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You are here: Home / SOCN Community / Barbara Entwisle, Ph.D.
Barbara Entwisle, PhD, is Kenan Distinguished Professor of Sociology and Vice Chancellor for Research at the University of North Carolina at Chapel Hill. As vice chancellor, Dr. Entwisle leads a campus-wide research program that attracted $788 million in contract and grant funding in fiscal 2011. She oversees the research support, compliance, communication, and technology transfer offices as well as a portfolio of 16 pan-university interdisciplinary research centers and institutes, including the Carolina Population Center, which she directed from 2002 to 2010. As a faculty member, Dr. Entwisle is currently Principal Investigator of the North Carolina components of the National Children's Study, a longitudinal cohort study focusing on environment and children's health. She is also leading an interdisciplinary study of migration and population-environment interactions in Northeast Thailand. Dr. Entwisle was elected Fellow of the American Association for the Advancement of Science (AAAS) in 2003 and named Mary Lily Kenan Flagler Bingham Distinguished Professor in 2007. She is Chair of the Expert Group on Social Science Data Infrastructure for the Organization of Economic Cooperation and Development's (OECD's) Global Science Forum and a member of the National Research Council's Board on Research Data and Infrastructure. Dr. Entwisle is past President of the Population Association of America and a former editor of Demography (2012).
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\section{Introduction}
Systems biology studies the complex systems which occur at many levels
of biology. Such systems involve large numbers of components and
interactions. We consider {\it metabolic networks}, that is, a set of
metabolites that can be interconverted by biochemical reactions. A
fundamental question about metabolic networks is to find knock out
strategies that block the operation of a given reaction or set of reactions.
A target reaction is
blocked if it cannot operate in a steady state. Some reactions can
be easily knocked out, while others may be expensive or impossible to
knock out directly. In this case, we consider the problem of blocking
target reactions by inhibiting other reactions so that the targets
cannot continue in a steady state.
Some applications of this problem are outlined in~\citep{KG04}
and~\citep{Kla06}, and an implementation has been included as part of
{{\small\tt CellNetAnalyzer}}~\citep{Klamt}, a {\small\tt MATLAB} package for analyzing
cellular and biochemical networks.
In this paper, we consider methods of computing
the minimal sets of reactions that need to be
disabled to block a given target. We call such a knock out
set a {\it cut set}. We remark that we focus only on the
(inclusion-wise) {\it minimal} cut sets since these are
the cheapest ways of blocking reactions in terms of
effort and impact on the system. The list of minimal cut sets
contains the same information as the full list
of cut sets but is much shorter.
We consider two main approaches.
The first is to build the hypergraph of elementary
modes and then compute the minimal cut sets as the
transversal of this hypergraph.
This strategy has been employed successfully in \citep{KG04},
however we observe we can substantially improve the
computation of the transversal hypergraph.
The second approach is to compute the minimal cut sets directly
using the ideas of Gurvich and Khachiyan \citep{GK99} and others
on generating monotone boolean formulae. This procedure
also generates the set of elementary modes employing a given set of
reactions as a by-product, which is a potentially useful feature.
We expect that these methods can be adapted to more of the
complex systems that are typical at many levels of biology.
Indeed, the question of finding minimal cut sets can be abstracted
to finding the minimal failure modes of a network, which
is a natural question arising in various contexts.
Some suggestions for using these methods in other
types of biochemical networks are presented in~\citep{KSRL+06}.
\section{Preliminaries}
We model a metabolic network as a number $m$ of metabolites
involved in a set $Q$ of $q$ reactions
(where $q$ is typically between $m$ and $2m$).
For our purposes, these reactions can be encoded in a
$m \times q$ matrix $N$ whose columns encode the metabolites
produced and consumed by a given reaction.
The matrix $N$ is known as the {\it stoichiometric} matrix.
The reactions may be divided into two types:
{\it reversible} reactions that can either
produce a given output from a given input or vice-versa;
and {\it irreversible} reactions which cannot operate in reverse.
Let $S$ be the index set of
the reversible reactions and $U=Q \setminus S$ be the index
set of the irreversible reactions.
We call our set of target reactions $T$, for simplicity we
will usually assume they are irreversible, i.e.~$T \subseteq U$.
Given such a network, we are interested in its potential steady-state
flux vectors. In
steady state, the reaction rates balance the metabolites, i.e. for
each metabolite it holds that the sum of the rates of all reactions
consuming the metabolite equals the sum of the rates of the reactions
producing it. We can represent such a steady state as a vector $x \in
\R^q$ s.t. $Nx=0$ and $x_i \ge 0$ for all $i \in U$. Then we can
formally define a {\it cut set} $C \subset Q$ as a subset such that
the system:
\begin{equation}\label{eq:cutset}
\{x \in \R^q | Nx=0, x_i \ge 0 ~ \forall i \in U, x_c=0 ~ \forall c \in C\}
\end{equation}
has only solutions with $x_t=0$ for all $t \in T$.
A {\it minimal cut set} (MCS) is simply a cut set none of whose proper
subsets is a cut set.
A concept closely related to minimal cuts sets is that of an
elementary mode. An {\it elementary mode} (EM) is a minimal
set of reactions that can exist in a steady state.
The importance of EM's in metabolic networks is discussed,
for example, in \citep{SFD00} and \citep{SKB+02}.
EM's are, up to a scaling factor, support minimal
solutions to the system:
\begin{equation}
\{x \in \R^q | Nx=0, x_i \ge 0 ~ \forall i \in U\}
\end{equation}
The problem of computing the EM's of a given system
has a nice geometric formulation: it reduces
to finding the
extreme rays of the cone $\{r | \hat{N}r=0, r \ge 0\}$, where
$\hat{N}$ is $N$ modified to represent reversible reactions as
opposite pairs of irreversible reactions.
This is described in \citep{GK04}. The EM's can then
be computed by applying the double description method
(see for example~\citep{FP96}) to this cone.
As observed in \citep{GK04}, EM's are
characterized up to a constant by their binary support patterns.
Hence we will also use the term EM to refer to this
support pattern, which we can view as a set of reactions.
For the purposes of finding cut sets for a given target $T$,
we consider only the EM's
that include at least one target reaction.
Note that cut sets are exactly
the sets of reactions that intersect each of these EM's.
The collection $\E$ of these EM's
defines a Sperner hypergraph $\H=(R,\E)$ on the set of reactions.
(A hypergraph is Sperner if it has no nested edges.)
The key observation
is that cut sets are exactly the sets
that intersect every edge of $\H$.
In the terminology of hypergraphs such sets are known as
{\it hitting sets} or {\it vertex covers}. The collection of all
minimal hitting sets for $\H$ is itself a hypergraph $\H'=(R,\E')$
which is dual to $\H$ in the sense that its minimal hitting sets
are the edges of $\H'$. The hypergraph $\H'$ is known as the
{\it transversal hypergraph} of $\H$ and is denoted $\operatorname{Tr}(\H)$.
The approach to computing minimal cut sets presented in
\citep{KG04} is to first compute the EM's hypergraph $\H$
via the double description method
and then compute $\operatorname{Tr}(\H)$. The computation of $\operatorname{Tr}(\H)$ is
done through an enumeration scheme.
This succeeds in solving four large scale networks arising in biomass
synthesis in {\it E.coli}. The computation benefits substantially
from effective preprocessing, but nevertheless consumes a lot
of time and memory. A faster and more memory efficient algorithm
is described in Section~\ref{se:berge}.
The double description method and the algorithms of
Section~\ref{se:berge} have uncertain complexity.
For this reason, we give in Section~\ref{se:joint} an algorithm
which generates both the EM's and the MCS's
directly from the stoichiometric matrix which has a
surprisingly good complexity bound of $m^{\operatorname{poly}(\log{m})}$ in
the output size. See~\citep{stougie:07} for an overview of the known
complexity results regarding EM and MCS algorithms.
\begin{remark}
We could consider weighting the reactions and looking for
a single minimum weight cut set.
This assumes that the costs are independent,
which is questionable - it may be possible to attain
some economies when knocking out multiple reactions.
Additionally, designing a weighting
function for blocking a metabolic reaction requires quantifying the costs to
disable various reactions, which is a labour intensive task.
Since we can in some interesting cases produce
the entire list of minimal cut sets, we view generating the
entire list as a suitable goal.
If we do want to find a minimum weight cut set, this problem
is the {\it minimum set cover}
problem on the dual hypergraph produced by interchanging the roles
of the vertices and edges of $\H$.
Minimum set cover is a classical NP-complete
problem, see for example \citep{ADP80}.
\end{remark}
\subsection{Characteristics of metabolic networks}\label{se:struct}
In our model the stoichiometric matrix $N$ is a real
$m \times q$ matrix. Typically, we would expect $N$ to
have many zero entries as any particular reaction will only involve
a few metabolites. Those non-zero entries will usually be small
integers since chemical reactions are discrete
rearrangements of molecules.
Note that a column describes the same reaction when it
is scaled by a positive factor.
Recall that reactions may be reversible or irreversible.
The hypergraphs of EM's and MCS's are 0-1 matrices $\H$ and $\H'$
determined by $N$. Each row of $\H$ ($\H'$) is an indicator
function for a given EM (MCS), that is, a row indicates the
complement of a maximal $C$
such that (\ref{eq:cutset}) has a solution with $x_t>0$ for some $t \in T$
(indicates a minimal $C$ such that (\ref{eq:cutset}) has only
solutions with $x_t=0$ for all $t \in T$).
Both $\H$ and $\H'$ have the same number of columns as $N$,
but in our applications they will have many more rows.
The orders of the rows and columns are arbitrary, but they can
affect the performance of the algorithms.
\subsubsection{Typical behaviour}
We understand from biological considerations that we would expect
to get many small hitting sets. The intuition is that
such networks have some important reactions whose loss
quickly impairs the operation of the network.
This can be quantified through ``fragility coefficients'' \citep{KG04},
which are an average of MCS sizes.
Their examples produced fragility coefficients in a narrow range.
\subsubsection{Test cases}\label{se:cases}
The motivating problems from \citep{KG04} are four networks
obtained from studying biomass synthesis in {\it E.coli}.
These are the growth modes for substrates acetate, succinate,
glycerol and glucose from the network presented in \citep{SKB+02}.
The objective is to block the single target reaction representing
growth in each of these networks.
For the purposes of this computation, a pair
of reactions corresponding to the same multifunctional
enzyme (transketolase) has been combined.
This modifies the input hypergraph by merging a pair of
vertices, creating some nested edges.
\section{Computing minimal cut sets via elementary modes}\label{se:berge}
The method proposed in \citep{KG04} to compute
minimal cut sets involves two steps. The first step is to compute the
set of EM's via polyhedral methods, and the second step
is to compute the transversal hypergraph of the EM's
hypergraph. Their method of computing the transversal hypergraph
requires enumerating many possible partial
solutions. As a result it consumes substantial time and memory,
and is ripe for improvement.
\subsection{Algorithms}
In this section we sketch algorithms
for the transversal hypergraph problem, including the enumeration
algorithm of \citep{KG04} and the algorithm
described by Berge in \citep{Ber89}.
For the latter, we describe several useful modifications.
\subsubsection{Enumeration algorithm}
This is the original algorithm implemented in {{\small\tt FluxAnalyzer}} \citep{KG04},
the predecessor to {{\small\tt CellNetAnalyzer}}.
Beginning at size 1, it tests for subsets of a given size. It maintains
a list of unused partial cut sets to avoid full enumeration
of the subsets.
The problem with this algorithm is that the list of partial cut sets
can grow quite quickly. Nevertheless, it can solve large problems.
A major reason for this is the
abundance of small cut sets, which keeps the list of partial cut sets
manageable.
\subsubsection{Berge's algorithm}\label{se:ourberge}
This algorithm~\citep{Ber89} orders the edges $e_1,e_2,\ldots,e_r$
of the hypergraph $\H$, and then computes in sequence
the transversal of each hypergraph $\H_i$ consisting
of edges $e_1$ through $e_i$. This can be done by taking all
the edges created by adding a vertex from $e_i$ to an edge in
$\H_{i-1}$ and keeping all the inclusionwise-minimal edges.
$\H_1$ has an edge consisting of a single vertex for
each vertex from $e_1$.
The performance of this algorithm depends on the size of the
intermediate transversals generated, which in turn depends on the
order of the vertices.
Intuitively we do not expect the size of a transversal
of a subgraph to substantially exceed the larger of the size of the
initial graph and the size of the final transversal hypergraph.
In practice this does often turn out to be the case, but Takata
exhibited an example where an intermediate transversal will have
size $\Theta(m^{\log(\log(m))})$ in the combined size of the input and
output for any ordering of the edges, see \citep{Hage07}. We
do not know of any non-trivial upper bounds for the size of
intermediate transversals or how to produce favourable edge
orderings when they exist.
A naive implementation of Berge's algorithm will be slow,
but there are a number of modifications that can make it more
effective.
The main bottleneck is the removal of superset rows from the
list generated from $\H_{i-1}$ and $e_i$. We do this in time
$O(n^2)$ in the length of the list using the simple algorithm
described below.
There are algorithms for superset removal that work in time
$O(n^2/\log{n})$ or slightly better, see \citep{Pri95} and \citep{SE96}.
It is known, that there is a lower bound of
$O(n^2/\log^2 n)$, if the complete subset lattice
is constructed by algorithm, see~\citep{Pri99}.
It appears that no non-trivial lower bounds are
known for superset removal alone, and we remark that it is an interesting
question.
When implementing Berge's algorithm, we can avoid generating
some edges that are clearly supersets before entering the removal
phase. For instance whenever an edge $f$ of $\H_{i-1}$ intersects $e_i$,
we can add only that edge to the list of candidates for $\H_i$
since $f$ will be generated as an edge using the common vertex with
$e_i$ and all edges generated using the other vertices of $e_i$ will
contain this edge. Additionally, we can see that the list of
edges retained in this way will itself be superset free
since it is a subhypergraph of $\H_{i-1}$.
Hence we only need to check which of the new edges generated
are supersets of these retained edges: newly generated edges cannot
be subsets of edges from $\H_{i-1}$ since they are edges from
$\H_{i-1}$ with an additional vertex, and if
$f_1 \cup v_1 \subset f_2 \cup v_2$ in the new edge list, then
$f_1$ must contain $v_2$ and hence would in fact be in the retained
edges list.
Our implementation of Berge's algorithm is now included in {{\small\tt CellNetAnalyzer}}.
\subsubsection{Others}\label{se:others}
Several algorithms have recently been proposed which build on
the Berge algorithm and are effective for some problems,
see for example \citep{BMR03} and \citep{KS05}.
A nice recent survey of methods for computing transversals is \citep{EMG06}.
It focuses on work based on the ideas of
Fredman and Khachiyan \citep{FK96} concerning generation of
logical formulae. These methods provide better theoretical
performance and we consider them in Section~\ref{se:joint}.
\subsection{Implementation} \label{se:impb}
While Berge's algorithm is well known, naive implementations
of it are quite slow. Modifications of Berge's algorithm,
such as the ones mentioned in Section~\ref{se:others} have
apparently yielded good results, but we know of no public
implementations of such an algorithm.
Thus we implemented our own version of the algorithm described in
Section~\ref{se:ourberge}. We used {\small\tt MATLAB} because it is the
platform for {{\small\tt FluxAnalyzer}} and {{\small\tt CellNetAnalyzer}}.
Our code is freely available for academic use \citep{BergeCode}.
We tested our code on the examples of \citep{KG04}.
These are obtained from the growth-related EM's calculated in
\citep{SKB+02} via simple modifications described
in Section~\ref{se:pre}.
Our results are presented in Section~\ref{se:compb}.
Below we describe some details of our implementation, as
well as the original {{\small\tt FluxAnalyzer}} implementation.
We used as much of the {{\small\tt FluxAnalyzer}} setup as possible,
including input data structures and pre- and post-processing code
to facilitate comparison between the core algorithms.
\subsubsection{Preprocessing}\label{se:pre}
Beginning from the computed full set of EM's, we first select only
those containing at least one of the target reactions.
As mentioned in Section~\ref{se:struct}, some groups of
reactions may be catalyzed by the same multifunctional enzyme.
These reactions are cut simultaneously by disabling such an enzyme.
We combine the reactions corresponding to such groups by a logical
``or'' operation as the corresponding elementary mode is
disabled if any of its constituent reactions are disabled.
This merges vertices in the input hypergraph;
the merged vertices in the new modes represent the set of
reactions blocked by the enzyme.
In our examples there is only one target reaction (biomass synthesis)
and one multifunctional enzyme (transketolase).
This gives us our initial hypergraph $\H$ of modes that include the target
reaction.
There are several further
preprocessing steps applied to $\H$.
The first is to scan for zero rows and columns, which should not occur.
The second is to find columns of ones,
which correspond to cut sets of size one and can be
noted and removed until postprocessing.
Next duplicate columns are identified.
They are treated as a single column
and then reexpanded during post-processing.
In terms of the hypergraph, they represent vertices that
are in exactly the same set of edges, and can thus be merged
during the calculation.
Finally, for Berge's algorithm,
we remove rows that are supersets of other rows.
Our current implementation does this in time $\Omega(m^2)$.
Superset or duplicate rows may be introduced when merging vertices
corresponding to multifunctional enzymes.
In our examples, about 20\% of the rows in $\H$ are supersets
of other rows, these come from the pairs of EM's that contain
only one of the transketolase reactions, each pair contains one
EM using transketolase1 and transaldo, and one EM instead using
transketolase2. The first of these becomes a superset of the
second upon merging the two columns.
Removing these supersets in preprocessing cut the observed
running time by about 30\%.
Removing superset rows does not speed up the algorithm in
{{\small\tt FluxAnalyzer}}, whose bottleneck is generating the possible cut sets
of a given size.
\subsubsection{Postprocessing}\label{se:post}
Following the application of either algorithm to the
hypergraph produced in Section~\ref{se:pre}, there is
a small amount of postprocessing that needs to be done.
Merged vertices are reexpanded: each cut set containing
a merged vertex $v$ will be replaced by cut sets containing
exactly one of the vertices that were merged into $v$.
Size one cut sets will be introduced for each column of
ones that was removed prior to the calculation.
Finally, the cut sets involving reactions corresponding
to a multifunctional enzyme are split into cut sets
containing the constituent reactions.
\subsubsection{Coding issues}
Because we are working with large matrices, memory use
is a key consideration, it is essential to use a bit-level
representation of the binary matrices describing
the EM's and MCS's. This is done in the implementation
of \citep{KG04}.
We had to accommodate {\small\tt MATLAB}'s strengths and
weakness: we obtained substantial time savings through
small changes in the main bottleneck routine.
This routine removes the rows from one list that
are supersets of rows from another.
Since memory allocation in {\small\tt MATLAB} is slow rather than
resizing the matrix when superset rows are identified,
we mark rows for removal in a pass through the matrix
and then we generate a single new matrix containing
those rows. It is also useful to take advantage of
{\small\tt MATLAB}'s internal parallelization. {\small\tt MATLAB} can quickly
check a single row for super- or sub-setness against an
entire matrix using a couple of bitwise comparisons.
We always cycle through the rows of the shorter of the
two lists and check its rows against the longer list.
\subsubsection{Computational results}\label{se:compb}
In this section we compare the performance of the transversal
hypergraph code written for {{\small\tt FluxAnalyzer}} to our implementation
of Berge's algorithm now included in {{\small\tt CellNetAnalyzer}}.
Our test base is the four EM
problems from~\citep{KG04} (see Section~\ref{se:cases})
and their hypergraph transversals.
The transversals are denoted by primes.
We first compare the algorithms from the point of view of the
largest intermediate lists generated. This tells
us how much memory each algorithm uses, and gives an idea of
how fast it can run. In the case of the {{\small\tt FluxAnalyzer}} algorithm,
the measure is the largest list of partial cut sets generated.
In the case of the Berge algorithm, the measure is the largest
intermediate transversal generated before removing nested subsets.
Due to the reductions of Section~\ref{se:pre}, this sometimes turned
out to be smaller than output it generated after postprocessing.
Results are in Table~\ref{ta:bsizes}. We used as inputs both the
EM's found for the networks described in Section~\ref{se:cases}
and the dual hypergraphs containing the MCS's that we computed
(denoted with a ').
\iftablestoend
\else
\begin{table}[ht!]
\begin{tabular}{l@{\quad}rr@{\quad}rr@{\quad}rr@{\quad}rr}
\toprule
Problem: & {\tt acet} & {\tt acet}'& {\tt succ} & {\tt succ}'& {\tt glyc} & {\tt glyc}' & {\tt gluc} & {\tt gluc}' \\
\midrule
Input columns & 104 & 104 & 104 & 104 & 105 & 105 & 105 & 105 \\
Preprocessed columns & 21 & 98 & 26 & 101 & 28 & 103 & 34 & 103 \\
Input rows & 363 & 245 & 3\,421 & 1\,255 & 9\,479 & 2\,970 & 21\,592 & 4\,225 \\
Preprocessed rows & 289 & 244 & 2\,722 & 1\,254 & 7\,472 & 2\,969 & 18\,481 & 4\,224 \\
Raw output rows & 54 & 280 & 159 & 2\,589 & 376 & 7\,047 & 918 & 18\,481 \\
Final output rows & 245 & 289 & 1\,255 & 2\,722 & 2\,970 & 7\,472 & 4\,225 & 18\,481 \\
{\tt FluxAnalyzer} largest & 3\,563 & -- & 69\,628 & -- & 342\,025 & -- & 902\,769 & -- \\
Berge largest & 94 & 296 & 304 & 2\,669 & 657 & 7\,047 & 1\,714 & 18\,569 \\
{\tt {{\small\tt FluxAnalyzer}}} time & 5.1 & -- & 633.5 & -- & 8\,696.2 & -- & 54\,099.1 & -- \\
Berge time & 0.7 & 1.1 & 7.1 & 35.8 & 29.6 & 215.0 & 206.5 & 727.4 \\
\bottomrule
\end{tabular}
\vspace{.2\baselineskip}
\caption{Sizes of intermediate lists generated in computing transversals
and computation times.}
\label{ta:bsizes}
\label{ta:btimes}
\end{table}
\fi
We record the number of columns and rows both before and after
preprocessing. The preprocessing reduces the problem size substantially
mainly by removing columns corresponding to reactions which form cut
sets by themselves.
The number of output rows is given before and after postprocessing.
The {{\small\tt FluxAnalyzer}} routine was not able to solve the dual problems due
to memory limitations. This routine is much better at converting
EM graphs to MCS graphs than vice versa because many of the MCS's are
small in these instances.
For the given examples, all the EM's are large, so the computation
begins by building a long list of partial cut sets.
While Berge's algorithm provides no complexity guarantees, it worked
very well for these problems. Of particular note is that, unlike
the {{\small\tt FluxAnalyzer}} algorithm, the information carried by Berge's algorithm
during its intermediate stages was never much larger than the
size of the final output prior to preprocessing.
We also give running times in seconds in Table~\ref{ta:btimes}.
Both codes are written in {\small\tt MATLAB}, take identical inputs, and
use the same preprocessing code as noted in Section~\ref{se:pre}.
These comparisons were run on a Sun Fire V890 with 32 GB
memory and 16 1200 MHz processors.
Note that running times include some preprocessing, such as
removing duplicate rows for Berge, but excludes postprocessing.
This method of reporting was used in \citep{KG04}.
\section{Computing Minimal Cut Sets Directly}\label{se:joint}
In this section, we consider methods of generating the
MCS's directly from the stoichiometric matrix.
The techniques outlined here are based on an algorithm of
Fredman and Khachiyan \citep{FK96} for dualizing boolean
functions. They offer better complexity guarantees than
the algorithms of Section~\ref{se:berge}. They work
directly from the stoichiometric matrix for a network
and generate the hypergraph of EM's containing the blocked
reactions as a byproduct of the computation.
\subsection{Algorithm}
The cut sets generated by a given stoichiometric
matrix define a boolean function, that is a function
that takes a binary pattern of included reactions as input, and
yields 1 if this set of reactions is a cut set, and 0
if it is not. Further, this is a {\it monotone} function in the sense
that if a given set is a cut set, then any superset of that set is
also a cut set. Thus the problem of finding {\it minimal} cut sets can
be viewed as a problem of representing such a boolean function via
its minimal true assignments. The support of an EM
is then the complement of a maximal false assignment.
A monotone boolean function can be represented uniquely
both by its minimal true
assignments and its maximal false assignments. The process of
converting from one representation to the other is sometimes
called dualization, and is equivalent to the hypergraph
transversal problem.
Fredman and Khachiyan \citep{FK96} proposed an algorithm that
generates the transversal incrementally using an algorithm
that is slightly superpolynomial in the size of the graph
and the transversal. This key idea in this algorithm is to
recurse on a variable that occurs with relatively high frequency.
As described in \citep{GK99}, this algorithm can be implemented from
a function evaluation oracle, producing both the hypergraph of
minimal true assignments and its transversal in $m^{o(\log(m))}$
oracle calls, where $m$ is the combined size of the
two hypergraphs.
This fits very well with our problem:
given the stoichiometric matrix,
we want to generate both the MCS's
and the EM's.
Note that the boolean function characterizing the cut set is monotone
because every superset of a cut set is again a cut set.
We remark that the amount of memory required to generate
the next clause is bounded by $m^{\operatorname{poly}(\log{m})}$,
while the worst-case memory blow up for Berge is unknown,
but not polynomial.
\subsection{Implementation}\label{se:impj}
The algorithm of \citep{FK96} gives remarkable theoretical results,
but the only implementation we know of is that of \citep{BEGK06}.
Their code is not public and uses hard-coded oracles different from
the one for our problem.
So we implemented a prototype of the Fredman-Khachiyan
algorithm with a suitable oracle, again in {\small\tt MATLAB} for easy
comparison to {{\small\tt FluxAnalyzer}} and the results in Section~\ref{se:impb}.
We emphasize, though, that this algorithm proceeds directly
from the stoichiometric matrix, in contrast to the Berge
algorithm, which requires the completed computation of the EM's
as input.
\subsubsection{Oracle}\label{se:oracle}
We test whether a given set $C$ is a cut set by checking
if the system (\ref{eq:cutset}) has any solutions with
$x_t > 0$ for some $t \in T$.
This is a linear programming feasibility problem, and thus
can be solved in polynomial time.
We do this via an external call to
{\small\tt CPLEX}~\citep{ilog-cplex-uuh}, which is known to have a fast and reliable LP solver.
We can test for non-trivial solutions by
maximizing the sum of the target variables
$\sum_{i \in T} x_i$ subject to (\ref{eq:cutset}).
If this is greater than zero or unbounded we have a non-trivial
solution.
\subsubsection{Duality checker}\label{se:checker}
Using this oracle, we implemented a version of the ``Algorithm A''
duality checker from \citep{FK96}. This checker either verifies
that our current collections of EM's and MCS's form dual hypergraphs,
in which case both sets are complete and we are done, or it
finds a set of reactions that is not a superset of any
current EM, and whose complement is not a superset of any
current MCS. Given such a clause we use the oracle to check if it
is a mode or if its complement is a cut set. If it is a mode, we
test whether it remains a mode upon removing in turn each of its
constituent reactions
-- if the resulting set is no longer a mode, we return
the removed reaction to the set, otherwise it stays out.
In this way we obtain an EM, which we add to our list.
Similarly, given a cut set, we obtain an MCS.
The essence of the algorithm is to recurse on a frequently
occurring reaction in one of the lists until we arrive at
a trivial case. The recursion then tests separately if
duality holds assuming that this variable is true and assuming
that it is false. By taking a frequently occurring
variable, Fredman and Khachiyan ensure that the sizes of
the lists decrease fast enough to guarantee that the
algorithm runs in time $m^{O(\log^2(m))}$, where $m$ is
the current joint length of the two lists.
As with Berge's algorithm, we find that the bottleneck is
removing supersets from lists: when we recurse on a reaction
some of the elements of the reduced lists, which omit this
reaction, will no longer be minimal. These must be removed,
and this takes $O(m^2)$ time.
Fredman and Khachiyan also provide an ``Algorithm B'' which
achieves further economy through observing some interdependencies
in the two subproblems. This reduces the time guarantee to
$m^{o(\log(m))}$, but the resulting algorithm is much
more complicated, so we did not implement a prototype.
We did try several variations of the simpler Algorithm A.
We found some useful corners to cut: it is helpful to
short circuit the recursion by treating more base cases
than suggested by the algorithm. We can substantially
reduce the number of superset removal calls required
by doing them only when necessary before recursing rather
than at the start of the checking routine.
We ran our code on the stoichiometric matrices that
are used to generate EM's in \citep{SKB+02}.
Our oracle tests if a set of reactions blocks the growth reaction,
the two transketolase reactions are treated as a single reaction for
the purposes of blocking. Thus the EM's we generate use
a single bit to indicate if either of the two reactions are active.
As with the preprocessing of Section~\ref{se:pre} this merges
certain EM's.
\iftablestoend
\else
\begin{table}[ht!]
\begin{tabular}{l@{\quad}rrrr}
\toprule
Problem &\multicolumn{1}{c}{{\tt acet}}& \multicolumn{1}{c}{{\tt succ}} & \multicolumn{1}{c}{{\tt glyc}} & \multicolumn{1}{c}{{\tt gluc}} \\
\midrule
EM's & 289 & 2\,722 & 7\,472 & 18\,481 \\
MCS's & 245 & 1\,255 & 2\,970 & 4\,225 \\
\addlinespace
Total recursive calls & 107\,781 & 11\,129\,110 & 122\,136\,668 & 764\,239\,195 \\
Time to generate & 194.8 & 10\,672.2 & 103\,511.2 & 677\,599.3 \\
\hline
\end{tabular}
\vspace{.2\baselineskip}
\caption{Call counts for the Fredman-Khachiyan algorithms.}
\label{ta:jsizes}
\end{table}
\fi
In Table~\ref{ta:jsizes}
we record the number of calls
to the (recursive) duality checker used in our implementation
of Fredman and Khachiyan's algorithm.
Each stoichiometric matrix has 89 metabolites and 105 or 106
reactions.
The algorithm is written in {\small\tt MATLAB} and the oracle uses external
calls to {\small\tt CPLEX} when necessary to solve linear programs.
Running times are included in Table~\ref{ta:jsizes} above.
We used the same computer as in Section~\ref{se:compb}.
We remark that if our objective is to produce the
EM's containing the target reactions rather than the MCS's,
we can compress the network as described in \citep{KGv06}.
This speeds up the computation, since it now needs to generate
only the few cut sets for the compressed modes
(which are hard to interpret), see Table~\ref{ta:compress_gen}.
In this case, since our objective is to produce the EM's,
we did not merge the two transketolase reactions.
\iftablestoend
\else
\begin{table}[ht!]
\begin{tabular}{l@{\quad}rrrrrc}
\toprule
Problem & {\tt acet} & {\tt succ} & {\tt glyc} & {\tt gluc} \\
\midrule
Compressed reactions & 40 & 40 & 42 & 42 \\
EM's & 363 & 3\,421 & 9\,479 & 21\,592 \\
Total recursive calls & 38\,503 & 3\,487\,200 & 20\,971\,005 & 217\,252\,316 \\
Time to generate & 45.8 & 2\,707.0 & 17\,202.0 & 210\,749.8 \\
\bottomrule
\end{tabular}
\vspace{.2\baselineskip}
\caption{Data for generating the EM's from the compressed network}
\label{ta:compress_gen}
\end{table}
\fi
In fact it is possible to do most of this compression in such a way
that the cut sets generated can be expanded to yield the MCS's for
the original system. Working with these partially compressed networks
takes less than 10\% longer than the fully compressed networks reported
in Table~\ref{ta:compress_gen}.
We can also produce the full set of EM's using this
method by blocking the full set of reactions.
This requires modifying the oracle to check for non-trivial
solutions to (\ref{eq:cutset}) via a rank check.
If we generate them from the compressed matrices, this is
somewhat tractable, but slower than generating only those
containing the target reactions.
\section{Conclusions}
As has been often seen in recent years, biological problems are
ripe for the application of mathematics.
In Section~\ref{se:berge}, we use a non-trivial, but simple
algorithm to compute MCS's from EM's
in minutes rather than hours in the context of a large metabolic
network problem. With such large data sets a careful implementation
of the algorithm was as essential to make it useful.
While Berge's algorithm is successful in practice, it
is poorly understood in theory, and thus must be considered
suspect. Additionally, it requires the precomputation of the EM's
via an algorithm which also has uncertain worst-case complexity.
Thus we also considered algorithms based on the
dual generation framework of Fredman and Khachiyan \citep{FK96}.
These offer a guaranteed theoretical performance which is
close to polynomial, i.e. $m^{\operatorname{poly}(\log{m})}$ where $m$ is
the joint size of the EM's and MCS's.
The Fredman-Khachiyan oracle-based algorithms also have the
advantage that, unlike Berge's algorithm, the lists are
generated {\it incrementally} - at each step a new EM or MCS is
added to the current collection, and in time and
memory $m^{\operatorname{poly}(\log{m})}$
in the current joint size of the lists.
Thus if we only have
the resources to do a partial computation, we will get a
partial answer. Indeed, much larger systems exists
for which the full sets of EM's and MCS's are too large
to store. In this situation, even if we could compute a
partial list of EM's, its dual is meaningless.
In contrast, the oracle-based algorithm can produce a
sampling of EM's and MCS's as resources permit.
The oracle-based algorithms also provide a method of computing
the EM's containing target reactions without computing the
full set of EM's. This is desirable since there may be an
enormous number of EM's, only a small fraction of which
contain the target reactions. The double description algorithm
can be modified to compute only the EM's containing a target,
however as noted in \citep{KGv06}, unless implemented carefully,
this may be slower than computing the full set of reactions.
However, there are several clear drawbacks to oracle-based
algorithms. The obvious ones are that they are more difficult
to implement and slower. The high number of recursive calls
used to solve the small {{\tt acet}} problem
(see Table~\ref{ta:jsizes}) suggests that it will be difficult
to make such an algorithm competitive with Berge, especially
in the case where the EM's are given. It is encouraging
that our simple implementation is able to solve even the
largest problem ({{\tt gluc}}), although the time required was
long. This gives us some hope that a more advanced implementation
of the oracle-based algorithm could be competitive in this
application. One place to start would be to implement
Fredman and Khachiyan's more intricate, but theoretically
faster $m^{o(\log(m))}$ algorithm.
The oracle-based methods could be improved
by additional preprocessing as is done when obtaining the EM's
from the MCS's. For example, single reaction cut sets are
treated separately in the algorithms of Section~\ref{se:berge},
this could also be implemented in the Fredman-Khachiyan
framework.
We expect this would yield a mild improvement in the running time.
\subsection{Acknowledgments}
We are grateful for fruitful discussions with Annegret Wagler and
Robert Weismantel
This work was partially supported by
DFG FG-468,
the Research Focus Program Dynamic Systems funded by the
Kultusministerium of Saxony-Anhalt, and by
a President's Research Grant at Simon Fraser University.
\bibliographystyle{amsalpha}
\newcommand{\etalchar}[1]{$^{#1}$}
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\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{%
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,407
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Q: Auto-refreshing of a portlet after another portlet finished its work I have two portlets:
*
*One is for displaying a list of files (and more)
*The second is to import files into Liferay's document library.
If I have both portlets on one page how can I achieve that the 'listing' portlet refreshes after the import portlet has finished its work? This is mandatory because the hyperlinks in the listing portlet will change after the import.
A: Here is a pointer on how to refresh a portlet using ajax call.
Liferay.Portlet.refresh("p_p_id_<targetportletnamespae>_");
Its upto you to decide when to call this js method, based on your requirement.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,668
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\section{Introduction}
Quantum anomalies considerably constrain the structure of chiral gauge theories in even dimensions.
Chiral matter is known to induce gauge and gravitational anomalies at the 1-loop level in perturbation theory \cite{AlvarezGaume:1983ig}, which jeopardize the consistency of the gauge theory.
In the presence of tensor fields the celebrated Green-Schwarz-Sagnotti-West mechanism \cite{Green:1984sg,Green:1984bx, Sagnotti:1992qw} can cancel such 1-loop anomalies provided the anomaly polynomial of the latter factorises suitably. A particularly interesting class of examples of such tensors are the self-dual tensor fields in $4k+2$ dimensions \cite{Witten:1996hc}.
The ramifications of the anomaly cancellation mechanism have been investigated in great detail, most notably in the context of 6d $N=(1,0)$ supergravity theories (see e.g. \cite{Schwarz:1995zw,Kumar:2009us,Kumar:2009ac,Park:2011wv,Monnier:2017oqd} and references therein).
A lower-dimensional analogue of these 6d $N=(1,0)$ supergravities, similar in many respects, are chiral theories in two dimensions with $N=(0,2)$ supersymmetry.
Such theories have sparked significant interest from various field theoretic perspectives, most notably concerning their RG flow to an SCFT point \cite{Benini:2012cz,Benini:2013cda,Benini:2015bwz,Gadde:2014ppa,Couzens:2017way}, in the context of computing elliptic genera and localisation \cite{Closset:2015ohf}, or with respect to novel types of dualities \cite{Gadde:2013lxa,Jia:2014ffa}.
Exploring the structure of anomalies of a class of 2d $N=(0,2)$ supergravities is the goal of this article.
If a supergravity theory is engineered by compactifying string theory, the consistency conditions from anomaly cancellation imply a rich set of constraints on the geometry defining the compactification. A prime example of this fruitful interplay between anomalies and geometry is provided by F-theory \cite{Vafa:1996xn,Morrison:1996na,Morrison:1996pp}.
In this framework, 6d $N=(1,0)$ supergravities arise via compactification on elliptically fibered Calabi-Yau 3-folds.
Anomaly cancellation then translates into various highly non-trivial relations between topological invariants of the latter \cite{Sadov:1996zm,Grassi:2000we, Grassi:2011hq, Park:2011ji, Park:2011wv}, which would be hard to guess otherwise, and some of which are even harder to prove in full generality.
Compactification of F-theory to four dimensions on a Calabi-Yau 4-fold gives rise to an $N=1$ supersymmetric theory which is chiral - and hence potentially anomalous - only in the presence of non-trivial gauge backgrounds.
This makes it perhaps even more intriguing that
the same types of topological relations \cite{Bies:2017abs} are responsible for the cancellation of gauge and mixed gauge-gravitational anomalies in six and four-dimensional \cite{Cvetic:2012xn} F-theory compactifications.
If one is able to
establish the cancellation of anomalies directly from a physical perspective, as has been achieved recently in \cite{Corvilain:2017luj} for four-dimensional F-theory vacua, such reasoning amounts to a physics proof of a number of highly non-trivial topological relations on elliptic fibrations of complex dimension three and four. One of the motivations for this work is to extend this list of topological identities to elliptic fibrations of higher dimension.
The 2d $(0,2)$ supergravity theories considered in this article are obtained by compatifying F-theory on an elliptically fibered Calabi-Yau 5-fold \cite{Schafer-Nameki:2016cfr, Apruzzi:2016iac}. As we will review in section \ref{sec_FonellCY5} the theories contain three different coupled sub-sectors: The structure of the gauge theory sector is similar to the 2d $(0,2)$ GLSMs familiar from the worldsheet formulation of the heterotic string \cite{Witten:1993yc,McOrist:2010ae}. It includes 2d $(0,2)$ chiral and Fermi multiplets charged under the in general abelian and non-abelian gauge group factors originating from
a topologically twisted theory on 7-branes \cite{Schafer-Nameki:2016cfr, Apruzzi:2016iac}.
D3-branes wrapped around curves on the base of the fibration give rise to additional degrees of freedom. These include a particularly fascinating, but largely mysterious sector of Fermi multiplets from the string excitations at the intersection of the D3-branes and the 7-branes \cite{Lawrie:2016axq}.\footnote{The theory on a D3-brane wrapping a curve \cite{Haghighat:2015ega,Lawrie:2016axq} or surface \cite{Martucci:2014ema,Assel:2016wcr} in F-theory is interesting by itself as an example of a gauge theory with varying gauge coupling. An AdS$_3$ gravity dual of an $N=(0,4)$ version has recently been constructed in \cite{Couzens:2017way}.}
These two sectors are coupled to a 2d $N=(0,2)$ supergravity sector \cite{Lawrie:2016rqe}.
The construction of 2d $N=(0,2)$ theories has received considerable attention also in other formulations of string theory, most notably via D1 branes probing singularities on Calabi-Yau 4-folds \cite{GarciaCompean:1998kh,Franco:2015tna, Franco:2016nwv,Franco:2016fxm,Franco:2017cjj,Closset:2017yte} and via orientifolds \cite{Forste:1997bd,Font:2004et}.
Various aspects of the non-abelian gauge and the gravitational anomalies in the chiral 2d $(0,2)$ theory obtained via F-theory have already been addressed in \cite{Schafer-Nameki:2016cfr, Apruzzi:2016iac,Apruzzi:2016nfr,Lawrie:2016axq,Lawrie:2016rqe}.
The non-abelian anomalies induced by the chiral fermions in the 7-brane brane gauge sector must be cancelled by the anomalies of the 3-7 modes, as indeed verified in globally consistent examples in \cite{Schafer-Nameki:2016cfr}. The cancellation of all gravitational anomalies for 2d (0,2) supergravities with a trivial gauge theory sector has been proven in \cite{Lawrie:2016rqe} with the help of various index theorems.
Such theories are obtained by F-theory compactification on smooth, generic Weierstrass models.
On the other hand, the structure of gauge anomalies in the presence of abelian gauge theory factors is considerably more involved, and the subject of this article.
As in higher dimensions, abelian anomalies induced at 1-loop level need not vanish by themselves provided they are consistently cancelled by a two-dimensional version of the Green-Schwarz mechanism. In general 2d $(0,2)$ gauge theories, the structure of the Green-Schwarz mechanism has been laid out in \cite{Adams:2006kb,Quigley:2011pv,Blaszczyk:2011ib} (see \cite{Mohri:1997ef,GarciaCompean:1998kh} for early work).
In the present situation, the Green-Schwarz mechanism operates at the level of real chiral scalar fields which are obtained by Kaluza-Klein reduction of the self-dual 4-form of Type IIB string theory. They enjoy a pseudo-action which is largely analogous to the pseudo-action of the self-dual 2-tensors in 6d $N=(1,0)$ supergravities and which we parametrise in general terms in section \ref{sec_Anomaliesin2d}.
As one of our main results we carefully derive this pseudo-action in section \ref{sec_GStermsderivation}, thereby identifying the structure (and correct normalisation) of the anomalous Green-Schwarz couplings. The latter depend on the non-trivial gauge background and imply a classical gauge variance of the right form to cancel the 1-loop abelian gauge anomalies.
A challenge we need to overcome to show anomaly cancellation is that in absence of a perturbative limit the abelian charges of the 3-7 sector modes are notoriously hard to determine in a microscopic approach.
Instead of computing the 3-7 anomaly from first principles we extract the anomaly inflow terms onto the worldvolume of the D3-branes in section \ref{sec_GStermsderivation}. To this end we start from the Chern-Simons terms of the 10d effective pseudo-action in the presence of brane sources.
Uplifting this result to F-theory allows us to quantify the contribution of the 3-7 modes in particular to the gauge anomalies and in turn also to deduce the net charge of the 3-7 modes.
One of our main results is to establish a closed expression for the complete gauge and gravitational anomalies of a 2d $(0,2)$ theory obtained by F-theory compactified on a Calabi-Yau 5-fold.
The resulting conditions for anomaly cancellation are summarized in
(\ref{gaugeanomalyFtheory}) and (\ref{gravitationalanomalies1}) of section \ref{sec_AnomalyEqu5folds}.
The structure of anomalies reflected in these equations interpolates between
their analogue in 6d and 4d F-theory vacua:
In 6d F-theory vacua the anomalies are purely dependent on properties of the elliptic fibration, while in 4d they vanish in absence of background flux and depend linearly on the flux background. In 2d F-theory vacua, we find both a purely geometric and a flux dependent contribution to the anomalies.
For anomalies to be cancelled, the flux dependent and the flux independent parts of the topological identities (\ref{gaugeanomalyFtheory}) and (\ref{gravitationalanomalies1}) must in fact hold separately, on any elliptically fibered Calabi-Yau 5-fold and for any gauge background satisfying the consistency relations reviewed in section \ref{sec_FonellCY5}.
We verify these highly non-trivial anomaly relations in a concrete example fibration for all chirality inducing gauge backgrounds in section \ref{sec_ExampleSU5U1}.
It has already been pointed out that, despite their rather different structure at first sight, the gauge anomalies in 6d and 4d boil down to one universal relation in the cohomology ring of an elliptic fibration over a general base, and similarly for the mixed gauge-gravitational anomalies \cite{Bies:2017abs}.\footnote{By contrast, the purely gravitational anomaly in 6d has no direct counterpart in 4d. See, however, \cite{Grimm:2012yq}.}
This prompts the question if the 2d anomaly relations (\ref{gaugeanomalyFtheory}) and (\ref{gravitationalanomalies1}) are also equivalent to this universal
relation governing the structure of anomalies in four and six dimensions.
As we will see in section \ref{sec_Comparison6d4d}, assuming the 4d/6d relation of \cite{Bies:2017abs} implies the flux dependent part of
(\ref{gaugeanomalyFtheory}) and (\ref{gravitationalanomalies1})
for a special class of gauge background.
However, it remains for further investigation whether the precise relations extracted in \cite{Bies:2017abs} on Calabi-Yau 3-folds and 4-folds follow in turn by anomaly cancellation on Calabi-Yau 5-folds in full generality.
\section{Anomalies in 2d $(0,2)$ supergravities} \label{sec_Anomaliesin2d}
Consider an $N=(0,2)$ supersymmetric theory in two dimensions with
gauge group
\begin{eqnarray} \label{eq:Gsplit}
G^{\mathrm{tot}}=\prod_{I=1}^{n_G} G_{I}\times \prod_{A=1}^{n_{U(1)}} U(1)_A
\end{eqnarray}
and matter fields in representations
\begin{eqnarray}
\mathbf{R}=(\mathbf{r}^1,\ldots,
\mathbf{r}^{n_G})_{\underline{q}} \,.
\end{eqnarray}
Here $\mathbf{r}^I$ denotes an irreducible representation of the simple gauge group factor $G_I$ and $\underline{q}=(q_1,
\ldots ,q_{n_{U(1)}})$ are the charges under the Abelian gauge group factors.
We are interested in the structure of the gauge and gravitational anomalies in such a theory.
These are induced by chiral matter at the 1-loop level. In a general $D$-dimensional quantum field theory, the gauge and gravitational anomalies can be described by a gauge invariant anomaly polynomial of degree $D/2+1$ in the gauge field strength $F$ and the curvature two-form $R$,
\begin{eqnarray}
\label{poly}
I_{D+2}=\sum_{\mathbf{R},s} n_s(\mathbf{R})I_{s}(\mathbf{R})|_{D+2} \,,
\end{eqnarray}
where the sum is over all matter fields with spin $s$ which have zero-modes in representation $\mathbf{R}$ with multiplicity $n_s(\mathbf{R})$.
In particular, a chiral fermion, corresponding to $s=1/2$, contributes with
\begin{eqnarray}
\label{anomaly}
I_{1/2}(\mathbf{R})=- {\rm tr}_\mathbf{R} \, e^{ -F} \, \hat A({\rm T}) \,,
\end{eqnarray}
where $\hat{A}({\rm T})$ is the A-roof genus and $F$ denotes the hermitian gauge field strength. An anti-chiral fermion contributes with the opposite sign. For more details on our conventions we refer to appendix \ref{convention}.
In $D=2$ dimensions, the 1-loop anomaly polynomial from the charged matter sector is hence a 4-form.
Correspondingly, the anomaly contribution from chiral and anti-chiral fermions in the theory sums up to
\begin{eqnarray} \label{I4total}
I_{4}=\sum_{\mathbf {R}} (n_+(\mathbf{R}) - n_-(\mathbf{R}) ) \left(-\frac{1}{2}{\rm tr}_{\mathbf R} ( F)^2 + \frac{1}{24} p_1({\rm T}) \, {\rm dim}({\bf R}) \right) \,,
\end{eqnarray}
where the first Pontryagin class of the tangent bundle is defined as $p_1({\rm T}) = - \frac{1}{2} {\rm tr} R^2$.
For future purposes we express the anomaly polynomial for the non-abelian, the abelian and the gravitational anomaly as
\begin{eqnarray}
I_4|_{G_I} &=& - {\cal A}_{I} \, {\rm tr}_{\bf fund} F_I^2 = - \frac{1}{2} \sum_{\mathbf{\mathbf{r}}^I} c^{(2)}_{\mathbf{r}^I} \, \chi(\mathbf{r}^I) \, {\rm tr}_{\bf fund} F_I^2 \label{AI-gen1} \\
I_4|_{ A B} &=& - {\cal A}_{AB} \, F^A F^B = - \frac{1}{2} \sum_{\mathbf{R}} q_{A}({\bf R}) \, q_{B}({\bf R}) \,\text{dim}(\mathbf{R}) \, \chi(\mathbf{R}) \, F^A F^B \label{AAB-gen1} \\
I_4|_{\rm grav} &=& \frac{1}{24} {\cal A}_{\rm grav}\, p_1(T) = \frac{1}{24} \sum_{\mathbf{R}}\chi(\mathbf{R}) \, \text{dim}(\mathbf{R}) \, p_1({\rm T}) \,,
\end{eqnarray}
with $\chi({\bf R})$ denoting the chiral index of zero-modes in representation ${\bf R}$. In the first line we have related the trace in a representation ${\bf r}^I$ of the simple gauge group factor $G_I$ to the trace in the fundamental representation
via
\begin{eqnarray} \label{cr2}
\text{tr}_{\mathbf{r}^I} F^2= c^{(2)}_{\mathbf{r}^I} \, \text{tr}_{\mathbf{fund}}F^2 \,.
\end{eqnarray}
In general, the 1-loop induced quantum anomaly need not be vanishing in a consistent theory provided the tree-level action contains gauge variant terms, the Green-Schwarz counter-terms, which cancel the anomaly encoded by $I_{D+2}$. For this cancellation to be possible,
the 1-loop anomaly polynomial $I_{D+2}$ of the matter sector must factorize suitably.
In two dimensions, the Green-Schwarz counterterms derive from gauge variant interactions of scalar fields.
The structure of the possible Green-Schwarz terms in a general 2d $N=(0,2)$ supersymmetric field theory has been analyzed in \cite{Adams:2006kb,Quigley:2011pv,Blaszczyk:2011ib} (see \cite{Mohri:1997ef,GarciaCompean:1998kh} for early work).
In this paper, however, we are interested in the specific 2d $N=(0,2)$ effective theory obtained by compactification of F-theory on an elliptically fibered Calabi-Yau 5-fold \cite{Schafer-Nameki:2016cfr,Apruzzi:2016iac}.
In these theories a gauge theory with gauge group (\ref{eq:Gsplit}) is coupled to a 2d $N=(0,2)$ supergravity sector.\footnote{The gauge theory in question arises from spacetime-filling 7-branes. In addition, the compactification contains spacetime-filling D3-branes, but the associated gauge fields are projected out due to $SL(2,\mathbb Z)$ monodromies along the D3-brane worldvolume \cite{Schafer-Nameki:2016cfr, Lawrie:2016axq}. }
The latter contains a set of
real axionic scalar fields $c^\alpha$ arising from the Kaluza-Klein (KK) reduction of the F-theory/Type IIB Ramond-Ramond
forms $C_4$ \cite{Lawrie:2016rqe}.\footnote{As discussed in \cite{Lawrie:2016rqe}, these scalars split into $n_+$ chiral and $n_-$ anti-chiral real scalars. Out of these $n_+$ pairs of real chiral and anti-chiral scalars form non-chiral real scalars, which constitute the imaginary part of the bosonic component of a corresponding number of 2d $(0,2)$ chiral multiplets. The remaining $\tau = n_- - n_+$ anti-chiral real scalars form 2d $(0,2)$ tensor multiplets and contribute, together with the gravitino, to the gravitational anomaly at 1-loop level according to the general formulae reviewed in appendix \ref{convention}. This contribution to the 1-loop anomaly is in addition to the classical gauge variance of the Green-Schwarz action discussed in this section.}
As we will derive in detail in section \ref{sec_GStermsderivation}, their pseudo-action can be parametrized as
\begin{eqnarray} \label{2daction1}
S_{\rm GS} = -\frac{1}{4} \int_{\mathbb R^{1,1}} g_{\alpha \beta} \, H^\alpha \wedge \ast H^\beta - \frac{1}{2} \int_{\mathbb R^{1,1}} \Omega_{\alpha \beta} \, c^\alpha \wedge X^\beta \,.
\end{eqnarray}
The structure of this action is completely analogous to the well-familiar generalized Green-Schwarz action \cite{Green:1984bx,Sagnotti:1992qw} of self-dual tensor fields in $D = 6$ (see e.g. \cite{Bonetti:2011mw}) and, in fact, $D=10$ dimensions, with the role of the gauge invariant self-dual field strengths being played here
by the 1-forms $H^\alpha = D c^\alpha$. These are subject to the self-duality condition
\begin{eqnarray} \label{selfduality2d}
g_{\alpha \beta} \ast H^\alpha = \Omega_{\alpha \beta} H^\beta \,.
\end{eqnarray}
The second term in the action constitutes the Green-Schwarz coupling, which is responsible for the non-standard Bianchi identity
\begin{eqnarray}
d H^\alpha = X^\alpha \,,
\end{eqnarray}
where we used that $\Omega_{\alpha \beta}$ is a constant matrix.
The Green-Schwarz couplings will be found to take the form
\begin{eqnarray} \label{Xbeta}
X^\beta = \Theta_A^\beta \, F^A
\end{eqnarray}
with $F^A$ the field strength associated with the gauge group factor $U(1)_A$ and with $\Theta_A^\beta$ depending on the background flux.
This identifies $H^\alpha$ as
\begin{eqnarray} \label{covariant}
H^\alpha = D c^\alpha = d c^\alpha + \Theta_A^\alpha A^A \,.
\end{eqnarray}
The axionic shift symmetry of the chiral scalars is gauged by the abelian vector $A^A$ according to the transformation rule
\begin{equation}
\begin{aligned} \label{A-gauging}
A^A &\rightarrow& A^A+d\lambda^A \cr
c^\alpha &\rightarrow& c^\alpha -\Theta^{\alpha}_{A} \lambda^A
\end{aligned}
\end{equation}
such that the covariant derivative $D c^\alpha$
is gauge invariant.
As a result, the pseudo-action picks up a gauge variation of the form
\begin{eqnarray}
\delta S_{GS} = \frac{1}{2} \int \Omega_{\alpha \beta} \, \Theta_A^\alpha \lambda^A \, X^\beta =: 2\pi \, \int_{\mathbb{R}^{1,1}} I_{2}^{(1), {\rm GS}}(\lambda) \,,
\end{eqnarray}
with $I^{(1),{\rm GS}}_{2}$ a gauge invariant 2-form. By the standard descent procedure, it defines an anomaly-polynomial $I^{\rm GS}_4$ encoding the contribution to the total anomaly from the Green-Schwarz sector.
Concretely, the descent equations
\begin{eqnarray}
I^{\rm GS}_4=d I^{\rm GS}_3, \qquad \quad \delta_\lambda I^{\rm GS}_3=d I_{2}^{(1), {\rm GS}}(\lambda)
\end{eqnarray}
imply
\begin{eqnarray}
\label{factor}
2 \pi I^{\rm GS}_4= \frac{1}{2} \Omega_{\alpha \beta} X^\alpha X^\beta = \frac{1}{2} \Omega_{\alpha \beta} \Theta_A^\alpha \Theta_B^\beta \, F^A F^B \,.
\end{eqnarray}
Consistency of the theory then requires that
\begin{eqnarray}
I_4 + I_4^{\rm GS} = 0 \,.
\end{eqnarray}
This is possible only if the non-abelian and gravitational anomalies vanish by themselves and the abelian anomalies factorise suitably.
The resulting constraints on the spectrum take the following form:
\begin{subequations} \label{AnomaliesF-theory}
\begin{empheq}[box=\widefbox]{align}
\text{Non-abelian} &: \qquad
& \frac{1}{2} \sum_{\mathbf{\mathbf{R}}^I} \chi(\mathbf{r}^I) \, c^{(2)}_{\mathbf{r}^I}=0 \label{Non-abelian1}\\
\text{Abelian} &:\qquad
& \frac{1}{2}\sum_{\bf R} {\rm dim}({\bf R}) \, \chi({\bf R}) \, q_{A}({\bf R}) \, q_{B}({\bf R})= \frac1{4\pi} \Omega_{\alpha \beta} \Theta_A^\alpha \Theta_B^\beta \label{Abelian1} \\
\text{Gravitational} &:\qquad
&\sum_{\mathbf{R}} \text{dim}(\mathbf{R}) \, \chi(\mathbf{R}) =0\,. \label{Gravity1}
\end{empheq}
\end{subequations}
Note that, unlike in higher dimensions, the 2d GS mechanism operates entirely at the level of the abelian gauge group factors:
In $(4k + 2)$ dimensions the analogue of (\ref{covariant}) is the gauge invariant field strength associated with the self-dual rank $(2k+1)$-tensor fields, and the correction term
in the covariant action involves the Chern-Simons $(2k+2)$-forms associated with the gauge and diffeomorphism group. In 2d the Chern-Simons form is proportional to the trace over the gauge connection and must hence be abelian. Therefore the 2d non-abelian and gravitational anomalies from the chiral sector at 1-loop must vanish by themselves; likewise there can be no mixed gravitational-gauge anomalies induced at 1-loop.
Furthermore, let us point out that in the 2d $(0,2)$ theories of the type considered here the gauging (\ref{A-gauging}) of the scalars is directly related to the anomalous Green-Schwarz coupling (\ref{Xbeta}). This is a notable difference to the implementation of the Green-Schwarz mechanism in the more general 2d $(0,2)$ gauge theories of \cite{Adams:2006kb}, where these two are in principle independent.
Before we proceed, we would like to comment on the scalar fields $c^\alpha$. In principle, all of the axionic scalar fields $c^\alpha$ obtained from the Type IIB RR fields $C_p$ can contribute to the Green-Schwarz mechansim. However, as in 6d and 4d F-theory compactifications, the gauging of the scalar fields from $C_2$ is encoded via a {\rm geometric} St\"uckelberg mechanism in terms of non-harmonic forms, at least in the description via the dual M-theory \cite{Grimm:2011tb}. In this work we will we will only focus on the Green-Schwarz mechanism associated with the scalar fields arising from the RR potential $C_4$, which will be seen to depend on the background flux.
\section{F-theory on elliptically fibered Calabi-Yau five-manifolds} \label{sec_FonellCY5}
In this section we provide some background material on $N=(0,2)$ supersymmetric compactifications of F-theory to two dimensions. The reader familiar with this type of constructions from \cite{Schafer-Nameki:2016cfr,Apruzzi:2016iac} can safely skip this summary.
\subsection{Gauge symmetries and gauge backgrounds, and 3-branes}
We consider a 2d $(0,2)$ supersymmetric theory describing a vacuum of F-theory compactified on an elliptically fibered Calabi-Yau 5-fold $X_5$ \cite{Schafer-Nameki:2016cfr,Apruzzi:2016iac} with projection
\begin{equation}
\pi: X_5\rightarrow B_4 \,.
\end{equation}
The base $B_4$ is a smooth complex 4-dimensional K\"ahler manifold, which is to be identified with the physical compactification space of F-theory. Via F/M-theory duality, F-theory on $B_4$ is related to the supersymmetric quantum mechanics \cite{Haupt:2008nu} obtained by compactification of M-theory on $X_5$.
For simplicity we assume that $X_5$ has a global section $[z=0]$ so that it can be described by a Weierstrass equation
\begin{eqnarray}
y^2=x^3+f\, x\, z^4+g\, z^6\,.
\end{eqnarray}
Here the projective coordinates $[x : y : z ]$ parametrise the fiber ambient space $\mathbb{P}_{2,3,1}$ and $f,g$ are sections of the fourth and sixth power of the anti-canonical bundle
$\bar K$ of the base.
The discriminant locus
\begin{eqnarray}
\Delta=4f^3+27g^2=0
\end{eqnarray}
specifies the location of the 7-branes.
The non-abelian gauge group factors $G_I$ in (\ref{eq:Gsplit}) are associated with 7-branes wrapping divisors $W_I$, which are complex 3-dimensional components of the discriminant locus $\Delta=0$ in the base.
We assume that the Kodaira singularities in the fibre above $W_I$ admit a crepant resolution\footnote{To avoid clutter we will mostly avoid the hat above $\pi$ in the sequel.}
\begin{equation}
\hat \pi: \hat X_5\rightarrow B_4 \,.
\end{equation}
The resolution replaces the singularity over $W_I$ by a chain of
rational curves. After taking into account monodromy effects, which appear for non-simply laced groups, this allows one to identify a collection $\mathbb{P}^1_{i_I}$, $i_I = 1, \ldots, {\rm rk}(\mathfrak{g}_I)$ of independent rational curves in the resolved fiber which can be associated with the simple roots $\alpha_{i_I}$ of the Lie algebra $\mathfrak{g}_I$ underlying $G_I$ in the following sense:
The fibration of $\mathbb P^1_{i_I}$ over $W_I$ - more precisely of the image of $\mathbb P^1_{i_I}$ under monodromies in the non-simply laced case - defines a resolution divisor $E_{i_I}$ with the property that
\begin{eqnarray}
[E_{i_I}]\cdot [\mathbb{P}^1_{j_J}]=-\delta_{IJ} \, C_{i_Ij_J} \,.
\end{eqnarray}
Here $[E_{i_I}]$ denotes the homology class of the divisor $E_{i_I}$ and unless noted otherwise, all intersection products are taken on $\hat X_5$. The matrix $C_{i_I j_I}$ is the Cartan matrix of $\mathfrak{g}_I$ (in conventions where the entries on its diagonal are $+2$).
Via duality with M-theory, M2-branes wrapping the fibral curves $\mathbb{P}^1_{j_J}$ give rise to states associated with the simple roots $ - \alpha_{i_I}$, and the Cartan $U(1)_{i_I}$ gauge field arises by KK reduction of the M-theory 3-form as
\begin{eqnarray}
C_3=A_{i_I}\wedge [E_{i_I}] + ....
\end{eqnarray}
In this sense the resolution divisors $[E_{i_I}]$ can be identified with the generators ${\cal T}_{i_I}$ of the Cartan subgroup of $G_I$ in the so-called co-root basis,
whose trace over the fundamental representation of $G_I$ is normalised such that
\begin{eqnarray}
{\rm tr}_{\rm fund} {\cal T}_{i_I} {\cal T}_{j_J} = \delta_{IJ} \, \lambda_I \, \mathfrak{C}_{i_I j_I} \qquad \quad {\rm with} \qquad \mathfrak{C}_{i_I j_I}=\frac{2}{\lambda_I} \frac{1}{\langle \alpha_{j_I}, \alpha_{j_I} \rangle} \, C_{i_I j_I} \,.
\end{eqnarray}
The quantity $\lambda_I$ denotes the Dynkin index in the fundamental representation and is tabulated in \autoref{Tab_Lambda}. Note that for simply-laced groups $\mathfrak{C}_{i_I j_I} = {C}_{i_I j_I}$.
The geometric manifestation of this identification is the important relation
\begin{equation} \label{piEiEj}
\pi_\ast( [E_{i_I}] \cdot [E_{j_J}])=-\delta_{IJ} \, \mathfrak{C}_{i_I j_I} \, [W_I] = - {\rm Tr} \, {\cal T}_{i_I} {\cal T}_{j_J} \, [W_I]\,
\end{equation}
where ${\rm Tr}$ is related to the trace in the fundamental representation via
\begin{eqnarray} \label{defTr}
{\rm Tr} = \frac{1}{\lambda_I} \, {\rm tr}_{\bf fund} \,.
\end{eqnarray}
The push-forward $\pi_\ast( [E_{i_I}] \cdot [E_{j_J}])$ to the base of the fibration is defined by requiring that
\begin{eqnarray}
[E_{i_I}]\cdot_{\hat X_5}[E_{j_J}]\cdot_{\hat X_5} [D_\alpha]\cdot_{\hat X_5}[D_\beta] \cdot_{\hat X_5} [D_\gamma]=\pi_\ast( [E_{i_I}]\cdot_{\hat X_5}[E_{j_J}])\cdot_{B_4} [D^{\rm b}_\alpha]\cdot_{B_4}[D^{\rm b}_\beta] \cdot_{B_4}[D^{\rm b}_\gamma]
\end{eqnarray}
for any basis of vertical divisors $[D_\alpha] = \pi^*[D^{\mathrm{b}}_\alpha] $, where $D^{\mathrm{b}}_\alpha$ is a divisor on $B_4$.
\begin{table}
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|}
\hline
$ {\mathfrak g}$ & $A_n$ & $D_n$ & $B_n$ & $C_n$ & $E_6$ & $E_7$ & $E_8$ & $F_4$ & $G_2$ \\ \hline
$ \lambda$ & $1$ & $2$ & $2 $ & $1$ & $6$ & $12$ & $60$ & $6$ & $2$ \\ \hline
\end{tabular}
\caption{Dynkin index of the fundamental representation for the simple Lie algebras. \label{Tab_Lambda}}
\end{center}
\end{table}
Each non-Cartan Abelian gauge group factor $U(1)_A$ is associated with a global rational section $S_A$ of $\hat X_5$ in addition to the zero-section $S_0$. To each $S_A$ one can assign an element $[U_A] \in \mathrm{CH}^1(\hat X_5)$ through the Shioda map
\begin{eqnarray} \label{ShiodaUA}
U_A=S_A - S_0- D_A+ \sum_{i_I}k_{i_I} E_{i_I} \,.
\end{eqnarray}
The vertical divisor $D_A$ and the in general fractional coefficients $k_{i_I}$ are chosen such that $U_A$ satisfies the transversality conditions
\begin{equation}
\begin{aligned}
&[U_A]\cdot_{\hat X_5} [D_\alpha]\cdot_{\hat X_5}[D_\beta]\cdot_{\hat X_5}[D_\gamma] \cdot_{\hat X_5} [D_\delta]=0 \qquad [U_A] \cdot_{\hat X_5}[S_0] \cdot_{\hat X_5} [D_\alpha]\cdot_{\hat X_5}[D_\beta] \cdot_{\hat X_5}[D_\gamma] =0 \cr
&[U_A] \cdot_{\hat X_5}[E_{i_I}] \cdot_{\hat X_5} [D_\alpha]\cdot_{\hat X_5}[D_\beta] \cdot_{\hat X_5}[D_\gamma] =0 \,,
\end{aligned}
\end{equation}
which must hold for every vertical divisor $[D_{\alpha}] = \pi^\ast D^{\mathrm b}_{\alpha}$.
In analogy with the relation (\ref{piEiEj}), one can define the so-called height pairing \cite{Park:2011ji,Morrison:2012ei}
\begin{eqnarray} \label{heightpairing}
\pi_\ast ( [U_A] \cdot_{\hat X_5} [U_B]) = - {\rm Tr} \, {\cal T}_A {\cal T}_B \, [D_{AB}] \, .
\end{eqnarray}
The objects ${\cal T}_A$, $ {\cal T}_B$ are the generators of $U(1)_A$ and $U(1)_B$ and $D_{AB}$ is a divisor on the base of the fibration.
Unlike the divisor $W_I$, even for $A=B$ this divisor is not one of the irreducible components of the discriminant $\Delta$ (in the sense that $\Delta$ would factorise into the union of various irreducible such $D_{AA}$). Nonetheless, we will see that it plays a very analogous role for the structure of anomalies also for F-theory compactifications to 2d.
A crucial ingredient in F/M-theory compactifications on Calabi-Yau five-folds is the gauge background for the field strength $G_4 =d C_3$ of the M-theory 3-form potential field.
As in compactifications to four dimensions, the full gauge background is an element of the Deligne cohomology group $H^4_D(\hat X_5, \mathbb Z(2))$ and can be parametrized by equivalence classes of rational complex codimension-2-cycles \cite{Bies:2014sra,Bies:2017fam}, which form the second Chow group $\mathrm{CH}^2(\hat X_5)$. The field strength of $G_4$ as such takes values in
$H^{4}( \hat{X}_5)$. It is subject to the
Freed-Witten quantization condition \cite{Witten:1996md}
\begin{equation}
G_4 + \frac{1}{2} c_2(Y_5) \in H^4(\hat{X}_5,\mathbb Z)\,.
\end{equation}
In order to preserve two supercharges in the M/F-theory compactification on $\hat{X}_5$, the $(3,1)$ and $(1,3)$ Hodge components of $H^{4}( \hat{X}_5)$ must vanish \cite{Haupt:2008nu} and hence
\begin{equation} \label{SUSYG422}
G_4 + \frac{1}{2} c_2(Y_5) \in H^4(\hat{X}_5,\mathbb Z) \cap H^{2,2}(\hat{X}_5)\,.
\end{equation}
By F/M-duality, the $G_4$ fluxes are subject to the transversality constraints
\begin{equation} \label{transverse1}
\int_{\hat X_5} G_4 \wedge S_0 \wedge \pi^\ast\omega_4 = 0 \quad
{\hbox{and}}
\qquad \int_{\hat X_5} G_4 \wedge \pi^\ast \omega_6 = 0\,, \qquad \forall \, \omega_4 \in H^4(B_4), \, \, \omega_6 \in H^6(B_4)\,.
\end{equation}
If this flux satisfies in addition the constraint
\begin{eqnarray} \label{gaugeinvariance-a}
\int_{\hat X_5} G_4 \wedge E_{i_I} \wedge \pi^\ast\omega_4 = 0
\end{eqnarray}
it leaves the gauge group factor $G_I$ unbroken.
Higher curvature corrections in the M-theory effective action induce a curvature dependent tadpole for the M-theory 3-form $C_3$.
In the dual F-theory these curvature corrections subsume the curvature contributions to the Chern-Simons action of the 7-branes (including, in the perturbative limit, the orientifold planes).
In a consistent M-theory vacuum this tadpole must be cancelled by the inclusion of background flux $G_4$ and/or by M2-branes wrapping a curve class on $\hat X_5$ determined by the tadpole equation \cite{Haupt:2008nu}. The projection of this curve class to the base $B_4$ describes\footnote{The M2-brane states along the fibral component of this class are related to momentum modes along the circle $S^1$ arising in F/M-theory duality \cite{Lawrie:2016rqe}.} , in the dual F-theory, the class wrapped by background D3-branes filling in addition the extended directions along $\mathbb R^{1,1}$. The projected class is given by \cite{Haupt:2008nu,Schafer-Nameki:2016cfr}
\begin{eqnarray} \label{Cclass}
[C] = \frac{1}{24} \pi_\ast c_4(\hat X_5) - \frac{1}{2} \pi_\ast (G_4 \cdot_{\hat X_5} G_4) \,.
\end{eqnarray}
\subsection{Matter spectrum from F-theory compactification on CY 5-folds}
The charged chiral matter fields whose contributions to the 1-loop anomalies we will be studying arise from three sources \cite{Schafer-Nameki:2016cfr,Apruzzi:2016iac}:
7-brane bulk matter propagating along the non-abelian divisors $W_I$, 7-brane codimension-two matter localised along the intersections of various discriminant components or self-intersections of the discriminant, and finally Fermi multiplets at the pointlike intersection of D3-branes with the 7-branes. Due to the chiral nature of the 2d $(0,2)$ theory, all three types of matter are chiral even for vanishing gauge backgrounds.
The bulk matter fields transform, in the absence of gauge flux, in the adjoint representation of $G_I$. In the dual M-theory quantum mechanics, this matter arises from M2-branes wrapping suitable combinations of resolution $\mathbb P^1_{i_I}$ in the fiber over $W_I$.
For non-vanishing gauge backgrounds,
which can be described by a non-trivial principal gauge bundle $L$, the original gauge group $G_I$ can be broken into a product of some sub-groups. The spectrum decomposea into irreducible representations $ \mathbf{R}$ of the unbroken gauge factors
\begin{eqnarray}
G_I &\rightarrow& H_I \\
\mathbf{Adj}(G_I) &\rightarrow& \mathbf{Adj}(H_I) \oplus \bigoplus_{\mathbf{R}}\mathbf{R}
\end{eqnarray}
Note that if ${\bf R} \neq {\bf \bar R}$, each representation is accompanied by its complex conjugate.
The matter fields organise into 2d $(0,2)$ chiral multiplets, which contain one complex boson and a complex chiral Weyl fermion, as well as Fermi multiplets, which contain one complex anti-chiral Weyl fermion. Each of these matter fields is counted by a certain cohomology group on $W_I$ involving the vector bundle $L_{\mathbf{R}}$.
The chiral index of massless matter in a given complex representation, defined as the difference of chiral and anti-chiral fermions in complex representation ${\mathbf{R}}$, is then given by \cite{Schafer-Nameki:2016cfr,Apruzzi:2016iac}
\begin{eqnarray} \label{chibulk1a}
\chi({\bf R}) = - \int_{W_I} c_1(W_I) \left( \frac{1}{12} {\rm rk}(L_{\mathbf R}) \, c_2(W_I) + \mathrm{ch}_2(L_{\mathbf R}) \right) \,.
\end{eqnarray}
For real representations, this expression is to be multiplied with a factor of $\frac{1}{2}$. In particular, the chiral index of the adjoint representation depends purely on the geometry and takes the form $\chi( \mathbf{Adj}(H_I)) = - \frac{1}{24} \int_{W_I} c_1(W_I) c_2(W_I) $.
Extra matter states in representation $\mathbf{R}$ of $G^{\rm tot}$ localizes on complex 2-dimensional surfaces $C_{\mathbf{ R}}$ on $B_4$.
This occurs whenever
some of the rational curves $\mathbb{P}^1_{i_I}$ in the fiber split over $C_{\mathbf{ R}}$.
Group theoretically, this signifies the splitting of the associated simple roots into weights of representation ${\bf R}$.
The associated charged matter fields arise from M2-branes wrapped on suitable linear combinations of fibral curves over $C_{\mathbf{ R}}$, which in fact span
the weight lattice of the gauge theory.
Hence to each state in representation $\mathbf{ R}$ we can associate a matter 3-cycle $S^a_{\mathbf{ R}}$ which is given by a linear combination of fibral curves over $C_{\mathbf{ R}}$ and carries a weight vector $\beta^a_{i_I}, a=1,...,{\rm dim}(\mathbf{R})$, such that
\begin{equation}
\pi_\ast ([E_{i_I}]\cdot [S^a_{\mathbf{ R}}])=\beta^a_{i_I}[C_{\mathbf{ R}}] \,.
\end{equation}
These matter states also organize both into chiral and Fermi multiplets and are counted by cohomology groups of a vector bundle $L_{\mathbf R}$ which derives from the gauge background.
If the surface $C_{\mathbf{R}}$ on $B_4$ is smooth, the chiral index of this type of matter follows from an index theorem as \cite{Schafer-Nameki:2016cfr,Apruzzi:2016iac}
\begin{equation}
\begin{aligned} \label{chiS2}
\chi({\bf R})
&= \int_{C_{\bf R}} \left(c_1^2(C_{\bf R}) \left(\frac{1}{12} - \frac{1}{8} {\rm rk}(L_{\bf R}) \right) + \frac{1}{12} c_2(C_{\bf R}) + \left(\frac{1}{2} c_1^2(L_{\bf R}) - c_2(L_{\bf R})\right) \right).
\end{aligned}\end{equation}
Otherwise one has to perform a suitable normalisation in order to be able to apply the index theorem, and this will lead to correction terms as exemplified in \cite{Schafer-Nameki:2016cfr}.
The third type of massless matter arises from 3-7 string states at the intersection of the 7-branes with the spacetime-filling D3-branes wrapping the curve class $[C]$ in (\ref{Cclass}).
Matter in the 3-7 sector comes in 2d (0,2) Fermi multiplets \cite{Schafer-Nameki:2016cfr,Apruzzi:2016iac}.
In purely perturbative setups, each intersection point of $[C]$ with one of the D7-branes carries a single Fermi multiplet in the fundamental representation of the D7-brane gauge group.
However, monodromy effects along the 3-brane worldvolume considerably obscure such a simple interpretation of the 3-7 modes in non-perturbative setups \cite{Schafer-Nameki:2016cfr, Lawrie:2016axq}. As one of our results, we will see how the structure of 2d anomalies sheds new light on the structure of 3-7 modes, including, in particular, their charges under the non-Cartan abelian gauge factors.
\section{Anomaly equations in F-theory on Calabi-Yau 5-folds } \label{sec_AnomalyEqu5folds}
In this section we present closed expressions for the anomaly cancellation conditions in 2d $(0,2)$ F-theory vacua. We begin in section \ref{sec_gaugean} by deriving a formula for the chiral index of charged matter states in the presence of 4-form flux $G_4$ in the dual M-theory, which is uniformly valid for the bulk and the localised 7-7 modes.
We also shed some more light on the counting of 3-7 modes. Together with the Green-Schwarz counterterms this leads to formula (\ref{gaugeanomalyFtheory}) for the cancellation of all gauge anomalies.
In section \ref{gravitationalanomaly} we extend the gravitational anomaly cancellation conditions of \cite{Lawrie:2016rqe} to situations with non-trivial 7-branes and fluxes, leading us to condition (\ref{gravanomA-general}).
\subsection{Gauge anomalies, Green-Schwarz terms and the 3-7 sector} \label{sec_gaugean}
Recall from the previous section that in this paper we assume the existence of a smooth crepant resolution $\hat X_5$, which describes the dual M-theory on its Coulomb branch. This forces us, as usual in this context, to restrict ourselves to Abelian gauge backgrounds $G_4$. In particular, the vector bundles appearing in the expressions (\ref{chibulk1a})
and (\ref{chiS2}) are complex line bundles.
For simplicity of presentation we first assume that the gauge flux $G_4$ does not break any of the non-abelian gauge group factors.
The chiral index (\ref{chiS2}) of the localised matter can be split into a purely geometric and a flux dependent contribution
\begin{eqnarray}
\chi({\bf R}) &=& \chi_{\rm geom}({\bf R}) + \chi_{\rm flux}({\bf R}) \nonumber \\
\chi_{\rm geom}({\bf R}) &=& - \frac{1}{12} \int_{C_{\bf R}} \mathrm{ch}_2(C_{\bf R}) = \frac{1}{12} \int_{C_{\bf R}} c_2(C_{\bf R}) - \frac{1}{2} c_1^2(C_{\bf R} ) \label{chigeom1}\\
\chi_{\rm flux}({\bf R}) &=& \int_{C_{\bf R}} \frac{1}{2} c_1^2(L_{\bf R}) \,. \nonumber
\end{eqnarray}
We stress that this expression is correct provided the matter 2-cycles $C_{\bf R}$ on $B_4$ are smooth. The line bundle $L_{\bf R}$ on $C_{\bf R}$ to which a state with weight vector $\beta^a({\bf R}))$ couples is obtained from $G_4$ by first integrating $G_4$ over the fiber of the matter 3-cycle $S^a_\mathbf{R}$ and then projecting onto the surface $C_{\bf R}$. This gives rise to a divisor class on $C_{\mathbf{R}}$ which is to be identified, similarly to the procedure in F-theory on Calabi-Yau 4-folds \cite{Bies:2014sra,Bies:2017fam}, with
\begin{eqnarray} \label{c1LR}
c_1(L_{\bf R}) = \pi_\ast (G_4 \cdot S^a_{\bf R}) \,.
\end{eqnarray}
Note that for gauge invariant flux, the result is the same for each of the matter 3-cycles $S^a_{\bf R}$ and hence correctly defines the line bundle associated with representation ${\mathbf{R}}$.
This allows us to rewrite $\chi_{\rm flux}({\bf R}) $ explicitly in terms of $G_4$ as
\begin{eqnarray}
\chi_{\rm flux}({\bf R}) = \frac{1}{2} \pi_\ast (G_4 \cdot S^a_{\bf R}) \cdot_{C_{\bf R}} \pi_\ast (G_4 \cdot S^a_{\bf R}) \,,
\end{eqnarray}
where $\cdot_{C_{\bf R}}$ denotes the intersection product on $C_{\bf R}$.
Next, consider the bulk modes.
For gauge invariant flux, this sector contributes only states in the adjoint representation of $G_I$ (which due to the quadratic nature of the anomalies nonetheless contribute to the anomaly), and according to (\ref{chibulk1a}) their chiral index is given by
\begin{eqnarray} \label{chibadj1}
\chi_{\rm bulk}({\bf R} = {{\rm \bf adj}_I}) = - \frac{1}{24} \int_{W_I} c_1(W_I) c_2(W_I) \,.
\end{eqnarray}
It is useful to note that $\chi_{\rm bulk}({\bf R})$ is formally identical to the flux-independent part of the chirality of a localised state whose matter locus is given by the canonical divisor on $W_I$, i.e. the complex 2-cycle on $W_I$ in the class
\begin{eqnarray}
[C_{\rm can}] = -c_1(W_I) = + c_1(K_{W_I}) \,. \,
\end{eqnarray}
Indeed, by adjunction, using the short exact sequence
\begin{eqnarray}
0 \rightarrow T_{C_{\rm can}} \rightarrow T_{W_I} \rightarrow N_{C_{\rm can}/W_I} \rightarrow 0
\end{eqnarray}
and the resulting relation
\begin{eqnarray}
c(T_{C_{\rm can}}) = c( T_{W_I}) / c(N_{C_{\rm can}/W_I}) = (1 + c_1(W_I) + c_2(W_I)) / (1 -c _1(W_I) ) ,
\end{eqnarray}
one computes
\begin{eqnarray}
c_1(C_{\rm can}) &=& 2 c_1(W_I) \\
c_2(C_{\rm can}) &=& c_2(W_I) + 2 c_1^2(W_I) \,.
\end{eqnarray}
This implies that
\begin{eqnarray}
\int_{C_{\rm can}} \frac{1}{12} (c_2(C_{\rm can}) - \frac{1}{2} c_1^2(C_{\rm can}) ) = -\frac{1}{12} \int_{W_I} c_1(W_I) \cdot c_2(W_I).
\end{eqnarray}
The additional factor of $\frac{1}{2}$ in (\ref{chibadj1}) is due to the fact that the adjoint is a real representation.
More generally, and in complete analogy to the description of bulk modes in compactifications on Calabi-Yau 4-folds \cite{Bies:2017fam}, we can associate to a bulk matter state associated with the root $\rho_I$ the 3-cycle
\begin{eqnarray} \label{Srho}
S^{{\rho}_I} = \sum_{i_I} \hat a_{i_I} E_{i_I} |_{K_{W_I}} \,.
\end{eqnarray}
The parameters $\hat a_{i_I}$ are related to the coefficients in the expansion of the root $\rho_I$ in terms of the simple roots $\alpha_{i_I}$.\footnote{For simply laced Lie algebras, $\rho_I = \sum_{i_I} \hat a_{i_I} \alpha_{i_I}$. For non-simply laced Lie algebras, fractional corrections must be included to take into account monodromy effects, as explained e.g. in appendix A of \cite{Park:2011ji}.}
Geometrically, the fiber of $S^{{\rho}_I} $ is given by the corresponding linear combination of fibral rational curves $\mathbb P^1_{i_I}$. An M2-brane wrapped along this linear combination of fibral curves gives rise to a state whose Cartan charges are given precisely by the root $\rho_I$.
For gauge invariant flux satisfying (\ref{gaugeinvariance-a}), the line bundle $\pi_\ast (S^{{\rho}_I} \cdot G_4)$ vanishes by construction. Hence the expression for the bulk and the localised chirality are completely analogous and both types of matter will from now on be treated on the same footing.
This conclusion persists if the gauge background breaks some or all of the simple gauge group factors $G_I$. In this case, the adjoint representation for the bulk matter or the representations associated with the localised matter decompose into irreducible representations of the unbroken subgroup.
The operation (\ref{c1LR}) now leads to a well-defined line bundle for each of these individual representations, for bulk and localised matter alike.
Next, we consider the contribution from the 3-7 modes. As it turns out, to each representation ${\bf R}$ one can associate a divisor $D_{37}({\bf R})$ on $B_4$ such that the chiral index of 3-7 states in representation ${\bf R}$ is given by
\begin{eqnarray} \label{D37notation}
\chi_{3-7}({\bf R}) = - \Big( \frac{1}{24} \pi_\ast (c_4(\hat X_5)) - \frac{1}{2} \pi_\ast(G_4 \cdot G_4) \Big) \cdot_{B_4} D_{37}({\bf R}) \,.
\end{eqnarray}
The expression in brackets is the curve class $[C]$, defined in (\ref{Cclass}), wrapped by the spacetime-filling D3-branes.
For instance, for a perturbative gauge group $G_I = SU(N)$, each intersection point of $[C]$ with the 7-brane divisor $W_I$ hosts a (negative chirality) Fermi multiplet in representation
${\bf R} = ({\bf N})$ \cite{Schafer-Nameki:2016cfr,Apruzzi:2016iac} and therefore $D_{37}({\bf R} = ({\bf N})) = W_I$.
For non-perturbative gauge groups and for Abelian non-Cartan groups $U(1)_A$ determining the representation and charge of the 3-7 strings from first principles is more obscure due to subtle $SL(2,\mathbb Z)$ monodromy effects on the worldvolume of the D3-brane along $C$ \cite{Lawrie:2016axq}.
However, in the next section we will derive that in the presence of extra $U(1)_A$ gauge group factors the net contribution to the $U(1)_A - U(1)_B$ anomaly (\ref{AAB-gen1})
from the 3-7 sector takes the form
\begin{eqnarray}
{\cal A}_{AB}|_{3-7} &=& \frac{1}{2} \sum_{{\bf R}, 3-7} q_A({\bf R}) \, q_B({\bf R}) \, {\rm dim}({\bf R}) \, \chi_{3-7}({\bf R}) \\
&=& \frac{1}{2} \, \Big( \frac{1}{24} \pi_\ast (c_4(\hat X_5)]) - \frac{1}{2} \pi_\ast(G_4 \cdot G_4) \Big) \cdot_{B_4} \pi_{\ast} (U_A \cdot U_B) \,. \label{37ABanomaly}
\end{eqnarray}
Here we recall that $U_A$ and $U_B$ generate the respective $U(1)$ factors via the Shioda map (\ref{ShiodaUA}) and that
the height-pairing $\pi_{\ast} (U_A \cdot U_B)$ had been introduced in (\ref{heightpairing}).
More generally, our results imply that the right-hand side correctly captures the contribution to the anomaly also of the Cartan $U(1)$ group for non-perturbative gauge groups.
Let us introduce the notation
\begin{eqnarray}
{\rm span}_{\mathbb C} \{ {\mathfrak F}_{\Sigma} \} = {\rm span}_{\mathbb C} \{E_{i_I}, U_A\}
\end{eqnarray}
to collectively denote set of divisors generating any of the Cartan $U(1)_{i_I}$ or non-Cartan $U(1)_A$ gauge symmetries.
Then our claim is that the contribution to the gauge anomaly due to 3-7 modes can be summarized as
\begin{eqnarray}
{\cal A}_{\Lambda \Sigma}|_{3-7} = \frac{1}{2} \sum_{{\bf R}, a}\beta^a_\Lambda({\bf R}) \, \beta^a_\Sigma({\bf R}) \, \chi_{3-7}({\bf R}) = \frac{1}{2} \, \Big( \frac{1}{24} \pi_\ast (c_4(\hat X_5)]) - \frac{1}{2} \pi_\ast(G_4 \cdot G_4) \Big) \cdot_{B_4} \pi_\ast (\mathfrak{F}_\Lambda \cdot \mathfrak{F}_\Sigma) \,.\label{37ABanomaly-b}
\end{eqnarray}
If the index $\Lambda = i_I$ refers to a Cartan $U(1)_{i_I}$, the object $\beta^a_{i_I}({\bf R})$ denotes the weights associated with representation ${\bf R}$ with respect this $U(1)_{i_I}$, and for $\Lambda = A$ we define $\beta^a_{A}({\bf R}) = q_A({\bf R})$.
We will come back to the interpretation of this formula at the end of this section.
As the final ingredient we will derive, in section \ref{sec_GStermsderivation}, the Green-Schwarz counterterms appearing on the righthand side of (\ref{Abelian1}).
These are found to be purely flux dependent and of the form
\begin{eqnarray} \label{GSclaim}
\frac1{4\pi} \Omega_{\alpha \beta} \Theta_\Sigma^\alpha \Theta_\Lambda^\beta = \frac{1}{2} \pi_\ast (G_4 \cdot \mathfrak{F}_\Sigma) \cdot_{B_4} \pi_\ast (G_4 \cdot \mathfrak{F}_\Lambda ) \,.
\end{eqnarray}
For instance, if we let $\mathfrak{F}_\Lambda = U_A$, $\mathfrak{F}_\Sigma = U_B$ refer to non-Cartan Abelian groups, then this describes the Green-Schwarz counterterms for the $U(1)_A- U(1)_B$ anomalies.
For $\mathfrak{F}_\Lambda = E_{i_I}$, $\mathfrak{F}_\Sigma = E_{j_I}$, the right-hand side is non-vanishing only if the gauge background $G_4$ breaks the simple gauge group factors $G_I$ and $G_J$, in which case it computes the counterterms for the $U(1)_{i_I} - U(1)_{j_J}$ anomaly. For gauge invariant flux, on the other hand, no such Green-Schwarz terms are induced, in agreement with expectations.
With this preparation we can now rewrite the gauge anomaly equations (\ref{Non-abelian1}), (\ref{Abelian1}) in a rather suggestive form.
Since the anomaly equations must hold for arbitrary gauge background $G_4$
and since the flux independent terms only give a constant off-set, the flux dependent and the flux independent contributions to the anomalies must vanish separately.
The requirement (\ref{Non-abelian1}), (\ref{Abelian1}) of cancellation of all gauge anomalies therefore results in two independent identities:
\begin{subequations} \label{gaugeanomalyFtheory}
\begin{empheq}[box=\widefbox]{align}
0 &=\sum_{{\bf R},a} \beta^a_\Lambda({\bf R}) \, \beta^a_\Sigma({\bf R}) \int_{C_{\bf R}} \mathrm{ch}_2(C_{\bf R}) - \frac{1}{2} \, \pi_\ast (c_4(\hat X_5)) \cdot \pi_\ast (\mathfrak{F}_\Lambda \cdot \mathfrak{F}_\Sigma) \label{gauge-geom1} \\
0 &= \sum_{{\bf R},a} \beta^a_\Lambda({\bf R}) \, \beta^a_\Sigma({\bf R}) \, \pi_\ast (G_4 \cdot S^a_{\bf R}) \cdot_{C_{\bf R}} \pi_\ast (G_4 \cdot S^a_{\bf R}) \nonumber \\
& - \big( \pi_\ast (G_4 \cdot G_4) \cdot_{B_4} \pi_\ast (\mathfrak{F}_\Lambda \cdot \mathfrak{F}_\Sigma) \label{gauge-flux1} \\
& \phantom{-(}+ \pi_\ast (G_4 \cdot \mathfrak{F}_\Sigma) \cdot_{B_4} \pi_\ast (G_4 \cdot \mathfrak{F}_\Lambda ) {+ \pi_\ast (G_4 \cdot \mathfrak{F}_\Lambda) \cdot_{B_4} \pi_\ast (\mathfrak{F}_\Sigma \cdot G_4) \big) \nonumber \,.}
\end{empheq}
\end{subequations}
The two terms in (\ref{gauge-geom1}) respectively represent the flux independent anomaly contribution from the 7-7 sector, (\ref{chigeom1}), and from the 3-7 sector, (\ref{37ABanomaly-b}).
In (\ref{gauge-flux1}) we have collected the flux dependent 3-7 and the Green-Schwarz contribution to the anomaly in the brackets in the second and third line to illustrate the striking formal similarity between them. We will understand this similarity in the next section.
Let us now come back to the interpretation of (\ref{37ABanomaly-b}).
For $\mathfrak{F}_\Lambda = E_{i_I}$, $\mathfrak{F}_\Sigma = E_{j_I}$ this equation allows us to deduce the net contribution to the anomalies due to 3-7 strings charged under the non-abelian gauge group factors, which, as noted already, can be rather obscure due to monodromy effects.
To interpret this expression, recall the crucial identity (\ref{piEiEj}).
If we assume that each geometric intersection point $[C] \cdot_{B_4} W_I$ hosts an (anti-chiral) Fermi multiplet in representation ${\bf R}$, then for consistency this representation must satisfy
\begin{eqnarray} \label{Cijbeta}
\sum_a \beta^a_{i_I}({\bf R} ) \beta^a_{j_I}({\bf R} ) \stackrel{!}{=} \mathfrak{C}_{i_I j_I} \,.
\end{eqnarray}
This is to be contrasted with the fact that for any representation ${\bf R}$ of a simple group $G_I$
\begin{eqnarray}
\sum_a \beta^a_{i_I}({\bf R} ) \beta^a_{j_I}({\bf R} ) = {\rm tr}_{\bf R} {\cal T}_{i_I} {\cal T}_{j_I} = \lambda_I \, c_{\mathbf{R}}^{(2)} \, \mathfrak{C}_{i_I j_I}
\end{eqnarray}
with ${\cal T}_{i_I}$ denoting the Cartan generators in the coroot basis.
The Dynkin index $\lambda_I$ for the fundamental representation of $G_I$ is collected, for all simple groups, in \autoref{Tab_Lambda}, and $c_{\mathbf{R}}^{(2)}$ normalizes the trace with respect to the fundamental representation as in (\ref{cr2}).
By definition, the smallest value of $c_{\mathbf{R}}^{(2)}$ occurs for the fundamental representation $c_{\mathrm{\bf fund}}^{(2)} =1$. Hence unless $\lambda_I = 1$ or $\lambda_I = 2$, the interpretation in terms of 3-7 modes necessarily involves 'fractional' Fermi multiplets.\footnote{The case $\lambda_I = 2$ requires, for consistency, that the fundamental representation be real and hence contributes with a factor of $\frac{1}{2}$ to compensate for $\lambda_I$. \autoref{Tab_Lambda} confirms that this is indeed the case for all simple algebras with $\lambda_I = 2$.} This is in agreement with the observation of \cite{Schafer-Nameki:2016cfr} that e.g. for $G_I = E_6$, the net contribution to the anomaly
from the 3-7 sectors corresponds to that of a $\frac{1}{6}$-fractional Fermi multiplet per intersection point.
\subsection{Gravitational Anomaly}
\label{gravitationalanomaly}
The gravitational anomaly for F-theory compactified on a smooth Weierstrass model $X_5$ without any 7-brane gauge group and background flux has already been discussed in \cite{Lawrie:2016rqe}.
The anomaly polynomial receives contributions from the moduli sector, from the 2d $(0,2)$ supergravity multiplet as well as from the 3-7 sector,
\begin{eqnarray}
I_{4, \rm grav} &=& \frac{1}{24} p_1(T) \left( {\cal A}_{\rm grav}|_{\rm mod} + {\cal A}_{\rm grav}|_{\rm uni} + {\cal A}_{\rm grav}|_{3-7} \right) \\
{\cal A}_{\rm grav}|_{\rm mod} &=& -\tau(B_4) + \chi_1(X_5) - 2 \chi_1(B_4), \label{Amod} \\
{\cal A}_{\rm grav}|_{\rm uni} &=& 24 \\
{\cal A}_{\rm grav}|_{3-7}& =& - 6 c_1(B_4) \cdot [C] \,.
\end{eqnarray}
Note that ${\cal A}_{\rm grav}|_{\rm mod}$ includes what would be called in Type IIB language the contributions from the closed string moduli sector, from the moduli associated with the 7-branes (which however by assumption carry no gauge group), and from $\tau(B_4)$ many 2d $(0,2)$ tensor multiplets.
Here
\begin{eqnarray}
\chi_q(M) = \sum_{p=1}^{{\rm dim}(M)} (-1)^p h^{p,q}(M)
\end{eqnarray}
and
\begin{equation}
\chi_1(X_5)=-\frac1{24}\int_{X_5} c_5(X_5)=\int_{B_4}(90c_1^4+3c_1^2c_2-\frac12c_1c_3)
\end{equation}
with $c_i = c_1(B_4)$.
Furthermore the signature $\tau(B_4)$ counts the difference of self-dual and anti-self-dual 4-forms on $B_4$ and is related to the Hodge numbers of $B_4$ as
\begin{eqnarray}
\tau(B_4) = b_4^+(B_4) - b_4^-(B_4) = 48 + 2 h^{1,1}(B_4) + 2 h^{3,1}(B_4) - 2 h^{2,1}(B_4) \,.
\end{eqnarray}
The D3-brane class appearing fixed by the tadpole on a smooth Weierstrass model without flux is $[C]=\frac1{24} \pi_\ast c_4(X_5)$.
As shown in \cite{Lawrie:2016rqe} with the help of various index theorems, the total anomaly can be evaluated as
\begin{eqnarray} \label{I4gravWeiersmooth}
I_{4, \rm grav} = \frac{1}{24} p_1(T) \left( -24 \chi_0(B_4) + 24 \right) \equiv 0 \,,
\end{eqnarray}
where the last equality holds because $h^{0,i}(B_4)$ for $i \neq 0$ if $B_4$ is to admit a smooth Calabi-Yau Weierstrass fibration over it.
Suppose now that the fibration contains in addition a non-trivial 7-brane gauge group and charged 7-7 matter, and let us also switch a non-trivial flux background $G_4$.
For simplicity assume first that the supersymmetry condition that $G_4$ be of pure $(2,2)$ Hodge type \cite{Haupt:2008nu} does not constrain the moduli of the compactification.
In analogy with $G_4$ flux on Calabi-Yau 4-folds, this is guaranteed whenever
$G_4 \in H^{2,2}_{\rm vert}(\hat X_5)$, the primary vertical subspace of $H^{2,2}(\hat X_5)$ generated by products of $(1,1)$ forms.\footnote{The space of $(2,2)$ forms on Calabi-Yau 5-folds deserves further study beyond the scope of this article. In particular it remains to investigate in more detail whether a similar split into horizontal and vertical subspaces exists as on Calabi-Yau 4-folds. In any event if $G_4$ is a sum of $(2,2)$ forms obtained as the product of two $(1,1)$ forms, the Hodge type does not vary.}
In this situation the gravitational anomaly generalizes as follows:
First, we must now work on the resolution $\hat X_5$ of the singular Weierstrass model describing the more general 7-brane configuration.
In particular the D3-brane curve class changes to $[C] = \frac{1}{24} \pi_\ast c_4(\hat X_5) - \frac{1}{2} \pi_\ast (G_4 \cdot G_4)$ with $c_4(\hat X_5)$ evaluated on the resolved space $\hat X_5$.
Second, we must add the anomaly contribution from the non-trivial 7-7 sector.
This sector includes the localised matter in some representation ${\bf R}$ of the total gauge group as well as the bulk matter in the adjoint representation (or its decomposition if the flux background breaks the non-abelian gauge symmetry). Each massless multiplet in the bulk sector contributes ${\rm dim}(\bf adj)$ many states to the anomaly. Of these, ${\rm rk}(G)$ many states are associated with the Cartan subgroup of the gauge group and are in fact encoded already in the contribution from the 'moduli sector'. More precisely, if we replace in (\ref{Amod}) the contribution $\chi_1(X_5)$ by $\chi_1(\hat X_5)$, the resulting expression ${\cal A}_{\rm grav}|_{\rm mod}$ now includes the anomaly from the ${\rm rk}(G) = h^{1,1}({\hat X_5}) - (h^{1,1}(B_4) -1)$
many vector multiplets associated with the Cartan subgroup as well as the 'open string moduli' in the Cartan, which enter the values of $h^{1,p}(\hat X_5)$.
As a result, the total gravitational anomaly polynomial is now
\begin{equation}
I_{4, \rm grav} = \frac{1}{24} p_1(T) \left( {\cal A}_{\rm grav}|_{\rm 7-7} + {\cal A}_{\rm grav}|_{\rm mod} + {\cal A}_{\rm grav}|_{\rm uni} + {\cal A}_{\rm grav}|_{3-7} \right)
\end{equation}
with the individual contributions
\begin{eqnarray}
{\cal A}_{\rm grav}|_{7-7} &=& \sum_{{\bf R}} {\rm dim}({\bf R}) \chi({\bf R}) - {\rm rk}(G) \chi({\bf adj}) \\
{\cal A}_{\rm grav}|_{\rm mod} &=& -\tau(B_4) + \chi_1(\hat X_5) - 2 \chi_1(B_4), \\
{\cal A}_{\rm grav}|_{\rm uni} &=& 24 \\
{\cal A}_{\rm grav}|_{3-7}& =& - 6 c_1(B_4) \cdot \left(\frac{1}{24} \pi_\ast (c_4(\hat X_5)) - \frac{1}{2} \pi_\ast (G_4 \cdot G_4) \right)\,.
\end{eqnarray}
Note that the topological invariants $\chi_1(\hat X_5)$ and $ c_4(\hat X_5)$
contain correction terms in addition to the base classes appearing for the case of a smooth Weierstrass model which depend on the resolution divisors and extra sections (if present).
The vanishing of the total gravitational anomaly implies that these individual contributions must cancel each other,
\begin{eqnarray} \label{gravanomA-general}
{\cal A}_{\rm grav}|_{\rm 7-7} + {\cal A}_{\rm grav}|_{\rm mod} + {\cal A}_{\rm grav}|_{\rm uni} + {\cal A}_{\rm grav}|_{3-7} = 0 \,.
\end{eqnarray}
This leads to a set of topological identities which must hold for every resolution $\hat X_5$ of an elliptically fibered Calabi-Yau 5-fold, and for every consistent configuration of background fluxes thereon, as specified above.
Note that the flux background enters not only through the 3-brane class in ${\cal A}_{3-7}$, but also because the chiral indices in the 7-brane sector split as $\chi({\bf R}) = \chi({\bf R})|_{\rm geom} + \chi({\bf R})|_{\rm flux}$ as in (\ref{chigeom1}).
In principle, if the Hodge type of $G_4$ were to vary over the moduli space, the supersymmetry condition $G_4 \in H^{2,2}(\hat X_5)$ would induce a potential for some of the moduli \cite{Haupt:2008nu} and hence modify the number of uncharged massless fields. According to our assumptions, this does not occur for the choice of flux considered here and the uncharged sector contributes to the anomaly as above.
Then the anomaly equations split into the independent sets of equations
\begin{subequations} \label{gravitationalanomalies1}
\begin{empheq}[box=\widefbox]{align}
&& \sum_{{\bf R}} {\rm dim}({\bf R}) \chi({\bf R}) |_{\rm geom} - {\rm rk}(G) \chi({\bf adj}) |_{\rm geom} -\tau(B_4) + \chi_1(\hat X_5) - 2 \chi_1(B_4) + 24 \nonumber \\
&& - \frac{1}{4} c_1(B_4) \cdot \left( \pi_\ast c_4(\hat X_5)\right) = 0 \label{gravgeom1} \\
&& -6c_1\cdot \pi_\ast(G_4\cdot G_4)= \sum_{\mathbf{R},a} \pi_\ast(G_4\cdot S^a_\mathbf{R})\cdot_{C_{\mathbf{R}}}\pi_\ast(G_4\cdot S^a_\mathbf{R}) \label{gravflux1}
\end{empheq}
\end{subequations}
In the second equation, which accounts for the flux dependent anomaly contribution, we do not need to treat the 7-brane states in the Cartan separately as their chirality is not affected by the flux background.
The flux independent contribution can be analysed further if the fibration $\hat X_5$ is smoothly connected to a smooth Weierstrass model $X_5$. In the terminology of \cite{Morrison:2014lca}, this means that the F-theory model does not contain any non-Higgsable clusters and hence after the blowdown of the resolution divisors the gauge symmetry can be completely Higgsed.
In that case we know already from (\ref{I4gravWeiersmooth}) that the anomalies on the resulting smooth Weierstrass model $X_5$ cancel for $G_4 =0$.
Let us therefore define
\begin{eqnarray}
\label{DeltaC}
\Delta[C] &=& \frac 1{24} \left(\pi_\ast c_4(\hat X_5) - \pi_\ast c_4(X_5) \right) = \frac 1{24} c_4(\hat X_5)|_{B_4}-(15c_1^3+\frac12c_1c_2) \\
\Delta\chi_1 &=& -\frac{1}{24} \left( \pi_\ast c_5(\hat X_5) - \pi_\ast c_5(X_5) \right) = -\frac{1}{24} \pi_\ast c_5(\hat X_5)-(90c_1^4+3c_1^2c_2-\frac12c_1c_3) \,.
\end{eqnarray}
The anomaly equations can then be rewritten as
\begin{subequations} \label{gravitationalanomalies}
\begin{empheq}[box=\widefbox]{align}
-6c_1\cdot \Delta[C] +\Delta\chi_1 =& -\frac1{12}\sum_{\mathbf{R}}\text{dim}(\mathbf{R})\int_{C_{\mathbf{R}}} {\rm ch}_2(C_{\mathbf{R}}) \label{curvgravanomalies}\nonumber \\
& + \frac{1}{12} {\rm rk}(G) \int_{C({\bf adj})} {\rm ch}_2(C({\bf adj})) \\
-6 c_1\cdot \pi_\ast(G_4\cdot G_4)=& \sum_{\mathbf{R},a} \pi_\ast(G_4\cdot S^a_\mathbf{R})\cdot_{C_{\mathbf{R}}}\pi_\ast(G_4\cdot S^a_\mathbf{R}) \label{gravitationalanomalies2}
\end{empheq}
\end{subequations}
It is interesting to speculate about the effect of $G_4$ fluxes which are not automatically of $(2,2)$ Hodge type.
The supersymmetry condition (\ref{SUSYG422}) is reflected in a dynamical potential which is expected to render some of the supergravity moduli massive \cite{Haupt:2008nu}.
The resulting change in the gravitational anomaly compared to the fluxless geometry must be compensated by a suitable modification of the remaining uncharged spectrum.
Indeed, the flux contributes at the same time to the D3-brane tadpole and hence changes the D3-brane curve class $[C]$ compared to the fluxless compactification.
This changes the number of massless Fermi multiplets in the 3-7 sector.
The net number of moduli stabilized in the presence of flux must equal the change in the number of 3-7 modes. This interesting effect has no analogue in 6d or 4d F-theory vacua: In 6d there is no background flux, and in 4d there is no purely gravitational anomaly.
\section{Derivation of the Green-Schwarz terms and 3-7 anomaly} \label{sec_GStermsderivation}
In this section we derive the two key results of this paper, the form and correct overall normalization of the 2d Green-Schwarz terms and the contribution to the gauge anomalies from the 3-7 string sector.
As we will see, both can be obtained in a very compact manner directly from the gauging of the Type IIB Ramond-Ramond 4-form in the presence of source terms.
The gauging of the Ramond-Ramond forms in the presence of brane sources is standard \cite{Green:1996dd,Cheung:1997az,Minasian:1997mm}, and a similar ten-dimensional approach to determining the gauging in a compactification has been taken in \cite{Martucci:2015oaa,Martucci:2015dxa}.
We will first derive this gauging in an orientifold limit and describe its implications for the Green-Schwarz terms and its relation to the 3-7 anomalies. We then uplift the result to F-theory on an elliptically fibered Calabi-Yau, which is valid beyond the perturbative limit. We close this section by making contact with the 2d effective action laid out in section \ref{sec_Anomaliesin2d}.
\subsection{10d Chern-Simons terms}
Consider a Type IIB orientifold compactification on a Calabi-Yau 4-fold $X_4$, with spacetime-filling D7-branes and O7-planes associated with a holomorphic orientifold involution $\sigma: X_4\rightarrow X_4$. To simplify the presentation we omit orientifold invariant D7-branes and only consider D7-branes as pairs D7$_a$, D7$_{a'}$ wrapping effective divisors $D_a$ and $D_{a'} = \sigma_{\ast}(D_a)\neq D_a$ on $X_4$. The cohomology class Poincar\'e dual to $D_a$ will be denoted by $[D_a]$. The field strength on the D$7_a$-brane is denoted as ${\mathbf F}_a$ with ${\mathbf F}_{a'}=-\sigma_{\ast} ({\mathbf F}_a)$. In addition we allow for spacetime-filling D3-branes and their image wrapping
curves $C_i$ and $C_{i'}$ on $X_4$. Our conventions for the effective action of the supergravity fields and the branes are summarized in appendix \ref{app_conventions}.
The 10d supergravity action in the presence of 7-branes and 3-branes and after taking the orientifold quotient takes the form\footnote{The overall factor of $\frac{1}{2}$ results from the orientifold quotient.}
\begin{equation}
\label{action}
S=\frac 12 \left( S_{\rm IIB}+ \sum_{a} (S_a^{\rm D7}+S_{a'}^{\rm D7}) + S^{\rm O7} + \sum_{i} (S_i^{\rm D3}+S_{i'}^{\rm D3}) \right) \,.
\end{equation}
We are interested in the gauging of the RR 4-form potential $C_4$. Prior to taking into account the source terms due to the branes, its associated field strength is\footnote{Strictly speaking, the $SL(2,\mathbb Z)$ invariant field strength in 10d is $\tilde F_5 = d C_4 + \frac{1}{2} B_2 \wedge F_3 - \frac{1}{2} C_2 \wedge H_3$ with $H_3 = d B_2$ and $F_3 = d C_2$, but since we are only interested in the gauging of $C_4$ these corrections play no role for us.} $F_5 = d C_4$. It is gauge invariant and satisfies the Bianchi identity $d F_5 = 0$.
Including the source terms, the relevant part of the action after taking the orientifold quotient becomes
\begin{eqnarray}
S|_{C_4} &=& 2\pi\int -\frac18 \, F_5\wedge \ast F_5+ \nonumber \\
&& 2 \pi \int C_4 \wedge \left( \frac12 \sum_{a} (Q_a({\bf F}_a) + Q_{a'}({\bf F}_{a'}) ) + \frac{1}{2} \, Q({\bf R}) + \frac{1}{2} \sum_i (Q({\rm D3}_i) + Q({\rm D3}_{i'}) ) \right) \label{C4action} \,.
\end{eqnarray}
The source terms linear in $C_4$ follow by summing up the $C_4$ dependent contributions to the Chern-Simons action of the 7-branes, the O7-plane and the D3-branes listed in Appendix \ref{app_conventions} as
\begin{eqnarray}
Q_a({\bf F}_a) &=& -\frac14 {\rm Tr} \, {\mathbf F}_a\wedge {\mathbf F}_a \wedge [D_a] \label{QaFa} \\
Q({\bf R}) &=& - \frac{1}{16} {\rm tr} \, {\mathbf R} \wedge {\mathbf R} \wedge [O7] \\
Q({\rm D3}_i) &=& \frac{1}{2} [C_i] \,.
\end{eqnarray}
Note the appearance of the trace ${\rm Tr}$, defined in (\ref{defTr}), in the expression (\ref{QaFa}). In the strict perturbative limit, in particular for gauge groups of type $SU(n)$, there is no difference compared to the trace in the fundamental representation. But more generally in F-theory, it is the object ${\rm Tr}$, rather than ${\rm tr}$, which appears in the Chern-Simons action.
As a result, the Bianchi identity for the field strength $F_5$ associated with $C_4$ now takes the non-standard form
\begin{equation}
\label{whole}
dF_5= \frac{1}{2} \sum_a \left( {\rm Tr} \, \mathbf F_a\wedge \mathbf F_a \wedge [D_a]+ {\rm Tr} \, \mathbf F_{a'}\wedge \mathbf F_{a'} \wedge [D_{a'}] \right) + {\rm tr} \, {\mathbf R} \wedge {\mathbf R} \wedge \frac{1}{8} [O7] - \sum_i ([C_i] + [C_{i'}]) \,.
\end{equation}
To proceed further, we introduce the Chern-Simons forms $\mathbf{w}_{3a}$ for the gauge group on the 7-brane along $D_a$ as well as $\mathbf{w}_{3Y}$ for the spin connection $\omega$ with the property
\begin{eqnarray}
{\rm Tr} \, \mathbf F_a\wedge \mathbf F_a = d \, \mathbf{w}_{3a}, \qquad {\rm tr} \, {\mathbf R} \wedge {\mathbf R} = d \, \mathbf{w}_{3Y} \,.
\end{eqnarray}
Similarly, one can define an Euler form $e_{5,i}$ associated with the 6-form $[C_i]$ Poincar\'e dual to the curve $C_i$ such that $d e_{5,i} = [C_i]$.\footnote{A careful definition can be found in \cite{Kim:2012wc}. A proper regularization of this term is necessary for a correct treatment of the normal bundle anomalies \cite{Cheung:1997az}, but this will play no role for us in this paper.}
This allows us to express (\ref{whole}) as
\begin{eqnarray} \label{dF5a}
d\left( F_5 - \frac{1}{2} \sum_a \big( \mathbf{w}_{3a} \wedge [D_a] + \mathbf{w}_{3a'} \wedge [D_{a'}]\big) - \frac{1}{8} \mathbf{w}_{3Y} \wedge [O7] + \sum_i ( e_{5,i} + e_{5,i'}) \right) = 0 \,,
\end{eqnarray}
which is solved by setting
\begin{equation} \label{F5def}
F_5= d C_4 + \frac{1}{2} \sum_a \big( \mathbf{w}_{3a} \wedge [D_a]+\mathbf{w}_{3a'} \wedge [D_{a'}] \big)
+ \frac{1}{8} \mathbf{w}_{3Y} \wedge [O7] - \sum_i ( e_{5,i} + e_{5,i'}) \, \,.
\end{equation}
Taking into account the backreation of the source terms means that it is now this form of $F_5$ which appears in the kinetic term in (\ref{whole}). The full action (\ref{C4action}) is equivalent to
\begin{eqnarray}
S|_{C_4} &=& 2\pi\int -\frac18 \, F_5\wedge \ast F_5 + \label{C4actionb} \\
&& 2 \pi \int F_5 \wedge \left( \frac18 \sum_{a} (\mathbf{w}_{3a} \wedge [D_a] + \mathbf{w}_{3a'} \wedge [D_{a'}] ) + \frac{1}{32} \,\mathbf{w}_{3Y} \wedge [O7] - \frac{1}{4} \sum_i (e_{5,i} + (e_{5,i'}) ) \right) \,,\nonumber
\end{eqnarray}
again with $F_5$ as in (\ref{F5def}).\footnote{Note that the cross-terms quadratic in the Chern-Simons terms vanish due their odd form degree.}
The form (\ref{F5def}) for the gauge invariant field strength $F_5$ implies that $C_4$ must transform non-trivially under gauge transformations associated with the 7-brane gauge group and the spin connection.
In absence of any background values for the fields,
if under a gauge and Lorentz transformation the gauge connection $\mathbf{A}_a$ and the spin connection ${\bf \omega}$ change as
\begin{eqnarray}
\mathbf{A}_a \rightarrow d \lambda_a + [\lambda_a,\mathbf{A}_a], \qquad \mathbf{\omega} \rightarrow d \chi + [\chi,{\bf \omega}] \,,
\end{eqnarray}
then the Chern-Simons forms vary as
\begin{eqnarray} \label{deltawomega}
\delta \mathbf{w}_{3a} = d (\lambda_a \, d {\mathbf A}_a), \qquad \quad \delta \mathbf{w}_{3Y} = d (\chi \, d {\bf \omega}) \,.
\end{eqnarray}
Since the field strength $F_5$ defined in (\ref{F5def}) is gauge invariant, this induces a corresponding gauge transformation of the potential $C_4$.
We are interested in situations in which both the gauge and the spin connection acquire non-trivial background values.
Correspondingly we can
decompose the field strength $\mathbf{F}$ into its fluctuation piece $F$ and a background component $\bar F$, and similarly for ${\bf R}$,
\begin{eqnarray} \label{Fbarfdecomp}
\mathbf{F}=F+ \bar F, \qquad \quad \mathbf{R}=R+ \bar R \,.
\end{eqnarray}
The gauge dependence of $C_4$ then becomes\footnote{Strictly speaking, we are not taking into account variations of the spin connection in the direction of the normal bundle, which are more subtle \cite{Cheung:1997az,Kim:2012wc} but play no role for us. Note also that, as we will argue momentarily, only abelian fluxes are of relevance for us so that we are writing $\bar F_a$ instead of $d \bar A_a$.}
\begin{eqnarray}\label{C4variation}
\delta_{\rm gauge} \, C_4 &=& - \sum_{a} {\rm Tr} \, \lambda_{a} \left( (\bar F_a \wedge [D_a]- \bar F_{a'} \wedge [D_{a'}]) +\frac{1}{2}( d A_a \wedge [D_a]- d A_{a'} \wedge [D_{a'}]) \right) \\
\delta_{\rm spin} \, C_4 &=& - {\rm tr} \, \chi {d \bar\omega} \wedge \frac{1}{4} [O7] - {\rm tr} \, d \omega \wedge \frac{1}{8} [O7] \label{Rterms} \,.
\end{eqnarray}
Here we have used $\lambda_{a}=-\lambda_{a'}$, relating the gauge group on each brane along $D_a$ and its orientifold image.
The relative factor of $2$ in the first terms involving the background field strength and curvature results from expanding ${\mathbf F}_a^2 = 2 F_a \bar F_a + F_a^2 + \bar F_a^2$, and similarly for ${\bf R}$.
As we will see next, the terms on the righthand side involving the internal background flux $\bar F_a$ induce the Green-Schwarz counterterms in the two-dimensional effective action, while the terms depending on the fluctuations $F_a$ and $R$ contribute to the anomaly inflow counterterms for the anomaly from the 3-7 string modes.
\subsection{Derivation of the GS term in Type IIB}
In order to derive the Green-Schwarz counterterms, we first consider the flux-dependent piece in the gauge variation of $C_4$, (\ref{C4variation}),
\begin{equation} \label{deltaC4flux}
\delta_{\rm gauge} \, C_4 |_{{\rm flux}}=- \sum_{a} {\rm Tr} \, \lambda_{a} \left( \bar F_a \wedge [D_{a}] - \bar F_{a'} \wedge [D_{a'}] \right) \,.
\end{equation}
Due to the appearance of $C_4$ in the action (\ref{C4action}), while $F_5$ by itself is gauge invariant, this induces a gauge dependence of the effective action, which is precisely the manifestation of a Green-Schwarz counterterm. As we will see, the only relevant terms contributing to the Green-Schwarz terms are the couplings to $Q_a({\bf F}_a)$ and $Q_{a'}({\bf F}_{a'})$.
If we focus on these, substituting the variation (\ref{deltaC4flux}) of $C_4$ into \eqref{action} gives
\begin{equation}
\begin{aligned}
\delta S_{\rm GS}=&\frac12 \left. \left(\sum_{b}\delta_{\rm gauge} \, S_b^{\rm D7}+\sum_{b'}\delta_{\rm gauge} \, S_{b'}^{\rm D7}\right) \right|_{\rm flux} \cr
=&\frac{2\pi}8 \int_{\mathbb{R}^{1,1}\times X_4}\sum_{a,b} {\rm Tr} \lambda_a \left( \bar F_a\wedge [D_a] - \bar F_{a'} \wedge [D_{a'}] \right) \wedge \big( {\rm tr} (\mathbf F_b\wedge \mathbf F_b)\wedge [D_b]+ {\rm tr}(\mathbf F_{b'} \wedge \mathbf F_{b'} )\wedge [D_{b'} ] \big) \,, \cr
\end{aligned}
\end{equation}
where we are indicating that after compactification the spacetime is of the form $\mathbb R^{1,1} \times X_4$. If we identify the fluctuations $F$ with the 2d field strength $F^{\rm 2d}$, we see that
for reasons of dimensionality only the last term in the decomposition
\begin{equation}
{\rm Tr}(\mathbf F_b\wedge \mathbf F_b)={\rm Tr}(F^{\rm 2d}_b\wedge F^{\rm 2d}_b)+{\rm Tr}(\bar F_b\wedge \bar F_b)+2 \, {\rm Tr}(F^{\rm 2d}_b\wedge \bar F_b)
\end{equation}
makes a contribution. We thus find
\begin{equation}
\delta S _{\rm GS}= \frac{2\pi}4\sum_{ab} {\rm Tr}_a {\rm Tr}_b \, \lambda_a F_b^{\rm 2d} \int_{X_4} \left(( \bar F_a \wedge[D_a]+\sigma^\ast(\bar F_a\wedge [D_a]) )\wedge ( \bar F_b \wedge[D_b]+\sigma^\ast( \bar F_b\wedge [D_b])) \right) \,.
\end{equation}
Here we have used the definition $\sigma^\ast({\rm Tr}\bar F_a\wedge [D_a])={\rm Tr} \bar F_{a'} \wedge [D_{a'}]$. Furthermore we have denoted the trace over the gauge group on brane $D_a$ with ${\rm Tr}_a$, and similarly for $D_b$.
Through the descent equations, this gauge variance yields the Green-Schwarz contribution to the anomaly polynomial
\begin{equation}
\label{GSterm}
I_4^{\rm GS}=\frac14\sum_{a, b} {\rm Tr}_a {\rm Tr}_b F^{\rm 2d}_a\wedge F_b^{\rm 2d}\int_{X_4} \left( ( \bar F_a\wedge [D_a]+\sigma^\ast(\bar F_a \wedge[D_a])) \wedge (\bar F_b\wedge [D_b]+\sigma^\ast(\bar F_b\wedge [D_b])) \right) \,.
\end{equation}
Note that the trace is taken simultaneously over the external and the internal components of the field strength, both for the gauge groups associated with $D_a$ and with $D_b$.
This implies that $I_4^{\rm GS}$ can only be non-vanishing for the abelian gauge symmetry factors in the two-dimensional effective action: Indeed, a contribution to a non-abelian gauge group would require at the same time non-abelian flux internally, but this would break the gauge group. The only option is that the flux is embedded along the direction of an abelian generator, which then acquires a Green-Schwarz anomaly term of the above form.
This is a notable difference from the Green-Schwarz mechanism in six dimensions, which is well-known to operate also at the level of non-abelian gauge groups.
For a similar reason, the other source terms in (\ref{C4action}) do not contribute to the gauge variance of the classical action. Also, there can be no Green-Schwarz contribution to the pure gravitational anomaly or even a mixed gauge-gravitational anomaly in two dimensions.
This can be seen explicitly if one proceeds along the same lines with the background terms in (\ref{Rterms}) and uses the direct product structure of the Lorentz group as $SO(1,1) \times G_{\rm int}$ upon compactification.
In summary, the complete effect of the gauge dependence associated with the background term in (\ref{C4variation}) is the Green-Schwarz anomaly polynomial (\ref{GSterm}), while the background term in (\ref{Rterms}) does not lead to any gauge dependence of the effective action.
The Green-Schwarz counterterm (\ref{GSterm}) and in particular its overall normalization will be checked in a prototypical brane setup in Appendix \ref{app_AnomaliesIIB}, where we will verify that it correctly cancels the 1-loop anomalies induced by the 3-7 and the 7-7 sector.
\subsection{ 3-7 anomaly from gauging in Type IIB}
Let us now analyze the effect of the
dependent piece of the gauging (\ref{C4variation}),
\begin{equation} \label{deltaCfluct}
\delta C_4|_{\rm fluct.}=- \frac{1}{2} {\rm Tr} \sum_{a} \lambda_a( d A^{\rm 2d}_a \wedge [D_{a}]- d A^{\rm 2d}_{a'} \wedge [D_{a'}] ) - {\rm tr} \, \chi \, { d \omega}^{\rm 2d} \wedge \frac{1}{8} [O7] \,.
\end{equation}
If we plug this expression into (\ref{C4action}) we receive a contribution only from the internal components of the source terms.
Summing over all source terms associated with the D3-branes, the D7-branes and the O7-plane gives a vanishing total result because
the total $C_4$ charge along the internal space $X_4$ vanishes as a result of D3-brane tadpole cancellation.
Nonetheless, each individual term by itself contains valuable information, namely (part of) the counterterms for the 1-loop gauge anomaly on the worldvolume of the respective branes.
By construction of the Chern-Simons brane actions, these counterterms \emph{locally} cancel the 1-loop anomaly associated with chiral modes on the worldvolume of the branes via the anomaly inflow mechanism \cite{Green:1996dd,Cheung:1997az,Minasian:1997mm}.
Tadpole cancellation then implies that the sum of all counterterms vanishes globally, which equivalent to the statement of anomaly cancellation.
To extract the full anomaly inflow counterterm cancelling the 7-brane gauge anomalies from the 3-7 sector as well as the tangent bundle anomalies along the D3-brane, we follow the standard procedure \cite{Green:1996dd,Cheung:1997az,Minasian:1997mm} and rewrite the non-kinetic terms in the action (\ref{C4action})
as
\begin{eqnarray}
S|_{C_4} &\supset & S_1 + S_2 \\
S_1 & = & \frac{2 \pi}{4} \int C_4 \wedge \sum_i ([C_i] + [C_{i'}] ) \\
S_2 &=& 2 \pi \int F_5 \wedge \left( \frac18 \sum_{a} (\mathbf{w}_{3a} \wedge [D_a] + \mathbf{w}_{3a'} \wedge [D_{a'}] ) + \frac{1}{32} \,\mathbf{w}_{3Y} \wedge [O7] \right) \,.
\end{eqnarray}
The anomaly inflow counterterms
now have two contributions.
The first contribution comes from plugging the gauge variation (\ref{deltaCfluct}) into $S_1$,
\begin{eqnarray} \label{deltaS1inflow}
\left.\delta \, S_1 \right|_{\rm inflow}
&=&- \frac{2\pi}8 \int_{\mathbb R^{1,1} } \sum_{a,i} {\rm Tr} \lambda_a \, d A_a^{\rm 2d} \int_{X_4}\big([D_a]+[D_{a'}])([C_i]+[C_{i'}]\big) \\
& & - \frac{2\pi}{32} \int_{{\mathbb R}^{1,1}} {\rm tr} \, \chi \, d \omega^{\rm 2d} \int_{X_4} [O7] \wedge \sum_{i} ([C_i] + [C_{i'}]) \label{deltaS1inflow-line2} \,,
\end{eqnarray}
where in the first line we have used that $A_{a'}^{\rm 2d}=-A_{a}^{\rm 2d}$.
In addition, the Chern-Simons forms appearing in $S_2$ vary according to (\ref{deltawomega}).\footnote{We are here only taking into account the contribution to (\ref{deltawomega}) from the fluctuations of the fields; the contributions involving the background fields enter the Green-Schwarz terms and have hence already been taken into account in the previous section.}
After integration by parts we find a non-zero contribution because of the Bianchi identity (\ref{whole}). The relevant terms describing the anomaly inflow are obtained by plugging in only the last terms in (\ref{whole}), i.e. using $d F_5 = - \sum_i ([C_i] + [C_{i'}]) + \ldots$. This gives a contribution of exactly the same form as (\ref{deltaS1inflow}) and hence altogether
\begin{eqnarray} \label{deltaS2inflow}
\left.\delta \, S \right|_{\rm inflow} = \left.\delta S_1 \right|_{\rm inflow} + \left.\delta S_2 \right|_{\rm inflow} = 2 \left.\delta S_1 \right|_{\rm inflow} \,.
\end{eqnarray}
The terms (\ref{deltaS1inflow}) cancel the contribution to the 7-brane gauge group anomaly from the sector of 3-7 strings.
By descent, the associated 1-loop anomaly polynomial is therefore
\begin{equation}\label{anomaly37}
I_{4, \rm gauge}^{3-7} =\frac{1}4 \sum_{a, i} {\rm Tr} F_a^{\rm 2d} \wedge F_a^{\rm 2d} \int_{X_4} \left([D_a]+[D_{a'}])([C_i]+[C_{i'}] \right) \,.
\end{equation}
Note that we have included a minus sign in $I_4^{3-7}$ because (\ref{deltaS2inflow}) represents the inflow counterterms to the actual 1-loop anomaly.
As the trace structure clearly shows, this contribution is non-vanishing also for simple gauge groups, in contrast to the Green-Schwarz terms derived earlier.
From the perspective of the effective 2d $(0,2)$ theory, the gauging (\ref{deltaCfluct}) translates into a gauging of the non-dynamical 2-forms obtained by dimensional reduction of $C_4$ in terms of internal 2-forms on $X_4$. This offers an interesting perspective on the contribution (\ref{anomaly37}) to the total anomaly polynomial: Rather than interpreting it as due to chiral localised defect modes we can view it as the effect of gauging these non-dynamical top-forms in the effective supergravity theory.
This makes the formal similarity between the Green-Schwarz terms, associated with the gauging of the scalars from $C_4$, and the 3-7 anomaly on the righthand side of (\ref{gauge-flux1}) more natural.
The remaining terms (\ref{deltaS1inflow-line2}) cancel the contribution to the gravitational anomaly from all modes on the D3-brane worldvolume. This includes the 3-7 modes as well as the 3-brane bulk modes analyzed in detail in \cite{Lawrie:2016axq}. The associated anomaly polynomial is
\begin{eqnarray}
I_{4, \rm grav}^{\rm D3} = {\rm tr} \, R^{\rm 2d} \wedge R^{\rm 2d} \, \int_{X_4}\frac{1}{16} [O7] \wedge \sum_{i} ([C_i] + [C_{i'}]) \,.
\end{eqnarray}
\subsection{F-theory lift} \label{sec_Ftheorylift}
It remains to uplift the perturbative results for the Green-Schwarz terms and the 3-7 anomaly to a description in fully-fledged F-theory, defined via duality to M-theory on an elliptic fibration $\hat X_5$. If a weakly coupled limit exists, the perturbative Type IIB Calabi-Yau $X_4$ is the double cover of the F-theory base $B_4$, with projection
\begin{eqnarray}
\pi_+: X_4 \rightarrow B_4 \,.
\end{eqnarray}
The cohomology classes even under the holomorphic involution $\sigma$ on $X_4$ uplift to cohomology classes of the same bidegree on $B_4$.
In particular, consider a divisor class $[D] \in H^{1,1}(X_4)$ and its image $\sigma^*[D]$ under the involution and define
\begin{eqnarray}
[D_+] := [D] + \sigma_\ast[D] =: \pi^\ast_+ [D^{\rm b}]
\end{eqnarray}
with $[D^{\rm b}] \in H^{1,1}(B_4)$. Then taking into account that $X_4$ is a double cover of $B_4$ the intersection numbers on both spaces are related as \cite{Krause:2012yh}
\begin{eqnarray} \label{IIBtoF}
[D_{a+}] \cdot_{X_4} [D_{b+}] \cdot_{X_4} [D_{c+}] \cdot_{X_4} [D_{d+}] = 2 \, D^{\rm b}_a \cdot_{B_4} D^{\rm b}_b \cdot_{B_4} D^{\rm b}_c \cdot_{B_4} D^{\rm b}_d \,.
\end{eqnarray}
With this in mind consider first the perturbative expression (\ref{anomaly37}) for the 3-7 anomaly, with the aim of uplifting the sum over all brane stacks and their image to F-theory.
A divisor on $X_4$ wrapped by a non-abelian stack of 7-branes on $X_4$ uplifts, together with its image under the involution, to a corresponding divisor on $B_4$ according to the above rule, and this divisor on $B$ is a component of the discriminant locus carrying the corresponding non-abelian gauge group.
More subtle are the non-Cartan abelian gauge groups. In Type IIB language, $U(1)$ gauge symmetries which are massless in the absence of background flux are supported on linear combinations of divisors which are in the same class as their orientifold image.
Hence each abelian gauge group factor $U(1)_A$ is associated with a linear combination of (typically several) divisor classes $[D_a] + \sigma^\ast [D_a]$ on $X_4$.
Let us assume first that a brane configuration gives rise to no massless (in absence of fluxes) abelian gauge symmetries, i.e. the gauge group is only a product of non-abelian factors $G_I$, Then the uplift of $\sum_{a} {\rm Tr} F_a^{\rm 2d} \wedge F_a^{\rm 2d} ([D_a]+[D_{a'}])$ to F-theory is
\begin{eqnarray}
\sum_{i_I, j_J} F^{\rm 2d}_{i_I} \wedge F^{\rm 2d}_{j_J} \, {\rm Tr}\, {\cal T}_{i_I} {\cal T}_{j_J} D^{\rm b}_{I} = \sum_{i_I, j_J} F^{\rm 2d}_{i_I} \wedge F^{\rm 2d}_{j_J} \, \left(-\pi_\ast(E_{i_I} \cdot E_{j_J}) \right) \,.
\end{eqnarray}
Here we used (\ref{piEiEj}) to express the correctly normalised trace to $\pi_\ast(E_{i_I} \cdot E_{j_J})$.
In the presence of non-Cartan abelian symmetries, we must include these in the sum.
In F-theory language, a non-Cartan gauge group factor $U(1)_A$ is generated by a 2-form $U_A$, defined via the Shioda-map as in (\ref{ShiodaUA}), but typically there is no separate component of the divisor $\Delta$ which we would associate with $U(1)_A$. This is because, form a 7-brane perspective, massless (in absence of gauge flux) $U(1)$s involve combinations of several divisor classes. However, the height-pairing (\ref{heightpairing}) defines a completely analogous object on the base $B_4$ including the information about the trace appearing in (\ref{anomaly37}).
Hence, the correct uplift of the expression for the 3-7 anomaly is
\begin{eqnarray}
\sum_{a} {\rm Tr} \, F^{\rm 2d}_a \wedge F^{\rm 2d}_a \, ([D_a]+ [D_{a'}]) & \longrightarrow & \sum_{\Lambda, \Sigma} F^{\rm 2d}_\Lambda \wedge F^{\rm 2d}_\Sigma \, \left(-\pi_\ast(\mathfrak{F}_\Lambda \cdot \mathfrak{F}_\Sigma) \right) \nonumber \\
\sum_{i} ([C_i] + [C_{i'}]) & \longrightarrow & [C] \\
\frac{1}{4} \sum_{a, i} {\rm Tr} F_a^{\rm 2d} \wedge F_a^{\rm 2d} \int_{X_4} \left([D_a]+[D_{a'}])([C_i]+[C_{i'}] \right) & \longrightarrow & \frac{1}{2} \, \sum_{\Lambda, \Sigma} F^{\rm 2d}_\Lambda \wedge F^{\rm 2d}_\Sigma \, \left(-\pi_\ast(\mathfrak{F}_\Lambda \cdot \mathfrak{F}_\Sigma) \right) \cdot_{B_4} [C] \nonumber
\end{eqnarray}
Here $[C]$ is the total class of the D3-brane on $B_4$ and we summing over all generators ${\mathfrak F}_\Sigma$, Cartan and non-Cartan. The last line in addition uses (\ref{IIBtoF}).
Hence
\begin{eqnarray}
I_{\rm 4, \rm gauge}^{3-7} = \sum_{\Lambda, \Sigma} F^{\rm 2d}_{\Lambda} F^{\rm 2d}_{\Sigma} \left( \frac{1}{2} \, \, \left(-\pi_\ast(\mathfrak{F}_\Lambda \cdot \mathfrak{F}_\Sigma) \right) \, \cdot_{B_4} [C] \right)
\end{eqnarray}
in precise agreement with our claim (\ref{37ABanomaly-b}) for the 3-7 gauge anomaly.
Note that in (\ref{anomaly37}), there appear no mixed anomaly contributions because in the perturbative limit the 3-7 strings can only be charged under the diagonal $U(1)_a$ gauge group of at most one D7-brane stack. On the other hand, if we sum over all massless (in absence of flux) $U(1)_A$ group factors (which are linear combinations of the $U(1)_a$ if a perturbative limit exists), mixed anomaly terms in general do result.
To uplift the 3-7 contribution to the gravitational anomaly polynomial, we recall from \cite{Krause:2012yh} the general rule that $\pi_+^\ast (c_{1}(B_4))= [O7]$, and therefore
\begin{eqnarray}
\int_{X_4} [O7] \wedge \sum_i ([C_i] + [C_{i'}]) & \longrightarrow& 2 c_1(B_4) \cdot_{B_4} [C] \,.
\end{eqnarray}
The resulting expression
\begin{eqnarray}
I_{4, \rm grav}^{\rm D3} = \frac{1}{2} {\rm tr} \, R^{\rm 2d} \wedge R^{\rm 2d} \left( \frac{1}{4} c_1(B_4) \cdot_{B_4} [C] \right)\,.
\end{eqnarray}
had already been derived in \cite{Lawrie:2016axq}.
It remains to uplift the Green-Schwarz anomaly polynomial (\ref{GSterm}) to F-theory.
Consider an internal flux background associated with a line bundle whose structure group is identified with either a Cartan or a non-Cartan $U(1)$ subgroup. Such fluxes uplift in F-theory to expressions of the form $G_4 = \bar F \wedge {\mathfrak F}_{\Lambda}$ for the corresponding divisor generator that $U(1)$ symmetry.
Employing once more (\ref{piEiEj}) and (\ref{heightpairing}), an expression of the form
${\rm Tr}_a \, F^{\rm 2d}_a ( \bar F_a \wedge[D_a]+\sigma^\ast(\bar F_a\wedge [D_a]) )$ uplifts to
\begin{eqnarray}
F^{\rm 2d}_\Sigma ( - \pi_\ast(G_4 \cdot {\mathfrak F}_\Sigma) ) \,.
\end{eqnarray}
This remains correct even if the flux, on the Type IIB side, is associated with a $U(1)$ that is geometrically massive even before switching on the flux. Such fluxes lift to more general elements of $G_4$ \cite{Krause:2012yh,MayorgaPena:2017eda}.
Taking again into account the factor of $2$ from (\ref{IIBtoF}) we therefore arrive at
\begin{eqnarray} \label{I4GS-F-theory}
I_4^{\rm GS} = \sum_{\Lambda, \Sigma} F^{\rm 2d}_{\Lambda} F^{\rm 2d}_{\Sigma} \left( \frac{1}{2} ( \pi_\ast(G_4 \cdot {\mathfrak F}_\Lambda) ) \cdot_{B_4} ( \pi_\ast(G_4 \cdot {\mathfrak F}_\Sigma) ) \right)
\end{eqnarray}
in agreement with our previous claim (\ref{GSclaim}).
\subsection{Relation to 2d effective action}
For completeness, we can express our findings in the language of the 2d effective action and make contact with the formalism introduced in section \ref{sec_Anomaliesin2d}.
Indeed, straightforward dimensional reduction of the action (\ref{C4action}) allows us to read off the kinetic metric $g_{\alpha \beta}$, the coupling matrix $\Omega_{\alpha \beta}$ and the gauging parameters $\Theta^\alpha_A$. This can be achieved by first performing the dimensional reduction in the language of Type IIB orientifolds and then uplifting the results according to the general rules described in section \ref{sec_Ftheorylift}.
We directly give the result in the language of F-theory: If we fix a basis $\omega_\alpha$ of $H^4(B_4,\mathbb R)$, the real scalar fields are obtained as
\begin{eqnarray}
C_4 = c^\alpha \, \omega_\alpha \,.
\end{eqnarray}
Matching the 10d and 2d kinetic terms in (\ref{C4action}) and (\ref{2daction1}), respectively, as well as the 10d self-duality condition $F_5 = \ast F_5$ with its 2d analogue (\ref{selfduality2d}) then fixes
\begin{eqnarray}
g_{\alpha \beta} &=& 2 \pi \int_{B_4} \omega_\alpha \wedge \ast \omega_\beta \\
\Omega_{\alpha \beta} &=& 2 \pi \int_{B_4} \omega_\alpha \wedge \omega_\beta =: 2 \pi \, \tilde \Omega_{\alpha \beta} \,.
\end{eqnarray}
Dimensional reduction of the interaction terms in (\ref{C4action}) finally identifies the gauging parameters
\begin{eqnarray}
\label{gaugingterm1}
\Theta^\alpha_\Gamma = \tilde \Omega^{\alpha \beta} \int_{B_4} \pi_\ast (G_4 \cdot \mathfrak{F}_\Gamma) \wedge \omega_{\beta}
\end{eqnarray}
in terms of the inverse matrix $\tilde \Omega$ satisfying $\tilde \Omega^{\alpha \beta} \tilde \Omega_{\beta \gamma} = \delta^\alpha_{\, \, \gamma}$.
As a check, plugging this expression into (\ref{factor}) correctly reproduces our result (\ref{I4GS-F-theory}) for the Green-Schwarz anomaly polynomial.
\section{Example: $SU(5)\times U(1)$ gauge symmetry in F-theory} \label{sec_ExampleSU5U1}
In this section we exemplify our general expressions for the anomaly relations in an F-theory compactification on a Calabi-Yau 5-fold with gauge group
$SU(5)\times U(1)$. The four-dimensional version of this model and its flux backgrounds has been studied in great detail in the literature \cite{Krause:2011xj,Krause:2012yh,Bies:2017fam,Bies:2017abs}, and its extension to Calabi-Yau five-folds has been discussed in \cite{Schafer-Nameki:2016cfr}.
The geometry is sufficiently intricate to exemplify all interesting aspects of abelian, non-abelian and gravitational anomaly cancellation, while at the same time it avoids extra complications which arise when the codimension-two matter loci on the base $B_4$ are singular.
\subsection{Geometric background and 3-7 states} \label{sec_geomback}
We will briefly recall the properties of this model relevant for our discussion, referring for more details to \cite{Schafer-Nameki:2016cfr} as well as to \cite{Krause:2011xj,Bies:2017abs}, whose notation we adopt.
We are considering the resolution of a Weierstrass model in Tate form defined by
\begin{equation}\label{Tate}
y^2 s e_3 e_4 + a_1 x y z s+ a_{3,2} y z^3 e_0^2 e_1 e_4= x^3 s^2 e_1 e_2^2 e_3+ a_{2,1} x^2 z^2 s e_0 e_1 e_2 + a_{4,3} x z^4 e_0^3 e_1^2 e_2 e_4 \,,
\end{equation}
where $[x: y : z]$ denote homogenous coordinates of the fibre ambient space $\mathbb P_{231}$ prior to resolution and $E_i: e_i=0$, $i=1,\ldots, 4$ represent the resolution divisors for the singularities associated with gauge group $SU(5)$. In addition to the zero section $S_0: z=0$, the fibration admits another independent rational section
$S_A: s=0$. The resolved $SU(5)$ singularity sits in the fibre over the divisor $W: w=0$ on $B_4$, with $\pi^{-1}W: e_0 e_1 e_2 e_3 e_4 = 0$. With the help of Sage, we find the projection of $c_5(\hat X_5)$ and $c_4(\hat X_5) $ of the resolved fibration ${\hat X_5}$ on the base $B_4$ and evaluate
\begin{eqnarray}
\label{Chernclass}
\pi_\ast (c_5(\hat X_5))&=&-576c_1^4+1464c_1^3 W-48 c_1^2 c_2-1410 c_1^2 W^2+46 c_1 c_2 W+12 c_1c_3+608c_1 W^3 \nonumber \\
&& -18 c_2 W^2-102 W^4 \label{c5hatX5} \\
\pi_\ast (c_4(\hat X_5))&=&144 c_1^3-264 c_1^2 W+12 c_1 c_2+162 {c_1} W^2-30 W^3 \,. \label{c4hatX5}
\end{eqnarray}
Here and in the sequel, the Chern classes $c_i$ without any specification denote $c_i(B_4)$ and all of the intersection numbers between the divisors are evaluated on $B_4$.
Finally, $a_{i,j}$ define the following divisor classes on the base $B_4$ with $c_1(B_4) =: c_1$,
\begin{equation}
\label{classtatecoefficient}
[a_1]=c_1\,,\quad
[a_{2,1}]= 2 c_1-W \,,\quad
[a_{3,2}]=3 c_1-2 W\,,\quad
[a_{4,3}]= 4 c_1-3 W\,.\quad
\end{equation}
The discriminant of the blowdown of this model (setting $e_i = 1$ for $i=1, \ldots, 4$) is
\begin{equation}
\Delta= w^5 \left( a_1^4 a_{3,2} \left(a_{2,1} a_{3,2}-a_1 a_{4,3}\right) w^5 + O(w) \right)
\end{equation}
and indicates that there are four codimension-two matter loci on $B_4$ with classes
\begin{equation}
\label{matterloci}
\begin{aligned}
C_{{\bf 10}_{1}}:\qquad &W \cdot [a_1]= W \cdot c_1\cr
C_{{\bf 5}_{3}}:\qquad& W \cdot [a_{3,2}]= W \cdot (3 c_1-2 W)\cr
C_{{\bf 5}_{-2}}:\qquad & W \cdot[a_1 a_{4,3} - a_{2,1} a_{3,2}] = W \cdot (5 c_1-3 W) \cr
C_{{\bf 1}_{5}}: \qquad & [a_3]\cdot[a_{4,3}] = (3 c_1 -2 W) \cdot (4 c_1 -3 W) \,.
\end{aligned}
\end{equation}
The subscripts denote the charges under the non-Cartan $U(1)_A$ associated with the divisor \cite{Krause:2011xj}\footnote{We are using the conventions of \cite{Bies:2017fam,Bies:2017abs}, where in particular the fibre structure and the resulting charge assignments are detailed.}
\begin{equation} \label{noncartan}
U_A =- \left( 5 (S_A - S_0 - c_1) + 2 E_1 + 4 E_2 + 6 E_3 + 3 E_4 \right) \,.
\end{equation}
Note that in this example all of the codimension-two loci are smooth, while in principle they could exhibit singularities. In this case the chirality formula \eqref{chiS2} would receive corrections \cite{Schafer-Nameki:2016cfr}.
The height pairing associated with $U_A$ is
\begin{eqnarray} \label{DA}
D_A = - \pi_* (U_A \cdot U_A) = - 30 W + 50 c_1 \,.
\end{eqnarray}
The D3-brane tadpole requires the inclusion of D3-branes wrapping a curve of total class $C$ constrained as in (\ref{Cclass}).
In the present example, each intersection point between $C$ and the $SU(5)$ divisor $W$ carries one Fermi multiplet in the fundamental representation ${\bf 5}_{q_1}$ of $SU(5)$ \cite{Schafer-Nameki:2016cfr} with $U(1)_A$ charge $q_1$.
The intersections with the remainder of the discriminant carry additional Fermi multiplets, whose determination is very subtle due to $SL(2, \mathbb Z)$ monodromies along $C$. In general some of these will have a non-zero $U(1)_A$ charge, while some may be completely uncharged under $SU(5) \times U(1)_A$.
Our knowledge of the net contribution (\ref{37ABanomaly}) of the 3-7 sector to the abelian anomaly together with its contribution to the gravitational anomaly \cite{Lawrie:2016rqe} allow us to constrain this matter as follows:
Let us adopt from the discussion around (\ref{D37notation}) the notation $D_{37}(\bf R)$ for the divisor class on $B_4$ such that the effective chiral index of 3-7 states in representation ${\bf R}$ is given by $\chi({\bf R}) = - [C] \cdot D_{37}({\bf R})$. Then $D_{37}({\bf 5}_{q_1}) = W$, and the remaining divisor classes are constrained by the abelian and gravitational anomaly as
\begin{eqnarray}
\label{3-7locis1}
5 \, q_1^2 \, D_{37}({\bf 5}_{q_1}) + \sum_i q_i^2 \, D_{37}({\bf 1}_{q_i}) &=& D_A = {-30 W + 50 c_1 } \\
5 \, D_{37}({\bf 5}_{q_1}) + \sum_i D_{37}({\bf 1}_{q_i}) &=& 8 c_1 \,.
\end{eqnarray}
These equations are consistent with the assertion that, in addition to the states ${\bf 5}_{q_1}$, there is only one further type of 3-7 Fermi multiplets in representation ${\bf 1}_{q_2}$ with charge assignments
\begin{eqnarray} \label{chargesU1A}
|q_1| = \frac{1}{2}, \qquad \quad |q_2| = \frac{5}{2}
\end{eqnarray}
such that
\begin{eqnarray} \label{D37152}
D_{37}({\bf 1}_{q_2}) = - 5 W + 8 c_1.
\end{eqnarray}
These values are in complete agreement with the perturbative limit of the compactification:
To see this, recall from \cite{Krause:2012yh} that the Type IIB limit consists of a brane stack (plus image) with gauge group $U(5)_a$ and another brane-image brane pair carrying gauge group $U(1)_b$. The geometrically massless $U(1)$ symmetry is given by the linear combination $U(1)_A = \frac{1}{2} (U(1)_a - 5 U(1)_b)$, where $U(1)_a$ is the diagonal $U(1)$ of $U(5)_a$ (cf. equ. 4.3 of \cite{Krause:2012yh}) and the normalization conforms with the definition (\ref{noncartan}) of the $U(1)_A$ generator. The 3-7 modes at the intersection of $C$ with the $U(5)_a$ stack hence carry charge $|q_1| = \frac{1}{2} |(1 + 0)|$ and transform as ${\bf 5}$ of $SU(5)_a$ and those at the intersection of $C$ with $U(1)_b$ are $SU(5)_a$ singlets with charge $|q_2| = \frac{1}{2}| (0 - 5) |$. The class (\ref{D37152}) furthermore coincides with the class of the $U(1)_b$ brane as dictated by the 7-brane tadpole cancellation condition.
We stress that more generally the pattern of singlets in the 3-7 sector can be more intricate. What is uniquely determined, however, is the net contribution of the 3-7 states both to the gauge and the gravitational anomalies.
Now we are in the position to check our proposal \eqref{AnomaliesF-theory} within this example. As we have discussed before, we expect that the curvature and the flux induced anomalies should each cancel among themselves. Therefore, in the following we split our proof into three parts: We begin with the flux independent contribution to the anomalies and verify their precise cancellation as a result of rather sophisticated relations between the topological invariants of the resolved 5-fold.
Next we consider the two different types of $G_4$ flux spanning the space of fluxes within $H^{2,2}_{\rm vert}(\hat X_5)$ with the purpose of verifying in particular our proposal for the Green-Schwarz term \eqref{gaugeanomalyFtheory}, and it will be shown that the anomalies induced by the two $G_4$ fluxes are cancelled very neatly.
\subsection{Curvature dependent anomaly relations}
In this section, we verify that the conditions \eqref{AnomaliesF-theory} for anomaly cancellation are satisfied in the absence of background flux, i.e. $G_4 = 0$. This amounts to evaluating (\ref{gauge-geom1}) for the gauge and (\ref{curvgravanomalies}) for the gravitational anomalies.
The various 7-brane codimension-two matter loci $C_{\bf R}$ have been listed in \eqref{matterloci}, and in the present example they are all smooth \cite{Schafer-Nameki:2016cfr} such that the index theorem can be applied as in (\ref{chigeom1}).
With the help of the adjunction formula we find the following flux independent part of the chiral indices for the matter surfaces,
\begin{equation}
\begin{aligned}
\chi({\bf 10}_{1})|_{\rm geom} &= \frac{1}{24} c_1 W \left(2 c_2+W^2\right)\cr
\chi({\bf 5}_{3})|_{\rm geom} &= \frac{1}{24} W \left(3 c_1-2 W\right) \left(-12 c_1 W+8 c_1^2+2 c_2+5
W^2\right)\cr
\chi({\bf 5}_{-2})|_{\rm geom} &= \frac{1}{12} W \left(5 c_1-3 W\right) \left(-15 c_1 W+12 c_1^2+c_2+5 W^2\right) \cr
\chi({\bf 1}_{5})|_{\rm geom} & = \frac{1}{24} \left(4 c_1 - 3 W\right) \left(3 c_1 - 2 W\right) \left(24 c_1^2 + 2 c_2 - 36 c_1 W + 13 W^2\right) .
\end{aligned}
\end{equation}
The first equation in \eqref{AnomaliesF-theory}, i.e. the purely non-abelian $SU(5)$ gauge anomaly, has been verified in \cite{Schafer-Nameki:2016cfr}. For this analysis to be self-contained, let us briefly recap the computation as a warmup. With the appropriate anomaly coefficients (\ref{cr2}), $c^{(2)}_{\bf 10} = {3}$, $c^{(2)}_{\bf 5}= 1$, the matter from the 7-brane codimension-two loci contributes to the non-abelian anomaly (\ref{AI-gen1})
\begin{equation}
{\cal A}_{SU(5)}|_{\rm surface, geom}= {\frac 3 2} \chi({{\bf 10}_{1}})|_{\rm geom} + {\frac1 2}\chi({{\bf 5}_{3}})|_{\rm geom} + {\frac 1 2}\chi({\bf 5}_{-2})|_{\rm geom} \,.
\end{equation}
The chiral matter from the 7-brane bulk transforms in the adjoint with $c^{(2)}_{\bf 24} =10$ and contributes
\begin{equation}
\mathcal{A}_{SU(5)}|_{\rm bulk, geom}= 5 \chi({\bf 24}_0)|_{\rm geom} = -\frac{5}{24} W \left(c_1-W\right) \left(W \left(W-c_1\right)+c_2\right) \,,
\end{equation}
where we have used (\ref{chibadj1}).
In addition, there is another contribution from anti-chiral fermions generated in the 3-7 sector. These modes transform in representation ${\bf 5}_{q_1}$ and their chiral index is given by minus the point-wise intersection number $ - [W]\cdot [C]$ with $[C]=\frac1{24} \pi_\ast(c_4(\hat X_5)$ in the absence of flux.
With the help of (\ref{c4hatX5}), their $SU(5)$ anomaly contribution follows as
\begin{eqnarray}
\mathcal{A}_{SU(5)}|_{3-7, {\rm geom}} &=&\frac12\chi_{3-7}(\mathbf{5}_{q_1}) |_{\rm geom} =-\frac12 W\cdot \frac1{24} \pi_\ast(c_4(\hat X_5))\cr
&=&-\frac1{48}W\cdot (144c_1^3-264c_1^2W+12c_1c_2+162c_1W^2-30W) \,.
\end{eqnarray}
Then the pure non-abelian $SU(5)$ anomalies, in the absence of $G_4$ fluxes, indeed cancel,
\begin{equation}
\mathcal{A}_{SU(5)}|_{3-7, {\rm geom}} + \mathcal{A}_{SU(5)}|_{{\rm bulk, geom}} + {\cal A}_{SU(5)}|_{\rm surface, geom} =0 \,.
\end{equation}
Now we switch gear to check the cancellation of the $U(1)_A$ gauge anomaly. As we have discussed above, there are two types of charged matter states in the 3-7 sector with different $U(1)_A$ charges. With the help of \eqref{3-7locis1}, their combined contribution to the abelian anomalies is
\begin{equation}
\mathcal{A}_{U(1)}|_{{3-7},\rm geom} = \frac{5}{2} q_1^2 \, \chi_{3-7}(\mathbf{5}_{q_1})|_{\rm geom} + \frac{1}{2} q_2^2 \chi_{3-7}({\bf 1}_{q_2})|_{\rm geom} =- \, \frac1{48} \pi_\ast (c_4(\hat X_5)) \cdot {(-30W+50 c_1)} \,.
\end{equation}
This perfectly cancels the anomalies from the 7-7 sector,
\begin{eqnarray}
\mathcal{A}_{U(1)}|_{\rm geom} &=&\frac{1}{2}\sum_{\bf R} q^2_{A}({\bf R}) {\rm dim}({\bf R}) \chi({\bf R})|_{\rm geom} \cr
&=&\frac{1}{2} \left[ 10\chi(\mathbf{10}_{1})+20\chi(\mathbf{5}_{-2})
+45 \chi(\mathbf{5}_{3})+25\chi(\mathbf{1}_5)
+( 5 q_1^2 \, \chi_{3-7}(\mathbf{5}_{q_1}) + q_2^2 \chi_{3-7}({\bf 1}_{q_2})) \right]|_{\rm geom}\cr
&=& 0
\end{eqnarray}
as it must since the Green-Schwarz counterterms vanish in absence of flux.
Finally, let us compute the gravitational anomalies. In absence of flux, gravitational anomaly cancellation is equivalent to (\ref{curvgravanomalies}) over a generic base $B_4$.
This equation involves the Chern class $c_5(\hat X_5)$ and $c_4(\hat X_5)$ of the resolved Calabi-Yau five-fold $\hat X_5$.
With the help of \eqref{Chernclass}, we find
\begin{eqnarray}
\Delta\chi_1(\hat X_5)&=&-66 c_1^4-61 c_1^3 W- {c_1}^2 {c_2}+\frac{235 c_1^2 W^2}{4}-\frac{23 c_1 c_2 W}{12}-\frac{76c_1 W^3}{3}+\frac{3 c_2 W^2}{4}+\frac{17 W^4}{4} \cr
-6c_1\Delta [C] &=&54 c_1^4+66 c_1^3 W-\frac{81 c_1^2 W^2}{2}+\frac{15 c_1 W^3}{2} \,.
\end{eqnarray}
Summing both terms up perfectly matches the RHS of (\ref{curvgravanomalies}),
\begin{eqnarray}
&& - (10 \chi(\mathbf{10}_{1})+5\chi(\mathbf{5}_{-2})+5\chi(\mathbf{5}_{3})+\chi(\mathbf{1}_5) +24 \chi({\bf 24}_0)-4\chi({\bf 24}_0))|_{\rm geom} \cr
&=& - (12c_1^4-5 c_1^3 W+c_1^2 c_2-\frac{73c_1^2 W^2}{4}+\frac{23c_1c_2 W}{12}+\frac{107c_1W^3}{6}-\frac{3 c_2 W^2}{4}-\frac{17 W^4}{4}) \,.
\end{eqnarray}
In summary, we have checked that in this example with the absence of $G_4$ fluxes, all types of anomalies are cancelled by themselves and in agreement with \eqref{AnomaliesF-theory}.
\subsection{Flux dependent anomaly relations} \label{subsec_fluxdepan}
In the $SU(5) \times U(1)_A$ model defined by (\ref{Tate}), there only exist two types of gauge invariant 4-form fluxes $G_4 \in H^{2,2}_{\rm vert}(\hat X_5)$ compatible with the $SU(5) \times U(1)_A$ gauge group \cite{Krause:2012yh}. We choose a basis of fluxes as
\begin{eqnarray} \label{fluxes-normalisation}
G_4^A &=& F \cdot [U_A] \label{G4A}\\
G_4^\lambda &=& - \lambda \left( E_2 \cdot E_4 + \frac{1}{5} (2 E_1 - E_2 + E_3 -2 E_4) \cdot c_1 \right) \,.
\end{eqnarray}
Here $[U_A]$ is the 2-form class dual to the non-Cartan $U(1)_A$ divisor $U_A$ defined in (\ref{noncartan}), $\lambda$ is a constant and $F \in H^{1,1}(B_4)$ is an arbitrary class parametrizing the flux. Both $\lambda$ and $F$ are to be chosen such that $G_4 + \frac{1}{2} c_2(\hat X_5) \in H^2(\hat X_5,\mathbb Z)$.
We now analyze the anomaly relations, including the Green-Schwarz terms, for both of these flux backgrounds in turn.
\subsubsection{$G_4^A$ flux}
We begin with the flux background (\ref{G4A}).
The cancellation of non-abelian $SU(5)$ gauge anomalies in the presence of $G_4^A$ has already been verified in \cite{Schafer-Nameki:2016cfr} so that we can focus on
evaluating \eqref{gauge-flux1}, or equivalently \eqref{Abelian1}, for the $U(1)_A$ anomaly.
To compute the flux dependent chiral index of the 7-brane various matter states, we need to extract the line bundle $L_{\bf R}$ defined in (\ref{c1LR}) on the 7-brane codimension-two matter loci. Since $G_4^A$ is simply the gauge flux associated with the non-Cartan factor $U(1)_A$, we know that $\pi_\ast(G_4^A \cdot S^a_{\bf R}) = q_A({\bf R}) F|_{C_{\bf R}}$. It follows that
\begin{eqnarray}
c_1(L_{{\bf 10}_1}) = \left. F \right|_{C_{\mathbf{10}_{1}}}, \qquad c_1(L_{{\bf 5}_3}) = 3\left. F \right|_{C_{\mathbf{5}_{3}}}, \qquad c_1(L_{{\bf 5}_{-2}}) = \left. -2 F \right|_{C_{\mathbf{5}_{-2}}}, \qquad c_1(L_{{\bf 1}_5}) = 5\left. F \right|_{C_{\mathbf{1}_{5}}}
\end{eqnarray}
and therefore
\begin{eqnarray}
\chi( \mathbf{10}_{1})|_{\rm flux}&=& \frac12 \int_{C_{{\mathbf{10}}_1}} F^2, \qquad \chi(\mathbf{5}_{-2})|_{\rm flux}=\frac12 \int_{C_{\mathbf {5}_{-2}}}(-2F)^2 \\
\chi({\mathbf 5}_{3})|_{\rm flux} &=& \frac12 \int_{C_{{\bf 5}_{3}}}(3F)^2,\qquad \chi(\mathbf{1}_{5})|_{\rm flux} = \frac12 \int_{C_{{\bf 1}_{5}}}(5F)^2 .
\end{eqnarray}
The anomaly contribution (\ref{37ABanomaly-b}) from the 3-7-brane sector is
\begin{eqnarray}
{\cal A}_{U(1)}|_{3-7,{\rm flux}} = - \frac{1}{4} F \cdot F \cdot_{B_4} \pi_\ast([U_A] \cdot [U_A]) \cdot_{B_4} \pi_\ast([U_A] \cdot [U_A])
\end{eqnarray}
with $\pi_\ast([U_A] \cdot [U_A]) = - D_A$ as in (\ref{DA}).
Altogether this gives for the LHS of \eqref{Abelian1}
\begin{eqnarray}
\mathcal{A}_{U(1)}|_{\rm flux}&=& \mathcal{A}_{U(1)}|_{7-7,{\rm flux}} + {\cal A}_{U(1)}|_{3-7,{\rm flux}} \cr
&=&\frac12 \left(10\chi(\mathbf{10}_{1})|_{\rm flux}+20\chi(\mathbf{5}_{-2})|_{\rm flux}
+45 \chi(\mathbf{5}_{3})|_{\rm flux} +25 \chi(\mathbf{1}_{5})|_{\rm flux} \right) - \frac{1}{4} F^2 D_A^2 \\
&=& \frac12 F^2 (50 c_1 - 30 W)^2 \,.
\end{eqnarray}
This is to be compared to the RHS of \eqref{Abelian1} given by the Green-Schwarz counterterms (\ref{GSclaim})
\begin{eqnarray}
\frac1{4\pi} \Omega_{\alpha \beta} \Theta_A^\alpha \Theta_B^\beta &=&\frac12 \pi_{\ast} (G_4 \cdot G_4) \cdot_{B_4} \pi_\ast ([U_A] \cdot [U_A]) =\frac12 F \cdot_{B_4} F \cdot_{B_4} \pi_\ast([U_A] \cdot [U_A])^2 \cr
&=& \frac12 F^2 \cdot (50 c_1 - 30 W)^2 \,.
\end{eqnarray}
Hence \eqref{Abelian1} and therefore \eqref{gauge-flux1} hold.
Finally, let us switch to cancellation of the purely gravitational anomaly. Given the above expressions, the LHS of \eqref{gravitationalanomalies2} yields
\begin{equation}
-6 c_1 \cdot \pi_{\ast}(G_4^A \cdot G_4^A) = - 6 c_1 \cdot F \cdot F \cdot (-D_A) = -6 c_1 F^2 (-50 c_1 +30 W)
\end{equation}
which perfectly matches the RHS of \eqref{gravitationalanomalies2} given by
\begin{equation}
2 \left(10\chi(\mathbf{10}_{1})|_{\rm flux}+5\chi({\mathbf 5}_{3})|_{\rm flux}+5\chi({\mathbf 5}_{-2})|_{\rm flux}+ \chi(\mathbf{1}_{5})|_{\rm flux} \right)=6 c_1 F^2 (50 c_1-30 W) \,.
\end{equation}
\subsubsection{$G_4^\lambda$ flux}
Verifying the anomalies in the presence of flux of the form $G_4^\lambda$ is slightly more involved. In the sequel we heavily build on the analysis of \cite{Bies:2017fam}, where this gauge background is described, in a compactification to four dimensions, as a 'matter surface flux'. Since the fiber structure is the same, we can extend these results to F-theory compactification on an elliptic 5-fold. Since we are now working over a base of complex dimension four, extra technical complications arise in the computation of the chiral index for the 7-brane ammeter, which we will solve in appendix \ref{app_chirality}.
Key to computing the 7-brane matter chiralities is again the induced line bundle $L_{\bf R} = \pi_\ast(G_4^\lambda \cdot S^a_{\bf R})$, given this time by
\begin{eqnarray}
c_1(L_{{\bf 10}_1})& =& \frac{-3\lambda}{5} [Y_1] + \frac{4 \lambda}{5} [Y_2], \qquad c_1(L_{{\bf 5}_3}) = \frac{-2\lambda}{5} [Y_2], \\
c_1(L_{{\bf 5}_{-2}}) &=& \frac{3 \lambda}{5} [Y_1] - \frac{2 \lambda}{5} [Y_2], \qquad
c_1(L_{{\bf 1}_5}) = 0 \,.
\end{eqnarray}
A derivation can be found in section 5 of \cite{Bies:2017fam}. By Poincar\'e duality, the objects $Y_i$ describe curve classes on the respective matter codimension-two loci on the base, defined as the intersection loci
\begin{align}
\begin{split} \label{transverseY}
C_{\mathbf{5}_{3}} \cap C_{\mathbf{10}_{1}}&= Y_2 \, , \\
C_{\mathbf{5}_{-2}} \cap C_{\mathbf{10}_{1}} &= Y_1 + Y_2 \, , \\
C_{\mathbf{5}_{-2}} \cap C_{\mathbf{5}_{3}} &= Y_2 + Y_3 \, .
\end{split}
\end{align}
The first Chern classes of the line bundles $L_{{\bf 10}_1}$ and $L_{{\bf 5}_3}$ can be expressed as the pullback of divisor classes from $W$ to the respective matter loci,
\begin{eqnarray}
c_1(L_{\mathbf{10}_{1}}) &=& \frac{\lambda}{5} \left( -3 ([Y_2] + [Y_1]) + 7 [Y_2] \right)|_{C_{\mathbf{10}_{1}}} = \frac{\lambda}{5} \left( 6 c_1 - 5 W \right)|_{C_{\mathbf{10}_{1}}} \label{L101lambdabundle}\\
c_1(L_{\mathbf{5}_{3}}) &=&\frac{\lambda}{5} \left( - 2 [Y_2] \right)|_{C_{\mathbf{5}_{3}}} = \frac{\lambda}{5} \left( - 2 c_1 \right)|_{C_{\mathbf{5}_{3}}} \,.
\end{eqnarray}
Hence we can straightforwardly compute the associated chiralities as integrals on $B_4$
\begin{eqnarray}
\chi(C_{\mathbf{10}_{1}})|_{\rm flux} &=& \frac{1}{2} \int_{C_{\mathbf{10}_{1}}} c_1^2(L_{\mathbf{10}_{1}}) = \frac{\lambda^2}{50}\, W \cdot c_1 \cdot ( 6 c_1 - 5 W )^2 \,, \\
\chi(C_{\mathbf{5}_{3}})|_{\rm flux} &=& \frac{1}{2} \int_{C_{\mathbf{5}_{3}}} c_1^2(L_{\mathbf{5}_{3}}) = \frac{\lambda^2}{50} \, W \cdot (3 c_1 - 2 W) \cdot 4 c_1^2 \,.
\end{eqnarray}
By contrast, $c_1(L_{{\bf 5}_{-2}})$ cannot be interpreted as the class of a complete intersection of a base divisor with $C_{\mathbf{5}_{-2}}$ \cite{Bies:2017fam}.
Each of the classes $Y_i$ defines a divisor class on $C_{{\bf 5}_{-2}}$, dual to a curve.
The technical difficulty is that $Y_1$ and $Y_2$ separately cannot be written as the pullback of a divisor class from the 7-brane divisor $W$ to $C_{{\bf 5}_{-2}}$.
Rather, on $W$, the curves $Y_i$ are given by intersections
\begin{eqnarray} \label{defYi}
Y_1 = a_1 \cap a_{2,1}|_W, \qquad Y_2 = a_1 \cap a_{3,2}|_W, \qquad Y_3 = a_{4,3} \cap a_{3,2}|_W \,,
\end{eqnarray}
where the class of these Tate coefficients have been listed in \eqref{classtatecoefficient}.
In appendix \ref{app_chirality} we will discuss how to evaluate the chirality of ${\bf 5}_{-2}$ despite this complication, our final result being
\begin{equation} \label{chi5-2lambda}
\chi(C_{\mathbf{5}_{-2}})|_{\rm flux} = \frac{1}{2} \int_{C_{\mathbf{5}_{-2}}} c_1^2(L_{\mathbf{5}_{-2}}) = -\frac{\lambda^2}{25}c_1 \cdot W \cdot \left(60 c_1^2-79 c_1W+25 W^2\right) \,.
\end{equation}
In light of the discussion of section (\ref{sec_geomback}), the chiral indices for the 3-7 matter states as induced by $G_4^\lambda$ take the form
\begin{eqnarray} \label{37lambda1}
\chi_{3-7}(\mathbf{5}_{q_1}) & =& - [C]|_{\rm flux} \cdot W \\
\chi_{3-7}({\bf 1}_{q_2}) & =& -[C]|_{\rm flux} \cdot (- 5 W + 8 c_1) \,,
\end{eqnarray}
where the flux dependent piece of the 3-brane class reads
\begin{eqnarray} \label{G4double}
[C]|_{\rm flux} = - \frac{1}{2} \pi_\ast(G_4^\lambda \cdot G_4^\lambda) = - \frac{\lambda^2}{10} W \cdot c_1 \cdot (6 c_1 - 5 W) \,.
\end{eqnarray}
To derive this latter result, recall from section 4.3 of \cite{Bies:2017fam} that up to irrelevant correction terms $G_4^\lambda$ for $\lambda=1$ is the class associated with one of the matter fibrations $S^a_{{\bf 10}_1}$. The result for $\pi_\ast(G_4^\lambda \cdot G_4^\lambda) = \lambda \, \pi_\ast( G_4^\lambda \cdot S^a_{{\bf 10}_1}$ can then be read off from (\ref{L101lambdabundle}).
We are finally in a position to check the cancellation of anomalies in the presence of $G_4^\lambda$, beginning with the pure non-abelian gauge anomaly. Note the $G_4^\lambda$ background does not induce any chirality for the 7-brane bulk matter. Together with the above explicit expressions for chiral indices in the 7-brane and the 3-7 sector, one can easily confirm that
\begin{eqnarray}
&&\mathcal{A}_{SU(5)}|_{\rm flux}= \frac32 \chi(C_{\mathbf{10}_{1}})|_{\rm flux} +\frac12\chi(C_{\mathbf{5}_{3}})|_{\rm flux} +\frac12 \chi(C_{\mathbf{5}_{-2}})|_{\rm flux} +\frac12 \chi_{3-7}(\mathbf{5}_{q_1})|_{\rm flux}=0 \,.
\end{eqnarray}
Next we turn to the $G_4^\lambda$ dependent part of the abelian gauge anomalies.
The combined 1-loop anomaly from the 7-7 and the 3-7 matter evaluates to
\begin{eqnarray} \label{U1lambdaanomaly}
\mathcal{A}_{U(1)_A}|_{\rm flux}&=&\frac{1}{2}\sum_{\bf R} \text{dim}({\bf R}) \, q_A^2({\bf R}) \, \chi({\bf R})|_{\rm flux} \cr
&=&\frac12 \left( 10\chi(\mathbf{10}_{1}) +20\chi(\mathbf{5}_{-2})
+45 \chi(\mathbf{5}_{3}) +25\chi(\mathbf{1}_{5}) + 5 q_1^2 \, \chi_{3-7}(\mathbf{5}_{q_1}) + q_2^2 \chi_{3-7}({\bf 1}_{q_2}) \right) |_{\rm flux} \cr
&=&\frac{1}{2} \lambda^2 \, c_1^2 \, W^2 \,.
\end{eqnarray}
For the 3-7 contribution we can either use (\ref{37lambda1}) with the charge assignments (\ref{chargesU1A}), or directly evaluate the $G_4^\lambda$ dependent component of (\ref{37ABanomaly}).
The combined 1-loop anomaly forms the LHS of (\ref{Abelian1}) and must be cancelled by the Green-Schwarz terms \eqref{GSclaim} appearing on the RHS.
To compute the latter, we make again use of the interpretation of $G_4^\lambda$ as one of the matter fibrations $S^a({\bf 10}_1)$. Intersection this with the $U(1)_A$ generator $U_A$ in the fiber reproduces the $U(1)_A$ charge of ${\bf 10}_1$ and therefore
\begin{eqnarray}
\pi_\ast (G_4 \cdot [U_A] )= \lambda \, C_{{\bf 10}_1} \cdot W = \lambda \, c_1 \cdot W \,.
\end{eqnarray}
With this the Green-Schwarz terms are
\begin{eqnarray}
\label{GSterm2}
&& \frac1{4\pi} \Omega_{\alpha \beta} \Theta_A^\alpha \Theta_B^\beta =\frac12 \pi_\ast(G_4^\lambda \cdot [U_A]) \cdot \pi_\ast(G_4^\lambda \cdot [U_A]) =\frac12\lambda^2 \, c^2_1 \cdot W^2 \,.
\end{eqnarray}
This perfectly cancels the 1-loop anomalies (\ref{U1lambdaanomaly}) and hence verifies the $G_4^\lambda$ dependent part of (\ref{Abelian1}) or equivalently (\ref{gauge-flux1}).
As for the cancellation of the gravitational anomalies, with the help of \eqref{G4double}, the LHS of \eqref{gravitationalanomalies2} becomes
\begin{equation}
-6 c_1 \cdot \pi_\ast (G_4^\lambda \cdot G_4^\lambda) = - \frac{6}{5} \lambda^2 \, c_1^2 \cdot W \cdot (6 c_1-5 W),
\end{equation}
which is again exactly equal to the RHS of \eqref{gravitationalanomalies2}
\begin{equation}
2 \, (10\chi(\mathbf{10}_{1})+5\chi({\mathbf 5}_{3}) +5\chi({\mathbf 5}_{-2}) + \chi(\mathbf{1}_{5}) )|_{\rm flux}=-\frac{6}{5} \lambda^2 \, c_1^2 \cdot W \cdot (6 c_1-5 W) \,.\end{equation}
\section{Comparison to 6d and 4d anomaly relations} \label{sec_Comparison6d4d}
In this final section we
compare the 2d anomaly relations (\ref{gaugeanomalyFtheory}) and (\ref{gravitationalanomalies1}) to their analogue in a 6d or 4d F-theory compactification on an elliptic fibration $ \hat X_3$ or $ \hat X_4$, respectively.
The cancellation of all gauge and mixed gauge-gravitational anomalies in both these classes of theories is captured by two relations, each valid in
$H^{4}(\hat X_3)$ or $H^{4}(\hat X_4)$, of the form
\begin{eqnarray} \label{4d6d1}
\sum_{{\bf R}, a} \beta^a_\Gamma({\bf R}) \, \beta^a_\Lambda({\bf R}) \, \beta^a_\Sigma({\bf R}) S^a_{\bf R} - 3 \, {\mathfrak F}_{(\Gamma} \cdot \pi^\ast \pi_\ast(\mathfrak{F}_\Lambda \cdot {\mathfrak F}_{\Sigma)}) &=& 0 \\
\sum_{{\bf R}, a} \beta^a_\Lambda({\bf R}) \, S^a_{\bf R} + 6 \, {\mathfrak F}_{\Lambda} \cdot c_1 &=& 0 \label{4d6d2} \,.
\end{eqnarray}
These two homological relations have been shown in \cite{Bies:2017abs} to be equivalent to the intersection theoretic identities derived from the requirement of gauge and mixed gauge-gravitational anomaly cancellation in 6d \cite{Park:2011ji} and 4d \cite{Cvetic:2012xn} F-theory vacua.
In addition the cancellation of purely gravitational anomalies in 6d F-theory vacua poses an extra constraint on the geometry of $\hat X_3$,
which has no direct counterpart in 4d.\footnote{\label{footnote6d}This relation is given, for example, as equation (3.8) in \cite{Park:2011ji}, and proven generally in \cite{Grassi:2000we}. } Interestingly enough, however, apart from this latter point anomaly cancellation in 6d and 4d F-theory vacua is based on the same type of homological relations.
While a general proof of these relations from first principles, and without relying on anomaly cancellation, is not yet available in the literature, these relations can be verified in explicit examples.\footnote{On the other hand, \cite{Corvilain:2017luj} proves anomaly cancellation in 4d F-theory vacua by comparison with the dual M-theory. Combined with the above statement this is a physics proof of (\ref{4d6d1}) and (\ref{4d6d2}) on elliptic Calabi-Yau 4-folds.} The details of such a verification appear to be completely independent of the choice of base of the elliptic fibration, including its dimension \cite{Bies:2017abs}.
This raises the question if the same type of relations also holds on elliptically fibred Calabi-Yau 5-folds and if they play any role in anomaly cancellation in the associated 2d (0,2) theories.
The situation in compactifications to two dimensions looks rather more involved at first sight:
As we have shown in section \ref{sec_AnomalyEqu5folds}, there are two types of independent anomaly relations, (\ref{gaugeanomalyFtheory}), associated with the cancellation of the gauge anomaly, and another two, (\ref{gravitationalanomalies1}), for the pure gravitational anomaly.
We will now see that the flux dependent part of these anomaly relations, (\ref{gauge-flux1}) and (\ref{gravflux1}), is in fact closely related in form to (\ref{4d6d1}) and (\ref{4d6d2}).
Consider first relation (\ref{gauge-flux1}) for the cancellation of the flux dependent part of the 2d gauge anomalies,
\begin{equation}
\label{GS1-v2}
\begin{split}
& \sum_{{\bf R},a} \beta^a_\Lambda({\bf R}) \beta^a_\Sigma({\bf R}) \pi_\ast (G_4 \cdot S^a_{\bf R}) \cdot_{C_{\bf R}} \pi_\ast (G_4 \cdot S^a_{\bf R}) \\
& = \pi_\ast (G_4 \cdot G_4) \cdot_{B_4} \pi_\ast (\mathfrak{F}_\Lambda \cdot \mathfrak{F}_\Sigma) + \pi_\ast (G_4 \cdot \mathfrak{F}_\Sigma) \cdot_{B_4} \pi_\ast (G_4 \cdot \mathfrak{F}_\Lambda ) {+ \pi_\ast (G_4 \cdot \mathfrak{F}_\Lambda) \cdot_{B_4} \pi_\ast (\mathfrak{F}_\Sigma \cdot G_4) \,.
}
\end{split}
\end{equation}
A priori (\ref{GS1-v2}) holds for every transversal flux $G_4$, i.e. for every element $G_4 \in H^{2,2}(\hat X_5)$ satisfying (\ref{transverse1}), including potentially non gauge invariant fluxes.
Our first observation is that this relation can be generalized to
\begin{equation}
\label{GS1-v3}
\begin{split}
& \sum_{{\bf R},a} \beta^a_\Lambda({\bf R}) \beta^a_\Sigma({\bf R}) \pi_\ast (G^{(1)}_4 \cdot S^a_{\bf R}) \cdot_{C_{\bf R}} \pi_\ast (G^{(2)}_4 \cdot S^a_{\bf R}) \\
& = \pi_\ast (G^{(1)}_4 \cdot G^{(2)}_4) \cdot_{B_4} \pi_\ast (\mathfrak{F}_\Lambda \cdot \mathfrak{F}_\Sigma) + \pi_\ast (G^{(1)}_4 \cdot \mathfrak{F}_\Sigma) \cdot_{B_4} \pi_\ast (G_4^{(2)} \cdot \mathfrak{F}_\Lambda ) {+ \pi_\ast (G_4^{(1)} \cdot \mathfrak{F}_\Lambda) \cdot_{B_4} \pi_\ast (\mathfrak{F}_\Sigma \cdot G^{(2)}_4) }
\end{split}
\end{equation}
valid for all transversal fluxes $G_4^{(1)}$ and $G_4^{(2)}$:
To see this, insert the ansatz $G_4 = G_4^{(1)} + G_4^{(2)}$ into (\ref{GS1-v2}). This gives three types of contributions, one depending quadratically on $G_4^{(1)}$ and on $G_4^{(2)}$, respectively, and a cross-term involving $G_4^{(1)}$ and $G_4^{(2)}$. Since the quadratic terms vanish by themselves thanks to (\ref{GS1-v2}), this is enough to establish the more general relation (\ref{GS1-v3}).
Let us now specialise one of the fluxes appearing in (\ref{GS1-v3}) to
\begin{eqnarray} \label{G41ansatz}
G_4^{(1)} = \pi^*D \cdot \mathfrak{F}_\Gamma \qquad {\rm with} \quad D \in H^{1,1}(B_4)
\end{eqnarray}
and analyze the resulting identity further by repeatedly using the projection formulae
\begin{eqnarray}
\pi_\ast (\pi^*A \cdot_{\hat X_5} B) &=& A \cdot_{B_4} \pi_\ast(B) \\
\pi_\ast(E) \cdot_{B_4} F &=& E \cdot_{\hat X_5} \pi^*(F)
\end{eqnarray}
for suitable cohomology classes on $B_4$ and $\hat X_5$.
In the sequel, unless specified explicitly, the symbol $\cdot$ denotes the intersection product on $\hat X_5$. Then with (\ref{G41ansatz}) the first term on the RHS takes the form
\begin{eqnarray}
\pi_\ast (G^{(1)}_4 \cdot G^{(2)}_4) \cdot_{B_4} \pi_\ast ({\mathfrak F}_\Lambda \cdot \mathfrak{F}_\Sigma) &=& \left(D \cdot_{B_4} \pi_\ast({\mathfrak F}_\Gamma \cdot G_4^{(2)})\right) \cdot_{B_4} \pi_\ast(\mathfrak{F}_\Lambda \cdot \mathfrak{F}_\Sigma) \\
&=& G_4^{(2)} \cdot {\mathfrak F}_\Gamma \cdot \pi^\ast ( D \cdot_{B_4} \pi_\ast(\mathfrak{F}_\Lambda \cdot \mathfrak{F}_\Sigma) ) \\
&=& \pi^\ast D \cdot G_4^{(2)} \cdot {\mathfrak F}_\Gamma \cdot \pi^\ast \pi_\ast(\mathfrak{F}_\Lambda \cdot \mathfrak{F}_\Sigma) \,.
\end{eqnarray}
Similar manipulations for the remaining two other terms on the RHS of (\ref{GS1-v2}) yield
\begin{eqnarray}
{\rm RHS \,\, of \, \, (\ref{GS1-v2})} = 3 \, \pi^*D \cdot G_4^{(2)} \cdot {\mathfrak F}_{(\Gamma} \cdot \pi^\ast \pi_\ast(\mathfrak{F}_\Lambda \cdot F_{\Sigma)}) \,.
\end{eqnarray}
As for the LHS, observe that
\begin{eqnarray} \label{G41SaR}
\pi_\ast(G_4^{(1)} \cdot S^a_{\bf R} ) = \pi_\ast( \pi^\ast D \cdot {\mathfrak F}_\Gamma \cdot S^a_{\bf R} ) = \beta^a_\Gamma({\bf R}) \left( D \cdot_{B_4} C_{\bf R} \right) \,.
\end{eqnarray}
Here we are using that in expressions of this form, the intersection of the divisor ${\mathfrak F}_\Gamma$ with the matter 3-cycle $S^a_{\bf R}$ in the fibre reproduces the charge $\beta^a_\Gamma$ of the associated state with respect to $U(1)_\Gamma$. As explained around (\ref{c1LR}), the expression on the right of (\ref{G41SaR}) is the first Chern class of the line bundle induced by the specific flux $G_4^{(1)}$ to which the matter states on $C_{\bf R}$ couple. For the special choice (\ref{G41ansatz}) this line bundle is the pullback of a line bundle from $B_4$.
With this understanding, the intersection product appearing on the LHS can be further simplified as
\begin{eqnarray}
\pi_\ast (G^{(1)}_4 \cdot S^a_{\bf R}) \cdot_{C_{\bf R}} \pi_\ast (G^{(2)}_4 \cdot S^a_{\bf R}) = \beta^a_\Gamma({\bf R}) \, \pi^*D \cdot G_4^{(2)} \cdot S^a_{\bf R} \,.
\end{eqnarray}
Altogether we have thus evaluated (\ref{GS1-v3}), for the special choice (\ref{G41ansatz}), to
\begin{eqnarray}
\pi^\ast D \cdot G_4^{(2)} \cdot \left( \sum_{{\bf R}, a} \beta^a_\Gamma({\bf R}) \, \beta^a_\Lambda({\bf R}) \, \beta^a_\Sigma({\bf R}) \, S^a_{\bf R} - 3 \, {\mathfrak F}_{(\Gamma} \cdot \pi^\ast \pi_\ast(\mathfrak{F}_\Lambda \cdot {\mathfrak F}_{\Sigma)}) \right) = 0 \,.
\end{eqnarray}
Repeating the same steps for the flux dependent gravitational anomaly relation (\ref{gravflux1}) leads to
\begin{eqnarray}
\pi^\ast D \cdot G_4^{(2)} \cdot \left( \sum_{{\bf R}, a} \beta^a_\Lambda({\bf R}) \, S^a_{\bf R} + 6 \, {\mathfrak F}_{\Lambda} \cdot c_1 \right) = 0 \,.
\end{eqnarray}
The terms in brackets are identical in form with the linear combinations of 4-form classes which are guaranteed to vanish on an elliptically fibered Calabi-Yau 3-fold and 4-fold by anomaly cancellation according to (\ref{4d6d1}) and (\ref{4d6d2}). We conclude that \emph{if} the relations (\ref{4d6d1}) and (\ref{4d6d2}) hold also within $H^4(\hat X_5)$, as suggested by the results of \cite{Bies:2017abs}, this implies cancellation of the flux dependent part of the anomalies in 2d F-theory vacua for the special choice of flux (\ref{G41ansatz}).
For more general fluxes, however, the constraints imposed on anomaly cancellation on a Calabi-Yau 5-fold seem to be stronger. In particular,
a direct comparison with (\ref{4d6d1}) and (\ref{4d6d2}) is made difficult by the fact that
(\ref{gauge-flux1}) and (\ref{gravflux1}) are quadratic in fluxes and a priori involve the intersection product on the matter loci $C_{\bf R}$, not on $B_4$. For general $G_4$ backgrounds, this makes a difference, as we have seen in section \ref{subsec_fluxdepan}. Furthermore, anomaly cancellation in 2d predicts the flux independent relations (\ref{gauge-geom1}) and (\ref{gravgeom1}). Condition (\ref{gravgeom1}) can be viewed as analogous, though very different in form, to the geometric condition on cancellation of the purely gravitational anomalies in 6d referred to in footnote \ref{footnote6d}.
It would be very interesting to investigate if a deconstruction of the topological invariants appearing in (\ref{gauge-geom1}) and (\ref{gravgeom1}), similar to the procedure applied for the Euler characteristic on Calabi-Yau 3-folds in \cite{Grassi:2000we,Grassi:2011hq}, can lead to a geometric proof of these identities.
\section{Conclusions and Outlook}
In this work we have provided closed expressions for the gravitational and gauge anomalies in 2d $N=(0,2)$ compactifications of F-theory on elliptically fibered Calabi-Yau 5-folds.
In particular, we have derived the Green-Schwarz counterterms for the cancellation of abelian gauge anomalies. The Green-Schwarz mechanism operates in a manner very similar to its 6d $N=(1,0)$ cousin: Dimensional reduction of the self-dual Type IIB 4-form results in real chiral scalar fields whose axionic shift symmetry is gauged and whose Chern-Simons type couplings hence become anomalous.
We have uplifted our results for the gauging and the couplings to an expression valid in the most general context of F-theory on elliptically fibered Calabi-Yau 5-folds.
Anomaly cancellation in the 2d $(0,2)$ supergravity is then equivalent to (\ref{gaugeanomalyFtheory}) for the gauge and (\ref{gravitationalanomalies1}) for the gravitational part. Each equation splits into a purely geometric and a flux dependent identity. These must hold separately on every elliptic Calabi-Yau 5-fold and for every consistent background of $G_4$ fluxes. We have verified this explicitly in a family of fibrations and for all vertical gauge fluxes thereon.
It is instructive to compare these 2d anomaly cancellation conditions to their analogue in 6d and 4d F-theory vacua in the form put forward in \cite{Park:2011ji} and \cite{Cvetic:2012xn}, respectively.
The structure of anomalies as such becomes more and more constraining in higher-dimensional field theories.
At the same time the engineering of the quantum field theory in terms of the internal geometry becomes more intricate as the dimension of the compactification space increases, and hence the number of large spacetime dimensions decreases.
Correspondingly, the topological identities governing anomaly cancellation on elliptic 5-folds contain considerably more structure compared to their analogues in 4d and 6d F-theory compactifications. For once, the anomaly relations in 6d $N=(1,0)$ F-theory vacua are only sensitive to the topology of the elliptic fibration, while in 4d $N=1$ theories they are linearly dependent on a gauge flux. In 2d $N=(0,2)$ F-theories, both a purely topological and a flux dependent contribution arises. The latter is, in fact, quadratic in the gauge background.
Despite differences in structure, the 6d and 4d gauge anomaly relations of \cite{Park:2011ji} and \cite{Cvetic:2012xn} can be reduced to one single identity \cite{Bies:2017abs}, valid in the cohomology ring $H^{2,2}(\hat X_{n})$ of an elliptically fibered Calabi-Yau $n$-fold, with $n=3$ and $4$, respectively. The same is true for their mixed gauge-gravitational counterparts. One motivation for the present work was to investigate these universal identities, (\ref{4d6d1}) and (\ref{4d6d2}), with respect to anomaly cancellation in 2d F-theories. The flux-dependent parts of (\ref{gaugeanomalyFtheory}) and (\ref{gravitationalanomalies1}) exhibit striking similarities to (\ref{4d6d1}) and (\ref{4d6d2}). We have shown that if the 6d and 4d universal relations hold also in the cohomology ring of an elliptic 5-fold, as suggested by the examples studied in \cite{Bies:2017abs}, they imply the flux dependent anomaly relations at least for the subset of gauge backgrounds associated with massless $U(1)$ gauge groups. It would be very interesting to study further if also the converse is true, i.e. if the 2d relations allow us to establish a relation in the cohomlogy ring of elliptic 5-folds governing the 4d and 6d anomalies as well.
The flux-independent anomaly relations, on the other hand, seem not to be related in a straightforward manner to the structure of anomalies in higher dimensions.
In fact, already in 6d $N=(1,0)$ F-theory vacua, cancellation of the purely gravitational anomalies implies another topological identity with no counterpart in 4d. This relation has been proven quite generally in \cite{Grassi:2000we} using a deconstruction of the Euler characteristic of elliptic 3-folds. It would be worthwhile exploring if a similar proof is possible on Calabi-Yau 5-folds.
The structure of anomalies in 6d and 4d F-theory vacua is closely related to the Chern-Simons terms in the dual M-theory in five \cite{Intriligator:1997pq,Bonetti:2011mw,Bonetti:2013cza} and three dimensions \cite{Aharony:1997bx,Grimm:2011fx,Cvetic:2012xn}. In \cite{Corvilain:2017luj} this reasoning has lead to a proof of anomaly cancellation in 4d $N=1$ vacua obtained as F-theory on an elliptic Calabi-Yau 4-fold. It would be very interesting to extend such reasoning also to the 2d case. The Chern-Simons terms in the dual 1d $N=2$ Super-Quantum-Mechanics have been analyzed in \cite{Schafer-Nameki:2016cfr} and expressed geometrically in terms of data of the Calabi-Yau 5-fold.
As expected, the similarities between the resulting identities such as (10.8) in \cite{Schafer-Nameki:2016cfr} and the 2d anomaly conditions are striking.
At a more technical level, the expressions for the anomalies presented in this work are valid under the assumption that the loci on the base hosting massless matter are smooth.
Quite frequently, this assumption is violated, and an application of the usual index theorems requires a normalization of the singular loci \cite{Schafer-Nameki:2016cfr}. We leave it for future investigations to establish the anomaly relations in such more general situations. Likewise, in the presence of $\mathbb Q$-factorial terminal singularities in the fiber the precise counting of uncharged massless states in terms of topological invariants will change. In 6d F-theory vacua, this leads to a modification of the condition for cancellation of the gravitational anomaly \cite{Arras:2016evy,AGTW}, and similar effects are expected to play a role in 2d models.
Our focus in this work has been on the implications of anomaly cancellation rather than on the structure of the effective 2d $N = (0,2)$ supergravity per se.
The axionic gaugings induced by the flux background, as derived in this context, give rise to a K\"ahler moduli dependent D-term, as noted already in \cite{Schafer-Nameki:2016cfr}.
What remains to be clarified is a careful definition of the chiral variables in the supergravity sector and a comparison of the Green-Schwarz action to the superspace formulation
put forward in 2d (0,2) gauge theories in \cite{Adams:2006kb,Quigley:2011pv,Blaszczyk:2011ib}. This will also determine the correct normalization of the D-term.
At the level of the supersymmetry conditions induced by the flux, we have made, in passing, an interesting observation: Extrapolating from the situation on Calabi-Yau 4-folds we expect the existence of $G_4$ backgrounds which are not automatically of $(2,2)$ Hodge type and would hence break supersymmetry \cite{Haupt:2008nu}. More precisely, whenever $H^{2,2}(\hat X_5)$ contains $(2,2)$ forms which are not products of $(1,1)$ forms, it is expected that the Hodge type of a 4-form varies over the complex structure moduli space. This would constrain some of the complex structure of the 5-fold \cite{Haupt:2008nu}.
This makes it tempting to speculate that the contribution of the supergravity sector to the purely gravitational anomaly should change compared to a background without flux.
At the same time, the flux dependent contribution to the D3-brane tadpole modifies the class of the D3-branes in the background and therefore also the anomaly contribution from the sector of 3-7 string modes. For consistency, both effects have to cancel each other, which is in principle possible due to the opposite chirality of the fields involved.
In this sense the net effect of complex moduli stabilization would be topological, in stark contrast to the situation in 4d $N=1$ compactifications. More work on elliptically fibered 5-folds is needed to flesh out the details behind this phenomenon.
\noindent {\bf Acknowledgements}
We thank Seung-Joo Lee and Diego Regalado for important discussions, and Martin Bies, Antonella Grassi, Craig Lawrie, Christoph Mayrhofer and Sakura Sch{\"a}fer-Nameki in addition for collaboration on related topics. The work of T.W. and F.X. is partially supported by DFG Transregio TR33 'The Dark Universe' and by DFG under GK 'Particle Physics Beyond the Standard Model'.
|
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| 9,596
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# BIG DATA
## USING SMART BIG DATA, ANALYTICS AND METRICS TO MAKE BETTER DECISIONS AND IMPROVE PERFORMANCE
**BERNARD MARR**
This edition first published 2015
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_Library of Congress Cataloging-in-Publication Data_
Marr, Bernard.
Big data : using smart big data, analytics and metrics to make better decisions and improve performance / Bernard Marr.
pages cm
Includes index.
ISBN 978-1-118-96583-2 (pbk.)
1. Information technology—Management. 2. Big data. 3. Management—Statistical methods.
4. Decision making—Statistical methods. I. Title.
HD30.2.M3744 2015
658.4′0380285574—dc23
2014040562
A catalogue record for this book is available from the British Library.
ISBN 978-1-118-96583-2 (pbk)
ISBN 978-1-118-96582-5 (ebk) ISBN 978-1-118-96578-8 (ebk)
Cover Design: Wiley
Cover Image: ©iStockphoto.com/marigold_88
To the most important people in my life:
My wife Claire and our three children Sophia,
James and Oliver;
as well as my brother Marc, Julie and Alan, all my wonderful
friends, and in memory of my parents.
# **CONTENTS**
1. Introduction: Welcome to a Smarter World
1. Smarter sport
2. Smarter health
3. Smarter homes
4. Smarter love
5. Smarter parenting
6. Notes
2. 1 SMARTER BUSINESS
1. Who is using Big Data?
2. How companies are using Big Data
3. Don't panic!
4. Focus to reap the rewards
5. Notes
3. 2 S = START WITH STRATEGY
1. Small is beautiful in a Big Data world
2. The SMART strategy board
3. SMART analytics and Google
4. Key points and call to action
5. Notes
4. 3 M = MEASURE METRICS AND DATA
1. Types of data
2. Datification: The new forms of data
3. The anatomy of Big Data
4. How to use metrics and data for strategic advantage
5. Metrics and data in action
6. Key points and call to action
7. Notes
5. 4 A = APPLY ANALYTICS
1. Evolution of analytics
2. Text analytics
3. Speech analytics
4. Video/image analytics
5. Combined analytics
6. Transparency
7. Prediction vs. privacy
8. Key points and call to action
9. Notes
6. 5 R = REPORT RESULTS
1. Data visualization
2. New data visualization
3. How to improve data visualization
4. Infographics
5. Beware the self-service business intelligence tools
6. The ingredients of successful data visualization and infographics
7. Management dashboards
8. Key points and call to action
9. Notes
7. 6 T = TRANSFORM BUSINESS
1. Better understand and target customers
2. Improve and optimize business processes
3. Improve people's health and well-being
4. Improve business security and reduce fraud
5. Drive business and people performance
6. Improve cities and other infrastructure
7. New business opportunities
8. Smart will transform employment too
9. Key points and call to action
10. Notes
8. Conclusion
9. About the Author
10. Acknowledgements
11. Index
12. End User License Agreement
## **List of Illustrations**
1. Chapter 1
1. **Figure 1.1** The SMART Model
2. Chapter 2
1. **Figure 2.1** All roads lead to Start with Strategy
2. **Figure 2.2** The SMART Strategy Board
3. **Figure 2.3** SMART Customer Questions: Deriving Key Performance Questions from your SMART Strategy Board
4. **Figure 2.4** SMART Finance Questions: Deriving SMART Questions from your SMART Strategy Board
5. **Figure 2.5** SMART Operations Questions: Deriving SMART Questions from your SMART Strategy Board
6. **Figure 2.6** SMART Resource Questions: Deriving SMART Questions from your SMART Strategy Board
7. **Figure 2.7** SMART Competition and Risk Questions: Deriving SMART Questions from your SMART Strategy Board
3. Chapter 3
1. **Figure 3.1** Data examples for the customer panel: Finding SMART Data to Answer your SMART Questions: http://www.marketingprofs.com/articles/2014/24670/little-data-vs-big-data-nine-types-of-data-and-how-they-should-be-used
2. **Figure 3.2** Data examples for the operations panel: Deriving SMART Questions from your SMART Strategy Board
3. **Figure 3.3** Data examples for the finance panel: Deriving SMART Questions from your SMART Strategy Board
4. **Figure 3.4** Data examples for the resource panel: Deriving SMART Questions from your SMART Strategy Board
5. **Figure 3.5** Data examples for the competition and risk finance panel: Deriving SMART Questions from your SMART Strategy Board
4. Chapter 4
1. **Figure 4.1** Example of positive and negative sentiment associated with words
2. **Figure 4.2** Example of sentiment for Starbucks using Twitrratr
5. Chapter 5
1. **Figure 5.1** Spreadsheet of sales (products and services)
2. **Figure 5.2** Pie chart of total sales (products and services)
3. **Figure 5.3** Bar chart of total sales (products and services)
4. **Figure 5.4** Example of maps available with PolyMaps.
5. **Figure 5.5** Word cloud of most common relationship advice
6. **Figure 5.6** Example of D3 charts
7. **Figure 5.7** Example of D3 plots
8. **Figure 5.8** Example of Crazy Egg heat map tool
9. **Figure 5.9** Minard's data visualization of Napoleon's 1812 March into Russia
10. **Figure 5.10** Ayadsi graphic on credit card fraud
11. **Figure 5.11** Map of the Internet
12. **Figure 5.12** World's biggest data breaches by variable
13. **Figure 5.13** Washington DC Metrorail map
14. **Figure 5.14** Infographic explaining the 'Left' and 'Right' in government
15. **Figure 5.15** Infographic of Twitter advice
16. **Figure 5.16** Example of an operational dashboard for social media
## Guide
1. Cover
2. Table of Contents
3. Chapter
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# INTRODUCTION: WELCOME TO A SMARTER WORLD
The world is getting smarter.
This evolution can be seen everywhere and no industry or sector is immune. Consider an industry as old and well established as fishing, for example. Although human beings have been fishing since the beginning it wasn't until the 16th century that fisherman had boats capable of going to sea. This advance radically changed the fortunes of fishing and made large, profitable catches possible for the first time. The ships would set out for the fishing grounds using little more than a compass, a sextant and some 'inside knowledge' passed down through the generations of fishing families. If they were sailing at night they would use celestial navigation techniques and plot a course by the stars in order to arrive in the right vicinity. When the fisherman arrived at the fishing grounds they would cast their nets and hope for the best.
By the late 19th century fishing had been commercialized. Small fishing boats gave way to massive trawlers with on-board processing capabilities, the discovery of longitude and latitude made navigation considerably easier and in the last few decades technology has transformed fishing from an art to a science. Modern fishing boats are technology rich, using high tech navigation systems and GPS. Often small sensors are attached to the fish to track where the shoals are at any given time and sonar is used to pinpoint the density of the shoal and where and when to cast the nets. Modern fisherman know where the fish are, they know where they will be tomorrow and when to cast their nets for the best possible catch of their target fish. Fishing has evolved to become smarter. And it is just one example. Today the world is smarter in everything from sport's performance to healthcare in the home. Even love and parenting is becoming smarter!
## Smarter sport
Smart technology is now widely used in sport to find and recruit talent as well as monitor and improve performance – both for the amateur and the professional. It's now possible to get a basketball with over 200 built-in sensors that provide player and coaches with detailed feedback on performance. In tennis a system called SlamTracker can record a player's performance providing real-time statistics and comprehensive match analytics. If you've ever watched rugby (union or league) you may have wondered what the bump is between the players' shoulder blades – it's a GPS tracking system that allows the coaching staff to assess performance in real time. The device will measure the players' average speed, whether the player is performing above or below their normal levels, and heart rate, to identify potential problems before they occur. All of which can help coaches avoid injury and assist in making appropriate substitution decisions. Similar technology exists in the English Premier League and is used by many Olympic sports such as cycling.
But the technology is not just for the professionals. There are many wearable devices that can monitor health and well-being on the go. For example I wear an 'Up' fitness band that tells me how many steps I have taken each day, how many calories I've burned and how well I've slept each night. It is synced to my bathroom scales so that if I put on weight it will tell me and prompt me to increase my activity or decrease my food intake.
## Smarter health
Healthcare is also becoming smarter and it's set to revolutionize our lives.
Professor Larry Smarr, one of the most influential computer scientists in the United States and the most monitored man on the planet was able to self-diagnose Crohn's disease – long before any symptoms emerged and early enough to be able to effectively manage the condition. Smarr states:
> 'In a world in which you can see what you are doing to yourself as you go along the hope is that people will take more personal responsibility for themselves, in keeping themselves healthy. So it's almost like we are at day zero of a whole new world of medicine, and what will come out the other end is a far healthier society that's focused on wellness rather than fixing sickness when it's way too late.'1
This ability to monitor our own health heralds a new and exciting frontier of preventative medicine based on data.
We have long understood that in theory prevention is better than cure but the collaboration of technology and health is turning that insight into practice. This year close to 42 million wearable wireless sports, fitness and wellness devices are expected to ship worldwide. According to ABI Research, 'Over the next five years spending on bringing these wearable wireless consumer activity device collected data will grow to a $52 million market by 2019.' Cloud services such as Ginger.io already allow care providers to monitor their patients through sensor-based applications on their smart phones.2 And Proteus manufactures an 'ingestible' scanner the size of a grain of sand, which can be used to track when and how patients are taking their medication. This gives providers information about 'compliance rates' – how often patients follow their doctor's orders – and can even alert a family member to remind them.
But it's not just the ability for us to monitor and manage our own health better; Big Data, analytics and the smart revolution are changing healthcare right now with innovations such as state-of-the-art brain injury scanners, premature baby units and cancer detection and diagnosis systems. The possibilities are endless.
## Smarter homes
Everything is also getting smarter at home. From the cars we drive to and from home, to the heating systems, gadgets, appliances and even the carpet!
The evolution from basic to smart is especially noticeable with cars. Initially the Model T Ford was black, stick shift, a few buttons and no seat belt. Today we have cars with dashboards that resemble an aircraft cockpit, with cameras and sensors for easy parking, alerting the driver if he or she gets too close to the kerb or another car. Some cars will parallel park themselves and brake automatically. Others will sync with traffic information and redirect you to a better route to avoid traffic black spots or an accident. Sensors on the engine will monitor how well you are driving, which will in turn potentially lower (or raise) your insurance and dynamically adjust your service intervals based on your driving style.
There are smart thermostats that monitor the home and only heat the areas that are being used. The temperature of your home can be changed while you are still at work so that when you arrive on a winter's evening the house is cosy. This ability to monitor and dynamically alter temperature can save energy and money. Obviously solving the energy crisis is not just about finding new energy sources such as wind and solar but also about saving the energy we have and using it more efficiently.
Smart TVs use face recognition to make sure your children don't ever watch anything unsuitable for their age and smart carpets can alert you should your elderly parent not make their usual morning coffee.
Considering all the toys, gadgets and smart appliances there are now more machines connected to the Internet than people. And all those smart things are gathering data and communicating with each other.
## Smarter love
Even something as personal and magical as falling in love is getting smarter. Everyone hopes to find their soul mate and yet, for many the search is far from straightforward. Online dating site eHarmony matches people based on twenty-nine different variables such as personality traits, behaviours, beliefs, values and social skills. Each person who joins eHarmony completes a comprehensive profile questionnaire, which provides the data for the analytics model to find potential matches.
US digital specialist, Amy Webb, even took the online data algorithms one step further. After one particularly terrible first date where her 'Prince Charming' ordered the most expensive items from the menu, enjoyed them and did a runner after excusing himself for the bathroom, Webb created her own personal scoring system based on what was important to her in a potential life partner. In addition she analysed other profiles to see what attracted attention, tested changes to her own profile to see what made a difference to the number and quality of enquires and would only agree to go on a date with someone if he scored above a certain number. And it worked... Amy Webb is now happily married and the couple have a daughter.3
## Smarter parenting
The complex art of parenting is also getting smarter. To identify and reduce potential pre- and postnatal risks, many babies around the world are being constantly monitored across a myriad of metrics and data points including heart rate and respiration. These vital measures are able to predict infections 24 hours before the baby shows any visible symptoms and can allow for early, often life-saving intervention.
Once your baby has arrived safely he or she can also sleep on a mattress full of sensors that monitor breathing patterns and heart rate and alerts parents if anything is wrong. Just imagine how many tragic cot deaths could be avoided with this smart technology. We can even buy digital diapers which will send a tweet to our smart phone when our baby needs changing! Obviously a good parent doesn't really need a tweet to tell them this information but the latest generation of these diapers automatically analyses the urine and alerts the parent of an increased sodium level, possible dehydration, as well as the onset of any infections – and all this even before any physical symptoms appear.
The marriage of data and technology is radically changing our world and making it smarter. And business must become smarter too.
Going back to the fishing analogy for a moment... When fishing emerged as an industry, the competition was sufficiently low and the stocks of fish sufficiently high that the fisherman didn't need to be in an exact location to enjoy a prosperous day at sea. Their experience, equipment and the number of fish in the oceans meant they would be successful unless they hit particularly bad weather. Today, with intense competition and finite fish stocks that need to be responsibly managed, fishermen have had to evolve and become smarter. And the same is true for all businesses in all sectors.
Today the really successful companies understand where their customers are and, perhaps more importantly, what they are doing and where they are going. They know what is happening as it's happening and they allow that information to guide their strategy and inform their decision-making.
Companies that won't embrace the SMART revolution will be left behind.
## Notes
1 BBC Two (2013) Horizon Monitor Me narrated by Dr Kevin Fong (2013). 2 Palmer, S., White, E., Romanski, P., Benedict, K. and Gardner, D. (2014) Integrating Consumer Wearable Health Devices Will Drive Healthcare Big Data Adoption, Says ABI Research. <http://bigdata.ulitzer.com/node/3058905> 3 <http://www.ted.com/talks/amy_webb_how_i_hacked_online_dating.html>
# 1
SMARTER BUSINESS
Big Data is at the heart of the smart revolution. The basic idea behind the phrase 'Big Data' is that everything we do is increasingly leaving a digital trace (or data), which we (and others) can use and analyse to become smarter. The driving forces in this brave new world are access to ever-increasing volumes of data and our ever-increasing technological capability to mine that data for commercial insights.
There is little doubt that Big Data is changing the world. It is already completely transforming the way we live, find love, cure cancer, conduct science, improve performance, run cities and countries and operate business. As a result there is a huge amount of hype and fuss over Big Data. Everyone is discussing it. It is THE hot topic discussed in every boardroom, every business publication from _The Economist_ to _Fortune_ to the _Harvard Business Review_. Big Data is even making its way into mainstream media.
But despite the noise around Big Data most people still don't really understand it and very few people know what to do about it. Personally, I don't like the term because it's too simplistic and potentially misleading. Granted, we are now tracking and storing data on everything so we potentially do have access to large volumes of data – hence the term Big Data. But the real value is not in the large volumes of data but what we can now do with it. It is not the amount of data that is making the difference but our ability to analyse vast and complex data sets beyond anything we could ever do before. Innovations such as cloud computing combined with improved network speed as well as creative techniques to analyse data have resulted in a new ability to turn vast amounts of complex data into value. What's more, the analysis can now be performed without the need to purchase or build large supercomputers. This means that any business, government body, or indeed anyone can now use Big Data to improve their decision-making.
Especially powerful is our ability to analyse so called 'unstructured data' (more on this in Chapter 3). Basically, unstructured data is the data we can't easily store and index in traditional formats or databases and includes email conversations, social media posts, video content, photos, voice recordings, sounds, etc. Combining this messy and complex data with other more traditional data is where a lot of the value lies. Many companies are starting to use Big Data analytics to complement their traditional data analysis in order to get richer and improved insights and make smarter decisions.
In effect what Big Data should really stand for is SMART Data and whilst I think the term Big Data will disappear in time, the increasing production and use of SMART Data is definitely here to stay.
## Who is using Big Data?
The big players in the space, including Amazon, Google, Walmart, and Facebook, are already making a splash. Walmart, for example, handles more than a million customer transactions each hour and imports those into databases estimated to contain more than 2.5 petabytes of data.1 The company is now able to combine data from a variety of sources including customers' past purchases and their mobile phone location data, Walmart internal stock control records, social media and information from external sources such as the weather, and initiate tailored sales promotions. For example, if you have bought any BBQ-related goods from Walmart, happen to be within a 3 mile radius of a Walmart store that has the BBQ cleaner in stock, and the weather is sunny, you might receive a voucher for money off a BBQ cleaner delivered to your smart phone!
In another example a client of mine, a leading telecom company, is using Big Data analytics to predict customer satisfaction and potential customer churn. Based on phone and text patterns as well as social media analytics, the company was able to classify customers into different categories. The analytics showed that people in one specific customer category were much more likely to cancel their contract and move to a competitor. This extremely useful information now helps the company closely monitor the satisfaction levels of these customers and prioritize actions that will prevent them from leaving and keep them happy.
Even mid-tier cars today have about 40 microprocessors that measure performance. These electronics usually account for about one-third of the cost of a new car. Of course, all this data that is being generated, collected and analysed by the car manufacturers offer them significant competitive advantages. One car maker working with an external analytics company noticed that a sensor in the fuel tank made by a German supplier was not working well at all. The manufacturer could have told the supplier and asked them to fix it but then the improvement would have been passed on to other car manufacturers that use that supplier. So instead the manufacturer invented a software patch that fixed the issue, received a patent on the fix and sold the patent to the supplier.2
Big Data is changing the very nature of business, from manufacturing to healthcare to retail to agriculture and beyond. The rate that data is and can be collected on every conceivable activity means that there are increasing opportunities to fine-tune procedures and operations to squeeze out every last drop of efficiency.
## How companies are using Big Data
Different industries have responded to the call in different ways. Retail and sales are seeking to collect as much information about their customers' lives as possible so as to fulfil their changing needs more effectively. Manufacturing are seeking to streamline operations. Equipment calibration settings can be recorded and refined, and product storage environments monitored to determine the optimum conditions that lead to minimum spoilage and waste.
For global companies this can mean collecting and analysing data from plants across the world, allowing minor variances to be studied and their results understood.
In 2013, for example, pharmaceutical giants Merck used analysis to dramatically cut the amount of waste caused by variance in manufacturing environment conditions. It took three months and involved 15 billion calculations on individual production data from 5.5 million vaccine batches. This allowed them to discover the optimum conditions during the fermentation process, and should greatly increase their yield, once the FDA has approved the proposed changes to the manufacturing process.
In the automotive industry a 2014 report by the Centre for Automotive Research stated that advances made possible through advanced IT solutions and Big Data represented 'an engine of innovation'. The report highlighted the growing complexity of cars and the industry as the biggest challenge faced by automotive manufacturers.
The efficiency of every machine – and human – involved in the manufacturing process can be recorded so companies know what is working, and can make improvements where they are needed.
And in agriculture, data analysis is helping the industry meet the challenge of increasing the world's food production by 60%, as forecasters have said will be necessary by 2050 due to the growing population. Tractor and agricultural machinery manufacturer, John Deere, already fits sensors to its machinery. The data that is available to the farmers via its myjohndeere.com and Farmsight services helps them to establish optimum conditions for their crops. Plus the data is also used by John Deere to forecast demand for spare parts.
Of course, in business once a product has been grown or manufactured it needs to be sold and distributed. The petabytes of customer data, including you and me, already gathered by big retailers tells them who will want to buy what, where and when. Amazon, for example, uses its S3 system to keep track of millions of stock items across dozens of warehouses and distribution centres scattered around the globe. Operatives can track deliveries in real-time to see what is where, and where it should be going.
At the point of sale, retailers can use data to determine where stock should be displayed, which stores will sell most of which particular product and track customer movements around stores. Loyalty cards are not new but ever more sophisticated analysis of customer habits will lead to an increase with which retailers can predict what you will buy. This has advanced to the point where Amazon believes it will soon be able to predict what you will buy accurately enough to despatch it toward you before you have even bought it!
The connectivity that is now possible is also changing business. In 2014 Cisco announced a $150 million fund for start-ups working on improving integration between the virtual and physical world. For a business, the ability to have its production, stock control, distribution and security systems all connected and talking to each other will mean greater efficiency and less waste. GE refers to this convergence of data and machinery as the 'Industrial Internet', and claims it can save global industry £150 billion in wastage.
Every area of industry is learning to reap the benefits of Big Data analysis, and it looks certain that finding innovative methods of gathering, recording and analysing data is going to play a big part of business in the foreseeable future.
Even something as subjective and 'human' as Human Resources is being transformed by Big Data and analytics. Finding and keeping the right people is a major bugbear for most businesses. Talent management is fraught with challenges and the cost of failed management and leadership is enormous. It is estimated that the average cost of executive failure is $2.7 million.3 Published estimates into the extent of poor leadership range from 33%4 to 67%.5 In other words between one- and two-thirds of all current leaders _will_ fail in their role.
But it's not just a financial cost. Unsuccessful executive appointments alone incur significant hidden costs, which can include lost opportunities, poor public relations, brand damage, poor productivity and employee disengagement and alienation. The impact of poor leadership on employee morale can be severe: 40 per cent of American workers classified their jobs as stressful and 75 per cent of working adults said the most stressful part of their job was their immediate supervisor.6
Getting the wrong person in any job can be a disaster. Get the wrong executive or leader and it can be catastrophic.
Considering that employees are the greatest asset of a business and, as the statistics confirm, potentially its greatest liability, it's easy to see how companies are getting excited by Big Data solutions such as Evolv.
Evolv is a software tool which helps assess and understand employees and candidates by crunching half a billion data points across 18 industries in 13 different countries on everything from gas prices, unemployment rates and social media use, to how long a person takes to travel to work, or to how often they speak to their managers. Although data collection methods include the controversial 'smart badges' that monitor employee movements and track which employees interact with each other, Evolv clients such as Bank of America are impressed.
Bank of America have reportedly improved performance metrics by 23% and decreased stress levels (measured by analysing workers' voices) by 19%, simply by allowing more staff to take their breaks together.7
The software is being used to predict a range of things including how long an employee is likely to stay in his or her job. Evolv has also gleaned some remarkable and unexpected insights such as the fact that in some careers, such as call centre work, employees with criminal records perform better than those without! Or the fact that employees who change the default browser on their computer to a nonstandard browser such as Firefox or Chrome perform better across the board than those who use a standard browser such as Internet Explorer and Safari.8 (Of course now this is public knowledge people could 'game' the predictor and change their default browser prior to interviews that will render the predictor useless.)
Although this type of Big Data analytics is currently focused on customer-facing roles it's only a matter of time before it reaches the upper echelons of management. Certainly improving the performance of top executives has a 'disproportionate effect on the company' so Big Data solutions are certain to be considered. According to the Economist Intelligence Unit more than half of HR departments have already reported an increase in data analytics since 2010.
## Don't panic!
The challenge of course is that when business leaders read stories like these or hear about the cool – and a little scary – things that Big Data Gods like Google, Amazon and Facebook are doing, they panic!
Most business leaders know about Big Data – they'd have to be living under a rock not to. They understand its inherent promise and they may even be fully aware of the fact that their business is data rich. But most business leaders have no idea what to do with it! We have been told for years that we live in the Information Age; we are reminded of the importance of information, knowledge and the role of knowledge workers. We know that we need to find a way to access and use the information we already have and manage the explosion of information we could have, or are being told we should have, moving into the future. Information is gathering momentum and pace, it's growing exponentially and yet our research suggested that less than 20 per cent of the data companies currently hold is used to inform decision-making. And this 20 per cent only took traditional structured KPI or financial data into account. If that is true of the structured data which is relatively easy to extract, insight from the unstructured data represents a rich untapped vein of information gold that is currently largely ignored.
Of course this escalation of data and endless information possibilities poses its own set of problems. If we are already drowning in data that we don't use then what on earth are we supposed to do with the rest?
Some stand on the sidelines feeling the pressure of inaction growing with every article they read about the Big Data revolution. The brave (or crazy) business leaders decide to dive in and work out what they can get access to and how they can use it but inevitably they get completely lost and end up drowning in their own information, unable to convert it into insight and meaning. Unfortunately, in this case the result of either action or inaction is the same – overwhelm and confusion!
This book is designed to help you change that outcome.
## Focus to reap the rewards
Big Data offers business an unparalleled opportunity to extract insight into the behaviour of their customer that can in turn transform business results. BUT just because we can measure, monitor and access everything doesn't mean we should. It is much too easy to get bamboozled by the proliferation of smart technology and endless possibilities that send business down resource-sapping rabbit holes without any definable or useful output. The danger therefore is that we get lost in a sea of data that delivers no value whatsoever.
So on one hand Big Data is changing the world because we now have so much more data and new data formats. But on the other nothing much has changed because we are still seeking to use data and information to inform corporate decision-making. The only real difference is that we now have new data formats that we can use and new technology to actually analyse that data and leverage it.
As business leaders we need to understand that lack of data is not the issue. Most businesses have more than enough data to use constructively; we just don't know how to use it. The reality is that most businesses are already data rich, but insight poor. It may be true that companies like Amazon, Google and Facebook enjoy a considerable competitive advantage because of the data they now have access to but they also have vast budgets and teams of data scientists whose only job is to analyse that data. For most businesses that is not possible, realistic or necessary. There is probably more than enough data in your business right now for you to tap into the power of Big Data without stellar tech or eye-watering budgets. And even if your business hasn't kept very good records or doesn't hold a huge amount of existing data there is definitely enough external sources to harness the power of Big Data in your business.
So essentially it doesn't matter whether you already have access to unfathomable amounts of information or your data collection systems have been a little sketchy up until now, Big Data can revolutionize your business – but only if we focus on SMART Data not Big Data. In order to do that we need a practical framework that can help us to wrestle the Big Data monster so that we can harness it in order to gain new insights that will guide the business into the future.
We need a way to navigate the oceans of data to find the pockets of meaning. Like the modern fisherman we need a sophisticated, but practical way of working out what customers we are trying to catch, finding out what we need to know to locate those customers, predict their behaviour and deliver bottom line results.
This book provides that urgently needed navigation system (see Figure 1.1) that will allow you to create a SMART business and harness the awesome power of Big Data regardless of your size or budget.
**Figure 1.1** The SMART Model
The SMART Model will mirror the structure of the book. Each chapter will unpack each part of the model and provide a practical structure that you can use to take advantage of Big Data in your business.
In order to cut through the chaos, confusion and sheer volume of data that can or does exist we must therefore ' **S** tart with strategy'. Instead of starting with the data, start with your business objectives and what you are specifically trying to achieve. This will automatically point you toward questions that you need to answer which will immediately narrow data requirements into manageable areas.
Once you know what you are trying to achieve you need to work out how you could access that information so you can ' **M** easure metrics and data'. Once you know what type of data is available and have accessed that data, you need to ' **A** pply analytics' to extract useful insights from the data that can help you to answer your strategic questions. Of course, the insights alone are useless unless you ' **R** eport results'. These three stages of SMART business are underpinned by technology. Technology will help you to collect the data that you need to measure, it will facilitate analytics in ways that you have probably never considered before and it will allow you to convert the insights into data visualizations that can be easily and quickly understood and acted on.
When you approach data (big and small) and analytics from this narrower more focused and practical perspective you can get rid of the stress and confusion surrounding Big Data, reap the considerably rewards and ' **T** ransform your business'.
## Notes
1 SAS Whitepaper (2012) Big Data Meets Big Data Analytics: Three Key Technologies for Extracting Real-Time Business Value from the Big Data That Threatens to Overwhelm Traditional Computing Architectures. 2 Mayer-Schonberger, V. and Cukier, K. (2013) _Big Data: A Revolution That Will Transform How We Live, Work and Think._ London: John Murray Publishers. 3 Smart, B. D. (1999) _Topgrading_. Upper Saddle River, NJ: Prentice-Hall. 4 Sorcher, M. (1985) _Predicting Executives Success._ New York: Wiley. 5 Hogan, R., and Hogan, J. (2001) Assessing leadership: A view of the dark side. _International Journal of Selection and Assessment_ , _9_ , 40-51. 6 Off the Rails: Avoiding the High Cost of Failed Leadership 7 Kuchler, H. (2014) Data pioneers watching us work. _Financial Times_. <http://www.ft.com/cms/s/2/d56004b0-9581-11e3-9fd6-00144feab7de.html#axzz2tdOLCswb> 8 Javers, E. (2014) Inside the wacky world of weird data: What's getting crunched'. CNBC. <http://www.cnbc.com/id/101410448>
# 2
S = START WITH STRATEGY
Smarter business starts with strategy. And that's true whether you're a Big Data giant like Amazon or Google or a small family run corner shop.
It is easy to get lost and overwhelmed by the science of data and analytics. Considering that was true _before_ Big Data it is exponentially so today. Business leaders all over the world are bamboozled. They read how Big Data giants never throw away any data, how everything in these data rich companies is captured and analysed because it's valuable and potentially offers unique and powerful insights for business development. Not even errors are discarded...Take misspelt names and search queries – surely that's data that can be deleted. Not to Google it's not. Instead of ignoring it they used it to compile the world's best spell checker!1
For most business leaders the very idea of collecting and storing _everything_ is genuinely terrifying. Not least because they already have a mountain of archive material that is lying in dusty folders in the basement never mind having to deal with all the new stuff that is generated every day! Even a moment's contemplation of the issues a business potentially faces in the Big Data world is exhausting and stressful... What constitutes everything, what sort of format, where will it be stored, how will it be stored, who will use it, who will own it, how will we pay for it, what will we do with it, where do we even start?
The thing is, for Big Data giants like Tesco, Walmart or Amazon every tiny piece of data may very well be valuable to some extent or another. But only because those businesses have the analytical expertise, money and technological capability to invest in sufficient storage capacity and mine those massive data sets to deliver insights. Plus, they are at the cutting edge of this new world and so often attract the very best talent.
But, at a guess, 99.9% of all the companies in the world will never be in that position. By far the majority of businesses will never have the time, money, expertise and/or inclination to crunch the data in the way that these giants can. But that doesn't mean Big Data is something you can ignore.
Besides there is still a huge amount of Big Data that smaller companies can use that has never been made available before. For example, if you run a small grocery shop you can now download the weather data from Met Office service and use it to make predictions about what you need to stock. So even smaller companies can tap into Big Data and use it to enhance their offering.
The really good news is that what you currently have access to or don't have access to doesn't really matter as evidenced by Figure 2.1.
**Figure 2.1** All roads lead to Start with Strategy
Whether your business has loads of analysis-ready data or doesn't have any data doesn't actually matter that much initially. It doesn't alter the validity of starting with strategy.
The only exception to this rule is if you already have a great deal of digitized data at your fingertips. In that case it may make sense to allocate a small part of the data budget and resources to data discovery.
If a business has access to a huge amount of data that can be mined and analysed then it's definitely worth spending 10% of their analytics efforts on data discovery. Data discovery is a process of looking at data from the other direction. When you start with strategy you work out what you need to know and therefore what data you need to collect to provide those answers. In data discovery you just look at the data with no questions or agenda to see what the data tells you about your business. This data discovery process can be a useful addition to the more tailored approach outlined in this book and can potentially throw up all sorts of data gems.
Facebook, for example, looked at all the data they had via millions of status updates and were able to decipher a pattern around relationships from that chaos. So much so that Facebook can now predict when you will change your status from 'Single' to 'In a relationship' and presumably vice versa. At the moment this is just a quirky insight but there may come a day when Facebook could license that data to companies who make products that could be particularly useful to someone who is newly in a relationship (couples holiday packages) or newly out of one (tissues and Ben & Jerry's ice cream!). Facebook invest in data discovery because they can – they have a huge amount of data not to mention the time, talent, tech and money to make it worthwhile but it's absolutely not the place for most businesses to start.
In the future these types of data discovery insight could certainly revolutionize business and could even end up changing your business model. But it's always just an addition to the SMART business approach and not a substitute for it.
## Small is beautiful in a Big Data world
In order to reap the benefits of Big Data you don't have to collect everything and produce the biggest, most complex database in the world. As I'll explain in this chapter your aim is actually the opposite – to get really clear about what data you need, what data you can and will use and build the smallest, most straightforward database in the world!
