the previous section. We stop the procedure simply when there is no gain anymore in terms of objective function values. The best cost values found before restarting the training procedures were 626.17, 479.68, and 478.61. Therefore, we stop restarting the training procedure after two retrainings. Figure \ref{fig:CaseStudyResult} provides the following insights:
\begin{itemize}
\item When the cost structures are more complex such as nonlinear (potentially very large) shortage penalties, the optimal solutions are biased towards ordering extra items in the upstream echelons. DNN-SMEIO sets the $(0\leftarrow1)$ OUL to $101.44$, while the average lead-time demand that node $1$ realizes is only $3\times 15.08=45.24$. The difference between the two values is considerable, and if a practitioner without this knowledge uses enumeration or an {\em ad hoc} approach, it is probable that the best value found by the DNN-SMEIO method would not even be in the interval selected by the practitioner.
\item In consensus with what we observed in the previous examples, the upstream echelons learn their OUL values more quickly than those downstream.
\end{itemize}
\section{Complex SCN Cost Comparison}
Table \ref{tabCostComparison} shows the structure details of the different experiments done for the Complex SCN studied in Section \textcolor{blue}{4.5} of the main paper as well as suggested OULs by DNNSMIO and DFO. H, local holding cost coefficients, stockout cost coefficients and salvage reward coefficients
\begin{table*}[]
\caption{Cost comparison cases: structure details and suggested OULs}
\label{tabCostComparison}
\hspace*{-1.5cm}
\setlength\tabcolsep{1.5pt}
\centering
\begin{scriptsize}
\begin{tabular}{lllllllll}
Instance 1 & & & & & & & & \\
& & & & & & & Suggested OUL & Suggested OUL \\
Node & Predecessor & H & S & R & LT & Demand & DNNSMIO & DFO \\
1 & -- & 2 & 4 & 1.25 & 1 & -- & 49.25 & 52.22203 \\
2 & 1 & 4-3 & 12-4 & 1.5 & 1 & -- & 17.75 & 18.48301 \\
3 & 1 & 4-3 & 12-4 & 1.5 & 1 & -- & 17.45 & 18.42489 \\
4 & 1 & 4-3 & 12-4 & 1.5 & 1 & -- & 17.36 & 18.40821 \\
5 & 2 & 7-6 & 36-12 & 1.25 & 1 & -- & 7.3 & 5.34762 \\
5 & 3 & 7-6 & 36-12 & 1.25 & 1 & -- & 6.7 & 5.29257 \\
5 & 4 & 7-6 & 36-12 & 1.25 & 1 & -- & 6.79 & 5.33311 \\
6 & 2 & 7-6 & 36-12 & 1.25 & 1 & -- & 7.55 & 5.32846 \\
6 & 3 & 7-6 & 36-12 & 1.25 & 1 & -- & 6.8 & 5.31856 \\
6 & 4 & 7-6 & 36-12 & 1.25 & 1 & -- & 6.99 & 5.30957 \\
7 & 2 & 7-6 & 36-12 & 1.25 & 1 & N(5,(1)\textasciicircum{}2) & 7.92 & 5.30319 \\
7 & 3 & 7-6 & 36-12 & 1.25 & 1 & N(5,(1)\textasciicircum{}2) & 6.64 & 5.30318 \\
7 & 4 & 7-6 & 36-12 & 1.25 & 1 & N(5,(1)\textasciicircum{}2) & 7.21 & 5.28193 \\
& & & & & & cost & 380.95 & 402.41 \\
& & & & & & & & \\
Instance 2 & & & & & & & & \\
& & & & & & & Suggested OUL & Suggested OUL \\
Node & Predecessor & H & S & R & LT & Demand & DNNSMIO & DFO \\
1 & -- & 2 & 4 & 2 & 1 & -- & 56.53 & 57.33026 \\
2 & 1 & 4-3 & 12-4 & 2 & 1 & -- & 18.64 & 20.50994 \\
3 & 1 & 4-3 & 12-4 & 2 & 1 & -- & 19.96 & 18.88558 \\
4 & 1 & 4-3 & 12-4 & 2 & 1 & -- & 19.07 & 19.12507 \\
5 & 2 & 7-6 & 36-12 & 2 & 1 & -- & 7.8 & 7.24254 \\
5 & 3 & 7-6 & 36-12 & 2 & 1 & -- & 6.91 & 4.22521 \\
5 & 4 & 7-6 & 36-12 & 2 & 1 & -- & 12.21 & 6.3352 \\
6 & 2 & 7-6 & 36-12 & 2 & 1 & -- & 8.36 & 6.94252 \\
6 & 3 & 7-6 & 36-12 & 2 & 1 & -- & 7.56 & 4.40232 \\
6 & 4 & 7-6 & 36-12 & 2 & 1 & -- & 11.83 & 7.46706 \\
7 & 2 & 7-6 & 36-12 & 2 & 1 & N(6,(1)\textasciicircum{}2) & 8.19 & 7.31975 \\
7 & 3 & 7-6 & 36-12 & 2 & 1 & N(4,(1)\textasciicircum{}2) & 5.97 & 4.39467 \\
7 & 4 & 7-6 & 36-12 & 2 & 1 & N(6,(2)\textasciicircum{}2) & 11.8 & 7.27793 \\
& & & & & & cost & 419.13 & 442.42 \\
& & & & & & & & \\
Instance 3 & & & & & & & & \\
& & & & & & & Suggested OUL & Suggested OUL \\
Node & Predecessor & H & S & R & LT & Demand & DNNSMIO & DFO \\
1 & -- & 2 & 4 & 1 & 1 & -- & 53.73 & 56.46892 \\
2 & 1 & 4-3 & 12-4 & 2 & 1 & -- & 18.87 & 19.30803 \\
3 & 1 & 4-3 & 12-4 & 2 & 1 & -- & 19.43 & 19.16313 \\
4 & 1 & 4-3 & 12-4 & 2 & 1 & -- & 18.75 & 19.61025 \\
5 & 2 & 7-6 & 36-12 & 1 & 1 & -- & 6.67 & 5.41518 \\
5 & 3 & 7-6 & 36-12 & 1 & 1 & -- & 7.28 & 5.61027 \\
5 & 4 & 7-6 & 36-12 & 1 & 1 & -- & 12.64 & 7.17493 \\
6 & 2 & 7-6 & 36-12 & 1 & 1 & -- & 7.49 & 5.40836 \\
6 & 3 & 7-6 & 36-12 & 1 & 1 & -- & 8.19 & 5.44619 \\
6 & 4 & 7-6 & 36-12 & 1 & 1 & -- & 10.67 & 6.90065 \\
7 & 2 & 7-6 & 36-12 & 1 & 1 & N(5,(1)\textasciicircum{}2) & 7.55 & 5.90935 \\
7 & 3 & 7-6 & 36-12 & 1 & 1 & N(5,(1.2)\textasciicircum{}2) & 8.59 & 4.75065 \\
7 & 4 & 7-6 & 36-12 & 1 & 1 & N(6,(1.5)\textasciicircum{}2) & 9.25 & 7.14491 \\
& & & & & & cost & 407.83 & 408.27 \\
& & & & & & & & \\
Instance 4 & & & & & & & & \\
& & & & & & & Suggested OUL & Suggested OUL \\
Node & Predecessor & H & S & R & LT & Demand & DNNSMIO & DFO \\
1 & -- & 2 & 4 & 2.5 & 1 & -- & 48.56 & 52.62784 \\
2 & 1 & 4-3 & 12-4 & 2.5 & 1 & -- & 17.56 & 17.88147 \\
3 & 1 & 4-3 & 12-4 & 2.5 & 1 & -- & 16.9 & 17.9517 \\
4 & 1 & 4-3 & 12-4 & 2.5 & 1 & -- & 17.62 & 16.90722 \\
5 & 2 & 7-6 & 36-12 & 2.5 & 1 & -- & 7.11 & 5.15377 \\
5 & 3 & 7-6 & 36-12 & 2.5 & 1 & -- & 7.85 & 7.34585 \\
5 & 4 & 7-6 & 36-12 & 2.5 & 1 & -- & 7.22 & 4.16108 \\
6 & 2 & 7-6 & 36-12 & 2.5 & 1 & -- & 7.19 & 5.48897 \\
6 & 3 & 7-6 & 36-12 & 2.5 & 1 & -- & 8.39 & 7.18886 \\
6 & 4 & 7-6 & 36-12 & 2.5 & 1 & -- & 6.39 & 4.39452 \\
7 & 2 & 7-6 & 36-12 & 2.5 & 1 & N(5,(1)\textasciicircum{}2) & 7.64 & 5.83787 \\
7 & 3 & 7-6 & 36-12 & 2.5 & 1 & N(6,(1)\textasciicircum{}2) & 8.75 & 6.75788 \\
7 & 4 & 7-6 & 36-12 & 2.5 & 1 & N(4,(1.5)\textasciicircum{}2) & 6.25 & 4.72713 \\
& & & & & & cost & 379.31 & 400.047 \\
& & & & & & & & \\
& & & & & & & &
\end{tabular}
\end{scriptsize}
\end{table*}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,900
|
{"url":"https:\/\/math.stackexchange.com\/questions\/546318\/prove-that-the-order-of-any-element-in-the-additive-group-of-integers-modulo-n-i","text":"Prove that the order of any element in the additive group of integers modulo n is a divisor of n.\n\nI have been on and off of this problem for three days and need to present the proof tomorrow. I am thinking that because I know for any element in the additive group of integers modulo n the order for that element is the ratio of n and the greatest common factor of that element and n then I can say,[...chirp, chirp, chirp...] and my mind goes blank. What can I say? I have also tried using the division algorithm to say the order of any element in the group can be written as a multiple of n and some other integer but I am coming up blank. There is something I do not fully understand and would like for you to point out my oversight. Thanks.\n\n\u2022 Lol, \"chirp chirp.\" You're funny! \u2013\u00a0Shine On You Crazy Diamond Oct 31 '13 at 0:57\n\u2022 This is just a direct application of Lagrange's theorem. \u2013\u00a0lhf Oct 31 '13 at 0:59\n\n$\\Bbb Z_n$ is a group and thus if $a\\in \\Bbb Z_n$ then $<a>$ is a cyclic subgroup of $\\Bbb Z_n$ with order equal to the order of the element $a$. By Lagrange's Theorem we have that $|\\Bbb Z_n|=[\\Bbb Z_n:<a>]|<a>|$ and thus $|<a>||n$\n\nLet $H$ be a subgroup of $G$. Let's count the cosets of $H$. They partition G. Look that up and see what it means. And there's a bijection between any two. Proof that. I'll wait...\n\nokay. So there's a bijection between $H$ and any other coset $aH$, therefore there are $k$ cosets of size $|H|$ (the size of the set $H$ sometimes called order of $H$ ), and since they partition $G$ you have $|H|k = |G|$. $k$ is usually denoted $[G : H]$ and called the index of $H$ in $G$. It's equal to the size of the group $G\/H$ when $H$ is normal.\n\nSo apply this result to any subgroups of $G$ including the cyclic ones you generate when you continually add an element to itself: $a, 2a, 3a, ..., na = 0$, so $n$ the order is equal to the size of that cyclic subgroup $H$. Apply theorem above.\n\nLet $G$ be any finite group and $g\\in G$, $g\\not=e$. Then the sequence $\\{g^n\\}_{n=1}^\\infty$ is an infinite subset of $G$ and it also is a subgroup of $G$. Since the group $G$ is finite, there must be integers $m$ and $n$ so that $g^m = g^n$. Hence there is a positive power $p$ so that $g^p = e$. The least such power is the order of $g$, and it is the order of the subgroup of $g$ defined by this sequence.\n\nBut the order of any subgroup of a finite group divides the order of the entire group. Hence $o(g)\\,\\, |\\,\\, |G|$.\n\nLet the order of our element $a$ be $m$. That means that $ma$ ($a$ added to itself $m$ times) is divisible by $n$, and there is no positive integer $k\\lt m$ such that $n$ divides $ka$.\n\nBy the Division Algorithm, we have $$n=qm+r$$ for some $r$, where $0\\le r\\lt m$.\n\nThus $na=q(ma)+ra$. Since $ma$ is divisible by $n$, we have that $n$ divides $q(ma)$, and therefore $n$ divides $ra$.\n\nThus $a$ added to itself $r$ times is the $0$-element of the group. If $r\\gt 0$, that contradicts the fact that $a$ has order $m$, since $m$ is the smallest positive integer such that $ma$ is the $0$-element of the group.\n\nThus $r=0$, and therefore $n=qm$, meaning that $m$ divides $n$.","date":"2019-12-08 19:33:48","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8659049272537231, \"perplexity\": 51.88845811379356}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-51\/segments\/1575540514475.44\/warc\/CC-MAIN-20191208174645-20191208202645-00411.warc.gz\"}"}
| null | null |
Court Orders Production of Backup Tapes and Provides for Hard Drive Access Based on Failure to Preserve Evidence
Renda Marine, Inc. v. United States, 58 Fed.Cl. 57 (2003)
Plaintiff moved to compel defendant to produce e-mail and related documents, asserting that, based on defendant's own admission, defendant did not search any hard drives or back-up tapes in preparing its response to plaintiff's document production requests. The requests specifically asked for back-up tapes. In addition, plaintiff sought access to the hard drive of a key player, based upon his admission that it is his practice to delete emails after sending or responding to them. 2003 WL 22427413, at *2.
As a threshold matter, the court discussed the duty to preserve evidence and when that duty first attaches. It noted that the defendant's records management policy "appears to the court to be inconsistent with its legal obligations to preserve evidence." Id. at *4. The policy provided that, after email has been read, it should be deleted or moved to a personal folder as soon as possible. The court stated it did not "believe that a records retention policy which is inconsistent with a party's obligations to a potential or actual adversary in litigation operates to excuse the party's failure to respond to discovery." Id. It determined that the trigger date for the defendant's duty to preserve was March 8, 2000, and concluded:
Accordingly, in view of Mr. Benero's practice of deleting relevant e-mail documents which continued even after the filing of this action, the court directs the defendant to produce at its expense those back-up tapes that were created on and after March 8, 2000 and to provide plaintiff with access to Mr. Benero's hard drive in accordance with the terms of the Protective Order Regarding Electronic Discovery filed with the court on July 29, 2003. Defendant shall also produce to plaintiff those back-up tapes that pre-date March 8, 2000 at plaintiff's expense and in accordance with the terms of the Protective Order Regarding Electronic Discovery filed with the court on July 29, 2003.
Id. at *5.
Court Finds Preservation Efforts Deficient
Court Abused Discretion by Allowing Direct Access to Databases Sans Evidence of Improper Conduct
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,887
|
Q: Facebook js sdk iPhone login I'm trying to login on my website with Facebook JS API from my iPhone. The login works fine on computer as well as iPad.
Here's the login screen:
On iPhone however, I get to a screen which looks like this:
When I close the Facebook screen with the "Back to previous page" message and reload the login page, I'm logged in successfully. It looks like the problem is, that iPhone is unable to close the login pop-up screen.
Here's my JavaScript code:
<div id="fb-root"></div>
<script type="text/javascript">
window.fbAsyncInit = function() {
FB.init({
appId : '', // the app id
status : true,
cookie : true,
xfbml : true
});
FB.getLoginStatus(function(response) {
if (response.status == "connected") {
// do something
}
});
FB.Event.subscribe("auth.login", function(response) {
if (response.status == "connected") {
// do something
}
});
};
function facebookRedirectLogin(redirect) {
// do something
}
function logout() {
FB.logout(function(response) {
// do something
});
}
(function(d){
var js, id = 'facebook-jssdk'; if (d.getElementById(id)) {return;}
js = d.createElement('script'); js.id = id; js.async = true;
js.src = "//connect.facebook.net/en_US/all.js";
d.getElementsByTagName('head')[0].appendChild(js);
}(document));
</script>
<div class="fb-login-button" style="margin-bottom: 20px;">Forbind med Facebook</div>
How can I fix this problem?
A: Checkout this URL,
please do the settings as mentioned in the above URL, this code is working in mobile devices.
<html>
<head>
<title>My Facebook Login Page</title>
</head>
<body>
<div id="fb-root"></div>
<div id="login">
<p><button onClick="loginUser();">Login</button></p>
</div>
<div id="logout">
<p><button onClick="FB.logout();">Logout</button></p>
</div>
<script>
function loginUser() {
FB.login(function(response) { }, {scope:'email'});
}
</script>
<script>
(function() {
var e = document.createElement('script'); e.async = true;
e.src = document.location.protocol + '//connect.facebook.net/en_US/all.js';
document.getElementById('fb-root').appendChild(e);
}());
</script>
<script>
window.fbAsyncInit = function() {
FB.init({ appId: 'xxxxxxxxxxxx',
status: true,
cookie: true,
xfbml: true,
oauth: true});
FB.Event.subscribe('auth.authResponseChange', handleResponseChange);
};
</script>
<script>
function handleReponseChange(response) {
document.body.className = response.authResponse ? 'connected' : 'not_connected';
if (response.authResponse) {
alert(response);
}
}
</script>
</body>
A: I have the exact same problem. This may (or may not) be an open Facebook bug: https://developers.facebook.com/bugs/280811108698311
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,447
|
# Copyright 2012, Red Hat, Inc.
#
# 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.
"""
Client side of the compute RPC API.
"""
from traffic import exception
from traffic import flags
from traffic.openstack.common import jsonutils
from traffic.openstack.common import rpc
import traffic.openstack.common.rpc.proxy
FLAGS = flags.FLAGS
def _compute_topic(topic, ctxt, host, instance):
'''Get the topic to use for a message.
:param topic: the base topic
:param ctxt: request context
:param host: explicit host to send the message to.
:param instance: If an explicit host was not specified, use
instance['host']
:returns: A topic string
'''
if not host:
if not instance:
raise exception.TrafficException(_('No compute host specified'))
host = instance['host']
if not host:
raise exception.TrafficException(_('Unable to find host for '
'Instance %s') % instance['uuid'])
return rpc.queue_get_for(ctxt, topic, host)
class ComputeAPI(traffic.openstack.common.rpc.proxy.RpcProxy):
'''Client side of the compute rpc API.
API version history:
1.0 - Initial version.
1.1 - Adds get_host_uptime()
1.2 - Adds check_can_live_migrate_[destination|source]
1.3 - Adds change_instance_metadata()
1.4 - Remove instance_uuid, add instance argument to reboot_instance()
1.5 - Remove instance_uuid, add instance argument to pause_instance(),
unpause_instance()
1.6 - Remove instance_uuid, add instance argument to suspend_instance()
1.7 - Remove instance_uuid, add instance argument to
get_console_output()
1.8 - Remove instance_uuid, add instance argument to
add_fixed_ip_to_instance()
1.9 - Remove instance_uuid, add instance argument to attach_volume()
1.10 - Remove instance_id, add instance argument to
check_can_live_migrate_destination()
1.11 - Remove instance_id, add instance argument to
check_can_live_migrate_source()
1.12 - Remove instance_uuid, add instance argument to confirm_resize()
1.13 - Remove instance_uuid, add instance argument to detach_volume()
1.14 - Remove instance_uuid, add instance argument to finish_resize()
1.15 - Remove instance_uuid, add instance argument to
finish_revert_resize()
1.16 - Remove instance_uuid, add instance argument to get_diagnostics()
1.17 - Remove instance_uuid, add instance argument to get_vnc_console()
1.18 - Remove instance_uuid, add instance argument to inject_file()
1.19 - Remove instance_uuid, add instance argument to
inject_network_info()
1.20 - Remove instance_id, add instance argument to
post_live_migration_at_destination()
1.21 - Remove instance_uuid, add instance argument to
power_off_instance() and stop_instance()
1.22 - Remove instance_uuid, add instance argument to
power_on_instance() and start_instance()
1.23 - Remove instance_id, add instance argument to
pre_live_migration()
1.24 - Remove instance_uuid, add instance argument to
rebuild_instance()
1.25 - Remove instance_uuid, add instance argument to
remove_fixed_ip_from_instance()
1.26 - Remove instance_id, add instance argument to
remove_volume_connection()
1.27 - Remove instance_uuid, add instance argument to
rescue_instance()
1.28 - Remove instance_uuid, add instance argument to reset_network()
1.29 - Remove instance_uuid, add instance argument to resize_instance()
1.30 - Remove instance_uuid, add instance argument to resume_instance()
1.31 - Remove instance_uuid, add instance argument to revert_resize()
1.32 - Remove instance_id, add instance argument to
rollback_live_migration_at_destination()
1.33 - Remove instance_uuid, add instance argument to
set_admin_password()
1.34 - Remove instance_uuid, add instance argument to
snapshot_instance()
1.35 - Remove instance_uuid, add instance argument to
unrescue_instance()
1.36 - Remove instance_uuid, add instance argument to
change_instance_metadata()
1.37 - Remove instance_uuid, add instance argument to
terminate_instance()
1.38 - Changes to prep_resize():
- remove instance_uuid, add instance
- remove instance_type_id, add instance_type
- remove topic, it was unused
1.39 - Remove instance_uuid, add instance argument to run_instance()
1.40 - Remove instance_id, add instance argument to live_migration()
1.41 - Adds refresh_instance_security_rules()
1.42 - Add reservations arg to prep_resize(), resize_instance(),
finish_resize(), confirm_resize(), revert_resize() and
finish_revert_resize()
1.43 - Add migrate_data to live_migration()
1.44 - Adds reserve_block_device_name()
2.0 - Remove 1.x backwards compat
2.1 - Adds orig_sys_metadata to rebuild_instance()
2.2 - Adds slave_info parameter to add_aggregate_host() and
remove_aggregate_host()
'''
#
# NOTE(russellb): This is the default minimum version that the server
# (manager) side must implement unless otherwise specified using a version
# argument to self.call()/cast()/etc. here. It should be left as X.0 where
# X is the current major API version (1.0, 2.0, ...). For more information
# about rpc API versioning, see the docs in
# openstack/common/rpc/dispatcher.py.
#
BASE_RPC_API_VERSION = '2.0'
def __init__(self):
super(ComputeAPI, self).__init__(
topic=FLAGS.compute_topic,
default_version=self.BASE_RPC_API_VERSION)
def add_aggregate_host(self, ctxt, aggregate_id, host_param, host,
slave_info=None):
'''Add aggregate host.
:param ctxt: request context
:param aggregate_id:
:param host_param: This value is placed in the message to be the 'host'
parameter for the remote method.
:param host: This is the host to send the message to.
'''
self.cast(ctxt, self.make_msg('add_aggregate_host',
aggregate_id=aggregate_id, host=host_param,
slave_info=slave_info),
topic=_compute_topic(self.topic, ctxt, host, None),
version='2.2')
def add_fixed_ip_to_instance(self, ctxt, instance, network_id):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('add_fixed_ip_to_instance',
instance=instance_p, network_id=network_id),
topic=_compute_topic(self.topic, ctxt, None, instance))
def attach_volume(self, ctxt, instance, volume_id, mountpoint):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('attach_volume',
instance=instance_p, volume_id=volume_id,
mountpoint=mountpoint),
topic=_compute_topic(self.topic, ctxt, None, instance))
# Added by YANGYUAN
def attach_image(self, ctxt, instance, image_id, mountpoint):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('attach_image',
instance=instance_p, image_id=image_id,
mountpoint=mountpoint),
topic=_compute_topic(self.topic, ctxt, None, instance))
# Added by YANGYUAN
def detach_image(self, ctxt, instance):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('detach_image',
instance=instance_p),
topic=_compute_topic(self.topic, ctxt, None, instance))
def change_instance_metadata(self, ctxt, instance, diff):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('change_instance_metadata',
instance=instance_p, diff=diff),
topic=_compute_topic(self.topic, ctxt, None, instance))
def check_can_live_migrate_destination(self, ctxt, instance, destination,
block_migration, disk_over_commit):
instance_p = jsonutils.to_primitive(instance)
return self.call(ctxt,
self.make_msg('check_can_live_migrate_destination',
instance=instance_p,
block_migration=block_migration,
disk_over_commit=disk_over_commit),
topic=_compute_topic(self.topic,
ctxt, destination, None))
def check_can_live_migrate_source(self, ctxt, instance, dest_check_data):
instance_p = jsonutils.to_primitive(instance)
self.call(ctxt, self.make_msg('check_can_live_migrate_source',
instance=instance_p,
dest_check_data=dest_check_data),
topic=_compute_topic(self.topic, ctxt, None, instance))
def confirm_resize(self, ctxt, instance, migration_id, host,
reservations=None, cast=True):
rpc_method = self.cast if cast else self.call
instance_p = jsonutils.to_primitive(instance)
return rpc_method(ctxt, self.make_msg('confirm_resize',
instance=instance_p, migration_id=migration_id,
reservations=reservations),
topic=_compute_topic(self.topic, ctxt, host, instance))
def detach_volume(self, ctxt, instance, volume_id):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('detach_volume',
instance=instance_p, volume_id=volume_id),
topic=_compute_topic(self.topic, ctxt, None, instance))
def finish_resize(self, ctxt, instance, migration_id, image, disk_info,
host, reservations=None):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('finish_resize',
instance=instance_p, migration_id=migration_id,
image=image, disk_info=disk_info, reservations=reservations),
topic=_compute_topic(self.topic, ctxt, host, None))
def finish_revert_resize(self, ctxt, instance, migration_id, host,
reservations=None):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('finish_revert_resize',
instance=instance_p, migration_id=migration_id,
reservations=reservations),
topic=_compute_topic(self.topic, ctxt, host, None))
def get_console_output(self, ctxt, instance, tail_length):
instance_p = jsonutils.to_primitive(instance)
return self.call(ctxt, self.make_msg('get_console_output',
instance=instance_p, tail_length=tail_length),
topic=_compute_topic(self.topic, ctxt, None, instance))
def get_console_pool_info(self, ctxt, console_type, host):
return self.call(ctxt, self.make_msg('get_console_pool_info',
console_type=console_type),
topic=_compute_topic(self.topic, ctxt, host, None))
def get_console_topic(self, ctxt, host):
return self.call(ctxt, self.make_msg('get_console_topic'),
topic=_compute_topic(self.topic, ctxt, host, None))
def get_diagnostics(self, ctxt, instance):
instance_p = jsonutils.to_primitive(instance)
return self.call(ctxt, self.make_msg('get_diagnostics',
instance=instance_p),
topic=_compute_topic(self.topic, ctxt, None, instance))
def get_vnc_console(self, ctxt, instance, console_type):
instance_p = jsonutils.to_primitive(instance)
return self.call(ctxt, self.make_msg('get_vnc_console',
instance=instance_p, console_type=console_type),
topic=_compute_topic(self.topic, ctxt, None, instance))
def host_maintenance_mode(self, ctxt, host_param, mode, host):
'''Set host maintenance mode
:param ctxt: request context
:param host_param: This value is placed in the message to be the 'host'
parameter for the remote method.
:param mode:
:param host: This is the host to send the message to.
'''
return self.call(ctxt, self.make_msg('host_maintenance_mode',
host=host_param, mode=mode),
topic=_compute_topic(self.topic, ctxt, host, None))
def host_power_action(self, ctxt, action, host):
topic = _compute_topic(self.topic, ctxt, host, None)
return self.call(ctxt, self.make_msg('host_power_action',
action=action), topic)
def inject_file(self, ctxt, instance, path, file_contents):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('inject_file',
instance=instance_p, path=path,
file_contents=file_contents),
topic=_compute_topic(self.topic, ctxt, None, instance))
def inject_network_info(self, ctxt, instance):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('inject_network_info',
instance=instance_p),
topic=_compute_topic(self.topic, ctxt, None, instance))
def live_migration(self, ctxt, instance, dest, block_migration, host,
migrate_data=None):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('live_migration', instance=instance_p,
dest=dest, block_migration=block_migration,
migrate_data=migrate_data),
topic=_compute_topic(self.topic, ctxt, host, None))
def pause_instance(self, ctxt, instance):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('pause_instance',
instance=instance_p),
topic=_compute_topic(self.topic, ctxt, None, instance))
def post_live_migration_at_destination(self, ctxt, instance,
block_migration, host):
instance_p = jsonutils.to_primitive(instance)
return self.call(ctxt,
self.make_msg('post_live_migration_at_destination',
instance=instance_p, block_migration=block_migration),
_compute_topic(self.topic, ctxt, host, None))
def power_off_instance(self, ctxt, instance):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('power_off_instance',
instance=instance_p),
topic=_compute_topic(self.topic, ctxt, None, instance))
def power_on_instance(self, ctxt, instance):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('power_on_instance',
instance=instance_p),
topic=_compute_topic(self.topic, ctxt, None, instance))
def pre_live_migration(self, ctxt, instance, block_migration, disk,
host):
instance_p = jsonutils.to_primitive(instance)
return self.call(ctxt, self.make_msg('pre_live_migration',
instance=instance_p, block_migration=block_migration,
disk=disk), _compute_topic(self.topic, ctxt, host, None))
def prep_resize(self, ctxt, image, instance, instance_type, host,
reservations=None):
instance_p = jsonutils.to_primitive(instance)
instance_type_p = jsonutils.to_primitive(instance_type)
self.cast(ctxt, self.make_msg('prep_resize',
instance=instance_p, instance_type=instance_type_p,
image=image, reservations=reservations),
_compute_topic(self.topic, ctxt, host, None))
def reboot_instance(self, ctxt, instance, reboot_type):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('reboot_instance',
instance=instance_p, reboot_type=reboot_type),
topic=_compute_topic(self.topic, ctxt, None, instance))
def rebuild_instance(self, ctxt, instance, new_pass, injected_files,
image_ref, orig_image_ref, orig_sys_metadata):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('rebuild_instance',
instance=instance_p, new_pass=new_pass,
injected_files=injected_files, image_ref=image_ref,
orig_image_ref=orig_image_ref,
orig_sys_metadata=orig_sys_metadata),
topic=_compute_topic(self.topic, ctxt, None, instance),
version='2.1')
def refresh_provider_fw_rules(self, ctxt, host):
self.cast(ctxt, self.make_msg('refresh_provider_fw_rules'),
_compute_topic(self.topic, ctxt, host, None))
def remove_aggregate_host(self, ctxt, aggregate_id, host_param, host,
slave_info=None):
'''Remove aggregate host.
:param ctxt: request context
:param aggregate_id:
:param host_param: This value is placed in the message to be the 'host'
parameter for the remote method.
:param host: This is the host to send the message to.
'''
self.cast(ctxt, self.make_msg('remove_aggregate_host',
aggregate_id=aggregate_id, host=host_param,
slave_info=slave_info),
topic=_compute_topic(self.topic, ctxt, host, None),
version='2.2')
def remove_fixed_ip_from_instance(self, ctxt, instance, address):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('remove_fixed_ip_from_instance',
instance=instance_p, address=address),
topic=_compute_topic(self.topic, ctxt, None, instance))
def remove_volume_connection(self, ctxt, instance, volume_id, host):
instance_p = jsonutils.to_primitive(instance)
return self.call(ctxt, self.make_msg('remove_volume_connection',
instance=instance_p, volume_id=volume_id),
topic=_compute_topic(self.topic, ctxt, host, None))
def rescue_instance(self, ctxt, instance, rescue_password):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('rescue_instance',
instance=instance_p,
rescue_password=rescue_password),
topic=_compute_topic(self.topic, ctxt, None, instance))
def reset_network(self, ctxt, instance):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('reset_network',
instance=instance_p),
topic=_compute_topic(self.topic, ctxt, None, instance))
def resize_instance(self, ctxt, instance, migration_id, image,
reservations=None):
topic = _compute_topic(self.topic, ctxt, None, instance)
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('resize_instance',
instance=instance_p, migration_id=migration_id,
image=image, reservations=reservations), topic)
def resume_instance(self, ctxt, instance):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('resume_instance',
instance=instance_p),
topic=_compute_topic(self.topic, ctxt, None, instance))
def revert_resize(self, ctxt, instance, migration_id, host,
reservations=None):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('revert_resize',
instance=instance_p, migration_id=migration_id,
reservations=reservations),
topic=_compute_topic(self.topic, ctxt, host, instance))
def rollback_live_migration_at_destination(self, ctxt, instance, host):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('rollback_live_migration_at_destination',
instance=instance_p),
topic=_compute_topic(self.topic, ctxt, host, None))
def run_instance(self, ctxt, instance, host, request_spec,
filter_properties, requested_networks,
injected_files, admin_password,
is_first_time):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('run_instance', instance=instance_p,
request_spec=request_spec, filter_properties=filter_properties,
requested_networks=requested_networks,
injected_files=injected_files, admin_password=admin_password,
is_first_time=is_first_time),
topic=_compute_topic(self.topic, ctxt, host, None))
def create_traffic(self, ctxt, instance_id, band, prio, host):
self.cast(ctxt, self.make_msg('create_traffic', instance_id=instance_id,
band=band, prio=prio), topic=_compute_topic(self.topic,
ctxt, host, None))
def set_admin_password(self, ctxt, instance, new_pass):
instance_p = jsonutils.to_primitive(instance)
return self.call(ctxt, self.make_msg('set_admin_password',
instance=instance_p, new_pass=new_pass),
topic=_compute_topic(self.topic, ctxt, None, instance))
def set_host_enabled(self, ctxt, enabled, host):
topic = _compute_topic(self.topic, ctxt, host, None)
return self.call(ctxt, self.make_msg('set_host_enabled',
enabled=enabled), topic)
def get_host_uptime(self, ctxt, host):
topic = _compute_topic(self.topic, ctxt, host, None)
return self.call(ctxt, self.make_msg('get_host_uptime'), topic)
def reserve_block_device_name(self, ctxt, instance, device):
instance_p = jsonutils.to_primitive(instance)
return self.call(ctxt, self.make_msg('reserve_block_device_name',
instance=instance_p, device=device),
topic=_compute_topic(self.topic, ctxt, None, instance))
def snapshot_instance(self, ctxt, instance, image_id, image_type,
backup_type, rotation):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('snapshot_instance',
instance=instance_p, image_id=image_id,
image_type=image_type, backup_type=backup_type,
rotation=rotation),
topic=_compute_topic(self.topic, ctxt, None, instance))
def start_instance(self, ctxt, instance):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('start_instance',
instance=instance_p),
topic=_compute_topic(self.topic, ctxt, None, instance))
def stop_instance(self, ctxt, instance, cast=True):
rpc_method = self.cast if cast else self.call
instance_p = jsonutils.to_primitive(instance)
return rpc_method(ctxt, self.make_msg('stop_instance',
instance=instance_p),
topic=_compute_topic(self.topic, ctxt, None, instance))
def suspend_instance(self, ctxt, instance):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('suspend_instance',
instance=instance_p),
topic=_compute_topic(self.topic, ctxt, None, instance))
def terminate_instance(self, ctxt, instance):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('terminate_instance',
instance=instance_p),
topic=_compute_topic(self.topic, ctxt, None, instance))
def unpause_instance(self, ctxt, instance):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('unpause_instance',
instance=instance_p),
topic=_compute_topic(self.topic, ctxt, None, instance))
def unrescue_instance(self, ctxt, instance):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('unrescue_instance',
instance=instance_p),
topic=_compute_topic(self.topic, ctxt, None, instance))
class SecurityGroupAPI(traffic.openstack.common.rpc.proxy.RpcProxy):
'''Client side of the security group rpc API.
API version history:
1.0 - Initial version.
1.41 - Adds refresh_instance_security_rules()
2.0 - Remove 1.x backwards compat
'''
#
# NOTE(russellb): This is the default minimum version that the server
# (manager) side must implement unless otherwise specified using a version
# argument to self.call()/cast()/etc. here. It should be left as X.0 where
# X is the current major API version (1.0, 2.0, ...). For more information
# about rpc API versioning, see the docs in
# openstack/common/rpc/dispatcher.py.
#
BASE_RPC_API_VERSION = '2.0'
def __init__(self):
super(SecurityGroupAPI, self).__init__(
topic=FLAGS.compute_topic,
default_version=self.BASE_RPC_API_VERSION)
def refresh_security_group_rules(self, ctxt, security_group_id, host):
self.cast(ctxt, self.make_msg('refresh_security_group_rules',
security_group_id=security_group_id),
topic=_compute_topic(self.topic, ctxt, host, None))
def refresh_security_group_members(self, ctxt, security_group_id,
host):
self.cast(ctxt, self.make_msg('refresh_security_group_members',
security_group_id=security_group_id),
topic=_compute_topic(self.topic, ctxt, host, None))
def refresh_instance_security_rules(self, ctxt, host, instance):
instance_p = jsonutils.to_primitive(instance)
self.cast(ctxt, self.make_msg('refresh_instance_security_rules',
instance=instance_p),
topic=_compute_topic(self.topic, ctxt, instance['host'],
instance))
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,199
|
{"url":"https:\/\/socratic.org\/questions\/how-do-you-find-the-inverse-of-a-1-1-2-2-2-2-2-1-1","text":"# How do you find the inverse of A=((1, 1, 2), (2, 2, 2), (2, 1, 1))?\n\nJan 7, 2017\n\n${A}^{-} 1 = \\left(\\begin{matrix}0 & - \\frac{1}{2} & 1 \\\\ - 1 & \\frac{3}{2} & - 1 \\\\ 1 & - \\frac{1}{2} & 0\\end{matrix}\\right)$\n\n#### Explanation:\n\nA matrix, $\\setminus A$, is invertible if and only if its determinant $| A | \\ne 0$. There are several ways to invert a matrix inclining finding the adjoint, row reduction or even a calculator. I will use the adjoint method, which has several steps;\n\n\u2022 Calculating the Matrix of Minors,\n\u2022 Form the Matrix of Cofactors, $c o f \\left(A\\right)$\n\u2022 Form the adjoint matrix, $a \\mathrm{dj} \\left(A\\right)$\n\u2022 Multiply $a \\mathrm{dj} \\left(A\\right)$ by $\\frac{1}{|} A |$ to form the inverse ${A}^{-} 1$\n\nAt some point we need to calculate $| A |$, or $\\det \\left(A\\right)$, and this can also be used to test if the matrix is actually invertible so I prefer to do this first;\n\n$A = \\left(\\begin{matrix}1 & 1 & 2 \\\\ 2 & 2 & 2 \\\\ 2 & 1 & 1\\end{matrix}\\right)$,\n\nIf we expand about the first row and \"strike out\" the row and column to form a smaller determinant and alternate signs we get;\n\n$| A | = + \\left(1\\right) | \\left(2 , 2\\right) , \\left(1 , 1\\right) | - \\left(1\\right) | \\left(2 , 2\\right) , \\left(2 , 1\\right) | + \\left(2\\right) | \\left(2 , 2\\right) , \\left(2 , 1\\right) |$\n$\\setminus \\setminus \\setminus \\setminus \\setminus = \\left\\{\\left(2\\right) \\left(1\\right) - \\left(1\\right) \\left(2\\right)\\right\\} - \\left\\{\\left(2\\right) \\left(1\\right) - \\left(2\\right) \\left(2\\right)\\right\\} + 2 \\left\\{\\left(2\\right) \\left(1\\right) - \\left(2\\right) \\left(2\\right)\\right\\}$\n$\\setminus \\setminus \\setminus \\setminus \\setminus = 0 - \\left(- 2\\right) + 2 \\left(- 2\\right)$\n$\\setminus \\setminus \\setminus \\setminus \\setminus = - 2$\n\nAs $A \\ne 0 \\implies A$ is invertible, so we now calculate the matrix of minors by systematically working through each element in the matrix and \"strike out\" that row and columns and form the determinant of the remaining elements, as follows:\n\nM=( ( |(2, 2), (1, 1)|, |(2, 2), (2, 1)|, |(2, 2), (2, 1)| ), ( |(1, 2), (1, 1)|, |(1, 2), (2, 1)|, |(1, 1), (2, 1)| ), ( |(1, 2), (2, 2)|, |(1, 2), (2, 2)|, |(1, 1), (2, 2)| ) )\n$\\setminus \\setminus \\setminus \\setminus = \\left(\\begin{matrix}0 & - 2 & - 2 \\\\ - 1 & - 3 & - 1 \\\\ - 2 & - 2 & 0\\end{matrix}\\right)$\n\nWe now form the matrix of cofactors, $c o f \\left(A\\right)$, by taking the above matrix of minors and applying the alternate sign matrix as in\n\n$\\left(\\begin{matrix}+ & - & + \\\\ - & + & - \\\\ + & - & +\\end{matrix}\\right)$\n\nWhere we change the sign of those elements with the minus sign to get;\n\n$c o f \\left(A\\right) = \\left(\\begin{matrix}0 & 2 & - 2 \\\\ 1 & - 3 & 1 \\\\ - 2 & 2 & 0\\end{matrix}\\right)$\n\nThen we form the adjoint matrix by transposing the matrix of cofactors, $c o f \\left(A\\right)$, so;\n\n$a \\mathrm{dj} \\left(A\\right) = c o f {\\left(A\\right)}^{T}$\n$\\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus = {\\left(\\begin{matrix}0 & 2 & - 2 \\\\ 1 & - 3 & 1 \\\\ - 2 & 2 & 0\\end{matrix}\\right)}^{T}$\n$\\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus = \\left(\\begin{matrix}0 & 1 & - 2 \\\\ 2 & - 3 & 2 \\\\ - 2 & 1 & 0\\end{matrix}\\right)$\n\nAnd then finally we multiply by the reciprocal of the determinant to get:\n\n${A}^{-} 1 = \\frac{1}{|} A | a \\mathrm{dj} \\left(A\\right)$\n$\\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus = \\left(- \\frac{1}{2}\\right) \\left(\\begin{matrix}0 & 1 & - 2 \\\\ 2 & - 3 & 2 \\\\ - 2 & 1 & 0\\end{matrix}\\right)$\n$\\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus = \\left(\\begin{matrix}0 & - \\frac{1}{2} & 1 \\\\ - 1 & \\frac{3}{2} & - 1 \\\\ 1 & - \\frac{1}{2} & 0\\end{matrix}\\right)$\n\nWe can easily check if this is the correct answer as $A {A}^{-} 1 = I$, So\n$A {A}^{-} 1 = \\left(\\begin{matrix}1 & 1 & 2 \\\\ 2 & 2 & 2 \\\\ 2 & 1 & 1\\end{matrix}\\right) \\left(\\begin{matrix}0 & - \\frac{1}{2} & 1 \\\\ - 1 & \\frac{3}{2} & - 1 \\\\ 1 & - \\frac{1}{2} & 0\\end{matrix}\\right)$\n \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ = ( (0-1+2\\ \\ \\ , -1\/2+3\/2-1\\ \\ \\ , 1-1+0), (-1+3-2,-1+3-1, -2+3-1 ), (1-1+0, 1-1+0, 2-1+0))\n$\\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus = \\left(\\begin{matrix}1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1\\end{matrix}\\right)$\n$\\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus \\setminus = I$ (or sometimes we write ${I}_{3}$).\n\nJan 7, 2017\n\n$\\left(\\begin{matrix}0 & - \\frac{1}{2} & 1 \\\\ - 1 & \\frac{3}{2} & - 1 \\\\ 1 & - \\frac{1}{2} & 0\\end{matrix}\\right)$\n\n#### Explanation:\n\nHere's the row reduction way, less to remember.....\n\n$\\left(\\begin{matrix}1 & 1 & 2 \\\\ 2 & 2 & 2 \\\\ 2 & 1 & 1\\end{matrix}\\right) \\left(\\begin{matrix}1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1\\end{matrix}\\right)$\n\n$R 2 \\to \\frac{R 2}{2}$\n$\\left(\\begin{matrix}1 & 1 & 2 \\\\ 1 & 1 & 1 \\\\ 2 & 1 & 1\\end{matrix}\\right) \\left(\\begin{matrix}1 & 0 & 0 \\\\ 0 & \\frac{1}{2} & 0 \\\\ 0 & 0 & 1\\end{matrix}\\right)$\n\n$R 2 \\to R 2 - R 1 , R 3 \\to R 3 - 2 R 1$\n$\\left(\\begin{matrix}1 & 1 & 2 \\\\ 0 & 0 & - 1 \\\\ 0 & - 1 & - 3\\end{matrix}\\right) \\left(\\begin{matrix}1 & 0 & 0 \\\\ - 1 & \\frac{1}{2} & 0 \\\\ - 2 & 0 & 1\\end{matrix}\\right)$\n\nswitch R2 and R3\n\n$\\left(\\begin{matrix}1 & 1 & 2 \\\\ 0 & - 1 & - 3 \\\\ 0 & 0 & - 1\\end{matrix}\\right) \\left(\\begin{matrix}1 & 0 & 0 \\\\ - 2 & 0 & 1 \\\\ - 1 & \\frac{1}{2} & 0\\end{matrix}\\right)$\n\n$R 1 \\to R 1 + 2 R 3 , R 2 \\to R 2 - 3 R 3$\n\n$\\left(\\begin{matrix}1 & 1 & 0 \\\\ 0 & - 1 & 0 \\\\ 0 & 0 & - 1\\end{matrix}\\right) \\left(\\begin{matrix}- 1 & 1 & 0 \\\\ 1 & - \\frac{3}{2} & 1 \\\\ - 1 & \\frac{1}{2} & 0\\end{matrix}\\right)$\n\n$R 1 \\to R 1 + R 2 , R 2 \\to - R 2 , R 3 \\to - R$#\n\n$\\left(\\begin{matrix}1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1\\end{matrix}\\right) \\left(\\begin{matrix}0 & - \\frac{1}{2} & 1 \\\\ - 1 & \\frac{3}{2} & - 1 \\\\ 1 & - \\frac{1}{2} & 0\\end{matrix}\\right)$\n\nJan 7, 2017\n\n${A}^{- 1} = \\left(\\begin{matrix}0 & - \\frac{1}{2} & 1 \\\\ - 1 & \\frac{3}{2} & - 1 \\\\ 1 & - \\frac{1}{2} & 0\\end{matrix}\\right)$\n\n#### Explanation:\n\nHere's one method:\n\nGiven:\n\n$A = \\left(\\begin{matrix}1 & 1 & 2 \\\\ 2 & 2 & 2 \\\\ 2 & 1 & 1\\end{matrix}\\right)$\n\nMake an augmented matrix by adding three columns containing the entries of a $3 \\times 3$ identity matrix:\n\n$\\left(\\begin{matrix}1 & 1 & 2 & | & 1 & 0 & 0 \\\\ 2 & 2 & 2 & | & 0 & 1 & 0 \\\\ 2 & 1 & 1 & | & 0 & 0 & 1\\end{matrix}\\right)$\n\nPerform a series of row operations to make the left half of this augmented matrix into an identity matrix.\n\nSubtract row 3 from row 2 to get:\n\n$\\left(\\begin{matrix}1 & 1 & 2 & | & 1 & 0 & 0 \\\\ 0 & 1 & 1 & | & 0 & 1 & - 1 \\\\ 2 & 1 & 1 & | & 0 & 0 & 1\\end{matrix}\\right)$\n\nAdd row 2 to row 3 to get:\n\n$\\left(\\begin{matrix}1 & 1 & 2 & | & 1 & 0 & 0 \\\\ 0 & 1 & 1 & | & 0 & 1 & - 1 \\\\ 2 & 2 & 2 & | & 0 & 1 & 0\\end{matrix}\\right)$\n\nDivide row 3 by $2$ to get:\n\n$\\left(\\begin{matrix}1 & 1 & 2 & | & 1 & 0 & 0 \\\\ 0 & 1 & 1 & | & 0 & 1 & - 1 \\\\ 1 & 1 & 1 & | & 0 & \\frac{1}{2} & 0\\end{matrix}\\right)$\n\nSubtract row 1 from row 3 to get:\n\n$\\left(\\begin{matrix}1 & 1 & 2 & | & 1 & 0 & 0 \\\\ 0 & 1 & 1 & | & 0 & 1 & - 1 \\\\ 0 & 0 & - 1 & | & - 1 & \\frac{1}{2} & 0\\end{matrix}\\right)$\n\nSubtract row 2 from row 1 to get:\n\n$\\left(\\begin{matrix}1 & 0 & 1 & | & 1 & - 1 & 1 \\\\ 0 & 1 & 1 & | & 0 & 1 & - 1 \\\\ 0 & 0 & - 1 & | & - 1 & \\frac{1}{2} & 0\\end{matrix}\\right)$\n\nAdd row 3 to row 1 and row 2 ro get:\n\n$\\left(\\begin{matrix}1 & 0 & 0 & | & 0 & - \\frac{1}{2} & 1 \\\\ 0 & 1 & 0 & | & - 1 & \\frac{3}{2} & - 1 \\\\ 0 & 0 & - 1 & | & - 1 & \\frac{1}{2} & 0\\end{matrix}\\right)$\n\nMultiply row 3 by $- 1$ to get:\n\n$\\left(\\begin{matrix}1 & 0 & 0 & | & 0 & - \\frac{1}{2} & 1 \\\\ 0 & 1 & 0 & | & - 1 & \\frac{3}{2} & - 1 \\\\ 0 & 0 & 1 & | & 1 & - \\frac{1}{2} & 0\\end{matrix}\\right)$\n\nNow we can read ${A}^{- 1}$ from the right hand $3$ columns:\n\n${A}^{- 1} = \\left(\\begin{matrix}0 & - \\frac{1}{2} & 1 \\\\ - 1 & \\frac{3}{2} & - 1 \\\\ 1 & - \\frac{1}{2} & 0\\end{matrix}\\right)$\n\nOne of the advantages of this method is that it works for square matrices of any size.","date":"2020-07-04 05:32:27","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 61, \"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\": 0.9807159900665283, \"perplexity\": 639.1523495388221}, \"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-2020-29\/segments\/1593655884012.26\/warc\/CC-MAIN-20200704042252-20200704072252-00593.warc.gz\"}"}
| null | null |
(489603) Kurtschreckling est un astéroïde de la ceinture principale.
Description
(489603) Kurtschreckling est un astéroïde de la ceinture principale. Il fut découvert le à Gaisberg par Richard Gierlinger. Il présente une orbite caractérisée par un demi-grand axe de 2,87 UA, une excentricité de 0,21 et une inclinaison de 1,1° par rapport à l'écliptique.
Articles connexes
Liste des planètes mineures (489001-490000)
Ceinture d'astéroïdes
Références
Planète mineure découverte en 2007
Astéroïde de la ceinture principale
Objet céleste découvert par Richard Gierlinger
Kurtschr
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,752
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Peachjar allows you to easily reach the parents of school-aged children by delivering school-approved digital flyers directly to their email inbox! Flyers are sent alongside important school information and are posted to the school's website to increase visibility.
Whether you're a local after school program or a national provider, you can start reaching parents more effectively with Peachjar, while also saving time and trees!
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The Peachjar platform exceeded our performance expectations and is now a vital ongoing communication channel at Kranse Institute.
Join the thousands of organizations that are using Peachjar to reach more parents!
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{
"redpajama_set_name": "RedPajamaC4"
}
| 274
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{"url":"https:\/\/proofwiki.org\/wiki\/Element_of_Universe","text":"# Element of Universe\n\n## Theorem\n\nLet $A$ be a class, which may be either a set or a proper class.\n\nThen:\n\n$\\forall A: \\paren {A \\in U \\iff \\exists x: x = A}$\n\nwhere $U$ is the universal class.\n\n## Proof\n\n $\\displaystyle \\forall A$ $:$ $\\displaystyle \\paren {A \\in U \\iff \\exists x: \\paren {x = A \\land x \\in U} }$ Definition of Class Membership $\\displaystyle \\forall x$ $:$ $\\displaystyle x \\in U$ Fundamental Law of Universal Class $\\displaystyle \\leadsto \\ \\$ $\\displaystyle \\forall A$ $:$ $\\displaystyle \\paren {A \\in U \\iff \\exists x: x = A}$ Conjunction with Tautology\n\n$\\blacksquare$","date":"2020-08-11 09:59:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.8791446685791016, \"perplexity\": 544.3735092422758}, \"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-2020-34\/segments\/1596439738746.41\/warc\/CC-MAIN-20200811090050-20200811120050-00336.warc.gz\"}"}
| null | null |
Storage - Household & Commercial in Boone, Iowa - Storage - Household & Commercial in Boone, IA: Yellow Pages Directory Inc.
'Storage', in Boone, is the stocking of goods, household or commercial. Self storage is an industry in which storage space like rooms, lockers, containers, and/or outdoor space, is rented to tenants in which they can store and access their goods. Self storage facilities lease space to individuals usually for storing household goods and the rented spaces are secured by the tenant's own lock and key.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 5,884
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Q: Issue with wp_schedule_event() I am trying to set up an event on a custom interval that will grab new posts from an Instagram search and create them as wordpress posts. I had the code working on its own page, and now I am trying to schedule it to happen automatically.
I have also installed Wp Crontroller to ensure the event is being recognized by Wordpress (which it is) but I cannot seem to get the code to run (nor can I figure out how to debug it.
add_action( 'wp', 'create_schedule' );
function create_schedule() {
wp_schedule_event(time(), 'quarter_hour', 'ig_fetch_start');
update_option('test', 'Close!');
}
add_action('ig_fetch_start','ig_fetch_new_posts');
function ig_fetch_new_posts() {
update_option('test', 'Fired');
// Run function
}
The option update is in there just to see if the event is firing. Wordpress simple never seems to make it to the ig_fetch_new_posts function.
A: You need to schedule the event differently. In your approach, you hook to wp to schedule the event, meaning that it is called everytime WordPress is called, setting your option back.
I am not quite sure if the schedule is postponed or if you create multiple schedules this way, but it is not correct.
You should check if this event is scheduled (using wp_next_scheduled()), and only schedule if this function returns false, to avoid multiple entries/postponing issues.
I would also use two different options to check if the functions get called, because due to the architecture of this system, as soon as your scheduled function is fired, the value changes - only to be overwritten when you next call WordPress, as no event is scheduled.
I also added a time to the optionvalue, to check when it was registered. I always like to know stuff like that on debugging.
The script (including the registration of a new schedule) would look something like this:
add_filter( 'cron_schedules', 'f711_add_quarter_hour_cron_schedule' );
function f711_add_quarter_hour_cron_schedule( $schedules ) {
$schedules['quarter_hour'] = array(
'interval' => 900, // Time in seconds
'display' => __( 'Quarter Hour' ),
);
return $schedules;
}
if ( ! wp_next_scheduled( 'ig_fetch_new_posts_hook' ) ) { // check if function is scheduled
wp_schedule_event( time(), 'quarter_hour', 'ig_fetch_new_posts_hook' ); // if not, schedule it
update_option('test_scheduled', 'scheduled' . time() ); // set your scheduleoption to scheduled
}
add_action( 'ig_fetch_new_posts_hook', 'ig_fetch_new_posts' ); // define the hook for the schedule
function ig_fetch_new_posts() { // your updatemagic
update_option('test_fired', 'Fired' . time() );
// Run function
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,716
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Robert Meglič (né le ) est un ancien sauteur à ski slovène.
Palmarès
Jeux Olympiques
Championnats du monde
Championnats du monde de vol à ski
Coupe du monde
Meilleur classement final: en 1995.
Meilleur résultat: .
Naissance en novembre 1974
Sauteur à ski slovène
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,448
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This month Shea and Derek discuss two great works that bring back some older, or classic, manga. They begin with Leiji Matsumoto's Captain Harlock: The Classic Collection, Vol 1 (Seven Seas Entertainment), a series that originally ran between 1977 and 1979. Captain Harlockis a classic science fiction work, and with a space pirate protagonist who can be seen as a romantic hero. The Two Guys discuss the figure of Harlock as a curious mixture, while at the same time trying to ferret out the his philosophy. In many ways, they see this manga anchored in its time, both aesthetically and politically. But this work definitely isn't limited to it's time and can be appreciated today.
Next they turn to Slum Wolf, another translated collection from Tadao Tsuge (New York Review Comics). An earlier collection of Tsuge's work, Trash Market, was released in 2015 by Drawn and Quarterly (which the guys reviewed on their June 2015 episode). Slum Wolfis a collection of nine stories originally published between 1969 and 1978 in various publications such as Garo and Yagyō. It also includes an essay by Tsuge, as well as a outstanding contextual essay by Ryan Holmberg, who also edited and translated the collection. The guys discuss all the stories, to greater or lesser degrees of depth, but they spend most of their time talking about the linking features that bind most of the narratives. As both Derek and Shea point out, this is one of the manga highlights of the year, so far.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,576
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Török László (1928–2010) Bedő-díjas erdőmérnök
Török László (1928–2013) a rádiházi ménes vezetője (1953-1989); Magyar Military Válogatott volt szövetségi kapitánya Somogyifoto.gportal.hu
Török László (1929–1997) író, újságíró, filmkritikus
Török László állatorvos, a rádiházi, Kabalaménes Kft. egyik tulajdonosa, fő lótenyésztője, Török László agrármérnök veje
Török László (1937) atléta, gátfutó
Török László (1941–2020) régész, történész, az MTA tagja
Török László (1947–2001) katonatiszt, ezredes, posztumusz dandártábornok
Török László (1948–2020) Balázs Béla-díjas fotóművész
Török László (1950) sportújságíró, sportvezető
Török László (1957) közgazdász, bankár
Török László (1962–) klasszika-filológus, történelemtanár
Török László (1967–) romániai magyar politikus (RMDSZ), Nagyszalonta polgármestere
Török László (1969) labdarúgó (Tiszakécske, Kispest-Honvéd), edző
Török László (1964) tanár, informatikus, kaktuszgyűjtő
Jegyzetek
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,265
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Controlled trials have found therapeutic plasma exchange (TPE) and intravenous immunoglobulin (IVIg) infusion therapy to be equally efficacious in treating Guillain-Barré syndrome (GBS). Due to increases in the price of IVIg compared to human serum albumin (HSA), used as a replacement fluid in TPE, we examined direct hospital-level expenditures for TPE and IVIg for meaningful cost-differences between these treatments.
Using financial data from our two institutions, hospital cost profiles for IVIg and 5% albumin were established. Reimbursement amounts were obtained from publicly available Medicare data resources to determine payment rates for TPE, non-tunneled central catheter line placement, and drug infusion therapy. A model was developed which allows hospitals to input cost and reimbursement amounts for both IVIg and TPE with HSA that results in real-time valuations of these interventions.
The direct cost of five IVIg infusion sessions totaling 2.0 grams per kilogram (g/kg) body weight was $10,329.85 compared to a series of five TPE procedures, which had direct costs of $4,638.16.
In GBS patients, direct costs of IVIg therapy are more than twice that of TPE. Given equivalent efficacy and similar severity and frequencies of adverse events, TPE appears to be a less expensive first-line therapy option for treatment of patients with GBS.
Guillain-Barré syndrome (GBS), the most common cause of acute neuromuscular paralysis in the U.S. with an incidence of 1 to 2 per 100,000, is an immune mediated demyelinating polyneuropathy mainly affecting motor and sensory peripheral nerves. Characterized by paresthesias, weakness, and ascending paralysis in a distal to proximal pattern, GBS patients may also demonstrate autonomic nerve dysfunction further complicating their recovery. The majority of severely affected patients require hospitalization, especially those with oropharyngeal and respiratory muscle involvement. Ventilator support may be needed in 30% of all GBS patients .
Efficacy of therapeutic plasma exchange (TPE) has been demonstrated in randomized trials comparing supportive care or corticosteroid therapy in GBS patients. TPE reduces ventilator support days and shortens time to unaided walking resulting in earlier hospital discharge . Trials comparing TPE with intravenous immunoglobulin (IVIg) have demonstrated equivalency of the two in shortening time to unaided walking and reducing length of ventilator support . In a summary of five trials with a combined enrollment of 582 patients, TPE when compared to IVIg was found to be equivalent with regard to improvement in disability grade with no significant differences in other outcome measures . Based on such data demonstrating equivalence of these two treatment options, the American Academy of Neurology has concluded that TPE and IVIg are equivalent and recommended either for the treatment of non-ambulatory patients .
While TPE and IVIg are equally effective in the treatment of GBS, concern has arisen over whether the safety profile of the two treatments is equivalent. The initial trial comparing IVIg to TPE found a lower overall adverse event rate with IVIg . A subsequent larger trial, however, found similar adverse event rates, with the majority being transient and mild . Subsequent meta-analysis of available data found a relative risk of complications with IVIg compared to TPE of 0.84 but the 95% confidence interval (0.54 to 1.30) crossed 1, indicating equivalency . These findings suggest that not only are the two treatments equivalent with regard to patient response but also with regard to potential risks.
Many physicians, however, prefer IVIg for the treatment of patients with severe GBS. This preference may be due partly to the 1997 Plasma Exchange/Sandoglobulin Guillain-Barré Syndrome Trial that reported, "On grounds of equal therapeutic benefit, greater convenience and similar overall cost, IVIg may be preferable to TPE for treatment of adult patients with Guillain-Barré syndrome...provided there are no contraindications to IVIg" . At the time of this trial, the average price of IVIg in the U.S. was approximately $30 per gram . In the decade following this study, prices for IVIg increased to >$60 per gram for liquid formulations and >$50 per gram for the powder formulation . Additionally, in 2005, manufacturers instituted allocation programs due to supply shortages that restricted IVIg availability. Some hospitals have had to establish triage plans allocating IVIg use only for approved indications whereas others have required consideration of alternate treatments, like TPE, for patients with neurologic disorders . Several factors contribute to IVIg shortages including consolidation in the blood product industry, the switch from the manufacturing of powder to liquid preparations, production cuts by some manufacturers and a lag time in production when firms change manufacturing processes . Additional supply pressures result from steadily increasing clinical demands for the off-label use of IVIg at a rate of 5-10% annually . During these periods, IVIg prices have risen more than 20% and as of April 2010, the average price for liquid formulations of IVIg had increased to $70.22 per gm.
During this same time, human serum albumin (HSA) prices averaged $30 for a 250 ml bottle. HSA prices reached a low in 2004-2005 and were less than one-half the1999 price in 2007. As of April 2010, the price of a 250 ml bottle of 5% HSA was $34.06.
Based on the above considerations, we re-examined the assumption of cost parity between a standard course of IVIg and a series of five TPEs by performing a cost-minimization analysis to determine whether meaningful differences in the direct cost currently exist between these two therapeutic options.
A cost-minimization analysis comparing a typical course of five TPEs against five IVIg infusions was performed. The model was developed from the study data included in Tables 1 and 2 and was created in a spreadsheet (Excel 2003, Microsoft Corporation, Redmond WA) and is available as additional file 1 to this manuscript. The model provides hospitals an opportunity to include their own direct costs and reimbursement amounts from any specific payer to arrive at real-time values.
a IVIg hospital cost $70.22 shown above is based on hospital contracted prices paid to manufacturers. Starting in Q1-2011 Medicare OPPS (outpatient prospective payment system) IVIg reimbursement is $73.30/gram based on average of five IVIg brands. Medicare payment rates for Q1-2011 are based on 106% of third quarter 2010 manufacturer's reported average sales price, which serves as a proxy for hospital acquisition cost, distributor mark-up and direct overhead including storage, preparation and disposal costs).
b Center for Medicare and Medicaid Services (CMS). CY 2010 Labor file. RUC source for CPT 96365 and 96366. Assumes an average two-hour infusion (one unit each of CPT 96365 + CPT 96366), which utilizes 59 minutes and 14 minutes (total 73 minutes) of nurse clinical labor time. This value understates typical labor costs for a PE nurse operator, which is better approximated by the RUC value of $0.79/minute applied for therapeutic intravenous infusions; we used that higher labor rate in our cost model.
c CMS. CY 2010 Supplies file. RUC source for CPT 96365 and 96366.
a CMS. CY 2010 Supplies file. PEAC (Practice Expense Advisory Committee) source for CPT 36514. March 2004 update.
b CMS. CY 2010 Labor file. PEAC source for CPT 36514. March 2004 update. For PE (CPT 36514), the CMS Labor file cites a labor cost of $0.42/minute, based on an RN/LVN operator. This value understates typical labor costs for a PE nurse operator, which is better approximated by the RUC value of $0.79/minute applied for therapeutic intravenous infusions; we used that higher labor rate in our cost model.
d 40 ml/kg 5% albumin (50 ml/kg total fluid volume replacement with 80% albumin:20% saline) × 70 kg patient × 5 procedures = 14,000 ml = 56 vials of 250 ml 5% albumin × $35.53/vial = $1,990 Albumin cost of $35.53 per vial represents the January 2011 provider contracted acquisition cost. Medicare payment rate is set at 95% of AWP for albumin, which serves as a proxy for hospital acquisition cost, distributor mark-up and direct overhead including storage, preparation and disposal costs.
e CMS. CPEP equipment file, March 2004 update (most recent data). Includes cell separator system, blood warmer and medical recliner chair, with useful lives of 6, 7 and 10 years (200 procedures per year) and costs of $59,320, $3,840 and $829, respectively.
f $4,450 annually per cell separator system (CaridianBCT device); assumes 200 procedures per year per device.
Basic assumptions about each treatment regimen are given in Table 3. With regard to the number of TPE procedures, in patients not requiring respiratory support, a course of two TPEs has been compared to four with the later being superior in multiple outcome measures. In more severely affected individuals, a course of four TPEs was compared to six and both were found to be equivalent with the exception of a higher frequency of hypotension with six TPEs . Based upon these findings, recommendations in published guidelines , and to maximize the costs of TPE, a course of five TPEs was selected. We conservatively selected the highest plasma exchange volume for this clinical indication based upon the medical literature . We assumed an average 70 kg adult and an 80:20 replacement fluid mix of 5% HSA and normal saline, resulting in 14,000 ml of HSA to complete all five TPE procedures. Alternate replacement colloids to 5% HSA, such as hydroxyethyl starch solutions, can be used and are less expensive than 5% HSA. In addition, a lower percentage of HSA, such as 60:40 or 50:50 HSA to saline could also be used and would further reduce the costs of TPE. However, these alternate replacement fluids and fluid mixtures are associated with a higher frequency of reactions and are not consistent with recommendations for replacement fluids in published guidelines . In addition, utilizing this replacement fluid mixture maximize the costs associated with TPE and represents a worst-case scenario. Central venous catheter placement for venous access in GBS patients treated with PE is not always necessary but we conservatively assumed that all patients would require this. We assumed that serious adverse events (AEs) attributable to TPE or IVIg occur infrequently at similar rates and that associated costs would not directly affect this analysis. Finally, we also assumed that the length of hospitalization would be equivalent between the two therapies given the equivalent time to response between the two [3, 4]. Both therapies can be performed as an outpatient once the patient has responded.
Direct labor costs were obtained from the Center for Medicare and Medicaid Services (CMS) Clinical Practice Expense Panel (CPEP) clinical labor database for plasma exchange and intravenous infusion . Supply costs for a TPE procedure, totaling just over $210, were identified in the CPEP supply database. The administration set and other supplies for an IVIg infusion cost less than $6 .
The direct hospital-level costs for HSA and IVIg are not published as prices and are driven by hospital contracts. CMS does, however, publish drug pricing files based on manufacturer reported average sales prices (ASP). Effective January 1, 2011 the agency uses these to develop reimbursement amounts by adding 6% to the manufacturer reported ASP for physician offices and the same 6% under outpatient prospective payment system (OPPS) for hospital outpatient reimbursement. Values used for reimbursement in this cost minimization analysis represent 106% of third quarter 2010 manufacturers' U.S. average sales prices for IVIg. For data used in the Budget Impact Model, we used the most current reimbursement rate for IVIg reimbursement from Medicare; the IVIg reimbursement of $73.226/gram equates with first quarter 2011 Medicare OPPS (outpatient prospective payment system) payment rate averaged across five liquid IVIg products (J1459, J1561, J1568, J1569, J1572) (Medicare reimbursement rates are for third quarter 2011 based on 106% of third quarter 2010 manufacturer's reported average sales price, which serves as a proxy for hospital acquisition cost, distributor mark-up and direct overhead including storage, preparation and disposal costs) .
HSA, however, is still reimbursed under the AWP (Average Wholesale Price) formula. These values serve as proxies for the hospital reimbursement and include acquisition cost, distributor mark-up, and direct overhead related to product storage, preparation and disposal . For hospital drug costs we used average prices paid by the two hospitals, based on contracts. The average hospital cost from two hospitals for IVIg was $ 70.22/gram and the average cost for 5% HSA was $35.35/250 ml bottle.
The surgeon's fee for placement of a central venous catheter is not a hospital cost and is not included in the analysis. However, for those interested in considering a global direct cost comparison, the average Medicare surgeon payment for non-tunneled central venous line placement is $130. Physician medical management and oversight fees for either TPE or IVIg infusions were not included in the analysis.
Direct hospital costs related to IVIg and TPE treatment are provided in Tables 1 and 2. The estimated direct cost for IVIg infusion therapy to treat a 70 kg adult is $10,305 (Table 1). This is approximately 159% more than the $3,980 direct cost of a course of TPE for the same 70 kg adult (Table 2).
Whereas IVIg prices have risen in recent years, in contrast, HSA prices have been more volatile. At its cost peak in 1999-2000, the price for HSA at our two institutions averaged between $55 and $58 per 250 ml bottle. As HSA represents the single largest TPE cost element, we calculated a 50% and 100% increase in the current cost of $36.48 per 250 ml bottle of HSA to determine how this might affect the comparative cost outcome. As of January 1, 2011, CMS reimburses $67.07 per 250 ml bottle based on the AWP formula. Hospital costs for HSA are significantly lower than this amount, $35.53 for 250 ml. The price of HSA would have to increase approximately five fold for TPE to match the cost of IVIg drug costs.
Given the higher cost of IVIg relative to TPE, a reassessment of their clinical benefits and associated AEs is relevant. The Guillain-Barré Syndrome Study Group first established the efficacy of TPE when 245 patients were randomized to receive five to six one plasma volume exchanges over 14 days or "best supportive care" at 21 North American treatment sites . In this study, the TPE group experienced a shorter median time to improve one grade (19 vs. 40 days; p < 0.001), better percentage of subjects who had improved one grade at 4 weeks (59% vs. 39%, p < 0.01), shorter time to walk unassisted (53 vs. 85 days; p < 0.001) and shorter mean time on ventilator support (24 vs. 48 days, p < 0.01). TPE and supportive care demonstrated no difference in complication frequency . A Cochrane Database review of six clinical trials (649 patients) confirmed that TPE is superior to supportive care across multiple functional outcome measures . TPE was associated with fewer severe sequelae after one year, fewer infectious events and fewer cardiac arrhythmias than supportive care. The reviewers concluded that TPE is "the first and only treatment that has been proven to be superior to supportive treatment alone," and should be regarded as the standard against which new treatments should be judged .
Consequently, TPE has served as the standard treatment arm for two large randomized trials examining IVIg use in patients with severe GBS. In 1992, the Dutch Guillain-Barré Study Group randomized 147 patients to five to six TPEs or five daily IVIg infusions of 0.4 grams/kg . The use of IVIg was associated with a shorter median time to improve one grade and a significantly higher percentage of subjects who improved one or more grades at four weeks. At three-month examination, however, no significant difference in this primary outcome was seen . Of note in this study those treated with TPE did no better than patients in the Guillain-Barré Syndrome Study Group trial who received supportive care. It was subsequently identified that the Dutch Guillain-Barré Study Group study did not appropriately compare the two therapies [18, 19]. The definition of functional grade 3 as "able to walk ≥10 m with a walker or support" resulted in the inclusion of patients in the study who would have been classified as functional grade 2 excluded by the Guillain-Barré Syndrome Study Group. The enrollment of less severely affected GBS patients may have masked the effectiveness of TPE. Of greater concern was inadequate matching in the study treatment arms. The TPE group was older and 41% of subjects had diminished compound muscle potentials at baseline. In comparison, the IVIg group was younger with diminished compound muscle potentials in only 27% of patients. Age and diminished compound muscle potentials are the two most reliable negative prognostic factors for response to TPE in GBS. Similar problems with equivalency of the TPE and IVIg groups were also seen in a more recent study by Alshekhlee et al where patients receiving TPE were older with higher complication rates and, not surprisingly, higher mortality rates .
The second larger trial by the Plasma Exchange/Sandoglobulin Guillain-Barré Syndrome Trial Group found equivalency between IVIg and TPE in patients across multiple disability parameters, with equivalent low rates of treatment complications and numbers of deaths . While TPE could, in theory, increase the risk of infection and hemorrhage through the removal of immunoglobulins and clotting factors, neither AEs occurred more frequently in the TPE group compared to the IVIg group in this trial or in trials of TPE tested against supportive care.
Table 4 shows AEs associated with TPE and IVIg treatment. The side-effect profiles are similar with regard to severity and frequency of mild and severe reactions. The profiles differ with regard to types of reactions and their underlying pathophysiology.
The frequency of adverse complications associated with TPE have been reported to occur in 4.75% to 36% of procedures [21–23]. The vast majority of reactions are mild, easily treated, and self-limited [21–23]. Severe reactions, defined as those with the potential to be life threatening, have been reported to occur in 0.12% of TPEs . Couriel and Weinstein observed a higher incidence of severe reactions, 6.15%, with all being related to central line placement complications where only 23% of the patients were treated via peripheral access . In contrast, Basic-Jukic et al noted a lower reaction rate in a study where 72% of the TPE procedures were performed with peripheral vascular access . Thus with TPE the frequency of reactions directly attributable to the procedure performed via peripheral access is less than those due to central line placement . While it is frequently assumed that all patients undergoing TPE for neurologic diseases require central line placement, studies demonstrating that a majority (72 to 96%) of patients can satisfactorily undergo TPE using peripheral vascular access suggest otherwise [22, 24].
Frequency of adverse effects of IVIg has been reported to occur in 11 to 81% of patients receiving infusions with more recent studies demonstrating reaction rates of 36% to 42% [25, 26]. As with TPE, the majority are mild, easily treated, and self-limited. The frequency of severe, life threatening, reactions is rare [25–28].
Reducing unnecessary costs is critical to addressing the growing lack of affordability of health care, now reflected in the fact that 15% of Americans do not have health insurance . Hospitals are motivated to find opportunities to reduce costs without compromising patient outcome, particularly in the inpatient setting where prospective payment arrangements with public- and private-sector insurers reward cost containment. Reimbursement in most instances is defined prospectively and remains fixed regardless of the actual costs of care for an individual patient. Medicare, for example, pays most hospitals a fixed payment amount based on the Diagnosis Related Group (DRG) to which the stay is assigned. Most private insurers pay a fixed per diem rate that is typically independent of how much is spent on procedures, drugs, supply items and other resources.
Since the Plasma Exchange/Sandoglobulin Guillain-Barré Syndrome trial, the presumption of similar costs may have influenced neurologists to use IVIg as a first-line for treatment of GBS rather than TPE. Similarly, the use of IVIg rather than TPE in the management of other neurological conditions (Table 5), including chronic inflammatory demyelinating polyneuropathy and myasthenia gravis may also be affected by this perspective [30–33].
The average price for IVIg since the Plasma Exchange/Sandoglobulin Guillain-Barré Syndrome trial has more than doubled from approximately $30 per gram to $70.22 as of April 6, 2010 . While the cost of HSA used for volume replacement in TPE is not inconsiderable, it is dwarfed by a 2.2 times higher average IVIg price. Use of less than an average of 2.8 liters of HSA per procedure due to smaller patient plasma volumes further widens this HSA-IVIg cost gap. Other resources required to perform a TPE procedure (e.g. the nurse operator, disposables including the apheresis tubing set, and amortization of the blood processing device) are less than 25% of the cost of a series of five IVIg infusions used to treat a 70 kg adult patient.
Little has been published comparing the costs of IVIg and TPE for GBS. Early studies found TPE to be cost-saving . However, in 1999; Canadian investigators published a cost minimization analysis that may have overstated costs for IVIg and albumin. While the Canadian group found that TPE was more cost effective than IVIg (TPE $6,204/patient versus IVIg $10,165/patient), they may have inadvertently incorporated published "list" prices for IVIg products, producing an averaged cost estimate roughly two-fold higher than the actual U.S. market price for IVIg at that time . An assumed average cost of $90 per 250 ml bottle of HSA was again dramatically higher than the U.S. hospital-level price of HSA at any time over the last decade. This group also applied an hourly nurse labor cost ($15.25) that was substantially lower than 1999 prevailing U.S. hourly wage ($21.38) for registered nurses . Such cost assumptions, coupled with an escalation in prices of IVIg products over the last several years, supports a re-examination and cost analysis of IVIg and TPE in GBS. Another notable exception to the lack of economic analyses is the work by Tsai, Wang and Liu . The authors retrospectively examined 24 patients with GBS who were admitted to the Taipei Veterans General Hospital, 10 of whom were treated with TPE, seven with IVIg, and seven who received supportive treatment and were treated in the study as a control group. They concluded that despite the significantly lower cost for TPE, total costs were lower in the IVIg group . In a response to the study, Buenz raised several critical questions . While the age difference between the two groups was not statistically significant, Buenz suggested that the difference between an 83-year-old and a 45 year-old-patient (statistically equally as probable according to the data presented where the 95% confidence interval for the age difference is -5.12 to 38.5 years) is clinically significant with regard to GBS prognosis, regardless of the therapeutic intervention . Second, the severity of disease and co-morbidities are not adequately described. Since patients receiving IVIg in Taipei must personally pay out-of-pocket expenses for their therapy it is reasonable to question whether wealthier and potentially younger patients with less co-morbidity seen earlier in their disease course may comprise and bias the cohort examined. Finally, the authors state that the incidence of complications is the same while the cost for treatment is higher for the TPE group. One patient in each group had pneumonia. Since there is no reason to assume that the cost of treating pneumonia would be higher in one group or the other, it is assumed that the sole difference in the cost of complications is from a septic patient. The assertion that TPE is more expensive than IVIg is based on this single patient with sepsis where the nature and source of the infection is unreported. While the study provides an important contribution to the literature on TPE and IVIg, it raises concerns that require additional research .
In a more recent detailed study of the economic cost of GBS in the U.S., the lifetime health burden in monetary terms for 5,500 GBS patients was examined . Such a study provides information to assess cost-effectiveness of health measures that affect GBS. The estimated annual cost of GBS was $1.7 billion (95% CI, $1.6 to 1.9 billion) including $200 million (14%) in direct medical costs and $1.5 billion (86%) in indirect costs. In our work, we focused solely on assessing the savings in drug costs. At $5,350.00 per course of five treatments multiplied by the number of treatments per year times the number of GBS patients treated at a given hospital, a significant budget impact on pharmacy costs could quickly accrue.
We intentionally did not consider "indirect cost" that hospitals may assign to the physical floor space required to perform a TPE procedure in our analysis. Space valuation is variable between institutions, reflecting differences in revenue and accounting practices. This valuation would be small on a per-procedure basis and therefore would not significantly affect this analysis. In addition, assignment of an indirect cost for physical space is financially relevant only if the treatment of this patient population requires addition of an apheresis medicine treatment unit. If existing apheresis medicine service capacity can absorb those procedures, it is difficult to argue that the hospital has incurred a new "cost."
Our study is limited in three significant ways. First, while AEs attributable to TPE and IVIg occurred with similar frequencies in the 1997 Plasma Exchange/Sandoglobulin Guillain-Barré Syndrome Trial, the nature of those events was different between the two groups. It is therefore possible that the overall costs of treating these AEs will also be dissimilar. This, however, may be of no importance since the complications for both treatments are typically mild and self-limited requiring minimal intervention. Second, we did not consider the influence of either abbreviated TPE or IVIg treatment courses on overall costs. In the Plasma Exchange/Sandoglobulin Guillain-Barré Syndrome Trial, less than 75% of the planned intervention was given to 13.8% of patients in the TPE group versus just 2.3% in the IVIg group. All subjects were included in the outcomes analysis. In addition, alternate replacement fluids and/or a lower percentage of HSA in the replacement fluid could be used, though this is not consistent with published guidelines. If similar under-dosing rates or use of alternate replacement fluids occur in current practice, this would result in further reductions of the average cost of TPE. Finally, our study examines only the direct costs of TPE and IVIg, not the total costs. In order to determine total costs it would be necessary to have data on the types and frequency of adverse effects of the two treatments as well the costs associated with treating these complications. Contemporary data is not available.
By treating GBS patients with a course of five TPE treatments instead of five IVIg infusions, U.S. hospitals can expect to realize a cost savings of more than $5,680.00. For a given hospital, the magnitude of this cost advantage will be only marginally affected by its average IVIg and albumin acquisition costs and local nursing wage rates. Considering the high cost of IVIg as a maintenance therapy, especially for other conditions like chronic inflammatory demyelinating polyneuropathy and myasthenia gravis, institutions may realize significant savings. Based not only on its efficacy and safety profiles but on its financial considerations as well, TPE should be strongly viewed as first line therapy in the treatment of GBS patients.
This study was sponsored by CaridianBCT.
Dr. Winters has received funding from CaridianBCT to investigate the use of the Spectra Optia in the collection of hematopoietic progenitor cells for stem cell transplantation. He is currently the president of the American Society for Apheresis and serves as an editorial board member for the Journal of Clinical Apheresis.
Dr. Brown reports no competing interests.
Dr. Hazard worked as a paid consultant (ACE Strategic Reimbursement) to CaridianBCT. She is a health economist with over 20 years experience consulting to the pharmaceutical, biotechnology and device industries.
Mr. Chainani worked as a paid consultant (ACE Strategic Reimbursement) to CaridianBCT until December 2009. Since January 2010, he has been an employee of CaridianBCT heading their HEOR, Pricing and Reimbursement Group. He has 30 years of diversified experience in pharmaceutical, biotechnology, device industries and consulting.
Dr Andrzejewski has served in a paid consultant capacity to the Gerson Lehrman Group as a subject matter expert on issues related to blood component therapy including the use of albumin and intravenous IgG products and apheresis medicine applications. He is a Past President of the American Society for Apheresis and serves as an editorial board member for the Journal of Clinical Apheresis.
JW and CA conceived of the study, participated in its design, collected institutional data, and helped draft the manuscript. DB participated in the study design, collected institutional data, and helped draft the manuscript. EH and AC collected government data, created the cost minimization model, and helped draft the manuscript. All authors read and approved the final manuscript.
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{
"redpajama_set_name": "RedPajamaC4"
}
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Q: Experiencing issues Changing form traits for users Im experiencing an issue being able to change user profile data traits for users once I set them with the initial form in forms.py to override the generic django settings.
class EditProfileForm(UserChangeForm):
class Meta:
model = User
fields=('username','first_name','last_name','email','street','city','state','zipcode','country',)
It fails when I try this, but works if I remove street, city, state, zipcode, and country, even though the registration form allows it to fill in all of this information.
The original form allows me to set all of this information with the registration form in forms.py without issue, and ive confirmed its working without flaw. However, it wont let me edit the setting once I set it.
Any insight would be absolutely appreciated. I believe it may be because the superuser doesnt have these fields, and if so, if I could set a default='' if thats possible? Im not aware of being able to do that though for previously existing users, and Ive also tried erasing all users and attempting it with no users, but that didnt work either.
A: Typically when I extend a django auth, I create a new model for the profile than override the user save method to also save a new creation into the extended profile table.
See this answer for more details... Creating a extended user profile
Another solution to this would be to use the new Django Signals to essentially do the same thing but with a more modern API.
https://docs.djangoproject.com/en/2.1/topics/signals/
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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NASA mission to probe moon's atmosphere may solve Apollo-era mystery
An artist's depiction of NASA's Lunar Atmosphere and Dust Environment Explorer (LADEE) spacecraft.
(Dana Berry / NASA )
Aug. 23, 2013 8:53 AM PT
NASA has sent a spacecraft to study the moon's surface and another to probe its "lumpy" gravity field, and now it's sending a new explorer to taste its atmosphere.
The Lunar Atmosphere and Dust Environment Explorer (LADEE) mission will seek to study the vanishingly thin lunar "air" before commercial space-goers in the coming years potentially start disrupting the near-pristine environment.
The $280-million mission is set for launch Sept. 6 at 11:27 p.m. EDT, LADEE scientists said Thursday during a news conference.
Yes, the moon does have an atmosphere, said program scientist Sarah Noble, "it's just really, really thin." This so-called exosphere is so sparse that its individual particles don't even collide with one another.
Earth has a thin exosphere too, she said, but it doesn't hug the surface, it hovers far above our thick atmosphere -- past even where the International Space Station orbits.
Planetary scientists are on the hunt for exoplanets with a dense, Earth-like atmosphere, but it's far more common for planets' surfaces to be hugged by a thin exosphere, Noble said. So studying the moon could teach researchers a lot about other rocky bodies, and even some in our own neighborhood such as Mercury, large asteroids and other planets' moons.
The scientists are also hoping to determine the nature of the bright rays that Apollo astronauts saw stretching across the lunar sky. These "streamers" are thought to be caused tiny grains of dust kicked tens of kilometers into the air, but researchers haven't been able to test that theory.
With this new mission, Noble said, the team can "go and solve this mystery that has been puzzling scientists for almost 50 years."
The spacecraft will study the lunar environment using a dust analyzer and two different spectrometers. It will also be testing out some fancy new equipment: a system to beam data back with near-infrared light, not radio waves, said Don Cornwell, the mission manager for the Lunar Laser Communications Demonstration. This will allow scientists to send data at six times the rate of current advanced radio communication systems -- with half the weight in equipment and using 25% less power.
If the new system is successful, perhaps future missions such as the 2020 rover that NASA's planning to send to Mars could use it to send back data as unheard of as 3-D video, said John Grunsfeld, NASA's associate administrator for science.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Wingen ist eine Ortschaft in Upper Hunter Shire, New South Wales, Australien. Das Dorf liegt am New England Highway und hat mehrere hundert Einwohner.
Wingen ist seit langem bekannt für seinen Burning Mountain, einen natürlichen Kohlebrand. Der Burning Mountain brennt nach wissenschaftlichen Schätzungen seit 6000 Jahren und ist damit der älteste bekannte natürliche Brand eines Kohlenflözes. Auf den Flözbrand geht vermutlich auch der Name des Ortes zurück; in der Sprache des regionalen Stammes der Aborigines, der Wonnarua, bedeutet das Wort "Feuer".
Weblinks
Einzelnachweise
Ort in New South Wales
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
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\section*{Acknowledgments}
K.D.~is supported by the ETH AI Center and the ETH Foundations of Data Science.
We would like to thank Geoffrey Chinot and Matthias L\"offler for insightful discussions.
Finally, we are very grateful to Sara van de Geer for her valuable advice.
\bibliographystyle{alpha}
\section{Future work}
\else
\section{FUTURE WORK}
\fi
\label{subsec:future_work}
Our main result gives tight bounds for BP on isotropic Gaussian features.
It would be interesting to extend the study to other connected settings, which we now motivate and for which we summarize key challenges.
Furthermore, we pose a research question which aims to give a better intuition for the proof.
\paragraph{Necessity of tightness at the localization step.}
As discussed in Section~\ref{subsec:l1proofsketch}, in order to obtain the right rate in Theorem~\ref{thm:main}, the localization step of our analysis needed to be very tight.
The expression we derive for a high-probability upper bound $M(n,d)$ (from Proposition~\ref{prop:main_norm}) on $\min_{Xw=\xi} \norm{w}_1$ needs to be precise up to relative error of no more than $\Theta \left( \frac{1}{\log(d/n)^2} \right)$.
This strikes us as an unusual feature of our derivation.
Yet, it is unclear whether this is an artifact of our analysis via the application of the GMT, or whether this is due to the nature of the statistical problem itself.
More specifically, we motivate future research to answer the question whether it is true that
\begin{enumerate}
\item for any $\tilde{M}(n,d) = c M(n,d)$ with $c>1$, we have that with high probability,
\begin{equation}
\max_{\substack{
\norm{w}_1 \leq \tilde{M}(n,d) \\
Xw=\xi
}} \norm{w}_2^2
\asymp \sigma^2.
\end{equation}
\item for any $\tilde{M}(n,d) = M(n,d)(1+\omega(\frac{1}{\log(d/n)}))$, we have that with high probability,
\begin{equation}
\max_{\substack{
\norm{w}_1 \leq \tilde{M}(n,d) \\
Xw=\xi
}} \norm{w}_2^2
= \omega\left(\frac{\sigma^2}{\log(d/n)}\right).
\end{equation}
\end{enumerate}
Resolving this question is challenging due to the non-concavity of the maximization objective. While we can still use the GMT to upper-bound this quantity (see Proposition~\ref{prop:CGMT_application}), we cannot use the CGMT to lower-bound it, and thus the methodologies
used in this paper fall short.
As a possible direction, we note that
this hypothesis is related to the question of finding tight lower bounds for the diameter of the intersection of the kernel of $X$ and the $\ell_1$-ball (see Theorem~3.5 in \cite{vershynin2011lectures}).
\paragraph{Non-isotropic features.}
Theorem~\ref{thm:main}
assumes isotropic features as we are interested in
showing consistency of BP for inherently high-dimensional input data.
By contrast, recently there has been an increased interest in studying spiked covariance data models (see e.g.\ \cite{bartlett_2020,muthukumar2021classification,chatterji_2021}). In such settings even
min-$\ell_2$-norm interpolators can achieve consistency.
The main obstacle
to extending our methodology to
non \iid\ features lies in
adapting the definition
of the path $\{\gamma(\alpha)\}_\alpha$.
Assuming a diagonal covariance matrix,
such an extension should be relatively straightforward.
We leave this task and the challenging non-diagonal case for future work.
\paragraph{Non-Gaussian features.}
The proof of Theorem~\ref{thm:main} crucially relies on the (Convex)
Gaussian Minimax Theorem \cite{thrampoulidis_2015,gordon_1988}, and
hence on the assumption that the input features are drawn from a
Gaussian distribution. In Figure~\ref{fig:distributions}, we include
plots of the prediction error $\norm{\what - \wgt}_2^2$ not only for Gaussian but also for Log Normal and Rademacher distributed features. We
observe that in all three cases, the prediction error
closely follows the trend line $\frac{\sigma^2}{\log(d/n)}$ (dashed curve). This leads
us to
conjecture that Theorem~\ref{thm:main} can be extended to a more general class of
distributions.
Generalizing our results in this direction appears to be a
challenging task since
the tools
used
in this paper are not directly applicable anymore.
Instead, for heavy-tailed distributions, a popular theoretical framework is the small-ball method
\cite{mendelson_2014,koltchinskii_2015}, which covers the Log Normal and Rademacher distributions.
\ifarx The authors of the paper
\cite{chinot_2021}
\else
\citet{chinot_2021}
\fi
apply this approach to
min-$\ell_1$-norm interpolation, and obtain the constant upper bound $O(\sigma^2)$, under more general assumptions than our setting
(in particular their analysis handles adversarial noise with magnitude controlled by $\sigma^2$).
Yet, it is unclear whether the looseness of their upper bound is an artifact of their proof, or whether the small-ball method itself is too general to capture the rates observed in Figure~\ref{fig:distributions}.
Finally,
we also leave
it as future work to adapt our proof technique for minimum-norm interpolators with general norms,
and for classification tasks.
\ifarx
\section{Conclusion}
\else
\section{CONCLUSION}
\fi
\label{sec:conclusion}
By introducing a novel proof technique, we derive matching upper and lower bounds of order $\frac{\sigma^2}{\log(d/n)}$ on the prediction error of
basis pursuit (BP, or min-$\ell_1$-norm interpolation) in noisy sparse linear regression.
Our result closes a gap in the minimum-norm interpolation literature,
disproves a conjecture from \cite{chinot_2021},
and is the first to imply asymptotic consistency of a minimum-norm interpolator
for isotropic features.
Furthermore, the prediction error decays with the amount of overparametrization $d/n$, confirming that BP also benefits from the regularization effect of high dimensionality, as suggested by the modern storyline on interpolating models.
\end{document}
\section{Main result}
\else
\section{MAIN RESULT}
\fi
\label{sec:main_result}
In this section we state our main result, followed by a
discussion of its implications.
We consider a linear regression model with input vectors $x \in
\RR^d$ drawn from an isotropic Gaussian distribution $x \sim
\Normal(0,I_d)$, and response variable $y = \innerprod{\wgt}{x} +
\xi$, where $w^*$ is the ground truth to be estimated and $\xi \sim
\Normal(0,\sigma^2)$ is a noise term independent of $x$.
Given $n$ \iid\ random samples $(x_i,y_i)_{i=1}^n$, the goal is to estimate $\wgt$
and obtain a small prediction error
(or risk)
for the estimate $\what$
\begin{equation}
\EE_{x,y} (\langle \what, x\rangle - y)^2 -\sigma^2
= \| \what - \wgt\|_2^2
\end{equation}
where we subtract the irreducible error $\sigma^2$. Note that this is also exactly the $\ell_2$-error of the estimator.
We study the min-$\ell_1$-norm interpolator (or BP solution)
defined by
\begin{equation}
\what = \argmin_w \|w\|_1
~~~\mathrm{such~that}~~~
\forall i,~ \langle x_i, w \rangle = y_i.
\end{equation}
Our main result, Theorem~\ref{thm:main}, provides non-asymptotic matching upper and lower bounds for the prediction error of this estimator:
\begin{theorem} \label{thm:main}
Suppose
${\norm{\wgt}_0 \leq \cwone \frac{n}{\log(d/n)^{5}}}$
for some universal constant $\cwone>0$.
There exist universal constants $\cn,\cl,\cd,\cone, \ctwo, \cprobgeoff>0$ such that,
for any $n,d$ with
$n \geq \cn$ and
$\cl n \log(n)^2 \leq d \leq \exp(\cd n^{1/5})$, the prediction error satisfies
\begin{equation}
\label{eq:mainbound}
\abs{
\| \what - \wgt\|_2^2
- \frac{\sigma^2}{\log(d/n)}
}
\leq \cone \frac{\sigma^2}{\log(d/n)^{3/2}}
\end{equation}
with probability at least
$1- \ctwo \exp\left(-\frac{n}{\log(d/n)^5}\right)$ $- d \exp\left(-\cprobgeoff n \right)$
over the draws of the dataset.
\end{theorem}
A proof sketch is presented in Section~\ref{sec:proof_sketch} and the full proof is given in \ifarx Section\else Appendix\fi~\ref{sec:proof_maintext}. We refer to Section~\ref{subsec:future_work} for a discussion on limitations of the assumptions.
This theorem proves an exact
statistical rate with respect to the leading factor of order
$\frac{\sigma^2}{\log(d/n)}$ for the prediction error of the BP solution.
Previous lower bounds
of order $\Omega \left(
\frac{\sigma^2}{\log(d/n)} \right)$
for the same distributional
setting (isotropic Gaussian features, \iid\ noise) only apply under
more restrictive assumptions, such as the zero-signal case $\wgt = 0$
\cite{muthukumar_2020}, or assuming $d >n^4$
\cite{ju_2020}.
On another note the best known upper bounds are of constant order $O(\sigma^2)$
\cite{chinot_2021,wojtaszczyk_2010}. Our result both proves the lower bound in more generality and significantly improves the upper bound that matches the lower bound, showing that the lower bound is in fact tight.
An important implication of the upper bound is that BP achieves high-dimensional asymptotic consistency when $d = \omega(n)$,
thus disproving to a recent conjecture by \cite{chinot_2021}.
\paragraph{Dependency on $\wgt$.}
We note that the bound for the risk in Theorem~\ref{thm:main} is independent of the choice of $\wgt$ assuming that it is sparse (i.e, has bounded $\ell_0$-norm). Essentially, this arises from the well known fact that in the noiseless case ($\sigma =0$) we can achieve exact recovery \cite{candes_2008} of sparse ground truths.
More generally, existing upper bounds for
the prediction error of the BP estimator
for general ground truths $\wgt$ are of the form%
\footnote{The notation $a \lesssim b$ means that there exists a universal constant $c_1>0$ such that $ a \leq c_1 b$, and we write
$a \asymp b$ for $a \lesssim b$ and $b \lesssim a$.}
\begin{equation}
\label{eq:chinotbound}
\norm{\what - \wgt}_2^2
\lesssim
\frac{\|\xi\|_2^2}{n} + \|\wgt\|_1^2 \frac{\log(d/n)}{n}
\end{equation}
(see \eg\ \cite[Theorem~3.1]{chinot_2021}).
That is, they contain a first term reflecting the error due to overfitting of the noise $\xi$ which is independent of $\wgt$
and a second term which can be understood as the noiseless error only depending on $\wgt$ but not on the noise $\xi$. In fact, the authors of both papers show that assuming the ground truth is hard-sparse (bounded $\ell_0$-norm), the second term on the RHS in Equation~\eqref{eq:chinotbound} can be avoided, resulting in the bound $ \norm{\what - \wgt}_2^2 \lesssim \frac{\|\xi\|_2^2}{n}$. Therefore, it is also not surprising that our tighter bound in Equation~\eqref{eq:mainbound} does not explicitly depend on $\wgt$.
\begin{figure*}[t]
\centering
\begin{subfigure}[b]{0.45\linewidth}
\centering
\includegraphics[width=\linewidth]{images/riskvslog-lognormal_precise.pdf}
\caption{Prediction error of BP vs.\ $\log(d/n)$} \label{fig:distributions}
\end{subfigure}
\begin{subfigure}[b]{0.45\linewidth}
\centering
\includegraphics[width=\linewidth]{images/riskvssimgatrend.pdf}
\caption{Prediction error vs.\ $\sigma^2$}
\label{fig:tradeoff}
\end{subfigure}
\captionsetup{width=.92\linewidth}
\caption{Prediction error as a function of
(a) $\log(d/n)$ with varying $d$ and $n=400$ fixed, and
(b) $\sigma^2=1$ with $d=20000,n=400$.
The features are generated by drawing from the isotropic
zero-mean and unit-variance
(b) \emph{Normal} and (a)~\emph{Normal}, \emph{Log Normal} and \emph{Rademacher} distributions.
For BP on Gaussian-distributed features (orange squares),
the plots correctly reflect the theoretical rate
${\frac{\sigma^2}{\log(d/n)}}$
(dashed curve).
See Section~\ref{subsec:numerical} for
further details.}
\label{fig:figure1}
\vspace{-0.1in}
\end{figure*}
\subsection{Numerical simulations}
\label{subsec:numerical}
We now present numerical simulations illustrating Theorem~\ref{thm:main}. Figure~\ref{fig:distributions} shows the prediction error of BP plotted as a function of $\log(d/n)$ with varying $d$ and $n=400$ fixed, for isotropic inputs generated from the
zero-mean and unit-variance
\emph{Normal}, \emph{Log Normal} and \emph{Rademacher} distributions. For all three distributions, the prediction error closely follows the trend line
$\frac{\sigma^2}{\log(d/n)}$ (dashed curve). While Theorem~\ref{thm:main} only applies for Gaussian features, the figure suggests that this statistical rate of BP holds more generally (see discussion in Section~\ref{subsec:future_work}).
Figure~\ref{fig:tradeoff} shows the prediction error of the min-$\ell_1$-norm (BP) and min-$\ell_2$-norm interpolators as a function of the noise $\sigma^2$, for fixed $d$ and $n$.
The prediction error of the former again aligns with the theoretical rate $\frac{\sigma^2}{\log(d/n)}$.
Furthermore, we observe that the min-$\ell_1$-norm
interpolator is sensitive to the noise level $\sigma^2$, while the min-$\ell_2$-norm interpolator has a similar (non-vanishing) prediction error across all values of $\sigma^2$.
For both plots we use $n=400$ and average the prediction error over 20 runs; in Figure~\ref{fig:tradeoff} we additionally show the standard deviation (shaded regions). The ground truth is $\wgt = (1,0,\cdots,0)$.
Finally, we choose $\sigma^2 =1$ in Figure~\ref{fig:distributions}, and $d = 20000$ in Figure~\ref{fig:tradeoff}.
\subsection{Implications and insights}
We now discuss
further
high-level implications and insights that follow from Theorem~\ref{thm:main}.
\paragraph{High-dimensional asymptotic consistency.}
Our result proves consistency of BP for any asymptotic regime $d \asymp n^{\beta}$ with $\beta >1$.
In fact, we argue that those are the only regimes of interest.
For $d$ growing exponentially with $n$, known minimax lower bounds for sparse problems
of order $\frac{\sigma^2 s\log(d/s)}{n}$ (with $s \leq n$ the $\ell_0$-norm of the BP estimator), preclude consistency \cite{verzelen_2012}.
On the other hand, for linear growth ${d \asymp n}$, i.e., $\beta = 1$ -- studied in detail
\ifarx in the paper \cite{li_2021}
\else by \citet{li_2021}
\fi
--, the uniform prediction error lower bound $\frac{\sigma^2 n}{d-n}$ holding for all interpolators \cite{zhou20,muthukumar_2020} also forbids vanishing prediction error.
Note that for $d \asymp n^\beta$ ($\beta>1$), asymptotic consistency can also be achieved
by a carefully designed ``hybrid'' interpolating estimator \cite[Section 5.2]{muthukumar_2020}; contrary to BP, this estimator is not a minimum-norm interpolator, and is not structured (not $n$-sparse).
\paragraph{Trade-off between structural bias and sensitivity to noise.}
As mentioned in the introduction,
our upper bound on the prediction error shows that, contrary to min-$\ell_2$-norm interpolation, BP is able to learn sparse signals
in high dimensions
thanks to its structural bias
towards sparsity.
However, our lower bound can be
seen as a tempering negative result: The prediction error decays only at a slow
rate of $\frac{\sigma^2}{\log(d/n)}$.
Compared to min-$\ell_2$-norm interpolation, BP (min-$\ell_1$-norm interpolation) suffers from a higher sensitivity to noise, but possesses a more advantageous structural bias.
To compare the two methods' sensitivity to noise, consider the case $\wgt = 0$, where the prediction error purely reflects the effect of noise.
In this case, although both methods achieve vanishing error, the statistical rate for BP, $\frac{\sigma^2}{\log(d/n)}$, is much slower than that of min-$\ell_2$-norm interpolation, $\sigma^2 \max(\frac{1}{\sqrt{n}},\frac{n}{d})$ \cite[Theorem 3]{koehler_2021}.
Contrariwise, to compare the effect of structural bias, consider the noiseless case with a
non-zero ground truth.
It is well
known that BP successfully learns sparse signals \cite{candes_2008},
while min-$\ell_2$-norm interpolation always fails to learn the ground truth due to the
lack of any corresponding structural bias.
Thus, there appears to be a trade-off between structural bias and sensitivity to noise: BP benefits from a strong
structural bias, allowing it to have good performance for noiseless recovery of
sparse signals, but in return displays a poor rate in the presence of
noise -- while min-$\ell_2$-norm interpolation has no structural bias
(except towards zero), causing it to fail to recover any non-zero signal even
in the absence of noise, but in return does not suffer from overfitting
of the noise. This behavior is also illustrated in
Figure~\ref{fig:tradeoff}.
\end{document}
\section{Introduction}
\else
\section{INTRODUCTION}
\fi
\label{sec:intro}
Recent experimental studies \cite{belkin_2019,zhang_2021} reveal that in the
modern high-dimensional regime, models that perfectly fit noisy
training data can still generalize well. The phenomenon stands in contrast to
the classical wisdom that interpolating the data results in poor
statistical performance due to overfitting. Many
theoretical papers have explored why, when, and to what extent
interpolation can be harmless for generalization, suggesting a coherent storyline:
High dimensionality itself can have a regularizing effect, in the
sense that it lowers the model's sensitivity to noise.
This intuition emerges from the fast-growing
literature studying min-$\ell_2$-norm interpolation in the
regression setting with input dimension $d$ substantially exceeding
sample size $n$ (see \cite{bartlett_2020,dobriban_2018} and
references therein). Results and intuition for this setting also
extend to kernel methods \cite{ghorbani_2021,mei_2019}.
\ifarx \else \\ \fi
However, a
closer look at this literature reveals that while high
dimensionality decreases the sensitivity to noise (error due to variance),
the prediction error generally does not vanish as $d,n\to\infty$.
Indeed, the bottleneck for asymptotic consistency is a non-vanishing bias term which can only be avoided
when the features have low effective dimension
$\effdim = \trace{\Sigma}/\spnorm{\Sigma} \ll n$, where $\Sigma$ is the covariance matrix \cite{tsigler_2020}.
Therefore, current theory
does not yet provide a convincing explanation
for why interpolating models generalize well for inherently high-dimensional input data.
This work takes a step towards addressing this gap.
When the input data is effectively high-dimensional (\eg\ isotropic and $d \gg n$),
we generally cannot expect any
data-driven estimator
to generalize well
unless there is
underlying
structure that can be exploited.
In this paper, we hence focus on linear regression on isotropic Gaussian features with the simplest
structural assumption: sparsity of the ground truth in the
standard basis. For this setting, the $\ell_1$-penalized regressor (LASSO, \cite{tibshirani_1996})
achieves minimax optimal rates in the presence of noise
\cite{vandegeer_2008}, while basis pursuit (BP, \cite{chen_1998}) -- that is
min-$\ell_1$-norm interpolation -- generalizes well in
the noiseless case but is known to be very sensitive to noise
\cite{candes_2008,donoho_2006}.
Given recent results on high dimensionality decreasing sensitivity of
interpolators to noise, and classical results on the low bias of BP for
learning sparse signals, the following question naturally arises:
\begin{center}
\emph{Can we consistently learn sparse ground truth functions
with minimum-norm interpolators on inherently %
high-dimensional features?}
\end{center}
So far, upper bounds on
the prediction error
of the BP
estimator
of the order of the noise level $O(\sigma^2)$
have been derived for isotropic Gaussian \cite{koehler_2021,ju_2020,wojtaszczyk_2010},
sub-exponential \cite{foucart_2014}, or heavy-tailed
\cite{chinot_2021,krahmer_2018} features.
In the case of isotropic Gaussian features, even though
\ifarx the authors of the paper \cite{chinot_2021}
\else \citet{chinot_2021}
\fi
show a tight matching lower bound for adversarial noise,
for \iid\ noise the best known results are not tight:
there is a gap between the non-vanishing upper bound $O(\sigma^2)$
\cite{wojtaszczyk_2010} and the lower bound $\Omega\left( \frac{\sigma^2}{\log(d/n)} \right)$ \cite{chatterji_2021,muthukumar_2020}.
For \iid\ noise,
\ifarx the authors of the paper \cite{chinot_2021}
\else \citet{chinot_2021}
\fi
conjecture that BP does not achieve consistency (see also \cite{koehler_2021}).
\paragraph{Contribution.}
We are the first to answer the above question in the affirmative.
Specifically, we show that for isotropic Gaussian features,
BP does in fact achieve asymptotic consistency
when $d$ grows superlinearly and subexponentially in~$n$,
disproving the recent conjecture by \cite{chinot_2021}.
Our result closes the aforementioned gap in the
literature on BP: We give matching upper and lower bounds of order~${ \frac{\sigma^2}{\log(d/n)}}$
on the prediction error
of the BP estimator, exact up to terms that are negligible when $d \gg n$.
Further, our proof technique is novel and may be of independent interest.
\paragraph{Structure of the paper.}
The rest of the article is structured as follows. In
Section~\ref{sec:main_result}, we give our main result and
discuss its implications.
In Section~\ref{sec:proof_sketch}, we present a proof sketch
and provide insights on why our approach leads to tighter bounds than previous works.
We discuss the
scope
of our assumptions and motivate future work in Section~\ref{subsec:future_work},
and conclude the paper in
Section~\ref{sec:conclusion}.
\end{document}
\section{Proof of main result}
\else
\section{PROOF OF MAIN RESULT}
\fi
\label{sec:proof_maintext}
In this section, we present the proof of our main result, Theorem~\ref{thm:main}.
In \ifarx Section~\ref{subsec:proof_of_thm}\else Appendix~\ref{subsec:proof_of_thm}\fi, we describe the main steps of the proof rigorously, in the form of three propositions which we then prove in \ifarx Section\else Appendix\fi~\ref{subsec:proof_of_props_phi}.
Full proofs for the intermediary Lemmas and Propositions are given in Appendix~\ref{apx:sec:appendix_proofs}.
\paragraph{Notation.}
On the finite-dimensional space $\RR^d$, we write $\norm{\cdot}_2$ for the Euclidean norm and $\innerprod{\cdot}{\cdot}$ for the Euclidean inner product.
The $\ell_1$ and $\ell_{\infty}$-norms are denoted by $\norm{\cdot}_1$ and $\norm{\cdot}_\infty$, respectively.
The vectors of the standard basis are denoted by $e_1, ..., e_d$, and
$\onevec \in \RR^d$ is the vector with all components equal to $1$.
For $s \leq d$ and $H \in \RR^d$, $\Hvecs$ is the vector such that $(\Hvecs)_i = H_i$ if $i \leq s$ and $0$ otherwise.
$\NNN(\mu, \Sigma)$ is the normal distribution with mean $\mu$ and covariance $\Sigma$, $\Phi(x)$ is
the cumulative distribution function of the scalar standard normal distribution, $\Phic(x) = 1-\Phi(x)$,
and $\log$ denotes the natural logarithm.
For all $s \leq d$, we denote by $t_s \in \RR$ the quantile of the standard normal distribution defined by
$2\Phic(t_s) = s/d$.
The $n$ samples $x_i \in \RR^d$
form the rows of the data matrix \mbox{$X = \left[ x_1 ~...~ x_n
\right]^\top$}, with $X_{ij} \sim \NNN(0,1)$ for each $i,j$. The
scalars $y_i$, $\xi_i$ are also aggregated into vectors $y, \xi \in
\RR^n$ with $\xi \sim \NNN(0, \sigma^2 I_n)$ and $y = Xw^* + \xi$.
With this notation, $\hw$ interpolates the data $X\hw = y$ which is equivalent to
$X(\hw-w^*) = \xi$.
To easily keep track of the dependency on dimension and
sample size,
we reserve the $O(\cdot)$ notation to contain only universal constants, without any hidden dependency
on $d$, $n$, $\sigma^2$ or $\norm{w^*}_0$.
We will also use $c_1,c_2,...$ and $\kappa_1,\kappa_2,...$ to denote positive universal constants reintroduced each time in the proposition and lemma statements, except for $\cgeoff$ and $\cbrho$ which should be considered as fixed throughout the whole proof.
\subsection{Proof of Theorem~\ref{thm:main}}
\label{subsec:proof_of_thm}
We proceed by a localized uniform convergence approach,
similar to
\ifarx the papers
\else \fi
\cite{chinot_2021,koehler_2021,ju_2020,muthukumar_2020}, and common in the literature, \eg, on structural risk minimization.
That is, the proof consists of two steps:
\begin{enumerate}
\item \emph{Localization.}
We prove that, with high probability, the min-$\ell_1$-norm interpolator $\hw$ satisfies
$\norm{\hw-w^*}_1 \leq c \sigma \sqrt{\sstar} + \min_{Xw=\xi} \norm{w}_1$ for some universal constant $c>0$.
We then derive a (finer than previously known) high-probability upper bound on the second term,
\begin{equation} \label{eq:Phi_N} \tag{$P_N$}
\min_{Xw=\xi} \norm{w}_1
~ =: \Phi_N
~ \leq M(n,d).
\end{equation}
Consequently, with high probability $\hw$ satisfies
\begin{equation}
\norm{\hw-w^*}_1 \leq c \sigma \sqrt{\sstar} + M(n,d) =: B(n,d).
\end{equation}
\item \emph{Uniform convergence.}
We derive high-probability uniform upper and lower bounds on the prediction error for all interpolators located no farther than $B(n,d)$ from $w^*$ in $\ell_1$ norm.
In symbols,
we find a high-probability upper bound for
\ifarxiv
\begin{align} \label{eq:Phi_+} \tag{$P_+$}
\max_{\substack{
\norm{w-w^*}_1 \leq B(n,d) \\
X(w-w^*)=\xi
}} \norm{w-w^*}_2^2
= \max_{\substack{
\norm{w}_1 \leq B(n,d) \\
Xw=\xi
}} \norm{w}_2^2
~
=: \Phi_+
\end{align}
\else
\begin{align} \label{eq:Phi_+} \tag{$P_+$}
\max_{\substack{
\norm{w-w^*}_1 \leq B(n,d) \\
X(w-w^*)=\xi
}} & \norm{w-w^*}_2^2 \\
= \max_{\substack{
\norm{w}_1 \leq B(n,d) \\
Xw=\xi
}} & \norm{w}_2^2
~
=: \Phi_+
\end{align}
\fi
and a high-probability lower bound for
\ifarxiv
\begin{align} \label{eq:Phi_-} \tag{$P_-$}
\min_{\substack{
\norm{w-w^*}_1 \leq B(n,d) \\
X(w-w^*)=\xi
}} \norm{w-w^*}_2^2
= \min_{\substack{
\norm{w}_1 \leq B(n,d) \\
Xw=\xi
}} \norm{w}_2^2
~
=: \Phi_-.
\end{align}
\else
\begin{align} \label{eq:Phi_-} \tag{$P_-$}
\min_{\substack{
\norm{w-w^*}_1 \leq B(n,d) \\
X(w-w^*)=\xi
}} & \norm{w-w^*}_2^2 \\
= \min_{\substack{
\norm{w}_1 \leq B(n,d) \\
Xw=\xi
}} & \norm{w}_2^2
~
=: \Phi_-.
\end{align}
\fi
\end{enumerate}
By definition of $B(n,d)$ in \eqref{eq:Phi_N}, with high probability the min-$\ell_1$-norm interpolator $\what$ belongs to the set of feasible solutions in \eqref{eq:Phi_+} and \eqref{eq:Phi_-},
and hence the second step yields high-probability upper and lower bounds on its prediction error $\norm{\what-w^*}_2^2$.
The key is thus to derive tight high-probability bounds for the quantities $\Phi_N, \Phi_+, \Phi_-$.
Our derivation proceeds in two parts, described below.
The first part uses the CGMT to convert the original optimization problem to an auxiliary problem, similar to \cite{koehler_2021}.
The second part, which contains the crucial elements for our proof of the vanishing upper bound and is the key technical contribution of this paper, consists in reducing the $d$-dimensional auxiliary problem to a scalar one using a path reparametrization.
\paragraph{Preliminary: Localization around $w^*$.}
The following fact shows how, as announced, $\Phi_N$ can be used to derive a localization bound for $\hw$.
\begin{lemma} \label{lm:triangle_ineq}
Suppose
${\norm{\wgt}_0 \leq \cwone \frac{n}{\log(d/n)^{5}}}$
for some universal constant $\cwone>0$.
There exist universal constants $\cn,\cl,\cd, \cgeoff, \cprobgeoff>0$ such that, if
$n \geq \cn$ and
$\cl n \log(n)^2 \leq d \leq \exp(\cd n^{1/5})$,
then
the min-$\ell_1$-norm interpolator $\hw$ satisfies
\begin{equation}
\norm{\hw-w^*}_1 \leq \cgeoff \sigma \sqrt{\sstar} + \min_{Xw=\xi} \norm{w}_1
\end{equation}
with probability at least
$1-d \exp\left( -\cprobgeoff n \right)$.
\end{lemma}
The proof of Lemma~\ref{lm:triangle_ineq} is given in Appendix~\ref{apx:subsec:triangle}.
Interestingly, it makes use of the loose upper bound $\norm{\hw-w^*}_2^2 \lesssim \sigma^2$, shown previously by \cite{wojtaszczyk_2010} and \cite{chinot_2021}, as an intermediate result.
\subsubsection{(Convex) Gaussian Minimax Theorem}
\label{subsec:CGMTproof}
Since each of the quantities $\Phi_N, \Phi_+, \Phi_-$ is defined as the optimal value of a stochastic program with Gaussian parameters, we may apply the (Convex) Gaussian Minimax Theorem ((C)GMT)
\cite{gordon_1988,thrampoulidis_2015}.
On a high level, given a ``primary'' optimization program with Gaussian parameters, the (C)GMT relates it to an ``auxiliary'' optimization program,
so that high-probability bounds on the latter imply high-probability bounds on the former. The following proposition applies the CGMT on $\Phi_N$ and the GMT on $\Phi_+$, $\Phi_-$.
\begin{proposition}
\label{prop:CGMT_application}
For $\Hgaussian \sim \NNN(0,I_d)$, define the stochastic auxiliary optimization problems:
\ifarxiv
\begin{align}
\tag{$A_N$} \label{eq:phidn}
\phidn(\cnoise) &= \min_w \norm{w}_1
\subjto \innerprod{w}{\Hgaussian}^2 \geq (1+\cnoise) n (\sigma^2 + \norm{w}_2^2) \\
\tag{$A_+$} \label{eq:phidp}
\phidp(\cnoise) &= \max_w \norm{w}_2^2
\subjto \begin{cases}
\norm{w}_1 \leq B(n,d) \\
\innerprod{w}{\Hgaussian}^2 \geq (1-\cnoise) n (\sigma^2 + \norm{w}_2^2)
\end{cases} \\
\tag{$A_-$} \label{eq:phidm}
\phidm(\cnoise) &= \min_w \norm{w}_2^2
\subjto \begin{cases}
\norm{w}_1 \leq B(n,d) \\
\innerprod{w}{\Hgaussian}^2 \geq (1-\cnoise) n (\sigma^2 + \norm{w}_2^2)
\end{cases}
\end{align}
\else
\begin{align}
\tag{$A_N$} \label{eq:phidn}
\phidn(\cnoise) &= \min_w \norm{w}_1 \\
&\subjto \innerprod{w}{\Hgaussian}^2 \geq (1+\cnoise) n (\sigma^2 + \norm{w}_2^2) \\
\tag{$A_+$} \label{eq:phidp}
\phidp(\cnoise) &= \max_w \norm{w}_2^2 \\
&\subjto \begin{cases}
\norm{w}_1 \leq B(n,d) \\
\innerprod{w}{\Hgaussian}^2 \geq (1-\cnoise) n (\sigma^2 + \norm{w}_2^2)
\end{cases} \\
\tag{$A_-$} \label{eq:phidm}
\phidm(\cnoise) &= \min_w \norm{w}_2^2 \\
&\subjto \begin{cases}
\norm{w}_1 \leq B(n,d) \\
\innerprod{w}{\Hgaussian}^2 \geq (1-\cnoise) n (\sigma^2 + \norm{w}_2^2)
\end{cases}
\end{align}
\fi
where $0<\cnoise<1/2$ can be any small enough quantity.
For any $t \in \RR$, it holds that
\begin{align}
\PP( \Phi_N > t ) &\leq 2\PP( \phidn(\cnoise) \geq t ) + 6 \exp\left(-\frac{n \cnoise^2}{100} \right) \label{eq:probeq1} \\
\text{and}~~~ \PP( \Phi_+ > t ) &\leq 2\PP( \phidp(\cnoise) \geq t ) + 6 \exp\left(-\frac{n \cnoise^2}{100}\right) \label{eq:probeq2} \\
\text{and}~~~ \PP( \Phi_- < t ) &\leq 2\PP( \phidm(\cnoise) \leq t ) + 6 \exp\left(-\frac{n \cnoise^2}{100}\right) \label{eq:probeq3} ,
\end{align}
where on the left-hand side $\PP$ denotes the probability distribution over $X$ and $\xi$, and on the right-hand side the distribution over $\Hgaussian$.
\end{proposition}
For the remainder of this proof, we choose%
\footnote{
This choice of $\cnoise$ is justified by the proof of Proposition~\ref{prop:main_norm}. Indeed, for an arbitrary choice of $\cnoise < 1/2$, one could still show the same bound with just an extra factor:
$(\phidn)^2 \leq (1+\cnoise) \Brho(n,d)$,
holding with still the same probability.
This would translate to a bound on $\Phi_N$ holding with probability
${1 - 12 \exp\left(-2\frac{n}{\log(d/n)^5}\right)} - 6 \exp\left(-\frac{n \cnoise^2}{100} \right)$.
So the choice $\cnoise = \frac{10}{\log(d/n)^{5/2}}$ ``comes at no cost'' in terms of the probability with which the bound holds,
while being sufficiently small to allow for a satisfactory bound (it only affects the constant $\cbrho$ appearing in $\Brho(n,d)$).
}
\begin{equation}
\cnoise = \frac{10}{\log(d/n)^{5/2}}.
\end{equation}
As such, from now on, we simply write $\phi_N$, $\phi_+$, $\phi_-$.
The proof of Proposition~\ref{prop:CGMT_application}, given in Appendix~\ref{apx:subsec:CGMT}, closely follows Lemmas~{3-7} in the paper \cite{koehler_2021}.
For clarity, note that the three pairs of stochastic programs
(\ref{eq:Phi_N}/\ref{eq:phidn}),
(\ref{eq:Phi_+}/\ref{eq:phidp}),
(\ref{eq:Phi_-}/\ref{eq:phidm})
are not coupled: Proposition~\ref{prop:CGMT_application} should be understood as consisting of three separate statements, each using a different independent copy of $h$.
As a result of the proposition, the goal of finding high-probability bounds on $\Phi_N, \Phi_+, \Phi_-$ now reduces to finding high-probability bounds on $\phi_N$, $\phi_+$, $\phi_-$, respectively.
\subsubsection{Bounds on \texorpdfstring{$\phidn, \phidp, \phidm$}{phiN,phi+,phi-}}
To obtain tight bounds on the auxiliary quantities $\phidn, \phidp, \phidm$, we adopt a significantly different approach from previous works.
The main idea is to reduce the
optimization problems \eqref{eq:phidn}, \eqref{eq:phidp} and \eqref{eq:phidm} to optimization problems over a parametric
path~$\{\gamma(\alpha)\}_{\alpha} \subset \RR^d$.
Here we only state the results and refer to \ifarx Section\else Appendix\fi~\ref{subsec:proof_of_props_phi} for their proofs and further intuition.
For the remainder of this proof,
we denote by $t_n \in \RR$ the quantile of the standard normal distribution defined by
$2\Phic(t_n) = n/d$.
\begin{proposition}
\label{prop:main_norm}
There exist universal constants $\cn,\cl,\cd, \cbrho>0$ such that, if
$n \geq \cn$ and
$\cl n \leq d \leq \exp(\cd n^{1/5})$,
then
\begin{align}
\phidn^2 &\leq
\frac{\sigma^2 n}{t_n^2} \left(
1 - \frac{2}{t_n^2} + \frac{\cbrho}{t_n^4}
\right)
\end{align}
with probability at least ${1 - 6 \exp\left(-2\frac{n}{\log(d/n)^5}\right)}$ over the draws of $\Hgaussian$.
Consequently, \eqref{eq:Phi_N} holds with
\begin{equation} \label{eq:def_Mnd}
M(n,d) := \sqrt{
\frac{\sigma^2 n}{t_n^2} \left(
1 - \frac{2}{t_n^2} + \frac{\cbrho}{t_n^4}
\right)
}
\end{equation}
with probability at least $1-18 \exp\left( -\frac{n}{\log(d/n)^5} \right)$
over the draws of $X$ and $\xi$.
\end{proposition}
Hence by Lemma~\ref{lm:triangle_ineq}, the min-$\ell_1$-norm interpolator is located close to the true vector $w^*$, namely the $\ell_1$ distance is bounded by the deterministic quantity
\begin{align}
\label{eq:bnequaiton}
\norm{\hw-w^*}_1 \leq c \sigma \sqrt{\sstar} + M(n,d) =: B(n,d)
\end{align}
with probability at least
$1 - 18 \exp\left(-\frac{n}{\log(d/n)^5}\right) - d \exp\left( -\cprobgeoff n \right)$ and with $\cgeoff, \cprobgeoff>0$ some universal constants.
We now establish high-probability upper resp.\ lower bounds for $\phidp$ resp.\ $\phidm$.
\begin{proposition}
\label{prop:main_upper_lower}
Suppose
${\norm{\wgt}_0 \leq \cwone \frac{n}{\log(d/n)^{5}}}$
for some universal constant $\cwone>0$.
There exist universal constants $\cn,\cl,\cd,\cbound, \cprobgeoff>0$ such that, if
$n \geq \cn$ and
$\cl n \leq d \leq \exp(\cd n^{1/5})$,
then
each of the two events
\ifarxiv
\begin{align}
\phidp &\leq \frac{\sigma^2}{\log(d/n)} \left(1+ \frac{\cbound}{\sqrt{\log(d/n)}}\right)
& &\text{and} &
\phidm &\geq \frac{\sigma^2}{\log(d/n)} \left(1 - \frac{\cbound}{\sqrt{\log(d/n)}}\right)
\end{align}
\else
\begin{align}
\phidm &\geq \frac{\sigma^2}{2\log(d/n)} \left(1- \frac{\cbound}{\sqrt{\log(d/n)}}\right) \\
\text{and}~
\phidp &\leq \frac{\sigma^2}{\log(d/n)} \left(1+ \frac{\cbound}{\sqrt{\log(d/n)}}\right)
\end{align}
\fi
happens with probability at least
${ 1 - 18 \exp\left(-\frac{n}{\log(d/n)^5}\right) }$
over the draws of $\Hgaussian$.
\end{proposition}
Theorem~\ref{thm:main} follows straightforwardly from
Lemma~\ref{lm:triangle_ineq} and Propositions~\ref{prop:CGMT_application}, \ref{prop:main_norm} and \ref{prop:main_upper_lower}.
\subsection{Proof of Propositions~\ref{prop:main_norm} and \ref{prop:main_upper_lower}}
\label{subsec:proof_of_props_phi}
In this section we detail our analysis of the auxiliary optimization problems \eqref{eq:phidn}, \eqref{eq:phidp} and \eqref{eq:phidm}.
We start by a remark that considerably simplifies notation:
The definitions of $\phidn, \phidp, \phidm$ are unchanged if,
in \eqref{eq:phidn}, \eqref{eq:phidp}, \eqref{eq:phidm},
$\Hgaussian$ is replaced by the reordered vector of its absolute order statistics, \ie, by $H$ such that $H_i$ is the $i$-th largest absolute value of $h$.
Throughout this proof, we condition on the event where $H$ has distinct and positive components:
\mbox{$H_1 > ... > H_d > 0$}, which holds with probability one.
Henceforth, unless specified otherwise, references to the optimization problems~\eqref{eq:phidn}, \eqref{eq:phidp} and \eqref{eq:phidm} refer to the equivalent problems where $\Hgaussian$ is replaced by $H$.
Also recall that
we choose
$\cnoise = \frac{10}{\log(d/n)^{5/2}}$.
The key steps in the proof of Propositions~\ref{prop:main_norm} and \ref{prop:main_upper_lower} are as follows.
\begin{itemize}
\item For each of the three optimization problems \eqref{eq:phidn}, \eqref{eq:phidp} and \eqref{eq:phidm},
we show that the argmax (or argmin) is of the form $\av \gamma(\alpha)$ for some $\av>0$ and a parametric path
$\Gamma = \lbrace\frac{\gamma(\alpha)}{\alpha}\rbrace_\alpha$
(which depends on $H$).
Hence we can restate \eqref{eq:phidn}, \eqref{eq:phidp} and \eqref{eq:phidm} as optimization problems over
a scalar variable $\alpha$ and a scale variable $\av>0$.
(\ifarx Section\else Appendix\fi~\ref{subsubsec:parametrizing_argmaxmin})
\item Still conditioning on $H$, we explicitly characterize the parametric path $\Gamma$.
In particular, we show that it is piecewise linear with breakpoints $\gamma(\alphathr{s})$ having closed-form expressions.
(\ifarx Section\else Appendix\fi~\ref{subsubsec:characterizing_path})
\item Thanks to the concentration properties of $H$ (\ifarx Section\else Appendix\fi~\ref{subsubsec:concentration_gammas}),
evaluating at one of the breakpoints yields the desired high-probability upper bound on $\phidn$ (\ifarx Section\else Appendix\fi~\ref{subsubsec:proof_bound_phidn}).
\item A fine-grained study of the intersection of
$\RR_+ \Gamma := \lbrace \av\frac{\gamma(\alpha)}{\alpha} \rbrace_{b \in \RR_+, \alpha}$
with the constraint set of \eqref{eq:phidp} and \eqref{eq:phidm}, as well as the concentration properties of $H$, yield the desired high-probability bounds on $\phidp$ and $\phidm$.
(\ifarx Section\else Appendix\fi~\ref{subsubsec:proof_bound_phidp_phidm})
\end{itemize}
\subsubsection{Parametrizing the argmax/argmin}
\label{subsubsec:parametrizing_argmaxmin}
Note that in the optimization problems \eqref{eq:phidn}, \eqref{eq:phidp} and \eqref{eq:phidm},
the variable $w$ only appears through $\norm{w}_2$, $\norm{w}_1$ and $\innerprod{w}{H}$. Thus, we can add the constraint that $\forall i,~w_i \geq 0$ without affecting the optimal solution.
We will show that the path $\Gamma = \left\lbrace\frac{\gamma(\alpha)}{\alpha}\right\rbrace_\alpha$ can be used to parametrize the solutions of the optimization problems, where
$\gamma: [1, \alpham] \to \RR^d$ is defined by
\begin{equation}
\gamma(\alpha) = \argmin_w \norm{w}_2^2
\subjto \begin{cases}
\innerprod{w}{H} \geq \norm{H}_\infty \\
\forall i, w_i \geq 0 \\
\onevec^\top w = \norm{w}_1 = \alpha
\end{cases}
\end{equation}
and $\alpham = d \frac{\norm{H}_\infty}{\norm{H}_1}$.
Specifically, the following key lemma states that (at least one element of) the argmax/argmin of \eqref{eq:phidn}, \eqref{eq:phidp} and \eqref{eq:phidm}
is of the form $\av \frac{\gamma(\alpha)}{\alpha}$ for some $\av > 0$ and $\alpha \in [1, \alpham]$.
This allows to reduce the optimization problems to a single scalar variable and a scale variable.
\begin{lemma}
\label{lm:parametrization}
Denoting for concision $\gennormboundp = \gennormboundp(n,d)$,
we have that:
\begin{enumerate}
\item The variable $w$ in \eqref{eq:phidn} can equivalently be constrained to belong to the set $\RR_+ \Gamma$, i.e.,
\begin{align}
\label{eq:phidn_reparam} \tag{$A_N'$}
\phidn
=
\min_{\av>0, 1 \leq \alpha \leq \alpham} \av
\subjto~
\av^2 \norm{H}_\infty^2 \geq (1+\cnoise) n (\sigma^2 \norm{\gamma(\alpha)}_1^2 + \av^2 \norm{\gamma(\alpha)}_2^2).
\end{align}
\item The variable $w$ in \eqref{eq:phidp} can equivalently be constrained to belong to the set $B\Gamma$, i.e.,
\begin{align}
\label{eq:phidp_reparam} \tag{$A_+'$}
\phidp
=
\max_{1 \leq \alpha \leq \alpham} \gennormboundp^2 \frac{\norm{\gamma(\alpha)}_2^2}{\norm{\gamma(\alpha)}_1^2}
\subjto~
\gennormboundp^2 \norm{H}_\infty^2 \geq (1-\cnoise) n (\sigma^2 \norm{\gamma(\alpha)}_1^2 + \gennormboundp^2 \norm{\gamma(\alpha)}_2^2).
\end{align}
\item The variable $w$ in \eqref{eq:phidm} can equivalently be constrained to belong to the set $(0,B] \Gamma$, i.e.,
\begin{align}
\label{eq:phidm_reparam} \tag{$A_-'$}
&\phidm
=
\min_{\substack{
0<\av \leq \gennormboundp \\
1 \leq \alpha \leq \alpham
}}
\av^2 \frac{\norm{\gamma(\alpha)}_2^2}{\norm{\gamma(\alpha)}_1^2}
\subjto~
\av^2 \norm{H}_\infty^2 \geq (1-\cnoise) n (\sigma^2 \norm{\gamma(\alpha)}_1^2 + \av^2 \norm{\gamma(\alpha)}_2^2).
\end{align}
\end{enumerate}
\end{lemma}
The proof of the lemma is given in Appendix~\ref{apx:subsec:param_argmaxmin}.
To give an intuitive explanation for the equivalence between \eqref{eq:phidn} and \eqref{eq:phidn_reparam}, consider a penalized version of \eqref{eq:phidn}:
$\min_w \norm{w}_1 - \lambda \left(
\innerprod{w}{\Hgaussian}^2
-
(1+\cnoise) n (\sigma^2 + \norm{w}_2^2)
\right)$
with $\lambda>0$.
For fixed values of $\norm{w}_1$ and $\innerprod{w}{\Hgaussian}$, minimizing this penalized objective is equivalent to minimizing $\norm{w}_2^2$.
Hence, we can expect the argmin to be attained at $\av \frac{\gamma(\alpha)}{\alpha}$ for some $\av>0,\alpha$.
\subsubsection{Characterizing the parametric path}
\label{subsubsec:characterizing_path}
As $\gamma(\alpha)$ is defined as the optimal solution of a convex optimization problem,
we are able to obtain a closed-form expression, by a straightforward application of Lagrangian duality.
The only other non-trivial ingredient is to notice that, at optimality, the inequality constraint $\innerprod{w}{H} \geq \norm{H}_\infty$
necessarily holds with equality.
Denote $\Hvecs$ the vector equal to $H$ on the first $s$ components and $0$ on the last $(d-s)$, and similarly for $\onevecs$.
Define, for any integer $2 \leq s \leq d$,
\begin{align}
\alphathr{s} =
\frac{
\left( \norm{\Hvecs}_1 - s H_s \right) \norm{H}_\infty
}{
\norm{\Hvecs}_2^2 - \norm{\Hvecs}_1 H_s
}.
\end{align}
Note that $\alphathr{2} = 1$.
Let $\alphathr{d+1} = \alpham$.
\begin{lemma}
\label{lm:charact_gamma}
For all $1 < \alpha \leq \alpham$,
denote $s$ the unique integer in $\{2,...,d\}$ such that
\mbox{$\alphathr{s} < \alpha \leq \alphathr{s+1}$}.
Then $\gamma(\alpha) = \lambda \Hvecs - \mu \onevecs$
(in particular it is $s$-sparse)
where the dual variables $\lambda$ and $\mu$ are given by
\begin{align}
\lambda &= \frac{1}{s \norm{\Hvecs}_2^2 - \norm{\Hvecs}_1^2}
\left( s \norm{H}_\infty - \alpha \norm{\Hvecs}_1 \right) \\
~~~\text{and}~~~
\mu &= \frac{1}{s \norm{\Hvecs}_2^2 - \norm{\Hvecs}_1^2}
\left( \norm{\Hvecs}_1 \norm{H}_\infty - \alpha \norm{\Hvecs}_2^2 \right).
\end{align}
\end{lemma}
The proof of the lemma is given in Appendix~\ref{apx:subsec:geometric_lemma}.
\subsubsection{Concentration of norms of \texorpdfstring{$\gamma(\alphathr{s})$}{gamma(alphas))}}
\label{subsubsec:concentration_gammas}
Given the explicit characterization of the parametric path, we now
study its breakpoints $\gamma(\alphathr{s})$
(\mbox{$s \in \{2,...,d\}$}),
and more precisely we estimate $\norm{\gamma(\alphathr{s})}_1 = \alphathr{s}$ and $\norm{\gamma(\alphathr{s})}_2$ as a function of $s$
(we have by definition $\innerprod{\gamma(\alphathr{s})}{H} = \norm{H}_\infty$).
Namely, we prove the following concentration result, where, analogously to $t_n$, we let $\ts \in \RR$ denote the quantity such that $2\Phic(\ts) = s/d$.
\begin{proposition}
\label{prop:concentration_gammas}
There exist universal constants $\cs,\cl,\cd,\cpropf>0$ such that for any $s,d$ with
$s \geq \cs$ and
$\cl s \leq d \leq \exp(\cd s^{1/5})$,
\ifarxiv
\begin{align}
\abs{
\frac{\norm{\gamma(\alphathr{s})}_1}{\norm{H}_\infty} - \left(\frac{1}{\ts} - \frac{2}{\ts^3} \right)
}
&\leq \frac{\cpropf}{\ts^5}
& &\text{and} &
\abs{
\frac{\norm{\gamma(\alphathr{s})}_2^2}{\norm{H}_\infty^2} - \frac{2}{s \ts^2}
}
\leq \frac{\cpropf}{s \ts^4}, \label{eq:concentration}
\end{align}
\else
\begin{align}
\abs{
\frac{\norm{\gamma(\alphathr{s})}_1}{\norm{H}_\infty} - (\frac{1}{\ts} - \frac{2}{\ts^3})
}
&\leq \frac{c}{\ts^5}
\\
\abs{
\frac{\norm{\gamma(\alphathr{s})}_2^2}{\norm{H}_\infty^2} - \frac{2}{s \ts^2}
}
&\leq \frac{c}{s t^4}, \label{eq:concentration}
\end{align}
\fi
with probability at least $1 - 6\exp \left( -2\frac{s}{\log(d/s)^5} \right)$
over the draws of $\Hgaussian$.
\end{proposition}
This proposition relies on and extends the literature studying concentration of order statistics \cite{boucheron_2012, li_2020}. An important ingredient for the proof of the proposition is the following lemma, which gives a tight approximation for $\ts$.
\begin{lemma}
\label{lm:tlogds}
There exist universal constants $\cl, \cone>0$ such that, for all $s \leq d/\cl$, $\ts$ satisfies
\begin{equation}
\olt_s^2 - \cone \leq \ts^2 \leq \olt_s^2
\end{equation}
where
\begin{equation}
\olt_s = \sqrt{2 \log(d/s) - \log \log (d/s) - \log(\frac{\pi}{2})}.
\end{equation}
Furthermore, $\cl$ and $\cone$ can be chosen
(\eg\ $\cl = 11$ and $\cone = 1$)
such that
$\log(d/s) \leq \ts^2 \leq 2 \log(d/s)$.
\end{lemma}
The proofs of Proposition~\ref{prop:concentration_gammas}
and of Lemma~\ref{lm:tlogds}
are given in Appendix~\ref{apx:subsec:concentration_gamma}.
\subsubsection{Localization: Proof of Proposition~\ref{prop:main_norm} (upper bound for \texorpdfstring{$\phidn$}{phiN})}
\label{subsubsec:proof_bound_phidn}
\label{apx:subsec:bound_phiN}
We now use the concentration bounds of Proposition \ref{prop:concentration_gammas} to obtain a high-probability upper bound for $\phidn$. Recall from Lemma~\ref{lm:parametrization} that it is given by \eqref{eq:phidn_reparam}:
\begin{equation}
(\phidn)^2 = \min_{\av>0, 1 \leq \alpha \leq \alpham}
\av^2
\subjto
\av^2 \norm{H}_\infty^2 \geq (1+\cnoise) n (\sigma^2 \norm{\gamma(\alpha)}_1^2 + \av^2 \norm{\gamma(\alpha)}_2^2).
\end{equation}
We may rewrite the constraint as
\begin{gather}
\av^2 \norm{H}_\infty^2 \left( 1- (1+\cnoise)n \frac{\norm{\gamma(\alpha)}_2^2)}{\norm{H}_\infty^2} \right)
\geq (1+\cnoise) n \sigma^2 \norm{\gamma(\alpha)}_1^2 \\
\iff
\av^2 \geq
\underbrace{
\frac{\norm{\gamma(\alpha)}_1^2 }{\norm{H}_\infty^2 }
\frac{
\sigma^2 n (1+\cnoise)
}{
1 - (1+\cnoise) n
\frac{\norm{\gamma(\alpha)}_2^2}{\norm{H}_\infty^2}
}
}_{=: \ftild(\alpha)^2}
~~~\text{and}~~~
(1+\cnoise) n \frac{\norm{\gamma(\alpha)}_2^2}{\norm{H}_\infty^2} < 1.
\end{gather}
Thus minimizing over $b$ shows that
$(\phidn)^2 = \min_{1 \leq \alpha \leq \alpham} \ftild(\alpha)^2
\subjto (1+\cnoise) n \frac{\norm{\gamma(\alpha)}_2^2}{\norm{H}_\infty^2} < 1$.
Since we want to upper-bound this minimum, it is sufficient to further restrict the optimization problem by the constraint
$\alpha \in \left\lbrace \alphathr{s} \middle\vert s \in \{2,...,d\} \right\rbrace$,
yielding
\begin{equation}
\label{eq:proofphilowerbound}
(\phidn)^2 \leq \min_{2 \leq s \leq d+1}
\ftild(\alphathr{s})^2
\subjto (1+\cnoise) n \frac{\norm{\gamma(\alphathr{s})}_2^2}{\norm{H}_\infty^2} < 1.
\end{equation}
We now show that for the choice $s=n$, the constraint is satisfied with high probability, and we give a high-probability estimate for the resulting upper bound $\ftild(\alpha_n)^2$.
See Remark~\ref{rk:choice_s_n} below for a justification of this choice. For the remainder of the proof of Proposition~\ref{prop:main_norm}, we condition on the event where the inequalities in Equation~\eqref{eq:concentration} hold for~$s=n$.
By the concentration bound for $\norm{\gamma(\alphathr{n})}_2^2$, a sufficient condition for the choice $s=n$ to be feasible is
\begin{equation}
(1+\cnoise)\frac{2}{t_n^2} \left(1+ \frac{c_1}{t_n^2}\right) < 1
\end{equation}
with $c_1>0$ some universal constant.
Now $t_n^2 \geq \log(d/n)$ by Lemma~\ref{lm:tlogds}, and recall that $\cnoise = \frac{10}{\log(d/n)^{5/2}}$.
For $\cl$ sufficiently large, the above inequality
holds for any $n,d$ with $\cl n \leq d$. Moreover, by the concentration bounds for $\norm{\gamma(\alphathr{n})}_2^2$ and $\norm{\gamma(\alphathr{n})}_1$,
$f(\alphathr{n})^2$ is upper-bounded by
\begin{align}
\ftild(\alpha_n)^2
&\leq
\frac{
(1-\frac{4}{t_n^2} + O(\frac{1}{t_n^4}) ) \sigma^2 n (1+\cnoise)
}{
t_n^2 - 2(1+\cnoise) (1+ O(\frac{1}{t_n^2}) )
}
\leq
\frac{\sigma^2 n}{t_n^2} (1+\cnoise) \left(
1 - \frac{2}{t_n^2} + O(\frac{1}{t_n^4})
\right).
\end{align}
Furthermore, $\cnoise = O \left( \frac{1}{t_n^5} \right)$ by Lemma~\ref{lm:tlogds}, so
$f(\alphathr{n})^2 \leq
\frac{\sigma^2 n}{t_n^2} \left(
1 - \frac{2}{t_n^2} + \frac{\cbrho}{t_n^4}
\right)
=: \Brho(n,d)^2$
for a universal constant $\cbrho>0$.
This concludes the proof of Proposition~\ref{prop:main_norm}.
\begin{remark} \label{rk:choice_s_n}
Let us informally justify why we can expect the choice $s=n$ to approximately minimize $\ftild(\alpha_s)$.
A first justification is that the min-$\ell_1$-norm interpolator $\what$, which is the solution of the optimization problem \eqref{eq:Phi_N}, is well-known to be $n$-sparse. Since the optimization problems \eqref{eq:Phi_N} and \eqref{eq:phidn} are intimately connected via the CGMT (Proposition~\ref{prop:CGMT_application}), we can expect the optimal solution of \eqref{eq:phidn} to have similar properties to $\what$ -- in particular, to have the same sparsity $s=n$. A second, more technical, justification is as follows.
Note that if we replace
$\norm{\gamma(\alphathr{s})}_2^2$
and
$\norm{\gamma(\alphathr{s})}_1$
by their estimates from Proposition~\ref{prop:concentration_gammas}
and ignore the higher-order terms,
we have
\begin{equation}
\ftild(\alphathr{s})^2
\approx \frac{1}{t_s^2}
\frac{
\sigma^2 n (1+\cnoise)
}{
1 - (1+\cnoise) n
\frac{2}{st_s^2}
}
= \frac{
\sigma^2 (1+\cnoise)
}{
\frac{t_s^2}{n} - (1+\cnoise)
\frac{2}{s}
}.
\end{equation}
Thus a good choice for $s$ is given by maximizing the denominator. By
using the estimate $\ts^2 \approx 2 \log(d/s)$ from Lemma~\ref{lm:tlogds},
we can approximate it by
\begin{equation}
\frac{\ts^2}{n} - (1+\cnoise)\frac{2}{s} \approx
\frac{2 \log(d/s)}{n} - (1+\cnoise)
\frac{2}{s} =: g(s).
\end{equation}
Interpreting $s$ as a continuous variable and setting $\frac{d}{ds}g(s) = 0$ yields the choice $s = (1+\cnoise) n \approx n$.
\end{remark}
\subsubsection{Uniform convergence: Proof of Proposition~\ref{prop:main_upper_lower} (bounds for \texorpdfstring{$\phidp$}{phi+} and \texorpdfstring{$\phidm$}{phi-})}
\label{subsubsec:proof_bound_phidp_phidm}
To obtain an upper bound for the maximization problem defining $\phidp$ (resp. lower bound for the minimization for $\phidm$),
a typical approach would be to find a tractable relaxation of the problem.
However, the more obvious relaxations already explored in the paper \cite{koehler_2021} turn out to be unsatisfactorily loose, as discussed in Section~\ref{sec:proofsketch}.
Here, thanks to the one-dimensional structure of our reformulations \eqref{eq:phidp_reparam} and \eqref{eq:phidm_reparam},
we take a different approach and study the monotonicity of the objectives.
We can decompose our proof in three steps.
Firstly, we describe our overall monotonicity-based approach.
Secondly, we find values $\ulalpha,\olalpha$ that allow us to unroll our approach.
Finally, we evaluate the bound that the first two steps give us, thus proving the proposition.
\paragraph{Step 1: Studying the feasible set of \eqref{eq:phidp_reparam} and \eqref{eq:phidm_reparam}.}
Recall
that
$\phidp$, $\phidm$
are respectively given by
Equations~\eqref{eq:phidp_reparam},
\eqref{eq:phidm_reparam}.
We can also write them in the following form, using the fact that $\norm{\gamma(\alpha)}_1 = \alpha$:
\begin{align}
\phidp
= \max_{1 \leq \alpha \leq \alpham} B^2 \frac{\norm{\gamma(\alpha)}_2^2}{\alpha^2}
&\subjto
B^2 \norm{H}_\infty^2 \geq (1-\cnoise) n (\sigma^2 \alpha^2 + B^2 \norm{\gamma(\alpha)}_2^2) \\
\phidm
= \min_{\substack{
0<\av \leq \gennormboundp \\
1 \leq \alpha \leq \alpham
}}
\av^2 \frac{\norm{\gamma(\alpha)}_2^2}{\alpha^2}
&\subjto
\av^2 \norm{H}_\infty^2 \geq (1-\cnoise) n (\sigma^2 \alpha^2 + \av^2 \norm{\gamma(\alpha)}_2^2).
\end{align}
We first study the sets of feasible solutions of \eqref{eq:phidp_reparam} and \eqref{eq:phidm_reparam}. Denote by $I$ the former set, \ie,
\begin{align}
I := \left\lbrace
\alpha \in \left[ 1, \alpham \right]
~\middle\vert~
B^2 \norm{H}_\infty^2 \geq (1-\cnoise) n \left( \sigma^2 \alpha^2 + B^2 \norm{\gamma(\alpha)}_2^2 \right)
\right\rbrace.
\end{align}
Also let $\alphad = \frac{\norm{H}_1\norm{H}_\infty}{\norm{H}_2^2}$; this choice of notation is purely symbolic, and is justified by the fact that $\alphathr{d} < \alphad < \alphathr{d+1}$.
\begin{lemma}
\label{lm:interval}
The following statements hold:
\begin{enumerate}
\item The mapping $\alpha \mapsto \norm{\gamma(\alpha)}_2^2$ is decreasing over $[1, \alphad]$ and increasing over $[\alphad, \alpham]$.
\item The mapping $\alpha \mapsto \norm{\gamma(\alpha)}_2^2$ is convex over $\left[1, \alpham \right]$, and $I$ is an interval.
\item The mapping $\alpha \mapsto \frac{\norm{\gamma(\alpha)}_2^2}{\alpha^2}$ is monotonically decreasing.
\end{enumerate}
\end{lemma}
These monotonicity properties lead us to a proof strategy that can be summarized as follows.
\begin{lemma}
\label{lm:bounds}
Denote $I = [\ulalpha_I, \olalpha_I]$ the endpoints of $I$.
For any $\ulalpha \leq \ulalpha_I$,
\begin{align}
\phidp &\leq
\gennormboundp^2 \frac{\norm{\gamma(\ulalpha)}_2^2}{\ulalpha^2}.
\end{align}
If $\olalpha_I < \alphad$, then for any $\olalpha_I \leq \olalpha \leq \alphad$,
\ifarxiv
\begin{align}
\phidm &\geq \frac{
\sigma^2 n (1-\cnoise)
}{
\norm{H}_\infty^2 - (1-\cnoise) n \norm{\gamma(\oalpha)}_2^2
} \norm{\gamma(\oalpha)}_2^2.
\end{align}
\else
\begin{align}
\phidm &\geq \frac{
\sigma^2 n (1-\cnoise)
}{
\norm{H}_\infty^2 - (1-\cnoise) n \norm{\gamma(\oalpha)}_2^2
} \norm{\gamma(\oalpha)}_2^2.
\end{align}
\fi
\end{lemma}
The proofs of Lemma~\ref{lm:interval} and Lemma~\ref{lm:bounds} are given in Appendix~\ref{apx:subsec:feasible_interval}.
\paragraph{Step 2: A tight admissible choice for $\ulalpha$ and $\olalpha$.}
To apply Lemma~\ref{lm:bounds} and obtain bounds on $\phidp$, $\phidm$, we need to find $\ulalpha$ and $\olalpha$ lying on the left, respectively on the right of the interval $I$, and such that $\olalpha \leq \alphad$.
By having a closer look at the way we derived the expression of $\Brho(n,d)$, we have by construction that with high probability, $\alphathr{n} \in I$.
In fact, we show that
there exist integers $\uls$ and $\ols$ very close to $n$
such that
$\alphathr{\uls}$ already falls to the left of $I$, and $\alphathr{\ols}$ to the right of $I$, with high probability.
\begin{lemma}
\label{lm:boundsonalpha}
Suppose
${\norm{\wgt}_1 \leq \cwone \sqrt{ \frac{\sigma^2 n}{\log(d/n)^{5}}}}$ for some universal constant $\cwone>0$.
There exist universal constants $ \cn, \cl,\cd,\lambda>0$
such that, for any $d,n$ with
$n \geq \cn$ and
$\cl n \leq d \leq \exp(\cd n^{1/5})$,
we can find integers $\us,\os \in \NN_+$ satisfying
\ifarxiv
\begin{align} \label{eq:boundsons}
\us =n\exp\left(-\frac{\lambda}{2t_n}\right)\left(1+ O\left(\frac{1}{t_n^2}\right)\right)
~~~\text{and}~~~
\os = n \exp\left(\frac{\lambda}{2t_n}\right)\left(1+ O\left(\frac{1}{t_n^2}\right)\right)
\end{align}
\else
\begin{align} \label{eq:boundsons}
\os &= n \exp\left(\frac{\lambda}{2t_n}\right)\left(1+ O\left(\frac{1}{t_n^2}\right)\right) \\
\text{and}~
\us &=n\exp\left(-\frac{\lambda}{2t_n}\right)\left(1+ O\left(\frac{1}{t_n^2}\right)\right)
\end{align}
\fi
and
\begin{equation} \label{eq:boundsonalpha}
\alphathr{\us} < \ulalpha_I \leq \alphathr{n} \leq \olalpha_I \leq \alphathr{\os} \leq \alphad,
\end{equation}
with probability at least ${1 - 18 \exp\left(-\frac{n}{\log(d/n)^5}\right)}$ over the draws of $\Hgaussian$.
Moreover, $t_{\us}^2 = t_n^2 + O(1)$ and $t_{\os}^2 = t_n^2 + O(1)$.
\end{lemma}
The proof of the lemma is given in Appendix~\ref{apx:subsec:bound_phip_phim}.
It relies in particular on the assumption that
$\norm{\wgt}_0 \leq \cwone \frac{n}{\log(d/n)^5}$
for some universal constant $\cwone$,
which implies that $\Brho(n,d)$ from Proposition~\ref{prop:main_norm} is the dominating term in $B$, and hence
${
B^2 = (\Brho(n,d) + c \sigma \sqrt{\sstar})^2
= \frac{\sigma^2 n}{t_n^2} \left(
1 - \frac{2}{t_n^2} + O\left( \frac{1}{t_n^4}\right)
\right)
}$.
Furthermore, the equations in the lemma hold true conditionally on the event where the inequalities in Equation~\eqref{eq:concentration} hold simultaneously for $s=n$, $s=\us$, and $s = \os$ -- which indeed occurs with
the announced probability.
These two elements of the proof will be reused in the following step.
\paragraph{Step 3: Applying Lemma~\ref{lm:bounds}.}
Lemma~\ref{lm:boundsonalpha} provides us with a choice of $\ulalpha = \alphathr{\uls}$ and $\olalpha = \alphathr{\ols}$ that satisfy the conditions of Lemma~\ref{lm:bounds} with high probability.
To conclude the proof of Proposition~\ref{prop:main_upper_lower}, all that remains to be done is to
compute the bounds given by Lemma~\ref{lm:bounds}, i.e.,
\begin{align}
\phidp &\leq
B^2 \frac{\norm{\gamma(\alphathr{\uls})}_2^2}{\norm{\gamma(\alphathr{\uls})}_1^2}
& &\text{and} &
\phidm &\geq \frac{
\sigma^2 n (1-\cnoise)
}{
\norm{H}_\infty^2 - (1-\cnoise) n \norm{\gamma(\alphathr{\ols})}_2^2
} \norm{\gamma(\alphathr{\ols})^2}_2^2.
\label{eq:upperlowerboundtemp}
\end{align}
For the remainder of the proof of Proposition~\ref{prop:main_upper_lower}, we condition on the event where the inequalities in Equation~\eqref{eq:concentration}
hold simultaneously for $s=n$, $s=\us$, and $s = \os$.
In particular, the conclusions of Lemma~\ref{lm:boundsonalpha} hold, as discussed just above.
We also recall that, because of the assumption on the growth of $\sstar$, we have
$B^2 = \frac{\sigma^2 n}{t_n^2} \left(
1 - \frac{2}{t_n^2} + O\left( \frac{1}{t_n^4}\right)
\right)$.
By applying the concentration inequalities from Equation~\eqref{eq:concentration}, and using the above estimate for $B^2$, we obtain
\begin{align}
\phidp
&\leq \frac{\sigma^2 n}{t_n^2}
\left(1 + O\left(\frac{1}{t_n^2}\right) \right)
\frac{2}{\uls t_{\uls}^2 } t_{\uls}^2
\left(1+O\left(\frac{1}{t_{\uls}^2}\right)\right)
& &\text{and} &
\phidm &\geq \frac{2\sigma^2 n}{\ols t_{\ols}^2}
\left(1 + O\left(\frac{1}{t_n^2}\right)\right).
\end{align}
By plugging in the approximate expressions of $\uls$ and $\ols$
from Equation~\eqref{eq:boundsons},
as well as the estimates $t_{\us}^2 = t_n^2 + O(1)$ and $t_{\os}^2 = t_n^2 + O(1)$ from Lemma~\ref{lm:boundsonalpha},
we further obtain
\begin{align}
\phidp
&\leq \frac{2\sigma^2 }{t_n^2}
\exp\left(\frac{\lambda}{2t_n}\right)
\left(1+ O\left(\frac{1}{t_n^2}\right)\right)
& &\text{and}~ &
\phidm &\geq \frac{2\sigma^2 }{ t_n^2} \exp\left(-\frac{\lambda}{2t_n}\right)
\left(1 + O\left(\frac{1}{t_n^2}\right)\right).
\end{align}
Finally, by the expansion $t_n^2 = 2\log(d/n) + O\left(\log\log(d/n)\right)$ from Lemma~\ref{lm:tlogds} and by the Taylor series approximation $\exp(x) = 1+ x + O(x^2)$ (for bounded $x$), we obtain the desired bounds
\begin{align}
\phidp &\leq \frac{\sigma^2}{\log(d/n)} \left(1+ O\left(\frac{1}{\sqrt{\log(d/n)}}\right)\right)
& &\text{and} &
\phidm &\geq \frac{\sigma^2}{\log(d/n)} \left(1+ O\left(\frac{1}{\sqrt{\log(d/n)}}\right)\right).
\end{align}
This concludes the proof of Proposition~\ref{prop:main_upper_lower}.
\end{document}
\section{Proof sketch}
\else
\section{PROOF SKETCH}
\fi
\label{sec:proof_sketch}
\label{sec:proofsketch}
In this section we present the main ingredients that are key to prove our risk upper bound of
$\frac{\sigma^2}{\log(d/n)} + O \left( \frac{\sigma^2}{\log(d/n)^{3/2}} \right)$.
The proof sketch is interleaved with remarks providing insights on how our technique allows to improve upon previous works.
For the sake of clarity, we omit the discussion of the matching lower bound, as its proof follows exactly the same ideas.
The full proof is given in
\ifarx Section\else Appendix\fi~\ref{sec:proof_maintext}.
The proof follows a standard localization/uniform convergence argument, where we first upper-bound the $\ell_1$-error $ \|\what - \wstar\|_1 $ (localization) and then uniformly upper-bound the risk (i.e.\ $\ell_2$-error) over all interpolators with bounded $\ell_1$-error (uniform convergence).
\subsection{Localization}
\label{subsec:l1proofsketch}
We derive a high-probability upper bound on the $\ell_1$-error: $\norm{\hw-\wstar}_1 \leq B(n,d)$,
implying that the estimator of interest $\hw$ is an interpolator located in the $\ell_1$-ball of radius $B(n,d)$ centered at $\wstar$.
To upper-bound $\norm{\hw-\wstar}_1$ we first observe that, by definition of the BP estimator $\hw$ and via a simple triangle inequality \cite{chinot_2021}, it holds that
\begin{equation}
\|\what - \wstar\|_1 \leq 2\sqrt{\sstar} \norm{\what - \wgt}_2 + \min_{X w = \xi} \|w\|_1
\end{equation}
which is proven in
Lemma~\ref{lm:triangle_ineq}.
To control the first term, we make use of the loose high-probability upper bound $\norm{\what-\wgt}_2 \lesssim \sigma$ from previous works \cite{chinot_2021, wojtaszczyk_2010}.
Thus we have $2\sqrt{\sstar} \norm{\what-\wgt}_2 \lesssim \sigma \sqrt{\sstar}$.
The second term,
$\Phi_N := \min_{Xw = \xi} \|w\|_1$,
reflects how enforcing the interpolation of noise affects the $\ell_1$-norm of the estimator.
To control it with high probability directly is challenging, due to the randomness of both data $X$ and noise $\xi$.
Instead, we bound it using the Convex Gaussian Minimax Theorem (CGMT) \cite{thrampoulidis_2015}; we postpone the sketch of this derivation to Section~\ref{subsec:proofsketch_phin} where we show how we derive a high probability bound $\Phi_N \leq M(n,d)$ (the precise expression can be found in Proposition~\ref{prop:main_norm}).
Having controlled the two terms separately, we get the high-probability bound
\ifarx
\begin{equation}
\norm{\hw-w^*}_1
\leq c \sigma \sqrt{\sstar} + M(n,d) =: B(n,d)
\end{equation}
\else
\begin{align}
&\norm{\hw-w^*}_1
\leq \cgeoff \sigma \sqrt{\sstar} + M(n,d) =: B(n,d)
\end{align}
\fi
for some universal constant $\cgeoff > 0$.
Note that by assumption on $\sstar$, the first term is of order at most $\sqrt{\frac{\sigma^2 n}{\log(d/n)^5}}$, and hence negligible compared to $M(n,d) \geq \sqrt{\frac{\sigma^2 n}{2\log(d/n)}}$.
\paragraph{How tightness of $M(n,d)$ affects the upper bound and comparison to \cite{chinot_2021,wojtaszczyk_2010}.}
Intriguingly, our analysis requires a very precise expression for the deterministic upper bound on
$\Phi_N = \min_{Xw=\xi} \norm{w}_1$.
Remember that $M(n,d)\approx \left(\frac{\sigma^2 n}{2 \log(d/n) - \log\log(d/n) - \log(\pi)}\right)^{1/2}$ is the precise upper bound used in our analysis, given in Proposition~\ref{prop:main_norm}.
We now discuss the repercussions of choosing a looser upper bound $\widetilde{M}(n,d)$ in our analysis:
\begin{enumerate}
\item for
$\widetilde{M}(n,d) = M(n,d)\left(1 + \frac{c}{\log(d/n)^2}\right)$ with $c>0$: We would still get exactly the precise upper bound of Theorem~\ref{thm:main}.
\item for $\widetilde{M}(n,d) = M(n,d)\left(1 + \frac{c}{\log(d/n)}\right)$ with $c>0$: We would obtain an upper bound of the correct order up to
a universal constant factor
$\norm{\what-\wgt}_2^2 \lesssim \frac{\sigma^2}{\log(d/n)}$.
\item for $\widetilde{M}(n,d) = M(n,d)\left(1 + c\right)$ with $c>0$:
The upper bound for $\Phi_N$ used in the analysis of \cite{chinot_2021} is of order $\sqrt{\frac{2\sigma^2 n}{\log(d/n)}} \geq \widetilde{M}(n,d)$. Using this loose upper bound, our analysis would only yield an upper bound of constant order
$\norm{\what-\wgt}_2^2 \lesssim \sigma^2$,
same as obtained by \cite{chinot_2021,wojtaszczyk_2010}.
\end{enumerate}
Yet, it is not clear whether the tightness of $M(n,d)$ is only needed due to our analysis, or whether any uniform convergence bound with the corresponding $B(n,d)$ would fail to yield tight bounds.
We leave this question as an interesting direction for future work, described further in Section~\ref{subsec:future_work}.
Note that the precise expression $M(n,d)$ alone
combined with previous analysis would only result in a constant upper bound of order $\sigma^2$ if inserted into the analysis
conveyed in \cite{chinot_2021,wojtaszczyk_2010,koehler_2021} (discussed in the next section).
In the following section we discuss the key steps that allow us to obtain the tight upper bound using our precise expression $M(n,d)$\footnote{Finally, we note that \cite{chinot_2021,wojtaszczyk_2010} both study the case where the noise can also be adversarial, for which the rate of order $\sigma^2$ for the risk is optimal.}.
\subsection{Uniform risk bound and reduction to auxiliary problem by GMT}
Given that $\hw$ belongs with high probability to the set of interpolators located in the $\ell_1$-ball of radius $B(n,d)$ centered at $\wstar$, we proceed to upper-bound the risk of all such interpolators.
Concretely, we find a high-probability upper bound on
\ifarx
\begin{align}
\Phi_+ :=& \max_{w} \norm{w-\wstar}_2^2
\subjto \norm{w-w^*}_1 \leq B(n,d) ~\text{and}~ X(w-w^*) = \xi \\
=& \max_w \norm{w}_2^2 \subjto \norm{w}_1 \leq B(n,d) ~\text{and}~ Xw = \xi.
\end{align}
\else
\begin{align}
\Phi_+ :=& \max_{w} \norm{w-\wstar}_2^2
\subjto \norm{w-w^*}_1 \leq B(n,d)\\
&\quad\quad\quad\quad\quad\quad\quad\quad ~\text{and}~ X(w-w^*) = \xi \\
=& \max_w \norm{w}_2^2 \subjto \norm{w}_1 \leq B(n,d) ~\text{and}~ Xw = \xi.
\end{align}
\fi
While directly bounding $\Phi_+$ is challenging
due to the randomness of both $X$ and $\xi$, we can instead make use of the Gaussian Minimax Theorem (GMT) which allows us to equivalently upper-bound the value of the so-called auxiliary problem
\ifarx
\begin{align}
\label{eq:proofsketch:phi+}
\phi_+ := \max_w \norm{w}_2^2 \subjto \norm{w}_1 \leq B(n,d) ~\text{and}~ \innerprod{w}{h}^2 \geq (1-\rho) n (\sigma^2 + \norm{w}_2^2)
\end{align}
\else
\begin{align}
\label{eq:proofsketch:phi+}
&\phi_+ := \max_w \norm{w}_2^2 \subjto \norm{w}_1 \leq B(n,d)\\
&\quad\quad\quad\quad\quad ~\text{and}~ \innerprod{w}{h}^2 \geq (1-\rho) n (\sigma^2 + \norm{w}_2^2)
\end{align}
\fi
with $h$ an \iid\ Gaussian random vector
and $\cnoise$ a vanishing parameter.
Indeed, the GMT ensures that $\PP_{X, \xi}(\Phi_+ > t) \leq 2 \PP_h(\phi_+ \geq t) + \epsilon_{\cnoise}$ (see Proposition~\ref{prop:CGMT_application} for the expression of $\epsilon_{\cnoise}$);
in words, a high-probability upper bound on $\phi_+$ gives a high-probability upper bound on $\Phi_+$.
Reformulating the optimization problem
\eqref{eq:proofsketch:phi+} as a one-dimensional optimization problem (see Section~\ref{subsec:pathapproach})
allows us to obtain the following tight upper bound
\begin{equation}
\phi_+ \leq \frac{\sigma^2}{\log(d/n)}\left(1 + \frac{c}{\log(d/n)^{1/2}}\right).
\end{equation}
We now discuss how a simpler relaxation of the constraints, used in previous work, leads to loose bounds.
\paragraph{Comparison to \cite{koehler_2021}.}
The optimization problem defining $\phi_+$ in Equation~\eqref{eq:proofsketch:phi+} is a maximization problem of a convex function over non-convex constraints.
We now show how to obtain a first loose upper bound
following
\cite{koehler_2021} and briefly discuss why this methodology fails to give tight bounds (see also
the paragraph
``Application: Isotropic features'' in that paper).
Using Hölder's inequality $\innerprod{w}{h} \leq \norm{w}_1 \norm{h}_{\infty}$, we obtain a proper relaxation of the problem~\eqref{eq:proofsketch:phi+} if we replace the constraints by
\begin{equation}
\locupper^2 \|h\|_\infty^2 \geq (1-\rho) n (\sigma^2 + \|w\|_2^2).
\label{eq:koehlereq}
\end{equation}
This immediately implies the upper bound
\begin{equation}
\phidp \leq \frac{\gennormboundp(n,d)^2
\|\Hgaussian\|_{\infty}^2}{n(1-\cnoise)} - \sigma^2.
\end{equation}
However this bound is loose, even when we plug in our tight localization bound for $\locupper^2 \approx \frac{\sigma^2 n}{2\log(d/n) - \log\log (d/n)}$.
Indeed, with this estimate and by Gaussian concentration results,
the above bound reads for $d\gg n$
\ifarx
\begin{align}
\phidp \leq
\frac{\locupper^2
\|\Hgaussian\|_{\infty}^2}{n(1-\cnoise)} - \sigma^2
&\approx \frac{\sigma^2~ 2\log(d)}{2\log(d/n)} - \sigma^2
= \sigma^2 \frac{\log(n)}{\log(d/n)}.
\label{eq:sometempeq}
\end{align}
\else
\begin{align}
\phidp &\leq
\frac{\locupper^2
\|\Hgaussian\|_{\infty}^2}{n(1-\cnoise)} - \sigma^2 \\
&\approx \frac{\sigma^2~ 2\log(d)}{2\log(d/n)} - \sigma^2
= \sigma^2 \frac{\log(n)}{\log(d/n)}.
\label{eq:sometempeq}
\end{align}
\fi
Note that this bound is constant in any polynomial growth regime $d \asymp n^\beta$, while we prove an upper bound in Theorem~\ref{thm:main} which vanishes in these regimes as $d,n \to \infty$.
This looseness points to the fact that using H\"olders inequality is too imprecise.
In the next subsection, we describe our refined analysis which better takes into account the relationship between $\innerprod{w}{\Hgaussian}, \|w\|_2$ and $\|w\|_1$.
\subsection{Path approach: reparametrizing the auxiliary problem as a one-dimensional problem}
\label{subsec:pathapproach}
The key observation that allows us to derive a tight bound for $\phi_+$,
is that we can cast the $d$-dimensional problem \eqref{eq:proofsketch:phi+}
into a one-dimensional problem,
which we can study explicitly.
Namely, we identify a path $\gamma: \RR \to \RR^d$ for which we show that the optimum
in~\eqref{eq:proofsketch:phi+}
is necessarily attained at $B(n,d) \gamma(\alpha)/\norm{\gamma(\alpha)}_1$ for some $\alpha \in \RR$.
Note that our reduction is exact,
not a relaxation.
More precisely, we define the path
$\gamma: [1, \alpham] \to \RR^d$ by
\begin{equation}
\gamma(\alpha) = \argmin_w \norm{w}_2^2
\subjto \begin{cases}
\innerprod{w}{h} = \norm{h}_\infty \\
\forall i, h_i w_i \geq 0 \\
\norm{w}_1 = \alpha
\end{cases}
\end{equation}
(in particular $\norm{\gamma(\alpha)}_1=\alpha$)
and we show that
\ifarx
\begin{align}
\label{eq:proof_sketch:pathformulation}
\phi_+ = \locupper^2 \max_{1\leq \alpha\leq \alpha_{\max}} \left(\frac{\|\gamma(\alpha)\|_2}{\alpha}\right)^2 \:\: \subjto \:\: &\frac{\|H\|^2_\infty}{(1-\rho)n } \geq \frac{\alpha^2\sigma^2}{\locupper^2} + \|\gamma(\alpha)\|_2^2
\end{align}
\else
\begin{align}
\label{eq:proof_sketch:pathformulation}
\phi_+ &= \locupper^2 \max_{1\leq \alpha\leq \alpha_{\max}} \left(\frac{\|\gamma(\alpha)\|_2}{\alpha}\right)^2 \:\:
\\\text{ s.t. } \:\: &\frac{\|h \|^2_\infty}{(1-\rho)n} \geq \frac{\alpha^2\sigma^2}{\locupper^2} + \|\gamma(\alpha)\|_2^2
\end{align}
\fi
(see \ifarx Section\else Appendix\fi~\ref{subsec:proof_of_props_phi}).
Because $\gamma(\alpha)$ is the argmin of a convex optimization problem, it is relatively easy to study, and we can even derive an exact expression for it (Lemma~\ref{lm:charact_gamma}). We now discuss the two key steps to study the optimization problem in Equation~\eqref{eq:proof_sketch:pathformulation}.
\paragraph{a) Monotonicity of the objective.} We observe that $\frac{\|\gamma(\alpha)\|_2}{\alpha}$ is monotonically decreasing and that $\norm{\gamma(\alpha)}_2^2$ is a convex function (Lemma~\ref{lm:interval}).
This has two important consequences.
Firstly, the set of $\alpha$'s which satisfy the constraints in~\eqref{eq:proof_sketch:pathformulation} is an interval, denoted $[\ulalpha_I, \olalpha_I]$.
Secondly, denoting
$\alpha^*$ ($\in [\ulalpha_I, \olalpha_I]$)
an argmax of~\eqref{eq:proof_sketch:pathformulation},
we have for any $\alpha < \ulalpha_I$
\begin{equation}
\label{eq:upperboundsimple}
\locupper^2 \left(\frac{\|\gamma(\alpha)\|_2}{\alpha}\right)^2 \geq \locupper^2 \left(\frac{\|\gamma(\alpha^*)\|_2}{\alpha^*}\right)^2 = \phi_+.
\end{equation}
So to obtain an upper bound on $\phi_+$, all we need is to find an $\alpha$ on the left of the feasible interval $[\ulalpha_I, \olalpha_I]$.
We do this by finding both an $\alpha_n$ such that ${ \ulalpha_I \leq \alpha_n \leq \olalpha_I }$ (i.e.\ $\alpha_n$ is feasible), and an $\alpha_s$ such that $\alpha_s < \alpha_n$ and $\alpha_s \not\in [\ulalpha_I, \olalpha_I]$ (i.e.\ $\alpha_s$ is not feasible).
\paragraph{b) Discretization of the path.}
We observe that there exist ``breakpoints'' $1 = \alpha_2 < ... < \alpha_{d+1} = \alphamax$ for which $\gamma(\alpha_s)$ has a special structure (in particular it is $(s-1)$-sparse).
Further, applying Gaussian concentration results to $h$ leads to high-probability estimates for
$\alpha_s$
and $\norm{\gamma(\alpha_s)}_2$
(Proposition~\ref{prop:concentration_gammas}).
Thanks to those estimates,
we show that $\alpha_n$ is feasible for \eqref{eq:proof_sketch:pathformulation}
and
we find a choice of $s < n$ such that $\alpha_s$ is not feasible, with high probability.
Thus, with high probability $\phi_+$ is upper-bounded by $\locupper^2 \left(\frac{\|\gamma(\alpha_s)\|}{\alpha_s}\right)^2$ -- for which we have high-probability estimates.
\paragraph{Intuition for the definition of $\gamma(\alpha)$.}
As discussed in the previous subsection, the relaxation of \eqref{eq:phidp} based on H\"older's inequality $\innerprod{w}{h} \leq \norm{w}_1 \norm{h}_\infty$, used
\ifarx in the paper \cite{koehler_2021},
\else by \citet{koehler_2021},
\fi
is too loose.
Informally, it effectively amounts to forgetting the direction of $h$ and only optimizing over the $\ell_1$ and $\ell_2$-norms of vectors.
The main idea of our refined analysis is to introduce a path, $\{\gamma(\alpha)/\alpha\}_\alpha$, allowing us to better take into account the relationship between $\innerprod{w}{\Hgaussian}, \|w\|_2$ and $\|w\|_1$.
To intuitively understand how the path achieves this goal, it may be easier to use its following form:
\begin{equation}
\olw(\beta) = \argmax_w \innerprod{w}{h} \subjto \begin{cases}
\norm{w}_2^2 \leq \beta \\
\forall i, h_i w_i \geq 0 \\
\norm{w}_1 = 1
\end{cases},
\end{equation}
which is equivalent to $\gamma(\alpha)$ up to linear reparametrization and rescaling (see Equation~\eqref{eq:def_olw_beta} in Appendix~\ref{apx:subsec:param_argmaxmin}).
Note that $\langle \olw(1), h \rangle = \norm{h}_{\infty} \norm{\olw(1)}_1$, which exactly recovers the equality case of H\"older's inequality (i.e.\ $\olw(1)$ is a subgradient of the $\ell_1$-norm at $h$).
More generally, $\langle w,h\rangle \leq \langle \olw(\beta),h\rangle \norm{w}_1$ for any $w$ such that $\norm{w}_2^2 \leq \beta \norm{w}_1^2$, which can be understood as a refined H\"older's inequality for limited $\ell_2$-norms.
\subsection{Obtaining a good estimate for \texorpdfstring{$\Phi_N$}{PhiN}}
\label{subsec:proofsketch_phin}
Finally, we unveil how we derive the high-probability upper bound $\Phi_N \leq M(n,d)$ in the localization step (1.):
The derivation actually uses the same tools as for the upper bound of $\Phi_+$.
Using the Convex Gaussian Minimax Theorem (CGMT) \cite{thrampoulidis_2015},
which is a variant of the GMT for convex-concave functions,
we can again introduce an auxiliary problem
\begin{equation}
\label{eq:proofsketch:phin}
\phi_N := \min_w \norm{w}_1 \subjto \innerprod{w}{h}^2 \geq (1+\rho) n (\sigma^2 + \|w\|_2^2)
\end{equation}
with $h$ an i.i.d.\ Gaussian vector and $\rho$ a vanishing parameter,
with the property that high-probability upper bounds on $\phi_N$ give high-probability upper bounds on $\Phi_N$.
Further,
we can again reduce this $d$-dimensional optimization problem to one over the same path $\{\gamma(\alpha)\}_\alpha$:
\begin{equation}
\label{eq:constarintsphin}
\phi_N = \min_{1\leq \alpha\leq\alphamax} f(\alpha)
\subjto
\norm{h}_\infty^2 \geq (1+\cnoise) n \norm{\gamma(\alpha)}_2^2
\end{equation}
(see \ifarx Section\else Appendix\fi~\ref{subsec:CGMTproof} for the expression of $f(\alpha)$).
Since we want to upper-bound this minimum, it is sufficient to find some $\alpha$ which satisfies the constraints in Equation~\eqref{eq:constarintsphin}.
In particular, we again
focus on
the breakpoints $\{\alpha_s\}_{s \in \{2,...,d+1\}}$,
and show that with high probability
$\alpha = \alpha_n$ is
a valid choice which is approximately tight (see Remark~\ref{rk:choice_s_n}).
\paragraph{Sparsity of $\gamma(\alpha_n)$.}
In summary, the proof is essentially based on the localization around a rescaled version of $\gamma(\alpha_n)$, which is a $(n-1)$-sparse vector.
This choice can also be motivated by a different argument:
It is well known that the minimizer of the optimization problem
$\min_{Xw=\xi} \norm{w}_1$
defining $\PhiN$ is $n$-sparse. Hence, due to the strong
connection between
the optimization problems defining $\phidn$ and $\PhiN$, we also expect the minimizer of Equation~\eqref{eq:constarintsphin} to be approximately $n$-sparse.
\end{document}
\subsection{Proof of Lemma~\ref{lm:triangle_ineq}: Preliminary}
\label{apx:subsec:triangle}
Let $\Sparseset := \{j:~ \wgt_j \neq 0\}$
(in particular, $\sstar = \vert \Sparseset \vert$).
Denote by $\whatS \in \RR^d$ the vector with entries
$(\whatS)_j = \what_j$ if $j \in \Sparseset$ and $0$ otherwise,
and let $\whatSm = \what - \whatS$.
Now recall the definition of $\what$,
and define $\nuhat$ by
\begin{align}
\what &= \arg\min_w \norm{w}_1 \subjto X(w-\wgt) = \xi \\
\nuhat &= \arg\min_v \norm{v}_1 \subjto Xv = \xi.
\end{align}
Clearly by definition of $\nuhat$ and $\what$, we have that ${\norm{\what}_1 \leq \norm{\nuhat}_1 + \norm{\wgt}_1}$.
Therefore
\begin{align}
0 &\leq \norm{\wgt}_1 - \norm{\what}_1 + \norm{\nuhat}_1\\
&= \norm{\wgt}_1 - \norm{\whatS}_1 - \norm{\whatSm} + \norm{\nuhat}_1\\
&\leq \norm{\wgt - \whatS}_1 - \norm{\whatSm}_1 + \norm{\nuhat}_1\\
&= 2\norm{\wgt - \whatS}_1 - \norm{\what - \wgt }_1 + \norm{\nuhat}_1\\
&\leq 2\sqrt{\sstar} \norm{\wgt - \whatS}_2 - \norm{\what - \wgt }_1 + \norm{\nuhat}_1.
\end{align}
Hence,
\begin{align}
\norm{\what - \wgt }_1
&\leq 2\sqrt{\sstar} \norm{\wgt - \whatS}_2 + \norm{\nuhat}_1 \\
&\leq 2\sqrt{\sstar} \norm{\wgt - \what}_2 + \norm{\nuhat}_1.
\end{align}
Finally, we bound $\norm{\wgt - \what}_2$
by applying Theorem~3.1 of \cite{chinot_2021}, noting that its assumptions are subsumed by the assumptions of our Theorem~\ref{thm:main}.
\end{document}
\subsection{Proof of Proposition~\ref{prop:CGMT_application}: Application of the (C)GMT}
\label{apx:subsec:CGMT}
Proposition~\ref{prop:CGMT_application} reduces the estimation of the quantities $\Phi_N, \Phi_+, \Phi_-$ in Equations~\eqref{eq:Phi_N}, \eqref{eq:Phi_+}, \eqref{eq:Phi_-} to the estimation of auxiliary quantities $\phidn, \phidp, \phidm$, using the (C)GMT.
As a first step, we apply the CGMT to $\Phi_N$ and the GMT to $\Phi_+$ analogously to \cite[Lemmas~4\&7]{koehler_2021}. We only restate the results here and refer the reader to that paper for details and proofs.
Note that the (C)GMT is applied on $X$ conditionally on $\xi$, so that the Gaussianity of the noise is not crucial.
\begin{lemma}[{\cite[Lemma~7]{koehler_2021}, Application of CGMT}]
Define
\begin{equation}
\tphi_N = \min_w \norm{w}_1 \subjto
\norm{\xi - \Ggaussian \norm{w}_2}_2 \leq \innerprod{w}{\Hgaussian},
\end{equation}
where $\Ggaussian \sim \NNN(0,I_n)$ and $\Hgaussian \sim \NNN(0,I_d)$ are independent random variables.
Then, for all $t \in \RR$,
\begin{equation}
\PP(\Phi_N>t) \leq 2 \PP(\tphi_N >t),
\end{equation}
where the probabilities
on the left and on the right
are over the draws of $X,\xi$ and of $\Ggaussian,\Hgaussian,\xi$, respectively.
\end{lemma}
\begin{lemma}[{\cite[Lemma~4]{koehler_2021}, Application of GMT}]
Define
\begin{equation}
\tphi_+ = \max_w \norm{w}_2^2
\subjto \begin{cases}
\norm{w}_1 \leq B(n,d) \\
\norm{\xi - \Ggaussian \norm{w}_2}_2 \leq \innerprod{w}{\Hgaussian}
\end{cases},
\end{equation}
where $\Ggaussian \sim \NNN(0,I_n)$ and $\Hgaussian \sim \NNN(0,I_d)$ are independent random variables.
Then, for all $t \in \RR$,
\begin{equation}
\PP(\Phi_+>t) \leq 2 \PP(\tphi_+ >t),
\end{equation}
where the probabilities
on the left and on the right
are over the draws of $X,\xi$ and of $\Ggaussian,\Hgaussian,\xi$, respectively.
\end{lemma}
Following the same argument as in \cite[Lemma~4]{koehler_2021}, we can show a corresponding lemma for $\Phi_-$ which we state without proof:
\begin{lemma}[{Application of GMT}]
Define
\begin{equation}
\tphi_- = \min_w \norm{w}_2^2
\subjto \begin{cases}
\norm{w}_1 \leq B(n,d) \\
\norm{\xi - \Ggaussian \norm{w}_2}_2 \leq \innerprod{w}{\Hgaussian}
\end{cases},
\end{equation}
where $\Ggaussian \sim \NNN(0,I_n)$ and $\Hgaussian \sim \NNN(0,I_d)$ are independent random variables.
Then, for all $t \in \RR$,
\begin{equation}
\PP(\Phi_-<t) \leq 2 \PP(\tphi_- <t),
\end{equation}
where the probabilities
on the left and on the right
are over the draws of $X,\xi$ and of $\Ggaussian,\Hgaussian,\xi$, respectively.
\end{lemma}
Next, by using Gaussian concentration results, we can formulate simpler versions of the above optimization problems defining the quantities $\tphi$'s.
\paragraph{Simplify $\tphi_N$.}
Following the same argument as in the first part of the proof of \cite[Lemma~8]{koehler_2021} (Equations~(68)-(70)),
we can show that for any $0<\rho<1/2$, with probability at least $1-6\exp(-n \cnoise^2/100)$,
uniformly over $w$,
\begin{equation}
\norm{\xi - \Ggaussian \norm{w}_2}_2^2
\leq
(1+\cnoise) n (\sigma^2 + \norm{w}_2^2).
\label{eq:apx1eventUpper}
\end{equation}
So on the event where Equation~\eqref{eq:apx1eventUpper} holds, we have that
\begin{equation}
\tphi_N \leq
\min_w \norm{w}_1 \subjto
\innerprod{w}{\Hgaussian}^2 \geq (1+\cnoise) n (\sigma^2 + \norm{w}_2^2)
= \phidn
\end{equation}
which proves the
first inequality
in Proposition~\ref{prop:CGMT_application}.
\paragraph{Simplify $\tphi_+,\tphi_-$.}
By the same argument as for $\tphi_N$,
we can show that for any $0<\rho<1/2$, with probability at least $1-6\exp(-n \cnoise^2/100)$,
uniformly over $w$,
\begin{equation}
\norm{\xi - \Ggaussian \norm{w}_2}_2^2
\geq
(1-\cnoise) n (\sigma^2 + \norm{w}_2^2).
\label{eq:apx1eventLower}
\end{equation}
So on the event where Equation~\eqref{eq:apx1eventLower} holds, we have that
\begin{equation}
\tphi_+ \leq
\max_w \norm{w}_2^2
\subjto \begin{cases}
\norm{w}_1 \leq M+2\|\wgt\|_1 \\
\innerprod{w}{\Hgaussian}^2 \geq (1-\cnoise) n (\sigma^2 + \norm{w}_2^2)
\end{cases}
= \phidp,
\end{equation}
and similarly $\tphi_- \geq \phidm$.
This proves the
second and third inequalities
in Proposition~\ref{prop:CGMT_application} and thus completes the proof.
\end{document}
\subsection{Proof of Lemma~\ref{lm:parametrization}: Parametrizing the argmax/argmin}
\label{apx:subsec:param_argmaxmin}
We now prove our first key lemma: we show that, up to scaling, the argmax/argmin in
\eqref{eq:phidn},
\eqref{eq:phidp}, and
\eqref{eq:phidm}
belong to a certain parametric path $\Gamma = \{ \frac{\gamma(\alpha)}{\alpha} \}_\alpha$.
Throughout this section and the next, we consider $H$ as a fixed vector such that
\mbox{$H_1 > ... > H_d > 0$}.
In other words, all of our statements should be understood as holding conditionally on $\Hgaussian$, and with $\Hgaussian$ in general position.
For all $\beta \in [\frac{1}{d}, 1]$, define
\begin{equation} \label{eq:def_olw_beta}
\olw(\beta) = \argmax_w \innerprod{w}{H} \subjto \begin{cases}
\norm{w}_2^2 \leq \beta \\
\forall i, w_i \geq 0 \\
\onevec^\top w = \norm{w}_1 = 1
\end{cases}.
\end{equation}
Importantly, note that the constraint
$\norm{w}_2^2 \leq \beta$
in the definition of $\olw(\beta)$ necessarily holds with equality at optimality.
Indeed, suppose by contradiction
$\norm{\olw(\beta)}_2^2 < \beta \leq 1 = \norm{\olw(\beta)}_1^2$.
This implies that $\olw(\beta)$ has at least two nonzero components; denote $i \neq 1$ such that $\olw(\beta)_i > 0$.
Then there exists some $\eps>0$ such that $\olw(\beta) + \eps e_1 - \eps e_i$ satisfies the constraints and achieves a higher objective value than $\olw(\beta)$, contradicting its optimality.
The first step of the proof is to show that (at least one element) of the argmax/argmin belongs to the set $\RR_+ \olGamma$,
where $\olGamma = \left\lbrace
\olw(\beta) ; \frac{1}{d} \leq \beta \leq 1
\right\rbrace$.
\begin{claim}
For each of the optimization problems
\eqref{eq:phidn},
\eqref{eq:phidp}, and
\eqref{eq:phidm},
there exist $b>0$ and
$\beta \in [\frac{1}{d}, 1]$
such that $b \olw(\beta)$ is an optimal solution.
\end{claim}
\begin{proof}
Let $v$ be an optimal solution of \eqref{eq:phidn}.
It is straightforward to check that we may assume \WLOG\ that $\forall i, v_i \geq 0$.
Choose $b = \norm{v}_1$
and $\beta = \frac{\norm{v}_2^2}{\norm{v}_1^2}$;
note that $\beta \in [\frac{1}{d}, 1]$.
By definition,
$\norm{b \olw(\beta)}_2 = \norm{v}_2$ and
$\norm{b \olw(\beta)}_1 = \norm{v}_1$,
and $v/b$ is feasible for \eqref{eq:def_olw_beta} so
$\innerprod{b \olw(\beta)}{H} \geq \innerprod{v}{H}$.
Therefore, $b \olw(\beta)$ satisfies the constraint of \eqref{eq:phidn} and achieves the optimal objective value, so is also an optimal solution of \eqref{eq:phidn}.
The statements for \eqref{eq:phidp} and \eqref{eq:phidm} follow by the exact same argument.
\end{proof}
Next, we show that $\{ \olw(\beta) \}_\beta$ and $\{ \frac{\gamma(\alpha)}{\alpha} \}_\alpha$ are two parametrizations of the same path.
\begin{claim}
We have the equality
\begin{equation}
\olGamma :=
\left\lbrace \olw(\beta) ; \beta \in \left[\frac{1}{d}, 1\right] \right\rbrace
~~=~~
\left\lbrace \frac{\gamma(\alpha)}{\alpha} ; \alpha \in [1,
\alpham] \right\rbrace
=: \Gamma
\end{equation}
where $\alpham = d \frac{\norm{H}_\infty}{\norm{H}_1}$.
\end{claim}
\begin{proof}
First note that we can characterize $\frac{\gamma(\alpha)}{\alpha}$ as the optimal solution of
\begin{equation} \label{eq:def_gammap_alphap}
\frac{\gamma(\alpha)}{\alpha} = \argmin_w \norm{w}_2^2
\subjto \begin{cases}
\innerprod{w}{H} \geq \frac{\norm{H}_\infty}{\alpha} \\
\forall i, w_i \geq 0 \\
\onevec^\top w = \norm{w}_1 = 1
\end{cases}.
\end{equation}
The optimization problems \eqref{eq:def_olw_beta} and \eqref{eq:def_gammap_alphap} are both convex and both satisfy the Linear Independence Constraint Qualification conditions.
So, denoting $\Delta_d = \left\lbrace
w \in [0,1]^d ; \onevec^\top w =1
\right\rbrace$
the standard simplex,
by the Lagrangian duality theorem (a.k.a.\ Karush-Kuhn-Tucker theorem) we have that for all $w \in \RR^d$,
\begin{align}
&\exists \beta>0; w = \olw(\beta)
&
&\iff
&
&\exists \lambda>0; w = \argmax_{w \in \Delta_d} \innerprod{w}{H} - \lambda \norm{w}_2^2 &&&& \\
&&
&\iff
&
&\exists \mu>0; w = \argmin_{w \in \Delta_d} \norm{w}_2^2 - \mu \innerprod{w}{H}
&
&\iff
&
&\exists \alpha>0; w = \frac{\gamma(\alpha)}{\alpha}.
\end{align}
Thus,
${
\left\lbrace \olw(\beta) ; \beta>0 \right\rbrace
=
\left\lbrace \frac{\gamma(\alpha)}{\alpha} ; \alpha>0 \right\rbrace
}$.
However it is straightforward to check that
${
\left\lbrace \olw(\beta) ; \beta>0 \right\rbrace
=
\olGamma
}$
and that
${
\left\lbrace \frac{\gamma(\alpha)}{\alpha} ; \alpha>0 \right\rbrace
=
\Gamma
}$,
which concludes the proof.
\end{proof}
Just as the first constraint in \eqref{eq:def_olw_beta} holds with equality at optimality, so does the first constraint in \eqref{eq:def_gammap_alphap}; that is,
$\innerprod{\gamma(\alpha)}{H} = \norm{H}_\infty$ for all $\alpha \in [1, \alpham]$.
This would follow from a careful study of the equivalence between the two problems, but here we give a more direct proof.
\begin{claim} \label{claim:saturate_in_gamma}
The inequality constraint
$\innerprod{w}{H} \geq \norm{H}_\infty$
in the problem defining $\gamma(\alpha)$ holds with equality, at optimality.
\end{claim}
\begin{proof}
Denote $w = \gamma(\alpha)$.
Suppose by contradiction
$\innerprod{w}{H} > \norm{H}_\infty$.
Let $i$ resp.\ $j$ the index of the largest resp.\ smallest component of $w$.
First note that if $w_i = w_j$, then $w \propto \bmone$, \ie, $w = \frac{\alpha}{d} \bmone$ and so $\innerprod{w}{H} = \frac{\alpha}{d} \norm{H}_1 > \norm{H}_\infty$, which would contradict $\alpha \leq \alpham$;
so we have the strict inequality $w_i > w_j$.
Now for some $\eps>0$ to be chosen, let $w' = w - \eps e_i + \eps e_j$.
Clearly $\eps>0$ can be chosen small enough so that $w'$ satisfies all three constraints in the optimization problem defining $\gamma(\alpha)$.
Furthermore, for small enough $\eps$,
$\norm{w}_2^2 - \norm{w'}_2^2 = w_i^2 - (w_i-\eps)^2 + w_j^2 - (w_j+\eps)^2 = 2\eps \left( w_i-w_j - \eps \right)$ is positive, i.e $\norm{w'}_2^2 < \norm{w}_2^2$, which contradicts optimality of $w = \gamma(\alpha)$.
\end{proof}
We now have all the necessary ingredients to prove Lemma~\ref{lm:parametrization}.
The equivalence between \eqref{eq:phidn} and \eqref{eq:phidn_reparam} follows immediately from constraining the variable $w$ (in the former) to belong to the set $\RR_+ \Gamma$.
The equivalence between \eqref{eq:phidm} and \eqref{eq:phidm_reparam} also follows immediately, noting that $\norm{b \frac{\gamma(\alpha)}{\alpha}}_1 = b$ by definition.
Finally, the equivalence between \eqref{eq:phidp} and \eqref{eq:phidp_reparam} follows by noticing that the inequality constraint $\norm{w}_1 \leq B$ (in the former) is necessarily saturated at optimality.
\end{document}
\subsection{Proof of Lemma~\ref{lm:charact_gamma}: Characterizing the parametric path}
\label{apx:subsec:geometric_lemma}
We now
give a precise characterization of the parametric path $\Gamma$, by studying the optimization problem defining $\gamma(\alpha)$.
Throughout this section (just as in the previous one), we consider $H$ as a fixed vector such that
\mbox{$H_1 > ... > H_d > 0$}.
In other words, all of our statements should be understood as holding conditionally on $\Hgaussian$, and with $\Hgaussian$ in general position.
Throughout the proof, consider a fixed $1<\alpha \leq \alpham$.
The goal is to derive a closed-form expression of $\gamma(\alpha)$.
We proceed by a Lagrangian duality approach, and first identify the dual variables $\lambda, \nu, \mu$ (a.k.a.\ Lagrangian multipliers, a.k.a.\ KKT vectors) of the optimization problem defining $\gamma(\alpha)$.
This first analysis yields an expression for $\gamma(\alpha)$ involving an unknown ``sparsity'' integer $s$, which depends on $\lambda, \nu, \mu$ and hence indirectly on $\alpha$.
We finish by showing how to determine $s$ explicitly from $\alpha$.
\paragraph{Karush-Kuhn-Tucker (KKT) conditions.}
Recall that $\gamma(\alpha)$ is defined by the following optimization problem (note that the additional factor $\frac{1}{2}$ in the objective does not change the $\argmin$):
\begin{equation}
\gamma(\alpha) = \argmin_w \frac{1}{2} \norm{w}_2^2
\subjto \begin{cases}
\innerprod{w}{H} \geq \norm{H}_\infty \\
\forall i, w_i \geq 0 \\
\onevec^\top w = \norm{w}_1 = \alpha
\end{cases}.
\end{equation}
This is a convex optimization problem with Lagrangian
\begin{equation}
L(w; \lambda, \mu, \nu) = \frac{1}{2} \norm{w}_2^2
- \lambda (\innerprod{w}{H} - \norm{H}_\infty)
+ \mu (\onevec^\top w - \alpha)
- \nu^\top w.
\end{equation}
The objective is convex and all the constraints are affine.
So by Lagrangian duality, $w = \gamma(\alpha)$ if and only if there exist
$\lambda, \mu \in \RR$ and $\nu \in \RR^d$
satisfying the KKT conditions:
\begin{itemize}
\item (Stationarity)
\mbox{$w - \lambda H + \mu \onevec - \nu = 0$}
~\ie~
\mbox{$w = \lambda H - \mu \onevec + \nu$}
\item (Primal feasibility)
\mbox{$\begin{cases}
\innerprod{w}{H} \geq \norm{H}_\infty \\
\forall i,~ w_i \geq 0 \\
\onevec^\top w = \alpha
\end{cases}$}
\item (Dual feasibility)
\mbox{$\lambda \geq 0$}
and
\mbox{$\forall i,~ \nu_i \geq 0$}
\item (Complementary slackness)
\mbox{$\lambda (\innerprod{w}{H} - \norm{H}_\infty) = 0$},
and
\mbox{$\forall i,~ \nu_i w_i = 0$}.
\end{itemize}
In the rest of this proof, denote $w = \gamma(\alpha)$, and let $\lambda, \mu, \nu$ as above.
\paragraph{Sparsity structure of $w$.}
Let $s$ denote the largest $s' \in \{1, ..., d \}$ such that $\lambda H_{s'} > \mu$.
Since $\lambda \geq 0$ and $H$ is ordered, we have
\begin{equation}
\lambda H_1 \geq ... \geq \lambda H_s > \mu \geq \lambda H_{s+1} \geq ...
\end{equation}
Consider the complementary slackness condition $\forall i,~ \nu_i w_i = 0$.
\begin{itemize}
\item If $w_i > 0$, then $\nu_i=0$ so $w_i = \lambda H_i - \mu > 0$, and so $i \leq s$.
\item If $\nu_i > 0$, then $w_i = \lambda H_i - \mu + \nu_i = 0$ so $\lambda H_i - \mu < 0$, and so $i > s$.
\\
So by contraposition, for all $i \leq s$, $\nu_i=0$ and $w_i = \lambda H_i - \mu$.
\end{itemize}
Thus, $\support(w) \subset \{1, ..., s\}$ and $w = \lambda \Hvecs - \mu \onevecs$,
where $\Hvecs$ is the vector equal to $H$ on the first $s$ components and $0$ on the last $(d-s)$, and similarly for $\onevecs$.
Furthermore, note that the case $s=1$ occurs only if $w \propto e_1$, and one can check that it implies $\alpha = 1$, which we excluded.
\paragraph{Closed-form expression of the dual variables $\lambda,\mu$.}
We can compute $\lambda$ and $\mu$ by substituting $w = \lambda \Hvecs - \mu \onevecs$ into the primal feasibility conditions.
\begin{itemize}
\item Since we know from Claim~\ref{claim:saturate_in_gamma}
(in Section~\ref{apx:subsec:param_argmaxmin}) that the first constraint in the problem defining $\gamma(\alpha)$ holds with equality at optimality, this means that the first primal feasibility condition holds with equality, \ie,
\begin{equation}
\innerprod{w}{H} = \lambda \norm{\Hvecs}_2^2 - \mu \norm{\Hvecs}
=
\norm{H}_\infty.
\end{equation}
\item By the last primal feasibility condition,
\mbox{$\onevec^\top w = \lambda \norm{\Hvecs}_1 - \mu s
=
\alpha$}.
\end{itemize}
So $\lambda$ and $\mu$ are given by
\begin{equation}
\begin{cases}
\lambda \norm{\Hvecs}_2^2 - \mu \norm{\Hvecs}
&=
\norm{H}_\infty \\
\lambda \norm{\Hvecs}_1 - \mu s
&=
\alpha
\end{cases}
\iff
\begin{cases}
\lambda &= \frac{1}{s \norm{\Hvecs}_2^2 - \norm{\Hvecs}_1^2}
\left( s \norm{H}_\infty - \alpha \norm{\Hvecs}_1 \right) \\
\mu &= \frac{1}{s \norm{\Hvecs}_2^2 - \norm{\Hvecs}_1^2}
\left( \norm{\Hvecs}_1 \norm{H}_\infty - \alpha \norm{\Hvecs}_2^2 \right)
\end{cases}.
\end{equation}
Note that the denominator is positive, since $\Hvecs$ has distinct components.
\paragraph{Closed-form characterization of $s$.}
We now show that there exists an increasing sequence $\alpha_2 = 1 < \cdots < \alpha_d <\alpha_{d+1} = \alpham $ such that for all $\alpha$, the sparsity $s$ of $w$ is exactly the index which satisfies $\alpha \in (\alpha_s,\alpha_{s+1}] $.
By plugging the expressions of $\lambda$ and $\mu$ into the condition defining $s$:
$\lambda H_s > \mu \geq \lambda H_{s+1}$,
we obtain
\begin{equation}
\left( s \norm{H}_\infty - \alpha \norm{\Hvecs}_1 \right) H_s
>
\norm{\Hvecs}_1 \norm{H}_\infty - \alpha \norm{\Hvecs}_2^2
\geq
\left( s \norm{H}_\infty - \alpha \norm{\Hvecs}_1 \right) H_{s+1}.
\end{equation}
Rearranging, this is equivalent to
\begin{align}
\alpha \left( \norm{\Hvecs}_2^2 - \norm{\Hvecs}_1 H_s \right)
&>
\left( \norm{\Hvecs}_1 - s H_s \right) \norm{H}_\infty \\
\text{and}~~
\alpha \left( \norm{\Hvecs}_2^2 - \norm{\Hvecs}_1 H_{s+1} \right)
&\leq
\left( \norm{\Hvecs}_1 - s H_{s+1} \right) \norm{H}_\infty.
\end{align}
One can check that
\mbox{$
\norm{\Hvecs}_2^2 - \norm{\Hvecs}_1 H_{s+1}
>
\norm{\Hvecs}_2^2 - \norm{\Hvecs}_1 H_s
> 0
$}.
So the above is equivalent to
\begin{equation}
\alphathr{s} :=
\frac{
\left( \norm{\Hvecs}_1 - s H_s \right) \norm{H}_\infty
}{
\norm{\Hvecs}_2^2 - \norm{\Hvecs}_1 H_s
}
< \alpha \leq
\frac{
\left( \norm{\Hvecs}_1 - s H_{s+1} \right) \norm{H}_\infty
}{
\norm{\Hvecs}_2^2 - \norm{\Hvecs}_1 H_{s+1}
}
=: \olalpha(s).
\end{equation}
A straightforward calculation shows that
$\olalpha(s) = \alphathr{s+1}$.
Thus, using the convention
$\alphathr{d+1} = \alpham$,
$s$ is uniquely characterized by
$\alphathr{s} < \alpha \leq \alphathr{s+1}$.
This concludes the proof of Lemma~\ref{lm:charact_gamma}.
\end{document}
\subsection{Proof of Proposition~\ref{prop:concentration_gammas}: Concentration of norms of \texorpdfstring{$\gamma(\alphathr{s})$}{gamma(alphas)}}
\label{apx:subsec:concentration_gamma}
In this section we prove Proposition~\ref{prop:concentration_gammas} and Lemma~\ref{lm:tlogds} which establish concentration inequalities for $\gamma(\alpha)$ at the breakpoints
$\alphathr{s}$ for $2 \leq s \leq d$.
More precisely, we give high-probability estimates (with respect to the draws of $\Hgaussian$) of their $\ell_1$ and $\ell_2$ norms, since those are the quantities that appear in
the stochastic optimization problems
\eqref{eq:phidn_reparam},
\eqref{eq:phidp_reparam} and
\eqref{eq:phidm_reparam}.
Plugging in $\alpha = \alphathr{s}$ into the closed-form expressions of $\lambda$ and $\mu$ in Lemma~\ref{lm:charact_gamma}, we obtain
${\gamma(\alphathr{s}) = \frac{\norm{H}_\infty}{\langle v_s, H\rangle} v_s}$
where $v_s := H_{[s]}- H_s \onevecs $.
Thus, to estimate the norms of $\gamma(\alphathr{s})$ it suffices to estimate the quantities
\begin{align}
\norm{v_s}_2^2 &= \norm{H_{[s]}}_2^2 - 2 \norm{H_{[s]}}_1 H_s + s H_s^2 \\
\norm{v_s}_1 &= \norm{H_{[s]}}_1 - s H_s \\
\innerprod{v_s}{H} &= \norm{H_{[s]}}_2^2 - \norm{H_{[s]}}_1 H_s.
\end{align}
Throughout the proofs in this section, we will use $\gencstpos>0$ to denote a universal constant (in particular, independent of $d$ and $s$) which may change from display to display.
Furthermore, in this section we let $Z$ denote a standard normal distributed random variable,
and recall that $\Phic(x) = \PP(Z>x) = \frac{1}{2} \PP(\abs{Z}>x)$ for $x>0$ denotes its complementary cumulative distribution function.
\subsubsection{Preliminary facts}
We start by stating some auxiliary facts about $\Phic$.
\begin{fact} \label{fact:estim_Phic}
Denote $h(x)$ the function such that
$\forall x>0,~ \Phic(x) = \frac{\exp(-x^2/2)}{x \sqrt{2\pi}} h(x)$.
We have the first-order and higher-order upper and lower bounds
\begin{align}
1-\frac{1}{1+x^2} &\leq h(x) \leq 1
~~~\mathrm{and}~~~
\abs{
h(x) - \left( 1- \frac{1}{x^2} + \frac{3}{x^4} - \frac{15}{x^6} \right)
}
\leq \frac{c}{x^8}
\end{align}
for all $x>0$.
\end{fact}
\begin{proof}
The first-order estimate follows from straightforward analysis.
The higher-order estimate follows from the exact asymptotic expansion of the \textit{complementary error function} $\text{erfc}$,%
since
$2\Phic(x) = \text{erfc}(x/\sqrt{2})$.
\end{proof}
\begin{fact} \label{fact:cond_expectations}
By straightforward calculations, we have
\begin{align}
\forall x>0,~
\EE [Z \vert Z \geq x]
&= \frac{1}{\Phic(x)} \frac{\exp(-x^2/2)}{\sqrt{2\pi}}
= \frac{x}{h(x)} \\
\mathrm{and}~~~
\EE[Z^2 | Z \geq x]
&= \frac{1}{\Phic(x)} \left(
\frac{x}{\sqrt{2\pi}} \exp(-x^2/2) + \Phic(x)
\right)
= 1+\frac{x^2}{h(x)}.
\end{align}
\end{fact}
We will also make repeated use of Lemma~\ref{lm:tlogds} (Section~\ref{subsubsec:concentration_gammas}), whose proof is deferred to Section~\ref{apx:subsubsec:concentration_gamma:tlogs}.
\subsubsection{Proof of Proposition~\ref{prop:concentration_gammas}}
We will show the following lemmas successively, in which $t \in \RR$ denotes the quantity such that ${2\Phic(t) = s/d}$
(we drop the explicit dependency on $s$ for concision in this section).
\begin{lemma}[Concentration of $H_s$] \label{lm:concentration_Hs}
Assume that $s < d/2$.
With probability at least $1-2\delta$, we have
\begin{equation}
\abs{H_s - t} \leq \gencstpos \left(
\frac{1}{\sqrt{s}}
+ \sqrt{\frac{\log(1/\delta)}{s}}
+ \frac{\log(1/\delta)}{s}
\right).
\end{equation}
\end{lemma}
\begin{lemma}[Concentration of $\norm{H_{[s]}}_2^2$] \label{lm:concentration_l2_Hs}
Assume $s < d/5$.
With probability at least $1-2\delta$, we have
\begin{equation}
\abs{
\norm{\Hvecs}_2^2 - s \EE[Z^2|Z \geq t]
} \leq
\gencstpos \sqrt{s} (1+\sqrt{\log(1/\delta)})
\left(
\frac{1}{\sqrt{s}} (1+\sqrt{\log(1/\delta)})
+ t
\right).
\end{equation}
\end{lemma}
\begin{lemma}[Concentration of $\norm{H_{[s]}}_1$] \label{lm:concentration_l1_Hs}
With probability at least $1-2\delta$, we have
\begin{equation}
\abs{ \norm{\Hvecs}_1 - s \EE\left[Z | Z \geq t \right] }
\leq
\gencstpos \left( \sqrt{s} + \sqrt{\log(1/\delta) s} \right).
\end{equation}
\end{lemma}
\begin{lemma}[Concentration of $v_s$] \label{lm:concentration_vs}
Assume $s < d/5$.
For
$\delta \geq \exp(-s)$,
with probability at least $1-6\delta$, we have
\begin{align}
\abs{
\norm{v_s}_2^2 - s \left( \frac{2}{t^2} - \frac{10}{t^4} \right)
}
&\leq s \left( \frac{\gencst}{t^6} +\Gencst{s,\delta} \right) ~\mathrm{and} \\
\abs{
\norm{v_s}_1 - s \left( \frac{1}{t} - \frac{2}{t^3} \right)
}
&\leq s \left( \frac{\gencst}{t^5} + \frac{\Gencst{s,\delta}}{t} \right) ~\mathrm{and} \\
\abs{\innerprod{v_s}{H} - s}
&\leq s \Gencst{s,\delta}
\end{align}
with
$\Gencst{s,\delta} = \gencstpos \frac{t+ t\sqrt{\log(1/\delta)}}{\sqrt{s}}$.
\end{lemma}
The proposition follows as a consequence of this last lemma:
\begin{proof}[Proof of Proposition~ \ref{prop:concentration_gammas}]
Let $c_1, c_2, \olt$ as in Lemma~\ref{lm:tlogds}, and assume $s \leq d/c_1$.
In particular, $\log(d/s) \leq t^2 \leq 2\log(d/s)$.
We apply
Lemma~\ref{lm:concentration_vs}
with
$\delta = \exp \left( - 2\frac{s}{\log(d/s)^5} \right)$.
Since $t^2 \leq 2 \log(d/s)$,
this choice ensures that
\mbox{$\frac{t\sqrt{\log(1/\delta)}}{\sqrt{s}} \leq 8/t^4$}.
Moreover, the assumption that $d \leq \exp\left( \cd s^{1/5} \right)$ ensures that
\mbox{$\frac{t}{\sqrt{s}} \leq \gencstpos/t^4$}.
So we have
\mbox{$\Gencst{s,\delta} \leq \gencstpos/t^4$}.
The proposition follows by substituting the estimates of $\norm{v_s}_2^2$, $\norm{v_s}_1$, $\innerprod{v_s}{H}$ into
${
\gamma(\alphathr{s}) = \frac{\norm{H}_\infty}{\innerprod{v_s}{H}} v_s
}$,
and making the appropriate simplifications.
\end{proof}
\subsubsection{Proofs of the concentration lemmas}
\paragraph{Proof of Lemma~\ref{lm:concentration_Hs}: Concentration of $H_s$.}
By observing that
the random variable ${\max \{s ; H_s > t\} }$
is binomially distributed with parameters $d$ and $p = \PP(\abs{Z}>t)$,
\cite{li_2020} show the following upper and lower tail bounds for $H_s$.
\begin{claim} \label{claim:con_hs_t}
Assume that $s < d/2$ and let $t$ be such that $2\Phic(t) = s/d$. Then for all $\eps>0$, we have the lower resp.\ upper tail bounds
\begin{align}
\PP(H_s \leq t - \eps)
&\leq \exp\left(
-\gencstpos s \eps^2 \log(d/s)
\right) \\
\mathrm{and}~~~
\PP(H_s \geq t + \eps)
&\leq \exp\left(
-\gencstpos s \eps^2 \log(d/s)
\exp\left( -2 \eps \sqrt{2\log(d/s) - \log\log(d/s) -\log(\frac{\pi}{2})} - \eps^2 \right)
\right).
\end{align}
\end{claim}
\begin{proof}
This follows straightforwardly from Lemma 2 of \cite{li_2020} and
from the estimate of $t$ in Lemma \ref{lm:tlogds}.
\end{proof}
The lower tail bound is already sufficiently tight to show our high-probability lower bound on ${H_s-t}$.
However we remark that the upper tail bound is too loose; indeed it is only reasonable when $\eps$ is sufficiently small.
So to prove our high-probability upper bound, we instead start from the following one-sided concentration inequality from \cite{boucheron_2012}.
\begin{claim} \label{cr:uppertailbound_Hs}
Assume
$d \geq 3$ and
$s < d/2$,
then
for all $z>0$,
\begin{equation}
\PP \left( H_{s} - \EE H_{s} \geq \gencstpos (\sqrt{z/s} + z/s) \right)
\leq \exp(-z).
\end{equation}
\end{claim}
\begin{proof}
The proof follows from the same argument as in Proposition 4.6 of \cite{boucheron_2012}.
\end{proof}
It only remains to bound the distance between $t$ and $\EE H_s$.
\begin{claim}
Assume that $s < d/2$ and let $t$ be such that $2\Phic(t) = s/d$. Then
\begin{equation}
\abs{ \EE H_s - t } \leq \gencstpos \frac{1}{\sqrt{s}}.
\end{equation}
\end{claim}
\begin{proof}
According to Proposition 4.2 of \cite{boucheron_2012},
\begin{equation}
\Var(H_s) \leq
\frac{1}{s \log 2}
\frac{8}{
\log\frac{2d}{s} - \log(1 + \frac{4}{s} \log\log \frac{2d}{s} )
}
\end{equation}
so by Chebyshev's inequality,
\begin{equation}
\PP \left( \abs{H_s - \EE H_s} > \eps' \right)
\leq \frac{\gencstpos'}{s} \frac{1}{(\eps')^2}.
\end{equation}
On the other hand, recall from Claim~\ref{claim:con_hs_t} that
\begin{equation}
\PP \left( \abs{H_s - t} > \eps \right)
\leq
2 \exp\left(
-\gencstpos s \eps^2 \log(d/s)
\exp\left( -2 \eps \sqrt{2\log(d/s) - \log\log(d/s) - \log(\frac{\pi}{2})} - \eps^2 \right)
\right)
\end{equation}
One can check that there exist universal constants $\cone, \ctwo$ such that, by picking $\eps = \cone/\sqrt{s \log(d/s)}$ and $\eps' = \ctwo/\sqrt{s}$,
the sum of the right-hand sides is less than $1$.
Thus, with positive probability we have
\begin{equation}
\abs{\EE H_s - t} \leq \abs{H_s - \EE H_s} + \abs{H_s - t} \leq \frac{\cone / \sqrt{\log(2)} +\ctwo}{\sqrt{s}}.
\end{equation}
\end{proof}
\paragraph{Proof of Lemma~\ref{lm:concentration_l2_Hs}: Concentration of $\norm{H_{[s]}}_2^2$.}
Let us first restate Proposition 2 of \cite{li_2020} in our notation.
We remark that their statement contained an additional $\log(d/s)$ factor due to a mistake in the proof. Correcting this mistake, we have that with probability at least $1-2\delta$,
\begin{equation}
\abs{
\frac{1}{\sqrt{s}} \norm{\Hvecs}_2 - \sqrt{\EE[Z^2|Z \geq t]}
} \leq \gencstpos \frac{1}{\sqrt{s}} ( 1+\sqrt{\log(1/\delta)} ).
\end{equation}
Since for all $a,b,\eps>0$,
$\abs{a-b} \leq \eps \implies \abs{a^2-b^2} \leq \eps (\eps + 2b)$,
this implies
\begin{equation}
\abs{
\frac{1}{s} \norm{\Hvecs}_2^2 - \EE[Z^2|Z \geq t]
} \leq
\gencstpos \frac{1}{\sqrt{s}} (1+\sqrt{\log(1/\delta)})
\left(
\frac{1}{\sqrt{s}} (1+\sqrt{\log(1/\delta)})
+ \sqrt{\EE[Z^2|Z \geq t]}
\right).
\end{equation}
Now
$\EE[Z^2|Z \geq t] = 1+\frac{t^2}{h(t)} \leq \gencstpos t^2$
whenever $t \geq 1$, which is ensured by our assumption that $s/d = \Phic(t) \leq 0.2$.
So
\begin{equation}
\abs{
\frac{1}{s} \norm{\Hvecs}_2^2 - \EE[Z^2|Z \geq t]
} \leq
\gencstpos \frac{1}{\sqrt{s}} (1+\sqrt{\log(1/\delta)})
\left(
\frac{1}{\sqrt{s}} (1+\sqrt{\log(1/\delta)})
+ t
\right).
\end{equation}
\paragraph{Proof of Lemma~\ref{lm:concentration_l1_Hs}: Concentration of $\norm{H_{[s]}}_1$.}
We use exactly the same argument as in the proof of Proposition 2 of \cite{li_2020}.
Namely, start by decomposing
\begin{equation}
\abs{
\frac{1}{s} \norm{\Hvecs}_1 - \EE[Z | Z \geq t]
}
\leq
\abs{
\frac{1}{s} \norm{\Hvecs}_1 - \frac{1}{s} \EE \norm{\Hvecs}_1
} + \abs{
\frac{1}{s} \EE \norm{\Hvecs}_1 - \EE[Z | Z \geq t]
}.
\end{equation}
For the first term, note that by rearrangement inequality,
\mbox{$Z \mapsto \frac{1}{s}\sum_{i=1}^s \abs{Z_{(i)}}$}
is $\frac{1}{\sqrt{s}}$-Lipschitz for the $\norm{\cdot}_2$ norm,
where $(Z_{(1)},...,Z_{(d)})$ is the nondecreasing reordering of the absolute values of $Z$.%
\footnote{
Proof: Denote $\tZ$ the reordering of $Z$ such that $\abs{\tZ_1}\geq...\geq\abs{\tZ_d}$ (but still with $\tZ$ signed).
Then
\begin{align}
\abs{ \sum_{i=1}^s \abs{Z_{(i)}}-\abs{Y_{(i)}} }
= \abs{ \norm{\tZ_{[s]}}_1 - \norm{\tY_{[s]}}_1 }
\leq \norm{\tZ_{[s]} - \tY_{[s]}}_1
\leq \sqrt{s} \norm{\tZ_{[s]} - \tY_{[s]}}_2
\leq \sqrt{s} \norm{Z - Y}_2
\end{align}
where the last inequality follows from the rearrangement inequality after taking squares and expanding.
}
So by concentration of Lipschitz-continuous functions of Gaussians,
\begin{equation}
\PP \left(
\abs{
\frac{1}{s} \sum_{i=1}^s H_i - \EE \frac{1}{s} \sum_{i=1}^s H_i
}
\geq \eps
\right)
\leq 2 \exp(-s \eps^2/2).
\end{equation}
For the second term, we can apply exactly the same arguments as in the proof of Proposition 2 of \cite{li_2020}, adapting equations (42) to (46), to obtain the bound
\begin{equation}
\abs{
\EE \frac{1}{s}\sum_{i=1}^s H_i - \EE[Z \vert Z \geq t]
}
\leq \gencstpos \EE \abs{H_{s+1} - t}
\leq \gencstpos \frac{1}{\sqrt{s}}.
\end{equation}
In particular, we use the fact that
$x \mapsto \EE [Z \vert Z \geq x]$
is a smooth function, which follows from its explicit expression given in Fact \ref{fact:cond_expectations}.
\paragraph{Proof of Lemma~\ref{lm:concentration_vs}: Concentration of $v_s$.}
For brevity of notation, let
$\Gencst{s,\delta} = \gencstpos \frac{t+ t\sqrt{\log(1/\delta)}}{\sqrt{s}}$.
Assume $\delta \geq e^{-s}$;
in particular,
$\Gencst{s,\delta} \leq \gencstpos t$.
Collecting and simplifying the above results, so far we showed that
\begin{align}
t \abs{H_s-t}
&\leq \Gencst{s,\delta} ~\mathrm{and}\\
\abs{
\frac{1}{s} \norm{\Hvecs}_2^2 - (1+\frac{t^2}{h(t)})
} &\leq \Gencst{s,\delta} ~\mathrm{and} \\
t \abs{
\frac{1}{s} \norm{\Hvecs}_1 - \frac{t}{h(t)}
} &\leq \Gencst{s,\delta}.