Think about data like your stuff and possessions at home. If you've lived in your home for more than five years then the chances are there is an accumulation of stuff – some of which you don't even remember having until you open a closet door and it lands on your head. Even when we decide to de-clutter, it can be a nightmare because we think, 'Hmm, I better not throw that away because it might come in handy in the future'. I've got a friend, for example, who has amassed a whole collection of small glass ramekins from bought deserts like crème brulee. Her husband keeps trying to throw them out but she keeps rescuing them because they might be useful. Sometimes they are, but there isn't really any need for 20 of them!
If you have lived in your home for 20 years the accumulation of this type of stuff from unwise purchases, hand-me-downs, unwanted presents or old outfits can be overwhelming. There are even TV shows about extreme hoarders who have so much stuff (mostly rubbish) that they can't get into parts of their home anymore! The sheer volume of stuff we accumulate is often only noticeable once we decide to move home or downsize. The challenge ahead can seem impossible. Often it's only when we finally bite the bullet and get into the loft to really look at the stuff do we realize it's completely out of date and obsolete!
The same is true of data. In most cases data has a life span. This is really important to appreciate because Big Data is really turning up the heat for businesses that are stressed about what to do with all their data and the stress is building exponentially with each passing year, as they perceive themselves to be getting buried under increasingly greater amounts of additional data. This isn't actually true because for most businesses customer data that is, say, older than five years is not going to be very useful anyway. The reason the Big Data giants are making Big Data work so well for them is because they have access to huge amounts of current customer data that helps them to profile their customers and improve performance. For the data giants with the tech and the talent to trawl through the older data it may produce some interesting insights or purchasing trends but it's not that relevant for most businesses. Consider it a 'fun to have' aspiration for the future not a mission critical objective of the present.
So rather than allowing this additional unrelenting accumulation of data to further amplify your stress levels just forget about data that is more than 5 years old and more importantly step back and ask yourself what really matters the most. And use those insights to direct your action and your data requirements.
Starting with strategy allows you to develop strategies that help you identify what data you really need and very often that will mean a combination of traditional 'small' data or existing data and new data formats, new faster moving data and Big Data.
For example researchers at Harvard and Northeastern University demonstrated that the Google Flu Trends project of 2009 had overestimated the number of cases for four years running. This project sought to identify flu outbreaks from Big Data-based Google search queries. And the logic is sound after all what do most people do when they start to feel unwell... they Google it! They will look up the symptoms usually before they even make an appointment with their doctor. Most people would only ever Google 'Flu' if they were feeling bad or a loved one was feeling bad. That said not everyone who was feeling bad and Googled 'flu' would have had flu so the overestimate is hardly a surprise. Many have sought to discount the relevance of Big Data using examples such as these. But even the researchers who identified the over-estimation acknowledged that combining the Big Data-driven Google Flu Trends with the traditional data from the Centres for Disease Control (CDC) improved the overall forecast.2
Big Data alone is not infallible, or some magic bullet, but when used strategically and combined with existing data sources it can offer up some extremely useful insights.
If you want to use Big Data to improve your business and carve out a competitive advantage then you need to understand what your big strategic objectives are. Once you've defined them you need to isolate the big questions you want to find answers to so you can find out what data – small or big – will give you the answers to those questions. Starting with strategy, therefore, allows you to identify your strategic information needs using the SMART strategy board.
## The SMART strategy board
In order to help companies get really clear on their data needs I've developed the SMART strategy board (see Figure 2.2). This template has already helped many of my clients to navigate the choppy waters of big (and small) data so they can reap the rewards without the stress.
**Figure 2.2** The SMART Strategy Board
The aim of the SMART strategy board is to help you step back and ask what are your strategic information needs. You can't identify your information needs if you are not clear about your strategy. Remember the value of data is not the data itself – it's what you do with the data. For data to be useful you first need to know what data you need, otherwise you just get tempted to know everything and that's not a strategy, it's an act of desperation that is doomed to end in failure. Why go to all the time and trouble collecting data that you won't or can't use to deliver business insights? You must focus on the things that matter the most otherwise you'll drown in data. Data is a strategic asset but it's only valuable if it's used constructively and appropriately to deliver results.
This is why it's so important to start with strategy. If you are clear about what you are trying to achieve then you can think about the SMART questions to which you need answers. For example, if your strategy is to increase your customer base, SMART questions that you will need answers to might include, 'Who are currently our customers?', 'What are the demographics of our most valuable customers?' and 'What is the lifetime value of our customers?'. When you know the questions you need answered then it's much easier to identify the data you need to access in order to answer those key questions. Your data requirements, cost and stress levels are massively reduced when you move from 'collect everything just in case' to 'collect and measure x and y to answer question z'. Big Data goes from 'impossible for us' to 'absolutely possible for us'.
For example, I worked with a small fashion retail company that had no data other than their traditional sales data. They wanted to increase sales but had no SMART Data to draw on to help them achieve that goal. Together we worked out that the SMART questions they needed answers to included:
* How many people actually pass our shops?
* How many stop to look in the window and for how long?
* How many of them then come into the shop, and
* How many then buy?
What we did was install a small, discreet device into the shop windows that tracked mobile phone signals as people walked past the shop. Everyone, at least everyone passing these particular stores with a mobile phone on them (which nowadays is almost everyone), would be picked up by the sensor in the device and counted, thereby answering the first question. The sensors would also measure how many people stopped to look at the window and for how long, how many people then walked into the store, and sales data would record who actually bought something.
By combining the data from inexpensive, readily available sensors placed in the window with transaction data we were able to measure conversion ratio and test window displays and various offers to see which ones increased conversion rate. Not only did this fashion retailer massively increase sales by getting smart about the way they were combining small traditional data with untraditional Big Data but also they used the insights to make a significant saving by closing one of their stores. The sensors were able to finally tell them that the footfall reported by the market research company prior to opening in that location was wrong and the passing traffic was insufficient to justify keeping the store open.
### The pear tree metaphor
The SMART strategy board is really quite intuitive and follows the laws of nature. If you look at a pear tree, for example, the purpose of a pear tree is to grow pears. But there is a lot going on inside the pear tree that makes that outcome possible. The same is true of your business. The purpose panel in your SMART strategy board is the visible output or desired outcome of your business, in the same way that pears are the visible output of the pear tree.
The resources panel at the bottom represents the stabilizing elements of your business. Like the roots of a tree these are often hidden from view and yet they provide the nutrients or ingredients that make the outcome possible. In the same way that the roots of the pear trees draw nutrients and water from the soil and directs it to the right part of the tree at the right time, your tangible and intangible business resources are directed to the right place to deliver the outcome.
In the middle, between resources and outcome are the customer, finance and operations panels in your SMART strategy board, which represent the core of your business operations that make the purpose possible – much like the trunk of the pear tree. Your products and services didn't arrive by accident or magic, just as the pears didn't arrive on the tree by magic, they are the natural outcome of a visible and invisible network of connections and interconnections – just like your product or service. Each of the main departments and functions within customer services, finance and operations represent the major branches of the pear tree and are crucial to the creation of the end product. Within those departments and areas there are core competencies in order to:
* **Develop Products & Services:** Market research, design and develop new products and services.
* **Generate Demand:** Analyse market information, develop and manage customer relationships, monitor customer trends, marketing and branding.
* **Fulfil Demand:** Source goods and services, produce goods and services, manage supply chains, manage business processes, manage delivery network, provide services and support.
* **Regulatory & Social:** Manage health and safety, corporate social responsibility, reduce environmental impact.
These core areas and core competencies give your business strength and connects the resources to the departments that make your purpose possible in the same way that the trunk gives the tree strength and connects the roots to the branches that make the pears possible.
Your business is a complete interdependent unit creating your product or service in the same way a pear tree is a complete interdependent unit creating pears. On its own the trunk couldn't create pears, nor could the branches in isolation of the trunk or the roots. Pears are only possible when the roots connect to the trunk that connects to the branches to create pears. In the same way your business will not produce your product or service efficiently and profitably unless your roots, trunk and branches are connected and integrated and everyone knows your strategy and how each part of the business connects and contributes to those objectives.
That said, to be successful you must understand more than just your product and service – which brings us to the competition and risk panel. This is the part of the strategic process that is often missed from traditional strategy maps or balanced scorecards. And yet anyone in business knows that there are risks and challenges that come from outside and have little to do with the business. Continuing the pear tree analogy...the competition and risk panel is like the weather or an unexpected disease. The pear tree has no control over the weather, or whether nearby trees have become infected with a new virus or through some miracle fertilizer are now producing pears that are significantly bigger and juicier. In business too there are internal and external risks that can impact on what is achieved and when.
### SMART strategy board in action
Before you even think about the data you could or should collect you need to work out what SMART questions you need to answer by considering each of the various panels from the SMART Strategy Board. Each panel provides a blueprint that will trigger four or five SMART questions per panel. These questions will then form the basis of your Big Data/analytics strategy. There are six panels in the Smart Strategy Board:
1. The Purpose Panel
2. The Customer Panel
3. The Finance Panel
4. The Operations Panel
5. The Resource Panel
6. The Competition and Risk Panel.
#### _1 The purpose panel_
The purpose panel sets the scene and provides an inspiring framework or overall context regarding your corporate strategy or what your business is aiming for or seeking to achieve. Please note that companies don't usually develop SMART questions for this panel; its role is more to set overall context and direction.
This can best be achieved by detailing your mission and vision statement – each doing a distinctly different job.
Your mission statement is a clear, concise statement of purpose setting out why your organization exists. A mission statement should communicate your intentions powerfully, providing a road map to guide action and decision-making as you strive toward the strategic goal or objective. It is primarily an internal document designed to motivate stakeholders and define the key measures of organizational success. As such it should include your target audience, what products or services you provide to that audience and what makes your product or service unique.
Your vision statement also defines purpose, but from the perspective of ambition or what you want your business to be in the future. As an inspiring picture of your aspirations the vision statement gives direction to internal and external stakeholders. Internal employees can be inspired to give their best by a strong and meaningful vision statement; customers can end up choosing you over the competition based on your vision statement and shareholders can be encouraged to invest. The vision statement gives direction about what the business values and therefore what behaviour it adheres to and expects from its stakeholders.
#### _2 The customer panel_
The customer panel prompts you to consider how much you currently know about the customers your strategy is targeting and what you may need to find out in order to deliver on your strategic objective. There are two parts to consider – target market and value proposition.
Considering your strategy (including your mission and vision) what is your target market? Are you planning to appeal to a particular segment – if so why and what do you know about that segment? Are you targeting a particular geographic region or specific demographic? If so, what do you need to know about those potential customers to improve the likelihood of success?
The second part of the customer panel encourages you to clarify your value proposition or what you are going to offer your target market. Why are these customers going to buy from you? Do you think they will value your quality, price, innovation, service, or something else? What will contribute to customer satisfaction and loyalty? Do you know?
Thinking about your customers in relation to your strategy will trigger SMART customer questions – or questions you need answers to (see Figure 2.3). These questions will then shed light on what type of data you will need to collect in order to answer those questions.
**Figure 2.3** SMART Customer Questions: Deriving Key Performance Questions from your SMART Strategy Board
#### _3 The finance panel_
The finance panel prompts you to consider how much you currently know about the financial implications of your strategy and what you may still need to find out.
How does your strategy generate money? What is the business model and are you confident it is accurate? What assumptions have you made about the revenue, profit and growth of your business as you implement the strategy? How much will it cost to produce and deliver your product and services? Do you know for sure or is it a guess? Obviously getting the customer panel right will help to get the finance panel right and drive revenue growth, profit and shareholder returns.
Thinking about your financial position in relation to your strategy will trigger SMART finance questions – or questions you need answers to (see Figure 2.4). These questions will then shed light on what type of data you will need to collect in order to answer those questions.
**Figure 2.4** SMART Finance Questions: Deriving SMART Questions from your SMART Strategy Board
#### _4 The operations panel_
The operations panel prompts you to consider what you actually need to do internally to deliver your strategy and what you may need to find out. Like the customer panel there are two components of the operations panel – partners and core competencies.
First you need to consider which suppliers, distributers, partners or other intermediaries are crucial in delivering your strategy. Do you currently work with these people or will you need to create the relationships? If the relationships already exist, how healthy are they right now?
In addition you need to consider what core competencies you need to excel in if you are going to execute your chosen strategy. Are there any gaps? If so, how easy is it going to be to fill those gaps? Do you know, or are you making assumptions? What processes are going to need to be perfected if you are to deliver what your target market wants?
Once you are clear on the individual elements highlighted in the customer, finance and operations panels you need to consider how they impact on each other. Remember, the customer, finance and operations are the core of the business and they need to work together.
Thinking about your operations in relation to your strategy and how it dovetails with your customers and finance will trigger SMART operations questions – or questions you need answers to (see Figure 2.5). These questions will then shed light on what type of data you will need to collect in order to answer those questions.
**Figure 2.5** SMART Operations Questions: Deriving SMART Questions from your SMART Strategy Board
#### _5 The resource panel_
The resource panel prompts you to consider what resources you need in order to deliver your strategy and what you may need to find out. There are four components of the resources panel: IT Systems and data; infrastructure; people and talent and cultures; and values and leadership.
Taking each in turn you need to consider: What IT systems and data sources are you going to need to deliver your strategy? What infrastructure – property, machinery or plant are you going to need? What are your people and talent requirements? Do you have the right people and if not, can you find them? Will you need to train your current staff or recruit new people? And finally, what are the key culture and leadership deliverables that will enable this strategy?
Thinking about the various resources you will need access to in relation to your strategy will trigger SMART resources questions – or questions you need answers to (see Figure 2.6). Again, these questions will then shed light on what type of data you will need to collect in order to answer those questions.
**Figure 2.6** SMART Resource Questions: Deriving SMART Questions from your SMART Strategy Board
#### _6 The competition and risk panel_
The competition and risk panel prompts you to consider what competition you will be up against as you seek to deliver your strategy and what risks you may face along the way.
This competition and risk perspective is the perspective that is most often missing from strategy maps and yet it poses a serious potential threat to successful strategic execution. Considering what you are seeking to achieve, who is your main competition and why? What is potentially threatening your success? Are there any specific market, customer, competition or regulatory risks that could derail your strategy? What are the operational, financial or talent risks you face?
Thinking about your competition and the various risks you could face will trigger SMART competition and risk questions – or questions you need answers to (see Figure 2.7). These questions will then shed light on what type of data you will need to collect in order to answer those questions.
**Figure 2.7** SMART Competition and Risk Questions: Deriving SMART Questions from your SMART Strategy Board
### Smart questions are the answer
Considering the furore that surrounds Big Data and analytics it is incredibly easy to get overwhelmed and intimidated. Every time you put on the TV or open a management journal there is another story of some awesome insight made possible by Big Data. But the reality is that most companies will never have the money, technological capability or talent to endlessly mine vast, messy datasets in the hope of unearthing a newsworthy nugget. And that's OK.
Focusing on SMART questions allows us to forget about Big Data and focus on SMART Data so that we work out exactly what we need to know and hone in on that.
Few businesses have the time, inclination or resources to collect endless amounts of data in order to answer questions they didn't need to ask or couldn't care less about answering. It's not productive or practical.
The truth is we are so mesmerized by the data that we've forgotten, that actually the question is much more important than the answer the data may, could or should provide. And you need to know what questions you need answers to _before_ you dive into the data – big or otherwise.
SMART questions allow you to articulate exactly what you need to know when it comes to each of your strategic objectives so that you can concentrate on what's going to be strategically important and discard the rest. These questions therefore help you identify your information needs so you can identify what Metrics and data (M in SMART) you need to collect to help you answer your SMART questions.
SMART questions help you and your executive team to:
* See the wood from the trees regarding what's important and what's not.
* Understand the relevance of the data sought because SMART questions indicate to everyone what your company's biggest concerns are.
* Open communication and guide discussion.
* Make better evidence-based decisions.
For each panel (except the purpose panel) in the SMART Strategy Board:
1. Identify a small number of SMART questions (usually two to five).
2. Engage key personnel from each 'panel' in the creation of the SMART questions to facilitate buy-in.
3. Where possible make your SMART questions clear and concise.
4. Use your SMART questions to guide your data needs so as to deliver relevant and meaningful information rather than being overwhelmed by the data.
## SMART analytics and Google
Really successful companies today are making decisions based on facts and data-driven insights, not opinion, gut feeling or even experience. Whether you have access to tons of data or not, if you start with strategy and identify the SMART questions you need answers to in order to deliver your outcomes then you will be on track to improve performance and harness the primary power of data.
Every manager now has the opportunity to use data to support their decision-making with actual facts. And one company that is brilliant at exploiting the analytic value chain is Google.
Google is one of the most successful companies in the world. It has 28,000 employees or 'Googlers' across 60 offices in over 30 countries. Unlike most HR departments Google HR's team states the objective that 'All people decisions at Google are based on data and analytics'.
Google knew that most HR decisions are based on opinion. 'I've just got a good feeling about that guy' gut feeling or an assumption of fit based on the unconscious bias of the person making the final decision usually play a bigger role in recruitment than any of us would care to admit.
By default HR has always been a data warehouse – who works in the company, how long they have been there, how many sick days have they taken, how much they make, when were they last promoted, how many people they manage, their qualifications, etc.
Like most companies Google also realized that the raw structured data alone wasn't enough. It was important to draw insight from that data so they applied key performance metrics. Metrics proved quite interesting because they provided even more information in terms of ratios, counts or trends of what was happening in the company over time. The challenge with metrics is that when the metric dashboard is distributed frequently enough to a sufficient number of people they become numb to dashboard. Google believes that metrics are important because they act as a starting point for understanding what's happening but they don't necessarily help to make decisions or drive action.
Google then analysed the metrics and data to identify relationships and correlations within the data, identify trends and special populations. And finally this led to insights which influence decision-makers and action. And those insights are then embedded into process, corporate policy or new initiatives so the benefit is baked into the business.
Project Oxygen is an example of this process of starting with opinion and ending up with action.
### Case Study: Project Oxygen
Project Oxygen was conducted by the People and Innovation Lab (PiLab), which is part of the Google people analytics group. PiLab is a group of social scientists, who would often team up with academic researchers to answer some of Google's longer-term questions or problems. They tackle challenges that are not necessarily immediate concerns for the business but could none the less improve performance and lead to breakthroughs.
PiLab would often look at issues that were important to productivity, performance and success, but not necessarily urgent. Their mission is 'To conduct innovative research that transforms our practice both within Google and beyond'.
One of the Google myths or opinions that emerged from the early days of the company was that managers didn't matter that much. As a tech company, the jobs with the highest status that everyone wanted were the tech jobs not the people management jobs. In fact Google co-founders Larry Page and Sergey Brin (both uber-computer science, engineering tech guys) were so sure that managers didn't matter that much to the business that they decided to get rid of them all and make everyone an individual contributor. It didn't work well and managers were brought back into the company. But the stigma remained that they really weren't doing that much and the managers were not nearly as valuable, or valued, as the tech guys. The opinion or assumption prevalent at this time was that there really was no proven impact about what the people managers brought to the table as opposed to the tech professionals.
So PiLab set out to establish whether the stigma was justified or not. And to do that they started with a question: 'Do managers actually make a positive impact at Google?'
The first thing they did was to look at data sources that already existed: performance reviews and employee surveys. Many businesses have looked at these types of data sources from two different perspectives: from the bottom up (employee survey) and from the top down (performance review). Plotting the results on a graph, all the managers appeared to be pretty tightly clustered and it looked like they were doing well. But the graph didn't answer the question, 'Do managers actually make a positive impact at Google?'
In order to answer the question they needed to look more closely at the data and cut it into sections, looking specifically at the top quartile (best managers) and the bottom quartile (worst managers). Further analysis was made on how the best and worst managers performed, based on their teams in terms of team productivity, how happy their employees were, how likely their employees were to stay with the company, etc. And the results were astonishing. Even though most of the managers appeared to be tightly clustered together on the graph further investigation, including regression analysis highlighted statistically significant differences between the best and worst managers in the cluster.
This analysis clearly answered the question. Managers do matter and they _can_ make a positive impact at Google.
But the information alone wasn't really going to change anything. So they came up with new questions they wanted to answer...'What makes a great manager at Google?'
If the team at PiLab could isolate the issues that made the difference those insights could then be used to help struggling managers to improve and direct future recruitment. They had no idea how the good managers became good or what made them so effective compared with others.
To answer the new question PiLab started two different qualitative studies. The first was to introduce an award called the Great Manager Award where direct reports could nominate a manager by writing up examples of behaviour that they felt qualified that manager for the award. That data was then coded and analysed to find out if there were common examples of great things that great managers did.
At the same time they introduced a double blind interview study where they interviewed managers from the top and bottom quartile, although neither the manager nor the interviewer knew which quartile the person being interviewed belonged to. Again the transcripts of the interviews were coded and analysed in an effort to understand what common things great managers were doing compared to the things the not-so-great managers were doing.
Based on this data they came up with eight behaviours that Google's best managers generally did and three pitfalls that the struggling managers tended to fall into.
In Google a great manager:
1. Is a good coach.
2. Empowers the team and does not micromanage.
3. Expresses interest/concern for team members' success and personal well-being.
4. Is productive and results orientated.
5. Is a good communicator – listens and shares information.
6. Helps with career development.
7. Has a clear vision/strategy for the team.
8. Has important technical skills that help him/her advise the team.
It's easy to look at this list of eight attributes and think they are obvious. And certainly if you asked most people to name attributes that characterize a great manager these eight would probably be among 20 or 30 behaviours mentioned. What the people analytics did was definitively identify what _eight_ behaviours make the biggest impact for managers at Google.
Pitfalls that might cause a manager to struggle is that he/she:
1. Had a tough transition (e.g. suddenly promoted or hired from outside with little training).
2. Lacks a consistent philosophy/approach to performance management and career development.
3. Spends too little time on managing and communicating.
What makes these insights so powerful at Google is that they took them even further to action and started measuring people on these attributes. Google implemented upward feedback surveys where every manager's direct reports rate their manager twice a year and the manager gets the results. He or she is then held to an expectation that they will take action to improve any areas of weakness – specifically around these eight positive traits and three pitfalls. Google has continued to run the Great Manager Award to recognize those that do these things particularly well, thus providing additional role models for other managers.
In addition Google has redesigned manager training programmes and created new communications plans to let people know what they are being judged on and what some of the best practices are at Google.
In this way Google went from the opinion: 'Managers don't impact performance' to using the data and metrics to prove that great managers had a statistically significant impact on team performance, employee engagement, employee churn and productivity. By extracting the insights from that analysis Google was then able to identify and articulate what made a great manager at Google and what caused less proficient managers to struggle. These insights were then embedded into Google culture through ongoing measurement of these factors, which acts as an early warning system to detect both great and struggling managers. For those that are struggling there is access to improved training and support as well as plenty of role models to learn from as identified and celebrated via the Great Manager Award.3
And all this was possible because they started with the right question; refining their question until they got to a practical, verifiable hypothesis that has improved management performance across Google.
When you start with a question or hypothesis and seek to find and analyse only the data that can directly answer that question, then you move away from the overwhelm of 'all data' and the panic that you are going to need to collect and analyse everything, to a much more manageable and sensible enquiry. That's the power of SMART questions.
A fact clearly not lost on Google's executive chairman, Eric Schmidt,4 who says: 'We run the company by questions, not by answers. So in the strategy process we've so far formulated 30 questions that we have to answer... You ask it as a question, rather than a pithy answer, and that stimulates conversation'.
According to the IDC Digital Universe Study in 2013 only 22% of the information in the digital universe was a candidate for analysis and only 5% was actually analysed. IDC predict that the useful percentage will grow to more than 35% by 2020 and more than 10% will be analysed to produce genuinely useful insights. It is therefore increasingly important that we learn to take advantage of new Big Data and analytics technologies as well as new data sources, types and forms of data and apply them to new parts of the business in a SMART way. This brave new world presents significant opportunities and significant threats. There is, as we will explore in the next chapter, a vast amount of valuable data in the digital universe, but it will take intelligence, determination and skill to find it and put it to use.5 If you start with strategy you make that possible.
Before launching your business into this brave new world start with strategy and articulate and clarify your purpose. Make sure you and your executive team understand what your business is seeking to achieve. Once you have this target identify four or five SMART questions for each of your SMART Strategy Board panels around customers, finance, operations, resources and competition and risk. Once you know what you need to know in these critical business areas then use those questions to guide what metrics and data you measure...
## Key points and call to action
* Smarter business always starts with strategy – regardless of size.
* Instead of starting with the big or small data you could or should access (which is a recipe for failure and overwhelm) start by working out what your business is seeking to achieve.
* Use the SMART Strategy Board to consider your strategic objectives for each key area of your business and therefore what information you need in order to answer your four or five SMART questions for each area.
* What is your customer strategy and what information do you need in order to help you achieve that strategy?
* What is your financial strategy and what information do you need in order to help you achieve that strategy?
* What is your operations strategy and what information do you need in order to help you achieve that strategy?
* What resources are you going to need in order to deliver on your customer, financial and operational objectives? And what information will you need in order to do that?
* What competition and risks will you face as you seek to deliver your strategy? And what information will you need in order to do that?
* The only exception to starting with strategy is if you already have a vast reservoir of digitized data. In this case use 10% of your budget on data discovery to see if the data throws up any unexpected correlations or business opportunities.
* However, remember that data has a shelf life. Spending huge amounts of time and money digitizing old customer records from 20 years ago is probably not the best use of your resources.
## Notes
1 Mayer-Schonberger V, Cukier K (2013) _Big Data: A Revolution That Will Transform How We Live, Work and Think._ London: John Murray Publishers. 2 The backlash against Big Data. (Apr 2014) _The Economist._ 3 Dekas, K. (2011) Strata Jumpstart: Kathryn Dekas, People analytics: Using data to drive HR strategy and action. You Tube <http://www.youtube.com/watch?v=l6ISTjupi5g> 4 Press, Gil (2012) Big Data News of the Week: Sexy and Social Data Scientists. _Forbes.com_ , 24. November 2012. 5 IDC The Digital Universe Study (April 2014) Sponsored by EMC2.
# 3
M = MEASURE METRICS AND DATA
In the furore about Big Data it's easy to forget that it's just data. There may be more data than ever before and there may be new forms of data but that data is still only really useful if we can use it to answer SMART questions.
However, the following might put the Big Data 'hype' into perspective. If you take all the data that was created in the world between the dawn of civilization until the year 2010, the same amount of data will soon be generated every minute. Take astronomy as an example: up until 2000 a great deal of data had been collected about the subject and yet when the Sloan Digital Sky Survey began in 2000 its telescope in New Mexico collected more data in its first few weeks of operation than had already been accumulated in the history of astronomy. In the decade that followed 140 terabytes of information was gathered. In case you're wondering how much data that is, it's equivalent to 700,000 movies which would take just under 160 years of continuous watching to see them all. When the successor to the New Mexico telescope comes online in 2016 it will be able to acquire 140 terabytes every five days!1 And the reason for this explosion of data is the datafication of the world and our ever-increasing ability to analyse that emerging data. And it is essentially this critical combination of factors that has created 'Big Data'.
The basic idea behind the phrase 'Big Data' is that everything we do in our lives is or soon will leave a digital trace (or data), which can be used and analysed.
The ability to harness the ever-expanding amounts of data is completely transforming our ability to understand the world and everything within it. And while business has been capturing and analysing data for years this global datification means that the rate at which we are generating new data is frightening. For example, there are 30 billion pieces of content uploaded to Facebook alone every day! The digital universe is doubling in size every two years. At that rate, by 2020 there will be nearly as many bits of information in the digital universe as there are stars in the physical universe.2
In addition, quantum leaps in computer storage and processing power has meant that for the first time we are able to analyse large, complex and messy data sets. Before looking at how to harness metrics and data we first take a look at the various types and forms of data, then discuss what Big Data really is as well as the privacy concerns around the collection of large data sets.
## Types of data
Big Data, data science and business analytics work with structured and unstructured data. But SMART business occurs when we combine existing data sets with unstructured or semi-structured data from both internal and external sources.
### Structured data
Structured data provides most of our current business insights but is often considered 'old hat' and a bit dull – especially in comparison to its rock star cousin, unstructured data – it is easy to ignore structured data. But that is a mistake as many Big Data insights are generated by combining structured and unstructured data.
Data that is located in a fixed field within a defined record or file is called structured data. This includes data contained in relational databases and spreadsheets.
Examples of structured data include:
* Point of sales data
* Financial data
* Customer data.
As the name would suggest structured data refers to data or information that has a predefined data model or is organized in a predetermined way.
A data model is a model of the types of business data that your business will record and how that data will be stored, processed and accessed. Within that data model the fields of data that you intend to capture need to be defined and any conventions set around how that data will be stored. For example, if you look at a standard customer database the fields that are defined will include name, address, contact telephone numbers, email address, etc. Within those fields conventions may also be set so, for example, the telephone number field will only accept numeric information. These conventions can also include drop down menus that limit the choices of the data that can be entered into a field, thus ensuring consistency of input. For example, a 'Title' field within a name structure may only give you certain options to choose from, such as Mr, Ms, Miss, Mrs, Dr, etc.
Structured data gives names to each field in a database and defines the relationships between the fields. As a result structured data is easy to input, easy to store and easy to analyse.
Up until relatively recently technology just didn't have the grunt to store, never mind analyse, anything other than structured data. Everything that didn't fit into the databases or spreadsheets was usually either discarded or stored on paper or microfiche in filing cabinets or storage facilities.
Structured data is often managed using Structured Query Language (SQL) – a programming language originally created by IBM in the 1970s for managing and querying data in relational database management systems. SQL represented a huge leap forward over paper-based data storage and analysis, but not everything in business will fit neatly into a predefined field.
### Unstructured and semi-structured data
Unstructured and semi-structured data are like the popular kids at school! Everyone is talking about them and they represent the sexy new frontier lauded by Big Data. It is estimated that 80% of business-relevant information originates in unstructured or semi-structured data.
It represents all the data that can't be so easily slotted into columns, rows and fields. It is usually text heavy, but may also contain data such as dates, numbers and facts or different types of data such as images. These inconsistencies make it difficult to analyse using traditional computer programs.
Examples of unstructured and semi-structured data include:
* Photos and graphic images
* Videos
* Websites
* Text files or documents such as email, PDF, blogs, social media posts, etc.
* PowerPoint presentations.
Semi-structured data is a cross between unstructured and structured. This is data that may have some structure that can be used for analysis but lacks the strict data model structure. In semi-structured data, tags or other types of markers are used to identify certain elements within the data, but the data doesn't have a rigid structure. For example, a Facebook post can be categorized by author, data, length and even sentiment but the content is generally unstructured. Another example is word processing software that includes metadata detailing the author's name, when it was created and amended but the content of the document is still unstructured.
### Internal data
Internal data accounts for everything your business currently has or could access.
This includes private or proprietary data that is collected and owned by the business where you control access.
Examples of internal data include:
* Customer feedback
* Sales data
* Employee or customer survey data
* CCTV video data
* Transactional data
* Customer record data
* Stock control data
* HR data.
Again, like structured data this isn't considered very sexy and everyone is almost fully focused on external data that they currently don't have. Again this is a mistake.
### External data
External data is the infinite array of information that exists outside your business.
External data is either public or private. Public data is data that anyone can obtain – either by collecting it for free, paying a third party for it or getting a third party to collect it for you. Private data is usually something you would need to source and pay for from another business or third party data supplier.
Examples of external data include:
* Weather data
* Government data such as census data
* Twitter data
* Social media profile data
* Google Trends or Google Maps.
A lot of the Big Data hype focuses on unstructured data and the allure and promise of external data, often at the expense or dismissal of internal or structured data.
It is really important to understand that no type of data is inherently better or more valuable than any other type. The key is to start with your strategy and establish your SMART questions so that those questions guide you to the best structured, unstructured, internal or external data to answer those questions and deliver the strategy.
But before we explore how to do that let's take a moment to appreciate the new forms of data that are now at your disposal as you seek to answer those questions.
## Datification: The new forms of data
Most human and computer based activities already leave a digital trace (or data) that can be collected and analysed to provide insights on everything from health to crime to business performance. Of the few activities that don't currently leave a digital trace – they soon will.
The world is being 'datafied' and there are now many forms of useful data. Some of the data forms are new such as social media posts; others have been around for a long time. For example, we've been able to record conversations for a long time but a lack of storage capacity or a way to really analyse those recordings limited their utility. But all that is changing.
Data is now being mined from:
* Our activities
* Our conversations
* Photos and video
* Sensors
* The Internet of Things.
### Activity data
More and more of the activities we engage in leave a data trail.
For example, when we go online our browser logs what we are searching for and what websites we visit. Most websites will log how many people visit the site, where those people are located (using the computer ISP), how long the person stayed on the site and how they navigated or clicked through the site. Often this information is used to assess website performance and delete areas that no-one visits while improving pages that seem to generate the most interest.
If we decide to go shopping online there is a record of what we share or like and of course what we buy, how much we paid for it, when we bought it, when it was delivered and often what we then thought of the product or service through user feedback.
If we decide to read a book chances are we will increasingly turn to a Kindle, iPad, smart phone or other e-reader. There are now millions of books available in a digital format. Some books such as technology text books which change rapidly are often never even released as a physical printed book.
It has been estimated that 130 million unique books have been published since the invention of the Gutenberg printing press in 1450. By 2012, just seven years into the Google Book Project, Google had scanned over 20 million titles or more than 15% of the world's entire written heritage!3 Amazon also gives us extensive access to old books in digital form.
When we use an e-reader we are usually not just reading a digital image of the page – the text is datafied. That means that we can change font size, add notes, highlight text or search the book. This datafication also means that data is gathered about what we read, how long we read for, whether we skip pages, what pages we annotate and what we choose to highlight. This information could certainly prove useful for authors and publishers. I would love to know how people use my books, which sections people skip, when readers stop reading a book. This would allow me – or indeed any author – to revise content in order to shorten or improve particular parts so that readers have a better experience. Furthermore, authors and publishers may be able to identify areas of interest from frequently highlighted passages across many books to identify new topic trends on which to commission new work.
If we listen to music using our smart phone or digital music player, data is also collected on what we are listening to, how long we are listening and what tracks we are skipping past. And artists like Lady Gaga are using this data to create playlists for live gigs and influence future song creation.
Even walking to work or going to the gym will generate data if we are wearing a smart device like the 'Up' band or are using an app on our smart phone. These apps and devices can measure how many steps we take each day, how many calories we burn, how well we sleep, log activity and exercise, deliver insights and celebrate milestones. Some devices also measure our heart rate and often our heart rate variation (HRV). HRV measures the tiny variations in the interval between each heart beat and has been proven to be a significant metric for predicting health problems. For example, since 1965 it has been common obstetric practice to monitor a baby's HRV during labour for early signs of foetal distress.4 In 1997 Jacqueline Dekker, Professor of Diabetes Epidemiology at the VU University Medical Centre in Amsterdam, along with her colleagues discovered that HRV was capable of predicting death, not only in babies or heart attack victims but it also predicted 'all cause mortality'.5 Clearly, data on HRV would be useful for us all to know and devices like smart watches will be able to collect such data.
Many of these wearable devices are now Internet-enabled so that they self-generate and share data. It is also almost inevitable that many of the current wearable devices and apps will be swallowed up by the smart watch in the same way iPods were swallowed up by iPhones.
The company that makes the 'Up' band, Jawbone, now collects sleep data from millions of people around the world (including me). This means they have unparalleled access to years' worth of sleep data – every night! No company on the planet has ever had that sort of data or that sort of volume of data. Jawbone is then able to analyse the data to understand more about sleep, our sleeping patterns and what disrupts those patterns. For example, Jawbone could look at the data and work out how many hours of sleep are lost, on average, when the Superbowl is broadcast in the US or how long it normally takes for travellers to get back to normal sleeping patterns if they fly between New York and San Francisco or between London and Sydney.
### Conversation data
Increasingly we also leave digital records of our conversations – either through text when we write an SMS message, on social media or an audio recording of a telephone call.
Just think of the billions of emails that are sent and stored every week. In fact, twenty million emails were written in the time it took to read this sentence.6
We are using social media to communicate and interact with each other, which is creating unfathomable amounts of data. Check out these stats:
* More than a billion tweets are sent every 48 hours.
* One million accounts are added to Twitter every day.
* Every sixty seconds, 293,000 status updates are posted on Facebook.
* Two new members join LinkedIn every second (172,800 per day)
* 72% of online adults use social networking sites.
* 25 percent of Facebook users never bother with any kind of privacy control.
* The average Facebook user creates 90 pieces of content including links, news stories, photo albums, notes, and videos each month.
* Incredibly, people in New York City received tweets about the August 2011 earthquake in Mineral, Virginia 30 seconds before they felt it.7
There are also already millions of website and blogs contributing to the conversation. An estimated 571 new websites are created every minute of the day. Every minute, Tumblr owners publish approximately 27,778 new blog posts and 3 million new blogs come online every month.8
Plus there is the data collected from our telephone conversations. If you call a customer service department we are always told the conversation _may_ be recorded. Often that data is being mined for content and sentiment and even analysed for stress levels in someone's voice to gauge how irritated the customers are!
Audio data is also being used to improve voice recognition and translation software. For example, Google decided to venture into translation in 2006 as part of its mission to 'organize the world's information and make it universally accessible and useful'. Most translation software utilize perfectly translated pages of text to create the algorithms but Google used the entire global Internet and more. Their system sucked in every translation – good and bad – that it could find in order to train the translation computers. As a result of the sheer volume of data that they could access and use Google translation is more accurate than any other system. By mid 2012 its dataset covered more than 60 languages and even accepts voice input in 14 languages for fluid translation.9 It's still not perfect but as the system learns from the correct translation and the incorrect translation chances are it will be in the future.
### Photo and video image data
Again the data being collected and stored is staggering. Digital cameras and smart phones are taking and sharing more photos and videos than ever before. Check out these stats:
* Each day 350 million photos are uploaded to Facebook, which equates to 4,000 photos per second.
* Flickr users upload 3.5 million photos to the site each day.
* Approximately 100 hours of video is uploaded to YouTube every minute.
* More than 45 million pictures are uploaded to Instagram every day.
* As of June 2013, Instagram users have shared more than 16 billion photos.10
Granted, sharing what we had for dinner or a picture of our new Labrador puppy won't change the world but this plethora of photo, video (and text data) is actually already saving lives in disaster areas.
When typhoon Hiayan hit the Philippines in 2013, for example, over 6,000 people were killed and 1.1 million homes were damaged or destroyed in hours. In the UK, a team of volunteers were creating a vital map of the damaged areas using just social media. Because it is now very common for people to share their experiences as they happen in almost real time, photos, tweets (#Hiayan) and videos about the disaster were being posted on social media. In the aftermath of Hiayan the volunteers were receiving on average a million photos, messages, tweets, videos, etc., every day!
After filtering the millions of messages using artificial intelligence to pick out the ones that could be important the team of volunteers then made an assessment of what they saw. For example, for a photograph they would be asked, 'How much damage do you see?' and they simply needed to click the appropriate button: 'none', 'mild', or 'severe'. For text based messages such as tweets or Facebook updates the volunteer was asked to decide if the text was 'not relevant', 'request for help', 'infrastructure damage', 'population displacement', 'relevant but other', etc. Each piece of data (picture, video or message) was then assessed by between three to five different people to make sure the assessment was consistent and therefore probably accurate.
By pinpointing where the data was coming from in the Philippines (using GPS sensors in the photos or through the text) the work of the volunteers then created an online map, not just of the disaster zone but of the needs in each area.
That meant that when the disaster relief effort arrived in the Philippines they didn't need to waste days working out what was happening and where the worst hit areas were. They already knew from the map – created by people half way around the world – who needed water, who needed food, where the dead bodies were and where people had been displaced, where the most damage was and what hospitals were least damaged, and therefore more able to help the injured.11
How cool is that?
In addition to all the photo and video data created by individuals via their digital tech or smart phone there is also all the CCTV camera footage. In days gone by companies may video record their premises or retail store and store the recording for a week or so before recording over older recordings. Now some of the larger data savvy stores are keeping all the CCTV camera footage and analysing it to study how people walk through the shops, where they stop, what they look at and for how long so they can make alterations to offers and boost sales. Some are even using face recognition software so it probably won't be long before a combination of data sources such as CCTV camera footage, loyalty card information and face recognition software will see us being welcomed to a store on our smart phones and directed to particular special offers or promotions that may be of interest to us based on our previous buying habits!
### Sensor data
There is also an increasing amount of data being generated and transmitted from sensors. There are sensors everywhere.
Have you ever wondered that makes your smart phone (or smart anything for that matter) smart? Basically what makes them smart is the inclusion of various sensors that capture data. In your smart phone for example there is a:
* GPS sensor
* Accelerometer sensor
* Gyroscope
* Proximity sensor
* Ambient sensor, and
* Near Field Communications (NFC) sensor.
The GPS (Global Positioning System) sensor lets us (and others) know where we are using the GPS satellite navigation system. The GPS sensors in our phone can pinpoint our location within a few meters (assuming we are with our phone of course!). The accelerometer sensor is a motion sensor and measures the acceleration or how quickly the phone is moving. It's this technology that allows you to take better photos with your smart phone because it's this sensor that triggers the shutter when it detects the camera is stationary or stable. The gyroscope sensor is used to maintain orientation and is used to rotate the screen. It is this sensor that is often utilized in gaming apps where you have to tilt the screen to direct the character or steer the car. As the name would suggest the proximity sensor senses proximity and how close we are to other objects or locations. Ambient sensors are the ones that detect changes in the ambience or atmosphere so it is this sensor that adjusts the backlight on your phone or saves power when it's not being actively used. And finally the NFC sensor is one of the latest communication protocols being utilized in smart phones. It is these NFC sensors that when enabled, allow you to transfer funds just by bumping phones or waving your phone close to an appropriate payment machine.
There are also sensors in the natural environment, for example, in the oceans for measuring the health, temperature and changes of the oceans in real time. Also in Japan there are sensors in the soil to collect data on how healthy the soil is and companies are combining that data with weather data. Farmers can then subscribe to the service to get information to optimize yield, including how much and when to put fertilizer on their crops.
Increasingly more and more machines are equipped with sensors to monitor performance and provide information on when best to service or repair the machines.
For example, Rolls Royce manufactures nearly half the world's passenger jet engines including the Trent 1000, the engine that powers many of our transatlantic flights. When in operation these engines reach incredibly high temperatures – half the temperature of the surface of the sun and 200 degrees _above_ that temperature when the metal used to make the engine melts! The only reason it doesn't melt is because the engines are being cooled through special passageways and channels that keep the heat away from the metal. Needless to say it's vital to know that everything is working and doing its job, as you wouldn't want the plane you are taking to visit your friends in New York to melt at 30,000 feet!
The engine is therefore full of vital components all engineered with absolute precision including an on-board computer that is the brains of the engine, controlling it and also collecting and monitoring data from sensors buried deep within the engine measuring 40 parameters 40 times per second including temperatures, pressures and turbine speeds.
All the measurements are stored in the computer and streamed via satellite back to Rolls Royce HQ in Derby, England. And that's true for the entire fleet of Rolls Royce engines, which is a lot of data when you consider that a Rolls Royce powered engine takes off or lands somewhere in the world every two and a half seconds.
Whenever those thousands of engines are in the air they are gathering data which is continuously sent back to HQ and constantly monitored using clever data analytics that are looking for anything unusual going on in the engine, or any sign that it may need to be serviced early or repaired. In Derby, computers then sift through the data to look for anomalies. If any are found they are immediately flagged and a human being will check the results and if necessary telephone the airline and work out what needs to be done – normally before the issue escalates into an actual problem.
These sensors therefore allow for dynamic maintenance based on actual engine-by-engine performance rather than some automatic rota system based on time alone. Instead of pulling an expensive piece of equipment out of service every three or six months these sensors allow the airlines to maintain their fleet much more cost effectively and, more importantly, these sensors make the planes much safer.12
Modern cars are also full of similar sensors that measure everything from fuel consumption to engine performance, which again allows for dynamic servicing and better long term performance. On-board sensors also alert the driver if they get too close to another car or object and can even parallel park the car without the driver having to do anything!
In the retail industry, data has long been collected via barcode; however, the sensors known as Radio Frequency Identification (RFID) systems increasingly used by retailers and others are generating 100 to 1,000 times more data than the conventional barcode system.13
There are sensors everywhere.
### The Internet of Things
The Internet of Things (IoT) is a result of more objects being manufactured with embedded sensors and the ability of those objects to communicate with each other.
IDC describes the IoT as:
> _'a network connecting – either wired or wireless – devices (things) that are characterized by automatic provisioning, management, and monitoring. It is innately analytical and integrated, and includes not just intelligent systems and devices, but connectivity enablement, platforms for device, network and application enablement, analytics and social business, and applications and vertical industry solutions. It is more than traditional machine-to-machine communication. Indeed, it is more than the traditional Information and Communications Technology (ICT) industry itself.'_14
This concept explores what is and will be possible as a result of advances in smart, sensor-based technology and massive advances in connectively between devices, systems and services that go way beyond business as usual. For example, research groups such as Gartner and ABI Research estimate that by 2020 there will be between 26 and 30 billion devices wirelessly connected to the IoT. And the resulting information networks promise to create new business models and improve business processes and performance, while also reducing cost and potentially risk.
The day will come, not far from now when your alarm will be synced to your email account and if an early meeting is cancelled your alarm will automatically reset to a later time, which will also postpone the coffee machine to the new wake-up time. Your fridge will know what's in it and place online orders to replenish stocks without you having to do anything. You'll put on your suit, with a payment chip in the sleeve so you can swipe payment for lunch without a credit card.
Your wearable device or smart watch will monitor your health through the day, watching your calorie intake and making sure you stay active and don't sit too long at your desk. As you get in your car to drive home at night the car will automatically check the route with traffic and weather information to get you home as quickly and safely as possible. On arriving home, the temperature will be perfect and your fridge will tell you what you can make for dinner based on what you currently have in stock.
As you settle down to watch TV with your family, you may be enjoying a film rated 18 when your 5-year-old child walks in and your smart TV will suspend the film and change channel. Oh and if your elderly mother is ever house sitting while you are away your smart carpet will measure and monitor her movements and patterns – perhaps she goes to the kitchen at 10.30 a.m. every morning to make a cup of coffee or always goes to bed at 11 p.m. Should those patterns change you will be alerted to get in touch and check everything is OK.
The wired and wireless networks that connect the Internet of Things often use the same Internet Protocol (IP) that connects the Internet – hence the name. These vast networks create huge volumes of data that's then available for analysis. When objects use sensors to sense the environment and communicate with each other, they become tools for understanding complexity and responding to it quickly. The resulting physical information systems are now beginning to be deployed, and some of them operate without human intervention.
Pill-shaped micro-cameras already traverse the human digestive tract and send back thousands of images to pinpoint sources of illness. Precision farming equipment with wireless links to data collected from remote satellites and ground sensors can take into account crop conditions and adjust the way each individual part of a field is farmed. There are even billboards in Japan that monitor passers-by, assess how they fit consumer profiles, and instantly change displayed messages based on those assessments. Advances in wireless networking technology and the greater standardization of communication protocols make it possible to collect data from these sensors almost anywhere at any time. Ever-smaller silicon chips are gaining new capabilities, while costs are falling. Massive increases in storage and computing power, some of it available via cloud computing, make number crunching possible on a very large scale and at declining cost.15
All coming together to create Big Data.
## The anatomy of Big Data
When we consider the types and forms of data that now exists it's easy to see how people become overwhelmed and bamboozled by the possibilities of Big Data. Although, as I've said I think the term will disappear and what we consider Big Data today will just be 'data' tomorrow.
For a start, what is uncommon and exciting now will become commonplace. Plus the term may be simple and easy to remember but it's overly simplistic and places far too much emphasis on the volume of data. But volume is just one of the four V's of Big Data:
* **_Volume_** – relating to the vast amounts of data generated every second.
* **_Velocity_** – relating to the speed at which new data is generated and moves around the world. For example, credit card fraud detection tracks millions of transactions for unusual patterns in almost real time.
* **_Variety_** – relating to the increasingly different types of data that is being generated from financial data to social media feeds; from photos to sensor data; from video footage to voice recordings.
* **_Veracity_** – relating to the messiness of the data being generated – just think of Twitter posts with hash tags, abbreviations, typos, text language and colloquial speech.
We don't need Big Data – we need SMART Data!
### Big Data backlash
As with any new frontier, the frontier of Big Data is also under attack. There are those that believe that it's a storm in a teacup and the theory of Big Data is so far removed from the reality for most businesses that it will never yield much, if any fruit for the vast majority of business.
Certainly there are some companies that already have these huge data sets; however, most businesses will never have access to the volume and variety of data that an Amazon, eBay or Facebook will have. But as I've said before that's OK because most businesses don't need access to oceans of data.
The other area of attack is around consumer data and privacy.
The reputation of Big Data has suffered with the revelations by whistleblower Edward Snowden that the US National Security Agency (NSA) has been systematically using Big Data analytics to 'spy' on everyone's communications as well as perform targeted surveillance of individuals and companies. We can all be certain that the US is not the only government agency in the world to collect and use Big Data. For example, former French foreign minister, Bernard Kouchner, stated, 'Let's be honest, we eavesdrop too. Everyone is listening to everyone else. But we don't have the same means as the United States, which makes us jealous.'
Despite high profile Snowden-type media stories, as I write this in 2014 most people are completely unaware of just how much data about them is freely available online. Even if someone takes the time to complete privacy settings on social media and is deliberately vague and cautious about over-sharing – there is still a phenomenal amount of information being collected, stored and analysed. Most of us are, for example, almost entirely oblivious to the fact that the GPS sensor in their smart phone makes it possible to identify where a picture was taken within a few metres, regardless of whether the person sharing the photo adds a tag, message or caption. They don't realize how open and freely available their social media sites are, how much of what they post is saved and analysed – even when the platform tells its users that the photo or video will self-destruct in 10 seconds! Those images may not be accessible to the user after a set time but they are saved. They have no idea that their web browser is monitoring their every move or even that people can easily hack into the camera on their laptop and watch them!
In 2013 a 19-year-old US student was charged with hacking Miss Teen USA's webcam. The FBI found that he had used malicious software to remotely operate webcams to get nude photos and videos of at least seven women as they changed clothes. Some of these women he knew personally and others he found by hacking Facebook pages.16 In the UK in 2014 another man received a suspended sentence for the same thing. Probably best to cover your webcam when you're not using it – just in case!
So far people have not really cottoned on to the dangers or the inherent value of their own data and are happy to freely share that data in exchange for services they want, such as Facebook.
Facebook is already a gigantic data mining paradise with unbelievable amounts of data at their disposal, all enthusiastically provided by the users of Facebook. Remember the stats from earlier – 350 million photos a day, 293,000 status updates a minute and 25% of users never bother with privacy!
Facebook knows what we look like, who our friends are, what our views are, what our interests are, when our birthday is, whether we are in a relationship or not, where we are, what we like and dislike, and much more. That is an awful lot of information (and power) in the hands of one commercial company.
People may start to get uncomfortable about the amount of data that is known and held amount them. But how much of a difference would it really make? Take Facebook again: even if we all stopped using Facebook today (which is very unlikely), the company would still have more information about people than any other private company on the planet. Google may come close but they don't have the plethora of detailed personal data that Facebook has. Of course it's not just Facebook.
The challenge is that once companies have access to the data they won't stop. And we don't have to be a loyalty card member for the companies to know about us: in addition to social media, they can also track our credit card use and use face recognition software to record what we are doing in store.
A recent study showed that it is possible to accurately predict a range of highly sensitive personal attributes simply by analysing the 'Likes' we have clicked on Facebook. The work conducted by researchers at Cambridge University and Microsoft Research shows how the patterns of Facebook 'Likes' can very accurately predict characteristics such as your sexual orientation, satisfaction with life, intelligence, emotional stability, religion, alcohol use and drug use, relationship status, age, gender, race and political views among many others.17
The fact is that the data collectively held on you by banks, credit card companies, insurance companies, supermarkets and social media is astonishing and it's growing all the time.
Even if people did become uncomfortable in enough numbers to bring about changes to legislation it may be too late. It would be like shutting the barn door once the horse had bolted. It may be that legislation may push for at least some of the most sensitive data to be anonymized, i.e. markers that identify an actual person to be removed, but it will still be used and the datifaction of the world will not stop.
## How to use metrics and data for strategic advantage
Whether we like it or not, or are ready for it or not, the future will involve Big Data. Our ability to harness that power with intelligence, common sense and practicality will see us turn it into meaningful SMART Data.
Having started with strategy and identified the SMART questions around customers, finance, operations, resources and risk you need to figure out what metrics and data you actually need access to in order to answer those questions, which in turn will help you to deliver your strategy.
### Identify your metric and data needs
The next step toward a SMARTer business is therefore to identify what data you need to access or acquire in order to answer the SMART questions from the previous chapter.
Knowing what you now know about structured, unstructured and semi-structured as well as internal data and external data there is a logical hierarchy of where you should first look when seeking to identify the metrics and data that will answer your SMART questions.
That hierarchy is:
1. Internal structured data
* This is easiest to find and easiest to analyse. It is also probably the least expensive to acquire.
2. Internal semi-structured
3. Internal unstructured
4. External structured
5. External unstructured.
Many people in business are too focused on the last port of call – external unstructured data. This is also an error. If you can effectively answer your SMART questions from internal structured data why on earth would you waste valuable time seeking the answers anywhere else?
Once you go through this process you will soon realize that some of the data is harder to get than others. Beyond internal and external, structured and unstructured there are seven main ways of collecting that data:18
1. Created data
2. Provoked data
3. Transaction data
4. Compiled data
5. Experimental data
6. Captured data
7. User-generated data.
Even though there might be some overlap between these categories, they provide a nice little framework. Let's now look at each of these data generation methods in a little more detail.
* 1 _Created data_ is 'created' because it wouldn't exist unless we asked people questions and put a mechanism in place to capture their answers. Examples of created data include data created by market research surveys, focus groups or employee surveys. People registering online for clubs or loyalty programmes are also examples of created data as the person is voluntarily providing information about themselves. Created data is usually structured or semi-structured and can be internal or external.
* 2 _Provoked data_ is 'provoked' because it wouldn't exist unless you invited people to express their views. Examples of provoked data are asking customers to rate and review a product or service. When you buy a product from Amazon, for example, you are provoked to rate both the product and the delivery of the product using a five star system. One star indicates that you were not very happy and five starts indicate you were extremely happy. Provoked data is usually structured or semi-structured and can be internal or external.
* 3 _Transaction data_ is generated every time a customer buys something. This is true online and off and it provides a powerful insight into what was bought, where it was bought, and when. Transaction data can also be very illuminating when combined with other data such as the weather. For example, a few years ago Walmart did some data discovery looking at past transaction data and cross referenced that data with weather data. What they found was that when a hurricane warning was issued sales for things like flashlights would increase. That seems expected. What they didn't expect however was to find that there was also a correlation between Pop-Tarts and hurricane warnings! Walmart didn't need to know why customers bought extra Pop-Tarts when a storm was approaching all they needed to do was stock boxes of them at the front of the store which further boosted sales. Transaction data is usually internal structured data.
* 4 _Compiled data_ is 'compiled' because it comes from the giant databases that companies like Experian and Axciom maintain on every household. These companies compile vast amounts of data from different sources often using your name and address as the common identifier. They provide a wealth of information for marketing companies to mine for a marketing advantage including credit scores, where you live, your purchase history, what cars you've registered in your name, insurance renewal dates and more. Compiled data is usually external structured data.
* 5 _Experimental data_ is really a hybrid of created and transacted data. It involves designing experiments in which different customer sets receive different marketing treatments (created) and observing the results in the real world (transaction). This is what we did with the small fashion retailer – once we knew passing footfall we could test various window displays to see which displays led to more people entering the shop and purchasing. Experimental data is usually structured or semi-structured and can be internal or external.
* 6 _Captured data_ is 'captured' because it refers to information gathered passively from an individual's behaviour, such as search terms you enter into Google or the location data that your phone generates through its GPS. It is this captured data that is exploding because of the Internet of Things (IoT). Most people are unaware of the data that is captured about them without their knowledge or permission. Captured data is usually unstructured and can be internal or external.
* 7 _User-generated data_ is 'user-generated' because it is the data that individuals and companies generate consciously – or at least knowingly. It includes the Facebook posts, tweets, videos posted on YouTube and comments made on an article or blog. Most user-generated data is not immediately attributable to an individual. For example, you may know the hashtag but not the person. It can be used to provide a context for product design and communications, but not for direct targeting. User-generated data is usually unstructured and can be internal or external.19
With this in mind consider your data needs for each of the panels in your SMART strategy board.
#### _Customer data needs_
When it comes to customer data there are many data source options of varying quality, complexity and expense. See Figure 3.1 for some of those options.
**Figure 3.1** Data examples for the customer panel: Finding SMART Data to Answer your SMART Questions: <http://www.marketingprofs.com/articles/2014/24670/little-data-vs-big-data-nine-types-of-data-and-how-they-should-be-used>
##### _Survey research data_
Survey research data, whether conducted in-house or through an experienced third party intermediary usually yields trustworthy data. Often the data is experimental in nature and can include research design, normative data, mathematical modelling, stimulus controls, statistical controls, historical experience and quality-assurance standards. Survey research is also relatively inexpensive.20
* Could you design a survey to help you answer your SMART questions?
##### _Sales and transactional data_
Sales data is useful for a whole host of reasons but it is rarely an exact measure of actual sales because it doesn't account for returns. It is a good measure of what has happened in the past but you will need to combine this data with other data sets to work out what it happened.
* Are your sales and transaction data going to help you answer your SMART questions?
##### _Eyeball tracking data_
There has been a steady improvement in the technology capable of measuring where your customers are looking. This means you can measure what is getting attention and what is not. Some of my customers have used eyeball tracking for their websites and another – a museum – uses the technology to understand how we look at some of the key paintings in their collection.
* Could you utilize eyeball tracking to answer your SMART questions?
##### _Operations Data Needs_
Your operational processes and procedures will generate, or could generate, a huge amount of data that could help you make better decisions and improve efficiency. Figure 3.2 identifies some of the options.
**Figure 3.2** Data examples for the operations panel: Deriving SMART Questions from your SMART Strategy Board
##### _RFID (Radio-frequency identification) data_
RFID data is currently used in consumer products, US passports, shipping packages, and credit and debit cards that employ 'touchless' transactions. It relies on microchips that emit radio waves containing particular information about a product or person.
* Could RFID data help you to answer some of the SMART questions?
##### _Supply chain data_
Supply chain data helps a business track where products are and where they have come from. This type of data is increasingly important in the food supply chain to measure and track traceability. Think of the UK horsemeat scandal of 2013 where several of the large supermarkets had to remove frozen beef burgers from their stores when they were found to contain horsemeat.
* Could supply chain data help you to answer some of the SMART questions?
##### _Biometric data_
Galvanic skin response, eye pupil dilation, heart rate, EEG (brainwave) measurements and facial emotions recognition are all mea-surable and present a very interesting and exciting area of exploration.
* Could biometric data help you to answer some of the SMART questions?
#### _Finance data needs_
Considering the most recognized purpose of business is to make money it's essential for every business to understand its financial position. Figure 3.3 details some of the data solutions for identifying financial data.
**Figure 3.3** Data examples for the finance panel: Deriving SMART Questions from your SMART Strategy Board
##### _Financial transaction data_
Financial transaction data includes the time and date of the transaction, a description of the event and a numerical value. For example, orders, invoices, payments, deliveries, travel records or storage records are all financial transaction data.
* Will your financial transaction data help you to answer your SMART questions?
##### _Stock market data_
Stock market data will include share price information, price movements and trends as well as numerous popular metrics used to measure performance in the stock market.
* Could your stock market data help you to answer your SMART questions?
##### _Cash flow data_
The biggest cause of business failure is lack of cash flow. Cash flow data therefore allows a business to monitor the incomings and outgoings of a business so as to maintain a cash flow positive position.
* Could your cash flow data help you to answer your SMART questions?
##### _Resource data needs_
Knowing what resources you have and working out how to utilize them most effectively is all part and parcel of SMART business. When it comes to finding data on your resources, Figure 3.4 suggests some of the more popular options.
**Figure 3.4** Data examples for the resource panel: Deriving SMART Questions from your SMART Strategy Board
##### _Interview data_
Interview data can be collected in both quantitative and qualitative formats. Quantitative involves the collection of data involving numbers and structured ranked responses. Qualitative is the collection of data that is not in numeric formats such as written feedback, open-ended question responses, observations or recordings, etc. Interview data is a common way to collect feedback from your most important resources – the people in your business.
* Could interview data help you to answer your SMART questions?
##### _Self-assessment performance data_
As the name would suggest this type of data provides answers to how someone sees their own performance or how they like a product – but it's provided by self-assessment rather than interactive interview.
* Could your self-assessment performance data help you to answer your SMART questions?
##### _Recruitment data_
Recruitment data will be held with HR and can help to tell you how successful or otherwise your recruitment is. What is the absenteeism in the business? What about staff turnover, training costs, sick days? This is all recruitment data.
* Could your HR or recruitment data help you to answer your SMART questions?
##### _Sensor and machine data_
Many machines and IT systems have inbuilt sensors and data collection capabilities that generate large volumes of often real-time data on performance, fault detection, capacity utilization and many other areas.
* Could your sensor or machine data help you to answer your SMART questions?
##### _Competition and risk data needs_
If you want to stay ahead of the competition and circumvent problems before they arise you need to understand your competition and risks. Figure 3.5 details some of the options open to you to uncover competition and risk data that can help you deliver on your strategy.
**Figure 3.5** Data examples for the competition and risk finance panel: Deriving SMART Questions from your SMART Strategy Board
##### _Fraud data_
Companies are building up data repositories on fraud activities. For example, insurance companies are collecting increasingly sophisticated data on fraudulent claims and how to predict them. Credit card companies and banks are also getting better at understanding fraud and the data behind it.
* Could fraud data help you to answer your SMART questions?
##### _Search engine data_
Apparently, each day 20% of Google searches have never been searched before.21 That indicates that people are thinking outside the norm and these searches could point to new products or services or ideas on how to improve business.
* Could search engine data help you to answer your SMART questions?
##### _Social media_
Without professional help you could waste a lot of time, effort and money on social media data. New tools are emerging all the time to measure and analyse this but social media data should always be backed up or triangulated with other data sources. Perhaps its most valuable use to every business is that if monitored properly it can act as an early-warning system. If customers are unhappy about a product or service they will usually vent on social media and this can alert the business to failings or areas to improve.
* Could social media data help you to answer your SMART questions?
## Metrics and data in action
When you first dip your toe into the Big Data and analytics universe it can be extremely overwhelming to know what can and is being measured and more than a little daunting to appreciate how some of the big players are so far ahead in many regards. The purpose of this book and the reason I wanted to write it is because none of that matters. Yes, there is an extraordinary array of internal, external, structured and unstructured data that can, could or even should be measured but if you let your SMART questions guide your approach then it's remarkably easy to manoeuvre through the overload and secure the exact, specific pieces of data that will answer those questions and deliver real business value.
In order to work out what data you are going to need access to you need to consider each of your SMART questions separately.
Go through the various 'panels' from the SMART strategy board and describe the data sets you could access that would help you answer each SMART question. Use the following data sheet to help you understand the data you are seeking answers from and where you will find it.
**SMART Question You Want To Answer:**
---
**Data sources that will help you answer the SMART question**
* Name of data set
* Describe type of data
* Location & Ownership: internal/external
* Format: Structured/Unstructured
* What is that data collection method?
* Where is the data stored or located?
* Describe Data Volumes
* Describe Data Velocity/Frequency/ Recency
* Describe Data Veracity/Quality
* How will the data be analysed?
* Costs associated with capturing, storing and analysing the data
| Data set 1 | Data set 2 | Data set 3
More than likely you will need to consider more than one data set. Make a note of which data sets you intend to use or could use. Describe the data for each data set and make a note of its location and who owns it. Is the data internal or external? The assumption is that internal data is easier and cheaper to access but this isn't always the case. It really depends on the data you are seeking and how readily available it is and in what format it currently exists. For example, if all your past customer records were on microfiche then it may be internal and you may own it but it could be very costly to get all that data converted to digital format. It may be that there is an alternative external solution which could prove cheaper in the long run.
Make a note of the data's format – is it structured or unstructured? What is the data collection method and where is the data stored or currently located?
Describe the data volume – how much data are you going to be looking at? Is the data changing rapidly? Is it collected frequently and how recent is it?
Make a note of how each data set will be analysed as well as the costs involved in the data capture, storage and analysis.
If it transpires that the cost of a particular data set is too high you may want to explore a different option or be really sure the answer you will derive from the data is strategically important enough to warrant the time, money and effort.
It is always better to have two data sets than one and always better to have three than two. Three data sets allow you to triangulate or verify the data from different perspectives. So if one data set is structured internal and another is unstructured internal and another unstructured external, then you will almost certainly get a much richer picture of what's happening so you can answer your SMART questions more effectively and accurately.
Remember: don't concern yourself with all the metrics and data that currently exist – only focus on the metrics and data that will help you answer your specific SMART questions, improve your performance and help you to fulfil your strategic objectives.
## Key points and call to action
* The basic idea behind the phrase 'Big Data' is that everything we do in our lives leaves, or soon will leave, a digital trace (or data), which can be used and analysed.
* That said, Big Data is just data and it is only useful if it answers questions you need answers to (SMART questions).
* In order to know what answers you could gain access to you need to first understand the various types and forms of data that can now be analysed for insight. These include:
* Structured Data: This is data that is located in fixed fields within a defined record or file such as spreadsheets or relational databases.
* Unstructured and Semi-structured Data: This is data that is not, or only partially, located in fixed fields or within a defined record or file such as images, text documents, social media posts.
* Internal Data: This is data that you can or could access from within your own business. You already own this data although it may currently be located in boxes in the basement!
* External Data: This is data that's been created or generated outside your business that you do not own or currently have access to. Some external data is free to access and some is not.
* In addition it is now possible to mine insights from:
* Activities
* Conversations
* Photos and video
* Sensors
* The Internet of Things.
* The next step is to marry your knowledge of what data now exists with your SMART questions. Consider what data you would ideally need access to in order to best answer each of your smart questions.
* Don't worry initially about whether you have access to that data – just identify the best metrics and data that could help you answer your most pressing questions, which in turn would help you to deliver on your strategic objectives.
* Use the SMART strategy board to consider each business area in turn and consider what metrics and data will be required to answer your resources, competition and risk questions too.
* Complete the data sheet for each data set you identify.
* Based on your data sheets choose the best data options to pursue based on how easy the data is to collect, how quick and how cost effective.
* As a rule of thumb start with internal data and structured data which is usually easier and cheaper to analyse than unstructured or semi-structured data.
* Remember: if you know exactly what you are looking for, i.e. what questions you need answers to then you avoid becoming overwhelmed by the vast amount of metrics and data that currently exists.
## Notes
1 Mayer-Schonberger, V. and Cukier, K. (2013) _Big Data: A Revolution That Will Transform How We Live, Work and Think._ London: John Murray Publishers. 2 IDC The Digital Universe Study (April 2014) Sponsored by EMC2. 3 Mayer-Schonberger, V. and Cukier, K. (2013) _Big Data: A Revolution That Will Transform How We Live, Work and Think_. London: John Murray Publishers. 4 Neilson, J.P. and Mistry, R.T. (2000) Fetal electrocardiogram plus heart rate recording for fetal monitoring during labour. _Cochrane Database of Systematic Reviews_ (2). 5 Dekker, J.M., Schouten, E.G., Klootwijk, P., Pool, J., Swenne, C.A. and Kromhout, D. (1997) Heart rate variability from short electrocardiographic recordings predicts mortality from all causes in Middle-aged and elderly men. The Zutphen Study, _American Journal of Epidemiology_ **145** (10). 6 IACP Centre for Social Media Fun Facts <http://www.iacpsocialmedia.org/Resources/FunFacts.aspx>. 7 IACP Centre for Social Media Fun Facts <http://www.iacpsocialmedia.org/Resources/FunFacts.aspx>. 8 IACP Centre for Social Media Fun Facts <http://www.iacpsocialmedia.org/Resources/FunFacts.aspx>. 9 Mayer-Schonberger, V. and Cukier, K. (2013) _Big Data: A Revolution That Will Transform How We Live, Work and Think_. London: John Murray Publishers. 10 IACP Centre for Social Media Fun Facts <http://www.iacpsocialmedia.org/Resources/FunFacts.aspx>. 11 BBC Two (2014) Bang goes the Theory, May 2014, Series 8: Big Data. 12 BBC 2 (2014) Bang Goes the Theory, March 2014, Series 8: Big Data. 13 SAS Whitepaper (2012) Big Data meets Big Data Analytics: Three key technologies for extracting real-time business value from the Big Data that threatens to overwhelm traditional computing architectures. 14 IDC (2014) The Digital Universe Study, April 2014. Sponsored by EMC2. 15 Chui M, Löffler M, and Roberts R (2010) The Internet of Things. _McKinsey Quarterly_ , March 2010. 16 BBC (2013) Miss Teen USA webcam hacker is charged. <http://www.bbc.co.uk/newsbeat/24303512> 17 Kosinski, M., Stillwell, D. and Graepel, T. (2013) Private traits and attributes are predictable from digital records of human behavior. Published online: <http://www.pnas.org/content/early/2013/03/06/1218772110.abstract> 18 Meer, D. (2013) What is 'Big Data' anyway? _strategy + business._ <http://www.strategy-business.com/blog/What-Is-Big-Data-Anyway?gko=28596> 19 Meer, D. (2013) What is 'Big Data' anyway? _strategy + business._ <http://www.strategy-business.com/blog/What-Is-Big-Data-Anyway?gko=28596.> 20 Thomas, J.W. (2014) Little Data vs. Big Data: Nine types of data and how they should be used. <http://www.marketingprofs.com/articles/2014/24670/little-data-vs-big-data-nine-types-of-data-and-how-they-should-be-used>. 21 Qualman, E. (2013) Social Media, 2013, You Tube, <http://www.youtube.com/watch?v=zxpa4dNVd3c>.
# 4
A = APPLY ANALYTICS
When you are clear about your strategy, then you know what target you are trying to hit. That target also brings clarity to what metrics and data you will need to collect in order to answer your SMART questions. Between the traditional internal structured data of financial records, databases and KPIs and the newer unstructured often-external data of weather data, CCTV video footage and sensor data, business is literally drowning in the stuff.
In the previous chapters I discussed how we collect the data, but having the data is not enough. You need to apply analytics so that you turn the data into meaningful business insights that can help you to execute your strategy and improve performance.
Data and analytics go hand in hand. After all, there is no point creating new and ever-expanding data collection opportunities, methods and capabilities if we don't then do something with that data or learn something new from it. As such the field of analytics is growing in line with the growth of data.
Remember in the last chapter we explored the 4 V's of Big Data: volume, velocity, variety and veracity. Analytics provide the fifth and perhaps most important V of Big Data – Value.
When we manage the volume, velocity, variety and veracity of data properly by applying analytics we can create unprecedented value that can help to shape strategy and inform decision-making.
## Evolution of analytics
Like Big Data itself, the analytics evolution has been made possible by a number of key innovations. Advances in storage and processing capabilities now mean we have access to vast data sets that we previously didn't have or couldn't use. There are also better networks to connect these data sets together for analysis and new software such as MapReduce, Big Table and Hadoop that allows us to break up the analysis of data.
In the past if we wanted access to data or wanted to be able to gain insights from that data it needed to be contained in a structured relational database and we needed to use SQL query tools to extract any value. That is now no longer the case – the data can be in just about any form, structured, unstructured, text, audio, video, sensor, imagines, messy or neat and we can extract value from it.
Bottom line: data is just information. And there are only a set number of ways that information exists and/or can be presented:
* Text data (including numbers)
* Sound data (audio files and music)
* Image data (photographs and graphics)
* Video data (combination of audio and visual)
* Sensor data.
Words, numbers, images, photographs, conversations, sounds and video are all well-known and obvious sources of data or information. Taken individually or collectively they offer a treasure trove of information. But there is also information being generated from sensors on stuff that we never even considered as being information before such as a person's location via GPS sensors in their phone, the vibration of an engine, the stress on a bridge during rush hour or the temperature of the ocean at 5 p.m. last Sunday. Someone somewhere knows all that stuff now or can know all that stuff now because the data from sensors is transformed into a quantified data format that can be analysed. And as sensors are increasingly found on everything from your car engine to your fridge to your TV, the data we can now analyse is exploding and it is often this data combined with one or a few of the more traditional datasets that is going to unlock latent value in business and society.
On their own those data sets or data points can be interesting but the real insights come when you apply analytics, combine them and extract more value than just the value of the original data. Let's look at some common types of analytics:
* Text analytics
* Speech analytics
* Video/image analytics.
## Text analytics
As the name would suggest text analytics or text mining is the process of extracting information and insights from text – often huge amounts of text.
In most businesses there are already huge amounts of text or word-based data in the form of documents, reports, internal and external communication, customer communication, emails, websites, social media updates, blogs, etc. And while all those words are structured to make sense to a human being they are unstructured from an analytics perspective, as they don't fit neatly into a relational database or rows and columns of a spreadsheet. But they still present a huge opportunity if we can just figure out how to use it.
Traditionally text documents may have been coded so that we could find those documents quickly later on. For example, a report may be given descriptors such as 'competitors' or 'shareholder meeting notes' so that a human being can figure out what the document is about without having to read it. When the text is digitized it is also possible to run queries to find certain pieces of information within the text. But this type of enquiry requires us to already know what we are looking for.
Text analytics is now capable of telling us things we didn't already know and perhaps more importantly had no way of knowing before. Access to huge text data sets and improved technical capability means we can now mine the text for patterns and trends that can be incredibly useful in business.
Typical text analytics tasks include:
* Text categorization
* Text clustering
* Concept extraction
* Sentiment analysis
* Document summarization.
### Text categorization
With the constantly expanding data sets of text it is increasingly important that we can categorize the data for future use. Text analytics assigns a document to one or more classes or categories according to the subject or according to other attributes such as document type, author, creation date, etc. Text categorization applies some structure to the text, which can then be used for analysis or query.
Spam filters use text classification to assess the text within incoming emails and decide if the email is legitimate or a message selling discount medication, mail order brides or some other uninvited messages. These spam filters are designed to automatically adapt as the types and content of junk emails change – effectively learning from the text and its various iterations. Email routing also uses this technique to re-route an email arriving at a general address to a more appropriate recipient based on the topic discussed in the text of the email.
Text classification also automatically determines what language the text is in, can assign a genre and deliver a readability assessment, which can be used to help find suitable material for different age groups.
### Text clustering
As the name would suggest text clustering allows you to automatically cluster huge repositories of text into meaningful topics or categories for fast information retrieval or filtering.
An online search engine uses clustering. If you enter 'cell' into a search engine the results would be clustered around 'biology', 'battery' and 'prison' – all of which use a different definition of the word 'cell'.
Often websites that contain a vast amount of information use text clustering to assist their visitors find what they want faster. For example usa.gov, the official portal for the US government, uses document clustering to automatically organize its search results into categories. If you search on 'immigration', before the list of results another list of further categories including 'visas', 'Green Card (Permanent Resident)', 'Diversity Visa Lottery' and 'Citizenship' help the user to refine the search and find what they are looking for more quickly.
### Concept extraction
This technique allows you to extract concepts from text. Language can be quite vague and mean different things depending on the context the language is used. Human beings are easily able to understand what is meant by the context and surrounding words. A computer can do something similar when it has a lot of data to process so as to even out the errors.
Concept extraction techniques are currently in commercial use in law firms where millions of legal documents exist in a business. Concept extraction analytics can hone in on the documents that are likely to be most relevant to the new case thus saving expensive personnel a huge amount of time trying to locate precedents, etc.
### Sentiment analysis
Sentiment Analysis also known as opinion mining seeks to extract subjective opinion or sentiment from text. In the English language alone there are about 615,000 words. If you include technical and scientific words there are millions more. In total about 200,000 words are in common use in English today.1 Some of those words are fairly neutral but others have a distinctly positive vibe while others are more negative. It is these variations in sentiment that text-based sentiment analytics seeks to identify.
The basic purpose of sentiment analysis is to classify the _polarity_ of any given text data as positive, negative or neutral. This can be applied to a whole document or particular paragraphs or sentences (see Figure 4.1).
This type of text polarity sentiment analytics seeks to determine whether the person writing the text is positive, negative or neutral. There are number of software tools (social mention, Twitter Sentiment, Yacktracker and Twitrratr) that can help you measure the sentiment around your product or service.
Twitrratr, for example, allows you to separate the positive tweets about your company, brand, product or service from the negative and neutral tweets so you can see how well you are doing in the Twitterverse (see Figure 4.2).
**Figure 4.1** Example of positive and negative sentiment associated with words
**Figure 4.2** Example of sentiment for Starbucks using Twitrratr
Advanced, 'beyond polarity' sentiment analysis can also go further by making a classification as to the emotional state of the person writing the text such as 'frustrated', 'angry' or 'happy'. This type of analytics is becoming increasingly popular with the rise of social media, blogs and social networks where people are sharing their thoughts and feelings about all sorts of things – including companies and products – much more readily.
When you consider that 53% of people on Twitter recommend products in their tweets, 93% of buyers' decisions are influenced by social media, and 90% of customers trust peer recommendations as opposed to just 14% that trust advertising, it's easy to see why measuring sentiment is so important.2
Millions of people are leaving reviews, rating products, making recommendations and expressing their opinion about businesses, products and services and other people are using those expressions to direct their decision-making. You need to know what people are saying about you and sentiment analytics can help you do that.
Online opinion and expression has turned into a kind of virtual currency for businesses looking to market their products, identify new opportunities and manage their reputations.
So much so that many companies are now creating social media monitoring departments or capabilities to assess and manage what is being said about them online.
For example, the sports drink company, Gatorade, has had a social media command centre inside its Chicago HQ since 2010. 'Mission Control' is a room inside the marketing department that acts like a war room that monitors the brand in real time. Gatorade measures blog conversations across a variety of topics and shows how hot those conversations are across the blogosphere. The company also runs detailed sentiment analysis around key topics and product and campaign launches. It also tracks terms relating to its brand, including competitors, as well as its athletes and sports nutrition-related topics. Basically, Gatorade knows what people are saying about the company and its products all over the world.
And knowing this stuff has made a big difference. Monitoring their 'Gatorade has evolved' campaign, which featured a song by rap artist David Banner, Mission Control were able to see that the song was being heavily discussed on social media. Within 24 hours, they had worked with Banner to put out a full-length version of the song and distribute it to Gatorade followers and fans on Twitter and Facebook, respectively. The company is also using the insights from Mission Control to optimize landing pages and ensure followers are being sent to the top performing pages. As an example, the company says it's been able to increase engagement with its product education (mostly video) by 250% and reduce its exit rate from 25% to 9%.3
Dell is another company that is taking sentiment and social media very seriously. Their social media ground control and command centre is located in Round Rock, Texas and has 70 employees monitoring social conversations (mostly text-based) around the globe 24 hours a day. The team process some 25,000 Dell-related messages via Twitter, Facebook, blogs and other social media every day – in 11 different languages responding to most queries or complaints within 24 hours.4
We all know that a happy customer will probably tell a few friends whereas an unhappy customer can really go to town and cause serious damage to the brand if they get any traction on social media. These command centres allow business to stop those issues before they escalate. Plus putting something right quickly can actually increase brand loyalty and customer satisfaction, so getting ahead of the curve and understanding what your customers are feeling and thinking as they are feeling it and thinking it can have huge commercial advantages.
By embracing social media, retail organizations are engaging brand advocates, changing the perception of brand antagonists, and even enabling enthusiastic customers to sell their products. They are also monitoring social media like Facebook and Twitter to get an unprecedented view into customer behaviour, preferences, and product perception. And manufacturers are monitoring social networks to detect aftermarket support issues before a warranty failure becomes publicly detrimental.
With the myriad of applications it's easy to see why more and more companies are investing in social media and seeking to tap into real-time sentiment and opinion around their products, services and brands.
### Document summarization
Again, as the name would suggest, this text analytic tool allows you to automatically summarize documents using a computer programme to retain the most important points from the original document.
This can be particularly useful for a busy executive who is suffering from information overload and the software can take length, writing system and syntax into account. Search engines also use this technology to summarize websites on result listings.
There are two approaches to automatic summarization: extraction and abstraction. Extraction works by selecting a subset of existing words, phrases, or sentences in the original text to form a summary. Alternatively, abstraction builds an internal semantic representation and then uses natural language generation techniques to create a summary that is closer to what a human might generate. Abstraction summaries might contain words not explicitly present in the original whereas extraction summaries would not.
These summarization tools can provide generic summaries of text or they can create query-relevant summaries, which can greatly speed up research. Imagine if you wanted to know what had been written on a particular topic you would direct the summarization to that topic and save potentially thousands of hours of research time.
Some systems will generate a summary based on a single source document, while others can use multiple source documents (for example, a cluster of news stories on the same topic). These systems are known as multi-document summarization systems.
### Summary
Text analytics is particularly useful for information retrieval, pattern recognition, tagging and annotation, information extraction, sentiment assessment and predictive analytics. In essence it's about getting more information from text and helping text to be even more useful over and above the actual meaning of the text.
For example, I know one organization that uses text analytics tools to scan and analyse the content of emails sent by their staff as well as the social media posts they make on Facebook or Twitter. This allows them to accurately understand the levels of staff engagement. Plus they no longer need the traditional staff surveys which can be expensive and time consuming to complete and analyse.
In another example of text analytics, a researcher at the Microsoft Research Labs in Washington discovered that it was possible to predict which women were at risk of postnatal depression just by analysing their Twitter posts. Instead of using an algorithm that looked at searches or purchases of the mother, the research focused on verbal cues that the mother would use weeks before giving birth.
Those who looked set to struggle with motherhood tended to use words that hinted at an underlying anxiety and unhappiness. There was more negativity in the language, an increase in the use of 'I' as well as an increase in words like 'disappointed', 'miserable' and 'hate' and various expletives that indicated a growing depression in the expectant mother.
As acknowledged by Eric Horvitz, co-director of Microsoft Labs, this type of information can be incredibly useful in helping women at this vulnerable time and also breaking down the stigma around postnatal depression. It would be a relatively simple step for a welfare group to create an app that could run on a smart phone and alert pregnant women to the onset of potential postnatal depression and direct them to resources to help them cope.5
Using the text entered into search engines, text analytics has also been used to discover the previously unknown side effects of taking two drugs together. For example, it was discovered that many people searching for the cholesterol-lowering drug 'Pravachol' and anti-depressant 'Paroxetine' alongside words such as 'tired', 'thirsty', 'dizzy', 'itchy' and 'out of breath', exposed a link to raised blood pressure, as a result of mixing those two particular drugs, that had not come out in clinical trials.6 And in another health example, the Carilion Clinic, in Virginia, says it used natural language processing algorithms to review more than 2 million patient encounters including medical records, clinician notes and discharge documents, which identified 8,500 patients at risk of heart problems with an 85% accuracy rate.7
## Speech analytics
Like text, audio recordings of conversations can now be analysed. As well as analysing the topics being discussed it is now possible to use speech analytics to analyse the emotional content of the speech.
By analysing the pitch and intonations of speech, call centres, for example, can gauge which of its customers are getting angry or frustrated. The amount of speech and location of speech versus silence, i.e. call hold times or periods of silence, can also help customer-facing businesses provide better service and keep their customers happier. As a result, the conversations we've been told 'may be recorded for training purposes' can actually be used for training _and_ provide useful insights instead of being lost or recorded over.
By analysing and categorizing recorded phone conversations between companies and their customers, it is now possible to discover strategically significant information about products, processes, operational issues, areas for improvement and customer service performance. This information gives decision-makers insight into what customers really think about their company so that they can quickly react. In addition, speech analytics can automatically identify areas in which contact centre agents may need additional training or coaching, and can automatically monitor the customer service provided on calls.
Speech analytics applications can spot spoken keywords or phrases, either as real-time alerts on live audio or as a post-processing step on recorded speech. This analytic ability can help programmes manage the unpredictability of people – radio and TV, for example, don't want to broadcast someone swearing if they can help it and speech analytics can help to recognize speech patterns that may be leading to that outcome and cut that person off before any damage is done.
As well as providing useful business insights speech analytics can be seen commercially in voice recognition software for Dictaphones or apps for your smart phone. Plus it is this capability to turn spoken words into text that allows us to speak your search request or commands to Siri on the iPhone, for example, who will listen to your voice and provide the results. Also many modern cars offer a text to voice feature so that if you get a text message to your phone the car will convert the text to speech so you can hear your message without disrupting your driving.
### Summary
Like text analytics, speech analytics provides value not only from what is actually said in the conversations but the way it was said and the emotion behind what was said. As such it is providing insights that simply were not possible even ten years ago and the benefits are significant.
If the analysis of someone's voice can tell us when they are stressed, scared, happy, sad, or even when they are lying, over and above the actual words they use when they are speaking then speech analytics has huge potential in crime and fraud prevention.
Imagine what could be done with speech analytics in police interviews. When the police interview a suspect or a witness those conversations are recorded. Voice analytics could therefore help to identify whether someone is overly stressed or possibly lying. Obviously being in a police station makes even the innocent nervous so the algorithms would need to find statistically significant patterns, but this type of innovation is not far away.
Imagine how it could be used in insurance companies where claims help lines or call centres could have voice analytic software running in the background of all calls to monitor and flag potential fraud.
In 2011, Verizon, a US communications company, filed a patent to allow it to watch you watching TV via your existing set top box. The set top box is the one that connects you to the Internet and allows you to access TV on demand etc., and it already collects data on what you are watching, for how long and how often you fast-forward through the ad breaks, but Verizon wants to listen into your conversations so that it can stream real-time applicable advertising to your TV. So, for example, if you and your spouse were having an argument the set top box would pick that up using voice analytics and stream an ad about marriage guidance. Or if you were discussing the desire to go on holiday an advert promoting Barbados may appear in the ad break of your favourite TV show. Potentially the set top box could contain sensors for both hearing and seeing what people are doing in their own homes and using that data to target advertising.
There are clearly good and disturbing uses of voice analytics and whether it is an appropriate tool for you will depend on what SMART questions you need answers to.
## Video/image analytics
In days gone by the only video data that was collected was security CCTV data. The purpose of that data was to monitor retail or business premises for shoplifting, malicious damage or employee wrongdoing. Most of the security systems would loop recordings which meant that they would record continuously onto video tapes or digital hard drives and then after a set number of days the recording would loop back and re-record over the old data.
If there was no incident in the area being recorded then the data was useless so it was erased over and over again. However, with the advance in video and image analytics all that is changing. The data is now being viewed as useful in ways that were not even considered before. Like all Big Data and analytic changes this has come about primarily because of the quantum leap in storage capability. Ten years ago it would have been unheard of to record and store all that CCTV footage – you'd have needed a warehouse just to keep the old tapes, which would degrade if not kept in a temperature controlled environment – all of which was expensive. Plus, as there was no way to really analyse it anyway there was no point.
When you consider that the amount of stored information grows four times faster than the world economy and the processing power of computers grows nine times faster8 it's easy to see how all that stored video data could now become useful.
In the past the only real analysis of video or image data was through the use of tags that described the video or image. So, for example, if someone uploads a video on YouTube there are descriptor tags attached to the file that are designed to describe what the video is about. So when you search YouTube for 'crazy squirrel' the search engine is searching the vast repository of clips using these tags as a way of hopefully identifying clips that match the search term 'crazy squirrel'. The search facility is not analysing (and has never analysed) all the video footage for instances of a crazy squirrel; they are searching the user uploaded tags for possible hits. YouTube is not actually looking at the images so whether the person finds what they are looking for is determined by how accurate the tags are at describing the content. Of course, those tags are subject to user interpretation so there is no unified way of tagging – it's personal choice.
The latest video analytics tools are changing all that because they now use algorithms that go through the video, scene-by-scene, shot-by-shot and actually capture what is in the video. And then they index that information and use it to identify patterns or cross reference with other analytic tools.
This video content analysis (VAC) is being used so far for:
* Identification (Face Recognition)
* Behaviour analysis
* Situation awareness.
### Face recognition
Face recognition is a computer application that can automatically identify or verify a person from a digital image or video frame.
Traditionally, facial recognition algorithms would pick out facial features and analyse their relative position, size, shape, etc. Or the algorithm would take a gallery of face images, normalize them and only save the distinct elements for face recognition. Recognition algorithms are either geometric, which looks at distinguishing features, or photometric, which is a statistical approach that distills an image into values and compares the values with templates to eliminate variances.
Of course, human beings grow older and change and they can also change features through surgery, so the newer trend which claims far greater accuracy is 3D face recognition. Using 3D sensors to capture information about the shape and topography of a face, the information is then used to identify distinctive features on the surface of a face, such as the contour of the eye sockets, nose, and chin.
The advantages of 3D facial recognition is that it can identify a face from a number of different angles, it doesn't matter if it's day or night the system works. 3D matching technique can be sensitive to facial expression although it's only a matter of time before that's no longer as issue. Another emerging trend called skin texture analysis uses the visual details on someone's skin to identify them. This process turns the unique lines, patterns, and spots apparent in a person's skin into a mathematical statement.
For face recognition to work the technology needs large data sets, called an image gallery, that contains photographs or video stills of faces already identified by name. Software then automatically converts the topography of each face in the gallery into a unique mathematical code, called a faceprint. This faceprint can then be used elsewhere to indentify someone from existing or subsequent photos or when they are caught on CCTV doing their grocery shopping.
Unsurprisingly, considering their existing data sets, companies like Google and Facebook are already ahead of the curve when it comes to face recognition. Using the photographs users willingly upload to Facebook, for example, they are able to create a 3D image of their face and then scan the rest of the Internet to find other pictures of those people to see where else they appear and who else they might know. So, for example, if you upload pictures on Facebook they will be able to match that photo to your corporate website, online dating profile or find you in news articles or blog posts. This way companies are able to triangulate the data they have on you and find out even more.
Face recognition allows these companies to suggest name tags for the group photographs you upload without you even having to tag the photo yourself. Google has applied for a patent on a method that would allow them to identify faces in video and one that will allow people to log on to devices by winking or making some other form of facial expression. And Facebook researchers have reported that their DeepFace pattern recognition system is achieving near-human face recognition accuracy. And if that makes you a little nervous what about the NameTag app, which was available in an early form to people trying out Google Glass. Utilizing face recognition technology amongst others, users would just need to glance at a stranger through the Google Glass and NameTag would instantly return a match complete with that stranger's name, occupation and public Facebook profile information!9
### Behaviour analytics
As well as identification video analytics can also be used to measure and monitor behaviour. Prior to the latest advances all video analytics could do was look at shapes and where those shapes move around but now it is possible to collect data from different cameras in a retail environment and analyse what people do and how they move through the store.
Say you own a store and there are five CCTV cameras located throughout the store. Instead of discarding that data after a week it can be uploaded to a cloud service provider somewhere without any need on your part for new infrastructure or equipment. An analytics company could then analyse that data by bringing all the camera data sets together so that you can understand how people move through your store and what promotions they stop at and pay more attention to.
One of my clients – a world-leading retailer – is currently using this type of technology. Originally they approached us to help them find data solutions to their most critical business questions. One of the customer-related business objectives was that they didn't want their customers to have stood in line for a long time at the checkout. A noble goal – but the problem was that they had little data (other than anecdotal) on how long the lines were and how long customers stood there.
They didn't have any data that could help them understand the current situation in their stores. Initially they asked the checkout supervisor to estimate how long the queues had been every hour but it wasn't very reliable. The supervisors were busy, they forgot and so they just guessed and the whole process was prone to mistakes.
We then changed it to reflect real time by adding a script to the checkout. So before the checkout operator started to scan products a prompt would appear on their screen to ask them how many people were in their queue. If there were 3 people in the queue at that time the operator would input '3' and then start to scan the items as normal. They would repeat this process before every transaction. This approach was obviously better because it was more accurate, although it was still prone to mistakes or the operator just typing anything or counting incorrectly. Plus it also took a lot of time – even if it was a few seconds per transaction, multiplied across thirty checkouts and thousands of stores it became expensive.
The retailer then moved to sensors that were installed over the tills that would detect how many bodies were in the queue. This approach was cheaper, more accurate and more reliable.
However, as video analytics has developed they now use the CCTV camera surveillance data that they already have. In the past each CCTV camera in every store had its own database and it would record the images for a week and then overwrite those images with new footage, but they realized that information was actually much more valuable than just helping to identify and stop theft. Now all the information from all the cameras in all the stores are connected to one Big Database that holds all the CCTV camera data and specialist software puts it all together to recognize movement and patterns as well as face recognition.
So basically the next time you walk into your local supermarket to buy a ready cooked BBQ chicken for dinner you will be caught on CCTV at various points around the store. The software will then combine all those images from different cameras to see how you walk through the store, what aisles you visit, what promotions you stop and look at and how long you are waiting in a queue.
Not only did this data allow the supermarket to track how long the queues were and take steps to reduce the wait time and meet their strategic objective, but it also gives them so much more information.
If you combine that macro surveillance CCTV data with micro video data or sensor data positioned at strategic points around special offers or promotions, the company can now see what people pick up, look at and buy and what they pick up, look at and put back on the shelf.
At the moment this data is not matched to an individual person – it's just recording what a person is doing but it's a small and very doable step to cross reference that information using face recognition and loyalty cards to access data about individual customers and potentially tailor promotions and offers based on that data.
### Situation awareness
Video and image analytics can also help where situation awareness is critical to decision-making in complex, highly fluid situations such as aviation, air traffic control, ship navigation, power plant operation and emergency services.
Using technology and data from video footage can help to alert personnel to any changes or anomalies can save lives and prevent crime.
For example, there is now software that allows you to automatically monitor a location or space 24/7; that video footage is then analysed using video and behavioural analytics solution and alerts you in real time to any abnormal and suspicious activity. Once installed and provided with the initial video feed, the software observes its environment, learns to distinguish normal behaviour from abnormal behaviour and sends relevant, real-time alerts to security personnel. The system is also self-correcting which means that it continuously refines its own assumptions about behaviour and no human effort is required to define its parameters.10
### How video/image analytics is already being used
Obviously video analytics have huge implication for consumer behaviour, security, anti-terrorism and also our own personal privacy.
This technology is, for example, already in use in law enforcement, crime prevention and it's also being used by casinos. Police departments in New York, Pennsylvania and California already use face recognition to cross reference a picture of a suspect, perhaps taken from CCTV to their image gallery of convicted criminals in order to find a match.
To a criminal, life must have been so much easier before Big Data and powerful analytics applications like face recognition. If you were not caught red-handed or your license plate or description noted down by an observant passer-by you didn't have a huge amount to worry about. Today intelligence-driven policing that utilizes smart technology serves to reduce both crime and the fear of crime.
In Victorian London, Dickens' Artful Dodger plied his trade as a pick-pocket and thief but he wouldn't last long on the streets of modern London. Even if he wasn't identified by the victim or a witness video footage available from any of the estimated 422,000 CCTV cameras in London alone would probably record his movements. This video footage is now routinely used to create a 3D faceprint of a suspect which is then used to compare to images available on the Internet or social media sites. Even criminals have Facebook profile pictures that can identify them using facial recognition technology. The detective of the future will be able to find out everything there is to know about a suspect, simply from one photographic image or video still. And considering there are estimated to be about 5.9 million CCTV cameras in the UK then that's got to be good for crime prevention.11
As well as video images, police forces have been compiling databases of offender DNA samples to compare with those taken from crime scenes. The Police National Computer in the UK now contains details of over 9 million individuals – nearly a sixth of the population. Police also have access to details of 50 million drivers taken from the records of the Driver and Vehicle Licensing Agency. Another UK initiative referred to by police as their 'Ring of Steel' involves positioning CCTV cameras with automated number plate recognition (ANPR) technology around every road into and out of a town. The cameras record the number plate, take a photo of the car and passengers and record the direction of travel. This is building up a huge database which police forces have started to interrogate using Big Data technology so they can correlate vehicle movements with crimes.
In the United States, addressing gun crime is a more urgent political priority and they are also turning to Big Data for answers. One tool, called ShotSpotter, uses a network of sensors positioned across an area or city to provide real-time GPS reports whenever a gun is fired. In essence the 'data' being analysed here is the entire soundscape of the city – when the distinctive sound of a gun being fired is recorded that data is collected and police are alerted. There is talk of expanding the system to include automatic video footage – instantly activating cameras to capture footage the second a trigger is pulled. The results in one New York area are said to show a 90% reduction in gun crime incidents since the system was installed.
Law enforcement is constantly finding new ways to use technology and Big Data in the fight against crime. In Silicon Valley, biometric technology is creating guns that will only fire when held by someone legally entitled to do so. CCTV cameras are no longer impotent, immobile observers – they are commonly used in police cars and carried by officers to create a permanent digital record of everything going on around them, and are increasingly taking to the skies, attached to remote controlled drone aircrafts.
All of this will make it harder for criminals to commit crimes and may even eradicate some forms of crime all together. However, criminals have always found new ways of doing their 'business' and there has been a huge increase in the amount of credit card and online identity fraud. But even there new Big Data algorithms are being developed to detect fraudulent behaviour in real time. Overall Big Data and analytics are making our world a safer place.
Casinos are also using this technology to identify high rollers for special treatment and presumably to identify people they want to keep out of their casinos too. In Japan, grocery stores even use face-matching to classify shoppers and blacklist serial complainers or shoplifters.12
### The dark side
While the applications are endless, so too are the concerns! The biggest challenges around faceprinting are that it can be used without the person's knowledge or consent. From a safe distance someone can covertly identify a live person by name which then connects the user to intimate details about that person like their home addresses, dating preferences, employment histories and religious beliefs. In 2011 researchers at Carnegie Mellon reported that this was not a hypothetical risk when they had used a face-recognition app to identify some students on campus by name, linking them to their public Facebook profiles and, in some cases, to their Social Security numbers.
Clearly, the potential to exploit this technology already exists – especially as there are currently no laws regulating the use of face recognition. In many ways face recognition is the new frontier. Because of these innovations companies increasingly don't even need to have their own customer data or rely on their own loyalty cards for information; they can use face recognition and video analytics to scan the Internet to find stuff out about you and use those insights to target promotions or offers. And although that isn't happening right now it's probably only a matter of time.
It's also possible, right now, for companies to not only analyse an individual image but how things have changed over a series of pictures. So, for example, if you upload a number of pictures of yourself on social media sites they can compare them to previous pictures of you and determine, for example, if you have put on weight over the summer. Potentially that data is valuable as it can be sold to companies like Weight Watchers who can target advertising to your profile!
## Combined analytics
Like data itself, the value is not just in one data set over another; the real value comes from the combination of data sets and the combination of analytics tools to analyse that data.
Remember the example of the fashion retailer I shared in Chapter Two. I was able to help them to improve sales and become much more efficient by running analytics on a combination of sensor data, traditional sales data and video data. That same retailer also wanted to know how to recruit the right people to their stores. As fashion is such a word-of-mouth industry they wanted to recruit people who were influential. In his book _Tipping Point_ author Malcolm Gladwell calls these individuals mavens. Mavens are influential individuals who accumulate knowledge. As a result they often hear about and spot trends quicker than everyone else. As trusted experts their opinion matters and these people can influence others and kick start a craze. Obviously in fashion mavens are useful people to have around because people listen to what they say.
The idea therefore was for this fashion retailer to employ all the popular kids in their shops because they would influence more people to visit. In the past they had to rely on their gut feeling and judgement when they were interviewing candidates but now they use data and analytics and look at their Klout score. The Klout score measures how popular or influential a person is, as measured by their social media presence and activity. How many responses or 'Likes' does the individual get when they post a tweet or update their Facebook status. How many followers do they have on Twitter or LinkedIn? Klout assigns a score out of 100 and this retailer uses that score to help pick the right applicant. As well as having the right personality, qualifications or experience they will seek to recruit individual with a high Klout score because those people will help to drive traffic to the store.
### Medical application of combined analytics
Traditional data sets of medical information together with new sensor data sets are combining to save lives. For example, the Hospital for Sick Children in Toronto uses a Big Data and analytic platform that alerts doctors to life threatening problems in premature babies.
Premature babies are particularly susceptible to late-onset neonatal sepsis, a blood infection that usually occurs several days after delivery. Obviously a premature baby's natural defences are low and are often underdeveloped. By monitoring everything from respiratory rate to heart rate to blood pressure and blood oxygen saturation and then analysing the vast data streams, doctors can monitor an infant's vital signs in real time and detect changes in their conditions.
Complex algorithms examine the data streams – about 1,200 data points every second – looking for features that are known to occur before the infection becomes clinically apparent. When found, the doctor is alerted and the baby receives life saving antibiotics _before_ they become ill. Premature babies are extremely fragile and this intervention is saving lives.13
Doctors in Kaiser Permanente's neonatal intensive care units, or NICUs, are using an innovative online calculator to determine whether preterm and newborn babies are at risk of the same life threatening condition.
Thanks to robust data sets maintained by the Division of Research (DOR) fewer babies are separated from their mothers and put on IV antibiotic drips. Instead the data is much smarter in identifying which babies need this intervention and which don't. Every year, the Kaiser Permanente Northern California Region provides care for 35,000 newborns and 350 very low birth weight infants under 1,500 grams. Since 1993, the DOR has been collecting demographic and clinical data on every infant born in the region, which currently amounts to more than 800,000 records. In addition, another data set aggregates information on infants admitted to NICUs – currently over 50,000 records. All of which provides unprecedented access to indispensable information that is saving lives and improving the quality of care. And other Kaiser Permanente regions, including Southern California, are also making use of neonatal data to improve care of newborn and premature infants.14
### Other applications of combined analytics
Combined analytics are also saving lives, reducing damage and saving money in a completely different way. Every now and again one of New York's hundreds of manhole covers explodes into the air like Old Faithful, the Yellowstone National Park geyser. However, unlike Old Faithful no one had any idea when a manhole would blow or how to stop it. And considering that a manhole cover weighs about 300 pounds, flying up to 50 feet in the air before crashing to the ground – this was a problem.
In May 2013, three manholes exploded in Brooklyn, setting cars on fire and sending people running for cover. The explosion also cut power to the area. In January 2014, three more exploded in the Upper East Side shattering a nearby brownstone window and scorching a Mercedes. In 2011, Consolidated Edison, the public utility company that provides the city's electricity and maintains the manholes, reported more than 60 instances of exploding manholes and in 2008 one of their employees was killed when the manhole he was working in exploded.15
What happens is that underground electricity cables become frayed with age or by corrosive chemicals (like salt on the roads in winter), or by rats biting through them. These cables, carrying about 13,000 volts of electricity heat up the cable insulation which then smoulders and releases gases. Pressure from the gas then builds up inside the manhole and the frayed electrical wires ignite the gases causing the manhole cover to explode into the air like a missile.
In the past, Con Edison conducted regular inspections and maintenance of the manholes every year. But with 94,000 miles of underground cables, 51,000 manholes and service boxes in Manhattan alone the job was difficult and unpredictable. To add to the challenge some of the infrastructure dated back to the days of Thomas Edison, the company's namesake. One in 20 cables were laid before 1930.
In an effort to reduce the incidents and predict where problems would occur so that the maintenance team could take appropriate action and avert the incident, Con Edison applied analytics.
Although the company had kept records since 1880s it was not digitized and the data was certainly not ripe for analysis. But with the help of statisticians at Columbia University they took all the raw, incredibly messy data they already had and applied a variety of analytic tools to help them establish which manholes were likely to blow. The team started with 106 predictors of a major manhole disaster and then condensed those down to a handful of the strongest signals as evidenced by the data, not opinion or assumption. The resulting algorithm then identified the top 10 per cent of manholes, which included a whopping 44 per cent of manholes that ended up having a severe incident.16 Just knowing how to narrow the focus has resulted in significantly less incidents year on year.
Because of combined analytics a computer can now do the job of a journalist and write stories! A company called Narrative Science launched a software product that can write newspaper stories about sports games directly from the games' statistics. The same software can now be used to automatically write an overview of a company's business performance using information available on the web. It uses algorithms to turn the information into attractive articles. Newspapers of the future could be fully automated. While these reports may lack the human touch it may well be that all stories are started in this way only to be edited and 'humanized' by the journalist. Considering the world's thirst for content this may well become normal in a few years!
### What your 'Likes' say about you
Today everything is potentially data and even the most innocuous piece of information can be turned into insight if you apply analytics to a large enough data set.
Did you know for example that your 'Likes' on Facebook are being used to expose intimate details about you as well as personality traits and preferences that you might not otherwise share? Most of us don't want to share personal details such as our religious beliefs, political views, sexual orientation or how much alcohol we drink. It's none of anyone's business!
And yet a study conducted by researchers at Cambridge University and Microsoft Research Labs showed how the patterns of Facebook 'Likes' can be used automatically to very accurately predict a range of highly sensitive personal attributes. Using the 'Like' data of 58,000 volunteers the study also illustrated that the 'Likes' can have little or nothing to do with the actual attributes they help to predict and often a single 'Like' is enough to generate an accurate prediction.
The study found that a 'Like' for:
* Curly Fries, Science, Mozart, Thunderstorms or The Daily Show predicted high intelligence.
* Harley Davidson, Lady Antebellum, and I Love Being a Mom predicted low intelligence.
* Swimming, Jesus, Pride and Prejudice and Indiana Jones predicted satisfaction with life.
* Ipod, Kickass, Lamb of God, Quote Portal and Gorillaz predicted dissatisfaction with life.
* So So Happy, Dot Dot Curve, Girl Interrupted, The Adams Family and Kurt Donald Cobain predicted being emotionally unstable or neurotic.
* Business Administration, Skydiving, Soccer, Mountain Biking and Parkour predicted being emotionally stable or calm and relaxed.
* Cup Of Joe For A Joe, Coffee Party Movement, The Closer, Freedomworks, Small Business Saturday and Fly The American Flag predicted that you were old.
* Body By Milk, I Hate My Id Photo, Dude Wait What, J Bigga and Because I Am A Girl predicted that you were young.
* Kathy Griffin, Adam Lambert, Wicked The Musical, Sue Sylvester, Glee and Juicy Couture predicted you were a homosexual man.
* X Games, Foot Locker, Being Confused After Waking Up From Naps, Sportsnation, WWE and Wu-Tang Clan predicted you were a heterosexual man.
When we click 'Like' we want to show our friends on Facebook that we feel positive or supportive of specific online content such as status updates, photos or products, books, music or other individuals such as celebrities. What many of us don't realize is that by doing so we openly share information about ourselves that can then be used to predict other, more personal, attributes that we would never dream of sharing so openly. We now live in a world where everything in digitalized and a great deal of what we do leave a digital trail about our life and our preferences, which in turn can make it easy to figure out our attributes and personality traits.
Predicting personality traits and attributes is nothing new and was around long before Big Data Analytics. What's changed, however, is that we had to complete a survey or give others permission to profile us in that way. What this study demonstrated is that permission is no longer needed and we can ascertain personality traits and predict behaviour based on publicly available data without us ever knowing about it. This means that the information you reveal by clicking on a 'Like' button can – by default – be used or 'exploited' by others, for good causes and not so good ones.
Commercial companies can use this type of Big Data analytics to dynamically customize the ads you see on your Facebook page (or in fact anywhere) based on your personality traits. Just think of an online ad for the latest car – for people that are classed as shy, reserved and married, the ad might highlight safety and family friendliness, while for an single, outgoing and active person it might highlight the attractive design and sporty drive. More worryingly, Governments could (and do) use this type of analysis to identify our political views and how they are shifting. Insights from this can then be used to target election campaigns etc.
One problem is that these predictive models are not perfect. No model ever is. Clearly not everyone who 'Likes' curly fries is automatically highly intelligent. Not everyone who 'Likes' The Addams Family is neurotic and yet this type of analytics could be used to wrongly label people and those labels could affect their ability to gain access to products and services.
Plus these predictive models can also be more than a little disconcerting as one Minneapolis father found out. Understandably furious that his local Target store had sent his 15-year-old daughter coupons for discounted maternity products, he visited the store. After demanding to see the store manager he said, 'My daughter got this in the mail! She's still in high school, and you're sending her coupons for baby clothes and cribs. Are you trying to encourage her to get pregnant?
The manager did know what the man was talking about, but could see that the coupons had been sent to his daughter so apologized on behalf of the company and put it down to an error. In fact he was so disturbed by the mailer that the manager called a few days later to apologize again to the father. Only this time the tables had turned: 'I had a talk with my daughter, it turns out there's been some activity in my house I haven't been completely aware of. She's due in August. I owe you an apology.'
Big Data and analytics meant that Target knew a high school girl was pregnant before her own father did. And the reason they did was they were able to identify 25 products that, when analysed together, allowed their statistician, Andrew Pole, to assign each shopper a 'pregnancy prediction' score.
Sending coupons congratulating people on their pregnancy was obviously going to freak them out and make them feel uncomfortable so what they ended up doing was mixing baby-related merchandise coupons in with other coupons that they knew the client would not be interested in so that it looked random – and it worked. In the eight years between 2002, when Pole was hired, and 2010, Target's revenues grew from $44 billion to $67 billion.17
When we hear about these predictive models and what companies are doing with Big Data and analytics there is a real danger that privacy will give way to probability. And there is also clearly an issue of transparency, which is going to be essential for the SMART business to manage.
## Transparency
You may be surprised and a little perturbed about some of the examples I've shared in this book. There is little doubt that Big Data is here to stay and the data we can now collect and store is going to grow exponentially. So too are the analytic tools that we will have access to in order to analyse that data.
The potential is enormous and the combination of data and analytics will change every aspect of business from customer service, to product development to R&D to marketing to logistics to operations and finance. But like most brilliant innovations the value can be derived for good and bad. The Internet for most people is an amazing thing– it's like having a library at your fingertips, it can speed up business and allows us to stay in touch with people we care about all over the world. But there is also a seedy underbelly that is so disturbing I don't even want to write about it.
The same is potentially true of Big Data and analytics.
The possibilities of face recognition software alone are more than a little frightening and whilst that software can help to prevent crime and thwart terrorist activities it can also be used to spy on ordinary people for commercial purposes. And therein lies one of the biggest challenges – most people have absolutely no idea what is going on in darkened rooms in places that don't officially exist or in the basements of giant corporations who have access to masses of data and futuristic technology.
We don't know what data is being compiled about us and even if companies or applications tell us in their terms and conditions most people don't read them, or even if they do read them they don't understand them or understand the implications of what they are agreeing to.
For example, did you realize that if you use Google's free email service Gmail they feel that you can't legitimately expect privacy. Basically Google believes it is okay to read and analyse the content of any and all of your private emails whether they are sent or received from a Gmail user.
This revelation was put forward in a brief that was filed in a federal court as part of a lawsuit against Google. Google is accused of breaking US federal and state laws by scanning the emails of Gmail users and in their defence put forward this statement (which was recently exposed by Consumer Watchdog):
> _Just as a sender of a letter to a business colleague cannot be surprised that the recipient's assistant opens the letter, people who use web-based email today cannot be surprised if their communications are processed by the recipient's ECS provider in the course of delivery. Indeed, 'a person has no legitimate expectation of privacy in information he voluntarily turns over to third parties.' Smith v. Maryland, 442 U.S. 735, 743-44 (1979)._
So essentially if you sign up to use Gmail then you waive all rights to privacy and Google can use what they discover using text analytics to better target their advertising. Only my guess is that most of the 400+ million users of the Gmail service don't currently realize this.
Everyone understands that companies need to make money and providing a free email service such as Gmail may be reward enough for some people. Many people may not care about this piece of information but if we are to navigate these murky waters safely then certainly I believe there needs to be much, much more transparency about what is being collected and how the data is being used or could be used. And we should have the right to be forgotten.
In 2014, Mario Costeja Gonzalez won a landmark court case against Google over his right to be forgotten. For years the Spaniard had been irritated by the fact that a simple search of his name would yield results showing two articles in the newspaper, _La Vanguardia_ – and indexed by Google – mentioning that his house had been repossessed due to social security debts. Gonzalez argued, quite legitimately, that the debts were paid and the information was out of date and irrelevant.
In a David and Goliath ruling, the European Union Court of Justice agreed with Gonzalez when it ruled that links to irrelevant or out-of-date information about individuals should be removed by search engines on request. The court ruling, which can't be appealed, appears to show that European law supports this right – and Google (as well as a lot of other Big Data hoarders) could soon find themselves faced with a flood of similar requests.
Google expressed disappointment in the ruling, arguing that it amounted to censorship but surely we should be allowed to take back control of our data – especially when that data is irrelevant, out of date or just plain wrong. It hardly seems fair that anyone can put anything online – true or false – and that person can't do anything to stop it. This ruling at least goes some way in putting someone's personal data back in their hands. In an interview with the Guardian newspaper, Gonzalez said, 'I was fighting for the elimination of data that adversely affects people's honour, dignity and exposes their private lives. Everything that undermines human beings, that's not freedom of expression.'
The 'right to be forgotten' proposals in Europe in many ways reflect the recently passed Californian 'eraser' law – requiring tech companies to remove material posted by a minor, if they request it – due to take effect next year. And these developments will not only apply to Google, but to all other search engines including Yahoo and Microsoft's Bing.
It is, however, unclear how it might apply to social media companies such as Facebook or Twitter. Does this mean that anyone who doesn't like something which has been said about them online can demand that it is struck from the record? Obviously this could potentially lead to censorship as the rich and powerful pay to have their dirty laundry removed from the Internet. But at the moment, it seems unlikely; the court was clear in its ruling that publishers will have various defences – including public interest – with which to resist requests for information removal.
Campaign group, Index on Censorship, has been forthright with its criticism of the decision – calling it: 'Akin to marching into a library and forcing it to pulp books'. Ultimately it all depends on how this new law (which currently only applies to citizens of the European Union) is implemented in practice. One thing for sure it will make life more complicated and expensive for search engine providers.
But it's not just Google. Facebook is famous – or rather infamous – for constantly tinkering with the privacy policy and privacy settings.
### Always add value
The whole idea of data protection and giving people back power over their data is a really important point. And for me the best way forward is not only to be really transparent about how the data is being used but also adding value to the user.
If a company is giving something useful back to me as a user then I don't mind them aggregating some of my data if it also helps them. For example, I have one of the latest Smart TVs from Samsung that allows me to programme the TV and using the inbuilt camera it detects the faces of my children and limits what they can watch. I don't mind Samsung knowing what I watch, when I watch and how long I watch my smart TV because they are helping me and my wife to protect our children from stuff they shouldn't see. Samsung did, however, get into trouble when it came out that they are actually counting the number of people watching TV but I think a lot of this could be avoided with greater transparency and by delivering value to the user.
In the same way I don't mind Jawbone, the manufacturers of my 'Up' band, analysing my sleeping patterns because the system helps me monitor my health and well-being in real time. I also use the data from my band to get better sleep and recover faster between time zones! One of the really cool features of the 'Up' band is that it's possible to set a smart alarm based on sleep patterns.
For example, I can set my alarm for 7 a.m. and instruct the 'Up' band to wake me at the best time anywhere from 6 am onwards. If I transition into a light sleep phase at 6.40 a.m. the 'Up', which is monitoring my sleep patterns throughout the night, will wake me up at 6.40 a.m. before I drift back into deeper sleep. Research has shown that if you are woken from deep sleep you feel tired for the rest of the day so this innovative feature solves the problem and ensures I'm more refreshed when I wake. The sleep functions are also fantastic when I'm travelling. Often I'm in two or even three different time zones in any given week so the 'Up' band can tell me when it would be optimal to have a power nap so that I can recalibrate my sleeping more quickly. It will even calculate the length of the power nap and wake me up appropriately so I can recharge quicker than I would normally.
But I want to know the truth about what they are doing with the data. If the data is aggregated and not necessarily connected to me as an individual I'm fine with that because it can help us understand more. For example, the data that Jawbone has collected on sleep alone is making huge inroads into our collective understanding of sleep, insomnia and how sleep is impacted by various factors. And this has the potential to help a lot of people.
If you want to be a SMART business, be open, honest and transparent about how you want to use the data. Operate ethically and offer genuine value to the customer in exchange for providing you with that data. If you provide value most people will be happy – especially if you remove personal markers that link you as an individual to the information. So in the case of Jawbone, the data they collect from my 'Up' band is not connected to me as an identifiable individual – it's just sleep data. This is known as anonymization: a process of turning data into a form that does not identify individuals and where identification is not likely to take place.
If you can demonstrate that you are using the data ethically people will respond. Plus this aggregate use of data will improve products and make them cheaper.
## Prediction vs. privacy
In addition to the issues of transparency, there are also people who believe that Big Data heralds the end to causality in favour of correlation.
Up to now human beings have been driven to know 'why' and will seek out causality to explain a whole range of phenomena from the weather to illness to human behaviour. In the past if you wanted to know something you developed a hypothesis and run experiments to establish if the hypothesis was correct or not. The experiments that took place would vary depending on what you were trying to find out but, regardless of whether you were seeking to understand consumer behaviour or the efficacy of a new drug, the experiment would always take a sample of data, people, ingredients or components in order to test the hypothesis. The sample was always therefore limited in size and the results were then extrapolated out to make assumptions or best and worst case scenario predictions for everything from the spread of disease to the accumulation of credit card debt.
This approach has worked well and is credited with many breakthroughs in just about every area of human endeavour. Big Data could change all that.
If you test a hypothesis on consumer behaviour, for example, the sample of people you test the hypothesis on is based on the assumption that the sample is representative of all consumers. It's not. But it was the best option considering the lack of data. The advent of Big Data and specifically the technology to store, collate and analyse that data means lack of data is now no longer the problem. Theoretically at least we will be able to use a sample where N equals All.
The danger of course is that N does not = All any more than the sample equals all.
Even if Target's really clever analytic algorithm identifies a woman as being pregnant that doesn't mean she is definitely pregnant. All the algorithm does is identify correlations and these correlations are being heralded as the end of causality. In other words Target doesn't need to know why a particular collection of product purchases indicates pregnancy, they just need to know it does so they can tailor offer and market to that audience in a way that increases sales. But what about the person who bought the products for someone else or the fluke purchase?
In the same way, if an insurance company was able to use all claim information and found a correlation between fraud and the amount of time taken to complete an online claim form then they don't need to know why time is an indication of fraud they just need to know it is so they can initiate an investigation of all claims over a certain time period. But what about the person who is not very computer literate or the person who was interrupted by her teenage daughter wanting a pair of trousers ironed or the man who went to make a cup of tea or answer the door bell?
The danger with these predictions is that each of us will be pigeon-holed by all sorts of organizations and businesses based on probability, not reality. Privacy and transparency is not the only ethical challenge with Big Data. What happens when someone is refused a mortgage because some algorithm identifies that person as a high risk even though he's never actually defaulted on a mortgage before? What happens if your insurance premium is increased based on your probability to claim in the future even though that future hasn't arrived yet? Does this mean that you and I could potentially be arrested for a crime we haven't committed before we actually do it because some analytic program has spat out our name based on a collection of other seemingly unrelated behaviours, purchases or situations?
### Do the right thing!
There are still significant moral and ethical dilemmas to be ironed out in this area. Big Data is a little like the gold rush – a lawless frontier of extraordinary opportunity for those willing to take the early risk. But the law _will_ catch up as more and more people become more and more uncomfortable about what's being collected, used and what's now possible. I watched a sci-fi TV programme recently where personal identification and tracking software was used in weapons so a bullet could be programmed to a person, fired from forty miles away and the bullet would track the individual and shoot them. It's only fiction but the component parts of that fantasy are already here and it's not actually such a huge leap to make this a reality!
Reality or fantasy aside, in business it just makes business sense to anticipate the day where greater privacy will be demanded and work ethically, responsibly and transparently from the start. Besides, many of the early wins have already been reaped by the pioneers such as Walmart, Target, and Amazon, etc.
Don't get bamboozled by N = All. Instead start with strategy, identify your SMART questions and what metrics and data can answer them and only apply analytics to those data sets. Offer value to your customers and make it beneficial for them to share their information, either through better or cheaper products or services. Use predictive analytics if appropriate but only to cluster people for further analysis –don't make blanket, automated untested assumptions about what people are going to do or may do. Aim for a win-win for you and your customers.
## Key points and call to action
* When you are clear about your strategy objectives they provide clarity to what metrics and data you will need to collect in order to answer your SMART questions. Starting with strategy therefore provides a target to hit that allows you to hone in on the right metrics and data to focus on.
* But it's not enough – you need to analyse the data in order to extract meaningful and useful business insights.
* Like metrics and data you need to understand what's possible before you can confidently decide what analytic techniques are best able to deliver answers to your smart questions.
* First you must appreciate that there are 5 key formats in which business data exist:
* Text data (including numbers)
* Sound data (audio files and music)
* Image data (photographs and graphics)
* Video data (combination of audio and visual)
* Sensor data.
* These data types in turn make it possible to conduct:
* Text analytics
* Speech analytics
* Video/image analytics
* Combination analytics.
* These analytic options allow you to identify patterns, better understand behaviour and learn more about your customers or employees through innovations such as sentiment analysis.
* Sentiment analysis is where you can assess everything from the words customers or employees use, to their tone of voice in order to figure out how happy or unhappy they are.
* What analytics you apply will depend on the questions you are seeking answers to around your strategic objectives.
* A word of warning: It's very easy to be seduced by some of the really cool analytic capabilities that currently exist. Don't be led astray – your job is to find the best, most accessible and inexpensive technique possible – regardless of how sexy it is.
* When it comes to data analysis be mindful of being honest and transparent over what you want data for and how you intend to use it – especially with your customers.
* Always seek to add value so that the people providing the data, be that customers, employees or other stakeholders, feel it's a fair and worthwhile exchange.
## Notes
1 Bryson, B. (1990) _Mother Tongue._ London: Penguin. 2 Qualman, E. (2014) Social media <http://www.youtube.com/watch?v=zxpa4dNVd3c>. 3 Ostrow, A. (2010) Inside Gatorade's Social Media Command Center. _Mashable,_ June 15, 2010. <http://mashable.com/2010/06/15/gatorade-social-media-mission-control/> 4 Holmes, R. (2012) NASA-style mission control centers for social media are taking off CNN Money <http://tech.fortune.cnn.com/2012/10/25/nasa-style-mission-control-centers-for-social-media-are-taking-off/> 5 Knapton, S. (2014) 'Postnatal depression can be predicted by monitoring women's twitter feed, scientists find'. _The Telegraph._ 6 Ibid. 7 Jackson, K., Panizar, J., Struckhoff, R. and Silva, P. (2014) 'IBM expands US federal healthcare practice', <http://gov.ulitzer.com/node/3071692>. 8 Mayer-Schonberger, V. and Cukier, K. (2013) _Big Data: A Revolution That Will Transform How We Live, Work and Think._ London: John Murray Publishers. 9 Singer, N. (2014) 'Never forgetting a face', _New York Times_ , <http://www.nytimes.com/2014/05/18/technology/never-forgetting-a-face.html?_r=0>. 10 <http://videoanalytics.com/?p=73> 11 Barrett, D. (2013) 'One surveillance camera for every 11 people in Britain, says CCTV survey', _The Telegraph,_ <http://www.telegraph.co.uk/technology/10172298/One-surveillance-camera-for-every-11-people-in-Britain-says-CCTV-survey.html>. 12 Singer, N. (2014) 'Never forgetting a face', _New York Times,_ <http://www.nytimes.com/2014/05/18/technology/never-forgetting-a-face.html?_r=0>. 13 Horowitz, B.T. (2013) 'IBM InfoSphere, Big Data help Toronto hospital monitor premature infants', <http://www.eweek.com/enterprise-apps/ibm-infosphere-big-data-help-toronto-hospital-monitor-premature-infants.html#sthash.fis2ibpX.ybxyNuGY.dpuf>. 14 Byron, J. (2014) Big Data Improves Care for Kaiser Permanente's Smallest Members <http://share.kaiserpermanente.org/article/big-data-improves-care-for-kaiser-permanentes-smallest-members/> 15 Cleri, C. (2011) 'Con Ed's manhole covers exploding all over NYC', <http://www.shopforenergy.com/ny/420-con-ed's-manhole-covers-exploding-all-over-nyc-.htm>. 16 Mayer-Schonberger, V. and Cukier, K. (2013) _Big Data: A Revolution That Will Transform How We Live, Work and Think_. London: John Murray Publishers. 17 Duhigg, C. (2012) 'How companies learn your secrets', _The New York Times._ http://www.nytimes.com/2012/02/19/magazine/shopping-habits.html?pagewanted=1&_r=2&hp&.
# 5
R = REPORT RESULTS
Big Data and analytics present a phenomenal opportunity for all business regardless of size or sector. But even if you start with strategy, identify the metrics and data that could help you answer your SMART questions and drive your strategy, and even if you then applied analytics to that data to identify insights, you still need to report insights in a way people understand.
Big Data and analytics may well pave the way to some really cool innovations, greater customer understanding and real time monitoring of what's actually happening in the business. But unless the results are presented to the right people in a meaningful way then the size of the data sets or the sophistication of the analytics tools won't really matter and the results will not inform decision-making and improve performance.
Of course Big Data and analytics are far sexier than humble reporting. Thankfully, reporting is in the middle of an extensive and exciting makeover that promises to help unleash the true potential of data.
Business leaders are already struggling to keep up with all the data they come across in the course of a normal week. They already receive floods of emails, countless reports they only ever skim read, if at all, so the idea of adding to that data explosion is not very welcome news for most business executives. All too often the real nuggets of information that could really impact strategy and tactics are lost within a 50-page report.
A perfect example of failing to communicate important information and insights is NASA's launch of the Challenger Space Shuttle. Some of the engineers at base had some serious concerns. Their calculations and tests showed that there was a serious problem with one of the components (an O-ring), which could potentially fail. Prior to the launch, the engineers reported all the detailed results of their tests to decision-makers at NASA with the assumption that they would take one look at the data and abandon the launch. The problem was that the messages were not clear; key facts were hidden in the detail of long reports. As a consequence, the Space Shuttle was launched, leading to a devastating disaster and the death of seven crew members.
It is important that any insights are reported in a way that focuses on making sure the right people get the right information, in the right format so they can make the right decisions more often.
## Data visualization
Analytics is only useful if the target audience understands the information and insights it creates.
According to research by the Advanced Performance Institute the most popular format for communicating results are tables and spreadsheets complemented by graphs and charts. The second most popular is purely numeric without the graphs and charts and the least common way to present results was through narrative commentary with supporting numeric data, and verbal communications.
Traditionally reports have used various types of graphs and charts to help visualize the results. The most common are:
* **Bar graph** – bars are positioned either vertically or horizontally side by side. This visualization is particularly useful for making an easy comparison between adjacent values.
* **Line graph** – ideal for displaying time-related data such as variations in share price over time, trends or revenue over the year.
* **Pie chart** – displays various segments of the pie, each representing the data as a percentage of the total data.
* **Scatter chart** – also known as scatter plots these are particularly useful for illustrating the correlation between two sets of data and illustrating the strength and direction of that relationship.
Visualizing the data through charts and graphics can not only make the data more accessible and meaningful but also can better illustrate the relationship between the data. For example, a lot of data is still presented in spreadsheets. Take Figure 5.1 for example.
Xeopm Corp. | SUBE Product | SHEE Product | YRBE Product | YFER Product | SUBE Service | SHEE Service | YRBE Service | YFER Service | TOTAL Overall
---|---|---|---|---|---|---|---|---|---
U.S. | $ 1,676,676.00 | $ 747,383.00 | $ 267,489.00 | $ 1,776,474.00 | $ 6,547.00 | **-$ 876,544.00** | $ 86,477.00 | $ 75,890.00 | **$ 3,760,392.00**
China | $ 1,738,494.00 | $ 375,834.00 | $ 876,543.00 | $ 1,387,435.00 | **-$ 678,987.00** | $ 73,839.00 | $ 36,666.00 | $ 76,453.00 | **$ 3,886,277.00**
Japan | $ 546,738.00 | $ 647,859.00 | $ 657,893.00 | $ 264,888.00 | **-$ 3,673.00** | $ 38,383.00 | $ 93,383.00 | $ 34,321.00 | **$ 2,279,792.00**
Germany | $ 746,838.00 | $ 635,747.00 | $ 836,365.00 | $ 978,433.00 | **-$ 38,933.00** | $ 83,944.00 | $ 83,944.00 | $ 43,567.00 | **$ 3,369,905.00**
France | $ 74,666.00 | $ 746,849.00 | $ 674,384.00 | $ 654,374.00 | $ 87,399.00 | $ 29,984.00 | $ 29,984.00 | $ 29,984.00 | **$ 2,327,624.00**
U.K. | $ 474,848.00 | $ 747,474.00 | $ 647,833.00 | $ 467,453.00 | $ 8,733.00 | $ 7,394.00 | $ 75,594.00 | $ 73,894.00 | **$ 2,503,223.00**
Brazil | $ 847,494.00 | $ 454,647.00 | $ 748,930.00 | $ 45,633.00 | $ 3,800.00 | $ 7,933.00 | $ 75,533.00 | $ 74,953.00 | **$ 2,258,923.00**
Italy | $ 786,595.00 | $ 846,748.00 | $ 847,646.00 | $ 23,883.00 | $ 3,883.00 | $ 89,333.00 | $ 18,333.00 | $ 37,721.00 | **$ 2,654,142.00**
Russia | $ 456,323.00 | $ 847,747.00 | $ 266,388.00 | $ 83,993.00 | $ 8,723.00 | $ 837,333.00 | $ 23,333.00 | $ 43,567.00 | **$ 2,567,407.00**
India | $ 232,567.00 | $ 987,574.00 | $ 146,636.00 | $ 71,663.00 | **-$ 23,999.00** | $ 73,833.00 | $ 34,663.00 | $ 34,457.00 | **$ 1,557,394.00**
**Total by Country** | **$ 5,904,563.00** | **$ 6,290,479.00** | **$ 5,702,618.00** | **$ 3,977,755.00** | **-$ 633,054.00** | **$ 1,241,976.00** | **$ 471,433.00** | **$ 448,917.00** | **$ 23,404,687.00**
**Figure 5.1** Spreadsheet of sales (products and services)
The information is there but it's not easily assimilated or understood and it would take a little while to figure out how each region was faring aginst the others. If, however, that data was converted into a pie chart, for example, it immediately becomes easier to understand (see Figure 5.2).
**Figure 5.2** Pie chart of total sales (products and services)
The pie chart is a marked improvement because it becomes much more obvious what divisions are generating the most sales. However, if a couple of segments are similar in size it can be quite difficult to know which is bigger. Plus if there are more than six segments it can get messy. If segments were similar it would be easier and quicker to understand if the data was represented in a linear way (see Figure 5.3) so that attention can be immediately drawn to the best and poorest performing units. The other problem is that negative data (e.g. losses in one of the service category (SUBE)) can't be correctly visualized in a pie chart.
**Figure 5.3** Bar chart of total sales (products and services)
Placed side-by-side it becomes much easier to see which products and services are generating (or indeed losing) the most revenue. But none of these visualization methods draw any attention to the various territories.
The chances are you already know all you need to know about the traditional data visualization techniques but technology is also delivering exciting new advances in this area.
## New data visualization
Big Data analytics have created a wave of new visualization tools capable of making the outputs of the analytics look pretty, and improving understanding and speed of comprehension.
Many of these tools are open-source, free applications that can be used independently or alongside another application or your existing design applications, often utilizing drag-and-drop functionality. Others are comprehensive business intelligence platforms that offer many ways to visualize data.
People don't want to search for the insights locked within the data, they want their insights provided to them, nicely packaged in a way that helps them understand the messages and these new tools can help in that challenge.
There are clearly many old and new ways to visualize data but visualization is not the only goal. Beautiful graphics can still be meaningless. We need to package the information in a way that tells a story. When considering how to best tell your story you may want to:
* Display maps
* Display text
* Display data
* Display behaviour and emotions
* Display connections.
### Displaying maps
Maps are already really strong visual representations but they can be presented in a variety of different ways with a variety of additional information overlaid across the map to provide additional insight.
If you need to display complex data sets where the story would be simplified and made clearer if those data sets were overlaid on maps, then there are a number of tools that can help. PolyMaps is a free JavaScript library and a joint project from SimpleGeo and Stamen. This complex map overlay tool can load data at a range of scales, offering multi-zoom functionality at levels ranging from country all the way down to street view (see Figure 5.4).1
Google Maps also offer several APIs for developers, such as Google Earth, Google Maps Images, and Google Places. These tools enable developers to build interactive visual mapping programs for any application or website.
### Displaying text
If you want to display text but don't necessarily want to get into the nitty-gritty of what individual people or sub-sets said, then a great way to illustrate sentiment or weighted opinion is to use word clouds.
There are many free software programs that will convert text data into a data visualization. For example, Figure 5.5 shows a word cloud of the most commonly given relationship advice taken from 25 of the most popular sites offering dating and relationship tips.
**Figure 5.4** Example of maps available with PolyMaps.
_Source:_ Sourced from <http://polymaps.org/>
**Figure 5.5** Word cloud of most common relationship advice
_Source:_ Sourced from http://www.informationisbeautiful.net/visualizations/good-relationships-most-commonly-given-relationship-advice/. Reproduced with permission of David McCandless
**Figure 5.6** Example of D3 charts
_Source:_ Reproduced with permission of Mike Bostock <https://github.com/mbostock/d3/wiki/Gallery>
The frequency with which the advice is given is represented by the size of the text, and sources include Cosmo, Elle, Huffington Post, Psychcentral and the Happy Wives Club. This form of visualization is also often called a consensus cloud.
This can be particularly useful for illustrating the qualitative information contained within a customer survey or employee engagement survey. The weighting allows you to see what most people think about your product, service, brand or company, which can offer up insights and avenues for improvement without reading every response.
### Displaying data
The ways of displaying data is as diverse as the data itself. And regardless of what type of data you have there will be an optimal way to display it. Below are just a few examples.
##### _D3.js_
D3.js is a JavaScript library for manipulating documents based on data and helps to bring that data to life using HTML, SVG and CSS. This free software can manipulate data in a mind-boggling array of ways from box plots and dendrograms (Figure 5.6) to hexagonal binding and interactive force layout. (Figure 5.7).
**Figure 5.7** Example of D3 plots
_Source:_ Reproduced with permission of Mike Bostock <https://github.com/mbostock/d3/wiki/Gallery>
### Displaying behaviour & emotions
There are even ways to display behaviour or data that wasn't even possible to know years ago. For example, we can now know hotspots on websites and the shape of a song!
##### _Crazy Egg_
Ever wondered if your website is working? Ever wondered where people go and what puts them off? If you have an analytics tool you may get click through rates and you may know how many people visit your site but you don't know why they leave.
Crazy Egg allows you to track visitor clicks, see where visitors stop scrolling down the page, connect clicks with traffic types and pinpoint hotspots using their heat map tool (see Figure 5.8). The heat map tool allows you to see what areas of your site are warm or hot with regular visitors and which parts are web wilderness. This type of tool can easily and very quickly illustrate user or customer behaviour online.
**Figure 5.8** Example of Crazy Egg heat map tool
_Source:_ Screenshot reproduced with permission of Crazy Egg.
There are even ways to visualize whether text from any source is happy, sad, angry or frustrated. And believe it or not you can even visualize the shape of a song! Shape of a Song is software that takes the data embedded in music and displays it as a visualization. Take a look at their website for some examples.2
### Displaying connections
Data by itself is often really interesting but when you can compile data and see the connections that exist between different data sets – then it really can deliver huge value to your business.
One of the earliest data visualizations depicting the connection between different variables was created in 1861 by Charles Joseph Minard (see Figure 5.9). Minard created a two-dimensional graphic that illustrated the four changing variables that contributed to Napoleon's downfall as he marched toward Moscow. The variables were the army's direction as they travelled, the location the troops passed through, the size of the army as troops died from hunger and wounds, and the freezing temperatures they experienced.
**Figure 5.9** Minard's data visualization of Napoleon's 1812 March into Russia
_Source:_ Sourced from <http://en.wikipedia.org/wiki/File:Minard>.png
In another example of how graphics can illustrate connections a financial services organization asked data analysts at Ayadsi for their insights into preventing credit card fraud. Fraud is a $3.5 trillion dollar problem. Big Data analytics are offering fresh insights into solving that problem.
This particular analysis looked at a network of 660,000 online transactions that took place over a one-month period (see Figure 5.10). Each transaction was automatically grouped using 1000 variables for each transaction, which then gave rise to clusters of different coloured transactions. The red transactions (circled in the diagram) illustrated where the percentage of fraudulent transactions were the highest. Further analysis of the similarities between these transactions gave new important insights. For example, the company was able to see that humans initiated a higher number of fraudulent transactions than had previously been thought. This became obvious when analysts noticed that the amount of time spent on the page where the fraud took place was longer than the average. This indicated that it was a person doing the fraud not a machine and they were taking their time to get everything right before perpetuating the fraud. This novel, subtle and previously unknown fraud signal was then automated by Ayadsi and has the potential to detect over 500,000 additional fraudulent charges every year that would otherwise not have been detected. Just because of data analytics.
**Figure 5.10** Ayadsi graphic on credit card fraud
_Source:_ Reproduced with permission of Ayasdi Inc. http://www.ayasdi.com/ solutions/use-case-financial-services.html Copyright 2014.
Data visualizations offer us an opportunity to see the big picture very quickly and understand things that text and numbers would take a long time to explain. The human brain is wired to see patterns and connections so visualization taps into a natural process and speeds up comprehension.
##### _The map of the Internet_
Imagine mapping the Internet. It's huge and constantly shifting and yet data visualization makes it possible. Figure 5.11 illustrates the map of the Internet.
**Figure 5.11** Map of the Internet
_Source:_ Sourced from <http://internet-map.net/>. Reproduced with permission of Ruslan Enikeev. internet-map.net © Ruslan Enikeev, 2013.
This enormous map is made up of data gathered from over 350,000 websites in 196 countries and lets users interactively examine the relationship between those sites. Every circle on the map represents a site, and the size of the site is determined by the amount of traffic.
The relationships (lines) between the sites are determined by links and visits between them – so sites with a large number of links between them, or visitors in common, appear closer together – creating the 'map' of the web. And if that isn't enough data crammed into one visualization, the sites are all colour-coded too; for example all US sites are light blue.
The foundations of the map were laid in 2011 – so newer sites aren't included – but all the data is kept up-to-date in real time.
### Endless possibilities
There are enough data visualization tools to warrant a book on their own but as the technology is evolving and developing all the time I just want to give you a flavour of what exists.
To get a fuller sense of everything that can now be done when it comes to data visualization, from the old to the new, check out the Periodic Table of Visualization Methods.3 Covering data, information, concept, strategy, metaphor and compound visualization this innovative display allows you to scroll over each 'element' and get and visual example and description. A brilliant resource for anyone keen to know what's possible...
The key message is that there is a vast amount of data that now exists and it provides us with a very real opportunity to find out things we simply didn't know before and often that knowledge allows us to squash unhelpful and inaccurate assumptions. A point well made by Swedish professor, Hans Rosling in his TED Talk.4
Rosling demonstrated brilliantly the power of data visualization to engage an audience and often turn accepted wisdom on its head. Using software he developed (which has since been acquired by Google) Rosling mined publicly funded data sources and turned complex global trends into lively animations, making decades of data incredibly interesting and informative. Using data visualization tools, back in 2006 he was able to make global trends on everything – from life expectancy to child mortality to distribution of wealth – clearer, simpler and intuitive. Far from lumping data together to make a sweeping generalization, this type of approach enables much more sophisticated storytelling and offers fresh and important insights into solutions.
The online, interactive element of data visualization is also making the data clearer, flexible and more insightful. For example, Figure 5.12 illustrates the data behind the world's biggest data breaches. The size of each 'bubble' relates to the number of records lost during each breach. It can also be filtered according to the sector of the organization responsible for the loss, and the cause of the loss – whether accidental or deliberate – due to acts of hacking. The data is taken from databreaches.net and was put together by informationisbeautiful.com.
**Figure 5.12** World's biggest data breaches by variable
_Source:_ Sourced from <http://www.informationisbeautiful.net/visualizations/worlds-biggest-data-breaches-hacks/>. Reproduced with permission of David McCandless.
## How to improve data visualization
We can learn a great deal about how to improve data visualization from the magazine publishing world.
The competition between magazines is fierce so publishers need to grab their audience's attention, especially in a newsagent where the scope of the competition is lined up against each other. The magazine must look appealing to encourage shoppers to pick it up, flick through and purchase it. This is usually achieved by using bright and vivid colour, powerful high-resolution photographs, striking images and graphics and smart, interesting or curious headings. If the shopper flicks through the magazine and confirms that the inside lives up to the promise of the cover they will buy.
In the same way, data analytics and the resulting visualizing needs to ensure it gets picked up, read and – most importantly –acted upon by key decision-makers. This requires the SMART combination of succinct presentation, aesthetics and meaningful, mission critical content.
People often talk about the 'data speaking for itself', only unless you understand the language of data then it won't speak for itself because not everyone can speak that language – as evidenced by the NASA story. If you want the data to speak for itself you have to convert it into a universally understood language using headlines, graphics and narrative to tell the story revealed by the data.
If you pick up any magazine or newspaper they follow a simple storytelling format that includes a headline, summary, photo, image or other graphic and the narrative of the story.
The reader scans the headline, image and summary in seconds, allowing them to quickly determine if the story is something he or she wants to read. High quality colour photographs or graphics are another staple in publishing because an image can convey a lot of information very quickly. They don't tell the whole story but they are a simple and effective way to increase interest and put the report in context. And finally a short summary encapsulates the story before going into more detail. Each stage allows the reader to quickly hone in on articles and stories that are of particular interest without having to read everything.
In business when we are already overwhelmed by data, reports, opinions, conversations and meetings this ability to hone in on what really matters is even more important.
And the lessons learned in publishing together with advances in technology and a deep need to cut through the clutter has led to a modern form of storytelling that is revolutionising data visualization... infographics.
## Infographics
Infographics is an area that has grown alongside Big Data, analytics and advancing technology. As the ability and opportunity to analyse more and more data has grown so too has the need to find ways to communicate and report the results effectively. Infographics – a hybrid of 'information' and 'graphics' – is a one-page visual representation intended to express a lot of information, data or knowledge quickly and clearly. An infographic of a detailed report, data analysis or employee survey, for example, can improve cognition by using a combination of headlines, graphics and narrative to tell the whole story through a one-page visual map.
Newspapers commonly use infographics to show the weather, illustrate statistics or survey results. Transit systems such as the London Underground and the following Washington DC Metrorail system are often depicted using an infographic (see Figure 5.13). These infographics integrate a variety of useful information in a very small area – station names, the various underground lines, conceptual layout of the transit network, transfer points, local landmarks and rail links, etc.
**Figure 5.13** Washington DC Metrorail map
_Source:_ Sourced from http://commons.wikimedia.org/wiki/File:Wash-dc-metro- map.png
Considering the avalanche of data and information that a typical executive is exposed to it's impossible to keep on top of it without meaningful data visualization. It's clearly unrealistic to expect busy professionals to wade through mountains of data with endless spreadsheet appendices and make sense of it all and extract the key messages. They won't do it! Besides, even if they did have the time or inclination to dive into the data themselves this individual approach leaves the data open to misunderstanding or misinterpretation. When the person looking at the information can pull out the key messages that suit their agenda or confirm their preferred decision it negates the whole point of data analysis as a tool for evidence-based decision-making.
Infographics presents an innovative solution to this dilemma because it synthesizes the key messages and tells the story of the data on one page. And everyone has time to look at one page!
There are three distinct parts of a successful infographic:
1. Visually attractive – use of colour, graphics and icons.
2. Useful content – use of time frames, statistics and references.
3. Impart knowledge – use of facts and deductions.
Together they combine to tell a story. Infographics is not just about finding the best most accessible way to present the data, it's about finding the best way to visually initiate conversations and interactions that can improve decision-making and performance. This is best achieved by focusing on the message and using whatever expresses that message best from a combination of words, numbers, pictures and colours. That way it looks good, engages the reader by simplifying comprehension and retention AND provides meaningful answers and insights into important SMART questions.
What makes infographics so effective is that they appeal to the way human beings receive and process information. We receive information from all five senses but for most of us the vast majority of our information is received visually. Half of our brain is dedicated to visual functions and images are processed and assimilated much faster than text. It is this fact that explains the adage a picture paints a thousand words. Whereas our brain processes text linearly, it processes an image all at once. We don't 'take in' a picture in the same way we 'take in' or read a book, which makes the uptake of information much quicker. Plus, it's estimated that 65 per cent of the population are visual learners as opposed to auditory or kinaesthetic. In other words most of us learn best by seeing something as opposed to hearing something or doing/experiencing something.5
Infographics provide us with a real opportunity to disseminate important information to the various parts of the business that need that information without overwhelming people. They can stand alone as the key story that executives need or they offer that snapshot and a link to a more complete report for those that want the detail. Either way the infographic primes the reader and makes the subject more accessible. It also helps the reader engage because they know in advance that they have the overview and are choosing to dive into the detail of the report because they want to rather than being forced to in an attempt to extract the golden nuggets of information they need.
Infographics acknowledge the fact that the Internet and the explosion of data has created apathy toward data and shortened our attention span. Both of which are solved by infographics. See Figures 5.14 and 5.15 for examples of infographics.
**Figure 5.14** Infographic explaining the 'Left' and 'Right' in government
_Source:_ Sourced from <http://www.informationisbeautiful.net/visualizations/left-vs-right-us/>. Reproduced with permission of David McCandless.
**Figure 5.15** Infographic of Twitter advice
Source: Sourced from <http://printmediacentr.com/2011/08/infographic-five-and-a-half-twitter-tips/.> Graphic reproduced with permission of NJI Media.
While Apache Hadoop and other technologies are emerging to support back-end Big Data analytics and address Big Data storage and processing issues, data visualization focuses on the front end of Big Data. Collecting the back-end data and getting it into a format that can be analysed is one thing – turning the results of that analysis into something that business leaders and executives understand and can use effectively is something else entirely.
## Beware the self-service business intelligence tools
Software that promises to turn the endless streams of data pouring into and out of businesses into nice sexy one-page infographics are the latest technological hotspot. Gartner has estimated that there will be a 30 per cent compound annual growth rate in data discovery tools through 2015.6
The problem is that these self-service business intelligence and data discovery solutions belong with the data scientists, not the business executives that need the data. Just as it's unrealistic to assume that business leaders and executives have the time to read every 30-page report that crosses their desk, it's also unrealistic to assume they will then be able to input that report along with a 'mash up' of data from social media, videos, clickstream data, blog posts, comments, surveys, machine sensors, etc., into some data discovery tool and miraculously create a useful infographic that synthesizes that data into a beautiful and functional one-page story.
It may happen in time, as tech savvy Gen Y'er and Digital natives increasingly move into executive roles, but it's never going to happen if your business is populated by Gen X'ers or Baby Boomers. Besides, each new generation will probably continue to leave the old one behind when it comes to technology.
Granted, that's a sweeping generalization but the point is if executives are already busy, do they really have the time or inclination to master a data discovery program? The answer is no – probably not. They want someone to tell them what it all means in a way that is simple, logical and visually appealing. They want a one-page solution. And the only way to get that is for the executives and the data scientists to work together on figuring out what is needed.
Infographics is the domain of the analysts and designers, not the decision-makers. But leaving it to the analysts and designers alone is as unproductive as leaving it to the executives. There needs to be two-way collaboration and interaction between the people creating the results and the people who need the results to make decisions. And in business, especially large business, there needs to be uniformity and commonality in the way the data is presented.
Take Procter & Gamble (P&G), for example. A global business with hundreds of brands, P&G has chosen to institutionalize data visualization as a primary tool of management. Working with a visual analytics software vendor the company put visual displays of key information on over 50,000 desktops that now provide access to a 'Decision Cockpit'.7
By establishing a common visual language for data, P&G have been able to radically upgrade how the data is used to direct decision-making and take action.
P&G have created 'Business Spheres', or meeting spaces with large screens on the walls. It doesn't matter what physical location the meeting is held, all business spheres have the same technology and data visualization protocols in place. And the meetings are attended by analysts from P&G's Information and Decision Solutions group who make those screens come to life with relevant data to aid the discussion and assist decision-making. And whilst the tools they have access to are undeniably funky and seriously creative, the real goal is to help everyone in the meetings to quickly understand the situations the group face and make timely, smart decisions.
Often what actually happens in business is that the people who need to make those decisions spend so much time looking at the data trying to figure out what has actually happened that they never get to why it happened or what they might be able to do to prevent it happening in the future. Data visualization and infographics can help by getting everyone involved quickly up to speed with what the data is actually saying.
Even if a business uses data visualization there is still room for confusion if the way data is visualized is constantly changing. By institutionalizing data visualization P&G have gone one step further to ensure that the information itself is presented in a common way across the whole company. P&G have initiated a set of seven 'business sufficiency models' that specify what information is used to address particular problem domains. For example, if a P&G executive is focused on supply chain issues the sufficiency models specify the key variables, how they should be displayed visually, and sometimes even the relationships between the variables and forecasts based on the relationships.8
This uniformity means that everyone is on the same page. Once understood, executives from any division or any brand, or in any country, can quickly and easily interpret the data they are given because it's always presented in a uniform manner. That way they spend much less time trying to understand the data and much more time putting the data to use and making better decisions.
Uniformity also prevents people from hijacking the data and presenting it in a way that supports their pet theory or hypothesis.
## The ingredients of successful data visualization and infographics
Just because you can visualize data beautifully and can create 20 different types of graphic to illustrate your point, doesn't mean you should use all 20 or even any of them.
If you want to report results successfully so that the metrics and data you've analysed can be effectively turned into commercially relevant insights that support evidence based decision-making then there are a few guidelines:
1. **Identify your target audience.** Whether you are creating a traditional report or a modern infographic, ask yourself who is going to see it and what do they already know about the issues being discussed? What do they need and want to know? And, what will they do with the information?
2. **Customize the data visualization.** Based on the answers to these questions be prepared to customize your data visualization to meet the specific requirements of each decision-maker. Too often in business reports are disseminated to everyone 'just in case' it's useful. Or parts of the report are sliced off and sent separately to different people. This just adds to the confusion and overload plus it increases the chance of key distinctions and insights that are relevant to one group being lost or missed amongst data that is useful for another group. Data visualizations should always be customized to the recipients and only include what _they_ need to know, putting the information into a context that is relevant or meaningful to them.
3. **Give the data visualization a clear label or title.** Don't be cryptic or clever, just explain what the graphic does. This helps to immediately put the visualization into context.
4. **Link the data visualization to your strategy.** If the data visualization is seeking to present data that answers one of your SMART questions then include the SMART question in the opening narrative. Linking back to the strategy you started with helps to position the data so the reader can immediately see the relevance and value of the visualization. As a result they are much more likely to engage and use the information wisely.
5. **Choose your graphics wisely**. Use whatever type of graphic best conveys the story as simply and succinctly as possible. That means:
* Use only relevant visuals that deliver important information that your target audience wants. Looking good is not a good enough reason to add a graphic, regardless of how clever or funky it is.
* Don't feel the need to fill every space on the page – too much clutter makes it harder to find the important information, harder to remember and easier to dismiss.
* Use colour appropriately to add depth to the information. And be mindful that some colours have unconscious meanings. Red, for example, is considered a warning or danger colour.
* Don't use too many different types of graph, chart or graphic. If it's going to be useful to compare various graphs with each other then make sure you use the same type of graph to illustrate the data so that comparison is as easy as possible.
* Make sure everything on the infographic serves at least one purpose.
6. **Use headings to make the important points stand out.** This allows the reader to scan the document and get the crux of the story very quickly.
7. **Add a short narrative where appropriate.** Narrative helps to explain the data in words and adds depth to the story while contextualizing the graphics. Numbers and charts may only give a snapshot; narrative allows you to embellish on key points, make observations or highlight implications.
Referred to as the 'da Vinci of Data' by _The New York Times_ , Edward Tufte suggests that graphical displays should:
* Show the data.
* Induce the viewer to think about the substance rather than about methodology, graphic design, the technology of graphic production or something else.
* Avoid distorting what the data has to say.
* Present many numbers in a small space.
* Make large data sets coherent.
* Encourage the eye to compare different pieces of data.
* Reveal the data at several levels of detail, from a broad overview to the fine structure.
* Serve a reasonably clear purpose: description, exploration, tabulation or decoration.
* Be closely integrated with the statistical and verbal descriptions of a data set.9
According to Tufte, 'Graphics reveal data. Indeed graphics can be more precise and revealing than conventional statistical computations.' Although written in 1983 before the advent of the Internet, Tufte's advice still holds true – especially in the field of infographics.
## Management dashboards
Some analytics that you run will be one-off, to answer a specific SMART question or questions. The results can then be reported via a traditional report using data visualization or through the new trend of infographics.
There are, however, other analytics that relate to ongoing strategic, tactical or operational issues. The results of those enquiries will need to be reported regularly and the best way to do that is to create a management dashboard.
Like P&G's 'Cockpit' dashboard, the management dashboard allows you to report relevant ongoing results that will help to keep the business on track toward objectives. Like the cockpit instruments in a fighter jet they allow the executive to know exactly where he or she is at any given time and focus on getting to the destination in one piece.
### Top gun for a day
As a teenager I must have watched Top Gun 50 times. I loved that movie and I'm sure I entertained fantasies of being a Top Gun pilot in my youth. So you can imagine my excitement when I did some work with the Ministry of Defence in the UK. They wanted my help in designing their strategy and cascading it into the air force. I vividly remember being in a meeting with the head of the Air Force and his team and they wanted to talk about KPIs, data, Big Data, analytics and strategy and the only thing I wanted to talk about was how they could get me into a fighter jet one day!
Over the coming months I asked the question so many times that eventually they agreed. I can't begin to explain how excited I was. I was invited to RAF Valley in Wales for my fast jet flight. To be honest at this point I didn't have a clue what was actually involved and just assumed that I would turn up, jump in the plane, fly about for a bit and then come back and tell everyone about it. Not quite.
I had to turn up the day before the flight to have a medical assessment; I needed to be weighed for the ejector seat settings. I had to learn about G-force and how it presses all the blood into your lower body, which means that you can pass out because there isn't enough blood in your brain. Needless to say, by the end of the first day I was pretty nervous. On the day of the flight the pilot sat me down and said we needed to plan our mission. Obviously with the cost of the aircraft and the pilot's time there needed to be some training value in the exercise otherwise they would never have agreed to the flight. We sat in front of the computer and mapped the flight path out which had us fly out to the sea from the airbase, do some interesting manoeuvres and then follow the mountain range and river up the Welsh countryside with the pretend mission goal of bombing one of the bridges in the Valleys.
All this data was then printed on a one-page document that was put in a clear plastic pocket on the flight dashboard. This is essentially the map of the mission and it's constantly in eyesight so that the pilot and co-pilot never forget it.
So we were almost ready. Just one last thing – I needed to watch a health and safety video. The video proceeded to explain that the plane was very safe; however, if something did go wrong and I heard the pilot say 'Eject' then I needed to pull the red level between my legs and I'd be automatically catapulted out of the aircraft. The parachute would then open and a tracker would pick up the signal and a helicopter would be dispatched to collect me. That made me even more nervous. But then the H&S video went on to say that although this would all happen automatically there was a possibility that it wouldn't, in which case I would have to do it manually which involved countless actions I needed to take in the proper sequence – the pilot would then fly the plane upside down and gravity would take care of the rest! By now I was very nervous. I knew full well that should the red ejector lever not work automatically the chances of me remembering what to unbuckle and unscrew were virtually zero and I would die!
Anyway I get strapped in and we took off. I'm sure everyone at the base had bets on how long I would last and certainly the first 15 minutes were very tough on my body. The manoeuvres out at sea were phenomenal – my head loved them but my stomach, which was a good 10 seconds behind, didn't.
We found the river and followed it up the valley to the hypothetical bombsite and dropped the bomb. But what the pilot had not told me in the briefing was that other aircraft would then be joining us in the skies to engage in practice evasive fighter manoeuvres. We darted about the sky, avoid each other for what seemed like an eternity, by this time I was hurting in ways I didn't even know were possible.
Luckily the pilot took pity on me and took us above the clouds, which he assured me, would make me feel much better. Obviously it's the clouds that cause turbulence and on a commercial airliner they can be tough, so imagine what it's like in a fighter jet.
We then talked about the dashboard, the dials on it and how to read them. He explained that it had five essential indicators on it that would allow you to know where you are in relation to where you should be as well as the flight map. Pilots have to know exactly where they are at all times, even if they are flying into a cloud or some bad weather. And they trust those five instruments completely.
Above the clouds I did feel better and the pilot gave me control of the plane for a few minutes. When he did, he informed mission control that I was in charge (even though I'm sure he had full control really). Mission control also played an important role by monitoring the airspace around us, assessing weather data and any unidentified aircraft in the area, as well as satellite data and other data from other sources.
I was reminded that day of a couple of really important points. The first is the evolution of anything. If you compare an aircraft that flew 100 years ago there were almost no navigation tools. The indicators where very rudimentary and simple and yet the latest aircraft don't even need pilots! And I can see why because after just one hour of flying I was completely exhausted. Clearly the human element is the weakest link in the chain – hence the development of unmanned aircraft or drones. Those drones are now packed with data gathering equipment, sensors and cameras. In fact, the sheer amount of data that is now being collected by drones is so vast that there aren't enough analysts to analyse it all! Despite what you may think about drones personally they demonstrate the perfect evolution of flight.
The other point is the importance of strategy and having a dashboard that can help you quickly assess whether you are on or off course without even looking out the window!
A management dashboard allows you to do that.
### Developing management dashboards
A management dashboard is simply the concise visual display of the most mission critical information needed to help executives and decision makers deliver on strategic and operational objectives.
Like the dials in the cockpit of the fighter jet, dashboards help everyone stay on track. They are best considered from an operational and strategic perspective. Operational dashboards monitor day-to-day processes and outputs to make sure expectations and performance are met consistently. They provide information that allows us to fix issues before they become problems and incrementally improve performance (See Figure 5.16).
**Figure 5.16** Example of an operational dashboard for social media
_Source:_ Sourced from <http://www.dashboardinsight.com/dashboards/live-dashboards/financial-operations-dashboard-dundas.aspx.> Image courtesy of Dundas Data Visualization, Inc – www.dundas.com
Strategic dashboards, on the other hand, look to the future and seek to identify obstacles and challenges that may occur on the way to the strategic destination. Both are important.
All of the tips for creating a successful infographic are also relevant to creating a successful dashboard. When I advise clients on their dashboards we always make sure they contain a mix of headlines and narratives as well as clear and well-designed graphs and charts.
Whether you decide to report results through traditional reporting that utilizes some data visualization techniques or whether you opt for management dashboards and/or infographics will very often depend on your in-house expertise.
For large companies like P&G, it's possible and practical to have an Information and Decision Solutions department. Data analyst and visualization experts attend the meetings to bridge the gap between the data and the decision-makers who need it. For smaller companies this may not be as practical. But one thing is for sure if you want to be a SMART business then you must develop these competencies either in-house or outsource to a trusted provider. Either way data analysis and data visualization are two sides of the same coin.
There is absolutely no point identifying metrics and data that can help you answer your SMART questions and applying the analytics to come up with those answers, if the answers are then buried in a 50-page report that no one reads or understands. Finding a way to report the results quickly, clearly and engagingly is crucial to any SMART business.
Remember you need to know who is going to use the information you have in your possession. What do they already know about the issues and what do they expect to see? Where possible focus on the story and bring the information to life through engaging visuals and infographics. Make it as short and as easy to access as possible so the peole that need the information can get it quickly and act on it.
## Key points and call to action
* Applying analytics is _still_ not enough. You need to report the insights extracted from the data in a way people understand.
* Business leaders are already struggling to keep up with all the data that comes across their desk so you need to report the results clearly – don't bury the insights in weighty reports.
* Big Data or any data analytics is only useful if you make sure the right people get the right information, in the right format, so they can make the right decisions more often.
* Once you've analysed the data you need to consider who needs the results in order to make better strategic decisions and tailor your data visualization to their needs.
* Data visualization can take many forms including:
* Graphs and charts
* Traditional reports
* Infographics that can display maps, text, data, behaviour, emotions and connections.
* Wherever possible use a one-page infographic to report the results and ensure the right people get quick and easy access to the insights.
* Data visualization is an extension of the analytics function and is not necessarily the role of the person seeking the answers to the SMART questions. You need to work collaboratively with the end user of the data and the data scientists to ensure you focus on the right data and present it in a way that is meaningful to the end user.
* Don't assume that Business Intelligence Software programs will solve this challenge. In most cases the people who need the answers will not have the time or inclination to work out how to use a software program regardless of how many brilliant and innovative ways it promises to slice and dice the data.
* If your data analytics identify a metric or data source that is going to be useful to continuously measure, then seek to convert that into a KPI and/or include it in the management dashboard.
## Notes
1 <http://polymaps.org/> 2 <http://www.turbulence.org/Works/song/gallery/gallery.html> 3 <http://www.visual-literacy.org/periodic_table/periodic_table.html> 4 <http://www.ted.com/talks/hans_rosling_shows_the_best_stats_you_ve_ever_seen> 5 Smiciklas, M. (2012) _The Power of Infographics: Using Pictures to Communicate and Connect with Your Audience_. London: Pearson Education. 6 Sommer, D., Sallam, R.L and Richardson, J. (2011) _Emerging technology analysis: Visualization-based data discovery tools_. Technical report. Gartner. 7 You can view this at <http://blogs.hbr.org/2013/04/how-p-and-g-presents-data/>. 8 Davenport, T. (2013) How P&G presents data to decision-makers HBR Blog Network. <http://blogs.hbr.org/2013/04/how-p-and-g-presents-data/>. 9 Tufte ER (1983) _The Visual Display of Quantitative Information_. Connecticut: Graphics Press.
# 6
T = TRANSFORM BUSINESS
Metrics, data and analytics are transforming the world including business. How much and how far you go in your business is up to you. But opportunities already exist to:
* Better understand and target customers.
* Improve and optimize business processes.
* Improve people's health and well-being.
* Increase security and reduce fraud.
* Drive business and people performance.
* Improve cities and other infrastructure.
## Better understand and target customers
By expanding on traditional data sets to include Big Data, structured and unstructured data such as social media, browser logs and/or sensor data and applying text analytics and other tools, companies are now able to better understand their customers and their behaviours and preferences.
The big objective, in many cases, is to create predictive models. Most large retailers from grocery chains to telecom providers to investment banks employ some form of predictive analytics in an attempt to get a jump on their competition.
Remember US retailer Target's pregnancy predictor model that alerts them to the probability that someone buying a specific combination of 25 products is pregnant. Social science has shown that most people make their buying decisions based on habit rather than choice – especially for the products sold by Target. When someone visits a supermarket they are not assessing the different types of butter on sale and weighing up the options; they are usually buying what they always buy. Occasionally they may be swayed to try something new if that product's on offer but usually they buy what they have always bought. There are, however, brief periods in a person's life where old routines and habits fall apart and buying behaviour is in a state of flux. Pregnancy is one of those periods. In fact Target knows THE moment when buying behaviour is up for grabs – around the time of the birth of a child, when both parents are exhausted and overwhelmed. They also know that timing is everything and by the time the baby is born it's too late to secure those parents as customers because the minute the baby is actually born the birth becomes a part of public record and the parents will receive a barrage of baby-related offers and incentives to add to their overwhelm. So what Target and many other large companies do is create a predictive model to identify the pregnancy before anyone else – including, as it turns out, the angry father of a 15-year-old girl.
By crunching data, Target were able to spot a pattern or cluster of buying behaviour that made it statistically likely that individual was pregnant. Things like a shift to fragrance-free products, the purchase of a large handbag that would hold nappies and baby paraphernalia, certain vitamin and supplements and obvious things like maternity clothing, triggered the predictive model and allowed Target to send baby-related offers to the mother so that when the change in buying behaviour is happening it would happen in Target.1 If an expectant mother received baby-related product offers in the run up to the birth, especially if they were disguised among other offers to appear random and not targeted, then they would be much more likely to visit the store and use the coupons to buy products they were going to need to buy anyway. And when in the store they were also much more likely to buy other things.
Target received a lot of flack when some of the details of their analytics program were revealed by _New York Times_ journalist Charles Duhigg but the truth is most large retailers are doing it. Maybe not back in 2002 when Target was just getting started but most are doing it now. And if the customers benefit from a better deal, better terms, cheaper products or discounted offers on things they want and need then surely that's a good thing for the customer and the business.
According to McKinsey, retailers willing to use Big Data analytics to increase operating margins can do so by as much as 60 per cent. In an industry that operates on razor thin margins, that's a huge advantage.2
Remember the telecom company I mentioned in Chapter Two. Using Big Data analytics to create a predictive model, they were able to reap huge commercial advantages from their data to help them better predict and manage customer churn. In the telecom industry churn is a serious issue as more and more people are happy to move around between providers looking for the best phones on the best deal. Understandably, the company wanted to understand their customer loyalty; who was moving and what they could do about it to stop the switch.
They knew they had a lot of existing data and they run analytics on some of that data but they had never looked at how people called each other. They didn't, for example, know whether their customers made mainly inbound or mainly outbound calls. They didn't know how long they spoke for or what times of the day were most popular. By mining that data and applying analytics they found that one particular calling pattern was much more likely to churn than all the rest.
This information is gold for a telecom company because all telecom companies are trying to pinch customers from each other and keep the ones they have from leaving. But they don't really know if there is a particular type of customer that moves more frequently than other types. The data analysis proved that there was a type and they were able to identify those customers most likely to leave so they could target that segment of the market with special offers and deals that would entice them to stay. By analysing traditional structured data a little differently they were able to extract commercially significant insights that massively reduced churn and increased profit.
Financial Services organizations are using data mined from customer interactions to slice and dice their users into finely-tuned segments. This enables these financial institutions to create increasingly relevant and sophisticated offers.
Insurance companies are also using data analytics to better understand their customers and deliver much more tailored insurance solutions based on actual customer behaviour rather than placing that customer into a broad category. Like all sectors, insurance is always looking for opportunities to remain competitive. Price comparison sites have changes the nature of insurance and they need to ask SMART questions around their customer base. Seeking to understand their market and the types of people seeking insurance rather than just understanding their products has led to new data collection and product innovation. For example, young drivers can opt to reduce their insurance premiums by having a black box fitted (or using a smart phone app), which monitors their driving and assesses their individual expertise and ability rather than making assumptions based only on their age. Insurance companies are also using Big Data analysis to see which home insurance applications can be immediately processed, and which ones need a validating in-person visit from an agent.
Web-based businesses are developing information products that combine data gathered from customers to offer more appealing recommendations and more successful coupon programs. Advertising and marketing agencies are tracking social media to understand responsiveness to campaigns, promotions, and other advertising mediums.
Hotels are also using analytics to better understand their customers and work out how to improve their offering. I have worked with a number of hotel chains that want to move away from the traditional in-house surveys, which are costly and questionably accurate, to using social media to analyse what people are saying and posting about their hotel. By running sentiment analysis on Facebook posts, tweets, etc., and reviews on TripAdvisor, and using that in addition to existing data, hotels are getting far more reliable information that they can action.
## Improve and optimize business processes
Big Data is also increasingly used to optimize business processes. Retailers are able to optimize their stock based on predictive models generated from social media data, web search trends and weather forecasts.
For example, Walmart is another big user of Big Data processing. Apart from knowing quirky things like the commercial advantage of stocking Pop-Tarts at the door when there is a hurricane warning, Big Data is changing this retail giant from the inside out. During the MIT CFO Summit in Boston in late 2013, Walmart's Vice President of Finance and CFO, Liz Coddington, stressed that the power of Big Data was moving Walmart forward and allowing them to avoid 'analysis-paralysis'.
Walmart use data to understand what's trending in social media, as well as buying patterns among similar types of customers and what competitors are charging in real time. For example, they learned via social media that 'cake pops' were popular with consumers and the company was able to respond quickly and get them into stores. They also changed their online shopping policy based on Big Data analytics, increasing the minimum online order from $45 to $50, while expanding the range, optimizing the business process and improving the online shopping experience.3
One particular business process that is seeing a lot of Big Data analytics is supply chain or delivery route optimization. Here, geographic positioning and radio frequency identification sensors are used to track goods or delivery vehicles and optimize routes by integrating live traffic data, etc. For instance, if a delivery driver has a schedule of optimized deliveries that schedule will interact in real time with weather data and traffic data so that if there is a traffic jam, accident or reports of delivery impacting weather such as snow or storm the schedule will automatically re-calibrate an alternate route.
Amazon is another company that is using Big Data analytics to improve business process and the retail experience. They already use algorithms to suggest and recommend other products you might like based on your previous buying behaviour and they have also recently patented something called 'anticipatory shipping'. Amazon have become so good at predictive analytics that they believe they know what you will buy before you buy it, so they will ship it toward you before the item is even in your shopping basket! Plus if Amazon get their way you could even opt for '30 Minute Amazon Prime Air' delivery in which your product would be delivered to your door within 30 minutes via a drone, or as CEO Jeff Bezos describes them, 'Octocopter'.
Big Data analytics also help machines and devices become smarter and more autonomous. For example, Big Data tools are used to operate Google's self-driving car. The Toyota Prius is fitted with cameras and GPS as well as powerful computers and sensors to safely drive on the road without the intervention of human beings.
Big Data tools are also used to optimize energy grids using data from smart meters. We can even use Big Data tools to optimize the performance of computers and data warehouses.
Manufacturers are monitoring minute vibration data from their equipment, which changes slightly as it wears down, to predict the optimal time to replace or maintain. Replacing it too soon wastes money; replacing it too late triggers an expensive work stoppage. Logistics and delivery companies can use sensors on pallets and handheld devices that record delivery to monitor where drivers are, while also monitoring the engines of the delivery vehicles to create dynamic servicing, etc.
## Improve people's health and well-being
Big Data is already revolutionizing health care. Take brain injury for example. When someone suffers brain injury it's extremely dangerous but it's never usually the initial injury that does the most damage. Electrical activity around the initial injury causes the surrounding brain cells to short circuit creating a secondary, often larger injury, which can be catastrophic. Clearly, if doctors were able to tell when this secondary injury was going to occur then they could intervene and potentially limit the damage. To that end, specialists at Kings College and Imperial College London created an early warning brain monitoring system that measures between 16 and 32 channels of data 200 times a second. That's a lot of data so they recruited the help of an analytics company to turn that data into useful insights that can then save lives. The result is a prototype brain monitor that measures brain activity in near real time and converts that data to useful information for the busy critical care staff to act on.4
In another amazing example of our smarter world, an American teenager with no medical training was able to use technology and vast amounts of data to create a breast cancer diagnosis program that correctly identifies cancer in breast tissue biopsy's 99% of the time. Brittany Wenger was a normal 15-year-old girl living in the US when she became interested in neural networks and computer programming. Then tragedy struck her family when her cousin was diagnosed with breast cancer. After school and in her spare time she created an artificial neural network that models the brain's neural network. Using a vast amount of different data points the network is able to learn and detect patterns that can't be detected by the human eye. For years doctors have found it incredibly difficult to diagnose breast cancer based on a biopsy but Wenger's program is set to change breast cancer diagnosis forever.5
The battle against cancer is also using gaming to advance research and in another fascinating insight into what's possible in a smarter world.
By some estimates 81 million people worldwide spend up to nine and a half hours a week playing mobile phone and online games like Candy Crush Saga, Flight Control or Angry Birds. Today scientists are tapping into that obsession in an effort to solve a whole host of important medical problems. New games are being created with the potential to pinpoint key information about killer diseases like cancer and diabetes. As people play along, the data is sent back to scientists to analyse. In one example game developers in Dundee created a mobile phone game similar to space invaders called 'Genes in Space' that could also help to cure cancer. Although to the gamer it looks like he or she is having to navigate through stars and galaxies what they are actually navigating through are graphics made up of the DNA information of thousands of tumour samples.
If you were to look at the data – a series of dots positioned across a computer screen – it looks like a sequence of peaks and troughs in differing concentrations. Prior to 'Genes in Space', finding the anomalies so that they could then be studied was hard enough and it was a very tedious and time-consuming process. Now people playing the game are helping scientists to identify these anomalies by steering their space ship through the galaxy and flying it through the areas that are most condensed. Every time a player completes a level it means that one DNA sample has been mapped and the data is automatically sent back to the lab at Cambridge University for analysis.
According to the leader of the research team, Professor Carlos Caldas, the lab had received 1.5 million analyses in just one month of the release of 'Genes in Space'. In other words gamers had generated their own interpretation of the data 1.5 million times. Considering that one analysis normally takes 5 minutes to map it would have taken the research team 125,000 non-stop hours or 14 years to cover the same amount of data that the gamers had covered in just one month! This innovative solution not only creates a game that people seem to enjoy playing but more importantly releases highly trained medical researchers to concentrate on studying the anomalies the gamers find so they can hopefully find answers to some of the big disease questions sooner and more cost effectively.
Premature baby units are now able to monitor thousands of data points to predict infection, intervene early and save precious tiny lives. Brain scanning sensors that can predict and better manage the secondary brain injury, which can often follow the initial injury. Often it is not the original injury that posses the biggest threat – it's the secondary short circuiting that can occur in the brain as a result of the first injury that often does most of the damage. And yet data analytics are helping hospitals solve that riddle.
In 2003, when scientists decoded the human genome it took a decade of intensive work to sequence three billion base pairs of DNA. Today the computing grunt of Big Data analytics enables us to decode that much DNA in a day!6 This data now allows us to predict the likelihoods of getting certain diseases, which in turn can lead to preventative actions and early interventions. It was this type of insight that prompted Angelina Jolie to have a preventative double mastectomy in 2013. This information also allows doctors to better customise treatments for diseases such as cancer because the DNA code will give physicians information about the most effective ways to treat tumours. When Steve Jobs was diagnosed with cancer his doctors were able to use his complete genetic code to select therapies most suited to his particular genetic make-up. Whenever one treatment lost its effectiveness the specialists would switch to another. Although Jobs did sadly succumb to cancer, this tailored approach that looked at all his DNA data and not just a section of it, gave him years of extra life.7 Coupled with mountains of data from clinical trials this DNA data is already providing crucial insights into disease patterns and pointing the way toward new cures. The clinical trials of the future won't be limited by small sample sizes but could potentially include everyone!
What's more, Big Data analytics allow us to monitor and predict the developments of epidemics and disease outbreaks. Integrating data from medical records with social media analytics enables us to monitor flu outbreaks in real time, simply by listening to what people are saying, i.e., 'Feeling rubbish today – in bed with a cold'.
Of course any advances in health affect us all. But personalized health monitoring through the use of apps and wearable devicess will hopefully over time see a reduction in stress-related illnesses brought on in the workplace.
Business life is challenging, especially in the C-suite. And yet the first time executives and business leaders think about their health is when they have a heart attack. We may read stories of senior executives stepping down because of stress but we never think it will happen to us. As the pressure of modern business mounts, more and more people are suffering from stress-related illness. In Japan, they even have a name for it – karōshi – which literally means 'death from overwork'. In China the same phenomenon is known as 'guolaosi'.
Obviously when people become ill at work or are absent for long periods of time it's stressful for the individual involved and his or her family, but it's also detrimental to the business and the people left picking up the extra workload. Personal analytics and health monitoring devices such as the 'Up' band, smart watches or smart phone apps are set to change all that and give us all a real-time insight into our own health and well-being. We are at the cusp of a new wave of preventative medicine based on data, where we can access that data to better understand links between lifestyles and diseases.
Hospitals are also analysing medical data and patient records to improve the business of medicine. For instance, they can now predict which patients are likely to seek readmission within a few months of discharge. The hospital can then intervene earlier for this segment in the hope of solving the issue for the patient and preventing another costly hospital stay.
## Improve business security and reduce fraud
Big Data is already applied heavily in improving business security through CCTV video footage analytics. Credit card and insurance companies are using data analytics to identify and prevent fraud.
One of my clients is an engineering and infrastructure services business that is experimenting with data analytic tools. People who work in potentially dangerous or stressful environments are being measured to monitor their fatigue levels and stress levels so that they can pull them off jobs before they get too tired and potentially cause an accident.
Insurance companies, for example, are using Big Data algorithms to check for fraudulent claims as well as anomalies in policy applications. Algorithms can now take into account the speed at which we complete a claim or application form – to spot those completed by machines versus people – as well as whether applicants have gone back and changed their initial application to reduce premiums by maybe not admitting a recent claim or decreasing the annual mileage.
Another way Big Data is used is to foil cyber attacks. Big Data algorithms and visualizations can detect cyber attacks as they happen and alert people as well as protect or shut down vital systems.
Of course it's not just business that uses analytics for security; law enforcement and governments also use it to foil terrorist attacks and prevent crime. Obviously the extent of the surveillance necessary to prevent these attacks raises issues of personal privacy and security but there is little doubt these programs do help to keep us safer. Police forces all over the world also use Big Data tools to catch criminals and even predict criminal activity.
## Drive business and people performance
Most elite sports have now embraced Big Data analytics. We have the IBM SlamTracker tool for tennis tournaments; we use video analytics that track the performance of every player in a baseball, rugby or football game.
Clubs including Manchester United and Chelsea are hiring data firms and data scientists to meticulously track every movement the players make in order to seek new ways to win. The stakes are high and football players are notoriously expensive – absorbing anywhere between 70 per cent and 94 per cent of an average club's takings. Any way of spotting talent earlier so a club can secure that talent without the financial burden of excessive wages, is clearly a benefit. As a result football is becoming smarter with pitch side analysts logging every tackle, pass and goal, typically collecting information on 2,000 or so 'events' per match. Overhead cameras track players' movements, logging their distance, speed and acceleration, all of which allows clubs to spot trends and correlations from a smorgasbord of data that a human being wouldn't be able to see. Armed with these insights into what combinations of skills and strengths separate a top player from an also-ran, the clubs are now scouring the lower leagues to spot unsigned, inexpensive players who demonstrate the same effective combinations identified by the data.8
Sensor technology in sports equipment such as basketballs or golf clubs also allows us to get feedback (via smart phones and cloud servers) on our game and how to improve it. Many elite sports teams also track athletes outside of the sporting environment – using smart technology to track nutrition and sleep, as well as social media conversations to monitor emotional well-being.
In competitive sport or athletics, when a fraction of a second can make all the difference between Silver and Gold, data analytics are transforming performance.
I worked with an Olympic cycling team who collect and analyse performance data gathered from sensors fitted to the pedals of the bikes. These sensors monitor how much acceleration or forward thrust every push on the pedal generates. This allows the team to analyse the performance of every cyclist in every race and every single training session. In addition, the team has started to integrate performance data with health and fitness data such as calorie intake, sleep quality, air quality, heart rate, etc., that is gathered from wearable devices. The latest innovation is even integrating further analysis of social media posts to better understand the emotional state of athletes and how this might impact track performance. Combing and analysing this treasure trove of data undoubtedly helped them to make the incremental improvements that led to Olympic glory in London 2012.
Sports teams are also using data for tracking ticket sales and even for tracking team strategies.
Of course this is now filtering into Human Resource business processes. Here, Big Data is used to optimize talent acquisition as well as the measurement of company culture and staff engagement using Big Data tools.
Remember Google's Project Oxygen from Chapter 2 – they used Big Data analytics to identify what makes a great manager at Google so they can tailor training programmes to help those that are struggling and so that they can recruit the right type of person from the start. Data is also being used to spot talent in sports like football and baseball.
This data-driven approach is not new and has been used in American sports for some time. In his book _Moneyball: The art of winning an unfair game_ ,9 author Michael Lewis documents the story of how baseball watcher and advisor, Bill James, changed baseball forever.
Prior to James, new baseball talent was spotted by talent scouts who would tour the country watching game after game of amateur baseball in the hopes of finding the next star. The assumption was that watching individuals was the only way to isolate the qualities that made a star a star. Once the would-be star was identified a bidding war would often ensue and clubs would need to pay vast sums of money to secure those players.
James argued that human observation wasn't enough to differentiate between players or even expert opinion. To test his hypothesis he created a new formula that crunched data based on a range of critical data points or separate, measureable component parts that would 'throw up' talented players for further investigation.
Billie Beane, general manager of the Oakland Athletics, known as the Oakland A's, heard about James' theory and was intrigued. Little wonder when you consider that Oakland A's had the third lowest payroll in the league so they simply couldn't compete with the big guns when the bidding wars began. Through experimentation James and Beane proved that there was another way to spot talent and the insights allowed the team to buy undervalued talent, which in turn took the club to the play offs in 2002 and 2003. Oakland A's got smarter and for the first time were able to successfully compete with legendary, deep-pocketed baseball clubs like the New York Yankees.
And this type of data-driven talent spotting is becoming more and more common in other sports and it's made its way into business too.
Companies can gain genuinely mouth-watering benefits when they use data well and apply analytics tools to turn the data into business critical insights. Take a look at these real world examples that show how collecting and analysing data can deliver impressive (and sometimes unexpected) insights:
* A bank was able to cut staff costs in one area by half, simply by analysing the performance of staff that were recruited from different universities. In the past, the bank assumed their best performing people would be those who have excellent degrees from Ivy League universities. Data analytics clearly showed the assumption was wrong. It turned out that candidates from non-prestige universities outperformed the top-university candidates allowing the business to recruit the right talent for less money.
* A retail company uses social media network analysis combined with other analytics tools to find the right candidates for their job. Simply by analysing social media profiles they can accurately predict the level of intelligence as well as the emotional stability of potential candidates.
* One of my clients wanted to recruit self-driven people that are able to take initiative. By analysing different data sets from the type of people they wanted to recruit and those they wanted to avoid, the company found that the type of browser used to complete the job applications was one of the strong predictors for the right candidate. Those candidates that used browsers that were not pre-installed on their computers and instead had to be installed separately (such as Firefox or Chrome) tended to be better for that particular job.
* A retail company I work with is now able to predict how key elements of staff engagement influence operational performance, customer satisfaction and ultimately financial performance. What's more, the company is now able to predict the extent to which a certain increase in specific elements of staff satisfaction drives a certain percentage increase in revenue by square foot in their shops.
* Another company found that call centre sales staff with a criminal record performed better than those without a criminal record and that rather surprisingly sales people with more Facebook connections performed poorer than those with few connections.
* One organization uses analytics tools to scan and analyse the content of emails sent by their staff as well as the social media posts they make on Facebook or Twitter. This allows them to accurately understand the levels of staff engagement and they no longer need the traditional staff surveys.
If we can use data analysis to find patterns and correlations between personality traits, behaviour and capabilities that turn out to 'fit' with particular roles, jobs or corporate cultures that would then translate into more of the right people in the right jobs, then productivity, employee engagement and happiness will increase and that's got to be a good thing for everyone.
## Improve cities and other infrastructure
Big Data is also used to improve many aspects of our cities and other infrastructure.
For example, it allows cities to optimize traffic flows based on real-time traffic information as well as social media and weather data. A number of cities are currently using Big Data analytics with the aim of turning themselves into Smart Cities, where the transport infrastructure and utility processes are joined together, communicating in real time. In this type of system a bus would automatically wait for a delayed train and where traffic signals predict traffic volumes and operate to minimize jams.
Another brilliant example is a little app developed by The City of Boston to detect where the potholes were in the road. In the past they would send survey vehicles out around the road in Boston maybe twice a year to do a complete survey. This involved someone driving along every road in the city to understand where the problems were and how bad they were in each location. So what they did instead was create an app and asked the people of Boston to download it to their smart phone, where it would run in the background. When all those people then went about their daily lives, driving to and from work, to the shopping mall or to the gym the accelerometer in the phone that says how fast someone is travelling would record when the person slowed down or braked. A little algorithm was then applied to this data to identify potential potholes. This is so clever and so simple. When you drive along a road and see a pothole you will usually slow down or swerve to avoid it and then speed up again. And even if you don't see it or can't avoid it and drive over the pothole the sensor in your smart phone will detect the little horizontal bump. The algorithm is then able to isolate potholes from the smart phone data. All this data is then going back to base and they have a live, almost real-time map of the city's streets and potholes. Instead of wasting huge amounts of time and money just to identify the worst potholes the city is now able to re-direct personnel and resources to fixing them.
## New business opportunities
Data analytics can transform your business from two different directions.
First you can create SMART businesses by using the framework described in this book to examine your existing business model and use the insights to improve the way you do business. There are huge benefits to be reaped for any business in any sector.
In addition there is also the possibility that the data will eventually change your business model or lead to diversification. If, for example, you realize that you have the ability to collect a lot of data then potentially that data will be valuable over and above what the data tells you about your business and how it might improve decision-making.
The main purpose of any Big Data initiative should be to find answers to your most critical unanswered business question or your SMART questions. However, once companies have found sources to answer their critical business questions they might then use Big Data to support ongoing operations and business processes.
For example, a credit card company might have started the enquiry with a question about new ways of detecting fraud but once it has found ways to identify fraudulent credit card usage through Big Data analytics it can then use these insights to put systems in place that continuously use those Big Data sources and tools to monitor their ongoing card operations.
Another example is an insurance company that set out to better monitor driving behaviour of their clients using inbuilt sensors or smart phone sensors. Once it found a reliable way of doing this, the focus shifted to putting this into commercial operation. New customers are now able to opt for dynamic insurance premiums based on their driving style. If a customer can prove safe driving and a high duty of care to other road users and the vehicle, then premiums may be reduced over time. Similarly if an insurance company uses Big Data to identify new ways of detecting fraud they will then roll that out across the business to prevent fraud in the future. For example, if a customer fills in an insurance claim form online a red flag will be raised if the customer completes a field and then changes their answer, as this may be an indication of fraud. Once they have found this fact they can keep collecting that data and use this new insight as an ongoing way of supporting and improving their day-to-day operations. So credit card companies and insurance companies then have the opportunity of running the insightful analytics along side to improve fraud detections in real time which also constantly seeking to improve.
The genuinely SMART business will therefore apply the data to their existing strategy and improve performance AND integrate those insights to improve day-to-day operational efficiency.
In addition, companies should also stay open to any new business opportunities the data may expose. Or to take this one step further you may want to make time and resources available for some data discovery – where analysts spend some time exploring the data and maybe finding new ways of leveraging it. This is especially relevant if you already have a lot of proprietary data.
It is never wise to start with the data and try and figure out what's in there but once you're more familiar with your data and once you have answered the pressing strategic questions, data discovery can sometimes trigger new SMART questions as well as discover new ways to answer existing ones.
The explosion in data types and our increasing ability to analyse it has also led to some major commercial shifts for some companies.
Take Jawbone, for example, the manufacturer of the 'Up' band. Jawbone used to just be a wearable device manufacturing company. But they realized that all the data their wearable device collected was actually more valuable that the device itself! As a result they are now moving into the world of Big Data where they are becoming a data company.
They still manufacture the products because they allow for the ongoing accumulation of the data, but the data is the primary focus. The 'Up' band collects data on calorie consumption, activity levels, and sleep patterns. While it gives individuals rich insights, the real value is in analysing the collective data. In Jawbone's case, the company now collects 60 years' worth of sleep data every night. Analysing such volumes of data will bring entirely new insights that it can feed back to individual users and be sold on to interested parties.
Aircraft engine manufacturer Rolls Royce is another brilliant example of how the data itself has changed their business model. The company has very successfully moved to a 'printer and ink' business model. There may be a little profit (or actually no profit) in the manufacture of the original printer but the real money comes from the ink that a customer needs to keep buying to make the initial product useful. Rolls Royce used to just make the engines but in an effort to stay competitive they sought to answer SMART questions about what their customer needed rather than just what they provided. Now they monitor the engines too, using thousands of sensors positioned throughout the engine. These sensors continuously monitor the performance of more than 3,700 jet engines worldwide to identify issues before they arise. As a result, Rolls Royce have moved their business model from solely manufacturing to the creation of recurring ongoing revenue streams over and above their manufacturing business. Now Rolls Royce sells the engines and offers to monitor them, charging customers based on engine usage time and repairs and replaces parts if there is a problem. So the client effectively buys a dynamic servicing option and this servicing now accounts for a massive 70 per cent of the civil-aircraft engine division's annual revenue.10
When Malaysian Airways flight MH370 went missing on March 8th, 2014 with 239 passengers and crew, rumour and speculation was rife. In the initial aftermath and confusion surrounding the disappearance reports circulated that the plane had flown on for an additional four hours from its last known location. It was the sensors embedded deep inside the two Trent 800 engines built by Rolls Royce that were able to confirm this was inaccurate.
As well as being able to provide important intelligence around disasters (of which there are thankfully few) the data that it collects, transmitted and analysed goes a long way in making the skies safer and more efficient. Take lightning strikes on passenger aircraft for instance. Lightning hits a plane a couple of times every hour somewhere in the world. In the pre-data days a strike would automatically trigger a full engine inspection once the plane landed – for obvious reasons. This would delay the return flight, irritate passengers and negatively impact 'On-time Departure' (OTD) – a significant metric in air travel. Now as soon as the engine so much as coughs, a torrent of data is automatically beamed back to HQ where screens jump into life, graphs are created and technicians assess the real-time impact of the incident. So much so that before the plane has even landed Rolls Royce can tell the airline with absolute confidence if the plane needs to be grounded or is cleared for the return journey.
Data, and Rolls Royce's strategic use of data, has transformed the company from a loss-making British firm into the world's second-biggest maker of large jet engines.11
General Electric (GE) is another company that's transformed their business away from their traditional manufacturing roots using Big Data analytics. GE makes products and services for aviation, healthcare, electrical distribution, lighting, energy, oil & gas, finance (business and consumer), rail and water. Those products now contain hundreds of sensors that collect data which is then analysed to improve efficiency for the client.
For example, GE's gas turbines are now smart gas turbines. In a press release issued by GE in October 2013 the company stated that the previous month:
> _'GE eclipsed 100 million hours of operational data documented on its globally monitored gas turbine fleet of more than 1,600 units, the world's largest. The insights derived from analysis of this operational "Big Data" can be applied to help customers expand their earning power while reducing operational costs and risk. As these "intelligent" machines communicate their operating statistics through an average of 100 physical sensors and 300 virtual sensors on each gas turbine, the GE team can help customers translate that information into actionable decisions. Armed with these data-driven insights, GE customers can more effectively identify potential barriers before they occur, treat minor issues before they lead to catastrophic events and dynamically adjust performance to improve efficiency and reduce parts wear and tear. GE is tapping into knowledge gained from this data analysis to develop new technology breakthroughs, both hardware- and software-based, that enable customers to unleash more potential from their existing gas turbine and balance of plant assets. Unlocking the full capacity of a 500-MW power plant could be worth more than $500,000 annually in increased revenue, while a public utility that could reduce its heat rate efficiency curve by 1 percent could save up to $1.25 million dollars annually in fuel costs.'_12
But it's not just gas turbines. Sensors on aircraft engines allow pilots to manage fuel efficiency and considering the airline industry spends $200 bn on fuel per year, even a 2 per cent saving equates to $4 billion in savings. Another fuel efficiency service, this time analysed from sensors on train engines assesses the terrain and the location of the train to calculate the optimal speed to run the train at maximum fuel efficiency. Software developed at GE is now being used by a Canadian electricity supplier to tackle the biggest cause of electricity outages – trees and branches falling on power lines. Using results and insights from analytics, vegetation along the electricity distribution lines are now pruned back preventing the outages before they occur.
Using operational data from sensors on a range of machinery and engines, GE applies analytics to identify patterns and deliver commercially relevant insights. And like Rolls-Royce, GE provides additional services tied to its products, designed to improve real-time efficiency and minimize downtime caused by parts failures. This servicing now accounts for one third of GE's business.13
## Smart will transform employment too
As technology transforms business it's also transforming employment. Many jobs will eventually disappear as smart technology improves our ability to capture and analyse data. As a result, more and more jobs will be automated. Clearly this has been happening for a while. Post industrial revolution we have seen the loss of thousands of manual or unskilled jobs but the smart revolution looks set to take that even further into employment areas previously considered highly skilled.
Consider the following professions that are already changing:
1. **Taxi-Drivers:** When I was in a taxi going from San Francisco airport to Silicon Valley I noticed one of Google's self-driving cars on the road. I said to the driver: 'Hey, check this out. The car we just passed has no driver in it. It's Google's self-driving car and it stays on the road safely by analysing a gigantic amount of data from sensor and cameras in real time.' His reply: 'So that means that Google will take away my job soon.' Actually yes – it probably does. Plus there is now an app called Uber which allows individuals to collect passengers who happen to be going the same place as them and charge for it.
2. **Border Control Agents:** Back at the airport to catch my plane to London I used the electronic passport machines. You put your passport it, it scans it, and then scans your face to see whether they match. Then the doors open and you go through immigration. No human contact and no need for border control agents any more. The machines do a better and more reliable job – plus they don't get surly when you forget to put the miniature toothpaste in the clear plastic bag!
3. **Pilots:** We know that autopilots have been assisting pilots to fly planes for many years. However, the latest commercial airlines are now able to fly the plane unaided. They can take off and land safely – possibly even safer than a pilot considering that most air disasters are down to 'human error'. Something I certainly related to in my short fighter jet experience. Unmanned drones are already pushing fighter pilots to the side and changing the aviation landscape forever.
4. **Doctors:** Robotic tools are already assisting surgeons to perform operations and doctors use large-scale databases of medical information to inform their decisions. However, I can imagine a scenario where a full body scanner takes a complete 3D image of you and where robots will perform an operation completely unassisted. We now have the technology and computing power to perform surgery without the need for humans. And therefore without the risk of human error. Supercomputers will be able to make a solid diagnosis based on all previous medical knowledge, as well as data from your own medical history, DNA code, etc. – all without the input from human doctors.
5. **Customer Support Agents:** We all know about the irritating automated answering systems in call centres that give you options and then supposedly route your call to the right person. What we are now seeing is the rise of natural language systems that are able to have a conversation with humans. IBM has developed Watson – a computer that recently challenged two of the all-time best _Jeopardy!_ players. Without access to the Internet, Watson won the game by interpreting natural language questions and answering back after analysing its massive data memory (that included a copy of the entire Wikipedia database). This means that when you ring any call centre you will always speak to the 'right person' – only that the person will be a machine instead.
I've already mentioned sports and the proliferation of technology that is helping coaches to improve performance and how technology is entering journalism. Narrative Science is a software product that can write newspaper stories about sports games directly from the games' statistics. In fact I can't think of many jobs that we can't automate to some extent using Big Data analytics, artificial intelligence and robots. It's therefore important to advance your career in a way that positions you at the forefront of these developments and that you stay away from jobs that will be the first to go. And if you want to be a modern day rock star or F1 racing driver but don't play and instrument or can't drive – what about a data scientist!
When starting out on the journey toward SMART business it is wise to focus your attention on your existing business model and find answers to your strategically significant SMART questions. Start with strategy so that you can improve performance and strengthen your existing position. At the same time be aware of the power of data and consider how the data you collect in the course of your business could be used in order to adapt or expand your business model and diversify your business. Data is already a new currency and it can and is changing the nature of many businesses. This secondary possibility will not be relevant to everyone but stay open to the possibilities as you progress and gain confidence in the power of data and analytics.
Ultimately access to data and the ability to analyse it allows us all to review evidence and make better decisions based on fact not assumption, 'experience' or 'gut feeling'.
## Key points and call to action
* Big Data and analytics are transforming the world including business. How much and how far you go in your business is up to you. But opportunities already exist to:
* Better understand and target customers
* Improve and optimize business processes
* Improve people's health and well-being
* Increase security and reduce fraud
* Drive business and people performance
* Improve cities and other infrastructure.
* You need to view data as an ongoing commitment to improvement that can allow you to implement your strategy quicker and more efficiently.
* Use the insights you gather – don't just sit on them or bury them. When you do you will encourage your people to shift their thinking to an evidence-based decision-making culture.
* Some answers you will be seeking will be one-off; some will be ongoing answers you want to keep an eye on. If the data will be useful ongoing, incorporate the collection and analysis into your regular reporting schedule.
* In addition stay vigilant to new business opportunities that emerge from the data.
* Use the insights gained from the SMART process to improve your decision-making, your customer experience, your employee brand and your business performance.
## Notes
1 Duhigg, C. (2012) How Companies learn your secrets. _The New York Times._http://www.nytimes.com/2012/02/19/magazine/shopping-habits.html?pagewanted=1&_r=2&hp& 2 Manyika, J., Chui, M., Brown, B., Bughin, J., Dobbs, R., Roxburgh, C. and Hung Byers, A. (2011) Big Data: The next frontier for innovation, competition, and productivity McKinsey & Co Insights and Publications. <http://www.mckinsey.com/insights/business_technology/big_data_the_next_frontier_for_innovation> 3 Knox, N. (2013) Now Trending: Big Data at Walmart.com. _CFO Journal._ <http://blogs.wsj.com/cfo/2013/11/22/now-trending-big-data-at-walmart-com/> 4 BBC One (2014) _Bang Goes the Theory_. Series 8: Big Data. 5 BBC Two (2013) _Horizon_ : Monitor Me, narrated by Dr Kevin Fong. 6 Mayer-Schönberger, V. and Cukier, K. (2013) _Big Data: A revolution that will transform how we live, work and think._ London: John Murray Publishers. 7 Mayer-Schönberger, V. and Cukier, K. (2013) _Big Data: A revolution that will transform how we live, work and think_. London: John Murray Publishers. 8 Ball-watching: The world's richest football league is embracing Big Data (Aug 2013) _The Economist._ 9 Lewis, Michael (2003) _Moneyball: The Art of Winning an Unfair Game_. New York: W.W Norton & Company Ltd. 10 Mayer-Schönberger, V. and Cukier, K. (2013) _Big Data: A revolution that will transform how we live, work and think._ London: John Murray Publishers. 11 Rolls Royce: Britain's Lonely High-Flier (2009) _The Economist._ <http://www.economist.com/node/12887368> 12 GE Press Release (2013) GE's New FlexEfficiency* Advantage and LifeMax* Advantage Platforms Unlock Full Performance, Value of Installed Gas Turbines. <http://www.genewscenter.com/Press-Releases/GE-s-New-FlexEfficiency-Advantage-and-LifeMax-Advantage-Platforms-Unlock-Full-Performance-Value-o-42e9.aspx>. 13 Saran, C. (2103) GE uses Big Data to power machine services business. _ComputerWeekly.com._ <http://www.computerweekly.com/news/2240176248/GE-uses-big-data-to-power-machine-services-business>
# CONCLUSION
Big Data and analytics are revolutionizing our lives.
Like all revolutions there are going to be winners and losers. But it's not just as simple as saying that those that have the largest amounts of data will win and those that have little, won't. At the same time, the hype around Big Data simply increases the stress levels for many business leaders because they are already fully aware that they either don't have masses of data, or they do have masses of data but absolutely no idea what to do with it. The problem is that the data revolution is happening alongside business as usual and that can be extremely overwhelming.
SMART business is a solution that encourages us all to step back from the hype and the noise around data – especially Big Data – and take stock of where we are, where we are trying to get to and what data and tools we can employ to help us get there.
Don't start with the data. If you do you will find yourself lost in an impossible rabbit warren of options. Start with strategy, get really clear about what you need to know and why and link that back to your strategic and tactical objectives. Just by starting with strategy and not data you will immediately focus in on your really important data requirements and what's needed instead of being overwhelmed by what's possible.
Once you know what you are trying to achieve and you are clear on what SMART questions you need answered then investigate the metrics and data that could potentially deliver the answers. If two or three sources of data could deliver the answers you want then focus on the data you already have or have easy access to. It's not just about Big Data – traditional data sources or 'small' data can be just as illuminating if not more so. Where possible use a combination of data sets and triangulate the data. In other words, see if each data set delivers the same result so that you can confirm and validate the answers. Always start from where you are so existing internal data is usually easier to access that external data and structured data is often easier to analyse than unstructured.
Once you have identified what metrics and data you want to use then you need to collect that data and apply analytics. What type of analytics will depend on the data and what answers you are seeking but Chapter 4 will have given you a good overview of what's possible. Remember the data itself is meaningless; it needs to be converted into insights. But even the insights are meaningless unless they are converted into reports or data visualizations that extract the key points and communicate those to the right people. No one has time to read 50-page reports – you must visualize the data so that the people that need the information to make better decisions get that information quickly and in a form that works for them. That means using data visualization tools, infographics and management dashboards to display the outputs instead of thumping a report on someone's desk and assuming they will be able to unearth the golden nuggets lying hidden within. It also means that the analytics and data visualization needs to belong together. Data visualization software may be terrific but the use of that software is the domain of the analysts not the executives that need to interpret the data. It's important that strong links are forged and maintained between the decision-makers and the data analyst and visualization experts.
When you can do all that you can, review the evidence and everyone in the business can move toward more fact-based decision-making and finally leverage data to gain real competitive advantage.
When it comes to data, Big Data and smart technology, there are still so many unanswered questions – not least the issue of privacy and transparency. Most of us have no idea just how much data about us already exists in one form or another. We have no clue how much companies know about us, not just by our buying habits but because of the ability to connect those things to other richer data sources such as Facebook and other social media activities. And chances are we'd be more than a little alarmed if we realized how much governments know about us.
Like most innovations, Big Data and analytics offer us a huge opportunity for good. Inroads are being made in every direction to improve health, performance, business, crime prevention, etc. But just as these tools can be used for the greater good they can also be used for the selective good and for the not-so-good at all. And of course who decides? Edward Snowden caused chaos by disclosing the level of surveillance the US Government (and almost certainly every Government) conducts on its citizens. If that surveillance stops people getting hurt, and reduces violence and terrorism then is it such a bad thing? But where does it stop? Who watches the watchers and holds a strong moral and ethical line? These are complicated questions.
At the time of writing the US is mourning another mass shooting. This time a 22-year-old man, angry at the fact that he was still a virgin killed six people and wounded 13 others before being killed himself. Amongst the many deep and disturbing questions that arise after these all-too-common tragedies it occurs to me that if Target can identify 25 products that when bought together predict pregnancy, then surely Big Data analytics can identify 25 online memberships, affiliations, clubs, books or other purchases that when bought by the same person predict mental instability, hatred toward a section of society and potential violence. If that could be done – should it be done?
At what point does privacy give way to probability? Target is not 100% right when it comes to pregnancy – estimates suggest if someone buys these 25 products there is an 87% chance they are pregnant. But getting that wrong isn't a major issue – the 13% who are not pregnant will just dismiss it as not relating to them and no harm is done. But if I were identified as a potential mass murderer it probably wouldn't be so harmful. That said, would I mind being interviewed and watched for a few weeks if it meant that these senseless killings ended? No probably not. But again – who decides? When does national or corporate interest give way to invasion of privacy? Companies are going to increasingly use predictive analytics to pigeonhole (or target) their customers and tailor offers. This could theoretically mean that someone is charged more for insurance or refused a loan just because they fall into one category or another. And that hardly seems fair either. Just because someone is likely to behave in a certain way doesn't mean they all will behave in that way.
These are complicated issues and there is no easy answer. But for me we need to drag Big Data and analytics out of the shadows. As I said earlier, too much of this revolution is occurring in darkened rooms in places that don't officially exist. It's too covert – too many people dedicated to harvesting as much data as possible before the general public realizes what's actually going on and governments and lawmakers step in. But by the time people realize just how much data is out there about them it's going to be too late. At the moment the lawmakers don't even understand the ramifications of this data revolution themselves and by the time they do it's going to be very hard to pull the worlds of data back toward privacy.
As individuals we need to be much more aware of the data we provide and pay much more attention to privacy settings online. As SMART businesses we need to be open and honest about what we intend to do with the data. If we give customers an opt-out and honour that – perhaps seeking permission to anonymize their data instead then we can shine a light on the revolution and take ownership of the good while more effectively managing the bad.
One thing is for sure, Big Data and analytics are here to stay and it's only going to get more sophisticated. We need to embrace it, operate ethically, deliver value in exchange for the data and apply its significant benefits for the betterment of our world.
# ABOUT THE AUTHOR
Bernard Marr is a best-selling business author, keynote speaker and consultant in Big Data, analytics, strategy management, performance management and KPIs. He helps companies collect and analyse the data to improve strategic decision-making and business performance. His leading-edge work with major companies, organizations and governments across the world makes him a globally acclaimed and award-winning consultant, researcher and teacher.
Bernard Marr is acknowledged by the CEO Journal as one of today's leading business brains and LinkedIn nominated him as one of World's top 100 business influencers. He has written a number of seminal books and over 300 high profile reports and articles. This includes the best sellers _The Intelligent Company_ , _25 Need-to-Know Key Performance Indicators_ , _Managing and Delivering Performance_ , _Key Performance Indicators for Dummies_ , _Strategic Performance Management_ , and _Doing More with Less_.
In his consulting work he helps executive teams to develop their Big Data and analytics strategies and trains teams in companies to better leverage data and metrics. He has worked with and advised many of the world's best-known organizations including Accenture, AstraZeneca, Bank of England, Barclays, BP, DHL, Fujitsu, Gartner, HSBC, Mars, Ministry of Defence, Microsoft, Oracle, The Home Office, NHS, Orange, Tetley, T-Mobile, Toyota, Royal Air Force, SAP and Shell, among many others.
Bernard Marr is the founder and CEO of the Advanced Performance Institute. If you would like to talk to Bernard about any Big Data project you require help with or if you are thinking of running a Big Data event or training in your organization, then contact him at: www.ap-institute.com or via email at: bernard.marr@ap-institute.com
You can also follow @bernardmarr on Twitter, where he regularly shares his ideas or connect with him on LinkedIn, where he writes a regular blog.
# ACKNOWLEDGEMENTS
I am so grateful to everyone who has helped me get to where I am today. All the great people in the companies I have worked with who put their trust in me to help them and in return give me so much new knowledge and experience. I must also thank everyone who has shared their thinking with me, either in person, in blog posts, books or any other formats. Thank you for generously sharing all the material I absorb every day! I am also lucky enough to personally know many of the key thinkers and thought leaders in the field and I hope you all know how much I value your inputs and exchanges. At this point I usually start a long list of key people but I always miss some off, so this time I want to resist that and hope your egos will forgive me. You are all amazing!
Finally, I want to thank the team at Wiley for all your support. Taking this book through production has been a particularly good experience and I really appreciate your input and help. Thank you Jonathan Shipley and Jenny Ng. And a big thank you to Karen McCreadie for the amazing editorial support!
# INDEX
* abstraction
* accelerometer sensors
* activity data
* adding value
* advertising
* agriculture
* algorithms
* Amazon
* combined analytics
* fraud detection
* Google translation
* online dating
* pothole detection
* prediction
* speech analytics
* text analytics
* video/image analytics
* Amazon
* analytics
* books
* competitive advantage
* strategy
* ambient sensors
* analytics
* Amazon
* behaviour analytics
* business insights
* combined analytics
* customer satisfaction
* evolution of
* fraud
* Human Resources
* 'Likes'
* management dashboards
* new business opportunities
* personal analytics
* prediction vs. privacy
* SMART
* speech analytics
* sports
* text analytics
* transparency
* video/image analytics
* anonymization of data
* apps
* Ayadsi
* baby monitoring
* Bank of America
* bar graphs
* Beane, Billie
* behaviour analytics
* Big Data
* accumulation of data
* anatomy of
* backlash against
* business process optimization
* business security and fraud
* cities and infrastructure
* collecting everything
* how companies use
* reporting
* SMART Model
* sports
* strategy
* types of
* who is using
* _see also_ analytics
* Big Table
* biometric data
* blogs
* books
* border control
* brain injuries
* breast cancer diagnosis
* Brin, Sergey
* browsers
* business intelligence tools
* business process optimization
* business security
* call centres
* cancer diagnosis
* captured data
* cars
* cash flow data
* categorization of text
* causality
* CCTV
* behaviour analytics
* law enforcement
* censorship
* charts
* Chrome
* Cisco
* cities
* cloud computing
* competition and risk
* compiled data
* concept extraction
* Con Edison
* connections, displaying
* consensus clouds
* conversations
* conversation data
* speech analytics
* core competencies
* costs
* Crazy Egg
* created data
* crime
* culture
* customers
* databases
* data needs
* offering value to
* privacy issues
* SMART strategy board
* social media
* speech analytics
* understanding and targeting
* vision statement
* customer support agents
* customization
## A
* D3.js
* data analysis _see_ analytics
* databases
* data breaches
* data collection
* employee monitoring
* SMART questions
* SMART strategy board
* data discovery
* data display
* datafication
* data models
* data, types of
* data visualization
* decision-making
* delivery routes
* Dell
* demand
* disaster relief
* distribution
* DNA information
* doctors
* document summarization
* e-books
* Economic Intelligence Unit
* eHarmony
* emails
* employees
* data collection
* performance data
* text analytics
* transformations in employment
* vision statement
* _see also_ recruitment
* energy efficiency
* ethical issues
* Evolv
* executive failure
* experimental data
* external data
* extraction
* eyeball tracking data
* Facebook
* competitive advantage
* data discovery
* Dell
* disaster relief
* employees' posts on
* face recognition
* Gatorade
* information held by
* Klout score
* 'Likes'
* photo data
* privacy issues
* sentiment analysis
* user-generated data
* face recognition
* finance
* financial services
* Firefox
* fishing
* Flickr
* fraud
* gaming
* Gatorade
* General Electric (GE)
* 'Genes in Space'
* Gladwell, Malcolm
* Gmail
* Gonzalez, Mario Costeja
* Google
* captured data
* competitive advantage
* data visualization
* face recognition
* Gmail
* Google Book Project
* Google Flu Trends project
* Google Glass
* Google Maps
* Project Oxygen
* right to be forgotten
* search engine data
* self-driving car
* strategy
* translation software
* government data
* GPS (Global Positioning System)
* graphs and charts
* gyroscopes
* Hadoop
* health monitoring
* heart rate variation (HRV)
* homes
* hospitals
* hotels
* Human Resources
* _see also_ employees
* IBM
* IDC Digital Universe Study
* image data
* infographics
* Information Age
* information processing
* infrastructure
* Instagram
* insurance companies
* internal data
* Internet
* activity data
* browsers
* conversation data
* good and bad aspects
* 'Industrial Internet'
* machines connected to the
* map of the
* online fraud
* search engine data
* text clustering
* video/image analytics
* _see also_ social media
* Internet of Things (IoT)
* interview data
* iPhones
* IT systems
* James, Bill
* Jawbone
* Jobs, Steve
* John Deere
* Jolie, Angelina
* Klout score
* law enforcement
* leadership
* legal issues
* 'Likes'
* line graphs
* LinkedIn
* love
* loyalty cards
* magazines
* management dashboards
* managers
* manufacturing
* MapReduce
* maps
* mavens
* measurement
* data needs
* new forms of data
* types of data
* medical information
* Merck
* metrics
* Microsoft Research Labs
* Minard, Charles Joseph
* mission statement
* Model T Ford
* music
* NameTag
* narrative
* Narrative Science
* NASA
* National Security Agency (NSA)
* Near Field Communication (NFC)
* networks
* new business opportunities
* new product development
* New York manhole explosions
* online dating
* operational dashboards
* operations
* Page, Larry
* parenting
* partners
* pear tree metaphor
* People and Innovation Lab (PiLab)
* Periodic Table of Visualization Methods
* personality traits
* Philippines disaster relief
* photo data
* pie charts
* piloting experience
* pilots
* Pole, Andrew
* PolyMaps
* postnatal depression
* pothole detection
* prediction
* pregnant women
* premature babies
* privacy
* Procter & Gamble (P&G)
* Project Oxygen
* provoked data
* proximity sensors
* purpose panel
* questions _see_ SMART questions
* Radio Frequency Identification (RFID)
* recruitment
* reporting
* data visualization
* infographics
* management dashboards
* self-service business intelligence tools
* uniformity of data
* resources
* retail
* behaviour analytics
* business process optimization
* combined analytics
* predictive analytics
* RFID systems
* transaction data
* right to be forgotten
* risk
* Rolls Royce engines
* Rosling, Hans
* Royal Air Force (RAF)
* sales
* Samsung
* scatter charts
* Schmidt, Eric
* search engines
* self-assessment performance data
* self-service business intelligence tools
* semi-structured data
* sensors
* baby monitoring
* business process optimization
* cars
* driving behaviour
* General Electric
* mobile phone signal detection
* pothole detection
* retail stores
* Rolls Royce engines
* sports
* sentiment analysis
* Shape of a Song
* ShotSpotter
* situation awareness
* skin texture analysis
* sleep data
* Smarr, Larry
* smart badges
* SMART Data
* SMART Model
* smart phones
* SMART questions
* competition and risk
* customers
* data discovery
* data needs
* data visualization
* finance
* operations
* resources
* SMART strategy board
* data needs
* description of panels
* pear tree metaphor
* Snowden, Edward
* social media
* athletes' use of
* conversation data
* data collection
* face recognition
* photo and video data
* recruitment based on
* sentiment analysis
* text analytics
* trending
* _see also_ Facebook; Internet; Twitter
* software
* data analysis
* data visualization
* employee monitoring
* face recognition
* self-service business intelligence
* sentiment analysis
* situation awareness
* voice recognition
* _see also_ analytics; apps
* sound data
* Space Shuttle disaster
* spam filters
* speech analytics
* sports
* spreadsheets
* stakeholders
* stock market data
* storage of data
* storytelling
* strategic dashboards
* strategy
* analytics
* data visualization
* metrics
* SMART strategy board
* stress
* structured data
* decision-making informed by
* hierarchy of data
* triangulation of data
* types of data
* Structured Query Language (SQL)
* supply chain data
* surveys
* tags
* Target
* target market
* taxi-drivers
* technology
* _see also_ Internet; sensors; software; wearable technology
* telecom industry
* telephone conversations
* Tesco
* text analytics
* text categorization
* text clustering
* text, displaying
* 3D face recognition
* Toyota Prius
* transaction data
* transformation
* business insights
* business process optimization
* business security and fraud
* employment
* health and well-being
* understanding and targeting customers
* translation software
* transparency
* triangulation of data
* TripAdvisor
* Tufte, Edward
* Tumblr
* TV set top boxes
* Twitrratr
* Twitter
* Dell
* employees' posts on
* Gatorade
* infographic of advice
* Klout score
* Uber
* uniformity of data
* unstructured data
* hierarchy of data
* insight from
* text analytics
* triangulation of data
* 'Up' fitness band
* user-generated data
* value
* value proposition
* variety of data
* velocity of data
* veracity of data
* Verizon
* video content analysis (VCA)
* video data
* vision statement
* visualization
* voice recognition software
* volume of data
* Walmart
* Watson (IBM computer)
* wearable technology
* weather data
* Webb, Amy
* webcams
* websites
* activity data
* Crazy Egg heat map tool
* map of the Internet
* text analytics
* _see also_ Internet
* Wenger, Brittany
* wireless networking
* word clouds
* YouTube
# **WILEY END USER LICENSE AGREEMENT**
Go to www.wiley.com/go/eula to access Wiley's ebook EULA.
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 9,259
|
Get immediate access to our Monthly EU Vanilla server and skip the queue for 30 days!
You will not be given ANY refunds if you are caught hacking.
You will not be given ANY refunds if you are banned for breaking any server rules.
If you have an alternate account w/ a game/VAC ban, WE WILL FIND IT and you will be banned without a refund.
After your purchase, please check your VIP Status here.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 267
|
using namespace v8::internal;
using v8::Just;
TEST(SequentialMarkingDeque) {
CcTest::InitializeVM();
SequentialMarkingDeque s(CcTest::i_isolate()->heap());
s.SetUp();
s.StartUsing();
Address original_address = reinterpret_cast<Address>(&s);
Address current_address = original_address;
while (!s.IsFull()) {
s.Push(HeapObject::FromAddress(current_address));
current_address += kPointerSize;
}
while (!s.IsEmpty()) {
Address value = s.Pop()->address();
current_address -= kPointerSize;
CHECK_EQ(current_address, value);
}
CHECK_EQ(original_address, current_address);
s.StopUsing();
CcTest::i_isolate()->cancelable_task_manager()->CancelAndWait();
s.TearDown();
}
TEST(Promotion) {
CcTest::InitializeVM();
Isolate* isolate = CcTest::i_isolate();
{
v8::HandleScope sc(CcTest::isolate());
Heap* heap = isolate->heap();
heap::SealCurrentObjects(heap);
int array_length = heap::FixedArrayLenFromSize(kMaxRegularHeapObjectSize);
Handle<FixedArray> array = isolate->factory()->NewFixedArray(array_length);
// Array should be in the new space.
CHECK(heap->InSpace(*array, NEW_SPACE));
CcTest::CollectAllGarbage();
CcTest::CollectAllGarbage();
CHECK(heap->InSpace(*array, OLD_SPACE));
}
}
HEAP_TEST(NoPromotion) {
// Page promotion allows pages to be moved to old space even in the case of
// OOM scenarios.
FLAG_page_promotion = false;
CcTest::InitializeVM();
Isolate* isolate = CcTest::i_isolate();
{
v8::HandleScope sc(CcTest::isolate());
Heap* heap = isolate->heap();
heap::SealCurrentObjects(heap);
int array_length = heap::FixedArrayLenFromSize(kMaxRegularHeapObjectSize);
Handle<FixedArray> array = isolate->factory()->NewFixedArray(array_length);
heap->set_force_oom(true);
// Array should be in the new space.
CHECK(heap->InSpace(*array, NEW_SPACE));
CcTest::CollectAllGarbage();
CcTest::CollectAllGarbage();
CHECK(heap->InSpace(*array, NEW_SPACE));
}
}
HEAP_TEST(MarkCompactCollector) {
FLAG_incremental_marking = false;
FLAG_retain_maps_for_n_gc = 0;
CcTest::InitializeVM();
Isolate* isolate = CcTest::i_isolate();
Heap* heap = CcTest::heap();
Factory* factory = isolate->factory();
v8::HandleScope sc(CcTest::isolate());
Handle<JSGlobalObject> global(isolate->context()->global_object());
// call mark-compact when heap is empty
CcTest::CollectGarbage(OLD_SPACE);
// keep allocating garbage in new space until it fails
const int arraysize = 100;
AllocationResult allocation;
do {
allocation = heap->AllocateFixedArray(arraysize);
} while (!allocation.IsRetry());
CcTest::CollectGarbage(NEW_SPACE);
heap->AllocateFixedArray(arraysize).ToObjectChecked();
// keep allocating maps until it fails
do {
allocation = heap->AllocateMap(JS_OBJECT_TYPE, JSObject::kHeaderSize);
} while (!allocation.IsRetry());
CcTest::CollectGarbage(MAP_SPACE);
heap->AllocateMap(JS_OBJECT_TYPE, JSObject::kHeaderSize).ToObjectChecked();
{ HandleScope scope(isolate);
// allocate a garbage
Handle<String> func_name = factory->InternalizeUtf8String("theFunction");
Handle<JSFunction> function = factory->NewFunction(func_name);
JSReceiver::SetProperty(global, func_name, function, SLOPPY).Check();
factory->NewJSObject(function);
}
CcTest::CollectGarbage(OLD_SPACE);
{ HandleScope scope(isolate);
Handle<String> func_name = factory->InternalizeUtf8String("theFunction");
CHECK(Just(true) == JSReceiver::HasOwnProperty(global, func_name));
Handle<Object> func_value =
Object::GetProperty(global, func_name).ToHandleChecked();
CHECK(func_value->IsJSFunction());
Handle<JSFunction> function = Handle<JSFunction>::cast(func_value);
Handle<JSObject> obj = factory->NewJSObject(function);
Handle<String> obj_name = factory->InternalizeUtf8String("theObject");
JSReceiver::SetProperty(global, obj_name, obj, SLOPPY).Check();
Handle<String> prop_name = factory->InternalizeUtf8String("theSlot");
Handle<Smi> twenty_three(Smi::FromInt(23), isolate);
JSReceiver::SetProperty(obj, prop_name, twenty_three, SLOPPY).Check();
}
CcTest::CollectGarbage(OLD_SPACE);
{ HandleScope scope(isolate);
Handle<String> obj_name = factory->InternalizeUtf8String("theObject");
CHECK(Just(true) == JSReceiver::HasOwnProperty(global, obj_name));
Handle<Object> object =
Object::GetProperty(global, obj_name).ToHandleChecked();
CHECK(object->IsJSObject());
Handle<String> prop_name = factory->InternalizeUtf8String("theSlot");
CHECK_EQ(*Object::GetProperty(object, prop_name).ToHandleChecked(),
Smi::FromInt(23));
}
}
// TODO(1600): compaction of map space is temporary removed from GC.
#if 0
static Handle<Map> CreateMap(Isolate* isolate) {
return isolate->factory()->NewMap(JS_OBJECT_TYPE, JSObject::kHeaderSize);
}
TEST(MapCompact) {
FLAG_max_map_space_pages = 16;
CcTest::InitializeVM();
Isolate* isolate = CcTest::i_isolate();
Factory* factory = isolate->factory();
{
v8::HandleScope sc;
// keep allocating maps while pointers are still encodable and thus
// mark compact is permitted.
Handle<JSObject> root = factory->NewJSObjectFromMap(CreateMap());
do {
Handle<Map> map = CreateMap();
map->set_prototype(*root);
root = factory->NewJSObjectFromMap(map);
} while (CcTest::heap()->map_space()->MapPointersEncodable());
}
// Now, as we don't have any handles to just allocated maps, we should
// be able to trigger map compaction.
// To give an additional chance to fail, try to force compaction which
// should be impossible right now.
CcTest::CollectAllGarbage(Heap::kForceCompactionMask);
// And now map pointers should be encodable again.
CHECK(CcTest::heap()->map_space()->MapPointersEncodable());
}
#endif
#if defined(__has_feature)
#if __has_feature(address_sanitizer)
#define V8_WITH_ASAN 1
#endif
#endif
// Here is a memory use test that uses /proc, and is therefore Linux-only. We
// do not care how much memory the simulator uses, since it is only there for
// debugging purposes. Testing with ASAN doesn't make sense, either.
#if defined(__linux__) && !defined(USE_SIMULATOR) && !defined(V8_WITH_ASAN)
static uintptr_t ReadLong(char* buffer, intptr_t* position, int base) {
char* end_address = buffer + *position;
uintptr_t result = strtoul(buffer + *position, &end_address, base);
CHECK(result != ULONG_MAX || errno != ERANGE);
CHECK(end_address > buffer + *position);
*position = end_address - buffer;
return result;
}
// The memory use computed this way is not entirely accurate and depends on
// the way malloc allocates memory. That's why the memory use may seem to
// increase even though the sum of the allocated object sizes decreases. It
// also means that the memory use depends on the kernel and stdlib.
static intptr_t MemoryInUse() {
intptr_t memory_use = 0;
int fd = open("/proc/self/maps", O_RDONLY);
if (fd < 0) return -1;
const int kBufSize = 20000;
char buffer[kBufSize];
ssize_t length = read(fd, buffer, kBufSize);
intptr_t line_start = 0;
CHECK_LT(length, kBufSize); // Make the buffer bigger.
CHECK_GT(length, 0); // We have to find some data in the file.
while (line_start < length) {
if (buffer[line_start] == '\n') {
line_start++;
continue;
}
intptr_t position = line_start;
uintptr_t start = ReadLong(buffer, &position, 16);
CHECK_EQ(buffer[position++], '-');
uintptr_t end = ReadLong(buffer, &position, 16);
CHECK_EQ(buffer[position++], ' ');
CHECK(buffer[position] == '-' || buffer[position] == 'r');
bool read_permission = (buffer[position++] == 'r');
CHECK(buffer[position] == '-' || buffer[position] == 'w');
bool write_permission = (buffer[position++] == 'w');
CHECK(buffer[position] == '-' || buffer[position] == 'x');
bool execute_permission = (buffer[position++] == 'x');
CHECK(buffer[position] == 's' || buffer[position] == 'p');
bool private_mapping = (buffer[position++] == 'p');
CHECK_EQ(buffer[position++], ' ');
uintptr_t offset = ReadLong(buffer, &position, 16);
USE(offset);
CHECK_EQ(buffer[position++], ' ');
uintptr_t major = ReadLong(buffer, &position, 16);
USE(major);
CHECK_EQ(buffer[position++], ':');
uintptr_t minor = ReadLong(buffer, &position, 16);
USE(minor);
CHECK_EQ(buffer[position++], ' ');
uintptr_t inode = ReadLong(buffer, &position, 10);
while (position < length && buffer[position] != '\n') position++;
if ((read_permission || write_permission || execute_permission) &&
private_mapping && inode == 0) {
memory_use += (end - start);
}
line_start = position;
}
close(fd);
return memory_use;
}
intptr_t ShortLivingIsolate() {
v8::Isolate::CreateParams create_params;
create_params.array_buffer_allocator = CcTest::array_buffer_allocator();
v8::Isolate* isolate = v8::Isolate::New(create_params);
{ v8::Isolate::Scope isolate_scope(isolate);
v8::Locker lock(isolate);
v8::HandleScope handle_scope(isolate);
v8::Local<v8::Context> context = v8::Context::New(isolate);
CHECK(!context.IsEmpty());
}
isolate->Dispose();
return MemoryInUse();
}
TEST(RegressJoinThreadsOnIsolateDeinit) {
intptr_t size_limit = ShortLivingIsolate() * 2;
for (int i = 0; i < 10; i++) {
CHECK_GT(size_limit, ShortLivingIsolate());
}
}
TEST(Regress5829) {
CcTest::InitializeVM();
Isolate* isolate = CcTest::i_isolate();
v8::HandleScope sc(CcTest::isolate());
Heap* heap = isolate->heap();
heap::SealCurrentObjects(heap);
i::MarkCompactCollector* collector = heap->mark_compact_collector();
i::IncrementalMarking* marking = heap->incremental_marking();
if (collector->sweeping_in_progress()) {
collector->EnsureSweepingCompleted();
}
CHECK(marking->IsMarking() || marking->IsStopped());
if (marking->IsStopped()) {
heap->StartIncrementalMarking(i::Heap::kNoGCFlags,
i::GarbageCollectionReason::kTesting);
}
CHECK(marking->IsMarking());
marking->StartBlackAllocationForTesting();
Handle<FixedArray> array = isolate->factory()->NewFixedArray(10, TENURED);
Address old_end = array->address() + array->Size();
// Right trim the array without clearing the mark bits.
array->set_length(9);
heap->CreateFillerObjectAt(old_end - kPointerSize, kPointerSize,
ClearRecordedSlots::kNo);
heap->old_space()->EmptyAllocationInfo();
Page* page = Page::FromAddress(array->address());
LiveObjectIterator<kGreyObjects> it(page, MarkingState::Internal(page));
HeapObject* object = nullptr;
while ((object = it.Next()) != nullptr) {
CHECK(!object->IsFiller());
}
}
#endif // __linux__ and !USE_SIMULATOR
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,039
|
Four Exiles named in England U20s squad for World Rugby U20 Championship
London Irish quartet Tom Parton, Ben Loader, Rory Brand and Josh Basham have all been named in Steve Bates' England U20s squad for this summer's World Rugby U20 Championship in France.