\end{align}
\begin{itemize}
\item
Substituting the deterministic estimates in the expression of $\norm{v_s}_2^2$ and carrying over the above concentration bounds, we obtain
\begin{equation}
\abs{
\norm{v_s}_2^2 - \left(
s(1+\frac{t^2}{h(t)})
- 2 \frac{st}{h(t)} t
+ s t^2 \right)
} \leq
s \Gencst{s,\delta} (1+\Gencst{s,\delta})
\end{equation}
and the deterministic estimate can be simplified to
\begin{equation}
s \frac{(1+t^2) h(t) - t^2}{h(t)}
= s \frac{
\frac{2}{t^2} - \frac{12}{t^4} + O\left(\frac{1}{t^6}\right)
}{
1 - \frac{1}{t^2} + O\left(\frac{1}{t^4}\right)
}
= s \left(
\frac{2}{t^2} - \frac{10}{t^4} + O\left(\frac{1}{t^6}\right)
\right)
\end{equation}
(where the $O(\cdot)$ hides a universal constant).
\item
Likewise for $\norm{v_s}_1$ we get
\begin{equation}
\abs{
\norm{v_s}_1 - \left( \frac{st}{h(t)} - st \right)
} \leq
\frac{s}{t} \Gencst{s,\delta}
\end{equation}
and the deterministic estimate can be simplified to
\begin{equation}
st \left( \frac{1}{h(t)}-1 \right)
= s \left( \frac{1}{t^2} - \frac{2}{t^3} + O\left(\frac{1}{t^5}\right) \right).
\end{equation}
\item
Likewise for $\innerprod{v_s}{H}$ we get
\begin{equation}
\abs{
\innerprod{v_s}{H} - \left( s \left( 1 + \frac{t^2}{h(t)} \right) - \frac{st}{h(t)} t \right)
} \leq
3 s \Gencst{s,\delta}
\end{equation}
and the deterministic estimate simplifies to $s$.
\end{itemize}
\subsubsection{Proof of Lemma~\ref{lm:tlogds}}
\label{apx:subsubsec:concentration_gamma:tlogs}
Using the upper bound
\mbox{$\Phic(x) \leq \frac{\exp(-x^2/2)}{x \sqrt{2\pi}}$} from the first part of Fact~\ref{fact:estim_Phic},
it is straightforward to check that
$2\Phic(\olt) \leq s/d = 2\Phic(t)$,
and so $\olt \geq t$.
Let $\ult^2 = \olt^2 - \cone$ for some constant $\cone>0$ to be chosen.
Using the lower bound
\mbox{$\Phic(x) \geq \frac{\exp(-x^2/2)}{x \sqrt{2\pi}} \frac{x^2}{1+x^2}$},
one can check that $\cone$ can be chosen
such that
$2\Phic(\ult) \geq s/d = 2\Phic(t)$, and so $\ult \leq t$.
Going through the calculations reveals
that $\cl \geq e^{2/\pi} \approx 2$ ensures $\olt^2 \leq 2\log(d/s)$,
that $2\Phic(\olt) \leq s/d$ is always true,
that
$\cone =
1 - \log(\frac{\pi}{2})
\approx 0.5$
ensures $\ult^2 \geq \log(d/s)$ for all $s, d$,
and that $\cl \geq e^{2.3415...} \approx 10.4$ ensures $2\Phic(\ult) \geq s/d$.
This concludes the proof of the lemma.
\begin{remark}
Tighter bounds for $t$ can be derived while still only using the first-order estimate of $h(x)$ (the first part of Fact~\ref{fact:estim_Phic}).
Namely, by similar straightforward calculations as above, one can check that
there exist universal constants $\kappa, \alpha_1, \alpha_2>0$ such that, for all $s \leq d/\kappa$, $t$ is bounded as
${
\ult \leq t \leq \olt
}$
where
\begin{align}
\ult^2 &= 2 \log(d/s) - \log\log(d/s) - \log(\pi)+ \frac{\log\log(d/s)}{2 \log(d/s)} - \frac{\alpha_1}{\log(d/s)} \\
\text{and}~~~
\olt^2 &= 2 \log(d/s) - \log\log(d/s) - \log(\pi) + \frac{\log\log(d/s)}{2 \log(d/s)} + \frac{\alpha_2}{\log(d/s)}.
\end{align}
\end{remark}
\end{document}
\subsection{Proofs of Lemmas~\ref{lm:interval} and \ref{lm:bounds}: Studying the feasible set of \texorpdfstring{\eqref{eq:phidp_reparam}}{A+'} and \texorpdfstring{\eqref{eq:phidm_reparam}}{A-'}}
\label{apx:subsec:feasible_interval}
\subsubsection{Proof of Lemma~\ref{lm:interval}
}
We give separate proofs for the statements 1-3 in the Lemma:
\newline\newline
\emph{First statement: The mapping $\alpha \mapsto \norm{\gamma(\alpha)}_2^2$ is decreasing over $[1, \alphad]$ and increasing over $[\alphad, \alpham]$.}
Using the notation of Section~\ref{apx:subsec:geometric_lemma}, the optimization problem defining $\gamma(\alpha)$ has Lagrangian
\begin{equation}
L(w; \lambda, \mu, \nu) = \frac{1}{2} \norm{w}_2^2
- \lambda (\innerprod{w}{H} - \norm{H}_\infty)
+ \mu (\onevec^\top w - \alpha)
- \nu^\top w,
\end{equation}
(up to the constant factor $\frac{1}{2}$ in the first term).
By the envelope theorem, the marginal effect on the optimal value of increasing $\alpha$, is equal to the associated Lagrangian multiplier at optimum:
$\frac{d \norm{\gamma(\alpha)}_2^2}{d\alpha} = -\mu$.
A straightforward computation using the expression of $\mu$ from Lemma~\ref{lm:charact_gamma} reveals that
$\mu>0$ for $1 < \alpha < \alphad$
and $\mu<0$ for $\alphad < \alpha \leq \alpham$;
hence the monotonicity of $\alpha \mapsto \norm{\gamma(\alpha)}_2^2$.
\newline \newline
\emph{Second statement: The mapping $\alpha \mapsto \norm{\gamma(\alpha)}_2^2$ is convex over $\left[1, \alpham \right]$, and $I$ is an interval.}
Recall that $\norm{\gamma(\alpha)}_2^2$ is given by
\begin{equation}
\norm{\gamma(\alpha)}_2^2
= \min_w \norm{w}_2^2
\subjto \begin{cases}
\innerprod{w}{H} \geq \norm{H}_\infty \\
\forall i, w_i \geq 0 \\
\onevec^\top w = \norm{w}_1 = \alpha
\end{cases}.
\end{equation}
Since $\alpha$ appears on the right-hand side of a linear constraint, it is straightforward to check directly that $\alpha \mapsto \norm{\gamma(\alpha)}_2^2$ is convex.
In detail: Let any $\alpha_0$ and $\alpha_1$,
let $w_i = \gamma(\alpha_i)$ for $i \in \{1,2\}$, and $\alpha_t = (1-t) \alpha_0 + t \alpha_1$
and $w_t = (1-t) w_0 + t w_1$ for $t \in [0,1]$; then
$w_t$ is feasible for the optimization problem defining $\gamma(\alpha_t)$,
so $\norm{\gamma(\alpha_t)}_2^2
\leq \norm{w_t}_2^2
\leq t \norm{w_0}_2^2 + (1-t) \norm{w_1}_2^2$
by convexity.
$I$ is the $(B^2 \norm{H}_\infty^2)$-sublevel set of the function
$\alpha \mapsto (1-\cnoise) n \left( \sigma^2 \alpha^2 + B^2 \norm{\gamma(\alpha)}_2^2 \right)$,
which is convex, so $I$ is an interval.
\newline \newline
\emph{Third statement: The mapping $\alpha \mapsto \frac{\norm{\gamma(\alpha)}_2^2}{\alpha^2}$ is monotonically decreasing.}
Note that for each $\alpha \in [1, \alpham]$, $\frac{\gamma(\alpha)}{\alpha}$ is the optimal solution of the optimization problem \begin{equation}
\frac{\gamma(\alpha)}{\alpha} = \arg\min_w \norm{w}_2^2
\subjto \begin{cases}
\innerprod{w}{H} \geq \frac{\norm{H}_\infty}{\alpha} \\
\forall i, w_i \geq 0 \\
\onevec^\top w = \norm{w}_1 = 1
\end{cases}.
\end{equation}
In particular, the constraint set is only increasing with $\alpha$, implying that $\alpha \mapsto \norm{\frac{\gamma(\alpha)}{\alpha}}_2^2$ is monotonically decreasing with $\alpha$.
\subsubsection{Proof of Lemma~\ref{lm:bounds}}
The upper bound for $\phidp$ immediately follows from Equation~\eqref{eq:phidp_reparam} and from the monotonicity of
$\alpha \mapsto \norm{\frac{\gamma(\alpha)}{\alpha}}_2^2$, which is the last statement of Lemma~\ref{lm:interval}.
For $\phidm$, there is an extra scale variable $0<\av \leq B$ which we first minimize out, similarly to the proof of Proposition~\ref{prop:main_norm} in Section~\ref{subsubsec:proof_bound_phidn}.
Starting from Equation~\eqref{eq:phidm_reparam}, first rewrite the constraint as
\begin{gather}
\av^2 \norm{H}_\infty^2 \geq (1-\cnoise) n \left( \sigma^2 \alpha^2 + \av^2 \norm{\gamma(\alpha)}_2^2 \right) \\
\iff
\av^2 \geq
\underbrace{
\frac{
(1-\cnoise) n \sigma^2 \alpha^2
}{
\norm{H}_\infty^2 - (1-\cnoise) n \norm{\gamma(\alpha)}_2^2
}
}_{=: \fun(\alpha)^2}
~~~\text{and}~~~
(1-\cnoise) n \norm{\gamma(\alpha)}_2^2 < \norm{H}_\infty^2.
\end{gather}
Then,
\begin{align}
\phidm &=
\min_{1 \leq \alpha \leq \alpham}
\min_{\av}~
\av^2 \frac{\norm{\gamma(\alpha)}_2^2}{\alpha^2}
\subjto
\fun(\alpha) \leq \av \leq B
~\text{and}~
(1-\cnoise) n \norm{\gamma(\alpha)}_2^2 < \norm{H}_\infty^2 \\
&= \min_{1 \leq \alpha \leq \alpham}
\fun(\alpha)^2
\frac{\norm{\gamma(\alpha)}_2^2}{\alpha^2}
\subjto
\fun(\alpha) \leq B
~\text{and}~
(1-\cnoise) n \norm{\gamma(\alpha)}_2^2 < \norm{H}_\infty^2 \\
&= \min_{1 \leq \alpha \leq \alpham}
\fun(\alpha)^2
\frac{\norm{\gamma(\alpha)}_2^2}{\alpha^2}
\subjto
\alpha \in I \\
&= \min_\alpha \frac{
(1-\cnoise) n \sigma^2 \norm{\gamma(\alpha)}_2^2
}{
\norm{H}_\infty^2 - (1-\cnoise) n \norm{\gamma(\alpha)}_2^2
}
\subjto
\ulalpha_I \leq \alpha \leq \olalpha_I.
\end{align}
This objective is increasing in $\norm{\gamma(\alpha)}_2^2$.
Now as shown in Lemma~\ref{lm:interval},
$\alpha \mapsto \norm{\gamma(\alpha)}_2^2$ is decreasing over $[1, \alphad]$.
Therefore, this objective is decreasing in $\alpha$ over
$[\ulalpha_I, \alphad]$,
and
$\phidm$ is lower-bounded by its value at any $\olalpha$ such that
$\olalpha_I \leq \olalpha \leq \alphad$.
\end{document}
\subsection{Proof of Proposition~\ref{prop:main_upper_lower}: Bounds for \texorpdfstring{$\phidp$}{phi+} and \texorpdfstring{$\phidm$}{phi-}}
\subsection{Proof of Lemma~\ref{lm:boundsonalpha}: A tight admissible choice for \texorpdfstring{$\ulalpha$}{underlinealpha} and \texorpdfstring{$\olalpha$}{overlinealpha}}
\label{apx:subsec:bound_phip_phim}
Recall that we defined
$ B(n,d) = c \sigma \sqrt{\norm{\wgt}_0} + M(n,d)$ where $\Brho(n,d)$ is given by Proposition~\ref{prop:main_norm} and $c>0$ some universal constant. For brevity of notation, abbreviate $B(n,d) = B$.
We seek integers $\us,\os$ such that $\alphathr{\us}$ and $\alphathr{\os}$ lie on the left, respectively on the right of the interval $I$, and such that $\alphathr{\os} \leq \alphad$, with high probability over the draws of $\Hgaussian$.
\subsubsection{Preliminaries}
We recall the notations
\mbox{$B = \Brho(n,d) + 2\norm{w^*}_1$} and
\begin{gather}
I = \left\lbrace
\alpha \in \left[ 1, \alpham \right]
\middle\vert
B^2 \norm{H}_\infty^2 \geq (1-\cnoise) n \left( \sigma^2 \alpha^2 + B^2 \norm{\gamma(\alpha)}_2^2 \right)
\right\rbrace \\
\text{and}~~~~~
\forall \alpha \in [1, \alpham],~
\fun(\alpha)^2 = \frac{
(1-\cnoise) n \sigma^2 \alpha^2
}{
\norm{H}_\infty^2 - (1-\cnoise) n \norm{\gamma(\alpha)}_2^2
}.
\end{gather}
Note that we have the equivalence
\begin{equation}
\label{eq:charactI_balpha}
\alpha \in I \iff
(1-\cnoise) n \norm{\gamma(\alpha)}_2^2 < \norm{H}_\infty^2
~~~\text{and}~~~
\fun(\alpha) \leq B.
\end{equation}
\paragraph{Reference point: $\alphathr{n} \in I$.}
We have by construction that, conditionally on the event where the inequalities in Equation~\ref{eq:concentration} hold for $s=n$,
$\alphathr{n} \in I$ --
in particular this holds with probability at least $1-6\exp\left( -2\frac{n}{\log(d/n)^5} \right)$.
Indeed, let us take a closer look at the way we chose $\Brho(n,d)$,
from the proof of Proposition~\ref{prop:main_norm} (Section~\ref{subsubsec:proof_bound_phidn}).
We showed that, conditionally on that event,
$\alphathr{n}$ satisfies
\begin{equation}
(1+\cnoise) n \norm{\gamma(\alphathr{n})}_2^2 < \norm{H}_\infty^2
~~~\text{and}~~~
\ftild(\alphathr{n}) \leq \Brho(n,d)
\end{equation}
where
$
\ftild(\alpha)^2
= \frac{
(1+\cnoise) n \sigma^2 \alpha^2
}{
\norm{H}_\infty^2 - (1+\cnoise) n \norm{\gamma(\alpha)}_2^2
}
$.
Since
$\fun(\alphathr{n}) \leq \ftild(\alphathr{n})$
and $\Brho(n,d) \leq B$,
clearly $\alphathr{n}$ satisfies the condition~\eqref{eq:charactI_balpha}, \ie, $\alphathr{n} \in I$.
\paragraph{Summary of (in)equalities to be used in the proof.}
For ease of presentation, let us recall some assumptions or definitions that we will use throughout this proof.
For each integer $s$, $t_s \in \RR$ denotes the quantity such that $2\Phic(t_s) = s/d$, and $t_s^2 \asymp \log(d/s)$ by Lemma~\ref{lm:tlogds}.
By assumption,
${
\norm{\wgt}_0 \leq \cwone \frac{\sigma^2 n}{\log(d/n)^{5}}
}$
for some universal constant $\cwone>0$.
By definition,
$\Brho(n,d)^2 =
\frac{\sigma^2 n}{t_n^2}
\left(
1 - \frac{2}{t_n^2} + \frac{\cbrho}{t_n^4}
\right)$
for some universal constant $\cbrho>0$.
In particular, this implies that
\begin{equation}
\label{eq:upperestim_B2}
B^2 = \left(
\Brho(n,d) + c \sigma \sqrt{\norm{\wgt}_0}
\right)^2
=
\frac{\sigma^2 n}{t_n^2}
\left(
1 - \frac{2}{t_n^2} + O\left(\frac{1}{t_n^4}\right)
\right).
\end{equation}
\subsubsection{Finding \texorpdfstring{$\us$}{underlines} such that \texorpdfstring{$\alphathr{\us} \leq \ualpha_I$}{alphaunderlines leq underlinealphaI}}
We want to find an $\us$ such that $\alphathr{\us}$ is on the left of the interval $I$, i.e., $\alphathr{\us} \leq \ualpha_I$.
Conditioning on the event where $\alpha_n \in I$, it suffices to have
$\alphathr{\us} < \alphathr{n}$ \ie\ $\us < n$,
and $\alphathr{\us} \not\in I$ \ie
\begin{align}
&\frac{1}{B^2 \|H\|_\infty^2}
(1-\cnoise) n
\left(
\sigma^2 \norm{\gamma(\alphathr{\us})}_1^2
+ B^2 \norm{\gamma(\alphathr{\us})}_2^2)
\right) \\
=&~ (1-\cnoise) n \frac{\sigma^2}{B^2} \frac{\norm{\gamma(\alphathr{\us})}_1^2}{\norm{H}_\infty^2}
+ (1-\cnoise) n \frac{\norm{\gamma(\alphathr{\us})}_2^2}{\norm{H}_\infty^2}
> 1.
\end{align}
Instead of working directly with $\us$, it is more convenient to define $\us$ implicitly through a condition on $t_{\us}$.
Namely, we choose $\us$ such that $t_{\us}^2 \approx t_n^2 + \frac{\lambda}{t_n}$ for some constant $\lambda>0$.
We now make this choice formal and show that $\us$ is very close to $n$; the following step will be to show that
this choice guarantees $\alpha_s \not\in I$.
\begin{claim} \label{claim:approxt_ulalpha_vgw}
Assume $\cl n \leq d$.
Let any fixed constant $0<\lambda\leq
\sqrt{\log(\cl)}$, and
let $\us$ be the largest integer $s$ such that $t_{s}^2 \geq t_n^2 + \frac{\lambda}{t_n}$.
Then,
\begin{gather}
\us = n \exp\left( -\frac{\lambda}{2t_n} \right) \left( 1+O\left(\frac{1}{t_n^2}\right) \right) \\
\text{and}~~~~~
\abs{t_{\us}^2 - \left( t_n^2 + \frac{\lambda}{t_n} \right)}
\leq O \left(
\frac{1}{n}
\right).
\end{gather}
\end{claim}
The first equation quantifies the fact that this choice of $s$ is very close to $n$;
the second equation controls the error due to rounding (due to the fact that there is no integer $s$ such that $t_s^2 = t_n^2 + \frac{\lambda}{t_n}$ exactly).
\begin{proof}
For concision, in this proof we write $s$ instead of $\us$. Denote $\olt_n^2 = t_n^2 + \frac{\lambda}{t_n}$.
By definition, $t_s^2 \geq \olt_n^2 > t_{s+1}^2$.
For the first part of the claim, we apply Fact~\ref{fact:estim_Phic} several times.
Firstly,
\begin{align}
2\Phic(t_s) = \frac{s}{d}
\leq \frac{2}{\sqrt{2\pi}} \frac{1}{t_s} e^{-t_s^2/2}
\leq
\frac{2}{\sqrt{2\pi}} \frac{1}{t_n} e^{-\olt_n^2/2}
=
\frac{2}{\sqrt{2\pi}} \frac{1}{t_n} e^{-t_n^2/2} \exp \left( -\frac{\lambda}{2 t_n} \right).
\end{align}
Secondly,
\begin{align}
\frac{t_n^2}{1+t_n^2} \cdot
\frac{2}{\sqrt{2\pi}} \frac{1}{t_n} e^{-t_n^2/2}
&\leq 2 \Phic(t_n) \\
\mathrm{and~hence}~~~~~
\frac{2}{\sqrt{2\pi}} \frac{1}{t_n} e^{-t_n^2/2}
&\leq \frac{n}{d} \left( 1 + \frac{1}{t_n^2} \right).
\end{align}
This proves the upper bound on $s$.
The lower bound can be proved in a similar fashion, by applying Fact~\ref{fact:estim_Phic}
to lower-bound $\frac{s+1}{d} = 2\Phic(t_{s+1})$,
and again to upper-bound $\frac{1}{t_n^2} e^{-t_n^2/2}$.
In this way we obtain
\begin{equation}
\frac{s+1}{d} \geq
\frac{n}{d} \exp\left( -\frac{\lambda}{2 t_n} \right)
\left( 1-\frac{1}{1+t_{s+1}^2} \right),
\end{equation}
and the bound $t_s^2 - t_{s+1}^2 \leq \frac{2}{s} \leq 2$ shown below implies that
$\frac{1}{1+t_{s+1}^2} = O \left(
\frac{1}{t_n^2}
\right)$.
We now turn to the second part of the claim.
By mean value theorem applied on $\Phic$, there exists $\xi \in [t_{s+1}, t_s]$ such that
\begin{align}
\frac{\Phic(t_{s+1}) - \Phic(t_s)}{t_{s+1}-t_s}
= \frac{1}{2d} \frac{1}{t_{s+1}-t_s}
&= (\Phic)'(\xi)
= -\frac{1}{\sqrt{2\pi}} e^{-\xi^2/2} \\ \mathrm{and}~~~~~
0 < t_s - t_{s+1}
&= \frac{\sqrt{2\pi}}{2d} e^{\xi^2/2}
\leq \frac{\sqrt{2\pi}}{2d} e^{t_s^2/2}.
\end{align}
Now, by Fact~\ref{fact:estim_Phic}, this can be further upper-bounded as
\begin{align}
\frac{\sqrt{2\pi}}{2d} e^{t_s^2/2}
\leq \frac{1}{2d} \frac{1}{\Phic(t_s)} \frac{1}{t_s}
= \frac{1}{s t_s}.
\end{align}
So we have the bound:
\begin{equation}
t_s^2 - \olt_n^2
\leq t_s^2 - t_{s+1}^2
\leq \frac{t_s+t_{s+1}}{s t_s}
\leq \frac{2}{s}.
\end{equation}
(This completes the proof of the first part of the claim, for the lower bound.)
We can conclude by substituting $s$ by its estimate from the first part of the claim,
noting that $\frac{\lambda}{t_n}$ is uniformly bounded by assumption since
$\lambda \leq \sqrt{\log(\cl)} \leq \sqrt{\log(d/n)} \leq t_n$
by Lemma~\ref{lm:tlogds} (for an appropriate choice of $\cl$).
\end{proof}
We now show that we can choose the constant $\lambda>0$ such that
$\alphathr{\us} \not\in I$ with high probability.
\begin{claim} \label{claim:ulalpha_notinI}
The constants $\cn,\cl,\cd, \lambda>0$ can be chosen such that for any $n,d$ with $n \geq \cn$ and
$\cl n \leq d \leq \exp(\cd n)$,
\begin{equation}
(1-\cnoise) n \frac{\sigma^2}{B^2} \frac{\norm{\gamma(\alphathr{\us})}_1^2}{\norm{H}_\infty^2}
+ (1-\cnoise) n \frac{\norm{\gamma(\alphathr{\us})}_2^2}{\norm{H}_\infty^2}
> 1
\end{equation}
with probability at least
$1 - 12\exp \left( -\frac{n}{\log(d/n)^5} \right)$
over the draws of $\Hgaussian$,
where
$\us$ is defined as in Claim~\ref{claim:approxt_ulalpha_vgw}.
\end{claim}
\begin{proof}
We will repeatedly use the following inequalities, which follow from Lemma~\ref{lm:tlogds} and Claim~\ref{claim:approxt_ulalpha_vgw}:
\begin{gather}
t_n^2 = \log(d/n) + O(\log \log(d/n)) \\
t_{\us}^2 = \log(d/n) + O(\log \log(d/n)) \\
\frac{t_n^2}{t_{\us}^2}
= \frac{1}{1+ \frac{\lambda}{t_n^3} + O\left(\frac{1}{t_n^2n}\right)}
= 1 - \frac{\lambda}{t_n^3} + O\left(\frac{1}{t_n^2n}\right) + O\left(\frac{\lambda^2}{t_n^6}\right).
\end{gather}
Note that for appropriate choices of $\cn,\cl,\cd$, Equation~\eqref{eq:concentration} in Proposition~\ref{prop:concentration_gammas} holds simultaneously for $s=n$ and for $s=\us$ with probability at least
$1 - 6\exp(-2\frac{n}{\log(d/n)^5}) - 6\exp(-2\frac{\us}{\log(d/\us)^5})
\geq 1 - 12 \exp(-\frac{n}{\log(d/n)^5})$.
We condition on this event throughout the remainder of the proof.
We begin with the first term, where we use the upper estimate of $B^2$ from Equation~\eqref{eq:upperestim_B2}:
\begin{align}
(1-\cnoise) n \frac{\sigma^2}{B^2} \frac{\norm{\gamma(\alphathr{\us})}_1^2}{\norm{H}_\infty^2}
&\geq
\frac{t_n^2}{1 - \frac{2}{t_n^2} + O\left(\frac{1}{t_n^4} \right)}
\frac{1}{t_{\us}^2} \left( 1- \frac{4}{t_{\us}^2} + O\left(\frac{1}{t_{\us}^4}\right) \right) \\
&\geq
\left(1 - \frac{2}{t_n^2} + O\left(\frac{1}{t_n^4}\right)\right)
\left( 1 - \frac{\lambda}{t_n^3} + O\left(\frac{1}{t_n^2 n} \right) + O\left(\frac{\lambda^2}{t_n^6} \right) \right) \\
&= 1 - \frac{2}{t_n^2}
- \frac{\lambda}{t_n^3} + O\left(\frac{1}{t_n^4}\right)
+ O\left(\frac{\lambda^2}{t_n^6}\right)
+ O\left(\frac{\lambda}{t_n^2n}\right).
\label{eq:proofphitmp1}
\end{align}
Next the second term:
\begin{align}
(1-\cnoise) n \frac{\norm{\gamma(\alphathr{\us})}_2^2}{\norm{H}_\infty^2}
&\geq n \frac{2}{\us t_{\us}^2}\left(1+ O\left(\frac{1}{t_{\us}^2}\right)\right) \\
&\geq \frac{2}{t_{\us}^2}
\exp\left(\frac{\lambda}{2t_n}\right)
\left(1 + O\left( \frac{1}{t_n^2}\right) \right) \\
&\geq \frac{2}{t_n^2} \left(1+ \frac{\lambda}{2t_n} + \frac{\lambda^2}{4t_n^2}+ O\left(\frac{\lambda^3}{t_n^3}\right)\right)\left(1 + O\left(\frac{1}{t_n^2}\right)\right).
\label{eq:proofphitmp2}
\end{align}
Summing up \eqref{eq:proofphitmp1} and \eqref{eq:proofphitmp2}, we get:
\begin{align}
(1-\cnoise) n \frac{\sigma^2}{B^2} \frac{\norm{\gamma(\alphathr{\us})}_1^2}{\norm{H}_\infty^2}
+ (1-\cnoise) n \frac{\norm{\gamma(\alphathr{\us})}_2^2}{\norm{H}_\infty^2}
\geq
1
+ \frac{\lambda^2}{2 t_n^4}
+ O\left(\frac{1}{t_n^4}\right)
+ O\left(\frac{\lambda^2}{t_n^6}\right)
+ O\left(\frac{\lambda}{t_n^2 n}\right). \label{eq:hereweneedwgt}
\end{align}
Clearly, we can choose the constants $\lambda, \cn,\cl,\cd>0$
such that the right-hand side
of the above equation
is strictly greater than $1$ for any $n,d$ with
$\cl n \leq d$, since $t_n^2 \asymp \log(d/n)$.
\end{proof}
\subsubsection{Finding \texorpdfstring{$\os$}{overlines} such that \texorpdfstring{$\oalpha_I \leq \alphathr{\os} \leq \alphad$}{overlinealphaI leq alphaoverlines}}
We take the exact same approach to find $\os \geq n$ such that $\alphathr{\os}$ is on the right of the interval $I$, i.e., $\oalpha_I \leq \alphathr{\os}$. The derivations can be straightforwardly adapted, and we get the analogous results:
\begin{claim} \label{claim:approxt_olalpha_vgw}
Assume $\cl n \leq d$.
Let any fixed constant $0<\lambda \leq \sqrt{\log(\cl)}$, and
let $\os$ be the smallest integer $s$ such that $t_{s}^2 \leq t_n^2 - \frac{\lambda}{t_n}$.
Then,
\begin{align}
\os &= n \exp\left( \frac{\lambda}{2t_n} \right) \left( 1+O\left(\frac{1}{t_n^2}\right) \right)
& &\text{and} &
\abs{t_{\os}^2 - \left( t_n^2 - \frac{\lambda}{t_n} \right)}
&\leq O \left(
\frac{1}{n}
\right).
\end{align}
\end{claim}
\begin{claim} \label{claim:olalpha_notinI}
The constants $\cn,\cl,\cd, \lambda>0$ can be chosen such that for any $n,d$ with $n \geq \cn$ and
$\cl n \leq d \leq \exp(\cd n)$,
\begin{equation}
(1-\cnoise) n \frac{\sigma^2}{B^2} \frac{\norm{\gamma(\alphathr{\os})}_1^2}{\norm{H}_\infty^2}
+ (1-\cnoise) n \frac{\norm{\gamma(\alphathr{\os})}_2^2}{\norm{H}_\infty^2}
> 1
\end{equation}
with probability at least
$1 - 12\exp \left( -\frac{n}{\log(d/n)^5} \right)$
over the draws of $\Hgaussian$,
where $\os$ is defined as in Claim~\ref{claim:approxt_olalpha_vgw}.
\end{claim}
It remains to check that this choice of $\os$ satisfies $\alphathr{\os} < \alphad$.
But this is clearly the case, because $\os<d$ (by the first part of Claim~\ref{claim:approxt_olalpha_vgw}, for appropriate choices of $\cl$) and $\alphathr{d} < \alphad$ by definition.
This concludes the proof of Lemma~\ref{lm:boundsonalpha}.
\end{document}
\section{Proof details}
\else
\section{PROOF DETAILS}
\fi
\label{apx:sec:appendix_proofs}
In this appendix, we provide details of the proof of our main result, Theorem~\ref{thm:main}, omitted in Section~\ref{sec:proof_maintext}.
We refer to that section for notation.
\subfile{0_prooftriangle}
\subfile{1_CGMT}
\subfile{2_param_argmaxmin_v2}
\subfile{3_geometric_lemma}
\subfile{4_concentration_gamma}
\subfile{6_feasible_interval}
\subfile{7_bound_phip_phim}
\end{document}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,492
|
{"url":"https:\/\/math.stackexchange.com\/questions\/1392314\/how-do-i-integrate-frac1-sin-x-cos-x4\/1392319","text":"# How do I integrate $\\frac{1}{(\\sin x + \\cos x)^{4}}$?\n\nHow do I integrate $\\dfrac{1}{(\\sin x + \\cos x)^{4}}$? I could not think of any way. I tried substitution but of no use.\n\n\u2022 use the $\\sin x + \\cos x =\\sqrt 2 \\sin(\\pi\/4+x)$ \u2013\u00a0Math-fun Aug 10 '15 at 19:27\n\u2022 i would use the tan-half angle substitution \u2013\u00a0Dr. Sonnhard Graubner Aug 10 '15 at 19:28\n\n$$\\int \\frac{1}{(\\sin x + \\cos x)^{4}}dx$$\n\nHint:\n\nMultiply numerator and denominator by $\\sec^4$\n\n$$=\\int \\frac{\\sec^4 (x)} { 1+4\\tan(x)+6\\tan^2 (x)+4\\tan^3(x)+\\tan^4(x)}dx$$\n\nUse $\\sec^2(x)=\\tan^2(x)+1$\n\n$$\\int \\frac{(1+\\tan^2(x))\\sec^2(x)}{(1+\\tan(x))^4}dx$$\n\nNow substitute $u=\\tan(x)$\n\n$$\\int \\frac{u^2+1}{(u+1)^4}du$$\n\nI hope that you can finish from here.\n\n\u2022 It might be easier to leave the denominator as $(1+\\tan x)^4$ and substitute $u = 1+\\tan x$. \u2013\u00a0JimmyK4542 Aug 10 '15 at 19:29\n\u2022 @Nehorai Yes Thanks \u2013\u00a0Taylor Ted Aug 10 '15 at 19:35\n\n$$\\sin x + \\cos x = \\sqrt{2} \\left( \\sin x \\cos \\frac{\\pi}{4} + \\cos x \\sin \\frac{\\pi}{4} \\right) = \\sqrt{2} \\sin \\left( x + \\frac{\\pi}{4} \\right).$$\n\n\u2022 what should i do next for integration, next step? \u2013\u00a0Taylor Ted Aug 11 '15 at 12:21","date":"2020-10-23 00:27:44","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.958117663860321, \"perplexity\": 1038.560235622682}, \"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\/1603107880401.35\/warc\/CC-MAIN-20201022225046-20201023015046-00415.warc.gz\"}"}
| null | null |
{"url":"https:\/\/bioinformatics.stackexchange.com\/questions\/7458\/what-is-a-quick-way-to-find-the-reverse-complement-in-bash","text":"# What is a quick way to find the reverse complement in bash\n\nI have a DNA sequence of which I would like to quickly find the reverse complement. Is there a quick way of doing this on the bash command line using only GNU tools?\n\necho ACCTTGAAA | tr ACGTacgt TGCAtgca | rev","date":"2020-02-21 16:26:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.19170641899108887, \"perplexity\": 830.8520184995537}, \"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-2020-10\/segments\/1581875145533.1\/warc\/CC-MAIN-20200221142006-20200221172006-00229.warc.gz\"}"}
| null | null |
Ed Oliver Announces Intentions To Enter 2019 NFL Draft
Ed Oliver reveals his 2019 professional intentions as early as possible.
Photo by Jack Gorman
Sean Pendergast | March 6, 2018 | 7:22am
Over the last few seasons, as the medical and financial welfare of college football players has become a bigger and bigger issue, experts have surmised whether or not it makes sense for certain ELITE players to sit out their collegiate junior seasons and wait for their NFL-mandated three year post-high school window to expire rather than play and risk injury during their junior season.
Former LSU running back Leonard Fournette and current Houston Texan Jadeveon Clowney, both eventual top four picks in their respective drafts, come to mind. The notion of their sitting out was only exacerbated by injuries each suffered during their sophomore seasons. Same goes for Houston's Ed Oliver, who was incredible in 2017 while playing through some nagging injuries.
A consensus All-American his first two seasons in college, Oliver will not be sitting out his junior year. However, he will make no bones about the business side of what he is doing, and to that end, Oliver made the earliest "I'm going pro" announcement that I can ever remember a college football player making, announcing after the Cougars' first spring practice that he will be entering the 2019 NFL Draft, two months in advance of the 2018 NFL DRAFT:
"This was a dream of mine coming in," Oliver said. "I knew I was going on a business trip, and whether my business trip was three years or four years it just depended on how early I got on the field and how effective I was."
Oliver's resume is about as good as it gets for an interior defensive lineman with two seasons of college football under his belt. In 2018, Oliver won the Outland Trophy winner and was named The American's Defensive Player of the Year. Oliver also leads the nation with a career average of 1.56 tackles for loss per game, totaling 39.5 tackles for loss in just 25 games played.
Football purists, a breed that seems to be losing more and more of its constituency with each passing hypocritical NCAA scandal, probably cringe at a player announcing intentions to go pro BEFORE the upcoming season, but Oliver's announcement and its timing make a lot of sense. It removes the specter of an obvious question hanging over the team during the 2018 season, and it actually provides a valuable tool on the recruiting trail in 2018 for head coach Major Applewhite.
In fact, you can see Applewhite practicing the pitch in this clip right here:
.@CoachApplewhite on @UHCougarFB having a player like @Edoliver_11 have the success he's had: "It kind of puts a hole in the whole (idea) you got to go to a Power Five or a certain school to get all those things done. It kind of ruins that theory." pic.twitter.com/KB8tkaI7v1
— Mark Berman (@MarkBermanFox26) March 6, 2018
The season begins in about six months, and it's worth noting, Ed Oliver is the only defensive lineman on the odds board for the Heisman Trophy. We will see how things go, but for now, we know that the Ed Oliver farewell tour begins with the Cougars' spring scrimmage.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 850
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Самоходни циркулар или цепач дрва је возило опремљено трачном или кружном тестером, које се користи у дрвној индустрији и намењено је за сечење већих трупаца на мање, погодне за пећи на дрва. Користи се углавном у јесен или рану зиму, када је потражња за огревним дрветом највећа.
Самоходни циркулар познат је и по народним називима немачког порекла (нем. Бандсаге: трачна тестера): банзек, бансек, бансег или банцек (не бонсек).
Први самоходни циркулар направљен је 1900. године, а за првог произвођача овог возила у Европи сматра се немачка компанија Каелбле из Бакнанга.
Ова возила користе углавном дизел моторе разних произвођача, најчешће "аран" и "ломбардини".
На Западу су ова возила била у честој употреби међу трговцима дрветом све до 1960-их. У источној Европи се још увек често може видети у јесен и рану зиму, када је потражња за огревним дрветом највећа. Занатлије у Србији, које се баве цепањем огревног дрвета, понекад користе своје ручно рађене самоходне циркуларе.
Референце
Спољашње везе
www.dhm.de: Самоходни циркулар из 1913. године у Немачком историјском музеју ()
Возила
Алати
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{
"redpajama_set_name": "RedPajamaWikipedia"
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| 9,006
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Böckweiler est un écart de la commune allemande de Blieskastel dans l'arrondissement de Sarre-Palatinat en Sarre.
Jusqu'au , le village était une commune indépendante de l'arrondissement de Hombourg.
Géographie
Localisation
Böckweiler se situe au sud-est de la région naturelle du Bliesgau et du Palatinat sarrois. Le village se trouve à environ de Blieskastel et de Gersheim.
Localités avoisinantes
Histoire
Böckweiler est mentionné pour la première fois en sous la forme Bickwilre dans des documents de l'abbaye de Hornbach.
En , le village est mentionné par Thierry Alix comme faisant partie de la mairie de Rimling du comté de Bitche.
Après la Révolution, Böckweiler devient français avec tout le reste de la rive gauche du Rhin. Sous administration française, le village appartient à la mairie de Mittelbach, au canton de Medelsheim, à l'arrondissement de Deux-Ponts et au département du Mont-Tonnerre.
Après le congrès de Vienne en , le village est attribué au royaume de Bavière. Sous son administration, le canton de Medelsheim est supprimé, et Böckweiler est alors incorporé au canton de Neuhornbach.
Böckweiler est incorporé à Blieskastel le .
Lieux et monuments
Annexes
Articles connexes
Blieskastel
Liens externes
Notes et références
Blieskastel
Localité de l'arrondissement de Sarre-Palatinat
Ancienne commune dans la Sarre
|
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| 795
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Q: Vuejs: Router links not correct after passing params router links are all working until I pass params to a component. The component loads correctly using the params but after that when I want to navigate to another link the route doesn't change back correctly.
For example:
I go to /error/123 and after that to /info -> what I get is /error/info
Somebody had similar issues?
routes: [
{
path: '/errors',
name: 'errors',
component: Errors
},
{
path: '/error/:id',
name: 'error',
component: Errors
},
{
path: '/info',
name: 'info',
component: Info
}]
// link in another component
<v-btn :to="{ name: 'error', params: { id: id } }">Go to</v-btn>
// in error compompent
if(this.$route.params.id) {
this.$store.dispatch("single_error", this.$route.params.id)
}
// navigation
<v-navigation-drawer
v-model="drawer"
app
>
<v-list dense v-for="(nav, i) in navs" :key="i">
<v-list-item :to=" { path: nav.path }">
<v-list-item-action>
<v-icon>{{ nav.icon }}</v-icon>
</v-list-item-action>
<v-list-item-content>
<v-list-item-title>{{ nav.title }}</v-list-item-title>
</v-list-item-content>
</v-list-item>
</v-list>
</v-navigation-drawer>
// data for navigation
export default {
data: () => ({
drawer: null,
navs: [
{ path: '/', icon: 'home', title: 'Home'},
{ path: 'errors', icon: 'error', title: 'Errors'},
{ path: 'info', icon: 'help', title: 'Info'},
]
}),
}
A: The URL path in your navs element do not have the initial /. Try this:
navs: [
{ path: '/', icon: 'home', title: 'Home'},
{ path: '/errors', icon: 'error', title: 'Errors'},
{ path: '/info', icon: 'help', title: 'Info'},
]
By the way, I suggest you to use the named routes in the navigator as well.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,426
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Q: Untar multipart tarball on Windows I have a series of files named filename.part0.tar, filename.part1.tar, … filename.part8.tar.
I guess tar can create multiple volumes when archiving, but I can't seem to find a way to unarchive them on Windows. I've tried to untar them using 7zip (GUI & commandline), WinRAR, tar114 (which doesn't run on 64-bit Windows), WinZip, and ZenTar (a little utility I found).
All programs run through the part0 file, extracting 3 rar files, then quit reporting an error. None of the other part files are recognized as .tar, .rar, .zip, or .gz.
I've tried concatenating them using the DOS copy command, but that doesn't work, possibly because part0 thru part6 and part8 are each 100Mb, while part7 is 53Mb and therefore likely the last part. I've tried several different logical orders for the files in concatenation, but no joy.
Other than installing Linux, finding a live distro, or tracking down the guy who left these files for me, how can I untar these files?
A: Install 7-zip. Right click on the first tar. In the context menu, go to "7zip -> Extract Here".
Works like a charm, no command-line kung-fu needed:)
EDIT:
I only now noticed that you mention already having tried 7zip. It might have balked if you tried to "open" the tar by going "open with" -> 7zip - Their command-line for opening files is a little unorthodox, so you have to associate via 7zip instead of via the file association system built-in to windows. If you try the right click -> "7-zip" -> "extract here", though, that should work- I tested the solution myself (albeit on a 32-bit Windows box- Don't have a 64 available)
A: 1) download gzip http://www.gzip.org/ for windows and unpack it
2) gzip -c filename.part0.tar > foo.gz
gzip -c filename.part1.tar >> foo.gz
...
gzip -c filename.part8.tar >> foo.gz
3) unpack foo.gz
worked for me
A: As above, I had the same issue and ran into this old thread. For me it was a severe case of RTFM when installing a Siebel VM . These instructions were straight from the manual:
cat \
OVM_EL5U3_X86_ORACLE11G_SIEBEL811ENU_SIA21111_PVM.tgz.1of3 \
OVM_EL5U3_X86_ORACLE11G_SIEBEL811ENU_SIA21111_PVM.tgz.2of3 \
OVM_EL5U3_X86_ORACLE11G_SIEBEL811ENU_SIA21111_PVM.tgz.3of3 \
| tar xzf –
Worked for me!
A: The tar -M switch should it for you on windows (I'm using tar.exe).
tar --help says:
-M, --multi-volume create/list/extract multi-volume archive
A: I found this thread because I had the same problem with these files. Yes, the same exact files you have. Here's the correct order: 042358617 (i.e. start with part0, then part4, etc.)
Concatenate in that order and you'll get a tarball you can unarchive. (I'm not on Windows, so I can't advise on what app to use.) Note that of the 19 items contained therein, 3 are zip files that some unarchive utilities will report as being corrupted. Other apps will allow you to extract 99% of their contents. Again, I'm not on Windows, so you'll have to experiment for yourself.
Enjoy! ;)
A: This works well for me with multivolume tar archives (numbered .tar.1, .tar.2 and so on) and even allows to --list or --get specific folders or files in them:
#!/bin/bash
TAR=/usr/bin/tar
ARCHIVE=bkup-01Jun
RPATH=home/user
RDEST=restore/
EXCLUDE=.*
mkdir -p $RDEST
$TAR vf $ARCHIVE.tar.1 -F 'echo '$ARCHIVE'.tar.${TAR_VOLUME} >&${TAR_FD}' -C $RDEST --get $RPATH --exclude "$EXCLUDE"
Copy to a script file, then just change the parameters:
*
*TAR=location of tar binary
*ARCHIVE=Archive base name (without .tar.multivolumenumber)
*RPATH=path to restore (leave empty for full restore)
*RDEST=restore destination folder (relative or absolute path)
*EXCLUDE=files to exclude (with pattern matching)
Interesting thing for me is you really DON'T use the -M option, as this would only ask you questions (insert next volume etc.)
A: Hello perhaps would help.
I had the same problems ...
a save on my web site made automaticaly in Centos at 4 am create multiple file in multivolume tar format (saveblabla.tar, saveblabla.tar1.tar, saveblabla.tar2.tar,etc..)
after downloading this file on my PC (windows) i can't extract them with both windows cmd or 7zip (unknow error).
I thirst binary copy file to reassemble tar files. (above in that thread)
copy /b file1+file2+file3 destination
after that, 7zip worked !!! Thanks for you help
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,715
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Characterization of leptospiral proteins that afford partial protection in hamsters against lethal challenge with Leptospira interrogans
Marina V. Atzingen1, Amane P. Gonçales2, Zenaide M. de Morais2, Eduardo R. Araújo3, Thales De Brito3, Silvio A. Vasconcellos2, Ana L. T. O. Nascimento1,4
Affiliations: 1 1Centro de Biotecnologia, Instituto Butantan, Avenida Vital Brazil 1500, 05503-900 São Paulo, SP, Brazil 2 2Laboratório de Zoonoses Bacterianas do VPS, Faculdade de Medicina Veterinária e Zootecnia, USP, Avenida Prof. Dr Orlando Marques de Paiva 87, 05508-270 São Paulo, SP, Brazil 3 3Instituto de Medicina Tropical, Departamento de Patologia, Faculdade de Medicina, USP, Avenida Dr Enéas Carvalho de Aguiar 470, 05403-000 São Paulo, SP, Brazil 4 4Interunidades em Biotecnologia, Instituto de Ciências Biomédicas, USP, Avenida Prof. Lineu Prestes 1730, 05508-900 São Paulo, SP, Brazil
CorrespondenceAna L. T. O. Nascimento
[email protected]
First Published: 01 September 2010 https://doi.org/10.1099/jmm.0.021485-0
Leptospirosis is a worldwide zoonosis caused by pathogenic Leptospira. The whole-genome sequence of Leptospira interrogans serovar Copenhageni together with bioinformatic tools allow us to search for novel antigen candidates suitable for improved vaccines against leptospirosis. This study focused on three genes encoding conserved hypothetical proteins predicted to be exported to the outer membrane. The genes were amplified by PCR from six predominant pathogenic serovars in Brazil. The genes were cloned and expressed in Escherichia coli strain BL21-SI using the expression vector pDEST17. The recombinant proteins tagged with N-terminal 6×His were purified by metal-charged chromatography. The proteins were recognized by antibodies present in sera from hamsters that were experimentally infected. Immunization of hamsters followed by challenge with a lethal dose of a virulent strain of Leptospira showed that the recombinant protein rLIC12730 afforded statistically significant protection to animals (44 %), followed by rLIC10494 (40 %) and rLIC12922 (30 %). Immunization with these proteins produced an increase in antibody titres during subsequent boosters, suggesting the involvement of a T-helper 2 response. Although more studies are needed, these data suggest that rLIC12730 and rLIC10494 are promising candidates for a multivalent vaccine for the prevention of leptospirosis.
Keyword(s): CD, circular dichroism , CDS, coding sequence , HRP, horseradish peroxidase , LAg, leptospiral antigen , OMP, outer-membrane protein and Th, T helper
/deliver/fulltext/jmm/59/9/1005.html?itemId=/content/journal/jmm/10.1099/jmm.0.021485-0&mimeType=html&fmt=ahah
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Nascimento, A. L., Ko, A. I., Martins, E. A., Monteiro-Vitorello, C. B., Ho, P. L., Haake, D. A., Verjovski-Almeida, S., Hartskeerl, R. A., Marques, M. V. & other authors ( 2004b; ). Comparative genomics of two Leptospira interrogans serovars reveals novel insights into physiology and pathogenesis. J Bacteriol 186, 2164–2172.[CrossRef]
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Yan, W., Faisal, S. M., McDonough, S. P., Divers, T. J., Barr, S. C., Chang, C. F., Pan, M. J. & Chang, Y. F. ( 2009; ). Immunogenicity and protective efficacy of recombinant Leptospira immunoglobulin-like protein B (rLigB) in a hamster challenge model. Microbes Infect 11, 230–237.[CrossRef]
Yan, W., Faisal, S. M., McDonough, S. P., Chang, C. F., Pan, M. J., Akey, B. & Chang, Y. F. ( 2010; ). Identification and characterization of OmpA-like proteins as novel vaccine candidates for leptospirosis. Vaccine 28, 2277–2283.[CrossRef]
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J. Med. Microbiol. 59, 1005 (2010); https://doi.org/10.1099/jmm.0.021485-0
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 7,812
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MEND Orthopedic is designed for tissue repair and recovery from injury, surgery or repetitive stress from exercise and sports.
Nutrition For Healing, the company behind MEND, was founded by world leading medical and nutrition experts who share a common philosophy on the critical importance of nutrition in healing following injury or trauma. Nutritional building blocks are utilized by the body to repair damaged tissue and enhance the healing process. A growing body of scientific evidence is showing a strong link between improved recovery following injury/surgery and nutritional supplementation.
For more almost 20 years, SupplySide has helped dietary supplement, food, beverage, personal care and cosmetic professionals find information to explore, discover, innovate and market their next best-selling product.
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{
"redpajama_set_name": "RedPajamaC4"
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22/04/2007 · Hey, I'm really stuck on this question. I need to find the equation of a plane passing through points (1,1,1) and (2,0,3) and perpendicular to the plane x+2y-3z=0.... 28/10/2010 · Curve with normal passing through a fixed point October 28, 2010 beni22sof Leave a comment Go to comments Suppose is a regular (arclength parametrized) curve with nonzero curvature.
(716, #21) Find an equation of the plane through the point ( 1, 2, 3 ) and with the normal vector n = 15i + 9j - 12k. Solution: The general equation of the plane pokemon how to get free primer balls If this line passes through the \(xz\)-plane then we know that the \(y\)-coordinate of that point must be zero. So, let's set the \(y\) component of the equation equal to zero and see if we can solve for \(t\). If we can, this will give the value of \(t\) for which the point will pass through the \(xz\)-plane.
What we want to calculate is the equation of the tangent at this point on the curve. The tangent must pass through the point and have gradient 10.
21/05/2018 · To find the equation for the normal, take advantage of the fact that (slope of tangent)(slope of normal) = -1, when they both pass through the same point on the graph. In other words: Find f'(x), the slope of the tangent line.
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"redpajama_set_name": "RedPajamaC4"
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PRAISE FOR MINA LOY
"[In _The Lost Lunar Baedeker_ ,] Mina Loy's wry, confident inquiries into the nature of men, women and sexuality are a great undiscovered treasure of modernism."
— _PUBLISHERS WEEKLY_
(STARRED REVIEW)
"Her utter absence from all canonical lists is one of modern literary history's most perplexing data."
—HUGH KENNER,
_THE NEW YORK TIMES_
"Among the great modernist poets, Mina Loy was surely the greatest wit, the most sophisticated commentator on the vagaries of love."
—MARJORIE PERLOFF
"Is there anyone in America except you, Bill [William Carlos Williams,] and Mina Loy who can write anything of interest in verse?"
—EZRA POUND, LETTER TO MARIANNE MOORE
"By divergent virtues these two women [Mina Loy and Marianne Moore] have achieved freshness of presentation, novelty, freedom, break with banality."
—WILLIAM CARLOS WILLIAMS
"You'd know a Loy poem when you read one; you'd recognize her art work as distinctively hers. And maybe that's the mark she would have most cared to leave on the world—literary and visual art made, unmistakably, by a true original."
— _THE NERVOUS BREAKDOWN_
# **INSEL**
**MINA LOY** (1882–1966) was born Mina Gertrude Lowy in London to a Hungarian father and an English mother. Originally trained as a painter, she was at the center of all the great artistic movements of the first half of the twentieth century: she wrote Futurist manifestos in Italy (including the "Feminist Manifesto," which denounced the misogyny and incipient fascist tendencies of Futurism); her poem "Brancusi's Golden Bird" appeared alongside T. S. Eliot's "The Waste Land" in _The Dial_ ; she starred in plays in Greenwich Village in the 1920s with William Carlos Williams; she was friends with Duchamp and Man Ray; she ran a lampshade business with Peggy Guggenheim; and in the 1940s, she lived on the Bowery, where she collected trash for found-art collages, as in the style of her friend Joseph Cornell, whose work she championed. During one of her earlier stints in New York, she met the love of her life, Arthur Cravan, the Dadaist poet and boxer who disappeared in mysterious circumstances shortly after their marriage. Only two collections of her work were published in her lifetime, _Lunar Baedecker_ (1923) and _Lunar Baedecker and the Time Tables_ (1958). She died in Aspen, Colorado, in 1966.
**ELIZABETH ARNOLD** , a scholar and poet, is the author of _Effacement_ and two other collections.
**SARAH HAYDEN** is Irish Research Council Postdoctoral Fellow at the School of English, University College Cork. Her monograph, _Curious Disciplines: Mina Loy and Avant-Garde Artisthood_ , is forthcoming in the Recencies series at University of New Mexico Press.
#
**THE NEVERSINK LIBRARY**
_I was by no means the only reader of books on board the_ Neversink. _Several other sailors were diligent readers, though their studies did not lie in the way of belles-lettres. Their favourite authors were such as you may find at the book-stalls around Fulton Market; they were slightly physiological in their nature. My book experiences on board of the frigate proved an example of a fact which every book-lover must have experienced before me, namely, that though public libraries have an imposing air, and doubtless contain invaluable volumes, yet, somehow, the books that prove most agreeable, grateful, and companionable, are those we pick up by chance here and there; those which seem put into our hands by Providence; those which pretend to little, but abound in much_.
—HERMAN MELVILLE, _WHITE JACKET_
INSEL
Copyright © 1991, 2014 by Roger L. Conover for the Estate of Mina Loy
Introduction copyright © 2014 by Sarah Hayden
Afterword copyright © 1991 by Elizabeth Arnold
With thanks to the Beinecke Library at Yale University for their assistance with the "Visitation" materials.
First Melville House printing: May 2014
Melville House Publishing
145 Plymouth Street
Brooklyn, NY 11201
and
8 Blackstock Mews
Islington
London N4 2BT
mhpbooks.com facebook.com/mhpbooks @melvillehouse
ISBN: 978-1-61219-354-0
eBook ISBN: 978-1-61219-354-0
Cover photograph: Mina Loy, ca. 1905.
Photo by Stephen Haweis, courtesy of Roger L. Conover
A catalog record for this title is available from the Library of Congress.
v3.1
# CONTENTS
_Cover_
_About the Author_
_Epigraph_
_Title Page_
_Copyright_
Introduction by Sarah Hayden
A Note on the Text
_Insel_
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 20
Chapter 21
Chapter 22
Chapter 23
Chapter 24
Chapter 25
Chapter 26
Visitation
Afterword by Elizabeth Arnold
Appendix A: Translation of Foreign Words and Phrases
Appendix B: Chronology of Mina Loy
Appendix C: Chronology of Richard Oelze
# **INTRODUCTION BY SARAH HAYDEN**
"It is the nature of individuality to look upon the rest of the world in the light of an audience. I got the idea that 'an impression is to be made.' "
—"Islands in the Air"
Born Mina Gertrude Lowy, a self-nominated "Anglo-mongrel," in London in 1882, Mina Loy died an American citizen in 1966. Between those dates, she was resident of _Jugendstil_ Munich (1900), Futurist Florence (1907–1916), Dada-enfevered New York (1916–1917) and Surrealist Paris (1923–1936), making interstitial appearances in Paris (1903–1906), Mexico (1918), Weimar Berlin (1922) and Freud's Vienna (1922), among other places. Relocating finally from Europe to New York in 1936, she was the unofficial artist-in-residence on the Bowery when New York was inaugurated as capital city of the postwar art world. Though the stamps in her passport furnish the coordinates to map the development of twentieth-century art, Loy was not just fortuitously "present" in these locations at the zero hour of their modernist fluorescence. She engaged with these art scenes neither as onlooker nor acolyte but as an acutely critical cross-media artist. Leaving New York in 1953, she made her last works in the barely nascent art town of Aspen; Loy's burial there in 1966 left her body at the site of yet another significant episode in the history of the transatlantic avant-gardes.
It has become something of a commonplace to populate the opening of any portrait of Loy with globe-wrapping litanies. Though the inventory device might be hackneyed, the listing of multiple cities, movements, artistic colocutors and art practices remains the only way to convey the multiplicities that characterize her formation and her work. There have been many Loys; more are emerging. On the event of their marriage in 1918, the Dada poet-pugilist Arthur Cravan declared: "Now I have caught you. I am at ease" ("Excerpts from 'Colossus' "). Cut short by his mysterious disappearance that same year, their romance was absolute. For the rest of us, capturing Loy is neither possible nor even advisable. Slipping between textual avatars, she remakes herself as Imna Loy, Goy, Ova, Sophia and, in _Insel_ , Mrs. Jones. Though much of her writing rehearses the contours of her biography, her predilection for imposture, anagram and factual distortion make slippery the lines between autobiography and fiction. Nowhere else are these distinctions so bewitchingly blurred as they are in this novel.
In life, the trajectory of Loy's career saw her play overlapping roles as painter, poet, model, actor, archetypal Modern Woman, playwright, novelist, autobiographer, inventor, polemicist, designer, gallery agent and assemblagist. Galvanized by inconstant affinities with myriad systems of thought (including Christian Science, Hindu mysticism, Theosophy, Bergsonism, sexology, psychoanalysis) and innumerable species of avant-garde art-making, Loy was variously feted and forgotten. Through poverty, alienation and heartbreak, she remained a practicing artist, ever in evolution, to the end. Striving to impress the intensely modern emanations of her "isolate consciousness" ("Anglo-Mongrels") upon the world, she proves herself, in _Insel_ and beyond, a sophisticated commentator on creativity—an expert theorist of modern artisthood.
Loy first came to public attention as a painter at the 1904 Paris Salon d'Automne; in 1906 she was elected _salonnier_. This early recognition of her prowess as a painter was superseded, however, by her reputation as a poet. The recovery of Loy's plastic arts production—which comprises drawings, paintings, sculptures and a remarkable body of assemblages—is at last underway. Perhaps our tardiness in coming to evaluate Loy's visual art can be attributed to her pursuit, alongside it, of such a variegate span of other creative pursuits—for she also designed lampshades, and patented inventions both prosaic (a device for cleaning windows "from the inside out") and fanciful (a corselet intended for the "alleviation of dowager's hump").
In her lifetime, what renown she attained derived predominantly from her activities as a poet. Between 1914 and 1962, her work was published in magazines and periodicals on both sides of the Atlantic, including _Camera Work, Trend, Rogue, Others, Blind Man, The Dial, The Little Review, Contact, Playboy, transatlantic review, Pagany, Accent_ and _View_ , and featured in anthologies such as the 1925 _Contact Collection of Contemporary Writers_ and in Kreymborg's 1930 _Lyric America: An Anthology of American Poetry_. Her first poetry collection, _Lunar Baedecker_ , was published by Robert McAlmon's Contact Editions in 1923, its title misspelled. In 1958, a new edition of her poems was published under Jonathan Williams's Jargon imprint. _Lunar Baedekerand Timetables_ corrected some of the 1923 edition's errors, but introduced others. For too long, she was overlooked, blighted by the critical amnesia that commonly affects women writers, and that is endemic in treatments of Loy's elusive, experimental ilk. In the last decades of the twentieth century, critics started to probe modernism's margins, questioning received histories of literary communities and recovering forgotten figures. With the publication of Roger Conover's editions of _The Last Lunar Baedeker_ (1982) and his revised and comprehensively annotated _The Lost Lunar Baedeker_ (1996), Loy's literary legacy came, convulsively, back to life. Loy-alists rejoiced to see her back in print; today, their ranks are great and still growing.
In his introduction to the first edition of _Insel_ , Roger Conover—Loy's longtime editor, literary executor and curator—traced Loy's posthumous evolution from " 'neglected' poet" to one "whose reputation and readership are very much on the rise." On the occasion of its re-publication, we find Loy securely instated among her rightful cohort—as a luminary of literary modernism. In 2011, Sara Crangle's edition of the _Stories and Essays of Mina Loy_ opened new vistas not only onto Loy's heretofore submerged prosodic imagination but onto the principal preoccupations of her poetry. Crosscurrents exist between her poetry and prose that weren't previously recognized: oppositions to censorship and social injustice implicit in her poetry are made explicit in her prose.
Where once she enjoyed only a refractory sort of fame-by-association, today the extent of Loy's intellectual interplay with figures such as Ezra Pound, F. T. Marinetti, Giovanni Papini, Marcel Duchamp, Djuna Barnes, Gertrude Stein and William Carlos Williams eclipses the anecdotal. We have come to recognize that Loy was acutely attuned to the ways in which cosmopolitan avant-garde movements were attempting to reconstitute the artist's role in modern society. Polemical texts such as "International Psycho-Democracy," "In... Formation" and "Aphorisms on Futurism" give us her position papers on contemporary notions of artisthood. Her short prose pieces—"O Marcel... Otherwise I Also Have Been to Louise's" and "Pas de Commentaires! Louis M. Eilshemius"—elucidate the depth of her ironic engagement with the performance politics of Duchampian New York Dada. Similarly, her conflicted response to the hypermasculine bombast of Florentine Futurism is rendered into the sharp satire of the early poems, "Brontolivido," "Lion's Jaws" and "The Effectual Marriage," Loy's investment in aesthetics led her to compose taut ekphrases—"Brancusi's Golden Bird," "Joyce's Ulysses," "The Starry Sky of Wyndham Lewis"—and acute portraits of her literary and artistic contemporaries, including Stein, Picasso, Pascin, Lewis and William Carlos Williams. The nature of genius fascinated her; in poetry, prose and drama, she anatomized its "curious disciplines" ("Apology of Genius"). _Insel_ is the product of one such investigation, a fantastical account of her experiences as "tout for a friend's art gallery, feeding a cagey genius in the hope of production" (this page).
Although the first flush of critical interest in Loy's oeuvre concentrated on her earlier writing, analysis of the prose and poetry that occupied her latter years is now well underway. These writings articulate a sustained attention to the nature of creative enterprise, the development of broad philosophical and spiritual curiosities and a burgeoning social conscience underwritten by a marked, if loose, affiliation to psychoanalytic theories of mind. From 1936 onwards, Loy's primary focus was on the composition "Islands in the Air": an ambitious, categorically unstable project to encompass the entirety of her autobiography in a fictionalized prose form. A sort of modernist case study, it would, as she envisioned it, offer "(m)y experience to yours for comparison" ("Islands in the Air"). _Insel_ was originally conceived as part of this immense experimental prose work.
In 1931, the art dealer Julien Levy appointed Mina Loy sole "Paris _représentante_ " of his newly opened Manhattan gallery. In her capacity as gallery agent (and Levy's mother-in-law), Loy represented a host of major artists, including Gris, Giacometti, Gorky, de Chirico, Dalí and Magritte. In 1933, Loy became acquainted with the isolated German Surrealist, Richard Oelze; three years later, they parted company. Leaving Oelze in Europe, Loy sailed for New York in 1936. As Elizabeth Arnold explains in her afterword, _Insel_ is Loy's prose-rendering of what transpired between those dates. Strung from a series of impossible happenings, furred with bizarre blooms and spasmodically fluctuating between revulsion and fascination, the story of Insel and Mrs. Jones is not a love story. Variously designated as Surrealist novel, _Künstlerroman_ and modernist _roman à clef, Insel_ is, like its eponymous anti-hero, a strange and beguilingly fugitive creation: a text that exists "at variance with" itself (this page).
Staged against the familiar backdrops of the Select, Dôme and Capoulards cafés, the Lutetia hotel, the Orangerie and Tuileries gardens and the Gare d'Orléans, _Insel_ is replete with references both concealed and transparent to historical inhabitants of Surrealist Paris. Dalí, Man Ray and Ernst appear undisguised, whereas the figures of Julien Levy and Arthur Cravan are manifest here, as elsewhere in Loy's writings, in the guises of Aaron and Colossus. Some come off better than others: the narrator refers admiringly to Joseph Cornell's boxes as "delicious" (this page) (the subject of Loy's laudatory review-essay "Phenomenon in American Art"), whereas Insel's sudden nausea at the sight of a painting by Raoul Dufy registers as a glancing blow of bad publicity (this page). Early drafts of the novel contain direct references to biographical figures that Loy later cut or obscured. Mary Reynolds—a fellow American expatriate then active in Surrealist Paris—appears once in the first draft, but was cut from later edits. The identity of Mlle. Alpha—introduced by Insel in chapter 4 as another benefactress "who was liable to feed him at crucial moments" (this page)—remains something of a mystery. Though Elizabeth Arnold suggests that we look to Peggy Guggenheim, the presence in one first-draft copy of a penciled-in (and then struck-out) qualifier, "the painter," complicates this identification. Indeed, in spite of her being—or perhaps because she is—so vital to the plot, we might surmise that she, like much else in the text, is a work of fiction.
In _Nadja_ , a novel to which _Insel_ is often compared, André Breton declares: "I insist on knowing the names, on being interested only in books left ajar like doors; I will not go looking for keys." In direct contravention of this call for transparency from the man christened the "pope" of Surrealism, the characters of "Acra" and "stiff Ussif the surrealist" in _Insel_ elude identification. Loy's biographer, Carolyn Burke, suggests that the character called "Sex" stands in for Max Ernst, and indeed that the replacement of Ernst's forename with "Sex" derives from a transcription error. Burke deduces from Loy's correspondence that the figure of Moto stands for Breton. Burke's assertion is corroborated by the fact that in the second draft, Breton's name has been scratched out and replaced with "Moto." However, Loy's insertion in pencil of the word "Dalí?" above the name "Acra" on an early typed draft implies that her approach was sometimes more oblique than we might presume. Even as we read _Insel_ as _roman à clef_ , it might ultimately prove more productive to consider what effect the author sought to achieve by this blending of recognizable cameos and enigmatic ciphers.
Perhaps most significantly of all, several suggestive inconsistencies are in evidence in Oelze's transposition into Insel. Against Oelze's relatively comfortable childhood and considerable education pursued across a number of cities (Magdeburg, Weimar, Dresden), Insel is presented as the product of an altogether more marginalized, less educated and less cosmopolitan life-story. In her portrayal of Insel's family, domestic circumstances and cultural consciousness, Loy adulterates Oelze's biography with myriad inventions, omissions and alterations—pushing this ostensible _roman à clef_ towards a parody of that form. Early in the novel, Loy's alter-ego, Mrs. Jones, is forced to abandon her plan to write Insel's biography when she realizes that her erstwhile subject had purloined the details of his own life story from a novel by Kafka. Jones is herself depicted as a figure blighted by creative impasses, troubled by an incapacity to distinguish between truth and fiction and prone to experiencing dramatic shifts in her perception. By littering the narrative with references to doors, keys and acts of obstructed and delayed ingress and egress, Loy draws a network of false and chimerical connections to the surface of this remarkably self-aware novel.
In recent decades, Loy has often been cast as a modernist feminist and _Insel_ is read as a radical _détournement_ of Surrealism's problematic modelling of gender and creativity. Notwithstanding the rhetorical force of her "Feminist Manifesto," the form of Loy's feminism remains hard to define. Figuring herself, in "Pazzarella," as an enemy of "the sacred and inalterable front of masculine solidarity," Loy writes of having evolved in adolescence "a weird strictly personal form of feminism of which the militant aspect consisted in being peculiarly benign to any woman who had been 'pushed' " ("Islands in the Air"). These idiosyncratic politics did not lead her into public activism on suffrage or education policy. Her feminism does, however, register in startling ways upon her work and, indeed, upon the economies of power and desire between Mrs. Jones and Insel in this novel. What makes Loy's feminism so fierce is its brutality—but perhaps only the revolutionary violence of her proposal for "the unconditional surgical destruction of virginity throughout the female population at puberty" ("Feminist Manifesto") is adequate riposte to the patriarchally straitjacketed society into which she had been born. Subtending all of Loy's writing is a furious resistance to the oppressive regulation of female embodied experience and a commitment to unsettling essentialist binarism. In _Insel_ , she plays out a narrative of reversed (and markedly unstable) gender polarities: making Insel the childlike, unstable and linguistically impoverished male muse to Jones's powerful, older female patron. The dynamic of their physically and psychically imbricated interaction is no more traditional than it is simple.
The story of _Insel_ 's composition and publication is a convoluted one. Loy is presumed to have started work on the novel in Paris some time after Oelze's arrival there in 1933, and to have developed and edited the novel after her relocation to New York in 1936. Folders in the Loy archive at the Beinecke Library containing multiple handwritten and typed drafts bearing substantial editorial marks, along with innumerable fragments and variants, attest to a prolonged and multiphased authorial process. In spite of these efforts, to Loy's disappointment, her editorial exertions on first one, and then the other side of the Atlantic, did not result in the novel's publication in her lifetime.
In her unpublished dissertation on the author, Marisa Januzzi relates how Loy's ambitions for the book's publication were confounded in her lifetime. In December 1953, James Laughlin, editor at New Directions, returned the manuscript of _Insel_ to its author, with regrets. By 1960, Loy's daughter Joella was also invested in the project to see _Insel_ realized in print but, though Laughlin wrote to the author again in that year, praising her larger "Islands in the Air" manuscript but cautioning that much "compression" would be required, their correspondence failed to yield a published text. In a letter of 1961 to Elizabeth Sutherland at Simon and Schuster, Laughlin identified _Insel_ as the most readily publishable component of "Islands." Though the editors were in agreement on its merit, Sutherland eventually returned the text in 1963. Here, the history of _Insel_ stalls.
In the course of archival research for her doctoral dissertation (University of Chicago, 1990), Elizabeth Arnold found the typescript manuscript for _Insel_ at the Beinecke library. Intrigued by the text, she was convinced that the manuscript had been left by Loy in a publication-ready state. After writing a chapter on the novel in her dissertation, she set out to pursue _Insel_ 's publication and approached John Martin at Black Sparrow Press. Arnold's editorial efforts—underwritten, as Conover points out in his original introduction to the Black Sparrow edition, by a recognition that "nothing had to be altered"—brought Loy's long prose, for the first time, into print. Conceived in Paris in 1936, carried as a collection of notes and ideas to New York, and there compiled, edited and (at least) twice rejected, _Insel_ finally saw publication in Santa Rosa, California, in 1991.
In October 2013, I came across the "Visitation of Insel" in a folder marked "YCAL MSS 6 SERIES 1 BOX 2, FOLDER 39: INSEL: FIRST DRAFT FRAGMENTS" in the Beinecke Library. In her 2013 essay on Loy, Amy Morris alludes briefly to the "Visitation of Insel," designating it "a prose fragment from the late thirties or early forties." A single sheet of paper, empty but for six words, is housed in the same folder at the archive. It reads: "End of Book Visitation of Insel." This note unambiguously identifies the "Visitation" as an intended addendum to the novel. Like the "Visitation" passage with which it is housed, it is materially congruent in terms of paper, writing materials and handwriting with the holograph fragments which constitute the novel's early draft notes. And yet the "Visitation of Insel" sits at a strange angle to the published text: the fossil of an authorial intention later revoked. Provoking questions as to authorial intent, working practices, and even the rightful designation of its own terminal point, the "Visitation" might complicate, but it does not compromise the validity of the posthumously published novel. The novel, as it was published in 1991 and is republished here, remains Loy's last edited and corrected draft of the text. Whatever authority once attached to Loy's note, the "Visitation of Insel" was excluded from subsequent drafts by the author herself. Nevertheless, its existence is exciting.
The end of _Insel_ finds Mrs. Jones taking leave of Insel in Paris as she prepares to join her daughters in New York. Their last meeting is bittersweet. While the narrator is exhilarated by the decision finally to sever their connection, her companion is vaguely stunned by this definitive dissolution of their relationship. He seems, nonetheless, to appreciate its inevitability. With Jones's earlier promise to deliver him into Manhattan either forgiven or forgotten, Insel's valedictory expression of globalized gratitude—" _Danke für alles_ —Thanks for everything"—closes the novel.
Set, we can surmise, some two years later, the "Visitation" opens with the narrator as alienated newcomer in the New York domain of her daughters Sofia (also referred to as "Sophia") and Alda, and the extra-diegetically invoked Aaron. Here, as in the contemporaneous prose piece, "Promised Land," these characters correspond to Loy's daughters Fabienne and Joella and the latter's husband, Julien Levy. The autonomy enjoyed by Jones in the novel is, in the "Visitation," greatly diminished. Her daughters have become her keepers and her gallery agent "business" is revealed as a latterly regretted stroke of largesse on the parts of Alda and Aaron. Most bruisingly, the narrator's inching progress on her book—already, in _Insel_ , the cause of much self-castigation—is described by her elder daughter as a ploy to "get more money out of" her resentful relatives. Alda's charge, " 'You're no good—never have been any good—' " is received by the beleaguered narrator as "blank truth." Resonating with her own low estimation of herself, this taunt acts as a sort of psychic propellent which sends her reeling.
Left alone, hungry and incapacitated by ulcer-induced pain, the narrator retires to a couch. Her solitude is soon ruptured by the "Visitation of Insel." An apparition manifests itself in her room, a surrealist "presence" which the narrator recognizes as Insel. Divested of all "shreds of flesh," Insel is now a palpable "invisibility." A mute ghostly force, he projects his thoughts directly into her brain. What follows is a disquisition on her onetime friend, charge, muse and occasional tormentor: an often bewildering exposition of Insel's character from which some astounding conclusions are drawn. At the end of the episode, the narrator is eventually recalled to reality by the voice of her younger daughter who, having returned, demands assistance with her preparty toilette. In a jarring shift, the "Beam controlling a surrealist man" collapses bathetically to "the high-light on a fallen curler." When the incorporeal "Sur-realist Being" is replaced by the very emphatically fleshed—"clammy," "honnied"—body of Sophia, a specifically maternal materiality reasserts primacy over the psycho-spiritual realm. By structuring the passage in this way, Loy bookends the fantastic "visitation" scene with the banal quotidian. The "Visitation"'s core of hallucinatory philosophy is set within this frame of unhappy domestic "reality." Echoing the effect of her earlier encounters with Insel in the novel, the narrator of the "Visitation" is left disoriented in his wake.
The "Visitation" presents Insel in terms that are at once familiar and significantly developed from his characterization in the novel. Imagery associated with light, divinity, electricity, technology, the manipulation of time, healing and doubling are carried over from _Insel_. Here, as before, peculiar powers are ascribed to his eyes. Some concepts relating to the pineal gland and "blind back"—charged though they are with intertextual allusions—are less immediately intelligible. The Insel of the "Visitation" is understood by the narrator to have undergone a further phase of evolution since his last appearance in the novel. While this advancement augments his powers, it also makes him vulnerable to explosion. By projecting himself into New York, Insel risks being dynamited with his "own force." In addition to the changes undergone by Insel, a major transformation is seen to have taken place in the narrator's perception of him. Whereas, in the novel, she shies from the suggestion that Insel's dissipation might, after all, be a product of morphine addiction, the narrator of the "Visitation" unshrinkingly salutes the revenant as "my drug addict."
Prior to addressing the reverberations of that revelation, I want to plot the lines by which the Insel who manifests so unexpectedly in Manhattan might be related to his characterization in the novel. In both book and addendum, Insel is figured as a semi-divine entity. In spite of his many unsavoury attributes (including lustfulness, mendacity, unproductivity and criminality) he retains a paradoxically angelic aura of sanctity. Repeatedly associated, in the novel, with an "especial clarity of the light" (this page), Insel's status as otherworldly "man-of light" is indexed to the degree of his material tangibility. Having arrived, in the "Visitation," at a state of absolute immateriality, he is perceived to hang from the "cosmic consciousness by a ring of light."
Among the most striking of Insel's attributes in the novel is his capacity to generate what he calls his _Strahlen_ , or rays. A curious amalgam of supernatural, spiritual and corporeal elements, these rays are generated by, and temporarily obscure, the revolting appearance of his flesh. In _Insel_ , the eponymous anti-hero's degree of fleshliness fluctuates constantly. Watching him swing between extremes of embodiment and immateriality, Jones comes to associate these fluctuations with the waxing and waning of his _Strahlen_. The physical environment of those in range of his rays is repeatedly seen to alter itself in sympathy with Insel's state of mind. Constituting a sort of fluid ephemeral exoskeleton, these rays provide the infrastructure for the psychic bridge between Insel and Mrs. Jones. These magnetic rays also possess capacities for healing; they nullify "the lightning hand of pain" (this page).
Insel retains this function as "magnetic healer" in the "Visitation." Indeed, there inheres a suggestion in the opening fragments that the narrator's ailment—which eventually "turned out to be a duodenal ulcer" (this page)—may have somehow summoned the surrealist spirit to her side. Upon Insel's arrival, "the pain lay dead among the shadows"; both physical and emotional suffering are immediately salved. As Carolyn Burke, Loy's biographer, notes, the author received a corresponding ulcer diagnosis in 1940. A psychosomatic explanation is posited by the author for the incidence of this longterm complaint. In "Islands in the Air" and the short essay, "Tuning in on the Atom Bomb," Loy associates the ulcer with atmospheres of culpability and condemnation, and with the operation of a pernicious internalized "Voice." A remnant of her mother's habitual recriminations, this "Voice" was incorporated into her own psyche in childhood and persists even into her own motherhood. Given Loy's interest in psychoanalysis, and her infamously troubled relationship with her mother, it bears highlighting that the "Visitation" pushes the role of ulcer-trigger onto the narrator's daughter.
In the poem, "Evolution," Loy marvels at Nature's progressive improvement of "increasingly/complex organisms/streamlined for survival," asking, at the end of the poem:
"what, in infinitude,
will be our contour,
our density,
our potency?"
Insel, as he appears in the "Visitation," is the disembodiment of these speculations. With its images of prototypical growth and musings about man's potential for future evolution, the novel prepares us for Insel's reappearance, in the "Visitation," as a new and improved version of himself. While the Insel of the novel remains a fundamentally corporeal being who absorbs and exudes electricity, the Insel of the "Visitation" is entirely, fleshlessly electric. Where once Jones was astonished at the extent of his arm's reach (this page), the evolved Insel of the "Visitation" reaches across continents. In her dialogic essay "Mi & Lo," Loy suggests that "a man aware of the fourth dimension, if enclosed within a room without exits, could get out of the room." In the "Visitation," Insel realizes this potential.
In _Insel_ , Jones wondered whether her pet Surrealist was somehow capable of existing in two states at once. In the "Visitation," she clearly perceives him as an innately binary construct. Fed by an internal circulatory system, he is supported by a quasi-internalised, exo-skeletal "phosphorescent circulation." This inconstant layer constitutes the aura glimpsed in the novel. A further comparison is made here between the nature of the normal man, in whom "good & evil are proportionately mixed," and that of the drug addict, in whom these opposites alternate. The description of Insel's "amazing dédoublement" is perhaps the most obscure component of the "Visitation." Building on the novel's frequent allusions to doublings, halvings and bisectionalities, Loy suggests that further evolution has enabled the surrealist artist, parthenogenetically, to split. By formulating his own simulacrum, Insel can exist bodily in Europe and spiritually in America.
Although, in "History of Religion and Eros," Loy had proposed that a "mystic" might, through training, gain control over atomic and electronic capabilities already latent within him, here it is the psychotropically enhanced surrealist artist who performs this feat. In the "Visitation" 's depiction of how Insel achieves "the electronic transfer of his person through space" ("History of Religion and Eros"), Loy employs a number of highly elusive concepts. The genealogy of her startling image of the spinal column acting as conductor for a dynamizing life-force of immense voltage can be traced through "Mi & Lo" and the short story "Incident." The "gland" which Insel suspects of enabling "the penetration of his mind by an extra-luminous radience" is, we can adduce from Loy's wider oeuvre, the pineal gland. More mystical than medical, Loy's notion of the pineal gland owes much more to the writings of Blavatsky than Bataille. The equally strange image of the "blind back" recurs in "Mi & Lo" and "The Child and the Parent." A sort of psychic stopper which exists forever in the past, the blind back shuts off the ordinary human body from an infinite "cosmic consciousness." It is "the shutter on the fourth dimension" which blocks off our entrance into that ordinarily unavailable zone ("Mi & Lo"). In the novel, Insel faces his blind back towards the future, thereby perverting this obstructive function (this page). Freed from the tri-dimensional incarceration that is the lot of the common man, the surrealist genius runs rampant in an invaded fourth dimension. This, we are given to understand, is how the "Visitation" came to pass.
Throughout _Insel_ , Loy connects the painter with machine technologies. Critics including David Ayers, Andrew Gaedtke, and Tyrus Miller have analyzed the operation of these technologies in the novel in relation to a complex of scientific and pseudo-scientific theories. They cite Loy's readings on mysticism, early experiences with Crowleyite occultism and a Christian Science vocabulary of mesmerism, magnetism and rays. In the novel, Mrs. Jones presents the telepathic connectivity she shares with Insel as a phenomenon no less magical and unfathomable than the latest developments in interwar communications technology. Whereas _Insel_ uses photographic terminology in its description of the artist's "development," the "Visitation" depicts the surrealist spirit being "relayed" through the air as though by radio. Insel's transatlantic manifestation is framed as an inter-psychic broadcast.
Pre-eminent among Insel's powers is his capacity to manipulate time. When he is at his most powerful, he becomes capable of enfolding his patron into his idiosyncratic timezone, applying himself variously in the novel to accelerate, decelerate and stop time. Reaching forward from 1938, the Insel of the "Visitation" remains paradoxically anchored, via an "antedeluvian tail," to the past. Appearing to the narrator in the present, he simultaneously inhabits both past and future. A "primordial soft-machine"—he is a being at once ancient and ultra-modern. Insel's anachronistic constitution is coded in a trail of bones, primitive tools and machines. Emitting "search-light shafts" from "future eyes," he beams a premonition of imminent war back from the future. Ocular analogue to the smile of Alice's Cheshire Cat, when the rest of him "fades," it is his eyes that linger.
We cannot know what caused Loy to resuscitate Insel, but we are certainly prompted to wonder. The narrator of the "Visitation" understands his appearance in New York to signal their "mutual forgiveness." She suggests two occasions, "his dope-ring duplicity" and her "written account of him," for which this forgiveness might be forthcoming. A third—that of Jones's abandonment of Insel—suggests itself for consideration. Inherent in the act of Insel's Manhattan transfer is the fact of his continued presence in Europe. At the end of the novel, in spite of the "pact" the pair draw up in Chapter Two to get him to America, Insel remains behind to live out the war in the continent from which Mrs. Jones promised to help him escape. As did Oelze. In _Becoming Modern: The Life of Mina Loy_ , Burke writes that "[i]n October 1936, after the failure of their friendship, Oelze made his way to Switzerland," and nothing in Loy's biography or writings suggests that she maintained correspondence with Oelze subsequent to that date.
The year 1933—the year of Oelze (and Insel)'s arrival in Paris—had seen the publication of the _Deutscher Kunstbericht_ (German Art Report), signaling the Nazis' dire intentions for German artists. In the same year, Goebbels established the _Reichskulturkammer_ (Reich Chamber of Culture), and the first _Schreckenskammern der Kunst_ (Horror Chambers of Art) assembled "degenerate art" for exhibition. However, notwithstanding a handful of curt references in _Insel_ to war, the recognition that Oelze's security is at stake is explicitly articulated only once by Mrs. Jones (this page). By 1936, the precarious position of German artists was abundantly and broadly apparent. In August 1937, _The New York Times_ ran a report headed: "Goering launches the Nazi Art Purge. Orders Broad Clean-Up of All Public Exhibits to Get Rid of 'Un-German' Works." Below it, the subtitle screamed "MODERNISTS ARE TARGET." A plethora of similarly stark portraits of an artistic community under attack presented America as a haven for German artists at risk of annihilation. If we can suppose that the "Visitation" dates to 1938, it might be worth considering what Loy had learnt in the two years since her solo flight from Paris. Could a dawning awareness of what had been at stake in their exchanges have spurred the author, in Oelze's absence, to resurrect Insel—the textual avatar of the German artist she had left to his uncertain fate?
While the "written account of him" to which the narrator of the "Visitation" alludes is, we might reasonably presume, _Insel_ itself, the matter of Insel's "dope-ring duplicity" is rather less straightforward. It has long been believed that Loy attempted to cure Richard Oelze of a debilitating drug addiction. In chapter 16 of the novel, Jones is dismayed to learn from Mlle. Alpha of Insel's heretofore undivulged history of morphine addiction. Her plaintive coda to this revelation, "m]oreover, was not Insel's morphinism a thing of the past?" ([this page), remains suspended over the remainder of the novel. The narrator's suspicion about the endurance of his habit remains, in the novel, remarkably contained—focused almost exclusively in this conversation with Mlle. Alpha. Admitting that she favoured her own idea of him, when confronted with Alpha's superior knowledge of Insel's insalubrious past, Jones admits to having "waived this information" (this page).
As the critic David Ayers has remarked, the novel suppresses its own troublingly prosaic suggestion that Insel's dissipation might, after all, be a product of morphine addiction. Ayers's diagnosis is lent further credence by the existence, among Loy's papers, of multiple handwritten fragments of the novel inscribed with the distinctly unambiguous abbreviation, "Morph." Most of these passages make the transfer into subsequent edits—but they do so divested of their header. Just as Jones denies the possibility that Insel might still be in thrall to his old addiction, _Insel_ 's readers are denied this narrow interpretation of Insel's behaviour. Or, given the success of _Insel_ as a novel, perhaps we are liberated from it. Insel's ability to enrich his personal, elastic, atmosphere with an array of sensory effects suggestive of certain psychopharmacopoeia is construed in the novel as a characteristic of his innate surrealism. In his 1924 "Manifesto of Surrealism," Breton likens Surrealism itself to a drug, writing: "There is every reason to believe that it acts on the mind very much as drugs do; like drugs, it creates a certain state of need and can push man to frightful revolts." By alternating rapidly between descriptors for Insel, calling him first "the surrealist man" and then "the drug addict," the "Visitation" suggestively conflates these terms.
We can only guess as to why Loy chose to minimize this aspect of the narrative. Perhaps, with an eye to a censorious market, the author decided that a novel about a shady surrealist with Black Magic propensities would be more readily publishable than one about a blatant "dope-fiend." Or, indeed, perhaps her motivation was more aesthetic than pragmatic. Several fragments which never make it in the "first draft" typescript further adumbrate the issue. In one, the narrator asks Aaron whether he could "tell" that Insel "was a morphomane." His response is blunt: "Naturally—he answered__ he looked queer__ he looked like garbage." In another example, the narrator addresses a character called Miriam, whose cameo appearances in _Insel_ 's draft notes were never translated into the novel itself: "Suppose I were sitting outside a Café with a man one would not touch with the tongs & he seemed to have some emanation on which one ascended to heaven—could it be—drugs?" Elsewhere, Miriam laments the "sheer perversity... that a lift to utter realisation of Beauty should leave the Body so ugly." All of these exchanges, like the "Morph" headers, were cut from later edits.
In this context, the "Visitation" achieves a heightened significance. Potentially reframing the events of the novel as the surreal story of what happened when chance "threw her] a dope-fiend," the "Visitation" gives us Jones's re-evaluation of the events of _Insel_. Whereas, in the novel, she admits that "drugs meant nothing to" her, the narrator of the "Visitation" regrets her hasty dismissal of opiates as mere "substitute for imagination" ([this page). Lamenting her former myopia, she writes: "We hear that a drug in impairing nerve tissue produces a vicious exaltation & our curiosity is no further intrigued." By explicitly announcing, "Here was my drug addict," she invites us to re-read the novel, attending more fully to Mrs. Jones's self-deception, and that of her implied audience. This re-reading delivers a parallel or supplementary, rather than a corrective interpretation of the book. For there is so much more going on in _Insel_ than could be attributed to even the most surrealistically potent narcotics.
The power of the "Visitation" 's last line pivots on "radium." That vibratory noun compels us to compare Loy's "fluctuant" conceptualizations of atomic energy with the equally uneven course of her fascination with Insel. Asked, in a _Little Review_ questionnaire of 1929, "What do you look forward to?," Loy answered: "The release of atomic energy." Throughout her writing life—from the poem "Gertrude Stein" of 1924 to the post–World War II prose of "Tuning in on the Atom Bomb"—Loy returned again and again to the concept of nuclear force. Her attitudes to atomic energy were heavily imprinted by the catastrophic inception, in that period, of the nuclear age. In its last lines, the "Visitation" elaborates on the significance of the "radio-activity" (this page) which Jones associates with Insel throughout the novel.
At the end of its retrospective analysis of Insel as "phosphorescent drug-addict," the "Visitation" concludes: "It is, in as far as I am aware, no particularly cleanly matter from which radium is extracted." The drugs that enable Insel to hook himself up to the cosmic consciousness are at once potent and poisonous. In this, their doubled potentiality, they resemble radioactive matter. The carcinogenic repercussions of experimentation with atomic energy were, in the 1930s, already widely known; the extraction of radium was understood to be a perilous process. Likewise, Insel's use of drugs to unlock untapped capacities in his mind affords him astonishing powers, but it also exposes him to considerable physical and psychic damage. Psychotropically enhanced and contaminated, his brain now "gives off a radium glow." In _Insel_ , the narrator experiences, albeit telepathically, the twinned paralysis and paradise of the surrealist artist's narcosis. Marvelling at the beauty of Insel's "increate" (this page) imaginings, Jones recoils, ultimately forever, from the horror of his disintegration. The "Visitation" explores the consequences of Insel's electric, surrealist, drug-assisted endeavours to amplify and extend Man's "dynamism." "Constructing, demolishing him kaleidoscopically," the "Visitation" seeks "to demonstrate how he 'worked.' " By cutting this one-time "End of Book" from future edits, Loy effectively rescinded the findings of her "research on the spirit"; this edition of _Insel_ recovers it from the archive for her readers.
# **A NOTE ON THE TEXT**
The base text for this edition of _Insel_ is a typescript manuscript labeled "Third draft, copy 1" at the Beinecke Rare Book and Manuscript Library at Yale University. This manuscript was prepared and corrected by Mina Loy. Her footnotes have been incorporated into Appendix A, which gives English translations of all foreign phrases except those which are clearly translated in the text of the novel. Where Loy indicated section breaks by triple spacing, we have numbered each section. A few minor corrections of punctuation and typing errors have been made, and foreign words and phrases have been italicized. In general, however, Loy's idiosyncrasies have been preserved. Throughout the manuscript, Loy used British and American spellings interchangeably. For the purposes of this edition, we have used all American spellings, following the ninth edition of _Webster's Collegiate Dictionary_.
Elizabeth Arnold
"Visitation of Insel," as it is published here for the first time, runs across forty-five small, handwritten and numbered pieces of plain paper. Most of the pages fit only one to three of Loy's characteristically long sentences, and they appear to have been torn to size along a roughly horizontal line. Many bear, on their versos, discarded drafts of these and related passages. Perhaps unsurprisingly, the product of this peripatetic, portable writing practice currently contains three outright lacunae. Unlike the manuscript of the novel itself, which exists in numerous typescript drafts, this handwritten text is rife with orthographical errors, typos and incompletions. It also bears evidence of substantial editing and correction by Loy. The punctuation of "Visitation," in line with the author's customary style, is dominated by indeterminate dashes of varying lengths, the intended position of which, on unlined paper, and in a wandering hand, is sometimes ambiguous; the published text offers the closest typographical correlates, as approximately as possible. The sequence is ordered by roman numerals from I–XXXIX, and a single arabic numeral that marks the fortieth fragment. The five subsequent pages were not assigned numbers by the author; they are represented here with roman numerals in brackets. All but five of the pages are dated; the dates they bear span the period from August 4–23. Fragments XXIV–XXVI and XXVIII A are undated, while the fragment numbered XXVII is marked with the curious interrogative: "Aug?" Loy presents these dates in diverse formats, variously representing the same date as "4th Aug.," "4th August," "August IV," "August IVth" and "Aug. IV." This striking degree of inconsistency, coupled with the highly irregular justification of the dates, problematizes the presumption that these dates correspond, quasi-diaristically, to dates of composition. In the interests of legibility, these dates have been erased. A single page, inscribed "End of Book / Visitation of Insel," is set off from the sequence with a pair of asterisks.
House publishing conventions demanded that various changes were made to the punctuation and layout of piece. In order to produce a readily readable text, Loy's haphazard systems of punctuation, indentation and capitalization have been substantially regularized. Words underlined in the manuscript have been italicized. A handful of compound words have been hyphenated or rendered as closed compounds. Some apparent spelling mistakes have been corrected; others, which have been deemed representative of Loy's deliberately unorthodox orthography, such as the onomatopoeic enunciation of "prove" as "proove," are retained. Two indecipherable words have been elided from the text and a single indefinite article has been inserted.
Due to the exigencies of space and formatting conventions established by the Neversink series, it was not possible to include my extended notes and critical apparatus. A comprehensively annotated version of the "Visitation of Insel," displaying and commenting upon its material particularities, along with all textual ambiguities, revisions, insertions and other markings, is available at www.mhpbooks.com/insel-visitation.
Sarah Hayden
#
**INSEL**
# **1**
THE FIRST I HEARD OF INSEL WAS THE STORY OF A madman, a more or less surrealist painter, who, although he had nothing to eat, was hoping to sell a picture to buy a set of false teeth. He wanted, he said, to go to the bordel but feared to disgust a prostitute with a mouthful of roots. The first I saw of this pathetically maimed celebrity were the tiny fireworks he let off in his eyes when offered a ham sandwich. What an incongruous end, my subconscious idly took note, for a man who must once have had such phenomenal attraction for women. And he wants them of the consistency of motor tires... my impression faded off. For, to my workaday consciousness, he only looked like an embryonic mind locked in a dilapidated structure. I heard plenty of talk about his pictures, but I was afraid to visit his studio as, to all accounts, his lunacy rendered him unsafe. It rather took me aback, when a few days after his casual introduction to me, he paid me a call. I had been giving tea to my little model after the pose when he arrived. Her Slavonic person was colored a lovely luminous yellow, owing to some liver complaint, and her sturdy legs, which _I supposed he could not see_ for she was already dressed for the street, were of such substance as sun-warmed stone. With the promptness of a magnet picking up a pin, he made a date with her for the following day.
Facing each other they possessed voluptuous attributes the poor will find in one another unmarred by an unwholesomeness which is mutual. The model, tremendously engaged in hoping to have a baby to persuade her lover to make her his wife, later decided it would not be politic to turn up. Not without regret, however, for "I _like_ him," she confided to me, squeezing her hands together in delight.
As for myself, he cleared my recollections of the prejudice for his madness as he sat disseminating in my amusing sitting room a pleasant neutrality, pulling one's sympathies in his direction. And as the afternoon wore out, it was as if a dove had flown through the window and settled upon a chair. Whenever his features obtruded on the sight some impulse of the mind would push them out of the way as if one obeyed an implicit appeal not to look at him but rather give in to the mischievous peace which seemed to enclose him in a sheath.
That evening I began a letter to a friend: "Aaron's latest surrealist is absolutely divine. He has painted a picture that's not so very hot in any particular detail—a gigantic back of a commonplace woman looking at the sky. It's here to be shipped with the consignment I am sending to Aaron, and I swear whenever I'm in the room with it I catch myself staring at that sky waiting, oblivious of time, for whatever is about to appear in it. Most eerie! The man himself is just like that. He did not say anything in particular, but you felt you were in the room with an invisible will-o'-the-wisp, and that any moment it might light up. He's the son, he said, of tiny working people and seems the most delicate and refined soul I've ever come across. He has an evening suit, but never an occasion to wear it, so he puts it on when he paints his pictures, first having meticulously cleaned everything in his studio. Now, I don't mean he's a delicate soul because he paints in evening dress—! That's just one of his stories I remember. I shall probably find this quality exists only in my imagination because there's something fundamentally black-magicky about the surrealists, and I feel that going in that direction, his face, that looks almost luminous from starvation, will turn out to be a death's head after all.
"It's funny how people who get mixed up with black magic do suddenly look like death's heads—they will grin and there is nothing but a skull peering at you, at once it's all over—but you remember. Sex is an exception. He is permanently a skull with ligaments attached, having the false eyes of an angel, and, at the back of them his cranium full of intellectual dust. Often they look like goat's. While Moto has the expression of an outrageous ram, his wife the re-animated mummy of an Egyptian sorceress. In fact, they are very, in their fantastic ways, expressive of their art, which after all takes on such shapes as would seethe from a cauldron overcast by some wizard's tortuous will."
The letter was never finished, for, and this was often to be the case, once he had left, his person would gradually gather together till at last one could normally see him as he really was, or had been when present. Tall, his torso concave, he was so emaciated that from his waist down he looked like a stork on one leg. His queer ashen face seemed veritably patched with the bruises of some physical defeat that had left him pretty repulsive. One's mind, released from the unaccountable influence of his nature's emanations, readjusted the time spent in his company to the rational proportion of an interview with a plain, eccentric, somewhat threadbare man, strangely pitiable in a premature old age. His manner alone remained unchanged by this surprising reverse—it was of an extreme distinction. Owing to this, before my letter drew to a close, I had lost the impression of whatever had inspired it, finding myself very much in the situation of Titania confronted with entirely meaningless donkey's ears.
It argued a certain good fortune, in Insel's timing of his next visit, that it should coincide with that of a German girl, as his absolute inability to acquire any language to add to his own must have made his inhabitation of a foreign land a somewhat lonely affair. But in how far I found it at last impossible to determine, so narrowly his unformulated existence seemed associated with itself. Her visit was fortunate for me also, for later on, in our checking up on the subject of this very Insel, she, so common-sensed and unimaginative, was able to clear my doubt as to whether it was I who must be mentally deranged. Wishing to get on familiar terms with an acknowledged surrealist, we took him to a cafe, and, in the embracing glare of a locality above all others conducive to the liberal exchange of confidences between the most heterogeneous people, the meager personality of this stranded German opened up.
I still retain wisps of the irreal crises in his footloose career that, as he related it, grew up, story by story, a frail edifice of lies and memory out of our marble tabletop. It was securely buttressed by groups of obese tradesmen who, in their agglomerated leisure, were playing _belote_. He told us, his gray eyes atwinkle with the inner security the possession of a strange surplus fortune, balancing destitution, gives to men of genius, he had solved the problem of keeping alive without any money and thus had lived for sixteen years.
A man who finds himself economically nude, should logically, in the thickset iron forest of our industrial structure, be banged to death from running into its fearfully rigid supports. He is again the primordial soft-machine without the protective overall of the daily job in which his fellows wend their way to some extent unbattered by this sphere of activity. For them, the atrocious jaws of the gigantic organism will open at fixed intervals and spit at them rations sufficient to sustain their coalescence with the screeching, booming, crashing dynamism of the universal "works." For the _révolté_ , for one incapable of taking it as it is, this metal forest of coin bearing machinery will partially revert to the condition of nature preserved in him, and show patches of moss as if he had projected there some of the verdure rooted in him. Oases of leisure, succorable, soft if ragged lining to the cage of practical mankind, these mossy refuges, along the life of a wayward spirit who refuses to do as he is told, preferring to find out for himself what to do, mostly materialize as the hospitalities of modest little women who find a temporary relief from their innate anxiety in association with an irresponsible man in whom the honest desire for survival of his creative impulse gets dishonestly mixed up with his amatory instinct.
Insel, as he talked, seemed to be recurrently emerging from predicaments of which, if some were lamentable, many were quite diaphanous as though nothing of him but the most subtle aspects of his peculiar temperament had got into them—He varnished his painting of the past with a gentle irritation of commentarial laughter. Unlike other men, he took delight in confessing that all his women had deserted him, divorced him, thrown him out. How he had pled with those women to have patience. "I am tired of supporting a waster," they would tell him at last. "But they were wrong," averred Insel. "While I appeared irretrievably idle, _Ich habe mich entwickelt_ —I was developing," he explained, the mischief dancing defiantly in his eyes. And this _Entwicklung_ I would not estimate blurred my view of Insel. I saw his image grown suddenly faint, imploring the shadow of woman "—to only wait—in the end—the end—I shall achieve glory."
These unfortunate separations, throwing him back upon the desert base from which he was ever setting out anew, formed part of the frieze of disaster through which he represented himself as forever fleeing under the vicious darts of his drastic horoscope.
Housing his poverty as animals tracked down enter abandoned holes or a honeyless bee might return to an empty hive—of all the makeshift burrows he found for himself in an unearned earth, so desolate and perilous seemed his escapade in some far away, dismantled villa he described to us that it has stuck in my mind. He had lived there one summer, with some caressive _Mädchen_ who, when she left the place, had forgotten to ask him to give up his key. And so it was that Insel, fallen once more under the heel of fate, crept back to that love-swept lair and, shutting himself up in one of the rooms, lived on the floor. "I could draw there," he pointed out to us thankfully. And one saw him, day alone, morrow alone, where the air was the breath of his own hunger, warily sneaking out at dark in search of a remnant of food, or, just as possibly, so complex is the status of the artist, dining with affable millionaires every other night. It comes back to me now that Insel had started this episode as the story of a haunted house.
"One day when I had hidden there so long," he said, "as to make one with the everlasting silence, I was startled by the sound of footfalls descending the stairs. 'Who could descend having never ascended?' I asked myself. 'What could have embodied itself under the roof to come down upon my isolation?' I at once turned the key in my lock and waited, listening to those fearful feet. At intervals they would halt, and at every halt I could hear the echoes of heavy fists pounding on a door. And the footsteps grew louder, the pounding nearer—to the pounding of my heart." Frau Feirlein and I hung on his words while, "Those heavy unreal fists fell in a rain of blows on the last door—the door," continued Insel, "that shut off the bearable emptiness of the room I was living in from the unbearable emptiness of the house as a whole."
I imagined his quivering breast receiving, through the transmission of his fear, the ghostly blows aimed at the door affording his outer defense as he stood face to face with it. "Not daring to stir," he proceeded, "I was almost choking for terror it would hear me breathe, when gradually I perceived a softer noise come to succeed that ominous pounding. I do not know what impulse this aroused in me, my one desire having been to remain undiscovered, that caused me so suddenly to fling myself on that door and to open it—outside stood a very severe looking _Huissier_. He was sealing up the rooms. And, as usual," added Insel in utter discouragement, " _They_ hauled me before a magistrate."
His father, he said, was a _Schlosser_ , which turned out to mean "blacksmith," and very early in Insel's life, as in the episode in the empty villa, his destiny appeared to me to get mixed up with keys. I cannot remember in detail the vague accidents in which his minor salvations depended upon keys except for a fleeting impression of Insel crawling under his destitute couch in search of some kind of key to a gas meter confused with an intermittent flitting in and out of gas men and electricity bill collectors who would come to cut him off and who, owing to some either mental or manual hocuspocus of Insel with the key, ended by turning the gas or electricity on. Thus leaving him at least two elements of life.
"Don't _Schlossers_ make keys?" I asked.
"Surely," he agreed.
"Well, you've _inherited_ the keys your father made. You'll see the whole of your life will turn on a key. Some people are accompanied throughout their career by a fixation of their destiny—yours is a key. While Dali, for instance, is fated to the most extravagant of publicities. He is inclined to accept my theory, for some of the shrillest gags in his already fabulous advertisement were the result of sheer coincidence, yet pointing out plainly the effective procedure to be followed thenceforth. He told me it all originated with his first lecture on surrealism in Spain, when the mayor, who was acting as chairman, fell down dead at his feet, almost as soon as he began—a windfall for journalists."
Insel, as if after this he feared his trips along the road to ruin might fall a little flat, changing his tempo, began to show off, surprising us with a burst of magnificence, he became so hilariously wealthy he juggled a fortune. "I spent ten millions in a year," he enthused. "Not one car, but bunches of cars I gave each of my friends, and the orchestras I ordered, the clusters of beautiful women I hung upon myself in those Berlin nights."
"Where did you get it all?"
"Forging," he replied, with the same elation with which he had dispensed for our entertainment his retrospective largess. " _Es war wirklich prachtvoll_ —we made bank notes in sheaves. You see, as a boy I was apprentice lithographer, and my technique was so remarkable I got raked in by a gang of crooks. We practically bought up Berlin before we were caught, and I was only in jail nine months."
"How was that?"
"Oh, one of the gang who escaped arrest used his influence. But I had time to reflect," he commented. "I saw other careers open to talent. In that long solitude I conceived of a greater wealth than the wealth of banks. Within myself I found the artist."
# **2**
ON THE GROUNDS THAT HE WAS STARVING TO death, he would exact from us the minutiae of advice on his alimentary problems to subsequently toss all advice aside in his audacious irresponsibility. Presenting himself as a pauper to the charitable organization of the Quakers, he had harvested, among other things, packages of macaroni and several pounds of cocoa, and as if these staple aliments were already consumed, he begged us to counsel him what to do now. He shook his head over the suggestion that he go there again. "My last supply is yet too recent," he objected. But, Frau Feirlein told me, on the morrow he presented himself at her flat with these same Quaker gifts intact as an offering preliminary to his indistinct courtship. "What is the use of cocoa to me," he argued with my bewilderment, "I have no sugar." And, for some vague reason, one took the opposition of his prodigality to his mendicancy as a matter of course. This reason consisted in an intuition, so deeply imbedded in one's subconscious it would not rise to the surface of the mind until the final phase in one's analysis of him—that this skeletal symbol of an ultimate starvation had need of a food we knew not of. Throughout his angling for compassion on behalf of his utter destitution, one never resented his open carelessness in throwing back the fish.
Meanwhile, his reserved distinction, as of an aristocrat who should in a lasting revolution have experienced yet unimaginably survived the guillotine, was so consistent it claimed one's respect for his nonsensical manner of being alive. But once was this impression dispelled when, in courteous haste to answer a question, he shifted the part of a hard roll sandwich he was eating, out of the way, horrifyingly developing a Dali-like protuberance of elongated flesh with his flaccid facial tissue. As if unexpectedly the _Schlosser_ one had hitherto been incapable of relating to him had at length intruded upon us with his anvil stuffed in his cheek.
Only towards the close of his reminiscences did he seem to have shared a responsibility with normal men: "They sent me to war," he told us wryly, voicing that unconvincing complaint against their perpetual situation in the ridiculous made by people who, pleasing to laugh at themselves, one suspects of aiding destiny in detaining them there, "in two left-foot boots, and," trotting his fingers along the table in a swerve, "the one _would_ follow the other," he explained as the mental eye also followed that earlier Insel—out of the ranks; on the march to a war that, at its blasting zenith, ceased to be war, for, in elaborating his martial adventures, Insel turned out to have been taking part in a film.
A wound up automaton running down, Insel ceased among the clatter of our amusement.
"I know how you can make money," I exclaimed agog with enthusiasm. "Write your biography."
"I am a painter," he objected. "It would take too long building a style."
"You'd only have to write the way you paint. Minutely, meticulously—like an ant! Can you remember every moment, every least incident of your life?"
"All," he replied decisively.
"Then start at once."
"It would need so much careful editing. In the raw it would be scandalous—"
"Scandalous," I cried scandalized— "the truth? Anyway you can write under a pseudonym."
"People would recognize me."
"Don't you know anything of the world? The artist's vindication does not lie in 'what happens to him' but in what shape he comes out."
"Oh," said Insel disinhibiting, "very well. It's not the material that is wanting," he sighed wearily, "the _stacks_ of manuscript notes I have accumulated!"
Then, "No," he reversed, "it's not my medium."
"Insel," I asked breathlessly, "would you let _me_ write it?"
"That would be feasible," he answered interested. "We will make a pact. Get me to America and you _have_ the biography."
"Done," I decided. "I'll write at once. America shall clamor for you."
"Don't overdo it," warned Insel, "it never works."
"You can have your dinners with me and tell me— Can you really remember—the minutest details?"
"Every one," he assured me.
"What a book," I sighed with satisfaction.
"Flight from Doom—every incident distorted to the pattern of an absurd destiny," Insel was looking delighted with himself.
He came out to dinner on a few evenings and I would talk with him for hours. The minute details were fewer than I had bargained for, his leitmotif being his strangeness in so seldom having spoken.
"My parents noticed it at once," he told me. "As a child I would remain absolutely silent for six months at a time."
He did not give a fig for heredity. All his relatives were chatty.
Another thing he had found in himself was his aptitude for housework. He had once married a stenographer, who simply _could_ not arrange the kitchen with the same precision as he.
"She tried so hard—for so long. She never came up to the mark. What I disliked was her plagiarism. Why," demanded Insel with retrospective annoyance, "could she not have worked out a system of her own?"
So they separated. Later, when Insel and I became uncannily intimate I understood what his unique orderliness had done to the girl—given her the jitters!
Nevertheless, he himself seemed sometimes to have difficulty in locating things. Once during coffee he drifted off to the lavabo and on his return took a seat some tables away from the one at which he had left me. In the same slightly deferent sociable concern he continued to "pay attention"—
The strain on this biography would consist in his too facile superposing of separate time—his reminiscences flitted about from one end of his life to the other.
"I saw an antique dealer carrying a picture to a taxi the other day—a portrait of some women. They were extraordinarily attractive to me; I was sure we would have been profoundly congenial. It was labeled 'The Brontë Sisters.' Do you know of anyone by that name?" asked Insel, who had not read Goethe nor heard of Shakespeare. "The dealer told me they were authoresses—I feel I should care for what they have written."
"The sister Emily wrote _Wuthering Heights_. I suppose it is one of the greatest novels ever written. I never remember for very long, after having read it, what it's about—yet whenever I think of it—I find myself standing on wild moors—alone with the elements—elements become articulate—. _You_ would care for it very much."
I began to think it improbable I should even find a basis for this biography. He was so at variance with himself, he existed on either side of a paradox. Even as he begged for food to throw away, forever in search of a haven, he preferred _any_ discomfort to going home. Constantly he thanked his stars for an iron constitution—while obviously in an alarming state of health.
# **3**
AT LAST THE BIOGRAPHY ABORTED AS HAD THE Quaker oats.
The first stage of Insel's intimacy completed, when he evidently intended to let you further "in on" his show, he insisted on your reading Kafka, just as on assisting at a foreign opera one is handed a book of the words.
Study this well he tacitly commended. It will give you an angle of approach. "In Kafka," he explained, "I found a foreshadowing of my hounded existence, recognized the relentless drive of my peculiar misfortune."
_Der Prozess_ was the volume he borrowed to lend me, and I lay awake reading on and on and on, curious for the book to begin, when, with one eye still open, I came upon the end to fall asleep in the unsatisfied certainty of having become acquainted with an undeniable, yet perhaps the most useless, genius who ever lived.
Enraged with bitter disappointment, _"Zum Teufel_ ," I berated Insel, when he appeared for our next session. If he was a lunatic, he was prodigious, dressing up his insanity in another man's madness. It was no use to me. Flight from Doom, with its pattern of absurd destiny, had already been written.
"You atrocious fake—you have no life to write—you're _acting_ Kafka!"
"And I," answered Insel, as I turned him out, "see clearly into you. Your brain is all Brontë." Flying the colors of his victory, he sauntered off.
# **4**
I _THOUGHT_ I _HAD_ DROPPED INSEL. I WAS MISTAKEN. Some weeks later I was writing letters when all of a sudden I stopped. An urgent telepathy impinging on my mind, I automatically dashed off a card. When I looked to see what I had so unpreparedly written—this is how it began:
"It is interesting," Insel was to remark significantly later on. "Your note to me was couched in flawless German."
For a while I sat wondering to _what_ appeal, and why, I had answered. _I did not care_ if Insel were in trouble. Obviously he fabricated trouble and far be it from me to deprive him of it—. I threw the card into the waste paper basket, and started for the post. When I had opened the front door I shut it again and retrieved the postcard. Before the letter-box I put it in my pocket and turned away, only to go back— with a relieved determination I posted it.
Insel must have crossed my message for in a couple of hours he panted into my place all undone, despairingly waving a sheet of blue paper.
" _Das blaue Papier_ ," he articulated hoarsely, ducking his head as if the _Papier_ was one of a shower of such sheets bombarding him in his dash for escape.
"Something the matter? Have a _porto_. Sit on a chair. Whatever it is—out with it!"
" _Das blaue Papier_ ," he reiterated, casting a haunted look over his shoulder. On its return that look fell in with some photographs of paintings lying on the table.
" _Whose_ pictures are these?" asked Insel, immediately collected, and staring at each in turn with entire attention. " _Who_ could have done these?"
"They are mine."
"You are an extraordinarily gifted woman," he said, still staring at them. "Oh, how I wish I could read your book."
"It's not like those pictures," I laughed and told him their brief history.
" 'Those' are my 'last exhibition' cancelled the moment the dealer set eyes on them."
"Good God," muttered Insel under his breath.
"I felt, if I were to go back, begin a universe all over again, forget all form I am familiar with, evoking a chaos from which I could draw forth incipient form, that at last the female brain might achieve an act of creation."
I did not know this as yet, but the man seated before me holding a photo in his somewhat invalid hand had done this very thing—visualized the mists of chaos curdling into shape. But with a male difference.
Well, it turned out that the blue paper was a summons for rent involving the evacuation of his studio.
Insel's system in such emergency was this:
Never to pay. To work himself into an individualistic kind of epilepsy whenever served with a summons or notified to appear in court to explain why the money was not forthcoming. Computing illusory accounts to find the exact sum he could promise to pay by a certain date, knowing full well he would not be able to pay anything at all, in order to scare himself into fits awaiting the fatal appointment.
Now one could watch him following the path of pursuit at an easy canter, having proved he had something definite to flee from.
His role was helplessness personified. So here he was without a roof. In spite of the ceiling a pitiless rain seemed to be falling upon him already.
Whenever I have seen poor people asleep on stone seats in the snow, like complementary colors in the eyes, there arise in my mind unused ballrooms and vacationers' apartments whose central heating warms a swarming absence. To the pure logician this association of ideas might suggest a possible trans-occupation of cubic space, while mere experience will prove that the least of being alive is transacted in space, so much does sheer individuality exceed it; that providing a refuge for a single castaway brings results more catastrophic than a state of siege.
So I kept saying to myself, "Remember, you don't care a damn what happens to this thin man." While what he did was to fill the room with all men who are over-lean. And the room fell open, extending to space—as such—to remind me of my futile superposition of stone benches on ballrooms. My lips opened automatically. "Don't be fools," I admonished them. "Keep out of this. You'll get me into an unnecessary jam." In the end I must have given in, for I heard myself telling him, to my despair, he could live in my flat when I had gone to the country. "If that's any help," I added dubiously. "It solves half my problem," he thanked me with appreciative warmth.
The result of this lapse of protective selfishness was days of agony. I had intended to run off to the country at once. But now—I sat looking at that apartment obsessed with the necessity of disencumbering it of personalia. The onus of trying to make up one's mind where to begin overpowered me.
The psychic effort of retracting oneself from the creative dimension where one can remain indefinitely—like a conscious rock—immovable—in intellectual transmutation of long since absorbed actualities, while the present actuality is let to go hang—was devastating.
The contemplation of a bureau whose drawers must be emptied—the idea of some sort of classification of manuscript notes and miscellaneous papers— that in habitual jumble are easily selectable by the remembrance of their subconscious "arrangement," the effort to concentrate on something in which one takes no interest, which is the major degradation of women, gives pain so acute that, in magnifying a plausible task to an inextricable infinity of deadly detail, the mind disintegrates. The only thing to do is to rush out of the house and forget it all. So disliking to leave one's work in favor of some practical imperative, in begrudging the time to undertake, one wastes triple the time in being averse to thinking.
Something would have to be done about it. Fortunately, after more than a week of this paralyzing resistance, I came across a long painting overall. Its amplitude made something click in my brain. I at once became animated with that operative frenzy which succeeds to such periods of unproductive strain. Sewing up its neck and sleeves on the Singer, I obtained a corpse-like sack, and stuffing it full of scribbles I tied it up, and, throwing it into a superfluous room, locked the door on it with a sigh of relief. I was once more myself.
In the meanwhile Insel had come to take me to see one of his rare paintings in the possession of a friend who was liable to feed him at crucial moments.
In the taxi I inquired, " _Was haben Sie schönes erlebt_ since I saw you?"
"I had two negresses at once," he answered, all aglitter.
"Two," I echoed anxiously. "I hope you didn't have to pay them."
"Oh, no," he assured me.
"So they liked the look of you," I teased with friendly disdain.
"Yes," he concurred apologetically.
"And—was it nice?"
"Well," he reflected, "I thought it was going to be nice. And now the trouble is to get rid of them. And what have you _erlebt_?" he commented.
"Not quite so _much_ —anyhow."
I saw the picture. Its various forms, at once embryonic and precocious, being half-evolved and of degenerate purpose, were overgrown with a hair that never grew anywhere else—it was so fine. And when our host had gone out of the room Insel stared at it amazed. His face became rigid with incredulity. "I cannot believe I ever painted anything so wonderful," he murmured. "How did I do it?" he begged himself to explain.
When we got out on the street again I walked some paces off parallel to him in order to observe him. Adverse remarks with ordinary men it is politic to keep to oneself, while to withhold one's comments from Insel would have appeared impolite. His very personality taking the form of a question mark, it would have shown a lack of perspicacity when intentionally confronted with a self-composed conundrum, not to attempt unobserved, the intriguer, underrated.
Curiosity he constrained to stand off to take his measure, mentality, to pivot him for noting whether there were any creases in his aural suit. As those who are of the body, whom other bodies have traffic with, slap each other on the back, with Insel intercourse depended on putting out feelers among the loose matter of psychologic nebulae.
"You walk so weirdly," I said. "Are you one of those surrealists who have taken up black magic?"
Totally bewildered, he exclaimed, "Whatever is that?" Yet, like all who have to do with any form of magic, he apparently had lost some of his specific gravity.
He was passing over the light-reflecting pavement in his shabby black as if a rigid crow, although with folded wings should skim.
"Aera," I said, "sends dreams across the Atlantic."
"He could not," protested Insel, off his guard. "He has not got the _power_."
There is a way of speaking that word peculiar to those alone who have wielded it—that way was his.
And he glided on, turning towards me his face hung with deflated muscles one felt could be blown about by the wind.
"You cannot glide," it defied me, and I noticed how I was keeping my distance in my effort to "get at him."
He had for the moment the stick-fast aloofness of an evil presentiment—the air of a priest of some criminal cult. All the same, this slight impression of criminality he gave off at intervals I did not receive as a direct impress on my own mind, but as a glimpse of a conviction he hid within himself.
"Aren't you rather bad?" I laughingly inquired.
"Everybody imagines I am the devil, and," he answered forlornly shrugging his shoulders, "there's no harm in me at all."
When we fell into line once more, he resumed the uniformity of all people making for a cafe.
I finally gave Insel the key. His mimicry of salvation convinced me my distress, after all, had not been in vain.
But, oh horror! On arriving at the country, I suddenly _seemed_ to remember the charwoman pouring some Normanol from an antique bottle I had told her to clean into an empty gin bottle. Normanol, being a dissolvent for rhodoid very much stronger than cutex which dissolves the cuticle around the fingernails, I had a shocking vision of Insel's diaphanous intestines entirely disappearing should he, as would be only natural, mistake it for a graceful token of absentee hospitality—and of myself arraigned for manslaughter.
"For God's sake, don't drink anything out of any kind of bottle in that flat," I wrote him immediately. "It might kill you."
"Dear Mrs. Jones: However do you think I am comporting myself in your home," Insel answered. "Were there thirty bottles of the finest schnapps I should not touch them. Rest assured you will find your apartment _exactly_ as you left it."
He sounded quite comfortably settled. I had also written him to get my charwoman to clean his suit with odorless gasoline at my expense, and inquired how much money he had left.
"Your suggestion for my suit is most kind—however, I am convinced that it is only on account of the dirt in it that it still holds together."
He had, he said, enough money to last him for a "little" week.
We had agreed that I should come every few days for a dressmaker's fitting at the further end of the flat where it would not disturb him. When I did go, a bewildered concierge informed me, "Madame, the artist who was to live in your apartment never came."
"That only means he has not yet been 'turned out,' " I explained to her, while to myself I reflected, "You will find your apartment exactly... the monkey!"
I felt that the end of his little week was no longer my concern, and I forgot all about him.
I would run into Paris for the dressmaker, a tea, a dinner and back to my little hotel in St. Cloud again, until at last the time drew near for exporting pictures—among them were to be included some of Insel's.
"Pictures, drawings, three o'clock," I wired him. At a quarter past, he had not arrived and I went to tea with a friend who hailed me from the courtyard, leaving a note on the door, "Will be back shortly."
When I returned the place was different—in the smoothed out air there was a suspicion of a collapse in time. As if by a magnet, I was drawn into the studio and up to the dark oak table. Upon it lay a flat packet. I could have sworn it emitted a faint phosphorescence that advanced from all the rest of the room. The wrapping paper was so strikingly creaseless it looked unusual. It had in some inexplicable manner become precious as ivory; its squareness was instinctively exact as the hexagons of wasps.
"He has left the drawings," I supposed, almost reluctantly undoing this magnetic focus of an uncanny precision. But once unfolded, I found it contained only a shabby block of writing paper that had been left lying there and from which I had torn the note I left for him.
"Did you find anything?" he asked when, later having resuscitated from the moribund state in which he preferred to arrive, he was able to articulate.
"Yes. What on earth—?"
"I wrapped it up," said Insel—an enormous intention fixed in his eyes.
It was at this moment that, for me, Insel, from a seedy man, dissolved into a strange mirage, the only thing in the world at that time to stir my curiosity.
On his arrival with the pictures he had appeared the phantom of himself as I had seen him last. He had so weakened, become so transparent.
Deeply bowed, he clutched his feeble fist in the emptiness where his stomach should have been. From this profound concavity arose a dying whimper of, "Water— _aspirine_ ," as out of the abdominal void rode the unclenching fist—his tremulous fingers, hovering over the bureau, grasped a cigarette.
"Well, you're in a nice state," I taunted him to cover my alarmed compassion. "Why didn't you write?"
He gulped his _aspirine_ as if to alleviate a death rattle. "I did write."
"Yes, a comic strip. I found my flat _exactly_ as I left it."
"I know," said Insel, gently abashed. "I ought to have told you 'I am not here.' "
"Even the least of philanthropists," I laughed, "has sensibilities—I thought I had been intrusive. You see, Insel, any possible gesture in the face of poverty must inevitably be insolent, its very necessity—in not being outs—makes poverty so aloof."
"And I thought you were _angry_ because I mentioned money."
"I had _told_ you to mention money. But all my sympathy for you was buried under that bunch of cheap flowers I put here to welcome the lonely _clochard_."
"But, after all, I have been here nearly every day," he almost sobbed, "to look for you. I could never find you. I knew when you had been here for where you trod there lay little fragments of stuff. I could trace your movements by the pins you shed on the floor. Think what it was like—to seek after a woman, a vanishing woman, and in her stead, to find nothing but pins," he implored. Then brightening, "I picked them all up. Look," said Insel, hurriedly reversing the lapel of his jacket. On the underside stuck in rows as precise as in packages from the factory, were my dressmaker's fallen pins. He dropped the lapel into place again as if too long he had bared this precious hoard of his compelling exactitude.
With an interminable cautiousness Insel had revived. " _Ich bin nicht fromm_ —I am not pious," he mused, deeply introspective. "And yet how I have prayed, I prayed," he burst out, a blind agony falling upon his eyes, "I prayed that you would come back!"
"You seem to have been thinking about me a good deal—hadn't you any steak?"
"I never _cease_ thinking of you," he muttered, as if fearful I should overhear—and aloud, "None," he answered flatly yet without reproach.
As mediums on becoming professional, obliged to continuate an intermittent condition, lapse to the most lamentable dupery, Insel would actually plagiarize his innate mediumistic quality of which he appeared to be but partially conscious.
It would seem unnecessary after the intrinsic wizardry of his simple packet to resort to the untenable mystery of a lie. Yet he did.
Awestricken, solemn, he recounted to me that while I had, as it were, struck myself off his menu, Mlle Alpha had sent him a card to know how he was getting on.
"The world is populated with people anxious to know how I am getting on. But when I tell them— the world immediately depopulates! I wrote her in answer, 'Am starving to death except for a miracle—three o'clock Tuesday afternoon will be the end.' —And then your telegram! for three o'clock. Today is Tuesday.
"Of course she did not answer," he commented, "I had rather thought she might be good for fifty francs. Nobody ever sends one fifty francs," he ended despondently.
"Oh, _what's_ the matter with you, Insel? That girl has more sex appeal than almost anyone in Paris. And all your reaction is that she might be good for fifty francs. I never could interpret, until I saw her, the French, _Elle n'a pas froid aux yeux_ —the Alpha's eyes are volcanic. All the men are in love with her."
"Not I," he boasted.
"No-o? I should say that clochards were hardly in her line."
"Grade—exactly," Insel concurred as if relieved of a responsibility.
Characteristically, after swearing he would ask her for his card as proof of a miraculous coincidence with his usual unconcern in breaking up his plots it was Insel who insisted on my meeting Mlle Alpha whom I knew only slightly.
# **5**
"FLEISCH OHNE KNOCHEN," INSEL ESPECIALLY hollow-voiced begged me when I took him to dine. This insistence on boneless pieces of meat was habitual with him.
"Do I look any fatter?" he inquired after he had eaten, as if consulting his doctor.
I thought it best to reply in the affirmative. As a matter of fact the disquieting thing about Insel was that however much food you sunk in him it no more seemed to amalgamate with him than would a concrete mass with a gaseous compound.
From now on Insel turned up regularly as soon as my fitting by the dressmaker was over.
Whenever I let him in he would halt on the threshold drawing the whole of his luminous life up into his smile. It radiated round his face and formed a halo hovering above the rod of his rigid body. He looked like a lamppost alight. Perhaps in that moment before the door opened he recreated himself out of a nothingness into which he must relapse when being alone his magnetism had no one to contact.
"I've brought 'it,' " his illusive grin seemed to be announcing, as if his visible person were a mannequin he operated on occasion. "Make what you can of it—you may wonder if I am sure of its nature myself—let us not be too precise as to what I am."
I led him down the corridor, feeling that he, so recently non-existent, was all-surprised at finding himself to be anything at all.
He shut the door, an act I have heard an authoress describe as so banal it is unfit for publication. But shutting the door, like all automatism we take for granted, is stupendous in its implications.
As the ancients built temples as isolators for the power of the Almighty, which their ritual focused on the altar, a force so dynamic that officiating priests, having evoked it, were constrained to descend the altar steps backwards without ceasing to face it; for the limitless capacity of the eyes could absorb such power, whereas if the blind back were turned upon it they would receive a shock that flung them to the ground.
So the shutting of doors is a concentration of our radiations in rectangular containers, to economize the essences of our being we dispense to those with whom we communicate.
Thus, when Insel shut the door infinitesimal currents ran out of him into the atmosphere as if he were growing a soft invisible fur that, when reciprocal conditions were sufficiently suave, grew longer and longer as the hair of the dead, it is maintained, will leisurely fill a coffin until it seemed with its measured infiltration even to interfere with Time. The mesmeric rhythm of a film slowed down conducted the tempo of thought and sentience in response to his half-petrified tepidity, for he moved within an outer circle of partial decease—a ring of death surrounding him—that reminded one of those magically animated corpses described by William Seabrook. Even before he came into one's presence, one received a draughty intimation of his frosty approach. He chilled the air, flattened the hour, faded color.
But if one could crash through this necrophilous aura, its consistency dissolved, one came to an inner circle where serial things floated in a semi-existent aquarium. Or, at times he, himself, would overflood it, as now when his coming close to me affected acclimatization, turning an irreal ice into a tenuous warmth.
"I was so terribly afraid I should miss you. I got to bed at seven this morning— (quite exceptional," he added hurriedly as if wishing to efface a bad impression, "I shall not do it again), and when I woke up my watch said twenty past six. I was convinced you would be gone, but—is it not astounding—a moment later it said half past four."
To these teeny nothings that marked out his life (as momentous events are the milestones of others) he imparted an interest peculiarly visual. You saw the watch in hallucinatory transformation, its dial advancing the gray diamonds of his eyes out of a murk more mysterious than darkness instead of correcting the eyes' mistake. He possessed some mental conjury enabling him to infuse an actual detail with the magical contrariness surrealism merely portrays. Perhaps it was the operation of this weird power that necessitated his speaking with such drilling intensity.
He had brought me a present— As he bowed his head over what he held in his hands, all the sweet-stuffs of the earth exuded from his nerves, in an exquisite music of a silence that is alive. He seemed to be sodden with some ineffable satisfaction, as if emerged drenched from some luxuriance requiring little tangible for its consummation. I had to hold myself in check. My charmed curiosity wanted to cry, "From what enchanted bed of love have you so lately arisen? What astral Venus has just receded from your embrace?"
It was a queer impulse, the idea of making such delicious inquiry of this bald and toothless man whose clothes were stiff with years of wear, yet deodorized by continuous exposure to the all-night air.
His voice, gone dim with a crushed emotion as he held out to me a black passe-partout, was saying, "I want to give you my own drawing; the only one I refuse to sell." The drawing in the passe-partout, like his atmosphere that clung to him as ours clings to the earth, seemed almost astir with that somnolent arrested motion revealing his nature.
It was so white, the flocking skies of a strangely disturbing purity drifted above vortices of snow-like mist in travail of taking shape, coiling the mind into following the spiral, eventual materialization of blindly virginal elementals.
"This," he continued, "is the first drawing of a new series—all my future work will be based on it. I intend my technique to become more and more minute, until, the grain becoming entirely invisible, it will look like a photograph. Then, when my monsters do evolve, they will create the illusion that they really exist; that they _have_ been photographed."
The while the drift of his words swept me together with the frozen drawing along a current of quiet reverence, expressing gratitude. As under his conjurative power of projecting images, I felt myself grow to the ruby proportions of a colossal beef steak.
I argued for some time over the idiocy of presents in the very jaws of economic death; proposed sending it to New York to be sold for him; but at length when he inquired sadly, "It doesn't please you? I will give you another," I promised to keep it.
# **6**
"I'M SO UGLY NAKED," HE TOLD ME MOST unexpectedly, in a tone of intense and anxious confidence. "I can't go to the public baths because I daren't walk down to the water."
"Your face is naked and you walk about with it."
"Yes," he assented miserably, "and it frightens the women. I used to be so beautiful. Is it imaginable?" he asked, peering expectantly into my face.
"I'm tired of your tirade as to how hideous you are."
"All women are terrified of me," he continued automatically.
"I said tired—not terrified, and I'll tell you why. I've never really seen you. You always give me the impression that you are _not there_. Sometimes you have no inside; sometimes no outside, and never enough of anything to entirely materialize. Like a quicksand, when one looks at you whatever one gets a glimpse of you immediately rush up from your own depths to snatch. _Your_ way of being alive is a sequence of disappearances. You're so afraid of actuality."
"I can materialize for _you_ ," he said raptly, " _forever_ —on the corner of this street."
Somehow we were sitting on the Terrasse of the Hotel Lutetia. It stood behind us dressed in its name of a pagan Paris. It might very well be actually surviving for our blind backs which, taking no part in the present, are carried around with us as if concrete in the past.
Darting about amazingly in the autumn vapor innumerable metal beetles of various species with which modern man, still unable to create soft-machines and so limited to the construction of heavy plagiarisms that sometimes crush him, had sprinkled the _carrefour_ facing us, where gasoline impregnating the dust had begotten a vitiated yet exhilarating up-to-date breath of life.
A distant gnat of a thousand horsepower hummed in the heavens.
"If only we could sit here eternally it would be _wunderbar_ —it's just like sitting in the kitchen."
"The kitchen?" I exclaimed.
"Ah, yes," he sighed. "All my youth I ate in the kitchen together with my _Vater und Mutter_ and the other five children." Then, inhaling the effluvia of the streets, "There was lots of steam," he said invigorated. "All the washing hung up to dry, the muddy boots stood in a row, and on the good hot stove—"
_"Die Suppe_ ," I joined in, entranced.
_"Ja, die Suppe_ ," he confirmed ecstatically. "You see, it is not so much a matter of materializing but of being able to speak. Before I found you I had never anyone I could speak to. I should never have been able to tell about all this—"
"All the boots?" I interrupted.
"—to anyone else," he concluded, his voice trailing off as if calling to me from another world.
_"Allein—allein_ ," he chanted forsakenly. Something was happening to this man's voice, the most musical modulations were stealing into it. "Always alone—alone in Berlin—alone in Paris—" The words floated out of him like wisps of a dream, "more than alone in prison."
"In prison," I responded, "where there is no one, _no one_ you can convince of your not being there. You might try it on the warden, but the moment he began to suspect would be the very moment he assured himself of your presence—I mean—let's leave that and have something more to eat!"
In sitting so close to Insel at the small terrace table all the filaments of what has been called the astral body, that network of vibrational force, were being drawn out of me towards a terrific magnet, while I sat unmoved beside the half-rotten looking man of flesh. My astral inclination, withheld by a counteractive physical repulsion, could not gain its presumable end of flying onto that magnet— It was as though he had achieved an impossible confusion of his positive and negative polarity— Out of a dim past echoed the din of a music hall refrain I had heard in Berlin: " _Du musst herüber—You_ must come over."
# **7**
THE LESS HE SEEMED TO BE "THERE," THE MORE HE spilled into the unknown, the more clearly I apprehended him, whereas Insel himself seemed ever to be seeking a reduction of focus through which to penetrate into the real world.
Suddenly he bowed his head over me in a wracking attentiveness. He had found such a focus. Darting, his constricted fingers cleaved to a white hair of my head which had fallen on my coat, he made a ritual of offering it to my eyes.
_"Je suis la ruine féerique_ ," I trilled in vanity.
"Ah, yes," sighed Insel, as I translated, churning me with his eyes into the colorless vapors of his creation.
The cloth of my coat, a FANTAISIE, was sewn with lacquered red setae—wisps, scarcely attached, which caught the light, and all through the evening unusual manifestations of consciousness occurring outside the Lutetia were punctuated by Insel's staccato spoliation of that hairy cloth. He could not desist. Like an adult elf insanely delousing a mortal, whenever I laughingly reprimanded him for ruining my coat, with an acrid cluck of refutation he would show me what he had instantly plucked from the cloth—it was always a _white_ hair— He did not trouble to contradict me—the evidence was clinching— But in the end the side of my coat sitting next to him was bare of all its fancy setae.
In accordance with the rules of sympathetic magic, so long concealing my one fallen hair in his palm augmented Insel's influence over me. An influence which, rather than having submitted to it, I purposely invaded, so urgent was my premonition of some treasure he contained. His voice now setting in a glowing duskiness haunted me with wonder as to where I had heard it before—
"Black as was the stain on my name—," I listened to Insel intoning as if he were celebrating mass, "even so white would I wash it in glory—."
Rising for a moment from the fantastic shallows of his cerebral proximity to my normal level, "This man is _fearfully_ banal," I said to myself, discerning in his confidences the prim hypocrisy of a wastrel bamboozling the patroness of some charitable institution. Any such patroness would have cried for help should she receive him as he was at present plunged in the depths of a subverted exaltation, so awesomely he stressed his lonely agony, his long starvation, the incidentless introspection of that enjailed jewel, his artist spirit. As for me, the fundamental lacuna in my experience was being "stopped up" with his moral man. The pattern held out to my early ethical life. The man who stones. He who unsuspectingly lingers in the world subconscious.
I did not care whom he bamboozled. Slipping back into his sensitized zone, I swallowed his platitudes gratefully. So seldom had I come across _anything_ sufficiently _condensed_ to satisfy my craving for "potted absolute." This man sufficed me as representing all the hungry errantry of the human race.
"What are you trying to be anyhow?" I asked bemused. _"La faim qui rode autour des palaces?"_
A sound of anguish was hovering above us but I scarcely registered it listening to his quiet soliloquy in reverence for the buried aspiration whence sprung the weedy heroism of his pretence.
"Dolefully trite in his insincerity," my common sense intervened.
"Inflexible is his moral will," countered the underside of my mentality which drawing comparisons to sociologists' deceptions in criminal reform preferred to remain impressed.
" _Der edler Mensch_ ," it breathed devoutly, "The noble creature."
Still looking so extraordinarily distinguished, Insel was illustrating a society by means of an empty plate, a diaphragm reducing the world to a white spot.
"There are _you_ —" pointing the tip of his nose toward the center—a comical almost four-cornered tip of a nose with the sudden sharpness of a (square tool, the name of which I have forgotten), "and here am I—on the outside—peeping over the edge at you," he said as he crept his fingers in their incipient movement up from under the rim.
I was disappointed. One thing about Insel that had struck me was this sporadic distinction I had often been "accused" of which I had always been eager to discover in anyone else who, like myself, had "popped up" from nowhere at all—as if all my life I had lacked a crony of my "own class."
I could not point this out. It would upset Insel's self-abasement which gave him some mysterious satisfaction—as of an Olympian in masquerade—
That sound above, once it hooked up with perception, became a squirling wail—soaring over the driving racket of the street corner.
"Do you hear?" asked Insel, "it's been going on for quite a while—in an aerial invasion people would sit on at their cafe tables just as we have done. Air raids," he shuddered, shrinking into himself, "and the French so proud of their Maginot Line. They have forgotten _their_ Stavisky was mixed up with it—Sugar," beamed Insel, a gentle delirium stealing into his eyes, "tons and tons of sugar poured into the foundations of the Maginot Line."
"Sugar's rather expensive," I ventured.
"Nevertheless," averred Insel, and I felt it might be perilous to contradict him.
His chuckle petered out as the siren insinuating to his brain the menace of a war which would cut off any chuckling, transformed it to a shudder.
Insel trembled with the cowardice of those whose instinct being to create even an iota, appear to slink into a corner before the heroic of destructive intent. This man, who, when he turned his face full on you, looked into your eyes with the great intensity of the hypnotist but with a force of concentration reaching inward and outward as if he must first subject himself to his own mysterious influence, this man in his terror had dropped from his own magnetic "line" soft as a larva. He cowered against the air. Inasmuch as it concerned him, war was not only imminent—he was already ripped open by its plough of anguish. Actually, he was in a fix—for, in the "event" being a German, here he was an enemy, whereas if he could return to Germany, there he was _Kultur Bolshewik_.
Man Ray came up and sat with us and went away. Tables filled and emptied. The dust grew denser and then lay down before the oncoming night.
I once heard somebody express surprise that instead of following it onward one should not take a cut across Time to secure a moment which, stretching out in line with oneself, would last indefinitely.
Time that evening lightly came to rest—an unburdened nomad let its three faces linger; the future and the past were with me at present: the whole of time—there was no more pursuing it, losing it, regretting it—while I sat almost shoulder to shoulder with this virtual stranger living the longest period of my life.
It is almost impossible to recover the sequence or the veritable simultaneity of the states of consciousness one experienced in the company of this uncommon derelict. It was so very much as if consciousness was performing stunts. Always in his vicinity one had the impression of living in or rather of being surrounded by an arid aquarium—filled with, not water, but a dim transparency: the procreational chaotic vapor in which all things may begin to grow.
Either he had a peculiar power of projecting his visualizations or some leak in his psyche enabled you to tap the half formulated concepts that drifted through his mind: glaucous shades dissolved and deepened into the unreal tides of an ocean without waves. Where in the bottom of slumber an immobile oncome of elementals formed of a submarine snow, and some aflicker, like drowned diamonds blew out their rudimentary bellies—almost protruded foetal arms over all an aimless baton of inaudible orchestra—a colorless water-plant growing the stumpy battlements of a castle in a game of chess waved in and out of perceptibility its vaguely phallic reminder—.
Projected effigies of Insel and myself insorcellated flotsam—never having left any land—never to arrive at any shore—static in an unsuspected magnitude of being alive in the "light of the eye" dilated to an all enclosing halo of unanalyzable insight, where wonder is its own revelation.
Even in the world of reality Insel's ideation was an introvert exploration of a brilliancy beneath his skull, an ever-crescent clarity which in the form of inspiration ripens creative fruit. But in him by reason of some interference I could not define, aborted as the introduction to an idea.
"I can see right into these people," he asserted, indicating the crowd gathered around the Hotel. "I know exactly what they are; I know what they do."
And that was all.
As if satisfied by his sense of insight, he needed not to perceive anything specifically, his mind exposed these people as brightly illuminated "whats." A reaction he accepted for entire comprehension.
His conceptions were like seeds fallen upon an iron girder. I noticed that I received them very much in the guise of photographic negatives so hollow and dusky they became in transmission, vaguely accentuated with inverted light—.
Thus, as he unfolded his ardent yearning to flee to New York from a threatening war, the transparencies his presence superposed on the living scene became crowded with flimsy skyscrapers. Up from among their floating foundations swam misty negresses, their limbs spread out at inviting angles, like starfish through the mirage of windows plunging in fathomless pools their reflections.
But this is not all that happened with him. The visions emitted by the organism of this truly congenital surrealist were only a wasted pollen drifting off from the nuclear flower of his identity. For my first unaccountable conclusion that he was "the most delicate—soul," my fascinated impression of his emergence from a goddess' embrace, the dove that, when he had been still for a while would seem to take his place, were conceptions fully justified by the lovely equilibrium his companionship conveyed to me. It was as if for an exceeding moment I could, rising above the distortion of life, hold inexpressible communion with Insel, where his spirit had no flaw.
Within range of the crystalline of his eyes become so brightly brittle, again I experienced the profound relief of the acute celerity rhythm that perpetually disintegrated me as I got out of watching a film in slow motion.
Imagining aloud the explorative kick of roaming the mountainous blocks of Manhattan "forever in New York—," Insel chanted, "we could have such a wonderful time together." He was not speaking. He was praying.
Idly I wondered with what he was communicating, when suddenly I felt myself sag; become so spineless, so raw—. I, a red island with its shores of suet, the most dependable substance in an aquarium-America not so very much dimmer than the Paris cars threading through it in the Rue de Sèvres.
I did not find it extraordinary that my condition as an undiminishable steak should make me feel almost sublime, or that the man intensely leaning towards me should pray to it.
There was another element in his unbelievable magnetism of recoil. His air of friability warning off contact lest he crumble. Not only was he preposterously emaciated, but even as his gravity seemed lightened, his body—what was left of it—seemed less ponderable than it should have been. Insel was made of extremely diaphanous stuff. Between the shrunken contour of his present volume his original "serial mold" was filled in with some intangible aural matter remaining in place despite his anatomical shrinkage. An aura that enveloped him with an extra external sensibility.
To investigate, I tapped him lightly on the arm in drawing his attention—and actually in a tenuous way I did feel my hand pass through "something." The surface of his cloth sleeve, like a stiff sieve, was letting that something through. The effect on Insel was unforeseeable—jerking his face over his shoulder, he twitched away from my fingers with the acid sneer of a wounded feline. This might be merely a reflex of physical repulsion to myself, so later I repeated the gesture, but as if my hand in its first contact had got coated with the psychic exudence it would seem there was no longer any hurt in it. He was calm under my touch.
# **8**
THE REVERSE OF HIS ALOOFNESS WAS A HOLLOW invitation to my intrusion. Urged to cross the frontier of his individuality, I got in the way of that faintly electric current he emitted. His magnetic pull steadily on the increase, the repulsion proportionately defined, threw me into a vibrational quandary, until as if it were imperative for me to make a connection with the emissive agency of my accidental clairvoyance, with a supernormal acumen he inspired, I located the one point of contact: the temple. Straightway I found myself possessed of an ability to form a "mental double" (for no portion of my palpable substantiality was in any way involved), a mental double of my own temple.
This was one manifestation of how in Insel's vicinity pieces of bodies would seem to break off as astral fractions and on occasion hang, visually suspended in the air. Quite apparently to my subconscious the bit of my skull encaving the fragile area flew off me, crashed onto his and stuck there.
On the spur of this subvoluntary cohesion to the telepathic center—I definitely penetrated (into) his mediumistic world where illusory experience which had so far escaped as scarcely whispered pictures took on a fair degree of resemblance to three-dimensional concretion: the sculpture of hallucination succeeding to the visionary film.
Insel straightened as a water level, his petrified eyes drilling the image of his coma into the ultimate ceiling, broke into a right angle of prostration and ascension.
Out of a torso of white ash arose iron rags as puffs of curling smoke, blocks of shadow crushed together in the outline of a giant. Dense as the dark, high as a tower, the almost imperceptible radiance of a will-o'-the-wisp shining from it—I crouched alongside encumbered with an enormous shell white as plaster which, having but partly taken shape, trailed to an end in a sail of mist.
And all the while Insel kept up his mixture of _Beggar's Opera_ and oratorio, showing a tragi-comic duality in his confidences. A coxcomb in purgatory, he enlivened a suppliant self-abasement with pranks he was proud to have played on a short-sighted mistress. Vast tracks of his barren universe were fixed in her monocle.
Our discussions of his tribulations had the light hilarity of conversation between clowns. Our shoulders almost touching, we seemed to have come within risible distance of each other. As if our imbecilic mirth were due to an assurance that suffering loses weight when tossed to and fro.
Intermittently his intriguing anxieties ceased to be actual. In his cerebral commotion, Trouble, attaining an inflation at which it burst, had no further existence except in the fragments constituting his exhortations for help which, at that, were his means of entertaining one.
Albert Londres tells of a lunatic who periodically would drop whatever he was doing to go up to the wall and say peculiar prayers to it. So Insel had two or three intimate anecdotes he had to "get off." He told them whenever I met him with an earnestness that, like a gentle gimlet, bored into my mind. The culminative point of his corporeal life had been his threat to shoot a girl who left him for a lesbian, and of his psychic life, his magnetic rays drawing some other girl out of bed on to her balcony whenever he passed below at night.
As a prayer, repeated over and over, becoming autohypnotic, attains to faith in each retelling, these stories grew vaster, lasted longer, reached farther into a kind of absolute of confidence. As if with incantations he must summon up his past because some unimaginable impediment withheld the present and the future from him. His mind besieged the barred outlet of today-into-tomorrow in an effort to break it down and gather fresh material, but on finding itself impotent revoked to memory, dilating his souvenirs until for him the story of the universe was blotted out by the gigantism of his meager individual experience.
Externally his aspect was vague as, internally, the rudimentary ideas stored in his cerebral cells. His person withdrawing in approaching, his eyes appeared to start their staring in advance of the brows that encaved them. Between his "expression" preceding his face and his speech which so often sounded as if issuing from a distance behind him, his person melted from view. In him everything seemed inverted. His voice in its drilling intensity getting softer, louder, would go up higher, lower.
My casual ability to partake of his moods evoked my own anxiety of the past which joining in his terror of the aerial omen made it doubly real. The nomistic warning which recurred to my mind, "He who looks back returns to the maze," I disobeyed; so intense was my intuitional curiosity as to the leak in Insel's magnetic coherence. I felt that giving in to a dislocation of my identity, which is usually perilous or demoralizing, must in this exceptional case, be finally vindicated by a revelation of what supremely lovely essence was being conveyed to me by this human wreck. In the light of this my certitude his corporeal mendicancy appeared fictitious. So surely it was an exquisite nucleus that in his somewhat comatose exaltation he struggled to save. On the instant I accepted this salvation as equally my affair. Memory in euthanasia will come to life when fed on the same sort of stuff as that which formed it—.
Insel, the animate cadaver, stretched with the pliancy of decay from the last war into the next—while walls of solid murder with soldiers for bricks came marching in on a living aspiration—to enclose it—waste it—it must not happen again.
" _Vielleicht verkaufen_ ," I could hear Insel muttering as I made this decision, obsessed by an impersonal responsibility. He was toting up imaginary accounts in payment of his passage to America.
"Promise to be my guide and companion?" he implored earnestly while staring straight before him as if it did not matter where I was.
"How tedious," my everyday self recoiled, the lovely essence evaporating, for whenever Insel turned his profile he sort of extinguished. It was only when both his eyes were fixed upon me I entered his Edenic region of unreasoning bliss to sway among immaterial algae.
In profile, as if he cut himself in half and in halving should leave himself evil, he became so alien, so very elfin, he induced aversion. The notch at the spring of the nose was further back than the drop of the upper lip. These angles of his pasty face were over-acute and out of plumb. A kink near the ear suggested the wire-hung jaw of a ventriloquist's dummy. In profile, this nitwit infused with the secret ghost, seemed to have been carved for a joke out of moldy wood.
" _Immer—immer spazieren_ —eternally taking a walk," he insisted, once more aware of my presence; his voice dwindled to a pathos so unearthly it could only converse with the unconscious. His eyes, for dusk had fallen, were phosphorescent as approaching fireflies.
As Insel consecrated our spontaneous comradeship with his tom-tom reiterations of how he delighted to talk to me, and I, nonplussed, would hazily inquire, "What about?" I kept on naming him to myself.
_"Die nackte Seele_ ," and again, _"Die nackte Seele."_
It seemed quite fortuitous that sitting beside him I should feel I was up against "the naked soul." Practically anything might substitute in my consciousness for a man, who, however long I looked at him I could never entirely "put together."
We had been sitting outside the Lutetia for six hours.
"Now," laughed Insel, "Man Ray should pass again."
"To conclude, we have no use for time."
"That is not what I mean—"
"You mean that eternity spins round and round?"
We arose. But our legs become paralyzed from sitting so long on the hard little chairs, we barely saved ourselves from toppling over and staggered across the pavement. Suddenly the Metro opened at my feet. In the midst of a sentence I dropped from sight as if impelled to conform with Insel's concepts by flickering out. One seldom took leave of him; (walking along with him one would unexpectedly drift sideways into a cab) as Insel, in an electric sputter, softly mumbled _"schade."_
So now I descended the stairway—Insel leaned over it in his disgraceful grace, "When shall I see you again?" he implored, clutching his concave breast. An awesome lunar reflection lit up his face from within.
# **9**
ON ONE OF MY VISITS TO TOWN INSEL DID NOT turn up, but when I left home to dine with friends I met him drifting round the corner with the wild concern of having too lately thrown off an unguessable inertia.
He greeted me with the relief of an object which, having fallen apart, should chance upon its other half again. His discomforting friable surface had slightly embodied; he even—I felt, but was not sure—took my arm: a brittle elbowing into the prelude of some _danse macabre_.
With a kind of epileptic cajolery he beseeched me to break my appointment. Finding this ghostly courtesy agreeable as any other, I decided to give him half an hour. I would drive to my friends instead of walking. So down we sat to suffuse another stray cafe with the ineffable haze of his contagion.
Insel, leaning back against his chair in a tall recline of felicity, groped in his pocket and took out something I could not see for he held it like a conjuror. Signing for me to hold out my hand he placed his over it as a cupola showering so discreet a sensitiveness my hand responded as a plot of invisible grasses grazed by an imperceptible breath.
"The girl," he whispered, and the grasses parted under a couple of atomies cupped in my palm; Insel and his girl embracing—or were they Adam and Eve? "The girl gave me this," he said, puckering his face in helpless incomprehension. "And it won't go."
I looked at what he had dropped in my hand—a sordid silver watch on a worn leather strap.
"Will you take it to be mended?" he wooed me. "You can speak French."
As soon as I was seated beside him I had reached the extremity of optimism. The landscape of a spattered hoarding across the street was too lovely to look at. I had to lower my eyelids. Insel already had lowered his on a face falling lower and lower into the excavation of his breast.
He started up, elated to impart what he had found there. Evidently a death warrant.
"I am to die," he rejoiced. "And will you weep one little tear for me?" he asked flirtatiously.
"Yum, yum," I jibed, intent on the beauty of the silver rivers he had loosened in the veins of the ugly marble table top. "Does ums want to be pitied?— you've struck a hunk of granite."
"You _won't_ weep?" he implored from a gust of sad laughter.
"Not a drop."
Insel tried again. " _Sterben_ ," he sighed in the voice of a weary archangel, an incommemorable voice burying the endlessness of death in two syllables. I was disturbed—if he should peter out on that annihilating refrain I would never know what was so weirdly, so wonderfully the matter with this exquisite scarecrow.
"Insel," I shook him gently, "you're much more likely to make people weep by remaining alive."
But Insel, passionately in love with Death, raved in a soft, a sublime sibilance, _"Sterben—man muss—man—mu—uss."_ This fair decease in which he infinitely delighted, vaster and more dimly distant than the lesser deaths of his usual aberrations, sailed with Insel on its wings to heights of a stratospheric purity.
At once the hoarding became abominable, the marble of the table the color of nausea, the whole of the world depressing, and Insel, a dilapidated suicide, hung aloft from a terrifyingly rusty nail together with all his unpainted pictures—. This was a recollection of the somber ambition which stirred him whensoever he became aware of his real life. It looked pretty bad—real life—so carelessly repaired by hand—that obscene, that relentless hoarding. Insel, his eyes closed upon it, induced by Death the absolute decoy, examined an integral vision lining the degeneracy of his brain.
His dirge still hummed on the air—.
Life without world, how starkly lovely, stripped of despair. The soul, inhabiting the body of an ethic, ascended to the sapphire in the attic. Here was no need for salvage. If he preferred to attain perfection, I would let Insel loose to die as he pleased. But my unconscious, with an inkling of what perfection was like after sharing to some degree in his increate Eden, squirmed with envy.
If Insel committed suicide—I could share in that, too. My envy at once supplanted by a flowering peace—filling with fragrance—space. Through a break in the cool white blot of its branches—I perceived the cafe clock. On that uncompromising dial all things converged to normal. I was a tout for a friend's art gallery, feeding a cagey genius in the hope of production. Insel's melodious ravings, an irritating whine— It was ten to eight.
Nevertheless, as Insel was going to _sterben_ —the word now sounded flatly banal—I promised to meet him at the Dôme after the cinema. "Take this," I said. "Be sure you eat a wholesome meal," with my usual mental ejection of the obvious man, to whom I was definitely averse.
This unreasonable nonchalant faith in Insel's alter ego was about to be greatly rewarded. After my amusing dinner and a good film which, when we came out, proved to have lasted much longer than usual, on our return in my friends' car the lightning hand of pain unexpectedly grabbed my internal organs and, twisting them in a grim convulsion, wrung out of them as from a dishrag a deathly inner perspiration—as if one were about to retch a nothingness poisoned with anguish. I was in for it, this being the preliminary to invasion by the tenacious rodent which would not cease from me for days.
It was one o'clock and Insel might have waited since half past eleven. He had. When my friends in some concern dropped me at the Dôme I could see him sitting outside.
Insel seemed unconscious of having waited for me for an hour and a half. After all it was ridiculous stopping to apologize to one who lived in that other time and space. My reflection immediately complicated, "When was he here? When was he there? Was he in a wavering way existing in both dimensions at once?" The distant aristo went about his simple social life with sufficient consecutiveness, save for long delays excused with mysterious illness and misplaced sleep, he visited anyone who would have him on the right day.
During my absence he had changed.
I had never seen him like this before—human—actually gay! As I tried to explain why I must go home, Insel, in laughing over something he wanted to tell me, laid a fluttering hand on my shoulder— the torture of my body ceased.
It was not only an interruption of pain. I was regalvanized. Straightening from top to toe, I inhaled a limpid air—the neon tubes caressed my eyes.
I looked at Insel amazed. In what unheard of parasitism had I drawn this vitality out of a creature half-disintegrated?
Evidently he was in good form. The sparks he seemed to emit in turn gave off smaller ones; an added superficial illumination induced by a few drinks, having much the same effect as the perspective confusion of traffic lights among electric signs.
Out of all this an intimate twinkle approached me. "Promise to sit here with me till seven o'clock tomorrow evening," Insel entreated.
"Naturally," I acquiesced.
There is no field of fantasy so rich as the financial promoting of failures. Weaving in and out of our conversation was a shuttle of money-making devices for Insel's relief, the most practical being to star him in a horror film. It is a poor horror which has to grime its face—the only face on the films that has true horror in it is Jouvet's—and that only an inkling—and so discreet.
Insel said he had been offered such a role. But again he had not been able, or wishful, to pursue anything that carried him into the future—a future that ebbed from him as from others the past. He would look forward with one eagerly—always at a certain point he reverted—turned his blind back on the forward direction—.
He said, "I have worn myself out tramping the city on fruitless quests—to show my good will."
Now I had found another profession for him— magnetic healer. Suddenly I foresaw the fear my physician would inspire nullifying his therapeutic value, and I did not suggest it to Insel.
In his unusual liveliness, words, like roomy cupboards, dipped into the reservoir of excited honey and flapping their open doors spilled it all over the place as they passed.
" _Unglaublich_ ," said Insel. "With you alone am I able to express myself. You tell me exactly what I am thinking. No one else has understood what we understand.
"You have such marvelous ideas—"
"But Insel," I protested conscientiously, "I have touched on my ideas so lightly— If I knew your language well enough to convey the subtlest shades of meaning—."
We decided to get a first-class dictionary. Henceforth _nothing_ was to be lost!
Summing it up, this unspecific converse whose savor lay in its impress of endlessness has left me an ineradicable visual definition of Insel with his whittled exterior jerking in tics of joy a pate too loosely attached and almost worn down to the skull—and myself expansive in some secondary glow from that icy conflagration strewing gray ashes over his face as it burnt itself out. Always at an instinctive interval of shoulder from shoulder, as two aloft on the same telegraph wire exchange a titter of godforsaken sparrows.
As night drew out—it got draftier and draftier— we removed, as if receding into a lair, from the terrace to further and further inside the cafe, from the open to the enclosed—each time ordering a new _consommation_ from a different waiter—till we reached an inaereate core of the establishment. Here we inexplicably came upon that friend whose hypothetic non-existence insured Insel's vaunted isolation. One after another the same Germanic wag would shuffle up to our table, each time wearing a different face.
One—projected that declamatory arm which in a certain condition present at the time falls with a forgetful plop before completing an indication. "Who is Insel," it challenged, "to monopolize this perfectly fascinating woman?"
Another—equally appreciative until he discovered the hair in the shadow of my hat to be undeniably white—apologized with a shudder, "I won't say it doesn't look all right on you—but I can't bear the sight. It reminds me that _I_ am old." He looked less old than Insel— He was one of the many unfortunates who have had nothing to "give off" but the bubbles of adolescence, whereas Insel's rattling pelvis was trotting the leather seat in the sitting leaps of an exuberant child.
"They are so surprised," he chortled repeatedly. "They are accustomed to seeing me _all alone_ —."
I ordered supper—got cigarettes at the counter and dumped them on our table on my way downstairs to buy some rouge (probably on a cue my subconscious had taken from my critic). When I returned it looked as if the empty space in our quiet comer had come alive, the leather padding had broken out in a parasitic formation, a double starfish whose radial extremities projected and retracted rapidly at dynamic angles.
It was Insel all cluttered up with his "private life." Draped with the bodies of two negresses, spiked with their limbs. They seemed, out of ambush, to have fallen upon him from over the back of the high seat. The waiter had laid a startling oblong of white cloth which knocked the milling muddle of polished black arms and faces round Insel's pallor into a factitious distance, although he and his mates were actually attached to my supper table.
The group being occupied it was difficult to know how to greet them. I swept an inclusive smile of welcome across them as I sat down and the waiter brought the food.
As I watched this virtually prohibited conjunction with a race whose ostracism "debunks" humanity's ostensible belief in its soul, I scarcely heard the scandalous din they were making; these negresses, with their fingers of twig, were tearing at some object—my scarlet packet of "High-Life"!— rapidly becoming invisible under Insel's touch—he clung to it with such constrictive tenacity, he might have been squeezing an atom.
_"Maquereau!" "Salaud!"_ shrieked the dark ladies to stress their pandemonium accounting of benefits bestowed.
"Insel," I addressed him authoritatively, not dreaming "pimp" and "skunk" were almost the only French words familiar to the poor dear, "if you could understand what they are calling you—you'd let go!"
Once more fallen sideways off himself like his own dead leaf in one of those unexpected carvings into profile; a zigzag profile of a jumping jack cut out of paper from an exercise book; shrunken to a strip of introvert concentration blind as a nerve among the women's volume, clenching his gums in a fearful sort of constipated fervor, as if hammering on an anvil, Insel thumped his closest negress with an immature fist. Every thump drove in my impression—as this black and white flesh glanced off one another—of their being totally unwed—that Insel, whom I often called _"Ameise"_ who was even now like an "ant," occupied with his problem of a load in another dimension, could never have worked on those polished bodies than with the microscopic function of a termite—unseeing, unknowing of all save an imperative to adhere—to never let go. He clung to my cigarettes conscious of nothing but his comic "tic."
There were onlookers peering under the brass rail topping the back-to-back upholstery—three heads left over from the crowded hours. One, the sharp mask of a Jew worn to a rudder with centuries of steering through hostile masses, lowered its pale eyelashes on the neighbors' insurrection as if closing a shop.
I paid the waiter, bought some more cigarettes, jumped into a taxi, undressed and went to bed, all with the delicious composure Insel instilled—not questioning the continuity of this "elevation of the pure in heart" even while he in whom it originated was being slapped by inexpensive harlots on their way home from work.
I was falling into blissful sleep when a hopeless S.O.S. vibrated on the air—an S.O.S. that sounded like a sobbing " _sterben_." I started up in horror of my selfishness. What could I have been thinking of to leave that delicate soul to his longing for suicide on the contemptible grounds that I was sick of the racket he had been causing.
What would he do, on emerging from a dimension where a packet of ten cigarettes encompassed a universe, to find that I, his very means of expression, had deserted him. With an aereal ease I must have "caught" from Insel, I threw on my clothes and more or less floating into the street together with the presage of dawn, the hoses of the street cleaners slushing my ankles, hounded by my ever growing obsession that Insel held a treasure to be saved at all costs. Damp and heroic I arrived at the Dôme. The piebald mix-up had disappeared.
"What happened to that skeleton I had with me an hour ago?" I asked the majordomo. "He got into a tangle with some negresses— Was he all right?"
"Oh, perfectly," he protested as if within his reach nothing could possibly go wrong. "You see, madame," confidentially, "the fellow lives off these women of the Dôme; there's bound to be a scrap every now and then!"
# **10**
"—ONE WHO HAS GREATLY SUFFERED," I WAS astounded to hear myself telling the man—like a nice old maid with illusions—in precisely the somber tones of Insel's "patroness drive." Equally astounded, he shrugged his shoulders.
"You'll find him in one of the little bars round here—he won't be far, madame."
I knew better. I had my own vision of him—it was the rustiness of that nail that haunted me. Or would I reach his attic only after an ebony vampire had sucked the last drop of blood from his corrupted carcass?
Nevertheless, on my swift passage I caught sidelong sight of Insel standing disproportionately at the end of a row of little men before a "zinc," his head, appearing enormous, shone with a muted gleam.
Without stopping I raised my hand. Insel, although he had his back to me, rushed into the street—he seemed to be continuing to run around.
In his gesture I could see a conclusion of distressful searching in which he had circled during my absence—beating his breast. " _Warum, warum, ist diese frau davon gegangen?_ —Why did this woman go away? I have not ceased to ask myself." Insel complained again and again in miserable bewilderment. "You went away— Why did you go away?"
"Only to fetch something I left at the other café."
Tenderly confidential he bent his neck—a gnarl in a stricken tree—I was about to learn what urgent anxiety had drawn me out of bed.
"There was a waiter," he whispered hoarsely into my hat, "who wouldn't let me out of the Dôme until I had paid for two _cafés fines_." (They had forgotten to include them when I paid for the supper.) "It isn't that I want you to pay me back," he protested with his so distinguished courtesy—."
I always had to order the same drinks for myself as for Insel, or he would not have taken anything— but I made him drink my _fine_. It would, I felt, have superfluous results were I to even sip alcohol in the company of this weirdly intoxicating creature. At the same time in accordance with my mission as a lifesaver, I begged him to take _café au lait_ —which roused a piteous opposition.
As if wound up he went on beating a _mea culpa_ on his absent breast.
I caught him by the arm.
Instantaneously he displaced to a distance. I was left with my own arm articulated at a right angle, holding in my hand a few inches of gray bone. It had come away with a bit of his sleeve, acutely decorated with the jagged edge of torn black cloth. At the same time, Insel laying his hand on my shoulder, the rag and the bone did a "fade-out."
"Promise me to stay here," he whispered, "while I go to the bar. These people would not like it if I did not pay."
Insel, who seemed to remember our pact, wanted to go back to the Dôme. But I refused.
"It's time for you to sleep," I commanded. That persistent teeter in my mind which was always tipping Insel up in a stiff horizontal straight line, his immovable eyes glued to infinity, was laying him out in state on no bed under an awesome canopy of poverty.
"No," I decided, "I shall put you back in your box—my pet clochard is going to lie in a row—under a bridge."
# **11**
WE WANDERED OFF IN SEARCH OF THE SEINE— IT was dawn.
Perhaps this showcase hung outside a _librairie_ was a prison and we, therefore, suspecting an isolation, dissolved its wire caging with the crafty focus of sight to set the content free.
We saw the primeval steam (whose last wisp straying endlessly had wreathed itself round Insel's brain) condense to stone in a frayed torso.
In the darkness it was blind. As the sky broke open, its outline entered the morning gently with the eyes of an animal. As daylight warmed the lids widened to the vision of a pagan.
In conception vast enough to absorb the centuries it survived, now in defiance of time to surpass it—the eternal Thing was looking at us with the fullness of the future. All we had ever understood that was less than itself peeled like spoiled armor.
What enormous foreboding, Insel, in his simplicity, I, in my complexity, recognized in its ideal expression, I cannot say. It was a recognition of something known which, in spite of life, we would know again. Insel, without speaking, turned to me staring at the re-impression of an impression on a book spread out for the passerby we had both, I could see, in identical silence found one significance in an early Greek fragment—I do not remember which.
I have heard that some philosophers assume reality to be absent without an audience. In empty streets the sun had a terrible excessive existence for ourselves alone. We walked together, yet repeatedly, as if having veered in an arc it took no time to describe, Insel would be coming towards me from far away.
"Go back!" he cried in gaunt derangement, "if it disgusts you to look at me." Shining uselessly, as an electric bulb "left on" by day, his face, unshaven, was partially clouded.
We came to a Raoul Dufy in a dealer's window; his charming "crook's technique" disintegrated my meticulous companion. I feared that, the shock reinforcing his perpetual cerebral fit, he was about to throw a physical one. Instead he became covered with verdigris.
We had to relapse at another cafe. Insel disappeared for quite a while.
"Have you been sick?" I asked solicitously. He was looking less green.
"Dufy," he explained.
I put down the money for the coffee and a twenty-five centime piece rolled to the ground.
"Would you pick that up?" I begged. Insel began pulling himself together but did nothing about it, so I picked it up myself.
"Oh, dear," he wailed forlornly. "I thought you pointed to me. For God's sake throw it down again—or I shall never forgive myself," he pled and pled—.
Nothing would induce me to. I foresaw him distinctly diminishing through the hole in the center of that tiny disc and I had to get him to the Seine.
At length we arrived at the gleaming water bearing so lightly its lazy barges with their drag of dancing diamonds. Whatever had been an "under-the-bridge" was all boxed in and the sun had crawled so far into the sky it was needless to look for another.
After that we seemed to be wandering in an aimless delight round and round the Orangerie. Insel's boots were hurting. His pain was impersonal; it might have been following him, snapping at his legs.
With some effort, having breakfasted all night, we conceived the idea of going to "lunch." Insel, who was on the point of allowing the air to lift him from the railed-in terrace of the Tuileries and set him down in the Rue de la Paix, appraised by normal standards, although it was just this "beauty of horror" I was sure should be worth such a lot of money to him, looked really terrifying. His being unshaven became a smoke screen. Always his self-illumination cast its own shadow. In shining he dragged an individual darkness into the world. I felt sure that as the thoroughfares refilled we would run less risk of being arrested for disturbing the public peace on the Left Bank.
"My friend _we_ are not dressed for going into town," I insisted, heading him off in another direction.
"Why?" asked Insel in bewildered politeness. "You look as lovely as you always do."
With a bizarre instinct for scenic effect the hazard presiding our senseless excursion drove us into the Gare d'Orléans.
In the almost gelatinous gloom of the great hall the enclosure before the Buffet Restaurant, its boundaries set by stifled shrubs, offered a stage for Insel to unroll his increate existence to the fitting applause of a dead echo, the countless scurry of departing feet.
This station, as he entered it, became the anteroom of dissolution, where the only constructions left of a real world were avalanches of newspapers, and even these aligned in a dusty perspective like ghosts of overgrown toys.
The place seemed deserted. There was no one to see Insel lay out hocus-pocus negresses on the table in apologetic sacrifice.
"They were _all wrong_ ," he brooded, as if he were a puritan with an ailing conscience. "I was going in the wrong direction!— I renounce," he sobbed hurling off the negresses, who, bashed against the dingy windows of the Gare, melted and dripped like black tears into limbo down a morbid adit leading to underground platforms—there to mingle with the inquietude of departure to be borne away on a hearse of the living throbbing along an iron rail which must be a solidified sweep of the Styx.
"The only thing wrong with those negresses was your beating one of them up!"
Insel denied this vehemently, and reproached me. I had, he said, inflamed their rebellion by smiling at them. That was no way to handle negresses.
"What? You can sleep with them, but I can't smile at them. How do you work that out?"
This muddled Insel, the theme of whose half-conscious theatricals must either be that his beefsteak shared jealous passions with less conclusively slaughtered meat or that prostitutes lay far beyond a patroness's permissions.
"Colored people are not—," he began, looking very Simon Legree.
"But Insel in your relationship she is entitled—"
"I only slept with her three times—"
"If she had slept with you _half a time_ I consider she has a right to everything you possess."
Insel, who had a fanciful ingenuity in extricating himself from any situation he felt to be awkward without very well understanding why, instructed me, "You know nothing of the etiquette of _my_ underworld—its _laws_. The rights of such women extend only to the level of the tabletop.
"It's like this—I am sitting at the Dôme—she comes along—"
"She dropped on you," I corrected— It was fun teasing him. Like tickling a dazed gnome with a spider's silk.
Ignoring my interruption, he continued, "She may take _anything_ under the table—she can grab a thousand francs from my pocket—it is hers. But to lift anything _off_ the t _able—ausgeschlossen!_ — impermissible!"
So exactly the logic on behalf of woman in the normal world that I squeaked, "You haven't got a thousand francs in your pocket."
What matter if we were trivial. We must find some excuse for our unending hazy laughter. Speech was an afterthought to that humorous peace as it fused with our incomparable exaltation. It was ridiculous to find ourselves, alone, in well-being so wide there was room for innumerable populations.
Insel harped back to _not_ having beaten the negress.
"Well," I temporized, relenting, "you thumped her—You did like this," clinching every nerve in my body I tried to imitate that excruciation which in him took the place of a sense of touch— But my fingers closed on an absence—incipience of all volume, Insel's volume. "Didn't you know?"
All he could remember was her stealing my cigarettes.
"Stealing," I exclaimed, "the waiter told me they support you—."
"Everybody," Insel reflected drearily, "thinks I am such an awful _maquereau_. I only had three meals with them."
"You don't have to exonerate yourself," I said dryly, overcome with compassion. "It's quite a feat—being a pimp and starving to death." Then laughing, "Whoever heard of a _maquereau_ without any money!" It made such a gorgeous sound when they were shouting—almost _macrusallo_. Like crucified mackerel—
"They stole my sheets," Insel interrupted sternly, "my six white sheets."
"Six sheets against three meals or three embraces! Whichever way you put it your honor is clear," I consoled him, "All the same, I shall not call you _clochard_ any more, but _macrusallo_."
Insel's luminous duality peculiar to this one night seemed to be forming a more domestic hallucination, an elfin attempt at flirtation, miraculously coy, which played all to itself against the greater glow and measure of his basic disarray—a tacit assumption of our having mutually renounced an inferior world in spite of his repulsiveness being, as he wailed, greater than I could bear.
I had once, to get a simple opinion, asked my dressmaker to take a look at him.
"Well, do you think he's mad?" I asked her.
"He looks so funny," she giggled. "He looks 'in love.' "
She was right, he had the air of being amorous of anything or everything in general which left him so rapt and gentle, or, taking an "inner" view, his astral Venus flowed in his veins. This was why, when he met a woman, Mme Feirlein or any other, he had an approach of continent rape, as if he were persuading her bemused, "See! It must be the more lovely for being already consummate."
For a moment I wondered if his unstaid mind had re-conceived in some unguessable aspect I assumed for him, its eerie durable passion in general—for myself. But apart from the likelihood of his having no idea as to whom he alternately bewailed and beamed upon, I remembered the only emotion I aroused in creative men was an impulse of "knock-out" (that any intuited opposition of the future stirs in the subconscious) which of course was _impossible_ with this delicate soul swimming so docilely along his astral stream under the thunder and lightnings of his distraction like a confiding duck as I scattered crumbs.
At the same time a worn down record of old-fashioned inflection clattered out of Insel's head:
"In spite of all—"
A lesson? A suggestion? A refrain to be taken up?
Instantly I knew this to be a touch-word on which some spring must snap, some wheel fly wild. That, as I watched, something horrible, in anguish, was _wanting to happen_ —a dangerous inertia waiting to be acted upon by some external irritant.
Our lake of peace was draining as Insel gathered himself together for some voluntary magnetic onslaught "in spite of all" had swollen on the air—
Shafts from his eyes became so penetrating I could feel myself dissolve to a transparent target, they pierced me, and, travelling to the further side, stared through my back on their return to his irises.
He seemed to collect electricity from the air (in the afternoon there was a violent storm). This crackling electricity flashed so nearby without attaining to me. It was as if I were _almost_ leaning up against a lightning conductor. I remembered his girl's watch was still in my handbag—it lay beside me—a kind of self-focus in his magnetic field.
He had always something about him of a lithe tree struck by its own lightning.
These magnetic tides would rise and ebb as we sat in felicity around an enormous _plat anglais_ , which I could not touch for my absorption in Insel and of which, as Insel ate of it, the rosy meats seemed to drop uselessly into void. And all the while Insel spasmodically kept up his bum's charade pleading for variable salvations. With his floppy pathos he implored me to take pity on him, to take him in—I would see how I would work with Insel keeping house for me with that precision he exercised in his own dimension—to put him in a nursing home and surround him with angelic choirs of pretty nurses "only to look at," he exclaimed—persuasive or timorous.
I had seen the actor Moisse by the light of a little candle remember some human tie in a prison cell; the humble flame drawing him into itself spread his reminiscent spirit over the callous walls to warm them. Such a candle was burning behind Insel's eyes as if he were his own narrow room. Yet the lines of its rays shining to infinite remoteness—a state of consciousness closing out the world—laid their ethereal carpets along the ceaseless levels of annihilation.
No rock, no root, no accident of Nature varied a virgin plain that had conceived no landscape, and I saw Insel reduced to the proportion he would have in the eye of a God—setting out—unaccompanied, unorientated, for here where nothing existed, no sound, no sun, reigned an unimaginable atmosphere he longed to breathe. I could see this, because he was seeing this, as still hanging back, he writhed to its lure. Although I promised solicitously to send him to a nursing home, we knew I could not come to his aid—. He had never told me _where he was_. His torment tantalized pity.
With that acrobatic facility he had for immeasurable leaps from despair to cajolery—he readjusted himself to the station buffet—as if to get down to some business.
# **12**
HIS EYES NOW PACIFIED IN A STEADY HUMAN mesmerism smiled cosily into mine.
_"An was denken Sie?_ " he asked in coquettish anticipation. "What are you thinking of?" Again I had that creepy impression of ultimate tension, of a cerebral elastic taut for the snap.
"—of you," wheezed the battered record turning on his brain to my sudden visualization of Insel as a gray tomcat having a fit in a cloud of ashes and lunar spangles.
I could not tell him, no thought coincided between one on the verge of dwelling among the levels he laid bare to me and one who remained outside.
Still he went on smiling a little vaingloriously. _"An was denken Sie?"_ he asked again, of God knows what girl, in God knows what decade, and all the same of me.
In my veritable séances with Insel, the clock alone retrieved me from nonentity—thrusting its real face into mine as reminder of the temporal.
Thus I saw how three whole hours went by while Insel asked me what I was thinking of. They passed off in a puff as though, for a change, he had contracted time into intensity.
All the whimsical nonsense ever conceived rotated on his eyeballs which seemed to convey "while I pretend to search for some secret in you the less danger is there of your being inquisitive as to mine."
With every question his eyes grew greater, thrust out longer spears, unctuous in the aromatic ooze from his brain.
"What are you thinking of?" urged Insel, and the softer fell his voice, the more inflexible he knit himself together—the more terrifically to disintegrate on some signal he invoked.
So I sat with as soothing and expressionless a smile as I could concoct and answered occasionally, "I am thinking of June 18th, 1931, or of nine o'clock on Tuesday of the week before last. —What are you thinking of?" His eyes converging on me, a yellow glow fused to a single planetary dilation rapped on the sun gong. _"—An was denken Sie?"_ Insel, discouraged, petrified his face before me—with a determination beyond all human power, in the "last expression" that death imposes on pain. Incredibly exact, rivalling even any original I had seen.
"I should have preferred," he said with his voice of dead lovers crying through the earth, "to be fit for you to look at." Then he deliberately set himself on fire.
In exact description—he did not consistently appear to the naked eye, as a bonfire, in a normal degree of comparison to the morning murk sifting through the glassed environment. As a thread in the general mass, he retained his depth of tone. But as if his astounding vibratory flux required a more delicate instrument than the eye for registration. Some infrared or there invisible ray he gave off, was immediately transferred on one's neural current to some dark room in the brain for instantaneous development in all its brilliancy. So one saw him as a gray man and an electrified organism at one and the same time—
—it was only the candle spluttering... preliminary to the most beautiful spectacle I have ever "seen."
Shaken with an unearthly anxiety, this creature of so divine a degradation, set upon himself with his queer hands and began to pull off his face.
For those whose flesh is their rags, it is not pitiable to undress.
As Insel dropped the scabs of his peculiar astral carbonization upon the table, his cheeks torn down, in bits upon the marble—one rift ran the whole length of his imperfect insulation, and for a moment exposed the "man-of-light."
He sat there inside him taking no notice at all, made of the first jelly quivering under the sun and some final unimaginable form of aereal substance, in the same eternal conviction as the Greek fragment—
Once at dark in the Maine woods, I had stumbled on a rotten log. The scabs of foetid bark flew off revealing a solid cellulose jewel. It glowed in the tremendous tepidity of phosphorescence from a store of moonlight similar to condensed sun in living vegetables.
At last Insel's eyes dying of hallucination, stared suddenly into the filtered day. Horrified almost to blindness he complained, " _Es ist zu hell_." He sounded as if deliberately quoting "it is too light"— That did not matter after all the ways he had been "happening."
"So you're starving, are you?" I mocked, exasperated with his total inability to estimate himself. "The greatest actor alive."
As I took him out, Insel suddenly blew hundreds of yards ahead. He was pirouetting perplexedly around himself when I caught up with him and we got into a cab.
In that small space he behaved like a fish on the end of a line, like a kite in the air entangled in its own tail—carrying about with him, in his awful unrest, my hand to which he clung—his own had clamped so fast to it, he could hardly get it off—when I dropped him at his door.
# **13**
I WAS READY TO LEAVE FOR SAINT-CLOUD WITH my little valise when there came a soft knocking on the door I was about to open, a knocking irreal as the fall of dusk. Insel had turned up again. He collapsed before me like a stricken gull having received some unavowable hurt in the unknown wastes where he belonged. The storm must have completely disintegrated his exceptional electrification.
_"Um Gottes Willen_ ," he panted almost inaudibly. "I cannot eat, I cannot sleep, and now my heart is ceasing to beat."
It was remarkable he should succeed in speaking—his body no longer showed much sign of life. He might be using this body—with its interwoven identity of the living remains of a dead man and the dead remains of a man once alive—as a medium, from a distance to which his fluctuant spirit had been temporarily released.
His face having lost its bruised appearance was set in the tidy waxen consistency that makes corpses look like sudden dolls.
I might be entertaining a ghost, so light a labor I found it to draw him through the glass doors into the studio. As I dropped him onto an enormous couch, my everyday self broadcast a panic.
"Mrs. Jones, her daughter having sailed for New York, is discovered alone in her flat with a dead tramp."
Briefly I thought of _blowing_ the thing out of the window. The seeming imminence of his death allowed for no other means of getting rid of him. But this was no solution. He would be found—sprawled in the courtyard.
Then it came to me that in spite of my willful descent into a forbidden psychology, I still had sufficient power to put him to rest. "Insel," I inquired, "can you hear me?" Then—very slowly—very distinctly, "You are going to sleep—sleep till the blue moon."
There surged out of Insel a whisper of horror, "But my heart isn't beating," he protested.
"That is only a neurotic illusion," I consoled him, believing I lied.
He lay on the couch and did not die. I began to arrange for a possible revival. "Is there anything you can drink that isn't alcoholic?" I asked him quietly.
After a while he murmured, _"Pfeffermintztee."_
"Try to be alive when I come back," I urged him in all sincerity.
"Where are you going?" he wanted to know, his voice a hoarse agony.
"To buy _pfeffermintztee_."
"No—no—you're not to leave me— _ever_."
With a strange grip of a limp vice his fingers clung to my wrist. I had to sit down beside him. Now he was staring as if bewitched, at the parquet of the floor.
"The peppermint won't grow out of the floor," I advised.
"It will," said Insel. "You're to stay here."
And I found myself staring together with him.
It was no peppermint growing out of the planes of polished oak. Only the creeping organic development of a microscopic undergrowth such as carpeted chaos in his work, almost as closely cramped as the creamy convolutions of a brain. Foliage of mildew it spread—and spread.
"An infusion of that fungus would be bad for you," I persuaded him, taking his fingers carefully apart, lifting their tentacular pulp from my wrist.
Escaping I rushed to the shop at the corner and back again. As Insel was still living, I made him his tea.
"And you will be able to sleep," he reproached me, oblivious of his drowsiness as he fell asleep.
I could watch over my invalid through a pane of glass incompletely covered by a curtain on the door at the far end of the studio.
A dense oppression stole through the flat all packed up in its iron shutters. Insel, who had no longer been able to bear a light, lay pallid and obscure in a faint reflection from a lantern in the hall, his slumber the extinction of a dim volcano. Lax as a larva, a glow worm "gone out," his head bared of its phosphorescent halo, seemed swollen in a meaningless hydrocephalus. As if, while conscious, electric emissions had diminished his cranial volume.
Around him the atmosphere was stale as an alcohol preserving a foetal monster he resembled in repose.
Insel was unpleasant bereft of his radiance.
His body had dwindled in distilling an immaterial essence to such concentration it was appreciable to the senses. One was aware of an effulgence, which, if it waxed and waned during his waking hours, had now altogether vanished. His body swept and garnished like the house in the Bible—devilishly invaded—was no longer human as it lay before me in the form of Insel.
There unshining and supine he seemed abandoned of all quality except the opaque. The gray inflated opacity of his unseeing head, which, should one lift it from the pillow, must surely loll on his shoulder—the head of an idiot.
The flat seemed emptier for his being there, until I found that further off it was filled to a weird expansion with emanations drifting away from Insel asleep. They crowded the air, minute horizontal icicles, with a tingling of frozen fire. In the room at the end of the corridor their force of vitalized nothingness was pushing back the walls. Why should Insel, less ponderable than other men, impart perceptible properties to the Air? Was he leaking out of himself, residuum of that ominous honey he stored behind his eyes into which it was his constant, his distraught concern to withdraw?
In his soaring, flagging excitations he might have spent a spiritual capital and going broke, be raising exhaustive loans on the steadily decreasing collateral of his vitality, until an ultimate bonfire in those eerie eyes should be extinguished in some unimaginable bankruptcy.
In him the intangible and tangible components of a human being had come apart. As if in some ruthless extraction of Supreme Good from a fallible pulp, the vibrancy interpenetrating normal muscular fiber had been indrawn from his physical structure to condense in a point of flame. When some mysterious fuel failed him, Insel remained—a mess of profane dross.
I thought of his pictures, those queerly luminous almost materializing projections. Curious creatures moving in levitation—frequently cerebral abortions of cats.
Any student of ancient occultism would recognize them for elementals. Imbecilic, vampiric—here and there an obsessive absence of a mouth implied an inconceivable constipation. A conspicuous liver, so personal he might have served as his own fluoroscope, clear as a pale coral was painted as only the Masters painted. He had no need to portray. His pictures grew, out of him, seeding through the inter-atomic spaces in his digital substance to urge tenacious roots into a plane surface.
I wondered in what psychic succession these monsters issued from a man, who himself when unlit or cut into profile, became so hauntingly animal, even insectile. Who, when asleep, being the makings of his own bestiary, was vilely void as an incubus—wondered why millenary monsters of a disreputable metaphysic should re-arise intact in a modern subconscious.
Insel slept for twenty hours. With one interruption. When I went in to see how he was I woke him up.
Through the slits in the shutters the outdoor lights laid narrow blades along the floor, above Insel's feet on the whitewashed wall they crossed and cast a double shadow of a hanging fern. Otherwise the room was a mausoleum.
Again I could have sworn I beheld the dead. Silence had hardened upon him in a stony armor, too heavy for the fluttering of breath.
I listened till the sound of his rigidity grew so shrill I was forced to make it mute. Terrified, I took hold of the door and crashed it to.
Insel—who after all must, of his nature, float quite lightly on the surface of a coma—easily lifted his lids.
"I'm sorry to have disturbed you," I gasped. "It was necessary to make a noise to know you are not dead."
With none of the daze of sudden rousing he excused me gently. And slept anew.
Those depths through which others plunge into sleep for him stretched shallow as moisture on a mirror.
In the morning I went out, off into the sunlight, shopping. Leaving far behind me that darkened room, and whatever it contained.
My major purchase was kilos of bright red beef.
When he awoke I fed him chunks from a great frying pan. Insel sat up and swallowed them with fairly bestial satisfaction.
"Why," I asked to make conversation, "do you always want _'Fleisch ohne Knochen'?_ "
I had taken it for granted he ordered boneless meat to avoid waste. But Insel began peering about shockingly as if suspicious of being overheard.
"When I am alone," he explained, in an unexpectedly vacuous voice, "I do not eat like this—I have to drag _bones_ into a corner—to gnaw."
I felt curious to know how—without teeth—. But Insel beginning to shine again put off the animal, to become the clown of an angel.
Through the row of glass doors the ornaments in the hall looked like fish under water as a celadon tide of pale lamplight sluiced into the studio. From the shutters on either side, entangled reflections flickered into the halo that was now re-forming round Insel's face.
Stark on the _sommier_ he floated up from the floor of a pool with the wavering fungus he had sown there clinging to his cover.
He told me he had found the secret of perpetual motion if only he had the money to buy the stuff. To me it seemed he had rather discovered a slow time that must result in eternity.
I told him I had for some while been conceiving a ballet.
"It is the story of a maiden seeing her life in a crystal— It would look exactly as it does here, everything translucent." I waved to Insel— "Yet as in the days when there were maidens they had no 'life,' what she sees is her future spouse sowing his 'wild oats.'
"All dancers are terribly ponderable after Nijinsky—yet once I came across one who possessed a dual _equipoise_ which threw him into a huddle with himself. That is how my youth would dance, with the wild oats springing up to the moon around him, whichever way he turned— But I should have to do maquettes—animated maquettes of the choreography—and _I_ can't make _anything_ grow out of the floor," I said deferentially.
"Of course he makes love to everything. A cocotte's eye. The woman in the litmus petticoat forecasting the weather. A rainbow," I continued, seeing Insel entranced. "The Queen of Fairyland— Mermaids and Medusae." Envy was stealing into Insel.
"I dance divinely," he said and I could see him crossing a ballroom floor propelled as if on invisible casters, as truly initiate acolytes, in reception and remittance of the Holy Book before the high altar.
"Always at the crucial moment the youth is intercepted. There comes floating in between him and the object of his concupiscence, a—" I stopped, as Insel, seemingly relieved by the frustration of a rival, closed his eyes, and waited till he came to. "Over and again I drop the idea in despair. Over and over again I find a solution so simple it constantly slips my mind. I have only to make some little people about five inches high and tell them what to dance." Insel nodded comprehendingly. "Yet whenever I get to work I come upon some fundamental obstacle. It takes me _hours_ ," I complained to Insel, "to remember it cannot be done. It is as if at the back of that memory stands another memory of having had the power to create whatever I pleased."
Insel's eyes enlarged in a ruminative stare. The stealthy oncreep of his visual lichen had reached the walls. We had no longer need of larynxes to converse. Insel _thought at me_. More precisely— vaguely conceived before me.
"To make things grow," he conveyed on his silence, "you would have to begin with the invisible dynamo of growth; it has the dimension of naught and the Power of Nature. As a rule it will only grow if planted in a woman— But _my brain_ is a more exquisite manure. In that time in which I exist alone, I recover the Oceanic grain of life to let it run through my fingers, multiple as sand."
Then the silence of Insel took on voice once more—A voice which as if returning from diffusion among the mists—might be coming from "anywhere," resumed his ever recurrent cries of horror on behalf of women who could no longer love him.
"For God's sake," I implored, for Insel returned to his "normal" state, I followed suit— "stop agonizing— Go to sleep— To negresses _every_ white man looks—white."
"It's the teeth," he groaned— _"Die Mädchen—"_
At least you'd have more chance with the girls if you got Bebelle to clean your suit—
"I'll tell you what," I said, overcome by my inherent conviction of personal blame for _anyone_ not being able to get _anything_ they want—. And in Insel it seemed his need was for something so sublime that over all his aspirations hovered crowns of glory— _Mädchen_ —something entirely outside my zone of attraction, in his regret for them, took on enchanting attributes—even those in a mountain village who ate such quantities of garlic it breathed from the pores of their skin—so much so that Insel, with the heartiest will in the world, had found it impossible to "hug them close enough."
"You'd better get _into_ that couch and leave your suit in the hall—when I come home I'll throw it into a bath of gasoline."
Insel was horrified. "I don't want anybody to see the dirt in that suit—let alone you—I've worn it for five years."
"All I shall see is the gasoline go dark—it would seem just as dark if I were cleaning something that had only been worn six months."
But Insel was actually writhing in a bitter determination to protect his own.
"Are you afraid," I asked, in a sudden concern for his "rays," "that it would interfere with your _Strahlen?_ —I'm not going to wash it. You can't short-circuit."
On the contrary—I anticipated him distinctly renewed in an intenser radiance—
"Please," I begged—enraptured as a nun seeking permission to lay fresh lilies before a shrine. " _Ich bitte Sie_."
He was obdurate—it would seem, in shame. It did not occur to me that in cleaning him up one would be cutting a slice from his "beggar's capital."
"It's not distinguished to be ashamed—"
Insel, in a way, gave in—.
"You try it," he warned me. "Before your eyes the suit will turn white."
"It won't, or if it does, I'll turn it black again."
"You may clean it forever," he intoned ominously— "the while it grows whiter—and whiter."
_"Mädchen,"_ I reminded him for bait— "or at least," as for an instant Insel's ravaged features showed through his ennobling aura— "better negresses."
Insel was pacified. But he did not go to sleep. He evaporated.
I recognized a vapor whose drifting suspension of invisible myriads he copied so passionately with the overfine point of his pencil.
When it cleared off it had left him again an effigy straightened as the level of water.
The world of the Lutetia had materialized. An infiltration of half-light softly bursting the dark, a thin cascade, the ferns dripping into a green gloom. Here, where dawn and noon and midnight were all so dim and Insel lay sensitive to clarity as a creature of the deep sea; the closely shuttered studio with its row of glass doors was a real replica of the irreal "aquarium."
Because I found the place somewhat chilly when sunless—I had thrown a great white blanket over my thin dress. This was due to no obsession for Insel's white miracles. Simply, everything being put away in naphthalene, this had returned from the cleaners and the _femme de ménage_ had not yet locked it up.
Fairly inflexible—it curved around me loosely, encaving me—its stiff corner trailing off like a sail.
I sat on the edge of the couch at the feet of that rigid flotsam—in a huge white shell.
Again I received a strictly lateral invitation to wholly exist in a region imposing a supine inhabitance. A region whose architecture, being parallel to Paradise, is only visible to a horizontal gaze. Should one stand up to it, it must disappear.
Somehow, unable to dissolve into mist, and thus too dense to enter a mirage, the nearest I could conform to the arid aquatic was in becoming crustacean.
Being an outsider did not interfere with my participation in the ebullient calm behind Insel's eyelids, where cerebral rays of imprecision, lengthening across an area of perfectibility, were intercepted by resonant images audible to the eye, visible to the ear; where even ultimate distance was brought within reach, tangible as a caress.
As all this "lasted forever" it seemed incompatible that Insel should slump back into a larva. Yet there he was—extinguished again in unregenerative sleep.
I turned to go. A scatter of objects on the table attracted my attention. Among some weary sous and tiny strangulated worms—broken shoe laces—lay evidently the bone of some prehistoric fish. A white comb shrill with the accumulated phosphorus of the ages. Insel had emptied out his pockets.
I went about my cleaning. Ordering several _bidons_ of gasoline, I poured them into an enamel tub, and suspending the suit by a wooden pincers, I dipped it in.
The sun was shining, the kitchen blazing white. Before the open windows that which seemed most substantial about Insel, like a corpse let down from a gallows, fell to its knees in the volatile fluid.
Then the ghastly thing began to turn pale. I set upon it in opposition and that white contorted outgrowth of a brain almost tangled in the whisk-brush—.
Had he really intended as much in his challenge—or did this Polar region of a mania— these maps of Himalayan anthills with their scabs of pure vegetation embossed upon the backdrop of his clothes, depend for their pictorial clarity on some accord between his cerebral vibrations and mine?
Being tired and bored, I went in to see if in exchange for some more food, I could make terms with Insel.
He appeared uneasy. Rolling his eyes like runaway wheels spoked with interrogations. His expression was such as I had never seen. Terror solidified.
"I am a prisoner here!"
It did not recur to me that his classic complaint is an echo in the corridors of asylums.
Tearing my visionary trappings of meat and such from me with his flippant accusation, Insel did not at all want to know, "When shall I see you again?"— He had never seen me before.
My everyday self shuddered— "Blackmail! Almost as awkward as dead tramps." I reflected, but I had become so nicely attuned to Insel's moods that my parasitic clairvoyance, of its very nature, being constrained to see eye to eye with him, immediately veered to his viewpoint, I concluded I must in a temporary aberration have kidnapped this gaunt guest whose snarl was unsociable.
"Beefsteak," I quavered, as if enticing a surly hound.
Insel, completely jammed between infinite walls, was not having any steak.
I must dislodge his attention.
Seemingly at hazard my dilemma linked up with one of the kind of infantile anecdotes Insel always greeted with glee.
"Have you heard about the Hungarian immigrant lost in London?" I inquired as engagingly as I could. "He wanted to find his consulate and could not understand why the policeman only shrugged his shoulders when he explained he was 'Hungry.' "
Strange how unerringly the unconscious picks its way. I had "found" Insel for himself again. To the Titan of Hunger—the policeman's shoulders heaved in the shrug of all humanity ignoring Insel. This recognition shook him with the most sophisticated laughter I have ever heard in my life.
"Your suit has turned white," I announced.
A gleam of crafty assurance stole into his transparent eyes.
"You will never 'get out' while your suit is white," I threatened, "all _die Mädchen_ are on the other side of the _wall_ —"
"Oh," said Insel with a conciliatory smile, "I only want them to look at."
"Well, they won't look at you until your suit is black; and as we're about it you'd better let me clean your shirt."
His shirt was of a dark gray design rather mellow. When I suggested renovation he clutched it by the open throat.
"See," he said, lifting it with a cautious yet ostentatiously offhand gesture, "the neckband is worn ragged inside—it's not worth it." He was cowering in some apprehension that constricted him, that even devitalized his hand. Become as the hand of a victim of infantile paralysis, it flopped over with the edge of the stuff. He had an air of shifting—just so far—the bandage of a wound.
Not for the first time, with Insel, I received a subliminal flash of an apostate Saint Sebastian writhing with arrows—in such privacy, it would be indelicate to intrude upon it with whatever assistance.
On looking back, it seems inconsistent, that once the elation he inspired in me died down— I should have continued in my obsession of conserving something very precious with an Insel changing to an incubus, playing his silly psychic tricks on his clothes—raving of imprisonment and the gnawing of _Knochen_. It had left me with the solicitude one might have for a valued friend with whom one has been on some glorious drinking bout, when he shows up next day at a disadvantage in a particularly nasty hangover.
One last struggle with the suit—and it turned black again. Insel must have forgotten about it.
# **14**
"BREAKFAST," I ANNOUNCED.
This time Insel did not stir.
His head, although returned to normal volume scarcely indented the pillow. He was set in the perfect quadrates of a couch, having no rumple anywhere. As he lay upon it without taking contact with it, the comfortable bulges of covers tucked under a mattress sharpened to corners of trigonometric exactitude.
The smoothing systemizing vibrations that straightened his surroundings, obviously did not issue from his frame, which had half-died for contributing vitality to some focus of force.
Perhaps they were transmitted by his hair. I have always presumed that hair with its electric properties will not remain unutilized in a future evolution of the brain.
His hair—what little was left—was so fine, that without amalgamating, it had the unity of surface of the horny plate with which hair furnishes the extremities in its aggregate form of a nail.
Tentatively—I touched that hair, repeating "Breakfast" on a cheerful note—to appear as if I were patting his head to wake him up.
In a decreased microscopic degree, my fingers encountered the same onslaught as had my whole person in the corridor. A sharp crackle of inconceivably minuscule machine guns carried to some psychic center of my ear.
The effect was astonishing as when I had tapped him on the arm. Insel did not awaken—he turned his head as if he were pushing it up into strata of delight above him. Which on contact melted upon his face in a slow smile.
He was smiling as if the tip of the wing of an angel had fanned him.
Again, as I watched, I had the sensation of "breaking point," an expectance of a spring flying loose to whirr insanely.
His face, like stale bread smeared with his private honey, stood still.
Then it broke.
With the unforeseen ugliness opening up suddenly emerging hippopotami _the gums_ in their hideous defenselessness _observed_ me—an obscene enjoyment of ill-will pleated his clamped lids.
His teeth had not decayed. They were _worn_ down.
_Der Totenkopf_ hung in my tract of vision like the last of _Alice in Wonderland's_ Cheshire cat.
Getting in touch with Insel was the whole itinerary of Good and Evil.
In the passing away of a miasma Insel awakened. Although never much the better for food, his temperament having relieved itself of some disproportionate impulse in that monomaniac gape, he now seemed normalized.
It was a serene creature who began to breakfast. Whatever introspective conflict usually engaged him, it had ceased.
"You really look rather well now. Why don't you just stay and have that rest cure _here_. I'll hire Bebelle to feed you—do everything for you while you lie down and drowse till you're quite fit. I _must_ get back to Saint-Cloud."
"Impossible," moaned Insel, instantly sagging, "I have to return to my troubles. You do not understand. They are my life. It waits for me."
"Nonsense, you spent the night in Montparnasse in one incessant gurgle of laughter."
"It was a _hollow_ laughter," he intercepted, sepulchrally. Insel had resumed his "line" which seemed so inadequate.
Should I risk an attempt to reveal to Insel those real-essences in Insel? Real-essences to a slight degree rationalized for my mind, they might be either the very symptoms of the so-called madness in him, or precisely the incognizable cause of his befuddlement.
"Insel," I set out determinedly. "You must get over your ugliness—it's an obsession! That's not all there is to you—you have some intrinsic quality I have never found in anyone else. It's difficult to tell you about it because I have no idea what it is. But it's something so valuable it's one's duty to keep you alive to discover its nature."
"Several alienists have offered to examine me— regularly—" said Insel, with self-complacence, "twice a week!"
"It's not pathological—only unprecedented. A kind of radio-activity you give off—. Insel," I asked puzzled, "how does the world look to you? Like an Aquarium?"
Insel looking no less puzzled than myself, I was taken aback. But I went on in the hope of striking common ground.
"It was the evening outside the Lutetia I experienced its effects. A sort of doubling of space where different selves lived different ways in different dimensions at once. Sitting on the sidewalk—floating in an Atlantic Ocean full of skyscrapers and ethereal cars. That was not particularly important— the wonder was the sense of timeless peace—of perfect happiness—"
# **15**
INSEL SAT BOLT UPRIGHT IN HIS COUCH AND LET out a thin screech like a mad cat; looking exactly as if he had caught a mouse he had watched for a long time.
"No." He wagged his poor bald head judiciously, " _It cannot be_ —I can only love forever."
I gave one gasp—then as always when taken unawares, my mother reproved me from my subconscious—a sophisticated middle-aged woman making immodest impressions on an innocent _Schlosser's_ son.
"You misunderstand. I had thought of you as a 'Will-o'-the-Wisp.' "
Insel took no heed, he was practically licking his chops. Quite as if it were an impulse habitual to me, I decided to slug him.
Then he began moaning again—of suffering, which one moment, he could allow me to share, and another, he refused to cause me.
"It would be too fearful for you—the Parting," he pointed out. "You see," he confided affectionately as if promising me a present, "I am going to get her back—"
A spiral craftiness wormed into his eyes as I asked, "Where is she?"
"In South Africa," he answered with some impatience, as if I should have remembered.
This girl in her role of "only beloved" was almost as unsettled as Insel himself. Only yesterday she lived with her Lesbian in Berlin—and now, "Since she left me she has married twice and borne four children." Before very long she actually split in two—
" _Es war eine Schwartze und eine Blonde_ —"
"When was she black and blond?" I exclaimed, intrigued.
"Last night, outside the Select—I saw at once they had fallen for me," said Insel, ignoring he had been fast asleep.
"But Insel," I laughed, "can't you remember how terribly miserable you are because you frighten the women?"
"All that is changed now," said Insel looking me over with sadistic compassion.
Quite forgetting my determination to slug him—I glowed with the satisfaction of a successful psychiatrist—"I have cured him of his fixed idea—" I congratulated myself—
Then with his lightning variability of mood, his eyes diluting in a difficult introspection,
"Outside the Lutetia," he pondered wonderingly. "That's funny. I had _exactly_ the same experience."
"You couldn't," I was about to retort, "it's not in Kafka," but checked myself, wishing to keep him on the subject of his radiation.
"That's why I adore talking to you—why I cannot allow you to suffer for me. I know too well what suffering is,"—and suddenly he threw up his head. The almost mummified chords of his throat vibrating in an ecstasy.
" _Die Liebe—wie schön—wie sch-ö öö -n_ —Love— the one beauty of Life." He gloated with the same singing inflection with which he had been wont to celebrate steak. There _is_ nothing else, he concluded. Evidently he was sane as any man in his therapeutic measures for saving woman from vain regret.
Without transition his fixed enchantment turned to a staggering stare. " _Die Liebe?—It's_ the _Strahlen_!" he hooted across to me in the haunted voice of the obsessed.
"Insel," I urged, bewildered, "don't look like that. Your _Strahlen_ are evidence of something in you—something noble."
" _So edel_ —" I trilled in remembrance of my contact with that flawless spirit. But as if leaping out of himself for once to take stock of an Insel I did not know and as if what he saw was horrible, Insel took that clear, that soaring word and wrapping it in bitterness, hurled it at himself.
_"So edel_ ," he echoed, infinitely disabused.
"At all events," I said as a pleasant jolt— "I am going to bring you your suit. You're going to look so fine."
As I passed the table I missed the phosphorescent bone.
"Hadn't you a comb?"
"It's here," said Insel—stretching out a skeletal arm toward the floor—there stood his shoes. In one was stored the white comb. The other was stuffed with a huge white handkerchief. They were his wardrobe.
A warm appreciation stole around my heart for that adorable domesticity of the tramp, which first attracted me, when in my childhood, a clown, taking off his tattered overcoat displayed a wash-hand stand built into the lining.
At that moment my friend Insel was very dear to me.
Then in a sudden I realized how always, and inevitably in attempting to follow it, I must run off the track of Insel's mind—himself unaware that nothing about him could ever stir from a so-mysteriously-appointed place, Insel had retrieved his comb _pour se faire une beauté_ , awaiting my arrival with his breakfast!
With a kick of tiny annoyance at a toe of his wardrobe, "Personally, I do not admit the power of the microbe, but if you do—I fear you'll be poisoned," I warned him.
Then I gave him his suit.
When I came in again, Insel was pacing the studio in stealthy meditation. His mischievous assurance was so much his axis and at once so exteriorized that his whole implication seemed to have contracted to the finger of old fashioned comedians pressed to a nose under a crafty eyelid—Insel was feeling so sly— Then, going into reverse—as it was time for him to leave, he began fiddling abstractedly with a gratitude he did not know what to do with. "You have been very good to me," he mumbled shamefacedly. "There is nothing I would not do for you—if ever," inspired, "you have a pair of boots which need cleaning."
"Insel," I exclaimed encouragingly, "you needn't say things like that about yourself."
With a jerk he pulled himself out of an underlying complex.
"That was a figure of speech." And inclining towards the couch with the bowing, palm-of-the-hand-drooping invitation with which saints in primitive pictures lead the eye to some sacred center, "Now you be ill, and go to bed so I can nurse you," he pleaded adoringly.
I had to refuse. As I came to think of it, I wouldn't know how to be nursed when ill.
So Insel, as if in prison or barracks, began folding up his sheets and blankets, I took a seat. With the stuff of my cape draping the chair, I felt like an emperor taking pride in a supreme buffoon.
There is no grace on earth to compare with a willowy man afflicted with levitation.
"It's pure selfishness my allowing you to do this. It's up to Bebelle—only I do so enormously enjoy your plastic geometry," I observed to Insel, who, as if fitting a label to perfection, swayed his dreary silhouette of aereal bones, against a lifted sheet bleached in the reflection of his phosphorescence.
"If you want to make a fortune," I advised him, "you should go on the Music Halls— Have you ever heard of Baggesson—one of the geniuses of the century? He broke white plates.
"You are even more wonderful folding white things up—"
"It would be utterly useless," Insel protested. "Nobody ever sees in me what you see in me—"
"Well, you frighten the 'people' out of their wits, that ought to give you a hold on your audience. Of course, you'd need to rehearse— Have someone sit in the back of the theater and tell you _where_ you get your effects. You should 'come on' in the fearful chatter of an earthquake and then all you'd have to do would be to leisurely tidy it up— I assure you you'd have the whole theater hallucinated."
When he had stacked up his covers like a deck of cards, there was still one ceremony to perform, I took him into the kitchen and gave him whatever food there was left.
Under my eyes, as he packed it up, it diminished and froze into a Chinese puzzle. The essential, he said, was a minimum bulk. It did not in the least concern me that it would all be thrown away. His tying of the string was the close of a linear symphony.
Insel left with a farewell flash from his cranium and his forlorn-howl-in-the-wilderness of when shall I see you again— Then he crept back to the doormat and whispered shyly, "I shall explain everything to you next time."
# **16**
ON MY WAY TO THE STATION I CALLED ON MLLE Alpha. In her slacks of rust colored linen—her coppery hair, blown into a fresh sunburn, she appeared to have just sailed in from a lagoon.
Her eyes like coals, continent, of their fire, were round as the eyes of the wooden negresses supporting the violet draperies of her day-bed. Her lacquered toenails played at hide-and-seek among the meshes of her sandals. Her whole body was impudent with a slightly crass adolescence; it centered in her little tummy, which dared to be round.
A hard young apple—it was immediately plain to see, how, had one been on the other side of the fence of sex—one would have wanted to bite into it.
It was Insel who had sponsored our meeting and I gave her his message—that he would keep an appointment at five o'clock.
"He's enthusiastic about you," she said—then, "Would you think me very indiscreet if I asked you what you find to talk to him about for six hours?"
"Oh!" I explained loyally, "we exchange our little anecdotes. There's the girl who went off with the Lesbian—it's stupendous—to halt the endlessness of drama in the mere contemplation of a couple of shots."
"So he tells you that one, too?"
"Look here," I confessed. "At first I was indignant with you for launching the opinion that Insel is mad—. Now I am not sure—. It occurs to me that I can't even make out what sanity is."
"Well, I find him such an awful bore, I am constantly having to turn him out—"
"That's because he's too surrealistic for the surrealists."
But when Mlle Alpha spoke of his work, it was with a profound veneration I could hardly share.
"I'm not so fond of elementals—I find that strata in the subliminal thin—. I know his work is a technical miracle and I submit to the active hypnosis with which he has the power to infuse dead paint—still—. There! That's one thing we're always talking about. His future work. He _shows_ me what he is going to do. Sometimes I feel he has found a short cut to consummation in defiance of the concrete. That he is filling the galleries of the increate. He seemed so worth helping, I've only just begun to notice he _never_ paints. If he ever does paint the things he sees—God knows _where_ —the result will be spectacular."
"Why? Haven't you heard about Insel?" asked Mlle Alpha. "He and the friend with whom he came to Paris took morphine together and two years ago this friend died. His death gave Insel such a jolt, he dropped the drug, and ever since has painted nothing of any account.
"Why on earth doesn't he take his old morphine?" she demanded of the universe in general, "and let himself die? At least he would have painted his pictures—while this way—where is the good in his remaining alive?"
Now drugs meant nothing to me. I had supposed they were a substitute for imagination in the unimaginative. I was prejudiced against the stories afloat of their awful destructiveness, ascribing them to one of those official dodges for preventing an exasperated humanity from having a little fun. Subconsciously, I waived this information. As if my mind were a jury refusing to be influenced by extraneous evidence. Being thus luckily prevented from putting two and two together, I was free to pursue my investigation of Insel in my own reactive way.
Moreover, was not Insel's morphinism a thing of the past?
# **17**
BEFORE I LEFT MLLE ALPHA TOLD ME THAT STIFF Ussif, the surrealist, had painted a picture I ought to see. Remembering (that under the influence of his feline screech) I had made no appointment with my strange boon companion, I arranged to go to Ussif's the following week on my return to town.
"By the way"— I exclaimed, "I forgot—. When Insel wrote to you—did he predict the day and hour when his resistance must give out—?"
"Nothing of the kind," she answered. "He wrote as usual, 'I am starving to death.' "
When the time came for me to return I arrived to find a telephone message from the dressmaker, who was ill. So I hurried off to do some shopping. Afterwards, on my way to the surealist's studio, I stopped the taxi at my flat to change my gloves.
As I ran up the one flight of stairs, I had to slow down. Surprisingly, on this warm day, an iciness was creeping up my ankles. I proceeded into a chill draught.
"Insel!" I realized.
There was nobody standing at my front door.
Although well lit by a staircase window, it was hung with a square curtain of black mist.
Slowly, this mist put forth an abstract sign of concavity, and still more slowly, a transparent diagram of my friend grew on to it.
Hunching into materialization, as a dead man who should vomit himself back to life, Insel, whose illness was dissolution, moaned to me in the voice of a wraith.
"I thought you would never come."
When I got him inside, we were already laughing—half apologetically—as if we found it absurd, this meeting in no man's land without explanations to offer.
"Why didn't you say you were coming?"
"But I thought—surely—" with an anguished grin, "Friday is my 'little afternoon.' "
"Of course it's your little afternoon, Insel," I laughed. "Only when you have turned the lady down is just when you have to specify the hour of your return; if she is to expect you—I've got an engagement."
"My little afternoon," he raved, collapsing, "I was going to take you to my room to see my picture."
# **18**
WHAT AM I TO DO WITH YOU? THE TAXI METER IS ticking, the surrealist's waiting. Pull yourself together—quick! I'll take you along.
"However did you get that hole in your trousers, it's new—" I demanded, detecting, as we got into the taxi, a perfect round of perforation letting out a tiny light from his thigh. I suspected him of replenishing his beggar's capital.
"It was there before," said Insel sanctimoniously, as if referring to a halo earned by excessive martyrdom.
"You might as well come up and see Ussif with me," I suggested.
"No," said Insel, "none of the surrealists will have anything to do with me. They know only too well, if they did, I should try to borrow money."
"I should have thought you'd be _worth_ a little money to a surrealist. He might learn what supereality is about—you are organically surreal—"
"I don't do it on purpose," said Insel dejected.
"I know you don't," I assured him warmly. "You only 'do' Kafka on purpose—you're _so_ much better in the original."
I kept my promise of going to his room on my way back. Strangely—the very name of the street he lived in had the sound of a ghostly exhaustion. His attic was on the seventh story.
Along the narrow open passage with its bare iron railing the _Chambres de Bonnes_ moved past me as I looked for his name on the doors, when, coming to a closed iron shutter fleeced with dust and cobwebs growing in patches like a moss of soot or hanging in gray festoons about its slits, I felt the liveness of the air decrease, and "Insel" written in the archaic hand of some automatic writings drew up my eyes—. To that darkened crack which outlines the magical versatility of a barrier measuring a yard across and with merely the touch of a hand diminishing to a strip three inches wide. That cover of a living book whose history may come to an end before you can get it open; or cut short your personal adventure by remaining shut; out of this oblong outline of Entrance and Exit there leaked a perceptible seepage of Insel's torpor.
Noiselessly, indolently, the door vanished. I walked into its chasm and Insel led me to his painting set in the pacific light of a large attic window.
_"Das ist die Irma?"_ he said with the secretive in-looking twinkle that lit up his eyes with recurrent delights. And suddenly it dawned upon me that one thing about this man that made him so different to other people was that contrary to our outrunning holding-up-the-mirror self-consciousness, his was constantly turning its back on the world and tiptoe with expectancy, peeping inquisitively into its own mischievous eyes. Or, in some cerebral acrobatic recoil, that being who is, in us, both outlooker and window, in him, astonishingly, was craning back to look _in_ at the _out_ looking window of himself, as if there were something there he might forget, some treasure as to whose existence he wished to remain assured, some lovely illusion inside him, he _must_ re-see to insure its continued projection.
" _Die Irma_ ," he repeated lovingly to introduce her to me, and the magnetic bond uniting her painted body to his emaciated stature—as if she were of an ectoplasm proceeding from him—was so apparent one felt as if one were surprising an insane liaison at almost too intimate a moment. He was glittering with a pleasure as dynamically compressed as the carbon of a diamond.
A narrow canvas, nigger-black, whose quality of shining obscurity was the effect of minutely painting in oil on some tempera ground, _die Irma_ stood knee-deep on an easel.
To her livid brow, rounded like a half-moon, clung a peculiarly clammy algaeic or fungoid substitute for hair. Beneath it a transparent mask of horizontal shadow was penetrated by the eyes of an hypnosis; flat disks of smoked mirror, having the selfsame semblance of looking into and out of oneself as her creator.
Perhaps in a superfine analysis, this is what all men really do, but as a natural interplay; whereas Insel and his picture were doing it with alternating intent. Indeed the great thin uninscribed coins of her gunmetal pupils, returning his fascinated gaze, were tilted at such an angle as to give a dimly illuminated reflection of an inner and outer darkness.
Her hands, as if nailed to her hips like crossed swords, jutted out from her body which seemed to be composed of rippling lava that here and there hardened to indentations like holly leaves growing from her sternum—her male hands that hardly made a pair, for the one had the bones of the back marked all of equal length and the other, one finger too long with an unmodeled edge which curved like paper against the background.
He hung over _die Irma_ like a tall insect and outside the window in the rotten rose of an asphyxiated sunset the skeleton phallus of the Eiffel Tower reared in the distance as slim as himself.
Beside the picture I noticed that the gutter of his upper lip was interrupted by a seam, a fine thread of flesh running from the base of the nose to his mouth that accentuated the compression of his lips in their continual retention of the one remaining tooth which, so thin as to be atavistic in an adult, was like a stump forgotten in a croquet ground, left over from the Game of Life. An incipience or reparation of harelip? And Irma? In this very same spot she puffed to a swollen convergence.
"But Insel," I asked, "her upper lip is about to burst with some inavowable disease. You have formed her of pus. Her body has already melted."
"Exactly," he answered with mysterious satisfaction.
"I don't care for it," I decided.
"And I," said Insel, with the reverent intonement with which he accompanied his tacitly implied admittance of myself to his holy-of-holies, "thought that _this_ picture would be just the one that you would like."
Time hovered, suspended in the attic air as the powders of life in the noxious mist of the exhausted city below. When suddenly the soporific lure he sowed in his magnetic field—shattered. Insel was snatching at the emptied flesh on his face in the recurrent anxiety inspiring his wilder gestures.
"She ought not to be," he cried out, "if you don't like her, I am going to destroy her."
His cerebral excitement seemed to inflate his head, rather as a balloon from which his wasted body hung in slight levitation.
"Come down to the floor, for God's sake," I said peremptorily. "What does my opinion matter? _I'm_ not the museum."
"But you're right," he insisted. "I have been going in the wrong direction. _Die Irma's out_."
"And don't use me as a sop for your terror of working."
"It's really not that—but a technical question. _Die Irma ist nass_."
"She isn't, she's bone dry. I felt her."
"I assure you, underneath—"
"Every time I've come to Paris you've said the same thing. Pull yourself together Insel, you've got to finish this for the museum. For you it's work or death. Can't you figure it out?" I urged helpfully— "When you have money and can eat you paint a picture so as to have more money— when you haven't any more money."
"It's more complicated than that," he objected again, " _die Irma_ is wet—"
I was getting exasperated— When the balls of our eyes caught each other, we both began to laugh.
"If you had heard the Lesbian's synopsis of Frank Harris's confessions, you wouldn't even trouble to mention it—."
"I shouldn't care to read this Lesbian's confessions—it is a Lesbian who has taken the love of my life away from me."
"Well now, I wouldn't mention that either. Of course, it does not matter with me—anybody can tell _me_ anything— you know what I mean—when you surrender your arms, chuck them onto neutral territory. I know it's a touch that modernizes your romanticism; all the same, I'd advise you never to make that particular confidence to a woman _'ou connait ça_.' "
But Insel was past advice. With a look of dogged emptiness he recited for the nth time the story of those _Mädchen_ "who shut themselves into the house for a fortnight for fear he would shoot them."
Mostly when speaking of his loves of the past he became quite normal; subnormal really, for his adventures in the actual world had been of an excruciating banality.
As I was also engaged for dinner, I asked the time. Insel who was sitting on a wooden stool stretched out his arm—it reached much further than its actual length would warrant.
Behind the curtain in the corner, carefully secreted under empty boxes, neatly stacked, was his wristwatch. He did not _bring_ it out—his arm seemed in some Einsteinian contraction to shorten the necessary distance for focusing the hands.
It was seven o'clock. I took my leave. Insel, astonished as if this were the first break in a timeless conversation, snapped in half; or at least bowed like a poplar in a sudden gale; his dessicated limbs the branches.
Staring vainly towards the door I was opening—he choked in the voice of a Robot, " _Morgen komm ich im Gericht_. Tomorrow I go to court—I am going mad!"
"Then don't forget your little afternoon," I reminded him— "I dote on madmen."
As I was leaving, he seized his palette and dripping an enormous brush into a pile of ebony pigment painted with a heinous neigh of victory, " _Die Irma_ —Out!"
# **19**
MY INTERMITTENT INTRUSTIONS ON INSEL'S inexplicable Eden of mischief had set their mark upon me. Some of his secretive twinkle had seeped into my eyes and lingered there, eliciting comments from my friends. I became more popular.
Insel, however, did not like it at all—as if I were a thief, a stark sternness shot with flashes of sadism replaced his usual intonations of abased tenderness while, awkwardly enough, I continued to feel myself elfin.
One day when I had returned from a lunch he came in to fetch his "Kafka." I had a good time and prattled to him sociably, "Alceste—the duchess— everyone was intrigued to know why I am so jolly."