Parton, 20, will embark on his second Junior World Cup in as many years this summer, the former Wellington College pupil having played eight times for the Exiles since moving into the senior academy set-up ahead of the 2016/17 season. That season, Parton notched up two tries in six appearances during the Exiles' British and Irish Cup exploits, his excellent form coinciding with an U20 Six Nations winners medal.
Fellow Exiles academy graduate Loader has also impressed club officials since making his first team bow against Edinburgh Rugby in the European Rugby Challenge Cup last autumn. Making seven appearances for Irish, the explosive winger dotted down on his Premiership debut at Bath on the final weekend of the campaign, and was recently on the score sheet for England U20s against the Junior Springboks in the build-up to this summer's tournament.
Meanwhile, scrum-half Brand, who like Parton was part of the Exiles side that was crowned Premiership Rugby U18 champions in 2016, has represented England U20s in the U20 Six Nations, picking up a Six Nations winners medal in 2017. Brand is joined in the squad by second-row Basham, who last year was part of the England U18 squad that toured South Africa. The former Wellington College talent made his Premiership debut during the Exiles' 35-5 triumph at Harlequins in April.
England meet Argentina in Narbonne on 30 May (kick-off 5.30pm BST), Italy on 3 June in Perpignan (kick-off 3.30pm BST) and Scotland at Beziers' Stade de la Méditerranée on 7 June (kick-off 8pm BST).
Narbonne and Perpignan will host the semi-finals and ranking play-offs on 12 June, with Beziers hosting all finals day matches on 17 June across two pitches.
England U20s have reached the final in five consecutive years, winning three (2013, 2014, 2016) and since 2008 they have reached eight out of the 10 finals.
The U20s are led by Bates, with support from Anthony Allen (backs), Richard Blaze (forwards) and James Ponton (defence) as part of the coach development agreement between the RFU and PRL.
"Selection has been incredibly difficult as there were a lot of players to choose from and in some positions it's been a really tight call," said England U20s boss Bates.
"We saw real glimpses of what this team is capable of during the Six Nations, and I was proud of the way they performed.
"The challenge for us now is to consistently play high-intensity, high-tempo rugby and to turn up the heat on our opposition for the full 80 minutes.
"The World Rugby U20 Championship is a fantastic tournament and for some this will be the pinnacle of their international careers. Some will go on to be very good professionals but not all will play international rugby again so they should look at this as a fantastic opportunity to play on the world stage and showcase their ability while representing England.
"The message going into this tournament is we are expecting much more of the players as individuals and a team and although we have a proud history in this tournament, we want to leave our own legacy."
England U20s squad for 2018 World Rugby U20 Championship:
Josh Basham (London Irish)
Ben Curry (Sale Sharks)
Beck Cutting (Worcester Warriors)
Joe Heyes (Leicester Tigers)
Ted Hill (Worcester Warriors)
Aaron Hinkley (Gloucester Rugby)
Ciaran Knight (Gloucester Rugby)
Joel Kpoku (Saracens)
Sam Lewis (Leicester Tigers)
Gabriel Oghre (Wasps)
Ehren Painter (Northampton Saints)
James Scott (Worcester Warriors)
Alex Seville (Gloucester Rugby)
Toby Trinder (Northampton Saints)
Henry Walker (Gloucester Rugby)
Tom Willis (Wasps)
Rory Brand (London Irish)
Will Butler (Worcester Warriors)
Ali Crossdale (Saracens)
Fraser Dingwall (Northampton Saints)
James Grayson (Northampton Saints)
Tom Hardwick (Leicester Tigers)
Gabriel Ibitoye (Harlequins)
Ben Loader (London Irish)
Jordan Olowofela (Leicester Tigers)
Tom Parton (London Irish)
Marcus Smith (Harlequins)
Ben White (Leicester Tigers)
Unavailable through injury:
Dino Lamb (Harlequins)
Sam Moore (Sale Sharks)
Marcus Street (Exeter Chiefs)
Unavailable as named in senior squad:
Tom Curry (Sale Sharks)
Ben Earl (Saracens)
Nick Isiekwe (Saracens)
Cameron Redpath (Sale Sharks)
England U20 fixtures:
England v Argentina
Wednesday 30 May, 5.30 KO BST – highlights on ITV Sport 00:00
Stade d'Honneur du Parc des Sports et de l'Amitié, Narbonne
Tickets available, here
England v Italy
Sunday 3 June, 3.30 KO BST - highlights on ITV Sport 23:00
Stade Aimé Giral, Perpignan
England v Scotland
Thursday 7 June, 8pm KO BST – highlights on ITV Sport 00:00
Stade de la Méditerranée, Beziers
Semi-finals and ranking play-offs:
Narbonne and Perpignan, 12 June
Finals day:
Beziers, 17 June
HOSKINS: THIS IS A REALLY KEY WINDOW
KIDNEY KEEN TO MAKE UP FOR LOST TIME
PARTON PROUD WITH QUINS COMEBACK
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,604
|
Lost in the Weekend è un singolo del cantautore italiano Cesare Cremonini, pubblicato il 4 settembre 2015 come secondo estratto dal secondo album dal vivo Più che logico (Live).
Descrizione
Cremonini ha così descritto il brano, scritto insieme a Davide Petrella:
Video musicale
Il videoclip, pubblicato il 4 settembre 2015, è stato girato a Los Angeles. Cremonini l'ha definito «molto bello e particolare, di cui andremo e andrete fieri». Scritto da Cremonini stesso e da Walter Mameli, è stato diretto da Gaetano Morbioli.
Tracce
Formazione
Cesare Cremonini – voce, chitarra acustica, tastiera
Nicola "Ballo" Balestri – basso
Alessandro Magnanini – tastiera, sintetizzatore, chitarra elettrica
Andrea Fontana – batteria
Classifiche
Note
Collegamenti esterni
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,485
|
NOTE: Comes as a cluster of various sizes.
The Edison Collection. Inspired by simple, yet functional antique lighting from the times of the Industrial Revolution.
A perfect addition to residential and commercial spaces such as living rooms, kitchens, work stations, restaurants, shops, and more! Feature multiples for a fabulous design.
We strive to offer you a wide range of fine-looking, high-quality light fixtures at reasonable prices along with great customer service.
SIZE: 5.5" W x 5.5" H / CEILING CANOPY: 5.5"
40W Edison style bulb(s) included.
60 Watt max, 110-120 Volts, UL Listed parts.
Installation by an electrician is recommended.
© 2018 OHR Lighting. Built & Managed with Love by Ecomclips.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,647
|
\section{Introduction}
\noindent With the renaissance of deep learning, tremendous breakthroughs have been achieved on various visual tasks \cite{he2016deep,fan2019shifting,ren2015faster}. However, the deep learning techniques typically rely on the availability of artificially balanced training data, which poses a significant bottleneck against building comprehensive models for the real visual world. In recent years, Zero-Shot Learning (ZSL)~\cite{lampert2009learning,frome2013devise,xian2016latent,Yu2018Stacked,xie2019attentive} has been attracting a lot of attention due to its potential to address the data scarcity issue.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.93\columnwidth]{fig1}
\end{center}
\caption{An illustration of the episode-based framework for ZSL. The training process consists of a collection of episodes, each of which randomly splits the training data into two class-exclusive subsets: one is used for training the base model, the other one is used for refining the model. The model generalized ability is progressively enhanced as the episodes go on. The test data are predicted with the final model.}
\label{fig1}
\end{figure}
Zero-Shot Learning (ZSL) aims at recognizing unseen classes that have no visual instances during the training stage. Such harsh but realistic scenarios are painful for the traditional classification approaches because there are no labeled visual data to support the parameter training for unseen classes. To tackle this task, the existing methods mostly resort to the transfer learning that assumes the model trained on the seen classes can be applied to the unseen classes, and focus on learning a transferable model with the seen data.
Although promising performances have been achieved, the most existing approaches \cite{akata2013label,akata2015evaluation,xian2016latent,romera2015embarrassingly,frome2013devise,shigeto2015ridge,song2018transductive,sariyildiz2019gradient} dedicated to designing visual-semantic interaction models with the seen classes cannot guarantee to generalize well to the unseen classes, as the seen and unseen classes are located in disjoint domains. Furthermore, the models trained with the seen data favorably guide the unseen test instances to be misclassified into the seen classes, which tends to produce a remarkable imbalanced classification shift issue. The existing generative approaches \cite{kumar2018generalized,li2019leveraging,xian2018feature} transfer the zero-shot classification task to a traditional classification problem via synthesizing some visual features for unseen classes, which can alleviate the above issues to some extent. However, they still struggle in the generalized ZSL task due to the instability in training and mode collapse issues.
Inspired by the success of meta-learning in the few-shot learning task \cite{snell2017prototypical,sung2018learning}, we introduce an episode-based training paradigm to learn a zero-shot classification model for mitigating the above issues. Specifically, the training process consists of a collection of episodes. Each episode randomly splits the training data into two class-exclusive subsets: one support set and one refining set. In this way, each episode mimics a fake zero-shot classification task. The support set is used to train a base model, which builds semantic interactions between the visual and the class semantic modalities. The refining set is used to refine the base model by minimizing the differences between the ground-truth labels and the predicted ones obtained with the base model in a pre-defined space. The model trained in the current episode is initialized with the model parameters learned from the previous episode. As the episodes go on, the base model progressively accumulates ensemble experiences on predicting fake unseen classes, which will generalize well to the real unseen classes. In this way, the gap between the seen and unseen domains can be reduced accordingly. The framework of the whole idea is illustrated in Fig.~\ref{fig1}.
Under the above episode-based training framework, the base model plays an indispensable role in the process of the prediction of unseen classes. In this work, we design an elegant Prototype Generating Network (PGN) as the base model to synthesize class-level visual prototypes conditioned on the class semantic prototypes. As a departure from the existing generative approaches that involve the minimax play games between a generator and a discriminator, our model consists of two generators that map the visual features and the class semantic prototypes into their counterparts and a discriminator that distinguishes between the concatenation of the real visual features and the real class semantic prototypes and the concatenation of the fake counterparts. To capture the discriminative information, we further devise a novel Multi-modal Cross-Entropy Loss to integrate the visual features, class semantic prototypes, and class labels into a classification network. Compared with the existing generative approaches that require an extra assisting classification network with a separate set of learning parameters, our classification network introduces no extra parameters, thus is more efficient.
In summary, our contributions are concluded into the following three-fold.
\begin{enumerate}
\item To enhance the adaptability of the model, we introduce an episode-based training paradigm for ZSL that trains the models within a collection of episodes, each of which is designed to simulate a fake ZSL task. Through training multiple episodes, the model progressively accumulates a wealth of experiences on predicting the fake unseen classes, which will generalize well to the real unseen classes.
\item We propose a well-designed prototype generating network to synthesize visual prototypes conditioned on the class semantic prototypes. It aligns the visual-semantic interactions by formulating both the visual prototype generation and the class semantic inference into an adversarial framework and captures the discriminative information with an efficient Multi-modal Cross-Entropy Loss.
\item Extensive experiments on four benchmarks show that the proposed approach achieves the state-of-the-art performances under both the traditional ZSL and the realistic generalized ZSL tasks.
\end{enumerate}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=14.2cm,height=6.2cm]{fig2}
\end{center}
\caption{Diagram of the proposed approach for one episode training step that consists of a training stage (top) and a refining stage (below). The training stage trains the base model by aligning the visual-semantic interaction (V.S. Interaction). The refining stage first initializes the label prediction of the test data with the trained model in a pre-defined space and then fine-tunes the model by minimizing the differences between the predicted results and the ground-truth labels.}
\label{fig2}
\end{figure*}
\section{Related Work}
In this section, we provide an overview of the most related work on ZSL and episode-based approaches.
\subsection{Generative ZSL}
Recently, the generative approaches predominate in ZSL by exploiting either the existing generative models \cite{goodfellow2014generative,kingma2014auto} or their variations \cite{schonfeld2019generalized,zhu2019learning,atzmon2019adaptive} to synthesize visual features from the class-level semantic features (e.g., attributes and text description embeddings) along with some noises. \cite{xian2018feature,zhu2018generative,li2019leveraging} introduce the Wasserstein generative adversarial network (WGAN) \cite{arjovsky2017wasserstein} paired with a classification network to synthesize visual features for unseen classes such that the ZSL task is transferred to a traditional classification problem. Differently, \cite{zhu2018generative} also introduces a visual pivot regularization to preserve the inter-class discrimination of the generated features while \cite{li2019leveraging} enhances the inter-class discrimination by enforcing the generated visual features to be close to at least one class meta-representations. In contrast to GAN-based approaches, \cite{wang2018zero,schonfeld2019generalized} formulate the feature generation into the variational autoencoder (VAE) \cite{kingma2014auto} model to fit the class-specific latent distribution and highly discriminative feature representations. To combine the strength of VAE and GAN, \cite{xian2019f} develops a conditional generative model to synthesize visual features, which is also extended to exploit the unlabeled instances under the transductive setting via an unconditional discriminator.
Our model is also a generative approach. Instead of synthesizing instance-level visual features, we synthesize class-level visual prototypes conditioned on the class semantic prototypes without extra noise input. Among previous generative approaches, several are closely related to our model.
For example, DEM \cite{zhang2017learning} trains a visual prototype generating network with a three-layer neural network by minimizing the differences between the synthesized visual prototypes and real visual features. In contrast, our approach formulates both the visual prototype generation and class semantic inference into a united framework. Different from \cite{felix2018multi,huang2019generative} that formulate the visual feature generation and class semantic inference in a cycle-consistent manner, our model formulates these two processes with two separable bidirectional mapping networks that are integrated by the discriminator and the classification network, which aligns the visual-semantic interactions better. Furthermore, our approach is trained in an episode-based framework to enhance the adaptability to the unseen classes.
\subsection{Episode-based approach}
Episode-based training strategy has been widely explored in the few-shot learning task \cite{finn2017model,ravi2017optimization,snell2017prototypical,vinyals2016matching} that divides the training process into extensive episodes, each of which mimics a few-shot learning task. However, few researches apply the episode-based training strategy to ZSL.
In this work, we introduce the episode-based paradigm to train the ZSL model. Different from the existing episode-based few-shot approaches, each episode in our approach mimics a zero-shot classification task, which requires to train a base visual-semantic interaction model to achieve the prediction of unseen classes. One related work to ours is RELATION NET \cite{sung2018learning} that also trains a ZSL model in an episode-based paradigm. However, RELATION NET \cite{sung2018learning} learns a general metric space to evaluate the relations between the visual instances and the class semantic features rather than simulating a zero-shot classification task. Another related work is 3ME \cite{felix2019multi} that improves the performance with an ensemble of two different models. Our approach can also be seen as a special ensemble approach that consists of a collection of models. Differently, the models are not equal to vote for the final classification instead of accumulating the previous experiences recursively.
\section{Methodology}
In this section, we first introduce the problem formulation and then report our approach in detail.
\subsection{Problem Formulation}
Suppose that we collect a training sample set
$\mathcal{S} =\{\mathbf{x}_i, \mathbf{a}_i,\mathbf{y}_i\}_{i=1}^N$ that consists of $N$ samples from $M$ seen categories, where $\mathbf{x}_i\in\mathbb{R}^D$ is the $D$-dimensional visual representation (e.g., CNN feature) for the $i$-th instance, $\mathbf{a}_i\in\mathbb{R}^K$ and $\mathbf{y}_i$ are its $K$-dimensional class semantic prototype (e.g., class-level attribute or text description vector) and one-hot class label, respectively. At the test time, in the traditional zero-shot classification setting, the task is to classify a test instance into one of the candidate unseen categories, and in the generalized zero-shot classification setting, the task is to classify the test instance into either a seen or an unseen category.
\subsection{Model}
At the training stage, we introduce an episode-based paradigm for training, which trains the model by simulating multiple zero-shot classification tasks on the seen categories. Each episode matches an individual zero-shot classification task. In each episode, the seen categories $\mathcal{S}$ are randomly split into two class-exclusive sets, one support set $\mathcal{S}^{tr}=\{\mathbf{X}_{tr},\mathbf{A}_{tr},\mathbf{Y}_{tr}\}$ and one refining set $\mathcal{S}^{te}=\{\mathbf{X}_{te},\mathbf{A}_{te},\mathbf{Y}_{te}\}$, where $\mathbf{Y}_{tr}$ and $\mathbf{Y}_{te}$ are disjoint.
As illustrated in Fig.~\ref{fig2}, each episode consists of a training stage and a refining stage. The training stage learns a base model to align the semantic consistency, which is used to predict the unseen classes from the corresponding class semantic prototypes. The refining stage updates the model parameters by minimizing the predicted results and the ground-truth labels. Training each episode can be seen as a process of accumulating experience on zero-shot classification. The experience will be carried forward to the next episode as the episode goes on. After training a collection of episodes, the model is expected to be an expert in predicting unseen classes such that it can generalize well to the real unseen classes. In the following, we introduce the base model and the refining model in an episode in detail.
\begin{figure}[t]
\begin{center}
\includegraphics[width=8.5cm,height=3.4cm]{fig3}
\end{center}
\caption{The basic model that aligns the semantic consistency across different modalities. The combination of both image feature $\mathbf{x}$ and class semantic prototype $\mathbf{a}$ takes as the real input, while the combination of both the synthesized visual prototype $\mathbf{\tilde{x}}$ and the projected class semantic feature $\mathbf{\tilde{a}}$ as the fake input of the discriminator $D$. Both $F$ and $G$ are mapping networks.}
\label{fig3}
\end{figure}
\subsubsection{Prototype Generating Network}
To address the zero-shot classification task, the learning agent requires learning a base model to infer unseen categories from the corresponding class semantic prototypes. In this paper, we devise a Prototype Generating Network (PGN) to achieve this goal.
For the visual modality, we learn a class semantic inference network $F: \mathbb{R}^D \rightarrow \mathbb{R}^{K}$ to project the image features into the class semantic space by regressing the image features to be close to the corresponding class semantic prototypes, which is formulated as:
\begin{equation}\label{equ1}
L_{\mathcal{V}\rightarrow \mathcal{A}} = \sum_i \|F(\mathbf{x}_i) - \mathbf{a}_i\|_2^2.
\end{equation}
Similarly, for the class semantic modality, we learn a visual prototype generating network $G: \mathbb{R}^K \rightarrow \mathbb{R}^D$ to project the class semantic prototypes into the visual space. Since each class usually consists of many image instances but corresponds to only one class semantic prototype, the mapping function $G$ can be seen as a one-to-many semantic-to-visual feature generator. The mapping function $G$ is learned by minimizing the distances between the synthesized visual feature $G(\mathbf{a}_i)$ (we call it visual prototype) and the real visual feature $\mathbf{x}_i$.
\begin{equation}\label{equ2}
L_{\mathcal{A}\rightarrow \mathcal{V}} = \sum_i \|G(\mathbf{a}_i) - \mathbf{x}_i\|_2^2.
\end{equation}
With $F$ and $G$, we can construct the relationships between the visual space and the class semantic space. However, they are independent to each other. To better align the semantic consistency, we introduce the adversarial mechanism to regularize both mapping networks, as illustrated in Fig.~\ref{fig3}. Specifically, we leverage a modified WGAN \cite{gulrajani2017improved} to integrate the projected class semantic vector $\mathbf{\tilde{a}}$ and the real class semantic prototype $\mathbf{a}$ separately to generator and discriminator. The loss is written:
\begin{equation}\label{equ3}
\begin{aligned}
L_{WGAN}= \mathbb{E}[D(\mathbf{x},\mathbf{a})] - \mathbb{E}[D(\mathbf{\tilde{x}},\mathbf{\tilde{a}})]-\\
\lambda\mathbb{E}[(\|\nabla_{\mathbf{\hat{x}}}D(\mathbf{\hat{x}},\mathbf{\hat{a}})\|_2-1)^2],
\end{aligned}
\end{equation}
where $\mathbf{\tilde{a}} = F({\mathbf{x}})$ is the inferential class semantic feature; $\mathbf{\tilde{x}} = G(\mathbf{a})$ is the synthetic visual prototype. $\mathbf{\hat{x}}=\tau \mathbf{x}+ (1-\tau) \mathbf{\tilde{x}}$ and $\mathbf{\hat{a}}=\tau \mathbf{a}+ (1-\tau) \mathbf{\tilde{a}}$ with $\tau \thicksim U(0,1)$, $\lambda$ is the penalty coefficient. $D$ denotes the discriminator network. In contrast to the existing GAN-based approaches, the proposed model can be seen as containing two generators and one discriminator, where the generators separately perform on the two different modalities while the discriminator integrates them.
The above model aligns the semantic consistency between the visual features and class semantics. However, training such a model neglects to exploit the discriminative information to distinguish categories, which is essential to the final class prediction. To address this issue, we further propose a multi-modal cross-entropy loss that interweaves the image features, class semantics, and the one-hot class labels into a united framework. With the above model, the class semantic prototypes of all training categories are projected into the visual space to obtain their corresponding class visual prototypes that are pre-stored in a visual feature buffer $G(\mathbf{A}_S)$, where $G(\mathbf{a}_i)$ denotes the class visual prototype of the $i$-th category. The affinities between a visual sample $\mathbf{x}$ and all class visual prototypes could be obtained with their inner products $\mathbf{x}^TG(\mathbf{A}_S)$. In this way, the probability of the input visual sample $\mathbf{x}$ belonging to the $i$-th category in the visual space can be evaluated with the affinity of visual sample $\mathbf{x}$ matching the $i$-th class semantic vector with the following cross-modal softmax function:
\begin{equation}\label{equ4}
p_i^\mathcal{V}(\mathbf{x})= \frac{exp(\mathbf{x}^TG(\mathbf{a}_i))}{\sum_{j}exp(\mathbf{x}^TG(\mathbf{a}_j))}.
\end{equation}
Similarly, in the class semantic space, all class semantic vectors are pre-stored in a class semantic buffer $\mathbf{A}_S$ and a visual sample $\mathbf{x}$ is represented as $F(\mathbf{x})$. Therefore, the probability of $\mathbf{x}$ belonging to the $i$-th category in the class semantic space can be defined as
\begin{equation}\label{equ5}
p_i^\mathcal{S}(\mathbf{x})= \frac{exp(F(\mathbf{x})^T\mathbf{a}_i)}{\sum_{j}exp(F(\mathbf{x})^T\mathbf{a}_j)}.
\end{equation}
Our goal is to maximize the above probabilities in both the visual and class semantic spaces, which can be formulated by minimizing the following Multi-modal Cross-Entropy (MCE) Loss,
\begin{equation}\label{equ6}
L_{MCE} = -\sum_{\mathbf{x}} \log p_i^\mathcal{V}(\mathbf{x})-\sum_{\mathbf{x}} \log p_i^\mathcal{S}(\mathbf{x}).
\end{equation}
By minimizing Eq.~(\ref{equ6}), the intra-class instances are forced to have higher affinities with their corresponding class semantic prototype than those with the other class semantic prototypes. In this way, the discriminative information can be effectively preserved in both the visual space and class semantic spaces. Compared with the existing generative approaches \cite{li2019leveraging,xian2018feature} that train a softmax classification model with both the real seen visual features and the synthesized unseen visual features, our classification model introduces no extra parameters, which is more efficient and feasible.
Overall, our full objective then becomes,
\begin{equation}\label{equ7}
\min_G \max_D L_{WGAN}+ \alpha L_{\mathcal{V}\rightarrow \mathcal{A}}+ \beta L_{\mathcal{A}\rightarrow \mathcal{V}}+\gamma L_{MCE},
\end{equation}
where $\alpha$, $\beta$, and $\gamma$ are hype-parameters to balance each terms.
\subsubsection{Refining Model}
With the trained $G$, the test instance could be classified by searching the nearest generated class visual prototype in the visual space with a pre-defined distance metric. For an unseen instance $\mathbf{x}_t$, its class label is predicted by,
\begin{equation}\label{equ8}
\hat{y}_t = \arg \min _k (d(\mathbf{x}_t,G(\mathbf{a}_k))),
\end{equation}
where $\mathbf{a}_k$ is the class semantic prototype of the $k$-th unseen class, $G(\mathbf{a}_k)$ is the corresponding generated class visual prototype. $d(\cdot,\cdot)$ denotes a certain distance metric, such as Euclidean or Consine distance.
The base model focuses on building the visual-semantic interactions on the seen classes, which cannot ensure that it generalizes well to the unseen classes in the pre-defined metric space. To enhance the model adaptability to the unseen classes, we refine the part parameters of the base model that are used for predicting unseen classes on the test set $\mathcal{S}^{te}$ in the pre-defined metric space. Specifically, given a distance function $d$, the base model produces a distribution over classes for a test instance $\mathbf{x}_t$ based on a softmax over distance to the class semantic prototypes in the visual space,
\begin{equation}\label{equ9}
p_G(y=k|\mathbf{x}_t)= \frac{exp(-d(\mathbf{x}_t,G(\mathbf{a}_k)))}{\sum_{k'}exp(-d(\mathbf{x}_t,G(\mathbf{a}_{k'})))},
\end{equation}
where $d(\cdot,\cdot)$ is the distance metric as the same as that in Eq.~(\ref{equ8}). By minimizing the negative log-probability $J(G) = -\log p_G(y=k|\mathbf{x}_t)$ of the true class $k$, the mapping function $G$ is improved for generalizing to the unseen classes in the defined metric space. We observe empirically that the choice of distance metric is vital, as the classification performances with Euclidean distance mostly outperform those with Cosine distance. In the experiments, we report the results with the Euclidean distance, if not specified.
\begin{table}[t]
\begin{center}
\centering
\begin{tabular}{|l@{\hspace{0.1cm}}|c@{\hspace{0.1cm}}|c@{\hspace{0.1cm}}|c@{\hspace{0.1cm}}|c@{\hspace{0.1cm}}|c@{\hspace{0.1cm}}|c@{\hspace{0.1cm}}|}
\hline
Dataset & $\mathcal{K}$ &$\mathcal{Y}_s$ &$\mathcal{Y}_u$ & $\mathcal{X}_a$ & $\mathcal{X}_s$ & $\mathcal{X}_u$ \\
\hline
\hline
AwA1 \cite{lampert2009learning} &85 &40 &10 &30,475 & 5,685 &4,958\\
AwA2 \cite{xian2018zero} &85 &40 &10 &37,322 &5,882 &7,913\\
CUB \cite{wah2011caltech} &1,024 &150 &50 &11,788 &2,967 &1,764\\
FLO \cite{nilsback2008automated} &1,024 &82 &20 &8,189 &5,394 &1,155\\
\hline
\end{tabular}
\end{center}
\caption{The statistics of four benchmark datasets, in terms of class semantic dimensionality $\mathcal{K}$, number of seen classes $\mathcal{Y}_s$, number of unseen classes $\mathcal{Y}_u$, number of all instances $\mathcal{X}_a$, number of test seen instances $\mathcal{X}_s$ and unseen instances $\mathcal{X}_u$. }
\label{tab1}
\end{table}
\begin{table*}
\small
\begin{center}
\begin{tabular}{|l@{\hspace{0.3cm}}|c@{\hspace{0.28cm}}c@{\hspace{0.28cm}}c@{\hspace{0.28cm}}c@{\hspace{0.28cm}}|c@{\hspace{0.1cm}}c@{\hspace{0.1cm}}
c@{\hspace{0.1cm}}c@{\hspace{0.1cm}}|c@{\hspace{0.28cm}}c@{\hspace{0.28cm}}c@{\hspace{0.28cm}}c@{\hspace{0.28cm}}|c@{\hspace{0.1cm}}
c@{\hspace{0.1cm}}c@{\hspace{0.1cm}}c@{\hspace{0.1cm}}|}
\hline
\quad &\multicolumn{4}{c|}{AwA1} &\multicolumn{4}{c|}{AwA2} &\multicolumn{4}{c|}{CUB} &\multicolumn{4}{c|}{FLO}\\
\cline{2-17}
Method &\textbf{T} &\textbf{u} & \textbf{s} &\textbf{H} &\textbf{T} &\textbf{u} & \textbf{s} &\textbf{H} &\textbf{T} &\textbf{u} & \textbf{s} &\textbf{H } &\textbf{T} &\textbf{u} & \textbf{s} &\textbf{H}\\
\hline
\hline
ALE \cite{akata2013label} & 59.9 &16.8 &76.1 &27.5 & 62.5 &14.0 &81.8 &23.9 & 54.9 &23.7 &62.8 &34.4 & 48.5 &13.3 &61.6 &21.9\\
SJE \cite{akata2015evaluation} & 65.6 &11.3 &74.6 &19.6 & 61.9 &8.0 &73.9 &14.4 & 53.9&23.5 &59.2 &33.6 & 53.4 &13.9 &47.6 &21.5\\
ESZSL \cite{romera2015embarrassingly} & 58.2 &2.4 &70.1 &4.6 & 58.6 &5.9 &77.8 &11.0 & 53.9 &12.6 &\textbf{63.8} &21.0 & 51.0 &11.4 &56.8 &19.0\\
DEM \cite{zhang2017learning} &68.4 &32.8 &84.7 &47.3 &67.1 &30.5 &86.4 &45.1 &51.7 &19.6 &57.9 &29.2 &77.8* &57.2* &67.7* &62.0*\\
GAZSL \cite{zhu2018generative} & 68.2&29.6 &84.2 &43.8 & 70.2 &35.4 &86.9 &50.3 & 55.8 &31.7 &61.3 &41.8 & 60.5&28.1 &77.4 &41.2\\
CLSWGAN \cite{xian2018feature} & 68.2& 57.9 &61.4 &59.6 & 65.3 &56.1 &65.5 &60.4 & 57.3 &43.7 &57.7 &49.7 & 67.2 &59.0 &73.9 &65.6\\
Cycle-UWGAN \cite{felix2018multi} & 66.8&56.9 &64.0 &60.2 & -&- &- &- & 58.6&45.7 &61.0 &52.3 & 70.3&59.2 &72.5 &65.1\\
SE-ZSL \cite{kumar2018generalized}& 69.5 &56.3 &67.8 &61.5 & 69.2&\textbf{58.3} &68.1 &62.8 & 59.6&41.5 &53.3 &46.7 & - &- &- &-\\
LisGAN \cite{li2019leveraging} & 70.6&52.6 &76.3 &62.3 & 70.4*&47.0*&77.6* &58.5* & 58.8&46.5 &57.9 &51.6 & 69.6 &57.7 &83.8 &68.3\\
f-VAEGAN-D2 \cite{xian2019f} & 71.1&57.6 &70.6 &63.5 & -&- &- &- & 61.0&48.4 &60.1 &53.6 & 67.7&56.8 &74.9 &64.6\\
CADA-VAE \cite{schonfeld2019generalized} &62.3 &57.3 &72.8 &64.1 &64.0 &55.8 &75.0 &63.9 &60.4 &51.6 & 53.5 &52.4 & -&-&-&-\\
ABP \cite{zhu2019learning} &69.3 &57.3 &67.1 &61.8 &70.4 &55.3 &72.6 &62.6 &58.5 &47.0 &54.8 &50.6 &- &- &- &-\\
RELATION NET \cite{sung2018learning} & 68.2 &31.4 &\textbf{91.3} &46.7 & 64.2 &30.0 &\textbf{93.4} &45.3 & 55.6&38.1 &61.1 &47.0 & 78.5* &50.8* &\textbf{88.5}* &64.5*\\
3ME \cite{felix2019multi} &65.6 &55.5 &65.7& 60.2 & - &- &- &- & 71.1 & 49.6 & 60.1 & 54.3 & 83.9 & 57.8 & 79.2 & 66.8\\
\hline
\textbf{E-PGN} (Ours) & \textbf{74.4}&\textbf{62.1} &83.4 &\textbf{71.2}& \textbf{73.4}& 52.6 &83.5 &\textbf{64.6} & \textbf{72.4} & \textbf{52.0} &61.1 &\textbf{56.2} & \textbf{85.7} &\textbf{71.5} &82.2 &\textbf{76.5}\\
\hline
\end{tabular}
\end{center}
\caption{\upshape Performance (in \%) comparisons for both traditional and generalized ZSL in terms of average per-class top-1 accuracy (\textbf{T}), unseen accuracy (\textbf{u}), seen accuracy (\textbf{s}), and their harmonic mean (\textbf{H}). $^*$ indicates the results obtained by ourselves with the codes released by the authors. The best results are marked in boldface.}
\label{tab2}
\end{table*}
The model PGN trained with episode-based framework is short for E-PGN. The training process of E-PGN is summarized in Algorithm~\ref{alg1}
\begin{algorithm}
\label{alg1}
\caption{Proposed E-PGN approach.}
\KwIn{The seen category set $\mathcal{S}$, the hyper-parameters $\alpha$, $\beta$, and $\gamma$.}
\KwOut{Visual prototype generating network $G$.}
Initialize the parameters of both $F$ and $G$. \\
\While{not done}
{
Randomly sample $\mathcal{S}^{tr}$ and $\mathcal{S}^{te}$ from $\mathcal{S}$\;
\For{samples in $\mathcal{S}^{tr}$}
{
Optimize $F$ and $G$ by Eq.~(\ref{equ7})\;
}
\For{samples in $\mathcal{S}^{te}$}
{
Calculate probability distribution by Eq.~(\ref{equ9})\;
Update $G$ by minimizing the negative log-probability.
}
}
return The parameters of $G$.
\end{algorithm}
\section{Experiments}
In this section, we conduct experiments to evaluate the effectiveness of the proposed model. We first document the datasets and experimental settings and then compare E-PGN with the state-of-the-art. Finally, we study the properties of the proposed E-PGN with a serious of ablation experiments.
\subsection{Datasets and Experimental settings}
\textbf{Datasets.} Among the most widely used datasets for zero-shot classification, we select two coarse-grained datasets, namely Animals with Attributes (AwA1) \cite{lampert2009learning}, Animals with Attributes2 (AwA2) \cite{xian2018zero}, and two fine-grained datasets, i.e., Caltech-UCSD Birds-200-2011 (CUB) \cite{wah2011caltech} and Oxford Flowers (FLO) \cite{nilsback2008automated}. AwA1 and AwA2 consist of different visual images from the same 50 animal classes, each class is annotated with 85-dimensional semantic attributes. CUB and FLO respectively contain 200 bird species and 102 flower categories. As for the class semantic representations of both CUB and FLO datasets, we average the 1,024-dimensional character-based CNN-RNN \cite{reed2016learning} features extracted from the fine-grained visual descriptions (10 sentences per image). We adopt the standard zero-shot splits provided by \cite{xian2018zero} for AwA1, AwA2, and CUB datasets. For FLO dataset, we use the splits provided by \cite{nilsback2008automated}. A dataset summary is given in Table~\ref{tab1}.
\textbf{Evaluation Protocol.} In this work, we evaluate our approach on both traditional ZSL and generalized ZSL tasks. For the traditional ZSL task, we apply the extensively used average per-class top-1 accuracy (\textbf{T}) as the evaluation protocol. For the generalized ZSL task, we follow the protocol proposed in \cite{xian2018zero} to evaluate the approaches with both seen class accuracy \textbf{s} and unseen class accuracy \textbf{u}, as well as their harmonic mean \textbf{H}.
\textbf{Implementation settings.} Following \cite{xian2018zero,xian2018feature}, we use the top pooling units of the ResNet-101 \cite{he2016deep} pre-trained on ImageNet-1K as the image features. Thus, each input image is represented as a 2,048-dimensional vector. As a pre-processing step, we normalize the visual features into [0,~1]. In terms of the model architecture, we implement $F$, $G$, and $D$ as simple three-layer neural networks with 1,800, 1,800, and 1,600 hidden units. Both $F$ and $D$ apply ReLU as the activation function on both the hidden layer and the output layer, both of which follow a dropout layer. While developing the model, we have observed that by applying the tanh activation function for the hidden layer of $G$ would obtain more stable and better results. In terms of the learning rate of the base model, we set $5e^{-5}$ for AwA1, CUB, and FLO datasets, and $2e^{-4}$ for AwA2 dataset. For all datasets, we set the learning rate of the refining model as $1/10$ of the original base model. In each episode, the base model is trained for 100 epochs by stochastic gradient decent using the Adam optimizer and a batch size of 128 for AwA1 dataset and 32 for the other datasets. The refining model in each episode is trained for 10 epochs using the same optimizer and batch size as the base model. Our model is implemented using TensorFlow framework. The code is available at \footnote {\url{https://github.com/yunlongyu/EPGN}}.
\begin{table*}[t]
\begin{center}
\centering
\begin{tabular}{|l@{\hspace{0.2cm}}|c@{\hspace{0.25cm}}c@{\hspace{0.25cm}}c@{\hspace{0.25cm}}c@{\hspace{0.25cm}}|c@{\hspace{0.25cm}}c@{\hspace{0.25cm}}
c@{\hspace{0.25cm}}c@{\hspace{0.25cm}}|c@{\hspace{0.25cm}}c@{\hspace{0.25cm}}c@{\hspace{0.25cm}}c@{\hspace{0.25cm}}|c@{\hspace{0.25cm}}
c@{\hspace{0.25cm}}c@{\hspace{0.25cm}}c@{\hspace{0.25cm}}|}
\hline
\quad &\multicolumn{4}{c|}{AwA1} &\multicolumn{4}{c|}{AwA2} &\multicolumn{4}{c|}{CUB} &\multicolumn{4}{c|}{FLO}\\
\cline{2-17}
Method&\textbf{T} &\textbf{u} & \textbf{s} &\textbf{H} &\textbf{T} &\textbf{u} & \textbf{s} &\textbf{H} &\textbf{T} &\textbf{u} & \textbf{s} &\textbf{H } &\textbf{T} &\textbf{u} & \textbf{s} &\textbf{H}\\
\hline
\hline
\textbf{PGN} & 72.2 &52.6&\textbf{86.3} &65.3 &71.2&48.0 &83.6 &61.0 & 68.3 &48.5 &57.2 &52.5 & 81.4 &63.6 &77.8 &70.0\\
\textbf{E-PGN} (5) & 72.2 &57.2 &83.8 &68.0 & 73.5 &51.2 &83.0 &63.3 & 70.4 &50.5 &59.0 &54.4 & 84.2 &67.7 &79.6 &73.2\\
\textbf{E-PGN} (10) & \textbf{74.4}&62.1 &83.4 &\textbf{71.2}& 73.4& \textbf{52.6} &83.5 &\textbf{64.6} & \textbf{72.4} & \textbf{52.0} &\textbf{61.1} &\textbf{56.2} &\textbf{85.7} &\textbf{71.5} &\textbf{82.2} &\textbf{76.5}\\
\textbf{E-PGN} (15) & 73.8&\textbf{62.2} &82.9 &71.1& \textbf{74.2}& 50.5 &\textbf{84.1} &63.1 & 69.6 & 51.5 &57.4 &54.3 & 85.3 &70.5 &80.4 &75.2\\
\hline
\end{tabular}
\end{center}
\caption{\upshape Performance (in \%) comparisons of the number of the selected mimetic unseen classes in each episode. PGN indicates the model trained without episode-based paradigm.}
\label{tab3}
\end{table*}
\subsection{Comparing State-of-The-Art Approaches}
Table~\ref{tab2} describes the classification performances of E-PGN and fourteen competitors including three discriminative approaches \cite{akata2013label,akata2015evaluation,romera2015embarrassingly}, nine generative approaches \cite{zhang2017learning,zhu2018generative,xian2018feature,felix2018multi,kumar2018generalized,li2019leveraging,xian2019f,schonfeld2019generalized,zhu2019learning}, one episode-based approach \cite{sung2018learning}, and one ensemble approach \cite{felix2019multi}.
From Table~\ref{tab2}, we observe that the proposed E-PGN achieves significant improvements over the state-of-the-art in terms of both \textbf{T} and \textbf{H} on four datasets. Specifically in \textbf{T} metric, the overall accuracy improvement on AwA1 increases from 71.1\% to 74.4\%, on AwA2 from 70.4\% to 73.4\%, on CUB from 71.1\% to 72.4\%, and on FLO from 83.9\% to 85.7\%, i.e., all quite significant. Remarkably, E-PGN achieves 71.2\% and 76.5\% for \textbf{H} metric on AwA1 and FLO datasets, which marginally improves the second-best performance by 7.1\% and 8.2\%. On AwA2 and CUB datasets, the proposed E-PGN also gains improvements from 63.9\% to 64.6\% and from 54.3\% to 56.2\%, respectively. Compared with the other episode-based approach RELATION NET \cite{sung2018learning}, our E-PGN achieves significant improvements on four datasets, which indicates that our mimetic strategy captures more discriminative transfer knowledge than learning the distance metric strategy. Compared with the other ensemble approach 3ME \cite{felix2019multi}, our E-PGN also has obvious improvements under different metrics across different datasets.
We also observe that the seen classification accuracy \textbf{s} is much better than unseen classification accuracy \textbf{u}, which indicates that the unseen test instances tend to be misclassified into the seen classes. This classification shifting issue is ubiquitous across all the existing approaches. From the results, we observe that the generative approaches alleviate this shift issue to some extent, resulting in the improvement of \textbf{H} measure. However, those approaches balance the differences between the seen class accuracy and the unseen class accuracy via decreasing the seen class accuracy while improving the unseen class accuracy, which is not desirable in practice. In contrast, our E-PGN is more robust than the competitors, which substantially boosts the harmonic mean \textbf{H} via improving the unseen class accuracy while maintaining the seen class accuracy at a high level. Our performance improvement is benefited from the progressive episode-training strategy paired with the effective base model.
\subsection{Further Analysis}
\subsubsection{Impact of episode-based paradigm} In the first experiment, we evaluate the impact of the episode-training scheme and how the number of selected mimetic unseen classes in each episode affects the performances on different datasets. To do so, we vary the number of selected mimetic unseen classes from 0 to 15 in intervals of 5. It should be noted that the case where the number of selected mimetic unseen classes equaling 0 indicates the approach trained without the episode-based paradigm and the optimization process degenerates to the traditional batch-based training strategy.
\begin{figure}[t]
\centering
\includegraphics[width=8.1cm,height=6.5cm]{fig4}
\caption{Traditional and generalized zero-shot classification results with traditional cross-entropy loss (short for CE) and multi-modal cross-entropy loss (short for MCE) on four datasets.}
\label{fig4}
\end{figure}
According to the results in Table~\ref{tab3}, we observe that E-PGN mostly performs better than PGN on four datasets under different metrics except \textbf{s} on AwA1 dataset, which indicates the effectiveness of the proposed episode-based training strategy. Compared with PGN, the E-PGN may spoil the whole training structure to some extent, but can progressively accumulate the knowledge on how to adapt to novel classes with the episode-based training paradigm, and thus better results are obtained. Besides, we also observe that the number of the selected mimetic unseen classes greatly impacts the classification performances. Specifically, E-PGN~(10) basically beats E-PGN~(5) on four datasets. However, with the further increase of the number, the performances tend to decrease, we speculate that the reason is that when more mimetic unseen classes are selected for refining, fewer training classes are left for training the base model, leading to unsatisfied initialization for the prediction of the mimetic unseen classes.
\begin{table*}[t]
\begin{center}
\begin{tabular}{|ccc| c@{\hspace{0.27cm}}c@{\hspace{0.27cm}}c@{\hspace{0.27cm}}c@{\hspace{0.27cm}}| c@{\hspace{0.27cm}}c@{\hspace{0.27cm}}c@{\hspace{0.27cm}}c@{\hspace{0.27cm}}| c@{\hspace{0.27cm}}c@{\hspace{0.27cm}}c@{\hspace{0.27cm}}c@{\hspace{0.27cm}}| c@{\hspace{0.27cm}}c@{\hspace{0.27cm}}c@{\hspace{0.27cm}}c@{\hspace{0.27cm}}|}
\hline
\multicolumn{3}{|c|}{\quad}& \multicolumn{4}{c|}{AwA1} & \multicolumn{4}{c|}{AwA2} & \multicolumn{4}{c|}{CUB} &\multicolumn{4}{c|}{FLO}\\
\cline{4-19}
$\alpha$ &$\beta$ &$\gamma$ &\textbf{T} &\textbf{u} & \textbf{s} &\textbf{H} &\textbf{T} &\textbf{u} & \textbf{s} &\textbf{H} &\textbf{T} &\textbf{u} & \textbf{s} &\textbf{H } &\textbf{T} &\textbf{u} & \textbf{s} &\textbf{H} \\
\hline
\hline
\quad &\checkmark &\checkmark & 73.1 &60.3&82.3 &69.6 & 72.6 &51.3 &81.6 &63.0 & 71.2 &50.9 &59.1 &54.7 & 85.0 &69.2 &79.7 &74.1\\
\checkmark &\quad &\checkmark & 73.8 &61.0 &83.1 &70.4 & 72.2 &52.5 &82.7 &64.3 & 67.2 &45.9 &55.9 &50.4 & 85.8 &69.4 &82.0 &75.2\\
\checkmark &\checkmark &\quad & 70.8&56.2 &82.2 &66.8& 70.9& 43.2 &79.9 &56.1 & 66.8 & 45.2 &52.5 &48.8 & \textbf{86.2} &70.0 &79.2 &74.4\\
\quad &\quad &\checkmark & 72.1 &56.2&81.5 &66.5 & 71.2 &48.5 &\textbf{84.0} &61.5 & 70.3 &50.0 &57.5 &53.5 & 85.6 &71.3 &80.5 &75.6\\
\checkmark &\checkmark &\checkmark &\textbf{74.4}&\textbf{62.1} &\textbf{83.4} &\textbf{71.2} & \textbf{73.4}& \textbf{52.6} &83.5 &\textbf{64.6} & \textbf{72.4} & \textbf{52.0} &\textbf{61.1} &\textbf{56.2} & 85.7 &\textbf{71.5} &\textbf{82.2} &\textbf{76.5}\\
\hline
\end{tabular}
\end{center}
\caption{Ablation study of the E-PGN components on four datasets. The best results are marked in boldface. }
\label{tab4}
\end{table*}
\subsubsection{Performance impacts of E-PGN components}
In this study, we quantify the benefits of the different components in E-PGN on the performances. In the proposed E-PGN model, except for the base adversarial loss, there are three components: two regression losses and one multi-modal classification loss. Each loss is controlled by a hyper-parameter, i.e., $\alpha$, $\beta$, and $\gamma$. We select the values of the hyper-parameters only from 0 and 1. When the value of a hyper-parameter equals 1, its corresponding component is ``switch on", otherwise is ``switch off". The performance differences between the two scenarios reveal the effects of the component.
From the results illustrated in Table~\ref{tab4}, we observe that the model with all three components mostly achieves the best performances for fourteen out of sixteen metrics, which indicates that the three calibration terms complement each other. Besides, we observe that the performances of the model without MCE loss ($\gamma=0$) degrade significantly on three out of four datasets, which reveals that the MCE loss contributes significantly to the classification performance.
\subsubsection{Impact of classification network}
To further validate the superiority of the proposed multi-modal cross-entropy loss, we compare our E-PGN against the method with the traditional cross-entropy loss. From the results illustrated in Fig.~\ref{fig4}, we observe that the proposed E-PGN with MCE loss performs much better than the counterpart with traditional Cross-Entropy (CE) loss on AwA1, AwA2, and CUB datasets, and performs neck to neck on FLO dataset. We argue that the superiority is due to that the MCE loss encodes with the class semantic information into the classification module, which both preserves the discriminative information and enhances the visual-semantic consistence. Besides, compared with the model with the traditional CE loss, the model with the MCE loss introduces no extra training parameters, which is more efficient.
\begin{figure}[t]
\centering
\includegraphics[width=8.1cm,height=6.3cm]{fig5}
\caption{Traditional and generalized zero-shot classification results with Euclidean distance (short for Euc) and Cosine distance (short for Cos) on four datasets.}
\label{fig5}
\end{figure}
\subsubsection{Impact of distance metric} In this experiment, we investigate how the distance metric affects the classification performance. In Fig.~\ref{fig5}, we compare Cosine vs. Euclidean distance under different metrics on four datasets. We observe that the performances obtained in the Euclidean space are significantly better than those obtained in the Cosine space under most cases, indicating that the Euclidean distance is more suitable to our approach. The inferior performances obtained in Cosine space may be due to that the Cosine distance is not a Bregman divergence~\cite{snell2017prototypical}.
\section{Conclusion}
In this paper, we have introduced an episode-based training paradigm to enhance the adaptability of the model for zero-shot learning. It divides the training process into a collection of episodes, each of which mimics a fake zero-shot classification task. By training multiple episodes, the model accumulates a wealth of ensemble experiences on predicting the mimetic unseen classes, which generalizes well on the real unseen classes. Under this training paradigm, we have proposed an effective generative model to align the visual-semantic consistency paired with a parameter-economic multi-modal cross-entropy loss. The comprehensive results on four benchmark datasets demonstrate that the proposed model achieves the new state-of-the-art and beats the competitors by large margins.
\noindent
\textbf{Acknowledgement.}
This work is supported in part by NSFC (61672456,U19B2043), Zhejiang Lab (2018EC0ZX01-2), the fundamental research funds for central universities in China (No. 2017FZA5007), Artificial Intelligence Research Foundation of Baidu Inc., the Key Program of Zhejiang Province, China (No. 2015C01027), the funding from HIKVision and Horizon Robotics, and ZJU Converging Media Computing Lab.
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News Film News
Harrison Ford calls Shia LaBeouf a 'fucking idiot' for criticising Indiana Jones
LaBeouf had claimed he 'dropped the ball' when making 'Kingdom Of The Crystal Skull'
By Daniel Martin
Harrison Ford has called his Indiana Jones co-star Shia LaBeouf a "fucking idiot" for publicly criticising the movie.
The pair appeared together in latter-day sequel Indiana Jones And The Kingdom Of The Crystal Skull, but LaBeouf later spoke out against the film, saying he and director Steven Spielberg "dropped the ball".
READ MORE: Star Wars: Where The Hell Is Luke Skywalker? These 5 Crazy Fan Theories Might Have The Answer
Now Ford has lambasted his screen son for his comments. He told Details: "I think he was a fucking idiot. As an actor, I think it's my obligation to support the film without making a complete ass of myself. Shia is ambitious, attentive, and talented – and he's learning how to deal with a situation which is very unique and difficult."
After poor critical reception to the film, LaBeouf said last May: "I feel like I dropped the ball on the legacy that people loved and cherished. I have a relationship with Steven that supersedes our business work. And believe me, I talk to him often enough to know that I'm not out of line. And I would never disrespect the man. But when you drop the ball you drop the ball."
He continued: "You get to monkey-swinging and things like that and you can blame it on the writer and you can blame it on Steven. But the actor's job is to make it come alive and make it work, and I couldn't do it. So that's my fault."
READ MORE: JJ Abrams says Luke Skywalker's absence from new 'Star Wars' trailer is 'no accident'
LaBeouf recently claimed that the current Transformers: Dark Of The Moon would be his final outing in the monster robot franchise.
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\section{Introduction}\label{sec:intro}
A trisection is a decomposition of a $4$-manifold into three $4$-dimensional $1$-handlebodies. It was introduced by Gay and Kirby \cite{GayKir16} as a $4$-dimensional analogue of Heegaard splittings of $3$-manifolds.
For compact $4$-manifolds with boundary, the notion of relative trisections was also introduced in \cite{GayKir16}.
Later, it was studied in \cite{Cas16}, \cite{CasGayPin18_1}, \cite{CasGayPin18_2}, \cite{CasOzb19}, \cite{KimMil20}, and \cite{CasIslMilTom21} for example.
A relative trisection diagram is a description of a relative trisection by three families of curves on a compact surface.
In \cite{CasGayPin18_1}, Castro, Gay, and Pinz\'{o}n-Caiced established a natural correspondence between relative trisections and relative trisection diagrams.
By this correspondence, one can represent smooth structures of $4$-manifolds with boundary by relative trisection diagrams.
A trisection genus is a fundamental invariant of smooth $4$-manifolds defined by trisections.
For a $4$-manifold $X$, the trisection genus of $X$ is the minimal integer $g$ such that $X$ admits a (relative) trisection with the triple intersection surface of genus $g$.
Meier and Zupan \cite{MeiZup17_1} classified closed, oriented, smooth $4$-manifolds with trisection genus at most $2$.
However, in general, it is difficult to determine the trisection genus of a $4$-manifold, and the following question naturally arises.
\begin{qst}\label{qst}
For a given smooth $4$-manifold, what is its trisection genus?
\end{qst}
In this paper, we answer the above question for the Akbulut cork.
\begin{thm}\label{mainthm:Ac}
The trisection genus of the Akbulut cork is $3$.
\end{thm}
As far as we know, this is the first example of contractible $4$-manifolds whose trisection genera are determined except for the $4$-ball.
A cork is a pair of a contractible $4$-manifold and a smooth involution on the boundary.
By using corks, one can construct exotic (i.e., homeomorphic but not diffeomorphic) smooth structures of $4$-manifolds.
It is well known that the Akbulut cork is the first example of a cork (\cite{Akb91_1}).
Details about corks will be described in Subsection~\ref{subsec:Corks}.
In addition, we show the following theorem.
\begin{thm}\label{mainthm:corks}
There are infinitely many corks with trisection genus $3$.
\end{thm}
This theorem is proved by constructing an infinite family $\{M_n\}_{n\in\NN}$ of trisected corks.
The genus $3$ relative trisection diagram of $M_n$ is shown in Figure~\ref{fig:D_n-rtd}.
We also give nice properties of $\{M_n\}_{n\in\NN}$ (see Theorems~\ref{thm:M_n-Mazur}, \ref{thm:M_n-cork}, and Proposition~\ref{prop:M_n-hyp}).
To prove that these corks cannot admit relative trisections of genus less than $3$, we give a lower bound for the trisection genus of a $4$-manifold with boundary.
\begin{thm}\label{mainthm:lowerbound}
Let $X$ be a compact, connected, oriented, smooth $4$-manifold with connected boundary, and let $\chi(X)$ denote the Euler characteristic.
If the boundary $\partial{X}$ is not a Seifert fiber space, then the trisection genus of $X$ is greater than $\chi(X)+1$.
\end{thm}
As an application, we obtain the following corollary.
\begin{cor}\label{maincor:bdryOB}
The minimal number of binding components of planar open book decompositions on each $\partial{M_n}$ is $4$.
\end{cor}
These results are obtained by the property that a relative trisection of a $4$-manifold induces an open book decomposition on the boundary. See section~\ref{sec:lowerbound} for details.
We finally discuss trisections of exotic $4$-manifolds.
In \cite{LamMei20}, Lambert-Cole and Meier conjectured that trisection genus is additive under connected sum, that is, for any $4$-manifolds $X$ and $Y$, the trisection genus of $X\# Y$ is the sum of the trisection genera of $X$ and $Y$.
They also showed that, if this is true, it follows that trisection genus is a homeomorphism invariant and there are no exotic $S^4$ or some $4$-manifolds (e.g., $\CC{P^2}$, $S^1\times S^3$, $S^2\times S^2$).
So it is interesting to construct trisections for exotic $4$-manifolds.
See \cite{BaySae17a}, \cite{MeiZup18}, \cite{CasOzb19}, and \cite{LamMei20}, for examples of such trisections.
A related problem is to find trisection diagrams for exotic pairs (\cite[Problem~1.26]{AimPL}).
In Section~\ref{sec:exotic}, we construct low genus relative trisection diagrams of an exotic pair of small $4$-manifolds (see Figures~\ref{fig:P-rtd} and \ref{fig:Q-rtd}). The diagrams give the following theorem.
\begin{thm}\label{mainthm:exotictris}
There exist an exotic pair of simply-connected compact $4$-manifolds $P$ and $Q$ with $b_2 = 1$ such that they admit relative trisections of genus $4$ and $5$, respectively.
In particular, the trisection genus of $Q$ is $4$, and the trisection genus of $P$ is either $4$ or $5$.
\end{thm}
A natural question is whether $P$ can admit a genus $4$ relative trisection.
However, we have not been able to find it.
If one can show that there is no such relative trisection, it follows that the trisection genus for $4$-manifolds with boundary is not homeomorphism invariant.
\section{Preliminaries}\label{sec:prel}
\subsection{Notation and conventions}
Throughout this paper, we assume that manifolds are compact, connected, oriented, and smooth unless otherwise stated. In addition, we will use the following notation.
\begin{itemize}
\item
If two manifolds $X$ and $Y$ are orientation-preserving diffeomorphic to each other, then we write $X\cong Y$.
\item
Let $\Sigma_{g,b}$ be a compact, connected, oriented surface of genus $g$ with $b$ boundary components.
\item
For a manifold $X$ and a submanifold $A\subset X$, we denote a tubular neighborhood of $A$ in $X$ by $\nu(A;X)$.
\end{itemize}
\subsection{Relative trisections}
We now introduce the definition of relative trisections rephrased by Castro, Gay, and Pinz\'{o}n-Caiced \cite{CasGayPin18_1}.
A relative trisection is a decomposition of a $4$-manifold with connected boundary into three $4$-dimensional $1$-handlebodies.
To describe how to glue these three pieces, we construct a model $Z_{k}$ of a genus $k$ $4$-dimensional $1$-handlebody.
Let $g$, $k$, $p$, and $b$ be integers satisfying $g,k,p\geq0$, $b\geq1$, and $2p+b-1\leq k\leq g+p+b-1$.
Note that the three integers $g-p$, $g-k+p+b-1$, and $k-2p-b+1$ are non-negative.
Ultimately, we will define $Z_{k}$ as a boundary connected sum of two $4$-manifolds ${U}$ and ${V}$, so we start by defining these.
First, we construct ${U}$ and give a decomposition of the boundary.
Let $D$ be the third of the unit disk defined as follows.
\begin{equation*}
D:=\{ (r,\theta)\in \CC \mid r\in[0,1], \theta \in[-\pi/3,\pi/3]\}.
\end{equation*}
Give a decomposition of the boundary as $\partial{D}=\partial^-D\cup \partial^0D\cup \partial^+D$, where
\begin{align*}
\partial^\pm D &:= \{ (r,\theta)\in D\mid r\in[0,1],\theta=\pm \pi/3\} \quad \textrm{and} \\
\partial^0D &:= \{ (r,\theta)\in D\mid r=1,\theta \in [-\pi/3,\pi/3] \}.
\end{align*}
Let $P:=\Sigma_{p,b}$ and ${U} := D\times P$.
We immediately see that ${U} \cong \natural^{2p+b-1}(S^1\times D^3)$.
Decompose the boundary of ${U}$ as $\partial{{U}}=\partial^-{U} \cup \partial^0U_{p,b} \cup \partial^+{U}$, where
\begin{align*}
\partial^\pm {U} := \partial^\pm{D}\times P \quad \textrm{and} \quad \partial^0{{U}} := (\partial^0{D}\times P)\cup(D\times \partial{P}).
\end{align*}
Next, we construct ${V}$ and give a decomposition of the boundary.
Consider a $4$-dimensional solid torus $S^1\times D^3$, and decompose the boundary as follows.
\begin{equation*}
\partial(S^1\times D^3) = S^1\times(S^2_-\cup S^2_+) = \partial^-(S^1\times D^3)\cup \partial^+(S^1\times D^3),
\end{equation*}
where $S_{\pm}^2$ are the northern and southern hemispheres of $\partial{D^3}$, and $\partial^\pm(S^1\times D^3):=S^1\times S_{\pm}^2$.
This decomposition is the standard genus $1$ Heegaard splitting of $S^1\times S^2$.
For $k-2p-b+1$ copies of $S^1\times D^3$ with the Heegaard splitting of the boundary, define $V_{k-2p-b+1}:= \natural^{k-2p-b+1}(S^1\times D^3)$, where the boundary connected sums are taken in neighborhoods of points in the Heegaard surfaces of copies of $\partial(S^1\times D^3)$ (see Figure~\ref{fig:connsum-S1xD3}).
Let $\partial^\pm{V_{k-2p-b+1}}:=\natural^{k-2p-b+1}{\partial^\pm(S^1\times D^3)}$, then $\partial{V_{k-2p-b+1}}=\partial^-{V_{k-2p-b+1}}\cup \partial^+{V_{k-2p-b+1}}$.
This is the standard genus $k-2p-b+1$ Heegaard splitting of $\#^{k-2p-b+1}(S^1\times S^2)$.
We denote the result of stabilizing $g-k+p+b-1$ times by $\partial{{V}}=\partial^-{{V}}\cup \partial^+{{V}}$, which has genus $g-p$.
Note that it is independent of the stabilizations (see Section~4 in \cite{Wal68}).
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/connsum-S1xD3.eps}
\caption{An image of the boundary connected sums $V_{k-2p-b+1}= \natural^{k-2p-b+1}(S^1\times D^3)$.}
\label{fig:connsum-S1xD3}
\end{figure}
Finally, we define $Z_{k} := {U}\natural {V}$, where the boundary connected sum is taken in neighborhoods of points in $\Int(\partial^-{U}\cap \partial^+{U})$ and $\partial^-{V}\cap \partial^+{V}$ (see Figure~\ref{fig:connsum-UV}).
We see that $Z_{k}$ is diffeomorphic to $\natural^k(S^1\times D^3)$.
Let $Y_{k} :=\partial{Z_{k}}$, and give a decomposition $Y_{k} = Y^-_{g,k;p,b}\cup Y^0_{g,k;p,b}\cup Y^+_{g,k;p,b}$, where
\begin{align*}
Y^\pm_{g,k;p,b} := \partial^{\pm}{U}\natural \partial^{\pm}{{V}} \quad \textrm{and} \quad Y^0_{g,k;p,b} := \partial^0U.
\end{align*}
See Sections~3 and 4 in \cite{CasGayPin18_1} for details of this model. Now we are ready to define a relative trisection.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/connsum-UV.eps}
\caption{An image of the boundary connected sum $Z_{k}={U}\natural {V}$.}
\label{fig:connsum-UV}
\end{figure}
\begin{defi}[Castro--Gay--Pinz\'{o}n-Caiced {\cite[Definition~10]{CasGayPin18_1}}]\label{def:rt}
Let $g$, $k$, $p$, and $b$ be integers satisfying $g,k,p\geq0$, $b\geq1$, and $2p+b-1\leq k\leq g+p+b-1$.
Let $X$ be a compact, connected, oriented, smooth $4$-manifold with connected boundary.
A decomposition $X=X_1\cup X_2\cup X_3$ is called a $(g,k;p,b)$-\textit{relative trisection} of $X$ if it satisfies the following conditions.
\begin{enumerate}
\item
For each $i\in\{1,2,3\}$, there is a diffeomorphism $\phi_i:X_i\to Z_{k}$.
\item
For each $i \in\{1,2,3\}$, taking indices mod $3$,
\begin{equation*}
\phi_i(X_i\cap X_{i\pm1})=Y^\mp_{g,k;p,b} \quad \textrm{and} \quad \phi_{i}(X_i\cap \partial{X})=Y^0_{g,k;p,b}.
\end{equation*}
\end{enumerate}
\end{defi}
In this paper, we sometimes denote a relative trisection $X=X_1\cup X_2\cup X_3$ by $\calT$.
The genus of the triple intersection surface $\Sigma := X_1\cap X_2\cap X_3$ is called the \textit{genus} of $\calT$.
The \textit{trisection genus} of $X$ is the minimal integer $g$ such that $X$ admits a (relative) trisection of genus $g$.
This is an invariant for smooth $4$-manifolds.
\begin{lem}[Castro--Gay--Pinz\'{o}n-Caiced {\cite[Lemma~11]{CasGayPin18_1}}]\label{lem:obd}
A $(g,k;p,b)$-relative trisection of $X$ induces an open book decomposition on the boundary $\partial{X}$ with pages of genus $p$ with $b$ boundary components.
\end{lem}
For a $(g,k;p,b)$-relative trisection $X=X_1\cup X_2\cup X_3$, each integer in the $4$-tuple $(g,k;p,b)$ has the following meaning.
The integer $g$ is the genus of the triple intersection surface, $k$ is the $4$-dimensional genus of each sector $X_i$, $p$ is the genus of the page of the induced open book decomposition, and $b$ is the number of binding components.
\begin{prop}[Castro--Ozbagci {\cite[Corollary~2.10]{CasOzb19}}]\label{prop:Euler}
Suppose that a $4$-manifold $X$ admits a $(g,k;p,b)$-relative trisection.
Then the Euler characteristic $\chi(X)$ is equal to $g-3k+3p+2b-1$.
\end{prop}
Next, we introduce the definition of a relative trisection diagram.
\begin{defi}[Castro--Gay--Pinz\'{o}n-Caiced{\cite[Definition~1]{CasGayPin18_1}}]
Let $\Sigma$ and $\Sigma'$ be compact, connected, oriented surfaces.
For $i\in \{1,\ldots,n\}$, let $\alpha^i$ and $\beta^i$ be families of $k$ pairwise disjoint simple closed curves on $\Sigma$ and $\Sigma'$, respectively.
Two $n+1$-tuples $(\Sigma;\alpha^1,\ldots,\alpha^n)$ and $(\Sigma';\beta^1,\ldots,\beta^n)$ are \textit{diffeomorphism and handle slide equivalent} if they are related by diffeomorphisms of $\Sigma$ and handle slides within each $\alpha^i$ (i.e., we are only allowed to slide curves from $\alpha^i$ over other curves from $\alpha^i$, but not over curves from $\alpha^j$ when $j\neq i$).
\end{defi}
\begin{defi}[Castro--Gay--Pinz\'{o}n-Caiced {\cite[Definition~2]{CasGayPin18_1}}]\label{def:rtd}
Let $g$, $k$, $p$, and $b$ be integers satisfying $g,k,p\geq0$, $b\geq1$, and $2p+b-1\leq k\leq g+p+b-1$.
Let $\Sigma$ be a surface diffeomorphic to $\Sigma_{g,b}$, and let $\alpha$, $\beta$, and $\gamma$ be families of $g-p$ pairwise disjoint simple closed curves on $\Sigma$.
A $4$-tuple $(\Sigma;\alpha,\beta,\gamma)$ is called a $(g,k;p,b)$-\textit{relative trisection diagram} if $(\Sigma;\alpha,\beta)$, $(\Sigma;\beta,\gamma)$, and $(\Sigma;\gamma,\alpha)$ are diffeomorphism and handle slide equivalent to the standard diagram $(\Sigma_{g,b};\delta,\epsilon)$ shown in Figure \ref{fig:std-rtd}, where the red curves are $\delta$ and the blue curves are $\epsilon$.
\end{defi}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/std-rtd.eps}
\caption{The standard diagram $(\Sigma_{g,b};\delta,\epsilon)$ of type $(g,k;p,b)$.}
\label{fig:std-rtd}
\end{figure}
In this paper, we sometimes denote a relative trisection diagram by $\calD$.
When considering a relative trisection diagram of the form $(\Sigma;\alpha,\beta,\gamma)$, we represent $\alpha$, $\beta$, and $\gamma$ curves by red, blue, and green curves, respectively (see Figure~\ref{fig:D_n-rtd} for example).
A pair of black disks with a number indicates an attaching of a cylinder (see Figure~\ref{fig:blackhole}).
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/blackhole-v2.eps}
\caption{A meaning of a pair of black disks with a number.}
\label{fig:blackhole}
\end{figure}
The following theorem gives a natural correspondence between relative trisections and relative trisection diagrams.
\begin{thm}[Castro--Gay--Pinz\'{o}n-Caiced {\cite[Theorem~3]{CasGayPin18_1}}]\label{thm:rt-rtd}
The following (i), (ii), and (iii) hold.
\begin{enumerate}
\item
For any $(g,k;p,b)$-relative trisection diagram $(\Sigma;\alpha,\beta,\gamma)$, there exists a unique (up to diffeomorphism) trisected $4$-manifold $X=X_1\cup X_2\cup X_3$ satisfying the following conditions.
\begin{itemize}
\item
$X_1\cap X_2\cap X_3\cong \Sigma$.
\item
Under the above identification, each of $\alpha$, $\beta$, and $\gamma$ curves bound compressing disks of $X_1\cap X_2$, $X_2\cap X_3$, and $X_3\cap X_1$, respectively.
\end{itemize}
\item
For any relative trisection $\calT$, there exists a relative trisection diagram $\calD$ such that $\calT$ is induced from $\calD$ by (i).
\item
Let $\calD$ and $\calD'$ be relative trisection diagrams. If the relative trisections corresponding to these diagrams are diffeomorphic, then $\calD$ and $\calD'$ are diffeomorphism and handle slide equivalent.
\end{enumerate}
\end{thm}
For a $4$-manifold $X$ and a relative trisection diagram $\calD$, if $X$ is diffeomorphic to the trisected $4$-manifold corresponding to $\calD$ by Theorem~\ref{thm:rt-rtd}, then we simply say that $\calD$ is a relative trisection diagram of $X$.
There is a transition between handlebody diagrams and relative trisection diagrams.
Castro, Gay, and Pinz\'{o}n-Caiced \cite{CasGayPin18_2} showed how to obtain a relative trisection from a handle decomposition of a $4$-manifold.
Moreover, they gave an algorithm to construct relative trisection diagrams from handlebody diagrams.
Conversely, Kim and Miller \cite{KimMil20} described a method for constructing handlebody diagrams from relative trisection diagrams.
\subsection{Corks}\label{subsec:Corks}
A pair of smooth manifolds are said to be \textit{exotic} if they are homeomorphic but not diffeomorphic. Corks are used to construct exotic smooth structures of $4$-manifolds.
\begin{defi}
Let $C$ be a compact, contractible, smooth $4$-manifold with boundary and $\tau:\partial{C}\to \partial{C}$ be a smooth involution on the boundary.
The pair $(C,\tau)$ is called a \textit{cork}, if $\tau$ extends to a self-homeomorphism of $C$, but cannot extend to any self-diffeomorphism of $C$.
\end{defi}
Let $X$ be a smooth $4$-manifold, and let $(C,\tau)$ be a cork.
Suppose that $C$ is embedded in $X$.
Let $X'$ be the $4$-manifold obtained from $X$ by removing $C$ and re-gluing it by $\tau$ (i.e., $X' := (X-C)\cup_{\tau}C$).
Then we say that $X'$ is obtained from $X$ by a \textit{cork twist} along $(C,\tau)$.
The $4$-manifolds $X$ and $X'$ are homeomorphic to each other, but they may not be diffeomorphic.
Conversely, it is known that any two simply-connected, closed, exotic $4$-manifolds are related by a cork twist (\cite{Mat96}, \cite{CurFreHsiSto96}).
We now introduce an example of corks.
\begin{defi}\label{def:ac}
Let $W_1$ be the smooth $4$-manifold given by the handlebody diagram in Figure~\ref{fig:Ac-Kd}.
Let $f_1:\partial{W_1}\to\partial{W_1}$ be the involution obtained by first surgering $S^1\times D^3$ embedded along the core of the $1$-handle to $D^2\times S^2$ in the interior of $W_1$, and then surgering $D^2\times S^2$ embedded along the core of the $2$-handle to $S^1\times D^3$ (i.e., replacing the ``dot'' and ``0'' in Figure~\ref{fig:Ac-Kd}).
\end{defi}
\begin{thm}[Akbulut \cite{Akb91_1}]
The pair $(W_1,f_1)$ is a cork.
\end{thm}
This is the first example of a cork and is called the \textit{Akbulut cork}.
Obviously the Akbulut cork is Mazur-type.
\begin{defi}
A contractible smooth $4$-manifold $X$ is called \textit{Mazur-type}, if $X$ admits a handle decomposition consisting of one $0$-handle, one $1$-handle, and one $2$-handle.
\end{defi}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/Ac-Kd0.eps}
\caption{A handlebody diagram of the Akbulut cork.}
\label{fig:Ac-Kd}
\end{figure}
\section{Genus $3$ relative trisections of corks}\label{sec:genus3}
In this section, we construct an infinite family of corks with genus $3$ relative trisection. First, we give $(3,3;0,4)$-relative trisection diagrams.
\begin{lem}\label{lem:rtd-prf}
For each positive integer $n\in\NN$, let $\calD_n=(\Sigma;\alpha,\beta,\gamma)$ be the diagram shown in Figure~\ref{fig:D_n-rtd}, where $\Sigma$ is the gray surface of genus $3$ with $4$ boundary components, and $\alpha$, $\beta$, and $\gamma$ are the families of three curves of red, blue, and green, respectively.
Then $\calD_n$ is a $(3,3;0,4)$-relative trisection diagram.
\end{lem}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/D_n-rtd.eps}
\caption{$\calD_n=(\Sigma;\alpha,\beta,\gamma)$.}
\label{fig:D_n-rtd}
\end{figure}
\begin{proof}
We prove that each triple $(\Sigma;\alpha,\beta)$, $(\Sigma;\beta,\gamma)$, and $(\Sigma;\gamma,\alpha)$ can be made the standard diagram in Figure~\ref{fig:std3304} by diffeomorphisms of $\Sigma$ and handle slides of each family of curves.
To prove this, we introduce the four operations shown in Figures~\ref{fig:modif12} and \ref{fig:modif345}.
The operation (iii) is obtained by two Dehn twists (see Figure~\ref{fig:modif3-prf}).
Combining the operations (iii) and (i) shown in Figure~\ref{fig:modif4-prf}, we obtain the operation (iv), which exchanges a meridian and a longitude.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/std3304.eps}
\caption{The standard diagram of type $(3,3;0,4)$.}
\label{fig:std3304}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/modif12.eps}
\caption{The operations (i) and (ii).}
\label{fig:modif12}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/modif34.eps}
\caption{The operations (iii) and (iv).}
\label{fig:modif345}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/modif3-prf.eps}
\caption{A proof of the operation (iii).}
\label{fig:modif3-prf}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/modif4-prf.eps}
\caption{A proof of the operation (iv).}
\label{fig:modif4-prf}
\end{figure}
A proof for the case $n=1$ given by Figures~\ref{fig:sigma-ab}, \ref{fig:sigma-ca}, and \ref{fig:sigma-bc}.
Note that the same operations can be performed for any $n\geq2$.
The triple $(\Sigma;\alpha,\beta)$ in Figure~\ref{fig:sigma-ab} is already diffeomorphic to the standard diagram.
$(\Sigma;\gamma,\alpha)$ can be made standard by only diffeomorphisms (see Figure~\ref{fig:sigma-ca}).
The second and third diagrams are obtained by dragging the black disks along the marked $\gamma$ curves.
Note that we can ignore the number of rotations of a $\gamma$ curve with respect to a black disk by the operation (i).
Applying the operation (iv) to the last diagram, we obtain the standard diagram.
$(\Sigma;\beta,\gamma)$ can be made standard by diffeomorphisms and handle slides shown in Figure~\ref{fig:sigma-bc}.
The third diagram is obtained by dragging the black disk with blue circle along the marked $\gamma$ curve. In this process, when the black disk approaches a $\beta$ curve, it can pass through by the operation (ii).
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/sigma-ab.eps}
\caption{$(\Sigma;\alpha,\beta)$.}
\label{fig:sigma-ab}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/sigma-ca.eps}
\caption{Diffeomorphisms proving $(\Sigma;\gamma,\alpha)$ can be made standard.}
\label{fig:sigma-ca}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/sigma-bc.eps}
\caption{Diffeomorphisms and handle slides proving $(\Sigma;\beta,\gamma)$ can be made standard.}
\label{fig:sigma-bc}
\end{figure}
\end{proof}
\begin{defi}\label{def:M_n}
For each positive integer $n\in \NN$, let $\calT_n$ be the $(3,3;0,4)$-relative trisection corresponding to $\calD_n$ by Theorem~\ref{thm:rt-rtd}, and let $M_n$ be the trisected $4$-manifold.
\end{defi}
\begin{lem}\label{lem:M_n-Kd}
Figure~\ref{fig:M_n-Kd} is a handlebody diagram of $M_n$.
\end{lem}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/M_n-Kd.eps}
\caption{The handlebody diagram of $M_n$ induced by $\calD_n$.}
\label{fig:M_n-Kd}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/D_n-rtd-cutarcs.eps}
\caption{A relative trisection diagram $\calD_n$ and a cut system $\eta$.}
\label{fig:D_n-rtd-cutarcs}
\end{figure}
\begin{proof}
By using the algorithm given by Kim and Miller \cite{KimMil20}, we can obtain a handlebody diagram of $M_n$ as follows (since $k-2p-b+1=0$, this is a simple case of the algorithm).
\begin{enumerate}
\item
Perform diffeomorphisms of $\Sigma$ and handle slides so that $(\Sigma;\alpha,\beta)$ is standard, and then embed $(\Sigma;\alpha,\beta,\gamma)$ into $S^3$.
The relative trisection diagram $\calD_n$ in Figure~\ref{fig:D_n-rtd} is already standard.
\item
Choose pairwise disjoint, properly embedded simple arcs $\eta\subset \Sigma$ that are disjoint from $\alpha$ and $\beta$ curves so that $\Sigma_{\alpha}-\nu(\eta;\Sigma) \cong \Sigma_{\beta}-\nu(\eta;\Sigma) \cong D^2$.
Each of $\Sigma_\alpha$ and $\Sigma_\beta$ denotes the result of surgering $\Sigma$ along $\alpha$ and $\beta$ curves, respectively.
We call such arcs $\eta$ a \textit{cut system} for $(\Sigma;\alpha,\beta)$.
In our case, choose $\eta$ as the three purple arcs in Figure~\ref{fig:D_n-rtd-cutarcs}.
\item
For each $\eta_i$, draw a dotted circle $C_i \subset \partial \nu(\Sigma;S^3)$ as shown in Figure~\ref{fig:choose-cutarcs} (i.e., under the identification $\nu(\Sigma;S^3)\cong \Sigma\times[-1,1]$, let $C_i:=\partial{(\eta_i\times[-1,1])}$).
\item
Consider the $\gamma$ curves as attaching circles of $2$-handles with the surface framing of $\Sigma$.
\item
Then $\{C_1,C_2,\ldots,C_{2p+b-1};\gamma_1,\gamma_2,\ldots,\gamma_{g-p}\}$ is a handlebody diagram of $M_n$.
\end{enumerate}
Figure~\ref{fig:KMalg} shows this algorithm for the case $n=1$.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/choose-cutarcs.eps}
\caption{How to obtain a dotted circle from a cut arc.}
\label{fig:choose-cutarcs}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/KMalg3.eps}
\caption{The algorithm of \cite{KimMil20} for the case of $\calD_1$.}
\label{fig:KMalg}
\end{figure}
\end{proof}
\begin{thm}\label{thm:M_n-Mazur}
The family $\{M_n\}_{n\in \NN}$ of $4$-manifolds satisfies the following conditions.
\begin{itemize}
\item Each $M_n$ is a Mazur-type $4$-manifold.
\item $M_1, M_2, \ldots$ are mutually non-homeomorphic.
\end{itemize}
\end{thm}
The following proof is inspired by the work of Oba \cite{Oba15}.
\begin{proof}
Performing the handle moves in Figure~\ref{fig:M_n-Kcalc}, we obtain the handlebody diagram of $M_n$ consisting of one $0$-handle, one $1$-handle, and one $2$-handle.
By using the Seifert--van Kampen theorem and the Mayer--Vietoris exact sequence, we see that $\pi_1(M_n)$ is trivial and $H_*(M_n) \cong H_*(\mathrm{pt.})$.
Thus $M_n$ is contractible.
\begin{figure}[tbp]
\centering
\includegraphics[scale=1.0]{figs/M_n-Kcalc.eps}
\caption{Handle moves of $M_n$.}
\label{fig:M_n-Kcalc}
\end{figure}
Next, we calculate the Casson invariant $\lambda(\partial{M_n})$.
Perform the handle moves in Figure~\ref{fig:bdryM_n-Kcalc} and, let $K_n$ be the knot of the last diagram.
We see that $\partial{M_n}$ is obtained by Dehn surgery along $K_n$ with coefficient $1$.
By the surgery formula for Casson invariants (see \cite{Sav02b}), the following holds.
\begin{equation*}
\lambda(\partial{M_n})=\lambda(S^3+\frac{1}{1} K_n) = \lambda(S^3)+\frac{1}{2}\Delta''_{K_n}(1),
\end{equation*}
where $\Delta''_{K_n}(1)$ is the second derivative of the Alexander polynomial $\Delta_{K_n}(t)$ at $t=1$.
By Lemma~\ref{lem:Alex-K_n} described later, $\Delta''_{K_n}(1) = -2n(n+1)$.
Hence we see that $\lambda(\partial{M_n}) = -n(n+1)$.
Thus, $\partial{M_1}, \partial{M_2}, \ldots$ are mutually non-homeomorphic, so $M_1, M_2, \ldots$ are also mutually non-homeomorphic.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/bdryM_n-Kcalc.eps}
\caption{Handle moves of $\partial{M_n}$.}
\label{fig:bdryM_n-Kcalc}
\end{figure}
\end{proof}
\begin{lem}\label{lem:Alex-K_n}
The Alexander polynomial of the knot $K_n$ is given by
\begin{equation*}
\Delta_{K_n}(t) = -n(n+1)t+(2n^2+2n+1)-n(n+1)t^{-1}.
\end{equation*}
\end{lem}
\begin{proof}
Let $D_n\subset \RR^3$ be the immersed disk in Figure~\ref{fig:ribbondisk}.
Since $\partial{D_n}=K_n$, the knot $K_n$ is ribbon and $D_n$ is its ribbon disk.
By Fox and Milnor \cite[Theorem~2]{FoxMil66}, there exists a polynomial $f(t)$ with integer coefficients such that $\Delta_{K_n}(t)=f(t)f(t^{-1})$.
We now calculate $f(t)$ by using the algorithm of Terasaka \cite{Ter59}.
First orient the knot $K_n$, and decompose $D_n$ to two disks $A,C$ and $2n+1$ bands $B_1, B_2, \ldots, B_{2n+1}$ as shown in Figure~\ref{fig:ribbondisk}, where full twists of bands are represented by Figure~\ref{fig:fulltwist}.
Then $f(t)$ is given by the following determinant of a $(2n+1)\times(2n+1)$ matrix.
\begin{equation*}
f(t) =
\begin{vmatrix}
-t^{\delta_1} & & & & & t^{\delta^c_1}-1 \\
1 & -t^{\delta_2} & & & \text{\huge{0}} & t^{\delta^c_2}-1 \\
& 1 & -t^{\delta_3} & & & t^{\delta^c_3}-1 \\
& & & \ddots & & \vdots \\
& \text{\huge{0}} & & & -t^{\delta_{2n}} & t^{\delta^c_{2n}}-1 \\
& & & & 1 & -1 \\
\end{vmatrix}.
\end{equation*}
Note that each of $\delta_i$ and $\delta^c_i$ is given as in Figure~\ref{fig:Terasaka_alg}.
In the case of Figure~\ref{fig:ribbondisk}, $\delta_i=\delta^c_i=-1$ if $i$ is odd, and $\delta_i=1$, $\delta^c_i=0$ if $i$ is even.
To compute this determinant, add all odd rows multiplied by $t$ and all even rows to the lowermost row. Then, all elements of the lowermost row become $0$ except for the rightmost one.
By the cofactor expansion along the lowermost row, we obtain $f(t)=-(n+1)t+n$, so the formula of the lemma follows.
\begin{figure}[tbp]
\centering
\includegraphics[scale=1.0]{figs/ribbondisk.eps}
\caption{A ribbon disk $D_n$ of $K_n$.}
\label{fig:ribbondisk}
\end{figure}
\begin{figure}[tbp]
\centering
\includegraphics[scale=0.9]{figs/fulltwist.eps}
\caption{Full twists of bands.}
\label{fig:fulltwist}
\end{figure}
\begin{figure}[tbp]
\centering
\includegraphics[scale=1.0]{figs/Terasaka_alg.eps}
\caption{How to determine $\delta_i$ and $\delta^c_i$.}
\label{fig:Terasaka_alg}
\end{figure}
\end{proof}
\begin{thm}\label{thm:M_n-cork}
Each $M_n$ is a cork and admits a Stein structure.
\end{thm}
\begin{proof}
First, we prove that $M_n$ admits a Stein structure.
Recall that the last diagram of Figure~\ref{fig:M_n-Kcalc} is a handlebody diagram of $M_n$, and perform the handle moves in Figure~\ref{fig:M_n-k.calc2}, where $(*)$ is the operation shown in Figure~\ref{fig:akmove} that was introduced in \cite{AkbKir79}.
For the last diagram of Figure~\ref{fig:M_n-k.calc2}, converting the $1$-handle notation, we obtain the Legendrian knot diagram in Figure~\ref{fig:M_n-Legendre}.
Since the Thurston--Bennequin number is $1$, $M_n$ admits a Stein structure (\cite[Proposition~2.3]{Gom98}).
\begin{figure}[tbp]
\centering
\includegraphics[scale=1.0]{figs/M_n-Kcalc2.eps}
\caption{Handle moves of $M_n$.}
\label{fig:M_n-k.calc2}
\end{figure}
\begin{figure}[tbp]
\centering
\includegraphics[scale=1.0]{figs/AKmove.eps}
\caption{The operation $(*)$.}
\label{fig:akmove}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/M_n-Legendre.eps}
\caption{A Legendrian knot diagram of $M_n$.}
\label{fig:M_n-Legendre}
\end{figure}
Next, we prove that $M_n$ admits a cork structure.
By the isotopies in Figure~\ref{fig:M_n-Kcalc3}, we obtain the diagram consisting of the $2$-component symmetric link.
Note that the second isotopy is obtained by repeating the operation of Figure~\ref{fig:M_n-induc} $n-1$ times.
Let $\tau_n:\partial{M_n}\to\partial{M_n}$ be the involution induced by $180^\circ$ rotation about the horizontal axis shown in Figure~\ref{fig:tau_n}.
Recall that $M_n$ is contractible and $\partial{M_n}$ is a homology $3$-sphere. By Boyer's theorem \cite{Boy86}, $\tau_n$ extends to a self-homeomorphism of $M_n$.
Since $M_n$ is a Stein $4$-manifold, we can use Corollary~2.1 in \cite{AkbMat97} to prove that $\tau_n$ cannot extend to a self-diffeomorphism of $M_n$.
For details of this argument, see the proof of Theorem~3.1 in \cite{AkbMat97}.
\begin{figure}[tbp]
\centering
\includegraphics[scale=1.0]{figs/M_n-Kcalc3.eps}
\caption{Isotopies of $M_n$.}
\label{fig:M_n-Kcalc3}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/M_n-induc.eps}
\caption{Isotopies of $M_n$.}
\label{fig:M_n-induc}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/tau_n.eps}
\caption{An involution $\tau:\partial{M_n}\to\partial{M_n}$.}
\label{fig:tau_n}
\end{figure}
\end{proof}
We can easily check that the diagram in Figure~\ref{fig:tau_n} for $n=1$ is isotopic to Figure~\ref{fig:Ac-Kd}, so $M_1$ is diffeomorphic to the Akbulut cork $W_1$.
\begin{rem}
The $4$-manifolds $M_1,M_2,\ldots$ are already known corks.
These were discovered by Dai, Hedden, and Mallick \cite[Theorem~1.12]{DaiHedMal20a}.
They proved that $M_n$ admits a cork structure by using Heegaard Floer homology.
\end{rem}
\begin{prop}\label{prop:M_n-hyp}
Each $\partial{M_n}$ is a hyperbolic $3$-manifold.
\end{prop}
\begin{proof}
Performing the handle moves shown in Figure~\ref{fig:bdryM_n-Kcalc2}, we see that $\partial{M_n}$ is homeomorphic to the $3$-manifold obtained by Dehn surgery along the $2$-bridge link $L_{[2,-2,2]}$ with coefficients $\{2-\frac{1}{n+1}, 2+\frac{1}{n} \}$.
For the notation $L_{[a_1,a_2,\ldots,a_k]}$, see Figures~1 and 2 in \cite{IchJonMas19a}.
Since the type of the $2$-bridge link $L_{[2,-2,2]}$ is $(5, 12)$, so it is not $(2,n)$-torus link.
By a result of Menasco \cite[Corollary~2]{Men84}, if a link $L$ is non-split, prime, alternating, and non-torus, then $L$ is hyperbolic.
Thus $L_{[2,-2,2]}$ is a hyperbolic link.
We prove that our surgeries are not exceptional for any $n\in\NN$.
An \textit{exceptional surgery} is a Dehn surgery along a hyperbolic link with coefficients such that the resulting $3$-manifold is non-hyperbolic.
In particular, an exceptional surgery along a link $L$ with coefficients is called \textit{complete}, if for any non-empty sublink $L'$, the $3$-manifold obtained from $S^3-\nu(L';S^3)$ by Dehn surgery along $L-L'$ is hyperbolic.
Ichihara, Jong, and Masai \cite{IchJonMas19a} gave a complete list of hyperbolic $2$-bridge links that can admit complete exceptional surgeries. They also listed candidates of surgery coefficients of them. In addition, Ichihara \cite{Ich12} classified exceptional surgeries along components of hyperbolic $2$-bridge links.
We verify that our surgeries are not included in these lists.
It is known that two non-trivial $2$-bridge links of types $(p, q)$ and $(p', q')$ are isotopic if and only if $q=q'$ and either $p\equiv p'$ or $pp'\equiv1 \pmod q$ (see Section~2.1 of \cite{Kaw96b}).
By using this, we see that $L_{[2,-2,2]}$ is not included in the list of \cite{Ich12}.
However, among the $2$-bridge links in the list of \cite{IchJonMas19a}, only $L_{[3,2,3]}$ is isotopic to $L_{[2,-2,2]}$, which has $11$ candidate coefficients that would be a complete exceptional surgery.
The first homology groups of the $3$-manifolds obtained by the surgeries with these coefficients are non-trivial except for one of the coefficients $\{ -3,-1 \}$.
We calculate the Casson invariant of this exception.
By blowing down a $-1$ framed circle shown in Figure~\ref{fig:5_2}, it can be represented by the mirror image of the knot $5_2$ with coefficient $1$.
It is known that the Alexander polynomial of the knot $5_2$ is $2t-3+2t^{-1}$ (see \cite{Rol76b}).
Hence the Casson invariant is $\frac{1}{2}\Delta''_{5_2}(1)=2$.
Thus, it could be homeomorphic to the boundary of the Akbulut cork $\partial{M_1}$, since $\lambda(\partial{M_n})=-n(n+1)$.
However, it is known to be hyperbolic (\cite[Example~4.2]{AkbRub16}).
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/bdryM_n-Kcalc2.eps}
\caption{Handle moves of $\partial{M_n}$.}
\label{fig:bdryM_n-Kcalc2}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/5_2.eps}
\caption{A blowing down of a $-1$ framed circle.}
\label{fig:5_2}
\end{figure}
\end{proof}
\begin{rem}\label{rem:Thurston}
Without using the result of the preprint \cite{IchJonMas19a}, we can show that there are infinitely many hyperbolic $3$-manifolds contained in $\{\partial{M_n}\}_{n\in\NN}$.
By Thurston's hyperbolic Dehn surgery theorem (\cite[Theorem~5.8.2]{Thu02b}), there exist finite subsets $E_1$ and $E_2$ of $\QQ$ such that, for any rational numbers $r_1\notin E_1$ and $r_2\notin E_2$, the $3$-manifold obtained by Dehn surgery along $L_{[2,-2,2]}$ with coefficients $\{ r_1,r_2 \}$ admits a hyperbolic structure.
Thus, $\partial{M_n}$ is hyperbolic for any positive integer $n$ satisfying $2-\frac{1}{n+1}\notin E_1$ and $2+\frac{1}{n}\notin E_2$, and the set of such $n$ is infinite.
\end{rem}
\section{A lower bound for the trisection genus of a $4$-manifold with boundary}\label{sec:lowerbound}
In this section, we prove that the trisection genus of $M_n$ is greater than $2$.
The following proposition narrows down the $4$-tuple $(g,k;p,b)$ of relative trisections.
\begin{prop}\label{prop:rtd-Euler}
Let $X$ be a compact, connected, oriented, smooth $4$-manifold with connected boundary.
Suppose that $X$ admits a $(g,k;p,b)$-relative trisection.
Let $A(g,k;p,b)$ be the number of pairs $(\delta_i,\epsilon_i)$ such that $\delta_i$ and $\epsilon_i$ intersect at one point in Figure~\ref{fig:std-rtd} (i.e., $A(g,k;p,b) := g+p+b-1-k$).
Then $g,k,p,b$, and $A(g,k;p,b)$ satisfy the following conditions.
\begin{itemize}
\item
$0\leq p \leq \mathrm{min}\{ \frac{g+1-\chi(x)}{3}, g\}$.
\item
$\frac{2g-1+\chi(X)}{3} \leq A(g,k;p,b) \leq g-p$.
\item
$k=1-\chi(X)-g+p+2A(g,k;p,b)$.
\item
$b=3A(g,k;p,b)-2g+2-\chi(X)$.
\end{itemize}
\end{prop}
\begin{proof}
The third and fourth conditions are obtained by combining the definition of $A(g,k;p,b)$ and the formula of Proposition~\ref{prop:Euler}.
Since $b\geq1$, we see that $\frac{2g-1+\chi(X)}{3} \leq A(g,k;p,b)$, and Figure~\ref{fig:std-rtd} shows that $A(g,k;p,b) \leq g-p$.
Hence the second condition holds.
It also follows that $\frac{2g-1+\chi(X)}{3} \leq g-p$, so we obtain $p \leq \frac{g+1-\chi(x)}{3}$.
Since $0\leq p \leq g$ (see Definition~\ref{def:rt}), the first condition holds.
\end{proof}
An open book decomposition is called \textit{planar}, if the genus of the page is $0$.
For $3$-manifolds that admit such open book decompositions of the number of binding components less than $4$, the following theorem holds.
\begin{thm}[Etnyre--Ozbagci \cite{EtnOzb08}]\label{thm:EO08}
Let $M$ be a closed, connected, oriented $3$-manifold.
Suppose that $M$ admits a planar open book decomposition with $b$ binding components.
If $b\leq3$, then $M$ is a Seifert fiber space.
In particular, $M$ is the $3$-sphere if $b=1$, and is a Lens space if $b=2$.
\end{thm}
Combining with Lemma~\ref{lem:obd}, we obtain the following corollary.
\begin{cor}\label{cor:SFS}
Suppose that $X$ admits a $(g,k;0,b)$-relative trisection.
If $b\leq3$, then the boundary $\partial{X}$ is a Seifert fiber space.
In particular, $\partial{X}$ is the $3$-sphere if $b=1$, and is a Lens space if $b=2$.
\end{cor}
We now prove the main results.
\begin{proof}[Proof of Theorem~\ref{mainthm:lowerbound}]
Assume that there exist integers $g$, $k$, $p$, and $b$ such that $X$ admits a $(g, k; p, b)$-relative trisection and $g\leq\chi(X)+1$.
Using Proposition~\ref{prop:rtd-Euler}, we easily see that $k\leq2$, $p=0$, and $b\leq3$.
Thus, it follows that $\partial{X}$ is a Seifert fiber space by Corollary~\ref{cor:SFS}.
This is a contradiction.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{mainthm:corks}]
By Proposition~\ref{prop:M_n-hyp}, each $\partial{M_n}$ is a hyperbolic $3$-manifold.
It is known that a Seifert fiber space cannot be hyperbolic (see \cite{Thu02b}), so we can use Theorem~\ref{mainthm:lowerbound}.
Since the Euler characteristic of $M_n$ is equal to $1$, the trisection genus of $M_n$ is greater than $2$.
On the other hand, $\calT_n$ is a genus $3$ relative trisection of $M_n$ (see Definition~\ref{def:M_n}).
Therefore, we conclude that the trisection genus of $M_n$ is $3$ for any $n\in\NN$.
\end{proof}
\begin{rem}
For the proof of the hyperbolicity of $\partial{M_n}$, we used the result of the preprint \cite{IchJonMas19a}.
Without using this, we can show that there are infinitely many corks with trisection genus $3$ by Remark~\ref{rem:Thurston}.
\end{rem}
\begin{proof}[Proof of Corollary~\ref{maincor:bdryOB}]
Recall that $\calT_n$ is a $(3,3;0,4)$-relative trisection of $M_n$.
By Lemma~\ref{lem:obd}, $\partial{M_n}$ admits an open book decomposition with page $\Sigma_{0,4}$.
If $\partial{M_n}$ admits a planar open book decomposition of the number of binding components less than $4$, then it also contradicts Proposition~\ref{prop:M_n-hyp}.
\end{proof}
Finally, we introduce a Heegaard splitting of $\partial{M_n}$ induced by our relative trisections.
\begin{lem}\label{prop:rt-Hspl}
Let $X$ be a compact, connected, oriented, smooth $4$-manifold with connected boundary.
If $X$ admits a $(g,k;p,b)$-relative trisection, then $\partial{X}$ admits a genus $2p+b-1$ Heegaard splitting.
\end{lem}
\begin{proof}
By Lemma~\ref{lem:obd}, there exists an open book decomposition $(B,\pi)$ on $\partial{X}$ with page $\Sigma_{p,b}$, where $B$ is a binding and $\pi:\partial{X}-B \to S^1$ is a fibration.
Under the identification $S^1\cong[0,1]/\sim$, we obtain the decomposition $\partial{X}=(\pi^{-1}([0,1/2])\cup B)\cup(\pi^{-1}([1/2,1])\cup B)$, which is a genus $2p+b-1$ Heegaard splitting.
\end{proof}
By Definition~\ref{def:M_n} and Lemma~\ref{prop:rt-Hspl}, we see that each $\partial{M_n}$ admits a genus $3$ Heegaard splitting.
\section{Relative trisections of an exotic pair of $4$-manifolds}\label{sec:exotic}
Recall that $M_1$ is diffeomorphic to the Akbulut cork $W_1$.
Thus the trisection genus of the Akbulut cork is $3$, that is, Theorem~\ref{mainthm:Ac} holds.
The diagram $\calD_1$ in Figure~\ref{fig:Ac-rtd} is a $(3,3;0,4)$-relative trisection diagram of $W_1$.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/Ac-rtd.eps}
\caption{A $(3,3;0,4)$-relative trisection diagram $\calD_1$ of the Akbulut cork $W_1$.}
\label{fig:Ac-rtd}
\end{figure}
In this section, we give relative trisection diagrams of an exotic pair.
Let $P$ and $Q$ be the $4$-manifolds with boundary given by Figures~\ref{fig:P-Kd} and \ref{fig:Q-Kd}, respectively.
According to subsection~9.1 in \cite{AkbYas13}, they are homeomorphic but not diffeomorphic and related by a cork twist along the Akbulut cork $W_1$.
\begin{figure}[!htbp]
\begin{tabular}{cc}
\begin{minipage}[t]{0.45\hsize}
\centering
\includegraphics[scale=1.0]{figs/P-Kd.eps}
\caption{$P$}
\label{fig:P-Kd}
\end{minipage} &
\begin{minipage}[t]{0.45\hsize}
\centering
\includegraphics[scale=1.0]{figs/Q-Kd.eps}
\caption{$Q$}
\label{fig:Q-Kd}
\end{minipage}
\end{tabular}
\end{figure}
\begin{proof}[Proof of Theorem~\ref{mainthm:exotictris}]
Let $\calD_P$ and $\calD_Q$ be the diagrams in Figures~\ref{fig:P-rtd} and \ref{fig:Q-rtd}, respectively.
First, we prove that $\calD_P$ and $\calD_Q$ are relative trisection diagrams.
As in the proof of Lemma~\ref{lem:rtd-prf}, we verify that $(\Sigma,\alpha,\beta)$, $(\Sigma,\beta,\gamma)$, and $(\Sigma,\gamma,\alpha)$ are diffeomorphism and handle slide equivalent to the standard diagram in Figure~\ref{fig:std-rtd}.
We omit proofs of easy parts $(\Sigma,\alpha,\beta)$ and $(\Sigma,\gamma,\alpha)$.
Proofs of the hardest part $(\Sigma;\beta,\gamma)$ are shown in Figures~\ref{fig:P-sigma-ca} and \ref{fig:Q-sigma-ca}.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/P-rtd.eps}
\caption{A $(5,4;0,5)$-relative trisection diagram $\calD_P$.}
\label{fig:P-rtd}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/Q-rtd.eps}
\caption{A $(4,3;0,4)$-relative trisection diagram $\calD_Q$.}
\label{fig:Q-rtd}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/P-sigma-bc.eps}
\caption{Diffeomorphisms and handle slides proving $(\Sigma;\beta,\gamma)$ of $\calD_P$ can be made standard.}
\label{fig:P-sigma-ca}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/Q-sigma-bc.eps}
\caption{Diffeomorphisms and handle slides proving $(\Sigma;\beta,\gamma)$ of $\calD_Q$ can be made standard.}
\label{fig:Q-sigma-ca}
\end{figure}
Next, we show that $\calD_P$ and $\calD_Q$ are relative trisection diagrams of the $4$-manifolds $P$ and $Q$, respectively.
Use the algorithm of \cite{KimMil20} to obtain the handlebody diagrams corresponding to these relative trisection diagrams, and perform the handle moves (see Figures~\ref{fig:P-KMalg} and \ref{fig:Q-KMalg}).
We then see that they coincide with $P$ and $Q$ by the handle moves in Figure~\ref{fig:PQ-Kcalc}.
For the last isotopy, see Figures~\ref{fig:M_n-k.calc2} and \ref{fig:M_n-Kcalc3} for the case $n=1$.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/P-KMalg.eps}
\caption{$\calD_P$ with a cut system and handle moves of the induced $4$-manifold.}
\label{fig:P-KMalg}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/Q-KMalg.eps}
\caption{$\calD_Q$ with a cut system and handle moves of the induced $4$-manifold.}
\label{fig:Q-KMalg}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/Ac-Kcalc2.eps}
\caption{Handle moves of the Akbulut cork with two circles.}
\label{fig:PQ-Kcalc}
\end{figure}
We now consider lower bounds for the trisection genera of $P$ and $Q$.
For the handlebody diagram of $Q$ in Figure~\ref{fig:Q-Kd}, performing the slam-dunk move, we see that $\partial{Q}$ is homeomorphic to the $3$-manifold obtained by Dehn surgery along the Mazur link with coefficients $\{ \frac{1}{2}, 0 \}$ (see Figure~\ref{fig:bdryQ}).
Exceptional surgeries along the Mazur link are classified by Yamada \cite[Theorem~1.1]{Yam18}.
According to this result, our Dehn surgery is not exceptional, so $\partial{Q}$ is hyperbolic.
Since $P$ and $Q$ are homeomorphic and $\chi(Q)=2$, we see that the trisection genera are greater than $3$ by Theorem~\ref{mainthm:lowerbound}.
Thus the trisection genus of $Q$ is $4$, and the trisection genus of $P$ is either $4$ or $5$.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1.0]{figs/bdryQ.eps}
\caption{A surgery diagram of $\partial{Q}$.}
\label{fig:bdryQ}
\end{figure}
\end{proof}
\begin{rem}
The trisection genus of $P$ is greater than or equal to $4$.
By Proposition~\ref{prop:rtd-Euler}, we see that $P$ could admit relative trisections of $(4,3;0,4)$ or $(4,2;1,1)$.
However, we have not been able to construct such relative trisections.
If $P$ cannot admit genus $4$ relative trisections, the trisection genus for $4$-manifolds with boundary is not homeomorphism invariant.
\end{rem}
\subsection*{Acknowledgements}
This paper is partially based on the author's master thesis.
The author would like to thank his adviser Kouichi Yasui for his encouragement and helpful discussions.
The author was partially supported by JST SPRING, Grant Number JPMJSP2138.
\bibliographystyle{amsplain}
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A recent poll by travel deals firm Travelzoo has revealed that the ¹UK is the top destination for British holidaymakers this year with 27% of those polled choosing a 'staycation' over Spain (21%) and France (10%), citing safety as a prime concern.
It's a trend that leading coach operator Epsom Coaches is seeing reflected in bookings this year as UK trips are up by over 13% compared to the same time last year with the top five destinations being Sidmouth, Chatsworth, Newcastle, 'island hopping' in the Hebrides and Cornwall.
And, according to Mintel it is short and sweet breaks that are top of the holiday charts with one quarter of holidaymakers (26%) saying they expect to take more short breaks i.e. within 1-3 nights over the next 12 months, compared to one in five (21%) who say they expect to take longer trips.
Epsom Coaches even offers a *Home to Holiday service on all their coach tours, with holiday goers collected from home and taken to the company's Departure Lounge at its HQ in Roy Richmond Way, Epsom.
Epsom Coaches recently launched its new Autumn/Spring 2016/17 brochure which includes some fantastic new excursions such as JRR Tolkien inspired Jewels, Canals and 'Middle Earth', a special Christmas version of the company's ever popular 'Mystery Breaks' and Nine Counties at Easter which explores the English Shires. The company has also added 40 new day trips to its schedule.
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You are here: Home Page > Science & Mathematics > Earth Sciences & Geography > A Guide to Countries of the World
This item has an extended shipping time. The typical delivery time is 2 weeks.
432 Pages | c.200 maps
5 X 7 3/4 inches
Available online as part of Oxford Reference - cross-search quality A-Z reference at the click of a button
Companion website
A Guide to Countries of the World
Peter Stalker
Revised and fully updated to reflect the very latest developments in every country
Concise overviews of every country in the world, covering social, economic, religious, and political issues
Features maps, web links, and key facts e.g. capital city, GDP, life expectancy, religion, etc.
Appendices include main social and economic indicators country-by-country as well as a list of regional groupings
Recommended web links are accessed and kept up-to-date via A Guide to Countries of the World companion website
This A-Z guide provides a wealth of information for every country in the world. Each entry offers a brief history and outlines contemporary social, economic, political and religious issues. For every country there is a map and a quick-reference fact box containing data and statistics including languages, population, GDP, capital city, life expectancy, and more. An appendix of indicator tables includes statistics on income and poverty, health and population and a second appendix features regional groupings, such as the EU and the African Union, with useful web links. Revised and fully updated, the Third Edition is ideal for students and teachers of geography, politics, economics, and world history, and it is a perfect home reference book. From major powers such as the United States or China, to hot spots such as North Korea or Iran, to little known nations such as Mayotte, Kiribati, or Nauru, this essential guide is the first place to turn for information on the world around us.
A-Z Countries of the World
Smaller Countries
Indicator tables
Peter Stalker is a former editor of the annual Human Development Report, produced by the United Nations Development Program.
A Dictionary of Agriculture and Land Management
Will Manley, Katharine Foot, and Andrew Davis
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The Oxford Dictionary of Philosophy
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Fourteenth Edition
Ashley Bowes
The Oxford Dictionary of Architecture
James Stevens Curl and Susan Wilson
Jorge Daniel Taillant
A Dictionary of Geography
Sixth Edition
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Snow Avalanches
Francois Louchet
A Dictionary of Shakespeare
Stanley Wells
Climate Change: A Very Short Introduction
Mark Maslin
A Dictionary of Marketing
Charles Doyle
The Arctic: A Very Short Introduction
Klaus Dodds and Jamie Woodward
Dictionary of British History
John Cannon
First Woman
James Rodger Fleming
Concise Oxford Dictionary of Archaeology
Timothy Darvill
Elements of a Sustainable World
Science & Mathematics > Earth Sciences & Geography
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{"url":"http:\/\/math.stackexchange.com\/questions\/65838\/fiber-bundle-m-x-m-diagonal","text":"# Fiber bundle M x M - diagonal\n\nUnder what conditions for a space $M$ does the projection map to the first factor $p: M \\times M - \\Delta \\rightarrow M$ has the local triviality condition, i.e. is a fiber bundle? Where $\\Delta$ denotes the diagonal $\\{(a,a) \\}_{a \\in M}$.\n\n-\nInteresting question. Certainly looks true for closed manifolds $M$. \u2013\u00a0 Grumpy Parsnip Sep 19 '11 at 18:55\nIf $M$ is a closed manifold we can do the following: Let $U_i$ be a chart in $M$, so we can assume $U_i\\cong \\mathbb{R}^n$. Now consider the neighborhood $\\mathbb{R}^n \\times \\mathbb{R}^n - \\Delta \\cong U_i \\times U_i - \\Delta \\subset M \\times M - \\Delta$. Showing the result for $\\mathbb{R}^n \\times \\mathbb{R}^n - \\Delta$ and using a bump function inside an open set of $U_i \\times U_i - \\Delta$ we can can conclude that $p^{-1}(U_i)\\cong \\mathbb{R}^n \\times (\\mathbb{R}^n- \\mathbb{R}^{n-1})$. \u2013\u00a0 Manuel Sep 19 '11 at 19:04\nOops, I meant $p^{-1}(U_i)\\cong \\mathbb{R}^n \\times (\\mathbb{R}^n-\\{0\\})$. And of course, we can ignore the $i$ index of the chart $U_i$... \u2013\u00a0 Manuel Sep 19 '11 at 20:51\n\nJim's comment that it's true for $M$ a manifold is an old result, perhaps first due to Richard Palais (and in greater generality). The result is usually attributed to Fadell and Neuwirth, but their result came later. In their set-up you call $M \\times M \\setminus \\Delta$ to be the configuration space of two points in $M$. In Palais's set-up, $M \\times M \\setminus \\Delta$ is the space of embeddings of a two-point set into $M$. Palais works in the generality of embedding spaces of manifolds, so the domain manifold does not have to be zero-dimensional like in this case. For example, if $Emb(S^2,M)$ denotes the space of embeddings of a 2-sphere in $M$, take any submanifold $X$ of $S^2$, then the restriction map $Emb(S^2,M) \\to Emb(X,M)$ is a locally-trivial fiber bundle. These proofs depend pretty heavily on the fact that $M$ is a manifold.\nFor example, if $M$ were not a manifold, say $M$ is the wedge of a finite collection of intervals (the cone on a finite set). Then your map isn't a fiber bundle, not even a fibration. Because the number of path-components of the fiber changes as you pass over the wedge \/ cone point.\nIn particular, if your map is a locally trivial fiber bundle, it means the space $X$ satisfies a weak type of isotopy extension theorem. Because given a path between any two points $x,y \\in M$ you can trivialize the bundle $p$ over that path. So if there is a path from $x$ to $y$, $M \\setminus \\{x\\}$ and $M \\setminus \\{y\\}$ are homeomorphic. If you were more ambitious you could turn this line of reasoning into an if and only if statement for $M \\times M \\setminus \\Delta \\to M$ to be a fiber bundle. It will amount to saying that the homeomorphisms $M \\setminus \\{x \\} \\to M \\setminus \\{y\\}$ can be chosen in a continuous fashion (details suppressed) as you vary $x$ (or $y$).","date":"2015-04-26 03:02:45","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.944496214389801, \"perplexity\": 108.28041229386874}, \"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-2015-18\/segments\/1429246652296.40\/warc\/CC-MAIN-20150417045732-00311-ip-10-235-10-82.ec2.internal.warc.gz\"}"}
| null | null |
{"url":"https:\/\/rosalind.info\/glossary\/failure-array\/","text":"# Glossary\n\n## Failure array\n\nA failure array (often called a failure function) is an array employed in the course of the KMP algorithm to speed up motif finding in strings.\n\nSpecifically, given a string $s$, its failure array $P$ is constructed so that $P[k]$ is equal to the length of the longest substring of $s$ ending at $k$ that isn't a prefix but is equal to some prefix $s[1:i]$, where $i$ must be less than $k$.\n\nFor example, given the string \"CAGTAAGCAGGGACTG\", its failure array is given by $[0 0 0 0 0 0 0 1 2 3 0 0 0 1 0 0]$.","date":"2022-05-27 21:38:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 9, \"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\": 0.6676856875419617, \"perplexity\": 344.6476640175997}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652663006341.98\/warc\/CC-MAIN-20220527205437-20220527235437-00073.warc.gz\"}"}
| null | null |
01/28/2023 Who Is Tyre Nichols' Girlfriend? He Allegedly Murdered By Memphis Police
Who Is Blake Martinez's Wife? Blake Announced His Retirement!
Blake Edmon Martinez was born on January 9, 1994. He used to play for the National Football League as an inside linebacker (NFL). He went to Canyon del Oro High School in Oro Valley, Arizona, before getting a scholarship to go to Stanford University.
He played linebacker for the Stanford Cardinal for two years and was named to the first-team All-Pac-12 team in 2015. In the 2016 NFL Draft, the Green Bay Packers picked him in the fourth round.
Martinez was a three-star inside linebacker. Schools like Stanford, Boise State, Oregon State, and San Jose State all offered him scholarships.
Blake has announced his retirement from football games. Yesterday, he shared a heartfelt post on his Instagram to announce this news.
A post shared by Blake Martinez (@blake_martinez50)
Blake Martinez is happily married to Kristy Martinez. To know more stick to the article.
Meet Blake Martinez's Wife, Kristy
Blake And Kristy are married for a long time. The couple shares two kids together. Kristy isn't well known, so not much is known about her. So, as soon as we find out anything new, we'll let you know. We'll tell you what happens.
Also Read: Who Is Matthew Stafford's Wife? Meet Kelly Stafford!
Blake asked his long-time girlfriend to marry him in a restaurant in New York City, in 2017. He then posted a picture of her wearing the ring with a very romantic caption. The two are very open about their relationship, and they post photos of each other on social media all the time.
Also, they often show each other pictures of their girls. They spend most of their time together and post pictures of their great times together on Instagram.
Blake Martinez's Wife Is Present On Instagram
Kristy Martinez's wife is on Instagram as @kristy leiigh_. She has 8,310 followers. In her bio, she put, "Kristy Martinez."
On August 3, Kristy posted a photo of her family to mark her daughter's birthday with the caption, "Happy birthday to the strongest, bravest, kindest, and most energetic soul in this world! We Thank god everyday that she is our daughter! :)."
On the other hand, Blake is currently present as @blake_martinez50. He has 153K followers. In his bio, he wrote, "Blake Martinez #StanU For any business inquires email me at @blakes.breaks www.whatnot.com/invite/blakemartinez54."
Know About Blake Martinez's Salary And Net Worth
His contract with the Raiders only gave him a base salary of $311,110. He wasn't making much on his second team this year, especially after leaving New York where he made over $1 million.
Idol Net Worth thinks that the linebacker, who is now retired, is worth a little more than $1 million. They think it will cost $1.2 million. He became a millionaire by getting paid, getting endorsements, and making money in other ways.
Also Read: Who Is Domenico Dolce's Husband? Complete Info!
The former Packer signed a total of three contracts during his time with the team. They had:
Green Bay: $4,723,393 for four years; New York: $30,750,000 for three years; Las Vegas: $1,120,000 for one year.
Spotrac says that he was guaranteed nearly $19.4 million over the course of his short career. Most of that came from New York, where a contract guaranteed $19 million.
Who Is Tyre Nichols' Girlfriend? He Allegedly Murdered By Memphis Police
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{"url":"https:\/\/zbmath.org\/?q=an%3A0746.58054","text":"# zbMATH \u2014 the first resource for mathematics\n\nBasins of Wada. (English) Zbl\u00a00746.58054\nThe authors describe situations where there are several regions with the Wada property, namely that each point that is on the boundary of one region is on the boundary of all. First, the classical example of the \u201cLakes of Wada\u201d is discussed. Then it is argued by numerical computations (and proved for a somewhat idealized situation) that for the forced damped pendulum three attractor regions (of attracting periodic orbits) may coexist and all three basins of attraction have the Wada property. The latter requires some kind of indecomposability of the sets. Indecomposable continua are discussed in the rest of the paper.\nReviewer:\u00a0G.Jetschke (Jena)\n\n##### MSC:\n 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 34C25 Periodic solutions to ordinary differential equations\nFull Text:\n##### References:\n [1] Barge, M., Homoclinic intersections and indecomposability, (), 541 \u00b7 Zbl\u00a00657.58022 [2] M. Barge and R. Gillette, Indecomposability and dynamics of invariant plane separating continua, preprint. \u00b7 Zbl\u00a00737.54020 [3] Barge, M.; Martin, J., Chaos, periodicity and snakelike continua, Trans. AMS, 289, 355, (1985) \u00b7 Zbl\u00a00559.58014 [4] Barge, M.; Martin, J., Dense orbits on the interval, Michigan math. J., 34, 3, (1987) \u00b7 Zbl\u00a00655.58023 [5] Barge, M.; Martin, J., Dense periodicity on the interval, (), 731 \u00b7 Zbl\u00a00567.54024 [6] M. Barge and J. Martin, The construction of global attractors, Proc. AMS, to appear. \u00b7 Zbl\u00a00714.58036 [7] Bing, R.H., Concerning hereditarily indecomposable continua, Pacific J. math., 1, 43, (1951) \u00b7 Zbl\u00a00043.16803 [8] Birkhoff, G.D., Sur quelques courbes ferm\u00e9es remarquables, Bull. soc. math. France, 60, 1, (1932) \u00b7 Zbl\u00a00005.22002 [9] Cartwright, M.L.; Littlewood, J.E., On non-linear differential equations of the second order: I. the equation $$y\u0308\u2212k(1\u2212y\\^{}\\{2\\})y\u0307+y=b\u03bbk cos (\u03bbt+\u03b1), k$$ large, J. London math. soc., 20, 180, (1945) \u00b7 Zbl\u00a00061.18903 [10] Cartwright, M.L.; Littlewood, J.E., Some fixed point theorems, Ann. math., 54, 1, (1951) \u00b7 Zbl\u00a00058.38604 [11] Charpentier, M., Sur quelques propri\u00e9t\u00e9s des courbes de M. Birkhoff, Bull. soc. math. France, 62, 193, (1934) \u00b7 Zbl\u00a00010.37701 [12] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, (), 6 [13] Handel, M., A pathological area preserving C^\u221e diffeomorphism of the plane, (), 163 \u00b7 Zbl\u00a00509.58031 [14] Kan, I., Strange attractors of uniform flows, Trans. AMS, 293, 135, (1986) \u00b7 Zbl\u00a00596.58025 [15] Kuratowski, C., Sur LES coupures irreducibles des plan, Fund. math., 6, 130, (1924) [16] McDonald, S.; Grebogi, C.; Ott, E.; Yorke, J.A., Fractal basin boundaries, Physica D, 17, 125, (1985) \u00b7 Zbl\u00a00588.58033 [17] Smale, S., Diffeomorphisms with many periodic points, (), 63 \u00b7 Zbl\u00a00142.41103 [18] Smale, S., Differentiable dynamical systems, Bull. amer. math. soc., 73, 747, (1967) \u00b7 Zbl\u00a00202.55202 [19] Williams, R.F., Expanding attractors, Publ. math. IHES, 43, 169, (1974) \u00b7 Zbl\u00a00279.58013 [20] Williams, R.F., One-dimensional nonwandering sets, Topology, 6, 473, (1967) \u00b7 Zbl\u00a00159.53702 [21] Williams, R.F., The structure of Lorenz attractors, Publ. math. IHES, 50, 101, (1979) \u00b7 Zbl\u00a00484.58021 [22] Yoneyama, K., Theory of continuous sets of points, Tohoku math. J., 11-12, 43, (1917) [23] Z.-P. You, E. Kostelich and J.A. Yorke, Computing stable and unstable manifolds, preprint. \u00b7 Zbl\u00a00874.58053\nThis reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.","date":"2021-02-24 18:38:57","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8978434801101685, \"perplexity\": 4697.50811472084}, \"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\/1614178347293.1\/warc\/CC-MAIN-20210224165708-20210224195708-00091.warc.gz\"}"}
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INFORMING THE LONG ISLANDER WITH NEWS AND INFORMATION MAINSTREAM MEDIA WON'T™
Clement: Roundup on State of Local News; Encapsulated by Preponderance of Conversations Throughout 2015
December 28, 2015 By Jaci Clement
LONG ISLAND, NY – To provide proper send-off to 2015, here's a roundup on the state of local news. We [Fair Media Council] simply encapsulated the preponderance of media conversations we had throughout the year. They're listed below, in no particular order other than stream of consciousness. The list is based on what people wanted to know most about when it comes to what's happening inside news here, there and everywhere.
1. It's here. Native advertising has arrived in newsrooms around the country … and no one really cares. Editors who should be up in arms about brands (read: PR agencies) working closely with journalists to develop what are supposed to be news stories are instead embracing the concept. Well, all except the WSJ's managing editor. Perhaps all the others are:
too tired of fighting the fight.
too terrified of not getting a paycheck.
too inexperienced to know they are, in fact, selling their souls to the devil.
Aligning news with advertisers is nothing new, it just has never really worked. In a previous life, native advertising was known as an advertorial (as in, part advertising, part editorial copy). Respected publications didn't run advertorials or, at least, not consistently. When they broke down and did, the worthy newsrooms would insist the content carried the label of 'advertorial' across the top, in an attempt to distance themselves from the crime that was about to be committed. Often, an advertorial would have a box around it, as an added security measure to prevent the advertorial from infecting the real news. Sometimes, it would even be in a different typeface, in the hopes that a reader would know something was radically amiss when what they were reading switched from a serif to a sans serif type.
Alas, today's native advertising comes with no such warning labels or subtle clues. It does nothing more than masquerade as news stories, which means it's up to the news consumer to know the diff. Here's the problem: Most don't. Net result? World. Hell. Handbasket.
2. Take me with you. Everyone is pushing mobile as the future, and that's fine, but it's only exacerbating our love-hate relationship with our phones. We want to get the news delivered faster than it actually occurs, but we really don't want to read too much about it.
YEAH, WE KNOW. YOU GET YOUR NEWS FROM TWITTER.
3. The great divide. Finally, after far too many years, it's cool to have a news program that doesn't look, act or talk like every other news program. Remember post-Sept. 11, when every newscast looked the same? Remember the crawl across the screen, meant to bring you breaking news, but vastly outstayed its welcome? Soon the crawl was giving you the time and temperature next to the terror level alert (which always stayed the same). Some overly ambitious news directors thought it a good idea to stack the crawls, so viewers were forced to read three lines of moving copy while trying to pay attention to the broadcast itself.
Now, however, things are getting nicely complicated. WPIX is launching a 6.30 p.m. local newscast in a time slot devoted to national news shows. WABC's 4 p.m. newscast provides local news before other channels go local at 5 p.m. WNBC is beefing up its investigative team. WWOR stopped its news format to give the TMZ version of what's happening in New Jersey. Whether these changes are good or bad makes for great debate over coffee, but compared to where we were a decade ago, viewers have more choices.
4. Waiting for the Fat Lady to Sing. Cable is a thing of the past and nothing signals that stronger than Cablevision's proposed sale to Altice. Smart of the Dolans to step aside when the industry they basically invented is now considered obsolete. But this sale to a company known for draconian measures when it comes to budget cuts doesn't bode well for the journalistic futures of News12 and Newsday. The proposed sale isn't slated to happen until second quarter, 2016 and Altice needs to borrow a boatload of moolah to make it happen. So, for now, we're taking a wait-and-see approach — and enjoying the rumors swirling over who may actually end up owning the news that owns Long Island.
5. The Disappearing Act of the Long Island Press. It's sorta still around, don't get me wrong, but it stopped print publishing in 2014. It's just that people are beginning to notice.
6. Twitter, Voice of God. People have gotten into the habit of saying they get their news from social media. Yeah, we know. You get your news from Twitter. But here's the thing: No, you don't. Twitter is just a delivery channel, not a content creator. What Twitter is putting in your newsfeed is coming from the media outlets you chose to follow. You know, the ones you keep referring to as dead. So please, stop that. It's just not nice.
7. Enough with Your Opinion, Let's Talk about My Opinion. The purported increased bias of news is what fills our inbox throughout the year. By random observation, it's funny how people complain about media bias, but will put on a full-throttle defense when the bias expressed matches their own. But here's the thing: Yes, there's a plethora of talk shows out there — and by plethora, we do mean excess — but those are supposed to be opinion drivers. They. Are. Biased. And. They. Are. Meant. To. Be. What's hard for viewers and listeners to deal with when it comes to talk formats is, even though the shows focus on what's in the news, they are not actually news programs. Our advice? Go to the gym.
8. When Local Goes Bad. Local broadcast news likes to show its local. That's its thing. But it actually hurts its own thing when it tries to show it's everywhere. Like when it leads the news with the random house fire or one-off crime. Now, to be clear: Public safety issues should top a newscast. But these aren't public safety issues as much as they are a person safety issue. Put another way: 30 million people in a broadcast coverage area don't care about a house burning in a neighborhood they've never heard of. Leave that coverage for the small local newspaper where that house fire is big news. Yes, we know timeliness is important, but so is relevancy. And when the news is not relevant to the majority of your viewers, they switch it off. And rightly so.
9. It's About the Few. Business news. It's a niche thing. It's never been about mass appeal and never will. Its specialized coverage is one of the things that makes it valuable. And, the last thing you want — trust us on this — is a mainstream media, general assignment reporter trying to report on financials.
10. This Death is Greatly Exaggerated. Newspapers are not dead. Neither is newspapering. Printing on dead trees? Well, that may be all but dead, but the paper thing is less important because newspapers using technology can deliver you a story while it's still relatively fresh. More to the point: You don't want newspapers to disappear. You need news for all sorts of things, not the least of which is the fact America would not be America without them.
11. As Local As Local News Gets? Here's something bubbling to the surface: Newsday's cut a deal with Anton Community Newspapers, to provide news on the hyperlocal level. It's currently in beta testing mode in a few zip codes on Long Island. If the results prove positive, then you need to wonder what happens next to a newspaper that relies on AP reports for national and international news and is now outsourcing for local content.
Jaci Clement is CEO of the Fair Media Council. Reach her at jaci@fairmediacouncil.org
FEATURED PHOTO: Jaci Clement, CEO of the Fair Media Council, at the 2012 FMC Folio Awards, April 20, 2012, Crest Hollow Country Club in Woodbury, N.Y.
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\section{Introduction}
Robotic systems are used in people's daily lives. They occur in various contexts, such as industry and manufacturing \cite{ifr:2020, Ross:2017}, military \cite{nath:2014, springer:2013}, household \cite{Prassler:2008, xu:2014}, and others \cite{bruno:2016}. Robotic systems can be classified as single robot or multi-robot. In this paper, we mainly focus on finding the optimal motion of a single robot in a dynamic environment.
Some of the problems of moving robots are path planning, collision avoidance, and map exploration, which are studied independently in the literature. In particular, when a robot is introduced to a new environment and given the task of moving an object from its position to another, the robot should explore the environment while moving to its final position and simultaneously detect and avoid all obstacles, including static and moving obstacles. To avoid obstacles, the robot should find a new route, but changing the robot's speed has been shown to be a better way to avoid collisions, \cite{Foka:2003}. Also, the movement of obstacles should be predicted, and the robot should adjust its movement according to the movement of obstacles nearby. In addition, the cost of the robot to complete the task should be minimized, i.e., the robot should complete the task in the shortest time and travel the shortest distance. Some studies consider a bounding box around an obstacle that a robot should not cross to avoid collisions, \cite{Xin:2018}. However, using bounding boxes is equivalent to reshaping obstacles and requires some estimation, and crossing a bounding box does not necessarily mean that the robot will collide with the obstacle, so the distance traveled and time may not be minimized. This is a complete list of the difficulties in studying robot's motion planning. As far as we know, there is no study that solves all these components simultaneously.
We have developed a mathematical model that can be used for exploration, path planning, and collision avoidance while predicting the obstacle's motion, controlling the robot's velocity, and ignoring bounding boxes. We show through designs that the proposed model minimizes the total distance traveled and the time to complete the task.
The paper is organized as follows. Section 2 discusses related work on motion planning and collision avoidance. Section 3 describes the mathematical model for optimal robot motion planning. And finally, Section 4 presents some conclusions and future lines of work.
\section{Related Works}
\cite{Hausler:2016} studied the motion planning of multiple vehicles in a simultaneous arrival problem with collision avoidance and minimization of total energy consumption by minimizing the temporal and spatial constraints. They consider the motions as a linear path between initial and final positions as straight lines. The total motion of each vehicle is converted into segments with elementary linear motions with constant velocities in each segment. For collision avoidance, they consider virtual circles around vehicles and obstacles. A vehicle may collide with obstacles or other vehicles if their virtual circles intersect. Then they translate their model into a multi-objective optimization where they find the maximum velocity vectors of the vehicles to find the optimal solution. In their formulation, the energy consumption is considered as the movement of the vehicles. Their model is a centralized optimization model and does not consider the communication between the vehicles. In their model, it is assumed that all vehicles arrive at the same final position. Figure \ref{fig1} shows an example with several vehicles arriving at different positions and colliding at one point. In this case, reducing the speed will not solve the problem. The vehicles have no priorities compared to each other. Since they are moving in a straight line at the collision point, all three vehicles will be stopped.
\begin{figure}[!h]\centering
\includegraphics[width=0.45\linewidth]{P1Copy.png}
\caption{Example of multiple vehicles arriving at different positions.}
\label{fig1}
\end{figure}
\cite{Gyenes:2018} studies the motion planning of multi-robots that ensures the avoidance of static and moving obstacles. In their method, they consider the motion of robots and obstacles in several smaller time periods. They propose an obstacle velocity prediction method that plans the next movement considering the positions and velocities of the robot and obstacles at the current time. They also proposed the safest obstacle velocity method, which not only finds the highest velocity in each time segment, but also finds the safest plan. This is because if only the fastest motion plan is considered, the robot may approach very close to the obstacles, which increases the collision risk due to the inaccuracy of the size, position, and velocity information. The velocity vector decision is made by choosing a velocity vector outside the set of velocity vectors of all moving obstacles, including the boundary information of all obstacles and vectors towards static obstacles and their boundaries. For this purpose, they consider the set of vectors called velocity-obstacle vector set starting from the position of the robot and choose the velocity vector outside this set. In their study, the prediction of the obstacle motion should be known in advance. The set of velocity obstacle vectors starting from the position of the robot does not work for the case when an obstacle is moving towards the robot. They have to include the set of vectors in the direction of the obstacle considering its expected position in the next time segment. Also, they have not considered the size of the robot. An example of this problem can be found in Figure \ref{fig3}.
\begin{figure}[!h]\centering
\includegraphics[width=0.55\linewidth]{P3Copy.png}
\caption{Example of determining the velocity vector $v$ of a robot based on the velocity of obstacles. $A$ is the robot, $B$ is a moving obstacle with velocity $v_{B}$. $VO$, the set of velocity obstacle vectors corresponding to obstacle $B$, is shown in the figure. One of the choices of velocity vector for the robot $v\notin VO$ is shown in the figure, but with this choice of $v$, the robot and the obstacle will collide in the next time segment, as shown in the figure.}
\label{fig3}
\end{figure}
\cite{Chu:2018} proposed polynomial interpolation for trajectory planning from initial to final state was used to find a continuous trajectory from initial to final state for each motion. To avoid obstacles, they decompose the trajectory from the initial to the final state into several shorter trajectory segments that do not intersect with obstacles, and use polynomial interpolation (polynomials of high degree or splines) to find smooth motions. The given points of the segments should be far away from obstacles to avoid collisions with obstacles at the endpoints. However, the polynomial path may intersect with obstacles regardless of how far the points are chosen. To find the points, we could select the points using Dijkstra's algorithm by considering the workspace as a grid.
In \cite{Yao:2020} the workspace is divided into grids. And the occupancy spaces of robots are identified as $0$ and $1$ values in their three-dimensional matrix representations. The movement of robots leads to changes in the values of their corresponding matrices, and the new matrices are compared to find possible collisions. Their method depends on the size of the grid. If the grid size is large, the current states of the robots can be considered as collisions, but if the grid size is smaller, they cannot.
\cite{Park:2020} studied the collision probability of two objects with uncertainty in their positions. They represented the uncertainty in the positions of the objects with non-Gaussian forms. They claimed that their proposed collision probabilities represent tight bounds on the convex shape (the safe region around the objects), and they used this for motion planning. To deal with the noise due to uncertainties, they used sensory information. The uncertainties of the objects are due to their geometric representation. To do this, they determine the probability distributions of the two objects and find the convolution distribution when one of the objects shifts. This creates the probability regions around the objects, which help to avoid collisions. They find the region in the Minkowski addition of two regions around the objects that have zero probability in the convolution distribution. They considered the truncated Gaussian mixture models for errors that increase the region around the objects and expand the collision probabilities from it.
\cite{Chen:2021a} describes a method for a path planning algorithm to coordinate neighbouring robots and avoid collisions. The problem is translated into a mixed observed Markov decision process and an optimization problem is extracted. Two robots are adjacent if they are in such states that they can reach a fixed state by certain actions. To avoid collisions, actions that cause the robots to reach the same state should be avoided. In this case, robots with possible collisions will perform actions that do not put them in the same state, and the total reward together with the total negative cost are maximal.
\cite{Behrens:2020} described an optimization problem for assigning tasks to robots in a sequence of tasks and motions of robots such that there are no collisions between robots, the predefined set of constraints is satisfied, and overall makespan is minimized. In their method, tasks are divided into confined tasks (a task where a robot's action is limited to a small part of the workspace, e.g., grasping and placing) and extended tasks (a task where a robot's action is limited to a large part of the workspace, e.g., welding along a line). The main focus is on the optimization of the extended tasks. For a given set of tasks that satisfy the set of constraints during execution, each task has multiple starting positions (called degrees of freedom), and the time interval in which a robot can perform each task is measured. The workspace is partitioned to identify regions that a robot occupies during the execution of a task in a sequence of time frames by discretizing the time interval of the task into smaller successive intervals. For solving tasks and motion planning for extended tasks, the problem is translated into a constraint satisfaction problem, which is a type of optimization problem modeled with triples $(X,D,C)$, where $X$ is a set of variables, $D$ is a set of domains where parameters take values, and $C$ is a set of constraints. The solution is to assign values from $D$ to the variables $X$ such that the set of all constraints $C$ is satisfied. To solve the problem, the gradient method and steepest descent by Cauchy \cite{Goldstein:1962} is used (backtracking search method). The optimization model is viewed from three perspectives: task layer, robot layer, and collision-free plan. The time intervals considered in the paper depend only on the task and the dependence on the robots is not considered, i.e., the robots should be identical. Moreover, the size of the region and the discretization of the time interval are not fixed and can be either a short or a long interval. Depending on the choice of the interval size and the size of the domain, different solutions can be obtained. Moreover, in the backtracking search method, in order to obtain a solution for the first upper bound, a random selection seems to be made in the solution space, so that the values in the domains that satisfy the constraints are selected. But the steps to reduce the upper bounds are not described. And the solution completely depends on the selection and reduction of the upper bounds. Moreover, different upper bounds may lead to different solutions. Also, the robot dependency is skipped but should be considered since the navigation codes, time intervals, constraints, and active components are robot-dependent, and their values may change when switching from one robot to another. In \cite{Behrens:2020}, tasks are translated into ordered visit constraints originally defined in \cite{Behrens:2019}. Here, the tasks are considered confined, so the start and end locations are considered identical, and the robot configuration is the same at the start and end. However, the new modified version includes different locations and different configurations to include extended tasks. Moreover, collisions may occur between components of a single robot, which is not considered. In addition, collision avoidance depends on the size of the regions (voxelization sizes). When the region size is large, the robots can have intersections in the configuration spaces, but when the region size is smaller, they have empty intersections. This is consistent with the well-known result that there are always infinitely many other real numbers between two distinct real numbers, see \cite{Gaughan:1993}. This means that if the two robots are not connected at any point, there will always be a pixel size where the intersection of their voxalizations is empty.
\cite{Sunkara:2019} studied the collision avoidance of an object with arbitrary shape and a deforming object, and proposed a nonlinear model for collision avoidance. In this study, the robot and the obstacle can have arbitrary shapes. It needs a constant velocity when the shape of the obstacle changes. If the deforming object has acceleration in a certain direction, collisions may occur because the guidance method does not consider the acceleration of the deforming object. In the proposed method, the boundary of the object is described by descritization and it is assumed that in each segment of the boundary, all points have the same velocity. The method uses Lyapunov function to guide the robot to avoid collision with the deformed object. Thus, to avoid collisions, the system is assumed to be locally Lyapunov-stable \cite{Lyapunov:1992} and the velocity vector of the robot is always correlated with the velocity of the deformed objects according to the collision avoidance guidance. Since the formulation considers the local region of the object near the robot, there are some examples where the robot is surrounded by the deforming object without the possibility of avoiding a collision, see Figure \ref{fig2}. Thus, without knowing the global pattern of the deforming object, the local guidance can also cause a collision.
\begin{figure}[!h]\centering
\includegraphics[width=0.9\linewidth]{P2Copy.png}
\caption{Example of a robot surrounded by a deforming object. The deformation appears with time from left to right. After time $t_3$, the robot can no longer avoid the collision.}
\label{fig2}
\end{figure}
\cite{Wing:2020} studied motion planning and collision avoidance without predicting the velocity of moving obstacles. In this method, the authors consider a safe distance around the obstacle and take the velocities of the obstacle and the robot to determine the angular velocity of the robot to avoid a collision. The method is a geometric approach that maps the motions of all objects in 2D space. To avoid a collision, the direction of the robot's velocity changes to an angle that is either opposite to the direction of the obstacle's velocity (when the obstacle moves in front of the robot) or to the direction that corresponds to the edge of the obstacle and has the shortest distance from the line generated by the robot's original direction (when the obstacle moves toward the robot). In the proposed method, the velocity of the obstacles should be measured at all times and sudden changes in the direction of the obstacles are not considered. Since the safety distance around all objects is considered, the method does not focus on the minimum total distance. Also, the method should consider the size of the robot, since larger robots may need a larger change in angular velocity than smaller robots to avoid collisions.
\cite{Zhang:2021} uses a topological approach to cover all objects with convex sets. In this way, the collision avoidance motion becomes a smooth function instead of a discrete function. The method penalizes the generated trajectory, which helps to find the least disturbing trajectory when a collision cannot be avoided. The method uses the property of the mathematical concept of compact set, which is defined as a set where each cover has a finite subcover, \cite{willard:2004}. This concept is purely theoretical and there is no unique way to find finite subcovers. Even for a given cover, the set of finite subcovers cannot be uniquely identified. After finding a finite cover, the union of all elements forms a bounding box around the robot. However, such a finite cover does not mean that the area covered by the union is minimal.
\cite{Hajiloo:2021} studied the threshold for direction change, braking, and acceleration of an autonomous vehicle with the minimum distance to obstacles that suddenly appear in the path of the vehicle, so that the vehicle remains stable. In this study, the friction of the road is considered as an important parameter that contributes to the stability of the vehicle motion and the controller handles the motion, angle and braking of all wheels. They translate the motion into a dynamical system and solve the optimal planning of the motion of all wheels in case an obstacle suddenly appears by changing the acceleration of each wheel.
In \cite{Nakamura:2020}, the authors assume that all vehicles send their local status and planned trajectory to a server, and the server determines whether there is a possible collision in a group of vehicles within a certain time. Then, the server sends a modified trajectory of the vehicles to avoid collisions. Their proposed method is a centralized trajectory planning method that reduces the computation time for collision detection and avoidance by removing vehicles without possible collisions. They assume that each vehicle has a rectangular area around it and consider a collision when the rectangular area around a vehicle intersects with the road boundary, obstacles, and the rectangular areas of other vehicles. And to avoid collision, they consider the change of acceleration vector of vehicles in case of possible collision. The proposed method considers the rules of passing a road, distinguishes the case of intersections and roundabouts, and treats them independently. The method of finding a rectangular box around a vehicle can only be applied to vehicles with regular shape and cannot be generalized to all vehicles with irregular shape. In this method, they find the center of gravity of the vehicle and create a rectangle by adding half the sizes of the length and width of the vehicle shape and assume that the resulting rectangle surrounds the vehicle. Figure \ref{fig4} shows an example of when the created rectangular area does not surround the vehicle. In these cases, part of the vehicle may be outside the rectangular area, which may intersect with other vehicles but is not considered in the collision avoidance.
\begin{figure}[!h]\centering
\includegraphics[width=0.8\linewidth]{P4Copy.png}
\caption{Example of irregularly shaped vehicles. The rectangular area created does not completely cover the vehicle. And there is a possibility of collision between the vehicles.}
\label{fig4}
\end{figure}
\cite{Rakita:2021} studied path planning with collision avoidance using combined graph theoretic and probabilistic approaches. In the proposed method, the workspace is divided into an obstacle space and an obstacle-free space. Then, first, the initial position is connected to the final position by a straight line. When the straight line intersects with the obstacle space, points near the first obstacle are randomly selected and a tree is created connecting the initial point with the selected points. Then select a point from the sample where the straight line from its position to the final point does not intersect the first obstacle, and select it as the first position of the trajectory from the initial point to the final point. This method only works if the obstacle space is completely known and all obstacles are static. If the robot has to explore the environment to detect obstacles and if the obstacles can move, the method does not work. The method also does not consider minimising the total distance the robot travels. See Figure \ref{fig5} for an example of when the shortest path that avoids collisions is longer than the path determined using random sampling.
\begin{figure}[!h]\centering
\includegraphics[width=1\linewidth]{P5Copy.png}
\caption{The shortest path cannot be determined by random sampling near the obstacle region. The shortest path (in blue) is 18, but the path by random sampling (in red) is 22.}
\label{fig5}
\end{figure}
\cite{Hai:2021} studied motion planning and collision avoidance for multi-robot in dynamic environment. The proposed method attempts to answer the question of what is the collision-free trajectory of each robot when the future goal of the robots is known. The proposed method is a decentralized model where the prediction of robot motion and behavior is learned through demonstration using a centralized sequential planner (based on recurrent neural networks (RNNs)). In the proposed model, by design, there is no communication between the robots and uncertainties in the motions of obstacles and robots are not considered.
The paper \cite{Fox:1997} described a simple geometrical model of the motions. The authors studied motion planning and collision avoidance by minimizing the travel distance and maximizing the speed of the robot. Their method uses what they call dynamic windows, which is a set of velocities that the robot can reach within a short period of time, allowing it to safely reduce or accelerate when it detects a potential collision. In this method, the robot's velocity is considered to be a constant value in each time segment, and it is assumed that the robot is the center of a circle whose radius is equal to the distance that the robot can travel at its current velocity (in \cite{Fox:1997}, they immediately switched from circle to rectangle). To model the robot's motion, angular acceleration was considered to allow the robot to change direction. Then, the velocity and angular acceleration are approximated while the position of the robot is updated using the information obtained from its wheels. In each time segment, the robot finds the rectangular area around the robot in which it can move. The area is obtained from the maximum speed that the robot can reach considering its predicted velocity and the angular acceleration of the robot. Then, each point of the obtained rectangular area is assigned three weights for the distance to an obstacle, the distance to the goal, and the velocity of the robot to reach that point, where the weight for the distance to an obstacle is higher for points farther from the obstacle, higher for points closer to the goal, and higher for points farther from the robot. The main optimization problem given in \cite{Fox:1997} is
$$G(v,w)=\sigma(\alpha.heading(v,w)+\beta.dist(v,w)+\gamma.vel(v,w)),$$
where $v$ is the velocity at which the robot moves straight, $w$ is the angular velocity at which the robot changes direction, $heading$ is the distance from the robot's next position to its final position, $dist$ is the distance from the robot's next position to the obstacle, and $vel$ is the speed at which the robot moves to its next position. The movement to the selected point with the highest weights may collide with an obstacle, see Figure \ref{f2}. Even if they ignore the history of the area under study and include only the points with the highest weights, their proposed method may enter an infinite loop of moving back and forth, for example, in Figure \ref{f3} and the second time segment, the optimal point to move to without considering history is the point closest to the obstacle, while the next time segment is the point where the robot was. It is also assumed that the obstacles are static.
\begin{figure}[!h]\centering
\includegraphics[width=1\linewidth]{p2p.png}
\caption{The robot moves to the points with the highest weights in two time segments and in the last time segment the movement to the point with the highest weight collides with the obstacle. The dashed area is the area that is not explored for possible collisions. The red circles are the points with the highest weights.}
\label{f2}
\includegraphics[width=0.5\linewidth]{p3p.png}
\caption{The robot moves infinitely back and forth to the point with the highest weight.}
\label{f3}
\end{figure}
\cite{Rosmann:2012} and \cite{Rosmann:2013} have studied motion planning with collision avoidance. In these methods, the path from the current position of the robot to the final point is considered as an elastic band, where after detecting each obstacle, the path is replaced by a curve from the current position of the robot to the final point that does not intersect with the obstacle, see Figure \ref{f4}. This method helps to improve the method in \cite{Fox:1997} by incorporating history (since the information about the band is stored in each time segment), avoiding an infinite loop, and avoiding scenarios where the robot moves through an obstacle by moving straight forward, see Figure \ref{f2}. The method in \cite{Rosmann:2012} does not optimize the robot's velocity along the path, and this problem is solved in \cite{Rosmann:2013}. The authors in \cite{Rosmann:2013} create a graph for the curved path by using the positions on the curved path as nodes and obtaining the edges by the sequential order of the positions on the path, taking into account the time differences in moving from one position to another. This time helps to change the acceleration of the robot and control the translational and angular velocity of the robot to further minimize the time to reach the final point. Since the methods in \cite{Rosmann:2012} and \cite{Rosmann:2013} use curves instead of direct lines, the magnitude of the distance traveled may not be minimal. Also, these methods do not account for moving obstacles.
\begin{figure}[!h]\centering
\includegraphics[width=0.5\linewidth]{p4p.png}
\caption{Intuition for elastic band. Starting from a straight line, at each time step where an obstacle is observed, a section of the line is replaced by a smooth curve.}
\label{f4}
\end{figure}
If the time steps are very short and the distance to the final position is long and there are many obstacles, the method requires a lot of memory to store the shape of the curve (all points on the curve).
The work \cite{Rosmann:2020} deals with motion planning with obstacle avoidance. They combine the Euclidean measure with a rotational component for motion planning and collision avoidance, where the goal is to minimize time. They model the trajectory as a function of time as a continuous nonlinear differential equation with bounded derivatives at each time point. Their goal is to minimize the cost of moving from the starting point to the end point. Since the robot can change direction (a rotation is applied) when moving to the next state (the next time step), the new operation should be defined to handle this non-Euclidean motion. For this purpose, in $2D$-space, they consider the motion at each time step as an operator in the special orthogonal group ($SO(2)$), which means that the velocity vector of the robot changes its direction with an angle. Since the rotation operation is only on a circle of radius $1$, they should use the special orthonormal group $\mathbb{S}^1$ instead of $SO(2)$. Later, they normalize the angle that maps the rotation to a circle of radius $1$. One of the constraints is that the dynamic error of the system should be small. In their formulation, the value of the error is considered to be equal to $0$ as one of the constraints. Moreover, for the finite difference kernel between two successive states, they used the implicit second-order Runge-Kutta (Crank-Nicolson) method, which is the average of the values of the state function on the two states. This can be replaced by the higher order implicit Runge-Kutta method to better describe the dynamics of the robot motion. If we use only the second-order difference, the effects of the motions at later time steps in the future are ignored. To avoid collisions, they specify the minimum distance between the robot and an obstacle and feed it as a constraint to find the region in which the robot can move without collisions while maintaining the specified minimum distance to obstacles. This is equivalent to considering a bounding box around all obstacles.
\section{Model}
The idea is to plan the motion of a robot in a dynamic environment with both deterministic and non-deterministic objects, creating a safe zone around all objects to avoid collisions, predicting the motions of non-deterministic objects, and taking into account the dimension of the objects. The main idea is to discretize all motions, define safe zones around all objects based on velocity vector probabilities, and find an optimal direction and velocity of the robot at each time step with the lowest cost (minimum travel distance at minimum time). To formulate the model, we use the following:
\begin{itemize}
\item\textbf{Object}: It is the set of all robots, humans, and obstacles that either change their position at a given time step or maintain their initial position at all times. Objects are identified as $i\in\{1,\ldots,N\}$.
\item\textbf{Map}: It is the environment $Env$ where all objects and the robot are located, and their movements are observed or predicted.
\item\textbf{Robot velocity limit}: It is the maximum velocity value of the robot, denoted by $\mathrm{v}$.
\item\textbf{State of an object}: For object $i$, at time step $t$, it is a tuple
\begin{equation}\label{eq:eq1}
(orien_t^i, dist_t^i,d_{v^i_t},P(d_{v^i_t},\-),v^i_t,P(v^i_t,\-)),
\end{equation}
where
$$(orien_t^i, dist_t^i)\in[0,2\pi)\times\mathbb{R}_{\geq0}$$
are the polar coordinates of the object $i$ at time $t$, $v^i_t\in\mathbb{R}_{\geq0}$ is the expected velocity value, $d_{v^i_t}\in[0,2\pi)$ is the expected direction of the object, $P(d_{v^i_t},\-)$ is the probability density function of the direction error of the object $i$ at time step $t$, and $P(v^i_t,\-)$ is the probability density function of the error of the velocity value of the object $i$ at time $t$.
\item\textbf{Deterministic object}: It is such an object whose state at each time step $t$, is
\begin{equation}\label{eq:eq2}
(orien_t^i, dist_t^i,d_{v^i_t},1,v^i_t,1).
\end{equation}
Obstacles and objects whose motions are known at each time step are deterministic objects.
\item\textbf{Non-deterministic object}: It is an object whose state at a given time step $t$, is given by \eqref{eq:eq1} with either $P(v^i_t,\-)\neq1$ or $P(d_{v^i_t},\-)\neq1$.
\item\textbf{Probability density function $P(v^i_t,\-)$}: Since for the cases where the expected velocity value is $v^i_t > 0$, the actual velocity should be $v^i_t+\Delta v$, this means that the object $i$ has a slightly higher or lower velocity than the expected velocity value ($\Delta v$ as an error). To increase the accuracy of the expected (predicted) velocity, the $\Delta v$ value should be close to $0$, with the same probabilities of higher or lower velocity. Now, if $v^i_t=0$, $\Delta v\geq0$. Therefore, $\Delta v$ should be considered as a truncated normal distribution in the interval $[-v_t^i,\infty)$, with $\mu=0$ and $\sigma_v > 0$. So
$$P(v^i_t,x)=\frac{\sqrt{2}}{\sqrt{\pi}\sigma_v}\frac{ \exp(-\frac{x^2}{2\sigma_v^2})}{1-\erf(\frac{-v_t^i}{\sigma_v})},$$
where $\erf$ is the Gaussian error function:
$$\erf(x)=\frac{2}{\sqrt{\pi}}\int_{0}^x\exp(-t^2)dt.$$
\item\textbf{Probability density function $P(d_{v^i_t},\-)$}: Since for the cases where the expected direction is $d_{v^i_t}\in[0,2\pi)$, the actual direction should be $d_{v^i_t}+\Delta\theta$, which means that the object $i$ has a slight angular deviation from the expected direction ($\Delta\theta$ as error), where to increase the accuracy of the predicted direction $\Delta\theta$ should be close to $0$, with the same probabilities for clockwise and counter-clockwise angles. If the object $i$ does not move at time step $t-1$, the motion at time step $t$ could be in any direction with uniform distribution $P(d_{v^i_t},x)=\frac{1}{2\pi}$, where $x\in[0,2\pi)$. If the expected direction is $d_{v^i_t}\in[0,2\pi)$, the error parameter $\Delta\theta$ should be a truncated normal distribution in the interval $[-\pi,\pi)$, with $\mu=0$ and $\sigma_d > 0$. So
$$P(d_{v^i_t},x)=\frac{\sqrt{2}}{\sqrt{\pi}\sigma_d}\frac{\exp(-\frac{x^2}{2\sigma_d^2})}{\erf(\frac{\pi}{\sigma_d})-\erf(\frac{-\pi}{\sigma_d})}.$$
\end{itemize}
The probability density functions $P(d_{v^i_t},\-)$ and $P(v^i_t,\-)$ describe the probability models for the errors in direction and velocity value. The magnitude of the errors in direction and velocity value at time $t$ are independent of the values at all other time steps.
\begin{itemize}
\item\textbf{State of the robot}: For the robot $r$, at time step $t$, it is a tuple
$$(orien_t^r, dist_t^r,d_{v^r_t},1,v^r_t,1),$$
where $(orien_t^i, dist_t^i)$ is the polar coordinate of the robot at time step $t$, $v^r_t\in\mathbb{R}_{\geq0}$ is the decision for the velocity value of the robot, $d_{v^r_t}$ is the decision for the direction of the robot at time step $t$.
\item\textbf{Safe zone}: It is an area around an object at time step $t$ that the robot cannot visit because of the risk of collision. The velocity value, direction, and size of the object are taken into account.
To find the safe zone for object $i$, at time $t$ in state
$$(orien_t^i, dist_t^i,d_{v^i_t},P(d_{v^i_t},\-),v^i_t,P(v^i_t,\-)),$$
find the errors of the angle $\Delta\theta$ and the velocity value $\Delta v$. Find the points $A$ and $B$, where the point $A$ is the last point (sometimes a set of points) of the object, so that when considering parallel lines to
$$\max\{d_{v_t^i}+\Delta\theta,d_{v_t^i}-\Delta\theta\}$$
and by increasing the width of the origin of the lines, the object and the line intersect, and after slightly increasing the width of the origin of the line, the line and the object do not intersect. And the point $B$ is the last point (sometimes a set of points) of the object, so that when parallel lines are viewed at
$$\min\{d_{v_t^i}+\Delta\theta,d_{v_t^i}-\Delta\theta\}$$
and by decreasing the width of the origin of the lines, there will be intersections between the object and the line, and after we decrease the width of the origin of the line a little, the line and the object will not intersect, see Figure \ref{lpoint}.
\begin{figure}[!h]\centering
\includegraphics[width=0.8\linewidth]{lastpoints.png}
\caption{Intuition for the method of determining the last points $A$ and $B$, using parallel lines to the directions of the maximum and the minimum expected velocity of the object.}
\label{lpoint}
\end{figure}
Now we should find either the curve $f(A,B)$ (the curve of the object $i$ from $A$ to $B$) and $g(B,A)$ (the curve of the object $i$ from $B$ to $A$) or their respective convex hulls enclosing $f(A,B)$ and $g(B,A)$. Then find the coordinates $A_c$ and $B_c$, respectively, for the motion of the object about the direction
$$\max\{d_{v_t^i}+\Delta\theta,d_{v_t^i}-\Delta\theta\}~~\text{and} ~~\min\{d_{v_t^i}+\Delta\theta,d_{v_t^i}-\Delta\theta\}$$
with scale $v_t^i+\Delta v$. Since we are working with a polar system, $g(B,A)$ is transformed with this scale to be
$$g_c(B_c,A_c)=(v_t^i+\Delta v)*Map(g(B,A))$$
and $f(A,B)$ is transformed to
$$f_c(A_c,B_c)=(v_t^i+\Delta v)*Map(f(A,B)),$$
where for each
$$\alpha\in[\min\{deg(A),deg(B)\},\max\{deg(A),deg(B)\}],$$
with $deg(X)$ is the degree of the point $X$ on the boundary of the object $i$ to the centroid of the object $i$ in $[0,2\pi)$, the mappings $Map(g(B,A))$ and $Map(f(A,B))$ can be found as functions of $\alpha$ as follows:
First find $\alpha_{\max}(h(X,Y))$ and $\alpha_{\min}(h(X,Y))$:
\begin{align*}
\alpha_{\max}(h(X,Y))=&\max\{\beta\mid dist(h(X,Y))\mid_{[\beta,\max\{deg(X),deg(Y)\}],0)} \\
&~~\text{ is strictly decreasing.}\}
\end{align*}
and
\begin{align*}
\alpha_{\min}(h(X,Y))&=\min\{\beta\mid dist(h(X,Y))\mid_{[\min\{deg(X),deg(Y)\},\beta],0)} \\
&~~\text{ is strictly increasing.}\}.
\end{align*}
Then we define for $Z\in h(X,Y)$:
\begin{align*}
&\Gamma_{1,\max}(Z,h(X,Y))=\\
&\left\{\begin{array}{ll}
\max\{h(X,Y)\mid_{\beta}\mid\beta\in[\alpha_{\min}(h(X,Y)),deg(Z)]\},°(Z)\geq\alpha_{\min}(h(X,Y))\\
h(X,Y),&otherwise,
\end{array}\right.&
\end{align*}
\begin{align*}
&\Gamma_{2,\max}(Z,h(X,Y))=\\
&\left\{\begin{array}{ll}
\max\{h(X,Y)\mid_{\beta}\mid\beta\in[deg(Z),\alpha_{\max}(h(X,Y))]\},°(Z)\leq\alpha_{\max}(h(X,Y))\\
h(X,Y),&otherwise,
\end{array}\right.
\end{align*}
\begin{align*}
&\Gamma_{1,\min}(Z,h(X,Y))=\\
&\left\{\begin{array}{ll}
\min\{h(X,Y)\mid_{\beta}\mid\beta\in[\alpha_{\min}(h(X,Y)),deg(Z)]\},°(Z)\geq\alpha_{\min}(h(X,Y))\\
h(X,Y),&otherwise,
\end{array}\right.
\end{align*}
and
\begin{align*}
&\Gamma_{2,\min}(Z,h(X,Y))=\\
&\left\{\begin{array}{ll}
\min\{h(X,Y)\mid_{\beta}\mid\beta\in[deg(Z),\alpha_{\max}(h(X,Y))]\},°(Z)\leq\alpha_{\max}(h(X,Y))\\
h(X,Y),&otherwise.
\end{array}\right.
\end{align*}
Then
$$Map(g(B,A))\mid_{\alpha}=\max\{\Gamma_{1,\max}(Z,g(B,A)),\Gamma_{2,\max}(Z,g(B,A))\mid Z\in g(B,A)\}$$
and
$$Map(f(A,B))\mid_{\alpha}=\min\{\Gamma_{1,\min}(Z,f(A,B)),\Gamma_{2,\min}(Z,f(A,B))\mid Z\in f(A,B)\}.$$
Intuitively, the maps $Map(g(B,A))$ and $Map(f(A,B))$ can be viewed as repeating local maxima and local minima within the interval of the first and last local maxima and the first and last local minima, respectively. Note that, by construction, the degree of points $A$ and $B$ with respect to the centroid of the object $i$ are the degrees between
$$\max\{d_{v_t^i}+\Delta\theta,d_{v_t^i}-\Delta\theta\}~~\text{and} ~~\min\{d_{v_t^i}+\Delta\theta,d_{v_t^i}-\Delta\theta\},$$
each with respect to the origin. So any point $X=(r,x)$, where $r$ is the radius to the centroid and $x$ is the angular degree, can be viewed as the vector $(r\cos(x),r\sin(x))$ with respect to the centroid in the Euclidean metric. Also, the current position of the centroid of the object $(R,\beta)$ can be viewed as the vector $(R\cos(\beta),R\sin(\beta))$ with respect to the origin in the Euclidean metric. This means that the coordinate of the point $X$ with respect to the origin in the Euclidean metric is the sum of two vectors:
\begin{align*}
(r\cos(x),r\sin(x))&+(R\cos(\beta),R\sin(\beta))=\\
&(r\cos(x)+R\cos(\beta),r\sin(x)+R\sin(\beta)),
\end{align*}
which can be easily converted into a polar metric as the point $Y=(\mathcal{R},\mathcal{\gamma})$, where
$$\mathcal{R}=\sqrt{(r\cos(x)+R\cos(\beta))^2+(r\sin(x)+R\sin(\beta))^2}$$
and
\begin{align*}
\gamma=&\tan^{-1}\left(\frac{r\sin(x)+R\sin(\beta)}{r\cos(x)+R\cos(\beta)}\right),\\
&\text{with}~~\gamma\in{[\min\{d_{v_t^i}+\Delta\theta,d_{v_t^i}-\Delta\theta\},\max\{d_{v_t^i}+\Delta\theta,d_{v_t^i}-\Delta\theta\}]}.
\end{align*}
So,
$$Map(X)=Y.$$
Note that the initial centroid has coordinates $(R,\beta)$ with respect to the origin, the new centroid of the object for the predicted position has coordinates $Predict\_Centroid=(v_t^i+\varepsilon,d_{v_t^i}+\delta)$ with respect to the initial centroid. Now suppose that $X$ is a point on the curve $f(A,B)$ (or on the curve $g(B,A)$), then its corresponding point $New\_X$ on the curve $f_c(A_c,B_c)$ (or on the curve $g_c(B_c,A_c)$) is obtained as follows:
\begin{align*}
New\_X&=Polar(Euclidean(Map(X),O)\\
&~~~~+Euclidean(Predict\_Centroid,(\mathcal{R},\gamma)))\\
&=Polar(Euclidean((\mathcal{R},\gamma),O)+Euclidean(X,(\mathcal{R},\gamma))\\
&~~~~+Euclidean(Predict\_Centroid,(\mathcal{R},\gamma)))
\end{align*}
which is a polar transformation of the sum of the vectors of the Euclidean coordinates of the points $X$ with respect to the origin and $Predict\_Centroid$ with respect to the initial centroid coordinate. See Figure \ref{figg1}.
\begin{figure}[!h]\centering
\includegraphics[width=0.5\linewidth]{P1.png}
\caption{Intuition for the method of determining the coordinates of each point of an object after predicting its motion. The dashed vector is the coordinate of interest, i.e., the coordinate of the point $X$ of the object $i$ after applying the predicted motion of the object. Dotted vectors are used to indicate parallel vectors.}
\label{figg1}
\end{figure}
Now, the safe zone of object $i$ at time step $t$ with state given by \eqref{eq:eq1} can be obtained by
$$SZ^i_{t+1}=\max\left\{\int_{d_{v^i_t}-\Delta\theta}^{d_{v^i_t}+\Delta\theta}(g_c(B_c,A_c)^2-f(A,B)^2),\int_{d_{v^i_t}-\Delta\theta}^{d_{v^i_t}+\Delta\theta}(f_c(A,B)^2-g(B,A)^2)\right\}.$$
See Figure \ref{figg2} for the intuition of obtaining the safe zone.
\begin{figure}[!h]\centering
\includegraphics[width=0.8\linewidth]{P2.png}
\caption{Intuition for the method of obtaining the safe zone. $g_c(B_c,A_c)$ ($g(B,A)$) is the boundary curve of the map of the object (the object) from $B_c$ ($B$) to $A_c$ ($A$), that is, the boundary curve of the map of the object (the object) to the right of Line 2 (Line 1). $f_c(A_c,B_c)$ is the boundary curve of the map of the object (the object) from $A_c$ ($A$) to $B_c$ ($B$), i.e., the boundary curve of the map of the object (the object) to the left of Line 2 (Line 1).}
\label{figg2}
\end{figure}
\item\textbf{The optimization objective}: Let the robot aims to move from point $A$ to point $B$. Let $Shortest\_Path (A,B)$ be the shortest path from $A$ to $B$ without considering moving objects and $|Shortest\_Path (A,B)|$ be the total distance of the shortest path and assume that the robot can travel the shortest path in time $t$. Let $act_t$ be the actual position of the robot at time step $t$. The actual position of the robot is a function of the initial position, the direction of the robot, and the velocity value of the robot. The positions are denoted as a sequence $P_0=A,P_1,\ldots,P_{T-1},P_{T}=B$. The objective is then to minimize the following
$$\min_{P_0=A,P_1,\ldots,P_{T-1},P_{T}=B~\text{ valid }}\sqrt{\left(\frac{T}{\mathrm{t}}\right)^2+\left(\frac{\left(\sum_{t=0}^T(\min_{P\in Shortest\_Path(A,B)}\{(P_t-P)^2\})\right)}{|Shortest\_Path(A,B)|^2}\right)^2},$$
where a path is considered valid if there are no collisions with other objects. Note that the time and distance are divided by the minimum time and length of the shortest path to remove scales.
\item\textbf{Robot direction and speed}: At time step $t$, find the union of all safe zones of all objects, $SZ_{t+1}=\bigcup_{i=1}^NSZ^i_{t+1}$. Then find the area $Env\setminus SZ_{t+1}$. For the direction of the robot, choose the point $Q\in Env\setminus SZ_{t+1}$ such that the length of the shortest path from $x$ to $B$ has the smallest value
$$Q=\argmin_{x\in Env\setminus SZ_{t+1}}\left\{|Shortest\_Path (x,B)|~\mid~Straight\_line (act_t,x)\cap SZ_{t+1}=\emptyset\right\},$$
where $Straight\_line (act_t,Q)$ is the straight line connecting the points $act_t$ and $Q$. Note that the velocity value of the robot is constrained to $\mathrm{v}$ and the point $Q$ may not be reached in one time step. Thus, the robot either moves toward point $Q$ at the maximum velocity $\mathrm{v}$ for one time step or it can travel to $Q$ at the velocity
$$v^r_{t+1}=\frac{|Shortest\_Path (act_t,Q)|}{\text{length of a time step}},$$
where $|Shortest\_Path (act_t,Q)|$ is a straight line.
So the state of the robot will be
$$(orien_{t+1}^i, dist_{t+1}^r,d_{v^r_{t+1}},1,v^r_{t+1},1),$$
where $d_{v^r_{t+1}}$ is the angle of the line $Straight\_line(act_t,Q)$ with the $x$-axis in gradient scale and
$$v^r_{t+1}=\min\left\{\mathrm{v},\frac{|Shortest\_Path (act_t,Q)|}{\text{length of a time step}}\right\}.$$
Note that after each time step, the current position of the robot and the shortest path should be updated. Since, by construction, for every two consecutive time steps $t$ and $t+1$ we have
$$0\leq|Shortest\_Path (act_{t+1},B)|\leq|Shortest\_Path (act_{t},B)|$$
the method converges.
The way the robot's direction works at each time step allows it to find the optimal path, which may use a different route than the initial shortest path from the initial position to the robot's target position.
\item\textbf{Next position}: If the state of the object $i$ at time step $t$ is given by \eqref{eq:eq1}, then the next position $(orien_{t+1}^i, dist_{t+1}^i)$ can be determined as follows:
$$\min\{orien_t^i,d_{v^i_t}\}\leq orien_{t+1}^i=\arctan(\frac{Y}{X})\leq\max\{orien_t^i,d_{v^i_t}\}$$
and
$$dist_{t+1}^i=\sqrt{X^2+Y^2},$$
where
$$X=dist_{t}^i\cos(orien_t^i)+v^i_t\cos(d_{v^i_t})$$
and
$$Y=dist_{t}^i\sin (orien_t^i)+v^i_t\sin (d_{v^i_t}).$$
\item\textbf{Velocity and direction pattern}: Suppose that at each time step $t$, the past $H$ time steps of the states of all $N$ objects are known. This means
$$(orien_s^i, dist_s^i,d_{v^i_s},P(d_{v^i_s},\-),v^i_s,P(v^i_s,\-)),~~s=t-H,\ldots,t-1.$$
We want to make a prediction of the directions $d_{v^i_s}$ and the velocity values $v^i_s$ for $s\geq t$. The following values can be natural choices:
\begin{itemize}
\item\textbf{Moving average}: For the future time step $t+k$, the velocity value of the object $i$ at time $t+k$ will be
$$v^i_{t+k}=mean(\{v^i_{s}\mid s=t+k-H,\ldots,t+k-1\})$$
and
$$d(v^i_{t+k})=mean(\{d(v^i_{s})\mid s=t+k-H,\ldots,t+k-1\}).$$
The advantage of this method is that the velocity values and directions change over time. However, it also has the disadvantage of becoming a linear function without updating if the number of time steps is long enough. This is a simple auto-regressive method for forecasting, see \cite{montgomery:2011}.
\item\textbf{Naive auto-regressive}:
For the future time step $t+k$, the velocity value of the object $i$ at time step $t+k$ is determined as follows: Let
$$W=(1,\frac{1}{2},\ldots,\frac{1}{2^{H-1}}).$$
Define
$$w=(w_{t+k-H},\ldots,w_{t+k-1}),$$
where $w_{t+k-u}=\frac{W_u}{\sum_{j=1}^{H}W_j}$ are $H$ weights for the past states.
Then
$$v^i_{t+k}=\sum_{j=1}^Hw_{t+k-j}v^i_{t+k-j}$$
and
$$d(v^i_{t+k})=\sum_{j=1}^Hw_{t+k-j}d(v^i_{t+k-j}).$$
Velocity value and direction are related to past velocity values and directions. However, past velocity values and directions must be weighted, i.e., the next velocity value and direction are more related to the first immediate past velocity value and direction than the velocity values and directions in the time steps before it.
The advantage of this method over the previous one is that the velocity values and directions change in a more realistic way over time. However, it also has the disadvantage that, without updating, the motion strategy is deterministic after a sufficiently long number of time steps, and also, the weights for the directions and velocity values are always the same all the time.
\item\textbf{Generalized auto-regressive}: For future time step $t+k$, the velocity value of object $i$ at time $t+k$ is determined as follows: Let
$$w^u=(w^k_{t+k-H},\ldots,w^k_{t+k-1}),$$
for $u=1,2$, be $H$ random variables in the interval $(0,1)$ such that for $u=1,2$, $\sum_{j=1}^Hw^k_{t+h-j}=1$ and
$$w^u_{t+k-H}\leq\ldots\leq w^u_{t+k-1}.$$
$w^u_{t+k-j}$'s are weights. Let
$$v^i_{t+k}=\sum_{j=1}^Hw^1_{t+k-j}a^i_{t+k-j}$$
and
$$d(v^i_{t+k})=\sum_{j=1}^Hw^2_{t+k-j}d(v^i_{t+k-j}).$$
The advantage of this method over the previous one is that the velocity values and directions change more realistically over time since the lists of probabilities are determined randomly. However, it also has the disadvantage that without additional information, some states may be invalid as future states of object $i$. This is a more general auto-regressive method for forecasting, see \cite{montgomery:2011}.
We could also introduce random noise at each step by adding weighted multipliers of white noise, see \cite{montgomery:2011}.
\end{itemize}
In all the above natural possibilities, the position of the object $i$ at time $t+k$ can be determined using the step (Next position).
\item\textbf{State update}: When at time step $t$ the states of all objects are observed, we need to update the states
$$(orien_t^i, dist_t^i,d_{v^i_t},1,v^i_t,1),~~ \forall i=1,\ldots,N.$$
The values $orien_t^i$ and $dist_s^i$ are updated by the current coordinate of the object $i$ at time $t$, and the direction and velocity value $d_{v^i_s}$ and $v^i_s$ will be updated by the current direction and velocity of the object $i$. Since the current values are observed, the probability densities of errors in the direction and velocity value of all objects can be avoided. Based on these new observations, all predictions for the states in the future should be updated using the methods described above by replacing the predicted states at time step $t$ with the actual values of the states at time step $t$. Note that if the current time step is $t$, this means that at time step $s$ with $s<t$, all states are already updated with the actual observed states.
\item\textbf{High risk scenarios (Run away)}: Let at time step $t$ $C_r(orien_t^r, dist_t^r)$ be the set of all points on the map that the robot can move from its current position, red circle in Figures \ref{f3p} and \ref{f4p}. Suppose that multiple objects, $i_1,\ldots,i_n$, are moving towards the robot such that
$$(orien_t^r, dist_t^r)\in SZ^{i_j}_{t+1},~~\forall j=1,\ldots,n.$$
This means that the sets
$$\left\{|Shortest\_Path (x,B)|~\mid~Straight\_line (act_t,x)\cap SZ_{t+1}=\emptyset\right\},~~\forall x\in Env\setminus SZ_{t+1}$$
are empty sets for all $x$. In such cases, the next state of the robot can be determined as follows:
\begin{itemize}
\item\textbf{With skip zone}: Assume that
$$D=C_r(orien_t^r, dist_t^r)\setminus\left(\bigcup_{j=1}^nSZ^{i_j}_{t+1}\right)\neq\emptyset.$$
The set $D$ is called the skip zone because if the robot moves to a point in $D$, it avoids possible collisions. For this case, find
$$Q=\argmin_{x\in D}\left\{|Shortest\_Path (x,B)|\right\}.$$
Note that the velocity value of the robot is constrained to $\mathrm{v}$ and the point $Q$ can be reached in a single time step. Thus, the robot moves towards the point $Q$ with the velocity value
$$v^r_{t+1}=\frac{|Shortest\_Path (act_t,Q)|}{\text{length of a time step}}\leq\mathrm{v}.$$
So the state of the robot will be
$$(orien_{t+1}^i, dist_{t+1}^r,d_{v^r_{t+1}},1,v^r_{t+1},1),$$
where $d_{v^r_{t+1}}$ is the angle of the line $Straight\_line(act_t,Q)$ with the $x$-axis in gradient scale, $(orien_{t+1}^i, dist_{t+1}^r)$ is the polar coordinate of $Q$, and
$$v^r_{t+1}=\frac{|Shortest\_Path (act_t,Q)|}{\text{length of a time step}}.$$
Figure \ref{f3p}, illustrate the scenario with skip zone.
\begin{figure}[!h]\centering
\includegraphics[width=0.8\linewidth]{P3.png}
\caption{Intuition to avoid collision, for the case with skip zone. The red bullet is the current position of the robot, the red vector indicates the maximum velocity value of the robot, and the red circle with the double dotted and dashed line indicates the set of positions that the robot can move with its limited velocity value. The blue dashed area is the skip zone $D$.}
\label{f3p}
\end{figure}
\item\textbf{Without skip zone}: Assume that
$$C_r(orien_t^r, dist_t^r)\setminus\left(\bigcup_{j=1}^nSZ^{i_j}_{t+1}\right)=\emptyset.$$
See figure \ref{f4p} for such a case.
\begin{figure}[!h]\centering
\includegraphics[width=0.8\linewidth]{P4.png}
\caption{The case without skip zone. The red bullet is the robot's current position, the red vector shows the robot's maximum velocity value, and the red circle with the double dotted and dashed line shows the set of positions the robot can travel to with its limited velocity value.}
\label{f4p}
\end{figure}
For this case, find the set
$$M=\argmin_{x\in C_r(orien_t^r, dist_t^r)}\left\{\sum_{j=1}^n\left(P(d_{v^{i_j}_t},x)+P(v^{i_j}_t,x)\right)\right\}.$$
The set $M$ is the set of all points that the robot can move to in a single time step with the least probability that objects $i_1,\ldots,i_n$ move to those points. Then, as before
$$Q=\argmin_{x\in M}\left\{|Shortest\_Path (x,B)|\right\}.$$
Note that the velocity value of the robot is constrained to $\mathrm{v}$ and the point $Q$ can be reached in a single time step. Thus, the robot moves towards the point $Q$ with the velocity value
$$v^r_{t+1}=\frac{|Shortest\_Path (act_t,Q)|}{\text{length of a time step}}\leq\mathrm{v}.$$
So the state of the robot will be
$$(orien_{t+1}^i, dist_{t+1}^r,d_{v^r_{t+1}},1,v^r_{t+1},1),$$
where $d_{v^r_{t+1}}$ is the angle of the line $Straight\_line(act_t,Q)$ with the $x$-axis in gradient scale, $(orien_{t+1}^i, dist_{t+1}^r)$ is the polar coordinate of $Q$, and
$$v^r_{t+1}=\frac{|Shortest\_Path (act_t,Q)|}{\text{length of a time step}}.$$
\end{itemize}
Note that throughout the formulation the robot is considered as a single point (centroid of the robot, $act_t$) at time step $t$. To generalize the formulation, assume that $OCP_t^r\subset Env$ are the positions in the environment that the robot occupies at time step $t$. Then all formulations involving the motion of the robot from $act_t$ to a point $w\in Env$ should be replaced by the mapping
$$\delta:OCP_t^r\rightarrow B(w,act_t,OCP_t^r),$$
which maps $act_t$ to $w$ with direction $d(v)$ (direction from $act_t$ to $w$) and velocity $v$ (the distance between $act_t$ and $w$ divided by the length of the time step), and $y\in OCP_t^r$ to the point $y_1\in B(w,act_t,OCP_t^r)$, where the line connecting $y$ and $y_1$ makes an angle $d(v)$ with the $x$-axis and the distance between $y$ and $y_1$ is equal to $v$. Then the whole formulation should be applied to the $(u,v)\ in OCP_t^r\times\delta(OCP_t^r)$, instead of only to $(act_t,w)$.
\end{itemize}
\section{Conclusion}
We have developed a simple model that allows optimal path planning in an unknown dynamic environment, minimizing the total distance the robot travels and the total time to complete a given task, while avoiding collisions. To avoid collisions, we consider no bounding box around obstacles to further minimize the distance traveled by the robot. For moving obstacles, we perform a simple prediction of the future motion of the obstacles and the motion of the robot changes accordingly. Over time, the robot updates the state of the moving objects to better estimate their future motion.
For the future work, to better estimate the next motion of the obstacles, we will replace the simple estimation with a better prediction method, such as more complex time series prediction and forecasting methods.
\section*{Acknowledgment}
This work was supported by operation Centro-01-0145-FEDER-000019-C4-Centro de Compet\^{e}ncias em Cloud Computing, cofinanced by the European Regional Development Fund (ERDF) through the Programa Operacional Regional do Centro (Centro 2020), in the scope of the Sistema de Apoio \`{a} Investiga\c{c}\~{a}o Cientif\'{i}ca e Tecnol\'{o}gica - Programas Integrados de IC\&DT. This work was supported by NOVA LINCS (UIDB/04516/2020) with the financial support of FCT-Funda\c{c}\~{a}o para a Ci\^{e}ncia e a Tecnologia, through national funds.
\bibliographystyle{unsrt}
\section{Introduction}
Robotic systems are used in people's daily lives. They occur in various contexts, such as industry and manufacturing \cite{ifr:2020, Ross:2017}, military \cite{nath:2014, springer:2013}, household \cite{Prassler:2008, xu:2014}, and others \cite{bruno:2016}. Robotic systems can be classified as single robot or multi-robot. In this paper, we mainly focus on finding the optimal motion of a single robot in a dynamic environment.
Some of the problems of moving robots are path planning, collision avoidance, and map exploration, which are studied independently in the literature. In particular, when a robot is introduced to a new environment and given the task of moving an object from its position to another, the robot should explore the environment while moving to its final position and simultaneously detect and avoid all obstacles, including static and moving obstacles. To avoid obstacles, the robot should find a new route, but changing the robot's speed has been shown to be a better way to avoid collisions, \cite{Foka:2003}. Also, the movement of obstacles should be predicted, and the robot should adjust its movement according to the movement of obstacles nearby. In addition, the cost of the robot to complete the task should be minimized, i.e., the robot should complete the task in the shortest time and travel the shortest distance. Some studies consider a bounding box around an obstacle that a robot should not cross to avoid collisions, \cite{Xin:2018}. However, using bounding boxes is equivalent to reshaping obstacles and requires some estimation, and crossing a bounding box does not necessarily mean that the robot will collide with the obstacle, so the distance traveled and time may not be minimized. This is a complete list of the difficulties in studying robot's motion planning. As far as we know, there is no study that solves all these components simultaneously.
We have developed a mathematical model that can be used for exploration, path planning, and collision avoidance while predicting the obstacle's motion, controlling the robot's velocity, and ignoring bounding boxes. We show through designs that the proposed model minimizes the total distance traveled and the time to complete the task.
The paper is organized as follows. Section 2 discusses related work on motion planning and collision avoidance. Section 3 describes the mathematical model for optimal robot motion planning. And finally, Section 4 presents some conclusions and future lines of work.
\section{Related Works}
\cite{Hausler:2016} studied the motion planning of multiple vehicles in a simultaneous arrival problem with collision avoidance and minimization of total energy consumption by minimizing the temporal and spatial constraints. They consider the motions as a linear path between initial and final positions as straight lines. The total motion of each vehicle is converted into segments with elementary linear motions with constant velocities in each segment. For collision avoidance, they consider virtual circles around vehicles and obstacles. A vehicle may collide with obstacles or other vehicles if their virtual circles intersect. Then they translate their model into a multi-objective optimization where they find the maximum velocity vectors of the vehicles to find the optimal solution. In their formulation, the energy consumption is considered as the movement of the vehicles. Their model is a centralized optimization model and does not consider the communication between the vehicles. In their model, it is assumed that all vehicles arrive at the same final position. Figure \ref{fig1} shows an example with several vehicles arriving at different positions and colliding at one point. In this case, reducing the speed will not solve the problem. The vehicles have no priorities compared to each other. Since they are moving in a straight line at the collision point, all three vehicles will be stopped.
\begin{figure}[!h]\centering
\includegraphics[width=0.45\linewidth]{P1Copy.png}
\caption{Example of multiple vehicles arriving at different positions.}
\label{fig1}
\end{figure}
\cite{Gyenes:2018} studies the motion planning of multi-robots that ensures the avoidance of static and moving obstacles. In their method, they consider the motion of robots and obstacles in several smaller time periods. They propose an obstacle velocity prediction method that plans the next movement considering the positions and velocities of the robot and obstacles at the current time. They also proposed the safest obstacle velocity method, which not only finds the highest velocity in each time segment, but also finds the safest plan. This is because if only the fastest motion plan is considered, the robot may approach very close to the obstacles, which increases the collision risk due to the inaccuracy of the size, position, and velocity information. The velocity vector decision is made by choosing a velocity vector outside the set of velocity vectors of all moving obstacles, including the boundary information of all obstacles and vectors towards static obstacles and their boundaries. For this purpose, they consider the set of vectors called velocity-obstacle vector set starting from the position of the robot and choose the velocity vector outside this set. In their study, the prediction of the obstacle motion should be known in advance. The set of velocity obstacle vectors starting from the position of the robot does not work for the case when an obstacle is moving towards the robot. They have to include the set of vectors in the direction of the obstacle considering its expected position in the next time segment. Also, they have not considered the size of the robot. An example of this problem can be found in Figure \ref{fig3}.
\begin{figure}[!h]\centering
\includegraphics[width=0.55\linewidth]{P3Copy.png}
\caption{Example of determining the velocity vector $v$ of a robot based on the velocity of obstacles. $A$ is the robot, $B$ is a moving obstacle with velocity $v_{B}$. $VO$, the set of velocity obstacle vectors corresponding to obstacle $B$, is shown in the figure. One of the choices of velocity vector for the robot $v\notin VO$ is shown in the figure, but with this choice of $v$, the robot and the obstacle will collide in the next time segment, as shown in the figure.}
\label{fig3}
\end{figure}
\cite{Chu:2018} proposed polynomial interpolation for trajectory planning from initial to final state was used to find a continuous trajectory from initial to final state for each motion. To avoid obstacles, they decompose the trajectory from the initial to the final state into several shorter trajectory segments that do not intersect with obstacles, and use polynomial interpolation (polynomials of high degree or splines) to find smooth motions. The given points of the segments should be far away from obstacles to avoid collisions with obstacles at the endpoints. However, the polynomial path may intersect with obstacles regardless of how far the points are chosen. To find the points, we could select the points using Dijkstra's algorithm by considering the workspace as a grid.
In \cite{Yao:2020} the workspace is divided into grids. And the occupancy spaces of robots are identified as $0$ and $1$ values in their three-dimensional matrix representations. The movement of robots leads to changes in the values of their corresponding matrices, and the new matrices are compared to find possible collisions. Their method depends on the size of the grid. If the grid size is large, the current states of the robots can be considered as collisions, but if the grid size is smaller, they cannot.
\cite{Park:2020} studied the collision probability of two objects with uncertainty in their positions. They represented the uncertainty in the positions of the objects with non-Gaussian forms. They claimed that their proposed collision probabilities represent tight bounds on the convex shape (the safe region around the objects), and they used this for motion planning. To deal with the noise due to uncertainties, they used sensory information. The uncertainties of the objects are due to their geometric representation. To do this, they determine the probability distributions of the two objects and find the convolution distribution when one of the objects shifts. This creates the probability regions around the objects, which help to avoid collisions. They find the region in the Minkowski addition of two regions around the objects that have zero probability in the convolution distribution. They considered the truncated Gaussian mixture models for errors that increase the region around the objects and expand the collision probabilities from it.
\cite{Chen:2021a} describes a method for a path planning algorithm to coordinate neighbouring robots and avoid collisions. The problem is translated into a mixed observed Markov decision process and an optimization problem is extracted. Two robots are adjacent if they are in such states that they can reach a fixed state by certain actions. To avoid collisions, actions that cause the robots to reach the same state should be avoided. In this case, robots with possible collisions will perform actions that do not put them in the same state, and the total reward together with the total negative cost are maximal.
\cite{Behrens:2020} described an optimization problem for assigning tasks to robots in a sequence of tasks and motions of robots such that there are no collisions between robots, the predefined set of constraints is satisfied, and overall makespan is minimized. In their method, tasks are divided into confined tasks (a task where a robot's action is limited to a small part of the workspace, e.g., grasping and placing) and extended tasks (a task where a robot's action is limited to a large part of the workspace, e.g., welding along a line). The main focus is on the optimization of the extended tasks. For a given set of tasks that satisfy the set of constraints during execution, each task has multiple starting positions (called degrees of freedom), and the time interval in which a robot can perform each task is measured. The workspace is partitioned to identify regions that a robot occupies during the execution of a task in a sequence of time frames by discretizing the time interval of the task into smaller successive intervals. For solving tasks and motion planning for extended tasks, the problem is translated into a constraint satisfaction problem, which is a type of optimization problem modeled with triples $(X,D,C)$, where $X$ is a set of variables, $D$ is a set of domains where parameters take values, and $C$ is a set of constraints. The solution is to assign values from $D$ to the variables $X$ such that the set of all constraints $C$ is satisfied. To solve the problem, the gradient method and steepest descent by Cauchy \cite{Goldstein:1962} is used (backtracking search method). The optimization model is viewed from three perspectives: task layer, robot layer, and collision-free plan. The time intervals considered in the paper depend only on the task and the dependence on the robots is not considered, i.e., the robots should be identical. Moreover, the size of the region and the discretization of the time interval are not fixed and can be either a short or a long interval. Depending on the choice of the interval size and the size of the domain, different solutions can be obtained. Moreover, in the backtracking search method, in order to obtain a solution for the first upper bound, a random selection seems to be made in the solution space, so that the values in the domains that satisfy the constraints are selected. But the steps to reduce the upper bounds are not described. And the solution completely depends on the selection and reduction of the upper bounds. Moreover, different upper bounds may lead to different solutions. Also, the robot dependency is skipped but should be considered since the navigation codes, time intervals, constraints, and active components are robot-dependent, and their values may change when switching from one robot to another. In \cite{Behrens:2020}, tasks are translated into ordered visit constraints originally defined in \cite{Behrens:2019}. Here, the tasks are considered confined, so the start and end locations are considered identical, and the robot configuration is the same at the start and end. However, the new modified version includes different locations and different configurations to include extended tasks. Moreover, collisions may occur between components of a single robot, which is not considered. In addition, collision avoidance depends on the size of the regions (voxelization sizes). When the region size is large, the robots can have intersections in the configuration spaces, but when the region size is smaller, they have empty intersections. This is consistent with the well-known result that there are always infinitely many other real numbers between two distinct real numbers, see \cite{Gaughan:1993}. This means that if the two robots are not connected at any point, there will always be a pixel size where the intersection of their voxalizations is empty.
\cite{Sunkara:2019} studied the collision avoidance of an object with arbitrary shape and a deforming object, and proposed a nonlinear model for collision avoidance. In this study, the robot and the obstacle can have arbitrary shapes. It needs a constant velocity when the shape of the obstacle changes. If the deforming object has acceleration in a certain direction, collisions may occur because the guidance method does not consider the acceleration of the deforming object. In the proposed method, the boundary of the object is described by descritization and it is assumed that in each segment of the boundary, all points have the same velocity. The method uses Lyapunov function to guide the robot to avoid collision with the deformed object. Thus, to avoid collisions, the system is assumed to be locally Lyapunov-stable \cite{Lyapunov:1992} and the velocity vector of the robot is always correlated with the velocity of the deformed objects according to the collision avoidance guidance. Since the formulation considers the local region of the object near the robot, there are some examples where the robot is surrounded by the deforming object without the possibility of avoiding a collision, see Figure \ref{fig2}. Thus, without knowing the global pattern of the deforming object, the local guidance can also cause a collision.
\begin{figure}[!h]\centering
\includegraphics[width=0.9\linewidth]{P2Copy.png}
\caption{Example of a robot surrounded by a deforming object. The deformation appears with time from left to right. After time $t_3$, the robot can no longer avoid the collision.}
\label{fig2}
\end{figure}
\cite{Wing:2020} studied motion planning and collision avoidance without predicting the velocity of moving obstacles. In this method, the authors consider a safe distance around the obstacle and take the velocities of the obstacle and the robot to determine the angular velocity of the robot to avoid a collision. The method is a geometric approach that maps the motions of all objects in 2D space. To avoid a collision, the direction of the robot's velocity changes to an angle that is either opposite to the direction of the obstacle's velocity (when the obstacle moves in front of the robot) or to the direction that corresponds to the edge of the obstacle and has the shortest distance from the line generated by the robot's original direction (when the obstacle moves toward the robot). In the proposed method, the velocity of the obstacles should be measured at all times and sudden changes in the direction of the obstacles are not considered. Since the safety distance around all objects is considered, the method does not focus on the minimum total distance. Also, the method should consider the size of the robot, since larger robots may need a larger change in angular velocity than smaller robots to avoid collisions.
\cite{Zhang:2021} uses a topological approach to cover all objects with convex sets. In this way, the collision avoidance motion becomes a smooth function instead of a discrete function. The method penalizes the generated trajectory, which helps to find the least disturbing trajectory when a collision cannot be avoided. The method uses the property of the mathematical concept of compact set, which is defined as a set where each cover has a finite subcover, \cite{willard:2004}. This concept is purely theoretical and there is no unique way to find finite subcovers. Even for a given cover, the set of finite subcovers cannot be uniquely identified. After finding a finite cover, the union of all elements forms a bounding box around the robot. However, such a finite cover does not mean that the area covered by the union is minimal.
\cite{Hajiloo:2021} studied the threshold for direction change, braking, and acceleration of an autonomous vehicle with the minimum distance to obstacles that suddenly appear in the path of the vehicle, so that the vehicle remains stable. In this study, the friction of the road is considered as an important parameter that contributes to the stability of the vehicle motion and the controller handles the motion, angle and braking of all wheels. They translate the motion into a dynamical system and solve the optimal planning of the motion of all wheels in case an obstacle suddenly appears by changing the acceleration of each wheel.
In \cite{Nakamura:2020}, the authors assume that all vehicles send their local status and planned trajectory to a server, and the server determines whether there is a possible collision in a group of vehicles within a certain time. Then, the server sends a modified trajectory of the vehicles to avoid collisions. Their proposed method is a centralized trajectory planning method that reduces the computation time for collision detection and avoidance by removing vehicles without possible collisions. They assume that each vehicle has a rectangular area around it and consider a collision when the rectangular area around a vehicle intersects with the road boundary, obstacles, and the rectangular areas of other vehicles. And to avoid collision, they consider the change of acceleration vector of vehicles in case of possible collision. The proposed method considers the rules of passing a road, distinguishes the case of intersections and roundabouts, and treats them independently. The method of finding a rectangular box around a vehicle can only be applied to vehicles with regular shape and cannot be generalized to all vehicles with irregular shape. In this method, they find the center of gravity of the vehicle and create a rectangle by adding half the sizes of the length and width of the vehicle shape and assume that the resulting rectangle surrounds the vehicle. Figure \ref{fig4} shows an example of when the created rectangular area does not surround the vehicle. In these cases, part of the vehicle may be outside the rectangular area, which may intersect with other vehicles but is not considered in the collision avoidance.
\begin{figure}[!h]\centering
\includegraphics[width=0.8\linewidth]{P4Copy.png}
\caption{Example of irregularly shaped vehicles. The rectangular area created does not completely cover the vehicle. And there is a possibility of collision between the vehicles.}
\label{fig4}
\end{figure}
\cite{Rakita:2021} studied path planning with collision avoidance using combined graph theoretic and probabilistic approaches. In the proposed method, the workspace is divided into an obstacle space and an obstacle-free space. Then, first, the initial position is connected to the final position by a straight line. When the straight line intersects with the obstacle space, points near the first obstacle are randomly selected and a tree is created connecting the initial point with the selected points. Then select a point from the sample where the straight line from its position to the final point does not intersect the first obstacle, and select it as the first position of the trajectory from the initial point to the final point. This method only works if the obstacle space is completely known and all obstacles are static. If the robot has to explore the environment to detect obstacles and if the obstacles can move, the method does not work. The method also does not consider minimising the total distance the robot travels. See Figure \ref{fig5} for an example of when the shortest path that avoids collisions is longer than the path determined using random sampling.
\begin{figure}[!h]\centering
\includegraphics[width=1\linewidth]{P5Copy.png}
\caption{The shortest path cannot be determined by random sampling near the obstacle region. The shortest path (in blue) is 18, but the path by random sampling (in red) is 22.}
\label{fig5}
\end{figure}
\cite{Hai:2021} studied motion planning and collision avoidance for multi-robot in dynamic environment. The proposed method attempts to answer the question of what is the collision-free trajectory of each robot when the future goal of the robots is known. The proposed method is a decentralized model where the prediction of robot motion and behavior is learned through demonstration using a centralized sequential planner (based on recurrent neural networks (RNNs)). In the proposed model, by design, there is no communication between the robots and uncertainties in the motions of obstacles and robots are not considered.
The paper \cite{Fox:1997} described a simple geometrical model of the motions. The authors studied motion planning and collision avoidance by minimizing the travel distance and maximizing the speed of the robot. Their method uses what they call dynamic windows, which is a set of velocities that the robot can reach within a short period of time, allowing it to safely reduce or accelerate when it detects a potential collision. In this method, the robot's velocity is considered to be a constant value in each time segment, and it is assumed that the robot is the center of a circle whose radius is equal to the distance that the robot can travel at its current velocity (in \cite{Fox:1997}, they immediately switched from circle to rectangle). To model the robot's motion, angular acceleration was considered to allow the robot to change direction. Then, the velocity and angular acceleration are approximated while the position of the robot is updated using the information obtained from its wheels. In each time segment, the robot finds the rectangular area around the robot in which it can move. The area is obtained from the maximum speed that the robot can reach considering its predicted velocity and the angular acceleration of the robot. Then, each point of the obtained rectangular area is assigned three weights for the distance to an obstacle, the distance to the goal, and the velocity of the robot to reach that point, where the weight for the distance to an obstacle is higher for points farther from the obstacle, higher for points closer to the goal, and higher for points farther from the robot. The main optimization problem given in \cite{Fox:1997} is
$$G(v,w)=\sigma(\alpha.heading(v,w)+\beta.dist(v,w)+\gamma.vel(v,w)),$$
where $v$ is the velocity at which the robot moves straight, $w$ is the angular velocity at which the robot changes direction, $heading$ is the distance from the robot's next position to its final position, $dist$ is the distance from the robot's next position to the obstacle, and $vel$ is the speed at which the robot moves to its next position. The movement to the selected point with the highest weights may collide with an obstacle, see Figure \ref{f2}. Even if they ignore the history of the area under study and include only the points with the highest weights, their proposed method may enter an infinite loop of moving back and forth, for example, in Figure \ref{f3} and the second time segment, the optimal point to move to without considering history is the point closest to the obstacle, while the next time segment is the point where the robot was. It is also assumed that the obstacles are static.
\begin{figure}[!h]\centering
\includegraphics[width=1\linewidth]{p2p.png}
\caption{The robot moves to the points with the highest weights in two time segments and in the last time segment the movement to the point with the highest weight collides with the obstacle. The dashed area is the area that is not explored for possible collisions. The red circles are the points with the highest weights.}
\label{f2}
\includegraphics[width=0.5\linewidth]{p3p.png}
\caption{The robot moves infinitely back and forth to the point with the highest weight.}
\label{f3}
\end{figure}
\cite{Rosmann:2012} and \cite{Rosmann:2013} have studied motion planning with collision avoidance. In these methods, the path from the current position of the robot to the final point is considered as an elastic band, where after detecting each obstacle, the path is replaced by a curve from the current position of the robot to the final point that does not intersect with the obstacle, see Figure \ref{f4}. This method helps to improve the method in \cite{Fox:1997} by incorporating history (since the information about the band is stored in each time segment), avoiding an infinite loop, and avoiding scenarios where the robot moves through an obstacle by moving straight forward, see Figure \ref{f2}. The method in \cite{Rosmann:2012} does not optimize the robot's velocity along the path, and this problem is solved in \cite{Rosmann:2013}. The authors in \cite{Rosmann:2013} create a graph for the curved path by using the positions on the curved path as nodes and obtaining the edges by the sequential order of the positions on the path, taking into account the time differences in moving from one position to another. This time helps to change the acceleration of the robot and control the translational and angular velocity of the robot to further minimize the time to reach the final point. Since the methods in \cite{Rosmann:2012} and \cite{Rosmann:2013} use curves instead of direct lines, the magnitude of the distance traveled may not be minimal. Also, these methods do not account for moving obstacles.
\begin{figure}[!h]\centering
\includegraphics[width=0.5\linewidth]{p4p.png}
\caption{Intuition for elastic band. Starting from a straight line, at each time step where an obstacle is observed, a section of the line is replaced by a smooth curve.}
\label{f4}
\end{figure}
If the time steps are very short and the distance to the final position is long and there are many obstacles, the method requires a lot of memory to store the shape of the curve (all points on the curve).
The work \cite{Rosmann:2020} deals with motion planning with obstacle avoidance. They combine the Euclidean measure with a rotational component for motion planning and collision avoidance, where the goal is to minimize time. They model the trajectory as a function of time as a continuous nonlinear differential equation with bounded derivatives at each time point. Their goal is to minimize the cost of moving from the starting point to the end point. Since the robot can change direction (a rotation is applied) when moving to the next state (the next time step), the new operation should be defined to handle this non-Euclidean motion. For this purpose, in $2D$-space, they consider the motion at each time step as an operator in the special orthogonal group ($SO(2)$), which means that the velocity vector of the robot changes its direction with an angle. Since the rotation operation is only on a circle of radius $1$, they should use the special orthonormal group $\mathbb{S}^1$ instead of $SO(2)$. Later, they normalize the angle that maps the rotation to a circle of radius $1$. One of the constraints is that the dynamic error of the system should be small. In their formulation, the value of the error is considered to be equal to $0$ as one of the constraints. Moreover, for the finite difference kernel between two successive states, they used the implicit second-order Runge-Kutta (Crank-Nicolson) method, which is the average of the values of the state function on the two states. This can be replaced by the higher order implicit Runge-Kutta method to better describe the dynamics of the robot motion. If we use only the second-order difference, the effects of the motions at later time steps in the future are ignored. To avoid collisions, they specify the minimum distance between the robot and an obstacle and feed it as a constraint to find the region in which the robot can move without collisions while maintaining the specified minimum distance to obstacles. This is equivalent to considering a bounding box around all obstacles.
\section{Model}
The idea is to plan the motion of a robot in a dynamic environment with both deterministic and non-deterministic objects, creating a safe zone around all objects to avoid collisions, predicting the motions of non-deterministic objects, and taking into account the dimension of the objects. The main idea is to discretize all motions, define safe zones around all objects based on velocity vector probabilities, and find an optimal direction and velocity of the robot at each time step with the lowest cost (minimum travel distance at minimum time). To formulate the model, we use the following:
\begin{itemize}
\item\textbf{Object}: It is the set of all robots, humans, and obstacles that either change their position at a given time step or maintain their initial position at all times. Objects are identified as $i\in\{1,\ldots,N\}$.
\item\textbf{Map}: It is the environment $Env$ where all objects and the robot are located, and their movements are observed or predicted.
\item\textbf{Robot velocity limit}: It is the maximum velocity value of the robot, denoted by $\mathrm{v}$.
\item\textbf{State of an object}: For object $i$, at time step $t$, it is a tuple
\begin{equation}\label{eq:eq1}
(orien_t^i, dist_t^i,d_{v^i_t},P(d_{v^i_t},\-),v^i_t,P(v^i_t,\-)),
\end{equation}
where
$$(orien_t^i, dist_t^i)\in[0,2\pi)\times\mathbb{R}_{\geq0}$$
are the polar coordinates of the object $i$ at time $t$, $v^i_t\in\mathbb{R}_{\geq0}$ is the expected velocity value, $d_{v^i_t}\in[0,2\pi)$ is the expected direction of the object, $P(d_{v^i_t},\-)$ is the probability density function of the direction error of the object $i$ at time step $t$, and $P(v^i_t,\-)$ is the probability density function of the error of the velocity value of the object $i$ at time $t$.
\item\textbf{Deterministic object}: It is such an object whose state at each time step $t$, is
\begin{equation}\label{eq:eq2}
(orien_t^i, dist_t^i,d_{v^i_t},1,v^i_t,1).
\end{equation}
Obstacles and objects whose motions are known at each time step are deterministic objects.
\item\textbf{Non-deterministic object}: It is an object whose state at a given time step $t$, is given by \eqref{eq:eq1} with either $P(v^i_t,\-)\neq1$ or $P(d_{v^i_t},\-)\neq1$.
\item\textbf{Probability density function $P(v^i_t,\-)$}: Since for the cases where the expected velocity value is $v^i_t > 0$, the actual velocity should be $v^i_t+\Delta v$, this means that the object $i$ has a slightly higher or lower velocity than the expected velocity value ($\Delta v$ as an error). To increase the accuracy of the expected (predicted) velocity, the $\Delta v$ value should be close to $0$, with the same probabilities of higher or lower velocity. Now, if $v^i_t=0$, $\Delta v\geq0$. Therefore, $\Delta v$ should be considered as a truncated normal distribution in the interval $[-v_t^i,\infty)$, with $\mu=0$ and $\sigma_v > 0$. So
$$P(v^i_t,x)=\frac{\sqrt{2}}{\sqrt{\pi}\sigma_v}\frac{ \exp(-\frac{x^2}{2\sigma_v^2})}{1-\erf(\frac{-v_t^i}{\sigma_v})},$$
where $\erf$ is the Gaussian error function:
$$\erf(x)=\frac{2}{\sqrt{\pi}}\int_{0}^x\exp(-t^2)dt.$$
\item\textbf{Probability density function $P(d_{v^i_t},\-)$}: Since for the cases where the expected direction is $d_{v^i_t}\in[0,2\pi)$, the actual direction should be $d_{v^i_t}+\Delta\theta$, which means that the object $i$ has a slight angular deviation from the expected direction ($\Delta\theta$ as error), where to increase the accuracy of the predicted direction $\Delta\theta$ should be close to $0$, with the same probabilities for clockwise and counter-clockwise angles. If the object $i$ does not move at time step $t-1$, the motion at time step $t$ could be in any direction with uniform distribution $P(d_{v^i_t},x)=\frac{1}{2\pi}$, where $x\in[0,2\pi)$. If the expected direction is $d_{v^i_t}\in[0,2\pi)$, the error parameter $\Delta\theta$ should be a truncated normal distribution in the interval $[-\pi,\pi)$, with $\mu=0$ and $\sigma_d > 0$. So
$$P(d_{v^i_t},x)=\frac{\sqrt{2}}{\sqrt{\pi}\sigma_d}\frac{\exp(-\frac{x^2}{2\sigma_d^2})}{\erf(\frac{\pi}{\sigma_d})-\erf(\frac{-\pi}{\sigma_d})}.$$
\end{itemize}
The probability density functions $P(d_{v^i_t},\-)$ and $P(v^i_t,\-)$ describe the probability models for the errors in direction and velocity value. The magnitude of the errors in direction and velocity value at time $t$ are independent of the values at all other time steps.
\begin{itemize}
\item\textbf{State of the robot}: For the robot $r$, at time step $t$, it is a tuple
$$(orien_t^r, dist_t^r,d_{v^r_t},1,v^r_t,1),$$
where $(orien_t^i, dist_t^i)$ is the polar coordinate of the robot at time step $t$, $v^r_t\in\mathbb{R}_{\geq0}$ is the decision for the velocity value of the robot, $d_{v^r_t}$ is the decision for the direction of the robot at time step $t$.
\item\textbf{Safe zone}: It is an area around an object at time step $t$ that the robot cannot visit because of the risk of collision. The velocity value, direction, and size of the object are taken into account.
To find the safe zone for object $i$, at time $t$ in state
$$(orien_t^i, dist_t^i,d_{v^i_t},P(d_{v^i_t},\-),v^i_t,P(v^i_t,\-)),$$
find the errors of the angle $\Delta\theta$ and the velocity value $\Delta v$. Find the points $A$ and $B$, where the point $A$ is the last point (sometimes a set of points) of the object, so that when considering parallel lines to
$$\max\{d_{v_t^i}+\Delta\theta,d_{v_t^i}-\Delta\theta\}$$
and by increasing the width of the origin of the lines, the object and the line intersect, and after slightly increasing the width of the origin of the line, the line and the object do not intersect. And the point $B$ is the last point (sometimes a set of points) of the object, so that when parallel lines are viewed at
$$\min\{d_{v_t^i}+\Delta\theta,d_{v_t^i}-\Delta\theta\}$$
and by decreasing the width of the origin of the lines, there will be intersections between the object and the line, and after we decrease the width of the origin of the line a little, the line and the object will not intersect, see Figure \ref{lpoint}.
\begin{figure}[!h]\centering
\includegraphics[width=0.8\linewidth]{lastpoints.png}
\caption{Intuition for the method of determining the last points $A$ and $B$, using parallel lines to the directions of the maximum and the minimum expected velocity of the object.}
\label{lpoint}
\end{figure}
Now we should find either the curve $f(A,B)$ (the curve of the object $i$ from $A$ to $B$) and $g(B,A)$ (the curve of the object $i$ from $B$ to $A$) or their respective convex hulls enclosing $f(A,B)$ and $g(B,A)$. Then find the coordinates $A_c$ and $B_c$, respectively, for the motion of the object about the direction
$$\max\{d_{v_t^i}+\Delta\theta,d_{v_t^i}-\Delta\theta\}~~\text{and} ~~\min\{d_{v_t^i}+\Delta\theta,d_{v_t^i}-\Delta\theta\}$$
with scale $v_t^i+\Delta v$. Since we are working with a polar system, $g(B,A)$ is transformed with this scale to be
$$g_c(B_c,A_c)=(v_t^i+\Delta v)*Map(g(B,A))$$
and $f(A,B)$ is transformed to
$$f_c(A_c,B_c)=(v_t^i+\Delta v)*Map(f(A,B)),$$
where for each
$$\alpha\in[\min\{deg(A),deg(B)\},\max\{deg(A),deg(B)\}],$$
with $deg(X)$ is the degree of the point $X$ on the boundary of the object $i$ to the centroid of the object $i$ in $[0,2\pi)$, the mappings $Map(g(B,A))$ and $Map(f(A,B))$ can be found as functions of $\alpha$ as follows:
First find $\alpha_{\max}(h(X,Y))$ and $\alpha_{\min}(h(X,Y))$:
\begin{align*}
\alpha_{\max}(h(X,Y))=&\max\{\beta\mid dist(h(X,Y))\mid_{[\beta,\max\{deg(X),deg(Y)\}],0)} \\
&~~\text{ is strictly decreasing.}\}
\end{align*}
and
\begin{align*}
\alpha_{\min}(h(X,Y))&=\min\{\beta\mid dist(h(X,Y))\mid_{[\min\{deg(X),deg(Y)\},\beta],0)} \\
&~~\text{ is strictly increasing.}\}.
\end{align*}
Then we define for $Z\in h(X,Y)$:
\begin{align*}
&\Gamma_{1,\max}(Z,h(X,Y))=\\
&\left\{\begin{array}{ll}
\max\{h(X,Y)\mid_{\beta}\mid\beta\in[\alpha_{\min}(h(X,Y)),deg(Z)]\},°(Z)\geq\alpha_{\min}(h(X,Y))\\
h(X,Y),&otherwise,
\end{array}\right.&
\end{align*}
\begin{align*}
&\Gamma_{2,\max}(Z,h(X,Y))=\\
&\left\{\begin{array}{ll}
\max\{h(X,Y)\mid_{\beta}\mid\beta\in[deg(Z),\alpha_{\max}(h(X,Y))]\},°(Z)\leq\alpha_{\max}(h(X,Y))\\
h(X,Y),&otherwise,
\end{array}\right.
\end{align*}
\begin{align*}
&\Gamma_{1,\min}(Z,h(X,Y))=\\
&\left\{\begin{array}{ll}
\min\{h(X,Y)\mid_{\beta}\mid\beta\in[\alpha_{\min}(h(X,Y)),deg(Z)]\},°(Z)\geq\alpha_{\min}(h(X,Y))\\
h(X,Y),&otherwise,
\end{array}\right.
\end{align*}
and
\begin{align*}
&\Gamma_{2,\min}(Z,h(X,Y))=\\
&\left\{\begin{array}{ll}
\min\{h(X,Y)\mid_{\beta}\mid\beta\in[deg(Z),\alpha_{\max}(h(X,Y))]\},°(Z)\leq\alpha_{\max}(h(X,Y))\\
h(X,Y),&otherwise.
\end{array}\right.
\end{align*}
Then
$$Map(g(B,A))\mid_{\alpha}=\max\{\Gamma_{1,\max}(Z,g(B,A)),\Gamma_{2,\max}(Z,g(B,A))\mid Z\in g(B,A)\}$$
and
$$Map(f(A,B))\mid_{\alpha}=\min\{\Gamma_{1,\min}(Z,f(A,B)),\Gamma_{2,\min}(Z,f(A,B))\mid Z\in f(A,B)\}.$$
Intuitively, the maps $Map(g(B,A))$ and $Map(f(A,B))$ can be viewed as repeating local maxima and local minima within the interval of the first and last local maxima and the first and last local minima, respectively. Note that, by construction, the degree of points $A$ and $B$ with respect to the centroid of the object $i$ are the degrees between
$$\max\{d_{v_t^i}+\Delta\theta,d_{v_t^i}-\Delta\theta\}~~\text{and} ~~\min\{d_{v_t^i}+\Delta\theta,d_{v_t^i}-\Delta\theta\},$$
each with respect to the origin. So any point $X=(r,x)$, where $r$ is the radius to the centroid and $x$ is the angular degree, can be viewed as the vector $(r\cos(x),r\sin(x))$ with respect to the centroid in the Euclidean metric. Also, the current position of the centroid of the object $(R,\beta)$ can be viewed as the vector $(R\cos(\beta),R\sin(\beta))$ with respect to the origin in the Euclidean metric. This means that the coordinate of the point $X$ with respect to the origin in the Euclidean metric is the sum of two vectors:
\begin{align*}
(r\cos(x),r\sin(x))&+(R\cos(\beta),R\sin(\beta))=\\
&(r\cos(x)+R\cos(\beta),r\sin(x)+R\sin(\beta)),
\end{align*}
which can be easily converted into a polar metric as the point $Y=(\mathcal{R},\mathcal{\gamma})$, where
$$\mathcal{R}=\sqrt{(r\cos(x)+R\cos(\beta))^2+(r\sin(x)+R\sin(\beta))^2}$$
and
\begin{align*}
\gamma=&\tan^{-1}\left(\frac{r\sin(x)+R\sin(\beta)}{r\cos(x)+R\cos(\beta)}\right),\\
&\text{with}~~\gamma\in{[\min\{d_{v_t^i}+\Delta\theta,d_{v_t^i}-\Delta\theta\},\max\{d_{v_t^i}+\Delta\theta,d_{v_t^i}-\Delta\theta\}]}.
\end{align*}
So,
$$Map(X)=Y.$$
Note that the initial centroid has coordinates $(R,\beta)$ with respect to the origin, the new centroid of the object for the predicted position has coordinates $Predict\_Centroid=(v_t^i+\varepsilon,d_{v_t^i}+\delta)$ with respect to the initial centroid. Now suppose that $X$ is a point on the curve $f(A,B)$ (or on the curve $g(B,A)$), then its corresponding point $New\_X$ on the curve $f_c(A_c,B_c)$ (or on the curve $g_c(B_c,A_c)$) is obtained as follows:
\begin{align*}
New\_X&=Polar(Euclidean(Map(X),O)\\
&~~~~+Euclidean(Predict\_Centroid,(\mathcal{R},\gamma)))\\
&=Polar(Euclidean((\mathcal{R},\gamma),O)+Euclidean(X,(\mathcal{R},\gamma))\\
&~~~~+Euclidean(Predict\_Centroid,(\mathcal{R},\gamma)))
\end{align*}
which is a polar transformation of the sum of the vectors of the Euclidean coordinates of the points $X$ with respect to the origin and $Predict\_Centroid$ with respect to the initial centroid coordinate. See Figure \ref{figg1}.
\begin{figure}[!h]\centering
\includegraphics[width=0.5\linewidth]{P1.png}
\caption{Intuition for the method of determining the coordinates of each point of an object after predicting its motion. The dashed vector is the coordinate of interest, i.e., the coordinate of the point $X$ of the object $i$ after applying the predicted motion of the object. Dotted vectors are used to indicate parallel vectors.}
\label{figg1}
\end{figure}
Now, the safe zone of object $i$ at time step $t$ with state given by \eqref{eq:eq1} can be obtained by
$$SZ^i_{t+1}=\max\left\{\int_{d_{v^i_t}-\Delta\theta}^{d_{v^i_t}+\Delta\theta}(g_c(B_c,A_c)^2-f(A,B)^2),\int_{d_{v^i_t}-\Delta\theta}^{d_{v^i_t}+\Delta\theta}(f_c(A,B)^2-g(B,A)^2)\right\}.$$
See Figure \ref{figg2} for the intuition of obtaining the safe zone.
\begin{figure}[!h]\centering
\includegraphics[width=0.8\linewidth]{P2.png}
\caption{Intuition for the method of obtaining the safe zone. $g_c(B_c,A_c)$ ($g(B,A)$) is the boundary curve of the map of the object (the object) from $B_c$ ($B$) to $A_c$ ($A$), that is, the boundary curve of the map of the object (the object) to the right of Line 2 (Line 1). $f_c(A_c,B_c)$ is the boundary curve of the map of the object (the object) from $A_c$ ($A$) to $B_c$ ($B$), i.e., the boundary curve of the map of the object (the object) to the left of Line 2 (Line 1).}
\label{figg2}
\end{figure}
\item\textbf{The optimization objective}: Let the robot aims to move from point $A$ to point $B$. Let $Shortest\_Path (A,B)$ be the shortest path from $A$ to $B$ without considering moving objects and $|Shortest\_Path (A,B)|$ be the total distance of the shortest path and assume that the robot can travel the shortest path in time $t$. Let $act_t$ be the actual position of the robot at time step $t$. The actual position of the robot is a function of the initial position, the direction of the robot, and the velocity value of the robot. The positions are denoted as a sequence $P_0=A,P_1,\ldots,P_{T-1},P_{T}=B$. The objective is then to minimize the following
$$\min_{P_0=A,P_1,\ldots,P_{T-1},P_{T}=B~\text{ valid }}\sqrt{\left(\frac{T}{\mathrm{t}}\right)^2+\left(\frac{\left(\sum_{t=0}^T(\min_{P\in Shortest\_Path(A,B)}\{(P_t-P)^2\})\right)}{|Shortest\_Path(A,B)|^2}\right)^2},$$
where a path is considered valid if there are no collisions with other objects. Note that the time and distance are divided by the minimum time and length of the shortest path to remove scales.
\item\textbf{Robot direction and speed}: At time step $t$, find the union of all safe zones of all objects, $SZ_{t+1}=\bigcup_{i=1}^NSZ^i_{t+1}$. Then find the area $Env\setminus SZ_{t+1}$. For the direction of the robot, choose the point $Q\in Env\setminus SZ_{t+1}$ such that the length of the shortest path from $x$ to $B$ has the smallest value
$$Q=\argmin_{x\in Env\setminus SZ_{t+1}}\left\{|Shortest\_Path (x,B)|~\mid~Straight\_line (act_t,x)\cap SZ_{t+1}=\emptyset\right\},$$
where $Straight\_line (act_t,Q)$ is the straight line connecting the points $act_t$ and $Q$. Note that the velocity value of the robot is constrained to $\mathrm{v}$ and the point $Q$ may not be reached in one time step. Thus, the robot either moves toward point $Q$ at the maximum velocity $\mathrm{v}$ for one time step or it can travel to $Q$ at the velocity
$$v^r_{t+1}=\frac{|Shortest\_Path (act_t,Q)|}{\text{length of a time step}},$$
where $|Shortest\_Path (act_t,Q)|$ is a straight line.
So the state of the robot will be
$$(orien_{t+1}^i, dist_{t+1}^r,d_{v^r_{t+1}},1,v^r_{t+1},1),$$
where $d_{v^r_{t+1}}$ is the angle of the line $Straight\_line(act_t,Q)$ with the $x$-axis in gradient scale and
$$v^r_{t+1}=\min\left\{\mathrm{v},\frac{|Shortest\_Path (act_t,Q)|}{\text{length of a time step}}\right\}.$$
Note that after each time step, the current position of the robot and the shortest path should be updated. Since, by construction, for every two consecutive time steps $t$ and $t+1$ we have
$$0\leq|Shortest\_Path (act_{t+1},B)|\leq|Shortest\_Path (act_{t},B)|$$
the method converges.
The way the robot's direction works at each time step allows it to find the optimal path, which may use a different route than the initial shortest path from the initial position to the robot's target position.
\item\textbf{Next position}: If the state of the object $i$ at time step $t$ is given by \eqref{eq:eq1}, then the next position $(orien_{t+1}^i, dist_{t+1}^i)$ can be determined as follows:
$$\min\{orien_t^i,d_{v^i_t}\}\leq orien_{t+1}^i=\arctan(\frac{Y}{X})\leq\max\{orien_t^i,d_{v^i_t}\}$$
and
$$dist_{t+1}^i=\sqrt{X^2+Y^2},$$
where
$$X=dist_{t}^i\cos(orien_t^i)+v^i_t\cos(d_{v^i_t})$$
and
$$Y=dist_{t}^i\sin (orien_t^i)+v^i_t\sin (d_{v^i_t}).$$
\item\textbf{Velocity and direction pattern}: Suppose that at each time step $t$, the past $H$ time steps of the states of all $N$ objects are known. This means
$$(orien_s^i, dist_s^i,d_{v^i_s},P(d_{v^i_s},\-),v^i_s,P(v^i_s,\-)),~~s=t-H,\ldots,t-1.$$
We want to make a prediction of the directions $d_{v^i_s}$ and the velocity values $v^i_s$ for $s\geq t$. The following values can be natural choices:
\begin{itemize}
\item\textbf{Moving average}: For the future time step $t+k$, the velocity value of the object $i$ at time $t+k$ will be
$$v^i_{t+k}=mean(\{v^i_{s}\mid s=t+k-H,\ldots,t+k-1\})$$
and
$$d(v^i_{t+k})=mean(\{d(v^i_{s})\mid s=t+k-H,\ldots,t+k-1\}).$$
The advantage of this method is that the velocity values and directions change over time. However, it also has the disadvantage of becoming a linear function without updating if the number of time steps is long enough. This is a simple auto-regressive method for forecasting, see \cite{montgomery:2011}.
\item\textbf{Naive auto-regressive}:
For the future time step $t+k$, the velocity value of the object $i$ at time step $t+k$ is determined as follows: Let
$$W=(1,\frac{1}{2},\ldots,\frac{1}{2^{H-1}}).$$
Define
$$w=(w_{t+k-H},\ldots,w_{t+k-1}),$$
where $w_{t+k-u}=\frac{W_u}{\sum_{j=1}^{H}W_j}$ are $H$ weights for the past states.
Then
$$v^i_{t+k}=\sum_{j=1}^Hw_{t+k-j}v^i_{t+k-j}$$
and
$$d(v^i_{t+k})=\sum_{j=1}^Hw_{t+k-j}d(v^i_{t+k-j}).$$
Velocity value and direction are related to past velocity values and directions. However, past velocity values and directions must be weighted, i.e., the next velocity value and direction are more related to the first immediate past velocity value and direction than the velocity values and directions in the time steps before it.
The advantage of this method over the previous one is that the velocity values and directions change in a more realistic way over time. However, it also has the disadvantage that, without updating, the motion strategy is deterministic after a sufficiently long number of time steps, and also, the weights for the directions and velocity values are always the same all the time.
\item\textbf{Generalized auto-regressive}: For future time step $t+k$, the velocity value of object $i$ at time $t+k$ is determined as follows: Let
$$w^u=(w^k_{t+k-H},\ldots,w^k_{t+k-1}),$$
for $u=1,2$, be $H$ random variables in the interval $(0,1)$ such that for $u=1,2$, $\sum_{j=1}^Hw^k_{t+h-j}=1$ and
$$w^u_{t+k-H}\leq\ldots\leq w^u_{t+k-1}.$$
$w^u_{t+k-j}$'s are weights. Let
$$v^i_{t+k}=\sum_{j=1}^Hw^1_{t+k-j}a^i_{t+k-j}$$
and
$$d(v^i_{t+k})=\sum_{j=1}^Hw^2_{t+k-j}d(v^i_{t+k-j}).$$
The advantage of this method over the previous one is that the velocity values and directions change more realistically over time since the lists of probabilities are determined randomly. However, it also has the disadvantage that without additional information, some states may be invalid as future states of object $i$. This is a more general auto-regressive method for forecasting, see \cite{montgomery:2011}.
We could also introduce random noise at each step by adding weighted multipliers of white noise, see \cite{montgomery:2011}.
\end{itemize}
In all the above natural possibilities, the position of the object $i$ at time $t+k$ can be determined using the step (Next position).
\item\textbf{State update}: When at time step $t$ the states of all objects are observed, we need to update the states
$$(orien_t^i, dist_t^i,d_{v^i_t},1,v^i_t,1),~~ \forall i=1,\ldots,N.$$
The values $orien_t^i$ and $dist_s^i$ are updated by the current coordinate of the object $i$ at time $t$, and the direction and velocity value $d_{v^i_s}$ and $v^i_s$ will be updated by the current direction and velocity of the object $i$. Since the current values are observed, the probability densities of errors in the direction and velocity value of all objects can be avoided. Based on these new observations, all predictions for the states in the future should be updated using the methods described above by replacing the predicted states at time step $t$ with the actual values of the states at time step $t$. Note that if the current time step is $t$, this means that at time step $s$ with $s<t$, all states are already updated with the actual observed states.
\item\textbf{High risk scenarios (Run away)}: Let at time step $t$ $C_r(orien_t^r, dist_t^r)$ be the set of all points on the map that the robot can move from its current position, red circle in Figures \ref{f3p} and \ref{f4p}. Suppose that multiple objects, $i_1,\ldots,i_n$, are moving towards the robot such that
$$(orien_t^r, dist_t^r)\in SZ^{i_j}_{t+1},~~\forall j=1,\ldots,n.$$
This means that the sets
$$\left\{|Shortest\_Path (x,B)|~\mid~Straight\_line (act_t,x)\cap SZ_{t+1}=\emptyset\right\},~~\forall x\in Env\setminus SZ_{t+1}$$
are empty sets for all $x$. In such cases, the next state of the robot can be determined as follows:
\begin{itemize}
\item\textbf{With skip zone}: Assume that
$$D=C_r(orien_t^r, dist_t^r)\setminus\left(\bigcup_{j=1}^nSZ^{i_j}_{t+1}\right)\neq\emptyset.$$
The set $D$ is called the skip zone because if the robot moves to a point in $D$, it avoids possible collisions. For this case, find
$$Q=\argmin_{x\in D}\left\{|Shortest\_Path (x,B)|\right\}.$$
Note that the velocity value of the robot is constrained to $\mathrm{v}$ and the point $Q$ can be reached in a single time step. Thus, the robot moves towards the point $Q$ with the velocity value
$$v^r_{t+1}=\frac{|Shortest\_Path (act_t,Q)|}{\text{length of a time step}}\leq\mathrm{v}.$$
So the state of the robot will be
$$(orien_{t+1}^i, dist_{t+1}^r,d_{v^r_{t+1}},1,v^r_{t+1},1),$$
where $d_{v^r_{t+1}}$ is the angle of the line $Straight\_line(act_t,Q)$ with the $x$-axis in gradient scale, $(orien_{t+1}^i, dist_{t+1}^r)$ is the polar coordinate of $Q$, and
$$v^r_{t+1}=\frac{|Shortest\_Path (act_t,Q)|}{\text{length of a time step}}.$$
Figure \ref{f3p}, illustrate the scenario with skip zone.
\begin{figure}[!h]\centering
\includegraphics[width=0.8\linewidth]{P3.png}
\caption{Intuition to avoid collision, for the case with skip zone. The red bullet is the current position of the robot, the red vector indicates the maximum velocity value of the robot, and the red circle with the double dotted and dashed line indicates the set of positions that the robot can move with its limited velocity value. The blue dashed area is the skip zone $D$.}
\label{f3p}
\end{figure}
\item\textbf{Without skip zone}: Assume that
$$C_r(orien_t^r, dist_t^r)\setminus\left(\bigcup_{j=1}^nSZ^{i_j}_{t+1}\right)=\emptyset.$$
See figure \ref{f4p} for such a case.
\begin{figure}[!h]\centering
\includegraphics[width=0.8\linewidth]{P4.png}
\caption{The case without skip zone. The red bullet is the robot's current position, the red vector shows the robot's maximum velocity value, and the red circle with the double dotted and dashed line shows the set of positions the robot can travel to with its limited velocity value.}
\label{f4p}
\end{figure}
For this case, find the set
$$M=\argmin_{x\in C_r(orien_t^r, dist_t^r)}\left\{\sum_{j=1}^n\left(P(d_{v^{i_j}_t},x)+P(v^{i_j}_t,x)\right)\right\}.$$
The set $M$ is the set of all points that the robot can move to in a single time step with the least probability that objects $i_1,\ldots,i_n$ move to those points. Then, as before
$$Q=\argmin_{x\in M}\left\{|Shortest\_Path (x,B)|\right\}.$$
Note that the velocity value of the robot is constrained to $\mathrm{v}$ and the point $Q$ can be reached in a single time step. Thus, the robot moves towards the point $Q$ with the velocity value
$$v^r_{t+1}=\frac{|Shortest\_Path (act_t,Q)|}{\text{length of a time step}}\leq\mathrm{v}.$$
So the state of the robot will be
$$(orien_{t+1}^i, dist_{t+1}^r,d_{v^r_{t+1}},1,v^r_{t+1},1),$$
where $d_{v^r_{t+1}}$ is the angle of the line $Straight\_line(act_t,Q)$ with the $x$-axis in gradient scale, $(orien_{t+1}^i, dist_{t+1}^r)$ is the polar coordinate of $Q$, and
$$v^r_{t+1}=\frac{|Shortest\_Path (act_t,Q)|}{\text{length of a time step}}.$$
\end{itemize}
Note that throughout the formulation the robot is considered as a single point (centroid of the robot, $act_t$) at time step $t$. To generalize the formulation, assume that $OCP_t^r\subset Env$ are the positions in the environment that the robot occupies at time step $t$. Then all formulations involving the motion of the robot from $act_t$ to a point $w\in Env$ should be replaced by the mapping
$$\delta:OCP_t^r\rightarrow B(w,act_t,OCP_t^r),$$
which maps $act_t$ to $w$ with direction $d(v)$ (direction from $act_t$ to $w$) and velocity $v$ (the distance between $act_t$ and $w$ divided by the length of the time step), and $y\in OCP_t^r$ to the point $y_1\in B(w,act_t,OCP_t^r)$, where the line connecting $y$ and $y_1$ makes an angle $d(v)$ with the $x$-axis and the distance between $y$ and $y_1$ is equal to $v$. Then the whole formulation should be applied to the $(u,v)\ in OCP_t^r\times\delta(OCP_t^r)$, instead of only to $(act_t,w)$.
\end{itemize}
\section{Conclusion}
We have developed a simple model that allows optimal path planning in an unknown dynamic environment, minimizing the total distance the robot travels and the total time to complete a given task, while avoiding collisions. To avoid collisions, we consider no bounding box around obstacles to further minimize the distance traveled by the robot. For moving obstacles, we perform a simple prediction of the future motion of the obstacles and the motion of the robot changes accordingly. Over time, the robot updates the state of the moving objects to better estimate their future motion.
For the future work, to better estimate the next motion of the obstacles, we will replace the simple estimation with a better prediction method, such as more complex time series prediction and forecasting methods.
\section*{Acknowledgment}
This work was supported by operation Centro-01-0145-FEDER-000019-C4-Centro de Compet\^{e}ncias em Cloud Computing, cofinanced by the European Regional Development Fund (ERDF) through the Programa Operacional Regional do Centro (Centro 2020), in the scope of the Sistema de Apoio \`{a} Investiga\c{c}\~{a}o Cientif\'{i}ca e Tecnol\'{o}gica - Programas Integrados de IC\&DT. This work was supported by NOVA LINCS (UIDB/04516/2020) with the financial support of FCT-Funda\c{c}\~{a}o para a Ci\^{e}ncia e a Tecnologia, through national funds.
\bibliographystyle{unsrt}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,803
|
{"url":"https:\/\/mathspace.co\/textbooks\/syllabuses\/Syllabus-812\/topics\/Topic-18336\/subtopics\/Subtopic-246426\/?textbookIntroActiveTab=guide&activeTab=worksheet","text":"# 4.08 Perimeter of land\n\nWorksheet\nPerimeter of land\n1\n\nA rectangular block of land is 12\\text{ m} long and 9\\text{ m} wide. A fence is to be constructed around the perimeter. What is the length of the fence?\n\n2\n\nAn outline of a block of land is pictured below.\n\nFind the perimeter in metres.\n\n3\n\nCalculate the outside perimeter of the plot of land on this site plan. All measurements are given in metres.\n\n4\n\nAn outline of a block of land is pictured below:\n\na\n\nFind the value of x.\n\nb\n\nFind the perimeter of the block of land.\n\n5\n\nConsider the following section of a street map:\n\nUse the scale to estimate the perimeter of the shaded block of land in metres.\n\nApplications\n6\n\nIn the following diagram, each square of the grid has a side length of 30\\text{ m}. A jogger runs the perimeter of the park four times each morning.\n\na\n\nCalculate the distance of one lap of the park.\n\nb\n\nHow far will the jogger run each week? Give your answer in kilometres to two decimal places.\n\n7\n\nIn the following diagram, each square of the grid has a side length of 15\\text{ m}. A jogger runs a lap along two sides of the park, then crosses to the opposite corner. He runs 5 laps each morning.\n\na\n\nWhat is the distance of one lap?\n\nb\n\nHow far will the jogger run each week?\n\n8\n\nA rectangular block of land is 25\\text{ m} long and 12\\text{ m} wide. A fence is to be constructed around the perimeter.\n\na\n\nCalculate the length of the fence.\n\nb\n\nFencing costs \\$47 per linear metre. Calculate the fencing cost, to the nearest dollar for this block of land. 9 An outline of a block of land is pictured below: a Calculate the perimeter. b Fencing costs \\$32 per linear meter. Calculate the fencing cost, to the nearest cent, for this block of land.\n\n10\n\nEllie is fencing a paddock. The length across the paddock is 90\\text{ m}. The boundary is as shown in the diagram:\n\na\n\nFind the value of x correct to two decimal places.\n\nb\n\nFind the length of fencing Ellie needs in total. Round your answer to two decimal places.\n\nc\n\nIf the cost of the fencing is \\$0.21 \\text{\/m}, how much will the total fencing cost? 11 The local council is investing in a low brick wall built around one of their parks. The known dimensions of the park are shown in the diagram: a Determine the length, l, of the unknown sides. Round your answer to two decimal places. b Hence, find the length of the brick wall correct to two decimal places. c If the wall costs on average \\$3.00 per metre, how much will the council spend on the wall?\n\n12\n\nThe diagram shows an aerial view of a swimming pool. The pool is surrounded by rectangular pavers.\n\nEach paver is 120\\text{ cm} long and 60\\text{ cm} wide.\n\na\n\nWhat is the perimeter of the pool? Use the outside edge and round your answer to two decimal places.\n\nb\n\nAs part of his daily exercise routine, Lachlan does 500\\text{ m} of walking.\n\nHow many laps around the pool will Lachlan have to walk, to reach his daily exercise goal?","date":"2022-01-25 11:28:40","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\": 0.42538371682167053, \"perplexity\": 1557.4076170261405}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-05\/segments\/1642320304810.95\/warc\/CC-MAIN-20220125100035-20220125130035-00135.warc.gz\"}"}
| null | null |
Isla Yerbas Buenas är en ö i Chile. Den ligger i regionen Región de Magallanes y de la Antártica Chilena, i den södra delen av landet, km söder om huvudstaden Santiago de Chile. Arean är kvadratkilometer.
Terrängen på Isla Yerbas Buenas är kuperad. Öns högsta punkt är meter över havet. Den sträcker sig 74,8 kilometer i nord-sydlig riktning, och 40,2 kilometer i öst-västlig riktning.
I övrigt finns följande på Isla Yerbas Buenas:
Sjöfartsrelaterade platser:
Paso Juan Bravo (en havskanal)
Insjöar:
Laguna Bella (en sjö)
Kullar:
Cerros Altos (kullar)
Pico Lecky (en kulle)
Halvöar:
Cabo Colworth (en udde)
Cabo Cuan (en udde)
Cabo Phillip (en udde)
Península Bueras (en halvö)
Punta Ancud (en udde)
Punta Ceres (en udde)
Punta Cutter (en udde)
Punta Gremi (en udde)
Punta Henry (en udde)
Punta Lavinia (en udde)
Punta Nason (en udde)
Punta Raby (en udde)
Punta Robert (en udde)
Punta Salmon (en udde)
Punta Terencio (en udde)
Punta Tirsa (en udde)
Berg:
Cerros Bajos (en bergskedja)
Monte Joy (ett berg)
Monte Souldrop (ett berg)
Pico del Ejército (ett berg)
Pico del Roble (ett berg)
Pico Notable (ett berg)
Pico Santa Agnés (ett berg)
Pico Santa Annes (en bergstopp)
Trakten runt Isla Yerbas Buenas består i huvudsak av gräsmarker.
Kommentarer
Källor
Öar i Región de Magallanes y de la Antártica Chilena
Öar i Chile större än 100 kvadratkilometer
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 3,015
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package org.auraframework.util.validation;
import java.util.Map;
import org.auraframework.util.javascript.JavascriptProcessingError.Level;
import org.auraframework.util.json.JsonEncoder;
import org.auraframework.util.json.JsonReader;
import org.auraframework.util.test.util.UnitTestCase;
import org.junit.Test;
public final class ValidationErrorTest extends UnitTestCase {
@Test
public void testJsonSerialization() {
ValidationError error = new ValidationError("tool", "/file/name", 11, 3, "message", "evidence",
Level.Error, "rule");
String json = JsonEncoder.serialize(error);
@SuppressWarnings("unchecked")
ValidationError dError = ValidationError.deserialize((Map<String, ?>) new JsonReader().read(json));
assertEquals(error.toCommonFormat(), dError.toCommonFormat());
}
@Test
public void testTextSerialization() {
ValidationError error = new ValidationError("tool", "/file/name", 11, 3, "message", "evidence",
Level.Error, "rule");
String text = error.toCommonFormat();
ValidationError dError = ValidationError.fromCommonFormat(text);
assertEquals(error.toCommonFormat(), dError.toCommonFormat());
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,515
|
The Complete Ravenscar Trilogy
Barbara Taylor Bradford (forfatter)
The Complete Ravenscar Trilogy ebok
Packaged together for the first time, THE RAVENSCAR TRILOGY is one of Barbara Taylor Bradford's best-loved series. Ravenscar: A house, a legacy and a dynasty... THE RAVENSCAR DYNASTY follows the fortunes of the Edward Deravenel and his family, as passions and power, ambition and treachery, love and loss combine in an epic tale of an extraordinary dynasty. In HEIRS OF RAVENSCAR, the First World War has drawn to a close, and troubles are looming for the Deravenel …
Undertittel The Ravenscar Dynasty, Heirs of Ravenscar, Being Elizabeth
Forfattere Barbara Taylor Bradford (forfatter)
Forlag HarperCollins
A Woman of Substance
Barbara Taylor Bradford 54,-
Hold the Dream
Barbara Taylor Bradford 162,-
The Emma Harte 7-Book Collection
Barbara Taylor Bradford's 4-Book Collection
Packaged together for the first time, THE RAVENSCAR TRILOGY is one of Barbara Taylor Bradford's best-loved series. Ravenscar: A house, a legacy and a dynasty... THE RAVENSCAR DYNASTY follows the fortunes of the Edward Deravenel and his family, as passions and power, ambition and treachery, love and loss combine in an epic tale of an extraordinary dynasty. In HEIRS OF RAVENSCAR, the First World War has drawn to a close, and troubles are looming for the Deravenel family. Blackmail, betrayal and jealously have caused problems from within and is it up to Edward's daughter, Bess, to secure the Ravenscar inheritance - whatever it takes... In the last part of the trilogy, BEING ELIZABETH, the Deravenel fortune has come to rest on the shoulders of Elizabeth Deravenel Turner after years of familial strife. As the twentieth century draws to a close, Elizabeth must fight for her birthright and inheritance in a family still divided by conflicting loyalties and politics.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 582
|
## _Contents_ | _Title_
---|---
|
_List of Maps_
|
_Acknowledgements_
|
_Prologue_
|
ONE | Fault Line
TWO | Saturday Night
THREE | And Sunday Morning
FOUR | First Week of the Rest of Your Life
FIVE | Cold as Ice
SIX | The Strangeness
SEVEN | The Bullet Run
EIGHT | Thaw
NINE | A Quiet Night Like This
TEN | Dawn
ELEVEN | Pieces
|
|
_Epilogue: Corridor of Emptiness_
|
_Notes_
|
_Bibliography_
|
_The Death Strip: The Toll_
|
_Copyright_
Mine is not a pleasant story, it does not possess the gentle harmony of invented tales; like the lives of all men who have given up trying to deceive themselves, it is a mixture of nonsense and chaos, madness and dreams.
_Demian_ ,
Hermann Hesse
* * *
_Demian_ , Hermann Hesse (Panther Books Ltd, London, 1969).
## _List of Maps_
1. The wall, cutting across central Berlin
2. The divided Germany
3. Districts surrounding Berlin
4. Bernauer Strasse
5. The Teltow Canal
6. The River Spree
7. Exits and Entrances
8. The death of Peter Fechter
9. Tunnel 57
10. The Steinstücken enclave
11. The gateways to the West
12. Checkpoint Charlie
## _Acknowledgements_
Any book like this, balancing political and historical background against the first-person testaments of the foreground, draws to itself a lot of people and a lot of sources of information. I pay my due tribute to them all and offer my sincere thanks.
First, Birgit Kubisch, who started off as an interpreter but became an enthusiast, an organiser, a translator, a provider, and handled some interviews herself. To her I must add my neighbour Inge Donnell who moved doggedly through translating the paragraphs of those who died at the wall even though she found the task upsetting; and to her I must add Viktoria Tischer, of Haynes, who set herself to add to the store of knowledge on Peter Fechter, the teenager whose fate – he was left to bleed to death in 1962 – still lives, if I may use that word, in the domain of the grotesque and the barbarous which the Berlin Wall created.
I list the others who helped in no particular order. The book is, I hope, greater than the sum total of its parts – and many, many good people provided those parts, each in their own way. To people who insist that civility, courtesy and cooperation are vanished echoes of a genteel past, I say: wrong. Almost _nobody_ I approached – in so many walks of life – was difficult. On the contrary, and for no reward of any kind, they went out of their way to help.
I owe a special debt to Hagen Koch, a former Border Guard who has, in his cosy East Berlin apartment, set up a museum to the wall because he feels there should be one place where people can find out about it. He has refused a great deal of money for some of the material he's gathered (including a unique set of photographs of the death strip) and I salute him for that as I thank him for maps, information, and advice. If you want to see for yourself, try <http://www.berliner-mauer.de>. The Rt Hon. Lord Hurd of Westwell CH, CBE, then British Foreign Secretary, sent specific memories which were gratefully received. Millie Waters, a bubbly public affairs specialist at the HQ US Army, Europe, proved to be a one-person army in finding people and arranging things.
And thanks to the interviewees: Chris Toft, British Military Police; Bernard Ledwidge, British Military Government, Berlin; the late Diana Loeser, who lived in East Berlin; Bernie Godek, Michael Raferty, Bill Bentz and Russ Anderson of the US military; former US Secretary of State Dean Rusk, Frank Cash, Dorothy Lightner (wife of Allan), the late John Ausland, George Muller, Al Hemsing, Richard Smyser and Frank Trinka of the US political and diplomatic services; Adam Kellett-Long (with particular thanks to his wife Mary who allowed me to use extracts from her diary), Peter Johnson (who also let me have his diary) and Erdmute Greis-Behrendt, of Reuters; Günter Moll and Roland Egersdörfer of the Border Guards; Peter Schultz and Ernest Steinke of the RIAS radio station, Berlin; Peter Dick, a Canadian and briefly a West Berlin resident; Peter and Daniel Glau of the Hotel Ahorn, which became a second home; Ekkehard Gurtz, Gerda Stern, Bodo Radtke, Nora Evans, Mateus, Kurt Behrendt, Heinz Sachsenweger, Uwe Nietzold, Erhard and Brigitte Schimke, Lutz and Ute Stolz, Rüdinger Hering, Klaus-Peter Grohmann, Martin Schabe, Horst Pruster, Birgit Wuthe, Jakob Burkhardt, Elli Köhn, Pastor Manfred Fischer, Marina Brath, Astrid Benner, Brita Segger, Katrin Monjau, Harald Jäger, Hartmut Richter, Janet and Jacqueline Burkhardt. E.L. Gordon kept watch on the US media. A friend and fellow enthusiast, John Woodcock, was a valued companion on trips to the city.
There is a Bibliography at the end, but for permission to quote I am indebted to: Edith Kohagen, Editor-in-chief of Presse- und Informationsamt des Landes Berlin for _The Wall and How it Fell (1994) and the invaluable Violations of human rights, illegal acts and incidents at the sector border in Berlin since the building of the wall (13 August 1961–15 August 1962)_ , published in 1962 on behalf of the government of the Federal Republic of Germany by the Federal Ministry for All-German Questions (Bonn and Berlin). She also demystified the time gap between Berlin and Washington in August 1961, something not as straightforward as one might imagine. The extracts from _Berlin Twilight_ by Lieutenant-Colonel W. Byford-Jones (Hutchinson), _Man Without a Face_ by Markus Wolf (Jonathan Cape), _Goodbye to Berlin_ by Christopher Isherwood (Hogarth Press) and _The Ugly Frontier_ by David Shears (Chatto & Windus) are courtesy of the Random House Group Ltd.
I sincerely thank the following for extracts: Aufbau-Verlag, Berlin for _Der Sturz_ by Reinhold Andert and Wolfgang Herzberg; Peter Owen publishers for the quotation from _Demian_ by Hermann Hesse (Panther); A.M. Heath & Co. Ltd for _The Ides of August_ by Curtis Cate (copyright © Curtis Cate, 1978); the Orion Publishing Group Ltd for Willy Brandt: _Portrait of a Statesman_ by Terence Prittie (Weidenfeld & Nicolson); Duke University Press for _We Were the People: Voices from East Germany's Revolutionary Autumn of 1989_ by Dirk Philipsen; Continuum for the _German Democratic Republic_ by Mike Dennis; the History Place (webmaster@historyplace.com) for the full text of John F. Kennedy's speech in Berlin in 1963; Rainer and Alexandra Hildebrandt for _Berlin: Von der Frontstadt zur Brücke Europas_ and _It Happened at the Wall_ ; the University of Massachusetts Press for _The Wall in My Backyard_ by Dinah Dodds and Pam Allen-Thompson; HarperCollins for _The Siege of Berlin_ by Mark Arnold-Forster; Sanga Music Inc. for the lines from _Where Have all the Flowers Gone?_ by Pete Seeger; Norman Gelb for _The Berlin Wall_ (Michael Joseph, 1986); George Bailey for _Germans: Biography of an Obsession_ (Free Press, New York) and _Battle-ground Berlin_ (with David E. Murphy and Sergei A. Kondraschev, Yale University Press); the British Army HQ in Germany (and thanks to Helga Heine for smoothing the way) for the Friday 17 November 1989 issue of _Berlin Bulletin_ , the magazine published by Education Branch, HQ Berlin Infantry Brigade for British Forces, Berlin; ITPS Ltd, on behalf of Routledge, for _Eastern Europe in the Twentieth Century_ by R.J. Crampton; Christian F. Ostermann, Director, Cold War International History Project, The Woodrow Wilson Center for Scholars Project, for _Khrushchev and the Berlin Crisis and Ulbricht and the Concrete 'Rose'_ ; the Associated Press for their reporting of escapes and escape attempts in the 1980s.
I owe special thanks to Yorkshire Television, for allowing me to quote verbatim from their emotive and emotional documentary _First Tuesday_ on relatives of those who died at the wall. The Bureau of Diplomatic Security of the Department of State, Washington, DC, has allowed me to use a memorandum exploring options before the wall was built, and thanks to Andy Laine for help there as well as the late John Ausland for providing it. Irene Böhme gave permission to quote from her book _Die da drüben_ (originally published by Rotbuch Verlag, Berlin, in 1982) in a charming letter.
## _Prologue_
Commander Günter Moll walked briskly across the concrete concourse which was carpeted by white light falling softly from the banks of tall arc-lamps. He was a small, neat man and, like many professional soldiers, his uniform seemed moulded to him. Five o'clock, almost to the minute, and his shift had ended.
He reached the car park at the far side of the concourse, eased himself into his Skoda, settled and fired the engine. He had no need to glance back at the checkpoint because he knew precisely how it was functioning, understood all the predetermined clockwork motions which kept it running as it had run twenty-four hours a day for twenty-eight years. It ticked at a slow, even, careful pace to regulations of great exactitude. He'd handed control to his deputy, Major Simon, a competent, reliable man, and as he drove away his mind was at peace. Another day had ticked by.
The checkpoint lay broad and deep, some 50 metres by 50, hewn out of a city centre and constructed in a clearing among ordinary streets. For the tourist who chanced upon it for the first time, the impression remained invariably the same: profound incomprehension that an armed encampment could be a few footsteps beyond shops and a corner café. At that first glance it all seemed bewildering – the wall, the watchtowers, the death strip – and only when you came to know it did the geography and the geometry make sense. The checkpoint was at the precise point where East and West met.
To Moll it was known as the crossing on Friedrichstrasse, the street it straddled. To most of the rest of the world it had another name altogether, although quite why the tourist would probably have been unable to say.
A generation before, the US Army decided that checkpoints should be designated in alphabetical order. To travel across East Germany one went through _Alpha_ and _Bravo_ , and now here was the third. The name had a simplicity, a resonance and an alliteration which made it and its connotation recognisable on every continent.
It was called Checkpoint Charlie.
The office Commander Moll left was on the second floor of a pastel-shaded building, and through a square, squat window he could survey the expanse of the checkpoint, his checkpoint, as it faced the West: successively the Customs offices under a vast roof covering the centre of the concourse; a plain area beyond that, then three watchtowers, then the wall itself, white, 12 feet high and of vertical slabs so smooth that they offered no grip to a hand. The wall flowed along the extremity of the checkpoint to the left and right like arms, flowed on mile after mile, making an encirclement so that it locked the Western half of the city in a military embrace.
The arms of the wall folded into the checkpoint and, lower here – only shoulder high – ran across in front of the watchtowers. Two gaps had been left in it like mouths, a narrow one for pedestrians, a broader one for vehicles. The gaps allowed the old road – Friedrichstrasse – to come through the checkpoint: same road, same width, same name but the different ends of it completely separated.
Even these fortifications had been tightened a month before by the erection of a hip-high forward barricade – three tiers of concrete laid horizontally and wire mesh fencing secured to the top – although it, too, had the two mouths for pedestrians and vehicles.
Daily, Moll controlled this geometry of division, and the leader of his country had, only months before, proclaimed proudly that – no matter how many people condemned it as primitive and inhuman – the division would endure for another hundred years, maybe more.
As Moll drove the small car to his apartment in the suburbs he was not only at peace, he was a man of certainties in a country of certainties. When he reached the apartment, while his wife Inge cooked the evening meal, he'd watch television.
And did.
And the world moved.
It was 9 November 1989.
## ONE
## _Fault Line_
The construction workers of our capital are for the most part busy building apartment houses, and their working capacities are fully employed to that end. Nobody intends to put up a wall.
Walter Ulbricht,
June 1961
Looking back on it, the mixture of madness and dreams seems logical, with each step leading inexorably to the next but, even so, dividing a major European city by a wall and for twenty-eight years killing anyone who tried to cross it without the right papers still stretches credulity and probably always will; but this is what happened to Berlin and this is what happened to ordinary human beings who lived and died with it.
The credulity is stretched even further because the division wasn't a neat thing: not Paris split either side of the Champs-Élysées, not London camped on either bank of the Thames, not New York riven into East and West of the Avenue of the Americas to make two separate, sovereign countries who hated each other. No: Berlin was bisected along ancient, interwoven, interlocking district boundaries so that the division zigged and zagged through sixty-two major roads, over tram tracks, round a church and through its cemetery, across the frontage of a railway station and, stretching the credulity to its absolute limit, clean through the middle of one house. Each day an estimated 500,000 people had circulated quite normally through what would become the two hostile, alien lands. To take a random year, every day in 1958 some 74,645 bus, tram and underground tickets were sold in the West to people from the East, and that didn't include the extensive overground rail network. Some 12,000 Eastern children went to school in the West.
The interlocking of the wall as it zigzagged across the belly of the city cutting main streets.
There was a street called Bernauer Strasse where the apartments stood in the East but the road in the West so that, by opening their front doors, residents stepped from one side to the other. There had been the subway network (U-Bahn), a spider's web, serving the whole city, and the overground train network (S-Bahn) fulfilling exactly the same function. Both were cut by the wall but still overlapped. There had been the sewage system, and electricity, and telephones, and postal districts as common to both sides as they would be in a capital city united since (depending on which date seems conclusive to which historian) at least 1230. There had been waste disposal and burying the dead and walking in the woods on a Sunday afternoon.
No city had ever been subjected to anything like this: a son being refused permission to attend his mother's funeral because he happened to live on the wrong side of the street.
Nor was that all. The dividing line represented the exact point where the two dominant political and financial systems of the twentieth century, each the opposite of the other, met. At Check-point Charlie in 1961 a jolly East German Border Guard called Hagen Koch had been given a bucket of white paint and a brush and told to paint a line across the road. It was, maybe, 6 inches wide but it held apart the equivalent of two tectonic plates: precisely _this_ side was where the power of the United States and its allies ended, precisely _that_ where it ended for the Soviet Union and its allies. That both were immensely armed with nuclear weapons, and that escalation towards their use could well begin with some trivial incident somewhere like here, made Hagen Koch's line a defining place. For three decades people came to gaze at it and what lay around it, and were frightened. They were not wrong.
What logic, what inescapable steps, led to this – both the beginning of it in 1961 and the end of it in 1989? The journey to the answer is itself tortuous and improbable but I am persuaded that without undertaking it – even short-stepping it, as we shall be – you cannot understand why the watchtowers went up or, those twenty-eight years later, why you could have owned one provided only you could take it away and give it a good home.
This book moves in two dimensions, background and foreground. The background is the story of the wall itself and the foreground is what it did to ordinary people. To understand that properly you need the context and the logic. Here it is.
When the war in Europe ended, at 2.41 a.m. on 7 May 1945, central Berlin resembled a moonscape, and that's as good a place as any to set out on the journey. There's a phrase which lingers down the years like an echo of the wrath which had been visited upon the city, _Year Zero_ , as if, now that everything had been destroyed, the trek back to civilisation must begin from here. Few Berliners thought like that yet, because the future meant surviving until tomorrow.
The bombing by the Royal Air Force and the U.S. Army Air Force broke Berlin – not the spirit of the people but most of the infrastructure and a high percentage of the buildings. Statistics are useless to convey the scale. Imagine, instead, gazing down on an area of 2 or 3 square miles and _every_ building there like a blackened, hollowed, broken tooth. Now imagine what that looked like from ground level.
The Soviet armies reached the city in the spring of 1945 and, fighting street by street, conquered whatever resistance remained. There are mute echoes of that still in the East, where the old stone buildings bear the scattered pockmarks of gunfire.
This Prussian city of broad avenues and heavy architecture, cathedrals and churches, embassies and opera houses, hospitals and universities, pavement cafés and naughty nightclubs – this widowed place – had been home to 4 million confident, cheeky people who'd walked the walk and talked the talk of a capital city; and now ate horsemeat if they could find it.
There's another phrase which lingers, _Alles für 10 Zigaretten_. It was the name of a stage review but it captured the plight of what remained of the 4 million: 'Everything for 10 cigarettes'. Money – the Reichsmark – had no value and the currency passed to cigarettes.
Hitler never liked Berlin but, from taking power in 1933, ruled from the Chancellery, a Prussian building – heavy and classical – in the city centre.1 Dictators and diplomats came to its courtyard and walked down its marble gallery to the intimidating office where Hitler would inform them what he had decided for the world. The Chancellery was a partial ruin now: not a broken tooth but a badly beaten face. Its landscaped gardens, where his body had been burnt on the afternoon of 30 April, were cratered and grotesquely strewn with debris like a mini-moonscape.
Now the Soviets were here in their baggy uniforms, and the absolute power had passed to them. Soon enough they'd bring the exiled German communists back from Mother Russia and exercise the absolute power through them.
Major-General Wilhelm Mohnke, who'd commanded the government area, would remember2 after his capture being driven out of the city and 'coming towards us column after column, endlessly, were the Red Army support units. I say columns, but they resembled more a cavalcade scene from a Russian film. Asia on this day was moving into the middle of Europe, a strange and exotic panorama. There were countless _Panya_ wagons, drawn by horse or pony, with singing soldiers perched high on bales of straw.' He added: 'Finally came the _Tross_ or quartermaster elements. These resembled units right out of the Thirty Years War [between Catholics and Protestants at the beginning of the seventeenth century]. All of those various wagons and carts were now loaded and overloaded with miscellaneous cumbersome booty – bureaux, and poster beds, sinks and toilets, barrels, umbrellas, quilts, rugs, bicycles, ladders. There were live chickens, ducks, and geese in cages.'
On the first days after the surrender, many soldiers in Berlin sampled the victors' spoils and no woman was safe. Four decades later, a sophisticated lady summed this up in a phrase: 'And the Russians had their fun.' She looked away when she said it. A shiver of fear had run through Berlin, and nobody knows for how many women – young, old – it was justified; but a lot.
The Germans had fought a barbaric war in the East and now the barbarism had come back to them.
The German communist leader was called Walter Ulbricht, a dour Saxon with a Lenin goatee beard who'd spent the war in Moscow. Several of his comrades had disappeared in the night as Stalin carried out mini-purges but he had survived by unquestioning loyalty and tacking in the wind. One of the men who would accompany Ulbricht, Wolfgang Leonhard, has described it graphically:3 'At six o'clock in the morning of 30th April, 1945, a bus stopped in a little side street off Gorky Street, in front of the side entrance of the Hotel Lux. It was to take the ten members of the Ulbricht Group to the airport. We climbed aboard in silence.' They were driven to Moscow airport and flown to Germany in an American Douglas aeroplane.
On the drive into Berlin 'the scene was like a picture of hell – flaming ruins and starving people shambling about in tattered clothing; dazed German soldiers who seemed to have lost all idea of what was going on; Red Army soldiers singing exultantly, and often drunk'. The Berliners, waiting in long queues to get water from pumps, looked 'terribly tired, hungry, tense and demoralised'.
Ulbricht, by nature a bureaucrat and organiser, set about creating an administration, and it would include non-communists to make it seem fully representative. He told Leonhard: 'It's quite clear – it's got to look democratic, but we must have everything in our control.'
The communism which the Ulbricht group brought was based exactly on the Soviet model and, with Stalin watching in all his suspicion and malevolence, would not deviate from that regardless of whatever happened. Leonhard put it this way. 'When he [Ulbricht] came to laying down the current political line, he did it in a tone which permitted no contradictions.'
The Soviet model which Ulbricht brought was already completed in every fundamental, because Stalin had done that through the 1920s and 1930s. It controlled _everything_. Built onto that was a further factor. No East European communist government came to power with the legitimacy of winning free elections, and each was pathologically suspicious of its own citizens. The logic of this, too, would play itself out.
Germany was no stranger to communism: at one point before Hitler seized power the Communist Party had 100 seats in the Reichstag. Many working-class Berlin districts remained solidly communist all the way to Ulbricht arriving. They believed that communism was a scientific path to peace, prosperity and justice for all and represented the only sane future. They had experienced market economics and had their life savings destroyed in the financial crash of 1929 (when inflation became so intense that diners paid for their meals in restaurants course by course because the price was rising as they ate). They had experienced democracy and it had brought them Hitler. Communism looked very attractive as the women of Berlin formed chains and began to clear the mountains of rubble by passing bricks from hand to hand, and the trek back to civilisation began.
Germany was carved into four Zones: Soviet, American, British and French; and, mirroring that, Berlin was carved into four Sectors. The Soviet Union took the east of the city (and its 1 million inhabitants), the Americans, British and French fashioning their Sectors out of the west (and its 2.2 million). In retrospect, and even knowing where the logic would go, it is extremely difficult to imagine how such an arrangement could have endured untroubled, because Berlin lay 130 kilometres inside the Soviet Zone. Air corridors were formally agreed but land links were not. The autobahn from West Germany to West Berlin stretched like an umbilical cord and Stalin could sever it at any moment he wished.
The Americans, British and French should have taken over their Sectors as soon as the war ended but the Soviets stalled them and it didn't happen until 4 July 1945. A Kommandatura was set up by the four occupying powers and the official statement said 'the administration of the "Greater Berlin" area will be directed by an Inter-Allied Governing Authority... and will consist of four Commandants, each of whom will serve in rotation as Chief Commandant. They will be assisted by a technical staff which will supervise and control the activities of the local German organs [organisations].'
The division of Germany immediately after the war.
The population had other concerns. A 10-year-old called Peter Schultz, who would go on to become a distinguished radio reporter, lived with an uncle.4 'I remember destroyed houses and no traffic at all except the military. I remember a jeep with a very big black American sergeant and he lifted me onto it and he gave me American-Canadian white bread. This is all I can remember about West Berlin in 1945. It was important that I got white bread: American soldiers lived downstairs and they gave me a lot of bread, for me and for the whole family. So we had bread and – this was the most important thing – white bread. I will never forget that. I can still taste it.'
In April 1946 the Communist Party merged with the Social Democrats and became the SED, which would govern East Germany throughout its life, but there were elections that autumn and the SED did badly against the other surviving parties. In Greater Berlin they finished third with 19.8 per cent of the vote and, from this moment on, would never allow another free election in the area under their jurisdiction. There would be further elections, however, on 18 March – 1990.
In retrospect, the Berlin Agreement carried too many anomalies and too many practical difficulties, heightened when the political differences between the Soviet Union and the Allies reasserted themselves as the warmth of shared purpose – defeating Hitler – iced over. That Berlin was an island deep inside the Soviet Sector made it a nerve centre between what would become NATO in the west and the Warsaw Pact in the east.
R.J. Crampton sums it up neatly:
By 1947 the British and American Zones had separated almost entirely from the Soviet and in March of that year the division of Europe into two hostile blocs became much more rapidly focused. The communists left the governing coalitions of France and Italy, the Truman doctrine warned against communist attempts to expand into Greece, and in the summer the Soviets insisted that the east European states should not take Marshall Aid. In May 1948 the new currency introduced into the three western zones of Germany provoked the Soviet blockade of Berlin and the Berlin airlift.5
To pass across this terrain citing the steps is all too tempting, but it misses the human element entirely. A British Lieutenant-Colonel, W. Byford-Jones, visited the city in 1947 and wrote:
Epidemics were rampant. The water supply was polluted – there were 521 major breaks in pipes of over 21 inches in the British Sector alone, and 80 per cent of the sewage was not reaching the sewage works. All but one of the 44 hospitals in the British Sector were badly damaged, and the 5,817 beds available were all filled, with long waiting lists. There were no medical supplies, not even anaesthetics, heart stimulants, or sulphonamides. Food was poor and at starvation level... .6
Bodo Radtke, a Berliner who was to become a leading East German journalist, says that 'it wasn't until 1948 that you felt things were getting back to normal. Every day you saw or heard something: one day, two U-Bahn stations are open again, _very good_ , then three stations, then the bridge over the river Spree is open, _oh very good_.'7
On 20 March 1948, the Soviet delegates walked out of the Allied Control Council and never went back. They were unhappy at how Marshall Aid was affecting Soviet influence throughout Germany and, in an effort to force the Allies from West Berlin, Stalin threatened to sever the umbilical cord: from 1 April road, rail and canal traffic was hindered crossing the Soviet Zone to West Berlin. On 18 June, to Stalin's fury, the west replaced the worthless Reichsmark with the new Deutschmark. Almost immediately it brought an end to the black markets and stimulated industry.
The British and principally the American air forces were able to sustain West Berlin by an astonishing airlift which lasted until May 1949. The Allied pilots, some of whom had been bombing the city barely four years earlier, were now keeping it alive. Enemies were becoming friends.
From Moscow, the perspective was very different. The Soviet Union had been invaded by Hitler in 1941 and almost torn apart by cruelty on a scale unimaginable. It may be that 20 million people died in the Soviet Union: Stalin, and all his successors up to Mikhail Gorbachev, regarded their primary duty as making sure that this never happened again. Stalin constructed a buffer zone of states – Romania, Bulgaria, Hungary, Czechoslovakia and Poland – between the motherland and Germany; and would go further.
That same May in 1949, the Federal Republic of Germany (FRG), embracing the American, British and French Zones, was born, but the Four Power status of Berlin remained unaltered. In October, the Soviet Zone became the German Democratic Republic (GDR), so that now two German states, both insisting they were sovereign, faced each other. The GDR government, however, claimed that East Berlin was their capital and called it simply Berlin. To them, West Berlin was _Berlin (West)_ or variations of that, a severed limb at first then eventually left blank on their maps: a dead limb.
From Moscow, the Deutschmark, Marshall Aid and possible FRG rearmament represented great danger. Compounding that, Stalin demanded reparations from the Germans and made the East pay them. He stripped his zone of perhaps 26 per cent of its industry, dismantling it and shipping it to the Soviet Union. In May and June 1945, about 460 Berlin enterprises were 'completely dismantled and transferred'.8 (Reparations, worth a total of 34.7 billion marks at 1944 prices were paid until 1953. It was a crippling disadvantage to the East just when the FRG's economy was racing.)
The postwar misery of Berlin could be expressed by many witnesses. One of them, Jacqueline Burkhardt, came to the city 'in 1949. I was born in Düsseldorf where my mother came from but my father was from Berlin. In my grandfather's will he left my father a big old apartment house in Schöneberg and we came to live in it. I was nine. Berlin was totally flat. I saw absolutely nothing here. It was total devastation, absolutely _down_. There were no cars, only military vehicles. Berlin had nothing to sell. There was nothing to eat. I was very small and always hungry – you couldn't buy food. I remember in the house was an old woman who had a ration card which got her extra rations and extra bread. For many weeks I ate the crust of the bread because this old woman had bad teeth and couldn't eat it. I lived on that and water. It was a horrible time. Then German marks were introduced and it was a chance to start anew.'
A flow of refugees began from East to West, some political and some economic. (For an explanation of the use of capital letters for East and West, please see the introduction to the Notes at the end of the book.) The total ran at over 2,000 a week in 1949, rising to 4,000 in 1950, dropping slightly in 1951 and 1952 then reaching 6,000 in 1953, when Stalin died and workers in East Berlin rose up over an increase in the work norms. That 'insurrection' had to be put down by Soviet tanks and for a moment the GDR's survival hung in the balance. It made Ulbricht's government even more pathologically suspicious of its own citizens.
By the time Nikita Khrushchev came to power the refugee flow had become self-generating. Professional, educated, trained people – the builders of the future – were leaving and their absence made conditions worse, and the flow increased. The East looked, and was, threadbare, the West looked, and was, increasingly affluent. _Berlin (West)_ beckoned magically from the far end of the street, across the park, a stop on the S-Bahn or U-Bahn. In Bernauer Strasse, you just opened your front door.
Because the city was under Four Power control, and open, whatever the GDR said, East Germans could make their way to the refugee centre in Marienfelde, a southern suburb of West Berlin relatively unhindered. From there they were flown to the FRG. And they kept on coming: 5,000 a week through 1955, 1956 and 1957. Despite that, Bodo Radtke says, 'this was the happiest time. You could see construction everywhere, apartment blocks going up for people to live in.'
The act of having fled remained sensitive for decades afterwards. The story of a man who would only describe himself as Mateus ('that should be enough') might be seen as typical.9 His father had worked in the southern GDR town of Carl Zeiss Jena and 'in 1955 the family decided to move to the West. He took the chance of a congress in the West to stay there, and he organised a new life. My mother started to sell what could be sold discreetly, and sending parcels to the West. Then in 1956 we went over, my mother, my sister and I. My sister was five years older than me – I was seven. Mother did not explain to me what was happening but probably my sister knew. It was better that I didn't know, because my mother was afraid I'd start talking if I was asked questions about where we were going. We first travelled to my grandparents just north of Jena and from there to Berlin by train with two suitcases. She left these suitcases with relatives in East Berlin.
'I can't remember the crossing point. It seemed a sort of open border with guards who were on patrol and who controlled the passports of the people trying to go West. We simply walked across – people were moving across. My mother later told me the story: just before we crossed she slapped me, I started crying like hell and the Border Guard was pretty annoyed about this kid making all the noise. "Get him out of here!" So that went OK. She dropped us at American friends who had contact with my father and went back to get the suitcases. We learned later that she was arrested. We waited for two days for her and that was terrifying. My sister fully understood what had happened and that we might not see our mother again.
'She was in jail for the two days but then they had to release her because there was nothing they could actually do. They didn't have anything against her. Finally she made it but without the suitcases and the three of us flew out to Frankfurt on an old American military machine with metal seats like bath tubs. We avoided going into a refugee camp there because my father had arranged everything.'
The absurdities and anomalies of the future were largely unforeseen, but not everywhere. A young man called Kurt Behrendt,10 just married in September 1957, needed an apartment. There were some in a scenic hamlet called Steinstücken, in the south of West Berlin not far from Potsdam and bordering Babelsberg, which had been the centre of German film-making before the war and was now in the East. He didn't know Steinstücken, he didn't know Babelsberg and he didn't much mind because an apartment was an apartment. 'I came on the day the Soviets sent up the first satellite: 4 October 1957.'
Although older residents naturally had family and friends in Babelsberg, Behrendt found himself living in what was now an enclave. 'It was difficult because, since the political division of Berlin in 1945, they argued whether it belonged to the East or the West. Then they recognised that it belonged to the West but one of the difficulties was getting to it. You came over a bridge which, prior to 1945, belonged to Potsdam but after 1945 belonged to the GDR. So you had to come through the GDR and only residents were allowed to do that. There wasn't even a street, only a path through the forest. You could get through by car but it was so narrow that if two cars met one had to pull onto the side. The school was in Wannsee [near a lake] and the children went along this path to the bridge and took the bus there. A little later the community had a school bus and my wife Helga drove it.'
The absurdities and anomalies would revisit Behrendt, becoming more serious each time.
In 1958 the refugees came at 4,000 a week, and in 1959 at 3,000. The overall total in 1959 stood at 143,917, of which 90,862 had passed through West Berlin and the rest from the GDR to the FRG. In 1960 the refugees came at 4,000 a week with a record 16,500 in May. The overall total stood at 199,188, of which 152,291 had passed through West Berlin.
This was the equivalent of losing a town a year, and no small country could survive it for long. Ulbricht saw this as clearly as everybody else and exhorted Khrushchev to staunch the flow. Khrushchev made several attempts to force the Allies out of West Berlin, threatening to sign a unilateral treaty with the East German government – but that would mean unilaterally ending the Four Power Agreement, and unilateralism was extremely problematical in a nervy, nuclear era.
And still the refugees came. After processing, most sat huddled shoulder to shoulder on mattresses in the centre at Marienfelde. Its interior was wide like an aircraft hanger, the children cradling their heads in their arms, already asleep. The women gazed ahead unseeing. Here and there a hand comforted a child, smoothing its hair in timeless rippling motions of warmth and reassurance. That masked anxiety. The Eastern Railway Police and Peoples' Police searched West-bound trains heavily now. Anyone with luggage might be hauled off, and any obvious family travelling together – father, mother, children – might be hauled off, too.
Friedrichstrasse station,11 which itself would become an absurdity and an anomaly, was the last stop before the West. Twelve policemen tried to check all the compartments as each train stopped but even on Saturdays, when the trains were more lightly filled, they struggled to check them all. Getting across became a matter of luck, chance, ill-luck, averting your gaze from a policeman's glance, keeping your nerve, looking innocent, literally sitting tight.
Some families took precautions and split up, the father going with a child one day, the mother following the next, perhaps alone – less suggestive like that – or perhaps with another child. Some left their suitcases behind, as the mother of Mateus had done, and went back to retrieve them only when the children were safely delivered to the West.
In January 1961 John Kennedy was sworn in as President of the USA and six months later he met Khrushchev in Vienna. In the background and far away, the mute, anxious, exhausted refugees still flowed in at 4,000 a week and Khrushchev's aides were joking that 'soon there will be nobody left in the GDR except for Ulbricht and his mistress'.12 In addition, the Soviets were concerned that their economic assistance to the GDR would 'end up in the pockets' of the Westerners because the Deutschmark had so much more purchasing power than the GDR's Ostmark.
At Vienna, Khrushchev went unilateral. Unless the Allies agreed to a German peace treaty within six months he would sign a treaty with the GDR 'normalising' the situation in Berlin. The Allied troops would have to go and West Berlin would become a free city with control of all access, including the air corridors, passing to the GDR.
'Mr. Khrushchev left no doubt as to his "irrevocable" determination to conclude a separate treaty with the GDR, with all attendant consequences for the west as already threatened. President Kennedy said the United States would regard any violation of vital rights of access and any encroachment on West Berlin as a breach of US rights and interests.'13
Frank Cash, who worked for the US State Department's Berlin Task Force, says14:
The bigger picture of interlocking Berlin: some of the districts on both sides of the wall and three Eastern towns, plus the Western enclave of Steinstücken.
I think Ulbricht pushed Khrushchev and Ulbricht was right because they were haemorrhaging with the refugee flow: all of these people who were coming out were those that he needed most, the young, active, bright engineers and professional people and he really had to stop it in some way. I also had the feeling, having attended the Vienna summit, that Kennedy really was pleading with Khrushchev to help him find a way out. I think that that was _Khrushchev's_ assessment of Kennedy and, OK, he – Khrushchev – could go ahead with a wall.
There's another thing that emerged from the Vienna meeting. Berlin came up [on the agenda] and the White House made much of the fact that the Allied response to Khrushchev's demands took 45 days to appear. What actually happened – and I know this very well – was: McGeorge Bundy [Kennedy's National Security Adviser] asked if we could prepare a policy and we did but when we rang the White House they said 'what document?' They'd lost it, so ten of the 45 days were really due to the White House. I'm not sure if Khrushchev took the delay as a sign of weakness.
We also looked back to other Soviet demands and our responses, and this was not an undue amount of time. All of Khrushchev's demands were a rehash of earlier demands and there was nothing new in it so we could go back to previous responses which had been agreed within the US government and say 'well, here's this point and here's our response'. But you want to get an answer out quickly. If you've got responses which have been previously cleared by the US government, the British and French, then you're going to have very little problem getting it cleared again and that's what we did.
I think the comment to our response from the White House, when it finally arrived, contained only one new idea – submission to the International Court of Justice of Berlin's rights – and from that point the answer as it went out was essentially the one we had submitted three or four days after we got the original request.
Did Khrushchev read much into this delay? Nobody knows.
From Sunday 4 June 1961, when the Vienna Summit broke up in bitter disarray, the logic became more insistent and the steps were closer, were taken more quickly and were moving towards an abyss. Even now, in retrospect, when the consequences of the logic should be clear, it is all but impossible to evaluate which of these steps increased the sense of crisis more. They have a cumulative feel to them, a sense of something very powerful in play.
When Kennedy returned to the United States two days after Vienna he made a television and radio address: 'Our most serious discussions dealt with Germany and Berlin. I made it clear to Mr. Khrushchev that the security of Western Europe, and with it our own security, is intimately interlinked with our presence in and our rights of access to Berlin, that these rights are based on legal foundation and not on sufferance, and that we are determined to maintain these rights at all cost and thus to stand by our commitments to the people of West Berlin and to guarantee their right to determine their own future.'
The flow of refugees rose to over 4,000 that first week after Vienna.
The Secretary of State, Dean Rusk, said15 that 'an attack on West Berlin would have moved rather quickly to a nuclear situation. Yes, I really think that. It was part of the Berlin planning all along, particularly since Khrushchev had given the president an ultimatum on Berlin at Vienna. So the British, the French and ourselves as well as the West Germans put our heads together to work out contingency plans and they were based on a gradual upping of the ante depending on the East German and Russian reaction. Kennedy had explained to Khrushchev that we could not accept an attack on West Berlin and, as a matter of fact, at one point during the conference Khrushchev said he was going to do this about Berlin and do that about Berlin and Kennedy said "there will be war. It's going to be a very cold winter." It was in the forefront of my mind that there was nothing we ought to try and do about East Berlin but any infringement into West Berlin would have set in motion the contingency plans.'
After Vienna, Khrushchev accepted Ulbricht's proposal that the Warsaw Pact countries should meet in Moscow as soon as practical.
The flow rose slightly above 4,000 in the second week.
On Saturday 10 June16 Yuli Kvitsinsky, an attaché in the Soviet Embassy in East Berlin, 'learned in a meeting with E. Huttner of the East German Foreign Ministry's department on the Soviet Union that many in the East German leadership felt that it was "high time, at last, to sign a peace treaty and on that basis resolve the West Berlin issue. This measure is connected with a certain risk, but there is even more risk in the further delay of the resolution of the issue, since any delay assists the growth of militarism in West Germany which increases the danger of a world war. Thus, the danger of conflict in connection with the conclusion of a treaty with the GDR is balanced by the other danger. In this connection... Comrade P. Florin, chairman of the SED CC International Relations Department, said that further dragging out the signing of a peace is a crime." It was also assumed that at some point "the sectoral border in Berlin would be closed."'
When Hope M. Harrison [see note 15] asked Kvitsinsky about this in an interview, he said: 'There was not then on our part a real readiness to conclude a [general overall] peace treaty with Germany. We were not impatient. But the GDR was impatient and in a weaker situation and Ulbricht used strong propaganda for a peace treaty.'
On Monday 15 June, Ulbricht called a press conference – itself highly unusual – at the House of Ministries (in another life it had been the headquarters of Hermann Goering's Luftwaffe) and, during it, restated the position outlined by Khrushchev's ultimatum. Annamarie Doherr, a journalist of the western _Frankfurter Rundschau_ , asked 'Does the formation of a Free City in your opinion mean that the state boundary will be erected at the Brandenburg Gate?'
'I understand by your question,' Ulbricht replied, 'that there are men in West Germany who wish that we [would] mobilise the construction workers of the GDR in order to build a wall. I don't know of any such intention. The construction workers of our country are principally occupied with home building and their strength is completely consumed by this task. Nobody has the intention of building a wall.'
Why Ulbricht said this is not known. One theory suggests that he was simply telling the truth as he perceived it at the time, another that he must have understood the effect his words would carry within the GDR, particularly when the press conference was extensively reported in Eastern newspapers and on Eastern television. Ordinary people, accustomed to reading the inner meanings of doublespeak, would assume he was going to build a wall – provoking a rush to Marienfelde. That would force Khrushchev's hand because Khrushchev could not afford to let the GDR die.
On Wednesday 28 June, a senior GDR International Department official returned from Moscow and told Ulbricht that the Soviet Presidium would discuss the request for the Warsaw Pact meeting on the following day. The steps within the logic were very short now.
In the third and fourth weeks of June the flow of refugees continued at over 4,000 and by the month's end, only halfway through the year, the overall total had reached 103,159, with 49.6 per cent under the age of 25. The future itself was flowing away.
Former Soviet diplomat Yuli Kvitsinsky wrote in his memoirs17 at the end of June or the beginning of July that Ulbricht invited him and Soviet Ambassador Mikhail Pervukhin to his country house. There Ulbricht told Pervukhin to inform Khrushchev that 'if the present situation of open borders remains, collapse is inevitable' and that 'he refuses all responsibility for what would then happen. He could not guarantee that he could keep the situation under control this time [the haunting of the 1953 uprising].' Kvitsinsky is not more specific about when this took place but notes that after it nothing seemed to change.
Then, according to Hope M. Harrison, a political scientist who has researched this period exhaustively, 'one day' Pervukhin told Kvitsinsky to find Ulbricht immediately and bring him to Pervukhin. Pervukhin informed Ulbricht that Khrushchev had agreed to close the border and that Ulbricht should begin preparations in great secrecy. The operation was to be executed very quickly so as to be a complete surprise to the West. 'Ulbricht immediately went into great detail about what must be done. He said that the only way to close the entire border quickly was to use barbed wire and fencing. He also said that the U-Bahn and S-Bahn to West Berlin must be stopped and that a glass wall should be put up at the main Friedrichstrasse train station so that East Berliners using the metro could not change over to the train to West Berlin. Kvitsinsky noticed that Pervukhin was quite surprised at how much Ulbricht had already thought through these details.'
On Monday 3 July, Ulbricht spoke to the 13th plenum of the SED. He said a peace treaty was imminent and assured the comrades that Khrushchev would act before the FRG had nuclear weapons. He outlined how action would be taken in Berlin against the so-called border-crossers: people who lived in one sector but profited by working in the other.
Next day, Pervukhin submitted a detailed sixteen-page report to Andrei Gromyko, the Soviet Union's Minister of Foreign Affairs, with his analysis of what signing a peace treaty with the GDR would bring. He wrote of 'the establishment of a regime over the movement of population' between the two halves of Berlin and felt 'it would be better at first not to close the border, since this would be difficult technically and politically'.18
The total that week was 4,000 again.
On Friday 7 July, Erich Mielke – head of the GDR secret service (the Stasi) – told his senior officials he was ordering immediate preparations so that 'operative measures can be carried out at a certain time according to a united plan' and demanded 'a strengthening of security of the western state border and ring around Berlin'.19
On Saturday 8 July, Khrushchev spoke at a military academy and said the demobilisation of Soviet armed forces had been halted. That second week the flow rose to 8,000.
On Saturday 15 July, an unsigned report to Ulbricht spoke of how, after the writer of the report had had conversations in Moscow, 'we should especially expect to deal with questions about West Berlin' at the Warsaw Pact meeting, now scheduled from 3 to 5 August.
Lutz Stolz and his fiancée Ute lived in East Berlin but went to visit Ute's aunt in the Black Forest. 'At this time you had to apply for an interzonal passport so I went to the authorities and applied for Lutz and myself,' Ute says. 'First, they only gave me a passport and wouldn't allow him to go. I applied a second time and, although I really don't know why, I succeeded. I remember the day well. I went along the little side street jumping up and down the kerb with joy waving the passport in my hand and saying "we can both go, we can both go".
'We went by train to Stuttgart where our relatives were waiting for us and they took us to their little village, Rohrdorf, a very romantic place in the mountains.' This happy, almost idyllic, time would haunt the couple. 'We had our work, our parents, the houses of our family all in the GDR, and the houses, even in the GDR, were worth something. We can never say whether we'd have stayed if we'd known. My parents-in-law could have sold up in the GDR and come as well, everything would have been done fast – but how could we know then? Would we really have stayed in the little village in the mountains, leaving behind the security of our family? Would we really have dared? When we were leaving we told my aunt and uncle that we'd try and visit them again the following year. It never occurred to us that...'
On Monday 17 July, the three Western Allies rejected Khrushchev's ultimatum to change the status of Berlin. That third week the flow rose to 9,000.
On Saturday 22 July, a secret telegram from the State Department in Washington went out to the London, Paris and Moscow Embassies as well as the US Embassy in Bonn and to Berlin. It was arranged in nine numbered paragraphs, phrased in a sort of cabalese, and it explored possibilities. It said that 'if refugee flood continues' the GDR could 'tighten controls over travel from Soviet Zone to East Berlin or by severely restricting travel from East to West Berlin', and added: 'We believe Soviets watching situation even more closely than we, since they are sitting on top volcano. Continued refugee flood could... tip balance towards restrictive measures.'
The telegram also contained a great truth: 'If GDR tightens travel controls between Soviet Zone and West Berlin there is not much US could do, other than advertise facts. If GDR should restrict travel within Berlin, US would favour counter-measures' – including economic.
Lutz Stolz and Ute came back from Rohrdorf. 'When I went to the authorities to get our identity cards back – we had had to leave them before our departure – I noticed that they were smiling in a strange way when they handed them over,' she says. 'It was like a scornful smile on their lips as if they were thinking: _these stupid people, if only they knew_...'
The German Protestant Church Synod, due to be held in West Berlin from 19 to 23 July, was banned by the East Berlin chief of police.
On Tuesday 25 July, Kennedy spoke on American television. He defended the Allies' rights in West Berlin and stressed that any unilateral action against West Berlin would mean war with the United States, but made no mention of _East_ Berlin, a deliberate nuance, perhaps, which Khrushchev would deduce as giving him a free hand there.
On Wednesday 26 July, Ulbricht sent to Moscow a summary of the speech he would be making at the Warsaw Pact meeting there nine days later. He proposed creating a state border between East and West Berlin.
On Sunday 30 July, Senator William Fulbright, chairman of the Committee on Foreign Relations, said publicly that he could not understand why the East Germans were not closing the border, because they had every right to do it. Under the Four Power Agreement, the GDR had no rights like this, of course.20 It may be, however, that Khrushchev and Ulbricht – who'd essentially spent the whole of their adult lives within totalitarianism – assumed Fulbright's words would have been sanctioned and were code for: do what you want in your own backyard.
Certainly Richard Smyser, a junior officer in the American Berlin Mission, thinks so. Smyser, an American born in Vienna (his father was a diplomat) had spent part of his childhood – 1939–41 – in Berlin and returned in 1960 to join the Mission. Smyser says that Khrushchev and Ulbricht 'took the Fulbright speech as the go-ahead. The Russians have told me that Ulbricht expected no military response.'21
That fourth week the flow fell slightly, to 8,000, but the total for the month reached 30,415, of which 51.4 per cent were under the age of 25. On 2 August alone the total was 1,322.
Many astute judges, particularly in the CIA and the State Department, had contemplated a division of the city as an abstract proposition and concluded that in practical terms it could not be done. At least one Stasi general, Markus Wolf – famed, fabled and feared spymaster – reached the same conclusion as he pondered dividing the East's 1,071,775 from the West's 2,207,984.
In the city, the halves interlocked intimately as they had for centuries and to prise them apart some sort of barrier would have to run for 28½ miles. From the north, and beginning in countryside, it would go westwards in a triangle at the first of the housing, follow a railway track with houses on both sides for 3 miles and embrace parallel railway tracks, one of them from the West but looping inside the line; then turn at a right angle into Bernauer Strasse; double back over the Western U-Bahn line beneath East Berlin, follow a narrow canal bank and bisect a bridge, run directly behind the Reichstag to the Brandenburg Gate and on to the broad expanse of Potsdamer Platz. It would then corkscrew left and bisect five streets, zigzag across seven more streets until it reached twin, curved roads with sunken gardens between them; and turn to follow the banks of the Spree to the district of Treptow, itself a maze. There the barrier would turn westward, bisecting half a dozen streets... .
To seal the GDR countryside looping round West Berlin, the barrier would have to run 22.30 miles through residential areas, 10.56 miles through industrial areas, 18.63 miles through woods, 14.90 miles through rivers, lakes and canals, and 34.15 miles along railway embankments, through fields and marshes.
Could such a thing really be done?
George Bailey, a veteran American journalist who knew Germany intimately, wrote a feature in a magazine called _The Reporter_ in which he said that the Soviet Union and 'its East German minions' have 'finally drawn the ultimate conclusion that the only way to stop the refugees is to seal off both East Berlin and the Soviet Zone by total physical security measures'. He foresaw 'searchlight and machine gun towers, barbed wire, and police dog patrols. Technically this is feasible.'22 The feature was billed as 'The Disappearing Satellite' and, in Bailey's words, 'attracted a good deal of attention'.
One account23 says Ulbricht flew to Moscow on the last day of July and was informed by Khrushchev that the peace treaty was not going to be signed yet but he could close the border. Ulbricht wanted to close the air corridors, too, which would prevent the refugees who did reach Marienfelde from flying on to West Germany, but Khrushchev weighed up the risks and said no. Ulbricht suggested building a wall around West Berlin _on East_ _German territory_ and Khrushchev said he would put that to the Warsaw Pact meeting. (For reasons of security, he cannot have done this in open session; rather, he sounded out various members in private.)
On Thursday 3 August, Khrushchev opened the meeting by accusing the 'Western powers' of receiving 'our proposal for a peace treaty with bayonets' and added: 'Kennedy essentially threatened us with war if we implement measures for liquidating the occupation regime in West Berlin.' He called for the meeting to work out detailed plans and gave the floor to Ulbricht.
In great and laboured detail, Ulbricht moved through the whole situation and made his pitch: how the peace treaty must be signed without delay and how the 'whole socialist bloc' must be ready to risk confrontation to protect the GDR, although he made no mention of a wall.
At some stage on 3 August,24 Khrushchev seems to have added a proviso to closing the border. Ulbricht must give an assurance that his government could deal with any civil unrest (the haunting of 1953 again). Ulbricht flew back to East Berlin on Friday 4 August and satisfied himself that such an assurance could be given. The _New York Herald Tribune_ prepared a front-page story headed ULBRICHT SAID TO SEEK ASSENT BY KHRUSHCHEV TO CLOSE BERLIN BORDER.
Their reporter Gaston Coblentz wrote that 'Ulbricht's move was... revealed by well-informed quarters.' He also noted: 'In addition to the highly reliable report of Mr Ulbricht's proposal to Mr Khrushchev, unconfirmed information said that he has made a secret flight to Moscow to put his plan before the Soviet leader.'25
That same Friday, the East Berlin Magistrat ordered that any Easterners working in the West – the border-crossers – had to register and pay their rent and bills in Deutschmarks. The three Western commanding officers protested about this to the Soviet Commander. The number of refugees for the day was 1,155, making a total of 11,000 for the week. Flood tide was near.
Dean Rusk flew from Washington to Paris for a meeting with the Foreign Ministers of Britain, France and West Germany. Before he left he spent an hour with Kennedy at the White House and said: 'There is peace in Berlin and there is no need to disturb it.'
On Saturday 4 August, Coblentz was reporting: 'The East German persistent "no comment" reply to all inquiries about Mr Ulbricht's movements since he was last seen in East Berlin Monday strengthened the belief he has been out of the country on a secret trip to the Kremlin.'
Ulbricht, as it seems, returned to Moscow on Sunday 5 August and gave the assurances. Khrushchev said the border could be closed with barbed wire – which could, if necessary, be removed as quickly as it was set down – and ordered the wire to be laid 'not one millimetre further' than the lip of East Berlin territory. There would be no incursion into the West. Ulbricht returned to East Berlin and summoned Erich Honecker, a trusted aide, and told him to draw up plans in the most laboured secrecy.
The total flow for the weekend was 3,268.
On Monday 6 August the West's Foreign Ministers in Paris called on the Soviet Union to provide a 'reasonable basis' for negotiating on Berlin. The meeting ended next day in what has been described as an atmosphere of calm confidence although, as they were sitting down to dinner in the private dining-room at the Quai d'Orsay (the French Foreign Ministry), news came through from Moscow of a 90-minute television and radio address by Khrushchev, during which he said he might call up military reserves.
The background to the Paris meeting remains important. Dorothy Lightner, wife of Allan – the minister at the American Berlin Mission – remembers26 that 'there was a meeting of the Foreign Ministers in Paris and the American ambassador, Walter Dowling, and Allan went there. Allan was to put in a plea for Berlin but they were given very short shrift. Nobody wanted to hear what they had to say. They wanted to tell them that the allies were not about to fight, bleed and die for the East Zone, so there was no question but that it was only about access. And once you know that, you can see that a policy had already been established. Allan was mad about that, yes. The decision perturbed him and it perturbed everybody in Berlin.'
John Ausland, who worked at the State Department on the Berlin Task Force, described how 'there is always tension between the State Department and the White House but this was the worst I have ever seen during my lifetime, and I've seen a number of examples – [John Foster] Dulles [who wanted to destroy communism] was pretty bad but it was nothing like the Kennedy situation. It was very hard to deal with people, and that went from top to bottom. That lessened as time went on.
'Part of the problem was the bureaucracy, which we overcame by organising the Berlin Task Force. When I returned to Washington in July of '61 from Australia I was shocked by the chaotic way business was being conducted – you know, people all over town were making decisions and not talking to each other. Right away I started talking with Martin Hillenbrand [of the State Department's German desk] and Foy Kohler [Assistant Secretary of State] and I said "we have got to get this thing organised". I am a kind of compulsive person who likes to organise things. I wrote a draft directive to take care of that but Kohler did not want to put anything in writing at that time. One day, as they were going off to Paris for the Foreign Ministers' meeting he called me in and said "I want you to go up to the Operations Centre and establish a Task Force." That was the beginning of it.'
Norman Gelb27 has written that 'this Operations Centre... was set up on the seventh floor of the State Department building, not far from Rusk's office. Among other things, it was to house a service special "task force" whenever one was organised for the purpose of gathering all pertinent facts bearing on a specific problem in foreign affairs and producing recommendations for action.'
The Berlin Task Force would consist of Kohler running it with Hillenbrand as his deputy, Frank Cash, Ausland, Arthur Day, David Klein, Jerry Holloway from the State Department, and [Wilbur] Bur Showalter, liaison officer to Assistant Secretary of Defence Paul Nitze.
On Tuesday 8 August, the flow reached 1,741, the next day 1,926.
Marshal Ivan Konev arrived from Moscow on Thursday 10 August to take command of all Soviet forces in the GDR and word of this spread. The _New York Herald Tribune_ carried the story and recorded: 'The East German news agency ADN made the move known shortly before midnight and reported that Marshal Konev conferred in East Berlin earlier today [this Thursday] with... Ulbricht.'
In East Berlin there were hints that special measures against the refugees might be taken at a meeting of the GDR People's Parliament the next day. Ulbricht, visiting a cable-making factory, warned his own people about spreading hostile propaganda. 'We will not tolerate revanchist agitation,' he said. All day, communist organisations within the GDR sent the government 'petitions' demanding action against the refugee flow and the ADN pumped this out. The total on this Thursday, most queuing in grey, sodden rain was 1,709 and the next day 1,532. It was Friday 11 August 1961.
The GDR Ministry of the Interior kept private statistics. Fifty-five members of the People's Police fled in 1959, sixty-one in 1960 and, an official report stated, 'this year up to now forty. It is a characteristic that the proportion of desertions is increasing.' Those who fled included all ranks from cadets to senior officers, some after fifteen years' service. Who knew who would be next, who could be guaranteed to stay? Within one more day the overall total of refugees would reach 16,167 so far in August. Taken all together, it was flood tide.
* * *
1. Hitler evidently disliked the city because of the Berliners' irreverence, indiscipline and delight in all carnal (and carnivorous) pleasures. Their wit was fast-moving and waspish, as befits any capital. Two examples:
The first: I met a single and independent woman who danced on the wall the night it came down even though she was pregnant. When she moved apartments, the concierge said – eyeing her enlarged stomach – 'But where is your man?' 'Man?' she replied, 'I'm the Virgin Mary.' And that, she reported to me, shut him up, all right.
The second: When Erich Honecker was ruling the GDR, he watched the sun rise in the East and asked it why it was smiling; and the sun said 'Because I'm going to the West...'
2. Quoted in _The Berlin Bunker_ , James P. O'Donnell (Arrow Books, London, 1979).
3. _Child of the Revolution_ , Wolfgang Leonhard (Collins, London, 1957).
4. Interview with author.
5. _Eastern Europe in the Twentieth Century_ , R.J. Crampton (Routledge, London and New York, 1994).
6. _Berlin Twilight_ , Lieutenant-Colonel W. Byford-Jones (Hutchinson & Co., undated).
7. Interview with author.
8. _German Democratic Republic_ , Mike Dennis (Pinter, London and New York, 1989).
9. Interview with author. The first time I went to Berlin to begin researching this book I travelled by train and, leaving on the way back, I sat next to a man in late middle age. We chatted and like all people touched by Berlin he had a tale to tell. He looked at me very hard, however, when my tape recorder came out. He said he'd call himself Mateus, and nothing more.
10. Interview with author.
11. Friedrichstrasse station will be fully examined later in the book, not least because in a city laden with symbolism it became, like Checkpoint Charlie, an almost _exquisite_ symbol, and in several different dimensions simultaneously. We shall see.
12. _Khrushchev and the Berlin Crisis (1959–1961)_ , Vladislav M. Zubok, Working Paper No. 6, Cold War International History Project (Woodrow Wilson Center, Washington, DC, 1993). The joke may have been the very idea of a pompous and portly bureaucrat like Ulbricht, who seems to have been one of the twentieth century's great bores, having a mistress at all.
13. _The Wall and How it Fell_ , Special Edition for the 5th Anniversary of the Fall of the Wall (Press and Information Office of the Land of Berlin, 1994).
14. Interview with author.
15. Interview with author.
16. _Ulbricht and the Concrete 'Rose': New Archival Evidence of the Dynamics of Soviet–East German Relations and the Berlin Crisis_ , Hope M. Harrison, Working Paper No. 5, Cold War International History Project (Woodrow Wilson Center, Washington, DC, 1993).
17. Ibid.
18. Ibid.
19. Ibid.
20. _The Berlin Wall_ , Norman Gelb (Michael Joseph, London, 1986).
21. Harrison, op. cit.
22. _Germans: Biography of an Obsession_ , George Bailey (Free Press, New York, 1991).
23. Harrison, op. cit.
24. Ibid.
25. _New York Herald Tribune_ , 4 August 1961.
26. Interview with author.
27. Gelb, op. cit.
## TWO
## _Saturday Night_
The Germans are a busy people
Peter Glau,
Owner, Hotel Ahorn, Berlin
## 12 AUGUST 1961
The first-floor apartment combined home and office. If the neat, busy man sitting at the typewriter gazed from the window over the rooftops opposite he saw an overcast morning, the light southerly wind pushing and prodding folds of cloud; saw at almost eye level the trains grunting and grumbling along the elevated rail in the middle of the road. Schönhauser Allee, a main artery from the centre of East Berlin, stretched long and broad and, like so many roads, remained tattered and exhausted all these years after 1945. The façades of darkened pre-war buildings loomed, guarding their memories.
Adam Kellett-Long worked for Reuters, a global news agency and the only one to have a bureau in East Berlin. He'd taken up the post in March and the GDR government, questing for international recognition, granted him more than usual access to themselves. Reuters had a bureau in West Berlin and for East Berlin to get one implied, to them, a step in the recognition process. Ulbricht called Kellett-Long 'my little shadow' as a term of endearment.
The day before, the People's Parliament had met in their chamber (it looked like an opera house) and Kellett-Long covered the meeting: 'A resolution was passed in effect giving the government the authority to take any action necessary to stop West Berlin being used as a propaganda hotbed ensnaring socialists and so forth.'1 On the way out, Kellett-Long ran into a contact, Horst Sindermann, a bespectacled senior communist official with the air of a businessman, his straight, short hair thinning over the crown of his head. Kellett-Long asked what the resolution really meant and Sindermann replied he couldn't say exactly 'but I would advise you not to leave Berlin this weekend'.
A newsman with the instincts of Kellett-Long didn't miss the possible significance, particularly since Konev's arrival.
Kellett-Long sat at his typewriter weighing up Sindermann's cryptic semi-sentence and, perhaps, subconsciously weighing up something else. After the Volkskammer meeting he called in to see 'the chap who used to get our coal and all our supplies from the Diplomatic Service Bureau. He was a member of the Factory Fighting Groups2 and he wore his uniform, he had his kit and his rifle.' Kellett-Long wondered why and the man said 'Oh, we are going off on an exercise this weekend.' The man didn't know where and, anyway, Kellett-Long thought, 'so what?' – but he'd never seen the man in his uniform before, and wearing it now seemed vaguely peculiar.
Kellett-Long reasoned that any steps to staunch the refugee flow would probably be taken during a weekend, when the tempo of the city was relaxed and most workers were at home. All else aside, that made a repetition of 1953 less likely. He composed a day-lead, standard procedure for Reuters correspondents. Usually a day-lead represented a synopsis of any situation overnight and a careful anticipation of what the day might bring. He wrote: 'Berlin is this weekend holding its breath for dramatic measures.' As he typed the words he understood 'I was sticking my neck out a hell of a long way but I trusted Sindermann.' The day-lead finished, he telexed it to London via the Reuters bureau in Bonn. London automatically processed it and fed it to subscribers all over the world.
The day-lead created an immediate effect. Reuters' News Editor in Bonn travelled to Berlin, arriving during the afternoon, and said 'We're right out on a limb, you'd better justify this.' Kellett-Long couldn't, because he'd constructed what he wrote on a tip and a hunch.
A news agency is, by definition, different to a newspaper. News agencies supply all branches of the media and are expected to provide probity, factual reporting and circumspection. Newspapers can write floral prose of their own and speculate as much as they want. Agency copy, as it is known, is used as a bedrock. Certainly newspapers from Australia to Singapore, Cape Town to Stockholm, Boston to Brazil would assume Kellet-Long's day-lead was based on hard fact – or he wouldn't have filed it. This is certainly what radio stations would have assumed, and broadcast it on the strength of that. If Kellett-Long had got this wrong, the error would be compounded by the fact that it was, quite literally, of global proportions.
The temperature climbed toward 20 degrees, and people took their leisure and their pleasure in the most ordinary way on this summer weekend which seemed so like any other. Heinz Sachsen-weger, who worked in a railway factory, spent agreeable hours with his wife and son at his brother-in-law's in Brandenburg, a solid town 50 kilometres away. Journalist Bodo Radtke and his wife relaxed in a chalet south of the city. Diana Loeser, a believer in communism from England, had travelled to Birmingham to see her family. Hagen Koch spent a day off relaxing with his wife. Erhard Schimke, a policeman who drove radio cars, spent a day off relaxing, too. None knew each other but they had something in common. All resided in East Berlin.
And still the refugees came. As they reached Marienfelde a flat, nasal female voice counted them in over loudspeakers: '763... 764... 765...'
Under the weight of these numbers Heinrich Albertz, town hall Chief of Staff to West Berlin Mayor, Willy Brandt, made a significant and worried telephone call. He rang George Muller, deputy political adviser at the American Mission, and asked if the US Army could provide field rations.
'When I got the call for food,' Muller says,3 'it contributed to the sense of crisis but we regarded our primary concern as safeguarding West Berlin and Allied access to it, so in a way the call was incidental to my other concerns. I did what I could to get food because the Reception Centre just couldn't handle the traffic. The refugees weren't allowed to eat in the dining halls until they'd been processed and with this stream of them the processing lines became so long people starved waiting. Part of Albertz's problem was to feed the people in line, especially the children.'
The nasal voice counted the refugees dispassionately in and the weight of numbers increased to a point where they overflowed into tents outside. Some wandered the centre's gardens taking the air, a father with a daughter hoisted over his shoulders walking and walking, the daughter constantly asking, 'When is Mummy coming?' To allay suspicion, she must have been travelling there by another route and was out there somewhere.
The dilemma in the West – of sensing that the East would have to respond to the situation but being unable to predict what that response might be – was a deep one. The CIA had seventy intelligence officers in their Berlin base. Donald Morris, who worked there from 1958 to 1963, has summed it up precisely:4 'We all knew something was going to happen. The border was too loose, there was a population drainage, a haemorrhage, from East Germany coming into the West. All of a sudden this haemorrhage assumed _arterial_ proportions. Something of a panic had set in, there were 2,000 refugees on a Monday, 4,000 on Tuesday, 6,000 on Wednesday. Ulbricht went to Moscow and got permission to lower the boom – I'm not using the wall – came back and did lower the boom. We knew the boom was about to fall, we didn't realise it was going to take the form of a physical wall surrounding West Berlin – but that is the form it took.'
To predict the building of the wall, 'you would have had to have had one of the top three men in the East German government to get even twenty-four hours' warning that a wall was going up. We had requirements to get agents in practically every sector of the eastern bloc, [but] to tell us that the logistical arrangements were being made would have involved recruiting several people in masonry supply and truck services. They probably weren't on the list... .'
Tom Polgar, a CIA man in Germany from 1947 to 1961, puts this into a broader perspective:5 'The United States government as a whole was not alerted to the possibility that something drastic was going to happen in Berlin which required an immediate American counter-action one way or another, whether physical or just oral [and] political. So in that sense it was certainly an intelligence failure but I think we have to be careful where we want to pinpoint the blame.'
To reinforce this, a British author – Terence Prittie – quotes Robert Lochner, head of the United States-sponsored radio in West Berlin (RIAS), as saying that no United States official had the faintest idea that a wall was about to be built through the middle of the city and adds an intriguing postscript of his own. On 30 July, Prittie had written to a 'very senior member of the British Intelligence Staff in Berlin' to voice concerns about a 'report that there was considerable movement on roads leading into East Berlin... mainly of goods packed in large lorries'. As a journalist, Prittie wanted to know if the report had any significance. The reply, written on 3 August, had reached Prittie on the 10th and assured him that there was nothing to worry about. The writer said 'I believe the present crisis will end in anti-climax after the [upcoming] Federal elections.'6
Lutz Stolz, the trainee civil engineer who'd been with his fiancée Ute to the Black Forest, fully intended to go across, but tomorrow. Lutz Stolz savoured the prospect of a football match. Although he and half the team lived in East Berlin the side was based in the West. They scarcely thought of the difference. Which city team would? Stolz tried never to miss a match in the crowded fixture list – matches were on Tuesdays, Thursdays and Sundays. The rendezvous for all eleven had been designated the most convenient place, the Lehrter station, first stop in the West after Friedrichstrasse in the East.
Willy Brandt was campaigning during the Federal elections and he had a party of people with him, including Klaus Schütz, his campaign manager. In the market square at Nuremberg, Brandt made a speech laden with emotion and foreboding which strayed from ordinary electioneering. He demanded a plebiscite for all Germans and added: 'Tonight refugee number 17,000 of the month will arrive in Berlin. For the first time we shall have taken in 2,500 refugees in twenty-four hours. Why are these people coming? The answer to this question is that the Soviet Union is preparing an attack against our people, the seriousness of which is apparent only to a few. The people in East Germany are afraid that the meshes of the Iron Curtain will be cemented shut. They are agonisingly worried that they might be forgotten, sacrificed on the altar of indifference and lost opportunities.'
Brandt had been toiling hard in this election – more than 500 speeches since May – and had hired two maroon railway carriages (a sleeper and a lounge car) as mobile headquarters. That evening they would be hitched to the Munich–Kiel express because Brandt's next destination was Kiel.
In the northern city of Lübeck, Chancellor Konrad Adenauer, also electioneering, appealed to East Germans not to panic and stampede, before returning to his home on the banks of the Rhine.
At 4.00 p.m., Ulbricht summoned Erich Honecker to the House of Ministries. A Politburo meeting had already been held in this building, Ulbricht chairing it. His level Saxon tones stroking the words, he gave a situation review and revealed the decision taken in Moscow. Reportedly no discussion took place and no dissenting voice rose. The fate of a city was sealed, and a decisive event of the century was enacted, in virtual silence.
Ulbricht signed an order and it assumed the authority of law. He instructed Honecker to enforce it. Self-doubt had been remote from either man for decades but it cannot be known if they suddenly felt it then. The Politburo meeting, the signing ceremony and the summoning of Honecker were simply a way of making the border sealing, already decided by Khrushchev, official. Since Honecker had previously been ordered to make preparations, what had just happened gave a cloak of legality and the sort of quasi-legitimacy GDR propagandists could wield like a sword in front of the cloak.
Even at this late moment the sealing was held tightly within a small circle of people. Some senior members of the Stasi did not know, Markus Wolf among them. 'At the risk of damaging my reputetion as the man who really knew what was going on in East Germany, I have to confess that the building of the Berlin wall was as much a surprise to me as everyone else... I can only conclude that Erich Mielke, who handled some of the covert planning of the operation, kept this information from me out of malice.'7
The CIA, with their payroll of seventy, also did not know. The American Mission did not know. The West Berlin police did not know. And Wolf did not know.
However, the FRG security service, led by the spider-like Reinhard Gehlen, claimed to know at least the outline. 'Many items of information', Gehlen subsequently wrote,8 'showed that it would not be long before the communists had to take vigorous steps. Then we learned from a reliable source that the Russians had given Ulbricht a free hand so that only the date was left open to conjecture. We received and passed on further reports of an imminent sealing of the sector boundary, particularly within Berlin itself, and of the stockpiling of light materials suitable for the construction of barriers. We could not predict the actual date they would start. It was known only to a handful of top Party officials.'
At Hyannis Port on the Massachusetts shore in the USA, Kennedy rested at his summer house with family and personal friends. White House Press Secretary, Pierre Salinger, said no business visitors were expected. Kennedy spent part of the morning reading official papers brought overnight from Washington, and he received his daily intelligence briefing from his naval aide, Commander Tazewell Shepard. A time difference, of course, held the United States' east coast and Berlin apart: the United States would be constantly and inescapably six hours behind.
While Ulbricht and Honecker spoke, Kennedy savoured the prospect of the cruise he and his wife Jacqueline would take off Nantucket Sound at midday, their time, in the family cruiser the _Marlin._
Honecker left the House of Ministries and was driven across East Berlin to the Police Headquarters, a tall, shorn, 1950s building behind the Alexanderplatz. He walked briskly in and along anonymous corridors. He'd taken over an office with secure communications during the week and laid the plans detail after detail. Now, shortly after 4.00 p.m., Honecker issued orders of his own, one to the Politburo member running the region from the Polish frontier to East Berlin, another to the Politburo candidate running the region which covered the arc round West Berlin, cumulatively a vice holding the whole city:
I ask you to arrange for the necessary measures to be taken on 13 August from 1.30 a.m. on as agreed. I will send the documents you already know about during the course of the evening. I enclose a draft of the alert order to the heads of operations.
With socialist greetings,
E. Honecker.
Instructions would spread out from this moment but still only at the highest, most secure and trusted level.
None of the 3.3 million Berliners heard the faintest whisper of a rumour, a single word of warning. Superficially the morning, afternoon and early evening of 12 August 1961 melted into another summer's day playing itself out in the innocence of ordinariness. The Reuters office secretary, Erdmute Greis-Behrendt, visited her aunt in West Berlin. Rüdinger Hering, a factory worker who lived in the countryside, went to see his uncle and cousin in the West Berlin district of Spandau, a couple of stops away along the S-Bahn. Why not?
Along the Ku-damm,9 girls in light summer dresses oscillated past the pavement cafés where patrons from East and West sipped a beer or mountaineered through cakes and cream. Easterners bought subsidised tickets to see the latest American films.
A woman (who many years later still preferred anonymity) sat in a sunlit garden in the West raking over the future with her son, her daughter and a lifelong friend. She lived in an apartment in Treptow, just beyond the line to the East. Her son had been arrested a few days earlier, helping a friend push a hand-cart across. The Stasi took him and when they released him he looked haggard, complained they'd treated him without dignity and announced he was going West, which he did. Now they all tried to persuade the woman to stay here in the West, too. Her friend even offered to go over and get the woman's papers from her apartment, eliminating the risk of anything happening if she went back to get them herself.
If only it could have been so easy. The woman felt attached to her parents over there, attached to the apartment over there, her friends, the familiar haunts of a life. Her son suggested that grandpa and grandma could follow later and they'd be reunited. The woman said no, she'd go home. After all, she could nip across to see her son whenever she wanted.
Some fulfilled domestic duties. A woman from the homely Eastern hamlet of Müggelheim helped repaper her sister's apartment in the West. If she'd time, she'd visit her mother nearby before going home. Her husband stayed in Muggelheim, gardening.
In a red-brick barracks south of Muggelheim, a manual typewriter clacked laboriously, its echo travelling down linoleum-laid corridors onto a parade ground fringed by lawns. A high, distempered boundary wall screened the barracks and almost concealed it. Pätz, a hamlet like Muggelheim, was hard to find even on a map, just a place down a country lane with a lake glimpsed through trees; quiet, rural, typical of so many hamlets beyond the tentacles of the city.
The typewriter clacked out (extract):
Command of the Border Police.
Order No. 002/61
Command Post Pätz
At X-hour + 30 minutes, an increased border defence has to be organised at the State border West to prohibit border violations in both directions and to avoid further provocation of the territory of the German Democratic Republic.
Staff officers are to be employed to control and instruct units about their tasks.
The heavy Border units are to be put on readiness for action and be prepared to move to trouble spots.
The greatest density of men and material is to assemble where the main flows of border trespassers take place and on the flanks of the border crossing points. In those sectors prone to provocation, camouflaged Border Police are to be installed. Motorised reserves are to be ready for back-up.
Political-ideological instruction is to be increased for all policemen, other ranks and officers by the Commanders and political units in conjunction with The Party.
It is to be ensured that all orders and instructions are obeyed correctly and conscientiously and the secrecy of all measures is maintained.
The reconnaissance units of the Border Police are to be given the task of identifying the enemy and his actions. All results are to be continually evaluated.
First situation report at X + 4 hours, further reports every six hours.
(Signed)
Peter, Colonel.10
Erich Peter commanded the three Border Police frontier forces, north, centre and south. X-hour remained unstated. The secrecy enveloped it. Further orders followed, some extremely specific: ninety-seven men, thirty-nine of them for construction, to go to the Friedrichstrasse crossing point and establish inter-state control. The ninety-seven gave a density of one man per square metre – the planning had been as specific as that.
Friedrichstrasse, a long street, remained battered and exhausted just like Schönhauser Allee. It was a backbone of East Berlin, stretching over the River Spree for half a mile due south to a crossroads, which was the sector boundary between East and West. The railway station which bore its name – a cold glasshouse of an arched roof, long platforms and long tiled corridors leading to them – was up near the Spree. Here twelve policemen searched the trains.
This Saturday afternoon, a cinema near the station showed _All Quiet on the Western Front._
People boated and rowed on the lakes around Berlin; many walked in the woods, a favourite pastime; Rüdinger Hering took coffee with his cousin and Erdmute Greis-Behrendt took coffee with her aunt.
Through these careless hours Adam Kellett-Long waited and waited. Reuters had the wire service of ADN, the official GDR Press Agency, and the ADN teleprinter chattered in his office, but it was normal traffic, nothing to justify the day-lead. The news editor from Bonn returned to his hotel in West Berlin and waited, too.
At the Police Headquarters a Lieutenant-General Schneider dictated a timetable and phrased it so that, step by step, it reached down in a pyramid:
**7.00 p.m.** : the summoning of leading Police chiefs to Room 5614 to be informed of the operation; **8.00** : a consultative meeting in the same room, sealed orders to be opened, senior officers to be briefed on specific tasks. Headquarters section heads ordered to meet in the room at **9.00** for instructions; **10.00** : the heads of special Police sections to be given instructions; **11.00** : officers running police stations to be told to attend the room at **12.00**.
It ended X + 30 minutes although X-hour remained unstated, the secrecy still enveloping it; X-hour would be added later, swiftly and by hand but, that aside, a timetable had been set out and a countdown had begun.
At 6.00 p.m. in Berlin, the city was sprucing itself up to take the pleasures of a Saturday night; it was noon in Hyannis Port, USA. Under a hot sun, Kennedy and his family set off on the 52-foot _Marlin_ , moving out into Nantucket Sound.
Around this time, in slightly scrawled writing, X-hour was added to the orders from Police Headquarters in Alexanderplatz: X-hour = 1.00 a.m., Sunday morning 13 August. That meant X + 30 would be 1.30 a.m.
The wallpapering complete, the woman from Muggelheim refused her sister's invitation to stay the night but she'd see her next week as usual. She travelled back to the apartment in the East, on the S-Bahn through to Friedrichstrasse station, and noticed it was strangely quiet.
A West Berliner, Klaus-Peter Grohmann – like Lutz Stolz a foot-ball fanatic although not a player – wrestled with a common human problem. His mother-in-law, also a West Berliner, had visited and 'We wanted to get rid of her, we wanted her to go home.' Grohmann and his wife began yawning very deliberately but the mother-in-law didn't go.
Stolz himself had a two-room apartment in his parents' house in the East. His fiancée Ute lodged with Lutz's grandparents across the courtyard. 'Usually I stole away to see Lutz carefully so that his grandparents didn't notice. We were not married yet.' She crept across the courtyard and they made love, but 'since his bed was too small for both of us I went in to the other room to spend the night there'.11
Around 11.00 p.m., Kellett-Long took the Reuters office car, a reddish-orange Wartburg, and drove to the Ostbahnhof, the East Berlin station serving the south of the country.
I had a system whereby for a few West marks I could get an advance copy of _Neues Deutschland_ [the official newspaper], which was useful if you suspected something was going to happen. All decrees and promulgations came out in _Neues Deutschland_. The paper didn't actually appear on the streets until the morning but they shipped off the provincial copies in the late evening, so in effect I got a first edition. I looked at the paper and to my horror – nothing. That really shook me. I was aware, however, of an awful lot of Railway Police at the station. They wore their black uniforms and the place swarmed with them. That was the only unusual thing. I reeled away from the station feeling a bit nervous and went back to the office thinking I'd write the day-lead for the morning.
Which, however he phrased it, would have to be an admission that he'd warned the world of dramatic measures and they hadn't happened.
Al Hemsing, who handled press relations for the American Mission, was at home.
Late in the evening I got a call from the European representative of _Reader's Digest_ based in Paris but on a trip to Berlin. He called to sort of check in with me and see what was going on. He added that he'd been to the Brandenburg Gate, he'd noticed a lot of milling around and something seemed to be afoot. Then I had a call from the son of one of the big shots in Reuters who'd been assigned to their West Berlin bureau. He said he'd been across and seen far more than the normal number of People's Police around; he felt some kind of tension in the air and what did I know about it? I said I knew nothing. I had a dedicated (secure) phone to the Mission and each time I had these calls I reported it, which ordinarily I wouldn't have done. We kept a permanent 24-hour staff in this period and I'd tell the duty officer and ask him what he knew. Each time, the duty officer said, 'No, we don't have any reports.'12
The man from the _Reader's Digest_ and the Reuters cub reporter experienced the very beginning – X-hour was now only 90 minutes away. What they'd glimpsed represented a discreet increase in police presence but not enough to announce the operation and set off serious alarms.
Rüdinger Hering said goodbye to his uncle and cousin and waited for the S-Bahn to pull into Spandau from its long loop across the city. He boarded the train and it crossed into the countryside of the GDR, delivering him to the administrative area of the prospective Politburo candidate who, in an office in Potsdam 12 miles away, had issued many orders of his own and waited for X-hour.
The train sighed and stopped at Falkensee, a worn, worked station in the middle of a solid village beside a lake – one of those favoured for boating, and surrounded by the woodland favoured for walks. The town itself consisted of substantial pre-war houses hugging narrow, cobbled streets. At the station, Hering noticed the controls were a bit harder than usual, more Railway Police in those black uniforms, but he thought no more of it. He lived in a little community outside Falkensee, on a cobbled roadway angled at a steep camber. It had pastures at the end and a farmhouse in the distance. The roadway narrowed to a track through the pastures and a small bridge took it over a brook where a couple of houses stood. He knew a girl in one of them because he went to school with her. The track wandered on towards the farmhouse.
Hering made his way home and went to sleep. He was 15 years old and would return to Spandau again quite normally to celebrate his uncle's 60th birthday; but that would be in 1988, when he was 42.
A small, wiry former prisoner of war, Martin Schabe was a farmer who'd built the farmhouse with his own hands. Although his land was in the West he'd regularly motored to Falkensee nearby to do his shopping. He had friends there, too. He hadn't been able to go to Falkensee for years, however, because a crude fence blocked the track as the first tightening of the borders began. Schabe was 41 years old; he would return to Falkensee again to do his shopping, but not until 1990, when he was 70. Some of the shops were still there and, although many of the people working in them had changed, some hadn't.
At midnight the Marienfelde Reception Centre drew up its total number of refugees for the day: 2,662, the second highest it had ever recorded. How many would tramp and shuffle in on the morrow? How many were already starting their journeys, as fugitives in the darkness? How many families would be reunited? How much more food would the centre need? The flow might rise to anything.
After midnight, Willy Brandt and his party turned in. Someone had brought bottles of Scotch and it softened the journey as the night train went north on its long journey to Kiel. The Nuremburg speech had gone well and the mood on the train had been slightly euphoric. They'd had a drink. Now they slept.
* * *
Quotation at head of chapter: a chance remark to the author about how much work would have to be done after the fall of the wall to rebuild East Berlin.
1. Interview with author.
2. The Factory Fighting Groups, formed in 1953 as 'Factory Combat Groups' were designed as Communist Party troops in the event of civil war ( _The East German Army_ , Thomas M. Forster (George Allen & Unwin, London, 1980)). They became a sort of territorial army, had a membership of 500,000 and were, as their name implies, based at, and drawn from, their workplaces.
3. Interview with author.
4. The CIA interviews with Morris and Polgar I recorded from a television programme on video. Unfortunately I didn't get the beginning or end of the programme and so I have no idea who broadcast it or when. What they had to say seemed so important historically that I have included it anyway. If I have tiptoed across somebody's copyright, sorry.
5. Ibid.
6. _Willy Brandt: Portrait of a Statesman_ , Terence Prittie (Weidenfeld & Nicolson, London, 1974).
7. _Man without a Face_ , Markus Wolf (Jonathan Cape, London, 1997). I have not, incidentally, dwelt on the role of spies for three reasons. First, it has already been covered, notably in George Bailey's extensive _Battleground Berlin_ , and _Stasi: The Untold Story of the East German Secret Police_ , John O. Koehler (Westview Press, Boulder, Colorado); second, to get at the truth is by definition extremely elusive in the espionage shadowland; and third because I wanted ordinary people's memories. With all possible respect, spies are not that. Let me just say that all these spies _on both sides_ put together did not know the wall was going up (even taking into consideration Gehlen's claim – see below) and they did not know it was coming down. Apart from that, I assume, they were gimlet-eyed on every nuance of everything... .
8. _The Gehlen Memoirs_ , General Reinhard Gehlen (Collins, London, 1972).
9. The Kurfürstendamm was, and is, a huge avenue stretching across West Berlin. It was famed for cafés, smart boutiques, salesrooms, restaurants and hotels. (The quality of the prostitutes who lined a certain section of it like sentries, equidistant and selecting their prey with subliminal accuracy, was famous; and by a law nobody understood, they became more beautiful – girl by girl – until you reached the intersection with Schluter Strasse, when they ceased altogether.) The avenue was frequently abbreviated to the Ku-damm and it was to this, on the night the wall fell, that the Easterners swarmed. _Everybody_ had heard of it.
10. Official GDR document, courtesy of Hagen Koch's archive.
11. Interview with Birgit Kubisch.
12. Interview with author.
## THREE
## _And Sunday Morning_
I am a camera with its shutter open, quite passive, recording, not thinking. Some day, all this will have to be developed, carefully printed, fixed.
Christopher Isherwood,
_Goodbye to Berlin_
Adam Kellett-Long sat in front of his typewriter trying to compose the day-lead for Sunday. 'I suppose I was in the third sentence when the phone rang. I picked it up and a man's voice I didn't recognise said in German "Don't go to bed this night." He hung up. At that moment the ADN service closed for the night as usual, _End of Transmissions_ , but because of this extraordinary call I stayed there wondering.' Kellett-Long couldn't know that the _Neues Deutschland_ he'd bought deliberately contained no news, or that a compact group of journalists prepared a special edition, or that it would fulfil a specific, pivotal function in the operation.
Al Hemsing's phone 'started to hot up but nothing firm, just enquiries'. George Muller's phone 'started to ring all the time. Around midnight Albertz called to tell me that something curious had happened to the S-Bahn system. At the Mission, we'd been concerned that because the East Germans owned the S-Bahn they could infiltrate large numbers of Factory Fighters through the stations in the West but this seemed contrary to that – a decrease in the traffic. The trains ran into the East and weren't coming back again. It created considerable confusion and one explanation might have been to reduce the number of trains refugees could use.'1
West Berlin taxi drivers caught the wind, passing and relaying a message among their fraternity, 'Don't accept fares to the East.'
The anonymous woman who'd sat in the garden got a lift back to Treptow with her daughter. They were dropped on a corner and walked the two streets to their apartment. They didn't see or hear anything unusual and went to sleep.
The policeman Erhard Schimke also slept, and so did a woman called Brigitta in her apartment near him. In time they would marry and live facing the twin curved roads and the sunken gardens, the dividing line running along the far side. She'd cross it very normally again to catch the bus to work – but not until 1990.
In Hyannis Port, Kennedy returned from his cruise and loaded a white golf cart with people: a friend, Jacqueline – chic in a blue blouse and red slacks – his 3½-year-old daughter Caroline and four of her cousins. He drove a block and a half to the neighbourhood candy store and the friend took the kids in to buy ice cream. Kennedy turned the cart around, inadvertently swerving onto some grass. He made a quip to a woman watching, collected the party outside the store and drove back with Caroline, who wore a blue and white bathing suit, sitting on his lap. It was an innocent day, as six hours before it had been in Berlin.
At 1.00 a.m., two orders chattered from the Police Headquarters, centralising the command structure:
(a) The first brigade of the Riot Police is subordinated to the Police Commander of the district of Berlin.
(b) The Security command, Berlin, is subordinated to the Commander of the First Brigade of Riot Police.
At 1.05 the lights at the Brandenburg Gate were switched off and Border Guards began to arrive there.
Grohmann's mother-in-law finally left for the Botanical Gardens station to take the S-Bahn home, a journey entirely within West Berlin. 'She came back after about ten minutes and said, "It's very peculiar but the S-Bahn is not running, it's all blocked up." She and we didn't understand why but we wondered.'
At 1.11, the ADN teleprinter in Kellett-Long's office 'suddenly opened up again and began to run a generalised Warsaw Pact communiqué from Moscow. We also had the Reuters service and I saw we were running it so I thought it was useless me filing the same stuff.' Kellett-Long studied this communiqué carefully:
The present traffic situation on the borders of West Berlin is being used by the ruling circles of West Germany and the intelligence agencies of the NATO countries to undermine the economy of the German Democratic Republic. Through deceit, bribery and blackmail, West German bodies and military interests induce certain unstable elements in the German Democratic Republic to leave for West Germany. In the face of the aggressive aspirations of the reactionary forces of West Germany and its NATO allies, the Warsaw Pact proposes reliable safeguards and effective control be established around the whole territory of West Berlin.
Kellett-Long rushed into the bedroom and said to his wife Mary, 'The Germans have closed the border – but don't worry.' She'd remember 'leaping' out of the bed.
This Sunday would be spread into a vast mosaic of images, of currents, of moments; but all that was in a sense governed by the time-table. East Berlin, slumbering, would become a prisoner of this time-table in a very few minutes as massive movement stirred in the battered, exhausted streets and overgrown bomb sites, at canals and bullet-riddled bridges, on the broad and brooding avenues and cosy little alleys, in fields and allotments and back gardens; movement played out under pallid street lamps making many thousands of uniformed men into flitting ghosts. And, as shadows within the shadows, the Stasi lurked, charting the moments, reading the currents.
Horst Pruster was an Eastern policeman who'd been in Berlin from January 1956:
That weekend I was working in the Interior Ministry in Mauer-strasse but I had some free time – I was alerted at midnight. I lived in Prenzlauer Berg with my wife and my little girl, who was two. I had a telephone and they rang – the Ministry commanded all the police in East Germany. The headquarters at Alexanderplatz was for East Berlin only.
I did not know the wall was going to be built. In the Ministry there had been about fifty officers – Majors, Lieutenants, Lieutenant-Colonels – and they were always in a locked room. We knew they were working on something but not what, and I can't remember for how long they had been working on it. I must say also I was a very little man [in terms of importance]!
After the telephone call, I came from my house and I saw a lot of movement, Soviet army, GDR army, police, Factory Fighters [ _Kampfgruppen_ ]. I had to get to the Ministry and I had no car so I hitch-hiked. I was there in thirty minutes. I was working in a centre where all the police reports came and were distributed, a very big room. We were eight and we had five, six telephones. The switchboard had buttons which lit up so you knew where the calls were coming from.
Within the coming hour and a half Pruster 'understood that the border was closed although at the moment that happened the police were not standing in the front line. They were the _Kampfgruppen_ and, in Berlin, the Riot Police. We thought the closure was temporary... .'2
At 1.30 a.m., senior officers of the Factory Fighting Groups in every district were alerted by Police Headquarters. The Fighters, organised in units of 100 subdivided into three platoons of three squads, could only be deployed by the police under instruction from the Ministry of the Interior.
At 1.40, a general alert went from the senior officers to every Factory Fighter in the districts of Mitte, Treptow, Prenzlauer Berg, Pankow and Friedrichshain. The general battalion at Weissensee and the reserve battalions at Lichtenberg and Köpenick were alerted, although these districts did not share a sector border with West Berlin. Cumulatively the timetable set in motion the massive movement.
The unkindest cut – Bernauer Strasse showing where Olga Segler jumped to her death, Border Guard Conrad Schumann jumped to freedom, Ida Siekmann jumped to her death, and the Church of Reconciliation, which was blown up.
The Fighters, activists drawn from each factory, were difficult to assemble quickly away from their workplace, so a system had been devised. If a Fighter lived in an apartment block but had no phone – an extremely scarce commodity – a designated contact in the block who did have one would be rung and told to go and wake him. Such a system could not be completely efficient and the police themselves went round and roused as many as possible. At the same time, the police moved towards many of the eighty-one crossing points, reinforcing officers who were already there.
Within the next hour 25,000 Fighters pulled on their baggy uniforms and joined the operation. Not one knew of it a minute before 1.40 a.m. and that they achieved what they did remains an enduring testament to their discipline and their belief. If they'd faltered, the operation risked disintegration – and that might have taken the GDR with it. An abortive attempt at sealing the border would compel hundreds of thousands to Marienfelde, and which East German would ever trust the government again?
The GDR army moved to positions back from the sector line except at carefully selected strategic and sensitive places like Bernauer Strasse. The GDR wanted the operation to appear as civilian as possible but the army served interlocking functions. Its presence intimidated any thoughts of insurrection, it stood poised if that happened, and it was in place if the Allies used troops against the operation.
Three divisions of the Soviet Army moved out of their barracks in Potsdam and a member of the American Military Liaison Mission there, hearing heavy metallic treads grinding, ventured out to investigate. A policeman held him motionless until a column of tanks moved into the darkness towards the Berliner Ring autobahn which snaked round the whole city.
North of Potsdam, at a barracks in a village called Rontgental, a platoon of the Border Police was woken and sealed orders read. They understood they were to take part in the division of Berlin and their Sergeant, Rudi Thurow, heard a member of the platoon wonder if the Allies would stand for it. The platoon was to go and take control of the mainline station at the town of Bernau, well beyond the city limits but a direct connection to West Berlin: East Germans boarding at Bernau arrived in the West very quickly. Thurow's platoon set off to be there before the first train of the morning came in.
The embrace of the operation spread, touching indiscriminately. In the Western district of Zehlendorf, a young radio reporter, Peter Schultz, and his wife Astrid – in the seventh month of pregnancy – slept. They looked forward to making her parents, who lived 50 kilometres into the GDR, grandparents. 'My telephone rang. It was RIAS calling. They told me to get up and dress, they said the radio car would be there in ten minutes and I should be ready to go by then.'3
So far away – now – in the other half of the city, Erhard Schimke, the policeman on a day off, was sleeping but 'a car came and I was told I had to serve, I had to do my duty. I was driven to the Police Station at Lichtenberg and they put me on standby.'4
Kellett-Long wanted to see for himself. On a hot night, and with the streets deserted, he drove towards the Brandenburg Gate. 'Up to then you'd been able to just drive through it into West Berlin. As I approached, a red torch waved at me and I stopped. A Border Policeman came up and said "I'm sorry, you can't proceed, it's closed." I didn't ask him too many questions. I turned the car round and started to dash home.' Kellett-Long had confirmation that the Moscow communiqué contained more than verbal sparring, and that his Saturday day-lead had been right after all. More than that, he had a world scoop on his hands.
At Marx-Engels-Platz, the broad, cleared area where the old Imperial Palace had stood (the GDR blew it up in 1950 because of its militaristic associations), 'another red torch waved and a policeman stopped me. I sat while a column of lorries went past, a very, very long column and the lorries seemed to be towing gun carriages. It was pitch dark because the street lighting in East Berlin was pretty minimal and I didn't find out until later that what I thought were army troops were in fact Factory Fighting Groups and the gun carriages were field kitchens. The column took about twenty minutes to pass as it moved towards the Brandenburg Gate, then the policeman waved me on. I didn't know what to do. I was half-tempted to follow but, if I did, I wasted time. I really needed to get to the office and put the snap message out that the border had been closed.'5 [A snap, in journalist jargon, is a blunt and staccato sentence announcing that something important has happened.] Because Kellett-Long did not follow, he missed the lorries fanning out to their appointed places along the line.
His wife Mary would remember: 'When Adam returned I think I was [back] in bed and he said, "Don't you worry about it, you go to sleep", that kind of thing, but I spent the night punching copy for him [onto the teleprinter].'
She'd write in her diary: 'Adam came back white to the lips and said the streets were full of soldiers, and lorries pulling guns. The soldiers and police all had sub-machine guns. Of course this frightened me but I managed to control myself and get over it in a short time. Soon after the announcements over the teleprinter, the streets filled with lorries and motorbikes. Police cars rushed along with their sirens sounding.' Mary Kellett-Long witnessed the direct result of the order from Pätz beginning: X + 30 minutes, _closure of the crossing points_.6
A reporter with another international news agency, United Press International (UPI) felt the consequences of the order. Three times he tried to drive east through the Brandenburg Gate and three times he was turned back. A West Berlin policeman told him 'they've shut it tight. Cars can't go through in either direction and neither can pedestrians.'
Kellett-Long filed the snap, 'by which time more stuff ran on ADN. We were lucky because within a hundred yards of our office in Schönhauser Allee a very large Police Station doubled as a headquarters for the Factory Fighting Groups. By then, that buzzed with activity, people coming and going, the whole place mad.'
The tempo quickened across the city as the timetable gripped. At 1.54 a.m., an S-Bahn train stopped at Staaken in the West – not far from Falkensee, but on a different line – and the Railway Police ordered it to turn back. Because the GDR controlled the whole S-Bahn system the police had the authority to do this; they also made all the passengers alight and returned their fares. The empty train reversed into the East. A minute later the police in the Western district of Wedding reported no S-Bahn trains coming through Gesundbrunnen, an important rail intersection, from the East.
At 2.10, a Factory Fighting Group reached the street linking the Brandenburg Gate to Potsdamer Platz, disgorged from lorries and hoisted down roll after roll of barbed wire. At the same moment the Factory Fighters in Treptow reported that they were ready.
At 2.15, the police chief of West Berlin put the entire police force on alert.
Already photographers caught images and froze them into permanence. Among the first: in the headlamps of a truck, uniformed men shifted into silhouettes as they unloaded the wire. A distant torch played over a young man wearing a camouflage jacket, an automatic weapon slung across his chest. Under the moulded steel helmet, his sallow face was startled by the sudden light. It looked morose, defiant, charged with mistrust.
Fifteen minutes after RIAS had rung him, Peter Schultz reached the line 'at a place where there had been a crossing point between Zehlendorf, a southern district of West Berlin, and Kleinmachnow, a small village in the East. Previously only a sign said you were leaving the American Sector: this was an ordinary street where people crossed and re-crossed normally. Now heavily armed men stood and they held torches. They'd laid barbed wire across the road but they weren't doing anything except line up behind it, looking.
'In a sense, they closed the crossing point with the barbed wire and their bodies. I saw they were Border Police and the Factory Fighting Groups. The light from their torches fell onto the street and that looked eerie. I couldn't imagine what was happening. _I could not_. From the radio car, I tried to explain this to the listeners, whilst I couldn't grasp it myself. I noticed no civilians on the Eastern side and the Factory Fighters appeared to be old men. I stayed for a few minutes making my report and moved to other places in Zehlendorf, and each time I saw the same thing.'
Bernard Ledwidge, political adviser to the British Mission, attended a party at a military mess near the Olympic Stadium.
They held dances on a Saturday night, that sort of thing. I'd come home and I was just getting ready for bed. I took a telephone call and the Military Police said some funny things were going on along the section border. 'We have heard from the West Germans that the late night trains have been stopped, communications with the west suspended and groups of People's Police at crossing points are closing the roads, so I thought I ought to report it.' I said, 'Well, ask the Commander of the Military Police to come and see me at my office at the Stadium.' I hadn't undressed so I went down there in my dinner jacket. In London a resident clerk is always on duty. I rang him and said, 'I am sending you an important telegram' and immediately that is what I did. I added 'Further Information Follows' and then telephoned my American and French colleagues in the city.
Ledwidge's telegram was the first official notification to the outside world that 'the balloon had gone up'.7
It had indeed.
The British Minister in Berlin, Geoffrey McDermott, acted, too. 'The first thing was to report these events by short, immediate telegram to the Foreign Office and (our embassies) in Bonn, Washington, Paris and Moscow. An alternative which I used on another critical occasion was a telephone call to the Ambassador in Bonn. The service through the GDR was quick and efficient and I hope they enjoyed listening in to what was said.'8
The operation did not take Ledwidge, an urbane and experienced man, particularly by surprise, nor perhaps McDermott. (The British Foreign Office, having in its time seen Napoleon and Hitler dispatched, operated on phlegmatic principles.)
'We felt something was bound to happen but I don't think anybody guessed exactly what,' Ledwidge says. 'We did get intelligence reports and we didn't quite believe them. They came from the Norwegian Military Mission in Berlin a few days earlier and the Norwegians told us they'd heard orders were issued for the closure points on 6 August but the order was cancelled at the last minute. We had not heard anything as specific as that from our own sources.'
Still the timetable gripped. At 2.20 a.m., the Factory Fighters at Prenzlauer Berg reported ready. Ten minutes later Albertz, as Willy Brandt's town hall Chief of Staff, telephoned the Allied commanders to ask 'What are you going to do?' At that moment, the West Berlin police at the Brandenburg Gate radioed that _twenty-three armoured personnel carriers have taken up position just to the East but are not under the columns of the Gate._
At 2.50, the Factory Fighters at Friedrichshain reported ready, at 2.55 those at Pankow, Köpenick and Treptow.
The GDR designated each of the eighty-one crossing points with a number, starting with 1 in the north. At 2.55, at crossing point 34 – the Brandenburg Gate, within clear sight of the West Berlin police who'd just reported the arrival of the armoured personnel carriers – a People's Policeman radioed that its force of 100 fellow policemen had closed it to traffic.
Five minutes later, the district of Weissensee reported that 10 per cent of its Factory Fighters had arrived and contact had been established with the Border Guards. As that came in, the district of Mitte reported ready and Police Headquarters made the first of their hourly reports to the Ministry of the Interior: two copies, one for the Ministry, the other for their own files.
In Kellett-Long's office 'ADN were running communiqués from the East German authorities about all sorts of rules and regulations, some underground stations closed and so on, although they described these as temporary measures. People couldn't, however, go to the West without special permits. On GDR radio the communiqués reeled out, too, in between western jazz which they never ordinarily played. You'd get a proclamation, then jazz, then another proclamation and all done while people slept. They'd wake to a _fait accompli_.'
Around 3.00 a.m., the Bonn news editor arrived from West Berlin so, Kellett-Long concluded, the border couldn't be completely sealed. The news editor added material to what Kellett-Long filed and returned to West Berlin.
Allan Lightner of the American Mission in Berlin got a call at the 'beautiful house' he and his wife Dorothy lived in. 'It came on the red phone, on my side of the bed,' Dorothy says. 'I picked it up and handed it to Allan. I didn't talk to whoever rang because it was the red phone. The Mission had tried to alert the political officer, Howard Trivers, an eccentric who didn't want his red phone by his bed because thinking about it being there would keep him awake. He'd put it out in the dressing room. His wife Millie would have heard it but she happened to be away for the weekend. Howard slept peaceably through the whole business! Allan immediately got up. I don't remember his exact words but he told me what was going on.'9
Lightner dressed and drove the ten minutes to the Mission. The tempo among the Americans quickened as the timetable embraced them. George Muller rang Richard Smyser, a junior officer, briefed him and said they needed first-hand information. Smyser and another junior officer, Frank Trinka, were to provide that. Trinka was bilingual: he'd been born in the United States but studied in Europe, particularly Austria. He'd come to Berlin in the summer of 1960 after a training programme in California.
Smyser rang Trinka, who'd already been woken by Muller, and they made their arrangements quickly. Smyser owned an open-top Mercedes and 'I took that because I judged we'd be able to see more. I picked up Frank. Being with the Mission, we didn't have a uniform, of course, and I was actually wearing casual clothes. We had a special licence plate, enough to guarantee us access to the East. If ever we were stopped we didn't show documents but pointed to the licence plate.'10
At 3.25, UPI felt confident enough to put out a snap, no doubt partly based on the three refusals at the Brandenburg Gate and partly on the mounting evidence: _Strong units of the Communist People's Police have blocked off the sector border between East and West Berlin during the course of the night._
'It was still dark when Richard picked me up,' Trinka says. 'We drove to Potsdamer Platz and when we reached it there were shadows around, trucks with their lights going. The East Germans unloaded the trucks and strung wire.' The night ebbed into the half-light although sunrise wouldn't come until 4.48. 'The effort was to erect some sort of barrier to pedestrian and automobile traffic. It appeared rather makeshift and improvised, there didn't seem to be anything permanent about it. We had an exchange with them... .'
Smyser remembers this exchange. 'People's Police – not Border Police – stood in a line and stretched coils of barbed wire. One stopped us at the wire and we demanded access. He was very reluctant. He consulted with what seemed to be an officer and they pulled the wire back and we drove through. I assume they were under orders to have no direct confrontation.' The open-top Mercedes motored across Potsdamer Platz. Other vehicles would motor across it quite normally, too – but not until 1990.
The half-light revealed the full scope of the operation.
At 4.00 a.m., Willy Brandt's train reached Hannover and the two maroon carriages were decoupled from the Munich–Kiel express. 'There a friend wakened us with the news,' Brandt would remember. Deputy Berlin Mayor, Franz Amrehn, had found a way to have Brandt woken; he 'dressed as quickly as he could' and shook campaign manager Schütz awake, 'telling him to get the others out of bed' while he went to call Berlin from the stationmaster's office.11 Brandt returned from making the call and found his party in the restaurant car eating rolls and drinking hot coffee. He looked composed and not hungover. He'd been informed during his telephone call that a special emergency meeting of West Berlin's Senate had been called for that morning and the Kommandatura – the Allied body – would meet afterwards. Schütz set off to get taxis to take them to Hannover airport.
French President Charles de Gaulle slept at his weekend retreat at Colombey-les-Deux-Eglises, far to the south of Paris. When his Foreign Minister, Maurice Couve de Murville, was informed of what was happening by telephone he said, 'Well, that settles the Berlin problem.'
British Prime Minister Harold Macmillan and his Foreign Secretary, Alec Douglas-Home – they'd been engaged in the traditional pursuit of shooting, although in different parts of the United Kingdom – were still sleeping.
Erdmute Greis-Behrendt, the Reuters office secretary, caught the first U-Bahn from her aunt's around 4.00 after a lot of talking. She sat, paying no particular attention to anything as the train rattled across the River Spree and stopped at Warschauer Strasse 100 metres further on. It was the first station in the East. She walked the 5 minutes to her apartment, noticing nothing unusual and went to sleep. Greis-Behrendt would always wonder about this, whether some administrative error temporarily overlooked Warschauer Strasse. Of the 4 million inhabitants, Greis-Behrendt and the others on that underground train must surely have been the last to cross the line quite normally. Others would again, too – but not until 1989.
At 4.00, all People's Police units established lines of communication, some using coded radio messages. The highest density along the border had been achieved and that allowed redeployment to support the Factory Fighting Groups. Within a few moments, jack-hammers beat holes in the roads to take concrete posts. Pneumatic drills gave Berlin its dawn chorus.
George Bailey, the American reporter, was 'awakened by a telephone call from my friend John Daly, who happened to be visiting Berlin. His message was short: "They have closed the sector boundary." I arrived at the sector boundary at 4.00. It was already light and the morning was clear. Members of the People's Police and East German troops had already strung barbed wire across Leipziger Strasse where it enters Potsdamer Platz from the east. To the left soldiers were sinking concrete pilings and stringing more barbed wire across them. Beyond, where the sector boundary runs through Ebert-Strasse to the Brandenburg Gate, People's Police and firemen were ripping up a strip six feet wide in the middle of the street. A police Captain in a plum-colored dress uniform was applying a pneumatic hammer to the pavement. With a cigarette in the corner of his mouth, he smiled as he worked. The rest of the city was still asleep... .'12
Smyser and Trinka toured the centre of East Berlin and noted a large number of police. 'We snaked in and out of about a dozen, maybe fifteen, crossing points trying to establish the policy line, what they were trying to do and how all of it affected Allied rights,' Trinka says. 'At the crossing points we saw plenty of activity, basically pretty much the same everywhere. The People's Police and Factory Fighting Groups were armed with automatic weapons, mostly sub-machine guns. We crossed at the most frequented places. Obviously we didn't go down side alleys or narrow streets. They were still open. We were looking at the main arteries. We went north to Bernauer Strasse, although we didn't notice as much activity there as at the other crossing points where dozens were busy stringing the wire.'
When Smyser and Trinka scanned Bernauer Strasse they saw a typical Berlin street: a terrace of dark stone apartments leaning together like elderly widows of the bombing so long ago. Some were seven storeys, their windows decorated by ornate pelmets and cornices; tiny balconies jutted like jaws from a few. Several shops, on the ground floors of the apartments, fed the community by selling groceries and milk and meat, and these shops had the same feel of survival about them. Their pre-war signs, painted on the stonework in old script, had faded. One said simply Lebensmittel – Food. It was the way the world had been, and the way it still was, here. Behind a white façade nearby a sort of boutique proffered porcelain.
Bernauer Strasse was a homely place. Members of the same families lived on both sides – a widow on one, her son on the other so he could tend her, carry coal across to her and help with the shopping. It had never mattered, and surely never could matter, that the house fronts on one side marked the precise boundary line of the district of Mitte in the East, or that as residents stepped from their front door they were in the Western district of Wedding. Bernauer Strasse was the sector boundary.
Tram tracks flowed up and down Bernauer Stasse and trams cranked along quite normally. At the southern end, the entrance to an S-Bahn station was in the West but the station itself and platforms were in the East.
Two cemeteries butted onto Bernauer Strasse from the East. Old people, particularly widows, often moved to the apartments to be near the graves – heavy, inscribed headstones in rows shaded by trees. Posies of flowers, constantly renewed, decorated the graves and racks of watering cans hung on hooks in special sheds to refresh them. At the rear of both cemeteries, family mausoleums charted the generations.
Six side-streets went across Bernauer Strasse: not unusual in a city centre street almost a kilometre long but, by definition, they bisected the sector boundary. It had never mattered, and surely could never matter.
Smyser and Trinka might have glimpsed a young army Sergeant, Conrad Schumann, who guarded the corner of one of the bisecting streets with a platoon of six men. 'Nothing was organised. We had to find ourselves somewhere to sleep in the empty houses,' Schumann would later say. 'Now and again we were able to take turns for a few hours' sleep on old blankets. We had nothing proper to eat, only a plate of soup.' Schumann would give the world one unforgettable image and Bernauer Strasse would give many.
At 4.30 a.m., Adenauer was woken at his home in Rohndorf, on the banks of the Rhine not far from Bonn, by the West German Minister for All German Affairs, Ernst Lemmer. He was told what was happening. Adenauer did not care for Brandt and did not care for Berlin. He fully intended to go to mass at 6.30 as usual.
Smyser and Trinka continued their reconnaissance and 'we got the impression of something major because they were out in such large numbers and they had brought up trucks,' Trinka says. 'When we penetrated further into East Berlin we could see quasi-military units in position with dozens of trucks and all kinds of engineering equipment – although nothing in the way of tanks or artillery indicating an intention to engage in combat.'
They went to Friedrichstrasse station. 'Dozens if not hundreds of people – women, children, the aged – tried to get up the stairs and out onto the platforms but the police pushed them back,' Trinka continues. 'They must have felt this was their last opportunity. A lot were in tears because they realised it was all over.'
'Plastered on the walls of the S-Bahn station,' Smyser says, 'was a decree from the Central Committee which contained a whole host of regulations. It was very carefully phrased. Frank and I read it while, around us, the most striking thing was the total confusion which reigned. These dozens of people, some with suitcases, had no idea what was going on and they were asking the Railway Police. I don't remember any anger, only bewilderment. I didn't notice any popular resistance or any shouting but I do remember the tears pouring out. They seemed to realise they were a day too late. They read the decree and tried to figure out what it meant because it did not say travel was completely prohibited. It did say there were a lot of procedures. We took note of all that.'
The announcement, dated 12 August and signed by the Minister of Transport, was set out in two columns of small, dense type under headings:
Long distance trains between East Berlin and West Germany run as per the timetable but the service now begins and ends on platform A at Friedrichstrasse station.
Direct S-Bahn lines between the East and West of the city are discontinued: Pankow-Gesundbrunnen, Schönhauser Allee-Gesundbrunnen, Treptower Park-Sonnenallee.
The S-Bahn remains open at Friedrichstrasse as a connection to and from West Berlin but the trains now begin and end on platform B. Suburban Eastern trains terminate at Friedrichstrasse and only on platform C.
It did not state that police manned controls barring platforms A and B to all East Germans without the right papers, which meant virtually the whole population. All the platforms faced each other like any station and would open again quite normally – but in 1989.
The announcement churned on, specifying which U-Bahn stations were closed and noting those stations on Western lines underneath East Berlin where trains no longer stopped, Bernauer Strasse among them. Paragraph after paragraph methodically forbade each outlet to the West. That's why so many studied it and, the police shoving them, why so many didn't have time to understand the immediate complexities.
'We drove back to Potsdamer Platz to try and go out,' Smyser says. 'By that time the wire lay all the way across and, although I didn't count them, there must have been ten rows, no way I could take my car across. The moment when they were going to part the wire, at least on Potsdamer Platz, had gone.'
Trinka expands on that. 'They were already _squeezing_ certain areas they had closed but the full implication only became clear later.'
This dawn, Potsdamer Platz already offered a stark image. The wire was attached to a crude post, and the street behind this post – where once the fashionable of Berlin shopped and drank and scanned the girls parading by – was empty save for a small, white car parked in the distance. The wire travelled in front of a group of low, saucer-like flower pots and a knot of policemen shuffled nearby. A Volkswagen moved gently up to the wire from the Western side, the driver expecting to continue as he always had. He stopped, then steered the car away while a policeman scrutinised him. The policeman's face betrayed no particular hostility, just traditional authority.
Smyser headed to the Brandenburg Gate. 'A uniformed man said "hold on". We always had difficulty there because somebody would try to score points, I guess to impress their superiors. We identified ourselves and they wanted the identification in their hand but we were members of the Allied presence and the Mercedes bore the licence plate. We had a verbal exchange and the man said "We are the German Democratic Republic, we are a sovereign state and the Soviet Union is not involved in controlling our borders." There are two side-pieces to the Gate a bit like folded wings. The man disappeared into the wing on the right, large enough to contain offices. I heard later a member of the Politburo was in there because the Gate was so sensitive. After a while the guy in uniform came out and said "OK, you can go" and we did.'
'At the Mission we told what we had seen,' Trinka adds. 'The Task Force gathered: operational staff and a large number of people called in, and I'm only talking about the State Department element. I wasn't aware of what others – the intelligence, the military – were doing.'
Al Hemsing, the Mission's press officer, was not among those present. 'I couldn't leave the phones that long. The other frustrating thing was that I kept asking, "Has the General in Berlin [Major General Albert Watson Jnr] been notified that barbed wire is being put up?" I couldn't get an answer but they hadn't called him. I said, "Dammit, if you don't I will!" They were reluctant to wake the General, especially if it proved to be nothing, but when I said I would they did. It was obvious he had to be notified.'
Smyser and Trinka presented a report orally then hand-wrote it. 'By then', Smyser says, 'a lot of information flowed in and of course everyone waited for instructions. We sat saying, "Well now, what is Washington going to do?" We were concerned about where the Russians were and their role. One report spoke of Russian tank movement – something not actually unusual, however.'
The report of that came from the man at the US Liaison Mission in Potsdam who'd been held motionless by a policeman while the column of tanks went by. It said:
The Soviet 19th Motorised Rifle Division, combined with 10th Guards Tank Division and possibly the 6th Motorised Rifle Division, moved out early this morning into position around Berlin. Elements of the 1st East German Motorised Rifle Division moved out of Potsdam and are presently unlocated. Soviet units deployed and moved off the autobahn, deploying the units into small outposts and roadblocks composed of three or four tanks, an armoured personnel-carrier and several troops. These outposts are established about 3 or 4 kilometres apart and appear to ring Berlin completely.
By moving to surround the city, Marshal Konev proclaimed how seriously the Soviet Union regarded the operation but, because the units remained in a ring away from the sealing, Khrushchev could claim that the GDR controlled its own borders as an independent state. This was the sovereignty thrust at Smyser and Trinka at the Brandenburg Gate.
Dawn spread purest blue sky over the city, and the first of the East Berliners who worked in the West made their way to their usual crossing points.
Sergeant Thurow, guarding the mainline railway station at Bernau, north of Berlin, found almost none of the people wanting to board a train knew anything about the official announcements. He refused them entry and one called out, 'Act like a human being, will you?' Thurow did allow a woman through whose face seemed familiar but, suspecting the Stasi watched him, made no further exceptions. He told the crowd to go and buy newspapers ( _Neues Deutschland_ had printed a special edition, remember) where everything would be explained. Some drifted away to do that.
The role of the little _Neues Deutschland_ team assumed its pivotal importance. Briefing every policeman, every Factory Fighter and every soldier on how to explain the operation to distraught, enraged and bemused fellow citizens hadn't been a practical possibility in the time available, the more so under the duress of the secrecy. A single instruction, a brilliant coup, solved it: tell the people to buy a newspaper. The Berliners, notoriously irreverent and independent, might prise a concession or wring a response from anyone in uniform but they couldn't do that to a newspaper bearing ponderous announcements which carried the force of law.
At 5.00 a.m. in Berlin, it was 11.00 on Saturday night in Washington. John Ausland of the State Department's Berlin Task Force, in bed early in his house just beyond the district line, slept blissfully. The timetable hadn't gripped him but it would within the hour and wouldn't release him for days, or years.
East Berlin seethed with movement, men working, trucks rumbling, jack-hammers beating, phones ringing in the Police Headquarters and, in every police station, orders being given and received. The swell of noise woke many East Berliners and they peered from their windows and wondered; and maybe went back to sleep thinking it was just some sort of night exercise for the military. Many slept clean through, the way people do.
At 5.30, Pioneer Platoons of the army in Potsdam, to be used for construction work, were alerted.
Outside Bernau station the crowd in front of Sergeant Thurow showed hostility and he requested reinforcements. Thirty reached him at 6.00 a.m.: men from a Factory Fighting Group and women from the Free German Youth. They threaded into the crowd to dilute the hostility. Thurow is reported to have heard someone call 'No-one can expect me to do a thing for the state if they are going to bar my way to my own mother.' A man, possibly a member of the Stasi, approached Thurow, reprimanded him for lacking resolution and ordered him to tell the crowd the sealing was really 'an anti-fascist protective wall'. Thurow said no, he'd do his duty but not use phrases like that when they kept a son from a mother. Later he'd remember one of the platoon washing his hands to cleanse them.
Bernard Ledwidge stayed in his office. 'Everyone came to the headquarters and enquiries came in from London. The head of the Central Department at the Foreign Office became involved and Alec Douglas-Home's private secretary did, too. Home was already up in Scotland shooting grouse, as Macmillan was in Yorkshire.'
Adenauer went to Mass.
Kellett-Long decided to have another look. 'Factory Fighting Groups and the police, miles and miles and miles of them with their guns at the ready, were putting up barbed wire. They lined the whole border. I only found out later that their guns weren't loaded, although their officers' guns were.' (The Factory Fighters and the police had clips of ammunition, but in their pockets.) A policeman told Kellett-Long nobody had been allowed through 'for a period of about an hour while we got into position, but now things are working as we mean them to and someone with papers like yours will not be prevented. You must take no East Germans with you.'
This initial and supposedly total sealing 'while we got into position' probably lasted no more than a few minutes, not an hour, and cannot have been total because the Brandenburg Gate remained open until 2.55 a.m. and Kellett-Long's Bonn news editor found a way across around 3.00. Erdmute Greis-Behrendt rode the U-Bahn at 4.00 to a station where no controls yet existed.
But that misses the point. The scale of the operation carried within it inevitable confusion and improvisation because so many men had been dragged from their sleep to seal off the two interlocking halves of the city, all 28.5 miles (45.8 kilometres) of it. However, the seemingly innocent fragment of conversation between the policeman and Kellett-Long implies that there was an attempt at a complete border sealing, an action in absolute violation of the Four Power Agreement. Would that have provoked a counteraction from Washington, London and Paris if they'd known? It remains academic because they didn't know. Smyser and Trinka had the wire pulled back around 4.00 a.m. when the operation was assuming its shape, and by then the thousands of soldiers, Factory Fighters and police were in position. The moment of being off-balance had gone and any physical counteraction now risked escalation to the nuclear level. Perhaps it always had.
Kellett-Long says that 'for the first few hours the British passport didn't matter and that was scary – it lasted from about from 2.00 to 6.00 – but after that they did let us through.'
While Kellett-Long drove on looking and noting, John Ausland's phone rang in Washington. It was midnight there. The call came from the State Department's Operations Centre and the caller said 'Press agency tickers indicate something is going on in Berlin but it's not clear what.' Ausland told him to 'call me when he had an official report and I went back to sleep'.
Those East Berliners who'd slept through woke into a nightmare. In the Stolz household Ute switched the radio on and tuned it to RIAS as she habitually did. 'RIAS said the border had been closed overnight and you could only cross at certain points.'
It might have been Peter Schultz broadcasting these words which Ute heard. Schultz had worked the whole night and 'I didn't leave the office because we had to do so many reports, Willy Brandt returning from the Federal Republic, calls from radio stations in Switzerland and Austria wanting information, colleagues at the Brandenburg Gate and at Potsdamer Platz sending in reports, _everything_.'
Ute 'rushed to the other room to tell my boyfriend, "The border is closed! The border is closed!" He told me off for waking him, and this after a wonderful night. "Don't talk rubbish," he said. I was very offended. I sat on the edge of his bed and I kept saying, "The border is closed; you can't cross anymore." He didn't believe me, he did not really grasp it. He was not keen to get out of bed and a little fight between us followed. He only grasped it when RIAS broadcast the next news bulletin.'13
Lutz felt 'stunned' as he heard that. Even when his disbelief hardened to recognition of the reality, 'I couldn't digest it, I couldn't judge the implications.' Lutz and Ute wondered about the direction their lives might have taken if they'd stayed in the West after their holiday in the Black Forest three weeks before.
'We talked with his parents about what had happened and they were filled with consternation, they couldn't digest it either,' Ute says. 'What really hurt Lutz immediately was that his sporting contacts had been broken.'
Stolz would remain a football fanatic and savour particular memories of a match, Stuttgart versus Bayern Munich in the West, which – of course – he watched on television. He'd remember a news flash at half-time and know he could now go quite normally to the Lehrter Stadium again – but that was 9 November 1989.
Around 7.30 a.m. the doorbell of the anonymous mother who'd returned to Treptow rang furiously. She opened the door and a neighbour gripped her arm, blurting, 'The border's closed, the border's closed.' The woman, sleepy, muttered, 'But that's impossible.' The neighbour led her to the window and she saw the barbed wire along the canal bank at the end of the street.
Also at 7.30, Colonel Ernest von Pawel of the US Army went to the Emergency Operations Center to report. He'd been out to Potsdam and seen the Soviet armaments on the move. Now he wanted to inform Major-General Watson that West Berlin was in effect surrounded. The noose went all the way round.
In Bernauer Strasse, Sergeant Schumann had three orders: 'In our sector we were not to allow anyone to proceed from East to West, we should not react to provocation from the West and we should not open fire with live ammunition.' He and his men patrolled behind the wire across the side-street. On the Western side a crowd hurled abuse, calling them concentration camp guards. A daughter on the Eastern side handed flowers to her mother over the wire. 'I heard the daughter wish her a happy birthday and say she wouldn't be able to come across any more because _those here_ [Schumann and his men] won't let me.' This disturbed Schumann more than the abuse. He started asking himself questions and wasn't getting any answers.
Brigitte Schimke awoke, gazed from her apartment to the twin-curved roads and sunken gardens, then over to the Western district where her parents lived. Between here and there she saw the barbed wire and she burst into tears. She'd cry for two days.
Klaus-Peter Grohmann woke up in the Western district of Lichterfelde and switched his television on. 'It cost a fortune, 998 marks, and only received what they called the Second Programme. I earned around 180 marks a week with the Berliner Bank so the TV really represented something. It was an ugly affair, enormous, a bit like "Big Brother Is Watching You" but I felt ever so proud because it was ours – although I had to pay for it on hire purchase over three years and the guarantee only lasted six months. We saw the first pictures of people at the border and troops and streets being blocked. It was genuinely incredible because all the streets connected right through the city. Try to think of that in terms of New York City or London or Paris. The men in uniform looked pretty grim. We wondered and we were scared. It seemed like war, you know.'14
Rüdinger Hering woke in his hamlet north of Falkensee and found he lived 150 metres the wrong side of a closed border. He went into the cobbled street with the angled camber and 'a lot of people stood watching the Factory Fighting Groups putting up barbed wire. They had come in trucks.' He turned away, embittered, I strongly suspected, and would never quite shed that bitterness.
At 8.00 a.m., Kellett-Long rang Erdmute Greis-Behrendt. 'Adam said, "Get out of bed and come to the office quickly because they've sealed off West Berlin, but before you do that tell your relatives." My grandparents lived close and I went there then I caught the tram. It stopped at Eberswalder Strasse, not very far from the office, because barbed wire had been rolled across the tramlines.' Eberswalder Strasse formed part of the northern tip of Bernauer Strasse. 'I got out and thought I was dreaming. People jumped over the one roll of wire and made fun of it like a game even though they were told not to. They thought it was a joke but as another roll uncoiled the seriousness dawned on them. Many hung around jeering as one lot of wire unrolled after another. People watched from windows. They thought the three Western allies would help them by doing something about it. I walked to the office.'15
By 8.00, several miles of wire had been laid and the noose tightened.
During the morning Greis-Behrendt found herself 'trapped but I couldn't believe it, I couldn't believe that that was it. All the people I spoke to said, "Wait for the Americans to come, they will help us. The Allies will help us."' Only in 1976 would she be able to cross to West Berlin again quite normally, and then by visa because she worked for Reuters. She wondered about staying in the West and talked it over very seriously with her husband but by then she had a six-month-old son. Of course she'd return East, and did.
A journalist called Peter Johnson had set up the Reuters bureau in Schönhauser Allee but, this Sunday, was finishing a tour of duty in Moscow. He'd learned of the border sealing from a Reuters service message alerting all correspondents, and by the BBC _World Service_ , which was broadcast on short wave and could be 'received without jamming'. Johnson lived in Bonn and 'I was dying to get back home to my family but at the same time sorry to be leaving Moscow after my first taste of life and work there.' A friend drove him to Sheremetevo airport to board an Aeroflot TV-104 jet for East Berlin where he'd catch a connection to Bonn. While he waited for the flight he bought a gramophone record of a popular Russian song, _Evening near Moscow_.16
From about 8.00, people began to gather at the wire, milling gently, staring at it, discussing it. At 8.15, Eastern bureaucracy began to function, officially recording the first escape – although that compressed the fate of human beings to lines on an incident sheet. The escape took place at crossing point 22, the street next to where Schumann patrolled:
_Date_ | _Time_ | _Command Post_ | _No. of Groups_ | _No. of People_
---|---|---|---|---
13.08 | 08.15 | 22 | 1 | 1
The second took place at
13.08 | 09.20 | 22 | 1 | 3
The bureaucracy added incidents during the day, however massively incomplete such an exercise would have to be. Many fled before 8.15 – over wasteland and allotments, or stepping from their front doors into Bernauer Strasse, or slipping into alleys and stealing across an unguarded corner. Some strolled over before the wire uncoiled and the police were too preoccupied to stop them. By contrast, a _Washington Post_ reporter saw a man visibly agonising for about fifteen minutes while the wire rolled towards him. He hesitated too long and the noose constricted. And further afield, in the countryside around West Berlin where, as Rüdinger Hering points out, woodland masked movement, 'in the first few days after 13 August many tried and 90 per cent made it'. How many did that before 8.15?
At 9.00, crossing point 48 – close to where Brigitte Schimke lived – reported that the number of people gathered on the Eastern side had increased to 100, but a platoon of police cleared the street. Crossing point 18 reported a crowd of 150 and requested reinforcements. The first crisis for the GDR government might be within the next few minutes or it might be now. Would the soldiers, police and Factory Fighters keep their nerve as more and more people, anger rising, were drawn to the line? Who knew what isolated squads of Fighters would do if an enraged mob advanced, or a daughter begged to be let through to her mother? Who knew what the population on the Western side would do? It might all turn on a horribly simple yet horribly brutel question: would Germans shoot Germans and, more specifically, would Berliners shoot Berliners?
At 9.11, a report from near Bernauer Strasse said 'three more persons went West at 9.02', implying previous escapes. The incident sheet represented a concerted and serious effort to record the number of East Berliners who, for whatever reason, came to somewhere along the 28.5 miles at any time after X + 30 hours, picked their moment and fled to the West.
Willy Brandt landed at Tempelhof. When he'd reached Hannover airport there'd been a delay to the PanAm flight he'd been able to get on and he'd phoned from the PanAm office to be given fuller details of the sealing. Brandt reported to his party that 'there are only a handful of people who know what's going on'. Now, from Tempelhof, 'I proceeded immediately to the sector border at Potsdamer Platz and at the Brandenburg Gate. The incredible and yet not totally surprising was under way. East Berlin was being cut off from us by a huge military force. Giant posts were being rammed into the ground and roll upon roll of barbed wire stretched between them.' It would become 'one of the saddest days I have ever lived through. Fear, desperation and rage were written on the faces of my fellow citizens.'17
The Eastern police monitored every move Brandt made, reporting he left the Brandenburg Gate at 9.05 and misspelt his name, Brand. The report added that 'the eighty persons who have gathered around him are about to disperse'.
At 9.15, crossing point 52 – the bridge Greis-Behrendt had crossed on the U-Bahn so long ago – noted 'the American officer who came from West Berlin and was reported at 7.00 had a look at the tanks in Warschauer Strasse and left again'. This may well have been a colonel captured by a photographer's camera as he walked, hand in pocket, past a policeman. Tanks, parked nose to tail, formed the backdrop to the print and, on the pavement opposite, passers-by glanced furtively at them.
Another crossing point recorded the defection of a riot policeman, reported at 6.27, but this proved to be untrue. At 9.20, a report from Bernauer Strasse said people were trying to smash the doors of houses so that they could reach Bernauer Strasse itself and step in to the West. Reinforcements had already been dispatched. At 9.20, Police Headquarters noted the arrival of a convoy of army heavy vehicles at a goods depot at Wriezen some 40 miles east. The commanding officer requested a police escort to bring them in, and a motorcycle left to do that at 9.30. The bureaucratic machine clicked out the strength of the Factory Fighting Groups:
_District_ | _Motorised_ | _General_
---|---|---
__ | __ | __
Friedrichshain | 82 | 102
Köpenick | 95 | 214
Lichtenberg | 187 | 0
Mitte | 274 | 782
Pankow | 0 | 120
Prenzlauer Berg | 69 | 81
Treptow | 127 | 103
Weissensee | 0 | 81
|
834 = 22.5 per cent | 1483 = 14.1 per cent
These totals constitute an immense force but the real significance is in the percentages of their full strengths. Some Fighters only heard on the radio – perhaps RIAS echoing over from the West – and hastened to their groups wearing sandals or ordinary shoes; some broke off holidays, some left the bedsides of pregnant wives, and their numbers rose during the morning, reaching 63 per cent. To assemble and deploy so many people without warning from a standing start remains an extraordinary achievement.
At 10.00 a.m., Kellett-Long's phone rang and this time he did recognise the voice, Ivan Shishkin, a Soviet Embassy spokesman enquiring how he was.
'I'm fine.'
'Not nervous?'
'No, not terribly, but it would be nice to know what's going to happen.'
'Well, don't worry because if things turn very nasty we'll get Mary out through Prague.'
Kellett-Long judged those words as being 'obviously part of the psychological warfare. Through me (and Reuters) he wanted to give the West the impression the Soviet Union really backed Ulbricht and this was a no-nonsense situation, this was for real. I'll never forget his words – _If things turn very nasty, we'll get Mary out through Prague_. He did not say he'd get me out through Prague!' Kellett-Long disregarded this psychology, although he found the ploy an 'interesting facet'.
While Kellett-Long fielded this call, in Washington John Ausland's phone rang again. 'The duty office at the State Department said a CIA message had arrived bearing an indicator that called for awakening the President in Hyannis Port.' It begs the questions: Why did the first notification to Ausland come from press agency ticker-tape and why, now, did the first official message reach him eight hours into the sealing of East Berlin from West? The answers demonstrate the difficulty of collating reliable information and communicating it in the era before computers, mobile phones, and satellite links; and before television channels like CNN were able to transmit live news globally and continously.
The timetable in play was in fact two timetables, one in Berlin and the other those six crucial hours further back in Washington.18 No doubt Ulbricht and Honecker had calculated this. The laying of the barbed wire began after 2.00 a.m. Trinka and Smyser didn't get back to the Mission to report until around 6.00 a.m. (midnight in Washington) and, when they did so, they had to observe procedures.
Frank Cash of the State Department's Task Force, explains it.
We had secure lines but they didn't always work and we could never compete with the wire services, who were sending everything in 'clear' [uncoded] and instantaneously. From an Embassy or Mission, a message needed clearance among a number of people in a number of offices and it would almost always be coded. That didn't involve an elaborate procedure because it was mostly done mechanically, but all of these things take time and, of course, a message had to be decoded at the other end and so on. That's why the news media can always beat any Department and why the Government is frequently behind the news media. Dick Smyser was in Berlin, he had to return to the Mission, make his report, put his message in writing, have it cleared by two or three people, then it's encoded...19
This does not alter the great truth that Berlin was essentially sealed while, as a phrase at the time went, Washinton slept.
Like Bernard Ledwidge and Willy Brandt, Ausland did not find himself totally surprised. Shortly before, he'd drafted a telegram on the theme that the Soviet Union would have to act but '"We're not quite certain whether they'll do it by cutting off all of Berlin or stopping movement within the city." Our Bonn and Washington and Moscow embassies commented on the telegram so it wasn't as if we didn't foresee something drastic, but there were no plans to cope with this because maintaining our access preoccupied everyone.'
And nobody knew. British minister McDermott wrote in his memoirs:20 'With hindsight it is clear our intelligence was not too good. British intelligence, which a few years back had greatly flourished in Berlin, had taken a hard knock as a result of the activities there of George Blake, the double agent, which had only recently been exposed. No doubt our allies' intelligence was affected to some extent, too, by his skilled treachery; but what is more surprising is that the West Berliners' own information was not more complete: with families divided between the West and East Sectors, and more than the usual supply of agents and informers in the city, one might have expected someone to have got wind of Ulbricht's intentions.'
The tempo in Washington quickened. Ausland phoned Cash, who lived in Maryland, and Cash said he was taking his wife to the airport but would join him at the State Department as soon as possible afterwards.
Cash remembers it clearly: 'Martin Hillenbrand, Director of the Office of German Affairs, Secretary of State Dean Rusk and I had just returned from a quadrapartite conference in Paris. Hillenbrand immediately took off on holiday and I was the guy left in charge of the Task Force. My wife and two children had scheduled a vacation in Florida with her mother and my parents, and their plane was due to leave at 7.00 that Sunday morning. It was somewhat after 4.00 a.m. when John called me. As I had to take my family to the airport and he was headed to the office I didn't see any point in going, then coming back to take them. It was at least a twenty minute drive each way. I told John that I'd be in shortly after 7.00.'
Ausland drove to the State Department.
In Moscow, Soviet diplomat Arkady N. Shevchenko sensed that 'in many offices of the Foreign Ministry a crisis atmosphere prevailed as we waited to see what kind of countermeasures Kennedy would take'.
Markus Wolf, running the Stasi department infiltrating the West, had heard rumours 'but nothing precise and above all no discussion. The decision had been taken in a military way and in the most absolute secrecy and, I swear, even I didn't know in advance. It created real difficulties because this barrier became uncrossable for us, too. We'd prepared people to be sent to the West under their real names as we always had but now that would be impossible. We couldn't furnish our agents with operational orders which the soldiers and police at the frontier would accept' – for security reasons.
Many others had more personal concerns. Bodo Radtke, the journalist on holiday, heard about the situation on the radio. 'I called my office and asked what's happening? What will happen tomorrow? They said, keep quiet, take your holidays and when you come back we will tell you.'
Heinz Sachsenweger, the railway worker on his family holiday in Brandenburg, also heard it on the radio and, as chairman of a Factory Fighting Group, knew he must go home. To get to Brandenburg, he'd ridden the S-Bahn from East Berlin as it looped across the Western half of the city and re-emerged in the East at Potsdam. There he'd changed to an inter-city train. Now he returned to Potsdam but had to work his way round the outskirts of West Berlin on the slow country line because the S-Bahn loop had been severed.
The timetable ticked. The police achieved more than 53 per cent of their full complement, although significantly higher (63 per cent) in Treptow where so many houses and apartments were close to each other and the word could be spread.
At 10.15, the Western newspaper _Morgenpost_ distributed a special edition, like a poster, with its headline screaming THE CITY IS CUT BY BARBED WIRE and a photographer froze an image of that: a lad in his Sunday best – white shirt, tie and braces, hair neatly combed – reading the poster in front of the wire. Concrete posts were in place and the wire was being stretched between them by two Factory Fighters.
Crossing point 55, in Treptow, reported that 'young people have swum the canal in the direction of West Berlin. A patrol has been brought into action.' The swimmers were two men who stripped down to trunks, so that they brought no possessions at all, not even the clothes they stood up in. Later they posed at the Marienfelde Centre draped in blankets. Both were smiling.
The American Mission required an update, and Smyser and Trinka drove to the East again. During their 4.00 a.m. visit, Trinka had judged it 'prudent for the East Germans to pull the wire back, give us a dressing down and, after telling us they were a sovereign state, allow us through – same thing on the way out. We'd had something to eat after making our report, now we'd gone back and we could see everything. The scenario had changed. Very obviously they were serious about putting something in the way of a permanent barrier up. They had barbed wire in place, we had to identify ourselves and in some places wait a while to pass.
'They were acting _de facto_ contrary to the Four Power Agreement but it's the old question: when do you decide your national interests are seriously involved? The confusion was so great and what did it all mean? Is it going to result in a military confrontation? Are we going to be overwhelmed in West Berlin? Will there be an uprising in East Berlin like 1953? What if it breaks down into civil war and thousands and thousands flee into West Berlin? How do we cope with that?'21
While Smyser and Trinka toured, the Police Headquarters received a report that 'about a hundred persons have gathered on both sides of the street at crossing point 48. They are provoking the Border Police and trying to break through the barrier. Operational staff in the Mitte district have sent two Groups of a hundred Factory Fighters each.' Five minutes later a telex chattered out to the police commanders of all Eastern districts: 'To maintain order and security, the battalions of the Factory Fighting Groups have to be brought into action in closed formation. It is once again pointed out that vigilance has to be increased.'
A loss of control might lead to Soviet Army intervention, bringing tanks back onto Potsdamer Platz like 1953, firing into a mass of civilians within full view of western cameras. Could Khrushchev survive that politically?
The Kommandatura met at 10.00 a.m. with Major-General Watson in the chair, and McDermott wrote that 'we discussed possible countermeasures, both local and further afield. The Americans would have liked to put on a show of force but we and the French questioned whether this would improve matters with large Soviet forces at the ready all round Berlin. I suggested that, as the Warsaw Pact countries had announced their support of the East Germans' action, reprisals might be taken against them or their allies, and it was agreed to refer this idea to our governments. Nothing came of it, so far as I know.'22
Willy Brandt went to the Kommandatura at 11.00 and was astonished to see an empty seat still there for a Soviet delegate, although none had attended a meeting since 1948. Brandt was preoccupied. Was, he wondered urgently to himself, the laying of the barbed wire 'only the first step? Would an attack on West Berlin follow? Would there be outbreaks of mass fury and attempts to break through to West Berlin? And what would the Allies do? Would they tolerate this violation of the Berlin treaty or merely react with protests? If they did not, would there be any way out except war?'23
George Muller remembers an atmosphere of 'consternation' throughout the meeting, nobody knowing GDR and Soviet intentions. 'Brandt was very, very taciturn although he wasn't a morning person anyway. He'd been taken off his train and flown to Berlin. He didn't say much of anything. My British counterpart and I were drafting the first cut of a protest but the Commandants couldn't say anything without instructions from their governments.'
'To be honest,' Brandt would say, 'we were very unhappy when it became obvious our friends, our protecting powers, were not able to change anything as far as the other part of the city was concerned.' The Allied representation told him they would issue a protest but he commented if that was all they were going to do 'people will laugh themselves sick from East Berlin to Vladivostok'. He said: 'scanning the faces of my American friends I could imagine what had happened. They had alerted the Pentagon, the State Department and the White House, only to be told that ungovernable reactions must be avoided at all costs.'
McDermott remembered:
Brandt joined us by invitation with Amrehn and other colleagues. He was grave but statesmanlike. He never demanded any rash actions nor reproached us for lack of firmness, though some of his colleagues later tried to make scapegoats of us. In reply to some criticism of bad Allied Intelligence which a Christian Democrat spokesman made I did not hesitate to express my surprise that the Berlin government's own information had not been better. But these disagreements came later and were never serious. On August 13 we were all concerned together to devise the best measures we could against the communist outrage which it was clear to us might have incalculable consequences. Far away in their capitals, officials and politicians began to think. In Berlin the first reaction was to call a committee meeting, which was hardly calculated to terrify the enemy. The Allied troops there, all 10,000 of them, were put on alert.
After Willy Brandt had arrived to join our deliberations we went on discussing for hours what effective practical counter-action we would launch. The fact is we were all stupefied and almost as much taken by surprise as everyone else. We decided to meet early the next day, and a French diplomat's suggestion that a quarter to eleven would be early enough was overruled.'24
The time-gap between Berlin and Washington is misleading. In fact Ausland arrived at the State Department, a ponderous and dignified building in the museum style of architecture near the Potomac River, around 5.00 a.m. his time – which was 11.00 in the morning in Berlin, as Brandt joined the Kommandatura meeting. There had been no time for the Pentagon, the State Department or the White House to say anything.
At the State Department, the Task Force occupied offices on the sixth floor where they had the secure phone, a detailed street map of Berlin and methods of seeing incoming communications traffic. Ausland felt very alone, a junior who'd returned from Australia only the month before. Dare he rouse Secretary of State Rusk at such an hour? He called Showalter, liaison officer to Paul Nitze (Assistant Secretary of Defence), but also a member of the Task Force. Showalter immediately phoned the US Army Command in Paris, and they said 'Well, we don't know any more than you.'
For a while, Ausland explains, 'we just lacked information, nothing we could do'. Meanwhile, he 'talked with the duty officer at the White House and gave him the points that he could pass along to the President's Press spokesman. I gave him three points... this, this and this. I was a rather junior officer. At that point Frank Cash is on his way to the airport and Martin Hillenbrand was on holiday. I tried to phone Foy Kohler, Assistant Secretary of State, and couldn't get a reply. I was too new back in Washington. If I'd been there longer I would have called Rusk directly, but I didn't feel at ease calling him. The irony is that I didn't hesitate to give guidance to the White House.
'A little later I had a call back from Walt Rostow [Deputy National Security Adviser] and he said the three points I had given to the White House looked all right but he wanted to add a fourth – so I know that both the duty officer at the White House and Rostow knew something was happening. I must assume they passed that on to Hyannis Port. Now what happened after it got to Hyannis Port is a matter for speculation... but the reality is that Kennedy had already decided in such a contingency not to do much, so in that sense it only mattered as far as appearances are concerned. It's this problem you always have: When do you wake the President? When do you bother the President? There was nothing he could do.' Ausland did manage to reach Kohler, who said, 'Well, you and Frank Cash go ahead and handle this.'25
At 11.00 a.m. in Berlin, the Aeroflot TU-104 jet landed at Schönefeld, the East's airport, with Peter Johnson of Reuters among the fifty passengers. He wrote cryptically in his diary: 'I should have arrived home after the flight via East Berlin but world events decided otherwise.' He saw a lot of soldiers with sub-machine guns in and around the airport, and he reacted like a newsman. He forgot his connecting flight and headed for the Reuters office in Schönhauser Allee as fast as he could.
A telex chattered out to police commanders that Factory Fighters could carry live ammunition but must be issued with blanks as well, and added that the use of water cannons and smoke canisters was permitted according to local situations. The noose was tightening further and Kellett-Long would witness the full consequences later because the Reuters office stood only about 300 metres from the line at Bernauer Strasse and 'as the morning went on people gathered and the Factory Fighters gradually pushed them to create a dead zone. The line of Factory Fighters came back very slowly during the morning.'
The timetable ticked and although it brought no insurrection, crowds were beginning to gather on both sides as the realisation settled into the reality. On the Western side near the Brandenburg Gate, some youths made noises and a (Western) policeman took their names. At 11.15, the crossing point at Wollankstrasse, two stops from Gesundbrunnen and where the line lay under a stout old bridge, reported 350 people on the Eastern side, adding that Westerners were 'trying to establish contact with them' – this meant, perhaps, waving to your parents, glimpsing your children, or saluting a lifelong friend. A photographer froze just such an image: a mother tentatively holding a baby across the wire so that the grandmother on the other side might lightly kiss the forehead.
A report near Wollankstrasse gave 200 people gathered on the Western side and 300 on the Eastern but the latter were being dispersed. At the same time, a crowd of 150 had assembled at Treptow on each side but didn't touch the wire and, again, the Easterners were being dispersed. A report said a senior West Berlin official arrived at the bridge opposite Warschauer Strasse and gazed across. He talked to people but on his own side and went away.
These were the last moments when the fury of a people might have prevailed. If enough thousands from the East had marched to the wire and begun tearing it down, what could anyone have done against them? Try and drive them back? Form cordons, arms linked? And what if the thousands were too strong, wanted it too much and were enraged by any attempt to deny them? But the moments were passing and each moment was a moment further away.
At 12.00, the Factory Fighting Groups achieved 40 to 45 per cent of their full complement but reports of desertions began. Eighty-two police officers and constables would also be punished for drunkenness and thirty-six discharged for refusing to obey orders. Did they let people through? Did they stand aside in disgust?
Also at 12.00, US press spokesman Hemsing felt able to leave his phones. He drove to the Brandenburg Gate and the Friedrichstrasse crossing point and found 'lots of police and Border Police patrolling the other side of the wire. Openings still existed, it wasn't shut off entirely. Some people showed their papers and mostly they'd be allowed through.' These included Easterners who'd finished the night shift at their jobs in the West and were going home as they'd done for years.
A male friend of the anonymous woman in Treptow arrived from the West around the same time and, as she remembers, 'he talked about something indescribable. All the U- and S-Bahn stations were crowded, the stairways and escalators packed, nobody could change trains and he'd taken hours to get here.' The talk turned, inevitably, to the conversation of the afternoon before in the garden, and the question: Why hadn't she stayed? It was too late. The shape of so many lives – and nobody can ever know how many – were forever altered by decisions which, it seemed, could wait a while longer on Saturday 12 August and had assumed a ghastly finality on Sunday 13 August.
Johnson arrived at the Reuters office and snatched something to eat with Kellett-Long, who he found 'slim and tired out after about thirty hours on the go'. Johnson took the 'new orange office Wartburg to have a look at the scene. Adam had earlier reported East German police and Factory Fighting Groups armed with rifles and sub-machine guns forcing back sullen crowds in some places away from the closed sector border.
'I went to view ten East German tanks parked in a square not far from the border in a drab working class district about a quarter of a mile from our office [the street was one of the six bisecting Bernauer Strasse]. I asked to see an officer and was presented with a stern-faced young blond Captain who stood with his battle overalls over his grey-brown uniform. I asked him what the tanks were doing and he said, "You'll find it in the newspapers and on the radio." The tanks had been roped off and small groups of people stood around talking about the situation. One man said to me, when I asked him what was going on, "It's unheard of." An elderly woman said, "I was absolutely staggered when I heard about it." Other people looked out of house windows.'
At 12.00, the Border Police reported two escapes in Treptow.
It was dawn in Washington, and Washington would begin to catch up. Frank Cash found the State Department very quiet when he arrived at 7.00 a.m. – 1.00 p.m. in Berlin.
As I recall, only John Ausland and the routine duty officer in the Operations Centre were present. We didn't have a clear idea but we knew something was happening because we kept getting reports from Berlin. John was concerned about contingency plans but he didn't have the combination of the safe. I did and I opened it. John pulled them out and it is true there were no specific plans to deal with a barrier across Berlin. All of us had concluded the East Germans would have to act but, almost to a man – well, I know I did – we thought they would do it by stopping the refugees coming from East Germany into East Berlin: do it where they exercised complete control. None of us really conceived they'd erect a wall down the centre of Berlin. We started drafting a statement to be released by either the Secretary of State or the President.
Dean Rusk got a call. 'What I was told was more precise than just something strange in Berlin. It was obviously a restraining barrier even though in those first stages it was simply a matter of barbed wire – although, anyhow, the barbed wire was there to keep people from crossing.'26
By now in Berlin, the Factory Fighters pushed the crowd – a 'pretty angry' crowd, Kellett-Long noted – 'back about quarter of a mile. It was a highly tense situation they absolutely had to handle with kid gloves. I did not see anybody being hauled away, because anything could have set it off. They were being, and I have to say this, extremely gentle, amazingly so because normally anyone speaking to an East German in uniform the way these people were speaking to them would have been manhandled out instantly. Around the middle of the afternoon the Fighters reached a crossroad in Schönhauser Allee and lined up. They'd pushed the people right back, they thought, until nobody in the West could see them. They forgot our office and us sitting watching it.' Kellett-Long went down to ask some questions.
His wife Mary watched from the window. 'Suddenly I saw masses of policemen and then three or four great flashes and clouds of smoke. My first thought was that they were grenades and Adam was in the middle of it.'
'They tear-gassed them and told them all to go home. It was a ten minute rumpus, absolute chaos,' Kellett-Long continues. 'When I first heard the bangs I thought they were shooting. I got to the chap who seemed to be in charge and I said, "What the hell's going on?" and he said, "We are dispersing the crowd. Everything is calm. Remain clam. We are not shooting."'
In Washington, Ausland was told that Dean Rusk would arrive at 10.00 a.m., his usual time, and speak to the President in due course. Rusk seems to have been a man disinclined to panic or nurture panic in others. Ausland waited for Rusk to come in.
Frank Trinka waited in Berlin but 'like the crowds, in anticipation that an event might take place, a confrontation of some sort or a breakout or an uprising. It never came. We'd gone over a third time and a psychological moment came in the early afternoon when the crowd could have gone either way after, I guess, they realised the Allies weren't going to do anything. These guys – the police and Factory Fighting Groups – had guns and the people weren't about to test them head on because the few that did were hauled off immediately. Violent verbal exchanges took place and then the people were grabbed or moved on. Whether they were taken to prison or put in holding pens I don't know, but the police were quick to focus in on and grab the outspoken ones, those trying to whip up some sort of opposition.
'We went to the Alexanderplatz [a big square in the East] where thousands of people remained until late afternoon, and government agitators, planted in the crowd, decried the West. The crowd waited and anticipated but the crowd-control was effective. If people said anything the Factory Fighters took them out and took them away. The Fighters also lined up with their guns and forced the people back. We visited various crossing points and saw thousands hoping, I guess, to flee. The women carried handbags and pushed baby carriages but I don't remember anyone holding suitcases. Maybe they'd liquidated everything into money.'
At 3.40 p.m., the Border Police reported a group of fifty had escaped between the Brandenburg Gate and Potsdamer Platz but gave no further details. At 3.45, they announced an arrest at Wollankstrasse. Ten minutes later a group of six escaped near Treptow and, ten minutes after that, two escaped north of Wollankstrasse, sprinting across the railway line in a strong, desperate burst of movement. Some escapes were only reported to the Western media: a man grasping a soldier's sub-machine gun and running over the line brandishing it; a man impulsively swimming a canal fully dressed; a man clambering a cemetery wall and dropping into the West.
Johnson and Kellett-Long decided to test their own rites of passage and drove to the Friedrichstrasse crossing point. They were allowed over but 'of course our office Wartburg had East German number-plates and the West Berliners were very, very angry – they didn't know who we were,' Kellett-Long says. 'They spat at me. We tried to hold up our British passports and we lived through tense moments until we got clear. These people shouted and shook their fists.'
He made another sweep during the afternoon and took Mary. 'We went round the border', she wrote in her diary, and 'by this time large crowds stood and watched and goaded the policemen. The police and Factory Fighters closed all the streets near the border and they all had guns. We thought that there might be demonstrations but on the whole the mood of the people was peaceful. They lacked a leader. Watching the border seemed to be the new Sunday afternoon pastime.'
One journalist noted that 'at one time, in a one-square-mile area... at the Brandenburg Gate and East German government headquarters, there were 25 tanks, 70 armoured troop carriers and 200 trucks filled with combat-clad police and soldiers. As they roared down Unter den Linden [the avenue in the centre of East Berlin which ran from the Brandenburg Gate] the catcalls of East Berlin crowds were so loud that they could be heard hundreds of yards away.'
Another journalist saw the road leading to a crossing point at Spandau 'jammed with West Berlin cars heading East for a look'. Easterners would form mighty queues for a look in much the same way, too – but that was on 10 November 1989.27
At 4.00 p.m. it was 10.00 in the morning in Washington. Kohler arrived at the State Department and so did Rusk, unhurried, certainly unflustered. 'I think I'm a calm man under pressure and I'd been through many crises in my day,' Rusk told me. 'My limousine came and picked me up at my house in Quebec Street and I was driven in.' He went to his office on the seventh floor – plush, as those of bureaucrats go, he insists. 'It had a sofa and paintings on the wall which came from the Museum of Modern Art in New York and were rotated by them.' Rusk gathered 'as much information as I could and I reminded myself that what the East Germans and Soviets did to their own people had never been a case of war and peace between NATO and the Warsaw Pact. Yes, I reminded myself of this.
'We anticipated the East Germans and Soviets would take some action but we didn't know exactly what. The erection of the wall came as a bit of a surprise to us all. It should have been included in the possible action they would take. The wall was a monstrous monument to the nature of the East German regime and we knew at once that its purpose was not to keep people out but to keep people in, that it was directed against the Easterners rather than the Westerners. So we did not take any military action against the wall – it was defensive on their part. No member of NATO, including West Germany, recommended that that action be taken because it might have been World War Three, oh yes. Moreover, we didn't take action because of the nature of Berlin's location and the nature of the forces surrounding it. A military response would have been futile and self-defeating and did not figure in our estimates.'
For a precise update, Rusk asked Kohler to ring the Mission in Berlin on an open line. 'Not a risk, really,' Rusk says, 'because the East Germans and Soviets knew what they were doing. We wouldn't be giving any information away.' Kohler spoke to Allan Lightner, who briefed him, but the telephone conversation did not end there – and the dividing of the Americans began.
Rusk forbade anyone in Berlin to issue a protest to the Soviet Union. 'What comment would come would come from the President. That was my decision. This was a serious situation and it would be perfectly natural for the President to be personally involved in any action we took or any statements we made. It was not a situation in which underlings – and in that I include the Secretary of State, myself – could do so on their own.'
Hemsing remembers Lightner getting the call. 'I was supposed to make some sort of statement and that was a terrible time,' he says. 'In addition to being the spokesman for the US Mission we had a system of rotating the Allied spokeman, and the US was in the chair. Therefore I'd also become the Allied spokesman. Rusk said, _Hold up on everything, no statements, the statements will be made here in Washington_.' Hemsing would have to live with that but, leaving statements aside, what about a protest to the Soviet Union? 'Altogether, I believe, it was fifty-six hours from the time the East Germans actually infracted the Four Power Agreement to the time a protest came, which as you may imagine had a terrible effect on the Berliners. They all knew the drill and previously an incursion by, say, an East German military vehicle would get a protest in a matter of hours and we'd present it in person at the Soviet headquarters at Karlshorst. Now a wall was going up in dead silence and Lightner and I were absolutely outraged.'
Muller also remembers Lightner getting the call. 'Foy Kohler phoned him with Rusk standing there dictating what Lightner could and could not do. My British counterpart and I had our protest drafted and Rusk said _no statements at all_ because it was so serious a matter. The Allies wanted to deliver the protest in Moscow and the West Berliners said, "My God, what's happening?" Before this, any little infringement met with an immediate protest to Karlshorst and now nothing's happening, now here is the real thing surrounded by silence. In Berlin we felt handcuffed.'
This friction ran like a theme into the days ahead. Washington took the global overview, with Rusk constantly reminding himself and Kennedy of nuclear conflict. The US community in Berlin took the emotive and personalised view, insisting the overall significance of the border sealing hadn't been appreciated and the mood of the West Berliners hadn't been appreciated, either. The friction built itself upon an already existing friction: Kennedy believed the situation in Berlin so emotive that it affected the judgement of Americans there: 'They go to the city', he'd say, 'and come back Berliners.'
George Muller of the Mission says 'the attitude of Washington was well nigh incomprehensible. We were always tainted with the brush of "Berlinitis" and there was inevitable tension between the diplomats on the spot and the foreign office at home, but in this case it was particularly striking because a number of our senior and otherwise highly respected diplomats argued nothing should be done. Their reasoning ran: "Khrushchev and Ulbricht cannot tolerate this blood-letting of East Germans, they had to do something, the wall was what they did and it will defuse the crisis." We in Berlin argued the opposite: that if Khrushchev got away with it, the operation would be a two-stage thing with much tighter controls to come.'28
The intensity of the friction only became clear after a couple of days. Now, on Sunday morning in Washington, 'Kohler turned up and disappeared into Rusk's office,' Ausland says, 'then he came back and told us "We've agreed to issue this" and it was basically the points I'd worked out, plus one Kohler added when I'd spoken to him on the telephone. Only later did I learn of the telephone conversation between Kohler and Lightner in which he told Lightner to lie low while we took care of this. Kohler read out this press release. I said, "Well, OK, now we know what we are going to say but some people are going to wonder what we are going to do" and Kohler said something to the effect of "Well, let's not be in a hurry on this. After all, that refugee flow is causing a lot of problems and we need to think about it. You call a meeting of the Task Force for tomorrow morning and let's take a look at the problem then."'
Rusk needed to talk to Kennedy 'because there was a chain of events that demanded I did. Cable communication was sent to the White House and then to Hyannis Port as a matter of course, as a matter of information.'
At 10.00 a.m., Kennedy attended mass in the Church of St Francis Xavier and changed for a cruise on the _Marlin._ As the cruise began, a message (presumably from Rusk) arrived at his mobile communications centre in a local yacht club. It was placed in a sealed envelope and taken to Kennedy's military aide, Major-General Chester V. Clifton, who opened and read it. Clifton used a walkie-talkie to reach Kennedy's secret service agent on the _Marlin_ and the yacht returned to shore. Kennedy telephoned Rusk, they reviewed the situation and Kennedy approved the statement.
In part it read:
The authorities in East Berlin and East Germany have taken severe measures to deny their own people access to West Berlin. The refugees are not responding to persuasion or propaganda from the west but to the failures of communism in East Germany. The pretence that communism desires only peaceful competition is exposed. The refugees, some half of whom are under 25 years of age, have 'voted with their feet'... . Limitation on travel within Berlin is a violation of the right of free circulation throughout the city.
To emphasise that, Rusk cited the Four Power Agreement signed in Paris on 20 June 1949; but the words seemed predictable and impotent to a population feeling isolated, violated, frightened and abandoned. Rusk had to address broader considerations. 'An uprising in the East and the unpredictable consequences of that was in my mind when Kennedy rang,' Rusk says. 'We simply did not know what the East German people thought and whether there would be any demonstrations or action which might force Soviet action to suppress it. We had to keep that in mind as one possibility.'
Rusk could give Kennedy dispassionate advice.
I saw Kennedy many hundreds of times but my relations with him were strictly business, they had to do with the business of government and did not involve his personal circumstances, his family circle or anything like that. He always called me Mr Secretary and I was the only member of the cabinet with whom he used that term. Jacqueline Kennedy once said at dinner, 'It's very significant my husband always calls you Mr Secretary' but she didn't explain the significance and I didn't ask her. It could have been a sign of respect or it could have been a sign of distance. I preferred it that way myself because I'd learned from George Marshall [of the Marshall Plan aid scheme fame] that you ought to keep at arm's length from the people who are working under you and are working over you.
How did Kennedy handle it over Berlin? Well, there wasn't much handling to be done except to fortify the morale of West Berlin and that was the real problem we had. Kennedy was pretty cool throughout the several crises through which we lived – he did not get flustered and he did not get emotional. He kept his eye on the main theme of the problem rather than be diverted by extraneous incidents of various sorts. He was a very impressive man, cool and calm in a crisis, very well informed, a rapid reader from a technical sense.
He was very conscious of the fact that he was the first President to be born in the twentieth century and he thought that gave him a mandate to review the bidding on all sorts of conventional policies to see whether they still made sense. We probed all sorts of conventional wisdom – and yet he was fun to work with. He had a sardonic sense of humour which he used first of all on himself so that, when he used it on you, you didn't mind. He'd look at a situation in a balanced way.
We knew that some action would be taken but we didn't know what: we didn't know that they were going to drive trucks up with barbed wire in them. We didn't get any indications of that at all ahead of time. I would have to say that tactically we were caught by surprise but strategically that wasn't the same thing.
Kennedy decided on no panic and no suggestion of panic. Rusk was due to attend a baseball game in the afternoon and he was advised to go to that. Kennedy would resume his cruise. The President's press spokesman said the President would not return to Washington until Monday morning as scheduled, when he'd have a pre-arranged meeting with the US Ambassador from Moscow, Llewellyn Thompson.
While this prudent but placid strategy unfolded in Washington, the timetable ticked in Berlin: incident sheets were filled, crowds gathered and dispersed, little flurries of movement stirred and died. The current of resentment did not, however, find a central, cohesive means of expression; did not find, as Mary Kellett-Long observed, a leader. How could it? All opposition had been rigorously suppressed since 1949. In the late afternoon Frank Trinka toured East Berlin again and judged the psychological moment for insurrection had gone.
Bernard Ledwidge journeyed to see from the Western side and noted the barbed wire but 'no signs of a wall. One did not know exactly what was going to happen at that stage. It was pretty clear they had sealed off East Berlin for German traffic but not Allied traffic – a very important point because, by doing that, they kept the crisis below boiling point. We could not have accepted exclusion from East Berlin. They took care to build all their obstructions on their own territory about five yards back from the actual line and we had no authority to attack those positions under the existing arrangements. However, if we'd been refused passage the fat would have been in the fire.
'The Brandenburg Gate looked quite different from normal because on the Western side the police erected barriers to keep the crowds back. The crowds were not allowed anywhere near the border line and the Soviet memorial [just on the Western side of the Gate] was given a special British guard so the locals couldn't get at it. We kept the temperature down in that way. Our Deputy Commandant, a new man, drove into the East in an official car to see if he actually could. Our Military Police were doing it all the time: they sent regular patrols round East Berlin and all those were admitted. The police looked them over but didn't prevent them.'
By coincidence, the famous and respected American broadcaster, Ed Murrow, was due to visit Berlin. 'As luck would have it,' Hemsing says, 'Murrow, then Director of the US Information Agency, had a long-planned trip. I asked someone to go and meet him at the airport and said I'd see him when he was installed in his guest quarters. Instead, Murrow immediately requested he be taken on a tour of the border, which he did for several hours.' Murrow's presence would assume particular significance.
Heinz Sachsenweger reached East Berlin after the difficult journey from Brandenburg.
I went to the factory and said here I am. The Fighting Group I commanded were in Treptow and we had to stand there for the first fourteen days after August 13, sleeping a few hours in the factory. I came from a very anti-Fascist family. I don't say that so people will admire me, it's just a fact. My father was a communist put in a concentration camp under Hitler. In 1961 we thought the wall was necessary. We stood in Treptow with machine guns and rifles because at the very beginning it wasn't even wire everywhere, it was only the men of the Fighting Groups. None of my hundred defected and I don't remember any people shouting at us.
We were stationed by some small gardens near a canal, and the gardens were typical of what Berliners had, chickens and rabbits. One member of the Group stole a rabbit to cook but we took him to the local police station. We arrested him because we did not want the people to think of us as thieves. We were convinced we were doing something good, we thought the GDR was a genuine alternative to German history and, of course, the international situation was very complicated then. Some in my Group even said 'Why not a barrier earlier?' in order, they thought, to calm that down.29
Such sentiments were scarcely reported in the Western media and have been barely covered in many subsequent studies. The coverage understandably pumped enormous moral indignation from the developed, fixed images recorded quite passively – that baby held tentatively across the wire and the grandmother's kiss bestowed so lightly on the forehead; the tanks along an ordinary street like Warschauer Strasse; handkerchiefs dabbing at swollen eyes, then waving to a loved one, then dabbing the eyes again – but, this late Sunday afternoon, many in East Berlin experienced an emotion far removed from the immediate and unavoidable suffering. They felt relief.
No understanding is complete without examining that, because it explains why the Factory Fighters and the police held steady. They were obedient but not blindly so, and although the number of ordinary citizens who shared their beliefs cannot be quantified, it may have been substantial. The whole of East Berlin, after all, did not fall enraged upon the wire and rend it asunder.
On an everyday level, East Berliners were tired of being humiliated by the power of their Western neighbours with their Deutschmarks which had to be balanced against the relative worthlessness of their own East mark. Mary Kellett-Long encapsulated that when she wrote in her diary a few days before: 'Hairdressers are plentiful and cheap. The great majority of clients had been West Berliners able to have a good hairdo at a fifth the price due to the exchange rate, but because of a clampdown any who come over now to the tailor or furriers must pay in D-marks at a rate of one to one. The majority of East Berliners are pleased by this move because it was annoying for them to have to wait weeks for a dress or hours in the hairdresser while Westerners got priority. On the black market, they got East marks for nothing, really. Within a week of 13 August you could buy butter, buy all sorts of things you couldn't before because – no doubt about this at all – I know lots of West Berliners would come over and do their weekly shopping.'
A 21-year-old, 'Jakob' was representative of this. 'We always went to the East side, my father also, to the gasoline station because it was very cheap.' He had experienced the clampdown before the wall. 'There were Stasi at parking places, you couldn't have any contact with the people and you could never offer someone a cigarette. The Stasi watched constantly. Someone might come up and try to talk but even traffic wardens watched out for that. I felt it very strongly. Once my car was broken into while they searched it for Western newspapers or anything else. They took the insides of the doors off.'
Gerda Stern, a Communist Party member since 1932, endured the frustrations of the hairdresser (something she emphasised to illustrate the humiliation). 'The Westerners came to buy everything because we were very cheap for them.' She welcomed the border sealing.
Elli Köhn, a Communist Party member since 1928, says that 'honestly I wished the wall would come because I saw the state of the area I lived in – terrible. Even the workers were divided. Some had jobs in the West, earned West-marks and lived like kings here compared to Eastern workers being paid in East marks. Like everybody else, I didn't know about the sealing but the question was, how could the situation be stabilised? We wanted action to staunch the brain drain and stop the young people, the highly qualified people, from leaving. We were trying to rebuild the country and this prevented us doing it.
'I lived in Karlshorst, I heard the news on the radio and I saw tanks in the street moving towards the city centre. My reaction? _Now we will be left alone, we will have our peace and we can continue to rebuild properly_. I don't know, of course, if most thought this but I do know a lot did because it had reached a point where it could not go on. We did not realise it would assume such tragic proportions. For example, part of my family lived in the West and I couldn't see my brothers and sisters.'
Horst Pruster, the policeman hard at work in the Interior Ministry, says at first 'we didn't recognise the whole extent of the measure but we knew things couldn't go on as they were. Many people worked in West Berlin, and they were millionaires – they lived here in the East in cheap flats, they didn't pay taxes because they didn't work here, but they took all the social advantages.'30
Sachsenweger re-emphasises his own words: 'Although it must have been difficult to keep the operation a secret, East Germany was very well organised, even in the first years of its existence. We had a lot of committed people and, to the majority of the politically active, the wall meant something good.'
Heiner Müller, a playwright, said retrospectively: 'We were pleased about the wall. We thought we'd be free to discuss our own problems at last.' And Stefan Hermlin, a writer, looking back, remarked, 'I repeat, the wall was not a crime. The wall was an action taken by the state in an extreme emergency. Don't forget the cold war, don't forget the postwar period. Millions in the GDR were in favour.'
At the time, however, Hermlin relayed a disturbing anecdote to Müller that 'while we had been happy about the wall Otto Gotsche, Ulbricht's secretary, was saying, "We'll crush anyone who's _against_ us, against the wall."' He concludes, 'We were so innocent, we never thought anyone could think like that.'
Honecker himself said later that 'the Cold War had reached its height. A mass exodus from the GDR was being organised. As a result the Political Consultative Committee of the Warsaw Pact met in Moscow in July. By unanimous decision the GDR was charged with taking the frontier with West Berlin and the FGR under its control. Of course there were human tragedies, but the main issue was to safeguard peace because instability in central Europe meant a danger to that.'
Easterners knew the haemorrhaging threatened the basic fabric of their lives but they also understood that the Soviet Union would not tolerate the country melting into West Germany, with the spectre of a vengeful, wrathful Fourth Reich rising someday. It simply didn't matter how uneasily this knowledge had to coexist with empty shelves in the shops. The knowledge recognised the fact. (By something of an irony, Lenin himself defended the violence of revolution by saying 'You can't make an omelette without breaking eggs', but he didn't mention what happened if you didn't have eggs... .)
On yet another level, every East German remained acutely aware of the desirability of peace at virtually any price and East Berlin, still trying to rise from the rubble of war, was a daily reminder of that. Each overgrown bombsite, each broken building, each street corner with its stonework gouged by the bullets of 1945 whispered what the alternative to peace looked like.31
Nor did every citizen regard the Soviets as oppressors, something else largely unreported, and Gerda Stern – the communist since 1932 – captures aspects of that. 'Many people worked with the Soviet Union to defeat Hitler. We thought we had to have the Soviet Union and only the Soviet Union to do that, to stop Hitler and then make a better world.'
Sometimes history seems to have been written in and by the West, and the East _seen from the East_ remains an unknown, darkened place – the way it goes.32 What went on from the 1940s through the early 1960s all too often gets lost in the refugee totals at Marienfelde and the kisses on the baby's forehead.
But a large number of people were communists because they were determined to make the world better. They saw themselves as liberators. Leaving aside innate German discipline, one cannot imagine _this_ quantity of Factory Fighters holding the line in the uncertain night unless they were believers. They could have turned over and gone back to sleep. The Factory Fighters were not seeing the future flow away, they were taking the first steps towards creating it, in public, on camera with, here and there, the most distressing abuse being shrieked at them; and they stood and they took it.
Nor were the communist leadership crazed tyrants, as they were portrayed in the West. If you seek power you go where power is, and this is exactly where the communists of the 1930s did not go. After 1933, if you sought this power, or even a taste of it, you joined the Nazi party and _ruled._ You did not go, as Ulbricht did, on his own tortuous journey from Berlin to Moscow where he'd spend each moment, especially the hours of darkness at the Hotel Lux, utterly at the whim of Stalin: Ulbricht, a German exile even as the German Army was tearing the Soviet Union to pieces. Consider it. Stalin, most malevolent of men, killed his friends as well as those who might be enemies; and in the last convulsive twist of paranoia thought that even he himself might be a counter-revolutionary, and logically _an enemy of himself._ What might he have done to Ulbricht, the German? He did it to other Germans in the Hotel Lux, the ones nobody saw again.
Yet still Ulbricht believed.
You did not go, as Erich Honecker did, to Nazi prison camps. There are disputes and contradictions about the evidence of this, and understandably so, but that does not disturb the central tenet. He was in the camps and if you sought power, and that was all you sought, you'd have been somewhere else, _anywhere_ else.
These communists, who ultimately did get the power, can be imputed with all manner of failings, and have been, and will continue to be. The failings are writ large, and larger now the documentary evidence is in and all the witnesses can be called, but they cannot in all conscience be accused of insincerity.
It's a different argument as to whether communism was inherently flawed. In a sense, _that_ logic had a chance to work itself out from 13 August 1961, and did, climaxing on 9 November 1989 by destroying itself across the white line Hagen Koch had painted, and thus ending precisely where it began.
To his last breath Honecker made no apologies because, in his eyes, he had nothing to apologise about and he remained sure that history would judge him well.
Until you understand this, you cannot understand the wall.
Perhaps the last word should go to Diana Loeser, the English-woman who'd gone to live in East Berlin. She was, as previously mentioned, visiting her home town of Birmingham, England when she heard the news of the wall going up and she prepared to return. 'If you lived in Berlin it was different to living in Frankfurt or wherever because the border was just up the road and you saw the smuggling, the black marketeering and the currency exploitation, and all of it was damaging the economy. We could have gone on trying to persuade people to stay through conviction but there was no chance. In West Berlin you saw the shop windows full, and you couldn't win economically on those grounds, so I was pleased when it went up. I thought at least we'd a chance of building socialism in this part of the world and, given a fair crack of the whip, it would go along at a galloping pace, we'd be all right and catch West Germany. The need for toing and froing across the border would go.'
Whatever, the toing and froing had gone now, gone for the dreamers, the believers, the disciples and the bloke in the corner bar wondering what it meant. He'd find out the hard way. They all would.
* * *
Quotation at head of chapter: _Goodbye to Berlin_ , Christopher Isherwood (Hogarth Press, 1939, rep. Minerva, London, 1989).
1. Interview with author.
2. Interview with author.
3. Interview with author.
4. Interview with author.
5. Interview with author.
6. Mary Kellett-Long's diary. She was kind enough to give me all the entries I wanted, but more than that she had a way of seeing the essentials and expressing them in vivid yet straightforward language.
7. Interview with author.
8. _Berlin: Success of a Mission?_ , Geoffrey McDermott (André Deutsch, London, 1963). (The title is a play on a book _Failure of a Mission_ written by Neville Henderson, about the British in Berlin up to and at the outbreak of the Second World War.)
9. Interview with author.
10. Interview with author.
11. _The Ides of August_ , Curtis Cate (Weidenfeld & Nicolson, London, 1978).
12. Bailey, op. cit.
13. Interview with Birgit Kubisch.
14. Interview with author.
15. Interview with author.
16. Interview with author.
17. _Willy Brandt, Portrait and Self-Portrait_ , Klaus Harpprecht (Abelard-Schuman, London, 1972).
18. The time gap from Berlin to Washington is today [December 2000] six hours but was it that in August 1961? Various books written subsequently are not unanimous. Some say five. I am indebted to Edith Kohagen of the _Presse und Informationsamt_ of the Landes Berlin. I asked her if she could find out and she replied: 'I called the _Physikalisch-technische Bundesanstalt_ in Braunsweig to get the solution. The Federal Institute is responsible for the time in Germany. They told me the time difference must have been six hours (I repeat: six hours) because we had no summer time. Between 1950 and 1980, in fact, there was no summer time in Germany.'
19. Interview with author.
20. McDermott, op. cit.
21. Interview with author.
22. McDermott, op. cit.
23. Harpprecht, op. cit.
24. McDermott, op. cit.
25. Interview with author.
26. Interview with author.
27. _Washington Post_ , 14 August 1961.
28. Interview with author.
29. Interview with author.
30. Interview with author.
31. The East Berlin journalist Bodo Radtke had, of course, the opportunity to travel. His family were originally from that part of Germany now become Poland, and he happened to be reporting a cycle race whose route passed near the village which had been home for generations. He found the street and, oh yes, people remembered the Radtkes. The race concluded, he got back to East Berlin and told his father the tale, and asked his father if he wasn't sad that he'd been driven from his home, it was in a foreign country and – Poland under Lech Walesa's counter-revolution now out of bounds – he could no longer go back even to have a look. 'Yes', his father said, 'yes, of course but if this is the price of peace, let's pay it.' I am persuaded that those who have known war, and where the logics of war go, all feel the same and if they don't they are very dangerous people.
32. And I am writing these words as a Westerner.
## FOUR
## _First Week of the Rest of Your Life_
When will they ever learn,
When will they ever learn?
Pete Seeger,
_Where have all the flowers gone?_ *
That Sunday afternoon, a small boy in a cardigan, arms upraised, beseeched that he be allowed to cross, and a policeman adjusted the top of the wire so that he could. The policeman's head twisted away to see if anyone was watching him – someone was, and he was arrested immediately.
Another fixed image, taken quite passively, was of two soldiers positioned at a broad crossroads beside a lamppost and a litter bin, with a rambling web of the wire in front of them. Behind, a crowd lingered on the pavement not daring to go near. They formed the eternal backdrop of history, curious and uncomprehending.
At another place, where the wire hadn't yet reached, a Western motorist emerged from his car, advanced and an Eastern policeman pushed him in the chest. A soldier, bayonet fixed to the barrel of his rifle, prodded it towards the motorist. The portals of a tall, broad church loomed behind all this, eternal against the foibles of men. Was the motorist going there to worship, as he always had, and now found his church in another, forbidden, country?
Two soldiers faced a crowd at a crossroads while a third had gone among the people to silence or intimidate them. Beyond the crowd, two elderly women in summer frocks chatted, arms folded. The other part of their own street was less than 40 metres away and now it, too, was in another, forbidden, country.
Two Factory Fighters in shirt sleeves walked with a coil of the wire wrapped in a ball round a stout stick, a Fighter holding each end of the stick. As they walked, the wire automatically uncoiled and another Fighter, kneeling, lifted a strand of it and hammered it into a concrete post using cleats, gathered another strand and hammered that in, then another – row upon row, neatly spaced – until the wire was head high.
An elderly couple were escorted from the wire by a policeman, the sadness and the powerlessness expressed by the stoop of their shoulders.
A family of four smiled and smiled and explained how they'd managed to scramble over because some incident diverted the police.
An Eastern policeman said to a photographer: 'This is Free Berlin, taking photographs is not allowed here.'
On a street bisected by knee-high wire, people milled on both sides. In the East, a young man had his hands in his pockets; nearby, a plump woman held a baby and two women stood next to her. A policeman strode by, automatic rifle on his back. On the Western side, a very young girl wearing a frock trampled the wire down with her foot, presumably to step over it. Her mother, one of the two women standing together, scolded her for risking tearing the frock. A man in shirt sleeves went by on the Western side and made a joke so that suddenly everyone was laughing. Did the child step over? Did the mother step over? Did the policeman stride back and hold them apart? The moving camera had stopped filming by then, and what it had taken would be developed, carefully printed, fixed into the montage but giving only the present, not past, not future.
'Toasts of good luck were drunk by some of the East Germans who arrived in West Berlin but some women in Marienfelde wept quietly', Kellett-Long filed to Reuters. 'They said they waited in vain for husbands or children who had arranged to cross by another route for safety.'
At 5.52 p.m., a report to Police Headquarters noted 'several young people rampaging near the National Council building but armoured cars dispersing them'. At 6.00, a police radio car announced, 'About 300 people are at the Unter den Linden– Friedrichstrasse intersection but do not show any negative attitude. Eighty per cent of them dispersed by the Factory Fighting Groups at 6.05 and the dispersing is being continued.' The Brandenburg Gate remained a place of potential insurrection, and that gave the Unter den Linden a particular importance.
Several reporters saw the dispersal at the intersection, Kellett-Long among them. 'A line of ten steel-helmeted black-booted policemen with rifles slightly down but at the ready faced the silent crowd. The atmosphere was tense in the centre of the crowd for a moment, but people gradually began to drift away while others took their place, and the moment was gone. The police, who also carried their usual pistols, were especially armed with automatic rifles.'
A _New York Times_ reporter sensed 'a note of fury building up. Hundreds of young men moved down Unter den Linden. Two troop carriers drew up to reinforce the police. A line of rifles was raised slightly towards the crowd who made a slow backward movement.'
A report to Police Headquarters said 'security forces and water cannon deployed in the passage under the Brandenburg Gate by 6.20. The water cannons are not in action.'
Armoured cars, which had arrived earlier, moved fractionally through the Gate. A long cordon of men from six different services – the six to demonstrate solidarity – arranged themselves shoulder to shoulder in front of the armoured cars because the sector line bulged there. One of these men, a Factory Fighter, wore his uniform but white socks and sandals, betraying the haste in which he had come.
The sector boundary was a white line painted across the roadway but none of the uniformed men in the cordon put a boot – or a sandal – on it; they stayed fractionally behind. Above them, by the old stone charioteer on top of the Gate, a soldier with field glasses scanned a British Military Police jeep parked 100 metres away. The officer behind the jeep's steering wheel spoke into a walkie-talkie. A helicopter from the West skimmed by far overhead and banked, monitoring.
The GDR did not pass up a chance to fix images. From their side a photographer moved behind the cordon and took a picture through it so that the cordon appeared as a protective bastion against the threatening mob of Westerners in the distance who bayed abuse, shook their fists and chanted. A middle-aged man, seeing this Eastern camera recording him, cupped his face in a hand to mask it. Did he have relatives over there, did he fear retribution being visited upon them if he was recognised? Was he from over there and he'd made it? The images posed questions as well as answered them.
Kellett-Long saw 'a working party of police erecting six-foot high concrete posts for a wire fence across open ground in Potsdamer Platz and then the Brandenburg Gate'. The noose was still tightening. A crane with a grab would clear this obstruction away in a straight-forward demolition job – but this would be on Sunday 12 November 1989.
At 6.10 p.m., a report to Police Headquarters said a West Berlin senator would make a speech at the Potsdamer Platz at 7.00 – a possible incitement to insurrection. A reserve Factory Fighting Group of 162, stationed just off the Unter den Linden, was dispatched to cover that and, at the same time, an order went out to deploy water cannons in Potsdamer Platz. At 6.20, the Factory Fighters cleared the road between the Brandenburg Gate and Potsdamer Platz – 'No people gathered on the western side at the moment but several little groups in West Berlin.'
At 6.30, a report from Treptow said young Westerners had trampled the barbed wire. Another from nearby announced that 'Citizens can still go to West Berlin unhindered. The situation cannot be changed with our own forces. Measures will be taken.' The noose had not tightened there yet.
At 6.40: 'The situation at Potsdamer Platz is unchanged as a whole although West Berlin television has put up a second camera.'
Two minutes later, a report noted 'about 500 persons gathered at Wollankstrasse, mostly young and undisciplined. Two platoons of Factory Fighters and one platoon of police brought into action.' Five minutes later, the number on the Western side grew towards 800. They shouted across, 'Have you a ticket for the Walter Ulbricht Stadium [a few streets away]? They're showing the last piece of butter there.'
At 6.50, a report from Treptow said a crowd of 150 in the West and 100 in the East 'whistled from both sides' and a battalion of Factory Fighters 'normalised the situation'.
Peter Johnson of Reuters spent 'much of the evening on different sides of the Brandenburg Gate. On the Western side a crowd of several thousand collected and for a time got partially out of hand because the West Berlin police precautions had not been strong enough. Some groups made dashes onto the road in front of the Gate – the road itself was East German territory – and were driven back by water cannons mounted on armoured cars. [These were the ones deployed at 6.20.] Later, the police confined these people behind ropes. A section of the crowd, perhaps two or three hundred strong and composed mainly of teddy-boy-type youths, shouted rhythmically such slogans as "Berlin stays free" and "Away with the Goatee Beard"— Herr Walter Ulbricht, the East German leader, sported one.
'A British Army Corporal keeping watch in a radio jeep was critical of the West Berliners who, he said, were causing the tense situation at this spot. The left-wing Social Democratic Mayor of the [Western] district of Kreuzberg asked a police officer to move back the crowd to reduce the danger of a clash. One police officer, I was told, said that a member of the West Berlin Senate had been on the spot and said, "Let the people stay, it's good like that." While I fully understand genuine feelings of anger at the East German action, I think it is wrong to let it be manifested in this dangerous way and mainly by a lot of louts who have little political sense anyway.'1
Mary Kellett-Long noted in her diary that 'later in the evening big crowds gathered at the Brandenburg Gate at both sides and reinforcements were called in and water cannons used and someone threw a Molotov cocktail. Eventually everything quietened down without any serious incidents.' She added: 'I've been worrying about the Berlin crisis ever since we came here but, funnily enough, apart from about five minutes I've been quite calm and relaxed. What is the point of worrying really? If there is a war we shall be as safe here as anywhere because both the Allies and the Russians have great numbers of troops. My own feeling is that there will be no war.'
The water cannon were framed into the images: a youth rushed forward to throw the Molotov cocktail and long jets of spray from the nozzle of the cannon engulfed him; someone else rushed forward and turned at the furthest limit of the spray then scampered back with his arm raised in the most temporal triumph.
Westerners called out, 'Why don't you go and dig your own graves?'
Round the House of Ministries, six tanks, three armoured cars and six trucks filled with troops stood guard. 'The helmeted tank drivers were perched in open turrets ready for action.'2
At Hyannis Port, Kennedy cruised gently on the _Marlin_. It's worth restating at least one of the reasons why he did not return to the White House. As Frank Cash says:
Berlin was a very special place, the only cosmopolitan city in West Germany. Munich might be big, Stuttgart might be big, Hamburg and Frankfurt might be big but none of them were the cosmopolitan centre which Berlin remained even though cut off from the West. I think Kennedy thought those who worked in the Mission and those of us in the State Department working on Berlin over-emphasised its importance, but to us it was the crucial point of confrontation between East and West. Perhaps we were gung-ho about it although, incidentally, nobody in the Department recommended knocking the wall down.
Kennedy never really committed himself to the whole of Berlin and you'll notice that all of his statements emphasise West Berlin. He constantly said, 'We will maintain our rights in West Berlin and our access to West Berlin.' Prior to that we, the Department, used the term Berlin because we felt that as an administrative area our rights were in the whole city. I think Kennedy felt if West Berlin was secure that was the main objective.3
The possibility that access might be cut in a second phase of the wall operation concerned Kennedy a great deal, but what could the United States physically do until that happened? So Kennedy waited and, in the meanwhile, did not intend to risk provoking an insurrection. It was an awesome power that one man, barely middle aged and on a cruiser off Nantucket Sound, might incite – by an inadvertent phrase, an innuendo, a hasty action – thousands upon thousands of total strangers a continent away to take to the streets and shake the world order; and it carried an inescapable conclusion. If they did, he would be powerless to help them.
He operated, too, within the nuclear constriction, that iron lung through which every United States President has had to try and breathe since September 1949 when the Soviet Union exploded their first atomic bomb. Each President would have to learn controlled breathing, even when he was running fast.4
Precisely to maintain visible calm, Dean Rusk prepared to attend his baseball game, Washington versus New York at Griffith Stadium in the north-west of the city, with his wife and 12-year-old daughter Peggy.
This measured approach found full endorsement and active promotion by the British Foreign Secretary, Douglas-Home, and also Prime Minister Macmillan, whose thinking had been conditioned by the slaughter of the First World War in which he served. The dictum was, better that grouse be shot on the Yorkshire moors than people in Berlin. The British did make a statement, a Foreign Office spokesman saying that 'restrictions which have been imposed on the movement between East and West Berlin are contrary to the Four Power status of the city and therefore illegal'. As an exposition of the obvious this is hard to fault, but it offers nothing more. The British, however, were in a harder situation than Kennedy because at least he could decide when to breathe out: they must hold their breaths until he did – unilateral action by the British belonged to the vanished imperial age. (GDR propaganda sensed the weakness of such statements and announced that, far from dismantling the barriers, they were here to stay.)
At 7.00, it was reported that 'a hooligan driver' burst through 'in a Trabant, colour white-red, number plate could not be seen and number of people in it could not be seen'. They may have been the first to escape using a vehicle, which was an obvious thing to do against barbed wire. Already, on some bridges, obstacles were being placed to prevent this.
Before dusk, Ulbricht – this portly man who'd been a believer since before Gerda Stern and Elli Köhn, who'd survived the Spanish Civil War and the Nazis, survived in Moscow at Stalin's whim and would survive into the decade beyond Khrushchev – made a tour of his own. He wore a white shirt, a tie and a light-coloured overcoat. Honecker, in a dark suit, walked behind. GDR television filmed the tour (although it was not shown for a week) and recorded a segment of dialogue:
Ulbricht: 'All the orders have been carried out on time, yes?'
Factory Fighter: 'Yes, yes sir.'
Ulbricht: 'Everybody was where he should have been at the right time, yes?'
Factory Fighter: 'Yes, sir. In the factories and our National People's Army.'
Ulbricht: 'The People's Army, yes...'
Policeman: 'The Factory Fighting Groups, their squads were there, too.'
Ulbricht: 'The Factory Fighters. And for support there are several tanks of the Soviet Army in reserve, yes, so that there won't be any misunderstanding on the part of our enemy, yes?' [Laughter]
Citizen: 'And we know those tanks...'
Ulbricht: 'Isn't that true. We know them, yes.'
Factory Fighter: 'Obviously the whole thing got to them [the West] because all day long a helicopter has been cruising around, they are burning up quite a bit of gas on behalf of us.'
Ulbricht: 'Yes, yes, ha, ha... well, I suppose everything is all right now, yes.'
Factory Fighter: 'I don't think they are going to say the "so-called GDR" any longer.'
Ulbricht: 'You see everything has simply been carried out on time, yes? Everything has simply been carried out. Everything is all right. There is no doubt everything is in order. The barbed wire is in place.'5
There is a certain awkwardness in these stilted words but the imperfections give authenticity, even as we hear ordinary people mouthing words they think Ulbricht wants to hear; but Ulbricht was under more pressure than anybody, including Khrushchev and Kennedy. They might lose a round in the long tactical struggle sprawling across the globe; he might lose everything. How did he withstand this pressure? How did he cope? What did he think, _really_ think? Did he have doubts and, more pertinently, self-doubts about a wall across a city? The answers are entombed with him in a Berlin cemetery and the only other person who could have told us lived in a plain, neat house with a modest garden a couple of streets from where the border, which her husband built, stood. She was called Lotte Ulbricht and when I asked for an interview in the early 1990s I got a polite refusal, typed on what seemed to be an ancient typewriter.
Darkness was drifting in and Al Hemsing spent a hectic evening because 'we'd planned a dinner with Ed Murrow and Mayor Brandt and, of course, Brandt got completely caught up. The phone kept ringing saying _he's coming, he's not coming, he's coming_ , and eventually he didn't come. You can hardly blame him. That night a group of us sat around and Murrow said he needed to draft a telegram direct to the President, telling him what was going on from his [Murrow's] point of view. Murrow reverted to being a reporter, so to speak. He sent the telegram, it had a clear message and I have always felt it was one of the ones that "got through". The brunt was _a terrible thing has happened, not only for a violation of the Agreements but because so many people go back and forth every day, and this is a tremendous blow to the morale of West Berlin. Something needs to be done_. I've subsequently had confirmation that that message was taken very seriously by the President.'6
(Earlier Murrow had given an interview to RIAS and forged words as only Americans can. 'There is no Berlin crisis, it is Khrushchev's crisis. He has, in the vernacular of our western movies, told us to get out of town by noon tomorrow. We do not choose to depart.')
At 9.00 p.m., a photographer froze an image on the corner of a street bisecting Bernauer Strasse: five uniformed men stood under one of the stone arches bearing the burst of bullet marks of 1945, men who looked morose and gazed in mistrust towards the photographer. One of the five angled his face away. What did he fear? Was it shame?
No photographer froze what would have been an equally telling image. Two young East Germans undressed completely and swam across the Teltow Canal.
By evening, a portrait of two cities began to emerge, not Berlin and Washington but Berlin and Berlin. On one side lay the worked-for affluence which was being dispensed down the Ku-damm, enjoyed in shady clubs, eaten in international restaurants, and sat through comfortably in the soft lighting of new apartments in futuristic buildings. Every smooth-tarmacked street was heavy with cars, many of them new cars. On the other side lay the worked-for poverty of waiting lists for accommodation along cobbled, potholed streets in pre-war apartment blocks so darkened and damaged they seemed to mourn what they had once been. People drank in little bars on the corner and played cards through the cigarette smoke across worn wooden tables.
These little bars, intimate by their cosiness, might be safe enough for a local to speak his mind and the natural forum for him to do it. If an insurrection came it would either be an impulse gripping a crowd at the wire or something explored, calculated and initiated in one of these little bars snuggled all over East Berlin. But as darkness drew its veil over the first day neither happened; and then it was too late.
At 10.30, a crowd of 4,000 lingered on the Western side of the Brandenburg Gate but for an insurrection to occur would mean bursting past their own police, then the cordon of Factory Fighters, then the armoured personnel carriers with the water cannons, then the Eastern police, before reaching Unter den Linden; and Unter den Linden had been cleared, anyway.
At 10.45, a report came in that 'a male person' had swum the Teltow Canal.
This is the Teltow Canal south of the city centre and, because of the green spaces on either side, a favourite place for early escapes.
Towards 11.00, the radio reporter Peter Schultz finished his shift of seventeen or eighteen hours. 'I went with a colleague to the Brandenburg Gate because I'd been reporting about it all day but I hadn't seen it. I got a very destructive impression because what I saw was even worse than I'd imagined, the light of arc-lamps and, nearby, wire and soldiers and the Factory Fighting Groups and the street torn up. They bored holes in the surface with pneumatic drills. The West Berlin police had to work hard to push back protesters.'7
At 11.45, a report to Police Headquarters said the situation at the Brandenburg Gate 'remains unchanged. There are still young people on the Western side in green spaces in the area of the Tiergarten who are agitating against us.' Between 11.20 and 11.40, a report announced 'about 3,000 young people who had gathered at the Gate were dispersed by Western police using truncheons. During this, the young people shouted "you are beating the wrong side".'
By midnight, the passions and pulses of the day were ebbing and a report caught that: a gathering of 150 people in the East dispersed. It isn't clear where in the East and somehow that doesn't matter any more. They melted into the cobbled streets, walked uneven pavements under pallid lamplight, and they were gone just as the moment had gone.
The true numerical scale of what had happened was subsequently expressed in dry language in a Western publication:8
Before August 13, 1961, the Sector Boundary dividing West Berlin and East Berlin was crossed by 500,000 Berliners every day. Eight to ten million inhabitants of East Berlin and the Soviet Occupation Zone visited cultural and sporting events in West Berlin every year. Until August 13, 1961, the metropolitan railway (S-Bahn) and the underground railway (U-Bahn) were public conveyances for inter-sector passenger traffic.
The communist sealing-off measures, however, ended the through-traffic of eight metropolitan and of four underground lines. In the Soviet Sector, all the 48 metropolitan railway stations were closed for inter-sector traffic, and, of the 33 underground stations in East Berlin, 13 were closed completely. For the inter-sector traffic of foreigners and citizens of the Federal Republic of Germany, a special platform has been established in East Berlin both at the metropolitan and at the underground station in Friedrichstrasse.
The sector and zonal border round West Berlin cuts 193 major roads and by-roads, 62 of them leading into East Berlin and 131 into the Soviet Zone. Before August 13, 1961, the sector boundary between West Berlin and the Soviet Sector could be crossed at 81 crossing points. On August 13, 1961, 69 crossing points were blocked by barbed wire or walled up. Twelve specially marked crossing points remained open for the purpose of entering the Soviet Sector.
By midnight, the reports coming in to Police Headquarters altered. An order went out that the Factory Fighters in Prenzlauer Berg must be relieved at 2.00 a.m. and replaced by two units of 100 each from the Mitte district. The order recorded those to be relieved: 'the second battalion motorised (101 comrades) and the seventh General battalion (46 comrades)'. A report said the Factory Fighter at crossing point 49, towards Treptow, 'who was injured by a smoke bomb has been taken to hospital and is being treated'. The Fighter thought he had time to pick it up and return it, and misjudged that.
A policeman recorded he'd picked up 180 cigarettes tossed to him from the West to entice or provoke him.
At crossing point 68, south of Treptow, a police inspector reported that the Factory Fighters 'have not received any provisions today. They do not have any torches and some of them have no blankets. They are in a bad mood.'
The incident sheet recorded a total for the day of sixty-six escapes and one caught; and although there must have been many more, the tightening of the noose produced an almost complete reduction in the number of refugees reaching Marienfelde, where the total of 150 registered compared to 2,662 on the Saturday. It is not known the numbers of those who got across and, bypassing Marienfelde, went to a mother, a father, a brother or sister, an aunt, a friend and slept there, and went looking for the rest of their lives on the Monday.
There must be 4 million postscripts. Here are just two. 'I can honestly tell you,' Lutz Stolz says, 'that Sunday 13 August 1961 passed normally for us, at least without us panicking. I tried to reach all my friends by telephone and they were as filled with consternation as Ute and I but we did not go anywhere near the border. We didn't panic because we were young and we hadn't realised what it all really meant.'
Geoffrey McDermott wrote: 'So August 13 ended after more than the usual quota of telegrams had been sent off to distant capitals asking for instructions.'9
Meanwhile, at 11.45 p.m. Michael Moore, a corporal in the 4th Royal Tank Regiment, had slotted his tank into position on the Western side of the border behind a railway station and wondered how he'd keep himself awake.
## MONDAY 14 AUGUST 1961
At 1.00 a.m., the Border Police stopped a party of two at crossing point 45 and entered it on the incident sheet using the same headings as the day before. At 1.30, X + 24 hours, 5.96 miles of wire barriers on concrete posts had been erected.
Before dawn, Corporal Moore could see nothing of the street where his tank stood – Invalidenstrasse, not far from the old disused Hamburg main line terminal. A canal curved to the right and a bridge lay directly ahead. The East began at the other end of the bridge and the buildings there stretched away like sentries guarding the past. 'Every now and again I would search around visually using the episcopes [optical projectors] and binoculars, but if the image is black, magnifying black seven times doesn't really improve it.' At dawn, a shape revealed itself, a tank which he identified as a T34-35 and bearing Soviet markings. Headquarters wanted to know how far away it was and he said, 'about fifteen yards'. Incredulous, headquarters queried that and he confirmed it: 'Our gun barrels were more or less five yards apart.'
Moore decided he'd have a mug of the tea which the radio operator had already brewed (egg sandwiches were coming in a minute). He took the mug, popped up from the observation hatch and placed it on the flap in front of him. 'Till then there had been no sign of life from the T34. It could have been an empty hull but – and I nearly spilled my tea – the Commander's hatch opened and a head appeared wearing one of those super Soviet helmets and a pair of binoculars. All I could see was him looking at me and me looking at him. I suppose he was probably intrigued about the tea more than anything else.' Corporal Moore leaned forward and gave him a 'gentle wave'. The Soviet commander ducked down, shut his hatch and was seen no more.
Invalidenstrasse was one of the twelve crossing points open and the Soviet tank was fully entitled to be there on the very lip of the Soviet zone, but Moore's experience provides a curious historial footnote. Very few Soviet tanks were deployed within the city and this is one of the few sightings.
Mary Kellett-Long wrote in her diary, 'I woke this morning feeling considerably rested to find that everything had quietened down somewhat. Of course everyone except the Grenzgängers [who lived in the East but crossed daily to their jobs in the West] was at work.'
An artificial normality had returned. Shops opened, factory shifts clocked on and off, and trams creaked down their tracks but no longer continued into the West. The S-Bahn did still run West, but only from that one platform at Friedrichstrasse station, of course. The _New York Times_ reported that: 'On the elevated train several hundred stiffly quiet people rode towards East Berlin. A blonde woman sobbed and tears creased her make-up. She breathed hard and forced herself to stop crying. At Friedrichstrasse, a middle-aged couple argued with mounting anger only to be pushed aside from the turnstiles by the policeman. As East Berliners, they cannot go West.' The same journalist noted 'thirteen tanks in an empty lot near the station. The crews were putting up tents and operating cook wagons.'
The GDR government methodically cleared up any overnight confusion. They issued a situation review and inserted into it brutelly simple instructions:
The measures taken were successful and the enemy so completely surprised he was not able to take effective countermeasures.
There are attempts at the moment to go to West Berlin on a legal basis. Yesterday, for example, several thousand people tried to get permission and applied at police stations. It is to be expected that the number of these persons will increase. It is not the task of the police to support these attempts by giving the relevant information.
The Ministry of the Interior announces that the issuing of such permits and the regulations governing that will be given in a special announcement. This demonstrates without question that the exact time for issuing these permits is not yet fixed.
Every person who appears with such an application has to be told with no possibility of misunderstanding that an application is useless at the moment. They must be told they have to wait for an announcement. No exceptions can be made.
It is no longer permitted to shunt applicants from one authority to another because nobody wants to take responsibility. It is also necessary for our comrades of the People's Police to show their class-related standpoint over this.
There is no reason to raise people's hopes of getting permits out of sympathy or indecision.
The blame for the harshness resulting from these measures rests with those who for years have rejected the serious proposals of our government to banish the danger to peace arising from West German militarism.
The measures taken by our government serve to protect democratic Berlin and its people. They are an effective blow against the illegal, subversive smuggling of human beings organised by the extremists in Bonn. This means that the security bodies have the task of preventing the citizens of our Republic and its capital being made into victims of unscrupulous slave traders. This entails preventing the entry of our citizens into West Berlin.10
The impact fell upon ordinary policemen behind counters in ordinary police stations who had to turn away distraught humanity, weeping no doubt, pleading no doubt, a mother to see son or daughter, a man to resume his job – the shock hit West Berlin hard because 53,000 Easterners couldn't report for work and never would again – a widow to visit a grave. The human consequences of wrenching Berlin apart fell upon these policemen now instructed to behave unlike human beings.
The police had been shunting people away because, up until the instructions, it must have been easier than refusing them outright. Whatever feeling the police had, the words 'no exceptions can be made' forbade equivocation. The noose was a tourniquet now.
During the morning an American journalist, Norman Gelb, went through the Brandenburg Gate. He recognised a policeman because when he'd gone through the previous Thursday they'd exchanged some banter about Gelb giving him a ride to the Ku-damm for a cup of coffee. This time, as Gelb says,11 'I stopped to be cleared and that same man, whose double-take indicated he recognised me, stiffly asked for my passport. He looked at it, handed it back, scanned the floor of the back seat of my car for any unauthorized passengers, saluted and briskly waved me on.'
Even now, the Brandenburg Gate remained a place for possible trouble. Peter Johnson wrote in his diary: 'Another example of how dangerous letting louts loose on the Western side can be came this morning when a youth in a small crowd, which the West Berlin police had ill-advisedly allowed to approach the border, again threw a Molotov cocktail onto the East German road in front of them. These weapons – a bottle filled with petrol – were used in profusion during the anti-communist revolt in June 1953 and could, in the mind of a communist East German policeman, have provoked a savage reaction. No wonder the West Berlin police used a certain amount of rough technique to hustle the youths away from the spot.'
Before Bodo Radtke, the journalist who'd returned from holiday, went to work 'I travelled to the border with my wife. We wanted to see with our own eyes. We saw barbed wire and at the Brandenburg Gate the people on both sides calling to each other "What's going on? Can we come? Can we go?" Nobody knew what the future held. Everybody was saying it's only for a couple of weeks, maybe a couple of months but not longer than that.'12
Mary Kellett-Long 'went through in a car and I must say it was quite a sight with Factory Fighting Groups all carrying sub-machine guns and about six armoured cars at strategic positions. There were also tanks in the side streets and water cannons on the Western side of the Gate. Apparently the number of forces has been increased and we wondered what was going on.'
Shortly after, an order chattered from Police Headquarters (extract):
Content: measures to be taken against provocation at the Brandenburg Gate.
Because of continued provocation, and particularly because of the agitation carried out at midday today by representatives of the West Berlin Senate and the government of Bonn as well as the irresponsible demands of the broadcasting stations Free Berlin and RIAS to violate the border at the Brandenburg Gate and prepare for other dangerous acts, I order
(1) The crossing point is to be closed with effect from 14 August 1961, 2.00 p.m.
(2) The traffic from democratic Berlin to the Gate is to be diverted at Friedrichstrasse. The drivers and passengers of these vehicles will have to be given the reason for this measure. The forecourt of the Gate, which belongs to democratic Berlin, has to be sealed using construction equipment. The passages under the Gate have to be kept open to guarantee our special vehicles and security forces can get through for swift and effective protection of the barriers which the construction equipment is putting up.
On the Eastern side, the closure was announced by loudspeakers on vans, the voices flat and nasal.
Mary Kellett-Long remembers coming back by train 'and that was all right. There were fears that you weren't going to be allowed to go through at all.' She'd write in her diary: 'I came back on the S-Bahn. It took ages. I had bought cigarettes and chocolate, newspapers and books. The trains now have to be emptied at Friedrichstrasse and everyone go through controls. I was stopped at least six times, once by a customs man who, I am glad to say, did not ask if I had any goods as I should almost certainly have lost all our cigarettes.
'We'd been wondering what the increases in forces meant and we found out when the Ministry of the Interior closed the Gate temporarily because of incidents. I don't know where all the people in cars come from. Usually the streets are empty but today a thick stream of traffic went up towards the Gate and hundreds of people just stood and looked. On the other side were tremendous crowds. The traffic was so awful I was glad I wasn't driving. The Berlin police were out in full force, including the riot police wearing tin helmets. There were film cameras and tourists gaping but everything quietened late in the day, and the West Berlin police moved the crowds right down to about half a mile away and the East Berlin police did the same. We think the Gate may be opened early in the morning.' It would be opened again, though not for traffic – and not until 1989. Now, in mid-afternoon, a curve of barbed wire snaked round the front of it in large, loose coils.
Brigadier Godfrey Hamilton, Commander of a British Infantry Group, stationed himself near the Gate on the Western side and watched the police keeping civilians away. When a reporter questioned him, he said as far as he was aware 'there are no Soviet forces in evidence in East Berlin but we know they alerted two motorized divisions in the area of Potsdam.' (Presumably he didn't know of the tank which Moore saw or perhaps regarded its solitary presence as legitimate and not worth putting into the equation.)
Al Hemsing dropped in on Reuters' West Berlin office near the Ku-damm to see the bureau chief, 'a wonderful guy. He and I had a very close personal relationship, we'd chat and say, "What do you know, what do you know?" A young lady cried at her desk and I asked her why. "I was away in West Germany and I gave my cat to my aunt in East Berlin to look after for the weekend and I'll never see it again."'
When Mary Kellett-Long reached the Reuters office in East Berlin she found that husband Adam, Peter Johnson and Erdmute Greis-Behrendt had been under arrest.
'Adam and I were slightly worried that Miss B. [Greis-Behrendt] hadn't returned from the Town Hall in Prenzlauer Berg where Adam ill-advisedly sent her to enquire about the numbers of East Berliners with jobs in West Berlin who had now registered for work,' Johnson would write.
He and Kellett-Long went to the Town Hall and 'at the wicket gate Adam and I presented our documents and explained who we were. A young man in plain clothes took us inside and showed us to a room where people were registering. He invited us to ask questions of them if we wished. Just then we saw Miss B. who told us she was waiting to see the top official, at present at lunch. Another man came along and offered to take us to the Major on the first floor. We walked upstairs and waited outside the Major's room. Three green-uniformed police officers came with several Factory Fighters and one of the police officers snapped, "You are temporarily under arrest."
'He immediately told us to get moving and to reinforce his order gave me a push. I protested at his rudeness. He took my passport and hit my hand when I said he had no right to take it and tried to get in back. He barked viciously, "Don't lay hands on me." We were taken back to the wicket gate, during which time Adam and I explained several times that we had declared our identity and that we were accredited to the East German authorities and all he need do was to ring the Foreign Ministry to check on that. Instead, after we had phoned from a cubicle, we were placed in the back of an elderly black BMW police radio car, where we giggled and joked for about ten minutes. Once I wound the window down a little and one of the policemen standing outside made me wind it up again.
'Then, with two policemen in the front seats, we were driven at breakneck speed to the borough police headquarters. After several furious corners I asked the driver if he was a racing ace. He replied with a grin, "I always drive like that." In the police headquarters, which is a few doors down the road from our East Berlin office, we were ushered into a barred room occupied by an artificial platinum blonde, a West German girl from Duisburg, who told us she'd been there since 8 a.m. – it was then 2.30 p.m. – after being picked up for entering East Berlin without a permit. They had not offered her a chair, but they brought three for us. I asked that they gave one to the girl, too, but she said she would rather stand. The door of the room was left open.
'Through the barred window we could see onto a wired-off yard, beyond which was a cemetery. I passed some time reading inscriptions on graves – _Meine liebe Frau_ [My Dear Wife] – and then Miss B. pointed out a coiled rope in the yard, which she opined was the hangman's rope. None of us had brought a nail file so we could not go to work on the bars! Instead, I spent some time convincing the police that they should ring up the Foreign Ministry, and asking them repeatedly to give us something to eat. First I was told, "You have to be here eight hours before you are fed" but after they became tired of my nattering, or had received instructions from higher quarters, we were told we would get something. Then, about an hour after we had been brought in, an elderly police officer entered with our passports and said, "You are free... . It was a mistake." We asked him what kind of mistake and he mumbled something about them having to be vigilant in the present situation. Just at that moment our lunch arrived – three large white porcelain bowls full of meat-and-noodle soup. We stayed to eat it up and Miss B. even congratulated the white-uniformed police cook who had been working overtime to supply the members of the _Einsatz-gruppen_ – the operational squads of police brought in to stop the refugee flow and stand by in case of disturbances.'13
Mary Kellett-Long wrote that 'as far as I can gather they took their arrest as a great joke and mystified the police by joking with them the whole time. Of course they came back and did a story about it and everyone was very amused. Someone from ADN rang Adam and congratulated him on his liberation.'
Johnson, also liberated, roved to the Brandenburg Gate. 'During the course of the day, apparently as a result of an order from the British occupation authorities in whose sector the road leading to the Brandenburg Gate lies, the West Berlin police pushed the crowds to about half a mile behind the barrier, thus relaxing the situation. Brigadier Hamilton, who made observations through battered field glasses, told me he thought the crisis was completely over.' As an aside, Johnson added in his diary: 'On the Eastern side there have been some clashes with police but no-one has been hurt as far as we know. They have happened when police have cleared away crowds who stood near the border looking sullen. There have also been minor brushes with people openly complaining about the new clampdown.'
The escapes went on, a group of four getting across in the south at 4.00 p.m. and, 90 minutes later, two more in the south. North of Bernauer Strasse, where the train ran along the border and no wire was yet in place, a youth sprinted but was caught by three policemen who wrestled him to the ground. He wriggled, seized one of their rifles, held them off with that and ran over the line. The policemen caught him again. A bayonet went into his knee and he lay in agony while the policemen backed off. He crawled into an allotment crying for help; fortunately someone heard and he was taken to hospital.
Klaus-Peter Grohmann, the young banker, felt drawn not to the Brandenburg Gate but the Potsdamer Platz. 'The barbed wire was already up, of course. I went to Potsdamer Platz because that used to be _the_ place in Berlin. It had been bombed during the war and there was still evidence of that, but somehow it remained the place to go.' He surveyed this vast, emptied expanse so full of memories for every Berliner. He had friends and relatives over there but 'telephones were cut, everything was cut, you couldn't communicate with anybody'. He would visit his relatives again – but not until 1972 and only for one day, the maximum time permitted. (It felt 'like an adventure, travelling into a strange country'. There'd be small talk and then many, many tears.)
Kennedy flew to Washington from Hyannis Port and met the Ambassador from Moscow, Thompson, for fifty-five minutes during the morning and Dean Rusk for seventy minutes during the afternoon. 'Secretary Rusk and I went over to see the President in the family quarters of the White House and talked to him about countermeasures we might take,' Frank Cash says. 'I suggested two: first to get the West Germans to stop inter-zonal trade with the East and second to stop East Germans travelling to the West. Kennedy immediately dismissed these and implied they were minor and ineffective. The actual word he used was Cajan [from a small French-speaking community in Louisiana] – _picayun._
'As I've subsequently read, the cessation of inter-zonal trade and the serious effect on the East German economy was one of the main concerns Khrushchev had in mind in letting Ulbricht put up the wall. The other measure was to cut off the only people the East Germans gave exit visas to, because they were the ones the government wanted to travel. The East Germans worked very hard to gain respectability and recognition and a lot of their efforts involved sending officials, musicians and athletes to Western countries.
'Kennedy appeared relatively relaxed because I think deep down he had made up his mind World War Three wasn't going to start in Berlin. Rusk and I recommended that he himself go to the city. We had had a lot of messages from Al Lightner saying how morale in West Berlin was plummeting. Kennedy said, "I'll think about it." I came away with the impression he was not overly upset with the situation as we all saw it but he also knew we had a morale problem and he'd consider somebody going to reassure the West Berliners. If we got to a direct confrontation with the Soviets, and nuclear weapons were to be used, he was the guy to have to do it and that's a very sobering responsibility.'14
Kennedy's thinking centred on Vice-President Lyndon Johnson and also on a resident of Chatham, Massachusetts, military (retd), Lucius D. Clay. There's a tale that Napoleon, when a young soldier, saw the storming of the Bastille and a biographer wrote that, had Napoleon been in charge of its defence, 'he would have known what to do'. If Lucius D. Clay had been in charge of the defence of West Berlin on Sunday 13 August 1961 he would have known what to do. Retrospectively, that terrified most of the State Department and many other people, too.
The Task Force met but moved among uncertainties. Would the barrier be permanent? Was this the initial move on a big board, with Allied access denied next? Was it only a clinical, surgical operation now completed?
That evening Peter Johnson spent an hour with his wife Elfi's relatives in West Berlin and 'found there Richard and Erna, Elfi's fiftyish cousins. We visited Richard's aged mother who is in hospital after a fall. We had a warm-hearted chat about the Berlin situation. They don't seem really worried. Usually the most worrying thing in such situations are the headlines in the London papers.'
That evening, too, a cameraman fixed an image which holds its poignancy: a hip-high stake in a pavement and those loose coils of wire undulating from a wooden door out across the street, the stake supporting it. A soldier stands, left arm folded over his chest nursing the muzzle of his rifle. He looks ahead, but away from the camera. The building behind him is typical East Berlin, two basement windows but the ornate façade decayed to expose the bricks. Faded script proclaims the place is a hairdresser's. A burst of machine-gun fire from 1945 has chattered into the masonry and the pockmarks remain. At the doorway a dark-haired, attractive young woman wearing a creamy dress has laid her canvas shopping bag against the wall. Her right hand is raised to her mouth, the index finger resting on her lower lip in the universal gesture of contemplation.
The photographer had chosen his angle carefully: in sequence the stake and the wire, the soldier, the woman fractionally to the right of the soldier. Was she looking over the wire for a husband, a lover? Was she in love with the soldier, waiting for him to finish his stint? Had she brought him bread and sausage in the shopping bag? Did she just stop and look and wonder? She must be old now and so must he, if they are alive, and whatever happened to them in between?
Ulbricht visited an army tank group, no overcoat this time but a panama hat and a light, baggy suit. He shook hands with a tank commander. Nearby, two gunners stood to attention in the hatches of their tanks. Other tank personnel, moving in around Ulbricht, wore pull-down headgear, the flaps drawn over their ears. They looked boyish and serious, he looked relaxed. He must have known he'd got away with it.
In the West there was a different image to be developed, fixed: a crowd behind a cordon of rope brandishing banners: GERMANY REMAINS GERMAN. THERE'S ONLY ONE GERMANY. It was true and untrue, all in the same moment.
In her diary Mary Kellett-Long wrote that 'telex and telecommunications have been cut with West Germany and we heard a story that post had been stopped but the East Germans denied it. We got into a great flap thinking about our teleprinter connection with Bonn but were able to telex and telephone London. In the evening we went to dinner and when we finished drove to the Brandenburg Gate. Things seemed to be quieter, not so many troops and all the tanks disappeared. When we got home we found a new decree issued saying West Berliners must get permits for their cars and motorcycles before driving them into East Berlin.'
The incident sheet clicked through its machine, adding and adding.
_Time_ | _Crossing Pt_ | _Fled the State_ | _Prevented from Fleeing_
---|---|---|---
| | _No. groups_ | _No. people_ | _No. groups_ | _No. people_
5.45 | 67 | 1 | 2 | |
6.15 | 11 | 1 | 2 | |
7.25 | 22 | 1 | 1 | 1 | 1
8.10 | 40 | 1 | 2 | 1 | 2
8.20 | 31 | 1 | 1 | |
8.30 | 68/69 | 1 | 3 | |
10.10 | 12 | 1 | 2 | |
The Ministry of the Interior fleshed out these statistics in a curiously rambling report to Ulbricht:
In the time between 7.45 a.m. and 11.00 p.m. about one thousand West Berlin people concentrated near the barriers in the area of Potsdamer Platz. Around 11.00 p.m. there were still about two hundred people on the square, who then gradually dispersed.
The riot police erected an iron barrier at crossing point Adalbert-strasse [the twin curved roads and sunken gardens] at 1.00 p.m. and prepared for similar measures at the crossing point at Köpernicker Strasse [further up the curve].
At the same time another barrier was erected at Kellerbrücke.
Around 11.15 p.m. an estate car, number could not be seen, did not respect the halt sign given by the Border Police and drove through to West Berlin at crossing point Wollankstrasse. Other hooligan drivers at this crossing point have been trying to break through. Measures to change the situation radically at Wollank-strasse are to be taken on 15 August.
A _Bild Zeitung_ reporter who made provocative statements and took photographs of military vehicles was arrested at crossing point Bornholmer Strasse at around 2.20 a.m. He was brought to the People's Police station at Prenzlauer Berg [and was no doubt at this moment in Kellett-Long and Johnson's barred room. Was the platinum blonde still here and, if she was, what did the reporter say to her?]
At crossing point 45, Heinrich-Heine-Strasse, a driver was told the new regulations and had to go back and the vehicles following him were not allowed through by the Riot Police.
One hundred and sixty-six persons have been brought to police stations for checking and sixty-one have been released.
All the offices of the Berlin police have received instructions from the district leadership of The Party to enable them to answer questions from the public [the order to refuse all applications without exception].
It has to be pointed out once again that the majority of the wives of our People's Policemen have a high degree of understanding of what their husbands are doing.
A battalion of police brought from outside Berlin have been accommodated in a schoolhouse in district Lichtenberg and another battalion in a slaughterhouse.
The mood of the population on the basis of previous reports: no new information resulted from the reports given during the night.
2,365 daily permits were issued to West Berlin citizens from 7 a.m. to 8 p.m. on 14 August. One hundred and twenty of the two hundred and fifty-five applications to travel out of Berlin into the area of the GDR by West Berliners have been granted. 9,851 permits to go to West Berlin have been applied for by the citizens of democratic Berlin on 14 August.
The incident sheet recorded its total for the day: fifteen escapes, seven caught. Marienfelde recorded its total for the day, forty-one. The noose had tightened.
## TUESDAY 15 AUGUST 1961
The GDR government used the night again, tightening and tightening.