" _So lustig_ ," Insel hissed—a maniac sadism flaring up in his eyes, and for the first time I saw him as dangerous. " _So lustig_ ," his hiss growing shriller and I could feel his hatred twining round my throat as he took a step towards me. But a step no longer the airy step of the hallucinated—it was the pounding tread of the infuriated male. " _Lustig_ ," he squeaked, his hiss exhausted.
He approached no nearer. Probably my absorbed interest in examining his insane pupils dominated him. Anyway, although it now surprises me—it seemed I could not be afraid of him—our "entente" in the visionary lethargy of that primeval chaos we were able to share was fundamental and secure. Confronted with his surface vagaries, I felt at once collected—as if I might have been his "keeper" since the dawn of creation.
"Insel," I said placatingly, "if it would improve your health were I to suffer a hopeless love for you, I'm quite willing. Not today—I have a cocktail party—but some other time, I promise" (thinking of my bouts with the _grand sympathique_ ), "—you shall see me suffer horridly."
Insel, unconvinced, let out a low growl which sounded like one more _lustig_ —while that strange bloom, as if he were growing feathers, spread over his face. He turned into a sugar dove. It flew about the sitting room, dropping from under its wings a three-ring circus. In one ring echoed the cracks of a whip; in one ring rotated an insane steed of mist; and in the other ring Insel's spirit astride an elemental Pegasus—.
"Horror," said Insel and I jumped. "Would I have to grow a _beard_ in order to make myself attractive to you?"
The _grand sympathique_ (which eventually turned out to be a duodenal ulcer) must inevitably go on the rampage again. Very soon it did. There was no resource to Insel's healing _Strahlen_. Since his screech of a vanquishing cat he had, as far as I was concerned, subconsciously thrown them into reverse.
For a while I was helpless; then one day when the pain calmed down somewhat, I crawled up to Insel's—still trusting he would finish _die Irma_ for America—to give him a hundred francs. That is, I never gave him anything. I am not generous. The few _billets_ necessary to keep him going were fully covered by the valuable drawing he had forced me to accept. It would be easy to sell if I needed the money.
It did not occur yet to me how unsuccessfully I had succored him, for when first I met him he had been merely a surrealist—his biography was coherent—steadily since I had "interfered" in his affairs he had grown hallucinatory.
"It's all very well," Frau Feirlein argued with me, "I was here when you advanced him five hundred francs for the Gallery—the very next day he hadn't a sou."
To me it appeared fitting Insel's finances should flicker in and out like himself. For the present no power on earth could dislodge from my mind that luminous effigy of generic hunger—or shake my serene unquestioning insistence on its preservation. Something unknowable had entered into a game with my intuition.
He let me in and returned straight to his needy couch, teetering on the end of his spine in a double triangle as he drew up his knees to replace his feet under cover. I was overcome by a rush of nervous sublimity carried by the air.
"If this is madness," I said to myself, breathing his atmosphere exquisite almost to sanctification, "madness is something very beautiful."
My relinquished conviction of his unutterable value returned as I looked up in the bare swept room. An especial clarity of the light I had noticed before to be associated with his presence was this evening so accentuated I could actually dissect it. Its softly bedazzling quality was not of any extra brightness, but of a penetrant purity that uplifted my eyes. I could discern among the unified flood of customary light an infiltration of rays as a rule imperceptible, filaments infinitessimally finer than the gossamer halo round a lamp in the fog—a white candescence that made the air look shinier, with the same soothing shimmer as candles at mass in sacred houses, only indescribably acute.
I was not unfamiliar with it. That different light I had seen etherealize the heavy features of Signora Machiabene an hour before she was stricken to death. That very essence of light I had begun to perceive during the prolonged moment when a dislocated vertebra had thrown me beyond the circumscription of bodily life.
There is no saying in what bliss consists, yet I could see it incorporated with Insel's face, bathed in that different light, as he lay under his only blanket, his limp hands clasped behind his head.
"I see through the wall," he said, his voice at peace. "I can lie here hour-long watching my neighbors live their dear little lives. Sometimes they play a gramophone and on its way to me the music has become miraculous."
"You have never known ennui," I laughed, forgetting as completely as he evidently had, if indeed he had ever been conscious of, the tortured glowworm of the Boulevard night, the inarticulate confidences of one cut off from mankind; the sleepless seeker after an unmentionable salvation who, blinded by his own unnatural glitter, was so wounded by the dawn; that distracted man who, terrified of isolation, hung onto my hand while he flopped and darted like a fish on the end of a line, stung by a mystifying despair.
"Never," he assented, beatified. "I am eternally content. My happiness is infinite. All the desires of the earth are consummated within myself."
"Aside from that—what are the people in the next room _doing?_ "
"Just being—I ask no more of anyone. _Being_ in itself is sufficient for us all," he answered enraptured.
Seeing I had taken up one of his drawings, he instantly arose. Always—there was something of the depths of the sea about him and his work, also of eventual evolution as in the drawing I was looking at where to a rock of lava a pale subaqueous weed clung in the process of becoming a small limp hand. The tips of its fingers were stealing into pink.
Insel himself had fearsome hands, narrow, and pallid like his face, with a hard, square ossification towards the base of the back, and then so tapering as if compressed in driving an instrument against some great resistance.
"You were a lithographer, not an engraver?" I had once asked him, puzzled by what his hands looked as if they must have been in the habit of doing, and we concluded this conformation could be an inheritance from the _Schlosser's_ driving power. But out of this atavistic base his fingers grew into the new sensibility of a younger generation, in his case excessive; his fingers clung together like a kind of pulpoid antennae, seemingly inert in their superfine sensibility, being aquiver with such minuscule vibrations they scarcely needed to move—fingers almost alarmingly fresh and pink for extremities of that bloodless carcass, the idle digits of some pampered daughter; and their fresh tips huddled together in collective instinct to more and more microscopically focus his infinitesimal touch. All the same, there was something unpleasantly embryonic about them. I had never seen anything that gave this impression of the cruel difficulty of coming apart since, in my babyhood, I had watched the freak in Bamum's circus unjoin the ominous limpness of the legs of his undeveloped twin.
"Let's have a look at your feet," I said as he came weightlessly towards me. He drew off his slippers, padding over the bare boards on the drained Gothic feet of a dying ivory Christ.
"What's this?" I teased, pointing to a lurid patch on his instep, "a chancre?"
"No, it's only where my shoe rubs me. I bought new shoes when I sold that picture and they hurt me," he explained, frowning helplessly.
"Why not try pouring water into them and wearing them till they 'adapt'? It often works."
A strange bruise. It shone with the eerie azure of a neon light. But once within range of Insel, nothing seemed unaccountable, as though he submitted to an unknown law enforcing itself through him. Each item of his furnishing, he having touched it, had undergone the precious transformation of the packet he had folded in my home. His hand, in passing over them, must have caused their simple structure to obtrude upon the sight in advance of their banal identity.
A row of powdered-soap cartons, set upon a shelf, he had stood up to the significant erectness of sentinels, their impressive uniforms consisting in the sufficiency of their sheer sides. He showed me they were empty. Altogether his place had an uncommon dignity. Within a stockade of right angles he had domesticated the steady spirit of geometry.
The room, with its two tiny matchboard tables, its curtains of washed-out cotton across an alcove, full of its supplementary radiance, had an air of illogical grandeur beyond commercial price.
# **20**
EVERY NOW AND THEN THE SHARP OF HIS FLICKERING sadism, a needle occasionally picking up the dropped thread of memory, would prick through his frayed conversation, woven of disjointed themes like an inconsistent lace eked out with stocking darnings.
He recalled my promise—to demonstrate unfortunate love whenever a twinge of pain contracted my features. He peered enthralled at the havoc pain played with me. His delirious peace expanded to full blossom in the smile of Buddha. One felt his utter joy at sight of my disablement had leapt to such a blaze he must melt off it, his fragile person dissolve in his delight, were it not for some mysterious source within him replenishing the exaggeration of his unabating intensity.
_"Gestatten Sie?"_ I inquired ceremoniously, unable to hold out any longer against the pathological rat gnawing at my entrails. And I subsided on his couch. Above a certain degree of agony, one is willing to subside anywhere. However, my slight repulsion dispersed as I lay down. Indeed, like the saints whose dead bodies did not decompose, Insel's electric exudation in some process of infinitesimal friction seemed to cleanse him of his grubbiness of the poor, to free him of any accretion natural to normal man. His couch was almost fragrant with that faint half-holy purity that hung about him.
"What color was this once?" I asked, as I drew up his gray blanket.
"White," said Insel.
It was incredible. That twilight sheer duration lowers upon all pale fabrics had so penetrated the thick wool, one could only believe with difficulty it had not been dyed—a perfect job at that—no spot, no smirch, no variation in tone disturbed the unity of its spread surface. For a moment I entertained the idea that Insel had worked all over it with the microscopic point of his lead pencil, for it seemed no earthly dust could defer to such patient order. Anyhow, I decided everything in the place is bewitched, and let it go at that.
Insel, intently keeping watch, had moved his stool some distance away as if to find his range for an inverted "Aim of Withdrawal." Spinning himself into a shimmering cocoon of his magnetic rays, introvert, incomparably aloof, "They're mine," he exulted as clearly as if he were crying aloud.
Too simple to fully imagine the effect of these rays, he had, it would seem, only an instinctive mesmeric use for them. He might even feel them as a sort of bodily loss compensated perhaps by rare encounters with one able to tune in.
"I shall make you some tea," said Insel affectionately, and hushed as a nurse, he began swimming about from his little sink to his wooden shelf—or as a panther softly pacing before a vanquished prey—. I noticed now, as always, whenever one encountered Insel at an angle of meals at home, there was appropriately just enough dust of tea leaves left at the bottom of a packet for brewing the last cup—he would open the door to you holding precisely the fag end of a loaf for the last bite. But today he served a minute carton from an automatic machine in the Metro. Out of it he rolled into my palm a bonbon, virulent green, less than a pea in size.
"Aren't you having any?" I invited, convivial as a gourmet, for in his dimension this was a spread feast and the hot unsavory tea had eased my pain. The very teenyness of his sweetstuff made it more seductive than a giant Christmas cake. But he shook his head impishly in abnegation. I looked inside the carton. There was nothing left.
In the concentrated one-sided luxuriance of our party he evoked his dazzling future—the work he no longer seemed able to do waxed so sublime in his visions (and also in mine as I watched all possible loveliness evolve from his elemental mists, and the creeping to maturity of the almost invisible herbage left from under the withdrawn tide of his hovering waves).
His fame was to be fabulous, his wealth extravagant—so that at last the great Insel would (with a gleam of furtive cruelty for me) marry, as far as I could make out, so increasingly incorporeal he grew in his grand exaltation, my daughter's photograph.
"That will be really nice," I responded genially. "As I come to think of it a son-in-law—."
"Exactly," he burst out wildly. "You and I, we could have such a wonderful time together."
"—a son-in-law with rays—" I brought up short, "—and what would my daughter be doing?" Then hurriedly, thinking to profit him by the occasion, I urged, "You must _paintthose pictures_—otherwise, you will grow to be too old to marry her."
"Ah, well," he waived, with that sudden doleful look he had of gazing into an abyss when confronted with whatever imperative of whatever consummation. "No matter! It would suffice me just to know her, to have the joy of _watching_ her evolve. It would be a very wonderful thing indeed to take part in the _Entwicklung_ of a young creature." And I realized there was nothing, nothing, in all the world elementary enough to serve as object for such simplified observation as his. Everything must henceforth for him drowse in an impotence of arrested development.
This very word, _Entwicklung_ , was so much Insel's word; its sound seemed to me onomatopoeic of his intellectual graph. For my alien ear it had a turn of the ridiculous as though a vast process had got twisted in a knot of tiny twigs, haply to unravel and root, and branch against the heavens.
# **21**
I REMEMBERED THOSE STACKS OF MANUSCRIPT HE had assured me were at my disposal in the days of "biography." "You promised to show me your notes," I reminded him. "May I not see them now?"
He was coy about his literature, sidling alongside himself in a sort of dual fidget impossible to describe, as if doing sentry duty before his own secrecy.
After a long persuasion he brought out a blotter, the kind for _écoliers_ sold in bazaars. Covered in black, stamped with a golden sailing ship, its funereal hue intended for neutralizing ink drops in a kindergarten, this unassuming blotter, the one thing in the place having any tradition, had a decorative air of intruding from a frivolous society. It contained a single sheet of paper which he handed me with great precaution. Very few lines were written upon it. They formed a square block in the center of the page covering little more than the area of a postage stamp.
Hardly had I caught a glimpse, when, "Can you see it?" he inquired suspiciously and snatched it away.
"How could I?" I demurred, "the words are scarcely visible—" Reassured, he gave it back to me.
"You've no idea," he sighed, "the pains I take when I write to you, forcing my hand to form letters big enough for you to read."
" _Liebe_ Herr Insel," I cajoled, "You read it to me."
At last he did so.
It was a beginning.
" 'My sister and I walked along the road. Coming to the town gate we gave it a good thump.'
"Do you know what a town gate is?" asked Insel professorially. "It's like a tower.
" 'All the townsfolk came out of the gate, swarming about us to look.' "
As ever, with Insel "to look" was a deadlock, he had written no more.
I proffered the necessary compliments. Agog with glee, he shimmered with satisfaction. This communication of an actual transcription of a mental process had reinforced his sociability. His contacts ordinarily depending almost entirely on his _Strahlen_ , for the moment our companionship was complete.
In reading aloud his manuscript he had formed an extra alliance with me—as _littérateurs_ , producing in Insel an enormous self-respect.
Nevertheless, it was a sympathization going on in some sphere to which I had no access.
Anything he perceived sufficiently to accept or that _thrust_ itself upon his attention (as in _my_ case) was instantly distilled in his precious essence. Behind his brow a void wraith, glorified, evaporate, dissociated from its originator, myself, to mix with his gaseous cerebration.
Insel let out a shrill crow. " _Es gibt nicht zwei vie Sie_ —There are not two like you."
Sparkling, entranced, he sat on his wooden chair as on the throne of the conquistador, for whatever I contributed to his transcendental enjoyment he was loath to let me go.
But I was beginning, myself, to feel unnatural. I distinctly detected my voice in ventriloquial emulation echo the wistful, surf-like swooning singing of his— _"Sterben—Man mu-u-uss—Man mu-u-uss"_ —as, worn out with pain, assuring him I must leave because I was tired, I said, " _Ich bin so mü-ü-de_."
Insel, responding to this bemused inflection, or rather, fusing with an ululation so singularly his own it almost obliterated our duality, I witnessed in him an inconceivable reversion of a standard transmutation. The _changing_ of _sadism_ into _love_. Not gallant love. The indiscriminate love of a savior.
Suffering, I had so gratified him, satiating his sadism—even to extinction, his gratitude refluent to me, enveloped me.
At last with a sacrificial decisiveness Insel consented to let me go. " _Ja_ ," he assented, bending over me in solicitude. "Go," he stretched out one of his thin branches in benediction, "Go and sleep." " _Schlafen_." His word was drowsy—so long drawn out. It did not cut the air as ordinary words do. Agelessly sailing, it passed across me—oblong and idle—spacious as an airship, its narcotic cargo a dream of a slumber unknown to normal man.
The rays that Insel so busily had been spinning around himself in an immeasurable tenderness released, attained once more to me.
Instantly all pain vanished. I sprang up elasticized. To an identical rhythm Insel and I, on a buoyancy, were danced toward the door bobbing and smiling good-bye in a mutual appreciation which I felt must be glittering off me as it did off him.
When I got to the train it was steaming out of the station; casually I skimmed onto it scarcely noticing this, for me, at other times, impossible achievement—I felt so airy. In the car, whenever I thought of Insel, I was shaken with a helpless laughter—a strange mixture of extreme friendliness and, inexplicably, derision.
The painless buoyancy lasted well into the night when, as I sat calmly at work in my hotel bedroom, I unexpectedly disintegrated. My body, which had hitherto made upon itself the impression of a compact mass, springing a multiplicity of rifts, changed to a fractional covering I can only compare to the spines of a porcupine; or rather vibrant streamers on which my density in plastic undulation was being carried away—perhaps into infinity. A greater dynamism than my own rushed in to fill the interstices. Looking down at myself I could _see_ my sensation. The life-force blasting me apart instead of holding me together. It set up a harrowing excitement in my brain. An atomic despair—so awful— my confines broke down. I lost contour. Once more I found myself in the "impossible situation" in which one cannot remain—from which there is no issue.
I cognized this situation as Insel's. A maddening with desire for a thing I did not know—a thing that, while being the agent of his—my—dematerialization alone could bring him together again. A desire of which one was "dead" and yet still alive—radial starfish underpattern of his life, it had communicated itself to me. I was being impelled to the pitiable serial choreography of Insel when in the closed cab he had chased himself along the incalculable itinerary of his dissolution.
In a darting anguish consciousness in pulverization peered from the ends of incontrollable antennae for something unattainable.
I had more space than had Insel in his cab, yet the in-slanting facade of my room under the eaves with the red glow of its wall paper got "in my way." Having no idea of what was happening to me, I seemed to have also unsuspected reserves of will power. I put up a pretty good fight against this incredible dematerialization—it took me hours to weave myself together—but at last, exhausted yet once more intact, I fell upon my bed and slept.
Next morning my face looked "destroyed" like Insel's.
Although I was all of a piece, my very bones were weak. I had to walk carefully. I found out why, when climbing slowly up the hill to the station to buy a newspaper, I was cleft in half. Like the witch's cat when cut apart running in opposite directions, suddenly my left leg began to dance off on its own. Thoroughly frightened at this bisectional automatism, I somehow hopped to the fence on my right and clung to it in an absurd discouragement.
The day after that I thought I was normal. Walking serenely at my habitual pace, I went to a shop in the village before keeping an appointment for lunch.
Without association (as usual) the idea of Insel rose in my mind. Quite different to thinking about someone. I was overcome with that imbecilic self-satisfied laughter, that Parnassian guffaw. It had nothing to do with any humor known to the intellect; being a sort of blank camouflage for all intellection. With me it was always filtered with a faint derision. But even this derision I took for granted. Brought to a halt under the full force of my mental hilarity, I felt constrained to _continue to share_ if—with what?— with whom? To do so I turned sideways. Whenever this _idea_ of Insel occurred I could not go straight ahead—I had to turn to it—as when I had tried to sum him up on the Boulevard.
My feet remembered that lightness, that skimming of the pavement—I was engrossed in a merriment beyond existence. When this merriment ceased, I found myself in a part of the country I had never seen before.
I had kept track of the time to avoid being late— as that gust of laughter caught me it had been twenty to one. I walked into an open yard where a man was washing a car. He informed me it was twenty to one and that it would take me half an hour to get back to the place where by all the laws of possibility I should still have been. He allowed me to telephone my friends, to begin lunch without me, that I had got lost—at the other end of the wire it was twenty to one.
All this was comparable to an incident that occurred when I made friends with a little girl whose intelligence was like a jewel in a case too tightly closed. A backward child, one of those partial imbeciles, who, not being "all there," showing only half their human nature, are either angelic or diabolic. Probably their own halves are all they respond to in other people, for Fifi, when she said _"Bonjour"_ with a smile of benediction, would discover, "Madame, you are as sweet as a rose."— "Monsieur, you are bright as gold." Her being subnormal lent an elfin prestige to her slow serenity among her associates, offspring of peasants and small tradesmen, who attended the informal court she held in the parlor behind her parents' shop.
Rigid as bygone queens in her orthopedic corset, she accepted the offering of every conceivable kind of toy duck from her wondering courtiers, with a lunar giggle that never precisely applied to anything. Her passion, her concretion of sublimity, took the form of a duck. "God is playing hide-and-seek," she would announce, "so the Virgin Mary has married a duck and they live in the top story of the Riviera." And once when I found her watching some live water fowl by a pond in a farmyard— "Why do you love the duck?" I asked her. " _Il dort dans son dos_ ," she perfectly replied.
A fearful future opened before her if she could never keep shop, and the medical specialists consulted on her behalf promised she would become like average children should they graft a bit of the bone in her leg as a wedge into her spine, thus rectifying her crookedness and relieving the pain. But this operation, successful they said in many other cases, failed with this half-wit angel, who, incidentally, had predicted the year of her own death.
So Fifi died most uncomfortably, lying very much like a trussed duck, only on her tummy—her leg being bent up behind her for the grafting and bound to her back—screaming in a nursing home until she had no more breath.
Only once, in talking to this little girl, had I seen her unhappy. An unhappiness intense as it was brief. A drip of anguished words revealing how she received as an awful animosity her mother's solicitous efforts to get her to "make sense."
While undressing to go to bed that night, as if a flash of sympathetic insight "put me in her place," I suddenly found myself imprisoned in Fifi's mind.
Strangely enough, it was analagous to my sensation of utter helplessness when dislocating my cervical vertebra, I had found myself without any instrument with which to contact the universe.
But now I was at the mercy of an imperfect instrument. The antennae of the contact with the world in some way crippled for their function seemed—like the umbilical cord in abnormal birth—to be wound round my brain in a fearful constriction, implacable as iron barriers.
My brain, like a bird in ceaseless hurt, beat its wings for the conscious liberation against a cage—or rather, a sort of immature sieve, which would spring a hole intermittently; here and there letting a glimpse of phenomena through—phenomena fitful and unrelated.
Caught in a horror of active impotence, I struggled in terror—unlike Fifi, I could get out.
This gratuitous experience was as nothing to that of disintegration when, on the contrary, one became aware of _forces_ inherent to phenomena, which, being beyond the range of registration by the normal instrument—the conscious organism as it exists in our present stage of evolution—resulted in a super-sensibility so acute it shattered itself to splinters.
The intuitional self is incapable of surprise, but my everyday self was amazed. I felt that for dabbling in the profane mysteries I had got more than was coming to me.
Less than anything on earth did I require a face destroyed as Insel's; for some while I should walk with misgivings—.
I racked my brain for an explanation of my soaring respect (respect being a sentiment foreign to me) for a loafer who in the light of common sense proved to be actually silly.
Insel, who, so sensible of his essential mystery—communicated that sense of imminent magic inevitably, just when one was in the "thick" of his influence would illustrate his "power" with a story such as how, after dining with married friends, he had predicted to his wife, " _That_ union will not last long."
" _Und Tatsächlich_ ," he concluded with an expression of awe, "they separated within the year."
He suffered, it would seem, from the incredible handicap of only being able to _mature_ in the imagination of another. His empty obsession somehow taking form in obsessing the furnished mind of a spectator.
From a distance one remembered him vaguely as an indulgence in a quaint innocuous vice. Still I could scarcely go further with him than dissolution. I decided it would be useless to see him again.
My brain still seemed to be vibrating out of time, when early one evening on leaving a library I wandered into an old church. Somebody up in the organ loft was playing Bach. A sublime repetitive patter of angels' feet soled with assuagement, giving chase to one another in a variable immobility of eternal arrival, they trod my cerebral vibrations from disarray into tempo once more.
I had not thought of my casual prediction that the whole of Insel's life would hang upon a key—when on mislaying my own key to my apartment he produced the duplicate I had lent him in the days of his "eviction"— and _forgot_ (the place was still at his disposal) to ask me for it again.
It was a long while before it occurred to me that his girl's watch was lying at the jeweler's. By then, all that remained of Insel was a vague impression of _trompe l'oeil_. I wrote him to call for this love relic, he having assured me that should I have it sent to his address his concierge would seize it towards arrears in rent.
Insel, an eroded scarecrow, greeted me with the somber dignity of a dejected god.
"Why did your girl give you such a rotten watch," I teased, "the jeweler won't guarantee it."
"One takes what one can get," said Insel with no trace of emotion as I had handed him the erstwhile "Adam and Eve in primeval embrace." His present concern was for getting back the key. Determined he should not have it, I pretended I was returning to Paris.
Lolling on either end of the great couch, supported by our elbows, our feet on the floor, we were at ease for conversation—the conversation would not begin—Insel being taken up with contracting to some intense concentration that gradually pushed out a sort of pallid ethereal moss to cover his ravaged face.
At last, to my raised eyebrows, " _Ah, liebe_ Frau Jones," he complained in prayerful peevishness, "it's not so easy for me—I don't mean anything to you anymore."
"I know," I said contritely, "I get these wild enthusiasms for things—they don't last."
"And we might have had such a wonderful time together," he sighed.
# **22**
I COULD NOT MAKE OUT WHY THIS FANTASTICALLY beautiful creature should have both hands round my throat, when Insel, shrunken to a nerve, his eyes fixed as blinded granite, sat at that distance with his fists so tightly clenched. Fingers of automatic pressure rapped their tonnage of abstract force on my jugular—the blood on my brain surged in a noisy confusion— "You are going to give in—obsessed by my beauty—having no hope—endlessly resigned—"
All the air wheezed in my exploding ears as a last breath, "—suffering—suffering—suffering— _choked by a robot!"_ This was not all that suffocated me—myriads upon myriads of distraught women were being strangled in my esophagus.
I had known exhaustive desperation but no such desperation as this—with its power of a universal conception—of limitless application: being impersonal made it the more overwhelming.
"You—are—going—to—give—in."
"To whom?" I wondered—my eyes closing. "To Insel? Or this incredibly lovely monster made of dead flesh."
"Thou art fair my beloved, thou—," rose from a subconscious abyss.
Not wholly convinced I wrenched my eyelids apart—my cerebral current, flowing an infinitessimal fraction of a second faster than the normal, registered Insel. I caught him at it. Swift as the leaves of the shutter on a camera when a snapshot is taken, there came together upon his concentric face a distinct enlargement of Colossus' photograph that always stood on the sitting room mantlepiece at the other end of the flat.
Simultaneously it came back to me how Insel, on his first visit, had taken that photo between his hands to stare at it inordinately as if for reproduction, for a long time, and at length bringing it nearer to his eyes.
"Such beauty as this," he said, "could scarcely happen more than once in a hundred years." He himself put it at two thousand, I had laughingly observed.
"Stop it," I commanded, letting fly a fearful kick at Insel's brittle shin. As if he were anaesthetized, the kick seemed not to hurt him—he received it with the smile of ultra-intimacy he had for me whenever we met on the unexplored frontiers of consciousness.
"The pet! The lamb!— it does television, too," I told myself delightedly.
"Insel," I laughed, enthusiastic over him once more.
" _Seien wir uns wieder gut_ —I give you the key—dinner—My man Godfrey—the loan on your picture—you go to the Balkans—you are the living confirmation of my favorite theories."
As for Insel, he emerged from his "raptness" babbling of Colossus—Colossus as he had himself foretold me having taken on an immortality as an evergrowing myth. Insel claimed him as a kindred spirit with ideas identical with his own.
"How entirely he would have accepted me— my art—We would have been as _one_ —"
I argued at length against this sudden conviction. "Do you know," I asked, "who, for the so-called precursor of surrealism, was the supreme painter?— Rubens—" Only then did Insel's illusions miserably dissolve.
# **23**
AFTER THE POWELL FILM, WE INSTINCTIVELY returned to Montparnasse—eating at a chic bar. The barman and Insel behaved as brothers—I vaguely noted a sort of ritual—the passing and repassing between them of half a cigarette. They addressed each other as "du." "No—for 'thee'—" Insel would say, placing the stump on a glass shelf as one handles a treasure.
Some days later I saw this barman out of doors wearing one of the richest overcoats I had seen in Paris. Evidently such acquaintances could hand out "leavings" superior to the plain nourishment Insel acquired from the Quakers.
We sat around the Dôme and Insel x-rayed. All the girls, as they giggled along the Boulevard, he disrobed—more precisely, he could not see that they were dressed. As if on an expedition for collecting ivory, he _handed_ me their variously molded thighs—weighed them with an indescribable sensitivity of touch.
"This one," he assured me, "in the summer is firmer—turns to gold—"
Recalling how terribly Mlle Alpha had said he dated, I presumed he was claiming my interest by indulging in what Boulevardiers of the old days called "undressing the women" in his own unbelievably tangible way. "I don't _need_ them to take off their clothes," he remarked.
In the Select Insel became actually involved with his watching of a red-haired girl he raved of as _"die Rothaarige"_ —her thighs were peculiarly long and agile. "She's a bit of a Lesbian," he sighed, filled with some inverted reminiscence of antagonism.
"Look here, Insel—you're crazy about that girl— and all you do is sit around x-raying her—Get up— go and speak to her—"
"She'd be too expensive—"
"Colossus never had any money."
"Colossus was beautiful—"
"What about it? You're looking unearthly. She might get a thrill out of it—try—forget the expense— I'll back you—Go along."
But Insel, subsiding in his inexplicable negativism, refused to stir.
"Listen," I admonished him, "all this is really unwholesome—and sitting boxed up in an attic adoring that canvas Irma all day—you'll become impotent—"
In a burst of the extravagantly sophisticated laughter I had heard him emit once before, "I only wish I could," he assured me.
"What a subject," I reflected, "the virility of the starving man." But the Select was undergoing change—opening out to aqueous space in darkling shadows of metallic liquidity as in the vision of the Lutetia, that strangely etiolate phallic ghost floated like the stem of a water lily. Before it had terminated in a battlement akin to that of the castle among chessmen; now it was topped with a little crown of thorns.
Through the chill shimmer of this unreal deep— the hallucinatory blue the Coupole had painted on the backs of dreary houses as a setting for its garden cafe—the blue I would wish the sky to be showed us another dawn.
"Look."
"There should," said Insel, extremely worried, "be a lighted lamppost there."
"There is," I reassured him, "lower your head—see it was cut off by the blind."
This was the last of the two or three nights I spent with Insel in Montparnasse.
We crossed again to the Dôme to have breakfast. Sitting beside him, I could see a man in white armor conduct a ballet. Serried rows of mustard pots drew up before him, their porcelain bellies burdened with amber. They moved to and fro as with a wooden spatula he lifted off their stale crust of night, filled and leveled them, and set each one down to be armed with a clean bone spoon.
_"Woher kommt diese halbe Mücke?"_ Insel grumbled, insanely hacking with his knife at a tiny aeronaut shade circling an inviolate orbit, because he could not make out "Where this half a fly comes from." I knew it was only a baby fly, yet all the same it loomed above him hugely as an insectile cherubim cut off from its entrails in a like unanatomical constipation to Insel's monsters.
The rest of the day till two o'clock when Insel, as usual, it seemed must "appear in court," we spent in an incredibly concentrated and somehow heartrending arithmetic, reckoning up whether Insel, out of the three thousand francs loaned on his picture, could possibly afford a new pair of boots. We had already decided he must have a warm overcoat when, although it was not particularly chilly, little muscles in the side of Insel's nose, self-animated, leapt up and shivered. "You are freezing," I discovered in startled concern, scanning his fragile flimsy features.
"I hardly feel it—I am used to it," said Insel, dolefully heroic. "It is only discomforting to those who are with me."
But I teased him a bit when we said good-bye, alluding to a lunch with the Alpha when to our mutual hilarity we had made out how only two hours after leaving my studio after that utter collapse, he had stumbled into hers.
"He looked _ghastly_ ," she told me. "He had not eaten, he had not slept—his heart had ceased to beat!"
Insel, whom I had seen so sly, had been vainly hoping to get his beefsteak fresher.
"How on earth," I inquired, "do you compose your _Totenkopf in_ so short a time? Pretending to Mlle Alpha—"
"Why," Insel answered pat, with the queerest inflection of intimacy, as if I were some virgin he had raped, "I thought you would not _like_ me to tell her I had _been with_ you."
"It's marvellous," I assured him in amused admiration, "your knack of dying on doorsteps. At will! At any moment! You might make a good thing out of it. Perhaps you do. Insel, I believe you put _lots_ of money in the bank!"
I could feel a distinct change in his aural temperature, but I was laughing too much to pay attention. An impression of a sacred stronghold "blowing up," that shadow-tower of iron rag the _clochard-deity_ Insel had built, like an ant of his wasted tissue, was so very, very faint—In view of America, I was constantly on the hop—busy with buyers of furniture—packers littering the place with straw.
Arriving for some appointment, I was unprepared to run into an Uneasiness in the vicinity of my home, although it remained closely sealed in its shutters and nothing by day ever went in or came out. _Les concierges_ , their aides and cronies, the grocers at the corner, all were under the apprehension of the place being haunted. Even Bebelle, whom I came across in the street, had, on going there to clean, turned and fled.
"Madame," she said in a hushed warning, "in there it is dark at noon. Terrible clothes have clotted on the floor—Never before have I seen what was lying on the bed."
Insel at last must have been evicted and at some unknown hour crept into the flat.
# **24**
SOMEONE WAS LIVING THERE.
On my throwing open a window, he hooked his arm round his neck, rubbing the mastoid. "I have lain here for two whole days," he said, ferocious with dignity. "I have a stiff neck."
A hard-eyed, low class German, his very existence an insolence, wearing a shirt from a cheap shop—Insel must have thrown himself away with his old black sweater above which his former face had risen like a worn, pocked moon.
Unquestionably, I had cured him. Here was the "normal" man. An Insel unobsessed. Someone "replacing himself," his mesmeric, melodic voice exchanged for a hostile creak.
This culminating phase of my eerie experience —Insel's _residence_ —remains confused, as I was busy directing packers.
Cavilling and bilious, whenever he caught sight of me he hardly refrained from spitting. Our relative positions entirely reversed, I had become for him a strange specimen, to whose slightest gesture he pinned an attention like that of a vindictive psychiatrist.
"Ha-ha!" he neighed irately, "I find little 'still life' in this flat. It would surely be of the greatest interest to Freud."
We had, in our "timeless conversation," with Insel's concurrence in my "wonderful ideas," superseded Freud. I must always have known he had never the slightest idea of what I was talking about—yet only now did this fact appear as negatory.
The still life that intrigued him was a pattern of a "detail" to be strewn about the surface of clear lamp shades. Through equidistant holes punched in a crystalline square, I had carefully urged in extension, a still celluloid coil of the color that Schiaparelli has since called _shocking_ pink. Made to be worn round pigeon's ankles for identification, I had picked it up in the Bon Marché.
Out of this harmless even pretty object an ignorant bully had constructed for me, according to his own conceptions, a libido threaded with some viciousness impossible to construe.
I was astounded.
It would be only natural that my jerky vibrational currents (which behave so much like a "poltergeist" that things when I touch them are apt to vanish, adding a superhuman difficulty to my work) should impinge on Insel's abnormal precision with the force of a shock, although in the hallucinatory dimension it was this very extreme of antithesis that must set up the telepathic, televisionary machinery of our reciprocity.
"What do you suppose," hissed my horrid guest, who somehow behaved like an alienated husband, "would happen to me if I were to lose anything?"
"Oh, I suppose," I countered rudely, "I'd buy you another."
Being the intrinsic complement of Insel's enmity, logically my loathing for the real man was unconcealed, while he must actually hold himself in check not to assassinate me, for no crueller abhorrence could ever issue from the human heart than Insel's for me.
There were brief abatements of his fantastic normality as when on coming up from the telephone I encountered only a creature of pathos in the hall.
"You would not notice, would you," wistfully, "that I have polished everything in the flat."
"No," I concurred, "I would not have noticed that."
Insel was long in swallowing his disappointment, then cryptically, " _Gut_ ," he snapped, "and I am always amorous when drunk."
And again, for fear I might forget the loan, Insel went limp as he had to the air raid siren. That unaccountable bloom he put forth when passing from one condition to another made his features appear to be of crumpled velvet.
Sitting on a chair of average height, he seemed to have sunk to bottomless depths, at the same time his imploring face peered at me—from the floor.
Craven to a degree that rendered his cowering august, of that meekness befitting a supplicant at the door of heaven, Insel was knowing an alibi so sublime—I again lost all knowledge of who he was.
"Here," I hailed the will-o'-the-wisp, "after all I will give you the little box." This box he desired, it was black, was a small object by the American surrealist, Joseph Cornell, the delicious head of a girl in slumber afloat with a night light flame on the surface of water in a tumbler, of bits cut from early _Ladies' Journals_ (technically in pupilage to Max Ernst) in loveliness, unique, in Surrealism— the tidal lines of engraving cooled its static peace. Under the glass lid a slim silver slipper and a silver ball and one of witch's blue came raining down on the gray somnolence when one lifted it up.
I should have preferred to keep it myself had I not suddenly realized she belonged in those idle hands to which the unreal Insel intermittently returned.
I only went twice to the flat while Insel was living there, but I flitted in and out so busily—those hours retain no sequence. As part of his loan I had arranged for a strictly non-negotiable ticket and brought him a first thousand to speed the acquisition of that overcoat.
Insel was completely cured of his obsession. I have never known any man to catch so many women. He seemed to be somehow barricaded with women. All my indulgence for human misdemeanors (which are so commendable when aesthetically good—such as the stellar combine of Insel and his ebony wives, his ivory eroticism in appraising thighs) was unavailing, confronted with this blatant lubricity of the normal Insel which, as he boasted, although in proper decency of word, seemed as did once an astral Venus to flow in his very veins: The dregs of all the secret gutters that carry off the unavowable residue of popular conceptions of physical life—
When I arrived with the rest of the loan, anxious to clear him out, my once luminous _clochard_ had composed himself in the kitchen holding his usual insignia, the heel of pumpernickel, this time one in either hand—extreme oval ends—unbitten—of an absent loaf. He looked forbidding as had they been bone.
"Did you get the overcoat," I inquired amiably.
"I may as well tell you," he snarled, "that I don't care for all this supervision—I had not the time. You understand—the last nights in Paris," he raved ecstatically. " _Es ist so schöndas Leben, wenn mann so leben kann_— It is so beautiful living, when one _can_ so live."
His emaciation no longer of flesh had become an exteriorized act of the flesh in which the last ooze of the spermatic juices might have been, in some fearful enervation, spent. Instead of being suffused with that liquidity of relief following upon embrace, his eyes, in some ultimate heat, were boiled to the creamy, soiled putrescence of stale oysters in a stew.
I did not reflect that this enormity of sensuous filth was probably as unreal as his nervous aromatics distilled from his astral collusions with a goddess. It was a mental impossibility to associate these opposite phenomena. Had I recalled the earlier iridescent Insel, it could only have been as a figment of _my_ insanity.
An alarming presentiment occurred to me. "Insel," I gasped, "you've blown that thousand francs."
"What are you?" he sneered venomously, "an inquisitor?"
"He has notions as to how white women should be handled, too," I laughed to myself as I hurried down the corridor to the dressmaker.
I was determined to take conventional leave of a guest who would be gone when I returned to Paris. It would put me to great pains, I supposed, breaking through animosity so unaccountable it left nothing intact but surprise. Still, it was pretty bad if I could not prevent the "epidemic quarrel with me" from spreading to even this lunatic whose essential void I had found so soothing.
After my fitting I invited him to come down to the cafe, intent on buttering him up, on bluffing him into forgetfulness of having allowed me to discover his awful alter ego (in cases of the sane, this alter ego seldom got to work until out of my sight), curious to see if we could part on good terms.
As we stood face to face with nothing in common, the last people on earth likely to become acquainted, I saw him force back his loathing, to accept. Our mutual distaste was noxious on the palate. We each had a pressing engagement for dinner.
I remembered Geronimo taunting me that I was "no psychologist." "You just walk into a man's brain, seat yourself comfortably in an armchair to take a look around—afterwards, you write down all you have found there," he had said. Then what the hell in Insel had I "walked into"? His complaint was true. Nobody saw in him what I saw in him. A kind of consciousness unconscious of its own potency. Even now he was disgusting to the point of revelation.
Insel had also the idea of bluffing a conformative wind-up to our illusory alliance. Resorting to his earlier priggish decency, once we were in the back of the cafe, he hung his head, apparently poisoning it with spurious shame, and mumbled:—
"The bad thing about me is that every now and then I come to a blind alley in my life—where somebody has to help me."
"Now look here, Insel," I persuaded him with stimulant hypocrisy, "if it were not for that basic something in you—no help would be forthcoming. That which is valuable one does not _help_ , one responds to a cosmic imperative."
He began to look as if he had been overdoing the shame.
"There was some mention," I added offhandedly authoritative, "of you busting a thousand francs. You seemed on the defensive. But _why?_ The artist requires _color_ in his life." This fallacious insight melted Insel's imitation shame, disclosing the very really wounded face of a child who has long been sulking for being misjudged.
"You told me," he burst out unhappily, writhing with reproach, "that I put lots of money in the bank."
So that was it. Insel, with his organic magnifications, had become a foul lout, because he was feeling—cross.
"I didn't," I fibbed, striking the suitable note. "I said you hid it under the carpet." Neither of us had a carpet—we immediately floated off as if on the magical mat of Baghdad, talking on—.
I could feel any word I was saying fit into Insel's brain appeasing as a missing piece in a jigsaw puzzle.
At once—it was growing late—he clamored for me to stay with him; for that period in which alone he seemed to recognize duration—forever.
Probably I was the collaborative audience to his finest act, the giving off of that calm equation that always reduced me to a hushed respect. He grew in power in his silent "role" in ratio to my reaction. The ultimate self, august in certainty, put forth a soporific bloom that covered his damaged face.
Only now I remarked that on the emergence of this ultimate self in its intangible armor of nobility depended that prolongation of time I so often experienced in the company of Insel, for at present there was no aquarium diffusion, none of that virtually giggling attainment to Nirvana. No x-ray excursion nor any fractionation. His medium-ship concentrated in a sole manifestation. This interference with time.
I could not make out whether the cause was a shift in the relative tempos of a cosmic and microcosmic "pulsation," whether _my_ instant—the instant of a reductive perceiver—passed through some preponderant magnifier and enlarged, or whether a concept (become gnarled in one's brain through restriction to the brain's capacity) unwinding at leisure, was drawing my perception—infinitely soothed—along with it. For again this novel aspect of time seemed, like light, to arrive in rays focusing on the brain at a minimum akin to images on retinas; and the further one projected one's being to meet it, the _broader_ one found it to be. Anyway, it was useless trying to analyze it. This alone was certain. It was absolutely engrossing to the mind, although nothing brief enough for us to cognize _happened_ in this longer time, which occurred commensurately with the bit of lingering I was wedging in for Insel between contiguous hours in defiance of occupational time.
Rarely, at intervals of aeons, Insel and I would look up at each other in an utter yet somehow communicative impersonality, the final relationship of distinct similars confronting the same phenomenon.
# **25**
INTIMATELY CONFIDENTIAL ONCE MORE, INSEL was trying to disentangle before me _the thousand directions_. He had shown them to me previously, in answer to my asking him why he did not work although I had left him materials in my studio.
"So often at dusk I come here to stare at that white canvas," he had told me dreamingly. "I see all the worlds I could paint upon it. But _um Himmels Willen_! Which one? I can create everything. Then what thing? A thousand directions are open to me, to take whichever I decide—I cannot decide."
I had long ago worn down in contemplation of that multiplicity of direction. How _far_ my mind had traveled; never to come to the _beginning_ of any route. Surely, for Insel it should have been different—starting with the spectral spermatozoa that seeped from his brain through his gardening hands.
The glare in Capoulards Cafe grew dim. Insel's brain floated up from his head, unraveled, projected its convolutions. They straightened in endless lines across a limitless canvas, a map of imminent direction. On the whole of space were only a few signboards on which grew hands, alive and beckoning.
"Of course," I was saying, "I don't know where you are—wherever it is is very far away. And I am just as far away. I have existed before my time."
"How true," said Insel.
"Whatever I have found out belongs to a future generation."
"How true," said Insel again, devoutly.
"And by the bye," I commented, "the sentiment of one generation is the neurosis of the next. All that stuff you have of 'suffering for love' is the most awful slush."
"I know—I know," he agreed with fervor.
—Indefinable lines of cerebral nerve marked on the map of inertia, unrealizable journeys. Along one route, _die Irma_ dissolved to a puddle of serum, to be absorbed by the all-pervasive whiteness. To travel there was difficult; that volatile fungoid lichen outcrept one as one picked one's way, grew tall until one must turn back.
My former trust in the ripening of Insel's work had had its foundation in that very "parting of the ways" he told me he had come to, where he turned assured toward something eternally immune to his host of elementals.
It had taken so short a time for this parting of the ways to subdivide into the thousand directions. Yet even now he was rich in postponement. While that commonplace back of a woman watching for signs on his painted firmament turned in anonymous patience to this chart of unarrival.
The curtain of the sky came down and she was not there— "If the painting no longer 'goes,' " Insel surprisingly was ruminating, "I shall do as you do. Write. What a profession. One carries one's studio about with one. A sheet of paper—"
Because it was only a brain that had been spilled, the blank of orientation faded—the thousand directions withdrew, leaving us at a destination.
Nothingness.
It was not black as night nor white as day, nor gray as death—only a nonexistent irritation as to what _purposed inconsequence_ had led us into the illusion of ever having come into being.
The haunting thing about this Nothingness was that it knew we were _still there_ — Two unmatched arrows sprung from its meaningless center—were surrounded by a numeral halo—I _had_ to leave Insel, it was ten to eight.
# **26**
"HOW GOES THE BOOK," HE ASKED WITH HIS FORMER appreciative intimacy as we passed out of the cafe. I was feeling exceptionally "good" about my work just then, vainly imagining I had criticized my last incompletion.
"It is going wonderfully," and with a flash of that exhibitionism of the spirit succeeding to inordinate periods spent with no means of communication—I threw out my hands—elatedly believing I had reached the stage prescribed by Colossus for creation, when all that one has collected rolls out with the facility of the song of a bird.
_"Sehen Sie_ , Insel," I explained, " _Man muss reif sein_ —One must be ripe."
I felt Insel crack as if he had been _shot_ alert.
"Can she possibly mean it," I could _"hear"_ him ask himself as he wheeled towards me, noticing me for the _first_ time; and then convinced, as I stood a little exalted on the corner of the street, decide, "Here is a woman with whom there is absolutely _nothing_ to be done."
I must have had my hands outspread, for Insel dropped like a soft moth into my open palm— On his face was a smile unlike all the fluctuant smiles of hallucinated angels I had watched there. It was a normal smile. Yet in the old abnormal voice of whispering emotion, laying his dried branch across my shoulder, he choked, " _Ich komme nach Hause_."
He was "coming home."
Across his gentle brow floated the will-o'-the-wisp trailing a pair of boiled oysters in its wake, _Mädchen_ , like missiles that have not gone off, he scattered abroad.
"But Insel," I reminded him, "you have an appointment for dinner."
Insel gaped at me.
The illocal foci of his pupils exploding incredulously, darted in all the directions of the radial underpattern of his life. It took some moments to sort these simultaneous impressions. When I had done so, I longed to get even with Insel, to say "I have absorbed all your _Strahlen. Now_ what are you going to do?"
I said nothing of the kind. Because firstly it was not true, and secondly, it might inspire in him a worse obsession; for one thing one feared as above all else menacing Insel was some climax in which his depredatory radioactivity must inevitably give out.
So all I said was "Good-bye."
"Good-bye," smiled Insel, his bittersweet stare both dazed and stoic, " _Danke für alles_ —Thanks for everything."
# **VISITATION**
I
In the organic continuity of family life, one is under the earth caught among bare roots of imperceptible plants whose flowers lean out smiling toward the solar stimulation of a heterogeneous society.
II
Women particularly—lose their lovely faces in private—so much so that it is only in the occasional hazard of a party one may gauge the effect of creatures, one has actually in some remote biological process given birth to.
III
Alda who, in a crowd, caused me to blink as at the too near approach of a brilliant star—
IV
totally extinguished on her filial visits. Her face almost blotched with a fundamental erosion my essence produced in her—developed a kind of set jowl. As she sat down before me she would clutch that soft white fist. I watched it grow rosy as it squeezed out the inadvertible tide of my futility.
"Aaron," she announced, "doesn't see why he should give you that hundred dollars"—and with that heinous crow I seemed to call up from the depths of so many of my intimates— _"Your book!"_ she sneered, "It's an excuse ___ to get money out of us!"
V
"You're no good—never have been any good—" This blank truth struck me with the finality of unconsciousness. It was from very far away in time & space I heard her aggravation hollow out a course for my second childhood.
"You _wanted_ the business—we gave you the business—You wanted an apartment—we gave you the apartment and you sell it for nothing & come over here!"
"But Aaron told me to sell it at that price — — —"
VI
I expostulated.
"Pooh—he was drunk," Alda retorted in a streak of decision.
It is the reverse of enlightenment to see oneself 'in reality'. Of the image & likeness that forms our inexpressible Being—in the metamorphosis of passing through other brains—all that appears to our companions is a chimney sweep.
VII
As she drew to a close, taunting me with my "painting _ _ _ that idleness where other artists prepared a whole exhibition in two months," Fact dilated for me. Alda's recriminations were identical with mine of myself. Incipient in my mother's womb their transcription effacing time in me they now reverberated from the lipsticked mouth of a child I loved.
VIIA
My year of psychic discipline of those recriminations had gone for naught. Returned, they dragged my frightened ears even in the direction of the grave.
VIII
"I can proove it" _ _ _ _ Alda was babbling her way to the door "with your awful belly-aching letters _ _ _ proove it to anybody. I've kept them all."
IX
Soon my breath grew regular again. "Now tell me," I asked Sofia who had been present—"am I a disgruntled old nitwit who imagines monstrous things being said to her _ _ or did you hear what I heard?"
"I heard," Sofia answered, "You imagined nothing"—
then with a flat neutrality—"she _intended_ to be _cruel __ _ _ _ __ So what? Do you think it's exceptional that a daughter should hate her mother—"
X
Sofia, after that prolonged séance with her make-up which condenses woman's life, returned in her hat & coat.
—– "Shan't be back this evening."
"Then _would_ you buy me a sandwich there's nothing to eat."
"No time"—she objected—
Bewildered, I reminded her she had asked to housekeep for me—
XI
"I have no intention of doing so—you're a beastly nuisance."
"But Sofia—I don't understand. You begged me to come—"
"I _had_ to have you here—to be able to get off on you all I dared not 'get off' on Alda __ __ I'm scared of her," she smiled engagingly.
XII
I also smiled as she left me alone. Intellectually it was refreshing, this ability of hers to express unabashed exactly what she felt with an honesty unveiling the ego. Ignoring distinctions between thee & me—she was with
XIII
precise calculation equally unbiased about the (rare) unpleasant or unfair reports of her made by other people.
XIV
Nevertheless my pain, itself behaving like an insupportable hunger, became grotesque when coupled with normal appetite, whereas, should I venture outside the cold would cleave it with a super-phenomenal blade.
XV
I ate a pat of butter & some dry corn-flakes left in the kitchen, then sickeningly relapsed to the depths of the divan. The pain stood out sharply as if in spite of the dim amber lamps it cast the impenetrable shadow of the gloomy sitting room.
XVI
I had lain there for a long while alternating that halfhearted squirm one opposes to agony & that unwilling patience imposed by agony, when, all at once the compact silence became curiously volatile. Drawn from my couch, I rose erect, walking, so far did my head turn sideways, rather like a crab. As if again I must 'take stock' of someone as I went my way.
XVII
There was no mistaking this ecclesiastic 'current'. Here was my drug addict; divested of those shreds of flesh, easily as an aria relayed across the Atlantic, a recognisable 'invisibility' come to visit me.
XVIII
As an automaton I returned his salute, with the same ecstatic, friendly yet clerical benediction whose significance I realised, as I inclined in _that_ direction, to be our mutual forgiveness. For his dope-ring duplicity? My written account of him?
XIX
His 'presence', conveying a solemn hilarity, declared in my brain "Ess ist doch nicht schlimm genüg __ _ _ Nothing they can do to you is bad enough _ _ _ _ you're a revenge on your unfair advantage __ _ _ they cannot see what we see."
And the pain lay dead among the shadows.
XX
This reminder of the strange attributes of the drug fiend renewed my curiosity as to the major factor in the human make-up.
Man's dynamism.
According to my experience in Geneva the force that drives us is of incalculable voltage conducted by the spinal column in the manner of a lightening rod.
XXI
If, as I suspect, we have our existence in an intelligential ether this force [flux] of life conveys to us not only our animation but also our intellectual concepts.
[ _MISSING XXI A_ ]
XXII
There are two modes in meditation, one in which the intellect functions with supernormal rapidity; one in which eased of even the normal staccato it slows down to the tempo of a prevalent wisdom at peace.
[ _MISSING XXIII_ ]
XXIV
Now I was engaged with a kind of surrealist man. Constructing, demolishing him kaleidoscopically, hoping to demonstrate how he 'worked.'
Made of that Shadow, beside me in Geneva, whose universe re-emerged as the omniprevalent ray struck him. What I seemed to be so intent on discovering was the nature of the
xxv
fusion of that Ray with himself.
An Island in the air sustained by unseen attributes, this man derived his form from the symmetric evidence of the one half of the man being a replica of the other half. Attached to his blind back, his antedeluvian tail anchored him in the past.
XXVI
Nuzzling the future, the features of more sharp-scented animals have dwindled to incomparable beauty in his face of pinkish pulp.
Behind this fragile front lies a delicate radio-raceiver of cosmic urges which canalised, intricated, misconstrued by his brain, compose the rhythm of his individuality.
XXVII
Become clair-voyable, whereas his body displays a crimson circulation, another half-extraneous phosphorescent circulation, some vortex in the intelligential ether spins through his head; as though he hung from the cosmic consciousness by a ring of light.
XXVIII
Taking on another aspect, emitting electric waves, he broadcast his thoughts which were returned to him conditioned by their effectiveness; ideas, operative as hands, shaping events.
XXVIII A
While, as directed by remote control above him in an 'atmosphere' enveloping his brain, shone the magnetic beam that guided him—the soul.
XXIX
I saw him submitted to opposite gravities, terrestial & celestial, pulling him downward & upward. When these were equal, he was in equilibrium. When he responded only to the terrestial, his body became heavy like lead; when more rarely, to the celestial, his spirit lightening, he diminished in weight.
[ _MISSING XXX_ ]
XXXI
So Manifold are the workings of the life-force _ _ so vast its resource_ _ _ Again [__] man appeared to me in the phenomenal world with his head at the same time in the eternally revealing cosmic consciousness.
In this consciousness lay strata of various inspirations __ _ _ somewhere among them a strata of _absolute felicity_ to which the majority of minds vaguely aspire. The clerical locate this Felicity in a region, the lay-man in a reaction, in this surrealist man the reaction derives from that region.
XXXII
Out of his head arose an ethereal dumb-waiter, stopped at the desired strata & having taken on the provision required descended to the intellectual laboratory __ __ __ __ __
The elevator falls apart, leaving antennal strands feeling their way into the stratal continuum. Up there where he is aware of the penetration of his mind by an extra-luminous radiance.
XXXIII
A cosmic obviousness everywhere _defined_ escapes him completely, intangible as God.
The destructible robot, soft machine, senses a mystery, & as if attempting to locate the 'genius' revealed in a work of art through the analysis of the chemical properties of paint digs ever more deeply into his island Base in search of the origin of his impeti _ _ _
For a moment, he imprisons the omniprevalent 'leaning' towards intercommunication in a gland _ _the last _ least co-operater becomes the initiator.
XXXIV
But when I watch this Sur-realist Being for long, I see him turn from his unfolding of concentrated distance; dropping his microscope in favour of an opposite lens which, contracting diffused distance, brings the unprecedented patterns of that cosmic obviousness he faces, within his view.__________
The surrealist man is very short, awakened by desire—eclipsed by ennui.
xxxv
The surrealist man is very long, stretching like a live wire from 1938 as far into the future & through equally numerous stages of evolution as he reaches into the past. His beginning is a speck of transparency, impinged upon by the sun. His ultimate presence would have been virtually invisible to a twentieth-century eye.
His way is strewn with stone implements, embedded bones & machinery he discarded as superannuated models of functions he slowly develops within himself. Transport telepathy, radio, & television together with surprising future facilities are effected by 'centres' in his cerebellum controlling the various potentials of the life-ray. The religious symbols of the precocious visionary in his early days, translated, become the 'scientific'
XXXVI
commonplace of his further condition.
Even as of old angels grew wings & emitted haloes, he is buoyant in defiance of atmospheric pressure, his brain gives off a radium glow become apparent. He has X-ray eyes.
XXXVII
Arduous is his transformation. While experimenting upon the regulation of his electo-atomic velocities & resistances, he must pass the danger point at which he takes the risk of the power that holds him together dynamiting him with his own force.
xxxvIII
Of this danger, as of every phase he passes through, he stages repeated rehearsals with his heavy mechanical toys.
Playing the role of a bombastic cell in an aggregate organism blasting surrounding cells to make room for his own inflation; his mind still bound by numeric (al?) restriction & geometric space waivers an infinite accommodation he imposes upon
XXXIX
himself a human menace—from without.
In an amazing 'dédoublement' he confronts himself with an 'Enemy'. Avid aggressor whose terrifying eyes are the eyes of an incontactable alien.
During the ensuing _horror_ any observer at large may witness a conjurous displacement: viewed from the opposite side the assailed becomes the assailant. He is blowing up his simulacrum.
40
"Mamma! I can't set the curls at the back of my neck."
In lightning metamorphoses, the clockwork of the surrealist man runs down.
At once an atom indistinguishable among a frontierless agglomeration and a tower of Babel built of all mankind _ _ _ he fades _ _ _ in ephemeral undulations to the etheric contour like a frame for training a fancy box-tree his substance clings to.
Now only the searchlight shafts of his future eyes __ __ __ __
[XLI]
For years, I had been submitted to the tedium of the imaginative living among races conceiving no final outlet for their dynamism but destruction, forced to inertia by the rush of intellect in the wrong direction, until the casual accident of chance threw me a dope-fiend—guinea-pig for experiment—in research on the _spirit_.
[XLII]
In the make-up of normal man, his good & evil are proportionately mixed. The outstanding characteristic of the drug addict is their separation; their awful alternation.
We hear that a drug in impairing nerve tissue produces a vicious exaltation & our curiosity is no further intrigued. Nevertheless I had come upon a creature of my own species intermittently enveloped in an aura identical with the atmosphere of some cathedrals in which one catches an actual detonation of a sanctity amassed through the ages.
[XLIII]
Sophia, rising from the incredible chaos she produced in the tiny bathroom, her arms white snakes 'before the fall', was weaving in the air the rhythm of her toilet.
Under my fingers the clammy tendrils clinging to her neck sizzled in the curling tongs. Her curses of procrastination crackled about my head while through the slab-like snow of her luminous back that faint electric 'comfort of life' conveyed her intrinsic aloofness of honnied marble. The silk, as if pleased to find no intervening fabric, slipped on the bare severity of her body.
I 'do her up'.
Five strass discs confined to the acute concavity of her waist, crests of soft rocks, the pyramidal folds of a taffeta the colour of dim coal.
"Why the hell must you go and marry a great cow of a man? I'm huge!" she exploded.
The glitter of a girl prepared for a
[XLIV]
party drew the depths of her eyes to the surface. A tinge of azure underlying the shadows & roses of her skin unfolded in the beauty of her face an ineffable magnolia.
With the deep velvet of her cloak, she doused the unbroken harmonies of a figure she could not 'see'. Her radiance flared in the slam of a door, leaving a scattered ash of toilet articles & undercloths.
I picked them up with the successive effort of manual acts performed while the brain is tracing a dissimilar diagram __ _ _ _
[XLV]
until as in the confusion of uneasy dreams I must identify that Beam controlling a surrealist man with the high-light on a fallen curler _ _ the scintilla assuming an intermediary significance __ the phosphorescent drug-addict, like a guinea-pig for experiment, flickers within range of my speculations. It is, in as far as I am aware, no particularly cleanly matter from which radium is extracted.
*
End of Book
Visitation of Insel
*
# **AFTERWORD**
_Insel_ is a novel written by a poet, with a poet's interest in the sounds of words. What is at first most striking, and of special interest to readers of Loy's poetry, is the adamantine, alliterative quality of the language here which, like the slow piling up of latinate diction and byzantine phrasing in her poems, makes Loy's novel difficult. But what may on first reading seem byzantine and unapproachable is the very quality which gives Loy's writing an austere beauty that repays the attentive reader. Choosing the most resistant subject matter, and employing language at once stony and visionary, she finds beatitude in the most unlikely places. Insel the _clochard_ , the ethereal bum, belongs to a long line of materially destitute characters in whom Loy located spiritual riches.
The arduous language with which she develops Insel's character reflects a decision to persist in the struggle to hammer the ineffable out of the hard physical matter of language, paint, stone and metal that were Loy's media as a poet and visual artist. The narrator's fear for Insel, and for herself as she comes under his spell, derives from Insel's disengagement from the physical world, which—in spite of its imperfection—provides the material for art. Although she is repeatedly tempted to join Insel on his flights into the "increate," lured by her glimpses of beauty in the perfect peace of his vision of the absolute, she is ultimately repelled by the way this vision turns one away from life rather than toward it. Her resolve to fight Insel is remarkable given the force with which his vision attracts her: "If Insel committed suicide—I could share in that, too." The promise of a blissful reprieve from life's suffering proves almost too great a temptation for the narrator, who later will need to weave her disintegrated self back together after an outing with Insel.
But just as the mantras of "timeless peace," "perfect happiness," blooming fragrance and space are about to pull her under, she happens to glance at a cafe clock, on whose "uncompromising dial all things converged to normal." "In my veritable seances with Insel, the clock alone retrieved me from nonentity—thrusting its real face into mine as reminder of the temporal." This periodic attention to the clock prevents her from merging with the otherworldly Insel, who seems to be on his way out of life, having relinquished his right to secular existence. Frequent appointments with friends and other artists—a relentless schedule—provide a structure within which the narrator can both experience Insel's world from a safe distance and maintain the balance necessary to record her experience of his "Edenic region of unreasoning bliss," which in spite of its destructiveness she values. In her description, Insel visualizes "the mists of chaos _curdling into shape_ ," just as she herself seeks to evoke "a chaos from which I could draw forth incipient form." The narrator consistently pushes the "procreational chaotic vapor" that threatens to destroy both Insel and herself in the direction of artistic form. It is no surprise, then, that the narrator's final victory over Insel—the definitive moment of the book—coincides with her success as a writer. By the end of the novel, she has reached the necessary compromise for the practicing artist: to make the most of the flawed human condition, to refine as much as possible the imperfect media available to the artist in this world. She encourages Insel to do the same, to get back to his painting in spite of both his financial worries and his precarious hold on reality. But she can only be sure that she herself will keep her balance; she leaves Insel at last to fend for himself.
Insel, the character, is modeled on the German surrealist painter, Richard Oelze, with whom Loy was acquainted in the mid-1930s in Paris. Rumor has it that Oelze was addicted to opium, and that Loy may have helped him recover from his addiction. Though in _Insel_ and many of her poems Loy focuses on how decadence incapacitates the artist, she also makes a point to cast in sharp relief the actual devastation of individual lives brought about by drug addiction, poverty, and madness. Throughout her writing career, Loy gravitated toward the rockbottom of human existence for her subject matter, always struggling to locate what beauty or hope might reside there, but without romanticizing the anarchy or squalor.
Mina Loy met Oelze in 1933. By this time, she had already written two other fictional accounts of avant-garde figures she knew, neither of which was ever published. _Brontolivido_ satirizes the Italian Futurist, F. T. Marinetti, and _Colossus_ describes her relationship with Arthur Cravan ("Colossus" in _Insel_ ), the proto-Dadaist poet whom she married in 1918. During the twenties, Loy had associated with several other expatriates living in Paris, including Ezra Pound, James Joyce, Natalie Barney, Djuna Barnes, Gertrude Stein, Constantin Brancusi, and Peggy Guggenheim (probably "Alpha" in _Insel_ ), who helped Loy financially, arranging exhibitions of her art work and backing her lamp shade business for a time.
From 1931 until she left Paris in 1936, Loy worked as Paris representative for her son-in-law, Julien Levy ("Aaron" in _Insel_ ), an art dealer and collector whose New York gallery introduced surrealist art to America. Her job was to commission paintings for the gallery from artists, such as Oelze, who were living in Paris. Earlier associations with Marcel Duchamp, Cravan, and Man Ray had given her entrée to André Breton's circle of surrealist artists in the twenties, and she successfully commissioned work for Levy's gallery from Max Ernst, Salvador Dali, René Magritte, Alberto Giacometti, Giorgio de Chirico, and other major figures of the movement. Chances are that it was in this capacity that Loy met Oelze, who arrived in Paris on the last train out of Hitler's Germany and, 33 years old and relatively unknown as a painter, continued an itinerant lifestyle that ended only after the war, when he settled in Worpswede. Around the time Loy knew him, he seemed always to be passing through the places he lived in, invariably choosing an apartment near the local train station.
In _Insel_ , she comments that Oelze did not speak a word of French, and that his "will-o'-the-wisp" behavior extended to his association with the French Surrealists, with whose work his own paintings have been grouped and among whom he might have found kindred spirits, or at least sympathetic colleagues. But Oelze assumed the pose of the reticent mingler rather than the blind conformist in Breton's regimented inner circle, just as Loy had assumed the role of critical observer in her associations with the Italian Futurists and the New York Dadaists. Oelze hid behind the language barrier and the identity of the transient.
Along with a mutual respect for each other as artists, it may have been this shared aversion to wholehearted membership in groups that drew Loy and Oelze together. In all of her associations with the avant-garde—she was well-connected with the important artistic and literary circles of the first decades of the century in Europe and America before she became a virtual recluse in the Lower East Side of New York—Loy fought to maintain her independence, and survival, as an artist. Likewise, Oelze seems to have developed a similar strategy with regard to the Surrealists. His first exposure to surrealist art came in 1921, when he saw reproductions of paintings by Max Ernst and Hans Arp in Ascona, Italy. The favorable impression they made on him eventually drew Oelze to Paris in 1933, where he soon met Ernst and struck up a friendship with Paul Eluard. He showed his paintings at a few of the Surrealists' exhibitions, but his contact with Breton's crew was sporadic at best, and when he did encounter them _en masse_ , he acted coy.
As time passed, Oelze moved farther and farther from the group, preferring to shut himself up in his sparsely furnished workroom to paint rather than to be seen at surrealist events. Though concerned about his psychological well-being and the precise direction in which he was headed as an artist, Loy apparently respected Oelze for his fundamentally surrealist nature and his independence from the surrealist group. She seems to have believed that, in spite of his periods of inactivity, this behavior was evidence of a more serious dedication to his art. Throughout her life, she struggled with the conflict between an attraction to centers of artistic and literary activity—meeting the Futurists in 1913 had jolted her out of a long debilitating isolation—and the need to stay at home and work. In a 1929 _Little Review_ questionnaire, she confessed that her greatest weakness was compassion, and her greatest strength was her "capacity for isolation."
The frequency with which social outcasts of every description appear in her poems and fiction reflects a concern about the possibility of maintaining one's integrity as an artist while part of a group, be it the middle class or the avant-garde. Her interest in Oelze continues this pattern of ambivalent feelings about avant-garde groups she had been associated with since she met Marinetti. Though she welcomed the heightened level of artistic activity and social life that surrounded avant-garde groups, she wasn't interested in collaboration; she couldn't abide by the tendency of the avant-garde to view works of art as means to political ends, for example; and there was no place for a serious woman artist in the elitist fraternities that these groups often became. Thus, it is not surprising that Loy was critical of the surrealist idea that the work of art is valuable only as a means of achieving the mental state of surreality, as well as of the Surrealists' tendency to view women as passive muses incapable of the work of the serious artist.
She takes her criticism of the Surrealists one step farther when she questions their very notion of what the surrealist state of mind actually is. Insel was, according to Loy, "more surrealistic than the Surrealists"; he "possessed some mental conjury enabling him to infuse an actual detail with the magical contrariness (that French) surrealism merely portrays." When Insel joked that the Surrealists wouldn't have anything to do with him because he'd ask them for money, Loy's narrator replies, "I should have thought you'd be _worth_ a little money to a Surrealist. He might learn what supereality is about— you are organically surreal—...."
In this way, Loy uses Insel to set herself not just apart from but far above the Surrealists while at the same time guarding against this quintessential Surrealist's instability and misogyny. The narrator's defiant farewell to Insel at the close of the novel sounds feminist but does not come across as hollow feminist dogma; her victory over his seductive aura and near violence is hard-won, and the tie to survival as an artist gives her victory more breadth. Loy's emphasis on preservation of the integral self or ego in _Insel_ affirms her life-long concern about her identity as a practicing artist. In this sense, _Insel_ can be read not only as an experiment in surrealist narrative, but as a satire on the whole surrealist endeavor. If this is true, the similarities between Loy's _Insel_ and André Breton's _Nadja_ bear more than a passing consideration. Loy may have actually structured her novel after Breton's in order to satirize him—as Victorian-styled middle class voyeur—and to express her indignation at the compromised role the Surrealists assigned to women.
Throughout her long career, Mina Loy kept a sober check on what glimpses of the other side the difficult and painful world can offer, partly because she recognized the futility of attempting to live in this world as if it were the next one, and partly because she was committed to producing an art with a measure of integrity. The limits Loy places on her narrator in _Insel_ reflect this commitment, as does the narrator's victory at the end of the novel, when she asserts her authority over what up to this point has been for her a vision of overwhelming, and mostly destructive, power, with Insel in control. Finally, she is able to draw Insel's attention to _her_ power. By transmuting his " _Sterben_ — _manmuss_" (Die, one must) to " _Man muss reif sein_ —One must be ripe," she shocks Insel into a new way of seeing; he notices " _me_ for the _first_ time." The narrator has surpassed the richness in postponement that paralyzes Insel, and Mina Loy has completed her novel.
Elizabeth Arnold
1991
# **APPENDIX A
TRANSLATION OF FOREIGN WORDS AND PHRASES**
this page | _belote_ : pinochle
---|---
this page | _Mädchen_ : girl
this page | _Huissier_ : sheriff [Loy's note]
this page | _Es war wirklich prachtvoll_ : It was really splendid
this page | _Der Prozess_ : _The Trial_
this page | _Zum Teufel_ : What the devil
this page | _Was haben Sie schönes erlebt_ : What beautiful experiences have you had [ _Loy's note_ ]
this page | _clochard_ : tramp, hobo, bum
this page | _Elle n'a pas froid aux yeux_ : She does not have cold eyes
this page | _Fleisch ohne Knochen_ : boneless meat
this page | _carrefour_ : intersection
this page | _Je suis la ruine féerique_ : I am an enchanting ruin
this page | _La faim qui rode autour des palaces?_ : Starvation prowling palaces? [Loy's note]
this page | I read later that sugar was used for strengthening concrete. [ _Loy's note_ ]
this page | _Vielleicht verkaufen_ : Perhaps to sell
this page | _Die nackte Seele_ : The naked soul [Loy's note]
this page | _schade_ : a pity [ _Loy's note_ ], i.e., too bad!
this page | _Sterben_ : To die
this page | _Sterben—man muss_ : Die—one must
this page | _Unglaublich_ : Incredible
this page | _consommation_ : drink, snack
this page | _Ameise_ : ant
this page | _cafés fines_ : coffees and brandy
this page | _librairie_ : bookshop
this page | _maquereau_ : pimp
this page | _macrusallo_ (i.e., maquereau and salaud blended together)
this page | _plat anglais_ : a plate of cold meats
this page | _Um Gottes Willen_ : For God's sake
this page | _Pfefferminztee_ : peppermint tea
this page | _sommier_ : divan
this page | _Strahlen_ : rays
this page | _Ich bitte Sie_ : I beg you
this page | _femme de ménage_ : housekeeper
this page | _bidons_ : cans
this page | _Der Totenkopf_ : The death's-head (In earlier manuscript versions and in letters, Loy called the novel _Der Totenkopf_. —Ed.)
this page | _pour se faire une beauté_ : to make himself up, to do his face
this page | _Chambres de Bonnes_ : Maids' Rooms
this page | _Das ist die Irma?_ : That's Irma?
this page | _Die Irma ist nass_ : Die Irma is wet [Loy's note]
this page | _ou connait ça_ : or knows that (obscure: perhaps a slip for " _qui connait ça_ , who knows that")
this page | _lustig_ : jolly [Loy's note]
this page | _grand sympathique_ : the sympathetic nerve
this page | _Gestatten Sie?_ : May I?
this page | _Entwicklung_ : development
this page | _écoliers_ : schoolchildren
this page | _Sterben—Man muss_ : One must die (see this page)
this page | _Ich bin so müde_ : I am so tired
this page | _Il dort dans son dos_ : It sleeps on its back
this page | _Und Tatsächlich_ : "And as a matter of fact" [ _Loy's note_ ]
this page | _trompe l'oeil_ : deceptive appearance, illusion
this page | The poet Arthur Cravan ("Colossus"), Loy's second husband, is considered a precursor of the Dadaists and a patron saint of the Surrealists. (Ed.)
this page | _Seien wir uns wieder gut_ : Let us like one another again, let's make up
this page | _die Rothaarige_ : the redhead
this page | _um Himmels Willen!_ : for Heaven's sake!
# **APPENDIX B**
**CHRONOLOGY OF MINA LOY**
1882 | b. December 27 as "Mina Gertrude Lowy," London, England.
---|---
1899 | Studies art with Angelo Jank at Kunstlerrinen Verein, Munich.
1901 | Studies with Augustus John in London.
1903 | Moves to Paris; marries Stephen Haweis.
1905 | Enters Gertrude Stein's circle of artists and writers.
1906 | Elected member of Salon d'Automne; moves to Florence.
1907 | Joella Synara Haweis is born.
1909 | Giles (John Giles Stephen Musgrove) Haweis is born.
1913 | Meets F. T. Marinetti and other Futurist artists and writers; separates from Haweis.
1914 | First poems published (Camera Work).
1916 | Sails for New York City, where she enters Walter Conrad Arensberg's circle of artists and writers; meets Man Ray, Marcel Duchamp, Arthur Cravan, etc.
1918 | Travels to Mexico; marries Cravan in Mexico City. She travels to Europe via Buenos Aires; he fails to meet her as planned; he is never seen again.
1919 | London; Lausanne; Florence Fabi (Jemima Fabienne) Cravan is born.
1920 | Returns to New York; deepens her ties with American avant-garde writers and artists.
1921 | Paris, Florence.
1922 | Florence, Vienna, Potsdam, Berlin.
1923 | Settles in Paris; first book of poems published (Lunar Baedecker [sic], Contact Pub. Co.).
1925 | Narrative poem, Anglo-Mongrels and the Rose, completed.
1927 | Gives talk on Stein and reads her own poems at Natalie Barney's salon.
1931 | Becomes Paris representative for her son-in-law's (Julien Levy's) New York gallery.
1933 | Meets Richard Oelze.
1936 | Moves to New York; lives in the Bowery; friendship with Joseph Cornell; begins revising prose works begun in Paris, including Insel, presumably.
1946 | Becomes American citizen.
1949 | Moves to heart of Bowery; less and less contact with old friends living in New York.
1953 | Lives near her daughters in Aspen, Colorado.
1959 | Exhibition of her work, Bodley Gallery, New York.
1966 | Dies, September 25, in Aspen.
(Adapted from _The Last Lunar Baedeker_ , ed. Roger L. Conover [Highlands, N.C.: Jargon Society, 1982].)
# **APPENDIX C**
**CHRONOLOGY OF RICHARD OELZE**
1900 | b. June 29, Magdeburg.
---|---
1921–25 | Studies at Bauhaus, Weimar; travels to Berlin, Hamburg, Leipzig, Köln, Düsseldorf.
1926–29 | Spends time in Dresden; more than a month-long visit at Dessau Bauhaus; travels to Essen.
1929–30 | Ascona.
1930–32 | Berlin; more than a month traveling in the Gardasee.
1932 | Late fall: Travels to Mainz and Frankfurt/Main.
1933 | March 31: Takes train across the border to France; settles in Paris; loose contact with Breton, Dali, Eluard, Ernst; October: exhibits paintings in Salon des Indépendents.
1936 | October 1: Leaves Paris for Switzerland.
1936–37 | Ascona.
1937–38 | Positano.
1938 | Returns to Germany: Essen, Mulheim/Ruhr, Magdeburg, Berlin.
1939 | Early in the year: settles in Worpswede.
1940 | Conscription and military service.
1945 | American imprisonment and release.
1945–62 | Worpswede.
1951 | Marries Hedwig Rohde.
1953 | Travels to Paris.
1962 | Moves to Porteholz bei Hameln.
1964 | Karl-Ernst-Osthaus-Preis der Stadt Hagen; other prizes.
1965 | Nomination as full member to Academy of Art, Berlin.
1980 | Dies, May 27, in Porteholz.
(Adapted from _Richard Oelze 1900–1980: Gemalde und Zeichnungen_ , hrsg. Wieland Schmied [Berlin: Akademie der Künste und Autoren, 1987], p. 185.)
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{
"redpajama_set_name": "RedPajamaBook"
}
| 4,533
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The style of architecture known as "mid-century modern" is a cousin to the "International style." A popular combination of European stylistic tendencies and domestic American influences, including furniture design, it has become an influential catch all term for distinguished post-World War II structures and commercial tract homes (like the Eichler Homes). While the style has become widely popular in lifestyle magazines like Dwell and even replicated in new suburban developments, the original homes are being regularly torn down and being replaced with bloated McMansions that have shoe closets the size of the former mid-century living rooms.
But the style has a huge following and a number of organizations to highlight and preserve is monuments. Docomomo has been in the lead highlighting these structures and Palm Springs was one of the first city to host a "modernism week." The latest city to create a week of activities devoted to the style is Sarasota, Florida, which along with Palm Springs and New Canaan, Connecticut, were experimental centers of the style. The Florida city also had a gifted number of architects working in the style: Paul Rudolph and his early mentor Ralph Twitchell, Gene Leedy, Victor Lundy, Tim Seibert, and Carl Abbott. The four day event of lectures, city and house tours that took place this fall was a model of how a community can highlight its unique but disappearing history.
The week was created the Sarasota Architectural Foundation (founded by Martie Lieberman, a realtor who specializes in the style of homes) which is trying to promote the city's modern architecture. It hopes to raise awareness of the style so its buildings can be preserved, updated, and even become a model of a future architecture that is more responsive to needs and demands than the typical McMansion.
Sarasota prides itself on its modern history and was a unique crossroads of culture, commerce, and environment after World War II that helped birth this style.
The week also highlighted the fascinating figure of Philip Hiss III who moved to the beach community in 1948 and became a major figure in the community. He was chair of its education department (which commissioned Paul Rudolph to design two high schools) and a developer of the modernist community Lido Shores. The Foundation is hoping to make their week an annual affair and the area has the modern assets to make it work.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,579
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\section{Introduction}
With the advent of pre-trained transformer-based language models (\citealp{Devlin2019BERTPO,Liu2019RoBERTaAR,He2021DeBERTaDB}), training models from scratch has been outperformed by fine-tuning pre-trained language models for several tasks in natural language processing, including text classification \citep{Howard2018UniversalLM}. However, fine-tuning these models still needs large labeled datasets to perform well on the downstream task \citep{Dodge2020FineTuningPL,Zhang2021RevisitingFB,Mosbach2021OnTS}. Collecting a large annotated dataset is a highly expensive and time-consuming process in specialized domains, where annotation can only be performed by the domain experts, such as the legal domain \citep{Hendrycks2021CUADAE}.
Active Learning (AL) has been proved effective for data-efficient fine-tuning of pre-trained language models in non-specialized domains like news, emotions, and movies \citep{EinDor2020ActiveLF,Margatina2022OnTI}. In addition, \citet{Margatina2022OnTI} have shown that the unlabeled data can be used to adapt the pre-trained language model to the downstream task, thereby improving the active learning performance with no extra annotation cost. On the specialized domains, \citet{Chhatwal2017EmpiricalEO} have evaluated multiple AL strategies in the legal domain before the emergence of pre-trained language models. Nevertheless, to the best of our knowledge, the effectiveness of active learning in fine-tuning pre-trained language models in the legal domain has been poorly studied.
In this work, we focus on efficient legal text classification with RoBERTa \citep{Liu2019RoBERTaAR} by leveraging existing AL strategies. We identify two challenges in deploying AL strategies in the legal domain; First, legal texts contain a specialized vocabulary that is not common in other domains, including the ones on which pre-trained language models are trained. Second, the annotation of legal texts is highly expensive and time-consuming due to the necessity of specialized training for understanding these texts. For example, \citet{Hendrycks2021CUADAE} reported a cost of over $\$2$ million for the annotation of the Contract Understanding Atticus Dataset (CUAD) consisting of around $500$ contracts.
To account for the specialized vocabulary, inspired by \citeposs{Margatina2022OnTI} work, we leverage the available \textit{unlabeled} data to adapt the pre-trained language model to the downstream task. In addition, considering the limitations of pre-trained language models like BERT and RoBERTa in capturing semantics \citep{Reimers2019SentenceBERTSE}, we use knowledge distillation to further improve the task-adapted model by mapping its embedding space to a semantically meaningful space. Our experiments demonstrate that AL strategies can benefit from semantically meaningful embeddings.
Concerning the cost and time constraints, we focus on the fact that many AL strategies \citep{Lewis1994ASA,Gal2016DropoutAA,Gissin2019DiscriminativeAL} require an annotated set of $N$ positive and negative samples to start with. In practice, acquiring this set is expensive for large and skewed datasets. We propose a strategy to make the first iteration more efficient by clustering the unlabeled samples and limiting the pull of candidates to the cluster medoids. Our experiments demonstrate we can achieve comparable results with the standard initial sampling approach with up to $63\%$, and $25\%$ fewer actions on the skewed Contract-NLI \citep{Koreeda2021ContractNLIAD}, and balanced LEDGAR benchmarks \citep{Tuggener2020LEDGARAL} respectively.
Our contributions can be summarized as follows:
\begin{itemize}
\item[1.] We design an efficient and effective active learning pipeline for legal text classification by leveraging the available unlabeled data using task-adaptation and knowledge distillation, which obtains comparable performance to fully-supervised fine-tuning with considerably reduced annotation effort.
\item[2.] We propose a strategy to reduce the number of actions in the first iteration of active learning by clustering the unlabeled data, and collecting the samples from cluster medoids, further increasing the efficiency of our approach.
\item[3.] We evaluate our approach over Contract-NLI and LEDGAR benchmarks. Our results illustrate an increase of $0.3346$, and $0.1658$ in the best obtained F1-score, compared to standard active learning strategies, for Contract-NLI and LEDGAR respectively.
\end{itemize}
\section{Related Work}
\paragraph{Active learning with pre-trained language models}
Multiple works have studied active learning for pre-trained language models like BERT. \citet{EinDor2020ActiveLF} have evaluated various AL strategies for fine-tuning BERT for text classification, and showed that AL can boost BERT's performance especially for skewed datasets. However, they do not leverage the available unlabeled data to adapt the pre-trained language model to the task at hand, and only focus on non-specialized domains like news and sentiment analysis that do not require experts' knowledge.
\citet{Gururangan2020DontSP} have shown that task-adaptive pre-training using the available unlabeled data leads to performance gain when using pre-trained language models like BERT. Following this observation, \citet{Margatina2022OnTI} demonstrated the importance of task-adaptation for active learning for non-specialized texts like news, movies and sentiment analysis.
Inspired by these works, we leverage the available unlabeled data to effectively adapt RoBERTa to legal text classification, where the annotation demands experts' knowledge. In addition, we propose an additional step to map the embedding space of the task-adapted RoBERTa to a semantically meaningful space using sentence transformers.
\paragraph{Sentence transformers}
\citet{Reimers2019SentenceBERTSE} have shown that the embedding space of off-the-shelf pre-trained language models like BERT \citep{Devlin2019BERTPO} and RoBERTa \citep{Liu2019RoBERTaAR} is not semantically meaningful, and thus, is not suitable for common sentence comparison measures like cosine similarity. To overcome this limitation, they propose sentence transformers, obtained by adding a pooling layer on top of pre-trained language models, and fine-tuning them in a Siamese network architecture with pairs of similar sentences. In this work, we use a RoBERTa-based sentence transformer as a teacher model and distill its knowledge to the task-adapted RoBERTa to produce sentence embeddings that capture the semantics and can be compared using cosine similarity.
\paragraph{Active learning strategies}
Numerous methods have been proposed to find proper labeling candidates for active learning. Majority of them belong to one or both of two categories: diversity-sampling, and uncertainty-sampling. Diversity-based methods (\citealp{Sener2018ActiveLF,Gissin2019DiscriminativeAL,Wang2017IncorporatingDA}) aim to find labeling candidates that best represent the dataset, whereas uncertainty-based methods (\citealp{Gal2016DropoutAA,Kirsch2019BatchBALDEA,Zhang2021CartographyAL}) target candidates about which the model is uncertain.
BADGE \citep{Ash2020DeepBA} is a cluster-based AL strategy that belongs to both of these categories. It transforms data into gradient embeddings that encode model confidence and sentence feature at the same time.
By applying kmeans++ on the gradient embeddings it can find samples that differ both in terms of semantics and predictive uncertainty. ALPS \citep{Yuan2020ColdstartAL} is another cluster-based AL strategy that leverages both uncertainty and diversity using the surprisal embeddings obtained by passing the sentences to the MLM head of the pre-trained language model, and computing the cross entropy loss for a random set of tokens against the target labels.
Existing AL strategies often require a set of labeled samples to start with, which is expensive to acquire. To overcome this high cost, we propose a clustering-based strategy to reduce the effort required to create the initial set of annotated samples.
\section{Notation and Setting}
In this section, we explain the structure shared between all AL strategies used in this work and fix the notation.
Active learning is an iterative process aiming to obtain a desired performance given an annotation budget. Here, we consider the annotation budget to be the number of actions performed by the annotator. In addition, we assume all annotators are legal experts, and that each annotator assigns perfect labels to text segments. Let $U_0$ and $L_0$ be the starting pool of unlabeled and labeled samples respectively. Initially, $L_0 = \emptyset$. At the first iteration, the annotator labels $N$ sample, $P$ positive and $N-P$ negative, to obtained $L_1$. Then, at each iteration $i$, the model is fine-tuned using $L_i$, and the AL strategy recommends a set of samples $C_i$ for annotation. These samples are labeled and $U_i$ and $L_i$ are updated as $U_{i+1} = U_i \setminus C_i$, and $L_{i+1} = L_i \cup C_i $. The procedure is repeated until the annotation budget is exhausted, or the desired performance is achieved.
We base our work on the Low-Resource Text Classification Framework introduced by \citet{EinDor2020ActiveLF}.
Following this work, we focus on binary text classification, given a small annotation budget and a potentially imbalanced dataset. This scenario matches common use cases in the legal domain, where the goal is to find phrases that correspond to a specific category, with the lowest possible number of actions, given a pool of unlabeled, imbalanced data. We perform $5$ AL iterations, and assume a more restricted annotation budget compared to \citet{EinDor2020ActiveLF}, allowing only $10$ annotations per iteration. For the first AL iteration, we assume that $5$ positive and $5$ negative samples need to be annotated.
\section{Methodology}
\begin{algorithm*}[ht]
\caption{AL pipeline for text classification}\label{alg:cap}
\hspace*{\algorithmicindent} \textbf{Input:} {unlabeled samples $U_0$, PT RoBERTa, PT Sentence-RoBERTa, AL strategy $\alpha$, \# iterations $T$}
\hspace*{\algorithmicindent} \textbf{Output:} {text classifier CLS RoBERTa, acquired labeled dataset $L_T$}
\begin{algorithmic}
\State $L_0 \gets \emptyset$
\State \textbf{\textit{Phase 1: Task-adaptation with Masked Language Modeling (MLM)}} \State TAPT RoBERTa $\gets$ MLM(PT RoBERTa, $U_0$)
\State \textbf{\textit{Phase 2: Knowledge distillation}}
\State DisTAPT RoBERTa $\gets$ Distill(TAPT RoBERTa, PT Sentence-RoBERTa, $U_0$)
\State \textbf{\textit{Phase 3: Initial sampling}}
\State cluster medoids $\gets$ KMeans(DisTAPT RoBERTa, $U_0$)
\State $L_1 \gets$ Sample(cluster medoids)
\State $U_1 \gets U_0 \setminus L_1$
\State \textbf{\textit{Phase 4: Active learning}}
\For{$i \gets 1$ to $T$ }
\State CLS RoBERTa $\gets$ Train(DisTAPT RoBERTa, $L_i$)
\State $C_i \gets \alpha$(CLS RoBERTa, $U_i$)
\State $L_{i+1} \gets L_i \cup C_i$
\State $U_{i+1} \gets U_i \setminus C_i$
\EndFor
\end{algorithmic}
\label{alg:algorithm}
\end{algorithm*}
We propose an efficient active learning pipeline for fine-tuning pre-trained language models for legal text classification. Our approach leverages available unlabeled data in three phases to adapt the pre-trained model to the downstream task (Sec.~\ref{sec:task-adaptation}), guide its embedding space to a semantically meaningful and comparable space (Sec.~\ref{sec:distillation}), and reduce the number of actions required to collect the initial labeled set (Sec.~\ref{sec:sampling}). Finally, it leverages existing AL strategies to efficiently fine-tune a classifier (Sec.~\ref{sec:meth_al}). We now explain each step in detail. An overview of this pipeline can be found in Algorithm~\ref{alg:algorithm}.
\subsection{Task-Adaptation}\label{sec:task-adaptation}
It has been shown that fine-tuning off-the-shelf pre-trained language models with standard approaches is unstable for small training sets \citep{Zhang2021RevisitingFB,Dodge2020FineTuningPL,Mosbach2021OnTS}. As shown by \citet{Margatina2022OnTI}, this can lead to poor performance when fine-tuning pre-trained language models with AL. In addition, existing pre-trained language models are often trained on texts that do not need specialized training to be understood. However, legal texts contain specialized words that are not common in other domains. Thus, task-adaptation is crucial for the effectiveness of active learning in legal text classification. In the first step of our proposed pipeline, we obtain the task-adapted pre-trained (TAPT) RoBERTa by continuing pre-training the model with unlabeled samples for the Masked Language Modeling (MLM) task, as suggested by \citet{Gururangan2020DontSP} and \citet{Margatina2022OnTI}.
\subsection{Knowledge Distillation}\label{sec:distillation}
Previous works \citep{Reimers2019SentenceBERTSE,Li2020OnTS,Su2021WhiteningSR} have shown that, without fine-tuning, the sentence embeddings produced by pre-trained language models poorly capture semantic meaning of sentences, and are not comparable using cosine similarity. To overcome this shortcoming, \citet{Reimers2019SentenceBERTSE} introduced sentence transformers by adding a pooling layer on top of pre-trained transformer-based language models, and training them in a Siamese network architecture with pairs of similar sentences. Compared to out-of-the-box RoBERTa, a RoBERTa-based sentence transformer drives semantically comparable sentence embeddings.
As we will explain in Sec.~\ref{sec:sampling}, we cluster the normalized sentence embeddings based on their Euclidean distance to efficiently acquire the labeled samples for the initial iteration of AL. The Euclidean distance between normalized embeddings can be driven from their cosine similarity. Hence, sentence embeddings that are comparable with cosine similarity can result in clusters with higher quality. In addition, semantically meaningful sentence embeddings give a better initialization of the \texttt{[CLS]} token, thereby obtaining better classification performance with a smaller training set.
We use a pre-trained RoBERTa-based sentence transformer (PT Sentence-RoBERTa) as a teacher model, and distill its knowledge to the TAPT RoBERTa. The resulting distilled task-adapted pre-trained (DisTAPT) RoBERTa produces semantically meaningful embeddings that are comparable via cosine similarity, and, as shown by our experiments (Sec.~\ref{sec:exp_distillaition}), benefit the classification task.
\subsection{Initial Sampling}\label{sec:sampling}
Many AL strategies (\citealp{Gissin2019DiscriminativeAL,Gal2016DropoutAA}) require an initial set of $N$ labeled samples containing $P$ positive and $N-P$ negative sentences, which is either assumed to be available, or obtained by randomly sampling the entire dataset until the desired number of positive and negative samples are found. This approach is highly expensive for large and skewed datasets. We propose a simple, yet effective, strategy to efficiently acquire the initial labeled set. To this end, we leverage the distilled task-adapted pre-trained RoBERTa to cluster the unlabeled samples using KMeans algorithm \citep{MacQueen1967SomeMF}. The labeled set for the initial iteration is then driven from the cluster medoids. As a result, we shrink the pool of candidates from the entire dataset to the cluster medoids, therefore, reduce the number of actions for obtaining the initial annotated set, while achieving comparable performance with the standard approach for initial sampling.
\subsection{Active Learning}\label{sec:meth_al}
In the last phase, we iteratively fine-tune the DisTAPT RoBERTa for the downstream task. The initial labeled set is used at the first iteration. Then, more samples are labeled in the following rounds using an AL acquisition strategy until the annotation budget is exhausted, or the classifier satisfies the expected performance.
Our proposed pipeline can be used with existing AL strategies and, as demonstrated by our experiments (Sec.~\ref{sec:exp_distillaition}), consistently outperforms standard AL approaches, regardless of the AL strategy used.
\section{Experimental Setup}\label{sec:setup}
We evaluate our approach against four standard active learning strategies provided in the Low-Resource Text Classification Framework \citep{EinDor2020ActiveLF}:
\begin{itemize}
\item \textbf{Random} At each iteration, this approach randomly chooses samples for annotation.
\item \textbf{Hard-Mining} Selects instances that the model is uncertain about, based on the absolute difference of prediction score and $0.5$.
\item \textbf{Perceptron Dropout} \citep{Gal2016DropoutAA} Also selects instances for which the model is least certain. The uncertainty is calculated using Monte Carlo Dropout on $10$ inference cycles.
\item \textbf{Discriminative Active Learning (DAL)} \citep{Gissin2019DiscriminativeAL} Deploys a binary classifier to select instances that best represent the entire unlabeled samples.
\end{itemize}
We consider pre-trained RoBERTa and LEGAL-BERT~\citep{Chalkidis2020LEGALBERTTM} as the baselines. Note that our goal is not to rely on domain-adapted models like LEGAL-BERT since they might not always be available. For example, if the data is in German, we can find a pre-trained RoBERTa in German, but the LEGAL-BERT is pre-trained on English text only.
\subsection{Datasets}\label{sec:datasets}
We evaluate our framework on Contract-NLI \citep{Koreeda2021ContractNLIAD} and LEDGAR \citep{Tuggener2020LEDGARAL} benchmarks.
Contract-NLI \citep{Koreeda2021ContractNLIAD} is a dataset for document-level natural language inference. It consists of $607$ documents with $77.8$ spans per document on average. Each span is checked against $17$ hypotheses and classified as contradiction, entailment, or not mentioned. In this work, we adapt this dataset to the classification task by considering each hypothesis as a category. If a span is classified as contradiction or entailment for a hypothesis, we label it with the corresponding category. Following this approach, we end up with a classification dataset with $4,371$ train, $614$ development, and $1,188$ test samples within $17$ classes.
LEDGAR \citep{Tuggener2020LEDGARAL} is a text classification benchmark consisting of a corpus of legal provisions in contracts. The entire dataset consists of $846,274$ provisions and $12,608$ labels. We only consider a subset of this dataset that corresponds to provisions with labels that appeared at least $10,000$ times in the corpus, resulting in $44,249$ train, $7,375$ development, and $12,907$ test samples across $5$ categories. Similar to \citet{Tuggener2020LEDGARAL}, we perform a $70\%-10\%-20\%$ random split to obtain the train, development and test sets.
The class distributions of both datasets can be found in the appendix (Sec.~\ref{sec:app_class_dist}). Compared to Contract-NLI, LEDGAR has fewer categories, is an order of magnitude bigger, and is more balanced.
\subsection{Implementation Details}
We base our implementation on the Low-Resource Text Classification Framework provided by \citet{EinDor2020ActiveLF}\footnote{\url{https://github.com/IBM/low-resource-text-classification-framework}}, and augment it with the task-adaptation, knowledge distillation, and initial sampling steps.
As the pre-trained model, we use \texttt{roberta-base}\footnote{\url{https://huggingface.co/roberta-base}} (with $125$M parameters), the RoBERTa \citep{Liu2019RoBERTaAR} language model trained on the union of $5$ datatsets: Book corpus \citep{Zhu2015AligningBA}, English Wikipedia\footnote{\url{https://dumps.wikimedia.org}}, CC-News \citep{Mackenzie2020CCNewsEnAL}, OpenWebText Corpus \citep{Gokaslan2019OpenWeb}, and Stories \citep{Trinh2018ASM}, none of which belong to the legal domain.
For LEGAL-BERT, we use the \texttt{nlpaueb/legal-bert-base-uncased}\footnote{\url{https://huggingface.co/nlpaueb/legal-bert-base-uncased}} (with $110$M parameters), trained on six datasets containing legal docments across Europe and the US.
For task-adaptation, we continue pre-training RoBERTa for the MLM task using the available unlabeled data. We train for $10$ epochs with batch-size $64$, and the learning rate set to $3\mathrm{e}{-4}$. The task-adapted model has perplexity $4.9706$ for Contract-NLI and $2.1628$ for LEDGAR.
For model distillation, we use \texttt{stsb-roberta-base-v2} (with $125$M parameters), a RoBERTa-based sentence transformer trained on the STS benchmark \citep{Cer2017SemEval2017T1}, as the teacher model, and the task-adapted RoBERTa as the student model. Mean Squared Error (MSE) is used as the loss function. The student model is trained for $10$ epochs, with $10$K warmup steps, $1\mathrm{e}{-4}$ learning rate and no bias correction. The final MSE ($\times 100$) is $6.8607$ for Contract-NLI, and $7.2003$ for LEDGAR.
For clustering the normalized sentence embeddings we use the KMeans implementation by \texttt{scikit-learn}. We cluster the Contract-NLI and LEDGAR sentence embeddings into $437$, and $442$ groups respectively. The number of clusters are chosen based on the dataset size, and the number of categories, and to make initial sampling with cluster medoids manageable for experts.
In all the active learning experiments, we perform $5$ AL iterations, starting with $10$ initial samples, and increasing the size of the annotated data by $10$ at each iteration. Adam optimizer \citep{Kingma2015AdamAM} is used with learning rate set to $5\mathrm{e}{-5}$. The model is trained for $100$ epochs and early stopping is used with patience set to $10$. To account for randomization, we repeat each experiment three times.
To compare our approach with standard AL methods, we use F1-score as the evaluation metric as it captures both precision and recall and is sensitive to data distribution.
\section{Results and Discussion}
In this section, we provide the results of our experiments and explain them in detail. We start by comparing our approach with and without the initial medoid sampling against standard AL strategies (Sec.~\ref{sec:ours-vs-baseline}). Then, we show the effectiveness of knowledge distillation on top of task-adaptation (Sec.~\ref{sec:exp_distillaition}). In addition, we demonstrate the efficiency of the initial sampling with cluster medoids (Sec.~\ref{sec:exp_initial-sampling}). Finally, we evaluate how well our approach performs for different AL strategies (Sec.~\ref{sec:exp_AL_effect}).
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{Contract-NLI_DAL_avg_with_legalbert.pdf}
\includegraphics[width=0.5\textwidth]{LEDGAR_DAL_avg_with_legalbert.pdf}
\caption{Test F1-score for \textbf{DAL} during AL iterations. The F1-score for the fully supervised fine-tuning is $0.6990$ for Contract-NLI and $0.9538$ for LEDGAR. The figure is best viewed in color.}
\label{fig:ours-vs-baseline}
\end{figure}
\subsection{Efficient AL Pipeline}\label{sec:ours-vs-baseline}
Figure~\ref{fig:ours-vs-baseline} compares our approach with and without the initial sampling phase (DisTAPT with IS, and DisTAPT) to standard DAL with pre-trained (PT) RoBERTa, LEGALBERT, and TAPT RoBERTa for Contract-NLI and LEDGAR benchmarks. We report the average F1-score over all categories. DAL is chosen due to its better performance, as shown in Figure~\ref{fig:all_ALs}.
The results for other AL strategies can be found in the appendix (Sec.~\ref{sec:app_tabs}).
Our experiments show the importance of task-adaptation and knowledge distillation for pre-trained language models prior to fine-tuning with active learning. Figure~\ref{fig:ours-vs-baseline} illustrates that, for the same size of annotated data, our pipeline consistently achieves better performance than standard AL approaches even for LEGAL-BERT.
For the Contract-NLI dataset, the F1-score obtained by fully-supervised fine-tuning (with $4,371$ labeled samples) is $0.6990$ for \texttt{roberta-base} and $0.7152$ for \texttt{legal-bert-base-uncased}. DisTAPT RoBERTa reaches a F1-score as high as $0.6508$ with only $40$ labeled samples. The best F1-score obtained using pre-trained RoBERTa is $0.3162$ with $30$ labeled samples, which is $0.3165$ lower than DisTAPT RoBERTa's F1-score for the same size of annotated data.
For the LEDGAR dataset, the F1-score obtained by the fully-supervised fine-tuning (with $44,249$ labeled samples) is $0.9538$ for \texttt{roberta-base} and $0.9588$ for \texttt{legal-bert-base-uncased}. DisTAPT RoBERTa reaches a very close performance of $0.9321$ F1-score with merely $60$ labeled samples. The highest F1-score that pre-trained RoBERT reaches is $0.7663$ with $20$ annotated samples, which is $0.0904$ lower than DisTAPT's performance with the same size of labeled data.
These results show that, for both datasets, there is only a small performance gap between our approach and the fully-supervised approach, indicating that our AL pipeline dramatically reduces the annotation cost, while achieving comparable performance with the fully-supervised fine-tuning.
In addition, It is observed that standard AL with off-the-shelf pre-trained RoBERTa is unstable. This is aligned with the previous works' observations \citep{Mosbach2021OnTS,Zhang2021RevisitingFB,Dodge2020FineTuningPL}. During fine-tuning, the pre-trained model should perform two tasks: adaptation to the legal domain with the new vocabulary, and classification. By performing task-adaptation and knowledge distillation before fine-tuning, we train the model in a curriculum learning approach, making the model stable even for small training sets.
\subsection{Effect of Knowledge Distillation}\label{sec:exp_distillaition}
To evaluate the effectiveness of knowledge distillation on the quality of obtained clusters, we compare the distribution of the Dunn Index of the clusters before and after knowledge distillation. For both datasets, after knowledge distillation, most of the clusters have higher Dunn Index which indicates that they are more compact and better separated than the clusters before knowledge distillation step. The results are provided in the appendix~\ref{sec:app_knowldge_distill} due to space constraints.
In addition, we evaluate the effect of knowledge distillation on the task-adapted pre-trained RoBERTa, and report the average F1-score over all classes for each dataset. Figure~\ref{fig:ours-vs-baseline} shows that, for both datasets, DisTAPT RoBERTa outperforms TAPT RoBERTa at early iterations of active learning, and as the size of the labeled set increases, the two models' performance converge. This can be explained by the fact that, initially, DisTAPT RoBERTa's embeddings better capture the semantics of sentences, and thus result in better classification performance. As the labeled data grows, TAPT RoBERTa is fine-tuned and can produce semantically meaningful embeddings as well. Hence, for a highly restricted annotation budget, distilling the knowledge of a sentence transformer to the TAPT language model can lead to performance gain.
\subsection{Efficiency of Initial Medoid Sampling} \label{sec:exp_initial-sampling}
It was shown in Figure~\ref{fig:ours-vs-baseline} that DisTAPT with IS obtains comparable performance with DisTAPT without IS. In this section, we evaluate the \textit{efficiency} of the proposed sampling strategy for the initial iteration of AL.
To this end, we simulate the standard sampling strategy by randomly sampling text segments from the full dataset until $5$ positive and $5$ negative samples are found. The number of iterations is then considered as the number of annotations required to collect the labeled set for the initial AL iteration. Similarly, to simulate our proposed initial sampling, we randomly sample from cluster medoids until $5$ positive and $5$ negative samples are obtained. To account for randomness, we repeat the simulations $1000$ times and report the median and the $90^{th}$ percentile over all runs.
Table~\ref{tab:exp_medoid} illustrates the results of our simulations for Contract-NLI and LEDGAR. Due to the high number of classes in Contract-NLI, only eight categories of this dataset are presented in this table, and the results for other categories can be found in the appendix (Sec.~\ref{sec:app_medoid}). For each class, in addition to the median and $90^{th}$ percentile over $1000$ runs, the difference in the $90^{th}$ percentile between standard approach and our strategy (in $\%$) is reported as the gain in annotation effort. For example, for the \texttt{Sharing with third-parties} class in Contract-NLI, the $90^{th}$ percentile is $62\%$ less when using medoids for initial sampling, meaning that, with $90\%$ confidence, the annotators perform $62\%$ fewer actions to acquire the initial labeled set using our approach.
It is observed that, for the skewed Contract-NLI dataset, our proposed initial sampling strategy reduces the number of actions performed by the annotator up to $63\%$. For LEDGAR however, which consists of balanced categories, the highest effort gain in sampling from cluster medoids is $25\%$. There are also few cases where using the entire dataset is more efficient than sampling from medoids. This happens when the class' frequency is higher in the full dataset than its frequency in the cluster medoids.
Overall, our results demonstrate the advantage of using the cluster medoids for collecting the initial annotated samples for a skewed dataset like Contract-NLI, which is a realistic use-case in the legal domain. It is noteworthy that the original version of LEDGAR dataset is also imbalanced, but as explained in Sec.~\ref{sec:datasets}, due to the drastically high number of classes, and for the sake of comparison with skewed datasets, only the most dominant categories are kept in this work.
Thanks to the semantically meaningful and comparable sentence embeddings obtained after the knowledge distillation step, the cluster medoids well represent the entire dataset, and thus sampling among them drastically reduces the annotation effort without harming the performance.
As a real life scenario, consider a company with hundreds of legal contracts aiming to classify their sentences into multiple categories, under a restricted budget. Reducing the annotation effort means lowering down the financial costs of annotation, which can be highly expensive in the legal domain (over $\$2$ million for annotating around $500$ contracts according to~\citet{Hendrycks2021CUADAE}).
\begin{table*}[h]
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{clccccc}
\hline
\multirow{2}{*}{Dataset} & \multirow{2}{*}{Category} & \multicolumn{2}{c}{full dataset} & \multicolumn{2}{c}{medoids} & \multirow{2}{*}{gain(\%)} \\
& & median & $90^{th}\%$tile & median & $90^{th}\%$tile & \\
\hline
\multirow{8}{*}{\small{\texttt{\rotatebox[origin=c]{90}{Contract-NLI}}}}
& \small{\texttt{Inclusion of verbally conveyed information}} & 75.0 & 125.0 & 35.5 & 59.0 & 52.8\\
& \small{\texttt{No licensing}} & 64.0 & 108.0 & 68.5 & 109.1 & -1.0 \\
& \small{\texttt{No reverse engineering}} & 342.0 & 568.0 & 144.0 & 209.1 & 63.2\\
& \small{\texttt{Notice on compelled disclosure}} & 74.5 & 122.0 & 99.0 & 155.0 & -27.0\\
& \small{\texttt{Sharing with employees}} & 57.0 & 90.0 & 21.0 & 34.1 & 62.1 \\
& \small{\texttt{Sharing with third-parties}} & 54.0 & 92.1 & 21.0 & 35.0 & 62.0\\
& \small{\texttt{Survival of obligations}} & 64.0 & 106.0 & 36.0 & 57.0 & 46.2\\
& \small{\texttt{Return of confidential information}} & 116.0 & 189.0 & 61.0 & 99.0 & 47.6\\
\hline
\multirow{5}{*}{\small{\texttt{\rotatebox[origin=c]{90}{LEDGAR}}}} & \small{\texttt{Amendments}} & 23.0 & 37.1 & 21.0 & 33.0 & 10.8\\
& \small{\texttt{Counterparts}} & 26.0 & 42.0 & 34.0 & 54.1 & -28.8 \\
& \small{\texttt{Entire agreements}} & 26.0 & 42.0 & 33.0 & 55.0 & -30.9 \\
& \small{\texttt{Governing laws}} & 17.5 & 28.0 & 14.0 & 21.0 & 25.0\\
& \small{\texttt{Notices}} & 29.0 & 49.0 & 26.0 & 44.0 & 10.2\\
\hline
\end{tabular}}
\caption{\label{tab:exp_medoid}
Number of actions to acquire the initial labeled set for $8$ categories of Contract-NLI, and LEDGAR when sampling from the full dataset (standard approach), and sampling from the cluster medoids (our approach).}
\end{table*}
\begin{figure}[h!]
\centering
\includegraphics[width=0.5\textwidth]{Contract-NLI_avg.pdf}
\includegraphics[width=0.5\textwidth]{LEDGAR_avg.pdf}
\caption{Comparison of four AL strategies when used with DisTAPT RoBERTa with IS.}
\label{fig:all_ALs}
\end{figure}
\subsection{Effect of AL strategy}\label{sec:exp_AL_effect}
Finally, we evaluate the generalizability of our approach over the four AL strategies mentioned in Sec.~\ref{sec:setup}: DAL, Random, Hard-Mining, and Perceptron Dropout. As shown in Figure~\ref{fig:all_ALs}, DAL results in the best performance with at most $0.08$ higher F1-score than other strategies with $60$ labeled samples for Contract-NLI, and less than $0.04$ higher F1-score with $40$ annotated samples for LEDGAR. The small performance gap of these four AL methods in our pipeline indicates the generalizability of this approach to various AL strategies.
\section{Conclusion}
We propose an efficient active learning pipeline for legal text classification. Our approach leverages the available unlabeled data to adapt the pre-trained language model to the downstream task, and guide its embeddings to a semantically meaningful space before fine-tuning. We use model distillation to produce semantically comparable embeddings. A future work can study the effect of other approaches like BERT-Flow \citep{Li2020OnTS} and whitening \citep{Su2021WhiteningSR} on AL with this pipeline. Moreover, we design a simple strategy to efficiently acquire a labeled set of positive and negative samples for the initial iteration of active learning.
Our experiments over Contract-NLI and LEDGAR benchmarks demonstrate the effectiveness of our approach compared to standard active learning strategies. Our results also show that our pipeline obtains very close performance to the fully-supervised approach with considerably less annotation cost. We test our methodology in the legal domain, and for four AL strategies, but we expect it to generalize to other strategies like ALPS and BADGE, and other specialized domains, like medicine. We leave this evaluation as a future work.
\section*{Limitations}
In this work, we have shown the importance of task-adaptation and knowledge distillation, and that we can leverage the available unlabeled data to perform efficient fine-tuning via active learning and obtain better performance. The price to pay for this performance gain is time and computational power. The time taken by task-adaptation and distillation scales with the size of unlabeled data. On the other hand, more unlabeled samples result in more effective adaptation to the downstream task. Therefore, the user of this approach needs to find the best trade-off given their data, annotation budget, time and computational power. For, LEDGAR, the larger dataset used in this work, we performed the adaptation and distillation steps in $4$ and $1$ hour(s) respectively, using a single Nvidia GeForce GTX TITAN X GPU.
Moreover, we showed that by clustering the sentence embeddings produced by DisTAPT RoBERTa, the initial labeled set can be acquired more efficiently. Nevertheless, this approach inherits the limitations of clustering. Namely, the time complexity of clustering the embeddings scales with the data, and the number of clusters should be empirically chosen. In our experiments we spent $10$ minutes to cluster the $44,249$ samples belonging to LEDGAR dataset into $442$ groups.
\section*{Ethics Statement}
Industries have hundreds of contracts with tens of thousands of sentences that belong to various topics. Labeling all of these samples is a highly expensive and time-consuming process. In this work, we aim to reduce the resources spent on this task by leveraging recent advances in natural language processing, while keeping the human expert in the loop. The goal is to reduce the human effort in annotation so that the legal experts' time and knowledge can be used in another task at which humans are better than machines.
\input{emnlp2022.bbl}
\bibliographystyle{acl_natbib}
\section{Introduction}
With the advent of pre-trained transformer-based language models (\citealp{Devlin2019BERTPO,Liu2019RoBERTaAR,He2021DeBERTaDB}), training models from scratch has been outperformed by fine-tuning pre-trained language models for several tasks in natural language processing, including text classification \citep{Howard2018UniversalLM}. However, fine-tuning these models still needs large labeled datasets to perform well on the downstream task \citep{Dodge2020FineTuningPL,Zhang2021RevisitingFB,Mosbach2021OnTS}. Collecting a large annotated dataset is a highly expensive and time-consuming process in specialized domains, where annotation can only be performed by the domain experts, such as the legal domain \citep{Hendrycks2021CUADAE}.
Active Learning (AL) has been proved effective for data-efficient fine-tuning of pre-trained language models in non-specialized domains like news, emotions, and movies \citep{EinDor2020ActiveLF,Margatina2022OnTI}. In addition, \citet{Margatina2022OnTI} have shown that the unlabeled data can be used to adapt the pre-trained language model to the downstream task, thereby improving the active learning performance with no extra annotation cost. On the specialized domains, \citet{Chhatwal2017EmpiricalEO} have evaluated multiple AL strategies in the legal domain before the emergence of pre-trained language models. Nevertheless, to the best of our knowledge, the effectiveness of active learning in fine-tuning pre-trained language models in the legal domain has been poorly studied.
In this work, we focus on efficient legal text classification with RoBERTa \citep{Liu2019RoBERTaAR} by leveraging existing AL strategies. We identify two challenges in deploying AL strategies in the legal domain; First, legal texts contain a specialized vocabulary that is not common in other domains, including the ones on which pre-trained language models are trained. Second, the annotation of legal texts is highly expensive and time-consuming due to the necessity of specialized training for understanding these texts. For example, \citet{Hendrycks2021CUADAE} reported a cost of over $\$2$ million for the annotation of the Contract Understanding Atticus Dataset (CUAD) consisting of around $500$ contracts.
To account for the specialized vocabulary, inspired by \citeposs{Margatina2022OnTI} work, we leverage the available \textit{unlabeled} data to adapt the pre-trained language model to the downstream task. In addition, considering the limitations of pre-trained language models like BERT and RoBERTa in capturing semantics \citep{Reimers2019SentenceBERTSE}, we use knowledge distillation to further improve the task-adapted model by mapping its embedding space to a semantically meaningful space. Our experiments demonstrate that AL strategies can benefit from semantically meaningful embeddings.
Concerning the cost and time constraints, we focus on the fact that many AL strategies \citep{Lewis1994ASA,Gal2016DropoutAA,Gissin2019DiscriminativeAL} require an annotated set of $N$ positive and negative samples to start with. In practice, acquiring this set is expensive for large and skewed datasets. We propose a strategy to make the first iteration more efficient by clustering the unlabeled samples and limiting the pull of candidates to the cluster medoids. Our experiments demonstrate we can achieve comparable results with the standard initial sampling approach with up to $63\%$, and $25\%$ fewer actions on the skewed Contract-NLI \citep{Koreeda2021ContractNLIAD}, and balanced LEDGAR benchmarks \citep{Tuggener2020LEDGARAL} respectively.
Our contributions can be summarized as follows:
\begin{itemize}
\item[1.] We design an efficient and effective active learning pipeline for legal text classification by leveraging the available unlabeled data using task-adaptation and knowledge distillation, which obtains comparable performance to fully-supervised fine-tuning with considerably reduced annotation effort.
\item[2.] We propose a strategy to reduce the number of actions in the first iteration of active learning by clustering the unlabeled data, and collecting the samples from cluster medoids, further increasing the efficiency of our approach.
\item[3.] We evaluate our approach over Contract-NLI and LEDGAR benchmarks. Our results illustrate an increase of $0.3346$, and $0.1658$ in the best obtained F1-score, compared to standard active learning strategies, for Contract-NLI and LEDGAR respectively.
\end{itemize}
\section{Related Work}
\paragraph{Active learning with pre-trained language models}
Multiple works have studied active learning for pre-trained language models like BERT. \citet{EinDor2020ActiveLF} have evaluated various AL strategies for fine-tuning BERT for text classification, and showed that AL can boost BERT's performance especially for skewed datasets. However, they do not leverage the available unlabeled data to adapt the pre-trained language model to the task at hand, and only focus on non-specialized domains like news and sentiment analysis that do not require experts' knowledge.
\citet{Gururangan2020DontSP} have shown that task-adaptive pre-training using the available unlabeled data leads to performance gain when using pre-trained language models like BERT. Following this observation, \citet{Margatina2022OnTI} demonstrated the importance of task-adaptation for active learning for non-specialized texts like news, movies and sentiment analysis.
Inspired by these works, we leverage the available unlabeled data to effectively adapt RoBERTa to legal text classification, where the annotation demands experts' knowledge. In addition, we propose an additional step to map the embedding space of the task-adapted RoBERTa to a semantically meaningful space using sentence transformers.
\paragraph{Sentence transformers}
\citet{Reimers2019SentenceBERTSE} have shown that the embedding space of off-the-shelf pre-trained language models like BERT \citep{Devlin2019BERTPO} and RoBERTa \citep{Liu2019RoBERTaAR} is not semantically meaningful, and thus, is not suitable for common sentence comparison measures like cosine similarity. To overcome this limitation, they propose sentence transformers, obtained by adding a pooling layer on top of pre-trained language models, and fine-tuning them in a Siamese network architecture with pairs of similar sentences. In this work, we use a RoBERTa-based sentence transformer as a teacher model and distill its knowledge to the task-adapted RoBERTa to produce sentence embeddings that capture the semantics and can be compared using cosine similarity.
\paragraph{Active learning strategies}
Numerous methods have been proposed to find proper labeling candidates for active learning. Majority of them belong to one or both of two categories: diversity-sampling, and uncertainty-sampling. Diversity-based methods (\citealp{Sener2018ActiveLF,Gissin2019DiscriminativeAL,Wang2017IncorporatingDA}) aim to find labeling candidates that best represent the dataset, whereas uncertainty-based methods (\citealp{Gal2016DropoutAA,Kirsch2019BatchBALDEA,Zhang2021CartographyAL}) target candidates about which the model is uncertain.
BADGE \citep{Ash2020DeepBA} is a cluster-based AL strategy that belongs to both of these categories. It transforms data into gradient embeddings that encode model confidence and sentence feature at the same time.
By applying kmeans++ on the gradient embeddings it can find samples that differ both in terms of semantics and predictive uncertainty. ALPS \citep{Yuan2020ColdstartAL} is another cluster-based AL strategy that leverages both uncertainty and diversity using the surprisal embeddings obtained by passing the sentences to the MLM head of the pre-trained language model, and computing the cross entropy loss for a random set of tokens against the target labels.
Existing AL strategies often require a set of labeled samples to start with, which is expensive to acquire. To overcome this high cost, we propose a clustering-based strategy to reduce the effort required to create the initial set of annotated samples.
\section{Notation and Setting}
In this section, we explain the structure shared between all AL strategies used in this work and fix the notation.
Active learning is an iterative process aiming to obtain a desired performance given an annotation budget. Here, we consider the annotation budget to be the number of actions performed by the annotator. In addition, we assume all annotators are legal experts, and that each annotator assigns perfect labels to text segments. Let $U_0$ and $L_0$ be the starting pool of unlabeled and labeled samples respectively. Initially, $L_0 = \emptyset$. At the first iteration, the annotator labels $N$ sample, $P$ positive and $N-P$ negative, to obtained $L_1$. Then, at each iteration $i$, the model is fine-tuned using $L_i$, and the AL strategy recommends a set of samples $C_i$ for annotation. These samples are labeled and $U_i$ and $L_i$ are updated as $U_{i+1} = U_i \setminus C_i$, and $L_{i+1} = L_i \cup C_i $. The procedure is repeated until the annotation budget is exhausted, or the desired performance is achieved.
We base our work on the Low-Resource Text Classification Framework introduced by \citet{EinDor2020ActiveLF}.
Following this work, we focus on binary text classification, given a small annotation budget and a potentially imbalanced dataset. This scenario matches common use cases in the legal domain, where the goal is to find phrases that correspond to a specific category, with the lowest possible number of actions, given a pool of unlabeled, imbalanced data. We perform $5$ AL iterations, and assume a more restricted annotation budget compared to \citet{EinDor2020ActiveLF}, allowing only $10$ annotations per iteration. For the first AL iteration, we assume that $5$ positive and $5$ negative samples need to be annotated.
\section{Methodology}
\begin{algorithm*}[ht]
\caption{AL pipeline for text classification}\label{alg:cap}
\hspace*{\algorithmicindent} \textbf{Input:} {unlabeled samples $U_0$, PT RoBERTa, PT Sentence-RoBERTa, AL strategy $\alpha$, \# iterations $T$}
\hspace*{\algorithmicindent} \textbf{Output:} {text classifier CLS RoBERTa, acquired labeled dataset $L_T$}
\begin{algorithmic}
\State $L_0 \gets \emptyset$
\State \textbf{\textit{Phase 1: Task-adaptation with Masked Language Modeling (MLM)}} \State TAPT RoBERTa $\gets$ MLM(PT RoBERTa, $U_0$)
\State \textbf{\textit{Phase 2: Knowledge distillation}}
\State DisTAPT RoBERTa $\gets$ Distill(TAPT RoBERTa, PT Sentence-RoBERTa, $U_0$)
\State \textbf{\textit{Phase 3: Initial sampling}}
\State cluster medoids $\gets$ KMeans(DisTAPT RoBERTa, $U_0$)
\State $L_1 \gets$ Sample(cluster medoids)
\State $U_1 \gets U_0 \setminus L_1$
\State \textbf{\textit{Phase 4: Active learning}}
\For{$i \gets 1$ to $T$ }
\State CLS RoBERTa $\gets$ Train(DisTAPT RoBERTa, $L_i$)
\State $C_i \gets \alpha$(CLS RoBERTa, $U_i$)
\State $L_{i+1} \gets L_i \cup C_i$
\State $U_{i+1} \gets U_i \setminus C_i$
\EndFor
\end{algorithmic}
\label{alg:algorithm}
\end{algorithm*}
We propose an efficient active learning pipeline for fine-tuning pre-trained language models for legal text classification. Our approach leverages available unlabeled data in three phases to adapt the pre-trained model to the downstream task (Sec.~\ref{sec:task-adaptation}), guide its embedding space to a semantically meaningful and comparable space (Sec.~\ref{sec:distillation}), and reduce the number of actions required to collect the initial labeled set (Sec.~\ref{sec:sampling}). Finally, it leverages existing AL strategies to efficiently fine-tune a classifier (Sec.~\ref{sec:meth_al}). We now explain each step in detail. An overview of this pipeline can be found in Algorithm~\ref{alg:algorithm}.
\subsection{Task-Adaptation}\label{sec:task-adaptation}
It has been shown that fine-tuning off-the-shelf pre-trained language models with standard approaches is unstable for small training sets \citep{Zhang2021RevisitingFB,Dodge2020FineTuningPL,Mosbach2021OnTS}. As shown by \citet{Margatina2022OnTI}, this can lead to poor performance when fine-tuning pre-trained language models with AL. In addition, existing pre-trained language models are often trained on texts that do not need specialized training to be understood. However, legal texts contain specialized words that are not common in other domains. Thus, task-adaptation is crucial for the effectiveness of active learning in legal text classification. In the first step of our proposed pipeline, we obtain the task-adapted pre-trained (TAPT) RoBERTa by continuing pre-training the model with unlabeled samples for the Masked Language Modeling (MLM) task, as suggested by \citet{Gururangan2020DontSP} and \citet{Margatina2022OnTI}.
\subsection{Knowledge Distillation}\label{sec:distillation}
Previous works \citep{Reimers2019SentenceBERTSE,Li2020OnTS,Su2021WhiteningSR} have shown that, without fine-tuning, the sentence embeddings produced by pre-trained language models poorly capture semantic meaning of sentences, and are not comparable using cosine similarity. To overcome this shortcoming, \citet{Reimers2019SentenceBERTSE} introduced sentence transformers by adding a pooling layer on top of pre-trained transformer-based language models, and training them in a Siamese network architecture with pairs of similar sentences. Compared to out-of-the-box RoBERTa, a RoBERTa-based sentence transformer drives semantically comparable sentence embeddings.
As we will explain in Sec.~\ref{sec:sampling}, we cluster the normalized sentence embeddings based on their Euclidean distance to efficiently acquire the labeled samples for the initial iteration of AL. The Euclidean distance between normalized embeddings can be driven from their cosine similarity. Hence, sentence embeddings that are comparable with cosine similarity can result in clusters with higher quality. In addition, semantically meaningful sentence embeddings give a better initialization of the \texttt{[CLS]} token, thereby obtaining better classification performance with a smaller training set.
We use a pre-trained RoBERTa-based sentence transformer (PT Sentence-RoBERTa) as a teacher model, and distill its knowledge to the TAPT RoBERTa. The resulting distilled task-adapted pre-trained (DisTAPT) RoBERTa produces semantically meaningful embeddings that are comparable via cosine similarity, and, as shown by our experiments (Sec.~\ref{sec:exp_distillaition}), benefit the classification task.
\subsection{Initial Sampling}\label{sec:sampling}
Many AL strategies (\citealp{Gissin2019DiscriminativeAL,Gal2016DropoutAA}) require an initial set of $N$ labeled samples containing $P$ positive and $N-P$ negative sentences, which is either assumed to be available, or obtained by randomly sampling the entire dataset until the desired number of positive and negative samples are found. This approach is highly expensive for large and skewed datasets. We propose a simple, yet effective, strategy to efficiently acquire the initial labeled set. To this end, we leverage the distilled task-adapted pre-trained RoBERTa to cluster the unlabeled samples using KMeans algorithm \citep{MacQueen1967SomeMF}. The labeled set for the initial iteration is then driven from the cluster medoids. As a result, we shrink the pool of candidates from the entire dataset to the cluster medoids, therefore, reduce the number of actions for obtaining the initial annotated set, while achieving comparable performance with the standard approach for initial sampling.
\subsection{Active Learning}\label{sec:meth_al}
In the last phase, we iteratively fine-tune the DisTAPT RoBERTa for the downstream task. The initial labeled set is used at the first iteration. Then, more samples are labeled in the following rounds using an AL acquisition strategy until the annotation budget is exhausted, or the classifier satisfies the expected performance.
Our proposed pipeline can be used with existing AL strategies and, as demonstrated by our experiments (Sec.~\ref{sec:exp_distillaition}), consistently outperforms standard AL approaches, regardless of the AL strategy used.
\section{Experimental Setup}\label{sec:setup}
We evaluate our approach against four standard active learning strategies provided in the Low-Resource Text Classification Framework \citep{EinDor2020ActiveLF}:
\begin{itemize}
\item \textbf{Random} At each iteration, this approach randomly chooses samples for annotation.
\item \textbf{Hard-Mining} Selects instances that the model is uncertain about, based on the absolute difference of prediction score and $0.5$.
\item \textbf{Perceptron Dropout} \citep{Gal2016DropoutAA} Also selects instances for which the model is least certain. The uncertainty is calculated using Monte Carlo Dropout on $10$ inference cycles.
\item \textbf{Discriminative Active Learning (DAL)} \citep{Gissin2019DiscriminativeAL} Deploys a binary classifier to select instances that best represent the entire unlabeled samples.
\end{itemize}
We consider pre-trained RoBERTa and LEGAL-BERT~\citep{Chalkidis2020LEGALBERTTM} as the baselines. Note that our goal is not to rely on domain-adapted models like LEGAL-BERT since they might not always be available. For example, if the data is in German, we can find a pre-trained RoBERTa in German, but the LEGAL-BERT is pre-trained on English text only.
\subsection{Datasets}\label{sec:datasets}
We evaluate our framework on Contract-NLI \citep{Koreeda2021ContractNLIAD} and LEDGAR \citep{Tuggener2020LEDGARAL} benchmarks.
Contract-NLI \citep{Koreeda2021ContractNLIAD} is a dataset for document-level natural language inference. It consists of $607$ documents with $77.8$ spans per document on average. Each span is checked against $17$ hypotheses and classified as contradiction, entailment, or not mentioned. In this work, we adapt this dataset to the classification task by considering each hypothesis as a category. If a span is classified as contradiction or entailment for a hypothesis, we label it with the corresponding category. Following this approach, we end up with a classification dataset with $4,371$ train, $614$ development, and $1,188$ test samples within $17$ classes.
LEDGAR \citep{Tuggener2020LEDGARAL} is a text classification benchmark consisting of a corpus of legal provisions in contracts. The entire dataset consists of $846,274$ provisions and $12,608$ labels. We only consider a subset of this dataset that corresponds to provisions with labels that appeared at least $10,000$ times in the corpus, resulting in $44,249$ train, $7,375$ development, and $12,907$ test samples across $5$ categories. Similar to \citet{Tuggener2020LEDGARAL}, we perform a $70\%-10\%-20\%$ random split to obtain the train, development and test sets.
The class distributions of both datasets can be found in the appendix (Sec.~\ref{sec:app_class_dist}). Compared to Contract-NLI, LEDGAR has fewer categories, is an order of magnitude bigger, and is more balanced.
\subsection{Implementation Details}
We base our implementation on the Low-Resource Text Classification Framework provided by \citet{EinDor2020ActiveLF}\footnote{\url{https://github.com/IBM/low-resource-text-classification-framework}}, and augment it with the task-adaptation, knowledge distillation, and initial sampling steps.
As the pre-trained model, we use \texttt{roberta-base}\footnote{\url{https://huggingface.co/roberta-base}} (with $125$M parameters), the RoBERTa \citep{Liu2019RoBERTaAR} language model trained on the union of $5$ datatsets: Book corpus \citep{Zhu2015AligningBA}, English Wikipedia\footnote{\url{https://dumps.wikimedia.org}}, CC-News \citep{Mackenzie2020CCNewsEnAL}, OpenWebText Corpus \citep{Gokaslan2019OpenWeb}, and Stories \citep{Trinh2018ASM}, none of which belong to the legal domain.
For LEGAL-BERT, we use the \texttt{nlpaueb/legal-bert-base-uncased}\footnote{\url{https://huggingface.co/nlpaueb/legal-bert-base-uncased}} (with $110$M parameters), trained on six datasets containing legal docments across Europe and the US.
For task-adaptation, we continue pre-training RoBERTa for the MLM task using the available unlabeled data. We train for $10$ epochs with batch-size $64$, and the learning rate set to $3\mathrm{e}{-4}$. The task-adapted model has perplexity $4.9706$ for Contract-NLI and $2.1628$ for LEDGAR.
For model distillation, we use \texttt{stsb-roberta-base-v2} (with $125$M parameters), a RoBERTa-based sentence transformer trained on the STS benchmark \citep{Cer2017SemEval2017T1}, as the teacher model, and the task-adapted RoBERTa as the student model. Mean Squared Error (MSE) is used as the loss function. The student model is trained for $10$ epochs, with $10$K warmup steps, $1\mathrm{e}{-4}$ learning rate and no bias correction. The final MSE ($\times 100$) is $6.8607$ for Contract-NLI, and $7.2003$ for LEDGAR.
For clustering the normalized sentence embeddings we use the KMeans implementation by \texttt{scikit-learn}. We cluster the Contract-NLI and LEDGAR sentence embeddings into $437$, and $442$ groups respectively. The number of clusters are chosen based on the dataset size, and the number of categories, and to make initial sampling with cluster medoids manageable for experts.
In all the active learning experiments, we perform $5$ AL iterations, starting with $10$ initial samples, and increasing the size of the annotated data by $10$ at each iteration. Adam optimizer \citep{Kingma2015AdamAM} is used with learning rate set to $5\mathrm{e}{-5}$. The model is trained for $100$ epochs and early stopping is used with patience set to $10$. To account for randomization, we repeat each experiment three times.
To compare our approach with standard AL methods, we use F1-score as the evaluation metric as it captures both precision and recall and is sensitive to data distribution.
\section{Results and Discussion}
In this section, we provide the results of our experiments and explain them in detail. We start by comparing our approach with and without the initial medoid sampling against standard AL strategies (Sec.~\ref{sec:ours-vs-baseline}). Then, we show the effectiveness of knowledge distillation on top of task-adaptation (Sec.~\ref{sec:exp_distillaition}). In addition, we demonstrate the efficiency of the initial sampling with cluster medoids (Sec.~\ref{sec:exp_initial-sampling}). Finally, we evaluate how well our approach performs for different AL strategies (Sec.~\ref{sec:exp_AL_effect}).
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{Contract-NLI_DAL_avg_with_legalbert.pdf}
\includegraphics[width=0.5\textwidth]{LEDGAR_DAL_avg_with_legalbert.pdf}
\caption{Test F1-score for \textbf{DAL} during AL iterations. The F1-score for the fully supervised fine-tuning is $0.6990$ for Contract-NLI and $0.9538$ for LEDGAR. The figure is best viewed in color.}
\label{fig:ours-vs-baseline}
\end{figure}
\subsection{Efficient AL Pipeline}\label{sec:ours-vs-baseline}
Figure~\ref{fig:ours-vs-baseline} compares our approach with and without the initial sampling phase (DisTAPT with IS, and DisTAPT) to standard DAL with pre-trained (PT) RoBERTa, LEGALBERT, and TAPT RoBERTa for Contract-NLI and LEDGAR benchmarks. We report the average F1-score over all categories. DAL is chosen due to its better performance, as shown in Figure~\ref{fig:all_ALs}.
The results for other AL strategies can be found in the appendix (Sec.~\ref{sec:app_tabs}).
Our experiments show the importance of task-adaptation and knowledge distillation for pre-trained language models prior to fine-tuning with active learning. Figure~\ref{fig:ours-vs-baseline} illustrates that, for the same size of annotated data, our pipeline consistently achieves better performance than standard AL approaches even for LEGAL-BERT.
For the Contract-NLI dataset, the F1-score obtained by fully-supervised fine-tuning (with $4,371$ labeled samples) is $0.6990$ for \texttt{roberta-base} and $0.7152$ for \texttt{legal-bert-base-uncased}. DisTAPT RoBERTa reaches a F1-score as high as $0.6508$ with only $40$ labeled samples. The best F1-score obtained using pre-trained RoBERTa is $0.3162$ with $30$ labeled samples, which is $0.3165$ lower than DisTAPT RoBERTa's F1-score for the same size of annotated data.
For the LEDGAR dataset, the F1-score obtained by the fully-supervised fine-tuning (with $44,249$ labeled samples) is $0.9538$ for \texttt{roberta-base} and $0.9588$ for \texttt{legal-bert-base-uncased}. DisTAPT RoBERTa reaches a very close performance of $0.9321$ F1-score with merely $60$ labeled samples. The highest F1-score that pre-trained RoBERT reaches is $0.7663$ with $20$ annotated samples, which is $0.0904$ lower than DisTAPT's performance with the same size of labeled data.
These results show that, for both datasets, there is only a small performance gap between our approach and the fully-supervised approach, indicating that our AL pipeline dramatically reduces the annotation cost, while achieving comparable performance with the fully-supervised fine-tuning.
In addition, It is observed that standard AL with off-the-shelf pre-trained RoBERTa is unstable. This is aligned with the previous works' observations \citep{Mosbach2021OnTS,Zhang2021RevisitingFB,Dodge2020FineTuningPL}. During fine-tuning, the pre-trained model should perform two tasks: adaptation to the legal domain with the new vocabulary, and classification. By performing task-adaptation and knowledge distillation before fine-tuning, we train the model in a curriculum learning approach, making the model stable even for small training sets.
\subsection{Effect of Knowledge Distillation}\label{sec:exp_distillaition}
To evaluate the effectiveness of knowledge distillation on the quality of obtained clusters, we compare the distribution of the Dunn Index of the clusters before and after knowledge distillation. For both datasets, after knowledge distillation, most of the clusters have higher Dunn Index which indicates that they are more compact and better separated than the clusters before knowledge distillation step. The results are provided in the appendix~\ref{sec:app_knowldge_distill} due to space constraints.
In addition, we evaluate the effect of knowledge distillation on the task-adapted pre-trained RoBERTa, and report the average F1-score over all classes for each dataset. Figure~\ref{fig:ours-vs-baseline} shows that, for both datasets, DisTAPT RoBERTa outperforms TAPT RoBERTa at early iterations of active learning, and as the size of the labeled set increases, the two models' performance converge. This can be explained by the fact that, initially, DisTAPT RoBERTa's embeddings better capture the semantics of sentences, and thus result in better classification performance. As the labeled data grows, TAPT RoBERTa is fine-tuned and can produce semantically meaningful embeddings as well. Hence, for a highly restricted annotation budget, distilling the knowledge of a sentence transformer to the TAPT language model can lead to performance gain.
\subsection{Efficiency of Initial Medoid Sampling} \label{sec:exp_initial-sampling}
It was shown in Figure~\ref{fig:ours-vs-baseline} that DisTAPT with IS obtains comparable performance with DisTAPT without IS. In this section, we evaluate the \textit{efficiency} of the proposed sampling strategy for the initial iteration of AL.
To this end, we simulate the standard sampling strategy by randomly sampling text segments from the full dataset until $5$ positive and $5$ negative samples are found. The number of iterations is then considered as the number of annotations required to collect the labeled set for the initial AL iteration. Similarly, to simulate our proposed initial sampling, we randomly sample from cluster medoids until $5$ positive and $5$ negative samples are obtained. To account for randomness, we repeat the simulations $1000$ times and report the median and the $90^{th}$ percentile over all runs.
Table~\ref{tab:exp_medoid} illustrates the results of our simulations for Contract-NLI and LEDGAR. Due to the high number of classes in Contract-NLI, only eight categories of this dataset are presented in this table, and the results for other categories can be found in the appendix (Sec.~\ref{sec:app_medoid}). For each class, in addition to the median and $90^{th}$ percentile over $1000$ runs, the difference in the $90^{th}$ percentile between standard approach and our strategy (in $\%$) is reported as the gain in annotation effort. For example, for the \texttt{Sharing with third-parties} class in Contract-NLI, the $90^{th}$ percentile is $62\%$ less when using medoids for initial sampling, meaning that, with $90\%$ confidence, the annotators perform $62\%$ fewer actions to acquire the initial labeled set using our approach.
It is observed that, for the skewed Contract-NLI dataset, our proposed initial sampling strategy reduces the number of actions performed by the annotator up to $63\%$. For LEDGAR however, which consists of balanced categories, the highest effort gain in sampling from cluster medoids is $25\%$. There are also few cases where using the entire dataset is more efficient than sampling from medoids. This happens when the class' frequency is higher in the full dataset than its frequency in the cluster medoids.
Overall, our results demonstrate the advantage of using the cluster medoids for collecting the initial annotated samples for a skewed dataset like Contract-NLI, which is a realistic use-case in the legal domain. It is noteworthy that the original version of LEDGAR dataset is also imbalanced, but as explained in Sec.~\ref{sec:datasets}, due to the drastically high number of classes, and for the sake of comparison with skewed datasets, only the most dominant categories are kept in this work.
Thanks to the semantically meaningful and comparable sentence embeddings obtained after the knowledge distillation step, the cluster medoids well represent the entire dataset, and thus sampling among them drastically reduces the annotation effort without harming the performance.
As a real life scenario, consider a company with hundreds of legal contracts aiming to classify their sentences into multiple categories, under a restricted budget. Reducing the annotation effort means lowering down the financial costs of annotation, which can be highly expensive in the legal domain (over $\$2$ million for annotating around $500$ contracts according to~\citet{Hendrycks2021CUADAE}).
\begin{table*}[h]
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{clccccc}
\hline
\multirow{2}{*}{Dataset} & \multirow{2}{*}{Category} & \multicolumn{2}{c}{full dataset} & \multicolumn{2}{c}{medoids} & \multirow{2}{*}{gain(\%)} \\
& & median & $90^{th}\%$tile & median & $90^{th}\%$tile & \\
\hline
\multirow{8}{*}{\small{\texttt{\rotatebox[origin=c]{90}{Contract-NLI}}}}
& \small{\texttt{Inclusion of verbally conveyed information}} & 75.0 & 125.0 & 35.5 & 59.0 & 52.8\\
& \small{\texttt{No licensing}} & 64.0 & 108.0 & 68.5 & 109.1 & -1.0 \\
& \small{\texttt{No reverse engineering}} & 342.0 & 568.0 & 144.0 & 209.1 & 63.2\\
& \small{\texttt{Notice on compelled disclosure}} & 74.5 & 122.0 & 99.0 & 155.0 & -27.0\\
& \small{\texttt{Sharing with employees}} & 57.0 & 90.0 & 21.0 & 34.1 & 62.1 \\
& \small{\texttt{Sharing with third-parties}} & 54.0 & 92.1 & 21.0 & 35.0 & 62.0\\
& \small{\texttt{Survival of obligations}} & 64.0 & 106.0 & 36.0 & 57.0 & 46.2\\
& \small{\texttt{Return of confidential information}} & 116.0 & 189.0 & 61.0 & 99.0 & 47.6\\
\hline
\multirow{5}{*}{\small{\texttt{\rotatebox[origin=c]{90}{LEDGAR}}}} & \small{\texttt{Amendments}} & 23.0 & 37.1 & 21.0 & 33.0 & 10.8\\
& \small{\texttt{Counterparts}} & 26.0 & 42.0 & 34.0 & 54.1 & -28.8 \\
& \small{\texttt{Entire agreements}} & 26.0 & 42.0 & 33.0 & 55.0 & -30.9 \\
& \small{\texttt{Governing laws}} & 17.5 & 28.0 & 14.0 & 21.0 & 25.0\\
& \small{\texttt{Notices}} & 29.0 & 49.0 & 26.0 & 44.0 & 10.2\\
\hline
\end{tabular}}
\caption{\label{tab:exp_medoid}
Number of actions to acquire the initial labeled set for $8$ categories of Contract-NLI, and LEDGAR when sampling from the full dataset (standard approach), and sampling from the cluster medoids (our approach).}
\end{table*}
\begin{figure}[h!]
\centering
\includegraphics[width=0.5\textwidth]{Contract-NLI_avg.pdf}
\includegraphics[width=0.5\textwidth]{LEDGAR_avg.pdf}
\caption{Comparison of four AL strategies when used with DisTAPT RoBERTa with IS.}
\label{fig:all_ALs}
\end{figure}
\subsection{Effect of AL strategy}\label{sec:exp_AL_effect}
Finally, we evaluate the generalizability of our approach over the four AL strategies mentioned in Sec.~\ref{sec:setup}: DAL, Random, Hard-Mining, and Perceptron Dropout. As shown in Figure~\ref{fig:all_ALs}, DAL results in the best performance with at most $0.08$ higher F1-score than other strategies with $60$ labeled samples for Contract-NLI, and less than $0.04$ higher F1-score with $40$ annotated samples for LEDGAR. The small performance gap of these four AL methods in our pipeline indicates the generalizability of this approach to various AL strategies.
\section{Conclusion}
We propose an efficient active learning pipeline for legal text classification. Our approach leverages the available unlabeled data to adapt the pre-trained language model to the downstream task, and guide its embeddings to a semantically meaningful space before fine-tuning. We use model distillation to produce semantically comparable embeddings. A future work can study the effect of other approaches like BERT-Flow \citep{Li2020OnTS} and whitening \citep{Su2021WhiteningSR} on AL with this pipeline. Moreover, we design a simple strategy to efficiently acquire a labeled set of positive and negative samples for the initial iteration of active learning.
Our experiments over Contract-NLI and LEDGAR benchmarks demonstrate the effectiveness of our approach compared to standard active learning strategies. Our results also show that our pipeline obtains very close performance to the fully-supervised approach with considerably less annotation cost. We test our methodology in the legal domain, and for four AL strategies, but we expect it to generalize to other strategies like ALPS and BADGE, and other specialized domains, like medicine. We leave this evaluation as a future work.
\section*{Limitations}
In this work, we have shown the importance of task-adaptation and knowledge distillation, and that we can leverage the available unlabeled data to perform efficient fine-tuning via active learning and obtain better performance. The price to pay for this performance gain is time and computational power. The time taken by task-adaptation and distillation scales with the size of unlabeled data. On the other hand, more unlabeled samples result in more effective adaptation to the downstream task. Therefore, the user of this approach needs to find the best trade-off given their data, annotation budget, time and computational power. For, LEDGAR, the larger dataset used in this work, we performed the adaptation and distillation steps in $4$ and $1$ hour(s) respectively, using a single Nvidia GeForce GTX TITAN X GPU.
Moreover, we showed that by clustering the sentence embeddings produced by DisTAPT RoBERTa, the initial labeled set can be acquired more efficiently. Nevertheless, this approach inherits the limitations of clustering. Namely, the time complexity of clustering the embeddings scales with the data, and the number of clusters should be empirically chosen. In our experiments we spent $10$ minutes to cluster the $44,249$ samples belonging to LEDGAR dataset into $442$ groups.
\section*{Ethics Statement}
Industries have hundreds of contracts with tens of thousands of sentences that belong to various topics. Labeling all of these samples is a highly expensive and time-consuming process. In this work, we aim to reduce the resources spent on this task by leveraging recent advances in natural language processing, while keeping the human expert in the loop. The goal is to reduce the human effort in annotation so that the legal experts' time and knowledge can be used in another task at which humans are better than machines.
\input{emnlp2022.bbl}
\bibliographystyle{acl_natbib}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Q: Instagram Login without oAuth I am trying again to login to instagram, but i have to do this without using their api v2 which uses oAuth for verifying.
The only thing i have is Username and Password to login.
So what i did was looked for the way, instagram application for Android and hopefully IOS does that.
I see that it creates a request at the following url:
https://instagr.am/api/v1/accounts/login/
If you would visit that link directly, you will more likely get a error saying that the page couldn't be find. After a little bit of googling, i came across a 2 years old post which states that in the Useragent, we have to contain the string "Instagram" for it to work.
I did that by faking the useragent and the result is as follows:
{"status":"fail","message":"Your version of Instagram is out of date. Please upgrade your app in the Play Store to log in to Instagram."}
Now, i am guessing that we also need to add something else or some other headers too, so i was looking for a way to grab the request being sended to instagram by either of their Android or IOS app.
So next i downloaded Bluestack and installed and ran instagram on my computer using this. I was able to install and login it successfully, but then i was unable to log it using Charles since the request is being sent to https:// server
I also tried to grab it through Wireshark but unfortuantely, i am not much experienced in using it and hence don't know it purpose.
So could anyone help me to get how to login to instagram with C# without using oAuth, cuz i just have username and password.
I would probably code the end part of requesting myself, but i am unable to capture the headers being sent to instagram.
Also, if there is anything like Charles/Wireshark which captures network traffic for Android, do let me know about it too.
A: You should be able to capture the request and see the request headers using Fiddler if you can tell your device to use your pcs internet connection by going through USB. It seems your getting close but might just need a version number somewhere in your request.
A: Yeah you can do this using Webview
Here you go
mWebView.setWebViewClient(new WebViewClient() {
public boolean shouldOverrideUrlLoading(WebView view, String url) {
view.loadUrl(url);
UrlQuerySanitizer.ValueSanitizer sanitizer = UrlQuerySanitizer.getAllButNulLegal();
// remember to decide if you want the first or last parameter with the same name
// If you want the first call setPreferFirstRepeatedParameter(true);
sanitizer.sanitize(url);
String value = sanitizer.sanitize("username"); // get your value
if(MyBridge.getUsername()!=null)username = MyBridge.getUsername();
return true;
}
@Override
public void onPageStarted(WebView view, String url, Bitmap favicon) {
super.onPageStarted(view, url, favicon);
// showLoading();
CookieManager.getInstance().removeAllCookies(null);
// CookieManager.getInstance().flush();
progressBar.setVisibility(View.VISIBLE);
}
@Override
public void onPageFinished(WebView view, String url) {
super.onPageFinished(view, url);
progressBar.setVisibility(View.GONE);
if (url.equalsIgnoreCase(mURL)) {
view.addJavascriptInterface(new MyBridge(InstagramOfficalLoginActivity.this), "bridge");
String javascript = "javascript: document.getElementsByClassName(\"_0mzm- sqdOP L3NKy \")[0].onclick = function() {\n" +
" var username = document.getElementsByName(\"username\").value;\n" +
" var password = document.getElementsByName(\"password\").value;\n" +
" bridge.saveData(username, password);\n" +
" };";
view.loadUrl(javascript);
}
if (isSessionid ) {
// username = MyBridge.getUsername();
//сохранение данных пользователя
Logins logins = new Logins();
logins.setUserId(InstaUtils.getUserId());
logins.setUserName("");
logins.setProfilePic("");
logins.setSession_id(InstaUtils.getSessionid());
logins.setCooki(InstaUtils.getCookies());
logins.setCsrf(InstaUtils.getCsrf());
long id = DataObjectRepositry.dataObjectRepositry.addNewUser(logins);
PreferencesManager.savePref(GlobalConstant.USERNAME,username);
PreferencesManager.savePref(GlobalConstant.USER_ID, InstaUtils.getUserId());
PreferencesManager.savePref(GlobalConstant.TOKEN, InstaUtils.getSessionid());
PreferencesManager.savePref(GlobalConstant.PROFILE_PIC,"");
Intent intent = new Intent(InstagramOfficalLoginActivity.this, MainActivity.class);
PreferencesManager.savePref("isLogin",true);
intent.addFlags(Intent.FLAG_ACTIVITY_CLEAR_TOP | Intent.FLAG_ACTIVITY_NEW_TASK);
intent.putExtra("user", InstaUtils.getUserId());
intent.putExtra("database_id",String.valueOf(id));
mWebView.destroy();
mWebView = null;
startActivity(intent);
}
}
@Override
public void onLoadResource(WebView view, String url) {
super.onLoadResource(view, url);
progressBar.setVisibility(View.GONE);
cookies = CookieManager.getInstance().getCookie(url);
try {
String session_id = getCookie(url, "sessionid");
String csrftoken = getCookie(url, "csrftoken");
String userid = getCookie(url, "ds_user_id");
if (session_id != null && csrftoken != null && userid != null) {
isSessionid = true;
InstaUtils.setSessionId(session_id);
InstaUtils.setUserId(userid);
InstaUtils.setCookies(cookies);
InstaUtils.setCsrf(csrftoken, cookies);
}
}catch (Exception e){
e.printStackTrace();
}
}
@SuppressWarnings("deprecation")
@Override
public void onReceivedError(WebView view, int errorCode, String description, String failingUrl) {
ToastUtils.ErrorToast(InstagramOfficalLoginActivity.this, description);
}
@TargetApi(android.os.Build.VERSION_CODES.M)
@Override
public void onReceivedError(WebView view, WebResourceRequest req, WebResourceError rerr) {
// Redirect to deprecated method, so you can use it in all SDK versions
onReceivedError(view, rerr.getErrorCode(), rerr.getDescription().toString(), req.getUrl().toString());
}
});
mWebView.loadUrl(mURL);
You can also download fully working code from my github profile Here is the link.
I hope it work,Thanks :)
A: you can always login through https://instagram.com/accounts/login/ having only login\password.
having ability to sniff traffic will not help you, as they sign all their messages now to prevent things you are trying to achieve.
A: You can find Instaguser library at here. https://github.com/ThinhVu/InstagramUser
Instaguser allow you login into instagram without OAuth.
At the moment, you can change user information: username, biography, ... etc.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,378
|
class PasswordsController < Devise::PasswordsController
layout false
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,882
|
01. Gathering Information
02. Making Our Voices Heard
01. Presenting the Vision
02. Going Public
03. The Reactions
Final Deliverables
How do you build a thriving, walkable City Center?
During the Spring of 2014 I was part of a group of students who researched, unearthed, and identified issues that we believed were holding back our city of Grand Rapids, Michigan. We worked on these issues for 4 months and then reported our findings in a presentation to faculty members, local stakeholders, and city officials.
We had a respectable amount of research, concepts, and ideas. And our presentation was well received, as we anticipated. But what we didn't expect was that our project would go on to become a viral topic of discussion, debate, and backlash for the entire city.
The Team.
Film and Project Development
Heather Seto
Adam Salois
Emily Nagy
Luc Fewer II
Matthew Vansweden
Rachael Fischer
For the start of our project we connected with 616 Development, a local urban development company, hoping to get some insight on what it would take to bring a fresh food grocery store to the City Center in response to the growing market rate housing options. At the time, the average resident was forced to drive 20 minutes outside of the city merely to obtain two to three weeks worth of groceries (subsequently sending their money elsewhere), when this city is perfectly capable of providing a fresh food culture, in the heart of the City Center, where a new housing development sprouts up practically every month.
As designers, our first instinct was to design a really amazing experience. However, we quickly realized that would be too premature. We needed to understand the problem better; we needed to raise awareness. This is not solely a development problem or an economic development problem, it is an urban planning problem, a public health problem, and a transportation problem. This means that there is no clear solution, there may not even be consensus that there is a problem to begin with.
In light of these revelations, we defined a goal.
Raise awareness of the need for essential basic services in the city center of grand rapids in order to inspire further actions within the community.
Gathering Information.
Before we could start to propose solutions, we had to fully understand the problem. And the obvious path to that understanding was to talk to the individuals that lived in and experienced the city everyday. We began a period of intense research, conversations, events, and surveys all aimed at the goal of understanding the problem more deeper and building empathy for our users.
We started this process by taking a look at the basic services offered in most towns and cities. We then mapped where they would fall on Maslow's Hierarchy of Needs, a theory of psychology that attempts to explain human motivation. And then based off this map, we surveyed the available resources in the Grand Rapids City Center and graphed them alongside Maslow's.
Maslow's Hierarchy
A comparison of the resources available in the City Center of Grand rapids compared to Maslow's Hierarchy of Needs.
Student Surveys.
Even though the information we had gathered so far was valuable, we wanted to hear from the experiences and opinions of the individuals that this would directly affect. So to gather these opinions, we set up an event and surveyed the student body of KCAD, which is located directly in the city Center. We collected surveys throughout the event and used this data to inform the rest of our project.
The number of miles students are traveling to purchase groceries.
The mode of transportation students use to travel to the grocery store.
The most popular places students said they purchased their groceries.
Whether a City Center grocer would be a beneficial addition to their lives.
Making our voice heard.
We knew that facts and numbers were an important part of the project. We needed to give quantitative data that would support the project we were proposing. But, we also wanted to communicate on an emotional and human level. We wanted to showcase prominent figures, locations, and influences in the community to speak to the heart and core motives of the project.
So I proposed the best way I know to communicate on this level, film. We gathered our core values and the opinions of the key outside voices we connected with to unite in a singular message. I directed and produced a short film that was designed to accompany our presentation and live as an artifact of the project's core ideas.
Presenting our vision.
As our project came to a close, our focus shifted towards the communication of our project. We wanted to make sure that the work we accomplished landed in the right hands and would be put to good use. We scheduled a presentation of our materials and invited community stakeholders to attend. We delivered a 45 minute presentation that covered every aspect of our project and proposed the request for a basic needs assesment of the City Center.
Releasing it to the public.
We all assumed that our presentation would be the end of the project. We fulfilled our goals and handed off the information into the right hands to take it forward. We were content.
I then took the video I created and uploaded it to Vimeo, just like I would with any video that I produce. And randomly, it got picked up by journalists. And much to our surprise, articles started sprouting up in every local new source. It even jumped mediums and landed on the front page of the printed newspaper.
Because of the various publicity and shareable / clickbait nature of the headlines and video, the viewership quickly shot straight up. The video received roughly 70,000 views over the course of two weeks, meager in the scope of viral videos, but massive compared to the intended audience.
70,000 views over the course of two weeks.
Reactions.
At first we were estatic about the focus on our project. We were excited to see everyone sharing and enjoying the thing we worked for months on. But then the feedback started to roll in...
WHERE IS THE AMAZING DIVERSITY THAT MAKES US WHO WE ARE?An Armchair Warrior.
Narcissistic, Pseudo, Self-righteousness, Closed-mindedness, Disillusioned, and Arrogant...An angry Facebook user.
I did not spot ONE person of color in this video....really?! Thanks, but I'll stay in Manhattan.An individual with a solid argument.
Please do not use this video to promote the city that I have come to love because of its diversity.A disappointed citizen.
frame by frame there are only 2 instances in this video where non-white people are shown, and in those frames they are out of focus and in the background. Overlooked again.A community member I failed to represent.
Research "journalistic integrity." You exaggerate like an adolescent.A fellow filmmaker I respect.
Mistakes & Reflection.
These were hard words to swallow from a community of individuals that I believed I was truly advocating for. At first I viewed these comments as fringe-cases. There will always be people who have a negative view of any message. I demeaned them to the "Armchair Warrior" and "Angry Facebook User". But as it became more and more apparent that this was the resounding consensus of the entire community, I began to see that these weren't just edge cases. These were instead concerned citizens, frustrated community advocates, and members of under represented groups who we didn't make feel valued.
Ultimately, these reactions didn't render our work useless or invaluable, because it helped to contribute to a broader dialogue. But, we learned a valuable lesson. Design is about context, we designed a video that served a specific purpose, and when it was re-contextualized its message was lost. We never purposely sought to exclude individuals, but it was the lack of purposeful inclusion that ultimately harmed our credibility.
In the end, this experience proved to be an invaluable learning experience about the ways in which good, or bad design, can have wide affects on culture and the broader dialogue, for better or worse.
The lesson.
Design is context.
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{
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Staten Island DA office makes black history, appoints first African-American prosecutor to serve at executive level
By Christina Carrega
Thomas C. Ridges enforced the law in the Brooklyn District Attorney's Office 17 years. (Jesse Ward/for New York Daily News)
The Staten Island District Attorney's office is making black history by appointing the first African-American prosecutor to serve at an executive level.
Thomas C. Ridges enforced the law in the Brooklyn District Attorney's Office 17 years. Now he'll kick off Black History Month in the newly-created post as Special Counsel to newly-elected DA Michael E. McMahon.
The Brooklyn native will advise McMahon on legal matters.
McMahon was sworn earlier this month and started cleaning up the office of his predecessor by letting go of seven prosecutors.
"Tom has a proven track record of success in the courtroom and brings with him a wealth of knowledge gained from working on some of the city's most high-profile crimes," said McMahon. "Adding Tom to our already impressive roster of prosecutors will help us to continue to keep Staten Island safe."
STACEY DASH CALLS FOR ELIMINATION OF BLACK HISTORY MONTH AND AWARDS THAT CELEBRATE AFRICAN-AMERICANS AMID OSCARS BLACKLASH
Prior to becoming a lead prosecutor, Ridges was a NYPD officer. Ridge was a Bureau Chief that oversaw crimes within Bensonhurst, Brownsville, Crown Heights, Park Slope as well as NYCHA and transit.
Ridges prosecuted about 50 homicide cases, including the murder trials of Imette St. Guillen, Police Officer Dillon Stewart and Detectives Patrick Rafferty and Robert Parker.
"I'm confident that Tom will now serve the residents of Staten Island with the same outstanding work ethic, unquestionable integrity, dedication to fairness and enormous skill that he demonstrated while in Brooklyn," said Brooklyn DA Ken Thompson.
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{
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| 2,807
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Artist Jonny Detiger: Passions of Family The artist and his family at home in NYC
Art — 03.27.17
Jonny Detiger is a Dutch, New York City-based artist and designer known for whimsical, optimistic body of work that includes intricate drawings, sculpture, furniture and soundwork. One of the most fascinating things about Jonny is that he does not keep his creativity hidden away; for this artist, art and life do not live in separate spaces. They are one-in-the-same. Inspiration and making are infused into the artist's everyday life, including the way he and his wife Lisa (a former textile colorist and described as the family's "glue") are raising their talented and inspired children—Blu, Rex and Luc. For this video, the artist let us into his stunning Greenwich Village pad to see how his family lives the art and music lifestyle everyday.
"Art, fashion, design and music have always been an integral part of my life and art. I move freely between these disciplines. Happiness, humor and positivity are common themes that run through all of my work."
-Jonny Detiger
If the Detiger's had an official family motto, it might be this statement from Jonny: "We believe that art = living and living = art. We approach all aspects of life with the same passion and creativity as making art." So, it's no surprise that the family home is an environment where creative experimentation is encouraged to take center stage. The space is both a reflection of Jonny's work—filled with his paintings, sculpture and furniture—and an open playground for the whole family to make music and art. "Our space is used as a living studio, performance space, a gallery and a recording studio. Experimentation and creativity are encouraged at every turn. Music is played, art is being made, photo shoots are in action. There is definitely a synergy there and an exchange of ideas that flows both ways," says the artist.
Jonny's artistic vibe and supportive spirit has undoubtedly rubbed off on his three children, Blu, Rex and Luc, who are already actively engaged as creatives as well. Blu has been a bassist since she was 7 years old and also DJs in some of NYC's top clubs. Rex is an artist, drummer, performer and songwriter. Additionally, Blu and Rex have a sister-and-brother band called BITS, which they intriguingly describe on their website as, "the idiosyncratix, melodramatix, honest-ironix, pop-romantix, show-antix, fauxmelodix, boom-pop-aesthetix." (They have some original music coming out soon—stay tuned!) Not to be left out, Jonny's youngest son, Luc, is a multihyphenate as well. A photographer, artist and actor, Luc already has an Instagram feed full of artistic photographs of his family and New York City life.
"The colors and the space and the vibe [were] all created to make art, to make design, to make music. To have a life filled with love and inspiration."
It's important to note that creativity is a two-way street in the Detiger household. It's not simply the parent "passing down" wisdom to the children. Blu, Rex and Luc fuel their father's work and his work inspires them. Jonny describes his children as "my greatest teachers" and says that, "their youthful optimism and work ethic is inspiring. They give me a window into their generation and a fresh way of thinking. My own work has a very optimistic, happy positive feel to it and that has a lot to do with them."
Similarly, Blu describes how her father's work impacts the spirit of their home and her music: "His paintings give off a very positive and light energy. The nature of the colors and materials he uses...creates a positive energy in the house at all times. Many of his pieces have music incorporated in them, both metaphorically and physically. As a musician, I'm always inspired by his ability to create spaces and environments that emote happiness and love."
"Having children opened up parts of myself that unveiled new channels creatively."
Jonny and Lisa describe their parenting philosophy as: "love, support, validation, empowerment and courage" and it's easy to see these values have played out in the intensely creative and happy hive that he's made with his family. When asked about a favorite family
memory, both Jonny and Blu mentioned listening to music while driving down the back roads of Ibiza. It's a scene that truly seems to encapsulate the Detiger ethos—a communal journey filled with music, art and joy.
Shop Jonny's Artwork
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{
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}
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Poop Culture can be found on Infirmary Media along with other great podcasts!
What does a Rock Star a Baldwin Brother and a Cult Cinema Icon all have in common? They have all been on The Poop Culture Podcast!
Check out some of our great guests!
Once a month we set sail to our 1980/1990's Utopia Island Studios to bring you retro episodes of Dueling Decades! Come back in time with us as we explore these most bodacious times! Join our Facebook group!
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{
"redpajama_set_name": "RedPajamaC4"
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| 6,485
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Pop Culture | TV
Forget Reboots, All Of Nickelodeon's Classics Are Coming To A Computer Near You
By Tanya | Aug 28, 2018
You know how everything and anything is being rebooted these days? Whether it's Roseanne, Alf, or even Frasier, a lot of us are getting really fed up with the attempts at bringing back our old favorites.
Why are we so upset about it? Well, these shows were perfect to begin with, so why bother remake them? I would rather just watch all my favorite episodes again and again, I don't need new stories.
If I wanted new stories I would watch something completely new and original, so just bring back the classics and let me live my nostalgia-based life.
But now, finally, after all of us begging for years, someone is finally listening. One of the biggest companies responsible for all of our favorite shows has finally heard our pleas and is giving us exactly what we want!
The original episodes of all your favorites Nickelodeon cartoons from your childhood are coming to a new streaming service called NickSplat!
The on-demand streaming channel will allow you to watch shows like All That, Are You Afraid Of The Dark?, Clarissa Explains It All, Legends of the Hidden Temple and more.
It's got a list of almost 30 that will be available, and the best part is, if your favorite isn't on there you don't have to worry because they are planning to add more as time goes on.
Now, it does cost a small fee to be able to watch all those childhood classics, but like most streaming services it's really not that expensive.
You can subscribe to NickSplat for only $5.99 per month, or if you already are subscribed to VRV Premium then good news for you, it's included in your $9.99 per month fee!
I don't know about you, but I am very excited to watch AAHHH Real Monsters! again. It was always my favorite when I was growing up, but there is so much to choose from.
Source - Entertainment Weekly / VRV
Which Nickelodeon show are you most excited to watch again?
Tags: nickelodeon, tv show, streaming, 90s
'Rugrats' Is Coming Back With A New TV Show, But It's The Movie That's Got Us Worried
By Tanya 2 years ago
Well, it was only a matter of time before one of the best shows was added to the reboot explosion. Get ready for Tommy, Chuckie, Angelica, and the rest of the Rugrats gang, because they are coming back in a big way.Rugrats first aired in 1991, and ever since we've all loved those goofy, little babies. Is it weird that Tommy kept a screwdriver in his diaper? Yes, of course it is, but how else is he going to escape? It's not like Angelica was going to help him out! The Nickelodeon classic is not just coming back to
Nickelodeon's Slime Is Now An Edible Sauce You Can Buy, And No We're Not Kidding
You know how you used to watch shows like Double Dare or The Splat and think, "OOO that slime looks so delicious, I wish I could eat it!" No? Right, because no one thought that. NickelodeonWell, even though it's not a substance that most people want to eat, Nickelodeon found a way to make kids want it. They turned their iconic slime into "Slime Sauce" and are now selling it at Walmarts all over the country. What is it? Well, don't worry, it's not the gross pudding and oatmeal mixture that Marc Summers revealed was a part of the original
'Clarissa Explains It All' Reboot Is Apparently In The Works, And Melissa Joan Hart Is Involved
We first met Melissa Joan Hart in 1991 when she stepped onto the scene as Clarissa Darling. She was cool, unique, and iconic in her time. She's credited as the first female lead of a Nickelodeon show, and because of her they were able to make many other hit series like The Secret World of Alex Mack and The Amanda Show. But Clarissa was truly one-of-a-kind. I mean, how many people do you know who owned a baby alligator named Elvis? Her show was different then a lot of others, because she would directly address the audience. She dealt
Millions Are Seeking to Rejuvenate After Reliving Their Youth with Disney+
People of all ages are reliving their youth by watching classic favorites from their childhoods on Disney's new subscription on-demand video streaming service, Disney+. And in their recollections of halcyon days and childhood movies, they're finding that age and time are catching up with them. The greatest television network in history is not only dusting off old classics and family favorites, but the phenomenon is also inspiring viewers to turn back the clock - if not on their lives, at least on their bodies and relationships. Read on to learn how Botox treatments are more popular than ever,
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Genre ▾ Business (20)
Radio drama (30)
Sci Fi and Fantasy (10)
German-speaking (100)
The Wonderful Wizard of Oz Audiobook (free download)
This popular and renowned Children audio book is now available for free download from Spotify, Deezer, und in high quality from Audible as well. It's one of the best-loved works by L. Frank Baum. Instead of downloading multiple mp3 files, you can now just listen to the original audio book for free and legally.
Audiobooks > Children > Bestseller
One of the best-known stories in American culture, "The Wonderful Wizard of Oz" has stirred the imagination of young and old alike for over 100 years. Academy Award-winning actress Anne Hathaway ("Rachel Getting Married, Alice In Wonderland"), fresh from filming one of this year's most anticipated films, "The Dark Knight Rises", lends her voice to this uniquely American fairy tale.
This week, the audio book version of L. Frank Baum's The Wonderful Wizard of Oz has made it into the top 50 bestsellers in the Children category. You can legally download or stream this audio book and listen to it for free at Spotify, Deezer, and in high quality at Audible.
by L. Frank Baum
Length: 3:49 h
Published: 03-08-12
You can download this book free of charge if you log in with your Amazon login data for a free 30-Day Trial Membership at the world's leading audiobook platform, Amazon Audible.
The Wonderful Wizard of Oz (By L. Frank Baum)
by Wordscape
Tracks: 07:16 Chapter 01 - The Cyclone12:01 Chapter 02 - The Council with the Munchkins11:54 Chapter 03 - How Dorothy Saved the Scarecrow08:43 Chapter 04 - The Road Through the Forest12:01 Chapter 05 - The Rescue of the Tin Woodman09:12 Chapter 06 - The Cowardly Lion11:12 Chapter 07 - The Journey to the Great Oz11:54 Chapter 08 - The Deadly Poppy Field08:41 Chapter 09 - The Queen of the Field Mice12:11 Chapter 10 - The Guardian of the Gate22:17 Chapter 11 - The Wonderful City of Oz22:29 Chapter 12 - The Search for the Wicked Witch07:02 Chapter 13 - The Rescue11:15 Chapter 14 - The Winged Monkeys16:39 Chapter 15 - The Discovery of Oz, The Terrible05:49 Chapter 16 - The Magic Art of the Great Humbug06:55 Chapter 17 - How the Balloon Was Launched07:10 Chapter 18 - Away to the South06:13 Chapter 19 - Attacked by the Fighting Trees09:25 Chapter 20 - The Dainty China Country05:54 Chapter 21 - The Lion Becomes the King of Beasts05:57 Chapter 22 - The Country of the Quadlings08:12 Chapter 23 - Glinda the Good Witch Grants Dorothy's Wish00:46 Chapter 24 - Home Again
▶ Listen now for free
(provided by Spotify Free)
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(provided by Napster)
Audioobook offers the day's hottest, funniest, most stunning and suspenseful audio books for download and streaming. If you're a fan of L. Frank Baum's "The Wonderful Wizard of Oz", then you may also be interested in the following Children audio books:
© 2019 Audioobook | Contact | Terms of Use | Privacy
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 9,546
|
It took me about an hour to get the goosebumps to go down.
Don't get me wrong, there are a lot of good reasons to encourage children to watch less TV, it was just the sheer puritanical unpleasantness of the logic that got to me. "You're basically worshipping the TV. That's time you should spend praying to God.". You could apply that to anything. If you read a book, you're basically worshipping the book. That's time you should spend praying to God. If you play outside, you're basically worshipping your garden. That's time you should spend praying to God. If you listen to music, you're basically worshipping the music. If you talk to your friends and family, you're basically worshipping your friends and family. If you think for yourself, you're basically worshipping your brain.
It's possible that the girl's pastor didn't intend to spread this message. It's possible that he was just condemning television in particular (since it was invented in the last hundred years and is therefore evil) and just didn't take his train of thought to the logical conclusion. Honestly, though, I doubt it. I think he wanted her to worry that she wasn't spending enough of her leisure time praying to God. I think he wanted her to feel guilty that she wasn't spending all of her leisure time that way. I think he wanted to teach her that the ideal spiritual life is spent doing nothing but praying, possibly with occasional breaks to eat (but possibly not, after all, when you eat dinner you're basically worshipping brussels sprouts).
What's more, I don't think he did this just because he believed in the value of an ascetic, cloistered lifestyle. I think he wanted to discourage his parishioners from listening to anybody other than him. When you hold people to an impossible standard and make them torture themselves with guilt ("When you breathe, you're basically worshipping oxygen…"), it becomes easier to control them. It becomes easy to manipulate people's faith and use it as a tool to keep them in their place. It's a lot like an abusive relationship, but instead of using violence and emotional blackmail to keep your victims in check, you use threats of excommunication and hellfire.
I admit, I don't know much about what God is like. Nobody does. But I have a really hard time believing that He (or She) put us in a massive, varied world, full of wonderful things to see and experience and horror and tragedy to fight against and intended for us to ignore all of it. And, honestly, I don't think there's anything in the Bible (or, say, the Qua'ran or the Bhagavat Gita) to suggest that's the case. Sooner or later you're going to do something other than pray, if only so you'll have something to pray about. That's not a bad thing. You're only in the world for 80 years or so, you might as well go out and have a look at it.
When a religious leader tries to guilt his congregation out of experiencing the world, it's usually because he wants to have a monopoly on their minds. Anything you do is something you're doing instead of praying… unless it's something that he wants you to do. It's never actually about God, or faith, or prayer- it's about the powers that be asserting their hold over those they deem inferior. When you do that, you're basically asking your congregation to worship you. And that's time they really could spend praying to God.
An indifferent magical transit child of the Earth. Here is my DeviantArt account, and here is a book what I wrote.
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\section*{References\markboth
{REFERENCES}{REFERENCES}}\list
{[\arabic{enumi}]}{\settowidth\labelwidth{[#1]}\leftmargin\labelwidth
\advance\leftmargin\labelsep
\usecounter{enumi}}
\def\newblock{\hskip .11em plus .33em minus .07em}
\sloppy
\sfcode`\.=1000\relax}
\let\endthebibliography=\endlist
\begin{document}
\vspace*{0ex}
\begin{center}
{\Large\bf
Solvability of the initial value problem \\[0.5ex]
to the Isobe--Kakinuma model for water waves
}
\end{center}
\begin{center}
Ryo Nemoto and Tatsuo Iguchi
\end{center}
\begin{abstract}
We consider the initial value problem to the Isobe--Kakinuma model for water waves and the
structure of the model.
The Isobe--Kakinuma model is the Euler--Lagrange equations for an approximate Lagrangian
which is derived from Luke's Lagrangian for water waves by approximating the velocity
potential in the Lagrangian.
The Isobe--Kakinuma model is a system of second order partial differential equations and
is classified into a system of nonlinear dispersive equations.
Since the hypersurface $t=0$ is characteristic for the Isobe--Kakinuma model,
the initial data have to be restricted in an infinite dimensional manifold for
the existence of the solution.
Under this necessary condition and a sign condition, which corresponds to
a generalized Rayleigh--Taylor sign condition for water waves, on the initial data,
we show that the initial value problem is solvable locally in time in Sobolev spaces.
We also discuss the linear dispersion relation to the model.
\end{abstract}
\section{Introduction}
\label{section:intro}
In this paper we are concerned with the solvability locally in time of the initial value problem
to the Isobe--Kakinuma model for water waves.
The water wave problem is mathematically formulated as a free boundary problem for
an irrotational flow of an inviscid and incompressible fluid under the gravitational field.
We consider the water filled in $(n+1)$-dimensional Euclidean space.
Let $t$ be the time, $x=(x_1,\ldots,x_n)$ the horizontal spatial coordinates,
and $z$ the vertical spatial coordinate.
We assume that the water surface and the bottom are represented as $z=\eta(x,t)$ and $z=-h+b(x)$,
respectively, where $\eta=\eta(x,t)$ is the surface elevation, $h$ is the mean depth, and $b=b(x)$
represents the bottom topography.
J. C. Luke \cite{Luke1967} showed that the water wave problem has a variational structure,
that is, he gave a Lagrangian in terms of the velocity potential $\Phi=\Phi(x,z,t)$ and the surface
elevation $\eta$ in the form
\begin{equation}\label{intro:Luke's Lagrangian}
\mathscr{L}_{\rm Luke}(\Phi,\eta) = \int_{-h+b(x)}^{\eta(x,t)}\biggl(\partial_t\Phi(x,z,t)
+\frac12|\nabla_X\Phi(x,z,t)|^2+gz\biggr){\rm d}z
\end{equation}
and the action function
\[
\mathscr{J}(\Phi,\eta)
= \int_{t_0}^{t_1}\!\!\!\int_{\Omega}\mathscr{L}_{\rm Luke}(\Phi,\eta){\rm d}x{\rm d}t,
\]
where $\nabla_X=(\nabla,\partial_z)=(\partial_{x_1},\ldots,\partial_{x_n},\partial_z)$,
$g$ is the gravitational constant, and $\Omega$ is an appropriate region in $\mathbf{R}^n$.
He showed that the corresponding Euler--Lagrange equation is exactly the basic
equations for water waves.
M. Isobe \cite{Isobe1994, Isobe1994-2} and T. Kakinuma \cite{Kakinuma2000, Kakinuma2001, Kakinuma2003}
approximated the velocity potential $\Phi$ in Luke's Lagrangian by
\[
\Phi^{\mbox{\rm\tiny app}}(x,z,t) = \sum_{i=0}^N\Psi_i(z;b)\phi_i(x,t),
\]
where $\{\Psi_i\}$ is an appropriate function system in the vertical coordinate $z$ and may depend on
the bottom topography $b$ and $(\phi_0,\phi_1,\ldots,\phi_N)$ are unknown variables,
and derived an approximate Lagrangian
$\mathscr{L}^{\mbox{\rm\tiny app}}(\phi_0,\phi_1,\ldots,\phi_N,\eta)
=\mathscr{L}_{\rm Luke}(\Phi^{\mbox{\rm\tiny app}},\eta)$.
The Isobe--Kakinuma model is the corresponding Euler--Lagrange equation for the
approximated Lagrangian.
Different choices of the function system $\{\Psi_i\}$ give different Isobe--Kakinuma models.
In this paper we adopt the approximation
\begin{equation}\label{intro:approximation}
\Phi^{\mbox{\rm\tiny app}}(x,z,t) = \sum_{i=0}^N(z+h-b(x))^{p_i}\phi_i(x,t),
\end{equation}
where $p_0,p_1,\ldots,p_N$ are nonnegative integers satisfying $0=p_0<p_1<\cdots<p_N$.
As we will see later, a natural choice of these exponents is given by
$p_i=2i$ in the case of the flat bottom, that is the case $b(x)\equiv0$,
and $p_i=i$ in the case of variable bottom topographies.
In order to treat the both cases at the same time, we consider such a general case.
Plugging this into Luke's Lagrangian \eqref{intro:Luke's Lagrangian} we obtain an
approximate Lagrangian
\begin{align*}
& \mathscr{L}^{\mbox{\rm\tiny app}}(\phi_0,\phi_1,\ldots,\phi_N,\eta) \\
&= \sum_{i=0}^N\frac{1}{p_i+1}H^{p_i+1}\partial_t\phi_i \\
&\quad\;
+\frac12\sum_{i,j=0}^N\biggl(
\frac{1}{p_i+p_j+1}H^{p_i+p_j+1}\nabla\phi_i\cdot\nabla\phi_j
-\frac{2p_i}{p_i+p_j}H^{p_i+p_j}\phi_i\nabla b\cdot\nabla\phi_j \\
&\makebox[6em]{}
+\frac{p_ip_j}{p_i+p_j-1}H^{p_i+p_j-1}(1+|\nabla b|^2)\phi_i\phi_j\biggr) \\
&\quad\;
+\frac12g(\eta^2-(-h+b)^2),
\end{align*}
where $H=H(x,t)$ is the depth of the water and is given by $H(x,t)=h+\eta(x,t)-b(x)$.
Here and in what follows we use the notational convention $0/0=0$.
Then, the corresponding Euler--Lagrange equation has the form
\begin{equation}\label{intro:IK model}
\left\{
\begin{array}{l}
\displaystyle
H^{p_i}\partial_t\eta+\sum_{j=0}^N\biggl\{\nabla\cdot\biggl(
\frac{1}{p_i+p_j+1}H^{p_i+p_j+1}\nabla\phi_j
-\frac{p_j}{p_i+p_j}H^{p_i+p_j}\phi_j\nabla b\biggr) \\
\displaystyle\phantom{ H^{p_i}\partial_t\eta+\sum_{j=0}^N\biggl\{ }
+\frac{p_i}{p_i+p_j}H^{p_i+p_j}\nabla b\cdot\nabla\phi_j
-\frac{p_ip_j}{p_i+p_j-1}H^{p_i+p_j-1}(1+|\nabla b|^2)\phi_j\biggr\}=0 \\
\makebox[27em]{}\mbox{for}\quad i=0,1,\ldots,N, \\
\displaystyle
\sum_{i=0}^NH^{p_i}\partial_t\phi_i+g\eta
+\frac12\sum_{i,j=0}^N\Bigl(H^{p_i+p_j}\nabla\phi_i\cdot\nabla\phi_j
-2p_iH^{p_i+p_j-1}\phi_i\nabla b\cdot\nabla\phi_j \\
\displaystyle\makebox[12em]{}
+p_ip_jH^{p_i+p_j-2}(1+|\nabla b|^2)\phi_i\phi_j\Bigr)=0.
\end{array}
\right.
\end{equation}
This is the Isobe--Kakinuma model that we are going to consider in this paper.
We consider the initial value problem to this Isobe--Kakinuma model \eqref{intro:IK model}
under the initial conditions
\begin{equation}\label{intro:initial conditions}
(\eta,\phi_0,\ldots,\phi_N)=(\eta_{(0)},\phi_{0(0)},\ldots,\phi_{N(0)}) \quad\makebox[3em]{at} t=0.
\end{equation}
Unique solvability locally in time of the initial value problem
\eqref{intro:IK model}--\eqref{intro:initial conditions} in the case $N=1$ and $p_1=2$
and fundamental properties of the model are presented in
Y. Murakami and T. Iguchi \cite{MurakamiIguchi2015}.
Therefore, this paper is a generalization of their results.
One of the interesting features of the model is its linear dispersion relation.
In the following section we will consider the linearized equations of the model around the rest state in the case
of the flat bottom and calculate the linear dispersion relation together with the phase speed
$c_{IK}(\xi)$ of the plane wave solution related to the wave vector $\xi\in\mathbf{R}^n$.
See \eqref{dispersion:phase speed}.
It is well known that the phase speed $c_{WW}(\xi)$ of the plane wave solution to the linearized equations
for water waves is given by
\[
c_{WW}(\xi)=\pm\sqrt{gh\frac{\tanh(h|\xi|)}{h|\xi|}}.
\]
If we choose $p_i=2i$, then we can show that
\begin{equation}\label{intro:Pade}
(c_{IK}(\xi))^2 = [2N/2N] \mbox{ Pad\'e approximant of } (c_{WW}(\xi))^2,
\end{equation}
which will be given in Theorem \ref{dispersion:theorem 1}.
Concerning Pad\'e approximants we refer to G. A. Baker and P. Graves-Morris \cite{BakerGraves-Morris1996}.
This relation \eqref{intro:Pade} implies that the Isobe--Kakinuma model gives a good approximation of the
basic equations for water waves in the shallow water regime $h|\xi|\ll1$.
In fact, T. Iguchi \cite{Iguchi2017} gave a mathematically rigorous justification of the Isobe--Kakinuma
model in the case of the flat bottom with the choice $N=1$ and $p_1=2$.
He showed that the Isobe--Kakinuma model gives a higher order shallow water approximation for water waves
with an error of order $O(\delta^6)$, where $\delta$ is a small nondimensional parameter defined as the
ratio of the mean depth $h$ to the typical wave length.
We note that the Green--Naghdi equations are known as a higher order shallow water approximation for water waves
with an error of order $\delta^4$.
Therefore, the Isobe--Kakinuma model gives a better approximation than the Green--Naghdi equations in the
shallow water regime.
Concerning the shallow water approximations and the rigorous justifications of the Green--Naghdi equations
we refer to T. Iguchi \cite{Iguchi2009, Iguchi2011}, B. Alvarez-Samaniego and D. Lannes
\cite{Alvarez-SamaniegoLannes2008}, Y. A. Li \cite{Li2006}, H. Fujiwara and T. Iguchi \cite{FujiwaraIguchi2015},
and D. Lannes \cite{Lannes2013-2}.
The relation \eqref{intro:Pade} anticipates that the Isobe--Kakinuma model \eqref{intro:IK model} in the
case of the flat bottom with the choice $p_i=2i$ would give an approximation with an error of order
$O(\delta^{4N+2})$. We postpone this subject in the future research.
If we choose $p_i=i$, we do not have such a beautiful relation as \eqref{intro:Pade} any more.
However, this choice of $p_i$ would be important in the case of the variable bottom topographies and
we still have a good approximation, which will be stated in Theorem \ref{dispersion:theorem 2}.
The Isobe--Kakinuma model \eqref{intro:IK model} is written in the matrix form as
\[
\left(
\begin{array}{cccc}
H^{p_0} & 0 & \cdots & 0 \\
\vdots & \vdots & & \vdots \\
H^{p_N} & 0 & \cdots & 0 \\
0 & H^{p_0} & \cdots & H^{p_N}
\end{array}
\right)
\partial_t
\left(
\begin{array}{c}
\eta \\
\phi_0 \\
\vdots \\
\phi_N
\end{array}
\right)
+\{\mbox{spatial derivatives}\}=\mbox{\boldmath$0$}.
\]
Since the coefficient matrix always has the zero eigenvalue, the hypersurface $t=0$ in the space-time
$\mathbf{R}^n\times\mathbf{R}$ is characteristic for the Isobe--Kakinuma model \eqref{intro:IK model},
so that the initial value problem \eqref{intro:IK model}--\eqref{intro:initial conditions} is not solvable in general.
In fact, if the problem has a solution $(\eta,\phi_0,\ldots,\phi_N)$, then by eliminating the time derivative
$\partial_t\eta$ from the equations we see that the solution has to satisfy the relation
\begin{align}\label{intro:compatibility}
& H^{p_i}\sum_{j=0}^N\nabla\cdot\biggl(
\frac{1}{p_j+1}H^{p_j+1}\nabla\phi_j
-\frac{p_j}{p_j}H^{p_j}\phi_j\nabla b\biggr) \nonumber \\
&= \sum_{j=0}^N\biggl\{\nabla\cdot\biggl(
\frac{1}{p_i+p_j+1}H^{p_i+p_j+1}\nabla\phi_j
-\frac{p_j}{p_i+p_j}H^{p_i+p_j}\phi_j\nabla b\biggr) \\
&\phantom{ =\sum_{j=0}^N\biggl\{ }
\displaystyle
+\frac{p_i}{p_i+p_j}H^{p_i+p_j}\nabla b\cdot\nabla\phi_j
-\frac{p_ip_j}{p_i+p_j-1}H^{p_i+p_j-1}(1+|\nabla b|^2)\phi_j\biggr\} \nonumber
\end{align}
for $i=1,\ldots,N$.
Therefore, as a necessary condition the initial date $(\eta_{(0)},\phi_{0(0)},\ldots,\phi_{N(0)})$ and
the bottom topography $b$ have to satisfy the relation \eqref{intro:compatibility} for the existence of
the solution.
As we will see in Proposition \ref{intro:prop},
such initial data are constructed from the three scalar functions $b$, $\eta_{(0)}$,
and $\phi_{(0)}$, where $\phi_{(0)}$ is the initial data for the trace of the velocity potential on the water surface.
The water wave problem has a conserved energy
\[
E_{WW}(t) = \frac{\rho}{2}\int_{\mathbf{R}^n}\biggl\{\int_{-h+b(x)}^{\eta(x,t)}\bigl|
\nabla_X\Phi(x,z,t)\bigr|^2{\rm d}z
+g\bigl(\eta(x,t)\bigr)^2\biggr\}{\rm d}x,
\]
where $\rho$ is the constant density of the water.
The first term in the right-hand side is the kinetic energy and the second one is the potential energy
due to the gravity.
The Isobe--Kakinuma model \eqref{intro:IK model} has also a conserved energy
\begin{equation}\label{intro:energy}
E(t) = \frac{\rho}{2}\int_{\mathbf{R}^n}\biggl\{\int_{-h+b(x)}^{\eta(x,t)}\bigl|
\nabla_X\Phi^{\mbox{\rm\tiny app}}(x,z,t)\bigr|^2{\rm d}z
+g\bigl(\eta(x,t)\bigr)^2\biggr\}{\rm d}x,
\end{equation}
where $\Phi^{\mbox{\rm\tiny app}}$ is given by \eqref{intro:approximation}.
See \eqref{proof:energy} for more explicit expression of this energy function $E(t)$.
Under a physically reasonable condition on the water surface $\eta$ and the bottom topography $b$
we have an equivalence
\[
E(t) \simeq \int_{\mathbf{R}^n}\biggl\{
|\nabla\phi_0(x,t)|^2+\sum_{i=1}^N\Bigl(|\nabla\phi_i(x,t)|^2+\bigl(\phi_i(x,t)\bigr)^2\Bigr)
+\bigl(\eta(x,t)\bigr)^2 \biggr\}{\rm d}x.
\]
Therefore, it is natural to work in the function space $\eta,\nabla\phi_0\in C([0,T];H^m)$ and
$\phi_1,\ldots,\phi_N\in C([0,T];H^{m+1})$, where $H^m=W^{m,2}(\mathbf{R}^n)$ is the standard $L^2$
Sobolev space of order $m$ on $\mathbf{R}^n$.
It is well known that the well-posedness of the initial value problem to the water wave problem may be
broken unless a generalized Rayleigh--Taylor sign condition $-\frac{\partial P}{\partial N} \geq c_0>0$
on the water surface is satisfied, where $P$ is the pressure and $N$ is the unit outward normal
on the water surface.
For example, we refer to S. Wu \cite{Wu1997, Wu1999} and D. Lannes \cite{Lannes2005}.
This sign condition is equivalent to $-\partial_z P\geq c_0>0$ because the pressure $P$ is equal to
the constant atmospheric pressure $P_0$ on the water surface.
By using Bernoulli's law
\[
\partial_t\Phi+\frac12|\nabla_X\Phi|^2+\frac{1}{\rho}(P-P_0)+gz\equiv0,
\]
the sign condition can be written in term of our unknowns $(\eta,\phi_0,\ldots,\phi_N)$ and $b$ as
$a(x,t)\geq c_0>0$, where
\begin{align}\label{intro:a}
a &= g+\sum_{i=0}^Np_iH^{p_i-1}\partial_t\phi_i \\
&\quad\;
+\frac12\sum_{i,j=0}^N\Bigl\{
(p_i+p_j)H^{p_i+p_j-1}\nabla\phi_i\cdot\nabla\phi_j
-2p_i(p_i+p_j-1)H^{p_i+p_j-2}\phi_i\nabla b\cdot\nabla\phi_j \nonumber \\
&\makebox[6em]{}
+p_ip_j(p_i+p_j-2)H^{p_i+p_j-3}(1+|\nabla b|^2)\phi_i\phi_j\Bigr\}. \nonumber
\end{align}
In fact, we have $-\frac{1}{\rho}\partial_z P^{\mbox{\rm\tiny app}}=g+\partial_z\partial_t\Phi^{\mbox{\rm\tiny app}}+
\nabla_X\partial_z\Phi^{\mbox{\rm\tiny app}}\cdot\nabla_X\Phi^{\mbox{\rm\tiny app}}=a$ on $z=\eta(x,t)$.
In this paper, we assume that this sign condition is satisfied at the initial time $t=0$.
Now, we are ready to give our main result in this paper.
\begin{theorem}\label{intro:theorem}
Let $g, h, c_0, M_0$ be positive constants and $m$ an integer such that $m>n/2+1$.
There exists a time $T>0$ such that if the initial data $(\eta_{(0)},\phi_{0(0)},\ldots,\phi_{N(0)})$ and
$b$ satisfy the relation \eqref{intro:compatibility} and
\begin{equation}\label{intro:conditions}
\left\{
\begin{array}{l}
\|\eta_{(0)}\|_m+\|\nabla\phi_{0(0)}\|_m+\|(\phi_{1(0)},\ldots,\phi_{N(0)})\|_{m+1}
+ \|b\|_{W^{m+2,\infty}}\leq M_0, \\[0.5ex]
h+\eta_{(0)}(x)-b(x)\geq c_0, \quad a(x,0)\geq c_0
\qquad\mbox{for}\quad x\in\mathbf{R}^n,
\end{array}
\right.
\end{equation}
then the initial value problem \eqref{intro:IK model}--\eqref{intro:initial conditions}
has a unique solution $(\eta,\phi_0,\ldots,\phi_N)$ satisfying
$$
\eta,\nabla\phi_0\in C([0,T];H^m), \quad
\phi_1,\ldots,\phi_N\in C([0,T];H^{m+1}).
$$
Moreover, the energy function $E(t)$ defined by \eqref{intro:energy} is a conserved quantity.
\end{theorem}
\begin{remark}\label{intro:remark}
{\rm
(1) \
If we impose an additional condition $\phi_{(0)}\in L^2(\mathbf{R}^n)$, then the solution satisfies
an additional integrability $\phi_0\in C([0,T];H^{m+1})$.
(2) \
In the sign condition $a(x,0)\geq c_0>0$ we use the quantities
$\partial_t\phi_1(x,0),\ldots,\partial_t\phi_N(x,0)$ which should be written in terms of the initial data.
Although the hypersurface $t=0$ is characteristic for the Isobe--Kakinuma model \eqref{intro:IK model},
we can express $\partial_t\phi_1(x,0),\ldots,\partial_t\phi_N(x,0)$ in terms of the initial data and $b$.
For details, we refer to Remark \ref{construction:remark}.
(3) \
Let $\phi$ be the trace of the velocity potential $\Phi$ on the water surface.
In view of our approximation \eqref{intro:approximation}, $\phi$ should be related to our variables
approximately by the formula
\begin{equation}\label{intro:relation}
\phi=\sum_{i=0}^NH^{p_i}\phi_i.
\end{equation}
Given the initial data $\eta_{(0)}$ and $\phi_{(0)} (=\phi|_{t=0})$ and the bottom topography $b$,
the necessary condition \eqref{intro:compatibility} and the relation \eqref{intro:relation} determine
uniquely the initial data $\phi_{0(0)},\ldots,\phi_{N(0)}$.
In fact, we have the following proposition.
}
\end{remark}
\begin{proposition}\label{intro:prop}
Let $h, c_0, M_0$ be positive constants and $m$ an integer such that $m>n/2+1$.
There exists a positive $C>0$ such that if the initial data $(\eta_{(0)},\phi_{(0)})$ and $b$ satisfy
\[
\left\{
\begin{array}{l}
\|\eta_{(0)}\|_m+\|b\|_{W^{m,\infty}}\leq M_0, \quad \|\nabla\phi_{(0)}\|_{m-1}<\infty, \\[0.5ex]
h+\eta_{(0)}(x)-b(x)\geq c_0 \qquad\mbox{for}\quad x\in\mathbf{R}^n,
\end{array}
\right.
\]
then the necessary condition \eqref{intro:compatibility} and the relation \eqref{intro:relation} determine
uniquely the initial data $\phi_{0(0)},\ldots,\phi_{N(0)}$, which satisfy
\[
\|\nabla\phi_{0(0)}\|_{m-1}+\|(\phi_{1(0)},\ldots,\phi_{N(0)})\|_m \leq C\|\nabla\phi_{(0)}\|_{m-1}.
\]
\end{proposition}
\medskip
The contents of this paper are as follows.
In Section \ref{section:dispersion} we consider the linearized equations of the Isobe--Kakinuma model
around the rest state in the case of the flat bottom and analyze the linear dispersion relation.
Especially, we show the beautiful relation \eqref{intro:Pade}.
In Section \ref{section:linear} we consider the linearized equations of the Isobe--Kakinuma model
around an arbitrary flow, reveal a hidden symmetric structure of the model, and give an idea to obtain
an energy estimate for the solution of the nonlinear equations.
In Section \ref{section:construction} we transform the Isobe--Kakinuma model to a system of equations
for which the hypersurface $t=0$ is noncharacteristic by using the necessary condition \eqref{intro:compatibility}
and construct the solution of the initial value problem to the transformed system by using a standard
parabolic regularization.
In Section \ref{section:proof} we show that the solution constructed in Section
\ref{section:construction} is the solution of the Isobe--Kakinuma model \eqref{intro:IK model}
if the initial data satisfy the necessary condition \eqref{intro:compatibility}.
\bigskip
\noindent
{\bf Notation}. \
We denote by $W^{m,p}(\mathbf{R}^n)$ the $L^p$ Sobolev space of order $m$ on $\mathbf{R}^n$.
The norms of the Lebesgue space $L^p(\mathbf{R}^n)$ and the Sobolev space $H^m=W^{m,2}(\mathbf{R}^n)$
are denoted by $|\cdot|_p$ and $\|\cdot\|_m$, respectively.
The $L^2$-norm and the $L^2$-inner product are simply denoted by $\|\cdot\|$ and $(\cdot,\cdot)_{L^2}$,
respectively.
We put $\partial_t=\partial/\partial t$, $\partial_j=\partial/\partial x_j$,
and $\partial_z=\partial/\partial z$.
For a multi-index $\alpha=(\alpha_1,\ldots,\alpha_n)$ we put
$\partial^{\alpha}=\partial_1^{\alpha_1}\cdots\partial_n^{\alpha_n}$.
$[P,Q]=PQ-QP$ denotes the commutator.
For a matrix $A$ we denote by $A^{\rm T}$ the transpose of $A$.
For a vector $\mbox{\boldmath$\phi$}=(\phi_0,\phi_1,\ldots,\phi_N)^{\rm T}$ we denote the last $N$
components by $\mbox{\boldmath$\phi$}'=(\phi_1,\ldots,\phi_N)^{\rm T}$.
We use the notational convention $0/0=0$.
\section{Linear dispersion relation}
\label{section:dispersion}
\setcounter{equation}{0}
\setcounter{theorem}{0}
In this section we consider the linearized equations of the Isobe--Kakinuma model \eqref{intro:IK model}
in the case of the flat bottom.
The linearized equations have the form
\begin{equation}\label{dispersion:IK model}
\left\{
\begin{array}{l}
\displaystyle
\partial_t\eta+\sum_{j=0}^N\biggl(
\frac{1}{p_i+p_j+1}h^{p_j+1}\Delta\phi_j
-\frac{p_ip_j}{p_i+p_j-1}h^{p_j-1}\phi_j\biggr)=0 \\
\makebox[18em]{}\mbox{for}\quad i=0,1,\ldots,N, \\
\displaystyle
\sum_{i=0}^Nh^{p_i}\partial_t\phi_i+g\eta=0.
\end{array}
\right.
\end{equation}
Putting $\mbox{\boldmath$\psi$}=(h^{p_0}\phi_0,\ldots,h^{p_N}\phi_N)^{\rm T}$ we can rewrite the above
equations as the following simple matrix form
\[
\left(
\begin{array}{cc}
0 & h\mbox{\boldmath$1$}^{\rm T} \\
-h\mbox{\boldmath$1$} & O
\end{array}
\right)
\partial_t
\left(
\begin{array}{c}
\eta \\
\mbox{\boldmath$\psi$}
\end{array}
\right)
+
\left(
\begin{array}{cc}
gh & \mbox{\boldmath$0$}^{\rm T} \\
\mbox{\boldmath$0$} & A(hD)
\end{array}
\right)
\left(
\begin{array}{c}
\eta \\
\mbox{\boldmath$\psi$}
\end{array}
\right)
= \mbox{\boldmath$0$},
\]
where $\mbox{\boldmath$1$}=(1,\ldots,1)^{\rm T}$ and $A(hD)=-A_0h^2\Delta+A_1$.
The $(N+1)\times(N+1)$ matrices $A_0$ and $A_1$ are given by
\[
A_0=\biggl(\frac{1}{p_i+p_j+1}\biggr)_{0\leq i,j\leq N}, \qquad
A_1=\biggl(\frac{p_ip_j}{p_i+p_j-1}\biggr)_{0\leq i,j\leq N},
\]
where we used a rather nonstandard notation for matrices.
Since it might not cause any confusion, we will continue to use this notation in the following.
Therefore, the linear dispersion relation is given by
\[
\det
\left(
\begin{array}{cc}
gh & \sqrt{-1}h\omega\mbox{\boldmath$1$}^{\rm T} \\
-\sqrt{-1}h\omega\mbox{\boldmath$1$} & A(h\xi)
\end{array}
\right)=0,
\]
where $\xi\in\mathbf{R}^n$ is the wave vector, $\omega\in\mathbf{C}$ is the angular frequency,
and $A(h\xi)=(h|\xi|)^2A_0+A_1$.
We can rewrite the above dispersion relation as
\begin{equation}\label{dispersion:dispersion}
h^2\omega^2\det\tilde{A}(h\xi)-gh\det A(h\xi)=0.
\end{equation}
Throughout this section we use the notation
\[
\tilde{A}=
\left(
\begin{array}{cc}
0 & \mbox{\boldmath$1$}^{\rm T} \\
-\mbox{\boldmath$1$} & A
\end{array}
\right)
\]
for a matrix $A$.
Concerning the determinants in the dispersion relation \eqref{dispersion:dispersion} we have the
following proposition.
\begin{proposition}\label{dispersion:prop 1}
\begin{enumerate}
\setlength{\itemsep}{-3pt}
\item
For any $\xi\in\mathbf{R}^n\setminus\{\mbox{\boldmath$0$}\}$, the symmetric matrix $A(h\xi)$ is positive.
\item
There exists $c_0>0$ such that for any $\xi\in\mathbf{R}^n$ we have $\det\tilde{A}(h\xi)\geq c_0$.
\item
$(h|\xi|)^{-2}\det A(h\xi)$ is a polynomial in $(h|\xi|)^2$ of degree $N$ and the coefficient of $(h|\xi|)^{2N}$
is $\det A_0$.
\item
$\det\tilde{A}(h\xi)$ is a polynomial in $(h|\xi|)^2$ of degree $N$ and the coefficient of $(h|\xi|)^{2N}$
is $\det\tilde{A}_0$.
\end{enumerate}
\end{proposition}
\noindent
{\bf Proof}. \
For any $\mbox{\boldmath$\psi$}=(\psi_0,\ldots,\psi_N)^{\rm T}\in\mathbf{R}^{N+1}$ we see that
\begin{align}\label{dispersion:identity}
(A(h\xi)\mbox{\boldmath$\psi$})\cdot\mbox{\boldmath$\psi$}
&= \int_0^1\biggl\{(h|\xi|)^2\biggl(\sum_{i=0}^N\psi_iz^{p_i}\biggr)^2
+\biggl(\sum_{i=1}^Np_i\psi_iz^{p_i-1}\biggr)^2\biggr\}{\rm d}z \\
&\simeq (h|\xi|)^2|\mbox{\boldmath$\psi$}|^2+|\mbox{\boldmath$\psi$}'|^2, \nonumber
\end{align}
where $\mbox{\boldmath$\psi$}'=(\psi_1,\ldots,\psi_N)^{\rm T}$.
This shows a positivity of $A(h\xi)$ for $\xi\ne\mbox{\boldmath$0$}$.
Let $\xi\ne\mbox{\boldmath$0$}$ and $C(h\xi)=(C_{ij}(h\xi))_{0\leq i,j\leq N}$ be the cofactor matrix of $A(h\xi)$.
Since $A(h\xi)$ is symmetric, $C(h\xi)$ is also symmetric and we have $C(h\xi)=(\det A(h\xi))A(h\xi)^{-1}$.
Then, expanding the first row of the matrix $\tilde{A}(h\xi)$ we see that
\[
\det\tilde{A}(h\xi)=\sum_{i,j=0}^NC_{ij}(h\xi)
=(C(h\xi)\mbox{\boldmath$1$})\cdot\mbox{\boldmath$1$}
=(\det A(h\xi))(A(h\xi)^{-1}\mbox{\boldmath$1$})\cdot\mbox{\boldmath$1$},
\]
which implies the positivity of $\det\tilde{A}(h\xi)$ due to the positivity of $A(h\xi)$.
We proceed to consider the case $\xi=\mbox{\boldmath$0$}$.
It holds that $\tilde{A}(\mbox{\boldmath$0$})=\tilde{A}_1$ and
\[
A_1=
\left(
\begin{array}{cc}
0 & \mbox{\boldmath$0$}^{\rm T} \\
\mbox{\boldmath$0$} & A_1'
\end{array}
\right), \quad
A_1'=\biggl(\frac{p_ip_j}{p_i+p_j-1}\biggr)_{1\leq i,j\leq N}.
\]
By the similar calculation as \eqref{dispersion:identity} we see the positivity of the symmetric matrix $A_1'$.
Moreover, we have $\det\tilde{A}(\mbox{\boldmath$0$})=\det A_1'>0$.
Therefore, we obtain the strict positivity of $\det\tilde{A}(h\xi)$.
It is easy to see that $\det A(h\xi)$ is a polynomial in $(h|\xi|)^2$ of degree less than or equal to $N+1$
and that the coefficient of $(h|\xi|)^{2(N+1)}$ is $\det A_0$.
By the similar calculation as \eqref{dispersion:identity} we see the positivity of the symmetric matrix $A_0$
so that the degree is in fact $N+1$.
Moreover, $A(\mbox{\boldmath$0$})$ has the zero eigenvalue with an eigenvector $(1,0,\ldots,0)^{\rm T}$,
so that $\det A(\mbox{\boldmath$0$})=0$.
Therefore, the term of degree $0$ in $\det A(h\xi)$ vanishes so that $(h|\xi|)^{-2}\det A(h\xi)$ is a
polynomial in $(h|\xi|)^2$ of degree $N$.
Similarly we see that $\det\tilde{A}(h\xi)$ is a polynomial in $(h|\xi|)^2$ of degree less than or equal to $N$
and that the coefficient of $(h|\xi|)^{2N}$ is $\det\tilde{A}_0$, which is also positive, so that the degree
is in fact $N$.
\quad$\Box$
\bigskip
Thanks of this proposition and the dispersion relation \eqref{dispersion:dispersion}, the linearized system
\eqref{dispersion:IK model} is classified into the dispersive system, so that the Isobe--Kakinuma model is
a nonlinear dispersive system of equations.
Therefore, we can define the phase speed $c_{IK}(\xi)$ of the plane wave solution to \eqref{dispersion:IK model}
related to the wave vector $\xi\in\mathbf{R}^n$ by
\begin{equation}\label{dispersion:phase speed}
c_{IK}(\xi) = \pm\sqrt{gh\frac{(h|\xi|)^{-2}\det A(h\xi)}{\det\tilde{A}(h\xi)}}.
\end{equation}
It follows from Proposition \ref{dispersion:prop 1} that
\[
\lim_{|\xi|\to+\infty}c_{IK}(\xi)=\pm\sqrt{gh\frac{\det A_0}{\det\tilde{A}_0}},
\]
which is not zero.
It is not consistent with the linear water waves:
$\lim_{|\xi|\to+\infty}c_{WW}(\xi)=0$.
This implies that the Isobe--Kakinuma model \eqref{intro:IK model} cannot give a good approximation
to the water waves in deep water.
However, as is shown by the following theorems the Isobe--Kakinuma model gives a very precise approximation
in the shallow water regime $h|\xi|\ll1$.
\begin{theorem}\label{dispersion:theorem 1}
If we choose $p_i=2i$ $(i=0,1,\ldots,N)$, then $(c_{IK}(\xi))^2$ becomes the $[2N/2N]$ Pad\'e approximant
of $(c_{WW}(\xi))^2$.
More precisely, there exists a positive constant $C$ depending only on $N$ such that for any
$\xi\in\mathbf{R}^n$ and any $h,g>0$ we have
\[
\biggl|\biggl(\frac{c_{WW}(\xi)}{\sqrt{gh}}\biggr)^2-\biggl(\frac{c_{IK}(\xi)}{\sqrt{gh}}\biggr)^2\biggr|
\leq C(h|\xi|)^{4N+2}.
\]
\end{theorem}
\noindent
{\bf Proof}. \
Without loss of generality it is sufficient to consider the case $h=1$ and to show that
\begin{equation}\label{dispersion:asymptotics}
|\xi|\tanh|\xi|=\frac{\det A(\xi)}{\det\tilde{A}(\xi)}+O(|\xi|^{4N+4})
\quad\mbox{as}\quad |\xi|\to0.
\end{equation}
Let $\hat{\Phi}(z)=\frac{1}{\cosh|\xi|}\cosh(|\xi|z)$.
In view of $\hat{\Phi}''(z)=|\xi|^2\hat{\Phi}(z)$, $\hat{\Phi}(1)=1$, $\hat{\Phi}'(1)=|\xi|\tanh|\xi|$,
and $\hat{\Phi}'(0)=0$, we have
\begin{equation}\label{dispersion:identity 1}
|\xi|\tanh|\xi|=\int_0^1\bigl\{|\xi|^2(\hat{\Phi}(z))^2+(\hat{\Phi}'(z))^2\bigr\}{\rm d}z.
\end{equation}
By the Taylor series expansion of $\cosh(|\xi|z)$ we have
\[
\hat{\Phi}(z)=\sum_{i=0}^{2N}\hat{\psi}_iz^{2i}+O(|\xi|^{4N+2}), \quad
\hat{\Phi}'(z)=\sum_{i=0}^{2N}2i\hat{\psi}_iz^{2i-1}+O(|\xi|^{4N+2})
\]
uniformly with respect to $z\in[0,1]$, where
\begin{equation}\label{dispersion:psi}
\hat{\psi}_i=\frac{|\xi|^{2i}}{(2i)!\cosh|\xi|}, \quad i=0,1,\ldots.
\end{equation}
Plugging these expansions into \eqref{dispersion:identity 1} we see that
\[
|\xi|\tanh|\xi|
= \sum_{i,j=0}^N\biggl(\frac{1}{2(i+j)+1}|\xi|^2+\frac{4ij}{2(i+j)-1}\biggr)\hat{\psi}_i\hat{\psi}_j
+ 2R + O(|\xi|^{4N+4}),
\]
where
\begin{align*}
R &= \sum_{i=0}^{N-1}\sum_{j=N+1}^{2N}\frac{|\xi|^2}{2(i+j)+1}\hat{\psi}_i\hat{\psi}_j
+ \sum_{i=0}^N\sum_{j=N+1}^{2N}\frac{4ij}{2(i+j)-1}\hat{\psi}_i\hat{\psi}_j \\
&= \sum_{i=0}^{N-1}\sum_{j=N+1}^{2N}\biggl(
\frac{|\xi|^2}{2(i+j)+1}\hat{\psi}_i\hat{\psi}_j
+\frac{4(i+1)j}{2(i+j)+1}\hat{\psi}_{i+1}\hat{\psi}_j\biggr) \\
&= \frac{|\xi|}{\cosh^2|\xi|}\sum_{i=0}^{N-1}\sum_{j=N+1}^{2N}
\frac{|\xi|^{2i+1}}{(2i+1)!}\frac{|\xi|^{2j}}{(2j)!} \\
&= \frac{|\xi|\tanh|\xi|}{\cosh|\xi|}\sum_{j=N+1}^{\infty}\frac{|\xi|^{2j}}{(2j)!}+O(|\xi|^{4N+4}).
\end{align*}
Here we used the explicit form \eqref{dispersion:psi} of $\hat{\psi}_i$.
Therefore, we obtain
\begin{align}\label{dispersion:identity 2}
& (|\xi|\tanh|\xi|)\biggl(1-\frac{2}{\cosh|\xi|}\sum_{j=N+1}^{\infty}\frac{|\xi|^{2j}}{(2j)!}\biggr) \\
&= \sum_{i,j=0}^N\biggl(\frac{1}{2(i+j)+1}|\xi|^2+\frac{4ij}{2(i+j)-1}\biggr)\hat{\psi}_i\hat{\psi}_j
+ O(|\xi|^{4N+4}). \nonumber
\end{align}
Up to now we did not use the specific choice of $p_i$.
From now on, we use the advantage of the choice $p_i=2i$.
Then, we can rewrite the above identity \eqref{dispersion:identity 2} as
\begin{equation}\label{dispersion:identity 3}
(|\xi|\tanh|\xi|)\biggl(1-\frac{2}{\cosh|\xi|}\sum_{j=N+1}^{\infty}\frac{|\xi|^{2j}}{(2j)!}\biggr) \\
= (A(\xi)\hat{\mbox{\boldmath$\psi$}})\cdot\hat{\mbox{\boldmath$\psi$}} + O(|\xi|^{4N+4}),
\end{equation}
where $\hat{\mbox{\boldmath$\psi$}}=(\hat{\psi}_0,\ldots,\hat{\psi}_N)^{\rm T}$.
On the other hand, we consider the following equations for $\hat{\eta}$ and
$\hat{\mbox{\boldmath$\phi$}}=(\hat{\phi}_0,\ldots,\hat{\phi}_N)^{\rm T}$.
\begin{equation}\label{dispersion:equation}
\left(
\begin{array}{cc}
0 & \mbox{\boldmath$1$}^{\rm T} \\
-\mbox{\boldmath$1$} & A(\xi)
\end{array}
\right)
\left(
\begin{array}{c}
\hat{\eta} \\
\hat{\mbox{\boldmath$\phi$}}
\end{array}
\right)
=
\left(
\begin{array}{c}
1 \\
\mbox{\boldmath$0$}
\end{array}
\right).
\end{equation}
Since the coefficient matrix $\tilde{A}(\xi)$ is nonsingular, we have a unique solution
$(\hat{\eta},\hat{\mbox{\boldmath$\phi$}})$ of these equations.
Moreover, we see that
\begin{equation}\label{dispersion:identity 4}
(A(\xi)\hat{\mbox{\boldmath$\phi$}})\cdot\hat{\mbox{\boldmath$\phi$}}
=\hat{\eta}=\frac{\det A(\xi)}{\det\tilde{A}(\xi)}.
\end{equation}
In view of this and \eqref{dispersion:identity 3}, it is sufficient to compare
$\hat{\mbox{\boldmath$\psi$}}$ with $\hat{\mbox{\boldmath$\phi$}}$.
Let $A(\xi)=(\mbox{\boldmath$a$}_0,\ldots,\mbox{\boldmath$a$}_N)$.
Then, by using the explicit form \eqref{dispersion:psi} of $\hat{\psi}_i$ we see that
\[
\mbox{\boldmath$a$}_i\cdot\hat{\mbox{\boldmath$\psi$}}
= \frac{1}{\cosh|\xi|}\biggl(\sum_{j=0}^{N-1}\frac{|\xi|^{2j+2}}{(2j+1)!}
+\frac{1}{2(i+N)+1}\frac{|\xi|^{2N+2}}{(2N)!}\biggr)
\]
for $i=0,1,\ldots,N$, and that
\[
\mbox{\boldmath$1$}\cdot\hat{\mbox{\boldmath$\psi$}}
=\frac{1}{\cosh|\xi|}\sum_{i=0}^N\frac{|\xi|^{2i}}{(2i)!}
=1+O(|\xi|^{2N+2}).
\]
Therefore, if we put $\hat{\zeta}=\frac{1}{\cosh|\xi|}\sum_{j=0}^{N-1}\frac{|\xi|^{2j+2}}{(2j+1)!}$,
then we have
\begin{equation}\label{dispersion:relation}
\left(
\begin{array}{cc}
0 & \mbox{\boldmath$1$}^{\rm T} \\
-\mbox{\boldmath$1$} & A(\xi)
\end{array}
\right)
\left(
\begin{array}{c}
\hat{\zeta} \\
\hat{\mbox{\boldmath$\psi$}}
\end{array}
\right)
=
\left(
\begin{array}{c}
1 \\
\mbox{\boldmath$0$}
\end{array}
\right)+O(|\xi|^{2N+2}).
\end{equation}
By Proposition \ref{dispersion:prop 1} the determinant of the coefficient matrix is strictly positive,
so that this together with \eqref{dispersion:equation} implies
$\hat{\mbox{\boldmath$\psi$}}=\hat{\mbox{\boldmath$\phi$}}+O(|\xi|^{2N+2})$.
Moreover, it holds that
\[
(A(\xi)\hat{\mbox{\boldmath$\phi$}})\cdot\hat{\mbox{\boldmath$\psi$}}
= \frac{\det A(\xi)}{\det\tilde{A}(\xi)}\frac{1}{\cosh|\xi|}\sum_{i=0}^N\frac{|\xi|^{2i}}{(2i)!},
\]
which together with \eqref{dispersion:identity 4} implies
\[
(A(\xi)\hat{\mbox{\boldmath$\phi$}})\cdot(\hat{\mbox{\boldmath$\phi$}}-\hat{\mbox{\boldmath$\psi$}})
= \frac{\det A(\xi)}{\det\tilde{A}(\xi)}\frac{1}{\cosh|\xi|}\sum_{i=N+1}^{\infty}\frac{|\xi|^{2i}}{(2i)!},
\]
so that
\begin{align}\label{dispersion:relation 2}
(A(\xi)\hat{\mbox{\boldmath$\psi$}})\cdot\hat{\mbox{\boldmath$\psi$}}
&= (A(\xi)\hat{\mbox{\boldmath$\phi$}})\cdot\hat{\mbox{\boldmath$\phi$}}
-2(A(\xi)\hat{\mbox{\boldmath$\phi$}})\cdot(\hat{\mbox{\boldmath$\phi$}}-\hat{\mbox{\boldmath$\psi$}})
+(A(\xi)(\hat{\mbox{\boldmath$\phi$}}-\hat{\mbox{\boldmath$\psi$}}))
\cdot(\hat{\mbox{\boldmath$\phi$}}-\hat{\mbox{\boldmath$\psi$}}) \\
&= \frac{\det A(\xi)}{\det\tilde{A}(\xi)}
\biggl(1-\frac{2}{\cosh|\xi|}\sum_{i=N+1}^{\infty}\frac{|\xi|^{2i}}{(2i)!}\biggr)
+O(|\xi|^{4N+4}). \nonumber
\end{align}
Plugging this into \eqref{dispersion:identity 3} we obtain the desired relation \eqref{dispersion:asymptotics}.
\quad$\Box$
\bigskip
As was shown by J. Boussinesq \cite{Boussinesq1872}, in the case of the flat bottom the velocity
potential $\Phi$ for the water wave problem can be expanded in a Taylor series with respect to
the vertical spatial variable $z$ around the bottom $z=-h$ as
$$
\Phi(x,z,t)=\sum_{i=0}^{\infty}\frac{(z+h)^{2i}}{(2i)!}(-\Delta)^i\phi_0(x,t),
$$
where $\phi_0$ is the trace of the velocity potential $\Phi$ on the bottom.
Therefore, it is natural to choose $p_i=2i$ in such a case.
However, in the case of variable bottom topographies the Taylor series with respect to $z$ around
the bottom $z=-h+b(x)$ contains terms of odd degree too, so that the choice $p_i=i$ would be important
to such cases.
If we choose $p_i=i$, we do not have such a beautiful result as Theorem \ref{dispersion:theorem 1} but
we still have the following theorem,
which asserts that the terms of odd degree in the approximation of the velocity potential $\Phi$
do not affect the precision of the linear dispersion relation in the shallow water regime $h|\xi|\ll1$.
\begin{theorem}\label{dispersion:theorem 2}
If we choose $p_i=i$ $(i=0,1,\ldots,N)$, then for any $\xi\in\mathbf{R}^n$ and any $h,g>0$ we have
\[
\biggl|\biggl(\frac{c_{WW}(\xi)}{\sqrt{gh}}\biggr)^2-\biggl(\frac{c_{IK}(\xi)}{\sqrt{gh}}\biggr)^2\biggr|
\leq C(h|\xi|)^{4[N/2]+2},
\]
where $C$ is a positive constant depending only on $N$ and $[N/2]$ is the integer part of $N/2$.
\end{theorem}
\noindent
{\bf Proof}.
We will give the proof in the case of even integer $N$.
As in the proof of the previous theorem, it is sufficient to show that
\begin{equation}\label{dispersion:asymptotics 2}
|\xi|\tanh|\xi|=\frac{\det A(\xi)}{\det\tilde{A}(\xi)}+O(|\xi|^{4[N/2]+4})
\quad\mbox{as}\quad |\xi|\to0.
\end{equation}
In place of \eqref{dispersion:psi}, we define $\hat{\psi}_i$ by
\[
\hat{\psi}_i=
\left\{
\begin{array}{ll}
\displaystyle
\frac{|\xi|^i}{i!\cosh|\xi|} & \mbox{if $i$ is even}, \\[0.5ex]
0 & \mbox{if $i$ is odd}
\end{array}
\right.
\]
and put $\hat{\mbox{\boldmath$\psi$}}=(\hat{\psi}_0,\ldots,\hat{\psi}_N)^{\rm T}$.