'Shortly after midnight,' Peter Johnson wrote, 'they announced that West German cars would be banned from East Berlin without special permits because such cars had been used for "spying". Doubtless some have smuggled refugees and goods, because so many cars cross from the West that only a minority are subjected to check, but this measure shows how worried the East Germans are about the atmosphere in their own population and are determined to go to extremes to stop people fleeing.
'Another restriction banned East Germans from holding identity cards issued by other countries or authorities, including West Berlin. Penalty: at least three years' imprisonment or a fine, or both. This is designed to prevent people fleeing by showing the identity card of another country because, at present, foreigners, West Germans and West Berliners can still move to and from East Berlin after showing documents.'
The clearing of woodland in the country began, swathes sometimes cut to a depth of 60 or 70 metres. It kept Rüdinger Hering far from the brook and the bridge and the farmhouse nestled over the meadows. 'They came at 3.00 in the morning with lorries, a Factory Fighting Group from Falkensee. The people living in the two houses on the border were told they had to leave immediately. I knew where some went but not others. Then bulldozers destroyed the houses completely.'
'Much quieter this morning', Mary Kellett-Long wrote, 'and lots of the troops and tanks have been moved. The Brandenburg Gate is still closed and barbed wire has been put up all along the front of it. Ulbricht says the West has seen that East Germany is quite capable of defending its borders and the East Germans have shown how quickly and well they can move without anyone getting any idea of it until afterwards.'
Half the tanks went back to their bases, more evidence that normality had returned and insurrection was no longer a concern. The FRG Minister of Posts and Telecommunications confirmed that the GDR had cut all links.
The Allied commandants delivered their protest to the Soviet commandant, Colonel Andrei Solovyev, who dismissed it by adopting the primary position: the Soviet Union did not interfere in the internal affairs of the sovereign state of the GDR. What else could Solovyev have said? But he must have known that if the Four Power Agreement could be negated unilaterally like this, then anything could, which itself was a nightmarish prospect in the nuclear age.
'It was a great shock for Berliners that such a division of the city was possible but they believed it would be removed by the Allies,' the radio reporter Peter Schultz says. 'The disappointment began when the Berliners realised the Allies weren't doing anything about it. The first Allied Ambassador who came to the city was the British and he invited the Press to meet him in the Royal Navy Officers' Club. I remember what he said, in English: "Ladies and gentlemen, whatever happens in East Berlin we are staying in West Berlin." He only spoke this one sentence and it had been coordinated with the other Allies. It became clear they wouldn't do anything against the wall.'15
The Western morning newspapers carried a photograph of Macmillan in country shooting uniform – plus fours and cap – on the moors near Swinton, Yorkshire, looking serene and unconcerned; and it did not please West Berliners.
Al Hemsing provides a telling aside. 'On the Monday and Tuesday, the West Berlin newspapers were treating it very much like a police story, you know, what was happening in street so-and-so. They had more or less given up on us.'
Allied access remained unaffected – as yet. 'We'd go over on the U-Bahn but Friedrichstrasse was the only station open and a control point there constantly checked identifications,' Frank Trinka says. 'If you were suspicious, or anything like suspicious, you were hauled off. Then the concrete blocks started to go up and you sensed a permanency, you sensed that this was really going to happen: a wall. It was a methodical but relatively slow-paced undertaking because they could only spare so many people to do the job and I guess they had to consult constantly with the Soviets.'
This was the beginning of the wall proper, something far more symbolic than the Brandenburg Gate, and a mystery endures: how did nobody notice the enormous stockpiling of material for it? The first barrier was the barbed wire, easy to store – one can put an awful lot in a warehouse – but the wall? The answer is both obvious and unexpected, and Adam Kellett-Long gives part of it: 'The wall was rubble at the beginning, any old rubble and there was plenty of rubble in East Berlin. It still makes me wonder if they thought they were going to get away with it. Apart from the barbed wire, they didn't seem to have any proper building materials for a wall but no mystery exists about assembling it because there was nothing to see.'
Not all the wall consisted of rubble, however. 'An American journalist came to Berlin later in August,' Hemsing says, 'and asked me how many apartments you could build with the material they'd made the wall from – the slabs were the same as the East Germans used for apartment blocks. I put him in touch with the Western building trades coordinator and they figured out about twenty thousand small apartments initially. It explains how they got their hands on so much so quickly. You could ask how they stacked up all this in East Berlin without anybody noticing but there were a lot of building sites.'
The Allied bombing had seen to that and by 1961 the immense task of rebuilding was simply continuing, as it would for many years afterwards. To build a wall, all one had to do was to go to the sites, load slabs onto trucks, ferry them to the line and lay them.
The crude, makeshift edifice was constructed out of whatever lay to hand: breeze blocks, concrete pillars one on another, the housing slabs, ordinary bricks, lintels; and that gave an image, such as a street bisecting Bernauer Strasse, a truck with a winch lowering a block and workmen steering it by hand into position. The wall would be laid on no foundation: its own weight would hold it. Elsewhere breeze blocks were cemented in tiers like bricks, and that also gave an image: a craggy old workman in overalls, cigarette clamped in his mouth, spreading the cement with a trowel while an armed soldier watched him carefully.
Another timetable had begun to tick, of long range and monumental. The events of the weekend of 12–13 August 1961, and the days immediately after, were only the beginning. Within a calendar year, 7,874 cubic yards of blocks would have made a wall 7.5 miles (12 kilometres) long where the city centre interlocking was closest and most intimate; the remaining 91.7 miles (137.5 kilometres) round West Berlin – the countryside, meadows, woodland and lakes – would be cut by twin rows of coiled wire. Sometimes they were as far apart as 70 metres, sometimes – depending on the contours of the land and the proximity of dwellings on either side – considerably narrower, but always fashioning a death strip between the rows once a chain of watchtowers had been put in place. Someone calculated that if all the strands had been stretched in a straight line they'd measure 6,313 miles. The total area of the death strip would measure between 49,000 and 55,000 square metres, the equivalent area of a town.
The wall would separate 10,000 Westerners from their allotments and weekend houses. These people were helpless victims of accidents of history and geography, and in time they'd have to find ways of accepting their losses. Many owners kept whatever legal documentation they had proving ownership, although that would become progressively more remote and more futile as the years solidified the division. Not until after 9 November 1989 would owners be able to come back to reclaim their property.
In Bernauer Strasse, Conrad Schumann challenged six men near the wire on his side and appealed to an officer to help him. The officer approached the men who produced Stasi identity cards and one said they 'just wanted to make sure you are awake, comrades'. Schumann judged the checking of one's own army a curious thing to be doing. Towards 12.00 midday, he heard sounds of protest and calls for 'freedom' from a square behind Bernauer Strasse. The protesters moved up towards him. He had orders not to open fire and thought he'd be overrun, but armoured vehicles and soldiers poured from two side streets. The soldiers drew their bayonets and forced the crowd back. 'I will never forget how some of the people stood their ground with their arms crossed and cried out, "Go on, shoot, you cowards."'
After 12.00 Schumann saw the first of the lorries lumber up with the slabs and someone told him a wall would be built. He could postpone the decision he'd been weighing up a little longer, but not much. Around 2.00, keeping a wary eye on the other soldiers, he repeatedly sidled towards the wire and fingered the top strand. A soldier noticed but Schumann passed it off saying 'the wire is starting to rust already'.
A Western photographer, who'd been taking images of Bernauer Strasse since Sunday, sat in his car and noticed Schumann. 'Every few minutes the soldier went back to the same piece of wire and pressed it down, a bit more at a time. _My God_ , I thought' – Schumann seemed to be preparing to jump the wire. The photographer emerged from his car and watched, his camera poised.
Schumann saw him but 'suddenly a young man came up right close to me on the Western side of the wire. I went towards him and shouted loudly so that my comrades could hear, "Get back immediately!" but I whispered, "I am going to make a run for it."' Schumann had no way of knowing if his comrades would shoot him in the back _after_ he'd jumped the wire.
The young man understood and walked briskly to the nearest police station. A moment or two later, four policemen arrived in a Volkswagen minibus and stationed it as if they were just patrolling and had stopped for a moment. Other photographers deliberately pointed their cameras at the other soldiers, who twisted their faces away.
Schumann selected an instant when the soldiers were at their furthest point from him and their faces still turned away. He was trembling. He flipped the magazine out of his rifle so it wouldn't go off if he fell. He sprinted the few paces to the wire and a film crew captured his right leg coming up in a vaulting motion, both arms out like wings for balance, rifle on its strap tight over his shoulder. The image of him, when one frame of him in mid-air had been taken from the film and developed, fixed, became globally symbolic and, like all great photographs distilling so much, it required no caption.
Schumann cleared the wire and landed awkwardly but on both feet, letting the rifle fall from him. He sprinted on, leaving behind him not just the soldiers who had been his comrades that instant before but, in the middle distance, three men and a woman who were chatting on the pavement and were now, also, startled witnesses. Schumann ducked into the minibus and a Western policeman in a white cap trotted to the rifle to pick it up.
Reports of escapes kept coming in, two at 3.47 p.m., and a flurry when people finished work: at 5.20 three, at 5.30 two, at 5.50 two caught. Shots were fired at a couple swimming the Teltow Canal – the first recorded instance of shoot-to-kill intent. As the echo of the shots died, everyone who could hear them understood the true nature of the wall. It was, in modern jargon, a defining moment as well as a hard one.
The shoot-to-kill policy haunted even Honecker, and nearly three decades later it pursued him to exile and the Chilean Embassy in Moscow, then back to Berlin and forty-nine charges of murder; then pursued him to Chile itself. Many dead at the wall – nobody can know the definitive total – lay in between.
'The border security installations between the socialist and capitalist worlds from the Baltic Sea to the Black Sea were all the same at the time,' Honecker would say in self-defence.16 'Incidentally, it is interesting that people talk much less about the border security installations, for instance, between the USA and Mexico than those which were normal between the Warsaw Pact and NATO. The shoot-to-kill order did not differ at all from that of the West German Border Guards. Basically it was the same order and anyway it was nothing secret.'17
In Washington, General Clay formally volunteered his services to Kennedy, although that might have served to heighten and broaden the friction because Clay was a Republican who'd thrown his weight behind Richard Nixon in the last election.
This Tuesday, the friction between Washington and Berlin rubbed, too. 'We didn't know what went on in Washington,' George Muller states. 'We kept sending in our reports but Kennedy said afterwards he hadn't heard from the Mission, which was really disingenuous because we'd reported the East German decree and by Tuesday, when morale plummeted, we sent in a recommendation that some high-level representative should come, the President himself or Dean Rusk. We didn't think of the Vice-President. We also suggested reinforcing the Berlin garrison or some other commensurate measure.'
Kennedy considered the latter option carefully because the dangers of dispatching reinforcements along the autobahn to West Berlin were immediately obvious and profound. If Soviet or GDR forces severed the umbilical cord, what then? They need only say a bridge was closed for repairs, as they had done in 1948 when Berlin had had to be sustained by the air lift. Worse, if Soviet or GDR forces deployed on the autobahn and blocked it, what would the American reinforcements do?
Kennedy waited.
That night Peter Johnson 'returned to Bonn at last' to see his family. 'I was captivated by the beauty of the clouds from the aeroplane but at the same time saddened as I looked at the lakes, fields and forests of East Germany thinking of the unhappiness suffered by people forced to live under an alien system. It was dark when we landed at Wahn, the home airport for Bonn. It took Elfi and the boys some time to recognise me – although I was the first in the queue beside the stewardess as we walked to the terminal building – but, when they did, I saw a flurry of waves from their shadowy figures. Soon we were all together chatting excitedly.'
How many Berliners wanted that, and only that?
At midnight the incident sheet recorded its total for the day: four escapes, fourteen caught.
## WEDNESDAY 16 AUGUST 1961
The timetable ticked on. At 3.20 a.m., Police Headquarters made their first entry of the day: one escaper caught in the south. The tightening tightened.
'We woke this morning to find everyone has to have new passes to cross and we got in a great state thinking we were shut in,' Mary Kellett-Long wrote, 'but it seems they only apply to East Germans because Adam went through without any trouble at all. We had an electricity failure which lasted half an hour and the water has been cut off. Really this place seems to be getting worse than ever now the border has been closed.'
All through the morning reports of escapes came in, at 9.00, 10.00, 10.15, 11.35, 11.45.
Mary Kellett-Long, who knew nothing of this, added in her diary:
The West Berlin police have kept people about half a mile back from the Brandenburg Gate but they are still making a nuisance of themselves at the other checkpoints. We find driving into West Berlin we get more stares than usual but not really any hostility. Yesterday we were waiting at some traffic lights, a big lorry drew up beside us and the driver leaned out of the cab and said to Adam, 'Oh good, you got a pass, then.' When Adam explained we were English he said, 'Oh good' again and we had quite a little chat.
West Berlin police at the border are funny, too. When they see our passports they smile and look rather puzzled, but they are charming. The East German police are still polite but thorough. I think they are all fresh and full of enthusiasm for the new rules and regulations but they will get tired, we hope, because it takes such a long time to get through on the way back. They searched thoroughly and looked in the car. There was a newspaper on the floor and they asked to see it. Luckily it was _Neues Deutschland_ so they couldn't say anything.
Rüdinger Hering, living out in the country immediately west of West Berlin, says that 'during the first few days many people tried to escape and about 90 per cent of those who tried made it. Around Falkensee there were forests' – ideal cover for a sprint across at night. 'Afterwards nobody tried because the space between the two walls was so wide and so open. At the beginning the electricity which the Border Guards used was connected to the local grid and when they switched on searchlights the picture on television went small because they were using so much electricity.'
In Bernauer Strasse, ground-floor residents were ordered to surrender their front door keys to the police. Because the road was in the West, no East German – whether army, police, Factory Fighter or bricklayer – was entitled to stand on it to guard the front doors. Police lurked in corridors and in the apartments to prevent escapes. On 17 August, workmen would begin bricking up all the 1,253 windows, starting with the ground floors. An evacuation of 2,000 residents would be forced through the following month, and – later – every building except the Church of Reconciliation would be demolished and cleared to create the death strip. The wall cut across the front of the church and elaborate patterns of tank traps would protect its rear.
But on this Wednesday, a resident of No. 47, within sight of where Conrad Schumann vaulted the wire, was caught letting the wife of a friend use his front door to escape. Police arrested him but somehow he persuaded them to release him, returned and with a spare key let himself and his family out, stepping into the other country and taking nothing with them.
Students at Bonn University sent Kennedy an umbrella echoing Chamberlain's pre-war appeasement and included a message. 'Sorry to say that because of your reserved reaction you have become the most worthy possessor of this symbol of fatal policy.'
The escapes went on, at 2.55 p.m., 3.30, 4.10, 4.15.
In the afternoon, 200,000 people gathered at West Berlin's town hall to hear Willy Brandt address them. Brandt looked almost a businessman in a dark suit, his hair swept back and perfectly in place. A thicket of microphones awaited him. Deputy Mayor Amrehn made the opening speech and then Brandt, using his right fist for emphasis, said:
The people of Berlin have a right to know the true situation. The people of this city are strong enough for the truth. Today we must be the voice of our fellow countrymen who can no longer speak out or come to us. In this hour I address myself particularly to those of my countrymen who work in government offices and organisations in the zone [the East]. Don't let them make fools of you! Conduct yourselves with manly restraint whenever possible. Above all, do not shoot at your fellow countryman.
I deem it necessary to lodge a protest with the world forum of the United Nations against the perpetrators of this act of inhumanity on the soil of Berlin and of Ulbricht's state. Our concern is not with the rights of the Western Powers but with human rights which must be restored. We Berliners have something to say to our protectors. This city wants peace but it is not capitulating. There can be no other city in the world that desires peace and calm and security more than Berlin, but peace has never been won through weakness. There is a point where you can't retreat even a single step. This point has been reached.
It was a speech of anger but its impotence could not be concealed.
Brandt had also written to Kennedy because Kennedy had invited him to do that whenever Brandt thought it necessary or important. The letter, telegraphed, included these words:
The danger above all is that inactivity and defensiveness might create a crisis of confidence vis-à-vis the Western Powers. We fear the Ulbricht government might interpret a wait-and-see attitude as a sign of weakness, a licence for further attacks. The Soviet Union has achieved half of its 'free city' proposals. If a second act should follow, what remains of Berlin would become a ghetto. Instead of refugees moving into West Berlin, a massive flight from Berlin would begin.
Brandt said that at the United Nations he wanted to accuse the Soviet Union of 'tampering in the crudest fashion with the Proclamation of Human Rights'. He finished:
After suffering an illegal Soviet _démarche_ , and considering the many tragedies taking place today in East Berlin and in the Soviet Zone, we must take the risk of assuming a firm stance. It would be desirable to reinforce the American garrison as a show of force. I consider the situation grave enough to write to you, Mr President, with the ultimate frankness that is possible only between friends who trust each other utterly.18
This letter evidently irritated Kennedy as an admonition – and some in Washington perceived it as an attempt to dictate American foreign policy – but its overall effect cannot be doubted.
George Muller says that, 'Ed Murrow called up Brandt and the criticial thing was that Brandt met Murrow on the Wednesday when Brandt was sending the telegram. Murrow had already sent his and Kennedy probably placed more weight on that than ours. It's not quite clear whether Murrow was the catalyst for action or Brandt, or whether it evolved in Washington during joint discussions.'
Kennedy waited.
In Bonn, Peter Johnson wrote:
There are only half our usual staff because of the Berlin crisis and the holiday period. As a result I had no day off but worked a long evening shift tonight with the Berlin story still humming along. Main thing was a televised speech by Brandt at a rally in West Berlin in which he announced he'd sent a letter to Kennedy demanding 'not merely words but political action'. Most of us regard this as an attempt to lift Berliners' morale because scarcely anyone thinks there is anything the West can do in the present crisis. I impressed colleagues by being able to take almost a verbatim of Brandt's speech (in English but with partly German word order, which was straightened out by a deskman) direct onto a typewriter.
Later I found myself listening to Adenauer, apparently as well as ever, speaking to a Bonn election meeting of several thousands in a trolley bus shed. He had rowed back from his immediate reaction to the East German measures, in which he called for an economic boycott of the communist countries, and is now talking of what has happened as only the pre-crisis. He spoke out in favour of negotiations and said he did not think there should be war – this seems to be his election line, to soothe people after he has opposed negotiations all along in the past. He repeated an earlier smear of Brandt by referring indirectly to his illegitimacy and directly to the fact that Brandt changed his name from Frahm – something being exploited as a suspicious act by the Christian Democrats.
Adenauer did not like Brandt, and almost incredibly did not visit Berlin until 22 August.
That evening a report of an escape came in at 6.45, two in different places at 6.50, one at 7.40 and the last at 9.45. The incident sheet recorded its total for the day: 21 escaped, 13 caught. However incomplete, the dehumanised statistics charted a trend: 66 gone over on Sunday, 15 on Monday, 14 on Tuesday and now 21. The Wednesday of the week before, 1,926 refugees had registered at Marienfelde.
The barbed wire, and the wall now rising, fulfilled a most primary Germanic objective. It worked.
## THURSDAY 17 AUGUST 1961
The GDR government used the night again for further tightening of controls. At 3.00 a.m., five truckloads of material drew up to Potsdamer Platz and workmen busied themselves constructing a 5-foot high wall. This would happen elsewhere, too, and where the wall already existed it would be heightened, sometimes with barbed wire strung along the top. The statistical trend shifted because, moment by moment, escape became more difficult and – with the shoot-to-kill order operating – became potentially fatal, too. Thirty-five minutes after the trucks reached Potsdamer Platz, police caught the first escapee of the day, at 4.20 two more, at 5.10 two more again, at 7.35 one. The first successful escape was reported at 8.23, where a cemetery wall formed the frontier.
In Bernauer Strasse workmen swarmed the ground floors of the apartment buildings, bricking up windows from the inside. Residents called to people in the West for help and clambered onto window ledges to drop to the pavement. The West Berlin Fire Brigade hurried there with safety nets anticipating that, when the bricklayers reached the upper floors, residents might jump.
Some windows already had necklaces of wire nailed over them and they framed another image: an old man in shirt sleeves and braces gazing through a window from the third floor, his face heavy with finality, his hands resting on a strand of the wire.
At the north end of Bernauer Strasse the police positioned tall sheets of wood in a screen behind the wall so that people on either side couldn't see each other. At the side roads with intersections on Bernauer Strasse, West Berliners scaled lampposts to wave. Others brought stepladders. Some pressed their faces against the breeze blocks, searching for a crevice to peer through.
Elsewhere, workmen pulled up S-Bahn tracks which straddled the line and that isolated a famous old terminal, the Görlitzer Bahnhof. It stood in the West but the track to it threaded through the East. Western trade unions retaliated by demanding a boycott of the whole network, which of course the GDR ran and was a source of hard currency. Over 260,000 people used the S-Bahn every day and now demonstrators carried placards saying 'Don't give your money to Ulbricht' outside Western stations. Posters reinforced the message, 'Don't travel the S-Bahn', and the 260,000 fell towards 25,000.
The GDR Ministry of the Interior issued a decree that 'no permits for West Berlin will be issued to East Berliners until the conclusion of a Peace Treaty' – the treaty to incorporate West Berlin into the GDR or declare West Berlin a 'free city', negating Allied rights.
The East German's Free German Youth called for 'voluntary service in armed units' and formed Regiments of Volunteers who took down television aerials which could receive Western broadcasts.
The Protestant bishop of the Brandenburg district and the Administrator of the bishopric of Berlin, both Westerners, were refused entry to East Berlin and an Easterner, the chairman of the Governing Council of the Protestant Church in Germany, was refused an exit visa.
Loudspeakers on vans and poles poured out the GDR's official line and within days the loudspeakers totalled 216. The West responded, Brandt speaking vehemently and vans relaying his words: 'The East German police and army units stationed at the West Berlin border have received orders concerning the use of their weapons. They have been ordered to fire aimed shots at anyone attempting to cross the border even if the bullets strike West Berlin territory. Nobody should believe that when he is one day brought to justice he will be able to say, "I was just following orders." Murder remains murder.'
People could no longer cross but words could, and in both directions.
A carefully printed photographic image showed Brandt in Berlin, a van relaying his speech through four speakers to a string of soldiers on a bridge in the East. They seemed mildly amused, mildly contemptuous, one poised to gesticulate towards the van but he changed his mind, smiled and turned to a colleague. They talked.
Was it this day, out in the country, that a man rushed the wire and wriggled under, a young woman running hard behind him? The top row of the wire caught her jumper and wrenched it up under her chin, felling her as if she'd been decapitated. Hands reached out, grabbed her and pulled her through.
Was it this day that, at a similar place, a family rushed the wire, the woman scrabbling through on all fours after hands from the West lifted the baby she'd held? What looked to be her husband, following, stumbled between the two rows of wire. A soldier rushed up and gave him a final prod with the butt of a rifle.
Was it this day that a photographer froze the image of a cobbled road, now decorated with stakes in the ground supporting the wire, and a group of children standing either side of it? One of them, in the West, steadied a bicycle. They gazed towards each other as pals do. Did they ever meet again?
If they did they'd be middle-aged then, because within the tortuous and improbable journey the wall acquired a logic of its own, which demanded that it be a complete border sealing. That led to 116 watchtowers within a year and successively refined walls numbered the first generation, second, third and finally fourth which was extremely difficult to scale. It was made of high-density reinforced concrete and its L-shaped segments were 3.60 metres (11.8 feet) high and 1.20 metres (3.9 feet) wide.
The fourth generation wall would be the one which fell.
The logic led to electrified fences, dogs on leashes trained to attack even their own handlers, scatter guns triggered by trip wire which flung 90 jagged iron fragments in a conical pattern (though between the two Germanies, not at Berlin), tank and vehicle traps, searchlights and arc-lamps, a constant chemical spraying of the dead zone so no blade of grass grew, jeeps patrolling twenty-four hours a day, special permits if you lived near it, and sometimes forced evacuation, as in Bernauer Strasse.
At 4.00 in the afternoon, the British made an exquisite move. Geoffrey McDermott wrote:
Although we did not find it easy to devise an effective riposte to Ulbricht's outrage, this is not to say we were powerless. It occurred to me that we had in the British Sector the Soviet war memorial, guarded always by Soviet soldiers and regarded by them as sacred. I suggested we should surround the memorial with barbed wire, inside which the Soviet guards might parade like animals in a cage, and station a small British contingent nearby. All this would be done, of course, in order to protect our Soviet ally.
Jumbo Delacombe [Major-General Rohan Delacombe, British commandant] warmly welcomed my little scheme. A Soviet Colonel came over and confronted our Chief of Military Police. Quivering with rage he asked what the hell we were doing to his memorial. The Chief of Military Police replied suavely that we wanted to ensure the safety and properly respectful treatment of the memorial and its guard against those who might wish to molest them. The Soviet Colonel said it was an outrage; no such measures should have been taken without consultation with the Soviet authorities. The Chief, gesturing at the Brandenburg Gate in its new condition, said that the Soviet authorities seemed to be taking quite a lot of action themselves without overmuch consultation. The Colonel departed, apparently heading for a coronary. This ploy proved a useful sanction on several occasions in bringing the Russians to heel when their provocations tended to go too far.19
The geometry of division found expression in Hagen Koch, the jolly young Eastern army volunteer from Dessau, a town south of Berlin. He joined up 'because I was nineteen and I wanted to wear a uniform'. Newly married, he and his wife lived with her parents near the twin curved roads and the sunken gardens. This Thursday he was on duty at Friedrichstrasse – the crossing point, not the railway station. He'd just been paid but hadn't given his wife his wages yet so a bundle of Ostmarks nestled in his breast pocket. He was issued with a brush, a bucket containing white paint and told to set down the precise line across Friedrichstrasse. Moving backwards he painted the line and the bulge of the banknotes rubbed against his arm as he did. The line, he reckons, took about twenty minutes to finish.20
In West Berlin, the loss of morale produced panic buying and hoarding of sugar, flour, canned food, oil, soap, and potatoes, but a spokesman for the Senate tried to calm this by announcing that ample stocks existed for between six months and a year. This was a legacy of the 1948–9 blockade and airlift. As a precaution against anything like that happening again, vast warehouses contained an inventory adequate for sustaining West Berlin: fuel for industry, caviar and castor oil, even hops, barley and malt for beer. The spokesman added that wholesalers had been instructed to speed distribution of all goods to retailers to 'mollify worried customers'.
In Washington, the full realisation came perhaps slowly. 'The key is that Foy Kohler and all these people had been off to Paris worrying about access to Berlin, what to do if the East Germans took over access,' John Ausland said. 'Their minds had not really focused on this problem of sealing East Berlin from West. Foy said in a Task Force meeting, "I hope we can get this out of the way so we can get back to planning for the crisis." There were about sixty people in the room and, me being junior, I wasn't going to say anything but afterwards I did say to Dick Davis, Foy's deputy, "You've got to explain to Foy Kohler that this is the crisis." They finally began to focus on it after the alarming telegram from Ed Murrow about the ugly mood and the angry letter from Brandt. There was a real danger West Berlin crowds would take matters into their own hands. On the Thursday there was a meeting at the White House and a decision was taken to send Vice-President Johnson and an armed Battle Group from West Germany.'21
Lucius Clay would go, too, and Frank Howley, a former US military commander in the city, and Charles Bohlen, a State Department Soviet expert.
Richard Smyser of the Mission in Berlin explores the context. 'Memories of 17 June 1953 haunted the East Germans but in a sense haunted us, too. What if we created a furore and the East German people responded, and rose, and we were able to do nothing to help them? I know it was a factor in Washington because it emerged when LBJ and Clay were coming. Young Kennedy had just gone through the Bay of Pigs [debacle] in Cuba and the Vienna Summit, and the feeling was we'd done something in Cuba which called for a popular uprising and it didn't happen – but in East Berlin we might have one, and Kennedy was very, very worried about it.'
George Muller in Berlin probes further into the context. 'Then we were told Johnson was coming and also General Howley, one of the old tank heroes who had a tremendous reputetion with the Berliners for standing up to the Soviets, but the real brains trust was Johnson, Clay and Bohlen.'
To dispatch a Battle Group was highly unlikely to provoke uprising in the GDR of itself, but it represented a decisive move which may or may not meet with an equally decisive response. Some 1,500 men of the First Battle Group of the Eighth Infantry Division, stationed in Mannheim, would move – exposed – along the umbilical cord from West Germany to West Berlin. At each moment they would be vulnerable and become like West Berlin, symbolic. Neither could be defended except by the implied threat that any attack on them would move towards the nuclear scale.
It created more friction in Washington. Ausland said:
No consideration was given to the rules of engagement. The Berlin Task Force met and Martha Mautner from the State Department's Intelligence leant over and said to me, 'Ask Kohler what we are going to do if the Russians won't let the Battle Group through.' It sounded like a good question and I did ask. The reply was that we would implement our contingency plans. The meeting went on and then Henry Owen from Policy Planning says, 'Foy, what are our contingency plans?' and Foy looks over at Dave Grey who represented the Joint Chiefs of Staff and asked, 'Dave, what are the rules of engagement?'
Grey said, 'I'll get them.' He later brought them over and I well remember we all looked at them and realised that if the Russians tried to stop the Battle Group there could be some shooting. I was sitting in on the meeting with Kohler and Paul Nitze when Rusk appeared at the door and said, 'Paul, have you seen these rules of engagement?' He said yes. Rusk said, 'Do you realise there could be some shooting if the Russians try and stop us?' Nitze said, 'Yes, we realise that but I'm satisfied they won't try.' Then there was a little discussion. Rusk said, 'Well, I just want to be sure you understand that' and went back to his office. What this makes clear is that, as so often, these sorts of decisions are made but the consequences are not examined.22
At least two generals cautioned against sending the Battle Group because it could be construed as a provocative act whose only practical effect was to feed a few more troops into a place which, by definition, was indefensible anyway. The 1,500 raised the US contingent to 6,500. The British had a permanent contingent of 4,000 and the French 3,000. Leaving the Soviet forces aside, the GDR army numbered 110,000, the police 78,000, the Border Police 45,000, the police reserves 30,000 and the Factory Fighting Groups 300,000. Rusk said, 'An attack on West Berlin would have moved rather quickly to a nuclear situation, yes, I really think that, it was part of the planning all along.'
Kennedy understood how much the Battle Group would reassure West Berlin and, at the same time, quieten the American right wing which vehemently objected to the United States being pushed around in front of the world audience by communism. He intended the Battle Group to cross three days hence, the Sunday, and by then Johnson would be present to welcome it, creating a fusion of America's military and political strengths for every Berliner and the world audience to see. All week Kennedy had said virtually nothing in public but now he justified sending Johnson and the Battle Group because 'recent developments including the movement of East German military forces into East Berlin' dictated it.
During the evening, Kennedy summoned Johnson from a dinner party and told him he was going to West Berlin the following day. Johnson initially refused but Kennedy explained he wasn't issuing an invitation. This was an order.
At midnight in Berlin the incident sheet recorded its total for the day: five escapes, fifteen caught.
## FRIDAY 18 AUGUST 1961
And still the escapers came, but more rarely, the first an hour after midnight, a couple at 2.15 a.m., someone caught at 3.50 but no further entry until 11.00 By then the houses along the Eastern side of Bernauer Strasse were being sealed. Behind each window workmen placed bricks and cemented them in, sometimes the moist cement spilling and falling in plump droplets onto the West.
Simultaneously, at Potsdamer Platz, the entrance to the S-Bahn station (which ran underground from the West) was sealed. Down there, the long platforms curved as they had always done and the station nameplate – in old, spiky script – remained set into tiles on the wall but the little kiosk, plastered in advertisements, shut. Shadows and semi-darkness fell upon this station which had once carried the same bustle as Times Square and Piccadilly Circus and the Place d'Etoile. Workmen cleaned it up and modernised it for a re-opening – but not until 1992. Until then the kiosk remained closed.
Outside the station on this Friday morning, a shop which had generated a handy livelihood for its owners by selling leather goods to Easterners – luxuries to them – shut its door forever.
Nearby, but on the other side, members of the Free German Youth manned sections of the border, releasing the Factory Fighters and demonstrating that the East German government judged the moment for insurrection absolutely gone.
In Bonn the Bundestag met and 'sharply' condemned the border sealing.
Alec Douglas-Home returned to London from Scotland 'interrupting' his holiday for, as one newspaper noted, 'emergency talks' with Western Allied ambassadors. It was very late for that.
Soviet commandant Solovyev restated the primary position, castigating 'the illegal provocative measures and subversion against the GDR', and damning protests as 'completely unfounded. As has already been stated several times the Commander of the garrison of Soviet troops in Berlin does not interfere in the affairs of the capital of the GDR.'
Noises off, really.
Kennedy, due to fly to Hyannis Port for the weekend, delayed that until Saturday and later delayed it again. He intended to be on hand as the Battle Group crossed. The time of sealed envelopes and walkie-talkies to the _Marlin_ in Nantucket Sound seemed, and was, already a long way in the past.
And still they came, one at 5.15 in the afternoon then a gap until three at 10.45. At midnight the incident sheet recorded its total for the day, twelve escapes, eleven caught.
## SATURDAY 19 AUGUST 1961
Lyndon Johnson took off from Andrews Air Force Base in the President's jet Air Force One at 9.14 in the evening of what was still Friday in Washington, 3.14 a.m. in Berlin. As the plane lifted and moved across the pastures of Maryland towards Chesapeake Bay and the Atlantic, the Battle Group – 300 trucks towing artillery but no tanks – lumbered from their barracks in Mannheim and strung out on the autobahn, settling to an average speed of between 30 and 40 miles an hour. The Soviet Union had been officially notified and, from Moscow, Pravda denounced it as 'military provocation'.
The first of the weekend civilian traffic was already on the move and flowed past the Battle Group which, travelling so slowly among it, faced a journey up past Frankfurt and out onto the plains to Hannover, then Brunswick [Braunschweig in German] where they'd pitch camp for the night. That was 20 miles from Helmstedt, the crossing point into East Germany.
On board Air Force One, now out over the Atlantic, Frank Cash found 'a big difference between this and a normal visit, very hectic but I'm not sure that wasn't easier than one of those visits pre-arranged six months ahead because then you have plan A and plan B and all the other plans. We simply made it up as we went along. On the plane over we worked on LBJ's speech and it proved to be exactly what was required. On a pre-arranged visit it can never be the same because speeches are written weeks in advance and everybody has to clear them.'
In Berlin, George Muller and Al Hemsing worked on this same speech all night.
Air Force One landed at Bonn in mid-morning and Adenauer greeted Johnson and the others. They travelled to Adenauer's official residence overlooking the Rhine. Johnson shook hands with some bystanders, went in and the men conferred for 90 minutes.
'We called on Chancellor Adenauer to discuss Berlin and we asked him would he like to come along,' Cash says. 'They were building up to an election in Germany, Willy Brandt was the opposition candidate and we did not want to get involved. It ended up with Adenauer not coming on Johnson's plane. The German Protocol Officer, Sigismund von Braun – Werner von Braun's brother, incidentally [the rocket man] – did, however, accompany us. The visit to Bonn was really one of protocol to check in with the Chancellor before continuing to Berlin. Adenauer and LBJ got on all right although there was no great rapport between them, I think. Of course Adenauer was an elder statesman and LBJ very much a practical American politician: they were from completely different backgrounds and with almost nothing in common.'
Johnson changed from the big jet to an Air Force Constellation for the 80-minute flight to Berlin. From it he saw what Peter Johnson had seen: the misty forests of the GDR, the patchwork of enormous collectivised fields, and traditional olde worlde villages sunk into the countryside, forever slumbering. Johnson landed at Tempelhof airport on a grey, overcast day threatening showers. He inspected a guard of honour and seven tanks fired a salute.
Brandt, welcoming him, said, 'It is a great day for Berlin. We are deeply grateful that you came at just this moment.'
Johnson immediately confirmed that the Battle Group was coming, too.
'We landed at Tempelhof Airport and we took a motorcade,' Cash says. 'That really was a very moving experience because up to that moment West Berliners had been so deeply concerned about what was going to happen to them. Mobs of people came out, old ladies up on balconies waving their handkerchiefs. It was a kind of triumphant motorcade.'
Johnson stood next to Brandt in a modest open-topped saloon car, the choice of vehicle removing any remoteness. Brandt waved his hat while Johnson flicked outstretched hands from the crowd. Muller rode 'in one of the cars behind. Johnson was delighted, he loved the adulation. Much to the consternation of the Secret Service people he stopped the car, got out, waved and shook hands as if he was running for office in Berlin. One of his bodyguards said, "You're taking a risk" but he said, "No, I'm among friends here." Johnson had his own photographer who took hundreds of pictures of him and always with the right profile... .'23
The motorcade moved much slower than the Battle Group was going, although it was nearly on the plains headed towards Hannover.
Johnson went first to see the wall at Potsdamer Platz. The saloon parked less than 10 metres from it and he and Brandt went walking beside the wall along roads glistening from a heavy shower. East German soldiers clambered onto the wall and took Johnson's picture, for official use.
Johnson did not reach the Town Hall until 6.20 p.m., 80 minutes late. Muller, meanwhile, tried to 'meld' the speech he and Hemsing had prepared with the speech created on Air Force One. 'Most of ours was discarded.'
Some estimates put the crowd filling the square in front of the Hall at 300,000. A line of children stood near the front, each supporting a giant white letter propped on the ground. _Freiheit_ , the letters said when they were arranged: Freedom. Johnson was introduced to the crowd and so was Clay who raised his right arm to greet a great roar. His face opened into a brief smile. Clay said, 'What you and I started together several years ago we will finish together and the world will be free.'
Geoffrey McDermott wrote that 'the vast crowd was in that emotional mood when it would cheer every mention of "freedom" and "independence" and all the cliches of public speech. When General Clay came to speak they nearly went berserk. Here was the man who had saved Berlin over ten years before come back to save its people from an even graver threat. They remembered; and they would not disperse for many hours.'24
'Willy Brandt made a speech,' Cash says, 'and he had tears running down his face... .'
Then Johnson, bespectacled, almost sombre, spoke. Clay was positioned behind him and certainly sombre. Brandt was slightly to one side and had regained his composure. Johnson placed his speech on his lectern below the microphones and his head dipped to reach for the words. A mood seized him.
'To the survival and the creative future of this city, we Americans have pledged what our ancestors pledged in forming the United States, our lives, our fortunes and our sacred honour.' These were the final words of the Declaration of Independence. Johnson's growl of a voice deepened. 'The President wants you to know that the pledge we have given to the freedom of West Berlin and of Western access is firm.' The faces of the 350,000 Berliners had been sombre themselves but at this another great roar went up, the faces consumed by release. 'This island does not stand alone. Your lives are linked not merely to those in Hamburg, Bonn and Frankfurt, they are also linked with those of every town in Western Europe, Canada and the United States and with those on every continent who live in freedom and are prepared to fight for it.'
'LBJ made a speech,' Cash says, 'and he ended with tears in his eyes. They played the German national anthem, a moment of absolute silence followed and then the Freedom Bell [which Clay had brought in 1950 after the airlift] rang out from the tower above and I must confess that is when I had tears in my eyes, too... .'
Johnson went in to address the Senate and after that, wearing a light summer overcoat against the showers, worked the crowds again. With Brandt at his side he moved through endless corridors of people, flicking more hands, beaming, beaming, beaming. And images were fixed: a child hoisted to see him; an old lady in a headscarf – she'd just had her hand flicked – looking suddenly serene; a young man laughing, his anxieties stilled.
_Pravda_ stabbed out its alternative view. 'Prussian marches rang out from amplifiers on special vehicles. From time to time appeals were heard from motorised propagandists. "Assemble to meet the American Vice-President Johnson, who is coming to defend Europe and Berlin."'
In the East on this Saturday, compulsory evacuation orders were served on the 2,000 residents of Bernauer Strasse – a turn of the tourniquet – and it brought the first of the killings:
19 August 1961 _Rudolf Urban, 47, and a friend were in number 1 Bernauer Strasse while workmen bricked up the doors and windows. When the workmen stopped for lunch, the two men lowered a rope from a first-floor window and their wives climbed down it to safety. People in the street called out in support and although Urban's friend made it, Urban himself evidently heard Border Guards or police rushing to the room, let go the rope and, the onlookers screaming, fell some 3 metres – almost 10 feet – and landed with what one report describes as 'an ugly thud'. He seemed to have no more than a broken ankle but was taken to hospital for examination and died there on 17 September._
In a technical sense, if you can describe a human life as technical, Urban was the first person to die at the wall or, more accurately, because of the wall. The killing ended with another escapee, Chris Gueffroy – but that would be 5 February 1989, and Gueffroy was not born until seven years after the building of the wall.25
In the chronological sense the first named person to die would be a frail-looking widow living on the third floor of No. 48 Bernauer Strasse. How did she spend her time this Saturday afternoon while West Berlin surrendered to LBJ? Shopping? Fussing about her apartment? She had four days to live.
The bricks encircled the front of the Church of Reconciliation and were laid high enough to mask its door.
Clay went to look at the wall at a point where it consisted of eight layers of horizontal concrete slabs. A photographer captured him levering himself up using both hands to peer over, his feet off the ground. It gave the GDR a propaganda opportunity, what you might call a reverse image. They reproduced the photograph on posters with a caption, 'Why so uncomfortable? You'd see more, you'd find out the truth, you'd be able to report objectively when you yourself visit the capital of the GDR, Berlin!' (The West had propaganda just as potent: a photograph of a labourer laying breeze blocks with a soldier covering him and – on the Western side, mounted on two metal supports – a placard showing Ulbricht and a quote from his speech at the press conference on 15 June: 'Nobody intends to build a wall.')
Clay did visit East Berlin, as he was fully entitled to do under the Four Power Agreement, and when he returned he did report objectively that almost no people were on the streets but 'I have never seen quite so many soldiers in a town' – more, he added, than when he'd been in Berlin at the end of the war.
That afternoon Ursula Heinemann, who'd been a waitress in a West Berlin hotel, went walking with her mother and they reached the Teltow Canal. Not far away Heinz Sachsenweger must have been on duty with his Factory Fighters, guarding the weekend chalets with their little gardens, their chickens and rabbits. The canal formed the border before it turned inwards to the Eastern district of Treptow. There, chalets clustered on either side of the line.
Heinemann and her mother reached them and Heinemann said the equivalent of 'Hold on a moment'. She opened the gate of a chalet and walked into the back garden. Wire had been uncoiled behind it. She crawled under the wire but found another coil in front of a drainage ditch. She crawled under that, crossed the ditch and saw 'a plume of cigarette smoke'. The smoker, a man, announced that she was in the West, congratulated her and gave her a cigarette. Added to her identity card and a handkerchief, that made her possessions three. A passing woman offered her a couple of marks and she caught the bus to the hotel to begin work.
The Battle Group crawled onto the plain towards distant Hannover and still, in constant profusion, the weekend civilian traffic flowed past it. Just another convoy, they must have thought.
Peter Johnson spent long hours at the Reuters office in Bonn. The day before, he'd covered a story about a British yacht sunk in the Elbe, 'a terrible story, three adults saved and their five children presumed drowned while their yacht was cut in two by a freighter in pitch darkness. I have the painful task of interviewing one of the survivors by telephone.' On this Saturday he was commended by Reuters' general manager for his 'indefatigable' work on the story and 'because LBJ was Berlin-bound I am being referred to in the office as the Vice-President'.
The first of the Battle Group stopped at Brunswick and began setting up an overnight camp. The rest were strung back down the autobahn.
At midnight, the incident sheet recorded its total for the day: three escapes, three caught.
## SUNDAY 20 AUGUST 1961
And still they tried to come, one caught at 15 minutes past midnight, a second at 1.00 a.m.. At almost exactly this moment the last of the Battle Group reached the camp, but they wouldn't be able to sleep for long.
Peter Johnson was woken by 'a call telling me that an American Battle Group which President Kennedy has ordered to West Berlin was leaving earlier than expected from its night camp near Brunswick. This meant I had to rise again at 4.00 to be in the office for 5.00 to start up the story by phoning the frontier just before Alfred Klühs [a Reuters reporter] arrived there from Berlin to travel with the convoy.'26
A report of the first escape came in at 2.40.
The Battle Group was roused from the darkness and ate their rations while army radio patrol trucks set off for Helmstedt and positioned themselves near the crossing point to act as a relay station for the convoy. Moment by moment the Battle Group would give a running report which, via the radio trucks, would go to the US Command in Heidelberg and, moving upwards and outwards, to the Supreme Allied Command near Paris, the Pentagon, the White House and US Army headquarters in West Berlin.
Major-General Clifton, Kennedy's military aide, went to the Situation Room in the White House around 10.00 in the evening Washington time (4.00 a.m., in Germany) and prepared to spend the night listening to the running commentary. Kennedy was to be woken immediately if any incident developed. Clifton reached the Situation Room pretty much as the Battle Group broke camp.
A reporter at Brunswick noted the American soldiers were armed with M1 rifles, recoilless rifles and 3½-inch rocket launchers although 'officers said the weapons would not be loaded' and the 49-year-old commander, Colonel Glover S. Johns Jnr, added: 'We are not anticipating any trouble. We told the Soviets we were coming.' He explained to the Battle Group that he'd travel at its head and deal with formalities at the crossing point. No one else must have any contact with East Germans or Soviets. The Battle Group set off along the autobahn again, through pleasantly rolling countryside and woodlands.
In West Berlin, George Muller established direct contact with the US Army headquarters and prepared to listen to the running commentary himself.
General Bruce Clarke, US Commander Europe in Heidelberg, had made the decision to give the Group live ammunition, albeit not to be loaded into the weapons. If the Soviets started shooting and the Battle Group couldn't return fire, well, what kind of a strategy was that?
Washington sources estimated the chances of an incident at 'a thousand to one'. Those odds might be short enough in a nuclear age.
The first couple of trucks carried bulldozers to remove any obstacles, but what would their use involve and would there be obstacles? Would the GDR or Soviets stand and watch as bulldozers cleared a path on their territory?
John Ausland gave the State Department thinking which, they believed, answered those questions. 'Khrushchev had problems on his hands, too. He and Ulbricht wanted to stop the refugee flow but they didn't want a big fat crisis about it and, having stopped it, they'd achieved what they wanted. Foy Kohler said "they've got that, they don't want to escalate it" and he was a Soviet expert, but I can tell you there were a lot of people listening to telephone conversations with Colonel Johns as the convoy went across, and a lot of nervousness. We faced the first potential flashpoint.'
Frank Cash, with the Lyndon Johnson party in West Berlin, emphasises that. 'The reinforcements crossing was a very emotional time because everyone wondered what would happen and it was essential from our point of view in reiterating our access to West Berlin, because without land access we faced another airlift. So we were all asking: Will the East Germans block the autobahn?'
The Battle Group reached Helmstedt at 5.00 a.m. Helmstedt was, in fact, a small town on the border with the autobahn flowing past its outskirts and on towards the Eastern hamlet of Marienborn, and the names betrayed the division: in the West the crossing point was known as Helmstedt, in the East as Marienborn. The crossing point itself was in open countryside with a small Allied hut on the Western side but formidable fortifications to the East.
A _New York Times_ reporter watched the Battle Group 'crawl down the two hundred yards of no man's land' and halt at the Eastern control huts. Colonel Johns, anxious to get through the formalities as quickly as possible, made a tactical error. Under the Four Power Agreement, he was obliged to inform a Soviet office how many men he had and he gave that, but when the Soviet officer counted them – they were in their vehicles – the number differed. The Soviet officer counted a second time and still the number differed. Johns ordered the Battle Group to dismount in order to simplify the count. Instantly it became a precedent and, in future, every convoy would be made to do likewise. The Soviets would use this as an irritant, spending as long as they wished counting, especially in the cold of winter.
Muller heard 'reports that the Battle Group was held up because of the size of it. That amount of men hadn't happened in a long, long time.'
The leading vehicle was cleared in 16 minutes and the Battle Group moved off along the same autobahn but now, of course, in the GDR. The autobahn was empty because far from erecting blockades the GDR had kept all other traffic off it. They, too, were nervous about incidents particularly, perhaps, civilian traffic getting in the way or people trying to board the American trucks as sanctuary.
A Soviet fighter plane flew over low and circled.
On the old bridges spanning the autobahn, police and soldiers watched the Battle Group move slowly beneath them.
This autobahn was a curious, eerie, slightly unreal place. Its surface was badly maintained so that vehicles bounced and rattled their way along it, even at slow speed. Many of the stone bridges had been destroyed in the war and not rebuilt, so that only stumps of the uprights remained, standing at either side of the road like sentries guarding the past. In many places, the autobahn was hemmed by woodland – the trees up to either side – and that gave the impression of screening out the light even in daytime; gave the impression of a country hidden, a country you couldn't penetrate. The autobahn did not go near towns: the spires of the cathedral at Magdeburg, the first town in the East, could be seen far away. It passed near a village or two but the old houses and streets in them seemed empty. It passed over country lanes of pre-war cobbles and sharp cambers. The fields seemed empty, too.
On this autobahn there was always a frisson of the forbidden, a sense you were taking a risk, a feeling of a deserted place locked into another time.
At 7.00 a.m., and with no fanfare, a British military train carrying eighteen armoured cars and sixteen scout jeeps pulled into West Berlin. The vehicles unloaded and drove to the barracks at the Olympic Stadium.
At 9.00, a report of one escape came in.
In East Berlin, Adam Kellett-Long went to St Mary's (the Marienkirche), the second oldest parish church in the city. 'Normally the Bishop of Berlin took the sermon but this was an unknown pastor. The church was absolutely packed. He said, "The theme of my sermon today will be 'God forgive them, they know not what they do'." He was one of the bravest men I have ever heard. I don't know what happened to him afterwards.'
At 10.55, a report of two escapes came in; at 11.00, a party of three got across; at 11.10, two but this time near Wollankstrasse.
The Battle Group crawled slowly on.
The _New York Times_ reported that West Berlin was 'like a boxer who had thrown off a heavy punch and was gathering stamina for another round'.
For a week Brigitte Schimke's only contact with her parents had been waving to them across the twin curved roads, the sunken gardens and the wire from her apartment. Her sister, who lived in West Berlin, had a simple idea, although dangerous. They looked alike. Her sister went to the police in the West, said she'd lost her identity card and they issued her with another so that now she was armed with two. On this Sunday, she crossed – Westerners still could, of course – and journeyed to Brigitta's apartment. She gave Brigitta one of the identity cards and her sister used it to cross. She stayed three days with her family before making the decision which she suspected would alter the direction of her life, perhaps irrevocably, although deep down she thought – as so many others did – that the wall could not be and would not be permanent. She decided to return to her husband and their apartment.
She would not re-cross to the West again until 1977 for her mother's funeral and, even then, only after she'd proved the deceased was her mother and was indeed deceased. Permission to attend her father's funeral two years later was refused. She would re-cross a second time, but that was in 1987. 'It was for my sister's fifty-fifth birthday. They made an exception because normally you could only go for the "ten" birthdays: forty, fifty, sixty and so on. I was allowed but not my husband. I had three children and they knew I'd come back because of them. I found West Berlin very exciting, hectic. I noticed a very big difference.' She re-crossed regularly – but after 1989, when she walked over ground where the inner wall, the death strip and the outer wall had been, to catch the bus to work.
At 1.30 p.m., a report came in of one escaper caught.
The Battle Group crawled slowly on. 'We spent,' Muller says, 'a very tense time because we didn't know what the Soviets might get up to. On the theory that you always fight the present crisis on the basis of the crisis before, much of our thinking was directed to a possible blockade.'
The first vehicles of the Battle Group neared Dreilinden, the checkpoint on the autobahn between the GDR and West Berlin. Thousands of Westerners waited to cheer them. Many among the thousands had picked bunches of wild flowers on the way and held them. Lyndon Johnson visited the wall again and set off for Dreilinden but the crowd became so great that his motorcade couldn't get through. It halted some distance back. As Colonel Johns led the Battle Group in after it had emerged from the checkpoint, the crowd closed on it, giving the flowers to the soldiers or casting them affectionately towards the trucks. Johns said, 'It's the most exciting and impressive reception I've seen all my life, with the possible exception of the liberation of France.'
The Battle Group stretched far, far back into the GDR but the vehicles were coming through surely and safely.
In Washington, just before dawn Major-General Clifton, Kennedy's military aide, left the Situation Room and drove to his house nearby. He changed clothes, returned and briefed Kennedy at 8.00 a.m. (2.00 p.m. in Berlin) that all was well.
At Dreilinden, Johnson stood on a very temporary reviewing stand to take the salute and at one point walked over to an armoured car and handed the driver a bouquet of the flowers he'd been given.
At 3.45, three reports came in from different sections of the border: two escaped, two caught. Ten minutes later a party of four escaped, again near Wollankstrasse.
The last Battle Group truck had come in by late afternoon. The US commander in Berlin, Major-General Watson, stopped it and said, 'The Vice-President of the United States wants to talk to you boys.' Johnson thrust his head through the window and said to the driver, 'My name is Johnson and I'm from Texas. Where are you from?' and the driver said, 'Brooklyn.'
At this moment Kennedy attended Mass at St Matthews Cathedral six blocks behind the White House and when the service finished he flew to Hyannis Port.
The Battle Group paraded down the Ku-damm past the broken-toothed Kaiser Wilhelm Memorial Church and another massive crowd celebrated their arrival.
The mood of West Berlin lightened and heightened.
The mood in East Berlin was reflected in silence, and in the reports which kept coming: one escaper caught at 7.35 that evening; a party of seven at 8.00; a couple at 10.00; three people at 11.45. The incident sheet recorded its total for the day: seventeen escaped, seventeen caught.
It was still afternoon in Hyannis Port. President John F. Kennedy and Jacqueline boarded the _Marlin_ and cruised Nantucket Sound just as they had on the Sunday before.
* * *
* © 1961 (renewed) by Sanga Music Inc. All rights reserved. Used by Permission.
1. Peter Johnson's diary.
2. _United Press International_ report.
3. Interview with author.
4. One of the lessons of studying the wall is that, even if you think you are open-minded and have breadth of vision, at some stage you'll probably discover you haven't. You'll be betrayed by your subconscious, just as I was in the paragraph about each American President since 1949 being constricted by the risk of nuclear war. It wasn't until I reflected, that I realised how instinctively and hopelessly Western the perspective was. After all, every Soviet leader has felt exactly the same constriction.
You can fall into this trap in other ways. In the early 1990s I was at a party in Marzahn, the new town on the edge of East Berlin, and a young woman was recounting her trip to Paris – still, then, a wondrous adventure for anyone who'd spent their lives behind the wall. I ventured that the trip must have been even better because, the currency union completed, she'd been able to take the mighty Deutschmark. She looked at me and said, 'Typical Westerner, seeing everything in terms of money.'
5. _Berlin: Von der Frontstadt zur Brücke Europas_ , Rainer Hildebrandt (Verlag Haus am Checkpoint Charlie, 1984).
6. Interview with author.
7. Interview with author.
8. _Violations of human rights, illegal acts and incidents at the sector border in Berlin since the building of the wall (13 August 1961–15 August 1962)_ , published on behalf of the government of the Federal Republic of Germany by the Federal Ministry for All-German Questions, Bonn and Berlin, August 1962.
9. McDermott, op. cit.
10. GDR document, courtesy of the Hagen Koch archive.
11. Gelb, op. cit.
12. Interview with author.
13. Interview with author.
14. Interview with author.
15. Interview with author.
16. _The Fall_ , Reinhold Andert and Wolfgang Herzberg (Aufbau-Verlag, Berlin and Weimar).
17. Honecker was being naturally protective of the GDR and thus trying to equate what happened at the wall in Berlin, and the inter-German border, as something quite normal in international terms. His words must be restricted to that context, not reality. The direction the European Union was taking was utterly different: the abolition of border controls altogether.
18. Harpprecht, op. cit.
19. McDermott, op. cit.
20. Interview with author.
21. Interview with author.
22. Interview with author.
23. Interview with author.
24. McDermott, op. cit.
25. Victims of the Wall may never be definitively known. I have leaned heavily on _Opfer der Mauer_ [Victims of the Wall] by Werner Filmer and Heribert Schwan (C. Bertelsmann Verlag GmbH, Munich, 1991). It is a thoroughly researched book using official (but previously highly secret) GDR reports of the incidents. Yet even here the record is incomplete and you come across entries like 'Unknown person, deceased 4.9.1962' because that is all the authors have been able to discover. Other published sources give victims which Filmer and Schwan don't, and vice versa. This leads me to think that, across the twenty-eight years that the wall stood, there may well have been fatal attempts at escape which were unreported in the West and for which no documentation survives in the East – or people who just drowned in their lonely anonymity. Nobody knew they were swimming for it and nobody ever came across their bodies. Hagen Koch, whose Berlin Wall archive is a treasure-trove of primary documentation, has compiled a list of 172 fatalities (not all attempting to escape: children playing and falling into the water by the wall then drowning. An explanation for this is in the text.). Koch's list begins with an unnamed East German Border Guard who died in an accident with a gun sometime after 13 August 1961. Koch also records someone unknown on 13 August itself but has no further details of any kind; and lists an unnamed 18-year-old who drowned on 16 April 1989 – making him the final victim. That the first and last victims are likely to be held forever in _their_ anonymity seems to reflect how the wall literally dehumanised.
26. Peter Johnson's diary.
## FIVE
## _Cold as Ice_
He who lives on an island should not make an enemy of the ocean.
GDR propaganda
By now, a week after the border sealing, a measure of clarity had settled over the confusion. The refugee flow had been almost completely staunched and, with the stability this brought, the GDR could begin to build a future. Part of that future would be in creating a distinctive society utterly unlike the West, part in establishing their right to run East Berlin as their capital, something precluded by the Four Power Agreement which, to the Western Allies, was still in full force. The GDR had already begun this process with the decrees of the previous Tuesday. They would increase them on the Wednesday of this second week. Here was another logic to work itself out: in the years to come East Germany would seek international recognition as a sovereign country and expend a considerable amount of money and effort to achieve that.
Inevitably, given so many complexities, that would create frictions of its own and one of them would not be long in coming – two months – when, it seemed, a single errant shot might have moved everything towards the nuclear escalator very quickly indeed.
Inevitably, too, the sealing was still imperfect this second week, and would be for months. The trauma of what amounted to captivity drove dozens to risk escape and the days were studded with the deaths of many of them. Shoot-to-kill became more than confirmed policy: it was happening with a terrible frequency. These escape attempts resembled a stampede and only when the wall had been refined to the point where it was extremely difficult to cross did this slow and virtually stop. To cover that, a lot of individual details are given which, cumulatively, become a broader picture. The slowing and stopping happened towards the end of 1964 and by then – in fact within a calendar year of the sealing – 116 watchtowers had been built, as we have seen, 84 round Berlin and 32 through it.
The attempts to cross still came, often using the full range of human ingenuity; and still the escapees died. Most, of course, did not try and for all the obvious reasons. Soon, East and West began to settle – however reluctantly – to their different normalities and one could feel the two halves moving at different rhythms and obeying different impulses; but not quite yet. When they did settle, the feeling would be that the Western half was searching out personal satisfaction in their lives, and the Eastern half was locked into some sort of vast communal endeavour. Each half would have its own centre of gravity and its own self-sufficiency; and soon enough the contrast would take your breath away, but not quite yet. Nor, in these comparisons, must anyone forget that the Western half was massively subsidised by the Federal Republic and the Eastern half had spent years paying reparations to the Soviet Union. The talk in the East, officially anyway, would be of overtaking the West but, as one swarthy Eastern worker said cryptically enough, 'Let's hope if we ever do they can't see the holes in our trousers... .'
22 August 1961 _Ida Siekmann, 58, jumped from the third floor of 48 Bernauer Strasse at 7.00 in the morning. Evidently she had been agonising over how to get to the West to celebrate her fifty-ninth birthday due the following day – the front door of the house was now bricked up. She heard knocking on apartment doors. Soldiers? Police? Stasi? She tried to make a rope out of bed sheets but couldn't. She threw her mattress out and jumped, aiming to land on it, but missed. She died of her injuries._
Thirteen people crawled through the municipal sewers to the West.
On 23 August, the GDR closed five of the twelve checkpoints, leaving: Friedrichstrasse, for foreigners, diplomats and Allied forces; Bornholmer Strasse; Heinrich-Heine-Strasse, for West German citizens and goods; Chausseestrasse; Invalidenstrasse; Oberbaum-brücke; Sonnenallee, for West Berliners who worked in the East and who had a special identity card – about 6,000 people. The GDR also introduced compulsory permits for West Berliners wanting to go to the East, and instructed people 'in the interests of their own safety to stay a hundred metres away from both sides of the borders between the capital of the GDR and West Berlin'. The three Western commandants protested over that and, to counter any notion that people could not go within the hundred metres on the Western side, drove military vehicles up to the wall.
George Muller, Deputy Political Adviser at the American Mission, says:
We in Berlin argued that if Ulbricht got away with this [building the wall] it would be a two stage thing, the 13th and the 23rd. The decrees of the 23rd were much more stringent because they realised they'd got away with it. They'd stuck a finger in our teeth and it went right through, and God knows where it would happen next. General Clarke said stay a baseball's throw from the wall and you know how far you can throw a baseball. It meant go easy on border patrolling. I argued the opposite, that maybe increased patrols wouldn't have made them take the wall down but it might have prevented the next Berlin crisis. We were a victim of our own propaganda and our own newspaper coverage, because Berlin was still a remnant of the fabric of a whole city before the wall.
The people in Washington and elsewhere saw it as a city which had been divided since the end of World War Two. They were saying, 'What's all the fuss? All they are doing is tightening up their zone of control in their sphere of influence. We're only there to protect the security of West Berlin. Why is it our business to go after them for what they are doing to the East Germans?' The other argument was that World War Three could start in Berlin. We felt the opposite – that it wouldn't because this was a very controlled action on the part of the Kremlin. And we were not holier than the Pope: we could not do for the Germans something which they themselves did not advocate, and neither Adenauer nor Brandt advocated any kind of strong countermeasures like tanks.1
Mary Kellett-Long wrote in her diary: 'Once the official announcement of the measures was out we felt much better as what they boil down to is East Germans are not allowed to go into West Berlin. The West Berliners and West Germans and foreigners are allowed in and out of East Berlin as before. A lot of the overhead and underground trains now stop at the border. I have never seen so many policemen and soldiers in my life. There are also the Factory Fighting Groups dressed up in uniform toting machine guns. Adam was the first person to drive into West Berlin in an East German car after the new regulations. Adam went rushing round all the borders to see what was going on.'
Al Hemsing remembers that 'early in the week after the border was sealed I crossed and my wife Esther crossed. It was getting dicey then because they [the _Volkspolizei_ , often abbreviated to _Vopos_ – People's Police] didn't know their orders. My wife told me she'd said to a _Vopo_ , "Now I'm going in, will you let me out?" and the _Vopo_ replied, "I can't tell you that" – in effect he had no orders. What I always thought was outrageous on our part, on the part of the Western Allies, was this: when there was an East German announcement which we got via radio – one of our sources of information was RIAS – saying that, in order to protect the GDR from chicanery and incursions from the West, henceforth the Allies would be restricted to crossing at what became Checkpoint Charlie, nothing was done. Now, the East Germans having accomplished their original purpose, I think that if we had insisted, we could have kept some of those other crossings open.'
24 August 1961 _Gunter Litwin, a 24-year-old tailor, was shot at 4.15 in the afternoon swimming the Humboldt Canal, after he had passed the middle of the dock basin, by a master sergeant of the Transport Police. His body was recovered at 7.10 and hauled crudely onto the quayside_ _by five uniformed policemen, while a group of soldiers watched._
There is a grotesque photograph of this. They are hauling him up by his left arm, the right lifeless at his side. The hauling distorts the jerkin he wears. His head hangs forward, the jerkin encircling his neck.
29 August 1961 _Roland Hoff, 27, was shot at 2.00 in the afternoon swimming across the Teltow Canal. He was fired on by rifles and automatic weapons and sank in the middle of the canal. An unnamed man swimming with him turned back and was arrested._
2 September 1961 _Hans-Joachim Rassmann, 21, died at the wall._
3 September 1961 _Axel Bruckner, 28, a lieutenant in the People's Police, was shot at the border trying to cross. His mother was told by the police only that he had had a fatal accident._
However, 'during the night between 9 and 10 September, a married couple swam across the Havel, pulling an 18-month-old child in a bath-tub with a rope behind them. They reached West Berlin territory unharmed in the neighbourhood of the Glienicker Bridge.'2 That same day three men broke through in a lorry at the sunken gardens. One man was slightly wounded in the right hand. A week later three men hijacked a postal truck and drove it through the barbed wire. They were shot at after they crossed.
By mid-September some 2,665 people had been escorted to police stations in East Berlin for 'state crimes', which were defined as 'rabble-rousing propaganda and state slandering' – euphemisms, presumably, for complaining about the wall. Preliminary proceedings had begun against 1,085 of them.
President Kennedy decided to appoint Clay as his special representative to the city, which was shrewd in terms of lifting West Berlin morale but might be more problematical if incidents developed and, the GDR pushing to assert what they claimed were their rights, incidents seemed inevitable. By nature, Clay preferred confrontation to compromise and his status was unclear, anyway. As Gelb writes: 'The White House was sending Clay to Berlin to reassure the Berliners; he was going there to take on the Russians.'3
Clay arrived on 19 September and moved quickly on three fronts. He ordered that a mock section of wall be built so American troops could practise knocking it down; he resumed military patrols along the autobahns in the GDR which had been suspended; and he turned his attention to the enclave of Steinstücken.
Kurt Behrendt, the amateur radio operator who'd moved there in 1957 soon after his wedding, remembers that 'General Clay came in September 1961. At that time, in fact, he wasn't a General any more. The first day he was in Berlin he came in a US military helicopter because they wanted to show him the American Sector and he was the inventor of the air bridge. They showed him the sector and, since the American helicopters were the only ones who were allowed to fly over the GDR, they came to Steinstücken as well and said, "OK, this belongs to the American Sector." There is a railway through Steinstücken and at the other side of it there is a big area, a field, and the helicopter landed there. He walked round Steinstücken and some of the residents accompanied him. At that time, I had a job where I worked at night so I was asleep. My wife woke me and I got my camera and took pictures of the whole thing.'
Next day the GDR began the process of forced evacuation of properties on the border line. This was at Harzer Strasse in the southern district of Treptow, a place which was as much an anomaly as Bernauer Strasse. The line followed one side of the road through a built-up area of four- and five-storey houses, running along Harzer Strasse, turning geometrically left into another street then geometrically right along the street after that.
Harzer Strasse had already provided a montage of its own photographic images: the wall made of stacked concrete lintels so close to the houses that, at one point, it passed under a balcony; two bricklayers positioning square stone blocks while three Border Guards monitored them; two young women holding hands and talking across a concrete block while a Border Guard surveyed them from three paces away; a crowd on the Western side watching a solitary bricklayer lift the stone blocks into place. The front door of No. 118 was the actual border line and, although the wooden door itself remained in place, the hall behind had been crudely bricked up. Now 20 houses and their 250 occupants were moved out.
25 September 1961 _Olga Segler, 80, jumped from the second-floor window of 34 Bernauer Strasse at 9.30. She landed in a blanket held by the West Berlin Fire Brigade but died of injuries._
There would be other images in this street, of a 77-year-old called Frieda Schulze4 who survived. Al Hemsing 'witnessed at Bernauer Strasse the old lady being held by the _Vopos_ and it was a tug-of-war until the people in the West finally got her'.
Wearing a plain dress, she climbed out of a window and stood on a little ledge. Someone from the West leapt up and tried to pull her onto the blanket which the Fire Brigade held, spread taut, below. He gripped her shoe and it came off. In all her fear, she held on to the window ledge with her right hand. Just then an arm came from the window and grasped her hand, trying to haul her back in. She was on her knees facing the window and now the man held her with both his hands while, just behind him, a capless policeman threw a small smoke bomb into the crowd below. The woman wriggled off the ledge so that the man trying to haul her back had to hold her whole bodyweight. He did, but someone from the West leapt up again and tried to seize her left ankle; someone else already had her right ankle and pulled.
This was the precise image which would be developed and carefully printed, fixed: two men at a window pulling the arm of a white-haired woman, and two men on the ground pulling her ankles. _This_ symbolism was so precise that it remains deeply evocative four decades later. She tumbled down into the blanket and the crowd raised their fists towards the window and shouted in triumph.
'I was standing right on the other side of the street and it was very emotional,' Hemsing says. 'Above all, my reaction was: can this be happening on an ordinary street? You just weren't ready to see that sort of thing. The people on the Western side were all shouting although I can't remember if I was shouting, too.'
Between 24 and 27 September, nearly 2,000 people living along Bernauer Strasse were forcibly evacuated. The GDR intended no more jumping from windows and ledges, but that would have to await complete evacuation. Elsewhere, however, a 35-year-old woman got across the border with two small children at 10.15 one night. They were fired at as they crawled under the wire but made it unhurt; and at the very end of the month further houses were evacuated in Treptow. The GDR resolved there would be no more holding hands over the concrete blocks down there.
4 October 1961 _Bernd Lunser, a 22-year-old student, died when he jumped from the roof of 44 Bernauer Strasse and missed the Fire Brigade blanket. He had been discovered and was shot at by the Border Guards. An unknown man planning to escape with him was reportedly beaten to death in the attic of number 44._
5 October 1961 _An unknown man tried to swim the River Spree at 11.50 at night. Border Guards shouted warnings and fired warning shots. A police boat pursued him and shots were fired from it when he was about 6 metres from the Western bank. He disappeared under the water and, at 1.02 a.m., the West Berlin Fire Brigade found a body in the water._
5 October 1961 _Udo Dullick, a 26-year-old engineer, tried to escape by swimming the Teltow Canal at 11.50 at night. His body was found hours later – he had been shot and then he'd drowned. An unknown man, making the escape attempt with him, was fatally hit and drowned, too. A third man met the same fate, although whether he was with the other two is unclear._
The River Spree, where tragic and very public escape attempts were made.
A pattern began to emerge, escapers working out that the waterways – the Humboldt basin, the Teltow Canal and the Spree – were weak points in the wall; and darkness might be their friend. If water equalled vulnerability so did the earth itself and that would bring the tunnellers. In these early days, however, the tightening of control was still incomplete. For example, an underground passage carrying clear rainwater ran under a street in East Berlin to the West and, to reach it, all anyone had to do was lift a grating and lower themselves down. So many escapers used this – twenty-eight in one night alone – that a quota system had to be instituted to maintain its secrecy.5 It was discovered after fifteen days and the grates replaced by bolted manhole covers.
12 October 1961 _Klaus-Peter Eich, 21, who'd hidden on a Westbound goods train, was seen by the Transport Police between the stations of Potsdam and Babelsberg. Two Border Guards fired four warning shots in the air and then four aimed shots. Eich died later in hospital. A man with him escaped by shaking off tracker dogs._
Bernauer Strasse was now evacuated further, and so were houses within 150 metres of the wall in the side streets off it. The area became a ghostly place of silences and blinded windows. The tramline along it was bisected by the wall so that the West Berlin trams had to stop and go back; and a television crew interviewed a conductress one day, framing her against the wall and the wooden screen behind it. (The screen had been erected so that people could not see each other, or wave to each other, over it. Bearing in mind that Bernauer Strasse had been one street with friends and families on both sides, the screen was almost more inhuman than the wall itself.) The conductress spoke a few words, gesticulated in a stunned sort of way and broke down. Bernauer Strasse lived on with its ghosts and its memories until the following August when the GDR began to demolish the houses. In time, they would all go, making a long, broad death strip with only the Church of Reconciliation left standing, and in time that would be demolished, too.
Between 23 September and 13 October 1961 an estimated 150 people escaped through a drainage canal before it was discovered.
14 October 1961 _Werner Probst, a 25-year-old transport worker, tried to swim the Spree. Twenty-one shots, including warning shots, were fired at him and he was hit in the head, heart and lungs._
18 October 1961 _An unknown man was shot by Transport Police between Potsdam and Babelsberg after boarding a Westbound goods train. He died of his wounds the next day._
Of necessity, many people tried to maintain normality as best they could and, as a consequence, Allan Lightner of the American Mission and his wife Dorothy prepared to enjoy themselves on the evening of the 22nd. In the background, however, the GDR were beginning their process of assertion. 'There had been a series of incidents with the _Vopos_ stopping American, British and French cars who were testing the free access to the East,' Dorothy Lightner says. 'These incidents were usually on the weekend. About a week before the 22nd I got a call from Allan's general services officer – named Earle Cleveland – inviting Allan and I to a Czechoslovakian performance in Eastern Berlin. I said to Allan, "Would you like to go?" and he said "Oh, Sunday evening, there are always the incidents." I said, "Earle has to know if we're going or not" and Allan said, "OK, we'll go."
'We had house guests [including] an army officer from Frankfurt up for that weekend. They were going back on the train which went from West Berlin to Frankfurt. We went down to see them off – they'd already been to East Berlin with George Muller [of the Mission] and they'd had some difficulty, but the _Vopos_ finally let them through. They reported to us that it seemed to them the _Vopos_ were trying to make trouble but would, in the end, chicken out.
'Cleveland thought the performance was something wonderful – an amazing combination of lantern slides and live action and projected pictures. I've never seen it, I only know it was evidently a terrific thing to watch and it was just something Cleveland thought we'd want to go to. We weren't in the business of provoking but we were not just about to give up anything, either. We'd dropped our friends off at the station. We debated whether we should get something to eat or go right to the show and we decided we were already eating too much, we were too fat and the Clevelands had invited us to have something afterwards – so we headed off to Friedrichstrasse.'
They reached Checkpoint Charlie and moved past the Allied control, a white hut in the middle of the road with US ARMY CHECKPOINT written above it. They reached the white sign as big as a hoarding with YOU ARE LEAVING THE AMERICAN SECTOR written on it. They continued to Hagen Koch's line, now just in front of them, and when they crossed that they were technically and physically in the East. The checkpoint had not yet grown into the finesse of fortifications it would later be: Friedrichstrasse, broad and straight, had, just over the line, a 'chicane' of concrete sleepers laid three deep to slow traffic, and then much further on a couple of sentry boxes. More than that, the area had a battered feel to it. The façades of the tall buildings – six storeys, some of them – still bore the familiar pockmarking of gunfire from 1945. There were vacant lots where the bombed buildings had been taken down. The road itself had been patched and so had the pavements along either side of it.
Dorothy continues, 'When we got there the _Vopos_ demanded to see our papers and, according to established procedure, my husband was supposed to demand a Russian officer. We were not negotiating with the _Vopos._ When, after an interval, there was no Russian officer and it was apparent that they hadn't even called one, Allan said, "I have a right to go ahead, I am taking advantage of my right."'
Al Hemsing, Press Officer at the Mission, explains it. Someone from the Mission had gone over 'in a dark Chevrolet and Al Lightner and his wife went over in their Beetle. The dark car was waved through just on a show of ID cards but Lightner, immediately after him, was held up and to this day I'm quite convinced that had he been in the Mission car he wouldn't have been, but the idea of the Chief of Mission going over in a little Volkswagen and driving himself probably struck the _Vopos_ as being strange. He said, "I don't show papers." The _Vopo_ was simply behaving like any Corporal would.'
Dorothy Lightner says that 'Allan was driving my car, which was a little Volkswagen and it had the licence plate one thousand. So although it was mine – well ours, Allan normally used an official vehicle and I was the one who really drove it – he was driving it at this point. He switched into first. Now if you know anything about those Volkswagens you know they make a terrible noise when they go into first gear – well, the 1959 version I had did. There were two _Vopos_ in front of the car and they sprang back. We were not about to run them down but they didn't know that. So they sprang back and we went ahead and it was a very short distance that we went ahead through this No Man's Land. They chased after us, stopped us and they thumped on the car with their batons – which really made me mad because they were denting it. They said, "This time you've really done it, now we're going to put you in jail." They made various and summary threats which we ignored. We sat in the car. Meanwhile the American lieutenant on duty at Checkpoint Charlie came walking over to us...'
Lieutenant: 'Is there anything I can do, Mr Lightner?'
Lightner: 'No, we're in no trouble, let's wait and see what's going to happen.'
The lieutenant clearly sensed the possibility of an incident and went off to telephone the American Operations Centre in Clayallee (named for General Clay, and in the west of the West). Clay himself was at the Operations Centre and he, too, sensed an incident. He fully intended to demonstrate how you dealt with incidents. He ordered soldiers, four tanks and two personnel carriers to the checkpoint.
Word had gone to Howard Trivers, Political Officer at the Mission. He and his wife were sitting in their living-room and the red phone rang, which was an alert. He answered it, spoke briefly, replaced the receiver and put his hat and coat on.
Mrs Trivers: 'Howard, what's going on?'
Trivers (turning and looking at her): 'They've got Al.' He headed for Clayallee.
## Operations Centre
Trivers arrived in time to hear Clay saying into a telephone, 'Tell Mrs Lightner to get out of the vehicle.' That would have been to the lieutenant at the checkpoint.
When Clay put the phone down Trivers said to him, 'General, I think you're going to have a little trouble with that.'
Clay, an old and good friend of Dorothy Lightner, said nothing.
## Checkpoint Charlie
She sat very calmly in the stationary Volkswagen and the lieutenant walked over to it again.
Lieutenant: 'Mrs Lightner, would you please come back with me? They've asked me to ask you to return to Checkpoint Charlie.'
Mrs Lightner: 'I certainly will not. My husband and I are in this together.'
She remembered they had a package of chewing gum, the only thing edible in the car, and they chewed it. The lieutenant walked to the checkpoint and telephoned the Operations Centre to report that Mrs Lightner refused to leave the vehicle.
## Operations Centre
Clay: 'Tell her I _order_ her to leave it.'
## Checkpoint Charlie
The 'poor lieutenant' (her phrase) came back a third time.
Lieutenant: 'Mrs Lightner, General Clay has ordered you to leave the car.'
Dorothy Lightner: 'Well, I guess I have to if it's General Clay, but tell him from me that I think he's very mean to ladies to order them out of vehicles.' Neither she nor her husband had felt in any danger except when, in the gloom, she could see _Vopos_ coming up with dogs. She walked back with the lieutenant, leaving her husband still sitting and the car still stationary. He was certainly not about to turn it round. When she reached the checkpoint she heard why she had been ordered out.
Lieutenant: 'Mrs Lightner, the reason this is happening is that they [the soldiers Clay had despatched] want to do something and what they are going to do is make a forced entry with fixed bayonets.'
By now, the _Vopos_ had reported the incident to their superiors. As Hemsing says, 'then it went up the line in the East and I think what happened is they thought this is as good a time as any, let's hold him up and make him show his ID. And it escalated. There were higher officers there by the time I arrived.'
Lightner didn't move until the lieutenant and the squad of Clay's soldiers came up. As they marched ahead of the Volkswagen, the lieutenant said, 'Boys, watch your throats', because of the dogs. Dorothy Lightner watched Clay's soldiers deploy to both sides of the Volkswagen with fixed bayonets. They all moved through the checkpoint into the East and Lightner drove symbolically for a couple of blocks before he came back. She waited and when he did come back they talked.
Lightner: 'We'll try again. For heaven's sake, the whole point of this is to prove our right of entry.'
Dorothy Lightner suggested somebody should go with him 'because it was a lonely thing in the car by yourself and he chose Al Hemsing to go. I felt that history had side-swiped me. I felt we should have been in it together because you can't fight Anglo-Saxon womanhood!'
Hemsing remembered her saying, 'No, it's not fair, I was in the car and it ought to be the way it was', but finally she agreed and Hemsing got in.
The Volkswagen set off again, the _Vopos_ challenged it again and the soldiers came out again. Hemsing says, 'He and I then set off, him driving and me on the passenger side. Members of the Berlin Brigade with rifles and drawn bayonets and hand grenades escorted us. The drill was that we'd go forward and they'd hang behind and we'd inch our way in. There was one _Vopo_ who always gave us a hard time and he did pull back a little bit. Lightner advanced but he let the clutch out too fast and hit this guy in the shin. Lightner and I were hardly as calm as we might be and this guy let out a string of curses. Lightner had German but he didn't have gutter German – as I did – and I was happy about that. I'm glad he didn't get the full brunt because I think he would have run the _Vopo_ down or something. The _Vopo_ was saying things like, "You bastard, I'll make mincemeat out of you" – maybe not those exact words but that sentiment. So we did go in. We went in maybe a couple of hundred feet, maybe a hundred yards, something like that, and then we turned around and came out again. We were emphasising our access to East Berlin.'6
Howard Trivers arrived. He was official liaison with the Soviets; he crossed and he spoke to7 the Soviet political adviser at the checkpoint, Lazarev (presumably Anatoly Ivanovich Lazarev, who was also involved in espionage).8 Lazarev expressed surprise at the incidents, leading to the supposition that Ulbricht had taken this further and started the assertion without Soviet knowledge.
But 'we had established our rights,' Dorothy Lightner says, 'and after the second time they didn't stop Allan.' Lightner went across a third time later on, unescorted and unmolested.
The next day, however, the GDR – no doubt this time with Soviet approval – announced that now uniformed Allied personnel could not go across without showing their identity papers. Hemsing says 'it was tried a number of times again next morning, this time with army officers as civilians doing the same drill, and that's what caused the tank confrontation'.
Clay decided to confront the challenge at Checkpoint Charlie and cleared it with Kennedy personally. This was delicate, Kennedy breathing inside the iron lung. Against that the niceties and formalities of showing ID papers at a distant checkpoint seemed fantastically _picayun_. Clay was thinking parochially, maybe, but he knew if you gave anything to Soviets they'd take that, and then take more, on and on. An evocative name had been coined for it: salami tactics. The difficult question was where and when you said 'no more slicing' and meant it. Clay was not troubled by such indecision. The slicing ceased here and now. The problem, as Hemsing explains, was that Clay had retired: 'Didn't belong in the military chain of command and he didn't give a damn. He was the personal representative of the President.'
On Wednesday 25 October, two American military police officers wearing civilian clothing drove an Opel with official licence plates across. The _Vopos_ flagged them down and they returned, picked up an escort, went back and went through. Ten American tanks arrived near the border. The Opel now made repeated passes at the checkpoint. Each time it was flagged down, each time it sought the escort. The Americans continued to maintain their rights.
On Thursday 26 October, they did this again. The Soviets decided that for them, too, it was enough.
'When they'd stopped Lightner coming through I wasn't there and I didn't know anything about it but I read it and in the subsequent days they began these forays through Checkpoint Charlie up to Unter den Linden. They'd turn round and come back again,' Kellett-Long says. 'Things had got quieter then and we had been to the cinema with our office secretary, Erdmute. Mary and I were driving her home after the cinema and we went along Warschauer Strasse. We crossed over one of the main roads leading into the middle of Berlin and she said to me, "Mr Kellet-Long, there's a soldier standing on that corner." I said, "Don't be silly, you're imagining things." We went on, we dropped her and on the way back I said to Mary, "I think we'd better have a look. Perhaps she wasn't seeing things after all." So we stopped the car on the crossroads and sure enough there was a soldier standing there.
'I went up to ask him what he was doing and at that point he stepped out into the middle of the road to halt the traffic coming both ways. A column of tanks roared by very fast bound for the middle of Berlin and I recognised them as Russian tanks because they had Russian markings on. I watched them go past and, being an agency man [Reuters filing to the world did not have deadlines], I rushed to the office and put out a "snap" saying "Column of Soviet tanks moved into the middle of East Berlin tonight". I sent out this one sentence, got back into the car and drove down to Unter den Linden to the Staatsoper [the opera house, bombed during the war and reopened in 1955] where I saw these tanks disappearing into a courtyard. Again they were shouting to each other in Russian as they were backing the last of them in, so that confirmed to me what and who they were. This was about eleven o'clock at night, because people were actually coming out of the Staatsoper – the opera performance had just finished – as the last of these tanks was being put in. The Pentagon issued a statement saying they were aware of this but they weren't Russian tanks they were East German tanks.'9
The Soviets were fully entitled under the Four Power Agreement to move their forces around their sector, but any movement of GDR tanks would have been a clear and serious violation.
Kellet-Long continues. 'I filed all that and overnight – again – somebody came up from Bonn to say, "We are in trouble here because you say they are Russian and the Pentagon says they are definitely not Russian." I said, "I'm sorry, but I speak Russian, I've seen them and they had Russian markings on them which I recognised and I've heard them shouting to each other in Russian. There is no question that they are Russian." They were quite invisible.'
It was Friday.
27 October 1961 _Gerhard Kayser, 21, was found at 4.25 in the morning caught in the barbed wire near a station in the north. He'd been shot. 'The severely injured man was brutelly dragged 30 metres through the barbed wire and was left lying for an hour before he was taken away.'_ _10_ _He died in hospital._
'About three o'clock that afternoon,' Kellett-Long says, 'the usual foray came through from Checkpoint Charlie, went back and five minutes later there was a rumble and the column of tanks went down Friedrichstrasse and stopped on the border. At this point I was standing at the border and I was very pleased to hear the American Commander on the other side, whoever he was, saying, "By God, they _are_ Russian." The Americans quickly marshalled their tanks and they were facing each other across the checkpoint. I can very much remember the relief on the American side at the fact that they were Russian but it was extremely tense because who the hell knew what was going to happen?
'If they had been East German tanks there would have been a real problem, and that's why I'm always convinced that the man who was running Checkpoint Charlie on the East German side was a Russian although he was in a _Vopo's_ uniform. Before this, I'd been down to the checkpoint and I was told to get the hell out of it – "You can't stay here, you've got to go back." I went to see the Soviet Second Secretary handling press and I said, "They've just told me I can't be near the border." His reaction was astonishing. He went out of the room, returned and said, "Come with me." We got into a Soviet Embassy car and we drove down to the border. I felt a bit embarrassed and I didn't know what was going to happen. At the checkpoint he spoke to this alleged East German who was in charge for a couple of minutes, and then he said, "All right" and he disappeared. That showed me who was in control of that operation. I mean, this was the Russian Second Secretary telling them, "Let this chap stay here." He'd been too far away for me to hear whether they were speaking in Russian. Anyway I was convinced that whatever the inital situation might have been the Russians had taken control of it.'
Hemsing says 'our tanks went up to the line at full speed, there was a great screeching, a bobbing of the guns – and at that point there was a man in his early sixties on the Eastern side. The tanks came eye-to-eye and he bolted to our side and went right past me. I said, "Hold it!" but he kept yelling, "I'm free! I'm free!" Some reporters caught up with him, but that was a street and a half past Checkpoint Charlie because only there did he stop running.'
Richard Smyser, as junior officer at the Mission, explains that 'it was always understood on our side that Western military would not go into the East except on patrols. That was proved during the Checkpoint Charlie confrontation where our tanks went up to the border but not across it. Curiously enough the East Germans were not sure and they put up an anti-tank barrier. Our principle was always that we would not go East just as we told the Russians that their military could not come to the West, and there were all kinds of arguments about that.
'During the confrontation I was in the Operations Centre situation room. Situation rooms have a kind of common appearance: no windows, a lot of tables, a lot of telephones and one large table where all the people sit. It was in the basement. Clay was in it, too. He was a very remarkable man, quite different from what I had expected. I'd been told he was a tough guy – which he was – but he was also a very human, sympathetic man.
'Actually, even as the tanks faced each other, it wasn't a tense time. One of the great mythologies of the Berlin confrontation, which has been permitted to go on and on, was that it was tense. It was tense _only_ until the Russians came, and when the Russians came Clay said, "OK, from now on they're in control. The status quo has been restored." The press wrote some of the most nonsensical stuff about this great confrontation. Garbage, utter garbage. We knew very well, Clay knew, I knew, everybody knew that the Russians had instigated this exercise. They had done it perhaps reluctantly – I was never sure and I'm still not sure – but they were certainly not the people who were pushing it. It was being pushed by Ulbricht and I think that Khrushchev had far different fish to fry. He had many things he wanted in Germany and I don't think sealing off Berlin had been an ambition, but – whatever it was – it became a situation where American and Russian tanks faced each other. Neither was going to shoot: we were certainly not going to start World War Three over some goddamned incident at Checkpoint Charlie and the Russians weren't going to start World War Three there, either.'
(The ordinary population didn't know this. Jacqueline Burkhardt, who'd moved to Berlin in 1949, was now 21. 'The tanks came to the demarcation line – Russian tanks, American tanks – and people who lived nearby went to see what was going on. I was nervous, I didn't know which tank would fire the first shot.')
Smyser accepts that 'it may seem a sophisticated interpretation but I think it is an accurate interpretation to judge that the wall was to consolidate – or save – East Germany and when the Check-point Charlie thing happened we were, in a sense, playing games. Clay figured that once we got our tanks there, "We'll see what happens." Through intelligence – and we had patrols over there, don't forget – we knew that the Russians were sending tanks. One of our patrols came back and told us that there was a platoon of Russian tanks.
'We were fascinated by what the Russians would do and, of course, when the Russian tanks came in it was calm but the wire services were churning out the confrontation. At that point Clay talked to the White House because Kennedy called – he must have had people running around saying, "What's happening?" Kennedy called and got Clay and must have said, "Now what's going on?" Clay replied – I can't remember the exact words – "The situation is calm, don't worry because the Russian tanks have arrived." The sentiment was "Relax Mr President, the status quo has been restored." [George Muller confirms this: 'I was with Clay when he called Rusk on the scrambler at some ungodly hour to report to him that Soviet tanks had drawn up and that was that.']
'Then we did a delicate manoeuvre in withdrawing because the rest was all face saving. When you have a confrontation you always have that problem and what you do is back off a little way. Then if the other guy doesn't do anything you go forward again, but if the other guy says "Ah ha, thank God" and also pulls back a little bit then you understand he's not trying to shoot you.' And you back off a bit more, and he backs off a bit more, and you back off a bit more... 11
In fact, 'Kennedy and Khrushchev quietly defused the situation by agreeing that the Soviet tanks would withdraw first.'12
Frank Cash of the Berlin Task Force in Washington, expands on the context:
Kennedy was far less than the public thought he was in terms of morality, principles, beliefs – in almost every respect. I never was as overwhelmed with Kennedy as the general public, because having dealt with him on Berlin I was somewhat sceptical. I didn't know about the amount of sex – God, he was almost an obsessive with it. His brother and his father, too. It all comes from old Joe [the father] who probably learned it in London!13
It was not evident to me. This was early on, you see, and there were some mild rumours but nothing like what is documented now. It's rather sad because Kennedy was such a hero to so many, and it really makes cynics out of people when their heroes are destroyed.
The nuclear aspect was what Kennedy had to keep in mind and that's what Rusk was constantly reminding him of when we were urging the President to be tough on Berlin, but he was the guy who was going to have to press the button if it came to it. I think the closest thing we had to Cuba was when the tanks pulled up to each side of Checkpoint Charlie. I remember we had a meeting in the White House and the Berlin Task Force were urging that our tanks stand firm and the White House staff were urging that they withdraw. We kept saying, 'If we stand firm the Soviets will back down' and Kennedy listened to this for about half an hour. Finally he got up, shook his head, turned to us and said, 'OK, but you'd better be right.' Then he walked out of the room.
We [Allied personnel] would hold our passports up to the window so the East Germans could look at them to see that we were indeed the guy in the picture but we refused to hand over our travel documents to the East Germans. It almost sounds ridiculous to make a point like that to the President of the United States: that we can't hand our documents over. Once again he'd say, 'That's _picayun_.'
At least the wall removed the threat of instability, and I think the strong stand we took – maintaining freedom in the West – wasn't only realised in East Berlin and East Germany but it went on back through the satellites and ultimately to the Soviet Union. I believe you can draw a direct line between the freedom in West Berlin and the end of the cold war. If we'd let West Berlin go, they'd have consolidated and maybe 1989 wouldn't have happened...'
John Ausland of the State Department gave another view from Washington:
The problem really became Clay. I think it's important to understand that no-one wanted to do anything against the division of the city. And another thing. I'd been watching the division of the city since 1951 so in a way it was only the completion of that ten-year process. Once it was agreed to accept that, our problem became one of dealing with the Germans, not the Russians, you know, because of the Germans' reactions. I'm talking now mainly about the Berliners but to some extent about the West Germans, too, because at that time Brandt and Adenauer were preoccupied with an election campaign and on different sides.
Our problem was how to calm down the West Berliners and to reassure them that we weren't going to desert them, and the [immediate] decision had been, of course, to send the Vice-President, the Battle Group and Clay. All right, now we'd got through the Battle Group problem [a possible challenge on the autobahn] very quickly, the Vice-President went and left, and then we had Clay. He stayed in Berlin for some time and he became the problem. Basically Kennedy was following a very cautious line and Clay and Lightner wanted to be very aggressive. We were caught in between. Clay couldn't afford to blame the President so he blamed us, and you have kind of a merry-go-round going on. Kennedy was warned he would have problems, although whether we had problems in the long run was not important, I think, in terms of reassuring the West Berliners. Clay being there and the tank incident were very reassuring.
The only thing that became a reality was the tank confrontation and that was not a thing that was welcomed in Washington. Everyone was very glad when it was over, and it was quite clear there was some communication between Kennedy and Khrushchev that ended it. No, we didn't have a hot line but we had ways of communicating. The useful thing, from my point of view, that Clay did was quiet down the Steinstücken problem that had been troubling us for years. He just took a couple of soldiers, put them on a helicopter, flew them out, put them there and the whole problem went away. I welcomed that one. So, in retrospect, having Clay there was very good but at the time it was a pain in the neck because we were caught between this cautious President and this aggressive bunch in Berlin.14
The Clay 'problem' became a strange thing to Geoffrey McDermott, the British minister in Berlin. Alec Douglas-Home, the Foreign Secretary, came on an official visit and only after it did Clay return, from a visit of his own to Washington. 'Others beside myself could not help wondering whether he had deliberately avoided Lord Home's visit. In my curiosity I asked to be allowed to call on him to exchange views. I was snubbed. From then onwards General Clay mingled very little with his Allies, even socially, and I have reason to believe that he also became increasingly difficult towards his compatriots. Reconstructing this curious episode, I can only conclude that Clay wanted to press on with the forward policy he had so clearly pursued in the October tank confrontation' – while, presumably, Douglas-Home would have advised extreme circumspection and caution.'15
Kurt Behrendt, the resident of Steinstücken, saw all this from ground level, as it were. 'When they started to build the wall, the Americans were passive but General Clay gave the order after the day he visited us that a company of American soldiers would be deployed here in tents. At this time we had the conflict at Checkpoint Charlie with the tanks and nobody wanted to risk a similar sort of conflict in Steinstücken. The order was that after two or three days the soldiers should be withdrawn but, three days later, Clay gave the order to deploy three American soldiers in order to demonstrate that Steinstücken was in their sector. These soldiers rotated after two or three days by helicopter and we called it the "little airbridge". [The 1948–9 airlift had been known as the 'airbridge'.] Except for residents, it was not allowed for anybody – even Americans, even politicians – to come here using the path [through the GDR] and so they had to come by helicopter. Even our relatives weren't allowed to use the path.'
31 October 1961 _An unknown man was fatally shot at the wall._
17 November 1961 _Lothar Lehmann, 20, tried to swim a lake to the west of West Berlin and was shot. His mother was told he died 'through an accident in the Army'._
After the tank confrontation, Berlin quietened into a sort of muted, choreographed dance. Frank Trinka of the Mission describes how 'we'd go to Karlshorst [the Eastern suburb where the Soviet headquarters was] to protest and they'd say that whatever had happened was in line with their policy and the GDR was a sovereign state. It was a Mexican stand-off, as we say. All people hold their positions and nothing happens. I was at Karlshorst frequently with Howard Trivers – he was the man who generally went over, the senior officer – to make the protests or to receive them. I accompanied him. It was a Soviet headquarters and there were troops everywhere so I'd say it was more a military headquarters. You couldn't compare it to the US Mission in Clayallee. On their side it was much more evident that the military were in charge.
'You had to go through a check-in procedure although there had already been prior contact, so they knew you were coming. We'd phone and they'd say, "Yes, we expect you at such and such a time." We'd go there and we'd register formal protests. The effect was probably that they'd tell the East Germans to back off and not be so aggressive, but it would not change their policy to us. We still maintained that freedom of access in Berlin encompassed all of Berlin and all of the civilian population. We generally had an interpreter along, and the way we did it was either English-to-Russian or German-to-Russian. Some of the Russians spoke German so German was fairly frequently used. Most of our people were quite competent in German, Smyser too.'16 (Trinka, incidentally, was transferred to Prague towards the end of 1961 and so his Berlin story ended then.)
And still they came. One mid-October morning at around 5.00, nine men in a lorry burst through two rolls of barbed wire at Kleinmachnow, where radio reporter Peter Schultz had watched in consternation and disbelief as the wire had been laid on the night the wall went up, but were halted by the third roll of it. They abandoned the beached lorry and, under a hail of fire, sprinted.
In early November a West Berliner smuggled his 19-year-old Eastern fiancée out in the boot of his car. When the customs officers at the checkpoint started to search it he accelerated away, breaking the wooden barrier, and although six shots hit the car they made it. Four days later a 28-year-old engineer and his wife duplicated this in a highly original and perceptive way. The date was the anniversary of the October Revolution (but falling in November in accordance with the Russian calendar). A Soviet military convoy moved up towards Checkpoint Charlie to cross and proceed to the war memorial in the West. The engineer bought a wreath and tied it to the roof of his car, then simply joined the convoy and was waved through with it. Two days afterwards, an unnamed 33-year-old went to a theatrical costume shop in East Berlin and hired a uniform which resembled that of an American soldier. He walked across Check-point Charlie. Four days after that, a car with two men and three women in it broke through near Bernauer Strasse. The Border Guards fired what one report suggests was about a hundred shots at it but the car had been reinforced with steel plating and concrete.17
17 November 1961 _An unknown person died at the wall, possibly drowning._
20 November 1961 _An unknown person, thought to be a man, may have been fatally shot and drowned._
That first Christmas was the bitterest, with families separated by a street or two and without any contact at all. The wooden screens at the end of Bernauer Strasse, as an example, had taken care of that. How those in the GDR government could justify such conduct to themselves and their families – at the same time as they were proclaiming to the world that they had established the first and only humane society the world had ever known – remains a complete mystery. Possibly, according to the size of their ambitions for humanity, they were prepared to sacrifice individuals for the collective whole. It was the most unChristian of ethics but then these were the people who would blow up the Church of Reconciliation and level it so the Border Guards in the watchtowers would have an unobstructed view of the death strip.
To liberate humanity.
The thought of the Christmas division seems to have kept the stampede moving and it produced one of the most extraordinary escapes.
On an early December evening towards 9.00, eight men, ten women and seven children crossed on a complete train of engine and eight carriages. The driver and fireman hatched the plan in conjunction with someone who threw a switch, diverting the train from the Eastern line to one leading to the West. The train thundered over the barbed wire across the track just beyond Falkensee and came to a halt in a field. Legend insists that while the escapees hugged in delight, some very startled passengers – who had known nothing of the attempt – sat bemused, and the guard, also unaware, marched up and said it was forbidden to stop at anywhere but a station... .
9 December 1961 _Dieter Wohlfahrt, a 21-year-old Austrian, and two other Westerners, cut through the wire at Staaken to help a relative get across at 7.15 in the evening. Wohlfahrt was hit by automatic fire and left for an hour with fatal injuries. The other two regained the West._
Wohlfahrt, a student at West Berlin's Technical University, had evidently grown impatient with what was known as the 'tour system' of escapes. It involved forged passports, go-betweens and involved many obvious risks. Wohlfahrt decided on a 'commando raid'18 instead but was cut down by gunfire as soon as he had cut the wire. He bled to death.
10 December 1961 _Ingo Krüger, 21, was seen swimming the Spree in a diving suit and oxygen mask. A police boat_ _tried to haul him up with grappling hooks and he was taken dead from the water._
Adam Kellett-Long would remember this Christmas. 'I'd been told about a terribly moving scene. At one point you could see over the wall and there were about a couple of hundred East Berliners waving white handkerchiefs to relatives on the other side. By Christmas that was the only point where people could still see each other. Shortly after Christmas – I hope not because I wrote about it, but I'm afraid probably because I did – they heightened the wall so that that gap was closed.
'Erdmute came in to the office one day and said, "I've got some oranges!" – a tremendous luxury. What had happened, the tram had been going along the street and the driver had spotted a shop selling oranges and stopped the tram. Everybody got off and rushed into the shop, stormed the shop. Oranges had been brought in to the shops for Christmas but not too many oranges and long queues formed.'
1 January 1962 _An unknown person died at the wall._
In January the tunnellers began. In the north, twenty-eight people aged between 8 and 71 emerged in the West at 2.00 one morning from a tunnel they had dug themselves.
12 January 1962 _Barbara Hildegard Blass, who had celebrated her nineteenth birthday on 12 December, died at the wall in unknown circumstances._
19 February 1962 _Doris Schmiel, a 20-year-old dressmaker, and another woman were among five people trying to escape in the north. Three rounds of machine-gun fire raked them and Schmiel fell, fatally wounded in the stomach._
On 22 February Robert Kennedy and his brother Edward visited Berlin. Robert made a short statement in German, 'reading it slowly and with some difficulty',19 and he reiterated the United States' determination to keep West Berlin free. He went in a motorcade to the wall at Potsdamer Platz and mounted a wooden observation platform to look over. 'It is even more shocking, even more shameful, than I had expected.' Edward was on a private visit and came in by ordinary commercial flight from Paris. It was his thirtieth birthday and city officials gave him a 'tree cake', evidently a Berlin delicacy. Edward visited the East, annoying John F. by showing his passport to an East German. He went walkabout, saw a queue outside a shop and spoke to a middle-aged woman who said she was waiting to get apples. He wondered if she did that every day and she replied, 'No, they don't have apples every day.'
14 March 1962 _Otto Müller, 55, was fatally shot at the Spree, although the circumstances are not clear._
27 March 1962 _Heinz Jercha, a 28-year-old West Berlin butcher, was shot by a Border Guard while helping people to escape through a tunnel he had dug. He died on the way to hospital from a haemorrhage._
3 April 1962 _An unnamed Border Guard tried to cross near the Dreilinden checkpoint at 1.45 in the afternoon. A police dog intercepted him and he was shot by automatic fire._
11 April 1962 _Philipp Held, 20, drowned in the Spree. His mother was informed by the Public Prosecutor in a letter on 30 April, which said only that her son had had 'a fatal accident'. She was told the body would be released for burial and she could make whatever arrangements she wished. A telephone call from the Public Prosecutor the same day, however, said Held had already been cremated._ _20_
_18 April 1962 Klaus Brueske, 23, drove a lorry which broke through the Heinrich-Heine-Strasse checkpoint. Around 20 shots were fired and Brueske was hit in the back of the neck. Two others with him were injured. The lorry got through._
18 April 1962 _At 7.30, two non-commissioned GDR officers tried to get across in the Potsdam area, taking their rifles and ammunition. One was Peter Böhme, 19. He and a Border Guard, Jörgen Schmidtchen, exchanged fire and both were killed._
29 April 1962 _Horst Frank, 20, and one other cut through the wire at a railway bridge 15 minutes after midnight. Horst was fatally shot and his body removed by the East Berlin Fire Brigade. The other got across._
29 April 1962 _Ernest Graupner, 49, died at the wall._
In May, five women and seven men aged between 19 and 81 emerged in the West from a 40-metre tunnel they had dug. The 81-year-old was called Max Thomas, a gardener who had a plot near the wall. In January, West Berlin officials had formally announced that twenty-eight people had escaped by cutting through the wire but an American news agency talked to them and discovered they'd come out through a tunnel. It ran from a basement in the East to a garden in the Western district of Frohnau. This was the first major escape by tunnel and, as it happened, Thomas had known about it. He lived four doors away from the house whose basement had been used. He'd wanted to go, too, but the organisers told him he and his friends were too old and 'your wives are too fat'.
Thomas resolved to have a tunnel of his own but he was too old to do the necessary digging. He teamed up with a 57-year-old truck driver and two 70-year-olds who moved 4,000 buckets of earth in sixteen days, starting from a small, wooden henhouse. Thomas rigged up an extension cable from his kitchen to light the tunnel and he used that as a warning because whenever a Border Guard patrol came by he switched it off. That alerted the diggers who stopped work until the patrol had gone by. The Thomas tunnel was extremely spacious at 5 feet 6 inches high (1.7 metres) because, as he said, 'We wanted to take our wives to freedom in comfort – and upright.'21
General Clay was withdrawn that May. Geoffrey McDermott wrote that after Lord Home's visit when Clay might have deliberately gone to Washington, 'from then on, when we met socially he was polite but certainly not warm' towards McDermott and his wife Elizabeth. 'No doubt he had been given a new brief in Washington. I did not let this curious behaviour lessen my respect for him and his work for Berlin; and both Elizabeth and I continued to enjoy the Clays' company.' Of Clay's withdrawal, McDermott added: 'I was told that he received orders to leave Berlin from the President at almost the same moment as his withdrawal was made public in Washington. He was furious. At any rate, he departed without any official Allied farewells. He saved some face with the announcement that he would continue to act as a special adviser to the President on Berlin.'22
23 May 1962 _At 5.25 in the afternoon a 'male person' escaped across the Invaliden Cemetery butting on to the Humboldt Canal – one shot was fired as he neared the wall, another as he clambered over it. He plunged into the canal beyond the cemetery but other Border Guards could see him and opened fire. He swam on and reached a 'sort of platform' leading to stone steps. The Border Guards continued to fire at him and cut him down. He lay on the platform for some three minutes while West Berlin policemen fired at the Border Guards to make them take cover; they hit and killed one of them, a Corporal Peter Göring. A West Berlin policeman went down the steps with a ladder to act as a stretcher, secured the 'male person' to it and he was lifted up._
The official GDR report, typed out on probably an old typewriter and dated the same day, is headed REPORT and sub-headed _betr_., which is an abbreviation for 'Concerning': Breaking through the border with use of firearms.
It covers the facts as they are set out above but what it does not say is far more revealing. Imagine looking up the canal. To the right was a deep stone embankment which was angled into the water and above it the old cemetery wall about the height of a man. A further wall had been built on top of that so that, overall, it would be 18 to 20 feet (6 metres). The 'male person' had somehow to scale that, scramble down the embankment into the still water and swim some 50 feet (15 metres) to the Western bank. Here the embankment, made of large stone blocks, was also deep – perhaps 12 feet (3.5 metres) – and vertical. The platform was a tiny 'jetty' with steps up from it and a metal handrail set into the stone.
The central drama of Berlin: Checkpoint Charlie and Friedrichstrasse station, conduits between the two halves for 28 years. The station mirrored the drama exactly – the line running underground which Easterners didn't know about and the overground line from the West which terminated at the station. Its continuation was a domestic Eastern line.
This was the view from, and of, the West.23 'The 15-year-old schoolboy T. from Erfurt (Soviet Zone) tried to escape to West Berlin by swimming through... in the Tiergarten district. Soviet zonal Border Guards shot at him after he had already reached the Western side of the canal. In order to save the severely wounded boy, West Berlin police were obliged to return the fire. One Soviet zonal policeman was fatally hit and another wounded.'
The question of whether Germans would shoot Germans and Berliners shoot Berliners had now been conclusively answered.
There is something the West Berlin report did not mention. The border itself was on the _Western_ bank so that, in the horror of Berlin's pedantry, the Border Guards were entitled to fire at the 15-year-old up to that. The Border Guards' own maps leave no room for doubt, and neither do anybody else's. It means that the photographer who so assiduously took pictures of where the GDR bullets chipped the stone blocks by the 'platform' and staircase was not freezing images of transgression – because that was short of the line.
There is a curious footnote to this, too, but it has been set in its context after another death at the wall three months away.
27 May 1962 _Lutz Haberlandt, 24, was shot dead at 4.40 in the afternoon, trying to cross a bridge beside the Charité Hospital in the city centre._
5 June 1962 _Axel Hannemann, 17, tried to swim the Spree towards the Reichstag at 5.15 in the afternoon. A Border Guard fired six shots at him from the bank. Hannemann's body was recovered some hours later._
10 June 1962 _Wolfgang Glöde, 13, died at the wall._
11 June 1962 _Erna Kelm, 54, drowned trying to swim the River Havel. She had been a nurse and, tucked into a swimming belt, had concealed 'important papers'._ 24
Two days later a tunnel collapsed and buried an escapee, although by then he had reached West Berlin. Three fellow escapees dug him out with the soup spoons and small shovels they'd used to dig the tunnel.
That same month a West Berliner, Peter Scholz, and friends dug a 60 foot (18 metres) tunnel some 9 feet (2.7 metres) underground to get his fiancée, her 4-month-old baby and nine others out. The tunnel stretched from a restaurant basement in West Berlin to a house in the East and was narrow (2 feet/ 0.6 metres). 'Progress was painful. Not only was there continual danger of cave-in because of the weight of the earth overhead, but the area was continually patrolled. The slightest noise might have been detected. To keep the infant from crying or being frightened and helplessly betraying them she was given a light sleeping pill, tucked in a metal washbasin and pulled to freedom by rope.'25 Scholz described this as 'the most difficult part' although, because the tunnel was so cramped, it had all been difficult and crawling it inch by inch took nearly three hours.
18 June 1962 _Reinhold Huhn, a 20-year-old corporal, was shot dead near Checkpoint Charlie._
In the East, Huhn would be glorified for his sacrifice by having a street named after him, but the contemporary views of the incident from both sides are revealing. To understand them you need to look at the geography and geometry. Checkpoint Charlie was, in fact, on an intersection, with the main thoroughfare, Friedrichstrasse, running through it north to south and a side street, Zimmerstrasse, from east to west. Where Huhn died, Zimmerstrasse consisted of some old buildings which had limped through the bombing in the war, then the road, then the wall. The far side of the wall was the West: a cleared area from Checkpoint Charlie some 150 metres to a skyscraper owned by the publishing house of Axel Springer, a fierce anti-communist who had positioned the building so the Easterners could see both it and the neon news bulletins on a big gantry on it.
At 2.55 in the afternoon in the East, Huhn and another corporal were on duty and noticed two civilians on the roof of the Springer building, watching them. Were they lookouts? Huhn also spotted a couple of workers repairing the roof of a Western barracks but didn't report whether they, too, seemed suspicious. Then a man emerged from one of the old buildings on Zimmerstrasse – No. 56 – and, evidently, walked along to the checkpoint and crossed. He must have been a Westerner. Later 'it was noted that several people came together and were talking', so something was going on. Huhn and his comrade went to investigate but, as they approached, these people quickly dispersed.
At 6.45 that evening the 'same male person' had re-crossed. He, two women and a child were making their way to No. 56. Huhn 'went towards' them and 'asked them to stop. They had to be asked several times before they did. The women and the child were about 10 metres away. The male person put his hand into an inner pocket and soon after a shot was fired. Comrade Huhn went down.' The other Border Guard drew his pistol and fired ten times at the 'male person' but in the confusion he, the women and the child 'disappeared' into the house. A subsequent search of the cellars revealed a tunnel going towards West Berlin. 'Thus it can be assumed that they had gone through it.' Huhn died in hospital.26
The 'male person' was called Rudolf Mueller, a young baker who'd gone to the West before the wall was built and spent weeks digging the 24-yard (22-metre) tunnel from under the Springer building in order to get his family out. He told West Berlin police that Huhn asked to see his documents and, as Huhn examined a bag, Mueller hit him so hard that he knocked him down. The other Border Guard fired – but hit Huhn. Mueller and his family then escaped through the tunnel.
At a news conference in the Springer building, however, Mueller was asked how many shots he'd fired (literally, how many times he had pulled the trigger) and reportedly said 'once. The man fell down immediately.' He had surrendered his weapon to the West Berlin police. The official West Berlin position was that Huhn had been shot by the other Guard.
The street was named Reinhold-Huhn-Strasse on 15 July 1966. It would be returned to its original name, Schützenstrasse, on 1 December 1991. Mueller was tried over Huhn in the normal way – but in December 1998. He was found guilty of manslaughter and given a one-year suspended sentence.
22 June 1962 _An unnamed man was shot down some 20 metres from the wall in the Neukölln district south of West Berlin. He was 'left to lie for about an hour beside the_ _roadway... before he was carried away, obviously dead, by the East Berlin Fire Brigade.'_ 27
28 June 1962 _Siegfried Noffke, a 23-year-old painter, was shot at midday at a tunnel he and two others had dug from West Berlin to try and get relatives out. A 23-year-old Eastern mechanic was shot and arrested._
In fact, the relatives were his wife and child. He selected a place in the Treptow district, where the streets and houses interlocked closely. He and two helpers dug about 30 metres, not knowing that simultaneously another tunnel was being dug two houses away. Together, the digging caused the land to subside and the People's Police waited. Noffke was shot dead and the helpers arrested.
The GDR logic, and self-confidence, reached out to the Church of Reconciliation in Bernauer Strasse in late July. They 'expropriated' the church itself, its property, a training centre for parish sisters and the community centre. Even given that the GDR was officially atheist and had an uneasy relationship with its religious bodies, to seize a church was in itself a statement: we can do whatever we deem necessary and we do not have Western inhibitions about a church being the ultimate untouchable sanctuary. The GDR was breaking Lenin's eggs.
To broaden the context further, the seizure (which likely would have brought a Western government down) was accepted in the East in silence. And the church would eventually be blown up, as we have seen, in order that Border Guards would have an unobstructed view up and down the death strip, the more efficiently to shoot people; and that was the ultimate insult to the ultimate sanctuary. That was also accepted in silence. The distance between East and West was growing and the Church of Reconciliation silently marked that fact, less than a year after the wall went up.
8 July 1962 _Herbert Mende, 23, died at the wall._
29 July 1962 _An unnamed man tried to get across the wire to the Western enclave of Eiskeller. He was some 3 metres from the wire when he was fatally shot._
Checkpoint Charlie remained a macho place where the face-off over Hagen Koch's line was a daily ritual. Bill Bentz, a second lieutenant in an infantry division, was in charge of it on the American side:
It was a very interesting assignment because you had all kinds of orders and instructions to carry out and one of the most interesting was to make sure that several very, very senior Soviet officers were not permitted to enter West Berlin. In the end, in fact, their pictures were posted inside the Checkpoint. I think it's kind of humorous but it also points out the contentious nature of the situation that somewhere in the time-frame before I got there a major incident had taken place where an American official attempted to go over into the Eastern Sector of the city and the Eastern authorities demanded to see this and that identification [Lightner, then the tank confrontation]. This was against the rules, it caused a tremendous problem and thereafter for a long period the instructions to the officer in charge were that if two certain Soviet officers should try to come into West Berlin they were not to be permitted until you had called American HQ and received certain instructions.
OK, that was the cause of that. The way we worked it was, up on the top floor – the very top floor of the building on the left-hand side above the American HQ – there was an observation post where an American military policeman was posted. He watched Friedrichstrasse and the Eastern checkpoint, and he had communications downstairs to our office and out to the hut [in the middle of the road]. If he saw a Soviet car or convoy coming, he'd ring an alarm, I would go out onto the street and the military police would pull a military police vehicle across Friedrichstrasse, blocking the way. It was my duty to observe who was in the car and make sure these two officers weren't. And if they weren't, then we would let the vehicle pass. That was the routine, that was the way it was, because uniformed people had every right to go through without identification. That is what caused this initial problem – the Soviets and East Germans demanded to see the identifications.
Here, too, was an acceptance of normality hewn from anomaly and abnormality. What had been a first step towards the nuclear escalator so few months before had now settled to an awkward but functioning _modus vivendi._ The possibility of friction could itself be hewn, however, from the delicate balance of regulations and understandings at the checkpoint.
Bentz continues:
I was on duty there on May Day, which was the big Soviet day for parades and also a day when they would come over to their War Memorial. Every year at that time they would come across and have their ceremony. This one year the alarm rings and it's the young military policeman upstairs. He says, 'Lieutenant Bentz, there is a 17-vehicle Soviet convoy coming down Friedrichstrasse.' So I go out, we pull the car across, these big limousines are coming down towards us and they're halted. I am checking each of them, I peer into this one car and there was this one guy who really looked like one of the guys in the photographs. I said, 'Just a minute please', actually went back inside, got the photographs and went out again. I looked at the photos and finally decided it was neither of the two officers. I waved them and the whole convoy went through.
It wasn't more than two minutes after that happened that the hot line from the American Headquarters [in Clayallee] rang. It was the senior officer of the Emergency Operations Center and he said, 'Lieutenant, you know that vehicle that you have just let through the Checkpoint?' I thought I'd got it wrong and I was going to have to throw my bars [insignia] down on the table, career over. 'Yes, sir,' I said. 'You know that convoy was observed doing 50 kph in a 30 kph zone,' he said. 'When it comes back through the Checkpoint you are to follow the instructions' – covering such and such. After the ceremony at the War Memorial, of course, they came back, I went out there and we stopped the lead vehicle. In my best English I said, 'You were observed doing 50 _clicks_ [kilometres] in a 35 _click_ zone – dangerous to the children and residents of West Berlin. Any future violations of this sort would be dealt with in an appropriate manner _di da di da_ _di da_. Do you understand this?' The guy sitting in the back seat said, 'No' in his _best_ English. I said, 'All right.' I went and got a Russian interpreter who went through the same thing in Russian. Then we allowed them to pass back into East Berlin.
Such was the delicate balance of regulations and understandings.
Nor was the daily ritual only that. 'I witnessed', Bentz adds, 'several successful and of course unsuccessful escapes across the wall near the Checkpoint. Very nasty business. I was always just amazed by the determination of the East Berliners – I just could not believe it. They'd escape by whatever route was possible, they'd get down off the wall and make a dash across the death strip, try and get up the outer wall against gun fire – across they'd come.'28
The first anniversary of the wall provoked angry demonstrations in West Berlin and Soviet vehicles were attacked. Then, four days later...
17 August 1962 _Peter Fechter, 18, and a friend hid in one of those limping old buildings on Zimmerstrasse roughly midway between Checkpoint Charlie and the Springer building. From it at 2.15 in the afternoon the teenagers ran across Zimmerstrasse to the wall. The friend got over but Fechter was hit in the lung or pelvis, or both, and fell back onto the Eastern side. It may be he was betrayed before he made his run, was surveyed and simply picked off._
Perhaps because it happened so close to the checkpoint – 50 metres – that this became a defining moment, captured by camera. Police and photographers and American soldiers were soon at the scene. Motorists who stopped could stand on their bonnets to watch. What they all saw was an image still powerful enough to shock in its brutality. Possibly it was because Fechter took a long time to die, and was doing so both in public and in circumstances which still remain difficult to set down without feeling rage. Maybe it was because he cried out for help and the cries became echoes for a whole city in all its agonies.
The shots came from a window which was an observation post.
Peter Fechter, a slim kid in cheap, dark clothing – his jacket thinly striped – lay with one arm folded under his body like a broken doll, a small pool of blood spreading by his forehead and, minute by minute, bleeding to death nestled against the other side of the breezeblocks. He was left like that from 2.15 to 3.10 p.m.
Dennis L. Bark, a 20-year-old preparing for a career in the theatre at Stanford University in California, was visiting Europe for the first time. He hitch-hiked to Berlin because 'my 18-year-old brother Jared was going to be there. A truck took me into the Kurfürstendamm – a huge German truck – and the guy drove right down the Kurfürsten-damm, dropped me and said, "You'll probably be able to find a little place down there that you can afford." And I did. One day my brother and I decided we wanted to go to East Berlin so we went through Checkpoint Charlie and we decided the best way to see that half of the city would be to get on a bus. So we got on an East Berlin bus and we rode to the end of the line. We figured: let's see what that's like.
'It was in the north and we got out. There was nothing there except a few buildings. There was a big empty lot and we started to walk across it to see what was on the other side and all of a sudden out of the ground popped a soldier. He just stood up and we hadn't seen him. He pointed his gun at us and he said, "Halt!" He asked what we were doing and he was speaking German. I didn't speak any German but my brother did. My brother told him what we were doing. The fellow was probably 20 years old himself. He said, "Turn around. You'd better get out of here." It must have been a border area although of course we didn't know it.
The bitter afternoon. Peter Fechter bleeds to death against the wall and in full view of the world.
'We turned around and went back to the bus stop and waited about 45 minutes for the bus and we finally got on it and made our way back to Checkpoint Charlie. We went through the East Berlin customs house and they looked at our passports. Then you walked out of the door to go towards Checkpoint Charlie and it was a huge area like a football field. They didn't have any barriers then or very few. So you just walked across this area, then across the road and you were at Checkpoint Charlie in the West. You walked diagonally across this area. It was a long walk. We got halfway out there – in the middle – and all of a sudden we heard these loud noises. _Bang-bang-bang-bang._ It didn't occur to us that it was a gun – I had never heard a gun before.'
It was 2.15 p.m.
'We looked up and as we looked up a Border Guard came up to us, pointed his gun at us and said, "Halt!" We looked and saw this body sort of flipping over the wall some distance away – it was so fast I wouldn't have recognised him if I'd seen him again. And then we saw a second man. He may have had his fingers up on top of the wall – I can't remember – but they hit him in the back. He fell down and they didn't go near him.
'There was a lot of hollering and a lot of shots – again I can't remember how many – but a lot, more than two or three. My brother in the meantime asked the Guard what was going on and he said, "Nothing, just stay here." He was as scared as we were. He knew they'd fired but he didn't know at what. I was never so frustrated in all my life because I couldn't speak German, and I kept asking my brother questions and asking what the Guard was saying. The Guard didn't say a hell of a lot but he kept us there. We stood for a long time, fifteen minutes maybe, and that is a long time to be standing facing a man with his gun out and pointing at you. We didn't move. We just didn't move. He finally said, "OK, you can go."'
It was about 2.30. Fechter had lain, bleeding, for those 15 minutes and would lie another 30.
'We walked towards Checkpoint Charlie and the white line. We tried to see what was happening but we couldn't because so many uniformed people were down there. I'll never forget putting my foot over that white line because I had a feeling in my stomach of such relief. I mean I can still _see_ this feeling [2001], so to speak. I can still see myself as I put that foot over the line. We then asked the American soldiers what had happened and they said, "Somebody tried to escape and they shot him." We walked down the Western side of the wall and by that time some West Berliners were gathering. There was a telephone pole and it had a post about three feet high next to it. I climbed up on that. We were all trying to look over the wall but we couldn't see because he was lying against it on the Eastern side.
'By that time the West Berlin police had come and some American soldiers and the American soldiers threw over, among other things, First Aid kits. I don't remember whether it was West Berlin policemen or American soldiers or both, but they threw over these smoke bombs and some of them were like firecrackers. They made huge noises. The West Berliners got angrier and angrier and started shouting. You could hear Fechter yelling, "Help me, then." What the translation actually means is "Why don't you help me?" My brother could hear it and he translated it. The wall was so tall that you could only see the tops of things. We knew when he was carried away but we couldn't see it done. Nobody could except the photographers who I think had brought ladders with them and that's how they got pictures, but none of the people standing there could see that. And it changed my life, changed my life 100 per cent.'
Al Hemsing 'did, alas, see Peter Fechter. I'd been notified that someone had been shot so I hot-footed it down. I got onto the observation platform and I heard him shouting "Help me!" but not very loudly. As I recall, I had to leave before his actual death.'
The West Berlin police clambered up the crude breeze-block wall and held on to the V-shaped metal rods which secured the strands of barbed wire along the top. A medical kit was thrown over towards where Fechter lay but it broke open and, anyway, he was too weak. All he could do was lie on his back, arms folded across his chest as a priest might have arranged them, then somehow turn onto his side as if trying to regain the foetal position. His left shoe was half off and you could see the white of his ankle.
The crowd chanted 'Murderers!' towards the Border Guards. Finally, four uniformed men, one clearly a People's Policeman, went to Fechter under a protective blanket of ten tear-gas shells. The West Berlin police in turn fired ten tear-gas shells back. The four uniformed men, working in this pall, lifted the body and carried it – limp, sprawled, awkward – across the road to the concrete posts and the wire between them and manoeuvred him over it at a point where the wire was waist high. As they did so, a photographer fixed that image: the men – three helmeted, two with machine-guns – hold Fechter high, and they have his arms splayed, and there is dried blood on his hands and wrists.
Two of these uniformed men trotted away along a connecting street into the East with Fechter hanging between them. Here a crowd had gathered, too, some standing in the long grass of bomb sites, some in a mute cluster further away. Fechter was put into the back of a vehicle and never seen again.
An American lieutenant, who'd been implored to help, replied that there was nothing he could do. His helplessness was governed by the Four Power Agreement, of course, as well as all the practical and logistical difficulties of doing anything, never mind the risk of escalation. His words were evidently translated as 'It's not my problem' which, when taken up by a West Berlin tabloid newspaper, provoked outrage.
The RIAS reporter Peter Schultz was there. 'What can I tell you about that? There was a young man lying on the other side of the wall who was crying out and suddenly he stopped. The police in the West tried to push the people back. The American military police were at the wall but they couldn't do anything and when he was dead a soldier carried him away. I can't answer the question why the Border Guards left him to die. I don't know. I don't know, I don't know. Maybe they were afraid because the distance between them and the US military police would have been 1 metre. There was an American military police officer who came some weeks after, and he said that if it had happened while he was there he would have helped Fechter. Some weeks after that he proved what he said, because there was a wounded person at Checkpoint Charlie and he did help him to the Western side. The man was not as badly wounded as Peter Fechter had been.'
The crowd on the Western side hung around until dusk on this chilled August day and then dispersed.
The GDR Ministry of the Interior's report, compiled the same day, distilled the incident into the language of bureaucracy, but even that could not dehumanise it completely:
On 17 August 1962, at 2.15 p.m., a violation of the border was effected from the capital of the DDR towards West Berlin by a male person at Zimmerstrasse/Charlottenstrasse, section l, 4th company, Guard 3, 4th Border section. In the course of this, a second border violator was taken to the VP [people's Police] hospital, having been seriously injured, where he passed away at around 3.15 p.m.
At about 2.15 p.m., Sergeant Friedrich and Lance-Corporal Schreiber noticed a male person climbing over the first fence at the corner between Zimmerstrasse and Markgrafenstrasse and making his way in the direction of the wall. At a distance of around 2–3 metres a second male person was following him. Sergeant Friedrich immediately opened fire at the two border violators. The distance between Border Guard to the border violators was about 50 metres. In total, the Sergeant fired 17 times, the guard Corporal 7 times.
At the neighbouring guard tower, Sergeant-Major Schönert and Lance-Corporal Buske heard the shots, and also saw the violators in front of the border protection, and opened fire. Sergeant-Major Schönert and Lance-Corporal Buske fired 11 times in total.
At this time, the lst person had already climbed the wall and was at such an angle to the Border Guards that further shooting would have targeted Western territory. The border violator managed to get over the wall.
The second person was hit and collapsed just in front of the wall. Immediately, Sergeant Friedrich and Lance-Corporal Schreiber took position, observed enemy territory. [Western] police and civilians carried a ladder to the wall to recover the injured border violator, presumably knowing that they would violate DDR territory in the course of their actions.
The report describes police and public gathering on the Western side, and tear-gas shells being thrown 'into DDR territory'. It continues:
In order to recover the injured border violator by our own means, a 'curtain of fog' was laid from the ruined house at Zimmerstrasse 72–74, under the protection of which Staff Sergeant Wursel and Corporal Lindenlaub were able to recover the injured border violator. They handed him over to a police car, which took him away to the VP hospital.
The report recounts 'provocation' from the West, and measures taken against it following the regulations.
After the recovery of the injured border violator, the normal situation was restored, water cannons were removed, special security guards were positioned and a Sergeant (for special observation) called. In addition a patrol to protect the hinterland was established.
Around 3.l5 p.m., the Brigade's headquarters was informed that the border violator had passed away at the VP hospital. The identity of the border violator is not known as no papers were found. The police have started an investigation.
The report stated that 'two citizens' who witnessed the incident were arrested. Evidently they had 'identified' one of the Border Guards who fired. 'It was reported that immediately after the event, one young male and one young female stayed at the scene of the crime [the crime, of course, was trying to escape] and that the female made the following remark to the male: "This is the Guard who fired, we have to take photographic evidence." Based on this remark, both people were arrested.'
It was noted that a wooden cross was erected in enemy territory, 4 metres away from the wall, opposite the scene of the 'crime', and flowers were laid.
The report came to the following conclusions:
l. The Border Guards' actions were correct, effective and determined. The use of weapons was justified. However, the question remains if the same effect would have been achieved if the Guards had fired single, targeted shots at the border violators.
2. Shots at the 2nd border violator would have been against the regulation/order of the Interior Ministry as they would have hit West Berlin territory around the Springerkonzern.
3. Establishing a 'curtain of fog' was a necessary measure, through which visibility for the enemy was obstructed.
Under 'Measures taken' was written: 'On my order, Sergeant Friedrich and Corporal Schreiber, Major Wursel and Lance-Corporal Buske received an award... . The incident won't be mentioned in the report in order to keep the names of the comrades secret.' It was signed by Commander Tschitschke.
Nobody knows who fired the fatal shots but, as it would seem, it was either Friedrich or Schreiber. Neither was responsible for allowing Fechter to bleed to death. Fechter had a sister, Ruth, and Friedrich would make a point of apologising to her – in March 1997.
Hagen Koch, former Stasi employee, now [in 2001] running an archive on the wall, provides a different and slightly conspiratorial context. 'On 23 May 1962 a Border Guard, Peter Göring, was shot by a Western policeman and the GDR made a hero out of Göring. The story is that although the Border Guards were ordered to stop shooting this Peter Göring continued. Only when the West tried to rescue the refugee did they – the West – shoot, and with the intention of rescue. But with Göring still shooting, they shot him.' This seems clear, no mystery at all. However 'all this was a kind of manipulation of what happened. Both sides _had_ to have a reason why a Border Guard was shot by a policeman from the West.' Simply put, the East was satisfied that one of their guardians at the frontier had been gunned down by the evil West, and made him into heroic propaganda; the West was satisfied that one of _their_ guardians had done his duty under fire. Koch continues:
Göring is declared a hero in the East and a criminal in the West, but an investigation found that Göring had all his ammunition left, and that meant he had refused to fire. That was something to do with his biography: his brother had been in West Germany since 1957, his mother was religious – although she lived in the East she was a member of a church – and Göring himself wanted to go to the West. It means he who was willing to escape was shot by a West German. So both sides lied. There are the two versions and they are not true.
Then on the 18th of June, the Border Guard Reinhold Huhn was shot by a West Berliner, although until 1998 the official version [in the West] was that he had been shot by his comrades. An investigation then found in the GDR archives that the GDR had been correct and he had indeed been shot by a West Berlin citizen. Then on 14 August a Border Guard was shot from the West at the inner-German border (between the two Germanies).
Now Peter Fechter tries to escape at the point where Huhn had been shot and the man who [allegedly] shot Fechter had been a friend of Huhn's. On the West Berlin side, those who shot Huhn and Göring say, 'Look what these Border Guards will do' but they don't say what happened. The conclusion among the Border Guards was that they were afraid they, too, would be shot from the West. 'If I help Fechter they will shoot me.'
The meaning of this is clear. Peter Fechter was not deliberately left to bleed to death but the Border Guards were frightened to go near him, so soldiers were ordered from the barracks at Rummelsburg, the southern suburb, and they took so long to arrive. It was these soldiers and the People's Policeman who bore Fechter away. (The soldier in charge is, says Koch in 2001, 'one of those who still today justify the dead of the wall. He won't talk to anybody.' Koch also points out that to counter the possibility that these soldiers might draw the wrong conclusions about Fechter's death, Ulbricht explained to them 'personally that it was right, because it was an example to others not to try and escape'. Ulbricht presented them with a flag, the only one he ever presented with his own face on it.)
A couple of days later, Dennis Bark left Berlin and prepared to change his life. 'I was the son of a professor of medieval history at Stanford and I was a theatre major at Stanford. I was about to enter my senior and last year and I was getting ready to go to New York. I came home at the end of August 1962 and went to see my father. I told him this young man had died at the wall trying to go from dictatorship to freedom and I didn't know why. I said I had never felt a feeling before in my stomach as I stepped over that line and could I change my major to history? My father said, "Yes, and this is how you're going to do it." So when I graduated in 1964 I graduated with a major in history.'
The delicate balance could so easily be disturbed by something unforeseen like the death of Fechter. John Ausland of the Task Force in Washington said:
We were having a problem after the Peter Fechter case, where the Russians were coming in to their War Memorial in armoured personnel carriers and we couldn't have them running around West Berlin [they were in these armoured vehicles because of the open hostility of West Berliners] 'so we worked out a thing where we would move them from coming through Checkpoint Charlie to an entry near the Brandenburg Gate and therefore near the Memorial.
Frank Cash and I went over with Rusk to see the President and get him to approve our approach to it. As usual he said 'Well, I don't know, I'm going to think about this.' We sat in silence on the way back and then Rusk – it was the only time he did this – invited us in for a drink. He said, 'I know you are very disappointed that the President didn't approve your proposal, and he will, but you've got to remember that whenever he deals with anything concerning the Russians he always has nuclear weapons in the back of his mind.' Now this was on a very minor thing – having them use another entry because the West Berliners were throwing stones at them – but it demonstrated to me, and it's the only one piece of evidence I have from my own experience, the extreme caution regarding nuclear weapons. Looking back, there is no question in my mind that he was right. In our wildest imaginations we couldn't see ourselves living under the Russians but I can imagine people in Western Europe might, especially West Berlin.
23 August 1962 _Hans-Dieter Weser, a 19-year-old policeman, deserted. About 10.13 at night he was seen trying to 'realise his criminal intentions' and was mortally wounded by a burst of eight rifle shots. A Border Guard shot him. Weser, using 'all his remaining strength just managed to reach a few metres inside the westberlin [sic] area'. He was retrieved, taken to hospital and declared dead._
That month Ausland presented a briefing to Kennedy in the Cabinet Room of the White House. It was based on a National Security Action Memorandum (nicknamed 'Poodle Blanket') which Kennedy had approved in October 1961. Ausland set out four phases if either the Soviet or East German forces interfered with Allied access to West Berlin: _Phase 1_ : Small military probes and air force pilots to fly commercial aircraft if the civilian pilots refused. Fighter protection could be deployed; _Phase 2_ : If the access denial is complete enough to be 'significant' there would be intense diplomatic activity, a NATO build-up, an airlift, naval counter-measures, economic sanctions and 'covert action designed to encourage passive resistance'. _Phase 3_ : If phase 2 doesn't work, the Allies will instigate 'offensive non-nuclear operations' into the GDR; _Phase 4_ : If that fails 'there could be a resort to nuclear weapons'. Ausland stressed that this was not an attempt to predict history and 'we have no idea of rushing from one phase to another'.29
4 September 1962 _An unknown person died at the wall._
4 September 1962 _At about 1.45 a.m., Ernst Mund, 41 was seen swimming a canal. A Border Guard approximately 50 metres away fired four shots and Mund could no longer be seen. It was unclear whether he had reached the West and six divers searched for him until 8.00 in the morning._
Twenty-nine people escaped through Tunnel 29, as it was known, in the Frohnau district of West Berlin. It was dug by students, was 125 metres (413 feet) long and some 6 metres (20 feet) below the wall but had to be abandoned because a broken water pipe flooded it. (In October 2000 the remains of the tunnel were discovered by researchers for a television documentary.)
30 September 1962 _Günter Seling, a junior officer in the People's Army, was fatally shot._
8 October 1962 _Anton Walzer, 60, tried to swim the Spree at 8.25 in the morning. Two warning shots were fired but he didn't react to them and continued towards the Western bank. A burst of seven shots killed him._
1 November 1962 _An unknown person died at the wall._
27 November 1962 _Ottfried Reck, 18. Two 'male civilians' were seen acting suspiciously in a U-Bahn station. Two soldiers approached them and they fled, ignoring warnings to halt. Four shots were fired and two hit Reck 'in the top half of the body'. The other man disappeared into the East and a manhunt began. Reck died in hospital at 9.30 that evening._
Towards the end of November _An unknown person died at the wall, possibly shot._
1 December 1962 _Hans-Joachim Nittmeier, 23, nationality unknown, died at the wall._
5 December 1962 _Two unnamed people were shot at 11.30 p.m. crossing a frozen lake. They fell through the ice and drowned. After the shots were fired there was silence and 'it is assumed this transgression did not succeed'. One of the men may have been called Günter Wiedenhöft, 20, nationality unknown._
18 December 1962 _Melita Hinz, 50, died at the wall._
1 January 1963 _Hans Räwel, 21, was seen at 6.15 in the morning 300 metres from a bridge in the Spree swimming West. A Border Guard patrol boat gave chase and Guards on it opened fire. The last shot was fired when it was 20 metres from him. He went under and did not resurface. 'It is assumed that the border crosser was mortally wounded.' The boat itself came under fire from the Western side and was hit twice and a Guard slightly injured. 'During this action on our part, the westberlin [sic] territory was not shot at.'_ 30
In January 1963 a young Border Guard, Fritz Hanke, was stationed at the Teltow Canal in January. He shot at and killed a man trying to swim through the icy water. 'It was only several weeks later that the bullet-riddled body was recovered. In the meanwhile, however, Hanke had escaped himself, and as a Guard, was naturally subjected to thorough interrogation by the [Western] security authorities. From this it transpired that he had been on duty that fateful night and he was put on trial for participation in a murder. Hanke was duly sentenced to fifteen months in prison, a light punishment but given as a warning to other Guards that they could not escape moral responsibility for their actions.'31
15 January 1963 _Horst Kutscher, 32, was shot at 12.10 a.m. Two people were noticed at the wall, a Border Guard called out a warning and then fired a warning shot. The two jumped up onto the wall. The Guard fired two aimed shots and hit Kutscher in his abdomen. The other man was arrested._
24 January 1963 _Peter Kreitlow, 20, was shot by a Soviet soldier. Five young men tried to cross in the north. Three were from Berlin, one from the small town of Henningsdorf nearby, and the fifth from the northern port of Rostock. They were stopped by a Soviet unit at 1.10 in the morning. They ignored a warning. Kreitlow died, another was injured and the remaining three arrested._
By now, the logic of the wall was well into working itself out, and the GDR even held exhibitions for Westerners, demonstrating that it was really an 'anti-fascist barrier' after all. That carried an intrinsic risk, because the GDR leadership found it almost impossible to admit they'd been forced to build it to stop thousands of their citizens from defecting. And if the leadership would not tell the truth about this, what would they tell the truth about? This absolute mistrust of the GDR's official pronouncements tracked them down the years, and the internal logic of the GDR itself produced a paranoia where almost _everything_ was a state secret. Reputable international financial organisations eventually gave up including the GDR and the other Eastern bloc countries in their studies because the figures they were given were unverifiable and wildly improbable.
What Ulbricht really understood of all this went to the grave with him but certainly throughout the 1950s he spoke in apocalyptical terms of events in West Germany, and this at a time when West Germany was becoming a stable democracy and the third strongest economy on the planet, with an astonishingly even spread of prosperity.
In 1963, for example, Ulbricht said, 'What we have long predicted is now a fact in West Germany... recurring government crises.'32 The author Carola Stern says: 'We do not know whether Ulbricht believed what he said about the Federal Republic.'
(There's a grotesque irony, too, in the fact that when the GDR finally disintegrated in 1989 its leadership still found it almost impossible to admit that it was being forced into responses to stop thousands of its citizens from leaving. The 'people's state' had no means of communication with its own people and arguably never really had. From 1961 the separation of the two Germanies accurately reflected the separation of Europe into blocs: the GDR felt it couldn't survive if it told the truth about itself, especially _to_ itself. West Germany felt it couldn't survive if it did _not_ tell the truth about itself, especially to itself.)
Sometime in March 1963 _Wolf-Olaf Muszinski, 16, drowned._
Sometime during April 1963 _Hedwig Forgert, 44, a female corporal, possibly tried to swim from the Pankow district to West Berlin. 'The reason for this escape has not been concluded yet.' She drowned in the Spree but her body was found and retrieved about 10.35 a.m. It may, however, have been suicide._
16 April 1963 _Two unknown people drowned._
26 April 1963 _Peter Mädler, 20, was seen at 4.45 in the morning swimming West in the Teltow Canal. A warning was shouted and then three shots fired. He disappeared under the water and a search for his body was unsuccessful until 4.45 p.m., when it was brought ashore. He had with him a cellophane bag containing all his documents including an army identification and driving licence._
An Austrian called Hans-Peter Meixner, who was studying in West Berlin, met a pretty Easterner called Margit Thurau at a wedding in East Berlin and soon enough they were in love. They imagined that, having become engaged, she would be entitled to leave as the future wife of a foreigner. Applications were refused and he began to contemplate how he could get her, and her mother, whom she lived with, across. He observed that the horizontal wooden pole at Checkpoint Charlie, which was raised to let each vehicle through, had no vertical struts to it and he noticed, too, one night as he queued to reach the pole, that the driver of a West German sportscar hadn't put his handbrake on. The car rolled towards the pole and part of the bonnet went under it. Interesting...
On his next visit Meixner estimated how high the pole came up on his own car, quickly marked that on the flank of the car with his finger and when he measured it later found the clearance was 0.9 metre (3 feet). There was a car hire on the Kurfürstendamm and they had a British sportscar, an Austin Healey Sprite, with a windscreen which could be detached. That took the vehicle just to below 0.9 metre.
The geometry of Checkpoint Charlie then was relatively unsophisticated. From the East a horizontal pole screened the entrance to it and once that had been raised a motorist went to a vehicle and document inspection area. After clearance there he threaded through a 'chicane' of three concrete walls, constructed to make vehicles move slowly and prevent anyone battering their way out in a bus or truck. Beyond that was the second pole, then the line which Hagen Koch had painted, and then the American Sector.
Meixner hired the Sprite and went over to the East quite normally. He would make his run in the early hours. He got Margit to lie across the cramped back seats and put her mother into the boot – she had to curl up to fit. He drove to the checkpoint and the first pole was raised almost on the nod: a glance at his passport, that was all. As he was flagged towards the vehicle inspection area he knew he'd reached the critical instant. The man at the pole might have been uninterested in the car but an inspection would instantly reveal Margit. Meixner hit the accelerator and weaved through the 'chicane', then ducked his head as the car went under the pole: he kept his head raised just enough to see with his left eye where he was going. The Guards, caught completely off-balance, did not even fire a shot.
Two weeks later, an Argentinian who also had a fiancée in the East, hired the same car and did the same thing. Legend insists that one of the Border Guards said 'Isn't that the car we had through the other day?'
_Then_ they put vertical struts on the poles...
On 26 June, John Kennedy arrived with his own mythology; came to deepen that. He was on a European tour and West Berlin surrendered to him. During the day an estimated 1.5 of the city's 2.2 million inhabitants came out to cheer him. He was young, very handsome and extremely charismatic, in direct contrast to the 'character' of the wall, and Ulbricht and Khrushchev behind it. Mayor Willy Brandt, who accompanied him, was also charismatic; and Adenauer, who flew in specially, had a certain dignity about him. General Clay was there and would, Kennedy promised, come back if he was ever needed.
The _New York Herald Tribune_ wrote of the 'gala mood that had many Berliners smiling, shouting, waving and weeping, sometimes all at the same time. From the moment Mr Kennedy left the plane that brought him from Wiesbaden at 9.45 a.m. it was evident that West Berliners set the highest store by his coming.' At the airport he told a crowd of 5,000 that he had not come to reassure West Berlin because that wasn't necessary. The Allied pledges were 'written on rock', he said. Police had to lock arms to hold back a crowd which stood six-deep at the exit to the airport.
As the motorcade moved away 'some onlookers threw flowers and so many tried to carry bouquets to the President that the escort of 125 white-jacketed motorcycle policemen formed a phalanx that encircled the car. About a dozen flower-carriers got through anyway.'33
Outside the city hall, people who had brought their mattresses the night before, and slept on them, in order to reserve a good position for Kennedy's speech, waited.
He went to a specially constructed observation platform at Checkpoint Charlie and he went to the Brandenburg Gate. The gaps between the columns had been draped with banners so that East Germans could not see him. A hoarding had been put against the bases of the columns and positioned so that any photographer taking pictures of Kennedy on the observation platform would inevitably have that as a backdrop. It had a propaganda message printed in English about how the GDR had uprooted German 'militarism and Nazism' and when will pledges like that be fulfilled by West Germany and West Berlin, President Kennedy?
At the city hall, Kennedy stepped onto a broad, deep balcony – large enough to accommodate a couple of dozen dignitaries. 'The roar was deafening when Mr Kennedy came forward to speak. The crowd was in a frantic state, chanting, waving and shouting. Red Cross helpers rushed in all directions to carry away fainting people. The crowd began a chant – Ken-ne-dee – that played counterpoint to the rumbling applause that greeted every sentence of his speech.'34
This speech became part of the mythology. He hammered it out: if people think communism is the future, let them come to Berlin. There are those who say we can work with the communists: let them come to Berlin. There are those who say communism is evil but it permits progress: let them come to Berlin – but this last sentence in German. 'Two thousand years ago the proudest boast was ' _civis Romanus sum_ [I am a Roman citizen]'. Today, in the world of freedom, the proudest boast is Ich bin ein Berliner [I am a Berliner].'35 There was an immortal simplicity to these few words, whose pronunciation he had rehearsed and rehearsed. Strong men wept and for an unnerving moment it seemed the crowd might lose control of itself.
Dean Rusk, the Secretary of State, had gone to Berlin with Kennedy. 'I looked at the wall and my views were the views of the consensus, of the government. There was a wall, it was put up to keep people in rather than exclude Westerners but it was still a shock to actually see it and a shock to see so many windows boarded up in the houses that fronted onto the wall to block escape routes.
'I was standing next to Konrad Adenauer after the speech was made. I asked him what he thought of the proceedings and, with a very solemn look on his face, he said "I am worried. Does this mean that Germany could have another Hitler?" He said it because of the emotional reaction from the crowd. Afterwards Kennedy thought he had probably overdone it a bit on the emotional side – rousing the emotion of the Berliners. He was concerned about that. I think we'd underestimated the strength of the Berliners' emotions. I was caught by surprise by the strength of it.'
Rusk could, moreover, put the speech into a historical context. 'Well, I had seen a million people at the Tempelhof airbase [south of the city] at a Hitler rally where you had something like the same thing. It was 1934 and I was a student at Oxford (England). I was studying the political situation in Germany and I went to Berlin. It was a massive city with big avenues, obviously an imposing place but in a Prussian style: not a very attractive city. It was heavy with its own seriousness.
'I stayed with an ordinary family in Babelsberg. The impression Hitler made on me was that he was mesmerising the German people. Of course in 1934 we did not know how much of his book _Mein Kampf_ would be carried out and how far he would go. No, he didn't mesmerise me, I never gave a Hitler salute and I never joined in with the chants. I just watched what was happening and I kept my American identity throughout. Berlin was becoming a wildly different city in 1933–34 as Hitler took charge. The Nazis had driven other political parties off the streets and assumed a political monopoly of the demonstrations in the streets. I wouldn't call them happy days, they weren't happy days in Germany while Hitler was coming to power. I'd thought of those days during the crisis when the wall went up... .'
Frank Cash of the State Department's Task Force echoes Rusk. 'The people involved with Berlin were known as the so-called Berlin Mafia.' That was because, Kennedy felt (as we have already seen), once they became involved they identified with the city and lost their judgement. 'But when people got there – and when Kennedy got there – they seemed to change because it was a very striking thing to go and see that this wall actually existed, and that it was built across the centre of a thriving modern city to keep people from coming out. There was nothing like walking up and seeing the wall to get the full impact. Until you confronted it, you could not get this full impact.'
* * *
1. Interview with author.
2. _Violations_ , op. cit.
3. Gelb, op. cit.
4. She is generally referred to as Schulze, without a first name, presumably because she survived and therefore does not appear (as far as I'm aware) in official records or lists. That is bizarre, because newsreel of her escape is so standard that it's almost impossible to find a documentary film on Berlin which does not contain at least some of it. Hagen Koch is fairly sure she was called Frieda and I've gone along with him.
5. _The Berlin Wall_ , Deane and David Heller.
6. I have taken certain liberties in re-creating this. The participants told me their tales, often repeating verbatim what people had said to them. As a result, they were themselves quoting others. I have lifted these verbatim quotes out and used them as if spoken by the people themselves. The result, I feel, is still as authentic as you can get over dialogue four decades before, and much easier to understand.
7. Gelb, op. cit.
8. _Battleground Berlin, CIA vs KGB in the Cold War_ , David E. Murphy, Sergei A. Kondrashev and George Bailey (Yale University Press, New Haven and London, 1997).
9. Interview with author.
10. _Violations_ , op. cit.
11. Interview with author.
12. _The Berlin Crisis_ (National Security Archive, Washington).
13. This was a little joke at my expense, an American gently teasing a Brit by suggesting that the British were responsible for Joe Kennedy discovering sex – in London. I suspect that the Kennedys discovered it for themselves, and most Americans, too, which is why there are 260 million of them. (Not all can have journeyed to London for lessons.) And that's my little joke in riposte.
14. Interview with author.
15. McDermott, op. cit.
16. Interview with author.
17. _Violations_ , op. cit.
18. Heller, op. cit.
19. _New York Herald Tribune_ , 23 February 1962.
20. _Violations_ , op. cit..
21. _The Ugly Frontier_ , David Shears (Chatto & Windus, London, 1970).
22. McDermott, op. cit.
23. _Violations_ , op. cit.
24. Ibid.
25. Heller, op. cit.
26. GDR Interior Ministry document.
27. _Violations_ , op. cit.
28. Interview with author.
29. Department of State document drafted by John Ausland on 20 July 1961 circulated to the US Ambassadors in London, Paris and Moscow, and to US offices in Berlin and the UN. Reproduced by kind permission.
30. GDR Ministry for National Defence document.
31. _Escape from Berlin_ , Anthony Kemp (Boxtree Limited, 1987).
32. _Ulbricht_ , Carola Stern (Pall Mall Press, London, 1965).
33. _New York Herald Tribune_ , 27 June 1963.
34. Ibid.
35. Kennedy's speech appears in full at The History Place, webmaster@historyplace.com and I am grateful to them for letting me have it. They also say 'Did you know the JFK Library has a big site with that speech and others?' Well, I know now – and so do you. It's www.ca.umb.edu/jfklibrary/main.html. Here is the speech:
I am proud to come to this city as the guest of your distinguished Mayor, who has symbolized throughout the world the fighting spirit of West Berlin. And I am proud to visit the Federal Republic with your distinguished Chancellor who for so many years has committed Germany to democracy and freedom and progress, and to come here in the company of my fellow American, General Clay, who has been in this city during its great moments of crisis and will come again if ever needed.
Two thousand years ago the proudest boast was ' _civis Romanus sum_ '. Today, in the world of freedom, the proudest boast is _Ich bin ein Berliner_. I appreciate my interpreter translating my German!
There are many people in the world who really don't understand, or say they don't, what is the great issue between the free world and the communist world. Let them come to Berlin. There are some who say that communism is the wave of the future. Let them come to Berlin. And there are some who say in Europe and elsewhere we can work with the communists. Let them come to Berlin. And there are even a few who say that it is true that communism is an evil system, but it permits us to make economic progress. _Lass' sie nach Berlin kommen._ Let them come to Berlin.
Freedom has many difficulties and democracy is not perfect, but we have never had to put a wall up to keep our people in, to prevent them from leaving us. I want to say, on behalf of my countrymen, who live many miles away on the other side of the Atlantic, who are far distant from you, that they take the greatest pride that they have been able to share with you, even from a distance, the story of the last eighteen years. I know of no town, no city, that has been besieged for eighteen years that still lives with the vitality and the force, and the hope and the determination of the city of West Berlin. While the wall is the most obvious and vivid demonstration of the failures of the communist system, for all the world to see, we take no satisfaction in it, for it is, as your Mayor has said, an offense not only against history but an offense against humanity, separating families, dividing husbands and wives and brothers and sisters, and dividing a people who wish to be joined together.
What is true of this city is true of Germany – real, lasting peace in Europe can never be assured as long as one German out of four is denied the elementary right of free men, and that is to make a free choice. In eighteen years of peace and good faith, this generation of Germans has earned the right to be free, including the right to unite their families and their nation in lasting peace, with good will to all people. You live in a defended island of freedom, but your life is part of the main. So let me ask you as I close, to lift your eyes beyond the dangers of today, to the hopes of tomorrow, beyond the freedom merely of this city of Berlin, or your country of Germany, to the advance of freedom everywhere, beyond the wall to the day of peace with justice, beyond yourselves and ourselves to all mankind.
Freedom is indivisible, and when one man is enslaved, all are not free. When all are free, then we can look forward to that day when this city will be joined as one and this country and this great Continent of Europe in a peaceful and hopeful globe. When that day finally comes, as it will, the people of West Berlin can take sober satisfaction in the fact that they were in the front lines for almost two decades.
All free men, wherever they may live, are citizens of Berlin, and, therefore, as a free man, I take pride in the words _Ich bin ein Berliner._
## SIX
## _The Strangeness_
The history of Germany is replete with blunders and missed opportunities involving all social and political factions.
Willy Brandt
Kennedy's visit could not mask the truth that the status quo had now been established in Berlin and would endure until the whole political climate of competition and confrontation between the two blocs changed. Nobody had any idea when, or if, that would happen, and each passing day deepened the divide.
By now, the impact of the wall was subtly increasing. As it became more difficult to penetrate, the number of escapes and deaths fell sharply. Statistics, that most inhuman measurement, tell it with their own exactitude. In 1962, thirty people died trying to cross; up to Kennedy's visit in 1963 the number was eight, and there would be another four but not until November and December. The days of impulsive dashes, of vaulting coils of barbed wire, of sneaking through back gardens, were all but over. The noose had finally tightened to the point where human ingenuity would be needed; and, in the sad streets of East Berlin where personal initiative was officially forbidden, there was enough of it – and enough desperation – to create legend.
By now, too, one could feel the two halves of the city moving to their different rhythms, although with its population stabilised the GDR embarked on its own economic miracle. In time it would give its citizens a certain pride in being East German, whatever frustrations they continued to endure. In time, also – the 1970s – the wall would be essentially rebuilt twice, each time making escape more difficult and each time requiring more ingenuity.
That would be accompanied by a constant, if diminishing, tension. John Ausland of the State Department said, 'We had planned if necessary to put fighters into the air corridors and then after Cuba – and people have forgotten this – we had problems with convoys along the autobahn through East Germany in, I think, September 1963. They started to make life difficult and I spent three weeks putting a solution together. I got everyone's approval in Washington, British government, the French government, the West German government. Then I got to the last guy – Kennedy. He said, "Well, do we really want to do that?" The problem in essence was this: I made the mistake of calling a meeting. Tommy Thompson, who was handling Berlin by that time, was out of town. He came back the next day and said, "For God's sake, John, you shouldn't have called a meeting. Come on with me." So we jumped in the car, went over and we caught Kennedy coming out of a meeting. He said, "Mr President, I think you really ought to approve this." Then someone came up, joined in the discussion and said, "Well, Tommy, you really think we ought to do this? Yes?" At that, Kennedy said, "OK, go ahead." It wasn't a big thing: for the first time we did alert our ground troop force and things like that – I don't think anything really awful could have happened. The net result was that we got an implicit agreement with the Soviets on convoy procedures and never had any problems after that.'
In September, the GDR drew up plans for clearing the houses at the border in the first six months of 1964. They did this methodically, setting it all out in nine vertical columns: House number; Map reference; Evacuated; Moved out; Number of families; Number of people; Demolition cost; Overall cost; Observations. Nos 10a, 12, and 13a Bernauer Strasse were among them. All three were to be evacuated on 28 February: Nos 10a and 13a (three families, fourteen people) to be demolished on 30 April and number 12 (six families, seventeen people) on 15 June. The overall cost for numbers 10a and 13a was 31,200 marks, and number 12 only 14,000. And so it went on, page after page, each line representing someone's home. (The precision of GDR thinking had already been established in these matters. An example of a job specification c. 1962 at the wall just along from Checkpoint Charlie shows this: 16 sq m hardboard; 7 kg nails; 30 sq m roofing felt; 1l creosote; 10 sq m glass; 1.5 cu m plants, the work to be completed in sixty hours.)
4 November 1963 _Klaus Schröter, 23, was seen swimming the Spree towards the Western bank at 4.01 a.m. After several warning shots he was fatally hit. His body was found at about 7.45._
25 November 1963 _Dietmar Schulz, 24, jumped out of a moving S-Bahn train at 10.20 at night as it passed close to the border. He suffered head injuries and died in hospital._
Kurt Behrendt, the resident of Steinstücken, watched the noose and its tightening. 'From the top of my house I had a very good view of the wall. They only began to build it in December 1963. Before that we had a special border, what are called Spanish riders: wooden crosses and barbed wire around them about a metre high – a sportsman could easily have jumped over. To reach Steinstücken you had to come through two controls. The first was from West Berlin – a little hut painted white with a window. It was like a mini-barracks for the Border Guards and had a pole across the way. It was manned by two Border Guards. When they raised the pole you walked about a kilometre along a path to the second control which was similar to the first. After that you were in Steinstücken.
'At first it was a very quiet place, almost lonely, although there were forty houses here. Of course it was a special situation since there were Border Guards all around us. We were the best supervised place in Berlin and we never needed to use our front doors to lock our houses! You could wander into anyone's living room. Women walked around in complete safety – nobody would harm them. We did, however, use to hear gunfire.'1
Because of its location as an enclave butting onto the East, and because an east-west railway line ran through it as well, Stein-stücken became and remained a potential weak link in the wall, and one evening more than a decade hence two lovers and an accomplice would try to exploit that, posing as GDR officers. Ingenuity there certainly was on those sad streets.
13 December 1963 _Dieter Berger, 24, was apprehended after shots were fired at 3.10 in the afternoon and died from his injuries._
That December, after seven preliminary meetings, a Pass Agreement was reached which enabled West Berliners to visit the East if they could prove they had relatives there. The GDR used the seven meetings in 'fruitlessly attempting extortions and trying to enforce a political recognition of the GDR'.2 More than 730,000 West Berliners took advantage of this, waiting up to twelve and fifteen hours in snow to go through a special entrance cut into the breeze blocks. They came with parcels and packages; they were of all ages, some children clutching dolls and teddy bears. In a city of so much pathos these humble, private meetings must have been almost unbearable. That was softened because the Pass Agreement was renewed until 1966.
25 December 1963 _Paul Schultz, 18, got over the wall at 4.30 in the afternoon despite being wounded from 13 shots fired at him. He died immediately afterwards on the Western side. A simple wooden cross would be erected there, his photograph – a young, open face, his hair bushy but neat – in a clear cellophane sachet attached to the cross. Wreaths were laid. His body was taken back to East Berlin in a hearse and there's a fixed image of that as it goes through the checkpoint, Border Guards watching it as if he might have been alive._
At that other, official level the anomalies reflected the status quo, if sometimes uneasily.
Bill Bentz had 'very interesting times at Checkpoint Charlie. After I completed the assignment as officer in charge down there I went back to my unit for a day or so. I got called by our group HQ and they informed me I was being considered as one of the officers on Flag Patrol. Would I be interested? I said, "What is that?" and they said, "Well, you go over into the Soviet Sector of Berlin in a car with the American flag on the side and drive around the whole of the Soviet Sector."
'The reason they were called Flag Patrols was because essentially they provided visible evidence we had every right to be in the Soviet Sector and we were carrying out that right. It also had various other missions tied to it which I am not going to go into – I will say it involved reporting anything unusual. The British and French had the same thing. The deal was I had a young enlisted man as a driver, I sat in the back seat and we would go over every other day during the day and the night because there were four or five other cars. Mine wasn't the only one for the mission.
'I had been doing this for some time. You know, we would enter at Checkpoint Charlie, they would open the gates and we would drive through because we were in uniform and the car was clearly marked. Then we'd drive around for two or three hours, come back out into the American Sector and write our reports. There were, however, certain areas posted by the Soviets and East Germans as restricted.
'I had been doing this for several weeks – or longer, really – with no problem and this one night was on mission. We went over. We were in an Opel, we carried out the normal mission and returned to pass through the Eastern side of the checkpoint. It was dark. We pulled up to the checkpoint and there were all kinds of Soviet officers around. At that time they basically just had wooden barricades that came down like a railroad crossing and they wouldn't raise it.
'A Soviet officer came over and said, "Get out of the car" and I said, "No, I am not getting out of the car." "Get out of the car!" "No!" So the Soviet officer went around to the back of the car and started jumping on it. I told my driver, "That's it, we're going." He put the car in low gear and we sped on through the wooden barrier with the wood flying, just like a movie. We raced on and got back to HQ where I was usually debriefed. After we drove in there an American Police Captain said, "Well, Lieutenant, you had a little trouble down there." I said, "Yes, sir." Everything was on film: the checkpoint was monitored [by a camera high up in a building and relayed to HQ]. They had seen what had happened. I said, "What's the problem? I have done this mission for a month or so and have never had any problems before." He said, "Come here, I want to show you something." He took me around to another spot and there sat an Opel exactly like mine: painted the same colours, the same numbers as mine and they had manufactured things like a radio cone on top out of tin cans. The only place they messed up was my licence plate' – they'd made a small mistake in the exact wording.
'What happened was that all the time I'd been going over I'd been watched by some East Berliners who wanted to escape. On this mission they rolled their car out with three people in it wearing made-up American uniforms like ours and there were more people in the trunk. They'd waited a couple of hours, which was the sort of time I'd be driving around East Berlin, and then they pulled up at the checkpoint. The barrier was raised and they crossed right over into West Berlin.'3
Perhaps the words 'absolute consternation' cover the reaction of the Border Guards when the second Opel came along; a consternation which can only have intensified when they realised Bentz was indeed Bentz; and increased further when their pole was shattered as he, too, was driven away at speed.
Capturing the uneasy status quo are the instructions for the United States military (albeit issued at a later date) to avoid incidents while using their access:
## CONTACT WITH COMMUNIST NATIONALS
## _I MPORTANT
YOU ARE TO READ THE FOLLOWING CAREFULLY_
**1.** **W EST BERLIN IS UNIQUE IN THAT IT IS THE ONLY LOCATION WHERE TRAVELLING TO AND FROM THE CITY BY ROAD, SERVICEMEN AND CIVILIANS HAVE DIRECT CONTACT WITH SOVIET MILITARY PERSONNEL. IT IS THEREFORE VITAL THAT ANY ATTEMPT TO ENGAGE YOU IN CONVERSATION OR THE TRADE/BARTER OF MILITARY ITEMS BE REPORTED.**
**2.** **S HOULD YOU BE SPOKEN TO BY A SOVIET OR EAST GERMAN NATIONAL IN ENGLISH, OR A LANGUAGE YOU BOTH**
**UNDERSTAND, DURING YOUR JOURNEY ON THE CORRIDOR, YOU SHOULD DO THE FOLLOWING:**
**a.** **R EMEMBER AS MUCH DETAIL ABOUT THE CONVERSATION AS YOU CAN, AS WELL AS THE PHYSICAL DESCRIPTION, DRESS AND RANK OF THE INDIVIDUAL;**
**b.** **REMAIN NON-COMMITTAL THROUGHOUT AND DO NOT AGREE TO ANYTHING;**
**c.** **DO NOT BECOME OVERLY NERVOUS OR AGGRESSIVE. ONCE IT IS REALISED THAT YOU ARE NOT RESPONDING, YOU WILL BE LEFT ALONE.**
**R EMEMBER, DO NOT ATTRACT ATTENTION TO YOURSELF BY SPEAKING IN RUSSIAN TO THE SOVIET CHECKPOINT PERSONNEL.**
**3.** **O N REACHING THE END OF YOUR CORRIDOR JOURNEY, YOU MUST REPORT THE INCIDENT TO THE RMP CHECKPOINT NCO AND COMPLETE THE PROFORMA PROVIDED. BE ASSURED THAT YOUR JOURNEY WILL NOT BE DELAYED, IF IN DOUBT ASK THE RMP CHECKPOINT NCO FOR ASSISTANCE.**
## **_HELP US TO HELP YOU_**
**Detentions by GDR police:**
**On the prescribed route of travel:**
**– Stay in your vehicle with the windows rolled up and the doors locked**
**– Do not show travel or identification documents**
**– Do not allow your person, your vehicle or your belongings to be searched**
**– Use your flashcard to demand your right to proceed**
**– If you are not allowed to proceed, use your flashcard to request the presence of a Soviet officer.**
**Off the prescribed route of travel – Follow above instructions.**
**Note: If you are detained by GDR police because you have left the authorized autobahn route, you may expect to be detained for one-half to four hours. Your vehicle will be physically blocked in and photographed. When GDR police release you, they will indicate the direction you should travel, or they will escort you back to the proper route.**
**Soviet involvement:**
**If Soviet authorities are at the scene of an incident or accident:**
**– Exchange military courtesies with them**
**– Cooperate with them**
**– Show them your travel and identification documents if they request you do so**
**– If you are off the prescribed route of travel, do not admit to any wrong doing other than to having made a navigational error.**
The precision of these guidelines was important because the Soviet Union and the GDR constantly sought to exploit the autobahn. The petrol and service station at Ziesar, about a third of the way between Dreilinden and Helmstedt, was a haunt of the Stasi, lurking, watching, monitoring. However, I don't intend to give the impression that everything lived in shadowland. Driving the autobahn in 1970, the author overtook a column of Soviet Army trucks with open backs so that the soldiers sitting there could see out. They returned waves enthusiastically and seemed a merry lot – quite different to the GDR military which, like the GDR as a whole, was taking itself very seriously indeed. There's an amusing anecdote4 about 'a smallish Russian soldier' at one of the autobahn check-points who demanded that a United States soldier show his travel papers and the American said, 'Get a taller Russian to see me!'
Long and deliberate delays to transit traffic were engineered whenever the FRG parliament met in West Berlin, bestowing a status on that half of the city which the GDR claimed it did not have. These delays could last up to fifty hours for the military, and civilian queues of 10 kilometres were not unknown.5
27 February 1964 _Walter Heyn, a 25-year-old divorced farmer, tried to get across at about 10.20 at night. He made his attempt through allotments. Two Border Guards saw him 50 metres away, called a warning and fired a warning shot. He was cut down by a burst of fire and three aimed shots. When he was examined it was found two of the shots had killed him._
26 March 1963 _An unknown person died at the wall._
5 May 1964 _Adolf Philipp, 20, a West Berliner. At about 1.45 a.m. a Border Guard noticed a man's tracks near the border at an unoccupied bunker. Two Guards went there and were challenged by Philipp, who held a revolver and said 'hands up'. They shot him in the chest. A subsequent investigation showed that he had cut through the barbed wire and walked backwards into the death strip so that anyone coming across his footprints would think he'd been walking Eastwards – and that would throw them off his trail._
22 June 1964 _Walter Heike, 30, tried to get across at the Invaliden cemetery beside the canal at about 5.40 a.m. He did not react to warning calls or shots. He got over the barbed wire but was wounded by a bullet above his buttock and a Border Guard carried him back from the death strip. Fifteen minutes later he was taken to hospital by ambulance and died there._
28 July 1964 _Rainer Gneiser, 18, and Norbert Wolscht, 17, may have tried to swim across. Both died, Wolscht of defective diving equipment._
18 August 1964 _Hildegard Trabant, 37, and Wernhard Mispelhorn, a 20-year-old Berliner, made a joint attempt at 6.53 in the evening. There are no further details._
On 13 September 1964, Michael Meyer, 21, tried at 5.20 in the morning to cross near Checkpoint Charlie. He ignored warning calls and shots, and the Border Guards opened fire. After having been hit several times Meyer collapsed near the wall and two Border Guards rushed up to where he lay. They aimed their weapons at him.
An American Military Policeman, Hans Werner Puhl, was on the second floor of a house overlooking the incident and headed downstairs to help. By then the Guards were dragging Meyer across the raked sand of the death strip. They noticed two armed West Berlin policemen, let go of Meyer and sprinted off. He tried to reach the wall again and was hit by five shots.
The West Berlin police, in the hall of a house, now sent across warning shots to the Guards who made a second attempt to drag Meyer away. Puhl shouted, 'Let go the boy' and hurled a tear-gas grenade over, forcing them to retreat. Then, aided by some civilians, he clambered over the wall to help.
'In doing so, the American was in breach of the state border', the GDR internal report would point out.
Puhl drew his pistol and pointed it at the two Border Guards while, at the same time, instructing Meyer to lie motionless. Civilians cut the barbed wire on top of the breeze blocks. Puhl came under fire from other Border Guards some 100 metres away but he wasn't hit. When the wire had been cut a rope was thrown over and Meyer was hauled over to the West.
The GDR report said that 'USA officials' had opened fire and 'more than 100 shots were "received"' on the Eastern side.
Later, Willy Brandt officially thanked Puhl.
The most celebrated tunnel rests in legend as 'Tunnel 57' because that was how many people came out through it on the nights of 3, 4 and 5 October 1964, a Saturday, Sunday and Monday. The man behind it, Wolfgang Fuchs, was an adventurer: 'large, generous, warm-hearted and outgoing'.6 Between 1963 and 1964 he built seven tunnels. Because the GDR had not yet demolished all the houses at the border, comparatively short tunnels could be dug across to them from cellars in the West. As the houses were demolished and the death strip widened, tunnel construction evolved into something approaching an industrial basis with shift work, financial support, electricity, artificial ventilation and ingenious ways of concealing all the soil which had to be removed.
'Tunnel 57' began in a bakery on the Western side of Bernauer Strasse which Fuchs had rented for 100 DM a month. It was known as Operation Tokyo because the 1964 Olympic Games were to be held in that Japanese city in October and digging began in April. The tunnel was to go 145 metres (158 yards) from the bakery, beneath Bernauer Strasse, following a route under the wall itself, the death strip, the inner wall, a Guard hut in one of the side streets and then come up in a ramshackle outdoor toilet in the backyard of a house.
Student volunteers from the Technical University approached the logistical problems of the tunnel in a highly professional way. They had many concerns: a previous tunnel ran nearby but had been discovered and blown up. Probably it was still being monitored. They also knew, because thirty-five people would be doing the digging, that they could not keep going in and out of the bakery. The Border Guards, not far away, would become suspicious. So they stayed there in ten-day shifts. A lookout was positioned on a roof to watch for any sign of the Border Guards using sound detectors.
The diggers sank a shaft 3 metres down from the bakery basement but it struck a well and flooded. When they'd dealt with that they began to dig a narrow passage horizontally out under Bernauer Strasse. The tunnel could only be 70 cm (27½ inches) high because no excavated soil could be taken from the bakery – alerting the Guards – and calculations showed that any more than 70 cm would be too large a quantity to heap into the various rooms of the bakery.
It meant, however, that only one student could dig at a time. He'd push the soil back to the student immediately behind him who loaded it onto a trolley. It travelled back along the tunnel and was hoisted up into the house, poured into a wheelbarrow and taken away. They needed six months to reach the backyard of No. 55 and by then the disparate group they'd bring across would have been selected and notified; by then, too, the basement and ground floor of the bakery would be completely filled by the soil. Their concern now was what would happen when they came up behind No. 55. Tunnels had been betrayed or discovered before, the Border Guards waiting until people emerged before arresting them. A student was sent to the East and he made his way to the house, positioned himself in the yard and stamped his foot to show the tunnellers where they were. They came up inside the outdoor toilet and had soon enlarged the entrance.
The famous 'Tunnel 57' stretching, from its very secret entrance in an Eastern backyard, out under Bernauer Strasse to its exit – and freedom – in a Western bakery basement.
The people who were to cross were contacted by Western couriers and coded telegrams. They made their way at intervals to the house – at least one via Friedrichstrasse station where he stood in front of a timetable. A stranger leaned forward and touched the D of Dresden, where the man had come from. The stranger took him to the house.7 Fuchs was on a roof with binoculars and a radio transmitter to make sure those arriving were not being followed. When he was satisfied that they weren't, he used the radio to contact the people in the house and they opened the door. The escaper was taken across the yard to the toilet where a wooden crate was pulled away and he (or she) went down into the tunnel. Reports vary between 11 and 30 minutes for the time it took to crawl to the West – at one point water was so deep that breathing was difficult. (A woman escapee was almost too large and got stuck. She began to cry, not least because she feared she'd have to go back. She was pushed through.)
'All the fugitives were shaking with fright. They had been told to cross the yard to the shack one by one, without shoes. One family did not want to be separated at this critical moment. Some walked as if in a trance, they were so afraid.'8
Raw emotion consumed the escapers when they emerged in the bakery basement: husband and wife reunited after three years; brothers reunited; a mother crawling past her son in a wide section of the tunnel, realising he had helped dig it to get her out, but they made no sound as she passed because that would have been so dangerous; a child of 3½; a 70-year-old with heart trouble whose lips went blue as, wearing stout gloves and an overcoat, he crawled forward; a 5-year-old lad who'd been told, to calm him, that he was going into a cave as a special treat to see some wild animals and who, when he was hoisted into the bakery, complained there weren't any wild animals down there at all... .
On the Saturday, twenty-eight crawled through; on the Sunday, twenty-nine, giving totals of twenty-three men, thirty-one women and three children. And then, in the early hours of Monday morning, it went wrong.
Four students were in the house when two men in plain clothes – Stasi, snooping around, no doubt – came in, one shining a torch. These plain-clothes men did not, of course, know the password ('Tokyo') but looked so terrified that the students naturally assumed they must be escapers. A student said, 'Are you crazy shining the torch? Turn it off and get going.' The plain-clothes men were instructed to remove their shoes so they could cross the courtyard silently, but one of them, thinking fast, said he needed to go and collect a third escaper who was hiding nearby and who'd lost his nerve. The other plain-clothes man said he'd accompany his friend to get him.
The students accepted this, and it was a crucial mistake.
The two plain-clothes men came back after about 15 minutes with a man in uniform although, evidently, the students did not immediately notice him in the darkness. From this moment the chronology is not entirely clear.
The student at the front door was informed he was under arrest. Either just before or just after this, the lookout on the rooftop in the West saw GDR soldiers arriving in a truck and on motorbikes. He shouted into his radio, 'Everybody back now.'
The uniformed man was called Egon Schultz and he was 21. One report says he was unsure of himself but he held a machine pistol and the men in plain clothes told him to release the safety catch: 'Load the pistol.' He went out into the courtyard. A student fired a warning shot. Schultz returned this with a burst from the machine pistol.
By now the soldiers had stormed into the house but, with no time to assess the situation, assumed the escape route must be in the basement and began firing down into it.
In the courtyard, the student fired seven shots towards Schultz aiming, in the darkness, at where the flashes of the machine pistol shots had come from; then all four students fled into the toilet. One, detailed to remain at the entrance guarding it, let the other three go first and followed. Schultz, shot in the chest, lay dying. Whether he'd been hit by the soldiers or the student was unknown.
Although the GDR subsequently shouted, 'Murder' (in their official report they described the death of Schultz as 'murder by Westberlin terrorists') and demanded the students' extradition; they didn't, however, produce the fatal bullet or bullets, which they surely would have done if they had been of Western calibre.
The GDR were clearly concerned about the whole incident. On 30 October an officer called Borning wrote to Erich Mielke, head of the Stasi since 1957, outlining the situation following the 'assassination of Sgt. Schultz by Westberlin terrorists'. An investigation uncovered weaknesses and 'these led to the tragic conclusion of the operation'. Erich Honecker, the letter said, had already involved himself in the investigation and the following conclusion and suggestions were agreed: 'The making sure of the safety of the borders with Westberlin is to be organised and carried out by one unit only – organs of the Ministry of State Security [the Stasi].'
By 20 November 1964, proposals had to be made to secure the border and submitted to Honecker. They must include getting the various bodies concerned to work together and liaise.
26 November 1964 _Hans-Joachim Wolff, 17, was shot swimming across a canal at 6.30 p.m._
In November the GDR government allowed pensioners to go to the West, to visit or stay. A total of 383,1819 visited the West from 1961 to 1988, and pensioners formed the bulk of them.
3 December 1964 _Joachim Mehr, 19, tried to get over the wall and was fatally shot at 2.40 in the morning. He made the attempt with a 23-year-old, Hans-Jürgen Kahl, whose manner of death is unknown._
During 1964, nine successful escapes were made using an Isetta bubble car, a vehicle with two seats and a front-opening door which was so small that the Border Guards scarcely bothered to check if anyone was concealed in it. However, if the the heater and air filter were removed, then the petrol tank taken out and replaced with a small canister, just enough room was created for a person to lie unseen on the engine and rear wheel. To maintain the Isetta's equilibrium the suspension had to be strengthened.10
1 January 1965 _An unknown person died at the wall._
19 January 1965 _An unknown man drowned trying to swim across the Spree. Before he crawled under the barbed wire he left a brown attaché case which contained a black shirt, a bottle of 'Goldwasser' alcohol (a liqueur), 12 copies of the periodical Magazin, an empty fountain pen case, and one sandwich wrapped up in a copy of the newspaper Freiheit, dated 21 September 1964. A pair of brown leather shoes was found near the water._ 11
3 March 1965 _Ulrich Krzemien, 22, died at the wall._
4 March 1965 _Christian Buttkus, 21, tried to cross with a female companion at 1.30 a.m. Two Border Guards fired 200 shots at them, killing Buttkus and slightly injuring the woman._
The whole essence of these chapters is one of captivity and trying to evade it. An Easterner might argue that that's just another biased (and selective) view from the West. Officially, of course, there was a view from the East, too, of an embattled, beleaguered sovereign state prey to gangsters, capitalistic robbers and West German militarists enticing their people away while they tried to build socialism. The GDR was meticulous in recording transgressions against their side of the frontier.
For example, on 1 April 1965, an internal report said a sergeant of the Royal Green Jackets 43rd and 52nd had been arrested on the territory of the GDR by a Border Guard. The map reference was given. 'The person arrested was handed over by the town commandant of the capital of the GDR, Berlin, Colonel Geier, to a person authorised by the British town Commander in WestBerlin at crossing point Staaken.'12
But still the escape attempts were made.
The House of Ministries, not far from Checkpoint Charlie, was a gaunt, stoneblock building five storeys high with its southern façade forming part of the inner wall itself. The ground-floor windows had been closed off by metal grilles. The death strip was narrow here: crude stone blocks cemented together, concrete lintels and barbed wire on V-shaped metal clasps running along the top. The immediate area on the West was wasteland – it had been a notorious SS interrogation centre, now completely levelled to the point where grass was growing.
A maintenance engineer worked in the building and his job included checking the lifts. He took his wife and son and they locked themselves in a toilet until evening, when the building was closed for the day. In the darkness they emerged and made their way onto the roof. The man anchored the rope (one report says fishing line) he'd brought then tied the other end to a hammer and cast it full across the wall to where Western 'helpers' waited. They had already secured a steel cable and now attached it to the rope. The man hauled it back across and in turn secured it at the roof. The cable now descended, as taut as he could make it, from the roof down over the wall to the grassy area in the West.
He'd made three metal wheels which would run on top of the cable and his wife had sewn three strong body harnesses which were fixed to the wheels. She put hers on and, the wheel running, disappeared into the darkness. She sailed over the wall and hands reached her before she hammered into the ground. The man hooked on the boy and he went too. When he'd landed, the man attached himself to the cable, which by now was so slack that his feet brushed the top of the barbed wire – but he made it.
9 June 1965 _An unknown person died at the wall. That same day Dieter Brandes, 19, was shot at the wall._
15 June 1965 _Hermann Döbler, 42, a West Berliner who, with a female companion, was in a paddleboat which strayed across into Eastern waters on the Teltow Canal at 1.55 in the afternoon. They were some 60 metres beyond the line. The Border Guards gave a warning shot and, when that was not heeded, opened fire from a watchtower. Due to the strength of the current the boat drifted back to the Western side and about 2.10 reached the bank. Dobler was dead and the woman critically injured._
Undated 1965 _Divers were carrying out exercises and at 4.05 in the afternoon found parts of a body._
Kurt Behrendt, who lived in Steinstücken, captures how sudden escape attempts were and how ill-defined. There was no harmony to them, just fear, and perhaps running, and hasty shots, and it all came from nowhere:
Once I saw somebody trying to escape in the night but he was arrested and taken away. I saw it from the roof of my house. I heard gunfire and I saw a man. I couldn't say how old he was because he was so far away. He was standing between the walls and he was almost paralysed because they had already seen him. Some soldiers from the People's Army came and took him to a watchtower and waited with him until a jeep arrived. They put him in the jeep and it went away. I think the gunfire I'd heard was only to warn the man. I don't think he was injured.
This was not the only man who tried to escape. Some reached Steinstücken but the tragic thing was that they were not in West Berlin – on West Berlin soil, yes, but they'd have had to get through the GDR controls where the main street is today and cross from our enclave to West Berlin itself. The American soldiers had a barracks and when somebody managed to get over the wall he was taken to the barracks, which was in a house, and they stayed there until the next helicopter came, after maybe three or four days. The helicopter took them.
Just over there was a street in Babelsberg which used to be called Red Cross Street, and what we supposed was a barracks was in fact the central building of the Red Cross in the eastern district of Brandenburg. It became an office for justice and administration.
This was the point where the whole of the wall round Berlin was narrowest: 5 metres [a corridor covered from both ends by watchtowers between the back gardens of houses]. In a technical sense, people were neighbours but we had no contact at all – that was prohibited to them. Because of the arc-lamps, all was light and very bright. The lights were not a problem because we had thick curtains and blinds. In the summer after dark it was not so bright because the foliage was on the trees but in winter yes, bright! We had our own street lights but we didn't need them... .
A man had an idea. He had been married for twenty-five years and he wanted to celebrate it by inviting many, many guests but it was not possible for them to come to Steinstücken because they didn't have residents' permits. He'd have to go to West Berlin to meet them in an hotel or restaurant. Some of those invited came from Western Germany and they'd have to stay the night. He had a big house and he wanted them to celebrate and be able to stay with him. That's normal – but they couldn't. Then he had a spark of an idea: every West German and West Berliner had the same rights he had, so if he went to the local [Western] police they would certainly have nothing against those people having their residence here. He sent these guests to the police and they were given a document, stamped and signed, which said that from a certain day on they had their second home in Steinstücken. It was normal and legal to have a second home.
He'd warned them that there may be some difficulties. The guests arrived at the control. There were two Border Guards on duty and they were very surprised at the documents because they'd never seen anything like them before. People from Steinstücken only had identity cards. The Guards did not know how to handle it. No regulations covered it and they may well have been afraid that something would happen to them if they did not let the people through. So they let them through and the celebration took place here, ten or twelve people with children, and some did spend the night. They all got back to West Berlin without any problems.
This was the starting point for many people in Steinstücken to do the same. I did, too, and for most of us it was the first time people could visit us. We did it for some years, but only with friends and relatives, not official people. We had 50 people living here, and after three months we had 1,200 residents! We didn't tell the trick to people we didn't like here.
We could do it like that up to 1967 and nobody did anything to stop it even though this was the time of the cold war. Nobody negotiated on such things: the West Berlin Senate did not negotiate with the GDR and maybe that explains it. We were surprised it wasn't stopped but equally we were aware that one day somebody would come and stop it. In 1967, the GDR authorities announced that from a certain day on – this was the l7th of June – they would not accept secondary residences as a valid reason to cross, but anyone who'd applied before could cross. The decision was not very effective because by that point everybody who wanted one already had one.
The road was built in 1972 and then everybody could get here [the road removed the anomaly because the wall now ran along either side of it (like a neck) and the controls were taken away].13
18 August 1965 _Klaus Garten, 24, was shot trying to get over the wall._
18 October 1965 _Walter Kittel, 23, and an unknown 21-year-old tried to get over the wall near the hamlet of Klein-Machnow. Seventy shots were fired and Kittel died immediately._
10 November 1965 _Heinz Cyrus, 29, jumped to his death from a window._
25 November 1965 _Heinz Sokolowski, 47, was shot at 4.58 a.m. It took eight bullets to stop him and he was severely wounded in the lower part of his body. He died at 6.10 in hospital. The official report added that four West Berlin policemen witnessed the incident._
26 November 1965 _Erich Kuhn, 62, tried to get across allotments at 7.35 in the evening. He was seen crawling about 40 metres from the border by a Guard and did not react to either warning call or shot. He tried to avoid arrest by making off into the allotments but the Guard fired a burst of six shots. He was hit in the stomach._
The essential problem with telling the story of the Berlin Wall is that it all happened on three levels: the political/nuclear, which is usually the subject of academic study; the silent backdrop of 4 million people finding their everyday normalities and living them, the hardest part to tell because it is so vast, silent and unknown; and those who became – some rainy night with the broken cobblestones glistening behind them and wrecked buildings rising round them – victims and martyrs and brave heroes with _causes_. They stood in their fear reading the shadows and the barbed wire and the distance between here and there. Maybe they heard footfalls. Perhaps the drizzle made the headstones glisten in the cemetery they'd chosen, because tactically the cemetery was a safer route from here to there than open ground or the water.
The most difficult aspect for any historian is holding the right balance between the silent backdrop of the 3.3 million, and the few who came – for whatever reason – out of the 3.3 million and chose the shadows and the bullet run. To set down these completed lives in no more than brief paragraphs, often without even the most basic facts, like their names, is a formidable task. It is easy to skim across the paragraphs and, inevitably, many are very similar. And it is easy to forget that each was a human being, loved and missed; not melodrama on black and white newsreels long ago, but real people running as hard as they could towards something or away from something.
26 December 1965 _Heinz Schöneberger, 27, was shot when a group attempted to escape by car. His three companions, a fellow West Berliner and two East Berlin women, were arrested. They were in a Ford Taunus, Schöneberger driving. It came into the checkpoint and at 12.55 a.m. was ordered into the vehicle control area. The second West Berliner was walking next to the car when the Guard at the control ordered Schöneberger to get out. The second West Berliner tried to jump in as the car accelerated away but he was pulled clear by the Guards. The Taunus drove into another West German car at the pole across the road, ramming it, Schöneberger sprang out and ran. Eleven shots were fired at him but he crossed the line into the West, severely wounded. He collapsed. Members of the West Berlin Fire Brigade took him to hospital but he later died of his wounds._
Sometime in 1965 _an unknown person drowned trying to escape._
In January 1966, an 18-year-old called Hartmut Richter tried to escape through Czechoslovakia to Austria. Richter's story, woven into the larger story of the wall, remains unusual. He grew up in Potsdam and was a bright pupil. He was also orthodox: he joined the _Jungpioniere_ (a children's organisation) because he wanted 'to become like Lenin'. By the next stage, the _Thälmannpioniere_ (a youth organisation), he had become critical. He was taught that people in the West were enemies but he had relatives over there and couldn't classify them as enemies. He was asked if any pupils in his class preferred watching _Bonanza_ on Western television to _Meister Nadelöhr_ , a popular children's programme on the GDR channel. He preferred Bonanza... .
He did not join the Free German Youth14 organisation in view of the realities, he states. 'I had lost his faith in babies being found under the gooseberry bush and Santa Claus' – a way of saying he was rejecting 'actually existing socialism', as the GDR described what they had created and were living within. He took the train to Czechoslovakia but was arrested on the way. It took almost a week to transport him back to Potsdam on the _Grotewohlexpress_ , a prisoner transport train disguised as a yellow postal wagon. He was held in custody but, shrewdly, wrote to his parents expressing his admiration for 'the heroes of socialism'. While he was in custody, somebody mentioned to him that there were weak points in the wall – at a certain point on the Teltow Canal. Interesting, he thought. He was put on probation in May.
7 February 1966 _Willi Block, 32, was hit by two bullets which grazed him and two which struck him 'with full effect' as the official report phrased it. Where this incident happened is not clear._
14 March 1966 _Lothar Schleussner, 23, was fatally shot at the wall._
15 March 1966 _Jörg Hartmann, 21, was fatally shot at the wall._
19 March 1966 _Willi Marzahn, of unknown age, was one of two members of the People's Army who tried to get across at 6.15 in the morning. The Border Guards opened fire on them with pistols and Marzahn, a sergeant, was hit in the head. The other soldier, also a Sergeant, made it to the West. They had been firing with machine pistols at the Border Guards, who were not injured._
30 March 1966 _Eberhard Schulze, 20, was fatally shot at the wall._
Perhaps this is as good a place as any to examine how the system of manning the border worked. It was, as one might imagine, carefully organised and structured, but not actually as complex as it looks at first glance.
Hagen Koch explains it. The Riot Police, the Border Police, the Border Guards, the Railway Police, the _Vopos_ (the People's Police) all guarded the border 'because there was what they called the National Border Protection System ( _Gesellschaftliches System der Grenzsicherung_ ). Directly at the border, there were the Border Guards ( _Grenztruppen der DDR_ ), and this existed from 15 September 1961. When the wall went up there were only the People's Police, the Riot Police ( _Bereitschaftspolizei_ ) and the Combat Groups ( _Kampf-gruppen_ ). The National People's Army ( _Nationale Volksarmee_ ) were not allowed to come closer to the frontier than 1,000 metres.
'On 15 September 1961, all forces were put under the command of the _Nationale Volksarmee_. Ulbricht said that the border was to defend the GDR – it was a protection against the exterior enemy. At all border crossings, the Ministry for State Security, Main Section VI (passport and search section), was present ( _Ministerium für Staatssicherheit, HA VI – Hauptabteilung VI – Pass und Fahndung_ ).
'The correct name of the Railway Police was _Transportpolizei_ – Transport Police. Their uniforms were blue. Their task was also to protect transport in West Berlin on the S-Bahn and interzonal trains between West Germany and the GDR.
'The Border Police were founded on 1 December 1946 and later became the Border Guards. Their task was, in cooperation with the Soviet Army, to arrest anyone trying to cross the border illegally. Their main focus was the border between the GDR and the FRG from June 1952, when it was closed, but at the same time they had to protect the border around Greater Berlin.
'The Riot Police were units of the People's Police who were quartered in barracks. They continued to exist even after the foundation of the _Nationale Volksarmee_ and were under the command of the Ministry of the Interior ( _Innenministerium_ ).
'The official relationship between the Border Guards and the People's Police: there were so-called measures of organization and cooperation (Maßnahmen des OZW – Organisation und Zusammen-wirken). The People's Police protected the border installations on the GDR side against the approach of refugees. Controls were coordinated with the Border Guards at both the national and regional levels.
'There were differing numbers of Border Guards round Berlin – governed by the increasing sophistication of the wall. As an example at the end, however, in the section between Pankow and the Brandenburg Gate, the personnel was: on duty per shift – 3 officers, 12 Sergeants, 94 soldiers. Technical equipment – 5 Border Trabants, 5 motorbikes, 10 radio sets, 4 walkie-talkies, 18 radio sets. That means there were about 1,000 Border Guards on duty at any one time. They worked in three shifts so that gives a total of approximately 3,000 Border Guards per day.
'In fact, for a better knowledge of the surroundings the soldiers had to draw sketches of what they could literally see from their viewpoint. They were, however, rotated [among themselves] in the watchtowers. They knew each other but there was no familiarity between them. Two Border Guards manned a normal watchtower (one of them, the _Postenführer_ , was in charge). On a _Führungsstelle_ – a command post like the one in Treptow – there were additionally drivers, dogs and an arrest group ( _Festnahmegruppe_ ). The Guards worked two different forms of shift: two of twelve hours per day or three of eight hours. The Guards were doing their eighteen-month military service. If they were officers that would be from three to ten years, and at least twenty-five for professional soldiers.'15
25 April 1966 _Michael Kollender, 21, was shot as he reached the vehicle trap in the death strip. He dived into the vehicle trap, crawled and was shot again. Kollender was a member of the National People's Army and had had a Kalashnikov with him, containing a magazine with 14 rounds. The safety catch was off and the weapon set for continuous fire. He also had a man's watch, a letter addressed to him, a letter from Oberlungwitz [a small place in the south], a purse with 0.32 Marks in it, one male ring and other small items._
On this same day _Peter Petermann, 58, died at the wall._
Border Guard Rudolf Loschek was 24 in 1966 and, many years later, explained how the system worked:
Every night at nine o'clock the guard changed. We arrived early to receive our weapons. Then we got our daily orders. I can still tell you exactly how it went. 'I order the securing of the border from nine o'clock until you are relieved. Border violators are to be arrested or destroyed.' And then the command to use weapons: 'You must shoot border violators when all means of arrest have been exhausted. You must not shoot at Allied vehicles, diplomatic vehicles, air targets or children.'16
It was my last day of service before leaving the army. Well, we'd celebrated, we'd had a few. Suddenly someone called, 'There's someone running over there!' I thought he was having me on. I didn't react. Suddenly I saw two of my men with machine pistols. They were already shooting. They were shooting continuously – everybody had sixty shots. You couldn't see a thing because of the dust. So I took my machine pistol as well. Suddenly I saw something so I just shot in that direction, without aiming. I was afraid in case they found out I hadn't fired. 'Why didn't you shoot?' It would have been difficult to talk myself out of it. When the shooting stopped you couldn't see anything, only clouds of dust. There was a person lying there, about 2 metres away from the barbed wire, on the control [death] strip. He was still breathing. He was gurgling. Then he was taken away. We didn't know if he was dead.
A day before my discharge we were commended. I got a watch. I can't say I was pleased with the watch. There was no engraving on it and nobody knew what it was. It did not say ' _For good service at the border_ '.
The postscript to this had to wait until the world had moved and the wall come down.
'We didn't get his wallet or his ring,' Michael Kollender's mother said when, those many years later, she could see the file. For twenty-five years the family had had virtually no information about the manner of his death. 'He was shot many times, many times. It happened early on Monday, between Sunday and Monday. My husband came up and said, "Michael..." and I said, "Is he over there?" He said, "It's a lot worse. He's dead." And the world collapsed for me. It was full of Stasi in the mortuary. I'd got some injections beforehand so I was really calm but numb. I pulled the cloth back and saw the shots. On top of his head he had a big bandage. There was blood on the pillow and here [indicating the right eye] a kind of vein was sticking out. It was all burst and bloody. The back of his head was all shot to bits. And I said farewell, and pressed him.
'They didn't let us out of their sight even at the funeral. Some school friends wanted to give flowers, but they were sent back. They went to every garden nursery and forbade them to supply wreaths. No wreath, you see. The one who shot him – I've thought about it so often in secret – could have shot without killing him, even if he was crippled. But not riddle him with holes from head to foot. I couldn't understand that. [Long pause while she composes herself.] Well, at least now we know. Twenty-five years and it seems like yesterday. [pause, voice lowers.] Twenty-five years.'
29 April 1966 _Paul Stretz, 31, tried to swim across behind a ship at 3.30 in the afternoon. He was shot and went under._
The GDR reports record two more incidents on the same day. At 10.50 that morning two people, one aged 77 and the other 78, were in a sailing boat some 20 metres from the border line. They were stopped, their papers examined, and they were released. The implication must be that they were a West German couple out enjoying themselves on the water and strayed too close.
Then Stertz made his attempt.
10 June 1966 _Elke Märtens, age unknown, died at the wall._
26 July 1966 _Eduard Wroblenski, 33, died at the wall._
Hartmut Richter had now resolved to try and escape again. As mentioned he had heard in prison about a place where potentially the wall was weak: the Teltow Canal, which had attracted many others as it meandered full across the southern part of the city. Richter seems to have reached a point where, although he was only 18, he preferred to risk his life rather than live it in the GDR. Originally he'd considered making the attempt across a lake between his home at Potsdam – he lived with his parents – and the West Berlin district of Wannsee, but other attempts had been successful there and he knew that, after each, the border was tightened. He'd trained and liked swimming but he decided against the lake.
On 26 August he selected a dark shirt and put it on, then put 700 East marks and some documents into a plastic bag and left home. He did not say goodbye to his parents because that would have been a risk for him and a risk for them, too, if they had known. He rode the S-Bahn to the stop at Teltow, a village about 2 kilometres from where he wanted to be. He walked to the canal there and slowly followed it towards the border. That meant passing under the main transit autobahn between West Berlin and Helmstedt, near the extensive GDR checkpoint of Dreilinden, on a country lane.
He had had no chance to recce the area. The weak point was where the canal went into the West and, in these comparatively early days, the area was not illuminated as it would have been in Berlin itself. There was the darkness of the wooded countryside on either bank of the canal and he'd picked a largely cloudy night. The geometry of this: an old S-Bahn bridge, then about 500 metres (546 yards) on the far side of it, the barbed wire on either bank for another 500 metres, then an obstacle across the canal. Beyond that lay the West. Watchtowers loomed, satanic, threatening shapes but too far apart to illuminate the whole area clearly. In the darkness there were shadowy pools on the water and he would use them. Better, it was drizzling and drizzle could only be extra cover hampering the visibility of the Guards in the watchtowers and anyone patrolling the banks.
He slipped into the water beside the old bridge because it screened him from the watchtowers. He was more anxious than afraid, and the water felt cold. The canal was about 40 metres (43 yards) wide and he intended to go towards the West zigzagging from bank to bank, diving and swimming underwater whenever he could, inhabiting the dark places – the searchlights from the watchtowers really weren't like in Berlin. He swam above water as little as possible because the movement of his arms would make a noise as they disturbed the water. When he reached a bank he paused to recover. Progress was agonisingly slow and he had a watch- tower to get past. He came to the point where the wire stretched along the banks, he zigzagged on and past the watchtower and reached an obstacle stretched across the canal. It was a steel-mesh fence and he summoned the strength to clamber up it – he estimates he had been in the water for four hours. As he slipped down the other side he saw a sign: 'YOU ARE LEAVING THE AMERICAN SECTOR' which, of course, warned people that the American Sector ended here, or in his case began here. He was in the West. He passed out – fainted is his word – and was taken to hospital in Wannsee, from where he sent a postcard to his parents telling them where he was.
His father went to the local police station to report Richter missing 'but they said, "You know what these young people are like. They just go off somewhere and they come back." My father said, "No, no, he's written me a postcard from Wannsee!" The policeman was just baffled.' Harmut Richter would see his parents again, but not until the early 1970s.
29 August 1966 _Heinz Schmidt, a 46-year-old Westerner, evidently jumped into a canal out at Spandau and began swimming East. To prevent this 'provocation' the Border Guards shot at Schmidt and he was hit. He managed to swim back. At this moment 'approximately' ten Western soldiers (unidentified) moved into position and fired at the Border Guards. This involved a total of about 50 people. 'About 4.00 the situation was normal again.'_
6 November 1966 _Gustav Lupke, 86, died at the wall._
21 November 1966 _Joachim Stephan, 29, died at the wall._
16 December 1966 _Karl-Heinz Kube, 17, was fatally shot at the wall._
27 January 1967 _Max Willi Sahmland, 37. At 11.07 p.m. he was shot swimming the Teltow Canal. Despite 53 shots being fired he 'arrived on westberlin [sic] territory without being hit'._
Were the East Germans wrong and he had been hit?
1 May 1967 _An unknown person died at the wall._
18 February 1968 _Dieter Weckeiser, 25, was fatally shot at the wall and so was Elke Weckeiser, 22. The National_ _People's Army report says only: 'At 22.50 hours one male and one female, 26 and 22 years old, living in Furstenwalde, were trying to get to the west.'_
6 July 1968 _Siegfried Krug, 29. '01.35 hours. One male person, living in Berlin, was killed trying to escape to the West.'_
15 November 1968 _Horst Körner, 21. At 10.55 at night two Border Guards – one called Rolf Henninger – patrolling in a Trabant jeep out towards Potsdam noticed a policeman hiding behind a tree approximately 10 metres from an old people's home. It was Körner. The jeep stopped and reversed. 'When the Trabant lights shone on the tree' Körner 'came forward with a machine pistol and opened fire. After the first shots', Henninger, hit in the heart and head, was killed. The other Border Guard leapt from the Trabant and shot Körner, who dropped the machine pistol. The Guard assumed that Körner was hit but 'since he did not fall, and since the Guard noted that his driver was dead, opened fire again.' He kept on until Körner went down. Altogether the Guard fired thirty shots. 'A further investigation showed that the culprit had fired fifteen times. Fourteen shots were noted on the Trabant. The culprit still had two full magazines of bullets with him. Due to the angle of the firing, no shots could have reached Westberlin.'_
In February 1969 President Richard Nixon visited Berlin as part of a European tour and for once large crowds were genuinely pleased to see him (his arrival in Rome provoked a riot among 5,000 demonstrators). In Berlin he said, 'Sometimes you must feel that you are very much alone. But always remember, we are with you and always remember that people who are free and who want to be free around the world are with you. In the sense that the people of Berlin stand for freedom and peace, all the people of the world are truly Berliners.' If it was an echo of Kennedy – who had been assassinated in November 1963 – that didn't matter, because it was what they wanted to hear. He visited the wall but at the Heinrich-Heine-Strasse checkpoint where houses from the West directly faced those in the East with only the death strip, narrow here, in between. He looked suitably sombre as he mounted the observation platform. Perhaps someone had told him of Heinz Schöneberger and those awful moments just after midnight three years before, moments enacted just _down there._ Perhaps someone hadn't.
(The geometry of this: Heinrich-Heine-Strasse was a broad avenue running north to south. It had been called Prinzenstrasse and the section in the West was still called that; the section in the East had been renamed after Heine, a poet. A side street bisected the avenue and it was here that the houses faced each other. Those in the West were five storeys and those in the East four storeys, so that all but those residents on the ground floors could see their neighbours clearly across the wall. Beyond the visual they had no relationship. (When it was finally all over, the author tried to interview one of the Eastern residents to hear what it had been like. She was a middle-aged woman who lived on the top floor, so she'd had an excellent view. By her body movements she implied reluctance but, in the Eastern way, was momentarily unsure whether she could refuse an interview. She did, saying, 'I'm very sorry but it's just too traumatic to talk about' and disappeared into the sanctuary of her apartment, taking care not to look back.)
Rüdinger Hering, who lived in the East near Falkensee, responded to the question 'Did you ever get used to the wall?' by replying, 'There were three rows of barbed wire and you can never get used to that. It always reminded me of the barbed wire round concentration camps. We had watchtowers, and arc-lamps every few metres which were very bright, totally bright. The wall (as opposed to the barbed wire) came at the end of 1968 and the beginning of 1969. I remember because I had to join the Army in 1968 and I had my first leave in May 1969.'
Bodo Radtke, the East Berlin journalist who was allowed to travel – it was necessary for his job – says that the situation was a psychological one. 'You could live all your days in East Berlin without the West being much of a factor [except TV] and then, when you were going on assignment, you'd take the train from Friedrichstrasse and, as it looped over the death strip and the watchtowers into the West, you'd suddenly think "It simply cannot be, this is not possible in the middle of a city, this is crazy." You'd go and do your work, you'd return to the East and somehow let the craziness drown into your normality – until the next time.'
1 April 1969 _Elmar Scholz, age unknown, was fatally shot at the wall._
9 April 1969 _Johannes Lange, 29, was fatally shot at the wall._
13 September 1969 _Klaus-Jürgen Kluge, 21. 'At 20.40 hours, one male person trying to get across the border...'_
20 September 1969 _Leo Lis, 45. 'At 20.00 hours a male person, 45 years old, living in Kamenz [a town in the south] was stopped trying to cross the border.'_
In October, Willy Brandt became Chancellor of the Federal Republic. Despite, or perhaps because of, his experiences as Mayor of Berlin he determined to approach relations with the East constructively and realistically. The wall stood, whether anybody liked it or not, and had now for eight years. He'd begin what became known as _Ostpolitik_ which, it seemed, marked the end of something and the beginning of something. Nobody knew what exactly, but at least it was a kind of movement across the geometry.
8 October 1969 _Wolfgang Puhlfüss, age unknown, died at the wall._
The first bitter decade was over.
* * *
Quotation at head of chapter: _Willy Brandt: Portrait and Self-Portrait_ , Klaus Harpprecht (Abelard-Schuman, London, 1972).
1. Interview with author.
2. _Berlin: Von der Frontstadt zur Brücke Europas_ , Rainer Hildebrandt (Verlag Haus am Checkpoint Charlie, 1984).
3. Interview with author.
4. Hildebrandt, op. cit.
5. Ibid.
6. _Escape from Berlin_ , Anthony Kemp (Boxtree Limited, 1987).
7. I have leaned heavily on _The Ugly Frontier_ , David Shears (Chatto & Windus, 1970), as well as _Escape from Berlin_ in re-creating this episode. Shears (ibid.) writes that 'one Dresden citizen who missed the last train to Berlin [after he received his message] took a taxi all the way from Leipzig. Completely exhausted, he reached the tunnel in time and made his escape.' Presumably this can't have been the same man who went to his rendezvous at Friedrichstrasse station.
8. Shears, ibid.
9. www.wall-berlin.org
10. Details of this particular escape are extremely vague, although the Isetta stands in the Museum at Checkpoint Charlie.
11. I'm indebted to Inge Donnell, the dogged translator, for shedding some unexpected light on this sad paragraph. The word used for sandwich was _Stullenpaar_ , a Berlin expression which could be translated as 'doorstep' and seemed discordant in an official document.
12. This report was given to me by Hagen Koch, most fair of men, who wanted to demonstrate that there were violations on both sides, not just East. He gave me a second example, but it is in the next chapter.
13. Interview with author.
14. The Free German Youth was founded in 1946 as an 'anti fascist movement of all young people' ( _German Democratic Republic_ , Mike Dennis), but by 1952 was firmly part of the SED's structure. Membership was open to anybody from 14 to 25 and in 1982 had 2.3 million members. Every 'school, enterprise and university' had a Free German Youth organisation (Dennis). It 'strove to mould young people as socialist personalities.'
15. Hagen Koch adds that: the following Border Regiments ( _Grenz-regimenter_ , abbreviated to GR) were deployed in the Border Command Mitte ( _Grenz-kommando Mitte_ , Berlin): GR34, GR38, GR33, GR35, GR36, GR42, GR44. There were two Border Training Regiments ( _Grenz-Ausbildungsregimenter_ , GAR 39 and GAR 40) and one temporary regiment protecting the official crossing points ( _Grenzübergangssicherungs-regiment_ ).
The main training area was called _Truppenübungsplatz Streganz_. The headquarters of the Border Guards was in Pätz, south of Berlin and the headquarters of the Border Command Mitte (Berlin) was in the Karlshorst district of East Berlin.
The barracks of the respective regiments were: GR33 in Berlin-Treptow; GR34 in Groß-Glienicke; GR35 in Berlin-Rummelsburg; GR36 in Berlin-Pankow; GR38 in Hennigsdorf; GR42 in Heinersdorf; GR44 in Potsdam-Babelsberg; GAR39 in Berlin-Wilhelmshagen; and GAR40 in Oranienburg.
16. I have unashamedly used the television programme _First Tuesday_ – which was made by Yorkshire Television and shown in November 1991 – for the whole of this episode, as well as one later in the book. The programme made me reflect on how ephemeral television is: professionally speaking, the contents were so moving and so profound that they deserved something more permanent. I hope the inclusion in this book goes some way towards that.
## SEVEN
## _The Bullet Run_
I thought that although all this was only 6 or 7 metres away I would never go there. It would have been easier to go to the moon. The moon was closer.
Roland Egersdörfer,
Border Guard
By now the two halves of the city were moving at different rhythms, obeying different impulses and it struck a visitor with immediate impact. West Berlin had become a bustling place, cosmopolitan, almost glamorous, a cocktail hour of draft-dodgers, revolutionary students and solid citizens making a lot of money; East Berlin had become austere, impersonal, a heavy, brooding place of wide, empty avenues and small, often empty, shops. Author John Ardagh coined a phrase about Dresden in the late 1980s: it still had a 'stunned, walking wounded feel'.1 In the early 1970s East Berlin had had that.
The wall subconsciously conditioned inhabitants and visitors to the truth that the other side of the city – East or West – was very different. The edifice, itself heavy and brooding, made this statement because, if the other side hadn't been different, the edifice wouldn't have been there. When you crossed, or even gazed at, the web of concrete lintels, tank traps, patrolling jeeps and looming watch-towers manned by Border Guards who really would shoot, you were being subconsciously prepared for the difference. That helped to accommodate the impact, but the impact remained hard enough because you might be on an 'ordinary' city street like Friedrichstrasse on both sides. More than that the division began to be reflected in the people, Westerners fashionably dressed and with a jaunty way of comporting themselves, Easterners in stout clothing, moving ponderously.
1 January 1970 _An unknown person died at the wall._
The tetchiness – perhaps the extreme sensitivity – in every nuance of the geometry of division remained in full force. A GDR report gives an account of what should have been a trivial incident:
10.06.1970, 17.45 hours, the soldier Peattie, William, Number I/A/SH-2373 0360, born on 21.01.1934 in Glasgow (according to his own words), belonging to the English2 troops stationed in West Berlin, went over the border of the GDR at Montgomery-Camp pumpstation.
General-Major Poppe, commandant of the capital of the GDR – Berlin – stresses that this is definitely a transgression of the state border made by the English soldier and a serious violation of the sovereignty of the GDR.
The town commandant points out again to the Commander of the troops stationed in West-berlin, General-Major F.C. Boyes Lyon, that the People's Army of the GDR can not permit such transgressions of the border.
For this, the Commander of the English troops stationed in Westberlin [sic] is fully responsible.
Today, the 11.06.1970, at [here there is a gap in the report] hour, the soldier Peattie, William was handed over. His kit and other personal belongings were not taken from him.3here there is a gap in the report
19 June 1970 _Heinz Müller, 27. 'At 01.50 hours, one male person got over the wall from WestBerlin to the GDR... .' There are no further details of why this particular man was going the other way, or what happened to him except that he died._
We went to Berlin as a foursome, the Hiltons and the Woodcocks (John and Christine), because we were curious to see what it was like. We drove and, although this book is in no sense a travelogue, getting there is interesting. Across the plain past Hannover to Helmstedt was a straightforward autobahn journey and at Helm-stedt – I can see this now, quite passively as I must have recorded it with the shutter open and memory developing it, fixing it – there was the last petrol station in the West then a little further on a hut in the middle of the road. It seemed to be unmanned.
Beyond it you crossed into an open jaw in the wall: watchtowers in the distance, a truck filled with boulders up a ramp (so it could be released and come down to block any escaping vehicle), a machine-gun emplacement behind sandbags in the middle of the road, uniformed men waving you to park the car here, get out and go into that Customs building.
The contrast between the unmanned hut and all this was making a statement no sane person could misinterpret.
We got the visas and stamps and moved along the umbilical cord towards West Berlin. You passed villages and hamlets and fields and never saw anyone. A plague might have been upon the land. Darkness fell and in the East there seemed to be few lights at all, until, all at once, a great saucer of neon – white, and rising into an aura – was making a statement, too. We were approaching West Berlin. It was as you might expect it to be, extensively rebuilt and bustling. It had a sheen, the sense of a frontier town during a goldrush where people were making money.
We went East the next day on foot.
Checkpoint Charlie was like Helmstedt, hitting you hard from nowhere. On the Western side were shops and shoppers who went about their business oblivious of the wall and the watchtowers. The Allied hut was in the middle of the road but you weren't even waved through. If you wanted to go over there you go over there, _buddy._
You crossed Hagen Koch's line into another jaw in the wall. The Border Guards with their fossilised faces were forbidding and instructed to be like that. The Customs, in a long hut, was deliberately mysterious (they handed your passport back through a hatch so you couldn't see what was happening to it) and equally fossilised. You needed more stamps on your visa and you had to change money in another long hut with queues and forms and receipts and written threats about not trying to take money in or out. The GDR was shaping you to its will and you hadn't even got beyond the check-point to find out what the place looked like – but you suspected it was all like this.
At this time Britain did not recognise the GDR and once you crossed the line you were on your own, _buddy_. Amid all these faces and uniforms you didn't forget that. You might not have had rights there and the cavalry certainly weren't coming to the rescue.
Friedrichstrasse on the Eastern side, once you'd left the check-point, pleaded with you to remember that before the war it couldn't have looked like this. It was a widowed street in eternal mourning, whole lots empty and overgrown or strewn with rubble, elderly buildings shorn off and patched up: buildings on crutches. The money which made West Berlin bustle _within sight_ over there, 50 yards away, was completely absent. You felt in the presence of the bombing raids last night, but they'd ended twenty-five years before.
On one of the sidestreets a small crowd had gathered and we advanced to see what it might be. A British company was shooting an advertisement for Polo mints and they'd hired a beautiful model who stood there wearing a white fur coat as the camera licked over her. The Easterners had gathered opposite her not, we sensed, to ogle but just because something was happening in a city where, clearly, apart from wearing heavy coats and walking with your head down, not much ever happened. We joined the gathering. A young girl – maybe in her late teens – marched up. She wore a plain raincoat. She scattered words across us like machine-gun fire and gesticulated in the way that Germans have. Clearly she was empowered to order us to go. Why? We spoke no German and we couldn't ask. Here, however, was the next statement: our standing there constituted in some pathological way a threat to something. She hustled us until we were gone.
We walked to Friedrichstrasse station and had some lunch in a basic café where the locals read the day's newspapers which had been clipped between two wooden sticks (acting as spines) so they would last longer, could be handled easily and passed on to the next customer. We'd seen exactly the same thing in West Berlin cafés... .
We walked down Unter den Linden to the square where the old _Schloss_ stood before they gave it the dynamite. A large number of chairs had been stacked for some imminent parade and a dozen soldiers guarded them. Everything here seemed to be under guard, even chairs; everything here seemed to be suspended animation caught in its own caution and its own exhaustion; and everything here seemed held down under some authority which was nowhere and everywhere. You kept thinking about the dead hand of that reaching out to you, and taking you, and you did have no rights and you did have no representation. You didn't need to have read Kafka to feel Kafka. And it was _real._
We walked to Alexanderplatz and went into a bookshop (which was still there twenty years later). A jam jar had been placed on a table with a card propped against it asking for donations for North Vietnam – it was 1970. The jam jar already contained coins and crumpled notes lying on them as autumn leaves might have done. It may be that the customers believed, and made their contribution, and were really going to defeat the Yankee imperialists and make the world a better place. It may be that the jam jar was brought out every morning with the same coins and notes in it to appear as if the customers believed and nobody ever put anything in.
Outside, one of those silly things happened. A square-set man with a briefcase was coming towards me and there was a moment where he'd go left or right but he didn't, ploughed on straight at me and I had to side-step him. I thought he'd come _for_ me and my nerve went. I said, 'We've got to get out of here.' We walked along Leipziger Strasse to Checkpoint Charlie and went through the wringer of all the controls again. I was ordered, alone, to the currency hut – a bad moment – to settle all our little group's money, however they decreed it must be settled. Then we stepped back across Hagen Koch's line and if we didn't feel as profoundly as Dennis L. Bark had felt at this instant, we did feel a lot better.
In the Allied hut nothing stirred – another statement.
We'd got visas for Poland and went for the day, and thereby hangs a tale, because the lady at the last petrol station in Poland thought we were German because we were in a VW, and filled the tank with diesel in order to demonstrate her dislike of Germans, and the VW struggled back to the GDR border crossing at Frankfurt-Oder and expired with naked flames ebbing from the exhaust pipes. By the time we got back to England we'd missed two night's sleep and had a _lot_ of stamps in our passports. We'd devised a system. Woodcock, in the front passenger seat, had a folder with all the documents in and whenever we were challenged (it seemed to happen hourly; even on the umbilical cord back to Helmstedt traffic police flagged you down) he proferred the documents, got them stamped, and we moved on to the next one.
Now, safe across the Channel and motoring through Kent, the garden of England, I needed to fill the car up. I pulled into a petrol station. Woodcock woke from a deep sleep, fingered the folder and murmured, 'What documents do they want?' I mention this because after only a week his subconscious had been altered, so that a forecourt and bright lights must mean an authoritatian inspection or visa stamping or currency exchange or something like that. What if he'd had that not for a week but all his adult life? A week had been enough...
18 July 1970 _Klaus Schmock, 19, died at the wall._
2 August 1970 _Friedhelm Ehrlich, 19, a corporal in the Border Guards, was prevented from escaping while not on duty. He was shot in the lower stomach and died in hospital._
7 August 1970 _Gerald Thiem, 42, a West Berliner, was fatally shot at the wall._
On 3 September (1971), what was known as the Quadripartite Agreement on Berlin was signed. _Ostpolitik_ was starting to happen although, by definition, the consequences had to be complicated – and were. Author Mick Dennis simplifies this as well as anybody: 'The favourable reception of Bonn's overtures among the GDR's Eastern partners also gave cause for concern. Bonn concluded an accord with the Soviet Union in August 1970 and with Poland three months later. Four-power negotiations on Berlin commenced in May 1970. Brandt made it clear that a settlement on the Berlin issue was necessary to ensure sufficient support in the Bundestag for the ratification of the Moscow and Warsaw treaties. Ulbricht's strategy was to make acceptance of the GDR's full sovereignty a precondition for entering into formal negotiations with Bonn. From Ulbricht's point of view the question of Berlin's status, and the GDR claim to the exclusive right to control the access routes to West Berlin, were intimately concerned with sovereignty. West Berlin should be treated as an autonomous political entity on "GDR sufferance", not on any four-power status.'
The Soviet Union, acting as a global power, had its own interests. As Dennis says, it and Poland 'completed negotiations on their treaties with Bonn without insisting on the prior recognition of the GDR. Ulbricht was also worried about the Soviet Union's concession, in the course of the Four-Power negotiations on Berlin in 1970, that certain economic and political ties existed between West Berlin and the FRG. At the TwentyFourth CPSU Congress in March 1971, Brezhnev decoupled agreement on the status of Berlin from the issue of the acceptance of the GDR's full sovereign rights.' It also meant that agreement was reached on the transit routes so that harassment along the autobahn virtually ceased, and, tying up one loose end, a road to the Steinstücken enclave was agreed. The problem of the other enclave, Eiskeller, was solved in 1972.
Dennis concludes that 'Ulbricht's determined opposition to Four-Power responsibility for West Berlin, and his obduracy on the transit routes, threatened not only a speedy settlement to the negotiations on Berlin but, because of Bonn's coupling of a Berlin agreement with the ratification of the Moscow and Warsaw treaties and the calling of a European Security Conference, also jeopardized Brezhnev's policy on détente. When, on 3 May, the SED announced Ulbricht was retiring as First Secretary, most observers assumed that the Soviet Union had been instrumental in his removal.'4 Within three years, sixty-eight countries would recognise the GDR as a sovereign country and it would be admitted to the United Nations in 1973.
The Steinstücken enclave after 1972 and unhindered road access. The East–West railway line was a potentially weak point in the wall.
25 December 1970 _Christian-Peter Friese, 21. 'At 00.03 hours, one male person, 22 years old, from Naumburg [a town in the middle of the country] trying to cross from the GDR to West Berlin...'_
7 January 1971 _Rolf-Dieter Kabelitz, 20, was fatally shot at the wall._
In late January 1971 telephone links were restored between the two Berlins. They had been cut in 1952 and few lines had remained before 1971: the Western Allies' headquarters and the Soviet Embassy was one, the Western and Eastern airports – Tempelhof and Schönefeld – another. The GDR needed to contact S-Bahn stations in the West because they ran the service. Evidently the two police forces had a teleprinter link and, among other things, it was used to try and identify bodies found at the wall. Initially ten lines were restored (there had been 4,000 in 1952) and from July were charged at the same rate as foreign calls.
Telephone links remained problematical between the two Germanies, however, and an anecdote illustrates that. A farm was bisected by the frontier and the two brothers running it decided that one would stay in the East, the other taking their mother to the West. The frontier cut the land into two distinct farms and the brothers farmed them, often doing the same tasks on the same days within sight of each other. The Eastern brother had children, and presents were always sent from the West on their birthdays, but phone calls could take up to nine hours to cover the 200 metres between the two farmhouses. The calls had to be booked and went on a circuitous route down to Bavaria, across and back up the other side.
In May, as has been seen, Ulbricht was replaced by Honecker.
15 July 1971 _Wolfgang Hoffmann, 29, died at the wall, possibly shot._
24 July 1971 _Werner Kühl, 22. At 10.45 p.m. a Border Guard in Britzer Allee, a wooded area in Treptow, noticed two people sprinting. They were 100 metres away and heading for a ditch before they reached the wall. The Guard opened fire, killing one and injuring the other in the left forearm. He was taken to hospital. 'The injured person was a Westberlin citizen, aged 22. An investigation is continuing. Further information is not available.'_
On 3 September the Quadripartite Agreement was formally signed, as we have already seen.
2 December 1971 _Dieter Beilig, 30, a West Berliner was arrested, possibly at a checkpoint, but escaped from where he was being detained and was fatally shot._
1 January 1972 _Horst Kullack, 23, was fatally shot at the wall._
13 January 1972 _Günter Semmler, 15, was fatally shot while trying to climb the wall._
The division did not spare friends and relatives. Klaus-Peter Grohmann, the man who on Saturday 12 August 1961 had yawned in an effort to make his mother-in-law go home, lived in the south of West Berlin not far from Potsdam. He'd understood 'only a few days afterwards, when they started to build the wall properly, that it was going to be permanent. Of course I had friends and relatives in the East, in Potsdam, but telephones were cut, everything was cut and you couldn't communicate with anybody over there. The next time I went to Potsdam was 1972 when the new contract was made between the East Germans and the West. I went to see my relatives there and in Dresden.
'I said to my wife that evening, "Tomorrow morning we're going, please take the music cassettes out of the car" – it was forbidden to take them into the East – and she put them in her handbag. In the morning I said, "Come on, let's go, we've got to be early at the border because it could take a long time to cross." We got to the border and they asked the usual questions: "Do you have weapons? Do you have cassettes?" I said no. They told my wife to open the handbag and there they were – she'd forgotten to take them out. But who the hell thinks like that, who the hell thinks about cassettes? It cost us three hours, by the way.
'The visit was a pretty tearful affair and at the beginning it was only for one day. You had to apply for a permit, you had to pay an "entrance" fee and, as I've said, when you reached the border you were heavily searched. Whenever we left West Berlin by car after the wall was built to go to West Germany [the transit route] it could take three hours at the Dreilinden checkpoint, two hours to drive to Helmstedt and then sometimes three or four hours at the Helmstedt checkpoint: they'd put a rod into the petrol tank, they'd look underneath the car...
'I went back to East Berlin in 1989 though, of course, I'd been to Potsdam that time. We'd drifted apart and, anyway, they weren't that close relatives. I don't know. People became different and we've only really been able to talk since reunification. I mean, we talked then as well but we tried to be careful with politics, and there was the question of getting them into trouble. That's why we didn't actually visit them for so long.
'I love soccer, I really love soccer and when the East Germans played and I heard the commentary I started to hate the team and in that team there were Berliners. If, for example, France won against the GDR I said to myself, "Great, that's the way I want it." This is how bad it got, and the same applied to the people across there.'5
14 February 1972 _Manfred Weylandt, 30. At Rummelsburg, an area in Treptow, about 11.30 p.m. a Border Guard at the Schilling Bridge noticed a person trying to swim across the Spree to West Berlin. He was shot and, in the official report, probably killed. The use of the weapon was 'parallel' to the border, meaning no shots entered West Berlin._
The division did not spare even the dead.
The Invaliden cemetery, perhaps the most famous in Berlin, was in the East beside a canal, and the canal was the border. The cemetery was subdivided into what have been termed 'burial fields' and three of these lay in what was now the death strip. Another three lay within the area designated as forbidden to ordinary Easterners because it was so close to the border.
One official publication subsequently said the cemetery was 'full of traditions and contradictions'. It had been neglected after the war and the GDR would have levelled it except that it contained the graves of General von Scharnhorst, a famous nineteenth-century military figure who created the Prussian General Staff, as well as other notables. They were considered too important and too emotive to permit the complete levelling. (The cemetery also contained the graves of General Werner von Fritsch, accused under Hitler of homosexuality and killed on active service in Poland; and a war minister whose headstone was riddled with machine-gun fire from 1945.) These, including the imposing Scharnhorst monument, designed by the famous Berlin architect Schinkel and regarded as classical, were only open to the public from a distance and then only twice a week. However, step by step, in 1962 and 1966, then between 1972 and 1975, about half of the 6.2 acres were cleared, until only 230 of 3,000 graves remained.
7 March 1972 _Klaus Schulze, 19, was killed at 9.45 p.m. 400 to 500 metres from some dwellings at Falkenhoh, a village in the countryside to the east of West Berlin. A second person escaped in the direction of Gross Kuhlake, a very small Western enclave. At 11.25 two West Berlin policewomen were between 400 and 1,000 metres away but the official Eastern report said that they did not involve themselves._
30 December 1972 _Cengiz Koc, a 5-year-old Turkish child living in West Berlin, drowned._
23 January 1973 _Peter Kreitloff, 30. Shot. No further details are known._
15 March 1973 _Horst Einsiedel, 33, was fatally shot at the wall._
27 April 1973 _Manfred Gertzki, 30, made his attempt near the Reichstag at about 5.45 p.m. The Border Guard immediately shouted 'halt' and fired warning shots but he ignored them. 'Aimed fire' opened up, he was hit and fell into the Spree. The Border Guards set off by boat – for a moment Gertzki resurfaced – but the current was too strong. After other boats and several divers joined the search the People's Police retrieved the body around 6.30. Some hundred West Berliners had gathered and, by the time the body was brought out, twenty of them were still there. Several took photographs._
14 May 1973 _Siegfried Krobot, a 5-year-old West Berliner, drowned._
20 July 1973 _Klaus Gomert, 18, died at the wall._
On 2 November, Roland Egersdörfer began his three-year National Service. It would take him into the Border Guards and I propose to give an extended portrait of him because Border Guards, those ominous figures with the binoculars in the watchtowers, were as human as anybody else. Egersdörfer was from Frankfurt Oder on the Polish border 100 kilometres from East Berlin, and a letter came telling him he had been called up and where he should report – Perleberg, a town in the north – for his six months of basic training. 'After these six months you did not know where [on the border] they would send you. The first day you'd get your equipment and uniform then you'd go to your room and arrange your things. Then you started training, quite normal training, although every second day it was political: one day you spent eight hours in the field – maybe marching 15 kilometres – next day you had lessons in Marx and Lenin and so on.'