Then, we have
\[
(A(\xi)\hat{\mbox{\boldmath$\psi$}})\cdot\hat{\mbox{\boldmath$\psi$}}
=\sum_{i,j=0}^{[N/2]}\biggl(\frac{1}{2(i+j)+1}|\xi|^2+\frac{4ij}{2(i+j)-1}\biggr)\hat{\psi}_{2i}\hat{\psi}_{2j},
\]
so that we can rewrite the identity \eqref{dispersion:identity 2} as
\begin{equation}\label{dispersion:identity 5}
(|\xi|\tanh|\xi|)\biggl(1-\frac{2}{\cosh|\xi|}\sum_{j=[N/2]+1}^{\infty}\frac{|\xi|^{2j}}{(2j)!}\biggr) \\
= (A(\xi)\hat{\mbox{\boldmath$\psi$}})\cdot\hat{\mbox{\boldmath$\psi$}} + O(|\xi|^{4[N/2]+4}).
\end{equation}
Let $(\hat{\eta},\hat{\mbox{\boldmath$\phi$}})$ be the solution of \eqref{dispersion:equation} as before.
Then, we have the identity \eqref{dispersion:identity 4}.
If we put $\hat{\zeta}=\frac{1}{\cosh|\xi|}\sum_{j=0}^{[N/2]-1}\frac{|\xi|^{2j+2}}{(2j+1)!}$,
then in place of \eqref{dispersion:relation} we have
\[
\left(
\begin{array}{cc}
0 & \mbox{\boldmath$1$}^{\rm T} \\
-\mbox{\boldmath$1$} & A(\xi)
\end{array}
\right)
\left(
\begin{array}{c}
\hat{\zeta} \\
\hat{\mbox{\boldmath$\psi$}}
\end{array}
\right)
=
\left(
\begin{array}{c}
1 \\
\mbox{\boldmath$0$}
\end{array}
\right)+O(|\xi|^{2[N/2]+2}).
\]
This together with \eqref{dispersion:equation} implies
$\hat{\mbox{\boldmath$\psi$}}=\hat{\mbox{\boldmath$\phi$}}+O(|\xi|^{2[N/2]+2})$.
Therefore, in place of \eqref{dispersion:relation 2} we have
\[
(A(\xi)\hat{\mbox{\boldmath$\psi$}})\cdot\hat{\mbox{\boldmath$\psi$}}
= \frac{\det A(\xi)}{\det\tilde{A}(\xi)}
\biggl(1-\frac{2}{\cosh|\xi|}\sum_{i=[N/2]+1}^{\infty}\frac{|\xi|^{2i}}{(2i)!}\biggr)
+O(|\xi|^{4[N/2]+4}).
\]
Plugging this into \eqref{dispersion:identity 5} we obtain the desired relation \eqref{dispersion:asymptotics 2}.
The case of odd integer $N$ can be proved in the same way, so we omit it.
\quad$\Box$
\section{Analysis of a linearized system}
\label{section:linear}
\setcounter{equation}{0}
\setcounter{theorem}{0}
In this section we consider the linearized equations of the Isobe--Kakinuma model \eqref{intro:IK model}
around an arbitrary flow $(\eta,\phi_0,\ldots,\phi_N)$, which is assumed to be sufficiently smooth.
The hypersurface $t=0$ is still characteristic for the linearized equations.
We will transform the equations to a symmetric positive system of partial differential equations for which
the hypersurface $t=0$ is noncharacteristic and give an idea to derive a priori estimates for the solution
to the nonlinear equations.
We introduce second order differential operators $L_{ij}=L_{ij}(H,b)$ $(i,j=0,1,\ldots,N)$ depending on
the water depth $H$ and the bottom topography $b$ by
\begin{align}\label{linear:L}
L_{ij}\psi_j
&= -\nabla\cdot\biggl(
\frac{1}{p_i+p_j+1}H^{p_i+p_j+1}\nabla\psi_j
-\frac{p_j}{p_i+p_j}H^{p_i+p_j}\psi_j\nabla b\biggr) \\[0.5ex]
&\quad\,
-\frac{p_i}{p_i+p_j}H^{p_i+p_j}\nabla b\cdot\nabla\psi_j
+\frac{p_ip_j}{p_i+p_j-1}H^{p_i+p_j-1}(1+|\nabla b|^2)\psi_j. \nonumber
\end{align}
Then, we have $L_{ij}^*=L_{ji}$, where $L_{ij}^*$ is the adjoint operator of $L_{ij}$ in $L^2(\mathbf{R}^n)$.
In addition to the function $a$ defined by \eqref{intro:a} we introduce the functions $\mbox{\boldmath$u$}$
and $w$ by
\begin{equation}\label{linear:uw}
\mbox{\boldmath$u$}=\sum_{i=0}^N(H^{p_i}\nabla\phi_i-p_iH^{p_i-1}\phi_i\nabla b), \quad
w=\sum_{i=0}^Np_iH^{p_i-1}\phi_i.
\end{equation}
Since $\mbox{\boldmath$u$}=\nabla\Phi^{\mbox{\rm\tiny app}}|_{z=\eta}$ and
$w=\partial_z\Phi^{\mbox{\rm\tiny app}}|_{z=\eta}$, where $\Phi^{\mbox{\rm\tiny app}}$ is the approximate
velocity potential defined by \eqref{intro:approximation}, $\mbox{\boldmath$u$}$ and $w$ represent
approximately the horizontal and the vertical components of the velocity field on the water surface,
respectively.
Then, the Isobe--Kakinuma model \eqref{intro:IK model} can be written simply as
\[
\left\{
\begin{array}{l}
\displaystyle
H^{p_i}\partial_t\eta-\sum_{j=0}^NL_{ij}\phi_j=0 \qquad\mbox{for}\quad i=0,1,\ldots,N, \\
\displaystyle
\sum_{j=0}^NH^{p_j}\partial_t\phi_j+g\eta+\frac12(|\mbox{\boldmath$u$}|^2+w^2)=0.
\end{array}
\right.
\]
Now, let us linearize the above equations around $(\eta,\phi_0,\ldots,\phi_N)$.
We denote by $(\zeta,\psi_0,\ldots,\psi_N)$ the variation from $(\eta,\phi_0,\ldots,\phi_N)$.
After a tedious but straightforward calculation we obtain the linearized equations
\begin{equation}\label{linear:linearized equations}
\left\{
\begin{array}{l}
\displaystyle
H^{p_i}(\partial_t\zeta+\nabla\cdot(\mbox{\boldmath$u$}\zeta))
+p_iH^{p_i-1}(\partial_t\eta+\mbox{\boldmath$u$}\cdot\nabla\eta-w)\zeta
-\sum_{j=0}^NL_{ij}\psi_j=f_i \\
\makebox[20em]{}\mbox{for}\quad i=0,1,\ldots,N, \\
\displaystyle
\sum_{j=0}^N\Bigl\{H^{p_j}(\partial_t\psi_j+\mbox{\boldmath$u$}\cdot\nabla\psi_j)
-p_jH^{p_j-1}(\mbox{\boldmath$u$}\cdot\nabla b-w)\psi_j\Bigr\}+a\zeta=f_{N+1},
\end{array}
\right.
\end{equation}
where $f_0,\ldots,f_{N+1}$ are given functions.
The hypersurface $t=0$ is still characteristic for these linearized equations.
In fact, by eliminating the time derivative $\partial_t\zeta$ from these equations we have
\[
\sum_{j=0}^N(L_{ij}-H^{p_i}L_{0j})\psi_j
=p_iH^{p_i-1}(\partial_t\eta+\mbox{\boldmath$u$}\cdot\nabla\eta-w)\zeta+H^{p_i}f_0-f_i
\]
for $i=1,\ldots,N$.
Now, we differentiate this with respect to the time $t$ and use the first equation in
\eqref{linear:linearized equations} to eliminate the time derivative $\partial_t\zeta$.
Then, we obtain
\begin{align}\label{linear:reduced 1}
\sum_{j=0}^N(L_{ij}-H^{p_i}L_{0j})\partial_t\psi_j
&= p_iH^{p_i-1}(\partial_t\eta+\mbox{\boldmath$u$}\cdot\nabla\eta-w)
\biggl(\sum_{j=0}^NL_{0j}\psi_j-\nabla\cdot(\mbox{\boldmath$u$}\zeta)\biggr) \\
&\quad\;
-\sum_{j=0}^N[\partial_t,L_{ij}-H^{p_i}L_{0j}]\psi_j
+[\partial_t,p_iH^{p_i-1}(\partial_t\eta+\mbox{\boldmath$u$}\cdot\nabla\eta-w)]\zeta \nonumber \\
&\quad\;
+\partial_t(H^{p_i}f_0-f_i)+p_iH^{p_i-1}(\partial_t\eta+\mbox{\boldmath$u$}\cdot\nabla\eta-w)f_0.
\nonumber
\end{align}
On the other hand, it follows from the second equation in \eqref{linear:linearized equations} that
\begin{equation}\label{linear:reduced 2}
\sum_{j=0}^NH^{p_j}\partial_t\psi_j
=-\sum_{j=0}^N\Bigl\{H^{p_j}\mbox{\boldmath$u$}\cdot\nabla\psi_j
-p_jH^{p_j-1}(\mbox{\boldmath$u$}\cdot\nabla b-w)\psi_j\Bigr\}-a\zeta+f_{N+1}.
\end{equation}
Here we note that the right hand sides of \eqref{linear:reduced 1} and \eqref{linear:reduced 2} do not include any
time derivatives of $(\zeta,\psi_0,\ldots,\psi_N)$.
In view of these equations we introduce linear operators $\mathscr{L}_i=\mathscr{L}_i(H,b)$ $(i=0,1,\ldots,N)$
depending on the water depth $H$ and the bottom topography $b$ and acting on
$\mbox{\boldmath$\varphi$}=(\varphi_0,\ldots,\varphi_N)^{\rm T}$ by
\begin{equation}\label{linear:L2}
\mathscr{L}_0\mbox{\boldmath$\varphi$}=\sum_{j=0}^NH^{p_j}\varphi_j, \qquad
\mathscr{L}_i\mbox{\boldmath$\varphi$}=\sum_{j=0}^N(L_{ij}-H^{p_i}L_{0j})\varphi_j
\quad\mbox{for}\quad i=1,\ldots,N,
\end{equation}
and put $\mathscr{L}\mbox{\boldmath$\varphi$}=(\mathscr{L}_0\mbox{\boldmath$\varphi$},\ldots,
\mathscr{L}_N\mbox{\boldmath$\varphi$})^{\rm T}$.
For a given $\mbox{\boldmath$F$}=(F_0,\ldots,F_N)^{\rm T}$ we consider the equation
\begin{equation}\label{linear:elliptic 1}
\mathscr{L}\mbox{\boldmath$\varphi$}=\mbox{\boldmath$F$}.
\end{equation}
Once we show the solvability of $\mbox{\boldmath$\varphi$}$ of this equation, we could express the
time derivatives $\partial_t\psi_i$ $(i=0,1,\ldots,N)$ in terms of the spatial derivatives so that
we could avoid the difficulty arising from the fact that the hypersurface $t=0$ is characteristic for
the linearized equations \eqref{linear:linearized equations}.
Let $\mbox{\boldmath$\varphi$}$ be a solution of \eqref{linear:elliptic 1}.
It follows from the first component of \eqref{linear:elliptic 1} that
\begin{equation}\label{linear:reduction}
\varphi_0=F_0-\sum_{j=1}^NH^{p_j}\varphi_j.
\end{equation}
Plugging this into the other components of \eqref{linear:elliptic 1} we obtain
\begin{equation}\label{linear:elliptic 2}
P_i\mbox{\boldmath$\varphi$}'= F_i-(L_{i0}-H^{p_i}L_{00})F_0 \quad\mbox{for}\quad i=1,\ldots,N,
\end{equation}
where $\mbox{\boldmath$\varphi$}'=(\varphi_1,\ldots,\varphi_N)^{\rm T}$ and
$P_j=P_j(H,b)$ $(j=1,\ldots,N)$ are second order differential operators defined by
\begin{equation}\label{linear:P}
P_i\mbox{\boldmath$\varphi$}'=
\sum_{j=1}^N\Bigl\{(L_{ij}-H^{p_i}L_{0j})\varphi_j-(L_{i0}-H^{p_i}L_{00})(H^{p_j}\varphi_j)\Bigr\}.
\end{equation}
We further introduce the operator $P\mbox{\boldmath$\varphi$}'=(P_1\mbox{\boldmath$\varphi$}',\ldots,
P_N\mbox{\boldmath$\varphi$}')^{\rm T}$.
Since $L_{ij}^*=L_{ji}$, we see easily that $P$ is symmetric in $L^2(\mathbf{R}^n)$.
Moreover, $P$ is positive in $L^2(\mathbf{R}^n)$ as shown in the following lemma.
\begin{lemma}\label{linear:lemma 1}
Let $c_0, c_1$ be positive constants.
There exists a positive constant $C=C(c_0,c_1)>0$ depending only on $c_0$ and $c_1$ such that
if $H,\nabla b\in L^{\infty}(\mathbf{R}^n)$ satisfy $H(x)\geq c_0$ and $|\nabla b(x)|\leq c_1$,
then we have
\[
(P\mbox{\boldmath$\varphi$}',\mbox{\boldmath$\varphi$}')_{L^2}\geq C^{-1}\|\mbox{\boldmath$\varphi$}'\|_1^2.
\]
\end{lemma}
\noindent
{\bf Proof}. \
Introducing $\varphi_0=-\sum_{j=1}^NH^{p_j}\varphi_j$, we have
\begin{equation}\label{linear:equality 1}
(P\mbox{\boldmath$\varphi$}',\mbox{\boldmath$\varphi$}')_{L^2}
=\sum_{i,j=0}^N(L_{ij}\varphi_j,\varphi_i)_{L^2}.
\end{equation}
Although this equality can be derived by direct calculation, it is also derived by the following argument
which help us to understand the structure of the equations.
Put $P\mbox{\boldmath$\varphi$}'=(G_1,\ldots,G_N)^{\rm T}$.
Then, we have
\[
\sum_{j=0}^NH^{p_j}\varphi_j=0, \qquad
\sum_{j=0}^N(L_{ij}-H^{p_i}L_{0j})\varphi_j=G_i \quad\mbox{for}\quad i=1,\ldots,N.
\]
If we further introduce $\zeta=\sum_{j=0}^NL_{0j}\varphi_j$, then the above equations can be written as
\[
\left(
\begin{array}{cc}
0 & \mbox{\boldmath$l$}^{\rm T} \\
-\mbox{\boldmath$l$} & L
\end{array}
\right)
\left(
\begin{array}{c}
\zeta \\
\mbox{\boldmath$\varphi$}
\end{array}
\right)
=
\left(
\begin{array}{c}
0 \\
\mbox{\boldmath$G$}
\end{array}
\right),
\]
where $\mbox{\boldmath$\varphi$}=(\varphi_0,\ldots,\varphi_N)^{\rm T}$,
$\mbox{\boldmath$G$}=(0,G_1,\ldots,G_N)^{\rm T}$, $L=(L_{ij})_{0\leq i,j\leq N}$, and
\begin{equation}\label{linear:l}
\mbox{\boldmath$l$}=\mbox{\boldmath$l$}(H)=(H^{p_0},\ldots,H^{p_N})^{\rm T}.
\end{equation}
By taking the $L^2$-inner product of the above equation with $(\zeta,\mbox{\boldmath$\varphi$}^{\rm T})^{\rm T}$,
we obtain $(L\mbox{\boldmath$\varphi$},\mbox{\boldmath$\varphi$})_{L^2}
=(\mbox{\boldmath$G$},\mbox{\boldmath$\varphi$})_{L^2}
=(P\mbox{\boldmath$\varphi$}',\mbox{\boldmath$\varphi$}')_{L^2}$, which implies \eqref{linear:equality 1}.
By direct calculation we have
\begin{align}\label{linear:equivalence}
I &:= \int_{\mathbf{R}^n}\!{\rm d}x\!\int_0^{H}\biggl\{
\biggl|\sum_{i=0}^N(z^{p_i}\nabla\varphi_i-p_iz^{p_i-1}\varphi_i\nabla b)\biggr|^2
+\biggl(\sum_{i=0}^Np_iz^{p_i-1}\varphi_i\biggr)^2\biggr\}{\rm d}z \\
&= \sum_{i,j=0}^N(L_{ij}\varphi_i,\varphi_j)_{L^2}. \nonumber
\end{align}
In the case $p_i=2i$ $(i=0,1,\ldots,N)$, $\{z^{p_i},z^{p_i-1}\}_{0\leq i\leq N}$ are linearly independent
so that
\begin{align*}
I &\simeq \int_{\mathbf{R}^n}\!{\rm d}x\!\int_0^{H}\sum_{i=0}^N\Bigl\{
\bigl(z^{2p_i}|\nabla\varphi_i|^2+p_i^2z^{2p_i-2}|\nabla b|^2\varphi_i^2\bigr)
+p_i^2z^{2p_i-2}\varphi_i^2\Bigr\}{\rm d}z \\
&\simeq \int_{\mathbf{R}^n}\sum_{i=0}^N\Bigl\{
H^{2p_i+1}|\nabla\varphi_i|^2+p_i^2H^{2p_i-1}(1+|\nabla b|^2)\varphi_i^2\Bigr\}{\rm d}x,
\end{align*}
which together with \eqref{linear:equality 1} and \eqref{linear:equivalence} implies the desired estimate
in that case.
We remark that the constant $C$ can be taken independent of $c_1$.
In the case $p_i=i$ $(i=0,1,\ldots,N)$, we see that
\begin{align*}
I &= \int_{\mathbf{R}^n}\!{\rm d}x\!\int_0^{H}\biggl\{
\biggl|\sum_{i=0}^{N-1}z^{i}\bigl(\nabla\varphi_i-(i+1)\varphi_{i+1}\nabla b\bigr)
+z^N\nabla\varphi_N\biggr|^2
+\biggl(\sum_{i=1}^Niz^{i-1}\varphi_i\biggr)^2\biggr\}{\rm d}z \\
&\simeq \int_{\mathbf{R}^n}\biggl\{\sum_{i=0}^{N-1}H^{2i+1}|\nabla\varphi_i-(i+1)\varphi_{i+1}\nabla b|^2
+H^{N+1}|\nabla\varphi_N|^2+\sum_{i=1}^NH^{2i-1}\varphi_i^2\biggr\}{\rm d}z \\
&\gtrsim \sum_{i=0}^{N-1}\|\nabla\varphi_i-(i+1)\varphi_{i+1}\nabla b\|^2
+\|\nabla\varphi_N\|^2+\sum_{i=1}^N\|\varphi_i\|^2
\end{align*}
which together with \eqref{linear:equality 1} and \eqref{linear:equivalence} implies the desired estimate
in that case.
The other cases can be treated in the same way, so we omit it.
\quad$\Box$
\bigskip
By this lemma, the explicit expression \eqref{linear:P} of the operator $P$ (see also \eqref{linear:L}),
and the standard theory of elliptic partial differential equations, we can obtain the following lemma.
\begin{lemma}\label{linear:lemma 2}
Let $h,c_0,M$ be positive constants and $m$ an integer such that $m>n/2+1$.
There exists a positive constant $C=C(h,c_0,M)$ such that if $\eta$ and $b$ satisfy
\[
\left\{
\begin{array}{l}
\|\eta\|_m+\|b\|_{W^{m,\infty}} \leq M, \\[0.5ex]
c_0\leq H(x)=h+\eta(x)-b(x) \quad\mbox{for}\quad x\in\mathbf{R}^n,
\end{array}
\right.
\]
then for $1\leq k\leq m$ we have
\[
\|P^{-1}\mbox{\boldmath$G$}'\|_k \leq C\|\mbox{\boldmath$G$}'\|_{k-2}.
\]
\end{lemma}
We proceed to consider equation \eqref{linear:elliptic 1}.
Thanks of this lemma, we see that for a given $\mbox{\boldmath$F$}$ there exists a unique solution
$\mbox{\boldmath$\varphi$}'=(\varphi_1,\ldots,\varphi_N)^{\rm T}$ of \eqref{linear:elliptic 2}.
If we define $\varphi_0$ by \eqref{linear:reduction}, then
$\mbox{\boldmath$\varphi$}=(\varphi_0,\ldots,\varphi_N)^{\rm T}$ is a solution of \eqref{linear:elliptic 1}.
More precisely, we have the following lemma.
\begin{lemma}\label{linear:lemma 3}
Under the hypothesis of Lemma {\rm \ref{linear:lemma 2}}, for any
$\mbox{\boldmath$F$}=(F_0,\ldots,F_N)^{\rm T}$ satisfying $\nabla F_0\in H^{k-1}$ and
$(F_1,\ldots,F_N)\in H^{k-2}$ with $1\leq k\leq m$ there exists a unique solution
$\mbox{\boldmath$\varphi$}=(\varphi_0,\ldots,\varphi_N)^{\rm T}$ of \eqref{linear:elliptic 1} satisfying
\[
\|\nabla\varphi_0\|_{k-1}+\|(\varphi_1,\ldots,\varphi_N)\|_k
\leq C(\|\nabla F_0\|_{k-1}+\|(F_1,\ldots,F_N)\|_{k-2}),
\]
where $C=C(h,c_0,M)>0$.
If, in addition, $F_0\in L^2(\mathbf{R}^n)$, then we have
\[
\|\mbox{\boldmath$\varphi$}\|_k \leq C(\|F_0\|_k+\|(F_1,\ldots,F_N)\|_{k-2}).
\]
\end{lemma}
\medskip
Now, we have established the solvability of $\mbox{\boldmath$\varphi$}$ to equation \eqref{linear:elliptic 1}.
We go back to consider the linearized equations in \eqref{linear:linearized equations}.
In the following of this section we denote lower order terms and terms related with the given functions
$f_0,\ldots,f_{N+1}$ by the same symbol $LOT$, which may change from line to line.
We introduce a symmetric matrix $A(H)$ depending on the water depth $H$ by
\begin{equation}\label{linear:A}
A(H)=(a_{ij}(H))_{0\leq i,j\leq N}, \quad a_{ij}(H)=\frac{1}{p_i+p_j+1}H^{p_i+p_j+1}.
\end{equation}
We also use a matrix $\tilde{A}(H)$ defined by
\begin{equation}\label{linear:tildeA}
\tilde{A}(H)=
\left(
\begin{array}{cc}
0 & \mbox{\boldmath$l$}(H)^{\rm T} \\
-\mbox{\boldmath$l$}(H) & A(H)
\end{array}
\right),
\end{equation}
where $\mbox{\boldmath$l$}(H)$ is defined by \eqref{linear:l}.
Since $L_{ij}(H,b)=-a_{ij}(H)\Delta+LOT$, it follows from \eqref{linear:reduced 1} and \eqref{linear:reduced 2}
that
\[
\left\{
\begin{array}{l}
\displaystyle
\sum_{j=0}^N(a_{ij}(H)-H^{p_i}a_{0j}(H))\Delta\partial_t\psi_j=\nabla\cdot(LOT)+LOT
\quad\mbox{for}\quad i=1,\ldots,N, \\
\displaystyle
\sum_{j=0}^NH^{p_j}\Delta\partial_t\psi_j
+\sum_{j=0}^NH^{p_j}(\mbox{\boldmath$u$}\cdot\nabla)\Delta\psi_j+\Delta(a\zeta)=\nabla\cdot(LOT)+LOT.
\end{array}
\right.
\]
We can rewrite these equations in a matrix form as
\begin{align}\label{linear:transformation 1}
\tilde{A}(H)
\left(
\begin{array}{c}
\sum_{j=0}^Na_{0j}(H)\Delta\partial_t\psi_j \\
\Delta\partial_t\mbox{\boldmath$\psi$}
\end{array}
\right)
+ &
\left(
\begin{array}{c}
\sum_{j=0}^NH^{p_j}(\mbox{\boldmath$u$}\cdot\nabla)\Delta\psi_j+\Delta(a\zeta) \\
\mbox{\boldmath$0$}
\end{array}
\right) \\
&=
\nabla\cdot(LOT)+LOT, \nonumber
\end{align}
where $\mbox{\boldmath$\psi$}=(\psi_0,\ldots,\psi_N)^{\rm T}$.
Since $\det\tilde{A}(H)=H^{2\sum_{i=0}^Np_i+N}\det\tilde{A}_0$, where $\tilde{A}_0$ is the matrix defined
in Section \ref{section:dispersion}, the matrix $\tilde{A}(H)$ is nonsingular and its inverse matrix can
be written as
\begin{equation}\label{linear:inverse}
\tilde{A}(H)^{-1}=
\left(
\begin{array}{cc}
q(H) & \mbox{\boldmath$q$}(H)^{\rm T} \\
-\mbox{\boldmath$q$}(H) & Q(H)
\end{array}
\right)
\end{equation}
with a symmetric matrix $Q(H)$.
If it causes no confusion, we omit the dependence of $H$ in the notation.
Then, it holds that
\begin{equation}\label{linear:q}
\mbox{\boldmath$l$}\cdot\mbox{\boldmath$q$}=-1, \quad
A\mbox{\boldmath$q$}=-q\mbox{\boldmath$l$}, \quad
q=\frac{\det A}{\det\tilde{A}}=H\frac{\det A_0}{\det\tilde{A}_0}.
\end{equation}
Therefore, it follows from \eqref{linear:transformation 1} that
\begin{equation}\label{linear:transformation 2}
-A\Delta\partial_t\mbox{\boldmath$\psi$}
= q\mbox{\boldmath$l$}\bigl\{\mbox{\boldmath$l$}^{\rm T}(\mbox{\boldmath$u$}\cdot\nabla)\Delta\mbox{\boldmath$\psi$}
+\Delta(a\zeta)\bigr\}+\nabla\cdot(LOT)+LOT.
\end{equation}
On the other hand, it follows from the first equation in \eqref{linear:linearized equations} that
\[
\mbox{\boldmath$l$}(\partial_t\zeta+\mbox{\boldmath$u$}\cdot\nabla\zeta)+A\Delta\mbox{\boldmath$\psi$}=LOT.
\]
Taking the Euclidean inner product of this equation with $-a\mbox{\boldmath$q$}$ and using the relations in
\eqref{linear:q} we obtain
\[
a(\partial_t\zeta+\mbox{\boldmath$u$}\cdot\nabla\zeta)
+a\Delta(q\mbox{\boldmath$l$}^{\rm T}\mbox{\boldmath$\psi$})=LOT.
\]
We can write this equation and \eqref{linear:transformation 2} in a matrix form as
\begin{align}\label{linear:positive system}
\left(
\begin{array}{cc}
a & \mbox{\boldmath$0$}^{\rm T} \\
\mbox{\boldmath$0$} & -A\Delta
\end{array}
\right)
\partial_t
\left(
\begin{array}{c}
\zeta \\
\mbox{\boldmath$\psi$}
\end{array}
\right)
+ &
\left(
\begin{array}{cc}
a\mbox{\boldmath$u$}\cdot\nabla & a\Delta(q\mbox{\boldmath$l$}^{\rm T}\,\cdot\,) \\
-q\mbox{\boldmath$l$}\Delta(a\,\cdot\,) &
-q\mbox{\boldmath$l$}\mbox{\boldmath$l$}^{\rm T}(\mbox{\boldmath$u$}\cdot\nabla)\Delta
\end{array}
\right)
\left(
\begin{array}{c}
\zeta \\
\mbox{\boldmath$\psi$}
\end{array}
\right) \\
&=
\left(
\begin{array}{c}
LOT \\
\nabla\cdot(LOT)+LOT
\end{array}
\right). \nonumber
\end{align}
Here the operator in the second term in the left-hand side is skew-symmetric in $L^2(\mathbf{R}^n)$
modulo lower order terms.
Since the matrix $A=A(H)$ can be written as
\[
A(H)=H{\rm diag}(H^{p_0},\ldots,H^{p_N})A_0{\rm diag}(H^{p_0},\ldots,H^{p_N})
\]
and we have shown the positivity of the matrix $A_0$ in Proposition \ref{dispersion:prop 1},
$A(H)$ is strictly positive under the strict positivity of the water depth $H$.
Therefore, under the sign condition $a>0$ \eqref{linear:positive system} forms a symmetric positive system,
so that the corresponding energy function is defined by
\[
E(\zeta,\mbox{\boldmath$\psi$})
=(a\zeta,\zeta)_{L^2}
+\sum_{k=1}^n(A\partial_k\mbox{\boldmath$\psi$},\partial_k\mbox{\boldmath$\psi$})_{L^2}
+\|\mbox{\boldmath$\psi$}'\|^2.
\]
In the following sections we use this symmetric structure of the Isobe--Kakinuma model
to construct the solution and derive an energy estimate.
\section{Construction of the solution to a reduced system}
\label{section:construction}
\setcounter{equation}{0}
\setcounter{theorem}{0}
In this section we transform the Isobe--Kakinuma model \eqref{intro:IK model} into a system of equations
for which the hypersurface $t=0$ is noncharacteristic by using the necessary condition \eqref{intro:compatibility}
for the existence of the solution and construct the solution of the initial value problem to the transformed
system by using a standard parabolic regularization.
By using the notation introduced in the previous section, the Isobe--Kakinuma model \eqref{intro:IK model}
can be written simply as
\begin{equation}\label{construction:IK model}
\left\{
\begin{array}{l}
\displaystyle
H^{p_i}\partial_t\eta-\sum_{j=0}^NL_{ij}\phi_j=0 \qquad\mbox{for}\quad i=0,1,\ldots,N, \\
\displaystyle
\sum_{j=0}^NH^{p_j}\partial_t\phi_j=F_0,
\end{array}
\right.
\end{equation}
where
\begin{equation}\label{construction:F0}
F_0=-g\eta-\frac12(|\mbox{\boldmath$u$}|^2+w^2).
\end{equation}
The necessary condition \eqref{intro:compatibility} can also be written simply as
\[
\sum_{j=0}^N(L_{ij}-H^{p_i}L_{0j})\phi_j=0 \qquad\mbox{for}\quad i=1,\ldots,N.
\]
We first derive an evolution equation for $\mbox{\boldmath$\phi$}=(\phi_0,\ldots,\phi_N)^{\rm T}$.
Since the left-hand side of the above equation does not contain the term $\nabla H$,
differentiating this with respect to the time $t$ we obtain
\[
\sum_{j=0}^N(L_{ij}-H^{p_i}L_{0j})\partial_t\phi_j=f_i\partial_t\eta \qquad\mbox{for}\quad i=1,\ldots,N,
\]
where
\begin{align}\label{construction:fi}
f_i &= -\frac{\partial}{\partial H}\sum_{j=0}^N(L_{ij}-H^{p_i}L_{0j})\phi_j \\
&= p_iH^{p_i-1}\biggl\{\nabla b\cdot\mbox{\boldmath$u$}-w-\sum_{j=0}^N\biggl(
\frac{1}{p_j+1}H^{p_j+1}\Delta\phi_j-\frac{p_j}{p_j}H^{p_j}\nabla\cdot(\phi_j\nabla b)\biggr)\biggr\}
\nonumber
\end{align}
for $i=1,\ldots,N$.
We use the first equation in \eqref{construction:IK model} with $i=0$ to remove the time derivative
$\partial_t\eta$ from the above equation and obtain
\begin{equation}\label{construction:reduced 1}
\sum_{j=0}^N(L_{ij}-H^{p_i}L_{0j})\partial_t\phi_j=F_i \qquad\mbox{for}\quad i=1,\ldots,N,
\end{equation}
where
\begin{equation}\label{consruction:Fi}
F_i=f_i\sum_{j=0}^NL_{0j}\phi_j \qquad\mbox{for}\quad i=1,\ldots,N.
\end{equation}
By using the operator $\mathscr{L}$ introduced by \eqref{linear:L2}, the second equation in
\eqref{construction:IK model} and \eqref{construction:reduced 1} constitute the equation
\[
\mathscr{L}\partial_t\mbox{\boldmath$\phi$}=\mbox{\boldmath$F$},
\]
where $\mbox{\boldmath$F$}=(F_0,\ldots,F_N)^{\rm T}$.
This is the evolution equation for $\mbox{\boldmath$\phi$}$.
We proceed to derive an appropriate evolution equation for $\eta$.
Let $\mbox{\boldmath$q$}=(q_0,\ldots,q_N)^{\rm T}=(q_0(H),\ldots,q_N(H))^{\rm T}$ be the
rational functions of $H$ defined by \eqref{linear:inverse}.
In view of the first relation in \eqref{linear:q} and the arguments in the previous section,
we multiply the first equation in \eqref{construction:IK model} by $q_i$ and adding the
resulting equations over $i=0,1,\ldots,N$ to obtain
\[
\partial_t\eta=F_{N+1},
\]
where
\begin{equation}\label{construction:FN+1}
F_{N+1}=-\sum_{i,j=0}^Nq_iL_{ij}\phi_j.
\end{equation}
To summarize we have reduced the Isobe--Kakinuma model \eqref{construction:IK model} to
\begin{equation}\label{construction:reduced 2}
\left\{
\begin{array}{l}
\mathscr{L}\partial_t\mbox{\boldmath$\phi$}=\mbox{\boldmath$F$}, \\
\partial_t\eta=F_{N+1}.
\end{array}
\right.
\end{equation}
We note that $\mbox{\boldmath$F$}$ and $F_{N+1}$ do not contain any time derivatives and that
$\mathscr{L}$ is invertible thanks to Lemma \ref{linear:lemma 3}.
Therefore, the hypersurface $t=0$ is not characteristic any more for the above reduced equations.
In the rest of this section we consider the initial value problem to \eqref{construction:reduced 2}
under the initial conditions
\begin{equation}\label{construction:initial conditions}
(\eta,\phi_0,\ldots,\phi_N)=(\eta_{(0)},\phi_{0(0)},\ldots,\phi_{N(0)}) \quad\makebox[3em]{at} t=0.
\end{equation}
\begin{remark}\label{construction:remark}
As mentioned in Remark {\rm \ref{intro:remark} (2)}, we have to express
$\partial_t\phi_1(x,0),\ldots,\partial_t\phi_N(x,0)$ in terms of the initial data and $b$.
Such an expression is given by
$
\partial_t\mbox{\boldmath$\phi$}(\cdot,0)=\mathscr{L}^{-1}\mbox{\boldmath$F$}|_{t=0}.
$
\end{remark}
We will construct the solution to the initial value problem for the reduced system
\eqref{construction:reduced 2}--\eqref{construction:initial conditions} by a standard
parabolic regularization of the equations, that is,
\begin{equation}\label{construction:regularized}
\left\{
\begin{array}{l}
\mathscr{L}(\partial_t\mbox{\boldmath$\phi$}-\varepsilon\Delta\mbox{\boldmath$\phi$})
=\mbox{\boldmath$F$}, \\
\partial_t\eta-\varepsilon\Delta\eta=F_{N+1},
\end{array}
\right.
\end{equation}
where $\varepsilon>0$ is a small regularized parameter.
By the definitions \eqref{construction:F0}, \eqref{consruction:Fi}, and \eqref{construction:FN+1} of
$\mbox{\boldmath$F$}=(F_0,\ldots,F_N)^{\rm T}$ and $F_{N+1}$ (see also \eqref{construction:fi},
\eqref{linear:L}, and \eqref{linear:uw}) we see that
\begin{equation}\label{construction:FF}
\left\{
\begin{array}{l}
F_0=F_0(\eta,H,\mbox{\boldmath$\phi$}',\nabla\mbox{\boldmath$\phi$},\nabla b), \\
F_i=F_i(H,\nabla H,\mbox{\boldmath$\phi$}',\nabla\mbox{\boldmath$\phi$},\Delta\mbox{\boldmath$\phi$},
\nabla b,\Delta b) \quad\mbox{for}\quad i=1,\ldots,N+1,
\end{array}
\right.
\end{equation}
where $\mbox{\boldmath$\phi$}'=(\phi_1,\ldots,\phi_N)^{\rm T}$.
Therefore, thanks to Lemma \ref{linear:lemma 3}, $\mathscr{L}^{-1}\mbox{\boldmath$F$}$ behaves as if
it is a function of $(\eta,H,\mbox{\boldmath$\phi$}',\nabla\mbox{\boldmath$\phi$},\nabla b)$,
so that we can show the following lemma.
\begin{lemma}\label{construction:existence 1}
Let $g, h, c_0, M_0$ be positive constants and $m$ an integer such that $m>n/2+1$.
Suppose that the initial data $(\eta_{(0)},\phi_{0(0)},\ldots,\phi_{N(0)})$ and $b$ satisfy the conditions
in \eqref{intro:conditions}, then for any $\varepsilon>0$ there exists a maximal existence time
$T_{\varepsilon}>0$ such that the initial value problem \eqref{construction:regularized} and
\eqref{construction:initial conditions} has a unique solution
$(\eta^{\varepsilon},\mbox{\boldmath$\phi$}^{\varepsilon})$ satisfying
\[
\eta^{\varepsilon},\nabla\phi_0^{\varepsilon}\in C([0,T_{\varepsilon});H^m), \quad
\phi_1^{\varepsilon},\ldots,\phi_N^{\varepsilon}\in C([0,T_{\varepsilon});H^{m+1}).
\]
\end{lemma}
We proceed to derive uniform estimates of the solution $(\eta^{\varepsilon},\mbox{\boldmath$\phi$}^{\varepsilon})$
with respect to the regularized parameter $\varepsilon\in(0,1]$ for a time interval $[0,T]$ independent of
$\varepsilon$.
To this end, we make use of a good symmetric structure of the Isobe--Kakinuma model
discussed in the previous section.
In order to simplify the notation we write $(\eta,\mbox{\boldmath$\phi$})$ in place of
$(\eta^{\varepsilon},\mbox{\boldmath$\phi$}^{\varepsilon})$ in the following.
Concerning the generalized Rayleigh--Taylor sign condition, a regularized version of the function $a$
defined by \eqref{intro:a} is given by
\begin{align}\label{construction:a}
a^{\varepsilon} &= g+\sum_{i=0}^Np_iH^{p_i-1}(\partial_t\phi_i-\varepsilon\Delta\phi_i) \\
&\quad\;
+\frac12\sum_{i,j=0}^N\Bigl\{
(p_i+p_j)H^{p_i+p_j-1}\nabla\phi_i\cdot\nabla\phi_j
-2p_i(p_i+p_j-1)H^{p_i+p_j-2}\phi_i\nabla b\cdot\nabla\phi_j \nonumber \\
&\makebox[6em]{}
+p_ip_j(p_i+p_j-2)H^{p_i+p_j-3}(1+|\nabla b|^2)\phi_i\phi_j\Bigr\}. \nonumber
\end{align}
Here, we have $\partial_t\mbox{\boldmath$\phi$}-\varepsilon\Delta\mbox{\boldmath$\phi$}
=\mathscr{L}^{-1}\mbox{\boldmath$F$}$ so that
$a^{\varepsilon}(x,0)=a(x,0)$. See also Remark \ref{construction:remark}.
Therefore, by the sign condition in \eqref{intro:conditions} we have
\begin{equation}\label{construction:sign}
a^{\varepsilon}(x,0)\geq c_0 \quad\mbox{for}\quad x\in\mathbf{R}^n.
\end{equation}
We will derive a regularized version of the symmetric positive system \eqref{linear:positive system}
for $(\eta,\mbox{\boldmath$\phi$})$.
First, we derive an evolution equation for $\mbox{\boldmath$\phi$}$.
The first equation in \eqref{construction:regularized} is written as
\begin{equation}\label{construction:regularized 2}
\left\{
\begin{array}{l}
\mbox{\boldmath$l$}\cdot(\partial_t\mbox{\boldmath$\phi$}-\varepsilon\Delta\mbox{\boldmath$\phi$})=F_0, \\[0.5ex]
\mathscr{L}_i(\partial_t\mbox{\boldmath$\phi$}-\varepsilon\Delta\mbox{\boldmath$\phi$})=F_i
\quad\mbox{for}\quad i=1,\ldots,N,
\end{array}
\right.
\end{equation}
where $\mbox{\boldmath$l$}=\mbox{\boldmath$l$}(H)=(H^{p_0},\ldots,H^{p_N})^{\rm T}$.
Here we remark that $F_i$ $(i=1,\ldots,N)$ is a correction of lower order terms and its explicit form has
no importance whereas $F_0$ contains principal terms and we have to treat it carefully.
In other words, the second equation in \eqref{construction:regularized 2} is quasilinear whereas
the first one is fully nonlinear.
Applying $\Delta$ to the first equation in \eqref{construction:regularized 2} we have
\begin{align*}
& \mbox{\boldmath$l$}(H)\cdot\Delta
(\partial_t\mbox{\boldmath$\phi$}-\varepsilon\Delta\mbox{\boldmath$\phi$})
+(\mbox{\boldmath$l$}^{(1)}(H)\cdot(\partial_t\mbox{\boldmath$\phi$}-\varepsilon\Delta\mbox{\boldmath$\phi$}))
\Delta H-\Delta F_0 \\
& = -([\Delta,\mbox{\boldmath$l$}(H)]-\mbox{\boldmath$l$}^{(1)}(H)(\Delta H))\cdot
\mathscr{L}^{-1}\mbox{\boldmath$F$},
\end{align*}
where $\mbox{\boldmath$l$}^{(1)}(H)=(0,p_1H^{p_1-1},\ldots,p_NH^{p_N-1})^{\rm T}$ is the derivative of
$\mbox{\boldmath$l$}(H)$ with respect to $H$.
Here we see that
\begin{align*}
-\Delta F_0
&= g\Delta\eta+\mbox{\boldmath$u$}\cdot\Delta\mbox{\boldmath$u$}+w\Delta w
+\frac12\bigl\{(\Delta(\mbox{\boldmath$u$}\cdot\mbox{\boldmath$u$})
-2\mbox{\boldmath$u$}\cdot\Delta\mbox{\boldmath$u$})
+(\Delta(w^2)-2w\Delta w)\bigr\},
\end{align*}
where
\begin{align*}
\Delta\mbox{\boldmath$u$}
&= \sum_{i=0}^N\bigl\{ H^{p_i}\nabla\Delta\phi_i
+p_i(H^{p_i-1}\nabla\phi_i-(p_i-1)H^{p_i-2}\phi_i\nabla b)(\Delta\eta) \\
&\qquad\quad
+\bigl([\Delta,H^{p_i}]-p_iH^{p_i-1}(\Delta H)-p_iH^{p_i-1}
(\Delta b)\bigr)\nabla\phi_i
-p_iH^{p_i-1}\Delta(\phi_i\nabla b) \\
&\qquad\quad
-p_i\bigl([\Delta,H^{p_i-1}]-(p_i-1)H^{p_i-2}(\Delta H)-(p_i-1)H^{p_i-2}
(\Delta b)\bigr)(\phi_i\nabla b)\bigr\},
\end{align*}
\begin{align*}
\Delta w
&= \sum_{i=0}^N\bigl\{ p_i(p_i-1)H^{p_i-2}\phi_i\Delta\eta
+p_iH^{p_i-1}\Delta\phi_i \\
&\qquad\quad
+p_i\bigl([\Delta,H^{p_i-1}]-(p_i-1)H^{p_i-2}(\Delta H)-(p_i-1)H^{p_i-2}
(\Delta b)\bigr)\phi_i \bigr\}.
\end{align*}
Therefore, we obtain
\begin{equation}\label{construction:quasilinear 1}
\mbox{\boldmath$l$}(H)\cdot\Delta
(\partial_t\mbox{\boldmath$\phi$}-\varepsilon\Delta\mbox{\boldmath$\phi$})
+ \mbox{\boldmath$l$}(H)\cdot(\mbox{\boldmath$u$}\cdot\nabla)\Delta\mbox{\boldmath$\phi$}
+ a^{\varepsilon}\Delta\eta
= G_{00},
\end{equation}
where
\begin{align}\label{construction:G00}
G_{00}
&= -([\Delta,\mbox{\boldmath$l$}(H)]-\mbox{\boldmath$l$}^{(1)}(H)(\Delta H))\cdot
\mathscr{L}^{-1}\mbox{\boldmath$F$}
+(\mbox{\boldmath$l$}^{(1)}(H)\cdot(\mathscr{L}^{-1}\mbox{\boldmath$F$}))
\Delta b \\
&\quad\;
-\frac12\bigl\{(\Delta(\mbox{\boldmath$u$}\cdot\mbox{\boldmath$u$})
-2\mbox{\boldmath$u$}\cdot\Delta\mbox{\boldmath$u$})
+(\Delta(w^2)-2w\Delta w)\bigr\} \nonumber \\
&\quad\;
-\mbox{\boldmath$u$}\cdot\sum_{i=0}^N\bigl\{
\bigl([\Delta,H^{p_i}]-p_iH^{p_i-1}(\Delta H)-p_iH^{p_i-1}
(\Delta b)\bigr)\nabla\phi_i
-p_iH^{p_i-1}\Delta(\phi_i\nabla b) \nonumber \\
&\qquad\qquad\quad
-p_i\bigl([\Delta,H^{p_i-1}]-(p_i-1)H^{p_i-2}(\Delta H)-(p_i-1)H^{p_i-2}
(\Delta b)\bigr)(\phi_i\nabla b)\bigr\} \nonumber \\
&\quad\;
-w\sum_{i=0}^N\bigl\{ p_iH^{p_i-1}\Delta\phi_i \nonumber \\
&\qquad\qquad\quad
+p_i\bigl([\Delta,H^{p_i-1}]-(p_i-1)H^{p_i-2}(\Delta H)-(p_i-1)H^{p_i-2}
(\Delta b)\bigr)\phi_i \bigr\}. \nonumber
\end{align}
In view of \eqref{linear:L2} and \eqref{linear:L} we divide the operator $\mathscr{L}_i$ into
its principal part and the remainder part $\mathscr{L}_i^{\rm low}$ as
\[
\mathscr{L}_i\mbox{\boldmath$\psi$}
= (-\mbox{\boldmath$a$}_{i}(H)+H^{p_i}\mbox{\boldmath$a$}_{0}(H))\cdot\Delta\mbox{\boldmath$\psi$}
+\mathscr{L}_i^{\rm low}\mbox{\boldmath$\psi$} \quad\mbox{for}\quad i=1,\ldots,N,
\]
where $A(H)=(\mbox{\boldmath$a$}_{0}(H),\ldots,\mbox{\boldmath$a$}_{N}(H))$ is the matrix defined by
\eqref{linear:A} and
\begin{align}\label{construction:Lilow}
\mathscr{L}_i^{\rm low}\mbox{\boldmath$\psi$}
&= \sum_{j=0}^N\biggl(-\frac{p_ip_j}{p_j(p_i+p_j)}H^{p_i+p_j}\nabla\cdot(\psi_j\nabla b)
-\frac{p_i}{p_i+p_j}H^{p_i+p_j}\nabla b\cdot\nabla\psi_j \\
&\phantom{ = \sum\biggl( }
+\frac{p_ip_j}{p_i+p_j-1}H^{p_i+p_j-1}(1+|\nabla b|^2)\psi_j\biggr). \nonumber
\end{align}
Therefore, we obtain
\begin{equation}\label{construction:quasilinear 2}
(\mbox{\boldmath$a$}_{i}(H)-H^{p_i}\mbox{\boldmath$a$}_{0}(H))\cdot\Delta
(\partial_t\mbox{\boldmath$\phi$}-\varepsilon\Delta\mbox{\boldmath$\phi$})
= G_{0i} \quad\mbox{for}\quad i=1,\ldots,N,
\end{equation}
where
\begin{align}\label{construction:G0i}
G_{0i}
&= -F_i
+\mathscr{L}_i^{\rm low}\mathscr{L}^{-1}\mbox{\boldmath$F$}.
\end{align}
We can rewrite \eqref{construction:quasilinear 1} and \eqref{construction:quasilinear 2}
in a matrix form as
\[
\tilde{A}(H)
\left(
\begin{array}{c}
\mbox{\boldmath$a$}_0\cdot\Delta(\partial_t\mbox{\boldmath$\phi$}-\varepsilon\Delta\mbox{\boldmath$\phi$}) \\
\Delta(\partial_t\mbox{\boldmath$\phi$}-\varepsilon\Delta\mbox{\boldmath$\phi$})
\end{array}
\right)
+
\left(
\begin{array}{c}
\mbox{\boldmath$l$}(H)^{\rm T}(\mbox{\boldmath$u$}\cdot\nabla)\Delta\mbox{\boldmath$\phi$}
+a^{\varepsilon}\Delta\eta \\
\mbox{\boldmath$0$}
\end{array}
\right)
=
\left(
\begin{array}{c}
G_{00} \\
\mbox{\boldmath$G$}_0
\end{array}
\right),
\]
where $\tilde{A}(H)$ is the matrix in \eqref{linear:tildeA} and
$\mbox{\boldmath$G$}_0=(0,G_{01},\ldots,G_{0N})^{\rm T}$.
Therefore, using the notation in \eqref{linear:inverse} and the relations in \eqref{linear:q} we have
\[
A(H)\Delta(\partial_t\mbox{\boldmath$\phi$}-\varepsilon\Delta\mbox{\boldmath$\phi$})
+ q(H)\mbox{\boldmath$l$}(H)\{ \mbox{\boldmath$l$}(H)^{\rm T}(\mbox{\boldmath$u$}\cdot\nabla)
\Delta\mbox{\boldmath$\phi$} +a^{\varepsilon}\Delta\eta\} = \mbox{\boldmath$G$},
\]
where
\begin{equation}\label{construction:G}
\mbox{\boldmath$G$}
= q(H)\mbox{\boldmath$l$}(H)G_{00}+A(H)Q(H)\mbox{\boldmath$G$}_0.
\end{equation}
This is the desired equation for $\mbox{\boldmath$\phi$}$.
Secondly, we derive an evolution equation for $\eta$.
In view of \eqref{linear:L} we divide the operator $L_{ij}$ into its principal part and
the remainder part $L_{ij}^{\rm low}$ as
\[
L_{ij}\phi_j=-a_{ij}(H)\Delta\phi_j-H^{p_i}(H^{p_j}\nabla\phi_j-p_jH^{p_j-1}\phi_j\nabla b)\cdot\nabla\eta
+L_{ij}^{\rm low}\phi_j,
\]
where
\begin{align}\label{construction:Llow}
L_{ij}^{\rm low}\phi_j
&= H^{p_i}(H^{p_j}\nabla\phi_j-p_jH^{p_j-1}\phi_j\nabla b)\cdot\nabla b
+ \frac{p_j}{p_i+p_j}H^{p_i+p_j}\nabla\cdot(\phi_j\nabla b) \\
&\quad\;
-\frac{p_i}{p_i+p_j}H^{p_i+p_j}\nabla b\cdot\nabla\phi_j
+\frac{p_ip_j}{p_i+p_j-1}H^{p_i+p_j-1}(1+|\nabla b|^2)\phi_j. \nonumber
\end{align}
Thanks to the relations in \eqref{linear:q}, the second equation in \eqref{construction:regularized}
can be written as
\[
\partial_t\eta-\varepsilon\Delta\eta
+\mbox{\boldmath$u$}\cdot\nabla\eta
+q(H)\mbox{\boldmath$l$}(H)^{\rm T}\Delta\mbox{\boldmath$\phi$}=G_0,
\]
where
\begin{equation}\label{construction:G0}
G_0 = -\sum_{i,j=0}^Nq_i(H)L_{ij}^{\rm low}\phi_j.
\end{equation}
This is the desired equation for $\eta$.
To summarize, we have derived the equations
\begin{equation}\label{construction:quasilinear 3}
\left\{
\begin{array}{l}
\partial_t\eta-\varepsilon\Delta\eta
+\mbox{\boldmath$u$}\cdot\nabla\eta
+q(H)\mbox{\boldmath$l$}(H)^{\rm T}\Delta\mbox{\boldmath$\phi$}=G_0, \\[0.5ex]
A(H)\Delta(\partial_t\mbox{\boldmath$\phi$}-\varepsilon\Delta\mbox{\boldmath$\phi$})
+ q(H)\mbox{\boldmath$l$}(H)\{ \mbox{\boldmath$l$}(H)^{\rm T}(\mbox{\boldmath$u$}\cdot\nabla)
\Delta\mbox{\boldmath$\phi$} +a^{\varepsilon}\Delta\eta\} = \mbox{\boldmath$G$}.
\end{array}
\right.
\end{equation}
Using this we can derive uniform estimate of the solution
$(\eta^{\varepsilon},\mbox{\boldmath$\phi$}^{\varepsilon})$ of the initial value problem
to the regularized equation \eqref{construction:regularized}.
\begin{lemma}\label{construction:uniform}
Under the hypothesis of Lemma {\rm \ref{construction:existence 1}},
there exist a time $T>0$ and a constant $C>0$ independent of $\varepsilon$ such that the solution
$(\eta^{\varepsilon},\mbox{\boldmath$\phi$}^{\varepsilon})$ obtained in Lemma {\rm \ref{construction:existence 1}}
satisfies the uniform estimate
\begin{align}\label{construction:uniform estimate}
& \sup_{0\leq t\leq T}\bigl(
\|\eta^{\varepsilon}(t)\|_m^2+\|\nabla\phi_0^{\varepsilon}(t)\|_m^2
+\|(\phi_1^{\varepsilon}(t),\ldots,\phi_N^{\varepsilon}(t))\|_{m+1}^2\bigr) \\
&\quad
+\varepsilon\int_0^T\bigl(\|\eta^{\varepsilon}(t)\|_{m+1}^2+\|\nabla\phi_0^{\varepsilon}(t)\|_{m+1}^2
+\|(\phi_1^{\varepsilon}(t),\ldots,\phi_N^{\varepsilon}(t))\|_{m+2}^2\bigr){\rm d}t \leq C
\nonumber
\end{align}
for $0<\varepsilon\leq 1$.
\end{lemma}
\noindent
{\bf Proof}. \
Once again we simply write $(\eta,\mbox{\boldmath$\phi$})$ in place of
$(\eta^{\varepsilon},\mbox{\boldmath$\phi$}^{\varepsilon})$.
Let $\alpha=(\alpha_1,\ldots,\alpha_n)$ be a multi-index satisfying $1\leq |\alpha|\leq m$.
Without loss of generality, we can assume that $\alpha_1\geq1$.
Applying the differential operator $a^{\varepsilon}\partial^{\alpha}$ to the first equation in
\eqref{construction:quasilinear 3} we have
\begin{equation}\label{construction:quasilinear 4}
a^{\varepsilon}\partial_t\partial^{\alpha}\eta-\varepsilon\nabla\cdot(a^{\varepsilon}\nabla\partial^{\alpha}\eta)
+a^{\varepsilon}\mbox{\boldmath$u$}\cdot\nabla\partial^{\alpha}\eta
+\sum_{k=1}^n\partial_k(a^{\varepsilon}q\mbox{\boldmath$l$}^{\rm T}\partial_k\partial^{\alpha}
\mbox{\boldmath$\phi$})=F_{0,\alpha},
\end{equation}
where
\begin{equation}\label{construction:F00}
F_{0,\alpha} = a^{\varepsilon}(\partial^{\alpha}G_0-[\partial^{\alpha},\mbox{\boldmath$u$}]\cdot\nabla\eta
-[\partial^{\alpha},q\mbox{\boldmath$l$}^{\rm T}]\Delta\mbox{\boldmath$\phi$})
-\varepsilon\nabla a^{\varepsilon}\cdot\nabla\partial^{\alpha}\eta
+\sum_{k=1}^n(\partial_k(a^{\varepsilon}q\mbox{\boldmath$l$}^{\rm T}))
\partial_k\partial^{\alpha}\mbox{\boldmath$\phi$}.
\end{equation}
Here and in what follows, for simplicity, we omit the dependence of $H$ in the notation.
Applying the differential operator $\partial^{\alpha}$ to the second equation in
\eqref{construction:quasilinear 3} we have
\begin{align}\label{construction:quasilinear 5}
-\sum_{k=1}^n\partial_k(A\partial_k\partial_t\partial^{\alpha}\mbox{\boldmath$\phi$})
+\varepsilon\Delta(A\Delta\partial^{\alpha}\mbox{\boldmath$\phi$})
-& \sum_{k=1}^n\partial_k\bigl\{q\mbox{\boldmath$l$}\mbox{\boldmath$l$}^{\rm T}(\mbox{\boldmath$u$}\cdot\nabla)
\partial_k\partial^{\alpha}\mbox{\boldmath$\phi$}
+a^{\varepsilon}q\mbox{\boldmath$l$}\partial_k\partial^{\alpha}\eta\bigr\} \\
&= \sum_{k=1}^n\partial_k\mbox{\boldmath$F$}_{k,\alpha}, \nonumber
\end{align}
where
\begin{align}\label{construction:F}
\mbox{\boldmath$F$}_{k,\alpha}
&= -\delta_{1k}\partial^{\alpha'}\biggl\{ \mbox{\boldmath$G$}
+\sum_{l=1}^n\bigl((\partial_lA)\partial_l\mathscr{L}^{-1}\mbox{\boldmath$F$}
+[\partial_l,q\mbox{\boldmath$l$}\mbox{\boldmath$l$}^{\rm T}(\mbox{\boldmath$u$}\cdot\nabla)]
\partial_l\mbox{\boldmath$\phi$}
+(\partial_l(a^{\varepsilon}q\mbox{\boldmath$l$}))\partial_l\eta\bigr)\biggr\} \\
&\quad\;
+\varepsilon(\partial_kA)\Delta\partial^{\alpha}\mbox{\boldmath$\phi$}
+[\partial^{\alpha},A]\partial_k\mathscr{L}^{-1}\mbox{\boldmath$F$}
+[\partial^{\alpha},q\mbox{\boldmath$l$}\mbox{\boldmath$l$}^{\rm T}(\mbox{\boldmath$u$}\cdot\nabla)]
\partial_k\mbox{\boldmath$\phi$}
+[\partial^{\alpha},a^{\varepsilon}q\mbox{\boldmath$l$}]\partial_k\eta.
\nonumber
\end{align}
Here, $\alpha'=(\alpha_1-1,\alpha_2,\ldots,\alpha_n)$ and $\delta_{1k}$ is the Kronecker delta.
\eqref{construction:quasilinear 4} and \eqref{construction:quasilinear 5} can be written in the
matrix form as
\begin{align}\label{construction:positive system 2}
& \left(
\begin{array}{cc}
a^{\varepsilon} & \mbox{\boldmath$0$}^{\rm T} \\
\mbox{\boldmath$0$} & -\sum_{k=1}^n\partial_k(A\partial_k\,\cdot\,)
\end{array}
\right)
\partial_t
\left(
\begin{array}{c}
\partial^{\alpha}\eta \\
\partial^{\alpha}\mbox{\boldmath$\phi$}
\end{array}
\right)
+\varepsilon
\left(
\begin{array}{cc}
-\nabla\cdot(a^{\varepsilon}\nabla\,\cdot\,) & \mbox{\boldmath$0$}^{\rm T} \\
\mbox{\boldmath$0$} & \Delta(A\Delta\,\cdot\,)
\end{array}
\right)
\left(
\begin{array}{c}
\partial^{\alpha}\eta \\
\partial^{\alpha}\mbox{\boldmath$\phi$}
\end{array}
\right) \\
&\quad +
\left(
\begin{array}{cc}
a^{\varepsilon}\mbox{\boldmath$u$}\cdot\nabla
& \sum_{k=1}^n\partial_k(a^{\varepsilon}q\mbox{\boldmath$l$}^{\rm T}\partial_k\,\cdot\,) \\
-\sum_{k=1}^n\partial_k(a^{\varepsilon}q\mbox{\boldmath$l$}\partial_k\,\cdot\,)
& -\sum_{k=1}^n\partial_k(q\mbox{\boldmath$l$}\mbox{\boldmath$l$}^{\rm T}(\mbox{\boldmath$u$}\cdot\nabla)
\partial_k\,\cdot\,)
\end{array}
\right)
\left(
\begin{array}{c}
\partial^{\alpha}\eta \\
\partial^{\alpha}\mbox{\boldmath$\phi$}
\end{array}
\right) \nonumber \\
&\quad =
\left(
\begin{array}{c}
F_{0,\alpha} \\
\sum_{k=1}^n\partial_k\mbox{\boldmath$F$}_{k,\alpha}
\end{array}
\right), \nonumber
\end{align}
which forms a symmetric positive system.
In view of \eqref{construction:positive system 2} we define an energy function $\mathscr{E}_m(t)$ and a
dissipation function $\mathscr{D}_m(t)$ by
\begin{align*}
& \mathscr{E}_m(t) = \sum_{|\alpha|\leq m}\biggl\{
(a^{\varepsilon}\partial^{\alpha}\eta(t),\partial^{\alpha}\eta(t))_{L^2}
+\sum_{k=1}^n(A\partial_k\partial^{\alpha}\mbox{\boldmath$\phi$}(t),
\partial_k\partial^{\alpha}\mbox{\boldmath$\phi$}(t))_{L^2}\biggr\}
+\|\mbox{\boldmath$\phi$}'(t)\|^2, \\
& \mathscr{D}_m(t) = \sum_{|\alpha|\leq m}\bigl\{
(a^{\varepsilon}\nabla\partial^{\alpha}\eta(t),\nabla\partial^{\alpha}\eta(t))_{L^2}
+(A\Delta\partial^{\alpha}\mbox{\boldmath$\phi$}(t),\Delta\partial^{\alpha}\mbox{\boldmath$\phi$}(t))_{L^2}
\bigr\},
\end{align*}
which will be equivalent to
\[
E_m(t) = \|\eta(t)\|_m^2+\|\nabla\phi_0(t)\|_m^2+\|\mbox{\boldmath$\phi$}'(t)\|_{m+1}^2, \quad
D_m(t) = \|\nabla\eta(t)\|_m^2+\|\Delta\mbox{\boldmath$\phi$}(t)\|_m^2,
\]
respectively,
where $\mbox{\boldmath$\phi$}'=(\phi_1,\ldots,\phi_N)^{\rm T}$.
In view of \eqref{intro:conditions}, \eqref{construction:FF}, \eqref{construction:a}, \eqref{construction:sign},
and Lemma \ref{linear:lemma 3}, we see that there exists a constant $C_0=C_0(g,h,c_0,M_0)>0$ such that
\[
c_0\leq H(x,0)\leq C_0, \quad c_0\leq a^{\varepsilon}(x,0)\leq C_0
\qquad\mbox{for}\quad x\in\mathbf{R}^n.
\]
Now, we assume that
\begin{equation}\label{construction:assumption}
E_m(t)+\varepsilon\int_0^tE_{m+1}(\tau){\rm d}\tau \leq M_1, \quad
\frac{c_0}{2}\leq H(x,t)\leq 2C_0, \quad
\frac{c_0}{2}\leq a^{\varepsilon}(x,t)\leq 2C_0
\end{equation}
for $0\leq t\leq T$ and $x\in\mathbf{R}^n$, where the constant $M_1$ and the time $T$ will be determined later.
In the following we simply write the constants depending only on $(g,h,c_0,C_0,M_0)$ by $C_1$ and the
constants depending also on $M_1$ by $C_2$, which may change from line to line.
Then, it holds that
\[
C_1^{-1}E_m(t) \leq \mathscr{E}_m(t) \leq C_1E_m(t), \quad
C_1^{-1}D_m(t) \leq \mathscr{D}_m(t) \leq C_1D_m(t)
\]
for $0\leq t\leq T$.
We are going to evaluate the evolution of the energy function $\mathscr{E}_m(t)$.
To this end, we make use of the symmetric form \eqref{construction:positive system 2} for the case
$1\leq|\alpha| \leq m$ and of \eqref{construction:regularized} directly for the case $|\alpha|=0$.
Then, by integration by parts we see that
\begin{align}\label{construction:energy}
&\frac{\rm d}{{\rm d}t}\mathscr{E}_m(t)+2\varepsilon(\mathscr{D}_m(t)+\|\nabla\mbox{\boldmath$\phi$}'(t)\|^2) \\
&= \sum_{|\alpha|\leq m}\biggl\{
((\partial_ta^{\varepsilon})\partial^{\alpha}\eta,\partial^{\alpha}\eta)_{L^2}
+\sum_{k=1}^n((\partial_tA)\partial_k\partial^{\alpha}\mbox{\boldmath$\phi$},
\partial_k\partial^{\alpha}\mbox{\boldmath$\phi$})_{L^2}\biggr\} \nonumber\\
&\quad\;
+\sum_{1\leq |\alpha|\leq m}\biggr\{
((\nabla\cdot(a^{\varepsilon}\mbox{\boldmath$u$}))\partial^{\alpha}\eta,\partial^{\alpha}\eta)_{L^2}
+\sum_{k=1}^n(\biggl(\sum_{l=1}^n\partial_l(u_lq\mbox{\boldmath$l$}\mbox{\boldmath$l$}^{\rm T})\biggr)
\partial_k\partial^{\alpha}\mbox{\boldmath$\phi$},
\partial_k\partial^{\alpha}\mbox{\boldmath$\phi$})_{L^2} \nonumber\\
&\phantom{\quad\; +\sum_{1\leq |\alpha|\leq m}\biggr\{ }
+2(F_{0,\alpha},\partial^{\alpha}\eta)_{L^2}
-2\sum_{k=1}^n(\mbox{\boldmath$F$}_{k,\alpha},\partial_k\partial^{\alpha}\mbox{\boldmath$\phi$})_{L^2}
\biggr\} \nonumber\\
&\quad\;
+2(a^{\varepsilon}F_{N+1},\eta)_{L^2}
+2\sum_{k=1}^n(\partial_k\mathscr{L}^{-1}\mbox{\boldmath$F$},A\partial_k\mbox{\boldmath$\phi$})_{L^2}
+2((\mathscr{L}^{-1}\mbox{\boldmath$F$})',\mbox{\boldmath$\phi$}')_{L^2} \nonumber\\
&\quad\;
-2\varepsilon(\nabla\eta,\eta\nabla a^{\varepsilon})_{L^2}
-2\varepsilon\sum_{k=1}^n(\Delta\mbox{\boldmath$\phi$},(\partial_kA)\partial_k\mbox{\boldmath$\phi$})_{L^2},
\nonumber
\end{align}
where $(\mathscr{L}^{-1}\mbox{\boldmath$F$})'$ is the last $N$ components of
$\mathscr{L}^{-1}\mbox{\boldmath$F$}$.
Here, we see easily that
$\|(\mbox{\boldmath$u$},w)\|_m\leq C_2$, $\|\mathscr{L}^{-1}\mbox{\boldmath$F$}\|_m\leq C_2$,
and $\|\nabla a^{\varepsilon}\|_{m-1}\leq C_2$.
It follows from \eqref{construction:F00} and \eqref{construction:F}
(see also \eqref{construction:G00}, \eqref{construction:Lilow}, and
\eqref{construction:G0i}--\eqref{construction:G0}) that
$\|F_{0,\alpha}\|\leq C_2(1+\varepsilon\|\nabla\eta\|_m)$ and
$\|\mbox{\boldmath$F$}_{k,\alpha}\|\leq C_2(1+\varepsilon\|\Delta\mbox{\boldmath$\phi$}\|_m)$
for $k=1,\ldots,n$.
We need to evaluate $\partial_ta^{\varepsilon}$.
Note that we can write formally as
$[\partial_t,\mathscr{L}]=(\partial_t\eta)(\frac{\partial}{\partial H}\mathscr{L})$
because the coefficients of the differential operator $\mathscr{L}$ do not depend on $\nabla H$ but on $H$.
By the relation
\[
\partial_t(\partial_t\mbox{\boldmath$\phi$}-\varepsilon\Delta\mbox{\boldmath$\phi$})
= -\mathscr{L}^{-1}[\partial_t,\mathscr{L}]
\mathscr{L}^{-1}\mbox{\boldmath$F$}+\mathscr{L}^{-1}\partial_t\mbox{\boldmath$F$}
\]
and Lemma \ref{linear:lemma 3} we see that
\begin{align*}
\|\partial_t(\partial_t\mbox{\boldmath$\phi$}-\varepsilon\Delta\mbox{\boldmath$\phi$})\|_{m-1}
&\leq C_1\biggl(\|[\partial_t,\mathscr{L}_0]\mathscr{L}^{-1}\mbox{\boldmath$F$}\|_{m-1}
+\sum_{i=1}^N\|[\partial_t,\mathscr{L}_i]\mathscr{L}^{-1}\mbox{\boldmath$F$}\|_{m-3} \\
&\qquad\quad
+\|\partial_tF_0\|_{m-1}+\|(\partial_tF_1,\ldots,\partial_tF_N)\|_{m-3}\biggr) \\
&\leq C_1(\|\partial_t\mbox{\boldmath$\phi$}\|_m+\|\partial_t\eta\|_{m-1}).
\end{align*}
Therefore, we have $\|\partial_ta^{\varepsilon}\|_{m-1}\leq
C_2(\|\partial_t\mbox{\boldmath$\phi$}\|_m+\|\partial_t\eta\|_{m-1})$.
In view of
$\|\partial_t\mbox{\boldmath$\phi$}-\varepsilon\Delta\mbox{\boldmath$\phi$}\|_m\leq C_2$ and
$\|\partial_t\eta-\varepsilon\Delta\eta\|_{m-1}\leq C_2$, we also have
$\|\partial_t\mbox{\boldmath$\phi$}\|_m\leq C_2(1+\varepsilon\|\Delta\mbox{\boldmath$\phi$}\|_m)$ and
$\|\partial_t\eta\|_{m-1} \leq C_2(1+\varepsilon\|\nabla\eta\|_m)$.
Thus, by \eqref{construction:energy} we have
$\frac{\rm d}{{\rm d}t}\mathscr{E}_m(t)+2\varepsilon\mathscr{D}_m(t)
\leq C_2(1+\varepsilon(\|\Delta\mbox{\boldmath$\phi$}\|_m+\varepsilon\|\nabla\eta\|_m))$,
which implies that
$\mathscr{E}_m(t)+2\varepsilon\int_0^t\mathscr{D}_m(\tau){\rm d}\tau \leq C_1+C_2(t+\sqrt{t})$.
To summarize, we have obtained the estimate
\[
\left\{
\begin{array}{l}
\displaystyle
E_m(t)+\varepsilon\int_0^tE_{m+1}(\tau){\rm d}\tau \leq C_1+C_2(t+\sqrt{t}), \\
|H(x,t)-H(x,0)|+|a^{\varepsilon}(x,t)-a^{\varepsilon}(x,0)|\leq C_2(t+\sqrt{t})
\quad\mbox{for}\quad x\in\mathbf{R}^n.
\end{array}
\right.
\]
Now, we define the constant $M_1$ by $M_1=2C_1$, and then the time $T$ sufficiently small
so that $C_2(T+\sqrt{T})\ll1$.
Then, we see that \eqref{construction:assumption} holds.
The proof is complete.
\quad$\Box$
\bigskip
Once we obtain this kind of uniform estimate \eqref{construction:uniform estimate},
we can pass to the limit $\varepsilon\to+0$ in the regularized problem \eqref{construction:regularized}
and \eqref{construction:initial conditions} and obtain the following lemma.
\begin{lemma}\label{construction:existence}
Under the hypothesis of Theorem {\rm \ref{intro:theorem}}, there exists a time $T>0$ such that the initial
value problem \eqref{construction:reduced 2}--\eqref{construction:initial conditions} has a unique
solution $(\eta,\mbox{\boldmath$\phi$})$ satisfying
\[
\eta,\nabla\phi_0\in C([0,T];H^m), \quad \phi_1,\ldots,\phi_N\in C([0,T];H^{m+1}).
\]
\end{lemma}
\section{Proof of the main theorem}
\label{section:proof}
\setcounter{equation}{0}
\setcounter{theorem}{0}
We will show that the solution to the transformed problem
\eqref{construction:reduced 2}--\eqref{construction:initial conditions} is the solution of the Isobe--Kakinuma
model \eqref{intro:IK model}--\eqref{intro:initial conditions} if the initial data
$(\eta_{(0)},\phi_{0(0)},\ldots,\phi_{N(0)})$ and the bottom topography $b$ satisfy the necessary condition
\eqref{intro:compatibility}.
Let $(\eta,\mbox{\boldmath$\phi$})$ be the solution of
\eqref{construction:reduced 2}--\eqref{construction:initial conditions} obtained in Lemma \ref{construction:existence}.
Then, we have $\mathscr{L}_0\partial_t\mbox{\boldmath$\phi$}=F_0$, which is exactly the second equation in
\eqref{construction:IK model}.
Therefore, it is sufficient to show that the first equation in \eqref{construction:IK model} holds for
$i=0,1,\ldots,N$.
To this end, putting
\begin{equation}\label{proof:R}
R_i=H^{p_i}\partial_t\eta-\sum_{j=0}^NL_{ij}\phi_j \quad\mbox{for}\quad i=0,1,\ldots,N,
\end{equation}
we are going to show $R_i(x,t)\equiv0$ for $i=0,1,\ldots,N$.
We also introduce auxiliary functions
\begin{equation}\label{proof:RR}
\tilde{R}_i=\mathscr{L}_i\mbox{\boldmath$\phi$} \quad\mbox{for}\quad i=1,\ldots,N.
\end{equation}
Then, we see that the necessary condition \eqref{intro:compatibility} is equivalent to $\tilde{R}_i=0$ for
$i=1,\ldots,N$.
We have also the relations
\begin{equation}\label{proof:relation 1}
\tilde{R}_i=H^{p_i}R_0-R_i \quad\mbox{for}\quad i=1,\ldots,N.
\end{equation}
Differentiate \eqref{proof:RR} with respect to the time $t$ and using \eqref{proof:R} with $i=0$ to eliminate
the time derivative $\partial_t\eta$ and
the equations $\mathscr{L}_i\partial_t\mbox{\boldmath$\phi$}=F_i$, we obtain
\begin{equation}\label{proof:ODE1}
\partial_t\tilde{R}_i=-f_iR_0 \quad\mbox{for}\quad i=1,\ldots,N,
\end{equation}
where $f_i$ is the function defined by \eqref{construction:fi}.
Let $\mbox{\boldmath$q$}=\mbox{\boldmath$q$}(H)=(q_0(H),\ldots,q_N(H))$ be rational functions of $H$ defined
by \eqref{linear:inverse}.
Multiplying \eqref{proof:R} by $q_i$, adding the resulting equations over $i=0,1,\ldots,N$, and using the
relation \eqref{linear:q} we obtain
\[
\sum_{i=0}^Nq_iR_i=-\partial_t\eta-\sum_{i,j=0}^Nq_iL_{ij}\phi_j=0,
\]
where we used the second equation in \eqref{construction:reduced 2}.
This together with \eqref{proof:relation 1} implies
\begin{equation}\label{proof:relation 2}
R_0=-\sum_{j=1}^Nq_j\tilde{R}_j.
\end{equation}
Here, we used the relation $\mbox{\boldmath$l$}\cdot\mbox{\boldmath$q$}=-1$ once again.
Plugging this into \eqref{proof:ODE1} we obtain a system of linear homogeneous ordinary differential
equations for $(\tilde{R}_1,\ldots,\tilde{R}_N)$:
\[
\partial_t\tilde{R}_i=f_i\sum_{j=1}^Nq_j\tilde{R}_j \quad\mbox{for}\quad i=1,\ldots,N.
\]
The necessary condition \eqref{intro:compatibility} for the initial data
$(\eta_{(0)},\phi_{0(0)},\ldots,\phi_{N(0)})$ and the bottom topography $b$ is equivalent to
$\tilde{R}_i(x,0)\equiv0$ for $i=1,\ldots,N$, so that the uniqueness of the solution to the
initial value problem implies $\tilde{R}_i=0$ for $i=1,\ldots,N$.
Then, by \eqref{proof:relation 2} and \eqref{proof:relation 1} we see in turn that
$R_0=0$ and $R_i=0$ for $i=1,\ldots,N$.
Therefore, we have shown that $(\eta,\mbox{\boldmath$\phi$})$ is the solution to the Isobe--Kakinuma
model \eqref{intro:IK model}--\eqref{intro:initial conditions}.
It remains to show that the energy function $E(t)$ defined by \eqref{intro:energy} is conserved in time.
The energy function can be written explicitly as
\begin{align}\label{proof:energy}
E(t) &= \frac{\rho}{2}\int_{\mathbf{R}^n}\biggl\{
\sum_{i,j=0}^N\biggl(
\frac{1}{p_i+p_j+1}H^{p_i+p_j+1}\nabla\phi_i\cdot\nabla\phi_j
-\frac{2p_i}{p_i+p_j}H^{p_i+p_j}\phi_i\nabla b\cdot\nabla\phi_j \\
&\makebox[6em]{}
+\frac{p_ip_j}{p_i+p_j-1}H^{p_i+p_j-1}(1+|\nabla b|^2)\phi_i\phi_j\biggr)+g\eta^2\biggr\}{\rm d}x.
\nonumber
\end{align}
Therefore, by the direct calculation we see that
\begin{align*}
\frac{\rm d}{{\rm d}t}E(t)
&= \rho\biggl\{\sum_{i,j=0}^N(L_{ij}\phi_j,\partial_t\phi_i)_{L^2}
+(g\eta+\frac12(|\mbox{\boldmath$u$}|^2+w^2),\partial_t\eta)_{L^2}\biggr\} \\
&= \rho\biggl\{\sum_{i=0}^N(H^{p_i}\partial_t\eta,\partial_t\phi_i)_{L^2}-(F_0,\partial_t\eta)_{L^2}\biggr\}
=0.
\end{align*}
The proof of Theorem \ref{intro:theorem} is complete.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,173
|
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{
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}
| 6,309
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North Korea Denounces South Korea-US Joint Military Exercises
The Koreas | Security | East Asia
Three days after the alliance's bilateral drills concluded, North Korea's state media published a statement denouncing the exercises.
By Mitch Shin for The Diplomat
Credit: Flickr/ (stephan)
North Korea's state-controlled media KCNA published a statement from the international secretary of the Communist Party in Denmark to denounce the joint military exercises of South Korea and the United States on Sunday.
"The joint military exercises, the outcome of the U.S.'s hostile policy toward the DPRK, is a dangerous nuclear war racket to stifle the DPRK militarily," the statement said. (DPRK is the acronym for North Korea's official name: Democratic People's Republic of Korea.)
Due to former South Korean President Moon Jae-in's peace process, under coordination with then-U.S. President Donald Trump until 2021, in the past few years the joint military drills had been scaled down to continue dialogue with North Korean leader Kim Jong Un. Even after talks with North Korea fell apart, the COVID-19 pandemic curtailed the drills.
However, South Korean President Yoon Suk-yeol and U.S. President Joe Biden reinvigorated the military drills recently. They took place from August 22 to September 1. In turn, North Korea again published statements to pinpoint the drills as the so-called "hostile policy."
"The situation on the Korean Peninsula is in a very unstable one [sic] by the U.S. imperialists and the South Korean puppets and is at a crisis in which a nuclear war breaks out anytime," the statement said.
Seoul and Washington have consistently called the joint military drills "annual defensive exercises" to prepare appropriate readiness against the rising threats of North Korea's nuclear and missile programs. However, North Korea has long deemed the drills a threat to the security of the country.
"The Korean People's Army will never remain an onlooker to the reckless nuclear war exercises, but firmly defend the sovereignty of the country and the socialist system from the aggressive maneuvers of the hostile forces," the statement said.
Despite North Korean leaders' previous provocative remarks over the joint military drills of the South and the U.S., KCNA used other sources to denounce this year's drills. On the same day that KCNA published the statement from the Communist Party in Demark, it also published a research report titled "The U.S.-South Korea joint military exercises are the cancer-like factor harassing peace and stability on the Korean Peninsula and in the region" from the Society for International Politics Study, the North's domestic research institute, on its website.
"The U.S. has steadily renewed the nuclear war scenarios aimed at the DPRK after setting it as the main means of its policy towards the DPRK to stifle our ideology and social system by strength, and has driven the situation on the Korean Peninsula to the brink of a war while training and perfecting the scenarios through all sorts of joint military drills," the report said.
The report also elaborated on Pyongyang's claim that South Korea-U.S. joint military drills as "a hostile policy" and "aggressive war drills."
Compared with the unexpectedly provocative and belligerent words from Kim Yo Jong, the powerful sister of the North Korean leader Kim Jong Un, toward Yoon and his "audacious initiative" to entice Pyongyang to pursue the denuclearization of his country last month, the two published statements from KCNA on Sunday seem to be Pyongyang's strategic decision to convince the international community and its unknown allies that the North faces serious and grave threats from South Korea and the U.S.
As Pyongyang is now well-positioned to show off either its advanced missile programs or nuclear power as a tit-for-tat response to the joint military drills of the South and the U.S., the published statements appeared to be a warm-up. Considering the possible damage from the powerful Typhoon Hinnamnor this week, and China's quest to keep the region calm in the lead-up to its important National Party Congress starting on October 16, however, North Korea might have difficulties conducting actions that can be considered a grave threat to the South and the U.S. in the short term.
In the wake of the Ukraine and Taiwan crisis, China and Russia are strengthening ties to confront the rising leverage of the U.S. and its allies in the region. As part of the Vostok 2022 exercises, they conducted their own drills jointly on the east coast of the Korean Peninsula. As China and Russia are gathering their power to prepare for the new Cold War in the region, North Korea might have been asked not to conduct a nuclear test for a while.
Under the North's five-year military modernization plan, however, it is a matter of time before Kim will approve his military to conduct the test to achieve the plan and his pledge to build more powerful nuclear weapons.
Mitch Shin
Mitch Shin is an assistant editor at The Diplomat and nonresident Korea Foundation fellow at Pacific Forum.
Hostile policy
North Korea diplomacy
U.S.-South Korea military exercises
Ulchi Freedom Shield
South Korea, US Return to Large-Scale Military Drills
By Troy Stangarone
The resumption of large-scale drills for the first time since 2017 will improve military readiness for the alliance.
What to Make of Kim Yo Jong's Verbal Attack of South Korea's Defense Minister
By Mitch Shin
Three days after Defense Minister Suh Wook underscored the South's preemptive strike capabilities, Pyongyang warned that Seoul was facing a serious threat.
North Korea Is Irrelevant Again
By Denny Roy
Continued missile testing by North Korea will not compel Seoul, Tokyo, or Washington to seek the negotiations Pyongyang could use to address its needs.
North Korea's 'Strategic Patience'
Recent developments suggest North Korea is adopting its own version of "strategic patience": ignoring diplomatic outreach until the situation turns in its favor.
|
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O povoamento da Islândia ), crê-se ter começado na segunda metade do século IX, quando os colonos nórdicos migraram ao longo do Atlântico Norte.
As razões da migração são incertas:
mais tarde, na Idade Média, os próprios islandeses tendiam a citar conflitos civis provocados pelas ambições do rei norueguês Harald I, da Noruega, mas os historiadores modernos centram-se em fatores mais profundos, como a escassez de terras aráveis na Escandinávia. Ao contrário da Grã-Bretanha e da Irlanda, a Islândia era uma terra não-habitada que podia ser ocupada sem conflitos.
Com base no Íslendingabók de Ari Þorgilsson e Landnámabók, histórias que datam dos séculos XII e XIII e contêm detalhes importantes sobre o povoamento da ilha, os anos 870 e 874 dC tradicionalmente considerados os iniciais.
Contudo, a exatidão dessas não é muito confiável, e pesquisas recentes guiam-se mais em evidências arqueológicas e genéticas.
Tradicionalmente, considera-se que a Era do Povoamento da Islândia durou de 874 a 930, altura em que a maior parte da ilha foi ocupada e Alþingi (Althingi), a assembleia da Comunidade da Islândia, foi fundada em Þingvellir (Thingvellir). A Islândia é, portanto, provavelmente a penúltima grande massa terrestre a ser colonizada por humanos (a Nova Zelândia é a última).
História da população
Povoamento pré-nórdico
Resultados controversos num recente trabalho de datação por carbono, publicado na revista Skírnir, sugerem que a Islândia pode ter sido habitada desde a segunda metade do século VII. O Íslendingabók de Ari Thorgilsson afirma que os colonos nórdicos encontraram monges gaélicos de uma missão Hiberno-escocesa quando chegaram à Islândia. Existem evidências arqueológicas de um local monástico da Irlanda na caverna Kverkarhellir, na localidade de Seljaland, no sul da Islândia. Depósitos de sedimentos indicam que as pessoas moraram lá à volta de 800 dC e cruzes consistentes com o estilo hiberno-escocês foram esculpidas na parede duma caverna próxima.
A fonte mais antiga conhecida, que menciona o nome "Islândia", é uma runa do século XI, esculpida na Gotland, enquanto as mais antigas descobertas arqueológicas indicam populações que datam do século IX. Existe uma possível menção anterior da Islândia no livro De mensura orbis terrae do monge irlandês Dicuil, datado de 825. Dicuil alegou ter conhecido alguns monges que viveram na ilha de Thule. Estes relataram que a escuridão reinava durante o inverno, mas que os verões eram claros o suficiente para catar piolhos das roupas, mas a veracidade dessa fonte pode ser questionada. Além disso, a Islândia dista apenas cerca de 450 Kms das Ilhas Faroé, que haviam sido visitadas pelos monges irlandeses no século VI, e colonizadas pelos nórdicos por volta de 650 dC.
Uma cabana em Hafnir abandonada entre 770 e 880, atestou a construção precede claramente o povoamento oficial de 874 dC. Pensa-se que tenha sido um posto avançado apenas habitado parte do ano, mas não se sabe se foi construído por pessoas da Escandinávia, Irlanda ou Escócia.
Povoamento nórdico
Fontes escritas consideram que a era da colonização na Islândia começou com a colonização de Ingólfr Arnarson por volta de 874, pois foi ele quem iniciou a navegação para a Islândia com o objetivo de colonizar a terra. Evidências arqueológicas mostram que a ocupação humana da ilha começou de facto nessa fase e "que o país inteiro foi colonizado nalgumas décadas no final do século IX". As estimativas do número de pessoas que migraram para o país durante a Idade da Liquidação variam entre 4.300 e 24.000, com estimativas do número de colonos iniciais variando entre 311 e 436.
Enquanto as fontes escritas enfatizam o populações da Noruega, as evidências genéticas mostram que a população fundadora da Islândia veio da Irlanda, Escócia e Escandinávia: estudos de DNA mitocondrial e cromossomas Y indicam que 62% da ancestralidade matrilinear dos islandeses deriva da Escócia e da Irlanda (sendo a maior parte do restante da Escandinávia), enquanto que 75% da ascendência patrilinear deriva da Escandinávia (sendo a maior parte do restante das Ilhas Irlandesas e Britânicas). Evidências arqueogenéticas sugerem que a população fundadora real incluiu novamente uma proporção maior de colonos das Ilhas Irlandesas e Britânicas: um estudo descobriu que a ascendência nórdica média entre os colonos da Islândia era de 56%, enquanto na população atual esse número era de 70%. Pensa-se que a maioria dos colonos da Irlanda e da Escócia tenha sido escrava e, portanto, reproduzida com menos sucesso do que os colonos de alto status da Escandinávia, tornando-os ancestrais de uma proporção menor da população moderna.
Embora a noção de que a pressão da população tenha impulsionado a migração para a Islândia permaneça sem suporte na literatura académica, várias razões foram apresentadas para o povoamento da Islândia:
A terra disponível teria sido apelativa para os escandinavos da era viking, especialmente devido ao clima relativamente quente da Islândia na época.
A observação de recursos valiosos, como o marfim da morsa, tornou a Islândia interessante para comerciantes.
Vada vez mais resistência às incursões vikings nas ilhas britânicas e na Europa continental no final do século IX, levou os vikings a buscar oportunidades mais pacíficas.
Fontes escritas medievais enfatizam como a centralização da Noruega por Harald Fairhair e a imposição de impostos onerosos aos agricultores, incentivaram classes agricolas a emigrar para a Islândia.
Fontes escritas relatam que alguns colonos escolheram terras livremente, outros compraram terreno de colonos anteriores, alguns foram agraciados com terras por colonos anteriores e que alguns colonos expropriaram outros das terras à força ou intimidação. As terras provavelmente não eram arrendadas durante a Era da colonização. O medievalista Hans Kuhn argumentou que as terras foram doadas ou tiradas livremente porque os colonos anteriores não precisavam de terras tão extensas. O historiador Gunnar Karlsson observa que poderia ser racional para os colonos iniciais, incentivar novos colonos a estabelecer terras nas proximidades, afim de facilitar a manutenção de gado e escravos e como património seguro em tempos de crise.
Ari Thorgilsson afirma em Íslendingabók que o país havia sido "totalmente repartido" em 930. Análogamente, Landnámabók sugere que em sessenta anos, toda a terra utilizável havia sido tomada; menciona 1.500 nomes de propriedades e locais e mais de 3.500 pessoas, organizadas de maneira geográfica.
Na periodização da história da Islândia, considera-se portanto, que a idade da colonização terminou no ano de 930, com o estabelecimento de Alþingi; ponto esse, considerado como o começo da Commonwealth da Islândia. Amostras arqueológicas revelam no entanto, "que os imigrantes continuaram a chegar à Islândia durante o século X". Os autores de um estudo especulam que "a imigração contínua pode ter sido necessária para sustentar a população".
Teorias
Naddoddr e Garðar
O Landnámabók afirma que o primeiro nórdico a pisar solo islandês foi um viking com o nome de Naddoddr. Naddoddr ficou por apenas um curto período de tempo, mas deu um nome ao país:Snæland (Terra da Neve). Foi seguido pelos suecos comandados por Garðar Svavarsson, que foi o primeiro a passar lá o Inverno. À volta de 860, uma tempestade empurrou o navio para o norte até chegar à costa leste da Islândia. Garðar atracou na ilha pelo leste, navegando para o oeste ao longo da costa e depois rumo a norte, construindo uma casa em Húsavík. Tendo circumnavegado a massa terrestre em questão, estabeleceram que se tratava de facto, duma ilha. Partiu no verão seguinte, para nunca mais voltar, mas não antes de dar à ilha um novo nome - Garðarshólmur (literalmente, a ilha de Garðar). Um dos seus homens, Náttfari, decidiu ficar na ilha com dois escravos. Náttfari estabeleceu-se no que é agora conhecido como Náttfaravík, perto de Skjálfandi. O Landnámabók no entanto, sistenta que Náttfari não foi um colono permanente.
Hrafna-Flóki
O segundo nórdico a chegar à Islândia foi nomeado Flóki Vilgerðarson, mas o ano da chegada não é claro. Segundo a história contada em Landnámabók, ele levou três corvos para ajudá-lo a encontrar o caminho. Assim, foi chamado de Raven-Flóki (islandês:Hrafna-Flóki). Flóki libertou os corvos perto das Ilhas Faroe. O primeiro corvo voou de volta para as Ilhas Faroé. O segundo voou no ar e depois voltou para o navio. No entanto, o terceiro voou em frente ao navio e eles seguiram-no rumo à Islândia.
Ele desembarcou em Vatnsfjörður nos Westfjords depois de passar o que é agora Reykjavík. Um dos seus homens, Faxi, observou que pareciam ter encontrado grandes terras - daí, a baía de Reykjavík ser conhecida como Faxaflói. Um inverno rigoroso fez com que todo o gado de Flóki morresse - ele amaldiçoou este país frio e, quando viu gelo no fiorde, decidiu chamar esta a "terra do gelo", "Ísland" (Islândia). Apesar das dificuldades em encontrar comida, ele e a sua tripulação ficaram mais um ano, desta vez em Borgarfjörður, mas voltaram para a Noruega no verão seguinte. Flóki retornaria muito mais tarde, e estabelecer-se-ia no que hoje é conhecido como Flókadalur.
Ingólfur Arnarson
Outro nórdico, chamado Ingólfur Arnarson, havia instigado um conflito de sangue na terra natal, a Noruega. Ele e o irmão adotivo Hjörleifur, foram a uma expedição exploratória para a Islândia e passaram o inverno no que é hoje Álftafjörður. Alguns anos depois, eles retornaram para estabelecer a terra com os seus homens. Quando se aproximaram da ilha, Ingólfur lançou os seus longos troncos de árvore ao mar e jurou que ficariam no local onde esses dessem à costa. Enviou os seus escravos Vífill e Karli para procurar os troncos flutuantes. Estes no entanto, encontraram o seu irmão adotivo Hjörleifur assassinado e abandonado pelos seus homens. Ingólfur deu ao irmão adotivo um funeral pagão no estilo nórdico e matou os assassinos, que tinham fugido para as ilhas Westman.
À medida que o inverno se aproximava, os escravos de Ingólfur encontraram os troncos de Arnarhvol. Quando chegou o verão, ele construiu uma quinta em Reykjavík e reclamou todas as terras a oeste dos rios Ölfusá, Öxará e Brynjudalsá. O seu escravo Karli não se importou com a localização e disse a Ingólfur: "Sofremos muito em boas terras até nos instalarmos nesta remota península".
Efeitos ambientais
Os académicos argumentaram que os colonos causaram erosão do solo pelos extensos desbravamentos e pastagens excessivas. Um estudo sugere que o principal motivo para desbravamento foi "explotação de pastagens e campos domésticos", não os "requisitos dos colonos para combustível e material de construção".
Ver também
Cronologia histórica da Islândia
Expansão Viking
Migrações humanas
Islândia
Vikings
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| 5,026
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Q: Adding an Item to the top of a LongListSelector I have a Windows Phone app that follows the Databound App template. It's basically a LongListSelector control manipulated by an ObservableCollection() thing.
Adding things is easy: I just do this:
App.ViewModel.Items.Add(new ItemViewModel() { ThingOne = "Blah", ThingTwo = "BlahBlahBlah"});
But my Item gets added to the bottom of the LongListSelector. I'd like to add it to the top.
How do I do this? Can I specify this behavior within the Items.Add() statement? Or is it something to do with the MainViewModel?
Thanks!
A: App.ViewModel.Items.Insert(0, new ItemViewModel() { ThingOne = "Blah", ThingTwo = "BlahBlahBlah"});
A:
Can I specify this behavior within the Items.Add() statement?
User Insert instead :Collection.Insert Method
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| 1,692
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Beauty and the Beast picoreview - the essential kit
Beauty and the Beast picoreview
It's late, so I'm intending this to be only a very brief review of Beauty and the Beast, although I have a great deal to say about it and may get carried away.
The very short version is, while I'm glad I went, it was an adequate performance of largely bad material, and as far as staging it is concerned, I would have done it *entirely* differently, just about from top to bottom. I think it suffered badly in comparison to the stage version of The Lion King, which is at least as good as the film, and possibly better due to the sheer technical wonder of it all. BatB opted for melodrama/caricature (which, admittedly, is in keeping with the film), and I didn't think it worked nearly as well.
For some *incredibly* bizarre reason, the actors all had their microphones between their eyebrows, like some kind of caste mark. It made everyone look like they had a giant zit thoroughly covered in makeup on their foreheads. The only principle who did not have a Zit Mic was Maurice, Belle's father, who was played by a balding man, and whose hair could presumably not hide the microphone cord. I cannot imagine what possessed them to do that.
One would think with the bizarre microphones and all, the sound balance between the orchestra and vocals would have been fantastic. It wasn't. It wasn't as bad as some shows I've gone to, but there was too much music and not enough voice in several of the songs. Professional theatre should not have that kind of problem, dammit.
The man who played Gaston was very tall (and made taller by heeled shoes with lifts in them) and very muscular, and while I didn't really enjoy him as a character, I couldn't help thinking that it really would've been an *awful* lot of fun to see Hugh Jackman play the part, which he did in the Australian version of the show. If they'd played it the same way, I still wouldn't have liked what they did very much, but Jackman would've been very, very fun to watch.
Among other things that I didn't like, he was costumed in pleather stuff with fake abs sewn in. I would not have done that. :) (There were two other men in the chorus who were quite tall, though still noticeably shorter than Gaston. Given any choice in the matter, I wouldn't have cast them, either, because without them, Gaston would have stood literally head and shoulders above everybody else he was on stage with, and the visual would have been even funnier.) While I realize even in the film, Gaston really is a caricature, I'd have played him much straighter; there was no sense of threat from him at all. He had more edge in the movie than he did on stage, and the fight between him and the Beast (and in fact anywhere else when he was supposed to be being Manly and Slightly Threatening) was so staged as to simply not work. As Jason Carter said one time after being told by someone that in his fight with Tybalt in Romeo & Juliet, "I really thought you were going to *kill* each other!", the whole *point* is that you think they're really going to kill each other. There was no such fear in this. In fact, I didn't think anybody was going to get killed.
That said, the "Kill the Beast" scene was really nicely staged. That's one of my favorite songs in the show (how can you not be a fan of a song with the line, Screw your courage to the sticking place?), and they pulled it off as a mob song really pretty well.
In fact, mostly, the songs from the film were done well. I was *extremely* dubious about "Be Our Guest", and again would not have made several of the staging choices they did, but in the end it was in fact a great enthusiastic number, easily the best in the show. The "Beauty and the Beast" number itself was prettily done (although I think they were too slavish in copying Belle's costume from the film: the 'sleeves' were so stiff that they billowed up and nearly hid Belle's head when she danced with the Beast. It would have been much better if they'd made them lacier and more forgiving.), but I would not have had Angela Lansb Mrs. Potts on stage at all to sing it. She was spotlighted and distracting from the story going on with Belle and the Beast. Worse, she was down stage left, which is where you traditionally put the leads for romantic scenes (romance plays best from there; my theory is that you're playing to people's hearts in a physical and literal sense), so just really lousy staging there.
Even "Human Again", which was cut from the film (with good reason) played all right on stage. However, the original music written for the stage show was incredibly bad. Really, really bad, and made worse by nearly every song (both from the film and not) being done as a Stand And Present, rather than any feel of incorporation into moving the storyline along. The Beast had a perfectly terrible Song That Goes Like This that actually made me *think* of The Song That Goes Like This while he was singing it, although later it occurred to me that Belle had also had her own Song That Went Like That.
I thought Lumiere was too raunchy, and Ted thought the Vegas showgirls style of "Be Our Guest" was too sexy for what's purportedly a kids' show (I just thought Lumiere bordered on tacky and wasn't especially sexy or charming, rather than being concerned with age appropriateness. He *was* funny, but it was a Bit Too Much). Cogsworth was pretty funny, and Babette (the feather duster) was not only very good but also had a really awesome costume that fit very snugly and had Ginger Rogers feathers from the knees down, but proved to have a zipped hip-height slit so it opened up into a quite full skirt for "Be Our Guest". I totally approved of that costume. :)
Belle was quite lovely, though I don't think her makeup should have tried so hard to bring to mind the big doe eyes the character had in the film. She did have a very nice and very understandable voice (unlike the three blondes, who were played as shrill and pushy instead of ditzy and cute like they were in the film).
I didn't think the Beast's normal voice had enough raar in it, but the problems with the show weren't, I thought, in the performances, but rather in the direction and staging choices. (Perhaps "an adequate performance" is being cruel, but I didn't feel they'd risen above the material, which the stage performance of "Rent" that I saw did.) It's a shame that the only decent song he had was "There's Something There", and only half of that is his, because he did have a gorgeous singing voice. I just got distracted by the Song That Goes Like This-ness of his other songs.
Clearly the girls behind me did not have that problem, as there was shrieking and squeeing over the Beast-turned-Prince at the end. :)
I could very probably go on, but my picoreview has now turned out to be an hour's worth of typing, and it's late, so I'm going to call it good at this point.
Really, that sounds like I didn't enjoy it, which isn't true. I did enjoy it, and I'm genuinely glad I went, because I've wanted for years to see it and I'd have felt hugely like I'd missed out if I hadn't gone. Besides, half the fun of theatre is being able to tear a show apart, and it's a rare damned thing when I can't. (That always stuns my family, when all we can say is, "Wow, that was really *good*!")
Also, more or less totally unrelated, as fantastically gorgeous and sexy as the high-stiletto-heeled Cadiz boots are, I'll be sending them back. They're just not wearable enough; four hours of them tonight, most of which wasn't on my feet, was too much. I love them, they're beautiful, they're gorgeous, but I really don't think I'll wear them again, and they cost too damned much to keep lying around. Someday when I'm rich perhaps I'll have an extremely sexy pair of boots like that custom made to fit my feet precisely. :)
miles to Dunharrow: 160
Tags: picoreviews, theatre
Current Mood: nerdy
Current Music: kill the beast/the song that goes like this
(Anonymous) on March 9th, 2007 08:16 am (UTC)
I think it suffered badly in comparison to the stage version of The Lion King, which is at least as good as the film, and possibly better due to the sheer technical wonder of it all.
Yah. That.
BatB actually makes Lion King look more impressive, in that BatB is quite a popular stage musical, so it makes you realize how little effort Disney actually would have had to put into a Lion King musical and still have it be a success... which makes me figure they did what they did (in Lion King) as a pure labor of love.
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org.apache.hadoop.lib.lang (Apache Hadoop HttpFS 2.6.0 API)
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Q: Java and stacks: correct implementation? I created this very simple stack concept's implementation.
Could you tell me if it is correct and clean? Do you see any bad coding habits?
public class MyStack
{
private static final int MAXELEMENTS = 10;
private int[] elements;
private int numElements;
public MyStack()
{
numElements = 0;
elements = new int[MAXELEMENTS];
}
public boolean isEmpty()
{
return (numElements == 0);
}
public boolean isFull()
{
return (numElements == MAXELEMENTS);
}
public void push(int e)
{
if (!isFull())
elements[numElements++] = e;
}
public int top()
{
if (!isEmpty())
return elements[numElements - 1];
else
return -1;
}
public void pop()
{
if (!isEmpty())
numElements--;
}
}
You can use it with the following code:
class MyStackTestDrive
{
public static void main(String[] args)
{
MyStack s1 = new MyStack();
MyStack s2 = new MyStack();
s1.push(2);
s2.push(4);
System.out.println(s1.top());
System.out.println(s2.top());
}
}
A: My comments are:
*
*This will be a stack for int only. You should make it generic
*This is a stack for only 10 elements. Not of much use
*numElements is a bad name for top of the stack variable
*top should throw and exception for empty stack (traditionally)
*pop should return the top of the stack. Right now it is void
A: There are some things I would do, some of them just a matter of preference :
*
*Rename "numElements" to "head" to better reflect its function.
*Add a constructor where the max size of the stack can be specified.
*The pop method should return the top element of the stack and remove it. That is the standard semantic.
*top and pop should throw an exception when called on an empty stack. This is to prevent errors where an empty stack is being operated without checking it first.
A: You should try to make your stack generic (if you want to allow any other type to be added to) also, you should try to make it grow automatically, by using a check within your push method and if it's full then you reallocate space for it (creating another array, maybe of twice previous space and realocating each by one). Any way you're using arrays instead of a linked list, for example, and that IMHO is a god idea.
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{"url":"http:\/\/math.stackexchange.com\/questions\/1292\/how-can-i-understand-and-prove-the-sum-and-difference-formulas-in-trigonometry\/1298","text":"# How can I understand and prove the \u201csum and difference formulas\u201d in trigonometry? (cos(a \u00b1 b) = \u2026, etc.)?\n\nThe \"sum and difference\" formulas often come in handy, but it's not immediately obvious that they would be true.\n\n\\begin{align} \\sin(\\alpha \\pm \\beta) &= \\sin \\alpha \\cos \\beta \\pm \\cos \\alpha \\sin \\beta \\\\ \\cos(\\alpha \\pm \\beta) &= \\cos \\alpha \\cos \\beta \\mp \\sin \\alpha \\sin \\beta \\end{align}\n\nSo what I want to know is,\n\n1. How can I prove that these formulas are correct?\n2. More importantly, how can I understand these formulas intuitively?\n\nIdeally, I'm looking for answers that make no reference to Calculus, or to Euler's formula, although such answers are still encouraged, for completeness.\n\n-\n\nThe key fact here is that rotation is a linear transformation, e.g. the rotation of $u + v$ is the rotation of $u$ plus the rotation of $v$. You should draw a diagram that shows this carefully if you don't believe it. That means a rotation is determined by what it does to $(1, 0)$ and to $(0, 1)$.\n\nBut $(1, 0)$ rotated by $\\theta$ degrees counterclockwise is just $(\\cos \\theta, \\sin \\theta)$, whereas $(0, 1)$ rotated by $\\theta$ degrees counterclockwise is just $(-\\sin \\theta, \\cos \\theta)$. (Again, draw a diagram.) That means a rotation by $\\theta$ is given by a $2 \\times 2$ matrix with those entries. (Matrices don't work here yet.)\n\nSo take a rotation by $\\theta$ and another one by $\\theta'$, and multiply the corresponding matrices. What you get is the sine and cosine angle addition formulas. (The connection to complex numbers is that one can represent complex numbers as $2 \\times 2$ real matrices.)\n\nAlso, if you believe that $a \\cdot b = |a| |b| \\cos \\theta$, this implies the cosine angle difference formula when $a$ and $b$ are unit vectors. Ditto for the cross product and the sine angle difference formula.\n\n-\n\nHere are my favorite diagrams:\n\nAs given, the diagrams put certain restrictions on the angles involved: neither angle, nor their sum, can be larger than 90 degrees; and neither angle, nor their difference, can be negative. The diagrams can be adjusted, however, to push beyond these limits.\n\nHere's a bonus mnemonic cheer (which probably isn't as exciting to read as to hear):\n\nSine, Cosine, Sign, Cosine, Sine!\nCosine, Cosine, Co-Sign, Sine, Sine!\n\nThe first line encapsulates the sine formulas; the second, cosine. Just drop the angles in (in order $\\alpha$, $\\beta$, $\\alpha$, $\\beta$ in each line), and know that \"Sign\" means to use the same sign as in the compound argument (\"+\" for angle sum, \"-\" for angle difference), while \"Co-Sign\" means to use the opposite sign.\n\n-\nDiagrams are neat; I had never seen them before. It was refreshing to see this related as relationships between triangles instead of relationships between unit circle angles. \u2013\u00a0Justin L. Aug 25 '10 at 3:09\nBTW: Apparently, someone copped my image and used an adapted version (plus a tangent rule variant) in the Wikipedia \"List of Trigonometric Identities\" page. I'm flattered, of course, but I don't think I was given proper credit according to the Creative Commons Attribution Share Alike license, or StackExchange's Terms of Service. Actually, I would've appreciated if the Wikipedia contributor had simply invited me to submit the image myself. (A note about the whole thing would've been nice.) \u2013\u00a0Blue Aug 20 '13 at 18:05\nI have now sourced you on Wikipedia. I have also asked you to be sourced in future from the original uploader. \u2013\u00a0Chris Sherlock Nov 28 '13 at 12:27\nFYI: These diagrams now appear on my Trigonography website. \u2013\u00a0Blue Nov 15 '15 at 1:03\n\nYou can use the complex representation,\n$\\cos(x) = \\frac{1}{2}(e^{ix} + e^{-ix})$\n$\\sin(x) = \\frac{1}{2i}(e^{ix} - e^{-ix})$\nand the rules for powers ($a^{x+y}=a^x a^y$)\n\n-\nThis is equivalent to diagonalizing the rotation matrices before multiplying them. \u2013\u00a0Qiaochu Yuan Jul 31 '10 at 17:35\n@QiaochuYuan Indeed, but in fewer words ;) \u2013\u00a0Tobias Kienzler Oct 6 '14 at 6:47\n\nThough the standard high-school derivations are not the most useful way to remember it in the long run, here's another one which I like because you can \"see\" it directly without much algebra.\n\nLet P be the point on the unit circle got by rotating (1,0) by angle \u03b1+\u03b2. Drop a perpendicular N to the \u03b1-rotated line, and R to the x-axis. So from the right triangle ONP, you see ON = cos \u03b2. You can see that the angle RPN is \u03b1 too: it's the complement of \u2220PNQ, and so is \u2220QNO = \u03b1. Now,\n\n$\\sin(\\alpha + \\beta) = \\mbox{PR} = \\mbox{PQ} + \\mbox{QR} = \\sin(\\beta)\\cos(\\alpha) + \\cos(\\beta)\\sin(\\alpha)$, and\n\n$\\cos(\\alpha + \\beta) = \\mbox{OR} = \\mbox{OM} - \\mbox{RM} = \\cos(\\beta)\\cos(\\alpha) - \\sin(\\beta)\\sin(\\alpha)$.\n\n-\nwith program did you draw this diagram? \u2013\u00a0seeker Jul 13 '14 at 17:33\n@Assad: If I remember correctly, I used TikZ, and this was in fact my first time using TikZ. I wish I had kept the source code of this figure; I haven't used TikZ much since then, and I'd have to re-learn it if I wanted to draw this again from scratch. :-) But it couldn't have been too hard, because I did learn enough to draw this. \u2013\u00a0ShreevatsaR Jul 13 '14 at 17:45\nThank you, this looks like a neat application \u2013\u00a0seeker Jul 13 '14 at 17:47\n\nThere are several typical derivations used in high school texts. Here's one:\n\nTake two points on the unit circle, one a rotation of (1,0) by \u03b1, the other a rotation of (1,0) by \u03b2. Their coordinates are as shown in the diagram. Let c be the length of the segment joining those two points. By the Law of Cosines (on the blue triangle), $c^2=1^2+1^2-2\\cdot1\\cdot1\\cdot\\cos(\\alpha-\\beta)$. Using the distance formula, $c=\\sqrt{(\\cos\\alpha-\\cos\\beta)^2+(\\sin\\alpha-\\sin\\beta)^2}$. Squaring the latter and setting the two equal, $1^2+1^2-2\\cdot1\\cdot1\\cdot\\cos(\\alpha-\\beta)=(\\cos\\alpha-\\cos\\beta)^2+(\\sin\\alpha-\\sin\\beta)^2$. Simplifying both sides, $2-2\\cos(\\alpha-\\beta)=\\cos^2\\alpha-2\\cos\\alpha\\cos\\beta+\\cos^2\\beta+\\sin^2\\alpha-2\\sin\\alpha\\sin\\beta+\\sin^2\\beta$ $=2-2\\cos\\alpha\\cos\\beta-2\\sin\\alpha\\sin\\beta$ (using the Pythagorean identity). Solving for $\\cos(\\alpha-\\beta)$, $\\cos(\\alpha-\\beta)=\\cos\\alpha\\cos\\beta+\\sin\\alpha\\sin\\beta$.\n\nFrom this identity, the other three can be derived by substituting $\\frac{\\pi}{2}-\\alpha$ for \u03b1 (gives sin(\u03b1+\u03b2)), then -\u03b2 for \u03b2 (gives the remaining two).\n\nAs to understanding the formulas intuitively, if you accept that multiplying by a complex number $z_\\theta$ for which |z|=1 rotates by \u03b8, then you can think about what happens when you multiply $z_\\alpha=\\cos\\alpha+i\\sin\\alpha$ and $z_\\beta=\\cos\\beta+i\\sin\\beta$ (by expanding the binomial product), which should result in $\\cos(\\alpha+\\beta)+i\\sin(\\alpha+\\beta)$.\n\n-\nPoint worth making: Isaac's last paragraph and my argument are the same. This is a point which is not often understood. \u2013\u00a0Qiaochu Yuan Jul 31 '10 at 7:53\n@Qiaochu Yuan: Yes, quite true. I think about it in complex numbers more naturally than in matrices, but it's equivalent. I don't think of it as a proof because my chain of derivations usually uses the sum\/difference identities to justify that complex multiplication (by a number of modulus 1) is geometrically a rotation. \u2013\u00a0Isaac Jul 31 '10 at 7:55\nAh. You don't need to do that: you just need to construct the isomorphism between 2x2 rotation matrices and the complex numbers. \u2013\u00a0Qiaochu Yuan Jul 31 '10 at 8:02\n@Isaac: You don't even need to involve rotation matrices. There is a wonderfully simple picture which explains why multiplication by $w$ scales the plane by $|w|$ and rotates by $\\arg w$. I can't draw it here, but it's Figure 6bc on p. 9 in Needham's Visual Complex Analysis. (You can view it on Google Books.) \u2013\u00a0Hans Lundmark Sep 3 '10 at 6:58\nyour diagram is gone \u2013\u00a0Tobias Kienzler Oct 19 '10 at 8:20\n\nI remember that $e^{i\\alpha}=\\cos\\alpha+i\\sin\\alpha$ and that $i^2=-1$. Both these relations are useful in many other situations and pretty fundamental to understanding complex numbers. Then your equalities are the real and, respectively, the imaginary part of $e^{i(\\alpha+\\beta)}=e^{i\\alpha}e^{i\\beta}$.\n\nThis is not very different from the other answers, but I actually prefer the algebra perspective. The only place where I think geometrically is in interpreting $e^{i\\alpha}=\\cos\\alpha+i\\sin\\alpha$ by thinking of the unit circle in the complex plane.\n\n-\nYou mean times. \u2013\u00a0Qiaochu Yuan Jul 31 '10 at 17:35\nSorry :( Fixed. Thanks. \u2013\u00a0rgrig Jul 31 '10 at 20:39\nI prefer the geometric perspective too, and it just seems like you shouldn't need to know anything about imaginary numbers to understand these identities. Which is why I asked the question. The big insight I'm getting of course is that the two ways of looking at it are really not that different. Thanks! \u2013\u00a0MatrixFrog Jul 31 '10 at 23:14\n\nI will prove the identity $\\cos(x+y)=\\cos x\\cos y-\\sin x\\sin y$, using with the following definitions of sine and cosine:\n\n$$\\sin x:= \\sum_{n=0}^{\\infty}(-1)^n\\frac{x^{2n+1}}{(2n+1)!} \\ \\ \\ \\ ;\\ \\ \\ \\cos x:= \\sum_{n=0}^{\\infty}(-1)^n\\frac{x^{2n}}{(2n)!}$$\n\nProof:\n\n$$\\cos (x+y)= \\sum_{n=0}^{\\infty}(-1)^n\\frac{(x+y)^{2n}}{(2n)!}$$\n\nUsing the Binomial theorem, we will have\n\n$$\\sum_{n=0}^{\\infty}(-1)^n\\sum^{2n}_{k=0}\\binom{2n}{k}\\frac{x^ky^{2n-k}}{(2n)!}=$$ $$=\\sum_{n=0}^{\\infty}(-1)^n\\sum^{2n}_{k=0}\\frac{x^ky^{2n-k}}{k!(2n-k)!}$$\n\nNow, separating the inner sum into two, for even $k$ and for odd $k$:\n\n$$=\\sum_{n=0}^{\\infty}(-1)^n\\sum^{n}_{k=0}\\frac{x^{2k}y^{2n-2k}}{(2k)!(2n-2k)!}+\\sum_{n=1}^{\\infty}(-1)^n\\sum^{n-1}_{k=0}\\frac{x^{2k+1}y^{2n-2k-1}}{(2k+1)!(2n-2k-1)!}$$\n\nNow, let us look on the first sum,\n\n$$\\sum_{n=0}^{\\infty}(-1)^n\\sum^{n}_{k=0}\\frac{x^{2k}y^{2n-2k}}{(2k)!(2n-2k)!}=$$ $$=\\sum_{n=0}^{\\infty}\\sum^{n}_{k=0}(-1)^k\\frac{x^{2k}}{(2k)!}(-1)^{n-k}\\frac{y^{2(n-k)}}{(2(n-k))!}=$$\n\nBy Cauchy product, we have:\n\n$$=\\sum_{n=0}^{\\infty}(-1)^k\\frac{x^{2k}}{(2k)!}\\sum_{n=0}^{\\infty}(-1)^k\\frac{y^{2k}}{(2k)!}=$$\n\n$$=\\cos x\\cos y$$\n\nFor the second sum,\n\n$$\\sum_{n=1}^{\\infty}(-1)^n\\sum^{n-1}_{k=0}\\frac{x^{2k+1}y^{2n-2k-1}}{(2k+1)!(2n-2k-1)!}=$$\n\nBy Cauchy product, we have:\n\n$$\\sum_{n=1}^{\\infty}\\sum^{n-1}_{k=0}(-1)^k\\frac{x^{2k+1}}{(2k+1)!}(-1)^{n-k}\\frac{y^{2((n-1)-k)+1}}{(2((n-1)-k)+1)!}$$\n\nAnd by substituting $t=n-1$, we will have:\n\n$$\\sum_{t=0}^{\\infty}\\sum^{t}_{k=0}(-1)^k\\frac{x^{2k+1}}{(2k+1)!}(-1)^{t+1-k}\\frac{y^{2(t-k)+1}}{(2(t-k)+1)!}$$\n\n$$=-[ \\sum_{k=0}^{\\infty}(-1)^k\\frac{x^{2k+1}}{(2k+1)!} ][\\sum_{k=0}^{\\infty}(-1)^k\\frac{y^{2k+1}}{(2k+1)!}]$$\n\n$$=-\\sin x\\sin y$$\n\nQ.E.D\n\n-","date":"2016-06-01 04:05:46","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 1, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9537052512168884, \"perplexity\": 597.4461250068257}, \"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-2016-22\/segments\/1464054526288.74\/warc\/CC-MAIN-20160524014846-00059-ip-10-185-217-139.ec2.internal.warc.gz\"}"}
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"Follow the money" is a line popularized by the film All the President's Men (1976), about the Watergate scandal. Deep Throat (Mark Felt, it was later revealed) tells Washington Post reporter Bob Woodward (played by Robert Redford) to "follow the money" to find the scandal. The line does not appear in the Bob Woodward-Carl Bernstein book, All the President's Men, and Woodward couldn't find anything like it in his notes.
The movie's screenwriter was William Goldman—an experienced writer who'd won a previous Academy Award for Butch Cassidy and the Sundance Kid (1969). Goldman said (see 1997 citation below): ''I can't believe I made it up. I was in constant contact with Woodward while writing the screenplay. I guess he made it up."
The line "follow the money" had been well-known Hollywood, cited in print since at least 1971. The camera and lights are told to always "follow the money"—either the highest paid star on the film or the expensive set. It's highly possible that William Goldman heard the line "follow the money" from his many years in Hollywood.
"Follow the money" remains a popular saying in business and government. "Policy follows the money"
Wikipedia: William Goldman
William Goldman (born August 12, 1931) is an American novelist, playwright, and screenwriter. He lives in New York City.
Goldman has won two Academy Awards: an Academy Award for Writing Original Screenplay for Butch Cassidy and the Sundance Kid, and an Academy Award for Writing Adapted Screenplay for All the President's Men.
Wikiquote: All the President's Men (film)
All the President's Men is a 1976 film about two journalists investigating the Watergate scandal for the Washington Post.
Directed by Alan J. Pakula. Written by William Goldman, based on the book by Bob Woodward and Carl Bernstein.
Deep Throat: You thought I'd help out on specifics? I'll confirm what you get, try to keep you on the right track, but that's all. Are you guys really working? [Woodward nods] How much?
Woodward: I don't know maybe sixteen, eighteen hours a day--we've got sources at Justice, the FBI, but it's still drying up.
Deep Throat: Then there must be something, mustn't there. Look, forget the myths the media's created about the White House--the truth is, these are not very bright guys, and things got out of hand.
Woodward: If you don't like them, why won't you be more concrete with me?
Deep Throat: Because the press stinks too--history on the run, that's all you're interested in. You come up with anything?
Woodward: John Mitchell resigned as head of CREEP to spend more time with his family. That doesn't exactly have the ring of truth. Howard Hunt's been found--there was talk that his lawyer had 25 thousand in cash in a paper bag.
Deep Throat: Follow the money. Always follow the money.
Woodward: What do you mean? Where?
Deep Throat: Oh, I can't tell you that.
Woodward: But you could tell me that.
Deep Throat: No, I have to do this my way. You tell me what you know, and I'll confirm. I'll keep you in the right direction if I can, but that's all. Just… follow the money.
22 August 1971, New York (NY) Times, "Don't Give My Regards To Broadway" by Gerald Hiken, pg. D1:
I know that stars go through hell to get where they are, but that doesn't make it easier to take their bullying, to sit through their tempers and nerves. To work with them is depressing. In Hollywood they tell the camera and lights to "follow the money." It's the same in New York. I learned to deal with them, to sympathize, and take their acting notes, and the notes from their agents, husbands or wives.
25 May 1975, Los Angeles (CA) Times, "Movies: Malcom McDowell at Mid Career" by Mary Blume, Calendar section, pg. Q36:
Follow the money. When you're filming, the operator will say to the director: "Where shall we go, guv?" And they say "follow the money" which means that when characters go two ways the camera follows the star.
Cinefantastique, Volume 5
Frederick S. Clarke
"It's a big tendency in Hollywood and has been in filmmaking since the very beginning. There's an old wise crack: 'follow the money' which means, when you're in doubt as to where to point the camera, point it at the most expensive item, whether it's the actor or the set."
29 April 1976, Bend (OR) Bulletin, "Watergate riddles remain unsolved" by Richard Street of The New Republic, pg. 6, col. 4 photo caption:
"Follow the money...who has the most to gain by shutting off presidential campaign funds?"
New York (NY) Times
On Language
Follow the Proffering Duck
By William Safire
But wait: thanks to Daniel Schorr, the National Public Radio commentator whose investigative credibility includes the credential of a place on the notorious ''Enemies List,'' we now have a new and disconcerting take on the origin of the famous phrase.
Schorr searched for the phrase in the journalists' book. It wasn't there. Nor was it in any of the Watergate reporting in the Washington Post. Follow the money first appeared in the movie ''All the President's Men,'' spoken by Hal Holbrook playing Deep Throat.
The screenplay was written by William Goldman. When Schorr called him, the famed screenwriter at first insisted that the line came from the book; when proved mistaken about that, he said: ''I can't believe I made it up. I was in constant contact with Woodward while writing the screenplay. I guess he made it up.''
Schorr then called Woodward, who could not find the phrase in his exhaustive notes of Watergate interviews. The reporter told Schorr he could no longer rely on his memory as to whether Deep Throat had said the line and was inclined to believe that Goldman had invented it.
Cameron will not break his vow on marriage
The Tory leader stumbled over his marriage tax plans. But he won't ditch them as they are at the heart of his beliefs
Daniel Finkelstein
Deep Throat didn't tell Bob Woodward to follow the money. William Goldman made it up for the screenplay of his Watergate film All the President's Men. The reason it became the most famous instruction ever (not) given to a journalist is because of its almost universal application.
New York City • Government/Law/Politics/Military/ Religion • (0) Comments • Monday, November 08, 2010 • Permalink
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Posted on July 30, 2014 by Neil
My Wollongong Library reading has lately included two very different biographies. Least controversial first:
Of course I knew about "The Father of Federation" – or thought I did! I had even stood outside the "witches' houses" in Annandale where he died, and of course knew of his connection with Centennial Park, the Education Act, and (more controversially perhaps) his anti-Chinese immigration activity. But in truth I knew very little, so I am obliged to this very readable account of his life.
So here are 10 facts you may not know about the man, Sir Henry Parkes:
Born into poverty in 1815, Parkes was the youngest of seven children. He had very little formal education after the age of eight.
Parkes came to Australia in 1839, aged 24, as a penniless English immigrant with wife Clarinda and a daughter born at sea two days before they arrived.
In the 1840s, he was briefly Sydney correspondent for the Launceston Examiner, and contributed occasional poems and articles on political and literary topics to the Sydney Morning Herald.
In 1854 Parkes was elected to the New South Wales Parliament, the start of half a century of almost unbroken – and predominantly unpaid – parliamentary membership.
Financial ruin was never far from Parkes' doorstep, and he sometimes lost his seat due to bankruptcy.
He rose to become Premier of New South Wales five times. Though not great at personal finance, his administrations were all sound, leaving the State in good shape on each occasion.
Parkes implemented a wide range of social reforms, and presided over the major achievement of his era: the introduction of a public education system.
Parkes was married on three occasions, and fathered 17 children!
Parkes died quietly on 27 April 1896 and his grave is in Faulconbridge, the Blue Mountains, NSW.
His image appears on the Australian one-dollar coin of 1996; and on the Centenary of Federation commemoration Australian $5 note issued in 2001.
The last of his children, Cobden Parkes, a WWI veteran and architect, died as recently as 1978! Sir Henry fathered him at the age of 77.
See also Who was the Father of Federation? We talk to Stephen Dando-Collins on Sir Henry Parkes.
…To mention merely a few of his numerous achievements, Parkes introduced the ground-breaking and highly controversial Public Schools Act which provided for the establishment of state schools, run by state-trained teachers. This had a huge impact on the funding and curriculum of religious schools. "The Church schools particularly on the Catholic side never forgave him", says Dando-Collins. Parkes also reformed the health system by contacting Florence Nightingale in London and asking her to handpick a team of four trained British nurses to be sent to Sydney to establish what became the foundation of a nurse training scheme in New South Wales. He also petitioned against convict transportation, actively supported women's suffrage, was a pioneer of mental health care and established the Royal Commission into Rabbit Destruction to tackle Australia's rabbit plague.
Aside from his stellar political career, Parkes was a fascinating man. "The personal side of the man intrigued and surprised me", says Dando-Collins. Before coming to Australia as a free immigrant, Parkes had a tough start going to work at a young age when his father was sent to debtor's prison. He had many different jobs, married three times, fathered seventeen children and took a mistress late in life who was many years his junior. He was a poet, journalist and editor and ran the Empire, a newspaper he established in direct competition with Fairfax. He corresponded with Dickens, Tennyson and Mark Twain and as a failed businessman, went bankrupt three times with debts in total of approximately twenty million dollars in today's equivalent…
Next a book that has proven very controversial: Mark Kurzem, The Mascot (2007). Sadly the author – also gay, by the way — died in 2010.
My partner, the writer, film-maker and academic Mark Kurzem, has died aged 52 from complications related to diabetes. His most enduring achievement was his voyage into his own and his father's past, detailed in an award-winning documentary, The Mascot (2002), which he wrote and co-produced, and a well-received 2007 book of the same name.
Both told the story of how Mark's father, Alex, had concealed his true identity from his wife and children. In the late 1990s, Alex had revealed to Mark that he was not, as his family had believed, the lost son of Russian pig farmers who had been adopted by a well-intentioned Latvian family after the second world war. Instead, in 1941, in a village in what is now Belarus, he had witnessed the mass murder of his mother and siblings, and other members of the local Jewish community.
After this massacre, he had run away into the woods where he was later found by a group of Latvian soldiers belonging to a Nazi police battalion. These soldiers, under the orders of their sergeant, eventually adopted the traumatised boy, dressed him in Nazi uniform, and declared him their "mascot". Alex went on to witness further atrocities and even ended up in Nazi propaganda newsreels. Mark's book examined the cost of this loss of identity in relation to his own family history…
I instantly thought of Europa, Europa and almost inevitably of the notorious The Hand that Signed the Paper hoax. I thought too of my Latvian classmate at Sutherland in the early 1950s and his mother and father, whom I came to know.
It is a good read, no doubt about that. However, shaped as it seems to be by its origins as almost a "Who Do You Think You Are?" episode, with the emphasis on the search, it admittedly blurs and simplifies. But is it true? Well some say this could be the last word on that.
Mr Kurzem was recently told he would continue to receive compensation after the ombudsman found in his favour. But, while the ombudsman was satisfied he was a Jewish Holocaust survivor, the report fell short of confirming his identity.
In his [sic] international bestseller, The Mascot, Mr Kurzem said he was a Russian Jew who survived the Holocaust by working closely with an SS extermination squad.
He claimed he was a member of the Galperin family, and that the Nazis gave him the name Kurzemnieks, which he had later shortened to Kurzem.
Mr Kurzem, 77, claims that as a five-year-old he watched the execution of his Jewish mother and siblings before becoming a child mascot for an SS unit in war-torn Europe.
His memoir was turned into an award-winning 2004 ABC documentary, and a French company is making a film about his life.
The ombudsman's report said evidence presented by Mr Kurzem's doubters was "purely circumstantial".
"None of these pieces of evidence, nor all of them together, can negate the feasibility of the story," the report said.
It said Mr Kurzem was eligible for compensation even if he were wrong in his belief that he was a Galperin.
The ombudsman was satisfied Mr Kurzem was Jewish; was separated from his parents during the war; lived under a false identity for at least 18 months, and; that his life had been in danger.
That 2013 Herald-Sun story does confuse father and son: the latter, Mark, wrote the book about his father, Alex. But note "…while the ombudsman was satisfied he was a Jewish Holocaust survivor, the report fell short of confirming his identity." So see also this 2012 piece on J-Wire: "The Mascot" – Truth or Fiction?. On the other hand ALEX KURZEM – MAKING PEACE WITH ILYA GALPERIN is more accepting, as is Jeffrey Shallit. That last one has a quite amazing – indeed hair-raising – comment thread.
What do I think? I think Mark Kurzem believed the story he eventually tracked down partly with his father and partly independently. I think there is a core of truth there, but it has probably been distorted over time, I suspect. But as I said, it is a good read, and certainly informative about conditions in Latvia and Belarus during WWII, not to mention about immigrant life in Australia in the post-war period and the way migrants adapted/integrated into Australia.
This entry was posted in Australia, history, immigration, reading, top read. Bookmark the permalink.
← Last night's sunset made the evening news…
Pig's guts! It's the end of the month! →
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Tchê é uma interjeição comumente utilizada no português, no castelhano (che) e no valenciano (xe), sendo usada como vocativo (significando "companheiro", "amigo") ou para expressar espanto ou surpresa. O revolucionário argentino Che Guevara foi apelidado como tal por seus companheiros cubanos devido ao seu uso frequente da expressão, percebida como estrangeira no país caribenho.
Uso
Em português, a expressão é utilizada no Rio Grande do Sul, sendo vocábulo característico do dialeto gaúcho. Em castelhano, a expressão é utilizada na Argentina, na Bolívia, no Paraguai, no Uruguai e em Valência. No inglês das Ilhas Malvinas, por influência castelhana, utiliza-se a expressão (grafada che, chay ou chey) com o mesmo significado.
Em alguns países latino-americanos o termo é usado como sinônimo de "argentino" ("jornais che", "obelisco che"). Na Espanha, che é utilizado para se referir a uma pessoa de origem valenciana. O time de futebol Valencia é conhecido como a "equipe che" ou simplesmente "os ches".
Variações
Em algumas regiões do interior de São Paulo, o "tchê" foi modificado até chegar ao "xé" ou "ché", possuindo a mesma pronúncia do castelhano, obtendo o mesmo significado. Isso se dá porque a região foi caminho de tropeiros, onde diversos comerciantes paulistas realizavam suas vendas e negócios, e muitos tropeiros e peões oriundos do Rio Grande do Sul passavam temporadas. A expressão é registrada como "ché!" em O Dialeto Caipira de Amadeu Amaral, como interjeição de dúvida do dialeto caipira.
Cultura da América do Sul
Língua espanhola
Língua portuguesa
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Thursday, 30 July 2020 / Published in Our Blog
Celebrating African Enterprises
African Business Magazine in its May 2020 issue ranked the top 250 African companies. The listing captured various sectors in a ranking that is often dominated by companies in Media, Banking, Metals & Mining. As expected, companies from Africa's leading economies of South Africa, Nigeria, Egypt and Morocco took a lead in these rankings with Kenya's Safaricom leading the Kenyan enterprises at position 10. The time is ripe for more and more African enterprises to tap into the opportunities within Africa and contribute to the continents growth and sharing with the rest of the world Africa's resilience in spite of the challenges that have challenged our economies, threatened our peace and put to question our political choices.
Nigeria's Aliko Dangote of Dangote Group said, "To build a successful business, you must start small and dream big. In the journey of entrepreneurship, tenacity of purpose is supreme". This statement rings true for many of these African enterprises of excellence and gives credence to those that are in the journey of rising amidst the challenges of starting a company in Africa.
Siginon Group is an African enterprise whose cargo handling potential spans across the African continent. Siginon is celebrating 35 Years since its inception in 1985. The transport and logistics company started off as a small clearing & forwarding company in the Port City of Mombasa which soon expanded to acquire 3 trucks and engaged 4 employees. The rest as they say is history, today, Siginon Group has grown to offer a wide spectrum of logistics services with over 200 trucks and engages 500 staff in its operations in Kenya, Uganda and Tanzania handling cargo across the globe. This indigenous Kenyan company prides itself in Powering Trade for their customers' enterprise to succeed through offering world-class logistics services.
Siginon' s corporate vision, "To Be A World-Class Logistics Company' points to a company that is ambitious and endeavours to position itself not only as Africa's leading logistics player but stretches further to cover the global scene. As African countries continue to be rich in supply of the world's raw material, logistics companies that are well established, efficient, reliable and well networked within Africa and across the globe are better positioned to meet the growing supply chain demands.
Siginon Group's humble beginnings tell of a company that started small yet bolstered by big dreams. In 1985, Siginon Group dared to venture into cargo logistics, a sector that was then defined as a multinational playing field. That notwithstanding, Siginon Group positioned itself to successfully compete for the customers hearts, mind and share of wallet with the global logistics 'big boys.' In spite of the entrepreneurial challenges faced while setting up, this African company focused on accessing and serving the cargo needs of regional and global markets at a time that many would have thrown in the towel and opted for other 'safer' sectors. Siginon' s focus and determination to succeed has paid off and with it left an indelible mark in the logistics sector by satisfying customer needs as well as keeping tabs on global trends to benchmark and deliver world-class services consistently.
Today, Siginon prides itself in its expertise as a global logistics solutions partner for cargo that needs to be shipped by air, sea, road or rail and providing all the intermediary cargo services such as; warehousing, container freight station, ground handling, customs clearance and distribution. Supported by Kenya's strategic location in the East Africa region, Siginon has played a major role in regional trade by handling cargo to and from Kenya's key entry points at the Port of Mombasa and Jomo Kenyatta International Airport (JKIA), Nairobi.
Siginon Group has supported the trade and development of African economies by offering logistics services to support national and regional programs such as; roads and infrastructure projects, delivering humanitarian relief to much needed communities in the region, supporting establishment of key manufacturing plants, aviation, hospitality and supporting government delivery in national projects such as distribution of vaccines, examinations and ballot papers, oil and gas as well as horticultural and tea exports.
Africa's latent potential is seen in unlocking barriers hindering Intra Africa trade. The launch of the African Continental Free Trade Area (AfCTA) that brings together 55 member states of the African Union covering a market of more than 1.2 billion people, including a growing middle class, and a combined gross domestic product (GDP) of more than US$3.4 trillion. AfCFTA has the potential of boosting intra-African trade by 52.3 percent by eliminating import duties, and doubling trade by reduction of non-tariff barriers. No one knows Africa better than a stable, sound well entrenched African company like Siginon.
All indications are that now is the time for African enterprises to play a role in making the African continent to shine even brighter. The opportunities for trade within Africa are limitless, with enablers of trade such as logistics companies playing a major role in ensuring there is efficient cargo movement from the source to consumer markets in Africa. The sky is the limit.
Let's encourage and celebrate all African enterprises that weather the storms, stand the test of time to unlock opportunities within Africa's treasure trove for it is through them that Africa rises.
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Pociąg – akwarela polskiego malarza Stanisława Masłowskiego (1853-1926) z około 1890 roku, znajdująca się (2022) w zbiorach Muzeum Narodowego w Warszawie, opatrzona u dołu po lewej sygnaturą: "1890, St. Masłowski, Łosijówka".
Opis
Namalowana w plenerze akwarela przedstawia rozległy, letni, w Polsce - zapewne lipcowy, wielobarwny krajobraz pól uprawnych przedzielonych po prawej stronie falistymi, szaro-zielonymi i białymi pasami drogi gruntowej. Letnią porę roku sugeruje szaro-słomkowy pas zboża po lewej i częściowo po prawej. Na dalszym planie, w głębi, zaznaczono poziomą linię szlaku kolejowego z szeregiem słupów telegraficznych/telefonicznych, z pociągiem napędzanym lokomotywą parową, z widocznym jedynie jej zadaszeniem i kominami i charakterystycznym białym obłoczkiem pary z lewej. U zbiegu drogi ze szlakiem kolejowym naszkicowano po prawej figurę ludzką (w kobiecym ubiorze?). Na jasnoszarym niebie zaznaczono malownicze, obfitujące w cumulusy, zachmurzenie.
Całość obrazu utrzymana jest w ciepłej, kremowej i jasnozielonej tonacji. Z lewej, u dołu znajduje się sygnatura postawiona ołówkiem co najmniej trzydzieści lat po jego powstaniu. Z tego względu data jego powstania jest niepewna
Na omawianym pejzażu z pociągiem towarowym, malowanym ze znaczną swobodą pędzla, graniczącą z nonszalancją, ukazano fragment częstego w Polsce w XIX i XX stuleciu, ale rzadko spotykanego w wieku XXI (z wielu zbieżnych przyczyn: rozdrobnienie upraw, nieutwardzona droga gruntowa, pociąg-retro...- chyba, że jeszcze w okolicach Wolsztyna, albo Chabówki) - nizinnego, równinnego krajobrazu "szachownicy pól" . Mimo to, może być on sugestywny i pociągający dla ludzi typu "turystycznych włóczęgów", ludzi "starszej daty", przewodników turystycznych, krajoznawców często kontaktujących się z przyrodą, a szczególnie, stałych mieszkańców wsi.
Omawiana akwarela jest reprezentatywna dla okresu twórczości malarza z lat 1884-1890 - okresu realizmu, czasów 'Wędrowca' i wczesnego pleneru, kiedy to - według jego syna historyka sztuki - [...] Masłowski "wszedł w nową fazę twórczości i w nowe środowisko sztuki" [...] nawiązując bliskie kontakty koleżeńskie z grupą malarzy i pisarzy związaną z tym czasopismem - z A. Gierymskim i A. Sygietyńskim, z młodymi J. Pankiewiczem i W. Podkowińskim.
Wymienione wyżej lata, obfitowały u autora niniejszego obrazu w ważkie innowacje malarskie. Masłowski bowiem - jak to podkreślał Henryk Piątkowski (1909) - [...]"jeden z pierwszych w malarstwie polskim wprowadził zupełną bezpośredniość wrażeń z natury. Cała jego młodzieńcza twórczość to fragmenty wyrwane gorąco z przyrody. Wśród pokolenia artystów, do których należał, stoi on na pierwszym planie, nie znika jak tylu innych w cieniu Chełmońskiego, lecz coraz bardziej rozszerza swą własną ścieżkę twórczości"
Wspomniane innowacje w twórczości artysty, zostały określone jako "nowe malarstwo natury", [...]"malarstwo oryginalnego, młodzieńczo bezpośredniego, subtelnego w skromnej kolorystycznej gamie 'wrażeniowego' realizmu"[...]
Prezentowany tu pejzaż z pociągiem, jest jednym z dokumentów tego najwcześniejszego pleneru ochrzczonych później mianem 'japońszczyzny'. Do tych "dokumentów" - prócz niego - syn artysty, historyk sztuki, Maciej Masłowski zaliczał również 'Lasek' i 'Moczary' (z dawnej kolekcji Feliksa Manggha-Jasieńskiego) - małe pejzaże akwarelowe ze zbiorów Muzeum Narodowego w Krakowie, a ponadto 'Łubin z wiatrakiem' - ze zbiorów rodziny.
Wspomniane obrazy (prócz innych, niewymienionych) - według Tadeusza Dobrowolskiego (1960) - świadczą o niezwykle silnym "odczuciu charakteru ziemi polskiej, szczególnie Mazowsza, umiejętności odszukania w przeciętnym, najzwyklejszym motywie cech ogólnych, istotnych dla płaskiego, pozornie jednostajnego krajobrazu, w związku z czym proste, do kilku głównych elementów sprowadzone pejzaże artysty urastały do rangi żywych, syntetycznych niezmiernie trafnych formuł przedstawienia."
Dane uzupełniające
Omawiana akwarela była reprodukowana barwnie w:
Stanisław Masłowski Akwarele 12 reprodukcji barwnych, wstęp opracował i dokonał wyboru materiału ilustracyjnego Maciej Masłowski, Wydawnictwo Sztuka, Warszawa 1956, jako tablica nr 9 - z objaśnieniem: "Pociąg, (akwarela wielk. oryg. 105 x 225 mm), własność Macieja Masłowskiego"
Obraz - zanim został zakupiony przez Muzeum - przez wiele lat dekorował mieszkanie syna artysty.
Literatura
Polski Słownik Biograficzny, Wrocław - Warszawa - Kraków - Gdańsk, 1975, wyd. "Ossolineum", tom XX/1, zesz.84
Tadeusz Dobrowolski: Nowoczesne malarstwo polskie, t. II, Wrocław - Kraków, 1960, wyd. Zakład Narodowy im. Ossolińskich - Wydawnictwo
Stanisław Masłowski - Materiały do życiorysu i twórczości, oprac. Maciej Masłowski, Wrocław, 1957, wyd. "Ossolineum"
Stanisław Masłowski Akwarele 12 reprodukcji barwnych, wstęp opracował i dokonał wyboru materiału ilustracyjnego Maciej Masłowski, Wydawnictwo Sztuka, Warszawa 1956
Przypisy
Malarstwo pejzażowe
Obrazy Stanisława Masłowskiego
Obrazy w Muzeum Narodowym w Warszawie
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\section{Introduction}
Eco-epidemiological models are ecological models that include infected compartments. In many situations, these models describe more accurately the real ecological system than models where the disease is not taken into account.
There is already a large number of works concerning eco-epidemiological models. To mention just a few recent works, we refer~\cite{Chakraborty-Das-Haldar-Kar-2015} where a mathematical study on disease persistence and extinction is carried out;~\cite{Bai-Xo-2018} where the authors study the global stability of a delayed eco-epidemiological model with holling type III functional response, and~\cite{Purnomo-Darti-Suryanto-2017} where an eco-epidemiological model with harvesting is considered.
One of the main concerns when studying eco-epidemiological models is to determine conditions under which one can predict if the disease persists or dies out. In mathematical epidemiology, these conditions are usually given in terms of the so called basic reproduction ratio $\mathcal{R}_0$, defined in~\cite{Diekmann-Heesterbeek-Metz-1990} for autonomous systems as the spectral radius of the next generation matrix.
In~\cite{Bacaer-Guernaoui-2006}, $\mathcal{R}_0$ was introduced for the periodic models, and later on, in~\cite{Wang-Zhao-JDDE-2008}, the definition of $\mathcal{R}_0$ was adapted to the study of periodic patchy models. In the recent article~\cite{Garrione-Rebelo-NARWA-2017} the theory in~\cite{Wang-Zhao-JDDE-2008} was used in the study of persistence of the predator in a general periodic predator-prey models.
When persistence is guaranteed, the obtention of conditions that assure the existence of periodic orbits for periodic eco-epidemiological models is an important issue in the deepening of the description of these models since these orbits correspond to situations where possibly there is some equilibrium in the described ecological system, reflected in the fact that the behaviour of the theoretical model is the same over the years. In~\cite{Silva-JMAA-2017} it was proved that there is an endemic periodic orbit for the periodic version of the model considered in~\cite{Niu-Zhang-Teng-AMM-2011} when the infected prey is permanent and some additional conditions are fulfilled, partially giving a positive answer to a conjecture in this last paper.
The models in~\cite{Niu-Zhang-Teng-AMM-2011} and~\cite{Silva-JMAA-2017} assume that there is no predation on uninfected preys. In spite of that, this assumption is not suitable for the description of many eco-epidemiological models. The main purpose of this paper is to present some results on the existence of an endemic periodic orbit for periodic eco-epidemiological systems with disease in the prey that generalize the systems in~\cite{Niu-Zhang-Teng-AMM-2011} and \cite{Silva-JMAA-2017} by including in the model a general function corresponding to the predation of uninfected preys. The proof of our result relies on Mawhin's continuation theorem. Following the approach in~\cite{Silva-JMAA-2017}, we begin by locating the components of possible periodic orbits for the one parameter family of systems that arise in Mawhin's result, allowing us to check that the conditions of that theorem are fulfilled. Although the main steps in our proof correspond to the ones in~\cite{Silva-JMAA-2017}, several additional nontrivial arguments are needed in our case. Additionally, there is also a substantial difference between our approach and the one in~\cite{Niu-Zhang-Teng-AMM-2011,Silva-JMAA-2017}.
In fact, we take as a departure point some prescribed behaviour of the uninfected subsystem, corresponding to the dynamics of preys and predators in the absence of disease: we will assume in this work that we have global asymptotic stability of solutions of some special perturbations of the bidimensional predator-prey system (the system obtained by letting $I=0$ in the first and third equations in~\eqref{eq:principal}). Thus, when applying our results to particular situations, one must verify that the underlying uninfected subsystem satisfies our assumptions. On the other hand, our approach allows us to construct an eco-epidemiological model from a previously studied predator-prey model (the uninfected subsystem) that satisfies our assumptions. This approach has the advantage of highlighting the link between the dynamics of the eco-epidemiological model and the dynamics of the predator-prey model used in its construction.
\section{A general eco-epidemiological model with disease in prey}
As a generalization of the model considered in~\cite{Silva-JMAA-2017}, a periodic version of the general non-autonomous model introduced in~\cite{Niu-Zhang-Teng-AMM-2011}, we consider the following periodic
eco-epidemiological model:
\begin{equation}\label{eq:principal}
\begin{cases}
S'=\Lambda(t)-\mu(t)S-a(t)f(S,P)P-\beta(t)SI\\
I'=\beta(t)SI-\eta(t)PI-c(t)I\\
P'=(r(t)-b(t)P)P+\gamma(t)a(t)f(S,P)P+\theta(t)\eta(t)PI
\end{cases},
\end{equation}
where $S$, $I$ and $P$ correspond, respectively, to the susceptible prey, infected prey and predator, $\Lambda(t)$ is the recruited rate of the prey population, $\mu(t)$ is the natural death rate of the prey population, $a(t)$ predation rate of susceptible prey, $\beta(t)$ is the incidence rate, $\eta(t)$ is the predation rate of infected prey, $c(t)$ is the death rate in the infective class ($c(t) \ge \mu(t)$), $\gamma(t)$ is the
rate converting susceptible prey into predator (biomass transfer), $\theta(t)$ is the rate of converting infected prey into predator, $r(t)$ and $b(t)$ are parameters related the vital dynamics of the predator population that is assumed to follow a logistic law and includes the intra-
specific competition between predators. It is assumed that only susceptible preys $S$ are capable of reproducing, i.e, the infected prey is removed by death (including natural and disease-related death) or by predation before having the possibility of reproducing.
Given a $\omega$-periodic function $f$ we will use throughout the paper the notations $f^\ell=\inf_{t \in (0,\omega]} f(t)$, $f^u=\sup_{t \in (0,\omega]} f(t)$ and $\bar{f}=\frac{1}{\omega} \int_0^\omega f(s) \, ds$.
We will assume the following structural hypothesis concerning the parameter functions and the function $f$ appearing in our model:
\begin{enumerate}[S$1$)]
\item \label{cond-1} The real valued functions $\Lambda$, $\mu$, $\beta$, $\eta$, $c$, $\gamma$, $r$, $\theta$ and $b$ are periodic with period $\omega$, nonnegative and continuous;
\item \label{cond-2} Function $f$ is nonnegative and continuous;
\item \label{cond-3} Function $x \mapsto f(x,z)$ is nondecreasing;
\item \label{cond-4} Function $z \mapsto f(x,z)$ is nonincreasing;
\item \label{cond-5} For all $(x,z)$ we have $\bar\beta \frac{\partial f}{\partial x}(x,z)+\eta \frac{\partial f}{\partial z}(x,z)\ge 0$;
\item \label{cond-6} For any $C_1,C_2>0$, function $x\mapsto f(C_1 x+C_2,x)$ is nondecreasing;
\item \label{cond-7} $\bar\Lambda>0$, $\bar\mu>0$, $\bar r>0$ and $\bar b>0$.
\item \label{cond-8} There is $\alpha\ge 1$ and $K>0$ such that $f(x,0) \le Kx^\alpha$.
\end{enumerate}
To formulate our next assumptions we need to consider two auxiliary equations and one auxiliary system. First, for each $\lambda \in (0,1]$, we need to consider the following equations:
\begin{equation}\label{eq:auxiliary-S(t)}
x'=\lambda(\Lambda(t)-\mu(t)x)
\end{equation}
and
\begin{equation}\label{eq:auxiliary-P(t)}
z'=\lambda(r(t)-b(t)z)z.
\end{equation}
Note that, if we identify $x$ with the susceptible prey population, equation~\ref{eq:auxiliary-S(t)} gives the behavior of the susceptible preys in the absence of infected preys and predator and identifying $z$ with the predator population, equation~\ref{eq:auxiliary-P(t)} gives the behavior of the predator in the absence of preys.
Equations~\eqref{eq:auxiliary-S(t)} and~\eqref{eq:auxiliary-P(t)} have a well known behavior, given in the following lemmas:
\begin{lemma}[Lemma 1 in~\cite{Niu-Zhang-Teng-AMM-2011}]\label{lm1-1dim-1}
For each $\lambda \in (0,1]$ there is a unique $\omega$-periodic solution of equation~\eqref{eq:auxiliary-S(t)}, $x_\lambda^*(t)$, that is globally asymptotically stable in $\ensuremath{\mathds R}^+$.
\end{lemma}
\begin{lemma}[Lemma 2 in~\cite{Niu-Zhang-Teng-AMM-2011}]\label{lm1-1dim-2}
For each $\lambda \in (0,1]$ there is a unique $\omega$-periodic solution of equation~\eqref{eq:auxiliary-P(t)}, $z_\lambda^*(t)$, that is globally asymptotically stable in $\ensuremath{\mathds R}^+$.
\end{lemma}
For each $\lambda \in (0,1]$, we also need to consider the next family of systems, which correspond to behavior of the preys and predators in the absence of infected preys (system~\eqref{eq:principal} with $I = 0$, $S = x$ and $P = z$):
\begin{equation}\label{eq:auxiliary-system-SP}
\begin{cases}
x'=\lambda(\Lambda(t)-\mu(t)x-a(t)f(x,z)z-\varepsilon_1 x)\\
z'=\lambda(\gamma(t)a(t)f(x,z)+r(t)- b(t)z+\varepsilon_2)z
\end{cases}.
\end{equation}
We now make our last structural assumption on system~\eqref{eq:principal}:
\begin{enumerate}[S$1$)]
\setcounter{enumi}{8}
\item\label{cond-9} For each $\lambda \in (0,1]$ and each $\varepsilon_1,\varepsilon_2\ge 0$ sufficiently small, system~\eqref{eq:auxiliary-system-SP} has a unique $\omega$-periodic solution, $(x^*_{\lambda,\varepsilon_1,\varepsilon_2}(t),z^*_{\lambda,\varepsilon_1,\varepsilon_2}(t))$, with $x^*_{\lambda,\varepsilon_1,\varepsilon_2}(t)>0$ and $z^*_{\lambda,\varepsilon_1,\varepsilon_2}(t)>0$, that is globally asymptotically stable in the set $\{(x,z)\in (\ensuremath{\mathds R}_0^+)^2: x\ge 0 \ \wedge \ z>0\}$. We assume that $(\varepsilon_1,\varepsilon_2) \mapsto (x^*_{\lambda,\varepsilon_1,\varepsilon_2}(t), \, z^*_{\lambda,\varepsilon_1,\varepsilon_2}(t))$ is continuous.
\end{enumerate}
Denoting $x^*_{\lambda}=x^*_{\lambda,0,0}$ and $z^*_{\lambda}=z^*_{\lambda,0,0}$, we introduce the numbers
\begin{equation}\label{def-R_0}
\overline{\mathcal R}_0=\frac{\bar \beta \bar \Lambda/\bar \mu}{\bar c+\bar \eta \bar r/\bar b}, \quad \mathcal R_0^\lambda=\frac{\overline{\beta x^*_{\lambda}}}{\overline{c}+\overline{\eta z^*_{\lambda}}} \quad \text{and} \quad \widetilde{\mathcal R}_0 =\inf_{\lambda \in (0,1]} \mathcal{R}_0^\lambda
\end{equation}
\section{Main result}
We now present our main result.
\begin{theorem}\label{the:main}
If $\widetilde{\mathcal R}_0>1$, $\overline{\gamma a}\,\overline{\beta}-\overline{\theta\eta}\overline{a}\le 0$ and
\begin{equation}\label{cond:teo}
\overline{R}_0>1+\dfrac{\overline{\beta}}{\overline{\mu}}\left(\dfrac{\overline{a}\,\overline{r}}{\overline{\eta}\,\overline{r}+\overline{c}\overline{b}}-\dfrac{\overline{\gamma a}}{\overline{\theta \eta}}\right) f\left(\frac{\overline{\eta}\,\overline{r}/\overline{b}+\overline{c}}{\overline{\beta}},\overline{r}/\overline{b}\right)
\end{equation}
then system~\eqref{eq:principal} possesses an endemic periodic orbit of period $\omega$.
\end{theorem}
Our proof relies on an application of Mawhin's continuation theorem. We will proceed in several steps. Firstly, in subsection~\ref{subsection:Proof-subsection1}, we consider a one parameter family of systems and obtain uniform bounds for the components of any periodic solution of these systems. Next, in subsection~\ref{subsection:Proof-subsection2} we make a suitable change of variables in our family of systems to establish the setting where we will apply Mawhin's continuation Theorem. Finally, in subsection~\ref{subsection:Proof-subsection3}, we use Mawhin's continuation Theorem to obtain our result.
\subsection{Uniform Persistence for the periodic orbits of a one parameter family of systems.}\label{subsection:Proof-subsection1}
In this section, to obtain uniform bounds for the components of any periodic solution of the family of systems that we can obtain multiplying the right hand side of~\eqref{eq:principal} by $\lambda \in (0,1]$, we need to consider the auxiliary systems:
\begin{equation}\label{eq:principal-aux}
\begin{cases}
S_\lambda'=\lambda(\Lambda(t)-\mu(t)S_\lambda-a(t)f(S_\lambda,P_\lambda)P_\lambda-\beta(t)S_\lambda I_\lambda)\\
I_\lambda'=\lambda(\beta(t)S_\lambda I_\lambda-\eta(t)P_\lambda I_\lambda-c(t)I_\lambda)\\
P_\lambda'=\lambda(\gamma(t)a(t)f(S_\lambda,P_\lambda)P_\lambda+\theta(t)\eta(t)P_\lambda I_\lambda+ r(t)P_\lambda-b(t)P_\lambda^2)
\end{cases}.
\end{equation}
We will consider separately each of the several components of any periodic orbit.
\begin{lemma}\label{lemma:subsection-persist-1}
Let $x_\lambda^*(t)$ be the unique solution of~\eqref{eq:auxiliary-S(t)}. There is $L_1>0$ such that, for any $\lambda \in (0,1]$ and any periodic solution
$(S_\lambda(t),I_\lambda(t),P_\lambda(t))$ of~\eqref{eq:principal-aux} with initial conditions $S_\lambda(t_0)=S_0>0$, $I_\lambda(t_0)=I_0>0$ and $P_\lambda(t_0)=P_0>0$, we have $S_\lambda(t)+I_\lambda(t)\le x^*_\lambda(t) \le \Lambda^u/\mu^\ell$ and $S_\lambda \ge L_1$, for all $t \in \ensuremath{\mathds R}$.
\end{lemma}
\begin{proof}
Let $(S_\lambda(t),I_\lambda(t),P_\lambda(t))$ be some periodic solution of~\eqref{eq:principal-aux} with initial conditions $S_\lambda(t_0)=S_0>0$, $I_\lambda(t_0)=I_0>0$ and $P_\lambda(t_0)=P_0>0$. Since $c(t) \ge \mu(t)$, we
have, by the first and second equations of~\eqref{eq:principal-aux},
$$(S_\lambda + I_\lambda)' \le \lambda\Lambda(t)-\lambda\mu(t) S_\lambda -\lambda c(t)I_\lambda \le \lambda\Lambda(t)-\lambda\mu(t)(S_\lambda + I_\lambda).$$
Since, by Lemma~\ref{lm1-1dim-1}, equation~\eqref{eq:auxiliary-S(t)} has a unique periodic orbit, $x^*_\lambda(t)$, that is globally
asymptotically stable, we conclude that $S_\lambda(t) + I_\lambda(t)\le x^*_\lambda(t)$ for all $t \in \ensuremath{\mathds R}$. Comparing equation~\eqref{eq:auxiliary-S(t)} with equation $x'=\lambda\Lambda^u-\lambda\mu^\ell x$, we conclude that $x^*_\lambda(t)\le \Lambda^u/\mu^\ell$.
Using conditions~S\ref{cond-3}) and S\ref{cond-4}), by the third equation of~\eqref{eq:principal-aux}, we have
$$P_\lambda'\le\lambda(r(t)+\gamma(t)a(t)f(x_\lambda^*(t),0)+\theta(t)\eta(t)x_\lambda^*(t)-b(t)P_\lambda)P_\lambda\le
(\Theta^u- b^\ell P_\lambda)P_\lambda,$$
where function $\Theta$ is given by $\Theta(t)=r(t)+\gamma(t)a(t)f(x_\lambda^*(t),0)+\theta(t)\eta(t)x_\lambda^*(t)$.
Thus, comparing with equation~\eqref{eq:auxiliary-P(t)} and using Lemma~\ref{lm1-1dim-2}, we get $P_\lambda(t)\le P_\lambda^*(t) \le \Theta^u/b^\ell$. Using
the bound obtained above, since $-\beta(t)S_\lambda(t)\ge-\beta(t)x^*_\lambda(t)$, we have, by conditions~S\ref{cond-3}), S\ref{cond-4}) and S\ref{cond-8}),
\[
\begin{split}
S_\lambda'
& =\lambda\Lambda(t)-\lambda\mu(t)S_\lambda-\lambda a(t)f(S_\lambda,P_\lambda)P_\lambda-\lambda \beta(t)S_\lambda I_\lambda\\
& \ge \lambda\Lambda^\ell-\left( \lambda\mu^u+\lambda a^u \frac{f(S_\lambda,0)}{S_\lambda}\frac{\Theta^u}{b^\ell} +\lambda \beta^u (x^*_\lambda)^u \right)S_\lambda\\
& \ge \lambda\Lambda^\ell-\left( \lambda\mu^u +\lambda a^u K((x^*_\lambda)^u)^{\alpha-1}\Theta^u/b^\ell +\lambda \beta^u (x^*_\lambda)^u \right)S_\lambda
\end{split}
\]
and thus
\[
S_\lambda(t) \ge \dfrac{ \lambda\Lambda^\ell}{\lambda\mu^u +\lambda a^u K((x^*_\lambda)^u)^{\alpha-1} \Theta^u/b^\ell +\lambda \beta^u (x^*_\lambda)^u}=:L_1.
\]
\end{proof}
\begin{lemma}\label{lemma:subsection-persist-2}
Let $z_\lambda^*(t)$ be the unique solution of~\eqref{eq:auxiliary-P(t)}. There is $L_2>0$ such that, for any $\lambda \in (0,1]$ and any periodic solution
$(S_\lambda(t),I_\lambda(t),P_\lambda(t))$ of~\eqref{eq:principal-aux} with initial conditions $S_\lambda(t_0)=S_0>0$, $I_\lambda(t_0)=I_0>0$ and $P_\lambda(t_0)=P_0>0$, we have $r^\ell/b^u\le z^*_\lambda(t)\le P_\lambda(t) \le L_2$, for all $t \in \ensuremath{\mathds R}$.
\end{lemma}
\begin{proof}
Let $\lambda \in (0,1]$ and $(S_\lambda(t),I_\lambda(t),P_\lambda(t))$ be any periodic solution of~\eqref{eq:principal-aux} with initial conditions $S_\lambda(t_0)=S_0>0$, $I_\lambda(t_0)=I_0>0$ and $P_\lambda(t_0)=P_0>0$. We have
$$P_\lambda'=\lambda P_\lambda(\gamma(t)a(t)f(S_\lambda,P_\lambda)+\theta(t)\eta(t) I_\lambda+ r(t)-b(t)P_\lambda)\ge (\lambda r(t)-\lambda b(t)P_\lambda)P_\lambda.$$
Comparing the previous inequality with equation~\eqref{eq:auxiliary-P(t)} and using Lemma~\ref{lm1-1dim-2}, we get $P_\lambda(t)\ge z^*_\lambda(t)$. Moreover, comparing equation~\eqref{eq:auxiliary-P(t)} with equation $z'=\lambda(r^\ell-b^uz)z$ we conclude that $z^*_\lambda(t) \ge r^\ell/b^u$.
Using the computations in proof of the previous lemma, we have $P_\lambda(t)\le L_1$ and we take $L_2=L_1$.
\end{proof}
\begin{lemma}\label{lemma:subsection-persist-3}
Let $\widetilde{\mathcal R}_0>1$. There are $L_3,L_4>0$ such that, for any $\lambda \in (0,1]$ and any periodic solution
$(S_\lambda(t),I_\lambda(t),P_\lambda(t))$ of~\eqref{eq:principal-aux} with initial conditions $S_\lambda(t_0)=S_0>0$, $I_\lambda(t_0)=I_0>0$ and $P_\lambda(t_0)=P_0>0$, we have $L_3 \le I_\lambda(t) \le L_4$, for all $t \in \ensuremath{\mathds R}$.
\end{lemma}
\begin{proof}
We will first prove that there is $\varepsilon_1>0$ such that, for any $\lambda\in (0,1]$, we have
\begin{equation}\label{eq:contradicao-persist-I}
\limsup_{t \to +\infty} I_\lambda(t)\ge \varepsilon_1.
\end{equation}
By contradiction, assume that~\eqref{eq:contradicao-persist-I} does not hold. Then, for any $\varepsilon>0$, there must
be $\lambda>0$ such that $I_\lambda(t)<\varepsilon$ for all $t \in \ensuremath{\mathds R}$. We have
\[
\begin{cases}
S_\lambda'\le \lambda\Lambda(t)-\lambda\mu(t)S_\lambda-\lambda a(t)f(S_\lambda,P_\lambda)P_\lambda\\
P_\lambda'\le \lambda(\gamma(t)a(t)f(S_\lambda,P_\lambda)+ r(t)-b(t)P_\lambda+\lambda\varepsilon\theta^u\eta^u)P_\lambda
\end{cases}
\]
and
\[
\begin{cases}
S_\lambda'\ge \lambda\Lambda(t)-\lambda\mu(t)S_\lambda-\lambda a(t)f(S_\lambda,P_\lambda)P_\lambda-\varepsilon\lambda\beta^uS_\lambda\\
P_\lambda'\ge \lambda(\gamma(t)a(t)f(S_\lambda,P_\lambda)+ r(t)-b(t)P_\lambda)P_\lambda
\end{cases}.
\]
By condition~S\ref{cond-9}), we conclude that
$$x^*_{\lambda,\varepsilon\lambda\beta^u,0}(t)\le S_\lambda(t)\le
x^*_{\lambda,0,\varepsilon\lambda\theta^u\eta^u}(t)$$
and
$$z^*_{\lambda,\varepsilon\lambda\beta^u,0}(t)\le P_\lambda(t)\le
z^*_{\lambda,0,\varepsilon\lambda\theta^u\eta^u}(t).$$
Thus, using condition~S\ref{cond-9}), we have
\begin{equation}
\begin{split}
I_\lambda'
& =\lambda(\beta(t)S_\lambda -\eta(t)P_\lambda-c(t))I_\lambda\\
& \ge(\lambda\beta(t)x^*_{\lambda,\varepsilon\lambda\beta^u,0}(t) -\lambda\eta(t)z^*_{\lambda,0,\varepsilon\lambda\theta^u\eta^u}(t)
-\lambda c(t))I_\lambda\\
& \ge(\lambda\beta(t)x^*_{\lambda}(t) -\lambda\eta(t)z^*_{\lambda}(t)-\lambda c(t)-\phi(\varepsilon))I_\lambda,
\end{split}
\end{equation}
where $\phi$ is a nonegative function such that $\phi(\varepsilon)\to 0$ as $\varepsilon \to 0$ (notice that, by continuity, we can assume that $\phi$ is
independent of $\lambda$ and, by periodicity of the parameter functions, it is independent of $t$).
Integrating in $[0,\omega]$ and using~\eqref{cond-9}, we get
\[
\begin{split}
0 & =\frac{1}{\omega}\left(\ln I_\lambda(\omega)-\ln I_\lambda(0)\right)
=\frac{1}{\omega}\int_0^\omega I_\lambda'(s)/I_\lambda(s)\, ds \\
& \ge
\lambda\left(\overline{\beta x^*_{\lambda}}-\bar c-\overline{\eta z^*_{\lambda}}\right)+\phi(\varepsilon)
= \lambda(\bar c+\overline{\eta z^*_{\lambda}})(\mathcal R_0^\lambda-1)+\phi(\varepsilon)
\end{split}
\]
and since
$$\mathcal R_0^\lambda\ge \inf_{\ell \in (0,1]} \mathcal R_0^\ell=\widetilde{\mathcal R}_0>1,$$
we have a contradiction. We conclude that~\eqref{eq:contradicao-persist-I} holds.
Next we will prove that there is $\varepsilon_2>0$ such that, for any $\lambda\in (0,1]$, we have
\begin{equation}\label{eq:contradicao-persist-I-2}
\liminf_{t \to +\infty} I_\lambda(t)\ge \varepsilon_2.
\end{equation}
Assuming by contradiction that~\eqref{eq:contradicao-persist-I-2} does not hold, we conclude that there is a sequence $(\lambda_n,I_{\lambda_n}(s_n),I_{\lambda_n}(t_n))\subset (0,1]\times\ensuremath{\mathds R}_0^+\times\ensuremath{\mathds R}_0^+$ such that $s_n<t_n$, $t_n-s_n\le \omega$,
$$I_{\lambda_n}(s_n)=1/n, \quad I_{\lambda_n}(t_n)=\varepsilon_2/2 \quad \text{and} \quad I_{\lambda_n}(t)\in (1/n,\varepsilon_2/2), \ \text{for all} \ t \in (s_n,t_n).$$
Since $\lambda_n\le 1$, by Lemma~\ref{lemma:subsection-persist-1} we have
$$I'_{\lambda_n}=({\lambda_n}\beta(t)S_{\lambda_n} -{\lambda_n}\eta(t)P_{\lambda_n}-{\lambda_n}c(t))I_{\lambda_n}\le \beta^u \Lambda^u I_{\lambda_n}/\mu^\ell$$
and thus
$$\ln(\varepsilon_2 n/2)=\ln(I_{\lambda_n}(t_n)/I_{\lambda_n}(s_n))
=\int_{s_n}^{t_n} I'_{\lambda_n}(s)/I_{\lambda_n}(s) \, ds \le \beta^u \Lambda^u \omega/\mu^\ell,$$
which is a contradiction since the sequence $(\ln(\varepsilon_2 n/2))_{n\in \ensuremath{\mathds N}}$ goes to $+\infty$ as $n \to +\infty$, and thus is not bounded.
We conclude that there is $\varepsilon_2>0$ such that~\eqref{eq:contradicao-persist-I-2} holds. Letting $L_3=\varepsilon_2$, we obtain $I_\lambda(t)\ge L_3$ for all $\lambda \in (0,1]$.
Since $I_\lambda(t)\le S_\lambda(t)+I_\lambda(t)$, by Lemma~\ref{lemma:subsection-persist-1}, we can take $L_4=L_2$ and the result is established.
\end{proof}
\subsection{Setting where Mawhin's continuation theorem will be applied.}\label{subsection:Proof-subsection2}
To apply Mawhin's continuation theorem to our model we make the change of variables: $S(t)=e^{u_1(t)}$, $I(t)=e^{u_2(t)}$ and $P(t)=e^{u_3(t)}$. With this change of variables, system~\eqref{eq:principal} becomes
\begin{equation}\label{eq:principal-aux-2}
\begin{cases}
u_1'=\Lambda(t)e^{-u_1}-a(t)f(e^{u_1},e^{u_3})e^{u_3-u_1}-\beta(t)e^{u_2}-\mu(t)\\
u_2'=\beta(t)e^{u_1}-\eta(t)e^{u_3}-c(t)\\
u_3'=\gamma(t)a(t)f(e^{u_1},e^{u_3})+\theta(t)\eta(t)e^{u_2}-b(t)e^{u_3}+r(t)
\end{cases}.
\end{equation}
Note that, if $(u_1^*(t),u_2^*(t),u_3^*(t))$ is an $\omega$-periodic solution of the system~\eqref{eq:principal-aux-2} then $(e^{u_1(t)},e^{u_2(t)},e^{u_3(t)})$ is an $\omega$-periodic solution of system~\eqref{eq:principal}.
To define the operators in Mawhin's theorem (see appendix~\ref{appendix:MCT}), we need to consider the Banach spaces $(X,\|\cdot\|)$ and $(Z,\|\cdot\|)$ where $X$ and $Z$ are the space of $\omega$-periodic continuous functions $u:\ensuremath{\mathds R} \to \ensuremath{\mathds R}^3$:
$$X=Z=\{u=(u_1,u_2,u_3) \in C(\mathbb{R},\mathbb{R}^3): u(t)=u(t+\omega)\}$$
and
$$\|u\|=\max_{t \in [0,\omega]}|u_1(t)|+\max_{t \in [0,\omega]}|u_2(t)|+\max_{t \in [0,\omega]}|u_3(t)|.$$
Next, we consider the linear map $\mathcal{L}: X \cap C^1(\mathbb{R},\mathbb{R}^3) \rightarrow Z$ given by \begin{equation}\label{eq:linear-map}
\mathcal{L}u(t)=\frac{du(t)}{dt}
\end{equation}
and the map $\mathcal{N}: X \rightarrow Z$ defined by
\begin{equation}\label{eq:linear-map-L-comp}
\mathcal{N}u(t)=\left[\begin{array}{l}
\Lambda(t)e^{-u_1(t)}-a(t)f(e^{u_1(t)},e^{u_3(t)})e^{u_3(t)-u_1(t)}-\beta(t)e^{u_2(t)}-\mu(t)\\
\beta(t)e^{u_1(t)}-\eta(t)e^{u_3(t)}-c(t)\\
\gamma(t)a(t)f(e^{u_1(t)},e^{u_3(t)})+\theta(t)\eta(t)e^{u_2(t)}-b(t)e^{u_3(t)}+r(t)
\end{array}\right].
\end{equation}
In the following lemma we show that the linear map in~\eqref{eq:linear-map} is a Fredholm mapping of index zero
\begin{lemma}\label{lemma:Fredholm}
The linear map $\mathcal{L}$ in~\eqref{eq:linear-map} is a Fredholm mapping of index zero.
\end{lemma}
\begin{proof}
We have
\[
\begin{split}
\ker \mathcal{L}
& =\left\{ (u_1,u_2,u_3) \in X \cap C^1(\mathbb{R},\mathbb{R}^3): \frac{du_i(t)}{dt}=0, \ \ i=1,2,3 \right\}\\
& = \left\{ (u_1,u_2,u_3) \in X \cap C^1(\mathbb{R},\mathbb{R}^3): u_i \ \ \text{is constant}, \ \ i=1,2,3 \right\}
\end{split}
\]
and thus $\ker \mathcal{L}$ can be identified with $\ensuremath{\mathds R}^3$. Therefore $\dim \ker \mathcal{L}=3$. On the other hand
\[
\begin{split}
\imagem \mathcal{L}
& =\left\{ (z_1,z_2,z_3) \in Z: \ \exists \ u \in X \cap C^1(\mathbb{R},\mathbb{R}^3): \frac{du_i(t)}{dt}=z_i(t), \ i=1,2,3 \right\}\\
& =\left\{ (z_1,z_2,z_3) \in Z: \int_0^\omega z_i(s)\, ds=0, \ i=1,2,3\right\}.
\end{split}
\]
and any $z \in Z$ can be written as $z=\tilde{z}+\alpha$, where $\alpha=(\alpha_1,\alpha_2,\alpha_3) \in \ensuremath{\mathds R}^3$ and $\tilde{z} \in \imagem \mathcal L$. Thus the complementary space of $\imagem \mathcal{L}$ consists of the constant functions. Thus, the complementary space has dimension 3 and therefore $\codim \imagem \mathcal{L}=3$.
Given any sequence $(z_n)$ in $\imagem \mathcal L$ such that
$$z_n=((z_1)_n,(z_2)_n,(z_3)_n) \to z=(z_1,z_2,z_3),$$
we have, for $i=1,2,3$ (note that $z \in Z$ since $Z$ is a Banach space and thus it is integrable in $[0,\omega]$ since it is continuous in that interval),
\[
\int_0^\omega z_i(s)\, ds=\int_0^\omega \lim_{n \to +\infty} (z_i)_n(s)\, ds=\lim_{n \to +\infty} \int_0^\omega (z_i)_n(s)\, ds=0.
\]
Thus, $z \in \imagem \mathcal L$ and we conclude that $\imagem \mathcal L$ is closed in $Z$. Thus $\mathcal L$ is a Fredholm mapping of index zero.
\end{proof}
Consider the projectors $P:X \rightarrow X$ and $Q:Z \rightarrow Z$ given by
$$Pu(t)=\frac{1}{\omega}\int_{0}^{\omega}u(s)ds \quad \text{ and } \quad Qz(t)=\frac{1}{\omega}\int_{0}^{\omega}z(s)ds.$$
Note that $\imagem P = \ker \mathcal{L}$ and that $\ker Q= \imagem (I-Q)= \imagem \mathcal{L}$.
Consider the generalized inverse of $\mathcal{L}$, $\mathcal K:\imagem \mathcal{L} \rightarrow D \cap \ker P$, given by $$\mathcal Kz(t)=\int_{0}^{t}z(s)ds-\frac{1}{\omega}\int_{0}^{\omega}\int_{0}^{r}z(s)\, ds\, dr$$
the operator $Q\mathcal{N}:X \rightarrow Z$ given by
\[
Q\mathcal{N}u(t)=\left[\begin{array}{l}
\frac{1}{\omega}\displaystyle \int_{0}^{\omega}\Lambda(s)e^{-u_1(s)}-a(s)f(e^{u_1(s)},e^{u_3(s)})e^{u_3(s)}-\beta(s)e^{u_2(s)}\, ds-\overline{\mu}\\[4mm]
\frac{1}{\omega}\displaystyle \int_{0}^{\omega}\beta(s)e^{u_1(s)}-\eta(s)e^{u_3(s)}\, ds-\overline{c}\\[4mm]
\frac{1}{\omega}\displaystyle \int_{0}^{\omega}\gamma(s)a(s)f(e^{u_1(s)},e^{u_3(s)})e^{u_3(s)}+\theta(s)\eta(s)e^{u_2(s)}- b(s)e^{u_3(s)}\, ds+\overline{r}
\end{array}\right]
\]
and the mapping $\mathcal K(I-Q)\mathcal{N}:X \rightarrow D \cap \ker P$ given by
\[
\mathcal K(I-Q)\mathcal{N}u(t)=B_1(t)-B_2(t)-B_3(t),
\]
where
\[
\begin{split}
B_1(t)=\left[
\begin{array}{l}
\displaystyle \int_{0}^{t}\Lambda(s)e^{-u_1(s)}-a(s)f(e^{u_1(s)},e^{u_3(s)})e^{u_3(s)}-\beta(s)e^{u_2(s)}-\mu(s)\, ds\\[4mm]
\displaystyle \int_{0}^{t}\beta(s)e^{u_1(s)}-\eta(s)e^{u_3(s)}-c(s)\, ds\\[4mm]
\displaystyle \int_{0}^{t}\gamma(s)a(s)f(e^{u_1(s)},e^{u_3(s)})e^{u_3(s)}+\theta(s)\eta(s)e^{u_2(s)}- b(s)e^{u_3(s)}dt+r(s)\, ds
\end{array}\right],
\end{split}
\]
\small{
\[
B_2(t)=\left[\begin{array}{l}
\frac{1}{\omega}\displaystyle \int_{0}^\omega \displaystyle \int_{0}^r\Lambda(s)e^{-u_1(s)}-a(s)f(e^{u_1(s)},e^{u_3(s)})e^{u_3(s)}-\beta(s)e^{u_2(s)}-\mu(s)\, ds\, dr\\[4mm]
\frac{1}{\omega}\displaystyle \int_{0}^\omega \displaystyle \int_{0}^r\beta(s)e^{u_1(s)}-\eta(s)e^{u_3(s)}-c(s)\, ds\, dr\\[4mm]
\frac{1}{\omega}\displaystyle \int_{0}^\omega \displaystyle \int_{0}^r\gamma(s)a(s)f(e^{u_1(s)},e^{u_3(s)})e^{u_3(s)}+\theta(s)\eta(s)e^{u_2(s)}- b(s)e^{u_3(s)}+r(s)\, ds\, dr
\end{array}\right],
\]
}
and
\[
B_3(t)=\left(\frac{t}{\omega}-\frac{1}{2}\right)
\left[\begin{array}{l}
\displaystyle \int_{0}^\omega\Lambda(s)e^{-u_1(s)}-a(s)f(e^{u_1(s)},e^{u_3(s)})e^{u_3(s)}-\beta(s)e^{u_2(s)}-\mu(s)\, ds\\[4mm]
\displaystyle \int_{0}^\omega\beta(s)e^{u_1(s)}-\eta(s)e^{u_3(s)}-c(s)\, ds\\[4mm]
\displaystyle \int_{0}^\omega\gamma(s)a(s)f(e^{u_1(s)},e^{u_3(s)})e^{u_3(s)}+\theta(s)\eta(s)e^{u_2(s)}- b(s)e^{u_3(s)}+r(s)\, ds
\end{array}\right].
\]
The next lemma shows that $\mathcal{N}$ is $\mathcal L$-compact in the closure of any open bounded subset of its domain.
\begin{lemma}
The map $\mathcal{N}$ is $\mathcal L$-compact in the closure of any open bounded set $U \subseteq X$.
\end{lemma}
\begin{proof}
Let $U \subseteq X$ be an open bounded set and $\overline{U}$ its closure in $X$. Then, there is $M>0$ such that, for any $u=(u_1,u_2,u_3) \in \overline{U}$, we have that $|u_i(t)| \leqslant M$, $i=1,2,3$. Letting $Q\mathcal{N}u=((Q\mathcal{N})_1u,(Q\mathcal{N})_2u,(Q\mathcal{N})_3u)$, we have
\[
\left|(Q\mathcal{N})_1u(t)\right|\le \e^M \left(\bar\Lambda+\bar a f(e^M,0)+\bar\beta\right)+\bar\mu,
\]
\[
\left|(Q\mathcal{N})_2u(t)\right|\le e^M(\bar\beta+\bar\eta)+\overline{c}
\]
\[
\left|(Q\mathcal{N})_3u(t)\right|\le \e^M\left(\overline{\gamma a}f(e^M,0)+\overline{\theta\eta}+ \bar b\right)+\overline{r}
\]
and we conclude that $Q\mathcal{N}(\overline{U})$ is bounded.
Let now $$\mathcal K(I-Q)\mathcal{N}u =\left((\mathcal K(I-Q)\mathcal{N})_1u,(\mathcal K(I-Q)\mathcal{N})_2u,(\mathcal K(I-Q)\mathcal{N})_3u\right).$$
Let $B \subset X$ be a bounded set. Note that the boundedness of $B$ implies that there is $M$ such that $|u_i|<M$, for all $i=1,2,3$, and all $u=(u_1,u_2,u_3) \in B$. It is immediate that $\{\mathcal K(I-Q)\mathcal{N}u:u \in B\}$ is pointwise bounded. Given $u=(u_1,u_2,u_3)_{n \in\ensuremath{\mathds N}} \in B$ we have
\begin{equation}\label{eq:equicont-1}
\begin{split}
& (\mathcal K(I-Q)\mathcal{N})_1u(t)-(\mathcal K(I-Q)\mathcal{N})_1u(v))\\
= &\displaystyle \int_v^t \Lambda(s)e^{-u_1(s)}-a(s)f(e^{u_1(s)},e^{u_2(s)})e^{u_2(s)}-\beta(s)e^{u_2(s)}-\mu(s)\, ds\\
&-\frac{t-v}{\omega}\displaystyle \int_{0}^\omega\Lambda(s)e^{-u_1(s)}-a(s)f(e^{u_1(s)},e^{u_2(s)})e^{u_2(s)}-\beta(s)e^{u_2(s)}-\mu(s)\, ds\\
\le & 2(t-v)\left[e^M(\Lambda^u+a^u f(e^M,0)+\beta^u e^M)+\mu^M\right],
\end{split}
\end{equation}
and similarly
\begin{equation}\label{eq:equicont-2}
\begin{split}
& (\mathcal K(I-Q)\mathcal{N})_2u(t)-(\mathcal K(I-Q)\mathcal{N})_2u(v)
\le2(t-v)\left[e^M(\beta^u+\eta^u)+c^u\right]
\end{split}
\end{equation}
and
\begin{equation}\label{eq:equicont-3}
\begin{split}
& (\mathcal K(I-Q)\mathcal{N})_3u(t)-(\mathcal K(I-Q)\mathcal{N})_3u(v))\\
& \le
2(t-v)\left[(\gamma^ua^u f(e^M,0)+\theta^u\eta^u+ b^u)e^M+r^u\right].
\end{split}
\end{equation}
By~\eqref{eq:equicont-1},~\eqref{eq:equicont-2} and~\eqref{eq:equicont-3}, we conclude that $\{\mathcal K(I-Q)\mathcal{N}u:u \in B\}$ is equicontinuous.
Therefore, by Ascoli-Arzela's theorem, $\mathcal K(I-Q)\mathcal{N}(B)$ is relatively compact. Thus the operator $\mathcal K(I-Q)\mathcal{N}$ is compact.
We conclude that $\mathcal{N}$ is $\mathcal L$-compact in the closure of any bounded set contained in $X$.
\end{proof}
\subsection{Application of Mawhin's continuation theorem.}\label{subsection:Proof-subsection3}
In this section we will construct the set where, applying Mahwin's continuation theorem, we will find the periodic orbit in the statement of our result.
Consider the system of algebraic equations:
\begin{equation}\label{eq:principal-aux-3}
\begin{cases}
\overline{\Lambda}e^{-u_1}-\overline{a}f(e^{u_1},e^{u_3})e^{u_3-u_1}-\overline{\beta}e^{u_2}-\overline{\mu}=0\\
\overline{\beta}e^{u_1}-\overline{\eta}e^{u_3}-\overline{c}=0\\
\overline{\gamma a}f(e^{u_1},e^{u_3})+ \overline{\theta\eta}e^{u_2}-\overline{b}e^{u_3}+\overline{r}=0
\end{cases}.
\end{equation}
By the second and third equations we get
\[
\e^{u_1}=\frac{\overline{\eta}\e^{u_3}+\overline{c}}{\overline{\beta}}
\quad \text{ and } \quad
\e^{u_2}=-\frac{\overline{\gamma a}}{\overline{\theta\eta}}f\left(\frac{\overline{\eta}\e^{u_3}+\overline{c}}{\overline{\beta}},e^{u_3}\right)
+\frac{\overline{b}}{\overline{\theta\eta}}e^{u_3}-\frac{\overline{r}}{\overline{\theta\eta}}.
\]
Therefore, using the first equation, we get
$$G_1(\e^{u_3})-G_2(\e^{u_3})f\left(\frac{\overline{\eta}\e^{u_3}+\overline{c}}{\overline{\beta}},e^{u_3}\right)
-G_3(\e^{u_3})=0,$$
where
$$
G_1(x)=\dfrac{\overline{\Lambda}\,\overline{\beta}}{\overline{\eta}x+\overline{c}},
\quad G_2(x)=\dfrac{\overline{a}\overline{\beta}x}{\overline{\eta}x+\overline{c}}-\dfrac{\overline{\beta}\overline{\gamma a}}{\overline{\theta \eta}} \quad \text{ and } \quad G_3(x)=\dfrac{\overline{\beta}\, \overline{b}}{\overline{\theta \eta}}x-\dfrac{\overline{\beta}\overline{r}}{\overline{\theta \eta}}+\overline{\mu}
$$
Consider the function $G:[\overline{r}/\overline{b},+\infty[\to \ensuremath{\mathds R}$ (notice that, by the third equation in~\eqref{eq:principal-aux-3},
we have $e^{u_3}\ge\overline{r}/\overline{b}$), given by
$$G(x)=G_1(x)-G_2(x)f\left(\frac{\overline{\eta}x+\overline{c}}{\overline{\beta}},x\right)
-G_3(x)$$
and observe that function $G_1$ is decreasing and functions $G_2$ and $G_3$ are increasing. Thus, since the function $[\overline{r}/\overline{b},+\infty[ \ni x \mapsto f((\overline{\eta}x+\overline{c})/\overline{\beta},x)$ is nondecreasing, we conclude that
$$[\overline{r}/\overline{b},+\infty[ \ni x \mapsto -G_2(x)f((\overline{\eta}x+\overline{c})/\overline{\beta},x)$$
is a decreasing function. It is immediate the function $[\overline{r}/\overline{b},+\infty[ \ni x \mapsto -G_3(x)$ is decreasing. Consequently $G$ is a decreasing function and equation~\eqref{eq:principal-aux-3} has, at most, one solution.
It is easy to verify that
$$\lim_{x \to +\infty} G(x)=-\infty$$
and, by the hypothesis in our theorem
\[
\begin{split}
G(\overline{r}/\overline{b})
& =\dfrac{\overline{\Lambda}\,\overline{\beta}}{\overline{\eta}\,\overline{r}/\overline{b}+\overline{c}}
-\left(\dfrac{\overline{a}\overline{\beta}\overline{r}/\overline{b}}{\overline{\eta}\,\overline{r}/\overline{b}+\overline{c}}-\dfrac{\overline{\beta}\overline{\gamma a}}{\overline{\theta \eta}}\right)f\left(\frac{\overline{\eta}\,\overline{r}/\overline{b}+\overline{c}}{\overline{\beta}},\overline{r}/\overline{b}\right)
-\overline{\mu}\\
& = \overline{\mu}\left(\overline{\mathcal R}_0-1-\dfrac{\overline{\beta}}{\overline{\mu}}\left(\dfrac{\overline{a}\,\overline{r}}{\overline{\eta}\,\overline{r}+\overline{c}\overline{b}} -\dfrac{\overline{\gamma a}}{\overline{\theta \eta}}\right) f\left(\frac{\overline{\eta}\,\overline{r}/\overline{b}+\overline{c}}{\overline{\beta}},\overline{r}/\overline{b}\right)
\right)>0.
\end{split}
\]
Thus we conclude that there is a unique solution of equation~\eqref{eq:principal-aux-3}. Denote this solution by $p^*(t)=(p_1^*,p_2^*,p_3^*)$.
By Lemmas~\ref{lemma:subsection-persist-1}, \ref{lemma:subsection-persist-2} and \ref{lemma:subsection-persist-3}, there is a constant $M_0>0$ such that $\|u_\lambda(t)\|<M_0$, for any $t \in [0,\omega]$ and any periodic solution $u_\lambda(t)$ of~\eqref{eq:principal-aux}. Let
\begin{equation}\label{eq:set-Mawhin}
U=\{(u_1,u_2,u_3) \in X: \|(u_1,u_2,u_3)\|<M_0+\|p^*\| \}.
\end{equation}
Conditions M1. and M2. in Mawhin's continuation theorem (see appendix~\ref{appendix:MCT}) are fulfilled in the set $U$ defined in~\eqref{eq:set-Mawhin}.
Using the notation $v=(\e^{p^*_1},\e^{p^*_3})$, the Jacobian matrix of the vector field corresponding to~\eqref{eq:principal-aux-3} computed in $(p_1^*,p_2^*,p_3^*)$ is
\[
J=
\left[
\begin{array}{ccc}
-\overline{a}\frac{\partial f}{\partial S}(v)\e^{p_3^*}-\overline{\beta}\e^{p_2^*}-\overline{\mu} & -\overline{\beta}\e^{p_2^*} &
-\overline{a}\frac{\partial f}{\partial P}(v)\e^{2p_3^*-p_1^*}-\overline{a}f(v)\e^{p_3^*-p_1^*}\\
\overline{\beta}\e^{p_1^*} & 0 & -\overline{\eta}\e^{p_3^*}\\
\overline{\gamma a}\frac{\partial f}{\partial S}(v)\e^{p_1^*} &
\overline{\theta \eta}\e^{p_2^*} & \overline{\gamma a}\frac{\partial f}{\partial P}(v)\e^{p_3^*}-\overline{b}\e^{p_3^*}
\end{array}
\right].
\]
Since we are assuming that $\overline{\beta}\overline{\gamma a}-\overline{\theta\eta}\overline{a}\le 0$, we have
\[
\begin{split}
& \det J(p_1^*,p_2^*,p_3^*)\\
=& -\overline{\beta}\e^{p_1^*} \left(-\overline{\beta}\e^{p_2^*}\left(\overline{\gamma a}\frac{\partial f}{\partial P}(v)\e^{p_3^*}-\overline{b}\e^{p_3^*}\right)+\left(\overline{a}\frac{\partial f}{\partial P}(v)\e^{2p_3^*-p_1^*}+\overline{a}f(v)\e^{p_3^*-p_1^*}\right) \overline{\theta \eta}\e^{p_2^*} \right)\\
&+\overline{\eta}\e^{p_3^*}\left(
\left(-\overline{a}\frac{\partial f}{\partial S}(v)\e^{p_3^*}-\overline{\beta}\e^{p_2^*}-\overline{\mu}\right)\overline{\theta \eta}\e^{p_2^*}
+\overline{\beta}\e^{p_2^*}\overline{\gamma a}\frac{\partial f}{\partial S}(v)\e^{p_1^*}\right)\\
=& -\overline{\beta}\e^{p_2^*+p_3^*} \left(-\overline{\beta}\overline{\gamma a}\frac{\partial f}{\partial P}(v)\e^{p_1^*}+\overline{\beta}\overline{b}\e^{p_1^*}
+\overline{\theta \eta}\overline{a}\frac{\partial f}{\partial P}(v)\e^{p_3^*}+\overline{\theta \eta}\overline{a}f(v)\right) \\
&+\overline{\eta}\e^{p_3^*+p_2^*}\left(
-\overline{a}\frac{\partial f}{\partial S}(v)\e^{p_3^*}\overline{\theta \eta}-\overline{\beta}\e^{p_2^*}\overline{\theta \eta}-\overline{\mu}\overline{\theta \eta}
+\overline{\beta}\overline{\gamma a}\frac{\partial f}{\partial S}(v)\e^{p_1^*}\right)\\
=& (\overline{\beta}\overline{\gamma a}-\overline{\theta\eta}\overline{a})\left(\overline{\beta}\frac{\partial f}{\partial P}(v)+\overline{\eta}\frac{\partial f}{\partial S}(v)\right) -\overline{\beta}\e^{p_2^*+p_3^*} \left(\overline{\beta}\overline{b}\e^{p_1^*}
+\overline{\theta \eta}\overline{a}f(v)\right) \\
&-\overline{\eta}\e^{p_3^*+p_2^*}\left(
\overline{\beta}\e^{p_2^*}\overline{\theta \eta}+\overline{\mu}\overline{\theta \eta}\right)<0.\\
\end{split}
\]
Let $\mathcal{I}: \text{Im} Q \rightarrow \text{ker} \mathcal{L}$ be an isomorphism. Thus
\begin{equation}\label{eq:cond-3-Mawhin}
\deg(\mathcal I\mathcal Q\mathcal N, U\cap\ker\mathcal L,0) = \det J(p_1^*,p_2^*,p_3^*) \ne 0
\end{equation}
and condition M3) in Mawhin's continuation theorem (see appendix~\ref{appendix:MCT}) holds.
Taking into account Lemma~\ref{lemma:subsection-persist-3}, the proof of Theorem~\ref{the:main} is completed.
\section{Examples.}\label{section:examples}
In this section we present some examples to illustrate our main result.
\subsection{A model with no predation on susceptible preys.}\label{subsection:example-no-predation-susceptible-prey}
Letting $a(t)\equiv 0$ or $f \equiv 0$ in system~\eqref{eq:principal}, and still assuming that the real valued functions $\Lambda$, $\mu$, $\beta$, $\eta$, $c$, $\gamma$, $r$, $\theta$ and $b$ are periodic with period $\omega$, nonnegative, continuous and also that $\bar\Lambda>0$, $\bar\mu>0$, $\bar r>0$ and $\bar b>0$, we obtain the periodic model considered in~\cite{Silva-JMAA-2017}:
\begin{equation}\label{eq:principal-example-no-predation-susceptible-prey}
\begin{cases}
S'=\Lambda(t)-\mu(t)S-\beta(t)SI\\
I'=\beta(t)SI-\eta(t)PI-c(t)I\\
P'=(r(t)-b(t)P)P+\theta(t)\eta(t)PI
\end{cases}
\end{equation}
Note that conditions S\ref{cond-1}) and S\ref{cond-7}) are assumed and conditions S\ref{cond-2}) to S\ref{cond-6}) and S\ref{cond-8}) are trivially satisfied since $f\equiv 0$.
Also note that system~\eqref{eq:auxiliary-system-SP} becomes in this context
\begin{equation}\label{eq:auxiliary-system-example-no-predation-susceptible-prey}
\begin{cases}
x'=\lambda(\Lambda(t)-\mu(t)x-\varepsilon_1 x)\\
z'=\lambda(r(t)- b(t)z+\varepsilon_2)z
\end{cases}
\end{equation}
and, by Lemmas 1 to 4 in~\cite{Niu-Zhang-Teng-AMM-2011} we conclude that condition~S\ref{cond-9}) holds in this setting.
Note also that condition~\eqref{cond:teo} becomes $\overline{\mathcal R}_0>1$ and condition $\overline{\gamma a} \,\overline{\beta}-\overline{\theta\eta}\overline{a}\le 0$ is trivially satisfied since we can take $\gamma=0$. We obtain the following corollary that recovers the result in~\cite{Silva-JMAA-2017}:
\begin{corollary}
If $\widetilde{\mathcal R}_0>1$ and $\overline{R}_0>1$ hold, then system~\eqref{eq:principal-example-no-predation-susceptible-prey} possesses an endemic periodic orbit of period $\omega$.
\end{corollary}
\subsection{A model with Holling-type I functional response.}\label{subsection:example-Holling-type I}
Letting $f(S,P) = S$ (Holling-type I functional response) in system~\eqref{eq:principal}, and assuming that the real valued functions $\beta$, $\eta$ e $c$ are periodic with period $\omega$ and that the real valued functions $\Lambda$, $\mu$, $\gamma$, $r$, $\theta$, $b$ and $a$ are constant and positive, we obtain the periodic model:
\begin{equation}\label{eq:principal-Holling-type I}
\begin{cases}
S'=\Lambda-\mu S-aSP+\beta(t) SI\\
I'=\beta(t) SI-\eta(t) PI-c(t)I\\
P'=(r - bP)P+\gamma aSP+\theta \eta(t) PI
\end{cases}
\end{equation}
Since $f(S,P)=S$, conditions S\ref{cond-2}) to S\ref{cond-6}) are trivially satisfied. Conditions S\ref{cond-1}) and S\ref{cond-7}) are assumed and S\ref{cond-8}) is satisfied with $K=\alpha=1$.
Notice additionally that system~\eqref{eq:auxiliary-system-SP} becomes in our context
\begin{equation}\label{eq:auxiliary-system-Holling-type I}
\begin{cases}
x'=\lambda(\Lambda-\mu x-a xz-\varepsilon_1 x)\\
z'=\lambda(r-bz +\gamma a x +\varepsilon_2)z
\end{cases}.
\end{equation}
System~\eqref{eq:auxiliary-system-Holling-type I} has two equilibriums: $E_1=(\Lambda/(\mu+\varepsilon_1),0)$ and
$$E_2=\left(\dfrac{\sqrt{V^2+4\Lambda\gamma a^2/b}-V}{2\gamma a^2/b}, \, \dfrac{\sqrt{V^2+4\Lambda\gamma a^2/b}-V}{2\gamma a^2/b}+r+\varepsilon_2\right),$$
where $V=\mu+\varepsilon_1+a(r+\varepsilon_2)/b$. It is easy to check that $E_2$ is locally attractive and that $E_1$ is a saddle point whose stable manifold coincides with the x-axis. If $0<\alpha<(r+\varepsilon_2)/b$ then, in the line $z=\alpha$ the flow points upward. Additionally, if $\Lambda<\mu(\mu+\varepsilon_1)/a$, in the line $x=\mu/a$ the flow points to the left and the $x$-coordinate of $E_1$ is less than $\mu/a$. Thus the region $R=\{(x,z) \in \ensuremath{\mathds R}^2: 0\le x \le \mu/a \ \wedge \ z \ge \alpha \}$ is positively invariant. Since the divergence of the vector field is given by $-\mu-\varepsilon_1+\varepsilon_2-(a+2b)z+\gamma ax$, we conclude that it is null on the line $z=\frac{-\mu-\varepsilon_1+\varepsilon_2}{a+2b}+\frac{\gamma a}{a+2b}x$. Thus the divergence of the vector field doesn't change sign on the region $R$ and this forbids the existence of a peridic orbit on $R$. There is also no periodic orbit on $(\ensuremath{\mathds R}_0^+)^2 \setminus R$ since there is no additional equilibrium in $(\ensuremath{\mathds R}_0^+)^2$. Since $E_2$ is locally asymptotically stable, there is no homoclinic orbit conecting $E_2$ to itself. Therefore, the $\omega$-limit of any orbit in $(\ensuremath{\mathds R}^2)^+$ must be the equilibrium point $E_2$ and the global asymptotic stability of~\eqref{eq:auxiliary-system-Holling-type I} for sufficiently small $\varepsilon_1,\varepsilon_2>0$ follows. We conclude that condition~S\ref{cond-9}) holds.
Notice that condition~\eqref{cond:teo} becomes $\overline{\mathcal R}_0>1+\dfrac{a(r \theta\bar{\eta}-\bar{\gamma}\bar{\eta} r - \bar{\gamma}\bar{c}b)}{\mu\bar{\eta}\theta b}$ and condition ${\gamma}\overline{\beta}-\theta\overline{\eta} \le 0$ is trivially satisfied. We obtain the corollary that generalizes the result in~\cite{Silva-JMAA-2017}:
\begin{corollary}\label{cor}
If $\Lambda<\mu^2/a$, ${\gamma}\overline{\beta}-\theta\overline{\eta} \le 0$, $\widetilde{\mathcal R}_0>1$ and
$$\overline{R}_0>1+\dfrac{a(r \theta\bar{\eta}-\bar{\gamma}\bar{\eta} r - \bar{\gamma}\bar{c}b)}{\mu\bar{\eta}\theta b}$$
then system~\eqref{eq:principal-Holling-type I} possesses an endemic periodic orbit of period $\omega$.
\end{corollary}
To do some simulation, we consider the following particular set of parameters: $\Lambda=0.1$; $\mu=0.6$; $\beta(t)=20(1+0.9\cos(2\pi t))$; $\eta(t)=0.7(1+0.7\cos(\pi+2\pi t))$; $c(t)=0.1$; $r=0.2$; $b=0.3$; $\theta=10$, $\gamma(t)=0.1$ and $a=3$. We obtain the model:
\begin{equation}\label{eq:Periodic-Holling-type I}
\begin{cases}
S'=0.1-0.6S-20(1+0.9\cos(2\pi t))SI-3SP\\
I'=20(1+0.9\cos(2\pi t))SI-0.7(1+0.7\cos(\pi+2\pi t))PI-0.1I\\
P'=(0.2-0.3P)P+7(1+0.7\cos(\pi+2\pi t))PI+0.3SP
\end{cases},
\end{equation}
Notice that, for our model, $\Lambda=0.1>0.012=\mu^2/a$, ${\gamma}\overline{\beta}-\theta\overline{\eta} =-5<0$, $\overline{R}_0\approx 5.88 >1 + 3.3$ and $\tilde{R}_0\approx 24.8 >1$, and thus the conditions in Corollary~\ref{cor} are fulfilled. Considering the initial condition $(S_0,I_0,P_0)=(0.03567,0.02047,0.88021)$ we obtain the periodic orbit in figure~\ref{fig_periodic2-Holling-type I}.
\begin{figure}[h]
\begin{minipage}[b]{.32\linewidth}
\includegraphics[width=\linewidth]{Figura1.eps}
\end{minipage}
\begin{minipage}[b]{.32\linewidth}
\includegraphics[width=\linewidth]{FiguraSI.eps}
\end{minipage}
\begin{minipage}[b]{.32\linewidth}
\includegraphics[width=\linewidth]{FiguraP.eps}
\end{minipage}
\caption{Periodic orbit for model~\eqref{eq:Periodic-Holling-type I}}
\label{fig_periodic2-Holling-type I}
\end{figure}
Although our theoretical result doesn't imply the attractivity of the periodic solution, the simulations carried out suggest that this is the case.
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Two dead in Australia floods as fresh warning issued
Two bodies were discovered in Australia's flood-ravaged northeast on Tuesday as continuing heavy rain prompted authorities to warn of further flash flooding in the hours ahead.
Prime Minister Scott Morrison toured the region where "once-in-a-century" floods have inundated thousands of homes and closed airports and schools, leaving cars and houses submerged.
As stormwaters receded in some areas, he warned of a difficult recovery ahead.
"It's just going to be tough, going to be tough for a while," Morrison said, expressing sympathy for young families who had lost their homes.
"The real work is to make sure that they can get through the clean-up and rebuild their lives."
In hard-hit Townsville, more than 650 people have been evacuated from their homes and about 11,000 houses remained without power on Tuesday. Around 9,000 more have asked for help.
Police said late on Tuesday that two bodies had been discovered in Townsville floodwaters following a search for two men in their 20s, who were last seen during the peak of the deluge early on Monday.
There seems little prospect of an immediate respite from what locals call the "Big Wet".
"We expect this active monsoon to remain active for the coming days, potentially easing over the weekend, so there is an end in sight," said Richard Wardle of Queensland's Bureau of Meteorology.
"But we are expecting further periods of heavy rainfall, some of it very heavy, about the north-east tropics for the next few days … with that, there is the real elevated risk of flash flooding."
Local gym instructor Rebecca Chilmaid described the scale of the devastation, of people left "without clothes, cars, their homes, everything".
"People and businesses have lost everything," she told AFP.
"I sit and struggle to keep tears from falling thinking about it."
Australia's tropical north typically experiences heavy rains during the monsoon season, but the recent downpour has seen some areas get a year's worth of rainfall in just a week.
"Last night it was crazy," Bluewater resident Michael Nieminen, some 30 kilometres (18 miles) northwest of Townsville, told national broadcaster ABC.
"We had another lot of flash flooding come through, we had water come right up in the backyard again, similar to last week, not as high, luckily, but a lot of rain and a lot of water upstream," he added.
The authorities were forced to open floodgates of a major dam late on Sunday, unleashing what they called "dangerous and high velocity flows".
As waters began to recede on Tuesday in some areas, authorities were warning people of the dangers of returning home, not least thanks to the risk of encountering crocodiles.
"Do not go swimming in those floodwaters," Queensland Premier Annastacia Palaszczuk said.
"And when the clean-up begins, you need to make sure that you are wearing closed-in boots and that you do not have any scratches or anything exposed."
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"redpajama_set_name": "RedPajamaCommonCrawl"
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Chris Kinsley
The Value of the Giveaway
A few years ago, (maybe more than a few; I can't remember; plus what number exactly constitutes a "few"?) while I was still employed at Student Life I wrote a spoken word piece. It was my job to write and create lot of things. That particular year this piece just happened to be one of them. I wrote it. Drew Francis directed it. Taylor Robinson shot it. Eric Chapman recorded the audio and mastered it. Jason Poole added some sweet graphics to it. And the incomparable Erin Moon performed it. The piece is entitled "Woman of No Distinction" but it's much more commonly known as "The Woman at the Well" since that's the story from the Bible upon which it's based and nowhere in the video is the title of the piece indicated. You can see the original below. (If you can't see the piece below, it's because your reader or device isn't displaying videos and you should click through directly to my blog.)
Woman of No Distinction from Chris Kinsley on Vimeo.
This video went viral, I guess. I mean, I don't know what are the precise criteria that have to be met in order for that to be true, but this video started floating around a lot of places and being seen and shared by a lot of people. Well, it wasn't long before people were simply transcribing the video and using the piece in other venues, even to make videos of their own. Below, I offer to you a selection of those videos.
[tentblogger-youtube 0mEfR3EideE]
[tentblogger-youtube UDfEyVphRqE]
[tentblogger-youtube 4Cqx0DcxrLA]
[tentblogger-youtube NxvNf9M1lnE]
[tentblogger-youtube K0aNg8RA8iE]
[tentblogger-youtube m1KkvTTAiXA]
[tentblogger-youtube dQ8ZS0hesXI]
[tentblogger-youtube BKastZ6yjF8]
[tentblogger-youtube Q7ue1gA9fb4]
[tentblogger-youtube i1eKMtdCfKw]
[tentblogger-youtube dzbNLB-uifQ]
[tentblogger-youtube X02I-cNC0i8]
[tentblogger-youtube alLe-bhbiGI]
As you can see, there's a decent amount of diversity other than all of the performers being women, but that's understandable. Some of these videos are well produced. Others are amateur shots of amateur performances. Some give credit to me as the author. Some credit myself and Drew, which is also understandable since the credits at the end of the original video are misleading. Some give no credit to authorship. One credits authorship to someone else entirely, namely Amena Brown, who is an extremely gifted spoken-word artist and sister in the faith (frankly, it's a little flattering to have people think she wrote it). However, none of these people own the copyright for this work. I don't even own it. Student Life does. As far as I know, none of them obtained permission from Student Life to perform the piece or to make a video of it.
Student Life could have gone after all of these people. They could have attempted to control and "protect" their content. But they didn't and they haven't. Whether intentionally or not, they've given it away. They've allowed it to spread as it would.
I think this is great. On some level I even think this is right. Student Life is a business, yes, but it's also a ministry. The original piece served it's purpose, and SL is able to continue to utilize it for ministry. However, by giving it away they have allowed it to become even more valuable. As it has struck a cord with people, those people have been able to freely share it so that it might minister to others, many if not most of whom Student Life would never have the opportunity to come in contact with.
This isn't an easy thing for creatives to do. But I wonder if it's an essential lesson for those creatives who are also Christian to learn. God has made us stewards of the creativity with which He has gifted us.
What kind of steward will you be?
What are you currently holding on to that you need to give away?
Christianity, Creativity, MinistryChris Kinsley June 14, 2011 Amena Brown, Artist, Creative, Creatives, Creativity, Drew Francis, Eric Chapman, Erin Moon, Giveaway, Jason Poole, Ministry, Spoken Word, Student Life, Student Life Camp, Taylor Robinson, Value, Video, Viral, Woman at the Well, Woman of No Distinction
WHAT?!?
VideosChris Kinsley October 3, 2011 Blu-Ray, Boy, Darth Vader, Father, Girl, Luke Skywalker, Reveal, Shock, Star Wars, The Empire Strikes Back, Video, YouTube
An Adoption Benefit
Family, FriendsChris Kinsley February 14, 2011 Adopt, Adopted, Adopting, Adoption, Benefit, Benefiting, Chris, Dessert Bar, Discomfort, Family, Finances, Financial, Friends, Fundraising, Gifts, Giving, Haven, Help, Helping, Home for Haven, How, Kinsley, Liza, Photo Booth, Silent Auction, Site, Thanks, Uncomfortable, What, When, Where, Who, Why
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction}\label{sec:introduction}
\subsection{Motivation}
Energy system planning models are broadly adopted around the world. They are used as instruments to inform policy and investment decision-making, such as operational, supply diversification, and long-term infrastructure planning studies. Inscrutable 'black-box' models, even though being under criticism in academia~\cite{Pfenninger2017EnergyWorkings, Pfenninger2017TheBehind}, are still the standard for high-impact modelling such as the African Continental Power System Plan \cite{IRENA2021}. This prevents transparent decision-making while having other major drawbacks, as described in \cite{Pfenninger2017EnergyWorkings, Pfenninger2017TheBehind}.
Open-source models evolved to overcome these typical black-box model problems and can perform equivalent or even more tasks, but at no charge, while additionally supporting transparent and robust analyses \cite{Brown2018PyPSA:Analysis}. In many examples, the European Commission applies open tools and requests their use in funded projects, proving its belief in the benefits of openness and transparency \cite{EC2020a}. Now with the encouraging rise of more than 31 models in 2019 \cite{Groissbock2019AreUse}, simultaneously concerns of failed collaboration and duplication are arising that cost taxpayer money \cite{Juha2022}.
As a result, it becomes increasingly important to avoid duplication and provide modelling solutions that allow global united efforts. For these reasons, in this study, we propose an open-source community-backed flexible energy system model able to represent any arbitrarily large region of the world energy system in high spatial and temporal resolution that leverages other existing open-source projects to serve industry, policymakers, and researchers.
\subsection{Literature analysis}
In general, models are idealised representations of real physical systems.
To ease building idealised systems, 'frameworks' have been developed to provide pre-compiled equations, algorithms, solver interfaces and/or input/output features. A framework becomes a 'model' only when data is added that describes real physical systems \cite{Pfenninger2018OpeningLearned}.
In this view, PyPSA is a framework and PyPSA\=/Eur and PyPSA\=/Earth are models for the European and Earth energy system, respectively.
Nowadays, the open-source community is rich in energy system modelling frameworks that can provide similar functionalities. Table \ref{tab:model comparison} compares some available functionalities across selected widely-adopted energy system modelling frameworks \cite{Brown2018PyPSA:Analysis, Victoria2020EarlyOff, Neumann2022, Poncelet2020UnitFlexibility}. Undoubtedly, each developer team might be capable of filling missing features, but the functionality of the frameworks is only one important part of models, the other one, often even more relevant, is the integration of data.
\begin{table*}[!ht]
\caption{Comparison of selected features for energy system modelling frameworks that are applied in Africa.}
\label{tab:model comparison}
\begin{tabularx}{\textwidth}{@{}l*{10}{C}c@{}}
Software &
\rotatebox[origin=c]{90}{Version} &
\rotatebox[origin=c]{90}{Citation} &
\rotatebox[origin=c]{90}{Language} &
\rotatebox[origin=c]{90}{Free and Open} &
\rotatebox[origin=c]{90}{Power Flow} &
\rotatebox[origin=c]{90}{Transport Model} &
\rotatebox[origin=c]{90}{LOPF} &
\rotatebox[origin=c]{90}{SCOPF} &
\rotatebox[origin=c]{90}{Unit Commitment} &
\rotatebox[origin=c]{90}{Sector-Coupling} &
\rotatebox[origin=c]{90}{Pathway Optimization}
\\ \midrule
Calliope & v0.6.8 & \cite{Pfenninger2018} & Python & \checkmark & & \checkmark & & & \checkmark & \checkmark \\
Dispa-SET & v2.4 & \cite{Dispa-SET} & GAMS & \checkmark & & \checkmark & & & \checkmark & & \\
GridPath & v0.14.1 & \cite{gridpath} & Python & \checkmark & & \checkmark & \checkmark & & \checkmark & & \checkmark \\
LEAP & 2020.1.63 & \cite{LEAPmodel} & NA$^b$ & & & & & & & \checkmark & \\
NEMO & v1.7 & \cite{NEMOmodel} & Julia & \checkmark & \checkmark & \checkmark & \checkmark & & \checkmark \\
OSeMOSYS & 2022 & \cite{HOWELLS20115850} & GNU$^a$ & \checkmark & & \checkmark & & & & \checkmark & \checkmark \\
PLEXOS & 9 & \cite{PLEXOSmodel} & NA$^b$ & & & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark \\
\textbf{PyPSA} & v0.20.0 & \cite{Brown2018PyPSA:Analysis} & Python & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark \\
SPLAT-MESSAGE & 2022 & \cite{SPLAT-MESSAGEmodel} & GAMS & & & \checkmark & & & & \\
TIMES & 2022 & \cite{TIMESmodel} & GAMS & & & \checkmark & \checkmark & & \checkmark & \checkmark & \checkmark\\
\bottomrule%
\multicolumn{12}{l}{$^a$ Is available in GNU Mathprog, Python and GAMS.}\\
\multicolumn{12}{l}{$^b$ NA = no information available.}
\end{tabularx}
\end{table*}
Existing models are often designed to implement data with limited geographical coverage, such as a specific province, country or continent \cite{Ringkjb2018ARenewables}.
Continental models with implemented high-resolution data have proven to be the most maintained and active, possibly by covering many regions of interest and giving the user options for the aggregation level \cite{Horsch2018PyPSA-Eur:System, PyPSA-Eur2021GithubPyPSA-Eur}. In contrast, there are several examples where single-country models have soon become outdated, poorly documented or inactive \cite{pypsa-za, Schlott2020, Liu2018, Calliope-Kenya}. While global energy system models exist, they currently have several shortcomings. 'GlobalEnergyGIS' \cite{Mattsson2021AnRegions}, an open source model which can create energy system model data for any arbitrary region, is used in the 'Supergrid' model \cite{Reichenberg2022}, but misses network data or a workflow management system that are important for flexible and reproducible data processing \cite{Kiviluoma2022}. Similarly, the GENeSYS-MOD model is a global open-source model, however, it is written in GAMS preventing free use and offers no data processing workflows \cite{genesys-mod-public}. Another promising candidate is the recently released OSeMOSYS-global model, which includes a workflow management system but misses network topology data as well as unit-commitment and power flow constraints that have been shown to strongly affect model results \cite{Barnes2022, Neumann2022}.
Similarly, existing PyPSA models are geographically limited. While PyPSA as a framework is adopted worldwide by many companies, non-profit organisations and universities (see example studies in \cite{PyPSAdocumenation}), there is no global model solution available yet. Providing a global energy system model ecosystem solution has the potential to unlock collaboration potentials that accelerate and improve the energy transition planning.
\subsection{Contributions}
In this paper, we present PyPSA\=/Earth, the first open-source global energy system model with data in high spatial and temporal resolution. Users can flexibly model the world or any subset of it with high spatial and temporal resolved data, including a GIS-tagged network representation based on OpenStreetMap. Using an automated workflow procedure it can (a) generate energy system model data and (b) perform energy planning studies. The novel contributions of PyPSA\=/Earth are detailed as follows:
\begin{enumerate}
\item New model creates arbitrary high spatial and temporal resolution representation of energy systems around the world
\item Automated workflow generates national, regional, continental or global model-ready data for energy planning studies based on open or optionally closed data
\item Integration and linking of multiple data sources and open-source tools to process raw data from multiple sources, e.g. OpenStreetMap
\item Provision of new spatial clustering strategies to simplify the high-resolution model
\item Data and model validation for the African continent and Nigeria
\item Development of 2060 net-zero energy planning study for Nigeria
\end{enumerate}
Following the open-source spirit, the model is not built from scratch but derived from the popular European transmission system model -- PyPSA\=/Eur \cite{Horsch2018PyPSA-Eur:System}. Novel additions that have not been part of PyPSA-Eur include the use and integration of global data by default (e.g. demand, generators, transmission grid) and building a meshed electricity network from OpenStreetMap. New features in PyPSA-Earth that can be optionally activated are the use of administrative zones instead of network-derived zones, consideration of DC meshed grids, and the automatic creation of line expansion options.
In this paper, the PyPSA\=/Earth model is presented and discussed with a focus on the African continent. This includes the data and model validation for the African continent and Nigeria. Finally, a 2060 net-zero energy planning study for Nigeria's electricity sector demonstrates the optimization capabilities.
All code and validation scripts are shared open-source under GPL 3.0 license. The data, often extracted by python script activation, is available under multiple open licenses. For a detailed license listing, see \cite{pypsaafrica}.
\subsection{Organization of the paper}
The rest of the paper is organized as follows. Section \ref{sec:methodology} introduces the novel PyPSA\=/Earth model that is able to perform large-scale energy system modelling studies. The data processing novelties are described in detail in Section \ref{sec:data_and_methods}. Data validation for the African continent is performed in Section \ref{sec: validation}, and a quantitative case study on Nigeria is discussed in Section \ref{sec: Nigeria case study}. Finally, the limitations of the model are discussed, and the conclusions are drawn.
\section{PyPSA\=/Earth model}\label{sec:methodology}
This section describes the scope of the PyPSA\=/Earth model, its features as well as the role of the initiative that is facilitating the model developments.
\subsection{Scope}
The PyPSA\=/Earth model is a novel open-source data management and optimization tool that aims to provide policymakers, companies and researchers with a shared platform for a wide range of macro-energy system analyses needed to achieve the energy transition together. The option to create a tailored country, continental or global model under a unique code repository maximises synergies and wider user-benefits. For instance, one user in Africa can implement new features and data, improve the documentation or implement bug fixes that immediately benefit all other users around the world.
Studies that were already demonstrated in the PyPSA ecosystem \cite{PyPSAdocumenation}, and are now globally available include:
\begin{enumerate}
\item energy system transition studies
\item power system studies
\item technology evaluation studies (e.g. energy storage, synthetic fuels and hydrogen pipelines)
\item technology phase-out plans (e.g. coal and nuclear)
\item supply diversification studies
\item electricity market simulations
\end{enumerate}
The PyPSA\=/Earth model currently focuses on the power system and does not yet provide capabilities for sector coupling studies such as co-optimization of e.g. power, heat, transport and industry sectors. However, these features are in active development building on top of PyPSA\=/Eur\=/Sec~\cite{Brown2018SynergiesSystem}.
\subsection{Features}
\label{sec: pypsa earth features}
The following features are implemented in PyPSA\=/Earth:
\begin{enumerate}
\item flexible model scope: from Earth to any subregion
\item high temporal and spatial resolution
\item model-ready data creation
\item co-optimization of investment and operation
\item single or multi-year optimization
\item flexible addition of arbitrary optimization constraints, e.g. socio-economic, technical, or economic
\end{enumerate}
Moreover, the PyPSA\=/Earth model has been developed with the following non-functional requirements:
\begin{enumerate}
\item easy to use and learn
\item highly customizable and flexible
\item modular to include new features and data
\item fully reproducible
\end{enumerate}
The proposed features of PyPSA\=/Earth are a novelty as compared to the literature in Section \ref{sec:introduction}. Furthermore, new features can be created in or adopted from other PyPSA-based models that share a similar backbone. Examples are the work on endogenous learning with pathway optimization and multiple investment periods \cite{ZeyenPathways2022}, dynamic line rating constraints based on spatially differing environmental conditions \cite{Glaum2022}, the implementation of generic constraint settings that enable equity constraints such as applied in \cite{Neumann2020CostsSystem} and uncertainty analyses by input parameter sweeps or by exploring the near-optimal solution space \cite{Neumann2021BroadNear-Optimality}.
The data and methods Section \ref{sec:data_and_methods} presents more details on the presented features.
\begin{figure}[h!]
\centering
\includegraphics[width=1\textwidth]{Illustrations/pypsa-africa-design.png}
\caption{PyPSA\=/Earth model design. After providing the configuration parameters and countries of interest, data is collected and processed to be then fed into the PyPSA model framework which enables to perform the desired optimization studies such as least-cost system transition scenarios.}
\label{fig:high level model design}
\end{figure}
\subsection{PyPSA meets Earth initiative}\label{sec:initiative}
The PyPSA meets Earth initiative is an independent research initiative that aims to improve energy system planning with open solutions. It supports, builds and maintains the PyPSA\=/Earth model and is therefore briefly introduced. The initiative's vision is to support transparent and debatable decision-making on the energy matter that cannot be achieved with the status quo ruled by commercial inscrutable closed-source \enquote{black-box} tools. Current research activities in the initiative can be categorised into three distinct groups:
\begin{itemize}
\item open data
\item open energy system model and
\item open source solver
\end{itemize}
First, the open data activities focus on open data creation, collection, fusion, modification, prediction and validation for energy system models. These data activities are not limited to aggregated country information but prioritise work on high spatial and temporal resolution data, which is fundamental for scalable and accurate mini-grid and macro-energy system model solutions. Second, the open energy system modelling activities focus on implementing new functions and data streams into the model, such as building a sector-coupled model with multi-horizon optimization that is useful across the globe. Third, open source solver-related activities deal with benchmarks and efficient interfaces that help to adopt and develop open source solvers.