In November or December _Anna Kirste, a 78-year-old West Berliner, drowned._
5 January 1974 _Burkhard Niering, 23-year-old reserve policeman, had a machine pistol and made his attempt at Checkpoint Charlie about 7.35 p.m. He took one of the men working in Passport Control hostage after firing two bursts and tried to make this man run with him as cover. The alarm sounded. Two other men from Passport Control shot Niering in the stomach and in the wrist. The incident lasted three minutes and he died in hospital at 9.05. During the incident there were approximately twelve cars, one an American military vehicle, inside the checkpoint and the checkpoint itself was closed from 7.35 to 7.45 – a serious contravention of the Four Power Agreement, of course._
On 12 April, Egersdörfer's training finished and he was a sergeant. 'I had ten days holiday, came back and they told me I was going to Berlin. They said the second company of Border Regiment 33. I didn't know what that meant.' He soon would. With his equipment he went by train via Potsdam to Köpenick 'and that was our destination. Everyone got out and our names were called. There were several lorries and when your name was called you were told which one to get into. You were driven to the barracks. The main building was of red brick and had a fence round it. Although it was only about 300 metres from the border, this was not the section we were to guard. During the first days I didn't think much about guarding the border because I was very happy to be in Berlin, not somewhere out in the country. In order to get to know our section of the border, I accompanied patrols for more than a week.'
Egersdörfer's section was 23.8 kilometres, and each company comprised seventy men. They had to man between thirty and forty control points – the watchtowers and underground stations where the Western trains never stopped. Intriguingly, Egersdörfer reveals that not all watchtowers were always manned. If there were three close together, like the three at Bernauer Strasse forming 'a relatively small section', one might be unmanned. 'The watchtowers in the 1970s were a bit different to those we knew later and in the daytime you couldn't really see if they were manned.'
A precise ritual had to be enacted before each shift. 'There was always a specific time when we were told we could get our weapons and then we gathered outside. We had a kind of rucksack but its contents were controlled. You could take food and a thermos flask but no bottles [which could be broken and used as a weapon] or cameras. If you were on the early shift you'd be given breakfast at the barracks. The _Postenführer_ [man in charge] had binoculars and a Very light pistol. The leader of the first group would say, "We are all present and we are prepared", then the leader of the second group would say that, too. The company Commander or sometimes his deputy would address us. Only then were we told where we had to go and with whom.' There were two Guards to a watchtower.
'The Commander had a piece of paper and he'd read out who was the _Postenführer_ of each watchtower and who was his companion, an ordinary Guard. During the first year the watchtowers had names but after that they were given numbers, so Ackerstrasse [at Bernauer Strasse] was 25. We received the "watch tower table" which was four pages full of technical instructions. For example, which order you should fire the Very lights, which colours, who should respond to them and so on. These were changed every four weeks. When you were in the watchtower and wanted to signal an escape you had a button, but if you happened to be patrolling between watchtowers you fired the Very light to signal it.
'The Commander might speak about the previous day, evaluating any incidents. Then the pairs of Border Guards stepped forward and were given a special order. "Border break-outs are to be avoided. People who try to damage the border are to be arrested or destroyed." There was no specific shoot-to-kill order written down – the order was always given orally.'
The Border Guards were required (in theory, certainly) to give a warning shot and if an escaper ignored that, to shoot to wound. 'Everybody knew that it is very difficult to target, say, someone's arm at a distance and we had Russian machine pistols, 30 bullets in the magazine and 30 in your pocket [and firing in bursts threatened precision].'
They were taken to their watchtowers by lorry, the pairs dropped off along the way. There was a metal gate at the inner wall and the _Postenführer_ opened it with a key, and locked it after they had gone through (but 'sometimes they forgot to lock it again and if the Commander came past and noticed, they were in trouble'). They walked to the watchtower and the handover required its own special ritual because at no moment must security be off-balance. They positioned themselves at each side of the watchtower ('one to secure this direction, one to secure that direction') while the ordinary Guard came down and replaced the _Postenführer_ who went up and had the watchtower formally handed over to him. The departing _Postenführer_ came down and replaced the newly arrived ordinary Guard, who went up inside the watchtower. The replaced pair walked to the gate, went through, locked it and waited for the bus to come back from its dropping-off points. 'It was said that if all four of them had been in the watchtower at the same time for the handover they'd talk to each other and not be so attentive' – being in fact, off balance, and any escaper watching diligently over days or weeks would observe that.
Once settled, the two Guards faced the eight hours. There was nothing to do except watch and talk. 'After some time you'd know each of the seventy people in the group and their lives. You'd also talk about possible situations and what you'd do. "What if an escaper came from just _there_?" I don't know of any Border Guard who said "I won't shoot" although I am sure some of them thought that.'
These strange, forced relationships had, of necessity, an under-current running through. Would any Guard broach the subject of escaping themselves? I asked Egersdörfer if, during these long hours of talking, he felt the Guard with him was looking for his reaction, probing a little. He only had that feeling once, he said, and then of course there was always the risk that they'd been put up to it by the Stasi to gauge _his_ reaction. It happened on patrol in Bernauer Strasse and the other Guard asked 'What would you do if I was going to escape?' Egersdörfer replied 'Rubbish. You couldn't get over the wall – it's 3 metres high.' Egersdörfer, who cut the conversation with the Guard at that point, is still not sure whether he was serious or not. Border Guards were known to kid each other about these matters, and sometimes even pretend they were going to make a run for it, but these were visible games and definitely not serious; they passed the time.
'Each group had a Stasi officer and he rated Guards for reliability. They always paired one who was reliable and the other who was not so reliable. If the Stasi officer noticed that one particular pair got on well together they were kept apart. The watchtowers didn't all have the same status – some were more important than others. They were termed "confirmed" and "unconfirmed".'
There was also the matter of communicating with headquarters, other watchtowers and signalling an escape attempt. 'We had three buttons you could push: one [the middle button] to talk to the section leader, one to talk to another watchtower and a third so you could listen to what everybody was saying. You could not speak to another watchtower without everyone hearing – although we did say funny things sometimes and make jokes. After two years, you could see on a board which watchtower was speaking. This was only within a section. If something had to be reported, it would go to headquarters who would contact the other sections.
'We had heating but it was cold in the winter. The heating was the same as in the old overground trains, metal heaters they put under the seats and which people took for their summer houses. They were quite good!'
The watchtowers had no toilets and the bodily functions might themselves have pitched security off-balance so a ritual for this had to be enacted, too. 'If someone wanted to relieve themselves, they called for the patrolling jeep. You had to wait for this to arrive. One Guard would go upstairs to replace the man who needed to go, he went down and did it against the inner wall. Then he went back up, replaced the replacement, and the patrol jeep went on its way.
'In my time, you had windows consisting of three panes of glass and the middle pane could be opened but if the situation was urgent and you didn't have time, you shot through the glass. In fact, in my time not all watchtowers had windows which could be opened.' This meant, of course, that the glass was not bulletproof, making the Border Guards themselves targets for any sniper in the West. 'When you are in a watchtower for eight hours, you think about these things. In a sense, you were a prisoner of the wall. And it was very hard at New Year because if you were patrolling they'd throw fireworks at you.
'When your shift was over, your feelings were always the same: "Thank God I didn't have to shoot." Another feeling was always the same, too, for all the Guards: "If you shoot, you may kill somebody and if you don't shoot you will be punished." Everybody hoped it wouldn't happen in their section on their shift.'
(As it happened, Egersdörfer never did have to shoot in his three years. He was asked at an interview in January 2001 if he would have done. 'I have thought about this a lot and today it is easy to say, "No, I wouldn't" but at that time I think I would. That was because of our education and our political training. We were told that everyone who tried to escape was a criminal and we were told this in a way which we really believed.')
'One of my first periods of duty as _Postenführer_ was during the night of 11/12 June on the watchtower _Humboldthafen_. It stood about 10 metres from the railway viaduct where trains and the S-Bahn came from Friedrichstrasse station and went over to the Lehrter station in the West. At about 22.00 we replaced the other pair of Border Guards. On a watchtower, there are always two directions that have to be guarded: always the main direction (areas with obstructions to a clear line of sight, difficult to overlook, with buildings on them, etc.) and this is always the direction for which the _Postenführer_ is responsible. In contravention of the regulations, I let the other Guard persuade me that he could do my job and he took the main direction, which was to the left. This section included the railway viaduct and the unlighted railway line which you could see at an angle of 20 degrees. I secured the direction to the right, a section of about 200 metres up to the checkpoint at Invalidenstrasse. No incidents happened during that night.
'The morning shift arrived at 6.00. While we were cleaning our weapons at about 7.30 in the barracks we were called to the company Commander. We were asked who was in charge and whether we had noticed anything. We said no. What could it have been? The Commander said that a ladder had been discovered leaning against the railway viaduct. He asked us again but we said we had not noticed anything and we were allowed to leave.
'What if a person really had escaped along the viaduct and my Guard hadn't noticed it? The most impossible thoughts went through my head. I was in charge, I was responsible. The punishment could easily be prison and for days I could hardly sleep – there was this permanent uncertainty – but we continued our guard duties.'
17 June 1974 _Giuseppe Savoca, believed to be a 6-year-old Italian living in West Berlin, drowned._
21 June 1974 _A man, possibly a Border Guard and between 30 and 40, was fatally shot at the wall._
'One afternoon, which must have been around 30 June,' Egersdörfer says, 'myself and the other Guard, plus the political education officer responsible for our company, were summoned to the Commander. Two civilians were present. We were asked again whether we had noticed something unusual that night of 11/12. He informed us that between 02.00 and 03.00, the night-time break when the S-Bahn wasn't operating, a male person had crawled under the railway lines in the direction of the Lehrter station. The incident became known because the person returned to the Chauseestrasse checkpoint on 15 June without having given himself over to any West Berlin authority.6 We were told to leave.
'Some days later, the incident was evaluated in front of the whole company and the company Commander was very upset that such a thing had happend in his company. From that time on, he would remind me of it at any opportunity he got.'
In May, Brandt resigned as Chancellor of West Germany, brought down after a GDR spy was discovered in his office. Paradoxically, by now the GDR was an accepted member of the international community, and the international community itself – or rather thirty-five European countries plus the United States and Canada – prepared to regulate its affairs. The postwar boundaries were recognised and human rights (including the free movement of people and ideas across frontiers) were agreed. It was known as the Helsinki Agreement and it was signed in August 1975.
The importance of this in the East remains problematical but there can be no doubt it generated hope and a subsequent disillusion when little or nothing changed. Frank Eigenfeld, an Eastern biologist, has said that 'during the seventies, hope blossomed again because of the Warsaw Pact treaties with the Federal Republic, and when those did not end up changing much, we began to hope again during the so-called Helsinki process.'7
On 25 August, with the episode of the viaduct escape still unresolved, Egersdörfer got five days' detention and, 'to be honest, I was happy not to have experienced something worse'.
The paranoia pervaded everything. Egersdörfer records this:
In the summer of 1974 Guards were in a street in the area of Bernauer Strasse, near the house that formed the inner wall.8 A woman of about 35 approached our Trabant Kübel9 and said, 'I do not know whether you are the right people to speak to but I live in this house and every evening – and during the night – I can hear knocking and scratching noises from the basement nearest to the border. It's the basement of a man who was in prison once for trying to escape. Maybe he is digging a tunnel to West Berlin10 because such things are said to have happened in the past.' Then she said goodbye and went away. I reported the incident and we were told to go back in one hour. When we got there a man in civilian clothing was waiting for us and he introduced himself as an employee of the Ministry of State Security. He asked us about the incident and showed the strongest interest in the woman who'd talked to me. I couldn't give him her name because we hadn't thought of asking her for it so a description was all I could do. I don't know if anything came of the incident.
Egersdörfer remembers something else:
Twice a year, our company was deployed in the section of area 35, which went from the bank of the Spree at the Reichstag via the Brandenburg Gate to the south – every six months every company had to go to a field camp for training [and therefore had to be replaced]. I was on the night shift in the section from the Brandenburg Gate to the checkpoint at Friedrich-Zimmer Strasse [Checkpoint Charlie]. I was patrolling on a motorbike. At about 8.30 in the evening it was already dark and the watchtower at the Brandenburg Gate informed us that 'a male person' stood at the inner wall. I was ordered to check this person. We went there and saw him. I opened the gate through the inner wall so that my driver could take the motorbike out. I approached the person and asked him for his identity card. There was no permission in the document for him to be in the border area and it gave Guben [on the Polish border] as his place of residence. He was dressed very normally, an anorak, jeans. I asked him what he was doing and he answered, 'I want to go over there!' I asked him, 'What do you mean by over there?' He said, 'Well, over there!' and he pointed towards West Berlin. I asked, 'Where is the other one?', on the assumption that there would generally be two making an escape attempt. 'I am alone!' I said, 'You are now under arrest. If you try to escape I will use my pistol.' He nodded – that was all. I checked him over and then fired my Very light – one green star. Shortly after, the company Commander came, questioned him once again and took him in a Trabant Kübel to a police station. I learned later that the man wanted to go to West Berlin because of family difficulties.
In 1975 Pastor Manfred Fischer, a Westerner and a Lutheran, assumed his duties in his parish on Bernauer Strasse. The church he ought to have taken over – the Church of Reconciliation – was on the Eastern side, actually within the death strip, and closed. The two cemeteries were on the Eastern side, also: the inner wall ran along the cemetery borders but the exclusion zone – beyond which people were forbidden to go without permission, as a way of further minimising escapes – ran through the cemeteries. To visit a grave, you had to apply for a special permit – a _Grabkarte_ : 'Permission to go to the exclusion zone between the GDR and Westberlin [sic].'
Each card, coloured green, was numbered and stated which cemetery could be visited. It carried full details of the applicant – name, age, where born, present address. It stated that the card was only valid in connection with a _Deutscher-Personalausweis-Nummer_ (identity card number). A Roman IV on it meant the card-holder was male, a V meant female. It was issued by the cemetery management and the card bore the formidable official designation of I/16.01 F250/67 KB 2070.
(There was also a border pass: 'Permission to enter the exclusion zone at the state border of the GDR and Westberlin.' For this the applicant had to provide a photograph as well as give personal details. There was also a _Passierschein_ , which allowed someone, for a limited period of time, to enter the exclusion zone at the state border of the GDR and West Berlin for either private or official reasons. People who were to spend longer than twelve hours within this exclusion zone had to report that. The _Passierschein_ was valid only with the person's identity card. Either 'private' or 'official' could be typed over using the traditional row of capital Xs.)
By 1975 a new church had been built on the Western side and that is where Pastor Fischer took up his duties. He had never met nor knew anything about the part of his congregation in the East, and had not, of course, been able to pay even a visit to the church.
Fischer understood the extraordinary history of Bernauer Strasse where, 'if the residents on the Eastern side leant out of their windows before 13 August 1961 they were in the West. Before that, for these people it was not a dramatic thing. They opened their front doors and stepped across without thinking about it.
'When I came the wall was up, but not the final kind of wall which the world remembers from 1989. It was being renewed and reconstructed all the time. The parish borders in Berlin are not accurate, if I can put it like that. There was a little bit of my parish in the district of Mitte [East] but most of it was in Wedding [West]. When Wedding was built up – and became a new district made up of three former districts – the parish boundary remained the same. I didn't know how many members of my parish were on the Eastern side and I didn't know their names. I had no information.
'We'd had the biggest urban renewal project in Europe so everything was torn down and rebuilt. In this way I was really used to seeing things blown up! The more modern wall was built in 1980 and from sometime in 1983 we got to know that the Church of Reconciliation would be blown up. The problem was that the explosion would break windows in the houses on the Western side and people could be injured by flying glass. These people would have to know when they should open their windows to prevent that.'
Talks began in East Berlin between the government and the religious leaders there, and then with politicians and the police in the West. 'They said the church would be blown up but I didn't know when. That was the situation in 1985.
'The Eastern government was a very, very accurate bureaucratic government. They did nothing until it was planned, permission sought, paper, paper, permission, planned, re-planned, paper, paper, permission. They told the West Berlin police of the whole operation so that the windows would be opened.
'I had a sabbatical and in January I was invited to the United States. I was in New York staying in a church apartment and I happened to be watching television. In the United States there is very little information about other countries, so I could not be forewarned. It was three o'clock. I watched the church being blown. In one sense I was not surprised because I knew it was going to happen at some time, but I was very surprised to see it on the television. This was my church although I had never been in it.
'Your feelings change between when something happens and when you become fully aware it has happened: these are two separate moments even when you see something and you are involved in it. It's the same when someone dies: the full impact is not always immediate. I needed one year to really get this mixture of feelings resolved. How do you call it when you say goodbye to someone who dies? A funeral. I tried to make a funeral sermon – a memorial service – for this church and I thought for a year how best to do it and first I asked a lot of people for their feelings.
'We did it one year later, in May, and it lasted three days. One man composed music for us and we had an open-air choir. There was a girl who organised dancing and she danced against the wall – which was not allowed. When we were going to do this I had to talk to the police and ask them not to notice! Every event which took place near the wall on the Western side was accompanied by the West Berlin police: you were not even allowed to touch the wall because it belonged to the East and was on Eastern land – when they wanted to work on it they built a wire pen on our side for their workers to come through into. Thinking about this sort of thing will last thirty years at least.'12
The division, this unkindest cut, was merciless on families. Brigitte Schimke, who lived in the city centre facing the twin curved roads and the sunken garden – which had now become the wall – still had a sister and brother on the other side in the West:
In general, living in front of the wall one became accustomed to it because the view you had of it was an everyday view. My brother Dieter was an engineer and when foreign businessmen visited his factory he showed them round West Berlin, including the wall. Opposite our apartment was an observation platform in the West and he made sure to take them there. If he knew a delegation was coming he'd write to me – we didn't have a telephone – saying which day and at what time he would be there. I'd stand on our little balcony and look at him while he told the delegation he had a sister over there. They couldn't really believe it. When I saw him I turned away from the watchtower in the death strip so I couldn't be observed, and I gave him tight little waves.
Very bright neon lights came on every night and if it was foggy or rainy they put them on during the day, too. The Border Guards patrolled outside on foot all day and night on the street outside my apartment which was in front of the inner wall. We gave them warm drinks when it was cold or it was Christmas. They were not supposed to go into the houses although they did when it was cold. They were never from Berlin, a deliberate tactic because then you wouldn't have Berliners shooting at Berliners, and also if they were Berliners people might know them.
One day a man made a run across the death strip and managed to reach the far wall but the roll-bar on top of it there stopped him. They shot him although I think he only had a slight injury to his knee. A very brave doctor from the hospital round the corner – a woman – rushed to the scene and demanded the Border Guards brought the man to her so she could treat him.
We did have problems living facing the wall. They installed lamps in the cellar so they could see if anybody was tunnelling there and they closed the attic – because we were so close, it might have been possible to string a rope across somehow. My husband had a 17-year-old cousin from Rostock who came to visit us and he was interrogated by the Border Guards because they thought it suspicious that someone would come from Rostock to so close to the wall. He had to tell them he was just on a visit.'12
The way the world is, and will always be, there was humour at the wall as well as grief. Egersdörfer recounts how, being so close to the West, with binoculars he could see across in the most vivid detail. I mention that I'd been told the Guards had a method of spreading the news if a pretty girl was undressing on the other side so they could have a look. 'Yes, and there were some girls who knew what they were doing because they would do it all the time. We could even see their buttons!' Were the women exercising power over the Guards, who could do nothing except watch? 'But we had to watch them! It was our task and our duty! There was one particular girl and she'd been given a nickname, _Long Tooth_ , although I don't know why. In German it's not actually a nice name – you'd call somebody you didn't like _Long Tooth_. I can't say whether she was attractive or not because if you've been in a watchtower for hours any woman might be. _Long Tooth_ was so popular that sometimes she was even mentioned in the Commander's briefing before our shifts. He'd say she'd been seen at such-and-such a time. Some people assumed she might have been paid to do it because she did it so regularly and professionally. There was a file on her.'
In the paranoia, this assumption was no light thing. If _Long Tooth_ turned her light on, drew her curtains back and began stripping, of course the Guards would watch – and be off-balance in their attention to duty. 'The Commanders were always afraid of such incidents.'
Egersdörfer could also see pedestrians in the West, and cars, and into a Western corner supermarket. He could even read the prices on the items. 'I knew the people were quite well off. I thought that although all this was only 6 or 7 metres away I would never go to it. In a way that was depressing. It would have been easier to go to the moon than cross the 6 or 7 metres. The moon was closer. But you had your family and your home and a good many people still work on the assumption of "Why didn't you come?" I can only speak for myself, and though I thought about it I never wanted to escape. And you must remember that we knew so little of what we know now. We knew there had been some resolutions in Helsinki [the Agreement on rights of travel] but we didn't really know what was in the texts because they were published only once in _Neues Deutschland_ and then taken out immediately.'
Hartmut Richter, then a resident of West Berlin for almost ten years, had been able to return to East Germany because, from 1972 when Honecker replaced Ulbricht and applied for United Nations recognition, people who had escaped before 1971 were pardoned and could make visits. Richter became involved in helping people escape but, in March 1975, was caught attempting to get his 21-year-old sister Elke to West Berlin in the boot of his car. He was held for a year in Stasi custody then sentenced to the maximum fifteen years in prison for 'trafficking in human beings'. He was transferred from one prison to another because he exercised a bad influence on his fellow inmates.
3 April 1975 _Norbert Halli, 22, was fatally shot – he may have been trying to get past Border Guards in disguise._
5 May 1975 _Mert Cetin, a 5-year-old Turk living in West Berlin, drowned._
'Although people had generally got used to the wall,' Thomas Flemming would record in _The Berlin Wall_ , 'there were still events which caused the West Berliners' emotions to rise. One such instance was when 5-year-old Cetin from Kreuzberg fell into the Spree when he was playing. Some West Berlin divers arrived within a few minutes but were not allowed into the water and had to watch helplessly as the boy drowned, because at this point the whole breadth of the Spree belonged to East Berlin. A GDR border boat arrived too late at the scene of the accident and could only recover the Turkish boy's body from the water.
'This incident released a storm of anger and indignation. Hundreds of people gathered on the Kreuzberg bank of the Spree and held up banners accusing the SED regime of "Child murder" and "Inhumanity". These kinds of events, which brought the Wall sharply into the public consciousness, did not fit into the GDR's plans. After the boy's death East and West came to an agreement about "unbureaucratic" emergency help with "accidents at the border".'13
The RIAS radio reporter Peter Schultz covered the wall going up and he, like Brigitte Schimke, explains the pain of division as it unfolded over the years. In August 1961 his wife Astrid had been pregnant:
Her parents lived in the GDR 50 kilometres from Berlin and it was a great shock for her when our daughter Martina was born two weeks later and she couldn't see her parents and they could not see their grandchild. We sent photographs but there was no other form of contact – no telephones – although we received letters from them and so we knew that they knew they were grandparents. The next time we saw them was Christmas 1963 when, for the first time, West Berliners got permits to cross.
We all crossed to East Berlin and then they could see Martina. We met at a bar which was run by my uncle in Senefelderplatz, near the U-Bahn station. It was very emotional. I had had no contact with my uncle except letters, but that was a normal situation after 1961. I can't imagine how it was normal – impossible. On one side we were so happy to see each other but on the other side we knew that we would part again: we were only allowed to stay that one day and you had to be back at midnight. We were there from nine in the morning until six or seven in the evening.
The next time we saw them was one year later. Every time we had Christmas or Easter you could get the permits. The bar at Senefelderplatz closed but my uncle still lived there. In 1965 my wife's parents got to West Berlin – it was what they called a gathering of families (they were pensioners) and you could apply for that, bring your furniture. Of course they had to leave their house and the ground it stood on. These were expropriated and the house had a new owner.
My uncle is dead. He died in 1966 – we heard by letter – and, no, I couldn't go to his funeral. I'd have been allowed to go if it had been my father or mother but it wasn't a first degree relative. All that... it's over now, thank God.
Because of my profession at RIAS, I've seen the whole world, the USA, Asia, Japan but not Potsdam where I studied. I saw American presidents here and in America but not my neighbours, not Potsdam. What is the feeling? I don't know. It's crazy. I've been back to Potsdam, I've friends there – colleagues from the school – and we met. I think [spring 1991] it will need a long time to get used to it mentally, to rediscover ourselves.14
17 May 1975 _Henry Weise, 23, died at the wall._
4 November 1975 _Lothar Hennig, 23, died at the wall._
The man who fled with his family, and who would only give the name Mateus ('that should be enough'), remembers 'my aunt's son worked for the Stasi. I only saw him when I went back to the East for my grandmother's funeral and he spoke no word. That was German family life for you. Pretty unbelievable.'
The division did not even spare those who worked for international companies.
Erdmute Greis-Behrendt of Reuters describes in a telling phrase that 'then came a very different time – I went back to West Berlin in 1976'. She had not been there, of course, since 13 August 1961 when she'd travelled home quite normally on the U-Bahn and noticed nothing unusual. Now 'it was the year after the Helsinki Agreement and the Reuters bureau chief in East Berlin applied for me to go to London for some training. It was turned down flatly. Since 1972 we had had to pay our salaries into a government department account and then we would get it back. It was for people working for foreigners or foreign companies. These were now the people who would issue the passports and they said, "No way." The correspondent said, "Right, I will raise this matter at the Helsinki follow-up conference in Vienna"' which was due imminently. 'He was really very good at it! A couple of days after that, the department phoned me up very excited and said, "Come to our office immediately and bring your nicest passport photo."'
Erdmute couldn't believe what she was hearing. To get the passport 'took a little while – but a couple of weeks later I held it. I had a small son, 6 months old, and we were sitting in our flat, my husband and I, wondering whether I should use that visa to go and stay in the West. I said I just couldn't, the baby was too small and I just couldn't.
'I hadn't been to West Berlin since 1961. It was absolutely... amazing, although of course I had known it before. West Berlin was brighter and it was louder and there was more traffic but it hadn't really changed. I crossed at Friedrichstrasse – I took the S-Bahn and got off at Savignyplatz because that was where the [Western] Reuters office used to be. I knew what the office looked like because before the wall I used to go over there. I wanted to try a trip across first. I went over secretly, so to speak, and thought, "Good God, I can!" I just had to sneak over and see some people and get the feeling, you know. I went back to East Berlin and told everybody I had been in the West and people just looked at me.
'Then, when the day really came for my flight to London, it was the correspondent who took me to Tegel. And then whoosh... London, which was lovely! I had never been before. When I was in London there were so many new impressions coming towards me but I never for a minute thought it was necessary to stay. Somehow, suddenly everything looked so normal and the East German problems were so far away. I phoned my husband every now and then and asked how my son was, and how things were going, and it didn't seem necessary to do anything drastic. It was only after I came back that it struck me it was a chance I hadn't used.'15
There was a fundamental misconception in the West that Eastern countries, having signed the Helsinki Agreement, would honour it however reluctantly. They were wrong. In 1976 the GDR started to build their Mark 111 wall, the most sophisticated of all. It replaced the old horizontal lintels laid one on top of another between vertical posts, with self-standing L-shaped slabs which could be joined to form a continuous, smooth surface. These were topped by a concrete cylinder so that anyone leaping up could gain no handhold because their hands would slip back over the cylinder. (The wall round the Brandenburg Gate was constructed in a different way, broader and lower, as an anti-tank barrier.) The slabs were 3.6 metres (11.81 feet) high minus the cylinder, 1.2 metres (3.9 feet) wide and weighed 2,750 kilograms (6,062 pounds). Each segment cost 359 Ostmarks and about 45,000 were put in place.
Erdmute Greis-Behrendt's father had remarried and reached retirement age. 'You could just apply to go to the West and he was permitted to do that. It was difficult because he asked each of us, his three children, whether we agreed to this or if we had any objection. I said, "Of course not. Why should I? It is your decision and you are old enough to make your own decisions. You don't have to ask me." He said he needed it in writing and we all wrote little letters saying, "I have no objection to my father moving to West Berlin." I couldn't go, of course not, but he could have come back to visit at any time. He came back but he didn't really visit us. It was his second wife. She didn't like his children.
'I didn't have relatives in the West but I did have lots of friends – I'd gone to school in West Berlin. When I got my first passport to go to England in 1976 – the invitation to do some advanced schooling was actually just a pretence for getting me out – I visited them all.'
Greis-Behrendt was in an almost unique position in that she was working for an international news agency and the only one which had offices in both parts. Yet, before that day in 1976 when she sneaked across, she couldn't go to the other one. _How did she cope?_
'We were so absolutely busy I hardly had a moment's time to consider it. Of course I noticed it when there was a free moment and I reflected on things. I knew that it was dreadful and of course I had so many friends in West Berlin.'
_But you were in daily contact with the forbidden West._
'Of course, but it was a nice contact.'
It was also a constant reminder, and most people in the East didn't by definition have that.
Incidentally, Reuters correspondents were changed every eighteen months because, in Greis-Behrendt's words, 'East Germany was regarded as a hardship country.' That meant a procession of correspondents came and each brought their 'new library of books and I read through them'.
_It remains astonishing how well she took it all._
'Well, what else was there to do?'
_Go crazy?_
'But I didn't want to go crazy! I was quite young when the wall went up, 23, 24, and I thought, "It can't be the end of the world, it can't be the end of my life, it won't last that long."'
But it did, and, the way the world is, abnormality becomes normal given a decade or two.
16 February 1977 _Dietmar Schwietzer, 19, tried to get across from some allotments at 7.07 a.m. The Border Guard noticed him after he had clambered over the inner wall_ _but triggered the alarm – in some places the border had acoustic and electric alarm systems. After a warning shot he was hit and died being transported to hospital._
The division between East and West Berlin did not spare the railway system, originally designed and built to service a whole city.
To see in one's mind's eye the geometry of Friedrichstrasse station is not at all easy, because here absurdity piled upon absurdity. Friedrichstrasse housed both a proper overground railway and an underground line, and was itself divided by internal walls. The underground line ran from the south of West Berlin under East Berlin, through disused stations bearing spidering pre-war gothic platform signs. These stations were guarded by GDR soldiers and the trains never stopped except at Friedrichstrasse, where only Westerners could board or alight if they had the right papers. The underground then continued under East Berlin until it surfaced in the north of West Berlin. Every day on this line commuters passed under the wall while they read their newspapers, barely noticing the ghost stations along the way, and re-emerged still reading them.
The railway station at Friedrichstrasse served as a terminal and a connection, and internal walls, partitions, separated these. The terminal was for domestic Eastern travellers: they arrived, got off and went down stairways into East Berlin. The connection was for international travellers coming in from, or going to, West Berlin and beyond to western Europe.
As at the other checkpoints, Border Guards controlled it and the Stasi were the customs, checking and examining and stamping papers. If you were a Westerner or you were an Easterner with a _Visum_ you could pass through the checkpoint, rise to the international platform and take a most normal train ride through the wall. A few minutes later you were at the Lehrter station in West Berlin – which Lutz Stolz had been due to go to on 13 August 1961 to meet the other members of his football team.
Friedrichstrasse had a tall room, coated in something not dissimilar to lavatory tiles, where Easterners waited for friends, lovers and relatives from the west to come through a big metal door from the international platform. Beside this same door they bade farewell when visits were over. The room was known as the Hall of Tears.
The underground, a potential escape route to West Berlin, was so prohibited to Easterners that it did not appear on their city maps and anyone born after 13 August 1961 had no idea it existed. Many of these young people who often frequented the terminal above it were openly shocked when, after 9 November 1989, they learned that the underground ran there, and had always run there.
Peter Dick was a Canadian who went to live in West Berlin and became fascinated by the absurdities:
I started to picture these dilapidated subway stops that had not been used since before the wall but nothing prepared me for the experience of riding through them. You emerged from a dark tunnel into a station/platform and the train slowed to a crawl as you went through (without stopping, of course). Unlike all the other stops on the western side, or the Friedrichstrasse 'official' stop, these unused stations were dimly lit, crusty and rough-finished in concrete. In some places light bulbs hung from the ceiling on their wires.
At either end of the platform stood a lone guard with rifle on shoulder. A friend informed me that these stations had to be guarded from potential Eastern citizens who might try to sneak into the station and somehow hop aboard a moving train. The guards were there to prevent any possible escape via the subway or unused stations. I looked at some of these stations and realised that at one time, when the city had been whole, they had been some of the busiest stations in Berlin, people flowing through them. I had dreams about this for months. Riding a different line one day – the S-Bahn line – there was an underground section where we went through the Unter den Linden station, which again must have been the heartbeat of Berlin before the wall and it, too, was now derelict.'16
The division did not spare those who, by geographical chance, lived in the historical enclaves like Steinstücken, and Eiskeller further north. In fact there was a sort of enclave in the city centre, the Lenné Triangle, too. This was a wedge of land butting onto Potsdamer Platz; it was technically in the Soviet Sector, but when the wall was built the GDR felt it would be easier to abandon it, and did. The triangle was allowed to grow wild until the West Berlin Senate bought it in 1988. Before 1988, it had never been checked for unexploded wartime bombs because the West wasn't allowed to go there and the East showed no interest in the place at all.
The division did spare Steinstücken, at least partially. Kurt Behrendt, the amateur radio operator who'd moved there in 1957, says that 'in 1971 there was a conference of the four Allies and they made the Berlin Agreement. They decided that since Steinstücken was the only enclave where people lived out of a total sum of ten enclaves [well, nearly], it should belong to West Berlin and should be given free access. A road had to connect it with West Berlin and was built in a relatively short period of time after the two German sides had decided exactly where it should go. They agreed that the strip should be 20 metres wide, plus a 6-metre path for bicycles and pedestrians, and that it should be incorporated into the district of Zehlendorf.'17
Eiskeller had one resident family, the Schabes. The father, Martin, says:
'Steinstücken was an enclosure, and Eiskeller – ice cellar, because there was one which kept the ice for Falkensee – was an enclosure. We had a normal way to go out, well, simply a path, no tarmac. There are photographs of my son Erwin cycling to school being escorted by a British armoured car. The Russians cut the road from here to Spandau and we couldn't cross but the British Army came and said 'please remove that'. Erwin's school was in Spandau and he was talked to several times by Soviet and GDR military. Sometimes he didn't want to go to school and pretended they'd been talking to him as an excuse! We always had to report the incidents to the British because it was their sector and they patrolled here. Sometimes the Soviets and East Germans wouldn't allow Erwin through so we decided to call for an escort. We even had a British military helicopter which was only removed in 1990 when the wall went.
I was born in Caputh, which is near Potsdam. My father was a gardener at the Sanssouci palace but in 1926 he went to the eastern side of the Oder, which is today Poland. I was a soldier for five years during the Hitler war and, from the beginning up to the end, I was in Russia. I was taken prisoner in 1945 – I was captured near Frankfurt-an-der-Oder and taken to Kiev by train.
I was a prisoner for three years and was one of the first to be released from the camps because I'd struck up a friendship with a Russian doctor. In this area they said _Schabe_ for Sunday and I was called Schabe so it was a kind of sympathy. I had to repair fences and we all ate out of one pot. We worked in the forest cutting trees – we had to cut a certain amount of trees at a certain length. We were allowed to collect mushrooms and I've never eaten so many in my life! During my time in the camp, I experienced that the ordinary Russian people were very friendly. They gave us bread.
When I was released I went to my parents, who lived in the district of Frankfurt-Oder. They didn't know I was alive. Then I came to Berlin and it was _kaputt_ , _kaputt_ , _alles kaputt_. I didn't recognise Berlin. Terrible.
Then I came here and built it all by myself. In 1948 you didn't see that it was virtually surrounded by the Soviet Zone. That started in 1953 with the controls. Before 1953 it was just an ordinary fence. In fact, when I first moved here there was nothing, just the farm and fields. In a sense, Eiskeller belonged politically to the east, to the district of Nauen, although in reality [on the maps] only the woodland in the middle actually belonged to the east. We'd go into the woods to gather mushrooms and it wasn't a problem. You only needed a piece of paper and nobody really cared.
After 1953 I helped a lot of people to escape – a young couple, who were engaged, got here. They asked me if they could get to West Berlin and I said, 'You certainly can't but the East German military know me and I will help you to do it.' I had a horse and cart. I put them on the cart, covered them with straw and got them through. There were no British soldiers yet.
The nearest shop we had to go to Spandau in the West – up to 1953 we went shopping in Falkensee in the Soviet Zone but after 1953 we couldn't even go to Schönwalde [a hamlet just up the road]. They built a fence, and they built it very quickly. Later they reinforced it with metal supports. Some people tried to escape over it and we'd hear shooting. We saw the Russians on patrol. [Ultimately, when the last generation of the wall was in place] at night you could read the newspapers by the light from the arc-lamps.18
The people who watched over this division – the Border Guards – were charged with sparing no one and ultimately would not be spared themselves. That ought to have included Honecker who, as head of the GDR, bore ultimate responsibility for the shoot-to-kill policy, but a court case against him collapsed on the grounds of ill-health. This is how Honecker thought.19
The regulations for securing the border are established in the border law. Fatal shots as occurred in the FRG have never been introduced in this country.
_Was it a shot with prior warning or what was it?_
You have to enquire about the rules on the use of firearms. They are the same in the FRG as in the GDR, and are as the present rules for the police.
_Haven't you talked about it with West German politicians again and again?_
I had several talks about it, among others with Strauss [Franz Josef Strauss, Minister-President of Bavaria and an opponent of Ostpolitik] and also with Social Democratic ministers. I asked them, 'Tell me, what about your rules on the use of firearms?' They said that suspects are required to stop. If these persons flee, there is a warning: 'Stop!' And if the person has not stopped by then, there is a warning shot. It was the same with our police in the GDR.
In connection with the development of the relations between the GDR and the FRG, I also endeavoured, as far as it was possible, to humanize the relations in this field. It took many years to reach a situation in which it was recommended to Border Guards not to make use of their firearms unless they had to defend their own lives – in other words, if they had to claim the right to self-defence. This was in 1986 and 1987.
It was also planned that warning shots should not be fired any more so that the other side could not launch campaigns in the media on the continued existence of the so-called shoot-to-kill command. This had been our policy from 1986/1987 onwards. We did not want to have ourselves 'disturbed' by shooting at the border. Therefore the then defence minister declared that a shoot-to-kill order did not exist at all.
_Don't you feel sorry for about 200 people killed at the wall?_
I feel sorry for our twenty-five comrades who have been treacherously murdered at the border. Our requests to the government of the Federal Republic to hand over these people were rejected.
_Did you know that many people saw the wall as a declaration of the moral bankruptcy of socialism because it could not make people stay in the country?_
What do you mean by 'many people'? The construction of the frontier had been a necessity in times of cold war. Thus we were able to stop the bleeding white of the GDR. You really have to stick to the decision taken by the members of the Warsaw Pact. It was done in order to safeguard peace and to secure the socialist society's construction.
These were not the words of a leader about to implement the Helsinki Agreement, or who had ever even considered implementing it, and the East made no attempt to disguise the fact that people were still trying to escape, with all the potential for tragedy that that carried. In July 1977, an attempt along the autobahn from Berlin to Helmstedt went horribly wrong. (The autobahn had become a favoured route to escape down because, since the transit agreement, fewer cars were searched.) A couple and their 6-month-old son were in the boot of an Opel, driven by a man hired for 3,000 DM. The car broke down and he spent a long time trying to flag down help before he got a tow to the checkpoint. By then the baby, who'd been tranquillised, had suffocated in the heat and lack of air. Discovered, a GDR court sentenced the driver to eight years in prison and the parents five years each.
12 October 1977 _Frank Neitzke, 19, died at the wall._
22 December 1977 _Gerd-Michael Frenk, 31, died at the wall._
Around 2 April 1978 _Hans-Joachim Manowski, a 34-year-old West German, died at the wall._
Some of the Border Guards found themselves caught out by an anomaly, because when the wall finally fell they sought rehabilitation for deliberately shooting to miss. Peter Kull was a case in point.20 He was posted to Berlin in October 1961, when he was 19, as a Guard. In February 1962 'I saw a man running across the death strip. As I outranked the soldier I was on duty with that day, I ordered that we go in the opposite direction the escapee was going. After I knew the man was through, I gave the alarm signal.'
Kull claimed that if he was being observed by other officers he 'simply aimed so that I would miss' the escapers and, once, fired at the wall rather than a man riding a bike across the death strip. He only just managed to explain that away. He was betrayed when he tried to help three people to get across, and was imprisoned and abused – nine months of solitary confinement when he had to stand facing a wall without moving for long periods. The abuse included being beaten unconscious. 'When I came to in my cell, I had a terrible pain between my legs. I thought they had only kicked me, but the nightmare was that there were burns all over my genitals. I guess they did it with a cigarette lighter.'
Kull, who kept as much documentation as he could, said 'I suffered and went to prison because I helped people try to flee a communist dictatorship. Afterwards everybody treated me like a criminal but my father told me, "Don't worry, Peter, some day people will look at you and say something different."' (After the wall fell Kull hoped 'this time had finally come'.)
None of this – the escapes, the failed escapes and the full apparatus to prevent the escapes – covers the breed of people everyone hated: those who pre-sold escapes and disappeared with the money. Wolf Quasner, a noted escape organiser has called them the 'rogues'. They came from the shadows, preyed on the yearnings of the vulnerable and merged back into the shadows. Quasner began organising escapes because he saw the misery the rogues were causing. Who they were, how many they were, how much they made and what devastation they wreaked cannot be quantified because the details still lurk in the shadows.
By 1978, one estimate suggested that more than 500 Western hire cars had been used to make escape attempts in the East. Karl Schwarz, speaking for the car rental industry in Düsseldorf, said: 'These escape operators never tell the car hire firm they are taking the vehicle into East Germany, let alone that they plan to use it to bring out refugees.'21
One case seems to encapsulate everything. West Berliner, Horst Poser, did a job for a celebrated escape organiser. He was smuggling a refugee family out through Czechoslovakia – the wall in Berlin and the German–German frontier were now as near impregnable as human planning could make them – and was caught. He was imprisoned in the GDR. He claimed he had been given amateurishly forged documents and sued the escape organiser for the 606 days he spent in prison. The Stasi supported Poser, as a ploy to get at – and destroy – the escape organiser. They cut his prison sentence and advanced him money towards paying his legal costs.22
The Western court, trapped in a truly desperate morass of the moral, the ethical and the legal, retreated to the only place which remained to it: the pragmatism of compromise. They awarded Poser damages, _small_ damages.
Far away from all this, a simplistic man was preparing to confront the world by making no compromises. He was an old man and one who saw the world in black and white, not colour. The fact that he was neither tactician nor theoretician, neither academic nor sophisticate wouldn't have mattered except that, as President of the United States, he would be the most powerful man in the world. He called the Soviet Union the 'evil empire' and didn't intend to coexist with communism, but defeat it.
Whatever, the second bitter decade was over.
* * *
Quotation at head of chapter, given to the author by Egersdörfer.
1. _Germany and the Germans_ , John Ardagh (Penguin Books, London, 1988). Apart from this lovely phrase, which I wish I'd thought of myself, he wrote that 'the area closest to the Wall has been left deliberately as a wasteland or semi-derelict jumble. So the first-time visitor arriving from West Berlin, by car at Checkpoint Charlie or by S-Bahn at the Friedrichstrasse station, receives the initial impression of a city more shabby and shattered, indeed sinister, than it really is.'
2. Just as many Western people thought of the Russians as the Soviets, plenty of people East and West could not distinguish between the four entities making up Britain (England, Scotland, Wales and Northern Ireland) and simply called them the English. Quite what Glaswegian Peattie would have made of being called English does not bear contemplation. On my first trip to Berlin in 1970 I gave a lift to a British soldier who turned out to be Scottish. As we ran along the wire in the countryside he pointed to the Border Guards, guns drawn, supervising some workmen in the death strip. 'I don't hate them,' he said, ' _I hate Rangers supporters... ._ '
3. Document from the Hagen Koch archive.
4. _German Democratic Republic_ , Mike Dennis (Pinter Publishers Ltd, 1988).
5. Interview with author.
6. Many of those who did escape, or emigrated legally, found themselves disorientated and sometimes lonely in the questing, more selfish West, as if they had found another subtly different no man's land. GDR papers sometimes published lists of those who had chosen to return home.
7. Quoted in _We Were The People_ , Dirk Philipsen (Duke University Press, Durham and London, 1993).
8. Referred to in the GDR as the hinterland wall.
9. The military patrol jeeps were in fact adapted Trabants, an old Wehrmacht term (according to Peter Johnson) for a type of jeep.
10. In anticipation of our interview (in January 2001) Egersdörfer had carefully written down some of the episodes he thought we'd be covering. He wrote Westberlin as one word quite naturally. The habits of a lifetime, literally, even if another lifetime altogether.
11. Interview with author.
12. _The Berlin Wall, Division of a City_ , Thomas Flemming (Be.bra.verlag, Berlin-Brandenburg, 2000).
13. Interview with author.
14. Interview with author.
15. Interview with author.
16. Letter to author.
17. Interview with author.
18. Interview with author.
19. _The Fall_ , Reinhold Andert and Wolfgang Herzberg (Aufbau-Verlag, Berlin and Weimar).
20. Drawn from a _UPI_ report by Leon Mangasarian, August 1991.
21. _Daily Telegraph_ , Sunday Magazine, 9 April 1978.
22. Ibid.
## EIGHT
## _Thaw_
I spent three wonderful weeks in Paris. As a result of all this, it became clear to me what an incredible crime the state had committed against us by never letting us travel. I stood in the cathedral at Notre Dame with tears in my eyes and could hardly believe that I was actually seeing such beauty.
Ursula Sydow,
East German Publishing Editor
Nobody knew. Nobody suspected. Nobody foresaw. The 1980s began with the GDR looking settled and relatively prosperous, and the FRG looking a model democracy and extremely prosperous. Neither country seemed to be living behind a façade. Both were welded into opposites: the GDR to the Soviet bloc, the Warsaw Pact and the COMECON trade organisation; West Germany to NATO and the European Union. More than that, each day which passed solidified this. West Germany's claim to be the only legitimate German state was demonstrably fiction and, although Western opinion polls suggested a certain reluctance to abandon the belief in eventual reunion, each passing day made that more remote, too. Nobody knew that the Soviet Union, for decades a secretive monolith, would begin to crack. Nobody suspected a thing.
In 1980, Ronald Reagan defeated Jimmy Carter for the American Presidency and the man of simple vision assumed the power. He understood, as so many Americans do, about winning and losing, and he mirrored America: he was a winner. He set about rearming to the point where, as someone observed, the United States could fight three major wars simultaneously.
Exact date unknown _A Border Guard called Walther was, it seems, a victim of an escape attempt._
On 2 October 1980, the West German government bought Hartmut Richter out of prison (or exchanged him). He went to West Berlin where, he insists, the Stasi constantly tried to smear and destroy him. 'I only have a victim's [Stasi] file, not an offender's file,' he says. He claims that Stasi agents in the West bugged his apartment and tried to penetrate his circle of friends.
4 November 1980 _Ulrich Steinhauer, a 24-year-old corporal, was shot through the heart by the Border Guard with him out in the country to the east of West Berlin. The bullet struck him in the back and went through him. It happened at 4.35 p.m. and therefore in the darkness of late afternoon. As it would seem, the other Border Guard wanted to desert and Steinhauer had been trying to prevent him – five shots were fired at him._
The GDR report of this incident lives on as a testament to a mind-set which insisted on detailing everything, however meaningless. Apart from the fatal shot, Steinhauer had also been struck a glancing blow but the investigators were unable to 'ascertain whether this happened before or after death. Furthermore, neither the direction nor the precise circumstances' could be 'ascertained by turning the body.' Five shots were missing from the weapon and five shells were found. 'It is not known yet whether this was in individual shots or in quick succession. On the corpse there are no signs of hitting or strangling.'
22 November 1980 _Marietta Jirkowski, 18. Three people tried to get across at 3.40 a.m. Each had a ladder and they selected a spot out in the country to the east of West Berlin. Jirkowski was accompanied by two men. They got into the death strip but that triggered the automatic alarm. After a warning was shouted and fired, she was hit in the abdomen. She died of the injuries at 11.30 that night. One of the men got back and disappeared into the Eastern hinterland. An investigation did not locate the other man or resolve whether he reached the West._
This book is not about the implosion of the GDR economy, or how fragile and trapped in a time warp that economy had really been, although the implosion would have a direct bearing on the wall. And if nobody knew, here are a couple of tantalising clues. A news item on a Western radio programme, broadcast as the 1980s began to unfold, said that one of the GDR's profitable exports to the West had been cheap and reliable washing machines. Now, however, Western markets were demanding more sophisticated machines and the GDR seemed unable to regenerate and re-tool their manufacture. If they couldn't do that for something which earned priceless hard currency, what state was the rest of their industry in? The second clue was provided by a GDR radio presenter. I asked him when he realised it was going wrong. Although he couldn't remember the year he did remember the moment. He'd been driving to work and saw a queue outside a food shop and he hadn't seen that for many, many years.
16 March 1981 _Dr. Johannes Muschol, mid-20s, was in the death strip at 11.07 a.m. and ignored both a shout to stop and a single warning shot. As he reached the wall itself the Guard opened fire and killed him. His body was transported to the nearest Guard tower and then taken away._
On 13 August 1981, the twentieth anniversary of the building of the wall, Hartmut Richter and some friends tried to put a wall round the Aeroflot offices in West Berlin.
The distance between here and there – between the two ends of Friedrichstrasse, if you like – and the cumulative effect which each passing day had had since 13 August 1961, is beautifully explored and defined by a piece written by author Irene Böhme.1 It is about a Westerner visiting family or a friend in the East and if it is couched in general terms it gains from that. This scene was replicated a thousand times:
For the West German, work is something you do with zip and verve. Work is the source of one's success and money supply, not a subject for social occasions. At most you make a passing remark about a successful deal, or a coup that came off. If you've got problems at work, you maybe admit it to your spouse but hardly to your best friends. If you didn't take early retirement entirely voluntarily, you keep that inside the immediate family circle.
Western man is programmed for success, he must always appear fighting fit and hide his weak points. If he then encounters someone who positively savours the opportunity to expound the problems he has at work, he is shaken and horrified. The only explanation he can think of is that in the GDR people are totally destroyed by the state. If he says as much, his host suddenly becomes a passionate defender of the system. He had been talking about how he was feeling, his low spirits, the way in which, deep within himself, he enjoyed conflict, he had wanted to show how the pleasure he took in fatalism was an optimistic attitude to life, he had uncovered the complexities of his self-image. His guest can't have been listening if he now talks about the state and tries to force a political interpretation on personal feelings.
Our tragic search for what we have in common obliges us to play down this situation too. Conversation is diverted to other areas, preferably to cars: that's something you can always agree about. But the West German will be unable to forget that he has had to look into the soul of a broken man, and how abruptly this downtrodden creature began defending the state. And the East German will not be able to forget that he has bared his soul to a philistine, and tried to discuss the meaning of life with someone who is unwilling to give it a moment's thought. Both repress how uncomfortable they feel.
Again it is the impression of similarity that prevails, as though the discrepancies between them came from outside and arose purely by chance. The one is unaware just how much the Slav has already entered into his character. The other is unaware of just how much, in certain matters, he thinks and feels like an American. Both have a sense of something alien, and brush it aside, without thinking or speaking about it. Each regards his behaviour as normal, and it doesn't occur to him that he is different because for thirty-five years he has been living a different everyday life in a different Germany.
Us and them: there shouldn't be this distinction. They cling desperately to each other, so restricting each other's normal range of movements, and the fraternal embrace threatens to suffocate them.
Astonished at what he has seen, the West German returns home, satisfied with his good deed. He never said that he found the streets stank, he turned a blind eye to dirt and disorder, he enjoyed the countryside, and the art and architecture of the past. If he was perturbed, he kept the feeling under control, adopting a mien of provincial imperturbability. He has been a good guest, though it was a strain on his nerves and his purse. He decides to visit the GDR again, because he has come to understand how important his visit is for his compatriots. He has shaken off a few prejudices, but his opinion of the country has been confirmed. He shudders when he recalls how careful he had to be in his dealings with people, how much he could not say openly, how alien everything was to him. He now knows, with greater certainty, that he prefers his own cares and worries, and that not just his standard of living, but his way of living is better. He crosses the frontier, takes a deep breath, he's home again. He's carrying no objects of value in his suitcase but a sense of value has stirred within him.
The Easterner waves goodbye to the parting visitor, contentedly feeling that he's been a good host, that he's offered something that cannot be found elsewhere. It has been a strain on his nerves and his purse. He has shaken off a few prejudices, his opinion of these Westerners has been confirmed, you can't speak freely and openly with them. After this visit he knows with greater certainty that although he is poorer, his way of living is better. He takes a deep breath, he is home again. He sees these wall-skimmers on their way, as the farmer does the migrant birds. They come once a year, pecking and chirping, and then fly on.
4 June 1982 _Lothar Fritz Freie, a 27-year-old West German, was shot in the left hip and left upper leg at 11.20 p.m. on a bridge opposite the Wedding district. He was given first aid and, at 12.03, was taken to hospital. From 11.30 to 11.50, ten West Berlin policemen observed the incident._
Dennis L. Bark had not only changed to studying history at Stanford, he had studied in Berlin and come to be fascinated by the city. A group of historians from all over the world, dedicated to private enterprise, met every two years and in 1982 held their gathering in Berlin. Bark says, 'I wanted to lay a wreath at the Fechter memorial and probably about 35 of them came. Important people were among them and it was a heavy deal. Afterwards Axel Springer gave a reception for all these people. It was quite something. I had been back to the spot many times and what I felt was, well, I'll tell you: 1982 I was forty and I thought, "Dennis, you are old enough to order your own wreath, to afford it, to decide what words you want to put on it and to get all these people to come. It's nice that you can do that."'
In October, Helmut Kohl became Chancellor of the Federal Republic, defeating Helmut Schmidt, who'd taken over when Willy Brandt resigned in 1974. Leonid Brezhnev died on 10 November at the age of 75, to be succeeded by Yuri Andropov, long-serving head of the KGB, who was 61. Andropov was ill with a kidney disease and, at the end, would exercise his power from a hospital bed. Cumulatively, it was the beginning of a decisive shift in generations and thinking, but nobody knew, nobody suspected, nobody foresaw the consequences.
April 1983 _Rudolf Burkert, a West German of unknown age, died at the wall._
25 December 1983 _Silvio Proksch, 20, climbed over the inner wall from a narrow side street in Pankow just before 7.30 p.m. and sprinted towards the outer wall. After two warning shouts and one warning shot, he came under 'aimed fire' and was hit. He was arrested about 30 metres from the outer wall, given first aid and taken to hospital. He died from_ _his injuries at 8.46. The official report added that the weapon was only used parallel to the West Berlin border._
Andropov died in February 1984, to be succeeded by Konstantin Chernenko, 72, a grey acolyte of Brezhnev and now the oldest man to become Soviet leader. Like Andropov he was ill. There is a union here, centred around a bitter sort of joke, between the fates of Freie and Proksch – both anonymous except in the instant of their deaths – and Brezhnev, Andropov and Chernenko with their special hospitals and teams of doctors and, if necessary, Western medical technology to keep them alive. The union is that the three Soviet leaders ruled half a world, and had to take responsibility for it. That included the wall and its victims. The bitter joke was that, because Soviet leaders didn't retire and the medicine kept them alive so long, 70 was only middle age in the Kremlin.
2 July 1984 _Unknown person. At 3.35 p.m. a male, between 60 and 70 years of age, was noticed lying on the ground about 250 metres north of a bridge over the Teltow Canal. He was 6 metres inside GDR territory. An investigation at 4.10 showed that he was dead. His body was passed to a Special Commission at 8.07 and taken to a medical centre for a post-mortem. The man was wearing a shirt and long trousers but no identification papers were found. The official report said that on the other side West Berlin policemen and firemen 'as well as one US Army person' were observing. This US Army person infringed the GDR territory by approximately 2 metres._
Just 2 metres? The pedantry of the geometry had become a yardstick for the moral questions of right and wrong.
Whether attempts to cross the wall had diminished because, by now, the wall itself was an awesome thing, or whether it was because life was more tolerable in the East, or whether a whole generation had grown up knowing nothing but the normality of the wall, nobody can say – but ordinary people kept nursing their reasons for trying, and they kept coming.
Peter Meyer, an East Berliner who worked on the railways, was one such person. Evidently he saw Westbound trains every day with Western passengers free to go where they wished, while his wife Hanni watched Western television and wanted the good life portrayed there. They had a son of four called Markus and no doubt they wanted the good life for him, too. It was spring 1984.
Meyer's mother knew a pensioner who had the right to travel to the West (as all pensioners did2) and asked him if, when he was over there, he could find somebody to help.3 He found an Arab who said he could get them out but wanted money in advance. Meyer raised that, the pensioner returned to the West and paid it. The Meyers were told to go to Prague and wait by a statue where they would be contacted – but nobody came. In a bar Meyer met a Westerner, a journalist, who knew Wolf Quasner, the escape organiser. The journalist rang Quasner who could, and did, get Meyer's wife and son out easily enough. Meyer was more of a problem and he stayed in Prague.
Quasner decided that Meyer needed a double who would donate his genuine passport, and found one – a Ghanian. That was intriguing because a black man was much less likely to be challenged. He dispatched a couple who were expert in make-up and they transformed Meyer into a Ghanian, cutting his hair and making it into African stubble. His face was covered in black dye and his nostrils enlarged. He was introduced to another Ghanian and together they flew to East Berlin then crossed by S-Bahn from Friedrichstrasse station.
1 November 1984 _Peter Böcker (or Boecker), 24, was fatally shot at the wall._
Yet still they came, or tried to come.
1 December 1984 _Michael Schmidt, 20, was fired at and injured by the Border Guards at 3.18 a.m. at Wollankstrasse. He'd scaled the inner wall on a wooden ladder and, using a second ladder, moved forward and over the alarm system – but he activated it. He kept going towards the wall but he'd been seen from a watchtower about 200 metres away as he climbed the inner wall. He was given a warning shout and warning shot but ran faster. Aimed shots brought him down. After he'd been given first aid he was taken to hospital at 4.11. From 3.40, a West Berlin police patrol, a Customs patrol and two French soldiers witnessed this; Wollankstrasse was opposite the French Sector._ 4
The Associated Press covered this death although, of course, the reporter had no way of knowing the name of the victim. The report said that 'a person trying to flee across the Berlin Wall early Saturday apparently was killed in a hail of bullets fired by East German guards. The shooting drew sharp protests from West German officials. West Berlin police, quoting residents of the Wedding district, said 20 to 30 shots were fired at 3.15 a.m. and then East German Border Guards searched an area near the wall for about 55 minutes before finding a body.
'The guards placed the body in a military vehicle, covered it with a tarpaulin and drove away in the pre-dawn darkness, according to the witnesses. Police said it was not known if the victim was a male or female.
'West Berlin Mayor Eberhard Diepgen said the shots "that apparently killed a person have once again revealed the inhumanity of the wall". West German government spokesman Peter Boenisch demanded that East Germany "once and for all stop using firearms against those who just want to exercise their human right of free movement".
'Gen. Olivier de Gabory, head of the French mission in West Berlin and speaking also for the U. S. and British allies in the divided city, accused the East Germans of "brutel force" and demanded they cease "this inhuman practice".
'The latest shooting came just one day after East Germany dismantled the last of some 55,000 automatic shrapnel-shooting guns along the border with West Germany, 110 miles or more to the west of Berlin. The guns, which were triggered by vibrations from anyone trying to climb border fences, were not deployed along the Berlin Wall.'
This report, couched in the vivid reportage demanded by the Western media, stands in direct contrast to the dry GDR report, of course, but more than that it captures the barbarism of still behaving like this in a European capital towards the end of the second Christian millennium. In its misguided way, the GDR report (and all their similar reports since 1961) reflected a whole way of thinking. It was not just that no human touch, however small and however fleeting, was permitted; it was bleaker than even that. The fallen might as well have been skittles.
10 December 1984 _Alice Paula Olga Gadegart, 70, died at the wall._
The pathological insecurity of the GDR leadership remained undiluted, and they were still prepared to kill for it. That touched on a question which has been already posed – if they were able to lie over why the wall was built, what would they tell the truth about? – and in these hard days, with the killing still going on and the wall frozen into permanence, who could know, or suspect, or foresee that the answer was coming before the end of the decade. When it did come, the answer to what they'd tell the truth about was: very little.
And still the escapees came, or tried to come.
Marina Brath and her husband Peter had to live through the abnormality of normality. Marina was born in Prenzlau, a town north of East Berlin towards the Polish border:
I decided to leave for the West in 1983–84. We made the decision together. We lived in an apartment in the Lichtenberg district, on the second floor of a small building. It's a long story, OK? My husband was hindered in his job – he worked in an hotel. Some office people said to him, 'If you don't join the Party you'll work as gardener or work as just anything.' We tried to get a visa for the West at the Office for Inner Relations in East Berlin. We didn't get a form: you only wrote a simple letter stating why you wanted to go, that's all, and you left the letter at this Office. That was rejected. We had to go back and they told us we couldn't leave. We complained in a letter to Honecker and three or four Ministries but nothing happened.
We knew it was a big risk but we also knew we had no future. That I would never see my parents again was a small risk because I thought my own life was more important than the relationship between my parents and me.
We'd been to the West German Consulate in East Berlin, to the American office and we tried to register ourselves as political prisoners. Anyway, there was no answer from our State so we surrendered our passports and declared ourselves stateless. Then we waited in our apartment because we knew there would be an official reaction. A few hours later the Stasi came in two cars, one of them a Trabant. They were not in uniform. There was a knock on the door and one of them said, 'Please come with us, we want to clear up some difficulties.' They showed us no identification. I went in one car and my husband in the other. I sat in the front seat and two men were in the back seat. I couldn't see my husband because the distance between the two cars was too great.
My car went to Keibelstrasse near the Alexanderplatz. I was taken to an office – a Stasi office but normal looking – and my husband was already there. They didn't say anything, they only asked questions. 'Why do you try and get out of Berlin? Why did you declare yourselves stateless?' Then they did say things, about our private lives but five or six years out of date. I don't know how they knew. I remember questions, questions and I was very frightened. I was questioned the whole night and I had no sleep. My husband had been taken away and the same man questioned me the whole night.
Then I was taken to a prison – not exactly a prison, because it's not clear yet whether you are guilty or not. It's a pre-prison and it was in the district of Pankow. I was put in a cell and I spent twelve weeks there. I was given old work clothes to wear. During that time I was able to communicate with my husband once. The same man who'd asked the questions came and we were brought together so we saw each other again – my husband was in the same pre-prison. No touches but we could speak.
Eventually we were tried and declared guilty. The trial was in a court nearby – a little room, one judge, one prosecutor and one defender – and the charge was Hindering State Offices. It was like a show. I could have spoken but I felt so pressured that there was no point. In reality the situation was that I could have spoken but they would not have _heard_ the words I was speaking.
The man 'defending' me was private and I could choose him, the state didn't give him to me. He had arranged some business between prisoners and the West German government, I think for money – the deal was the better the prisoner the higher the price. We never spoke of this. Everybody who wanted to go to the West knew this defender. He wasn't sympathetic, just a dealer in the East. He advised me to plead guilty because then it might not be so hard to get to West Germany eventually.
Before it started they all spoke together and it was decided I was guilty so the actual trial was like a show. My husband had been put under pressure. They told him that if he said anything wrong I'd be sent to a psychiatric ward – so they _assaulted_ him by suggesting they would hurt me. The trial lasted half an hour. It wasn't only like a show, it was a sort of joke thing. The judge said, 'You are guilty and you go to prison, you are not a good citizen.'
I was sentenced to one year and four months and my husband one year and eight months. We received the judgment together and then we were separated again. We couldn't say goodbye. He was taken through one door and I was taken through another directly to prison – Stollberg (in the south of the GDR). The building had three floors, beds and only cold water. All the other prisoners were political.
I sewed pillows to be sold to the West – you couldn't buy them in the East. I didn't know the time, the day, the hour, anything, but I'd been there eleven months. One day someone knocked on the door, came in and said, 'You, you and you, come!' We were taken in a van to another prison in Chemnitz5 and there we had to declare why we wanted to leave. We had to fill out some forms, then we got our private things. In the courtyard there was a bus from the West and at this point my husband and I saw each other for the first time – in front of the bus. We sat together on the bus and at the moment we crossed the border I felt _angst_ and at the same time happiness. There was no freedom to travel in the Eastern bloc. The bus went to Giessen [a reception centre for refugees] near Frankfurt without stopping and from there we could decide which town we wanted to go and live in. We got tickets and went to West Berlin. We flew to Tegel because we didn't want to travel across the GDR again.
My first impression of West Berlin? Astonished, shocked, yes. I was still in my own city but the difference was like day and night. I still [1990] don't regard West Berlin as normal. I like to buy strawberries in winter; I am _still_ astonished that you can buy strawberries in winter.
Husband Peter was then working in computers.6
In March 1985 Chernenko died and Mikhail Gorbachev came to power at the age of 54. The dinosaurs in the Soviet Union were either extinct, or soon to be, even if across the bloc as a whole they lumbered on. Nor is the analogy lightly chosen: dinosaurs possessed large bodies and were unable to survive a sudden climate change. Honecker's thinking had been shaped an aeon before and he was simply old – born in 1912, five years before the October Revolution brought communism to Soviet Russia. Gustav Husak, the Czechos-lo-vakian leader, was born in 1913; Janos Kardar (Hungary) 1912; Nicolae Ceaus¸escu (Romania) 1918; and Todor Zhivkov (Bulgaria), 1911. Moreover, writing an introduction to a book on the devout communist and spy Kim Philby (born, incidentally, 1912)7 John le Carré penned this phrase about the formation of Philby's beliefs: 'all Kim's life was early'. It was true of Honecker, true of all of them.8
22 August 1985 _Wolfgang Behnke, 46, died at the wall._
Gorbachev took power with few of the inhibitions which held his predecessors so tightly. He knew change was essential for survival and new words drifted into the international language: _glasnost_ (more open government) and _perestroika_ (reform). To the dinosaurs this was worse than incomprehensible, it was heresy, and events at the wall in 1986 showed the distance between _glasnost_ , _perestroika_ and the pre-war people running the GDR.
And still they came, or tried to come.
On 26 June the Associated Press reported that 'an East German soldier apparently trying to escape to the West was sent sprawling by gunfire just short of the Berlin Wall and carried away in a truck by other border guards, witnesses said.
'West Berlin police spokesman Hans-Heiner Salbrecht said witnesses in the Frohnau neighbourhood, in the northern part of the partitioned city, heard a volley of shots Wednesday evening and saw a uniformed soldier sprawled on the ground near the wall in communist East Berlin. They told police the man "was still making movements" when East German guards loaded him into a truck that took him away, Salbrecht said.
'Another police spokesman, Klaus Roennebeck, said the soldier was still alive when Guards took him to the truck. Roennebeck said police had nothing beyond the witness reports because East German authorities provide no information on incidents at the wall. At least 73 people have been killed trying to cross the wall, but more than 4,900 have succeeded in escaping, according to West Berlin police figures.'
In August, two Border Guards – a 21-year-old private and a 24-year-old corporal – escaped by abandoning the watchtower they were in and scaling the wall. Reportedly they threw their guns back over and walked to the nearest inn (The Wallflower) where locals bought them a beer. As has been seen, such escapes were extremely uncommon because the Border Guards worked in twos but were rotated so that they did not know each other, making raising the topic of escape extremely dangerous.
To emphasise this point – and echoing Roland Egersdörfer – Jens Bernhardt, a Guard who did escape, reflected, 'There were always two Guards but your partner and the tower you served in were changed regularly so you didn't get to know each other well enough to build up trust.'
3 September 1986 _Rainer Liebke, 35, died at the wall._
Two men successfully crossed on 12 November despite being shot at; eight days later a 33-year-old used what appeared to be a home-made ladder and clambered to the top of the wall at Bernauer Strasse. At least seven shots were fired at him, hitting him in both legs, and he fell forward into the West and was taken to hospital. Protests were lodged with the GDR government.
21 November 1986 _Rene Gross, 21, and Manfred Mader, 38, made their attempt in Treptow at 5.04 a.m. The official report said 'it was noticed that somebody was trying to get through the border using force'. They approached in a lorry along a side street which ran straight to the wall. The plan was evidently to ram their way through the gate the Border Guards used, to get to the death strip, which they succeeded in doing. But the Border Guards fired forty-nine shots and both were hit. The lorry was, the report added, '50 centimetres from the wall' when it was stopped. They were given first aid, but died on the way to hospital. The lorry was driven away at 5.58._
24 November 1986 _Michael Bittner, 25, was arrested at 1.19 a.m. to the east of West Berlin. He'd approached the inner wall with a 3-metre high ladder and clambered over it, then faced the alarm system. He scaled that but triggered it and was shot. At 2.15 people on the Western side were observing._
The Associated Press covered Bittner's death:
East German border guards today shot a man who climbed to the top of the Berlin Wall from the Communist-ruled Eastern Sector, officials said. The Intra-German Relations Ministry in Bonn said the would-be escapee was a man and that he probably was dead.
Police in West Berlin said a witness reported hearing shouts of "Halt, stand still!" from the East side of the Berlin Wall at about 1:30 a.m., followed by 30 to 50 shots. A person on top of the 14-foot-high wall who apparently was trying to escape to the West fell back into East German territory, police quoted the witness as saying. According to the police account, the witness peered through a break in the wall in West Berlin's Frohnau district and saw what was "apparently a man", sprawled on the ground covered with blood.
The witness heard an East German Guard say to the person, 'I got you, you pig', then heard another Guard complain loudly. The complaining Guard, who threw his hat on the ground, was disarmed by colleagues and escorted away. The person who was shot was covered with a tarpaulin and carried off. Police did not explain what the Guard complained about or why.
The witness saw a ladder on the ground near where the person lay, police said. Police said they had no other witnesses to the incident. In Bonn, chief West German government spokesman Friedhelm Ost said it was the fourth time in 12 days that East German Guards shot at people trying to escape. Some of the escapes were successful.
The escape brought to ten the number of East Germans who crossed the border – to West Berlin or West Germany – within seventeen days.
Michael 'Micha' Bittner's mother, Irmgard, a woman with a strong, open, almost plain face, came home from work that morning and listened for his alarm clock to ring. She heard it every morning. For some reason, she felt she needed to hear it now – and she didn't. She made her way up to the little balcony of their apartment to water the flowers and noticed that Micha wasn't there. She was confused and searched for any note he might have left explaining where he'd gone. She found nothing.
A Western witness to where Micha had been, Wolfgang Vogt, would come forward and say these words: 'I saw somebody trying to get over to our side, over the wall. I saw a hand, and a head, and then he fell back. I wanted to get over but more shots were fired. It was a salvo of shots from a machine gun. It sounded like _drrrr_. Then I heard someone coming from the watchtower. He was shouting very loudly, "I got the bastard! I got him!" He was murdered. And with malice, because he had already been shot when he fell back. So why were more shots fired? They shot him again. He must have been full of holes. The amount of bullets must have left him unrecognisable.'
Irmgard Bittner waited for three and a half years to discover Micha's fate. 'I had to guess where my son was.' With each day that passed, whatever hopes she had diminished because, as a mother, she knew her son would have contacted her somehow, wherever he'd gone. To herself, she could not accept that he had died and even as hope faded, it somehow remained. 'That's why for three and a half years he died a little bit at a time.' As she spoke those words, two years after she found out the truth (1991) she was holding herself hard from breaking up. ' _They_ [the State]' – the functionaries in the monolith, the Kafka people – 'didn't admit they'd killed him,' she said.
When asked who was responsible for his death, she seemed suddenly emptied of emotion, her blue eyes gazing down; then her whole face twisted in exhaustion. 'A good question,' she said, softly. Then, from a deep place: 'All of them.'
A wooden cross was placed in front of a single section of the wall out at Frohnau on the Western side, in commemoration. There was a metal plaque in the tall upright and it bore the simplest of inscriptions.
'I've often dreamt of Micha – of his death – and that ugly old cross there in Frohnau. That cross scares me somehow. I go there because I haven't got a grave, because I can put a few flowers there. But I don't feel closer to my son because I know he isn't there. I just don't know where he is. I'm sure they have burnt him.'
12 February 1987 _Lutz Schmidt, 28, made his attempt at 11.34 p.m. in Treptow. 'A male person was arrested after a weapon had been used.' The man did not have personal documentation but was 'probably' Schmidt. Although the visibility was 50 metres – mist or fog or lashing rain, perhaps – he was seen and warned. He didn't react. Then they fired and he was hit. After first aid he was taken to hospital and handed over to the 'appropriate authorities'._
There is the chill of authoritarian anonymity to that phrase, and when set into a historical Germanic context it becomes very frightening indeed; this was 1987, not 1937.
Reagan visited Berlin in mid-June 1987 and stepped onto a platform in front of the Brandenburg Gate. Unlike Kennedy, who walked wreathed within a certain mystique, Reagan walked outside it. He was slightly stilted in presence and delivery, a man with the common touch at one remove from common people. It was not as if he was playing the role of President, more that the role was playing him. This audience had been carefully selected but reacted quietly to what were described in the Western media as the provocative parts of his speech. He said the wall was 'a gash of barbed wire, concrete, dog runs and guard towers', as if nobody except his speech writers had noticed, but in a stirring call he added, voice amplified: 'Mr Gorbachev, open this gate. Mr Gorbachev, tear down this wall.' These words are remembered, and they take their place as an attempt at an update of _Ich bin ein Berliner_. The words with which he ended the speech, however, have been largely forgotten. 'This wall will fall. It cannot withstand faith. It cannot withstand truth. The wall cannot withstand freedom.' They might have sounded glib political rhetoric, even to the speech writers who actually wrote it, especially after the wall had withstood everything fot the last twenty-six years, but there can be no doubt that Reagan believed them; and no doubt, now, that the stilted man from the black-and-white age was right, and was going to win.
18 August 1987 _An unknown man died at the wall._
29 September 1987 _Falk Schröder, 25 – a West Berliner or a West German – committed suicide in circumstances which are unclear at the wall._
18 November 1987 _The body of Peter Urban, 29, a West Berliner, was found. He was estimated to have been dead for between four and five weeks._
Nobody knew. Nobody suspected. Nobody foresaw. Between 1987 and 1989, however, 'as the Moscow–East Berlin conflict steadily grew in intensity, East German dissidents and reformers within the Communist party gained confidence'.10
At least in June 1988, Honecker confirmed that the shoot-to-kill order at the wall had been rescinded, although that would be tested in the most direct way eight months later when a waiter from the East felt he wished to open a restaurant in the West and went across the death strip to try it.
There had been, too, an increase in the number of Easterners who were given permission to go across on visits. Lutz Stolz, the trainee engineer who'd been going to play football in the West on 13 August 1961, had 'an aunt in West Berlin whom I was allowed to visit between Christmas and New Year's Eve in 1988. I had not seen her for twenty-eight years and she was very ill so I couldn't stay with her. I had to stay with one of my friends.'
His wife Ute says:
He had to be back in the East at midnight of December 31. At 11.30 I got a call from him telling me that he was going to finish his cognac and come back. I went by S-Bahn to Friedrichstrasse station, I sat on a bench and waited all alone. The station was deathly silent. There were two Border Guards scrutinising me constantly while I scrutinised the iron door' – from which Lutz would emerge.
At 12 o'clock on the dot the door opened and I saw my husband with a special kind of hat on his head that he had always wanted to buy for himself. I had organised a taxi and we hurried out of the station. He'd been liberated by the cognac and wanted to express all his impressions immediately. 'This visit has opened my eyes to see what is wrong in the East,' he said.
I had seen the West before, in March 1987 and March 1988, when I was allowed to visit an aunt who actually was the wife of my blood-related uncle. Then my uncle died and, according to the Eastern police, she wasn't my aunt any more. So I simply made a unmarried sister of my mother's out of my aunt! She had the same name, so things worked out: they let me visit her. I came back with a lot of luggage and struggled through this damned iron door that only had a handle on the inside. After that second visit, they fired my husband as trainer of the regional soccer league 'because his wife had Western contacts'.
Lutz continues:
We had heard of the development in the West, we knew that they were getting on much faster, but as things [propaganda] were really rammed into our heads all these years we did not really believe it. For me it was worse when I came back from my journey in 1988. From that time on I did not believe in anything I heard or read in the Eastern media. They told us that, in the West, people would only stay outside the shops and look at the window displays because they could not afford things, they told us look at the growing unemployment rate...
I experienced something very different. I went through the big department stores, I saw the big purchasing power and their will to buy things. The department stores were sometimes so crowded that they almost tore the buttons off your coat. And among the 8 or 10 per cent unemployed were 4 or 5 per cent who did not want to work, who were satisfied with their unemployment. They were unskilled, they were lazy and lived on their unemployment money – which was more than somebody in the East who worked all day long would ever earn.
This is why I was in such a rage when I came back, it wasn't only the alcohol I had consumed in West Berlin that day. When I came out of the 'hatch' at Friedrichstrasse [the iron door] to see all the grey I was furious. It felt like going to Poland from East Germany. Measured by all the glitter in West Berlin... until this point I had always believed somehow... .11
This should be contrasted with the recollections of Birgit Kubisch.
I was born in [East] Berlin in 1969, so eight years after the wall was built. My parents worked in North Africa – the GDR embassy in Algeria – for four years. My sister and I went with them. I think I had seen almost all the Socialist countries except Albania and Cuba and North Korea. Of West Berlin I had seen what the TV showed us, because we had the Western TV. What do you see then? You see shops. I had never been to the end of Unter den Linden, actually not until I was eighteen or nineteen. [It faces the Brandenburg Gate and, of course, the West]. Strange. I don't know why, I can't tell you why.
You know, you have a certain sort of education and until you are maybe fourteen, fifteen you don't doubt much about what you have learned. Only at a certain age do you begin to think about some of the facts, to question. I always had one problem, in that I could not believe the wall was the solution to having a capitalist town in a socialist country.
I only saw the wall when I was visiting friends who lived near it. From certain places in East Berlin you could see some of the West – if you went, for instance, by S-Bahn near the district of Köpenick. I saw houses. I had an image. Maybe I had a different sort of image to other people because my parents worked in foreign countries and we knew people who lived in other countries. Maybe I had a more realistic picture, not sort of black and white between the East and the West.11
The GDR was her whole world and the West held no particular interest for her. After all, why should it? But now normality itself was uncertain, something which had not happened since 13 August 1961. Gorbachev had made it clear to his satellites that the Soviet Union would not interfere in their internal affairs.
Still the escape attempts were made. In March 1988 three men in a truck drove through the barriers across the Glienicker Bridge just before dawn. Reportedly, they had loaded the 7.5-ton vehicle with empty propane gas tanks and calculated that the Border Guards, not knowing they were empty, wouldn't shoot and risk an immense explosion. The three men guessed correctly. The truck battered a path through two steel gates and an electrified fence on the Eastern side, and continued across the bridge, smashing through the West Berlin control. (The bridge had long been used to swap spies, among them American pilot Gary Powers in 1962 and Soviet dissident Anatoli Sharansky in 1986.)
5 February 1989 _Chris Gueffroy, 20. Gueffroy, the waiter who wanted to open a restaurant in the West, made a_ _night-time attempt with Christian Gaudian, 21, in Treptow. The two men, without what was described as 'special help' – ladders, presumably – got over the inner wall. At 11.39 the alarm sounded when they were about 5 metres from the outer wall. The Border Guards reacted immediately. Gueffroy was shot through the heart and died. Gaudian was shot through the foot._
These two young men were the last victims of bullets at the wall, although Gueffroy was not the last to die because of it. Like the first victim – Rudolf Urban in 1961 – it is neither simple nor straight-forward. Urban, remember, made his attempt on 19 August but died later, so he was and was not the first. The last named escaper to die would make a balloon and try to cross in it. The last one of all would have no name and no details, a horribly apt epitaph for the wall.
The death of Gueffroy provoked international outrage but, in small and almost subtle ways, it demonstrated that the winds of change had begun to whisper. The Gueffroy family were allowed to place a death notice in the GDR press, something which had not happened before. It listed family and friends and announced that the funeral would be on 23 February. Some 120 people attended, young faces expressing grief in public, and a photograph of the image of that would appear in the Western media. Gaudian was subsequently tried and sentenced to three years in prison but West Germany bought him out.
3 March 1989 _Winfried Freundenberg, 33_.
At 1.50 a.m. on 3 March 1989, the Ministry of the Interior was informed by a man that he had seen somebody in the gasworks in the district of Weissensee filling a balloon. The man said that he had seen this at 1.35 as he was riding past on a bus. Immediately, a car was dispatched to the gasworks. At 2.10 a balloon of approximately 5 metres in diameter was seen rising with one person in it. The balloon hit a power cable of 380 volts which plunged the area into darkness. Due to this, it could not be ascertained in which direction the balloon was flying. When investigating the situation at the gasworks, a plastic tube was found, connected to the gas supply, and it had obviously been used to fill the balloon; the gas was still escaping from it.
The person was Winfried Freudenberg. When his wife was arrested she confirmed that her husband had committed this crime. She had helped him to make and fill the balloon. It had been planned for a long time and to achieve it Freudenberg got a job at the gasworks. Both of them were due to go in the balloon but a car had passed close to the gasworks, frightening them as they filled the balloon. The woman abandoned any idea of going and Freudenberg immediately started to ascend as fast as he could.
Futher investigations were carried out in the areas of Frankfurt-Oder and Potsdam because they didn't know where the balloon had gone. At 9.15 Freudenberg's car, a Trabant, was discovered and impounded. According to the weather station at Schönefeld airport, when the balloon ascended a north-east wind was blowing. As soon as it was daylight the search was resumed by helicopter.
But they were searching in the wrong place. Freudenberg reached the Western district of Zehlendorf, but the balloon fell from what police described as a great height, killing him.
On 8 April, two youths tried to escape through the Chaussee-strasse checkpoint but were halted by a warning shot. This was the last shot fired at the wall.13
16 April 1989 _An unknown man, about 18, drowned._
Nobody knew that that was it, the last of them, even though another seven months would pass during which, at any moment day and night, someone might have died because, for whatever reason, they wished to be free to go wherever they wanted and the journey to that began only on the other side of the wall. This book is the people's story, often told in the vividness of memory, but the italic paragraphs marking each death are dotted through like tombstones in their silence: each death, however shorn of detail, has not been a person's story. None spoke except perhaps, like Peter Fechter, to cry out in their fear and their agony. Each death has been another sort of story: of how anonymous we are, and sometimes how brave and how cruel and how obedient; of unimaginable circumstances leading ordinary people to the sprint into arc-lamps and watchtowers and gunfire; and of those ordinary people suddenly cast up upon a deadly stage. I wish I could give you a definitive list of every name and, perhaps, a little human anecdote for each but the italic paragraphs are all I can do. It remains the ultimate violation of these ordinary people that many have no tombstones of their own, because they were buried or cremated in official anonymity. So the paragraphs in italics here will have to serve.
In May, Hungary dismantled its wire border with Austria, and Hungary was a favoured holiday destination for East Germans, who drove down through Czechoslovakia towing their little caravans to camp round Lake Balaton. All they had to do now was go to the Hungarian border instead, select a quiet stretch of woodland and walk across. Once in Austria they could continue to West Germany where they would automatically, and instantly, become West German citizens. Nobody knew, but some did foresee it: this was the beginning of the final steps of the logic. By 1 October 30,000 had gone.
The rest was a straightforward end game played out to its own logic which intensified week by week into day by day and then, like the last convulsive spasm of its existence, into moment by moment. The actual process of this would run full across the fortieth anniversary celebrations of the GDR's founding, due at the beginning of October.
Bernie Godek, a 36-year-old major from Huntington, New York, was assigned to Headquarters Company, United States Command Berlin 'in July 1989 – I can remember it was the fourth of July. I had been stationed in West Germany before that and never visited Berlin. Some people did. They used to travel the autobahn from Helmstedt to Dreilinden but many of us perceived that it was just too difficult a thing to do – the processing of paperwork and getting permission. We were down in Bamberg and we enjoyed it there – in fact we loved it down there. We didn't have much interest to go to Berlin at the time.' By 'we', Godek means himself, his wife Donna, daughter Jennifer, who'd celebrate her twelfth birthday in late October, and son Christopher, who'd celebrate his ninth on 9 November.
Of course when in the army you go where they decide you'll go, and the Godeks moved into a detached house which was technically 'just in the British Sector. The houses were intermingled with some of the local German homes.' It was in the Dahlem district.
Godek's duties would be at Checkpoint Charlie but 'as a matter of fact, before I actually went up to visit the Checkpoint for the first time I had received briefings from several different individuals within the command. The most trivial thing there could suddenly become enormous and people wanted me to be sure that I didn't do anything wrong when I did go up there and that I was familiar with the procedures of everything. There were a number of dos and don'ts and I needed to be familiar with those.
'Having never seen it before, my first impression was surprise at how simple our checkpoint looked. I think I'd expected to see something a little bit bigger, something a little bit more elaborate on our side. There was nothing except a little metal building [the hut] and that was it. But on their side I was simply amazed – amazed at the contrast between the simplicity of Checkpoint Charlie and that metal building and the complexity of the crossing point on the East Berlin side. A monstrous complex. In general things were still very rigid.' All unknowing, Godek's posting would do more than introduce him to this rigidity which now spanned generations – much more.
On 8 August the FRG Consulate in East Berlin closed because it was swamped by refugees, and five days later the Consulate in Budapest closed for the same reason. Six days after that a symbolic Pan-European Picnic was held on the Austro-Hungarian border and 900 East Germans went over.
Godek remembers that 'once the official announcement had been made that Gorbachev was coming to East Berlin for the anniversary celebrations, several nights before he was due to arrive I received a call at home late in the evening, maybe about midnight, from the checkpoint. The Military Policeman on duty said, "Sir, you need to come out here right away. There is something going on." I asked him what and he said, "Well, there appears to be some engineering vehicles mustering on the opposite side of the checkpoint." We knew that because we had cameras positioned so that we could see over the checkpoint [into the East].'
From three storeys up on the side of a building a camera captured every movement beyond the white line and relayed the pictures to a screen in the hut and an observation office on the floor above the Café Adler. The office was named after General John Mitchell, the US commander in the city for four years, and a brass plate on the door announced that: the Mitchell Suite.
The Military Policeman added that, 'There seem to be some Border troops there, too.'
Godek said, 'OK, I'll be right there.' He called 'my driver, Sergeant Yount – my assistant – and he picked me up at home.' Michael Yount was an acting NCO. Godek recounts:
We went to the checkpoint. Sure enough, I looked at the camera and after that I went up into the Mitchell Suite with a pair of binoculars. I observed a collection of engineering vehicles – there were fork lifts, a mini crane of some type, some dump trucks and a couple of trucks of the Border Guards. We really didn't know what to expect because we hadn't seen anything like this before. I started to notify people within the Command as to what was going on. There were procedures set up where I would make notifications to certain individuals based on what activities I was observing, especially if strange things were happening – and this was very strange. Very, very strange.
I would say that after about an hour of observing them, they proceeded through their checkpoint and started progressing towards Checkpoint Charlie. We watched and we watched and we watched, and they stopped at the end of the roadway: they stopped about 5 metres before the demarcation line [Hagen Koch's line]. Remember the wall never sat on that line, it was always back from it. Then they walked up very close, probably within a metre of the line and they were talking and looking. They appeared to be officers from both the Border Guards and the engineering unit that was there.
Sgt Yount was fluent in German so he went outside and I walked up to within a metre of the line. I was very close and I was watching everything they were doing, which made them a little nervous. I tried to mirror their activity. According to what they were talking about, Sgt Yount said, 'It appears as if they are going to come up and extend the wall to the demarcation line.' That surprised me. After a period of time they started bringing up some of the engineering vehicles and, along with that, we noticed construction material in the back of some of their trucks. There was wood, there was some cement and blocks, things of that nature.
As they came forward with their equipment, I instructed the Military Policeman at the checkpoint to get out the movie camera so we could start videotaping this. This he did. Again this made the East Germans there a little nervous and after about fifteen minutes of us being out there with a video camera they then produced a soldier with a video camera who started videotaping us videotaping them...
They started bringing up the materials and they started extending the wall from its current location using this wood and cement blocks. They extended it right up to the demarcation line and they continued to do this. They took this new wall straight across the front of Checkpoint Charlie leaving just enough room for two lanes [so cars could enter and exit at the same time]. They knew that we would demand our right to access into the Soviet Sector. As the construction went on we were still trying to figure out what it was, what they were up to, and they fortified it pretty well. The cements blocks they were putting down were in fact more than that, they were cement slabs. They produced a formidable barricade. They didn't leave a lot of room for people. They'd have to go into the vehicle lanes.
This is now about two o'clock in the morning. We're starting to gain a little bit of media attention and of course your late-night pubs in West Berlin are clearing out – and this was the place to go. Next thing you know, we had a crowd of West Berliners gathering at the checkpoint, yelling at the Guards, yelling at the construction workers. So now we have a little bit of a commotion going on.
The really interesting thing was, and as far as I can tell this had never occurred before, their senior officer in charge walked down to within inches of the demarcation line. He was inspecting the construction work. I was walking the line – even – with him, and I was within inches of it. Of course I didn't go over it although I was authorised to do so – he wasn't. At one point I let him go ahead. I stopped. Then he turned around and came back. As he came back and he was shoulder to shoulder with me he leaned out and he bumped me rather hard, which surprised me and surprised Sgt Yount. He said, 'I can't believe he just did that to you.' I said, 'I can't believe it either.' So I walked back, caught up with him and from my side of the line I bumped him. At that point he walked away from the line. Sgt Yount said, 'You do realise that we have just had a serious international incident' and I said, 'Yes, but let's keep it between ourselves!' I walked away.
Normally an East German would have kept away from anything like that, and that's what was so unusual about it. He made an attempt to have physical contact with the American officer in charge of Checkpoint Charlie and, I repeat, I don't think it had ever occurred before. It seemed pointless, and not a word was spoken, but I assume he was trying to tell me that he was very annoyed that I was out there and watching so closely what they were doing. That was his way of telling me that 'You're really annoying me. I've got to get this job done and you're out here scrutinising me.' That was the message I got.
We figured out why they put the new wall up. In fact I think it was explained to us through official channels afterwards, and the reasoning was twofold. The main reason was that they expected riots and demonstrations on the West Berlin side against Gorbachev's presence in East Berlin and that turned out to be true. We had some very terrible demonstrations, and some very physical demonstrations the day after that wall was put up. Secondly it was another means of protection at a point where the wall was open, at Checkpoint Charlie. I think not only were they trying to prohibit the West Berliners from getting over that demarcation line and demonstrating inside their checkpoint but using a bit of precaution on _their_ side against East Berliners. It was one more barrier that Easterners would have had to go through: they had Guards positioned at those little vehicle lanes, armed Guards. They were there for the entire time that Gorbachev visited.
By now opposition groups were forming within the GDR, demonstrations were taking place in Leipzig and the FRG Embassy in Prague was itself swamped by refugees. They were allowed to emigrate but, to save face, the trains had to pass through the GDR to the West. In Dresden some 5,000 people gathered to try and board them. A day later the GDR closed the border with Czechos-lovakia and had now cut itself off from the twentieth century.
A day after that Honecker was at Schönefeld airport to greet Gorbachev, guest of honour at the fortieth anniversary celebrations. Honecker, hands deep in the pockets of his dark coat but looking somehow neat and dapper, walked across to where a bank of the media hovered. He was asked how he felt. 'Those who are condemned live a long time,' he replied – the notion made him smile – and added that on such a morning he felt like a Berliner. A group of senior politicians stood around him and one, Egon Krenz, stood behind him, hands in coat pockets, too. His smile was drawn between obligation and genuine amusement. Honecker was asked what he'd be saying to Gorbachev; he turned and said, 'Wouldn't you like to know!' That made everybody laugh.
Moments later the big white and blue Aeroflot jet landed and taxied towards where the group waited for it. A dutiful crowd waved GDR flags as Gorbachev, overcoat unbuttoned, came down the gangway with his wife Raisa beside him. Honecker advanced and they shook hands, then Gorbachev, himself smiling, bent forward and the two men embraced, kissed three times on the cheeks in the Russian way. GDR television described this as 'hearty greetings between two men who know each other well'.
The Gorbachev visit became a juxtaposition, with demonstrators on the streets of East Berlin calling, 'Gorby, help us! Gorby, help us!' It was utterly unthinkable even five years earlier for any Eastern bloc population to plead genuinely with a Soviet leader to liberate them.
Perhaps it can be explained in this way. Stalin had tried to shape the twentieth century and had been responsible for the deaths of uncounted millions. Gorbachev was allowing it to reshape itself without killing anyone at all. The transition between the two mentalities, from the Middle Ages to Enlightenment, was fundamental and it had happened.
At Checkpoint Charlie the mood was different. Godek resumes the story: 'As I've said, the demonstrations that went on from the West Berliners were just terrible. I couldn't help but feel a little compassion for the Border Guards, the armed Guards that they'd positioned at the new wall. They had quite a number of them there, had them in depth. The West Berliners were throwing beer bottles and eggs and anything they could get their hands on across that wall, that temporary partition, to hit those Guards who were there. I specifically remember the Commander – he was a little guy, really short, which surprised me a little bit... .'
He was called Günter Moll. 'He got pelted with eggs at one point but he stood there firm even though the egg yolk was pouring down from his hat and his shoulder and his chest where he'd been hit with the eggs. He didn't flinch in front of his troops. He eventually did, trying not to make too much motion, wipe some of it away so it wouldn't fall on his face too much. He wasn't the one who bumped into me. The individual who did that was a rather robust guy.'
Gorbachev met the GDR Politburo on 7 October and listened to Honecker extol the virtues of the country without even mentioning the refugees. Gorbachev rolled his eyes as if to say 'OK, Comrades, that's it, what are you going to do about it?' It meant: 'The Soviet Union will not interfere in you regulating your own affairs.' There must have been an unsuspected beauty about the moment: the wall which was so solid and so lethal would be destroyed by a minimal gesture – the eyes of the man.
As it happened, the new wall at Checkpoint Charlie was dismantled when Gorbachev had gone, but that must have been pre-planned and seemed to presage nothing.
Krenz and Berlin Party secretary Günter Schabowski overcame what has been described as their mutual mistrust and sounded out as discreetly and delicately as they could, other Politburo members. On 18 October the Central Committee met. At 9.59 a.m. Honecker entered the room as usual and shook hands with all twenty-six members. He was just about to read out the first point on the agenda when Willy Stoph, the Prime Minister, said, 'Erich, allow me.'
Honecker, taken by surprise, said, 'Well, yes.'
'I propose that Comrade Honecker abdicates his function as General Secretary.' Stoph added that two other comrades should go, one of them being Günter Mittag, the economics supremo.
Honecker, obviously stunned, made a fleeting attempt to return to the agenda. That was protested. 'All right then, let's discuss it.' He searched the room among the most sympathetic of these old comrades for allies, and found none. In accordance with communist practice all decisions had to be unanimous, obliging Honecker to vote against himself to maintain that. Ultimately, the logic was as unforgiving as that.
Krenz took over a country breaking up, technically bankrupt and deep in an identity crisis, something it was unlikely to survive given its history and especially with Big Brother the other end of Friedrichstrasse waiting to help. Krenz tried to buy time and bought less than two weeks of it.
It had reached the day-by-day stage.
Demonstrations increased, the Politburo met the opposition and announced that from 1 November every East German could travel to the West for thirty days, but these were gestures, holding measures, concessions from the planned place which had no plans. And everybody saw, and knew. The monolith could neither disintegrate nor adapt to changing climates, and certainly not at this pace. On 1 November Krenz met Gorbachev in Moscow and said that demolishing the wall was 'unrealistic', a statement which itself would become _unreal_ inside eight days.
The FRG Embassy in Prague was swamped again and a million people demonstrated in East Berlin. On 8 November the Politburo resigned (some members were re-elected but the currency for those things was long exhausted). Krenz was clinging on and the logic was preparing to consume him.
Next morning, Commander Günter Moll got in his Skoda and drove from his apartment in the suburbs to the checkpoint where he'd enact the usual rituals of control. It was Thursday and, he assumed, just another day of clockwork within the unchanging geometry.
At 9.00 that morning four men met at the Ministry of the Interior in Mauerstrasse behind Checkpoint Charlie, two of them Stasi. The Politburo wanted new travel regulations to solve the 'Czech problem' because the Czechs were unhappy that Prague, flooded with refugees, was still a staging post to the West. The four men decided to tackle the problem 'head on'14 by recommending that Easterners should be allowed to travel, although with the understanding there would be procedures for obtaining visas. At 12.00 the proposal was given to Krenz in a Politburo meeting. It was approved at 12.30 and passed on to the Council of Ministers (a symbolic body). At 3.00 work was going on to resolve the details of how the new regulations would function and at 3.30 the proposal came back from the Council of Ministers, rubber-stamped. Although the regulations constituted another concession the intention was to control the process.
Krenz read the regulations again. 'Whatever we do in this situation we're bound to make a mistake but it's the only solution that spares us the problem of dealing with this through third countries, which harms the international image of the GDR.' Schabowski wasn't in the room at this moment. As the government's press spokesman he was constantly being called away to talk to journalists: not that that could have mattered. Could it?
* * *
Quotation at head of chapter: _The Wall in My Backyard_ , ed. by Dinah Dodds and Pam Allen-Thompson (University of Massachusetts Press, Amherst, 1994).
1. Irene Böhme, _Die da drüben_ (Rotbuch Verlag, Berlin, 1982). I have used the translation in _Opposition in the GDR under Honecker 1971–1985_ by Roger Woods and Christopher Upward (Macmillan Press Ltd, London, 1986).
2. Pensioners were not only allowed to cross but were not prevented from emigrating West if they wanted. In this way the FRG had to pay their pensions, relieving the GDR of that burden. See Chapter Six.
3. This account is based on the description in _Escape from Berlin._
4. Associated Press.
5. Monica Brath used the old German name Chemnitz for the town in the south of the GDR which had been renamed, reflecting the communist desire to impose themselves, Karl-Marx-Stadt on 10 May 1953. To use Chemnitz was itself a statement of rejection – and she used it twice. Incidentally, in the 1970s and 1980s I used to goad GDR journalists working abroad by referring to it loudly as 'Chemnitz', to which they would reply, in unison, 'Karl-Marx-Stadt', almost as a reflex action.
6. Interview with author.
7. _Philby: the Spy who Betrayed a Generation_ , Bruce Page, David Leitch and Phillip Knightley (André Deutsch, 1969).
8. Long after his and the GDR's fall, Birgit Kubisch heard Honecker interviewed and was astonished at how he used the old terminology as if nothing had happened. She could scarcely believe it. From every utterance he made then, and elsewhere, in his mind nothing _had_ happened to disturb his creed.
9. I am entirely indebted to Yorkshire Television and their programme _First Tuesday_ in November 1991 for this account.
10. _The Hidden Hand_ , Jeffrey Gedmin (AEI Press, Washington, DC, 1992).
11. Interview with author.
12. Interview with author.
13. _The Berlin Wall_ , op. cit.
14. _Unchained Eagle_ , Tom Henegham (Reuters, Pearson Education, Harlow, 2000).
## NINE
## _A Quiet Night Like This_
The wall will still be standing in 50 or 100 years if the reasons for its existence are not removed.
Erich Honecker,
January 1989
Beyond the barricade Günter Moll could see Hagen Koch's faded, hand-painted white line bisecting Friedrichstrasse and still marking where the tectonic plates met. Beyond that he could see the West clearly, the alien place, the hostile land which was known to him only by reputetion and image. Under no circumstances would he step across the white line.
Beyond the line Moll could see the big roadside board informing travellers that they were leaving the American Sector and the Allied hut in the middle of Friedrichstrasse, long and creamy in colour, decorated with the Stars and Stripes, and the Union Jack and Tricolour: the British and French had offices there, too. He could see the window in the hut from where American Military Police watched the three watchtowers across the line and the Border Guards in the watchtowers gazed back. They were forty paces apart.
Beside the hut Moll could see a corner café and, further away, a shop but not the bank because the angle from his office was wrong.
The safety valves which blew open on 9 November 1989:
1) Bornholmer Strasse;
2) Chausseestrasse;
3) Invaliden Strasse;
4) Friedrichstrasse (Checkpoint Charlie);
5) Heinrich-Heine-Strasse;
6) Oberbaumbrücke;
7) Sonnenallee.
Often enough from the turret of the watchtower on the right, its roofing like a pelmet, binoculars lingered on a 29-year-old waitress in the Café Adler on the corner just the other side of the line. She was financing her studies to become an art teacher. Every evening she saw the binoculars sweeping her through the tall window of the café and she wondered what the Guards thought. She did not regard them as remote, only as people of her own age – and probably younger – who she could not reach. She was called Astrid Benner.
They surveyed her ferrying coffee and beer and schnapps – _korn_ in the dialect – and sometimes she felt sorry for them. 'We'd have parties, it was colourful, it was fun and over there it was so sad. I sensed the Guards in the tower were interested and I would like to have made contact with them but, of course, I couldn't.'1 If any one of these Guards tried to come to her he'd risk being shot, and she couldn't go to them because nobody ever approached a watchtower. You risked being shot yourself and, even if you weren't, you'd certainly be hauled away; and who knew what would happen to you then?
Astrid Benner worked within hailing distance of Moll but she knew nothing of him. The line held their worlds apart, too.
As Moll drove off she had already been serving for an hour. She'd taken the underground from her university class and emerged at the station beside the café as the day ebbed into darkness. No doubt the Guards swept her as she went into the café. They watched everything, especially a pretty girl. It helped pass the time. They'd evolved the system about that. Any chance of using the binoculars to watch a girl undressing in a bedroom on the Western side, like _Long Tooth_ , and the word went from watchtower to watchtower like an electric current – but she was only serving the way she did each evening.
Moll took his Skoda round the gentle curve of buildings flanking the checkpoint's rear, some sharply modern, some riddled with staccato bullet marks from 1945. Any journey into the East was always movement through time and you could read the story of the city through its buildings and its streets renamed across the years to commemorate heroes of socialism.
Moll reached traffic lights with red figurines on their stems, arms outstretched to hold pedestrians from crossing, and green figurines who made walking motions to release them. This was the pre-determined civilian clockwork. He was already deep into the East now and out of sight of the checkpoint. At these traffic lights a postal museum stood, a brooding, sombre stone edifice. The stone wasn't merely chipped by the rifle fire of '45 but gouged deep, each mark so vivid that the mortar shells might have burst only a few moments before.
When the lights changed he turned into Leipziger Strasse. Before the war it had been a shopping promenade but the Allied bombing broke it and no doubt the street fighting finished it off. So little usable survived that the whole lot was demolished to make a broad avenue lined by workers' apartments, each a cliff-face of socialist reality. Their balconies, small as birdcage perches, rose in vertical ranks floor by floor.
As Moll drove, a cluster of little Trabant cars putt-putted like lawnmowers around him, square Wartburg saloons hiccoughed forward and an occasional pugnosed lorry lumbered under the weight of its load. He was in the evening rush hour going home, same as every evening.
Far down Leipziger Strasse he passed the twin spires of the Nikolai Church, the oldest church in the city and, restored from the bombing, beautiful in its simplicity. The church was drawn in great contrast to the cliff-faces around it. Then he passed the rear of the SED's headquarters building, where Krenz and the Central Committee were meeting, discussing economics. He drove towards the park dominated by the Soviet war memorial, turned towards the suburb of Treptow. He'd be home nicely in time for dinner.
At 5.30 p.m., Chris Toft, a 36-year-old British Military Police sergeant, lined his duty men up in the barracks near the Olympic Stadium far to the Western side and gave them a normal briefing, then issued them with pistols as he did each evening. 'We went to the loading bay and I supervised the loading of the pistols. The bay was surrounded by sandbags and they pointed the pistols at them while they were loading in case one went off by accident. They got into our white Volkswagen Combi with Military Police written on the side and drove off.'2 They headed across the city towards a small control room near the wall in the British Sector where they'd take over patrolling from the day shift while, at 5.45, Toft settled in the main control room at the barracks. Desks were arranged in a square and each had a little bank of a switchboard. Another evening, that was all. Toft had the usual personnel around him, a desk NCO, an interpreter, a German liaison officer from the city's riot squad. At night there might be trouble, likely as not involving drunks, and a liaison officer was always present to help coordinate action between the civilian authorities and the British military.
Some time around 5.30 Schabowski spoke to Krenz about a press conference scheduled for 6.00 which he – Schabowski – would give. Krenz handed him the piece of paper which had the new law on it. 'Announce this,' Krenz said. 'It will be a bombshell.' Schabowski did not read what was written on the paper.
In the Café Adler, a place of wooden chairs and tables and a semicircular bar nestling around one corner, Astrid Benner put a cassette into the tape recorder on a shelf. It was music to take the dull edge off the place, music to soothe the boredom. 'Because of its situation the café was not very popular after nightfall and I was working alone the whole evening, only me. One person was enough. In the daytime the café was popular with tourists looking at the checkpoint but in the evenings not many people came, never more than twenty. It wasn't very interesting to be there then.'
Moll reached his apartment, small and cosy, parked the Skoda and settled in front of the television. His wife Inge began to prepare the meal.
At 5.45 Bernie Godek left his 'very simple office, beige walls with a drab green carpet and a military couch covered with artificial leather look-alike' at a barracks not far from the Olympic Stadium and drove the ten minutes to his four-bedroomed house in the wooded suburb of Dahlem. It was son Christopher's ninth birthday and they were having a party. Every year Godek tried to be home for that, wherever his duties took him.3
He knew Commander Moll, although by sight, not by name. In a sense, Moll did not exist because if any incident developed, Godek would demand the presence of a Soviet officer.
The party was going well. Godek took off his uniform shirt but 'I still had my fatigue pants on and my boots and a green tee-shirt. My wife asked me if I'd go ahead and cook up some hot dogs in the microwave for the kids so I started heating them and putting them in buns.' Ordinarily he might have switched the TV on, but not now.
Godek bore the responsibility of his life calmly although at moments he found ensuring Allied access at Checkpoint Charlie 'a hell of a thing'. While, of course, the checkpoint lay within the American Sector it was an official crossing place for all the Allies, hence the presence of the British and French in the hut. Allied military vehicles still went daily trawling in the East as they were entitled to do, just as Soviet vehicles could – and did – still enter the West.
The party was due to go on for another hour or so and Godek was good at serving hot dogs, the way dads are.
At 5.50, Schabowski travelled the hundred yards or so to the International Press Centre where he would give the press conference. He had a heavy face, nearly Slavonic, all cheeks and jowls. It was about to become extremely famous. He wore a light but formal suit in soft tartan, a white shirt, a dark blue tie with a pattern angled down it. The press conference room was full: maybe a hundred journalists.
At 6.00, the British Military Police Volkswagen arrived at the small control room near the wall – it was about a mile from Checkpoint Charlie. The men in the Volkswagen duly relieved the dayshift, clambered into their vehicles and began to patrol the wall in this, the British Sector. The Volkswagen continued on to Check-point Charlie to also relieve the dayshift in the hut before it headed back to the barracks.
At 6.00, Schabowski entered the press conference, a tall room of polished wooden walls and rows of red seats, and walked ponderously towards the platform at the far end where he'd sit behind a long desk. Television cameras probed at him – GDR television was covering the event live. He sat and began talking about the day's developments.
They were all in place now, or nearly in place. Godek sliced more buns and Toft waited by his little switchboard for the first radio reports – routine traffic, as it would surely be – to come in. Astrid Benner listened to the taped music and flitted to and fro serving seven customers.
Maybe the Border Guards observed her, maybe they didn't.
Moll watched television, and there was Schabowski wandering along a great verbal path which probably wouldn't lead anywhere.
Far up Friedrichstrasse on the Eastern side a dark-haired student of journalism made her way to an evening class. In her twenty-one years Brita Segger had never been to the other end of Friedrichstrasse nor imagined she ever would. She could go to Vietnam because that was approved, but not the Café Adler. Even geographically the Café Adler just down the road lay much further away than Hanoi, or the moon.
She worked on a people's factory newspaper as an apprentice and the evening classes were to teach her the art of writing. She needed to pass her _Abitur_ , the examination which would take her to university and an assured future for the rest of her life. All she'd have to do was write the Party line for ever and ever and ever. She walked the few minutes to her class.
Inge Moll flitted out from the kitchen to catch moments of the television. Günter watched with interest but no particular sense of anticipation. He was 47 and far too old for speculating, because his domain was one of certainties.
Schabowski had seven microphones in a thicket before him. He'd put on his half-lens spectacles and they gave him the air of a schoolmaster. He dipped his head and droned on. At no moment did he raise his voice to generate anything but a monologue in monotone.
At 6.53 an Italian journalist sitting on the rim of the platform, microphone in hand, broached the subject of travel. Schabowski announced that 'today' new regulations had been drawn up to allow East Germans to travel. 'Private trips abroad can be applied for without questions, and applications will be dealt with at high speed. The People's Police have been instructed to hand out long-term exit visas without delay, and the conditions which have applied up until now are redundant.'
That brought a sharper question from the floor: 'When? Immediately?'
Schabowski stroked his chin with his right hand, a contemplative gesture, and peered over the half-lenses towards the questioner. He reached down and turned the paper he had been reading from, searching for something and, as he searched, he murmured, 'As far as I know immediately, without delay.' Twice, three times he turned the piece of paper, abandoned the search. He pursed his lips. He had not, after all, read it when Krenz gave it to him.
It was 6.57.
Now the questioning moved to West Berlin and if East Germans could go there – an obviously, and perhaps profoundly, delicate thing. Schabowski wondered if anybody had told the Soviet Union about the new rules. It was too late for that now. He consulted the paper and found what he needed: crossing was permitted to West Germany and West Berlin.
Only once, and then only slightly, did he raise his voice. He might have been working through a litany about tractor factory production or announcing a friendship visit by the fraternal comrades of the People's Republic of Mongolia. Schabowski was unaccustomed to altering the tonality of his voice, no doubt because that had never been necessary. In an authoritarian state, when people in authority spoke others just listened.
It was 54 seconds after 7.00 p.m.
The timetable which had begun by being measured in months, taking as its starting point 2 May when Hungary opened its borders and the East German exodus began, and then in weeks since the fortieth anniversary of the GDR on 6 October, had now descended to hours; and not many hours remained – about four and a half.
Within four minutes, the news agencies – Reuters, DPA and Associated Press – were sending newsflashes. In the confusion, Reuters and the DPA concentrated on the fact that there would be new travel arrangements but Associated Press went hard that the borders were going to open.
Erdmute Greis-Behrendt 'was in the office, actually. The Press Conference was in the building where our office was – the International Press Centre – and we didn't know what to expect. Some Reuters people had gone down to it but I watched on television. Schabowski used to come over for Press Conferences and they were always held at the Centre. The Conference was on the ground floor in a huge big room.
'We had no warning of what he was going to say although we thought something might happen. We had a little argument after he said, "As far as I know immediately." There was a bit of confusion about what it meant and nobody seemed to be sure. However, we made up our minds that we had understood it correctly and that he just opened the wall: we replayed a video of it to make sure we weren't getting it wrong. And then, of course, everybody was very busy writing and writing and writing.'4
Greis-Behrendt remembers a certain confusion over whether people would need identity cards or visas and it transpired they'd need visas, which would be issued by travel offices. She was dispatched into East Berlin to visit these travel offices to see what was happening there – all else aside, it would give confirmation of whether the wall was really opening if visas were being issued. They were closed, 'all dark', in her phrase.
## TREPTOW, EAST BERLIN, 7.00 P.M.
Moll heard what Schabowski said but felt no sense of urgency, even at the word _immediately_ 'because things had always been the way they were and there would have to be a process for any change to be made, it would all have to be worked out legally. Perhaps the following day, perhaps the day after that, I would get some instructions but probably they'd take longer. I didn't think about it any more.'5 Still Inge flitted, dipping in and out of the news.
## BORNHOLMER STRASSE CHECKPOINT, 7.00 P.M.
It was a deep place, slightly sloping up towards an old bridge. It was an extensive place, too, with a wide awning under which vehicles were checked before they could continue to the bridge. The West began on the other side. The man in charge of the Border Guards, Manfred Sens, watched the Schabowski press conference on television and made his way to the opening in the inner wall. He wanted to warn the Guards there that people might come and ask if they could cross but by the time he arrived a few people had already gathered.
The man in charge of the passport control, Harald Jäger, watched the Schabowski press conference, too. They were working 24-hour shifts and he'd begun at 6.00 that morning. He was eating his supper in the canteen – bread and sausages – and saw the conference on the black and white television set there, but when Schabowski said, 'As far as I am aware, immediately', he stopped eating because he almost choked. Rubbish! he thought instantly. What does this Schabowski mean by immediately? He's had a moment of verbal diarrhoea! Immediately – that is simply not possible.
This division between Border Guards and passport control must be examined because, as the hours tightened into minutes, it assumed particular importance. The Border Guards, by definition, secured the border, including the seven checkpoints – here at Bornholmer Strasse, Chausseestrasse, Invalidenstrasse, Checkpoint Charlie, Heinrich-Heine-Strasse, Oberbaumbrücke and Sonnenallee. The passport controls were part of the state security service, the Stasi, and there was no doubt in Jäger's mind that he was effectively in overall charge of Bornholmer Strasse. He was 48, had been in passport control for twenty-eight years and knew the form. For the twenty-eight years all checkpoints had functioned normally although this was disturbed by the weight of traffic at holidays when they had 'a lot of trouble'.
Jäger describes the relationship between passport control and the Border Guards as 'a division of tasks. Everything which happened within the checkpoint – citizens coming in, citizens going out – was our responsibility. The Border Guards were responsible for the outer security, which meant regulating who could get in to the checkpoint.' Jäger entertained no doubts about his position or responsibilities because these things were 'regulated, fixed in writing'.
In the canteen were others, but chatting to each other, not really listening. When they saw Jäger's reaction they asked him what his problem was. He thought he'd better go and telephone headquarters and find out what was going on. He left his supper and walked to his office, which, because of the size of the checkpoint, took a few minutes. He got there at about 7.15 and already by then the number of people who'd gathered had risen to about ten.
He rang his headquarters to speak to his boss 'who was responsible for all the controls within the capital [East Berlin] and he was informed of everything that happened at the checkpoints. I wanted to know if it was right that GDR citizens were allowed to travel abroad, because it was clear that Schabowski could not just give such an order. There was a chain of command, measures that would have to be prepared and of course we would need enough extra personnel at the checkpoint to cope with such a measure: stamps, technical equipment, everything. There would have to be a regulation about which document they were allowed to travel abroad with. We didn't know which document we would have to ask citizens to present.'
At 7.17 the West German television channel ZDF's programme _heute_ ( _Today_ ) carried the news, but as their sixth item. Reflecting the confusion, they spoke only of new travel arrangements. At 7.30, the GDR news programme _Aktuelle Kamera_ carried the news as their second item.
Jäger rang his boss, who had seen the press conference. 'I have never heard such rubbish in my life,' his boss said. 'Did you hear that rubbish, too?'
'Yes.'
The boss asked how many people had gathered, and Jäger phoned what was called the _Vorkontrolle_ , the point where people entered the checkpoint. Between ten and twenty people, he was told. He reported that to his boss, who said, 'You must wait: sit and wait.'
Jäger made his way to the _Vorkontrolle_ , where Sens was. That was about 7.30, and even as he walked there the pace quickened. He estimated there were some fifty to a hundred people there now 'and they were asking if they could travel. They had seen the Press Conference and come to see if it was true. I told them it was not possible because according to our regulations they needed a passport and visa and without them they couldn't go' – you had, of course, to apply for a visa and ordinarily if you wanted to go to the West you wouldn't get one, you'd get trouble instead. 'I told them to come back the following day and some of them went away. There was a fence to which they could go, a little fence, and only those with a passport and visa could go beyond it – it was like a little dead area. It belonged to the Border area: they were already on the area of the checkpoint but not in the checkpoint itself, so now they stood on an area which was normally prohibited.'
## DAHLEM DISTRICT, WEST BERLIN
The telephone rang at Godek's home and he heard a voice he knew well, Sergeant Nathaniel (Nate to his friends) Brown, the American duty man in the Allied hut. Brown told Godek he had 'an indication that something was going on because local news media were starting to show up. They said there was talk of the wall opening but, being the kind of guy he was, Sergeant Brown doubted it at first. Then he got a call from one of the local media asking him if he knew anything about it. He decided it was time to ring me to let me know what he had been hearing. I told my wife that based on that I needed to go to the checkpoint right away. I never really told her why, I never shared that information with her and that's the way she probably preferred it. Her attitude was 'as long as you have something important to do you just go do it'. I felt a bit sad because I always enjoyed doing the hot-dogs.'
Godek called Sergeant Yount and relayed what Brown had told him. Godek asked Yount to 'meet me at my office. Our Military Police vehicle was parked there so we used that as a central point.' When Godek arrived he called Colonel John Greathouse, a political military adviser who was working late. Greathouse asked Godek to '"Come by and pick me up." He'd like to go there with us.'
Brita Segger continued to learn the art of writing.
## CHECKPOINT CHARLIE, BETWEEN 7.30 AND 8.00 P.M.
Of Astrid Benner's seven customers five were women who worked on a newspaper round the corner. They used the Café Adler as a haunt because it was conveniently close to mull over the doings of the day. They were talking about the woman's page. A photographer came in and said that 'in the next hour something will happen here'.
Astrid Benner wondered what he was talking about.
'Don't you know? They will open the border.'
The women didn't know either because the taped music had insulated them. One said 'Can you believe it? Let's listen to the radio.' Benner flipped the tape out, turned on the big box radio with its strong speakers and everyone heard it together, a recording of the laden voice of Schabowski seemingly coming at them from wherever Benner twisted the dial. All channels were carrying it and trying to wring a meaning from those final words about the lifting of travel restrictions.
The women, according to Benner, 'got excited and said, "We'd better get back to the office" and they started to run. I thought to myself, "What will I do now? I am by myself here." I called my boss, the owner of the café, at his home. "Hell," I said, "you have to get here because I am totally alone and thousands of people may be coming at any moment. This is the first place they'll reach... ."'
The owner, Albrecht Raw, made it inside 15 minutes with his wife Nellie.
Outside, Benner saw 'a few people beginning to arrive from our side, the Western side. They stood round and some came into the café for a drink. A feeling grew like it was New Year's Eve, a sort of mounting excitement but nothing happened. We were listening to the radio, commentators were talking and talking but nothing happened. We felt it had to, and now, but it didn't.'
Across the line the checkpoint ticked evenly. The white light fell on its carpet, the binoculars tracked from the watchtowers. There was minimal movement under the Customs awning. The checkpoint presented itself as it had always done, set in its terrible permanence.
'When my boss and his wife arrived we wanted to drink champagne,' Astrid Benner says. 'Then I had an idea. _It's not just we who ought to be celebrating. The Border Guards should be celebrating, too. Let's give them some champagne_. My boss thought it was a good idea. I got a tray out and set about twenty glasses on it, he opened two bottles.'
Checkpoint Charlie (not to scale).
Benner and Raw came down the three shallow steps of the café and turned towards the white line. An hour before this would have been unthinkable and extremely dangerous: the Guards in the tower lurked like menacing presences behind the windows, trained to shoot. Benner and Raw crossed the white line, walking at a stately pace towards the watchtower. Benner didn't 'think of the possibility the Guards would shoot us because we were making such a friendly, open action and everybody could see it was not dangerous'. Something fundamental, she felt, had altered. As they walked, two Guards emerged from the watchtower and advanced urgently towards them. 'They wore grey uniforms. One stood in the background and the other came directly to us – always two Guards, one covering the other.'
The first intercepted Benner and Raw midway between the line and the tower. Astrid offered him champagne.
'Go back,' he said.
'But we have to celebrate this exciting day, don't you want to celebrate with us?'
'No, no, we don't want that, please go back again.'
'If you don't want to drink now take this bottle. You can drink it later.'
The Guard, Benner would remember, was 'severe just like the Border Guards were but I could sense he didn't know what to do. I assumed he had heard the news from the Press Conference but of course he might not. When it was obvious he wasn't going to accept any champagne we re-crossed the line and drank with the people standing there. By now, slowly, more and more people had gathered. Then we returned to the café and began to prepare for a big party. We went into the cellar to get glasses and telephoned anybody who might be able to lend a hand but either they didn't have time or they weren't at home. My boss's wife had never worked in this line of business but she said, "OK, I'll help." The Guards were back in their tower watching us through their binoculars.'
After this innocent little flurry the checkpoint re-set itself in concrete.
Moll's telephone rang. Simon informed him that 'a waitress and a man from the Café Adler had come across the line and wanted the Border Guards to drink with them. The Guards had told them they were on duty and not allowed to drink, the waitress and the man had returned to the café. I told Major Simon it didn't seem a serious situation but to keep his eyes open and watch everything. If it did become serious he should call again.'
Far to the other side of the city Godek picked up Greathouse who told him about the press conference. It was a lengthy drive. They reached the checkpoint and 'there were just a few people in the area, as a matter of fact. It really didn't look much different to the way it looked on any evening. A few tourists always milled around. Some media personnel were there, some people were just lingering. We walked to the hut and spoke to Sgt Brown. The _Imbiss_ [the snack bar beside the Allied hut] was open and one or two people stood eating. Nothing unusual.'
Godek had those unseen eyes, the camera up on the side of the building above the café relaying every movement on the Eastern side to a screen in the hut as well as the Mitchell Suite. 'Colonel Greathouse asked me to take him up there because he wanted to use the telephone. I unlocked the door and let him in so he could make some calls. I went back down to the checkpoint to monitor and sense what was going on. More and more people started showing up, more media started showing up.'
They were selling champagne by the bottle in the Café Adler now.
Sergeant Michael Raferty was on a couple of days leave and during the afternoon he'd picked up his wife and daughter from the airport. They were returning from a trip to the States. Raferty sat watching the news in his pleasant flat near where Godek lived. He was due to take over from Brown in the hut the following morning, 5.30 sharp, but when he heard Schabowski's voice chewing through the statement, heard the question and answer he rang the checkpoint. Brown said, 'Nothing was going on. That kind of shook my head because of what had been on the news. I thought I must be blowing this out of proportion.' Raferty woke his wife and said East Germans were being given the right to travel. From deep within her jet-lag she said gently, 'That's nice' and drifted back to sleep. He decided to get a good night's sleep himself. He was due, after all, to start at 5.30.6
At 7.42 p.m. one of the telephones in front of Toft rang. Major Watson, his commanding officer, at home, had heard from someone that 'the BBC World Service stated the border would be opening and did I know anything about it. I said no, I didn't. He said, "Can you find out?" I asked the interpreter and he said he knew nothing, he phoned the police and they didn't know, either, then he phoned the city's customs service and they did know. They said, "Yes, it's opening at midnight."' In a very British way Toft said to the interpreter, 'Oh, thank you very much' and phoned Watson, who said he'd be there at eleven. Toft had a ruled piece of paper in front of him which he used as an incident sheet. On it in careful handwriting he set down at the top of the page using military time: '1942 hrs. Telephone message from Maj. Watson stating BBC World Service had reported East Germany has lifted all travel restrictions to West Germany. All Royal Military Police informed.'
Silent as thieves, cautioned by great fear, lured by the press conference and a fascination they could no longer resist, people started to come from the dark and ghostly side streets of the East, each footfall a trepidation. They came to these side streets from Leipziger Strasse and hesitated. From their side the checkpoint looked more forbidding still, a crossing point for foreigners. No East German without a visa on his passport – vetted by the secret police, the Stasi, and rarely granted – ventured near any of the city's checkpoints but especially not this one. It was a criminal offence – 'unwarranted intrusion into a border area'. That was not a nicety. Something very nasty would happen to you, probably three years in jail, and GDR jails were not necessarily places you wanted to go.
In the twenty-eight years of the wall and the checkpoints, so many had died and the communal, unstated memory of that lingered. But now they began to come and not one of them knew if the press conference was a confidence trick, an attempt to buy breathing space. Every one of them knew the word 'immediately' might mean now, tomorrow, a week, a month, a year. Every one of them knew about the shoot-to-kill policy.
## BORNHOLMER STRASSE CHECKPOINT, 8.00 P.M.
The fifty to a hundred had grown to several hundred. Jäger had rung headquarters again to describe the situation but his boss repeated that there were no new instructions and he should 'send these people back'. Jäger began, however, to form an impression that something was going to have to give.
At 8.00, the West German television channel ARD's programme _Tagesschau_ – a daily review watched by many Easterners – led with 'GDR opens border' and this can only have increased the curiosity of thousands upon thousands of East Berliners to go and see if it was true. There were even rumours that Bornholmer Strasse had actually opened. However, whether you went to a checkpoint to try your luck, disbelieved the whole thing, or simply continued with your life regardless, seems to have been a matter of chance as well as choice: where you were, what you heard, what you didn't hear.
## FRIEDRICHSHAIN DISTRICT, EAST BERLIN, 8.00 P.M.
Uwe Nietzold was 'a student and I had never been to West Berlin. I'd seen it from the S-Bahn and from gliders, because I had tried that. You could approach the border line up to 5 kilometres and the allowed height was 2,500 metres, and that means you can see a lot. So I'd seen the wall, and forests and houses in the south of West Berlin.
'That night I went to a poetry reading with my club, a culture club but everybody wrote their own thing and read it out to the others. They discussed their writing. It was in the building of Neues Deutschland because the printing department had financed it. They tried to encourage young people to write. It was usual that every factory and enterprise had a cultural department. This meeting was in a little canteen and began at 6.00.
'I read a short story about a little boy with a ball which he'd discarded – he was sitting in a playground alone playing games on a little computer. The moral was that children weren't playing with other children and conventional toys any more, they play alone with computers.
'At 8.00 somebody came in – a member of the club, but late – who'd seen the press conference. They said, "If you can believe what was said, we can go to West Berlin and everywhere else tomorrow morning." I didn't believe it. I thought it must have been a mistake. Some people thought it would happen but not tomorrow morning. The meeting continued and we didn't take it seriously... .'
## CHECKPOINT CHARLIE, 8.00–8.30 P.M.
There was an inherent sense of theatre about this place and always had been. Since August 1961, it resembled a stage in the structure of its background, its foreground and its personnel who, in a sense, were actors moving in their allocated parts across it, never deviating from the script. This night the last act was being played out, obeying the formalised structure in which it was set.
By 8.00 only a few were congregating on the Eastern side of the checkpoint, but along the street at the rear of it because fear kept them from approaching. By 8.30, several hundred people had gathered on the Western side.
Elsewhere, as this was happening, the night began to touch people at random, and in differing ways.
Ernest Steinke was chief editor of RIAS 1, the broadcasting station founded in 1946, but since 1968 a German and American broadcasting station. 'I worked in the Eastern section of this station and I saw the press conference on TV. I went to the studio afterwards to tell the people there what had happened. Somebody had asked Schabowski when people could go over to the Western side and he said, "We have a new law, it's OK." Someone asked, "What time?" and he said, "I don't know but I think the law is ready and they can go now." He also said they had to have a passport and a visa. I went up two flights of stairs to the studio and people asked me what had happened. I said the people could come over and someone said, "There will be thousands coming, a hundred thousand." I pointed out that they had to have a visa, had to go to the police for a stamp and "the police will open tomorrow morning at eight o'clock, maybe, and there will be a lot of people going there. I guess about ten o'clock they'll start coming." Then I went home and went to bed to go back to RIAS early next morning... .'
RIAS radio reporter Peter Schultz felt 'the impression it made on me was even stronger than the night it had gone up – throughout the years, we had seen people dying at the wall, we'd seen people jumping out of the windows at Bernauer Strasse; I'd covered Peter Fechter. After twenty-eight years of these experiences, there was nobody in Berlin who believed that the wall could come down again after a sudden and comparatively short period: no journalist, no politician, _nobody._ When it happened it was – again – like a shock, but this time a positive shock. When I saw the press conference on TV I had an idea something would happen but I thought it would be the next morning. A football match was being televised [on Western television] and there was a flash that the wall was open and thousands of people were marching in the direction of the checkpoints. So I went to the radio station... .'
## KÖPENICK DISTRICT, EAST BERLIN, 8.30 P.M.
On the other side of this wall, Lutz Stolz and his wife Ute watched the match, too. 'I am a great sports fan', says Lutz, 'and we were watching the quarter final in the Stuttgart Neckar-stadium on TV. There wasn't just anybody playing but Bayern Munich which has been our top team since the war.7 It was during half-time that they announced that something very surprising had happened in Berlin, that presumably they had opened the wall. We switched over to the East channel at once and it was broadcasting [a recording of] Schabowski's Press Conference. He did not seem to be very convinced of what he was saying.'
Ute continues: 'He acted as if it was only an error, as if it was not a fact. We did not take it as an absolute truth because there were so many things happening those days – every day something different. Schabowski had the sheet of paper, and he read it as if out of context. Actually it was not completely clear what was meant and, in fact, he did not say clearly that the borders were open: from that moment on, travel permits would be issued without applying in advance – which didn't really mean that you could simply go to the wall and cross, something not possible for nearly thirty years. And it took twenty to thirty minutes to understand. We switched over to the West channel to see what was happening. People simply wanted to understand, and when they did they rushed off... .
'We had had some cherry brandy while we watched the soccer game because my husband was enjoying the game so much. When we switched over again to watch the second half we drank a lot because of the message. And when the game was over, they confirmed the message. I remember it very well. I took my Persian cat in my arms and went down to the courtyard8 where, also because of the cherry brandy of course, I wanted to sing – any song that came to mind. I don't know why, but I remembered one from school, _Brüder, zur Sonne, zur Freiheit_ , but I did not remember much of it.9 I was swaying from side to side with the cat in my arms.'
Lutz adds: 'Standing in the courtyard, I shouted, "We are free, we are free." Of course, we had drunk a lot. We went to bed at about half past eleven, inebriated (and, as usual, I got up at 5 o'clock and went to work at 6.00–6.30)... .'
## TEMPELHOF, WEST BERLIN, 8.30 P.M.
Daniel Glau worked in his father's hotel, the Ahorn, just off the Ku-damm. He manned the switchboard in the office on the first floor and dealt with whatever problems arose. He lived in his own apartment in the district of Tempelhof and was there now. He had no relatives in the East, something he did not think unusual. 'As time went on, people died so the number of people with relatives went down. I had never been to East Berlin. I'd seen the wall but it was not interesting for me. Sometimes I looked over it, yes, it was very depressing to see the same people as here, in the same town but we had much money, we had everything we liked to have and they had nothing. I could tell that just by looking. I was at home. I heard it on the radio, they said only the wall falls and I didn't believe it because I had lived here twenty-four years and I had known the wall for twenty-four years. I didn't know how it was before the wall. I know thousands in the East made for the wall but my reaction was _keep cool_. It didn't affect me so much.'
He was the same age as Uwe Nietzold.
## BORNHOLMER STRASSE CHECKPOINT, 8.30 P.M.
The hundreds gathering had grown into thousands. This was unlike Check-point Charlie because it was within easy walking distance of a vast residential area. Some of the tall apartment buildings actually overlooked the checkpoint. Cars were drawing up in numbers now, too, blocking the long, wide, straight avenue towards the check-point. 'The police came,' Jäger says, 'and demanded that the citizens leave the area. They were told to go to the police stations and ask for papers to travel abroad. Some left and did that but the police stations didn't know anything about it and the citizens came back angry because they felt betrayed – and they thought it had all come from the checkpoint.'
At some undefined moment the mood began to change in the dead area: this corridor flanked on the left by the side of a building and to the right by head-high metal railings. The corridor led to the little fence (actually of wire mesh and about 10 feet high, but little in the sense that it was not really designed to withstand assault). The corridor was wide enough to allow about ten people to stand shoulder to shoulder. It was full, all the way back. The mood – impossible to quantify with any precision – changed slowly from the habitual subservience before uniformed men to insisting on the rights Schabowski had said they now possessed. Jäger caught that mood.
Street lights fell on the columns of waiting cars, shimmered on the moist cobblestones of the road. Drivers got out, chatted, and one held a map towards a television camera. He laughed in embarrassment that he might ever be able to use it _da drüben_ – over there – in the impossibly distant West, which lay up the slope and across the steel-spanned bridge. A pretty girl in a short coat got out of a Trabant and loosened up by wandering into the lamplight. And everybody wondered.
Jäger was now ringing headquarters every twenty minutes with situation reports and always his boss said the same thing: no new instructions, sit it out.
## CHECKPOINT CHARLIE, 8.30–9.00 P.M.
In the Mitchell Suite, Yount and Greathouse watched. Yount phoned Godek in the hut and reported that he had noticed 'some vehicles coming into an open cement area at the back of the checkpoint and it looked as if they might be carrying East German soldiers. The vehicle movement brought about a lot of concern because the last time we had seen that they'd built the extension to the front of the check-point. We saw Guards coming out and although we really didn't know what to expect we figured it might turn into a nasty situation.'
They were Guards not soldiers. Moll had sensed days before that, with the government truly floundering and demonstrations on the streets, he might need reinforcements and had requested sixty of them. They were armed.
## BORNHOLMER STRASSE CHECKPOINT, 9.00 P.M.
The few thousands had become many thousands. The waiting cars now stretched back to Schönhauser Allee, a main road several hundred metres away. They choked the side streets all the way back to Schönhauser Allee, too. Jäger surveyed the men he had available, sixteen or eighteen, and knew how hopelessly inadequate that was. He telephoned for reinforcements and about fifty were dispatched to 'protect the checkpoint'.
Major Sens had telephoned _his_ headquarters to ask for instructions but no orders had reached them and so they could give him none.
In the corridor people shuffled to and fro and, through the wire mesh, watched the normal passport control going on. The lights were bright there, almost a glare.
At 9.00, Jäger rang headquarters again, the press of people mounting and growing. He gave a situation report and his boss said that what Jäger and his men must do was to pick out the most 'aggressive' among the people waiting and let them through. That would calm things, take some of the pressure off. These 'aggressive' people were, unknown to them, to be punished, because the photographs in their identity cards were to be stamped _on the photograph_. Anyone trying to return with a stamp like that would be refused re-entry. Without being told, they were being deprived of their citizenship, and the decisions would have to be made by the passport controllers on sight. Jäger was ordered to keep a list of those allowed through – their identity card number, their date of birth and so on. He was also ordered to pick out a few non-aggressive people and let them through, so the waiting mass wouldn't realise that aggression was the way to get through.
It was the bureaucracy of madness.
If Harald Jäger could see full down to Schönhauser Allee he'd be looking at 20,000 people, with more coming.
He ordered three passport huts to be opened and at 9.20 the people they picked out were filtered through them. Jäger estimates the number as between 200 and 250; some got the stamp over their photographs and some didn't. And they came through, in single file, emerged from the huts and – some dancing, some scurrying, some close to tears, some shaking their heads in disbelief – they went past the final watchtower and over the bridge to dreamland.
## CHECKPOINT CHARLIE, 9.30 P.M.
Moll's deputy rang again and said about a hundred people, some foreigners, had gathered on the other side of the line and were imploring the Guards to please let the people from the East come through. 'I told him to make sure that everything remained quiet, that law and order was maintained. I told him it was his duty to ensure that normal traffic – the cars and buses – should continue to cross as usual. I added, "I will be there as soon as possible..."' Within a few minutes Moll got into his Skoda and set off on the journey back.
Brita Segger finished her evening class and was very tired. She walked to Marx-Engels-Platz and waited for the train to the suburb where she lived. Somewhere, as the old train rattled through the seven stations to it on the elevated rail, Moll in the Skoda passed her going the other way. He was driving quickly. On that journey Moll made a decision: no Tiananmen Square, no raking gunfire across civilians, no single death, never mind a massacre.
## BORNHOLMER STRASSE CHECKPOINT, 9.30–10.00 P.M.
Jäger had the flow of people more or less under control while, in the corridor, so many waited as they had done for so many years, for so many things. There was an honesty about them, and their stout leather jackets and tartan shirts and heavy shoes, working men's honesty, but they wanted their rights and they would wait for them, too. They had time. They had the rest of their lives. They were noisy, but mostly football crowd noises, good-natured, semi-banter, some jokes; but the fact they were making the noises at all suggested that the fear of men in uniform was, perhaps for the first time in their lives, going away.
At some unremembered moment the tone of Jäger's conversations with headquarters itself began to change from situation reports to a request to let everybody go through. During one conversation he found his boss in the midst of talking to the general responsible for all security in Berlin. Jäger was allowed to eavesdrop and he heard the to and fro although he was forbidden to say anything himself.
His boss: 'The situation is exceptional.'
General: 'Is this correct? Jäger: has he panicked or has he given a real description of the situation?'
Boss: 'I have known Comrade Jäger for many years and if he says it is like that, it is like that. Rely on it.'
General: 'I have my doubts...'
At this point, Jäger could stand no more. He shouted into the phone, 'If you don't believe me, I'll hold the receiver out of the window and you can hear it for yourselves.'
The line went dead.
Of equally immediate concern, letting the more aggressive people through had not really calmed the situation. Relentlessly, by weight of numbers and weight of expectation, the pressure kept increasing.
## CHECKPOINT CHARLIE, 10.00 P.M.
Moll turned off Leipziger Strasse past the postal museum and was astonished to find the street leading to the checkpoint choked with Trabants and Wartburgs. He had to park where he could, some distance away. As he walked people noticed his uniform and asked 'How do we get through, how do we get through?' He couldn't tell them because he had no idea himself and, anyway, his sworn duty was to stop them using whatever force necessary. He had a lot of that. Only he knew he wouldn't use it.
Greathouse asked Godek if he'd mind taking him through the checkpoint 'for a feeling of what was going on over there'. It was a way of maintaining access to the East and also a way of keeping himself informed. Godek decided he'd be better monitoring from the hut but told Yount to drive Greathouse. They went on a routine sweep which lasted thirty minutes.
Moll arrived – perhaps he glimpsed Yount and Greathouse going by, perhaps not – and 'informed myself about the situation. There must have been about two hundred and fifty people on the western side and they wanted to cross. I decided to go and speak to these people. I did not go alone, I had three or four Border Guards with me. The Guards in the watchtowers wouldn't shoot me, I was in charge, it was my area, my men wouldn't think I was trying to get across.
'I talked to the crowd and I explained that there were no new regulations yet. I went to a watchtower and called my superiors at the Border Force's Headquarters.' This was a plain and typical barracks off a long, straight street near Treptow, a horizontal steel pole blocking the entrance, a reception hut with shifty Guards of its own, a parade area, basic buildings – their corridors laid with cheap linoleum which creaked underfoot – taut military offices, phones, in-trays, out-trays, and an occasional potted plant to soften the impact. Khaki vehicles stood parked in clearings, ready. The headquarters ticked to the same rhythm as the Border Guards at the checkpoint who it ultimately controlled.
Moll asked about the new regulations. No new regulations. Law and order should be maintained. 'More and more people came from the underground station on the other side. I deployed my reinforcements in a line to hold these people back.' This deployment was behind the forward barrier and Godek watched it, a soldier monitoring the formation of other soldiers, watched the Guards being arranged, their boots stroking the concrete quietly as they made the line, paradeground power tight as a Roman phalanx and governed by the same ethos: order, stand, obey, be prepared to fight. The Guards had revolvers but no machine-guns.10
Greathouse and Yount returned from their sweep in the East where they'd seen movement but nothing really unusual. Still the crowd grew all around the hut, the _Imbiss_ and the Café Adler.
Brita Segger reached her 'little flat' far away in the East and turned the radio on. She walked into the bathroom to clean her teeth and distantly heard from the radio a voice saying the border was opening. She thought, 'No, no, that's a joke.' She emerged from the bathroom, switched it off, went to bed and fell asleep immediately.
The single phone in the American part of the hut came alive. 'We started getting calls from different radio and TV stations in the United States. They could call directly to the hut because it was an unclassified number; it was published in our telephone directory,' Godek says. He found himself 'on line with a radio station in New York City, there was another call from a radio station in San Francisco and Colonel Greathouse did an interview. I took a call from Sydney, Australia asking us if we would mind confirming what they had just heard, then we got a call from Canada. The thing they were interested in was, "Tell us what you see outside." They weren't so concerned about what we had heard or what we thought was going to happen, but, "Tell us what you can actually see out of the window." We were getting so many calls Colonel Greathouse told me to deal with them. He had notifications to make and because our phone was constantly ringing he went up to the Mitchell Suite to do that. The phone there was not a published line. Before he went he said, "This is what you can talk about and these are the things you are not prepared to talk about." It was clearly understood we must not speculate.'
Below the Mitchell Suite, in the Café Adler, the little team of three worked hard serving but braced themselves for a total onslaught, the place overrun from the East. Perhaps it would come in a moment, perhaps it would never come.
Godek studied Moll's reinforcements. 'They were acting as they normally would. They stood back from the white line in a typical formation, very cold faces, stone faces. They seemed almost not to be bothered about what was going on in front of them. We had seen all that before but again we were really concerned about potential developments because we didn't know what they were going to try.
'Rumours started heating up about the checkpoint opening, mostly coming from the press. I would walk around outside still trying to get a sense of developments. People asked me if I knew when the checkpoint was opening and I'd tell them I didn't, I'd tell them I hadn't heard anything. I had two concerns. The first: was the scene going to turn ugly, would there be physical altercations between the Border Guards and the crowd on our side? The second: were the Border Guards going to try and seal the checkpoint in case a crowd did try to cross from East to West?'
Any sealing, even momentarily, would, of course, be a direct violation of the Four Power Agreement and news of it would travel fast to Washington and London and Paris and Moscow, while Godek requested the presence of a Soviet officer, as he had to do in case of incidents. If a Third World War ever did start it would probably start like this, with the sealing of the checkpoint, and escalate all the way up.
At around 10.00 p.m., some sixty to seventy Westerners crossed the white line in front of the checkpoint so that technically they were on East German territory.
At the same time Godek estimated the total number of people in the West as 2,000 and 'in a narrow street like the one where the hut was that really packs them in. We were still able to maintain our movement – our movement being Allies wanting to go over for dinner or returning from a day of shopping or whatever in the East. There were Allied people coming through, some on foot, mostly by that time in vehicles. These were either military or military-related personnel and they registered with us before they went so we knew who was over there. By 10.30 the large majority were back.'
At 10.35 some 100 people had crossed the white line, but the Border Guards pushed them back. Moll monitored the crowd on the Eastern side and estimated it at between 70 and 100. He monitored the crowd on the Western side and 'there were hundreds, maybe thousands. We had had these problems every seventeenth of June on the anniversary of the "revolt" in 1953 [the workers' uprising put down by Soviet tanks at the end of Leipziger Strasse beyond the postal museum] and in most cases the people who came to the white line were our former citizens who had fled. They were very aggressive. This evening I looked into the faces of the crowd and they were somehow peaceful. They were normal people, not people who wanted to provoke anything. They were drinking champagne and what they wanted was the checkpoint opened. They were polite to the Border Guards. I called my superiors again and there were no new regulations.'
## MITTE DISTRICT, EAST BERLIN, 10.30 P.M.
At 10.30, Erdmute Greis-Behrendt 'phoned my husband at home and said, "I'm checking that you're still there" and he said, "What do you mean?"' She was in the Reuters office and had been watching television, if only to get confirmation from their news bulletins that Reuters had interpreted the press conference correctly. She asked him if he'd been watching because 'the border's going to be open tonight'.
He replied, 'Yes, I've been watching television but I really haven't been able to put it into perspective.'
'Where is our son?'
The son, Maximilian, had just gone to bed.
'Get him out of it and go in the Trabbi to the first Western transit point and you cross over. It's the night of nights tonight.'
'Do you think that?'
'Of course. You must be there. Go, go, go!'
## CHECKPOINT CHARLIE, 10.30
Greathouse asked Godek if 'we would be willing to venture through the crowds with our vehicle to exercise our right of free access because by then it was essentially a mass of people out in front of the hut. Sgt Yount and someone else attempted to go through but they had to stop. The crowd was solid. Not far behind Yount was a Soviet vehicle with four uniformed soldiers in it – they had been over in the West for whatever reason – and they were trapped maybe four, maybe five vehicles behind ours. Finally this Soviet vehicle caught the attention of some of the crowd who started rocking it, which to us seemed rather unusual because they didn't show that reaction to Sgt Yount in ours.'
Inside the Café Adler, Astrid Benner heard 'screaming begin, just screaming. I couldn't tell what they were screaming.' It was the crowd, and the screaming actually a chant deep from the pits of many stomachs.
Moll moved to the front of the checkpoint and tried to calm the 2,000, 3,000. A flashbulb burst nearby and the moment was captured for the morning newspaper. 'Already I had a feeling that this could not go on. Something had to happen. I knew my men would not be able to hold them back, it was not possible, so they withdrew behind the wall.'
This 'totally surprised' Godek. Border Guards did not withdraw in the face of a crowd. Quite the opposite, they fronted it out.
By 10.45, ZDF and ARD were broadcasting that the checkpoints at Bornholmer Strasse, Sonnenallee and Invalidenstrasse would open. What was happening at Bornholmer Strasse and Checkpoint Charlie was being duplicated at each of the other checkpoints; and at each, as the structures of obedience broke down, Border Guards and passport controls were facing extraordinary decisions unaided. Their predicament was not helped by the singular fact that they had spent their careers not really making any decisions at all, never mind anything like this.
A photographer at Checkpoint Charlie noted that 'by now the Guards didn't look aggressive and they had not a clue what was going on. You could see that on their faces. They didn't pull back behind the wall in formation. They looked very undecided among themselves. First one or two pulled back, then a few more, then a few more...'
They vacated the wedge of road between the white line and the wall itself. In a few moments around ten Westerners moved across the white line through the barricade and up to the wall. Within seconds thirty or forty more followed and then, in a vast slow-rippling wave, more followed.
Moll walked briskly to the watchtower and telephoned headquarters a third time. No new regulations. At this instant, with so many people so near and more and more arriving, Commander Günter Moll was the loneliest of men. He did not know that what was happening here was happening at the six other checkpoints simultaneously, that a great current flowed through the city. The checkpoints had no direct communication with each other for security reasons, so that no one could coordinate a mass exodus or indeed coordinate anything at all. He gave the order to close the small pedestrian gate. Sealing the vehicle gate would have represented violation of the Four Power Agreement and risked escalation of the situation.
Sergeant Yount struggled through the crowd and moved past the line of Border Guards, crossed the checkpoint, followed by the Soviet vehicle, returned after what seemed a long time and reported that yes, a crowd had gathered in the East, they were very quiet but he sensed anticipation.
Obedience had been studiously bred into them, waiting was what they did, mute and uncomplaining. It was never wise to complain. None of them put a foot inside the checkpoint, none made an intrusion.
In the hut Greathouse took a decision and informed Godek of it. 'He decided that we did not need to be accessed. We accepted a point in time where our access would be completely blocked – by ordinary people. I was with the Colonel when he made the decision and it was based on two factors: the emotions of the people on our side and the reactions of the Border Guards. Their conduct changed shortly after 11.00. The crowd started crossing the line in a mass from our side, something completely unheard of before. People meandered over, people tried to climb up onto the wall. The first couple were initially asked by the Guards to "please come off it" but there was no physical contact at all, they weren't dragged off. Eventually enough people were up on the wall that the Guards just let them sit there.'
The current had reached here, unspoken, unregulated, spontaneous – but here. These people on the wall sat with their legs draped down, almost lolled, casual as you like.
Godek continues: 'Just in front of one of the watchtowers, the one to the left as we looked at it, there was a little grassy area and the crowd started milling on that and the Guards let them sit there. Then people sat on the wall next to the watchtower actually looking in at the Guards, something else completely unheard of. The mood was jovial, almost carnival. When Sgt Yount came back from his last sweep Colonel Greathouse said, "We don't need to press our rights, we need to just let this happen. This is a moment for the German people."'
The current caught the Guards and embraced them: a lover's embrace.
'Some of the Guards took their hats off and threw them into the crowd. I couldn't believe it. People were talking with them, they were talking with the people which they had always been prohibited from doing. We saw smiles on their faces.' Major Godek was a calm and conscientious man. He read the implications of these little gestures perfectly. He knew.
One of the Westerners grabbed the cap of a Guards officer, put it on and stood laughing. The officer said, 'Please may I have it back, I need it or I'll be in trouble.' The cap was returned. At this instant, the wedge of road was a solid sea of Westerners, the full 2,000, and they pressed against the low wall, although gently, no push and shove, almost no jostling. Behind it the Guards – nearly all capless – stood milling. Hands reached across to try and shake theirs. This low wall was of wood and three Guards pressed their hands against it to stop the pressure of the 2,000 from tipping it over. Someone wearing trainers and an anorak clambered onto it from the West and stood precariously beside the pedestrian gate, turned back to the West and, arms outstretched, gestured as if to say 'Well, what are you waiting for?' That was greeted by reverberating laughter among the 2,000.
Two stout supports flanked the pedestrian gate and Moll – he had his cap, he still looked as if his uniform had been moulded to him – clambered onto one of them, surveyed the sea which had flowed to just in front of him. His arms were slack at his sides. As he surveyed, he knew. Nearby a photographer asked a Guard to haul him up onto the wall so he could take pictures from both sides and was deeply astonished that the Guard did.
## BORNHOLMER STRASSE CHECKPOINT, 11.00 P.M.
The crowd in the corridor were waving their arms and they could see the ones who'd been picked move into the entrance to a passport control hut. A gate in the fence had been opened to let them through. Many people – waiting or going through – joked self-consciously. People tried to ebb into a gateway for vehicles. Five passport officers tried to make sure they didn't because, like a dam, once a crack appeared the whole edifice would be swept away by a tremendous torrent.
'Although I was their superior,' Jäger says, 'the people I was working with that night were experiencing what I was. They kept demanding that I do something but I was not sure what to do. So I kept asking them, "What shall I do? Order you to shoot?"' It was a rhetorical question, of course, but born because he had neither the rank nor the authority to order the checkpoint opened, but there seemed no alternative.11 The thousands were not going to go away.
At 11.15, Rudower Chaussee in the extreme south became the first checkpoint to open.
'We kept discussing what to do,' Jäger says, 'discussing what to do, discussing what to do.' The geometry still held: to one side of the fence the thousands herded like so many cattle, growing increasingly restless; to the other side were empty tarmac lanes, marked by red and white plastic cones, leading up to the inspection awning. The tarmac glistened as the cobblestones did. Here, uniformed men moved through their motions but it was evident that they were now moving faster. The crowd began to remonstrate with anybody in uniform they could reach. 'We're coming back,' they kept mouthing. 'We're coming back.' It deepened into a chant. Someone slapped a customs officer on the shoulder, said 'Stop being so silly' and everyone smiled.
Someone else said, 'There aren't any more cars coming' – so why can't we go through the car lane? Hands were hoisted with fingers splayed in the inverted V for victory. The chant became 'Open it! Open it!' and those in the middle of the crowd began waving their arms aloft. The chant deepened to an insistence – 'Open it! Open it! Open it!' – and continued rhythmically. A bearded man at the very front tried to reason with four or five customs officers. The chant melted into the crowd and died there, was renewed at the same rhythm as 'We're coming back! We're coming back! We're coming back!' Someone pushed at a customs officer, who reacted by pushing him back with both hands and those around shouted, 'No violence, no violence.'
Beyond the fence was a pole across the road and Jäger 'had been standing there all the time. I could telephone from a little hut. The situation was now so hot that we'd reached a point where we could go no further. All I was thinking about now was to avoid bloodshed. There were so many people and they didn't have the space to move. If a panic developed, people would have been crushed.' He was thinking something else: if any of the uniformed men, armed with pistols, fired – for whatever reason, and who could know what reasons there might be? – anything, _anything_ , could be unleashed. 'I just did not want anybody to die. We had the pistols, there were instructions not to use them but what if any of the men had lost his nerve? Even if he had shot into the air I cannot imagine what reaction that might have provoked.' He called headquarters again and 'I said, "We will have to let all of them out." The boss replied, "You know your instructions and you must do only what they say."'
The decision was passed deftly back to Jäger.
He said, 'It cannot be held any longer. We have to open the checkpoint. I will discontinue the checks and let the people out.'
He went from the telephone, and might have surveyed the entrances and exits of this checkpoint – the entry to the huts, the coned-off lanes, the gate in the fence, the vehicle gate, the pole across the road. He said, 'Open them all.' He had a feeling something important, perhaps profound, was at hand and, although he didn't exactly know what, he sensed that 'something very bad would happen to the GDR'. Later, when it was being talked over, someone said to Jäger, 'That's it, that's the end of the GDR', but now he had time only for something approaching a gut reaction to his own action.
He was at the pole, 'eight to ten' fellow officers beside him against 'hundreds of people just in front of us; no, thousands'. These officers turned from the pole but the press of the people forced it back; and they came, 20,000 strong, came in a torrent, came with no violence, came stepping sprightly. A young man in a jerkin engulfed in the torrent, clenched his fists as he moved past. A young woman danced herself clean off the ground, her hands clapping above her head. An inscruteble man with his hands in his pockets might have been going for a stroll.12 A child rose above the torrent on father's shoulders.
For a moment Jäger turned and faced them and was suddenly lost among them. Instinctively he directed a man over there: that's the way across. For another moment, when he emerged, he stood alone, surveying his creation, this mighty mosaic of movement.
And the 20,000 denied people went up the incline past the final watchtower, where a stone-faced Border Guard watched them impotently, and crossed the bridge to dreamland.
The checkpoint was a natural funnel but even so they were through in 45 minutes.
Jäger felt 'my knees begin to tremble and I had a very bad feeling in my stomach. I went to the telephone and rang headquarters. I said, "Comrade _Oberst_ [the boss's rank], I opened the border. I couldn't hold it any longer. I let them all out." He said, "It's OK, _junge_ "' [ it's OK, guy].
With hindsight, the opening seems natural and inevitable and sanitised by the fact that it did not go wrong: no shot was fired anywhere. Harald Jäger could not know that, and what he had done might have put him in jail for the rest of his life and brought lasting official vengeance on his family. That's why his knees trembled and his stomach turned over. 'Normally, you'd expect something to happen to you. I had refused to carry out – well, implement – the orders.' The fact that he had done this because he needed new orders and didn't get them might not have been any defence at all. Nor did he know, because each checkpoint was isolated from the rest, that mighty torrents were flowing there, too, and the same decisions were being reached.
Jäger phoned his wife when he had a minute. She'd been at work all day and partially isolated from events. 'I won't be home at 6.00 a.m.,' he told her, 'I've opened the checkpoint.'
'You're kidding,' she said, and made fun of him.
In the control room at the Olympic Stadium, Toft heard voices from the patrol jeeps 'starting to get excited'. He felt that himself. The radio crackled in front of him, a report from the British section of the hut. He wrote: '2325 [11.25] hours. Large crowd of East Germans at checkpoint on East German side. Crowd building up on western side.'
At 11.35, the Heinrich-Heine-Strasse checkpoint opened.
## CHECKPOINT CHARLIE, TOWARDS MIDNIGHT
The impression that 'something had to happen' was growing stronger and stronger on Moll. 'The crowd were calling to each other, "Let us go! Let us go!"' He was under an arch of sound, a chant rising from the West – 'Come! Come! Come!' – and echoing back from the East – 'We're coming! We're coming! We're coming!' He made a fourth and final call.
In the hut Godek watched. Greathouse departed to see developments elsewhere in the city but, before he did, he said the word was that the checkpoint would open, sometime. 'He also informed us that we had been receiving reports of crowds forming at some of the other crossing points. My British counterpart, Major Ross Mackay, stopped by. He'd been to the Brandenburg Gate – which was in the British Sector – and some of his crossing points and he reported the same kind of atmosphere.
'If anything impressed me it was the completely different attitude of the Guards, their change in behaviour. It was the fact that they tossed their hats into the crowd, these same Guards I had seen during previous demonstrations with their stone faces. They were being offered champagne, they were being offered beer. They didn't accept anything but they were laughing. One Guard had his hat removed by someone sitting on the wall and didn't make any attempt to get it back.'
The current exposed the Border Guards for what they were: people.
In the Café Adler Astrid Benner heard 'the noise grow and grow. We could see the Guards in the watchtower still using their binoculars but you knew they didn't know what to do except keep doing that.'
Brita Segger slept on.
Sergeant Michael Raferty slept on.
They were still precisely a world apart.
At 11.40, the Oberbaumbrücke and Chausseestrasse checkpoints opened.
Moll made his call. No new regulations. 'I felt pressure from the people calling to each other but I also felt pressure from history.' He had an immediate problem, itself one of geometry. Nominally he commanded the whole of the checkpoint including the area under the awning where (as we have seen) the Stasi manned the customs, but this had never been tested. The Stasi, that state within a state, wielded stark and immense power even at this minute to midnight. They employed more than 30,000 people; they had more than 250,000 agents and informers in every nook and cranny of the land; they had compiled at least 6 million excruciatingly detailed files on their own citizens, one for every three of the population, sometimes right down to how many times a grandmother telephoned her granddaughter. It was not unknown for fathers to inform on sons and sons on fathers. Love affairs were particularly charted, the where and the when of them. Any political deviation provoked their wrath. The Stasi were the custodians of the revolution and the guardians of it. Their armaments were second only to those of the GDR Army, their power used to strike at any enemy of the state real or imagined. They could take a life, keep anyone in their own prisons, destroy a career, extend their retribution to relatives of the guilty, and there was no law, no legal process to prevent any of this except the most vague. The very existence of the Stasi covered only two paragraphs of the Constitution and the paragraphs were deliberately ambiguous, granting them autonomy and no accountability. The Stasi were feared, and rightly so.
Moll walked briskly from the watchtower to the Customs and spoke to the most senior officer. Moll knew nothing of this officer, not even his name. Moll risked his career and quite possibly his life when he said, 'I am opening the border.' He could not do it without the Stasi. What might happen if he said the people could come through and the Stasi, occupying the strategic middle ground under the awning, said they could not? It was the Stasi, not Moll, who would have to consent, would have to open the rear gate of the checkpoint to allow the people to shuffle forward in their obedience; the Stasi who would have to stamp all the passports with exit visas; and the Stasi were still undiluted. What if they struck at Moll? He faced essentially the same dilemma as Jäger, although Jäger was Stasi himself. Someone had to make the decision.
Moll waited this last, final moment for the passport controller to reply. The passport controller said, 'Yes.'
That was all.
Godek sensed the current and knew it was now. Why, he asked himself, interfere with it? He heard the shouting in the East but 'from the distance away that we were, we couldn't tell what they were shouting.'
Moll walked from the awning to where his reinforcements stood; walked to the pedestrian gate and gave the order very quietly, without any sense of theatre.
'Open it.'
The Guard manning the gate obeyed. It was a tactical move which would allow people from the West an entry point. At the same time Moll's men lined up again but this time in a different formation: a diagonal.
Toft's radio crackled and he wrote on his piece of paper: '2359 hours. Small build up of East Germans at Checkpoint. Approx. twenty East German Border Guards present.'
In a certain sense this was one of the last messages transmitted when the cold war was still cold.
Godek, watching his TV monitor intently, saw the crowd in the East move from the road towards and through the narrow gate at the back of the checkpoint, saw them entering the Customs area. This crowd, orderly and in line, were now where Godek's TV could no longer follow them. Customs was under the awning and enclosed.
It was, truly, a minute before midnight on Thursday 9 November 1989 or a minute or two into Friday 10 November. There is no precision about the timing because there is no exact moment, no instant which would have to be developed, carefully printed, fixed. The current was a progression of moments flowing one to another.
At midnight the Invalidenstrasse checkpoint opened. East Berlin itself was open for the first time since midnight, 13 August 1961.
## INVALIDENSTRASSE CHECKPOINT, AROUND MIDNIGHT
Jacqueline Burkhardt, who'd lived in East Berlin for many years, saw Schabowski's press conference on the family's black and white television but it must have been a recording because that had been at 8.00 p.m.. 'I couldn't believe it. I woke my daughter and said, "The wall is open" and she didn't believe it either. I went into the street and hundreds of people were walking towards the wall. Everybody watched television – it was not "normal" to be on the streets after 8.00.' Eventually a decision was made to try and go over. 'Some people were in night clothes with trousers over them and shawls – because they couldn't believe it.' (At one checkpoint an elderly lady wearing an overcoat insisted she had come from her bed and hadn't wasted time dressing. She threatened to lift the overcoat so that her nightie could be seen underneath, just in case anyone doubted her. That was greeted by a roar of laughter, and she was laughing, too.)
There were no buses around and so people walked. At Invalidenstrasse the Border Guards appeared bemused and couldn't seem to grasp what was unfolding, especially when confronted by about 500 people determined to cross, which they did. 'Total strangers on the other side were waiting and they kissed us.' The rest dissolved into the valley of emotions, although there was a sister in the West who'd seen the news and – unable to deduce which checkpoint they'd use but assuming they'd come – went to the Reichstag as a sort of meeting point. They missed each other by 500 metres.
## KARL-MARX-ALLEE, EAST BERLIN, MIDNIGHT
Birgit Kubisch, who lived along this broad Stalinesque avenue of heavy-pile stone buildings, found it 'quite funny because I was listening to the BBC World Service lying in bed. We had some guests and I was sleeping in another room in our flat, and somebody said, "Oh, there are people at the Brandenburg Gate and they are for the first time in West Berlin" and then they interviewed the people and I really thought I could not understand English! In those days you heard many, many things and everywhere there were rumours, and after one or two days you'd hear something completely different. So I heard it and I thought, "Tomorrow we'll find out that this is also a rumour." I was quite excited but we'd got used to false messages. I thought maybe some part of it was true but I was questioning the whole thing. I couldn't believe that, after everything, they would simply open the wall. I fell asleep... .'13
## GOSEN, OUTSIDE EAST BERLIN, MIDNIGHT
Uwe Neitzold had listened to the readings at the culture club at _Neues Deutschland_ ; about 10.15 a newspaper worker had come in and said that people were at the border. 'We wanted to see it – eight or nine from the poetry group – and we went to Friedrichstrasse on the S-Bahn.' At the station he noticed the underground. 'We didn't even know that the underground went under our own city. How could I know? I was 25 and nobody had told us. If you went to Friedrichstrasse you only saw the S-Bahn station and only the Berliners who had lived there before 1961 knew – my family came to the area in 1965. Anyway, Friedrichstrasse was totally crowded so we walked to the Invalidenstrasse checkpoint, we stood and we looked because we couldn't make up our minds. There were at least 2,000 people in the street and we couldn't _see_ the checkpoint! I suppose we were 200 metres away and all we _could_ see was the checkpoint lights. We couldn't see that people were actually crossing. We still didn't believe it was open. We thought they'd keep the border closed. We stood there for around twenty minutes, the time was late and we agreed to go home. We thought it was all a demonstration like so many in the past few days.
'I lived with my parents on the outskirts of Berlin – a village called Gosen – and if I missed the last bus I'd have had to wait until five or six in the morning for another one. I got home at midnight.'
Gosen was a small, quiet place, a slumbering Eastern village and the house where Nietzold lived had been built by his father with his own hands: someone with the intrinsic dignity of a working man. Gosen was silent, as it was every night, and Nietzold was very tired. Apart from the illusion at Invalidenstrasse, it was all much ado about nothing. His parents were asleep. He went to sleep, too.
* * *
1. Interview with author.
2. Interview with author.
3. Interview with author.
4. Interview with author.
5. Interview with author.
6. Interview with author.
7. Interview with Birgit Kubisch. It is surely revealing that Stolz uses the word 'our' for a West German team, implying, as it does, that he regarded Germany as still one country. How many others did, and how many others didn't? And how many were in no man's land, not sure either way?
8. Courtyard does not carry the connotation it would in English: there were small courtyards with apartment buildings round them.
9. Birgit Kubisch explains: 'A typical GDR song that every child learned at school, approximate translation _Brothers, soar up to the sun, to freedom, a star is shining bright out of the dark past._ It refers to the hard times after the war.'
10. Interview with author.
11. There's a curious story that this evening of all evenings all the passport control commanders had been summoned to a meeting in the Ministry of the Interior, leaving their deputies in charge. In other words, the people who should have been taking the decisions weren't, and the people who shouldn't, were... .
12. I came across an amusing episode. A family lived near Bornholmer Strasse – they could see a tiny segment of the West if they leaned out of their window – and when the border seemed to be opening they decided that, since they walked their dog every evening, they might as well go that way. They were so sure the border wasn't really open that they took no identification papers with them. When they arrived and saw it was, they went to the passport control and explained that the only one of the three with any papers was the dog – they'd brought its licence – and therefore it was the only one of them entitled to go over. 'Tonight,' one of them announced, 'the dog is going to do its business in West Berlin.' They were waved through and the dog no doubt did.
13. Interview with author.
## TEN
## _Dawn_
They built the wall to keep people in and they took it down to keep people in.
American joke
Some 5 minutes ticked by, 10, 15 and no person emerged from the Customs at Checkpoint Charlie. The Stasi were in unknown territory. The people before them and those stretching back had driving licences or identity cards or passports; navy blue passports bearing the title of the German Democratic Republic and valid only for travel to 'friendly' countries – Cuba, Vietnam, Mongolia, North Korea, Eastern Europe. No person had authorisation to be granted a visa to the West; but the current had flowed under the awning.
Checkpoint Charlie did not fall to an assault from American tanks; no bomb tumbled into its heart from a dark-laden sky; no shooting rang out across the concourse; not even a punch was thrown. Checkpoint Charlie fell into the valley of emotion, as did both halves of Berlin. At 2 minutes past midnight the Eastern police announced that all checkpoints were open and the biggest party in the history of Germany was under way.
At Checkpoint Charlie the Stasi began stamping. They stamped anything, possibly even the driving licences. The current was travelling too fast for pedantry, too fast to be stopped in any way including, perhaps, by armed intervention. Everything got a plump, sharp _Visum_ on it, thump, next please.
Godek saw a group emerge from the Customs under the awning, saw them being directed diagonally across the concourse away from the pedestrian gate to the vehicle exit. By stringing them out there would be no rush, no crush, no stampede. Order was being maintained full up to the white line. In a certain sense this was one of the last authoritarian gestures of the cold war.
Moll watched 'the people pass through the Customs and hesitate at the vehicle exit leading to the white line because they did not know how to react. They saw the Guards there and the Guards waved that they could go through.'
Godek said to Sergeant Yount, 'This is it. There's no stopping them. There's no turning back.' Godek (continuing): 'I'll be very honest, we knew what our mission was and that was to be a symbol of freedom. We tried not to get wrapped up in the emotion but we looked at each other and as two grown men we had a tear in our eye.'
A group came through, maybe 150 strong, maybe 200. Cautiously, sheepishly, they reached the line and kept on coming. In the end, after the twenty-eight years of fear at and around this forbidding place, it was the simplest thing in the world.
Toft's radio crackled and he wrote, '0028 hours. East Germans now crossing to West'.
Perhaps that was the moment which can be, and ought to be, developed, fixed.
In Bernauer Strasse there was no checkpoint, just the wall. However, Pastor Manfred Fischer 'was coming back from visiting a friend at about midnight and I passed Chausseestrasse', where there was a checkpoint, two streets away from Bernauer Strasse. 'I saw a crowd of what looked like families and I thought, "What's this?" People were lying face down and, although I didn't know what had happened, it seemed there had been an accident. I reasoned that I wasn't needed because there were so many people there. When I got home the phone rang immediately and people from the parish said, "Have you heard?" I said, "No, I haven't heard anything." They said, "We want to go to Chausseestrasse. Do you want to come with us?" I said I'd just passed it and thought there had been an accident. So I went back and the people lying face down were Easterners and they were looking at a map of West Berlin which they'd spread out! All they'd had was an image in their minds, they didn't know where things were exactly. It was as if they'd dug a tunnel and emerged on the other side... .'1
In the Café Adler, Astrid Benner was very busy and suddenly a man exploded into the place He was a little bit fat, around fifty, small, a typical worker wearing a typical worker's cap. He shouted, 'I am the first! I am the first!' The whole café burst into applause. He ran to the bar and said, 'Give me a stamp of the café – otherwise nobody will believe I was here.' Benner stooped, brought out the inky pad in a tin and pressed the tiny wooden handle of the stamp onto it. These stamps were – and are – used on bills all over Germany to signify they've been paid: cafés and bars each have their own stamp. The man held his hand out rigid and she stamped the back of it.
He had proof he'd been, because being there was the only way he could have got the stamp.
Says Astrid, 'His eyes were wide open, he thought it was all a mistake, all a kind of dream and at any moment he would wake. I gave him a glass of beer and he drank. We gave free beer to the first twenty, thirty who came in but then it was too much to know what we were doing. After ten minutes the café was completely full. People were calling out "I'm from the East and I'm really here" and people were crying...'2
Godek watched professionally. 'After the first group, for some reason it stopped and nobody came for five or ten minutes, then another group of about the same size, and it all happened again like that a third time. After that it was a continuous flow.'
Moll had the most vivid impression of 'people... people... people'.
Godek felt the full current. 'These people were received like long-lost family members, patted on the back, embraced, given glasses of champagne, given flowers. It was fascinating to see their reaction to our military presence in the middle of the roadway. Initially it seemed to be to ignore that. They were very cautious of glancing at us or making any reaction to us. We smiled, we waved, we stuck our heads out of the window, we said hello. They'd glance at us but keep on going. They were very hesitant to make any contact although they were obviously curious. I got the impression they were asking themselves, "Are these friendly men, are these not friendly men?"'
The Easterners had spent all the twenty-eight years being taught that anyone in an American uniform must be an imperialist warmonger who was trying to smash their socialist homeland and enslave the globe; they had lived within seconds of an American nuclear first strike; and they'd spent all twenty-eight years being very circumspect about anyone in uniform, even their own kind.
A humble, almost private moment among all this was when a young woman came through the checkpoint, preceded by a friend. Of necessity she moved diagonally behind Moll's reinforcements; her face was full of incredulity. As she reached the exit she looked ahead for the reassurance of her friend and, all in the moment, gave a subtle yet profound sideways glance towards the very last Border Guard, bespectacled, his arms limp at his side. Then her face craned up, grinning wildly, as if she was beseeching, 'Can I really, can I truly?', and then was over the white line into the melee – the patting and the tears and the phosphorescent flashbulbs of the cameras stabbing at the darkness – to find the rest of her life which would never be the same again now.
The 'siege' inside the Café Adler held fast. 'A friend of mine had arrived and I asked if he could take care of the entrance because too many people wanted to enter and there was no room for any more, none, so when some left he let exactly that number in. The chief editor of a newspaper arrived and he started to serve.' Astrid Benner felt her certainties breaking up.
Godek watched the crowds at the checkpoint. 'When they reached the line they stopped then they took a step over it. People cheered for them as they made that step over, a deliberate step over.'
And still they came.
Toft's radio cracked and he wrote, '0050 hours. No vehicles allowed across Checkpoint Charlie'. It meant no vehicles could _get_ across Checkpoint Charlie.
The 'siege' outside the Café Adler held fast. Something approaching delirium danced in the faces and the eyes of the people; complete strangers embraced and any bottle was a bottle to drink from. It didn't matter whose hand held it. A morass of ordinary folk jostled shoulder to shoulder, and somewhere in the middle of this, just by the Allied hut, a live bear, red muzzle over its nose, stood on its hind legs looking benign and bemused. The official symbol of Berlin was a bear and the owner of a children's circus had brought it: the perfect gesture. Someone put a Guard's hat on its head and that got a reverberating cheer of its own. At 0135 Toft recorded people 'chipping away at wall with hammers'.
In the growing confusion the British thought they still had one of their vehicles out in the East where it had been all night. It ought to have re-crossed at Checkpoint Charlie but could get nowhere near. Toft recorded the message from the British section of the hut: '0149 hours. Possible military vehicle now moved to a southern crossing.'
Some 11 minutes later – at 2.00 a.m. – deep from within her slumber Brita Segger heard her doorbell shrill. She wondered who it was at such an hour. The bell rang and rang. 'I got up and went to the door and it was my boyfriend Mehmet, a Kurd living in the West. He said, "I've been waiting at the checkpoint for you for hours." I wondered what he was talking about. He said, "The border's open. I've been waiting but you didn't come." He assumed I'd seen it on TV and assumed I'd be coming', and Checkpoint Charlie, because that was where foreigners crossed, where she'd have deduced that Mehmet would be waiting for her.
Mehmet: 'Get dressed, we're going to the West.'
Brita: 'It's the middle of the night, I've got to work early in the morning and I'm very tired. No.'3
Mehmet: 'You have to come.'
She gave in, dressed and they went in his car, a new Audi 80. To someone from the two-stroke Trabant culture this was luxury itself. They drove from the sombre suburbs of the East down a long highway which thrust towards the city centre like a rod. Just before it ended, they passed a complex of square and oblong buildings with antennae and aerials, a giant grouping stretching a long way back resembling a web: the headquarters of the Stasi. No doubt a telephone had rung there from the Customs at Checkpoint Charlie reporting the opening and getting the reply 'No new regulations.'
The highway became Karl-Marx-Allee, its architecture on such a scale that people were incidental to it; and the road itself had been Stalin Allee before the dictator's disgrace, before the timeless safety of renaming it Karl-Marx-Allee. If Mehmet and Brita looked over to the right they'd have seen the statue of Lenin in an oval-shaped platz; Lenin, the foreigner who was prophet and architect to their own country, up there on a big plinth in his waistcoat and suit, arm outstretched, beckoning and guiding. A minute before midnight he had still been that, prophet and guide. Not now, not as Mehmet and Brita passed him. If they looked over to the left, at the apartment block just before a square, protruding building called the Café Moskau, they'd have seen a darkened ground-floor window. Birgit Kubisch, having dismissed the BBC World Service news bulletin as relaying just another rumour, was sound asleep there.
They might have seen, also, the twin spires of the Nikolai Church which had outlasted Lenin and all his successors, but by then they were into Leipziger Strasse and wondering about the checkpoint. Mehmet parked the Audi 50 metres away – like Moll the nearest he could get – and, like Moll, they walked the rest of the way. Brita Segger would gain no precise impression of the geometry because 'there were so many people, I remember a lot of lights and it looked to me like a very big party going on'.
Brita and Mehmet moved into the Customs area but the stamping had evidently been abandoned, the Customs overrun, just like the Café Adler, now so near. Brita clutched her passport but found it irrelevant. In her twenty-one years that was as astonishing as anything she had ever known. 'We went across and a couple of girls came through at the same time and they were very happy. At this moment I didn't understand what I was doing. Without thinking I was in my boyfriend's arms. We danced for a little but in the road and people were shouting, "Hello, how are you?" People were drinking champagne and wine and singing and crying.'
Brita stood only a few paces on the Western side of the white line and amid the delirium began to think. She hadn't heard Schabowski's press conference about exit and re-entry visas because she'd been at evening class, hadn't heard the analyses of that because, cleaning her teeth, she knew perfectly well it was a joke, and she'd gone to sleep instead. Now here she was beyond the line of no return. 'I thought, "I can't get back. They're allowing people to leave but they won't allow them to re-cross." I could not comprehend that it was possible to cross the border normally in both directions.' Her complete possessions might well prove to be what she stood up in, and if she couldn't go back she was cut adrift from all she knew, her flat, her family, her career. She would be a penniless orphan in this alien land where, apart from curiosity and affairs of the heart, she had no great wish to be. If the checkpoint re-set itself in concrete and she did go back, she'd be regarded as having fled the state, a serious criminal offence.
She reasoned that 'if I went further up the street away from the border that would somehow mean I'd gone fully into the West and I really wouldn't be able to get back.' She and Mehmet lurked in the crowd, but near the line so that she could skip across it. 'I wanted to have a look at the West but not live there, I wanted to look and go home. We met three West boys and they said, "Come on, we've a car, we're going to the Ku-damm."' Brita and Mehmet decided it was a nice offer but, Mehmet said, their car was only just over there and why not take that?
'No, no, don't go back,' Brita said.
Mehmet pointed out that, while these boys were offering a lift, once he had the Audi over here they'd be free to go to the Ku-damm, to the Brandenburg Gate or wherever else they wanted. Something else had become true, too. With all the borders open in Berlin _and_ between East and West Germany, the car represented freedom to go anywhere else, Stockholm or Madrid, Paris or Istanbul, Rome or Amsterdam and all places in between; but so far Brita had remained near the line.
Then she said, 'OK', the way people do, and Mehmet set off into the bowels of Checkpoint Charlie to fetch the Audi. When he'd gone she did move away from the line, to a point facing the underground station. She stood outside the bank and 'many people spoke to me'. The current made strangers curious about strangers, made strangers want to clasp hands and – perhaps – become family again. '"What are you doing?" they asked and I said I was waiting and that I hoped my boyfriend would be able to manage it with the car. He arrived after about twenty minutes. He'd come through another checkpoint although I don't know which because I didn't ask. We drove to a little restaurant and ate a Turkish meal. It was the first time I had eaten Turkish food and it was very good. We drank a little bit of champagne and I smoked Western cigarettes. Now it was nearly three in the morning and we took the car to the Ku-damm.'
Toft wrote, '0225 hours. No vehicles crossing but people allowed to cross'. Eleven minutes later the radio crackled again: '0236 hours. 250+ people on the Western side right up to the barrier, no vehicles allowed through, crowd still in good mood'.
Brita had images of the Ku-damm in her mind already, as so many Easterners must have done because they'd all seen Western television for years. The Ku-damm represented capitalist realism overlaid by a multicoloured neon backdrop of galactic proportions.
The opulence of the shops cascaded over the eye in a caress: mannequins in designer creations, caves of jewellery, luxury car showrooms, porno kinos, perfumeries, chic restaurants – French, Italian, Chinese, Greek, an Argentine steak house – international bookshops, fast-food places and slow-food places, stores specialising in delicacies, whole windows heavy with chocolates; and, further up, curious glass display squares mounted in rows on the pavement containing more jewellery, leather handbags and shoes and gloves and Parisian silk scarves; and so it went, on and on.
In the early hours of this morning the Ku-damm presented a freak spectacle: it was completely overrun. You could pick out the Easterners by their hardy jerkins and coats, their stout shoes made to endure, their old-style caps and hats that Westerners hadn't worn for years, their eyes darting in curiosity and wonder.
'The clothes in the shops were not so important to me, it was interesting to look, yes, but that was all,' Brita Segger says. 'For me, it was important I was there, there with my boyfriend, and I could go on the same streets as he went on, see the same buildings he saw every day. We went to a restaurant on a corner – the Café Kranzler – and people were dancing and drinking and we had another glass of champagne. Outside there was a flower shop which was open and the man in it gave me a present, a black rose with a long stem.'
Erdmute Greis-Behrendt says her husband Thomas and son had gone to the wall straightaway after she'd rung home at 10.30 p.m. Thomas 'phoned me at the office about half past three in the morning. My son said, "Mummy, we have just come back from West Berlin. It's really absolutely super. I'm going to bed now because I'm dead tired." This was so touching to me because he had never been before, of course. They'd driven towards the Invalidenstrasse checkpoint in our Trabbi to cross there and when they couldn't go any further they'd parked the car and walked. They were pushed through the checkpoint with thousands of other people and they found themselves on the other side. They took a taxi to the Ku-damm, and that's what they did. As my son told me how wonderful it had been I was trying not to cry because there are so many moments in your life when you should be with your child and this thing happened and I wasn't.'
Toft wrote, '0346 hours. Checkpoint Charlie reports lost military vehicle still on east side... . 0405 hours. East German authorities starting to clear Checkpoint Charlie by moving them into the West. East German authorities not wearing hats.'
The Café Adler closed some little time later. Since 4.00 a.m. people had been drifting away and in any case 'we had no more to sell. The wine and champagne and beer was all gone – well, we had just a little bit of beer left. It had been so crazy we were exhausted. We had to close. My friend who had been on the door drove me home.' Astrid Benner couldn't sleep. The excitement had been so strong that as she lay she could feel her heart beating.
The Easterners were re-crossing and that posed a particular problem for Harald Jäger at Bornholmer Strasse, if they had the stamp over the photographs in their identity cards. He points out that Bornholmer Strasse was the only checkpoint which had operated that policy because 'there was no other checkpoint where so many GDR citizens had gathered'. And now here they were, unaware that they were no longer citizens. He remembers a couple who must have been about 35 and they had the fatal stamps. The woman started to cry: she had left her children sleeping in their apartment nearby while she and her husband had a sneak look at dreamland. Jäger made the only decision a good, honest man can make. He waved them back through. (Others, turned away, no doubt discovered that the various checkpoints hadn't been operating this system and re-crossed there; and, anyway, the days of such bureaucratic madness were to end soon enough.) Jäger also points out that Easterners discovered the other checkpoints had been opened because, fanning out into West Berlin, they would re-cross at them. Jäger himself, isolated and working so hard, only discovered this on the Friday afternoon.
Toft wrote, '0458 hours. Three missing soldiers plus missing vehicle have returned to West Berlin... 0503 hours. Guardroom informed of soldiers' return... 0507 hours. Checkpoint Charlie. East Germans slowly crossing border on foot from East. East German vehicles returning to east. Four East German Border Guards on border line sending back West Berlin citizens from barrier.'
They were trying, perhaps, to re-set it in concrete, but they had come far too late to change history. From this moment on, a measure of crowd control would be exercised but the right of any Eastern citizen to cross and re-cross would never again be seriously challenged.
Raferty had woken in the most blissful ignorance and called a Military Police patrol to take him to the checkpoint. During the journey, they told him of the night's events. The vehicle couldn't get near the checkpoint itself and had to drop Raferty way down Friedrichstrasse because the street was still blocked solidly with Trabants, Wartburgs, Western cars, people... people... people. When Raferty reached the checkpoint it resembled a 'madhouse. I had to elbow my way to the hut. Most of the people were Westerners coming to have a look, crowding around just looking.'
Mehmet and Brita drove to the Brandenburg Gate shortly after 5.00 a.m. The old stone charioteer still rode the stone horses up on top of colonnades that still bore the chipped marks of fire and counter fire from 1945, but here the wall was so broad that Border Guards could stroll two abreast on its curved circumference as it looped round the front of the Gate. Not this night. The top of the wall presented a fresh spectacle of its own. Some 200 or 300 young Westerners had taken possession and stood or sat or danced.
Many times Brita had seen the Gate and the wall from her side although no close approach could be made. You had to stand 100 yards away behind a metal fence and gaze. Now she was curious to see quite what it looked like from the other perspective. On this Western side an observation platform of tubular scaffolding had stood for many years and it was much favoured by tourists because they could peek through the Iron Curtain from a safe distance.
To this place Kennedy had come and two decades later Reagan. On 10 November 1989 they were succeeded by many ordinary folk, among them the 21-year-old trainee journalist holding a black rose. She looked across at her own half of the city, looked at this wall which cut through with the brutality of a plough and she expressed in a single word what millions of other ordinary people felt: _Madness_.
Those on the platform naturally assumed she was a Westerner. 'No,' she said, 'I'm from the East.' Persistently they asked her if she wanted to go back and she said, 'Yes, of course.' They said if they were her they'd stay, they couldn't understand why she wanted to go back; and that was the real division of their city and hers, not this wall but the distance between the ordinary people of East and West understanding each other.
Brita kept looking at her watch. Dawn was creeping in from the East. Those around her, still incredulous that any sophisticated young woman would voluntarily re-cross, were informed that she had to go to work and that was where she was going.
She and Mehmet drove to Checkpoint Charlie and got out of the car. Raferty or Godek might have glimpsed them but we'll never know because there'd been people... people... people. Brita said a little nervously and a little mischievously to Mehmet, 'I'll see you later on this afternoon – in the West.' Mehmet had been her boyfriend for two years but she had never, of course, visited his apartment. 'I kissed him good-bye politely because we were in public. I said, "See you."' During the night she'd been 'losing my fear of whether I'd be allowed back across but it was still there. I took my passport out and stood in a queue with some workmen. They had their helmets and overalls on and I stayed with them. We chatted a bit while we waited. We went into the room where the Customs officers were and the workmen said they had to go to work. I said that, too. Then we just walked through.'
RIAS reporter Peter Schultz says, 'We made a programme with a lot of colleagues and then I went to Invalidenstrasse [where Uwe Nietzold had tried to cross the day before] at four or five in the morning when a colleague relieved me. There were thousands and thousands of people and Trabbis were coming. People were embracing each other, drinking champagne and knocking on the cars. My first daughter was born 29 August 1961 and my second in 1965 so the whole of their lives the wall had been up. I went to the Brandenburg Gate, that was my Gate, _my_ Gate. I saw people dancing on the wall and the soldiers with water cannon...'.4
Toft wrote, '0632 hours. Checkpoint Charlie: East German vehicles going East to West and vice versa. About twenty people milling around checkpoint area. Four East German Border Guards on border line.'
Brita Seggar worked 'about ten minutes away and when I reached my office I put the black rose in a jar of water'. Her colleagues wondered where she'd got it. She told them, the Ku-damm, you know.
Astrid Benner woke after three hours' sleep. Normally a couple of waitresses could handle the daytime in the café but not this Friday and anyway 'I wanted to be there.' The Café Adler had run out of bread as well as everything else and she went to the shop and bought some, then took the underground and started work again. Albrecht Raw had already re-stocked the café with wine, champagne and beer. 'People,' as Benner says, 'are very quick.'
That Friday itself fell into a valley of emotion.
The Trabants putt-putted in an endless column past Raferty in the hut and Astrid sensed it as it happened: 'The traffic in the whole area suddenly broke down.'
Raferty had much else on his mind. 'You've got to remember the Friday was also an American holiday (Veterans Day, for Federal employees including the military) and a lot of our travellers showed up to go over. We processed hundreds and hundreds and at least three hundred and fifty vehicles, too. Sgt Brown ended up staying over.' By 1.00 p.m. the street was so full that Raferty had to 'climb through the window of the hut because I couldn't get in the door. A Military Policeman was standing there, I asked him to cup his hands and I used that as a step-ladder.'5
Raferty noted what Godek had noted. 'You'd see the people from the East pause and take a deep breath before they crossed the white line because they understood what it meant. Almost everybody did and immediately they were cheered.'
During the afternoon Brita Segger returned, although through the Invalidenstrasse checkpoint in the north – where Germans ought to have gone, not Checkpoint Charlie. This time she went to Mehmet's apartment.
Between 3.00 and 4.00 Godek surveyed a 'tremendous line of pedestrians and vehicles waiting to get back over [to the East]. Half the pedestrians had shopping bags or plastic bags with oranges, bananas, all sorts of things.'
For the twenty-eight years the people of East Berlin had known only the most minimal consumerism. The whole shift of the society had gone the other way, towards subsidised rents and basic food which everyone could afford and which never rose in price, free medicine, a massive university programme on an absolutely egalitarian principle. The cost was to the consumer because the money the government had available remained finite. To get a telephone might involve a wait of eleven years; a Trabant (amazingly made of compressed brown paper) as much as fifteen; a colour television took a year's wages to buy; and, while the staples of life were often abundant, much of East Berlin still looked threadbare and meagre and morose, almost brooding on the old wounds, when it was measured against the West, as it now was by the line of pedestrians.
They carried what they had never touched before: exotic fruits – a woman brandished aubergines – and 200 packs of Marlboro cigarettes and electrical playthings; and often, with a terrible tenderness, nothing but presents for their children, the pocket-calculator games you could hold in the palm of your hand, Snoopy versus the Red Baron, whatever that might be.
These people, if one may risk a generalisation, had suspected they were relatively impoverished before they crossed and saw for themselves, but that only sharpened the depth of irony. They or their parents just happened to be on the wrong side of the road on 13 August 1961.
Some went the whole way and nursed ghetto blasters and video recorders in the cardboard boxes stamped Tokyo, Japan; held them with the same tenderness that they held new-born babies, in both hands.
None of these people resembled refugees. They were what they had become, strong and solid and privately not _un_ proud of what they had collectively hewn with their bare hands across all these hard years from _Year Zero_ in 1945; but now here they were in dreamland and they were humping and heaving as much of its luxuries home as they decently could. Most were joyous, intoxicated, held by feelings of release, some were humbled and baffled by the choice available, ten different brands of everything. Some were frankly ashamed it had come to this but had to go and look all the same. Some wondered about the human condition which could still be bought so easily. Some already wondered about the future because, in the political sense, there could be no going back.
At 4.00 on that Friday afteroon, Birgit Wuthe, like Astrid Benner a part-time waitress, arrived at the Café Adler. She was 22, a student of politics and the Border Guards in the tower must have enjoyed sweeping her with their binoculars because she liked France and had acquired the Parisian way of moving her tight _derrière_ , which oscillated as she flitted from table to table:
I'd been working at the café for five months but I wasn't there on the Thursday night. I was sleeping at home in my apartment in Wedding and I heard the news at seven o'clock in the morning. My father telephoned and said 'Turn on the TV!' I did and I was quite surprised. They were showing pictures from the night: lots of people crying and shouting and jumping, beating the roofs of the cars.
That morning, the Friday, I had an appointment with a girlfriend – she was moving and I said that I would help her. I jumped into my car and tried to get to her where she lived in Tempelhof but that was difficult because the streets were full. I still didn't realise the full extent of the situation. And my friend wasn't very interested in this whole thing because she was so busy with the move. After I'd helped her I went to the café by tube. That was in the afternoon and when I went up the steps from the U-Bahn station just before 4.00 it was too much. Checkpoint Charlie was only for diplomats and foreigners – I wasn't allowed to cross there, for instance – but I realised very quickly that _masses_ of people were crossing it. The checkpoint was _full_ of people. Zimmerstrasse was packed with people and they wanted to hear the sound of the wall peckers [people chipping off bits of the wall as souvenirs]. Cars couldn't get through the crowds easily.
I started working at 4.00 and this was the day [Chancellor] Kohl was in front of the café and there were so many people I couldn't get out to see what was happening in front of it! One of the owners served for the first time. A girl and a boy asked if they could help and we said, 'Well, it would be nice.' We were charging for drinks – that was possible – and most of the people drank beer. We were open until four o'clock in the morning and we only closed because the glasses ran out, not the beer. Customers took the glasses outside [to watch] and left them. There were _millions_ of bottles.
Those drinking the beer were at last drawing it from the same well.6
Late in the afternoon Godek and Yount were standing outside the _Imbiss_. A man from the East approached and 'just kind of stared at me. Sgt Yount, was fluent in German, and when the man spoke and I asked what he'd said. "He's saying hello to us, that's all."'
Yount mused 'Isn't this a great day?' and the man said yes, then put his arm around Yount who said, 'We are very happy for you.' The Easterner offered Godek and Yount a drink from the bottle of beer he was carrying. There were at this moment half a million Soviet soldiers in the GDR, officially there as liberators and guardians against the Yankee peril. What courage did it take for an Easterner to approach two American servicemen in public and offer them a drink?
In the early evening Raferty himself crossed the line. 'We had a bus full of travellers over there, we had heard there was a problem and I wanted to make sure everything was OK. Usually there was not a soul at that time but now I had to dodge people who were coming my way. When I wanted to re-cross, the Border Guards tried to get me in back of this line full of East Germans waiting to cross. They were checking passports and stamping them. I said, "No way" and I walked back right down the middle of the checkpoint. I told Major Godek when I got back and he just laughed.'
What would have been one of those potentially serious incidents at this time the day before was now so trivial you truly could laugh about it. The Border Guards had abandoned any pretence at protocol and were simply trying to keep thousands of people on the move.
The world had gone sane.
'To be honest,' Raferty says, 'the Border Guards and the Customs were dumbfounded. They didn't know what to do. One day they're guarding the checkpoint real tight, the next day they're just looking at passports and stamping them.'
Raferty finished his shift some time around 8.00 that evening.
I had to ride the subway because no vehicle could come to get to me. I had a 104-degree temperature. It was such a long day compounded with all the travellers we'd had to look after and the Trabants coming through right by the hut. We'd had the window open so I'd been breathing the fumes from their exhausts. I got a real bad fever. I had to wait for like three or four trains at the underground station before I could finally squeeze into one. By the time I changed lines heading for home I was real sick, I thought I was going to throw up. I looked over and there was this little girl and a man. I speak a bit of German and I asked him where he was from. He said from the East, from Potsdam and this was his first time in the West. I was just trying to keep the conversation going. The girl was his daughter and it was her first time, too. I took my Berlin Brigade badge off my sweater and pinned it on her scarf. Her father started crying and all I wanted to say was 'Get me off this thing, I feel sick!' When I did get off, everyone was waving at me. All night I dreamed about Trabants... Trabants... Trabants.
And that was the Friday, or at least fragments from it. Many are the tales told: at least 3.3 million, because that was the combined population of Berlin.
Godek worked thirty-six hours straight, full on into Saturday morning, and he didn't feel tired because adrenalin took care of that. He was reluctant to leave but he had an eye appointment.
Astrid Benner received a telegram from a friend, Harriet, who lived in Dresden in deepest GDR and to whom Vietnam and the moon had been closer than the Café Adler. The telegram said she was coming to East Berlin and would be at the Friedrichstrasse station far up the street in the East.
This station had, as has been seen, its own geography and geometry. The underground was so prohibited to Easterners – because it was a potential conduit to West Berlin – that it did not appear on their city maps. Young people, many regularly using the domestic mainline station immediately above it, were openly shocked – like Uwe Nietzold and Birgit Kubisch – when they learned after 9 November that an underground line ran there and had always run there, north–south and linking one part of West Berlin with another.
That Saturday, Harriet arrived at the mainline station and Benner was there to meet her but 'we had a problem finding each other because we were both in streams of people. You had to go up some steps to the checkpoint and the Border Guards were still there. Harriet hadn't been able to find her identity card, she only had one from when she'd been a child and she thought she wouldn't be able to come over with it but the Guards didn't care: they took a quick look and we were through. We got on the underground' – the north-south line. They caught a southbound train through the empty stations which had been blocked off for the twenty-eight years – Französische Strasse, Stadtmitte – and, passing directly under Checkpoint Charlie, reached the underground station by the café. Harriet knew she'd reached the West because this station sign was not in old gothic, it was new. Unconsciously she was reflecting her expectations of the West.
Benner says, 'Harriet hadn't much money so I said she would be able to work in the café to earn some. She spent her first day in the West making sandwiches and looking at Checkpoint Charlie from the Western side.'
This was the new normality and Harriet, making the sandwiches for a little pocket money, had found a measure of it already. People, as Astrid Benner has said, are quick.
* * *
Quotation at the head of the chapter may well have originated as a German joke and been borrowed by an American, but either way that does not diminish how apt it is, and how perceptive. Wish I'd thought of it.
1. Interview with author.
2. Interview with author.
3. Interview with author
4. Interview with author.
5. Interview with author.
6. Interview with author.
## ELEVEN
## _Pieces_
When I went to a home appliance store in the West for the first time, I walked through the aisles and just mumbled to myself like a senile old man, 'This can't be true, this is unbelievable, I must be dreaming.'
Bernd K, skilled worker
quoted in _We Were The People_
The day after the wall fell there was a private moment at Check-point Charlie, and in a way it was more touching and more profound because nothing happened, nothing at all.
For years a man called Helmut came daily to the checkpoint and everyone remembers Helmut – Godek, Raferty, Yount, Brown, the British. Helmut was a legend and the legend grew as it was handed carefully down from one generation of soldiers to the next in the Allied hut.
This is how it was described to Raferty: 'Helmut was from the East but worked in the West and one day he walked across to go to work' – it can only have been Saturday 12 August 1961. 'When he tried to get back to where he lived in the East they wouldn't let him so he decided to relocate himself in the West. His family were on the other side. As he got older he had a couple of strokes and he was in a home somewhere near the Checkpoint but he came every day.'1
Raferty handed the legend on to Godek and this is how he describes it:
Helmut was a very old man, probably in his early eighties. He would come religiously and stand maybe two or three feet short of the white line and look through the Checkpoint. He would watch and he would watch and he would watch. After he was done, he would stop in at the little sliding window on the side of the Hut and, in German – he never spoke a whiff of English – talk with the American soldiers. One day Sgt. Raferty asked him why he always came. Helmut relayed the story that he worked in a little factory in the West and when the wall went up he was never permitted to return to the East.
So every day for years and years Helmut came. He wore the same dark blue suit, brown shoes, white shirt, a tie canted over to the side a little bit. We would give him badges, we would give him ranks and his whole lapel on his left side was like a decorated war hero. He was so proud of that stuff. The first time I went to Checkpoint Charlie he was talking to Sgt Raferty and I was introduced. The next time I saw him I took the little Major's oak leaf off my hat because he didn't have one and pinned it on his chest and he wore that proudly, too.
Well, I guess it was the day after the night the wall opened up that Helmut came as usual and I assume he did his usual thing, walked near the line, started over towards us when he'd done that. He came to the sliding window and he talked briefly and he took off. We didn't see Helmut again for a month. As a matter of fact we had such a relationship with him that we became a little bit worried about him. A month later he returned but this time we didn't notice him go look. He just walked up to the window and we asked him if he'd been sick. He said no. We said, 'But we haven't seen you up here – what's the matter?' He said, 'There is no reason to come.' We asked if he had been over to the East and he said, 'No, there's no reason to do that either _because I can_.' It brought tears to our eyes.2
Helmut took off and never did come back to the place which, soon now, would no longer exist.
Maybe, in the seething multitudes who bodily seized dreamland – overrunning and overwhelming the transport system, packing the bars, blocking the roads with their put-put Trabants, dancing along the Ku-damm, drinking a lot of their own beer and a lot of other people's – every moment was somehow private. In the first weekend after the fall, 4.3 million Easterners got exit stamps. In round figures, 2 million visited West Germany (and caused a 60-kilometre traffic jam at Helmstedt), and 2 million visited West Berlin. You could pick any of these people to understand what that privacy mean. Take a 22-year-old student, Katrin Monjau, who was studying languages at the Humboldt University, the country's best. She had never been to West Berlin 'although every day when I took the train from the city centre to Köpenick where I lived I could see it: new blocks of flats'.
I had relatives and acquaintances in the West and I always thought it was one town, Berlin. When I looked across I thought I wanted to visit that part, too, visit my relatives. My grandmother, who was born in 1910 and died three years before the opening, could [as a pensioner] and I know she was very sorry for us. As a young girl, my mother always went to the cinema in West Berlin. She was studying in Dresden when the wall was built and it was a horror for her.
I heard the news on Friday morning. I came home very late on the Thursday because I was with friends. My mother had been watching the Schabowski press conference on TV but it was too boring for her and she switched it off [before he said the fateful words]. I don't even know that if she had heard the news she would have believed it.
So I got home, we didn't know anything about it and we only found out next morning on the radio. We heard that last night the wall was open and thousands of people went to West Berlin. We also heard that you could only go to West Berlin until eight o'clock this Friday morning – they'd try to close the wall again to have everything back under control. I called my friend – I had a date with him – and he said, 'OK, it doesn't matter that the wall is open.' We could not really believe it, you see. He had to go to work and I had to go to the University so we couldn't get to a crossing point before 8.00. I thought I'd go to the University and see what could happen next, because if they had opened the wall I didn't think it would be possible for them to close it again.
At 12.00, I went from the University to the S-Bahn at Friedrichstrasse Station on foot with a girlfriend and there were not so many people at the checkpoint there. Before this, to take a train to the West you had to go out of Friedrichstrasse and under the bridge, go through a control, a body search and then you went up into another S-Bahn platform [the one forbidden to ordinary Easterners]. I asked two building workers, 'How does it work with the documents?' and one said, 'It's not very difficult. You only get a stamp and you can travel to West Berlin for three days.' I had my identity card with me and it was stamped each time you crossed – although sometimes the place was so crammed full that they stopped stamping the cards and just let the people through.
I stood on the platform with my girlfriend and the S-Bahn arrived in the station. Masses got out and by now there were masses on the platform, too. We were among the first to get on and we got a seat. The train was so crowded that we didn't see the wall as we went across it. We were busy with ourselves. We were talking to other people because everybody was happy and everybody was talking to each other – but we did see the Reichstag. We got off at the Zoo station and we changed trains: we went on the U-Bahn to Nollendorfplatz because a friend of ours had left East Berlin shortly before and I thought I'd never see him again. He wasn't allowed to come back, even for a visit.
We couldn't find the place where he lived, although I'd looked at the city map, and we hadn't so much time because we had to be back at seven o'clock in the evening. It was the time of carnival and we had to dance. We decided to go to the bank [to get the 100 DM welcome money offered to every Easterner] and we waited ten minutes, then we took the underground to Wittenberg-platz to look at the KDW [the famous luxury food emporium]. It was the first department store of this kind I had seen and it was, I thought, very dear so I didn't buy anything. I only looked at everything to get the whole impression.
Katrin was looking, for the first time in her life, at the pleasure of plenty – something normal and unremarkable in the West for going on three decades. She can have had little true comprehension as she stood there of what it took to create the plenty, which was approaching life from a completely different direction. The Westerners had little understanding of her and hers, either, assuming that they would surrender everything before the altar of this plenty they'd coveted. The reality was always going to be a messy balance between these two extremes, this gain and loss – for the West, too.
Then already we had to return to East Berlin. I called the cousin of an acquaintance of my mother who lived in the West and she said, 'Where are you? When are you coming' I had to tell her I was sorry but I had to dance that evening. I said we'd come the following day.
I went back [to East Berlin] and danced, then after the show I went with my friend to Eberswalder Strasse, which is near Bernauer Strasse, because we had heard on the radio that they were breaking down the wall there to make a checkpoint. There were very many people, and also television and radio. Some building workers were breaking the wall down very slowly and a lot of Army officers watched. That must have been ten or eleven in the evening and the people who were standing had hammers and they began battering the wall down. They found a big piece of steel, shaped like a pole but big and heavy, and five of them – ordinary people – used it as a battering ram. They broke down the wall with it. My friend helped. I couldn't – it was too heavy for a woman, I think. The people watching counted out 'One... Two... Three' and the five hit the wall with the pole. Their hands were already bloody so others took over. I gave an interview in French to a French TV station. Afterwards, when we left, we took a piece of the wall with us from the East, which was very nice because it was totally white.3
The graffiti, the clever slogans, the adolescent messages, the artwork had all been on the Western side. No East German had ever done anything like that because you were shot if you tried.
In the morning Birgit Kubisch 'went to university and had the feeling that people were very excited. We were sitting in the canteen discussing it and somebody said, "I can't believe it but it's true, the wall has come down." I still didn't believe that one could simply go through without showing one's passport. Then a girl said, "You know, I've been to West Berlin. I've been there. I've been to the Ku-damm." People made plans to drop their lessons or go after their lessons. I thought how strange it all was because the whole street was covered with people rushing, the buses were crammed full.'
So far away on the other side, and now so physically near, Daniel Glau went to work at the Hotel Ahorn. 'Of course the Ku-damm was full of people and very under pressure. You could tell the people from the East because of the way they dressed, their shoes, their behaviour. They were very... I will not say unfriendly, but it was very hard for them: another culture, forty years of looking over their shoulders. I had no experience of anything like that. I left earlier for work because I knew I'd need more time to get there.'4
Uwe Nietzold 'slept late. I got up at eight o'clock. I should have been up at five to get to my lectures and now it was half past seven. I had to hurry to get the bus so I didn't hear the radio or watch any TV. When I reached Friedrichstrasse it seemed quite normal. I walked to Humboldt University and there should have been a lecture on the sixth floor about the penal code, starting at 9.15. I got there at 9.30 and the University was empty, no people. For the lecture not even the professor was there and I couldn't get into the lecture room. I saw some people standing outside it and they were very excited, speaking, talking, shouting. I spoke to somebody who said I could go to West Berlin without any problems. I couldn't say a word. I was speechless. Then some of the department staff came and said that every lecture was cancelled because nobody was there.
'A friend came and said, "I don't believe it but let's go." We went to Friedrichstrasse station and joined the queue but it was so crowded we decided to go to Invalidenstrasse. There we saw the queue – a tremendously long queue – and joined the end of it. It was so long we were just about to turn away when a West German car came – a Ford – with a woman driving and it was moving very slowly. She brushed against my friend, got out and said she was very sorry. She'd hit his leg. She apologised a thousand times and asked how badly he was hurt. He said, "You could do us a favour and take us to West Berlin." We got into the car in the back seat – we didn't know whether it would work to go in a West Berlin car with a GDR identification pass and no visa. We came slowly to the control because there was also a queue of cars. We gave the officer our identification from the window, waited five minutes, got the papers back and we could go through.
'She drove to Wedding because she lived there and we went to the nearest bank. We were almost the first so we didn't have to join a queue for our 100 DM [welcome money]. To be _given_ money was a bit embarrassing. The big impression came in the late afternoon when they switched on the lights – everywhere neon adverts. I'd seen something like it once when I'd been in Vietnam. Before, I'd been free to go there but not the Ku-damm. That's a paradox. I didn't go into the shops. I was afraid. There were too many people, too many goods and I didn't know how to move. I looked around and then I saw a man selling watches for 50 DM and I saw one I liked very much and I bought it. I didn't know what else to buy! I didn't feel I was still in Germany, I knew it, but it was in my head not in my heart because it was so different. I re-crossed at about six o'clock because there was a big demonstration in front of the Old Museum in East Berlin and I was interested in it.'5
The parents of Marina Brath, the girl who'd been bought out of the GDR prison, came to see her in West Berlin 'one day after the wall came down' – this Friday. 'They didn't telephone first because they didn't have one. They came to the hotel where I worked but I wasn't there, I was at home. They went to my flat, I opened the door and there they were. I was really shocked: my brother Alexander, who'd been eleven when I left, was now 18 and a young man! My mother cried a little. They weren't interested in the shops, only in seeing me.
'Mother hadn't believed it the night the wall came down when she watched it on television. I made a meal and we talked the whole time. My mother saw the apartment and said wow! My brother only looked at me and he said, "Oh Marina, my sister." I don't know what they thought of me leaving. My mother was in the Party and she believed in the good things of the system because she was a child of the war. That's the difference between the generations.
'I went over to see them after four weeks. I felt angry and frightened. I felt strange. I felt like a stranger. I felt an inner pressure and I wasn't really relaxed – no, I wasn't actually frightened but _different_. I still feel it [1990]. After that I had my first dream about the prison – no, nightmares about the prison. I went to see a doctor and he said the feeling will stay with me for a minimum of between ten and twenty years and, if I'm unlucky, maybe forever.' (To the author: 'Feel my hands. They're cold when I talk about it, that's the problem.') 6
Brigitte Schimke, the former Westerner who'd married an Easterner – Erhard – and lived in the apartment overlooking the wall, and what had been the sunken gardens, 'saw Schabowski on television and we heard the famous sentence. We managed to get the telephone from the people on the first floor. We had a plug. My brother [who she'd waved to so discreetly from her little balcony while he stood on the observation platform over there in the West] had already called and asked when we were coming and said the champagne was on ice. My youngest son was at home. I said he should go on his motorbike and visit my brother who lived in the district of Rudow (in the south-east of West Berlin). I went across the next day' – on the Friday.
Erhard 'didn't really dare to cross. I could not really believe we could, but I did cross. We drank champagne and it was very touching, people were giving roses and beer to the Border Guards. People were walking in the death strip, East people, West people, everybody together.'
(In 1992 Brigitta crossed what was now a green strip 'every day to get to the bus to work. I am very happy but it's not yet a normal feeling.' Brigitta met someone who, 'we found out later', had been in the same church congregation in the West 'and we both realised we were Protestant and we'd been confirmed there. The church was called the South Star but the wall divided the parish and afterwards we'd had no contact.')7
The British Army caught the mood of this Friday perfectly and if there is a slight schoolmasterish feel to what they set down afterwards, in their _Berlin Bulletin_ publication, then never mind.
It is not every day that you find yourself standing in the midst of history as it unfolds around you. The staff at the Havel School felt that the event... presented too good an opportunity to miss and were determined that they would do whatever possible to ensure that the children would always remember they were among those fortunate few who were in Berlin when 'The Wall' was breached.
Following a couple of telephone calls we were grateful that 62 Transport and Movement Squadron RCT [Royal Corps of Transport] could provide us with a coach for the morning without our normal prior booking procedures.
Parties of our youngest pupils, this year called Year 5 under the National Curriculum Council's recommendations, eagerly boarded the bus. In total, over 130 children were to have an experience of a lifetime.
Pausing only to buy flowers at the Garden Centre, where a further 50 were donated to our cause by a tearful sales assistant, we were off to the Staaken Crossing Point between the GDR and West Berlin. When they arrived, their teachers described the background and set the scene. The plan was that each child would have the opportunity to present a flower of greeting to a person coming across what had been until fifteen hours earlier a closed border.
The reality proved to provide an even more moving experience. We were privileged to witness many scenes of tremendous joy and beauty as visitors streamed across the once-daunting boundary. One man stopped his car in the car park just behind where we stood – the first place he could legally stop after crossing the line. Once our children had overcome their initial reticence, he found himself swamped with flowers. And all the while he stood staring at his bleak surroundings as if he were in the finest palace saying over and over again – 'This is my first time!'
Another Trabant responded to our waves and stopped beside us and as the children joyously gave their flowers, the moment proved too much for the young couple in the car, and they both wept openly.
Later we went right down to the actual crossing point and joined in the celebrations with the many Berliners who had come to meet friends and relations as they poured through.
Meanwhile, a very small number of senior pupils were taken by private car to the area near the Brandenburg Gate. Here they witnessed similar scenes of joy. We have precious memories which we will never forget.8
Katrin Monjau, the language student who'd promised to go to the West again on the Saturday morning to see the 'cousin of an acquaintance of my mother', did go, 'together with my mother, and this was the greatest thing for me: the first time my mother had been because on the Friday she was working and didn't feel like going to West Berlin alone. My parents are divorced.
'We left at six in the morning. The acquaintance was not very far away [as the crow flies] – we lived _here_ and she lived _there_. Today it takes us fifteen minutes by car but then you had to go all the way round on the S-Bahn and U-Bahn and it took an hour and a half. We had to change on the U-Bahn several times but we had a map so we knew where. She was a distant cousin and I'd seen her only once or twice in my life when she'd come to East Berlin. So I knew her although there had not been very much contact. We have no telephone and I couldn't get letters from Western countries because it was not allowed.
'She lived in an apartment, we arrived, we had breakfast and it was impossible for us to think! We were so happy. I was crying, not too much I think, and my mother was crying too. It was too great for us. After breakfast we went to her daughter and we spent the whole day together. Later we walked a lot. In the evening it was the same for me: I had to return for dancing, there was a show and people had bought the tickets in advance. At the checkpoint I thought, "What if I can't return?" But I could because when the show was over I returned and we spent the night there.'9
Rüdinger Hering, who lived in the East near Falkensee, 'didn't have television or the radio on so we didn't really notice what was happening' on the Thursday. On the Friday, 'I had to start work at 5.30 in the morning – I worked in an agricultural machine factory in Falkensee. I went there and at the entrance to the factory the woman on the gate asked me where I was going.'
Bemused, Hering said he'd come to work, of course.
'What?!' she said.
It begs the question of how many of the factory workforce were now in the West or at home with hangovers from the West. The night shift had left at 10.00 the previous night and, evidently, not been told what had happened, but they'd have heard soon enough and you could walk to the checkpoint at Staaken.
Hering's wife Ingelore telephoned. 'Everybody would need a visa stamp and she had already been to Staaken to get hers. I worked until four, then I went home and picked up my identity card and driving licence. My son [14-year-old Marko] came from school and we went to Staaken together. They charged me 15 marks to stamp the visa!'
Everything was now in order and, on the Saturday, the family went to Spandau to see Hering's uncle, taking an overnight bag. Hering's wife and son did not know Spandau at all. It had always been a kilometre away, but the wrong kilometre.10
Birgit Kubisch, another language student,11 contemplated crossing on the Friday but found the crowds too claustrophobic and waited twenty-four hours until Saturday:
It was a very strange feeling. We went at nine in the morning – my step-sister Connie and her husband Manfred, who were on an ordinary visit from Dresden and now absolutely wanted to see West Berlin. We took the S-Bahn to Friedrichstrasse and we wanted to cross there but we saw all the people so we decided to go to Invalidenstrasse where, actually, there were fewer people – a lot, but not so many. They didn't control the passports any longer. The Border Guards were drinking champagne and some of the people crossing had little bottles and gave them to them. The Guards said, 'Thank you very much' and drank. There were so many people I couldn't see the checkpoint itself. A Guard was standing, a chair was 5 metres away from him and we went straight between them. It took us maybe a quarter of an hour.
The first impression was the Guards with their bottles of champagne not controlling the passports, and the people very excited, some of them laughing but others being very quiet, silent. They were just looking and I can't say what they were thinking, but not everybody was loud and funny.
If you are there for the first time you want to see. I sort of looked into myself. It may be strange but I was somehow sad because even today [1990] I still cannot understand that it had lasted so long. I can understand that somebody wants to build a society which is an alternative to capitalism, but the more you think of it the more you think you can't build such a society by putting a wall round it.
I was shocked by the amount of traffic in the West – so many cars. We went to the Ku-damm because my sister and her husband wanted to buy a radio for their car. First they wanted to get their money because every GDR citizen got 100 Deutschmarks [the welcome money]. I got mine only later, in December or January, because I felt ashamed. I did not want charity. I felt like a beggar. Anyway, this first time the queue to the bank was 500 metres and I was walking along this queue, looking at the people there and I felt shocked. No, ashamed. And what shamed me even more was people from the West distributing cakes as if the Easterners were so poor they didn't have the money to buy cakes themselves. Maybe it was not meant as a humiliation. Somebody offered me one of these cakes and I said, 'No, thank you.'
But you know what was even more humiliating? The people from the East humiliated themselves. They were so greedy they took three of the cakes – they held two in one hand while they were eating the third. Maybe I was too proud. I had 30 Deutschmarks which I'd collected. Relatives had sent it and I got some from my grandmother.
I didn't like the Ku-damm because I do not like advertising and anyway you couldn't enter the shops because they were so crowded. Even my sister and her husband said 'oh no' but they absolutely wanted to have this radio. We found an electrical shop which was also very crowded but we went in and they bought the radio for 59 marks. We had some french fries and Coca-Cola in the street because you couldn't get into a café.
On the Saturday Lutz and Ute Stolz went over, crossing at 11.11 in the morning (people remember important moments like that). 'I did not intend to go immediately,' Lutz says, 'because I needed some time to digest it, to realize the new and different situation. We crossed at the Warschauer Bridge and when we arrived at the border there we met a friend who'd queued for four hours to get his Welcome Money. I noticed how friendly the Border Guards suddenly were. We weren't sure how all these new regulations would work.'
'I thought somehow they would manage to control it again,' Ute adds, 'and that all this was a wonderful ghost story which wouldn't last for long, I did not really believe that something would develop from that.'
The Stolzes didn't go to get the welcome money. 'We wanted to visit some friends who helped me a lot when I studied medicine at the Charité [East Berlin's leading hospital],' Ute continues. 'This was in the 1950s and I studied medicine for two and a half years. I lived in a furnished room. My grant was 95 Marks, 35 of which I had to pay for the room. These friends helped me with money and food, and I spent many weekends with them. We went to visit them – I hadn't seen them for twenty-eight years. They could not remember my name but when they saw me they recognized me.'
'We also went to the sports grounds where I had played soccer when I was a child,' Lutz says. 'All the changes in West Berlinwere somehow frightening for us because of course we hadn't seen them over the years. Take for instance the _Stadtautobahn_ [the dual carriageway which is an inner ring road]. It has facilitated the traffic but it has also led to the destruction of many buildings. We tried to find old buildings and places we remembered. Before the wall went up, about 80 per cent of the members of my soccer club went to the Western part of the city to play so I was always going there. I knew West Berlin almost better than East Berlin. I had many friends there and I spent most of my spare time there. We went looking for that.'
'This was', Ute states, 'our first day in West Berlin. It ended with a surprise at home. I had forgotten to turn the washing machine off. Chaos. The wallpaper came off the walls. And that was it, our first contact with freedom.'
'When,' Lutz remarks, 'we went for the first time by car we visited my cousin who lived in Rudow [a district in the south of West Berlin], although we had seen her several times through these years. The worst was for people who went to the West in the 1950s and had no contact at all after that.'
'The fact that the gap between East and West became bigger and bigger was due in part to the way in which West Berlin and West German people were treated by the Border Guards,' Ute says. 'This is one of the reasons why they said they did not want to have anything to do with the East anymore.'
Lutz reflects that 'my wife and I have remained the people we were and we are happy that we can go wherever we want. I keep saying that the freedom we have obtained is the greatest thing. But you can only judge what this freedom means if you were not free before. Nobody else can. What can a Westerner know or tell anybody about freedom? He cannot judge this. In the East we were stultified and spied on and these were the classic characteristics of an Easterner, the characteristics that we got from our government. This is the conclusion I have drawn.'
The RIAS radio reporter Peter Schultz covered the opening of a section of the wall at Bernauer Strasse on the Monday. 'There were a lot of press people, media people. Professionally it was a good job, not easy but a very good job. Highly emotional? Tears? Yes – there were three walls, an older one and another two. It was a cloudy day but there were masses of people, like an avalanche, and it became one street again. And the people who went over were from the Wedding district [in the West] – the people who had been living alongside the wall – and the moment people got back to this part of what had been twenty-eight years before was the moment when even a news reporter couldn't hold back tears. There were girls from the East who were the same age as my daughter and they were embracing me and saying, "We only know Westerners from what our grandparents have told us and now we can see for ourselves." This was the moment when I started to cry.'12
One of these early days, Erdmute Greis-Behrendt says, a couple of brothers – aged 31 and 32 – went over for the first time. Their parents, pensioners, had been able to go back and forth whenever they wished, giving them the importance of exclusivity within the family. They'd bring chocolate and coffee back. Now the brothers went into a shop and found that their parents had always bought the cheapest chocolate. The brothers, evidently, were not happy and a family row developed, although there must be more to it than this. Pensioners, hoarding precious Western currency, may have only been able to afford the cheapest or, just as likely, the family dynamics – stretched by who knows what – found this strange little episode a catalyst. The point is that the fall of the wall was not a simple event, not a television spectacular before the next television spectacular came along, but something of unending consequences and the most subtle nuances for many millions of people – as well as the world moving, which it was.
Harald Jäger worked three days straight at Bornholmer Strasse, the Thursday to the Saturday, with virtually no sleep, and his wife couldn't come because 'she was not allowed to'. The border area was officially still forbidden – especially, no doubt, to wives of members of the Stasi. Jäger survived on adrenalin and the rucksack they all had which contained provisions for a week. 'We were quite fit but I became so tired I could fall asleep while walking.' Saturday morning he went home and Saturday afternoon, because it was his duty, he returned and worked through to the Monday morning then sleep-walked straight to bed. (He only crossed to West Berlin in January 1990 and it made an 'impression'.)
Martin Schabe, who'd farmed for so long in Eiskeller, found it very easy to go to the neighbouring town of Falkensee again when the wall round the enclave came down: 'Some of the same shops were still there. Some of the people were my age and we met again and we were very happy. And people came over from the East and had ice cream and beer. They felt very much at home. The British military was still here that Christmas and they brought whisky and sweets. We had a lot of whisky and soup and white bread.'13
Douglas Hurd visited Berlin a week after the fall:
Of course, as Foreign Secretary (appointed a fortnight earlier) I followed hour by hour events in Berlin on 9 November. Although no-one supposed then that German unification would follow as quickly as it did, nevertheless there was no doubt about the extraordinary and dramatic significance of the breakdown of the Wall. I flew to Berlin on 16 November and wrote in my diary that night: 'In helicopter above the Wall, old and new crossings, one or two crowded, but on the whole freedom after a week is becoming normal. To City Hall, acting Mayor, Golden Book, to the Wall at Potsdamer Platz, Brandenburg Gate (still closed), walk to Reichstag. There is genuine happiness and excitement, school children, British soldiers offering coffee, visitors from Hannover, but all this swamped by thousands of journalists whom the German police try very efficiently to keep at bay.'
I remember that in the huge press of people I stepped into East Germany and shook hands with an East German Border Guard. The Foreign Office lawyers were dismayed at this because it seemed to them to have a legal significance, which I must say escaped me. No harm came of it.14
A man called Heinz Sachsenweger, who'd come to Berlin in 1964 to study, was director of the Berlin City Museum in the East:
I waited a very long time before I went over – two weeks. There were so many people crossing and there were still controls at the border, you know. You'd have to wait an hour, and I had no family in West Berlin. Really, TV was much more efficient and you could see more than if you went. Not everybody in the GDR was so enthusiastic at that moment, either, because they thought it was too quick and we knew reunification would bring a lot of problems, above all psychological. Some of the Western people behaved as if they had acquired a colony.
I crossed and walked to the Berlin Museum in West Berlin. That was my first stop: a permament exhibition of old Berlin plans. If you are interested in Berlin, and work in this field, you must go there. The actual Berlin Museum had been in the Eastern part of the city for about 100 years, incidentally. [After the division of the city, each half had to have its own.] I didn't want to see the shops because I didn't have money and I knew there were so many people in the department stores – more people than goods. You had to join the queues at the banks for more than two hours to get your Welcome Money and while I understood why people wanted to have the money I personally thought it was primitive to stand there.
It was a beautiful museum in a palace and I spent two hours looking at it. Then I walked to Checkpoint Charlie, because I knew a bit how the streets were. I'd an old map – 1951 – with the whole of Berlin on it [Eastern maps only showed West Berlin as a blank space, remember] and I had studied it before I went. I wanted to see this Checkpoint Charlie, the most famous in Berlin.
The next time I crossed I went to other museums like Spandau. There was a plan in East Berlin to knock down the 100–150-year-old buildings from Alexanderplatz to Friedrichstrasse but since we had no money to build modern buildings it didn't happen, so in one way it was good not to have had the money.
It was not really policy, more a fact that after the war people liked to throw everything old away and wanted to have something modern, something new. For example, a very beautiful old wooden table would be thrown out in order to have a plastic table. Then there were plastic lamps, too, trashy things... .15
There are two ironies in these words. The first is that a museum director would have, and need, a 1951 map to find his way to another museum within the same city. The second is that soon enough East Germans would be throwing so much of what they had into heaps in the streets and replacing them, yet again fifty-four years later, with things modern and things new.
Erdmute Greis-Behrendt had already been to the West with her husband. 'When the people started getting out through Hungary [in summer 1989] my husband and I got permission because it was my father's eightieth birthday. However, we couldn't get permission to take our son. In the West we read the newspapers and saw the news about events in Hungary. We bought a lot of very nice presents for our son and after we came back home he was sitting in our garden. He said, "Thank you for the presents but I think I will have to be 65 before I can go and see it myself", and I said, "Never! I swear to you that you will not have to wait until you are 65." The flight through Hungary had started and I knew it couldn't be stopped.'
She crossed again 'about three weeks after the wall opened because I was working the whole time. I had been very close to it once before, about a week after the opening. The East Berlin entry visa of one of our visiting correspondents had expired and he needed to go across the border to the West and return: as he returned they'd give him a new one. We drove to Checkpoint Charlie and they said, "Oh, if that's all you need, just drive round one of the huts, get out and we'll give you a visa." That's what we did – so I didn't [actually] get to West Berlin, only the other side of the Checkpoint... then we had an evening free and I went across with my family and that was very touching.'16
Ekkehard Gurtz was serving in the GDR Army and, born in East Berlin, had never been to the West although 'we all watched Western TV and we saw another life, a better life'.17
The night the wall fell, Gurtz, then 20, 'was in woods on the border line in the south of the GDR and I had a little walkie-talkie. The operator said, "A lot of people are on the wall, there are demonstrations and they want to pull it down." There must have been some sort of live communication because I heard a voice saying, "Let the wall be pulled down." Two hours later I heard that the government had ordered the Border Guards in Berlin to open all checkpoints [which they hadn't]. It was crazy. Then a jeep came up to the other side and the soldiers in it looked out of their windows. They spoke to us across the border. They shouted, "The wall has fallen, you are free. We are a united Germany – Come, Come, Come." It was very difficult for us. We had standing orders not to speak to the Western Army and it was very dangerous to speak. So we stayed in the woods and we did not speak.
'My mother wrote me letters about the situation in Berlin and my heart was beating. Everybody could cross the border in Berlin except the military, who were not allowed to. It was three weeks before I could get leave. I went on the train from Meiningen [the nearest town to where he was serving: it was in the south of East Germany] to Berlin and the people looked down on me because I was wearing my Border Guard uniform. They said to me, "Go home, go home", and I said, "I am going home!"
'They had a strong look in their eyes. Some people were happy and friendly to me but a lot of others were very angry. We went to the U-Bahn and people started to shout at us. We tried to hide ourselves. Someone asked me, "What will you do now? The army's finished for you" and I said, "It's not finished. I have one week's leave and then I have to go back."
'When I got home my mother repeated that the border was open but "you can't cross". I was very angry. The order from the Army chief was "you don't cross the border". People could cross with ordinary passports but I had a military passport. However, I had an idea: _my brother Uwe looks like me and maybe I can go across using his passport._ Uwe was 28, but he had an old picture in his passport. He came to mother's apartment to visit and we had a conversation there. He said, "Everybody can cross, but what about you?" I _was_ very angry. I explained what I had in mind and he said, "It's dangerous for you, but it's your decision." I said, "Yes, it's my decision and my problem." He said, "Here, you can have the passport. If I'm asked, I'll say I've lost it."
'I went towards Checkpoint Charlie, my heart was beating and I was sweating a hundred metres before I reached it. I thought, _it's too dangerous for me, too dangerous_. If this went wrong it would have been big trouble – I had disobeyed orders and I would have gone to prison for a long time. I waited until a lot of people were crossing together, I went to a Customs officer checking the passports and I showed him my brother's quickly. I went past him and suddenly I was in West Berlin. I went to the first shop which was selling souvenirs about the wall [the 'Museum at Checkpoint Charlie', a collection of memorabilia about the wall and escapes]. I looked at them but I didn't buy any – I'd just come from the wall! The question I started to ask myself was, what happens when I try and re-cross to go home?
'I had no money but I wanted to see the Kurfürstendamm. That was my ambition. I asked people directions and they told me bus number so and so. I got on the bus and you didn't need to pay, you only had to show the GDR passport. I got out at Kurfürstendamm and I walked, just walked – just walked in streets I had never been in before, walked for seven hours. When I got back to Checkpoint Charlie it was very dark. The Border Guards there had by now a very good life – the right mentality for controlling people! There was a queue, a lot of people, and I could feel my heart beating again as I waited. I was sweating again. It was the first time in my life I had the feeling of fear – I never had this feeling before.'