For instance, we created a benchmark that became a successful public funding proposal and attracted sufficient funding for the open-source solver HiGHS \cite{Parzen2022OptimizationSolvers}
. This activity pushes breakthroughs in large-scale optimization performance required for energy system models, which were until now reserved only for people that can afford commercial proprietary solvers.
In order to assure a continuous inflow of people that maintain, improve and use the software, as needed by open-source software \cite{Steinmacher2019}, the initiative supports a free and open community where anyone can contribute. The initiative adopts:
\begin{itemize}
\item \textit{GitHub} to publicly record issues, requests, solutions or source code-based discussions
\item \textit{Discord} as a voice channel and messaging social platform for regular public meetings and exchanges
\item \textit{Google Drive} to publicly store files and meeting notes
\end{itemize}
Together, these tools provide the backbone of the open community supporting the initiative goals and activities (data, model, solver).
\section{Data and methods}\label{sec:data_and_methods}
In this section, the PyPSA\=/Earth methodology sketched in Figure \ref{fig:high level model design} is described in detail. First, we introduce the workflow management tool that supports the model user experience. Then, data creation and processing approaches are discussed considering the main data blocks used by the PyPSA\=/Earth model: power grid topology and spatial shapes, electricity demand, renewable potential and power plant locations. Further, we describe some advanced pre-processing techniques such as clustering and line augmentation used to introduce data into the model in a robust and efficient way. Finally, we describe the energy system modelling and optimization framework with its solver interfaces.
\subsection{Workflow management tool}\label{subsec: workflow}
First of all, inherited from PyPSA\=/Eur \cite{Horsch2018PyPSA-Eur:System}, PyPSA\=/Earth relies on the 'Snakemake' workflow management \cite{Koster2012} that decomposes a large software process into a set of subtasks, or 'rules', that are automatically chained to obtain the desired output. Accordingly, 'Snakemake' helps sustainable software design that enables reproducible, adaptable and transparent science, as described in \cite{Moelder2021}.
For example, Figure \ref{fig: workflow} represents a workflow of PyPSA\=/Earth automatically created by 'Snakemake' for which the user can execute any part of the workflow with a single line of code.
The implications of such design on the energy system model are described in the following.
Starting with essential usability features, the implemented Snakemake procedure enables the user to flexibly execute the entire workflow with various options without writing a single line of code. For instance, the user can model the world energy system or any subset of countries only using the required data. Wildcards, which are special generic keys that can assume multiple values depending on the configuration options, help to execute large workflows with parameter sweeps and various options. Examples are shown in Figure \ref{fig: workflow}, where the function 'build\_renewable\_profile' is executed with the wildcards 'offwind-ac', 'solar', 'onwind', 'hydro' and 'offwind-dc' to estimate renewable potentials and create synthetic time series for renewable feed-in availability by technology. Lastly, workflows allow easy scaling by parallel execution. A few more benefits also exist from the developer's perspective. For instance, the modular design of rules allows parallel code development and deployment with continuous integration tests of the complete workflow that guarantee the stability of PyPSA\=/Earth.
\begin{figure}[h!]
\centering
\includegraphics[trim={0cm 0cm 0cm 0cm}, clip, width=1\textwidth]{Illustrations/workflow_dag.png}
\caption{Directed acyclic graph of a PyPSA\=/Earth example workflow. Each box represents a rule that modifies or creates the input, resulting in outputs. Already computed rules are shown as boxes with dashed borders, solid bordered rules indicate rules yet to be computed during the workflow execution. The workflow creates here four least-cost scenarios with varying spatial and temporal resolution and is triggered by a single line of code. Wildcards define options for the spatial resolution with 50 and 100 clusters and the temporal resolution with 1 and 3 hours (Co2L-1H and Co2l-3H).}
\label{fig: workflow}
\end{figure}
\subsection{Network topology and model}
The electricity network topology is one of the main inputs needed to build an energy system model which accounts for realistic power flow approximations across regions. The most comprehensive and accurate data on power grids are curated by the transmission system operator.
In practice, the availability of open power grid data is still relatively low for many parts of the world, with the situation in Africa being extremely sparse.
A natural way to address the lack of power grid data provided by transmission system operators' is to utilize open geospatial datasets. Currently, a few open source packages have been published to extract and build networks from such datasets (e.g. Gridkit \cite{Medjroubi2014}, Transnet \cite{Rivera2018}, SciGrid \cite{Medjroubi2014}). However, each of these packages focuses on applications for a particular world region rather than on the global coverage and there is still no ready-to-use solution which could be implemented into a global model. To fill this gap, we have developed an original approach which reconstructs the network topology by relying solely on open globally-available data. The developed approach is based on the OpenStreetMap (OSM) datasets that are a crowd-sourced collection of geographic information, which is daily updated and includes geolocation references \cite{OpenStreetMap}.
The electricity network topology is created in three novel steps: i) downloading, ii) filtering and cleaning the data, and iii) building a meshed network dataset with transformer, substation, converter and high voltage alternating current (HVAC) as well as high voltage direct current (HVDC) components. Figure \ref{fig:clustered and augmented meshed network} shows sample raw and cleaned networks along with the options for clustering and line augmentation that are introduced in Section \ref{subsec: clustering} and \ref{subsec:augmented line connections}, respectively.
The following explains the most critical steps to building the electricity network topology and model.
First, the model implements an interface to download the OSM database. From this interface, substations and network data of any available voltage level are retrieved with essential operating details of the infrastructure. The interface thereby runs with the \textit{esy-osm} tool that allows fast retrieval of OSM data through multi-threaded processing \cite{esy-osm}.
To select only power-related data, the raw OSM database is filtered using the "power" key. By doing so, electrical substations, generators, high-voltage transmission lines and underground cables information can be obtained. A further data cleaning procedure is designed to restore the missed pieces of information and ensure that the overall data quality conforms with the requirements of the PyPSA modelling framework. In particular, the power infrastructure elements are being filtered by the validity of their geometry and the desired voltage level. Available data on the electrical overhead and underground cables are used to improve the representation of the high-voltage transmission lines, while the frequency values are applied to identify the HVDC transmission systems.
The cleaned dataset is pre-processed to build the network topology. The pre-processing includes:
\begin{enumerate}
\item extracting additional data on electrical substations from the transmission line dataset,
\item cleaning duplicated records,
\item splitting substations by the voltage level to represent each voltage by a separate substation,
\item deriving data on transformers and AC/DC converters from the information on the transmission lines.
\end{enumerate}
Beyond that, an approach has been developed to improve the quality of the OSM-extracted grid topology: substations within a given tolerance (few kilometres) from power transmission lines with the same voltage level are merged and connected to the power lines. Otherwise, the network may result in incomplete and unexpected distortions because of small missing information, such as short underground cables connecting the last mile of a transmission line with a transformer in a substation.
\begin{figure}[h!]
\centering
\subfigure[]{\includegraphics[width=\textwidth]{Illustrations/continent_image.png}}
\subfigure[]{\includegraphics[width=0.48\textwidth, trim={0cm 0cm 3.4cm 0cm}, clip]{Illustrations/420-clustered-network-africa.pdf}}
\subfigure[]{\includegraphics[width=0.48\textwidth, trim={0cm 0cm 3.0cm 0cm}, clip]{Illustrations/420-augmented-line-africa.pdf}}
\caption{Representation of transmission networks and shapes produced by PyPSA\=/Earth show: (a) a sample Open Street Map transmission network, (b) a clustered 420 node African transmission network and (c) its augmented version with additional line connections to test the benefits of additional interconnections. In case of c), the applied k-edge augmentation guarantees that every node has at least a certain number of connections, three in the case of the figure. The augmented lines are connected by a minimum spanning tree algorithm to the nearest neighbour.}
\label{fig:clustered and augmented meshed network}
\end{figure}
\subsection{Fundamental shapes}\label{subsec: fundamental shape}
Fundamental shapes represent the smallest defined regions that gather various data types to characterise the energy system (see Figure \ref{fig:fundamental data shape}). Before being ready for the model-framework execution, data is often provided in many different ways. For instance, it can be represented as geo-referenced point location for power plants, raster in many different resolutions such for population and GDP zones or lines for country boundaries. The data needs to be combined so that each fundamental shape represents a network node from which meshed energy systems can be built. These shapes are used multiple times across the workflow. For instance, one can apply network meshing strategies and clustering approaches, as illustrated in Figure \ref{fig:clustered and augmented meshed network}, to explore or reduce the complexity of optimization problems depending on the user requirements. Another example is to use the fundamental shapes to spatially distribute the national electricity demand or as catchment areas to aggregate renewable sources (see Figure \ref{fig:atlite demonstration}).
For onshore regions, the model provides two ways to build fundamental data shapes. The first retrieves the so-called Global Administrative Areas (GADM) that represent administrative zones at various levels of detail (e.g. national, regional, province, municipality) \cite{gadm2022}. The second one uses the substation GIS location to create Voronoi partitioned areas for each substation, which boundary is defined as equidistant to the centroid of the nearest sites \cite{Frysztacki2021TheSolar}. The latter approach is beneficial to replicating the network accurately, while the former helps communicate results.
For offshore regions, the model uses only Voronoi partitioned areas to create fundamental shapes. These Voronoi areas are built from high voltage onshore nodes and are limited to the offshore extent by the Maritime Boundaries and Exclusive Economic Zones (EEZ) data for each country \cite{marineregions2019}. The same Voronoi approach is applied to the offshore zones splitting them into separate regions and linking each to the closest onshore shape.
\begin{figure}[h!]
\centering
\subfigure[]{\includegraphics[width=0.32\textwidth]{Illustrations/NG_GADM.pdf}}
\subfigure[]{\includegraphics[width=0.32\textwidth]{Illustrations/NG_voronoix.pdf}}
\subfigure[]{\includegraphics[width=0.32\textwidth]{Illustrations/NG_Offshore.pdf}}
\caption{Fundamental shapes of Nigeria in PyPSA\=/Earth: (a) shows the onshore regions represented by the GADM zones at level 1, (b) shows the onshore regions represented by Voronoi cells that are derived from the network structure, and (c) shows the offshore regions also represented by Voronoi cells based on the closest onshore nodes.}
\label{fig:fundamental data shape}
\end{figure}
\subsection{Electricity consumption and prediction}\label{subsec: demand}
The model currently provides globally hourly demand predictions for 2030, 2040, 2050 and 2100. Each year includes demand predictions for the ERA5 reanalysis weather years of 2011, 2013 and 2018 which ERA5 provides in principle between 1959-2021. The demand time series is created using the new contributed \textit{synde} package~\cite{Synde2022} which implements a workflow management system to extract the demand data created with the open source Global-Energy GIS (GEGIS) package \cite{Mattsson2021AnRegions}.
In principle, GEGIS produces hourly demand time series by applying machine learning methods \cite{Mattsson2021AnRegions}. It uses information such as temperature profiles, population, GDP, and predicted values from the 'Shared Socioeconomic Pathways' \cite{SSP2017} to forecast the annual and hourly demand over the following decades. This approach is not new as it was already applied and tested in \cite{TOKTAROVA2019160}.
The observed absolute error of GEGIS in the validation test is considered acceptable for energy studies as it is 8\% across 44 countries, yet with generally worse performance in low-income countries \cite{Mattsson2021AnRegions}.
The coverage of the \textit{synde} package is currently limited. Figure \ref{fig:demand prediction coverage} shows that there are no data outputs for especially low-demand countries. A heuristic creates data for the countries with missing data by scaling the Nigerian demand time series proportionally to population and GDP. We validate this approach in Section \ref{subsec: demand validation}.
\begin{figure}[h!]
\centering
\includegraphics[trim={0cm 0cm 0cm 0cm}, clip, width=0.99\textwidth]{Illustrations/world_annual-demand_2030_v3.png}
\caption{
Demand predictions created per country using the \textit{synde} workflow based on \textit{GEGIS}.
For grey-coloured countries, synde does not provide data, however, a heuristic creates representative time series as described in Section \ref{subsec: demand}.
}
\label{fig:demand prediction coverage}
\end{figure}
\subsection{Renewable Energy Sources}\label{subsec: renewable}
Renewable energy sources such as solar, wind and hydro time series are modelled with the open-source package \textit{Atlite} \cite{HofmannAtlite:Series}. \textit{Atlite} i) creates cutouts that define spatio-temporal boundaries, ii) prepares cutouts, which means that environmental and weather data is added to geospatial boundaries by matching various datasets (ERA5 reanalysis data \cite{Hersbach2020}, SARAH-2 satellite data \cite{Sarah2}, and GEBCO bathymetry \cite{GEBCO}), and finally, iii) applies conversion functions to produce technology-specific spatially resolved time series and potentials \cite{HofmannAtlite:Series}. Currently, the \textit{PyPSA\=/Earth} model framework implements solar photovoltaic, on- and offshore wind turbines, hydro-runoff, reservoir and dam power resources. In the case of hydro, the runoff time series are obtained by \textit{Atlite} for each powerplant location, as described in Section \ref{subsec: generators}. As our new contribution, the hydro power output is thereby proportionally rescaled to match the reported total energy production of existing plants as reported per country by the open US Energy Information Administration (EIA) platform \cite{EIA2022}.
At the time of writing, available in \textit{Atlite} but not yet implemented in PyPSA\=/Earth are potentials and time series for concentrated solar power, solar thermal collectors, heat demand and dynamic line rating with a wide range of technology options. For details on the model implementation for each technology, we refer the reader to the \textit{PyPSA\=/Eur} publication which the presented model mostly builds-upon \cite{ Horsch2018PyPSA-Eur:System}.
A brief concept demonstration of \textit{Atlite} is provided in the Figure \ref{fig:atlite demonstration}.
\begin{figure}[!h]
\centering
\subfigure[]{\includegraphics[width=0.25\textwidth]{Illustrations/cutout-earthplot-nigeria.pdf}}
\subfigure[]{\includegraphics[width=0.39\textwidth]{Illustrations/weather-data-cutout-cell.pdf}}
\subfigure[]{\includegraphics[width=0.32\textwidth]{Illustrations/rasterized-eligible-areas.pdf}}
\subfigure[]{\includegraphics[width=0.32\textwidth]{Illustrations/solar-gridcell.pdf}}
\subfigure[]{\includegraphics[width=0.32\textwidth]{Illustrations/solar-cell.pdf}}
\subfigure[]{\includegraphics[width=0.32\textwidth]{Illustrations/solar-power-profile.pdf}}
\caption{A concept demonstration of Atlite for Nigeria. (a) Shows that environmental and weather data is extracted in a \textit{cutout} for the region of interest. (b) The cutout is split in a raster of $(0.25^{\circ})^2$ or roughly $(27.5km)^2$ (length varies along latitude), whereby each cell contains static or hourly time series data. The example wind speed and direct irradiation influx time series are shown for one cutout cell that contains an \textit{ERA5} extract of the Copernicus Data Store \cite{Hersbach2020}. (c) Shows the eligible area raster, which is built by excluding protected and reserved areas recorded in \textit{protectedplanet.net} and excluding specific land-cover types from \textit{Copernicus Global Land Service} whose eligibility can vary depending on the technology. (d) Illustrates the maximal installable power raster, which is calculated by the eligible area and the socio-technical power density of a technology e.g. $4.6MW/km^2$ for solar photovoltaic. (e) The raster is then downsampled to the region of interest or fundamental shape by averaging the proportion of the overlapping areas. (f) Finally, by applying a PV technology model to (b) and combining it with (e) we can define per region the upper expansion limit and the maximal hourly availability constraint for a given technology.
}
\label{fig:atlite demonstration}
\end{figure}
\subsection{Generators}\label{subsec: generators}
Given the limitation of reliable datasets for power plants for the African region, the existing powerplantmatching tool \cite{Gotzens2019PerformingDatabases} has been extended to include additional datasets, such as OpenStreetMap, to fine-tune the African model and validate the results with the final goal of maximizing accuracy and quality of the result.
\textit{Powerplantmatching} has been successfully proposed to estimate the location and capacity of power plants in Europe. The validation performed with respect to the commercial World Electric Power Plants Database (WEPP) by Platts and the dataset by the Association of European Transmission System Operators (ENTSO\=/E) reaches an accuracy of around 90\% using only open data \cite{Gotzens2019PerformingDatabases}. By default various open data sources are included such as CARMA (discontinued)~\cite{CFGD2012}, ENTSO\=/E~\cite{ENTSOE2022}, ESE~\cite{Sandia2021}, GEO \cite{GEO2018}, OSPD~\cite{OSPD2020}, GPD (discontinued) \cite{WRI2021}, JRC database on hydro powerplants~\cite{JRC2019} and renewable statistics by IRENA~\cite{IRENA2022}.
The approach applied for \textit{powerplantmatching} is based on the procedure depicted in Figure \ref{fig: ppl_method}, where the raw datasets are first downloaded, then filtered to remove missing or damaged data, and aggregated. Once the refined data are obtained, the datasets are pairwise compared to identify duplicated entries. Finally, non-duplicated data are merged into a unique dataset and used as a source for PyPSA\=/Earth. Only a few of these datasets have global scope (GEO, GPD and IRENA) and have been validated for Africa. In particular for Africa, where data is lacking, including all available open data can be critical to maximizing the accuracy of the results. Therefore, inspired by future work suggested in \cite{Powerplantmatching2019}, we have extended the \textit{powerplantmatching} tool to optionally include and process OpenStreetMap data to improve the quality of outputs
\begin{figure}[h!]
\centering
\includegraphics[trim={0cm 0cm 0cm 0cm}, clip, width=.95\linewidth]{Illustrations/ppl_scheme.png}
\caption{Flowchart of the \textit{powerplantmatching} procedure, including the novel OSM input (in bold) which was developed for PyPSA\=/Earth.}
\label{fig: ppl_method}
\end{figure}
\subsection{Spatial clustering approach}\label{subsec: clustering}
In order to tackle the computational complexity of solving a co-optimisation problem of transmission and generation capacity expansion, the model offers state-of-the-art spatial clustering methods which are imported from the PyPSA package and PyPSA\=/Eur model \cite{Brown2018PyPSA:Analysis, Horsch2020PyPSA-Eur:Code} and adapted for the model to simulate future scenarios. Spatial clustering allows finding sets of nodes that are similar and aggregating them to a single node to represent the original set. This way, the network is reduced to a smaller number of nodes to manage the model's computational complexity.
The available clustering methods provide a focus on (i) conserving the representation of renewable potentials as well as the topology of the transmission grid, (ii) accurately representing the electrical parameters to improve estimates of electrical power flows in an aggregated model, (iii) aggregating spatially close nodes disregarding other a-priori information of the network, or (iv) according to their location with regards to the country's subdivisions facilitating results interpretation for policy recommendations. An analysis of suitable clustering methods that depend on the modelling application is provided in \cite{Frystacki2022comparison}.
In summary, (i) the clustering approach that focuses on a better representation of variable sources or sinks of the model is inspired by \cite{SIALA201975}. It includes variable potentials, i.e. capacity factors or full load hours for solar and wind, or the variable electricity demand as a distance metric between nodes. This is combined with a hierarchical clustering approach, similar to the suggestions provided in \cite{Kueppers2020Data-DrivenOptimization}. However, we only allow nodes to be aggregated when a physical transmission line connects them instead of assuming a synthesised grid in contrast with \cite{Kueppers2020Data-DrivenOptimization}.
(ii) The clustering method that focuses on a better representation of the transmission grid was initially suggested by \cite{BIENER2020106349} to be applied for the case of electricity system modelling. It is a density-based hierarchical clustering operating on the line impedance.
(iii) The network can also be reduced using a weighted k-means algorithm on the locations of the network nodes as explained in detail in \cite{Frysztacki2021TheSolar}. (iv) Finally, using the GADM shapes allows aggregating all nodes in the same shape.
Any of these methods can be applied in a single or two distinct iterations, as displayed in Figure \ref{fig: clustering} for Nigeria. In each of these two iterations, a different method can be applied, choosing from (i)-(iv). In the first iteration, all nodes are clustered to a desired number of representative nodes, aggregating generators, flexibility options (electricity storage and transmission lines) and electrical demand. The second iteration is optional and allows the remaining nodes to be clustered again. However, now only the transmission network is effectively reduced such that the representation of renewable resources is fixed to the resolution of the previous iteration (compare the first row and second row of Figure \ref{fig: clustering}). The spatial resolution of the transmission network must always be larger or equal to the resource resolution, i.e. the clustering of the first iteration sets an upper bound.
\begin{figure}[h!]
\centering
\includegraphics[trim={0cm 0cm 0cm 0cm}, clip, width=.95\linewidth]{Illustrations/visualise_clustering.pdf}
\caption{Illustration of the clustering methodology applied for the transmission network (red nodes and edges) and resource resolution (grey nodes and edges). In the first row, we show how nodal data (i.e. generators, storage units, electrical loads etc.) is aggregated in tandem with the resolution of the transmission network. Three exemplary resolutions of the network for Nigeria are displayed here: 4, 14 and 54 nodes from left to right. The second row shows how the clustering also allows modelling the transmission grid at a different resolution than the resources. In this example, the transmission network contains 4 nodes connected by 4 lines (all in red) at every resolution, while 4, 14 and 54 generation sites become available (left to right). The background colour represents exemplary capacity factors in shades of red, for an arbitrary technology. The darker the colour, the higher the capacity factor.}
\label{fig: clustering}
\end{figure}
\subsection{Augmented line connection}\label{subsec:augmented line connections}
The African network is often not well interconnected. This is due to isolated national planning data or the presence of isolated mini-grids that are popular electrification measures \cite{Fioriti2021CouplingMicrogrid}. Therefore, we propose an algorithm to mesh a given network and assess different grades of connectivity. To investigate the benefits of meshed networks, PyPSA\=/Earth can perform a k-edge augmentation algorithm that guarantees every node has a modifiable number of connections to other nodes. Only if nodes do not already fulfil the connectivity condition, the algorithm will create new lines to the nearest neighbour by a minimum spanning tree. The new 'augmented' lines can be set to an insignificant size (e.g. 1 MW) to create new options for line expansion in the investment optimization. For example, Figure \ref{fig:clustered and augmented meshed network} shows the comparison between the standard clustered network with 420 nodes and its augmented version. Only the model that includes augmented line connections can explore an interconnected continent.
\subsection{Model framework and solver interface}\label{subsec: pypsa and solver}
The PyPSA-Earth model integrates the PyPSA model framework with its solver interfaces to perform energy system planning studies. Using PyPSA has several benefits compared to other tools that are briefly introduced in the following. First, PyPSA enables large-scale optimization in Python.
Python is well known for being user-friendly \cite{PythonOliphant}, but when analysing the memory consumption and speed for building optimization problems it was considered non-competitive compared to tools based on the programming language `Julia` or `C++` \cite{Linopy} -- a bottleneck which also hinders large-scale optimization required for PyPSA\=/Earth. As reaction, developers in the PyPSA ecosystem built \textit{nomopyomo} overcoming the bottlenecks \cite{Groissbock2019AreUse}. More recently, the same group is working on a general package called \textit{Linopy} that promises a 4-6 runtime speed up and a 50\% improvement in memory consumption compared to the optimization problem formulator Pyomo, possibly making it also more memory efficient than the Julia alternative \textit{JuMP} as indicated in \cite{Linopy}. Another point making the PyPSA dependency attractive is that it is one of the most popular tools, as suggested by GitHub stars in the GPST benchmark \cite{GpstBenchmark}, possibly due to its standard component objects and the continuously maintained documentation \cite{PyPSAdocumenation}. Finally, the framework offers several solver interfaces (HiGHS, Cbc, GLPK, Gurobi, among others) providing flexibility in solving various optimization problems with open-source and proprietary solutions.
\section{Validation}\label{sec: validation}
The data validation section aims to assess the data quality with publicly available data: at a continental level in Africa and a country level in Nigeria.
\subsection{Network topology and length}\label{subsec: validation. network topology and length}
Validating the African power grid is challenging. Unlike in Europe, where ENTSO-E \cite{ENTSO-E2020ENTSO-EPlatform} provides reliable open data with continental scope, such a transparent data source is lacking in Africa, and only a few utilities release open data. The self-proclaimed most complete and up-to-date open map of Africa's electricity network is offered by the World Bank Group, which implements Open Street Map data, as well as indicative maps data from multiple sources \cite{worldbank-network}. However, the World Bank data should not be used as a single validation set, because it may report outdated data, given that it has not been updated after 2020, and is partially based on indicative maps rather than on geo-referenced data, making the post-validation time-consuming.
Conversely, PyPSA\=/Earth builds its grid topology directly from daily updated Open Street Map data. Finally, the World Bank data also provides less detailed information than Open Street Map; for instance, it does not give any information on the frequency, circuit or cable number, limiting the information that can be used for validation. In the following, grid statistics and topology are compared on a Nigerian and African scale. This also includes nationally reported data from the Nigerian energy commission.
First, the transmission lines are validated by comparing the total circuit lengths at different alternate current (AC) voltage levels. Transmission lines can carry one or more 3-phase circuits, whereby each circuit has at least three cables. Instead of looking only at the line length, which is the distance between high voltage towers, it is common to report the total circuit length, which multiplies each line length, e.g. distance from tower to tower, with the number of circuits \cite{Horsch2018PyPSA-Eur:System}.
Table \ref{tab:AC line length validation} indicates that the Nigerian network length reported at the World Bank aligns approximately with the official transmission company statistics \cite{NigerianTSO}, suggesting that the World Bank data is either accurate in Nigeria or used as a reference by the transmission system operator. This official reported total circuit length is approximately 35\% longer than the original Open Street Map data or the modified and cleaned PyPSA\=/Earth derivative. Conversely, on a continental scale, Open Street Map provides approximately a 117\% longer total circuit length than the reported World Bank data. To summarise, while Open Street Map data is qualitatively less available in Nigeria by looking at the statistics, it offers significantly more data on a continental scale.
\begin{table}[!h]
\centering
\caption{HVAC and HVDC circuit line lengths of Nigeria and Africa from different sources}
\label{tab:AC line length validation}
\resizebox{\columnwidth}{!}{%
\begin{tabular}{|l|ccc|ccc|c|}
\hline
\multicolumn{1}{|c|}{\multirow{2}{*}{Circuit lengths in 1000km}} &
\multicolumn{3}{c|}{Nigeria} &
\multicolumn{3}{c|}{Africa} &
\multicolumn{1}{c|}{\multirow{2}{*}{Ref}} \\
\multicolumn{1}{|c|}{} &
110-220kV &
220-380kV &
\textgreater{}380kV &
110-220kV &
220-380kV &
\textgreater{}380kV &
\multicolumn{1}{c|}{} \\ \hline
World Bank Group$^a$ & 9.3 & 12.1 & 0.0 & 59.4 & 63.5 & 41.0 & \cite{worldbank-network}\\ \cline{1-1}
Open Street Map (OSM) & 6.3 & 9.1 & 0.0 & 87.9 & 180.7 & 76.7 & \cite{OpenStreetMap} \\ \cline{1-1}
Transmission Company of Nigeria & \multicolumn{3}{c|}{More than 20} & - & - & - & \cite{NigerianTSO} \\ \cline{1-1}
PyPSA\=/Earth (cleaned OSM) & 6.7 & 9.1 & 0.0 & 88.3 & 183.7 & 82.9 & \\ \hline
\multicolumn{7}{l}{$^a$ Information about circuits is missing.}
\end{tabular}%
}
\end{table}
To further compare and validate the data, Figure \ref{fig:network topology} highlights good agreement between the network topology in Nigeria and Africa of Open Street Map and World Bank sources. However, in central and south Nigeria, the World Bank covers more power lines. On the African scale, the opposite is observed. Open Street Map covers more network structures in East and North Africa.
\begin{figure}[h!]
\centering
\subfigure[]{\includegraphics[width=0.32\textwidth]{Illustrations/nigeria-network-comparison-osm.pdf}}
\subfigure[]{\includegraphics[width=0.32\textwidth]{Illustrations/nigeria-network-comparison-worldbank.pdf}}
\subfigure[]{\includegraphics[width=0.32\textwidth]{Illustrations/Nigeria_network_map.png}}
\subfigure[]{\includegraphics[width=0.48\textwidth]{Illustrations/africa-network-comparison-osm.pdf}}
\subfigure[]{\includegraphics[width=0.48\textwidth]{Illustrations/africa-network-comparison-worldbank.pdf}}
\caption{Network topology of open available transmission network data (above 110kV) from (a) \& (d) Open Street Map, (b) \& (e) World Bank Group and (c) the Nigerian Transmission Company. On the African scale, the voltage ranges from 110-765 kV in both data sets. The line format varies with the voltage level and includes transparency, thickness and colour.}
\label{fig:network topology}
\end{figure}
\subsection{Electricity consumption}\label{subsec: demand validation}
This subsection validates the demand prediction on the example year 2030 for every country in Africa by comparing the individual country consumption for 2020 and 2030 with official continental annual electricity consumption used in PyPSA\=/Earth. Figure \ref{fig: annual electricity consumption} shows 2020 reported electricity consumption data per country, published from \textit{Our World in Data} that is additionally refined by data from the global energy think-tank Ember and BP's statistical review of world energy \cite{OurWorldInDataDemand}. The used electricity demand data in PyPSA\=/Earth roughly doubles from 2020 to 2030, indicating demand growth. While national demand predictions are often not available, the demand prediction is further validated by comparing it to other - more common - continental demand predictions. In Africa, \textit{Our World in Data} reported an electricity consumption in 2020 of $782$~TWh/a. For 2030, \citet{IRENA} predicted $1924$TWh/a, \citet{Alova2021} $1877$~TWh/a and the PyPSA\=/Earth model data $1866$~TWh/a, predicting more than a doubling of Africa's electricity consumption by 2030. In summary, looking at the total African electricity consumption suggests that the data used in the global PyPSA model is in the range of others.
\begin{figure}[h!]
\centering
\includegraphics[width=1\textwidth]{Illustrations/demand-validation-part.pdf}
\caption{Comparison of reported \cite{OurWorldInDataDemand} and predicted annual electricity consumption data across African countries indicate in every country demand growth. For 2030, the African total electricity consumption of PyPSA aligns with other predictions from \citet{IRENA} and \citet{Alova2021}}.
\label{fig: annual electricity consumption}
\end{figure}
\subsection{Solar and wind power potentials}\label{subsec: validation solar and wind}
The validation of solar and wind potential is performed by comparing statistics by international organizations, such as IRENA, with the outputs of the PyPSA\=/Earth model, both including total generation capabilities and the specific power densities per unit of available land.
Solar and wind potentials are well reported across the African continent. In 2021, the Global Wind Energy Council estimated for Africa a technical potential for wind generation of $180.000 TWh$ (PyPSA-Earth: $108.700 TWh$), which is sufficient to electrify the continent 250 times relative to the 2019 demand \cite{WindpotentialIFC}. Similar, the International Renewable Energy Agency estimated in 2014 that the technical potential in Africa is $660.000 TWh$ (PyPSA-Earth: $122.200 TWh$), which is sufficient to electrify the continent 916 times \cite{IRENA2014}. The discrepancy between the technical potentials observed in the PyPSA-Earth model and the institutional reports is due to the underlying assumptions. In fact, how many renewables can be installed in a region depends on two main assumptions: the excluded areas $[km^2]$ and the power density per technology $[MW/km^2]$, both discussed in the following.
While we define the available areas in a data-driven way similar to \cite{IRENA2014} and \cite{Horsch2018PyPSA-Eur:System} (see details in Section \ref{subsec: renewable}), the remaining eligible area quantifies the technical potential per technology through the technical power density factor. However, this density applies only to land specifically and uniquely allocated to renewable production, yet this cannot easily be generalized to all non-protected land areas at the country level. In fact, land areas are also necessary for non-technical activities such as economic activities, industries, farming, well-being, and housing, among others. Accordingly, in PyPSA\=/Earth we considered a more conservative power density coefficient to account for such socio-economic considerations.
Focusing on solar photovoltaic power plants, we assessed the power density of the 41 largest installations in the world \cite{Bolinger2022, WikipediaSolar}: the average power density is $46.4~MW/km^2$, the minimum $10.41~MW/km^2$, and the maximum $150.0~MW/km^2$.
The type of solar module and the solar photovoltaic plant design are driving factors for this extensive range of values.
For instance, the Cestas Solar Park in France uses high-performing solar modules and additionally contains a compact east-west orientation solar field design leading to the $150.0~MW/km^2$ extreme.
Similarly to \cite{Horsch2018PyPSA-Eur:System}, we reduced the technical power density to $10\%$ of the average power plant density to represent the socio-technical limit: $4.6~MW/km^2$ for solar photovoltaic.
For onshore and offshore wind farm technologies, we verify power density assumptions by analysing seven existing utility-scale wind farms. The observed average technically feasible power density for onshore wind farms is 6.2~$MW/km^2$, and for offshore wind farms, 4~$MW/km^2$ \cite{GithubWindList}. Using the same approach as \cite{Horsch2018PyPSA-Eur:System}, we reduced these values from 6.2 to 3~$MW/km^2$ and from 4 to 2~$MW/km^2$ for onshore and offshore wind farms, respectively, to represent socio-technical power densities and give wind farms space to lower generation reducing wake-effects.
Currently, we apply the same socio-technical power density irrespective of the land cover type. However, roughly about $43\%$ of the continent is characterised as extreme deserts \cite{Reich2004}, giving the opportunity in these regions to be less conservative about the social-technical power density.
\subsection{Power plant database}\label{subsec: powerplant data validation}
In this section, we compare the site-specific power plant database used in PyPSA-Earth to national statistics provided by IRENA and USAID.
Data on existing power plants is critical for accurate energy simulations as they affect long-term investments, dispatch, and stability of the energy systems. For validation purpose
relevant country statistics are provided by IRENA \cite{IRENAPxWeb2022} and USAID \cite{Usaid2022}. While the PyPSA\=/Earth data is geo-referenced, hence including the location, type and nominal capacities of each power plant, the other sources only provide country statistics. Therefore, data used in PyPSA\=/Earth is of higher quality, especially for energy system modelling in a high spatial resolution that would be impossible to perform with the IRENA and USAID sources only.
Figure \ref{fig: capacity_technology_validation} shows that the PyPSA-Earth model matches the largest fraction of the installed capacity of existing databases, with 165~GW out of the 229~GW reported by IRENA. Most technologies are matched with adequate accuracy (2--15\% error), yet larger differences occur especially for coal and gas power plants, partially due to the recent installation of power plants over the last 3-4 years, whose data has not been updated by the sources described in Section \ref{subsec: generators}. In future work, adding more recent data sources may improve the data situation \cite{GlobalEnergyMonitor}. Furthermore, we note that, although the current PyPSA-Earth procedure does not include geothermal and CSP technologies, their capacity can be relevant for certain countries (e.g. Kenya, Morrocco and South Africa), but at an African scope, these technologies still represent a small fraction of the installed capacity. Therefore, the proposed validation is considered of good accuracy, supporting the appropriateness of the PyPSA\=/Earth model.
\begin{figure}[h!]
\centering
\subfigure[\label{fig: capacity_country_validation}]{\includegraphics[width=\textwidth]{Illustrations/capacity-validation-by-country.pdf}}
\subfigure[\label{fig: capacity_technology_validation}]{\includegraphics[width=0.48\textwidth]{Illustrations/capacity-validation-all.pdf}}\hfill
\subfigure[\label{fig: capacity_technology_validation_ng}]{\includegraphics[width=0.48\textwidth]{Illustrations/capacity-validation-ng.pdf}}\\
\caption{Total installed generation capacity in Africa by (a) country and (b) technology, including a focus on (c) the installed capacities in Nigeria.}
\label{fig: capacity_validation}
\end{figure}
\section{Demonstration of optimization capabilities in Nigeria}\label{sec: Nigeria case study}
At COP26, Nigeria's president Buhari committed to net zero emissions by 2060 \cite{ClimateActionTracker}. To demonstrate that the presented model can be useful for Nigeria's energy planning activities, we showcase the optimization capabilities of PyPSA\=/Earth. In particular, this section covers two least-cost power system optimizations, one for 2020 to reproduce the historical system behaviour and one representing a decarbonised 2060 scenario (see Figure \ref{fig: nigeria 2020} and \ref{fig: nigeria 2060}).
\subsection{Nigeria 2020 - Dispatch validation}
The 2020 scenario applies a dispatch optimization with linear optimal power flow constraints to simulate and validate the optimization results for Nigeria. Accordingly, only the operation of existing infrastructure is optimized for the lowest system cost, excluding any infrastructure expansion e.g. generation or transmission line expansion (see Figure \ref{fig: nigeria 2020}).
Starting with the scenario design. The power grid retrieved from OpenStreetMap is clustered to 54 nodes, representing the aggregation zones for the demand and supply. Since the existing network is more meshed than the OpenStreetMap based PyPSA-Earth network (see Figure \ref{fig:network topology}), a few augmented line connections with a negligible minimal capacity of 1~MW are added such that every node has at least two line connections, see (b) in Figure \ref{fig: nigeria 2020}, and overcome short missing network data
A total demand of 29.5~TWh is considered for 2020 using the national demand profiles provided in PyPSA-Earth. The magnitude aligns with reports from \textit{Our World in Data} (28.2 TWh) \cite{OurWorldInDataDemand}. The demand profiles are distributed across all nodes proportional to GDP and population. With the available hourly electricity demand time series and the existing 2020 power plant fleet (validated in Section \ref{subsec: powerplant data validation}), the model calculates the optimal generator dispatch considering power flow constraints.
The dispatch validation shown in Table \ref{tab:nigeria_2020_comparison} compares the generation shares of the PyPSA\=/Earth results to those reported at \textit{Our World in Data} \cite{OurWorldInDataGeneration}. The comparison highlights that PyPSA\=/Earth adequately represents the total electricity production shares by source in Nigeria with acceptable accuracy. Model results for the solar generation have a $100\%$ accuracy compared to data provided by \textit{Our World in Data}, gas generation is $2TWh$ ($10\%$) higher than the benchmark, while hydro generation is $0.3~TWh$ ($5\%$) lower. These deviations could be explained by the $1.3~TWh$ ($4\%$) higher assumption of total electricity demand and differences in the specific marginal costs of resources. Using the cost assumptions from \cite{PyPSA-Eur2021GithubPyPSA-Eur}, we derive an average marginal price for electricity of 59~\euro/MWh, which aligns with reported production costs in the range of 45-70~\euro/MWh \cite{nesg2017}.
\begin{table}[!htbp]
\centering
\caption{Nigeria 2020 dispatch comparison}
\begin{tabular}{c|c|cccccc}
& Total & Hydro & Coal & Gas & Wind & Solar \\\hline
PyPSA\=/Earth [TWh] & 29.5 & 5.8 & - & 23.6 & 0 & 0.04 \\
Our World in Data [TWh] & 28.2 & 6.1 & 0.6 & 21.4 & 0 & 0.04
\end{tabular}
\label{tab:nigeria_2020_comparison}
\end{table}
The computational needs for this scenario in terms of total solving time (computational time times the average load of the processors) and memory, are shown in Figure \ref{fig: solver_comparison}. We used 4 threads with Gurobi 9.5.1 solver while only a single-core with HiGHS 1.2.1, since its parallel solving capabilities are currently limited. While the commercial Gurobi solver is very efficient, the results in Figure \ref{fig: solver_comparison} confirm that the open-source HiGHS solver can also optimize the network below one day with memory requirements that are available for laptops. Given the expected improvements for open-source solvers, the computational requirements are likely to decrease significantly \cite{Parzen2022OptimizationSolvers}.
\begin{figure}[h!]
\centering
\includegraphics[width=0.7\textwidth]{Illustrations/solver_comparison.pdf}
\caption{Solution time and memory requirements for the 2020 Nigeria dispatch optimization for HiGHS 1.2.1 and Gurobi 9.5.1 solver at different spatial resolution; solution time is weighted by threads.}
\label{fig: solver_comparison}
\end{figure}
\subsection{Nigeria 2060 - Net-zero study}
In the 2060 net-zero scenario, we perform a brownfield capacity expansion optimization. This means that new renewable energy and transmission capacity can be built on top of existing infrastructure. Simultaneously, a dispatch optimisation is performed subject to linear optimal power flow constraints. To explore new transmission grid structures, the meshing strategy is increased such that each node connects HVAC lines to at least three nearest neighbouring nodes and that a random selection of far distance nodes above $600km$ connects HVDC lines, see b) in Figure \ref{fig: nigeria 2060}). Using a random selection for long-distance HVDC can help identify valuable line connections before applying any heuristic that might not find these. Additional to the net-zero emission constraint, the 2060 total demand has been calibrated in agreement to \cite{Kim2021} to about 250 TWh by linear interpolation of the Stated Energy Policies of IEA for Nigeria \cite{IRENA2022}.
Observing the optimized infrastructure in Figure \ref{fig: nigeria 2020}, the overall optimal least-cost power system can be mostly supplied with solar energy and a mix of battery energy storage. Hydrogen energy storage with steel tanks is included as an expansion option. However, it is not significantly optimized, probably because we ignore fuel trade with other countries and the unique geo-location of the country. Nigeria lies close to the equator, where solar irradiation is homogenous across the year, requiring less seasonal energy storage. The battery storage consists of an inverter component $[\euro/MW]$ and a Li-Ion battery stack $[\euro/MWh]$ that can be independently scaled by the model such as applied in \cite{Parzen2021BeyondSystems}. The energy to power ratio (EP) indicating the sizing between these storage components is optimized in the range $4.6~h$ -- $15.1~h$ with an average of $7.4~h$. The optimal solar capacity distribution is spatially uneven. Most solar is expanded in the country's north, where the solar potential is significantly higher \cite{Umar2021}. It is also cost-optimal to build new transmission routes in the north and east of Nigeria, enabling the spatial distribution of the electricity. The HVDC options are not used significantly, indicating it is not cost-optimal in the scenario. Notably, to be conservative, with cost assumptions for 2050 \cite{lisazeyen_2022_6885392}, the average marginal prices reach only 51~\euro/MWh, compared to 59~\euro/MWh in the 2020 scenario. As a result, the optimized renewable energy future for Nigeria can be more economical than the current energy system design.
\begin{figure}[h!]
\centering
\subfigure[]{\includegraphics[width=0.65\textwidth, trim={0cm 0cm 0cm 0cm}, clip,]{Illustrations/nigeria-2020-optimization.pdf}}
\subfigure[]{\includegraphics[width=0.3\textwidth, trim={0cm 0cm 0cm 0cm}, clip,]{Illustrations/nigeria-2020-loadmap.pdf}}
\caption{Optimization results of Nigeria's (a) 2020 power system. The coloured points represent installed capacities. (b) Shows all network options on a different scale as (a) with the total electricity consumption per node.}
\label{fig: nigeria 2020}
\end{figure}
\begin{figure}[h!]
\subfigure[]{\includegraphics[width=0.65\textwidth]{Illustrations/nigeria-2060-optimization.pdf}}
\subfigure[]{\includegraphics[width=0.3\textwidth, trim={0cm 0cm 0cm 0cm}, clip,]{Illustrations/nigeria-2060-loadmap.pdf}}
\caption{Optimization result represent Nigeria's (a) 2060 power system. The coloured points represent installed capacities. Light grey and dark grey lines are existing and newly optimized transmission lines, respectively. (b) Shows all network options on a different scale as (a) with the total electricity consumption per node.}
\label{fig: nigeria 2060}
\end{figure}
\section{Limitations and future opportunities}\label{sec: limitations and future developments}
\subsection{Missing network topology data}
\label{subsec: satellite image detection}
Modelling can only be as good as underlying data -- the same applies for PyPSA\=/Earth. By relying on open sources to model energy systems, their data quality is a concern that we also acknowledge for the present paper. Yet, we also describe possible procedures based on image recognition to not only improve the data situation in PyPSA\=/Earth but potentially all energy models. This effort may complement the traditional effort by public institutions that disclose data of public relevance, such as installed network infrastructure, as performed by ENTSO\=/E \cite{ENTSOE2022}.
Compared to Europe or North America, institutions that provide infrastructure data with geolocation for modelling have no analogues in Africa. The missing network data situation is limiting the use of energy system models.
However, other types of data from which energy system components can be inferred exist on a much larger scale.
Satellite imagery is one such data type.
As part of the PyPSA meets Earth initiative, we are exploring opportunities to use neural network based object detection applied to satellite imagery to enrich the existing datasets on energy infrastructure.
In the past, such efforts have either been hard to apply on larger scales due to high requirements on manual input \cite{devseed2017} or are coarse approximations to the true grid structure \cite{Arderne2020PredictiveData}.
Under the umbrella of this initiative, we aim to develop precise and scalable methods and base our efforts on recent advances in the field \cite{huang2021gridtracer, talukdar2018transfer, deng2021unbiased, maxar}.
\subsection{Missing demand time series and prediction biases}\label{subsec: missing demand}
Demand is a significant uncertainty factor in Africa due to the growth in magnitudes over the following decades that has implications on results created by PyPSA\=/Earth. Therefore, improving demand predictions is essential. We acknowledge this limitation in the data fed into our model and highlight research opportunities to address this challenge.
PyPSA-Earths demand data is limited, as indicated in Section \ref{subsec: demand}, by poor prediction performance for low-income countries due to input data biases and by missing machine learning output data for some low-demand countries due to software bugs. Additionally, while the open-source \textit{synde} package~\cite{Synde2022} used in PyPSA-Earth extended the original GEGIS package for demand prediction by a workflow, there are opportunities to create a package focusing only on demand prediction substituting the GEGIS design that provides all energy model data in one package \cite{Mattsson2021AnRegions}.
Developing a package focusing on electricity demand forecast for macro-energy system modelling worldwide is an opportunity to improve the status quo of existing tools. Its scope can also be generalised to other energy carriers. Adopting the findings from \cite{Synde2022}, \cite{Mattsson2021AnRegions} and \cite{TOKTAROVA2019160}, the output of such package can be static and time series information for the demand of any year beyond 2000, depending on the available features, for arbitrary regions on a sub-national and national scale. Creating such a package is feasible but needs considerable work and a sustainably funded group of people who enhance the software.
\subsection{Imprecise global data}\label{subsec: linkers}
PyPSA\=/Earth relies on open data with global scope. This means that sometimes data is used that approximate country-specific details needed for national energy planning studies.
While improving the global open data situation is one opportunity \cite{Ritchie2021}, another is to enable the integration of national and regional more precise data that can also be used as a source for validation.
Therefore, PyPSA\=/Earth does not only use global data as default but also allows the integration of national or regional more precise data. Accordingly, specialized functions here coined as "linkers", can enable the fetching of these region-specific to replace the default global data.
These linkers allow, for instance, local transmission system operators to execute models with national data. PyPSA\=/Earth does not include this national potentially more precise data by default but workflow contributions to make these optional are welcome and can help maximise benefits for the society by better planning studies.
\subsection{Additional technologies}
\label{sec: additional techs}
PyPSA\=/Earth includes the major transmission, generation and storage technologies, however, some are not yet included. Examples of not implemented generation technologies are Concentrating Solar Power (CSP), location-based geothermal, and other secondary technologies such as wave/tidal energy harvesting. While at a global scale, these technologies represent a minor fraction, for country-specific analyses, they may have substantial implications, such as in the case of Kenya for geothermal or Morocco for CSP. Moreover, while currently only lithium-ion batteries and hydrogen energy storage are considered, additional technologies may be considered and tested, such as the well-known Redox Flow batteries, Compressed-Air Energy Storage (CAES), Liquified-Air Energy Storage (LAES), that can have a large market in the future. Moreover, the dynamic calculation of the transmission capacity as a function of weather conditions \cite{Horsch2018PyPSA-Eur:System}, also known as Dynamic Line Rating (DLR) \cite{Poli2019}, is not yet included.
These limitations, at the time of writing, represent future opportunities to improve the model and capture relevant technologies to perform detailed energy studies for all countries.
\section{Conclusions}\label{sec: conclusion}
This paper presents the PyPSA\=/Earth model, which is the first open-source global energy system model in high spatial and temporal resolution. It is making high-resolution modelling accessible to countries which so far had not detailed energy planning scenarios developed.
Using a novel comprehensive workflow procedure PyPSA\=/Earth automatically downloads open data, provides model-ready data and integrates optimization features to address large scale energy system planning.
In agreement with the open-source spirit, the model is not built from scratch but derived from the European-focused PyPSA\=/Eur model adding global data as well as several new features.
The methodology is confirmed to be flexible and accommodate a high temporal and spatial resolution energy model for national and regional energy planning with global scope. The validation performed for the African continent highlighted that PyPSA\=/Earth successfully provides power network and installed generation data that match trustworthy third-party national data with adequate accuracy, hence suggesting PyPSA\=/Earth to be a reliable model for energy planning. The 2020 and 2060 planning studies for Nigeria have further confirmed that net-zero emission scenarios for the electricity sector can be performed using PyPSA\=/Earth, leading to realistic results comparable with similar studies but in higher spatial detail. That further stresses the robustness of the approach and the flexibility of the methodology to be used in practical projects.
Given the need for reliable tools to foster the energy transition and the need for the efficient use of resources, PyPSA\=/Earth can successfully support policymakers, utilities, and scholars in providing reliable, transparent, and efficient decision-making on energy studies. While several open source projects are developed but discontinued, the authors of this paper and PyPSA\=/Earth developers aim to foster collaborative energy system modelling on the same code-base to provide a well-maintained and robust tool, rather than disperse resources across multiple models that get easily outdated. Given the flexibility of the approach, additional improvements can be integrated, and scholars interested in contributing are invited to contact the PyPSA\=/Earth team to join forces. Accordingly, this paper and the proposed tool can serve as a backbone for further research and business activities built on top of PyPSA\=/Earth, to meet various energy transition planning needs that must be cheap and fast to develop for every nation and community on Earth.
Further studies, may address the sector-coupled version of PyPSA\=/Earth, to account for sectors beyond power (e.g. industry and/or transport), the interface of energy modelling with economics modelling for better energy policy decisions, the improvement of the demand forecasts also in alignment to climate change scenarios, the improvement of imprecise global network data using object detection on satellite images or the validation of the model in other regions.
\section*{Code and Data availability}\label{sec:codedata}
Code and data to reproduce results and illustrations are available by using PyPSA-Earth v0.1 \cite{pypsaafrica}. Further, instruction and configurations to reproduce the scenarios and plots are provided here: {\color{blue}\href{ https://github.com/pz-max/pypsa-earth-paper}{https://github.com/pz-max/pypsa-earth-paper}}.
\section*{Credit authorship contribution statement}
M.P, D.F., F.N., conceptualised the study; M.P., D.F administrated the project; M.P., D.F., H.A., E.F., M.M., M.F., J.H., L.S. contributed to the software development, validation and figure production,
A.K. acquired the funding; all authors contributed to writing and revising the manuscript.
\section*{Declaration of Competing Interest}
All authors declare that there are no competing interests.
\section*{Acknowledgements}
This research was supported by UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/P007805/1 for the Centre for Advanced Materials for Renewable Energy Generation (CAMREG) and EPSRC grant EP/V042955/1 DISPATCH. Maximilian Parzen would like to thank Tom Brown, Aminu Haruna Isa, Dahunsi Okekunle, Matija Pavičević, Michael Dioha and every one of our continuous supporters for their helpful comments and inspiring discussion.
\section{Appendix}
\bibliographystyle{elsarticle-num-names}
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\section{Introduction}
Convex functions in $\rm CAT(0)$ spaces were first considered during the 1990s. In particular, Jost \cite{MR1360608} and Mayer \cite{MR1651416} independently studied the proximal operators of convex functions in complete $\rm CAT(0)$ and applied them to investigate harmonic functions and also gradient flows. The proximal operators were later used by Ba\v{c}\'ak \cite{MR3047087} to study proximal algorithms for minimizing convex functions. He also proved a semigroup generation theorem from the exponential formula applied to these proximal operators. Further explanations can also be found in \cite{MR3241330}. In 2017, Khatibzadeh and Ranjbar \cite{MR3679017} as well as Chaipunya and Kumam \cite{MR3691338} generalized the results of Ba\v{c}\'ak \cite{MR3047087} by introducing the monotone operators and examined proximal algorithms using the dual space concept introduced earlier by Ahmadi Kakavandi and Amini \cite{MR2680038} (also see \cite{MR3003694}). The dual space used in \cite{MR2680038} is known to generalize the usual dual space when the linear structure is provided. However, it is still unknown (as also posted in \cite{MR2680038}) what relationship this dual space has with the Riemannian tangent spaces when the space in question is an Hadamard manifold. Consequently, the monotone operators introduced in this aspect barely have obvious or simple relationships with monotone vector fields on Hadamard manifolds as introduced by N\'emeth \cite{MR1694468} and later enriched by Li et al. \cite{MR2506692}.
In this present paper, we adopt a new approach to generalize and unify the concept of monotonicity into complete $\rm CAT(0)$ spaces and ultimately prove a semigroup generation theorem. Instead of applying the dual space of \cite{MR2680038}, we make use of the tangent spaces (also called tangent cones) of Nikolaev \cite{MR1317739} which is consistent with the contexts of Hilbert spaces or Hadamard manifolds. With this nice attribute of tangent spaces, the concept of monotone vector fields introduced in this paper is a natural complement to the theory in Hilbert spaces as well as in Hadamard manifolds. In addition to the main results, we introduce and deduce several fundamental properties of resolvents and Yosida approximations which become the major machinery in proving the generation theorem and are also important in their own rights.
Note that the convergence procedure applied in our generation theorem (Theorem \ref{thm:S(t)x}) was used by Jost \cite[Theorem 1.3.13]{MR1652278}, Mayer \cite[Theorem 1.13]{MR1651416}, Stojkovic \cite[Theorem 2.18]{MR2879568}, and Ba\v{c}\'{a}k \cite[Theorem 1.5]{MR3047087} for proper, convex, lower semicontinuous functions in the context of complete $\rm CAT(0)$ spaces, and also by Iwamiya and Okochi \cite[Theorem 4.1]{MR1979730} for monotone vector fields in the context of Hilbert manifolds. Our generation theorem elevates the abovementioned results of \cite{MR1652278,MR1651416,MR3047087} to monotone vector fields and also improves that of \cite{MR1979730} when reduced to an Hadamard manifold, taken into account the equivalent formulation studied by Wang et al. \cite[Corollary 3.8]{MR2720608}.
The rest of this paper is organized as follows. In Section 2, we collect background materials which are required for our main results in subsequent sections. Especially, the definitions and properties of a $\rm CAT(0)$ space and its tangent spaces are given here. Moreover, several useful inequalities and estimates are also derived thereof. In Section 3, we give the definition of a monotone vector field and provide fundamental observations accordingly. Moreover, we investigate a convex subdifferential as an example for a monotone vector field. In Section 4, the notion of resolvents for a monotone vector field is given and several tools are obtained. We then further construct the Yosida approximations, which will be used as a central equipment in the generation theorem. Important results for this section include the resolvent identity, the convexity of closed domains for monotone vector fields, asymptotic convergence on both ends for the resolvent operators, and the Yosida approximation estimate. The final Section 5, the generation theorem is proved and a simple convergence for the corresponding trajectories are derived.
\section{Preliminaries}
Recall that a metric space $(X,d)$ is said to be \emph{geodesic} if for each two points $x,y \in X$, there exists a mapping $c : [0,\ell] \to X$ (with $\ell \geq 0$) and a constant $K \geq 0$ such that $c(0) = x$, $c(\ell) = y$, and $d(c(t),c(t')) = K\abs{t-t'}$ for any $t,t' \in [0,\ell]$. In this way, $c$ is called a \emph{geodesic} joining $x$ and $y$, and it is said to \emph{issue from} $x$ and \emph{end} at $y$. We say that $c$ is a \emph{zero geodesic} at $x$ if $x = y$, otherwise we say that it is \emph{nonzero}. Let us adopt the zero geodesic indicator $\zeta(c) := 0$ if $c$ is a zero geodesic and $\zeta(c) := 1$ otherwise. We say that $c$ is \emph{normalized} if $\ell = 1$ and that it is of \emph{unit speed} of $K = 1$. In the latter case, we also have $K = \rho(x,y)$. On the other hand, a mapping $c : \mathbb{R} \to X$ is called a \emph{geodesic line} if there exists a strictly positive constant $K > 0$ such that $\rho(c(t),c(t')) = K\abs{t-t'}$ for all $t,t' \in \mathbb{R}$.
In the sequel where the choice of geodesics is insignificant or where the geodesic is unique, we write $\gamma_{x,x}$ to denote the zero normalized geodesic at $x$, and we write $\gamma_{x,y}$ to denote the nonzero normalized geodesic joining $x$ and $y$ with $x \neq y$. We also write $\llbracket x,y \rrbracket$ to denote the image of $\gamma_{x,y}$ over $[0,1]$.
Here and henceforth, let $(\mathbb{E}^{2},\product{\cdot}{\cdot})$ be the Euclidean plane with usual inner product $\product{u}{v} := u^{\top}v$ and the Euclidean norm $\norm{u} := \sqrt{\product{u}{u}}$, for $u,v \in \mathbb{E}^{2}$. For each points $p,q,r \in X$, the geodesic triangle $\Delta \subset X$ is defined by $\Delta(pqr) := \llbracket p,q \rrbracket \cup \llbracket q,r \rrbracket \cup \llbracket r,p \rrbracket$. The triangle defined by $\overline{\Delta}(\bar{p}\bar{q}\bar{r}) := \Delta(\bar{p}\bar{q}\bar{r})$ with $\bar{p},\bar{q},\bar{r} \in \mathbb{E}^{2}$ is said to be a \emph{Euclidean comparison triangle}, or simply a \emph{comparison triangle}, if $\norm{\bar{p} - \bar{q}} = d(p,q)$, $\norm{\bar{q} - \bar{r}} = d(q,r)$, and $\norm{\bar{r} - \bar{p}} = d(r,p)$. Note that the triangle inequality of $d$ implies the existence of such comparison triangle. Moreover, the comparison triangle of each geodesic triangle in $X$ is unique up to rigid motions. Suppose that $\Delta(pqr) \subset X$ is a geodesic triangle whose comparison triangle is $\overline{\Delta}(\bar{p}\bar{q}\bar{r})$. Given $u \in \llbracket p,q \rrbracket$, the point $\bar{u} \in \llbracket \bar{p},\bar{q} \rrbracket$ is said to be a \emph{comparison point} of $u$ if $\norm{\bar{p} - \bar{u}} = d(p,u)$. Comparison points for $u' \in \llbracket q,r \rrbracket$ and $u'' \in \llbracket r,p \rrbracket$ are defined likewise.
\begin{dfn}
A geodesic metric space $(X,d)$ is said to be a \emph{$\rm CAT(0)$ space} if for each geodesic triangle $\Delta \subset X$ and two points $u,v \in \Delta$, the following $\rm CAT(0)$ inequality holds:
\[
d(u,v) \leq \norm{\bar{u} - \bar{v}},
\]
where $\bar{u},\bar{v} \in \overline{\Delta}$ are the comparison points of $u$ and $v$, respectivly, and $\overline{\Delta} \subset \mathbb{E}^{2}$ is a comparison triangle of $\Delta$. A complete $\rm CAT(0)$ space is also called an \emph{Hadamard space}.
\end{dfn}
The following proposition gives useful characterizations of the $\rm CAT(0)$ inequality.
\begin{prop}\label{Prop:CAT(0)Char}
Suppose that $(X,d)$ is a geodesic metric space. Then, the following conditions are equivalent:
\begin{enumerate}[label=\rm{(\roman*)}]
\item $X$ is a $\rm CAT(0)$ space.
\item\label{cdn:CN} For all $v \in X$ and a normalized geodesic $\gamma : [0,1] \to X$, the following \eqref{eqn:CN} inequality holds for any $t \in [0,1]$:
\[\label{eqn:CN}
d^{2}(\gamma(t),v) \leq (1-t)d^{2}(\gamma(0),v) + td^{2}(\gamma(1),v) - t(1-t)d^{2}(\gamma(0),\gamma(1)). \tag{CN}
\]
\item\label{cdn:quad-characterization-CAT(0)} For all $x,y,u,v \in X$, the following inequality holds:
\begin{equation}\label{eqn:quad-characterization-CAT(0)}
d^{2}(x,v) + d^{2}(y,u) \leq d^{2}(x,u) + d^{2}(y,v) + 2d(x,y)d(u,v).
\end{equation}
\end{enumerate}
\end{prop}
Unless otherwise specified, always assume that $(H,\rho)$ is a complete $\rm CAT(0)$ space. Note that $H$ is always uniquely geodesic and a subset $C \subset H$ is called \emph{convex} if $\llbracket x,y \rrbracket \subset C$ for all $x,y \in C$.
\begin{thm}[\cite{MR3241330}]\label{thm:projection}
Let $C \subset H$ be nonempty, closed, and convex, and $P_{C} : H \multimap C$ be a mapping defined by
\[
P_{C}(x) := \argmin_{y \in C} \rho(x,y) \quad (\forall x \in H).
\]
Then, the following assertions hold:
\begin{enumerate}[label=\rm{(\roman*)}]
\item $P_{C}$ is defined for all $x \in H$ and is single-valued.
\item\label{cdn:ProjIneq} If $x \in H$ and $z \in C$, then
\[
\rho^{2}(x,P_{C}(x)) + \rho^{2}(P_{C}(x),z) \leq \rho^{2}(x,z).
\]
\item\label{cdn:ProjNX} $P_{C} : H \to C$ is nonexpansive.
\end{enumerate}
\end{thm}
\subsection{$\Delta$-Convergence}
If $(x^{k})$ is a bounded sequence in $H$, then the functional $H \ni x \mapsto \limsup_{k} \rho(x,x^{k})$ is finite and admits a unique minimizer \cite{MR2232680}. Such unique minimizer is called the \emph{asymptotic center} of $(x^{k})$. In this case, it is immediate that if $z = \argmin_{x \in H} \limsup_{k} \rho(x,x^{k})$, then the following \emph{Opial property} holds: $\limsup_{k} \rho(z,x^{k}) < \limsup_{k} \rho(u,x^{k})$ for all $u \in H \setminus\{z\}$. This asymptotic center is used in defining the so-called $\Delta$-convergence in the following definition.
\begin{dfn}[\cite{MR2416076}]
A bounded sequence $(x^{k})$ in $H$ is said to be \emph{$\Delta$-convergent} to a point $x \in H$ if $x$ is the (unique) asymptotic center of all subsequences of $(x^{k})$. In this case, $x$ is said to be the \emph{$\Delta$-sequential limit} of $(x^{k})$ \cite[Corollary 3.2.4]{MR3241330}.
\end{dfn}
We say a function $F : H \to \mathbb{R}$ is called \emph{$\Delta$-lower semicontinuous} at $\bar{x} \in H$ if
\[
F(\bar{x}) \geq \liminf_{k \longrightarrow \infty} F(x^{k})
\]
for any sequence $(x^{k})$ in $H$ with $\Delta$-sequential limit $\bar{x}$. In particular, the function $\rho(\cdot,x)$ is $\Delta$-lower semicountinuous on $H$ for each fixed $x \in H$.
Given any bounded sequence $(x^{k})$ in $H$, we write $\omega_{\Delta}(x^{k})$ to denote the set of all \emph{$\Delta$-subsequential limits} of $(x^{k})$, i.e., the set of all possible $\Delta$-sequential limits of subsequences of $(x^{k})$. In every $\rm CAT(0)$ spaces, such a set is nonempty for all bounded sequence \cite{MR3047087}. It is evident that strong convergence (i.e., convergence in metric) implies $\Delta$-convergence, but the converse implication is not true in general.
\begin{prop}\label{prop:singletonOmegaDelta}
Let $z \in H$ and $(x^{k})$ be a bounded sequence in $H$ with $\omega_{\Delta}(x^{k}) = \{z\}$. Then, $(x^{k})$ is $\Delta$-convergent to $z$.
\end{prop}
\begin{proof}
Let us suppose to the contrary that $(x^{k})$ is not $\Delta$-convergent to $z$. Hence, there exists a subsequence $(u^{k})$ of $(x^{k})$ such that the asymptotic center $u \in H$ of $(u^{k})$ is different from the point $z$.
Let $(v^{k})$ be a subsequence of $(u^{k})$ such that $\limsup_{k} \rho(z,v^{k}) = \limsup_{k} \rho(z,u^{k})$ and that $(v^{k})$ is $\Delta$-convergent to some point $v \in H$. Since $(x^{k})$ has only one $\Delta$-subsequential limit, it must be the case that $v = z$. Using the Opial property, the fact that $(v^{k}) \subset (u^{k})$, and the definition of asymptotic center, we obtain
\begin{align*}
\limsup_{k \longrightarrow \infty} \rho(z,u^{k}) &= \limsup_{k \longrightarrow \infty} \rho(z,v^{k})
< \limsup_{k \longrightarrow \infty} \rho(u,v^{k}) \\
&\leq \limsup_{k \longrightarrow \infty} \rho(u,u^{k})
< \limsup_{k \longrightarrow \infty} \rho(z,u^{k}),
\end{align*}
which is a contradiction. Therefore, $(x^{k})$ must be $\Delta$-convergent to $z$.
\end{proof}
\begin{lem}[Kadec-Klee property \cite{MR2754196}]\label{lem:Kadec-Klee}
Suppose that $(x^{k})$ is $\Delta$-convergent to $x \in H$ and $\lim_{k} \rho(x^{k},p) = \rho(x,p)$ for some $p \in H$, then $(x^{k})$ is strongly convergent to $x$.
\end{lem}
Combining the $\Delta$-convergence with the strong convergence implies the following Demiclosedness Principle, which is an important result in fixed point theory.
\begin{thm}[\cite{MR2416076}]\label{thm:demiclosed}
Suppose that $C \subset H$ is nonempty, closed, and convex, and $T : C \to C$ is nonexpansive. If $(x^{k})$ is a sequence in $H$ that is $\Delta$-convergent to $x$ and that $\lim_{k} \rho(x^{k},Tx^{k}) = 0$, then $Tx = x$.
\end{thm}
The following concept of convergence and its properties are very essential in our works.
\begin{dfn}
A sequence $(x^{k})$ (resp. net $(x_{t})_{t \geq 0}$) in $H$ is said to be \emph{Fej\'er monotone} with respect to a nonempty set $V \subset H$ if for each $x \in V$, we have $\rho(x^{k+1},x) \leq \rho(x^{k},x)$ for all $k \in \mathbb{N}$ (resp. $\rho(x_{t},x) \leq \rho(x_{s},x)$ for all $0\leq s \leq t$).
\end{dfn}
\begin{prop}[\cite{MR3241330}]\label{prop:Fejer}
Suppose that $(x^{k}) \subset H$ is Fej\'er monotone with respect to a nonempty set $V \subset H$. Then, the following are true:
\begin{enumerate}[label=\rm{(\roman*)}]
\item $(x^{k})$ is bounded.
\item $(\rho(x,x^{k}))$ converges for any $x \in V$.
\item If every $\Delta$-accumulation point lies within $V$, then $(x^{k})$ is $\Delta$-convergent to an element in $V$.
\end{enumerate}
\end{prop}
\begin{rmk}
We can replace the sequence $(x^{k})$ in the above proposition also with a net $(x_{t})_{t \geq 0}$ and still obtain similar results.
\end{rmk}
\subsection{Tangent Spaces}
Tangent spaces (also called tangent cones) to a given $\rm CAT(0)$ space were introduced earlier in \cite{MR1317739} (see also \cite{MR1744486,MR1835418}). However, we make a slight modification on their representations in this paper for the future technical convenience in our studies.
In order to introduce the tangent space and related notions subsequently, we first recall the notion of comparison angle with respect to $\mathbb{E}^{2}$.
\begin{dfn}
Suppose that $p,q,r \in H$. The \emph{comparison angle} between $q$ and $r$ at $p$, denoted with $\overline{\angle}_{p}(q,r)$, is given as follows: If $q,r \in H \setminus \{p\}$, we set
\[
\cos \overline{\angle}_{p}(q,r) := \frac{\product{\bar{q}-\bar{p}}{\bar{r}-\bar{p}}}{\norm{\bar{q}-\bar{p}}\norm{\bar{r}-\bar{p}}},
\]
where $\overline{\Delta}(\bar{p},\bar{q},\bar{r}) \subset \mathbb{E}^{2}$ is the comparison triangle of the geodesic triangle $\Delta(p,q,r) \subset H$. On the other hand, we set $\overline{\angle}_{p}(p,p) := 0$, and $\overline{\angle}_{p}(p,r) = \overline{\angle}_{p}(r,p) := \frac{\pi}{2}$ for $r \in X\setminus\{p\}$.
\end{dfn}
Given two geodesics $\gamma_{1},\gamma_{2}$ on $H$ issuing from a common point $p \in H$. The \emph{Alexandrov angle} between the two geodesics is then defined by
\[
\alpha_{p} (\gamma_{1},\gamma_{2}) := \lim_{s,t \longrightarrow 0^{+}} \overline{\angle}_{p} (\gamma_{1}(s),\gamma_{2}(t)).
\]
To effectively compute the Alexandrov angle, the \emph{First Variation Formula} is available in the following form:
\begin{lem}[First Variation Formula]\label{thm:FVF}
Suppose that $p \in H$, $u \in H \setminus \{p\}$, and $\gamma$ is a nonzero unit-speed geodesic issuing from $p$. Then the following identity holds:
\[
\lim_{s \longrightarrow 0^{+}} \frac{\rho(u,p) - \rho(u,\gamma(s))}{s} = \cos \alpha_{p}(\gamma_{p,u},\gamma).
\]
\end{lem}
Recall that the \emph{metric identification} of a pseudometric space $(\tilde{M},\tilde{d})$ is a metric space $(M,d)$, where $M$ consists of equivalence classes $[x] := \{y \in X \,|\, \tilde{d}(x,y) = 0\}$ of $x \in X$ and $d([x],[y]) := \tilde{d}(x,y)$ for all $[x],[y] \in M$.
Denoted by $\tilde{S}_{p}$ the set of all normalized geodesics issuing from $p \in H$. Then $\tilde\angle_{p} := \alpha_{p}$ defines a pseudometric on $\tilde{S}_{p}$. The metric identification of $(\tilde{S}_{p},\tilde\angle_{p})$, denoted by $(S_{p},\angle_{p})$, is called the \emph{space of directions at $p$}. In the sequel, we write $\gamma \equiv [\gamma]$ for elements of $S_{p}$. Suppose that $\sim$ is an equivalence relation on $[0,\infty) \times S_{p}$ such that $(t_{1},\gamma_{1}) \sim (t_{2},\gamma_{2})$ if and only if one of the following conditions is satisfied:
\begin{enumerate}[label=(T\arabic*)]
\item\label{cdn:T1} $t_{1}\zeta(\gamma_{1}) = t_{2}\zeta(\gamma_{2}) = 0$ or
\item\label{cdn:T2} $t_{1}\zeta(\gamma_{1}) = t_{2}\zeta(\gamma_{2}) > 0$ and $\gamma_{1} = \gamma_{2}$.
\end{enumerate}
Put $T_{p}H := \left([0,\infty) \times S_{p}\right) / \sim$ and whenever there is no ambiguity, let us write $t\gamma \equiv [(t,\gamma)]_{\sim}$ for elements of $T_{p}H$ to simplify the notions.
Next, we endow $T_{p}H$ with a metric $d_{p}$ defined for each $t_{1}\gamma_{1},t_{2}\gamma_{2} \in T_{p}H$ by
\[
d_{p}(t_{1}\gamma_{1},t_{2}\gamma_{2}) := \sqrt{t_{1}^{2}\zeta(\gamma_{1}) + t_{2}^{2}\zeta(\gamma_{2}) - 2t_{1}t_{2}\zeta(\gamma_{1})\zeta(\gamma_{2})\cos \angle_{p}(\gamma_{1},\gamma_{2})}.
\]
To see the metric properties of $d_{p}$, we first note that the inequalities
\begin{equation}\label{eqn:aux_trig}
\abs{t_{1}\zeta(\gamma_{1}) - t_{2}\zeta\gamma_{2})} \leq d_{p}(t_{1}\gamma_{1},t_{2}\gamma_{2}) \leq t_{1}\zeta(\gamma_{1}) + t_{2}\zeta(\gamma_{2})
\end{equation}
hold for any $t_{1}\gamma_{1},t_{2}\gamma_{2} \in T_{p}H$. Let $\mathbb{T}_{p}H := \left([0,\infty) \times S'_{p}\right)/\approx$, where $S'_{p} := \{\gamma \in S_{p} \,|\, \zeta(\gamma) = 1\}$ and $\approx \,:=\, \sim \upharpoonright _{\mathbb{T}_{p}H}$ is the restricted equivalence relation. One can see easily that $[(t,\gamma)]_{\approx} = [(t,\gamma)]_{\sim}$ whenever $t > 0$ and $\gamma \in S'_{p}$. Let us put
\[
\left\{\begin{array}{l}
X_{+} := \{[(t,\gamma)]_{\sim} \,|\, t > 0, \zeta(\gamma) = 1\}; \\[.5em]
X_{0} := \{[(0,\gamma)]_{\sim}\} = \big\{\{(t,\gamma) \,|\, t = 0 \vee \zeta(\gamma) = 0\}\big\}; \\[.5em]
X_{0}' := \{[(0,\gamma)]_{\approx}\} = \big\{\{(t,\gamma) \,|\, t = 0, \zeta(\gamma) = 1\}\big\}.
\end{array}
\right.
\]
We can then write $T_{p}H$ and $\mathbb{T}_{p}H$ with the following representations:
\begin{equation}\label{eqn:representation}
\text{$T_{p}H = X_{+} \cup X_{0}$ \,\, and \,\, $\mathbb{T}_{p}H = X_{+} \cup X_{0}'.$}
\end{equation}
According to \cite{MR1744486} and \cite{MR1835418}, $\mathbb{T}_{p}H$ is a metric space with respect to a metric $D_{p}$ given by
\[
D_{p}([(t_{1},\gamma_{1})]_{\approx},[(t_{2},\gamma_{2})]_{\approx}) := \sqrt{t_{1}^{2} + t_{2}^{2} - 2t_{1}t_{2}\cos \angle_{p}(\gamma_{1},\gamma_{2})}, \quad \forall [(t_{1},\gamma_{1})]_{\approx},[(t_{2},\gamma_{2})]_{\approx} \in \mathbb{T}_{p}H.
\]
We will show now that $(T_{p}H,d_{p})$ is a metric space and it is isometry to $(\mathbb{T}_{p}H,D_{p})$.
\begin{prop}
$d_{p}$ is a metric on $T_{p}H$.
\end{prop}
\begin{proof}
Let $t_{1}\gamma_{1},t_{2}\gamma_{2} \in T_{p}H$. By \eqref{eqn:aux_trig}, we can see that
\begin{align*}
d_{p}(t_{1}\gamma_{1},t_{2}\gamma_{2}) = 0 &\iff \text{$t_{1}\zeta(\gamma_{1}) = t_{2}\zeta(\gamma_{2})$ and $t_{1}^{2}(1 - \cos \angle_{p} (\gamma_{1},\gamma_{2})) = 0$} \\
&\iff \text{either \ref{cdn:T1} or \ref{cdn:T2} holds} \\
&\iff t_{1}\gamma_{1} = t_{2}\gamma_{2}.
\end{align*}
Next, since $\angle_{p}$ is a metric on $S_{p}$, we obtain $d_{p}(t_{1}\gamma_{1},t_{2}\gamma_{2}) = d_{p}(t_{2}\gamma_{2},t_{1}\gamma_{1})$ immediately. It remains to show that the triangle inequality
\[
d_{p}(t_{1}\gamma_{1},t_{3}\gamma_{3}) \leq d_{p}(t_{1}\gamma_{1},t_{2}\gamma_{2}) + d_{p}(t_{2}\gamma_{2},t_{3}\gamma_{3})
\]
holds for any $t_{i}\gamma_{i} \in T_{p}H$, with $i = 1,2,3$. If $t_{i} > 0$ and $\zeta(\gamma_{i}) = 1$ for all $i = 1,2,3$, we can use the fact that $d_{p}$ is a metric on $X_{+}$ to obtain the triangle inequality. If $t_{1} = 0$ or $\zeta(\gamma_{1}) = 0$, then it follows from \eqref{eqn:aux_trig} that
\begin{align*}
d_{p} (t_{1}\gamma_{1},t_{3}\gamma_{3}) &= t_{3}\zeta(\gamma_{3}) \\
&\leq t_{2}\zeta(\gamma_{2}) + \abs{t_{2}\zeta(\gamma_{2}) - t_{3}\zeta(\gamma_{3}) } \\
&\leq t_{2}\zeta(\gamma_{2}) + d_{p}(t_{2}\gamma_{2},t_{3}\gamma_{3}) \\
&= d_{p}(t_{1}\gamma_{1},t_{2}\gamma_{2}) + d_{p}(t_{2}\gamma_{2},t_{3}\gamma_{3}).
\end{align*}
Similar procedure also works when $t_{3} = 0$ or $\zeta(\gamma_{3}) = 0$. Finally, if $t_{2} = 0$ or $\zeta(\gamma_{2}) = 0$, then \eqref{eqn:aux_trig} implies
\[
d_{p}(t_{1}\gamma_{1},t_{3}\gamma_{3}) \leq t_{1}\zeta(\gamma_{1}) + t_{3}\zeta(\gamma_{3}) = d_{p}(t_{1}\gamma_{1},t_{2}\gamma_{2}) + d_{p}(t_{2}\gamma_{2},t_{3}\gamma_{3}).
\]
Hence, we now conclude that $d_{p}$ is a metric on $T_{p}H$.
\end{proof}
\begin{prop}
There is a bijection which preserves distances between $T_{p}H$ and $\mathbb{T}_{p}H$.
\end{prop}
\begin{proof}
We may see from \eqref{eqn:representation} that a mapping $\varphi_{p} : \mathbb{T}_{p}H \to T_{p}H$ given by
\[
\varphi_{p} \left([(t,\gamma)]_{\approx}\right) := \left\{\begin{array}{ll}
[(t,\gamma)]_{\sim} &\text{if $[(t,\gamma)]_{\approx} \in X_{+}$;} \\[.5em]
[(0,\gamma)]_{\sim} &\text{if $[(t,\gamma)]_{\approx} \in X_{0}'$}
\end{array}
\right.
\]
is a bijection. That is, $\varphi_{p}$ is identity on $X_{+}$ and maps $X_{0}'$ onto $X_{0}$. The fact that $\varphi_{p}$ preserves distances is trivial.
\end{proof}
The metric space $(T_{p}H,d_{p})$ is henceforth called the \emph{tangent space} of $H$ at $p$. The \emph{tangent bundle} of $H$ is then defined by $TH := \bigcup_{p \in H} T_{p}H$. The isometry result above ensures that it is consistent with the classical notion of tangent spaces of a complete $\rm CAT(0)$ space as was given by \cite{MR1317739}. Moreover, we also have further implication in cases of $H$ being a Hilbert space or an Hadamard manifold. For instance, if $\mathcal{H}$ is a Hilbert space and $p \in \mathcal{H}$, then $T_{p}\mathcal{H}$ is isometric with $\mathcal{H}-\{p\}$ (and hence to $\mathcal{H}$) by the canonical map $t\gamma \mapsto \gamma(t) - p$. On the other hand, let $M$ be an Hadamard manifold with Riemannian tangent space at $p \in M$ denoted by $\mathds{T}_{p}M$. We know, in this case, that the exponential map $\exp_{p} : \mathds{T}_{p}M \to M$ is well-defined on the whole tangent space and is a diffeomorphism. Then $T_{p}M$ is isometric with $\mathds{T}_{p}M$ by the canonical map $t\gamma \mapsto \exp_{p}^{-1} t\dot\gamma(0)$, where we use the convention $\dot\gamma_{p,p}(0) = 0_{\mathds{T}_{p}M}$. See also \cite{MR1744486,MR1835418} and references therein for further information.
On each tangent space $T_{p}H$ we write $0_{p} := 0\gamma = t\gamma_{p,p}$ (here, $\gamma \in S_{p}$ and $t > 0$) and $\norm{t\gamma}_{p} := d_{p}(0_{p},t\gamma) = t\zeta(\gamma)$. For convenience, we invoke the notion $\mathbf{0} := \{0_{p} \,|\, p \in H\}$ the \emph{zero section} of $TH$. Moreover, we adopt the product
\[
g_{p}(t_{1}\gamma_{1},t_{2}\gamma_{2}) := \frac{1}{2} \left[ \norm{t_{1}\gamma_{1}}_{p}^{2} + \norm{t_{2}\gamma_{2}}_{p}^{2} - d_{p}^{2}(t_{1}\gamma_{1},t_{2}\gamma_{2})\right]
\]
for any $t_{1}\gamma_{1},t_{2}\gamma_{2} \in T_{p}H$. By a direct calculation, we can deduce that
\[
g_{p}(t_{1}\gamma_{1},t_{2}\gamma_{2}) = t_{1}t_{2}\zeta(\gamma_{1})\zeta(\gamma_{2})\cos \angle_{p}(\gamma_{1},\gamma_{2}) \leq \norm{t_{1}\gamma_{1}}_{p}\norm{t_{2}\gamma_{2}}_{p},
\]
which is an analogue of the Cauchy-Schwarz inequality. It is easy to see that
\[
g_{p}(t_{1}\gamma_{1},t_{1}\gamma_{1}) = \norm{t_{1}\gamma_{1}}_{p}^{2}, \quad g_{p}(t_{1}\gamma_{1},t_{2}\gamma_{2}) = g_{p}(t_{2}\gamma_{2},t_{1}\gamma_{1}), \quad \text{and \,\,} g_{p}(t_{1}\gamma_{1},t_{2}\gamma_{2}) = t_{1}g_{p}(\gamma_{1},t_{2}\gamma_{2}).
\]
The following inequality is of fundamental importance in this present paper. Here, we adopt the notation
\[
\product{t\vec{px}}{s\vec{py}} := \frac{ts}{2} \left[\rho^{2}(p,x) + \rho^{2}(p,y) - \rho^{2}(x,y)\right]
\]
for $p,x,y \in H$ and $t,s \geq 0$.
\begin{prop}\label{Prop:gp-ql}
For each $s,t \geq 0$ and $p,x,y \in H$, the following inequality holds:
\[
g_{p}\left(t\rho(p,x)\gamma_{p,x},s\rho(p,y)\gamma_{p,y}\right) \geq \product{t\vec{px}}{s\vec{py}}.
\]
\end{prop}
\begin{proof}
The case where one of $\gamma_{p,x}$ and $\gamma_{p,y}$ is a zero geodesic is obvious. Suppose that both of them are nonzero geodesics. Since $\angle_{p}(\gamma_{p,x},\gamma_{p,y}) \leq \overline{\angle}_{p}(\gamma_{p,x},\gamma_{p,y})$, we have
\begin{align*}
\lefteqn{g_{p}\left(t\rho(p,x)\gamma_{p,x},s\rho(p,y)\gamma_{p,y}\right)} \\
&= st\rho(p,x)\rho(p,y) \cos \angle_{p}(\gamma_{p,x},\gamma_{p,y}) \\
&\geq st\rho(p,x)\rho(p,y) \cos \overline\angle_{p}(x,y) \\
&= st\rho(p,x)\rho(p,y) \left[ \frac{1}{2\rho(p,x)\rho(p,y)} \left( \rho^{2}(p,x) + \rho^{2}(p,y) - \rho^{2}(x,y) \right) \right] \\
&= \product{t\vec{px}}{s\vec{py}}. \tag*\qedhere
\end{align*}
\end{proof}
\section{Monotone vector fields}
In this section, we give a systematic study of the class of monotone vector fields on a $\rm CAT(0)$ space $H$ together with the two main properties, the maximality and the surjectivity condition. To emphasize the practical use of this class of vector fields, we also dedicate an especial observation of the subdifferential for a proper convex lower semicontinuous function as a monotone vector field.
By a \emph{(set-valued) vector field on $H$}, we mean a mapping $A : H \multimap TH$ satisfying $Ap \subset T_{p}H$ for all $p \in H$.
\begin{dfn}
A vector field $A : H \multimap TH$ is said to be \emph{monotone} if
\[
g_{p}(\eta,\gamma_{p,q}) \leq -g_{q}(\nu,\gamma_{q,p})
\]
holds for every $(p,\eta),(q,\nu) \in \gr(A)$, where $\gr(A) := \{(x,u) \in H \times TH \,|\, u \in Ax\}$ denotes the \emph{graph} of $A$. In addition, if $\gr(A)$ is not properly contained in the graph of any other monotone vector fields, then $A$ is said to be \emph{maximally monotone}.
\end{dfn}
\begin{prop}
Let $A$ be a monotone vector field on $H$ and $(p,\eta),(q,\nu) \in \gr(A)$. Suppose that the representations $t\rho(p,x)\gamma_{p,x} = \eta$ and $s\rho(q,y)\gamma_{q,y} = \nu$ are satisfied for some $s,t \geq 0$, and $x,y \in H$. Then the following inequality holds:
\[
\product{t\vec{px}}{\vec{pq}} + \product{s\vec{qy}}{\vec{qp}} \leq 0
\]
\end{prop}
\begin{proof}
The inequality is obvious if $p = q$. Thus we assume that $p \neq q$. It yields immediately from the monotonicity of $A$ and Proposition \ref{Prop:gp-ql} that
\begin{align*}
0 &\geq g_{p}(\eta,\gamma_{p,q}) + g_{q}(\nu,\gamma_{q,p}) \\
&= g_{p}(t\rho(p,x)\gamma_{p,x},\gamma_{p,q}) + g_{q}(s\rho(q,y)\gamma_{q,y},\gamma_{q,p}) \\
&\geq \frac{1}{\rho(p,q)}\left[ \product{t\vec{px}}{\vec{pq}} + \product{s\vec{qy}}{\vec{qp}} \right].
\end{align*}
The desired inequality then follows by rearrangements.
\end{proof}
The following definition is central in the studies in the rest of this paper.
\begin{dfn}
A vector field $A : H \multimap TH$ is said to satisfy the \emph{surjectivity condition} if for any $t > 0$ and $x \in H$, there exists a point $p \in H$ such that $t\rho(p,x)\gamma_{p,x} \in Ap$.
\end{dfn}
The monotonicity and surjectivity conditions are the two main ingredients for our theory developed henceforth in this paper.
\begin{prop}\label{prop:uniqep}
If $A$ is a monotone vector field on $H$ with surjectivity condition, then for any $t > 0$ and $x \in H$, there exists a unique $p \in H$ such that $t\rho(p,x)\gamma_{p,x} \in Ap$.
\end{prop}
\begin{proof}
Let $t > 0$, $x \in H$, and $p,p' \in H$ be points where $t\rho(p,x)\gamma_{p,x} \in Ap$ and $t\rho(p',x)\gamma_{p',x} \in Ap'$. By the monotonicity of $A$ and Proposition \ref{Prop:gp-ql}, we have
\begin{align*}
0 &\geq \rho(p,p'
)\left[g_{p}(t\rho(p,x)\gamma_{p,x},\gamma_{p,p'}) + g_{p'}(t\rho(p',x)\gamma_{p',x},\gamma_{p',p})\right] \\
&= g_{p}(t\rho(p,x)\gamma_{p,x},\rho(p,p')\gamma_{p,p'}) + g_{p'}(t\rho(p',x)\gamma_{p',x},\rho(p,p')\gamma_{p',p}) \\
&\geq t \left[ \product{\vec{px\vphantom{p'}}}{\vec{pp'}} + \product{\vec{p'x}}{\vec{p'p}} \right] \\
&= t \left[ \product{\vec{px\vphantom{p'}}}{\vec{pp'}} + \product{\vec{xp'}}{\vec{pp'}} \right] \\
&= t\product{\vec{pp'}}{\vec{pp'}} \\
&= t\rho^{2}(p,p').
\end{align*}
Hence $p = p'$ and the statement is proved.
\end{proof}
\begin{prop}\label{prop:surj-mm}
If $A$ is a monotone vector field over $H$ with surjectivity condition, then it is maximally monotone.
\end{prop}
\begin{proof}
Suppose that $p \in H$ and $p^{\ast} \in T_{p}H$ satisfy the inequality
\begin{equation}\label{eqn:premax}
g_{p}(p^{\ast},\gamma_{p,q}) + g_{q}(q^{\ast},\gamma_{q,p}) \leq 0,
\end{equation}
for any choice of $(q,q^{\ast}) \in \gr(A)$. Suppose that $p^{\ast}$ has the representation $p^{\ast} = t\rho(p,u)\gamma_{p,u}$ for some $t > 0$ and $u \in H$. By the surjectivity condition of $A$, there is a unique point $z \in H$ such that $t\rho(z,u)\gamma_{z,u} \in Az$. In view of \eqref{eqn:premax} and Proposition \ref{Prop:gp-ql}, we have
\begin{align*}
0 &\geq \rho(p,z)\left[ g_{p}(t\rho(p,u)\gamma_{p,u},\gamma_{p,z}) + g_{z}(t\rho(z,u)\gamma_{z,u},\gamma_{z,p}) \right] \\
&= g_{p}(t\rho(p,u)\gamma_{p,u},\rho(p,z)\gamma_{p,z}) + g_{z}(t\rho(z,u)\gamma_{z,u},\rho(p,z)\gamma_{z,p}) \\
&\geq t \left[ \product{\vec{pu}}{\vec{pz}} + \product{\vec{zu}}{\vec{zp}} \right] \\
&= t \left[ \product{\vec{pu}}{\vec{pz}} + \product{\vec{uz}}{\vec{pz}} \right] \\
&= t \product{\vec{pz}}{\vec{pz}} \\
&= t\rho^{2}(p,z).
\end{align*}
Since $t > 0$, the above inequalities imply $p = z$ so that $(p,p^{\ast}) \in \gr(A)$. Hence the maximality is obtained.
\end{proof}
\subsection{Subdifferential of a convex function}
In this subsection, we study a particular example of a monotone vector field with surjectivity condition, namely the subdifferential of a proper, convex, lower semicontinuous function. For simplicity, we write $\Gamma_{0}$ to denote the class of proper, convex, lower semicontinuous functions $F : H \to (-\infty,+\infty]$.
\begin{dfn}
Let $F \in \Gamma_{0}$ be given. At each $p \in H$, a tangent vector $p^{\ast} \in T_{p}H$ is called a \emph{subgradient} of $F$ at $p$ if
\[
F(x) \geq F(p) + \rho(p,x)g_{p}(p^{\ast},\gamma_{p,x})
\]
for every $x \in H$. The \emph{subdifferential} of $F$ is the vector field $\partial F : H \multimap TH$, where $\partial F (p)$ is the set of all subgradients of $F$ at $p$ for each $p \in H$.
\end{dfn}
\begin{prop}
$\partial F$ is monotone for each $F \in \Gamma_{0}$.
\end{prop}
\begin{proof}
Suppose that $F \in \Gamma_{0}$ and $(p,p^{\ast}),(q,q^{\ast}) \in \gr(\partial F)$. We thus have
\[
F(q) \geq F(p) + \rho(p,q)g_{p}(p^{\ast},\gamma_{p,q})
\]
and also
\[
F(p) \geq F(q) + \rho(p,q)g_{q}(q^{\ast},\gamma_{q,p}).
\]
Rearraging yields
\[
-\rho(p,q)g_{q}(q^{\ast},\gamma_{q,p}) \geq F(q) - F(p) \geq \rho(p,q)g_{p}(p^{\ast},\gamma_{p,q}),
\]
and so the monotonicity is obtained.
\end{proof}
The following result characterizes elements in $\partial F$ and can also be regarded as a generalized Fermat rule.
\begin{prop}\label{prop:genFermat}
Let $F \in \Gamma_{0}$, $p,\bar{p} \in H$ and $\lambda > 0$ be given. Then
\[
\lambda^{-1}\rho(\bar{p},p)\gamma_{\bar{p},p} \in \partial F(\bar{p}) \iff \bar{p} = \prox_{\lambda} (p) = \argmin_{y \in H} \left[ F(y) + \frac{1}{2\lambda}\rho^{2}(y,p) \right].
\]
\end{prop}
\begin{proof}
Let us first show the `only if' part. Assume that $\lambda^{-1}\rho(\bar{p},p)\gamma_{\bar{p},p} \in \partial F(\bar{p})$. For any $y \in H$, we get
\begin{align*}
\lambda F(y) &\geq \lambda F(\bar{p}) + \rho(\bar{p},p)\rho(\bar{p},y)g_{\bar{p}}(\gamma_{\bar{p},p},\gamma_{\bar{p},y}) \\
&\geq \lambda F(\bar{p}) + \product{\vec{\bar{p}p}}{\vec{\bar{p}y}} \\
&= \lambda F(\bar{p}) + \frac{1}{2} \left[ \rho^{2}(\bar{p},p) + \rho^{2}(\bar{p},y) - \rho^{2}(p,y) \right] \\
&\geq \lambda F(\bar{p}) + \frac{1}{2} \left[ \rho^{2}(\bar{p},p) - \rho^{2}(p,y) \right].
\end{align*}
Therefore, we have $\bar{p} = \prox_{\lambda} (p)$.
Next, we show the `if' part. Suppose that $\bar{p} = \prox_{\lambda}(p)$, whose definition gives
\[
F(y) - F(\bar{p}) \geq \frac{1}{2\lambda} \left[ \rho^{2}(\bar{p},p) - \rho^{2}(y,p) \right],
\]
for all $y \in H$. If $p = \bar{p}$, then $\bar{p}$ is a minimizer of $F$ (see \cite[Proposition 6.5]{MR3206460}). It follows from the definition of $\partial F$ that $\lambda^{-1}\rho(\bar{p},p)\gamma_{\bar{p},p} = 0_{\bar{p}} \in \partial F(\bar{p})$. Next, suppose that $p \neq \bar{p}$. Let us fix any $y \in H\setminus\{\bar{p}\}$ and for each $t \in [0,1]$, put $y_{t} := (1-t)\bar{p}\oplus ty$. Further, if we set $Y(s) := y_{s/\rho(\bar{p},y)}$ for each $s \in [0,\rho(\bar{p},y)]$, then $Y$ is a unit-speed geodesic.
For $t \in (0,1)$, putting $y = y_{t}$ in the above inequality and applying the convexity of $F$ yield
\begin{align*}
F(y) - F(\bar{p}) &\geq \frac{1}{2\lambda t} \left[ \rho^{2}(\bar{p},p) - \rho^{2}(y_{t},p) \right] \\
&= \frac{1}{2\lambda}[\rho(\bar{p},p) + \rho(y_{t},p)]\left[\frac{\rho(\bar{p},p) - \rho(y_{t},p)}{t}\right] \\
&= \frac{\rho(\bar{p},y)}{2\lambda}[\rho(\bar{p},p) + \rho(y_{t},p)]\left[\frac{\rho(\bar{p},p) - \rho(Y(t\rho(\bar{p},y)),p)}{t\rho(\bar{p},y)}\right].
\end{align*}
Letting $t \longrightarrow 0^{+}$ and taking into account the First Variation Formula (Theorem \ref{thm:FVF}), we have
\[
F(y) - F(\bar{p}) \geq \frac{\rho(\bar{p},p)\rho(\bar{p},y)}{\lambda} \cos \angle_{p}(\gamma_{\bar{p},p},\gamma_{\bar{p},y}) = \rho(\bar{p},y) g_{p}\left(\lambda^{-1}\rho(\bar{p},p)\gamma_{\bar{p},p},\gamma_{\bar{p},y}\right).
\]
It is trivial that the above inequality holds for $y = \bar{p}$. Therefore, we may conclude that $\lambda^{-1}\rho(\bar{p},p)\gamma_{\bar{p},p} \in \partial F(\bar{p})$.
\end{proof}
\begin{thm}
$\partial F$ is maximally monotone for $F \in \Gamma_{0}$.
\end{thm}
\begin{proof}
It is clear by the property of the operator $\prox$ and the previous theorem that $\partial F$ satisfies the surjectivity condition. The conclusion follows in view of Proposition \ref{prop:surj-mm}.
\end{proof}
Finally, we show the density of the domain of $\partial F$ in that of $F$. In other words, a proper, convex, lower semicontinuous function is subdifferentiable almost everywhere in reasonable measures.
\begin{thm}
$\overline{\dom \partial F} = \overline{\dom F}$.
\end{thm}
\begin{proof}
The inclusion $\overline{\dom \partial F} \subset \overline{\dom F}$ is immediately implied from the definition. Hence we only need to show the inclusion $\overline{\dom F} \subset \overline{\dom \partial F}$. Suppose that $x \in \overline{\dom F}$. According to \cite[Proposition 2.2.26]{MR3241330}, we know that $x = \lim_{\lambda \longrightarrow 0^{+}} \prox_{\lambda} (x)$. By Proposition \ref{prop:genFermat}, we may see that
\[
\lambda^{-1}\rho(\prox_{\lambda}(x),x)\gamma_{\prox_{\lambda}(x),x} \in \partial F (\prox_{\lambda} (x)),
\]
for any $\lambda > 0$. This shows that $(\prox_{\lambda}(x))_{\lambda > 0}$ is a net in $\dom \partial F$. Therefore, $x$ as a limit point of this net must lies within the closure $\overline{\dom \partial F}$.
\end{proof}
\section{Resolvents and Yosida approximations}
Now, we shall define the resolvent for a given vector field $A : H \multimap TH$ and derive some of its fundamental properties. Results in this section are considered to be the main auxiliary tools used in the final section.
\begin{dfn}
Given $\lambda > 0$, the \emph{$\lambda$-resolvent} of $A$ is the mapping $J_{\lambda} : H \multimap H$ defined by
\[
J_{\lambda}x := \{z \in H \,|\, \lambda^{-1}\rho(z,x)\gamma_{z,x} \in Az\} \quad (\forall x \in H).
\]
Moreover, we define $J_{0}$ to be the identity mapping.
\end{dfn}
\begin{dfn}
A mapping $T : C \to C$ is called \emph{firmly nonexpansive} if for any $x,y \in C$, the function
\[
\varphi_{x,y}(t) := \rho(\gamma_{x,Tx}(t),\gamma_{y,Ty}(t))
\]
is nonincreasing on $t \in [0,1]$.
\end{dfn}
\begin{prop}\label{prop:properties-J}
Suppose that $A$ is a monotone vector field on $H$ satisfying the surjectivity condition. Then the following facts hold true:
\begin{enumerate}[label=\rm{(\roman*)}]
\item\label{cdn:singlevalued} $J_{\lambda}$ is well-defined on $H$ and is single-valued.
\item\label{cdn:J-NX} $J_{\lambda}$ is nonexpansive.
\item\label{cdn:Fixed=Stationary} $\Fix(J_{\lambda}) = A^{-1}(\mathbf{0})$.
\item\label{cdn:J-ResolventIden} If $0 < \mu \leq \lambda$, then $J_{\lambda}(x) = J_{\mu} (u)$ with $u := \left(1 - \frac{\mu}{\lambda}\right)J_{\lambda}(x) \oplus \frac{\mu}{\lambda} x$, for each $x \in H$.
\item\label{cdn:J-FirmNX} $J_{\lambda}$ is firmly nonexpansive.
\end{enumerate}
\end{prop}
\begin{proof}
\ref{cdn:singlevalued} The well-definition as a single-valued mapping follows from the surjectivity condition and Proposition \ref{prop:uniqep}.
\ref{cdn:J-NX} Suppose that $x,y \in H$. By the monotonicity of $A$, we have
\begin{align*}
0 &\geq \rho(J_{\lambda}x,J_{\lambda}y) \left[ g_{J_{\lambda}x}(\lambda^{-1}\rho(J_{\lambda}x,x)\gamma_{J_{\lambda}x,x},\gamma_{J_{\lambda}x,J_{\lambda}y}) + g_{J_{\lambda}y}(\lambda^{-1}\rho(J_{\lambda}y,y)\gamma_{J_{\lambda}y,y},\gamma_{J_{\lambda}y,J_{\lambda}x})\right] \\
&\geq \product{\lambda^{-1}\gamma_{J_{\lambda}x,x}}{\gamma_{J_{\lambda}x,J_{\lambda}y}} + \product{\lambda^{-1}\gamma_{J_{\lambda}y,y}}{\gamma_{J_{\lambda}y,J_{\lambda}x}} \\
&= \frac{1}{2\lambda} \left( \rho^{2}(J_{\lambda}x,x) + \rho^{2}(J_{\lambda}x,J_{\lambda}y) - \rho^{2}(x,J_{\lambda}y) \right) \\
&\qquad + \frac{1}{2\lambda}\left(\rho^{2}(J_{\lambda}y,y) + \rho^{2}(J_{\lambda}y,J_{\lambda}x) - \rho^{2}(y,J_{\lambda}x) \right).
\end{align*}
Rearranging and applying \eqref{eqn:quad-characterization-CAT(0)}, we get
\begin{align}
2\rho^{2}(J_{\lambda}x,J_{\lambda}y) &\leq \rho^{2}(x,J_{\lambda}y) + \rho^{2}(y,J_{\lambda}x) - \rho^{2}(J_{\lambda}x,x) - \rho^{2}(J_{\lambda}y,y) \label{eqn:firmlynonspreading}\\
&\leq 2\rho(x,y)\rho(J_{\lambda}x,J_{\lambda}y), \nonumber
\end{align}
which implies the nonexpansivity of $J_{\lambda}$.
\ref{cdn:Fixed=Stationary} The result is simply obtained from
\[
x \in \Fix(J_{\lambda}) \iff x = J_{\lambda}x \iff 0_{x} = \lambda^{-1}\gamma_{x,x} \in Ax \iff x \in A^{-1}(\mathbf{0}).
\]
\ref{cdn:J-ResolventIden} Let $x \in H$ be arbitrary and $0 < \mu \leq \lambda$. Set $u := \left(1-\frac{\mu}{\lambda}\right)J_{\lambda}x \oplus \frac{\mu}{\lambda}x$. If $x = J_{\lambda}x$, then $u = x \in A^{-1}\mathbf{0}$. It follows from \ref{cdn:Fixed=Stationary} that $J_{\lambda} x = x = u = J_{\mu} u$. Thus we suppose that $x \neq J_{\lambda}x$. In this case, we have $\angle_{J_{\lambda}x}(\gamma_{J_{\lambda}x,x},\gamma_{J_{\lambda}x,u}) = 0$ and $\zeta(\gamma_{J_{\lambda}x,x}) = \zeta(\gamma_{J_{\lambda}x,u}) = 1$. Also note that
\[
\lambda^{-1}\rho(J_{\lambda}x,x) = \mu^{-1}(\mu\lambda^{-1})\rho(J_{\lambda}x,x) = \mu^{-1}\rho(J_{\lambda}x,u).
\]
By the definition of a tangent space, we get
\[
\mu^{-1}\rho(J_{\lambda}x,u)\gamma_{J_{\lambda}x,u} = \lambda^{-1}\rho(J_{\lambda}x,x)\gamma_{J_{\lambda}x,x} \in AJ_{\lambda}x,
\]
which leads to the conclusion that $J_{\lambda}x = J_{\mu}u$.
\ref{cdn:J-FirmNX} Let $x,y \in H$. Since $t \mapsto \varphi_{x,y}(t)$ is convex on $[0,1]$ (see \cite[Proposition 1.1.5]{MR3241330}), it is sufficient to show that $\varphi_{x,y}(1) \leq \varphi_{x,y}(t)$ for all $t \in (0,1)$. Indeed, for $t \in (0,1)$, we obtain from \ref{cdn:J-ResolventIden} and \ref{cdn:J-NX} the following:
\begin{align*}
\varphi_{x,y}(1) &= \rho(J_{\lambda}x,J_{\lambda}y) = \rho\left(J_{(1-t)\lambda}(tJ_{\lambda}x \oplus (1-t)x),J_{(1-t)\lambda}(tJ_{\lambda}y \oplus (1-t)y)\right) \\
&\leq \rho(tJ_{\lambda}x \oplus (1-t)x,tJ_{\lambda}y \oplus (1-t)y) = \varphi_{x,y}(t).
\end{align*}
Therefore, $J_{\lambda}$ is firmly nonexpansive.
\end{proof}
Henceforth in this paper, we need to assume the geodesic extension property on $H$ in order to define a \emph{negative geodesic}. Recall that a $\rm CAT(0)$ space $H$ has the \emph{geodesic extension property} if each unit speed geodesic $\gamma : [0,L] \to H$, there exists an isometry $\eta : \mathbb{R} \to H$ such that $\eta(t) = \gamma(t)$ whenever $t \in [0,L]$. It is clear that $H$ has the geodesic extension property if and only if for each $p,q \in H$, there exists $\eta : \mathbb{R} \to H$ such that $\rho(\eta(s),\eta(t)) = \rho(p,q)\abs{s-t}$ for all $s,t \in \mathbb{R}$ with $\eta(0) = p$ and $\eta(1) = q$. (See \cite{MR1744486} for a detailed description).
\begin{dfn}
Suppose that $H$ has the geodesic extension property, and that $p,x \in H$. By the geodesic extension property, there is a point $y \in H$ such that $\gamma_{x,y}(\frac{1}{2}) = p$. The \emph{negative geodesic} of $\gamma_{p,x}$ is then defined by $-\gamma_{p,x} := \gamma_{p,y}$.
\end{dfn}
\begin{rmk}
Notice that the point $y \in H$ appeared in the above definition is not necessarily unique. In this case, we fix one of such points $y$ for any given $x,z \in H$.
\end{rmk}
\begin{prop}\label{prop:negativeineq}
Suppose that $H$ has the geodesic extension property. Then
\[
g_{p}(-\gamma_{p,x},\gamma_{p,q}) \leq - g_{p}(\gamma_{p,x},\gamma_{p,q})
\]
for all $p,q,x \in H$.
\end{prop}
\begin{proof}
Suppose that $p \in H$ and $q,x \in H \setminus\{p\}$, otherwise there is nothing to be proved. Suppose that $y \in H$ satifies $\gamma_{x,y}(1/2) = p$. By the triangle inequality, it follows that
\[
\pi = \angle_{p}(\gamma_{p,x},\gamma_{p,y}) \leq \angle_{p}(\gamma_{p,x},\gamma_{p,q}) + \angle_{p}(\gamma_{p,y},\gamma_{p,q}).
\]
Rearranging the inequality yields $\angle_{p}(\gamma_{p,x},\gamma_{p,q}) \geq \pi - \angle_{p}(\gamma_{p,y},\gamma_{p,q})$, and we further have
\[
\cos \angle_{p}(\gamma_{p,x},\gamma_{p,q}) \leq \cos \left(\pi - \angle_{p}(\gamma_{p,y},\gamma_{p,q})\right) = - \cos \angle_{p}(\gamma_{p,y},\gamma_{p,q}).
\]
Using this fact, we obtain
\begin{align*}
g_{p}(-\gamma_{p,x},\gamma_{p,q}) &= g_{p}(\gamma_{p,y},\gamma_{p,q}) \\
&= \cos \angle_{p}(\gamma_{p,y},\gamma_{p,q}) \\
&\leq - \cos \angle_{p}(\gamma_{p,x},\gamma_{p,q}) \\
&= -g_{p}(\gamma_{p,x},\gamma_{p,q}). \tag*{\qedhere}
\end{align*}
\end{proof}
Now that we have defined the negative geodesic, we use it in the definition of the so-called complemenatary vector field.
\begin{dfn}
Given a mapping $T : H \to H$. The \emph{complementary vector field} of $T$, denoted by $A_{T} : H \to TH$ is defined by
\[
A_{T}x := \rho(x,Tx)(-\gamma_{x,Tx}) \quad (\forall x \in H).
\]
\end{dfn}
It is immediate to observe that $\Fix(T) = A_{T}^{-1}\mathbf{0}$. The next proposition shows that a complementary vector field is monotone if applied to a nonexpansive mapping.
\begin{prop}
Suppose that $H$ has the geodesic extension property. If $T : H \to H$ is nonexpansive, then $A_{T}$ is monotone.
\end{prop}
\begin{proof}
Let $x,y \in H$. In view of \eqref{eqn:quad-characterization-CAT(0)}, the nonexpansivity of $T$, Propositions \ref{Prop:gp-ql} and \ref{prop:negativeineq}, we have
\begin{align*}
\lefteqn{2\rho(x,y)[g_{x}(A_{T}x,\gamma_{x,y}) + g_{y}(A_{T}y,\gamma_{y,x})]} \\
&\leq -2\rho(x,y)[g_{x}(\rho(x,Tx)\gamma_{x,Tx},\gamma_{x,y}) + g_{y}(\rho(y,Ty)\gamma_{y,Ty}),\gamma_{y,x})] \\
&\leq -2[g_{x}(\rho(x,Tx)\gamma_{x,Tx},\rho(x,y)\gamma_{x,y}) + g_{y}(\rho(y,Ty)\gamma_{y,Ty}),\rho(x,y)\gamma_{y,x})] \\
&\leq -2[\product{\vec{xTx}}{\vec{xy}} + \product{\vec{yTy}}{\vec{yx}}] \\
&= \rho^{2}(y,Tx) + \rho^{2}(x,Ty) - \rho^{2}(x,Tx) - \rho^{2}(y,Ty) -2 \rho^{2}(x,y) \\
&\leq 2\rho(x,y)\rho(Tx,Ty) - 2\rho^{2}(x,y) \leq 0.
\end{align*}
Therefore, $A_{T}$ is monotone.
\end{proof}
By using the complementary vector field defined above, we can construct an important device called the Yosida approximation.
\begin{dfn}
Suppose that $H$ has the geodesic extension property, $A : H \multimap TH$ is a monotone vector field with surjectivity condition. Let $\lambda > 0$. The \emph{$\lambda$-Yosida approximation} of $A$, denoted by $A_{\lambda}$, is defined by
\[
A_{\lambda}x := \lambda^{-1}A_{J_{\lambda}}x, \quad \forall x \in H.
\]
\end{dfn}
One may see in the following proposition that Yosida approximations can produce a useful estimate for each $x \in \dom A$.
\begin{prop}\label{prop:HeartWarmingIneq}
Suppose that $H$ has the geodesic extension property and $A$ is a monotone vector field with the surjectivity condition. Then, the inequality
\[
\norm{A_{\lambda}x} \leq \abs{Ax} := \inf_{x^{\ast} \in Ax} \norm{x^{\ast}}
\]
holds for every $\lambda > 0$ and $x \in H$, where the convention $\inf \emptyset = +\infty$.
\end{prop}
\begin{proof}
If $x \not\in \dom A$, the inequality always holds. Suppose now that $x \in \dom A$. If $x \in A^{-1}\mathbf{0}$, then $\norm{A_{\lambda}x} = \norm{0_{x}} = 0 = \abs{Ax}$. On the other hand, if $x \not\in A^{-1}\mathbf{0}$ we pick $u^{\ast} \in Ax$ arbitrarily. For some $t > 0$ and $u \in H\setminus\{x\}$, we have $u^{\ast} = t\rho(x,u)\gamma_{x,u}$ and thus $\norm{u^{\ast}} = t\rho(x,u)$.
Since $H$ has the geodesic extension property, there exists a map $\gamma : \mathbb{R} \to H$ such that $\rho(\gamma(s),\gamma(t)) = \rho(x,u)\abs{s-t}$ for all $s,t \in \mathbb{R}$ with $\gamma(0) = x$ and $\gamma(1) = u$. Let $\lambda > 0$. Put $v := \gamma(\lambda t)$ and $v^{\ast} := \lambda^{-1}\rho(x,v)\gamma_{x,v}$. Observe that $\angle_{x}(\gamma_{x,u},\gamma_{x,v}) = 0$ and that
\[
\norm{v^{\ast}} = \lambda^{-1}\rho(x,v) = \lambda^{-1}\lambda t \rho(x,u) = t\rho(x,u) = \norm{u^{\ast}}.
\]
By the definition of a tangent space, we have $\lambda^{-1}\rho(x,v)\gamma_{x,v} = v^{\ast} = u^{\ast} \in Ax$. This means $x = J_{\lambda}v$. As a consequence, we obtain
\[
\norm{A_{\lambda}x} = \lambda^{-1}\rho(x,J_{\lambda}x) = \lambda^{-1}\rho(J_{\lambda}v,J_{\lambda}x) \leq \lambda^{-1}\rho(v,x) = \norm{v^{\ast}} = \norm{u^{\ast}}.
\]
Since $u^{\ast} \in Ax$ is chosen arbitrarily, we finally have $\norm{A_{\lambda}x} \leq \abs{Ax}$.
\end{proof}
\begin{prop}\label{prop:convexity}
If $H$ satisfies the geodesic extension property and $A : H \multimap TH$ is a monotone vector field with the surjectivity condition, then $\overline{\dom A}$ is convex.
\end{prop}
\begin{proof}
First, we make a claim that $\lim_{\lambda \longrightarrow 0^{+}} J_{\lambda}x = x$ for any $x \in \overline{\dom A}$. For the moment, consider $x \in \dom A$. For each $\lambda > 0$, we have
\[
\rho(x,J_{\lambda}x) = \lambda\norm{A_{\lambda}x} \leq \lambda \abs{Ax} < +\infty.
\]
Letting $\lambda \longrightarrow 0^{+}$, we get $\lim_{\lambda \longrightarrow 0^{+}} J_{\lambda}x = x$. Next, suppose that $x \in \overline{\dom A}$ and let $(x^{k})$ be a sequence in $\dom A$ which converges to $x$. Since $J_{\lambda}$ is nonexpansive, we have
\begin{align*}
\rho(x,J_{\lambda}x) &\leq \rho(x,x^{k}) + \rho(x^{k},J_{\lambda}x^{k}) + \rho(J_{\lambda}x^{k},J_{\lambda}x) \\
&\leq 2\rho(x,x^{k}) + \rho(x^{k},J_{\lambda}x^{k})
\end{align*}
and hence
\[
\limsup_{\lambda \longrightarrow 0^{+}} \rho(x,J_{\lambda}x) \leq \limsup_{\lambda \longrightarrow 0^{+}} [2\rho(x,x^{k}) + \rho(x^{k},J_{\lambda}x^{k})] = 2\rho(x,x^{k}).
\]
Letting $k \longrightarrow \infty$ on the right hand side then proves the claim.
Put $Q := \{x \in H \,|\, \lim_{\lambda \longrightarrow 0^{+}} J_{\lambda}x = x\}$. Sicne $J_{\lambda} x \in \dom A$ for all $\lambda > 0$ and $x \in Q$, its limit $x$ is included in $\overline{\dom A}$. Hence $Q \subset \overline{\dom A}$. On the other hand, we have $\overline{\dom A} \subset Q$. We thus have $Q = \overline{\dom A}$.
Now, take $x,y \in Q$ and $t \in (0,1)$. Put $z := (1-t)x \oplus ty$. For any $\lambda > 0$, we obtain
\begin{equation}\label{eqn:est-clD(A)}
\rho(J_{\lambda}z,x) \leq \rho(J_{\lambda}z,J_{\lambda}x) + \rho(J_{\lambda}x,x) \leq \rho(x,z) + \rho(x,J_{\lambda}x) = t\rho(x,y) + \rho(x,J_{\lambda}x),
\end{equation}
and similarly have
\begin{equation}\label{eqn:est-clD(A)2}
\rho(J_{\lambda}z,y) \leq \rho(J_{\lambda}z,J_{\lambda}y) + \rho(J_{\lambda}y,y) \leq \rho(y,z) + \rho(y,J_{\lambda}y) = (1-t)\rho(x,y) + \rho(y,J_{\lambda}y).
\end{equation}
Take any sequence $(\lambda_{k})$ in $(0,\infty)$ such that $\lambda_{k} \longrightarrow 0$. Then it follows from \eqref{eqn:est-clD(A)} and $J_{\lambda_{k}}x \longrightarrow x$ that the sequence $(J_{\lambda_{k}}z)$ is bounded. So, $(J_{\lambda_{k}}z)$ contains a $\Delta$-convergent subsequence $(J_{\lambda_{k_{i}}}z)$ with $\Delta$-limit $p \in H$. Putting $\lambda = \lambda_{k_{i}}$ in \eqref{eqn:est-clD(A)} and \eqref{eqn:est-clD(A)2} and letting $i \longrightarrow \infty$, we obtain
\begin{equation}\label{eqn:liminf1}
\liminf_{i \longrightarrow \infty} \rho(J_{\lambda_{k_{i}}}z,x) \leq t\rho(x,y)
\end{equation}
and also
\begin{equation}\label{eqn:liminf2}
\liminf_{i \longrightarrow \infty} \rho(J_{\lambda_{k_{i}}}z,y) \leq (1-t)\rho(x,y).
\end{equation}
By \eqref{eqn:CN}, the $\Delta$-lower semicontinuity of $\rho(x,\cdot)$, and ineqalities \eqref{eqn:liminf1} as well as \eqref{eqn:liminf2}, we have
\begin{align*}
\rho^{2}(p,z) &\leq (1-t)\rho^{2}(x,p) + t\rho^{2}(y,p) - t(1-t)\rho^{2}(x,y) \\
& \leq (1-t)\left[\liminf_{i \longrightarrow \infty} \rho(x,J_{\lambda_{k_{i}}}z)\right]^{2} + t\left[\liminf_{i \longrightarrow \infty} \rho(y,J_{\lambda_{k_{i}}}z)\right]^{2} - t(1-t)\rho^{2}(x,y) \\
& \leq (1-t)t^{2}\rho^{2}(x,y) + t(1-t)^{2}\rho^{2}(x,y) - t(1-t)\rho^{2}(x,y) \\
& = 0.
\end{align*}
Hence $(J_{\lambda_{k_{i}}}z)$ is $\Delta$-convergent to $z$, which implies that $\omega_{\Delta}(J_{\lambda_{k}}z) = \{z\}$. By Proposition \ref{prop:singletonOmegaDelta} we conclude that $(J_{\lambda_{k}}z)$ is $\Delta$-convergent to $z$.
Next, observe that
\[
\rho(J_{\lambda_{k}}z,x) \leq \rho(J_{\lambda_{k}}z,J_{\lambda_{k}}x) + \rho(J_{\lambda_{k}}x,x) \leq \rho(z,x) + \rho(J_{\lambda_{k}}x,x).
\]
This inequality together with the $\Delta$-lower semicontinuity of $\rho(\cdot,x)$, we obtain
\[
\rho(z,x) \leq \liminf_{k \longrightarrow \infty} \rho(J_{\lambda_{k}}z,x) \leq \limsup_{k \longrightarrow \infty} \rho(J_{\lambda_{k}}z,x) \leq \rho(z,x),
\]
and so $\lim_{k} \rho(J_{\lambda_{k}}z,x) = \rho(z,x)$. By the Kadec-Klee property (Lemma \ref{lem:Kadec-Klee}), we get $\lim_{k} J_{\lambda_{k}}z = z$. Since $(\lambda_{k})$ is an arbitrary sequence of positive numbers with $\lambda_{k} \longrightarrow 0$, we can conclude that $\lim_{\lambda \longrightarrow 0^{+}} J_{\lambda}z = z$ and so $z \in Q$. This shows the convexity of $Q = \overline{\dom A}$.
\end{proof}
\begin{thm}\label{thm:Asymp0}
Suppose that $H$ satisfies the geodesic extension property, and $A : H \multimap TH$ is a monotone vector field with the surjectivity condition. Then,
\[
\lim_{\lambda \longrightarrow 0^{+}} J_{\lambda}x = P_{\overline{\dom A}} x,
\]
for every $x \in H$.
\end{thm}
\begin{proof}
Put $P := P_{\overline{\dom A}}$. Let $x \in H$ and $\lambda > 0$. By the monotonicity of $A$, we have
\begin{align}
0 &\geq 2\lambda\rho(J_{\lambda}x,J_{\lambda}Px) \left[ g_{J_{\lambda}x}(\lambda^{-1}\rho(J_{\lambda}x,x)\gamma_{J_{\lambda}x,x},\gamma_{J_{\lambda}x,J_{\lambda}Px}) \right. \nonumber\\
&\qquad\qquad \qquad \qquad\qquad + \left. g_{J_{\lambda}Px}(\lambda^{-1}\rho(J_{\lambda}Px,Px)\gamma_{J_{\lambda}Px,Px},\gamma_{J_{\lambda}Px,J_{\lambda}x}) \right] \nonumber\\
&= 2 \left[ g_{J_{\lambda}x}(\rho(J_{\lambda}x,x)\gamma_{J_{\lambda}x,x},\rho(J_{\lambda}x,J_{\lambda}Px)\gamma_{J_{\lambda}x,J_{\lambda}Px}) \right. \nonumber\\
&\qquad\qquad\qquad + \left. g_{J_{\lambda}Px}(\rho(J_{\lambda}Px,Px)\gamma_{J_{\lambda}Px,Px},\rho(J_{\lambda}x,J_{\lambda}Px) \gamma_{J_{\lambda}Px,J_{\lambda}x}) \right] \nonumber\\
&\geq 2 \left[\product{\gamma_{J_{\lambda}x,x}}{\gamma_{J_{\lambda}x,J_{\lambda}Px}} + \product{\gamma_{J_{\lambda}Px,Px}}{\gamma_{J_{\lambda}Px,J_{\lambda}x}}\right] \nonumber\\
& = \rho^{2}(x,J_{\lambda}x) + 2\rho^{2}(J_{\lambda}x,J_{\lambda}Px) - \rho^{2}(x,J_{\lambda}Px) + \rho^{2}(Px,J_{\lambda}Px) - \rho^{2}(Px,J_{\lambda}x). \label{eqn:est-zero-1}
\end{align}
We recall from \ref{cdn:ProjIneq} in Theorem \ref{thm:projection} that
\[
\rho^{2}(x,J_{\lambda}x) \geq \rho^{2}(x,Px) + \rho^{2}(Px,J_{\lambda}x).
\]
Substitute this inequality in \eqref{eqn:est-zero-1}, we get
\begin{align*}
0 &\geq \left(\rho^{2}(x,Px) + \rho^{2}(Px,J_{\lambda}x)\right) + 2\rho^{2}(J_{\lambda}x,J_{\lambda}Px) - \rho^{2}(x,J_{\lambda}Px) \\
& \qquad + \rho^{2}(Px,J_{\lambda}Px) - \rho^{2}(Px,J_{\lambda}x) \\
& = \rho^{2}(x,Px) + 2\rho^{2}(J_{\lambda}x,J_{\lambda}Px) - \rho^{2}(x,J_{\lambda}Px) + \rho^{2}(Px,J_{\lambda}Px).
\end{align*}
From the proof of the previous proposition and the fact that $Px \in \overline{\dom A}$, letting $\lambda \longrightarrow 0^{+}$ in the above inequality yields
\begin{align*}
0 & \geq \rho^{2}(x,Px) + 2\limsup_{\lambda \longrightarrow 0^{+}} \rho^{2}(J_{\lambda}x,J_{\lambda}Px) - \lim_{\lambda \longrightarrow 0^{+}} \rho^{2}(x,J_{\lambda}Px) + \lim_{\lambda \longrightarrow 0^{+}} \rho^{2}(Px,J_{\lambda}Px) \\
& = \rho^{2}(x,Px) + 2\limsup_{\lambda \longrightarrow 0^{+}} \rho^{2}(J_{\lambda}x,J_{\lambda}Px) - \rho^{2}(x,Px) + \rho^{2}(Px,Px) \\
& = 2\limsup_{\lambda \longrightarrow 0^{+}} \rho^{2}(J_{\lambda}x,J_{\lambda}Px) \geq 2\liminf_{\lambda \longrightarrow 0^{+}} \rho^{2}(J_{\lambda}x,J_{\lambda}Px) \geq 0,
\end{align*}
and so $\lim_{\lambda \longrightarrow 0^{+}} \rho^{2}(J_{\lambda}x,J_{\lambda}Px) = 0$. This equality and the triangle inequality further imply
\begin{align*}
0 &= \lim_{\lambda \longrightarrow 0^{+}} \rho(J_{\lambda}x,J_{\lambda}Px) \geq \limsup_{\lambda \longrightarrow 0^{+}} \rho(J_{\lambda}x,Px) - \lim_{\lambda \longrightarrow 0^{+}} \rho(Px,J_{\lambda}Px) \\
&\geq \limsup_{\lambda \longrightarrow 0^{+}} \rho(J_{\lambda}x,Px) \geq \liminf_{\lambda \longrightarrow 0^{+}} \rho(J_{\lambda}x,Px) \geq 0.
\end{align*}
We finally conclude that $\lim_{\lambda \longrightarrow 0^{+}}J_{\lambda}x = Px$.
\end{proof}
\begin{prop}
If $H$ satisfies the geodesic extension property, then the mapping $\lambda \mapsto J_{\lambda} x$ is continuous on $(0,+\infty)$ for every $x \in H$. If $x \in \overline{\dom A}$, then it is continuous on $[0,+\infty)$.
\end{prop}
\begin{proof}
Let $z \in \dom A$ and $\lambda \geq \mu > 0$. By \ref{cdn:J-NX} and \ref{cdn:J-ResolventIden} in Proposition \ref{prop:properties-J} and Proposition \ref{prop:HeartWarmingIneq}, we get
\begin{align}\label{eqn:est-cont-J}
\rho(J_{\mu}z,J_{\lambda}z) &\leq \rho\left(J_{\mu}z,J_{\mu}\left(\left(1-\frac{\mu}{\lambda}\right)J_{\lambda}z \oplus \frac{\mu}{\lambda}z\right)\right) \nonumber\\
&\leq \rho\left(z,\left(1-\frac{\mu}{\lambda}\right)J_{\lambda}z \oplus \frac{\mu}{\lambda}z\right) \nonumber\\
& = \left(1 - \frac{\mu}{\lambda}\right)\rho(z,J_{\lambda}z) \leq (\lambda - \mu)\abs{Az}.
\end{align}
Therefore, $\lambda \mapsto J_{\lambda}z$ is continuous on $(0,\infty)$ for $z \in \dom A$.
Now, let $x \in H$ and $I := [a,b]$ be a bounded closed subinterval of $(0,+\infty)$. We have
\[
\rho(x,J_{t}x) \leq \rho(x,J_{t}z) + \rho(J_{t}z,J_{t}x) \leq \rho(x,J_{t}z) + \rho(z,x)
\]
for all $t \in I$. By the continuity of $J_{t}z$, we know that $\rho(x,J_{t}z)$ is bounded on $I$. The above inequality implies that $\rho(x,J_{t}x)$ is also bounded on $I$. Thus there exists $M > 0$ such that we may similarly obtain
\[
\rho(J_{\mu}x,J_{\lambda}x) \leq \left(1 - \frac{\mu}{\lambda}\right)\rho(x,J_{\lambda}x) \leq \left(1 - \frac{\mu}{\lambda}\right)M
\]
whenever $\mu,\lambda \in (a,b)$ and $\mu \leq \lambda$. This proves the desired continuity. Moreover, Theorem \ref{thm:Asymp0} shows the right continuity at $\lambda = 0$ for $x \in \overline{\dom A}$.
\end{proof}
\begin{thm}\label{thm:asymp_infty}
Suppose that $H$ satisfies the geodesic extension property, $A : H \multimap TH$ is a monotone vector field with the surjectivity condition, and $A^{-1}(\mathbf{0}) \neq \emptyset$. Then,
\[
\lim_{\lambda \longrightarrow \infty} J_{\lambda}x = P_{A^{-1}(\mathbf{0})} x,
\]
for every $x \in H$.
\end{thm}
\begin{proof}
Let $x \in H$ and $(\lambda_{k}) \subset (0,\infty)$ be a sequence with $\lambda_{k} \longrightarrow \infty$. For convenience, we put $u := P_{A^{-1}(\mathbf{0})} x$ and $x^{k} := J_{\lambda_{k}} x$ for each $k \in \mathbb{N}$. In view of the inequality \eqref{eqn:firmlynonspreading}, we have for each $k \in \mathbb{N}$:
\begin{equation}\label{eqn:bound-1}
\rho^{2}(x^{k},x) \leq \rho^{2}(x^{k},x) + \rho^{2}(x^{k},u) \leq \rho^{2}(x,u) =: K.
\end{equation}
Therefore, $(x^{k})$ is bounded and so it admits a subsequence $(x^{k_{i}})$ which is $\Delta$-convergent to some point $z \in H$. Recall that $\lambda_{k_{i}}^{-1}\rho(x^{k_{i}},x)\gamma_{x^{k_{i}},x} \in Ax^{k_{i}}$, and so
\[
\abs{Ax^{k_{i}}} \leq \norm{\lambda_{k_{i}}^{-1}\rho(x^{k_{i}},x)\gamma_{x^{k_{i}},x}} = \lambda_{k_{i}}^{-1}\rho(x^{k_{i}},x) \leq \lambda_{k_{i}}^{-1}K.
\]
Fix any $\mu > 0$. From Proposition \ref{prop:HeartWarmingIneq}, we have
\[
\mu^{-1}\rho(x^{k_{i}},J_{\mu}x^{k_{i}}) = \norm{A_{\mu}x^{k_{i}}} \leq \abs{Ax^{k_{i}}} \leq \lambda_{k_{i}}^{-1}K.
\]
It follows that
\[
\rho(x^{k_{i}},J_{\mu}x^{k_{i}}) \leq \mu\lambda_{k_{i}}^{-1}K,
\]
which yields $\lim_{k}\rho(x^{k_{i}},J_{\mu}x^{k_{i}}) = 0$. By the Demiclosedness Principle (Theorem \ref{thm:demiclosed}), we have $z \in \Fix(J_{\mu}) = A^{-1}\mathbf{0}$. Since $\rho^{2}(\cdot,x)$ is $\Delta$-lower semicontinuous and $(x^{k_{i}})$ is $\Delta$-convergent to $z$, we obtain from \eqref{eqn:bound-1} the following:
\[
\rho^{2}(u,x) \leq \rho^{2}(z,x) \leq \liminf_{i \longrightarrow \infty} \rho^{2}(x^{k_{i}},x) \leq \limsup_{i \longrightarrow \infty} \rho^{2}(x^{k_{i}},x) \leq \rho^{2}(u,x).
\]
Therefore, $z = u$ and so $\omega_{\Delta}(x^{k}) = \{u\}$. By Proposition \ref{prop:singletonOmegaDelta} we conclude that $(x^{k})$ is $\Delta$-convergent to $u$. With similar procedure above, we obtain $\lim_{k} \rho(x^{k},x) = \rho(u,x)$. By the Kadec-Klee property (Lemma \ref{lem:Kadec-Klee}), we have the strong convergence $\lim_{k} J_{\lambda_{k}}x = Px$. Since $(\lambda_{k})$ is an arbitrary sequence with $\lambda_{k} \longrightarrow \infty$, the convergence is attained.
\end{proof}
\section{The generation of a nonexpansive semigroup}
In this section, we state and prove the generation theorem applied to a monotone vector field on a complete $\rm CAT(0)$ space. Our results in this section extend several results in the literature. For instance, the generation theorems for proper, convex, lower semicontinuous functions in complete $\rm CAT(0)$ spaces of Jost \cite[Theorem 1.3.13]{MR1652278}, Mayer \cite[Theorem 1.13]{MR1651416}, Stojkovic \cite[Theorem 2.18]{MR2879568}, and Ba\v{c}\'{a}k \cite[Theorem 1.5]{MR3047087} are successfully amplified to use with monotone vector fields. Moreover, the results of Iwamiya and Okochi \cite[Theorem 4.1]{MR1979730} are lifted to a greater generality where the previous requirement for the smoothness of class $C^{3}$ is completely removed.
Recall that for a nonempty subset $C \subset H$, a family $\{S(t)\}_{t \geq 0}$ is said to be a \emph{nonexpansive semigroup} on $C$ if the following conditions are satisfied:
\begin{enumerate}[label=(\roman*)]
\item $S(t) : C \to C$ is nonexpansive for each $t \geq 0$.
\item $S(s+t) = S(s) \circ S(t)$ for all $s,t \geq 0$.
\item $S(0) = I$, where $I$ denotes the identity mapping on $C$.
\item $t \mapsto S(t)x$ is continuous on $[0,+\infty)$ for each $x \in C$.
\end{enumerate}
The following technical lemma can be extracted from the proof of \cite[Theorem 1 of Chapter XIV.7]{MR1336382}. It also appears in a Japanese textbook \cite{MR840217} in the present form. Here, we adopt the notation $\mathbb{N}_{0} := \mathbb{N} \cup \{0\}$. \begin{lem}[\cite{MR840217,MR1336382}]\label{lem:doubleSeq}
Let $\lambda > 0$. For each $k \in \mathbb{N}_{0}$, let $\mu_{k} \in (0,\lambda]$, $\alpha_{k} := \frac{\mu_{k}}{\lambda}$, $\beta_{k} := 1 - \alpha_{k}$, and
\[
t_{k} := \left\{
\begin{array}{ll}
0 &\text{if $k = 0$;} \medskip\\
\sum_{i=1}^{k} \mu_{i} &\text{if $k \in \mathbb{N}$.}
\end{array}
\right.
\]
Let $(A_{j,k})$ be a double sequence such that
\[
\left\{
\begin{array}{l}
A_{0,k} \leq k\lambda, \quad A_{j,0} \leq t_{j} \quad (\forall j,k \in \mathbb{N}_{0}); \smallskip\\
A_{j,k} \leq \alpha_{j}A_{j-1,k-1} + \beta_{j}A_{j-1,k} \quad (\forall j,k \in \mathbb{N}).
\end{array}
\right.
\]
Then the following inequality holds:
\[
A_{j,k} \leq \sqrt{(k\lambda - t_{j})^{2} + k\lambda^{2}} + \sqrt{(k\lambda - t_{j})^{2} + \lambda t_{j}} \quad (\forall j,k \in \mathbb{N}_{0}).
\]
\end{lem}
Now, we are ready to prove the main result of this section. The theorem extends a similar result of Crandal and Liggett \cite{MR0287357} to the setting of a complete $\rm CAT(0)$ space.
\begin{thm}\label{thm:S(t)x}
Suppose that $H$ satisfies the geodesic extension property, $A : H \multimap TH$ is a monotone vector field with the surjectivity condition. Assume that $x \in \overline{\dom A}$. Then, the following limit exists and is uniform on each bounded subinterval of $[0,+\infty)$:
\begin{equation}\label{eqn:S(t)x}
S(t)x := \lim_{k \longrightarrow \infty} J_{t/k}^{k} x \quad (\forall t \in [0,+\infty)).
\end{equation}
Moreover, the following estimate holds for any $x \in \dom A$:
\begin{equation}\label{eqn:semigroupestimate}
\rho(J_{t/k}^{k} x,S(t)x) \leq \abs{Ax} \cdot \frac{2t}{\sqrt{k}}
\end{equation}
for any $t \geq 0$ and $k \in \mathbb{N}$.
\end{thm}
\begin{proof}
Let $t \geq 0$ be given. Then we first show the convergence on \eqref{eqn:S(t)x} by showing that $(J_{t/k}^{k})$ is Cauchy. Let $x \in \dom A$ and $\lambda > 0$ be given. Let $(\mu_{k})$ and $(t_{k})$ be sequences given in Lemma \ref{lem:doubleSeq}. Put
\[
\tilde A_{j,k} := \left\{
\begin{array}{ll}
\rho(J_{\mu_{j}}J_{\mu_{j-1}}\cdots J_{\mu_{1}}x,J_{\lambda}^{k}x) & (\forall j \in \mathbb{N}, \forall k \in \mathbb{N}_{0}); \medskip\\
\rho(x,J_{\lambda}^{k}x) & (j=0, \forall k \in \mathbb{N}_{0}).
\end{array}
\right.
\]
For $k \in \mathbb{N}_{0}$, it follows from Propositions \ref{prop:properties-J} and \ref{prop:HeartWarmingIneq} that
\begin{align*}
\tilde A_{0,k} &= \rho(x,J_{\lambda}^{k}x) \\
&\leq \rho(x,J_{\lambda}x) + \rho(J_{\lambda}x,J_{\lambda}^{2}x) + \cdots + \rho(J_{\lambda}^{k-1}x,J_{\lambda}^{k}x) \\
&\leq k \rho(x,J_{\lambda}x) = k\lambda \norm{A_{\lambda}x} \leq k\lambda\abs{Ax}.
\end{align*}
For $j \in \mathbb{N}$, we have
\begin{align*}
\tilde A_{j,0} &= \rho(J_{\mu_{j}}J_{\mu_{j-1}}\cdots J_{\mu_{1}}x,x) \\
&\leq \rho(J_{\mu_{j}}J_{\mu_{j-1}}\cdots J_{\mu_{1}}x,J_{\mu_{j}}J_{\mu_{j-1}}\cdots J_{\mu_{2}}x) + \rho(J_{\mu_{j}}J_{\mu_{j-1}}\cdots J_{\mu_{2}}x,J_{\mu_{j}}J_{\mu_{j-1}}\cdots J_{\mu_{3}}x) \\
&\quad + \cdots + \rho(J_{\mu_{j}}J_{\mu_{j-1}}x,J_{\mu_{j}}x) + \rho(J_{\mu_{j}}x,x) \\
&\leq \rho(J_{\mu_{1}}x,x) + \rho(J_{\mu_{2}}x,x) + \cdots + \rho(J_{\mu_{j}}x,x) \\
&= \mu_{1}\norm{A_{\mu_{1}}x} + \mu_{2}\norm{A_{\mu_{2}}x} + \cdots + \mu_{j}\norm{A_{\mu_{j}}x} \\
&\leq (\mu_{1} + \cdots + \mu_{j})\abs{Ax} = t_{j}\abs{Ax}.
\end{align*}
Now, let $j,k \in \mathbb{N}$. By using \ref{cdn:J-NX} and \ref{cdn:J-ResolventIden} of Proposition \ref{prop:properties-J}, we get
\begin{align*}
\tilde A_{j,k} &= \rho(J_{\mu_{j}}J_{\mu_{j-1}}\cdots J_{\mu_{1}}x,J_{\lambda}^{k}x) \\
&= \rho\left(J_{\mu_{j}}J_{\mu_{j-1}}\cdots J_{\mu_{1}}x,J_{\mu_{j}}\left(\frac{\mu_{j}}{\lambda}J_{\lambda}^{k-1}x \oplus \left(1 - \frac{\mu_{k}}{\lambda}\right)J_{\lambda}^{k}x\right)\right) \\
&\leq \rho\left(J_{\mu_{j-1}}\cdots J_{\mu_{1}}x,\frac{\mu_{j}}{\lambda}J_{\lambda}^{k-1}x \oplus \left(1 - \frac{\mu_{k}}{\lambda}\right)J_{\lambda}^{k}x\right) \\
&\leq \frac{\mu_{j}}{\lambda} \rho(J_{\mu_{j-1}}\cdots J_{\mu_{1}}x,J_{\lambda}^{k-1}x) + \left(1 - \frac{\mu_{k}}{\lambda}\right)\rho(J_{\mu_{j-1}}\cdots J_{\mu_{1}}x,J_{\lambda}^{k}x) \\
&= \alpha_{j}\tilde A_{j-1,k-1} + \beta_{j}\tilde A_{j-1,k}.
\end{align*}
If $\abs{Ax} = 0$, then $\rho(x,J_{\lambda}^{k}x) = \tilde A_{0,k} = 0$ and so $(J_{t/k}^{k} x)$ is a constant sequence. If $\abs{Ax} > 0$, we put $A_{j,k} := \tilde A_{j,k}/\abs{Ax}$ for all $j,k \in \mathbb{N}_{0}$. Then, $(A_{j,k})$ satisfies the hypotheses of Lemma \ref{lem:doubleSeq} so that we have
\[
\tilde A_{j,k} \leq \abs{Ax} \left(\sqrt{(k\lambda - t_{j})^{2} + k\lambda^{2}} + \sqrt{(k\lambda - t_{j})^{2} + \lambda t_{j}}\right) \quad (\forall j,k \in \mathbb{N}_{0}).
\]
Fix $t > 0$ and $n,m \in \mathbb{N}$ with $m < n$. Take $\lambda := \frac{t}{m}$ and for each $k \in \mathbb{N}_{0}$, take $\mu_{k} := \frac{t}{n}$. So, we have $t_{0} = 0$ and $t_{n} = t$. Hence, we have
\begin{align*}
\rho(J_{t/n}^{n}x,J_{t/m}^{m}x) &= \rho(J_{\mu_{n}}J_{\mu_{n-1}}\cdots J_{\mu_{1}}x,J_{\lambda}^{m}x) = \tilde A_{n,m} \\
&\leq \abs{Ax}\left(\sqrt{(m\lambda - t_{n})^{2} + m\lambda^{2}} + \sqrt{(m\lambda - t_{n})^{2} + \lambda t_{n}}\right) \\
& = \abs{Ax}\cdot\frac{2t}{\sqrt{m}}.
\end{align*}
This shows that $(J_{t/k}^{k}x)$ is Cauchy and so the limit $S(t)x$ is defined, by the completeness of $H$. The above inequality also yields the estimate \eqref{eqn:semigroupestimate}. Take any $T > 0$, we further have
\[
\sup_{t \in [0,T]}\rho(J_{t/k}^{k} x,S(t)x) \leq \abs{Ax} \cdot \frac{2T}{\sqrt{k}} \quad (\forall k \in \mathbb{N}),
\]
which guarantee that the convergence is uniform on compact intervals of $[0,\infty)$.
Next, we show convergence for any $x \in \overline{\dom A}$. For instance, let $(x^{i})$ be a sequence in $\dom A$ such that $x^{i} \longrightarrow x$. For any $m,m \in \mathbb{N}$, we have
\begin{align*}
\rho(J_{t/n}^{n}x,J_{t/m}^{m}x) &\leq \rho(J_{t/n}^{n}x,J_{t/n}^{n}x^{i}) + \rho(J_{t/n}^{n}x^{i},J_{t/m}^{m}x^{i}) + \rho(J_{t/m}^{m}x^{i},J_{t/m}^{m}x) \\
&\leq 2\rho(x^{i},x) + \rho(J_{t/n}^{n}x^{i},J_{t/m}^{m}x^{i}).
\end{align*}
Since $x^{i} \longrightarrow x$ and the sequence $(J_{t/k}^{k}x^{i})_{n \in \mathbb{N}}$ is Cauchy for any $i \in \mathbb{N}$, we conclude that $(J_{t/k}^{k}x)$ is a Cauchy sequence. The convergence is again obtained by the completeness of $H$.
It remains to show that $S(t)x$ is attained uniformly on each bounded interval $[0,T]$ for $x \in \overline{\dom A}$. Again, let $(x^{i})$ be a sequence in $\dom A$ convergent to $x$. Since each mapping $J_{t}$ is nonexpansive for any $t \geq 0$, it follows that $S(t)$ is also nonexpansive for each $t \geq 0$. Consider for $0 \leq t \leq T$, we have
\begin{align*}
\rho(J_{t/k}^{k}x,S(t)x) &\leq \rho(J_{t/k}^{k}x,J_{t/k}^{k}x^{i}) + \rho(J_{t/k}^{k}x^{i},S(t)x^{i}) + \rho(S(t)x^{i},S(t)x) \\
&\leq 2\rho(x,x^{i}) + \rho(J_{t/n}^{n}x^{k},S(t)x^{k}) \\
&\leq 2\rho(x,x^{i}) + \sup_{0 \leq t \leq T} \rho(J_{t/k}^{k}x^{i},S(t)x^{i}) \\
&\leq 2\rho(x,x^{i}) + \abs{Ax^{i}}\cdot\frac{2T}{\sqrt{k}}.
\end{align*}
Letting $\varepsilon > 0$ be arbitrary, we can choose $i_{0} \in \mathbb{N}$ such that $2\rho(x,x^{i_{0}}) < \varepsilon$. This further implies
\[
\sup_{0 \leq t \leq T} \rho(J_{t/k}^{k}x,S(t)x) \leq \varepsilon + \abs{Ax^{i_{0}}}\cdot\frac{2T}{\sqrt{k}}
\]
and hence
\[
\lim_{k \longrightarrow \infty} \sup_{0 \leq t \leq T} \rho(J_{t/k}^{k}x,S(t)x) \leq \varepsilon.
\]
Since $\varepsilon$ is chosen arbitrarily, then convergence is uniform on $[0,T]$.
\end{proof}
In the above theorem, we have in fact defined a family $\{S(t)\}_{t \geq 0}$ of mappings $S(t) : \overline{\dom A} \to \overline{\dom A}$. We next show that this family is a nonexpansive semigroup. Note that the semigroup constructed in this way is said to be \emph{generated by $A$}.
\begin{thm}
Under the assumptions of Theorem \ref{thm:S(t)x}, the family $\{S(t)\}_{t \geq 0}$ generated by the formula \eqref{eqn:S(t)x} is a nonexpansive semigroup.
\end{thm}
\begin{proof}
The fact that $S(0) = I$, $S(t)$ is nonexpansive and $t \mapsto S(t)x$ is continuous on $[0,\infty)$ follows from $J_{0} = I$, $J_{t}$ is nonexpansive, $t \mapsto J_{t}x$ is continuous on $[0,\infty)$ and the uniform convergence of $\{J_{t/k}^{k} x\}$ on each bounded subinterval of $[0,\infty)$ for $x \in \overline{\dom A}$. It is therefore sufficient to show only $S(s+t) = S(s)S(t)$.
Let $t \geq 0$. We first show that $S(nt) = S(t)^{n}$ for all $n \in \mathbb{N}$. For $n =2$, we have
\begin{align*}
\rho(S(t)^{2}x,J_{t/k}^{2k}x) &\leq \rho(S(t)S(t)x,J_{t/k}^{k}S(t)x) + \rho(J_{t/k}^{k}S(t)x,J_{t/k}^{k}J_{t/k}^{k}x) \\
&\leq \rho(S(t)^{2}x,J_{t/k}^{k}S(t)x) + \rho(S(t)x,J_{t/k}^{k}x).
\end{align*}
Letting $k \longrightarrow \infty$, we have $S(t)^{2}x = \lim_{n} J_{t/k}^{2k} x = \lim_{n} J_{2t/2k}^{2k} x = S(2t)x$. Next, let $n \geq 3$ and suppose that $S((n-1)t) = S(t)^{n-1}$ holds true. We have
\[
\rho(S(t)^{n}x,J_{t/k}^{nk}x) \leq \rho(S(t)S(t)^{n-1}x,J_{t/k}^{k}S(t)^{n-1}x) + \rho(S(t)^{n-1}x,J_{t/k}^{(n-1)k}x).
\]
Again, letting $k \longrightarrow \infty$, we similarly have $S(t)^{n} = S(nt)$. The claim that $S(nt) = S(t)^{n}$ for all $n \in \mathbb{N}$ is thus proved.
Now, for two positive rationals $s := c/d$ and $t := p/q$ with $c,d,p,q \in \mathbb{N}$, we have
\begin{align*}
S(s+t) & = S\left(\frac{c}{d} + \frac{p}{q}\right) = S\left(\frac{cq + dp}{dq}\right) = S\left(\frac{1}{dq}\right)^{cq + dp} \\
& = S\left(\frac{1}{dq}\right)^{cq}S\left(\frac{1}{dq}\right)^{dp} = S\left(\frac{c}{d}\right)S\left(\frac{p}{q}\right) = S(s)S(t).
\end{align*}
The continuity of $t \mapsto S(t)$ at each $x \in \overline{\dom A}$ implies that $S(s+t) = S(s)S(t)$ for $s,t \geq 0$.
\end{proof}
The trajectory $(S(t)x)_{t \geq 0}$ can behave very unstably. However, if a stationary point exists, the trajectory is bounded. As is known from Hilbert space theory, $(S(t)x)_{t \geq 0}$ is not necessarily weakly convergent at all. The following result is our final result, and it shows that the mentioned trajectory is bounded if a stationary point exists and if all the accumulated points are contained in $A^{-1}\mathbf{0}$, we have its $\Delta$-convergence to a stationary.
\begin{thm}\label{thm:S(t)x-convergent}
Suppose that all the assumptions of Theorem \ref{thm:S(t)x} hold, $A^{-1}\mathbf{0} \neq \emptyset$, and $\{S(t)\}_{t \geq 0}$ be the semigroup generated by $A$. Let $x \in \overline{\dom A}$. Then, $(S(t)x)_{t \geq 0}$ is bounded. Moreover, if all $\Delta$-accumulated points of $(S(t)x)_{t \geq 0}$ is contained in $A^{-1}\mathbf{0}$, then $(S(t)x)_{t \geq 0}$ is $\Delta$-convergent to a stationary point as $t \longrightarrow \infty$.
\end{thm}
\begin{proof}
Suppose that $p \in A^{-1}\mathbf{0}$. Then, it is a fixed point to all $S(t)$'s. Therefore, we have
\[
\rho(p,S(t)x) \leq \rho(p,x), \quad \forall t \geq 0.
\]
This shows the boundedness, and it implies that a $\Delta$-accumulation point of $(S(t)x)_{t \geq 0}$ exists. Assume that every $\Delta$-accumulation points of $(S(t)x)_{t \geq 0}$ is contained in $A^{-1}\mathbf{0}$. Then, we have
\[
\rho(p,S(t)x) = \rho(S(t-s)p,S(t-s)S(s)x) \leq \rho(p,S(s)x),
\]
for any $0 \leq s \leq t$. Hence, $(S(t)x)_{t \geq 0}$ is Fej\'er monotone with respect to $A^{-1}\mathbf{0}$. By the hypothesis and Proposition \ref{prop:Fejer}, we conclude that $(S(t)x)_{t \geq 0}$ is $\Delta$-convergent to a stationary point of $A$ as $t \longrightarrow \infty$.
\end{proof}
\section*{Conclusions}
In this paper, we utilized the concept of tangent spaces to develop a theory on monotone vector fields and generalized gradient flows over complete $\rm CAT(0)$ spaces. This approach is a natural extension of the theory from the frameworks of Hilbert spaces and Hadamard manifolds. Important instruments in our studies include resolvents and Yosida approximations. Among others, we obtained the resolvent identity (see in Proposition \ref{prop:properties-J}), the convexity of $\overline{\dom A}$ for a monotone vector field $A$ (Proposition \ref{prop:convexity}), asymptotic convergence for the resolvents (Theorems \ref{thm:Asymp0} and \ref{thm:asymp_infty}) and derived a useful estimate for the Yosida approximations (Proposition \ref{prop:HeartWarmingIneq}). Finally, we used such devices to establish a generation theorem of nonexpansive semigroup (Theorem \ref{thm:S(t)x}) which improves and generalizes, to some extents, results of Jost \cite[Theorem 1.3.13]{MR1652278}, Mayer \cite[Theorem 1.13]{MR1651416}, Stojkovic \cite[Theorem 2.18]{MR2879568}, Ba\v{c}\'{a}k \cite[Theorem 1.5]{MR3047087}, and Iwamiya and Okochi \cite[Theorem 4.1]{MR1979730}. The error estimate for the generation is also given in terms of Yosida approximations up to any given accuracy and time.
We also propose the following open questions which are yet to be considered from a viewpoint of our paper.
\begin{enumerate}[label=Q\arabic*.]
\item Does the surjectivity condition always hold for a maximally monotone vector field?
\item Is it possible to drop the geodesic extension property in all definitions and results, where the Yosida approximations are involved?
\end{enumerate}
\section*{Acknowledgements}
The second author was supported by JSPS KAKENHI Grant No. 17K05372. The third author was supported by he Thailand Research Fund (TRF) and the King Mongkut's University of Technology Thonburi (KMUTT) under the TRF Research Scholar Award (Grant No. RSA6080047).
\renewcommand\bibname{References}
\bibliographystyle{abbrv}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,172
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Der Begriff Bundware bezeichnet in der Floristik Blumen, in der Aquaristik dagegen Wasserpflanzen, die vom Käufer selbst verarbeitet werden.
Bundware in der Floristik
Als Bundware bezeichnen Floristen im engen Sinne einen Bund an Schnittblumen, die nicht aufgebunden sind, d. h. nicht zu einem Strauß arrangiert wurden. Der Käufer selbst arrangiert die Blumen daheim in der Vase.
Durch den ersparten Arbeitsaufwand des Floristen kann die Ware knapper kalkuliert werden, was sie preislich attraktiv macht für Menschen, die ihren Blumenschmuck gerne selbst gestalten.
Besonders häufig in dieser Form angebotene Blumen sind Tulpen, Narzissen, Rosen und Chrysanthemen.
Solche Bundware wird mittlerweile auch häufig in Supermärkten angeboten.
Bundware in der Aquaristik
Bundware werden in der Aquaristik die Wasserpflanzen genannt, die nicht eingetopft angeboten werden. Typischerweise handelt es sich dabei um Pflanzen, die mit vielen Haltebedingungen im Aquarium zurechtkommen und sehr schnell wachsen. Sie bieten Aquarianern im Gegensatz zu der getopften Ware die Möglichkeit, verhältnismäßig günstig das Aquarium zu bepflanzen, um für die dort zu haltenden Fische geeignete Lebensvoraussetzungen zu schaffen.
Die Pflanzen sind in der Regel am unteren Ende mit einem Plastikring oder einem Metallstreifen zusammengehalten. Zarte Stängel wie die der Dichtblättrigen Wasserpest oder der Glattblättrige Valisnerie können nach einem leichten Beschneiden von Wurzeln direkt so in den Aquariengrund gepflanzt werden. Alternativ wird die Befestigung entfernt und jeder einzelne Spross für sich in den Aquarienboden gepflanzt. Die meist noch nicht besonders gut bewurzelten Sprosse schwimmen allerdings leicht auf. Sie können mit Pflanzennadeln im Boden befestigt werden, um sie besser zu verankern. Diese arbeitsintensive Methode hat den Vorteil, dass die Einrichtung neuer und großer Aquarien preisgünstiger ist. Bis sich der Kreislauf im Aquarium so eingespielt hat, dass Fische und Wirbellose eingesetzt werden können, ist das Aquarium meist schon verhältnismäßig dicht bewachsen.
Zu den regelmäßig als Bundware angebotenen Arten zählen Dichtblättrige Wasserpest, Glattblättrige Valisnerie, Falscher Wasserfreund, Indischer Wasserfreund und Indischer Wasserwedel.
Floristik
!
Produkt von Baumschulen
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,128
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Dairy Australia is the Australian national body for the dairy industry, and a Research and development corporation (RDC). It is mainly funded by the Dairy Services Levy, a tax paid by farmers based on milk production. It also acts as a funding body through which the federal Department of Agriculture provides funding for rural research and development in Australia. Dairy Australia also attracts funding at project level from state governments, universities, research organisations and other dairy support organisations. It was previously known as the Dairy Research & Development Corporation.
Industry figure Pat Rowley acted as Chairman of Dairy Australia from 2003 until his retirement in 2006.
Dairy Australia have argued that adding dairy products to plant-based diets enhances the nutritional adequacy of the diet and the taste.
References
External links
Australian dairy organisations
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,649
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Q: React Native Text Input Insert and Update I have a screen it has text input's. I am trying to insert the TextInput value into the database and retrieve the data from database and display on the TextInputwithout any problem. But the problem is to edit the data on the TextInput. I try to edit the TextInputvalue the text cannot be changed.
if (this.state.mode == 'edit') {
customer.updateCustomer(JSON.stringify(customerregVOObj));
} else {
customerregVOObj._id = 'abc';
customer.createCustomer(customerregVOObj);
}
updateCustomer = () => {
customerregVOObj.shopName = this.state.shopName;
}
<TextInput
underlineColorAndroid='transparent' style={styles.input}
returnKeyType={"next"} autoFocus={true} placeholder="Shop Name"
onChangeText={(text) => this.setState({ shopName: text })}
value={this.state.shopDetail.shopName}
/>
this.state.mode == 'new' ? <Button onPress={this.updateCustomer} title="Submit" /> :
<Button onPress={this.updateCustomer} title="Update" />
A: You are passing the wrong state in value. Kindly replace below code:
<TextInput
underlineColorAndroid='transparent' style={styles.input}
returnKeyType={"next"} autoFocus={true} placeholder="Shop Name"
onChangeText={(text) => this.setState({ shopName: text })}
value={this.state.shopName}
/>
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,001
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\section{Introduction}
\label{sec:Introduction}
The basic goal of supervised learning is to determine a function\footnote{In what follows, the input domain $[0,1]^d$ could be replaced by more general domains (for example Lipschitz domains) without any change in the later results.
The unit cube $[0,1]^d$ is merely chosen for concreteness.}
$u:[0,1]^d\to \mathbb{R}$ from (possibly noisy) samples $(u(x_1),\dots , u(x_m))$.
As the function $u$ can take arbitrary values between these samples, this problem is, of course, not solvable without any further information on $u$.
In practice, one typically leverages domain knowledge to a priori estimate the structure and regularity of $u$, for instance in terms of symmetries, its smoothness or compositionality.
Such additional information can be encoded via a suitable \emph{target class} $U\subset C([0,1]^d)$ that $u$ is known to be a member of.
We are interested in the optimal accuracy for reconstructing $u$ that can be achieved by any algorithm which utilizes $m$ point samples.
To make this mathematically precise, we assume that this accuracy is measured by a norm $\|\cdot\|_Y$ of a suitable Banach space $Y\supset U$.
Formally, an algorithm can thus be described by a map $A:U\to Y$ that can query the function $u$ at $m$ points $x_i$ and that outputs a function $A(u)$ with $A(u) \approx u$ (see Section~\ref{sec:IBC} for a precise definition that incorporates adaptivity and stochasticity).
We will be interested in \emph{upper and lower bounds} on the accuracy that can be reached by any such algorithm --- equivalently, we are interested in the minimal number $m$ of point samples needed for any algorithm to achieve a given accuracy $\varepsilon$ for every $u\in U$.
This $m$ would then establish a fundamental benchmark on the sampling complexity (and the algorithmic complexity) of learning functions in $U$ to a given accuracy (for instance, if $m$ is larger than the estimated number of atoms in our universe, the reconstruction problem turns out to be intractable).
The choice of the Banach space $Y$ --- in other words \emph{how we measure accuracy} --- is very crucial here. For example,
statistical learning theory provides upper bounds on the optimal accuracy in terms of an expected loss, or with respect to $Y=L^2([0,1]^d,d\mathbb{P})$, where $\mathbb{P}$ is a (generally unknown) data generating distribution;
see \cite{mohri2012, devroye2013probabilistic,shalev2014understanding,kim2021fast}.
This offers a powerful approach for achieving a small reconstruction error in an averaged sense, but there are many important scenarios where such bounds on the accuracy are not sufficient: For several applications in the sciences (for example in the context of the numerical solution of partial differential equations \cite{raissi2019physics,han2018solving}), any security-critical application (for example, facial ID authentication schemes~\cite{guo2019survey}),
as well as any application with a \emph{distribution-shift} (i.e., the data generating distribution is different from the distribution in which the accuracy is measured~\cite{quinonero2008dataset}) one would like to obtain an approximation of $u$ that is close to $u$ not only on average, but that can be guaranteed to be close \emph{for every} $x\in [0,1]^d$.
Such applications can only be efficiently solved if there exists an efficient algorithm that achieves a small error with respect to $Y=L^\infty([0,1]^d)$, i.e., the uniform norm.
Inspired by recent successes of \emph{deep learning} across a plethora of tasks in machine learning \cite{lecun2015deep} and also increasingly the sciences \cite{jumper2021highly,pfau2020ab}, we will be particularly interested in the case where the target class $U$ consists of --- or contains --- realizations of (feed-forward) neural networks of a specific architecture\footnote{By \emph{architecture} we mean the number of layers $L$, as well as the number of neurons in each layer, typically denoted $(N_0,N_1,\dots , N_L)$.}. Neural networks have been proven and observed to be extremely powerful in terms of their expressivity, that is, their ability to accurately approximate large classes of complicated functions with only relatively few parameters \cite{elbrachter2021deep}. However, it has also been repeatedly observed that the \emph{training} of neural networks (e.g., fitting a neural network to data samples) to high \emph{uniform} accuracy presents a big challenge: conventional training algorithms (such as SGD and its variants) often find neural networks that perform well on average (meaning that they achieve a small generalization error), but there are typically some regions in the input space where the error is large; see Figure \ref{fig:unstableNN} for an illustrative example. This phenomenon has been systematically studied on an empirical level in \cite{adcock2021gap}. It is also at the heart of several observed instabilities in the training of deep neural networks, including \emph{adversarial examples} \cite{szegedy2013intriguing, goodfellow2015explaining} or so-called \emph{hallucinations} emerging in generative modeling, e.g., tomographic reconstructions \cite{bhadra2021hallucinations} or machine translation \cite{muller2020domain}.
In a security-critical context such instabilities can be exploited by an adversary, for example to hack mobile phones by bypassing its (deep learning based) facial ID authentication \cite{shmelkin2021generating}.
These examples strongly indicate that many deep learning based systems are prone to being fooled, posing a crucial problem in any security-critical context.
\begin{figure}
\centering
\includegraphics[width = \textwidth]{figures/sinusoidal.pdf}
\caption{Even though the training of neural networks from data samples may achieve a small error \emph{on average}, there are typically regions in the input space where the pointwise error is large. The target function in this plot is given by $x \mapsto \log(\sin(50x) + 2) + \sin(5x)$ (based on~\cite{adcock2021gap}) and the model is a neural network of width $N_1=\dots=N_{L-1}=50$ and depth $L=22$. It is trained on $m=1000$ uniformly distributed samples with batch-size $20$ according to the hyperparameters in Tables~\ref{table:hp} and~\ref{table:hp_unstable} and achieves final $L^1$ and $L^\infty$ errors of $2.8\cdot 10^{-3}$ and $0.19$. The middle and right plot are zoomed versions of the left plot.}
\label{fig:unstableNN}
\end{figure}
In the present paper we show that these observations are not a coincidence:
\begin{theorem}\label{thm:main_intro}
Suppose that $U$ contains all neural networks with $d$-dimensional input, ReLU activation function, $L$ layers of width up to $3d$, and coefficients bounded by $c$ in the $\ell^q$ norm.
Assume that there exists an algorithm that reconstructs all functions in $U$ to uniform (i.e., $L^\infty([0,1]^d)$) accuracy $\varepsilon$ from $m$ point samples.
Then, with
\begin{equation*}\label{eq:constant_intro}
\Omega
= \begin{cases}
\frac{1}{4\cdot 3^{\nicefrac{2}{q}}} \cdot c^L \cdot d^{1-\frac{2}{q}}
& \text{if } q\le 2 \\[0.2cm]
\frac{1}{24} \cdot c^L \cdot ((3d)^{1-\frac{2}{q}})^{L-1}
& \text{if } q\geq 2
\end{cases}
\end{equation*}
we have
$$
m\geq \left(\frac{\Omega}{64d}\right)^d \cdot \varepsilon^{-d}.
$$
\end{theorem}
Theorem~\ref{thm:main_intro} is a special case of Theorem~\ref{thm:main} (covering $Y=L^p([0,1]^d)$ for all $p\in [1,\infty]$) which will be stated and proven in Section~\ref{sub:MainResultProof}.
To give a concrete example, we consider the problem of training neural networks with ReLU activation function, $L$ layers of width at most $3d$, and coefficients bounded by $c$ to uniform accuracy $\varepsilon = \nicefrac{1}{1024}$. According to our results we would need at least
$$
m \geq 2^d \cdot c^{dL} \cdot (3d)^{d (L-2)}
$$
many samples --- \emph{the sampling complexity thus depends exponentially on the input dimension $d$, the network width, and the network depth}, becoming intractable even for moderate values of $d,c,L$ (for $d=15$, $c=2$, and $L=7$, the sample size $m$ would already have to exceed the estimated number of atoms in our universe).
If, on the other hand, reconstruction only with respect to the $L^2$ norm were required, standard results in statistical learning theory (see, for example, \cite{berner2020analysis}) show that $m$ only needs to depend polynomially on $d$.
We conclude that uniform reconstruction is vastly harder than reconstruction with respect to the $L^2$ norm and, in particular, intractable.
Our results are further corroborated by numerical experiments presented in Section~\ref{sec:num} below.
We also complement the lower bounds of Theorem~\ref{thm:main_intro} with corresponding upper bounds.
\begin{theorem}
\label{thm:upper_bound_intro}
Suppose that $U$ consists of all neural networks with $d$-dimensional input, ReLU activation function, $L$ layers of width at most $(N_1,\dots,N_{L-1},1)$, and coefficients bounded by $c$ in the $\ell^q$ norm.
Then, there exists an algorithm that reconstructs all functions in $U$ to uniform (i.e., $L^\infty([0,1]^d)$) accuracy $\varepsilon$ from $m$ point samples with
\begin{equation*}
\varepsilon
\le \begin{cases}
2\sqrt{d}\cdot c^L\cdot m^{-\frac1d} & \text{if } q\le 2 \\
2\sqrt{d}\cdot c^L\cdot (\sqrt{d}\cdot N_1 \cdots N_{L-1})^{1-\frac{2}{q}}\cdot m^{-\frac1d}
& \text{if } q\geq 2 .
\end{cases}
\end{equation*}
\end{theorem}
Theorem~\ref{thm:upper_bound_intro} follows from Theorem~\ref{thm:upper_bound} that will be stated and proven in Section~\ref{sub:ProofUpperBound}.
\begin{remark}
Our setting allows for an algorithm to choose the sample points $(x_1,\dots , x_m)$ in an adaptive way for each $u\in U$; see Section~\ref{sec:IBC} for a precise definition of the class of adaptive (possibly randomized) algorithms.
This implies that even a very clever sampling strategy (as would be employed in active learning) cannot break the bounds established in this paper.
\end{remark}
\begin{remark}
Close inspection of the proof of Theorem~\ref{thm:main} shows that one can replace the point samples $u(x_i)$ by $T u (x_i)$, where $T : U \to C([0,1]^d)$ is any local operator\footnote{This means that if $f = g$ on a neighborhood of $x \in [0,1]^d$, then $(T f)(x) = (T g)(x)$.}.
Since any differential operator is a local operator, our lower bounds also hold if we measure point samples of a differential operator applied to $u$, as it is commonly done in the context of so-called physics-informed neural networks \cite{raissi2019physics}.
\end{remark}
\begin{remark}
Our results also shed light on the impact of different regularization methods applied to the network coefficients. While picking a stronger regularizer (e.g., a small value of $q$) yields quantitative improvements (in the sense of a smaller $\Omega$), for small $\varepsilon$ the sampling size required for approximation in $L^\infty$ always scales exponentially in the input dimension.
However, this scaling is only visible for very small $\varepsilon$.
\end{remark}
\paragraph{Related Work.}
Several other works have established ``hardness'' results for neural network training. For example, the foundational work \cite{blum1992training} shows that for certain specific architectures the training process can be NP-complete. By contrast, our results do not directly consider algorithm runtime at all; our results are much stronger in the sense of showing that even if it were possible to fit a neural network to data efficiently, the necessary number of data points would be too large to be tractable.
Our results are most closely related to recent results in \cite{grohs2021proof} where target classes consisting of neural network approximation spaces are considered.
The results of \cite{grohs2021proof}, however, are purely asymptotic.
Since the asymptotic behavior incurred by the rate is often only visible for very fine accuracies, the results of \cite{grohs2021proof} therefore cannot be applied to obtain concrete lower bounds on the required sample size.
By contrast, our results are completely explicit in all parameters and readily yield practically relevant bounds. In addition, our results elucidate the role of adaptive sampling and different coefficient regularization methods.
\paragraph{Notation.}
\label{sub:Notation}
For a finite set $I$ and $(a_i)_{i\in I}\in \mathbb{R}^I$, we write $\mathop{\mathpalette\avsuminner\relax}\displaylimits_{i\in I}a_i\coloneqq\frac{1}{|I|}\sum_{i\in I}a_i$.
For $m\in \mathbb{N}$, we write ${[m]\coloneqq\{1,\dots , m\}}$.
For $d\in \mathbb{N}$ and $A \subset \mathbb{R}^d$, we denote by $\interior{A}$ the set of interior points of $A$.
For a matrix $W \in \mathbb{R}^{n \times k}$ and $q \in [1,\infty)$, we write $\|W\|_{\ell^q} \coloneqq \big( \sum_{i,j} |W_{i,j}|^q \big)^{1/q}$, and for $q = \infty$ we write $\|W\|_{\ell^\infty} \coloneqq \max_{i,j} |W_{i,j}|$.
For vectors $b \in \mathbb{R}^n$, we use the analogously defined notation $\|b\|_{\ell^q}$.
\section{Main Results}
\label{sec:MainResults}
This section contains our main theoretical results. We introduce the considered classes of algorithms in Section \ref{sec:IBC} and target classes in Section \ref{sec:NNDef}.
Our main lower and upper bounds are formulated and proven in Section \ref{sub:MainResultProof} and Section \ref{sub:ProofUpperBound}, respectively.
\subsection{Adaptive (randomized) algorithms based on point samples}
\label{sec:IBC}
As described in the introduction, our goal is to analyze how well one can recover an unknown
function $u$ from a target class $U$ in a Banach space $Y$ based on point samples.
This is one of the main problems in \emph{information-based complexity} \cite{traub2003information},
and in this section we briefly recall the most important related notions.
Given $U \subset C([0,1]^d) \cap Y$ for a Banach space $Y$, we say that a map $A:U\to Y$ is an \emph{adaptive deterministic method using $m\in \mathbb{N}$ point samples} if there are $f_1\in [0,1]^d$ and mappings
\begin{equation*}
f_i:\left([0,1]^{d}\right)^{i-1}\times \mathbb{R}^{i-1}\to [0,1]^d, \quad i = 2,\dots , m,
\quad
\mathrm{and}
\quad
Q: \left([0,1]^{d}\right)^m\times \mathbb{R}^m \to Y
\end{equation*}
such that for every $u\in U$, using the point sequence $\mathbf{x}(u)= (x_1,\dots , x_m)\subset [0,1]^d$ defined as
\begin{equation}\label{eq:points}
x_1 = f_1,\quad x_i = f_i(x_1,\dots , x_{i-1},u(x_1),\dots , u(x_{i-1})),\quad i = 2,\dots , m ,
\end{equation}
the map $A$ is of the form
\begin{equation*}
A(u) = Q(x_1,\dots , x_m , u(x_1),\dots , u(x_m)) \in Y.
\label{eq:DeterministicAlgorithm}
\end{equation*}
The set of all deterministic methods using $m$ point samples will be denoted $\mathrm{Alg}_m(U,Y)$.
In addition to such deterministic methods, we also study randomized methods defined as follows:
A tuple $(\mathbf{A},\mathbf{m})$ is called an \emph{adaptive random method using $m\in \mathbb{N}$ point samples on average}
if $\mathbf{A} = (A_\omega)_{\omega \in \Omega}$ where $(\Omega,\mathcal{F},\mathbb{P})$ is a probability space,
and where $\mathbf{m}:\Omega \to \mathbb{N}$ is such that the following conditions hold:
\begin{enumerate}
\item $\mathbf{m}$ is measurable, and $\mathbb{E}[\mathbf{m}]\le m$;
\item $\forall \, u\in U: \ \omega \mapsto A_\omega(u)$ is measurable with respect to the Borel $\sigma$-algebra on $Y$;
\item $\forall \, \omega \in \Omega:\ A_\omega \in \mathrm{Alg}_{\mathbf{m}(\omega)}(U,Y)$.
\end{enumerate}
The set of all random methods using $m$ point samples on average will be denoted $\mathrm{Alg}^{MC}_m(U,Y)$, since such methods are sometimes called
\emph{Monte-Carlo algorithms}.
For a target class $U$, we denote the \emph{optimal (randomized) error} as
\begin{equation*}
\label{eq:mc_err}
\mathrm{err}_m^{MC}(U,Y)
\coloneqq \inf_{(\mathbf{A},\mathbf{m})\in \mathrm{Alg}^{MC}_m(U,Y)} \,\,
\sup_{u\in U} \,\,
\mathbb{E}\left[\|u - A_\omega(u)\|_Y\right].
\end{equation*}
We note that $\mathrm{Alg}_m (U,Y) \subset \mathrm{Alg}_m^{MC}(U,Y)$,
since each deterministic method can be interpreted as a randomized
method over a trivial probability space.
\begin{comment}
{We remark that each adaptive deterministic method using $m \in \mathbb{N}$ point samples
can also be interpreted as an adaptive randomized method using $m$ point samples on average
(over a trivial probability space);
hence, $\mathrm{err}_m^{MC}(U,Y) \leq \mathrm{err}_m^{\mathrm{det}}(U,Y)$.}
\end{comment}
\subsection{Neural network classes}
\label{sec:NNDef}
We will be concerned with target classes related to ReLU neural networks.
These will be defined in the present subsection.
Let $\varrho : \mathbb{R} \to \mathbb{R}$, $\varrho(x) = \max \{0, x\}$ be the \emph{ReLU activation function}.
Given a \emph{depth} $L\in \mathbb{N}$, an \emph{architecture} $(N_0,N_1,\dots , N_L)\in \mathbb{N}^{L+1}$, and \emph{neural network coefficients}
\[
\Phi = \left((W^i,b^i)\right)_{i=1}^L
\in \bigtimes_{i=1}^L \left( \mathbb{R}^{N_i\times N_{i-1}}\times \mathbb{R}^{N_i}\right) ,
\]
we define their \emph{realization} $R(\Phi)\in C(\mathbb{R}^{N_0},\mathbb{R}^{N_L})$ as $R(\Phi)(x) = x^L$, where
\begin{align*}
x^0 &= x \in \mathbb{R}^{N_0},\\
x^i &= \varrho\left(W^i x^{i-1}+b^i\right) \in \mathbb{R}^{N_i}\, \text{ for }i \in [L-1], \\
x^L &= W^L x^{L-1} + b^L \in \mathbb{R}^{N_L} ,
\end{align*}
with $\varrho$ applied componentwise.
Given $c > 0$ and $q \in [1,\infty]$, define the class
\begin{equation*}
\mathcal{H}_{(N_0,\dots , N_L),c}^q
\coloneqq \left\{
R(\Phi)
:\,
\Phi \in \bigtimes_{i=1}^L
\left( \mathbb{R}^{N_i\times N_{i-1}}\times \mathbb{R}^{N_i}\right)
\text{ and } \|\Phi\|_{\ell^q} \le c
\right\},
\end{equation*}
where $\|\Phi\|_{\ell^q} = \max_{1 \leq i \leq L} \max \{ \|W^i\|_{\ell^q}, \|b^i\|_{\ell^q} \}$.
For studying target classes related to neural networks, the following definition will be useful.
\begin{definition}
Let $U,\mathcal{H} \subset C([0,1]^d)$.
We say that \emph{$U$ contains a copy of $\mathcal{H}$ (attached to $u_0 \in U$ with constant $c_0 > 0$)} if
\begin{equation*}
u_0 + c_0\cdot\mathcal{H}\subset U.
\end{equation*}
\end{definition}
\subsection{Proof of our main lower bound}
\label{sub:MainResultProof}
The following result constitutes the main result of the present paper.
Theorem~\ref{thm:main_intro} readily follows from it as a special case.
\begin{theorem}\label{thm:main}Let $L\in \mathbb{N}_{\geq3}$, $d,B\in \mathbb{N}$, $p,q\in [1,\infty]$, and $c\in (0,\infty)$.
Suppose that the target class $U\subset C([0,1]^d)$ contains
a copy of $\mathcal{H}_{(d,B\dots , B,1),c}^q$ with constant $c_0$, where the $B$ in $(d,B,\dots , B,1)$ appears $L-1$ times.
Then, for any $s\in \mathbb{N}$ with $s\le \min\left\{\frac{B}{3},d\right\}$ we have
\begin{equation}
\mathrm{err}_m^{MC}(U,L^p([0,1]^d))
\geq c_0\cdot \frac{\Omega}{(64s)^{1+\frac{s}{p}}}\cdot m^{-\frac{1}{p} - \frac{1}{s}},
\end{equation}
where
\begin{equation}
\Omega
= \begin{cases}
\frac{1}{4 \cdot 3^{\nicefrac{2}{q}}} \cdot c^L \cdot s^{1-\frac{2}{q}}
& \text{if } q\le 2 \\
\frac{1}{24} \cdot c^L\cdot (B^{1-\frac{2}{q}})^{L-1}
& \text{if } q\geq 2 .
\end{cases}
\end{equation}
\end{theorem}
The proof of the above theorem follows by first showing that the class
$\mathcal{H}_{(d,B,\dots,B,1),c}^q$ contains a large class of ``hat functions''
(see Section~\ref{subsub:HatFunctionConstruction}),
and then showing that the lower bound stated in Theorem~\ref{thm:main} holds whenever
the class $U$ contains such a large class of hat functions (see Section~\ref{subsub:GeneralLowerBound}).
\subsubsection{Construction of hat functions implemented by ReLU networks}
\label{subsub:HatFunctionConstruction}
For $d \in \mathbb{N}$, $M > 0$, $\sigma \in \mathbb{R}$, $s \in [d]$, and $y \in \mathbb{R}^d$, define
\begin{equation}
\label{eq:lambda}
\Lambda_{M,\sigma} : \quad
\mathbb{R} \to (-\infty,1], \quad
t \mapsto \begin{cases}
0 & \text{if } t \leq \sigma - \frac{1}{M} \\
1 - M \cdot |t - \sigma| & \text{if } t \geq \sigma - \frac{1}{M}
\end{cases}
\end{equation}
and furthermore
\begin{alignat*}{5}
&& \Delta_{M,y}^{(s)} : \quad
& \mathbb{R}^d \to (-\infty,1], \quad
&& x \mapsto \left(\sum_{i=1}^s \Lambda_{M,y_i}(x_i)\right) - (s - 1) \\
&& \vartheta_{M,y}^{(s)} : \quad
& \mathbb{R}^d \to [0,1], \quad
&& x \mapsto \varrho(\Delta_{M,y}^{(s)}(x)) ,
\end{alignat*}
where, as before, $\varrho : \mathbb{R} \to \mathbb{R}, \, x \mapsto \max \{ 0,x \}$ denotes the ReLU activation function.
A plot of $\Lambda_{M,\sigma}$ is shown in Figure~\ref{fig:LambdaPlot}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=\linewidth]{figures/lambda.pdf}
\end{center}
\caption{\label{fig:LambdaPlot}
Plots of the function $\Lambda_{M,\sigma}$ in Equation~\eqref{eq:lambda}
for $(M,\sigma)\in \{(2,1),(4,1),(4,\frac32)\}$.}
\end{figure}
With these definitions, the function $\vartheta_{M,y}^{(s)}$ satisfies the following properties:
\begin{lemma}\label{lem:BumpProperties}
For $d \in \mathbb{N}$, $s \in [d]$, $M \geq 1$, $y \in [0,1]^d$, and $p \in (0,\infty]$, we have
\[
\operatorname{supp} \vartheta_{M,y}^{(s)}
\subset y + (M^{-1} [-1,1]^s \times \mathbb{R}^{d - s})
\]
and
\[
\frac12 \cdot (4s)^{-s/p}\cdot M^{-s/p}\le \| \vartheta_{M,y}^{(s)} \|_{L^p([0,1]^d)}
\le 2^{s/p} \cdot M^{-s/p} .
\]
\end{lemma}
\begin{proof}
See Appendix \ref{app:lower}
\end{proof}
Finally, we show that the functions $\vartheta_{M,y}^{(s)}$
can be represented using neural networks of a suitable architecture and
with a suitable bound on the magnitude of the coefficients.
We begin with the (more interesting) case where the exponent $q$
that determines the regularization of the weights satisfies $q \geq 2$.
\begin{lemma}\label{lem:ImplementingBadFunctionsDeepBigQ}
Let $d \in \mathbb{N}$, $L \in \mathbb{N}_{\geq 3}$, $B \in \mathbb{N}_{\geq 3}$, $c > 0$, $q \in [2,\infty]$,
and $s \in \mathbb{N}$ with $s \leq \min \{ \frac{B}{3}, d \}$. Then, there exists a constant
\[
\lambda \geq c^L \cdot (B^{1 - \frac{2}{q}})^{L-1} / 12
\]
such that
\[
\nu \cdot \frac{\lambda}{M s} \cdot \vartheta_{M,y}^{(s)}
\in \mathcal{H}_{(d,B,\dots,B,1),c}^q
\qquad \forall \, M \in \mathbb{N}, \nu \in \{ \pm 1 \}, \text{ and } y \in [0,1]^d,
\]
where the $B$ in $(d,B,\dots,B,1)$ appears $L-1$ times.
\end{lemma}
\begin{proof}See Appendix \ref{app:lower}
\end{proof}
Now, we also consider the case $q \leq 2$.
We remark that in the case $q = 2$, the next lemma only agrees with
Lemma~\ref{lem:ImplementingBadFunctionsDeepBigQ} up to a constant factor.
This is a proof artifact and is inconsequential for the questions we are interested in.
\begin{lemma}\label{lem:ImplementingBadFunctionsDeepSmallQ}
Let $d \in \mathbb{N}$, $L \in \mathbb{N}_{\geq 3}$, $B \in \mathbb{N}_{\geq 3}$, $c > 0$, $q \in [1,2]$, and
$s \in \mathbb{N}$ with $s \leq \min \{ d, \frac{B}{3} \}$.
Then, we have
\[
\nu \cdot \frac{c^L s^{1 - \frac{2}{q}} / (2 \cdot 3^{2/q})}{M s} \vartheta_{M,y}^{(s)}
\in \mathcal{H}_{(d,B,\dots,B,1),c}^q
\quad \forall \, M \in \mathbb{N}, \nu \in \{ \pm 1 \}, \text{ and } y \in [0,1]^d
,
\]
where the $B$ in $(d,B,\dots,B,1)$ appears $L-1$ times.
\end{lemma}
\begin{remark*}
A straightforward adaptation of the proof shows that the same statement holds
for $\mathcal{H}_{(d,B,B_2,\dots,B_{L-1},1),c}^q$ instead of
$\mathcal{H}_{(d,B,\dots,B,1),c}^q$, for arbitrary $B_2,\dots,B_{L-1} \in \mathbb{N}$.
\end{remark*}
\begin{proof}See Appendix~\ref{app:lower}.
\end{proof}
\subsubsection{A general lower bound}
\label{subsub:GeneralLowerBound}
The next theorem shows that any target class containing a large number of (shifted) hat function has a large optimal error.
\begin{theorem}\label{thm:GeneralLowerBound}
Let $d,m \in \mathbb{N}$, $s \in[d]$, and $M := 8 \lceil m^{1/s} \rceil$.
Assume that $U \subset C([0,1]^d)$ satisfies
\[
u_0 + \nu \cdot \frac{\lambda}{M s} \vartheta^{(s)}_{M,y}
\in U
\qquad \forall \, \nu \in \{ \pm 1 \} \text{ and } y \in [0,1]^d
\]
for certain $\lambda > 0$ and $u_0 \in C([0,1]^d)$. Then,
\[
\mathrm{err}_m^{MC} (U; L^p([0,1]^d))
\geq \frac{\lambda / 2}{(64 s)^{1 + \frac{s}{p}}} \cdot m^{-\frac{1}{p} - \frac{1}{s}}
\qquad \forall \, p \in [1,\infty]
.
\]
\end{theorem}
\begin{proof}See Appendix \ref{app:lower}.
\end{proof}
\subsubsection{Completing the proof}
\label{subsub:LowerBoundProofCompletion}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
This follows by combining Theorem~\ref{thm:GeneralLowerBound}
with Lemmas~\ref{lem:ImplementingBadFunctionsDeepBigQ}
and \ref{lem:ImplementingBadFunctionsDeepSmallQ}.
\end{proof}
\subsection{Proof of our main upper bound}
\label{sub:ProofUpperBound}
In this section we present a proof of the following result.
\begin{theorem}
\label{thm:upper_bound}
Let $L,d\in \mathbb{N}$, $q\in [1,\infty]$, $c\in (0,\infty)$, and $N_1,\dots , N_{L-1}\in \mathbb{N}$.
Then, we have
\begin{equation}
\begin{split}
& \mathrm{err}_m^{MC}\left(\mathcal{H}_{(d,N_1,\dots , N_{L-1},1),c}^q,L^\infty([0,1]^d)\right) \\
& \le \begin{cases}
2\sqrt{d}\cdot c^L\cdot m^{-\frac1d} & \text{if } q\le 2 \\
2\sqrt{d}\cdot c^L\cdot (\sqrt{d}\cdot N_1 \cdots N_{L-1})^{1-\frac{2}{q}}\cdot m^{-\frac1d} & \text{if } q\geq 2 .
\end{cases}
\end{split}
\end{equation}
\end{theorem}
For the proof, we first show that each neural network
$R (\Phi) \in \mathcal{H}_{(N_0,\dots,N_L),c}^q$ is Lipschitz-continuous,
where the Lipschitz constant can be conveniently bounded in terms of the
parameters $N_0,\dots,N_L, c, q$.
Here, we compute the Lipschitz constant with respect to the $\ell^2$-norm,
i.e., we consider $\mathrm{Lip}_{\ell^2}$, where we generally define
\begin{equation}
\mathrm{Lip}_{\ell^q} (f)
:= \sup_{x,y \in \mathbb{R}^d, x \neq y}
\frac{\|f(x) - f(y)\|_{\ell^q}}{\| x - y \|_{\ell^q}}
\quad \text{for} \quad
f : \mathbb{R}^d \to \mathbb{R}^k
.
\label{eq:LipschitzConstantDefinition}
\end{equation}
\begin{lemma}\label{lem:LipschitzContinuityForNN}
Let $L \in \mathbb{N}$, $q \in [1,\infty]$, $c > 0$, and $N_0,\dots,N_L \in \mathbb{N}$.
Then, each $R (\Phi) \in \mathcal{H}_{(N_0,\dots,N_L),c}^q$ satisfies
\[
\mathrm{Lip}_{\ell^2} ( R(\Phi) )
\leq \begin{cases}
c^{L} & \text{if } q \leq 2 \\[0.2cm]
c^{L} \cdot (\sqrt{N_0 N_L} \cdot N_{1} \cdots N_{L-1})^{1 - 2 / q} & \text{if } q \geq 2 .
\end{cases}
\]
\end{lemma}
\begin{proof}See Appendix \ref{app:upper}.
\end{proof}
Next we show that any function with moderate Lipschitz constant can be reconstructed from samples by piecewise constant interpolation.
\begin{lemma}\label{lem:LipschitzRecovery}
Let $d\in\mathbb{N}$. Then, for every $m \in \mathbb{N}$, there exist points $x_1,\dots,x_m \in [0,1]^d$ and a map
\({
\Theta_m :
\mathbb{R}^m \to L^\infty ([0,1]^d)
}\)
satisfying
\begin{equation}
\big\| \Theta_m \bigl(u(x_1),\dots,u(x_m)\bigr) - u \big\|_{L^\infty([0,1]^d)}
\leq \mathrm{Lip}_{\ell^2} (u) \cdot 2 \sqrt{d} \cdot m^{-1/d}
\label{eq:LipschitzReconstruction}
\end{equation}
for every function $u : [0,1]^d \to \mathbb{R}$ with $\mathrm{Lip}_{\ell^2}(u) < \infty$.
\end{lemma}
\begin{proof}See Appendix \ref{app:upper}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:upper_bound}]
This follows immediately by combining Lemmas~\ref{lem:LipschitzContinuityForNN}
and \ref{lem:LipschitzRecovery}.
\end{proof}
\section{Numerical Experiments}
\label{sec:num}
Having established fundamental bounds on the performance of any learning algorithm, we want to numerically evaluate the performance of commonly used deep learning methods.
To illustrate our main result in Theorem~\ref{thm:main}, we estimate the error in~\eqref{eq:mc_err} by a tractable approximation in a student-teacher setting. Specifically, we compute the minimal error over neural network target functions (\enquote{teachers}) $\widehat{U}\subset \mathcal{H}^q_{(d,N_1,\dots,N_{L-1},1),c}$ for deep learning algorithms $\widehat{A} \subset\mathrm{Alg}^{MC}_m(U,L^p)$ via Monte-Carlo sampling, i.e.,
\begin{equation}
\label{eq:min_max}
\widehat{\mathrm{err}}_m\left(\widehat{U},L^p;\widehat{A}\right)
\coloneqq \inf_{(\mathbf{A},\mathbf{m})\in \widehat{A}} \,\,
\sup_{u\in \widehat{U}} \,\,
\mathop{\mathpalette\avsuminner\relax}\displaylimits_{\omega \in \widehat{\Omega}}\left(\mathop{\mathpalette\avsuminner\relax}\displaylimits_{j\in [J]} \big(u(X_j) - A_\omega(u)(X_j)\big)^p\right)^{1/p},
\end{equation}
where $(X_j)_{j=1}^J$ are independent evaluation samples uniformly distributed on\footnote{To have centered input data, we consider the hypercube $[-0.5,0.5]^d$ in our experiments. Note that this does not change any of the theoretical results.} $[-0.5,0.5]^d$ and $\widehat{\Omega}$ represents the seeds for the algorithms.
We obtain teacher networks $u\in \mathcal{H}^\infty_{(d,N_1,\dots,N_{L-1},1),c}$ by sampling their coefficients $\Phi$ componentwise according to a uniform distribution on $[-c,c]$. For every algorithm $(\mathbf{A},\mathbf{m})\in \widehat{A}$ and seed $\omega\in\widehat{\Omega}$ we consider point sequences $\mathbf{x}(u)$ uniformly distributed in $[-0.5,0.5]^d$ with $\mathbf{m}(\omega)=m$. The corresponding point samples are used to train the coefficients of a neural network (\enquote{student}) using the Adam optimizer~\cite{kingma2015adam} with exponentially decaying learning rate. We consider input dimensions $d=1$ and $d=3$, for each of which we compute the error in~\eqref{eq:min_max} for $4$ different sample sizes $m$ over $40$ teacher networks $u$. For each combination, we train student networks with $3$ different seeds, $3$ different widths, and $3$ different batch-sizes. In summary, this yields
$2 \cdot 4 \cdot 40 \cdot 3 \cdot 3 \cdot 3 = 8640$ experiments each executed on a single GPU. The precise hyperparameters can be found in Tables~\ref{table:hp} and~\ref{table:hp_min_max} of Appendix~\ref{app:hyperparameters}.
Figure~\ref{fig:min_max} shows that there is a clear gap between the errors $\widehat{\mathrm{err}}_m(\widehat{U},L^p;\widehat{A})$ for $p \in \{1,2\}$ and $p=\infty$. Especially in the one-dimensional case, the rate w.r.t.\@~the number of samples also seems to stagnate at a precision that might be insufficient for certain applications.
Figure~\ref{fig:teacher_student} illustrates that the errors are caused by spikes of the teacher network which are not covered by any sample.
Note that this is very similar to the construction in the proof of our main result, see Section~\ref{sub:MainResultProof}.
In general, the rates worsen when considering more teacher networks $\widehat{U}$ and improve when considering further deep learning algorithms $\widehat{A}$, including other architectures or more elaborate training and sampling schemes. Note, however, that each setting needs to be evaluated for a number of teacher networks, sample sizes, and seeds. We provide an extensible implementation\footnote{The code can be found at ~\url{https://github.com/juliusberner/theory2practice}.} in PyTorch~\cite{paszke2019pytorch} featuring multi-node experiment execution and hyperparameter tuning using Ray Tune~\cite{liaw2018tune}, experiment tracking using Weights \& Biases and TensorBoard, and flexible experiment configuration. Building upon our work, research teams with sufficient computational resources can provide further numerical evidence on an even larger scale.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\textwidth]{figures/min_max.pdf}
\end{center}
\caption{Evaluation of the error in~\eqref{eq:min_max} for input dimensions $d \in \{1,3\}$, sample sizes ${m\in \{10^2,10^3,10^4, 10^5\}}$, $p\in\{1,2,\infty\}$, and hyperparameters given in Tables~\ref{table:hp} and~\ref{table:hp_min_max}.}
\label{fig:min_max}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=\textwidth]{figures/teacher_student.pdf}
\end{center}
\caption{Target function (\enquote{teacher}), samples, and model of the deep learning algorithm (\enquote{student}) attaining the min-max value for $m=100$ and $p=\infty$ in the experiment depicted in Figure~\ref{fig:min_max}. The middle and right plots are zoomed versions of the left plot. The $L^\infty$ error $(2.7\cdot 10^{-3})$ is about one magnitude larger than the $L^2$ and $L^1$ errors $(3.9\cdot 10^{-4}$ and $2.4\cdot 10^{-4})$ caused by spikes of the teacher network between samples.}
\label{fig:teacher_student}
\end{figure}
\section{Discussion and Limitations}\label{sec:discussion}
\paragraph{Discussion. }
We derived fundamental upper and lower bounds for the number of training samples needed for any algorithm to reconstruct an arbitrary function from a target class consisting of (or containing) realizations of neural networks with ReLU activation function of a given architecture and subject to $\ell^q$ regularization constraints on the network coefficients. These bounds are completely explicit in the network architecture, the type of regularization, and the norm in which the reconstruction error is measured. We observe that our lower bounds are severely more restrictive if the error is measured in the uniform $L^\infty$ norm rather than the (more commonly studied) $L^2$ norm. Particularly, the training of neural networks with ReLU activation function with moderately high accuracy in the $L^\infty$ norm is intractable for moderate input dimensions, as well as network width and depth. We anticipate that our results contribute to a better understanding of possible circumstances under which it is possible to design reliable deep learning algorithms and help explain well-known instability phenomena such as adversarial examples. To the best of our knowledge, we provide the first practically relevant bounds on the number of samples needed in order to completely prevent adversarial examples (on the whole domain). Such an understanding can be beneficial in assessing the potential and limitations of machine learning methods applied to security-critical scenarios and thus have a positive societal impact.
Our contribution increases our understanding of the limitations of a class of popular machine learning methods; thus, we do not anticipate any potential negative societal impacts of our work.
\paragraph{Limitations and Outlook. }
We finally discuss some possible implications and also limitations of our work.
First of all, our results are highly specific to neural networks with the ReLU activation function. We expect that obtaining similar results for other activation functions will require substantially new methods. We plan to investigate this in future work.
Our analysis is a worst-case analysis in the sense that we show that for any given algorithm $A$, there exists at least one $u$ in our target class $U$ on which $A$ performs poorly. The question of whether this poor behavior is actually generic will be studied in future work. One way to establish such generic results could be to prove that our considered target classes contain copies of neural network realizations attached to many different $u$'s.
Finally, we consider target classes $U$ that contain realizations of neural networks subject to different regularizations. This is justified because whenever a deep learning method is employed to reconstruct a function by representing it approximately by a neural network of a given architecture (and without further knowledge about $u$), a natural minimal requirement is that the method should perform well if the sought function is in fact equal to a neural network. In future work we plan to extend our results to other architectures, such as convolutional neural networks, transformer networks, and graph neural networks.
\begin{ack}
The research of Julius Berner was supported by the Austrian Science Fund (FWF) under grant I3403-N32. The computational results presented have been achieved in part using the Vienna Scientific Cluster (VSC).
Felix Voigtlaender acknowledges support by the DFG in the context of the Emmy Noether junior research group VO 2594/1-1.
\end{ack}
\bibliographystyle{plain}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 457
|
Q: Calling an overloaded C# method from F# There are some similar questions on SO but I can't seem to find quite what I'm looking for.
There is a C# library (OpenCVSharp) declaring an overloaded method like this:
public static void CalcHist(Mat[] images,
int[] channels, InputArray mask,
OutputArray hist, int dims, int[] histSize,
Rangef[] ranges, bool uniform = true, bool accumulate = false)
{
....
}
public static void CalcHist(Mat[] images,
int[] channels, InputArray mask,
OutputArray hist, int dims, int[] histSize,
float[][] ranges, bool uniform = true, bool accumulate = false)
{
....
}
i.e. varying only by the type of the "ranges" parameter.
I can't seem to call this method, even when using tupled argument style, including the optional parameters and adding a whole bunch of type annotations:
let images = [|new Mat()|]
let hist = OutputArray.Create(new Mat());
let hdims = [|256|];
let ranges = [| new Rangef(0.f,256.f) |];
Cv2.CalcHist<Mat [] * int [] * InputArray * OutputArray * int * int [] * Rangef [] * bool * bool>
(images,
[|0|],
null,
hist,
1,
hdims,
ranges,
true,
false)
The error is "Error 4 The member or object constructor 'CalcHist' taking 9 arguments are not accessible from this code location. All accessible versions of method 'CalcHist' take 9 arguments"
Is there some way I can call this method from F#?
A: The syntax you have chosen to specify the types of arguments does not actually work for specifying types of arguments. That syntax is for specifying generic arguments of generic functions or types:
let imGeneric<'t, 'u> (x: 't, y: 'u) = ...
let callGeneric = imGeneric<int, string> (5, "abc")
But of course, most of the time, F# compiler can infer generic arguments for you, so you don't really need to explicitly specify them that often:
let imGeneric (x, y) = ...
let callGeneric = imGeneric (5, "abc")
Your Cv2.CalcHist method is not generic. Remove generic arguments, and it should work just fine:
Cv2.CalcHist( images, [|0|], null, hist, 1, hdims, ranges, true, false )
The above works, because the compiler already knows that ranges has type Rangef [], so it's able to choose the correct overload without you doing any extra work.
But sometimes it is really necessary to explicitly specify types of some (or all) arguments. In these cases, you can specify types right in place:
Cv2.CalcHist(
images, [|0|], null, hist, 1, hdims,
(ranges : Rangef []),
true, false )
Note the extra parentheses around (ranges : Rangef []). Without them, the type annotation will apply not just to ranges, but to the whole tuple of images, [|0|], null, hist, 1, hdims, ranges, which will cause an compile-time error, because such tuple clearly cannot have type Rangef [].
Alternatively, you can fix the type of your values at any point before the call:
let ranges: Rangef [] = getRanges()
Cv2.CalcHist( images, [|0|], null, hist, 1, hdims, ranges, true, false )
This will have the same effect. The compiler really only needs to know the type, it doesn't care much where exactly it's defined, as long as it's before the point where it's needed.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,522
|
import random
import re
import string
import subprocess
import unittest
from textwrap import dedent
from backsplash import Layout, Tile
def format_labels_py(width, tiles):
m = Layout(width)
for t in tiles:
m.add(t)
lines = []
lines.append('order: ' + ''.join(t.label for t in m.linearize()))
lines.append('')
for r in range(len(m._holes)):
labels = []
for c in range(m._width):
t = m._tiles.get((r, c))
labels.append(t.label if t else ' ')
lines.append(''.join(labels))
return '\n'.join(lines)
def format_labels_js(width, tiles):
cmd = ['node', 'test_backsplash.js', width]
for tile in tiles:
cmd.extend((tile.label, tile.row or 0, tile.col or 0, tile.height, tile.width))
cmd = map(str, cmd)
out = subprocess.check_output(cmd)
out = re.sub(r'\n\s+$', '', out)
return out
class TestCase(unittest.TestCase):
def assertEqualLayout(self, width, tiles, manual=None):
py = format_labels_py(width, [t.copy() for t in tiles])
js = format_labels_js(width, [t.copy() for t in tiles])
print py
print
if manual:
self.assertEqualLines(py, manual)
self.assertEqualLines(py, js)
def assertEqualLines(self, a, b, *args):
a = dedent(a).strip().splitlines()
a = [x.rstrip() for x in a]
b = dedent(b).strip().splitlines()
b = [x.rstrip() for x in b]
self.assertEqual(a, b, *args)
class TestLayout(TestCase):
def test_basic_layouts(self):
self.assertEqualLayout(5, [
Tile(2, 2, label='A'),
Tile(2, 2, label='B'),
Tile(2, 2, label='C'),
Tile(1, 1, label='D'),
], '''
order: ABDC
AABBD
AABB
CC
CC
''')
def test_leaving_gaps(self):
self.assertEqualLayout(5, [
Tile(2, 2, label='A'),
Tile(1, 4, label='B'),
Tile(3, 1, label='C'),
Tile(1, 1, label='D'),
], '''
order: ADCB
AAD C
AA C
BBBBC
''')
def test_regression_1(self):
# This was actually an issue with the Layout constructor changing, but
# it is always helpful to have more tests.
self.assertEqualLayout(6, [
Tile(2, 2, label='0'),
Tile(1, 2, label='1'),
Tile(1, 1, label='2'),
Tile(1, 1, label='3'),
Tile(1, 2, label='4'),
Tile(2, 1, label='5'),
Tile(1, 2, label='6'),
Tile(2, 1, label='7'),
Tile(1, 1, label='8'),
Tile(2, 2, label='9'),
], '''
order: 0123457689
001123
004457
668 57
99
99
''')
def test_random_layouts(self):
for width in range(5, 11):
for max_size in range(2, width + 1):
tiles = []
for _ in range(random.randrange(width, width * 2)):
tiles.append(Tile(
random.randrange(1, max_size + 1),
random.randrange(1, max_size + 1),
label=string.printable[len(tiles)]
))
self.assertEqualLayout(width, tiles)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,273
|
/*
* Do not modify this file. This file is generated from the elasticmapreduce-2009-03-31.normal.json service model.
*/
using System;
using Amazon.Runtime;
namespace Amazon.ElasticMapReduce
{
/// <summary>
/// Constants used for properties of type ActionOnFailure.
/// </summary>
public class ActionOnFailure : ConstantClass
{
/// <summary>
/// Constant CANCEL_AND_WAIT for ActionOnFailure
/// </summary>
public static readonly ActionOnFailure CANCEL_AND_WAIT = new ActionOnFailure("CANCEL_AND_WAIT");
/// <summary>
/// Constant CONTINUE for ActionOnFailure
/// </summary>
public static readonly ActionOnFailure CONTINUE = new ActionOnFailure("CONTINUE");
/// <summary>
/// Constant TERMINATE_CLUSTER for ActionOnFailure
/// </summary>
public static readonly ActionOnFailure TERMINATE_CLUSTER = new ActionOnFailure("TERMINATE_CLUSTER");
/// <summary>
/// Constant TERMINATE_JOB_FLOW for ActionOnFailure
/// </summary>
public static readonly ActionOnFailure TERMINATE_JOB_FLOW = new ActionOnFailure("TERMINATE_JOB_FLOW");
/// <summary>
/// Default Constructor
/// </summary>
public ActionOnFailure(string value)
: base(value)
{
}
/// <summary>
/// Finds the constant for the unique value.
/// </summary>
/// <param name="value">The unique value for the constant</param>
/// <returns>The constant for the unique value</returns>
public static ActionOnFailure FindValue(string value)
{
return FindValue<ActionOnFailure>(value);
}
/// <summary>
/// Utility method to convert strings to the constant class.
/// </summary>
/// <param name="value">The string value to convert to the constant class.</param>
/// <returns></returns>
public static implicit operator ActionOnFailure(string value)
{
return FindValue(value);
}
}
/// <summary>
/// Constants used for properties of type ClusterState.
/// </summary>
public class ClusterState : ConstantClass
{
/// <summary>
/// Constant BOOTSTRAPPING for ClusterState
/// </summary>
public static readonly ClusterState BOOTSTRAPPING = new ClusterState("BOOTSTRAPPING");
/// <summary>
/// Constant RUNNING for ClusterState
/// </summary>
public static readonly ClusterState RUNNING = new ClusterState("RUNNING");
/// <summary>
/// Constant STARTING for ClusterState
/// </summary>
public static readonly ClusterState STARTING = new ClusterState("STARTING");
/// <summary>
/// Constant TERMINATED for ClusterState
/// </summary>
public static readonly ClusterState TERMINATED = new ClusterState("TERMINATED");
/// <summary>
/// Constant TERMINATED_WITH_ERRORS for ClusterState
/// </summary>
public static readonly ClusterState TERMINATED_WITH_ERRORS = new ClusterState("TERMINATED_WITH_ERRORS");
/// <summary>
/// Constant TERMINATING for ClusterState
/// </summary>
public static readonly ClusterState TERMINATING = new ClusterState("TERMINATING");
/// <summary>
/// Constant WAITING for ClusterState
/// </summary>
public static readonly ClusterState WAITING = new ClusterState("WAITING");
/// <summary>
/// Default Constructor
/// </summary>
public ClusterState(string value)
: base(value)
{
}
/// <summary>
/// Finds the constant for the unique value.
/// </summary>
/// <param name="value">The unique value for the constant</param>
/// <returns>The constant for the unique value</returns>
public static ClusterState FindValue(string value)
{
return FindValue<ClusterState>(value);
}
/// <summary>
/// Utility method to convert strings to the constant class.
/// </summary>
/// <param name="value">The string value to convert to the constant class.</param>
/// <returns></returns>
public static implicit operator ClusterState(string value)
{
return FindValue(value);
}
}
/// <summary>
/// Constants used for properties of type ClusterStateChangeReasonCode.
/// </summary>
public class ClusterStateChangeReasonCode : ConstantClass
{
/// <summary>
/// Constant ALL_STEPS_COMPLETED for ClusterStateChangeReasonCode
/// </summary>
public static readonly ClusterStateChangeReasonCode ALL_STEPS_COMPLETED = new ClusterStateChangeReasonCode("ALL_STEPS_COMPLETED");
/// <summary>
/// Constant BOOTSTRAP_FAILURE for ClusterStateChangeReasonCode
/// </summary>
public static readonly ClusterStateChangeReasonCode BOOTSTRAP_FAILURE = new ClusterStateChangeReasonCode("BOOTSTRAP_FAILURE");
/// <summary>
/// Constant INSTANCE_FAILURE for ClusterStateChangeReasonCode
/// </summary>
public static readonly ClusterStateChangeReasonCode INSTANCE_FAILURE = new ClusterStateChangeReasonCode("INSTANCE_FAILURE");
/// <summary>
/// Constant INTERNAL_ERROR for ClusterStateChangeReasonCode
/// </summary>
public static readonly ClusterStateChangeReasonCode INTERNAL_ERROR = new ClusterStateChangeReasonCode("INTERNAL_ERROR");
/// <summary>
/// Constant STEP_FAILURE for ClusterStateChangeReasonCode
/// </summary>
public static readonly ClusterStateChangeReasonCode STEP_FAILURE = new ClusterStateChangeReasonCode("STEP_FAILURE");
/// <summary>
/// Constant USER_REQUEST for ClusterStateChangeReasonCode
/// </summary>
public static readonly ClusterStateChangeReasonCode USER_REQUEST = new ClusterStateChangeReasonCode("USER_REQUEST");
/// <summary>
/// Constant VALIDATION_ERROR for ClusterStateChangeReasonCode
/// </summary>
public static readonly ClusterStateChangeReasonCode VALIDATION_ERROR = new ClusterStateChangeReasonCode("VALIDATION_ERROR");
/// <summary>
/// Default Constructor
/// </summary>
public ClusterStateChangeReasonCode(string value)
: base(value)
{
}
/// <summary>
/// Finds the constant for the unique value.
/// </summary>
/// <param name="value">The unique value for the constant</param>
/// <returns>The constant for the unique value</returns>
public static ClusterStateChangeReasonCode FindValue(string value)
{
return FindValue<ClusterStateChangeReasonCode>(value);
}
/// <summary>
/// Utility method to convert strings to the constant class.
/// </summary>
/// <param name="value">The string value to convert to the constant class.</param>
/// <returns></returns>
public static implicit operator ClusterStateChangeReasonCode(string value)
{
return FindValue(value);
}
}
/// <summary>
/// Constants used for properties of type InstanceGroupState.
/// </summary>
public class InstanceGroupState : ConstantClass
{
/// <summary>
/// Constant ARRESTED for InstanceGroupState
/// </summary>
public static readonly InstanceGroupState ARRESTED = new InstanceGroupState("ARRESTED");
/// <summary>
/// Constant BOOTSTRAPPING for InstanceGroupState
/// </summary>
public static readonly InstanceGroupState BOOTSTRAPPING = new InstanceGroupState("BOOTSTRAPPING");
/// <summary>
/// Constant ENDED for InstanceGroupState
/// </summary>
public static readonly InstanceGroupState ENDED = new InstanceGroupState("ENDED");
/// <summary>
/// Constant PROVISIONING for InstanceGroupState
/// </summary>
public static readonly InstanceGroupState PROVISIONING = new InstanceGroupState("PROVISIONING");
/// <summary>
/// Constant RESIZING for InstanceGroupState
/// </summary>
public static readonly InstanceGroupState RESIZING = new InstanceGroupState("RESIZING");
/// <summary>
/// Constant RUNNING for InstanceGroupState
/// </summary>
public static readonly InstanceGroupState RUNNING = new InstanceGroupState("RUNNING");
/// <summary>
/// Constant SHUTTING_DOWN for InstanceGroupState
/// </summary>
public static readonly InstanceGroupState SHUTTING_DOWN = new InstanceGroupState("SHUTTING_DOWN");
/// <summary>
/// Constant SUSPENDED for InstanceGroupState
/// </summary>
public static readonly InstanceGroupState SUSPENDED = new InstanceGroupState("SUSPENDED");
/// <summary>
/// Constant TERMINATED for InstanceGroupState
/// </summary>
public static readonly InstanceGroupState TERMINATED = new InstanceGroupState("TERMINATED");
/// <summary>
/// Constant TERMINATING for InstanceGroupState
/// </summary>
public static readonly InstanceGroupState TERMINATING = new InstanceGroupState("TERMINATING");
/// <summary>
/// Default Constructor
/// </summary>
public InstanceGroupState(string value)
: base(value)
{
}
/// <summary>
/// Finds the constant for the unique value.
/// </summary>
/// <param name="value">The unique value for the constant</param>
/// <returns>The constant for the unique value</returns>
public static InstanceGroupState FindValue(string value)
{
return FindValue<InstanceGroupState>(value);
}
/// <summary>
/// Utility method to convert strings to the constant class.
/// </summary>
/// <param name="value">The string value to convert to the constant class.</param>
/// <returns></returns>
public static implicit operator InstanceGroupState(string value)
{
return FindValue(value);
}
}
/// <summary>
/// Constants used for properties of type InstanceGroupStateChangeReasonCode.
/// </summary>
public class InstanceGroupStateChangeReasonCode : ConstantClass
{
/// <summary>
/// Constant CLUSTER_TERMINATED for InstanceGroupStateChangeReasonCode
/// </summary>
public static readonly InstanceGroupStateChangeReasonCode CLUSTER_TERMINATED = new InstanceGroupStateChangeReasonCode("CLUSTER_TERMINATED");
/// <summary>
/// Constant INSTANCE_FAILURE for InstanceGroupStateChangeReasonCode
/// </summary>
public static readonly InstanceGroupStateChangeReasonCode INSTANCE_FAILURE = new InstanceGroupStateChangeReasonCode("INSTANCE_FAILURE");
/// <summary>
/// Constant INTERNAL_ERROR for InstanceGroupStateChangeReasonCode
/// </summary>
public static readonly InstanceGroupStateChangeReasonCode INTERNAL_ERROR = new InstanceGroupStateChangeReasonCode("INTERNAL_ERROR");
/// <summary>
/// Constant VALIDATION_ERROR for InstanceGroupStateChangeReasonCode
/// </summary>
public static readonly InstanceGroupStateChangeReasonCode VALIDATION_ERROR = new InstanceGroupStateChangeReasonCode("VALIDATION_ERROR");
/// <summary>
/// Default Constructor
/// </summary>
public InstanceGroupStateChangeReasonCode(string value)
: base(value)
{
}
/// <summary>
/// Finds the constant for the unique value.
/// </summary>
/// <param name="value">The unique value for the constant</param>
/// <returns>The constant for the unique value</returns>
public static InstanceGroupStateChangeReasonCode FindValue(string value)
{
return FindValue<InstanceGroupStateChangeReasonCode>(value);
}
/// <summary>
/// Utility method to convert strings to the constant class.
/// </summary>
/// <param name="value">The string value to convert to the constant class.</param>
/// <returns></returns>
public static implicit operator InstanceGroupStateChangeReasonCode(string value)
{
return FindValue(value);
}
}
/// <summary>
/// Constants used for properties of type InstanceGroupType.
/// </summary>
public class InstanceGroupType : ConstantClass
{
/// <summary>
/// Constant CORE for InstanceGroupType
/// </summary>
public static readonly InstanceGroupType CORE = new InstanceGroupType("CORE");
/// <summary>
/// Constant MASTER for InstanceGroupType
/// </summary>
public static readonly InstanceGroupType MASTER = new InstanceGroupType("MASTER");
/// <summary>
/// Constant TASK for InstanceGroupType
/// </summary>
public static readonly InstanceGroupType TASK = new InstanceGroupType("TASK");
/// <summary>
/// Default Constructor
/// </summary>
public InstanceGroupType(string value)
: base(value)
{
}
/// <summary>
/// Finds the constant for the unique value.
/// </summary>
/// <param name="value">The unique value for the constant</param>
/// <returns>The constant for the unique value</returns>
public static InstanceGroupType FindValue(string value)
{
return FindValue<InstanceGroupType>(value);
}
/// <summary>
/// Utility method to convert strings to the constant class.
/// </summary>
/// <param name="value">The string value to convert to the constant class.</param>
/// <returns></returns>
public static implicit operator InstanceGroupType(string value)
{
return FindValue(value);
}
}
/// <summary>
/// Constants used for properties of type InstanceRoleType.
/// </summary>
public class InstanceRoleType : ConstantClass
{
/// <summary>
/// Constant CORE for InstanceRoleType
/// </summary>
public static readonly InstanceRoleType CORE = new InstanceRoleType("CORE");
/// <summary>
/// Constant MASTER for InstanceRoleType
/// </summary>
public static readonly InstanceRoleType MASTER = new InstanceRoleType("MASTER");
/// <summary>
/// Constant TASK for InstanceRoleType
/// </summary>
public static readonly InstanceRoleType TASK = new InstanceRoleType("TASK");
/// <summary>
/// Default Constructor
/// </summary>
public InstanceRoleType(string value)
: base(value)
{
}
/// <summary>
/// Finds the constant for the unique value.
/// </summary>
/// <param name="value">The unique value for the constant</param>
/// <returns>The constant for the unique value</returns>
public static InstanceRoleType FindValue(string value)
{
return FindValue<InstanceRoleType>(value);
}
/// <summary>
/// Utility method to convert strings to the constant class.
/// </summary>
/// <param name="value">The string value to convert to the constant class.</param>
/// <returns></returns>
public static implicit operator InstanceRoleType(string value)
{
return FindValue(value);
}
}
/// <summary>
/// Constants used for properties of type InstanceState.
/// </summary>
public class InstanceState : ConstantClass
{
/// <summary>
/// Constant AWAITING_FULFILLMENT for InstanceState
/// </summary>
public static readonly InstanceState AWAITING_FULFILLMENT = new InstanceState("AWAITING_FULFILLMENT");
/// <summary>
/// Constant BOOTSTRAPPING for InstanceState
/// </summary>
public static readonly InstanceState BOOTSTRAPPING = new InstanceState("BOOTSTRAPPING");
/// <summary>
/// Constant PROVISIONING for InstanceState
/// </summary>
public static readonly InstanceState PROVISIONING = new InstanceState("PROVISIONING");
/// <summary>
/// Constant RUNNING for InstanceState
/// </summary>
public static readonly InstanceState RUNNING = new InstanceState("RUNNING");
/// <summary>
/// Constant TERMINATED for InstanceState
/// </summary>
public static readonly InstanceState TERMINATED = new InstanceState("TERMINATED");
/// <summary>
/// Default Constructor
/// </summary>
public InstanceState(string value)
: base(value)
{
}
/// <summary>
/// Finds the constant for the unique value.
/// </summary>
/// <param name="value">The unique value for the constant</param>
/// <returns>The constant for the unique value</returns>
public static InstanceState FindValue(string value)
{
return FindValue<InstanceState>(value);
}
/// <summary>
/// Utility method to convert strings to the constant class.
/// </summary>
/// <param name="value">The string value to convert to the constant class.</param>
/// <returns></returns>
public static implicit operator InstanceState(string value)
{
return FindValue(value);
}
}
/// <summary>
/// Constants used for properties of type InstanceStateChangeReasonCode.
/// </summary>
public class InstanceStateChangeReasonCode : ConstantClass
{
/// <summary>
/// Constant BOOTSTRAP_FAILURE for InstanceStateChangeReasonCode
/// </summary>
public static readonly InstanceStateChangeReasonCode BOOTSTRAP_FAILURE = new InstanceStateChangeReasonCode("BOOTSTRAP_FAILURE");
/// <summary>
/// Constant CLUSTER_TERMINATED for InstanceStateChangeReasonCode
/// </summary>
public static readonly InstanceStateChangeReasonCode CLUSTER_TERMINATED = new InstanceStateChangeReasonCode("CLUSTER_TERMINATED");
/// <summary>
/// Constant INSTANCE_FAILURE for InstanceStateChangeReasonCode
/// </summary>
public static readonly InstanceStateChangeReasonCode INSTANCE_FAILURE = new InstanceStateChangeReasonCode("INSTANCE_FAILURE");
/// <summary>
/// Constant INTERNAL_ERROR for InstanceStateChangeReasonCode
/// </summary>
public static readonly InstanceStateChangeReasonCode INTERNAL_ERROR = new InstanceStateChangeReasonCode("INTERNAL_ERROR");
/// <summary>
/// Constant VALIDATION_ERROR for InstanceStateChangeReasonCode
/// </summary>
public static readonly InstanceStateChangeReasonCode VALIDATION_ERROR = new InstanceStateChangeReasonCode("VALIDATION_ERROR");
/// <summary>
/// Default Constructor
/// </summary>
public InstanceStateChangeReasonCode(string value)
: base(value)
{
}
/// <summary>
/// Finds the constant for the unique value.
/// </summary>
/// <param name="value">The unique value for the constant</param>
/// <returns>The constant for the unique value</returns>
public static InstanceStateChangeReasonCode FindValue(string value)
{
return FindValue<InstanceStateChangeReasonCode>(value);
}
/// <summary>
/// Utility method to convert strings to the constant class.
/// </summary>
/// <param name="value">The string value to convert to the constant class.</param>
/// <returns></returns>
public static implicit operator InstanceStateChangeReasonCode(string value)
{
return FindValue(value);
}
}
/// <summary>
/// Constants used for properties of type JobFlowExecutionState.
/// </summary>
public class JobFlowExecutionState : ConstantClass
{
/// <summary>
/// Constant BOOTSTRAPPING for JobFlowExecutionState
/// </summary>
public static readonly JobFlowExecutionState BOOTSTRAPPING = new JobFlowExecutionState("BOOTSTRAPPING");
/// <summary>
/// Constant COMPLETED for JobFlowExecutionState
/// </summary>
public static readonly JobFlowExecutionState COMPLETED = new JobFlowExecutionState("COMPLETED");
/// <summary>
/// Constant FAILED for JobFlowExecutionState
/// </summary>
public static readonly JobFlowExecutionState FAILED = new JobFlowExecutionState("FAILED");
/// <summary>
/// Constant RUNNING for JobFlowExecutionState
/// </summary>
public static readonly JobFlowExecutionState RUNNING = new JobFlowExecutionState("RUNNING");
/// <summary>
/// Constant SHUTTING_DOWN for JobFlowExecutionState
/// </summary>
public static readonly JobFlowExecutionState SHUTTING_DOWN = new JobFlowExecutionState("SHUTTING_DOWN");
/// <summary>
/// Constant STARTING for JobFlowExecutionState
/// </summary>
public static readonly JobFlowExecutionState STARTING = new JobFlowExecutionState("STARTING");
/// <summary>
/// Constant TERMINATED for JobFlowExecutionState
/// </summary>
public static readonly JobFlowExecutionState TERMINATED = new JobFlowExecutionState("TERMINATED");
/// <summary>
/// Constant WAITING for JobFlowExecutionState
/// </summary>
public static readonly JobFlowExecutionState WAITING = new JobFlowExecutionState("WAITING");
/// <summary>
/// Default Constructor
/// </summary>
public JobFlowExecutionState(string value)
: base(value)
{
}
/// <summary>
/// Finds the constant for the unique value.
/// </summary>
/// <param name="value">The unique value for the constant</param>
/// <returns>The constant for the unique value</returns>
public static JobFlowExecutionState FindValue(string value)
{
return FindValue<JobFlowExecutionState>(value);
}
/// <summary>
/// Utility method to convert strings to the constant class.
/// </summary>
/// <param name="value">The string value to convert to the constant class.</param>
/// <returns></returns>
public static implicit operator JobFlowExecutionState(string value)
{
return FindValue(value);
}
}
/// <summary>
/// Constants used for properties of type MarketType.
/// </summary>
public class MarketType : ConstantClass
{
/// <summary>
/// Constant ON_DEMAND for MarketType
/// </summary>
public static readonly MarketType ON_DEMAND = new MarketType("ON_DEMAND");
/// <summary>
/// Constant SPOT for MarketType
/// </summary>
public static readonly MarketType SPOT = new MarketType("SPOT");
/// <summary>
/// Default Constructor
/// </summary>
public MarketType(string value)
: base(value)
{
}
/// <summary>
/// Finds the constant for the unique value.
/// </summary>
/// <param name="value">The unique value for the constant</param>
/// <returns>The constant for the unique value</returns>
public static MarketType FindValue(string value)
{
return FindValue<MarketType>(value);
}
/// <summary>
/// Utility method to convert strings to the constant class.
/// </summary>
/// <param name="value">The string value to convert to the constant class.</param>
/// <returns></returns>
public static implicit operator MarketType(string value)
{
return FindValue(value);
}
}
/// <summary>
/// Constants used for properties of type StepExecutionState.
/// </summary>
public class StepExecutionState : ConstantClass
{
/// <summary>
/// Constant CANCELLED for StepExecutionState
/// </summary>
public static readonly StepExecutionState CANCELLED = new StepExecutionState("CANCELLED");
/// <summary>
/// Constant COMPLETED for StepExecutionState
/// </summary>
public static readonly StepExecutionState COMPLETED = new StepExecutionState("COMPLETED");
/// <summary>
/// Constant CONTINUE for StepExecutionState
/// </summary>
public static readonly StepExecutionState CONTINUE = new StepExecutionState("CONTINUE");
/// <summary>
/// Constant FAILED for StepExecutionState
/// </summary>
public static readonly StepExecutionState FAILED = new StepExecutionState("FAILED");
/// <summary>
/// Constant INTERRUPTED for StepExecutionState
/// </summary>
public static readonly StepExecutionState INTERRUPTED = new StepExecutionState("INTERRUPTED");
/// <summary>
/// Constant PENDING for StepExecutionState
/// </summary>
public static readonly StepExecutionState PENDING = new StepExecutionState("PENDING");
/// <summary>
/// Constant RUNNING for StepExecutionState
/// </summary>
public static readonly StepExecutionState RUNNING = new StepExecutionState("RUNNING");
/// <summary>
/// Default Constructor
/// </summary>
public StepExecutionState(string value)
: base(value)
{
}
/// <summary>
/// Finds the constant for the unique value.
/// </summary>
/// <param name="value">The unique value for the constant</param>
/// <returns>The constant for the unique value</returns>
public static StepExecutionState FindValue(string value)
{
return FindValue<StepExecutionState>(value);
}
/// <summary>
/// Utility method to convert strings to the constant class.
/// </summary>
/// <param name="value">The string value to convert to the constant class.</param>
/// <returns></returns>
public static implicit operator StepExecutionState(string value)
{
return FindValue(value);
}
}
/// <summary>
/// Constants used for properties of type StepState.
/// </summary>
public class StepState : ConstantClass
{
/// <summary>
/// Constant CANCELLED for StepState
/// </summary>
public static readonly StepState CANCELLED = new StepState("CANCELLED");
/// <summary>
/// Constant COMPLETED for StepState
/// </summary>
public static readonly StepState COMPLETED = new StepState("COMPLETED");
/// <summary>
/// Constant FAILED for StepState
/// </summary>
public static readonly StepState FAILED = new StepState("FAILED");
/// <summary>
/// Constant INTERRUPTED for StepState
/// </summary>
public static readonly StepState INTERRUPTED = new StepState("INTERRUPTED");
/// <summary>
/// Constant PENDING for StepState
/// </summary>
public static readonly StepState PENDING = new StepState("PENDING");
/// <summary>
/// Constant RUNNING for StepState
/// </summary>
public static readonly StepState RUNNING = new StepState("RUNNING");
/// <summary>
/// Default Constructor
/// </summary>
public StepState(string value)
: base(value)
{
}
/// <summary>
/// Finds the constant for the unique value.
/// </summary>
/// <param name="value">The unique value for the constant</param>
/// <returns>The constant for the unique value</returns>
public static StepState FindValue(string value)
{
return FindValue<StepState>(value);
}
/// <summary>
/// Utility method to convert strings to the constant class.
/// </summary>
/// <param name="value">The string value to convert to the constant class.</param>
/// <returns></returns>
public static implicit operator StepState(string value)
{
return FindValue(value);
}
}
/// <summary>
/// Constants used for properties of type StepStateChangeReasonCode.
/// </summary>
public class StepStateChangeReasonCode : ConstantClass
{
/// <summary>
/// Constant NONE for StepStateChangeReasonCode
/// </summary>
public static readonly StepStateChangeReasonCode NONE = new StepStateChangeReasonCode("NONE");
/// <summary>
/// Default Constructor
/// </summary>
public StepStateChangeReasonCode(string value)
: base(value)
{
}
/// <summary>
/// Finds the constant for the unique value.
/// </summary>
/// <param name="value">The unique value for the constant</param>
/// <returns>The constant for the unique value</returns>
public static StepStateChangeReasonCode FindValue(string value)
{
return FindValue<StepStateChangeReasonCode>(value);
}
/// <summary>
/// Utility method to convert strings to the constant class.
/// </summary>
/// <param name="value">The string value to convert to the constant class.</param>
/// <returns></returns>
public static implicit operator StepStateChangeReasonCode(string value)
{
return FindValue(value);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,343
|
**LENNART BERNTSON**
**SVANTE NORDIN**
**Efter
revolutionen**
**Vänstern i svensk
kulturdebatt sedan 1968**
**Natur & Kultur**
info@nok.se
www.nok.se
© Lennart Berntson & Svante Nordin 2017
Natur & Kultur, Stockholm
Omslag, layout och grafisk form: Lars Paulsrud
E-boksproduktion: Axiell Media 2017
ISBN 978-91-27-15227-4
Förlaget Natur & Kultur är en stiftelse som utan ägare kan agera självständigt och långsiktigt. Vårt mål är att genom stöd, inspiration, utbildning och bildning verka för tolerans, humanism och demokrati.
## **FÖRORD**
DENNA BOK ÄR resultatet av ett forskningsprojekt om idéarvet efter vänstervågen 1968. Projektet inleddes 2013 med utgivningen av boken _Arvet efter 1968. Studier kring ett problemområde_ (red. Lennart Berntson och Svante Nordin), där forsknings- och debattläge kring temat diskuterades mera ingående och där de olika delprojekten beskrevs. Projektgruppen har därefter träffats i Engelsberg två gånger om året för diskussion av bidrag och idéer. Sommaren 2016 publicerades den första boken i projektet, nämligen Inger Enkvists _De svenska skolreformerna 1962_ – _1985 och personerna bakom dem_ (Gidlunds förlag). Vintern 2017 kom Johan Sundeens _68-kyrkan. Svensk kristen vänsters möte med marxismen 1965–1989_ (Bladh by Bladh) och David Anderssons _Med skuldkänslan som drivkraft: Om svenska Israelvänner och västfiender_ (Timbro). Vidare planeras utgivning om 68-rörelsen och Miljöpartiet (Charlotta Seiler Brylla), om 68-aktivismen i den svenska utrikespolitiken (Gunnar Åselius), om maoismen i Sverige (Ingrid Wållgren) och om vänsterns kultursyn (Sara Cederberg).
_Efter revolutionen_ är resultatet av ett gemensamt arbete, där Svante Nordin har skrivit den del som omfattar perioden 1968–1984 samt dessutom kapitlet »1968 och filosofin i Sverige» och »68-vänsterns eftermäle». Lennart Berntson har skrivit den del som täcker åren 1987–2000, samt kapitlet »1968 i historiens ljus». Förord och inledning har skrivits gemensamt. Att de båda författarna främst genom sitt engagemang i tidskriften Zenit en gång varit del av den 68-vänster de nedan beskriver framgår på och mellan raderna i det som följer.
Vi tackar Axel och Margaret Ax:son Johnsons stiftelse för allmännyttiga ändamål för dess generösa stöd till detta projekt.
Brantevik/Lund i januari 2017
Lennart Berntson och Svante Nordin
## **INLEDNING**
DENNA BOK har sitt upphov i en undran över den svenska vänsterns utveckling efter revolutionen 1968. Redan vid 2000-talets inledning var det tydligt att vänstern börjat omfatta idéer som stod i bjärt kontrast till det klassiska socialistiska arvet – det arv den anknutit till med upproret 1968. Upplysningens tro på förnuft och framsteg, som historiskt sett främst angripits av högern, utsattes allt oftare för vänsterns kritik. Under inflytande av ekologiska idéer framställdes nu ekonomisk tillväxt, teknisk utveckling och avancerad vetenskap som ett hot mot mänsklig välfärd. Visserligen pläderade vänstern som tidigare för social jämlikhet och jämställdhet mellan könen, feminismen hade till och med blivit en del av Vänsterpartiets identitet under 1990-talet, men samtidigt hade de liberala idéerna om mänskliga rättigheter fått en alltmer framträdande roll i vänsterns ideologi. Intrycket av en djupgående förändring stärktes av att kraven på multikulturalism och identitetspolitik i hög grad övertagit den politiska roll som klasskamp och planhushållning tidigare spelat.
Vad hade egentligen hänt med idéerna som symbolåret 1968 representerade? Å ena sidan hade de revolutionära strömningar och grupper som formligen vällt fram våren 1968 försvunnit från den politiska scenen redan efter något årtionde. Å andra sidan hade de radikala idéerna, om än i förändrad form, levt vidare inom en rad områden som kultur och media, etik och estetik, vid universiteten och inom den statliga byråkratin. Hur hade denna transformation ägt rum och hur hade sammanbrottet för socialismen i öst 1989 påverkat idéklimatet i Sverige?
Såväl i Sverige som internationellt finns en myckenhet av forskning kring student- och ungdomsrevolten 1968 och vad som hängde samman med den. Det finns emellertid inte på samma sätt någon omfattande forskning kring de långsiktiga verkningarna av denna revolt eller av vänsterns idéutveckling efter realsocialismens fall 1989. Visserligen var händelserna 1968 inte i linje med omvälvningarna 1789 eller 1848 eller 1917 – realpolitiskt hade ju vänstergrupperna i väst varit tämligen betydelselösa, men efterverkningarna var ändå tydligt märkbara. Paris skakades förvisso i maj 1968, men inte heller Paris gick under. I Tyskland och Italien blev förvisso terrorismen ett betydelsefullt följdfenomen, men till och med den var av övergående natur. Påverkade den amerikanska studentrevolten utgången av Vietnamkriget? Förmodligen inte mer än marginellt.
I Sverige är efterverkningarna långt ifrån kartlagda. Ändå har det gjorts en hel del forskning som tangerar vårt ämne – arvet från 1968 – även om detta sällan har definierats som den centrala frågeställningen. Till studier som i varierande grad tar upp 68-vänsterns idéutveckling hör Johan Svedjedal _Ner med allt? Essäer om protestlitteraturen och demokratin, cirka 1965–1975_ (2014), i vilken han diskuterar hur den nya vänsterns protestlitteratur påverkade svensk litteratur. Lundahistorikern Johan Stenfeldt analyserar i sin avhandling _Dystopiernas seger. Totalitarism som orienteringspunkt i efterkrigstidens svenska idédebatt_ (2013) tankeutbytet mellan främst liberaler och vänsterradikaler i synen på 1900-talets totalitära erfarenheter. Martin Wiklund tar i sin _Historia som domstol. Historisk värdering och retorisk argumentation kring »68»_ (2012) upp frågan hur 68-företeelsen framträdde inför historiens domstol. Jens Ljunggren utforskar i _Inget land för intellektuella. 68-revolten och svenska vänsterintellektuella_ (2009) 68-rörelsen som en del av de svenska intellektuellas historia. Marie Cronqvist, Lina Sturfelt & Mats Wiklund (red.) samlar ett antal uppsatser kring tematiken i _1973. En träff med tidsandan_ (2008). Svängningarna i kulturdebatten, sedda ur ett vänsterperspektiv, har studerats av Anders Frenander i _Debattens vågor. Om politisk-ideologiska frågor i efterkrigstidens svenska kulturdebatt_ (1998).
En väsentlig del av litteraturen har journalistisk eller polemisk form eller har självbiografiskt innehåll. David Brolins _Omprövningar. Svenska vänsterintellektuella i skiftet från 70-tal till 80-tal_ (2015) är sålunda en uppgörelse med vänsterintellektuella som i författarens ögon avfallit från den rätta tron. Håkan Arvidssons _Vi som visste allt. Minnesbilder från 1960-talets vänsterrörelse_ (2008) är en uppgörelse med författarens eget förflutna och samtidigt med vänsterrörelsen i sin helhet. Lennart Berntson & Svante Nordins »När vi stod i Zenit» (i Tove Krause & Lisbeth Larsson red. _Ett historiskt tillfälle: en festskrift till Håkan Arvidsson_ (2003) innehåller författarnas minnen från en vänstertidskrifts uppgång och fall.
Vad gäller perioden från Berlinmurens fall hösten 1989 till millennieskiftet är den forskningsbaserade litteraturen om den svenska vänsterns omprövningar och nya ansatser av begripliga skäl mindre omfattande. Den hittills mest grundliga studien är Petter Bergners avhandling om Vänsterpartiets utveckling fram till 2006, _Med historien som motståndare. SKP/VPK/V och det kommunistiska arvet 1956–2006_ (2013). En mer polemisk och journalistisk bok om Vänsterpartiet, vilken behandlar perioden fram till 2000-talets början, är Kjell Albin Abrahamsons _Låt mig få städa klart! Om kommunister, kryptokommunister och antikommunister_ (2014). Under 1990-talet utkom en rad andra tillbakablickande och kritiska skrifter om den svenska vänstern. Flera av dem diskuteras i det följande och nämns därför inte här. Vad gäller den internationella litteratur som sätter efterverkningarna av 1968 i fokus kan särskilt framhållas Richard Wolins läsvärda _The Wind from the East. French Intellectuals, the Cultural Revolution and the Legacy of the 1960s_ (2010). Två andra studier vilka söker fånga de nya tendenserna inom den samtida vänstern och se dem i ljuset av utvecklingen från 1968 är dels Göran Therborns _From Marxism to Post-Marxism_ (2008) och dels fransmannen Razmig Keucheyans _The Left Hemisphere. Mapping Critical Theory Today_ (2014). Också Paul Hollander i _The End of Commitment. Intellectuals, Revolutionaries, and Political Moralists_ (2006) och Jean-François Revel i _La grande parade. Essai sur la survie de l'utopie_ (2000) jämför 68-vänsterns engagemang med det post-kommunistiska perspektivet.
Det vi försökt göra i denna bok skiljer sig något både från de nämnda forskningsinsatserna och från den polemiska och självbiografiska litteraturen. Vår ambition har varit att teckna den svenska vänsterns intellektuella historia sådan den framträder i den allmänna kulturdebatten. Svante Nordins avsnitt handlar om den ursprungliga 68-vänstern, dess framväxt, kulmination, ansträngningar att omdefiniera sig själv och dess slutliga sönderfall under sent 1970- och tidigt 1980-tal. Vänstern framträdde här med expressiva krav och upproriska uttrycksformer, ofta företrädd av radikala författare. Nordins genomgång skildrar denna på en och samma gång eruptiva och utebbande period.
Lennart Berntsons del däremot handlar om debatten efter murens fall 1989. Nu stod vänsterns förflutna, i synnerhet dess hållning till den parlamentariska demokratin, i fokus. Perspektivet var tillbakablickande och kritiken dominerades av liberala debattörer. Vänstern befann sig på defensiven och sökte efter en ny identitet. 1968 års revolterande författare hade ofta antingen tystnat eller valt nya ämnesområden. Sett i efterhand kan övergången – 1980-talet – framstå som ett mellanspel, en period av relativ stiltje. I verkligheten var den ett genombrott för nyliberala idéer.
## **1**
## **Vänstern 1968–1984**
## **I. FRÅN DEN NYA VÄNSTERN TILL 68-VÄNSTERN**
OM 68-VÄNSTERNS GENEALOGI såväl internationellt som i Sverige har skrivits åtskilligt. Termen »68» etablerades inte omedelbart utan tog ett antal år på sig innan den vann allmän acceptans. Valet av just detta år som ikoniskt och symboliskt hade också en viss godtycklighet över sig. För svensk del kunde det kopplas till kårhusockupationen i Stockholm, till uppträdena kring universitetsjubileet i Lund, till tennismatchen mellan Sverige och Rhodesia i Båstad, till »U68», den kontroversiella universitetsreform som lades fram detta år. Tidigare hade man vanligen talat om »den nya vänstern». Det var namnet på en bok av Alvar Alsterdal ( _Den nya vänstern_ , 1963) som gav utblickar över den europeiska scenen. Det gav namnet till en bok, _En ny vänster_ (1966), som redigerades av Göran Therborn. Det var denna nya vänster som blev föremål för Daniel Tarschys och Carl Thams polemik i _Den nygamla vänstern_ (1967). Men hur förhöll sig denna tidiga »nya vänster» till den senare »68-vänstern»? Jag tillåter mig trots att det inte finns något konsekvent vedertaget bruk att med termen »den nya vänstern» avse en position mitt emellan socialdemokratin och VPK (Vänsterpartiet Kommunisterna), sålunda en vänstersocialistisk position i släkt med bland annat Aksel Larsens danska, men också med britternas och amerikanernas »New Left», tyskarnas »Neue Linke» och fransmännens »Nouvelle Gauche». Denna typ av vänster försökte undvika å ena sidan stalinism och Moskvakommunism enligt de gängse kommunistpartiernas modell, men revolterade å andra sidan mot socialdemokratins reformism. Den var angelägen om att ge de kulturella och existensiella frågorna större utrymme än de brukade ha i den kommunistiska eller socialdemokratiska retoriken. Den fäste större vikt vid »tredje världen» och dess problem. Den vädjade till en ny generation. Ur radikaliseringen av denna nya vänster växte sedan »68-vänstern» fram. Den ställde krav på en leninistisk ideologi, den underströk alliansen med de kommunistiska länderna i tredje världen och med kommunistiska gerillarörelser, den ropade på en tydligare teori och på en mer disciplinerad politisk organisering.
Hur såg denna för 68-vänsterns uppkomst avgörande förändring ut för svenskt vidkommande? En utgångspunkt kan man finna i Alsterdals nyss nämnda bok.
Den socialdemokratiske tidningsmannen Alsterdal ägnade en serie reportage, senare publicerade i bokform, åt den nya vänsterns framväxt i Europa. Ett avsnitt om fenomenet Aksel Larsen i Danmark inledde för att följas av kapitel om Norge, England, Italien, Västtyskland och Frankrike. Ett nytt spöke gick genom Europa, nyradikalismens spöke. Men hur förhöll det sig med nyvänstern i Sverige? Till den frågan konstaterade Alsterdal kort och gott: »Till detta kan man bara säga som den gamla geografiboken under rubriken 'Elefanterna på Grönland: finns inga'.»
Alsterdal trodde inte på ryktena att någon sorts vänstersocialistiskt parti skulle vara på väg i Sverige. Det fanns egentligen ingen fråga ett sådant parti skulle kunna ta fasta på. Sverige var inte med i Nato och hade ingen atombomb. Här fanns inga missnöjda kommunister att rekrytera och inga missnöjda socialdemokrater: »Viktigast av allt är kanske dock att svensken är sällsynt allergisk mot den politiska retorik, som alla dessa vänsterriktningar flitigt odlar.» Så bedömde en erfaren journalist på vänsterkanten läget 1963. Han skulle inte ha kunnat ha mera fel. Under de kommande femton åren skulle den politiska vänsterretoriken blomma i Sverige med en växtkraft som aldrig förr (eller senare). De missnöjda socialdemokraterna skulle visa sig vara åtskilliga och de missnöjda kommunisterna så många att ingen riktigt kunde hålla reda på alla de nya grupper och partier de gav upphov till. År 1966, när man ännu bara stod i början av denna utveckling, redigerade Kjell E. Johanson, ordförande för VPK:s ungdomsförbund, boken _Unga kommunister – aktuellt alternativ_ , som kom ut i en debattbokserie på Bonniers förlag. Baksidestexten sammanfattade bokens tema. Det handlade om att locka över socialdemokrater på vänsterkanten till kommunismen. Men boken hade också »en kraftig udd mot den egna äldre partibyråkratin och mot stalinismen». Skribenterna tillhörde alla »den kritiska yngre grupp inom Sveriges kommunistiska parti (SKP) som har engagerat sig för en ny kurs, med större frihet till debatt inom partiet och en brytning med åtskilliga av de gamla principerna». Flera hörde till undertecknarna av »de 29:s brev», ett brev till SKP:s centralkommitté från 29 av partiets yngre medlemmar som var missnöjda med tillståndet inom partiet. I det de skrev kunde man utläsa vissa ansatser till ett politiskt program för ett nytt socialistiskt vänsterparti. Kjell E. Johanson hade hört till initiativtagarna till detta brev. I _Unga kommunister_ ansåg han sig kunna konstatera: »Det är 'inne' att vara vänster. En massa människor står och trängs i kapprummet till den nya vänsterns gyllene salar.»
Johanson hade fullkomligt rätt. Det var »inne» att vara vänster. Men vad exakt innebar det? Två unga liberaler, Daniel Tarschys och Carl Tham, grubblade över det i boken _Den nygamla vänstern_ , som kom ut, likaledes på Bonniers, påföljande år. De kontrasterade Alsterdals uttalande från 1963 om de obefintliga elefanterna med Johansons uttalande från 1966 om trängseln i kapprummet. De konstaterade att läget under bara dessa korta år helt hade förändrats:
Den nya vänstern omfattar nu en ansenlig del av den unga opinionen i Sverige. Den kulturella klimatförskjutningen är påtaglig. I skuggan av de båda arbetarpartierna har en tät undervegetation av socialistiska grupper och kotterier vuxit fram. Den intellektuella vänsterdebatten breder ut sig, både i den etablerade pressen och i en rad nya tidskrifter av mer eller mindre marxistisk färgning. Med andra ord: det har blivit alldeles tjockt med grönländska elefanter. Var kommer de ifrån?
Något alldeles tydligt svar på den sista frågan hade inte Tarschys och Tham, även om de nämnde ett antal bidragande faktorer som Moskvakommunismens kris efter 1956, kritiken mot en förstelnad svensk socialdemokrati och protesterna mot Vietnamkriget. Men de kunde åtminstone peka ut en del företrädare för den nya vänstern. Två av dem prydde vid sidan av bilder på Karl Marx omslaget på deras bok. De var Göran Therborn och C. H. Hermansson.
Therborn, vid denna tid fil.kand. och lärare vid sociologiska institutionen vid Lunds universitet, var inte bara den ledande personligheten på tidskriften Zenit, som efter en syndikalistisk period omstöpts till att bli en viktig företrädare för den nya vänstern. Han hade som nämnts redigerat debattboken _En ny vänster_ , som 1966 utkommit på Rabén & Sjögren. Medarbetarna var unga socialdemokrater, men socialdemokrater som var missnöjda med sitt parti och önskade en mera radikal linje. Det handlade om en vänstersocialistisk politik, men också om att lansera »ett mänskligt vidperspektiv», där gränserna mellan praktisk politik, samhällsanalys, idéteoretisk diskussion och kulturdebatt skulle upphävas. Gramsci, Lukács och andra filosofiska förebilder åberopades. Begrepp som »hegemoni» och »alienation» ställdes i centrum. Men det handlade inte om kommunism. Tvärtom ansåg sig Therborn kunna slå fast att både »kommunismen och antikommunismen har blivit passé». Leninismen var irrelevant och vad gällde den kinesiska kommunismen var så mycket klart, »att maoismens bonderevolution naturligtvis är Västeuropa ännu mer fjärran än det leninistiska proletariatets diktatur». Närmast var det en radikalisering av socialdemokratin som boken syftade till.
_En ny vänster_ väckte intensiv debatt. Men det visade sig snart att Therborn och hans vänner missbedömt vart debatten skulle leda, vart den skulle leda opinionen men också vart den skulle leda dem själva. En del av debatten utspann sig naturligt nog i Zenit. De flesta reaktioner som publicerades i Zenits »Enkät och debatt kring _En ny vänster_ » var positiva till boken. Men det var de negativa reaktionerna som var signifikanta, som indikerade åt vilket håll vinden blåste. En sådan reaktion kom från Gunnar Bylin, ordförande i svenska Clartéförbundet. Clarté, ursprungligen en socialistisk organisation utan partianknytning, hade under den senaste tiden befunnit sig i hastig förändring i maoistisk riktning. Bylin var företrädare för en sorts maoistiskt maktövertagande inom tidskriften och förbundet och skulle senare bli ett framträdande namn inom KFML/SKP (Kommunistiska förbundet marxist-leninisterna/Sveriges kommunistiska parti). Det märktes på vad han nu skrev (Zenit nr 4 1966). Resonemangen i _En ny vänster_ avfärdades som pseudorevolutionära men i grunden borgerliga fraser. Boken hade på idealistiskt vis framför allt uppehållit sig vid kulturen och ideologin i stället för att angripa den politiska och ekonomiska makten. Mot denna Don Quijotes strid mot väderkvarnar ställde Bylin det verkliga alternativet: »Till dags dato är den marxistisk-leninistiska teorin den enda som visat sig kunna vägleda massorna till seger i kampen för socialismen, den enda som visat sig kunna vägleda revolutionärer i Västeuropa att rätt värdera situationen i det egna landet.»
Therborn svarade (i samma nummer av Zenit), men hans svar blev förvånansvärt lamt. De »ortodoxa leninisterna» var fast i dogmerna, ansåg han, utan att någon skillnad gjordes mellan de gamla Moskvakommunisterna och de nya unga maoisterna. Men bara några år senare finner vi att Therborn gått över till den nya positionen och anammat leninismen. I en stor artikel i Zenit (nr 13 1969) skriven tillsammans med Lennart Berntson under rubriken »Lenin om parti och revolution», finner vi honom beklaga att »det här i landet saknas ett revolutionärt parti». Konklusionen blev: »På längre sikt är naturligtvis ett marxistisktleninistiskt parti absolut nödvändigt men innan ett sådant bildas krävs politisk klarhet och styrka.»
Innebar detta att Therborn numera var fullkomligt enig med Bylin? Knappast. Det fanns marxist-leninister och marxist-leninister. En viktig skillnad fanns i själva den teoretiska hållningen. Inom KFML hade man inget intresse för den typ av teoretisk debatt som fördes på exempelvis Zenit. Björn Erik Rosin framhöll detta i den artikel om KFML/SKP, som han skrev för antologin _Utanför systemet. Vänstern i Sverige 1968–1978_ (1978). Det hette där: »Marxister som Althusser, Poulantzas, Mandel, Perry Anderson, Paul Sweezy ägnades föga intresse inom KFML. Kort sagt stod förbundet nästan helt vid sidan av den marxistiska debatt, som började flöda ungefär vid denna tidpunkt. KFML hade axlat andra uppgifter än att problematisera marxismen.»
Inom KFML hade man Mao. Det räckte. Problemet uppstod när Mao dog, men det är en senare historia.
Om den senare Therborn stod för den intellektuella, problematiserande leninismen och Bylin för den dogmatiska fanns C. H. Hermansson någonstans mitt emellan. Han var partiman först som sist, sedan 1964 ordförande för Sveriges Kommunistiska Parti (vars namn KFML senare skulle stjäla när det ursprungliga SKP bytt namn till VPK). Men Hermansson framträdde också som intellektuell. Han skrev böcker. En sådan kom ut 1965 och hette _Vänsterns väg. Ett debattinlägg_. Här var det inte Hermanssons ambition att göra ett snävt partipolitiskt inlägg, men däremot att försöka fånga in den nya vänsterströmningen in i det egna partiet och samtidigt placera in VPK i en bred vänstersamling. Det egna partiet karaktäriserade han samtidigt som »gammalmodigt», med tillägget att de övriga riksdagspartierna också var gammalmodiga.
I _Den nygamla vänstern_ menade Tarschys och Tham att »C. H. Hermansson har åtskilliga orsaker att jubla över den nya vänstervågen». Kanske. Men en partiledares lott är inte alltid lycklig. Revolutionsåret 1968 gjorde Hermanssons parti ett rekorddåligt val. Det var inte hans fel utan ryssarnas som i augusti det året, strax före den svenska valdagen, marscherade in i Tjeckoslovakien. År 1969 blev inte avsevärt bättre ur Hermanssons synpunkt. Det var vid partikongressen det året han myntade sitt mest (och kanske enda) kända bevingade ord: »Någon djälva ordning får det vara i ett parti!» I detta parti fanns Moskvaorienterade gammalkommunister och nyleninister. Där fanns, eller hade intill helt nyligen funnits, maoister, vänstersocialister och trotskister. Hermanssons skicklighet att hoppa från stock till stock sattes på hårda prov.
C. H. Hermansson framträdde som partiledare till en början som allierad till en vänstersocialistisk tendens – eller till en tendens som ville förhindra en utveckling liknande den i Danmark där Aksel Larsens Socialistisk Folkeparti brutit sig ur kommunistpartiet. I stället ville han reformera SKP på ett sådant sätt att den vänstersocialistiska tendensen skulle kunna fångas in och få rum inom partiet. När den Moskvatrogne Hilding Hagberg fördes undan till förmån för Hermansson som partiordförande var detta en bakomliggande tanke. De idéer som Hermansson 1965 formulerade i _Vänsterns väg_ låg i förlängningen av denna tanke. Ett gammalmodigt kommunistparti skulle reformeras i vänstersocialistisk riktning. Man skulle befria sig från den barlast som den absoluta troheten mot Sovjetunionen hade inneburit. Man skulle öppna för bred samverkan med nya vänsterkrafter. Vid sidan av Hermansson framträdde Sven Landin och Bo Hammar, i början även Kjell E. Johanson, som företrädare för denna linje. Tidskriften Tidsignal blev dess organ. När SKP 1967 bytte namn till VPK fanns en underförstådd avsikt att snart nog låta K:et falla till förmån för det enkla Vänsterpartiet. Så blev det inte. K:et följde med fram till tiden för kommunismens fall i Europa. Det blev inte VPK som övergav kommunismen. Det blev kommunismen som övergav VPK. Ännu många år efter kommunismens fall framhärdade för övrigt Vänsterpartiledaren Lars Ohly med att kalla sig »kommunist». En bidragande orsak till att namnbyte till Vänsterpartiet aldrig skedde under Hermanssons ordförandetid var nykommunismens (eller nyleninismens) uppdykande på scenen.
Till en början hade detta uppdykande framför allt maoismens form. Splittringen mellan SUKP (Sovjetunionens kommunistiska parti) och KKP (Kinas kommunistiska parti) på världsscenen förde med sig att först maoistiska tendenser inom de gamla Kominternpartierna, sedan maoistiska utbrytningar från dessa förekom överallt i världen. Inom det svenska partiet fördes den kinesiska tendensen fram i en allians mellan gammalstalinister som Nils Holmberg och unga Kinatrogna som Bo Gustafsson. Så länge denna grupp fanns kvar inom moderpartiet blockerade den en utveckling i vänstersocialistisk riktning. Den bröt sig ut och bildade KFML (Kommunistiska förbundet marxist-leninisterna) 1967. Men dess roll att driva VPK mot vänster övertogs av företrädare för en nyleninistisk tendens som inte hade en lika ensidigt maoistisk orientering. Lars Herlitz och kanske framför allt Jörn Svensson kan ses som företrädare för en sådan nyleninism. Att Gustafsson och Herlitz var akademiskt verksamma och att Svensson var fil.dr – alla inom ekonomisk historia – var kännetecknande för de nya riktningarnas bas bland universitetsanknutna och intellektuella.
Inför trycket av nyleninismen, som i vissa stycken samverkade med gammalkommunisterna, lämnade Sven Landin 1968 VPK och gick över till SAP (Sveriges socialdemokratiska arbetareparti). Bo Hammar blev kvar och tog striden genom vad han själv kallat »ett långt farväl till kommunismen», fram till början av 1990-talet. I fortsättningen kom motsättningen mellan nyleninisternas i huvudsak sovjetkritiska och gammalkommunisternas sovjettrogna linje att spela en viktig roll fram till »Flamman»-falangens utbrytning och bildande av APK (Arbetarpartiet kommunisterna) 1977.
Striderna mellan de tre falangerna – particentern, nyleninisterna och gammalkommunisterna – kom i övrigt att prägla mycket av VPK:s utveckling mellan 1967 och 1977. Frågan om demokratin och om socialismens genomförande på demokratisk väg eller genom en våldsam revolution var central. Kring den fäktades vid varje kongress och varje ny programskrivning utan att någon riktig klarhet uppstod om var partiet egentligen stod. Den därmed förknippade frågan om förhållandet till Sovjetunionen, Kina och övriga socialistiska länder och deras kommunistiska partier var på ett liknande sätt aktuell under hela perioden och fram till 1991.
Kan en utveckling från »nyvänster» till »68-vänster» spåras också i den allmänna kulturdebatten? Jag tror det. Åke Lundqvist framhåller med all rätt i sin _Från sextital till åttital. Färdvägar i svensk prosa_ (1981): »Kulturdebatten i början av sextitalet präglas av en radikal anda, estetiskt radikal, kulturpolitiskt radikal, moraliskt radikal. Den polemiska udden riktas mot den borgerlighet och borgerliga kulturmiljö som de unga författarna självklart tillhör, mot dess stelnade och föråldrade former och normer.»
Omständigheter tillstötte som bidrog till att radikalisera denna radikalism, inte minst Vietnamkrigets utveckling. De författare som här gick i spetsen var i motsats till studentrevoltens unga inga 40-talister. De var födda på 1920-talet eller 1930-talet (Sara Lidman 1923, Jan Myrdal 1927, Lars Forssell 1928, Göran Palm 1931, Sven Lindqvist 1932, Per Olov Enquist 1934, Staffan Beckman 1934, P. C. Jersild 1935) och hade debuterat under 1950-talet eller i början av 1960-talet. Att de skönlitterära författarna spelade en framträdande roll var ett tecken i tiden. Redan under den första stora Vietnamdebatt som fördes på DN:s kultursida 1965 hade de dominerat. Kultursidorna och kulturtidskrifterna fick en ny betydelse för den politiska debatten. Det hade samband med en förändring i själva debattklimatet. Tarschys och Tham kommenterade kritiskt i _Den nygamla vänstern_ hur engagemanget nu blivit det väsentliga i debatten: »Argument, analys, diskussion och slutsats ersätts av suggestivt formulerade vädjanden, engagemang och inlevelser.»
Kanske fanns det också i detta lyrikernas och romanförfattarnas maktövertagande i debatten en sorts alibinism, en ursäkt för den dramatiska radikaliseringen.
År 1968 utkom Staffan Beckmans roman _Tycker du att Georg är en galning?_ Den handlar om en väpnad stalinistisk kupp som genomförs i Sverige. Sensmoralen tycks vara att en sådan kupp, avsedd att avskaffa demokratins »skenkonstruktion», är önskvärd. Johan Svedjedal kommenterar i sin _Ned med allt? Essäer om protestlitteraturen och demokratin, cirka 1965–1975_ (2014) ett inslag i denna roman: »På några ställen finns utdrag ur 'Blå pärmen': listor över makthavare, medlemmar i högerorganisationer, borgerlig press med mera (det lämnas åt läsarens fantasi att avgöra om listan redovisar kunskapsinhämtning, planer för husarrest eller för önkvärda likvidationer).»
Denna lekfulla inställning till exempelvis likvidationer var ett väsentligt inslag i tidsmiljön. Den hade något att göra med de skönlitterära författarnas ställning. Den hade något att göra med att revolutionen i Sverige låg så långt borta att den mest föreföll vara en fantasi. Men den lämnade öppet var gränsen mellan fantasi och verklighet borde gå. Terrorgrupperna Röda brigaderna i Italien och Röda arméfraktionen i Västtyskland förverkligade plötsligt vad som dittills hade gällt för att vara fantasi. I Sverige behölls den poetiska ambivalensen. Men radikalismen var inte desto mindre verklig.
Oklarheten i förhållandet mellan fantasi och verklighet var grunden för de unga revolutionärernas hopp. År 1968 fick VPK 3 % av rösterna vid riksdagsvalet, onekligen ett uselt resultat för ett revolutionsår. Men hade inte bolsjevikerna i Ryssland vid krigsutbrottet 1914 bara varit en handfull? Hade de inte likväl kunnat ta makten 1917? Vem kunde med säkerhet säga att något likande inte kunde upprepas i Västeuropa, kanske till och med i Sverige? I Frankrike hade studentrevolten i maj 1968 lyckats destabilisera landet, visserligen bara för en kort stund. Så varför kasta yxan i sjön? Allt var möjligt, inte minst det omöjliga.
Uppkomsten av »68-vänstern» måste på detta sätt förstås mot bakgrunden av en internationell politisk konjunktur. Det handlade om ett generationsuppror, javisst. Det handlade om studenter under en period när tillströmningen till universiteten svällde ut till en flod, för all del. Men det som gav denna vänster det politiska innehållet var inte krav på den ena eller den andra studiesociala reformen. Inte heller hade denna studentvänster och strömning bland de intellektuella någon motsvarighet i djupare sociala strömningar hos arbetarklassen eller andra bredare skikt av befolkningen, inte just då.
Det hela började, som berörts, genom framväxten av en »ny vänster». Den var livaktig i åtskilliga länder innan den sent kom till Sverige vid mitten av 1960-talet. Denna nya vänster hade sin bakgrund i en kris inom den kommunistiska rörelsen, men också hos strömningar inom socialdemokratin. Kommunismens kris inleddes 1956 genom Chrusjtjovs »avstaliniserings»-tal vid SUKP:s tjugonde partikongress, genom Ungernhändelserna, genom dispyten med Kina. Inom kommunistpartierna i Västeuropa uppstod ett behov att frigöra sig från Moskvakommunismen, att hitta en självständig linje, mer anpassad till det egna landets politiska traditioner, mer »demokratisk». Samtidigt bidrog Algerietkriget, avkoloniseringen, uppkomsten av ett icke-allierat »block» av länder till en vänsterriktning inom socialdemokratin. Tanken på en ny »vänstersocialism» uppstod och förverkligades exempelvis i Danmark i en partibildning. Till Sverige kom »den nya vänstern» dels som en ung vänsterrörelse inom socialdemokratin, dels som ungas önskemål om frigörelse från Moskva inom SKP.
Denna nya vänster blev emellertid för Sveriges vidkommande snabbt överflyglad. Den »nya vänstern» transformerades till »68-vänstern». Intrycken från det pågående Vietnamkriget och från den kinesiska kulturrevolutionen var det avgörande för denna utveckling. Studentvänstern blev i ökande utsträckning »leninistisk» och på sina håll »maoistisk». Uppmärksamheten drogs allt starkare bort från Europa mot »tredje världens kamp». Vad som börjat som u-landsengagemang och sympatier för den koloniala »frigörelsekampen» slutade som ett ställningstagande för kommunismen sådan den utformades i Kina, Nordvietnam, Nordkorea och på Kuba.
De följande kapitlen skall handla om denna »68-vänster» och dess utveckling under det dryga årtionde som följde på märkesåret 1968.
## **II. VÄNSTERNS LÅNGA SJUTTIOTAL**
## **1. 1968 – revolutionsåret**
VÄNSTERNS UTVECKLING I SVERIGE under vad jag här kallar »det långa sjuttiotalet» kan uppfattas som en offensiv följd av en defensiv. Vänstervinden friskade under de tidiga åren i om inte till orkan så åtminstone till styv kuling för att sedan mojna och ersättas av en vind i motsatt riktning. Tidvattnet – för att använda en annan metafor – steg till synes oemotståndligt för att sedan kulminera, vända och sjunka tillbaka. Hade en sentida Josef i slutet av 1968 kunnat tyda vänsterns drömmar och mardrömmar skulle han ha förutspått sju goda år följda av sju svåra. Denna rytm skall här återges. Men först måste vi börja med det stora symboliska året 1968, studentrevoltens år, när startskottet avlossades. Det året hade majrevolten i Paris som sitt mest spektakulära event och kårhusockupationen i Stockholm som en, visserligens ganska beskedlig, svensk motsvarighet.
För att ta pulsen på vänsterns ideologiska utveckling detta märkesår väljer jag två böcker, som då kom ut på svensk bokmarknad. Den ena är Herbert Marcuses _Den endimensionella människan_. Marcuses _One-Dimensional Man_ hade i originalversionen kommit ut 1964. Men 1968 publicerades _Den endimensionella människan_. _Studier i det avancerade industrisamhällets ideologi_ på svenska i översättning av Sven-Eric Liedman som originalutgåva i pocketbokserien Aldus (Bonniers). Att denna bok, ofta betraktad som studenrevoltens bibel, skulle komma på svenska just studentrevoltens år var kanske inte förvånande. Överhuvud blev 1968 Marcuseåret i svensk bokutgivning. Det året utkom, likaledes i pocketbokens form, ytterligare viktiga Marcusetexter på svenska. Det var författarens »Repressiv tolerans», utgiven tillsammans med två texter av Robert Paul Wolff (»Bortom toleransen») och Barrington More Jr (»Tolerans och vetenskaplig syn») under den gemensamma titeln _Kritik av den rena toleransen_. Vidare utkom _Protest, demonstration, revolt_ samt _Eros och civilisation_ , den senare inte på Aldus utan Arcana förlag.
Den andra boken är Göran Palms _Indoktrineringen i Sverige_ , som kom ut i en pocketutgåva på Norstedts förlag. Denna bok flankeras av andra av samme författare. Två år tidigare, 1966, hade Palms _En orättvis betraktelse_ givit upphov till intensiv debatt. Påföljande år, 1969, skulle hans _Vad kan man göra?_ dra de så att säga operativa slutsatserna för vänsteraktivisternas vidkommande av den analys som lagts i de två tidigare böckerna. Det kan kanske sägas att ingen annan svensk författare lika tydligt som Göran Palm representerade stämningarna just 1968 när nyvänstern ännu befann sig i sin Sturm- und Drangperiod av revolutionär eufori. Palm skrev intressanta och uppmärksammade böcker före _En orättvis betraktelse_ och efter _Indoktrineringen i Sverige_. Men inga som fick samma inflytande eller nådde liknande upplagor. »5:e upplagan, 50:e tusendet» ropar förslagsreklamen på baksidan av den nu skrivande författarens exemplar av boken. Göran Palm var principiell motståndare till reklam, men här gjorde han uppenbarligen ett undantag.
Herbert Marcuse var väl skickad att tala till studenter i alla länder. Han var modernare än Engels, sexigare än Marx, roligare än Ulbricht. Hans marxism – om det nu var marxism – var tillskuren för det postindustriella samhället. Han vädjade inte till den försoffade och förborgerligade arbetarklassen utan till studenterna. Han anklagade inte kapitalismen för att skapa fattigdom och nöd utan för att skapa överflöd, ett »konsumtionssamhälle». Han anklagade den för att förtrycka folket med demokrati och tolerans som försåtliga verkningsmedel.
De svenska kommentarerna till Marcuses böcker framhöll det som gjorde hans budskap aktuellt i 1968 års svenska sammanhang. Baksidestexten till _Den endimensionella människan_ poängterade att den västliga demokratin rymde en »totalitär tendens»: »Detta samhälle tycks kunna förhindra en kvalitativ social förändring som skulle kunna medföra helt andra institutioner, en ny inriktning av produktionsprocessen och nya mänskliga levnadssätt.»
Den västliga demokratins motståndskraft mot en socialistisk revolution var enligt detta resonemang tillräckligt bevis för en »totalitär tendens», onekligen alldeles i Marcuses anda. Baksidestexten till _Kritik av den rena toleransen_ ställde på liknande sätt frågan om inte tolerans blivit en »i djupaste mening antidemokratisk faktor». Frågan förklarades ha »ett viktigt samband med studenternas revolt på olika håll i världen mot det etablerade samhället». Av bokens författare prisades i synnerhet Marcuse som avslöjat den »förtryckande tolerans» som accepterar »det onda likaväl som det goda» och därmed hämmar utvecklingen mot socialismen. »Den revolutionära minoritetens» betydelse framhölls i denna kontext. Baksidestexten till _Eros och civilisation_ underströk Marcuses kamp för »ett samhälle som i långt större utsträckning än hittills skulle kunna ge uttryck åt mänsklighetens strävan efter lust och öppenhet mot andra».
_Protest, demonstration, revolt_ var försedd med ett förord av vänstersociologen Joachim Israel (professor i Lund med tysk-judisk bakgrund). Israel fann det utmärkt att Marcuse »äntligen presenteras på svenska». Marcuse hade blivit »något av en ideologisk ledare för den vänsterradikala studentoppositionen i Västtyskland, där den gamla auktoritära andan frodas bland universitetsprofessorerna». Bland Marcuses böcker lyfte Israel fram tre: _Soviet-Marxism, a Critical Analysis_ med dess kritik mot den förlegade Sovjetmarxismen, _Eros and Civilization_ med dess plädering för lustprincipen, och _One-Dimensional Man_. Den sistnämnda betecknades som »Marcuses viktigaste bok» och dess centrala tema refererades med gillande: »Masskonsumtionssamhället har förvandlat människan till en slav som lever i lyx, för vilken all opposition mot samhällsskicket ter sig meningslös därför att den riskerar att avskaffa de fördelar som finns.»
Detta var en god sammanfattning av människans tragiska tillstånd under de kapitalistiska demokratierna sådant också den svenska 68-vänstern betraktade det. Hon var helt enkelt en »slav» som levde i lyx och som fegt avstod från att genomföra den kommunistiska revolutionen eftersom denna skulle riskera att »avskaffa de fördelar som finns». Den enda glimten av hopp i detta mörker kom från »den revolutionära minoriteten». Detta var, som Israel tydliggjorde, inte marxism i traditionell mening utan något annat, något som tedde sig nytt och hoppingivande för de revolterande studenterna i många länder, bland dem Sverige.
En författare, denna gång inhemsk, som i likhet med Marcuse på ett särskilt sätt var förknippad med året 1968 var Göran Palm. Tidigare främst känd som poet i »nyenkelhetens» tecken hade Palm redan 1966 lagt grunden till sitt rykte som profet genom _En orättvis betraktelse_. Nu blev hans ställning till synes orubblig genom _Indoktrineringen i Sverige_.
_En orättvis betraktelse_ har en medvetet rapsodisk karaktär och är svår att sammanfatta. Redan i en artikel i Expressen 23.6.1964 hade Palm tagit avstånd från den rationellt argumenterande stil som den svenska kulturdebatten antagit under professorerna Herbert Tingstens och Ingemar Hedenius spiror. Han gick till angrepp mot föreställningen att »den påverkan som varje kulturartikel syftar till huvudsakligen måste bygga på förnuftsskäl för att kunna bli framgångsrik». Mer känsloengagemang, mer personligt utspel var hans recept: »Man börjar tvivla på den strikta och opersonliga argumentationslunken. Man börjar längta efter mera emotionell frihet i kulturdebatten.» Den väldiga framgången för _En orättvis betraktelse_ utgjorde verkligen i sig en effektiv dementi av föreställningen att varje inlägg i kulturdebatten »huvudsakligen måste bygga på förnuftsskäl för att kunna bli framgångsrik». Det var i stället författarens litterära och retoriska talang liksom det högt uppskruvade känsloläget som gjorde intryck. Det mer intellektuella innehållet i Palms tidiga debattböcker sammanfattades pregnant av Kjerstin Norén i en litteraturvetenskaplig framställning: »Han vill visa att det västerland vi otvetydigt tillhör och som flertalet menar vilar på demokratisk, humanistisk och frihetlig grund i själva verket har en historia av mord och grymhet, kolonial och imperialistisk utsugning och förtryck.»
Hon påpekade helt korrekt att Palm arbetade utifrån ett »vi», utifrån en tillhörighet till en kultur: »Men hans bok är en uppvisning av ett heroiskt försök att fjärma sig från denna kultur, att skaffa sig ny identitet. Den nya identiteten finner Palm hos tredje världens kämpande folk, de som är förfördelade av oss.» Kjerstin Norén gör ytterligare två klarsynta påpekanden. Hon konstaterar att en läsning av Palms bok år 1977 – när hon själv skrev – »innebär att återkalla den medvetandenivå som var till städes vid den tidpunkt när den nya vänstern började utvecklas». Och hon ser att _En orättvis betraktelse_ inte utan vidare kan kallas en marxistisk bok. I stället är andra inspirationskällor aktuella för Palm: »Inte minst spelar den religiösa (kristna) humanismen en stor roll i hans argumentering.»
Som motto för _En orättvis betraktelse_ står ett citat från Bergspredikan. Som modern motsvarighet till Bergspredikan använder Palm gärna den algeriske revolutionären Frantz Fanons _Jordens fördömda_ , som ofta citeras. Det som Palm tar till sig är Fanons »bild av Europa som proletärländernas bödel och dödsfiende».
I _Indoktrineringen i Sverige_ vandrade Göran Palm vidare på den väg han anträtt i _En orättvis betraktelse_. Boken var »till hälften en exempelrik granskning av vår samhällsbevarande indoktrinering till hälften en socialistisk stridsskrift».
Huvudmotståndaren i Palms nya bok blev vad författaren kallar »den borgerliga demokratin». Exempel på den indoktrinering han granskar blir »den faktabeljugande antikommunismen i skolornas läroböcker i samhällskunskap och historia», västindoktrineringen i TV:s amerikanska spionfilmer, den »klassöverslätande indoktrinering» som bedrivs när klasskampen inte kommer till sin rätt i familjeserierna, den »egendomsindoktrinering» som återfinns i Tio Guds bud med mera. Det finns två sidor av denna kamp: »När den svenska indoktrineringen rör sig på det faktaförvrängande planet kan man avvisa den med hjälp av fakta, men när den rör sig på värderingsplanet kan man bara avvisa den med hjälp av socialistisk motindoktrinering.»
Det visar sig mycket riktigt att »den socialistiska motindoktrineringen» spelar en väsentlig roll i Palms bok.
Om de olika åskådningar som fanns representerade i det svenska samhället skrev Palm: »Den ram inom vilken man finner dessa åskådningar samsade, liksom rester av andra, brukar kallas den demokratiska värdegemenskapen. Det tycks huvudsakligen vara denna värdegemenskap som indoktrineringen i Sverige avser att värna och befästa.»
Här hade han från sina utgångspunkter rätt. Det var demokratin som sedan 1917 och än tydligare sedan 1945 blivit den bärande svenska överideologin. Ifrågasättandet av demokratin, uppfattad som »borgerlig» eller »formell» demokrati, blev en huvudsak för anhängarna av 1968 års idéer. Det var nära kopplat till ifrågasättandet av marknadsekonomin eller kapitalismen. Detta var också något som Palm starkt betonade. Det typiska för svenska skolläroböcker och för svensk »indoktrinering» överhuvud var att man ville »upprätthålla illusionen om att det är kampen mellan två politiska system (diktatur och demokrati) som är den fundamentala i världen i dag – och inte kampen mellan två ekonomiska system, det socialistiska och det kapitalistiska».
Men man kunde enligt Palm inte begripa något av 1900-talets historia »om man inte ser demokratin som underordnad kapitalismen». I stället för att uppfatta demokratin som det goda och diktaturen som det onda borde man så att säga gå över till att uppfatta socialismen som det goda och kapitalismen som det onda. Därmed blir för Palm också kritiken mot de kommunistiska staterna, »antikommunismen», själva dödssynden och huvudfelet med »indoktrineringen i Sverige». Han ondgör sig över svartmålningen av Sovjetunionen, som i Sverige ofta utmålas som ofritt: »frågan är om ryssarna ens får gå på toaletten utan att begära polistillstånd». Inte minst upprör det Palm när vänsterdiktaturer jämförs med högerdiktaturer så att man »i det kalla krigets namn» vill »fösa in Kosygin, Castro, Mao och Nasser i samma diktaturcell som Hitler, Salazar, Tubman och Barrientos».
I det svenska sammanhanget ifrågasätts om kommunisterna inte bara är de bästa socialisterna utan också noga besett de bästa demokraterna »högern medräknas bland 'de fyra demokratiska partierna' medan det enda parti som kräver folkstyre på det ekonomiska området inte medräknas».
Socialisterna, förklaras det, vill – i motsats till socialdemokraterna – ha »folkvälde». Hotet mot demokratin kommer heller inte som det kalla krigets propaganda velat göra gällande från Sovjetunionen eller Kina utan från den inre fascism som ständigt lurar på de enbart »borgerliga» demokratierna. De som inte förstått detta är illa ute: »De har därmed blivit reaktionärer, ty det är bara mot fascismen man i ett land som vårt kan försvara demokratin utan att samtidigt bejaka kapitalism och imperialism.»
Kampen mot »indoktrineringen» måste enligt Palm föras ständigt och i alla sammanhang.
I det roande och lärorika kapitlet »Barnen indoktrinerar föräldrarna» skildrar Palm hur det kan bli i de progressiva familjerna:
För varje fotografi av Che Guevara eller Mao Tse-tung som fadern sätter upp på väggen sätter sönerna upp ett fotografi av John Wayne, Robert Kennedy eller Bosse Högberg. [---]
Varje fredagskväll avbryter sönerna faderns högläsning av Hubermans 'Människans rikedomar' genom att sätta på Kvällstoppen i radio, berätta innehållet i en Vilda Västern-film, räkna sina sparpengar eller gå ut och leka Påven bannlyser. [---]
Ju äldre barnen blir desto oftare undrar de varför föräldrarna inte slutar med att gå i demonstrationer, varför det har ett så litet och konstigt umgänge, varför de tycker så annorlunda än andra föräldrar, och ber dem allt enträgnare att tycka mera lika. [---]
Kvar sitter föräldrarna i sin grånade värdegemenskap.
Det var ett framsynt mardrömsscenario som skulle komma att förverkligas för många föräldrar ur 68-generationen.
Just 1968 var värdegemenskapen emellertid varken grånad eller uttunnad. _Indoktrineringen i Sverige_ blev en stor försäljningssuccé. I lätt självironiska ordalag har Palm senare kommenterat: »Vid denna tid spreds vänstervridna debattböcker i större upplagor än deckare och kärleksromaner, varför Palms ultrademokratiska socialism från början gick med vinst.»
Det som gjorde _En orättvis betraktelse_ och _Indoktrineringen i Sverige_ så typiska för tidens »medvetandenivå» (för att begagna Kjerstin Noréns uttryck) var inte minst att deras hållning var protomarxistisk snarare än marxistisk, protokommunistisk snarare än kommunistisk. Vänsterförkunnelsen hade inte stelnat till dogmatik, engagemanget hade inte formats till partibyggande militans. Men rörelseriktningen fanns där. Vagheten och känslosamheten innebar en fördel, men kunde också kritiseras som »flummig». Den självironi som här och var fick sticka fram var avväpnande, men kunde också uppfattas som oseriös. Prästsonen Palm, som lirade än med Jesus, än med Marx, blev profet för en dag. Men inte mer. Palm hade sin språkliga talang, sitt dåliga samvete, sin moralism. Men ingen organisation, ingen strategi, ingen teori. Detta skulle snart nog bli det som krävdes.
Året 1968 hade varit studenternas år. Myndigheter och politiker utarbetade U68 för att reformera universiteten. Revolutionära studenter kolliderade med 300-årsjubileet i Lund. I Stockholm ockuperade man kårhuset. Året 1969 skulle bli arbetarnas år. Nu om någonsin hade allvaret kommit. Revolutionen tycktes stå för dörren.
## **2. Sju feta år**
### _1969 – gruvstrejkens år._
I Frankrike hade studentrevolten tänt en storstrejk bland arbetarna. För ett ögonblick hade revolutionen förefallit vara nära. Kanske skulle scenariot upprepas i Sverige med till och med större framgång. Många hoppades det. Den stora gruvstrejken i Malmfälten 1969–1970 tycktes blåsa revelj för ett revolutionärt uppvaknande hos den svenska arbetarklassen. Det faktum att LKAB, där strejken utspelades, var ett statligt bolag var betydelsefullt. Arbetarna tycktes gå till kamp i vild strejk inte bara mot Kapitalet utan mot socialdemokratin, statsminister Palme, Gruvförbundet, LO och den sedan årtionden av reformisterna styrda svenska staten, med andra ord mot just de krafter som studentupproret framför allt ville ställa vid skampålen. En sådan syn på saken var inte helt felaktig. Strejken i Malmfälten skulle visa sig signalera slutet på en era. Men precis som eran de Gaulle i Frankrike inte följdes av socialistisk omstörtning utan av Pompidou skulle det visa sig att slutet på eran av socialdemokratisk hegemoni i Sverige inte följdes av revolutionen utan av jämviktsriksdag och därefter av regeringen Fälldin. År 1969 var denna framtid emellertid barmhärtigt fördold för vänstern.
En bok som kanske kunde uppfattas som ett varsel om gruvstrejken (även om miljön var Svappavaara och inte Malmberget) var Sara Lidmans _Gruva_ , illustrerad med suggestiva bilder av Odd Uhrbom. Boken kom redan 1968, men med en utökad upplaga i Aldusserien i november 1969. Strejken bröt ut första december samma år, så förlagets timing kunde tyckas perfekt.
Att ett viktigt inlägg i samhällsdebatten nu åter kom från en skönlitterär författare var ett tecken i tiden. Om professorerna dominerat på Tingstens och Hedenius tid var det numera i stor utsträckning skönlitterära författare som blev de mest framträdande. Göran Palm, Sara Lidman, Jan Myrdal, Karl Vennberg, Sven Lindqvist, Peter Weiss och många andra blev nu viktiga i nya sammanhang. Kanske hade det något att göra med viljan att överge »den strikta opersonliga argumentationslunken» och ersätta den med »mera emotionell frihet». Detta innebar en ny roll i offentligheten för den skönlitterära författaren. Men också att många skönlitterära författare fick en ny syn på sin egen uppgift, ja, att de i många fall kom att åtminstone momentant överge skönlitteraturen för att i stället författa debattböcker och rapportböcker. Göran Palm kan ses som ett exempel, Sara Lidman som ännu ett. »Varför skriver Sara Lidman inte romaner längre?» blev titeln på ett bidrag av Torgny Schunnesson och Lars Wickman till den tidigare nämnda antologin _Linjer i nordisk prosa. Sverige 1965–1975_. Varför hade författaren till kritikerrosade romaner med Norrlandsmotiv som _Tjärdalen_ (1953), _Hjortronlandet_ (1955), _Regnspiran_ (1958) och _Bära mistel_ (1960) nu övergått till att skriva först indignerade romaner om Sydafrika, sedan en resebok och aktuellt debattinlägg som _Samtal i Hanoi_ (1966). Den senare tillkom efter en månads vistelse i Nordvietnams huvudstad och speglar Lidmans allt starkare engagemang i FNL-rörelsen.
Svaret fanns, som Schunnesson och Wickman framhåller, i en ny syn på förhållandet mellan litteratur och politik, en syn som har sin plats i Sara Lidmans personliga utveckling men också hos den nya vänsterns stämningar. Den estetiska sfären, med dess experiment och grubbel över den litterära formen, tedde sig inte lika viktig som under 50-talet. I stället handlade det om samhällsanalys och politiskt engagemang. För Lidmans del vidgades perspektivet från hembygden i Missenträsk till att omfatta hela världen från Sydafrika till Vietnam.
_Gruva_ börjar med att Sara Lidman är på väg till ett FNL-möte i Svappavaara. Odd Uhrbom ringer och berättar att han tagit bilder från gruvarbetarnas vardag i Malmberget, Kiruna och just Svappavaara. Ville hon skriva en text och ge ut det hela som bok? Det ville hon. Hon talar sedan en mörk oktoberdag i Folkets hus i Svappavaara: »Egentligen hade jag velat lämna manuskriptet och tala fritt ur hjärtat med dessa gruvarbetarfamiljer om deras kamrater i Vietnam. Men krypande för alla domar över 'känslotänkandet i Vietnamfrågan' höll jag mig benhårt till historiken.»
Citatet föranleder ett par kommentarer. Den första gäller den självklara utgångspunkten att gruvarbetarna i Svappavaara är utsatta för samma förtryck från kapitalism och imperialism som folket i Vietnam och känner sig vara det. Strax efteråt berättas om en gruvarbetare som skänker en slant till det goda kriget: »Om du känner nån FNL-grabb i Vietnam så kan du väl skicka han dom här tjugo kronorna från mig och hälsa.» Den andra synpunkten gäller Sara Lidmans behov att värja sig mot beskyllningar för »känslotänkande». Kritik för sådant tänkande hade kommit henne till del efter _Samtal i Hanoi_. Det kan, som Schunnesson och Wickman förmodar, ha bidragit till att hon i _Gruva_ utvecklar en »dokumentär teknik». Här består huvuddelen av texten av gruvarbetares berättelser om sitt liv och arbete. Läsaren får intrycket att Lidman helt enkelt låtit bandspelaren rulla och sedan fäst resultatet på pränt. Men så förhöll det sig inte. Lidman hade skrivit texterna även om hon använt sig av bandinspelat material. De arbetare som framträder i _Gruva_ är fiktiva. Det socialistiska budskapet framstår som mera objektivt genom att det läggs i arbetarnas mun snarare än i författarens. Alla redovisar de negativa erfarenheter från produktionen, alla upplever främlingskap i arbetet, alla dras med yrkesskador. Schunnesson och Wickman som framhållit detta menar att den kritik mot förfarandet som framfördes av enstaka recensenter mer berodde på »deras allmänt borgerliga verklighetsuppfattning än på brister i boken». Själva ger de en mer lojal kommentar:
Om innehållet i _Gruva_ är sant eller falskt beror inte på i vilken grad Sara Lidman bearbetat intervjuerna med gruvarbetarna. Motsättningen mellan subjektiv och objektiv verklighet existerar, men den är inte absolut [...] den är dialektisk. Det innebär att den verklighet som beskrivs i _Gruva_ inte på något sätt behöver avvika från verkligheten så som den objektivt ser ut för gruvarbetare i det kapitalistiska samhället.
Tvärtom menar vi att Sara Lidman genom sin bearbetning trängt innanför empirin och skärpt förståelsen för verkligheten.
Måhända kan man även mera allmänt säga att de radikala »rapportböcker», som nu blir en så populär genre bland både författare och läsare, syftade till att tränga »innanför empirin» och på så sätt skärpa »förståelsen för verkligheten». Också Odd Uhrboms fotografier vill bidra till detta. Hans gruvarbetare ser genomgående dystra och sammanbitna ut, låt vara att ett foto faktiskt visar en leende gruvarbetardotter. I det fallet har fotografens instruktioner kanske varit bristfälliga.
Några av Sara Lidmans direkta kommentarer bidrar till intrycket av gruvarbetarnas helvete och dess samband med andra lidanden. Planer på framtida processtyrd teknik som skulle kunna delvis ta över gruvarbetarnas hårda arbete hälsas inte av Lidman som en möjlig befrielse utan förses med den ironiska rubriken »Personalfrågans slutliga lösning». Avsikten är uppenbarligen att framkalla en jämförelse med Förintelsen. På ett annat ställe parallellställs två uttalanden. Det ena kommer från LKAB:s verkställande direktör och handlar om planerade investeringar. Det andra kommer från Frantz Fanons _Jordens fördömda_ och handlar om byggande av en bro under socialistiska villkor. Åberopandet av Fanon bidrar till att framhålla det som också återkommer på många andra ställen i _Gruva_ , nämligen att de svenska gruvarbetarna är offer för ett globalt förtryck som drabbar också folket i Vietnam, Afrikas folk och alla andra folk.
Fanons _Jordens fördömda_ kom för övrigt ut just 1969 i svensk nyöversättning av Per-Olov Zennström med Jean-Paul Sartres berömda förord, i vilket han hyllar våldet: »Att döda en europé är att slå två flugor i en smäll, att samtidigt avskaffa förtryckaren och den förtryckte. Kvar blir en död och en fri man.»
I boken beskriver Fanon ett socialistiskt program för tredje världens fullständiga brytning med det onda Europa: »Men vi vet i dag med vilka lidanden mänskligheten fått betala för envar av den europeiska andens segrar. Nej, kamrater, vi är färdiga med att leka européer. Vi måste hitta på något nytt. Vi har alla möjligheter så länge vi inte apar efter Europa, bara vi inte blir besatta av tanken att hinna ifatt Europa.»
Fanon, som avled 1961 innan kriget i Algeriet avslutats, kunde inte ana att algerier och andra afrikanska och arabiska folk i fortsättningen mera skulle ägna sig åt att mörda varandra än åt att mörda européer. Inte heller att de i hundratusental skulle fly till Europa för att där finna skydd undan sina landsmän. Kärnan i det hela formulerade Sartre väl i sitt förord när han manade läsaren av _Jordens fördömda_ : »Tag dig mod att läsa den! I första hand därför att du då kommer att skämmas, och därför att skamkänslan enligt Marx är en revolutionär känsla.»
Detta är på flera sätt konstigt. Internationalen hade manat »jordens fördömda» (eller »trälar uti alla stater») att resa sig mot ett förtryck som förutsattes drabba dem själva. För Marx hade vreden långt mera än skamkänslan varit en revolutionär känsla. Men den nya vänstern drevs i hög grad just av skamkänslan och dess ideologi kan i långa stycken te sig som en sekulär utgåva av syndabekännelsens »Jag, fattig syndig människa...». Man skämdes över att vara europé eller vit och förtrycka de färgade folken. Eller över att vara medelklass och förtrycka arbetarna. Eller över att vara man och förtrycka kvinnorna. Eller över att vara människa och förtrycka djuren. Eller över att vara en tärande varelse och förtrycka naturen, jorden och miljön. »Jag skäms över att existera», hade Sartre skrivit i sitt existentialistiska mästerverk _Varat och Intet_. När han blev marxist ersatte han denna existentiella skam med en politisk. Han vann gehör. Det stycke som citerats anfördes på omslaget av 1969 års svenska utgåva (Rabén & Sjögrens Temaserie) av _Jordens fördömda_. Den nya vänstern var i långt högre grad än den gamla ett uttryck för de välbärgades dåliga samvete, för samma känslor som förr tog sig uttryck i välgörenhet eller stöd till Afrikamissionen, men som nu gavs utlopp i revolutionärt engagemang. I detta avseende är Göran Palm och Sara Lidman typiska. Men kanske inte vår näste författare, Jan Myrdal. I tidigare böcker som _Samtida bekännelser av en europeisk intellektuell_ (1964) hade visserligen även Myrdal ägnat sig åt att bekänna sin skuld och odla sitt dåliga samvete. Men sedan dess hade han vänt sig bort från det privata känslolivet.
### _1970 – ultrakommunismens år._
Gruvstrejken kom och gick. I september 1970 undertecknades ett avtal av alla i strejkdelegationen utom de två mest revolutionära, Harry Isaksson och Elof Luspa. Strejken var slut. Medan den pågick som bäst hade en av periodens tongivande poeter, Göran Sonnevi, skrivit att strejken inte handlat om pengar utan om »att bli människa», om skapandet av en ny människa:
Ingenting kan gå tillbaka till det gamla
Tillsammans har vi makt att upphäva makten
Nu hade dock allting gått tillbaka till det gamla. Några hade till och med insett att makten att upphäva makten inte fanns, åtminstone inte om man hade för avsikt att fortsätta bryta malm. Andra trodde tvärtom att man genom att stegra makten skulle kunna få den att försvinna definitivt. De menade att vägen till det klasslösa samhället gick genom proletatiatets diktatur.
Vilken bok borde man helst välja för att illustrera året 1970? Flera alternativ erbjuder sig. Där finns Maja Ekelöfs _Rapport från en skurhink_ , där författarinnan/städerskan varvar glimtar ur en förtryckt arbetarkvinnas vardag med revolutionära kommentarer till världsläget. Eller P. C. Jersilds uppmärksammade _Vi ses i Song My_ , en roman utgiven på det nybildade Författarförlaget, som kunde läsas som en berättelse om »maktmekanismerna i dagens samhälle» och som en satir över reformismens elände. Eller kanske Sven Delblancs ännu mer framgångsrika _Åminne. En berättelse från Sörmland_ , som enligt Karl Erik Lagerlöf i Göteborgs Handels- och Sjöfartstidning (GHT) med större kraft än något annat litterärt verk från senare år teckande »ihåligheten i vårt samhällssystem». Jersilds och Delblancs respektive böcker är exempel på att författare som egentligen hade en annan inriktning kunde svepas med av vänstervågen och åtminstone stundtals lägga ett offer på den nya trons altare. Men jag stannar ändå vid Gun Kessle och Jan Myrdals _Albansk utmaning_ , där Gun Kessle stod för fotografierna och Jan Myrdal för texten. Ett skäl för detta val är att Jan Myrdal tillsammans med Göran Palm och Sara Lidman tillhörde den trio av författare som kanske tydligare än några andra gick i spetsen för den nya vänstern, tillsammans bärande en sorts litterär röd fanborg. När i en senare tid Palm sackade efter och Lidman och Myrdal gick åt olika håll blev det tecken på att det inte längre fanns någon tät att följa. Men 1970 gick man ännu något så när i takt.
Ett annat skäl är att Jan Myrdal trots allt representerade något annat. Hos honom fanns till synes inget känslotänkande, bara analys och handlingsberedskap. Inte heller fanns det något osäkert eller ursäktande grävande i egna själstillstånd. Bara visshet och beslutsamhet. Inget år var detta tydligare än 1970. Detta var samtidigt ultrakommunismens år när de revolutionära sekterna förökade sig genom delning och tävlade om vem som stod längst till vänster. Sålunda bröt sig 1970 KFML(r) (Kommunistiska förbunder marxist-leninisterna – revolutionärerna) ur det Myrdal närstående KFML. KFML var en smula mer maoistiskt, KFML(r) en smula mer stalinistiskt. Myrdal var framför allt känd som en vän av Maos Kina. Men han framstod samtidigt i jämförelse med de mest yrvakna kulturrevolutionsmaoisterna som tydligare förankrad i Kominterntraditionen och som mer befryndad med stalinismen. Inte minst hans hyllning till Albanien vittnade om detta. Albanien var ett litet land med en stor ledare, Enver Hoxha, vars klargörande uttalanden i olika frågor Myrdal flitigt citerade i _Albansk utmaning_. Hoxha var vid tidpunkten Maos allierade. Men den som läser Hoxhas texter slås av deras fullkomligt okinesiska karaktär. Här är det Stalins språk, inte Maos, som talas. Det var ett språk som Myrdal fann sig väl tillrätta med – manligt, kraftfullt, övertygat och betvingande. Det hade hos Hoxha en smula träsmak, men den hade Myrdal förmågan att trolla bort med diskreta medel. Kominternspråket kunde inte användas för den löpande texten, men det tillhandahöll en hemlig kraftkälla och inspirerade till en sorts stenstil som kunde tas fram när Myrdal kom till refrängen, där alla borde klämma i. Ytterst härrörde säkerheten från vissheten om att ha historiens vind i ryggen. Historien behövde noga taget inte ens analyseras. Den analyserade sig själv och dess dom kunde inte överklagas. Myrdal hade bara uppdraget att uttala den, vid behov vädjande till Maos eller Hoxhas auktoritet.
_Albansk utmaning_ är en reseskildring och rapportbok i en genre som Jan Myrdal varit med om att uppfinna genom sin _Rapport från kinesisk by_ (1963), vilken förresten 1967 kommit ut i ny upplaga på PAN/Norstedts med nya fotografier av Gun Kessle. Redan där hade det handlat om reseintryck som givits en hård politisk vinkling. I _Albansk utmaning_ var denna politiska vinkling om möjligt ännu tydligare. Boken var dessutom mer av en introduktion till det skildrade landet, med inte minst långa historiska avsnitt. Dessa avsnitt bidrog dessutom till att göra det politiska budskapet mera utmejslat. Det handlade inte längre mer allmänt om bygget av socialismen/kommunismen i ett fattigt land som i _Rapport från kinesisk by_. Det handlade om hotet från »revisionismen» och från Sovjetunionen.
När först Kina och sedan Albanien bröt med Sovjetunionen och när KFML bröt sig ur VPK hade det handlat om just »revisionismen» och om den nya linje som Sovjetledarna efter Stalins död hade slagit in på. För Albaniens del var det fråga om både ett yttre och ett inre hot. Ett yttre hot från Jugoslavien och Sovjetunionen. Ett inre hot från den »revisionism» som när som helst kunde bryta ut, förgöra socialismen och föra landet in på en väg tillbaka till kapitalismen.
_Albansk utmaning_ är försedd med ett motto från lord Byron, där den engelske barden kallar Albanien »thou rugged nurse of savage men». En sorts romantisk förkärlek för det primitiva, det stolta, oberoende och vilda delade Jan Mydal med Byron. Han hade manifesterat den redan i _Kulturers korsväg_ , boken om Afghanistan från 1960. Liksom hos Byron kan sympati för bondementaliteten kombineras med sympati för bondebefolkningars kamp för nationell självständighet. I _Albansk utmaning_ skildras albanernas historia som en oavbruten kamp för just detta med nationalhjälten Skanderbeg (1400-talet) som centralgestalt. Enver Hoxha blir Skanderbegs rättmätige arvtagare. Men därmed förändras också tematiken inte så litet. Byrons hyllning till dem som slogs mot det osmanska imperiet och Myrdals hyllning till dem som bjuder supermakterna USA och Sovjetunionen spetsen har likheter, men framför allt skillnader. Hos Myrdal ingår hyllningen i en marxist-leninistisk (eller stalinistisk-maoistisk) berättelse.
Maoismen hade ingen hegemoni i den svenska 68-rörelsen. Men i Sverige (liksom i många andra länder) hade maoisterna en strategisk nyckelroll. KFML/SKP var ledande i den betydelsfulla Vietnamrörelsen och dominerade DFFG (De förenade FNL-grupperna). Maoisterna gick i spetsen när det gällde solidaritetsarbetet för Palestina. De hade genom Clartéförbundet från början haft initiativet i studentrörelsen och särskilt i kampen mot U68. De styrde viktiga tidskrifter som Clarté och FIB/Kulturfront. Myrdals betydelsefulla roll berodde inte bara på hans obestridliga litterära talang utan utspelades mot fonden av maoisternas hela organisatoriska styrka. För de rättrogna var han alltid Sveriges störste författare.
Skälen för maoisternas strategiska ställning var flera. Som de tyngst vägande kan framhållas:
1)Brytningen mellan Sovjetunionen och Kina hade öppnat möjligheten till en kritik mot Sovjetunionen och Moskvakommunismen »från vänster», till vad som sågs som ett nytt och fräscht revolutionärt alternativ, fjärran från all »revisionism».
2)Detta alternativ emanerade inte från Europa utan från »tredje världen» och föreföll därför särskilt obefläckat av Västerlandets försyndelser.
3)Kina manade till total kamp mot USA-imperialismen utan korrumperande tal om »fredlig samexistens» och det understödde – villkorslöst som det antogs – vietnameserna och förtryckta folk i deras revolutionära gerillakamp.
4)Den stora kulturrevolutionen på 1960-talet hade tillfört en ny och för studenter speciellt intressant dimension där det handlade om att göra uppror mot lärare, professorer och auktoriteter, mot gammal lärdom, mot byråkrati och föråldrade strukturer, mot allt som var gammalt (utom Mao Zedong själv).
5)Maoisterna var »ortodoxa», höll fast vid Lenins och Stalins tradition i en form av vänsterfundamentalism. De stod för en absolut vänster.
För Jan Myrdal fanns det 1970 två ljusgestaler på den internationella scenen, två ofelbara förvaltare av den revolutionära sanningen. Vid sidan av den store rorsmannen Mao Zedong fanns den lille rorsmannen Enver Hoxha.
I slutavsnittet av _Albansk utmaning_ skriver Myrdal: »Men att ta ställning mot revisionismen, att ta ställning mot hela den utveckling som nu kännetecknar Sovjetunionen och som visar sig inom alla områden från konstpolitik till ockupation av grannländer och samarbete med Förenta Staternas imperialism innebär ju inte bara att ta ställning mot något. Det innebär inte bara att vara 'ortodox'.»
Det handlade också om vad man bejakade.
Myrdal kommer in på det stöd Kina gav Albanien efter brytningen med Sovjetunionen. Han kommer ihåg diskussioner med kinesiska vänner:
Jag minns en kväll i Kunming i december 1962 när vi hade diskuterat Albanien. Efteråt talade Gun och jag med varandra. Vi tog en kvällspromenad genom Kunming för att i lugn och ro kunna diskutera utan att andra var närvarande. [---]
– Du vet, sade jag till Gun, Hoxha lät ju skjuta Xoxe och något rätt har väl Chrustjev.
När vi första gången kom till Albanien var vi inte bara okunniga, vi var fördomsfulla. Våra fördomar utgjorde en hel serie av historisk felinformation, kulturell isolering och sovjetiska propagandaklyschor. [---]
Vi reste i landet. Hörde, såg, diskuterade. Sedan började vi läsa. Vi återvände. Vi reste igen. Vi satt på nationalbiblioteket i Tirana (och man bar boktravar till mitt bord). Vi hade haft fel. Det var albanerna som hade haft rätt.
Tanken att Jan Myrdal vid något tillfälle skulle kunna ha haft fel kan te sig svindlande. Men han kunde ibland medge det i efterhand, för den retoriska effektens skull. Hur kom han till klarhet? Boktravarna på hans bord i Tirana bevisade åtskilligt. Men det fanns också högre auktoriteter. Mao. Och Hoxha själv, som strax därpå citeras för att vederlägga allt dumt som Chrusjtjov sagt om Stalin: »Stalin var en anspråkslös människa. Som marxist-leninist gjorde han en riktig bedömning av den roll som massorna och individen spelar.»
Personkulten, argumenterar Hoxha, hade i stället blåsts upp av Stalins underhuggare, desamma som senare skulle förråda socialismen och slå in på den revisionistiska vägen. Här hade Stalin inte vidtagit de nödvändiga åtgärderna, det måste till och med Hoxha medge: »I synnerhet om man tar hänsyn till att det stora rykte som Stalin vunnit genom sin kamp och sitt handlande och det obegränsade förtroende och den obegränsade kärlek partiet och folket hyste för honom hade varit tillräckliga för att han skulle ha kunnat utdela ett kännbart slag mot de byråkratiska element som hotade proletariatets diktatur.»
Stalins fel var sålunda att han inte skjutit tillräckligt många politbyråmedlemmar. Med tanke på Stalins imponerande meriter i just detta avseende kan Hoxhas beskyllning synas småaktig. Men rätt skall vara rätt. Stalin hade i viktiga fall försummat att utdela ett kännbart slag. Kontrasten mot Hoxha själv låg implicit i resonemanget. Han hade ju – för att nu bara nämna ett exempel – på sin tid skjutit Xoxe, före detta organisationsskreterare i Albaniens kommunistiska parti. Myrdal hade som vi sett enligt egen uppgift, vilseledd av sin kulturella isolering, haft invändningar mot detta. Numera hade han dock kommit på bättre tankar. _Albansk utmaning_ slutar i ljust tonläge: »Det som är så hoppfyllt med Albanien är att man klart ser och öppet diskuterar alla dessa möjligheter till en ond utveckling. Ty därigenom kan den övervinnas och Albanien gå vidare på revolutionens väg.»
Inom KFML och den svenska smågruppsvänstern i övrigt delade man dessa insikter. Det gällde att ständigt vara vaksam mot förrädare som när som helst kunde dyka upp inom organisationerna själva, ofta inom deras högsta ledning. Visserligen kunde man i Sverige inte arkebusera men man kunde utesluta dem ur organisationen. Metoden tillgreps ofta, inte minst inom KFML/SKP och Clarté.
Baksidestexten till _Albansk utmaning_ anslog något mildare tongångar. Det heter om Albanien: »Det var det fattigaste landet i Europa: nu går var fjärde alban i skola och landet industrialiseras snabbt. I Afrika, Asien och Latinamerika spelar det albanska exemplet stor roll. Dess sociala reformer, dess lokaliseringspolitik och dess könsrollsdiskussion är radikalare än i något skandinaviskt land.»
Om dessa omdömen skall uppfattas som utfärdade av Norstedts förlag eller om de helt och hållet står för Myrdals räkning framgår inte. Man vill förmoda en blandning av bådadera.
Baksidan framhöll också att Albanien blivit ett turistmål: »Turistströmmen ökar för varje år.» För unga revolutionära entusiaster i Sverige låg Albanien närmre än Kina. De hårt reglerade och övervakade resorna bjöd inte bara på bad, kulturminnen, landskap och goda viner utan även på en politisk upplevelse. Det var ju på sätt och vis framtiden man reste till, där allting på ett hänförande sätt var både mer primitivt och mycket mer avancerat och radikalt än i något skandinaviskt land. Myrdal/Kessles bok var på en gång resehandbok och politisk vägvisare.
### _1971 – den begynnande besvikelsens år._
Visst var 1971 ett radikalt år. Det var ju exempelvis det året som Suzanne Osten tillsammans med Margareta Garpe kom med kvinnopjäsen _Tjejsnack_. Det var där man framförde kvinnokampssången »ÅÅÅ tjejer, vi måste höja våra röster för att höras» som sedan sjöngs på varje kvinnorörelsemöte. Den nya kvinnorörelsen hade uppstått ur 68-rörelsen, till exempel genom Grupp 8 som bildats just 1968. Den bröt med den gamla kvinnorörelsen av Fredrika Bremer-typ just genom att vara röd, socialistisk och radikal. Den var en del av vänstern, men en dynamisk del. Ännu hade inte kvinnokampstjejerna överröstat alla andra, men deras växande roll inom vänstern innehöll både löften och hot. Löften om en revitalisering av projektet. Hot om en underminering eller åtminstone brist på fokusering. Den ännu latenta konflikten mellan feminism och marxism/kommunism skulle efterhand komma att bli tydligare.
Överhuvud var 1971 på många sätt ett självtvivlets år. En viktig formulering av den begynnande besvikelsen kom det året i Karl Vennbergs diktsamling _Sju ord på tunnelbanan_.
Karl Vennberg hade som kulturredaktör för Aftonbladet från början varit en av den nya vänsterns mest framträdande kritikerröster. I 1965 års stora Vietnamdebatt hade han spelat en ledande roll. Som kulturchef symboliserade han vid sidan av Dagens Nyheters Olof Lagercrantz den kultursidornas snabba omorientering mot vänster som var ett omdebatterat tidens tecken. Den nya typen av vänsterengagemang gav också hans poesi ny vitalitet. Hans diktsamling _Tillskrift_ från 1960 hade följts av en lång poetisk tystnad. Men 1971 kom _Sju ord på tunnelbanan_. I dikten »Decennieskifte» stod dess oftast citerade rader, rader som föreföll kunna vara 70-talets ingångsmotto: »Låt oss gå rakt på sak: 70-talets utfall kan bero av hur vänstern förstår att hantera sin besvikelse.»
Vilka skäl för besvikelse hade vänstern 1971? Det mest uppenbara skälet har berörts. Gruvstrejken i Malmfälten hade tagit slut utan att saker och ting för den skull hade blivit radikalt annorlunda. Andra strejker hade visserligen brutit ut, men fastän vänstern gjort det mesta möjliga av dem hade inga haft samma omfattning eller chockverkan som Kirunastrejken. Dessutom fanns det andra, mera subtila anledningar till besvikelse. Den revolutionära euforin från 1968–1969 var ett minne blott. Vänsterns former hade stelnat. Den kommunistiska smågruppsvänstern hade inneburit ett försök att ge den nya vänstern stadga och organisatorisk kraft. Men ultrakommunismen med dess inbördes stridande sekter hade också förskräckt många vänstersympatisörer genom sin dogmatism. En ytterligare diktsamling från året 1971 är belysande för läget. Jag tänker på Göran Palms _Varför har nätterna inga namn?_ Palm har själv skrivit om den i det biografiska lexikonet _Författaren själv_ : »Att ständigt avkrävas journalistiska barrikadinsatser präglade av 'den rätta linjen' blev dock neurotiserande i längden, se depparboken _Varför har nätterna inga namn?_ »
En väsentlig del av diktsamlingens tematik fångas i dikten »Marxistiskt dilemma»:
Natten är inte lätt att väcka till politisk insikt.
Det är svårt att tjäna folket när man sover.
Det har sina sidor att styra sina drömmar i revolutionär riktning.
Det är besvärligt att lägga sig på den rätta klasståndpunkten.
Måste vi alltså vaka dygnet runt?
I dikten »Stormötet» dyker en schäfer upp genom en fallucka i golvet och blandar sig i ett stormigt stormöte där de revolutionära fraserna haglar som projektiler i luften.
Dikten »25 invånare i Sverige» börjar:
Det finns inga genvägar.
Andersson Margret Aff.bitr. Folkungag. 164 Stockholm
Andersson Nils Arne Arrend. Prästgdn Odensala
Andersson Allan Bag.arb. Tre Kronors väg 3 Nacka
och slutar:
Andersson Paul Överfurir Frihetsv. 54 Jakobsberg
Det finns inga genvägar till folket.
Intog Göran Palm i _Varför har nätterna inga namn?_ en felaktig klasståndpunkt (småborgerlig)? Eller deppade han bara? Eller var inte det ena och det andra samma sak? Olika bedömare såg olika på detta. Säkert är att diktsamlingen markerade en kris, en kris hos författaren men också en begynnande kris hos den svenska vänstern. Ännu var denna kris emellertid inte värre än att den kunde övervinnas. Själv hade Palm haft en idé om hur. I _Författaren själv_ berättar han att han »praktiskt taget flydde till LM Ericssons verkstadsgolv». Det var en upplevelse som han med tiden dokumenterade i två rapportböcker – _Ett år på LM_ (1972) och _Bokslut från LM_ (1974). Den så kallade proletariseringen blev på en gång terapi och sökande efter nytt litterärt stoff.
En tredje text från 1971 – denna gång en roman – skall lyftas fram. Det handlar om Per Olov Enquists _Sekonden_ , som blev en av 1971 års mest uppmärksammade romaner.
Per Olov Enquist tillhörde – det har redan berörts – i likhet med Göran Palm, Sara Lidman och Jan Myrdal en betydligt äldre generation än de »40-talister» som bar upp den egentliga studentrevolten. I likhet med bland andra Palm, Lidman, Anders Ehnmark, Sven Delblanc och Lars Gustafsson hade han studerat i Uppsala. Själv har han sagt: »Jag tvivlar på att vår litteratur särskilt ofta fått uppleva en så extremt intellektuell och sammansvetsad litterär grupp författare som de som fanns i Uppsala under sextiotalets förra hälft.» De hade vandrat samma vägar från 50-talets Uppsala, präglat av Tingsten/Hedenius-epoken med dess kombination av kulturradikalism och avståndstagande från kommunismen, till det tidiga 60-talets begynnande vänsterradikalisering. Lars Gustafsson skulle senare komma att skildra miljön i romanen _Sorgemusik för frimurare_ (1983). Enquist hade debuterat 1961 med romanen _Kristallögat_. År 1968 hade han haft stor framgång med _Legionärerna_ – en roman om baltutlämningen, som även blev en framgångsrik och omdebatterad film. Dess budskap var, som Åke Lundqvist noterat, glasklart. Sverige hade gjort rätt i att utlämna de baltiska flyktingarna till Stalin: »Utlämningsbeslutet var riktigt, den hysteriska opinionen mot utlämningen var centralstyrd och manipulerad, uppgifterna om de rysliga straff som väntade balterna vid återkomsten till Sovjet var grovt överdrivna, det var den uppiskade opinionsstormen snarare än själva utlämningsbeslutet som drev flera balter till självmord och självstympning.»
Att bli utlämnad till Stalin borde enligt denna analys vara ett ringa bekymmer jämfört med att vara utlämnad åt den svenska borgerliga pressen.
Som Lundqvist också noterar dolde den »dokumentära» formen flera fiktiva inslag. Redovisandet av motsägelser och komplikationer var delvis »ett retoriskt grepp, avsett att inge läsaren förtroende».
Men när Enquist kommit fram till _Sekonden_ hade motsägelserna och komplikationerna ätit sig in i honom själv. Huvudperson i boken, tillika berättarjaget, är Christian Lindner. Hans pappa, med förebild i verkligheten, var släggkastare som ung och vann på sin tid framgång genom att vid kast använda en urholkad slägga, fyra hekto för lätt. Bedrägeriet avslöjades, det hela blev en omskriven storskandal. Själv är Christian administratör. Han är socialdemokrat och socialist. Men han lever i en tid när socialismen komprometterats bland annat genom inmarschen i Tjeckoslovakien i augusti 1968, samtidigt med att romanen tar sin början. Dock är DDR ett ideal, men ett tvetydigt ideal. Christian har tidigare förälskat sig i Gisela, en idrottsflicka från DDR. Hon erbjuder en dröm om befrielse och gemenskap. När hon sjunger en socialistisk sång »Du hast ja ein Ziel vor den Augen / damit du in der Welt dich nicht irrst...», ser Christian på henne med blandade känslor: »Och jag avundades och misstrodde samtidigt hennes renhet, övertygelse och hetta: som en röd kil slog hennes ord in i mitt pansar.»
Kan Gisela tina upp Christians känslomässiga vinter? Kan hon ge honom del av de goda och ädla känslornas förförelse? Men det lurar något farligt i denna typ av känslosvall. Jan Stenkvist har kommenterat detta i sin studie av _Sekonden_ i _Flykt och motstånd. Fyra studier i politisk dikt_ (1978): »Här har vi alltså känslofloden, den ljuva och mäktiga, som kan utnyttjas för vilket ändamål som helst, till att lyfta fram undergörande magnetisörer eller extaser av politisk eller religiös art.»
I väst och öst dyrkar man enligt _Sekonden_ samma falska gud – utvecklingen. Det är denna dyrkan som leder oss alla vilse:
Vilken utveckling då? Är den ekonomiska utvecklingen ofrånkomlig? Är den utvecklingen? Är det den som den stora tävlingen mellan det kapitalistiska och kommunistiska systemet egentligen gäller? Är det så? [---]
Nya rekord var på gång. Utvecklingen stod ej att hejda, och den tredje tävlanden, den fulltoniga socialistiska människan, måste på ett tidigt stadium utgå ur tävlingen på grund av mjälthugg och andnöd, varpå slutstriden står mellan den kapitalistiska och kommunistiska teknologiska utvecklingen. Nu går det framåt
_Sekonden_ blev en bok där självtvivlen och ironierna gröpte ur alla övertygelser. Hade 68-vänstern – och dess mest uppskattade författare – vunnit sina tidiga segrar med en urholkad slägga? Det blev den fråga som hängde kvar i luften. Än var inte allt förlorat. Marschriktningen skulle kanske en dag komma att stå klar. Men som Christian säger: »Gud skall veta att jag inte är säker på min sak.»
Skulle de följande åren medföra större klarhet? Vi skall se.
### _1972 – det vaknande folkets år._
År 1972 var Per Gahrton livligt verksam på den litterära fronten. Under året hade han givit ut _Upp till kamp – pensionärer!_ och _Kan folkpartiet spela någon roll?_ Många läsare torde ha darrat. Hade nu revolutionen spridit sig till pensionärerna och till och med till Folkpartiet så var kapitalismen onekligen illa ute. Så kom, liksom för att göra slag i saken, Gahrtons _Revolution på svenska_ (Författarförlaget).
Gahrton gick till storms mot det rådande kapitalistiska systemet. Kapitalismen alstrar tillväxt som i sin tur i Sverige i icke ringa utsträckning går till välfärd. Men just tillväxten är bekymret. Liksom Enquist störs Gahrton av »utvecklingen». Utan tillväxten och allt det elände den för med sig skulle kanske ingen större välfärdsapparat behövas. Tillväxten slår ut människorna, stressar ihjäl dem och skapar på så vis vårdbehov. Konklusionen ligger mer eller mindre i öppen dag: »Det går inte att exakt i siffror visa förhållandet mellan produktionens företagsekonomiska vinst och sociala och mänskliga förlust; och det går inte att i tiden exakt fastställa när den totala produktionen i Sverige nådde en sådan takt att alla eller många hastighetsökningar får en negativ totaleffekt.» Ökad ekonomisk tillväxt är inte önskvärd »i varje fall inte förrän alla de baksidor som nu plågar människorna har blivit undanröjda». Välfärden måste mätas efter helt andra normer.
Slutsats: 'Välfärdskapitalismen' är varken möjlig eller önskvärd.
På så sätt står den svenska blandekonomin vid skärningspunkten. Det går inte att fortsätta som hittills med socialdemokratiska styrningsförsök; och det går inte att återvända till en mer kapitalistisk blandning. Det är nödvändigt att försöka hitta en helt ny väg mot framtiden.
Den enda fråga som återstår är: Vem skall göra det? Vem skall genomföra revolutionen på svenska? Den utomparlamentariska vänstern har visat sin oduglighet, menar Gahrton. De etablerade riksdagspartierna likaså. Samtidigt finns å andra sidan »tusentals öppna radikaler som stöts bort såväl av vänsterns fanatiska sekterism som av riksdagspartiernas samhällsbevarande maktstrukturer. Man kan lugnt konstatera: rollen som förtrupp för en revolution på svenska väntar på sin aktör».
Den som läste Gahrtons efterord till boken behövde inte tveka om var denne aktör fanns: »Min adress är: Östra Mårtensgatan 16, 223 61 LUND.» Läsaren uppmanades ta kontakt: »Aktivera Dej politiskt!» och »om Du någonsin skall kunna få makt över Din egen tillvaro måste Du själv medverka till att skapa nya förutsättningar. [---] Väl mött på 'barrikaderna'.»
Generalen på Östra Mårtensgatan i Lund väntade på att trupperna skulle höra av sig. Det märkliga är att de till sist gjorde det. De hade dessutom ett program. De ville bilda Miljöpartiet som Gahrton skulle leda.
Med tanke på detta är det märkligt att _Revolution på svenska_ knappast innehåller ett ord om miljöfrågan. Slutsatsen att tillväxten måste avskaffas vilar på helt andra argument. Det som med tiden skulle bli själva huvudargumentet – att tillväxten är till skada för miljön – tillkom i efterhand. Miljöideologin var redan i sina grunddrag utvecklad 1972, delvis inom ramarna för 68-rörelsen, men den måste smältas samman med Gahrtons vision innan Miljöpartiet kunde uppstå.
Om _Revolution på svenska_ skriver Gahrton i _Författaren själv_ att den innehöll »en alternativ samhällsmodell som förebådar G:s brytning med folkpartiet». Det visade sig att avskaffandet av välfärdskapitalismen trots allt inte rymdes inom Folkpartiets ramar. Först i boken _Det behövs ett framtidsparti_ (1980) hade Gahrton tagit ett avgörande nytt steg på vägen och »lanserade idén om ett nytt ekologiskt parti».
Gahrton var för övrigt verksam som debattör också på andra fält och framförde även där åsikter som knappast hade någon verklig framtid inom Folkpartiet. Hans _Kampen om Palestina_ (1970) utvecklade en radikal kritik mot Israel och sionismen. Sin expertis i fråga om Mellanöstern skulle Gahrton senare demonstrera också i _Egypten – arabisk demokrati_ (1987), enligt _Författaren själv_ »ett av G:s mest genomarbetade arbeten, en historisk-sociologisk analys av Egypten från Nasser-revolutionen 1952 till Mubaraks demokratiexperiment».
Per Gahrton tillhörde 68-vänsterns kärngeneration. Hans utveckling är ett vittnesbörd om denna vänsters styrka, men också om dess svaghetern. Som ledare för FPU (Folkpartiets Ungdomsförbund) på sextiotalet blev Gahrton den främsta symbolen för den snabba radikalisering som ägde rum bland unga folkpartister, sålunda bland från början borgerligt inställda ungdomar. När han i _Revolution på svenska_ talade om en revolution som skulle avskaffa kapitalismen visade det onekligen i hur hög grad vänsterns jargong kunde attrahera beundrare även på andra sidan mittfåran i svensk politik. Samtidigt tar Gahrton redan här avstånd från vänsterns »fanatiska sekterism». När han sedan frammanar en vision om »det vaknande folket» är det inte den traditionella arbetarklassen han tänker på utan exempelvis »anstaltsfolket» (framträdande på scenen bland annat vid »tjyvriksdagen» i Strömstad 1966), pensionärerna, de arbetslösa ungdomarna, studenterna, de utslagna, låglönegrupperna. Kanske oroad av känslan att en koalition mellan exempelvis pensionärerna och de kriminella skulle kunna visa sig instabil avstår Gahrton dock från alltför skarpa preciseringar. Han har redan nu en taktiskt riktig känsla av att det skulle gå att uppfånga missnöje på en gång från många olika håll. Revolutionen var nära. Men den behövde inte nödvändigtvis bli röd.
Andra exempel på vänsterlitteratur från 1972 och på Författarförlagets utgivning kan nämnas. Där fanns till exempel den lilla boken _Klartext. Marxistisk litteraturkritik_ , utgiven av Kritiska seminariet. Här handlade det om en produkt av den akademiska vänster, i detta fall litteraturvetare från Göteborg, som dessa år var flitigt verksam. Bokens förord betonade just detta. Debatten om universitetsreformen UKAS hade skapat en progressiv svensk studentrörelse, menade man. Uppsvinget 1968 blev visserligen kortvarigt. Men vid sidan av intressekampen »växte en ideologikritik fram på universiteten, baserad på studier i marxism. På både samhällsvetenskapliga och humanistiska fakulteter började grupper av studerande att kritisera gängse teoribildningar och vetenskapsideal som systemimmanenta och borgerliga. 'Kritiska seminarier' etablerades vid sidan av de lojalt nationalekonomiska, sociologiska, litteraturvetenskapliga. 'Kritiska studier' påvisade hur heliga femtitalsbegrepp som vetenskap, objektivitet, oavhängighet dolde fundamentala värderingar och ideologibindningar.»
Denna rörelse, som från 1968 fick sitt viktigaste organ i Häften för Kritiska Studier skulle med tiden komma att få stor framgång och bidra till en vänsterideologisering av stora delar av svensk samhälls- och humanvetenskap, en vänsterideologisering som till stor del fortfarande består. Universiteten blev en bastion för 68-vänstern och dess följdfenomen. Därvid omvärderades också själva vetenskapsbegreppet. Kraven på vetenskap, objektivitet och forskningens oavhängighet uppfattades nu som »systemimmanenta och borgerliga». Det gällde att ställa forskning och undervisning i arbetarklassens (eller om man så vill i vänsterns) tjänst.
De specifika studier som Göteborgslitteraturvetarna lade fram i _Klartext_ har mindre räckvidd än själva den principiella vetenskapssynen. Intressantast är kanske studien »Litteraturkritik i borgarklassens tjänst» som »är en undersökning av hur litterära verks politiska innehåll tenderar att smusslas undan av en estetiserande kritikerkår, som är mer skolad i att se berättartekniska grepp än politiska ställningstaganden». Det kan väl tilläggas att en ny kritikerkår just var i färd med att växa fram och ta över, i vissa fall skolad just vid de »kritiska seminarierna», en kår av kritiker som skulle göra just det politiska ställningstagandet till det avgörande kriteriet på god litteratur.
Antologin _Klartext_ andades idel marxistisk förvissning. Tveksamhet kännetecknade däremot den flitige Göran Palm som 1972 kom ut med sin _Ett år på LM_. Palms tveksamhet var ett litterärt karaktärsdrag som funnits med i hans författarskap från begynnelsen. Men i den nya boken kunde en del av denna tveksamhet underbyggas genom de erfarenheter som författaren gjort under ett år på verkstadsgolvet och som han nu redovisade. Den viktigaste iakttagelsen gällde kanske själva arbetarklassen, den som Palm hade tagit anställning på LM för att stifta bekantskap med, den klass som skulle genomföra revolutionen enligt de marxistiska handböckerna. Palm gjorde en obehaglig upptäckt beträffande denna klass – den fanns inte.
Stående ute på Telefonplan i Hägersten under lunchrasten försökte Palm urskilja vilka av de flanerande LM-arna som var arbetare och vilka som var tjänstemän. Han fann att det var omöjligt: »För ett tag var det som om hela den arbetarklass jag kommit till LM för att se i verksamhet plötsligt hade försvunnit genom en fallucka och bara lämnat en brokig samling individer kvar.»
Palm försöker bemanna sig och övervinna detta intryck. Men det han lägger märke till är ofta nedslående. Den revolutionära andan finns bland vänsterungdomarna som står utanför fabriksgrindarna och delar ut flygblad. Men inte inne på arbetsplatsen. Kontrasten befanns vara »enorm mellan de vältaliga flygblad som delades ut vid porten på mornarna och de fåordiga kommentarer som ägnades dessa flygblad inne på fabriken».
Palm tvivlade inte alls på de skildringar av de förtryckta arbetarna som Sara Lidman och andra signerat. Men han tvingades för egen del med någon besvikelse konstatera: »I övrigt var min arbetsmiljö skäligen hygglig.»
Romaner med politiska ställningstaganden av den typ som borgerliga kritiker enligt _Klartext_ inte begrep sig på fortsatte att komma ut. År 1972 publicerades exempelvis Lars Ardelius _Kronprinsarna_ som var en sorts politisk experimentroman.
_Kronprinsarna_ handlar om ett par enäggstvillingar, Arne och Bengt, som tidigt blir föräldralösa och därför placeras i var sitt fosterhem. Arne hamnar hos en fin läkarfamilj, blir akademiker, fördelaktigt gift och villaägare. Bengt blir arbetare och alkoholist. Allting beror på de sociala villkoren, tycks Ardelius mena, ingenting på det biologiska arvet. Människan är vad hon matas med, vad samhället gör henne till. Det var en av vänsterns trossatser. I denna roman föreföll Ardelius höra till dess anhängare.
Skulle framtidens människor kunna formas genom radikalt förändrade samhälleliga omständigheter? Det återstod att se.
### _1973 – brödratvisternas år._
När den kommunistiska världsrörelsen under 1960-talet splittrades i en sovjettrogen och en kinatrogen del hade detta som berörts påverkat också det svenska kommunistpartiet. När KFML 1967 bröt sig ur VPK blev detta början till en »smågruppsvänster», en flora av organisationer som aspirerade på att utgöra kärnan i ett nytt, icke-revisionistiskt kommunistparti. Under 1968 utvecklades den s.k. rebellrörelsen med ultramaoistiskt program ur Clartéföreningarna i Stockholm och Uppsala. Den uttömde snart sina krafter på vapenövningar, Maodyrkan och revolutionära handgemäng i studentkorridorerna. Men år 1970 lämnade en framför allt Göteborgsförankrad grupp med tidigare vice ordförande Frank Baude i spetsen KFML och bildade KFML(r), en organisation som skulle få längre varaktighet. Olika syn på taktiken i förhållande till Kirunastrejken hade varit den ursprungliga anledningen till brytningen. Schematiskt kunde KFML, med sin tidning Gnistan, betecknas som mer maoistiskt och Kinatroget, KFML(r), med sin tidning Proletären, som mer stalinistiskt. I januari 1971 bildades det trotskistiska Revolutionära Marxisters Förbund (RMF), 1975 ombildat till Kommunistiska Arbetarförbundet (KAF). Vänsterns Ungdomsförbund (VUF) bröt sig 1970 ur moderpartiet VPK, men bara för att självt splittras. En majoritet slog in på en maoistisk väg och blev Marxist-leninistiska kampförbundet (MLK), karaktäriserat genom den hyllning man ständigt ägnade minnet av Set Persson, som på sin tid fört kampen mot revisionismen inom SKP. Minoriteten trädde ut ur MLK och bildade Förbundet Kommunist (FK). Artrikedomen nådde en kulmen 1977 när den Moskvatrogna gruppen kring tidningen Norrskensflamman bröt sig ur VPK och bildade Arbetarpartiet Kommunisterna (APK). Alla dessa grupper ägnade sig åt en livlig inbördes polemik, präglad av förhållandet att de alla var varandras dödsfiender samtidigt som de delade samma kommunistiska ideologi och samma lärofäder i form av Marx, Engels och Lenin. Alla grupperna försökte i största möjliga utsträckning ge ut egna tidskrifter och tidningar, broschyrer och böcker, detta genom egna förlag och tryckerier. En avsevärd subkultur som innefattade tidnings- och bulletinförsäljning, bokhandlar och bokkaféer florerade, till stor del med FNL-rörelsen som ursprungligt mönster.
Vid sidan av synen på den internationella politiken och frågan om vilka kommunistiska stater man borde se upp till var det frågan om demokratin som skilde de olika organisationerna åt. Kunde kommunister acceptera borgerlig demokrati med yttrandefrihet, organisationsfrihet, flerpartisystem, fria allmänna val, riksdagens makt? Eller borde detta ersättas av proletariatets diktatur i Lenins mening? »Den vetenskapliga termen 'diktatur'», hade Lenin framhållit, »betyder ingenting annat än auktoritet ohindrad av lagar, absolut obegränsad av några som helst regler och baserad omedelbart på våld.» Lenin hade också sagt: »Marxist är endast den, som utsträcker erkännandet av klasskampen till erkännandet av proletariatets diktatur.»
Det kan förefalla som om Lenin uttryckt sig med all önskvärd klarhet. Inte heller hade hans praktik i något avseende avvikit från vad han rent vetenskapligt yttrat om proletariatets diktatur. Ändå gav dessa frågor upphov till oändliga diskussioner inom den svenska vänstern, först inom SKP/VPK, därefter, sedan KFML brutit sig loss, inom den allt rikare förgrenade smågruppsvänstern. VPK:s nya program 1972, där eftergifter gjordes för den radikala 68-tendensen inom partiet, och valrörelsen 1973, gav ny fart åt debatten. Några inlägg från denna period skall beröras.
KFML(r):s broschyr _Vänd VPK ryggen!_ , skriven av Sigvard Åkervall, kom i sin andra upplaga i april 1973, utgiven av förlaget Proletärkultur i Göteborg. Redan omslagsbilden är talande. VPK:s ordförande, C. H. Hermansson, klamrar sig hukande fast vid sin talarstol. Han ansiktsuttryck är lidande och sammanbitet. Förbi honom rör sig, vändande honom ryggen, en till synes aldrig sinande ström av arbetare. Uppenbarligen är de på väg att ansluta sig till KFML(r).
Förordet till den nya upplagan av _Vänd VPK ryggen!_ framhåller det uppenbara, nämligen att VPK visserligen under sin 23:e kongress 1972 och genom det då antagna programmet tyckts slå in på den revolutionära vägen, men allt eftersom valrörelsen 1973 närmat sig återvänt till den parlamentariska demokratin. Flirten med vad KFML(r) kallar »småborgarvänstern», till exempel inom Vietnamrörelsen, var då avslutad. VPK:s politik beskrivs som hycklande, ja, med Stalins ord som »en väg till hjärnornas mörkläggning». Mot den vetenskapliga marxismen »väger de vänsterpartistiska fraserna lika lätt som medicinmannens besvärjelser mot läkarvetenskapen».
_Vänd VPK ryggen!_ beskriver på ett inte alldeles missvisande sätt VPK:s plågsamma och slingrande väg åren 1967–1972. Ännu 1967 föreföll partiet vara på väg i »vänstersocialistisk» riktning, med andra ord på väg mot att fullt ut acceptera den parlamentariska demokratin och dess spelregler. Men denna tendens kolliderade med 68-vänsterns tendens i rakt motsatt riktning, som under intryck av studentradikaliseringen återvände till Lenin och den kommunistiska traditionen. Dess främste företrädare inom VPK blev Jörn Svensson, av KFML(r) alltid kallad »byrådirektören Jörn Svensson» (eftersom detta var Svenssons position före riksdagsmannaskapet). Svensson och hans anhängare hade vunnit vissa segrar när partiprogrammet diskuterades vid kongressen 1972, framför allt genom att den tidigare markerade anslutningen till demokratin nu luckrades upp och ersattes av oklara skrivningar som tillät tolkningar också i leninistisk riktning. Denna korrigering hade välkomnats av »Gnistangruppen» under ledning av »professor Gustafsson» som just 1973 sökte valsamarbete med VPK (ehuru utan att lyckas). Bo Gustafssons illusioner blir föremål för åtskilliga ironiska kommentarer i _Vänd VPK ryggen!_
Broschyren slår fast att den springande punkten helt enkelt gäller »det revolutionära våldet»: »I ett program för ett parti som är eller säger sig vara kommunistiskt finns det, och kan det finnas endast en kärnfråga och det är målsättningen för den proletära klasskampen. För kommunister är det och har det alltid varit den väpnade socialistiska revolutionen och proletariatets diktatur.»
Vad hjälpte det då att Jörn Svensson försökte blanda bort korten genom att hävda att man kan erkänna revolutionens »våldskaraktär» utan att förneka den fredliga vägen till socialismen? Svensson förespråkade »fredligt våld». KFML(r) fann det svårt att förstå hur ett sådant våld kunde vara beskaffat. Kanske fanns här någon dialektisk finess som undgått förbundet?
Vad KFML(r) saknade i fråga om dialektik ägde de i fråga om klarhet. Signaturen K.K. beskrev sålunda både klargörande och humoristiskt i förbundets tidskrift Klasskampen nr 1 1973 hur pressfriheten skulle komma att te sig i det socialistiska samhället:
Under proletariatets diktatur kommer det inte att råda någon allmän yttrande- och pressfrihet. Kapitalisterna kommer att mista hela sin propagandaapparat. Papperslager, tryckerier, tidningar och tidskrifter kommer att tillfalla staten. Bourgeoisin kommer att berövas friheten att torgföra sina lögner och fördomar.
Om bourgeoisin är ytterst svag medan proletariatet är starkt och enigt, kan man möjligen tänka sig att proletariatet som en ämabel gest skulle erbjuda bourgeoisin att skicka en och annan insändare, som starkt beskurna skulle publiceras under små käcka signaturer.
SKP publicerade för sin del 1973 broschyren _Vad vill VPK? Vad tänker dom på?_ (Bokförlaget Oktober, Göteborg). Huvudförfattare var Bo Gustafsson. Också han gjorde en återblick på VPK:s utveckling från brytningen med KFML/SKP och framåt. Lenins ord om erkännande av proletariatets diktatur som proberstenen för en äkta marxist återkom i Gustafssons framställning: »Om vi använder denna 'prövosten' på VPK:s program 1967 kommer det ohjälpligt till korta.»
Hade det blivit bättre i 1972 års program? Gustafsson erkände att det nya programmet »gjort vissa eftergifter för den marxist-leninistiska kritiken». I det nya programmet stod det tydligt att partiets grundval var »Marx och Lenins revolutionära teori». Men fullständig klarhet hade inte uppnåtts. Fortfarande spökade föreställningar om allmänna val och liknande även i 1972 års program. Helt hade man inte tillgodogjort sig Lenins ord att vägen till socialismen helt beror på »de väpnade folkmassornas omedelbara styrka».
MLK:s Kommunistisk Tidskrift framförde liknande synpunkter på VPK, dess program och dess utveckling, förtydligade genom hänvisningar till Set Perssons revolutionära kamp under 1940- och 1950-talen.
Hur utspelades då diskussionen om demokratin inom VPK självt? Den bästa redogörelsen finns i Håkan Holmbergs doktorsavhandling _Folkmakt, folkfront, folkdemokrati. De svenska kommunisterna och demokratifrågan 1943–1977_ (1982). Den kompletteras genom en annan doktorsavhandling från samma år – Kent Lindkvists _Program och parti. Principprogram och partiideologi inom den kommunistiska rörelsen i Sverige 1917–72_ (1982). Ett särskilt intresse tilldrar sig som berörts den position som företräddes av Jörn Svensson, den person i VPK:s ledning som stod 68-vänstern närmast. Sina uppfattningar kodifierade han 1974 i boken _Du skall ta ledningen och makten_. Också C. H. Hermansson åtnjöt betydande aktning bland »68:orna» under det att man allmänt tog avstånd både från den Moskvatrogna »Flammanfalangen» och från den vänstersocialistiska riktningen. Den »eurokommunistiska» tendens som var under utveckling i Sydeuropa hade ännu inte fått något genomslag inom det svenska partiet.
Enligt Jörn Svensson innebar den borgerliga demokratin ingenting annat än »en klassdiktatur med vissa demokratiska rättigheter för den behärskade klassen». Dess värde var sålunda inte så stort. Därför måste revolutionen också med nödvändighet få en våldsam karaktär. Riksdagens roll skulle närmast bli att i efterhand sanktionera kommunisternas åtgärder: »Parlamentet kan endast formellt bekräfta den revolutionära tvångsakten mot den gamla härskarklassen.»
Detta betyder inte att revolutionen nödvändigtvis behöver bli blodig. Borgarklassen kunde tänkas ge upp utan strid: »Självklart måste socialismens krafter äga tillgång till vapen, de segrar givetvis inte genom sin moraliska kraft. Men våldet, tvångsakten, kan utövas utan att vapnen behöver användas. Hotet är mer än tillräckligt.»
Detta blev »byrådirektör Svenssons» konklusion.
Termen »proletariatets diktatur» började nu åter bli populär inom VPK, närmast under tryck från nyvänstern. Men hur skulle denna diktatur komma att se ut? Hermansson gick i en intervju (Expressen 29.3.1973) så långt att han kunde tänka sig ett flerpartisystem, väl närmast efter modell från DDR: »Det starkaste partiet måste givetvis ha marxism som bas. Sen kan man tänka sig ett parti som företräder småborgarna och ett som företräder bönderna.»
Jörn Svensson sade inte emot, men var framför allt angelägen att inskärpa begränsningarna i den politiska friheten under socialismen. I det socialistiska Sverige skulle det »vara olagligt att väcka frågan om inskränkning eller raserande av det beslutssystem socialismen byggt upp och om ersättande av detta med ett beslutssystem i kapitalistiska former. Varje försök att rubba folkegendomen och arbetarsjälvstyret skall vara olagligt.»
Partier som drev propaganda mot socialismen skulle enligt Svensson helt enkelt inte tolereras utan mötas av »organisationsförbud».
De inom VPK som främst gick emot Jörn Svenssons ståndpunkter var den Moskvatrogna falangen. Den förre partiledaren Hilding Hagberg försvarade den svenska demokratin nästan lika envetet som han försvarade den sovjetiska diktaturen. Bakom detta fanns en taktisk linje. Det var sålunda enligt Hagberg helt riktigt att se nödvändigheten av proletariatets diktatur, men man borde inte göra väljarna nervösa genom att i onödan använda uttrycket. Här skilde han sig från 68-rörelsens nyleninister som fann stor tillfredsställelse i att använda uttryck som »proletariatets diktatur», »våldsmakt» och »väpnat uppror». En återgång till Leninepokens oförblommerade språkbruk utgjorde en inte oväsentlig del av denna vänsters avsikt och framtoning.
Brödratvisten mellan nyvänsterorganisationerna fick därmed sin speciella karaktär av ett gräl om vem som var »vänstrast». Motståndaren stod definitionsmässigt alltid till höger. Han företrädde alltid borgerlighetens (eller någon gång småborgerlighetens) intressen, sprang kapitalismens och imperialismens ärenden, förrådde revolutionen och socialismen. Mao Zedong hade talat om att »motsättningar inom folket» borde kunna få förekomma och lösas i vänskapliga former. Men i själva verket tenderade såväl i Maos Kina som i den svenska vänsterns vokabulär alla motsättningar att vara motsättningar gentemot klassfienden. Själva den marxist-leninistiska ideologin gjorde det svårt att tänka sig en meningsskiljaktighet som inte hade sin bas i skilda klasståndpunkter. Den ena parten måste alltid företräda proletariatet, den andra antagonistiska klassintressen. Tanken på ett socialistiskt land med ett genuint flerpartisystem blev därför en omöjlighet. Att något sådant land inte fanns på kartan var därför inte bara en empirisk tillfällighet utan begreppsmässigt oundvikligt. I den utanförståendes ögon kunde kampen mellan vänstergrupperna te sig som ett slagsmål mellan tvillingar eller åtminstone bröder. Men bröderna hette Kain och Abel, något som man kunde hämta kunskap om i världskommunismens historia.
### _1974 – den onödiga samtidens år._
År 1974 utkom en bok som föranledde en hel del debatt. Det var Jan Myrdals och Lars Gustafssons _Den onödiga samtiden_. Denna samtalsbok mellan en kommunist och en liberal var förvånande redan genom att innehålla ett sådant »omöjligt» tankeutbyte. Inte kunde en kommunist tala med en »borgare»? Å andra sidan hade Mao skakat hand med Nixon. Varför skulle Myrdal då inte kunna skaka hand med Gustafsson? Det höga föredömet i Peking höll uppenbarligen på att förändra något även i svensk debatt.
Men förvånande var också bokens titel. Att samtiden var onödig – vad skulle det kunna betyda?
Den som läst Jan Myrdals verk dittills måste ha fått intrycket att författaren alltid var i pakt med historien, att det var genom honom som historien talade. Det föreföll som om Myrdal så att säga alltid fick läsa historiens manuskript innan det publicerades och alltid sände det tillbaka med endast obetydliga rättelser. Men nu verkade han plötsligt ha refuserat hela texten. Så här skriver han bland annat i boken:
Ja, den som inte på en och samma gång är privilegierad, okunnig, dum och fantasilös måste – om han vågar se sig om – uppleva samtiden som ohygglig. Nu går denna ohygglighet att förklara. Karl Marx gjorde det. [---] revolutionen blev en verklig möjlighet i Europa vid mitten av förra århundradet. Industrikapitalismens utveckling hade skapat proletariatet och den 'fullkomligare samhällsordning' de stora utopisterna diskuterat slog därmed över från social kritik och politisk vision till politisk nödvändighet. [---] Det som är nödvändigt och verkligt möjligt har i och med sin nödvändiga och verkliga möjlighet dock inte blivit verklighet. Vi känner de senaste 125 årens ohyggliga historia. Om Marx hade rätt så var denna vår samtid icke nödvändig.
Om den kommunistiska revolutionen genomförts 1848 skulle vi ha sluppit samtiden. Hela historien under de senaste 125 åren har varit fel. På något sätt föreföll Myrdal ha gett upp hoppet:
Ty om samtiden är onödig så innebär det att det vid ett flertal tillfällen varit möjligt att fullfölja det nödvändiga 1848 i Europa. Men 1973 är inte något magiskt årtal. Ohyggligheterna har varit i tilltagande under de senaste 125 åren. Det kommande skedet av vår onödiga historia kan bli ett av sådana ohyggligheter att de redan utförda illdåden framstår som relativt oskyldiga.
Det bör påpekas att Myrdal inte när han talar om »de redan utförda illdåden» avser exempelvis Stalins eller Mao Zedongs illdåd. Inte minst Stalin anförs i texten som innehavare av djup visdom och klarsyn. Nej, det är kapitalismens illdåd Myrdal tänker på. Därvid tycks han medge den »ohyggliga» möjligheten att kapitalismen och den liberala demokratin skulle kunna tänkas bestå. Revolutionen ter sig inte oundviklig. Framtiden kommer inte att med absolut säkerhet medföra kommunismens seger. Det ohyggliga – revolutionens fortsatta uteblivande – kan komma att inträffa. Det som är »nödvändigt» kanske inte kommer att äga rum. Tydligen är något på väg att brista i Myrdals världsbild.
Lars Gustafssons svar på dessa tankar håller sig inom den vänskapliga konversationens ramar. Också han är uppenbarligen på väg från ett synsätt till ett annat. Gustafsson hade omfattat någon sorts socialism, som han nu var på väg bort ifrån. Men hans samhällskritik hade fortfarande i mycket en vänsterprägel. Dock är det tydligt att både Myrdal och Gustafsson distanserade sig från de gängse vänsterkretsarna. Myrdal skrev sålunda på ett ställe: »Både Gun och jag har ju sett hur vissa intellektuella miljöer i Sverige lever tätt intill varandra. Vi finner det mindre behagligt. Sådana människor vill vi inte umgås med.»
Fann Jan Myrdal stugan för trång? Var han på väg att avfalla från vänstern? Men i så fall bara till följd av ett storpolitiskt skeende. Hans position i svensk offentlighet berodde till stor del på hans speciella relation till Maos Kina. Sedan Nixons besök 1972 var kineserna uppenbarligen på väg någonstans. Men vart?
Någonting är på väg att brista i _Den onödiga samtiden_. Reaktionerna på boken speglade perplexitet. Olof Lagercrantz beskyllde i Dagens Nyheter (11/4) författarna för fikonspråk och läsarförakt. Sven Delblanc menade i samma tidning (13/6 och 21/6) att boken var ett intellektuellt pekoral och att de båda samtalsdeltagarna aldrig svarade någonting annat än »God dag – yxskaft» på varandras inlägg. Ett antal år senare skulle Delblanc dock i Svenska Dagbladet komma att göra gemensam sak med både Gustafsson och Myrdal i debatten om »mediavänstern». Fröet till den debatten fanns redan i _Den onödiga samtiden_.
Vart var historien på väg? Hade den redan kört förbi samtiden och lämnat denna att rosta sönder på ett stickspår? Var inte folket på väg att definitivt segra i Sydvietnam som av USA nu lämnats åt de nordvietnamesiska kommunisternas omsorger? Och om så var förhållandet, varför var Myrdal inte gladare? Kanske var alltsammans återigen som 1848, när ingen riktigt kunde bli klok på vad som var seger och vad som var nederlag?
### _1975 – kvinnornas och strömkantringens år._
Det var 1975 som kommunisterna – eller »folket» – defintivt segrade i Indokina. FNL och de nordvietnamesiska förbanden intog Saigon. De röda khmererna intog Phnom Penh. Detta år hade också av FN utsetts till »det internationella kvinnoåret». Kanske fanns något samband? Det tyckte åtminstonen två av vänsterns främsta kvinnokämpar, Maria Bergom-Larsson och Kajsa Ohrlander, som bidrog till Bonniers Litterära Magasin (BLM) med artikeln »Kvinnoåret 1975 – i vems intresse?»
Frågeställningen låg nära till hands. Den danska feministen Suzanne Brögger hade i en artikel i Aftonbladet (4/1) manat sina medsystrar att inte »låta oss kuvas av kvinnoåret». Olof Lagercrantz hade menat att »kvinnoåret» var ett försök att dra bort uppmärksamheten från svälten och krigen. Men Bergom-Larsson och Ohrlander ville nyansera debatten. De påpekade att FN numera hade blivit ett organ för de fattiga länderna: »Det kommer nu in länder för vilka orden och deklarationerna har djupare innebörd i verkligheten än de har för USA-imperialisterna.»
Med FN som forum kunde dessa länder nu föra ut sina problem, sin ideologi och sin kamp, såsom Arafats tal i New York föregående år hade visat. Det var i detta perspektiv man borde se kvinnofrågan. Bergom-Larsson och Ohrlander illustrerade sin tanke genom en talande jämförelse mellan kvinnornas situation i Nordvietnam och i Latinamerika. I Nordvietnam deltog kvinnorna aktivt i det produktiva arbetet. De hade »lika lön för lika arbete som sina manliga kamrater». De hade tillgång till regelbunden mödravård liksom till preventivmedel och aborter. De var väl representerade i de politiska organen. Gigantiska framsteg hade gjorts: »Att dessutom varken kvinnorna eller männen svälter, att de på lika villkor deltar i uppbyggnadsarbetet av landet för ekonomisk jämställdhet är naturligtvis den grund som kvinnorna står på och som betyder lika mycket för dem som för männen.»
Att även kvinnorna i Sydvietnam genom de nordvietnamesiska vapnens framgångar snart skulle få del av dessa välsignelser utsades inte men måste ha varit en glädjande tanke. Kontrasten var enorm mot förhållandena i Latinamerika där den amerikanska imperialismen höll kvinnorna i svält, förtryck och arbetslöshet. Mot denna bakgrund kunde man inte annat än hålla med Nguyen Thi Dinh, ordförande för de sydvietnamesiska kvinnornas befrielsefront, när hon påpekat att den antiimperialistiska kampen frigör kvinnan: »Exemplet från Vietnam visar att endast folkens egen kamp kan förändra den ekonomiska världsordningen. Endast i den kampen kan kvinnoförtrycket besegras.»
Slutsatsen kunde utan svårighet överföras även till svenska förhållanden: »De slutsatser man kan dra av ovanstående är väl helt enkelt att kampen för kvinnans frigörelse knappast går via kvinnoåret utan genom klasskampen.»
Ibland är det enkelt. Män och kvinnor hade uppenbarligen samma skäl att jubla över den stora folksegern i Vietnam senare den våren.
Men jublande verkligen alla? KFML(r):s Frank Baude tyckte sig med sina skarpa öron kunna uppfatta uteblivet jubel. Vaksam som alltid beslöt han sig för att slå larm. Tillsammans med förbundskamraten Lasse Carlsson skrev han broschyren _Borgerlig eller proletär politik – en kritisk granskning av SKP_. Skriften avslutades enligt förordet i december 1975 och kom i början av 1976 ut i regi av Proletärkultur. Vad Baude och Carlsson hade att förmäla i denna skrift var onekligen ganska allvarligt. Alla deras tidigare misstankar i fråga om SKP – och Gud skall veta att de hade haft misstankar – var nu på väg att bekräftas, nej överträffas. I samma förord hette det om avsikten bakom den lilla skriften: »Vår strävan är inte att ge en mängd färdiga scheman för den framtida kampen, men väl att gentemot den intellektuella högeropportunismens principlösa kryperi för bourgeoisin lyfta fram några grundläggande marxistleninistiska teser – framför allt den att tänka med eget huvud.»
Inte att ge färdiga scheman? Rent av att tänka med eget huvud? Fanns möjligen i sådana formuleringar en smygande ängslan att SKP:s kris också kunde visa sig vara KFML(r):s kris? I så fall doldes denna ängslan skickligt i det som följde.
Men låt oss gå direkt på huvudpunkten, som är följande: »Efter FNL:s och PRR:s [Republiken Sydvietnams provisoriska revolutionära regering] totala militära seger i södra Vietnam, har den svenska organisationen SKP totalt slutat med rapporter om det segerrika uppbygget i landet. Istället för ett stegrat solidaritetsarbete, vilket borde vara det naturliga när landet skall byggas upp igen, tiger man som muren i tidningen Gnistan, SKP:s organ.»
Samt vidare: »Detta får också direkta återverkningar på frontorganisationen DFFG i vilken SKP haft ett stort inflytande och denna organisation verkar nu dö sotdöden trots att arbetsuppgifter och solidaritetsuppgifter sannerligen inte behöver saknas.»
I segerns ögonblick sviker SKP. Det är vad man kunde vänta sig. Men ligger inte bakom detta ett långt mera förskräckande förhållande. I segerns ögonblick sviker Kina och dess »ärorika» parti, KKP. Baude och Carlsson gör allt i sin makt för att undvika den slutsatsen. Men gång på gång smyger sig misstankar på dem. De slår dem tillbaka, men de återkommer. De prisar Maos och KKP:s politik i det förgångna. De hävdar att SKP missuppfattat KKP:s nuvarande linje: »De har lyckats med att totalt förvränga KKP:s ståndpunkter.» Men ändå, ändå...
Det går inte riktigt att förneka att Mao och KKP numera betraktar den sovjetiska supermakten som ett större hot än den amerikanska, ja, att det i själva verket efter Nixons besök upprättats en sorts allians mellan Kina och USA. Det går inte riktigt att förneka att KKP betraktar alla krafter som kan stå emot Sovjetunionen som goda krafter. Men Baude och Carlsson säger aldrig detta rakt ut. De nöjer sig med att beskylla SKP för att flirta med Chile under Pinochet, med Iran under shahen, ja, med den svenska militären i dess krav på förstärkt försvarsbudget. Man vill inte påstå att »Kina skulle föra en felaktig utrikespolitik», däremot vill man påstå att de maoistiska grupperna (och i första hand Sveriges SKP) »faktiskt framstått som reaktionens vänstra flank i sina respektive hemländer». De knöt numera an till »det kalla krigets mest rabiata 'antisovjetism'».
Termen »antisovjetism», som annars mest användes av socialdemokrater, sattes visserligen inom citattecken. Men det är ändå tydligt att en viss uppluckring i KFML(r):s tidigare totalt avvisande hållning till Sovjetunionen var på väg.
En förändring mot en tydligare pro-sovjetisk hållning kunde man skönja även inom VPK och i synnerhet inom ungdomsförbundet, Kommunistisk Ungdom (KU), där denna stämning var stark. Man kände behov av att polemisera mot SKP:s tes att Sovjetunionen skulle vara en kapitalistisk och imperialistisk makt. Broschyren _Vart går SKP?_ , som KU gav ut 1975, bär vittne om detta. Försvaret av Sovjetunionen och SUKP mot SKP:s beskyllningar blev här det stora temat. I uppgörelsen med konkurrentpartiets »antisovjetism» gjordes visserligen vissa medgivanden. Det erkändes sålunda »att demokratin och kritiken underifrån ofta inte fungerade under Stalintiden» samt att Stalin rent av »bröt mot de demokratiska principer som borde styra partilivet». Allt hade numera blivit mycket bättre, men KU var även i fråga om nuläget beredd till medgivanden: »Ändå kan man inte säga att den socialistiska demokratin i Sovjetunionen är fullständig.»
I väntan på korrigering av de detaljer som ännu kunde återstå att förbättra i den sovjetiska demokratin slog KU emellertid fast att SKP:s syn på läget var helt felaktig: »De progressiva och socialistiska befrielserörelserna delar sällan denna syn på världsläget; därmed inte sagt att de inte kan ha kritik mot Sovjetunionens politik i vissa avseenden.»
Många förändringar var på väg 1975, inte bara inom VPK:s ungdomsförbund. Litteraturkritikern Karl Erik Lagerlöf gav det året ut sin bok _Strömkantringens år och andra essäer om den nya litteraturen_ (PAN/Norstedts). Med »strömkantringens år» avsåg Lagerlöf 1965–1967, när den nya vänstern gick segrande fram inom litteraturen. Han ville visa hur 1950-talets och det tidiga 1960-talets formexperiment fått träda tillbaka först för rapportböcker och politiska uppgörelser, sedan för en ny realism inom romankonsten. Han skrev: »Men när nymarxismen etablerat sig och dess tolkningar blivit gängse fick vi de nya regelrätt realistiska romanerna. Ty vi hade fått en ny 'verklighet'. Den nya realismen är en vänster-rörelse.»
Det blev den genom att tolka människan som en samhällsvarelse, som en skapelse av sociala och ekonomiska motsättningar. Man gjorde realism utifrån en bestämd ideologi, menade Lagerlöf: »Den har falska anspråk om ideologin har falska anspråk. Men få inom vänstern intresserar sig i dag för om nymarxismen innehåller den slutgiltiga sanningen om människan eller inte. Den är en modell som gör skeendet i världen begripligt, avläsbart.»
Det handlade om en pragmatisk sanning som samtidigt var en moralisk uppfordran till kamp mot det förtryck som strukturerna utövade: »Den realistiska romanen är utomordentligt väl ägnad att skapa en medvetenhet som i dag är viktigare än filosofisk skepsis.»
Hur menar Lagerlöf här? Spelar det ingen roll om den nya realismen har falska anspråk eller om den nymarxistiska ideologin har det, detta eftersom det viktiga inte är vad som är sant utan att skapa en ny medvetenhet? Men skulle en felaktig ideologi verkligen kunna göra skeendet i världen »begripligt» och »avläsbart»? Det menade Lagerlöf förmodligen inte. Men begripligheten var på något vis viktigare än »den slutgiltiga sanningen», det ville han ha sagt.
Alla var inte lika ståndaktiga som Karl Erik Lagerlöf. Hans term »strömkantringens år» skulle i själva verket kunna användas också om publiceringsåret för hans bok, 1975. Det året – eller någonstans därikring – föddes, kunde man mena, en ny filosofisk skepsis, en skepsis som till en del uppstod ur tvivel på marxismen, en skepsis som ibland skulle komma att kallas »postmodernistisk» och som bidrog till att underminera »den nya realismen». En ny litterär era var på väg, en era av formexperiment och av tveksamhet inför den nya »verklighet» som nyvänstern hade levererat. Här hade Lagerlöf halkat efter en aning.
Den nya strömkantringen hade mycket att göra med nymarxismens svårigheter att göra skeendet i världen begripligt. Någon sorts krångel tycktes ha uppstått med själva avläsbarheten. Ty det jublades inte som det borde under vänsterns segerår 1975. Med folkens seger i Indokina borde väl allting ha varit frid och fröjd? Särskilt som befrielserörelserna i Angola och Moçambique också gått segrande fram (med en smula hjälp från Kuba) och revolutionen gjort framsteg i Portugal. Men någonting fattades. Från det väldiga Kina hördes tystnad – i bästa fall. Från de svenska maoisterna, som en gång utgjort själva den hårda kärnan i 68-vänstern, hördes tystnad – i bästa fall. Tiden var ur led. Jan Myrdal hävdade att den spårat ur redan 1848. I vart fall någon gång senare.
Med segern i Vietnam var de goda åren slut. De svåra åren vidtog.
## **3. Sju svåra år**
### _1976 – terapiernas år._
Bonniers Litterära Magasin publicerade 1976 (1976/4) en författarenkät under rubriken »Halvvägs in i 70-talet – riktpunkter». Ett antal vänsterförfattare – och snart sagt alla författare var vid denna tidpunkt vänsterförfattare – fick lägga fingret på tidens puls och ge sitt utlåtande.
Sivar Arnér formulerade en dysterhet som knappast visste några gränser: »Engagemanget är mindre spektakulärt än för tio år sen; men världen har blivit än kusligare, och allt fler blir medvetna om det.»
Terrorbalansen var på väg att brista, kärnvapenkriget närmade sig, det var »kyliga vetenskapsmän» som sagt det. Välfärden blev allt skevare fördelad i världen. Den »lilla terrorismen» var, enligt Arnér, (Röda brigaderna?) en »mycket begriplig reaktion» på giganternas terrorism. Tilltron till de vanliga vänsterrecepten sviktade: »Men ingenstans står nån konstruktiv anvisning att upptäcka; även den som tror på socialismen har svårt att tro att den skall hinna före, hinna åstadkomma den nödvändiga nya ekonomiska världsordningen. Under tiden fortgår nedräkningen. [---] rädslan kramar allt värre.»
Författarens uppgift blir att försöka bemästra ångesten, att lägga en blomma på kistan innan den sjunker ned i graven.
Claes Engström var inte fullt lika pessimistisk. Hans minns 60-talet med glädje, »främst den då så vitt spridda optimismen». Men han hade inte givit upp nu. Den »demokratiska socialismen» var fortfarande vad han trodde på.
Jan Gudmundsson berättade om sin radikalisering på 1960-talet. Under 1970-talet hade han inte ändrat sig. Han skrev pjäser och romaner »utifrån en socialistisk grundsyn». Han trodde på politiken och på dikten.
Björn Håkanson diskuterade sitt vänsterengagemang i dess samband med författarrollen. Sedan lång tid tillbaka hade han avsvurit sig den »trolöshet» som han en gång i tiden blivit känd för att företräda. Vid mitten av 1960-talet hade han rent av beskyllts för att vara »engagemangsterrorist». Men med sin diktsamling _Mellan två val_ (1969) hade han nått en sorts slutpunkt: »Längre kunde jag inte gå som socialistisk tendensdiktare med bevarande av min konstnärliga identitet. Plakatpoet klarade jag inte av att bli, och jag ville det inte.» Världsutvecklingen var inte desto mindre förfärande: »En krypande fascism med rasistiska inslag runtom i Europa. [---] Klyftan mellan i-land och u-land ökar för varje dag. I Sverige vinner småborgerliga värderingar mark.» Det gällde att söka efter en motbild, ett utopiskt alternativ: »Min uppgift som socialistisk författare är att försöka ge socialistisk laddning åt denna motbild.»
Reidar Jönsson hade ungefär samma åsikt om författarens uppgift. Men kampen mot kapitalismen måste ske tillsammans med arbetarna: »Att ta upp kampen för vad som vunnits under det plötsliga andningshål som ändå inträffade under senare delen av 60-talet verkar vara den radikaliserade arbetarklassens uppgift.»
Kjell Sundberg hade också upplevt det sena 1960-talet, men inte uteslutande som ett andningshål: »Den enda vitala kritiken var den marxistiska. Dagsrecensionerna kunde vara frånstötande i sin dogmatism.» Inte desto mindre hade Sundberg dragit sina egna slutsatser av marxism och maoism. De handlade om gemenskap, som han ansåg vara »socialismens själva hjärteblod».
Hur det nu än må ha förhållit sig med den saken så lämnade BLM sista ordet till Werner Aspenström. Aspenström valde den lyriska formen för sitt inlägg där han konstaterade:
Hoppet står inte till framtiden
utan till förseningen.
Det blev debattens avslutning.
Hans Isaksson, BLM:s nytillträdde redaktör, ansåg sig kunna konstatera att »sextiotalet» definitivt var slut. Upphetsningen och förhoppningarna från 1968 var på upphällningen. Hos vissa författare hade de förbytts i nattsvart pessimism. Rädslan kramade.
Inom vänstern upplevde många som inte nödvändigtvis var kända författare en ny osäkerhet. Kanske behövde vänstern rent av gå i terapi? Det undrade två unga journalister, Jan Josefsson, 24, och Mats Zetterberg, 25, som 1976 gav ut intervjuboken _Behöver vänstern gå i terapi?_ (Forum).
I ett inledande avsnitt, avslutat i maj 1976, ställde Josefsson och Zetterberg frågan: »Vad händer i vänstern?» De skrev: »Vänstern har hamnat i en återvändsgränd. Uppsvinget och entusiasmen från slutet av sextiotalet har ersatts med nedgång, förtvivlan och uppgivenhet inför det nuvarande läget. Såväl mänskligt som politiskt står det illa till.»
En stor del av förändringen hade enligt författarna utspelats just på det »mänskliga» planet: »Engagemanget, upproret och känslosvallet från 1968 känns plötsligt avlägset. Mycket har hänt sedan den franske studentledaren Daniel Cohn-Bendit utropade: Ta ägg och gå till handling! Säg nej till allt. Finn ett nytt förhållande till dina vänner. Älska på ett annat sätt. Det är för dig själv du gör revolution, här och nu!»
Studentrevolten hade från början varit ett uttryck för ungdomar som velat förändra världen och göra upp med sin egen borgerliga uppväxt. Det var i mycket en kamp för personlig »frigörelse» hos en generation universitetsstudenter. Under den nya vänsterns första år stod arbetarklassen så gott som helt utanför: »Ett totalt moraliskt förkastande av det borgerliga samhället i kombination med en fullständig isolering från arbetarklassen skapade ett tomrum.»
Man försökte fylla detta tomrum genom att återanknyta till den kommunistiska traditionen, genom att skapa organisationer efter leninistisk modell. Men isoleringen bröts inte, menade Josefsson och Zetterberg:
Början av sjuttiotalet skulle bli en brytningstid. Det var då vänstern formerade sig i de grupper som finns idag. Men de hängde länge i luften utan några nämnvärda förgreningar och kontakter på arbetsplatserna. Insikten om detta tvingade vänstern att rikta sin politik ut mot proletära bostadsområden och fabriker – där arbetarna fanns. Det blev allt vanligare att radikala studenter tog anställning i industrin, vad man kallar proletarisering.
Denna utveckling ledde emellertid till hårda påfrestningar:
Det krävs utan tvivel mycket av den enskilde medlemmen i en vänsterorganisation. [---] Man skall krossa det nuvarande samhället men även kunna formulera ett alternativ i ett läge där allt färre vill hänvisa till Kina och Sovjet. Som aktiv skall man vara genomsyrad av politik och på kort tid kunna ta ställning i alla möjliga frågor, ofta utan egna erfarenheter. Ett vänsterengagemang är inte en hobby. Det påverkar hela ens framtid och villkor. Man får göra upp med mycket.
Det var till stor del genom den socialistiska kvinnokampen som förhållandet mellan person och politik kommit i fokus. Det var svårt att klara av det hårda politiska arbetet. Man kunde se att somliga som tidigare varit aktiva »nu sviker och i värsta fall t.o.m. försvinner in i psykoterapin». De som intresserade sig för personliga problem riskerade att stämplas som »flummiga». Samtalen inom grupperna kretsade ständigt kring politik, senaste nytt på strejkfronten, möten, demonstrationer och hur många flygblad som återstod att dela ut. Det gav upphov till frågor: »Är politiken ett medel att förverkliga sig själv? Är det människor som går in med den föreställningen som drabbas av problemen? Fanatismen och arbetsdisciplinen har otvivelaktigt skördat sina offer.»
Jan Myrdal hade vid telefonkontakt med Josefsson och Zetterberg hävdat att »de som passiviserats har fått borgerliga griller på hjärnan», att de var offer för falska föreställningar och borgerliga fördomar. Men författarna till _Behöver vänstern gå i terapi?_ höll inte med Myrdal om att de personliga problemen borde ses som ovidkommande: »Med denna bok vill vi visa att det inte är så.»
Deras konklusion efter de nio intervjuer med unga vänsteraktivister som utgör huvuddelen av boken gick i samma riktning: »Att vänstern börjar intressera sig för privat- och vardagslivet är inte bara nödvändigt för att ta sig ur den nuvarande nedgångsperioden utan överhuvudtaget för att socialismen någon gång skall kunna genomföras och förverkligas i människors liv.»
Frågan var bara om inte det nya intresset för»privat- och vardagslivet» inom vänstern i sig var ett symptom på »den nuvarande nedgångsperioden» och oskiljaktigt förknippat med denna. Skulle ens psykoterapi kunna hjälpa vänstern att övervinna sin molokenhet om hela dess politiska projekt gått i kras? Det var den underliggande frågan som bara skymtade i bakgrunden i _Behöver vänstern gå i terapi?_ Hur skulle man svara på frågorna om det socialistiska alternativet? Skulle man hänvisa till Kina eller till Sovjet? Eller skulle man nöja sig med att förbanna kapitalismen? Vad ville all denna vänsteraktivism och vänsterretorik egentligen åstadkomma? Tänk om arbetarklassen hade rätt när den ryckte på axlarna? Inte ens Jan Myrdal hade något riktigt övertygande svar på de frågorna. Ja, många inom vänstern började nog tycka att Myrdal och hans meningsfränder var en del av problemet snarare än en del av lösningen.
I september 1976 skärptes problemet. Då dog Mao Zedong. Hela världens nyvänster stod faderlös. Ty mer än någon annan gestalt på världsscenen hade Mao med sin kulturrevolution, sitt »Det är rätt att göra uppror» och sitt »Bombardera högkvarteret!» förkroppsligat den nya vänstern. Mao hade visserligen till synes själv förrått maoismen när han gjort slut på kulturrevolutionen och sedan slutit förbund med Nixon och USA. Men Mao var oberäknelig. Hur lätt skulle han inte ha kunnat lägga om rodret en gång till? Nu var han död. Ingen visste vad som skulle komma efter honom. Världen – och vänstern med den – höll andan.
### _1977 – omprövningarnas år._
Debatterna inom vänstern spretade åt olika håll 1977. Ett sätt att få en bild av läget är att gå till tidskrifterna.
Ord och Bild hade sedan 1890-talet varit Sveriges ledande tidskrift för litteratur och konst. Som man kunnat vänta hade den i likhet med BLM sedan slutet av 1960-talet tagits över av vänstern. Debattämnena sammanföll sedan länge i stor utsträckning med vänsterns debattämnen. Samhällsfrågorna upptog stort utrymme. Årgång 1977 var det i synnerhet två debatter som dominerade. Den ena handlade om Kina. Den andra om kvinnorörelsen.
Vart var Kina på väg efter Maos död och efter det att de ultraradikala ledarna i »de fyras gäng» arresterats? Var maoismen nu definitivt slut? Förestod en »avmaoisering» i Kina? En av bidragsgivarna till en enkät om dessa ämnen var Bo Gustafsson, länge ett av SKP:s främsta namn.
Gustafsson gjorde en öppenhjärtig återblick. Gång på gång hade utvecklingen i Kina försatt maoisterna, inte minst de svenska, i chocktillstånd. Egentligen hade det börjat redan 1971: »Det som skapat tveksamhet är ytterst Lin Biao-affären. Alla Kina-vänner betraktade det som en självklarhet att Lin Biao var Maos 'nära vapenbroder'.»
Hade det inte rent av skrivits in i KKP:s stadgar att denne vapenbroder skulle efterträda Mao? Men så kom plötsligt Lin Biaos påstådda statskupp och dramatiska död i september 1971. Det visade sig att Lin Biao planerat att mörda Mao. Vad skulle man tro på efter en sådan sak? Gustafsson drar för sin del slutsatsen att man borde vara försiktig med att tro på någonting alls. Man kunde inte utan vidare utgå från vad den kinesiska ledningen nu sa om »de fyras gäng». Alla beskyllningar var kanske inte sanna. Å andra sidan trodde även Gustafsson att Kina nu behövde avsluta »den period av ständig revolution, omprövning och delvis anarki som kulturrevolutionen inledde 1966». Slutsatsen blir ett särdeles garderat stöd till Maos efterträdare Hua Gufeng och hans linje: »Men även i det fallet bör man nog ha i åtanke, att en i och för sig riktig tendens alltid bär inom sig fröet till sin motsats.»
Detta var onekligen långt ifrån den självsäkerhet som Jan Myrdal uttalat i Expressen (26.12.1976): »'De fyras' nederlag är en av de viktigaste händelserna i Kinas historia sedan 1949. [---] Det är inte utan skäl som lättnaden över 'de fyras' nederlag är så stor över hela Asien.»
Det förvånande här var inte själva omdömena utan att de kom från just Jan Myrdal, vars förmåga att följa med i de snabba omkastningarna därmed triumferade över alla svårigheter.
Ett tidigare nummer av Ord och Bild detta år (2–3) hade handlat om »Vänstern i Skandinavien». Där gav Anu Mai Köll ett bidrag som också handlade om maoismens utveckling, men från en utomordentligt kritisk utgångspunkt. Köll var en FNL-aktivist som funnit det svårt att anpassa sig till förändringarna i den kinesiska utrikespolitiken.
Artikeln hette »Från anti-imperialistisk kamp till supermaktsteori». Den följde FNL-rörelsen från den euforiska begynnelsen till det dystra nu-läget. Från början hade allt varit jublande tillförsikt: »Vietnam, Guinea-Bissau, Cuba, alla de andra, gav oss aningar om en ny och bättre värld bortom kapitalism och Sovjet-kommunism, en värld där solidaritet och kollektivt arbete är grundstenar i samhället.»
Kampen i Vietnam hade nu gått till seger. Men själva solidaritetsrörelsen hade gått i stå. Det var SKP:s fel. KFML hade från början haft en stark ställning inom DFFG. Men på sistone hade man alltmera använt denna ställning för att sabotera arbetet. Detta hade gjorts i namn av den kinesiska »supermaktsteorin» som riktade kritiken mot i första hand Sovjetunionen. Därmed hade den ursprungliga linjen – stöd till FNL mot USA-imperialismen – fullständigt övergivits: »Efter 1974 har det varit FNL-grupperna som bytt ståndpunkt. För dem är USA-imperialismen inte längre huvudsak och därmed har sprickan i solidaritetsrörelsen blivit omöjlig att överbrygga.»
Sprickan i solidaritetsrörelsen var också en spricka i vänstern, som nu inte längre existerade i den gamla meningen. Omprövning stod på dagordningen. Vänsterns kvinnorörelse debatterades sålunda intensivt detta år. I samma nummer av Ord och Bild stod en artikel av Margot Bengtsson, »Det personliga är politiskt», som innehöll en personlig återblick på den vänsterradikala kvinnorörelsens utveckling. I början gick man från klarhet till klarhet: »Vi väcktes politiskt, liksom killarna vid universitetet, av Vietnamkriget och engagerade oss liksom de i studentrörelsen, som nådde sin höjdpunkt 1968. Det var där och i samband med våra studier som vi först började ana kvinnoförtrycket.»
Snart nog visade det sig emellertid att kvinnoförtrycket fanns också inom vänstern, och Kvinnoligan startade 1970 som reaktion på »den manligt dominerade vänstern i Lund». Inom kort bröt det emellertid ut en strid inom Kvinnoligan. Man konfronterades med representanter för Grupp 8 i Stockholm som ville att feminismen skulle underordnas den socialistiska kampen: »8:orna framstod på exakt samma sätt för oss som killarna inom vänstern: allvarstyngda, teorispäckade, människofientliga och handlingsförlamade.»
Kvinnoligan sprack och en mindre grupp kvinnor bröt sig ur och bildade Socialistiska kvinnogrupperna. Ibland kunde utvecklingen gå alldeles för långt åt andra hållet så att vänsterkvinnorna i stället för att likna männen utvecklade en beslutsam kvinnlighet: »Att klassas som mansidentifierad var det värsta man kunde råka ut för. Om många tidigare haft skuldkänslor för sina 'kvinnliga' sidor, utvecklade de flesta nu en rädsla för att på något sätt verka 'manliga'.»
Det blev fritt fram för att »pynta sig, gå i långkjolar och färgsprakande kläder, att sjunga, dansa och måla».
Ett problem var också att männen inom vänstern utvecklade »subtila förtrycksmekanismer». Det gjorde de genom att »de kastade sig över oss, höjde oss till skyarna och bokstavligen dränkte oss i förtroendeuppdrag – en och en». En mer separat kvinnoorganisering hade blivit nödvändig: »Jag upplever starkt behovet av en separat militant kvinnorörelse.»
Margot Bengtsson var medlem av redaktionen för den socialistiska tidskriften Zenit, ett av den nya vänsterns ledande teoretiska organ. Hon upplevde de manliga medlemmarna där på samma sätt som dessa torde ha upplevt de främsta feministerna, nämligen som »allvarstyngda, teorispäckade, människofientliga».
Bläddrar man i Zenits årgång för 1977 finner man ungefär samma debattämnen som i Ord och Bild. Det handlar om kvinnorörelsen, om Kinas utrikespolitik, om Sovjetunionen, om APK:s utbrytning ur VPK. Men den dominerande debatten det året startades av en artikel om demokratifrågan, Håkan Arvidssons och Lennart Berntsons »Socialismen och demokratin».
De båda författarna hade från början hört hemma i 68-vänstern. I likhet med Zenit-redaktionen i övrigt hade de tidigare företrätt en »nyleninistisk» syn på demokratifrågorna. Inom Zenit hade den »vänstersocialistiska» tendens som framträtt inom VPK vid början av 1960-talet utsatts för skarp kritik, inspirerad av den marxistiske filosofen Louis Althusser och hans skola, av den brittiska vänstertidskriften New Left Review samt till någon del av den kinesiska kritiken mot den sovjetiska »revisionismen». Men nu gick Arvidsson och Berntson i bräschen för det motsatta synsättet. Demokratin – varmed avsågs det som tidigare brukade kallas »borgerlig» demokrati – borde accepteras fullt ut. Moskvakommunismen kritiserades, men inte från »vänster» för sitt knäfall för den fredliga vägen till socialismen, utan från »höger» för sin hyllning till diktaturerna i öst. De båda författarna föreföll plädera för en syn på demokratin liknande den som några år tidigare fått Sven Landin att lämna VPK och skriva boken _Uppbrott från stalinismen_ (1973). Eftersom just denna syn hade varit anatema inte bara inom redaktionen för Zenit utan inom hela 68-vänstern var deras utspel utomordentligt djärvt och föranledde många skarpa motinlägg från medlemmar av Zenit-redaktionen och dess omkrets som företrädde mer traditionella synsätt.
Artikeln, förklarade författarna, var en uppgörelse med »de västeuropeiska kommunistpartiernas, och särskilt VPK:s, hållning till demokratin» samt med »de politiska förhållandena i Sovjet och Östeuropa». Inte minst angrep de tesen om »proletariatets diktatur» som tidigare inom Zenit och stora delar av 68-vänstern i enlighet med Lenins åsikt hade betraktats som själva kriteriet på sann marxism. För säkerhets skull klargjorde de att de inte i »Kina eller 'Mao Tse-Tungs tänkande'» såg något alternativ för den västeuropeiska arbetarklassen.
Arvidsson och Berntson gjorde klart att det de försvarade var den »borgerliga eller liberala demokratin», vilken förklarades ha varit »ett resultat av arbetarklassens och den progressiva småborgerlighetens kamp». Deras kritik mot den ryska oktoberrevolutionen och mot Lenins politik var dock dämpad. Visserligen uttalade de förståelse för »socialistiska länder som Kuba, Kina, Vietnam och Nordkorea», trots avsaknaden av demokrati. Visserligen framhöll de att den socialistiska omvälvningen aldrig skulle kunna genomföras »om arbetarklassens kamp begränsas till strikt parlamentariska åtgärder» utan måste bygga på »de arbetande massornas resning». Men huvudtesen i deras artikel var en annan. Den innebar en uppgörelse med hela det kommunistiska arvet i demokratifrågan från Komintern och framåt. Socialism utan demokrati förklarades vara en omöjlighet. Resonemanget anknöt till VPK:s partisprängning samma år när APK lämnat partiet: »För VPK:s vidkommande menar vi att det nu, efter den 'bresjnevitiska' flamman-gruppens utbrytning, öppnas en möjlighet till en mer definitiv uppgörelse med den prosovjetiska och ortodoxa synen på socialismen och demokratin. Frågan är om partiet kommer att ta tillvara den möjligheten?»
Svaret på Arvidssons och Berntsons avslutande fråga blev tydligt. Utvecklingen inom VPK gick inom de närmaste åren snarast i »nybresjnevitisk» riktning, inte minst inom ungdomsförbundet. Sovjets stöd till Vietnam i konflikten med Kina, dess stöd till MPLA i Angola och andra »befrielserörelser» i Afrika, liksom till Kuba, bidrog inom dessa år till att många inom vänstern fick förnyat förtroende för Sovjetunionen. Polariseringen mellan Sovjet och Kina framtvingade ett val. Eftersom Kina allt tydligare allierade sig med dödsfienden USA drevs många inom nyvänstern att liera sig med Sovjet på ett sätt som tidigare inte varit vanligt. Också KFML(r):s utveckling är intressant i detta avseende.
Inom den intellektuella vänster som Zenit representerade fortsatte emellertid avståndstagandet från Sovjet och Östeuropa att vara massivt. Dissidentrörelsena i Tjeckoslovakien och Polen, som Arvidsson och Berntson åberopade, spelade en roll. Det gjorde även »eurokommunismen», som artikelförfattarna visserligen kritiserade för att innehålla »en klar högertendens». Det nya hos de båda författarna var inte att de kritiserade Sovjet utan att denna kritik nu utvecklades från rakt motsatt håll jämfört med den som varit vanlig under 68-vänsterns första skede. Den sovjetiska linjen ansågs nu fela genom att alltför mycket godta idén om »proletariatets diktatur» snarare än genom att avvisa den.
Den omvärdering av demokratifrågan som diskussionen i Zenit var ett symptom på hade fundamental betydelse för vänstern. Som Göran Palm påpekat i sina tidiga böcker var just brytningen med den »borgerliga» demokratisynen avgörande för uppkomsten av en ny vänster. Skulle denna brytning omintetgöras var hela nyvänstern försatt i gungning. Situationen komplicerades genom att även maoisterna (eller postmaoisterna) bytt fot. Försvar för demokrati och rättsstat ingick numera i exempelvis Jan Myrdals repertoar av argument mot den prosovjetiska vänstern. Den protoliberala kritik som nu började växa fram inom bland annat Zenit fick därmed stöd från en senmaoistisk eller postmaoistisk tendens. Lägg därtill att under dessa år kom oppositionsgruppen Charta 77 i Tjeckoslovakien, fackföreningsrörelsen Solidaritet i Polen, Aleksandr Solzjenitsyns _Gulagarkipelagen_ och de franska så kallade nya filosoferna och det blir tydligt att hela 68-konceptet var på väg att gå under. Tioårsjubileet skulle därmed komma att firas i förvirringens och splittringens tecken.
### _1978 – nytänkandets år._
För vänstern präglades 1978 av förvirring och splittring, men också av nytänkande. Den danske filosofen Peter Kemp sammanfattade situationen för den europeiska vänsterns vidkommande i sin bok _Marxismen i Frankrike. Apropos de»nye filosofer»og marxismens krise_ (1978). Han skrev bland annat (varvid jag citerar den svenska översättningen): »Om man vill göra sig hörd i samhällsdebatten måste man säkert fortfarande framstå som vänsterorienterad, eller i varje fall klä ut sig till sådan. Men det är inte längre fråga om att hitta en marxism som kan passa. Nu måste det heta: vi är alla vänsterorienterade; det avgörande är hur vi är det. Vi är inte alla utan vidare marxister. Marxismen har blivit ett problem.»
Utopierna från Kina, Kuba och Albanien hade spruckit. Marx analys av kapitalet framstod heller inte längre som tidsenlig: »Vårt samhälle är visserligen fortfarande kapitalistiskt, men dels är det i allt högre grad kapitalistiskt på ett annat sätt än Marx föreställde sig, t.ex. i en ständigt alltmer komplicerad allians mellan staten och monopolkapitalet, dels är det mycket annat än kapitalistiskt, t.ex. mer byråkratiskt, mer teknologiskt likriktat, mer ödeläggande för den ekologiska balansen än någonsin förr.»
När kommunismen och de socialistiska utopierna svek gällde det för vänstern att hitta andra mer lovande föremål för agitation. Teknologin och miljön framstod som sådana förhållandevis oexploaterade områden, som tidigare visserligen uppmärksammats men kommit i skymundan för den socialistiska kampen. Byråkratin likaså, även om det här gällde att akta sig för att göra gemensam sak med nyliberalismen som just dessa år vann segrar.
En bok som kom 1978 och som förde en diskussion i den nya andan var _Utopien och dess skugga_ (Norstedts), skriven av Madeleine Gustafsson, kritiker i DN och då gift med Lars Gustafsson. Den var kritisk mot vänstern, men innehöll också en vilja att förnya projektet. De franska nya filosoferna diskuterades i boken, men också Michel Foucault som just då framstod som dessas allierade, samt dessutom historiker ur den franska Annales-skolan.
Två framstående historiker ur Annales-skolan uppmärksammas. Där finns Emmanuel Le Roy Ladurie med hans _Montaillou_ (1975), boken om kättarbyn i Sydfrankrike (Occitanien) på 1300-talet som utsätts för inkvisitionens ingripande. I Gustafssons tolkning: »Occitanien är slaget, det är de kvardröjande formerna för lokalt självstyre som nu stampas ut.» Där finns Philippe Ariès som skrivit barndomens historia men också dödens. Båda studierna utmynnar i en kritik mot det moderna samhället »som självt var byggt som en maskin, så genomkontrollerat att detaljerat samtycke inte längre behövs, så tekniskt fjärrstyrt att greppet om individen kunde lossas. Frihetens gränser hade sjunkit under synlighetens horisont».
Ariès sympatiserade personligen med fransk ytterlighetshöger, men hans budskap sådant Madeleine Gustafsson tolkade det kunde lätt samsas med den mer vänsterorienterade Michel Foucault, vars undersökning av »vansinnets historia» hon refererar. Också här blir moderniteten boven i dramat: »Ty det yttre behärskandet av vansinnet, den sjukes kropp och hans handlingar, ses inte längre som det viktiga: det viktiga är hans inre underkastelse under normen.»
Modern psykvård – liksom all annan sjukvård – är framför allt maktutövning, disciplinering, det är tesen. Gustafsson förknippar denna maktutövnig med kapitalism och kommersialism. Men också med kommunism och marxism. Med inlevelse och instämmande återger hon vad de franska »nya filosoferna» haft att säga om detta. Hon framhåller hur förutsättningen för de nya filosofernas framgång varit att de nästan alla haft en bakgrund inom vänstern (i första hand inom maoismen):
De nya franska filosoferna har varit marxister, allesammans. För dem liksom för deras publik, den våg som bär dem, är maj 68 den självklara utgångspunkten. Deras nya hållning är inte en återgång, inte ett ångerköpt bekännande av den kapitalistiska pluralismen som den bästa av alla världar. [---] Om sjukan har de inte ändrat mening, det är botemedlet de ifrågasätter: tänk om det i själva verket bara är mer av samma sak?
Solzjenitsyn och _Gulagarkipelagen_ förs in i sammanhanget på samma sätt som de franska nya filosoferna: »Solsjenitsyns böcker om Gulagarkipelagen har sent om sider börjat få en verkan liknande den som rapporterna från de tyska förintelselägren fick på en tidigare generation.» Anknytningen till Foucault betonas likaledes. _Utopien och dess skugga_ slutar med att författarinnan tecknar bilden av en ny motståndsfront:
På det praktiska planet finner man plötsligt garvade marxister kämpa för yttrandefriheten tillsammans med konservativa jurister ur Högsta Domstolen, nordtyska lantarbetare sammanträder mot kärnkraften på adliga slott sida vid sida med maoistiska miljöaktivister, och SKP-are och pingstvänner slåss i enad front mot porrindustrin och den sexuella utarmningen av människan.
Det är vackert så. Motståndet är nödvändigt. Utopierna kan vänta.
Misstron mot utopin och dess »skugga», dess förmåga att rättfärdiga våld och totalitära lösningar var ett tema i boken. Idén om den partiella revolten som skulle ersätta den allt omstörtande revolutionen likaså. Men ändå? Förutsätter inte motstånd tanken på ett system som man gör motstånd mot? Hos Madeleine Gustafsson finns alldeles tydligt ett sådant ont system som det gäller att bekämpa. Men vad är det? Inte längre rätt och slätt kapitalismen. Inte heller socialismen. Inte utan vidare den teknologiska utvecklingen eller moderniteten. Kan det vara tillvaron som sådan, själva jordelivet? Håller hon med Bernard-Henri Lévy om att »Fursten är ett annat namn på världen»? Eller är tanken rätt och slätt reformistisk, att samhället alltid kan förbättras här och var genom begränsade reformer? Men i så fall – varför denna apokalyptiska ton?
Många av »de nya filosoferna» var besvikna maoister. På en punkt kommenterar Madeleine Gustafsson deras syn på utvecklingen i Kina: »Varje gång upproret bryter ut, lyckas Fursten vända dess spets tillbaka mot folket, skriver de: så förvandlades kristendomen från 'kulturrevolution' till 'ideologisk revolution': efter Lin Piao kommer alltid Teng Hsiao Ping. [---] Anakoreten blev munk och upprorsmannen samhällsbevarare; tillvaron gick vidare.»
Det låg något i detta. I Sverige tycktes mycket riktigt många maoister vara beredda till omvärderingar, däribland Bo Gustafsson som detta år blev professor på allvar och höll sin installationsföreläsning. I den vände han den ekonomiska teorin upp och ned. I stället för Lin Biaos syn på världens ekonomi och förutsättningarna för ekonomisk utveckling i fattiga länder kom Dengs.
Under 1960- och 1970-talen hade vänstern hållit sig till »beroendeskolan» som hävdade att »imperialismen» och beroendet av världsmarknaden blockerade möjligheterna till ekonomisk utveckling i den så kallade tredje världen (Asien, Afrika och Latinamerika). I dessa delar av världen kunde kapitalismen bara leda till »utveckling av underutvecklingen». Enbart genom att göra sig fria från världsmarknaden kunde sådana länder utvecklas. Maos Kina, Kim Il Sungs Nordkorea, Castros Kuba blev för en tid de stora exemplen på de ekonomiska mirakel som en sådan oberoende socialistisk utveckling kunde leda till.
Under det senare 1970-talet och under 1980-talet började dessa teorier emellertid att utsättas för kritik också inom vänstern. Bo Gustafssons installationsföreläsning »Imperialismen, Tredje världen och historiens list» (publicerad i Ekonomisk debatt 1978:5) ger exempel på en sådan kritik.
Gustafsson refererar till de marxistiska ekonomerna Andre Gunder Frank och Samir Amin och deras teori om »utveckling av underutvecklingen» – båda var väl kända i Sverige och deras viktigaste arbeten hade översatts till svenska och studerats noga inom vänstern. Han påpekade emellertid att just »vid den tid då teorin spreds och omfattades som mest – och den tiden är ännu inte förbi – inleddes faktiskt den industriella revolutionen i Tredje Världen». I själva verket hade bruttonationalprodukten i tredje världens länder vuxit betydligt snabbare än i de utvecklade länderna. Detta hade även medfört en ökad social och medicinsk standard liksom en kraftig reduktion av analfabetismen. Gustafsson fann att det som pågick i tredje världen var »en ekonomisk revolution: den kapitalistiska industrialiseringen». Denna revolution hade till stor del haft »imperialismen» som sin orsak, den hade spridits från de redan utvecklade kapitalistiska och industrialiserade länderna, något som Gustafsson uppfattade som en »historiens list» i Hegels mening.
»Tigerekonomierna» (Taiwan, Sydkorea med flera) hade vederlagt beroendeskolans teorier. Nu hade Deng dessutom i Kina börjat lägga om hela utvecklingsstrategin så att den i nästan alla avseenden blev motsatsen till kulturrevolutionens, byggd på den öppna dörrens politik, på handelsutbyte och delaktighet i världsmarknaden, på forskning, kunskaper och respekt för experter.
För Kinas befolkning blev detta en väg ur fattigdom, ohälsa och svält. Men för den västerländska vänstern var det en olycka. En utopi gick förlorad. 68-vänstern hade ju gjort uppror mot överflöds- och konsumtionssamhället. Maos och Lin Biaos Kina hade framstått som ett alternativ till detta. Dengs Kina, på väg att hinna upp Västerlandet i ekonomisk utveckling, gjorde det inte. Detta Kina framstod tvärtom som mera av det onda. Om anakoreten blev munk och upprorsmannen samhällsbevarare – vad var då egentligen vunnet? Vilken sorts list var detta från historiens sida? Det var som när vid slutet av maskeradbalen sjörövarkaptenen demaskeras som herr Svensson och geishan som fru Andersson. Om Kina bara skulle bli likt USA eller Västtyskland eller Sverige – var fanns då »det vidunderliga»?
### _1979 – de socialistiska krigens år._
En teaterpjäs som försökte sammanfatta den svenska vänsterns utveckling dittills uruppfördes 1979 på Dramaten i Stockholm. Det var _Mannen på trottoaren_ , skriven av Per Olov Enquist och Anders Ehnmark. Den senare var en av vänsterns mest uppmärksammade journalister. Han hade varit Expressens Romkorrespondent på 1960-talet och hade bakom sig reportage om bland annat Italien, Kuba och Afrika.
I en senare gemensam bok, _Protagoras sats. På spaning efter det politiska förnuftet_ (1987), berättar författarna att pjäsen påbörjats våren 1978: »Det var vid tiden för mordet på den italienske konseljpresidenten Aldo Moro, tryckvågen från terrorismens vårkampanj' sändes ut över Europa.»
Pjäsen fick undertiteln _Förundersökning angående mordet på doktor Plehve_ och gavs formen av en rättegång mot terroristen Sasonov som utfört mordet på Plehve i Petersburg 1904. Anklagelsen mot honom förs fram av dubbelagenten Azeff. Men det hela utspelas i modern svensk miljö. Stycket inleddes av ett »Förspel» där vissa teser slogs fast: »Socialism utan demokrati är inte socialism. Den politiska demokratin är en nödvändig del av socialismen, inte bara ett taktiskt medel på vägen dit.»
Gav man upp demokratin gav man även upp socialismen. Detta var emellertid ett kontroversiellt påstående inom vänstern: »Problemet gäller elitismen, eller leninismen om man så vill. I den radikalismens kris vi bevittnar finner man det vid hjärtpunkten.»
Ett extremt utslag av denna elitism var terrorismen som börjat som idealistisk terrorism men slutat som systemstabiliserande terrorism: »Terrorismens utveckling: från bräckjärn till armeringsjärn.»
Utgångspunkten för pjäsen var den progressiva rörelsens nedgång med avstamp från »ett av nedgångens fenomen», terrorismen: »Vi söker oss tillbaka till de åren då nedgången förbereddes, uppsvingets år, till de mekanismer och felslut som isolerade rörelsen inom mellanskikten från rörelsen inom det arbetade foket, och tvärtom.»
Terrorismens ideologi berörs – att driva samhället till fascism för att på så vis bereda väg för den socialistiska revolutionen: »Vägen till socialismen går närmare bestämt över fascismen. Fascismen är ett etappmål.»
Det gällde att hitta rötterna: »Vad beror det på att de senaste femton årens så oerhört omskrivna vänsterframgångar inte ledde till framgång?»
Därefter följer själva pjäsen, en fars i sju tablåer och ett slutspel. Den första tablån handlar om den sexuella befrielsen fram till 1968. Den andra om den nyleninistiska sektvänsterns och maoismen år. Där finns också exmaoisten Ekfält som numera anslutit sig till de nya filosoferna. Den tredje tablån handlar om gruvstrejkens år. Och så vidare via uteslutningsraseriet inom vänstern, almstriden i Stockholm mellan demonstranter och polis 1971 och fram till den västtyska terrorismen. Det hela mynnar ut i en återvändsgränd. Vad som behövs för vänstern är en återanknytning till »mannen på trottoaren», till de vanliga människorna som hela tiden stått vid sidan av vänsterns olika upptåg.
_Mannen på trottoaren_ blev en träffsäker och småelak kavalkad över vänsterns felslut och absurditeter. Pjäsen kritiserade utifrån en vänsterpostion. Det hela hade karaktär av insideraffär och upphovsmännen sågs som renegater och fick möta anklagelser om trolöshet mot huvudman. Men just det de initierade var framställningens styrka och roade åtminstone en del av publiken (medan den retade gallfeber på andra).
Demokratifrågan stod i centrum för _Mannen på trottoaren_. En annan stor debatt 1979, även den i omprövningens tecken, handlade om imperialism, krig och socialism. Detta var »de socialistiska krigens» år.
I december 1978 hade Vietnam inlett sin invasion av Kampuchea/Kambodja efter några år av intensifierade konflikter. Redan den 7 januari 1979 intogs Phnom Penh av de invaderande styrkorna. Pol Pots regering flydde hals över huvud. En Vietnamvänlig regim installerades i huvudstaden. Pol Pot, Kinas allierade hade lidit nederlag. Nordvietnam, nu Vietnam, var i allians med Sovjetunionen på väg att nå målet att ena hela Indokina under sin hegemoni. Hur skulle Kina svara på detta? Svaret kom den 17 februari när Kina påbörjade en massiv »straffexpedition» in i Vietnam. En sovjetisk svarsaktion på Kinas angrepp på Vietnam uteblev och efter några veckor drog sig kineserna programenligt tillbaka ur landet. Händelseförloppet reste frågor som angick hela världen. Men det reste också mer speciella frågor som i synnerhet angick vänstern, också den svenska vänstern. Inom vänstern hade man brukat säga att krig var en effekt av kapitalism och imperialism. Något sådant som ett krig mellan socialistiska stater hade tyckts uteslutet. Men nu hade det skett. De två länder som framför andra hade inspirerat 1968 års vänster, Vietnam och Kina, hade gått i krig med varandra. Hur borde vänstern reagera på denna utmaning?
Åtminstone till en början reagerade man med fullkomlig förvirring. En del av inläggen i den debatt som följde sammanställdes av Kent Lindkvist och mig själv (Svante Nordin) i den lilla volymen _De socialistiska krigen. En antologi om konflikten i Sydostasien_ , utgiven i serien Zenit Häften 1979.
Den svenska debatten om konflikten Kambodja–Vietnam–Kina fördes framför allt inom vänstern. De officiella reaktionerna från exempelvis Olof Palme och utrikesminister Hans Blix var dämpade. Sverige hade goda förbindelser med både Vietnam och Kina. Men inom vänstern fördes debatten. Därvid kom SKP att stå tämligen isolerat i sitt stöd till Kina. Den övriga vänstern, representerad av SAP, VPK, APK, KPML(r), FK och KAF, stödde med varierande grad av intensitet Vietnam och tog avstånd från Kina. Men deras argumentering och sätt att föra fram sitt ställningstagande uppvisade åtskilliga variationer.
VPK försökte till en början undvika ett ställningstagande till Vietnams invasion av Kambodja genom att förneka att den överhuvud taget ägt rum. Detta var i överensstämmelse med den officiella vietnamesiska linjen enligt vilken en spontan folkresning i Kambodja utan vietnamesisk inblandning hade fått Pol Pots regim på fall, ett exempel sålunda på ett utomordentligt snabbt revolutionärt förlopp. Ordföranden för Kommunistisk Ungdom, Lars Johansson, svarade i förbundets tidning Stormklockan på frågan hur han såg på det aktuella läget i landet: »Situationen är ju den, att imperialismens nyhetsbyråer tillsammans med Kina tutat ut att Vietnam ockuperat Kampuchea. Vietnam å sin sida har förklarat att så inte skett utan det som hänt är en folklig resning mot Pol Pot-regimen.»
Johansson ställde sig skeptisk till uppgiften om en vietnamesisk ockupation. Men han sade också: »Om detta verkligen hänt så är vår inställning helt klar. Vi försvarar varje lands nationella självbestämmanderätt och en Vietnam-ockupation kan utifrån det inte accepteras. Om en ockupation inträffat skulle vi omedelbart kräva tillbakadragande av trupperna.»
Fram emot sommaren, när till och med Björn Ahrén, VPK-organet Ny Dags Vietnamkorrespondent, erkänt att den vietnamesiska interventionen varit av »troligen avgörande» betydelse för Pol Pot-regimens fall, hade VPK/KU emellertid bytt fot och försvarade inmarschen med hänvisning till Pol Pots grymheter.
VPK:s vägval, för Vietnam och mot Kina, blev ännu tydligare när Kinas straffexpedition mot Vietnam ägde rum och föranledde upprörda avståndstaganden från VPK:s sida.
Den svenska vänstern ställde sig – med ett undantag – i stort sett mangrant på Vietnams sida. Det Moskvatrogna APK, som något år tidigare brutit sig ur VPK, var självfallet än mer militant och kompromisslöst i sitt ställningstagande för Vietnam och mot Kina. KPML(r) hade från början beslutsamt slutit upp bakom Kina i den sino-sovjetiska konflikten men hade vid denna tidpunkt gått igenom en fullkomlig nyorientering. Kina och Pol Pot-regimen förkastades, den nya ordningen i Phnom Penh ansågs av Proletären öppna »en bättre framtid åt Kampucheas folk». FK och KAF tog visserligen avstånd från Vietnams invasion men sparade sina skarpaste fördömanden för Kinas straffexpedition.
Det stora undantaget från denna enighet var SKP.
De svårigheter som KPML(r) eller VPK möjligen kan ha känt när det gällde att förklara varför Pol Pot, som 1976 fortfarande byggde socialismen med det kampucheanska folkets entusiastiska stöd, 1978 var en blodig tyrann, mötte SKP i skärpt form då det gällde Vietnam. Solidaritetsarbetet för Vietnam hade ju givit KFML mycket av dess ursprungliga rekryteringsbas. Vietnam hade i SKP-pressen fram tills helt nyligen skildrats som ett socialistiskt föregångsland. Inte förrän gränsstrider bröt ut mellan Vietnam och Kampuchea och den vietnamesisk-kinesiska konflikten trätt fram i dagen ändrades tonen i Gnistan. Å andra sidan erbjöd Vietnams invasion av Kampuchea ett tacksamt tema för SKP:s agitation. Föreställningen om Sovjet som den aggressiva och expansionistiska makten och USA som den undfallande föreföll få stöd i händelseförloppet. Försvaret för små länders nationella självbestämmanderätt hade också länge spelat en viktig roll i SKP:s framtoning. Det kinesiska anfallet på Vietnam gav problem, men kunde förklaras som solidarisk hjälp åt det kampucheanska folk som nu i de röda khmerernas motståndsrörelse åter kämpade mot en ockupant. Inte heller fällde man några tårar över den svenska vänsterns splittring. I januari 1979 skrev Gnistan:
Gamla vänner blir nu fiender i det nya Indokinakriget. Så beskriver Aftonbladet det faktum att två 'vietnamdemonstrationer' gick i Stockholm i lördags – den ena för Kampucheas oberoende och den andra för Vietnams angrepp.
Samma tema har med små variationer upprepats i andra media. 'Vänstern splittras', etc.
Men faktum är att 'vänstern' länge varit splittrad. Och det är ingen olycklig splittring. Det är nödvändigt att dra en skarp skiljelinje gentemot de politiska krafter som i imperialismens epok inte kan skilja på angripare och angripen.
Själva begreppet »vänster» måste numera sättas inom citattecken. Det var inte längre på något uppenbart sätt mellan »vänster» och »höger» som skiljelinjen gick utan mellan olika vänsterorganisationer.
Schismen mellan organisationerna hade sin motsvarighet i ett bråk mellan några kulturpersonligheter. Sara Lidman krävde ovillkorlig solidaritet med Vietnam: »Att i dag demonstrera mot Vietnam är att (oavsiktligt) tjäna CIAs makt i världen. Det är att gynna USA-imperialismen.» Peter Weiss framförde liknande synpunkter. Han förnekade att Vietnams arméförband anfallit Kampuchea. Han förklarade att om Vietnam slagit tillbaka var det i nödvärn. Han såg »båtflyktingarna» från Vietnam som offer inte för den vietnamesiska regimen utan för amerikansk antikommunistisk propaganda, som »sådde denna fruktan och vantro». Staffan Beckman skrev i liknande anda om dessa människor, mestadels tillhörande den kinesiska minoriteten, som flytt från Vietnam i öppna båtar.
Den främste kämpen på den motsatta sidan var Jan Myrdal. Redan tidigare hade han skildrat Pol Pots Kampuchea i lyriska ordalag: »Sannerligen detta är tidens tredje ålder! Här skulle Thomas Münzer och hans bröder i femtonhundratalets Tyskland ha känt igen sig. Här har Wat Tyler segrat. Här härskar Jacques Bonhomme. Ty här reste sig jordens fördömda och fattigbönderna tog upp kampen mot en härskara fiender och fördrev dem alla och började bygga sina drömmars välde av rättvisa och jämlikhet och gemenskap. Allt stämmer.»
I detta kolorerade träsnitt blandades den pittoreska primitivismen med en underliggande apologi för våldet.
Hur Myrdal skulle betrakta Vietnams invasion var självklart: »Nu är det nytt år och åter brinner Kampucheas byar. Mig-planen anfaller i våg på våg. De tvångsrekryterade soldaterna från svältens Vietnam har beordrats ut att erövra Kampucheas bördiga rismarker.»
Myrdal såg de debattörer som försvarade Vietnam som »femtekolonnfigurer». De var helt enkelt potentiella quislingar som vid ett sovjetiskt anfall mot Sverige skulle stå beredda att gå ockupanten till mötes.
Att i det opinionsläge som uppstått försöka medla var en vansklig uppgift. Journalisten Einar von Bredow försökte i Aftonbladet inta en »tredje ståndpunkt» men fick mothugg av de tidigare Vietnamaktivisterna Jan Hinderson och Jörgen Widsell, som från motsatta utgångspunkter tog avstånd från alla förmedlande resonemang. Bland de sällsynta försöken att dra mer övergripande och teoretiska slutsatser kan nämnas en artikel av statsvetaren Mats Dahlkvist i Ny Dag, »Gud och djävulen i Peking, Moskva och Hanoi».
»Det är öppet krig mellan två socialistiska stater», konstaterade Dahlkvist och påpekade hur illa detta stämde med grundkursen i marxism där kapitalismen uppgavs vara orsaken till krig och socialismen en garanti för fred: »Det kan inte vara fel på ett socialistiskt land. Partiet kan aldrig ha fel, har aldrig haft fel. Moskva har aldrig fel. Peking har aldrig fel. Hanoi har aldrig fel.»
Dahlkvist uppmanade Ny Dags läsare att vänja sig vid tanken att socialistiska stater för krig mot varandra, hur avskyvärd den än är: »Vi måste försöka lära oss att komma ifrån det religiösa tänkandet som uppdelar världen i djävulska krafter och gudliga.»
Men med detta och ett fåtal andra undantag var den svenska vänstern inte beredd att rucka på sina principer. Alla utgick från att det inte kan vara fel på ett socialistiskt land. Men där de tidigare varit ense om att se inte minst Kina och Vietnam som socialistiska länder betraktade en grupp nu Kina som socialistiskt men Vietnam som imperialistiskt (med Sovjetunionen som dirigent) medan en annan grupp såg Vietnam som socialistiskt men Kina som imperalistiskt (med USA som dirigent). Själva teorin var den samma, men tillämpningarna varierade på ett sätt som skadade trovärdigheten i argumenteringen. Det var mer än en splittring inom vänstern. Det var, som redan framhållits, en klyfta som omöjliggjorde själva termen »vänster» annat än inom citattecken. Förändringen hade gått snabbt från självsäkerhet till rådlöshet och kaos, lika snabbt som utvecklingen i Indokina från seger till inbördeskrig. Och även om renlärigheten upprätthölls utåt var den underminerad. Inte heller i Jan Myrdals ögon var Dengs Kina riktigt det absoluta ideal som Maos Kina en gång varit. Mao hade haft rätt i sin kritik mot Deng. Inte heller Sara Lidman kan ha känt sig alldeles tillfreds med båtflyktingarna. Inte heller hade hon alldeles lätt att förklara sin egen snabba positionsförflyttning i synen på Pol Pots Kampuchea. Det fanns sprickor i fasaden på ömse håll. De socialistiska länderna var helt enkelt inte längre brukbara som föredömen, som realexisterande utopier, som garantier för socialismens genomförbarhet eller för att de ljuva drömmarna inte skulle sluta som mardrömmar.
Slutsatserna skulle komma att dras (utan att enighet för den skull uppnåddes) i det följande årets stora debatt om »mediavänstern».
### _1980 – mediavänsterns år._
Det var Jan Myrdal som startade debatten och det är betecknande för läget att han gjorde det i Svenska Dagbladet (SvD), obundet moderat.
Också kultursidorna hade förändrats. När Per Wästberg 1976 tog över efter Olof Lagercrantz på Dagens Nyheters kultursida var kulturmaoismens glansdagar slut. Vänstersocialismen var inte längre lika självklart den flagga man seglade under. I bakgrunden förberedde Horace Engdahl och andra skribenter från kulturtidskriften Kris redan ett »postmodernistiskt» maktövertagande. På Svenska Dagbladet hade man å andra sidan missat vänstervågen och fått nöja sig med opportunistiska anpassningsrörelser. Nu var det dags att ta revansch, att överta initiativet. Med Mats Gellerfelt randades en ny dag. Gellerfelt hörde till en yngre generation än 68-generationen, men hade liksom dess medlemmar ett frejdigt polemiskt humör. Hans närvaro på Svenska Dagbladets kultursida var en av förutsättningarna för den debatt om »mediavänstern» som bröt ut sommaren 1980 och som förändrade styrkebalansen i den svenska mediadebatten – till nackdel för både vänstern och DN (något som bland annat framhållits av Anders Frenander i _Debattens vågor: om politisk-ideologiska frågor i efterkrigstidens svenska kulturdebatt_ (1998) och av Åke Lundqvist i _Kultursidan. Kulturjournalistiken i Dagens Nyheter_ (2012).
Det bör kanske poängteras att även om styrkebalansen förändrades så var debattatmosfären tills vidare densamma. Den apokalyptiska tonen i många av inläggen var inte att ta fel på. Redan det faktum att debatten initierades av Jan Myrdal var en garanti för den saken.
Myrdals första artikel om de intellektuellas ansvar stod i SvD den sista maj 1980. Utgångspunkten var att man gick mot »slutet av en mellankrigstid». Ett nytt storkrig stod för dörren. Kriget var visserligen inte oundvikligt men sannolikt. En av de faktorer som gjorde det troligt var de »intellektuellas roll som producenter av falskt medvetande». Myrdal gjorde en snabbskiss av de svenska intellektuellas utveckling »sedan ungefär 1964»: »Ena dagen gick de i smalrandiga kostymer och slips och talade om konst och nästa dag gick de i tjocka tröjor och talade om Afrika, Asien och samhället. Sedan deltog de i protesterna mot kriget i Indokina. Det var bra.»
Myrdal var inte omedveten om sin egen verksamma del i denna utveckling. 1964 hade ju varit året för den självbiografiska _Samtida bekännelser av en europeisk intellektuell_. Sedan hade strejkvågen brutit ut. De intellektuella hade enligt Myrdal då ställt sig utanför. Men värre skulle komma. De intellektuella och mediafolket lyckades organisera sig för sina egna syften:
Deras organisationer kontrollerar mediavärlden och bestämmer genom medinflytande i ledningen över både rekrytering och program. Det är kanske inte någon klass, men det är i varje fall ett starkt korporativt skrå vilket organiserar sig själv som kast. [---] Och medan det tidigare var deras samhälleliga roll att producera ideologi – ett särkilt medvetande – som tjänade verkliga samhällsklasser producerar de nu, efter att ha organiserat sig själva materiellt, ideologi för att stärka sina egna syften; dvs ett samhälleligt medvetande som legitimerar deras eget behov av att försörjas av det arbetande folket.
Detta kunde enligt Myrdal »förklara varför de leder kampen för 'nolltillväxt' och propagerar för detta mål samtidigt som både Arbetsgivarföreningen och Landsorganisationen av bitter nödvändighet försöker tala om produktionen».
Han framhöll kärnkraftsfrågan som exempel och tecken i tiden: »Men som grupp, som skrå, som underkast var de intellektuella och mediafolket emot kärnkraften. Och deras argument var ofta av det mest primitiva och känslomässiga slag. Som tur var förlorade de. Men utanför naturvetarna var det bara en handfull intellektuella som stod för förnuft, rationalitet, vetenskap och tillväxt.»
Denna antiintellektualism hade kommit att prägla även den svenska skolan och universitetsundervisningen: »På universitetsnivå är studierna nu grundare, mer perifera och ofta mytiska.»
Det Myrdal kallar glidningen »bort från verkligheten» präglade även den utrikespolitiska debatten. Hotet från Sovjetunionen hade sålunda inte alls uppfattats av svenska media eller i svensk kulturdebatt.
Det Myrdal på ett verkningsfullt – om än långt ifrån självklart – sätt förde samman var några iakttagelser. Dels att vänstern i ökande utsträckning ägnade sig åt en skarp kritik riktad mot industri, tillväxt, modern teknik, modern naturvetenskap, »det västerländska förnuftet», mot allt tal om »fakta», »verklighet» och »sanning». Dels att denna strömning ibland samsades med ett »fredstänkande» som tycktes berett att acceptera om inte sovjetisk ockupation av Sverige så dock en sovjetisk hegemoni i Europa, med andra ord vad somliga debattörer kallade »finlandisering». Efter USA:s nederlag i Vietnam och Sovjetunionens framstötar på olika fronter kunde ett sådant tänkade rent av te sig som en oundviklig säkerhetspolitisk nyorientering. Myrdal manade, i likhet med ledarna i Peking, till motstånd. Mediavänstern fick i hans scenario rollen av »femtekolonnare». Klerkernas förräderi blev samtidigt ett förräderi mot fosterlandet.
Myrdals artikel föranledde ett inlägg av författaren Sven Fagerberg (SvD 12.6.1980). Det bar rubriken »Sveriges vänsterfascism» och blev »Fagerbergs sannolikt mest uppmärksammade artikel någonsin». Fagerberg med bakgrund i näringslivet ville inte höra talas om Myrdals motsättning mellan kapital och arbete. Men han instämde i kritiken mot »den nya klassen». En klass av nya herrar hade uppstått, bestående av politiker, ombudsmän, byråkrater, massmediamän, akademiska lärare, som ansågs parasitera på företagare och arbetare och företräda en »vänsterfascistisk» ideologi. Inte minst hade de tagit över massmedia som blivit till megafoner för deras socialistiska dagordning.
Sven Delblanc vände sig i sin tur mot Fagerberg, samtidigt som han delvis instämde (SvD 20.6.1980). Trots att han inte höll med om allt skrev Delblanc: »Men visst är det sant att svenska intellektuella verkar i en likriktning som är nästan total, och att de få avvikande rösterna borde få allt stöd man kan ge dem. Fagerberg, Myrdal, Lars Gustafsson, de är inte många, och deras utgångspunkter är som synes vitt skilda. I en kvävande konformism öppnar de ändå ett andningshål.»
I mycket instämde Delblanc i Myrdals och Fagerbergs alarmism. Han talade om »det korporativa Sverige, som snart har gjort vår parlamentariska demokrati till en tom formalitet». Han talade om »vår eländiga utbildningspolitik». Han vände sig mot nedmonteringen av försvaret och mot att »svenska intellektuella sprider defaitism och underminerar vår tro på Sveriges suveränitet». Sverige hade »en feg och konformistisk intelligentia».
Lars Gustafsson trädde in i debatten (SvD 6.7.1980) med kompletterande synpunkter. Hans inlägg kom senare i volymen _För liberalismen. En stridsskrift_ (1981) under rubriken »Mediavänstern: den nya klassens vakthund». Gustafsson skrev bland annat att »den nya klass i det svenska samhället som den offentliga sektorns jättetillväxt har skapat» kunde beskrivas »både med Sven Fagerbergs klassiska liberalism och i Jan Myrdals marxistiska premisser». Han hänvisade till nyklassteoretiker som Galbraith, Djilas, Gramsci och Bahro och framhöll: »Mediavänstern har inte i första hand med åsikter att göra. Att tillhöra mediavänstern är att utföra en bestämd social roll: nämligen att avleda emotionella spänningar och skapa gemenskapskänslor av ett slag som är nödvändigt för att den offentliga sektorns problemformuleringar inte skall kunna ifrågasättas.»
Gustafsson hade därmed introducerat den idé om »problemformuleringsprivilegiet», som sedan skulle komma att ofta anföras i debatten. Han hade dessutom tillämpat teorin genom att (tillsammans med Myrdal, Fagerberg och Delblanc) rycka åt sig problemformuleringsinitiativet. Det var nu de som förklarade att mediavänstern var problemet snarare än lösningen. Därmed lyckades de som framhållits i anmärkningsvärd utsträckning vända debatten som sedan inte blev sig riktigt lik.
Från »mediavänsterns» sida stod man inte svarslös. Kritikern Jörgen Eriksson anklagade medlemmarna av »de fyras gäng» (Myrdal, Fagerberg, Delblanc och Gustafsson) för att ha »en elitistisk människosyn». Myrdal hade »smitt om sig till ett vapen för den kapitalism som förordar tillväxt till varje pris», menade Karl Erik Lagerlöf. Idéhistorieprofessorn Sven-Eric Liedman skrev en uppmärksammad artikel i tidskriften Zenit (5/1980, senare i bearbetad form i _Frihetens herrar, frihetens knektar. Ideologier på 1980-talet_ , 1981). Där kallade han de fyra fåfänga »påfåglar». Mot dessas position som isolerade intellektuella ställde han en vänster som spelade »en icke obetydlig roll i opinionsbildningen på åtskilliga tidningar», som startat tidskrifter, skaffat sig inflytande i fackförbund: »Motsvarande förändringar, större och mindre, kan konstateras i vänsterpartierna, i ämbetsverken, i skolor och högskolor osv. I sanning – 'den långa marschen genom institutionerna', som Rudi Dutschke talade om, har pågått i flera år redan.»
Påfågelsuppvisningar på den intellektuella scenen spelade i längden mindre roll än den penetrering av ämbetsverk och andra institutioner som vänstern var på väg att genomföra, tycktes Liedman mena. Detta hindrar inte att 68-vänstern 1980 framstod som en smula tillbakaträngd på just den arena – kulturdebattens – där den skördat sina tidiga lagrar.
Den nya situationen avspeglades i en författarenkät med öppna svar som BLM publicerade det året (BLM 1980/1). Det hade gått fem år sedan mitten av 1970-talet då en tidigare avstämning gjorts. Från början hade rubriken varit »In i 80-talet». Men redaktören Hans Isaksson förklarade i sitt förord att man ändrat sig: »Men en vändning i Carl-Göran Ekerwalds svar – 'eller borde vi säga ut i 80-talet, ut i snålblåsten' – tyckte vi var så träffande att enkäten i stället fick heta så.»
När 1980-talet började kände sig vänsterförfattarna sålunda komma ut i snålblåsten. Den skulle komma att vara decenniet igenom och blåsa upp till storm mot slutet av årtiondet när kommunismen föll i Europa. Men stämningarna av pessimism och förtvivlan fanns där redan nu. Detta var årtiondet när vänstern kämpade i motvind men högern tycktes ha medvind, när socialismen krackelerade men marknadsekonomin tycktes gå från framgång till framgång då Kina, Indien och stora delar av den tidigare »tredje världen» anträdde vägen mot snabb tillväxt, för att till sist utbreda sig också i det tidigare socialistiska Östeuropa. I början av 1980-talet handlade det bara om föraningar. Men dessa var smärtsamma. Vänstern var splittrad. Tillförsikten och framtidstron från tio år tidigare var som bortblåst. Isaksson betonade detta i sitt förord. Han satte som ingress ett citat från Expressen (16.12.1979) där Kerstin Ekman talade om sin »vanmakt och förtvivlan» inför utvecklingen. Själv skrev han: »De allra flesta bedömningar av läget nu i början av 80-talet präglas ju av en långtgående osäkerhet och ovisshet.»
Tobias Berggrens enkätsvar vittnade vältaligt om dessa känslor inför decennieskiftet:
In i åttitalet: Fortsatt utbyggnad av högenergisamhället under ett ständigt upptrappat kärnvapenkrigshot. [---] 'Efterkrigstiden' har ju varit en av historiens värsta krigstider. [---] Klyftan mellan rika länder och fattiga har ökat betydligt på sjuttiotalet [---] Världssvälten visar inga som helst tendenser att minska. [---] Den världsrevolution som t ex Mao talade om gick i graven under sjuttiotalet; [---] Den kvarvarande socialistiska kritiken möts endera med totalitära och fascistiska metoder eller genom att man helt enkelt finansierar den, som hos oss. Mediajippon, stipendier, anställningar, förmåner... allt möjligt som gör en kritisk kultur fullständigt meningslös.
Det fanns som synes ingen ände på eländet. Monopolkapitalets lömska förfarande att finansiera sina kritiker fick Berggren att slutgiltigt misströsta. Den svenska vänstern var »splittrad, splittrorna yr omkring i någon sorts centrifugeringsprocess». I periferin fanns till på köpet »halvliberala okunniga mysvänsterister, jagsvaga passiva uppropspåskrivare, infantila reduktionister». Summa summarum: »Sverige är ett smygfascistiskt samhälle; jag måste konstatera, att jag inte så sällan faktiskt hatar det land som jag borde uppleva som mitt, det språk jag diktar på.»
Alla var inte fullt ut lika pessimistiska, ens bland vänsterförfattarna. Carl-Göran Ekerwald var i sitt svar hoppfull. Han trodde att spänningarna växte mellan överheten och folket: »Det är på sätt och vis 60-talets studentrörelse som kommer tillbaka fast nu på en bredare bas.» Ekerwald varnar för de p-piller som makthavarna vill få kvinnorna att äta och som ofta leder till att de »som äter dem faller döda ner på golvet». Makthavarna vill också tvinga på medborgarna fluor i dricksvattnet som ger upphov till cancer. Men nu var en ny proteströrelse på väg!
Författaren Rita Tornborg manade till »gerillakamp mot uppgivenheten». Det fanns ingen »gränspåle mellan sjuttiotalet och åttiotalet», påpekade hon, ingen ridå som går upp för ett scenbyte: »Den första januari står vi där lika förvirrade, lika illa till mods som dagen innan, dignande under samma börda av på varandra staplade 'anti'. Se här bara: antikärnkraft, antimanssamhälle, antikommunism, antiimperialism, antikonsumtionssamhälle.»
Hon var dock optimistisk: »Vi äger makten att omskapa både oss själva och världen.»
Pessimistisk var däremot Birgitta Trotzig. Beträffande socialismen föreföll hon misströsta. Kärnkraften användes av makthavarna på ett cyniskt sätt för att avsiktligt utsätta civilbefolkningen för risker. Striden stod mellan humanism och fascism.
Lars Andersson (BLM 1980/2) ger likaledes uttryck för stark ovilja mot den moderna civilisationen. Vi bär på »förbannelsen av tusen generationers auktoritära arv».
I samma nummer varnade Artur Lundkvist för »det ohejdade tillväxtsamhället med kärnkraft, befolkningsökning och stegrad förbrukning av naturtillgångar». Per Olof Sundman talade om tidens »skepticism och pessmism», men ville motarbeta dem.
Alla tillfrågade författare i BLM:s enkät företrädde inte vänstern, ett begrepp som dessutom, som framhållits, hade hunnit bli oklart. En distanserad betraktelse kom från Lars Gyllensten, som reflekterade över kulturdebattens oberäkneliga svängningar:
Ingen hade kunnat tro att marxismen, som på 30- och 40-talen bland de flesta intellektuella i västerlandet betraktades som en obsolet filosofi, en ideologi som inte höll för den tidens filosofiska, kritiska apparat, skulle bli en dominerande modefilosofi 30 år senare. Få ville sedan tänka sig att marxismen, sådan den florerade i samma västerland på 1960- och 1970-talet, för senare generationer skulle komma att te sig ointressant, oinspirerande, och bland några s k intellektuella ersättas av motsatta strömningar, individualistiska, »ny-konservativa».
Ingen ridå gick upp för ett scenbyte. Men scenbytet var på väg. Ännu några år skulle det dock dröja innan den nya tablån var helt färdiginstallerad.
### _1981 – året när motståndet blev estetiskt._
Sista delen av Peter Weiss stora romantrilogi _Motståndets estetik_ utkom 1981, i svensk översättning från det tyska originalet av Ulrika Wallenström, på VPK-förlaget Arbetarkultur. Första delen hade kommit 1976, andra delen 1979. Peter Weiss var svensk författare i lika hög grad som tysk, och han hade samarbetat nära med översättaren. _Motståndets estetik_ hade, som litteraturvetaren Magnus Bergh framhållit, »en speciell karaktär som tvåspråkigt, svensk-tyskt litterärt verk».
Peter Weiss hade sedan länge en stark närvaro på den svenska litterära scenen och i svensk debatt. Författaren kom till Sverige som flykting 1939, och slog senare igenom även internationellt med pjäsen _Mordet på Marat_ (1964), där revolutionens problematik diskuterats. Senare arbeten hade inkluderat bland annat en pjäs om Vietnamkriget och en om Trotskij. Bland Vietnamaktivister på den svenska parnassen hade Weiss varit en av de mest synliga. I konflikten Vietnam–Kambodja–Kina hade han som nämnts häftigt tagit Vietnams parti. Det hade ställt honom skuldra vid skuldra med Sara Lidman, men i motsatsställning till exempelvis Jan Myrdal. Jan Myrdal hade också tagit itu med Weiss med anledning av dennes ställningstaganden i fråga om Vietnam. Weiss hade försvarat Vietnams invasion av Kampuchea, men vänt sig mot båtflyktingarna från Vietnam. Sympatierna för dessa flyktingar hade han betecknat som en antivietnamesisk psykos spridd »av en press som inte tar under övervägande varifrån flyktingarna fått de tusentals amerikanska dollar, de guldtackor för vilka de köper sig en båtplats». Han hade anklagat Myrdal för att tillhöra »de notoriska bekämparna av Sovjetunionen». Myrdal å sin sida jämförde hans ställningstagande med den opinion mot de judiska flyktingarna från Nazityskland som spreds i vissa kretsar under 1930-talet. Weiss »använder sig nu av samma antisemitiska metoder när det gäller att försvara den vietnamesiska regimens rasistiska politik». Konklusionen blev amper för att handla om en före detta vän: »Jag finner inte bara hans skriverier föraktliga; jag finner hans karaktär föraktlig.»
Inom den svenska vänstern rådde som synes fullt inbördeskrig. Ändå fanns det många likheter mellan Weiss och Myrdal, bland dem upptagenheten av Kominterntraditionen. För dem båda var kommunisternas alla stridsfrågor i det förflutna på ett absolut sätt aktuella, mönsterbildande.
Peter Weiss djupa förhållande till Kominterntraditionen står klart för varje läsare av _Motståndets estetik_. Romanens huvudperson upplever skiftande öden bland annat i inbördeskrigets Spanien och bland Stockholms radikala kretsar på 1930- och 1940-talen. Berättartekniken är invecklad. Händelseförloppet återberättas i andra och tredje hand i ett motsägelsefyllt spel av speglingar.
Det kan sägas – och har sagts av många kommentatorer – att denna estetik röjer Weiss känsla av att verkligheten är komplicerad och undflyende. Även om den raka socialistiska realismen undviks och utsätts för en implicit kritik finns en kommunistisk grundtendens. Romanen kunde uppfattas som riktad mot »tidsandan». Men den diskussion som förts under senare delen av 1970-talet hade inte desto mindre påverkat Weiss. Den marxistiska övertygelsen om att revolutionen skall kunna omgestalta världen till överensstämmelse med förnuftet motverkas av en tilltagande skepsis, en kritik mot det »manliga» förnuftet och en växande pessimism:
Förnuftets lösning [...] hade, på ett kusligt sätt, skymts för honom, och när han nu förnam tomheten i sina ord, så var detta inte bara en följd av insikten att två fientliga system stod emot varandra, att det numer bara handlade om oerhörda makter, styrda enbart av tillväxtprincipen, varvid det inte längre frågades efter folken, snarare var det en följd av vetenskapen om hela den ohejdbara, atavistiska uppladdning som hade bedrivits under en av män uttänkt, av män ända in i den yttersta katastrofen ledd ordning. Ingen tankemöda räckte längre till för att åstadkomma en förändring.
Tilltron till Sovjetunionen och de kommunistiska lösningarna var uppenbarligen inte längre grundmurad hos Weiss. Den trosvisshet han visade upp i sina polemiska tidningsinlägg försvann eller komplicerades i det oändliga i romanverkets labyrinter. En insiktsfull bedömare, Lars Wendelius, har skrivit:
Understrykas bör att det är just den ensidiga positivismen och rationalismen som författaren vill åt. Weiss var marxist och får själv betraktas som exponent för de aktuella strömningarna. Men likt många andra vänsterintellektuella omkring 1980 kände han ett allt starkare tvivel på den teknologifrälsta och tillväxtorienterade socialism som praktiserades i både öst och väst. Mot denna bakgrund avtecknar sig romanens markeringar.
I ett brev till Sara Lidman i januari 1979 framhöll Weiss att han officiellt och utåt alltid stödde Vietnam. Samtidigt uttalade han i detta privatbrev till en nära meningsfrände oro över den chauvinism som kom till uttryck från vietnamesiskt håll. Han befarade att den vietnamesiska regeringen rymde »krafter som faktiskt har en läggning som i långa loppet kanske visar sig vara ödesdiger». Peter Weiss kunde säga nej, men hans nej riktades in i väggen där ingen skulle kunna höra det. Eller kom till utlopp i ett romanbygge där en estetik av oändlig komplikation skapade en frizon, ett bygge där den lyhörde utan svårighet kunde höra ljudet av knackningar, men där ingen med fullkomlig säkerhet kunde lokalisera varifrån de kom. Dock stod det envar fritt att tro att ljuden härrörde från en instängd som bultade.
### _1982 – testamentets år._
När Arne Ruth 1982 blev kulturchef på DN efter Per Wästberg markerades en scenförändring, liksom Per Wästbergs och Olof Lagercrantz tillträden på sin tid hade gjort. Åke Lundqvist skriver härom i sin _Kultursidan. Kulturjournalistiken i Dagens Nyheter 1864–2012_ (2012): »Arne Ruths första år på DN är en liberal rekyl efter femton års hegemoni för socialistiska vänsteranalyser.»
Ruth var i allt väsentligt en vänsterman. Men han släppte in andra synpunkter. En del av dessa kom från dissidentrörelsen i Östeuropa som under dessa år var livlig och starkt påverkade tankeklimatet också i väst. Och under hans tid rekryterade DN kritiker som mindre företrädde marxism än »postmodernism». Horace Engdahl, Anders Olsson, Ingela Lind, Lars O. Ericsson och andra introducerade de nya tankarna. En del av dessa hade redan introducerats i kulturtidskriften Kris, där Engdahl varit den ledande personligheten.
Men även de gamla vänsterskribenterna sökte nya utvägar. Eller gjorde upp med de gamla. En sorts 68-vänsterns testamente kom med Anders Ehnmarks och Per Olov Enquists »detektivroman» _Doktor Mabuses nya testamente_. Den fortsatte satiren från _Mannen på trottoaren_ men i nya vändningar. Vänsterns absurditeter skärskådades från olika infallsvinklar.
En bild av den intellektuella scenen i Sverige sådan den tedde sig 1982 fick man genom en serie intervjuer för tidskriften Zenit (»Den intellektuella scenen. Samtal med Anders Ehnmark, Gunnar Fredriksson, Lars Gustafsson, Per Lysander och Agneta Pleijel» av Gunnar Andersson, Lennart Berntson och Svante Nordin, Zenit nr 74 1/1982). Avsikten var att skildra förändringarna i den svenska opinionen under efterkrigstiden med tonvikt på vänsterns framgång och tillbakagång sedan mitten av 1960-talet samt att försöka teckna det innevarande läget i början av 1980-talet. Det blev en form av slutlikvid eller åtminstone dellikvid från de inblandades sida.
Anders Ehnmark kommenterade vänsterns tillbakagång och de nya liberalernas framgångar i den intellektuella debatten. Men han framhöll också en svaghet i liberalernas position: »Men det finns inte så många liberala intellektuella som förr: liberalerna har en historisk möjlighet som de av personliga skäl kanske tvingas hoppa över. Eller hur man skall säga. Vänstern har behållit sitt starka grepp om de intellektuella, även om man inte har grepp om så mycket annat, och det präglar det offentliga rummet.»
Ehnmark poängterade också den tröghet i svängningarna som gjorde att 40-talisterna med sin vänsteråskådning satt kvar på de positioner som de en gång erövrat och att den »högervind» som just blåste kunde förväntas få effekter först på 1990-talet. Samtidigt kritiserade han vänsterns reflexmässiga avvisande av allt som kunde påstås »komma från höger», även där det egentligen borde kunna ses som vänsterståndpunkter. Ett exempel var skoldebatten, där vänstern vände sig mot de »reaktionära» kraven på att skolan skulle förmedla kunskaper. Ett annat var kultursynen där den »kulturdemokratiska» linje som företräddes av vänstermän som Bengt Nerman eller Göran Palm gav »en kultursyn som kan illustreras med bildningsförbundens programhäften: fiske, poesi etc. om vartannat», inte den kultursyn som borde vara vänsterns, nämligen synen på »kulturen som en erövrad överblick».
Hade vänstern numera försvunnit, gått sin väg? undrade intervjuarna. Ehnmark svarade: »Gått sin väg är väl fel att säga, passiviserats snarare. Vi har fått en vänster i disponibilitet, där vi förut hade en arbetande vänster, som syntes på gatorna. I grunden beror det på att vänstern saknar ett politiskt projekt.»
Det gamla projektet – den socialistiska revolutionen – hade försvunnit när svärmeriet för Kina, Kuba, Albanien, Vietnam inte längre kunde hållas uppe.
Vänstern hade också fått en ny hjälte, inte längre den gamla revolutionsberedde proletären. Ny Dags hjälte, sådan bilden framtonade i spalterna, var numera »en akademiker, välbetald men juste, offentliganställd och en kompromisslös försvarare av den offentliga sektorn, cirka 35 år, kvinna, pacifist, miljövän på gränsen till teknikfiende, ungefär så».
Vänsterns debatteknik när det gällde exempelvis Sovjetunionen beskrev Ehnmark med anledning av den aktuella diskussionen om den sovjetiska ubåt som strandat i Blekinge skärgård: »Gör Sovjet något galet, så säger man att 'nu jublar reaktionärerna', och så flyttas uppmärksamheten bort från det pinsamma området till de gamla vanliga, trygga fienderna. Likafullt ligger ubåten där.»
»Mediavänster» ville Ehnmark inte tala om, men om ett politiskt styrande skikt som även innefattade det offentliga rummets personal. »Den nya politiska klassen» vore kanske en bättre benämning, framhöll han.
Ehnmark var en företrädare för 1960-talets nyvänsterkonjunktur. Gunnar Fredriksson var tidigare chefredaktör för Aftonbladet med stark förankring i socialdemokratin, men samtidigt med nära band till den socialdemokratiska vänsterflygel som vuxit fram delvis under intryck av den nya vänsterns stämningar. Han utvecklade initierade synpunkter på socialdemokratin i dess förhållande till 68-vänstern. Under 1960-talet skedde en tillströmning av kulturarbetare till SAP: »Ja fram till 1968, som markerar höjdpunkten. Då hade partiet den största dragningskraften på de intellektuella, efter 68 minskar det igen.»
Som orsaker till denna minskning angav Fredriksson dels att nästan hela Vietnamrörelsen stod utanför socialdemokratin, dels de impopulära universitetsreformerna. Han framhöll hur missen i fråga om Vietnamrörelsen oroat socialdemokratins ledning, särskilt Olof Palme: »Palme vill uppenbarligen till varje pris undvika att hamna i samma läge i förhållande till fredsrörelsen. Jag tror han ser en parallell mellan Vietnamrörelsen och fredsrörelsen. Invändningar mot t ex en kärnvapenfri zon väger lätt, eftersom han ville vara med politiskt.»
Samtidigt medgav Fredriksson svårigheterna: »Det är en stor skillnad i politisk kultur mellan de spontana rörelserna och socialdemokratin.»
Den västtyska socialdemokratins betydelse för Palme och SAP, särskilt i internationella frågor, var något som Fredriksson betonade. Willy Brandt var beredd att stödja fredsrörelsen, distansera sig från USA och närma sig Sovjetunionen. Fredriksson hade själv sympatier för dem som ville möta det eventuella hotet från Sovjet genom tillmötesgående: »Man kan agera som t ex Willy Brandt och Finland, eller som Reagan och Haig, och i det senare fallet blir hotet större.»
För egen del trodde Fredriksson också på att 1980-talet skulle komma att se ett förbättrat förhållande mellan socialdemokratin och de intellektuella i ljuset av bland annat fredsfrågan.
En lång intervju med Lars Gustafsson öppnade perspektiv som skilde sig från såväl Ehnmarks som Fredrikssons. Gustafsson befann sig numera gott och väl i det liberala lägret, men hade under ett par decennier spelat en viktig roll som deltagare också i vänsterdebatten. Intervjun rörde sig till en del om Gustafssons egen ideologiska utveckling och om 1950-talets debatter. Men några intressanta observationer beträffande 68-vänstern blev det också. Liksom Fredriksson framhöll Gustafsson den snabba förändring i inställningen till socialdemokratin som ägde rum bland de intellektuella under 1960-talet: »Man kan i riksdagsvalet -64 uppleva något som vi aldrig upplevt tidigare, hur de intellektuella i stora annonser i tidningarna stöder, förordar röstning på socialdemokratin. Det är alldeles omöjligt i riksdagsvalet därpå, för då har de intellektuella som ville stödja socialdemokratin -64, övergått till – jag vill inte säga kommunistsympatier [...] Men då ligger sympatierna redan till vänster om socialdemokratin.»
Under den följande perioden hade också Gustafsson uppfattat sig som vänster, något som inte minst Jan Myrdal botat honom från: »Ingen människa har mer hjälpt mig att bli – vad skall vi kalla det? – bli antisocialistisk, antiutopisk än Jan Myrdal.»
Detta hade Myrdal uppenbarligen åstadkommit både genom sin kritik mot den sovjetvänliga vänstern och genom sina egna snabba förflyttningar från position till position. Gustafsson kom fram till ett avvisande av marxismen och av det utopiska tänkandet: »Totalitetsanspråken är ett utopiskt tänkande, jag talar inte längre bara om marxismen, det finns ju andra utopier. Totalitetsanspråken utesluter just arbetet med konkreta delar av verkligheten, därför att detaljerna blir ju oviktiga om du emotser Messias ankomst.»
Samtidigt betecknade Gustafsson de stora utopiska ideologiernas sammanbrott som en »besvikelsekatastrof».
Hur 68-vänstern genererat följdfenomen beskrev Gustafsson utifrån en påfallande mekanism: »Kring varje allvarlig konflikt uppstår skuggfenomen. Frantz Fanon säger att de undertryckta folken måste resa sig genom sin egen kraft. Strax är någon där och säger att de homosexuella är ett undertryckt folk som på liknande sätt måste resa sig mot sina kolonisatörer, vilket är en vansinnig analys av problemen. Strax är någon där och säger att kvinnorna är en klass som är förtryckta av männen.»
Dessa frågor om 68-rörelsens skugg- eller svallvågsfenomen kom upp till diskussion också i den avslutande intervjun med Agneta Pleijel och Per Lysander. Båda var starkt kritiska mot vänsterns utveckling, detta utifrån en position inom vänstern och med argument som ter sig karaktäristiska för det tidiga 1980-talet. De såg nyvänstern som en kort period präglad av »drömmen om en verklig socialistisk frihet», en dröm som kvävdes när nyvänstern splittrades upp i sekter »som i sitt bekämpande av varandra utförde en snabb repetition av 1900-talets kommunistiska teoristrider» (Pleijel). En stalinistisk form hade gradvis och alltmer förkvävande slutit sig »runt det anti-auktoritära och frihetliga innehållet» (Lysander).
Både Pleijel och Lysander var eniga om att en skarp kritik mot Sovjetkommunismen, delvis i maoistisk eller castroistisk anda, från början präglat nyvänstern, men att denna kritik steg för steg dragits tillbaka. Lysander talade om »en sorts kräftgång» där man »givit upp position efter position i kritiken mot Sovjet» och att man därför stod »väldigt oförberedd inför det som skedde i Polen». Pleijel betonade att den kulturrevolutionära maoismen liksom den oberoende castroismen fallit »bort som alternativ och kritiken mot Sovjet underminerades». Solidaritets framträdande i Polen hade därmed, menade Lysander, kommit att beröra »en bultande och öm punkt, som man hittills inte förmått bearbeta på något vettigt sätt».
Inom teaterområdet, där både Lysander och Pleijel varit verksamma, ville de urskilja en liknande motsättning mellan en antiauktoritär impuls och framväxten av auktoritära åsiktspoliser inom vänstern, vars uppfattningar till på köpet spridit sig långt in i den offentliga byråkratin. Lysander gav ett talande exempel: »Man kan bli mörkrädd, när man återser argument och resonemang, som formulerades i vänsterns tidskrifter för 5–10 år sedan, nu utklädda till statliga dekret och påbud. Instanser som Barnfilmrådet t ex hävdar att Pippi Långstrump är olämplig för barn och gör det med stöd av våra vällovliga analyser.»
De nya civilisationskritiska rörelserna, kvinno-, miljö- och fredsrörelserna, hälsades av båda som en förnyelse av vänstern, men med vissa reservationer. Lysander ansåg det exempelvis naivt av fredsrörelsen att »gå i fredsmarsch till Paris arm i arm med Moskvakommunisterna».
Den intellektuella scenen 1982 befolkades som synes till stor del av en vänster som tappat tilltron till sina ursprungliga idéer och som befann sig i ett läge av villrådighet och desorientering. Men den var inte död. Den befann sig som Ehnmark uttryckte det »i disponibilitet», men hade fortfarande ett starkt »grepp om de intellektuella, även om den inte har grepp om så mycket annat». Den långa marschen genom institutionerna hade anträtts. Det man samlat in i visthuset under de goda åren hade möjliggjort överlevnad under de svåra åren. Vänstern hade tappat det mesta av problemformuleringsinitiativet, men dess institutionella förankring var stark. En påminnelse om detta kom i början av 1983 genom Gösta Hulténs och Jan Samuelssons bok _Mediavänstern. En närbild av dendolda åsiktsproduktionen i Sverige_. Den gav – närmast inspirerad av fyrklövern Myrdal–Fagerberg–Gustafsson–Delblanc – en bild av vänsterns grepp om svensk offentlighet. Författarnas hållning till mediavänstern var liksom de fyras skarpt avvisande: »Vi ser mediavänstern som vår nya överhet. [---] Mediavänstern utgör ett hot mot demokratin, därför att den driver en politik som inte överensstämmer med befolkningsmajoritetens uppfattningar och därför att den strävar efter kontroll över massmedia och därmed monopol på åsiktsproduktion.»
Författarna framhöll att mediavänstern var en utpräglad storstadsföreteelse: »Dess starkaste fäste är den stockholmska massmediavärlden, men förgreningarna når också till övriga större städer.»
Den hade ett viktigt stöd inom den offentliga förvaltningen. Hultén och Samuelsson citerar en entusiastisk DN-journalist: »Jag möter just de här engagerade människorna: avdelningsdirektörer, byråchefer och avdelningschefer: Lite över byrådirektörer, där börjar engagemanget komma.»
De båda författarna gör en inventering över mediavänsterns viktigaste stödjepunkter sådana de såg ut vid 1980-talets början: »Televisionen – särskilt TV2:s fakta-, utrikes-, kultur- och barnredaktioner. Regional-TV. Riksradion – särskilt dess samhälls-, kultur- och nyhetsredaktioner. Lokalradion. Utbildningsradion.»