Gurtz moved past the Border Guards to Passport Control and two officers stood there. 'One waved his finger and I showed the passport and he said, "Go!" and I walked through again. I hadn't told my mother – maybe it could have been dangerous for her, too, if I'd been found out – and my brother hadn't told her anything either. My brother could have been in trouble. I got home and said, "I have been in West Berlin" and my mother said, "You're mad!" It was a beautiful thing my brother did. I think about it all the time.'18
Klaus-Peter Grohmann, who'd visited Potsdam and Dresden in 1972, ruminates on his feelings when he returned to East Berlin in 1989 for the first time since 1961:
I thought _my goodness_! I was so happy to be there again and I thought, _this is part of my town as well_. Actually it was a fantastic feeling to see the Dom [cathedral] and be able to walk down Unter den Linden again, but somehow it didn't seem to have that much to offer. And it's strange. We still [early 1990s] travel around in circles and they are still the Western circles, just as in the East they travel in Eastern circles. I'd had friends in the East – everybody had – but after that length of time there was no more contact, no more writing. People started to get married and have children. People built up new lives on both sides and the distance between us got worse.
We have to get over this, we have to start enjoying common things again and learn to be the same. It's difficult. In the East they missed forty years of evolution and I'll give you a practical example of that. I've bought a farmhouse in the East, at Leitzkau near Magdeburg. There, the walls are still painted with chalk and I recall that the last time I had to wash a wall was at the age of 23 [in 1959!] when I had my first flat, but even then we had powder to which you just added water and you didn't need to wash the wall again. So that's a small example of what they missed.19
Pastor Manfred Fischer, who had inherited the Church of Reconciliation and never put foot inside it before it was blown up all those years ago, now faced reuniting his congregation on both sides of Bernauer Strasse:
It was different from East to West. We had a religious town initiative with people from Mitte and Prenzlauer Berg in the East, Wedding and Tiergarten in the West. We talked about it. Let me give you an example of the differences: one of our members lived in the East and he could look onto Bernauer Strasse but he needed months to take it all in. First he went to the wall, then he went through it but only a little bit, then he went past the [deserted] watchtower. He did all this slowly – _until psychologically he occupied this area._
There is the word Zone and it is a political word in our German language but it also carries very, very different meanings between the East and West: same word but altered by context. In the West, nobody talked about the American, British and French Zones but they did use it for the GDR, calling it the Occupied Zone, the Russian Zone.
To writers and artists and philosophers, zone means a forbidden area where you cannot go and the man really had that feeling: _this is a zone – you can never even touch this._ He was a normal man and I want to describe how it wasn't an easy thing for him to go through an open street after twenty-eight years. It was _not_ just going through.
Now look at the Western people. They'd shrug and go straight through. But me, too, I had troubles going over this border because we lived here [and saw it every day]. I couldn't cross here, I couldn't.
I can't describe how these people from the East came to this church step by step. It was a very, very unique situation and it changed day by day. The Chausseestrasse checkpoint [south of Bernauer Strasse] was 'opened' on the night of 9 November. Here [Bernauer Strasse] the wall itself was opened on 18 June 1990. These were the steps: first on 9 November all the checkpoints were open but they were still checkpoints. Then new checkpoints were to be added. Two months later the first was built at the end of Bernauer Strasse, but still controlled.
The next step was a checkpoint in Bernauer Strasse but only occasionally controlled. Then the U-Bahn station which was exactly on the wall was opened in the summer – a completely new step, because you couldn't go there before. The first opening of a street without any controls was Ackerstrasse [a side street] – no controls, no police, nothing. On 30 June they stopped the controls all over Berlin, because by then it made no sense to control the other crossing points. People don't remember this. A year later, they'd arrive and ask, 'Where was it? Where was the wall?'
Slowly people came to us. I recognised them because somehow you know your own people. You'd say, 'Hello, what's your name?' We are very open. We had a coffee bar every Sunday and we invited people and we offered them a cup of coffee. We discussed. They'd say, 'Yes, I'm from this house over there, I've come to look' and so on. It wasn't a lot of people from the other side of the parish because there were not many members there any more: first only very few houses are left, then the percentage of Christians was very low and the percentage of Lutherans even lower.
However, people came from other churches – the Elizabeth-kirche, the Sophienkirche, all these churches over there. And before, my people couldn't be buried in the cemetery and couldn't even visit it. Well, they could visit but they had to obtain permission weeks ahead and then go to the right checkpoint. It was half a day to get to the other side from here. It was also very difficult for the old people – for someone in a wheelchair, say, who wanted to see the grave of their husband. One of these old ladies told me, 'I came to this house to be nearer to the cemetery where my husband is buried' but she could hardly go.
After the church was blown up, people who'd moved away would come back and say, 'Where is Ackerstrasse?' [you couldn't see over the wall]. They came to a very strange, unknown place and they had no point of reference. I tried to maintain links because otherwise it's like a balloon which has burst and nothing is left. Always you make the past nicer than it was and that is why I fought to keep the section of the wall here standing as a reminder of what the past really was.20
_Come together_ ; that was the slogan of the moment. Only gradually did people from both sides fully begin to understand how far apart they still were and how, as Pastor Fischer says, even physically it had to be done step by step.
Kurt Behrendt, so long a resident of the Steinstücken enclave, says:
When the wall came down nobody knew what was going on. Even before the road was built in the 1970s there had been two factions. One said it was good we were enclosed because it was so quiet. The other said, 'OK, but we can't see our families.' When it was obvious the wall was coming down the situation was the same, two factions. Then the wall peckers came, too!
The wall came down in the spring and summer of 1990. It had been opened at some places in order to combine the streets from here to Babelsberg. I could walk across the strip and meet my neighbours and it was a very good feeling. I was always in favour of the wall being opened. It was also a very strange feeling, and many of the people sort of hesitated but – since I had my camera and I was so curious to see and take pictures of everything taking place – I was not so hesitant. I took my camera and observed the people.
Walking along the streets over there I didn't know where I was. There were two phases: first we could go to Babelsberg but we were hesitant, unsure at the moment we stepped onto the death strip. It _was_ a very strange feeling because the border authorities were still in power and, although they let the people through, you were afraid they'd suddenly come and say, 'It's over, you can't go there.' That's why they were hesitant.
I was very sad when I saw Babelsberg. I knew some places in the GDR and East Berlin and I had assumed that there would be lots of buildings which were very well maintained. At first glance everything looked very good in Babelsberg but when I looked closer I saw many of the buildings were rotten. This does not refer to the people. On the contrary, I felt a kind of admiration for them because they were able to live with what they had – the few things they had. When I saw Babelsberg I realised what it meant for the people to have lived there. The general feeling was that they were sad. And when they demolished the watchtowers and removed the wall my general feeling was that people in the East did not really know what would happen.21
A reminder of how strange, perhaps sinister, this past had been, is given by Birgit Wuthe, who worked at the Café Adler, had a boyfriend in the East and went to visit him so regularly – two or three times a week – that she was well known to the Border Guards at Friedrichstrasse railway station:
Lots of things happened at the checkpoint. One day they stole my passport – I turned my back and it was gone. I don't know who stole it and a week later a woman called me. She worked in a government office in West Berlin and she said, 'I found your passport in my bag. Could you please pick it up?' I went to her office and I thought, 'Well, this wasn't an accident, I don't believe this.' I mean, I'm not stupid.
In May 1991 my boyfriend's parents [the father was a diplomat] had moved to Guatemala and their flat was empty. My boyfriend wanted to see how the flat was and he went there. He still had the key. He opened the door, went in and telephoned his mother in Guatemala. He spoke to her for one and a half minutes. He said, 'I'm now in our flat and it's nice that, for the first time, I can use the telephone and the Stasi are not listening.' At this moment the line was cut. That's really strange... .22
Gerda Stern, the lady who'd joined the Communist Party in 1932, watched the opening of the wall on television from her East Berlin apartment:
I felt it was right because I thought that the idea that people can travel is itself right. When I saw the thousands of people crossing I thought it was the end of everything I had worked for in my life, yes, of course. It was very hard. I am still a communist [1990], and socialism is not dead. We won't be dead!
I am very glad that many intelligent people, having been members of the Party, think the same – even Christians. It was not communism, anyway, it was not even a good socialism: but the overall idea of socialism I believe in. Why was I a communist? Because I wanted everybody to live a good life. There shouldn't be some very rich people and others who are hungry. The world can't always be like that.
I went to West Berlin after the wall came down. I was of course interested because I was born there and spent my youth there. I wanted to see some of these places but I didn't find them – they weren't there any more. I went by car with a friend, not even a whole day, just a trip over. I went to see where my parents had lived in Wilmersdorf. They'd had a little villa but it wasn't there any more, either. I didn't like West Berlin – it's a different town.23
The problems of growing together had barely surfaced and when they did the distance asserted itself again. Lutz Stolz's wife Ute says that 'after the change [the _Wende_ , which is how East Germans summed up the events of 1989–90] many of the old friends from childhood wanted me to come to West Berlin, and they invited us several times, but when this first euphoria was over, relations with many of them became much cooler. Actually there are only two left with whom I have regular contact and who did not regard us as mentally less gifted, as stupid Easterners. One has,' she states, referring to herself but speaking for many Easterners, 'a certain pride, of course.'24
Rainer Hildebrandt opened the Museum at Checkpoint Charlie in the 1960s. It commemorated, and still commemorates, those who escaped but has a wider feeling about what it wants to say: all abuse of human rights is unacceptable. Hildebrandt was awarded Germany's highest civilian honour, the Federal Merit Cross, when he was almost 80. A perceptive man – the first time we met he said, 'I can read you straight away, I know you, you are a dreamer!' – he used to tell the tale of an East German who was arrested for distributing unauthorised leaflets and taken to the police station. The leaflets were blank and when the police asked him what was going on he replied, 'Anything I would write on a protest leaflet, everyone already knows.' He had to be released, as he knew he would have to be, but he had made his point: protest is always possible.
When the Stasi files were eventually opened, and so-far unknown deaths at the frontier were revealed, Hildebrandt said, 'We have to find ways to find the truth. Then those who are guilty can grapple with their responsibility and those who are not guilty can grapple with the task of forgiveness.' As a philosophical statement, it beats the hell out of watchtowers, Border Guards and shots in the night.
In 1990 the GDR government, now non-communist, gave a watchtower from Checkpoint Charlie to Hildebrandt's museum and, for good measure, gave him another from Stallschreiberstrasse, a street near the Heinrich-Heine-Strasse checkpoint. Hildebrandt said in 1993 that 'there's no real interest in Germany' in such things any more. 'It's a form of repression. For some, it's because they suffered too much under the old regime. For others, it's because they did nothing against it.' He offered to give the towers to anyone who could find a good home for them. In December 2000 one tower remained and it was demolished 'under cover of darkness.'25 A story of capitalism triumphant, evidently: the person who owned the land it stood on wanted to build offices there and was uninterested in having a watchtower instead.
Dennis L. Bark went back to Berlin in January 1990 and went to the place where Peter Fechter died. He hammered a chunk out of the wall, took it back to California and 'had a case made for it. It's in my office.'
During the Second World War, Leningrad withstood a ferocious and prolonged Nazi siege, during which the population was decimated by starvation; but somehow it clung on and survived. In a Leningrad cemetery there's an inscription: _Nobody is forgotten, nothing is forgotten_.
There should be one in Berlin, too, where Fechter lay calling out, ' _Help me, help me_ ', but – post the wall – they put up a building where his simple wooden cross used to stand and where, invariably, people had come to lay fresh flowers any day one went to it. Although they did construct a monument round the corner, it is not the same thing and never will be.
The two Germanies were reunited on 3 October 1990.
Checkpoint Charlie has gone now, gone as if it had never been, although for years after 1989 the bullet marks of 1945 remained scattered across the flanks of buildings and gouged deep into the dark stone blocks of the postal museum, also gone.
For the tourists going to it after the wall fell, the watchtower still stood beside a small segment of the wall. The watchtower's door was padlocked against vandals, and the padlock made in West Germany. The awning, the Customs offices, the lanes for traffic and the path for pedestrians were taken down and taken away. Check-point Charlie was so emptied that almost no sense of poigancy and no sense of its passing lingered. Checkpoint Charlie became a car park. Then, of course, they erected buildings on it so that, with a breath-taking narrowness, one of the most famous and emotive places on earth became just another construction or two.
The statue of Lenin was taken from Leninplatz, the broad area surrounded by apartments off Karl-Marx-Allee. As it was being taken down it was cordoned off so people couldn't get to it, which provoked some wry Berliner comment about the purpose of walls.
It frightens many, including Germans, that the places of their past and the people of their past can be taken away so easily.
At Checkpoint Charlie, _c_. 1990, Turkish and Polish vendors sold medals, caps, insignia, Soviet generals' hats and any other Eastern memorabilia they could lay hands on; sold them as vendors do, from trestle tables and car boots or they arranged them on the concrete where Moll's men had stood. A nation was for sale and they were selling it. They did brisk trade in chippings from the wall encased in polythene with an authenticity certificate for good measure, and charged more for bits which had been daubed with graffitti, especially anything multicoloured.
The tourists, disgorging from buses, would gather round their guides and hear, in summary, tales that are told: of the shoot-to-kill orders _just about here where you're standing_ ; of the wooden platform _just over there_ , built to give JFK his feel of the forbidden East; of Peter Fechter being left to bleed to death _just over there, a bit further on_. You could read on the faces of these tourists, wherever in the world they'd journeyed from, a question: Can it all have been true? Did it all really happen?
Bernie Godek remembers that some time after the wall fell there was a celebration at Checkpoint Charlie; a mass of people, and Commander Moll was 'doing his best to try and keep it open for vehicles. A marching band came and I was about 5 metres away from him. He looked at me and I looked at him and we both smiled and kind of shrugged our shoulders as if to say, "Well, what are we going to do? Let it happen." He actually smiled and he was a completely different person to that night when the wall came down.'
On 22 June 1990 a crane lifted the Allied hut clean away while bands played and the Foreign Ministers of the United States, France and Great Britain made speeches. Douglas Hurd remembers 'this ceremony was essentially devised by my American colleague Jim Baker. The hut was lifted away by crane after the ceremony of closure. We all made speeches. Jim Baker acknowledged that mine contained the neatest phrase ("Charlie has come in from the cold"). Then we went on to a more serious negotiating meeting of the 2 plus 4 Group of Foreign Ministers. The Soviet Foreign Minister Shevardnadze made some proposals which I described in my diary as "ridiculous" for imposing conditions on the Germans after unification. It seemed to me then, and is certain now, that he did not expect these proposals to be accepted, but made them in order to placate his critics at home in Moscow. Indeed at lunch that day, attended only by Ministers, he told us that he had stayed awake all night because of the criticisms made of him by a particular Russian General and the applause which they received.
'It has always seemed to me that the main point about the Wall was its brutel ugliness. It is nothing like the other great walls of history – the Chinese or Hadrian's. It was essentially cheap and insignificant, which characterises the nature of the regime which it protected. I am glad that bits of it are being preserved, so that its real character will not be forgotten.'26
But there was no plaque to mark the position the hut occupied, in the road or in the world. Raferty was pretty sure the hut was stored somewhere and, when he returned to the States years ago, said he'd be upset if he discovered that it wasn't. That hut, he insisted, was 'my second home'.
This has been a story about people caught within the logic of the wall.
In the early 1990s I went over into the countryside west of West Berlin exploring where the wall had run, trying to follow its course. It was surprisingly difficult in places now that the actual wall, the watchtowers, the tank traps and the dog runs had all gone, and the swathe of no man's land between, where inner and outer walls stood, were being reclaimed by long grass. From a distance the cleared strip blended into the countryside.
The main road in one hamlet ran to where the inner wall bisected it. Old, detached houses lingered on either side of the road, and they'd been improved by improvisation: building materials were scarce. The houses were substantial and sombre. Little Trabants stood under home-made lean-to garages, gardens had been tended but, somehow, not landscaped in any way. Trees overhung the street, keeping the daylight away. A _Bierhaus_ , very near the wall, provided something to do in the empty afternoons: the menfolk, now unemployed, smoked and drank and played cards and made earthy jokes.
I sat down next to a swarthy man nursing his beer. He had eyes which did not look into yours but always towards the bar, the card school, the window, somewhere else. Sure he remembered the night the wall went up and the days which followed. There used to be two houses over there at the end of the street but they fell on the other side, so one dawn trucks came and the families were moved out. It wasn't very nice to see and you don't forget that, especially the children. We never saw them again and the houses were torn down to make the death-strip. He put a paw of a hand round the beer and drank, now looking straight ahead past me.
_Escapes?_
I don't know. It was quiet around here. You'd hear shots from time to time, always at night and over by the woods, but you never found out what happened. That was the same with everything: you didn't find out.
_Big changes now?_
He gripped his beer again and he was looking full at me, a deep and sudden anger alive in his eyes sweeping my question aside.
You know what I want? Twenty-eight years of my life back.
He was called Rüdinger Hering and he has been quoted several times already, but that last sentence seemed to say what so many would have said, a whole generation of them.27
Later that afternoon I drove further north because, judging by the map, another hamlet had been reconnected with the West. I found where the wall had bisected it here and, as luck would have it, the old patrol road in the death strip was intact. It meandered away into the distance, following the contours of what was now returning to the green and pleasant meadowland. A new sign said 'Cycling Only', but I ignored that and set off. A family of three came cycling along the strip towards me, obviously West Berliners by the cut of their clothes and the quality of their bikes, and they flagged me down. They saw the car had British registration plates and the man said in English, and in that impossibly prissy Germanic way, 'It is forbidden to drive here!'
'Listen,' I said, 'this is the last place on mother earth where anybody can forbid anybody anything now. Haven't you heard?'
He wasn't looking for twenty-eight years you see, just a cycle ride cocooned in law and order.
That was the difference, and it wouldn't be going away for another generation, maybe two. And anyway, after what Hering had said, I felt better.
A gentler example or two of such things. Each visit I made I stayed at a small hotel – the Ahorn – on Schlüterstrasse, one of the streets off the Ku-Damm, and the proprietor's son, Daniel Glau (who's been quoted already) played an active part in running the place. He was 24. I was curious about his reactions to the East because he had grown up with the wall. (It was, people would tell you, like living at the seaside and the wall was the sea: always there, nothing you could do about it. What lay on the other side was lost in the darkness of the deep.) Says Glau:
The first time I decided to go over was in spring 1990. I was not curious before that. I didn't feel it was my town. My parents have a different relationship with this town. My relationship was with West Berlin and West people. East Germany was not Germany for me, it was like Czechoslovakia, Hungary, Bulgaria, so to say. Because I have lived twenty-four years with this I didn't say, 'I am going to another area of Berlin.' I'd say, 'I go to East Berlin' in the same way I'd have said, 'I'm going to Prague or Budapest.'
I crossed at the Glienicker Bridge and I went to Oranienburg and Potsdam by car. As I drove across I thought I liked the bridge very much. I stopped and looked across the water to the other side. I went just because I thought I'd like the park near the bridge. I had a strong feeling, an unusual feeling: It didn't look like West Berlin and the people were not the same.
In one way there's more culture than we have here, more friendly feelings between the people, but in another way the people are very hard and very different. It was depressing, yes. I had a feeling this was not a free place.
On the first visit I was only looking for the green places, for the surroundings of the town, not for the town itself – because I had never seen the surroundings of Berlin, the parks and the lakes. I think the surroundings are very nice and I thought this is where my parents would have gone on a hot summer's day.28
A young journalist from Munich came along one day, doing work experience, interviewing onto a big tape recorder she had slung over her shoulder, and I ran her around in the car a bit. We were out on the road by Potsdam, a rutted road, patched and cobbled and sinking, the grass verges gouged where lorries had parked and mud gathered in the tyre tracks. The buildings to either side had that uncoordinated 'we survived' feel to them, a bleakness, a paucity wrapping round them.
_This is your country now_ , I said.
_No, no_ , she said. _This is Romania._
Yes, a generation away, maybe two.
Conrad Schumann, who'd jumped the wire on Bernauer Strasse at 4.00 on the Tuesday after the wall went up, became symbolically famous round the world because in Christopher Isherwood's words a camera recorded it – Schumann fixed forever in mid-air over the barbed wire – and it was developed, carefully printed, fixed.
Schumann was, of course, a simple, ordinary man: taken from the scene in a police van to be debriefed. All he'd wanted was a sandwich.
After that it was a story of alcohol, modest jobs and a confusion about the strange fame which one moment had given him. That brought a truly terrible irony of its own because Schumann had escaped communism, the GDR and the wall, but he could never escape the image of him escaping.
When the wall came down he made guest appearances at the Museum at Checkpoint Charlie but 'he was no longer recognisable from the photo: now he was a podgy middle-aged man with tattoos on both arms'. He'd lived in Bavaria for years, working on Audi's assembly line. In June 1998, after a family argument, his wife found him hanging from a tree.29
Roland Egersdörfer went to the West, but the corner supermarket he had gazed into so intimately with his binoculars had gone so he never did get inside.
They carried the body of Chris Gueffroy away that February midnight in 1989, but they could not carry Chris Gueffroy away. He'd keep coming back long after they – the pall-bearers – had gone from the place.
His mother Karin, a woman with an open, saddened face and soft, sympathetic eyes, would recount how the Public Prosecutor came and told her that her son was dead. She would say that publicly there was a policy of official denial about death at the wall but, with condemnation ringing round the world, this route had been blocked to them. Denial was an obscenity too far. That was only a technical aspect, though. The simple truth was that, as a mother, she had nothing left to lose and consequently that liberated her to seek justice. She told the Stasi that, some day, she would see them charged with the murder of her son and they laughed in her face.
That 'some day' turned out to be a September day in 1991 when four former Border Guards were charged with killing Gueffroy. In court they were frightened, avoided eye contact and wept often. They pleaded the only sane defence available – _obeying orders_ – although those grounds had been dismissed at the Nuremburg trials after the Second World War when _obeying orders_ had been rejected absolutely. The simple charge of _crimes against humanity_ overrode everything.
According to one witness, the four accused 'mumbled' answers to questions in Eastern accents so heavy that the Western judges couldn't always understand them. And in a corner Karin Gueffroy watched – relentlessly, as someone noted.
The man who fired the deadly shot, Ingo Heinrich, was sentenced to three and a half years and his comrade, Andreas Kühnpast, two years suspended. Two years later, Heinrich's sentence was reduced to two years on probation. Heinrich, who in civilian clothing became a most ordinary clean-cut young man, subsequently took part in a British television documentary where he explained the context of what had happened. He acted within the GDR's laws and, at the time of the shooting, could not imagine ever being tried for acting _legally._
The interviewer, Joan Bakewell, wondered if he felt 'trapped by history'.
He said: 'As you grew up from kindergarten it was driven in to you what you had to do. The West was always condemned right from kindergarten and that carried on through school and into the army. And somehow at that point in time you believed that the West wasn't the true way: that the Eastern socialism, or more specifically communism, was going to be the future.' He'd add that if you refused to fire you faced court martial and military prison.
Karin Gueffroy demanded that anyone who picks up a gun accept the consequences of that and, if they fire it, they look deep into their conscience to see what is there.
The logic of the wall, which began so long, long before, was almost played out but sharp fragments remained. The men who gave the orders to shoot were acquitted at the Gueffroy trial, and there was an extreme sense of unease that – higher up still – those who introduced the shoot-to-kill policy, and demanded under harsh penalty that it be carried out, were absent from the courts and courtrooms as justice was meted out to the hapless footsoldiers. There were other senses of unease. Westerners were doling it out to Easterners – victor's justice – and these were Westerners who had never known the grip of totalitarianism on the whole of your life. To the GDR citizens, the GDR was not a freak show run by criminals, it was _life_. Worse, since the West had recognised the GDR as a sovereign state, those who acted within its laws were answerable only to it, weren't they? It was murky territory and only dimly through it could you glimpse a 20-year-old running hard, hard, hard across a death strip at 11.39 one night because he wanted to open a restaurant, and going down, shot through the heart.
But he was there.
Egon Krenz was sentenced to six and a half years in prison in 1997 – Schabowski got three – because, a judge ruled, 'the Politburo was responsible for border security. The Guards were in fact given an ideological order to shoot.'
They carried the body of Peter Fechter away that August afternoon in 1962, but they could not carry Peter Fechter away. He'd keep coming back long after the pall-bearers had gone.
On the thirtieth anniversary of the wall going up there was a ceremony where the simple wooden cross to Fechter's memory stood. The German Interior Minister attended, as did the Mayor, and Fechter's sister Gisela Geue. (Fechter had two sisters. The other was called Ruth.) 'He was the only son, the darling of the family,' Geue said. She explained that the Stasi had 'hounded' the family. Her parents had been hauled from their apartment to identify their son, and the Stasi searched it, looking for anything suggesting his 'political crimes'. They found nothing. 'After that, for the whole week up to the burial, we were practically besieged and watched,' she said. 'They didn't even stop at the cemetery. They took the flowers away.'
She was crying. 'Now anyone can go through, and our brother, our Peter, he was shot only because he wanted to go from Germany to Germany, and then these inhuman people did not help him.'
Her parents became 'miserable mental cases'. Her father died, brokenhearted, in 1968 and her mother in February 1991. 'She did not really understand the opening of the wall.'30
Still Peter Fechter would not go away. In August 1997 a Berlin court convicted two former Border Guards of manslaughter, although the judge, Hans-Juergen Schaal, explained that the trial hadn't found which of them shot him. Both were given suspended sentences: Rolf Friedrich, aged 61, twenty-one months and Erich Schreiber twenty.
'Joint manslaughter was committed here,' Schaal said, 'because neither of you consciously tried to miss the target.' He stressed that neither Guard had been responsible for allowing Fechter to bleed to death but also stressed that the fact it happened was not acceptable.
Friedrich said afterwards 'I am sorry and would like to apologise to his sister.'31
The wooden cross was taken into the Museum at Checkpoint Charlie to accommodate (I've used this word very deliberately) the new building going up where it stood. The new monument round the corner is circular and quietly dignified. There is no sense of drama about it, no striving to stir your emotions, no gesturing to personalise the past. It stands to record a fact, and it records it. The cross, painted a reddish colour which resembles dried blood, is quite different.
Hartmut Richter lived on Bernauer Strasse, so that he could see the wall every day. These days he works at the Church of Reconciliation, a new structure which has been built where the old one stood in the death strip. It's circular and the exterior is of wood. It has clean lines and a great timeless dignity in a modernistic way. Beside it there's a statue of a couple clasping each other, donated by Sir Richard Branson, three enormous bells rescued from the original church and what seems, from a distance, to be an anchor. In fact it's the ironwork from the top of the original spire. When the church was blown up the spire fell and was spread over some 60 or 70 metres – the ironwork fell into the adjacent graveyard and workers there hid it, kept it safe for the day which surely would never come, but did.
Richter is not the sort of man you'd expect to deal in irony, yet he does. He feels that those Stasi members who dealt with him up to 1989 as a 'hostile' entity owe him, and others like him, a real debt. As he says, the Stasi earned a lot of money in their jobs, more than normal workers did, and get high pensions now.
Harald Jäger owns a newsagent's shop in the old East, and thereby hangs a tale. Birgit Kubisch and I tracked him to there, but when we visited the shop his son-in-law, serving, said Jäger was away for the weekend and didn't give interviews. 'He's had some bad experiences.' That I believed. Ex-Stasi don't, as a rule, like the light of day anyway and they have their reasons. We said we'd come back on the Monday and if Jäger refused, he refused. We went back. He is a tall man carrying, as befits one of his size and age, a comfortable semi-paunch. He has, though, a certain presence about him: not respectability but something more. It is the same presence as Günter Moll.
The geometry has been reduced, here, to a neighbourhood community centre (which is what any self-respecting newsagent's really is): cigarettes and tobacco, sweets, magazines, newspapers. One customer can't, for some reason, pay for his morning paper and gets it on tick. Of course. Much laughter about that. You know, the way it goes in your local corner shop, too.
So I make my pitch and yes, he's had bad experiences, been _turned over_ , as we say in the media. That story only emerges later: a female Wessie who wrote that by the tone of his voice she could hear a Stasi hissing diabolical orders. He was so angry that he faxed off a challenge to her facts, refuting them, and, most unforgivable sin of the media and _our_ power without responsibility, she did not even reply.
He doesn't exactly say he doesn't give interviews, but he isn't giving any today. Tomorrow? But I'm leaving early tomorrow. All right, he says, this afternoon; 2.00 p.m. at the checkpoint, that OK? A particular man: he says there's a monument there with a metal plaque and he gives quasi-military directions with the requisite points of reference so we can't miss it, and we don't. He poses before the monument and we walk down the incline past the domain where, during those brief, teeming hours on Thursday 9 November 1989, he held great power over 20,000 people. It's a car sales area now, the models arranged in rows where the awning used to be and, on lines stretched between poles, small plastic triangular flags – part decoration, part advertising – hang limp. There's no breeze today.
I expect whoever owns it has a deal for you. But I bet he doesn't have the deal that Harald Jäger had on 9 November 1989.
We find a café, a couple of youngish men playing snooker, a prim madame supervising the whole thing, the slot machine silent, a sense of mid-afternoon emptiness hanging over it. We sit and he tells the tale. When he's done I'm curious because Ulbricht and Honecker and Krenz went looking for the power and the fame and found it but, as the logic of 1945 finally played itself out so deep into Bornholmer Strasse, all this devolved onto Jäger, who hadn't sought anything, really. He seemed ambivalent about that, didn't care one way or the other. What happened happened.
I'm still curious about how, on a whim and just over there where the cars are arranged, he could have stamped a human being's identity card and potentially changed their lives forever by whether he chose to stamp over the photograph or not.
'Now I am on the good side because it turned out to be good for me but I often think about what might have happened if it hadn't happened like that.'
Like Moll, at decisive moments in a dialogue he swivels his eyes towards yours. You have the recurring thought, within those eyes, about the Tiananmen Square option which was theoretically open to both but which neither considered. That, unstated, is in the eyes, also. These two men were never going to do anything like that.
'I am getting sick of thinking about this, I am still afraid today because I would have been the one responsible... .'
Just for a moment, in memory and imagination, the car park is cleared and it is all going on again, the awning, the excluded area, the uniformed officers striding the glistening tarmac like emperors, everything regulated and controlled, and dreamland too far away to be even seen over the metal-spanned bridge. Ordinary traffic is passing across it today because it's just an ordinary road.
Nothing else.
In January 2001 I asked Birgit Kubisch to read again the interview she had given ten years before and reflect – as she wished – on what she had said.
'I want to comment on two things. Crossing for the first time is exactly the way I remember it although there are some things I'd forgotten, like the bottle of champagne and the Border Guards. One of the reasons I hadn't crossed earlier than I did was my fear of the crowds but the other reason was that I just wasn't sure whether I should go or not.' That would mean confronting actually existing capitalism, which might well destroy a great deal of what she believed in and had worked towards.
As I've already mentioned, deep misgivings jostled silently with the jubilation and the momentary triumphalism.
'I think that what I thought then was quite realistic. Somehow it was not just a cheering moment, it was also a moment of contemplation. I was thinking what the future would be – the immediate future. Obviously I couldn't predict what would happen in ten years. I could understand somebody wanting to build a society which is an alternative to capitalism – capitalisim as I knew it then.'
_What do you think now of this capitalisim you do know?_
'Well, I still think that in the beginning the GDR really had been an attempt to do something different and I still believe it.'
_Is an alternative possible, a viable alternative that gives you all this?_ (Ms Kubisch has a modernised apartment within walking distance of the East Berlin city centre, has a computer – which she complains is too slow – and email and a mobile phone. She likes Indian food. This is a complete culture away from waiting fifteen years for a Trabant.)
'You could write a different book on that.'
_I repeat, you have lived in both systems. Is there any alternative that could give you all this?_
'Not at this time. And the more I think of it the more I think you could not have built an alternative by putting a wall around it. That hasn't changed at all. But I also happen to think that you have to understand historically why they built this wall and ask if they thought about how long it would last. I don't think they did. I think they were reacting.'
_If you are going to build an alternative, people have to want that alternative, and then you don't need a wall – but people here had no choice._
'But you know it was an immediate consequence of the war.'
_What do you miss?_
'I miss the things that everyone would miss in their lives, not so much to do with the GDR but with my childhood. I don't really miss anything else because I always followed events and thought about why they were happening. I might miss a kind of certain security, you know, for planning your career but then, on the other side, there are so many things I could not have done if it hadn't happened. I am very split.'
_What I meant was that when East Germany existed there was_ _a communal feeling of everybody doing something together. They had to do it and they were all in it whether they wanted to be or not and it created a sense of community._
'I have worked in some schools and I miss the way children were treated in schools: everything was prepared. I miss this sense of group feeling and maybe I miss it now because everything is so much more individual.'
_Precisely the opposite of before._
'But what strikes me most is that there are still huge differences in the mentality of people in the East and West after ten years. In one way they have to do with really technical things – salaries, finance, economy – but in another way they have to do with your history and in the way in which the reunification took place.'
_Henriette, your sister's baby girl, won't know any of these differences as she grows up._
'She won't. We can tell her but will she be interested? We talk about things very freely, we don't close our eyes to what was wrong and right before, but she will lead her life quite normally and if it is anything to her it will be just history.'
_Take countries like Britain, America, France. There is a continuity – however strained – between successive generations._
'What we have here is a generation in the East which feels a stranger in two distinct ways. One is that they have lost their home country – whatever the GDR might have been, it gave a sense to their lives. The other is that they won't have the understanding of the next generations because those generations simply won't be able to understand what happened. So they feel a stranger in the two distinct ways.'
This is made more poignant because, on a damp, grey, soulless Saturday afternoon we all go – myself, Birgit, her sister and husband and Henriette – in search of a remaining watchtower in the Treptow district. The watchtower stands in a broad triangle of grass – the death strip, now a park – _but_ there are tall trees, obviously older than ten years, near it and wouldn't they have been cut down because they afforded cover? We fall to discussing whether the watchtower has been moved from somewhere else and placed here and we can't say. (It hadn't.)
Little Henriette is being pushed along in her buggy, chewing biscuits and sweets.
We search for where the wall went but what with the grass growing and roads resurfaced and bushes planted and buildings built and rebuilt we can't tell that, either. What will be left for the Henriettes to find, twenty years on?
Virtually nothing.
When I drove to Berlin in 1991, Ms Kubisch liked my Ford Sierra but wondered how a company as big as Ford could make a car without handles to wind the windows up and down. I introduced her to the electronic buttons – as a child of the Trabant culture, she had never seen them before. She was delighted and spent the rest of the day working them. I hired a VW Lupo on my last visit in 2001 and she exclaimed: 'They've got handles. You wouldn't think anybody made cars like that any more!' Both moments were innocent in their way, and of no consequence when they are set against the weight of the tectonic plates, the enormity of the wall or the fate of such as Peter Fechter, but they do measure the subconscious movement from one normality towards another.
The world had moved all right although you were constantly forced to ask yourself a question: moved where?
That's another story altogether.
* * *
Quotation at head of chapter: _We Were The People_ , Dirk Philipsen (Duke University Press, Durham and London, 1993).
1. Interview with author.
2. Interview with author.
3. Interview with author.
4. Interview with author.
5. Interview with author.
6. Marina Brath worked in the Hotel Ahorn and was one of those Easterners who seemed to have adapted quickly and conclusively to the supposedly insurmountable problems encountered in the West. When she and her husband were taken to the FRG they flew on to West Berlin. I wondered what her first impression was. _All the lights_ , she said, promising freedom and plenty. She glowed at the recollection of the moment.
7. Interview with author.
8. _Berlin Bulletin_ (Education Branch, HQ Berlin Infantry Brigade for British Forces).
9. Interview with author. To recapture now the sense of the forbidden in crossing, and the disbelief that it had really happened and would become normality, was elusive then and has become more so as time has cemented the normality. The idea that it could be withdrawn as quickly as it had been granted is caught nicely here when she wondered if she really would be able to go back.
As an aside, when the wall had been down for about three years I asked Birgit Kubisch how she felt about the new normality and she was ambivalent. She accepted that, physically, she could go whenever she wished but it still felt as if she was journeying _somewhere else._
10. Interview with author. That Hering did not rush to the border but enacted the procedures to get the correct documentation reveals, perhaps, how deep the sense of such things was in the GDR (and the FRG, too...).
11. Interview with author. Birgit Kubisch was typical of many East Germans in that she knew what was wrong with her own country but regarded it as home, was not a little proud of it, and was very uneasy at abandoning everything to the West. Once, when we were walking across Alexanderplatz, she said with sudden vehemence, 'The sun shone here, too! People went on holiday here, too! People made babies here, too! It wasn't a zoo!'
12. Interview with author.
13. Interview with author. After all he had seen in the war and after it, Schabe became an absolute pacifist and we had a heated discussion about the possible consequences if, say, your country or your home was invaded. He remained adamant that his pacifism was non-negotiable and smiled because I didn't seem to understand.
14. Letter to author.
15. Interview with author.
16. Telephone interview with author. When Erdmute reached the point where she recounted promising her son that he would not have to wait until he was 65, her voice raised to a pitch of what I can only describe as angry sincerity.
17. Emphasising how everybody had a fascinating tale to tell, Gurtz was in a hotel I happened to be staying at.
18. Interview with author.
19. Interview with author.
20. Interview with author.
21. Interview with author. Steinstücken was more than an enclave with the unavoidable anomalies, it was frankly very odd indeed because the houses East and West were so close. At one point the rear gardens backed on to each with only the twin walls and the death strip separating them – about, I suppose, 12 feet wide. When the wall came down, people discovered that they had had neighbours for years just the other side of the street and, in one case, the same side of the street. When I went there in 1990, an American officer was showing a couple of guests round a broader area of the death strip and a GDR Border Guard patrol jeep approached. 'Don't worry,' he said. 'They won't shoot, they'll probably try and sell you their uniforms.'
22. Interview with author. The activities of the Stasi have been well documented, and this book isn't about that, but what they did cut so deep that new wounds are being diagnosed all the time. For example, as I was writing this chapter in November 2000, the London _Sunday Telegraph_ carried a news story headed SECRET STASI DOSSIER SET TO SHAKE BERLIN. It began: 'A hitherto secret CIA catalogue of East German Stasi agents threatens to embarrass Berlin's political establishment after the disclosure that at least 120 spies worked, or are still working, for Germany's main parties.'
23. Broadening what Gerda Stern has just said, Diana Loeser, the Briton who embraced the GDR and lived there for so long, was utterly sincere about her beliefs and articulate in discussing them. I interviewed her where she was living in Potsdam: a heavy old stone house, darkened within. At the end of the interview she said that 'communism has failed _this time round_... .'
24. Interview with author.
25. _Daily Telegraph_ , 9 December 2000.
26. Letter to author.
27. Interview with author.
28. Interview with author.
29. _The leap of hope that ended in despair_ , Imre Karacs, _The Independent_ , 24 June 1998.
30. Associated Press.
31. _The Herald_ , Glasgow, 6 March 1997.
## EPILOGUE
## _Corridor of Emptiness_
... to be perfectly frank, if you're hoping for an idea of what the Berlin Wall was actually like, you are going to be disappointed. For something that loomed so large, physically and in the minds of a generation, there is amazingly little of it left.
Jon Henley1
They are almost all gone now or they have become simply old this half a century later, and the brave new world they were creating with their bare hands has gone, too. Even the fragments of the wall they built to insulate this new world are so few and so violated that they hardly constitute a memory. It has left a corridor of emptiness running through Berlin which measures 28 miles but cannot be measured in any of the more important ways.
The potentates of the East, who might have done the measuring, went many years ago: Ulbricht and Khrushchev dead these four decades; General Heinz Hoffmann, Minister of National Defence, two and a half decades; Honecker a decade and a half; spidery Erich Mielke, demanding total knowledge, a decade; and who even remembers those such as Alfred Neumann, holder of a clutch of senior posts from the 1940s to 1989, or Hermann Axen, Auschwitz survivor and the most senior East German to visit the United States, or Kurt Hager proffering calcified ideology from a lost age and socialist-realist culture nobody wanted?
It is as if they had never been and the events of August 1961 had never been, either.
If you stand within sight of Potsdamer Platz, or within Potsdamer Platz itself, this feeling becomes overwhelming. In Ulbricht's time and in Honecker's time it was a broad death strip much photographed from the West because it had a wooden observation tower overlooking it. The bustling centre of the 1930s, with its railway station, department stores and, as every guidebook tells you, the first traffic lights in Europe, had been badly mauled by the Second World War bombing. The East Germans levelled it and only ghosts, watchtowers, vehicle traps, anti-tank obstacles and patrolling jeeps knew it until November 1989. If you saw television images of a section of the wall being lifted away by a grabbing crane, saw the gap like missing teeth and East Berliners streaming through that first weekend, saw East and West policemen linking arms to try to control them, saw the two mayors meet without ceremony as crowds milled round them in delirium, it all happened exactly here at Potsdamer Platz.
That weekend is as if it had never been, too, because these days Potsdamer Platz resembles a version of Tokyo. A thicket of immense, sheer, glass skyscrapers, each shaped in the modern way, each an atrium of capitalism, curve and carve towards the sky. Even if you knew it before, Potsdamer Platz has changed so completely that just trying to recapture the emptied flatland with the patrol jeeps and the watchtowers challenges your own credulity.
If you did not know it before, I am persuaded that _nothing_ – not photographs, not maps, not guided tours, not videos, not border guards reminiscing, not even the three sections of wall which remain – can bring back to you the mundane daily normality of arguably the most abnormal structure ever constructed by human beings. The world is full of monuments and with a little imagination you can evoke what was there, even Roman ruins. The wall has become impossibly elusive and, fifty years on from its construction, the singular fact that thousands upon thousands of people travel from everywhere to visit Berlin expecting to have some experience of it simply heightens the sense of elusiveness.
They all find out that the political symbol distilling the whole twentieth century has essentially ceased to exist.
I know, I know: the talk is of 'the wall in the head' because that's where it lingers now, and we'll be coming to that.
A government armed with totalitarian powers, like the one which ruled East Germany from cradle to grave, has certain inherent advantages. It can do what it wants without reference to its own citizenry – like building a wall to keep the citizenry in2 and, realising that a single wall would be porous, building a second, inner wall with a death strip between (hence Potsdamer Platz). The government decreed that everything in the death strip be cleared away. To the north, in Bernauer Strasse, that meant – as we have seen – forcibly moving families from their homes to let the bulldozers come and eventually blowing up the Church of Reconciliation as well as digging up the graveyard.
This, in turn, created the 28-mile swathe of what would otherwise have been priceless real estate, running fully through the middle of Berlin. When the wall fell, sensitive mechanisms were put in place3 to restore the property to its previous owners or pay them compensation and, as Berlin embarked on a tremendous rebuilding programme (hence Potsdamer Platz), the swathe was too central, too valuable and too symbolic to lie fallow, although – even moving to the fiftieth anniversary, and more than twenty years after the fall – you could still trace some of where the death strip ran because the builders had not reached it and, growing freely, it returned to nature. In those twenty years, saplings became trees, small bushes became big bushes and multiplied, and grass grew in every crevice. One place not far from the site of the Heinrich-Heine-Strasse checkpoint was so overgrown you could barely penetrate it.
That and other places like it transformed the death strip into something unique. Berlin became the only European city with areas of jungle in the middle of it.
In many more places, of course, apartments and office blocks had gone up, were going up and would continue to go up, burying it. The whole death strip will be consumed soon enough, all of it. The developers, planners and property imperatives will see to that.
Nor, in the initial enthusiasm of 1989 to shed the past and reunify, did the population express any desire for the wall to remain. Who would?
'You have to understand, the thought in everybody's mind was, _let's get shot of it_. Nobody wanted to see it any more, not the least trace of it. Nobody,' Pastor Manfred Fischer of the Church of Reconciliation says. 'Die Mauer muss weg – the wall must go: that was the cry before it came down, and for months afterwards, too. We couldn't get rid of it fast enough.'4
Berlin was intent on dismantling something which ought to have been declared a world heritage site. The consequences remain of historical proportions because none of the three surviving sections gives an authentic impression of the wall. They are quite literally fragments which must constitute tantalising and perhaps slightly misleading glimpses to the curious who come in search. No reasonable person could expect the wall to have been left as it was but, given time to reflect, any reasonable person might have expected more than the three short, incomplete sections to be preserved so that generations of Germans – never mind everyone else from everywhere else – might have a chance of understanding the most abnormal structure.
The longest of the three is called the East Side Gallery. It stretches for 1.3km (0.8 miles) and was, in fact, the inner wall beside the River Spree. 'After the Wall came down in 1989, hundreds of artists from all over the world gathered and transformed the eastside of the Wall that had been untouchable up to now, with their paintings, giving the Wall a new face in a new time.'5 This may be admirable, and some of the paintings are almost iconic, especially a Trabant breaking through and Brezhnev kissing Honecker (as it seems, on the lips), but some have been defaced. What the wall really wanted, and will always want, was its old face in its old time because every self-respecting town on the planet has an art gallery but only Berlin had the wall. The very act of painting it means camouflaging it, which means, in turn, concealing its scale, its brooding menace and its breathtakingly stark functionality. Every brush stroke necessarily softened the wall to the point where, if it hadn't been defaced by the graffiti bores, it would have been positively pretty.
That is far more than role-reversal; it is a shocking violation and distortion of the past.
The next section is just along from Checkpoint Charlie. It stretches for approximately 200 metres beside Goering's Air Ministry – a heavy, grey-stone, modernistic building which survived the bombing – and along what was the Gestapo HQ. The street is Niederkirchnerstrasse. Only the outer wall remains and the wall peckers have been there, although now it is protected by fencing.6 You can stand on the corner of the street and have one Nazi monument in front of you, the Communist monument next to it and a second Nazi monument next to that. The curiosity is that you have no sense of the weight of this historical junction. The Air Ministry is so ugly, it's uninteresting; the Gestapo HQ is now a few foundations and an exhibition centre; and, standing alone, robbed of its context, the wall looks like a severed limb. Everyday traffic and pedestrians pass quite normally the way they do in all the other Berlin streets.
The most northerly section is at Bernauer Strasse which, as we saw in the chapters covering the wall going up, produced images so powerful that they don't need a gallery to display them. Konrad Schumann jumping the wire, Ida Siekmann jumping to her death, the windows being sealed brick by brick and the tunnel are still firmly embedded in international consciousness. That they are black and white is symbolic and enhances their power: nobody can camouflage them, nobody can soften them and nothing can make them pretty.
The Bernauer Strasse of these images is unrecognisable. The buildings in the West are modern and of course the tall, terraced row of houses in the East was bulldozed to create the death strip. The new Church of Reconciliation is on it and so is a photographic exhibition showing what was where, but, further up, building is under way.
The outer wall remains at the bottom of Bernauer Strasse but a lower, inaccurate inner wall has been added and, compounding that, a metal wall – evidently architects competed with ideas – links the two walls. On the already abnormal, irrelevant abnormality has been imposed, creating for those who come to experience it a perplexing ensemble. They stand in the corridor of emptiness and if you read their faces you can tell.
There are other remnants and fragments7 as there would have to be with something so pervasive, of such scale and stretched across such a distance. They are almost collectors' items: bits of the patrol roads, a watchtower surrounded by apartments, floodlights, painted road markings to direct vehicles, railings, marking posts, patterns in the road showing where early wall supports were sunk, a panel or two of the inner wall.
The exact path of the outer wall is marked by a twin row of cobblestones which are cut into roads and pavements as they zig and zag, but they don't cover the full 28 miles and, because you can walk and drive quite normally across them, they are very one-dimensional. The wall is too elusive now for them to catch it in any but a decorative sort of way.
The wall in the head is as elusive as the wall itself. The physical wall did much more than hold East and West apart; it created the circumstances where they would have to grow away from each other, each passing year lengthening and deepening that divergence. This was not just an absence of normal, everyday contact but something more far-reaching: the societies on both sides were quite different in many fundamental ways and were constantly evolving away from each other.
This was partially obscured by the fact that they all spoke German, although in the East the language was considered old-fashioned as it was missing the newer marketing and technology expressions, which were often a blend of English and German (Denglish).8 And anyway they were all German. The evolution was also partially obscured because in Berlin they might live on the same street – Friedrichstrasse, for example – with the same name and within sight of each other; and partially obscured because so many of their mannerisms and customs were unaltered. After twenty-eight years, however, nobody knew how far the evolution had taken them and how much remained in common. Nobody had much interest in finding out, either, because it no longer seemed to matter. Reunification was entirely theoretical and unimaginable for generations to come, if ever.
The experiences of the wall mirrored East and West.
To the Westerners it was like living beside the sea: it was just there, a screen obscuring a lost national identity, and although they could cross, why bother? Who needed the border guards' performance, the exchanging of hard currency, the Trabant fumes thick in your nostrils, the threadbare shops? West Berlin truly was an island entire of itself, anyway, and the only bell likely to toll was the Liberty Bell donated by General Lucius Clay so long ago.
To the Easterners it was like living with the clearest demarcation line beyond which all was forbidden. They did not spend their days dreaming of the beyond and (although they had a window on it by watching western television) had little understanding of it.
That is why the evening of 9 November 1989 administered such a shock – because without warning here they were in the beyond – and why, fifty years on, the shockwaves continued to be felt, because here they still were in the beyond, and always would be. East Germany was as dead as Ulbricht, as dead as Honecker, except in memory. That is where the wall lives now, because every East German has had to become a West German. There are plenty of consolations – not least that the wall itself is dead as well as the shoot-to-kill at it9 – but, especially for the older generation, the adjustment has made profound demands and sometimes proved impossible, leaving all manner of casualties in its wake. Commander Moll's country of certainties was replaced with the capitalist casino and some Easterners had no idea how to play in it because nothing prepared them for it. They didn't know the rules and maybe didn't know there were rules.
Their mentalities had been shaped in a quite distinct way and at a distinct tempo.
By the fiftieth anniversay you can still tell elderly Easterners from Westerners by their comportment, their body language and sometimes even their clothing. Risking a generalisation, the middle-aged were young and flexible enough to adjust – however uncomfortably – and the generation born afterwards don't know what you're talking about.
As each of the elderly Easterners dies, a little bit of the wall in the head will die with them and, when the Grim Reaper has finally finished, the wall will be as dead as Ulbricht and Honecker except for the three mystifying sections (if they survive) and as a subject in history classes. Given how tough human beings are, this may take well into this century. In the meanwhile, the elderly inhabit an eternal No Man's Land, although no attack dogs are going to bare their fangs at them there, no armed guards in watchtowers search for them with binoculars, no scatter guns will be activated if they tread on a trigger, no People's Court will mete out savage punishments for trying to flee.
That's dead, too; dead as Ulbricht and Honecker.
* * *
1. _The Guardian_ , London, 27 October 2009.
2. And, of course, taking it down.
3. For a full description of the process and the problems it encountered, see Chapter 4 of _After the Berlin Wall_ , Hilton (The History Press, 2009).
4. _The Guardian_ , ibid.
5. www.eastsidegallery.com/historyesg.htm.
6. For connoisseurs of irony Berlin is fertile ground, and this is but one example: that a wall 11.8ft high (3.6m) and composed of slabs weighing 2.7 tons, designed and refined to be impregnable, should need protection by an ordinary fence.
7. For an exhaustive examination of everything that remains, see _Wall Remnants – Wall Traces_ , Axel Klausmeier and Leo Schmidt (Westkreuz-Verlag, Berlin, 2004). The website of the Berlin Senate Department for Urban Development has an interactive map of remaining border installations (www.stadtentwicklung.berlin.de/ denkmal/ denkmale_in_berlin/ de/ berliner_mauer/).
8. I didn't find that at all, but what was really striking, was the warm-hearted openness they had cultivated in a very small, close circle of friends they could trust and now they could apply this talent to all people – in sharp contrast to the Germans running after careers and money, and holidays and expensive coffee dispenser must-haves, who in big cities had lost a genuine sense of the other and seemed cold and insensitive by contrast.
9. More for the irony connoisseurs: killing the shoot-to-kill policy, and killing it very, very dead.
## _Notes_
By definition, Berlin is a convoluted and complicated subject. For reasons of simplicity, I give West Berlin and West Germany always with a capital W, and East Berlin and East Germany always with a capital E, except when I'm making general references to east and west. Officially East Germany called itself the Deutsche Demo-kratische Republic (DDR) – the German Democratic Republic (GDR) – and I have used this abbreviation more or less throughout. West Germany called itself the Bundesrepublik Deutschland (BRD) – the Federal Republic of Germany (FRG). The status of Berlin remained disputed throughout but, in practical terms, East Berlin was regarded by the GDR as its capital, and referred to only as Berlin; West Berlin was regarded by West Germany as a part of itself but not as a capital. Bonn was that.
The East Germans were dismissive of West Berlin when they were obliged to mention it and referred to it as Westberlin or Berlin (West).
There was widespread misuse (and perhaps misunderstanding) in the whole western world about the Soviet Union, which constituted a group of republics – for example, Georgia and the Ukraine. [The USSR = Union of Soviet Socialist Republics.] Russia was the largest and dominated. Frequently in this book, people say Russia and the Russians when they mean the Soviet Union. By a happy coincidence, the USSR existed throughout the period this book covers and so whenever anyone speaks of the Russians it always means the Soviets – I have not felt free to alter direct quotations from the people I interviewed, so when they say Russian you will know what they (really) mean... .
(I got the dirtiest look of my life when, in a Moscow park in the early 1970s, a charming couple took the chance to practise their English on me and I complimented them on the Russian educational system. 'Russian? We're Georgian,' they said with obvious annoyance, and walked away.)
## _Bibliography_
Alter, Reinhard and Peter Monteath. _Rewriting the German Past. Humanities Press_ , New Jersey, 1997.
Andrew, Christopher and Vasili Mitrokhin. _The Mitrokhin Archive_. Allen Lane/Penguin Press, London, 1999.
Annan, Noel. _Changing Enemies: the Defeat and Regeneration of Germany_. HarperCollins, London, 1995.
Ardagh, John. _Germany and the Germans_. Penguin Books, London, 1987.
Bailey, George. _Germans: Biography of an Obsession_. Free Press, New York, 1991.
——, with David E. Murphy and Sergei A. Kondraschev. _Battleground Berlin_. Yale University Press, New Haven and London, 1997.
Balfour, Alan. _Berlin: The Politics of Order 1737–1989_. Rizzoli International Publications Inc., New York, 1990.
Bearend, Hanna. _German Unification: The Destruction of an Ecomony_. Pluto Press, London and East Haven, CT, 1995.
Borneman, John. _After The Wall_. __BasicBooks (a division of HarperCollins), 1991.
Bornstein, Jerry. _The Wall Came Tumbling Down_. Outlet Book Company Inc., 1990.
Bourke-White, Margaret. _"Dear Fatherland, Rest Quietly."_ Simon and Schuster, New York, 1946.
Brandt, Willy. _My Road to Berlin_. Peter Davies, London, 1960.
Brogan, Patrick. _Eastern Europe 1939–1989_. Bloomsbury Publishing Ltd, London, 1990.
Byford-Jones, Lieutenant-Colonel W. _Berlin Twilight_. Hutchinson & Co. Ltd, London, undated.
Cate, Curtis. _The Ides of August._ Weidenfeld & Nicolson, London, 1978.
Childs, David. _The GDR: Moscow's German Ally_. Unwin Hyman, London, 1988.
Crampton, R.J. _Eastern Europe in the Twentieth Century._ Routledge, London and New York, 1995.
Davidson, Eugene. _The Death and Life of Germany_. Jonathan Cape, London, 1959.
Deloffre, Jacqueline and Hans Joachim Neyer. _Berlin Capitale_. Éditions Autrement, Paris, 1992.
Dennis, Mike. _German Democratic Republic._ Pinter Publishers, London and New York, 1988.
Dodds, Dinah and Pam Allen-Thompson. _The Wall in My Backyard._ University of Massachusetts Press, 1994.
Echikson, William. _Lighting the Night._ Pan Books, London, 1990.
Fest, Winfried. _The Wall, 13 August 1961–1987_. Press and Information Office of the Land of Berlin, 1986.
Flemming, Thomas. _The Berlin Wall, Division of a City_. Be.bra.verlag, Berlin-Brandenburg, 2000.
Forster, Thomas M. _The East German Army._ George Allen & Unwin Ltd, London, 1980.
Gablentz, O.M. von der. _Documents on the Status of Berlin, 1944–1959_. R. Oldenburg Verlag, Munchen, 1959.
Gay, Peter. _My German Question_. Yale University Press, New Haven and London, 1998.
Gedmin, Jeffrey. _The Hidden Hand_. The AEI Press, Washington, DC, 1992.
Gehlen, General Reinhard. _The Gehlen Memoirs_. William Collins Sons & Co. Ltd., London, 1972.
Gelb, Norman. _The Berlin Wall_. Michael Joseph, London, 1986.
Government of the Federal Republic of Germany. _Violations of Human Rights, illegal Acts and Incidents at the Sector/Border in Berlin since the Building of the Wall (13 August 1961–15 August 1962)._ Published on behalf of the Government of the Federal Republic by the Federal Ministry for All-German Questions, Bonn and Berlin, 1962.
Grant, R.G. _The Rise and Fall of the Berlin Wall._ Magna Books, Leicester, 1991.
Gray, Richard T. and Wilkie, Sabine. _German Unification and its Discontents_. University of Washington Press, Seattle and London, 1996.
Gympel, Jan and Wernicke, Ingolf. _The Berlin Wall._ Jaron Verlag GmbH, Berlin, 1998.
Hafner, Katie. _The House at the Bridge_. Scribner, New York, 1995.
Harpprecht, Klaus. _Willy Brandt: Portrait and Self-Portrait._ Abelard-Schuman, London, 1972.
Harrison, Hope M. _Ulbricht and the Concrete 'Rose': New Archival Evidence of the Dynamics of Soviet–East German Relations and the Berlin Crisis, 1958–1961._ Cold War International History Project, The Woodrow Wilson Center, Washington, DC, 1993.
Heller, Deane and David. _The Berlin Wall._ Frederick Muller Ltd, London, 1964.
Henderson, the Rt Hon. Sir Neville. _Failure of a Mission_. Readers Union Ltd by arrangement with Hodder & Stoughton, London, 1941.
Heneghan, Tom. _Unchained Eagle (Germany after the Wall)_. Reuters, Pearson Education, Harlow, 2000.
Hertle, Hans-Hermann. _Chronik des Mauerfalls_. Ch. Links Verlag, Berlin, 1999.
Hildebrandt, Dr Rainer. _Es Geschah an der Mauer_. Verlag Haus am Checkpoint Charlie, 1984.
——. _Berlin: Von der Frontstadt zur Brücke Europas._ Verlag Haus am Checkpoint Charlie, 1984.
——. _Von Ghandi bis Walesa._ Verlag Haus am Checkpoint Charlie, 1993.
Isherwood, C. _Goodbye to Berlin._ Minerva, London, 1989.
Jarausch, Konrad H. _The Rush to German Unity_. Oxford University Press, New York and Oxford, 1994.
——, and Volker Gransow. _Uniting Germany_. Berghahn Books, Providence, RI, 1994.
Kemp, Anthony. _Escape from Berlin_. Boxtree Ltd, London, 1987.
Laufer, Peter. _Iron Curtain Rising_. Mercury House, San Francisco, 1991.
Leonhard, Wolfgang. _Child of the Revolution_. William Collins Sons & Co. Ltd, London, 1957.
Le Tissier, Tony. Berlin _Then and Now. After the Battle_ , London, 1992.
Lippert, Barbara and Rosalind Stevens-Ströhmann. _German Unification and EC Integration_. The Royal Institute of International Affairs, Pinter Publishers, London, 1993.
Lippmann, Heinz. _Honecker and the New Politics of Europe_. Macmillan Co., New York, 1972.
Maaz, Hans-Joachim. _Behind the Wall_. W.W. Norton & Co., New York and London, 1995.
MacDonogh, Giles. _Berlin._ Sinclair-Stevenson, London, 1997.
Marsh, David. _The Germans: Rich, Bothered and Divided._ Century, London, 1989.
McCauley, Martin. _The German Democratic Republic since 1945._ Macmillan Press, London, 1986.
McDermott, Geoffrey. _Berlin: Success of a Mission?_ André Deutsch, London, 1963.
McDougall, Ian. _German Notebook_. Elek, London, 1953.
Mercer, Derrik (ed.-in-chief). _Chronicle of the 20th Century_. Longman, London, 1988.
Millar, Peter. _Tomorrow Belongs to Me._ Bloomsbury Publishing Ltd, London, 1992.
Möbius, Peter and Helmut Trotnow. _Mauern sind nicht für ewig gebaut._ Verlag Ullstein GmBh, 1990.
Mount, Ferdinand (ed.). _Communism_. A Times Literary Supplement Companion, Harvill, London, 1992.
Naimark, Norman M. _The Russians in Germany._ Belknap Press of Harvard University Press, 1995.
Nelson, Walter Henry. _The Berliners_. Longmans, Green and Co. Ltd, London, 1969.
Newman, Bernard. _Behind the Berlin Wall_. Robert Hale Ltd, London, 1964.
Oberdorfer, Don. _The Turn._ Jonathan Cape, London, 1992.
O'Donnell, James P. _The Berlin Bunker_. Arrow Books, London, 1979.
Petschull, Jürgen. _With the Wind to the West_. Hodder & Stoughton, London, 1981.
Philipsen, Dirk. _We Were the People._ Duke University Press, Durham and London, 1993.
Pond, Elizabeth. _Beyond the Wall_. Brookings Institution, Washington, DC, 1993.
Press and Information Office of the Land of Berlin. _The Wall and How it Fell_. 1994.
Prittie, Terence. _Willy Brandt: Portrait of a Statesman_. Weidenfeld & Nicolson, London, 1974.
Read, Anthony and David Fisher. _The Fall of Berlin._ Hutchinson, London, 1992.
——. Berlin: _The Biography of a City._ Pimlico, London, 1994.
Schaffer, Gordon. _Russian Zone_. Published for the Co-operative Press by George Allen & Unwin, London, 1947.
Scholze, Thomas and Falk Blask. _Halt! Grenzgebiet!_ BasisDruck Verlag GmbH, Berlin, 1992.
Shears, David. _The Ugly Frontier_. Chatto & Windus Ltd, London, 1970.
Shirer, William. _Berlin Diary_. Hamish Hamilton, London, 1941.
——. _The Rise and Fall of the Third Reich_. Pan Books Ltd, London, 1971.
Smith, Ken. _Berlin: Coming in from the Cold_. Hamish Hamilton, London, 1990.
Steele, Jonathan. _Socialism with a German Face_. Jonathan Cape, London, 1977.
Steinberg, Rolf (ed.). _Berlin im November_. Nicolaische Verlagsbuchhandlung, Berlin, 1990.
Stern, Carola. _Ulbricht, a Political Biography_. Pall Mall Press, London, 1965.
Wolf, Markus, with Anne McElvoy. _Man without a Face._ Jonathan Cape, London, 1997.
Woods, Roger and Christopher Upward. _Opposition in the GDR under Honecker, 1971–1985_. Macmillan Press Ltd, London, 1986.
Zubok, Vladislav M. _Khrushchev and the Berlin Crisis (1958–1962)._ Cold War International History Project, The Woodrow Wilson Center, Washington, DC, 1993.
## _The Death Strip: The Toll_
Note: known fatalities are indicated by f following the date in brackets.
Behnke, Wolfgang (22.8.85f) (1)
Beilig, Dieter (2.12.71f) (1)
Berger, Dieter (13.12.63f) (1)
Bittner, Michael (24.11.86f) (1)
Blass, Barbara Hildegard (12.1.62f) (1)
Block, Willi (7.2.66f) (1)
Böcker/Boecker, Peter (1.11.84f) (1)
Böhme, Peter (18.4.62f) (1)
Brandes, Dieter (9.6.65f) (1)
Bruckner, Axel (3.9.61f) (1)
Brueske, Klaus (18.4.62f) (1)
Burkett, Rudolf (Apr. 1983f) (1)
Buttkus, Christian (4.3.65f) (1)
Cyrus, Heinz (10.11.65f) (1)
Döbler, Hermann (15.6.65f) (1)
Dullick, Udo (5.10.61f) (1)
Ehrlich, Friedhelm (2.8.70f) (1)
Eich, Klaus-Peter (12.10.61f) (1)
Einsiedel, Horst (15.3.73f) (1)
Fechter, Peter (17.8.62f) (1), (2), (3), (4), (5)
Forgert, Hedwig (April 1963f) (1)
Frank, Horst (29.4.62f) (1)
Freie, Lothar Fritz (4.6.72f) (1)
Frenk, Gerd-Michael (22.12.77f) (1)
Freundenberg, Winfried (3.3.89f) (1)
Friese, Christin-Peter (25.12.70f) (1)
Gadegart, Alice Paula Olga (10.12.84f) (1)
Garten, Klaus (18.8.65f) (1)
Gaudian, Christian (5.2.89) (1)
Gertzki, Manfred (27.4.73f) (1)
Glöde, Wolfgang (10.6.62f) (1)
Gneiser, Rainer (28.7.64f) (1)
Gomert, Klaus (20.7.73f) (1)
Graupner, Ernest (29.4.62f) (1)
Gross, Rene (21.11.86f) (1)
Gueffroy, Chris (5.2.89f) (1), (2)
Haberlandt, Lutz (27.5.61f) (1)
Halli, Norbert (2.4.75f) (1)
Hanke, Fritz (1963) (1)
Hannemann, Axel (5.6.62f) (1)
Hartmann, Jörg (15.3.66f) (1)
Heike, Walter (22.6.64f) (1)
Heinemann, Ursula (19.8.61) (1)
Held, Philipp (11.4.62f) (1)
Hennig, Lothar (4.11.75f) (1)
Heyn, Walter (27.2.64f) (1)
Hinz, Melita (18.12.62f) (1)
Hoff, Roland (29.8.61f) (1)
Hoffmann, Wolfgang (15.7.71f) (1)
Jercha, Heinz (27.3.62f) (1)
Jirkowski, Marietta (22.11.80f) (1)
Kabelitz, Rolf-Dieter (7.1.71f) (1)
Kahl, Hans-Jürgen (3.12.64f) (1)
Kayser, Gerhard (27.10.61f) (1)
Kelm, Erna (11.6.62f) (1)
Kirste, Anna (Nov/Dec. 1973f) (1)
Kittel, Walter (18.1.65f) (1)
Kluge, Klaus-Jürgen (13.9.69f) (1)
Kollender, Michael (25.4.66f) (1)
Körner, Horst (15.11.68f) (1)
Kreitloff, Peter (23.1.73f) (1)
Kreitlow, Peter (24.1.63f) (1)
Krug, Siegfried (6.7.68f) (1)
Krüger, Ingo (10.12.61f) (1)
Krzemien, Ulrich (3.3.65f) (1)
Kube, Karl-Heinz (16.12.66) (1)
Kühl, Werner (24.7.71f) (1)
Kuhn, Erich (26.11.65f) (1)
Kullack, Horst (1.1.72f) (1)
Kutscher, Horst (15.1.63) (1)
Lange, Johannes (9.4.69f) (1)
Lehmann, Lothar (17.11.61f) (1)
Liebke, Rainer (3.9.86f) (1)
Lis, Leo (20.9.69) (1)
Litwin, Gunter (24.8.61f) (1)
Lunser, Bernd (4.10.61f) (1)
Lupke, Gustav (6.11.66f) (1)
Mader, Manfred (21.11.86f) (1)
Mädler, Peter (26.4.63f) (1)
Manowski, Hans-Joachim (c2.4.78f) (1)
Märtens, Elke (10.6.66f) (1)
Marzhan, Willi (19.3.66f) (1)
Mehr, Joachim (3.12.64f) (1)
Meixner, Hans-Peter (1963) (1)
Mende, Herbert (8.7.62f) (1)
Meyer, Michael (13.9.64) (1)
Mispelhorn, Wernhorn (18.8.64) (1)
Mueller, Rudolf (18.6.62) (1)
Müller, Heinz (19.6.70f) (1)
Müller, Otto (14.3.62f) (1)
Mund, Ernst (4.9.62) (1)
Muschol, Dr Johannes (16.3.81f) (1)
Muszinski, Wolf-Olaf (Mar. 1963f) (1)
Neitzke, Frank (12.10.77f) (1)
Niering, Burkhard (5.1.74f) (1)
Nittmeier, Hans-Joachim (1.12.62) (1)
Noffke, Siegfried (28.6.62) (1)
Petermann, Peter (25.4.66f) (1)
Philipp, Adolf (5.5.64f) (1)
Probst, Werner (14.10.61f) (1)
Proksch, Silvio (25.12.83f) (1)
Puhlfüss, Wolfgang (8.10.69f) (1)
Rassmann, Hans-Joachim (2.9.61f) (1)
Räwel, Hans (1.1.63f) (1)
Reck, Ottfried (27.11.62) (1)
Richter, Hartmut (Jan. 1966) (1), (2)
Sahmland, Max Willi (27.1.67) (1)
Schleussner, Lothar (14.3.66f) (1)
Schmidt, Heinz (29.8.66f) (1)
Schmidt, Lutz (12.2.87) (1)
Schmidt, Michael (1.12.84f) (1)
Schmiel, Doris (19.2.62f) (1)
Schmock, Klaus (18.7.70f) (1)
Scholz, Elmar (1.4.69f) (1)
Scholz, Peter (June 62) (1)
Schöneberger, Heinz (26.12.65f) (1), (2)
Schröder, Falk (29.9.87f) (1)
Schröter, Klaus (4.11.63f) (1)
Schultz, Paul (25.12.63f) (1)
Schulz, Dietmar (25.11.63f) (1)
Schulze, Eberhard (30.3.66f) (1)
Schulze, Frieda (Sept. 1961) (1)
Schuman, Conrad (Aug. 1961) (1), (2)
Schütz, Klaus (7.3.72f) (1)
Schwietzer, Dietmar (16.2.77f) (1)
Segler, Olga (25.9.61f) (1)
Seling, Günter (30.9.62f) (1)
Semmler, Günter (13.1.72f) (1)
Siekmann, Ida (22.8.61f) (1)
Sokolowski, Heinz (25.11.65f) (1)
Steinhauer, Ulrich (4.11.80f) (1)
Stephan, Joachim (21.11.66f) (1)
Stretz, Paul (29.4.66f) (1)
Tharau, Margit (1963) (1)
Thiem, Gerald (7.8.70f) (1)
Thomas, Max (May 1962) (1)
Trabant, Hildegard (18.8.64) (1)
Urban, Peter (?18.11.87f) (1)
Urban, Rudolf (19.8.61f) (1), (2)
Walzer, Anton (8.10.62f) (1)
Weckeiser, Dieter (18.2.68f) (1)
Weckeiser, Elke (18.2.68f) (1)
Weise, Henry (17.5.75f) (1)
Weser, Hans-Dieter (23.8.62f) (1)
Weylandt, Manfred (14.2.72f) (1)
Wiedenhöft, Günter (5.12.62f) (1)
Wohlfahrt, Dieter (9.12.61f) (1)
Wolff, Hans-Joachim (26.11.64f) (1)
Wolscht, Norbert (28.7.64f) (1)
Wroblenski, Eduard (26.7.66f) (1)
**Unnamed escapees:**
(23.5.62) (1), (2)
(57 tunnellers: 3–5.10.64) (1)
(2 Border Guards: Aug. 1985) (1)
**Unnamed fatalities:**
(5.10.61) (1)
(18.10.61) (1)
(31.10.61) (1)
(17.11.61) (1)
(20.11.61) (1)
(1.1.62) (1)
(3.4.62) (1)
(22.6.62) (1)
(29.7.62) (1)
(4.9.62) (1)
(Nov. 62) (1)
(1.11.62) (1)
(16.4.63) (1)
(26.3.64) (1)
(1.1.65) (1)
(19.1.65) (1)
(9.6.65) (1)
(1.5.67) (1)
(1.1.70) (1)
(21.6.74) (1)
(2.7.84) (1)
(18.8.87) (1)
(16.4.89) (1)
**Other casualties:**
**Children (drowned)**
Cetin, Mert (5.5.75) (1)
Koc, Cengiz (30.12.72f) (1)
Krobot, Siegfried (14.5.73f) (1)
Savoca, Guiseppe (17.6.74) (1)
**East German Border Guards**
Göring, Peter (23.5.62) (1), (2)
Huhn, Reinhold (18.6.62) (1), (2)
Schmidtchen, Jörgen (18.4.62) (1)
Schulz, Egon (5.10.64) (1)
## _Copyright_
_Cover illustrations_ : _front lower and spine_ : Berlin, Brandenburger Tor, Silvesterfeier (Bundesarchiv, Bild 183-1990-0101-007 / Reiche, Hartmut / CC-BY-SA).
First published by Sutton Publishing Ltd, 2001
This edition published by The History Press, 2011
The History Press
The Mill, Brimscombe Port
Stroud, Gloucestershire, GL5 2QG
www.thehistorypress.co.uk
This ebook edition first published in 2011
All rights reserved
© Christopher Hilton, 2001, 2002, 2003, 2011
The right of Christopher Hilton to be identified as the Author of this work has been asserted in accordance with the Copyrights, Designs and Patents Act 1988.
This ebook is copyright material and must not be copied, reproduced, transferred, distributed, leased, licensed or publicly performed or used in any way except as specifically permitted in writing by the publishers, as allowed under the terms and conditions under which it was purchased or as strictly permitted by applicable copyright law. Any unauthorised distribution or use of this text may be a direct infringement of the author's and publisher's rights, and those responsible may be liable in law accordingly.
EPUB ISBN 978 0 7524 6698 9
MOBI ISBN 978 0 7524 6699 6
Original typesetting by The History Press
Ebook compilation by RefineCatch Limited, Bungay, Suffolk
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{
"redpajama_set_name": "RedPajamaBook"
}
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A1 Basket Greece
A blog in English about basketball news from Greece.
Basket League, Euroleague, Greek players abroad and the Greek National Team.
AEK signed Jordan Theodore
AEK signed Jordan Theodore. The 29-year-old American guard penned a contract until the end of this season with the Greek club. Theodore graduated from Seton Hall and after going undrafted in the 2012 NBA Draft he started his career in Turkey with Antalya BSB. He has also played in Puerto Rico for Mets de Guaynabo, in Dominican Republic for Huracanes del Atlántico, back in Turkey for Mersin BSB and Banvit, in France for JL Bourg, in Germany for Skyliners Frankfurt, and most recently in Italy for Olimpia Milano.
Αναρτήθηκε από Spyros στις 3:03 PM
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Olympiacos BC roster 2016-2017
Olympiacos BC roster 2016-2017 1. Erick Green 2. Kem Birtch 4. Patric Young 5. Vasilis Toliopoulos 6. Giannis Papapetrou 7. Vasili...
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Olympiacos BC roster 2013-2014 ( update with the addition of Jamario Moon and the release of Jamario Moon) (update with November tran...
Panathinaikos BC 2017-2018 roster
Panathinaikos BC 2017-2018 roster Chris Singleton KC Rivers Lefteris Bochoridis Nikos Pappas Marcus Denmon James Gist Ian Vougio...
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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This Preschool Graduation Diploma is bursting with lots of Smiley Faces that will match your students' smiles on their big day! The unique Preschool Graduation Diploma measures 11 inches x 8 1/2 inches. 30 Preschool Diplomas per package.
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{
"redpajama_set_name": "RedPajamaC4"
}
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package org.agrona.concurrent;
/**
* Implementation that calls {@link System#currentTimeMillis()}.
*/
public class SystemEpochClock implements EpochClock
{
public long time()
{
return System.currentTimeMillis();
}
}
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{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,043
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The Investment Bank Tipping Gold to Hit $1,400
by GTI Research · August 14, 2017
Gold prices are set to jump to a four-year high of $1,400 an ounce by the end of the year over mounting tensions between North Korea and the U.S., and surging demand in the world's biggest consumers, according to the head of precious metals at a Russian investment bank.
Bullion could rise to $1,360 within three months before climbing higher, fueled by global political risks and buying from China and India, said Evgeny Ananiev at VTB Capital JSC, the investment-banking unit of Russia's second-largest lender VTB Group. "We may see some correction, but I don't think gold will drop below $1,200 as it's well supported," he said in a weekend interview in Goa. The metal traded at $1,285.73 on Monday.
Prices have climbed 12 percent this year, driven by worries over a potential nuclear conflict between the U.S. and North Korea, and subdued inflation in the U.S., which is cooling chances of a further increase in interest rates. President Donald Trump has intensified warnings to North Korea, promising a massive response to any strike against the U.S. or its allies. Hedge fund billionaire Ray Dalio recommends investors place 5 to 10 percent of their assets in gold.
The upbeat sentiment was shared by other participants at the conference. "Fundamentally, we have been very bullish on the market," said Chirag Sheth, an analyst at Metals Focus Ltd., an independent precious-metals research firm based in London. "What North Korea has done is given gold the legs to go above the $1,300 level and sustain above that level," he said in an interview.
Sheth expects prices to advance to $1,400 in six to nine months as the situation in North Korea sees investors coming back to the market in search of a haven. The U.S. Federal Reserve, which was hawkish on interest rates, has now softened its stance, providing further support to bullion, he said.
'Hellacious War'
As Trump and Kim Jong Un traded barbs, Dalio wrote in a LinkedIn post last week that "the world is watching to see which one will be caught bluffing, or if there will be a hellacious war." Dalio, who manages Bridgewater Associates, also said he sees rising odds of Congress failing to increase the U.S. debt ceiling, leading to a technical default.
The U.S. consumer-price index rose 1.7 percent in July from year earlier, a report showed Friday, trailing the 1.8 percent median estimate of economists surveyed by Bloomberg. Cooling price pressures could make it tougher for the U.S. central bank to raise interest rates again this year.
Indian demand has also recovered after a poor performance in 2016 and jewelry consumption may climb by about 6 percent this year, said Sheth, who provides supply and demand data to the World Gold Council. Imports may jump about 30 percent to as high as 800 metric tons in 2017, he said.
Demand for gold bars in China, the world's biggest bullion market, soared by more than half in the first six months of the year, while overall gold consumption climbed almost 10 percent to 545.2 tons, according to the China Gold Association.
Asian shares post modest gains as trade fears keep investors cautious
FG recorded N66.51bn deficit in October – CBN
Tomato squeeze: U.S. sanctions begin to distort Iran's economy
Next story Libyan Oil Supplies Disrupted by Security Threats, Shut Port
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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In Greek mythology, Eupalamus (Ancient Greek: Εὐπαλάμου means "handy, skilful, ingenious") was an Athenian prince. There are two versions of his genealogy: Eupalamus was called (1) the son of Metion (son of King Erechtheus), and the father by Alcippe of Daedalus, Perdix and Metiadusa, wife of King Cecrops II or instead (2) the son of Erechtheus and possibly Praxithea, and became the father of Metion, father of Daedalus.
Notes
References
Apollodorus, The Library with an English Translation by Sir James George Frazer, F.B.A., F.R.S. in 2 Volumes, Cambridge, MA, Harvard University Press; London, William Heinemann Ltd. 1921. ISBN 0-674-99135-4. Online version at the Perseus Digital Library. Greek text available from the same website.
Diodorus Siculus, The Library of History translated by Charles Henry Oldfather. Twelve volumes. Loeb Classical Library. Cambridge, Massachusetts: Harvard University Press; London: William Heinemann, Ltd. 1989. Vol. 3. Books 4.59–8. Online version at Bill Thayer's Web Site
Diodorus Siculus, Bibliotheca Historica. Vol 1-2. Immanel Bekker. Ludwig Dindorf. Friedrich Vogel. in aedibus B. G. Teubneri. Leipzig. 1888-1890. Greek text available at the Perseus Digital Library.
Gaius Julius Hyginus, Fabulae from The Myths of Hyginus translated and edited by Mary Grant. University of Kansas Publications in Humanistic Studies. Online version at the Topos Text Project.
Maurus Servius Honoratus, In Vergilii carmina comentarii. Servii Grammatici qui feruntur in Vergilii carmina commentarii; recensuerunt Georgius Thilo et Hermannus Hagen. Georgius Thilo. Leipzig. B. G. Teubner. 1881. Online version at the Perseus Digital Library.
Suida, Suda Encyclopedia translated by Ross Scaife, David Whitehead, William Hutton, Catharine Roth, Jennifer Benedict, Gregory Hays, Malcolm Heath Sean M. Redmond, Nicholas Fincher, Patrick Rourke, Elizabeth Vandiver, Raphael Finkel, Frederick Williams, Carl Widstrand, Robert Dyer, Joseph L. Rife, Oliver Phillips and many others. Online version at the Topos Text Project.
Tzetzes, John, Book of Histories, Book I translated by Ana Untila from the original Greek of T. Kiessling's edition of 1826. Online version at theio.com
Princes in Greek mythology
Attican characters in Greek mythology
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
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Q: How to proper way of using Diff utils when updating view in adapter kotlin Hey I am new in DiffUtil in adpater. I read some articles from stack overflow, google docs and some articles. I am trying to understand callback of DiffUtil areItemsTheSame and areContentsTheSame but, I am not clear what that means. I am adding some code, please have a look. If I am doing wrong please guide me.
GroupKey
data class GroupKey(
val type: EnumType,
val sender: Sender? = null,
val close: String? = null
)
EnumType
enum class EnumType {
A,
B
}
Sender
data class Sender(
val company: RoleType? = null,
val id: String? = null
)
RoleType
data class RoleType(
val name : String?= null
val id: String? = null
)
Group
data class Group(
val key: GroupKey,
val value: MutableList<Item?>
)
I am passing my list to adapter which is a Group mutableList
var messageGroupList: MutableList<Group>? = null
..
val adapter = MainAdapter()
binding.recylerview.adapter = adapter
adapter.submitList(groupList)
Using DiffUtil in adapter
MainAdapter.kt
class MainAdapter :ListAdapter<Group, RecyclerView.ViewHolder>(COMPARATOR) {
companion object {
private val COMPARATOR = object : DiffUtil.ItemCallback<Group>() {
override fun areItemsTheSame(oldItem: Group, newItem: Group): Boolean {
return oldItem == newItem
}
override fun areContentsTheSame(oldItem: Group, newItem: Group): Boolean {
return ((oldItem.value == newItem.value) && (oldItem.key == newItem.key))
}
}
}
.....
}
1. Here do I need to compare key other property like type, sender etc. also inside this DiffUtil.ItemCallback.
2. when to use == or === and what about equals()
3. If we compare int, boolean or String we use == or something else ?
Inside this adapter I am calling another Recyclerview with passing list of Item inside that adapter.
Item
data class Item(
val text: String? = null,
var isRead: Boolean? = null,
val sender: Sender? = null,
val id: Int? = null
)
NestedRecyclerView.kt
class NestedRecyclerView : ListAdapter<Item, IncomingMessagesViewHolder>(COMPARATOR) {
companion object {
private val COMPARATOR = object : DiffUtil.ItemCallback<Item>() {
override fun areItemsTheSame(oldItem: Item, newItem: Item): Boolean {
return oldItem.id == newItem.id
}
override fun areContentsTheSame(oldItem: Item, newItem: Item): Boolean {
return ((oldItem.isRead == oldItem.isRead) &&
(oldItem.sender == newItem.sender) &&
(oldItem.text == oldItem.text))
}
}
}
}
Again Same question Do I need to compare sender's other property here as well.
4. In areItemsTheSame do I need to compare id or just oldItem == newItem this?
5. How to proper way to update my adapter items. In normal reyclerview we use notifiyDataSetChanged. But in diffutil do I need to call again submitList function and it will take care of everything?
adapter.submitList(groupList)
A: Questions 1 and 4:
areItemsTheSame means that the two instances represent the same data item, even if some of the contents might be different. Suppose you had a list of contacts, and Jane's middle initial has been changed, but the row should still represent the same person Jane. There might be distinct instances of your model class, with some different values, but they are supposed to represent the same row.
So, usually you will compare only one thing between the old and new items that will be the same for each of them in this case. Usually, if you're getting data from a database or API, there will be some unique ID that represents a data point, and that's all you need to compare in areItemsTheSame. For example, oldItem.id == newItem.id.
areContentsTheSame means that if the two instances were each displayed in your list, they would look identical. So if you are using a data class, it is sufficient to use oldItem == newItem because a data class has an equals function that compares every property.
In your Item callback code, it looks like your areItemsTheSame is correct, but your areContentsTheSame is overly complex. Since Item is a data class, you only need to compare the two items directly.
override fun areContentsTheSame(oldItem: Item, newItem: Item) = oldItem == newItem
In your Group callback code, maybe you can compare the GroupKeys of the old and new items if that is a valid way to determine the items are the same. Since you are using only a direct == comparison, when items change partially you might have some visual defects like views disappearing and reappearing instead of simply having some of their text change.
Question 2
You should rarely ever have to use === in Kotlin. It does not only check if two items are equivalent, but it checks if the two items refer to the exact same instance in memory. It is not appropriate for DiffUtil.ItemCallback at all.
Question 3
== is the correct way to compare any two objects. In Kotlin, even primitives should be compared this way because they behave like objects.
Question 5
With ListAdapter, you should always use submitList instead of notifyDataSetChanged. notifyDataSetChanged would cause a pointless refresh of all the views and defeat the purpose of using ListAdapter and DiffUtil.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,704
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export default function isInFlowTypedRepo() {
return /\/flow-typed/.test(process.cwd());
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 447
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La Circoncisione di Gesù è un dipinto a olio su tavola (36,7x79,36 cm) di Tiziano, databile al 1510 circa e conservato nella Yale University Art Gallery di New Haven.
Descrizione e stile
Nonostante il cattivo stato di conservazione dell'opera, l'attribuzione è stata riferita quasi concordemente a Tiziano, nella fase giovanile prima del 1511. Sebbene siano leggibili ancora influssi giorgioneschi, lo schema iconografico tradizionale appare già rinnovato, con un'originale impostazione in orizzontale in cui le figure principali si dispongono, tagliate all'altezza del bacino, attorno all'altare su cui avviene il rito ebraico della circoncisione.
Fa da sfondo una finestra aperta su un cielo chiaro, messa in posizione asimmetrica per assecondare il gusto "moderno", più spigliato e informale, del primo XVI secolo.
Bibliografia
Francesco Valcanover, L'opera completa di Tiziano, Rizzoli, Milano 1969.
Voci correlate
Madonna Bache
Dipinti di Tiziano
Tiziano
Dipinti nella Yale University Art Gallery
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
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{"url":"https:\/\/cs.stackexchange.com\/questions\/72299\/bin-redistribution-problem","text":"# Bin Redistribution Problem\n\nI'm trying to figure out what this NP-hard problem is most similar to and therefore which solutions are available to me.\n\nThere are a fixed number k of bins, each with the same initial capacity c1. There are a number of items of different sizes packed into each of the bins, such that the sum of the sizes of the items in each bin is less than c1. In other words, we are given a viable solution to the bin packing problem.\n\nIf we now change the capacity of each of the bins to c2 < c1, can we redistribute the items so that the sum of the sizes of the items in each bin is now less than c2, in such a way that minimises the total amount of size moved. There is no limit on the number of items in each bin, just the sum of their sizes.\n\nIllustrated as an example:\n\nc1 = 20\n\n| | | | | | | 4 |\n| 3 | | 2 | | 1 | | 2 |\n| 1 | | 1 | | 7 | | 3 |\n| 4 | | 7 | | 4 | | 4 |\n|__3__| |__4__| |__4__| |__5__|\n\nFirst bin total = 11 < 20\nSecond bin total = 14 < 20\nThird bin total = 16 < 20\nFourth bin total = 18 < 20\n\nBy changing c2 to 16, a (potentially sub-obtimal) solution is:\n\n| 4 | | 1 | | | | |\n| 3 | | 2 | | | | 2 |\n| 1 | | 1 | | 7 | | 3 |\n| 4 | | 7 | | 4 | | 4 |\n|__3__| |__4__| |__4__| |__5__|\n\nFirst bin total = 15 < 16\nSecond bin total = 15 < 16\nThird bin total = 15 < 16\nFourth bin total = 14 < 16\n\nWe moved a 4 from the fourth bin to the first, and a 1 from the third bin to the second - giving a total movement of 5.\n\nThere are some similarities with a view problems I'm aware of, but nothing which quite is the same.\n\nFormally, as a decision problem:\n\nGiven a capacity $C$, and a set of $B$ bins $\\{b_1, b_2, ..., b_B\\}$ each containing a number of items with varying sizes. Can we redistribute the items such that $\\sum$ (contents of bin) $< C$ for each bin $\\in \\{b_1, b_2, ..., b_B\\}$ where the total size moved is at most $M$.\n\n\u2022 Can you formulate this as a decision problem? It's not clear what happens if the items cannot be redistributed at all. Depending on the answer, you could reduce SUBSET-SUM to your problem. \u2013\u00a0Yuval Filmus Mar 31 '17 at 13:49\n\u2022 Given a capacity $C$, and a set of $B$ bins $\\{b_1, b_2, ..., b_B\\}$ each containing a number of items with varying sizes. Can we redistribute the items such that $\\sum$ (contents of bin) $< C$ for each bin $\\in \\{b_1, b_2, ..., b_B\\}$ and the total amount of size moved is at most $M$. \u2013\u00a0Matt Lane Mar 31 '17 at 15:44\n\u2022 You can reduce PARTITION to this problem. \u2013\u00a0Yuval Filmus Mar 31 '17 at 18:29\n\u2022 You first have to solve the problem whether fitting all items into k bins of size c2 is possible at all, and that would already be NP-complete (if you turn this into a decision problem where the limit for amount moved is very large). \u2013\u00a0gnasher729 Mar 31 '17 at 21:53","date":"2019-08-23 11:56:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.3473933935165405, \"perplexity\": 272.35260116507374}, \"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-2019-35\/segments\/1566027318375.80\/warc\/CC-MAIN-20190823104239-20190823130239-00280.warc.gz\"}"}
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{"url":"https:\/\/web2.0calc.com\/questions\/help-thanks-for-your-help-ep","text":"+0\n\n# help! Thanks for your help EP!\n\n+1\n193\n1\n+823\n\nIf $$f(x) = \\begin{cases} x^2-4 &\\quad \\text{if } x \\ge -4, \\\\ x + 3 &\\quad \\text{otherwise}, \\end{cases}$$then for how many values of x\u00a0is f(f(x)) = 5?\n\nJul 25, 2020","date":"2021-05-15 12:10:09","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9991188645362854, \"perplexity\": 3390.405889046787}, \"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-2021-21\/segments\/1620243991801.49\/warc\/CC-MAIN-20210515100825-20210515130825-00434.warc.gz\"}"}
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\section{Introduction}
\vspace*{-0.3cm}
In recent years there has been a growing interest in the magnetic properties of 4d and 5d transition metal compounds with tri-coordinated lattices and bond-directional exchange anisotropies, broadly known as Kitaev materials~\cite{Jackeli2009,Jackeli2010,BookCao,Rau2016,Trebst2017,Knolle2017,Winter2017,Takagi2019,Motome2019}. Among these, the most extensively studied are the iridates A$_2$IrO$_3$ (A = Li, Na)~\cite{Singh2010,Liu2011, Singh2012,Ye2012,Biffin2014a,Biffin2014b,Takayama2015,Chun2015, Williams2016,Modic2014,Breznay2017,Ruiz2017,Veiga2017,Takayama2019,Majumder2019,Majumder2019b} and H$_3$LiIr$_2$O$_6$~\cite{Kitagawa2018,Pei2019}, and the ruthenate $\alpha$-RuCl$_3$~\cite{Plumb2014, Sears2015, Majumder2015, Johnson2015}. The main interest in these materials has been triggered by the realization~\cite{Jackeli2009,Jackeli2010} that the dominant exchange interaction between the effective spin-orbit-entangled $j_{\rm eff}\!=\!1/2$ moments is the so-called Kitaev anisotropy which is known to stabilize a variety of quantum spin liquid phases~\cite{Kitaev2006,Mandal2009,Kimchi2014,Hermanns2016,IoannisClassKitaev}.
Besides the dominant Kitaev anisotropy, the above materials feature additional weaker interactions which generally give rise to a wealth of nontrivial phases competing with the quantum spin liquids~\cite{Jackeli2009,Jackeli2010,BookCao,Rau2016,Trebst2017,Knolle2017,Winter2017,Takagi2019}.
A central goal in the field is therefore to map out the various instabilities and identify the distinctive experimental signatures of the most relevant interactions. One of the most promising ways to achieve this goal experimentally is to subject the Kitaev materials in external magnetic fields along different directions.
Apart from controlling the interplay of various zero-field competing phases, such external fields can also stabilize new collective spin states. There are, for example, many reports for possible magnetic field-induced quantum spin liquids~\cite{Baek2017,Zheng2017,Wolter2017,Kasahara2018}, and a variety of complex multi-sublattice, single- and multi-${\bf Q}$ phases~\cite{Janssen2016,Chern2017,Janssen2017,LiChern2019,Janssen2019}.
Remarkably, all experimental data reported so far for Kitaev materials show that their response to the magnetic field depends very strongly on its direction. This is true for the layered compounds Na$_2$IrO$_3$~\cite{Singh2010}, $\alpha$-Li$_2$IrO$_3$~\cite{Freund2016}, and $\alpha$-RuCl$_3$~\cite{Sears2015,Johnson2015,Majumder2015,Kelley2018}, as well as for the three-dimensional (3D) iridates $\beta$-Li$_2$IrO$_3$~\cite{Ruiz2017,Majumder2019} and $\gamma$-Li$_2$IrO$_3$~\cite{Modic2014,Modic2018}.
Here we revisit the case of the hyper-honeycomb $\beta$-Li$_2$IrO$_3$ and show that its strongly anisotropic response signifies a large separation of energy scales between the relevant microscopic interactions, and can thus be used to extract information about the relative strength of these interactions in a direct way.
The main features of $\beta$-Li$_2$IrO$_3$ that are known so far are as follows~\cite{Biffin2014a,Takayama2015,Veiga2017,Ruiz2017,Majumder2019}.
At zero field, the system orders magnetically below $T_N\!=\!38$~K, with the spins forming a non-coplanar, incommensurate (IC) modulation, with propagation wavevector ${\bf Q}\!=\!(0.57,0,0)$ in the orthorhombic frame, and two counter-rotating sets of moments~\cite{Biffin2014a}, similar to those in $\gamma$-Li$_2$IrO$_3$~\cite{Biffin2014b} and $\alpha$-Li$_2$IrO$_3$~\cite{Williams2016}.
A magnetic field along ${\bf b}$ destroys the IC order at a characteristic field $H_{\bf b}^\ast\sim2.8$~T, beyond which the spins show a uniform ${\bf Q}\!=\!0$ coplanar phase, comprising a ferromagnetic (FM) component along the field and a robust zigzag component along ${\bf a}$~\cite{Ruiz2017}. These components are also present below $H_{\bf b}^\ast$, but are too small to be detected at zero field~\cite{Ducatman2018,Rousochatzakis2018}.
For {${\bf H}\!\parallel\!{\bf a}$} and {${\bf H}\!\parallel\!{\bf c}$}, the system shows a much weaker response, with the IC order remaining robust and the magnetization being linear up to the maximum fields measured (see supplemental material in \cite{Ruiz2017} and also \cite{Majumder2019}).
On the theory side, it has been established that the magnetism of $\beta$-Li$_2$IrO$_3$ can be accurately described by the nearest neighbor (NN) $J$-$K$-$\Gamma$ model~\cite{Lee2015,Lee2016,Ducatman2018,Rousochatzakis2018}, where $K$ denotes the Kitaev coupling, $J$ the Heisenberg coupling and $\Gamma$ the so-called symmetric exchange anisotropy which is present in many Kitaev materials~\cite{Katukuri2014,Rau2014,Lee2015,Lee2016,KimKimKee2016,IoannisGamma}.
In particular, $\beta$-Li$_2$IrO$_3$ is believed to be in the regime of large negative $K$, large negative $\Gamma$ (with $|\Gamma|\!<\!|K|)$ and small positive $J$ (with $J\!\ll|\Gamma|$), see detailed discussion in \cite{Ducatman2018,Rousochatzakis2018}.
Remarkably, in this parameter regime, the critical field $H_{\bf b}^\ast$ depends only on $J$, specifically~\cite{Rousochatzakis2018} $\mu_B H_{\bf b}^\ast\!\sim\!0.46 J \left(4S/g_{bb}\right)$, where $S\!=\!1/2$ denotes the classical spin length of the $j_{\text{eff}}\!=\!1/2$ degree of freedom, $\mu_B$ is the Bohr magneton and $g_{bb}$ is the diagonal element of the electronic ${\bf g}$-tensor along ${\bf b}$. The small value of the experimentally measured $H_{\bf b}^\ast$ is therefore a signature of the smallness of $J$ ($J\!\sim\!4$~K).
It has also been shown~\cite{Ducatman2018,Rousochatzakis2018} that the IC order of $\beta$-Li$_2$IrO$_3$ can be treated as a long-distance twisting of a nearby commensurate period-3 state with ${\bf Q}\!=\!\frac{2}{3}\hat{\bf a}$ (in units $\frac{2\pi}{a}$). This state is amenable to a semi-analytical treatment of the problem, with results that are consistent with almost all experimental findings so far, both in zero field and at finite fields along ${\bf b}$~\cite{Ducatman2018,Rousochatzakis2018}. This analysis explains, for example, the presence of a uniform zigzag component along ${\bf a}$ on top of the modulated order, and the intensity sum rule of the corresponding Bragg peaks~\cite{Ruiz2017}.
Here we show that this semi-analytical description can be naturally extended to the cases where ${\bf H}$ is along ${\bf a}$ and ${\bf c}$. The results, which are cross-checked with classical Monte Carlo simulations, show that the response along ${\bf a}$ and ${\bf c}$ directions shares many qualitative features with that along ${\bf b}$. Specifically, we find that the period-3 order disappears at a critical field $H^\ast$, whose value depends strongly on the field direction. Importantly, none of the critical fields depends on the Kitaev interaction $K$, and moreover $H_{\bf a}^\ast$ are mainly controlled by $\Gamma$.
A realistic set of coupling parameters,
\small\begin{equation}\label{eq:JKGammaPars}
J\!=\!0.4~\text{meV}, ~~
K\!=\!-18~\text{meV} ~~ \text{and}~~
\Gamma\!=\!-10~\text{meV}\,
\end{equation}\normalsize
delivers $H_{\bf b}^\ast\!\sim\!2.88$~T and $T_N\!\sim\!35.5$~K (in good agreement with corresponding experimental values of 2.8~T and 38~K~\cite{Biffin2014a,Ruiz2017}), and also gives $H_{\bf a}^\ast\!\sim\!102$~T and $H_{\bf c}^\ast\!\sim\!13$~T. This means that at least the transition at $H_{\bf c}^\ast$ should be accessible experimentally, and the measured value of $H_{\bf c}^\ast$ can provide the value of $\Gamma$.
The same semi-analytical approach provides a number of additional qualitative findings:
(i) The period-3 order is always intertwined with a set of uniform orders, some of which give rise to a finite torque that can be measured experimentally.
(ii) Among these uniform orders, there is always a zigzag component which remains robust above $H^\ast$ and coexists with the FM order along the field. Classically, the zigzag component disappears (but only for $g_{ab}\!=\!0$, see below) at $H^{\ast\ast}\!\to\!\infty$ for fields along ${\bf a}$ and ${\bf b}$, but for {${\bf H}\!\parallel\!{\bf c}$} the corresponding field $H_{\bf c}^{\ast\ast}$ is finite. In particular, $H_{\bf c}^{\ast\ast}$ is governed mostly by $\Gamma$, with $H_{\bf c}^{\ast\ast}\!\sim\!45$~T for the parameters of Eq.~(\ref{eq:JKGammaPars}).
(iii) The intensity sum rule between the Bragg peaks at ${\bf Q}\!=\!\frac{2}{3}\hat{\bf a}$ and ${\bf Q}\!=\!0$, which has been observed experimentally for {${\bf H}\!\parallel\!{\bf b}$}~\cite{Ruiz2017}, is actually fulfilled for all field directions. As it turns out, this rule is an experimental fingerprint of the spin length constraints.
(iv) While the transitions at $H_{\bf a}^\ast$ and $H_{\bf b}^\ast$ are continuous, the transition at $H_{\bf c}^\ast$ is of first order. Moreover, this transition does not restore all broken symmetries, which leads to the prediction of a second thermal phase transition at high enough fields along ${\bf c}$. This transition will be demonstrated explicitly by classical Monte Carlo simulations.
One shortcoming of our semi-analytical classical approach is that it overestimates the magnetization at $H^\ast_{\bf b}$ by approximately a factor of two compared to the experimental value. This has led to the assertion~\cite{Rousochatzakis2018} that the spin lengths are strongly renormalized by quantum fluctuations due to the close proximity to the special $K$-$\Gamma$ line in parameter space, where the system is highly-frustrated~\cite{Ducatman2018}.
This assertion is now demonstrated explicitly by a semiclassical $1/S$ expansion. The results confirm that the magnetization correction can be as large as 50\%, and a direct comparison with published experimental data shows good agreement for all three field directions.
\begin{figure}[!t]
{\includegraphics[width=0.9\linewidth]{Figure1}}
\caption{Sketch of a hyperhoneycomb lattice. The five NN bonds of the $J$-$K$-$\Gamma$ model are shown by solid (dashed) red lines for $t\in\{x,x'\}$, solid (dashed) green lines for $t\in\{y,y'\}$, and solid blue lines for $t\in\{z\}$.}\label{fig:lattice}
\end{figure}
The rest of the paper is organized as follows. In Sec.~\ref{sec:LatticeSymmetry}, we recall the structural and symmetry aspects of $\beta$-Li$_2$IrO$_3$ that are most relevant for this study. In Sec.~\ref{sec:Model}, we review the minimal $J$-$K$-$\Gamma$ model~\cite{Lee2015,Lee2016} and the symmetries of the corresponding spin Hamiltonian. In Sec.~\ref{sec:ansatze}, we present the unified semi-analytical description for all three orthorhombic directions. This includes the basic spin-sublattice structure of the various configurations (Sec.~\ref{sec:SpinSublattices}), their parametrization in terms of Cartesian components and the associated symmetry-resolved static structure factors (Secs.~\ref{sec:parametrization1}-\ref{sec:parametrization2}), the dependence of the critical fields $H^\ast$ on the model parameters (Sec.~\ref{sec:Hstar}), and a discussion of the symmetries that are broken in each regime (Sec.~\ref{sec:Symmetries}).
The role of quantum fluctuations is then addressed in Sec.~\ref{sec:QFs}, along with the direct comparison of the predicted magnetizations with available experimental data.
In Sec.~\ref{sec:Torque} we discuss how the various transitions can be detected experimentally via measurements of the magnetic torque, for which we provide predictions with and without harmonic spin-wave corrections.
In Sec.~\ref{sec:FiniteT}, we cross-check our ans\"atze with classical Monte Carlo simulations and compute the $H$-$T$ phase diagram for all three orthorhombic directions. Here we also highlight the qualitative difference between the zigzag orders for {${\bf H}\!\parallel\!{\bf a}$} and {${\bf H}\!\parallel\!{\bf b}$} versus the spontaneous high-field zigzag order for {${\bf H}\!\parallel\!{\bf c}$}.
A summary and a general discussion is given in Sec.~\ref{sec:Discussion}.
Auxiliary information and technical details are provided in Appendices~\ref{app:SSF}-\ref{app:MC}.
\vspace*{-0.3cm}
\section{Lattice structure, symmetries \& conventions}\label{sec:LatticeSymmetry}
\vspace*{-0.3cm}
$\beta$-Li$_2$IrO$_3$ crystallizes in a hyperhoneycomb structure (shown in Fig.~\ref{fig:lattice}) and has the F$ddd$ space group. Its conventional orthorhombic unit cell is set by the crystallographic axes $\{\hat{{\bf a}}, \hat{{\bf b}},\hat{{\bf c}}\}$, which are related to the Cartesian axes $\{\hat{{\bf x}}, \hat{{\bf y}}, \hat{{\bf z}}\}$ appearing in the spin Hamiltonian below [Eqs.~(\ref{eq:Hamiltonian0})-(\ref{eq:Hamiltonian})] by
\small\begin{equation}\label{eq:xyzframe}
\hat{{\bf x}}=(\hat{{\bf a}}+\hat{{\bf c}})/\sqrt{2}\,,~~~
\hat{{\bf y}}=(\hat{{\bf c}}-\hat{{\bf a}})/\sqrt{2}\,,~~~
\hat{{\bf z}}=-\hat{{\bf b}}\,.
\end{equation}\normalsize
We note here that we stick to the $xyz$-frame convention of Refs.~[\onlinecite{Lee2015,Lee2016,Ducatman2018,Rousochatzakis2018}], which is different from the one used in Ref.~[\onlinecite{Biffin2014a}].
The two $xyz$-frames are related to each other by a two-fold rotation around the ${\bf x}$-axis. This is important as the choice of the frame affects the overall sign structure of the $\Gamma$ interactions.
The orthorhombic unit cell contains four primitive unit cells, of four $\text{Ir}^{4+}$ ions each (labeled by Ir$_1$-Ir$_4$ in Fig.~\ref{fig:lattice}).
The $\text{Ir}^{4+}$ ions form a hyperhoneycomb structure, which can be viewed as a stacking of two types of zigzag chains, which we will denote by $xy$- and $x'y'$-chains. The $xy$-chains run along the direction ${\bf a}\!+\!{\bf b}$ and are shown in Fig.~\ref{fig:lattice} by the alternating red and green solid bonds, denoted by $x$ and $y$ respectively. The $x'y'$-chains run along ${\bf a}\!-\!{\bf b}$ and are shown in Fig.~\ref{fig:lattice} by the alternating red and green dashed bonds, denoted by $x'$ and $y'$ respectively. The two types of chains are interconnected with vertical NN Ir-Ir bonds denoted in Fig.~\ref{fig:lattice} by $z$ (blue solid lines). In total, there are five types of NN Ir-Ir bonds, $x$, $y$, $x'$, $y'$ and $z$.
Apart from translations, the crystal structure is invariant under the following point group operations~\cite{Ruiz2017}: (i) Inversion $\mathcal{I}$ through the center of every $x$- or $y$- or $x'$- or $y'$-type of bond, such as the center of the Ir$_2$-Ir$_4$ bond of Fig.~\ref{fig:lattice}.
(ii) Three $\pi$-rotations in combined spin-orbit space, $C_{2\bf a}$, $C_{2\bf b}$, and $C_{2\bf c}$, around the axes ${\bf a}$, ${\bf b}$ and ${\bf c}$, respectively, passing through the middle of the $z$ bonds, as shown in Fig.~\ref{fig:lattice}. In particular, $C_{2\bf a}$ maps $x$-bonds to $y'$-bonds and $y$-bonds to $x'$-bonds in real space, and $[S_x,S_y,S_z]\rightarrow[-S_y,-S_x,-S_z]$ in spin space.
Similarly, $C_{2\bf b}$ maps $x$-bonds to $x'$-bonds and $y$-bonds to $y'$-bonds in real space, and $[S_x,S_y,S_z]\rightarrow[-S_x,-S_y,S_z]$ in spin space.
Finally, $C_{2\bf c}$ maps $x$-bonds to $y$-bonds and $x'$-bonds to $y'$-bonds in real space, and $[S_x,S_y,S_z]\rightarrow[S_y,S_x,-S_z]$ in spin space.
(iii) Three glide planes which arise by reflections across the $\bf ab$-, $\bf bc$- and $\bf ac$-planes passing through an inversion center, followed by nonprimitive translations by $(\frac{1}{4}\frac{1}{4}0)$, $(0\frac{1}{4}\frac{1}{4})$ and $(\frac{1}{4}0\frac{1}{4})$, in orthorhombic units, respectively.
At this point it is also worth introducing some terminology that we will need later in the analysis of the static structure factors. Following Ref.~[\onlinecite{Biffin2014a}], we define four-component symmetry basis vectors,
\small\begin{equation}\label{eq:symmetrybasis}
A\!=\!\begin{pmatrix} 1\\-1\\-1\\1\end{pmatrix},~~~
C\!=\!\begin{pmatrix} 1\\1\\-1\\-1\end{pmatrix},~~~
F\!=\!\begin{pmatrix}1\\1\\1\\1\end{pmatrix},~~~
G\!=\!\begin{pmatrix}1\\-1\\1\\-1\end{pmatrix}\,.
\end{equation}\normalsize
These vectors represent, respectively, the relative amplitudes of the four sites of the primitive unit cell in the N\'eel (A), stripy (C), ferromagnetic (F) and zigzag (G) order.
Note that, for consistency, our 4-site labeling Ir$_1$-Ir$_4$ of Fig.~\ref{fig:lattice} follows the convention of Fig.~7 of Ref.~[\onlinecite{Biffin2014a}].
For the various components of the static structure factor, we follow the convention of Ref.~[\onlinecite{Ducatman2018}] and denote the modulated components with ${\bf Q}\!=\!\frac{2}{3}\hat{{\bf a}}$ by the letter $M$ and the uniform components with ${\bf Q}\!=\!0$ by $M'$. Therefore, $M_a(A)$ denotes the modulated N\'eel ($A$) component along ${\bf a}$, $M'_b(F)$ denotes the uniform ferromagnetic ($F$) component along ${\bf b}$, and so on. The definitions of these components in terms of the Fourier transform of the spin configuration are given in Appendix~\ref{app:SSFconventions}.
\vspace*{-0.3cm}
\section{The minimal $J$-$K$-$\Gamma$ model} \label{sec:Model}
\vspace*{-0.3cm}
Following earlier works~\cite{Lee2015,Lee2016,Ducatman2018,Rousochatzakis2018}, we consider here the minimal microscopic $J$-$K$-$\Gamma$ model mentioned above, supplemented with a Zeeman term $\mathcal{H}_\text{Z}$ to describe the coupling to the external field ${\bf H}$. The total Hamiltonian then reads
\small\begin{equation}\label{eq:Hamiltonian0}
\mathcal{H}\!=\!\sum_t\sum_{\langle ij\rangle\in t}\mathcal{H}_{ij}^t+\mathcal{H}_\text{Z}\,,
\end{equation}\normalsize
where
\small\begin{equation}\label{eq:Hamiltonian}
\begin{array}{c}
\mathcal{H}_{ij}^t=J\mathbf{S}_i\cdot\mathbf{S}_j+K S_i^{\alpha_t}S_j^{\alpha_t}+\sigma_t\Gamma(S_i^{\beta_t}S_j^{\gamma_t}+S_i^{\gamma_t}S_j^{\beta_t})\,, \\[1ex]
\mathcal{H}_\text{Z}=-\mu_B\mathbf{H}\cdot\sum_{i}\mathbf{g}_i\cdot \mathbf{S}_i \,.
\end{array}
\end{equation}\normalsize
Here ${\bf S}_i$ denotes the pseudo-spin $j_{\text{eff}}\!=\!1/2$ operator at site $i$, $t\in\{x,y,z,x',y'\}$ labels the five different types of NN Ir-Ir bonds and $(\alpha_t,\beta_t,\gamma_t)\!=\!(x,y,z)$, $(y,z,x)$, and $(z,x,y)$ for $t\in\{x,x'\}$, $\{y,y'\}$, and $\{z\}$, respectively. The prefactor $\sigma_t$ equals $+1$ for $t\in\{x,y',z\}$ and $-1$ for $t\in\{y,x'\}$, see $\pm$ symbols in Figs.~\ref{fig:lattice} and \ref{fig:cartoon}. This overall sign structure of the $\Gamma$ interactions derives from the symmetries mentioned above~\cite{Lee2015} and our choice of the $xyz$-frame in Eq.~(\ref{eq:xyzframe}).
Finally, $\mathbf{g}_i$ stands for the ${\bf g}$-tensor of the $i$-th Ir ion. As discussed by Ruiz {\it et al}~\cite{Ruiz2017}, these tensors carry a site-dependent, staggered off-diagonal element $g_{ab}$. Specifically, in the orthorhombic frame,
\small\begin{equation}\label{eq:gtensor}
\mathbf{g}_{i}={\bf g}_{\rm\small diag} + p_i {\bf g}_{\rm\small off-diag} \equiv
\begin{pmatrix}
g_{aa} & 0 & 0\\
0 & g_{bb} & 0\\
0 & 0 & g_{cc}\\
\end{pmatrix}
+p_i
\begin{pmatrix}
0 & g_{ab} & 0\\
g_{ab} & 0 & 0\\
0 & 0 & 0\\
\end{pmatrix},
\end{equation}\normalsize
where $p_i=+1$ for spins on the $xy$ chains and $-1$ for spins on the $x'y'$ chains. Here we take $g_{aa}\!=\!g_{bb}\!=\!g_{cc}\!=\!2$ and $g_{ab}\!=\!0.1$.
We note that, depending on the direction of the field, some of the discrete symmetries mentioned above may or may not be preserved, see Table~\ref{tab:symmetries}. A field along $\bf a$, for example, breaks both $C_{2\bf b}$ and $C_{2\bf c}$, but still respects $C_{2\bf a}$, $\Theta C_{2\bf b}$ and $\Theta C_{2\bf c}$, where $\Theta$ is the time reversal operation.
In the following we restrict ourselves to the so-called $K$-region of the parameter space with dominant Kitaev interaction, which is believed to be relevant for $\beta$-Li$_2$IrO$_3$~\cite{Ducatman2018}, and fix the parameters to the representative set given in Eq.~(\ref{eq:JKGammaPars}).
\begin{figure}
\includegraphics[width=0.7\columnwidth]{Figure2}
\caption{General structure of the two field-induced phases of $\beta$-Li$_2$IrO$_3$ for ${\bf H}$ along ${\bf a}$, ${\bf b}$ or ${\bf c}$. (a) The six-sublattice low-field phase ($0\!\le\!H\!<H^\ast$). (b) The two-sublattice high-field phase ($H^\ast\!\le\!H\!<H^{\ast\ast}$), where $H_{{\bf a},{\bf b}}^{\ast\ast}\!=\!\infty$ and $H_{\bf c}^{\ast\ast}$ is finite. The Cartesian components of the various sublattices for each field direction are given in Table~\ref{tab:ansatze}.}
\label{fig:cartoon}
\end{figure}
\vspace*{-0.3cm}
\section{Unified description of $\beta$-Li$_2$IrO$_3$ for ${\bf H}$ along ${{\bf a}}$, ${{\bf b}}$ and ${{\bf c}}$ axes}\label{sec:ansatze}
\vspace*{-0.3cm}
\subsection{General spin sublattice structure}\label{sec:SpinSublattices}
\vspace*{-0.3cm}
The behavior of $\beta$-Li$_2$IrO$_3$ under a magnetic field along the three orthorhombic directions can be described in a unified manner as shown in Fig.~\ref{fig:cartoon}. For all three directions, ${\bf a}$, ${\bf b}$ and ${\bf c}$, the system goes through a low-field phase ($0\!\le\!H\!<\!H^\ast$) with six spin sublattices [${\bf A}$, ${\bf B}$ and ${\bf C}$ along the $xy$-chains, and ${\bf A}'$, ${\bf B}'$ and ${\bf C}'$ along the $x'y'$-chains, see Fig.~\ref{fig:cartoon}\,(a)], followed by a high-field canted phase ($H^\ast\!<\!H\!<\!H^{\ast\ast}$) with two spin sublattices [${\bf F}$ along the $xy$-chains and ${\bf F}'$ along the $x'y'$ chains, see Fig.~\ref{fig:cartoon}\,(b)].
The high-field phase terminates at $H^{\ast\ast}\!=\!\infty$ for {${\bf H}\!\parallel\!{\bf a}$} and {${\bf H}\!\parallel\!{\bf b}$} (with a small zigzag component remaining if $g_{ab}\!\neq\!0$, see Appendix~\ref{app:ansatze}), whereas $H_{\bf c}^{\ast\ast}$ is finite and the classical state reached at $H_{\bf c}^{\ast\ast}$ is the fully polarized state.
Figure~\ref{fig:spinConfig} shows a series of representative snapshots, of various ground state configurations for different field directions and strengths (obtained from numerical minimization of the classical ans\"atze discussed below). As discussed in Ref.~[\onlinecite{Ducatman2018}], in the zero-field state, the three sublattices ${\bf A}$, ${\bf B}$ and ${\bf C}$ along the $xy$ chains form a nearly coplanar 120$^\circ$ state, and the three sublattices ${\bf A}'$, ${\bf B}'$ and ${\bf C}'$ along the $x'y'$ chains form another such nearly 120$^\circ$ structure, on a different plane, see dotted blue triangles at the top left panel of Fig.~\ref{fig:spinConfig}. Under a magnetic field, the three sublattices of each given chain cant toward each other and eventually get aligned at the characteristic field $H^\ast$ where ${\bf A}\!=\!{\bf B}\!=\!{\bf C}\!\equiv\!{\bf F}$ and ${\bf A}'\!=\!{\bf B}'\!=\!{\bf C}'\!\equiv\!{\bf F}'$. For fields along ${\bf a}$ and ${\bf b}$, this intra-chain alignment happens continuously, whereas for fields along ${\bf c}$ it happens abruptly. Above $H^\ast$, ${\bf F}$ and ${\bf F}'$ cant toward the field in a non-uniform way and at a pace that is strongly dependent on the field direction.
\vspace*{-0.3cm}
\subsection{Basic characterization of the low-field phase ($H\!<\!H^\ast$)}\label{sec:parametrization1}
\vspace*{-0.3cm}
The individual Cartesian spin components of the various configurations are related to each other in a specific way, see the parametrization in Table~\ref{tab:ansatze}. For each given spin sublattice, a spin length constraint must be imposed, for example $x_1^2+y_1^2+z_1^2\!=\!1$ for the ${\bf A}$ sublattice, $2x_3^2+z_3^2\!=\!1$ for the ${\bf C}$ sublattice of the {${\bf H}\!\parallel\!{\bf a}$} case, etc.
The field dependence of the Cartesian components can be obtained by a numerical minimization of the total energy of the system [see Eqs.~(\ref{eq:energy_b1}), (\ref{eq:energy_a1}) and (\ref{eq:energy_c1})], and the results are shown in Figs.~\ref{fig:AnsatzResults}\,(a-c) as a function of the field.
Equivalently, the spin configurations can be described in terms of the associated symmetry-resolved static structure factors, and the same is true for the total energy [see Eqs.~(\ref{eq:energy_b2}), (\ref{eq:energy_a2}) and (\ref{eq:energy_c2})]. The structure factors obey the same number of constraints as the Cartesian components [the relations between the two are given in Eqs.~(\ref{eq:SFvsCartesianb}), (\ref{eq:SFvsCartesiana}) and (\ref{eq:SFvsCartesianc})], and their evolution with field are shown in Figs.~\ref{fig:AnsatzResults}\,(d-f) and \ref{fig:SmallComponents}.
\begin{figure}[!t]
{\includegraphics[width=\columnwidth]{Figure3}}
\caption{Snapshots of representative spin configurations for ${\bf H}$ along $\bf a$ (first row), $\bf b$ (second row), or $\bf c$ (third row). Each color represents one of the six sublattices of the zero-field state. The dashed lines in the upper left panel depict the nearly coplanar, 120$^\circ$ order of the ${\bf A B C}$ and ${\bf A}'{\bf B}'{\bf C}'$ sublattices~\cite{Ducatman2018}.}\label{fig:spinConfig}
\end{figure}
\begin{table}[!b]
\renewcommand{\arraystretch}{1.3}
\centering
\begin{tabular}{L{0.5cm} L{0.5cm} C{2.2cm} C{2.2cm} C{2.2cm}}
\toprule[1.pt]
&& {${\bf H}\!\parallel\!{\bf a}$} & {${\bf H}\!\parallel\!{\bf b}$} & {${\bf H}\!\parallel\!{\bf c}$} \\
\midrule[1.pt]
&${\bf A}$ & $S [x_1, y_1, z_1]$ & $S [x_1, y_1, z_1]$ & $S [x_1, y_1, z_1]$ \\
&${\bf A}'$ & $S [y_2, x_2, z_2]$ & $S [y_1, x_1, z_1]$ & $S [y_1, x_1, z_1]$ \\
&${\bf B}$ & $S [-y_1, -x_1, z_1]$ & $S [-y_1, -x_1, z_1]$ & $S [-y_2, -x_2, z_2]$ \\
&${\bf B}'$ & $S [-x_2, -y_2, z_2]$ & $S [-x_1, -y_1, z_1]$ & $S [-x_2, -y_2, z_2]$ \\
\rot{\rlap{~~$0\!\le\!H\!<\!H^\ast$}}&${\bf C}$ & $S [-x_3, x_3, -z_3]$ & $S [-x_2, x_2, -z_2]$ & $S [-y_3, x_3, -z_3]$ \\
&${\bf C}'$ & $S [x_4, -x_4, -z_4]$ & $S [x_2, -x_2, -z_2]$ & $S [x_3, -y_3, -z_3]$ \\
\midrule[1.pt]
&${\bf F}$ & $S [x_1, -x_1, z_1]$ & $S [x_1, -x_1, z_1]$ & $S [x_1, y_1, z_1]$ \\
\rot{\rlap{\!$ H\ge\!H^{\ast}$}}&${\bf F}'$ & $S [x_1, -x_1, -z_1]$ & $S [-x_1, x_1, z_1]$ & $S [y_1, x_1, z_1]$\\
\bottomrule[1.pt]
\end{tabular}
\setlength{\tabcolsep}{3em}
\caption{Cartesian components of the spin sublattices of the three ans\"atze analyzed here (see Appendix~\ref{app:ansatze} for more details). There are six spin sublattices for $0\!\le\!H\!<\!H^\ast$ (${\bf A}$, ${\bf B}$ and ${\bf C}$ along the $xy$-chains and ${\bf A}'$, ${\bf B}'$ and ${\bf C}'$ along the $x'y'$-chains), and two sublattices for $H\!>\!H^\ast$ (${\bf F}$ along the $xy$-chains and ${\bf F}'$ along the $x'y'$-chains), see Fig.~\ref{fig:cartoon}.}
\label{tab:ansatze}
\renewcommand{\arraystretch}{1}
\end{table}
\begin{figure*}[!t]
{\includegraphics[width=\textwidth]{Figure4}}
\caption{Field evolution of Cartesian spin components (a-c), dominant symmetry-resolved static structure factors (d-f) [see Fig.~\ref{fig:SmallComponents} for the remaining much weaker components], and Bragg peak intensities $I_I$, $I_{V}$ and $I_{\text{tot}}$ (g-i). The insets in (b) and (e) show the high-field behavior.}\label{fig:AnsatzResults}
\end{figure*}
The low-field phase for {${\bf H}\!\parallel\!{\bf b}$} is described by five Cartesian components ($x_1$, $y_1$, $z_1$, $x_2$ and $z_2$) or, equivalently, by five structure factors~\cite{Rousochatzakis2018}: Three modulated ${\bf Q}\!=\!\frac{2}{3}\hat{{\bf a}}$ components $M_a(A)$, $M_b(C)$ and $M_c(F)$, and two uniform ${\bf Q}\!=\!0$ components $M'_a(G)$ and $M'_b(F)$.
This precise combination is in fact present for all three orthorhombic directions for $0\!\le\!H\!<\!H^\ast$, as it is a property of the zero-field state.
In particular, the uniform components $M_a'(G)$ and $M_b'(F)$ of the zero-field order reflect the deviation from the perfect 120$^\circ$ coplanar order mentioned above, see detailed analysis in Ref.~[\onlinecite{Ducatman2018}].
Note further that the modulated components $M_a(A)$, $M_b(C)$ and $M_c(F)$ belong to the $\Gamma_4$ irreducible representation, in agreement with experiment~\cite{Biffin2014a}.
The five structure factors satisfy two constrains which, when normalized appropriately [see Appendix~\ref{app:SSFconventions}], can be combined to give the Bragg peak intensity sum rule observed experimentally~\cite{Ruiz2017}. Namely,
\small\begin{equation}\label{eq:ISR0}
\!\!I_{\text{tot}} = 2 I_I + I_V = S^2\,,
\end{equation}\normalsize
where
\small\begin{equation}\label{eq:ISRb}
\renewcommand{\arraystretch}{1.2}
\begin{array}{c}
\!I_I=\!\!|M_a(A)|^2\!+\!|M_b(C)|^2\!+\!|M_c(F)|^2\!\equiv\!I_{I,\Gamma_4}\,,\\
\!I_V=|M'_a(G)|^2\!+\!|M'_b(F)|^2\,.
\end{array}
\end{equation}\normalsize
\begin{figure}[!b]
\includegraphics[width=0.7\columnwidth]{Figure5}
\caption{Field dependence of the structure factors $M_a(C)$, $M_b(A)$, $M_c(G)$, $M'_a(G)$ and $M'_b(F)$ generated for ${\bf H}$ along ${\bf a}$ (a), and $M_a(G)$, $M_b(F)$ and $M_c(C)$ generated along ${\bf c}$ (b).}\label{fig:SmallComponents}
\end{figure}
Turning to the low-field phase for {${\bf H}\!\parallel\!{\bf a}$}, here we have ten Cartesian components (see Table~\ref{tab:ansatze}) or, equivalently, ten structure factors: six modulated components (the three zero-field components plus three induced by the field) and four uniform components (the two zero-field components plus two induced by the field):
\small\begin{equation}
\renewcommand{\arraystretch}{1.2}
\!\!\!\!\begin{array}{ll}
H\!\geq\!0: & M_a(A),~M_b(C),~M_c(F)~\text{and}~M'_a(G),~M'_b(F)\,,\\
H\!>\!0: & M_a(C),~M_b(A),~M_c(G)~\text{and}~M'_a(F),~M'_b(G)\,.
\end{array}
\end{equation}\normalsize
Hence, a field along ${\bf a}$ induces a finite FM component $M_a'(F)$ (which couples explicitly to the Zeeman field), and a finite zigzag component $M_b'(G)$ along ${\bf b}$. The latter can become relatively large with field [see Fig.~\ref{fig:AnsatzResults}\,(d)] and should be observable experimentally, unlike the components $M'_a(G)$ and $M'_b(F)$ which remain at least one order of magnitude smaller, see Fig.~\ref{fig:SmallComponents}\,(a).
The same is true for the field-induced modulated components $M_a(C)$, $M_b(A)$ and $M_c(G)$, which belong to the irreducible representation $\Gamma_2$ (see Table II of Ref.~[\onlinecite{Biffin2014a}]).
Altogether, the ten structure factors satisfy four constraints, and one combination of them gives the Bragg peak intensity sum rule of Eq.~(\ref{eq:ISR0}), where now
\small\begin{equation}\label{eq:ISRa}
\renewcommand{\arraystretch}{1.2}
\!\!\!\!\begin{array}{c}
I_{I}=I_{I,\Gamma_4}\!+\!I_{I,\Gamma_2} \simeq I_{I,\Gamma_4}\,,\\
I_{I,\Gamma_2}=|M_a(C)|^2\!+\!|M_b(A)|^2\!+\!|M_c(G)|^2 \ll I_{I,\Gamma_4}\,,\\
I_V=|M'_a(G)|^2\!+\!|M'_b(F)|^2\!+\!|M'_a(F)|^2\!+\!|M'_b(G)|^2\,.
\end{array}
\end{equation}\normalsize
The low-field phase for {${\bf H}\!\parallel\!{\bf c}$} is described by nine Cartesian components (see Table~\ref{tab:ansatze}) or by nine structure factors:
\small\begin{equation}
\renewcommand{\arraystretch}{1.2}
\!\!\!\!\!\begin{array}{ll}
H\!\geq\!0: & M_a(A),~M_b(C),~M_c(F)~\text{and}~M'_a(G),~M'_b(F)\,,\\
H\!>\!0: & M_a(G),~M_b(F),~M_c(C)~\text{and}~M'_c(F)\,.
\end{array}
\end{equation}\normalsize
Here the field induces three modulated components ($M_a(G)$, $M_b(F)$ and $M_c(C)$) and one uniform component $M'_c(F)$ (which couples directly to the Zeeman field). The modulated components belong to the irreducible representation $\Gamma_3$ (see Table II of Ref.~[\onlinecite{Biffin2014a}]), and, as it turns out, they remain at least one order of magnitude smaller than the dominant $\Gamma_4$ components, see Fig.~\ref{fig:SmallComponents}\,(b).
Altogether, the nine structure factors satisfy three constraints, and one combination of them gives the Bragg peak intensity sum rule of Eq.~(\ref{eq:ISR0}), where now
\small\begin{equation}\label{eq:ISRc}
\renewcommand{\arraystretch}{1.2}
\!\!\!\!\begin{array}{c}
I_{I}=I_{I,\Gamma_4}\!+\!I_{I,\Gamma_3} \simeq I_{I,\Gamma_4}\,, \\
I_{I,\Gamma_3}=|M_a(G)|^2\!+\!|M_b(F)|^2\!+\!|M_c(C)|^2 \ll I_{I,\Gamma_4}\,,\\
I_V=|M'_a(G)|^2\!+\!|M'_b(F)|^2\!+\!|M'_c(F)|^2\,.
\end{array}
\end{equation}\normalsize
Let us emphasize that the fulfilment of the intensity sum rule Eq.~(\ref{eq:ISR0}) for all field directions and strengths is a direct fingerprint of the local spin length constraints. The numerical prefactor of $2$ in the definition $I_{\text{tot}}\!=\!2I_I\!+\!I_V$ reflects the fact that there are twice as many Bragg peaks characterizing the modulated order (${\bf Q}\!=\!\pm\frac{2}{3}\hat{{\bf a}}$) compared to the peaks characterizing the uniform order (${\bf Q}\!=\!0$), see detailed analysis and a general proof of Eq.~(\ref{eq:ISR0}) in Appendix~\ref{app:Constraints}.
Note finally that some of the uniform components generated for ${\bf H}$ along ${\bf a}$ and ${\bf c}$ give rise to a finite magnetic torque signal, which will be examined separately in Sec.~\ref{sec:Torque}.
\begin{figure*}[!t]
{\includegraphics[width=0.85\linewidth]{Figure6}}
\caption{(a-c) Evolution of the various contributions to the energy ($E_J$, $E_K$, $E_\Gamma$ and $E_H$ denote the contributions from $J$, $K$, $\Gamma$ and the Zeeman field, respectively) with $H$, for ${\bf H}$ along ${\bf a}$, ${\bf b}$ and ${\bf c}$. (d-f): Evolution of the first derivates of $E_J$, $E_K$, $E_\Gamma$ and $E_H$ with respect to the field $H$. All energies are given in meV.}\label{fig:Energy}
\end{figure*}
\vspace*{-0.3cm}
\subsection{Basic characterization of the high-field phase ($H^{\ast}\!<\!H\!<\!H^{\ast\ast}$)}\label{sec:parametrization2}
\vspace*{-0.3cm}
For $H\!>\!H^\ast$, all modulated components vanish identically, and we are left with uniform structure factors only. In particular, for ${\bf H}$ along ${\bf a}$ and ${\bf b}$, there are only two uniform components, a FM component along the field and a zigzag component perpendicular to the field. For {${\bf H}\!\parallel\!{\bf c}$}, there is an additional FM component $M'_b(F)$ perpendicular to the field. In terms of the two spin sublattices ${\bf F}$ and ${\bf F}'$ of Fig.~\ref{fig:cartoon}\,(b), the FM component is proportional to ${\bf F}+{\bf F}'$ and the zigzag component is proportional to ${\bf F}-{\bf F}'$.
The direction of the zigzag component depends on the direction of the field. When {${\bf H}\!\parallel\!{\bf a}$}, ${\bf F}-{\bf F}' \!=\! 2S z_1 \hat{{\bf z}}$, see Table~\ref{tab:ansatze}, and therefore the zigzag component is fixed along ${\bf b}$.
By contrast, the zigzag component is fixed along ${\bf a}$ when ${\bf H}$ points along ${\bf b}$ or ${\bf c}$, with ${\bf F}\!-\!{\bf F}'\!=\!2\sqrt{2}S x_1\hat{{\bf a}}$ and ${\bf F}\!-\!{\bf F}'\!=\!\sqrt{2}S(x_1\!-\!y_1)\hat{\bf a}$, respectively, see Table~\ref{tab:ansatze}. Note also that, for $H\!\ge\!H^\ast$, the spins lie on the $ab$-plane for {${\bf H}\!\parallel\!{\bf a}$} and {${\bf H}\!\parallel\!{\bf b}$}, but for {${\bf H}\!\parallel\!{\bf c}$} the spin plane changes continuously. This is related to the fact that the uniform components of the zero-field state all lie in the $ab$-plane, and so a field applied in this plane will merely reorganize these components and not rotate them out of the plane, unlike what happens for {${\bf H}\!\parallel\!{\bf c}$}.
The zigzag component disappears at a characteristic field $H^{\ast\ast}$. As mentioned above, $H^{\ast\ast}$ is infinite for ${\bf H}$ along ${\bf a}$ and ${\bf b}$ but finite for {${\bf H}\!\parallel\!{\bf c}$}, with (see Appendix~\ref{app:Halongc})
\small\begin{equation}\label{eq:Hcstarstar}
\mu_B H_{\bf c}^{\ast\ast}
=\Big( \Gamma+2J+\sqrt{(\Gamma-2J)^2+8\Gamma^2} ~\Big)\frac{S}{2g_{cc}} \,,
\end{equation}\normalsize
which, for $J\!\ll\!|\Gamma|$, reduces to
\small\begin{equation}\label{eq:Hcstarstar2}
\mu_B H_{\bf c}^{\ast\ast}\!\simeq\!\Big(\frac{4}{3}J\!+\!|\Gamma| \Big)\frac{S}{g_{cc}} \,.
\end{equation}\normalsize
According to this relation, $H_{\bf c}^{\ast\ast}$ depends mostly on $\Gamma$, with $H_{\bf c}^{\ast\ast}\!\sim\!45$~T for the coupling parameters of Eq.~(\ref{eq:JKGammaPars}).
\vspace*{-0.3cm}
\subsection{Robustness of high-field zigzag orders}\label{sec:Robustness}
\vspace*{-0.3cm}
We now discuss why the various high-field zigzag orders remain robust up to very high fields, for all three orthorhombic directions. The most direct way to see this is to express the total energies $E_{\bf b}$, $E_{\bf a}$ and $E_{\bf c}$ in terms of the various static structure factors, see Eqs.~(\ref{eq:energy_b3}), (\ref{eq:energy_a3}) and (\ref{eq:energy_c3}), respectively. It turns out that $E_{\bf b}$ and $E_{\bf c}$ contain an explicit cross-coupling term between $M'_a(G)$ and $M'_b(F)$,
\small\begin{equation}\label{eq:cross-coupling1}
-\sqrt{2}\Gamma M'_a(G) M'_b(F)\,,
\end{equation}\normalsize
while $E_{\bf a}$ contains an explicit cross-coupling term between $M'_a(F)$ and $M'_b(G)$,
\small\begin{equation}\label{eq:cross-coupling2}
-\sqrt{2}\Gamma M'_a(F) M'_b(G)\,.
\end{equation}\normalsize
The presence of these terms reveal that the qualitative reason why it is energetically favorable for the system to sustain appreciable zigzag orders up to high fields is the strong $\Gamma$ interaction.
Of course, the actual quantitative details for each field direction derive from the minimization of the total energies under the given constraints. For example, the analytical expression Eq.~(\ref{eq:Hcstarstar}) for $H_{\bf c}^{\ast\ast}$ can be derived by minimizing $E_{\bf c}$ in Eq.~(\ref{eq:energy_c3}) under the single constraint $|M'_a(G)|^2\!+\!|M'_b(F)|^2\!+\!|M'_c(F)|^2\!=\!S^2$.
\vspace*{-0.3cm}
\subsection{Dependence of $H^\ast$ on microscopic coupling parameters}\label{sec:Hstar}
\vspace*{-0.3cm}
The characteristic field $H^\ast$ marks the disappearance of the modulated components (and $M_a'(G)$ and $M_b'(F)$ for {${\bf H}\!\parallel\!{\bf a}$}). As mentioned earlier, this transition is continuous for {${\bf H}\!\parallel\!{\bf a}$} and {${\bf H}\!\parallel\!{\bf b}$}, but of first-order for {${\bf H}\!\parallel\!{\bf c}$}, see Figs.~\ref{fig:AnsatzResults}\,(a- f).
Furthermore, the value of $H^\ast$ depends strongly on the direction of the field. For the coupling parameters of Eq.~(\ref{eq:JKGammaPars}), $H_{\bf a}^\ast\!\sim\!102$~T, $H_{\bf b}^\ast\sim2.88$~T and $H_{\bf c}^\ast\!\sim\!13$~T. This large difference between the critical fields along different directions is related to the strongly anisotropic character of the Hamiltonian, and the different role of the various couplings in each case. For example, as we discussed in Ref.~[\onlinecite{Rousochatzakis2018}], in the parameter regime of interest, $H_{\bf b}^\ast$ depends only on $J$, which is why $H_{\bf b}^\ast$ is very small.
We will now show that $H_{\bf a}^\ast$ and $H_{\bf c}^\ast$ do not depend on $K$ but only on $J$ and $\Gamma$, and that the inequality $J\ll |\Gamma|$ explains why these critical fields are larger compared to $H_{\bf b}^\ast$.
To this end, we will vary the parameters of the model and take a closer look at the evolution of the various contributions to the total energy with the field. Figure~\ref{fig:Energy}\,(a-c) shows the field-driven evolution of $E_J$, $E_K$, $E_\Gamma$ and $E_\text{Z}$, which denote the contributions from $J$, $K$ and $\Gamma$ interactions and the Zeeman energy, respectively.
The corresponding derivatives of these energies with respect to $H$ are shown in Fig.~\ref{fig:Energy}\,(d-f). The main finding is that, in the parameter regime of interest, $E_K$ remains almost insensitive to $H$, and this is true for all field directions. This means that the Zeeman field does not act against $K$, which explains why none of the critical fields $H^\ast$ depends on the dominant coupling of the theory.
The results also show that, unlike $H_{\bf a}^\ast$ and $H_{\bf c}^\ast$, the critical field $H_{\bf b}^\ast$ depends only on $J$ and not on $\Gamma$; this is the consequence of the fact that $E_\Gamma$ does not change with $H$ in this direction.
These arguments can be formulated mathematically by the following relations that arise from a classical version of Feynman-Hellmann theorem (see Appendix~\ref{app:FeynmanHellmann}):
\small\begin{equation}\label{eq:FeynmanHellmann}
\begin{array}{c}
\mathcal{N} \frac{\partial}{\partial J} m_\parallel(J,K,\Gamma,H) = -\frac{\partial}{\partial H} E_J(J,K,\Gamma,H)/J \,, \\[1.5ex]
\mathcal{N} \frac{\partial}{\partial K} m_\parallel(J,K,\Gamma,H) = -\frac{\partial}{\partial H} E_K(J,K,\Gamma,H)/K \,, \\[1.5ex]
\mathcal{N} \frac{\partial}{\partial J} m_\parallel(J,K,\Gamma,H) = - \frac{\partial}{\partial H} E_\Gamma(J,K,\Gamma,H)/K \,,
\end{array}
\end{equation}\normalsize
where $\mathcal{N} $ is the total number of spins and $m_\parallel(J,K,\Gamma,H)$ is the magnetization per site along the field. According to these relations, the fact that $\partial E_K/\partial H\!\approx\!0$ implies that $\partial m_\parallel/\partial K\!\approx\!0$, i.e., that the whole magnetization process does not depend on $K$. Likewise, the fact that $\partial E_\Gamma/\partial H\!\approx\!0$ for {${\bf H}\!\parallel\!{\bf b}$} implies that $\partial m_\parallel/\partial \Gamma\!\approx\!0$, and therefore the whole magnetization process depends only on $J$ in this field direction.
\begin{figure*}[!t]
{\includegraphics[width=0.95\textwidth]{Figure7}}
\caption{Variation of critical fields $H^\ast$ in the $J$-$\Gamma$ plane around the relevant parameter regime for $\beta$-Li$_2$IrO$_3$.}\label{fig:Hstar}
\end{figure*}
We can go one step further and extract the actual dependence of the critical fields on the relevant couplings by computing these fields for a wider range of parameters. The results are shown in Fig.~\ref{fig:Hstar} and demonstrate that the critical fields $H_{\bf a}^\ast$ and $H_{\bf c}^\ast$ depend almost perfectly linearly on $J$ and $\Gamma$. Fitting the numerical data for $H^\ast$ gives, in particular,
\small\begin{equation}\label{eq:Hacstar}
\begin{array}{l}
\mu_B H_{\bf a}^\ast \simeq \Big(0.54 J + 0.57 |\Gamma| \Big) \frac{4S}{g_{aa}},\\[3pt]
\mu_B H_{\bf b}^\ast \simeq 0.42 J ~\left(\frac{4S}{g_{bb}}\right)\,,\\[3pt]
\mu_B H_{\bf c}^\ast \simeq \Big(0.94 J + 0.04 |\Gamma| \Big) \frac{4S}{g_{cc}}\,.
\end{array}
\end{equation}\normalsize
Thus, besides the independence of $H_{\bf b}^\ast$ on $K$ and $\Gamma$,~\footnote{Note that the coefficient $0.42$ in the second relation is slightly different from the corresponding coefficient $0.46$ reported previously~\cite{Rousochatzakis2018}, which was obtained from fitting the numerical data up to larger values of $J$, where the assumption of $H_{\bf b}^\ast$ being independent on $K$ and $\Gamma$ breaks down.}, we find that $H_{\bf a}^\ast$ is controlled mainly by $\Gamma$ (given that $J\!\ll\!|\Gamma|$), whereas $H_{\bf c}^\ast$ is controlled by both $J$ and $\Gamma$. Note that the coefficients appearing in Eqs.~(\ref{eq:Hacstar}) correspond to the value $g_{ab}\!=\!0.1$ whose sign and magnitude is chosen arbitrarily here. However, the coefficients do not depend much on this choice. For example, for $g_{ab}\!=\!0$ we get $\mu_B H_{\bf a}^\ast\!\simeq\!\Big(0.54 J \!+\! 0.59 |\Gamma| \Big) \frac{4S}{g_{aa}}$, $\mu_B H_{\bf b}^\ast\!\simeq\!0.45 J ~\left(\frac{4S}{g_{bb}}\right)$, while $H_{\bf c}^\ast$ remains unchanged.
\vspace*{-0.3cm}
\subsection{Symmetries}\label{sec:Symmetries}
\vspace*{-0.3cm}
Table~\ref{tab:symmetries} shows the symmetry properties of the various field-induced configurations for different field directions. The primitive translations (denoted by $\mathcal{T}$) are broken spontaneously in the low-field phase ($0\!<\!H\!<H^\ast$) due to the modulating components of the order. This symmetry is restored above $H^\ast$ with the disappearance of these components. Furthermore, the low-field phases preserve the inversion symmetries $\mathcal{I}$ around the centers of the FM dimers ${\bf A}{\bf A}$, ${\bf B}{\bf B}$, ${\bf A}'{\bf A}'$ or ${\bf B}'{\bf B}'$ of Fig.~\ref{fig:cartoon}\,(a), while the high-field phases preserve the inversion centers on all $x$, $y$, $x'$ and $y'$ bonds.
\begin{table*}[!t]
\renewcommand{\arraystretch}{1.3}
\centering
\begin{tabular}{L{2.7cm} C{0.8cm}C{0.8cm}C{0.8cm}C{0.8cm}C{0.8cm} C{0.8cm}C{0.8cm}C{0.8cm}C{0.8cm}C{0.8cm} C{0.8cm}C{0.8cm}C{0.8cm}C{0.8cm}C{0.8cm}}
\toprule[1.pt]
\multicolumn{1}{l}{field direction}
&
\multicolumn{5}{c}{{${\bf H}\!\parallel\!{\bf a}$}} &
\multicolumn{5}{c}{{${\bf H}\!\parallel\!{\bf b}$}} &
\multicolumn{5}{c}{{${\bf H}\!\parallel\!{\bf c}$}} \\
\cmidrule(lr){2-6}
\cmidrule(lr){7-11}
\cmidrule(lr){12-16}
\multicolumn{1}{l}{Hamiltonian $\mathcal{H}$} &
\multicolumn{1}{c}{$\mathcal{T}$} &
\multicolumn{1}{c}{$\mathcal{I}$} &
\multicolumn{1}{c}{$C_{2\bf a}$} &
\multicolumn{1}{c}{$\Theta C_{2\bf b}$}&
\multicolumn{1}{c}{$\Theta C_{2\bf c}$}&
\multicolumn{1}{c}{$\mathcal{T}$} &
\multicolumn{1}{c}{$\mathcal{I}$} &
\multicolumn{1}{c}{$\Theta C_{2\bf a}$} &
\multicolumn{1}{c}{$C_{2\bf b}$}&
\multicolumn{1}{c}{$\Theta C_{2\bf c}$}&
\multicolumn{1}{c}{$\mathcal{T}$} &
\multicolumn{1}{c}{$\mathcal{I}$} &
\multicolumn{1}{c}{$\Theta C_{2\bf a}$} &
\multicolumn{1}{c}{$\Theta C_{2\bf b}$}&
\multicolumn{1}{c}{$C_{2\bf c}$}\\
\midrule[1.pt]
state at $0\!<\!H\!<\!H^\ast$ & ${\color{red}\times}$& $\surd$ & ${\color{red}\times}$&${\color{red}\times}$&$\surd$ & ${\color{red}\times}$& $\surd$ & $\surd$ & $\surd$ & $\surd$ &${\color{red}\times}$& $\surd$ & $\surd$&${\color{red}\times}$&${\color{red}\times}$ \\
state at $H^\ast\!<\!H\!<\!H^{\ast\ast}$ & $\surd$&$\surd$& $\surd$ &$\surd$ & $\surd$ & $\surd$ & $\surd$ & $\surd$ & $\surd$ & $\surd$& $\surd$ & $\surd$& $\surd$ &${\color{red}\times}$&${\color{red}\times}$\\
state at $H\!>\!H^{\ast\ast}$ & $\surd$&$\surd$& $\surd$ &$\surd$ & $\surd$ & $\surd$ & $\surd$ & $\surd$ & $\surd$ & $\surd$& $\surd$ & $\surd$& $\surd$ &$\surd$&$\surd$\\
\bottomrule[1.pt]
\end{tabular}
\setlength{\tabcolsep}{3em}
\caption{Discrete symmetries of the Hamiltonian (see Sec.~\ref{sec:LatticeSymmetry}) and the various states discussed in the text. $\mathcal{T}$ denotes the primitive translations of the crystal, $\Theta$ is time reversal, and $\mathcal{I}$ denotes the inversion centers of the ferromagnetic dimers for $0\!<\!H\!<H^\ast$, or any inversion center of the structure for $H\!>\!H^\ast$. Note that $H_{\bf a}^{\ast\ast}\!=\!\infty$ and $H_{\bf b}^{\ast\ast}\!=\!\infty$, whereas $H_{\bf c}^{\ast\ast}$ is finite.}
\label{tab:symmetries}
\renewcommand{\arraystretch}{1}
\end{table*}
Let us now turn to the $C_2$-rotation symmetries discussed in Sec.~\ref{sec:LatticeSymmetry} or their combinations with time reversal $\Theta$. For {${\bf H}\!\parallel\!{\bf b}$}, the symmetries $\Theta C_{2{\bf a}}$, $C_{2{\bf b}}$ and $\Theta C_{2{\bf c}}$ of the model are all preserved in both the low- and the high-field phases, emphasizing once again the special role of the ${\bf b}$ axis~\cite{Biffin2014a, Ruiz2017,Rousochatzakis2018}.
For {${\bf H}\!\parallel\!{\bf a}$}, on the other hand, among the three symmetries $C_{2{\bf a}}$, $\Theta C_{2{\bf b}}$ and $\Theta C_{2{\bf c}}$, the first two are broken spontaneously in the low-field phase due to $M'_a(G)$ and $M'_b(F)$. This symmetry breaking is associated with the choice of the overall sign of $M'_a(G)$ and $M'_b(F)$. One can see this more directly from the cross-coupling term of Eq.~(\ref{eq:energy_a2}), according to which the relative signs of $M'_a(G)$ and $M'_b(F)$ are fixed by the sign of $\Gamma$, but one can still change both signs at the same time without changing the energy.
Note that, while a similar cross-coupling term appears between $M'_a(F)$ and $M'_b(G)$, see Eq.~(\ref{eq:cross-coupling2}), the individual signs of these two components are fixed by the Zeeman field which couples directly to $M'_a(F)$, see Appendix~\ref{app:Halonga}.
The symmetries $C_{2{\bf a}}$ and $\Theta C_{2{\bf b}}$ are restored at $H_{\bf a}^\ast$ with the disappearance of the $M'_a(G)$ and $M'_b(F)$.
The situation for {${\bf H}\!\parallel\!{\bf c}$} has one qualitative difference (besides the abrupt transition at $H_{\bf c}^\ast$). Here, among the three symmetries $\Theta C_{2\bf a}$, $\Theta C_{2\bf b}$ and $C_{2\bf c}$ of the model, the last two are broken spontaneously in both the low- and the high-field phase, and only get restored at $H\!\geq\!H_{\bf c}^{\ast\ast}$.
The symmetry breaking occurs again due to $M'_a(G)$ and $M'_b(F)$, which couple via Eq.~(\ref{eq:cross-coupling1}). As above then, $\Gamma$ fixes the relative signs of $M'_a(G)$ and $M'_b(F)$, but the overall choice of the global sign remains arbitrary.
Altogether, unlike what happens along ${\bf a}$, the transition at $H_{\bf c}^\ast$ does not restore all broken symmetries, and one thus expects a second thermal phase transition at high fields, even after the disappearance of the modulated order. This will be shown explicitly in Sec.~\ref{sec:FiniteT}.
For completeness, let us recall that the zero-field state breaks $C_{2{\bf a}}$ and $C_{2{\bf c}}$, but respects $\Theta C_{2{\bf a}}$, $\Theta C_{2{\bf c}}$ and $C_{2{\bf b}}$~\cite{Ducatman2018}.
\vspace*{-0.3cm}
\section{Magnetization process \& the effect of quantum fluctuations}\label{sec:QFs}
\vspace*{-0.3cm}
We now focus on the magnetization per site ${\bf m}$, defined as
\small\begin{equation}\label{eq:magn}
{\bf m} = \frac{1}{\mathcal{N}_{\text{m}}} \mu_B \left( {\bf g}_{\rm\small diag} \cdot \sum_\mu \langle {\bf S}_{\mu} \rangle + {\bf g}_{\rm\small off-diag} \cdot \sum_\mu p_\mu \langle {\bf S}_{\mu} \rangle \right)\,.
\end{equation}\normalsize
Here $\mathcal{N}_{\text{m}}$ is the number of spins inside the magnetic unit cell ($\mathcal{N}_{\text{m}}\!=\!48$ for $H\!<\!H^\ast$ and $\mathcal{N}_{\text{m}}\!=\!2$ for $H\!>\!H^\ast$), $\mu\!=\!1$-$\mathcal{N}_{\text{m}}$, $\langle{\bf S}_{\mu}\rangle$ is the expectation value of the spin on the $\mu$-th sublattice, and ${\bf g}_{\rm\small diag}$, ${\bf g}_{\rm\small off-diag}$ and $p_\mu$ are defined in Eq.~(\ref{eq:gtensor}).
Recalling that $p_\mu\!=\!+1$ for spins along the $xy$ chains and $-1$ for spins along the $x'y'$ chains [see Fig.~\ref{fig:lattice}], we see that the second contribution of Eq.~(\ref{eq:magn}) comes from the zigzag component of the order. This contribution vanishes for $g_{ab}\!=\!0$ and is about 5\% of the first term of Eq.~(\ref{eq:magn}) for $g_{ab}\!=\!0.1$.
More explicitly, we have
\small\begin{equation}\label{eq:magnetization}
\!\!\!\!{\bf m}\!=\!\!\left\{\!\!
\renewcommand{\arraystretch}{1.3}
\begin{array}{cl}
\Big[g_{aa} M'_a(F) \!+\! g_{ab} M'_b(G)\Big] \hat{{\bf a}}\!+\!\Big[g_{bb} M'_b(F) \!+\! g_{ab} M'_a(G)\Big] \hat{{\bf b}}, & \!\! {\bf H}\!\parallel\!{\bf a}\\
\Big[g_{bb} M'_b(F) \!+\! g_{ab} M'_a(G)\Big] \hat{{\bf b}}, & \!\!{\bf H}\!\parallel\!{\bf b}\\
g_{cc} M'_c(F) \hat{{\bf c}} \!+\!\Big[g_{bb} M'_b(F) \!+\! g_{ab} M'_a(G)\Big] \hat{{\bf b}}, & \!\!{\bf H}\!\parallel\!{\bf c}
\end{array}
\right.
\end{equation}\normalsize
The magnetizations along the field $m_\parallel$ (denoted by $m_a$, $m_b$ and $m_c$ for ${\bf H}$ along ${\bf a}$, ${\bf b}$ and ${\bf c}$, respectively) are given by $m_a\!=\!g_{aa} M'_a(F)\!+\!g_{ab} M'_b(G)$, $m_b\!=\!g_{bb}M'_b(F)\!+\!g_{ab}M'_a(G)$, and $m_c\!=\!g_{cc}M'_c(F)$. Their evolutions with field are shown by the orange solid lines in Figs.~\ref{fig:magnetization}\,(a-c), and follow the general trend of $M'_a(F)$, $M'_b(F)$ and $M'_c(F)$, see Figs.~\ref{fig:AnsatzResults}\,(d-f).
In agreement with experiment, $m_b$ rises much faster than $m_a$ and $m_c$.
Furthermore, the magnetizations $m_a$ and $m_b$ first increase monotonously with the field, then show a kink at $H_{\bf a}^\ast$ and $H_{\bf b}^\ast$, respectively, and then increase at a much slower pace towards a limiting value that is determined by the ratios $g_{ab}/g_{aa}$ and $g_{ab}/g_{bb}$, respectively (see Appendix~\ref{app:ansatze}).
By contrast, $m_c$ shows a finite jump (instead of a kink) at $H_{\bf c}^\ast$, reflecting the corresponding jumps in Fig.~\ref{fig:AnsatzResults}\,(f). At higher fields, $m_c$ shows a kink at $H_{\bf c}^{**}$ and then saturates. Note that here the exact saturation is only true for classical spins, and the kink in the classical magnetization will be smoothed by quantum fluctuations (as the spin Hamiltonian does not conserve rotations around the field axis, and the fully polarized state is not a true eigenstate).
\begin{figure*}
{\includegraphics[width=0.9\linewidth]{Figure8}}
\caption{Main panels: Magnetization process up to $7$~T, obtained from the classical ans\"atze (orange line), the linear spin wave approximation (blue line), and classical Monte Carlo simulations (red crosses). For comparison, we also show published experimental data (see supplementing material of Ref.~[\onlinecite{Ruiz2017}]).
The insets at the upper-right corners show the computed magnetization curves up to much higher fields (up to $150$~T, $15$~T and $60$~T for panels (a), (b) and (c), respectively).}\label{fig:magnetization}
\end{figure*}
Let us now compare these classical predictions for $m_\parallel$ with available experimental data published by Ruiz {\it et al} (see supplementing material in \cite{Ruiz2017}), which are shown in Fig.~\ref{fig:magnetization} by black lines.
Quite generally, while the classical ans\"atze capture the observed magnetization processes qualitatively, there is a large quantitative discrepancy. For {${\bf H}\!\parallel\!{\bf b}$}, for example, the classical prediction for the magnetization at $H_{\bf b}^\ast$ is about two times larger than the measured value. This deficiency has been recognized previously~\cite{Rousochatzakis2018}, and has led to the assertion that the system must feature strong quantum fluctuations due to the close proximity to the highly-frustrated $K$-$\Gamma$ line~\cite{Ducatman2018}.
Here we confirm this hypothesis by calculating the leading $1/S$ corrections to the magnetization from quantum fluctuations. The details of this calculation are provided in Appendix~\ref{app:SW} and the renormalized magnetization curves are shown by the solid blue lines in Fig.~\ref{fig:magnetization}. The results show that already the leading $1/S$ corrections reduce the magnetization quite strongly, bringing the curves much closer to the measured data. While subleading higher-order corrections will reduce the magnetization even further, providing a better comparison between theory and experiment, a final quantitative agreement will also require an appropriate re-adjustment of the microscopic couplings.
Importantly, our semiclassical results show further that, for {${\bf H}\!\parallel\!{\bf c}$}, the magnitude of the magnetization jump at $H_{\bf c}^\ast$ is significantly reduced by quantum fluctuations, almost to the point that there is no visible change, including the overall slopes of the curves below and above $H_{\bf c}^\ast$. This renders the detection of this feature in magnetization measurements more challenging and probably explains the absence of the kink in recent measurements~\cite{Majumder2019}. The detection is even more challenging for powder samples given that $m_c\!\ll\!m_b$. Nevertheless, as we will discuss next [Sec.~\ref{sec:Torque}], the transition at $H_{\bf c}^\ast$ should be still visible via the kink in the corresponding magnetic torque.
\begin{figure}[!b]
{\includegraphics[width=0.95\linewidth]{Figure9}}
\caption{Field dependence of the torques computed with the classical ans\"atze (orange) and within the linear spin wave approximation (blue).}
\label{fig:torque}
\end{figure}
\vspace*{-0.3cm}
\section{Magnetic torque}\label{sec:Torque}
\vspace*{-0.3cm}
According to Eq.~(\ref{eq:magnetization}), when {${\bf H}\!\parallel\!{\bf a}$} and {${\bf H}\!\parallel\!{\bf c}$}, the magnetization ${\bf m}$ develops a component perpendicular to ${\bf H}$. This implies the presence of a finite torque,
\small\begin{equation}\label{eq:Torque}
\renewcommand{\arraystretch}{1.2}
\begin{array}{c}
{\bf H}\!\parallel\!{\bf a}\!: \boldsymbol{\tau}= - \xi H ~\hat{{\bf c}},~~~
{\bf H}\!\parallel\!{\bf c}\!: \boldsymbol{\tau}= \xi H ~\hat{{\bf a}},\\
\xi \equiv g_{bb} M'_b(F)+g_{ab}M'_a(G)\,.
\end{array}
\end{equation}\normalsize
Interestingly, the expression for $\xi$ that gives the transverse components for {${\bf H}\!\parallel\!{\bf a}$} and {${\bf H}\!\parallel\!{\bf c}$} coincides with the expression for $m_b$, see Eq.~(\ref{eq:magnetization}).
Note that, as we discussed in Sec.~\ref{sec:Symmetries}, the overall signs of $M'_a(G)$ and $M'_b(F)$ are chosen spontaneously by the system for both {${\bf H}\!\parallel\!{\bf a}$} and {${\bf H}\!\parallel\!{\bf c}$}, and therefore the sign of the torque (or $\xi$) is arbitrary for both directions. This aspect has further observable consequences, which will be discussed in Sec.~\ref{sec:Discussion}.
Figure~\ref{fig:torque}\,(b) shows the evolution of $\tau/H$ with $H$ for fields along ${\bf a}$ and ${\bf c}$, with and without harmonic spin-wave corrections. First of all, the torque for {${\bf H}\!\parallel\!{\bf a}$} is about 40 times weaker than the torque for {${\bf H}\!\parallel\!{\bf c}$}. This reflects the smallness of $M'_a (G)$ and $M'_b (F)$ components for {${\bf H}\!\parallel\!{\bf a}$}, as shown in Fig.~\ref{fig:SmallComponents}\,(a).
Second, the torque for {${\bf H}\!\parallel\!{\bf a}$} remains non-zero up to $H_{\bf a}^{\ast}$, whereas the torque for {${\bf H}\!\parallel\!{\bf c}$} remains non-zero up to $H_{\bf c}^{\ast\ast}$. This again stems from the associated behaviors of $M'_a(G)$ and $M'_b(F)$ [see Figs.~\ref{fig:SmallComponents}\,(a) and \ref{fig:AnsatzResults}\,(f)].
Third, both torques show a non-monotonic behavior as a function of the field. The torque for {${\bf H}\!\parallel\!{\bf c}$}, in particular, shows a characteristic sharp kink at $H_{\bf c}^\ast$, reflecting the first order transition between the low-field six-sublattice and the high-field two-sublattice state.
Importantly, this kink remains sharp even after we include the leading $1/S$ spin-wave corrections (blue line, see Appendix~\ref{app:SW}). A measurement of the torque can therefore give direct evidence for the transition at $H_{\bf c}^\ast$, and thus provide information for the value of $\Gamma$ via Eq.~(\ref{eq:Hacstar}).
Finally, for {${\bf H}\!\parallel\!{\bf c}$}, the torque in the high-field phase scales as
\begin{equation}
H_{\bf c}^\ast \leq H\leq H_{\bf c}^{\ast\ast}:~~ \tau/H \propto \sqrt{1-\left(H/H_{\bf c}^{\ast\ast}\right)^2}\,.
\end{equation}
Thus, a measurement of the torque at high fields can also be used to extract $H_{\bf c}^{\ast\ast}$ and, in turn, an independent constraint on the microscopic parameters $J$ and $\Gamma$ via Eq.~(\ref{eq:Hcstarstar}).
\vspace*{-0.3cm}
\section{Effect of thermal fluctuations \& classical $H$-$T$ phase diagram}\label{sec:FiniteT}
\vspace*{-0.3cm}
To cross-check the above zero-temperature results from the classical ans\"atze and confirm the high-field thermal transition for {${\bf H}\!\parallel\!{\bf c}$} mentioned above, we have performed classical Monte Carlo simulations using the standard Metropolis algorithm combined with the over-relaxation algorithm~\cite{Metropolis1953,Creutz1987}. The simulations were performed on finite-size clusters with a total number of sites ${\mathcal N}\!\in\!\{48,96,144,192,240,288\}$ and periodic boundary conditions. All considered systems, spanned by the unit vectors of the orthorhombic lattice, have at least three periods in the orthorhombic ${\bf a}$-direction in order to accommodate ${\bf Q}=2\hat{\bf a}/3$ order, see more details in Appendix~\ref{app:MC}.
The results obtained by a thermal annealing down to $T\!=\!5$~K show that the total magnetization is almost indistinguishable from the predictions of the semi-analytical approach, lending strong support that the latter delivers quantitatively accurate results for the local physics of the problem.
Let us now turn to the classical $H$-$T$ phase diagrams, which are shown in Fig.~\ref{fig:phaseD} for the three orthorhombic directions. The boundary lines of the counter-rotating order (denoted by `IC') have been extracted by a finite-size analysis of the so-called Binder cumulant~\cite{Binder1993} (see Appendix~\ref{app:MC}),
\small\begin{equation}
B_{\mathcal{O}_{{\bf Q}=2\hat{\bf a}/3}}= 1-\langle \mathcal{O}^4_{{\bf Q}=2\hat{\bf a}/3}\rangle \Big{/} \Big( 3\langle \mathcal{O}^2_{{\bf Q}=2\hat{\bf a}/3}\rangle^2 \Big)\,,
\end{equation}\normalsize
of the equally-weighted combination of the three modulated static structure factor components (for all field directions):
\small\begin{equation}
\mathcal{O}_{{\bf Q}=2\hat{\bf a}/3}=\sqrt{|M_{\bf a}(A)|^2+|M_{\bf b}(C)|^2+|M_{\bf c}(F)|^2}\,.
\end{equation}\normalsize
For {${\bf H}\!\parallel\!{\bf b}$}, the phase diagram contains two distinct phases, the high-$T$ paramagnetic phase and the low-$T$ counter-rotating order, which persists up to $H_{\bf b}^\ast\!\sim\!2.8$~T, see Fig.~\ref{fig:phaseD}\,(b).
For {${\bf H}\!\parallel\!{\bf a}$}, the counter-rotating order persists up to very high fields ($H_{\bf a}^\ast\!\sim\!102$~T), see Fig.~\ref{fig:phaseD}\,(a), and is accompanied by the uniform orders $M'_a(G)$ and $M'_b(F)$ which are however extremely weak, see Fig.~\ref{fig:SmallComponents}\,(a). While these orders onset at the same field $H_{\bf a}^\ast$ as the modulated order at $T\!=\!0$, it is unclear whether this remains true for finite $T$. In fact, symmetry considerations alone tell us that the boundaries of the two types of orders can in general be different, as the they break different symmetries (the modulated order breaks translations whereas the uniform orders break $C_{2{\bf a}}$ and $\Theta C_{2{\bf b}}$, see Table~\ref{tab:symmetries}). Unfortunately, the smallness of $M'_a(G)$ and $M'_b(F)$ does not allow for an accurate numerical determination of their transition temperature line.
For {${\bf H}\!\parallel\!{\bf c}$}, there are three distinct phases, see Fig.~\ref{fig:phaseD}\,(c). Apart from the paramagnetic and the modulated phase, there is a robust high-field order associated with $M'_a(G)$ and $M'_b(F)$, and the spontaneous breaking of $C_{2\bf c}$ and $\Theta C_{2\bf b}$ (see Table~\ref{tab:symmetries}). This phase coexists with the modulated order at low $H$ and $T$, but extends up to very high fields ($H_{\bf c}^{\ast\ast}\!\sim\!45$~T). Its boundary line has been extracted from the Binder cumulant $B_{\mathcal{O}_{{\bf Q}=0}}$ associated with
\small\begin{equation}
\mathcal{O}_{{\bf Q}=0} = \sqrt{|M_{\bf a}'(G)|^2+|M_{\bf b}'(F)|^2}\,.
\end{equation}\normalsize
Finally, the yellow shading in Figs.~\ref{fig:phaseD}\,(a) and (b) represents the variation of the magnitude of $|M_{\bf b}'(G)|$ and $|M_{\bf a}'(G)|$, respectively, from high values (intense yellow) at low $T$ to vanishing values (blue) at higher $T$.
\begin{figure}[!t]
{\includegraphics[width=0.95\columnwidth]{Figure10}}
\caption{The field-temperature phase diagram obtained from MC simulations for field applied along (a) $\bf a$-, (b) $\bf b$-, and (c) $\bf c$-axis.} \label{fig:phaseD}
\end{figure}
\vspace{-0.3cm}
\section{Discussion}\label{sec:Discussion}
\vspace{-0.3cm}
The study presented here provides a semi-analytical framework for the anisotropic response of $\beta$-$\text{Li}_2\text{IrO}_3$ under a magnetic field along the three orthorhombic directions. This framework is based on the minimal nearest-neighbor $J$-$K$-$\Gamma$ model~\cite{Lee2015,Lee2016,Ducatman2018,Rousochatzakis2018} and the hypothesis that the local correlations of the low-field incommensurate order can be captured by its closest commensurate approximant with the right symmetry~\cite{Ducatman2018}. The results are in qualitative agreement with almost all experimental facts collected so far, and we have shown how a quantitative agreement can also be reached by including quantum fluctuations.
In addition, our analysis delivers a number of predictions which await experimental verification.
First, the critical fields $H^\ast$ that mark the disappearance of the modulated order are highly anisotropic, in particular, $H_{\bf b}^\ast\!<\!H_{\bf c}^\ast\!\ll\!H_{\bf a}^\ast$. Such an anisotropic response, which is also evidenced in susceptibility~\cite{Ruiz2017,Majumder2019}, signifies a large separation of energy scales between $J$ and $\Gamma$. An explicit dependence of $H^\ast$ on these interactions is derived in this work [Eq.~(\ref{eq:Hacstar})] and can be used to extract the actual strength of $\Gamma$ (the value of $J$ is estimated $\sim 4$~K from the value of $H_{\bf b}^\ast$~\cite{Rousochatzakis2018}). Importantly, the dominant Kitaev coupling $K$ does not affect any of the critical fields, partly because it is ferromagnetic.
Second, for all orthorhombic directions our analysis reveals the presence of various intertwined uniform zigzag and FM orders, some of which remain robust far above $H^\ast$. The physical origin of this robustness is related to the cross-coupling terms of Eqs.~(\ref{eq:cross-coupling1}-\ref{eq:cross-coupling2}). Some of the uniform orders give rise to a finite torque signal and can thus be detected in a direct way. Alternatively, they can also be observed by magnetic X-ray diffraction~\cite{Ruiz2017}, or by local probes like NMR or $\mu$SR.
Third, we have shown that the high-field response for {${\bf H}\!\parallel\!{\bf c}$} is special, in that the disappearance of the modulated order at $H_{\bf c}^\ast$ restores only the translational symmetry and leaves some of the discrete symmetries broken. This implies the presence of a second thermal transition above $H_{\bf c}^\ast$, which is associated with the onset of the uniform orders $M'_a(G)$ and $M'_b(F)$. This transition can then be detected with thermodynamic measurements at high enough fields.
A natural extension of the present study is the investigation of the field-induced behavior of $\beta$-Li$_2$IrO$_3$ for general field directions, i.e., away from the orthorhombic axes. As it turns out, a semi-analytical description can be also obtained for fields in the $ab$- and $bc$- planes~\cite{Mengqun2}. The emerging picture reveals a remarkable interplay of the various modulated and uniform orders and rich anisotropic phase diagrams, the details of which will be given elsewhere~\cite{Mengqun2}. We can, however, comment on one particular aspect related to the torque signal discussed in Sec.~\ref{sec:Torque}. As mentioned there, the predicted torque signals are proportional to the quantity $\xi = g_{bb} M'_b(F)+g_{ab}M'_a(G)$, whose sign is chosen spontaneously by the system for {${\bf H}\!\parallel\!{\bf a}$} or {${\bf H}\!\parallel\!{\bf c}$}. However, adding an infinitesimal field along ${\bf b}$ will actually fix the sign of $\xi$, since the two are directly coupled to each other. This simple argument shows that, as a function of the angle in the $ab$- or $bc$-planes, the torque will show an abrupt reversal when the field passes through the ${\bf a}$ and ${\bf c}$ axes, respectively. Such a first-order transition scenario could also be relevant for the explanation of the sawtooth-like torque anomalies observed experimentally in the closely related compound $\gamma$-Li$_2$IrO$_3$~\cite{Modic2017,Modic2018} (see also \cite{Winter2019}).
Finally, we would like to touch upon an aspect that may be relevant for the interpretation of the phase transition reported recently around 100 K~\cite{Ruiz2019}. As discussed in Ref.~\cite{Ducatman2018}, the zero-field and zero-temperature configuration contains the uniform orders $M'_a(G)$ and $M'_b(F)$, in addition to the modulated order. Given that the two types of order break different symmetries (the modulated order breaks translations whereas the uniform orders break $C_{2{\bf a}}$ and $C_{2{\bf c}}$~\cite{Ducatman2018}) one generally expects that the two types of order onset at different temperatures. In particular, we have checked numerically (unpublished) that the modulated period-3 six-sublattice order carries a pseudo-Goldstone low-energy mode, similar to other incommensurate phases in related models~\cite{Mengqun2019,Choi_2013,Choi2013}. On the other hand, the energy barrier associated with flipping the signs of the uniform orders $M'_a(G)$ and $M'_b(F)$ gives rise to a finite energy gap. It is then plausible that the uniform orders onset at a higher temperature $T_{\text{uni}}$ compared to $T_N$. While the smallness of the uniform orders does not allow to check this numerically with Monte Carlo, the cross-coupling term of Eq.~(\ref{eq:cross-coupling1}) suggests that $T_{\text{uni}}$ could scale with $\Gamma$. In such a scenario, a field along the ${\bf b}$ axis will turn the zero-field line extending from $T\!=\!0$ up to $T\!=\!T_{\text{uni}}$ into a line of first order transitions, because the field couples directly to $M'_b(F)$ (and to $M'_a(G)$ via $g_{ab}$). For very low fields, the proximity to this first-order line would then give rise to hysteresis effects, similar to those observed in Ref.~ \cite{Ruiz2019}. The actual details of this scenario (in particular, the connection of the measured torque signals with the ones we report here at zero-temperature), remain to be explored.
\vspace*{0.3cm}
\noindent{\it Acknowledgments:}
We thank J. Analytis, J. Betouras, A. Ruiz and A. A. Tsirlin for helpful discussions. This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award No. DE-SC0018056. We also acknowledge the support of the Minnesota Supercomputing Institute (MSI) at the University of Minnesota. N.B.P. also acknowledges the hospitality of the Aspen Center for Physics supported by National Science Foundation grant PHY-1607611, where part of the work on this manuscript has been done.
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"redpajama_set_name": "RedPajamaArXiv"
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Before Pathaan releases, here is an analysis of Shah Rukh Khan starrer Zero
Mumbai: Shah Rukh Khan is all set to make his comeback as the lead role with Pathan, which is scheduled to release on 25 January 2023. The film directed by Siddharth Anand also stars Deepika Padukone and John Abraham in lead roles. All eyes are on Pathan as Bollywood is going through a rough phase.
But before the release of Pathan, let's take a look at the box office analysis of Shah Rukh Khan's last release 'Zero'. The film was directed by Aanand L. Rai and also starred Anushka Sharma and Katrina Kaif.
Read also: Budget Vs Box Office Collection: A Look At The Analysis Of Akshay Kumar Starrer Samrat Prithviraj
Zero opened well and collected Rs 500. 19.35 crores on the first day, but later it started declining. The film collected Rs 500 crore in its first weekend. 54.60 crore, and saw a further decline on weekdays. In India, the film's lifetime collection stood at Rs. 90.28 crore nett, and the film's worldwide gross collection was Rs. 186 crores.
OK, Rs. The collection of 90.28 crores is surely going to be pretty good for a mid budget film. But Zero was made on a much larger scale and according to reports, the film was made on a budget of Rs 500 crore. 200 crores. Even if we just ignore the Indian collection and look at the Worldwide gross collection of Zero, Rs. 186 crores, it is still less than the film's budget. So zero flopped.
While the official budget of Pathan is yet to be revealed, reports suggest that the film was made on a whopping budget of Rs. 250 crores. Let's see what response Pathan gets at the box office.
Read also: Budget Vs Box Office Collection: A Look At The Box Office Analysis Of Shahid Kapoor And Mrunal Thakur Starrer Jersey
For more news and updates from the world of Television, Bollywood and OTT, stay tuned with Telly Chakkar.
Categories box office collection
Buzz: Bollywood actor in Prabhas's next
How Tom Hanks' New Movie A Man Called Otto Beat The Box Office Odds
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,489
|
Mesclun () is a mix of assorted small young salad greens that originated in Provence, France. The traditional mix includes chervil, arugula, leafy lettuces and endive, while the term mesclun may also refer to a blend that might include some or all of these four and baby spinach, collard greens, Swiss chard (silver beet), mustard greens, dandelion greens, frisée, mizuna, mâche (lamb's lettuce), radicchio, sorrel, or other fresh leaf vegetables.
Origins
On July 10, 1924, in Paris, Philippe Tiranty and Paul Gordeaux, reunited with many friends at the Cochon d'Or (a famous restaurant in La Villette), decided to create the foyer des Amitiés niçoises, and to call it Lou Mesclun. For these comedians and humanists, this expression meant "real living together".
The term mesclun for a mixture of young salad greens is quite recent, according to the Merriam-Webster Dictionary first used in 1976. Of Provençal dialect origin, it derives from the verb mesclar, to "mix thoroughly" and literally means "mixture". According to local lore, mesclun originated with the farmers around Nice, who would each bring their own unique and prized mix of baby greens to the farmers' markets. One of the most representative and authentic versions combined baby dandelion, lettuce and rocket (arugula).
Noted chef Alice Waters comments, "Outdoor markets in Provence display mesclun in profusion, a melange of the first tender young leaves which appear in the garden. Mesclun can be an extraordinary lettuce mixture: rocket, much like the rugola (arugula) found in Italian markets, chervil, mâche or lamb's lettuce and oakleaf lettuce. On occasion, baby curly endive (chicory) or young dandelion greens find their way into the medley, depending solely upon the grower's personal preferences combined with the reality of whatever else might send up shoots in the spot where mesclun grows."
Spring mix
In the North American foodservice industry, the first appearance of mesclun traces to restaurants and farm stands in the early 1980s, with a rise in popularity since. A mesclun mix can be described as comprising baby leaves of lettuces and other greens (and often herbs) in a wide range of leaf shapes, colors, textures and tastes. While the overwhelming amount of mesclun sold approximates the traditional blend of chervil, arugula, leafy lettuces and endive, depending on the season, anywhere from a dozen to three dozen different varieties of baby greens, including red and green oak leaves, romaine and lolla rossa lettuces, frisée, tatsoi, bok choy (joi choi), arugula, spinach, orach, mizuna, dandelion, mustard greens and garden cress may compose what is commercially referred to as a "spring mix". When available, locally grown, direct-from-the-farmer sourcing is recommended over commercial bulk packs for best flavor and freshness.
See also
List of salads
References
Salads
Cuisine of Provence
French cuisine
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 965
|
Xie Wei Ready To Continue Knockout Streak On Biggest Stage
Bear Frazer
After a year of stunning results in ONE Hero Series (OHS), "The Hunter" Xie Wei is prepared to bring his incredible winning streak to the global stage.
The 23-year-old's reward for a run of five straight TKOs was a chance to compete on the main ONE Championship roster, and he will jump straight in at the deep end against Danny "The King" Kingad at ONE: FIRE & FURY.
Xie's upcoming opponent may be one of the most talented and experienced flyweights in The Home Of Martial Arts, but the Chinese athlete believes he is more than a match for him.
If there is one area that "The Hunter" believes he eclipses Kingad, it is his finishing ability. While the Filipino has failed to finish any of his last nine opponents, there is no doubt about the man from Hunan's knockout power after his incredible recent run.
Though Xie does not want to say whether he expects to hit another stoppage in Manila, he has confidence in his kickboxing and believes his striking skills will do the talking for him.
"My standing skills are better, but I won't make a prediction for this match," he says.
"I think all I have to do is focus on myself, and give what I have been trained. I will show my skills and my spirit."
With that said, Xie has a tremendous amount of respect for Kingad. The Team Lakay representative may not have won many bouts by knockout or submission, but he has beaten a long list of elite opponents. He is also arguably his division's most exciting performer.
Danny Kingad 'Hungry' To Stand And Trade With Xie Wei
Xie Wei – From Shaolin To Breakout Star Of ONE Hero Series
Top 5 Performances From The Heroes Of ONE: FIRE AND FURY
"I think he is a very good fighter, I have watched his fights, he has almost never lost in his career," says Xie.
"His strengths are his punching, wrestling, jiu-jitsu. I feel he is a very comprehensive fighter."
The same could be said about Xie after his success in ONE's home for Chinese rising stars. While he displayed slick stand-up skills that showed why he is 5-0 in kickboxing, he also demonstrated that his all-around mixed martial arts game is on point.
Along with his crisp punching, hard clinch strikes, and flying attacks, he showed good grappling to control skilled opponents and unleash dangerous ground and pound.
"I think ONE Hero Series is a very good stage, it helped me to be prepared and fight with so many good Chinese fighters, and that makes me happy," the Sunkin International Fight Club representative says.
"I learned from each of my fights and I improved. I think this is a good way to let more Chinese fighters show their talent."
However, he knows the main roster of ONE Championship is an entirely different level to the OHS ring.
"For fighting with ONE main fights I need to be more focused. Also, I need to work harder," he adds.
Xie admits he is also feeling a few jitters ahead of his main card match-up in the Mall Of Asia Arena, but overall, he cannot wait to get out there and show the world what he can do.
He knows he cannot underestimate the challenge he has to overcome, but he seems to be approaching it with the right mindset.
"I am excited because I can finally walk into the international stage and fight with those international fighters," Xie says.
"I am nervous because he is a very good fighter I need to give 100 percent of my strength to face him. I understand his ability and that's why I need to respect him."
Read more: Joshua Pacio 'In Awe' Of Alex Silva's Grappling
Xie Wei Earns First Win In ONE Championship With TKO Of Rothana
The Chinese flyweight emerged victorious after a gritty battle at ONE: COLLISION COURSE on Friday night.
Chan Rothana vs. Xie Wei | ONE Championship Full Fight
Chan Rothana vs. Xie Wei from ONE: COLLISION COURSE.
Home » News » Xie Wei Ready To Continue Knockout Streak On Biggest Stage
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,129
|
By Cassandra Clare
Review By Crea Grennan
Fifteen year old Clary Fray is in a teenage nightclub with her friend Simon when she witnesses the murder of a blue-haired boy. After talking to Simon, she realizes that only she can see the "murderers" : Jace, Alec and Isabelle.
A few days later, Jace finds her and explains that he, Alec abd Isabelle are "Shadowhunters", trained to kill Demons and naught downworlders,.
A worrying call from Clary's mother leads to Clary coming face to face with a demon, which she kills with her bare hands. Jace realizes there is something special about her, but Clary has bigger problems. Her mother had been kidnapped, and Clary knows it's up to her to save her mother.
Along the way, Clary faces many difficulties, including falling in love with the wrong person, finding out that her mother has lied to her all her life and ,the most shocking, what Clary is herself.
Thie first in a series of six books, City of Bones is an amazingly written story, full of mystery and surprises.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,886
|
Q: How to find phone device is reachable or not using GCM I am using GCM and my server code is in PHP.
I am sending request to device which is not reachable in network; still I got replay as success.
Replay is {"multicast_id":6818856885806573730,"success":1,"failure":0,"canonical_ids":0,"results":[{"message_id":"0:1345696774899998%096341d0f9fd7ecd"}]}
Is it possible to find out whether device(android phone) is reachable or not..?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,499
|
Isabelle Marie Françoise (Fanny) Geefs-Corr (Brussel, 2 mei 1807 – Schaarbeek, 23 januari 1883) was een Belgische schilderes en tekenares.
Leven en werk
Fanny of Fi Geefs was een dochter van Mathieu Corr en Monique Landy en een zuster van de graveur Erin Corr en de schilderes Mathilde Corr. Haar ouders hadden zich in 1802 vanuit Ierland in Brussel (aan de Hofberg 685) gevestigd, waar ze een luxe schoenmakerij voor "op maat te maken laarzen" openden. Ze trouwde in 1836 met beeldhouwer Willem Geefs (1805-1883).
Fanny Geefs was een leerlinge van François-Joseph Navez. Ze maakte genrevoorstellingen en portretten, waaronder enkele van koningin Louise. Ook maakte ze een tekening van haar mans standbeeld van Leopold I van België, waarna haar broer Erin er een prent van maakte. Geefs exposeerde onder meer bij de Parijse salons en op de tentoonstellingen van Levende Meesters. Haar werk werd diverse malen bekroond. In 1845 werd ze honorair lid van de Koninklijke Academie voor Schone Kunsten van Brussel. Immerzeel schrijft over haar: (zij) "bekleedt ontegenzeggelijk eene eerste plaats onder de kunstenaressen van haren tijd".
In de episode voorafgaand aan haar overlijden, woonde ze met haar echtgenoot te Schaarbeek, in de Paleizenstraat 20-22. Geefs overleed op 75-jarige leeftijd, vier dagen na haar man.
Na haar overlijden kwam het Prentenkabinet van de Koninklijke Bibliotheek van België in het bezit van zowat vijfhonderd door haar gemaakte schetsen en tekeningen.
Enkele werken
Zie ook
Belgisch kunstschilder
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,981
|
Mario Baccini (born 14 December 1957 in Rome) is an Italian politician, former member of the Union of Christian and Centre Democrats and promoter of the White Rose.
Biography
He started his political activity as town councilor in Rome, for the Christian Democracy party (DC). In 1994, when DC disbanded, he joined the Christian Democratic Centre, which entered the Pole of Freedoms.
In the same year, he became president of the Christian Democratic Centre (CCD) parliamentary group at the Chamber of Deputies. Then he was secretary and national co-ordinator of the Christian Democratic Centre. In 2001 he was re-elected at the Parliament; in 2002 he joined the Union of Christian and Centre Democrats, the result of a merger between CCD, United Christian Democrats, and European Democracy.
In the Berlusconi II Cabinet he was sub-secretary of the Foreign Ministry from 2001 to 2004. On 3 December 2004, he became Minister of the Public Function. He was reconfirmed minister in Berlusconi III Cabinet from 2005 to 2006.
From 2006 to 2008 he was vice-president of the Senate.
On 30 January 2008, he left UDC and he founded the White Rose movement with Bruno Tabacci. He was candidated as mayor of Roma, taking just 0.8% of the votes. At Italian general election, the White Rose was in alliance with UDC and other Christian democrats movements into the Union of the Centre, and Baccini was elected to the Chamber of Deputies: he didn't join White Rose's parliamentary group, but rather he adhered to the mixed group.
On 14 May, during the parliament discussion about the motion of confidence to Berlusconi IV Cabinet, he announced to vote the confidence to the cabinet, leaving the project of the White Rose.
See also
White Rose
References
External links
Official website
1957 births
Living people
Politicians from Rome
Christian Democracy (Italy) politicians
Christian Democratic Centre politicians
Union of the Centre (2002) politicians
The Rose for Italy politicians
The People of Freedom politicians
New Centre-Right politicians
Deputies of Legislature XII of Italy
Deputies of Legislature XIII of Italy
Deputies of Legislature XIV of Italy
Senators of Legislature XV of Italy
Deputies of Legislature XVI of Italy
Libera Università Maria SS. Assunta alumni
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,993
|
There are dozens — if not hundreds — of freight audit and payment companies, 4PL, 3PL, and 3PP companies focused on the financial supply chain for the shipper. Daycos is the sole provider exclusively focused on you — the carrier.
Forty years of serving as the leading trusted provider to transportation service providers, freight carriers and agents have taught us one thing — listening to your customers first and foremost allows you to develop products and services that deliver real value. Daycos is not a value-add. Daycos is a value delivery partner.
Our commitment to the market is simple — we exist to do good and be good. We want our products and services to do the same for your business. Our Revenue Solutions have contributed to the bottom line of customers for decades. Contact us today to see how we can deliver value to yours!
Daycos offers solutions at each stage of the revenue, from initiation to recovery, to meet your company's needs.
The easiest way to get your earned revenue quickly is with the efficient and accurate presentation of an invoice. Daycos offers solutions from full-service to our Accubill software to allow you to process accurate invoices for HHG moves.
The shipper is able to write rules on how invoices are held up; now you are able to write rules that dictate which ones you really need to worry about. Manage your exceptions your way and keep your revenue flowing.
We locate the revenue left behind. Experts at DOD HHG post-audit for nearly 40 years, we have recovered hundreds of thousands of dollars on behalf of our customers.
Extend the service to include the agents you pay to eliminate headaches and disputes. Customizable compensation schedules provide visibility, clarity and accuracy — works in conjunction with our distribution systems or with your internal payables systems.
We have the power to make a choice. We can sit at our desks and focus on the "whats" that need to get done and the "how" it gets done and a whole lot of "when" things get done, but we have a choice as to "why" things get done.
At Daycos, we decided that delivering value to our company was only one of the "whys". We wanted to serve our company well, but we also wanted to serve our incredibly talented team and the community that team lives in, too. We felt that by investing in where we live, we get better employees … which means we deliver the best value to our customers.
Daycos4Good represents our methodology. We weigh our decisions on the impact the outcome would have on our four stakeholders — customers, employees, community and company. If something is great for our company but bad for our customers, we don't do it. If something is good for all four stakeholders, it is a sound decision.
As we grow, our commitment to investing in ourselves, our community, and our customers will result in value for our company, too.
Daycos is a force in the industry because we are a force for good!
Why have providers relied on Daycos for nearly 40 years?
Daycos is a very important business partner — they do all our billing for us. Daycos representatives are very responsive to my questions. They take a very complicated process and simplify it; they have our back and it saves us a lot of money.
Daycos is innovative, honest, and always willing to help. Our internal IT department didn't even want to touch the EDI program — we knew that Daycos had a sophisticated system and there was no need to re-invent the wheel. Their staff is great — they are always there and available when I call. Daycos makes our life easy.
The billing services offered by Daycos are second-to-none and they help us get the money we are owed quickly.
Daycos is technically savvy and that helps us get our money fast and accurately. Integrity is one word I would use to define Daycos. Actual people involved and they are the finest people in the business and we enjoy working with them. Daycos is extremely responsive and they know what they need to do in order to get the job done.
Daycos has a great ability to stay with the ever-changing rules and regulations of invoicing. They are very responsive if we ever need anything. If we have a problem with anything, Daycos finds a solution for us.
Daycos is very well known in the industry and well-respected for what they do; very customer oriented. Daycos is saving us a ton of money … the cost savings and their knowledge base is much more effective than bringing someone in-house to do our billing. They understand the tariffs better than anyone in the industry.
I've been working with Daycos for 20 years, and they are always willing to work with me to ensure that all of our invoices are submitted timely and accurately. There isn't any other solution or comparison out there to Daycos.
Daycos takes the weight off of our shoulders by taking care of our billing for us. The number one factor affecting our decision to partner with Daycos was the service, hands down.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,698
|
Gunnar "Gurra" Krantz is a Swedish sailor. He skippered the Swedish America's Cup Challenge at the 1992 Louis Vuitton Cup, Swedish Match in the 1997–98 Whitbread Round the World Race and Team SEB in the 2001–02 Volvo Ocean Race
Achievements
References
External links
Swedish male sailors (sport)
Volvo Ocean Race sailors
Volvo Ocean 60 class sailors
1992 America's Cup sailors
Living people
Year of birth missing (living people)
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,087
|
#region License
//
// Author: Nate Kohari <nkohari@gmail.com>
// Copyright (c) 2007-2008, Enkari, Ltd.
//
// 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.
//
#endregion
#region Using Directives
using System;
using Ninject.Core.Activation;
using Ninject.Core.Behavior;
using Ninject.Core.Creation;
using Ninject.Core.Infrastructure;
using Ninject.Core.Parameters;
using Ninject.Core.Selection;
#endregion
namespace Ninject.Core.Binding
{
/// <summary>
/// The stock implementation of a binding.
/// </summary>
public class StandardBinding : DebugInfoProvider, IBinding
{
/*----------------------------------------------------------------------------------------*/
#region Properties
/// <summary>
/// Gets the kernel that created the binding.
/// </summary>
public IKernel Kernel { get; private set; }
/*----------------------------------------------------------------------------------------*/
/// <summary>
/// Gets the service type that the binding is associated with.
/// </summary>
public Type Service { get; private set; }
/*----------------------------------------------------------------------------------------*/
/// <summary>
/// Gets or sets the provider that can create instances of the type.
/// </summary>
public IProvider Provider { get; set; }
/*----------------------------------------------------------------------------------------*/
/// <summary>
/// Gets or sets the behavior that decides whether to re-use existing instances or create new ones.
/// </summary>
public IBehavior Behavior { get; set; }
/*----------------------------------------------------------------------------------------*/
/// <summary>
/// Gets or sets the condition under which this binding should be used.
/// </summary>
public ICondition<IContext> Condition { get; set; }
/*----------------------------------------------------------------------------------------*/
/// <summary>
/// Gets or sets the selection heuristics defined on the binding.
/// </summary>
public IHeuristicCollection Heuristics { get; set; }
/*----------------------------------------------------------------------------------------*/
/// <summary>
/// Gets any parameters that have been defined for the binding.
/// </summary>
public IParameterCollection Parameters { get; set; }
/*----------------------------------------------------------------------------------------*/
/// <summary>
/// Gets or sets the component container for the binding.
/// </summary>
public IComponentContainer Components { get; set; }
/*----------------------------------------------------------------------------------------*/
/// <summary>
/// Gets a value indicating whether the binding was implicitly created by the kernel.
/// </summary>
public bool IsImplicit { get; set; }
/*----------------------------------------------------------------------------------------*/
/// <summary>
/// Gets a value indicating whether the binding is conditional (that is, whether it has
/// a condition associated with it.)
/// </summary>
/// <value></value>
public bool IsConditional
{
get { return (Condition != null); }
}
/*----------------------------------------------------------------------------------------*/
/// <summary>
/// Gets a value indicating whether the binding is the default binding for the type.
/// (If a binding has no condition associated with it, it is the default binding.)
/// </summary>
public bool IsDefault
{
get { return (Condition == null); }
}
#endregion
/*----------------------------------------------------------------------------------------*/
#region Disposal
/// <summary>
/// Releases all resources held by the object.
/// </summary>
/// <param name="disposing"><see langword="True"/> if managed objects should be disposed, otherwise <see langword="false"/>.</param>
protected override void Dispose(bool disposing)
{
if (disposing && !IsDisposed)
{
DisposeMember(Provider);
DisposeMember(Behavior);
DisposeMember(Condition);
Kernel = null;
Service = null;
Provider = null;
Behavior = null;
Condition = null;
}
base.Dispose(disposing);
}
#endregion
/*----------------------------------------------------------------------------------------*/
#region Constructors
/// <summary>
/// Creates a new StandardBinding.
/// </summary>
/// <param name="kernel">The kernel that is creating the binding.</param>
/// <param name="service">The service type to bind from.</param>
public StandardBinding(IKernel kernel, Type service)
{
Ensure.ArgumentNotNull(kernel, "kernel");
Ensure.ArgumentNotNull(service, "service");
Kernel = kernel;
Service = service;
Heuristics = new HeuristicCollection();
Parameters = new ParameterCollection();
Components = new StandardComponentContainer(kernel, kernel.Components);
}
#endregion
/*----------------------------------------------------------------------------------------*/
#region Public Methods
/// <summary>
/// Determines whether the specified context matches this binding.
/// </summary>
/// <param name="context">The context in question.</param>
public bool Matches(IContext context)
{
Ensure.ArgumentNotNull(context, "context");
Ensure.NotDisposed(this);
return Condition.Matches(context);
}
#endregion
/*----------------------------------------------------------------------------------------*/
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 856
|
Compare opinions and buy at the best price!
25,610,633 reviews
5,181 brands
Diplotop - product comparison - gathers SONY A290 users reviews, tests and opinions.With a data base of unprecedented wealth, 8 reviews for the Digital Camera SONY A290, Diplotop compares the Digital Camera SONY A290 with its competitors in order to find the best.
Download your SONY A290 user guide or user manual
Brand or manufacturer:
All Digital Camera - SONY reference numbers
The best Digital Camera - SONY products
Liste of brands and manufacturers
Find the best products
Digital Camcorder & Video Recorder
Average score for the 8 opinions:
User vote distribution:
High-performance Reliability
Ease of use Value for money
SONY A290 Reviews
Its users find that the SONY A290 doesn't have any particular problems of user-friendliness.They find it relatively fragile., Moreover, most of them share the same opinion You can look at the SONY A290 forum to identify problems that users have come across and the suggested solutions.
According to its users, it is very efficient., They mostly agree on this point. On average they find that it is very good value for money You can download the SONY A290 user manual to ensure that its features correspond to your needs.
The users were asked the following question :
Is the A290 easy to use?
8 users answered questions and rated the product on a scale of 0 to 10. The rating is 10/10 if the SONY A290 is very user-friendly.
The reviews (rough results) are presented in the following graph :
By leaving the mouse on a column for a few seconds, you can see the number of people who voted to make up the score that appears in the horizontal axis.
Statistical data :
The average score balanced by the number of reviews is 7.25 and the standard differential is 1.4.
Is the A290 highly efficient?
8 users answered questions and rated the product on a scale of 0 to 10. The rating is 10/10 if the SONY A290 is, in its domain, the best on a technical level, the one offering the best quality, or offering the largest range of options.
Is the A290 reliable, sturdy?
8 users answered questions and rated the product on a scale of 0 to 10. The rating is 10/10 if you think that the SONY A290 is a sturdy product, which will last a long time before breaking down.
Is the A290 good value for money?
8 users answered questions and rated the product on a scale of 0 to 10. The rating is 10/10 if you think that the SONY A290 is really not expensive considering its features.
The average score balanced by the number of reviews is 8.5 and the standard differential is 1.2.
Contact Diplotop team
The main brands
The best products
The most reviewed products
The new products
References starting with letter A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 0 1 2 3 4 5 6 7 8 9
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,090
|
package org.spongepowered.common.mixin.core.network;
import static com.google.common.base.Preconditions.checkNotNull;
import com.google.common.base.Charsets;
import io.netty.buffer.ByteBuf;
import io.netty.handler.codec.DecoderException;
import io.netty.handler.codec.EncoderException;
import net.minecraft.nbt.NBTTagCompound;
import net.minecraft.network.PacketBuffer;
import org.spongepowered.api.data.DataView;
import org.spongepowered.api.network.ChannelBuf;
import org.spongepowered.asm.mixin.Final;
import org.spongepowered.asm.mixin.Implements;
import org.spongepowered.asm.mixin.Interface;
import org.spongepowered.asm.mixin.Intrinsic;
import org.spongepowered.asm.mixin.Mixin;
import org.spongepowered.asm.mixin.Shadow;
import org.spongepowered.common.data.persistence.NbtTranslator;
import java.io.IOException;
import java.nio.ByteOrder;
import java.util.UUID;
@Mixin(PacketBuffer.class)
@Implements(@Interface(iface = ChannelBuf.class, prefix = "cbuf$"))
public abstract class MixinPacketBuffer extends ByteBuf {
@Shadow @Final private ByteBuf buf;
@Shadow protected abstract NBTTagCompound readNBTTagCompoundFromBuffer() throws IOException;
@Shadow protected abstract void writeNBTTagCompoundToBuffer(NBTTagCompound compound);
@Shadow public abstract int readVarIntFromBuffer();
@Shadow public abstract void writeVarIntToBuffer(int input);
@Shadow public abstract String readStringFromBuffer(int maxLength);
@Shadow public abstract PacketBuffer writeString(String string);
@Shadow public abstract byte[] readByteArray();
@Shadow public abstract void writeByteArray(byte[] array);
private ChannelBuf oppositeOrder;
public int cbuf$getCapacity() {
return this.buf.capacity();
}
public int cbuf$available() {
return this.buf.writerIndex() - this.buf.readerIndex();
}
public ChannelBuf cbuf$order(ByteOrder order) {
checkNotNull(order, "order");
if (this.buf.order().equals(order)) {
return (ChannelBuf) this;
}
if (this.oppositeOrder == null) {
this.oppositeOrder = (ChannelBuf) new PacketBuffer(this.buf.order(order));
}
return this.oppositeOrder;
}
public ByteOrder cbuf$getByteOrder() {
return this.buf.order();
}
public int cbuf$readerIndex() {
return this.buf.readerIndex();
}
public ChannelBuf cbuf$setReadIndex(int index) {
this.buf.readerIndex(index);
return (ChannelBuf) this;
}
public int cbuf$writerIndex() {
return this.buf.writerIndex();
}
public ChannelBuf cbuf$setWriteIndex(int index) {
this.buf.writerIndex(index);
return (ChannelBuf) this;
}
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| 2,030
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\section{Introduction}
Algebraic cobordism is a bigraded cohomology theory of smooth schemes represented by the Thom spectrum $\mathbf{MGL}$ in the stable motivic homotopy category.
Algebraic cobordism was introduced by Voevodsky as an analogue to complex cobordism to help solve the Milnor conjecture \cite{Voevodsky:ICM}. This paper investigates the bigraded motivic homotopy groups $\pi_\star(\mathbf{MGL})$ of the algebraic cobordism spectrum over the real numbers, number fields and rings of $\mathcal S$-integers.
The algebraic cobordism spectrum $\mathbf{MGL}$ is a $\P^1$-spectrum constructed analogously to the complex cobordism spectrum $\mathbf{MU}$ in topology \cite[6.3]{Voevodsky:ICM}.
Consider the Grassmann scheme $\operatorname{Gr}(m,n)$ of dimension $m$-planes in $\mathbb{A}^n$, and its canonical $m$-vector bundle $\gamma_{m,n}$.
Taking the colimit over $n$ we get the infinite Grassmannian $\operatorname{Gr}(m,\infty)$
with canonical bundle $\gamma_{m,\infty}$.
The embeddings $\operatorname{Gr}(m,\infty) \to \operatorname{Gr}(m+1, \infty)$ induce maps $\epsilon^1\oplus\gamma_{m,\infty} \to \gamma_{m+1,\infty}$, where $\epsilon^1$ is the trivial rank one bundle. In the unstable motivic homotopy category the infinite Grassmannians are classifying spaces for the isomorphism classes of $m$-vector bundles over smooth schemes \cite{Morel:A1}.
Taking the Thom spaces of the bundles $\gamma_{m,\infty}$ we get the constituent spaces of the $\P^1$-spectrum $\mathbf{MGL} = (\operatorname{Th}(\gamma_{0,\infty}), \operatorname{Th}(\gamma_{1,\infty}), \dots)$ with structure maps
\[
\P^1 \wedge \operatorname{Th}(\gamma_{m,\infty})
\cong \operatorname{Th}(\epsilon^1\oplus\gamma_{m,\infty})
\to \operatorname{Th}(\gamma_{m+1,\infty}).
\]
In motivic homotopy theory the spheres are bigraded
$S^{p,q} = (S^1_s)^{p-q}\wedge \mathbb{G}_m^{\wedge q}$, where $S^1_s$ is the simplicial circle and $\mathbb{G}_m$ is the punctured affine line pointed at 1.
Hence motivic homotopy groups are bigraded as well, similar to how $C_2$-equivariant homotopy groups are indexed by the trivial representation and the sign representation.
The bigraded homotopy groups of the algebraic cobordism spectrum are known along some special lines.
The Hopkins-Morel isomorphism implies that for fields in characteristic 0 there is an isomorphism $\pi_{2n,n}(\mathbf{MGL}) \cong L_n$,
where $L_*$ is the Lazard ring \cite[Proposition 8.2]{Hoyois:fromto}.
More generally Levine and Morel \cite{Levine-Morel} defined a cohomology theory $\Omega^*(-)$ on smooth schemes in geometric terms such that for a smooth scheme $X$ there is an isomorphism $\Omega^n(X)\cong \mathbf{MGL}^{2n,n}(X)$ \cite[Corollary 8.15]{Hoyois:fromto}.
It is an open problem to give a geometric description of all of $\mathbf{MGL}^\star(X)$.
For a field $F$ Spitzweck showed that $\mathbf{MGL}_{2n+1,n}(F;\mathbb{Z}) \cong F^\times \otimes L_{n+1}$ \cite[Corollary 7.5]{Spitzweck:mixed}, because the slice spectral sequence collapses in this range.
For the same reason $\mathbf{MGL}_{2n+2,n}(F;\mathbb{Z}) \cong K_2(F) \otimes L_{n+2}$,
and along the diagonal there is an isomorphism with Milnor $K$-theory $ K^M_{-n}(F) \cong \pi_{n,n}(\mathbf{MGL})$.
Furthermore, $\pi_{p,q}(\mathbf{MGL}) = 0$ for $p < q$ or $2p < q$.
We compute all of the bigraded motivic homotopy groups of the algebraic cobordism spectrum over number fields in terms of the integral motivic cohomology of the number fields, up to extension, in \Cref{thm:MGLF}.
Over rings of $\mathcal{S}$-integers and the real numbers we compute the homotopy groups of the 2-completed algebraic cobordism spectrum in \Cref{thm:OFS} and \Cref{thm:MGLR}.
As an application we relate the order of
the algebraic cobordism groups of the ring of $2$-integers in a totally real abelian number field $F$
to special values of the Dedekind $\zeta$-function of $F$ in \Cref{thm:zeta}.
Associated to the algebraic cobordism spectrum there is the motivic Brown-Peterson spectrum $\mathbf{BPGL}$, its truncations $\mathbf{BPGL}\langle n \rangle$, and motivic Morava $K$-theory $K(n)$.
The topological realizations of these motivic spectra play an important role in chromatic homotopy theory.
The homotopy groups of these motivic spectra can be computed by the same techniques used for $\mathbf{MGL}$.
We compute $\pi_\star(\mathbf{BPGL})$, $\pi_\star(\mathbf{BPGL}\langle n\rangle)$ over the real numbers in \Cref{thm:BPGL}, and $\pi_\star(K(n))$ over fields with virtual cohomological dimension less than $2^{n+1}-2$ in \Cref{thm:Kn}.
Our main tool is the slice spectral sequence and the slice filtration introduced by Voevodsky in \cite{Voevodsky:open}.
We also make use of the interplay between motivic and $C_2$-equivariant stable homotopy theory. This interplay is facilitated by the $C_2$-equivariant complex realization functor induced by sending a smooth scheme over $\mathbb{R}$ to its complex realization equipped with the $C_2$-action given by complex conjugation.
This allows us to use results of Hill, Hopkins and Ravenel \cite{HHR} in the motivic setting. Some of our proofs are both inspired by and heavily dependent on those of \cite{HHR}.
The results over rings of $\mathcal S$-integers relies on the work of Levine \cite{Levine99} and Spitzweck \cite{Spitzweck:commutative} on motivic cohomology of Dedekind domains. Our results over rings of integers are somewhat similar to those of Rognes and Weibel \cite{Rognes-Weibel} on algebraic $K$-theory of rings of 2-integers. They use the Bloch-Lichtenbaum spectral sequence to compute 2-primary algebraic $K$-theory of number fields and rings of $2$-integers.
\subsection*{Previous work}
Yagita \cite{Yagita:atiyah} uses the slice spectral sequence to compute the associated graded of $\pi_{*,*}(\mathbf{BPGL}/2)$ over the real numbers.
Even earlier Hu and Kriz \cite{Hu-Kriz:real} computed the coefficients $\pi_{*,*}^{C_2}(BP\mathbb{R})$ of the Brown-Peterson spectrum associated to $C_2$-equivariant complex cobordism. A careful proof of this is also in the appendix of \cite{Greenlees-Meier}.
Their answer has essentially the same form as $\mathbf{MGL}_{*,*}(\mathbb{R};\mathbb{Z}_2)$,
and they explore relations to $\mathbf{MGL}$ in \cite{Hu-Kriz:remarks},
where they also compute $K_\star(n)(\mathbb{R})$.
In \cite{Hill} Hill proves that the motivic Adams spectral sequence of $\mathbf{BPGL}\langle n \rangle$ over $\mathbb{R}$ collapses, and notes that $\pi_\star(\mathbf{BPGL})$ can be obtained from the computation of $\pi_\star^{C_2}(BP\mathbb{R})$ in \cite{Hu-Kriz:real}.
Ormsby uses the motivic Adams spectral sequence to compute $\pi_\star(\mathbf{BPGL}\langle n \rangle)$ over $p$-adic fields with the motivic Adams spectral sequence in \cite{Ormsby}.
Similar techniques are used by Ormsby and \O{}stv\ae{}r \cite{OP:BP} to compute $\pi_{*,*}(\mathbf{BPGL}\langle n \rangle)$ and $\pi_{*,*}(\mathbf{MGL}_2^{\wedge})$ over the rational numbers.
Ellis considers a motivic $C_2$-equivariant version of $\mathbf{MGL}$ and computes its coefficient over the complex numbers in his thesis \cite{Ellis}. %
Recently Heard \cite{Heard} compared the motivic slice filtration of $\mathbf{MGL}$ to the equivariant slice filtration of $\mathbf{MU}$, and computed $\pi_\star(\mathbf{BPGL})$ and $\pi_\star(\mathbf{BPGL}\langle n \rangle/2)$ over the real numbers.
\subsection*{Organization of this paper}
We begin with three short sections recalling theory and results used later in the paper.
In \Cref{sec:intro} we give a quick review of the relation between motivic and $C_2$-equivariant stable homotopy theory.
\Cref{sec:motcoh} follows with a description of (2-complete) motivic cohomology of the real numbers, number fields and rings of $\mathcal{S}$-integers in number fields.
In \Cref{sec:slice} we review the slice spectral sequence with particular attention to the case of $\mathbf{MGL}$.
Next are the computations, which are the most technical part of the paper.
We compute $\mathbf{MGL}_{*,*}(\mathbb{R};\mathbb{Z}_2)$ in \Cref{sec:MGLR} by first running the slice spectral sequence for $\mathbf{MGL}/2^n$ and then passing to the limit over $n$.
We determine the multiplicative extensions in $\mathbf{MGL}_{*,*}(\mathbb{R};\mathbb{Z}_2)$ by elementary means in contrast to \cite{Hu-Kriz:real}, cf.~\Cref{rmk:elementary}. This is similar to the proof in \cite{Greenlees-Meier}.
In \Cref{sec:MGLF} we use the computations over $\mathbb{R}$ to determine the differentials and the $E^\infty$-page of the slice spectral sequence over number fields.
In \Cref{sec:zeta} we relate the cardinality of $\mathbf{MGL}_{*,*}(\mathcal{O}_F[\frac{1}{2}];\mathbb{Z}_2)$ to special values of the Dedekind $\zeta$-function of a totally real abelian number field $F$.
This is a corollary of a result of Manfred Kolster and the computation of $\mathbf{MGL}_{*,*}(\mathcal{O}_F[\frac{1}{2}];\mathbb{Z}_2)$ in terms of motivic cohomology.
For completeness we summarize some easy results on the motivic homotopy groups of (2-completed) algebraic cobordism over fields of $2$-cohomological dimension less than or equal to $2$ in \Cref{sec:final}. In this case the slice spectral sequence collapses for degree reasons.
In \Cref{sec:morava} we compute the motivic homotopy groups of the (truncated) Brown-Peterson spectra $\mathbf{BPGL}$ and $\mathbf{BPGL}\langle n\rangle$ over the real numbers, and of Morava K-theory $K(n)$ over fields with virtual cohomological dimension $\text{vcd}(F) < 2(2^n - 1)$.
The techniques are the same as for the algebraic cobordism spectrum.
\subsection*{Acknowledgments}
I would like to thank Lorenzo Mantovani for interesting discussions which inspired the questions considered in this paper.
I would also like to thank John Rognes, Oliver R\"{o}ndigs, Glen Wilson and Paul Arne \O{}stv\ae{}r for helpful discussions and comments.
The author is grateful to Lennart Meier for pointing out an error in the statement of \Cref{thm:MGLR}. The author thanks the Hausdorff research Institute for Mathematics in Bonn for the hospitality and the delicious cakes during the Hausdorff Trimester Program on ``K-Theory and Related Fields'' in the summer of 2017, where this work was initiated. The author is also grateful for the hospitality of the Faculty of Mathematics of the University of Duisburg-Essen for a week during the autumn of 2017.
\subsection*{Notation}
Throughout the paper we will work over a number field $F$, with the exception of the final section on motivic Morava $K$-theory.
We write $\mathcal{O}_{F,\mathcal{S}}$ for the ring of $\mathcal S$-integers in $F$, for $\mathcal S \supset \{2, \infty \}$ a set of places.
We will work in the stable motivic homotopy category $\mathcal{SH}(F)$.
The following table summarizes the notation used in the paper:
\begin{center}
\footnotesize
\begin{tabular}{l|l}
$\mathcal{SH}(F)$, $\mathcal{SH}^{C_2},\mathcal{SH}$ & motivic, $C_2$-equivariant and ordinary stable homotopy category \\
$\mathbf{E}$, $\S$ & motivic spectrum, motivic sphere spectrum \\
$\mathbf{MGL}$, $\mathbf{M}\Z$ & algebraic cobordism, motivic cohomology spectrum \\
$\mathbf{BPGL}$, $\mathbf{BPGL}\langle n \rangle$ & motivic (truncated) Brown-Peterson spectrum \\
$K(n)$ & $n$th motivic Morava $K$-theory \\
$\mathbf{E}_2^{\wedge}$, $\mathbf{M}\Z_2$ & $\operatorname{holim}_n \mathbf{E}/2^n$, $\operatorname{holim}_n \mathbf{M}\Z/2^n$ \\
$\pi_{p,q}(\mathbf{E})$ & $[(S^1)^{\wedge p-q}\wedge \mathbb{G}_m^{\wedge q}, \mathbf{E}]_{\mathcal{SH}(F)}$ \\
$\mathbf{E}_{*,*}(F;\mathbb{Z})$, $\mathbf{E}_{*,*}(F;\mathbb{Z}/2^n)$ & $\pi_{*,*}(\mathbf{E})$, $\pi_{*,*}(\mathbf{E}/2^n)$ \\
$\mathbf{E}_{*,*}(F;\mathbb{Z}_2)$ &
$\pi_{*,*}(\mathbf{E}_2^{\wedge})$ \\
$H^{p,q}(F;A)$ & $\pi_{-p,-q}(\mathbf{M}A)$, $A \neq \mathbb{Z}_2$ an abelian group \\ %
$\wt{H}^{p,q}(F;\mathbb{Z})$ & $\ker(H^{p,q}(F;\mathbb{Z}) \to \oplus^{r_1} H^{p,q}(\mathbb{R};\mathbb{Z}))$, $F$ a field with $r_1$ real embeddings \\
$\mathsf{f}_q$, $\mathsf{s}_q$ & $q$th effective cover, $q$th slice \\
$E^r(F;\mathbb{Z})$, $E^r(F;\mathbb{Z}/2^n)$ & $E^r$-page of the slice spectral sequence of $\mathbf{MGL}$ and $\mathbf{MGL}/2^n$ over $F$ \\
$E^r(F;\mathbb{Z}_2)$ & $\lim_n E^r(F;\mathbb{Z}/2^n)$ \\
$E^r$, $E^r(\mathbb{Z})$, $E^r(\mathbb{Z}/2^n)$ & $E^r(F;\mathbb{Z})$, and so on \\
$r_1$, $r_2$ & real and complex conjugate pairs of embeddings of a number field $F$ \\
$\mathcal{O}_{F,\mathcal{S}}$ & ring of $\mathcal{S}$-integers in a number field $F$, for a set of places $\mathcal{S} \supset \{2,\infty\}$ \\
$\nu_2(n)$ & the $2$-adic valuation of $n$ \\
$L_*$, $K^M_*(F), k_*(F)$ & the Lazard ring, integral and mod 2 Milnor $K$-theory of $F$ \\
$R_\mathbb{C}$, $R_\mathbb{C}^{C_2}$, $R_\mathbb{R}$, & complex realization, $C_2$-equivariant complex realization, real realization \\
$\Phi^{C_2}$, $\rho$, $\sigma$ & geometric fixed points, $\{\pm 1 \} = S^{0} \hookrightarrow \mathbb{G}_m \in \pi_{-1,-1}(\S)$, sign representation of $C_2$ \\
$\mathcal{A}^\star, \mathcal{A}_\star$, $\Delta$ & motivic Steenrod algebra, dual, coproduct of dual \\
$\tau$, $\rho$, $\xi_i$, $\tau_i$ & algebra generators of $\mathcal{A}_\star(\mathbb{R})$ \\
$\operatorname{Sq}^i$, $Q_i$ & motivic Steenrod squares, Milnor primitive, the dual of $\tau_i$ \\
$\text{vcd}(F)$ & $\text{cd}_2(F(\sqrt{-1}))$, virtual cohomological dimension of a field $F$
\end{tabular}
\end{center}
\section{Motivic and $C_2$-equivariant stable homotopy theory}
\label{sec:intro}
In this section we recall the construction of some functors between the motivic, $C_2$-equivariant and topological stable homotopy category, and the image of some of the spectra we are interested in by these functors.
The main observation is \Cref{lem:fixed-points-realization}, which tells us how to compute real realizations.
Recall that there are realization functors
\begin{align*}
R_\mathbb{C} : \mathcal{SH}(\mathbb{C}) \to \mathcal{SH}, \qquad
R_\mathbb{C}^{C_2} : \mathcal{SH}(\mathbb{R}) \to \mathcal{SH}^{C_2}, \qquad
R_\mathbb{R} : \mathcal{SH}(\mathbb{R}) \to \mathcal{SH}.
\end{align*}
The $C_2$-equivariant complex realization functor $R_\mathbb{C}^{C_2}$ is induced by
\[
\operatorname{Sm}/\operatorname{Spec}(\mathbb{R}) \ni X \mapsto X(\mathbb{C}) \in \mathcal{T}op^{C_2}
\]
with the analytic topology and $C_2$-action given by complex conjugation,
while $R_\mathbb{R}$ is induced by
\[
\operatorname{Sm}/\operatorname{Spec}(\mathbb{R}) \ni X \mapsto X(\mathbb{R}) \in \mathcal{T}op.
\]
This give rise to well defined functors of the stable homotopy categories since the motivic stable homotopy category is generated by suspension spectra of schemes, the functors map motivic spheres to ($C_2$-equivariant) spheres, and are compatible with $\mathbb{A}^1$-invariance and Nisnevich descent.
Both realization functors are left adjoints.
See \cite{HO:galois} for a more careful construction and thorough discussion of the realization functors.
Recall that the geometric fixed points functor $\Phi^{C_2} : \mathcal{SH}^{C_2} \to \mathcal{SH}$ is characterized by the following properties \cite[Remark 7.15]{Schwede}, \cite[Proposition 2.45]{HHR}:
\begin{enumerate}
\item $\Phi^{C_2}(\Sigma^\infty_+ A) = \Sigma^\infty_+ (A^{C_2})$,
\item $\Phi^{C_2}$ is monoidal,
\item $\Phi^{C_2}$ preserves filtered homotopy colimits.
\end{enumerate}
Hence
\[
R_\mathbb{R}\Sigma_+^\infty (X) = \Sigma_+^\infty (X(\mathbb{R})) = \Sigma_+^\infty (X(\mathbb{C})^{C_2}) = \Phi^{C_2}\Sigma_+^\infty(X(\mathbb{C})) = \Phi^{C_2}R_\mathbb{C}^{C_2}\Sigma^\infty_+(X).
\]
Since $\Phi^{C_2}$ preserves filtered colimits and $\mathcal{SH}(\mathbb{R})$ is generated by suspension spectra of schemes we get the following lemma, which seems to be folklore.
\begin{lemma}
\label{lem:fixed-points-realization}
Real realization is naturally isomorphic to the geometric fixed points of the $C_2$-equivariant complex realization. That is,
$
R_\mathbb{R} \cong \Phi^{C_2}R_\mathbb{C}^{C_2}.
$
\end{lemma}
Bachmann proved an equivalence of $\mathcal{SH}(\mathbb{R})[\rho^{-1}]$ with $\mathcal{SH}$.
\begin{theorem}[\protect{\cite{Bachmann:real}}]
\label{thm:tom}
Real realization induces an equivalence
\[
\mathcal{SH}(\mathbb{R})[\rho^{-1}] \to \mathcal{SH}.
\]
\end{theorem}
To summarize, we have a commutative square
\[
\begin{tikzcd}
\mathcal{SH}(\mathbb{R}) \ar[r, "R_\mathbb{C}^{C_2}"]\ar[d] & \mathcal{SH}^{C_2} \ar[d, "\Phi^{C_2}"] \\
\mathcal{SH}(\mathbb{R})[\rho^{-1}] \ar[r, "R_\mathbb{R}"] & \mathcal{SH}
\end{tikzcd}
\]
where the bottom horizontal map is an equivalence.
We have $R^{C_2}_\mathbb{C}(\mathbf{MGL}) = \mathbf{MU}$, where $\mathbf{MU}$ is considered as the $C_2$-equivariant complex cobordism spectrum defined by Landweber \cite{Landweber}.
Indeed, $\mathbf{MGL} \simeq \operatorname{colim}_q \Sigma^{-2q,-q}\operatorname{Th}(\gamma_{q,\infty}),$
and the $C_2$-equivariant complex realization of $\operatorname{Th}(\gamma_{q,\infty})$ are the Thom spaces defining $\mathbf{MU}$ as a $C_2$-spectrum.
Heller and Ormsby showed that the $C_2$-equivariant complex realization of $\mathbf{M}\Z$ is $C_2$-equivariant Bredon-cohomology \cite[Theorem 4.17]{HO:galois}.
That is, $R_\C^{C_2}(\mathbf{M}\Z) = \mathbf{H}\Z$, for $\mathbb{Z}$ the constant Mackey functor.
\section{Motivic cohomology of the real numbers and number fields}
\label{sec:motcoh}
In this section we state some results on the structure of motivic cohomology of the real numbers, number fields and rings of $\mathcal{S}$-integers in number fields. For a number field $F$ with $r_1$ real embeddings, let $\mathcal S \supset \{2, \infty\}$ be a (not necessarily finite) set of places in $F$. We denote the ring of $\mathcal S$-integers by $\mathcal{O}_{F,\mathcal{S}}$.
The Bloch-Kato-conjecture (\cite{Voevodsky:Z2}, \cite{Voevodsky:Zl}) gives an isomorphism of motivic cohomology and
\'{e}tale-cohomology with finite coefficients above the diagonal.
That is,
\begin{equation}
\label{eq:BK}
H^{p,q}(F;A) \cong H^{p,q}_{\acute{e}t}(F; A), \text{ when } p \leq q,
\end{equation}
and $A$ a finite abelian group.
We have, cf.~\cite[2.1]{DI:adams},
\[
H^{*,*}(\mathbb{R};\mathbb{Z}/2) \cong \mathbb{Z}/2[\rho,\tau],
\]
where $\rho = [-1] \in \mathbb{R}^\times/(\mathbb{R}^\times)^2 \cong H^{1,1}(\mathbb{R};\mathbb{Z}/2)$,
and $\tau = [-1] \in \mu_2(\mathbb{R}) \cong H^{0,1}(\mathbb{R};\mathbb{Z}/2)$.
\begin{lemma}
\label{lem:HR}
Let $n > 1$.
As a $\mathbb{Z}/2^n$-algebra mod $2^n$ motivic cohomology of the real numbers is
\[
H^{*,*}(\mathbb{R};\mathbb{Z}/2^n) \cong \mathbb{Z}/2^{n}[\rho,\tau, u]/(2\rho, 2\tau, \tau^2).
\]
The elements $\rho$, $\tau$ and $u$ are the generators of the cohomology groups in bidegrees $(1,1)$, $(0,1)$ and $(0,2)$, represented by $-1$, $-1$ and $\zeta_{2^n}$, respectively.
As a $\mathbb{Z}_2$-algebra $2$-complete motivic cohomology of the real numbers is
\[
H^{*,*}(\mathbb{R};\mathbb{Z}_2) \cong \mathbb{Z}_2[\rho, u]/(2\rho).
\]
\end{lemma}
Note that $\rho$ maps to $\rho$ and $u$ maps to $\tau^2$ via the projection $H^{*,*}(\mathbb{R};\mathbb{Z}/2^n) \to H^{*,*}(\mathbb{R};\mathbb{Z}/2)$.
The element $\tau \in H^{*,*}(\mathbb{R};\mathbb{Z}/2^n)$ is the image of $\tau \in H^{*,*}(\mathbb{R};\mathbb{Z}/2)$ through $H^{*,*}(\mathbb{R};\mathbb{Z}/2) \to H^{*,*}(\mathbb{R};\mathbb{Z}/2^n)$. We use the same name for $\tau$ and $\rho$ in the various motivic cohomology groups.
\begin{proof}
With \eqref{eq:BK} this reduces to a computation in \'{e}tale cohomology.
That is, we must compute
\[
H^{p}_{\acute{e}t}(\mathbb{R};\mu_{2^n}^{\otimes q}) = \operatorname{Ext}^p_{\mathbb{Z}[C_2]}(\mathbb{Z}, \mu_{2^n}^{\otimes q}),
\] where $\mathbb{Z}[C_2] = \mathbb{Z}[x]/(x^2-1)$.
Use the resolution
\[
\dots \xrightarrow{\cdot (x+1)} \mathbb{Z}[C_2] \xrightarrow{\cdot (x-1)} \mathbb{Z}[C_2] \xrightarrow{x \mapsto 1} \mathbb{Z},
\]
and that $x$ acts on $\mu_{2^n}^{\otimes q} = \mathbb{Z}/2^n$ as $1$ when $q$ is even and $-1$ when $q$ is odd.
Products are formed by forming tensor products of functions.
To obtain the $2$-adic description we take the limit.
The structure maps in the inverse system are induced by the projection maps of the coefficients.
The structure map of $H^{p}_{\acute{e}t}(\mathbb{R};\mu_{2^n}^{\otimes q})$ is multiplication by $2$ when $p - q$ is odd. Hence, all multiples of $\tau$ vanish in the limit.
\end{proof}
\begin{theorem}[\protect{\cite[Theorems 14.5, 14.6]{Levine99}}]
\label{thm:hOFS}
Let $\mathcal{O}_{F,\mathcal{S}}$ be the ring of $\mathcal{S}$-integers,
$\mathcal{S} \supset \{2, \infty\}$,
in a number field $F$ with $r_1$ real embeddings.
Let $\mathbb{Z}_{\mathcal S}$ be the localization of $\mathbb{Z}$ such that the primes not in $\mathcal S$ are invertible.
The motivic cohomology groups $H^{p,q}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_{\mathcal S})$ are trivial outside the range $1 \leq p \leq q$ except possibly in the bidegrees $(0,0)$ and $(2,1)$.
We have $H^{0,0}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_{\mathcal S}) = \mathbb{Z}_{\mathcal S}$ and $H^{2,1}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_{\mathcal S}) = \operatorname{Pic}(\mathcal{O}_{F,\mathcal{S}})\otimes\mathbb{Z}_{\mathcal S}$.
For $3 \leq p\leq q$ the real embeddings induce an isomorphism
\[
H^{p,q}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_{\mathcal S})
\cong
\oplus^{r_1} H^{p,q}(\mathbb{R};\mathbb{Z}_2)
\cong
\begin{cases}
(\mathbb{Z}/2)^{r_1} & p \equiv q \bmod 2 \\
0 & p \not\equiv q \bmod 2.
\end{cases}
\]
\end{theorem}
\begin{proof}[Proof sketch]
We only prove the final statement for number fields following \cite[Theorem 14.5]{Levine99}.
By the Beilinson-Lichtenbaum conjecture \cite[Theorem 6.1]{Voevodsky:Z2}, \cite[Theorem 6.17]{Voevodsky:Zl} there is an isomorphism
$
H^{p,q}(F;\mathbb{Z}/\ell) \cong H^{p}_{\acute{e}t}(F;\mu_{\ell}^{\otimes q}).
$
A number field $F$ has $\ell$-cohomological dimension 2 when $\ell$ is odd \cite[Proposition 8.3.17]{NSW}.
When $\ell = 2$ we have an isomorphism $H^{p,q}(F;\mathbb{Z}/2) \cong K^M_{p}/2\{\tau^q \} \cong \mathbb{Z}/2$ for $q \geq p \geq 3$ by \cite[Theorem A.2]{Milnor:K-theory}.
Levine shows $H^{p,q}(F;\mathbb{Q}) = 0$ for $q \neq 1$, except for $p=q=0$ \cite[Theorem 14.5]{Levine99}. Combining the torsion part and the rational part implies the statement.
\end{proof}
More detailed descriptions of $H^{1,q}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_{\mathcal S})$ and $H^{2,q}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_{\mathcal S})$ are given in \cite[14]{Levine99}.
\begin{corollary}
\label{lem:H3R}
Let $F$ be a number field.