Till stödjepunkterna räknades också Aftonbladets, Stockholms-Tidningens, Sydsvenska Dagbladets och Expressens kultursidor liksom Svenska Dagbladets familjesida. Vidare ETC, »ett exempel på kolorerad veckopress av en ny typ som säljer på modeåsikter». Dessutom gamla kulturtidskrifter som Ord & Bild och BLM, som tagits över av vänstern, kvinnopolitiska tidskrifter. Så ock en flora av specialtidskrifter inom film, konst, teater och undervisning. Tidskrifter för sjukvårdsdebatt, barnkultur eller socialpolitiska frågor. Bulletiner för Afrika-, Chile-, Kuba- och andra rörelser. Fredsrörelsens olika bulletiner och tidskrifter, delar av fackförbundspressen:
Till mediavänstern bör också räknas den del av den offentliga byråkratin som främst är åsiktsproducerande, t ex Statens Kulturråd, Statens Ungdomsråd, Statens Institut för Läromedelsinformation, Svenska Barnboksinstitutet, Handelsflottans Kultur- och Fritidsråd, Arbetslivscentrum, Riksutställningar, Riksteatern samt delar av Socialstyrelsen, Skolstyrelsen, Konsumentverket, Sida och Sipri.
Vidare de olika kulturarbetarförbunden, delar av förlagsvärlden med mera.
Universiteten och Svenska Kyrkan nämndes inte i denna uppräkning. Däremot framhöll författarna i hur hög grad hela journalistkåren – och därmed indirekt hela nyhetsförmedlingen – var präglad av vänsteråsikter. En undersökning från 1980 anfördes: »Bland eleverna vid journalisthögskolan röstade vid samma tid 55 % på VPK. Bland de Göteborgsjournalister som gått igenom journalisthögskolan röstade 60 % på VPK.»
I den mån Hultén och Samuelsson hade rätt i sin genomgång – och det torde vara svårt att bestrida riktigheten av den allmänna bilden av nyvänsterns ställning inom offentligheten – hade den nya vänstern trots sin splittring och desorientering i många frågor vid denna tid en stark institutionell förankring. Det var en dubbelhet som skulle komma att bestå under de följande decennierna.
## **III. FARVÄL TILL VÄNSTERN?**
## **Sammanfattning och slutsatser om perioden 1968–1984**
ÅR 1984 SAMLADE SIG Svenska Dagbladet till en artikelserie kring temat »Farväl till vänstern?». Inläggen utkom samma år i bokform. Brännpunktredaktionens Fredrik Braconier skrev i sitt förord bland annat:
För femton år sedan dominerade vänstern opinionsbildningen i Sverige. [---] Idag förefaller situationen radikalt annorlunda. Vänstern märks föga i den allmänna debatten. En del av 60-talets och 70-talets mer framträdande opinionsbildare på vänsterkanten har bytt sida. Under de senaste åren har istället högern – eller vad man nu skall kalla den – befunnit sig på offensiven.
Braconier ställde frågan om orsaken till opinionsförändringen och undrade vart vinden skulle blåsa nästa gång. Var det dags att säga farväl till vänstern?
De inbjudna debattörerna gav naturligt nog skiftande svar på de frågorna. Anders Ehnmark ansåg det förhastat att säga farväl. Men han medgav att vänstern i det nya läget saknade idéer om framtiden och att dess kris var verklig och sannolikt skulle bli långvarig. Sven-Eric Liedman uppfattade ingen kris. I stället hyllade han vänsterns stora projekt, »den kollektiva friheten». Stefan Lindgren med sin SKP-förankring menade att kapitalismens problem kvarstod och att tvärtemot vad hans forna partikamrat Bo Gustafsson hävdade var utvecklingsländerna på väg att bli allt fattigare mätt i exempelvis exportsiffror. Historieprofessorn Kristian Gerner förvånades över vänsterns fortsatta stöd för Sovjetunionen. Vänsterjournalisten Fritiof Haglund var självkritisk. Han tog avstånd från Che Guevaras och de nicaraguanska sandinisternas våldsromantik. Den latinamerikanska vänster som länge hyllat folkkriget och den heroiska gerillan var inne i en period av omprövning som även påverkade de journalister i Europa som likt Haglund noga följde utvecklingen i Sydamerika. Journalisten Göran Rosenberg menade att roten till det onda för vänsterns del fanns i själva den utopiska vision som tycktes kunna legitimera de mest förfärliga handlingar. Joachim Israel välkomnade en mer okonventionell vänster medan Göran Therborn försäkrade Svenska Dagbladets läsare att »historiens hjärta slår till vänster» (även om han medgav att pulsen just då var svår att få fatt på). Kjell E. Johansson, gammal företrädare för en vänstersocialistisk linje inom VPK, krävde en definitiv uppgörelse med leninismen. Vänsterförläggaren Bo Cavefors fann uppdelningen i vänster och höger meningslös men hyllade en tradition av »uppror», »förvandling» och »utveckling» som rymde inte bara Josef Stalin, Rosa Luxemburg, Ulrike Meinhof och Karl Marx utan också Per Engdahl och Ernst Jünger. Det var ett intressant inlägg som förebådade inte bara Cavefors personliga utveckling, utan också en utveckling hos vänstern som skulle få den att mer och mer överge vänsterradikalismens ikoner till förmån för högerradikalismens som Nietzsche, Heidegger, Jünger och Carl Schmitt. C. H. Hermansson efterlyste »hållbara och realistiska visioner» inom vänstern. Filosofen och fysiologen Germund Hesslow trodde att vänstern skulle kunna återvinna sin styrka, något som han beklagade. Han fann att suddigheten beträffande vad »socialism» egentligen var och det vaga talet om alternativ till marknaden och staten gav vänstern möjlighet att »fly undan ansvaret för sina idéer».
Detta sista skulle, som vi skall återkomma till, senare under 1980-talet komma att underlättas genom kommunismens fall, som befriade den socialistiska utopin från omedelbar konfrontation med den socialistiska verkligheten.
År 1984 var det, som Anders Ehnmark framhöll, förhastat att säga farväl till vänstern. Den skulle komma att överleva i nya former och med delvis starkt förändrad ideologi. Men 1980-talets början var ett passande tillfälle att säga farväl till 68-vänstern, till den nyvänster som vuxit fram under 1960-talet, som haft sitt euforiska ögonblick 1968, som haft sitt starkaste grepp över debatten och sin starkaste hegemoni bland de intellektuella under det tidiga 1970-talet och som sedan under 1970-talets senare och 1980-talets tidiga år allt tydligare framstått som splittrad, ideologiskt desorienterad och debattmässigt stadd på reträtt. Denna vänster försvann nu för att aldrig återkomma i sin gamla form. Den hade föregåtts av andra »vänstrar» och skulle efterföljas av senare. Men den hade i sig själv en unik karaktär som inte kunde upprepas. Den utkämpade sin sista strid, formerad i de gamla förbanden, i samband med »de socialistiska krigen» 1978–1979, varvid dock de förband som tidigare gemensamt marscherat mot borgarklassen nu gjorde upp inbördes. Beteckningar som »sextioåttor», om dess adepter, hade vunnit burskap. De omgavs av en nimbus och en folklore som blev bestående. Det finns skäl att i några punkter försöka ringa in det utmärkande för denna »68-vänster» och dess utveckling så som den såg ut i Sverige.
1. Vad skilde »den nya vänstern» från »den gamla vänstern»? Vad skilde revolten 1968 från tidigare revolter och revolutioner i historien? Ett svar faller i ögonen – dess klasskaraktär. Vad som gjorde detta till (med Raymond Arons ord) en »säregen revolution» var att den genomfördes av studenter, akademiker och författare, därmed av en av de mest välmående och privilegierade grupperna i länder som själva var mer välmående och privilegierade än några andra historiskt kända samhällen (vare sig förflutna och samtida). Det rörde sig med andra ord inte här om en revolution motiverad av nöd och förtryck, utan om en revolution stimulerad av överflöd och privilegier. Det handlade inte om en revolt mot fattigdomen utan om en revolt mot rikedomen, mot »överflödssamhälle», »konsumtionssamhälle» och »prylsamhälle». Det var inte en revolt av de utarmade, okunniga och förtryckta utan av de välnärda och välutbildade, av personer som genom härstamning och framtidsutsikter tillhörde samhällets toppskikt. I det förflutna hade visserligen sådana personer ofta tillhört de revolutionära rörelserna. Personer som Lenin, Trotskij, Mao och Pol Pot hade visserligen härstammat från sina länders överskikt. Men deras anhängare hade varit hårt arbetande proletärer och fattiga bönder. I 68-rörelsen var både ledare och efterföljare studenter och akademiker. Arbetarklassen inte bara lyste med sin frånvaro. Den tog i stor utsträckning ställning mot studenterna och betraktade i det stora hela 68-rörelsen med en klassmässigt motiverad motvilja. Arbetarromantiken frodades inom 68-rörelsen. Men den hade till revolutionärernas sorg ingen förankring i verkligheten.
En blick på 68-rörelsens idéer ger också upphov till slutsatsen att dessa idéer i allt väsentligt bar prägel av rörelsens klasskaraktär, det vill säga var idéer av en typ som man skulle vänta sig att återfinna hos humanistiskt, samhällsvetenskapligt eller teologiskt inriktade studenter och akademiska skikt men knappast hos arbetare. I detta ligger också en av nycklarna till rörelsens utveckling. Ju mer tiden gick, desto tydligare trädde klasskaraktären fram.
Den klassiska revolutionen eller revolten vänder sig mot missförhållanden, diskriminering eller förtryck av något slag som drabbar de revolterande. De privilegierade ungdomarnas 68-revolution hade däremot en väsentligen ställföreträdande karaktär. Det gällde att revoltera mot tredje världens lidanden eller de lidanden som antogs drabba arbetarklassen eller utsatta grupper i det egna landet. Det var med andra ord fråga om en filantropisk revolution, driven av deltagarnas dåliga samvete, »solidaritet» med de förtryckta och önskan att göra gott.
Men det handlade också om en leda vid själva överflödet sådan den kan drabba de välmående. Anblicken av överflöd och lyx, även när dessa utvecklades inom arbetarklassen i form av bilar eller olika prylar, gav upphov till motvilja bland puritanskt inställda ungdomar. En revolt inom arbetarklassen riktad mot konsumtionssamhället skulle vara otänkbar. Likaså en arbetarklassrevolt riktad mot tillväxtsamhället. Arbetarklassen kan knappast tänkas strejka eller demonstrera för lägre lön eller minskat välstånd. Att göra så är ett privilegium för de privilegierade. Det liknar Marie Antoinette i Trianon mer än sansculotterna. Det liknar ett rousseauanskt uppror mot civilisationen, odlat av unga herrar och damer ur överklassen, mer än ett proletärt uppror. Den litterära intelligentian utgör enligt C. P. Snows berömda analys av »de två kulturerna» av en grupp av »naturliga maskinstormare», fiender till teknik och utveckling. Snow syftade på högerradikala brittiska intellektuella 1959, men något liknande kunde sägas om de vänsterradikala svenska intellektuella ett årtionde senare. »Ludditerna», maskinstormarna, fanns där fortfarande. Där fanns också gränserna för deras »solidaritet». När hundratals miljoner människor i framför allt Asien lyftes ur svält och fattigdom genom en industriell och teknisk revolution på världsmarknadens villkor blev vänsterns reaktion surmulet fientlig. Varför skulle man också egentligen glädjas om människorna i »tredje världen» genom ekonomisk tillväxt kom att dela det välstånd och det »konsumtionssamhälle» som 68-revolten ursprungligen riktade sig mot?
Motviljan mot den moderna civilisationen sådan den tedde sig för ungdomar inom de högst utvecklade ländernas mera välbärgade samhällsskikt var från första stund på detta sätt ett väsentligt inslag i 68-rörelsen. Det handlade om en »livsstilsrevolt».
2. Men om 68-rörelsen bara hade varit en rousseauansk maskerad bland över- och medelklassungdomar skulle dess mer sinistra aspekter ha uteblivit. Där fanns emellertid från början också något annat, en vilja att knyta an till den gamla vänstern, framför allt i form av den kommunistiska traditionen. Och där fanns en vilja att alliera sig med kommunistiska rörelser, partier och stater i samtiden. Detta räddade 68-rörelsen från att vara enbart ungdomsuppror eller studentspex. Det gav den samtidigt karaktär av stöd och hjälp åt totalitära krafter och åt några av historiens värsta massmördande regimer. Och det förde in element av avsevärd sprängkraft i dess ideologi.
Redan tidigt tillhandahöll marxismen den nya vänsterns viktigaste teori och ideologiska schema. Men det marxistiska arvet var mångtydigt. Där fanns exempelvis den så kallade »västmarxismen». Den innehöll gestalter som Lukács, Benjamin, Sartre, Frankfurtskolans frontfigurer som Marcuse, Habermas, och många andra. Denna västmarxism var högintellektuell, filosofisk och litterär. Den tenderade mot pessimism och mot kritik av framstegsidén. Den var obegriplig för arbetarklassen men vädjade till intellektuella och akademiskt bildade. Den producerade böcker och artiklar snarare än strejker och revolutioner. Den passade 68-vänsterns klasskaraktär som hand i handske.
Därtill fanns »östmarxismen», utvecklad av Lenin, Stalin, Mao, Castro, Ho Chi Minh med flera. Den gav tillgång till proletariatets kamp, till tredje världens befrielsefronter, till den förverkligade utopin i Kina och Nordvietnam, Nordkorea och Kuba. Den innehöll även annat, som järnhård diktatur, väldiga militärapparater och ledarkult. Den bar på en övertygelse om en socialistisk framtid som skulle byggas genom stålverk, kraftverk (inklusive atomkraftverk), maskiner, ständigt stegrade mål för produktionen.
Förbindelsen mellan dessa två traditioner visade sig i längden svår att upprätthålla trots att det fanns förmedlare: Gramsci som varit både kulturfilosof och kommunistledare, Althusser som visat på förbindelsen mellan den kinesiska revolutionen och en strukturalistisk vetenskapsteori, Trotskij som varit bolsjevikledare för att sedan bli rebell i exil.
3. Hos svärmeriet för den kommunistiska traditionen fanns en annan sprickbildning av stor betydelse, inte mellan västmarxism och östmarxism utan mellan östmarxism och östmarxism, det vill säga mellan Moskvamarxism och Pekingmarxism. Chrustjovs tal mot Stalin vid det sovjetiska kommunistpartiets tjugonde partikongress blev upptakten till den sino-sovjetiska schism som sedan rev sönder världskommunismen. Den blev också upptakten till vad som med tiden skulle bli den nya vänstern. Den maoistiska riktningen skilde ut sig från kommunismen i övrigt. Den erbjöd möjligheten att kritisera Sovjetunionen »från vänster», den erbjöd möjligheten att kritisera Moskvakommunismen som revisionism i namn av en renare kommunism och marxism, en möjlighet som bland andra althusserianismen tog fasta på. När sedan den kinesiska kulturrevolutionen utspelades under 1960-talet kunde kritiken vidgas till att gälla hela den sovjetiska utvecklingsmodellen med dess storskalighet och inriktning på tung industri. Kulturrevolutionen kom på så sätt att spela en avgörande roll för den nya vänsterns utveckling i Västeuropa och USA. En om möjligt än större roll spelade Vietnamkriget. Både maoismen och FNL:s kamp kunde tolkas som en del av den så kallade tredje världens kamp mot »den rika världen». Detta passade den nya vänsterns »ställföreträdande» klasskaraktär. Det var mot sig själva, mot sitt eget samhälle man revolterade. Det handlade om en sorts ädelmodets revoltörer som drevs av solidaritet och dåligt samvete. Det handlade inte som i den klassiska marxismen om en arbetarklass vars befrielse är dess eget verk utan om en idealistisk, oegoistisk, filantropisk revolution där belöningarna var gott samvete och övertygelse om den egna moraliska överlägsenheten. Att de rörelser, partier och stater man idealiserade befann sig långt borta, på andra kontinenter, blev en fördel ur denna synpunkt. Den gamla kommunismen hade inspirerats av händelser i närområdet – Ryssland, Tyskland, Finland, Östeuropa. Nu handlade det om Kina, Vietnam, Nordkorea, Kuba, något som bidrog till att ge det hela en exotisk, lätt overklig karaktär som passade en rörelse för studenter och skönlitterära författare. Verkligheten var blodig för revolutionernas offer, men inte för de västerländska studenterna som revolterade och uttryckte sin »solidaritet» skyddade av rättsstaten i de egna hemländerna.
Den speciella arten av 68-vänsterns »solidaritet» med tredje världens folk bör i sammanhanget noteras. Det fanns ingen solidaritet med offren för Mao Zedongs utrensningar och svältkatastrofer, för Kim Il Sungs eller Pol Pots terror eller för flyktingvågorna från Vietnam eller Kuba. När tredje världens folk så småningom lyftes ur svält, sjukdomar och misär genom en utveckling på världsmarknadens villkor och genom ökad tillgång till modern vetenskap och teknologi, hördes inga applåder från vänstern. »Solidariteten» gällde egentligen aldrig de fattiga i fjärran världsdelar, som man i grunden inte visste mycket om. Den var avigsidan av det hat mot det egna samhället, den vantrivsel i kulturen, som drev 68-revoltörerna till ungdomlig frenesi.
4. Samtidigt fanns på närmre håll en tydlig, ehuru inte alltid fullt medveten geopolitisk bakgrund. Den nya vänsterns antiamerikanism utspelade sig mot bakgrund av USA:s nederlag i Vietnam och försvagade position och mot Sovjetunionens stärkta militära ställning och territoriella framstötar i olika delar av världen. Maktbalansen kunde under 1970-talet se ut att vara på väg att tippa till Sovjetunionens fördel. Detta utlöste inom exempelvis de svenska och västtyska socialdemokratierna anpassningsrörelser som ingalunda var motiverade enbart av behovet att fånga upp en ny vänsterradikal och USA-fientlig ungdomsgeneration. Det handlade också om att finna sin plats i ett nytt läge där Sovjetunionen kunde bli den förstärkta makten i Europa och USA den försvagade. En nyordning som ibland döptes till »finlandisering» var påtänkt, varvid svensk socialdemokrati under Olof Palme framstod som en intressant exponent.
Sovjetunionens expanderande makt utlöste emellertid motreaktioner. En del kom efterhand med allt större kraft från den amerikanska högern (Reagan) och från den brittiska högern (Thatcher). En del kom från Kina med den åldrande Mao Zedong och senare Deng Xiaoping. Ur överenskommelsen mellan Nixon och Mao utvecklades en triangulering av världspolitiken. Men det som hände i den geopolitiska stratosfären påverkade också opinionsvindarna på marken, till och med i ett så perifert land som Sverige. Den svenska vänstern sprack itu under påfrestningarna. Den från början tongivande Kinasympatiserande delen av den svenska nyvänstern lämnade vänsterburen och korsade strecket mellan vänster och höger. Jan Myrdals utveckling ter sig representativ. Därmed var den ursprungliga 68-vänstern sprängd i bitar.
5. Men också på andra sätt förändrades och försämrades villkoren för en vänster längs de gamla linjerna. Nyliberala och nykonservativa rörelser gjorde sig gällande även i Sverige, delvis som del av en rekyl mot 68-vänsterns framgångar. Tankesmedjan Timbro är ett exempel på en form av intellektuell höger som inte kan tänkas utan en föregående intellektuell vänster. Samtidigt rörde sig opinionen i Sovjetunionen och Östeuropa på ett sätt som innebar något nytt. Gestalter som Solzjenitsyn, Sacharov, Havel, Kuron, Konrád, rörelser som Solidaritet i Polen och Charta 77 i Tjeckoslovakien påverkade även svensk opinion och ställde till svårigheter för den sovjetvänliga »fredsopinionen», svårigheter som förstärktes genom den maoiserande kritiken. De franska nya filosoferna och deras inflytande i Sverige måste ses i detta ljus liksom till viss del »eurokommunismens» Sovjetkritik. Den sovjetiska framstöten alstrade såväl friktion som motkrafter. Palmes brobyggande till Sovjetunionen uppfattades inte längre som självklart progressivt. Den sovjetiska invasionen av Afghanistan liksom de föregående krigen i Indokina bidrog till att förändra perspektivet.
6. Efter mitten av 1970-talet var egentligen varken Kina eller Vietnam brukbara för vänstern som realexisterande utopier eller som garanter för att socialismen verkligen bär lycka till alla. Bortfallet av utopin innebar ett bortfall av Alternativet. Det innebar också bortfallet av garantin för att historien skulle komma att sluta lyckligt med det klasslösa samhället. Därmed sveks tron på historien, tron på framsteget, tron på förnuftet, tron på verkligheten själv. Den pessimism och brist på tilltro till vetenskap och tillväxt som funnits latent redan i den tidiga nyvänstern kunde nu framträda, ohämmad av den avlagda marxistiska historieteorin.
1980-talet blev mycket riktigt ett snålblåstens årtionde ur vänsterns synpunkt, precis som Carl-Göran Ekerwald fruktat. De nyliberala och nykonservativa vindarna blåste. Margaret Thatcher i Storbritannien och Ronald Reagan i USA satte sin prägel på skedet. I Kina slog man alltmera beslutsamt in på Deng Xiaopings marknadsorienterade ekonomiska politik. I Sovjetunionen valdes Michail Gorbatjov till generalsekreterare för landets kommunistiska parti och han slog in på den reformkurs som skulle komma att leda till kommunismens fall. I Sverige kunde vänstern göra föga mer än klaga över »högerkrafternas» framfart. Postmodernismen, som nu blomstrade, blev framför allt en tillflykt undan den politiska verkligheten (eller verkligheten överhuvud taget). För en tid åtminstone hade vänstern förlorat problemformuleringsinitiativet.
Antiimperialismens ekonomiska teori (beroendeskolan) hade gått i kvav. Inte bara »tigerekonomierna» (Taiwan, Sydkorea med flera) växte nu på kapitalismens mark. Också Indien hade slagit upp portarna mot världsmarknaden och odlade en framgångsrik, delvis exportdriven tillväxt. De fattiga hade blivit »tillväxtekonomierna». Det var en kollaps för vänsterns ekonomiska tänkande som bara nödtorftigt kunde kompenseras genom postkolonialismens övningar inom litteraturteorin. Att demolera exempelvis Jane Austens romaner blev en ringa kompensation för att demolera världsmarknaden.
Ändå fortlevde vänstern. Dess utveckling fram till kommunismens fall och ett stycke därefter blir föremålet för nästa del.
## **2**
## **Vänstern 1987–2000**
## **IV. PROLOG**
VID 1980-TALETS MITT präglades den svenska vänstern av misströstan och stagnation. Den radikala agendan från tidigt 1970-tal hade sedan länge trängts undan och den politiska debatten i Sverige dominerades av nyliberala krav på avregleringar och privatiseringar och av strävan efter penningpolitisk stabilitet i stället för keynesiansk arbetslöshetsbekämpning.
Den internationellt mest uppmärksammade förändringen ägde rum i Sovjetunionen där Michail Gorbatjov 1985 övertog det politiska ledarskapet. Från att ha varit den korrekta lärans högborg blev Moskva plötsligt det kommunistiska nytänkandets centrum. Gorbatjov proklamerade förnyelse och öppenhet i inrikespolitiken samt sökte samförstånd med Reagans USA. Allmänt sett hälsade den europeiska vänstern förändringen välkommen. Så var fallet även för VPK:s vidkommande. Partiet hade 1985 blivit parlamentariskt underlag för den socialdemokratiska regeringen. Det ökade politiska inflytandet hade emellertid inte stärkt utan snarare söndrat partiet. Striden stod mellan en mer demokratisk och reforminriktad falang och en mer ideologiskt renlärig som ville säkerställa partiets kommunistiska profil. Den motsättningen hade sett dagens ljus redan 1964 och bars nu vidare av 68-generationens aktivister. Trots förändringens vind i VPK:s omvärld segrade ortodoxin vid partikongressen 1987. Partimedlemmen och professorn i sociologi Göran Therborn som tillhörde den förlorande sidan menade att konflikten hade stått mellan en »bred historisk vänstertradition och en mer marxist-leninistisk».
I det nya partiprogrammet betecknade sig VPK som ett marxistiskt parti med rötter i den revolutionära vänstertraditionen: »VPK är ett marxistiskt parti. Det bygger på den revolutionära teori som grundlades och formades av Karl Marx och Friedrich Engels [...] Marxismen är ett rikt flöde av idéer [...] Ett viktigt bidrag är Lenins utveckling av den revolutionära teorin och hans kritik av revisionisternas vantolkning av Marx.»
Programmet utgjorde framför allt en markering mot dem som företrädde en mer demokratisk socialism och önskade vända ryggen åt den kommunistiska traditionen: arvet från 1917 skulle inte skingras, den socialistiska uppbyggnaden i Sovjet och Östeuropa inte förkastas och de bärande idéerna i marxism och leninism inte överges. Bara ett par år innan socialismen bröt samman karakteriserades den med följande formulering i programmet: »Socialismen som ekonomi uppvisar storslagna resultat när det gäller att resa länder ur underutveckling och fattigdom och omskapa dem till industrialiserade välfärdsstater. Den socialistiska samhällsomvandlingen har frigjort folkliga krafter och därigenom kunnat förverkliga en framgångsrik ekonomisk, social och kulturell utveckling.»
Historikern Petter Bergner har i sin avhandling om Vänsterpartiet karakteriserat programmet med följande ord: »I ljuset av realsocialismens sönderfall i Östeuropa framstod dock 1987 års program som hopplöst föråldrat.» Saken är väl att programmet var överspelat redan när det skrevs – i varje fall mot bakgrund av de kunskaper om socialismen som då förelåg.
Sommaren 1989 öppnades gränsen mellan Ungern och Österrike, vilket fick som oförutsedd konsekvens att tiotusentals människor från främst Östtyskland reste till Ungern för att därefter ta sig till Österrike och väst. Regimen i DDR kom i gungning och den 9 november 1989 revs muren mellan östra och västra Berlin. Därefter följde händelserna slag i slag. Det till synes mäktiga kommunistpartiet i Östtyskland föll på kort tid samman. De ledande personerna flydde eller fängslades. I mars 1990 hölls det första fria valet i östra Tyskland sedan början av 1930-talet och den 3 oktober återförenades de båda tyska landsdelarna. I maj samma år höll VPK en kongress där marxismen, leninismen och kommunismen ströks ur programmet.
## **V. SAMMANBROTTET I ÖST**
## **1990-talets första år**
## **1. Omvändelse under galgen**
VPK:S DÅVARANDE TIDNING Ny Dag kommenterade för första gången händelserna i Östtyskland i samband med DDR-statens 40-åriga jubileumsfirande i oktober 1989. I en ledare hoppades man att DDR-ledningen skulle ta efter Sovjetunionen och att »förnyelsens vindar» skulle börja blåsa också i den tyska staten. Tidningen förutspådde avslutningsvis att det skulle ske stora personförändringar vid 1990 års kommunistiska partikongress i DDR: »Om Honecker inte avgått dessförinnan kommer kraven på hans avgång att växa sig starka, också i partiet.» Ny Dag skulle bli bönhörd långt snabbare än vad någon kunnat föreställa sig. Honecker lämnade scenen bara några dagar senare och hela partiledningen avgick mindre än en månad efter murens fall.
Den dramatiska scenförändringen i öst ledde till ett nästan lika snabbt kursskifte från VPK:s sida. Det östtyska broderpartiets upplösning och det socialistiska systemets fall framställdes i Ny Dag som en framgång för den svenska vänstern:
Förvisso ställer utvecklingen i Östeuropa den svenska vänstern inför många svåra frågor, inte bara om framtiden utan också om historien. Men för vpk som länge kritiserat de demokratiska begränsningarna i de socialistiska staterna, måste ändå den demokratiska revolutionen i Östeuropa vara en välkommen källa till glädje. Partiet har fått rätt i sin hårda kritik och i sina demokratiska krav. Att de demokratiska landvinningarna har fått till resultat att de kommunistiska partierna lidit nederlag och som i Polen tvingats överlämna regeringsmakten till oppositionen, är hur underligt det än kan låta, en framgång för vpk.
En motsvarande tankegång, men hävdad med mer bombastisk retorik, formulerades ett par månader senare av Göran Therborn i en kommentar i Dagens Nyheter:
Så kom den till slut, den som vi väntat så länge på. Och segrade. Lätt som en vårvind, när den äntligen kom. Revolutionen i Europa. De surögda kan ju säga att revolutionen var väntad väster om Elbe. Men rätt kontinent var det i alla fall, och tidpunkten kom att falla inom den tidrymd som revolutionärerna från 1968–70 räknade med, till och med 80-talet. Få stora politiska förutsägelser har varit tillnärmelsevis lika träffsäkra.
Therborn karakteriserade förändringarna i Östeuropa som en »demokratisk revolution» och jämförde den med 1968 i Västeuropa och framför allt med 60-talsradikalismen i USA. Han tycktes uppfatta omvälvningarna som en radikalisering av socialismen. Det var inte borgarklassens och kapitalismens intressen som stod på dagordningen, utan de breda folkgruppernas. Ändå menade Therborn att den största faran låg i införandet av marknadsliberalism, en modell vilken han benämnde ML för att kunna associera den med den komprometterade marxism-leninismen.
Försöken att få socialismens kollaps i öst att framstå som en seger för VPK och 68-vänstern vann dock ingen spridning. Tvärtom ställdes en rad kritiska frågor rörande partiets ideologiska arv och relationer till regimerna i öst.
På nyåret 1990 frågade Ny Dag hur det kunde ha gått så snett i Östeuropa. »Varför blev det diktatur i stället för demokrati? Var det fel på ledarna eller ideologin?» Ledarskribenten själv hävdade att de demokratiska revolutionerna avslöjat att det hade rått svåra »systemfel i öststatskommunismen» som också reste besvärande frågor »om hela den kommunistiska ideologin och demokratin».
Samtidigt höjde vederbörande ett varnande finger mot dem som enbart ville hålla »de kommunistiska ledarna och ledningarna ensamma ansvariga för den urartning och diktatur som nu avslöjas». Det var systemet och ideologin i sig som resulterade i 1989 års kris. De som på den andra sidan främst pekade på ledarnas ansvar såg inget fel på den marxistiska eller kommunistiska ideologin. Några gick ytterligare ett steg längre och påstod att de östeuropeiska samhällena, inklusive Sovjetunionen, överhuvud aldrig präglats av vare sig socialism eller kommunism i marxistisk mening. Hur kunde ideologin bära ansvar för något ledarna själva i förening med ogynnsamma historiska omständigheter lagt hinder i vägen för att förverkliga?
Under trycket från händelseutvecklingen i Östeuropa uppstod det efterhand även inom VPK en insikt om att de demokratiska revolutionerna i öst inte lett till någon förnyelse underifrån av socialismen, utan i stället till dess fullständiga sammanbrott. I februari 1990 fastslog partiledaren Lars Werner att kommunismens praktik har »nedvärderat demokrati och frihet och lyft upp partiet som ett enväldigt maktcentrum». Den tidigare förespråkaren för proletariatets diktatur Torbjörn Tännsjö, som ingick i programkommissionen inför kongressen i maj 1990, menade att det var dags att rensa ut marxismen och analysen av socialismens historia ur programmet, eftersom det ändå inte gick att uppnå någon samsyn i dessa frågor. Tännsjö ansåg också att marxismens »totalitära anspråk på vetenskaplighet har varit oerhört skadliga». Han hävdade vidare att »de resultat som uppnåtts [i de socialistiska länderna] ofta vunnits med medel som strider mot grundläggande värderingar inom arbetarrörelsen».
Ett annat överraskande inlägg gjordes av partisekreteraren Kenneth Kvist i början av januari 1990 i ett slags upptakt till den inre partidebatten inför kongressen. Kvist skrev följande: »Det som händer i Öst- och Centraleuropa, de breda folkliga kraven och rörelserna för demokrati är dödsstöten åt tanken på en socialism utan demokrati.»
Den politiska ståndpunkt partisekreteraren här formulerade som en självklarhet var, i ett kommunistiskt perspektiv anatema. Som strategisk doktrin betraktades den liberala demokratin med största skepsis inom partiet, även efter det att proletariatets diktatur övergivits. Den ansågs ju utgöra ett försvar för kapitalismen och ett hinder för införandet av socialismen. En inte ovanlig uppfattning, som kanske var mer uttalad på 1970- än 1980-talet, var att den liberala demokratin och socialismen var oförenliga storheter. Så exempelvis fastslogs så sent som i början av 1980-talet i ett internt dokument att det under socialismen skulle vara förbjudet att besluta om en återgång till kapitalismen. När partistyrelsen våren 1990 diskuterade de nya grundsatserna, ledde återigen frågan om den parlamentariska demokratin till en häftig debatt. En opinion i styrelsen hade reagerat mot att värnandet av den parlamentariska demokratin skulle skrivas in i de nya grundsatserna. Programmakarna Torbjörn Tännsjö och Herman Schmid förklarade reaktionen med dels en rädsla bland många att partiet därmed skulle »högervridas» genom att fokusera på det parlamentariska arbetet, dels att det fanns en »äldre» syn på den parlamentariska demokratin som »borgerlig». Och arkitekten bakom 1987 års partiprogram, Jörn Svensson, karakteriserade grundsatserna som ett intellektuellt hån mot arbetarrörelsens bildningstradition och tillade syrligt att de speglade nivån bland partiets ledande kader.
Efter årsskiftet eskalerade kritiken såväl inifrån partiet som från personer inom övriga vänstern. Per Kågesson som tidigare varit partimedlem men gått över till socialdemokratin fokuserade i en debattartikel i Dagens Nyheter på VPK:s långvariga och nära förbindelser med de kommunistiska partierna och regimerna i öst:
Fram växer bilden av ett vpk som visserligen protesterat mot vissa övergrepp i öst, men som systematiskt och med stor konsekvens under hela efterkrigstiden på ideologisk grund utvecklat nära och vänskapliga förbindelser med förtryckarna. Något egentligt stöd till den demokratiska rörelsen har man däremot inte givit – i varje fall inte under den tid då kampen pågick.
Enligt Kågesson hade de nära relationerna med regimerna i öst bidragit till att legitimera förtrycket av medborgarna och därmed försvårat uppkomsten av en demokratisk rörelse. Retoriskt frågade han vad C. H. Hermansson och Lars Werner skulle ha sagt om borgerliga partiledare som Bohman, Adelsohn och Bildt hade besökt Franco, Botha, Pinochet och Marcos och antagit gemensamma uttalanden? Avslutningsvis inplacerade Kågesson VPK bland demokratins historiska motståndare: »Vpk kommer för evigt att vara förknippat med stödet till Stalin och det nära samarbetet med efterkrigstidens kommunistiska diktaturer. En omvändelse under galgen kan aldrig utplåna minnet av det förgångna.»
Några dagar senare publicerade också den tidigare partisekreteraren Bo Hammar, som under många år varit Lars Werners närmaste man, en debattartikel i Dagens Nyheter. Han försvarade här visserligen en del av kontakterna med de östliga kommunistpartierna, men han var samtidigt ångerfull:
Sammanbrottet i de s.k. reellt existerande socialistiska länderna berör oss djupt. Det går inte att vifta bort med att vi i olika sammanhang kritiserat bristen på demokrati eller med att fel personer har stått i spetsen för regimerna. Det handlar i stället om fundamentala brister i den kommunistiska praktiken. Den praktiken förknippas i dag självklart med kommunismens ideologi. Kommunismens klasslösa och paradisiska lyckorike lever bara bland de troende.
Slutsatsen, om än inte explicit, var att partiet nu måste överge den kommunistiska traditionen och övergå till en bred och internationalistisk vänstersocialism. Hammar lyfte dessutom fram en annan aspekt som kom att vålla en hel del debatt inom partiet, nämligen frågan om planhushållning eller planekonomi. Hammar som varit en flitig resenär i de realsocialistiska länderna karakteriserade deras ekonomiska system med följande ord: »Den klassiska planhushållningsmodellen har brutit samman. Ekonomier som för inte så länge sedan betecknades som krisfria fungerar över huvud taget inte. Varken arbetslöshet eller inflation har avskaffats – tvärtom. Vad vi eufemistiskt betecknar som miljöproblem är i verkligheten ofattbara miljökatastrofer i flera av de s k socialistiska länderna.»
Just uppfattningen att de realsocialistiska ekonomierna i Östeuropa och Sovjetunionen, alla andra brister till trots, varit effektiva och välfungerande var en utbredd föreställning, inte bara inom VPK utan inom vänstern i stort. Ny Dag framhöll detta synsätt bara några veckor före murens fall: »Länge framstod DDR som den kanske mest 'västerländska' av de östeuropeiska staterna. Med en modern och utvecklad industri, ett fungerande jordbruk. Den östtyska ekonomin har också, till skillnad mot andra socialistiska stater, varit i god ordning.»
Frågan om planekonomi kan kanske tyckas teknisk för en nutida läsare, men hade i själva verket en central politisk betydelse i den marxistiska teorin. Planekonomin eller planen var det institutionella uttrycket för det förhållandet att socialismen till skillnad från kapitalismen byggde på en genomtänkt idé om hur resurserna i samhället skulle prioriteras och fördelas. Planen var kronan på verket i socialismens/kommunismens medvetna ansträngning att befria människan från historiens godtycke, frigöra henne från ett system där den enskilde entreprenören, konkurrensen och profitjakten var för sig och utan inre samordning – helt planlöst – fick bestämma vad som skulle tillverkas och säljas. Likväl kom det efter murens fall omgående signaler från ledande partimedlemmar att »planhushållningen måste ifrågasättas». Annika Åhnberg som vid denna tid var riksdagsledamot för VPK menade: »Centralt planerande lägger band på människors kreativitet och utveckling. Man kan aldrig planera fram ett bra samhälle.» Från många andra däribland Lars Werner själv kom förslag att planekonomin skulle ersättas med marknadssocialism – något filosofen och partimedlemmen Ingvar Johansson karakteriserade som »lite av en omvändelse under galgen».
Våren 1990 blev en hektisk period för VPK, inte bara för att socialismens sammanbrott i Östeuropa fortsatte och nya frigörelsesträvanden visade sig i Sovjetunionen, utan främst därför att partiet inte kunde enas i en rad viktiga frågor inför den stundande kongressen i maj. Hur långt skulle förändringsprocessen gå? Skulle partiet byta namn eller ej, det vill säga skulle ordet »kommunisterna» slopas i det dåvarande partinamnet? Skulle det skrivas ett helt nytt partiprogram eller räckte det med ett antal nya grundsatser? Skulle den kommunistiska traditionen helt överges och partiet bli ett demokratiskt vänsterparti som det danska Socialistisk Folkeparti? En Sifo-undersökning visade att det vägde jämt mellan VPK:s väljare som ville behålla respektive överge begreppet kommunism. En medlemsenkät visade däremot att en majoritet av medlemmarna ville behålla termen i partinamnet. Avgörandet föll på kongressen som med ytterst smal marginal (136 mot 133) beslöt att ändra namnet till enbart Vänsterpartiet.
1987 års marxist-leninistiskt influerade partiprogram hade nu övergivits – inte bara leninismen utan också marxismen hade försvunnit ur grundsatserna och i partinamnet fanns inte längre någon referens till kommunismen. Men som partiideologen Herman Schmid fastslog på kongressen hade ordet kommunism slopats därför att »det _missuppfattas_ av omvärlden och splittrar oss i två läger». I sak hade dock enligt Schmid inget ändrats, »partiet höll fast vid sin gamla vision». Det var ett uttalande som i ett nötskal sammanfattade partiets kursbyte. Både namn och idéer hade utskiftats av yttre tvång och mot partiets inre vilja. Precis som vid partikongresserna 1964 och 1977 hade partiet 1990 sökt att både anpassa sig till det nya och behålla delar av det gamla. Man hade under åren efter 1968 sökt dra till sig den nya vänsterradikalismen men samtidigt inte velat stöta bort den traditionella marxism-leninismens anhängare (även om de mest Sovjettrogna ändå lämnat 1977). Dessa försök till överbryggning hade till viss del varit framgångsrika, men de hade samtidigt alstrat ständigt återkommande inre slitningar och vidmakthållit släktskapet med Sovjetkommunismens djupt komprometterade idéarv. Kovändningen 1990 hade mot denna bakgrund karaktären av en omvändelse under galgen och var till betydande delar en läpparnas bekännelse.
## **2. Kommunismen kastas på historiens sophög**
Under våren 1990 fördes en debatt i vissa medier om vänsterns framtida ideologiska orientering och politiska utsikter. I Ny Dag publicerades flera inlägg av både vänsterintellektuella och partimedlemmar. De senare var alla mer eller mindre indragna i striden om partiets framtida politik, något som påverkade såväl formuleringar som ideologiska ståndpunkter. Tongången i tidningen följde den allmänna tendensen. Den realsocialistiska modellen i öst ansågs passerad, de Moskvaorienterade kommunistpartiernas politik betraktades som förkastlig och planekonomin uppfattades ha spelat ut sin roll. Det fanns emellertid andra drag i debatten som var av större intresse. Ett sådant var det tomrum som plötsligt uppstått kring begreppet arbetarklass. Denna juvel i den marxistiska teorins krona hade efter murens fall förlorat sin lyster. Kanske hängde det samman med en annan omständighet, nämligen den utbredda önskan att distansera sig från marxismen, i varje fall i den form som Vänsterpartiet hade omfattat den fram till murens fall några månader tidigare.
Den som mest explicit argumenterade för att överge »partimarxismen» var partimedlemmen och dåvarande professorn i filosofi vid Umeå universitet Ingvar Johansson. Han menade att det fanns en rad punkter i marxismen som nu var helt överspelade, till exempel historiefilosofin, synen på demokrati, på kvinnornas ställning, på miljön och marknaden. Men akilleshälen i partiets marxism, hävdade Johansson, var »övertron på planekonomin». Framför allt uppfattades ekonomisk utveckling inte som ett problem partiet var tvunget att förhålla sig till:
Ekonomisk utveckling ses inte som ett problem. De flesta inom vänstern behandlar ekonomisk utveckling som man tidigare behandlade miljön. Den är något som sköter sig självt och som inte behöver tas med i diskussionen. Trots att detta efter Polens konkurs borde vara lika omöjligt som att blunda för miljöfrågan.
Johansson pläderade för en »marknadssocialistisk» lösning på planekonomins problem. Samtidigt tycks han ha menat att en sådan modell inte bara var oförenlig med traditionell kommunism utan också med vänstersocialism. Hur ser en marxistisk vänstersocialism ut, frågade han retoriskt. Han avslutade med krav om ett mer realistiskt förhållningssätt till ekonomiska problem och skrev polemiskt att de ekonomer inom partiet som intagit en sådan ståndpunkt har blivit hånade som omarxistiska: »Idag står i viss mån också dom som ideologiska segrare. Det borde alla partimedlemmar inse.»
Johansson fick ett något överraskande svar på tal av en av dem han (förmodligen) utpekat som ideologisk segrare, nämligen Johan Lönnroth, som menade att »vi kan vara ett snäpp mer optimistiska» och således behålla marxismen. Efter ett inledande försvarstal för olika inslag i den klassiska marxismen konkluderade Lönnroth: »En förnyad marxism kan mycket väl vara grundvalen för ett vänstersocialistiskt parti i Sverige på 1990-talet.» Samtidigt var det tydligt att den marxism Lönnroth ville behålla var svår att skilja från den marxism som Johansson ville avföra ur partiprogrammet. För det första skulle den inte ha något att göra med leninismen och den vetenskapliga socialismen, det vill säga med den traditionella kommunismen eller Kominterntraditionen. Vidare ansåg Lönnroth att den nya marxismen inte skulle ha karaktären av statlig planhushållning utan av »demokratisk arbetarmakt» och att staten inte skulle vara centrum i en kommande »socialistisk omvandling».
En med Johansson och Lönnroth likartad hållning till marxismen intog Herman Schmid, en av programkommissionens ledande företrädare. Också Schmid menade att det fanns en avgörande skiljelinje mellan klassisk marxism och marxism-leninismen. Den senare var främst färgad av Sovjetunionens historiska erfarenheter, medan den klassiska varianten var ett svar på den dåtida kapitalismens utvecklingsproblem: »Detta är alltså den ena sidan av den marxistiska förnyelsen: att göra upp med marxism-leninismen och att återknyta till den levande kritiska marxismen och dess djupa insikter i kapitalismens natur.»
Framför allt hade den traditionella kommunismen inga svar på det aktuella samhällets problem med kvinnoförtryck, miljöförstöring och stor offentlig sektor. Bättre var då, enligt Schmid, att återknyta till en ännu äldre men också mindre komprometterad teoribildning, nämligen den klassiska marxismen.
Också författaren Anders Ehnmark och idéhistorikern Sven-Eric Liedman vände nu unisont det sovjetiska och östeuropeiska arvet ryggen. Ehnmark inledde med att fastslå att det inte längre kunde råda många tvivel om att socialismen »i den leninistiska varianten» hade resulterat i »en i teorin oförutsedd kombination av misär, förtryck och miljöförstöring». Men vad var det »som tog slut när kommunismen tog slut»? Enligt Ehnmark var det inget fel på arbetarrörelsens ideal om rättvisa, jämlikhet och frihet. Dessa ideal hade sedan franska revolutionen styrt alla radikala utopier, men hade komprometterats av leninismen i allmänhet och planekonomin i synnerhet. Problemet med den senare var framför allt att företagen inte varit självstyrande eller självförvaltande. Ehnmark tycktes här mena att hade bara företagen varit »samhälleligt eller demokratiskt» ägda skulle socialismens utveckling i öst blivit en annan. Bristen på lokal demokrati hade således, enligt Ehnmark, varit roten till det onda. Med andra medel skulle således målen ha kunnat uppnås.
Sven-Eric Liedmans inlägg löpte på ett par punkter parallellt med Ehnmarks. Precis som denne uppfattade Liedman att kommunismens epok var till ända, och på samma sätt som Ehnmark sökte han förklara debaclet med en distinktion mellan mål och medel, fast han benämnde dem utopin och rörelsen. Utopin var de skiftande betydelserna i begreppet kommunism. »Utopin», skriver Liedman, »är inte död, men att låta utopin ge namn åt något som vill vara en rörelse är orimligt.» I motsats till Ehnmark verkade inte Liedman anse att bristen på självförvaltning var problemet, utan det faktum att »kommunismens hela begreppsapparat», under framför allt Stalin, »låses fast. Ja, hamnar i ett skruvstäd.» Den öppenhet som Liedman menade sig se hos Marx, Engels och även Lenin hade gått förlorad i och med Stalins kodifiering av ideologin till marxism-leninism.
Kommunismen med stort K är givetvis den kommunistiska internationalen, grundad 1919, upplöst 1943. Dess grundläggande principer – inget samarbete och inga kompromisser med andra politiska rörelser och en långtgående om också inte total samordning av de olika nationella kommunistpartierna är alltför välkända för att vi ska ta upp tiden med dem här.
Ehnmark och Liedmans inlägg är lika också på en annan punkt, nämligen att de anser kommunismen för att ha varit heroisk. Ehnmark talar obestämt om leninismens »heroiska höjdpunkter» och Liedman menar att kommunismens »heroiska period» inföll vid andra världskrigets slut då den »överallt spelar en avgörande roll i motståndet mot fascisterna». Huruvida kommunismens uppgörelse med fascismen överhuvud bör förknippas med ord som heroism kan diskuteras, men det är ändå anmärkningsvärt att två av Sveriges ledande vänsterintellektuella i sitt avståndstagande från den kommunism de som alla andra nu menar tillhör historien lyfter fram det påstått heroiska i kommunismen samtidigt som de inte diskuterar kommunismens mest utmärkande drag – ofriheten och förtrycket. När kommunismen misslyckades att etablera sig som ett emancipatoriskt alternativ till kapitalismen, beror det nog mindre på bristande företagsdemokrati och teoretisk förkalkning och mer på dess ytterst brutala maktutövning mot de egna folken med arbetsläger, likvideringar och inre folkfördrivningar, mer på dess ockupation av grannländer med omfattande utrensningar som följd och inte minst på dess under en period nära samarbete med nazismen.
Partidebatten innehöll också mer defensiva och förbittrade synpunkter. Partimedlemmen Bitte Engzell frågade retoriskt varför kritikerna »inte har en positiv stavelse att säga om vpk» och om det verkligen är nödvändigt för partiets medlemmar att klä sig i tagelskjortor. Engzell var tydligt upprörd över kritiken av Marx och Lenin: »Och jag måste fråga: är man ortodox för att man vidhåller kommunismen som vision? Innebär ett missbruk till exempel av Lenins texter att det han skrivit är diktatoriskt och dåligt? Eller platsar Marx bara på sophögen för att exempelvis polska arbetarpartiet förtryckt massorna med Marx som tillhygge?»
Hon ställde sig vidare tveksam till de anklagelser som framkommit mot kommunismen och hävdade att så länge det inte fanns en »rejäl historieforskning» var det lämpligast att avstå från tvärsäkra uttalanden. En liknande tankegång, nämligen att det var svårt att veta vad som skett av förtryck och korruption i de realsocialistiska länderna, framförde partimedlemmen Dick Urban Vestbro i sitt inlägg:
En fråga som måste ställas är om vpk:s ledning verkligen varit ovetande om den urartning som de östeuropeiska partierna genomgått. De partikamrater som besökt dessa länder måste väl ha sett något av korruptionen där! Som partimedlem känner jag mig lurad. Jag vill att det skall utredas om någon gjort fel genom att framföra hälsningar till de korrupta partiernas kongresser.
Vestbros förvåning och indignation över att ha svävat i okunskap om förhållandena i öst verkar äkta, men ter sig ändå märklig med tanke på att han engagerade sig i vänstern redan på 1960-talet. Han var dock säker på en sak, nämligen att det inte fanns något samband mellan Marx och Lenins kommunism och »det som avslöjats på senare tid» om de socialistiska staterna. Vestbro menade till och med att Vänsterpartiet inte hade behövt överge begreppet proletariatets diktatur, hade bara partiet på ett tillräckligt tidigt stadium i sin historia kritiserat stalinismens likställande av proletariatets diktatur med kommunistpartiets diktatur. Vestbro verkar mena att den kommunistiska idén om diktaturens företräde framför demokratin varit lättare att acceptera utan stalinismens »horribla förvrängning» av Lenins lära – som exempel framhåller han att den demokratiska innebörden i teorin om proletariatets diktatur då hade kunnat framstå i ett mer sympatiskt ljus. Så motsägelsefullt kunde debatten föras när vänsterns gamla landmärken slocknat och skeppet skulle navigeras i okänt farvatten.
Vid en jämförelse mellan Ehnmarks och Liedmans analys å ena sidan och Vestbros å andra sidan är likheten påfallande. Alla har de det gemensamt att de söker avgränsa och värna de vänsterideal de håller högt från den komprometterande historiska verkligheten. Lika lite som kristendomen menat att människans synder befläckat Jesu lära, lika lite ansåg flera av debattörerna att realsocialismens brott belastade den kommunistiska utopin eller som Ehnmark benämnde det, »arbetarrörelsens ideal». Det var kyrkan och kardinalerna som varit korrupta, inte läran som vilselett kyrkan.
## **3. Nostalgi och klagorop. DN Kultur**
De realsocialistiska samhällssystemens fall gav upphov till en hel del triumfatoriska reaktioner bland dess motståndare. Det togs som bevis på socialismens inneboende uselhet och kapitalismens historiska överlägsenhet. Att den liberala demokratin med lagstyre och konstitutionell maktdelning visat sig mer fri, jämlik och rättvis än folkdemokratierna i öst tog vänstern 1990 till sig, men att marknaden övertrumfat planen, att föreningen av enskilt ägande och konkurrens var en mer effektiv metod för innovation och resursfördelning än politisk-ideologiska beslutsprocesser var svårare att acceptera. På Dagens Nyheters kultursida publicerades under våren och sommaren 1990 en rad ofta korta kommentarer som i vissa fall präglades av tillbakablick och saknad efter det som brutit samman i öst. I andra fall gav de uttryck för vrede över den liberala kapitalismens utbredning i Östeuropa.
Debatten fördes även på ledarsidan där den tidigare folkpartistiske statssekreteraren Sten Westerberg skrev i en polemisk kolumn att de realsocialistiska planekonomiernas kollaps också drabbade den reformistiska socialdemokratin. Denna hade definierat sig som en tredje väg mellan kommunism och kapitalism, men när nu den förra hade försvunnit var frågan var socialdemokratin stod. Westerbergs huvudtes var att »de systemfel som ekonomiskt knäckt kommunismen återfinns i viktiga sektorer i Sverige». Han pekade på undervisningen, vården och bostadssektorn. Dessa sektorer styrdes av politiker och byråkrater efter ideologiska idéer och löften: »Konkurrens, alternativ och konsumentstyrning motarbetas. Resultaten är de klassiska: byråkrati, slöseri, dålig kvalité, köer, svarta marknader och korruption.»
Westerberg avslutade med påståendet att de länder (i väst) som hade de största planstyrda sektorerna också hade den sämsta standardutvecklingen. »Den tredje vägen leder till den tredje världen.»
Under rubriken »McCarthy härjar i Sverige» publicerade kultursidan några dagar senare en krönika av medarbetaren Karl Erik Lagerlöf, som menade att Westerberg, liksom ekonomen Anders Åslund och författaren Lars Andersson, manipulerade med begreppet socialism i syfte att associera den demokratiska socialismen med öststatskommunismen. Lagerlöf hävdade att Westerbergs resonemang rymde en »uppsjö förutsättningar», som inte gavs någon hederlig redovisning. Men framför allt såg Lagerlöf kritiken som uttryck för en förföljelse av oliktänkande:
Ytterligheterna berör varandra. Extremister är lika. Drömmen om renhet och tendensen att rensa ut oliktänkande delar McCarthyismen med öststatskommunismen [...] Det var i USA på 1950-talet. Masshysteri uppstod. Folk drogs inför rätta. Det liknade häxprocesser. Människor som vagt sympatiserat med kommunism av något slag brännmärktes och fick sitt liv förstört.
En liknande process pågick nu tydligen i Sverige, ledd av nyliberaler med tillgång till Dagens Nyheters ledarsidor. Samma intryck gav en annan kommentar på kultursidan ett par månader senare. Journalisten Synnöve Clason rapporterade under rubriken »Klappjakt på intellektuella» om den kritik som den tidigare uppburna östtyska författarinnan Christa Wolf hade utsatts för i västtyska medier. Clason talade om kritikerna som »huvudjägare» och hon vände sig med den tyske publicisten Walter Jens formulering mot »alla dem i väst som anser sig ha rätt att döma de få modiga i Honeckers Tyskland». Hon framhöll avslutningsvis, återigen med Jens ord, att DDR-kulturens ledmotiv hade varit en »dröm om frihet».
Den västtyske författaren Peter Schneider, som ibland publicerade sig på kultursidan, vände sig liksom Clason mot de västtyska kultursidornas »jaktsport» på östtyska författare. Om det tidigare hade varit rent litterära och estetiska värderingar som låg till grund för hållningen till den östtyska litteraturen, var det nu alltigenom politiska kriterier som styrde bedömningarna. Detta hade särskilt drabbat Christa Wolf vars böcker nu reducerades till politiska utsagor. Huvudpoängen i Schneiders inlägg var att de västtyska »litteraturpåvarna» fram till murens fall strukit DDR-litteraturen medhårs, men att de sedan återföreningen sällat sig till kritikerna av den östtyska realsocialismen i allmänhet och DDR-författarna i synnerhet.
Det som särskilt provocerade flera debattörer på Dagens Nyheters kultursida var som sagt marknadsekonomins eller kapitalismens återkomst i Östeuropa. Läraren vid Dramatiska institutet, Leif Dahlberg, kommenterade i ett inlägg en skrift om Östeuropa utgiven av Utrikespolitiska institutet. Enligt Dahlberg bestod den av alltför många bidrag från nyliberala företrädare: »får de nyliberala intellektuella i öst bara råda kommer marknadens osynliga hand – den fria konkurrensen – med tiden att ställa allt till rätta. Släpp företagarna loss och det blir vår.»
Dahlberg ansåg att nyliberalernas »ensidighet och misstag» när det gällde att förstå Östeuropa var uppenbar – den västliga »1800-talsliberalismen» hade ingen plats i det postsocialistiska Östeuropa. En besläktad syn på »de fria marknadskrafterna» kom till uttryck i ett kort inlägg av filosofen och författaren Åke Löfgren, som vid ett besök i Budapest hade funnit att en av världens största permanenta affischutställningar på museet för den ungerska arbetarrörelsens historia hade monterats ned och att museet var stängt. Löfgren frågar sig besviket: »Vad i övrigt av det socialistiska samhällsbygget kommer att behöva demonteras för att det icke-socialistiska Ungerns ansikte skall kunna anses vara återställt? Vet vi i Väst vad vi egentligen applåderar? För de fria marknadskrafterna lär frågan enbart vara av akademiskt intresse.»
Det Löfgren avsåg med frågan om man i väst visste vad man applåderade var det faktum att Ungern vid det första fria valet efter kriget, i april 1990, hade fått en konservativt ledd koalitionsregering med en – i Löfgrens ögon – revanschistisk nationalism på dagordningen. Men var den nationalistiska revanschismen, som Löfgren menade, verkligen bara av »akademiskt intresse» för de »fria marknadskrafterna» i väst? Det Löfgren anspelar på är väl snarast att liberalismen, i den radikala vänsterns ögon, aldrig varit någon egentlig motståndare till högerextremism och att »marknadskrafterna» kanske till och med har intresse av en sådan rörelse. Grundläggande uppfattar ju vänstern högerextremism och fascism som ett av kapitalismen betingat problem.
En annan infallsvinkel på kapitalismen presenterades i en krönika av kultursidans medarbetare Jörgen Eriksson under rubriken »Kapitalismens Gulag urholkar människors liv». Eriksson beskriver här ögruppen Truk i Mikronesien där det funnits en lång tradition av självmord bland unga män. Kring 1970 hade emellertid självmordstalet kraftigt ökat, vilket Eriksson förklarar med att USA efter världskriget har »öst in dollar i området». Som det beskrivs i krönikan handlade det om att övergången till penningekonomi slog sönder den traditionella familjeoch auktoritetsstrukturen, vilket i sin tur resulterade i en ökning av självmordsfrekvensen. För Eriksson var tragedin inte de oavsiktliga och välkända verkningarna av mötet mellan den moderna västliga civilisationen och en traditionell ursprungskultur, utan följden av den kapitalistiska grottekvarnens malande: »Kapitalismens gulag har en världsvid utbredning. Den dirigerar smutsiga contras-gerillor och bygger oöverstigliga Berlinmurar mellan fattiga och rika. Och den styr och ställer med människors liv och urholkar dem så att ynglingarna i Truk måste söka sig ett lämpligt träd.»
Erikssons tanke, nämligen att kapitalismen är ett system som fungerar lika, om inte mer, ondskefullt än en gång stalinismen, kom att bli ett återkommande tema i den postkommunistiska vänsterns kritik. Gulagsystemets avsiktliga och medvetna metod att andligt och fysiskt krossa all opposition i Sovjetunionen, hade enligt vänsterkritikerna sin motsvarighet i det sätt på vilket kapitalismen styrde och ställde med människors liv. Vad Stalin var för Sovjetunionens folk skulle således det kapitalistiska »systemet» vara för världens fattiga massor.
Att vänsterintellektuella i Sverige och andra länder reagerade mer eller mindre häftigt på kapitalismens återinförande i östra Europa är inte ägnat att förvåna. Det finns en inbyggd konflikt mellan å ena sidan den vänsterintellektuelles tro på och engagemang i humana, solidariska och universella värden, och å andra sidan den kommersiella verksamhetens konkurrensutsatta kamp för överlevnad och utveckling på en marknad styrd av tvingande avkastningskrav. Denna konflikt har under årtionden manifesterat sig, inte minst inom den konstnärliga modernismen, som ömsom ett vänster-, ömsom ett högeruppror mot liberalism och kapitalism.
Mer förvånande var emellertid den reservation som vid några tillfällen, om än indirekt, kom till uttryck i synen på de demokratiska besluten i öst. En av kultursidans fasta medarbetare, Hans Axel Holm, kunde inte hålla tillbaka sin besvikelse inför åsynen av östtyskarnas överväldigande stöd till Helmut Kohls förslag om Tysklands återförening:
och jag får till svar att östtyskarna själva valt det, då tänker jag att det kan väl ändå inte vara riktigt: förutom arbetslösheten och bristen på pengar till löner och betalningar, dessa västtyskar köper upp deras gamla städer och struntar i vad de själva tycker, bryter avtal med östtyska företag redan för egen vinnings skull och som dessutom utpekar dem som moraliskt misstänkliga för att de aldrig flydde eller opponerade sig högljutt, då. Valde de detta? Kanske det. Men de trodde så godtroget att de hade bröder och systrar i väst. Så hade det ju alltid hetat.
Holm skriver till sina vänner i den före detta folkdemokratin och frågar hur de förhåller sig till sin nyvunna frihet. Det svar han förmedlar är att nu kan man som medborgare i det nya östra Tyskland svära och skälla utan att bli fotograferad av Stasi. En annan som på kultursidan uttryckte sitt ogillande över framgångarna för Helmut Kohls återföreningspolitik var Sven-Eric Liedman: »Frigörelsen i Östeuropa var åtminstone fram till Kohls seger en härlig sak: men fy fan för den inbilska okritiska självgodhet som den gett upphov till i det första Europa.»
Med det »första Europa» förstår Liedman här Västeuropa och »självgodheten» är något han menar sig se i dess nyhetsförmedling. Medierna i väst »instruerar» medborgarna att inte säga något kritiskt om kapitalismen, skulle så ändå vara fallet får de till svar: »Det som till äventyrs är dåligt hos oss är sämre i Sovjet.» Kritikerna av det västliga systemet tystas eller skräms med att socialismen visat sig värre – »se så illa tåld den är i Polen och Ungern». Liedmans tes att medierna i väst skulle »instruera» medborgarna att inte kritisera kapitalismen är uttryck för ett konspiratoriskt perspektiv på det demokratiska samhället. Detta att framställa kapitalismen och liberalismen som centrala och dirigerande subjekt, på linje med kommunismens centrala parti- och statsorgan, vilka explicit sökte styra människors handlingar och idévärld är en grundtanke i vänsterns åskådning. »Mirror-imaging» kallas i dag företeelsen med en anglosaxisk term: som man själv tänker och agerar förutsätter man att också motståndaren tänker och agerar. Utgår man från att medierna under socialismen ska instruera (»top-down») medborgarna om hur sakernas tillstånd ter sig, ligger det nära till hands att betrakta medierna i ett liberalt marknadssamhälle på samma vis. Men medan socialismen var centralstyrd är den kapitalistiska demokratin, idealtypiskt sett, mångfaldig, decentral och konkurrenspräglad.
Dagens Nyheter kultursida innehöll det första året efter murens fall även andra inlägg och debatter om förändringarna i Östeuropa och dess återverkningar på det politiska och ideologiska klimatet i Västeuropa. Hit hörde dödsdomen mot Salman Rushdie, postmodernismen och Sverige under andra världskriget. Därutöver inbjöd kultursidans chef Arne Ruth under våren en rad internationellt kända vänsterintellektuella att diskutera frågan om »Nästa vänster?». Själv gjorde Ruth ett rasande angrepp, inte på de härskare som störtats av folkresningen i öst, utan på den demokratiskt valda premiärministern i Storbritannien Margaret Thatcher för att hon hotade yttrandefriheten i landet och hade förpestat det intellektuella klimatet med sin »egennyttans revolution».
Ett tema lyste dock i stort med sin frånvaro på kultursidan, nämligen skildringar och analyser av den ofrihet och det förtryck de östeuropeiska medborgarna fått uppleva under en dryg generation som följd av den sovjetryska ockupationen 1945 och den därefter påtvingade omvälvningen av samhällsförhållandena. Det sovjetkommunistiska systemets sammanbrott i Östeuropa var ju trots allt en händelse av historisk betydelse, precis som omvälvningarna 1917/1918 eller vid andra världskrigets slut. Allt det som under årtionden plågat människor i östra Europa var nu borta: partidiktaturen, hemliga polisen, utrensningarna av »borgerliga element» inom ämbetsverk, universitet, skolor och kulturinstitutioner, tvångsförstatliganden av företag, fastigheter och jordbruk, kommunistpartiets kontroll av konsten, forskningen och medierna, judeförföljelserna och undertryckandet av den nationella identiteten.
Saken var inte den att detta förtryck fortsatt förnekades eller försvarades på DN Kultur, utan att fokus för kritiken konsekvent låg på det system som skapat en rimlig frihet för sina medborgare om än med varierande grad av sociala klyftor, nämligen den liberala demokratin och den kapitalistiska marknadsekonomin. Visserligen var vänstern nu (nästan) överens om att folkdemokratin och planekonomin lett till ofrihet och misär, men den liberala marknadsekonomi som i väst skapat välstånd ansågs likväl fortfarande utgöra ett avskräckande exempel. Visst var den nyvunna friheten för östtyskarna välkommen, men att se Västtyskland och Helmut Kohl presidera över den tyska återföreningen var ändå outhärdligt – och hade verkligen östtyskarna valt återförening om de vetat vad som väntade. Störst indignation väckte ändå den kritik vänsterintellektuella i väst nu fick utstå för sina sympatier för den östeuropeiska kommunismen. Denna kritik liknades vid »McCarthyism» och »klappjakt» och jämfördes med »öststatskommunismens» likvideringar och utrensningar. Klappjakten var inte värre än att vänsterintellektuella med sympatier för kommunismen hade fri tillgång till kultursidan i Nordens största liberala dagstidning.
## **4. Inga lik i garderoben. Självrannsakan som uteblev**
Sett på ett par årtiondens avstånd är kanske det mest märkliga med det svenska Vänsterpartiets brytning med sitt kommunistiska arv hur snabbt och friktionsfritt brottet till en början sker. Visst fanns det inom och kring Vänsterpartiet både ovilja och olust inför scenväxling – saknaden efter förlorade kontakter och fallna institutioner i den försvunna kommunistiska världen var stor och bitterheten över att den »borgerliga klassfienden» gått segrande ur det kalla kriget var djup – ressentimenten fanns inte bara inom partiet utan kom också till uttryck inom bredare vänsterkretsar, som genomgången av DN:s kultursida visat.
Svallvågorna från Roms fall trängde emellertid tillbaka allt inre motstånd. De tvingade vänsterns företrädare att uttala sig som om de alltid önskat socialismens sammanbrott. »Härligt, härligt utropade Gudrun Schyman och Lars Werner när partidiktaturerna i öst föll och broderpartierna upplöstes. Det här har vpk väntat och hoppats på länge fick vi höra», kommenterade Vänsterpartiets internationelle sekreterare Bo Hammar i Expressen. Samma svallvågor begränsade också uppslutningen bakom den tidigare dominerande marxismen och leninismen till de betydelselösa och fanatiska sekterna i vänsterns utkant. Framför allt kom sammanbrottet för 1917 års idéer att kompromettera den radikala (kommunistiska) vänsterns ideologi och retorik. Det gällde inte bara nyckelbegrepp som »proletär revolution» och »proletär diktatur», vilka sedan länge mist sin strålglans, nu utmönstrades också termer som »det socialistiska lägret», »folkdemokrati», »partiets ledande roll» och »klasskamp» – de forna broderstaterna i öst hade i partivokabulären snabbt reducerats till »partidiktaturer». Trots det snabba fotbytet från en kommunism byggd på marxism och leninism, till en mer allmän och oklar vänsterradikalism, skavde emellertid ett sårigt problem: Hur kunde en ideologisk rörelse som hade syftat till människans frigörelse från fattigdom och ofrihet, överallt där den kommit till makten ha resulterat i förföljelser och likvideringar? Hade inte Vänsterpartiet under årtionden, också efter 1968, varit del av en internationell rörelse som velat avskaffa den liberala demokratin i Västeuropa och införa ett planekonomiskt system – ett system man nu dömde ut som orättfärdigt, ineffektivt och miljöförstörande?
Om den omedelbara reaktionen på realsocialismens försvinnande närmast hade prägel av chock, kom de följande åren mer att kännetecknas av ett försök till självrannsakan och uppgörelse. Debatten om den svenska vänsterns demokrati- och samhällssyn var emellertid inte begränsad till vänstern själv. Också i det borgerliga lägret ägde det rum en häftig sammandrabbning. I fokus för denna stod Per Ahlmark och de skrifter han publicerade, framför allt mellan åren 1992 och 1994. Problematiken här var inte helt olik den som återfanns inom vänstern. Om den senare frågade sig hur den kunde ha varit en del av den totalitära sovjetkommunistiska rörelsen, frågade Ahlmark hur den svenska vänsterliberalismen hade kunnat ge vika för nyvänsterns extrema syn på i synnerhet demokrati, äganderätt och marknadsekonomi.
Redan 1991 hade DN-journalisten Staffan Skott utkommit med en bok om den svenska kommunismens historia, _Liken i garderoben_. Den innehöll en provkarta på djupt komprometterande ställningstaganden från det svenska kommunistpartiets sida – från de devota hyllningarna av Stalin, över stödet till Molotov–Ribbentrop-pakten i augusti 1939, kuppen i Prag 1948 och krossandet av Ungernupproret 1956, till förbindelserna med de Moskvastyrda regimerna i Östeuropa fram till hösten 1989. Skotts tes var att Vänsterpartiet i grunden inte hade gjort upp med sitt totalitära arv och att det inte ens efter Sovjetunionens upplösning hade sökt besvara frågan hur partiets tidigare medlemmar – »gårdagens kommunister» – hade kunnat »stödja världshistoriens största massmördare»?
Skotts bok kommenterades i flera tidningar. Den fick många lovord, men nästan alla framhöll att tonen i Skotts framställning var hård och oförsonlig. Det är en riktig karakteristik, boken är en stridsskrift i den klassiska radikalismens tradition – träffande, brutalt ironisk och med sarkastiska överdrifter. Men hade liknande invändningar rörande tonläget kunnat framföras mot en kritisk genomgång av den svenska nazismens internationella anknytningar? Knappast. Kommentarerna till Skotts upprördhet över kommunismens brott och Vänsterpartiets försvar för dessa speglar det förhållandet att det i svensk offentlig debatt, i varje fall sedan början av 1970-talet, inte varit god ton att visa avsky eller indignation inför vänsterns komprometterande förflutna. Som tidigare påtalats och som framgår av debatten nedan riskerade i stället den som kritiserade den yttersta vänstern att själv anklagas för »McCarthyism» eller reaktionär »antikommunism».
Ett annat belysande bidrag till uppgörelsen med den svenska vänsterns historia framlades 1992 av Vänsterpartiet. En arbetsgrupp hade fått i uppdrag av partistyrelsen att utvärdera »hela den historiska utvecklingen i partiet».
Resultatet presenterades i en skrift med titeln _Lik i garderoben?_ (även kallad _Vitboken_ ). Det var första gången i partiets historia som det gjordes ett kritiskt försök att förstå dess idéutveckling och agerande i såväl ett nationellt som ett internationellt sammanhang.
Utvärderingen av Vänsterpartiets förflutna bör sättas in i ett bredare historiskt sammanhang. Sedan årtionden var förtrycket i Sovjetunionen känt och dokumenterat. Det gällde också de brutala sovjetiska maktövertagandena i Östeuropa, där lydregimer upprättats efter 1945. Partiet hade från sitt bildande 1919/1921 och fram till och med 1956 lojalt stött den sovjetiska politiken, både nationellt och internationellt. Lika välkänt i den svenska offentligheten var att partiet antingen tigit om eller helt förnekat de övergrepp som Sovjetmakten begått i de områden den behärskade, vilket inte hindrat partiets företrädare att framställa de politiska förhållandena i de socialistiska länderna som mer demokratiska och jämlika än i väst. Samtidens Sverige förvånades dock inte av VPK:s förfalskade självbild. Det offentliga Sverige visste att partiet förkastade både demokratin och blandekonomin, att det i varje fall fram till 1950-talets slut hyste större lojalitet med den »första arbetarstaten» än med det svenska samhället och att dess mål var ett kommunistiskt maktövertagande genom »massmobilisering av folket» och att den fredliga väg partiet proklamerade efter 1945 förutsatte att motståndarna – socialdemokratin och borgerligheten – kapitulerade inför kommunisternas maktanspråk. Här ska dock tillfogas att vänstern efter 1968 hade stor framgång med att stämpla denna beskrivning av sitt förflutna som antikommunistisk i betydelsen ovederhäftig, illasinnad och reaktionär.
Hur förhöll sig då _Lik i garderoben?_ till partiets förflutna? Det mest genomarbetade bidraget i _Vitboken_ författades av Lundahistorikern Lars-Arne Norborg. Hans uppsats ger en habil översikt över Vänsterpartiets förhållande till Sovjetunionens kommunistiska parti. Norborg visar att den svenska vänstern, de första 45 åren av sin historia, gick i takt med de politiska direktiven från Moskva – kanske kan man skönja en liten avvikelse här eller där, men i allt väsentligt var partiet som också statsvetaren Jörgen Hermansson understrukit i sin avhandling obrottsligt lojalt mot den politik som fördes både inrikes- och utrikespolitiskt av Sovjetunionen. Först när den sovjetiske ledaren Nikita Chrusjtjov i februari 1956 i ett hemligt tal inför partikongressen i Moskva hade avslöjat det som redan var känt i väst, nämligen att Stalins Sovjetunionen varit en diktatur byggd på hemliga polisens utrensningar och likvideringar och på ett omfattande system av tvångsarbete i interneringsläger, först därefter började lojalitetsbanden sakta och mycket försiktigt upplösas mellan Moskva och de västliga kommunistpartierna.
Med C. H. Hermanssons tillträde som partiordförande 1964 skedde en viss nyorientering, bland annat tonades kontakterna med Sovjetunionen ned något och Vänsterpartiet deklarerade för första gången sin principiella anslutning till den parlamentariska demokratin. På kongressen tre år senare markerades den nya linjen genom att partiet bytte namn till Vänsterpartiet kommunisterna (VPK) och att en viss kritik av Sovjetunionen infördes i det nya programmet. Men som Norborg skriver satte samtidigt Hermanssons kompromiss med de traditionella Moskvakommunisterna stopp för en brytning med Sovjetunionen.
Sett i ett längre perspektiv kan det sägas att C. H. Hermansson försatt det historiska tillfället att kasta partiets kommunistiska barlast överbord och låta det segla vidare som ett fritt demokratiskt vänsterparti. Partiets raison d'être sedan 1917–1921 hade ju varit att socialdemokratisk reformism och liberal parlamentarism aldrig skulle kunna leda till framsteg för de »arbetande massorna». Ändå var det just stegvisa sociala reformer och politisk stabilitet som präglat Sverige sedan första världskrigets slut. Men i stället för att erkänna sitt historiska misstag och helt lämna den kommunistiska rörelsen, som SF (Socialistisk Folkeparti) i Danmark gjorde, vidhöll alltså det svenska partiet sin marxistleninistiska ideologi och upprätthöll sina relationer till kommunistpartierna i öst. Den försiktiga »modernistiska» linje Hermansson valde låg dessutom helt inom ramen för de riktlinjer Nikita Chrusjtjov dragit upp efter 20:e partikongressen 1956. Chrusjtjov proklamerade nu fredlig samlevnad med de kapitalistiska staterna i väst och han underströk att övergången till socialism kunde ske på en rad olika vis. Varje parti kunde välja en egen nationell väg till socialismen, en väg som kunde vara både fredlig och parlamentarisk. Någon brytning med Sovjetunionens kommunistiska parti ägde således aldrig rum under Hermanssons »modernistiska» period.
Denna period skulle dessutom, som Norborg och andra har påpekat, bli kortvarig. Med 68-vänsterns inträde i partiet skedde åter en antiliberal radikalisering i synen på reformism, demokrati och parlamentarism. Såväl Norborg själv som Jörgen Hermansson betecknar den ideologi som avlöste »modernismen» som nyleninistisk. Och även om »Sovjetmodellen» kritiserades från vänster – för byråkratisk stelhet, nomenklaturans privilegier och bristen på arbetarmakt i företagen – växte samtidigt en ny respekt och beundran fram inom den unga vänstergenerationen för Sovjetunionens roll och historia.
Norborg karakteriserar stämningarna i VPK vid 1970-talets mitt med följande ord:
Oktoberrevolutionen och 'den unga sovjetstaten' heroiserades och det drömdes drömmar om massmobilisering och mer eller mindre våldsam revolution även i Sverige. Återigen skulle den borgerliga staten krossas i stället för att användas som arena i klasskampen. Sovjetunionens betydelse som motmakt mot det imperialistiska USA och stödjepunkt för tredje världens befrielserörelser betonades.
I stora drag dominerade denna hållning VPK:s ideologi ända fram till murens fall 1989, även om 1980-talet såg framväxten av en inre opposition som lade vikt vid en liberal eller icke-marxistisk tolkning av demokratibegreppet – en tendens som i Norborgs ögon stärkte hans huvudtes om Vänsterpartiets eurokommunistiska orientering.
Det är riktigt att den sorts kommunism VPK företrädde efter nyvänsterns ankomst till partiet var annorlunda än den fram till 1956 eller under Hermanssons »modernism»; partiet blev dessutom mer ideologiskt vildvuxet och aktivistiskt efter 68-generationens inträde. Den radikalisering som således sker under 1970- och 1980-talet stöder därför knappast tesen om eurokommunism. Termen eurokommunism debuterade emellertid först vid 1970-talets mitt och eurokommunismen representerade i motsats till det svenska partiets »modernism» en mer långtgående brytning med Sovjetunionen, främst från det spanska och italienska partiets sida. Ändå är denna tes inte det mest remarkabla med Norborgs bidrag utan i stället det faktum att han – liksom Kent Lindkvist och Sven-Eric Liedman – inte reser frågan om sambandet mellan den kommunistiska ideologin och förtrycket i öst. Norborg och Lindkvist gör ett stort nummer av partiets minsta avvikelse från den kommunistiska rörelsens linje, såväl under som efter Kominternperioden. Här framhålls exempelvis att partiet såväl i slutet av 1930-talet, som de första åren efter kriget, sökte formulera en mer nationellt präglad och oberoende »högerlinje»; här citeras en amerikansk forskare som framhäver att partiet var bland de första »som alltmer bestämt lösgjort sig» från den kommunistiska världsrörelsen och hur det samtidigt hade sökt utveckla en egen »demokratisk-parlamentarisk ideologi med utgångspunkt i svensk politisk tradition»; här hänvisas till redaktören Hans Arvidsson som ansåg att partiet var det första som ifrågasatte »det stalinistiska systemet» och att det »konsekvent protesterat mot sovjetiska övergrepp»; här refereras Göran Therborn som fastslog att »kommunismen i Sverige» redan före 1964 »representerade en äkta radikal demokratisk tradition» och här skriver Sven-Eric Liedman att partiet inte var en »främmande fågel i svensk politik» utan ett typiskt svenskt »kompromissparti» – och Norborg sammanfattar själv med orden att det »trots allt förekom ganska mycket kritik mot Sovjetunionen inom Vpk».
Partiet framställs närmast som ett dissidentparti inom den kommunistiska rörelsen, som om det hade påtvingats sin totalitära ideologi av en yttre makt och som om det egentligen alltid velat frigöra sig från rörelsens marxist-leninistiska politik. Fokus i _Vitboken_ ligger på partiets relationer med de realsocialistiska regimerna i öst, i synnerhet då det sovjetiska kommunistpartiet. Det är partiets beroende av Sovjetunionen kontra dess förankring i en svensk tradition som framställningen kretsar kring. Vänsterpartiets problem reduceras till dess politiska och organisatoriska förhållande till partidiktaturerna i öst – deltagandet i de östliga partikongresserna, de servila gratulations- och hyllningstelegrammen, de egna partiskolorna i Sovjet och DDR, de återkommande besöken på öststatsambassaderna i Stockholm, delegations- och semesterresorna till Östeuropa, de ekonomiska bidragen etc. Det är riktigt att allt detta skadade partiet, ändå är det uppenbart för en utomstående att det som främst belastade partiet fram till murens fall var det faktum att det företrädde en odemokratisk ideologi – och att denna ideologi överallt där dess företrädare kommit till makten resulterat i ofrihet.
Sett mot denna bakgrund är det motiverat att påstå att den svenska vänsterradikalismens tragedi var att den åren efter första världskriget försvor sig åt en utopisk emancipationsidé och drogs in i en våldsorienterad internationell rörelse vars udd var riktad inte bara mot det borgerliga samhället i Sverige utan i väl så hög grad mot den socialdemokratiska arbetarrörelsen. Denna extremism skulle i Sverige gång efter annan leda till att partiet sprängdes och att rader av enskilda hoppade av – i partiets ögon var dessa avhoppare renegater, förrädare och opportunister och de utsattes genomgående för hat och smutskastning. Kvar blev en sektliknande kärna av kommunister som i synnerhet under andra världskriget och det första årtiondet efter kriget av offentligheten uppfattades som det de faktiskt var, nämligen en inre fiende mot den svenska demokratin. Fram till början av 1960-talet betraktades Sveriges kommunistiska parti – föregångare till dagens Vänsterpartiet – därför som ett främmande element i den svenska politiska kulturen.
Hur ska denna tystnad i _Vitboken_ om kommunismens koppling till omfattande brott mot de mänskliga rättigheterna förstås? Det finns faktiskt en ledtråd i flera av bidragen. Kent Lindkvist skriver att det inom Vänsterpartiet förelåg »en brist... på samhällsvetenskaplig analys av de socialistiska samhällena» och Lars-Arne Norborg menar att den unga VPK-generationens kunskaper om de realsocialistiska länderna »förefaller på det hela taget ha varit begränsade». Längst går Sven-Eric Liedman som förvånas över att partiet inte lyckades knyta till sig »någon med gedigna kunskaper om Sovjetunionen» och att »vi inte tillräckligt ingående studerade Sovjetsystemet sådant det faktiskt var». De som i första hand borde klandras för detta var »de som i långa stycken hade genomskådat det sovjetiska eländet men ändå lät spelet fortsätta». Och Liedman fortsätter: »De som här har huvudansvaret är rimligen de inom VPK:s ledande skikt som genomskådade hur det var ställt med kommunistpartierna i öster och deras regemente men som ändå av olika taktiska skäl lät trafiken med diverse hövligheter och 'kamratligheter' fortsätta.»
Liedman formulerar sig som om kunskapen om övergreppen och missförhållandena i Sovjetunionen hade varit svårtillgänglig eller begränsad till en exklusiv elit. Sanningen är ju att kunskapen sedan lång tid tillbaka var vida känd och spridd. Särskilt från 1950-talet och framöver fanns en betydande historisk forskning om förhållandena i Sovjetunionen och folkdemokratiernas Östeuropa, en forskning som också framhöll sambandet mellan det realsocialistiska förtrycket och regimernas marxist-leninistiska ideologi. _Vitbokens_ författare – alla utom Ulf Nymark var sedan årtionden universitetsforskare inom humaniora – måste rimligtvis ha haft vetskap om detta. Eller var det kanske ändå så att de inte ville ta denna kunskap till sig eftersom den kom från »höger» och därför uppfattades som illegitim? Norborg talar i sin forskningsöversikt om »fiendelitteraturen» och nämner här framför allt Staffan Skott. Lindkvist efterlyser enbart marxistiska analyser av realsocialismen och när Liedman frågar efter mer kunskap om Sovjetunionen dömer han samtidigt ut de svenska Sovjetexperterna som »internationellt uppmärksammade» för sin »extrema högerprofil».
När Liedman vidare skall förklara varför VPK inte ens under 1980-talet förmådde skaffa sig relevanta kunskaper om realsocialismen är det återigen »högern» och »renegaterna» som bär skulden. Decenniet var:
den aggressiva nyliberalismens, den hejdlösa spekulationens, den ohämmade girighetens glansperiod. Många som stått långt till vänster under 70-talet dök nu upp på högerflanken, skoningslösare i sin uppgörelse med allt de själva nyss stått för än någon gammal högerman. De som fortsatte att se sig som socialister trängdes opinionsmässigt upp i ett hörn [---] Motvinden var så stark att det var svårt att sätta fötterna rätt. Vad partiet än gjorde sattes det i samband med kommunismens värsta avarter.
Man måste här fråga sig om inte den radikala vänsterns ideologi – oavsett om den kallas eurokommunistisk, Kominternkommunistisk, marxist-leninistisk eller nyleninistisk – utgjorde en mental och intellektuell fängelsemur som utestängde all extern eller icke-kommunistisk kunskap genom att delegitimera den som »höger», »nyliberal» eller »antikommunistisk».
## **VI. UPPGÖRELSERNAS TID**
## **1. Om vänsterns moraliska skuld**
I AUGUSTI 1990 INVADERADE Saddam Husseins Irak grannlandet Kuwait. De efterföljande månaderna fördes en intensiv debatt i medierna. Den kom i stora drag att handla om USA:s rätt, att med FN i ryggen, befria Kuwait från den irakiska ockupationen av landet. Fastän ingen försvarade Saddam Husseins erövring fanns likväl en stark svensk opinion som av olika grunder motsatte sig ett väpnat FN-ingrepp under USA:s ledning. Debatten fortsatte även efter det att Irak i början av 1991 besegrats och Kuwait återfått sin självständighet. Den kom då emellertid att vävas samman med en ny diskussion som startats på initiativ av den nyliberala tankesmedjan Timbro vid en konferens i maj 1991 under rubriken »Vänsterns moraliska skuld». Konferensen kom att utlösa en intensiv och långvarig strid om arvet efter kommunismen i allmänhet och arvet efter 68 i synnerhet. Denna strid utkämpades i allt väsentligt mellan dem som redan stått på barrikaderna – ofta på motsatta sidor sedan 1968.
Debatten kom att spänna över flera ämnen, men i fokus stod frågan om den svenska vänsterns ansvar för sin hållning till de realsocialistiska samhällssystemen i Sovjetunionen, Kina och Östeuropa. Hade den genom sina idéer och sitt handlande legitimerat ofriheten? Varför hade Vänsterpartiet upprätthållit nära relationer till broderpartierna i öst utan att rapportera om det förtryck av oliktänkande som man nu fördömde? Hur kom det sig att så många vänsterintellektuella rest till diktaturer som Kina, Kuba, Kambodja, Nordkorea och Albanien och återvänt med antingen tystnad eller lovsång? Det var uppenbart att den kommunistiska modellens sammanbrott hade satt hård press på såväl Vänsterpartiet som vänsterintellektuella utanför partiet.
Författaren och journalisten Kay Glans kom att bli en av centralfigurerna i de följande månadernas sammandrabbning. På Timbros konferens i maj hade han tagit upp ett tema som länge diskuterats internationellt, nämligen de så kallade pilgrimsresorna till socialistiska länder. Härmed förstods resor där västliga intellektuella, oftast med icke-kommunistisk profil, inbjudits som gäster till regimerna i till exempel Sovjetunionen eller Kina, och återvänt med mer eller mindre positiva beskrivningar av livsvillkoren i diktaturerna, beskrivningar som tillbakavisat den liberala och konservativa kritiken av dem. Glans lyfte fram ett antal svenska pilgrimer, med inbördes olika politisk hållning, som Artur Lundkvist, Olof Lagercrantz, Jan Myrdal, Lars Gustafsson samt Jan Guillou och Marina Stagh, vilka publicerat idealiserande reseberättelser från Sovjetunionen, Kina respektive Irak. Han ställde frågan hur man skulle se på »ansvarsfrågan och frågan om moralisk skuld». Glans avvisade tanken att de skulle bära ansvar för det förtryck som utövats i de länder de hade besökt och skildrat, men menade att de genom sin välvilliga beskrivning och sin offentliga ställning skänkt regimerna legitimitet inför det egna landets offentlighet.
Det var enligt Glans förkastligt att stödja totalitära regimer, men det moraliska ansvaret varierade från fall till fall beroende på omständigheterna. Viktigare än att utmäta skuld i de enskilda fallen var dock, menade Glans, att ställa krav på en öppen revision av de åsikter som pilgrimsfärderna var ett uttryck för.
Vilka åsikter var det då Glans menade borde omprövas? Något klart och uttömmande svar på denna fråga gav han aldrig, men en ståndpunkt han särskilt kritiserade var »vänsterns dubbla måttstockar». Utgångspunkten i en understreckare i Svenska Dagbladet var hållningen till Saddam Husseins Baathregim i Irak efter angreppet på Kuwait. Vänstern hade länge, enligt Glans, framställt den som en av tredje världens progressiva regimer – den hade som flera andra regeringar i arabvärlden varit både socialistisk och antikolonialistisk. Av denna anledning hade man, hävdade Glans, inte velat se att regimen under Saddam Husseins ledning hade utvecklats i alltmer tyrannisk riktning. När Irak genom erövringen av Kuwait nu direkt utmanade amerikanska intressen visade flera inom vänstern förståelse för Saddam Husseins handlande, eftersom det betraktades som antiimperialistiskt. De västliga demokratierna däremot, med USA i spetsen, ansågs rätt och slätt agera utifrån ett materiellt egenintresse – för dem påstods interventionen enbart handla om kontrollen av Mellanösterns oljetillgångar.
När många vänsterintellektuella, enligt Glans mening, inte ville inse omfattningen av Saddam Husseins brutala styre hade det sina skäl: »uppgörelsen med gamla hållningar [hade] varit så halvhjärtad. Under detta sekel har intellektuella från väst rest runt i Sovjet och tredje världen, lovprisat framåtskridandet och bagatelliserat förtrycket hos s k progressiva regimer. Man har konsekvent arbetat med dubbla måttstockar.»
Progressiva regimer i tredje världen, hur brutala och korrupta de än var, hade vänstern genomgående urskuldat med hänvisning till kolonialismens historiska verkningar. En skribent som särskilt drabbades av Glans kritik var DN:s kulturchef Arne Ruth som kritiserat den USAledda FN-koalitionen för dess militära ingrepp i konflikten i januari 1991 och i stället hävdat att sanktionerna mot Irak borde ha givits tid att verka innan angreppet sattes in. Ruth hade därutöver menat att Irak genom erövringen av Kuwait bara hade spelat efter samma maktpolitiska regler som USA själv brukade göra. Mot denna »förståelse» för Iraks handlande hade Glans invänt: »Inte för ett ögonblick tycks Ruth och hans meningsfränder vara beredda att diskutera huruvida andra kulturkretsar kan ha destruktiva egenskaper av liknande slag som dem vi sett föröda Europa under nuvarande sekel.»
I Glans ögon intog den postkommunistiska vänstern en kulturrelativistisk ståndpunkt, vilken han menade härrörde ur dess reaktion på nazismens grymheter – efter Auschwitz ansåg vänstern att europeiska värden inte hade någon moralisk företrädesrätt överhuvud. Varje försök att hävda en sådan rätt var för vänstern uttryck för en herrementalitet och för eurocentrism. Glans fann denna inställning ohållbar och påminde om att nazismen, med sitt förkastande av den liberala idén om en internationell rättsordning byggd på universella värden, hade intagit samma hållning som vänstern nu gjorde.
I Aftonbladet svarade ett par veckor senare kulturredaktören Karl Vennberg på Kay Glans understreckare. Vennberg tog utgångspunkt i den dåtida författarstriden om bland annat Tysklands deltagande i första världskriget och menade att då som nu såg många »ensamma famlande esteter» – här avsågs med viss säkerhet den unge poeten Kay Glans – den moderna krigshänförelsen som »en väg till gemenskap». En rad av dåtidens stora tyska författare hade ju med entusiasm ställt sig bakom kriget, medan de som manat till förnuft och moderation hade ringaktats och marginaliserats. Enligt Vennberg var Glans en av dem som hade »sitt heroiska hjärta hos kriget» och som låtit sig ryckas med i den aktuella krigshysterin: »När ett högteknologiskt krig nu har öppnat missionsvägen tänker han antagligen sekularisera islam, så att araberna i en nära framtid kan leva i samma fredliga och lyckliga gudlöshet som de västerländska folken.»
Udden i Vennbergs artikel var inte bara riktad mot Glans utan också mot det liberala samhället. Vennbergs tes var att liberalismen – i frågan om Irakkriget – kännetecknades av en fanatism som tog sig uttryck i publicistisk likriktning. Liberalismens företrädare ville fösa undan vänstern för att bana väg för Sveriges inträde i EG, som dagens EU då benämndes. Den förändring som skett i Sverige och den framtid som väntade landet vid ett eventuellt inträde i EU:s »fria kapitalism» beskrev Vennberg i följande termer: »där varje missnöje är bundet till händer och fötter, också marginella sociala experiment är uteslutna, arbetslösheten kan permanentas, de utstötta förblir utstötta och en demokrati med anarkistiska svagheter i bästa fall kan övergå i en fascism med demokratiskt ansikte.»
En som delade Ruths och Vennbergs kritiska inställning till USA:s militära ingrepp i Irak var den vasse skribenten Gunnar Fredriksson, Aftonbladets chefredaktör. Han menade att den så kallade tredje ståndpunkten var ett pragmatiskt sätt att ge sanktionerna och diplomatin en verklig chans och han raljerade över den amerikanska och västliga dubbelmoralen. Å ena sidan, skrev han, gav väst ett frikostigt stöd till allsköns diktatorer, schejkfamiljer och korrumperade militärdiktatorer, å andra sidan tog man till »brösttoner om frihet och demokrati» så snart de egna intressena stod på spel. Fredriksson tycktes mer allmänt uppfatta de stora västländernas utrikespolitik som blott och bart ett uttryck för snöda kommersiella intressen. Bakom varje principiell konflikt såg Fredriksson det materiella egenintressets bockfot, eller som han formulerade orsaken till Irakkriget: »hur kunde en trivial tvist inom Opec om oljepriserna bli till en global fråga om internationell lag.»
Glans återkom med ett inlägg i vilket han skärpte tonen, inte bara mot Vennberg som han menade »håller på att bevisa att hans stämma är klart umbärlig», utan mot vänstern mer allmänt. Enligt Glans förstod inte vänstern – och i synnerhet inte Vennberg som en gång i tiden inte ville ta ställning mellan Nazityskland och Churchills England – att sanktioner var verkningslösa mot regimer som var »beredda att föröda sitt land och i vilka befolkningens uppfattning totalt saknar betydelse». När Vennberg inte ville diskutera denna sakfråga utan i stället insinuerade att Glans var krigsromantiker var förmodligen orsaken följande:
Jag betecknar de intellektuellas apologier för kommunistiska despotier som ett skamligt och alltför litet diskuterat problem. Vänstertotalitarismen är ett ohyggligt kapitel i historien och ett bokslut över kommunismen är ofrånkomligt. Numer torde det stå klart för nästan alla att dessa blodiga experiment var återvändsgränder.
Glans avvisade vidare att det var företrädare för liberala idéer som dominerade debatten. Tvärtom menade han att kulturvärlden och då i synnerhet debatten i medierna dominerades av ett »vänsteretablissemang» som efter 1989 fruktade att förlora sina positioner:
Detta etablissemang är starkt ambivalent till att kommunismens förbrytelser kommer i centrum för diskussionen. Bortträngningen som råder påminner en del om den som rådde i Förbundsrepubliken före 60-talet, innan en ny generation på allvar tvingade fram en uppgörelse med det nationalsocialistiska förflutna.
Vad gällde Vennberg själv var han, enligt Glans, förmodligen orolig för att en granskning av kommunismens övergrepp skulle ställa hans »tredje ståndpunkt» under 1950-talet i dålig dager.
En ytterst kränkt Karl Vennberg svarade i Aftonbladet och karakteriserade där Glans replik som »en enda gemen insinuation» och menade att Glans genom påståendet om hans nazistiska sympatier på 1930-talet (något han bestämt tillbakavisade) hade sökt att »moraliskt förlama mig». Och Vennberg avslutade sitt inlägg med att Glans möjligen tillhörde dem »som beklagar att kärnvapenkriget inte blev av. I varje fall är han en svuren anhängare av engångsinflation i människoblod.»
Kay Glans publicerade cirka en månad senare, i juni 1991, en understreckare i Svenska Dagbladet med titeln »Vänsterns kultursfär och de brända ordens taktik». Här hävdade han att vänsterns idéer efter murens fall mist sin vitalitet och att den hade förlorat sitt intellektuella initiativ. Samtidigt påstod han att den inom kultursfären utgjorde en »diskret nomenklatura» med fortsatt makt över de viktigaste opinionsorganen. Här tillämpades den »de brända ordens taktik», som innebar att »om inte vänstern kan ha debattinitiativet, så skall heller ingen annan ha det». Glans kontrasterade vänsterns faktiska hegemoni över kultursfären med det förhållandet att den gärna framställde sig som »en trängd sanningssägande minoritet».
Detta argument är värt att notera, då det de följande åren skulle bli ett återkommande tema i 1990-talets vänsterdebatt, inte minst inom en yngre inte alltid så kritisk vänstergeneration. Tankegången var att kommunismens borgerliga motståndare visserligen hade varit förtryckta i de realsocialistiska staterna, men att de svenska kommunisterna hade på motsvarande vis varit förföljda i det borgerliga Sverige: övervakade av Säpo, registrerade, utestängda från en rad arbeten, satta i läger under andra världskriget och dessutom belagda med censur under samma period. Många vänsterföreträdare drog därför slutsatsen att den liberala demokratin var lika intolerant mot oliktänkande som en gång det sovjetkommunistiska systemet varit.
Glans undrade också varför den unga generationens skribenter – exempelvis Kristoffer Leandoer, Nina Lekander, Åsa Beckman, Magnus William-Olsson – inte utmanade den äldre vänstergenerationen med tanke på att den socialism de hade företrätt nu var diskrediterad. De unga ville gärna framstå som om de förde vänsterradikalismen vidare, men i själva verket tycktes de sakna politiska perspektiv. I stället för politiska analyser riktade man »sin opposition mot tänkandets hierarkier och språkliga ordningar»: »Att vara vänster på ett till intet förpliktigande sätt har faktiskt blivit en ny form för brackighet bland de yngre intellektuella – man undviker att tänka själva och håller sig inom ramen för det som anses passande.»
Glans kritik av vänstern vilade på en utgångspunkt som han bara tillfälligt tangerat i sina första artiklar, men formulerade mer tydligt i ett av sina avslutande inlägg i Svenska Dagbladet. Han hävdade här med eftertryck att vänsterintellektuella som stött de totalitära regimerna i öst belastades av en moralisk skuld och han jämförde dem med de författare som hade sympatiserat med eller till och med varit anhängare av nazismen. Samtidigt menade Glans att det på denna punkt fanns en uppenbar skillnad – medan det rådde stor enighet om att fördöma nazismen fanns det en betydande förståelse för den sovjetiska kommunismens förtryck.
Denna diskrepans har blivit tydlig under debatten om vänsterns moraliska skuld. Diskussionen har karakteriserats som moraliserande [...] eller som mindre angelägen [...] Inga egentliga argument för denna drastiska skillnad i hållningssätt gentemot seklets två största massmord har hittills presenterats och att så många föredrar att driva en skendebatt beror förmodligen på att de är svåra att finna.
Jämförelse var i hög grad relevant. Nazismen och kommunismen uppvisade inte bara ideologiska likheter, de hade dessutom båda iscensatt förbrytelser i en omfattning som saknade historiska motsvarigheter. Jämförelsen var relevant också i ett annat avseende, nämligen att förhållandet mellan de två ideologierna hade varit föremål för omfattande studier och debatter under flera årtionden.
Glans fick aldrig något svar på sin jämförelse mellan kommunism och nazism. I stället valde ett par av kulturvänsterns centrala personer – Aftonbladets kulturchef Gunder Andersson och DN:s kultursidas 'starke man' vid denna tid Lars-Olof Franzén – andra linjer för sin argumentation. Andersson inledde sin artikel med den teatraliska formuleringen: »Nu slipas knivarna. Nu stundar domens dag. Nu skall den gamla vänstern strimlas.» Bakom »högerns» kritik av vänsterns kontakter med den sovjetiska kommunismen dolde sig en helt annan agenda, nämligen att »söka kompromettera allt kritiskt, allt rationellt tänkande. [...] När nu Östeuropa fallit ihop så tar högern chansen att försöka göra sig av med allt som på minsta vis har med jämlikhetstänkande och solidaritetsidéer att göra».
Och Gunder Andersson avslutade sin artikel med förmodandet att Kay Glans gärna hade sett ett amerikanskt kärnvapenanfall på Sovjetunionen i slutet av 1940-talet, så att det som misslyckades 1921 kunde ha avslutats. Andersson syftade här på den brittiska och amerikanska interventionen i det ryska inbördeskriget med syfte att söka störta bolsjevikregimen.
Flera debattörer antydde att Glans hade låtit sig köpas av Timbro och Svenskt Näringsliv (SAF) och att han genom sin kritik av vänstern sökte förminska dess insatser för det svenska välfärdssamhällets utveckling. En variant av Anderssons tema formulerades av Franzén som bestämt förnekade att vänstern eller socialismen lidit något nederlag i och med 1989. För nederlaget stod enbart den statsbärande kommunismen i öst. Enligt Franzén hade det alltid varit en »önskedröm för de konservativa och 'nyliberala' krafter som Kay Glans ställer sig i tjänst hos att vänstern entydigt ska kunna identifieras med ett totalitärt samhällsskick».
Det både Franzén och Andersson verkade vilja säga med detta var att Glans och andra liberala kritiker av kommunismen i grund och botten gick »högerns» ärende och att deras egentliga mål var att kompromettera den socialdemokratiska reformismen – att Glans i den aktuella debatten aldrig riktat någon kritik mot vare sig socialdemokratin eller den demokratiska socialismen låtsades varken Andersson eller Franzén om.
Det är svårt att värja sig mot intrycket att de båda redaktörerna ville undvika att få sin vänsterradikalism identifierad med kommunismen och i stället sökte skydd bakom ett vänsterbegrepp som lade tyngdpunkten vid den socialdemokratiska arbetarrörelsen. Likväl är det uppenbart att Kay Glans och andra liberala debattörer aldrig refererade till socialdemokratin när de talade om vänsterns moraliska skuld. Fokus låg uteslutande på kommunismen och i synnerhet på de vänsterradikala revoltörerna från 1968 som glidit in under maoismens eller Sovjetkommunismens ideologiska auspicier.
I motsats till Andersson var Franzéns tonläge mer modest. Han nöjde sig med att kalla Glans okunnig och totalt utan »sofistikation» och ansåg att denne ville »att de skyldiga ska naglas fast och fördömas». Inte desto mindre är det värt att notera att såväl Andersson som Franzén gav Glans rätt på en rad punkter. Det gällde i synnerhet Franzén som höll med om att »författare och intellektuella ofta visat en oacceptabel svaghet för ideologiska extremister». Han delade även Glans uppfattning att »uppgörelsen med stalinismen aldrig fått samma omfattning som uppgörelsen med nazismens brott». Också Gunder Andersson medgav, om än motvilligt, att det »fanns inom vänstern naiva föreställningar om förhållandena i vissa socialistiska länder». Han framhöll här särskilt Albanien, men nämnde inget om Sovjetunionen och öststaterna, eftersom han menade att »60-talets nyvänster» redan från början hade gjort upp med Sovjetkommunismen. Andersson medgav dessutom att Vennberg och de andra företrädarna för den tredje ståndpunkten under 1950-talet »säkert» underskattat »omfattningen av Stalins utrotningar».
Till skillnad från några år tidigare fanns det inte längre många inom vänstern som 1991 försvarade de realsocialistiska staterna och de sovjetkommunistiska partierna. Vänsterns politiska retorik om Sovjetkommunismen hade helt enkelt förlorat sin historiska legitimitet. Lika plötsligt som överraskande fanns bland svenska vänsterintellektuella en konsensus om att »realsocialismen» varit ett misslyckat samhällssystem, präglat av misär och förtryck. Skiljelinjen i debatten hade förflyttats till en helt annan terräng, nämligen mellan de liberala kritiker som fördömde den svenska vänsterns anknytning till de totalitära systemen i öst och de vänsterföreträdare som tillbakavisade kritiken som orättvis och överdriven. De senare menade att »högern» bedrev en McCarthyistisk klappjakt på vänsterintellektuella, att den moraliserade och fördömde på godtyckliga grunder och att den förfalskade historien om de vänsterintellektuellas pilgrimsresor i österled. De tillbakavisade fullständigt att vänstern skulle inta en hegemonisk position inom kultursfären, tvärtom var den i ideologiskt avseende en trängd minoritet och för övrigt hade vänstern i grund och botten aldrig stått för annat än jämlikhet, solidaritet och demokrati. Värden som omfattades av såväl socialdemokratin som progressiva liberaler, men som »högern» alltid hade bekämpat.
## **2. Vänsterliberalismens svek. Herbert Tingsten går igen**
År 1968 står som sagt inte bara symbol för en radikalisering av den politiska vänstern. Idéförskjutningen i slutet av 1960-talet påverkade därutöver socialdemokratin och i hög grad också det borgerliga Sverige, i synnerhet den borgerliga vänsterns företrädare som exempelvis den liberala pressen, Folkpartiet (Liberalerna) och dess ungdomsförbund, liksom delar av Centern. Framför allt det senare partiet influerades starkt av 1970-talets vänster – den gröna vågen med decentralisering, lokaldemokrati, antikärnkraft, ekologiskt jordbruk och hållbar utveckling på programmet. I den liberala strömfåran kom man i högre grad att ta till sig den nya vänsterns »antiimperialistiska» syn på tredje världen och de nationella befrielserörelserna – vilka med få undantag var marxistiskt och leninistiskt styrda – liksom idén att det fanns andra former av demokrati än den liberala som var bättre anpassade till utvecklingsländernas behov.
Det faktum att socialdemokratin i Sverige samtidigt med vänsterrevolten bytte partiledare 1969 (Olof Palme ersatte Tage Erlander) och inledde en ny och radikalare fas i sin utveckling spelade en viktig roll – oavsett hur man bedömer vad som var orsak och verkan. Radikaliseringen innebar en hårdare konfrontation med näringsliv och borgerlighet i synen på jämlikhet, äganderätt och styrning av näringslivet, men fick kanske ändå störst genomslag vad gällde Sveriges hållning till de revolutionära krafterna i tredje världen och kraftmätningen mellan västmakterna och östblocket.
Året efter Sovjetunionens upplösning publicerade den förre folkpartiledaren Per Ahlmark _Tingsten. Tyranniet begär förtroende_. Herbert Tingsten dominerade svensk efterkrigsdebatt i kraft av sin ställning som chefredaktör för Dagens Nyheter (1946–1959), sin vidsträckta beläsenhet, polemiska skärpa och stilistiska briljans. Efter sin avgång från DN utkom han med en självutlämnande, personlig och ytterst läsvärd memoarsvit. Per Ahlmark tillhörde en generation unga intellektuella liberaler som tog starka intryck av Herbert Tingstens kritik av kommunism och nazism, liksom hans syn på demokrati och kultur – Ahlmark blev dessutom personlig vän med Tingsten och stod honom särskilt nära efter det att han lämnat Dagens Nyheter. I förordet till boken riktar Ahlmark en förbittrad kritik mot Dagens Nyheter i synnerhet och indirekt mot svensk vänsterliberalism i allmänhet för undfallenhet mot 68 års vänster.
Tingstens ideologiska utveckling rymmer flera aspekter som också kan kasta ett visst, om än indirekt, ljus över 60-talets vänster. En utvikning i ämnet är därför på sin plats. Tingsten växte upp i en oscariansk ämbetsmannamiljö. Efter studier i juridik och statskunskap blev han först docent i Uppsala 1924 och därefter professor i statskunskap vid Stockholms högskola 1935. I memoarerna framställer Tingsten sin politiska utveckling som en bana från gymnasietidens nationalkonservatism, över studentårens liberalism till socialdemokratin vid 1920-talets mitt och åter till liberalism vid krigsslutet 1945.
Likväl tyder mycket på att Tingsten attraherades av den radikalisering som kom att prägla i synnerhet 1930-talets unga författare, akademiker, arkitekter och konstnärer. Tingsten diskuterade i memoarerna sin anslutning till socialdemokratin och han utgav under kriget en banbrytande analys av socialdemokratins idéutveckling. Tingsten skriver själv att han tog starka intryck av marxismen och av den ryska revolutionen. Som så många andra unga intellektuella som upplevt känslan av värdeförlust och intighet efter första världskriget vände han sin kritik mot borgerskapet, kapitalismen, kyrkan och monarkin, men också mot de konservativa och fascistiska idéerna. Han blev av mycket att döma del av en vänsterströmning som återfanns inom den radikala delen av socialdemokratin, inom den bolsjeviksvärmande organisationen Clarté och i tidskrifter som Spektrum, acceptera, Mänsklighet, Kulturfront, Kontakt och Karavan. De aktiva här var vänsterradikaler som pläderade för en ny människa, en ny livskänsla och ett nytt samhälle. Få av dem var kommunister men alla verkade under intryck av den dramatiska oktoberrevolutionens utopiska och modernistiska idéer. Redan under 1930-talet kom artiklar i tidskrifter som till exempel Clarté, acceptera och Spektrum om den nya framtida människan och världen. Här krävdes en funktionalistiskt inspirerad stadsplanering, en demokratisk skola utan betygssättning, en barnuppfostran i offentlig regi, en liberalare strafflagstiftning och psykiatri, en friare sexualsyn och en planmässig styrning av det privata näringslivet. (Rader av de vänsterradikaler som var aktiva i exempelvis Clarté med omnejd under 1930-talet gick efter kriget vidare till ledande befattningar inom den svenska staten och organisationsväsendet.)
Det var i denna intellektuella och politiska miljö som Herbert Tingsten levde sitt liv under 1930-talet. Framför allt hade han av personliga skäl en ytterst nära kontakt med den socialdemokratiske politikern och advokaten Georg Branting, son till Hjalmar Branting. Denne var gift med Tingstens första fru Rina Lind och styvfar till Herberts och Rinas gemensamma barn Kaj Tingsten. Branting var under 1930-talet den socialdemokratiska riksdagsgruppens gossen Ruda. Enligt den framlidne journalisten Anders Isakssons lysande Per Albin-biografi hörde han till »den nya tidens monarki... Han blev aktiv antifascist och arbetade mycket med flyktingfrågor... Hans bördshögfärd inför dem som nu kontrollerade partiet ledde honom till Zäta Höglund, en intellektuell och politisk frasmakare som han själv».
Branting var vänsteroppositionens förgrundsfigur, svärmade för det bolsjevikiska Ryssland, hade utmärkta relationer med Sovjetunionens sändebud i Stockholm, Alexandra Kollontaj, och hade en uttalad fiende i partiledaren Per Albin Hansson.
När Tingsten tillträdde som Dagens Nyheters chefredaktör 1946 var det inte bara en av landets ledande statsvetare och idéanalytiker, utan dessutom en av 1930-talets främsta vänsterintellektuella och kulturradikaler – som under kriget blivit liberal antikommunist – som tog över tidningen. Från samma radikala kretsar kom efterhand flera medarbetare på Dagens Nyheter som exempelvis Ingemar Wizelius, Ingemar Hedenius och Torsten Gårdlund. Något tillspetsat kan det sägas att Dagens Nyheter såväl åren efter krigsslutet 1945 som åren efter 1968 kom att domineras av den tidens segrande vänsterradikalism. En avgörande skillnad mellan de två »vaktavlösningarna» var dock att Tingstens DN tog lika mycket avstånd från kommunism som nazism, medan 68-grupperingen på Dagens Nyheter, framför allt dess kultursida, kom att idealisera de nya revolutionära strömningarna i främst tredje världen. Fanns det ett samband mellan 1930-talets radikalism och 68-generationens uppror? En idémässig förbindelse via efterkrigstidens samhällsomvandling? Frågan är svår att besvara eftersom forskningen på området är begränsad. I varje fall på ett område är ny forskning nu på väg och pekar på ett sådant samband, nämligen vad gäller den svenska skolans radikala förändring från slutet av 1950-talet, en förändring som bland annat hade sin upprinnelse i 1930-talets diskussioner.
I inledningen till _Tingsten. Tyranniet begär förtroende_ går alltså Ahlmark till frontalattack mot 68-vänsterns inflytande. Men han inleder sitt angrepp med en kringgående rörelse. Han startar med den svenska utrikespolitiken under socialdemokratins tid vid makten efter 1945 – först nästan två årtionden med Östen Undén som utrikesminister och därefter ytterligare två årtionden med i praktiken Olof Palme vid rodret. Sverige hade under hela denna period stått för en undfallande och selektiv hållning mot världens diktaturer. »Långt vanligare än fördömande hade varit avmätta uttalanden [om kommunistregimerna i öst], som genom sin så kallade balans bidragit till en djupgående svensk förljugenhet.» Östen Undén hade varit alltför försiktig i hållningen till diktaturerna i Sveriges närhet. »Han hade en romantisk syn på ryska revolutionen och vägrade medge att den blivit ett skräckvälde.» Dessutom hade Sverige, genom Undén, givet sitt »stöd åt den ryska ockupationen av de baltiska staterna». Mot denna anpassningslinje hade Tingsten hävdat att varje utrikespolitiskt ställningstagande borde utgå från försvaret av de mänskliga fri- och rättigheterna. Denna hållning hade, menade Ahlmark, fört Tingsten till en »permanent opposition mot den svenska utrikeslinjen». Genom hans återkommande kritik av regeringens utrikespolitik hade det, enligt Ahlmark, skapats ett slags balans mellan Undéns undfallenhet och Dagens Nyheters fördömande av varje form av diktatur. Denna balans hade helt gått förlorad efter det att Tingsten lämnat tidningen och Olof Lagercrantz börjat teckna dess ideologiska profil:
Under Undéns tid som utrikesminister skapade Dagens Nyheters motstånd en sorts balans i debatten. Men när Olof Palme var statsminister och partiledare blev Dagens Nyheter istället en allierad i utrikes ärenden... Denna gemenskap i utrikessyn mellan socialdemokratins ledning och den största morgontidningen (främst dess kultursida) rubbade den tidigare balansen.
Dagens Nyheter hade kommit att bli en bärare av det nya politiska klimat som dominerades av vänstern och som resulterat i en ensidig kritik av »väst» och »kapitalismen» och en »romantisering av många diktaturer och enpartistater samt till mer förståelse för, än kritisk granskning av åtskilliga marxistiska/socialistiska regimer och rörelser». Där Herbert Tingsten hade stått upp till försvar för demokrati och mänskliga rättigheter reste Olof Lagercrantz 1970 till Maos Kina och lovordade, enligt Ahlmark, kulturrevolutionen utan att vilja se dess tyranni. I synnerhet Dagens Nyheters kultursida hade under Lagercrantz och senare Arne Ruths ledning »förtigit, förvrängt eller förlöjligat Tingstens insatser».
Ahlmarks angrepp på Dagens Nyheter väckte stor uppmärksamhet och bemöttes omgående i tidningen av framför allt den politiske chefredaktören Svante Nycander och kultursidans Arne Ruth. Flera av deras, och andras, kommentarer erinrar om den starka personliga animositeten mot Tingsten och är i detta sammanhang mindre intressanta. Det saken gällde var ju tidsskiftet eller idéskredet med 1968 som symbolår, huruvida den liberala vänstern – det reforminriktade frisinne – som Dagens Nyheter ansåg sig företräda, hade svikits under »stormarnas år»?
Hur förhöll sig tidningens företrädare till Ahlmarks häftiga anklagelser? Både Nycander och Ruth honorerade Tingstens lärdom, skärpa och stil och menade att mycket av det han skrivit fortfarande var aktuellt, men sedan gled deras argumentation och bedömningar isär. Nycander ansåg att det från de sista åren av 1960-talet och framöver fanns »medarbetare i DN» som hade befunnit sig »i trakten av det tidigare vänsterpartiet kommunisterna». När dessa hade kritiserat Tingsten var »det uppenbart att idémotsättningen berör just demokrati–diktatur». Till denna kategori på Dagens Nyheter – som pläderade för en eller annan form av socialistisk diktatur framför »borgerlig demokrati» – hade emellertid inte Olof Lagercrantz hört. Enligt Nycander emanerade hans motiv ur något annat:
i likhet med flertalet författare, journalister, studenter i stora delar av västvärlden såg han kommunismen i ett annat perspektiv än Tingsten. Det berodde främst på att den tredje världen under sextiotalet kom att stå i centrum för idédebatten och att 'kommunism' därmed upphörde att vara något entydigt och Moskvastyrt.
Nycander menade att därmed hade tilltron till den västliga demokratin försvagats och fått många att – i tredje världen – föredra en utvecklingsmodell byggd på en socialistisk enpartistat framför någon reaktionär militärjunta som åtnjöt USA:s stöd. I ett läge där det saknades ett demokratiskt alternativ hade man, menade Nycander, rätt att välja ett icke-demokratiskt. Härav följde dock inte att man svikit demokratins idéer, även om den regim man stött också skulle visa sig vara korrupt – avgörande var på vilka bevekelsegrunder man tagit ställning. »Resonemanget gäller tusenden som betraktat det kommunistiska Kina med förståelse, bland dem Olof Lagercrantz.» Det är tydligt att Svante Nycander vidgår att Dagens Nyheter, i varje fall dess kultursida, skiftade perspektiv i synen på kommunism och diktatur kring 1968 – däremot tycks han anse att detta skifte var del av en allmän tendens och att den omfattade flertalet »författare, journalister, studenter i stora delar av västvärlden».
Arne Ruth ägnade Per Ahlmarks bok tre stora artiklar på Dagens Nyheters kultursida. De är mångordiga och tar upp skiftande ämnen, men tecknar samtidigt en position, ett ideologiskt mönster. I den första artikeln med rubriken »Mångsidig Tingsten i ensidigt urval» sätter Ruth fokus på den radikala sidan av Herbert Tingstens skriftställning. Särskilt lyfter han fram _Dagbok från Amerika_ som utgavs 1968 och i vilken Tingsten enligt Ruth ger en sympatisk bild av den radikala svarta rörelsen, kritiserar konkurrensen mellan de privata universiteten, pläderar för offentligt ägd TV och ger en »balanserad beskrivning» av den amerikanska Vietnamdebatten. Ruth berömmer också Tingsten för att han definierar de mänskliga rättigheterna oberoende av tid och rum och menar självkritiskt att Vietnamrörelsen i sitt stöd till FNL och det vietnamesiska folkets rättigheter inte såg att Hanoi »rymde sin egen form av förtryck». En särskild poäng gör Ruth av Tingstens avståndstagande i memoarerna från det »gammalliberala» Mont Pèlerin-sällskapet, grundat av bland andra Friedrich von Hayek – »om detta är liberalism är jag nog fortfarande socialist» – hade Tingsten sagt. Sist men inte minst framhåller Ruth Tingstens klarsyn i maktfrågor, där han inte väjde för tanken att »demokratin» också måste reglera och kontrollera den ekonomiska makten. »Koncentrationen av makt, rikedom och prestige hos ett fåtal är en kränkning av flertalet», skrev Tingsten och Ruth tillägger: »Så talar en liberal värd namnet.»
Genom att understryka Tingstens sociala och kulturella radikalism och nedtona hans motstånd mot diktaturer av alla slag sökte Ruth vrida bilden av Tingsten ur händerna på Per Ahlmark. Ruth medgav att »vänstern kastade ut åtskilliga barn med badvattnet», här ibland tydligen Tingstens sociala kritik och kulturella radikalism. Ahlmarks försök att återuppliva Tingstens kritik av kommunismen och att vända denna kritik mot Dagens Nyheter såg Ruth som ett uttryck för »politisk fundamentalism». Ja, mer än så: Genom att lyfta fram Tingstens kalla krigetretorik ställde Ahlmark sig i grund och botten till förfogande för den nyliberala högern med stöd av »SAF:s propagandaavdelning Timbro».
I motsats till Svante Nycander medgav inte Arne Ruth att Dagens Nyheters kultursida från och med 1968 mer allmänt hade blivit ett språkrör för den nya vänsterradikalismen. Ruth mejslade ut sin ståndpunkt i sin andra replik på Ahlmarks bok. Här hävdade han att »när 68-vänstern tog avstånd från Herbert Tingsten var det tolkningen av hans roll i tredje ståndpunkts-debatten 1951 som fällde utslaget». Det var hans påstått svartvita syn på konflikten mellan öst och väst som marginaliserade honom. Som dokument för denna bedömning hänvisade Ruth till ett nummer av Ord & Bild och en debattbok utgiven av Tomas Forser och Per Arne Tjäder. Ruths huvudargument var följande:
Tingsten blev en symbolfigur i den uppgörelse med det kalla krigets tankemodeller som följde med Vietnamdebatten. Lagercrantz stödde en ny världsbild som vägrade se alla politiska motsättningar i termer av 'världskommunismen' mot den fria världen'.
Lagercrantz hade, menade Ruth, rest till Kina 1971 för att nyansera bilden av Kina som »den kommunistiska världserövringens centrum». Att han inte såg de mänskliga kostnaderna för kulturrevolutionen berodde på att dessa inte var kända. Något knäfall för Mao var det bestämt inte fråga om. Arne Ruth framställde Lagercrantz som en föregångare i strävan att bryta med den stereotypa synen på kommunismen. Lagercrantz hade först och främst frigjort Dagens Nyheters kultursida från en reaktionär antikommunism, men utan att falla för yttersta vänsterns revolutionsromantik och idealiserande självbedrägeri – och därmed försvarat en klassisk radikal liberalism. I den politiska världsbild Ruth tecknade fanns med andra ord tre politiska krafter. Längst till vänster stod de revolutionära drömmarna (som ingen tog på allvar) och till höger återfanns den verkliga fienden nyliberalismen, moderaterna, storkapitalisterna, arbetsgivarföreningens propagandister och den amerikanska imperialismen. Mellan dessa stod en progressiv liberalism, »värd namnet», bland annat företrädd av Dagens Nyheters kultursida. Under Lagercrantz ledning, var Ruths budskap, hade kultursidan bekämpat såväl romantisk vänsterideologi som egoistiska och individualistiska högeridéer.
Ahlmark blev inte svaret skyldigt. Han sköt in sig på Ruths och Nycanders försvar av Lagercrantz artiklar om kulturrevolutionens Kina, publicerade i Dagens Nyheter 1970–1971. Dessa utgjorde en reseberättelse från det Kina som efter Maos brytning med Stalin och den sovjetiska kommunismen hade seglat upp som en ny förebild för den av marxism och leninism influerade 68-rörelsen i väst. Detta visste säkert Lagercrantz och han var förmodligen också klar över att Kina var en kommunistisk enpartidiktatur utan mänskliga fri- och rättigheter. Han måste också rimligen ha varit bekant med de anklagelser för övergrepp som kommit från såväl flyktingar i Hong Kong som västliga Kinakännare. Om detta fanns emellertid ingenting i Lagercrantz artikelserie. I stället skildrades ett samhälle på frammarsch, fyllt av ungdomlig entusiasm, ekonomisk tillväxt och vilja till fred. Om Mao skrev Lagercrantz:
I kraft av sin unika ställning som på en gång stor skribent och beslutsam handlingsmänniska [...] var Mao Tse Tung den ende som kunde föras fram som samlande symbol [...] Främst tack vare honom kunde det förhindras att den kinesiska kommunismen misstänkliggjordes som främmande tyranners påhitt för att förslava landet ännu en gång.
I sitt försvar för Olof Lagercrantz ignorerade Arne Ruth de komprometterande omdömen som artikelserien innehöll flera exempel på och som i Per Ahlmarks ögon hade förvandlat Lagercrantz till »språkrör för Maos Kina under några av detta lands mest förnedrande år». För att visa att Lagercrantz inte var »en allierad till 60-talets nymarxism och vänsterromantik» lyfte Ruth fram en rad mot vänstern kritiska inlägg och handlingar från Lagercrantz sida: han hade förmedlat ett brev från Solzjenitsyn till den franska dagstidningen Le Monde, han hade tagit ställning mot Che Guevara-kulten i polemik mot Peter Weiss och han hade släppt fram kritiska Kinaartiklar på Dagens Nyheters kultursida.
Ruth ville emellertid inte se vad Svante Nycander indirekt vidgick, nämligen att Lagercrantz omfamning av en vänsteropportun Kinabild inte var något unikt för Dagens Nyheter. Precis som Nycander underströk tillhörde Lagercrantz de »tusenden» som i brist på demokratiska alternativ hade »betraktat det kommunistiska Kina med förståelse». Mot Nycander kan väl här invändas att det faktum att ett sådant alternativ saknades, knappast var något argument för en liberal redaktör att inta en välvillig syn på kommunismen eller att låta tidningens kultursida bli ett språkrör för den nya vänsterns synsätt. Det är riktigt att vänstervridningen i delar av den etablerade borgerligt-liberala opinionen var ett allmänt fenomen i västländerna. De internationellt uppmärksammade forskarna David Caute och Paul Hollander har i olika studier klarlagt den omfattande trafiken av både liberala och marxistiska författare och journalister till Kina i samband med kulturrevolutionen – och nästan alla återvände de med djupt förljugna och tillrättalagda bilder av maoismens påstådda välgärningar. Denna prokommunistiska panegyrik av ledande liberala opinionsbildare köptes aldrig av Caute och Hollander. I stället kartlade de den våg av förståelse och medhåll som kom nyvänstern till del bland många västliga intellektuella och studenter under 1960- och 1970-talet.
Vad var det då för mer »nyanserad» bild av världskommunismen som Olof Lagercrantz, enligt Ruth, hade utvecklat och som kom att ersätta Tingstens kalla kriget-schematik? Synar man denna bild närmare i sömmarna är det dock påfallande hur väl den överensstämmer med Cautes och Hollanders beskrivning av de politiska pilgrimernas eller medresenärernas syn på kommunismen. I den bok av Forser och Tjäder, som Ruth hänvisade till i sitt inlägg, fanns en sammanfattning av de nya perspektiven på efterkrigstidens stora konflikt och en redovisning av den tredje ståndpunkten. Författarna hävdade här med referens till amerikanska vänsterhistoriker att det inte var kommunismens Sovjetunionen utan det kapitalistiska USA som bar huvudansvaret för det kalla kriget. Den grundläggande orsaken var, menade de, att den amerikanska ekonomin drevs av »krav på fria avsättningsmarknader och framför allt fria råvarutillgångar som USA:s alltmer expanderande moderna kapitalism krävde för att kunna överleva».
Tankegången, som hade hämtats från den leninistiska imperialismteorin, var att kapitalismen som följd av konkurrensen och »profitjakten» till sin natur var aggressiv, medan ett land som Sovjetunionen vilket avskaffat det privata ägandet och den kapitalistiska konkurrensen saknade expansiva impulser till och motiv för territoriell utvidgning. De ledande västmakterna, tidigare anförda av Storbritannien och nu av USA, var imperialistiska stater som överallt stödde reaktionen mot den »verkliga» demokratins krafter. Antikommunismen var inget annat än en täckmantel för »kamp mot radikala tendenser inom den inhemska arbetarklassen». Den nya »världsbild» Ruth såg i uppgörelsen med Tingstens syn på kalla kriget var helt enkelt övergången från ett antitotalitärt och liberal-demokratiskt till ett radikalt marxistiskt perspektiv på de internationella relationerna. Detta synsätt fick en mer allmän spridning från slutet av 1960-talet genom att den formulerades av den nya vänsterns samhällsvetare och historiker.
Detta avsnitt har handlat om en liberal kritik av en liberal tidning och delvis även av en liberal offentlighet, för att den under 68 års vänsteruppsving gav vika för vänsterns kritik och till viss del anammade dess radikalt marxistiska syn på till exempel den tredje världen och den kinesiska kommunismen och för att den dessutom lånade ut »juvelen i det liberala imperiets krona» – DN:s kultursida – till den nymarxistiska vänsterns opinionsbildning. Hur ska dessa Ahlmarks anklagelser bedömas? Vad var sant och vad var falskt? Värderingen avhänger i hög grad hur begreppen vänster, liberal och höger definieras.
Att medlemmar i och sympatisörer till det dåvarande Vänsterpartiet kommunisterna var kommunister, marxister och leninister i en eller annan tappning ligger i sakens natur. Det samma gäller den skara som fylkades i och kring de vänsteristiska smågrupperna. Det klassifikatoriska problemet uppstår när vi betraktar de breda, ganska passiva, »68-kretsar» som i efterhand har kallats eller kallade sig själva antingen socialister, vänsterradikaler eller enbart kulturradikaler, men som likväl kom att inta extrema vänsterståndpunkter, i varje fall tillfälligt och i enskilda frågor. Å ena sidan gäller det här att vara observant på nyanser, å andra sidan att ändå dra rimliga demarkationslinjer. Ett exempel från Åke Lundqvists tidigare nämnda bok om Dagens Nyheters kultursida kan klargöra vad jag avser. I boken betecknar Åke Lundqvist konstkritikern Per-Olov Zennström som »socialist». Zennström, en kunnig och begåvad kulturskribent, var organiserad kommunist och kulturredaktör på Ny Dag bland annat under åren 1943–1947 och 1956–1964 och alltså aktiv medlem i ett stalinistiskt orienterat kommunistparti. Varför kalla denna vänsterextremism för socialism, om inte tanken är att släta över eller trivialisera hans kommunism?
När eftervärlden ser tillbaka på 1968 års vänster och dess föregångare finns det som ovanstående visar, en tendens att släta över vad den nya vänstern egentligen representerade. Denna tendens förstärks när blicken faller på vänsterliberalism och vänstersocialdemokrati under samma period – på de personer, rörelser och publika organ som egentligen var del av det borgerligt-liberala samhället, men ändå rycktes med av vänstervågen. Varken liberaler eller socialdemokrater vill i dag kännas vid sin långtgående förståelse för och ibland till och med entusiasm inför de marxist-leninistiska »frihetskämpar» i exempelvis södra Afrika som de försåg med miljontals svenska skattekronor.
Hur kunde detta idéklimat uppstå? Hur kunde vänsterextrema, i grund och botten Komintern-kommunistiska teorier om imperialism och fascism, dyka upp i ny form och dupera dem inom arbetarrörelse och borgerlighet som sett faran både under 1930-talet och under den första efterkrigstiden? Att de förförde många unga och naiva studenter på 1960-talet är faktiskt lättare att förstå. Det som gör Dagens Nyheters kultursida till ett intressant och relevant exempel är inte främst de enskilda medarbetarnas personliga sympatier, även om det är genom den enskildes göranden och låtanden som historien talar. Nej, sidans utveckling åskådliggör först och främst hur extrema idéer kunde smälta in i och legera sig med ett liberal-demokratiskt perspektiv – när garden sänktes.
Av de kända intellektuella från väst som exempelvis beskrev kulturrevolutionen i positiva ordalag var få uttalade kommunister. Snarare var de allmänt vänsterorienterade eller liberala intellektuella som blivit desillusionerade av västmakternas kolonialkrig under 1950- och 1960-talet och som radikaliserats under intryck av klyftan mellan den rika och den fattiga världen. De riktade främst sin besvikelse, sin cynism och sitt främlingskap mot det egna liberala och kapitalistiska samhället. Det är ingen överdrift att påstå att de i »högern», också den demokratiska högern, såg ett större hot mot demokratin än den extrema 68-vänstern. Av idealism eller opportunism bidrog de såväl indirekt som direkt till att ge den nya vänstern vind i seglen.
Arne Ruth har likväl en poäng i sin referens till en »ny världsbild». Den liberala kritiken av kommunismen hade vid mitten av 1960-talet nått en utmattningspunkt, en bristningsgräns. De upptinade relationerna mellan väst och öst och avstaliniseringen i öst hade lyft fram nya sidor av de så kallade folkdemokratierna – nya östliga författare hade debuterat och nya filmer nådde den västliga marknaden. 1950-talets kritik framstod mer och mer som förgrovad och mekanisk. Till detta kom att avkoloniseringen i Afrika och Asien försköt perspektivet på de internationella relationerna och satte fokus på den »tredje världen» i stället för på försvaret av »den fria världen». Den unga vänstergenerationen beskrev och analyserade – till en början – inte heller världen i den sovjetkommunistiska marxism-leninismens förkalkade terminologi. Den marxism de inledningsvis utvecklade hade i högre grad sina rötter i en västlig filosofi och företräddes av intellektuella som Perry Anderson, Raymond Williams, Nicos Poulantzas, Paul Baran, Louis Althusser och Jürgen Habermas. Den liberala kritiken av kommunismen och det kalla kriget hamnade opinionsmässigt på mellanhand, utan att i sak ha blivit felaktig. Frånvaron av demokratiska och reformorienterade krafter i tredje världen fick många liberaler att tro att bara revolutionära omvälvningar ledda av marxist-leninistiska befrielserörelser kunde lösa utvecklingsproblemen. Den stegvisa och fredliga reformism dessa vänstersinnade liberaler och socialdemokrater såg som en välsignelsebringande tillgång för det egna landet avvisade de som modell för länderna i tredje världen.
Alltför många kom på alltför kort tid att anamma detta perspektiv. Några få bröt tidigt med det när de insåg dess auktoritära innebörd och blev omgående stämplade som reaktionärer, förrädare och högeragenter. Andra var oförmögna att se den totalitära kärnan i vänsterns budskap om jämlikhet och solidaritet och rannsakade aldrig sin självrättfärdigande övertygelse. Till den senare skaran hörde delar av Dagens Nyheters kulturredaktion, också efter Olof Lagercrantz avgång 1977.
Det finns en intressant epilog till 1992 års Ahlmarkdebatt. Sommaren 2010 publicerade Svante Nycander, långt efter det att han gått i en intellektuellt ytterst produktiv pension, två inlägg på DN Debatt vars udd var riktad mot »antiliberalismens starka ställning i svenskt kulturliv». Nycander inledde sin första artikel med att återknyta till diskussionen om Dagens Nyheters kultursidas politiska inriktning cirka två årtionden tidigare. Hans omdöme om sidan var nu, arton år efteråt, oförblommerat kritiskt. Dagens Nyheter, skriver han, gav genom Lagercrantz »ett mäktigt stöd åt 68-radikalismen, som inom kort blev marxistisk. _Den politiska tendensen inom DN:s kulturredaktion har sedan dess varit vänsterbetonad_.» Nycander gick till och med längre i sin kritik. »Antiliberalismen» menade han var ett utbrett fenomen i det samtida Sverige, »inte bara inom marxistisk och socialdemokratisk vänster.» Han pekade härutöver på den »gröna» rörelsen, där det fanns en »civilisationskritisk variant och en postmodern ogiltigförklaring av övergripande tolkningar och tankesystem, 'stora berättelser', bland dem liberalismen».
Också Arne Ruth har blickat tillbaka på sina ideologiska preferenser och på sin tid som chefredaktör för kultursidan på Dagens Nyheter. I en samtalsbok med författaren Lars-Åke Augustsson som utfrågare och redaktör beskriver han sin tid på tidningen. Främst sätter han, med all rätt, de stora debatter i viktiga frågor som han tog initiativ till liksom återinförandet av det berättande reportaget, framför allt exemplifierat av Maciej Zarembas artikelserier. Men hur ser han på kultursidans idémässiga profil – den profil som Nycander i sina artiklar 2010 kallade »vänsterbetonad»? Om detta finns det inte mycket explicit att hämta i Ruths svar på Augustssons frågor. Alltså inte mycket, men likväl något om man beaktar de många kringgående rörelser Ruth gör när han reflekterar över sina politiska uppfattningar.
På tal om Peter Weiss säger Ruth att »han anslöt sig från mitten av 1960-talet till samma vänster som jag också räknade mig till. Vietnam var den avgörande faktorn. Vi tog ställning mot USA-imperialismen från ett globalt rättviseperspektiv». I början av 1970-talet, när Ruth bodde i Göteborg, sökte KFML(r) värva hans flickvän, själv anser han sig ha varit mindre intressant som värvningsobjekt, men »en del av våra vänner hade anslutit sig». Vi talar alltså här om den mest extrema vänstergruppen under hela 1968-perioden. Under 1970-talet tog han i motsats till många av sina vänsterkamrater, bland andra Peter Weiss, ställning för demokratirörelsen i Tjeckoslovakien och solidaritetsrörelsen i Polen, ett engagemang han behöll under 1980-talet. Under början av 2000-talet beskriver Ruth sig i Åke Lundqvists bok som »vänstersocialdemokrat i synen på välfärdsstaten och liberal i synen på mänskliga rättigheter». På en punkt är Ruth dock tydlig, Dagens Nyheters kultursida blev under hans chefredaktörstid »den främsta plattformen för den här trenden». Det handlar om poststrukturalismen och postmodernismen och årtiondet är 1980-talet – och han tillfogar att »själv var jag inte frälst för vad som också ibland kallades poststrukturalismen».
Det finns en genomgående ståndpunkt i Ruths idévärld – från ungdomsåren och framåt – och det är aversionen mot marknaden, kapitalismen och antikommunismen. Han talar om »marknadsevangeliet som frälsningslära» och befarar att vi går mot en tidsålder där »kapitalmakten går i allians med politiska maktstrukturer som hejdar folkligt inflytande». Han raljerar över att »marknadsekonomins profeter lovar evigt stabil tillväxt» och oroas över att om »detta löfte sviks kan det rubba tilltron till demokratin». Han uppfattar tanken »på evig materiell tillväxt som ett evigt tvång» och han pläderar 1978 för att vi skall »trappa ned vårt utlandsberoende» och finna nya resurser »i syfte att bibehålla chansen att själva bestämma vad som ska hända med Sverige» – må vara att han senare finner detta förslag orealistiskt och hävdar att han alltid varit för EU. Ruth menar vidare att den svenska välfärdsstaten sedan mordet på Palme 1986 befinner sig i »upplösning» och hävdar att »nu trasas skolsystemet sönder av marknadstänkandet». På äldre dar har han »blivit pacifist på nya premisser». Ruth anser att det Kommunistiska Manifestets paroll om föreningen av arbetare i alla länder, »kan tillämpas på nytt – men inte för att genomföra en kommunistisk världsrevolution utan för att bygga ett solidariskt civilt motstånd underifrån». Han vidhåller »dårskapen i USA:s invasion av Irak år 2003» – trots att han säger sig ha underskattat »det irrationella draget hos Saddam Hussein». Denna bedömning hindrar inte Ruth att några år senare »stödja Ship to Gaza», ett projekt initierat av turkiska islamister – som vanligt var motivet det ädlast tänkbara, nämligen solidaritet med de mest utsatta. Som många andra äldre intellektuella tenderar Ruth avslutningsvis till en viss pessimism och tillfogar att »optimism kan jag bara hysa på lång sikt. Där är kvinnorörelsen mitt främsta hopp».
## **3. Kampen mot antikommunismen. Fallet Gerner**
Fortfarande under 1980-talet betraktades det som en belastning för en forskare eller författare att tillvitas beteckningen »antikommunist». Därmed inrangerades man ofta bland grupperingar som stod den yttersta högern nära. Som François Furet understryker i sin tidigare nämnda bok bestod in i det sista kommunisternas avsky för dem som riktade kritik mot eller belyste förföljelserna i Sovjetunionen och de andra öststaterna från en icke-kommunistisk synvinkel. Furet formulerar det på följande vis: »Alldeles innan den regim som Lenin grundat imploderar fördömer man i väst antikommunismen lika mycket som under antifascismens framgångsrika dagar.»
Med beteckningen »antikommunist» lyckades partierna i väst att framgångsrikt brännmärka liberala och konservativa kritiker av kommunismen. Om de så bara påstods sympatisera med antikommunister ansågs de gå reaktionens ärenden och vara ute efter att skada vänstern och arbetarklassen. Från försommaren och långt in på hösten 1991 ägde en polemik rum i Sydsvenska Dagbladet som i hög grad bekräftade den bild Furet ger. Den grep emellertid tillbaka på en debatt i Dagens Nyheter från 1988 som är väl värd att här belysa.
Den inleddes med att ekonomen Gunnar Adler-Karlsson under rubriken »en ny bok» på DN:s kultursida recenserade _The Harvest of Sorrow. Soviet Collectivization and the Terror-Famine_ av historikern Robert Conquest. Boken som utkom 1986 handlar om kollektiviseringen av jordbruket i Sovjetunionen i allmänhet och hungersnöden i Ukraina vintern 1932–1933 i synnerhet. Conquest hade 1968 utgivit den internationellt uppmärksammade studien av Stalintiden, _The Great Terror. Stalin's Purge of the Thirties_. De uppgifter Conquest här presenterade var i de flesta fall inte nya i sig, men han hade sammanställt en mängd av tidigare spridd information till den första helhetsbilden av förtrycket och likvideringarna under det kommunistiska styrets inledande tre årtionden vid makten. I den senare publicerade studien gav han en förödande bild av terrorn mot landsbygdsbefolkningen och visade hur Stalin i början av 1930-talet avsiktligt isolerade Ukraina och beslagtog all spannmål i området i syfte att genom en utbredd hungersnöd krossa den ukrainska nationalismen. Två till tre miljoner föll offer för folkmordet. Conquests provocerande slutsats var att terrorn inte berodde på Stalins person utan var en följd av det marxist-leninistiska systemet. Conquest utsattes för en omfattande och ihärdig förtalskampanj, inte bara från kommunistisk sida utan i lika hög grad och på ett mer försåtligt vis från kommunismens »fellow-travellers» vid akademier och i borgerlig press.
Några dagar efter Adler-Karlssons recension publicerar kultursidans fast anställde medarbetare Torkel Rasmusson en kort kommentar med rubriken »Kan man lita på Conquest?» Med hänvisning till en artikel i New York-tidningen Village Voice beskriver Rasmusson hur Conquests bok initierades av Ukrainian Research Institute och sponsrades av Ukranian National Association, som med Rasmussons ord var »en New Jersey-baserad exilgrupp långt ute på högerkanten». Över huvud hade exilukrainare i USA, enligt Rasmusson, bedrivit en kampanj för att göra allmänheten uppmärksam på folkmordet, en kampanj som varit särskilt framgångsrik »inom konservativa kretsar». Men Robert Conquest – framhöll Rasmusson i sin kommentar – var mer belastad än så. Han var inte bara sponsrad av exilukrainare utan dessutom antikommunist: »Conquest framställs här [i Village Voice] som en under åren alltmer rabiat antikommunist som inte drar sig för att utnyttja de grumligaste källor för att hitta graverande material om bolsjevikerna. Hans historiska forskning av senare datum utdöms av flera namngivna experter på samtida Sovjethistoria.»
Så här i efterhand är de så kallade experterna glömda medan Conquest allmänt betraktas som en pionjär inom forskningen om den stalinistiska terrorn, inte minst bland ryska historiker. Den kritik som har riktats mot hans studier är att han underskattade antalet offer för det kommunistiska samhällsexperimentet i det dåvarande Sovjetunionen.
Debatten i Dagens Nyheter fortsatte under mars månad med flera något disparata inlägg av såväl Staffan Skott som Kristian Gerner och Christian Palme. Den avslutades i början av april med att Gunnar Adler-Karlsson bland annat påpekade att Ukranian Research Institute tillhörde Harvarduniversitetet och att Conquest i inledningen till sin bok tackat Ukranian National Association för bidraget till boken. Som DN-medarbetare fick Torkel Rasmusson sista ordet. Han påpekade inledningsvis att hans referat inte var en partsinlaga utan bara en neutral redogörelse för ett inlägg i den amerikanska debatten, men han tillade samtidigt: »Att hungerkatastrofen i Ukraina utnyttjas av politiska krafter med ofta mycket grumliga motiv är ovedersägligt.» Rasmusson försvarade sin kommentar med att det var ett »ordinärt kulturjournalistiskt jobb». Men hans neutralitet var inte större än att han i avslutningen av sin replik skrev: »Bland annat berättade jag om hur Conquest utnyttjas, låter sig utnyttjas och själv också drar ekonomisk nytta av den kampanj som exilukrainare i USA och Kanada bedriver.»
Rasmusson framställer det som om han här inte själv tar ställning utan bara hänvisar till mer kunniga experter. Till synes opartiskt och neutralt framhåller han att forskaren tycks vara associerad med krafter som har illasinnade motiv och låter sig utnyttjas av privatekonomiska skäl. Dessutom antyder han att mycket pekar i riktning mot att forskarens metoder och källor kan vara tvivelaktiga och hans slutsatser överdrivna. Som nästa exempel visar representerade Torkel Rasmusson ett bara alltför utbrett förhållningssätt till seriös kritik av samhällssystemet i Sovjetunionen och Östeuropa, ett förhållningssätt som under 1970-talet flyttade ut från mer eller mindre obskyra vänstertidskrifter och vann hemortsrätt även på de liberala rikstidningarnas kultursidor.
Det mest komprometterande exemplet för de inblandade rör historikern Kristian Gerner. Gerner som i början av 2000-talet blev professor i historia i Lund, var och är en av Sveriges främsta experter på sovjet-rysk och östeuropeisk historia. I början av 1990-talet var han dessutom gästskribent på Sydsvenskans ledarsida. Han hade under 1980-talet i olika skrifter fäst uppmärksamhet på såväl de svårartade miljöproblemen i öst som de tilltagande etnisk-nationella spänningarna inom främst Sovjetunionen. Vid upprepade tillfällen hade han hävdat att det sovjetkommunistiska systemet led av olösliga inre motsättningar och omöjligen kunde överleva på sikt. Detta ståndpunktstagande hade gjort honom till en, milt uttryckt, ytterst kontroversiell person bland vänsterintellektuella, men också mycket uppskattad av en stor läsekrets. Till kontroverserna bidrog förmodligen att Gerner var en mycket engagerad, skarp och skicklig debattör.
I en krönika på Sydsvenskans ledarsida uppmärksammade Gerner att tidningens filmkännare Jan Aghed, också känd för sin hetlevrade vänsterhållning, hade karakteriserat McCarthyismen i USA som »en amerikansk variant av åsiktsförtrycket i Sovjetunionen». Enligt Gerner saknade denna jämförelse fullständigt proportioner. I ena fallet handlade det om brutala likvideringar av miljontals invånare, i det andra fallet om några tusen medborgare som i många fall visserligen helt orättmätigt utpekats som kommunister, offentligt brännmärkts för detta, men som mest förlorat sina arbeten och fått ekonomiska bekymmer – ingen hade sänts till arbetsläger och ingen hade avrättats i någon CIA-källare. Till detta skall läggas att Gerner påpekade att också Aghed tidigare hade avfärdat den främste kännaren av stalinismens brott, Robert Conquest, med att han var antikommunist.
Aghed replikerade omgående på Sydsvenskans kultursida och hänvisade till att han i samband med 1988 års DN-debatt om Conquests bok _The Harvest of Sorrow_ bara hade refererat till den amerikanska tidskriften Village Voice där amerikanska sovjetologer hade hävdat att de »betraktar Conquest som mindre av en seriös och objektiv forskare än en ideolog i det förnyade kalla krigets tjänst». I likhet med Torkel Rasmusson i Dagens Nyheter tre år tidigare, som sagt sig bara opartiskt vilja återge en amerikansk debatt, påstod Aghed nu att han »inte var kapabel att hysa någon kategorisk uppfattning» om värdet av vad som skrevs i Village Voice, han hade endast velat lyfta fram en motbild mot Gunnar Adler-Karlssons »oreserverade mottagande» av Conquests bok. Precis som Torkel Rasmusson 1988 försvarade sig här Aghed med att han bara gjort ett ordinärt journalistiskt arbete – han hade endast lyft fram intressanta amerikanska kommentarer i en uppmärksammad tidskrift.
Det är väl i dag knappast någon överdrift att påstå att såväl Dagens Nyheters Torkel Rasmusson som Sydsvenskans Jan Aghed var ute i syftet att misstänkliggöra Robert Conquest och ifrågasätta hans ambitioner genom att låta till synes neutrala forskare förknippa honom med mörka, antikommunistiska, högerkrafter – de försvarade inte förtrycket i Sovjetunionen men ömsom ifrågasatte, ömsom fördömde dem som kritiserade förtrycket. Som Gerner framhävde i en replik till Aghed hade denne redan 1988 fastslagit att »bara intellektuellt korrupta skojare» som Mats Gellerfelt och Lars Gustafsson tog Robert Conquest på allvar.
Ändå var det tydligt att Agheds tonläge var något lägre 1991 än 1988. Han gjorde inga nya utfall mot Robert Conquest, snarare var han förnärmad av att Gerner påminde honom om inläggen från 1988. Han hänvisade till den »stalinistiska terrorstaten» som en självklarhet och han underströk att Gerner på flera sätt säkert var »en framsynt, kunnig och utmärkt sovjetforskare». Men han avslutade med att Gerner ägnade sig åt en »oljigt självbelåten» sovjetologi med inslag av »revanschism och nymccarthyism».
Innan murens fall hade vänstern som sagt upprörts av och förkastat liberal eller konservativ kritik av de realsocialistiska planekonomierna. Nästan all form för icke-socialistisk systemkritik hade avfärdats som »sovjetfientlig», »reaktionär» eller »antikommunistisk». Efter murens fall hade denna typ av argumentation mer eller mindre upphört – den hade till stora delar mist sin legitimitet även i vänsterns egna ögon. Nu riktades i stället avståndstagandet mot de liberala debattörer som påvisade vänsterns ignorans inför ofriheten i öst eller lyfte fram och kritiserade enskilda vänsterintellektuella för deras misstänkliggörande av vad icke-socialistiska forskare och författare skrivit om realsocialismens förtryck. Nu fick de liberala debattörerna höra att de var »hämndlystna», »inkvisitoriska» eller »McCarthyistiska» – den förföljelse som liberaler utsatts för i Sovjetunionen och Östeuropa drabbade nu radikala vänsterintellektuella i väst, sökte man göra gällande. Och visst kunde det finnas ett segervisst drag i vissa liberala inlägg – historien hade ju visat vem som »haft rätt». Kristian Gerner kritiserades också för att hans kritik ibland träffade alltför brett och att hans polemiska tonläge kunde vara alltför högt.
Likväl var Gerners polemik ofta träffande, inte minst när han framhöll att han blivit bortdefinierad i debatten om förhållandena i öst, som exempel nämnde han mottagandet av den bok, _Arvet från det förflutna. Sovjet på tröskeln till 1980-talet_ som han utgav 1980. I Aftonbladet hade boken recenserats av litteraturvetaren och författaren Göran Hägg under rubriken »Sovjethatare, se hit» och karakteriserats »som ett symptom på en bred våg av rå spekulation i det nyreaktionära klimatet». På kultursidan i Dagens Nyheter hade boken först fått en seriös recension av Disa Håstad, för att därefter ha blivit frontalt attackerad i ett inlägg under rubriken »Narrbilder av Sovjet», där Gerner hade kallats »sovjetforskare» inom citationstecken och påståtts vara en i raden av »kapitalistiska imperialister, exilryssar, fascister, nazister och amerikaner» som sökte skrämma offentligheten i väst med bilden »av röda horder som kommer att översvämma landet, inta ditt hem, ta dina pengar och våldta din hustru och dina döttrar».
Det fanns också ett annat spår i Gerners polemik som, visade det sig, satte sinnena i brand. Det spåret rörde frågan om skuld och ansvar. Gerner var här mer offensiv i sina formuleringar än Kay Glans. Som exempel på medansvariga till stalinismens brott och apologeter för kommunismen utpekade han exempelvis Göran Therborn, Olof Lagercrantz, Sara Lidman och Sven-Eric Liedman. De hade som många andra omfattat marxistiska ståndpunkter. Problemet, i Gerners ögon, var att de »valde att hålla fast i sin övertygelse även sedan de fått möjlighet att skaffa faktiska kunskaper».
Gerner anknöt vidare till en distinktion som Håkan Arvidsson gjort i skuldfrågan, nämligen mellan dem som var verkliga medlöpare och de »hållningslösa opportunisterna». I sin artikel kommenterade Gerner att problemet egentligen var alla de som fanns där emellan, som vare sig var medlöpare eller opportunister, men som ändå hade uppträtt »som apologeter för diktaturer och bidrar till att legitimera förtryck och förnedring». Till dessa hörde de som »reagerar mot våldet och orättvisorna i det västliga samhället genom att gå till marxismen och socialismen för att finna svar och frälsning».
Just denna formulering, vilken i efterhand ter sig sakligt sett oskyldig – hur ställer sig offentligheten i dag till dem som sympatiserade med nazismen även efter det att de fick kännedom om judeutrotningen? – fick ett antal kolleger på Kristian Gerners institution i Lund att reagera genom att i Sydsvenskan publicera ett öppet och kollektivt avståndstagande från honom. De 16 menade att Gerners formuleringar om skuld och ansvar hade »forskningsetiska implikationer» och upprördes över att han förknippade historiker som »ställer sina problem inom ramen för en historiematerialistisk vetenskapstradition» med »stalinismens brott». De hävdade vidare att ett sådant skuldbegrepp var »främmande för den västerländska liberala rättstraditionen» och att Gerners sätt att argumentera hotade den »pluralism och demokrati» de själva anslöt sig till. Kristian Gerner ville uppenbarligen inte skilja mellan »statsformer» och »idétraditioner» och de frågade retoriskt om det fanns annat än »demagogiska skäl» till detta. Skulle Gerners skuldbegrepp vinna spridning måste ju också de som sökte svar på frågor inom en liberal idétradition vara »ansvariga för brott som begåtts i demokratins namn eller av demokratiska stater».
Resonemanget kan te sig överraskande med tanke på att det fanns och finns en omfattande forskning, både inom statsvetenskap och inom idéhistoria, som just analyserat det positiva sambandet mellan idéer och politiska uttrycksformer såsom partier och statsbildningar. Det ligger med andra ord inget obskyrt i att påstå att en stats uppbyggnad och handlingar är uttryck för den ideologi den bygger på; snarare är ett sådant påstående ganska trivialt. Marx och Lenin hade ju dessutom, som politisk doktrin, explicit förordat proletariatets diktatur och undertryckande av den borgerliga klassen genom expropriering av dess egendom – att de samhällen som byggde på deras lära både resulterade i diktatur och förtryck kan väl knappast förvåna.
Liberalismen däremot hade, ur många och heterogena källor, formats till en ideologi för den enskildes frihet, för allas lika värde och för en på folksuveränitet grundad konstitutionell styrelseform – må vara att liberalismen länge ignorerat det nya industrisamhällets sociala klyftor. En sådan styrelse hade, även till den kommunistiska vänsterns fromma, etablerats i de länder som var demokratiska. Att övergrepp, misstag och dumheter i demokratiska samhällen därmed skulle vara avförda från den historiska dagordningen hade nog inte många föreställt sig – ej heller hade den liberala ideologins fäder någonsin gjort anspråk på att så skulle bli fallet.
De 16 kollegerna, bland vilka återfanns institutionens ledande vänsterprofiler och marxistiska forskare, vände sig i avslutningen på sin artikel direkt till Gerner: »Men, genom att ha utslungat dessa våldsamma anklagelser på det sätt du gjort, Kristian Gerner, måste du, för att diskussionen skall bli meningsfull och du själv bli tagen på allvar, på ett mer dialogiskt sätt än hittills, tala om vad du menar. Som intellektuella har vi ett ansvar både inför oss själva och mot varandra att föra en hederlig vetenskaplig debatt; det ansvaret vill vi att du tar.»
Gerners slutreplik som också kan ses som slutreplik i debatten om vänsterns skuld publicerades några få dagar innan Sovjetunionen bröt samman – detta samhälle som hade fått så många intellektuella i väst att tro på en ny människa och en ny värld. I sitt svar gjorde Gerner en jämförelse och beskrev hur vetenskapsmän i de sovjetkommunistiska länderna brukade »underteckna fördömanden av för makthavarna misshagliga intellektuella». Samtidigt vidhöll Gerner att de 16 genom att välja en historiematerialistisk synvinkel också hade valt att utgå från vissa värdepremisser, till exempel att analysera den sociala strukturen utifrån tesen att industriarbetarna har en historiskt progressiv funktion att fylla, medan den borgerliga klassen »är moraliskt mindervärdig». Eftersom den socialhistoriska forskningen i marxistisk tappning stod stark vid institutionen i Lund, påpekade Gerner dessutom att i den klassteoretiska modell de 16 valt ingick idéer om klasshat och oförsonlig kamp mot klassfienden – ett budskap som framförts från både vänstern och Sovjetkommunismen. Och Gerner tillade:
Dagens svenska historiematerialistiska fotfolk har inte någon juridisk skuld till eländet i de f d socialistiska samhällena, men de har ett ideologiskt ansvar. Historiematerialisterna legitimerar klasshatet som organisationsprincip i samhällslivet och de har inte tillräckligt fördömt det socialistiska systemet som sådant. Det är slående att de nu, 1991, träder fram och förklarar att de riktat kritik mot Sovjetunionen.
Den underkastelse de 16 forskarna på historiska institutionen i Lund krävt av Gerner – för att bli tagen på allvar av sina kollegor behövde han argumentera på ett mer »dialogiskt» vis med dem – väckte uppmärksamhet, inte minst därför att denna form för kollektiv brännmärkning av en enskild kollega var synnerligen ovanlig vid akademiska institutioner i Sverige. Likväl utgör den ett belysande exempel på hur hårt klimatet kunde vara vid de »fria akademierna» när nu den ideologiska och politiska balansen mellan kommunism/socialism respektive liberalism försköts på grund av yttre omständigheter.
Det finns ett efterspel till de 16:s försök att 1991 stöta ut Kristian Gerner ur den akademiska gemenskapen, ett efterspel som belyser den nya akademiska vänsterelitens maktutövning mot dem som bryter mot den postkommunistiska hegemonin. Vid millennieskiftet sökte Gerner, som efter brännmärkningen hade sökt sig till Uppsala, en ordinarie professur vid sin gamla institution i Lund. Av sju sökande blev Gerner placerad i första rummet av samtliga sakkunniga. Ärendet gick därefter, i laga ordning, vidare till den dåvarande lärarförslagsnämnden som leddes av en professor vid namn Inger Lövkrona. Hon lät cirkulera en _skriftlig_ PM till de betrodda i nämnden där taktiken för att hindra Gerner från att få professuren lades upp. I motsats till 1991 skulle Gerner inte brännmärkas för att han var liberal och borgerlig. Nu gällde nya tider. I det protokoll Lövkrona signerade framhölls att nämndens majoritet »upplevt» Gerner som arrogant och att han brast i medvetenhet om jämställdhet, etnisk mångfald och genusperspektiv och därför inte var kvalificerad till tjänsten. När rundskrivelsen läckte ut i pressen protesterade emellertid ett stort antal lärare och kulturskribenter, vilket fick rektorn vid Lunds universitet att efter mycken tvekan till slut tilldela Kristian Gerner professuren.
Historien är emellertid inte slut här. Någon tid senare beslöt ett annat organ inom den nybysantiska universitetsbyråkratin, nämligen »områdesstyrelsen för humaniora och teologi vid Lunds universitet» att beställa och finansiera en medial studie av »Gerneraffären» – som ordförande i styrelsen när beslutet klubbades satt professor Inger Lövkrona. Uppdraget gick till docenten vid Institutet för mediastudier i Stockholm Gunilla Jarlbro som lade fram en kortfattad rapport med titeln »Manliga snillen och tokiga feminister», i vilken hon konkluderade att mediabevakningen hade varit »mansdominerad» och att den kvinnliga majoriteten i lärarförslagsnämnden hade utsatts för »symboliskt våld» – v.s.b. (vilket skulle bevisas). För uppdraget erhöll docent Jarlbro av Lunds universitetsområdesstyrelse för humaniora och teologi 100 000 kronor.
## **VII. KULTURKAMPENS HÖJDPUNKT**
## **Vid 1990-talets mitt**
## **1. Till demokratins försvar**
DEN MEST KOMPROMISSLÖSA kritiken av 1968 års vänster och dess samhälleliga inflytande kom att formuleras av Per Ahlmark. Efter utgivningen av antologin _Tingsten. Tyranniet begär förtroende_ (1992), som framför allt var ett polemiskt sabelhugg mot Dagens Nyheters politiska profil från 1968 och fram till 1989, kom Ahlmark de följande åren att publicera ytterligare två uppgörelser nu riktade mot 1968 års vänster och dess medlöpare, dels _Vänstern och tyranniet. Det galna kvartsseklet_ (1994), dels _Det öppna såret. Om massmord och medlöperi_ (1997). Det råder ingen tvekan om att Ahlmarks böcker representerar en imponerande intellektuell kraftsamling som övertygande visar hur djupt den nya vänstern, från 1970-talet och framöver, kom att prägla idéklimatet inom främst kultur och media:
Men vänsterns dominans över ett antal kulturredaktioner och avdelningar på radio-TV hade vuxit fram förbluffande snabbt och lätt. Anpassningen från en stor del av kulturvärlden och det skrivande Sverige hade varit så omfattande att skiftet tycktes permanent.
Det råder heller ingen tvekan om att Ahlmark inte bara visade att den marxistiska och leninistiska vänstern under ett par årtionden idealiserat förtrycket i de kommunistiska samhällena – vilka den uppfattade som ett nytt och emancipatoriskt steg i mänsklighetens framåtskridande – utan också dokumenterade hur rader av icke-kommunistiska intellektuella hade uttryckt förståelse och sympati med kommunistiska diktaturer eller försvarat dem som visat sådan sympati. Till de senare hade han som tidigare framgått ett särskilt gott öga:
Inte heller tar jag upp de kommunistiska partiernas knäfall inför direktiv utfärdade i Kreml, deras elände är så uppenbart att det nästan blir ointressant. Jag är mer intresserad av varför människor som var demokrater, eller ansåg sig vara det, eller åtminstone verkade i en demokratisk miljö, kunde bli förstående inför Sovjetunionen.
Vilka var då de som Ahlmark betecknade »medlöpare»? Vad förstod han med begreppet? Jag har tangerat frågan tidigare, men den kan här förtjäna en närmare diskussion. Termen har en lång historia och går tillbaka till mellankrigstiden, då rader av kända författare och politiker reste till Sovjetunionen och kom tillbaka med ofta lyriska beskrivningar av den stalinistiska regimens så kallade framsteg, rättvisa och fredsvilja. Dessa intellektuella och politiker var merendels kulturradikaler och vänstersocialdemokrater och kom av sina kritiker att betecknas som medlöpare. Härmed förstods som den brittiske historikern David Caute har klarlagt att de ansågs främja den kommunistiska diktaturens sak utan att nödvändigtvis själva vara eller vilja framstå som organiserade företrädare för Stalins och Sovjetunionens politik. Begreppet rymmer således en dualism mellan det medlöparen »egentligen» var och det han eller hon framträdde som. Kring medlöparna uppstod därför återkommande dispyter om var de »i grund och botten» hörde hemma politiskt. Gick de det kommunistiska förtryckets ärende eller omfattade de demokratins ideal? Efter andra världskrigets slut utgav flera av dem självkritiska uppgörelser där de tog avstånd från sina tidigare ståndpunkter.
Paul Hollanders bok _Political Pilgrims_ utkom bara åtta år efter David Cautes studie, men handlar i allt väsentligt om ett helt nytt fenomen, nämligen 1968 års vänsterradikaler och deras engagemang och resor till kommunismens nya favoritländer: Kina, Kuba, Vietnam och Nordkorea. Det fanns, som Hollander understryker, flera viktiga skillnader mellan mellankrigstidens medlöpare och pilgrimerna under 1960- och 1970-talen. Pilgrimerna bestod mest av unga marxistiska aktivister utan uppburen ställning i offentligheten – från detta fanns dock som vi skall se flera undantag. En annan olikhet var att medlöparna ofta kom att ta avstånd från såväl kommunismen i öst som den egna totalitära vänstern. Pilgrimerna däremot förblev, även efter det att de tappat tilltron till sina drömländer, vanligtvis sin vänsterradikalism trogen, även om det också här fanns flera undantag. Hollander medgav att det också fanns likheter mellan de båda grupperingarna, exempelvis att mellankrigstidens resenärer och 68 års pilgrimer kunde ses som två olika vågor med likartad bakgrund, nämligen ett slags desillusion och främlingskap inför det västliga samhällets tendenser till avideologisering, kommersialism och maktkoncentration. Han landade emellertid i slutsatsen att medlöpare utgjorde ett äldre och snävare begrepp än politisk pilgrim.
Ahlmarks målgrupp var 1960- och 1970-talens vänsterradikalism med dess romantisering av socialismen, bland annat i Kina och på Kuba. Han diskuterar Hollanders distinktion, men avvisar den i allt väsentligt och formulerar i stället en egen definition, som rymmer paradoxen att den fokuserar på pilgrimerna, men karakteriserar dem som medlöpare. Som framgår av denna definition ingår det fler grupper bland Ahlmarks medlöpare än Hollanders pilgrimer. I hans eget begrepp ryms inte bara unga vänsterextrema aktivister, utan egentligen alla som främjar eller försvarar totalitära system och läror, oavsett var de anser sig höra hemma ideologiskt och politiskt. Av mer underordnad betydelse är kanske att Ahlmarks definition samtidigt är snävare än Hollanders, därför att han främst, men inte enbart, fokuserar på hållningen till dimensionen demokrati–diktatur och inte inkluderar den kommunistiska ideologin mer allmänt som Hollander gör.
Det ligger en moralisk anklagelse i begreppet medlöpare – det handlar om någon som svikit sina ideal eller förrått sin tillhörighet, någon som utan att vara marxist eller kommunist deltagit i en påstådd »försköning av», »tigande om» eller »försvar för» kommunistiska diktaturer. Ett problem härvidlag är att Ahlmark inte skiljer mellan avsikt och konsekvens och inte diskuterar att olika motiv kan få likartade effekter. Artur Lundkvist och Karl Vennberg var två av 1950-talets medlöpare, som både direkt och indirekt visade förståelse för och/eller sympati med Sovjetunionen och som företrädde svensk undfallenhet visavi Moskva. Enligt Ahlmark intog också den socialdemokratiske utrikesministern vid samma tid, Östen Undén, en anpasslig hållning till stormakten i öst. Men var Undéns hållning en följd av ideologisk Sovjetvänlighet eller betingad av en lång svensk tradition av försiktig och undfallande diplomati visavi Ryssland? Var Undén medlöpare eller blott och bart en småstatsrealist? Skillnaden är helt avgörande för bedömningen av Undén, ändå berör Ahlmark inte den. Flertalet av de intellektuella och politiker som Ahlmark häftigast kritiserade för medlöperi var icke-kommunister som till exempel Olof Palme, Sverker Åström, Pierre Schori, Gunnar Fredriksson, Alf W. Johansson, Olof Lagercrantz, Svante Nycander, Bo Strömstedt och Per Olov Enquist.
Ahlmark rannsakade också personer som öppet och aktivt tog ställning för kommunismens sak som Jan Myrdal, Anders Ehnmark, Jan Guillou och Göran Palm. Det var kommunismen som ideologi och politiskt system som Ahlmark såg som det avgörande hotet – varje totalitär idé var i hans ögon oförenlig med ett fritt samhälle, med mångfald, mänskliga rättigheter och lagstyre. Men när han betraktar 1968 års idéskifte är det ändå den liberala Bonnier-pressens och socialdemokratins överslätande hållning till och understödjande av den nya vänsterradikalismen han uppfattar som det värsta sveket mot demokratins värden.
Ahlmark vrider och vänder, precis som i den tidigare debatten om vänsterliberalismen, på frågan om vänstervågens genomslag. Han pekar på följande paradoxala förhållande:
Hur ska man förklara att borgerliga partier i Sverige vann flera val från 1970 och framåt (och tre gånger erövrade eller behöll regeringsmakten) samtidigt som en dominerande del av kultur- och massmediadebatten byggde på marxistiska eller socialistiska tolkningar av ekonomi och utrikespolitik?
Han lyfter fram olika förklaringar till vänstervågen som idealitet, okunnighet och okänslighet. Han framhäver den ideologiska snedrekryteringen till massmedierna och det västliga självhatet. Han reflekterar över Hollanders tes om de intellektuellas desillusion med och främlingskap inför det liberalt-borgerliga samhället i väst. Och han frågar sig om denna förklaring verkligen kan gälla framträdande icke-kommunistiska ledare som Palme och Schori. I Ahlmarks ögon utgjorde vänstervridningen först och främst en intellektuell och moralisk kollaps för det demokratiska Sverige, en kollaps där bristen på civilkurage (underförstått: hos ledande personer inom media och politik) utgjorde den springande punkten. Medlöperiet skriver han är »inte en gemensam, sammanhållen och genomtänkt ideologi. Här handlar det mer om svek på olika plan».
Per Ahlmark återkom i sina två böcker till de tidigare anklagelserna mot Olof Lagercrantz, Arne Ruth och Svante Nycander med flera, för att de löpt med vänsterextremisternas syn på Kina, Kuba och den tredje världens befrielserörelser generellt. I _Det öppna såret_ placerade han dessutom Expressens chefredaktör Bo Strömstedt på de anklagades bänk, då han tagit såväl journalisten Anders Ehnmark som författaren Per Olov Enquist i försvar mot den hårda kritik Ahlmark bestått dem med för deras uppslutning bakom totalitära vänsterdiktaturer i tredje världen. »Den semantik han [Strömstedt] ägnar sig åt i boken [ _Löpsedeln och insidan_ ] är inte alls avsedd att skapa klarhet. Dess syfte är att försköna ett förfall. Det berättar något om Strömstedt att han blir särskilt demagogisk och engagerad i sin polemik när han ska försvara medlöpare och angripa deras kritiker.»
Ahlmark utvidgade således i _Det öppna såret_ den personkrets han kritiserade för att ha svikit demokratins värden, men argumenten förblev i stora drag desamma. När det gäller Olof Lagercrantz, Anders Ehnmark och Per Olov Enquist måste man nog ge Ahlmark rätt. De gick under 1970-talet i varierande grad och former kommunismens ärende; de utmålade förhållandena i de västliga demokratierna som präglade av människans lägsta drifter, girighet och maktlystnad, medan de förskönade förtrycket i några av de värsta kommunistiska diktaturerna. För detta blev de av sina samtida kolleger hyllade och uppburna.
Både i _Vänstern och tyranniet_ och _Det öppna såret_ riktade Ahlmark ytterligare tre förebråelser mot den yttersta vänstern. Den första rörde den enligt honom tilltagande antisemitismen inom vänstern efter 1968. Den andra handlade om det Ahlmark uppfattade som socialdemokratins återfall i den tredje ståndpunktens neutralism i kraftmätningen mellan demokrati och diktatur. Den tredje slutligen gällde den amerikanske forskaren Rudolph J. Rummels tes att demokratier är fredssökande och ej går i krig med varandra.
Vad först angår antisemitismen tog Ahlmark upp »affären» Ahmed Rami. Denne drev under flera år en radiostation med grovt antisemitiska inslag, för vilka han också blev dömd för hets mot folkgrupp av Svea hovrätt 1990. Det uppseendeväckande i denna affär var att rader av vänsterorienterade intellektuella slöt upp bakom Rami och försvarade eller urskuldade hans judehets. Bland dem som stödde Ramis sak återfanns personer som religionshistorikern Jan Hjärpe, Göran Palm, läkaren Sven Britton och mindre överraskande Per Gahrton och Jan Guillou, vilka var öppna motståndare till staten Israel. En aktiv roll bakom Rami spelade också framträdande företrädare för Svenska kyrkan, den socialdemokratiska Broderskapsrörelsen (i dag: Socialdemokrater för tro och solidaritet) och ledande teologer vid Uppsala universitet som Sigbert Axelsson och Jan Bergman. Ahlmarks utpekande fick senare belägg i historikern Håkan Arvidssons prisbelönta _Affären Rami-Bergman._ 12
I ett längre perspektiv framstår dock ett annat förhållande som kanske mer ödesdigert för uppkomsten av den antisemitism som i dag på nytt plågar judarna i Sverige. År 1982 anföll Israel Libanon i syfte att sätta stopp för den terror som Arafats PLO, med sitt högkvarter i Beirut, utövade mot civila israeler både i och utanför Israel. Interventionen fick nästan allt motstånd mot antisemitism att kollapsa. Att den yttersta vänstern slöt upp bakom PLO och fördömde Israel var inte förvånande, men att ledande företrädare för socialdemokratin som Olof Palme och Carl Lidbom började göra jämförelser mellan Israels roll i Mellanöstern och Nazitysklands uppträdande i Europa under kriget öppnade för en demonisering av staten Israel. Palme framträdde sommaren 1982 inför TCO-kongressen och beskrev den »oerhörda förbrytelse» som de judiska barnen utsattes för i de nazistiska koncentrationslägren, varefter han tillfogade att smärtan var densamma vid åsynen av de palestinska barn som dött under striderna i Libanon – »precis på samma sätt förföljda». Palme fick några dagar senare stöd för sitt tal av Aftonbladets tidigare chefredaktör Gunnar Fredriksson som skrev att »palestinierna skall tydligen utrotas [...] Nog påminner detta om judeförföljelserna i Europa.» Och riksdagsledamoten Carl Lidbom talade i DN om att israelerna uppträdde som ett »herrefolk» och betraktade araber som »undermänniskor».
I dag betraktas jämförelsen mellan Israels handlande och nazismen som öppet antisemitisk inom EU. Bakom det nya antijudiska språkbruket fanns säkert flera motiv som bidrog till att svänga opinionen så pass kraftigt mot Israel. Vänstern hade tidigt inplacerat konflikten i ett tredje världen-schema: palestinierna var koloniserade offer och bedrev en nationell självständighetskamp mot en imperialistisk judisk stat. Tredje världen-romantiken sträckte sig efterhand långt in i arbetarrörelsen och även in i en del socialliberala kretsar och kom palestinierna till godo. Den »antiimperialistiska» opinion som brett ut sig efter Vietnamkriget påverkade synen på Israel, i synnerhet efter junikriget 1967 och ockupationen av Västbanken. Till detta kom att den traditionella antijudiska hållningen inom delar av svenska kyrkan, förstärkt av en ny generation vänsterradikala präster och teologer, åter blev legitim.
Ahlmark uppmärksammade således tidigare än många andra den antijudiska och antisionistiska hållning inom delar av vänstern som ett par årtionde senare skulle övergå i en mer elakartad antisemitism. Lika tidigt lyfte han fram den muslimska radikalism som långt senare skulle skapa dels kaotiska uppror och revolutioner i Mellanöstern, dels en svårhanterlig politisk-religiös extremism bland vissa muslimska grupper i Europa. Som Ahlmark skriver i _Vänstern och tyranniet:_ »Min gissning är att fanatismen i stora delar av den muslimska världen kommer att få ännu fler ursäktare och skönmålare i Sverige än hittills.» Dessvärre skulle Ahlmark få helt rätt också på denna punkt. Först när terrorn drabbade civila i Västvärlden började vänsterns skönmålning av den islamska extremismen att utmanas av insikten att islamismen är en ny form för antiliberal totalitarism.
Vänstervridningen efter 1968 kom enligt Ahlmark också att påverka den svenska utrikespolitiken och synen på det kalla kriget. Redan i inledningen till Tingsten-antologin hade han riktat kritik mot socialdemokratins utrikespolitik för anpasslighet till de kommunistiska diktaturerna. Kritiken stegrades i _Vänstern och tyranniet_ , där Gulfkriget stod i förgrunden för analysen och kulminerade i _Det öppna såret_ med en uppgörelse med Olof Palmes utrikespolitiska linje under främst 1980-talet.
Under tidigt 1980-tal skärptes spänningarna kraftigt mellan de två supermakterna. Sovjetunionen hade nämligen vid 1970-talets mitt utplacerat kärnvapenbärande medeldistansraketer (SS 20) i Östeuropa, vilket kraftigt minskade förvarningstiden vid en eventuell kärnvapenattack. Som svar beslöt NATO 1979 att installera motsvarande amerikanska raketer (Pershing 2) i Västtyskland och Storbritannien. Mot detta beslut startade socialdemokratin i Västeuropa, framför allt i Västtyskland, Sverige och Danmark, en aggressiv opposition, främst riktad mot USA och president Reagan. Denna opposition leddes av de tyska socialdemokraterna med dess talesman i säkerhetsfrågor Egon Bahr i spetsen. Politiken gick, med hänvisning till kärnvapenhotet, i allt väsentligt ut på att söka samförstånd med Sovjetunionen och kommunistregimerna i Östeuropa, ett samförstånd som hade sin udd riktad mot USA. Olof Palme kom här att spela en viktig roll genom upprättandet av den så kallade Palmekommissionen som krävde en kärnvapenfri zon i Europa och i vilken såväl Egon Bahr som ledande representanter för Sovjetdiktaturen ingick.
Ahlmark visade att det knappast låg någon ny realpolitisk kalkyl bakom den socialdemokratiska regeringens utrikespolitik. I högre grad speglade den en stark ideologisk aversion mot president Reagans skarpa retoriska antikommunism och hans hårda säkerhetslinje visavi Moskva. »Gång på gång attackerade Olof Palme borgerliga politiker, som uttalat kritik mot Sovjetunionen. Teckna inte 'djävulsbilder', sade han, bedriv inte 'hets mot Sovjet'. 'Vi sysslar inte med antisovjetism', meddelade Palme sin partikongress 1984. » Och när folkpartiledaren Bengt Westerberg i januari 1986 i ett tal menade att ingenting skulle gynna freden i världen så mycket som en demokratisering av Sovjetunionen kritiserade den socialdemokratin närstående f.d. ambassadören Sverker Åström Westerberg för hans »extrema» ståndpunkt. Åström hävdade dessutom att uttalandet stred mot Helsingforsavtalet från 1975 och att det inte var möjligt att förena med »korrekta» relationer till Moskva. Lika anmärkningsvärt, menade Ahlmark, var det faktum att Palme 1983, efter att Carl Bildt besökt Washington och mött amerikanska säkerhetsexperter, fick regeringen att kollektivt – i riksdagen – brännmärka Bildt för hans resa och kontakter med företrädare för den amerikanska administrationen.
Det skulle här kunna invändas att Palmes utrikespolitiska linje, precis som Östen Undéns, mer speglade en traditionell svensk undfallenhet i förhållande till Ryssland än några ideologiska sympatier för Moskvas så kallade fredspolitik. Något ligger det säkert i detta, men intrycket kvarstår likväl att den vänsterradikala antiamerikanismen kom att påverka socialdemokratins inställning. De som menar att denna inställning skall uppfattas som medlöperi med den sovjetiska kommunismen har i mina ögon sakligt fel, vilket inte utesluter att politiken kan ha spelat Moskva i händerna.
Ahlmarks angrepp på svensk neutralism fick i _Det öppna såret_ en fortsättning genom hans kritik av historikern Alf W. Johanssons bok om Herbert Tingstens antikommunism under det kalla kriget. Johansson utvecklade i sin bok en revisionistisk syn på det kalla kriget. Härmed förstås en doktrin som vände sig mot tesen om sovjetisk expansionism som orsak till konflikten mellan öst och väst. Tvärtom menade revisionisterna, som främst företräddes av amerikanska historiker, att Sovjetunionens utrikespolitik efter andra världskriget kännetecknades av en realpolitiskt betingad återhållsamhet och försiktighet. Revisionisterna hävdade att den huvudsakliga drivkraften bakom det kalla kriget var den amerikanska kapitalismens expansionsbehov. Mot denna bakgrund framställde Johansson Tingstens bild av Moskvas aggressivitet och den svenska regeringens undfallenhet som grovt överdriven, närmast hysterisk – i stora drag såg han Tingstens kritik som uttryck för en oresonlig antikommunism och ett häftigt publicistiskt temperament. Från 1960-talets slut och fram till Sovjetunionens sammanbrott 1991 dominerade den revisionistiska synen den internationella akademiska diskussionen. Alf W. Johansson uppfattade revisionismen som ett resultat av forskningsmässiga framsteg och tycktes inte se att den i lika hög grad var resultatet av de vänsterradikala forskarnas antiamerikanska perspektiv. Efter Sovjetunionens fall skulle emellertid en mer nyanserad bild av det kalla kriget framträda, en bild som med stöd i bland annat sovjetiska källor till stora delar bekräftade den liberala och antitotalitära hållning som Tingsten företrätt.
Per Ahlmark upprördes av det han uppfattade som Johanssons tendens att förminska Tingsten som publicist och tänkare – att reducera honom till en intellektuell som bländade sin omgivning men egentligen var i otakt med sin tid. Ahlmark hade dock två mer principiella invändningar. För det första menade han att Johansson inte ville erkänna Tingstens insats som kritiker av kommunismen, eftersom han inte var benägen att ta till sig det som faktiskt hände i Östeuropa mellan 1945 och 1948. För det andra ansåg Ahlmark att Johanssons kritik av antikommunismen, i förening med den långa svenska traditionen att se Ryssland som mer osäkert än aggressivt, mer defensivt än offensivt i sin säkerhetspolitik, drev honom till en ideologisk förståelse för den totalitära diktaturen i öst.
Det är svårt att förstå att en klok och kvalificerad forskare som Alf W. Johansson vid 1990-talets mitt inte fullt ut såg att den ryska revolutionen, bolsjevismen och efterkrigstidens sovjetiska imperium hade givit upphov till en konvulsion i det europeiska statssystemet och att kommunismens allmänna strävanden medförde ett djupgående och långsiktigt hot mot den samhällsform som vuxit fram i den västliga delen av Europa, ett hot som fick en särskild relief av att imperiet i öst hade stöd av stora skaror anhängare i väst som var mer lojala med den totalitära maktstaten och dess ideologi än med det egna demokratiska samhället. Det var just kombinationen av Förenta staternas geopolitiska intressen och dess tradition av liberal internationalism som låg till grund för dess vilja att konfrontera Sovjetunionen, inte den amerikanska efterkrigsekonomins behov.
Även om man kan ge Ahlmark rätt på en central punkt i polemiken, är det dock lika klart att de frågor som Alf W. Johansson ställde och de konklusioner han drog var fullt motiverade och helt legitima ur historieforskningens synvinkel. Framför allt var det inte, som Ahlmark sökte göra gällande, uttryck för ett svekfullt ideologiskt ställningstagande.
Det fanns emellertid en annan provocerande sida med Ahlmarks polemik gentemot Johansson. Ahlmark gjorde, precis som med flera andra antagonister, kritiken av Johansson till en fråga om personlig moral. Han kallade Alf W. Johansson en »skojare som är ute och går». Han karakteriserade honom som en »man utan inre kompass» och en »karikatyr på människor utan inlevelse». Det fanns ingen grund för dessa överdrifter, allra minst fanns det anledning att utmåla Johansson som en forskare vilken strök kommunistiska härskare medhårs. Om Stalin skrev till exempel Johansson att han var »en brutal och skoningslös diktator; hans välde en form av polis- och militärstyre där all opposition slogs ned med hänsynslös hårdhet». Och om författaren Artur Lundkvist som mottog Leninorden från Moskva skrev Johansson:
Hos Artur Lundkvist möter man de flesta av de element som skulle bli bestämmande för vänstervågen på sextiotalet: kulturrelativismen, antiimperialismen, amerikahatet, anti-parlamentarismen, den flagellantiska hållningen till den västerländska kulturen, en vagt antydd ekologism, kritik av en ansvarslös konkurrenskapitalism som hotade att leda världen till en katastrof. I Lundkvists fall var Tingstens beteckning 'halvkommunist' helt befogad.
Per Ahlmark hade som sagt, på en viktig punkt, saklig grund för sin kritik av Alf W. Johanssons karakteristik av Tingsten och det kalla kriget, men Johansson fick också klä skott för Ahlmarks allmänna aversion mot Sveriges neutralitetspolitik och den tredje ståndpunkten, samt hans emotionella idealisering av Herbert Tingsten.
Den tredje huvudinvändningen mot vänstern utvecklade Ahlmark framför allt i _Det öppna såret_. Med utgångspunkt i den amerikanske forskaren Rudolph J. Rummels arbeten hävdade Ahlmark att den enda garantin mot förtryck, krig och massmord är demokratin. Rummel hade myntat tesen att demokratier inte går i krig med varandra och Ahlmark framställde detta som en »empirisk lag»: »Vågen av undersökningar på 80-talet och början av 90-talet visade alltså med stor tydlighet att demokratier inte går i krig mot varandra [...] I mitten av 90-talet kom en reaktion mot denna tes. Läser man de forskare, som nu rest invändningar, ser man efter ett tag _att de i praktiken bekräftar att demokratier inte bekrigar varandra_.»
Ahlmark citerade vidare samhällsvetaren Jack Levys tes om att frånvaron av krig mellan demokratier »är det närmste vi har en empirisk lag i internationella relationer». Förmodligen var tanken bakom Ahlmarks presentation av Rummels arbeten att stärka dels sitt försvar för den liberala demokratins värde, dels sin kritik av 1900-talets totalitära diktaturer. På denna avgörande punkt hade och har de två forskare Ahlmark stödde sig på nog i huvudsak rätt – diktaturer tenderar till krig, demokratier till fred. Som alla teser kan emellertid också denna problematiseras. Första världskriget utkämpades mellan två konstitutionella rättsstater – Tyskland och Storbritannien – med utbredd rösträtt och betydande friheter. Den demokratiska Weimarrepubliken förvandlades i korrekta parlamentariska former till en totalitär diktatur; framför allt hade de ledande västliga imperiemakterna, också under demokratiska regimer, fört brutala krig vid uppror i de tidigare kolonierna. Märkligt nog uppfattade Ahlmark den kritik som riktats mot Rummels huvudtes som politiskt suspekt. Han beskrev hur forskningen om demokratiernas fredlighet vuxit fram under stort ideologiskt motstånd från framför allt vänsterorienterade och pacifistiska forskare. Så var det förmodligen, men det fanns också sakligt rimliga invändningar mot Rummels schematiska teser som upprörde Ahlmark. Det är tydligt att Per Ahlmarks värld var besatt av politik och ideologi, vilket också färgade hans syn på samhällsforskningen – också denna såg han som en strid, inte främst mellan sant och falsk, snarare mellan gott och ont.
Per Ahlmark ägde civilkurage och insikt nog att påvisa den svenska 68-vänsterns ställningstagande för socialistiska vänsterdiktaturer mot liberal demokrati, dess hyllande av våld och förtryck riktat mot borgerliga grupper och intressen. Han genomskådade dessutom den socialdemokratiska och socialliberala anpassligheten gentemot de, i tredje världen, revolutionära rörelsernas och staternas maktvilja – inte minst uppmärksammade han tidigt den antisionistiska/antijudiska agitationen och stegrande antisemitismen inom vänstern i bred mening. Men hans perspektiv var samtidigt begränsat. Det framträder tydligast i hans syn på det ideologiska klimatomslaget under 1960-talet. Lika klarsynt som Ahlmark var i att genomskåda vänsterns fascination för våld och förtryck under socialistiskt styre, lika oförmögen var han att betrakta den nya motsättningen mellan avancerade västländer och tredje världens underutvecklade samhällen annat än i termer av det kalla krigets konflikt mellan demokrati och diktatur.
Ahlmark tycktes inte vilja se att de två europeiska inbördeskrigen, vars grymhet kulminerade i Förintelsen, under 1960-talet kom att intellektuellt belasta – orätt i mångas ögon – hela den borgerligt-liberala civilisationen. Legeringen av kolonialism och Förintelsen gav upphov till ett västligt självförakt, som märkligt nog har förstärkts efter Sovjetkommunismens undergång. Precis som Svante Nycander påpekade i 1992 års Tingstensdebatt medförde fokuseringen på den tredje världen att »kommunismen» upphörde att vara identisk med sovjetisk stalinism. Vänsterns dröm om ett nytt kollektivt och egalitärt utopia projicerades nu på Kina och Kuba. När klimatet under 1960-talet svängde hamnade Ahlmark och en hel generation av unga liberaler, intellektuellt på mellanhand. Det var därför ingen tillfällighet att Ahlmark först återkom med sin kritik av vänsterradikalismen efter det att den sovjetiska kommunismen kollapsat och den svenska vänstern hamnat i ett tillstånd av förvirring och reträtt.
I de tre böcker om 68-vänstern som Ahlmark utgav under 1990-talet värjer han sig emellertid mot denna övergripande perspektivförskjutning. I stället tilldelar han Dagens Nyheter ett oproportionerligt stort ansvar för idéskiftet. Hade tidningen bara försvarat demokratins värden med samma frenesi som under Tingstens tid hade den borgerligt-liberala kollapsen inför yttersta vänsterns idéer inte inträffat eller i varje fall hade skadeverkningarna avsevärt kunnat begränsas – så löd Ahlmarks återkommande mantra.
Det fanns i hans liberala kritik av den nya vänsterns idévärld en hel del brister. Ahlmarks analyser fokuserar till exempel nästan enbart på relationen mellan demokrati och diktatur, det vill säga på den politiska styrelseformen. Däremot är det högst sällan Ahlmark skjuter in sig på exempelvis vänsterns bild av den privata äganderätten och näringsfriheten. Dessa rättsområden var och är av minst lika stor betydelse för ett fritt och demokratiskt samhälle som mänskliga rättigheter och konstitutionell maktdelning. Lika sällan analyserar Ahlmark vänsterns avståndstagande från fristående civila institutioner och vikten av att dra en demarkationslinje mellan politiken/staten och det civila samhället. Inte heller finner man hos Ahlmark någon problematisering av vänsterns – till sitt ursprung främst Lenins – totala fördömande av 1800-talets västliga globalisering eller imperialism. Precis som Tingsten tog Ahlmark avstånd från kolonialismens många missdåd och övergrepp, men vänsterns nattsvarta syn på den västliga civilisationens inflytande, som han säkert avvisade, skärskådade han aldrig.
## **2. Ahlmarkfejden**
Både _Vänstern och tyranniet_ och _Det öppna såret_ ledde till en rad recensioner och kommentarer. Framför allt den första boken utlöste en omfattande debatt som sträckte sig över flera månader och inbegrep ett stort antal skribenter. Trots att polemiken stundtals var grov och ofta fastnade i formella dispyter, innehöll den samtidigt partier och element som bidrog till att kasta ljus över den svenska vänsterns förändrade inriktning och ställning.
En viktig problemställning som återkom i båda debatterna gällde bakgrunden till och konsekvenserna av vänstervridningen från 1968 och framåt. Ahlmarks fokus var här tydligt: han frågade vilka motiv som drev de unga att ömsom hylla och ömsom ursäkta kommunistiska diktaturer. »Gång på gång samma bild: vänsterns debattörer förvanskar eller nonchalerar totalitära hot, läror och regimer. Vad har drivit dem under det senaste kvartsseklet?» Och längre fram i samma artikel frågar han: »Varför blev de [medlöparna] hypnotiserade av doktriner som byggde på människors underkastelse?» Ahlmark avvisade »en enda och sammanfattande teori» och hänvisade i stället till svenska omständigheter:
Den specifikt svenska förklaringen ligger närmast till hands. Vårt land ockuperades aldrig av nazisterna och kände sig fritt att stå utanför alliansen mot Sovjetväldet. De europeiska katastroferna rusade Sverige förbi. Kanske ledde det till att driften _att inte se_ förstärktes, känslan av att omvärlden inte är verklig [...] Eller blev Vietnamdebattens genomslag i Sverige särskilt dramatiskt eftersom det inte balanserades av Natoländernas insikt om USA:s avgörande roll för Europas frihet? Bidrog bristen på samlat och envist borgerligt motstånd att slå sönder den balans som tidigare fanns?
Just den senare punkten är av central betydelse. Det var sveket, ja, den intellektuella kollapsen i borgerligt-liberala medier som stod i fokus för hans förklaring av vänsterfenomenet i Sverige. Han talar exempelvis om den »oerhörda oturen» att Palme och Lagercrantz samtidigt råkade »dominera svensk utrikesdebatt», i motsats till 1950-talet då Östen Undéns sovjetvänliga utrikespolitik balanserades av Herbert Tingstens kritik. Och Ahlmark frågade:
Men vilka politiska ideal fick de unga efter 1968? Vilka var förebilderna? Göran Palm eller Jan Myrdal, Olof Lagercrantz eller Jan Guillou? Gissningarna här blir svidande tidskritik. De ger konturerna av det galna kvartsseklet. Ty vad som då hände var att den tidigare _balansen_ bröts sönder. Den ersattes av hegemoni i viktiga delar av utrikesdebatten för dem som drogs till marxistiska regimer och samtidigt godtog de flesta fördömanden av kapitalism och västmakter.
Det är viktigt att här understryka att Ahlmark med formuleringen »dem som drogs till marxistiska regimer» inte främst avsåg organiserade eller öppet deklarerade kommunister utan »medlöpare», det vill säga demokrater av i huvudsak borgerligt eller socialdemokratiskt slag. Framför allt _Vänstern och tyranniet_ har sin udd riktad mot just denna grupp. En recensent som uppmärksammade detta var ledarskribenten i Svenska Dagbladet Håkan Hagwall: »Ahlmarks bok handlar inte alls om svenska kommunistpartier och mycket lite om partikommunister. Den handlar om makthavare och debattörer och författare som format 'tidsandan'.»
Hagwall instämde helt i Ahlmarks kritik: »Naturligtvis ligger en mycket stor del av ansvaret hos dem som kom att ingå i den tredje gruppen, medlöparnas medlöpare. Och därvid måste man bittert sätta ljuset på massmedierna, dvs på folk i massmedierna, på alla dem som bara släppte fram, aldrig ställde kritiska frågor, alltid bemötte med respekt.» Samma tankegång kom till uttryck i en kommentar av Åke Ortmark som frågade: »När ska familjen Bonnier ta sitt ansvar? Hur länge ska ägarna – och samhället – acceptera att viktiga medier blir instrument för en godtycklig journalistisk maktutövning?» Ortmark vände sig egentligen mot den hårda kritik som Ahlmark fått utstå i flera av de medier han menade hade svikit sitt uppdrag under vänsterns kvartssekel. Särskilt uppmärksammade Ortmark »journalistkårens underliga sammansättning, som i sin tur förklaras av ägarnas bristande civilkurage och ovilja att ta ansvar för rekryteringen». Precis som Hagwall bekräftade dessutom Ortmark bilden av den mediala vänsterns dominans:
Vi är många som bör ta åt oss av Ahlmarks kritik, antingen för att vi tvekat och tvivlat i ställningstagandet för demokratin eller därför att fegheten, inställsamheten eller vänskapskorruptionen frestat oss till medlöperi i en tid då vänstervindarna blåst med förödande kraft. Det har inte varit lätt att överleva i Sverige utan att då och då sända insmickrande leenden vänsterut.
Enligt Åke Ortmark hade inte heller situationen förbättrats: »Det kan sägas att perioden var extrem. Men sanningen är ju att läget på våra redaktioner fortfarande är extremt.» I denna karakteristik instämde Kay Glans: »Det gamla vänstergardet kan genom sitt starka grepp om viktiga kulturinstitutioner fortfarande diktera diskussionens villkor... Det är fortfarande ett slags vänster som avgör vad som är korrekt och vem som är extremist.»
De som slöt upp på Ahlmarks sida i debatten bekräftade i stora drag hans bild av bakgrunden till vänsterns dominans nämligen, _att_ ett antal ledande personer inom media och politik (medlöparna), förmodligen mer av opportunism än övertygelse, strök den militanta vänsterns antivästliga och antikapitalistiska ideologi medhårs och därigenom banade väg för dess genombrott inom centrala institutioner, _att_ ledarna av och/eller ägarna till dessa institutioner svek sitt ansvar, _att_ svensk borgerlighet vek undan i stället för att bjuda motstånd. Kay Glans gav i sitt debattinlägg den kanske mest träffande förklaringen till att efterkrigstidens liberalism dukade under för 1960-talets vänstervåg:
En viktig lärdom från vänstervågen är att en relativt liten grupp med ett militant ideologiskt system kan få stort inflytande över opinionen. Sådana grupper kan i kraft av sin sammanhållning och känslomässiga övertygelse skrämma företrädare för mer sansade uppfattningar. Inför en eventuell ny vänstervåg är den massmediala situationen sämre idag än den var i början av 60-talet.
Mot denna syn på drivkrafterna bakom vänstervridningen ställdes i debatten ett helt annat perspektiv. I sin ampra recension av Ahlmarks bok menade historikern Håkan Arvidsson att klimatomslaget 1968 var en följd av den »reella förskjutningen i det världspolitiska läget». Arvidsson hävdade att de nationella befrielserörelserna i tredje världen mycket tidigt hade formulerat sina krav på politiskt oberoende och ekonomisk utveckling i socialistiska termer – här kan tillfogas att ett stort antal hade sin upprinnelse i eller grep tillbaka på mellankrigstidens marxismleninism i Kominterns utformning. Arvidsson konkluderade att den svenska vänsterns genomslag kring 1970 hade sin grund i denna internationella förändring och inte var en följd av Palmes och Lagercrantz samfällda agerande. Mot denna bakgrund hävdade han att Ahlmark hade en alltför konspirativ syn på orsakerna till vänsterns uppsving. Likartade invändningar mot Ahlmark framfördes av flera andra debattörer. I Dagens Nyheter skrev Sverker Sörlin: »Det var Västerlandets svek vänsterns revolt gällde. Och denna vänster var internationell, inte unikt svensk, som Ahlmark vill suggerera oss att tro.» Samma hållning intog Tomas Lappalainen i Aftonbladet: »Vänsteruppsvinget byggde på en kritik av västmakternas politiska inkonsistens: de hävdade de demokratiska värdena på hemmaplan men struntade i dem i tredje världen.»
Att den yttersta vänstern skulle varit något slags solidaritetsrörelse med tredje världens fattiga ville Ahlmark emellertid inte lyssna på. Han erkände utan reservation att de ledande västmakterna ofta förtryckt och förgripit sig på folk i de tidigare kolonierna, men menade att det funnits en liberal kritik av dessa övergrepp och att vänstern därför hade fel i fråga om västvärldens påstådda svek. Framför allt frågade sig Ahlmark varför upprördheten över västmakternas agerande i tredje världen måste leda till accept av totalitära idéer och kommunistiska tyranner.
Om bakgrunden till vänsteruppsvinget kan urskiljas som det ena av Ahlmarkfejdens centrala teman, var det andra temat frågan om medlöparnas och vänsteraktivisternas ansvar och skuld inför sina ställningstaganden. Här öppnades debatten av litteraturprofessorn Knut Ahnlund som särskilt angrep biskopen Claes-Bertil Ytterberg för dennes uttalande att »människor bör dömas efter nuet» – det förgångna däremot bör få ligga begravet. Knut Ahnlund som hade en särdeles välslipad penna hävdade:
Den av honom [C-B Ytterberg] begärda preskriptionstiden för blamager tycks vara rekordkort. Idealtillståndet vore kanske att allt det ideologiska dravel som framväller här i landet sades i vinden och skrevs i vattnet och att ingen funnes som småaktigt mindes någonting. Åtskilliga av vänsterns pratmakare beter sig oavbrutet som om detta tillstånd rådde.
Och Ahnlund fortsatte i samma inlägg: »Det verkligt skamliga i den här härvan är ju att så många åsiktsbildare profiterade på sin okunskap om detaljerna i de stora och små förtryckarstaternas inre tragedier.» Mot denna inställning reagerade Dagens Nyheters chefredaktör Svante Nycander, som något raljerande sammanfattade Ahnlunds hållning: »Med andra ord: vissa personer har ostraffat kunnat göra klandervärda saker; deras gärningar får inte anses preskriberade; de kan inte förlåtas i första taget, eftersom de inte har bättrat sig eller ångrat sig.» Nycander hävdade att detta var ett från kriminalrätten hämtat repressivt synsätt och frågade i vilket syfte de skulle utsättas för någon slags reaktion? Han ansåg sig kunna se en motsvarighet bland de anklagade »som menar sig vara utsatta för något slags vedergällning; självutnämnda domare utdelar skamstraff för förmenta åsiktsbrott». Nycander lyfte vidare fram en formulering av Kay Glans om de politiska pilgrimerna där denne skrivit att den »enda meningsfulla hållningen är att fordra att de reviderar sina åsikter». Ur Nycanders liberala perspektiv stred detta mot åsiktsfrihetens idé och han frågade med en underton av anklagelse: »Hur kan man fordra att andra människor ska revidera sina åsikter?»
I en replik till Nycander sökte Knut Ahnlund precisera sin ståndpunkt. Hans utgångspunkt var att 1968 års skribenter hade »misskött sitt värv»; de hade felinformerat allmänheten om vad som pågick i de kommunistiska diktaturerna och de hade hämtat mycket av sin »propagandaarsenal» från den kommunistiska ideologin. Om dessa skribenter inte kunde ställas till ansvar för vad de sagt »blir allt meningsutbyte i dagspress, massmedier och böcker en ren fars». Svante Nycander svarade att naturligtvis skulle politiska idéer kritiseras, men »kritikens väsen är inte att vara repressiv» – dess metod var i stället en rationell process som syftade till »gemensam upplysning».
Men hur långt sträckte sig de unga pilgrimernas och de makthavande medlöparnas ansvar? I _Det öppna såret_ frågade Ahlmark: »Kan 'idealitet' se ut hur som helst?» Anledningen till hans fråga var Göran Rosenbergs bok _Da Capo al Fine_ i vilken denne diskuterar motivet till sitt ställningstagande för maoismen åren kring 1968. Rosenberg skrev: »I sextiotalets revolutionära vänsterrörelse rymdes, bland mycket annat, drömmen om den självständiga människan, förmågan att råda över sitt eget öde och bygga sitt eget samhälle. Det var min utgångspunkt då, och är min utgångspunkt nu.»
Ahlmark vägrade att ta Rosenbergs självförståelse för god vara och frågade ironiskt varför han hade sökt stöd för sin dröm om den självständiga människan »i en rörelse, som i både teorin och praktiken utplånat just människors självständighet». Med denna elegant formulerade florettstöt demaskerade Ahlmark Rosenbergs självbild. Gåtan är och var ju varför unga begåvade studenter sökte lösningen på aktuella samhällsproblem i en svärmisk kommunistisk utopi? I motsats till Knut Ahnlund ansåg Ahlmark inte att 68 års vänsteraktivister »profiterade på sin okunskap». Även om de inte kände till alla detaljer om förtrycket under maoismen borde de haft kunskap nog om 1900-talets totalitära rörelser för att förstå att kulturrevolutionens masshysteri måste leda till misshandel och dödande.
Ahlmark ställde därutöver frågan hur den moraliska skulden skulle delas mellan rödgardisten i Kina »som dödade på platsen» och den svenska författare som »hejade på»? Det kan tyckas vara en överdriven och provokativ vinkling av skuldfrågan, men Ahlmark menade att »frågor om moralisk skuld har länge tillhört de centrala i västerlandet [...] Att just när det handlar om 1900-talets största katastrof – massmorden och massförtrycket – utesluta spörsmål om 'skuld' är otänkbart.»
Ahlmark hade rätt i att frågor av denna art måste kunna diskuteras. Men det är lika viktigt att i en sådan diskussion dra nyanserade skiljelinjer, till exempel mellan direkta gärningsmän, aktiva och medvetna påhejare, medlöpare och sympatisörer – skuld och ansvar är inga odelbara storheter och det är stor skillnad mellan att vara gärningsman och utanförstående åsiktsinnehavare. Ahlmark drog aldrig några sådana skiljelinjer. Han anklagades, med viss rätt, för att hellre fälla än fria och han delade knappast Harry Scheins åsikt: »Absolut inte glömma, kanske förlåta, men i varje fall förstå.»
Ett helt annat synsätt på det förflutna formulerades av socialpsykologen Lars Dencik som kritiserats av Ahlmark i _Det öppna såret_. I ett par större artiklar i Svenska Dagbladet angrep han Ahlmark för bristande inlevelseförmåga, enögdhet och böjelse för rättshaveri. Framför allt ansåg han att Ahlmark inte tog hänsyn till 1960-talets tidsförhållanden. »Tidsandan», skrev Dencik, »är den luft som vi genom våra göranden och låtanden både inandas och utandas.» Och han fortsatte: »Att leva med i samhället och inte andas med tidsandan kan vara lika svårt som att lyfta sig i håret.» Som exempel på en sådan anda pekade Dencik på alliansen mellan västmakterna och Sovjetunionen under andra världskriget – som »alla anständiga människor» stödde. Betydde det, frågade Dencik polemiskt, att Ahlmark i efterhand ville anklaga dem som stödde Sovjetunionens kamp mot nazismen för sympati med en totalitär ideologi och moraliskt medansvariga för förtrycket i Gulag? Dencik tillade att »en befrielse» – som den på Kuba 1960 – kan »föra till oanade och oönskade konsekvenser. Blir det därför moraliskt förkastligt att stödja den?» En med Dencik likartad tankegång hade Sverker Sörlin framfört i sin recension av _Vänstern och tyranniet_ 1994, där han skrev: »Vi kan i varje läge bara handla i enlighet med vår moraliska övertygelse och utifrån de villkor som gäller. När ytterligare omständigheter uppdagas kan det bli nödvändigt att ändra åsikt. Bedömningen av det förflutnas handlingar måste göras på det förflutnas villkor, inte på nuets.»
Men vilken var »tidsandan» i slutet av 1960-talet? Den definierades knappast av den nya vänstern, som Sörlin och Dencik antyder. Idealiseringen av Kina, Kuba och Nordvietnam var ännu begränsad till små och extrema grupper. Den radikala vänstern hade vid denna tid inte nått några maktpositioner inom media och kultur. Den representerade fortfarande en revolt mot det etablerade samhället. De som protesterade mot kriget i Vietnam, reste till Kina eller till Kuba gjorde det i opposition mot tidsandan. De representerade en hållning som fördömdes av dåtidens makthavare inom socialdemokrati, fackföreningsrörelse och borgerlighet. De kunde inte heller hänvisa till okunskap om kommunismen, i varje fall inte om förhållandena i Sovjetunionen och Östeuropa. Men å andra sidan är det riktigt att det, under intryck av frigörelsesträvandena i tredje världen och det eskalerande Vietnamkriget, hade börjat blåsa en förändringens vind både i Sverige och i övriga västvärlden. Denna vind drog med sig många unga sinnen in i en upprorsrörelse med rötter i marxismens och leninismens idévärld.
Men var det verkligen solidaritet med »jordens fördömda» och indignation över förtryck och orättvisor som låg till grund för vänsterupproret? I så fall blir det, som Ahlmark gång på gång påpekade, obegripligt varför revoltörerna anslöt till en ideologi som skapade en ny ofrihet och nya orättvisor. Eller var den självbild som exempelvis Göran Rosenberg gav uttryck för idealiserad och självgod? Detta hävdade Kay Glans i ett inlägg 1994: »Man behöver inte tro på religiösa fostrare när de försäkrar att de tuktar av kärlek och man måste inte heller tro på alla de motiv som 68-generationen själv anför. Om inte andra drivkrafter än altruism var verksamma blir det svårt att förstå den snabba urartningen till aggressiv sekterism.»
Glans menade, med hänvisning till den tyske forskaren Fritz Stern, att det fanns likheter mellan mellankrigstidens högerrevoltörer och 60-talsvänstern. 1968 var nämligen inte bara en revolt mot auktoriteter, det handlade lika mycket om en »revolt för auktoriteter»: »Det moderna samhället förkastades inte därför att det var extremt förtryckande... utan därför att det inte var tillräckligt omslutande. Bakom intresset för det väpnade upproret – en helt föråldrad strategi i det moderna samhället – dolde sig också en fascination inför våldet.»
Kay Glans lyfter här fram en avgörande punkt. Det hävdas ofta – som ett slags försvar – att kommunismen med sin påstådda dröm om det klasslösa samhället representerar en värdeuniversalism i 1789 års anda, i motsats till nazismens partikularistiska rasideologi. Det ligger bestämt mycket i detta; men denna distinktion döljer samtidigt att kommunismen också rymde en cynisk maktfilosofi i Machiavellis, Hobbes och Robespierres anda. Såväl Marx som Lenin betonade genomgående följande budskap: det borgerliga samhället måste »krossas» och det måste krossas med »våld», helst inbördeskrig. Borgerliga grupper skulle »fråntas» sin egendom, »avlägsnas» från makt och inflytande; satte de sig till motvärn föreslog Lenin att de skulle mötas med »terror». Den nya makten, alltså »upprorets» och »befrielsens» makt skulle vara »obunden av lagar», vilket i realiteten innebar att rättsstaten, oberoende domstolar och konstitutionell maktdelning skulle elimineras.
Den tidigare nämnde historikern François Furet myntade uttrycket »den revolutionära passionen». Härmed menade han att det sedan den franska revolutionen, för att inte säga redan under högmedeltidens kristendom, funnits en aversion mot privat egendom, handel, konkurrens och berikande, en passion som inte minst kommunismen och nazismen underblåste och utnyttjade. Furet skriver att det egendomligaste av allt med 1900-talets totalitära massrörelser var att de »framkallat så starka känslor och framfött så många fanatiska individer». Den fascination inför våldet som den nya vänstern uppvisade – och som var en del av indignationen och solidariteten – hade sin grund i de känslor av revanschism, hämnd och maktlystnad som den revolutionära passionen närde. Extremismen inom 68-vänstern är omöjlig att förstå utan insikt om den roll dessa passioner spelade som drivkraft för vänsterns angrepp på det borgerligt-liberala samhället. I motsats till idéer som fästs på papper – skrivna i svart på vitt – är emellertid känslor och stämningar skrivna i vatten och formulerade i vinden och nästan omöjliga att i efterhand, när även de deltagande själva glömt eller förträngt dem, återskapa och belägga.
Jag har här sökt rekonstruera den rationella kärnan i Ahlmarkfejden. Varje bild av fejden som främst betonar rationaliteten blir emellertid missvisande, eftersom debatterna var fyllda av ömsesidiga överdrifter, polemiska förenklingar och personliga förolämpningar. När _Vänstern och tyranniet_ utkom 1994 skrev Sverker Sörlin i Dagens Nyheter (5/2) att »citattekniken är löjeväckande, överdrifterna bisarra»; boken som helhet betraktade han som en »parodisk förenkling». I Svenska Dagbladet (3/2) menade Håkan Arvidsson att Ahlmark inte kunde skilja på stort och smått – han angrep »allt med samma frenesi». Arvidsson ansåg att han släpade sina motståndare inför »domarskranket, där de konfronteras med löst och självsvåldigt sammanfogade citat». Det är, skrev Arvidsson, »i bokstavlig mening en McCarthyistisk teknik».
Angreppen fortsatte. I Aftonbladet (11/2) hävdade Tomas Lappalainen att Ahlmarks sätt att hantera konflikten mellan demokrati och diktatur »var djupt anti-intellektuellt» och att han konstruerade »en fiktiv fiende» i den vänster han angrep, eftersom alla inom den inte haft samma uppfattning om exempelvis Sovjet och Kina. I det liberala flaggskeppet Expressen (21/2) skrev ledarskribenten Cecilia Garme att hon betvivlade att Ahlmark annat än »i något enstaka fall (Myrdal?) skulle kunna belägga en sammanhängande diktaturromantisk läggning» inom 68-vänstern. Hon tillade att det inte var så märkligt att definitionen av demokrati hade »hamnat i gungning, när 60-talets skribenter konfronterades med fattigdomen i USA:s slum respektive glada barn i kinesiska mönsterskolor». Diplomaten Ingmar Karlsson hade i Svenska Dagbladet (24/2) det goda omdömet att jämföra Per Ahlmarks citatteknik med den för antisemitiska hatbrott dömde Ahmed Rami: »Ahlmark har bara en like. Ahmed Rami arbetar med samma metoder.» Karlssons inlägg fick ordförandena i Stockholms judiska församling respektive Svenska kommittén mot antisemitism att i samma tidning (8/3) ifrågasätta hans lämplighet som utrikespolitisk expert åt svenska regeringen. »Vi protesterar», skrev de tillsammans med professor Georg Klein, »mot Ingmar Karlssons nedsmutsning av det offentliga samtalet.» Ahlmark blev inte svaret skyldigt. I Svenska Dagbladet (18/2) invände han »att de som det senaste kvartsseklet löpt med tyranner vill idag bara mumla om saken». När debatten började ebba ut frågade han i Göteborgsposten (23/4) varför alla medlöparna teg och han räknade bland annat upp Olof Lagercrantz, Gunnar Fredriksson, Jan Guillou, Pierre Schori, Karl Vennberg, Sten Andersson och Jan Myrdal: »Har de som tiger äntligen insett att debattörer i väst som skönmålat tyrannier bär på ett indirekt, moraliskt delansvar för fasor som drabbat miljoner människor i diktaturerna?»
I samma upprörda tonläge fortsatte det vid utgivningen av _Det öppna såret_ 1997. I Expressen (7/2) skrev Gabi Gleichmann: »Ahlmarkstribunalens arbete är fullbordat. Tyranniets sanna ansikte är blottlagt. Dess svenska lakejer, vilka anmärkningsvärt nog alla har hjärtat till vänster, är avslöjade.» Ett par veckor senare (18/2) menade Peter Englund i samma tidning att Ahlmark hade en tendens »att tricksa med statistiken för att få den att stämma överens med sin tes» om diktaturernas inneboende tendens till massmord. Ahlmark svarade (26/2) att Englund, vad gällde statistiken i boken, »skalar lök». Lars Dencik karakteriserade i Svenska Dagbladet (20/2) Ahlmark som rättshaverist och en tragisk skepnad, varvid Ahlmark replikerade (26/3) att Dencik »saknar moral som skribent» och med sin bok om _Kuba – revolutionärt exempel_ från 1968 bidragit till att »fördumma» offentligheten i Sverige. Knappt en månad senare tillade Ahlmark (20/4) att »Lars Dencik har hyllat Castroregimen och missuppfattat nästan alla fakta och strukturer om Kuba. I dag vänder han ryggen åt sitt fiasko genom att utnämna några av oss, som alltid varit demokrater till 'fascistoida personligheter'».
En intressant omständighet finns avslutningsvis att notera, nämligen Per Ahlmarks påpekande 1994 att nästan inga av huvudfigurerna i hans böcker deltog i debatten, någon kommenterade den men ingen gick i en saklig debatt om 68-vänsterns ståndpunkter och agerande. Än mer frapperande var den tystnad varmed 1990-talets etablerade vänster bemötte böckerna. Det är riktigt att några tidigare vänsterintellektuella som exempelvis Håkan Arvidsson, Göran Rosenberg och Lars Dencik medverkade med inlägg, men de hade alla reviderat sin vänsterhållning och omfattade när Ahlmarks kritik kom sedan länge demokratiska ideal. Men precis som Ahlmark påpekade rådde i övrigt en talande tystnad.
Ahlmark återkom till ämnet i sin brevväxling med Georg Klein år 2001. I boken _Motståndet – med mera_ påpekar han att vid utgivningen av _Vänstern och tyranniet_ var »kultursidorna i huvudstadspressen... totalt nedgörande». Och han tillfogar med polemisk träffsäkerhet:
Eftersom chefredaktör Arne Ruth på Dagens Nyheters kultursida den här gången var lika ivrig att refusera mig som han tidigare varit att vitmåla Olof Lagercrantz dyrkan av Mao kom Peter Luthersson, kulturchef på Svenska Dagbladet, att spela en avgörande roll. Under flera månader öppnade han sin sida för en stor, hård och klargörande debatt om tjugofem år av svenskt medlöperi och hur den perioden skildrades i min bok.
Ahlmark tillfogade med tillfredsställelse att _Vänstern och tyranniet_ var den bok – »har man sagt mig» – som bröt »socialisternas hegemoni». Det är säkert riktigt att boken och debatten, mot bakgrund av den östliga socialismens debacle, bidrog till att tvinga 68-vänstern och dess arvtagare på reträtt i fråga om synen på Sovjetkommunismen och den marxistiska och leninistiska idétraditionen. Men reträtten var tillfällig och hindrade inte en ideologiskt delvis nyskrudad vänster att framträda under 1990-talets senare år.
## **VIII. POSTKOMMUNISMENS GENOMBROTT**
## **1990-talets slut**
## **1. Från marxism till feminism, ekologi och rättigheter**
UNDER 1990-TALETS SENARE DEL framstod den radikala, icke-socialdemokratiska, vänstern i Europa som historiskt överspelad. De planekonomiska systemen var avskaffade, den privata äganderätten och marknadsekonomin var inte längre ifrågasatta som grundval för industriell utveckling och av den en gång inflytelserika kommunistiska arbetarrörelsen återstod mest bara ruiner. I Italien och Frankrike hade de stora kommunistpartierna snabbt skiftat kurs efter 1989. Majoriteten i det italienska partiet bytte namn till Partiet för den Demokratiska Vänstern (PDS) och antog ett socialdemokratiskt reformprogram, medan det franska partiet visserligen bevarade sitt namn och sin organisation, men tog avstånd från såväl marxism-leninismen som den demokratiska centralismen. Marxismen som världsåskådning och teori, denna väldiga idékälla under 1900-talets första hälft, attraherade efter kommunismens fall enbart grupper på den mest extrema vänsterflygeln. Till bilden hör att också de socialdemokratiska partierna flyttade sina ideologiska positioner åt »höger», något som främst symboliserades av den brittiske labourledaren Tony Blairs idé om »the new labour», men också av den socialdemokratiske statsministern Göran Perssons pragmatiska åtstramningspolitik i Sverige och den tyske förbundskanslern Gerhard Schröders Agenda 2010 som innebar en motsvarande »liberalisering» av SPD:s politik några år senare. Den »nya» socialdemokratin accepterade fullt ut marknadsekonomin, ville inte kännas vid sina tidigare krav på förstatligande av privata företag, valde bort keynesiansk sysselsättningspolitik till fördel för monetaristisk penningpolitik, avskrev helt korporativistiska idéer om ett näringsliv styrt av stat och fackförbund, men fortsatte samtidigt att eftersträva social utjämning, jämställdhet och trygghet.
Med kommunismens fall försvann således inte bara en militär utmaning mot de västeuropeiska demokratiernas suveränitet, borta var också det ideologiska och politiska hotet mot ett liberalt näringsliv som hängt över den västliga kapitalismen sedan det tidiga 1900-talet. Den liberala demokratins seger fick den amerikanske filosofen Francis Fukuyama att redan 1989 i tidskriften National Interest mynta begreppet »historiens slut», varmed han förstod att den västliga demokratin och de idéer på vilka den vilade utgjorde en slutpunkt i mänsklighetens ideologiska och politiska utveckling. Som de kommande årens utveckling skulle visa var emellertid detta ett ytterst förhastat triumfrop. Vad liberala tänkare inte beaktade var att Sovjetblockets upplösning inte bara var ett nederlag för vänstern, utan också en befrielse. Den radikala vänstern, i varje fall huvuddelen av den, tog tillfället i akt att i viss mening göra sig kvitt den belastning som den sovjetkommunistiska traditionen länge utgjort. Frigörelsen var förvisso inte entydig och rätlinjig, snarare var den konfliktfylld och ambivalent, men sammanbrottet i öst skapade likväl ett ideologiskt tomrum som banade väg för nya idéer.
Det svenska Vänsterpartiets utveckling under 1990-talet belyser väl scenförändringen. Redan innan Sovjetunionen upplösts övergav partiet termen kommunism i såväl namn som program. På kongressen 1993 övergick makten till en yngre generation med feministen Gudrun Schyman i spetsen. Föga överraskande skrevs feminismen in i partiprogrammet på den efterföljande kongressen 1996; kravet på en radikal miljöpolitik hade funnits sedan tidigare och i valmanifestet från 1994 karakteriserar sig partiet som »röd-grönt socialistiskt parti». Den traditionsbevarande strömningen i partiet hade dock återhämtat sig så pass att den redan 1994 fick in den öppet kommunistiske Lars Ohly som partisekreterare och två år senare återinfördes marxismen som partiets teoretiska grundval. Socialismens trots allt avtagande attraktionskraft kan också spåras bakom det ideologiskifte som ägde rum inom socialdemokratin i och med Göran Perssons lansering av parollen om det Gröna folkhemmet vid sitt tillträde som ny partiledare. Överhuvud markerar mitten av 1990-talet feminismens och ekologismens definitiva genomslag, inte bara inom den radikala vänstern, utan också, om än i mer modest form, som betingelse för borgerlig politik.
Men det ideologiska tomrummet efter kommunismens fall skulle också öppna för mer långtgående idéförskjutningar inom den svenska vänstern, förskjutningar som varit på gång innan 1989 men som fick ett mer definitivt genomslag under 1990-talet. Den första och mest övergripande förändringen gällde den radikala vänsterns syn på förnuft och framsteg. I ett historiskt perspektiv brukar kommunismen, liksom socialdemokratin, inrangeras på den europeiska upplysningens sida – de bejakade modernitetens syn på vetenskap, teknik och sekulär rationalism. Men med den nya vänstern 1968 – och framför allt med postmodernismens genomslag under senare delen av 1980-talet – sker en gradvis förändring som får sitt definitiva genombrott efter 1989. Medan kapitalismen tidigare hade kritiserats för sin påstådda brist på tillväxt och produktivitet och för att det enskilda ägandet hindrade teknisk och institutionell förnyelse, intogs från 1990-talet en rakt motsatt ståndpunkt. Faran med kapitalismen ansågs nu vara dess tillväxtfilosofi och effektivitet, dess spridning och globalisering, såväl inom den svenska vänstern som internationellt. Medan kapitalismen tidigare uppfattats som ett hinder för sociala reformer, anklagades den nu för att hota naturen – naturen framställdes som god och harmonisk, samhället (kapitalismen) som ont och destruktivt. För Vänsterpartiets del är detta särskilt tydligt mot slutet av 1990-talet. I programmet från kongressen år 2000 talas om tillväxtekonomiernas »blinda satsningar på massiv kvantitativ tillväxt» och här hävdas också att en »blind tro på teknikens och vetenskapens möjligheter leder till ekologiska katastrofer». Sociologen Göran Therborn framhåller i en internationell översikt över vänsterns idéutveckling att den i stora stycken gjort den ekologiska tillväxtkritiken av kapitalismen till sin egen och att det numera är nyliberalismen som står för begreppet modern.
För det andra bortföll den marxistiska historiefilosofin ur den radikala vänsterns begreppsvärld. Denna hade byggt på idén om en progression från lägre till högre produktionssätt, från feodalism till kapitalism och socialism/kommunism. Den hade utgjort den grundläggande referensramen för vänsterns bedömning av egna och andras beslut och handlingar. Grundkriteriet hade varit vad som främjade övergången till socialism, alternativt försvagade kapitalismen, på kort eller på lång sikt. Den postmoderna kritiken av de »stora berättelserna» var först och främst riktad mot marxismens historiefilosofi, vilket blev särskilt tydligt efter 1989 när socialismen inte längre existerade, vare sig som realitet eller utopi. Men kritiken var inte bara riktad mot marxismen utan mot alla former för framstegstro och drabbade därför också den liberala varianten, som ibland kallas för »the whig interpretation of history». Den radikala vänstern ersatte på relativt kort tid marxismens historiesyn med föreställningen om naturliga rättigheter, som ny måttstock för sitt politiska handlande. Rättighetsidén hade länge funnits i vänsterns retorik, men mer spelat rollen av en kontrapunkt till borgerlig politik – borgerliga rättigheter ansågs vara formella och sakna reellt socialt innehåll. Nu kom idén om naturgivna rättigheter att stegvis inta en dominerande ställning i den radikala vänsterns perspektiv och vändas mot det västliga samhället. Den borgerliga kapitalismen, hävdades det, förnekade i själva verket människor deras rättigheter. I synnerhet, påstods det, kränkte och diskriminerade den de svaga, de utanförstående – minoriteterna.
Det ideologiska kvantsprånget, om uttrycket tillåts, hade för det tredje sin grund i arbetarklassens »frånfälle». Under hela vänsterns existens hade konflikten mellan arbete och kapital varit den axel kring vilken all politik kretsade. Såväl sociologiska som idémässiga förändringar hade emellertid urholkat (det marxistiska) begreppet arbetarklass – få eller ingen såg längre »arbetarklassen» som bärare av ett nytt och högre produktionssätt, som historisk hävstång till en universell frigörelse av mänskligheten. Tendensen hade varit märkbar redan under 1970- och 1980-talen, men blev mer synlig efter 1989. Nu ersattes kampen för arbetarklassens »intressen» med krav på sexuella och etniska minoriteters rätt till egen identitet – arbetarklassens frigörelse blev till de homosexuellas frigörelse, kvinnans frigörelse, invandrarnas frigörelse. Den radikala vänstern kan sägas ha genomfört en lång ideologisk rockad efter kommunismens fall. Den flyttade sig, i stora drag, från den europeiska upplysningens sida till en mer postmodern syn på natur, rättigheter och identitet. Den förde den egna nationella arbetarklassen åt sidan till fördel för nya etniska, sexuella och religiösa grupper. Detta var inte helt synligt vid 1990-talets mitt, men korten i den idémässiga patiensen var på väg att falla på plats.
När triumfen efter murens fall lagt sig såg inte heller liberala tänkare och politiker, eller ville inte se, att en ny radikal vänster var i vardande. Flera omständigheter spelade här in. Först och främst uppfattades nyorienteringen i hög grad som en seger för det liberala lägret. Vänstern intog ju ståndpunkter som inte minst företrätts av de nyliberala strömningarna inom svensk borgerlighet, hit hörde främst rättighetstänkandet. Betecknande nog kom ofta vänsterradikala intellektuella som hade övergått eller kom att övergå till en mer rättighetsorienterad ideologi att karakteriseras som (vänster)liberaler. Vidare hade den nya radikala synen på jämställdhet, miljö och identitet fått ett allt starkare inflytande inom medier och vid delar av universiteten. Vid 1990-talets mitt var identitets- och rättighetstänkandet, tillsammans med feminism och ekologism, på väg att bli en hegemonisk referensram i svensk offentlighet. Traditionellt borgerliga dygder – krav på ansvar och disciplin, respekt för bildning och auktoritet, belöning av flit och förkovran, vaktslående om oberoende civila institutioner och nationell sammanhållning, allt sådant som i dag upprör den förhärskande politiska och mediala korrektheten men i ett längre perspektiv är avgörande för varje utvecklat samhälles stabilitet och välgång – blev till stora delar främmande också för en yngre generation av borgerliga politiker och debattörer.
Allt inom vänstern var dock inte nytt och mycket av det nya var svårt att fånga eftersom kritiken av kapitalismen, liberalismen, det borgerliga samhället och västvärlden var till stora delar densamma som tidigare och kom delvis från de kvarvarande resterna av 1968 års bokstavsvänster, som efter kommunismens fall reducerats till perifera sekter i den mån de inte redan var upplösta. Den postkommunistiska svenska vänstern var med andra ord inte mindre antikapitalistisk och antiliberal än den tidigare traditionellt kommunistiska. Den anklagade det borgerliga samhället för rasism, sexism och fascism. Den rasade mot globalisering och social olikhet – en kampanj som nådde sin kulmen med Göteborgskravallerna i samband med EU:s toppmöte i staden 2001. Det är mot denna bakgrund man skall se det som kan kallas den pro-kommunistiska motreaktionen inom Vänsterpartiet under senare delen av 1990-talet, med andra ord det självhävdande och känslosamma försvaret för den svenska kommunismens historia och dess marxist-leninistiska idétradition – ett försvar som väcktes till liv av en yngre generation av aktivister och initierades av publiceringen av _Kommunismens svarta bok_ (se VIII 3. Brott och förnekelse). Som jag tolkar det var avsikten aldrig att återupprätta det marxist-leninistiska perspektivet på samhälle och politik. I stället handlade det om en arvsstrid. Ytterst gällde striden hur den radikala vänsterns förflutna skulle tolkas, hur dess roll i svensk politik skulle framställas. Hade den varit en främmande fågel, en samhällsomstörtande representant för en totalitär stormakt, eller hade den varit en integrerad, inhemsk, del av den demokratiska svenska arbetarrörelsen?
## **2. En äkta demokratisk tradition?**
Vänsterpartiets ungdomsförbund, Ung Vänster, fick 1996 en ny, ytterst radikal och frispråkig ledare i Jenny Lindahl. Hon representerade tillsammans med bland andra Ali Esbati, sedermera hennes efterföljare som ordförande i Ung Vänster, och den blivande riksdagsmannen Kalle Larsson en generation som inte hade några rötter i 1968 års vänster. De var dock inte tillfreds med den självkritiska förändring som kännetecknat Vänsterpartiets ambitioner på kongresserna 1990 och 1993. De uppfattade dessa som en ideologisk reträtt och som eftergifter för borgerlig antikommunism. I uttalanden och intervjuer sände Lindahl, liksom Ohly och flera andra, ut helt andra signaler; signaler som vittnade om en återgång till en mer positiv syn på den svenska kommunismen, på oktober-revolutionen och på Lenin. Med Lindahls tillträde som ordförande återgavs kommunismen en viktig roll i ungdomsförbundets politik. Hon vände sig vidare mot beskrivningen av Lenin som »massmördare»: »Han var inte en värre massmördare än någon annan av statsledarna vid den här tiden.» Här måste Lindahl ha avsett demokratiskt valda politiker som Branting, Ebert, Wilson, Clemenceau och Lloyd George.
Några månader efter att ha nått ordförandeposten i Ung Vänster framträdde Lindahl på partiets kongress med ett militant tal. »Makthavarna» i Sverige, menade hon, kommer aldrig att släppa makten frivilligt, därför krävs en folklig rörelse som är beredd »att ta det avgörande klivet den dag det krävs.» Inför denna traditionellt kommunistiska demokratisyn – ger inte borgarklassen frivilligt upp inför de förmenta massornas raseri får den skylla sig själv – gav kongressen Lindahl stående ovationer – »och applåderna ville aldrig ta slut» skrev Dagens Nyheters reporter. Om feministen Gudrun Schymans tillträde som partiledare 1993 hade utgjort en form av avvärjningsmanöver i ett läge där Vänsterpartiet, som radikalt socialistiskt parti, var tillbakaträngt och demoraliserat, markerade den unga kommunisten Jenny Lindahls framträdande 1996 en ny, mer offensiv vändpunkt. Medvetet eller ej representerade hon en frustrerad strömning inom den radikala vänstern som ville övervinna det tidiga 1990-talets apati och återupprätta självrespekten inom Vänsterpartiet.
I motsats till tonläget i partiet våren 1990 och de närmast följande åren var hållningen nu en annan. Vi har inget att skämmas för påstod såväl Ohly som Schyman inför 80-årsjubileet våren 1997, ett jubileum som avhölls under parollen »demokrati och socialism». Ohly framhävde dessutom två andra argument som skulle återkomma i debatten, dels att Vänsterpartiet var det enda parti i Sverige som öppet gjort upp med sitt förflutna, dels att partiet i motsats till andra varit förföljt och förtryckt av den svenska staten – åsiktsregistrering, internering och yrkesförbud. Ohly förnekade vidare, i ett annat sammanhang, att den kommunistiska ideologin gav upphov till förtryck. Aron Etzler (från 2012 partisekreterare i Vänsterpartiet) påstod ett par år senare att brotten i Sovjetunionen och andra länder hade begåtts i kommunismens namn och inte hade något med Vänsterpartiets ideologi att göra. Jenny Lindahl hävdade 1998 att den svenska kommunismen alltid stått för »demokratiska reformer».
När programkommissionen inför 1999 års kongress ville stryka ned de självkritiska delarna från 1990 och 1993 hänvisade den till att Vänsterpartiet och dess föregångare aldrig varit beroende av Sovjetunionen – i kommissionen ingick bland andra Lindahl och Ali Esbati.
Det handlade alltså inte om att göra Vänsterpartiet till en retrokommunistisk gruppering. Men i kontrast till Gudrun Schyman som helt och hållet ville förkasta partiets historia, är det tydligt att Lindahl och Ohly med följe främst ville försvara Vänsterpartiets kommunistiska förflutna och få det att framstå som del av en fredlig, oberoende och demokratisk vänstertradition inom svensk arbetarrörelse. Perspektivet var ingalunda nytt. I partiets idétidskrift Socialistisk Debatt hade Göran Therborn redan 1981 markerat att C. H. Hermanssons kritik av Sovjetkommunismen på 1964 års kongress »nu [var] både slutförd och konsoliderad». Det fanns inte längre behov av några nya markeringar österut för att vinna demokratisk trovärdighet. Visserligen hade partiet trott på myterna om Stalins Sovjetunionen, men dessa myter hade »partiet nu gjort upp med». Therborn sammanfattade sin huvudpoäng med följande ord: »Men under mytbildningen om förhållandena på andra sidan Östersjön har den kommunistiska politiken i Sverige utgjort en radikal demokratisk tradition inom arbetarklassen och arbetarrörelsen.»
Therborn hade vid flera senare tillfällen, som jag visat, upprepat sitt påstående. En liknande hållning hade Sven-Eric Liedman givit uttryck för i vitboken om partiets historia:
VPK har av sina kritiker ofta utmålats som en främmande fågel i svensk politik. _Jag skulle tvärtom vilja hävda att VPK var ett mycket svenskt parti: ett kompromissparti_. Det ville samla många stridande viljor till en bräcklig enhet. Det yppade kritik mot Sovjet men bevarade sina traditionella vänskapliga relationer till dess kommunistparti. I kompromissens anda balanserades varje kritisk synpunkt österut av hårdare kritik västerut; USA var värst, med Sovjet fanns – så antyddes det – trots allt ett slags värdegemenskap.
Under 1980-talet hade detta synsätt förkastats av den kommunistiska ortodoxins företrädare som ett sätt att utsudda partiets marxist-leninistiska profil. Nu var tiderna emellertid andra, Sovjetkommunismen bortblåst och partiets prokommunistiska strömning i behov av en ny historisk legitimitet. Vad passade bättre än att relatera 1990-talets vänster till 1917 års brytning inom socialdemokratin? De unga nostalgikerna ville se Vänsterpartiet som del av en bred radikal tradition som alltid bekämpat, det de uppfattade som, kapitalismens förtryck, utsugning och våld, en tradition som de påstod alltid stått för demokrati, rättvisa och frihet. Att en sådan rörelse fanns, eller kan inskrivas i det historiska förloppet särskilt under 1800-talets senare del och början av 1900-talet, är mer än välkänt.
Det är också lätt att förstå att den unga och ny-kommunistiska generationen i Vänsterpartiet gärna tolkade sina företrädares idéer och handlande i detta välvilliga ljus. Det fanns bara en hake i resonemanget, nämligen att den svenska vänsterradikalism som Therborn – liksom Lindahl, Ohly och flera andra – åsyftade vände ryggen till demokratin och den demokratiska arbetarrörelsen vid första världskriget slut. Då sprängdes ju socialdemokratin på frågan om det framtida samhället skulle genomföras med våld eller allmän rösträtt. En gren förespråkade det revolutionära våldets väg, till denna hörde Vänsterpartiets föregångare, en annan valde den allmänna rösträttens och parlamentarismens väg. Den förra blev från 1921, som Sverige Kommunistiska Parti, en underordnad och följsam sektion av den Moskvastyrda Tredje internationalen. Det är riktigt att det utanför socialdemokratin kortare eller längre perioder existerade små vänstersocialistiska partier, i opposition mot såväl kommunismen som socialdemokratin. Alla hade de dock en begränsad livslängd och räckvidd och oftast sögs de relativt omgående upp av socialdemokratin efter att ha stötts bort av kommunismen. Att tala om dessa smågrupper som en bred vänsterradikalism är dock lika missvisande som att påstå att kommunismen representerade en demokratisk vänstertradition.
Faktum är ju att den svenska kommunismen förkastade och på olika vis bekämpade den demokrati som genomförts av den borgerliga vänstern och socialdemokratin 1917/1918. I synnerhet var detta fallet under 1920-talet och fram till folkfrontsperioden 1936. Också på det internationella planet gav partiet sitt stöd till diktaturens företrädare. Från augusti 1939 till juni 1941 slöt det upp bakom Stalins pakt med Hitlertyskland och bejakade Sovjetunionens undertryckande av de baltiska republikernas självständighet. Några månader senare gav det sitt stöd till Stalins anfall på Finland – efter detta betraktades det på goda grunder av en bred opinion som potentiellt landsförrädiskt såväl under resten av världskriget som under en lång period av det kalla kriget. När det senare väl tog sin början understödde partiet Moskvas utrensningar, likvideringar och deportationer i Östeuropa och ansåg att »folkdemokratierna» i öst var en högre form för demokrati än den liberala styrelsen i väst. Det är riktigt att partiet under en ny ledare, C. H. Hermansson och ett nytt namn, Vänsterpartiet kommunisterna, uttalade kritik mot »stalinismen» och i viss mån distanserade sig i förhållande till Sovjetunionens kommunistiska parti. Men detta var del av en mer allmän linje bland de västeuropeiska kommunistpartierna efter den 20:e partikongressen i Moskva 1956. Faktum kvarstår dock: partiet bröt först banden med den sovjetiska och östeuropeiska kommunismen då denna föll samman 1989–1991.
Hur kommer det sig att Vänsterpartiets föregångare – den rörelse och idétradition som partiet är sprunget ur och fortfarande identifierar sig med – intog dessa extrema ståndpunkter? Vad i den kommunistiska eller marxist-leninistiska demokratisynen ledde till dessa ställningstaganden? Och hur förhöll sig ledande företrädare för 68-vänstern till partiets historiska plädering för proletariatets diktatur och dess återkommande försvar av övergreppen i de socialistiska staterna?
Den yttersta vänsterns perspektiv på demokratin i västvärlden var ingalunda statiskt utan genomgick flera förändringar. I en första fas, präglad av oktoberrevolutionen och oron i Europa efter första världskriget betonades särskilt att parlamentarismen utgjorde ett sken eller en förklädnad för den härskande ägarklassens intressen. Den reformistiska idén om parlamentarism och allmän rösträtt som vägen till en fredlig och stegvis övergång till socialism, förkastades som uttryck för »legalistiska illusioner» och ansågs vara ett svek mot proletariatets intressen. I stället för demokrati och parlamentarism krävde den kommunistiska rörelsen under hela 1920-talet – fram till 1933 – en revolutionär omgestaltning av kapitalismen och ett politiskt styre kännetecknat av proletariatets diktatur, det vill säga en enpartistat i vilken all opposition mot den förhärskande politiken var förbjuden. Kommunismen var således aldrig på demokratins sida och det svenska Vänsterpartiet har sålunda sina rötter i en rörelse som bekämpade införandet av demokrati och parlamentarism i Sverige.
Den andra fasen inträdde efter att Hitler hade tagit makten i Tyskland. Den kommunistiska politiken gentemot den parlamentariska demokratins krafter lades då radikalt om. Efter denna omsvängning, som påbörjades kring 1934–1935, stod inte längre den socialistiska revolutionen och proletariatets diktatur i fokus, utan försvaret av den »borgerliga demokratin». Kommunistpartierna vädjade nu till de partier som de tidigare av all kraft fördömt, nämligen socialdemokrater och liberaler, om att upprätta en »antifascistisk folkfront» till försvar för den existerande liberala demokratin. Anspråket på att plötsligt agera demokratins försvarare kan te sig paradoxalt, med tanke på att kommunisterna fram till 1933 bekämpade Weimardemokratin med lika våldsamma medel som nazisterna, men var en direkt följd av deras katastrofala misslyckande och av att Sovjetunionen uppfattade Hitlertyskland som ett dödligt hot. Den nya politik som utgick från Moskva skulle vara fredlig och nationell, inte revolutionär och internationalistisk. Även om folkfronten i efterhand kan ses som ett första steg emot ett accepterande av fredlig reformism och liberal demokrati, betingades den i grund och botten bara av taktiska skäl. Den infördes som följd av Hitlers maktövertagande och var inte resultatet av en ändrad ideologisk syn på den kapitalistiska staten och den liberala demokratin. Tesen om proletariatets diktatur lades så att säga i malpåse i väntan på gynnsammare tider för den proletära revolutionen.
En tredje fas i kommunismens hållning till demokratin inleddes efter det att Sovjetunionen vid 1940-talets slut inordnat de östeuropeiska staterna i sin maktsfär. År 1947 upprättade Sovjetunionen en ny internationell organisation – Kominform – för koordineringen av de kommunistiska partiernas politik. Kominform hävdade, liksom tidigare Komintern efter 1936, möjligheten av en nationell väg till socialismen. Samtidigt som Sovjet med utomordentlig brutalitet avskaffade varje form för folkstyre i Östeuropa och Baltikum, påstod man från Moskva att kampen i väst nu syftade till att försvara och fördjupa de demokratiska fri- och rättigheterna, vilka framför allt, ansågs det, hotades av storfinansen men också av socialdemokratins samarbete med densamma.
Också de svenska kommunisternas mål var att upprätta en »folkregering» som samlade hela »folket» och som införde en »folkdemokrati», vilken vare sig uppfattades som en borgerlig demokrati eller en proletär diktatur. Förebilden var de nya staterna i Östeuropa som alltså benämndes »folkdemokratier» och som påstods vara en högre och mer utvidgad form för demokrati än den borgerliga. I den nya fredliga »folkdemokratin» var – precis som under proletariatets diktatur – uppdelningen mellan stat och civilt emellertid avskaffad och de politiska besluten kunde därmed utsträckas till att omfatta samhällets samtliga områden och inte som under kapitalismen (delvis) tvingas stanna vid fabriksporten och privatsfären. Denna tolkning av demokratibegreppet som en successiv utvidgning till samhällslivets alla områden och aspekter kom från 1950-talet att bli den dominerande inom dåvarande kommunistpartiet – i dag Vänsterpartiet – och stärkte självbilden att partiet var minst lika eller till och med mer demokratiskt än dess borgerliga motståndare. Mindre sällan framhölls det att också denna demokratisyn byggde på ett upphävande av den liberala demokratins konstitutionella maktdelning, dess civilrättsliga lagstiftning och privata äganderätt.
Det är viktigt att understryka att de kommunistiska partierna, inklusive SKP/VPK, ända fram till 1989 fasthöll att övergången till socialism måste ha karaktären av en social revolution. Härmed förstod man en samhällsomvandling som, i första hand, förutsattes ske utanför de parlamentariska institutionerna; en omvandling som fråntog ägarna av produktionsmedlen deras egendom, överförde statsapparaten i den nya »folkregeringens» händer och om nödvändigt med våld förhindrade »borgarklassen» att återta sin maktposition. Det nya i denna position var inte så mycket att begreppet proletariatets diktatur fallit bort som att kommunistpartiet inte längre ställde krav på att ensamt leda »revolutionen» – omvälvningen skulle ha formen av en utomparlamentarisk »folkallians» av olika krafter såsom arbetare, bönder, socialdemokrater, fackföreningar och kristna reformvänner.
Som exempel på hur starkt den marxist-leninistiska teorin i sovjetkommunistisk tappning påverkade 68-vänstern skall jag här lyfta fram två belysande exempel. Det första gäller Göran Therborns 1978 utgivna bok _What does the Ruling Class do when it Rules?_ Här sökte han utveckla och fördjupa Marx och Lenins viktigaste politiska doktriner genom att utnyttja den moderna sociologins metodologiska angreppssätt och begreppsapparat. Therborns framställning rör sig samtidigt mycket lojalt inom den kommunistiska traditionens ram, om än inspirerad av den nyleninism som formulerats av den så kallade Althusserskolan i Paris. Så exempelvis definierar Therborn den västliga demokratin på konventionellt marxistiskt vis som den politiska form genom vilken borgerskapet utövar sitt herravälde över arbetarklassen. Den borgerliga demokratin är »en liten minoritets härskande genom institutioner som allmän rösträtt och fria val».
Med uppsvinget i »klasskampen» efter 1968 ansågs möjligheten till socialism ha ryckt närmare i de avancerade kapitalistiska samhällena. Therborn betonade att själva övergången inte behövde följa den ryska revolutionens modell. Den kunde i stället ske genom en utdragen massmobilisering eller till och med genom allmänna val. Oavsett vilken historisk form transformationen tog, måste den emellertid medföra att den borgerliga staten »krossades», att kapitalisterna fråntogs sin egendom, att åtskillnaden mellan stat och samhälle upphävdes, att de civila institutionerna och privatsfären politiserades och att en kollektiv plan för produktionen upprättades. Eftersom arbetarna också i den nya proletära staten till en början var i en underordnad position – de flesta byråkrater var ju »borgerliga specialister» – måste staten ledas av arbetarklassens eget parti, som Therborn med hänvisning till erfarenheterna från Sovjetunionen och Östeuropa definierade som ett kaderparti. Precis som Lenin byggde Therborns proletära stat således inte på allmänna val, inte ens inom arbetarklassen, utan på ett traditionellt kommunistiskt enpartivälde.
I Therborns perspektiv hade 1968 års ungdomsrevolt alltså aktualiserat möjligheten till en socialistisk revolution i de västliga demokratierna, men grundvalen för denna möjlighet låg i den första »arbetarstatens» styrka: »Det är i princip de kommunistiska partierna, den internationella avspänningen och den maktbalans som garanteras av Sovjetunionen som gjort att den demokratiska socialismen har blivit ett verkligt alternativ i de avancerade kapitalistiska länderna.»
Hur såg då den demokratiska socialism ut som Therborn i slutet av 1970-talet föreställde sig? Föga överraskande betecknar han proletariatets diktatur som den mest avancerade formen för demokrati. Som Marx och Lenin hade definierat diktatur var den en högre form för demokrati – »den utgjorde i själva verket den bredaste formen av demokrati – en stat i vilken folkmajoritetens fria vilja är garanterad». Therborn gick till och med ett steg vidare. Han menade att Lenins strategi om väpnat uppror inte kunde ställas i motsats till demokrati. Tvärtom hävdade han: »Väpnat uppror kan, där det ger uttryck för majoritetens vilja, vara minst lika demokratiskt som den parlamentariska majoritetens maktutövning.»
Denna tanke, nämligen att inbördeskriget skulle kunna vara en väl så demokratisk procedur som ett parlamentariskt val, är åtminstone retrospektivt sett anmärkningsvärd. Inte minst därför att den formuleras av en forskare som i ett brett panorama över statsmakten i de dåvarande socialistiska länderna inte med ett ord nämner utrensningarna, massmorden, fördrivningarna, arbetslägren och säkerhetspolisens roll i upprätthållandet av det proletära herraväldet. Therborn påpekar visserligen att proletariatets diktatur i Sovjetunionen under Stalin hade en auktoritär karaktär, men han relaterar aldrig detta till Sovjetstatens grundstruktur. Om Lenin, som ju var arkitekten bakom denna stat, skriver Therborn att han var »den utan jämförelse överlägset störste politiske teoretikern inom den historiska materialismen». Sett mot denna bakgrund kan det knappast förvåna att Therborn i sina analyser av den borgerliga demokratin reducerar den till ett klassherravälde och helt ignorerar betydelsen av konstitutionell maktdelning, oberoende domstolar, lagstyre, enskild äganderätt, civil avtalsrätt och individuella rättigheter för såväl legitimiteten som stabiliteten i de västliga demokratierna.
Det andra exemplet rör professorn i praktisk filosofi och tillika medlemmen i VPK och Vänsterpartiet Torbjörn Tännsjö, som mot slutet av 1970-talet sökte precisera den proletära diktaturens förutsättningar i boken _Demokrati och proletär revolution._ 27 Framställningen har formen av en begreppsanalytisk diskurs, i vilken Tännsjö utgår från några klassiska demokratimodeller som han tillämpar på en diskussion av några tänkta varianter av proletariatets diktatur. Den borgerliga eller västliga demokratin karakteriserade Tännsjö på konventionellt marxistiskt vis som ett förklätt klassherravälde, en styrelseform som kapitalistklassen »manipulerar» genom att den behärskar det ekonomiska systemet. Skillnaden mellan en fascistisk diktatur och en borgerlig demokrati beskrev Tännsjö på ett något överraskande vis. Han menade att läget »åtminstone momentant kan vara mycket mer demokratiskt i en söndervittrande fascistiskt diktatur med en konstitution utan några som helst demokratiska fri- och rättigheter, än i en s.k. utvecklad demokrati av Sveriges typ».
Vilken fascistisk diktatur Tännsjö 1977 ansåg mer demokratisk än Sverige framgick dock inte av resonemanget. Däremot var Tännsjös bild av övergången från borgerlig demokrati till socialism desto tydligare. För det första handlade den inte, här delade han Therborns åsikt, om en förändring som byggde på allmänna val, även om en sådan i princip inte kunde uteslutas. Tännsjö förutsatte i stället att övergången fick karaktären av en långdragen massmobilisering i kombination med en djupgående samhällskris:
Under kampens gång kommer arbetarklassen att utveckla allt mer avancerade metoder: från strejker och blockader till ockupationer. Från strejkvakt till beväpnad arbetarmilis [...] Om en rörelse av den här typen utvecklas så att den kommer att omfatta folkets majoritet, om den är beväpnad och om den borgerliga statsapparaten och särskilt militärapparaten är i upplösning eller passiviseras, har ett maktövertagande utsikter att ske fredligt.
Tännsjö formulerade här en tanke som återfinns redan under Kominterns första år och sedan återkom under såväl folkfrontstiden under 1930-talet som under efterkrigstiden: om det borgerliga samhällets krafter kapitulerar inför de kommunistiska maktanspråken sker övergången fredligt, annars blir det väpnad strid, det vill säga i praktiken inbördeskrig. Skulle så bli fallet vilar ansvaret uteslutande på de krafter som vägrat acceptera den proletära sidans kapitulationserbjudande. Vad skulle då ske med enskilda företagare efter maktövertagandet? Först och främst skulle de utan ersättning fråntas sin egendom, därutöver skulle de politiskt motarbetas och exkluderas. Som Tännsjö skrev apropå Bonnierkoncernen kunde han tänka sig »att vi ger Lukas Bonnier, H-son och grabbarna en stencileringsapparat under proletariatets diktatur – så får vi se vad de kan åstadkomma!»
Accepterade borgarklassen inte den proletära diktaturen förlorade den alltså sina demokratiska rättigheter och skulle den i praktisk handling motsätta sig diktaturen förbjöds den helt. Tännsjö understryker i detta sammanhang en avgörande skillnad – han kallar det en paradox – mellan demokrati under borgerlig respektive proletär överhöghet. Medan utomparlamentariska vänsteraktioner är tillåtna i det förra fallet, är de oacceptabla i det senare fallet. Under den proletära diktaturen, då det inte längre finns någon motsättning mellan staten och arbetarna, krävs en sträng respekt för lag och ordning. Självständiga fackföreningar är uteslutna och den proletära staten »kommer att uppfatta bråkiga arbetarkollektiv som främmande och endast som objekt för 'dess' egen politik».
Den bild som framträder av det proletära styret bär visserligen Tännsjös signum men följer helt den Komintern- eller sovjetkommunistiska traditionen. Den liberala, västliga demokratin beskrevs i upprörda ordalag som ytlig och skenbar. Den konstitutionella maktfördelningen påstods vara rent formell och manipulerad av bourgeoisien. Dessutom lades det hinder i vägen för den revolutionära vänstern att mobilisera massorna genom utomparlamentariska massaktioner och att organisera beväpnade arbetarförband för att störta den parlamentariska demokratin. Den som av denna karakteristik av demokratin i väst drog slutsatsen att friheten skulle bli större när proletariatet väl hade segrat tog dock, enligt Tännsjö, grundligt miste.
Tännsjö menade att det under proletariatets diktatur inte skulle finnas någon konstitutionell maktdelning. Den centrala makten skulle över huvud inte begränsas av formella regler. Tvärtom skulle den ha både rätt och plikt att ingripa i varje enskild sak och mot varje misshaglig individ. Fri- och rättigheter skulle bara finnas för dem som i praktiken accepterade den nya statsmaktens politik. Självständiga civila organisationer skulle ej tillåtas och någon skyddad privatsfär var det bara inte tal om. De utomparlamentariska strejker och demonstrationer som var tillåtna under den liberala demokratin, skulle vara strängeligen förbjudna under det proletära klasstyret, i varje fall om de uttryckte kritik av den proletära politiken.
Till detta ska läggas att det i Tännsjös kommunistiska drömstat inte skulle finnas några ekonomiska resurser utanför och oberoende av den proletära makten – inga fristående institutioner baserade på egna medel, inga tidningar, tidskrifter eller tankesmedjor som ej finansierades av staten eller arbetarkollektiven. Och vad den enskilde medborgaren angick skulle livet komma att bestå i ett pärlband av möten: på arbetsplatsen, i hyreskollektivet, i det lokala kooperativet, i den proletära ungdomsklubben, i den folkägda skolan och det folkägda daghemmet. Detta sakernas tillstånd under proletariatets herravälde betecknade såväl den kommunistiska traditionen som 68-vänstern som frihetens och jämlikhetens fullbordan, medan den liberala demokratins främsta kännetecken ansågs vara godtycke och förtryck.
Det hävdas ibland från vänsterns sida att kommunismen representerar idén om ett klasslöst och frihetligt samhälle, och det är riktigt att enstaka sådana formuleringar går att finna hos främst Marx, men som praktisk utopi var marxism-leninismen i första hand genomsyrad av hat och revanschism: den borgerliga klassen – och socialdemokratin – skulle besegras och en gång för alla utraderas. Idén om proletariatets diktatur hos Marx och Lenin, liksom hos Therborn och Tännsjö, präglas långt mer av maktlystnad och hämndbegär än av demokratins strävan efter kompromisser och samhällsfred.
## **3. Brott och förnekelse**
_Kommunismens svarta bok_ (»den svarta boken») skrevs av en grupp franska historiker och sociologer, varav de mest kända var Stéphane Courtois, författare till bokens inledning, och Nicolas Werth. Tyngdpunkten i boken är en genomgång av det kommunistiska förtrycket i främst Sovjetunionen från 1917 till Stalins död. Vidare diskuteras Kominterns, i realiteten den sovjetiska underrättelsetjänstens, våldsaktioner i olika länder, främst mellankrigstidens Spanien. Genomgången av de kommunistiska våldshandlingarna i efterkrigstidens Östeuropa fokuserar övervägande på Polen, även om förhållandena i central- och sydeuropeiska länder också berörs. Den sista tredjedelen av den historiska analysen ägnas kommunismen i Asien, Latinamerika och Afrika. I centrum för framställningen står alltså de kommunistiska partiernas förföljelser, likvideringar och inte minst det lägersystem (Gulag) som i Sovjetunionen byggdes upp för organiserandet av det statliga tvångsarbetet. Till viss del bygger framställningen på arkivmaterial som kom fram efter sammanbrottet 1989–1991, men väsentligen handlar det om insamling och systematisering av tidigare publicerat material. _Kommunismens svarta bok_ är en empirisk dokumentation av förbrytelserna i de länder som kommunismen bemäktigade sig från 1917 och årtiondena därefter.
Ändå var det inte själva dokumentationen som väckte störst uppmärksamhet vid bokens utgivning 1997, utan det förhållandet att en av huvudförfattarna, Stéphane Courtois, i bokens inledning mer eller mindre jämställde kommunismens förbrytelser med nazismens. Kontroversen ledde bland annat till att två av medförfattarna, Nicolas Werth och Jean-Louis Margolin, förklarade sig oenig med Courtois jämförelse. Jag skall här begränsa mig till det svenska mottagandet av boken, till vilket jag också räknar Arne Ruths förord.
_Kommunismens svarta bok_ utgavs på Dagens Nyheters förlag bara två år efter den utkommit i Frankrike. Därmed fanns för första gången någonsin på svenska en samlad översikt över de kommunistiska regimernas våldshandlingar, en översikt som inte bara belyste terrorn under kommunismen, utan också kastade ett komprometterande ljus över den västliga vänsterradikalismens stöd till regimerna i öst – må vara att detta stöd efter Moskvas invasion av Tjeckoslovakien 1968 i många fall ackompanjerades av öppen kritik av de östliga kommunistpartierna.
Hur förhöll sig då Ruth till bokens budskap? En enkel sammanfattning är att han kretsade som katten kring het gröt. Han bejakade visserligen utan reservationer bokens framställning av terrorn och förtrycket i de stater där kommunismen grep makten. Han beklagade också, om än indirekt, att den nya vänstern vid 1960-talets slut »ersatte _en_ svartvit tankemodell med en annan». Den liberala antikommunismen hade avlösts av en romantisk och idealiserande bild av de revolutionära regimerna och rörelserna i Asien och Afrika. »Vi som trodde på FNL som en självständig kraft insåg inte att Hanoi rymde sin egen form för förtryck.» Han till och med accepterade den kontroversiella jämförelsen mellan kommunismen och nazismen som Stéphane Courtois gjorde, samtidigt som han anförde ett antal invändningar.
Tyngdpunkten i Ruths förord ligger emellertid inte på kommunismens brott utan på den västliga kolonialismen och den »rovgiriga kapitalismens» skuld gentemot de fattiga i tredje världen. I Ruths ögon framstår den västliga och liberala antikommunismen som en reaktionär och destruktiv kraft – detta gäller i synnerhet USA:s roll i Latinamerika. Han frågar med polemisk underton om »bokens beskrivning av kommunismen» hjälper honom att förstå sociala och politiska konflikter som de i Guatemala, Colombia och Chile, och svarar att han inte tror att så är fallet. Både frågan och svaret kan te sig märkligt. »Den svarta boken» var som sagt den första övergripande framställningen om kommunismens förbrytelser som någonsin publicerats. Även om Ruth inte förnekade att dessa brott, moraliskt sett, var på samma nivå som nazismens var det ändå som att han inte vill förhålla sig till de centrala frågor boken reser. Vilket samband fanns det mellan den marxistiska och leninistiska ideologin och övergreppen? Varför var förtryckets mönster så likartat i nästan alla socialistiska samhällen? Varför hade inte bara de västliga kommunistpartierna utan också oberoende progressiva krafter i väst både blundat för övergreppen och fördömt de som kritiserat dem? Hur hade den liberala pressen och de fria akademierna i väst förhållit sig till den kunskap om terrorn som kommit till kännedom i varje fall från 1950-talet och framåt? Hur skulle den aktuella boken värderas i ett historiografiskt perspektiv, jämfört med tidigare framställningar om förtrycket?
Ruth kommenterade inte någon av dessa frågor. Framför allt ville han inte tillerkänna de liberala demokratiernas motstånd någon del i kommunismens nedgång och fall. I stället gav hans framställning intryck av att drivkraften bakom den östliga socialismens sammanbrott stod att söka inom de kommunistiska samhällena själva. Han framhöll Titos brytning med Stalin 1948 och han lyfte fram den kritik som dissidenterna i Tjeckoslovakien och Solidaritet i Polen riktat mot systemet i öst. Det sägs aldrig rakt ut men budskapet blir likväl, kanske oavsiktligt att det var oberoende vänsterkrafter i Östeuropa som underminerade Sovjetkommunismen, inte det västliga motståndet. Att döma av Ruths förord verkade det kalla kriget ha stått mellan de repressiva kommunistiska regimerna i öst och den inre opposition som här och där kom till ytan de sista årtiondena av dess herravälde.
Det fanns en ytterligare tystnad i Ruths förord. Han förbigick intressant nog helt den återkommande kritik som Courtois riktade mot framför allt den västliga vänstern, den vänster Ruth själv i sitt förord sade sig tillhöra. Om kommunismens roll i framför allt västvärlden påstod Courtois bland annat:
Från 1920-talet till 1950-talet applåderade kommunister i hela världen och många andra människor entusiastiskt först Lenins och sedan Stalins politik [...] Från 1950-talet till 1970-talet hyllade hundratusentals andra människor hänfört den kinesiska revolutionens 'Store rorsman' och sjöng lovsånger till stora språnget och kulturrevolutionen. Ännu senare var det många som gladde sig åt Pol Pots maktövertagande.
Courtois gick vidare och pekade på de många modiga forskare och författare som lyfte fram kommunismens systematiska övergrepp, som till exempel Robert Conquest, Annie Krigel och Alexander Soljzenitsyn. Han frågade även varför så lite forskning hade bedrivits om de socialistiska staterna: »Och framför allt: varför denna 'akademiska' tystnad om den kommunistiska katastrofen som sedan åttio år tillbaka har omfattat omkring en tredjedel av människosläktet, på fyra kontinenter?»
Även om Courtois inte sade det explicit var det tystnaden vid de fria akademierna i den västliga världen han avsåg. Han sökte efter och framlade olika förklaringar. Han underströk den »enastående blindhet» som Västerlandet uppvisat inför den kommunistiska propagandan. Och Courtois tillade:
Denna blindhet förstärktes och nästan legitimerades av den tro som hystes av kommunister och många vänstermänniskor i väst och som gick ut på att de här länderna höll på att 'bygga socialismen' och att den utopi som närde de sociala och politiska konflikterna i demokratierna höll på att bli verklighet 'därborta'.
Som tidigare framkommit fanns lovsångerna för Maos Kina och aversionen mot dem som visade på förtrycket under den sovjetiska kommunismen också bland ledande svenska vänsterintellektuella. Arne Ruth nämner dock inte i sitt förord hur en av hans företrädare som kulturchef på DN – Olof Lagercrantz – i en rad artiklar på kultursidan hyllade maoismen och han undanhåller att sidan under hans chefskap misstänkliggjorde Robert Conquest som person och forskare.
Arne Ruth var emellertid inte ensam på barrikaden att distansera sig visavi »högerkritiken» av kommunismens övergrepp. Sven-Eric Liedman försökte, som redan visats, delegitimera de svenska Sovjetforskarnas kunskaper med hänvisning till att de »var internationellt uppmärksammade för sin extrema högerprofil». Härmed menade han – helt i linje med den tidigare sovjetkommunistiska propagandans huvudlinje – att de forskare, författare eller journalister som ansågs vara »höger» gick det inte att lita på.
Det finns en viktig lärdom i detta. Den sovjetkommunistiska propagandaapparatens brännmärkning av alla former för »antisocialistisk» kritik gav så sent som på 1970- och 1980-talen ekon i väst – bland vänsterintellektuella, inom liberal press och vid oberoende akademier. Det skall här understrykas att denna »kritik» var självgående – den fungerade utan någon konspiration, utan någon central eller internationell dirigering av åsikter och inlägg. Även efter Sovjetunionens sammanbrott var avståndstagandet från »högeridéer» ett »förgivet-tagande», en outtalad utgångspunkt vid bedömningen av kritik mot det som uppfattades som progressiva och emancipatoriska idéer och strävanden. Men det dominerande idéklimatet förklarar inte allt. Varje individ har ett eget val och ansvar för detta val. Det förefaller som att det fanns en medveten vilja bland många vänsterintellektuella att stänga ute och stigmatisera forskare och journalister av mer liberal inriktning, i synnerhet de som kritiserade marxistiska och »progressiva» idéer.
Likväl hade hållningen till kommunismen och synen på Sovjetunionen ändrat sig avsevärt sedan sammanbrottet för ett knappt årtionde sedan. Detta framgår tydligt av de flesta recensioner som utgivningen av _Kommunismens svarta bok_ föranledde i landets ledande tidningar. Helt borta var nu varje ord om socialismens »storslagna resultat» och kommunismens »heroiska höjdpunkter», borta var också i allt väsentligt påståenden om att de brott som begåtts var en följd av att Sovjetkommunismen varit utsatt för yttre, »imperialistiska» angrepp och hot, att ansvaret i realiteten låg på den »aggressiva» borgerliga kapitalismen eller att kritik av förtrycket var uttryck för en »reaktionär antikommunism».
I Sydsvenska Dagbladet vände sig Anders Ehnmark mot Stéphane Courtois påstående att det i väst rått en »tystnad» om förtrycket under kommunismen i öst:
Det är en egendomlig historieskrivning. Man kan med större fog hävda att omvärlden inte talat om annat än kommunismens brott de senaste femtio åren. Antikommunismen har varit efterkrigstidens mest kreativa och framgångsrika ideologi. Så länge jag minns tillbaka har det dånat i det offentliga rummet. Ett berg av vittnesmål och forskning om brotten under Stalin föreligger. Insynen i KGB:s hemligaste vrår har till och med varit god.
Ehnmark tyckte sig i »den svarta boken» se en »i huvudsak tidigare känd bild» få skarpare konturer och antydde att Courtois genom att »låtsas» bryta tystnaden hade försökt »återlansera antikommunismens ideologiska vinster», vilket inte lät sig göras när kommunismen nu var död. Ehnmark vore inte Ehnmark om han här inte hade rejält hårdragit Courtois resonemang. Denne hade nämligen främst talat om den tystnad som hade rått inom vänstern och de »fria akademierna», inte om någon tystnad i allmänhet. På andra sidan tillstod Ehnmark utan förbehåll att boken var viktig, intressant och klargörande, frågan var bara hur övergreppen under kommunismen skulle förklaras. Var de avsiktliga, frågade sig Ehnmark, fanns det en plan? Han gav inget eget svar utan lutade sig i stället mot Nicolas Werths tanke att mycket var svårtolkat och att det fanns »synnerligen många dunkla områden kvar».
Ehnmark reste alltså frågan om bolsjevikernas terror var medveten. Fanns det en plan, frågade han. Med hänvisning till Nicolas Werth verkade han svara negativt på frågan. I stället skrev han att »mycket skedde slumpmässigt», samtidigt som han tillade att »samma misstag begicks gång på gång». Vän av ordning kan här invända att ett misstag som begås gång på gång konstituerar ett mönster, medan slumpen till sin natur är en oförutsebar avvikelse från etablerade förhållanden. Ehnmark tillade att »omnipotent planering och sömnighet» uppträdde sida vid sida under det ryska bolsjevikväldet. Ehnmark reste dock aldrig frågan om inte själva grundidén i den radikala socialism som Marx och Lenin förfäktade, nämligen att med tvång beröva de samhällsbärande »borgerliga» individerna deras privata egendom och omyndigförklara dem som medborgare, ofrånkomligen måste leda till ett sammanbrott för stabilitet och ordning, där allsköns destruktiva krafter, vare sig politiska eller rent kriminella, får fritt spelrum för sina materiella, maktmässiga och sexuella habegär – och om inte en sådan strävan med nödvändighet måste resultera i ett brutalt förtryck. Kommunismens egen historia utgör väl här det bästa exemplet på att just »samma misstag begicks gång på gång».
På Dagens Nyheters kultursida recenserade Sven-Eric Liedman »den svarta boken». När realsocialismen i öst kollapsat ett årtionde tidigare hade han formulerat sig som om kunskapen om »det sovjetiska eländet» varit begränsad till ett fåtal insiktsfulla. Nu framställde han det som om kunskapen var en följd av tillgång till de öppnade ryska arkiven: »De gamla sovjetiska arkiven har öppnats. Den fasansfulla Gulaghanteringen håller på att kartläggas.»
Precis som Ehnmark hade Liedman inga invändningar mot »den svarta bokens» resonemang om kommunismen som en brutal form av förtryck. Tvärtom menade han att kommunistregimer är »usla och de goda avsikterna har blivit till tomma fraser». Det var den senare aspekten – förhållandet mellan avsikter och konsekvenser – som stod i fokus i Liedmans recension. Som många andra ansåg han följande: »Kommunismens teori stämmer inte med dess praktik, så mycket är tydligt. Vägen från Hitlers _Mein Kampf_ till världskriget och Förintelsen är däremot rak och konsekvent [...] Kan man då säga: Kommunismen är bättre än nazismen eftersom dess avsikter är goda?»
Liedman menade att frågan tangerade den klassiska problematiken om konflikten mellan pliktetik och konsekvensetik. Han ansåg att kommunismen moraliskt sett var kluven. Ingen kunde längre bestrida att konsekvenserna av den var usla, men ingen kunde heller förneka att »många kommunister... gjort aktningsvärda, självuppoffrande insatser som motståndsmän. Någon nazist av det slaget finns inte inom synhåll.»
På kommunismens pluskonto stod också, enligt Liedman, att dess ideologi utgick från en av 1800-talets »stora samhällstänkare» – Karl Marx – och att den i motsats till nazismen utvecklat en sammanhängande teori om samhället och historien. Samtidigt tillfogade Liedman att det fanns en »mörk punkt också hos Marx». Han hade nämligen förutsatt att de sociala konflikterna – klasskampen – »bara kunde lösas med strid». Denna våldsinspirerade syn på lösningen av sociala och politiska konflikter hade, menade Liedman, förstärkts av första världskrigets blodsutgjutelser – våldet hade blivit »normen för civiliserat beteende». Liedman menade att det var de ledande ländernas krigsledare som hade satt den nya normen för »civiliserat beteende» och att Lenin blott »gjorde normen till sin när han kom till makten».
Det är riktigt att första världskriget eskalerade våldsanvändningen och att krigsslutets kaos innebar ett politiskt genombrott för det gångna sekelskiftets revolutionära rörelser – antidemokratiska, antikapitalistiska och antiliberala. I det maktvakuum som uppstod lyckades ibland vänster- och ibland högerextrema grupper tillfälligt gripa makten i ett antal enskilda städer och i vissa regioner i Europa, med likvideringar, våldtäkter och plundringar som förutsebar följd – men bara i två stater förblev herraväldet permanent i extremisternas händer, i Ryssland 1917 och i Italien 1922. Faktum är att de ryska bolsjevikernas terrorvälde, under först Lenin och sedan Stalin, utgjorde en både dramatisk och historisk avvikelse från det civiliserade politiska beteende i form av konstitutionalism, rättsstatlighet och maktdelning som från 1800-talets mitt gradvis hade fått allt fastare form i främst västra Europa och som vid första världskrigets slut kulminerade med införandet av parlamentarism och allmän rösträtt, en kulmen i vilken socialdemokratiska och liberala ledare spelade en central roll för samhällsfreden. Dessutom ignorerar Liedman att Lenin, liksom en rad andra vänster- och högerextremister, redan innan kriget hade hyllat våldet som vägen till makten – idén om det väpnade upproret föregick med andra ord kriget.
I Svenska Dagbladet diskuterade Kay Glans boken i en understreckare. Också han satte fokus på sambandet mellan kommunismens idé och dess praktik, men hans analys var, i all korthet, både mer inträngande och mer kritisk än Ehnmarks och Liedmans. Glans ställde frågan varför »kommunismen har urartat överallt». Han avvisade det traditionella kommunistiska försvaret att urartning var en följd av »olyckliga omständigheter eller misstag». I stället undrade han om »förtrycket inte är inneboende i det kommunistiska projektet». Han tog upp den vanligt förekommande invändningen, som exempelvis Liedman framhöll, nämligen att kommunismens övergripande avsikter var goda och emancipatoriska. Kommunismen önskade ju upprätta ett samhälle där alla skillnader mellan människor hade upphävts, där friheten var en följd av en gemensam strävan efter hela samhällets bästa, där ingen satte sina egna intressen framför gemenskapens intressen. Glans menade att det var i denna önskan om perfektion som roten till kommunismens ondska stod att söka.
Han lyfte fram ett betecknande citat från »den svarta boken», ett citat som författaren Nicolas Werth hämtat från bolsjevikernas säkerhetspolis i Kiev och som på grund av sin representativitet för tidens mentalitet är värt att återge. Den hemliga polisens – tjekans – företrädare legitimerade där sina ingrepp på följande vis: »Vår moral saknar förebild, vår humanitet är absolut ty den vilar på ett nytt ideal: att förinta alla former av förtryck och våld. För oss är allt tillåtet, ty vi är de första i världen som reser svärdet inte för att förtrycka och förslava utan för att befria mänskligheten från dess bojor.»
Glans ansåg att den kommunistiska ideologin var en dröm om ett samhälle där allt var kontrollerat; ja, där själva friheten för den enskilde låg i känslan av identitet med den perfekta helheten. Hotet mot kommunismen kom från dem som inte ville delta, inte inordna sig; från dem som ville behålla sin egendom, som ville gå sin egen väg, som hade en annan tro eller åskådning. Och Glans ställde frågan:
I vilken utsträckning är kommunisternas klasslösa utopi också präglad av ett hat mot differentiering och social mångfald, mot det okontrollerbara i tillvaron som marknaden är ett så centralt exempel på? Kommunismen är en dröm om en fantastisk kontroll och kontrollsträvan och sadism brukar aldrig ligga särskilt långt från varandra. Strävan att skapa det perfekta förutsätter ju att förstöra det befintliga och det är inte alltid lätt att veta var tyngdpunkten skall läggas, vid skapandet eller förstörandet. Jämlikhetssträvanden infiltreras också lätt av avunden, en av människans mest destruktiva krafter.
Som Glans påpekade finns en omfattande forskning om den auktoritära, nazistiska personlighetstypen, men däremot nästan ingen forskning alls om den kommunistiska mentaliteten och dess betydelse för alstrandet av våld och förtryck mot icke-kommunister.
I Expressen recenserade journalisten och författaren Per Svensson _Kommunismens svarta bok_. Han tog upp en annan aspekt på den fråga Ehnmark och Liedman diskuterat, nämligen om det fanns någon skillnad mellan kommunistiskt våld och våld i »kommunismens namn». Denna distinktion mellan en verklig kommunism (»realsocialism») och en tänkt kommunism var vanlig inom de vänsterkretsar som vid 1990-talets slut åter började svärma för kommunismen som ideologi. Per Svensson replikerade ironiskt att det tydligen var: »En bedragarkommunism [som] stulit den äkta kommunismens ID-handlingar och i 'dess namn' skapat Gulag.» Det är uppenbart att Svensson betraktade distinktionen som nonsens och hans omdöme var inte nådigt: »Överallt och alltid har kommunismen drivit fram det sämsta hos, och för, människan. Skulle det bero på att ideologin aldrig förverkligats? Ligger det inte närmare till hands att anta att det förhåller sig tvärtom? Är det inte mer sannolikt att det finns ett kausalt samband mellan ideologin och praktiken.»
Som exempel på detta samband lyfte han bland annat fram skådeprocesserna, utrensningarna, massdeportationerna och svältkampanjerna i Sovjetunionen – de hade ju alla det gemensamma temat, precis som Lenins och Stalins program föreskrev, att en gång för alla göra upp med det kommunistiska framtidsrikets borgerliga, kapitalistiska och nationalistiska motståndare – liksom »förrädare» och »utländska agenter» i de egna leden. Medan idéhistorikern Sven-Eric Liedman inte såg något samband mellan den marxist-leninistiska teorin och övergreppen i de kommunistiska staternas praktik, menade således Per Svensson att där fanns en tydlig förbindelse. I själva verket var Svensson – indirekt – ytterst skoningslös mot Ruths, Ehnmarks och Liedmans undanglidande kommentarer:
Det avgörande skälet till att 'Kommunismens Svarta bok' – som ju i huvudsak är en sammanställning av tidigare kända fakta – fått så förvånansvärt många vänstersinnade debattörer att se rött, måste vara att den så tungt och övertygande både implicit och explicit förnekar 'skillnaden' – skillnaden mellan teorin och praktiken, drömmen och mardrömmen, kyrkan och tron.
Det fanns flera andra som kommenterade »den svarta boken», men utan det försök att problematisera relationen mellan ideologi och praktik som ändå de ovan refererade inläggen hade försökt. Den socialdemokratiske idédebattören Per Wirtén uppfattade att boken var skriven från vänster. Han förknippade den inte med det kalla krigets antikommunism. I stället såg han boken som en bekräftelse på att »revolutionerna 1989 befriade radikalismen från kommunismens ok». Mauricio Rojas, historieforskare och tidigare liberal riksdagsman, reflekterade också kort över »den paradoxala kopplingen mellan en bländande utopi och den mest förfärliga ondskan» under kommunistiska regimer. Han var emellertid inte förvånad: »Att de som ville skapa en helt ny värld befolkad av helt nya människor har tagit sig rätten att utrota miljontals reellt existerade människor är tragiskt men inte ologiskt.»
Rojas konkluderade att i synnerhet de som ansåg sig företräda godheten här i världen borde ha ett särskilt intresse att söka förstå »länken mellan den 'barmhärtiga' drömmen om ett kommande harmoniskt lyckorike och kommunismens brott».
Också Per Ahlmark kommenterade boken i en kolumn i Dagens Nyheter. Han blickade tillbaka på sin ungdoms diskussioner mellan och med Vilhelm Moberg, Eyvind Johnson, Ingemar Hedenius och Herbert Tingsten åren efter andra världskriget:
Jag kan knappast påminna mig något enda utrikespolitiskt samtal... då de inte drog paralleller mellan Nazi-Tyskland och sovjetväldet. Inte så att dessa... ansåg att de totalitära imperierna var varandras kopior. Men likheterna och skillnaderna ansågs uppenbara och avgörande. Att diskutera dem var lika naturligt som att andas.
Det Ahlmark säger är egentligen självklart, nämligen att »jämföra» och att »jämställa» är två olika saker och att det ena inte följer av det andra. Jämförelsen är en metodisk grundförutsättning i all historisk forskning, utan jämförelsen är det vare sig möjligt att kunna urskilja olikheter eller likheter mellan olika fenomen. Att jämställa däremot är en politiskmoralisk bedömning, som mycket väl kan vara resultatet av jämförelsen, men som bestämt inte har något nödvändigt eller logiskt samband med den.
Samma år som »den svarta boken» utkom publicerade Vänsterpartiets ungdomsförbund, Ung vänster, en motskrift _Utan heder_ som redigerats av Aron Etzler. I sitt förord menade Etzler att »högerns» intresse för kommunismens brott ingick i en svensk och europeisk offensiv mot vänstern. I Sverige hade Vänsterpartiet gått framåt i riksdagsvalet 1998 och i Europa kunde allt fler se de »katastrofala» resultaten av »nyliberalismens» framfart. Trots att vänstern sedan länge tagit avstånd från alla brott i »kommunismens namn» och markerat att den inte ville »ha det som i Sovjetunionen» fortsatte kritiken. I Etzlers ögon var »den svarta boken» kvasivetenskap och de som ställt sig bakom den var lögnaktiga och saknade heder. Han skrev bland annat: »Hela denna kampanj mot kommunismen har inslag som borde förkastas av historiker och samhällsvetare... Demoniseringen av Lenin, som framställts som en av historiens värsta massmördare, är en tillbakagång till ett sätt att skriva historia som hör 1800-talet till.»
Trots att denna inledning, liksom flera andra inlägg i antologin, lät som ett eko från den sovjetkommunistiska propagandans dagar, innehöll boken åtminstone något bidrag som hade mer principiell karaktär. Den tidigare maoisten Stefan Lindgren satte frågetecken vid dels Stéphane Courtois påstående att kommunismen krävt 100 miljoner dödsoffer dels Rudolph J. Rummels beräkningar. Lindgren använde delvis de sedvanliga argumenten mot till exempel Conquest och de siffror han själv kom fram till är förmodligen inte mer trovärdiga än de han avfärdade. Men han satte likväl fingret på en öm punkt när det gäller beräkningen av antalet dödsoffer för i synnerhet de avsiktliga (»man-made») svältkampanjerna i till exempel Ukraina och Kina, av antalet likviderade i krigsårens Sovjetunionen eller i det närmast hermetiskt slutna Kampuchea under Pol Pots terrorvälde. Någon adekvat folkbokföring finns inte tillgång till i dessa länder, officiella dokument är otillförlitliga och beräkningar som grundas på »utebliven» befolkningstillväxt är osäkra. Att det handlar om sammanlagt flera tiotals miljoner är odiskutabelt, liksom att regimerna i fråga styrdes av en ideologiskt betingad brutalitet och total mänsklig hänsynslöshet. Men några siffror med samma tillförlitlighet som de vilka rör till exempel Förintelsen lär aldrig komma fram.
Boken innehöll vidare ett par bidrag som berörde totalitarismteorin. Historikern Werner Schmidt tillstod något överraskande att doktrinen »kan lämna ett visst positivt bidrag till förståelsen av de komplexa och i de flesta avseenden olika fenomenen kommunism och fascism, så länge den begränsas till den uppgift som den faktiskt har analytisk förmåga att lösa, nämligen att undersöka och beskriva likheterna vad gäller kommunismens och fascismens maktmetoder. Tillämpad på detta område är doktrinen utan tvekan ägnad att påvisa de katastrofala konsekvenserna som avsaknaden av demokrati kan få.»
Det tycks som en bra karakteristik av teorins styrka – och det är långt mer än man kan säga om de marxistiskt influerade teorierna om fascismen vilka menar att denna ytterst har sitt upphov i kapitalismen och ser den kommunistiska rörelsen som den enda garanten mot de högerextrema hoten. Svagheten menade Schmidt var att totalitarismdoktrinen ignorerade de »verkliga faror som hotar vår tid». Vilka dessa faror är nämndes aldrig, men Schmidt hänvisade till ett citat från Walter Benjamin om att den nuvarande civilisationen i sig är en katastrof – »nuet är katastrofen». Det Schmidt egentligen verkade irritera sig på var knappast nuet som katastrof, utan det faktum att totalitarismteorin utgår från att demokrati är att föredra framför kommunistiska diktaturer. Schmidt menade nämligen att doktrinen »idealiserade» det demokratiska systemet och den gav den borgerligt-kapitalistiska civilisationen »en oförtjänt glans och oskuld». Teorin om totalitarism har helt klart begränsad räckvidd och lämnar flera frågor kring kommunism och nazism obesvarade. Men som Werner Schmidts inlägg visar är det inte av metodisk omtanke utan av ideologiska motiv som den vanligtvis förkastas av vänstern.
Även om det således fanns förnuftiga överväganden, och även en viss självkritik i några artiklar, genomsyrades ändå boken av en påfallande aggressivitet i försvaret av kommunismen. Visst hade Stalin, Mao och Pol Pot begått fruktansvärda brott i kommunismens namn och självfallet skulle dessa misstag inte begås igen, vänstern visste hur Sovjetunionen hade fungerat och ingen önskade upprepa experimentet i Sverige. Och för övrigt, skrev Jenny Lindahl som formulerat den nyss nämnda försäkran, hade ju den socialistiska kampen »alltid handlat framför allt om att försvara människors rättigheter». Samtidigt fastslogs i boken att den kommunistiska rörelsen och den ryska revolutionen, alltså Lenins och Stalins regim, hade »bidragit till många av 1900-talets viktigaste framsteg: koloniernas frigörelse, välfärdsstaternas framväxt i Europa och rösträttens genombrott». Kanske var det denna »djupa» självkritik av kommunismens förflutna som fick flera skribenter att teckna en alltför välkänd bild av socialismen: den privata äganderätten skulle upphävas, bortsett från vissa personliga ägodelar och de stora och långsiktiga investeringarna skulle beslutas av regeringen och parlamentet. I motsats till tidigare skulle metoden nu emellertid vara demokratisk, småskalig, närdemokratisk och miljövänlig.
_Utan heder_ är en vittnesbörd om att Vänsterpartiet visserligen var snabbt att ta avstånd från kommunismen i öst när den plötsligt bröt samman 1989–1991, men att partiet i grunden aldrig förmådde att göra upp med den kommunistiska traditionen – att man i motsats till vänsterpartier i flera andra europeiska länder, missade möjligheten att efter kommunismens fall rensa bordet från totalitära idéer och i stället lät dessa fortsätta att existera i partiets undervegetation för att återigen blomma ut vid 1990-talets slut.
## **IX. SLUTET PÅ EN ILLUSION**
## **Slutsatser om perioden 1987–2000**
I UNDERSÖKNINGEN AV perioden 1987 till 2000 har Vänsterpartiet och debatten om 68-vänsterns karaktär och kopplingar till kommunismen i öst stått i förgrunden. Händelserna i Berlin och Östeuropa senhösten 1989 var en fullständig överraskning för alla, men de var kanske särskilt chockartade för det svenska Vänsterpartiet som så sent som 1987, mitt under Gorbatjovs perestrojka, upprepat sin tilltro till Lenins revolutionära projekt. Den nyleninistiska vågen i partiet fick emellertid ett abrupt slut i samband med DDR-regimens fall. Det följande årtiondet var för vänstern i viss mening en vandring i dödsskuggans dal. Inget ont emellertid utan något gott. Redan vid millennieskiftet hade vänstern i bred mening ömsat ideologiskt skinn. Följande iakttagelser är särskilt viktiga att understryka.
För det första att vänstern som följd av 1989 års händelser hamnade i ett tillstånd av ideologisk chock och förvirring. Märkligt nog hindrade detta inte partiet och rader av dess intellektuella sympatisörer att närmast i panik vända kommunismen ryggen, när regimerna i öst föll samman. Marxismen och leninismen ströks genast ur programmet och partiet bytte namn. Plötsligt fanns inte längre någon enighet om oktoberrevolutionen hade varit på det godas eller ondas sida, om kommunismen varit en progressiv eller destruktiv kraft eller om det inom vänstern funnits vetskap om förtrycket under socialismen eller ej.
Vidare visar genomgången att Vänsterpartiet aldrig genomförde någon riktigt djupgående självrannsakan eller omprövning av sitt kommunistiska förflutna, snarare handlade det om taktiska anpassningar till en ny tidsanda. Efter chockvågen 1989–1991 återtog de mer ortodoxa och traditionalistiska krafterna makten i partiet. Men utvecklingen var paradoxal och motsägelsefull. Samtidigt som marxismen återinfördes i programmet blev partiet feministiskt och ekologiskt. Samtidigt som en nostalgisk sympativåg för kommunismen sköljde över partiet mot 1990-talets slut ersatte den postmoderna identitetsideologin klasskampen som vänsterpolitikens gravitationspunkt. Vänsterpartiet försköt efter 1989 kommunismen men det bröt aldrig med den.
En tredje viktig slutsats är att den hårda liberala kritiken av 68-vänstern som bland andra Per Ahlmark och Kay Glans svarade för, hade en avgörande betydelse för att tvinga ledande företrädare för 68 års idéer, däribland Sven-Eric Liedman, Anders Ehnmark, Arne Ruth, Göran Therborn och Torbjörn Tännsjö att överge sin tidigare omfamning av eller positiva syn på kommunismen. Therborn och Tännsjö pläderade efter 1989 för den »borgerligt-liberala» demokratins företräden, från att ett drygt årtionde tidigare ha pläderat för proletariatets diktatur. Arne Ruth skrev att han nu kommit till insikt om de marxist-leninistiska befrielserörelsernas odemokratiska karaktär. Liedman och Ehnmark, som båda omgående förkastade kommunismen när den väl föll, menade vid 1990-talets slut att kommunismen i öst varit ett förskräckligt system. Utan 1990-talets välformulerade och träffande liberala kritik av kommunismen hade dessa vänsterintellektuella förmodligen inte omvärderat sin syn på liberal demokrati kontra kommunistiska styrelseformer.
Slutligen visar genomgången att antiliberalismen och antikapitalismen, liksom avståndstagandet från antikommunism, har förblivit bestående och bärande inslag i den postkommunistiska vänsterns ideologi. Kommunismens historiska nederlag och den liberala demokratins seger rubbade inte de intellektuella 68-ornas – och deras arvtagares – förakt för och avståndstagande från kapitalismen som ekonomiskt system och från den liberala idétraditionen. Kanske är detta den mest bestående delen av arvet efter 68.
## **X. 1968 OCH FILOSOFIN I SVERIGE**
## **1. Ett mål i sikte**
ÅR 1968 STARTADES Häften för Kritiska Studier, en tidskrift som skulle komma att få utomordentligt stor betydelse för den akademiska vänstern i Sverige. Det första numret kom ut i september och ägnades framför allt åt två teman, dels en programförklaring för tidskriftens del, dels en diskussion av en doktorsavhandling, Gerard Radnitzkys _Contemporary Schools of Metascience_ , som lagts fram samma år vid institutionen för vetenskapsteori i Göteborg. Dessa båda teman hade i själva verket ett nära samband med varandra.
Programförklaringen för tidskriften var skriven av Roger Fjällström, som förde ordet för gruppen Unga filosofer. Unga filosofer hade bildats 1965 bland studerande i praktisk filosofi i Stockholm och stod bakom utgivandet av Häften för Kritiska Studier. I presentationen av den nya tidskriften hette det:
Under 60-talet – särskilt genom det ökade internationella och sociala engagemanget – har det skett en ansamling av komplicerade och politiskt brännbara problem. De etablerade samhällsvetenskaperna har inte utformats för att göra reda för denna nya situation. På ett ensidigt sätt har de dominerats av och mer eller mindre utformats i enlighet med borgerligt konservativa värderingar.
Inom samhällsvetenskaperna, hette det i fortsättningen, hade en anglosaxisk vetenskapstradition varit förhärskande medan »kontinentala riktningar, t.ex. marxismen, har ignorerats». Socialistiska värderingar och marxistisk teori var emellertid nödvändiga »för att vi på ett fruktbart sätt skall kunna gripa oss an problemen». Det fanns redan vid de olika svenska universiteten »institutionsgrupper» i olika ämnen som arbetade på att förändra vetenskapen i denna anda. Tidskriften hade tillkommit i anslutning till verksamheten inom bland annat dessa grupper: »Vi hoppas att Kritiska studier skall fylla ett behov genom att vara ett organ för en framväxande, radikal student- och forskargeneration som kräver ett nytt socialt och vetenskapligt klimat.»
Programförklaringen bars på detta sätt upp av en motsatsställning mellan en dålig och på konservativa värderingar baserad, anglosaxisk vetenskapssyn och en god, kontinental sådan, som rymde bl. a. marxismen och som förknippades med socialistiska värderingar. Den senare bars upp av en ung student- och forskargeneration som nu – 1968 – krävde att »nytt socialt och vetenskapligt klimat».
Det framgick i fortsättningen att den förra vetenskapssynen kunde karaktäriseras som »positivism» eller »logisk positivism», medan den senare var »kritisk».
Många av dem som läste första numret av Häften för Kritiska Studier hade förmodligen redan tidigare träffat på motsatsställningen mellan de två olika sorternas filosofi, men då med omvända förtecken och värderingar. En första början hade gjorts genom den föreläsning som Ingemar Hedenius höll vid sin installation som professor i praktisk filosofi i Uppsala 1948. Han framhöll vid detta tillfälle att ingen annan vetenskap i så hög grad som filosofin var offer »för osakliga krav, begär och affekter». Detta illustrerade han med en genomgång av de former av ovetenskaplig filosofi som var i svang i samtiden. Där fanns marxismen och nythomismen, som beskyddades av auktoritära makter (partiet i kommuniststaterna, kyrkan i den romerskt katolska världen). Han nämnde också existentialismen, som hade sin rot i »människornas, som det förefaller, djupt rotade begär efter andra och känslomässigt mer stimulerande bilder av verkligheten än dem som vetenskapen ger». Dock fanns det en filosofi, den analytiska, som var vetenskaplig och som byggde enbart på förnuftsmässig argumentation. Hedenius sa bland annat: »Och dock är det en tröst, att om förhållandena överallt medgåve full frihet åt filosofien, så skulle troligen den filosofiska tradition, som nu framstår som den mest vetenskapliga, bli den enda härskande.»
Han tillade: »Till sist vill jag peka på den kraft, som är filosofins enda bundsförvant i denna prekära situation: den västerländska människans sanningskrav. Detta krav tillhör i högre grad än någon politisk eller religiös dogmatik själva livsnerven i vår kulturtradition.»
För egen del menade Hedenius att vetenskapen överhuvud inte kan läggas till grund för några sammanhållande livsåskådningar eller helhetssyner. I uppsatsen »Livsåskådningar» från 1949 inskärpte han detta: »Vad den moderna vetenskapen ger oss är inte en totalåskådning utan blott en rad olika betraktelsesätt, lämpade efter vitt skilda aspekter på tillvaron. Och för övrigt en nästan fullkomlig skepsis i alla frågor om hur verkligheten är i största allmänhet. Det totala avvisandet av varje helhetssyn.»
Hedenius syn på den moderna filosofins situation bekräftades av hans kollega Konrad Marc-Wogau i studiehandboken _Att studera filosofi_ (1961). Redan i förordet till boken slog Marc-Wogau fast: »Bokens kanske viktigaste uppgift är att varna för filosofiskt tänkande som enligt mitt sätt att se befinner sig på avvägar.»
Under rubriken »Ovetenskaplig filosofi» avhandlades sedan marxismen, nythomismen och existentialismen som exempel på »hur filosofin inte bör studeras». Marxismen diskuterades med rubriken »I politikens våld. Filosofin i Sovjetryssland». Marxistisk filosofi utanför Sovjetunionen nämndes inte. Mot dessa riktningar ställdes sedan den enda vetenskapliga filosofin, den analytiska.
Både Hedenius och Marc-Wogau arbetade sålunda med en fyrdelning av den moderna filosofin – marxism, nythomism, existentialism och analytisk filosofi. Men bakom denna fyrdelning låg en tvådelning mellan ovetenskaplig filosofi (marxism, nythomism och existentialism) och vetenskaplig filosofi (analytisk filosofi). Det var samma tvådelning som Häften för Kritiska Studier arbetade med. Också Hedenius och Marc-Wogau förutsatte att det fanns inte bara en intellektuell skillnad – den mellan vetenskapligt och ovetenskapligt – utan också en politisk – den mellan demokratiskt och auktoritärt. Marxismen var nära förknippad med kommunismen och med Sovjetunionen, nythomismen med den auktoritära påvekyrkan, existentialismen med nazismen. Hedenius menade sålunda att existentialismen förr eller senare skulle komma att sjunka undan och övergå till att bli »en detalj i det sönderslitna Hitlerskedets litteraturhistoria». Hans ord om den västerländska traditionen ger också en ledtråd. Den analytiska filosofin är förbunden med upplysnings- och modernitetssträvanden, de andra filosofierna med motupplysning och obskurantism. Det är i viktiga avseenden samma politiska skillnad som Fjällström urskilde, ehuru värderad på motsatt sätt. Västerländsk demokrati associeras med analytisk filosofi, de andra filosofierna med antivästliga eller åtminstone icke-västliga strömningar. Åtskillnaden börjar redan vid Engelska kanalen – västerut analytisk filosofi, österut kontinental.
Såväl för Hedenius och Marc-Wogau som för Häften för Kritiska Studier fanns föregångare att ta fasta på när denna dikotomi formulerades.
En tänkbar förebild för de analytiska filosoferna fanns i den så kallade Wienkretsens broschyr från 1929 – _Wissenschaftliche Weltauffassung – Der Wiener Kreis_ (»Vetenskaplig världsupfattning – Wienkretsen»). Den arbetade med en motsatsställning mellan vetenskaplig filosofi, i samtiden företrädd framför allt av Wienkretsen, och ovetenskaplig sådan, som inbegrep nästa all annan filosofi. En skillnad gentemot Hedenius och Marc-Wogau var att Wienkretsens företrädare hänförde marxismen (samt även psykoanalysen) till den vetenskapliga sidan av skiljelinjen. Det gjorde å andra sidan inte Karl Popper, som bland annat i verket _The Open Society and its Enemies_ brännmärkte olika typer av frihetsfientlig filosofi, bland dem sådan som stammade från Platon, Hegel eller Marx.
Den ursprungliga logiska positivismen hade haft sina centra i Wien och Berlin. Men efter andra världskriget kom gränsen mellan analytisk och icke-analytisk filosofi att gå vid Engelska kanalen. Vid det laget hade också USA i stor utsträckning vunnits för den analytiska filosofin. Fransk och tysk filosofi tenderade numera av de engelskspråkiga analytiska filosoferna (liksom av deras allierade i bland annat Skandinavien) att betraktas som ovetenskaplig gallimatias.
Från andra sidan staketet fanns också en tradition av demarkeringar. Tidigt gjordes en sådan av Georg Lukács i hans _Geschichte und Klassenbewusstsein_. Den marxistiske, starkt Hegelinspirerade ungerske filosofen drog där upp skiljelinjen mellan det »borgerliga» tänkandet, som byggde på reifierade, avhumaniserade, atomistiska begrepp, och det marxistiska, som sökte en helhetssyn, en totalitetskategori. Boken översattes 1968 till svenska som _Historia och klassmedvetande_ försedd med ett förord av Gunnar Gunnarsson (Bo Cavefors förlag). Gunnarsson skrev bland annat: »Den ungdom i vårt land, som 'upptäckt' marxismens möjligheter, kan i detta arbete [...] slipa sin tanke och förbereda sig för nästa stadium av det svenska samhällets utveckling: den socialistiska revolutionen.»
Lukács tankar plockades senare upp av den så kallade Frankfurtskolan. I en programförklaring förklarade skolans ledare, Max Horkheimer, skillnaden mellan »traditionell» och »kritisk» teori. Den traditionella teorin var borgerlig och flyttade över naturvetenskapernas vetenskapsbegrepp till samhällsvetenskaperna. Den kritiska teorin ville ha förändring, den betraktade den existerande världen utifrån ett kritiskt och socialistiskt perspektiv.
Efter andra världskriget kom den kritiska teorin att återetableras i Tyskland. Här utspelades också 1961 den så kallade »Positivismusstreit». Där framförde företrädare för »den kritiska teorin» som Theodor Adorno och Jürgen Habermas sin kritik mot »positivismen», vari de inbegrep all »borgerlig» och »konventionell» vetenskapssyn, inte minst Poppers.
Detta ger bakgrunden till hur Häften för Kritiska Studier nu och senare använde ord som »kritisk» och »positivism», en användning som var helt olik exempelvis Poppers. Bakgrunden ger också en nyckel till den diskussion kring Radnitzkys avhandling, som inleddes i tidskriftens första nummer.
Redaktionen började med en presentation: »Boken är till stor del en uppgörelse med den logiska empirismen som haft ett avgörande inflytande i Sverige, och för i stället fram en kontinental, dialektisk tankeriktning med centrum i frankfurtskolan.»
Den logiska empirismen i egentlig mening hade visserligen aldrig haft några företrädare i Sverige. Men redaktionen avsåg förmodligen att tala allmänt om den analytiska filosofin, inom vilken den logiska empirismen var en riktning (eller åtminstone hade varit några decennier tidigare). Dock handlade Radnitzkys bok inte alls om Sverige, den ende svenska forskare som togs upp var Radnitzkys handledare, Håkan Törnebohm.
I stället ställde avhandlingen mot varandra två skolor (eller grupper av skolor) inom vetenskapsteorin (»metascience»). Å ena sidan fanns »Anglo-saxon schools of metascience», å den andra »Continental schools of metascience». De förra identifierades med framför allt den logiska empirismen, de senare med en »dialektisk-hermeneutisk» approach. Radnitzky tog tydligt parti för den senare riktningen, men ville samtidigt skapa en dialog.
Diskussionen kring avhandlingen, vars betygssättning skapat kontrovers vid universitetet i Göteborg, inleddes i Häften för Kritiska Studier genom presenterande artiklar av Aant Elzinga, en kollega, och av Radnitzky själv.
Under rubriken »anglosaxisk kontra kontinental vetenskapssyn» skrev Elzinga om kapitalismens framväxt och om den vetenskapliga revolutionen. Via Bacon, Descartes, Newton och andra kom man fram till den moderna situationen. Den anglosaxiska vetenskapssynen likställs med »den logiska empirismen». Efterföljare till Bacon, Locke och Hume sägs fortfarande »bedriva sin häxjakt på 'metafysik'». Men särskiljandet av »vetenskap» och »metafysik» har framför allt en politisk innebörd, något som empiristerna dock inte är medvetna om, menade Elzinga. Deras uppfattning att vetenskap och värderingar, forskning och politik, bör hållas isär bedöms som ett borgerligt försvar för det bestående samhället. Yngre vetenskapsmän ifrågasatte emellertid nu »de empiristiska auktoriteternas åsikter om objektivitet» och fann därvid stöd i kontinentala vetenskapsteoretiska traditioner som »härstammade från Hegel och Marx».
Dessa formuleringar från Elzingas sida kompletterades av Radnitzky som framhöll att den logiska empirismens vetenskapsideal var monistiskt, reduktionistiskt och »fysikalistiskt». Från kontinentala utgångspunkter måste empirismens syn innebära ett förtingligande, en reifikation av det vetenskapliga arbetet. Radnitzky hade med sin avhandling velat motverka den logiska empirismens inflytande och göra den kontinentala vetenskapsfilosofin mera tillgänglig bland exempelvis svenska humanvetare. En praktisk konsekvens av det hela blev enligt Elzinga (med ett djärvt språng i tankegången) att man inte borde sända medicin utan vapen till det kämpande FNL. Radnitzky hade, hette det, genomfört ett »teoretiskt traditionsbrott».
En del av den KFML(r)-anslutne Elzingas synpunkter torde i verkligheten ha varit ganska främmande för Radnitzky. Deras gemensamma front bröts också senare när Elzinga förklarade att han egentligen inte delade Radnitzkys krav på vetenskapens frihet. Elzinga skrev: »Felet är inte att vetenskapen är ofri eller bunden. Det avgörande felet är för mig att så mycket av vetenskapen i dag är i fel händer. Den tjänar reaktionära i stället för progressiva krafter.»
För att komma tillrätta med detta missförhållande krävdes emellertid en proletär revolution, något som Elzinga villigt erkände.
Att det fanns olika linjer inom 68-vänstern framkom också med tydlighet när Göran Therborn i Häften för Kritiska Studier (1969:5–6) publicerade vad som i praktiken var ett generalangrepp på hela den kritiska teorin. Therborns utgångspunkter var i huvudsak althusserianska, med den tolkning av althusserianismen som florerade inom New Left Review och i Sverige inom tidskriften Zenit, där Therborn var något av chefsideolog.
Therborns polemiska vapen var knölpåken snarare än floretten. Hans huvudtes formulerades redan i rubriken för hans artikel, »Från revolutionär teori till akademisk metafysik. Till den kritiska teorins historia.» Redan den kritiska teorins utveckling från de ursprungliga lärofäderna till Habermas innebar en urartning, ansåg Therborn. I den svenska debatten hade denna urartning förts fram till sina yttersta konsekvenser. Radnitzkys position karaktäriserades av Therborn som »eklektisk labilitet och beskäftighet samt accentuerad politisk konservatism». Den kritiska teorin och dess upplösningstendenser hade för övrigt präglat ett flertal författare i Häften för Kritiska Studier, förutom Fjellström, Radnitzky och Elzinga även bland andra Joachim Israel. Hela debatten präglades enligt Therborn av ideologisk och teoretisk osäkerhet, vilken bidragit till det märkligt positiva mottagande inom vänstern för Radnitzkys svaga avhandling.
Två teoretiska tendenser inom 68-vänstern kan urskiljas. Häften för Kritiska Studier, som i kommande nummer och årgångar skulle fortsätta att bedriva »positivismkritik», uppfattade en »anglosaxisk» filosofi, som var nära förknippad med idéer om förnuft, vetenskap, teknik, upplysning, modernitet och framsteg. Mot denna stod en »kontinental» tradition, som inbegrep marxismen men också rymde andra riktningar, och där kritik mot den gängse vetenskapen och den förhärskande moderniteten var det centrala. Therborn och med honom många i kretsen runt Zenit tenderade att se marxismen som del av ett upplysnings- och modernitetsprojekt, där den borgerligt-demokratiska kapitalismen uppfattades som ett etappmål på vägen till det klasslösa samhället och där den moderna naturvetenskapen och tekniken godtogs som i grunden progressiva företeelser, vilka dock i framtiden måste ställas i arbetarklassens tjänst. För denna senare gruppering framstod många former av »kontinental» filosofi som minst lika belastad som den »anglosaxiska».
Therborn skulle senare (1976) ge ut en liten skrift, _Frankfurtskolan.Till kritiken av den kritiska teorin_, i vilken han fullföljde sin kritik mot Frankfurtskolans tolkning av marxismen. Therborns tolkningar skulle i sin tur bli utsatta för en kritisk granskning när en annan marxistisk sociolog, Alvin W. Gouldner, ägnade honom ett kritiskt appendix i sin bok _The Two Marxisms. Contradictions and Anomalies in the Development of Theory_ (1980), Gouldner ville visa just att det fanns två marxistiska traditioner, båda med rötter hos Marx och Engels, där den ena betonade marxismen som objektiv vetenskap, den andra marxismen som revolutionär kritik. Gouldner sympatiserade med den senare tolkningstraditionen och beskyllde Therborn och hela althusserianismen för att dogmatiskt ansluta sig till den förra. I Sverige kan väl dessa båda tendenser inom internationell marxism under årtiondet efter 1968 ha representerats av tidskrifterna Zenit respektive Häften för Kritiska Studier.
Meningsutbytena kring 1968 ger en sorts ingångsvärden för svensk filosofisk nyvänsterdebatt. En del av positionerna kom med tiden att bilda en sorts nyvänsterortodoxi i synen på vetenskapen. Det märktes tio år efteråt när boken _Humaniora på undantag?_ kom ut 1978 som resultatet av ett forskningsprojekt om humanistiska forskningstraditioner under ledning av idéhistorikern Sven-Eric Liedman, litteraturforskaren Kurt Aspelin och musikvetaren Jan Ling. I ett programmatiskt inriktat bidrag framhöll Liedman marxismens uppgifter samtidigt som han gav erkännade också åt andra teorier som ur marxistisk synpunkt kunde godkännas:
I kritiken av en socialt oansvarig, ytligt sett neutral vetenskap är marxismen emellertid inte ensam. Ensam och unik är den i sin strävan att göra kunskapen till ett revolutionärt vapen och i sitt påstående att en kunskap som verkar omvälvande ger en bättre bild av verkligheten än den som håller sig på försiktigt, neutralt avstånd. [...] Låt oss alltså inskärpa att ett levande marxistiskt alternativ i humanistisk forskning är beroende av att det finns revolutionärt sinnade humanister: marxistiska forskningsframgångar är inte nog. Strukturalist eller semiotiker eller hermeneutiker kan man bli vid seminarieborden, inte marxist.
Det fanns med andra ord två sorters kritik mot den tidigare etablerade, »borgerliga» vetenskapen. Liedman skrev: »Dominerande humanistisk forskning är [...] principellt teorilös.» Teori skulle till, men också revolutionär praxis. Teorilöshet och avsaknad av revolutionär praxis gick hand i hand för den dominerande vetenskapens del. Marxismens överlägsenhet bestod i att den – i motsats till en del andra i och för sig hedervärda teorier – förenade teorin med revolutionär aktivism.
Ytterligare några år senare finner vi Liedman i färd med att formulera en långsiktig strategi för vänstern inom universiteten (Zenit 1980:5). Han knöt an till 68-ledaren Rudi Dutschke och dennes tal om »den långa marschen genom institutionerna» och skrev:
Lite förenklat skulle jag vilja påstå att det bland dem som lät sig ryckas med under 60-talets slut går en gräns mellan dem som nöjde sig med manifestationer, proklamationer, protester, stora ord, ledarfigurer osv, och dem som strax påbörjade ett långvarigt, ofta motigt och plågsamt arbete att komma till större klarhet.
Bland de senare har en mindre grupp skaffat sig positioner i och genom intellektuellt arbete. Jag skulle vilja påstå att de också är nödvändiga både därför att det intellektuella arbetet måste gå vidare och för att det är en möjlighet att genomsyra t o m institutionerna med nya tankar.
År 1980 var den långa marschen mot professurerna i själva verket på gång sedan länge. Även så den långa marschen mot byrådirektörstjänsterna, skolledartjänsterna, programledaruppdragen, sakkunniguppdragen, biskopsämbetena och så vidare. De ideologiska statsapparaterna i Sverige var redan på god väg att penetreras av 68-vänsterns folk och »nya tankar». Den nya filosofin skulle på så vis komma att genomsyra hela samhället, oftast i uttunnad men ändå fullt igenkännbar form.
## **2. Ett mål fördolt**
Marxismen öppnade porten för andra »teorier» och »metoder», bland dem de av Liedman nämnda, strukturalism, semiotik och hermeneutik. Kravet på »teori och metod» blev efterhand allt starkare och mer etablerat inom human- och samhällsvetenskaperna. Därvid gjorde sig en intressant förskjutning snart gällande. Från början var tanken att »teorier» inom dessa fält liksom inom naturvetenskaperna skulle förklara fakta och gjorda observationer, testas mot dessa, verifieras eller falsifieras av dessa. Med tiden uppstod emellertid en sorts teoriernas och metodernas primat. Om fakta och observationer inte överensstämde med den valda teorin och metoden ansågs detta närmast diskvalificera »fakta» och »observationer», inte teorin eller metoden. Förhållandet till verkligheten sågs alltmera så att »verkligheten» betraktades som en social konstuktion. Teorin gav upphov till verkligheten, inte verkligheten till teorin. Namn som »postmodernism», »poststrukturalism», »postmarxism», »social konstruktivism» med flera användes för att referera till detta nya synsätt som alltmera blev det dominerande inom dessa vetenskaper. Det kunde lätt kopplas till idén om det politiska engagemanget som avgörande inom vetenskapen. Tanken på »filosofi från en revolutionär klasståndpunkt» kunde vidgas till att handla om exempelvis »historia underifrån» eller »genusvetenskap». Inom »genusvetenskapen» betraktades könet som en social konstruktion och vetenskapen sågs som omedelbart förknippad med feminism, »kvinnokamp» eller offentligt »jämställdhetsarbete». På motsvarande sätt utgick forskningen om etnicitet och invandring från den politiska aktivismens eller den offentliga politikens målsättningar och konstruerade utfrån dessa sina objekt. De gamla föreställningarna om vetenskapens »objektivitet», »opartiskhet» och politiskt-ideologiska neutralitet avfärdades med förakt. Detta blev 68-vänsterns kanske mest betydelsefulla långsiktiga insats för att förändra synen på human-, samhälls- och religionsvetenskapen och på sättet att bedriva dessa vetenskaper.
Övergången från marxism till »postmarxism» ägde framför allt rum under 1970-talet. Det är under detta årtionde som vänstern tappade tron på de förverkligade utopierna (Kina, Vietnam med flera), på utopierna som sådana, på socialismen som alternativ och kommande verklighet. Därmed går själva tron på framtiden förlorad och med den tron på historien och förnuftet. Marxismens optimistiska schema, där historien förverkligar en förnuftig framtid, ersätts av pessimism. Föreställningen om ett framsteg som kontinuerligt för mänskligheten närmare målet med teknik, vetenskap och produktion som drivkrafter, ersätts av ett fördömande av hela civilisationen och dess verk. Det nya avståndstagandet från vetenskapen och den moderna civilisationen som denna vetenskap bidragit till att skapa, transformerar naturligt nog hur man uppfattar den »vetenskap» man själv bedriver. Human- och samhällsvetenskaperna ersätts av »civilisationskritik». Ibland går detta under namnet »cultural studies». Ibland kallas det i USA »French Theory». Kärnpunkter är en strävan efter en postmarxistisk vänsterposition som kan tjäna som fana för dem som företar den långa marschen genom institutionerna, en marsch vars mål, bortsett från målet att nå egna positioner, efterhand blir alltmera fördolt även för de marscherande.
En centralfigur också i Sverige i det nya skedet var Michel Foucault. Som ingen annan filosof representerade han 1970-talets mentala läge och intellektuella utveckling. Vill vi se honom som navet kan vi räkna ekrarna och iaktta hur de löper åt olika håll. Det bör framhållas att Foucaults verk i stor utsträckning översattes till svenska under 1970- och 1980-talen. _Vansinnets historia under den klassiska epoken_ kom på svenska 1973, _Vetandets arkeologi_ 1972, _Sexualitetens historia 1–3_ 1980, 1986, _Övervakning och straff_ 1987. Dessförinnan hade verken översatts till engelska _–_ mottagande av Foucault i USA var en förutsättning för hans påverkan i Sverige.
Låt oss dela upp inflytandet från Foucault på ett antal punkter:
1. Den första boken, _Vansinnets historia_ , tolkades ofta som del av den »antipsykiatri» som florerade inom vänstern och som kritiserade psykiatri och mentalvård, vilka betraktades som led i ett kapitalistiskt förtryck.
2. Såväl _Vansinnets historia_ som i synnerhet den senare _Naissance de la clinique_ kunde inordnas i en motsvarande antimedicinsk riktning som kritiserade medicin och sjukvård som led i ett klassförtryck.
3. Men i båda de nämnda fallen fanns en svävning. Var det enbart mot kapitalism och borgarvälde som kritiken riktades? Eller i lika hög grad mot vetenskapen, mot tekniken, mot det moderna samhället, mot Västerlandet, mot den för detta Västerland utmärkande förnufts- och upplysningstraditionen? Var det fråga om en »reaktionär» motupplysning och handlade det i så fall verkligen om en kritik från »vänster»? Eller borde man anse att kritiken mot modernitet och upplysning framför allt var en angelägenhet för vänstern? Hade den »progressiva», framstegsvänliga vänstern efterträtts av en utvecklingsfientlig? Var Foucault en av denna utvecklingsfientliga vänsters profeter? Eller var kanske hela idén om »vänster» och »höger» föråldrad och oanvändbar?
4. Frågorna är desto mer relevanta som Foucault blev en av de viktiga krafterna bakom en ny motupplysning. Den historiska berättelse han berättade handlade om en ständigt mörknande himmel, en ständigt tilltagande disciplinering, en Makt som ständigt växte och i vars hand kunskap och vetenskap var vapen. Denna Makt hade hos Foucault närmast metafysiska dimensioner. Den var en Världsfurste som överallt var närvarande, som kunde identifieras med kapitalismen, men som kanske snarare var moderniteten själv, som sammanföll med historien på ett sätt som inte verkade tillåta någon möjlighet till befrielse. Likheterna med Nietzsche eller Heidegger var mer uppenbara än likheterna med Marx eller Mao, åtminstone hos den senare Foucault. Ett kort ögonblick framskymtade den iranska revolutionen som en möjlig frälsningsväg. Men den stängdes snart. Inte ens ayatollorna kunde besegra Makten.
5. Det var det genialiska hos Foucault att han kunde få dessa gnostiska fantasier, detta kompromisslösa angrepp på vetenskap och modernitet, att själva framstå som exempel på modern vetenskap. En viktig anledning till detta, vid sidan av fascinationen hos en betvingande retorik, var ett tidsläge. Här fanns en vänster på jakt efter ett projekt. Det gamla projektet, den socialistiska revolutionen, övertygade inte längre någon när 1970-talet gick mot sitt slut och 1980-talet började. Ljusen hade slocknat. Världsfursten tycktes ha tagit över. Foucault formulerade åtminstone protesten med en vältalighet som glödde av mörk intensitet.
6. Foucaults betydelse för de så kallade nya filosoferna var uppenbar. Att Makten var tillvaron, att Fursten var världen, att Gulag brett ut sig över hela jordens yta, det stod klart för dessa tänkare som förlorat tron på marxism och maoism. Också hos dem likställdes kunskap med makt, i synnerhet i kritiken av Mästertänkarna, bland vilka Marx bara var en.
7. Men Foucault gjorde gemensam sak också med den så kallade Annalesskolans tredje generation. Deras läromästare, Fernand Braudel, hade en gång, efter Frankrikes nederlag 1940, vänt sig bort från »händelsehistorien»:
»Att vägra ta befattning med händelserna och med händelsernas tid, det var att placera sig i marginalen, att söka skydd för att betrakta dessa händelser på en smula avstånd, för att bedöma dem bättre och inte tro alltför mycket på dem.»
I besvikelsen över världshändelser som gått fel bredde liknande stämningar ut sig på 1970-talet. Man sökte tröst hos en »orörlig historia», hos den franska medeltidens magi, hos Annaleshistorikernas rika gobelänger, fyllda av figurer från ett försvunnet bondesamhälle. Samtidigt såg man överallt en utveckling som förtrampade friheten, som trängde ut alla protester i marginalen av en ständigt fortskridande normalisering.
8. Sexualitetens historia sådan Foucault framställde den var likaledes eminent brukbar. Som en väldig Prideparad, gestaltad av en man, drog denna berättelse fram. Återigen var protesten mot »normaliseringen» det centrala. Men samma berättelse kunde tvångslöst utnyttjas av feminister. Gay Lib protesterade mot en normalisering som likställde det normala med det heterosexualla, feministerna mot en normalisering som likställde det normala med det manliga.
9. Vänstern kunde använda sig av både feminismen och gayrörelsen. Men det hela kunde också ges en mer liberal utformning. Från Foucault kom incitamentet till en ny kritik mot välfärdsstaten, mot (för svensk del) folkhemmet, mot planekonomi och planering av alla slag. Det var en kritik som vann genklang både till höger och till vänster. Eftersom samhällsvetenskaperna var nära förknippade med socialstatens framväxt i de flesta länder kom kritiken mot kunskapen/makten till pass även här.
10. En aspekt av kritiken mot den moderna staten som kom i fokus genom boken _Övervakning och straff_ var kritiken mot rättsväsendet och fängelserna. Politiskt svarade detta mot organisationer som KRUM (Riksförbundet för kriminalvårdens humanisering) i Sverige (Foucault var själv engagerad i motsvarande rörelser i Frankrike) samt mot idealiseringen av kriminalitet som en form av »motstånd». Foucaults berömda kritik mot Jeremy Benthams »panoptikon» hade nära samband med synen på straffsystemet som en del av ett allmänt förtryck.
11. Den sene Foucaults kritik mot själva idén om en möjlig »befrielse» stämde väl med en desillusionerad vänster som givit upp hoppet om utopin, om revolutionen, om det radikala Alternativet. Med marxismen föll för vänsterns del det mesta av sådana förhoppningar bort. Foucaults pessimism (som för övrigt låg i linje med antiutopiska inslag hos Lévi-Strauss, Althusser och Derrida) passade denna nya situation.
12. Foucaults kritik mot det västerländska förnuftet gick väl ihop med en tiermondism, en tredjevärldenromantik, som till en början hade maoismens form för att sedan för ett ögonblick ta sig uttryck i entusiasm för Khomeinis islamska revolution. Edward Said och andra som konstruerade det »postkoloniala» betraktelsesättet var starkt inspirerade av Foucault i detta avseende, liksom i sin kritik mot ett västerländskt förnuft och en västerländsk diskurs.
13. Vändningen inom vänstern bort från Marx och till Nietzsche och Heidegger som filosofiska inspirationskällor främjades i hög grad av Foucault, som blev en av »vänsternietzscheanismens» arkitekter.
14. Filosofiskt spelade Foucaults kritik mot »humanismen» också en viktig roll. När humanismen och tron på Människan avslöjades som en västerländsk maktideologi var det inte bara veganismen som drog fördel av detta. Hela subjektsfilosofin – till och med Sartres – kom att betraktas som del av en kolonial maktutövning. Detsamma gällde historismen, nu föremål för ett av Foucault och andra poststrukturalister formulerat förakt, med nietzscheanska och heideggerianska förtecken.
15. Foucaults neognosticism kan även sägas ha förebådat en nykommunistisk historien-är-slut-filosofi av den typ som senare företräddes av Agamben, Badiou, Žižek, Negri eller Hardt. Foucaults inflytande tog inte slut med hans död 1984 utan det har i olika former framför allt inom vänstern, inte minst den svenska vänstern, varit starkt alltsedan dess.
I en ofta citerade passage ur _Vetandets arkeologi_ skrev Foucault:
Det finns säkert fler än jag som skriver för att inte längre ha ett ansikte. Fråga mig inte vem jag är och säg inte åt mig att förbli den jag är: det är en mantalsskrivningsmoral, det är den som bestämmer över våra identitetshandlingar. Men den skall lämna oss i fred när det gäller att skriva.
Just denna Proteusartade karaktär kom att spela en väsentlig roll för Foucaults inflytande. Politiskt engagerade han sig för Maos kulturrevolution och den mest långtgående maoismen, för Khomeinis islamism liksom för hjälpen till båtflyktingarna från Vietnam eller protesterna mot den franska vänsterregeringens försummelse att uttala sig mot Jaruzelskis kupp i Polen 1981. För vänstern i det läge man befann sig efter Maos död måste en sådan figur vara tilltalande. Marxismen, kommunismen, den socialistiska utopin hade förlorat sin trovärdighet. Vänstern sökte efter något nytt. Att Foucault också på vissa punkter kunde tilltala liberaler och konservativa behövde inte minska hans användbarhet ur vänstersynpunkt. Att forma allianser i enskilda frågor var tvärtom en del av en ny vänsterstrategi som Foucault varit med om att formulera. De stora revolutionerna skulle ersättas av små revolter liksom marxismen stora berättelse skulle ersättas av små berättelsefragment, som likväl skulle kunna mobilisera. Det var inte längre fråga om en lång marsch mot en slutgiltig triumf utan om plötsliga protestaktioner som likt uppflammande tändstickor skulle kunna ge ljus i ett tilltagande världsmörker. Eller så kunde det hela tolkas mer reformistiskt som en väg mot partiella förbättringar utan storslagna anspråk.
Ett gott exempel på hur Foucault mottogs i Sverige kan hämtas ur socialvetaren Sune Sunessons förord till en svensk nyutgåva 1983 av _Vansinnets historia under den klassiska epoken_. Foucault i svensk översättning hade först kommit på Cavefors förlag, men tagits över av ett annat vänsterförlag, Arkiv, där Sunesson var engagerad och som stod tidskriften Zenit nära.
Sunesson parallellställer _Vansinnets historia_ med böcker av Erving Goffman, Thomas Szasz och Ronald Laing. Alla hänförs de till en tradition av antipsykiatri med ett synsätt »som befästs i de nya psykiatriformer som betonar att mentalvården skall bedrivas utanför sjukhusen, utan inspärrning. I stället för den stora interneringen kommer nu psykoterapi och olika psykodynamiska behandlingsformer».
Om _Vansinnets historia_ heter det även att boken blev »inledningen till det tjugoåriga angrepp mot vetenskap, institutioner, konventionellt förnuft och makt, som Michel Foucault har drivit».
Sunesson går sedan igenom Foucaults utveckling under betonande av vänsterhållningen. Det framhålls att Foucault avslutade en tidig bok om psykiatri »med en uppskattande del om Pavlov och den sovjetiska psykologin». Samtidigt poängteras Foucaults uppbrott »från den partitrogna kommunismen». Foucaults en smula yrvakna reaktioner på maj 68 skildras liksom hans försök att senare hinna ifatt. Likheterna mellan Foucault och »de nya filosoferna» avfärdas som obetydliga. Foucault förs emellertid till vänsterlägret mindre som systembyggare än som »den alltid misstänksamme undermineraren».
I förordet till Foucaults _Övervakning och straff_ för Arkivs utgåva 1987 brottas Sunesson likaledes med frågan om Foucaults förhållande till vänstern. Det gäller att rädda Foucault från den »högerbur» där han ibland placerats. Han var nietzschean, ja, men det handlade om »ett slags 'vänsternietzscheanism'».
Sunessons ibland plågade, ibland entusiastiska kommentarer illustrerar hur den svenska vänstern kämpade med Foucault. Den franske filosofen erbjöd en räddning när marxismen tonat bort. Också en annan vänsterman från samma generation, Bengt Olle Bengtsson, hyllade Foucault, i detta fall i förordet till Arkivs nyutgåva av _Vetandets arkeologi_ (2002). Böcker av Foucault låg alltid framme hemma i förordsförfattarens vänsterpartistiska hem, berättade han. Han var visserligen marxist, men bekände att _Kapitalet_ normalt inte låg på hans nattduksbord. Foucault blev på detta sätt en räddning för många vars marxism börjat svalna. Men en flertydig räddning. Var detta en eldstod eller bara ett irrbloss? Det förlovade landet vägrade på något vis att uppenbara sig, åren i öknen blev många och långa.
Skillnaderna mellan Foucault och Marx var också påfallande.
Hos Foucault fanns ingen framstegstro, ingen tro på historien, ingen tro på förnuftet eller vetenskapen, tvärtom, ingen föreställning om att den moderna utvecklingen med nödvändighet (eller ens sannolikhet) skulle kunna föra fram till ett lyckligt slut, ingen optimism. Men dessa nackdelar var också fördelar. Foucault avfärdade under sina senare år Marx och Lenin. Men han anammade alltid Nietzsche och Heidegger. Varför var dessa tänkare, som tidigare dömts ut som högertänkare, nu mer gångbara än vänsterns klassiker? Kanske helt enkelt därför att de lät bli att tala om den socialistiska revolutionen och det kommande klasslösa samhället, som numera få ens inom vänstern trodde på. Kanske därför att de nöjde sig med en svartsyn, en pessimism beträffade moderniteten och hela den västerländska utvecklingen, en pessimism som vänstern numera delade. Kanske därför att Heidegger, Carl Schmitt och andra sedan Hitlerrevolutionen misslyckats befann sig en smula i samma läge som vänstern efter det att den kommunistiska revolutionen misslyckats.
Foucault var förvisso subversiv. Men han trodde inte att undergrävandet av det bestående skulle kunna leda till befrielse. Han var en mullvad som – i likhet med naturens mullvadar – saknade hopp om att få skåda ljuset utan nöjde sig med att förstöra gräsmattorna. I första delen av sin _Sexualitetens historia_ gjorde han detta klart genom ett angrepp mot freudomarxismen, alltså mot den ideologi som i exempelvis Marcuses utformning en gång hade varit så karaktäristisk för 68-vänstern. Makten tycktes nu finnas överallt, varje försök att befria sig från den måste stärka den. Vi måste inse »att makten kommer underifrån: det vill säga att det inte finns någon binär och genomgående motsättning mellan de härskande och dem de härskar över såsom grundläggande allmänt mönster för relationerna».
Detta synsätt lämnade öppningar för mindre vänsterprojekt, för revolter, för protester, för aktioner i enskilda frågor. Men den stora drömmen om »befrielsen» fanns inte längre. När teleologin försvann, försvann också tron på historien och med den tron på moderniteten, på förnuftet, på vetenskapen, på framtiden. Det emancipatoriska förnuftet ersattes i vänsterns världsuppfattning av ett förtryckande förnuft.
För svensk del kunde man förvåna sig över de hyllningar från den unga vänstergenerationen som kom den i flera avseenden högst borgerlige finländske filosofen Georg Henrik von Wright till del. När han förkunnade mänsklighetens och den tekniskt-vetenskapliga västliga civilisationens nära förestående undergång lyssnade den unga vänstern med andakt. För sin del åberopade von Wright gärna Oswald Spengler, en tänkare som förmodligen få av de unga hade läst och som politiskt varit tämligen fast förankrad inom den tyska högern. Men detaljerna spelade mindre roll. Det var det allmänna civilisationskritiska och civilisationspessimistiska perspektivet som fängslade. På nytt väntade man – och hoppades kanske – på Västerlandets undergång. Om den skulle komma att äga rum genom kärnvapenkrig, genom överbefolkning, genom klimatets nedkylning till en ny istid eller uppvärmning till en ny syndaflod spelade mindre roll.
Men det fanns också en möjlighet att svepa undergångsvisionerna åt sidan och att i stället för en ände med förskräckelse tänka sig en förskräckelse utan ände, en långsam apokalyps.
Det hade varit möjligt att lyfta fram någon annan tänkare än Foucault som exempel på 1970-talets utveckling från marxism till postmarxism (poststrukturalism, postmodernism). Derrida, Baudrillard, Lyotard eller Deleuze kunde ha framhållits. Men i just de specifika avseenden som vi här velat belysa skulle bilden ha blivit snarlik. Även i de nämnda fallen rörde det sig om tänkare som var tydligt positionerade inom den franska vänstern men som undvek marxismen, med dess framstegs-, förnuftsoch vetenskapstro och med dess förankring i en bestämd syn på historiens gång och på ett kommande klasslöst samhälle. Även i de nämnda fallen handlade det om filosofer som mera inspirerades av Nietzsche eller Heidegger än av Marx, men som inte för den skull försummade vänsterrevolutionär gestikulation vid olika tillfällen. I Sverige torde de för övrigt ännu tydligare än i sitt hemland ha uppfattats som hemmahörande i vänsterlägret. Deras inflytande på kulturdebatt samt på human- och samhällsvetenskaper blev också i vårt land utomordentligt stort.
De ståndpunkter som vid 1970-talets slut och 1980-talets början intogs av de nämnda författarna fortsatte att vara aktuella i svensk debatt under de följande årtiondena. Nyutgåvorna av Foucaults till svenska översatta böcker har nämnts liksom Sunessons och Bengtssons förord. Man kan också nämna publikationer som den av Kenneth Hultqvist och Kenneth Petersson ederade _Foucault. Namnet på en modern vetenskaplig och filosofisk problematik. Texter om maktens mentaliteter, pedagogik, psykologi,medicinsk sociologi, feminism och bio-politik_ (1995) samt Roddy Nilssons _Foucault – en introduktion_ (2008).
Under senare år kan man kanske urskilja några olika attityder till det marxistiska filosofiska arvet. Där finns de som betraktar marxismen med en sorts nostalgi. Som exempel kan nämnas litteraturkritikern Stefan Jonsson i ett inlägg »Till frågan om marxismen». Han skrev där bland annat:
Vare sig vi kallar dagens tillstånd för senkapitalism, globalisering, postmodernitet eller postfordism så är en sak klar: vi saknar en slutgiltig teoretisk beskrivning av det, av det slag marxismen tillhandahöll en gång för sin tids kapitalism. Vi anar heller inte vilket politiskt subjekt som skulle kunna förändra situationen och organisera människans arbete mer förnuftigt. Det ser alltså ut som om dagens generationer av systemkritiska rörelser saknar vad marxismen representerade för gårdagens generationer.
Här finns en längtan efter en förlorad visshet, teoretisk och praktisk, som 68-generationen tycktes ha haft och som marxismen representerade på ett filosofiskt plan. Jonsson skriver också:
Jag är för ung för att ha kunnat tillhöra 1968 års nya vänster, men just därför tycker jag mig också kunna se nyktert att den i grova drag förstod världen rätt. Marxismen är för mig ett kunskapsteoretiskt och etiskt val. Med marxismens glasögon på näsan ter sig världen mer begriplig och meningsfull, men i samma stund märker man hur fasansfull och orättvis den är – eländig och vanställd skrev Adorno – och det framstår som ansvarslöst, ja, hänsynslöst att inte viga sig åt en annan.
I Jonssons text ser man på detta vis olika tankemotiv brottas. Marxismen tillhör det förgångna, men är ändå ett nödvändigt val i nutiden. Det går inte att se något subjekt som kunde genomföra revolutionen, men man måste ända tro på denna revolution eftersom den existerande världen är outhärdlig, kanske så outhärdlig att varje alternativ värld är bättre. Jonsson anför tidigare i texten med gillande olika »postkoloniala» tänkare som filosofiskt sett är allt annat än marxister. Men uppenbarligen är det för honom den etiska dimensionen som är avgörande. Marxismen är ett »etiskt» val eftersom världen är ruskig. Den är enligt Jonsson också ett »kunskapsteoretiskt» val, men detta senare är en uppfattning som inte alls motiveras i hans text. Ganska uppenbart finns hos honom en sorts viljans primat. Tänkare som Michael Hardt, Antonio Negri, Giorgio Agamben och Slavoj Žižek med deras bagage från Heidegger och Carl Schmitt är det som gäller nu, det är Jonsson fullt medveten om, skolad i den moderna litteraturkritiken och litteraturforskningen som han är. Men en känsla oroar honom att Marx ändå var rejälare.
Hårdare i greppen, mindre subtil än Jonsson, är Göran Fredriksson när han i ett sent nummer (200/201, 2012) av Häften för Kritiska Studier, där han är redaktör, gör upp med den postmarxistiska vänsterns utveckling. »Det börjar bli tröttsamt med den postmoderna lanseringen av högerradikala tänkare som bot för en vilsen vänster», skriver Fredriksson med anledning av att efter Nietzsche och Heidegger nu även Carl Schmitt hyllats från svenskt vänsterhåll (närmast av idéhistorikern Anders Burman i Svenska Dagbladet 20/12 2011). Fredriksson avslutade sitt inlägg med följande tirad: »Varje anhängare av frihet, jämlikhet och humanitet, som vill förstå nyliberalismens avpolitiserande framfart och radikalkonservatismens existentiella lockelser, bör ta sitt förnuft till fånga och tacka nej till hela den postdiskursiva raden av kittlande erbjudanden.»
Vid Fredrikssons sida uppträdde den honom ideologiskt närstående David Brolin med en vidräkning med den svenska vänsterns omfamnande av Nietzsche, vilken enligt Brolin inte låter sig tvångslöst vänsteriseras på det sätt som under senare år blivit vanligt.
På senare tid har dock vänsterns kamp mot »globaliseringen» och »Mc World» mattats. Här har vänstern konkurrerats ut av högerpopulistiska rörelser som med en helt annan kraft och förankring fört denna strid. Även EU-motståndet har mattats när andra krafter tagit över.
Marxismen innehöll från början möjligheten till två filosofiska och praktiska allianser. Antingen med den framstegsvänliga borgerligheten. Den alliansen hörde ihop med bejakande av upplysning och förnuft, av vetenskap och teknik, av ekonomisk tillväxt, av den västerländska civilisationstypen. Den betydde att den kommande socialistiska revolutionen betraktades som en fortsättning på den demokratiskt-borgerliga vari denna på en gång upphävdes och bevarades. Den andra alliansmöjligheten innebar att gå i förbund med alla som var fientligt inställda till kapitalismen, till borgerligheten, till moderniteten i den utformning den fått i Västerlandet. Denna senare allians förutsatte förnekande av allt som den förra förde med sig, upplysning, förnuft, vetenskap, teknik, tillväxt, borgerlig demokrati, modernitet. Allianser, vare sig nu teoretiska eller praktiska, bestäms alltid i någon mån av motståndarens styrka. Allt eftersom den västerländska moderniteten under åren efter 1989 tycktes uppnå en alltmera fullständig global seger blev det för vänstern naturligt att gå i allians med en kritik som kom från »höger». Där Engels en gång i tiden hade hämtat uppslag från Carlyle hämtade den moderna vänstern sina idéer från Nietzsche, Heidegger, Jünger och Carl Schmitt och allierade sig i »postkolonial» anda med Usama bin Ladin eller Hamas.
På sistone har situationen emellertid förändrats. USA är inte längre som för några år sedan den helt dominerande supermakten. Tendensen mot globalisering har brutits. Vänstern söker nya allianser. Kanske söker den till och med plats i en liberal konsensus.
Alldeles tillfredsställande kan detta dock aldrig bli. Vänstern behöver ett eget projekt.
Sökandet efter det revolutionära projektet och det revolutionära subjektet fortsätter därför. Marxismen i sin klassiska form tillhandahöll på sin tid bådadera (socialismen/kommunismen respektive proletariatet/världsproletariatet). Därför kommer nostalgin för marxism (och kommunism) att bestå. Men så länge ett nytt revolutionärt tillfälle inte gör entré befinner sig vänstern med Ehnmarks ord såväl praktiskt som filosofiskt i fortsatt disponibilitet.
## **XI. 1968 I HISTORIENS LJUS**
VAD UTMÄRKTE DÅ 68 ÅRS omvälvning? Vilka var de mest karakteristiska dragen? Frågorna har varit föremål för en omfattande debatt, både bland svenska och utländska forskare. Här skall följande aspekter framhållas.
Först och främst var 68-vänstern _tidsmässigt_ en kortvarig men kraftfull stormvind. _Rumsmässigt_ hade den sin upptakt i det existerande samhällets och kulturens utkanter, men svepte trots detta snabbt in i dess centrum. Även om stormen blåste förbi efter något årtionde skulle dess följdverkningar visa sig utbredda och långvariga. Det förhållandet att idéer som ena ögonblicket avvisas av en bred allmänhet på relativt kort tid kan vinna offentlig accept är en av 68-upprorets både viktigaste och mest oroande lärdomar – ett memento att beakta när i dag extrema idéer på nytt vinner insteg i det politiska livet. Det kan också hävdas att rörelsens ideologiska utveckling rymde två olika steg. Det inledande kan kallas för _förtätning/koncentration_ och det senare för _förtunning/utbredning_. När 68-vänstern slog igenom chockade den offentligheten: dess idéer var extrema, dess företrädare var ofta exalterade och bombastiska, dess revolutionära smågrupper var exklusiva, auktoritära och sekteristiska. 68-upproret både attraherade och stötte bort genom sin radikalism. I den efterföljande fasen, när rörelsen växte bortom sekternas kontroll och det politiska engagemanget ackompanjerades av en mer individualistisk livsstil, mattades radikalismen av och blev mer samhällstillvänd, vilket i sin tur gjorde många av 68 års vänsteridéer, som jämställdhet, decentralisering och miljötänkande, till allmän egendom.
Ett andra utmärkande drag vad gäller 68-vänstern var att den anknöt till och tog utgångspunkt i en radikal marxistisk och kommunistisk tradition, samtidigt som den kom att skilja sig från denna i flera avseenden: dels såg sig 1968 års vänster till en början som progressiv i upplysningens mening, men den gav samtidigt och i tilltagande grad uttryck för en kritik av det moderna industrisamhället och banade i detta avseende väg för den postmodernistiska revolten mot modernismen; dels kom 68-vänstern att fokusera på ideologi, medvetenhet och kultur, där den klassiska marxistiska vänstern betonat samhällets ekonomiska och sociala sidor. Vidare lyfte 68-vänstern och särskilt då maoismen fram studenterna, ungdomen och folken i de fattiga länderna som den revolutionära kampens viktigaste drivkraft, där Marx tidigare utpekat arbetarklassen som historiens avgörande aktör. Slutligen hade 68-vänstern framför allt fokus på imperialismen och tredje världen, där kommunismen traditionellt varit koncentrerad på förhållandena i det avancerade Europa.
68-vänstern lät sig följaktligen inspireras av den kommunistiska ideologin, samtidigt som den i desillusion vände sig mot främst Sovjetunionen för att ha svikit den (och i stället svärmade för Kina och Kuba). Det är viktigt att notera att desillusionen inget hade att göra med förtrycket i de socialistiska länderna. 68-vänstern gjorde aldrig någon affär av att »arbetarklassens fiender» förföljdes och avrättades i Sovjetunionen, Kina, på Kuba och i andra socialistiska samhällen. Mänskliga rättigheter dömde den ut som »borgerliga» och »formella».
68-vänstern hade förmodligen aldrig kunnat nå det inflytande den fick om den inskränkts till ett traditionellt kommunistiskt angrepp på det borgerliga samhället. Men genom att på en och samma gång distansera sig från den byråkratiserade och förstelnade Sovjetkommunismen och bli plattform för en rad radikala tendenser i 1960-talets samhälle – som en ny musikkultur, den sexuella frigörelsen, det växande fredsoch miljöintresset och ett nymornat svärmeri för tredje världen – fick 68-vänstern ett långt större genomslag än vad som annars varit möjligt. Den blev ett kraftcentrum – inte som centraliserat parti, men som en vittförgrenad rörelse, med starka inslag av toppstyrda och rivaliserande sektpartier – för idéer och krav som i många fall hade sitt upphov bland liberaler, socialdemokrater och borgerliga kulturradikaler, som exempelvis kraven på fri abort, ökad u-hjälp, radikal nedrustning, en jämlik skola och ett nytt formspråk inom teater, film och litteratur.
För det tredje utmärktes 68-vänstern av två olika men inbördes förbundna sidor. Den ena sidan utgjordes av en extrem och upprorisk _politisk ideologi_. Den andra sidan bestod av provokativa _kulturellt-estetiska_ uttrycksformer eller en kognitiv respektive emotionell aspekt om man så vill. I det förra fallet handlade det om ett politiskt engagemang i de kommunistiska idéernas tecken, men med en förnyad marxistisk begreppsvärld influerad av samtida kontinental filosofi för att tolka nuet och det förflutna. I det senare fallet rörde det sig om ett expressivt sökande efter en ny livsstil, nya uttrycks- och umgängesformer och en dröm om att både uppleva individuell frihet och ingå i en själslig/ideologisk gemenskap. Medan 68:ornas politiska engagemang efter något årtionde ebbade ut (även om de centrala idéerna fördes vidare i delvis annan skepnad) slog sextiotalets nya värde- och livsorientering i fråga om klädstil, barnuppfostran och äktenskap, i fråga om musik, konst och litteratur, igenom på bred front. Under 68-vänsterns »grundargeneration» smälte de båda sidorna samman på ett helt naturligt sätt. Ett illustrativt exempel är det uppmärksammade Tältprojektet från 1970-talets början, en kulturrevolutionär politisk teaterform vars ledande medlemmar var maoistiska kommunister.
Den fjärde och sista aspekten som kännetecknade 68 års omvälvning, eller kanske snarare dess efterverkningar, avser dess inflytande på samhället. Detta inflytande var både _utbrett_ och _begränsat_. Det har varit utbrett inom kultur, akademier och media, men samtidigt ytterst begränsat inom samhällsekonomin och ekonomisk politik, där en marknadsliberal omvälvning slog igenom från 1980-talets slut. 68-vänstern gav alltså upphov till en inflytelserik motreaktion, nämligen det som brukar kallas nyliberalism. Denna hade sitt ursprung i en kritik av keynesianismen och fick inflytande till följd av att den keynesianska politiken visade sig helt oförmögen att finna ett svar på 1970-talets djupa ekonomiska kris. Nyliberalismen var i grunden en ekonomisk teori, men den var samtidigt förankrad i ett klassiskt försvar för individuell frihet och privat äganderätt. Idéutvecklingen sedan 1960-talets slut har sett en ofta intensiv strid mellan liberala och vänsterorienterade idéer. I mer än många andra länder har arvet efter 68-vänstern i Sverige haft ett dominerande inflytande i synen på vår historia och kultur, i perspektivet på jämlikhet, kön, familj och sexualitet, men också på uppfattningen om nationen, etnicitet och mångfald. De områden där borgerliga och liberala idéer däremot i allt väsentligt hållit ställningarna rör exempelvis synen på marknadsekonomi, konkurrens, vinst, effektivitet, enskilt ägande, kontraktsrätt, entreprenörskap, frihandel, demokrati, rättsstat, lagstyre och civilsamhälle. Det svenska samhället är med andra ord märkligt delat mellan en utbredd accept av marknadsekonomin eller kapitalismen som ekonomiskt system och en kultur- och mediasfär i vilken idéarvet efter 68-vänstern har ett markant övertag. Detta är desto märkligare som det före socialismens fall var just kapitalismen och den privata äganderätten som stod i fokus för konflikten mellan vänster och höger, medan kulturområdet i bred mening låg tryggt förankrat i borgerliga händer, i varje fall till långt in på 1950-talet.
I vilket historiskt ljus eller perspektiv är det då rimligt att inplacera revolten 1968? Flera historiker har jämfört den med tidigare eruptioner i Europa, som 1789, 1848, 1917 och 1956. Mot en sådan jämförelse mellan 68 och »långa vågor» av återkommande uppror finns det flera invändningar. För det första inträffade 68 i västvärldens stabila demokratier och i samhällen med för sin tid goda sociala villkor. Aktivisterna kom för det andra inte från någon förtryckt grupp som gjorde uppror av nöd och armod, snarare var det studerande från medelklassen som lockades av de röda fanorna. Slutligen var 68 i motsats till de andra stora händelserna ett i allt väsentligt fredligt upptåg, även om det kunde ha spårat ur utan aktiva insatser från ordningsmakt och domstolar.
Andra har sett 68-åren som en övergångsperiod, antingen dess inledning eller dess avslutning, till en ny typ av samhälle eller en ny fas i en lång historisk moderniseringsprocess – den franske filosofen Lyotard myntade begreppet post-modernism, den spanske samhällsvetaren Manuel Castells talade om informationssamhället, den amerikanske sociologen Daniel Bell om det post-industriella samhället och den tyske sociologen Ulrich Beck formulerade teorin om risk-samhället. I dessa perspektiv blir 68 inte så mycket en revolt mot det bestående som en länk till det nya.
Ett tredje perspektiv framlades på 1970-talet av den amerikanske sociologen Alvin W. Gouldner som i motsats till den spanske filosofen José Ortega Y Gassets idé om »massornas uppror», lanserade tanken att de »intellektuella» i västvärlden var på väg att etablera sig som en ny härskande klass, en ståndpunkt som för övrigt presenterades redan under mellankrigstiden av den amerikanske trotskisten James Burnham. Följdverkningarna av studentupproret 1968, kan mot denna bakgrund ses som ett påskyndande av eller ett energitillskott till de intellektuellas marsch mot makten.
När detta är sagt skulle vi ändå vilja återknyta till idén om »långa vågor». Historikern François Furet lyfter i sin bok _Slutet på en illusion_ fram den jakobinska strömningen i europeisk politik. Härmed förstår han en strömning karakteriserad av idén om extrem jämlikhet och av militant revolt mot det borgerliga samhällets lagar, institutioner och ideal; en strömning som alltid förkastat föreställningen att människans vilja och handlande skulle vara begränsat av traditioner, av den knapphet som driver konkurrensen och av naturens ofrånkomliga tvång. Denna upproriska underström har främst kommit till politiskt uttryck i samband med de stora omvälvningarna – 1789, 1848 och 1917 – i det moderna Europas historia. Vid krigsslutet 1918 uppenbarade sig dessutom ett annat extremt följe: en jakobinsk höger, en revolutionär högerradikalism besatt av ras och nation, men med mål likartade yttersta vänsterns, nämligen att krossa demokratin och kapitalismen, liksom adel, borgare och kyrka.
Till omvärldens stora överraskning kom de antiliberala och antikapitalistiska rörelserna till makten vid första världskrigets slut, först i Ryssland och sedan i Italien. De utgjorde ett existentiellt hot mot den västliga civilisationen. I nordvästra Europas länder lyckades ett samarbete mellan liberaler och reformistiska socialdemokrater stabilisera det borgerliga samhället och integrera de breda folklagren i ett fredligt parlamentariskt styre, byggt på allmän rösträtt, men också på respekt för enskilt ägande, civila institutioner, lagstyre och konstitutionell maktdelning. Det ödesdigra undantaget, bland dessa länder, visade sig bli Tyskland, Europas hjärta och hjärna.
Intressant nog återfanns emellertid mellankrigstidens radikalism inte bara på politikens ytterflyglar. Tidens progressiva modernism var en bred rörelse. Den attraherade konstnärer, författare, samhällsforskare och publicister ur samhällets olika skikt. Denna kulturradikalism hade många drag gemensamt med den jakobinska traditionen. Hit hörde oppositionen mot den borgerliga kulturen, vurmen för det upproriska och egalitära, fascinationen för utopiska och anarkistiska reformidéer. Den radikala modernismen påverkade arkitekturen, bildkonsten och litteraturen, liksom synen på sexualitet, könsroller, barnuppfostran och psykiatri på ett i flera avseenden framsynt och positivt vis. Den utgjorde också en strävan efter mer jämlik utbildning, en mer humanistisk människosyn och en förbättring av kvinnans situation.
Den långa och stabila epok i västra Europas historia som nu tycks på väg att bryta samman var ett resultat av den västliga liberalismen och sovjetiska kommunismens seger i andra världskriget. Den traditionella högern och kulturkonservatismen var efter kriget ohjälpligt komprometterade. Tvärtemot vad som ofta hävdas var dock de två årtiondena efter 1945 präglade av radikala reformer. Marknadsekonomin underställdes statlig planhushållning, viktiga industrier och banker förstatligades, fackföreningarnas inflytande stärktes. Den högre utbildningen breddades socialt. Den kulturella kritiken mot etablerade institutioner och sociala hierarkier fortsatte med stor styrka. Reformtakten och inriktningen mellan Västeuropas olika länder varierade, men den övergripande tendensen var likartad. Svensk efterkrigstid kom i hög grad att präglas av ett osvuret förbund mellan socialdemokratins egalitära ideal och den kulturradikala elitens sociala ingenjörskonst – kvarlevorna av det gamla samhället, oavsett om det gällde de äldre stadskärnorna eller undervisningen i antikens historia – skulle bort. Denna efterkrigstidens modernism, också den som företräddes av Tingstens DN, bidrog i hög grad till att – indirekt och oavsiktligt – bereda vägen för 1960-talets ungdomsrevolt. Sett i detta ljus är det knappast någon tillfällighet att Dagens Nyheters kultursidesansvariga, först Olof Lagercrantz och sedan Arne Ruth, lät sig ryckas med i det vänsterextrema följetåget.
Om det västliga samhället således bär på en underström av jakobinsk radikalism, som återkommande genererar en inre revolt, inställer sig likväl frågan varför 1968 års politiska uppror gick i den maoistiska kommunismens tecken. Här är 1956 års händelser avgörande. Det var detta år Nikita Chrusjtjov avslöjade stalinismens brott och det var detta år Moskva slog ned det nationella upproret i Ungern. Båda dessa händelser komprometterade Sovjetkommunismen. Socialismen i öst kritiserades därefter öppet av den radikala vänstern i väst för att vara auktoritär, byråkratisk och hierarkisk. Sovjet upphörde att vara vänsterns föredöme och blev i stället för många ett avskräckande exempel. Den »nya vänstern», kritisk mot både kommunism och socialdemokrati, debuterade vid 1960-talets mitt. Den dansade en synnerligen kort sommar. I östern uppenbarade sig nämligen en ny socialistisk utopi. Den kinesiska kulturrevolutionen förkroppsligade för västvärldens unga jakobinska revoltörer drömmen om ett helt jämlikt, solidariskt och rättvist samhälle. Maos Kina fyllde det tomrum som uppstått när oktoberrevolutionen hade förlorat sin heder. 1968 års studentuppror sögs nu in i den kommunistiska idévärldens torktumlare.
Ungdomsupproret och studentrevolten i väst ägde rum samtidigt som de sista resterna av den europeiska imperialismen avvecklades. Med avkolonialiseringen följde också en ideologisk avförtrollning av den västliga världens självförståelse – inte minst dess framtidstro och utvecklingsoptimism. Om nazismens nederlag hade svärtat den kulturkonservativa högern, komprometterade i 68-vänsterns ögon kolonialismens historia hela den västligt liberala civilisationen. Kritiken av imperialismen hade sedan första världskrigets slut främst drivits av kommunismen. Inom 68-vänstern blev den nu allmän egendom. Tredje världen och dess marxist-leninistiska befrielserörelser blev de unga europeiska radikalernas kärleksbarn.
Som framställningen ovan visat blev 68-vänsterns resa kort men intensiv, likväl fick dess idéer brett genomslag inom medier, kulturliv och akademier. VPK blev från 1980-talet 68-generationens främsta företrädare. Vid murens fall 1989 övergav partiet i panik kommunismen. Därefter följde en ytterst segdragen och motvillig nedtoning av det historiska arvet, i hög grad framtvingad av den liberala kritiken. Samtidigt som feminism, ekologism och rättigheter blev bärande inslag i vänsterns nya ideologi, återkom mot 1990-talets slut ett nostalgiskt svärmeri för kommunismen. Även om Vänsterpartiet i dag har vänt kommunismen ryggen kom det således aldrig att göra upp med sitt komprometterade förflutna.
## **XII. 68-VÄNSTERNS EFTERMÄLE**
SOM TIDIGARE FRAMKOMMIT publicerades år 1997 i Frankrike _Le livre noir du communisme_ ( _Kommunismens svarta bok_ ), en genomgång av de kommunistiska regimernas massmord och andra förbrytelser. Ett uppmärksammat förord av Stéphane Courtois, chef för ett franskt forskningsinstitut, tog upp en väsentlig fråga. Varför hade kommunismens förbrytelser aldrig fördömts på samma sätt som nazismens utan snarare ursäktats och förringats? Courtois skrev bland annat:
Det är inte min avsikt att här anställa något makabert jämförande räknestycke, att göra någon sorts dubbel bokföring för fasor eller upprätta en hierarki för grymheter. Men fakta är envisa och visar att de kommunistiska regimerna har begått brott som omfattar omkring 100 miljoner människor, mot omkring 25 miljoner människor för nazismen. Detta enkla konstaterande borde åtminstone leda till vissa reflexioner om likheten mellan den regim vilken år 1945 betraktades som århundradets brottsligaste och det kommunistiska system som ända fram till 1991 behöll hela sin internationella legitimitet, som än i dag har makten i vissa länder och som har adepter i hela världen.
Courtois skrev vidare:
Den enastående uppmärksamhet som ägnas de hitleristiska brotten är fullkomligt berättigad. [---] Men varför detta svaga gensvar i opinionen på vittnesbörd om de kommunistiska brotten? Varför denna generade tystnad bland politikerna? Och framför allt: varför denna 'akademiska' tystnad om den kommunistiska katastrofen som sedan åttio år tillbaka har omfattat omkring en tredjedel av människosläktet, på fyra kontinenter? Varför denna oförmåga att i centrum av analysen av kommunismen sätta in en så pass väsentlig faktor som brottet, brottet i masskala, det systematiska brottet, brott mot mänskligheten?
Courtois ger själv ett antal förklaringar till varför kommunistiska massmord alltid mäts med en annan måttstock än nazistiska. För svensk del kan framhållas att de nazistiska organisationerna alltid till och med under sin storhetstid på 1930-talet var marginella i förhållande till de kommunistiska, åtnjöt långt mindre sympati och i denna rent kvantitativa mening var mer »extrema» än till och med det stalinistiska kommunistpartiet. Det väsentliga är att faktum kvarstår, måttstockarna är olika. Det finns goda eller åtminstone ursäktliga massmord (kommunistiska) och det finns onda (nazistiska). I boken _De svenska maoisterna_ (2001) säger vänsterdebattören Sven-Eric Liedman a propos jämförelsen mellan kommunism och nazism: »Jag tycker den är helt absurd och jag blir väldigt upprörd över det lättköpta jämförandet. Kommunismens verkningar i Sovjet och Kina är förfärliga, men det är bara i det avseendet man kan säga att där finns något som påminner om nazismen.» Bara? Bara i avseende på »verkningarna»? Men om dessa »verkningar» inbegrep tiotals miljoner döda? Urskuldas detta av att de dödades för att skapa »det klasslösa samhället» (snarare än tusenårsriket)?
De olika måttstockar som Courtois observerade gäller inte bara kommunismens respektive nazismens förbrytelser. De gäller också dess anhängare. Att en gång ha varit anhängare till nazismen uppfattas som ett brännmärke, en outplånlig skam. Att en gång ha varit anhängare av kommunism/leninism/maoism uppfattas som vittnesbörd om att man haft känsla för rättvisa och solidaritet. Per Svensson har i sin på flera sätt förtjänstfulla bok _Frostviken. Ett reportage om Per Olof Sundman, nazismen och tigandet_ (1998) skrivit om författaren Sundman som i sin ungdom varit medlem i en nazistisk organisation och sedan gått i skräck för att detta skulle upptäckas. Svensson ger beskrivningar av Sundmans situation. Sundman bar, heter det, sin hemlighet som en Tolkienring »som först får sin bärare att växa men sedan gröper ur honom och gör honom till en skugga, en gollum. [...] Ringen blir en boja som äter sig in i köttet och vidare in i själen.» Längre fram i boken återkommer Svensson i liknande ordalag till beskrivningen av Sundmans skamliga hemlighet: »Till sist har giftet gröpt ur människan och förvandlat henne till en skråpuk, skramlande skelett, ett tomt skal.» Men på en punkt leder resonemangen Per Svensson fram till en mycket underlig reflexion. Han påpekar att många »av oss som idag diskuterar Per Olof Sundmans förflutna har ju själva under kortare eller längre tid, som tonåringar eller tjugoåringar, attraherats eller förförts av andra former av totalitarism än nazismen. Lenin, Stalin, Mao och Pol Pot. Kastar vi inte sten i glashus?» Svensson finner dock en viktig skillnad: »Och utan att därför ge sig in i diskussionen om vilken ideologi och praktik – nazismen eller kommunismen – som gjort sig skyldig till de grövsta brotten mot mänskligheten, kan man väl ändå konstatera en skillnad mellan 30-talsnazisterna och 70-talsmaoisterna ändå är att de senare i långt högre grad än de förra varit redo att offentligt diskutera och bearbeta sitt politiska förflutna.»
Huruvida 70-talsmaoisterna verkligen varit redo att diskutera sitt politiska förflutna kan ifrågasättas. Men den avgörande skillnaden är att medan nazisternas politiska förflutna i offentligheten betraktats som den yttersta skam har maoisternas politiska förflutna betraktats som glada ungdomssynder, kanske rent av som vittnesbörd om en ungdomlig känsla för rättvisa och ställningstagande för de förtryckta.
Åtskillnaden mellan nazisternas ofattbart onda massmord och kommunisternas goda eller åtminstone begripliga och ursäktliga massmord har blivit en åtskillnad även i hur man bedömt sympatisörer i de båda lägren. Ett karaktäristiskt vittnesbörd för svenskt vidkommande finns i Tage Danielssons och Hasse Alfredsons text »Var blev ni av, ljuva drömmar?», sjungen av Monica Zetterlund i Svenska Ords revy Svea Hund på Göta Lejon 1976, där första och sista strofen lyder: »Svara, du med röda stjärnan på din vårkavaj: alla tåg som går mot lyckans land på första maj – svara på en fråga från en vän som tappat tron: när är de framme vid sin slutstation? [---] Var blev ni av, ljuva drömmar om en rimligare jord, ett nytt sätt att leva? Var det bara tomma ord? Var är han nu, våra frihetsdrömmars junker Morgonröd? Han rör ju på sej, så han är nog inte riktigt död...» Hade Monica Zetterlund kunnat sjunga lika ömt och nostalgiskt om de – visserligen mycket färre – svenska ungdomar som på 1930- och 1940-talen bar hakkorset på sin vårkavaj? Eller kunnat begråta att de tappat tron eller upphört att drömma ljuva drömmar om en bättre och rättvisare värld i folkgemenskapens tecken? Knappast.
68-nostalgin har emellertid färgat inte bara populär journalistik utan också stora delar av den forskning som med anspråk på vetenskaplighet ägnats fenomenen, inte minst i Sverige, där 68-generationens ideologi i så hög grad präglat årsklass efter årsklass av forskare inom humaniora och samhällsvetenskap. Vilken är då den rätta attityden till 68-rörelsen som historiskt fenomen? Det är inte historikerns uppgift att etablera sig som domstol. Följden av det blir i allmänhet en lynchjustis där offrens politiska hudfärg kommer att betyda mer än arten av de begångna brotten. Däremot är det en uppgift för historikern att hämta fram det material, de förhållanden som kan ligga till grund också för en rättvis moralisk bedömning. Kravet på »opartiskhet» kommer in här. Men den moraliska bedömningen är inte den enda, vanligen inte alls den huvudsakliga. Det förflutna kan man inte ändra. Men man kan förstå det. I vissa fall kan man hämta lärdomar för framtiden. Också detta är en senare uppgift. Den primära handlar om att kartlägga, analysera och ställa in i sammanhang.
Vilken har då vänsterns roll och position varit under de senaste tio till femton åren – under de år som ligger efter vår undersökningsperiod?
Det förefaller oss tydligt att denna roll och position även under denna korta tid genomgått betydande förändringar såväl internationellt som i Sverige. Från 1989 och tjugo år framåt rådde i hela västvärlden en sorts eufori. Slagordet »Historiens slut», lanserat sommaren 1989 av den amerikanske statsvetaren Francis Fukuyama, var ägnat att ge stämningen ett namn. Den tidigare 1900-talshistoriens kamp mellan olika ideologier (kommunism, fascism, nazism, liberalism) menades vara över. Den demokratiska marknadsekonomin hade segrat som världsomspännande koncept. En era av globalisering var för handen. Murarna hade fallit och gränserna mellan nationer skulle snart förlora sin gamla betydelse.
Mot denna liberala feststämning protesterade åtminstone en hård kärna inom vänstern. Globaliseringen föreföll i det nya läget vara liktydig med världskapitalismens totala seger. Inga motståndare av någon verklig betydelse kunde skönjas längs horisonten. Attac-rörelsen, protesterna mot EU, manifestationer mot möten mellan den kapitalistiska världens ledare, allt detta föreföll lönlöst. »McWorld» hade tagit över. »Imperiet» regerade. Inom vänstern härskade en stämning av »Vänstern contra mundum», man befann sig på den sista barrikaden.
Men efter 2009 har situationen snabbt förändrats. USA har i betydande utsträckning dragit sig tillbaka i rollen som den globaliserade världsordningens garant. Opponenterna har blivit fler, efterhand också starkare och djärvare. Putins Ryssland, det nya Kina, Den islamska staten, olika EU-fientliga partier i Europa, dessa har med ny kraft utmanat den ordning som inträdde efter 1989. Det handlar inte om uppkomsten av nya ideologier av typen kommunism eller fascism. Snarare handlar det (bortsett från islamismen) om en ny nationalism. Gränserna har återfått sin betydelse. Omsorgen om den nationella säkerheten har ersatt den tidigare sorglösheten. Geopolitiken har kommit tillbaka i ett tänkande som nyss behärskades av föreställningar om »mänskliga rättigheter» i en värld utan gränser.
Därmed har också vänsterns position förändrats. Vänstern har nu blivit en del av en bred allians som formats mot de nya populistiska och nationella rörelserna. Det har handlat om försvaret av det »mångkulturella» samhället. Det har handlat om försvaret av universella »mänskliga rättigheter». Det har handlat om att bekämpa nya former av nationalism. Vänstern har på köpet givit upp eller åtminstone tonat ned det mesta av sin tidigare kritik mot globaliseringen, mot EU, mot olika övernationella institutioner och överenskommelser. Fronten har blivit en annan.
Frontförändringar av denna typ är ingenting nytt i vänsterns historia. Alltsedan Marx och Engels dagar (och kanske tidigare) har vänstern kunnat välja mellan att bekämpa kapitalismen i visst samförstånd med konservativ, ibland romantiskt färgad antikapitalism. Eller att samverka mot »högern» i samförstånd med liberal, »framstegsvänlig» borgerlighet. Under de senaste tio till femton åren har en sådan frontförändring ägt rum under trycket från en ny globaliseringsfientlig populism. Detta är något som blivit märkbart såväl i Sverige som i resten av världen. Den politiska kartan är ånyo på väg att ritas om.
## **NOTER**
I. FRÅN DEN NYA VÄNSTERN TILL 68-VÄNSTERN
.Alvar Alsterdal: _Den nya vänstern_ (1963) s. 155.
.op.cit., s. 158.
.Kjell E. Johanson red.: _Unga kommunister – aktuellt alternativ_ (1966) s. 13.
.Daniel Tarschys & Carl Tham: _Den nygamla vänstern_ (1967) s. 9.
.Göran Therborn red.: _En ny vänster_ (1966) s. 9f.
. _Utanför systemet. Vänstern i Sverige 1968–1978_ (1978) s. 59.
.C. H. Hermansson: _Vänsterns väg. Ett debattinlägg_ (1965) s. 9ff.
.Tarschys & Tham: _Den nygamla vänstern_ , s. 15.
.Åke Lundqvist: _Från sextital till åttital. Färdvägar i svensk prosa_ (1981) s. 29.
.Tarschys & Tham: _Den nygamla vänstern_ , s. 17.
.Johan Svedjedal: _Ned med allt? Essäer om protestlitteraturen och demokratin, cirka 1965–1975_ (2014) s. 111.
II. VÄNSTERNS LÅNGA SJUTTIOTAL
.Herbert Marcuse: _Protest, demonstration, revolt_ (1968), Joachim Israels förord, s. 5ff.
.Kjerstin Norén red.: _Linjer i nordisk prosa. Sverige 1965–1975_ (1977) s. 11ff.
.Göran Palm: _En orättvis betraktelse_ (2003) s. 102.
.Göran Palm: _Indoktrineringen i Sverige_ (1968) s. 9f.
.op.cit., s. 6.
.op.cit., s. 33.
.op.cit., s. 31.
.op.cit., s. 32.
.op.cit., s. 43.
.op.cit., s. 46.
.op.cit., s. 213f.
.Bo Heurling red.: _Författaren själv_ , s. 269.
.Sara Lidman: _Gruva_ (1969) s. 59.
.op.cit., s. 60.
.Norén: _Linjer i nordisk prosa_ , s. 291f.
.Lidman: _Gruva_ , s. 156.
.op.cit., s. 61f.
.Frantz Fanon: _Jordens fördömda_ (sv. övers. av Per-Olof Zennström 1969), Sartres förord, s. 20.
.op.cit., s. 239f.
.op.cit., Sartres förord, s. 13.
.Jan Myrdal: _Albansk utmaning_ (1970) s. 208ff.
.Heurling: _Författaren själv_ , s. 269.
.Lundqvist: _Från sextital till åttital_ , s. 25.
.op.cit., s. 167.
.Per Olov Enquist: _Sekonden_ (1971) s. 218.
.Jan Stenkvist: _Flykt och motstånd. Fyra studier i politisk dikt_ (1978) s. 151.
.Enquist: _Sekonden_ , s. 186ff.
.op.cit., s. 269.
.Per Gahrton: _Revolution på svenska_ (1972) s. 95.
.op.cit., s. 223.
.op.cit., s. 247.
. _Klartext. Marxistisk litteraturkritik_ (1972) s. 5.
.op.cit., s. 6.
.Göran Palm: _Ett år på LM 1_ (1972) s. 92.
.op.cit., s. 96.
.op.cit., s. 62.
. _Vänd VPK ryggen!_ (1973) s. 6f.
.op.cit., s. 19.
. _Vad vill VPK? Vad tänker dom på?_ (1973) s. 34.
.ibid.
.op.cit., s. 42.
.Håkan Holmberg: _Folkmakt, folkfront, folkdemokrati. De svenska kommunisterna och demokratifrågan 1943–1977_ (1982) s. 179ff.
.op.cit., s. 199.
.Lars Gustafsson & Jan Myrdal: _Den onödiga samtiden_ (1974) s. 7ff.
.op.cit., s. 12.
.op.cit., s. 39.
.BLM 1975, s. 163ff.
. _Borgerlig eller proletär politik – en kritisk granskning av SKP_ (1976) s. 5.
.op.cit., s. 25f.
.op.cit., s. 120.
.op.cit., s. 94ff.
.op.cit., s. 40.
. _Vart går SKP?_ , s. 20f.
.Karl Erik Lagerlöf: _Strömkantringens år och andra essäer om den nya litteraturen_ (1975) s. 37.
.ibid.
.Jan Josefsson & Mats Zetterberg: _Behöver vänstern gå i terapi?_ (1976) s. 9.
.op.cit., s. 10.
.op.cit., s. 12.
.op.cit., s. 19.
.op.cit., s. 216.
.Ord och Bild nr 4 1977, s. 65ff.
.Ord och Bild nr 2-3 1977, s. 156.
.op.cit., s. 163.
.op.cit., s. 58f.
.op.cit., s. 60ff.
.Zenit nr 51 1977, s. 73ff.
.Peter Kemp: _Marxismen i Frankrike_ (sv. övers. av Svante Hansson 1981) s. 12.
.Madeleine Gustafsson: _Utopien och dess skugga_ (1978) s. 17.
.op.cit., s. 42.
.op.cit., s. 101.
.op.cit., s. 137ff.
.op.cit., s. 163f.
.op.cit., s. 149.
.op.cit., s. 160f.
.Anders Ehnmark & Per Olov Enquist: _Protagoras sats. På spaning efter det politiska förnuftet_ (1987) s. 89.
.op.cit., s. 95ff.
.Kent Lindkvist & Svante Nordin: _De socialistiska krigen. En antologi om konflikten i Sydostasien_ (1979) s. 28f.
.op.cit., s. 31.
.op.cit., s. 45.
.op.cit., s. 56.
.op.cit., s. 38.
.op.cit., s. 69.
.op.cit., s. 70.
.op.cit., s. 69.
.op.cit., s. 89ff.
.Jan Myrdal: _Dussinet fullt. Skriftställning 12_ (1982) s. 158ff.
.Carl Johan Ljungberg: _Fagerberg. Sven Fagerberg som romanförfattare och polemiker_ (1999) s. 169.
.Artikeln även publicerad i Lars Ahlbom & Crister Enander (red.) Sven Delblanc: _Det är du själv som bestämmer! Politik, samhällskritik och litteraturdebatt från tre decennier_ (2009) s. 181ff.
.Magnus Bergh: _Mörkrets litteratur. Peter Weiss i Motståndets estetik_ (1991) s. 8.
.Jan Myrdal: _Dussinet fullt_ (1981) s. 26ff.
.op.cit., s. 33.
.Peter Weiss: _Motståndets estetik III_ (sv. övers. av Ulrika Wallenström 1981) s. 312.
.Lars Wendelius: _Form och tematik i Peter Weiss Motståndets estetik_ (1991) s. 13.
.op.cit., s. 134.
.Åke Lundqvist: _Kultursidan. Kulturjournalistiken i Dagens Nyheter 1864–2012_ (2012) s. 360.
.Gösta Hultén & Jan Samuelsson: _Mediavänstern. En närbild av den dolda åsiktsproduktionen i Sverige_ (1983) s. 9.
.op.cit., s. 16.
.op.cit., s. 17f.
.op.cit., s. 61.
IV. PROLOG
.Med begreppet vänster avses, om formuleringen tillåts, politiska rörelser till »vänster» om den reformistiska delen av arbetarrörelsen, d.v.s. socialdemokratin eller den demokratiska socialismen. I första delen av denna bok görs en distinktion mellan 1960-talets »nya vänster» och »68-vänstern». Den förstnämnda var ett kort försök, cirka 1965 till 1968, att etablera ett vänsterperspektiv som vare sig var socialdemokratiskt eller kommunistiskt. Den senare, som föddes av och genom studentrevolten 1968, sögs snabbt in i den kommunistiska traditionen. I Sverige handlade det främst men långtifrån enbart om maoismen. Med den kommunistiska traditionen/den traditionella kommunismen/Komintern-kommunismen/marxism-leninismen förstår jag här det samhälls- och historieperspektiv som först formulerades av Marx och Lenin och som fick sitt historiska genombrott i samband med ryska revolutionen.
68-vänstern i snäv mening ebbade (som tidigare visats) ut i slutet av 1970- och början av 1980-talet. Men vänstergenerationen från 1968 vidareförde i hög grad sina idéer även efter 1989, dels i de kvarvarande smågrupperna, dels och i huvudsak inom Vänsterpartiet. Minst lika viktigt var dock att 68 års idéer fick ett brett genomslag inom medier, universitetens humanistiska institutioner och olika gräsrotsrörelser.
.Kommentar av Göran Therborn i Ny Dag nr 5/1990.
.Se 1987 års partiprogram: snd.gu.se/sv/vivill/party/v/program/1987
.ibid.
.Petter Bergner: _Med Historien som motståndare. SKP/VPK/V och det kommunistiska arvet 1956–2006_ (2013) s. 282.
V. SAMMANBROTTET I ÖST
.Ny Dag, ledarkommentar, nr 41/1989. (Ny Dag var en veckotidning och siffran 41 refererar till motsvarande vecka under året.)
.Ny Dag nr 46/1989. En liknande ståndpunkt intog partisekreteraren Kenneth Kvist i ett inlägg i Ny Dag nr 1/1990.
.Dagens Nyheter 8/2 1990.
.Ny Dag nr 1/1990.
.ibid.
.ibid.
.op.cit., nr 6/1990.
.op.cit., nr 5/1990. Om Torbjörn Tännsjös plädering för proletariatets diktatur se diskussionen av hans bok _Demokrati och proletär revolution_ (1977) i avsnitt VIII.2.
.Ny Dag nr 1/1990.
.Den inom dåvarande VPK etablerade definitionen av socialism hade innan 1989 överhuvud inget med demokrati att skaffa. C. H. Hermansson hade i tidskriften Zenit tydligt markerat detta: »... det socialistiska produktionssättets grundläggande drag är hela samhällets ägande av produktionsmedlen och i den meningen är staterna i Östeuropa socialistiska.» (Zenit nr 50, april/maj 1977).
.Per Kågessons inlägg i Dagens Nyheter 14/1 1990. Som Kågesson påpekar hade samma ståndpunkt formulerats av Jörn Svensson i hans bok _Du ska ta ledningen och makten kamrat_ (1974). Svensson var ledande ideolog i partiet under hela 1980-talet och en av arkitekterna bakom 1987 års slutliga partiprogram.
.Ny Dag nr 23/1990.
.op.cit., nr 12/1990.
.Dagens Nyheter 14/1 1990 (DN Debatt). Annika Åhnberg vittnar i sin självbiografi om hur hon överraskades av de starka sympatierna för kommunismen i partiets riksdagsgrupp när hon själv kom med i denna efter valet 1988. Se _Isprinsessa_ (1999) s. 115.
.Dagens Nyheter 14/1 1990.
.op.cit., 18/1 1990 (DN Debatt).
.ibid.
.Ny Dag nr 41/1989 (12–18 oktober). Se också 1987 års program. Faktum är att den positiva synen på DDR:s ekonomi var utbredd i väst; denna uppfattning inkluderade också amerikanska CIA. I själva verket var landet i praktiken bankrutt hösten 1989. Se Robert Service (2015).
.Ny Dag nr 42/1989.
.op.cit., nr 8/1990.
.op.cit., nr 19 och 22/1990. Se också Bergner (2013), kapitlet 'En ny tid, en ny vänster', s. 283–369.
.Bergner (2013) s. 292, min kursiv.
.Samtliga citat som refererar till Ingvar Johansson är hämtade från hans inlägg i Ny Dag nr 8/1990.
.Johan Lönnroths inlägg återfinns i Ny Dag nr 15/1990.
.Ny Dag nr 17/1990.
.op.cit., nr 1/1990.
.Ehnmark bortsåg här från att valet mellan demokrati och diktatur träffades redan vid första världskrigets slut och att det således under kommunismens hela existens funnits en arbetarrörelse som både varit demokratisk och just sagt sig företräda »arbetarrörelsens ideal». I den kritik som åren efter 1917 riktades mot leninismen hävdade dessutom många kritiker att både proletariatets diktatur och den rena partidiktaturen ofrånkomligen skulle medföra förtryck och misär för folkets breda lager. Att förtrycket var i teorin oförutsett håller alltså inte. Såväl Karl Kautsky, tysk socialdemokratisk ledare, som Rosa Luxemburg, polsk-tysk vänstersocialist, skrev och varnade för Lenins våldsstrategi. De tog båda uttryckligen avstånd från Lenin och bolsjevismen.
.Citaten är från Sven-Eric Liedmans artikel i Ny Dag nr 21/1990.
.op.cit., nr 2/1990.
.op.cit., nr 9/1990.
.Dagens Nyheter 16/5 1990.
.op.cit., 23/5 1990.
.op.cit., 10/7 1990.
.op.cit., 29/9 1990.
.op.cit., 30/5 1990.
.Löfgrens kommentar återfinns i Dagens Nyheter 19/9 1990. En likartad reflektion publicerades av Hans Axel Holm några dagar senare (DN 28/9) under rubriken »DDR-kulturen sprids för vinden». Holm beskriver både sorgset och bittert hur nedläggningen av DDR-Kulturcentrum i Stockholm medför att dess samling upplöses utan att någon tar ansvar för allt viktigt material, istället sprids den ut på en rad olika institutioner som Goetheinstitutet, Musikaliska akademin m.fl.
.Dagens Nyheter 22/5 1990.
.Robert Griffins ger en insiktsfull analys av hur nära förknippade de tidiga modernistiska kulturyttringarna var med såväl fascism som med kommunism. Se Robert Griffin: _Modernism and Fascism_ (2007).
.Dagens Nyheter 27/8 1990.
.op.cit., 24/9 1990.
.Dagens Nyheter 17/3 1990.
.Lars-Arne Norborg karakteriserar i partiets _Vitbok_ förändringen med följande ord: »Den marxistiska ortodoxins sista skansar utrymdes i snabb takt. På högsta partinivå ifrågasattes planhushållningen och experimenterades med teorier om marknadssocialism. Marxismens skarpa konturer suddades ut till förmån för en sympatisk men vag humanism.» _Vitboken_ s. 38. En annan av författarna i boken, Kent Lindkvist, formulerar ett likartat omdöme: »Efter 1989 drabbades Vänsterpartiet av ord- och handlingsförlamning där 1964 års linje bröt samman i samma takt som de folkliga upproren störtade de realsocialistiska regimerna. Vänsterpartiets språkbruk om de f d socialistiska regimerna blev nu sådant som man aldrig tidigare hört om dessa regimer – och som partiet t o m genom kongressbeslut avvisat.» _Vitboken_ , s. 72.
.Expressen 3/5 1991.
.Staffan Skott: _Liken i garderoben. Vänsterpartiets förflutna 1917_ – _1989_ (Tidens förlag 1991). Ett knappt årtionde senare, år 2000, utgav Skott _Liken i garderoben lever än. Undanflykternas mästare. Om Vänsterpartiers förflutna 1918_ – _1998_. (Hjalmarson & Högbergs förlag). Utgåvan från 2000 innehåller hela den första boken samt en uppdatering. Det är till den senare boken av Skott som jag här hänvisar. Se också Kjell Albin Abrahamson: _Låt mig få städa klart! Om kommunister, kryptokommunister och antikommunister_ (2014).
.Skott (2000) s. 203.
.Bo Hammar talade om ett »polemiskt raseri», Expressen 3/5 1991, Lars-Arne Norborg betecknade den som en bok i genren »orättvisa betraktelser», Sydsvenskan 12/5 1991, Olle Svenning menade att undertexten »vibrerar av känslomässig upprördhet», Aftonbladet 22/5 1991, Stellan Lindqvist karakteriserade framställningen som en »ursinnig uppgörelse» med VPK, Göteborgsposten 24/5 1991. Mest positiv var Thomas Nordengren i Dagens Nyheter 18/6 1991, som hävdade att det behövdes fler studier av Skotts typ.
.På Vänsterpartiets kongress våren 1990 beslöts att utarbeta en redogörelse för partiets relationer med kommunismen i Sovjet/Östeuropa. Först ett år senare tillsattes en arbetsgrupp som presenterade en vitbok – _Lik i garderoben? En rapport om SKP/VPKs internationella förbindelser_ – på en presskonferens i Malmö i december 1992. Enda närvarande representant för »medierna» var en John Andersson från Socialistiska Partiets tidskrift Internationalen (trotskistisk). Dokumentet togs upp på kongressen 1993 och lades till handlingarna nästan utan debatt. En andra utökad upplaga utkom 1996, vilken är den jag här använt under beteckningen _Vitboken_. Medlemmar i arbetsgruppen var Lars-Arne Norborg, Kent Lindkvist som ersatte Bengt Schüllerqvist, Ulf Nymark och Sven-Eric Liedman, samtliga medlemmar i eller närstående Vänsterpartiet. Se inledningen till _Vitboken_ , samt www.marxistarkiv.se/index.php?option=com_content&task=view&id=54&Itemid=134.
. _Vitboken_ (1996).
.Se Lars-Arne Norborgs avsnitt om »Den partiinterna debatten. Om relationerna till SUKP och andra maktägande 'realsocialistiska' partier». _Vitboken_ (1996) s. 2-52.
.Jörgen Hermansson skriver i sin avhandling _Kommunism på svenska? SKP/VPK:s idéutveckling efter Komintern_ (1984) att »Det svenska kommunistpartiet... utvecklades så småningom till en lydig och följsam sektion av den Kommunistiska internationalen. Det utmärktes på intet sätt av ideologiskt nytänkande och självständighet. När Komintern lades ned på våren 1943 och partiet stod inför uppgiften att självt svara för sig, var det aldrig fråga om att SKP skulle utforma någon alldeles egen och annorlunda ideologi. Den folkfrontsideologi som anammades, vilken för övrigt hade förberetts redan under Kominterns sista år, stämde i allt väsentligt överens med internationella förebilder.», s. 333.
.I sina memoarer skriver partiets tidigare ordförande C. H. Hermansson att första gången partiet över huvud riktade en programmatisk kritik mot villkoren i Sovjet var i samband med kongressen 1967: »För första gången framfördes i våra program också kritiska synpunkter på förhållandena i Sovjetunionen och övriga stater som betecknade sig som socialistiska.» _CH Minnen_ (1993) s. 241. Se också _Vitboken_ (1996) s. 40.
.Jfr Lars-Arne Norborg: »Å andra sidan fick den kompromiss mellan traditionsbevarare och förnyare som bl.a. det nya partinamnet Vänsterpartiet kommunisterna är ett uttryck för till följd att uppkomsten av ett SF-parti och därmed ett fullständig frigörande från Moskva förhindrades.» _Vitboken_ (1996) s. 41.
.Den brittiske historikern Robert Service formulerar Chrusjtjovs politik på följande vis: »Furthermore, he /Chrusjtjov/ allowed for a variety of ways, including peaceful ones, of making the 'transition to socialism'. Communist parties did not have to copy the Soviet historical experience.» Se _Comrades. Communism: A World History_ (2008) s. 319. Samma sak framhåller David Priestland i _The Red Flag. Communism and the making of the Modern World_ (2009) s. 342-343. Norberg ger uttryck för samma hållning i _Vitboken_ (1996) s. 18, liksom Kent Lindkvist i samma skrift, s. 58. Se också Jörgen Hermanssons avhandling (1984) s. 334. Historikern Håkan Holmberg betonar i sin avhandling _Folkmakt, folkfront, folkdemokrati. Den svenska kommunismen och demokratifrågan 1943_ – _1977_ (1982) att Vänsterpartiet bibehöll grundelementen i den kommunistiska synen på politisk makt och demokrati under hela den undersökta perioden. Se s. 52, 218.
. _Vitboken_ (1996) s. 41. Norborg talar också om den »nyleninistiska» renässans som präglade 70-talets VPK, s. 22. Se också avsnittet om Therborns och Tännsjös plädering för proletariatets diktatur i kapitel VIII.2.
.Jfr Lars-Arne Norborg: »I själva verket torde partiets linje ha stått i god överensstämmelse med huvudströmningen inom de 'eurokommunistiska' västeuropeiska partierna, i synnerhet då PCI» [Italiens kommunistiska parti, LB]. _Vitboken_ (1996) s. 31. Kent Lindkvist är inne på samma linje som Norborg.
.Tre av författarna – Lars-Arne Norborg, Ulf Nymark och Kent Lindkvist – var eurokommunistiskt orienterade, kritiska till partiets tidigare Sovjetkommunism och förmodligen också till partiets nyleninistiska orientering under 1980-talet.
. _Vitboken_ (1996) s. 2, 12f, 36, 30, 87 samt 43.
.Lars-Arne Norborg menar att den centrala frågan är hur »de starka emotionella bindningarna till Sovjetunionen» skall förklaras historiskt. _Vitboken_ (1996) s. 4.
.Se Ulf Nymarks bidrag »Vad sa man om Östeuropa på kongresserna», _Vitboken_ (1996) s. 75–86. Här listas kontakter och utbyten med de östliga kommunistpartierna under 1980-talet. Vad angår de ekonomiska bidragen från Moskva till Vänsterpartiet finns dessa redovisade i historikern Lars Björlins uppsats »Russisk guld i svensk kommunisme», s. 42–127 i _Guldet fra Moskva. Finansieringen af de nordiske kommunistpartier 1917_ – _1990_ , red. Morten Thing (2001).
. _Vitboken_ (1996) s. 73 respektive s. 41.
.op.cit., s. 89.
.Exempel på tidiga standardverk är Merle Fainsods böcker: _How Russia is Ruled_ (1953) och _Smolensk under Soviet Rule_ (1958), Zbigniew K. Brzezinski: _The Soviet Bloc_ (1960). Robert Conquest började sin utgivning av böcker om Sovjetunionen redan på 1960-talet: _Soviet Deportation of Nationalities_ (1960), _The Great Terror_ (1968). Inifrån realsocialismens läger skrev Roy A. Medvedev: _Let History Judge. The Origins and Consequences of Stalinism_ (1971) och Artur London: _Bekännelsen. I Prag-processens maskineri 1951_ – _1952_ (1971). Härutöver fanns rader av memoarer av flyktingar från öst och en oändlig mängd journalistiska reportage om förtrycket därstädes.
. _Vitboken_ (1996) s. 3, 73, 89.
.op.cit., s. 88.
VI. UPPGÖRELSERNAS TID
.En polemisk analys från liberala utgångspunkter publicerades hösten 1991 av Per T. Ohlsson: _Landet utanför. Saddam och ståndpunkterna_ (1991).
.Debattinläggen från Timbrokonferensen finns publicerade i _Vänsterns moraliska skuld_ (1991). Någon redaktör för antologin finns ej angiven, men den brukar tillskrivas Per Ahlmark och Kay Glans vilka står som första namn bland flera medförfattare.
. _Vänsterns moraliska skuld_ , s. 46–47.
.Se Kay Glans understreckare i Svenska Dagbladet 7/3 1991.
.ibid.
.Karl Vennbergs artikel var införd i Aftonbladet 28/3 1991.
.op.cit., 23/1 1991.
.Svenska Dagbladet 3/5 1991.
.ibid.
.I den politisk-ideologiska striden, under främst 1950-talet, mellan de liberala demokratierna i väst och de kommunistiska enpartidiktaturerna i öst, fanns debattörer som menade att ingetdera systemet var perfekt och att båda systemen egentligen var lika goda »kålsupare». Av denna anledning önskade de inta en neutral eller tredje ståndpunkt mellan demokrati och diktatur. Se Tomas Forser och Per Arne Tjäder: _Tredje ståndpunkten_ (1972). Forser och Tjäder var två unga litteraturvetare från Göteborg, lärjungar till den begåvade marxistiske litteraturforskaren Kurt Aspelin och kring 1970 politiskt hemmahörande i trakten av Vänsterpartiet kommunisterna. Se också not 135.
.Aftonbladet 10/5 1991.
.Svenska Dagbladet 9/6 1991.
.De unga kommunisternas självömkande syn på partiets historia hade enligt Bo Hammar fått näring av C. H. Hermanssons intervjuböcker – _Kommunister I–II_ , utgivna 1977 respektive 1980 – med gamla ortodoxa och sovjetnostalgiska partikommunister. Hammar kallar dessa intervjuböcker för »kaninbiblar för nya generationer kommunister». Se Bo Hammar: _Ett långt farväl till kommunismen_ (1992) s. 23.
.Svenska Dagbladet 9/6 1991.
.op.cit., 22/8 1991.
.ibid.
.Aftonbladet 16/6 1991.
.Dagens Nyheter 24/8 1991.
.ibid.
.En bok som kom att få betydande inflytande på relativiseringen av demokratibegreppet var den vänsterliberala kanadensiska statsvetaren C. B. Macphersons _Den mångtydiga demokratin_ (1966, 1968). Macpherson skiljer här mellan tre former för demokrati av vilka han betecknade två som icke-liberala, nämligen de kommunistiska styrelserna i öst och de nya enpartistaterna i tredje världen, vilka alla strävade efter att demokratisera ekonomin före politiken, och slutligen den liberala vilken hade demokratiserat politiken, men inte ekonomin. Att människor förföljdes och likviderades i de två förra när ekonomin skulle »demokratiseras», men inte i den senare fäste Macpherson, utifrån sitt egalitära perspektiv, ingen vikt vid. Så kunde under 1970-talet begreppen vändas upp och ned även utanför den marxistiska 68-vänstern.
.Per Ahlmarks _Tingsten. Tyranniet begär förtroende_ (1992) – fortsättningsvis enbart kallad _Tyranniet begär förtroende_ – är ett urval Tingstens artiklar, med en skarpt formulerad inledning av Ahlmark. Artiklarna är grupperade under följande rubriker: Vägar till despoti, Sovjet: makten utan mål, Fred och motstånd, Förtryckets ansikten, Förräderi, Medlöpare, Tredje ståndpunkten – en orimlighet och Fyra tal.
.Herbert Tingsten _Mitt Liv. Ungdomsåren_ (1961), _Mellan trettio och femtio_ (1962), _Tidningen_ (1963), _Tio år_ (1964).
.Se Herbert Tingsten: _Socialdemokratins idéutveckling I_ – _II_ (1941).
. _Mitt liv. Ungdomsåren_ (1961) s. 273–290.
.Se den lundensiske litteraturforskaren Claes-Göran Holmberg, _Upprorets tradition. Den unglitterära tidskriften i Sverige_ (1987).
.Inom stora delar av socialdemokratin sågs Clarté och dess medlemmar som influerade av den sovjetiska bolsjevismen. I synnerhet SSU bekämpade »kommunisterna» inom Clarté och ansåg clartéism som oförenligt med socialdemokratiskt medlemskap, men flera ledande socialdemokrater som Wigforss och Erlander förhöll sig mer positivt avvaktande. Frågan om den socialdemokratiska kritiken av vänstersocialismen och intellektualismen inom Clarté diskuterades återkommande i tidskriften. Se _Clarté_ nr 10-11/1930, 4–5/1933 och 7–8/1939.
.Alf W. Johansson refererar i sin _Herbert Tingsten och det kalla kriget_ (1995) till ett uttalande av Gunnar Myrdal från 1946 vari denne karakteriserade Tingstens hållning vid 1930-talets början som »varm Sovjetvän, kommunistsympatisör och allmän kulturbolsjevik». Johansson menar att uttalandet är alltför tendentiöst, färgat av Myrdals konflikt med DN åren efter krigsslutet 1945, för att vara trovärdigt. Jag är tveksam till detta avfärdande. Myrdals motiv att söka chikanera Tingsten var visserligen uppenbara, men karakteristiken stämmer samtidigt väl överens med hållningen i de intellektuella vänsterkretsar som Tingsten färdades i vid denna tid. För Myrdals uttalande se Alf W. Johansson (1995) s. 356.
.Herbert Tingsten: _Mitt liv. Mellan trettio och femtio_ (1962) s. 64: »Från omkring 1932 blev umgänget mycket intensivt [med Georg och Rina Branting]... Sedan var Brantings under mer än tio år våra kanske närmaste vänner...» Förhållandet mellan de båda familjerna under trettiotalet beskrivs ingående, men från dottern Kajs synvinkel, i _I Skuggan av Tingsten. En släktberättelse_ (2010) av Jill Tingsten Klackenberg, dotter till Kaj och dotterdotter till Herbert. Se särskilt s. 38–76.
.Anders Isaksson: _Per Albin III. Partiledaren_ (1996) s. 266.
.Tingsten har själv beskrivit sin ideologi i samband med tillträdet som chefredaktör: »Jag var exempelvis extrem antikommunist och antisocialist (om man i socialism inlägger vilja till socialisering) men samtidigt ivrig för socialpolitiska reformer och ökad jämlikhet, motståndare till monarki och statskyrka, med inrotad ovilja mot politiska, religiösa, sociala och ekonomiska hierarkier. Min antisocialism hade inte ett spår av respekt för kapitalismen; snarast hade jag större aversion mot förmögenhetens makthavare än mot andra former av 'överhet'.» _Mitt liv. Tidningen 1946_ – _1952_ (1963) s. 46.
.Under hela 68-perioden förblev dock ledarsidan på Dagens Nyheters ledarsida liberalt orienterad. Kultursidan däremot, som under Olof Lagercrantz ledning sedan slutet av 1960-talet var landets ledande opinionsbildande sida, kom att präglas av de nya vänsterströmningarna. Detta gällde i hög grad också den allmänna redaktionen. Om detta finns många dokument, så bl.a. skriver den framstående DNjournalisten Karl Anders Adrup levande och detaljerat i _Stormarnas hus. En krönika om DN 1964_ – _1989. Del 2_ (1989). Som Adrup formulerar det framställdes trots de inre konvulsionerna [som stormöten, marginaliserade äldre liberala medarbetare och chefskonflikter] en tidning av hög standard, men han tillägger: »tillräckligt mycket trängde ändå ut för att ge näring åt ryktena om vänstervridning och rödstrumperi.» (s. 182). Redaktören på DN:s kultursida Åke Lundqvist är lika tydlig i sin _Kultursidan. Kulturjournalistiken i Dagens Nyheter 1864_ – _2012_ (2012).
.Se Inger Enkvists _De svenska skolreformerna 1962–1985 och personerna bakom dem_ (2016). Boken är en del av forskningsprojektet Arvet efter 68.
. _Tingsten. Tyranniet begär förtroende_ , s. 11. I motsats till t.ex. USA och Danmark erkände Sverige snabbt inkorporeringen av de baltiska länderna i Sovjetunionen år 1940.
.op.cit., s. 10. När Ahlmark talar om socialdemokratins selektiva syn på diktaturer är det för att understryka att vissa diktaturer, som de i Tjeckoslovakien, Sydafrika och Chile, tog Palme i skarpa ordalag avstånd från.
.op.cit., s. 27.
.op.cit., s. 28. Citationstecknen är Per Ahlmarks egna.
.op.cit., s. 9.
.De tre olika citaten från Svante Nycanders inlägg är från Dagens Nyheter 31/1 1992.
.op.cit., 26/2 1992.
.Citaten är från Arne Ruths artikel i Dagens Nyheter 13/2 1992. Ruths hänvisning till Timbro och SAF syftar på det faktum att Ratio förlag, där Ahlmark utgivit sin bok, var del av Timbro AB, som i sin tur ägdes av Näringslivets Fond.
.Ord & Bild-numret var från 1971 och Forser och Tjäders bok _Tredje ståndpunkten_ utgavs, som jag tidigare skrivit, 1972.
.Detta, ovanstående och följande citat är från Arne Ruths andra inlägg publicerade i Dagens Nyheter 15/2 1992.
.Jämför här dels David Caute: _The Fellow-Travellers. Intellectual Friends of Communism_ (1973, 1988) i vilken det finns flera hänvisningar till denna kritik från tiden innan Lagercrantz resa. Se kapitlet 'Into China', s. 347–391; dels Paul Hollander: _Political Pilgrims. Travels of Western Intellectuals to the Soviet Union, China and Cuba 1928_ – _1978_ (1981, 1990). Se kapitel 'The Pilgrimage to China', s. 278–346.
.Olof Lagercrantz skrev fjorton artiklar om Kina. De första tio publicerades i Dagens Nyheter mellan den 10 november och den 13 december 1970, de resterande fyra från den 24 januari till den 14 februari 1971. Citatet är från Dagens Nyheter 31/1 1971. I andra artiklar i samma serie talade Lagercrantz om »partiets hängivenhet och osjälviskhet» och Maos »levande medkänsla med det kinesiska folket». Per Ahlmark citerar honom för följande: »det är en väldig tillgång för ett land att ha Mao Tsetung som lärofader därför att han är praktiskt erfaren och fylld av levande medkänsla med det kinesiska folket.» Se Per Ahlmark: _Tyranniet begär förtroende_ , s. 22.
.op.cit., (1992), s. 23.
.Se Svante Nycanders inlägg i Dagens Nyheter 31/1 respektive 26/2 1992.
. _The Fellow Travellers_ , s. 383. Se också _Political Pilgrims_ , s. 325–331. Både Caute och Hollander nämner rader av andra pilgrimer som lät sig förföras av de Potemkinkulisser regimen i Kina visade upp. Så exempelvis Simone de Beauvoir, Basil Davidson, René Dumont och Felix Green.
.Thomas Forser & Per Arne Tjäder: _Tredje ståndpunkten_ (1972).
.op.cit., s. 13.
.op.cit., s. 16. Se också den inflytelserika vänstertolkning av det kalla kriget som David Horowitz utgav, _Från Jalta till Vietnam_ (1966); särskilt s. 58–119.
.Efter det kalla krigets slut och med tillgång till nya källor har en mindre ideologisk, mer balanserad och realistisk bild av efterkrigstiden vuxit fram, som indirekt stöder Tingstens antitotalitära skiljelinje mellan de västliga demokratierna och det Sovjetryska väldet. Norrmannen Geir Lundestad talar i _East, West, North and South. Major Developments in International Politics since 1945_ (1999) om det sovjetiska väldet i öst som »empire by imposition» och den västliga alliansen som »empire by invitation». Den amerikanska kalla krigsforskningens 'grand old man' John Lewis Gaddis har sammanfattat den nya bilden i _We Now Know. Rethinking Cold War History_ (1997) och _The Cold War_ (2005). Något stöd för Olof Lagercrantz och Arne Ruths »nytänkande» i frågan står inte att finna i den nya forskningen.
. _Kultursidan_ (2012), s. 299.
.Exempel på vänsterintellektuella vakthundar som med förkärlek ägnade sig åt stigmatisering av kommunismens kritiker är idéhistorikern Sven-Eric Liedman och den dåvarande filmrecensenten i Sydsvenska Dagbladet Jan Aghed.
.Dagens Nyheter 18/7 2010, min kursiv. Nycander riktade dagen efter på DN Debatt ett angrepp på de svenska universiteten för att de i betydande grad blivit vänsterbastioner. Fyra år senare utkom Svante Nycander med en minnesbok som rör striden mellan de två chefredaktörer som efterträdde Herbert Tingsten. Se Svante Nycander: _Makten över åsikterna. DN under Olof Lagercrantz och Sven-Erik Larsson_ (2014).
.Lars-Åke Augustsson: _Arne Ruth talar ut. Minnen, medier, moral_ (2013).
.op.cit., s. 105. Vad angår Peter Weiss bör väl i rättvisans namn sägas att han visserligen var en man med många strängar på sin lyra, men politiskt hörde han knappast hemma i ett »globalt rättviseperspektiv», utan i en folkfrontskommunism med sympatier för diktaturerna i öst och i denna tradition fanns ingen »global rättvisa». Se beskrivningen av Weiss vänsterpolitiska miljö och engagemang i Gunilla Palmstierna-Weiss, _Minnets spelplats_ (2013). I _Arne Ruth talar ut_ pekar för övrigt Augustsson på det faktum att Peter Weiss hade en heroiserande syn på både (Nord) Vietnam och Kuba och att han intog en apologetisk hållning till den nya vietnamesiska regimen efter 1975, t.ex. vad gäller fånglägren och båtflyktingarna (s. 106). Ruth kommenterar inte dessa påpekanden.
.op.cit., s. 77, 107, 141, 143. DN-medarbetaren Lars Linder säger i samma bok att den postmodernism som »DN:s kultursida under 1980-talet varit med om att lansera hade slagit igenom...» (s. 229).
.op.cit. Citaten återfinns i tur och ordning på s. 106, 107, 130, 155, 154, 163.
.op.cit. Citaten återfinns i tur och ordning på s. 116, 214, 190, 185, 189, 334.
.Francois Furet, _Slutet på en illusion_ (fr. 1995; sv. 2011) s. 720.
.Dagens Nyheter 29/2 1988.
.op.cit., 3/3 1988.
.ibid.
.Dagens Nyheter 10/4 1988.
.ibid.
.Sydsvenska Dagbladet 29/4 1991.
.op.cit., 4/5 1991.
.op.cit., 11/5 1991. Aghed vidgick själv denna formulering i en replik till Gerner. Se Sydsvenska Dagbladet 11/5 1991.
.op.cit., 4/5 1991.
.Till dem som kritiserade Gerner för detta hörde jag själv och Peter Luthersson, då kulturredaktör på Sydsvenska Dagbladet. Den liberale Luthersson var upprörd över att han av Gerner räknades in bland kultursidornas »velourstalinister». Sydsvenskan 10 oktober 1991. Själv menade jag att Gerner borde sänka tonläget, även om jag skrev att jag i sak delade hans syn på vänsterns hållning till Sovjetunionen. Se Sydsvenskan 30/5 1991.
.Göran Hägg publicerade sin recension i Aftonbladet 30/5 1980. Disa Håstads recension återfinns i Dagens Nyheter 4/7 1980 och det följande inlägget av en Sven Ekström infördes i samma tidning 16/7 1980. Det fanns hatinlägg, även i etablerade medier, också innan internet.
.Sven-Eric Liedman reagerade på denna karakteristik genom att tala om »Den oförtröttlige Kristian Gerner som i inkvisitoriskt nit brännmärkt alla dem som han nu vill ta bort från den offentliga debatten». Sydsvenskan 22/11 1991.
.Håkan Arvidssons inlägg fanns i Sydsvenskan 26/10 1991.
.op.cit., 8/11 1991.
.Brevet är publicerat på Sydsvenskans kultursida 5/12 1991 och undertecknat av följande kolleger vid historiska institutionen i Lund: Peter Aronsson, Lars Berggren, Peter Billing, Per Bolin-Hort, Lars Edgren, Monika Edgren, Mats Greiff, Jón-Gunnar Grjetarsson, Hans Hägerdahl, Kenneth Johansson, Marion Leffler, Ingrid Millbourn, Lars Olsson, Ulla Rosén, Bengt Sandin, Hans Wallengren.
.Sydsvenskan 5/12 1991.
.op.cit., 17/12 1991
.Redogörelsen bygger dels på en artikel i Sveriges universitetslärarförbunds tidskrift, SULF, nr. 8/2002 och dels på en artikel av den politiske redaktören för Kvällsposten, Peter J. Olsson i _Svensk Tidskrift_ , nr 2/2003. (www.svensktidskrift.se/2003-2/2003-2-20.html)
VII. KULTURKAMPENS HÖJDPUNKT
. _Vänstern och tyranniet_ , s. 173.
.op.cit., s. 50. Ahlmark utpekar här särskilt Sovjetunionen, men hans undran gäller hållningen till de kommunistiska systemen i allmänhet.
. _The Fellow-Travellers_ , s. 1–16.
.Se t.ex. _The God that Failed_ från 1949, med bidragsgivare som André Gide och Arthur Koestler.
.Faktum är att Ahlmark inte var de sofistikerade distinktionernas man. Teorier uppfattade han av allt att döma som en intellektuell börda som sällan hade praktisk relevans.
.Om vänsterns inflytande i Sverige skriver Ahlmark: »Marxistiska partiers betydelse för svensk politik har nästan alltid varit starkt begränsad.» _Vänstern och tyranniet_ , s. 198.
.op.cit., s. 196.
.op.cit., s. 10.
. _Det öppna såret_ , s. 141.
.Han anförde här flera nya och i eftervärldens ljus bisarra exempel. Så exempelvis refererar han följande om Olof Lagercrantz som hemkommen från Kina deltog i ett möte på Kårhuset i Stockholm. Han fick här en fråga om det fanns en »personkult» i Kina och svarade följande: »Det finns ett abstrakt drag hos Mao Tse-tung som motverkar det... Dessutom är han (Mao) ju en vettig, en mogen karl. de Gaulle framstår som en pompös soptunna, Churchill som en tolvårig scoutledare i jämförelse med honom.» _Det öppna såret_ , s. 307.
. _Vänstern och tyranniet_ , s. 201–218.
op.cit., s. 83–124. Historikern Håkan Arvidsson har utgivit dokumentsamlingen (med ett eget längre förord) _Affären Rami-Bergman_ (1994), som ingående påvisar judefientligheten vid Uppsala universitets teologiska fakultet.
. _Vänstern och tyranniet_ , s. 96f respektive 117.
.Se: <https://www.holocaustremembrance.com/media-room/stories/workingdefinition-antisemitism>. Beslutet om en »working-definition» av antisemitism togs inom IHRA (International Holocaust Remembrance Alliance) som är en organisation för upplysning och utbildning om antisemitism. Den består av regeringsföreträdare för nästan samtliga EU-länder, liksom representanter för EU-kommissionen och Europarådet, samt ett antal stater utanför EU. På sitt möte i Bukarest den 26 juli 2016 antog organisationen den nämnda »arbetsdefinitionen». I denna fastslås att jämförelser »of contemporary Israeli policy to that of the Nazis» kan, beroende på sammanhanget, vara uttryck för antisemitism. Samtliga exempel på antisemitism som nämns, även förnekandet av Förintelsen, är försedda med formuleringen »kan, beroende på sammanhanget, vara».
.Israels eget agerande spelade också en roll för reaktionerna. Det gäller främst massakern på palestinier i flyktinglägren Sabra och Shatila, som de internationella medierna ensidigt tillskrev Israel. Övergreppen utfördes av den kristna libanesiska milisen, men en israelisk undersökningskommission menade likväl att dåvarande försvarsminister Sharon, genom underlåtenhet att ingripa, bar ett indirekt ansvar för händelsen.
.Se Per Ahlmarks skrift _Det eviga hatet: om nynazism, antisemitism och Radio Islam_ (1993).
. _Vänstern och tyranniet_ , s. 60
. _The End of the Cold War_ , s. 50, 93, 97.
.Ahlmark refererar intressant nog en artikel av den socialdemokratiske journalisten Olle Svenning som 1993 skrev att: »SPD-politiken under oppositionsåren kom i praktiken att tjäna som skydd för härskarna i DDR och som hot mot medborgarrättsrörelserna i Östeuropa.» Se _Det öppna såret_ , s. 184. Med »oppositionsåren» avses här perioden 1982 till 1989.
.Den tidigare Moskvaambassadören Örjan Berner har, enligt citat i Ahlmarks bok, karakteriserat Palmes linje på följande vis: »I själva verket har det varit så att Stockholms utrikespolitiska linje på för Moskva viktiga områden, det vill säga rustningsfrågor i Norden och Europa, sällan löpt så parallellt med de sovjetiska ståndpunkterna som under 1980-talet.» _Det öppna såret_ , s. 171.
.op.cit., s. 163.
.op.cit., s. 168–169.
. _Vänstern och tyranniet_ , s. 17.
. _Herbert Tingsten och det kalla kriget_ (1995).
.op.cit. Om Tingstens manikeiska världsbild, s. 50 och 148; om hans »oblidkeliga antikommunism», s. 58; om hans syn på Sovjetunionen som »allmänt expansivt», s. 153 och om Tingstens »endimensionella världsbild», s. 157.
.Till Johanssons försvar skall dock sägas att det vetenskapliga underlaget för uppgörelsen med revisionismen växte fram långsamt. Se de tidigare nämnda böckerna av historikern John Lewis Gaddis och hans inträngande kritik av revisionismen, _We Now Know. Rethinking Cold War History_ (1997) och _The Cold War_ (2005).
.Jag citerar här Ahlmark ordagrant: »Den första stora skevheten i Johanssons bok på 375 sidor är alltså att han VÄGRAR ATT TYDLIGT BESKRIVA VAD VÄSTMAKTERNA FAKTISKT SÅG: en diktaturs militära och politiska erövring av nation efter nation i östra och centrala Europa.» _Det öppna såret_ , s. 92. (Ahlmarks versaler).
. _Det öppna såret_ , s. 109, 121, 124.
. _Herbert Tingsten och det kalla kriget_ , s. 287.
. _Det öppna såret_ , s. 35. Ahlmarks kursiv.
.op.cit., s. 31.
.op.cit., s. 23–38.
.Här skall tillfogas att t.o.m. Marx ställde sig positiv, men inte okritisk, till den västliga kolonialismen, ja, till det borgerliga samhället i allmänhet. Han menade att det senare stegrade produktivkrafterna och därmed spelade en progressiv historisk roll för att påskynda övergången till socialism.
.I Ahlmarkfejden inkluderar jag här såväl 1994 som 1997 års debatter. Jag skiljer dem dessutom inte åt kronologiskt, med något undantag, utan lyfter i stället fram ett antal teman som gick igen vid bägge tillfällena; inte heller när jag analyserar de olika positionerna härvidlag går jag nödvändigtvis kronologiskt tillväga. Snarare rekonstruerar jag olika debatt-teman, utifrån skribenternas egna utsagor.
.Svenska Dagbladet 9/3 1994.
.ibid., Ahlmarks kursiv. I samma artikel skrev Ahlmark att han kombinerade Paul Hollanders tes om främlingskap och desillusion med svenska förhållanden. Men i vare sig hans böcker eller debattartiklar spelar Hollanders teori någon roll. Fokus ligger på Sveriges påstådda särprägel.
.op.cit., 6/4 1994, Ahlmarks kursiv.
.Citaten av Håkan Hagwall är från Svenska Dagbladets ledarsida 4/2 1994. Här bör tilläggas att Ahlmark själv skrev i Expressen 12/3 1994, att: »Min bok vill belysa svenskt medlöperi under det senaste kvartsseklet.»
.Citaten av Åke Ortmark är från Svenska Dagbladet 3/3 1994.
.op.cit., 31/3 1994.
.ibid.
. _The Communist International, 1919–1943 Documents. Vol. I 1919–1922_ , red. Jane Degras s. 138–145, 382–393 (1971).
.Svenska Dagbladet 3/2 och 15/2 1994.
.Dagens Nyheter 5/2 1994.
.Aftonbladet 11/2 1994.
.Samtliga citat från Ahnlund är hämtade ur Svenska Dagbladet 2/3 1994.
.Dagens Nyheter 7/3 1994.
.Göran Rosenberg: _Da Capo al Fine_ (1994).
.Citatet är hämtat från Per Ahlmarks _Det öppna såret_ , s. 77–79.
.op.cit., s. 74.
.op.cit., s. 75.
.Dagens Nyheter 16/2 1997.
.Svenska Dagbladet 20/2 respektive 17/3 1997.
.Dagens Nyheter 5/2 1994.
.Svenska Dagbladet 31/3 1994.
.För en dokumentation av detta synsätt se Håkan Arvidsson & Lennart Berntson: _Makten, socialismen och demokratin_ (1980), Del I.2, del 2.3 och 2.4.
. _Slutet på en illusion_ , s. 28. Furet framhävde också att: »Om man radade upp alla kända europeiska författare som under 1900-talet, vid ett eller annat tillfälle, varit kommunister eller prokommunister, fascister eller profascister skulle man få fram en veritabel tankens adelskalender över vetenskap och litteratur.» (s. 27).
.Nathan Shachar visar i sin lysande skildring av det spanska inbördeskriget, _Sin egen värsta fiende_ , (2016) på den brutalitet och blodtörst med vilken de vänsterextrema miliserna kastade sig över oskyldiga »borgare» – hur de ohämmat mördade, våldtog och plundrade. De överträffades bara av Francosidans mördarligor. Den revolutionära passionen – »klasshatet» – odlas fortfarande inom dagens vänsterextrema strömningar och bestås med överslätande och förstående tolkningar i flera liberaldemokratiska medier. Göteborgskravallerna utgör här ett lärorikt exempel.
.Per Ahlmark & Georg Klein: _Motståndet – med mera. Tjugo brev om död och liv_ (2001) s. 293. Boken utkom första gången 1991 på Bonniers (med undertiteln _Arton brev om död och liv_ ). 2001 års utgåva publicerades på Natur & Kultur.
VIII. POSTKOMMUNISMENS GENOMBROTT
.Vänsterpartiets valmanifest 1994: snd.gu.se/sv/vivill/party/v/manifesto/1994
. _Med historien som motståndare_ , s. 320f.
.Se Göran Persson, _Min väg, mina val_ (2007).
.Vänsterpartiets program för år 2000 _En solidarisk värld_ : www.snd.gu.se/vivill
.Göran Therborn: _From Marxism to Post-Marxism_ (2008, 2010), s. 32f, 37f.
.Denna typ av kritik kom givetvis också från de kvarvarande resterna av 1968 års bokstavsvänster, som efter kommunismens fall reducerats till perifera sekter i den mån de inte redan var upplösta. I sina huvuddrag bestod yttersta vänstern en bit in på 1990-talet av följande grupperingar: De två huvudgrenarna av 68-vänstern hade båda utgått från Sveriges Kommunistiska Parti (SKP), alltså det parti som bildades 1921 och var en del av den Moskvastyrda Komintern-internationalen fram till 1943 och som C. H. Hermansson var ordförande för mellan 1964 och 1975; partiet bytte 1967 namn till VPK. Samma år bröt sig en »maoistisk» (1) fraktion ut ur partiet under beteckningen KFML (Kommunistiska Förbundet Marxist-Leninisterna), vilken kom att helt dominera den svenska Vietnamrörelsen och även överta den klassiska studentorganisationen Clarté. KFML delades 1970 då en mer »stalinistisk» (2) och Sovjetorienterad grupp, KFML(r), bildade eget. Medan KFML huvudsakligen var baserat i Uppsala-Stockholm, med Gnistan som huvudorgan, var KFML(r) en Göteborgsbaserad gruppering med tidningen Proletären som sitt huvudorgan. Den maoistiska strömningen (1) antog 1973 namnet SKP och var länge 68-vänsterns huvudgruppering. Utvecklingen i Kina fragmentiserade stegvis de svenska maoisterna och partiet upplöstes år 1990. Det stalinistiska KFML(r) (2), antog 1977 namnet KPML(r) och 2005 beteckningen KP (Kommunistiska Partiet). I valet 2010 fick det ca 8 000 röster. Till detta skall läggas att VPK, också under C. H. Hermanssons tid, rymde en hårdför stalinistisk och Sovjettrogen gruppering med säte i Norrbotten kring tidningen Norrskensflamman. Falangen var inflytelserik i VPK men bröt sig ur 1977 under namnet APK (Arbetarpartiet Kommunisterna) och antog 1995 partibeteckningen SKP, som blivit ledigt efter det maoistiska SKP:s upplösning 1990. Sist men inte minst fanns trotskisterna som efter flera bokstavsändringar ställde upp i valet 2002 under namnet Socialistiska Partiet (SP) och samlade 3 200 röster.
. _Liken i garderoben lever än_ , s. 278. Uttalandet är från radioekots lördagsintervju med Jenny Lindahl den 14/11 1998.
.Dagens Nyheter 17/5 1996.
. _Liken i garderoben lever än_ , s. 230.
. _Med historien som motståndare_ , s. 326f.
. _Liken i garderoben lever än_ , s. 231.
. _Utan heder. En bok om ohederliga debattörer och högerns rädsla för förändring_ , red. Aron Etzler (1999) s. 12.
. _Liken i garderoben lever än_ , s. 274.
. _Med historien som motståndare_ , s. 355f.
. _Socialistisk Debatt_ nr 2–3/1981, s. 6f. Lars-Arne Norborg hade i _Vitboken_ kommenterat Therborns inlägg på följande vis: »Anslaget gavs av Göran Therborn som i Socialistisk Debatt 1981/23 (sic!) fann hela diskussionen vara en skendebatt. VPK:s kritiska och oberoende hållning till de existerande efterrevolutionära samhällena hade varit klar och oåterkallelig alltsedan 20:e partikongressen 1964. Men även dessförinnan representerade kommunismen i Sverige en äkta radikal demokratisk tradition, trots att den tro på oktoberrevolutionen som gav pionjärerna kraft visade sig vara en myt som var blind för Stalins grymhet.» (s. 30).
. _Vitboken_ , s. 87, min kursiv.
.Till denna breda tradition hade Göran Therborn appellerat i DN:s serie »Nästa vänster»; se Dagens Nyheter 10/7 1990, men också Ny Dag nr 5/1990.
.Se Franz Borkenau: _World Communism_ (1939/1962), kapitel XXIII. Med en ironisk kapitelrubrik betecknar Borkenau kommunisterna – efter 1936 – som »Champions of democracy». Först när Hitler tog makten 1933 insåg kommunisterna i väst att de kunde ta skydd bakom den liberala demokratin de medverkat till att underminera.
.Se Håkan Holmbergs _Folkmakt, folkfront, folkdemokrati_ (1982) s. 54–78.
.Göran Therborn: _What does the Ruling Class do when it Rules_? (1978) min översättning. Se exempelvis s. 16 där Therborn skriver följande: »Still, it should not be forgotten that the mainstream of the current [68-vänstern, LB] resurgence of Marxist theory and research is flowing out of the Communist tradition.»
.Therborn (1978) s. 76.
.op.cit., s. 123–124.
.op.cit., s. 254.
.op.cit., s. 259.
.op.cit., s. 254.
.op.cit., s. 249. Se också s. 123.
.Torbjörn Tännsjö: _Demokrati och proletär revolution_ (Doxa) 1977.
.op.cit., s. 100.
.op.cit., s. 128f.
.op.cit., s. 133.
.ibid.
.op.cit., se avsnittet »Den marxistiska demokratidoktrinen», s. 94–149.
.Som framgår av avsnitt V.1 och V.2 tillhörde både Therborn och Tännsjö, i varje fall från 1980-talets senare del, den strömning inom VPK som ville bryta med den traditionella kommunismen och övergå till en mer demokratisk socialism.
.Stéphane Courtois (et.al.), _Kommunismens svarta bok_. Bokförlaget DN (1999). Förord av Arne Ruth.
.En utmärkt översikt över forskningslitteraturen om kommunismens övergrepp utgavs 2008 av Klas-Göran Karlsson och Michael Schoenhals på uppdrag av Forum för levande historia, _Brott mot mänskligheten under kommunistiska regimer. Forskningsöversikt_. Klas-Göran Karlsson publicerade redan 2003 boken _Terror och tystnad. Sovjetregimens krig mot den egna befolkningen_.
.Citaten från Ruths förord i _Kommunismens svarta bok_ , i forstättningen KSB, s. 8, min kursiv.
. _KSB_ (1999), Ruths förord, s. 5.
.op.cit., Courtois inledning, s. 28.
.op.cit., s. 34.
.op.cit., s. 37.
. _Vitboken_ , s. 89.
.Samtliga citat som rör Ehnmarks recension är från Sydsvenska Dagbladet 18/4 1999. Ehnmark anmälde också boken i Expressen 21/4 – kritiken var i allt väsentligt den samma.
.Samtliga citat som refererar till Liedmans recension är från Dagens Nyheter 4/5 1999.
.Se _Vitboken_ , s. 89.
.Jag antar att Liedman med »motståndsmän» här avsåg de som deltagit aktivt i kampen mot nazism och tysk ockupation under andra världskriget.
.Samtliga citat som refererar till Glans understreckare är från Svenska Dagbladet 13/6 1999.
.Citaten ur Per Svenssons recension är hämtade från Expressen den 21/4 1999.
.Per Wirtén skrev i Expressen 21/4 1999.
.Samtliga Rojas citat är från hans krönika i Göteborgsposten 15/5 1999.
.Dagens Nyheter 29/4 1999.
. _Utan heder_ (1999).
.op.cit., s. 13.
.op.cit., s. 63.
.op.cit., s. 63f.
.op.cit., s. 253, 262. Lindahl var uppenbarligen okunnig om att begreppet »rättigheter» inom den kommunistiska rörelsen betraktades som ett uttryck för borgerlig ideologi och under flera årtionden var anatema inom densamma.
.op.cit., s. 19.
X. 1968 OCH FILOSOFIN I SVERIGE
.Ingemar Hedenius: _Om människans moraliska villkor_ (1972) s. 17ff.
.Op.cit., s. 320f.
.Konrad Marc-Wogau: _Att studera filosofi_ (1961) s. 19.
.Georg Lukács: _Historia och klassmedvetande_ (1968), Gunnar Gunnarssons förord, s. 28.
. _Humaniora på undantag?_ (1978) s. 68ff.
.Michel Foucault: _Vansinnets historia_ (1983), Sune Sundessons förord, s. If.
.op.cit., s. IIIf.
.op.cit., s. XVII.
.op.cit., s. XIX.
.Michel Foucault: _Vetandets arkeologi_ (2002), Bengt Olle Bengtssons förord, s. 7.
.Michel Foucault: _Sexualitetens historia. Band 1: Viljan att veta_ (sv. övers. av Britta Gröndahl 2002) s. 104.
.Stefan Arvidsson red.: _Marxismens filosofi_ (2007).
XII. 68-VÄNSTERNS EFTERMÄLE
.Stéphane Courtois m.fl.: _Kommunismens svarta bok_ (sv. övers. av Jan Stolpe 1999) s. 31f.
.op.cit., s. 34.
.Lars Åke Augustsson & Stig Hansén: _De svenska maoisterna_ (2001) s. 194.
.Per Svensson: _Frostviken. Ett reportage om Per Olof Sundman, nazismen och tigandet_ (1998) s. 76.
.op.cit., s. 172.
.op.cit., s. 189.
## **LITTERATUR**
**För _Vänstern 1968–1984_ samt kapitel X och XII**
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HOLLANDER, PAUL: _Political Pilgrims. Travels of Western Intellectuals to the Soviet Union, China and Cuba_ (1981; 1990)
HOLLANDER, PAUL: _The End of Commitment. Intellectuals, Revolutionaries and Political Moralists_ (2006)
HOLMBERG, CLAES-GÖRAN: _Upprorets tradition. Den unglitterära tidskriften i Sverige_ (1987)
HOLMBERG, HÅKAN: _Folkmakt, folkfront, folkdemokrati. Den svenska kommunismen och demokratifrågan 1943–1977_ (1982)
HOROWITZ, DAVID: _Från Jalta till Vietnam_ (1966)
ISAKSSON, ANDERS: _Per Albin III. Partiledaren_ (1996)
JOHANSSON, ALF W.: _Herbert Tingsten och det kalla kriget_ (1995)
KARLSSON, KLAS-GÖRAN: _Terror och tystnad. Sovjetregimens krig mot den egna befolkningen_ (2003)
KARLSSON, KLAS-GÖRAN & SCHOENHALS, MICHAEL: _Brott mot mänskligheten under kommunistiska regimer. Forskningsöversikt_ (2008)
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LUNDQVIST, ÅKE: _Kultursidan. Kulturjournalistiken i Dagens Nyheter 1864–2012_ (2012)
MACPHERSON, C.B.: _Den mångtydiga demokratin_ (1966; 1968)
MEDVEDEV, ROY A.: _Let History Judge. The Origins and Consequences of Stalinism_ (1971)
NORBORG, LARS-ARNE, m.fl.: _Lik i garderoben? En rapport om SKP/VPK:s internationella förbindelser, »Vitboken»_ (1992, 1996)
NYCANDER, SVANTE: _Makten över åsikterna. DN under Olof Lagercrantz och Sven-Erik Larsson_ (2014)
OHLSSON, PER T.: _Landet utanför. Saddam och ståndpunkterna_ (1991)
PALMSTIERNA-WEISS, GUNILLA: _Minnets spelplats_ (2013)
PRIESTLAND, DAVID: _The Red Flag. Communism and the Making of the Modern World_ (2009)
REVEL, JEAN-FRANÇOIS: _La grande parade. Essai sur la survie de l'utopie_ (2000)
ROSENBERG, GÖRAN: _Da Capo al Fine_ (1994)
SERVICE, ROBERT: _Comrades. Communism: A World History_ (2008)
SERVICE, ROBERT: _The End of the Cold War. 1985–2001_ (2015)
SHACHAR, NATHAN: _Sin egen värsta fiende. Essäer om spanska inbördeskriget_ (2016)
SKOTT, STAFFAN: _Liken i garderoben. Vänsterpartiets förflutna 1917–1989_ (1991)
SKOTT, STAFFAN: _Liken i garderoben lever än. Undanflykternas mästare. Om Vänsterpartiets förflutna 1918–1998_ (2000)
STRÖMSTEDT, BO: _Löpsedeln och insidan_ (1994)
SULF. _Sveriges universitetslärarförbunds tidskrift_. Nr 8/2002
SVEDJEDAL, JOHAN: _Spektrum. Den svenska drömmen_ (2011)
SVENSSON, JÖRN: _Du skall ta ledningen och makten kamrat_ (1974)
THERBORN, GÖRAN: _What does the Ruling Class do when it Rules?_ (1978)
TINGSTEN, HERBERT: _Socialdemokratins idéutveckling I–II_ (1941)
TINGSTEN, HERBERT: _Mitt liv. Ungdomsåren_ (1961), _Mellan trettio och femtio_ (1962), _Tidningen_ (1963), _Tio år_ (1964)
TINGSTEN KLACKENBERG, JILL: _I skuggan av Tingsten. En släktberättelse_ (2010)
TÄNNSJÖ, TORBJÖRN: _Demokrati och proletär revolution_ (1977)
ÅHNBERG, ANNIKA: _Isprinsessa_ (1999)
## **PERSONREGISTER**
Adelsohn, Ulf
Adler-Karlsson, Gunnar , ,
Adorno, Theodor ,
Agamben, Giorgio ,
Aghed, Jan ,
Ahlmark, Per , , –, –, , ,
Ahnlund, Knut –
Ahrén, Björn
Alfredson, Hans
Alsterdal, Alvar –
Althusser, Louis , , , , , , , ,
Amin, Samir
Anderson, Perry ,
Andersson, Gunder ,
Andersson, Lars ,
Andersson, Sten
Arafat, Yassir ,
Ardelius, Lars
Ariès, Philippe
Arnér, Sivar
Aron, Raymond
Arvidsson, Hans
Arvidsson, Håkan , , , , , ,
Aspenström, Werner
Augustsson, Lars-Åke
Axelsson, Sigbert
Bacon, Francis
Badiou, Alain
Bahr, Egon ,
Baran, Paul
Baude, Frank , ,
Baudrillard, Jean
Beck, Ulrich
Beckman, Staffan ,
Beckman, Åsa
Bell, Daniel
Bengtsson, Bengt Olle
Bengtsson, Margot ,
Benjamin, Walter ,
Bentham, Jeremy
Berggren, Tobias ,
Bergh, Magnus
Bergman, Jan
Bergner, Petter ,
Bergom-Larsson, Maria
Berntson, Lennart , –,
Bildt, Carl ,
bin Ladin, Usama
Blair, Tony
Blix, Hans
Bohman, Gösta
Bonnier, Lukas
Botha, Pieter Willem (P. W.)
Braconier, Fredrik
Branting, Georg
Branting, Hjalmar
Braudel, Fernand
Bredow, Einar von
Britton, Sven
Brögger, Suzanne
Burman, Anders
Bylin, Gunnar ,
Byron, lord
Carlsson, Lasse
Carlyle, Thomas
Castells, Manuel
Castro, Fidel , , , ,
Caute, David ,
Chrusjtjov, Nikita , , , ,
Clason, Synnöve
Clemenceau, Georges
Conquest, Robert –, ,
Courtois, Stéphane –, , ,
Dahlberg, Leif
Dahlkvist, Mats
Danielsson, Tage
Delblanc, Sven , , , , ,
Deleuze, Gilles
Dencik, Lars , , ,
Derrida, Jacques ,
Descartes, René
Dutschke, Rudi ,
Ebert, Friedrich
Ehnmark, Anders , , –, , , –, , , –, ,
Ekelöf, Maja
Ekerwald, Carl-Göran , ,
Ekman, Kerstin
Elzinga, Aant ,
Engdahl, Horace ,
Engdahl, Per
Engels, Friedrich , , , , , ,
Englund, Peter
Engström, Claes
Engzell, Bitte
Enquist, Per Olov , , , , , ,
Ericsson, Lars O.
Eriksson, Jörgen ,
Esbati, Ali ,
Etzler, Aron ,
Fagerberg, Sven , ,
Fanon, Frantz , , , ,
Fjellström, Roger
Forssell, Lars
Foucault, Michel , , –
Franco, Francisco
Frank, Andre Gunder
Franzén, Lars-Olof ,
Fredriksson, Gunnar , , , , ,
Fredriksson, Göran
Frenander, Anders ,
Fukuyama, Francis ,
Furet, François , ,
Gahrton, Per –,
Garme, Cecilia
Garpe, Margareta
Gaulle, Charles de
Gellerfelt, Mats ,
Gerner, Kristian , –
Glans, Kay –, , , , , , ,
Gleichmann, Gabi
Goffman, Erving
Gorbatjov, Michail , ,
Gouldner, Alvin W. ,
Gramsci, Antonio , ,
Gudmundsson, Jan
Guillou, Jan , , , ,
Gunnarsson, Gunnar
Gustafsson, Bo , , , , , ,
Gustafsson, Lars , , , , , , , , ,
Gustafsson, Madeleine –
Gyllensten, Lars
Gårdlund, Torsten
Habermas, Jürgen , , ,
Haglund, Fritiof
Hagwall, Håkan ,
Hammar, Bo , , , ,
Hansson, Per Albin
Hardt, Michael ,
Havel, Václav
Hayek, Friedrich von
Hedenius, Ingemar , , , , , –
Hegel, Friedrich , ,
Heidegger, Martin , –
Herlitz, Lars
Hermansson, Carl-Henrik (C. H.) , , , , , , , , ,
Hermansson, Jörgen ,
Hesslow, Germund
Hinderson, Jan
Hjärpe, Jan
Ho Chi Minh
Hobbes, Thomas
Hollander, Paul , , –
Holm, Hans Axel
Holmberg, Nils ,
Honecker, Erich ,
Horkheimer, Max
Hoxha, Enver –
Hua, Gufeng
Hultén, Gösta –
Hultqvist, Kenneth
Hume, David
Håkanson, Björn
Håstad, Disa
Hägg, Göran
Höglund, Zäta
Isaksson, Anders
Isaksson, Hans ,
Isaksson, Harry
Israel, Joachim , ,
Jarlbro, Gunilla
Jaruzelski, Wojciech
Jens, Walter
Jersild, Per Christian (P. C.) ,
Johanson, Kjell E. , ,
Johansson, Alf W. , –
Johansson, Ingvar ,
Johnson, Eyvind
Jonsson, Stefan
Josefsson, Jan
Jünger, Ernst ,
Jönsson, Reidar
Karlsson, Ingmar ,
Kemp, Peter
Kessle, Gun , ,
Khomeini, Ruhollah ,
Kim Il Sung ,
Klein, Georg ,
Kohl, Helmut ,
Kollontaj, Alexandra
Konrád, George ,
Krigel, Annie
Kuroń, Jacek
Kvist, Kenneth
Kågesson, Per
Köll, Anu Mai
Lagercrantz, Olof , , , , , , –, , , , , , , , , , , ,
Lagerlöf, Karl Erik , , , ,
Laing, Ronald
Landin, Sven , ,
Lappalainen, Tomas ,
Larsen, Aksel , ,
Larsson, Kalle
Le Roy Ladurie, Emmanuel
Leandoer, Kristoffer
Lekander, Nina
Lenin, Vladimir Iljitj Uljanov , –, , , , , –, , , , , , , , , , , , , , , , ,
Lévy, Bernard-Henri
Levy, Jack
Lévi-Strauss, Claude
Lidbom, Carl
Lidman, Sara , –, , , , , , ,
Liedman, Sven-Eric , , , , , , , , –, , , , –, , , , ,
Lin, Biao , ,
Lind, Ingela
Lind, Rina
Lindahl, Jenny , ,
Lindgren, Stefan ,
Lindkvist, Kent , , , ,
Lindqvist, Sven ,
Lloyd George, David
Locke, John
Lukács, Georg , , ,
Lundkvist, Artur , , ,
Lundqvist, Åke , , , , ,
Luspa, Elof
Luthersson, Peter
Luxemburg, Rosa
Lyotard, Jean-François ,
Lysander, Per , ,
Löfgren, Åke ,
Lönnroth, Johan
Lövkrona, Inger ,
Machiavelli, Niccolò
Mao Zedong (Mao Tse-Tung) , , , , , , , , , , , , , , , , , , , , , , , , , , ,
Marcos, Ferdinand
Marcuse, Herbert –, ,
Marc-Wogau, Konrad ,
Margolin, Jean-Louis
Marx, Karl , , , , , , , , , , –, , , , , , , , , , , , , , , , ,
Meinhof, Ulrike
Moberg, Vilhelm
More, Barrington Jr.
Myrdal, Jan , , –, , –, , –, , –, , , , , , , , ,
Negri, Antonio ,
Nerman, Bengt
Newton, Isaac
Nietzsche, Friedrich , , –
Nixon, Richard , , , ,
Norborg, Lars-Arne –
Nordin, Svante ,
Norén, Kjerstin ,
Nycander, Svante –, , ,
Ohly, Lars , , –
Ohrlander, Kajsa
Olsson, Anders
Ortega Y Gasset, José
Ortmark, Åke ,
Osten, Suzanne
Palm, Göran , –, –, , , , , , , ,
Palme, Christian
Palme, Olof , , , , , , , , , –, ,
Persson, Göran ,
Petersson, Kenneth
Pinochet, Augusto ,
Platon
Pleijel, Agneta , ,
Pol Pot –, , , , , ,
Pompidou, George
Popper, Karl ,
Poulantzas, Nicos ,
Putin, Vladimir
Radnitzky, Gerard , –
Rami, Ahmed , ,
Rasmusson, Torkel –
Reagan, Ronald , , , , ,
Robespierre, Maximilien
Rojas, Mauricio
Rosenberg, Göran , –,
Rosin, Björn Erik
Rummel, Rudolph J. , , ,
Rushdie, Salman
Ruth, Arne , , , , –, , , –, , ,
Sacharov, Andrej
Saddam Hussein –,
Said, Edward
Samuelsson, Jan –
Sartre, Jean-Paul , ,
Schein, Harry
Schmid, Herman , ,
Schmidt, Werner ,
Schmitt, Carl , , ,
Schneider, Peter
Schori, Pierre ,
Schröder, Gerhard
Schunnesson, Torgny –
Schyman, Gudrun , , ,
Skott, Staffan , , ,
Snow, Charles Percy (C. P.)
Solzjenitsyn, Aleksandr , , ,
Spengler, Oswald
Stagh, Marina
Stalin, Josef –, , , , , , , , , , , , , , , , , , , , , , –, , , ,
Stenkvist, Jan
Stern, Fritz
Strömstedt, Bo ,
Sundberg, Kjell
Sundman, Per Olof , ,
Sunesson, Sune , –
Svensson, Jörn , –,
Svensson, Per , , ,
Szasz, Thomas
Sörlin, Sverker , ,
Tarschys, Daniel , , ,
Tham, Carl , , ,
Thatcher, Margaret , ,
Therborn, Göran , , –, , , , , , , , , –, , ,
Tingsten, Herbert , , , –, –, ,
Tingsten, Kaj
Tjäder, Per Arne ,
Tornborg, Rita
Trotskij, Lev , ,
Trotzig, Birgitta
Tännsjö, Torbjörn , , –,
Törnebohm, Håkan
Uhrbom, Odd –
Ulbricht, Walter
Undén, Östen , , , ,
Vennberg, Karl , , –, , ,
Vestbro, Dick Urban ,
Wallenström, Ulrika
Weiss, Peter , , –, ,
Werner, Lars –,
Werth, Nicolas , , ,
Westerberg, Bengt
Westerberg, Sten
Wickman, Lars –
Widsell, Jörgen
William-Olsson, Magnus
Williams, Raymond
Wilson, Woodrow
Wirtén, Per
Wizelius, Ingemar
Wolf, Christa
Wolff, Robert Paul
Wright, Georg Henrik von
Wästberg, Per ,
Xiaoping, Deng ,
Xoxe, Koçi ,
Ytterberg, Claes-Bertil ,
Zaremba, Maciej
Zennström, Per-Olov ,
Zetterberg, Mats ,
Zetterlund, Monica
Žižek, Slavoj ,
Åhnberg, Annika
Åkervall, Sigvard
Åslund, Anders
Åström, Sverker ,
## **INNEHÅLL**
| FÖRORD
---|---
| INLEDNING
DEL ETT
VÄNSTERN 1968–1984
I.| FRÅN DEN NYA VÄNSTERN TILL 68-VÄNSTERN
---|---
II.| VÄNSTERNS LÅNGA SJUTTIOTAL
| 1. 1968 – revolutionsåret
| 2. Sju feta år
| 3. Sju svåra år
III.| FARVÄL TILL VÄNSTERN? SAMMANFATTNING OCH SLUTSATSER OM PERIODEN 1968–1984
DEL TVÅ
VÄNSTERN 1987–2000
IV.| PROLOG
---|---
V.| SAMMANBROTTET I ÖST. 1990-TALETS FÖRSTA ÅR
| 1. Omvändelse under galgen
| 2. Kommunismen kastas på historiens sophög
| 3. Nostalgi och klagorop. DN Kultur
| 4. Inga lik i garderoben. Självrannsakan som uteblev
VI.| UPPGÖRELSERNAS TID
| 1. Om vänsterns moraliska skuld
| 2. Vänsterliberalismens svek. Herbert Tingsten går igen
| 3. Kampen mot antikommunismen. Fallet Gerner
VII.| KULTURKAMPENS HÖJDPUNKT. VID 1990-TALETS MITT
| 1. Till demokratins försvar
| 2. Ahlmarkfejden
VIII.| POSTKOMMUNISMENS GENOMBROTT. 1990-TALETS SLUT
| 1. Från marxism till feminism, ekologi och rättigheter
| 2. En äkta demokratisk tradition?
| 3. Brott och förnekelse
IX.| SLUTET PÅ EN ILLUSION. SLUTSATSER OM PERIODEN 1987–2000
X.| 1968 OCH FILOSOFIN I SVERIGE
| 1. Ett mål i sikte
| 2. Ett mål fördolt
XI.| 1968 I HISTORIENS LJUS
XII.| 68-VÄNSTERNS EFTERMÄLE
| NOTER
---|---
| LITTERATUR
| PERSONREGISTER
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 2,406
|
Today at our annual Microsoft Cloud and Hosting Summit, we released a new Microsoft-commissioned study by 451 Research, titled Digital Transformation Opportunity for Service Providers: New Paths to "Beyond Infrastructure." The study reveals an enormous opportunity for our cloud partners to help customers with both hybrid cloud implementation and with managed services for the next phase of digital transformation.
Microsoft's ecosystem of cloud and managed service provider (MSP) partners are uniquely positioned to help customers with the critical managed services and capabilities to achieve their top business goals.
According to Gartner, hybrid will be the most common usage of the cloud — but this will require public cloud to be part of the overall strategy by 2020. As digital transformation projects move forward, nearly 90% of surveyed customers are willing to pay a hefty premium to service providers to help them implement and manage their hybrid cloud environment.
The survey also found, that for the third year running, Microsoft Azure is the top pick among hybrid cloud users for IaaS Platforms. At Microsoft, we are working with tens of thousands of partners to joint deliver not only hybrid offerings but a full portfolio of cloud services. We have seen double-digit growth in Hosting & Managed Services over the past five years straight, with no signs of a slowdown.
Our 2016 study revealed that for the first time, companies were relying more on digital infrastructure rather than physical. That trend continued to manifest in customer responses this year, demonstrating increased opportunity for service providers. In fact, as the "beyond infrastructure" shift continues, services will account for 74% of hosting/cloud spend in 2017, up from 71% as reported in 2016.
Half of all organizations surveyed consider service providers as vital for future digital transformation projects. Even better, 60% of those would be willing to pay twice as much as they currently spend to have a single trusted advisor solution to manage all their digital transformation-related sourcing, implementation, and management needs.
Additionally, the new research shows that 62% of cloud/hosting infrastructure spending comes bundled with value-added services, rising to 84% for the next hosting/cloud infrastructure engagement. By owning the customer relationship end to end it provides the perfect platform for partners to build value-added services for their customers that will create stickiness and differentiation from the competition.
When it comes to vertical markets, the survey indicated that standout opportunities for digital transformation projects in the coming year include financial services, retail, and healthcare sectors.
In the short term, customers will need anything and everything related to data and site protection, including backup and recovery, disaster recovery/site protection, and archiving. All of these areas have shown substantial increases in current and planned usage for 2016-2017.
The evidence is clear: hybrid is here to stay and the opportunity for service providers has never been greater. If your business offers solutions and services that can benefit customers and their digital transformation projects, please join us by becoming a Microsoft partner today. To learn more about how to maximize the opportunity with Azure, and take your cloud business to the next level, check out our comprehensive Cloud Practice Development Playbooks.
Are you already implementing cloud or managed services? How do you see these new opportunities as helpful to your customer and impactful for your business? Share your thoughts with us in the comments below.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,491
|
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