Then for $p \geq 3$ the canonical maps
\[
H^{p,q}(F;\mathbb{Z}_{\mathcal S})
\xrightarrow{\cong} H^{p,q}(F;\mathbb{Z}_2)
\xrightarrow{\cong} \oplus^{r_1} H^{p,q}(\mathbb{R};\mathbb{Z}_2)
\]
are isomorphisms.
\end{corollary}
\begin{lemma}
\label{lem:HFR}
\label{lem:H12R}
Let $\mathcal{O}_{F,\mathcal{S}}$ be a number field or its ring of $\mathcal{S}$-integers.
Then the canonical map
\[
H^{2,q}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_{\mathcal S}) \twoheadrightarrow \oplus^{r_1} H^{2,q}(\mathbb{R};\mathbb{Z}_2)
\]
induced by the real embeddings
is surjective.
\end{lemma}
\begin{proof}
By \Cref{lem:HR} we may assume $q$ is even.
Consider the commutative diagram
\[
\begin{tikzcd}
H^{2,q}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_{\mathcal S}) \ar[r]\ar[d, "\operatorname{pr}"] & \oplus^{r_1}H^{2,q}(\mathbb{R};\mathbb{Z}_2)\ar[d, "\operatorname{pr}"] \\
H^{2,q}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}/2) \ar[r]\ar[d, "\rho"] & \oplus^{r_1}H^{2,q}(\mathbb{R};\mathbb{Z}/2) \ar[d, "\rho"] \\
H^{3,q+1}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}/2) \ar[r] & \oplus^{r_1}H^{3,q+1}(\mathbb{R};\mathbb{Z}/2).
\end{tikzcd}
\]
The vertical maps in the right column are isomorphisms by \Cref{lem:HR}.
The bottom horizontal map is an isomorphism by \Cref{thm:hOFS}. %
Multiplication by $\rho$ induces a surjective map $H^{2,q}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}/2) \to H^{3,q+1}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}/2)$ by \cite[Lemma 7.18]{KRO}.
The projection $\operatorname{pr}: H^{2,q}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_{\mathcal S}) \to H^{2,q}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}/2)$ has cokernel $H^{3,q}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_{\mathcal S}) = 0$.
Hence the top horizontal map is surjective.
\end{proof}
\begin{remark}
In general, the map $H^{1,q}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_{\mathcal S}) \to \oplus^{r_1} H^{1,q}(\mathbb{R};\mathbb{Z}_2)$ has a nontrivial cokernel,
cf.~\cite{Rognes}.
However, the cokernels are isomorphic for $q \geq 1$ of the same parity.
Indeed, for $q$ odd this follows from the commutative diagram
\[
\begin{tikzcd}
H^{1,q}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_{\mathcal S}) \ar[r]\ar[d, "\operatorname{pr}"] & \oplus^{r_1}H^{1,q}(\mathbb{R};\mathbb{Z}_2)\ar[d, "\operatorname{pr}"] \\
H^{1,q}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}/2) \ar[r]\ar[d, "\tau^2"] & \oplus^{r_1}H^{1,q}(\mathbb{R};\mathbb{Z}/2) \ar[d, "\tau^2"] \\
H^{1,q+2}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}/2) \ar[r] & \oplus^{r_1}H^{1,q+2}(\mathbb{R};\mathbb{Z}/2) \\
H^{1,q+2}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_{\mathcal S}) \ar[r]\ar[u, "\operatorname{pr}"] & \oplus^{r_1}H^{1,q+2}(\mathbb{R};\mathbb{Z}_2)\ar[u, "\operatorname{pr}"]
\end{tikzcd}
\]
since $\operatorname{pr}$ is an isomorphism over $\mathbb{R}$ and $\tau$ is an isomorphism on mod 2 motivic cohomology.
\end{remark}
\section{The slice spectral sequence of algebraic cobordism}
\label{sec:slice}
In this section we construct the slices spectral sequence of $\mathbf{MGL}$ and discuss its multiplicative and strong convergence properties.
The slice filtration was originally introduced by Voevodsky in \cite{Voevodsky:open}. In \cite{RO:hermitian}, \cite{RSO:April1} and \cite{KRO} there are slice spectral sequence computations similar in spirit to ours.
The slice filtration is obtained by considering the triangulated subcategory $\mathcal{SH}^{\text{eff}}(F) \subset \mathcal{SH}(F)$ generated by suspension spectra of smooth schemes.
We then filter by $i_q : \Sigma^{2q,q}\mathcal{SH}^{\text{eff}}(F) \hookrightarrow \mathcal{SH}(F), q \in \mathbb{Z}$. Each inclusion $i_q$ has a right adjoint $r_q$,
and the composite $\mathsf{f}_q = i_q \circ r_q$ is defined to be the $q$th effective cover.
The cofiber $\mathsf{s}_q(-) = \operatorname{cofib}(\mathsf{f}_{q+1}(-) \to \mathsf{f}_q(-))$ is the $q$th slice functor,
and these functors assemble to a tower of cofiber sequences which gives rise to a spectral sequence in the usual way. Since $\mathcal{SH}^{\text{eff}}(F)$ is a triangulated category, the functors $\mathsf{f}_q$ and $\mathsf{s}_q$ are exact.
It is possible to consider a finer filtration of $\mathcal{SH}(F)$ by considering the subcategory $\mathcal{SH}^\text{veff}(F) \subset \mathcal{SH}^\text{eff}(F)$ consisting of the spectra which are connective in Morel's $t$-structure \cite{Bachmann:veff}. For $\mathbf{MGL}$ these filtrations agree since $\mathsf{f}_q(\mathbf{MGL})$ is $q$-connective, see \Cref{lem:veff-agree}. %
Recall that the slices of $\mathbf{MGL}$ are \cite{Voevodsky:open}, \cite[Theorem 4.7]{Spitzweck:slices}, \cite[Theorem 8.5]{Hoyois:fromto}
\[
\mathsf{s}_q(\mathbf{MGL}) = \Sigma^{2q,q}\mathbf{M}\Z \otimes L_q,
\]
compatible with the ring map $L_* \to \mathbf{MGL}_{2*,*}$.
Here $L_* = \mathbb{Z}[x_1, x_2, \cdots]$ is the Lazard ring,
with generators $x_i$ in degree $i$ (i.e., half of the usual indexing).
Since the slices are modules over $\mathsf{s}_0(\S) = \mathbf{M}\Z$ \cite[Theorem 3.6.22]{Pelaez} the multiplicative structure on the slices $\mathsf{s}_*(\mathbf{MGL})$ is the one induced from the product on $\mathbf{M}\Z$ and $L_*$.
The slice spectral sequence of $\mathbf{MGL}$ is obtained by taking homotopy groups in the slice tower of $\mathbf{MGL}$.
\[
\begin{tikzcd}
\dots \ar[r] & \mathsf{f}_{q+1}(\mathbf{MGL}) \ar[r]\ar[d] & \mathsf{f}_{q}(\mathbf{MGL}) \ar[r]\ar[d] & \mathsf{f}_{q-1}(\mathbf{MGL}) \ar[r]\ar[d] & \dots \ar[r] & \mathsf{f}_0(\mathbf{MGL}) = \mathbf{MGL} \\
& \mathsf{s}_{q+1}(\mathbf{MGL}) & \mathsf{s}_{q}(\mathbf{MGL}) & \mathsf{s}_{q-1}(\mathbf{MGL}) &
\end{tikzcd}
\]
This spectral sequence has $E^1$-page
\[
E^1_{p,q,w} = \pi_{p,w}\mathsf{s}_q(\mathbf{MGL}),
\]
and $d^r$-differential
\[
d^r : E^r_{p,q,w} \to E^r_{p-1,q+r,w}.
\]
By the work of Pelaez the diagrams
\[
\begin{tikzcd}
\mathsf{f}_{q+1}(\mathbf{MGL})\wedge \mathsf{f}_{q'}(\mathbf{MGL}) \ar[r]\ar[d] & \mathsf{f}_{q+q'+1}(\mathbf{MGL}) \ar[d] \\
\mathsf{f}_{q}(\mathbf{MGL})\wedge \mathsf{f}_{q'}(\mathbf{MGL}) \ar[r] & \mathsf{f}_{q+q'}(\mathbf{MGL})
\end{tikzcd}
\]
in $\mathcal{SH}(F)$ are induced by a zigzag of commutative diagrams in motivic symmetric spectra \cite[Theorem 3.6.16]{Pelaez}.
Hence the slice filtration gives rise to a multiplicative Cartan-Eilenberg system,
and the slice spectral sequence is a multiplicative spectral sequence, see also \cite{GRSO}.
For formal reasons the slice spectral sequence is conditionally convergent to the homotopy groups of the slice completion (\cite[8.4]{Hoyois:fromto}, \cite[Definition 3.1]{RSO:April1}) of $\mathbf{MGL}$.
Since $\mathbf{MGL}$ is slice complete \cite[Lemma 8.10, Corollary 2.4]{Hoyois:fromto}, i.e., $\mathbf{MGL}$ is equivalent to its slice completion, the slice spectral sequence is conditionally convergent to the homotopy groups of $\mathbf{MGL}$.
This is also true over rings of $\mathcal S$-integers for $\mathbf{MGL}$ localized at $\mathcal S$, $\mathbf{MGL}_{\mathcal S}$, see \cite{Spitzweck:mixed}.
Since in a fixed tridegree $(p,q,w)$ there are only finitely many entering and exiting differentials of $E^1_{p,q,w}$, the conditional convergence is in fact strong. That is, we have a strongly convergent spectral sequence
\[
E^1_{p,q,w} = \pi_{p,w}\mathsf{s}_q(\mathbf{MGL}) = H^{2q-p,q-w}(F;\mathbb{Z})\otimes L_q \implies \pi_{p,w}(\mathbf{MGL}).
\]
See \Cref{fig:E1-page} for a picture of the spectral sequence.
Similarly we also have strong convergence of the slice spectral sequence for $\mathbf{MGL}/2^n$.
Its slices are
\[
\mathsf{s}_q(\mathbf{MGL}/2^n) = \Sigma^{2q,q}\mathbf{M}\Z/2^n \otimes L_q.
\]
Consider the system of maps
\begin{equation}
\label{eq:2n-sys}
\begin{tikzcd}
\mathbf{MGL} \ar[r, "2^n"] & \mathbf{MGL} \ar[r] & \mathbf{MGL}/2^n \ar[r] & \Sigma^{1,0}\mathbf{MGL} \\
\mathbf{MGL} \ar[r, "2^{n+1}"]\ar[u, "2"] & \mathbf{MGL} \ar[r]\ar[u] & \mathbf{MGL}/2^{n+1} \ar[r]\ar[u] & \Sigma^{1,0}\mathbf{MGL} \ar[u, "2"]
\end{tikzcd}
\end{equation}
and the induced limit of spectral sequences
\[
E^r_{p,q,w}(\mathbb{Z}_2) = \lim_n E^r_{p,q,w}(\mathbb{Z}/2^n).
\]
In general we make no claims about $\{E^r_{p,q,w}(\mathbb{Z}_2)\}_r$ being a spectral sequence or its convergence.
However, if the groups $E^\infty_{p,q,w}(\mathbb{Z}/2^n)$ are finite
then $E^\infty_{p,q,w}(\mathbb{Z}_2)$ is the associated graded of an exhaustive, Hausdorff and complete filtration of $\pi_{*,*}(\mathbf{MGL}_2^{\wedge})$.
This is the case over the real numbers and rings of $\mathcal S$-integers when $\mathcal S$ is finite.
\begin{figure}
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{\draw (2,4) -- (2,2);}
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{\draw (6,6) -- (4,6);}
{\draw (4,6) -- (4,4);}
{\node[above right=0pt] at (12,6.4) { $0,3$};}
{\draw (12,6) -- (14,6);}
{\draw (14,6) -- (14,8);}
{\draw (14,8) -- (12,8);}
{\draw (12,8) -- (12,6);}
{\node[above right=0pt] at (10,6.4) { $1,3$};}
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{\draw (6,6) -- (8,6);}
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{\draw (8,8) -- (6,8);}
{\draw (6,8) -- (6,6);}
{\node[above right=0pt] at (-5.0,0) {\normalsize $L_{w} \otimes $};}
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{\node[above right=0pt] at (9.8,10.6) { $q,$};}
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{\draw (26,14) -- (24,14);}
{\draw (24,14) -- (24,12);}
{\node[above right=0pt] at (21.8,12.6) { $1,$};}
{\node[above right=0pt] at (21.8,12) { $q + 1$};}
{\draw (22,12) -- (24,12);}
{\draw (24,12) -- (24,14);}
{\draw (24,14) -- (22,14);}
{\draw (22,14) -- (22,12);}
{\node[above right=0pt] at (19.8,12.6) { $2,$};}
{\node[above right=0pt] at (19.8,12) { $q + 1$};}
{\draw (20,12) -- (22,12);}
{\draw (22,12) -- (22,14);}
{\draw (22,14) -- (20,14);}
{\draw (20,14) -- (20,12);}
{\node[above right=0pt] at (17.8,12.6) { $3,$};}
{\node[above right=0pt] at (17.8,12) { $q + 1$};}
{\draw (18,12) -- (20,12);}
{\draw (20,12) -- (20,14);}
{\draw (20,14) -- (18,14);}
{\draw (18,14) -- (18,12);}
{\node[above right=0pt] at (19.8,20.6) { $2q,$};}
{\node[above right=0pt] at (19.8,20) { $2q$};}
{\draw (20,20) -- (22,20);}
{\draw (22,20) -- (22,22);}
{\draw (22,22) -- (20,22);}
{\draw (20,22) -- (20,20);}
{\node[above right=0pt] at (21.8,20.6) { $2q - 1,$};}
{\node[above right=0pt] at (21.8,20) { $2q$};}
{\draw (22,20) -- (24,20);}
{\draw (24,20) -- (24,22);}
{\draw (24,22) -- (22,22);}
{\draw (22,22) -- (22,20);}
{\node[above right=0pt] at (21.8,22.6) { $2q + 1,$};}
{\node[above right=0pt] at (21.8,22) { $2q + 1$};}
{\draw (22,22) -- (24,22);}
{\draw (24,22) -- (24,24);}
{\draw (24,24) -- (22,24);}
{\draw (22,24) -- (22,22);}
{\node[above right=0pt] at (23.8,24.6) { $2q + 2,$};}
{\node[above right=0pt] at (23.8,24) { $2q + 2$};}
{\draw (24,24) -- (26,24);}
{\draw (26,24) -- (26,26);}
{\draw (26,26) -- (24,26);}
{\draw (24,26) -- (24,24);}
{\node[above right=0pt] at (23.8,22.6) { $2q,$};}
{\node[above right=0pt] at (23.8,22) { $2q + 1$};}
{\draw (24,22) -- (26,22);}
{\draw (26,22) -- (26,24);}
{\draw (26,24) -- (24,24);}
{\draw (24,24) -- (24,22);}
{\node[above right=0pt] at (17.8,18.6) { $2q - 1,$};}
{\node[above right=0pt] at (17.8,18) { $2q - 1$};}
{\draw (18,18) -- (20,18);}
{\draw (20,18) -- (20,20);}
{\draw (20,20) -- (18,20);}
{\draw (18,20) -- (18,18);}
{\node[above right=0pt] at (19.8,18.6) { $2q - 2,$};}
{\node[above right=0pt] at (19.8,18) { $2q - 1$};}
{\draw (20,18) -- (22,18);}
{\draw (22,18) -- (22,20);}
{\draw (22,20) -- (20,20);}
{\draw (20,20) -- (20,18);}
{\node[above right=0pt] at (13.0,18) {\normalsize $L_{w+2q - 1} \otimes $};}
{\node[above right=0pt] at (15.0,20) {\normalsize $L_{w+2q} \otimes $};}
{\node[above right=0pt] at (17.0,22) {\normalsize $L_{w+2q + 1} \otimes $};}
{\node[above right=0pt] at (19.0,24) {\normalsize $L_{w+2q + 2} \otimes $};}
{\node[above right=0pt] at (0,-2) {\footnotesize $\pi_{2w, w}$};}
{\node[above right=0pt] at (2,-4) {\footnotesize $\pi_{2w - 1, w}$};}
{\node[above right=0pt] at (4,-2) {\footnotesize $\pi_{2w - 2, w}$};}
{\node[above right=0pt] at (6,-4) {\footnotesize $\pi_{2w - 3, w}$};}
{\node[above right=0pt] at (8,-2) {\footnotesize $\pi_{2w - 4, w}$};}
{\node[above right=0pt] at (10,-4) {\footnotesize $\pi_{2w - 5, w}$};}
{\node[above right=0pt] at (12,-2) {\footnotesize $\pi_{2w - 6, w}$};}
{\node[above right=0pt] at (16,-3.0) {\large $\cdots$};}
{\node[above right=0pt] at (18,-4) {\footnotesize $\pi_{2w-2q - 1, w}$};}
{\node[above right=0pt] at (20,-2) {\footnotesize $\pi_{2w-2q, w}$};}
{\node[above right=0pt] at (22,-4) {\footnotesize $\pi_{2w-2q + 1, w}$};}
{\node[above right=0pt] at (24,-2) {\footnotesize $\pi_{2w-2q + 2, w}$};}
{\draw[->] (-5.0,-0.5) -- (-5.0,26.5);}
{\draw[->] (-5.0,-0.5) -- (29.0,-0.5);}
{\node[above right=0pt] at (-5.0,25.5) { $q$};}
{\node[above right=0pt] at (28.0,-0.5) { $p$};}
\end{tikzpicture}
\caption{The $E^1$-page of the slice spectral sequence of $\mathbf{MGL}$ with Adams grading in weight $w$.
Each box with indices $(p, q)$ represents a copy of $H^{p,q}(F;\mathbb{Z})$ tensored with $L_{w + q}$.
The $d^r$-differentials goes one step to the left and $r$-steps upwards.
The filtration degree is along the vertical $q$-axis, so the abutment is read off from the $E^\infty$-page vertically.
The motivic homotopy group each column contributes to is indicated below the $E^1$-page.
}
\label{fig:E1-page}
\end{figure}
\section{Algebraic cobordism of the real numbers}
\label{sec:MGLR}
In this section we compute $\mathbf{MGL}_{*,*}(\mathbb{R};\mathbb{Z}_2)$.
We first consider the slice spectral sequence which converges to $\mathbf{MGL}_{*,*}(\mathbb{R};\mathbb{Z}/2^n)$, $n > 1$, and calculate its $E^\infty$-page.
Then we pass to the limit to obtain the homotopy groups of the $2$-complete algebraic cobordism spectrum $\mathbf{MGL}_{2}^{\wedge}$. Throughout this section we always assume $n > 1$.
\subsection*{Mod $2^n$ calculations}
Since $\mathbf{MGL}/2^n$ is a ring spectrum for $n > 1$, cf.~\cite{Oka},
the slice spectral sequence is multiplicative,
and we have a Leibniz rule. %
Hence, it suffices to determine the differentials on the algebra generators.
As an algebra, $E^1_{*,*,*}$ is generated by
$E^1_{-1,0,-1} = H^{1,1}\otimes L_0$,
$E^1_{0,0,-1} = H^{0,1}\otimes L_0$,
$E^1_{0,0,-2} = H^{0,2}\otimes L_0$
and
$E^1_{2i,i,i} = H^{0,0}\otimes L_i$, for $i \in \mathbb{Z}$.
The algebra generators are $\rho$, $\tau$, $u$ and $x_i$ in degree $(-1, 0, -1)$, $(0, 0, -1)$, $(0, 0, -2)$ and $(2i, i, i)$.
For degree reasons, the only algebra generators on the $E^r$-page which can support differentials are powers of $u$ in $E^r_{0,0,2*} \cong H^{0,2*}\otimes L_0$.
Their differentials are described in \Cref{thm:diffs} below.
We need some results of \cite{HHR} for the group $G = C_2$.
In this case the norm map $N = N^{C_2}_{C_2}$ is the identity \cite[Proposition 2.27]{HHR},
and the norm construction on $\mathbf{MU}$ is simply $\mathbf{MU}$, that is $\mathbf{MU}^{((C_2))} = \mathbf{MU}$ \cite[5.1]{HHR}.
We have a factorization $L_* \to \pi_{2*,*}(\mathbf{MGL}) \to \pi_{*(1+\sigma)}^{C_2}(\mathbf{MU})$,
and choose algebra generators $x_i$ of $L_*$ which map to the $\bar{r}_i \in \pi_{i + i \sigma}^{C_2}(\mathbf{MU})$ defined in \cite[5.4.2]{HHR}. %
We let the image of the $x_i \in \pi_{2i,i}(\mathbf{MGL})$ (also denoted $x_i$) be the canonical choice of algebra generators for $\pi_{2*,*}(\mathbf{MGL})$.
The geometric fixed points of $\mathbf{MU}$ is the unoriented cobordism spectrum $\mathbf{MO}$.
\begin{proposition}[\protect{\cite[Proposition 5.50]{HHR}}]
\label{lem:HHR1}
The canonical map
\[
\pi_{*(1 + \sigma)}(\mathbf{MU}) \to \pi_*(\Phi^{C_2}(\mathbf{MU})) \cong \pi_*(\mathbf{MO})
\]
maps $x_i$ to zero if and only if $i = 2^k - 1$ for some $k$.
\end{proposition}
\begin{proposition}[\protect{\cite[Proposition 7.6]{HHR}}]
\label{lem:HHR2}
The canonical map
\[
\pi_*(\mathbf{MO}) \to \pi_*(\Phi^{C_2}(\mathbf{H}\Z_{(2)}))
\]
is zero for $* > 0$.
\end{proposition}
The next lemma limits the possible targets of the differentials.
\begin{lemma}
\label{lem:rhotors}
The canonical map
\[
\pi_{*,*}(\mathbf{MGL}) \to \pi_{*,*}(\mathbf{MGL}[\rho^{-1}]) \cong \pi_*(\mathbf{MO})
\]
sends
$x_i \in \pi_{2i,i}(\mathbf{MGL})$ to $0$ if and only if $i = 2^k - 1$, for some $k$.
In particular, only the elements $x_{2^k - 1}$ are $\rho$-torsion.
\end{lemma}
\begin{proof}
Consider the diagram
\[
\begin{tikzcd}
\pi_{*,*}(\mathbf{MGL}) \ar[r]\ar[d] & \pi_{*,*}(\mathbf{MGL}[\rho^{-1}]) \ar[d, "\cong"] \\
\pi_{*,*}^{C_2}(\mathbf{MU}) \ar[r] & \pi_{*,*}^{C_2}(\mathbf{MU})[a^{-1}] \cong \pi_{*}(\mathbf{MO})
\end{tikzcd}
\]
where $a$ is the Euler class.
The left vertical map sends $x_i$ to $x_i$.
The bottom map sends $x_i$ to $0$ if and only if $i = 2^k - 1$ by \Cref{lem:HHR1}.
The right vertical map is an isomorphism by \Cref{thm:tom}.
Hence the assertion is proven
(unoriented cobordism has homotopy groups $\pi_*(\mathbf{MO}) = \mathbb{Z}/2[h_j, j \neq 2^{k} - 1]$, where the $h_j$'s are classes in degree $j$ defined as the coefficients of a certain power series in the Euler class $a$ given in \cite[Lemma 5.21]{HHR}).
\end{proof}
\begin{theorem}
\label{thm:diffs}
The $d^i$-differential in the slice spectral sequence of $\mathbf{MGL}/2^n$ on
$u^{2^{k-1}}$
is trivial for $i < r = 2^k - 1$ and
$
d^r(u^{2^{k-1}}) = \rho^{2^{k+1} - 1} x_{2^k - 1}.
$
\end{theorem}
\begin{proof}
We use the proof in \cite[Theorem 9.9]{HHR} adapted to the slice spectral sequence of $\mathbf{MGL}/2$.
The projection $\mathbf{MGL}/2^n \to \mathbf{MGL}/2$ induces a surjection on the slices $H^{0,q}(\mathbb{R};\mathbb{Z}/2^n) \to H^{0,q}(\mathbb{R};\mathbb{Z}/2)$ when is $q$ even.
Let $k \geq 1$ and do induction on $k$.
The differentials of $u^{2^{k-1}}$ in $\mathbf{MGL}/2^n$ determines the differentials on $\tau^{2^k}$ in $\mathbf{MGL}/2$.
\Cref{lem:rhotors} and induction implies that the only possible nonzero differential on $\tau^{2^k}$ in $\mathbf{MGL}/2$ is
\[
d^r(\tau^{2^k}) = \rho^{2^{k+1}-1} x_{2^k - 1}.
\]
We must show that the $d^r$-differential on $\tau^{2^k}$ is nonzero, i.e., $\tau^{2^k}$ is not a permanent cycle.
Assume $\tau^{2^k}$ is a permanent cycle.
That is, it represents some element in $\pi_{0,-2^k}(\mathbf{MGL}/2)$.
Consider the commutative diagram
\[
\begin{tikzcd}
\mathbf{MGL}/2 \ar[r]\ar[d] & \mathbf{M}\Z/2 \ar[d] \\
\mathbf{MGL}/2[\rho^{-1}] \ar[r] & \mathbf{M}\Z/2[\rho^{-1}],
\end{tikzcd}
\]
and the induced map of slice spectral sequences.
On $E^1$-pages, the $\tau^{2^k}$ in $E^1(\mathbf{MGL}/2)$ maps to $\tau^{2^k}$ in $E^1(\mathbf{M}\Z/2)$.
The slice spectral sequence of $\mathbf{M}\Z/2$ collapses on the $E^1$-page.
If $\tau^{2^k}$ survives the slice spectral sequence of $\mathbf{MGL}/2$, it represents a nonzero element $[\tau^{2^k}]$ in $\pi_{*,*}(\mathbf{MGL}/2)$ which maps to $\tau^{2^k}$ in $\pi_{*,*}(\mathbf{M}\Z/2)$.
Note that $\tau^{2^k}$ is not $\rho$-torsion in $\pi_{*,*}(\mathbf{M}\Z/2)$,
hence $[\tau^{2^k}]$ is not $\rho$-torsion, and thus survives $\rho$-localization.
That is,
$[\tau^{2^k}] \in \pi_{0,-2^k}(\mathbf{MGL}/2[\rho^{-1}])$ is nonzero and maps to
$\tau^{2^k} \in \pi_{0,-2^k}(\mathbf{M}\Z/2[\rho^{-1}])$.
Identifying $\mathbf{MGL}/2[\rho^{-1}]$ with $\Phi^{C_2}(R_\C^{C_2}(\mathbf{MGL}/2)) = \mathbf{MO}/2$,
and $\mathbf{M}\Z[\rho^{-1}]$ with $\Phi^{C_2}(R_\C^{C_2}(\mathbf{M}\Z/2)) = \Phi^{C_2}(\mathbf{H}\Z/2)$ we conclude that
the map $\pi_*(\mathbf{MO}) \to \pi_*(\Phi^{C_2}(\mathbf{H}\Z_{(2)}))$ is nonzero for $* > 0$,
contradicting \Cref{lem:HHR2}.
Indeed, $\pi_*(\mathbf{MO}/2)$ is an extension of
$\operatorname{Tor}^\mathbb{Z}_1(\pi_*(\mathbf{MO}), \mathbb{Z}/2)$
by $\pi_*(\mathbf{MO})/2$, and similarly for $\pi_*(\Phi^{C_2}(\mathbf{H}\Z/2))$.
Hence, in the induced map of extensions we get that
$\pi_*(\mathbf{MO}) \to \pi_*(\Phi^{C_2}(\mathbf{H}\Z_{(2)}))$ is nonzero for $* > 0$, since $k \geq 1$.
\end{proof}
The differentials in \Cref{thm:diffs} together with the Leibniz rule determine all possible differentials.
We proceed to describe the $E^\infty$-page of the slice spectral sequence of $\mathbf{MGL}/2^n$ over $\mathbb{R}$.
Consider the subalgebra
\[
B_{*,*} = \bigoplus_{l,p\geq 0}E^1_{2l-p,l,l-p}
\]
of the $E^1$-page
(i.e., $B_{*,*}$ is everything on the $E^1$-page not a multiple of $\tau$ or $u$, it is suggestive to write $B_{*,*} = H^{*=*}\otimes L_*$).
Above we observed
that $d^r(B_{*,*}) = 0$ for degree reasons for all $r$.
Hence, each $E^r$-page is a $B_{*,*}$-module.
The $E^1$-page is the $B_{*,*}$-module
\begin{equation}
\label{eq:E1}
\bigoplus_{i\geq0} u^i B_{*,*}
\oplus
\bigoplus_{i\geq0} \tau u^i B_{*,*}.
\end{equation}
\begin{theorem}
\label{thm:E8R}
Let $I_l$ be the ideal of $B_{*,*}$ generated by
\[
(\rho^3 x_1, \dots, \rho^{2^{l+1}-1}x_{2^l-1}),
\]
and let $I_0 = (0)$.
Let $I$ be $I_\infty$.
Let $J_l$ be the ideal $(2, x_1, \dots, x_{2^l-1})$ in $B_{*,*}$,
and let $J_0 = (0)$.
Let $A(l)$ denote the quotient $J_l/I_l$.
Then the $E^\infty$-page is
\begin{align}
E^\infty(\mathbb{R};\mathbb{Z}/2^n)
\cong &
B_{*,*}/I \oplus \tau B_{*,*}/I \nonumber\\
&\oplus \bigoplus_{i \geq 1} u^i A(\nu_{2}(i)) L_*
\oplus \bigoplus_{i \geq 1} \tau u^i A(\nu_{2}(i)) L_*. \label{eq:Einfty}
\end{align}
Here $L_*$ is inserted into $E^\infty(\mathbb{R};\mathbb{Z}/2^n)$ via the canonical embedding
induced by $L_k \cong E^1_{2k,k,k}$.
\end{theorem}
\begin{proof}
Let $l_2(r)$ be the length of the binary expansion of $r$, i.e., the minimal integer $l_2(r)$ such that $r < 2^{l_2(r)}$.
By induction on $r$ we get
\begin{align}
E^r(\mathbb{R};\mathbb{Z}/2^n) =
&\bigoplus_{\nu_2(i)\geq l_2(r)-1} u^i B_{*,*}/I_{l_2(r)-1}
\oplus
\bigoplus_{\nu_2(i)\geq l_2(r)-1} \tau u^i B_{*,*}/I_{l_2(r)-1} \nonumber \\
&\oplus \bigoplus_{l_2(r)-1 > i \geq 1} u^i A(\nu_{2}(i)) L_*
\oplus \bigoplus_{l_2(r)-1 > i \geq 1} \tau u^i A(\nu_{2}(i)) L_*.
\label{eq:Er}
\end{align}
Indeed, the base case is \eqref{eq:E1}.
Assume inductively \eqref{eq:Er} is true for $r$.
If $l_2(r) = l_2(r+1)$ then $E^r = E^{r+1}$,
since there are only differentials when
$l_2(r) < l_2(r+1)$.
So we may assume $r=2^k-1$.
Then we have a $d^{r}$-differential
\[
d^{r}(u^{2^{k-1}})
=
\rho^{2^{k+1} -1}x_{2^{k}-1},
\]
and more generally $d^{r}$-differentials on the $u^{i}$ with $2$-adic valuation $\nu_2(i) = k-1$.
We analyze the differential on each summand in \eqref{eq:Er}:
Note that the summands do not interact since the differentials are $u^{2^k}$- and $\tau$-linear.
The differential on the last two summands are zero for degree reasons, cf.~\Cref{fig:E1-page}.
The contribution to the $E^{r+1}$-page of the differential on the summand
\[
\bigoplus_{\nu_2(i) = l_2(r)-1} u^i B_{*,*}/I_{l_2(r)-1}
\]
is
\[
\bigoplus_{\nu_2(i)=l_2(r)-1} u^i A(\nu_{2}(i)) L_*.
\]
Indeed, the differential of an element
$a = u^i \rho^j y \in u^i B_{*,*}/I_{l_2(r)-1}$ is
\[
d^{r}(a)
=
\rho^{2^{k+1} -1 + j}x_{2^{k}-1} u^{i - 2^{k-1}} y.
\]
This is zero in the image if and only if
$\rho^{2^{k+1} -1 + j}y \in I_{l_2(r)-1}$.
That is, $\rho^j y$ is an element of $J_{l_2(r)-1}$
(note that $\rho^{2^{k+1} -1} y \equiv 0 \bmod I_{l_2(r)-1}$ for $y \in J_{l_2(r)-1}$).
Similarly the summand
\[
\bigoplus_{\nu_2(i) = l_2(r)-1} \tau u^i B_{*,*}/I_{l_2(r)-1}
\]
contributes
\[
\bigoplus_{\nu_2(i)=l_2(r)-1} \tau u^i A(\nu_{2}(i)) L_*.
\]
The Leibniz rule implies that summands with $\nu_2(i) > l_2(r)-1$ have zero $d^r$-differential.
Passing to the limit over $r$ in \eqref{eq:Er} we obtain \eqref{eq:Einfty}.
\end{proof}
\begin{remark}
Determining possible extensions and the multiplicative structure in $\mathbf{MGL}_{*,*}(\mathbb{R}; \mathbb{Z}/2^n)$ seems to be fairly hard.
The simplest extensions $2[\tau] = [\rho^2 x_1]$ and $2^{n-1}[2u] \neq 0$ can be determined by comparison with $K$-theory of the real numbers.
There are extension problems for $x_1 u, x_3 u, \dots, 2u, 2u^2, \dots$ and their $\tau$-multiples.
Unfortunately, $K$-theory does not detect the ones not a multiple of $x_1$.
For motivic Morava $K$-theory of the real numbers we are generally unable to determine the extensions, see \Cref{rmk:Kn-exts}.
\end{remark}
\subsection*{Two-complete algebraic cobordism of $\mathbb{R}$}%
\label{sec:R2}
Consider the filtration of $\mathbf{MGL}_{*,*}(\mathbb{R};\mathbb{Z}/2^n)$ associated to $E^\infty(\mathbb{R};\mathbb{Z}/2^n)$.
Since all the groups in the filtration are finite $\lim_n$ is exact and we may pass to the limit over $n$ to obtain a filtration of $\mathbf{MGL}_{*,*}(\mathbb{R};\mathbb{Z}_2)$,
the motivic homotopy groups of $\operatorname{holim}_n \mathbf{MGL}/2^n$.
We write $E^\infty(\mathbb{R};\mathbb{Z}_2)$ for the associated graded of this filtration.
This associated graded is simpler than $E^\infty(\mathbb{R};\mathbb{Z}/2^n)$ since all the $\tau$-multiples disappear.
\begin{lemma}
The associated graded of the filtration of $\mathbf{MGL}_\star(\mathbb{R};\mathbb{Z}_2)$ described above is
\begin{align}
E^\infty(\mathbb{R};\mathbb{Z}_2)
\cong &
\lim_n \left( B_{*,*}/I \oplus \bigoplus_{i \geq 1} u^i A(\nu_{2}(i)) L_* \right) \nonumber \\
\cong &
\frac{\mathbb{Z}_2[\rho, z_{j,l}, x_h \vert l, j \geq 0, 2^{j}-1 \neq h \geq 1]}
{(2\rho, \rho^{2^{j+1} - 1}z_{j,l},
z_{k,l}z_{a,b} = z_{k,l+b2^{a-k}}z_{a,0}, \text{ for } a \geq k)}.
\end{align}
Here we write $z_{j,l} = u^{l2^{j}}x_{2^{j}-1}$,
which explains the relations (with the convention $x_0 = 2$).
\end{lemma}
\begin{proof}
This reduces to the computation in \Cref{lem:HR}:
The limit is induced by the system \eqref{eq:2n-sys},
and the structure maps in the inverse system are the same as in motivic cohomology. The multiples of $\tau$ all have multiplication by $2$ as structure maps.
Since these are all $2$-torsion they vanish in the limit.
Hence we are left with computing the limit of
\begin{align}
B_{*,*}/I \oplus \bigoplus_{i \geq 1} u^i A(\nu_{2}(i)) L_*. \nonumber
\end{align}
Anything not a multiple of $\rho$ is part of a system which is $\mathbb{Z}_2$ in the limit.
Otherwise the structure map is the identity of $\mathbb{Z}/2$ and in the limit we get $\mathbb{Z}/2$.
\end{proof}
\begin{theorem}
\label{thm:MGLR}
There are no hidden additive or multiplicative extensions on the $E^\infty$-page.
That is, as an algebra
\[
\bigoplus_{p,w} \bigoplus_q E^\infty_{p,q,w} \cong \mathbf{MGL}_{*,*}(\mathbb{R};\mathbb{Z}_2).
\]
Hence,
\begin{equation}
\mathbf{MGL}_{*,*}(\mathbb{R};\mathbb{Z}_2)
=
\frac{\mathbb{Z}_2[\rho, z_{j,l}, x_h \vert l, j \geq 0, 2^{j}-1 \neq h \geq 1]}
{(2\rho, \rho^{2^{j+1} - 1}z_{j,l}, z_{k,l}z_{a,b} = z_{k,l+b2^{a-k}}z_{a,0}, \text{ for } a \geq k)}.
\end{equation}
Here $z_{j,l} = [u^{l2^{j}}x_{2^{j}-1}]$ (with the convention $x_0 = 2$).
That is, $z_{j,l}$ is a particular representative of $u^{l2^{j}}x_{2^{j}-1}$
(during the proof we will see that it is a particular representative).
\end{theorem}
\begin{proof}
Note that any element on the $E^1$-page is a sum of monomials $u^i\rho^jx_K$ for some numbers $i, j$ and multi-index $K$. The degrees of $u$, $\rho$ and $x_K$ are linearly independent.
It is helpful to keep in mind that a specific element of $\mathbf{MGL}_{*,*}(\mathbb{R};\mathbb{Z}_2)$ is obtained by going downwards along $u$, $(2,1)$-diagonally upwards along $x_K$ and $(-1,-1)$-diagonally downwards along $\rho$. We will perform induction on the power of $u$.
We choose representatives $z_{j,l} = [u^{l2^{j}}x_{2^{j}-1}]$ in $\mathbf{MGL}_{*,*}(\mathbb{R};\mathbb{Z}_2)$ of the algebra generators $u^{l2^{j}}x_{2^{j}-1}$ on the $E^\infty$-page inductively such that $\rho^i[u^{l2^{j}}x_{2^{j}-1}] = [\rho^i u^{l2^{j}}x_{2^{j}-1}]$, in particular $\rho^{2^{l+1} -1}z_{l,j} = 0$ (here $[\rho^i u^{l2^{j}}x_{2^{j}-1}]$ is some element representing $\rho^i u^{l2^{j}}x_{2^{j}-1}$).
Indeed, in general
\[
\rho^i[u^{l2^{j}}x_{2^{j}-1}] = [\rho^i u^{l2^{j}}x_{2^{j}-1}] + [\rho^{k} u^{l'2^{j'}}x_K] + \dots,
\]
where $K > 2^j - 1$ (so it lies in a higher filtration), which implies $k > i$ and $l2^j > l'2^{j'}$ (hence the second term is already defined by induction on $l$).
Hence, we could as well have chosen
\[
[u^{l2^{j}}x_{2^{j}-1}] - [\rho^{k-i} u^{l'2^{j'}}x_K]
\]
as a representative for $u^{l2^{j}}x_{2^{j}-1}$. Inductively we get
\[
\rho^i[u^{l2^{j}}x_{2^{j}-1}] = [\rho^i u^{l2^{j}}x_{2^{j}-1}]
\]
(this is really only a condition for $i = 2^{j+1} - 1$).
It remains to show
$z_{k,l}z_{a,b} = z_{k,l+b2^{a-k}}z_{a,0}$, $a \geq k$.
For the remainder of the proof we use the notation $z_{k,l}= [u^{l2^{k}}x_{2^{k}-1}]$.
In general,
\[
[u^{l2^{k}}x_{2^{k}-1}][u^{b2^{a}}x_{2^{a}-1}]
= [u^{(l + b2^{a - k})2^{k}}x_{2^{k}-1}][x_{2^{a}-1}] + \rho^i[u^{n2^m}x_{2^m-1}][x_K] + \dots.
\]
Without loss of generality we may assume $a \geq k$, and that there is no $m' < m$ such that $x_{2^{m'} - 1}$ divides $x_K$.
Note that if we multiply the left hand side by $\rho^{2^{k+1}-1}$ we get 0.
For the right hand side to be zero we must have $2^{k+1} -1 + i \geq 2^{m+1} - 1$.
Furthermore, for $\rho^i[u^{n2^m}x_{2^m-1}]x_K$ to be nonzero we must have $i < 2^{m+1}-1$.
Both sides are in the same bidegree of $\mathbf{MGL}_{*,*}(\mathbb{R};\mathbb{Z}_2)$, and the term $\rho^i[u^{n2^m}x_{2^m-1}]x_K$ is in a higher filtration.
This gives a system of equations and inequalities on $k,l,a,b,m,n,K,i$ (by abuse of notation $K$ denotes either a multi-index or the degree of the multi-index).
This system has no solutions. We now carry this out in detail.
Comparison of the left and right hand side gives the following relations
\[
\begin{matrix}
2(2^k - 1) + 2(2^a - 1) \\
2^k - 1 + 2^a - 1 \\
(1 - 2l)2^k - 1 + (1 - 2b)2^a - 1
\end{matrix}
\begin{matrix}
= \\
\leq \\
=
\end{matrix}
\begin{matrix}
2(2^m - 1) + 2K - i \\
2^m - 1 + K \\
(1 - 2n)2^m - 1 + K - i
\end{matrix}
\]
Additionally we have
\[
a \geq k,\qquad 2^{k+1} + i \geq 2^{m+1},\qquad 2^{m+1}-1 > i > 0.
\]
Solving for $i/2$ ($i$ is a multiple of 4) this gives
\[
2l2^k + 2b2^a - 2n2^m = (2l + 2b2^{a-k})2^k - 2n2^m = i/2
\]
which must satisfy
\[
2^k + 2^{m} > 2^k + 2l2^k + 2b2^a - 2n2^m \geq 2^m
\]
and $i/2 > 0$.
Equivalently,
\begin{equation}
\label{eq:linineq}
2^{m} > 2l2^k + 2b2^a - 2n2^m = (l + b2^{a-k})2^{k+1} - n2^{m+1} \geq \max(2^m - 2^k,1).
\end{equation}
If $k \geq m$ there are clearly no solutions.
If $k < m$ this is equivalent to
$(1 + 2n)2^{m} > (l + b2^{a-k})2^{k+1} \geq (1 + 2m)2^m - 2^k$,
i.e., $(l + b2^{a-k})2^{k+1}$ is in the half open interval $[(1 + 2n)2^m - 2^k, (1 + 2n)2^{m})$. But the endpoint $(1 + 2n)2^{m} = (1 + 2n)2^{m-k-1}2^{k+1}$ is a multiple of $2^{k+1}$, i.e., the interval never contains $(l + b2^{a-k})2^{k+1}$.
\end{proof}
\begin{remark}
\label{rmk:elementary}
As an alternative to the above proof it is possible to copy the proof of \cite[Theorem 4.11]{Hu-Kriz:real} (or rather \cite[Theorem 7.4]{Hu-Kriz:real}) verbatim.
That is, use the motivic Adams spectral sequence to determine the multiplicative extensions (in fact, the computation of Hu and Kriz takes place in an $\operatorname{Ext}$-group of a subalgebra of the $C_2$-equivariant Steenrod algebra. This subalgebra is isomorphic to the mod 2 motivic Steenrod algebra).
A second alternative is to take $C_2$-equivariant complex realization, and compare directly with the $C_2$-equivariant homotopy groups of $BP\mathbb{R}$.
Conversely the above proof applies to the setting of Hu and Kriz, since \eqref{eq:linineq} has no solutions even if $b, l, n$ are allowed to take negative values.
This provides an alternative route to the $C_2$-equivariant homotopy groups of $BP\mathbb{R}$ without use of the Adams spectral sequence.
\end{remark}
\section{Algebraic cobordism of real number fields}
\label{sec:MGLF}
In this section we compare the slice spectral sequence of $\mathbf{MGL}$ over a number field $F$ with the slice spectral sequence over $\mathbb{R}$. This determines all the differentials in the slice spectral sequence over $F$ and we compute the associated graded of $\mathbf{MGL}_{*,*}(F;\mathbb{Z})$.
Consider the map of $E^r$-pages of the slice spectral sequence for $\mathbf{MGL}$ induced by the real embeddings of $F$
\[
f^r_{p,q,w} : E^r_{p,q,w}(F;\mathbb{Z}) \to \oplus^{r_1} E^r_{p,q,w}(\mathbb{R};\mathbb{Z}_2).
\]
This gives a commutative diagram
\begin{equation}
\label{eq:diff-diag}
\begin{tikzcd}[column sep=70pt]
E^r_{p,q,w}(F;\mathbb{Z}) \ar[r, "f^r_{p,q,w}"]\ar[d, "d^r(F)"] & \oplus^{r_1} E^r_{p,q,w}(\mathbb{R};\mathbb{Z}_2)\ar[d, "\oplus^{r_1}d^r(\mathbb{R})"] \\
E^r_{p-1,q+r,w}(F;\mathbb{Z}) \ar[r, "f^r_{p-1,q+r,w}"] & \oplus^{r_1} E^r_{p-1,q+r,w}(\mathbb{R};\mathbb{Z}_2).
\end{tikzcd}
\end{equation}
Whenever the source of $d^r(F)$ is nonzero, the target $E^r_{p-1,q+r,w}(F;\mathbb{Z})$ is isomorphic to $(\mathbb{Z}/2)^{\oplus k}$ for some $k$ and $f^r_{p-1,q+r,w}$ is an isomorphism.
That is, the differential $d^r(F)$ is determined by the differential $d^r(\mathbb{R})$ and the map $f^r_{p,q,w}$, i.e., of the sum of the real embeddings.
By \Cref{lem:HFR} and \Cref{lem:H3R} the map $f^r_{p,q,w}$ is surjective except possibly when restricted to a summand $H^{0,q'}(F;\mathbb{Z})$ or $H^{1,q'}(F;\mathbb{Z})$ of $E^r_{p,q,w}(F;\mathbb{Z})$.
We define
\[
\ol{H}^{p,q,(l)}(F;\mathbb{Z}) = \operatorname{coker}(
g: H^{p-(2^{l+1}-1),q-(2^l-1)}(F;\mathbb{Z}) \to H^{p,q}(F;\mathbb{Z})
)
\]
where $g$ is the unique arrow making the diagram
\begin{equation}
\label{eq:H-bar}
\begin{tikzcd}[column sep=70pt]%
H^{p-(2^{l+1}-1),q-(2^l-1)}(F;\mathbb{Z}) \ar[d]\ar[r, dashed, "g"] & H^{p,q}(F;\mathbb{Z})\ar[d] \\
\oplus^{r_1}H^{p-(2^{l+1}-1),q-(2^l-1)}(\mathbb{R};\mathbb{Z}_2) \ar[r, "\rho^{2^{l+1}-1}u^{-2^{l-1}}"] & \oplus^{r_1} H^{p,q}(\mathbb{R};\mathbb{Z}_2)
\end{tikzcd}
\end{equation}
commute (note that either $H^{p-(2^{l+1}-1),q-(2^l-1)}(F;\mathbb{Z}) = 0$, or the right vertical map in \eqref{eq:H-bar} is an isomorphism).
We define
\[
\wt{H}^{p,q}(F;\mathbb{Z}) = \ker(H^{p,q}(F;\mathbb{Z}) \to \oplus^{r_1}H^{p,q}(\mathbb{R};\mathbb{Z}_2)).
\]
Note that
$\ol{H}^{p,q,(l)}(F;\mathbb{Z}) = 0$ if $p-(2^{l+1}- 1) > 1$
and $\wt{H}^{p,q}(F;\mathbb{Z}) = 0$ if $p \geq 3$.
\begin{theorem}
\label{thm:MGLF}
The $E^\infty$-page of the slice spectral sequence is
\[
E^\infty(F;\mathbb{Z}) \cong \bigoplus_{p,q,K} A^{p,q,K}
\]
where $K$ runs over all multi-indices of monomials in $L_*$.
Here
\begin{equation}
\label{eq:Apqk}
A^{p,q,K} = \begin{cases}
\ol{H}^{p,q,(l)}(F;\mathbb{Z}) x_K & \text{if } l = \min\{l \colon x_{2^l-1} \vert x_K, 1 \leq l < \nu_2(q-p) \} < \infty \\
\wt{H}^{p,q}(F;\mathbb{Z}) x_K & \text{if } x_{2^l - 1} \nmid x_K \text{ for all } 1 \leq l \leq \nu_2(q-p) \text{ and } 0 < \nu_2(q - p) < \infty \\
H^{p,q}(F;\mathbb{Z}) x_K & \text{otherwise}.
\end{cases}
\end{equation}
By definition $\min \{ \} = \infty$.
\end{theorem}
\begin{proof}
We will prove inductively that
\begin{equation}
\label{eq:Erind}
E^r_{p,q,w}
\cong
\bigoplus_{p,q,K} A^{p,q,K,(r)}
\end{equation}
where
\begin{equation}
\label{eq:Apqk}
A^{p,q,K,(r)} = \begin{cases}
\ol{H}^{p,q,(l)}(F;\mathbb{Z}) x_K & \text{if } l = \min\{l \colon x_{2^l-1} \vert x_K, 1 \leq l < l_{p,q,r} \} < \infty \\
\wt{H}^{p,q}(F;\mathbb{Z}) x_K & \text{if } x_{2^l - 1} \nmid x_K \text{ for all } 1 \leq l \leq l_{p,q,r}, \text{ and } 0 < \nu_2(q - p) < l_2(r)\\
H^{p,q}(F;\mathbb{Z}) x_K & \text{otherwise,}
\end{cases}
\end{equation}
and $l_{p,q,r} = \min\{l_2(r)-1, \nu_2(q-p)\}$.
When $r = 1$ we are in the last case of \eqref{eq:Apqk}, so \eqref{eq:Erind} holds by definition.
Assume \eqref{eq:Erind} to be true inductively for $r$.
We must show that \eqref{eq:Erind} is true for $r+1$.
From \eqref{eq:diff-diag} we see that there are only $d^r$-differentials when $l_2(r) < l_2(r+1)$, so we may assume $r = 2^{k}-1$.
The only terms $A^{p,q,K,(r)}$ which can support a $d^r$ differentials are the terms with $\nu_2(q-p) = k$.
The target of a $d^r$-differential on $A^{p,q,K,(r)}$ is $A^{p',q',K,(r)}x_{2^k-1}$, for $q' = p + 2^{k+1}-1, q' = q+2^k - 1$.
We note that $\nu_2(q' - p') = \nu_2(q - p - 2^k) > k$.
Hence all terms with $\nu_2(q - p) < k$ are unchanged when passing from the $E^r$-page to the $E^{r+1}$-page.
Consider a term $A^{p,q,K}$ on the $E^r$-page.
Assume $\nu_{2}(q-p) = k$, and that $A^{p,q,K,(r)}$ is nonzero and supports a $d^r$-differential to $A^{p',q',K,(r)}x_{2^k-1}$.
Then $A^{p,q,K,(r)}$ is of the form $H^{p,q}(F;\mathbb{Z})x_K$ or $\ol{H}^{p,q,(l)}(F;\mathbb{Z})x_K$ for some $l < k$.
In the latter case the $d^r$-differential is zero, indeed, $\rho^{2^{k+1}-1}x_K = 0$.
In the former case the $d^r$-differential is nonzero if and only if
$\rho^{2^{k+1}-1}x_K$ is nonzero, that is, if and only if $x_{2^l - 1} \nmid x_K$ for all $1 \leq l < k$.
Hence $H^{p,q}(F;\mathbb{Z})x_K$ is replaced by $\wt{H}^{p,q}(F;\mathbb{Z})x_K$ if and only if $x_{2^l - 1} \nmid x_K$ for all $1 \leq l < k = \nu_{2}(q-p) = l_2(r+1)-1$.
Assume next $\nu_2(q-p) > k$.
That is, $A^{p,q,K,(r)}$ is potentially the target of a $d^r$-differential.
If we are in the first case of \eqref{eq:Apqk} there is nothing to show,
so assume that we are in the last case of \eqref{eq:Apqk}.
If $x_{2^k-1} \nmid x_K$ there can be no differential to $A^{p,q,K,(r)}$, so $A^{p,q,K,(r)} = A^{p,q,K,(r+1)} = H^{p,q}(F;\mathbb{Z})x_K$.
If $x_{2^k-1} \vert x_K$ there is a differential from
$A^{p-(2^{k+1}-1),q - (2^{k}-1),K,(r)}/x_{2^k-1}$ and $A^{p,q,K,(r)}$ is replaced by $\ol{H}^{p,q,(k)}(F;\mathbb{Z})x_K$.
This completes the inductive step.
Passing to the limit over $r$ we get the $E^\infty$-page.
\end{proof}
\begin{remark}
It is possible to give a presentation of $E^\infty(F;\mathbb{Z})$ in terms of generators and relations.
There is a generator for every pair $(x_{2^k-1}, y), y \in \oplus_j H^{1,j2^k}(F;\mathbb{Z})$,
and we must re-encode the multiplication in $H^{*,*}(F;\mathbb{Z})$.
This can be compared with giving a presentation of
the subalgebra $k\{1, x^iy^j \vert i \geq 0, j > 1\} \subset k[x, y]$.
\end{remark}
\Cref{thm:MGLF} and comparison with $\mathbf{MGL}_\star(\mathbb{R};\mathbb{Z}_2)$ determines $\mathbf{MGL}_\star(F;\mathbb{Z})$ up to some indeterminacy in the additive and multiplicative structure in the part of $\mathbf{MGL}_\star(F;\mathbb{Z})$ coming from $H^{1,*}(F;\mathbb{Z})$ and $H^{2,*}(F;\mathbb{Z})$.
For $2$-regular number fields with exactly one prime dividing $2$ (e.g., $\mathbb{Q}$) it might be possible to determine the extensions as in \cite{John-Paul}.
\section{Algebraic cobordism and $\zeta$-functions}
\label{sec:zeta}
Motivic cohomology of number fields and rings of $\mathcal S$-integers are qualitatively the same. In this section we exploit this likeness to compute $2$-complete algebraic cobordism over rings of $\mathcal S$-integers, $\mathcal S \supset \{2, \infty\}$ a finite set of places, and give a formula relating the order of the algebraic cobordism groups of rings of $2$-integers to special values of Dedekind $\zeta$-functions.
This is an analogy to Lichtenbaum's conjecture relating the order of algebraic $K$-theory of rings of integers to special values of Dedekind $\zeta$-functions, see \cite[Theorem 0.2]{Rognes-Weibel}.
An analogy to Lichtenbaum's conjecture for hermitian $K$-theory was proven in \cite[Theorem 1.10]{KRO}.
We use the motivic cohomology spectrum of Spitzweck over the ring of $\mathcal S$-integers in a number field $F$ \cite{Spitzweck:commutative}.
When $\mathcal{S} \supset \{2, \infty \}$ the residue fields of $\mathcal{O}_{F,\mathcal{S}}$ all have odd characteristic so the characteristics are invertible in $\mathbb{Z}/2^n$.
Then the slices of $\mathbf{MGL}/2^n$ are
$\mathsf{s}_q(\mathbf{MGL}/2^n) = \Sigma^{2q,q} \mathbf{M}\Z/2^n \otimes L_q$ \cite[Theorem 11.3]{Spitzweck:commutative}.
When the set of places $\mathcal S$ is finite the 2-completed motivic cohomology groups are finitely generated outside of $(0,0)$, the Picard group $H^{2,1}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_2)$ can be nonzero, and we have an isomorphism $H^{p,q}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_2) \cong H^{p,q}(\mathbb{R};\mathbb{Z}_2)$ for $p \geq 3$.
Hence the differentials in the slice spectral sequence over $\mathcal{O}_{F,\mathcal{S}}$ are determined by the same procedure as over $F$ by comparison with the real embeddings.
For degree reasons the extra groups $H^{2,1}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_2)\otimes L_*$ do not interact with any terms in the slice spectral sequence or support any differentials.
They give rise to the extra summand $H^{2,1}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_2)\otimes L_{n+1}$ in $\mathbf{MGL}_{2n,n}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_2)$.
This summand has zero multiplication with motivic cohomology of positive weight.
The $2$-local motivic cohomology groups of $\mathcal{O}_{F,\mathcal{S}}$ are finitely generated, hence $\lim_n$ is exact on the mod $2^n$ motivic cohomology groups, and completion before or after taking homotopy groups is the same for $\mathbf{M}\Z$ and $\mathbf{MGL}$.
As in \Cref{sec:R2} we first run the slice spectral sequence for $\mathbf{MGL}/2^n$ over $\mathcal{O}_{F,\mathcal{S}}$.
This gives filtrations of $\mathbf{MGL}_{*,*}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}/2^n)$ for each $n$. Taking the limit over $n$ of the filtrations we obtain a filtration of $\mathbf{MGL}_{*,*}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_2)$.
\begin{theorem}
\label{thm:OFS}
\label{thm:MGLO}
The associated graded of the 2-complete algebraic cobordism groups of a ring of $\mathcal S$-integers in a number field $F$, $\mathcal S \supset \{2, \infty\}$, $\mathcal S$ finite, is
\[
E^\infty(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_2) \cong \bigoplus_{p,q,K} A^{p,q,K}
\]
where $K$ runs over all multi-indices of monomials in $L_*$.
Here
\[
A^{p,q,K} = \begin{cases}
\ol{H}^{p,q,(l)}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_2) x_K & \text{if } l = \min\{l \colon x_{2^l-1} \vert x_K, 1 \leq l < \nu_2(q-p) \} < \infty \\
\wt{H}^{p,q}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_2) x_K & \text{if } x_{2^l - 1} \nmid x_K \text{ for all } 1 \leq l \leq \nu_2(q-p) \text{ and } 0 < \nu_2(q - p) < \infty \\
H^{p,q}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_2) x_K & \text{otherwise}.
\end{cases}
\]
The groups $\ol{H}^{p,q,(l)}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_2)$ and $\wt{H}^{p,q}(\mathcal{O}_{F,\mathcal{S}};\mathbb{Z}_2)$ are defined similarly as in \Cref{sec:MGLF}.
\end{theorem}
Recall the following theorem of Manfred Kolster:
\begin{theorem}[\protect{\cite[Theorem A.1]{Rognes-Weibel}}]
\label{thm:Kolster}
Let $F$ be a totally real abelian number field and $k \geq 1$.
Then
\[
\zeta_{F}(1 - 2k) \sim_2 \frac{\# H^{2}_{\acute{e}t}(\mathcal{O}_F[\frac{1}{2}]; \mathbb{Z}_2(2k)) }{\# H^{1}_{\acute{e}t}(\mathcal{O}_F[\frac{1}{2}]; \mathbb{Z}_2(2k)) }.
\]
Here $a \sim_2 b$ means that $a$ and $b$ have the same $2$-adic valuation.
\end{theorem}
Define the subgroups
$L_t' = \mathbb{Z}\{x_K \in L_t \colon \exists n, x_{2^n - 1} \mid x_K \}$
and
$L_t'' = \mathbb{Z}\{x_K \in L_t \colon \forall n, x_{2^n - 1} \nmid x_K \}$
of $L_t$,
such that $L_t' \oplus L_t''= L_t$.
We write $n^{L_t}$ for $n^{\operatorname{rk}_{\mathbb{Z}}L_t}$ and so on.
Combining \Cref{thm:MGLO} with \Cref{thm:Kolster} we get the following relation between algebraic cobordism of $\mathcal{O}_F[\frac{1}{2}]$ and special values of the Dedekind $\zeta$-function of $F$.
\begin{corollary}
\label{thm:zeta}
Let $F$ be a totally real abelian number field with $r_1$ real embeddings and $n \geq 1$ an integer.
Then
\begin{equation}
\label{eq:zeta}
2^{r_1L_{2n+w}''}
\frac{\# \mathbf{MGL}_{4n+2w-2,w}(\mathcal{O}_F[\frac{1}{2}];\mathbb{Z}_2) }
{\# \mathbf{MGL}_{4n+2w-3,w-1}(\mathcal{O}_F[\frac{1}{2}];\mathbb{Z}_2) }
\sim_2
(\zeta_{F}(1-2n))^{L_{2n+w}}.
\end{equation}
\end{corollary}
\begin{proof}
The cardinality of $\mathbf{MGL}_{p,w}(\mathcal{O}_F[\frac{1}{2}];\mathbb{Z}_2)$ is the product of the cardinalities of the groups
\[
A^{-p + 2\vert K \vert, -w + \vert K \vert, K},\quad x_K \in L_*.
\]
The cardinality of $\mathbf{MGL}_{p-1,w-1}(\mathcal{O}_F[\frac{1}{2}];\mathbb{Z}_2)$ is the product of the cardinalities of the groups
\[
A^{-p+1 + 2\vert K \vert, -w+1 + \vert K \vert, K},\quad x_K \in L_*.
\]
From now on we assume $p$ and $(p+2)/2 - w$ to be even.
Then multiplication by $\rho$ induces an isomorphism
\begin{equation}
\label{eq:rho-mult}
A^{-p + 2\vert K \vert, -w + \vert K \vert, K}
\to
A^{-p +1 + 2\vert K \vert, -w + 1 + \vert K \vert, K}
\end{equation}
when $-p + 2\vert K \vert > 2$.
Indeed, this follows from parity considerations: %
Let $p' = -p + 2\vert K \vert, q' = -w + \vert K \vert$.
If $A^{p', q', K}$ is given by the two last cases of \eqref{eq:Apqk} then \eqref{eq:rho-mult} is clear,
so we may assume
\[
A^{p', q', K}
=
\ol{H}^{p',q',(l)}(F;\mathbb{Z})x_K,
\qquad
A^{p'+1, q'+1, K}
=
\ol{H}^{p'+1,q'+1,(l)}(F;\mathbb{Z})x_K,
\]
for some minimal $l$ such that $x_{2^{l}-1}\vert x_K$ and $q' - p' = j2^{l+1}$ for some $j$.
The statement is then equivalent to the map of cokernels
\[
\begin{tikzcd}
\operatorname{coker}\Bigl(H^{p'-(2^{l+1} - 1), q' - (2^l - 1)}(F;\mathbb{Z}) \ar[r]\ar[d, "\rho"] & \oplus^{r_1}H^{p'-(2^{l+1}-1),q'-(2^l - 1)}(\mathbb{R};\mathbb{Z}_2)\Bigr)\ar[d, "\rho"] \\
\operatorname{coker}\Bigl(H^{p'+1-(2^{l+1} - 1), q'+1 - (2^l - 1)}(F;\mathbb{Z}) \ar[r] & \oplus^{r_1}H^{p'+1-(2^{l+1}-1),q'+1-(2^l - 1)}(\mathbb{R};\mathbb{Z}_2)\Bigr)
\end{tikzcd}
\]
being an isomorphism.
In general this map is not an isomorphism if $p' - (2^{l+1} - 1) = 0, 1$,
but this possibility is excluded by the assumptions on $p$ and $w$.
Indeed, $p' - (2^{l+1} -1) \neq 0$,
since $p' = -p + 2 \vert K \vert$ is even by assumption.
If $p' - (2^{l+1} -1) = 1$,
then $-p + 2\vert K \vert = 2^{l+1}$ and $-w + \vert K \vert = (j+1)2^{l+1}$. Hence, $p - 2w = (j+1)2^{l+2} - 2^{l+1}$, contradicting $(p+2)/2 - w$ being even.
With the above assumptions on $p$ and $w$ we get
\begin{align*}
\frac{\# \mathbf{MGL}_{p,w}(\mathcal{O}_F[\frac{1}{2}];\mathbb{Z}_2) }{\# \mathbf{MGL}_{p-1,w-1}(\mathcal{O}_F[\frac{1}{2}];\mathbb{Z}_2) }
&=
\left(\frac{\# H^{2,(p+2)/2-w} }
{\# H^{1,(p+2)/2-w} }\right)^{L_{(p+2)/2}'}
\left(\frac{\# \wt{H}^{2,(p+2)/2-w} }
{\# \wt{H}^{1,(p+2)/2-w} }\right)^{L_{(p+2)/2}''} \\
&=
\left(\frac{\# H^{2,(p+2)/2-w} }
{\# H^{1,(p+2)/2-w} }\right)^{L_{(p+2)/2}'}
\left(2^{-r_1}\frac{\# H^{2,(p+2)/2-w} }
{\# H^{1,(p+2)/2-w} }\right)^{L_{(p+2)/2}''} \\
&=
2^{-r_1 L_{(p+2)/2}''} \left(\frac{\# H^{2,(p+2)/2-w} }
{\# H^{1,(p+2)/2-w} }\right)^{L_{(p+2)/2}}.
\end{align*}
Here we use \Cref{lem:H12R} for the second equality.
Hence,
\[
2^{r_1L_{(p+2)/2}''}\frac{\# \mathbf{MGL}_{p,w}(\mathcal{O}_F[\frac{1}{2}];\mathbb{Z}_2) }{\# \mathbf{MGL}_{p-1,w-1}(\mathcal{O}_F[\frac{1}{2}];\mathbb{Z}_2) }
\sim_2
(\zeta_F(1+w-(p+2)/2))^{L_{(p+2)/2}}.
\]
A change of variables yields \eqref{eq:zeta}.
\end{proof}
\section{Algebraic cobordism of totally imaginary number fields and fields of low $2$-cohomological dimension}
\label{sec:final}
For completeness we state the following results on algebraic cobordism of totally imaginary number fields and fields with $2$-cohomological dimension less than or equal to $2$.
For such fields the slice spectral sequence collapses on the $E^1$-page for degree reasons.
\label{sec:MGLC}
\label{sec:lowcoh}
\begin{theorem}
\label{thm:MGLC}
Let $F$ be a totally imaginary number field. Then
\[
\mathbf{MGL}_{*,*}(F;\mathbb{Z}) \cong H^{*,*}(F;\mathbb{Z})\otimes L_*.
\]
Here $H^{p,q}(F;\mathbb{Z})$ is a summand of $\mathbf{MGL}_{-p,-q}(F;\mathbb{Z})$,
and $L_k$ is identified with $\mathbf{MGL}_{2k,k}(F;\mathbb{Z})$.
\end{theorem}
\begin{proof}
The motivic cohomology $H^{p,q}(F;\mathbb{Z})$ is zero for $p=0, q \neq 0$ or $p > 2$.
Hence, the slice spectral sequence collapses at the $E^1$-page, cf.~\Cref{fig:E1-page}.
There are no hidden extensions since in any fixed bidegree of $\mathbf{MGL}_{*,*}(F;\mathbb{Z})$ the slice filtration has length $1$.
\end{proof}
\begin{theorem}
\label{thm:2coh}
Let $F$ be a field of characteristic not 2 with $2$-cohomological dimension less than or equal to 2,
such that $H^{p,q}(F;\mathbb{Z}/2)$ is finite for all $p,q$, and $H^{0,q}(F;\mathbb{Z}_2)$ is zero for $q \neq 0$.
Then
\[
\mathbf{MGL}_{*,*}(F;\mathbb{Z}_2) \cong H^{*,*}(F;\mathbb{Z}_2)\otimes L_*.
\]
Here $H^{p,q}(F;\mathbb{Z}_2)$ is a summand of $\mathbf{MGL}_{-p,-q}(F;\mathbb{Z})$,
and $L_k$ is identified with $\mathbf{MGL}_{2k,k}(F;\mathbb{Z})$.
\end{theorem}
In particular \Cref{thm:2coh} applies to local fields in characteristic $0$, cf.~\cite[Corollary 5.10]{Ormsby}, and finite fields of odd characteristic.
\section{The motivic Brown-Peterson spectrum and motivic Morava $K$-theory}
\label{sec:morava}
In this section we compute the motivic homotopy groups of the 2-completed (truncated) motivic Brown-Peterson spectrum $\mathbf{BPGL}$ over $\mathbb{R}$ at the prime 2, and of mod 2 Morava $K$-theory $K(n)$ over fields with low virtual cohomological dimension of characteristic not 2. The techniques are analogous to the previous sections. As part of the computation we give a description of the action of the mod 2 motivic Steenrod algebra on powers of $\tau$. Except when we discuss the general construction of $\mathbf{BPGL}$ and $K(n)$ we work at the prime $2$, so $\mathbf{BPGL}$ will be a $2$-local spectrum.
Most of the results over the real numbers in this section have been obtained previously by Yagita in \cite{Yagita:atiyah}.
For the construction of $\mathbf{BPGL}$ and $K(n)$, we follow \cite{Hu-Kriz:remarks}, \cite{Vezzosi} and \cite[Chapter 4]{Ravenel}. See also \cite[Remark 5.3]{Spitzweck:slices}.
Localizing the bijection
$
[\mathbf{MGL}, \mathbf{MGL}]^\star \cong \mathbf{MGL}^{\star}[\![ b_1, b_2, \dots]\!]
$
\cite[Proposition 6.2]{NSO}
at $\ell$ we get the Quillen idempotent $e : \mathbf{MGL}_{(\ell)} \to \mathbf{MGL}_{(\ell)}$ \cite[Theorem 4.1.12]{Ravenel}.
The $\ell$-local motivic Brown-Peterson spectrum is defined as
\[
\mathbf{BPGL} = e\mathbf{MGL}_{{(\ell)}} = \operatorname{hocolim}(\mathbf{MGL}_{(\ell)} \xrightarrow{e} \mathbf{MGL}_{(\ell)} \xrightarrow{e} \dots).
\]
Since slices commute with homotopy colimits we get
\begin{align*}
\mathsf{s}_q(\mathbf{BPGL})
& = \mathsf{s}_q(\operatorname{hocolim}(\mathbf{MGL}_{(\ell)} \xrightarrow{e} \mathbf{MGL}_{(\ell)} \xrightarrow{e} \dots)) \\
& = \operatorname{hocolim}(\mathsf{s}_q(\mathbf{MGL}_{(\ell)}) \xrightarrow{\mathsf{s}_q(e)} \mathsf{s}_q(\mathbf{MGL}_{(\ell)}) \xrightarrow{\mathsf{s}_q(e)} \dots) \\
& = \operatorname{hocolim}(\mathbf{M}\Z_{(\ell)} \otimes L_q \xrightarrow{\mathsf{s}_q(e)} \mathbf{M}\Z_{(\ell)} \otimes L_q \xrightarrow{\mathsf{s}_q(e)} \dots) \\
& = \operatorname{hocolim}(\mathbf{M}\Z_{(\ell)} \otimes L_q \xrightarrow{1 \otimes e} \mathbf{M}\Z_{(\ell)} \otimes L_q \xrightarrow{1\otimes e} \dots) \\
& = \mathbf{M}\Z_{(\ell)} \otimes \operatorname{colim}(\mathbb{Z}_{(\ell)} \otimes L_q \xrightarrow{1 \otimes e} \mathbb{Z}_{(\ell)} \otimes L_q \xrightarrow{1 \otimes e} \dots) \\
& = \mathbf{M}\Z_{(\ell)} \otimes (\mathbb{Z}_{(\ell)}[v_1, v_2, \dots])_q,
\end{align*}
where $\vert v_i \vert = \ell^i - 1$. %
This computation has been carried out in more detail in \cite{Levine-Tripathi}.
The truncated Brown-Peterson spectra are defined to be
\[
\mathbf{BPGL}\langle n \rangle = \mathbf{BPGL}/(v_{n+1}, v_{n+2}, \dots).
\]
The quotient is obtained by inductively forming the cofiber $\mathbf{BPGL}/(v_{n+1}, \dots, v_{n+k-1}, v_{n+k})$ and then taking a homotopy colimit, see \cite{Spitzweck:slices}.
In particular $\mathbf{BPGL}\langle 0 \rangle = \mathbf{M}\Z_{(\ell)}$,
and $\mathbf{BPGL}\langle 0 \rangle = \mathbf{kgl}_{(\ell)} = \mathsf{f}_0(\mathbf{KGL}_{(\ell)})$ is the (very) effective cover of $\ell$-local algebraic $K$-theory \cite{Spitzweck:slices}.
The slices are
\[
\mathsf{s}_q(\mathbf{BPGL}\langle n \rangle) \cong \mathbf{M}\Z_{(\ell)} \otimes (\mathbb{Z}_{(\ell)}[v_1, \dots, v_n])_q.
\]
The $n$th mod $\ell$ Morava $K$-theory is defined to be
\[
K(n) = \mathbf{BPGL}/(\ell, v_i \vert i \neq n)[v_n^{-1}].
\]
The slices are
\[
\mathsf{s}_q(K(n)) = \mathbf{M}\Z/\ell \otimes (\mathbb{Z}/\ell[v_n^{\pm 1}])_q.
\]
Hence for $\mathsf{s}_q(K(n))$ to be nonzero $q$ must be a multiple of $\ell^{i} - 1$.
From now on we fix $\ell = 2$.
By adapting the proof of \Cref{thm:MGLR} we obtain the following calculation of the motivic homotopy groups of $\mathbf{BPGL}_{2}^{\wedge}$ and $\mathbf{BPGL}\langle n\rangle_{2}^{\wedge}$ over the reals.
This has been stated previously by Hu and Kriz in \cite{Hu-Kriz:real}, \cite{Hu-Kriz:remarks}.
\begin{theorem}
\label{thm:BPGL}
The motivic homotopy groups of $2$-complete $\mathbf{BPGL}$ and $\mathbf{BPGL}\langle n\rangle$ over the real numbers are
\[
\mathbf{BPGL}_\star(\mathbb{R};\mathbb{Z}_2) \cong
\frac{\mathbb{Z}_2[\rho, z_{j,l} \vert l, j \geq 0]}
{(2\rho, \rho^{2^{j+1} - 1}z_{j,l}, z_{k,l}z_{a,b} = z_{k,l+b2^{a-k}}z_{a,0}, \text{ for } a \geq k)},
\]
\[
\mathbf{BPGL}\langle n \rangle_\star(\mathbb{R};\mathbb{Z}_2) \cong
\frac{\mathbb{Z}_2[\rho, z_{j,l}, u^{2^n} \vert l,j \geq 0, n \geq j ]}
{(2\rho, \rho^{2^{j+1} - 1}z_{j,l}, z_{k,l}z_{a,b} = z_{k,l+b2^{a-k}}z_{a,0}, \text{ for } a \geq k, z_{j,l}u^{2^n} = z_{j,l+2^{n-j}})}.
\]
\end{theorem}
\begin{proof}
Since $\mathbf{BPGL}$ is a summand of $\mathbf{MGL}_{(2)}$, the differentials take the same form.
The differentials of $\mathbf{BPGL}\langle n \rangle$ are determined by the differentials of $\mathbf{BPGL}$.
\end{proof}
Next we compute the $n$th mod 2 Morava $K$-theory of fields not of characteristic 2 with virtual cohomological dimension $\text{vcd}(F) < 2(2^n-1)$.
\begin{lemma}[\protect{\cite[Lemma 8.11]{Hoyois:fromto}}]
\label{lem:veff-agree}
The effective and the very effective slice filtrations \cite{Bachmann:veff} are the same on
$\mathbf{MGL}$, $\mathbf{BPGL}$, $\mathbf{BPGL}\langle n \rangle$ and $K(n)$.
That is,
$
\mathsf{f}_q(\mathbf{E}) = \widetilde{\mathsf{f}}_q(\mathbf{E}),
$
for $\mathbf{E}$ any of the above spectra.
\end{lemma}
\begin{proof}
By \cite[Lemma 8.11]{Hoyois:fromto}, $\mathsf{f}_q(\mathbf{E})$ is $q$-connective.
Hence, $\mathsf{f}_q(\mathbf{E})$ actually lies in $\mathcal{SH}^{\text{veff}}(q)$,
but then it is equal to $\widetilde{\mathsf{f}}_q(\mathbf{E})$ by uniqueness of adjoints.
\end{proof}
\begin{lemma}
\label{lem:veff-real}
Let $\mathbf{E}$ be a motivic spectrum in $\mathcal{SH}(\mathbb{C})$ whose complex realization
$R_\mathbb{C}(\mathbf{E})$ has only cells in even dimensions.
Then the complex realization of the very effective slice filtration of $\mathbf{E}$ is twice the Postnikov filtration of the complex realization of $\mathbf{E}$.
That is,
$
R_\mathbb{C}(\widetilde{\mathsf{f}}_q(\mathbf{E})) = (R_\mathbb{C}(\mathbf{E}))_{\geq 2q}.
$
Hence complex realization induces a map from the very effective slice spectral sequence of $\mathbf{E}$ to the Atiyah-Hirzebruch spectral sequence of $R_\mathbb{C}(\mathbf{E})$ \cite[Theorem 3.3]{Maunder}.
\end{lemma}
\begin{proof}
By construction \cite[Section 4]{Bachmann:veff}
\[
\mathcal{SH}(\mathbb{C})^\text{veff}(q) = T^{\wedge q} \wedge \mathcal{SH}(\mathbb{C})^\text{eff}_{\geq 0} =
T^{\wedge q} \wedge \{\mathbf{E} \in \mathcal{SH}(\mathbb{C})^\text{eff} \vert \underline{\pi}_{i, 0}(\mathbf{E}) = 0, i < 0\},
\]
where $\underline{\pi}_{i,0}(\mathbf{E})$ is the Nisnevich sheaf associated to the presheaf $X \mapsto [\Sigma^\infty X_+ \wedge S^{i,0}, \mathbf{E}]$.
Since $R_\mathbb{C}(T) \simeq S^2$, we have
$R_\mathbb{C}(\mathcal{SH}(k)^\text{veff}(q)) = \mathcal{SH}_{\geq 2q}$, i.e., the usual Postnikov filtration of $\mathcal{SH}$ at twice the usual speed.
If $R_\mathbb{C}(\mathbf{E})_{\geq 2q} = R_\mathbb{C}(\mathbf{E})_{\geq 2q + 1}$,
then $R_\mathbb{C}(\widetilde{\mathsf{f}}_q(\mathbf{E})) = (R_\mathbb{C}(\mathbf{E}))_{\geq 2q}$ by definition.
\end{proof}
Note that the condition of \Cref{lem:veff-real} is satisfied by the spectra
$\mathbf{MGL}, \mathbf{BPGL}, \mathbf{BPGL}\langle n \rangle$ and $K(n)$.
The homotopy groups of their complex realizations are only nonzero in even degrees.
Hence for each of these spectra we have a well defined map from their slice spectral sequence to the Atiyah-Hirzebruch spectral sequence of their complex realizations.
\begin{lemma}
\label{lem:Kn-diffs}
Over any field $F$ of characteristic not 2 the $d^i$-differential in the slice spectral sequence for $K(n)$ is trivial for $i < r = 2^n-1$ and
$d^{r} = Q_n + \Psi$. Here $Q_n$ is the $n$th motivic Milnor primitive, the dual of $\tau_n$ in the mod 2 dual motivic Steenrod algebra, while $\Psi$ is some element of the motivic Steenrod algebra which acts as zero on $H^\star(F;\mathbb{Z}/2)$.
\end{lemma}
\begin{proof}
Since the distance between the nonzero slices are $2^n-1$ the first possibly nonzero differential is $d^{2^n-1}$
which is an element of $\mathcal{A}^{2^{n+1}-1,2^n-1}$.
Elements in this bidegree are of the form
\[
aQ_n + bQ_0\hat{f} + \phi \Phi,
\]
where $a, b \in \mathbb{Z}/2$, $\phi \in H^{1,1}(\mathbb{R};\mathbb{Z}/2) = \mathbb{Z}/2\{\rho\}$, $\Phi \in \mathcal{A}^\star$, $f$ is a polynomial in the dual motivic Steenrod algebra and $\hat{f}$ its dual.
By complex realization $a = 1$ and $b = 0$ by \Cref{lem:veff-real}, \Cref{lem:veff-agree} and \cite[Lemma 2.1]{Yagita:Morava} (note that \cite[Lemma 2.1]{Yagita:Morava} is only stated for odd primes, but because of the (noncommutative) ring structure on $K(n)$, cf.~\cite[p.~1756]{Nassau}, Yagita's proof works for $p=2$). %
It remains to show that $\Psi = \phi \Phi$ acts as zero on $H^\star(F;\mathbb{Z}/2)$.
Using \Cref{thm:BPGL}, the map $\mathbf{BPGL} \to K(n)$ implies that $d^{2^n-1}(\tau^{2^n}) = \rho^{2^{n+1}-1} = Q_n(\tau^{2^n})$, where the last equality is by \Cref{cor:Qn-action}. Note that \Cref{thm:BPGL} is only stated for $\mathbb{R}$, but by base change from $\mathbb{Q}$ the differential takes the same form over any field of characteristic 0, since $H^{p,q}(\mathbb{R};\mathbb{Z}/2) \cong H^{p,q}(\mathbb{Q};\mathbb{Z}/2)$ when $p > 2$. For fields of positive characteristic there is nothing to show, since then $\rho^3 = 0$.
More generally we get that $d^{2^n-1}(\tau^{u2^n}) = Q_n(\tau^{u2^n})$, so $\Psi(\tau^{u2^n}) = 0$.
Since $\Psi$ has bidegree $(2^{n+1}-1, 2^n-1)$ we have $\Psi(\tau^k) = 0$, since $H^{2^{n+1}-1, 2^n-1 + k} = 0$ when $k < 2^n$.
Next we use the $\mathbf{BPGL}$-module structure $\mathbf{BPGL} \wedge K(n) \to K(n)$ and the Leibniz rule to conclude that
$\Psi(\tau^{k + u2^n}) = 0$ for $0 \leq k < 2^n$ and all $u$.
Indeed,
\begin{align*}
Q_n(\tau^{k + u2^n}) + \Psi(\tau^{k + u 2^n})
&= d^{2^n-1}(\tau^{u2^n}\wedge \tau^k)
= d^{2^n-1}(\tau^{u2^n})\wedge \tau^k + \tau^{u2^n} \wedge d^{2^n-1}(\tau^{k}) \\
&= Q_n(\tau^{u2^n})\wedge \tau^k + 0.
\end{align*}
\end{proof}
\begin{remark}
The condition $(d^{2^n-1})^2 = 0$ is not sufficient to determine if $\phi = 0$.
Indeed, for $n=2$ we could have $d^3 = Q_2 + \rho Q_0 Q_1\operatorname{Sq}^2$.
One approach to decide if $d^{2^n-1} = Q_n$ would be to compute the motivic Morava $K$-theory of products of the motivic classifying space $B\mu_2$.
\end{remark}
Over the real numbers the next result have been obtained in \cite[p.~10]{Hu-Kriz:remarks} and \cite[Theorem 6.4]{Yagita:atiyah}.
\begin{theorem}
\label{lem:Kn-coeff}
\label{thm:Kn}
The associated graded of the motivic homotopy groups of $K(n)$ over a field $F$ of characteristic not 2 with virtual cohomological dimension $\text{vcd}(F) < 2(2^n-1)$ is
\begin{align*}
E^\infty(K_\star(n)) \cong &
\bigoplus_{i\geq0, {i \choose 2^n} \equiv 0}k_*(F)[v_n^{\pm 1}]/\rho^{2^{n+1}-1}\{\tau^i\}\\
&\oplus
\bigoplus_{i\geq0, {i \choose 2^n} \equiv 1}\ker(\rho^{2^{n+1}-1} : k_*(F)[v_n^{\pm 1}] \to k_*(F)[v_n^{\pm 1}])\{\tau^i \}.
\end{align*}
Here $\rho_{} \in E^\infty_{-1,0,-1}(K(n)), \tau \in E^\infty_{0,0,-1}(K(n)), v_n \in E^\infty_{2(2^{n}-1), 2^n-1, 2^{n}-1}(K(n))$.
When $p - 2w < 0$, $K_{p,w}(n) = 0$.
See \Cref{fig:K(n)} for a picture of $K_{\star}(n)(\mathbb{R})$.
\end{theorem}
\begin{proof}
As a module over mod $2$ Milnor $K$-theory $k_*(F)$ we have
\[
E^1(K(n)) = \pi_{*,*}\mathsf{s}_*(K(n)) = \oplus_{i\geq0}k_*(F)[v_n^{\pm 1}]\{\tau^i\}.
\]
The differentials are $k_*(F)$- and $v_n$-linear,
the module structure being induced by the $\S$-module and $\mathbf{BPGL}$-module structure on $K(n)$.
By \Cref{lem:Kn-diffs} we may assume $d^{2^n-1} = Q_n$, so \Cref{cor:Qn-action} implies that the $E^{2^n}$-page is
\begin{align*}
E^{2^n}(K(n)) =&
\bigoplus_{i\geq0, {i \choose 2^n} \equiv 0}k_*(F)[v_n^{\pm 1}]/\rho^{2^{n+1}-1}\{\tau^i\}\\
&\oplus
\bigoplus_{i\geq0, {i \choose 2^n} \equiv 1}\ker\left(\rho^{2^{n+1}-1} : k_*(F)[v_n^{\pm 1}] \to k_*(F)[v_n^{\pm 1}]\right)\{\tau^i \}.
\end{align*}
Over a field $F$ with $\text{vcd}(F) < 2(2^n-1)$ the spectral sequence collapses at the $E^{2^n}$-page for degree reasons,
i.e., $E^{2^n}(K(n)) = E^\infty(K(n))$.
Indeed,
the $d^r$-differential is a map
$d^r : E^{r}_{p,q,w} \to E^{r}_{p-1,q+r,w}$,
where $E^r_{p,q,w}$ is a subquotient of $H^{2q-p,q-w}(F;\mathbb{Z}/2) \otimes (\mathbb{Z}[v_n^{\pm}])_q$.
At an earlier stage the group $E^{r'}_{p-1,q+r,w}$ could be the target of a nonzero $d^{r'}$-differential from $E^{r'}_{p,q+r-r',w}$.
If $2(q+r-r') - p = 2q - p + 2(r-r') \geq 0 + 2(r - r') \geq 2(2^n-1) > \text{vcd}(F)$ then the nonzero $d^{r'}$-differential is an isomorphism, hence the target of the $d^r$-differential is zero (note that $r$ is necessarily a multiple of $2^n-1$ and $r'=2^n-1$). Hence there are no nonzero $d^r$-differentials for $r > 2^n-1$.
\end{proof}
\begin{remark}
For $K(1) = \mathbf{KGL}/2$ a finer analysis can be carried out over number fields, cf.~\cite{Rognes-Weibel}, \cite[Section 4]{KRO}.
\end{remark}
\begin{remark}
\label{rmk:Kn-exts}
Over the real numbers we are unable to determine the additive extension
\[
E^\infty_{0,*,-(2^n-1)} = \mathbb{Z}/2\{\tau^{2^n-1}, \rho^{2^{n+1}-2}v_n \}.
\]
Based on the $n=0,1$ cases, we may guess that multiplication by $2$ on $K(n)$ induces the map $v_n\rho^nQ_0Q_1\dots Q_{n-1}$ on slices, which would detect that the extension is nonsplit.
\end{remark}
\begin{figure}
\centering
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{\node[above right=0pt,font=\tiny] at (18.0,17.0) {$v_n$};}
{\draw[fill] (18.0,17.0) circle (4pt);}
{\node[above right=0pt,font=\tiny] at (6.0,11.0) {$v_n^{-1}$};}
{\draw[fill] (6.0,11.0) circle (4pt);}
{\node[above right=0pt,font=\tiny] at (10.5,13.0) {$\rho$};}
{\draw[fill] (11.0,13.0) circle (4pt);}
{\node[above right=0pt,font=\tiny] at (12.0,13.0) {$\tau$};}
{\draw[fill] (12.0,13.0) circle (4pt);}
{\node[above right=0pt,font=\tiny] at (6.1,7.7) {$\rho^{2^{n+1}-2}$};}
{\draw[fill] (6.0,8.0) circle (4pt);}
{\node[above right=0pt,font=\tiny] at (10.8,9.7) {$\{\tau^{2^{n}-1}, \rho^{2^{n+1}-2}v_n \}$};}
{\draw[fill] (12.0,11.0) circle (4pt);}
{\node[above right=0pt,font=\tiny] at (12.0,6.0) {$\tau^{2^{n+1}}$};}
{\draw[fill] (12.0,6.0) circle (4pt);}
{\draw[fill] (0.0,5.0) circle (6pt);}
{\draw[fill] (6.0,8.0) circle (6pt);}
{\draw[fill] (6.0,0.0) circle (6pt);}
{\draw[fill] (12.0,11.0) circle (6pt);}
{\draw[fill] (12.0,3.0) circle (6pt);}
{\draw[fill] (18.0,14.0) circle (6pt);}
{\draw[fill] (18.0,6.0) circle (6pt);}
\end{tikzpicture}
\caption{$K_{\star}(n)$ of the real numbers for $n=2$. The pattern is repeated horizontally by multiplication by $v_n$ and vertically by multiplication by $\tau^{2^{n+1}}$.}
\label{fig:K(n)}
\end{figure}
Below are some lemmas on the action of $\mathcal{A}^\star$ on $H^\star(F;\mathbb{Z}/2)$ used in the proof of \Cref{lem:Kn-coeff}.
The following lemma for $k = 2^j$ is \cite[Lemma 6.3]{Yagita:atiyah}.
\begin{lemma}
\label{lem:Sqtau}
The action of the motivic Steenrod squares on $\tau^n$ is given by
\begin{align*}
\operatorname{Sq}^{2k}(\tau^n) &= {\lfloor\frac{n+k-1}{2}\rfloor \choose k}\rho^{2k}\tau^{n-k}, k > 0, \\
\operatorname{Sq}^{2k+1}(\tau^n) &=
\left({\lfloor\frac{n+k+1}{2}\rfloor \choose k + 1}
+ {\lfloor\frac{n+k}{2}\rfloor \choose k + 1}\right)\rho^{2k+1}\tau^{n-k-1}.
\end{align*}
Here we define $\tau^j = 0$ for $j < 0$.
\end{lemma}
\begin{proof}
We use the notation $P^k = Sq^{2k}$ and $\operatorname{Sq}^{2k+1} = B^k$,
and define $P^k(\tau^n) = P^{k,n}\rho^{2k}\tau^{n-k}$, where $P^{k,n}\in\mathbb{Z}/2$, and similarly for $B^{k,n}\in\mathbb{Z}/2$.
The Cartan formula \cite[Proposition 9.6]{Voevodsky:power}, \cite[Proposition 4.4.2]{Riou}, \cite[p.~32]{Riou:slides} implies
\begin{align*}
P^{k}(\tau^n) &= \tau P^{k}(\tau^{n-1}) + \rho\tau B^{k-1}(\tau^{n-1}) \\
B^{k}(\tau^n) &= \tau B^{k}(\tau^{n-1}) + \rho P^{k-1}(\tau^{n-1}) + \rho^2 B^{k-1}(\tau^{n-1})
\end{align*}
which gives the recursion \eqref{eq:PBrecursion} in \Cref{lem:PBrecursion} with the solution \eqref{eq:PBsolution}.
\end{proof}
\begin{lemma}
\label{lem:PBrecursion}
Modulo 2
\begin{align}
\label{eq:PBsolution}
P^{k,n} &= \begin{cases}
1 & k = n = 0 \\
{\lfloor\frac{n+k-1}{2}\rfloor \choose k} & \text{otherwise},
\end{cases} \\
B^{k,n} &= {\lfloor\frac{n+k+1}{2}\rfloor \choose k + 1}
+ {\lfloor\frac{n+k}{2}\rfloor \choose k + 1},
\nonumber
\end{align}
solves the recursion
\begin{align}
\label{eq:PBrecursion}
P^{0,n} &= 1, B^{0,2n} = 0, B^{0,2n+1} = 1, n \geq 0, \\
P^{0,0} &= 1, B^{k,0} = 0, P^{k+1,0} = 0, k > 0, \nonumber\\
P^{k,n} &= P^{k,n-1} + B^{k-1,n-1}, \nonumber\\
B^{k,n} &= B^{k,n-1} + P^{k,n-1} + B^{k-1,n-1},\nonumber
\end{align}
for $k, n \geq 0$.
\end{lemma}
\begin{remark}
By adding correction terms of the form ${0 \choose k}{n \choose 0}$ to $P^{k,n}$ and $B^{k,n}$ in \eqref{eq:PBsolution} we obtain closed forms for $P^{k,n}$ solving \eqref{eq:PBrecursion} for all integers $k, n$.
\end{remark}
As a consequence of \Cref{lem:Sqtau} we get the following lemma.
This is also stated in \cite[Lemma 6.2]{Yagita:atiyah}.
\begin{lemma}
\label{cor:Qn-action}
The action of the Milnor-primitives on $\tau^n$ is given by
\[
Q_k(\tau^n) = {n \choose 2^k}\rho^{2^{k+1}-1}\tau^{n-2^k}.
\]
\end{lemma}
\begin{proof}
Do induction on $k$ and $n$, use \cite[Corollary 4]{Kylling} and the identity
\begin{align*}
{n \choose 2^{k+1}} =&
{n \choose 2^{k}}{\lfloor\frac{n-1}{2}\rfloor \choose 2^{k}}
+
{n-2^k \choose 2^{k}}{\lfloor\frac{n+2^k-1}{2}\rfloor \choose 2^{k}}\\
&+
{n-2^k \choose 2^{k}}{n \choose 2^{k-1}}{\lfloor\frac{n-1}{2}\rfloor \choose 2^{k-1}} \\
=&
{n \choose 2^{k}}{\lfloor\frac{n-1}{2}\rfloor \choose 2^{k}}
+
{n-2^k \choose 2^{k}}{\lfloor\frac{n+2^k-1}{2}\rfloor \choose 2^{k}} \bmod 2.
\end{align*}
The identity is easy to show using that ${n \choose 2^k} = 1$ if and only if the $k$th bit in the binary expansion of $n$ is $1$, and similar expressions for the other binomial coefficients.
\end{proof}
\printbibliography
\end{document}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,595
|
Retro Gaming The 70s & 80s: Dungeon!
In 1975, just a year after the release of Dungeons & Dragons, TSR Hobbies released the boardgame Dungeon!. The game mirrored many aspects of D&D's dungeon-crawling framework while offering a much lower point of entry for players either new to the genre or looking to just play a quick game.
I got a copy of Dungeon! after a couple years of spending hours playing D&D and painting my first lead miniatures from the likes of Ral Partha. Dungeon! offered a traditional boardgame format and came as an occasional break from the open-ended role-playing my friends and I were used to with D&D. Ads for the game at the time (left) were also pitching the game to families and casual gamers as a point of entry and enticement to the larger world D&D offered.
Playing as an Elf, Hero, Superhero or Wizard, players moved through six progressively-difficult levels, encountered random monsters and collected treasure. Each character class carried specific strengths and powers, with more powerful characters such as the Wizard or Superhero requiring greater treasure to win the game.
Large chambers such as the Kitchen, Crypt, King's Library or Queen's Treasure Room were surrounded by smaller rooms, color-coded to indicate the level. In each large chamber, a stack of three randomized monster cards specific to that level were placed at the beginning at the game. In each smaller room, one monster and a treasure was placed. When a space was entered, the top card was drawn and combat between the player and the monster ensued with a simple die roll. Winning combat won the treasure while ties or losses might result in a retreat or loss of treasure.
For players familiar with D&D like myself, Dungeon! mirrored a lot of the established canon. Monsters fell along the lines of hobgoblins, werewolves, dragons, mummies and snakes. Dungeon pitfalls such as traps might be encountered. Characters like Wizards held special spell cards such as Fireball or Teleport, and found magic items included an ESP Medallion and a Magic Sword, offering certain characters bonuses to their play. Treasure ranged from meager bags of glod found in the easy levels to the covtted Huge Diamond (worth 10,000 points) hidden deep in the sixth level.
Several editions of Dungeon! followed through the years, the most recent in 2012 from Wizards of the Coast, now-owners of the D&D franchise. Longtime players have also created a rich universe of house rules, customized boards and even hand-painted figures to supplement the basics of the game. Despite its relative ease of play, Dungeon! has remained a classic bit of fun among even the most serious gamers today.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,262
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Formica dakotensis är en myrart som beskrevs av Carlo Emery 1893. Formica dakotensis ingår i släktet Formica och familjen myror. Inga underarter finns listade i Catalogue of Life.
Bildgalleri
Källor
Externa länkar
Myror
dakotensis
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,245
|
\section{Introduction}
Gravitational-wave (GW) astrophysics is in full bloom since the first direct detection of gravitational radiation from the infamous GW150914 black hole (BH) merger \cite{LIGOScientific:2016aoc}. The Laser Interferometer Gravitational-Wave Observatory (LIGO) and Virgo continue detecting GW events with imperious momentum \cite{LIGOScientific:2021djp}, which has lead to a vast database of incoming merger signals from the edges of our Cosmos.
Our current knowledge regarding the evolution of gravitationally captured compact objects indicates that the merger's constituents undergo a decaying circumrotation around a center of mass, know as the inspiral stage, till they merge in a violent manner to unite into a single remnant which vibrates till its relaxation to a final stable state is achieved.
The final phase is known as the ringdown and can be, in principle, characterized by the vibrational spectra of the final perturbed compact object; the quasinormal mode (QNMs) frequencies \cite{Kokkotas:1999bd,Berti:2009kk,Konoplya:2011qq}. The extraction of QNMs from realistic GW ringdown data, in order to characterize the externally observable properties of the final object, the underlying spacetime geometry and test the no-hair theorem of General Relativity (GR) is currently one of the most thriving topics of data-analysis exploration, known as BH spectroscopy \cite{Echeverria:1989hg,Dreyer:2003bv,Berti:2005ys,Berti:2007zu,Baibhav:2017jhs,Giesler:2019uxc,Isi:2019aib,Bustillo:2020buq,Isi:2021iql,Cotesta:2022pci}.
Besides the fact that BHs appear as `gravitational' bells, they exhibit a myriad of interesting phenomena which have yet to be detected experimentally. BHs are not completely dark and emit Hawking radiation \cite{Hawking:1974rv,Hawking:1975vcx}. Most importantly, BHs are sources of energy. They can amplify incident waves that scatter off them, under certain circumstances, in the expense of their rotational or electromagnetic energy \cite{Penrose:1971uk,Bekenstein:1973mi}. This phenomenon is similar to the Penrose process \cite{Penrose:1971uk} and is known as superradiance \cite{Zeldovich}. The vast literature of BH superradiance covers rotating and charged BHs, as well as stars and compact objects in GR and various modified theories of gravity (see the review \cite{Brito:2015oca} for further details).
The epitome of superradiance in GR are Reissner-Nordstr\"om (RN), Kerr and Kerr-Newman BHs \cite{DiMenza:2014vpa,Benone:2014qaa,Benone:2015bst,Balakumar:2020gli,Glampedakis:2001cx,Dolan:2008kf,Benone:2019all}. In principle, the occurence of superradiance does not necessarily imply that the spacetime itself will be destabilized. Nonetheless, BH superradiant instabilities do appear when these waves are confined in negative potential wells \cite{Dolan:2007mj,Dolan:2012yt,Vitor5,RN1,Kerr3,KN1,Konoplya:2008au,Konoplya:2013sba,Vieira:2021nha,Franzin:2021kvj} or when the boundary conditions are altered \cite{Vitor1,Vitor3,Vitor4,RN5,Kerr1,Kerr2,Kerr4,Kerr5,Li:2012rx,Li:2014fna,Li:2014gfg,Li:2015mqa,Herdeiro:2013pia,Herdeiro:2018wub,Sanchis-Gual:2015lje,Sanchis-Gual:2016tcm,Cuadros-Melgar:2021sjy}. Under such circumstances, perturbations grow exponentially and lead to the formation of BH bombs \cite{Press:1972zz}.
Interestingly, RN BHs embedded in de Sitter space can superradiantly amplify perturbations and when particular conditions are met the corresponding charged scalar QNMs become unstable \cite{Zhu:2014sya,Konoplya:2014lha,Cardoso:2018nvb,Dias:2018ufh,Mo:2018nnu,Destounis:2019hca} inciting charged-de Sitter BH bombs. This superradiant instability appears in a small parameter space of the system when the cosmological constant of spacetime and the charge of spherically-symmetric scalar perturbations are sufficiently small \cite{Dias:2018ufh,Destounis:2019hca}. de Sitter space provides a physical representation of the current cosmological evolution of the Universe, with the positive cosmological constant, although minuscule, supplying ample driving force for the accelerated expansion of our Universe.
The superradiant amplification of incident waves scattering off of Reissner-Nordstr\"om-de Sitter (RNdS) BHs has not been consistently studied yet, in contrast to superradiance in Kerr-de Sitter BHs \cite{Maeda:1993}. The analysis in \cite{Maeda:1993} has shown that despite the fact that the superradiant frequency range is shortened when a positive cosmological constant is incorporated, the amplification factors of incident waves are significantly elevated with respect to those occurring in Kerr BHs.
Our work aims to analyze the phenomenon of superradiance of charged massless and massive scalar incident waves impinging RNdS BHs with numerical scattering techniques, compare the amplification factors with those of RN BHs and interpret the effect of QNM resonances which belong in the superradiant regime.
We find that the existence of superradiant monopolar QNMs induces strong amplification of incident waves, when the wave's frequency coincides with the real part of the QNM, and is loosely related to the imaginary part of the QNM \cite{ChandraFerrari1991,ChandraFerrari2,Kokkotas1994}. The `hyperradiant' peaks formed around QNMs are oblivious to the linear stability properties of spacetime and are consistent with Breit-Wigner resonances \cite{Thorne1969,Berti:2009wx}. When monopolar scalar waves are highly charged their amplification factors approach $100\%$ from above, since the lifetime of QNMs becomes shorter, in direct contrast to scattered waves in RN which tend to $100\%$ amplification from below \cite{Brito:2015oca}.
We further find that higher multipole incident waves still possess superradiant frequency regimes but no QNMs, and subsequent resonant peaks, appear in such range. Even so, these waves can reach amplification factors up to $100\%$ from below at the large scalar charge limit. \emph{In finality, we demonstrate that the addition of mass to the scalar wave acts as a friction medium which rapidly diminishes resonant peaks and minimizes the amplification factors, since the increment of mass expels QNMs from the frequency domain of superradiance} \cite{Cardoso:2018nvb,Destounis:2019hca}. In what follows, we assume geometrized units such that $G=c=1$.
\section{The Reissner-Nordstr\"om-de Sitter spacetime}
RNdS BHs are spherically-symmetric static solutions to the Einstein-Maxwell field equations with a positive cosmological constant \cite{Hawking:1973uf}. They are described by the line element
\begin{equation}
ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2\left(d\theta^2+\sin^2\theta\,d\phi^2\right),
\end{equation}
where
\begin{align}
f(r)&=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\Lambda r^2}{3},
\end{align}
with $M,\, Q$ the mass and charge of the BH and $\Lambda>0$ the cosmological constant. The associated potential sourced by the electric field is $A_\mu=\left(-Q/r,0,0,0\right)$.
The causal structure of a subextremal RNdS spacetime possesses three distinct null hypersurfaces, namely the Cauchy $r=r_-$, event $r=r_+$ and cosmological horizon $r=r_c$, where $r_-<r_+<r_c$. Obviously, the BH's interior lies in $r\leq r_+$, with the Cauchy horizon being the boundary of maximal globally-hyperbolic development of initial data on a Cauchy hypersurface \cite{Hawking:1973uf}. The static (observable) region lies between the event and cosmological horizons, such as $r_+<r<r_c$. In the cosmological region, where $r\geq r_c$, the accelerated expansion of the Universe is so rapid that all events are infinitely redshifted and causally disconnected with the static region.
\section{Charged massive scalar wave equation}
We are interested in the propagation of linear massive and charged scalar fields $\psi$ on a fixed RNdS background. Their motion is governed by the Klein-Gordon equation
\begin{equation}\label{KG}
(D^\nu D_\nu-\mu^2)\psi=0,
\end{equation}
where $\mu$, $q$ the mass and charge of the field, respectively, and $D_\nu=\nabla_\nu-iqA_\nu$ the covariant derivative which incorporates the effects of curvature and electromagnetism. By expanding the scalar field $\psi(t,r,\theta,\phi)$ in terms of spherical harmonics (due to spherical symmetry) with a harmonic time dependence (due to a Fourier decomposition) as
\begin{equation}
\psi(t,r,\theta,\phi)=\sum_{l,m}\frac{\Psi_{l m}(r)}{r}Y_{lm}(\theta,\phi)e^{-i\omega t},
\end{equation}
we reduce Eq. \eqref{KG} to a one-dimensional Schr\"odinger-like equation
\begin{equation}
\label{master_eq_RNdS}
\frac{d^2 \Psi}{d r_*^2}+\left[\omega^2-2\omega\Phi(r)-V(r)\right]\Psi=0\,,
\end{equation}
where $\Phi(r)=q Q/r$ is the electrostatic potential energy, $qQ$ is the charge coupling and $dr_*=dr/f(r)$ is the tortoise coordinate. The effective potential for massive and charged scalar perturbations is then
\begin{equation}
\label{RNdS_general potential}
V(r)=f(r)\left(\mu^2+\frac{\ell(\ell+1)}{r^2}+\frac{f^\prime(r)}{r}\right)-\Phi(r)^2,
\end{equation}
where $\ell$ is the multipolar degree (angular number) of spherical harmonics which correspond to the eigenvalue $\ell(\ell+1)$ of the squared orbital angular momentum operator. Here, primes denote derivatives with respect to the radial coordinate $r$.
\begin{figure*}[t]
\includegraphics[scale=0.5]{Figure1}
\caption{Left: Effective potential of $\ell=0$ massless charged scalar perturbations with charge coupling $qQ=0.4$ for a RNdS BH with charge $Q=0.5Q_\text{max}$ and varying cosmological constants $\Lambda M^2$. Right: Effective potential of $\ell=0$ massless charged scalar perturbations for a RNdS BH with charge $Q=0.5Q_\text{max}$, cosmological constant $\Lambda M^2=0.025$ and varying charge couplings $qQ$.}
\label{potentials}
\end{figure*}
\section{Superradiant amplification of monochromatic waves}
Let a massive and charged monochromatic incident wave from the cosmological horizon, with amplitude coefficient $\mathcal{I}$, that scatters off of the RNdS photon sphere. The wave will be partially reflected back toward the cosmological horizon, with reflection coefficient $\mathcal{R}$, and partially transmitted through the potential barrier and into the event horizon, with transmission coefficient $\mathcal{T}$. The above scattering experiment translates to the following boundary conditions in (\ref{master_eq_RNdS}):
\begin{equation}
\label{scat}
\Psi \sim
\left\{
\begin{array}{lcl}
\mathcal{T} e^{-i (\omega-\Phi(r_+))r_* },\,\,\,\,\,\,\quad\quad\quad\quad\quad\quad r \rightarrow r_+, \\
&
&
\\
\mathcal{I}e^{-i(\omega-\Phi(r_c))r_*} + \mathcal{R} e^{i(\omega-\Phi(r_c))r_*},\, r \rightarrow r_c.
\end{array}
\right.
\end{equation}
Due to the fact that the Wronskian of $\Psi$ and its linearly-independent complex conjugate counterpart $\Psi^\dagger$ is $r_*$-independent, we immediately realize that the Wronskian at both boundaries coincide. This equality leads to the following relation
\begin{equation}\label{Wronskian}
|\mathcal{R}|^2=|\mathcal{I}|^2-\frac{\omega-\Phi(r_+)}{\omega-\Phi(r_c)}|\mathcal{T}|^2.
\end{equation}
From relation \eqref{Wronskian}, we observe that when the incident wave's frequency satisfies
\begin{equation}
\label{suprad}
\Phi(r_c)<\omega<\Phi(r_+),
\end{equation}
then the amplitude of the reflected wave is larger than the amplitude of the incident wave. Equation \eqref{suprad} is known as the superradiance relation. Physically, this translates to the amplification of incident massive and charged monochromatic waves under the expense of the BH's electromagnetic energy. Notice that the presence of a non-zero mass to the incident wave does not affect Eq. \eqref{suprad} in contrast to the case when $\Lambda M^2\rightarrow 0$, where Eq. (\ref{suprad}) reduces to the respective one for asymptotically flat RN BHs and the scalar mass becomes the lower frequency bound for superradiant amplification \cite{Bekenstein:1973mi,Brito:2015oca}.
In what follows, we use the energy fluxes of scalar fields at the cosmological horizon to define the amplification factor \cite{Brito:2015oca}
\begin{equation}\label{amplification_factor}
Z_\ell=\frac{|\mathcal{R}|^2}{|\mathcal{I}|^2}-1,
\end{equation}
as a function of the frequency $\omega$. It is obvious that when $Z_\ell< 0$ superradiance does not occur while when $Z_\ell> 0$ the incident wave is superradiantly amplified. In turn, $Z_\ell=0$ holds at the bounds of the superradiant relation \eqref{suprad}.
To numerically integrate Eq. \eqref{master_eq_RNdS} with boundary conditions \eqref{scat}, we expand its solutions at the event and cosmological horizons to arbitrary order (till we reach numerical convergence) and then match the two asymptotic solutions at an intermediate point by imposing regularity of the solutions and their derivatives. Through this numerical process we extract the amplification factor for numerous monochromatic incident waves with varying $\omega$. We have performed immense convergence tests by increasing the order of expansion of solutions at the boundaries of integration and also find very good agreement with the amplification factors of RN shown in \cite{Brito:2015oca}.
\section{Quasinormal modes}
\begin{figure*}[t]
\includegraphics[scale=0.43]{Figure2}
\caption{Left: Amplification factors of massless $\ell=0$ monochromatic waves with $qQ=0.5$ for a RNdS BH with $Q=0.5Q_\text{max}$ and varying cosmological constant $\Lambda M^2$. The horizontal black dashed line designates the onset of superradiant amplification while the black dot-dashed line designates amplification factors that equal $100\%$. Right: Same as left but with $Q=0.999Q_\text{max}$.}
\label{varL}
\end{figure*}
\begin{figure*}[t]
\includegraphics[scale=0.43]{Figure3}
\caption{Left: Amplification factors of massless $\ell=0$ monochromatic waves for a RNdS BH with $Q=0.5Q_\text{max}$, $\Lambda M^2=0.025$ and varying charge coupling $qQ$. The horizontal black dashed line designates the onset of superradiant amplification while the black dot-dashed line designates amplification factors that equal $100\%$. Right: Same as left but with $Q=0.999Q_\text{max}$.}
\label{varqQ}
\end{figure*}
To obtain the characteristic QNM frequencies $\omega_\text{QNM}$ of RNdS spacetime, we impose ingoing waves at the event horizon and outgoing waves at the cosmological horizon by neglecting the incident wave in Eq. \eqref{scat}, that is $|\mathcal{I}|=0$ for QNMs. From the asymptotic behavior of Eq. \eqref{master_eq_RNdS} these boundary conditions translate to~\cite{Berti:2009kk}
\begin{equation}
\label{bcs}
\Psi \sim
\left\{
\begin{array}{lcl}
e^{-i (\omega-\Phi(r_+))r_* },\,\,\,\quad r \rightarrow r_+, \\
&
&
\\
e^{+i(\omega-\Phi(r_c))r_*},\,\,\,\,\quad r \rightarrow r_c.
\end{array}
\right.
\end{equation}
The QNM frequencies are characterized, for each $\ell$, by an integer $n\geq 0$ labeling the mode number, with the fundamental mode $n=0$ corresponding, by definition, to the non-vanishing frequency $\omega_R$ with the smallest (by absolute value) imaginary part $\omega_I$ and $n>0$ denotes higher harmonics of oscillation (overtones). Due to the underlying symmetry $\omega\rightarrow-\omega$ and $\Phi(r)\rightarrow-\Phi(r)$ of Eq. (\ref{master_eq_RNdS}), we will consider only cases where the charge coupling satisfies $qQ>0$.
The QNMs of RNdS spacetime has been extensively analyzed in the current literature. Their spectra consist of three distinct families of modes which govern the early and late-time dynamics of perturbation evolution (see \cite{Cardoso:2017soq,Cardoso:2018nvb,Destounis:2018qnb,Liu:2019lon,Destounis:2019omd,Dias:2018etb,Dias:2018ufh,Mo:2018nnu,Joykutty:2021fgj} for further details). Even though a QNM analysis is not the main point of this work, any QNMs discussed henceforth will be obtained via the Mathematica package of \cite{Jansen:2017oag} (based on collocation methods developed in \cite{Dias:2010eu}), and checked in various cases with a WKB approximation \cite{Iyer:1986np} and a code developed based on the matrix method in \cite{KaiLin1}.
\subsection{Superradiant instability and the role of the effective potential}
The fact that incident waves can be superradiantly amplified when scattered off of charged BHs is closely connected with the properties of the underlying effective potential. In RNdS spacetime, spherically-symmetric ($\ell=0$) charge scalar perturbations form quasibound states \cite{Dolan:2007mj,Vieira:2016ubt,Vieira:2021doo,Vieira:2021xqw,Vieira:2021ozg} localized at a potential well which is present right outside the photon sphere (see Fig. \ref{potentials}). For a particular subspace of the parameter space of the system, it has been shown that such modes become unstable due to confinement and simultaneously satisfy the superradiant relation \eqref{suprad} \cite{Zhu:2014sya,Konoplya:2014lha,Cardoso:2018nvb,Destounis:2019hca}. This does not occur in RN geometries, since there are no unstable or superradiant QNMs. Thus RNdS BHs withstand an alternative charged BH bomb mechanism, under charged scalar perturbations, without the need of AdS asymptotics or artificial mirrors placed around charged BHs to confine perturbations and trigger instabilities.
\section{Amplification factors, superradiance and resonances}
In what follows we present the amplification factors $Z_\ell$ of incident charged massless and massive scalar waves scattering off of RNdS BHs.
\subsection{$\ell=0$ scalar waves}
\begin{figure}[t]
\includegraphics[scale=0.42]{Figure4}
\caption{Amplification factors of massless $\ell=0$ monochromatic waves for a RNdS BH with $Q=0.5Q_\text{max}$, $\Lambda M^2=0.025$ and large charge couplings $qQ$. The horizontal black dashed line designates the onset of superradiant amplification while the black dot-dashed line designates amplification factors that equal $100\%$.}
\label{largeqQ}
\end{figure}
\begin{figure}[t]
\includegraphics[scale=0.42]{Figure5}
\caption{Amplification factors of massless $\ell=0$ monochromatic waves with charge coupling $qQ=0.5$ for a RNdS BH with $\Lambda M^2=0.025$ and varying charge $Q/Q_\text{max}$. The horizontal black dashed line designates the onset of superradiant amplification while the black dot-dashed line designates amplification factors that equal $100\%$.}
\label{varQ}
\end{figure}
\begin{table}[t]
\centering
\scalebox{1.1}{
\begin{tabular}{||c| c | c ||}
\hline
\multicolumn{3}{||c||}{$Q=0.5Q_\text{max}$, $qQ=0.5$} \\
\hline
$\Lambda M^2$ & $\omega_\text{QNM}$ & $\omega_\text{peak}$ \\ [0.5ex]
\hline
$0.015$ & 0.04652 + 0.00074 i & 0.04655 \\
\hline
$0.025$ & 0.06175 -- 0.00057 i & 0.06177 \\
\hline
\multicolumn{3}{||c||}{$Q=0.999Q_\text{max}$, $qQ=0.5$} \\
\hline
$\Lambda M^2$ & $\omega_\text{QNM}$ & $\omega_\text{peak}$ \\ [0.5ex]
\hline
$0.005$ & 0.02599 + 0.00169 i & 0.02626 \\
\hline
$0.025$ & 0.06217 + 0.00004 i & 0.06217 \\
\hline
\multicolumn{3}{||c||}{$\Lambda M^2=0.025$, $qQ=0.5$} \\
\hline
$Q/Q_\text{max}$ & $\omega_\text{QNM}$ & $\omega_\text{peak}$ \\ [0.5ex]
\hline
$0.2$ & 0.06173 -- 0.00078 i & 0.06176 \\
\hline
$0.7$ & 0.06182 -- 0.00034 i & 0.06183 \\
\hline
\multicolumn{3}{||c||}{$\Lambda M^2=0.025$, $Q=0.5Q_\text{max}$} \\
\hline
$qQ$ & $\omega_\text{QNM}$ & $\omega_\text{peak}$ \\ [0.5ex]
\hline
$0.8$ & 0.09876 -- 0.00518 i & 0.09947 \\
\hline
$1$ & 0.12174 -- 0.00925 i & 0.12382 \\
\hline
\end{tabular}
}
\caption{Superradiant $l=0$ massless charged scalar QNMs $\omega_\text{QNM}$ in RNdS spacetime with various parameters and the respective frequency peak position $\omega_\text{peak}$ of the amplification factor $Z_0$.}
\label{table}
\end{table}
First, we focus on $\ell=0$ waves which generally lead to superradiantly stable and unstable QNMs \cite{Zhu:2014sya,Konoplya:2014lha,Destounis:2019hca}. In Figs. \ref{varL}, \ref{varqQ}, \ref{largeqQ} and \ref{varQ} we show the amplification factors for $\ell=0$ waves, in various scenarios of the available parameter space. By fitting the amplification factor curves we find when $Z_0=0$ in order to further validate the accuracy of our numerical analysis. We find that the onset and termination of superradiance agrees very well with Eq. \eqref{suprad} for every case we explored.
\begin{figure*}[t]
\includegraphics[scale=0.43]{Figure6}
\caption{Left: Amplification factors of $\ell=1$ monochromatic waves for a RNdS BH with $Q=0.5Q_\text{max}$, $\Lambda M^2=0.025$ and varying charge coupling $qQ$. The horizontal dashed black line designates the onset of superradiant amplification. Right: Same as left but with larger charge couplings $qQ$.}
\label{varqQl1}
\end{figure*}
\begin{figure}[t]
\includegraphics[scale=0.42]{Figure7}
\caption{Amplification factors of monochromatic waves with charge coupling $qQ=1$ for a RNdS BH with $\Lambda M^2=0.025$, $Q=0.5Q_\text{max}$ and varying angular index $\ell$. The horizontal black dashed line designates the onset of superradiant amplification while the black dot-dashed line designates amplification factors that equal $100\%$.}
\label{varl}
\end{figure}
In Fig. \ref{varL} we observe that even though the frequency domain of superradiance is shortened when a positive cosmological constant is present, in agreement with the findings in Kerr-de Sitter BHs \cite{Maeda:1993}, the amplification is significantly elevated beyond $100\%$ due to the existence of hyperradiating resonances. Here we have to note that the cosmological constants chosen to demonstrate our results are far from its physical values. This is due to the limitations of a numerical evolution with such a small value of $\Lambda$. Nevertheless, we can safely expect that the amplification factors $Z_{0}$ of RNdS BHs with arbitrarily small $\Lambda$ will be identical to that of RN BHs (see Fig. \ref{varL}). For any other sufficiently small but non-zero positive value of $\Lambda$, a resonant peak should be present in the superradiant regime with a height that
depends on the value of $qQ$ (the highest peaks will be present around the zeroes of imaginary part of the superradiant QNM in accord with the Breit-Wigner formula discussed below). The important aspect of arbitrarily small cosmological constants is the fact that the resonant frequency (real part of superradiant QNM) is placed arbitrarily close to $\omega M=0$ thus resonances should be visible (depending on the choice of $qQ$) at the infrared radiation regime.
The increment of the charge coupling $qQ$ shifts the superradiant regimes in larger frequency onsets as seen in Figs. \ref{varqQ}, \ref{largeqQ}. Our analysis demonstrates that at the large charge coupling limit $qQ>>1$, the amplification factors $Z_0$ approach $100\%$ from above, with ultraviolet monochromatic wave resonant peaks, in contrast to $Z_0$ in RN BHs which tends to $100\%$ from below \cite{Brito:2015oca}. Preexisting QNM studies (see e.g. Refs. \cite{Cardoso:2018nvb,Destounis:2019hca}) have shown that the increment of $qQ$ increases the oscillation frequency of QNMs monotonously, while when $qQ>>1$ then the absolute value of the imaginary part also increases. We can therefore conclude that at the large coupling limit the behavior of superradiance may be falsely assumed to be similar to that of RN BHs if one is unaware of the results presented on the whole parametric space involved in our analysis. In turn, the increment of the BH's charge enlarges the superradiant frequency regime and the amplification factors (see Fig. \ref{varQ}).
A careful fitting on the amplification curves, similar to that initially performed in \cite{Thorne1969}, unveils the underlying connection between the peak position, and its sharpness, and the $\ell=0$ scalar QNM resonance. The peaks appearing in the superradiant regime correspond to frequencies which lie close to the real part of the corresponding fundamental QNM of the system \footnote{Similar behavior has been observed in \cite{Khodadi:2020cht} where resonant peaks appear in the superradiant domain of scattered waves off of Kerr-Newman BHs in $f(R)$ modified theories of gravity.}. The extent of amplification and sharpness of the peak is analogous to how close (in absolute value) the imaginary part of the corresponding QNM is to the real axis. Therefore, even when the spacetime is stable under $\ell=0$ charged scalar perturbations, which generically occurs for sufficiently large cosmological constants and charge couplings \cite{Destounis:2019hca}, the superradiant amplification curve becomes sharper close to long-lived resonances and can lead to factors that are orders of magnitude larger than $100\%$, in accord with \cite{Thorne1969,ChandraFerrari1991,ChandraFerrari2}. Numerical tests have revealed that when the monochromatic wave's frequency hits a `sweet spot' for which the underlying spacetimes admits QNMs with extremely large lifetimes (which are practically normal modes up to numerical accuracy for specific charge couplings $qQ$) \cite{Cardoso:2018nvb,Destounis:2019hca}, the resonant hyperradiation peaks become delta functions (see also \cite{Kokkotas1994}).
The above picture is akin to Breit-Wigner resonances for which the incident waves that trigger trapped modes localized in the potential well behave, close to QNM resonances, as $|\mathcal{I}|^2\sim C\left[ (\omega-\omega_R)^2+\omega_I^2\right]$ \cite{Berti:2009wx}, where $C$ is a constant. Thus, when the frequency of the incident monochromatic wave matches that of the corresponding QNM resonance, the amplification factor is analogous to the lifetime of the resonance $\omega_I^{-2}$ (see Eq. \ref{amplification_factor}).
In Figs. \ref{varL}, \ref{varqQ}, \ref{largeqQ} and \ref{varQ}, all cases depicted (besides the ones with vanishing $\Lambda M^2$) possess superradiant QNMs in their respective frequency domain. For completeness, we demonstrate the aforementioned resonant behavior in Table \ref{table}, where the peak position and the respective fundamental QNMs are shown for selected cases.
\subsection{Higher multipole scalar waves}
Higher angular index perturbations are not known to induce superradiant instabilities \cite{Zhu:2014sya}, neither support QNM resonances in the superradiant frequency regime. In this section we investigate if superradiance exists at all for $\ell>0$ in RNdS spacetime. As shown in Figs. \ref{varqQl1} and \ref{varl}, superradiant amplification of charged massless scalar waves can indeed occur for particular frequency ranges that strongly depend on the charge coupling. Our numerical analysis demonstrates that at the large coupling limit, where $qQ>>1$, the superradiant amplification tends to $100\%$ from below as in RN BHs.
We further demonstrate how suppressed superradiance is for $\ell>0$ incident waves with the quadrupole ones reaching a menial amplification of $Z_2\sim 10^{-3}$ with respect to $Z_0,\, Z_1$ for a particular set of parameters (see zoomed region in Fig. \ref{varl}).
\subsection{Massive $\ell=0$ scalar waves}
The role of the mass of scalar perturbations in RNdS is quite clear through the effective potential \eqref{RNdS_general potential}, as well as its effect on QNMs \cite{Cardoso:2018nvb,Destounis:2019hca}. The scalar mass antagonizes the charge coupling, which physically translates to a competition between the gravitational interaction of the BH mass and the massive field $\mu M$ and the electromagnetic repulsion of the BH's electric source and the charge of the field $qQ$ \cite{Konoplya:2014lha,Destounis:2019hca}. When the gravitational coupling $\mu M$ is very mild, the electromagnetic repulsion overcomes gravity and the superradiant instability remains, though is significantly suppressed. On the other hand, when the coupling $\mu M$ is strong enough, superradiantly unstable modes are completely absent from the QNM spectrum and, for $\mu M>>0$, no superradiantly stable QNMs exist at all \cite{Cardoso:2018nvb,Destounis:2019hca}. Therefore, one should expect that for sufficiently large $\mu M$, the amplification factors will never exceed $100\%$ since there will be no QNM resonances in the superradiant regime of monopole monochromatic incident waves.
In Fig. \ref{varmass} we have depicted two cases for which the respective fundamental QNMs satisfy the superradiant relation ($\mu M=0, \,0.03$) and two more cases for which the corresponding fundamental QNMs are not superradiant ($\mu M=0.09, \,0.012$). It is noteworthy that the superradiant relation \eqref{suprad} does not depend on the mass of the field and that is clearly demonstrated in Fig. \ref{varmass} where the onset and termination of superradiance occurs exactly at the same frequencies regardless of the mass. It is clear that when there are no resonances in the superradiant domain, the amplification does not exceed $100\%$, but still occurs. Further numerical analysis unveils that even larger scalar masses diminish $Z_0$ quickly.
\begin{figure}[t]
\includegraphics[scale=0.42]{Figure8}
\caption{Amplification factors of $\ell=0$ massive monochromatic waves with charge coupling $qQ=0.5$ for a RNdS BH with $\Lambda M^2=0.025$, $Q=0.5Q_\text{max}$ and varying mass $\mu M$. The horizontal black dashed line designates the onset of superradiant amplification while the black dot-dashed line designates amplification factors that equal $100\%$.}
\label{varmass}
\end{figure}
\section{Conclusions}
In this work, we have explored the superradiant amplification of charged scalar incident waves on RNdS BHs. In generality, RNdS BHs superradiate in a similar manner as RN and Kerr BHs. Even so, new phenomena arise in RNdS due to the negative wells formed at the effective potential of charged scalar perturbations. Supperadiance in RNdS is severely amplified with respect to RN BHs, despite the fact that the frequency domain of superradiant amplification is lessened due to the introduction of a positive cosmological constant.
Ultralight and massless spherically-symmetric scalar waves can extract large amounts of electromagnetic energy from RNdS BHs (assuming an unrealistic cosmological constant) when the incident wave's frequency matches the real part of the corresponding fundamental QNM oscillation that satisfies the superradiant relation. The amplification factors in these cases are orders of magnitude larger than $100\%$ and loosely depend on the lifetime of the respective QNM which is dictated by its imaginary part, in accord with Breit-Wigner resonances \cite{Thorne1969,Kokkotas1994,Berti:2009wx}. Interestingly, the larger the lifetime of the superradiant QNM the higher the resonant amplification peak is regardless of the linear stability of spacetime. On the other hand, when the scalar field is massive enough, QNMs are expelled from the superradiant regime \cite{Destounis:2019hca} and superradiance is rapidly suppressed.
Higher multipole incident waves do not exhibit resonant amplification since no QNMs reside in the superradiant regime. This leads to similar amplification factors as in RN, which asymptote to $100\%$ at the physical limit $qQ>>1$, in contrast to $\ell=0$ waves for which the amplification factors tend to $100\%$ from above at the same limit, even though ultraviolet resonances still occur but with small lifetimes. We can conjecture that the amplification of massive $\ell>0$ incident waves will also be highly quenched, therefore the most prominent amplifiers of scattered waves are linearly stable RNdS BHs with particular sets of parameters that lead to long-lived superradiant $\ell=0$ QNM resonances with arbitrarily large decay timescales. In any case, a physical choice of the cosmological constant should render the intriguing resonant hyperradiation phenomenon exhibited by RNdS BHs extremely ambitious, if not fictitious, in the majority of the parameter space involved expect when the charge coupling $qQ$ conspires to elicit extremely long-lived superradiant QNMs.
A very interesting extension of this work would be to perform an analysis of BH superradiance in charged accelerating spacetimes described by the $C$-metric \cite{Kinnersley:1970zw,Hawking:1997ia,Griffiths:2006tk}. Such spacetimes have a very similar causal structure with RNdS BHs, with the cosmological horizon being replaced by an acceleration horizon beyond which events are causally disconnected with the respective static region. Even though the $C$-metric is not spherically-symmetric, the neutral massless QNMs have been recently calculated \cite{Destounis:2020pjk,Destounis:2020yav}. Hence, the generalization to a charged QNM analysis, as well as the exploration of scattered charged scalar waves off charged accelerating BHs is of particular interest. We plan to pursue this work in the near future \cite{DestounisMascher}.
\begin{acknowledgments}
The authors would like to warmly thank Vitor Cardoso for helpful discussions.
\end{acknowledgments}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,354
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Q: setOnCancel and setOnDimiss Listeners doesn't execute Face of Codes:
I have a n Object of Dialog that defined below.
dselect=new Dialog(SuraSurface.this);
dselect.requestWindowFeature(Window.FEATURE_NO_TITLE);
dselect.setContentView(R.layout.dialogstatus);
dselect.getWindow().setBackgroundDrawable(new ColorDrawable(android.graphics.Color.TRANSPARENT));
and I implemented Listeners for it below.
dselect.setOnDismissListener(new DialogInterface.OnDismissListener() {
@Override
public void onDismiss(DialogInterface arg0) {
Log.d("DIMISSED", "");
}
});
Problem
setOnDimissListener
of dialog doesn't execute.
Question
How can I solve this problem?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 655
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The Andean Amazon is being rapidly altered by a wave of hydropower development and the consequences have been underestimated, according to newly released research.
A team led by professor Elizabeth Anderson from the FIU Institute of Water and Environment documented 142 existing dams and 160 proposed dams for rivers draining Andean headwaters of the Amazon. The researchers are concerned the proposed dams could result in significant losses in river connectivity, threatening fish populations and forever changing river channels and floodplains.
This hydropower project in Ecuador is one of many impacting the Andean Amazon.
The greater Amazon is one of the most prolific, biodiverse habitats in the world and Andean rivers have a disproportionate influence on the lowland areas. Andean rivers also predominantly define the formation of rivers downstream, controlling their curves and sediment. With as many as 5,000 species of fish inhabiting the waters, the rapid development of hydropower is a potential threat to these species that are a primary source of food and income for the 30 million people who live along the Amazon basin. If the proposed development continues, only one of the eight major Andean Amazon river systems would be left unimpeded.
There are also cultural implications for native people who live along the rivers, some of whom believe the waters are sacred.
The team collaborated with local governments and conservation organizations to compile international data on the dams, assembling the most comprehensive database of dams in the Andean-Amazon region. The result is a system-wide look at the overall impact, which presents a very different picture than when assessing impact of individual dams on individual systems.
"I hope that by showing the regional trends and that there is widespread river alteration happening, this research can lead to more coordinated development and help to highlight the importance of keeping some rivers free-flowing in the region," Anderson said.
The research team's efforts are already paying off. The international collaboration behind this research has led to the formation of a new initiative, Rios Vivos Andinos, which aims to facilitate more regional scientific analyses that examine the linkages between river flows, freshwater biodiversity, and human well-being. Anderson and her collaborators recently received funds from the MacArthur Foundation to support these efforts.
The research was published this week in Science Advances and is a collaboration between 15 institutions spanning eight countries. It was supported, in part, by grants from USAID, the MacArthur Foundation, and the Amazon Fish Project.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,465
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Q: Attempting to call a method from another method within a class, getting a TypeError So, I have two methods here for adding/updating a card on a Trello board. They are both declared in a class, with the updateWatchlistCard being declared before the second method addWatchlistCard. I am getting the following error in my code when I make a call to the updateWatchlistCard in addWatchlistCard: TypeError: undefined is not a function.
I'm not sure what exactly is wrong here or why it is doing this. I've tried rearranging the code, rewriting the function, and none of it has worked.
async updateWatchlistCard([cardData], card){
console.log('fired updatewatchlistcard');
//* Attempt to grab user data.
try{
var userId = await noblox.getIdFromUsername(cardData.Username);
var properUsername = await noblox.getUsernameFromId(userId);
}catch(error){
return false;
}
//* Adds the check to the card.
await Trello.addCommentToCard(card.id,
`**Moderator:** ${cardData.ModeratorName}:${cardData.ModeratorId}` +
`\n**Suspicion:** ${cardData.Suspiscion}` +
`\n**Evidence:** ${cardData.Evidence}` +
`\n**Comments:** ${cardData.Comments}`
);
return {value: '**Card updated successfully!**', cardName: card.name, cardUrl: card.url};
}```
``` async addWatchlistCard([cardData]){
// First search and see if the card already exists.
var cardExists = await this.getModerationCard(cardData.Username, cardData.WatchlistId, cardData.BoardId);
if(!cardExists){
//Tries getting new data for the card.
try{
var userId = await noblox.getIdFromUsername(cardData.Username);
var properUsername = await noblox.getUsernameFromId(Number(userId));
}catch(error){
return false;
}
// Data for card fields.
var cardTitle = (`${properUsername}:${userId}`);
var cardDesc = (`Moderator: ${cardData.ModeratorName}:${cardData.ModeratorId}` + `\nSuspected of: ${cardData.Suspicion}` + `\nEvidence: ${cardData.Evidence}` + `\nComments: ${cardData.Comments}`);
// Create the new card.
var newCard = await Trello.addCard(cardTitle, cardDesc, cardData.WatchlistId);
return {value: '**Card created successfully**', cardName: newCard.name, cardUrl: newCard.url};
}else{
console.log('got here.');
let response = this.updateWatchlistCard(cardData, cardExists);
return response;
}
} ```
A: Update: I figured out it had to do with the way I was writing the parameters in the updateWatchlistCard method. Thanks!
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,828
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Q: What does having the v in the center, with the positive near the top wire and negative near the bottom wire mean in this circuit diagram? This is the diagram that I am asking about:
I understand that the symbol on the left hand side of that middle branch is a current controlled current source, and the symbol on the right is a current source.
What does it mean to have the CCCS on the left, the CS on the right, with the v in the middle, and + and - on top and bottom, respectively?
Does the whole branch, having a v in the middle describe some kind of circuit that I don't know about? Can this circuit be redrawn? Perhaps as:
I'm sorry for the poor redrawing.
Thanks for any clarification.
A: Circular you redraw is correct meaning of v at the center and + above and - below means drop in voltage with mentioned polarities there
A: I think it's just to clarify that the top wire is considered to have a higher potential than the bottom wire. So for a simulation tool, you would connect the bottom wire to ground.
The v is just to indicate the voltage between top and bottom wire.
A: I hope you are familiar with the concept of independent and dependent sources. Still, let me explain it here. Voltage sources and current sources are basically classified as independent and dependent:
A dependent source is one whose value depends on some other parameter in the circuit. In your circuit, the current source on left side, with a diamond shaped symbol is a dependent current source. The current supplied by this source will be 2Ix (two times the value of current flowing in the branch on top right side)
On the other hand, an independent source always maintains a value which does not depend on any other parameter in the circuit. The 24mA source on right side will constantly supply 24mA, irrespective of the activities happening in other sections of the circuit.
Now, coming to your question, this is nothing but four elements in a simple parallel connection. There is nothing mysterious about the voltage v. As we know, the voltage across each element in a parallel connection will be same, this v can be considered as the voltage drop across R1, which is same as the voltage drop across R2. + and - symbols indicate the polarity of voltage drop. After solving, if v turns out to be positive, it means that top side of R1 and R2 is at a higher voltage than the bottom side.
And yes, you can redraw the circuit as you have done. And for practical purposes, you can consider this as a single sources of value (24mA+2Ix).
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,299
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From long time ago, dogs are considered as friendly pets. People love to spend time with them, they receive and give back love and affection. In order to maintain the health of your beloved pet you need to make an arrangement for a dog run, where your pet will be able to run and play freely. In this regard, we provide artificial synthetic grass turf, which look clean and attractive, non-toxic for dogs and safe. Dogs can run on this particular type of surface without any hassles. We know how much you love your dog, and by keeping this in mind, we offer secure outdoor environment services.
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,771
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\section{Introduction}
We consider compact Lagrangian correspondences ${L_{01} \subset M_0^- \times M_1}$ and $L_{12} \subset M_1^- \times M_2$, where $M_\ell=(M_\ell,\omega_{M_\ell})$
are symplectic manifolds that are either compact or satisfy appropriate boundedness conditions (see Remark~\ref{rmk:noncompact}), and where \(M_\ell^- := (M_\ell, -\omega_{M_\ell})\).
The {\bf geometric composition} of such Lagrangian correspondences is
$L_{01} \circ L_{12} := \pi_{02}( L_{01} \times_{M_1} L_{12}) $, the image under the projection $ \pi_{02}\colon M_0^- \times
M_1 \times M_1^- \times M_2 \to M_0^- \times M_2 $ of the fiber product
\begin{align*}
L_{01} \times_{M_1} L_{12} := (L_{01} \times L_{12})
\cap (M_0^- \times \Delta_1 \times M_2) .
\end{align*}
Here \(\Delta_1 \subset M_1 \times M_1^-\) denotes the diagonal.
If \(L_{01}\times L_{12}$ intersects $M_0^- \times \Delta_1 \times M_2\) transversely then \( \pi_{02}\colon L_{01} \times_{M_1} L_{12} \to M_0^-\times M_2\) is a Lagrangian immersion, in which case we call \(L_{01}\circ L_{12}\) an {\bf immersed composition}.
In the case of {\bf embedded composition}, where the projection is injective and hence a Lagrangian embedding,
some strict monotonicity and Maslov index assumptions allowed Wehrheim--Woodward \cite{isom} to establish an isomorphism of quilted Floer cohomologies
\begin{equation} \label{eq:HFiso}
HF(\ldots, L_{01},L_{12}, \ldots ) \cong HF(\ldots , L_{01} \circ L_{12}, \ldots) .
\end{equation}
A {\bf{strip-shrinking degeneration}}, in which a triple of pseudoholomorphic strips coupled by Lagrangian seam conditions degenerates to a pair of strips via the width of the middle strip shrinking to zero, formed the analytic core of the proof. All bubbling --- which was conjectured to include a novel {\bf{figure eight bubbling}} that (unlike disk or sphere bubbling) could be an algebraic obstruction to \eqref{eq:HFiso} --- was implicitly excluded by monotonicity and embeddedness assumptions.
The aim of this paper is to achieve a geometric understanding of figure eight bubbling toward encoding its effect algebraically.
In particular, in \S\ref{ss:algebra1} we propose a direct generalization of \eqref{eq:HFiso} to the nonmonotone case as an isomorphism of quilted Floer homologies with twisted differentials,\footnote{ %
Twisted differentials are obtained by adding to the Floer differential (that may not square to zero) further contributions arising from an $A_\infty$-structure applied to repetitions of a fixed cochain.
}
\begin{align*}
HF\bigl( \ldots , (L_{01},b_{01}), (L_{12},b_{12}), \ldots \bigr)
\; \simeq\;
HF\bigl( \ldots , (L_{01}\circ L_{12}, 8(b_{01},b_{12}) ), \ldots \bigr),
\end{align*}
in which the cochain $8(b_{01},b_{12})$ for the composed Lagrangian is obtained from moduli spaces of figure eight bubble trees with inputs $b_{01}$ and $b_{12}$.
Here even $8(0,0)$ is a generally nonzero count of figure eight bubbles.
Our proposal extends to geometric compositions $L_{01} \circ L_{12}$ that may only be
\textbf{cleanly immersed}, i.e.\ immersed in such a way that the local branches of \(L_{01} \circ L_{12}\) intersect cleanly, and leads us to formulate Conjecture~\ref{conj:bifunctor}:
Geometric composition defines an {\bf \(\mathbf{A_\infty}\)-bifunctor}
\(C^2\colon (\on{Fuk}(M_1^- \times M_2), \on{Fuk}(M_0^- \times M_1)) \to \on{Fuk}(M_0^- \times M_2)\)
between Fukaya categories of immersed Lagrangians with clean self-intersections,
by adding input marked points to the seams of a figure eight bubble and viewing the quilt singularity as an output marked point.
In the special case $M_0=\pt$, such a bifunctor in particular induces an \(A_\infty\)-functor
\(\on{Fuk}(M_1^- \times M_2) \to A_\infty{\rm Fun}\bigl(\on{Fuk}(M_1) , \on{Fuk}(M_2)\bigr)\) that would provide a more general version of the functor developed in \cite{mww} between extended Fukaya categories of monotone Lagrangians. More generally, our proposal naturally extends to a symplectic $A_\infty$ 2-category \cite{bt,bw:bigkahuna}.
\begin{figure}
\centering
\def6in{\columnwidth}
\input{tree.pdf_tex}
\caption{
A prototypical configuration of bubbles arising in the limit of strip shrinking.
The colors red/yellow/blue of patches indicate the target manifolds $M_0$/$M_1$/$M_2$ of the corresponding pseudoholomorphic maps.
More generally, we allow the boundary seams of the yellow strip to be squiggly as specified in \S\ref{sec:rescale}.}
\label{fig:tree}
\end{figure}
Toward these proposals we study families of pseudoholomorphic quilts in which the width of a
strip or annulus shrinks to zero, where it is replaced by the composition of the two correspondences $L_{01},L_{12}$ associated to the adjacent seams (see Figure~\ref{fig:tree}).
The main results of this paper, described in more detail in \S\ref{ss:overview},
are a Gromov compactness theorem for this strip-shrinking,
a precise description of the singular quilt bubbling phenomena --- figure eights and squashed eights --- and a lower bound on their bubbling energy.
The proofs rely on a collection of width-independent elliptic estimates established in \cite{b:singularity}. In turn, our Gromov compactness theorem is a key ingredient for \cite[Thm.\ 2.2]{b:singularity}, a removal of singularity for figure eight bubbles.
In the latter application to a singular quilt, the shrinking strips arising from reparametrization to cylindrical coordinates near the singularity have nonstandard complex structure:
The quilted surface is locally biholomorphic to a part of the complex plane with nonstraight seams \(\Im z = \pm \arcsin(\tfrac 1 2\Re z)\). Since essentially only \(\cC^0\)-convergence and \(\cC^k\)-bounds on the width functions \(f^\nu(s) := \arcsin(\tfrac 1 2s)|_{[-\nu,-\nu+1]}\) for \(\nu \in \mathbb{N}\) affect the analysis, we formulate our results for general ``squiggly strip shrinking''.
\medskip
\noindent
{\bf Philosophical Remark:}
{\it
In the early days of pseudoholomorphic quilts, the relevance of figure eight bubbles was doubted. We hope that this paper puts those doubts to rest. As we summarize in \S\ref{ss:algconsequences} and explain in \S\ref{s:propaganda}, figure eight bubbling cannot be \emph{a priori} excluded for dimension reasons, and will contribute to the algebra.
So even if e.g.\ the isomorphism \eqref{eq:HFiso} of Floer homology under geometric composition was to be proven with methods other than strip shrinking, one must in general expect figure eight type obstructions.
However, figure eight bubbles should be viewed as a tool rather than an inconvenience: For instance, they provide a cochain for \(L_{01} \circ L_{12}\) so that \eqref{eq:HFiso} continues to hold in more general situations than \cite{isom} considered. More generally, fully embracing figure eight bubbles will yield a natural 2-categorical structure on the collection of all compact symplectic manifolds, which will unify and extend a wide variety of currently-known algebraic structures.
While this paper only provides substantial evidence for these algebraic results (or ``proofs up to technical details" depending on ones standards of rigour), it does demonstrate in full detail that figure eight bubbling is in fact analytically manageable: Section~\ref{sec:bubbles} gives rigorous definitions of ``squiggly strip shrinking'' and the novel bubble types --- figure eight bubbles and squashed eight bubbles --- and establishes lower bounds as well as topological controls on their energy. A full removal of singularities for the new bubbles is established in \cite{b:singularity}. Section~\ref{sec:rescale} shows how the full diversity of bubble types appears in the Gromov compactification for ``squiggly strip shrinking''.
Moreover, Appendix~\ref{app:ex} in collaboration with Felix Schm\"{a}schke provides the first nontrivial example of a general figure eight bubble.
}
\subsection{Analytic Results} \label{ss:overview}
\smallskip
\noindent
The compactness analysis in \S\ref{sec:rescale} will for ease of notation be performed in the special case of quilted squares with seam conditions in \(L_{01}, L_{12}\) and the width of the strip mapping to $M_1$ converging to zero. However, it generalizes directly to the following result for strip or annulus shrinking in pseudoholomorphic quilts.
(For an introduction to quilts see \cite{quilts}.)
\medskip
\noindent{\bf Gromov Compactness Theorem~\ref{thm:rescale}:}
Let $\underline{Q}^\nu$ be a sequence of quilted surfaces containing a patch \(Q_1^\nu\) diffeomorphic to an annulus or strip,
equipped with complex structures in which the width of \(Q_1^\nu\) tends to zero as \(\nu \to \infty\).
(For allowable squiggliness --- i.e.\ variation of width --- see Definition~\ref{def:obedience}.)
Label the patches of $\underline{Q}^\nu$ with a tuple $\underline{M}$ of closed symplectic manifolds
(or noncompact ones without boundary which come with a priori \(\cC^0\)-bounds as discussed in Remark~\ref{rmk:noncompact}), let $M_1$ and $M_0,M_2$ be the labels of $Q_1^\nu$ and the adjacent patches, and fix compatible almost complex structures over each patch.
Fix compact Lagrangian seam conditions for each seam of $\underline{Q}^\nu$ so that the Lagrangian correspondences $L_{01}, L_{12}$ associated to the seams bordering $Q_1^\nu$ have immersed composition \(L_{01} \circ L_{12}\).
Now suppose that $(\underline{v}^\nu)_{\nu\in\mathbb{N}}\colon \underline{Q}^\nu \to \underline{M}$ is a sequence of pseudoholomorphic quilts of bounded energy with the given seam conditions.
Then there is a subsequence (still denoted $(\underline{v}^\nu)_{\nu\in\mathbb{N}}$) that converges up to bubbling to a punctured pseudoholomorphic quilt $\underline{v}^\infty\colon \underline{Q}^\infty {\smallsetminus} Z \to (\underline{M}{\smallsetminus} M_1)$. Here $\underline{Q}^\infty$ is the quilted surface obtained as limit of the $\underline{Q}^\nu$ by replacing $Q_1^\nu$ with a seam labeled by \(L_{01} \circ L_{12}\), $Z$ is a finite set of bubbling points, $\underline{v}^\infty$ satisfies seam conditions in the fixed Lagrangian correspondences and for the new seam in \(L_{01} \circ L_{12}\) (in the generalized sense of \eqref{gen bc}), and convergence holds in the following sense:
\begin{itemlist}
\item
The energy densities $|\delta\underline{v}^{\nu}|^2$ are uniformly bounded on every compact subset of $\underline{Q}^\infty {\smallsetminus} Z$, and at each point in $Z$ there is energy concentration of at least $\hbar>0$, given by the minimal bubbling energy from Definition~\ref{def:bubble energy}.
\item
The quilt maps $\underline{v}^{\nu}|_{\underline{Q}^\nu {\smallsetminus} (Q^\nu_1\cup Z)}$ on the complement of $Z$ in the patches other than $Q^\nu_1$ converge with all derivatives on every compact set to $\underline{v}^\infty$.
\item
At least one type of bubble forms at each point $z\in Z$ in the following sense: There is a sequence of (tuples of) maps obtained by rescaling the maps defined on the various patches near $z$, which converges $\cC^\infty_\loc$ to one of the following:
\begin{itemize}
\item[--]
a nonconstant, finite-energy pseudoholomorphic map $\mathbb{R}^2 \to M_\ell$ to one of the symplectic manifolds in $\underline{M}$ (this can be completed to a nonconstant pseudoholomorphic sphere in $M_\ell$);
\item[--]
a nonconstant, finite-energy pseudoholomorphic map $\H \to M_k^-\times M_\ell$ to a product of symplectic manifolds associated to the patches on either side of a seam in $\underline{Q}^\nu$, that satisfies the corresponding Lagrangian seam condition (this can be extended to a nonconstant pseudoholomorphic disk in $M_k^-\times M_\ell$, in particular including the cases of disks with boundary on $L_{01}\subset M_0^-\times M_1$ or $L_{12}\subset M_1^-\times M_2$);
\item[--]
a nonconstant, finite-energy figure eight bubble in the sense of \eqref{eq:8} below;
\item[--]
a nonconstant, finite-energy squashed eight bubble in the sense of \eqref{gen bc} below,
with generalized seam conditions in $L_{01}\circ L_{12}$.
\end{itemize}
\end{itemlist}
\medskip
\noindent {\bf Gromov compactification of strip-shrinking moduli spaces:}
Note that the above partial Gromov compactness statement only requires the composition $L_{01}\circ L_{12}$ to be immersed.
If the self-intersections of this immersion are locally clean, then the results of \cite{b:singularity} allow us to remove the singularities in the limits of the main component as well as the figure eight and squashed eight bubbles.
Moreover, the techniques of \cite{b:singularity} also provide ``bubbles connect'' results for long cylinders, so that a ``soft rescaling iteration'' (guided by capturing all energy) will yield the following full Gromov compactification of a moduli space with squiggly strip- (or annulus-) shrinking in terms of bubble trees:
On the complement of the new seam, these trees are made up of trees of disk bubbles\footnote{\label{foot:fold}
Here we identify spheres with a single circle as seam and two patches labeled by \(M_k\) and \(M_\ell\) with disk bubbles in \(M_k^- \times M_\ell\) by ``folding'' across the seam as in the portion of the proof of Theorem~\ref{thm:rescale} treating the (D01) case.}
attached to the seams, with additional trees of sphere bubbles attached to the disks, seams, or interior of the patches. On the new seam, as indicated in Figure~\ref{fig:tree}, starting from the root, every bubble tree starts with a (possibly empty or containing constant vertices) tree of squashed eight bubbles.
Attached to this are figure eight bubbles (possibly constant) in such a way that between any leaf and the root of the complete tree there is at most one figure eight. Trees of disk bubbles with boundary on $L_{01}$ resp.\ $L_{12}$ can then be attached to the corresponding seams of the figure eight bubbles. Finally, trees of sphere bubbles can be attached to the interior of each patch or the seams in this bubble tree.
The hierarchy in this compactification is illustrated in Figure~\ref{fig:prediction} (including the additional complication of Morse flow lines).
We will not provide a detailed construction of this compactification in the present paper, whose point is to establish the foundational analysis. The full compactification (including Morse flow lines) will be part of the polyfold setup in\cite{bw:bigkahuna}.
\medskip
\noindent
{\bf Singular quilt bubbling phenomena:}
Beyond the standard bubbling phenomena (holomorphic spheres and disks) our Gromov compactification for quilts with strip-shrinking involves two new types of bubbles:
A {\bf figure eight bubble} is a tuple of finite-energy pseudoholomorphic maps
\begin{equation} \label{eq:8}
w_0\colon \mathbb{R}\times(-\infty,-\tfrac 1 2]\to M_0, \qquad
w_1\colon \mathbb{R}\times[-\tfrac 1 2,\tfrac 1 2]\to M_1, \qquad
w_2\colon \mathbb{R}\times[\tfrac 1 2, \infty)\to M_2
\end{equation}
satisfying the seam conditions
\begin{equation*}
(w_0(s,-\tfrac 1 2),w_1(s,-\tfrac 1 2))\in L_{01} , \quad
(w_1(s,\tfrac 1 2),w_2(s,\tfrac 1 2))\in L_{12}
\qquad\forall \: s\in \mathbb{R} ,
\end{equation*}
while a {\bf squashed eight bubble} is a triple of finite-energy pseudoholomorphic maps \begin{align*}
w_0\colon \mathbb{R}\times(-\infty,0]\to M_0, \qquad w_1\colon \mathbb{R} \to M_1, \qquad w_2\colon \mathbb{R}\times[0, \infty)\to M_2
\end{align*}
satisfying the seam condition
\begin{equation}\label{gen bc}
(w_0(s,0), w_1(s), w_1(s), w_2(s,0)) \in L_{01} \times_{M_1} L_{12} \quad \forall \: s \in \mathbb{R}.
\end{equation}
\begin{remark}
Both of these bubbles are of singular quilt type in the sense that we cannot generally expect a smooth extension to a quilted sphere. For the squashed eight bubble this results from the boundary condition $L_{01} \circ L_{12}$ generally just being a Lagrangian immersion. For the figure eight bubble this is due to the way in which the two seams intersect at infinity: After stereographic compactification to a quilted sphere, they touch tangentially rather than intersect transversely, which would allow a description of the singularity in terms of striplike ends.
Nevertheless, if the composition $L_{01}\circ L_{12}$ is cleanly immersed, then the removable singularity result in \cite{b:singularity} shows that $w_0(s,t)\to p_0\in M_0$ and $w_2(s,t)\to p_2\in M_2$ have uniform limits as $s^2+t^2\to\infty$, whereas $w_1(s,t)\to p_1^{\pm}\in M_1$ has two possibly different limits as $s\to \pm \infty$, both of which are lifts of $(p_0,p_2)\in L_{01}\circ L_{12}$, that is $(p_0,p_1^{\pm},p_1^{\pm},p_2)\in L_{01}\times_{M_1} L_{12}$.
\end{remark}
Recall that even smooth removal of singularity generally does not provide lower bounds on the energy of bubbles except in situations with simple topology; see Remark~\ref{rmk:htop}.
In \S\ref{sec:bubbles} we establish this lower energy bound for figure eight and squashed eight bubbles by purely analytic means.
\medskip
\noindent{\bf Lower Energy Bound Lemma~\ref{lem:hbar}:}
For fixed almost complex structures and Lagrangians with immersed composition \(L_{01} \circ L_{12}\), the energy of nontrivial figure eight and squashed eight bubbles is bounded below by a positive quantity.
\medskip
Finally, Appendix~\ref{app:ex} in collaboration with Felix Schm\"{a}schke explains how pseudoholomorphic disks and strips can be viewed as special cases of figure eight bubbles, and we provide an example of a nontrivial figure eight bubble with embedded composition $L_{01}\circ L_{12}$ and target spaces $M_0=M_2=\mathbb{CP}^3$, $M_1=\mathbb{CP}^1 \times \mathbb{CP}^1$.
\subsection{Algebraic consequences of figure eight bubbling} \label{ss:algconsequences}
We establish our analytic results in settings that will allow us to describe compactified moduli spaces of pseudoholomorphic quilts with shrinking strips as zero sets of Fredholm sections in polyfold bundles.
This will put the universal regularization theory of \cite{hwz:fred2} at our disposal. In particular, there will be no need to prove a separate gluing theorem for exhibiting configurations with figure eight bubbles as boundaries of the compactified moduli space:
Pre-gluing constructions outlined in \S\ref{ss:boundary} will provide polyfold charts with boundary for bubble trees of (not necessarily pseudoholomorphic) quilted maps, and the proof of the nonlinear Fredholm property of the quilted Cauchy--Riemann operator in this polyfold setup will be essentially a version of the quadratic estimates in the classical gluing analysis (which in our case should follow from combining the results of \cite{isom} and \cite{b:singularity}).
Moreover, the boundary stratification of the ambient polyfold will directly induce the boundary stratification of the (regularized) moduli spaces. This offers a {\bf semi-rigorous method for predicting algebraic consequences:} If the considered moduli spaces can be cut out from ambient polyfolds, then the algebraic identities are given by summing over the top boundary strata of the polyfold. Furthermore, if the local charts for the polyfold arise from pre-gluing constructions (as has been the case in all known examples), then the boundary stratification can be read off from the gluing parameters.
We thus analyze in \S\ref{ss:HF} the boundary strata predicted by our Gromov-compactification in the case of strip-shrinking used to prove \eqref{eq:HFiso}.
Based on that, \S\ref{ss:Morse} gives a fair amount of detail on the extension of these moduli spaces by Morse trajectories. This is desirable for easy polyfold implementation as well as reducing algebraic headaches by working with finitely generated chain complexes.
Finally, we use this analysis of boundary strata to predict in \S\ref{ss:algebra1} a generalization of the isomorphism \eqref{eq:HFiso} of Floer homology under geometric composition.
Besides a Fredholm description of figure eight moduli spaces and generalization of \eqref{eq:HFiso}, another motivation for developing the analysis described in \S\ref{ss:overview} is to obtain a new approach to the construction of the $A_\infty$-functors associated to monotone Lagrangian correspondences in \cite{mww}. Whereas the latter requires a technically cumbersome construction of quilted surfaces with striplike ends and regular Hamiltonian perturbations, and is heavily restricted by monotonicity requirements, a polyfold setup for figure eight moduli spaces will provide a direct construction of curved\footnote{
Disk bubbling gives rise to $m_0$ terms in all $A_\infty$-relations, so that in particular squares of differentials can be nonzero. This can be thought of as allowing curvature, hence we follow e.g.\ \cite{a:intro} and denote this generalized type of $A_\infty$-relations by the prefix ``curved''.
}
$A_\infty$-functors for general Lagrangian correspondences. We explain in \S\ref{ss:algebra2} that we will be able to work directly with the singular quilted surfaces that realize the multiplihedra in \cite{mw}, since these are special cases of figure eights with $M_0=\pt$ and marked points on boundary and seams.
Analyzing moreover the boundary stratification of general figure eight moduli spaces, we arrive at the conjecture that geometric composition is encoded in terms of a curved \(A_\infty\)-bifunctor, which in turn specializes to the desired generalization of the $A_\infty$-functors in \cite{mww}.
In fact, these methods can be extended to generalizations of figure eight moduli spaces, leading to a conjectural symplectic $(\infty,2)$-category that we will further investigate in future work. While the complete polyfold construction of these new algebraic structures in \cite{b:thesis,bw:bigkahuna} will be lengthy since we aim to provide a technically sound and easily portable basis for all future use of quilt moduli spaces, its rough form and algebraic consequences are already so apparent from our current understanding that it seems timely to give this semi-rigorous exposition.
We do so in order to motivate the development of this theory and enable investigations of its future applications.
\subsection{Acknowledgements}
The first ideas of studying figure eight bubbling, and preliminary results toward Theorem~\ref{thm:rescale}, were obtained by the second author during her collaboration with Chris Woodward.
We would like to thank Mohammed Abouzaid, Sheel Ganatra, Tim Perutz, Anatoly Preygel, and Zachary Sylvan for helpful conversations about \S\ref{ss:algebra2}.
We gratefully acknowledge support by an NSF Graduate Research Fellowship, a Davidson Fellowship, and an NSF Career Grant, and would like to thank the Institute for Advanced Study, Princeton University, and the University of California, Berkeley for their hospitality.
\section{Squiggly strip quilts and figure eight bubbles}
\label{sec:bubbles}
The purpose of this section is to establish a general setup for squiggly strip shrinking in quilted surfaces and introduce the new bubbling phenomena with their basic properties.
Besides sphere and disk bubbling, two novel sorts of bubbles may appear: figure eight bubbles and squashed eight bubbles, both of which are introduced in Definition~\ref{def:8}. In Lemma~\ref{lem:hbar}, we show that the energy of the figure eight and squashed eight bubbles is bounded below,
which will be a key ingredient in our proof of the Gromov Compactness Theorem~\ref{thm:rescale}.
The proof of Lemma~\ref{lem:hbar} relies on a \(\cC^\infty\)-compactness statement for squiggly strip shrinking from \cite{b:singularity}, which we restate in Theorem~\ref{thm:nonfoldedstripshrink}.
\medskip
In this section and the next we will be working with symplectic manifolds and with pseudoholomorphic curves with seam conditions defined by compact Lagrangian correspondences
\begin{equation}\label{eq:lag}
L_{01} \subset M_0^- \times M_1, \qquad
L_{12} \subset M_1^- \times M_2.
\end{equation}
When the following intersection in \(M_0^- \times M_1 \times M_1^- \times M_2\) is transverse, we follow \cite{quiltfloer} and say that \(L_{01}\) and \(L_{12}\) have {\bf immersed composition}:
\begin{equation} \label{eq:comptrans}
(L_{01} \times L_{12}) \pitchfork (M_0^- \times \Delta_1 \times M_2) \; =: \,
L_{01}\times_{M_1} L_{12}.
\end{equation}
Indeed, the transversality implies that \(L_{01} \times_{M_1} L_{12} \subset M_0^- \times M_1 \times M_1^- \times M_2\) is a compact submanifold, and the projection \(\pi_{02}\colon L_{01} \times_{M_1} L_{12} \to M_0^- \times M_2\) is a Lagrangian immersion by e.g.\ \cite[Lemma~2.0.5]{quiltfloer} (which builds on \cite[\S4.1]{gu:rev}).
We will denote its image, the {\bf geometric composition} of $L_{01}$ and $L_{12}$,
by \begin{align*}
L_{01} \circ L_{12} \,:= \; \pi_{02}( L_{01} \times_{M_1} L_{12} ) \;\subset\; M_0^- \times M_2 .
\end{align*}
In some contexts we will assume \textbf{cleanly-immersed composition}, that is an immersed composition such that any two local branches of the immersed Lagrangian \(L_{01} \circ L_{12}\) intersect cleanly.
\smallskip
\begin{center}
\fbox{\parbox{0.9\columnwidth}{
Throughout \S\ref{sec:bubbles} and \S\ref{sec:rescale} we will work with fixed
symplectic manifolds \(M_0, M_1, M_2\) without boundary which are either compact or satisfy boundedness assumptions as detailed in Remark~\ref{rmk:noncompact}, and compact Lagrangians \(L_{01}, L_{12}\) as in \eqref{eq:lag} with immersed composition.}}
\end{center}
\medskip
We will consider pseudoholomorphic quilts with respect to compatible almost complex structure:
\begin{equation}\label{eq:J}
J_\ell\colon [-\rho,\rho]^2 \to\mathcal{J}(M_\ell,\omega_\ell) \qquad\text{for}\; \ell = 0,1,2 .
\end{equation}
These are allowed to be domain-dependent\footnote{
Note in particular that we do not require \(J_\ell\) to be constant near
the seam as in \cite{isom}. This is necessary because the corrected proof of transversality in \cite{striptrans} does not guarantee regular almost complex structures in this class.}
but are \(\cC^k\) as maps \([-\rho,\rho]^2\times {\rm T} M_\ell\to {\rm T} M_\ell\), where \(k\) will be either a positive integer or infinity.
Then compatibility means that
\begin{align} \label{eq:metrics}
g_\ell(s,t):= \omega_\ell (- , J_\ell(s,t) - )
\end{align}
are metrics on \(M_\ell\) that are \(\cC^k\) in \((s,t)\in[-\rho,\rho]^2\).
The underlying quilted domains of our pseudoholomorphic quilt maps will be open squares with two seams. One should imagine these as part of the domain of a larger pseudoholomorphic quilt with compact domain or with quilted cylindrical ends.
The basic (localized and rescaled) examples studied in \cite{isom} are squares $(-1,1)^2$ with seams $(-1,1)\times\{\pm\delta\}$, whose main feature is a middle strip $(-1,1)\times[-\delta,\delta]$ of constant width $2\delta>0$. The following definition generalizes the underlying quilted surfaces to allow middle domains $\{ |t|\leq f(s) \}$ of local widths $2f(s)>0$ varying with $s\in (-1,1)$. Since diffeomorphically such domains are still strips, we call them ``squiggly strips''.
\begin{definition} \label{def:squigglystrip}
Fix \(\rho>0\), a real-analytic function \(f\colon [-\rho,\rho] \to (0, \rho/2]\), almost complex structures \(J_\ell\), \(\ell = 0,1,2\) as in \eqref{eq:J}, and a complex structure \(j\) on \([-\rho,\rho]^2\). A {\bf \(\mathbf{(J_0,J_1,J_2,j)}\)-holomorphic size-\(\mathbf{(f,\rho)}\) squiggly strip quilt for \(\mathbf{(L_{01},L_{12})}\)} is a triple of smooth maps
\begin{align} \label{eq:squigglymaps}
\underline{v}=\left(
\begin{aligned}
v_0\colon \{(s,t) \in (-\rho,\rho)^2 \: | & \; t \leq -f(s) \} \to M_0 \\
v_1\colon \{(s,t) \in (-\rho,\rho)^2\: | & \; |t | \leq f(s) \} \to M_1 \\
v_2\colon \{(s,t) \in (-\rho,\rho)^2\: | & \; t \geq f(s) \} \to M_2
\end{aligned}
\right)
\end{align}
that satisfy the Cauchy--Riemann equations
\begin{align} \label{eq:squigglyCR}
\delta v_\ell(s,t) \circ j(s,t) - J_\ell(s,t,v_\ell(s,t)) \circ \delta v_\ell(s,t) = 0 \qquad \forall\: \ell = 0,1,2
\end{align}
for \((s,t)\) in the relevant domains,
fulfill the seam conditions
\begin{align} \label{eq:squigglyseams}
\bigl(v_0(s,-f(s)),v_1(s,-f(s))\bigr)\in L_{01}, \qquad \bigl(v_1(s,f(s)), v_2(s,f(s))\bigr)\in L_{12} \qquad \forall \: s \in (-\rho,\rho),
\end{align}
and have finite energy\footnote{
Throughout we will make use of the standard energy identity \cite[Lemma 2.2.1]{ms:jh} for pseudoholomorphic maps.
}
\begin{align*}
E(\underline{v})
\,:= \; {\textstyle\int} v_0^*\omega_0 + {\textstyle\int} v_1^*\omega_1 + {\textstyle\int} v_2^*\omega_2
\;= \tfrac 12 \Bigl( {\textstyle\int} |\delta v_0|^2 +{\textstyle\int} |\delta v_1|^2 +{\textstyle\int} |\delta v_2|^2 \Bigr)
\;<\; \infty .
\end{align*}
\end{definition}
\noindent When \(j\) is the standard complex structure \(i\), \eqref{eq:squigglyCR} reduces to the equation
\begin{align*}
\partial_s v_\ell(s,t) + J_\ell(s,t, v_\ell(s,t))\partial_t v_\ell(s,t)=0.
\end{align*}
\noindent
When considering a \((J_0,J_1,J_2)\)-holomorphic squiggly strip quilt \(\underline v^\nu\),
it will be useful to consider the {\bf energy density functions}
\begin{align} \label{eq:density}
\bigl| \delta\underline{v} \bigr|:(-1,1)^2\to[0,\infty), \qquad
\bigl|\delta\underline{v}(s,t)\bigr| := \left( \bigl|\delta v_0(s,t )\bigr|_{J_0}^2
+ \bigl|\delta v_1(s,t)\bigr|_{J_1}^2 + \bigl|\delta v_2(s, t)\bigr|_{J_2}^2 \right)^{\frac 12},
\end{align}
where the norms \(|\delta v_\ell(s,t)|\) are induced by the metrics defined by \(\omega_\ell\) and \(J_\ell\) as in \eqref{eq:metrics} and set to be zero on the complement of the domain of \(v_\ell\). If \(\underline v\) is a squiggly strip quilt, then \(|\delta \underline v|\) is upper semi-continuous on \(\{ (s, \pm f(s)) \}\), continuous elsewhere, and satisfies \(E(\underline{v})= \tfrac 12\int |\delta\underline{v} |^2\).
The goal of this paper is to generalize and strengthen the strip shrinking analysis in \cite{isom}, which considers sequences of \((J_0^\nu,J_1^\nu,J_2^\nu)\)-holomorphic squiggly strip quilts of width \(f^\nu\equiv\delta^\nu\to 0\).
For that purpose we consider varying width functions $f^\nu$ that uniformly converge to zero as follows.
\begin{definition} \label{def:obedience}
Fix \(\rho>0\). A sequence \(\bigl( f^\nu \bigr)_{\nu\in\mathbb{N}}\) of real-analytic functions \(f^\nu\colon [-\rho,\rho]\to (0,\rho/2]\)
\textbf{obediently shrinks to zero}, \(\mathbf{f^\nu \Rightarrow 0}\),
if
\(\max_{s\in[-\rho,\rho]} f^\nu(s) \underset{\nu\to\infty}{\longrightarrow} 0\)
and
\begin{align*}
\sup_{\nu\in\mathbb{N}}\; \frac{\max_{s\in[-\rho,\rho]} \bigl| \tfrac{{\rm d}^k}{{\rm d} s^k}f^\nu(s) \bigr|}{\min_{s\in[-\rho,\rho]} f^\nu(s)} =: C_k <\infty \qquad\forall \: k\in \mathbb{N}_0,
\end{align*}
and in addition there are holomorphic extensions \(F^\nu\colon [-\rho,\rho]^2 \to \mathbb{C}\) of \(f^\nu(s) = F^\nu(s,0)\) such that \((F^\nu)\) converges \(\cC^\infty\) to zero\footnote{To see why the last condition is necessary, consider the sequence of functions \((F^\nu\colon [-1,1]^2 \to \mathbb{C})\) defined by \(F^\nu(z) := \exp(-\nu)\sin(2\nu z) + 1/4\). For \(x \in \mathbb{R}\), we have the formulas \(F^\nu(x) = \exp(-\nu)\sin(2\nu x) + 1/4\) and \(F^\nu(ix) = i \exp(-\nu)( \exp(2\nu x) - \exp(-2 \nu x) ) / 2 + 1/4\), so the restrictions to \([-1,1] \times \{0\}\) converge in \(\cC^\infty\) to zero but \(F^\nu(3i / 4)\) diverges to \(i\infty\).}.
\end{definition}
We will see in Theorem~\ref{thm:rescale} that any sequence \((\underline v^\nu)\) of pseudoholomorphic squiggly strip quilts of bounded energy and obediently shrinking widths \(f^\nu\Rightarrow 0\) has a subsequence that --- up to finitely many points where energy concentrates --- converges to a degenerate strip quilt, in which the middle domain mapping to \(M_1\) is replaced by a single straight seam mapping to the immersed Lagrangian \(L_{01} \circ L_{12}\).
Here bubbling near the middle squiggly strip may lead to limit maps whose seam values switch between the sheets of \(L_{01}\circ L_{12}\). Thus we need to allow for singularities in the degenerate strip quilts as follows.
\begin{definition}
Fix \(\rho>0\), almost complex structures \(J_\ell\), \(\ell \in \{0,2\}\) as in \eqref{eq:J}, and a complex structure \(j\) on \([-\rho,\rho]^2\). A {\bf \(\mathbf{(J_0,J_2,j)}\)-holomorphic size-\(\mathbf{\rho}\) degenerate strip quilt for \(\mathbf{(L_{01} \times_{M_1} L_{12})}\) with singularities} is a triple of smooth maps
\begin{align} \label{eq:degenmaps}
\underline{v}=\left(
\begin{aligned}
u_0\colon & \, (-\rho,\rho)\times(-\rho,0] \;{\smallsetminus}\; S\times\{0\} \to M_0 \\
u_1\colon & \, (-\rho,\rho) \;{\smallsetminus}\; S \to M_1 \\
u_2\colon & \, (-\rho,\rho)\times[0,\rho) \;{\smallsetminus}\; S\times\{0\} \to M_2
\end{aligned}
\right)
\end{align}
defined on the complement of a finite set \(S\subset\mathbb{R}\) that satisfy the Cauchy--Riemann equation \eqref{eq:squigglyCR} for \(\ell \in \{0,2\}\) and
\((s,t)\) in the relevant domains,
fulfill the lifted seam condition
\begin{align} \label{eq:degenseams}
\bigl( u_0(s,0), u_1(s), v_1(s), u_2(s,0) \bigr) \in
L_{01}\times_{M_1} L_{12}
\qquad\forall \: s\in(-\rho,\rho){\smallsetminus} S,
\end{align}
and have finite energy
\begin{align*}
E(\underline{u}) \, := \; {\textstyle\int} u_0^*\omega_0 + {\textstyle\int} u_2^*\omega_2
\;= \tfrac 12 \Bigl( {\textstyle\int} |\delta u_0|^2 + {\textstyle\int} |\delta u_2|^2 \Bigr)
\;<\; \infty.
\end{align*}
\end{definition}
\begin{remark} \label{rem:singset}
If \(u_1\) in the above definition
continuously extends to a point in \(S\), then --- by the standard removal of singularity result with embedded Lagrangian boundary conditions --- all \(u_i\) extend smoothly to this point.
Hence one can prescribe \(S\) to be the set of discontinuities of \(u_1\).
In fact, the removal of singularity for squashed eights established in \cite[Appendix A]{b:singularity} shows that \(u_0\) and \(u_2\) extend continuously to any point in \(S\) under the hypothesis that \(L_{01}\) and \(L_{12}\) have cleanly-immersed composition. In this case, the only map with any discontinuities is \(u_1\).
\end{remark}
At the points of energy concentration, we will see that
four types of bubbles may occur: the familiar sphere and disk bubbles, and the novel figure eight and squashed eight bubbles. These novel types of
bubbles result from energy concentrating on the limit seam \((-\rho,\rho)\times\{0\}\) in such a way that after rescaling (to achieve uniform gradient bounds), the middle squiggly strip converges to a straight strip of constant width, or zero width in the case of a squashed eight bubble.
Note here that limit maps of this rescaling will be pseudoholomorphic with respect to the almost complex structures at the point of energy concentration.
\begin{definition} \label{def:8}
Fix domain-independent almost complex structures \(J_\ell \in \mathcal{J}(M_\ell, \omega_\ell)\) for \(\ell = 0,1,2\).
\vspace{-4mm}
\begin{itemlist}
\item
A {\bf figure eight bubble between \(\mathbf{L_{01}}\) and \(\mathbf{L_{12}}\)} is a triple of smooth maps \begin{align*}
\underline w = \left( \begin{aligned}
w_0&\colon\mathbb{R} \times (-\infty, -\tfrac 1 2] \to M_0 \\
w_1&\colon\mathbb{R} \times [-\tfrac 1 2, \tfrac 1 2] \to M_1 \\
w_2&\colon\mathbb{R} \times [\tfrac 1 2, \infty) \to M_2
\end{aligned} \right)
\end{align*}
that satisfy the Cauchy--Riemann equations \(\partial_s w_\ell + J_\ell(w_\ell)\partial_t w_\ell = 0\) for \(\ell= 0,1,2 \), fulfill the seam conditions
\begin{gather*}
(w_0(s, -\tfrac 1 2), w_1(s, -\tfrac 1 2)) \in L_{01}, \quad
(w_1(s, \tfrac 1 2), w_2(s, \tfrac 1 2)) \in L_{12} \qquad \forall \: s\in\mathbb{R},
\end{gather*}
and have finite energy
\begin{align*}
{\textstyle\int} w_0^* \omega_0 + {\textstyle\int} w_1^* \omega_1 + {\textstyle\int} w_2^* \omega_2
\;= \tfrac 12 \Bigl( {\textstyle\int} |\delta w_0|^2 + {\textstyle\int} |\delta w_1|^2 + {\textstyle\int} |\delta w_2|^2 \Bigr)
\;<\; \infty .
\end{align*}
\item
A {\bf squashed eight bubble with seam in \(\mathbf{L_{01} \times_{M_1} L_{12}}\)} is a triple of smooth maps
\begin{align*}
\underline w = \left( \begin{aligned}
w_0&\colon\mathbb{R} \times (-\infty, 0] \to M_0 \\
w_1&\colon\mathbb{R} \to M_1 \\
w_2&\colon\mathbb{R} \times [0, \infty) \to M_2
\end{aligned} \right)
\end{align*}
that satisfy the Cauchy--Riemann equations
\(\partial_s w_\ell + J_\ell(w_\ell)\partial_t w_\ell = 0\) for
\(\ell\in\{0,2\}\),
fulfill the generalized seam condition
\begin{equation*}
(w_0(s,0), w_1(s), w_1(s), w_2(s,0))\in L_{01} \times_{M_1} L_{12}
\qquad\forall \: s\in \mathbb{R} ,
\end{equation*}
and have finite energy
\begin{align*}
{\textstyle\int} w_0^* \omega_0 + {\textstyle\int} w_2^* \omega_2
\;= \tfrac 12 \Bigl( {\textstyle\int} |\delta w_0|^2 +{\textstyle\int} |\delta w_2|^2 \Bigr)
\;<\; \infty .
\end{align*}
\end{itemlist}
\end{definition}
The name ``figure eight" for the first type of pseudoholomorphic quilt comes from an equivalent description via stereographic projection (as explained in the following remark), while the name ``squashed eight'' indicates that the second type of quilt can occur as limits of figure eights whose entire energy concentrates at infinity, corresponding to shrinking the middle strip. Alternatively, squashed eights can be viewed as punctured disk bubbles $D{\smallsetminus}\{1\}\to M_0^-\times M_2$ with boundary mapping to the immersed
Lagrangian $L_{01}\circ L_{12}$ in such a way that it has a smooth lift to \(L_{01} \times_{M_1} L_{12}\). As explained below, the singularity cannot necessarily be removed.
\begin{remark}\rm \label{rmk:stereographic} \hspace{2em}
\begin{itemlist}
\item
Recall that a pseudoholomorphic map $\mathbb{R}^2 \to M$ gives rise to a punctured pseudoholomorphic sphere $w:S^2{\smallsetminus}\{(0,0,1)\} \to M$ via stereographic projection $S^2{\smallsetminus}\{(0,0,1)\}\to\mathbb{R}^2$, where we identify
$S^2=\{(x,y,z)\in\mathbb{R}^3 \,|\, x^2+y^2+z^2 = 1\}$ with the unit sphere in \(\mathbb{R}^3\).
If the energy $\int w^*\omega_M$ is finite, then $w$ extends smoothly to the puncture $(0,0,1)$ by the standard removal of singularity theorem.
\item
Similarly, one can view a pseudoholomorphic disk with boundary on $L_{01}$ as a quilt on $S^2$
arising from a quilt on $\mathbb{R}^2$, given by a $J_0$-holomorphic patch $w_0: \mathbb{R}\times(-\infty,0]\to M_0$ and a $J_1$-holomorphic patch $w_1: \mathbb{R}\times[0,\infty)\to M_1$ satisfying the seam conditions $(w_0(s,0), w_1(s,0)\bigr) \in L_{01}$, as follows:
Stereographic projection lifts these to pseudoholomorphic maps
$w_0: S^2{\smallsetminus}\{(0,0,1)\} \cap \{ y \leq 0 \}\to M_0$ and $w_1: S^2{\smallsetminus}\{(0,0,1)\} \cap \{ y \geq 0 \} \to M_1$ defined on the two punctured hemispheres, which map
the common boundary to $L_{01}$. The standard removal of singularity
can be interpreted to say that $w_0$ and $w_1$ extend smoothly to the puncture $(0,0,1)$, thus
forming a pseudoholomorphic quilted sphere with one seam --- the equator $\{y=0\}$.
The two hemispheres are conformal to disks, so that the extended maps $w_0, w_1$ can be combined to a single pseudoholomorphic map from the disk to $M_0^- \times M_1$
w.r.t.\ the almost complex structure $(-J_0)\times J_1$,
with boundary values in $L_{01}$.
\item
A squashed eight bubble gives rise to a quilt on \(S^2\) as in the previous item, but due to the generalized nature of the seam condition, the removal of singularity is less standard.
Under the hypothesis that \(L_{01}\) and \(L_{12}\) have cleanly-immersed composition, \cite[Appendix A]{b:singularity} yields continuous extensions of \(w_0\) and \(w_2\) across \(\{(0,0,1)\}\), thus giving rise to a continuous but not necessarily smooth map from the disk to \(M_0^- \times M_2\) with boundary values in \(L_{01}\circ L_{12}\).
\begin{figure}
\centering
\def6in{0.7\columnwidth}
\input{8reparam.pdf_tex}
\caption{The left figure illustrates a figure eight bubble, while the right figure illustrates its reparametrization as a pseudoholomorphic quilt whose domain is the punctured sphere. The domain of the left figure is \(\mathbb{C}\), and the point at infinity corresponds to the puncture in the right figure.}
\label{fig:viewsof8}
\end{figure}
\item
In the case of the figure eight bubble, \((w_0,w_1,w_2)\) is a pseudoholomorphic quilt with total domain \(\mathbb{R}^2\), which maps the seam \(\mathbb{R}\times\{-\mbox{\fontsize{10}{10}\selectfont $\frac 12$}\}\) to \(L_{01}\) and the seam \(\mathbb{R}\times\{\mbox{\fontsize{10}{10}\selectfont $\frac 12$}\}\) to \(L_{12}\).
Pulling these maps back to the sphere by stereographic projection, we obtain a pseudoholomorphic quilt as shown in Figure~\ref{fig:viewsof8} whose domain is the punctured sphere, and which consists of the following patches:
\begin{align*}
w_0\colon& S^2{\smallsetminus}\{(0,0,1)\} \cap \{ y \leq -\mbox{\fontsize{10}{10}\selectfont $\frac 12$} (1-z) \}\;\to\; M_0 , \\
w_1\colon& S^2 {\smallsetminus}\{(0,0,1)\} \cap \{ -\mbox{\fontsize{10}{10}\selectfont $\frac 12$} (1-z) \leq y \leq \mbox{\fontsize{10}{10}\selectfont $\frac 12$} (1-z)\} \;\to\; M_1 ,\\
w_2\colon& S^2{\smallsetminus}\{(0,0,1)\} \cap \{ \mbox{\fontsize{10}{10}\selectfont $\frac 12$} (1-z) \leq y \} \;\to\; M_2 .
\end{align*}
This quilt maps the seam \(\{y = -\mbox{\fontsize{10}{10}\selectfont $\frac 12$} (1-z)\}$ to \(L_{01}\) and the seam \(\{y = \mbox{\fontsize{10}{10}\selectfont $\frac 12$} (1-z)\}\) to \(L_{12}\).
The union of these two seams \(y=\pm\mbox{\fontsize{10}{10}\selectfont $\frac 12$}(1-z)\) on the sphere looks like the figure eight when viewed from the positive \(z\)-axis: two circles that intersect tangentially at \((0,0,1)\).
\cite{b:singularity} establishes a continuous removal of singularity for \((w_0, w_1, w_2)\) at this tangential intersection when \(L_{01}\) and \(L_{12}\) have cleanly-immersed composition.
\end{itemlist}
\end{remark}
We now turn to the definition of, and lower bounds on, the minimal bubbling energy \(\hbar\), which we will need to control the number of bubbling points in the proof of Theorem~\ref{thm:rescale}.
\begin{definition} \label{def:bubble energy}
The {\bf minimal bubbling energy for almost complex structures $\mathbf{J_0,J_1,J_2}$} as in \eqref{eq:J} is the minimum $\hbar := \min\{\hbar_{S^2},\hbar_{D^2},\hbar_{L_{01}\circ L_{12}},\hbar_{8} \}$ of the following types of bubble energies.\footnote{\label{foot:bubblesinstrata}
For noncompact manifolds as in Remark~\ref{rmk:noncompact}, spheres, disks, and figure eights touching or contained in the boundary strata
of a compactification
have to be considered here.}
\begin{itemlist}
\item
The \textbf{minimal sphere energy \(\hbar_{S^2}\)} is the minimal energy of a nonconstant\footnote{ %
If there are no nonconstant pseudoholomorphic spheres, e.g.\ because the symplectic manifolds are exact, then we set $\hbar_{S^2}=\inf\emptyset:=\infty$; and similarly in the following.}
\\ $J_\ell(s_0,t_0)$-holomorphic sphere in $M_\ell$ for any $\ell = 0,1,2 $ and $(s_0,t_0)\in[-\rho,\rho]^2$.
\item
The \textbf{minimal disk energy \(\hbar_{D^2}\)} is the minimal energy of a nonconstant pseudoholomorphic disk in $(M_0\times M_1, (-J_0(s_0,0) )\times J_1(s_0,0))$ with boundary on $L_{01}$ or in $(M_1\times M_2, (-J_1(s_0,0) )\times J_2(s_0,0))$ with boundary on $L_{12}$ for any $s_0\in[-\rho,\rho]$.
\item
The {\bf minimal figure eight energy \(\hbar_8\)} is the minimal energy of a nonconstant \\
\((J_0(s_0,0), J_1(s_0,0), J_2(s_0,0))\)-holomorphic figure eight bubble between \(L_{01}\) and \(L_{12}\) for any $s_0\in[-\rho,\rho]$.
\item
The {\bf minimal squashed eight bubble energy \(\hbar_{L_{01} \circ L_{12}}\)} is the minimal energy of a nonconstant \((J_0(s_0,0), J_2(s_0,0))\)-holomorphic squashed eight with seam in \(L_{01} \times_{M_1} L_{12}\) for any $s_0\in[-\rho,\rho]$.
\end{itemlist}
\end{definition}
In the remainder of the section, we prove two results related to the minimal figure eight energy. We begin by establishing positivity \(\hbar_8>0\) in Lemma~\ref{lem:hbar}, which we will need in \S\ref{sec:rescale} to bound the number of bubbles during strip shrinking.
This considerably strengthens the bubbling analysis in \cite{isom}, which merely proves that the number of bubbling points must be finite.
The final result, Proposition~\ref{prop weak remsing}, is a weak removal of singularity for any figure eight and squashed eight bubble. It applies even when the geometric composition \(L_{01}\circ L_{12}\) is not immersed and yields a tuple of smooth maps with compact quilted domain that approximately capture the energy of the bubble and thus can be used in Remark~\ref{rmk:htop} to give a topological understanding of the possible bubble energies.
\begin{lemma} \label{lem:hbar}
Fix \(\rho>0\) and sequences \((J_0^\nu,J_1^\nu,J_2^\nu)_{\nu \in \mathbb{N}_0}\) of \(\cC^3\) almost complex structures on \([-\rho,\rho]^2\) as in \eqref{eq:J}, such that \(J_\ell^\nu\) is locally bounded in \(\cC^3\) and such that the \(\cC^2_\loc\)-limit of \(J_\ell^\nu\) is a \(\cC^\infty\) almost complex structure. Then \(\inf_\nu \hbar(J_0^\nu,J_1^\nu,J_2^\nu)\) is positive, where \(\hbar\) is the minimum bubbling energy as in Definition~\ref{def:bubble energy}.
\end{lemma}
\noindent
We will prove this energy bound
by contradiction: Given a sequence of figure eight or squashed eight bubbles with positive energy tending to zero, we rescale to produce a nonconstant tuple of maps, which is a contradiction to the scale-invariance of energy. Here the
convergence of the rescaled maps will be deduced from the following result of \cite{b:singularity},
which establishes \(\cC^\infty\)-compactness given a uniform gradient bound.
It uses the notion of a {\bf symmetric complex structure} on \([-\rho,\rho]^2\), which is a complex structure \(j\) such that the equality \begin{align*}
j(s,t) = -\sigma \circ j(s,-t) \circ \sigma
\end{align*}
holds for any \((s,t) \in [-\rho,\rho]^2\), where \(\sigma\) is the conjugation \(\alpha\partial_s + \beta\partial_t \mapsto \alpha\partial_s - \beta\partial_t\). (The standard complex structure, for instance, is symmetric.)
\begin{theorem}[Thm.\ 3.3, \cite{b:singularity}] \label{thm:nonfoldedstripshrink}
There exists \(\epsilon > 0\) such that the following holds: Fix \(k \in \mathbb{N}_{\geq 1}\), positive reals \(\delta^\nu \to 0\) and \(\rho > 0\), symmetric complex structures \(j^\nu\) on \([-\rho,\rho]^2\) that converge \(\cC^\infty\) to \(j^\infty\) with \(\| j^\infty - i \|_{\cC^0} \leq \epsilon\), and \(\cC^{k+2}\)-bounded sequences of \(\cC^{k+2}\) almost complex structures \(J_\ell^\nu\) on \([-\rho,\rho]^2\) as in \eqref{eq:J} such that the \(\cC^{k+1}\)-limit of each \((J_\ell^\nu)\) is a \(\cC^\infty\) almost complex structure.
Then if \((v_0^\nu,v_1^\nu,v_2^\nu)\) is a sequence of size-\((\delta^\nu,\rho)\) \((J_0^\nu,J_1^\nu,J_2^\nu, j^\nu)\)-holomorphic squiggly strip quilts for \((L_{01}, L_{12})\) with uniformly bounded gradients, \begin{align*}
\sup_{\nu \in \mathbb{N}, \: (s,t) \in [-\rho,\rho]^2} |\delta v^\nu|(s,t) < \infty,
\end{align*}
then there is a subsequence in which \((v_0^\nu(t - \delta^\nu))\), \((v_1^\nu|_{t=0})\), \((v_2^\nu(t + \delta^\nu))\) converge \(\cC^k_\loc\) to a \((J_0^\infty, J_2^\infty,i)\)-holomorphic size-\(\rho\) degenerate strip quilt \((v_0^\infty, v_1^\infty, v_2^\infty)\) for \(L_{01} \times_{M_1} L_{12}\).
If the inequality \(\liminf_{\nu\to\infty, (s,t) \in [-\rho,\rho]^2} | \delta v^\nu |(s,t) > 0\) holds, then \(v_0^\infty, v_2^\infty\) are not both constant.
\end{theorem}
\begin{proof}[Proof of Lemma~\ref{lem:hbar}]
We begin by proving energy quantization for the figure eight bubble. Suppose by contradiction that there is a sequence $\underline{w}^\nu=(w_0^\nu,w_1^\nu,w_2^\nu)$ of \((J_0^\nu(\sigma^\nu,0), J_1^\nu(\sigma^\nu,0), J_2^\nu(\sigma^\nu,0))\)-holomorphic nonconstant figure eight bubbles for some \((\sigma^\nu) \subset [-\rho,\rho]\), with energy ${E(\underline{w}^\nu)\to 0}$.
Then, despite dealing with a quilted domain, we can deduce \(\lim_{\nu \to \infty} \sup_{\ell \in \{0,1,2\}} \sup | \delta w^\nu_\ell | = 0\) from the mean value inequality $|{\rm d} u(z)|^2 \leq C r^{-2} \int_{B_r(z)} |{\rm d} u|^2$
for pseudoholomorphic maps (see e.g.\ \cite[Lemma~4.3.1]{ms:jh} or \cite[Theorem~1.3, Lemma A.1]{w:quant}).\footnote{
Here we use the metric on \(M_\ell\) that is induced by \(\omega_\ell\) and \(J_\ell^\infty\) by \eqref{eq:metrics}. Note that for any fixed \(\nu\), the convergence \(J_\ell^\nu \to J_\ell^\infty\) implies that the norm induced by this metric is equivalent to the norm induced by \(\omega_\ell\) and \(J_\ell^\nu\);
furthermore, the constant of equivalence can be chosen to be independent of \(\nu\).
This in particular yields uniform constants in the mean value inequalities.
}
Indeed, it applies to each of the maps \(w_0, w_1, w_2\) on balls of radius \(\tfrac 1 2\) that do not intersect seams, and it applies to the folded maps \((w_0(s, -\frac 12 -t), w_1(s,-\frac 12 + t))\) and \((w_1(s,\frac 12 -t), w_2(s,\frac 12 + t))\) on partial balls of radius \(\tfrac 1 2\) that intersect the boundary of the domain $\mathbb{R}\times[0,1]$, where these maps are defined, only in $\mathbb{R}\times\{0\}$, where we have Lagrangian boundary conditions in $L_{01}$ resp.\ $L_{12}$. Together, these balls cover the entire domain of the figure eight, and thus prove the uniform gradient convergence.
Next, since each triple is nonconstant we can find a subsequence (still denoted $(\underline{w}^\nu)_{\nu\in\mathbb{N}}$), an index $\ell_0 \in\{0,1,2\}$, and points $(s^\nu,t^\nu)$ in the domain of $w^\nu_{\ell_0}$ such that $\delta^\nu:=|\delta w^\nu_{\ell_0}(s^\nu,t^\nu)|\geq \mbox{\fontsize{10}{10}\selectfont $\frac 12$} \sup_{\ell\in\{0,1,2\}} \sup| \delta w^\nu_\ell |$.
We just showed \(\delta^\nu\to 0\), and we claim that in fact \(\delta^\nu t^\nu \to 0\). Indeed,
this only requires a proof in the case \(|t^\nu| \to \infty\). In that case we may apply the mean value inequality on balls of radius \(t^\nu - 1\) to obtain \(\delta^\nu t^\nu \to 0\). By shifting each triple of maps in the \(s\)-direction, we may moreover assume \(s^\nu = 0\) for all $\nu\in\mathbb{N}$.
Now rescale \(v_\ell^\nu(s, t) := w_\ell^\nu(s / \delta^\nu, t/\delta^\nu)\) to obtain maps \begin{align*}
v_0^\nu : \mathbb{R} \times (-\infty, \tfrac 1 2\delta^\nu] \to M_0, \qquad v_1^\nu: \mathbb{R} \times [-\tfrac 1 2\delta^\nu, \tfrac 1 2\delta^\nu] \to M_1, \qquad v_2^\nu: \mathbb{R} \times [\tfrac 1 2\delta^\nu, \infty) \to M_2.
\end{align*}
These maps are \(J_\ell(\sigma^\nu,0)\)-holomorphic and satisfy the following seam conditions: \begin{align*}
(v_0^\nu(s,-\tfrac 1 2\delta^\nu), v_1^\nu(s, -\tfrac 1 2\delta^\nu)) \in L_{01}, \qquad (v_1^\nu(s, \tfrac 1 2\delta^\nu), v_2^\nu(s, \tfrac 1 2\delta^\nu)) \in L_{12} \qquad \forall \: s \in \mathbb{R}.
\end{align*}
The rescaling was chosen to ensure an upper bound on the gradient,
\(\sup_{\ell \in \{0,1,2\}} | \delta v_\ell^\nu | \leq 2\), as well as a lower bound
\(| \delta v_{\ell_0}^\nu(0, \tau^\nu) | \geq 1\) for \(\tau^\nu: = \delta^\nu t^\nu \to 0\). Theorem~\ref{thm:nonfoldedstripshrink} implies that the restrictions of \(v_0^\nu(s, t - \tfrac 1 2\delta^\nu)\) resp.\ \(v_2^\nu(s, t + \tfrac 1 2\delta^\nu)\) to \((-1,1) \times (-1,0]\) resp.\ \((-1,1) \times [0,1)\) converge \(\cC^1_\loc\) to maps \(v_0^\infty\) resp.\ \(v_2^\infty\), and that at least one of the limit maps is nonconstant.
This is in contradiction to the scale-invariant energy converging to 0:
\begin{align*}
0 &< \int_{(-1,1) \times (-1,0]} (v_0^\infty)^*\omega_0 + \int_{(-1,1) \times [0,1)} (v_2^\infty)^*\omega_2 \\
&= \lim_{\nu \to \infty} \left( \int_{(-1,1)\times[0,1)} (v_0^\nu)^*\omega_0 + \int_{(-1,1)\times[0,1)} (v_2^\nu)^*\omega_2 \right) \\
&= \lim_{\nu \to \infty} \left(\int_{B_{(-\delta^\nu,\delta^\nu)\times[0,\delta^\nu)}} (w_0^\nu)^*\omega_0 + \int_{B_{(-\delta^\nu,\delta^\nu)\times[0,\delta^\nu)}} (w_2^\nu)^*\omega_2\right) \;\leq\; \liminf_{\nu\to\infty} E(\underline w^\nu) = 0.
\end{align*}
Hence we have proven the existence of a positive lower bound \(\hbar_8 > 0\).
A similar argument establishes energy quantization for squashed eights.
One difference between the two arguments is that the mean value inequality as stated in the literature requires the boundary to map to an \emph{embedded} Lagrangian, so we cannot deduce uniform gradient convergence to zero. Hence we consider two cases, depending on whether the limit \(L := \lim_{\nu \to \infty} \sup_{\ell \in \{0,1,2\}} \sup | \delta w_\ell^\nu| \) (which exists after passing to a subsequence) is finite or infinite. \begin{itemize}
\item \(\mathbf{L \in [0, \infty):}\) Center and rescale as in the proof of \(\hbar_8 > 0\).
To deal with the immersed boundary condition, choose a finite open cover \(L_{01} \times_{M_1} L_{12} = \bigcup_{i=1}^N U_i\) such that \(\pi_{02}: L_{01} \times_{M_1} L_{12} \to M_0^- \times M_2\) restricts to an embedding on each \(U_i\). Since the rescaled maps have uniformly-bounded gradient, and since their boundary values have smooth lifts to \(L_{01} \times_{M_1} L_{12}\), we can pass to a subsequence and bounded domain to work with embedded boundary conditions in some \(\pi_{02}(U_i)\).
Depending on whether \(L\) is finite or infinite, we can then appeal to either standard bootstrapping techniques (e.g.\ \cite[Theorem 4.1.1]{ms:jh}) or Theorem~\ref{thm:nonfoldedstripshrink} to obtain convergence and hence a contradiction.
\item \(\mathbf{L = \infty:}\) Choose points \((s^\nu,t^\nu)\) and \(\ell_0\) so that \(|\delta w_{\ell_0}^\nu(s^\nu,t^\nu)| \to \infty\). As in the proof of Theorem~\ref{thm:rescale}, we can apply the Hofer trick to vary the points \((s^\nu,t^\nu)\) slightly and produce numbers \(R^\nu, \epsilon^\nu\); rescaling by \(v_\ell^\nu(s,t) := w_\ell^\nu(s^\nu + s/R^\nu, t^\nu + t/R^\nu)\) produces a sequence of maps which has nonconstant limit.
\end{itemize}
Finally, $\hbar>0$ follows from the above since we have standard lower bounds $\hbar_{S^2}, \hbar_{D^2}>0$, which can be proven by a single mean value inequality applied to balls resp.\ half balls of large radius, see e.g.\ \cite[Proposition 4.1.4]{ms:jh}.
\end{proof}
\begin{remark} \label{rmk:htop}
The minimal bubbling energies
\(\hbar_{S^2}, \hbar_{D^2}, \hbar_8, \hbar_{L_{01} \circ L_{12}}\) in Definition~\ref{def:bubble energy} can also be bounded below by concrete topological quantities
\begin{align*}
\hbar_{S^2} \geq \hbar_{S^2}^{\rm top}, \qquad \hbar_{D^2} \geq \hbar_{D^2}^{\rm top}, \qquad \min\{\hbar_8, \hbar_{L_{01}\circ L_{12}}\} \geq \hbar_8^{\rm top}
\end{align*}
rather than the abstract analytic lower bound from Lemma~\ref{lem:hbar}.
For sphere and disk bubbles, this topological quantity is the minimal positive symplectic area of spherical or disk (relative) homotopy classes.
Proposition~\ref{prop weak remsing} bounds the minimal energy of squashed eight and figure eight bubbles by the minimal positive symplectic area of ``quilted homotopy classes''
\begin{equation*}
\hbar^{\rm top}_8 := \inf\left\{ \sum_{\ell \in \{0,1,2\}} \langle [\omega_\ell], [u_\ell] \rangle \,\left|\,
\begin{matrix}
u_0:D^2\to M_0, \\
u_1:[0,2\pi]\times [-\frac12, \frac12]\to M_1,\\
u_2:D^2\to M_2 ,
\end{matrix}
\quad\eqref{eq: htop 8 seam},
\eqref{eq: htop 8 w1 ends}
\right.\right\}
\end{equation*}
for which $u_0, u_1, u_2$ are continuous and satisfy the seam conditions
\begin{equation}\label{eq: htop 8 seam}
(u_0(e^{-i\theta}), u_1(\theta,-\tfrac12))\in L_{01} , \quad
(u_1(\theta,\tfrac12), u_2(e^{i\theta}))\in L_{12}
\qquad\forall \: \theta\in[0,2\pi]
\end{equation}
and the ``constant limit conditions''
\begin{equation}\label{eq: htop 8 w1 ends}
u_1(0, t_1) = u_1(0, t_2), \quad u_1(2\pi, t_1) = u_1(2\pi, t_2) \quad \forall \: t_1, t_2 \in [-\tfrac12, \tfrac12].
\end{equation}
We differentiate such quilt maps by the relation between the two constants $u_1(0,-)$ and $u_1(2\pi,-)$:
\begin{itemize}
\item A {\bf (non-switching) homotopy figure eight} is a tuple of maps $(u_0, u_1, u_2)$ as above with $u_1(2\pi, -)= u_1(0, -)$.
That is, $u_1: S^1\times [-\frac12, \frac12]\to M_1$ is in fact defined on an annulus with seam conditions that identify $S^1\times \{\pm\frac12\}$ with the boundaries of the two disk patches.
\item
A {\bf sheet-switching homotopy figure eight} is a tuple of maps $(u_0, u_1, u_2)$ as above with \(u_1(0, -) =: u_1^- \neq u_1^+ := u_1(2\pi, -)\), where \(u_1^-, u_1^+\) represent two different lifts of $\bigl( u_0(1), u_2(1)\bigr)\in L_{01}\circ L_{12}$ to \(L_{01} \times_{M_1} L_{12}\).
\end{itemize}
Note that sphere homotopy classes as well as disk homotopy classes for \(L_{01}\) and \(L_{12}\) can be represented by non-switching homotopy figure eights with one or two constant patches.\footnote{
Given a sphere or disk bubble, we can attach it to a constant homotopy figure eight under mild hypotheses (e.g.\ that \(L_{01}, L_{12}\) are nonempty and \(M_1\) is connected): Given any two points on \(L_{01}\) and \(L_{12}\), make a zero-energy homotopy figure eight with these values on the seams, \(u_0\) and \(u_2\) constant, and \(u_1\) a path between the two projections.}
However, $\hbar^{\rm top} := \min\{\hbar^{\rm top}_{S^2}, \hbar^{\rm top}_{D^2}, \hbar^{\rm top}_8\}$ is not generally positive unless the symplectic and Lagrangian manifolds have very simple topology.
For example, we have $\hbar^{\rm top}_{S^2} = 0$ as soon as $\langle [\omega_\ell] , \pi_2(M_\ell) \rangle\subset\mathbb{R}$ contains two incommensurate values for some $\ell\in\{0,1,2\}$.
\end{remark}
The possible homotopy classes of figure eight bubbles in the above remark can be deduced from the removal of singularity theorem in \cite{b:singularity}. However, this also follows from the following weaker result which requires fewer estimates.
It yields not a pseudoholomorphic quilt on \(S^2\) but a smooth quilt map with domain \(S^2 \cong (D^2)^- \cup (S^1 \times [0, 1]) \cup D^2\) that approximately captures the energy of the bubble.
This result was first announced in \cite{w:chekanov}, but we include it here for convenience. It is the only point in this paper where we will not assume \(L_{01}\) and \(L_{12}\) to have immersed composition.
\begin{proposition} \label{prop weak remsing}
Let \(L_{01} \subset M_0^- \times M_1\), \(L_{12} \subset M_1^- \times M_2\) be compact Lagrangian correspondences, and let \((w_0,w_1,w_2)\) be either (1) a figure eight bubble between \(L_{01}\) and \(L_{12}\) or (2) a squashed eight bubble with seam in \(L_{01} \times_{M_1} L_{12}\), where \(w_1\) is the pullback \(w_1(s,t) := \overline w_1(s)\).
Then for any \(\epsilon > 0\) there exist smooth maps
\(u_0:D^2\to M_0\),
\(\widehat u_1:[0,2\pi]\times [-\frac12,\frac12]\to M_1\),
\(u_2:D^2\to M_2\)
satisfying the seam conditions
\begin{align*}
\bigl(u_0(e^{-i\theta}),\widehat u_1(\theta,-\tfrac12)\bigr)\in L_{01} , \quad
\bigl(\widehat u_1(\theta,\tfrac12),u_2(e^{i\theta})\bigr)\in L_{12}
\qquad\forall \: e^{i\theta}\in\partial D^2\cong \mathbb{R}/2\pi\mathbb{Z} ,
\end{align*}
and whose energy is \(\epsilon\)-close to that of \((w_0,w_1,w_2)\),
\begin{align*}
\Bigl| \Bigl( {\textstyle\int} u_0^*\omega_0 + {\textstyle\int} \widehat u_1^*\omega_1 + {\textstyle\int} u_2^*\omega_2 \Bigr)
- \Bigl( {\textstyle\int} w_0^*\omega_0 + {\textstyle\int} w_1^*\omega_1 + {\textstyle\int} w_2^*\omega_2 \Bigr) \Bigr|
\leq \epsilon.
\end{align*}
Moreover, \(\widehat u_1\) is constant on the two lifts of the line \(\{[0]=[2\pi]\}\times[-\frac12,\frac12]\subset S^1\times [-\frac12,\frac12]\), so that \(\widehat u_1|_{\{0\}\times[-\frac12,\frac12]}\equiv p_1^+\), \(\widehat u_1|_{\{2\pi\}\times[-\frac12,\frac12]}\equiv p_1^-\) form together with \(p_0:=u_0(e^{i0})\), \(p_2:=u_2(e^{i0})\) two lifts \((p_0,p_1^\pm,p_1^\pm,p_2)\in L_{01}\times_{M_1}L_{12}\) of the same point \((p_0,p_2)\in L_{01}\circ L_{12}\).
In particular, if \(\pi_{02} : L_{01} \times_{M_1} L_{12} \to L_{01}\circ L_{12}\) is injective, then \(\widehat u_1\) can be chosen such that it induces a smooth map \(u_1: S^1\times[-\frac12,\frac12] \to M_1\).
\end{proposition}
\noindent For the proof of Proposition~\ref{prop weak remsing}, we will need an extension result.
To state it we will use the following notation for partitioning the closed unit ball \(\overline B_1(0)\subset\mathbb{R}^2\) into four quadrants:\begin{gather} \label{eq:quadrants}
U_0 := \{(x, y) \in \overline B(0,1) \: | \: y \leq x, \: y \leq -x\}, \qquad U_1 := \{(x, y) \in \overline B(0,1) \: | \: x \geq y, \: x \geq -y\}, \\
U_2 := \{(x, y) \in \overline B(0,1) \: | \: y \geq x, \: y \geq -x\}, \qquad U_3 := \{(x, y) \in \overline B(0,1) \: | \: x \leq y, \: x \leq -y \}. \nonumber
\end{gather}
The resulting partition of the boundary circle $\partial \overline B_1(0)$ will be denoted by \(A_i := U_i \cap \partial \overline B_1(0)\) for \(i = 0,1,2,3 \), and we denote the intersections of these arcs by \(p_{i(i+1)} := A_i\cap A_{i+1}\) for \(i \;{\rm mod}\; 4\).
We denote the length of a path $\sigma_i:A_i \to X_i$ with respect to $g_i$ by $\ell(\sigma_i):=\int_{A_i} |\delta \sigma_i| $.
\begin{lemma} \label{lem:extension}
Let \((X_i,g_i)\) be Riemannian manifolds equipped with closed $2$-forms $\omega_i$
for $i=0,1,2$, and let \(Y_{01} \subset X_0 \times X_1\), \(Y_{12} \subset X_1 \times X_2\) be closed submanifolds. Then for every \(\epsilon > 0\) there exists \(\delta > 0\) such that the following extension property holds: Suppose that \(\sigma_i:A_i\to X_i\) for $i=0,1,2,3$ are smooth arcs that satisfy
\begin{gather} \label{eq:condsforextension}
\ell(\sigma_i) \leq \delta, \qquad (\sigma_i(p_{i(i+1)}), \sigma_{i+1}(p_{i(i+1)})) \in Y_{i(i+1)} \qquad \forall \: i \;{\rm mod}\; 4.
\end{gather}
Here we denote \(X_3 := X_1\), \(Y_{23} := Y_{12}^T\), and \(Y_{30} := Y_{01}^T\), with $(\cdot)^T$ denoting the interchange of factors in \(X_i \times X_{i+1}\). Then there exist smooth extensions \(\widetilde \sigma_i:U_i\to X\) of \(\widetilde\sigma_i|_{A_i}=\sigma_i\) such that
\begin{gather*}
\int_{U_i} \widetilde\sigma_i^*\omega_i \leq \epsilon , \qquad
(\widetilde\sigma_i(p), \widetilde\sigma_{i+1}(p)) \in Y_{i(i+1)} \quad \forall \: p \in U_i \cap U_{i+1} \qquad \forall \: i \;{\rm mod}\; 4.
\end{gather*}
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{lem:extension}]
Set \(a_i := \sigma_i(p_{(i-1)i})\), \(b_i := \sigma_i(p_{i(i+1)})\) for \(i \;{\rm mod}\; 4\). For a constant \(\epsilon' > 0\) that we will fix later in the proof, let us show that if \(\delta\) is chosen small enough, there exist \(x^\pm = (x_0, x_1^\pm, x_1^\pm, x_2) \in Y_{01} \times_{X_1} Y_{12}\) (two lifts of the same point in $Y_{01}\circ Y_{12}\subset X_0\times X_2$) such that the following distances with respect to the metric \(g_0 \oplus g_1 \oplus g_1 \oplus g_2\) are bounded,
\begin{align} \label{eq:closepoint}
\max\bigl\{ d( (b_0, a_1, b_1, a_2), x^+ )\, ,\, d( (a_0, b_3, a_3, b_2), x^-) \bigr\} \leq \epsilon' .
\end{align}
Suppose by contradiction that the sequences \((\sigma_0^\nu,\sigma_1^\nu,\sigma_2^\nu,\sigma_3^\nu)\) satisfy \eqref{eq:condsforextension} for a sequence \(\delta^\nu \to 0\) but
\begin{equation}\label{eq:contradicte}
\min_{x^\pm=(x_0, x_1^\pm, x_1^\pm, x_2) \in Y_{01} \times_{X_1} Y_{12}}
\max\bigl\{ d( (b^\nu_0, a^\nu_1, b^\nu_1, a^\nu_2), x^+ )\, ,\, d( (a^\nu_0, b^\nu_3, a^\nu_3, b^\nu_2), x^-) \bigr\} > \epsilon'
\end{equation}
for all $\nu\in\mathbb{N}$ with \(a_i^\nu := \sigma_i^\nu(p_{(i-1)i})\), \(b_i^\nu := \sigma_i^\nu(p_{i(i+1)})\).
Since $(b_0^\nu,a_1^\nu)\in Y_{01}, (b_1^\nu,a_2^\nu)\in Y_{12}, (a_3^\nu,b_2^\nu)\in Y_{12}, (a_0^\nu,b_3^\nu)\in Y_{01}$ and \(Y_{01}, Y_{12}\) are compact, we may pass to a subsequence and assume that \(a_i^\nu\), \(b_i^\nu\) have limits \(a_i^\infty, b_i^\infty\) as \(\nu \to \infty\).
These limits have to coincide \(a_i^\infty = b_i^\infty\) since they are the limits of endpoints of the paths $\sigma_i^\nu$ whose length $\ell(\sigma_i^\nu)\leq \delta^\nu$ goes to zero with $\nu\to\infty$.
This gives rise to two lifts $x^+ := (a_0^\infty, a_1^\infty, a_1^\infty, a_2^\infty), x^- := (a_0^\infty, a_3^\infty, a_3^\infty, a_2^\infty)\in Y_{01} \times_{X_1} Y_{12}$ since
$(a_0^\infty, a_1^\infty) =\lim_{\nu\to\infty}(b_0^\nu, a_1^\nu)$ and
$(a_0^\infty, a_3^\infty) =\lim_{\nu\to\infty}(a_0^\nu, b_3^\nu)$ are limits in the closed submanifold $Y_{01}$ and
$(a_1^\infty, a_2^\infty) =\lim_{\nu\to\infty}(b_1^\nu, a_2^\nu)$ and
$(a_3^\infty, a_2^\infty) =\lim_{\nu\to\infty}(a_3^\nu, b_2^\nu)$ are limits in the closed submanifold $Y_{12}$,
and they contradict \eqref{eq:contradicte} since both distances converge to zero, e.g.
$$
d( (b^\nu_0, a^\nu_1, b^\nu_1, a^\nu_2), x^+ )
\leq d(b^\nu_0,a_0^\infty=b_0^\infty) + d(a^\nu_1,a_1^\infty) + d(b^\nu_1,a_1^\infty=b_1^\infty) + d(a^\nu_2,a_2^\infty)
\;\underset{\nu\to\infty}\longrightarrow\; 0 .
$$
With that we may assume to have lifts \(x^\pm=(x_0, x_1^\pm, x_1^\pm, x_2) \in Y_{01} \times_{X_1} Y_{12} \) satisfying \eqref{eq:closepoint} and begin to construct the extensions \(\widetilde \sigma_i\) by
\begin{align*}
\widetilde\sigma_0(0) := x_0, \qquad \widetilde\sigma_1(0) := x_1^+, \qquad \widetilde \sigma_2(0) := x_2, \qquad \widetilde \sigma_3(0) := x_1^-.
\end{align*}
To construct $\widetilde\sigma_i\times \widetilde\sigma_{i+1}: U_i \cap U_{i+1} \to Y_{i(i+1)}$, note that the given values on both ends of this line segment are at distance at most $\epsilon'$ in $Y_{i(i+1)}$. Hence for sufficiently small \(\epsilon'\) we may use local charts of the submanifolds \(Y_{01}, Y_{12}\) to choose each extension $\widetilde\sigma_i :\partial U_i \to X_i$ of $\widetilde\sigma_i|_{A_i}=\sigma_i$ such that they satisfy the seam conditions $(\widetilde\sigma_i(p), \widetilde\sigma_{i+1}(p)) \in Y_{i(i+1)}$ and length bound $\ell(\widetilde\sigma_i|_{\partial U_i})\leq 2\epsilon' + \delta$.
By choosing $\epsilon'$ and $\delta$ sufficiently small, we can moreover ensure that each of these loops lies in contractible charts of $X_i$.
On the one hand, that allows us to extend the given $\widetilde\sigma_i:\partial U_i\to X$ to a smooth map $\widetilde\sigma_i: U_i \to X_i$. On the other hand, in each such contractible chart $V\subset X_i$ the given $2$-form $\omega_i|_V = {\rm d} \eta_{V}$ has a uniformly bounded primitive $\eta_V\in\Omega^1(V)$, which gives the desired bound
$$
\int_{U_i} \widetilde\sigma_i^*\omega_i \;=\; \int_{\partial U_i} \widetilde\sigma_i^*\eta_V
\;\leq\; \|\eta_V\|_\infty \ell(\widetilde\sigma_i|_{\partial U_i})
\;\leq\; \|\eta_V\|_\infty (2\epsilon' + \delta)\;\leq\;\epsilon
$$
for sufficiently small $\delta>0$.
In fact, we can cover the projections of the compact Lagrangians to the factors $X_i$ with finitely many contractible charts $V$ so that $\|\eta_V\|_\infty$ is uniformly bounded. This ensures that the choice of sufficiently small $\delta>0$ for given $\epsilon>0$ is independent of the arcs~$\sigma_i$.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop weak remsing}]
To simplify notation we shift domains so that $w_0$ and $w_2$ in case (i) as well as (ii) are parametrized by $\mathbb{R}\times (-\infty,0]$ and $\mathbb{R}\times[0,\infty)$, respectively.
On these domains we rewrite the figure eight energy integral in polar coordinates
$\widetilde w_\ell (r,\theta) = w_\ell (r\cos\theta, r\sin\theta)$
to obtain
\begin{align*}
E& := {\textstyle\int} w_0^* \omega_0 + {\textstyle\int} w_1^* \omega_1 + {\textstyle\int} w_2^* \omega_2 \\
& =\int_{\mathbb{R}\times (-\infty,0]} r^{-2} |\partial_\theta \widetilde w_0 |^2 \delta s \delta t
\; + \int_{\mathbb{R}\times[-\frac12,\frac12]} |\partial_t w_1 |^2 \delta s \delta t
\; + \int_{\mathbb{R}\times[0,\infty)} r^{-2} |\partial_\theta \widetilde w_2|^2 \delta s \delta t \\
& = \lim_{R\to\infty} \int_0^R r^{-1} A(r) \, \delta r
\end{align*}
with integrand
\begin{align*}
A(r) :=
\int_\pi^{2\pi} |\partial_\theta \widetilde w_0 (r,\theta)|^2 \delta\theta
+\int_0^\pi |\partial_\theta \widetilde w_2(r,\theta)|^2 \delta\theta
+ \int_{-\frac12}^{\frac 12} r |\partial_t w_1 (-r,t) |^2 \delta t
+ \int_{-\frac12}^{\frac 12} r |\partial_t w_1 (r,t) |^2 \delta t .
\end{align*}
The same holds for squashed eights if we drop the terms involving $w_1$.
By assumption $\int_0^R r^{-1} A(r) \delta r$ converges as $R\to\infty$ although $A(r)\geq 0$ and $\int_0^R r^{-1} \delta r \to\infty$ as $R\to\infty$. Hence there exists a sequence $r_i \to \infty$ such that $A(r_i)\to 0$.
Depending on a $\delta>0$ to be determined and the $\epsilon>0$ given, we now choose $r_0>1$ sufficiently large such that $A(r_0)\leq\delta$ and $\bigl| E - \int_0^{r_0} r^{-1} A(r) \delta r \bigr| \leq \mbox{\fontsize{10}{10}\selectfont $\frac 12$}\epsilon$.
Denoting by $B_{r_0}^\pm$ the ball of radius $r_0$ around the origin in the halfplanes $\H^+=\mathbb{R}\times[0,\infty)$ resp.\ $\H^-=\mathbb{R}\times(-\infty,0]$, we now have approximated the energy
\begin{align*}
\biggl| E - \biggl( \int_{B_{r_0}^-} w_0^* \omega_0 + \int_{[-r_0,r_0]\times[-\frac12, \frac 12]} w_1^* \omega_1 + \int_{B_{r_0}^+} w_2^* \omega_2 \biggr) \biggr| \leq \mbox{\fontsize{10}{10}\selectfont $\frac 12$}\epsilon
\end{align*}
and bounded lengths of arcs
\begin{align*}
\ell\bigl( w_1|_{\{\pm r_0\} \times [-\frac12, \frac12]} \bigr) \leq \sqrt{\delta/{r_0}} ,
\qquad
\ell\bigl( \widetilde w_\ell |_{|z|= r_0} \bigr) \leq \sqrt{\pi \delta} \quad\text{for}\; \ell\in\{0,2\}.
\end{align*}
Here the latter for \(\widetilde w_0\) (and analogously for \(\widetilde w_2\) and \(w_1\)) follows from the estimate
\begin{align*}
\ell\bigl( \widetilde w_0|_{|z|= r_0}\bigr)
\leq \int_\pi^{2\pi} |\partial_\theta \widetilde w_0 (r_0,\theta)| \delta\theta
\leq \sqrt{\pi A(r_0)} .
\end{align*}
Then for sufficiently large \(r_0>1\) and small \(\delta>0\), the maps \((u_0,\widehat u_1,u_2)\) will be constructed as extensions of \((w_0|_{B_{r_0}^-},w_1|_{[-r_0,r_0]\times[-\frac12, \frac 12]},w_2|_{B_{r_0}^+})\). We first pull them back to the quilted sphere by stereographic projection as in Remark~\ref{rmk:stereographic}, to define a quilted map \((v_0,\widehat v_1, v_2)\) on the complement of a neighborhood \(N\subset S^2\) of the puncture \((0,0,1)\).
Thus $N$ is a slightly-deformed ball with diameter of order \(r_0^{-1}\), which we identify with $\overline{B}_1(0)$ in Lemma~\ref{lem:extension} so that the arcs $\sigma_0=v_0|_{\partial N}$ and $\sigma_2=v_0|_{\partial N}$ are reparametrizations of the short paths \((w_0|_{B_{r_0}^-},w_2|_{B_{r_0}^+})\), and $\sigma_1,\sigma_3$ are the two connected components of $v_1|_{\partial N}$, given by reparametrizations of the short paths $w_1|_{[-r_0,r_0]\times[-\frac12, \frac 12]}$.
For sufficiently small $\delta>0$, Lemma~\ref{lem:extension} then provides a smooth extension of \((v_0,\widehat v_1,v_2)|_{\partial N}\) by a quilted map \((\widetilde\sigma_0,\widetilde\sigma_1,\widetilde\sigma_3,\widetilde\sigma_2)\) on $N$ with total symplectic area bounded by \(\frac12 \epsilon\). After smoothing these extensions near $\partial N$, we finally construct \((u_0,\widehat u_1,u_2)\) by pullback of these extended maps. More precisely, we construct $u_0$ (and similarly $u_2$) by precomposition of $v_0,\widetilde\sigma_0$ with a smooth bijection from $D^2$ to ${\rm dom}(v_0)\cup U_0$ which maps $1\in\partial D^2$ to the corner of $U_0$. (This is not possible by a diffeomorphism, but there is a smooth map with vanishing derivatives at $1$.)
To construct \(\widehat u_1:[0,2\pi]\times [-\frac12,\frac12]\to M_1\) we pull back $v_1,\widetilde\sigma_1,\widetilde\sigma_3$ by a smooth map from $[0,2\pi]\times [-\frac12,\frac12]$ to ${\rm dom}(u_1)\cup U_1\cup U_3$ which on the boundary components $[0,2\pi]\times \{\pm\frac12\}$ (in polar coordinates) coincides with the bijections from $\partial D^2$ to $U_0\cap (U_1\cup U_3)$ resp.\ $U_2\cap (U_1\cup U_3)$ used in the construction of $u_0,u_2$, thus guaranteeing the seam conditions.
It can moreover be chosen as bijection with the exception of mapping the two edges $\{0,2\pi\}\times [-\frac12,\frac 12]$ to the common corner point $U_1\cap U_3$.
Smoothness of these maps guarantees smoothness of the pullbacks \((u_0,\widehat u_1,u_2)\), and bijectivity on the complement of a zero set guarantees that they have the same symplectic area as the extension of \((v_0,\widehat v_1,v_2)|_{\partial N}\). Finally,
$\widehat u_1$ by construction is constant equal to $\widetilde\sigma_1(0)$ on $\{0\}\times [-\frac12,\frac 12]$ and equal to $\widetilde\sigma_3(0)$ on $\{2\pi\}\times [-\frac12,\frac 12]$, and extends smoothly to an annulus if $\widetilde\sigma_1(0)=\widetilde\sigma_3(0)$.
The latter is guaranteed by the seam conditions on the extensions $\widetilde\sigma_i$ if \(\pi_{02} : L_{01} \times_{X_1} L_{12} \to L_{01}\circ L_{12}\) is injective.
\end{proof}
\begin{remark}
Under the hypothesis that \(L_{01}, L_{12}\) have immersed composition, Proposition~\ref{prop weak remsing} can be modified to show that a squashed eight can be approximated by a homotopy squashed eight, rather than a homotopy figure eight. In this situation, the minimum squashed eight energy \(\hbar_{L_{01} \circ L_{12}}\) can be bounded below by the minimum positive symplectic area of ``homotopy squashed eights'': \begin{align*}
\hbar_{L_{01} \circ L_{12}}^{\rm top} := \inf\bigl\{ \langle [(-\omega_0) \oplus \omega_2] , [u] \rangle >0 \,\big|\, u\in\cC^0(D,M_0^-\times M_2) , u(\partial D) \subset L_{01}\circ L_{12} \bigr\} .
\end{align*}
\end{remark}
\section{Toward Gromov compactness for strip shrinking}
\label{sec:rescale}
In this section we state and prove the Gromov Compactness Theorem~\ref{thm:rescale}, which is the main result of this paper.
In order to focus on the relevant effects, rather than deal with complicated notation,
Theorem~\ref{thm:rescale} is stated in the setting of squiggly strip quilts, with the width of the middle strip shrinking obediently to zero.
However, the results of this section directly generalize to a sequence of pseudoholomorphic quilt maps whose domains are quilted surfaces which vary only by the width of one patch --- diffeomorphic to a strip or annulus --- going to zero.
Theorem~\ref{thm:rescale} is a refinement and generalization of
\cite[Theorem 3.3.1 and Lemma 3.3.2]{isom}, where compactness up to energy concentration is proven for strip shrinking in the special case of embedded composition, though only in an $H^2\cap W^{1,4}$-topology and with a lower bound on the energy concentration that has no geometric interpretation but arises by contradiction from mean value inequalities.
(In fact, the $H^2\cap W^{1,4}$-convergence does not even suffice to deduce nontriviality of the weak limit of rescaled solutions near a bubbling point.)
We establish full $\cC^\infty_{\rm loc}$-convergence in the most general natural case, with the further generalization to noncompact manifolds being discussed in Remark~\ref{rmk:noncompact}. The proof will moreover illuminate the origin of the different bubbling phenomena. Analytically, it relies on Theorem~\ref{thm:nonfoldedstripshrink}, a result from \cite{b:singularity}. A further generalization is that we will allow the two seams bordering the middle strip to not be straight, so that Theorem~\ref{thm:rescale} allows the first author to establish a removal of singularity theorem for figure eight bubbles in \cite{b:singularity} and a ``bubbles connect'' result in \cite{b:thesis}.
\begin{theorem} \label{thm:rescale}
Fix \(\rho > 0\), sequences \(J_0^\nu ,J_1^\nu ,J_2^\nu\) of smooth almost complex structures on \([-\rho,\rho]^2\) as in \eqref{eq:J} that converge \(\cC^\infty_\loc\) to \(J_\ell^\infty\colon [-\rho,\rho]^2 \to \mathcal{J}(M_\ell,\omega_\ell)\) for \(\ell = 0,1,2 \), and a sequence \((f^\nu\colon[-\rho,\rho] \to (0,\rho/2])\) of real-analytic functions shrinking obediently to zero as in Definition~\ref{def:obedience}.
Then for any sequence \((\underline{v}^\nu)_{\nu\in\mathbb{N}}\) of \((J_0^\nu,J_1^\nu,J_2^\nu)\)-holomorphic size-\((f^\nu,\rho)\) squiggly strip quilts for \((L_{01},L_{12})\) as in Definition~\ref{def:squigglystrip}
with bounded energy \(E := \sup_{\nu \in \mathbb{N}} E(\underline v^\nu) < \infty\) there exist finitely many blow-up points \(Z=\{z_1,\ldots,z_{N}\}\subset (-\rho,\rho)^2\) and a subsequence that Gromov-converges in the following sense:
\begin{enumerate}
\item
There exists a \((J_0^\infty, J_2^\infty)\)-holomorphic degenerate strip quilt \(\underline v^\infty\) for \(L_{01} \times_{M_1} L_{12}\) with singularities, whose singular set \(S \subset (-\rho,\rho) \cong (-\rho,\rho)\times\{0\}\) is contained in \(\{z_1, \ldots, z_N\}\cap (-\rho,\rho)\times\{0\}\),
such that
\((v_0^\nu(s,t - f^\nu(s)))\) resp.\ \((v_1^\nu(s,0))\) resp.\ \((v_2^\nu(s,t + f^\nu(s)))\) converge \(\cC^\infty_\loc\) on the domains \((-\rho,\rho) \times (-\rho,0] {\smallsetminus} Z\) resp.\ \((-\rho,\rho) \times \{0\} {\smallsetminus} Z\) resp.\ \((-\rho,\rho) \times [0, \rho) {\smallsetminus} Z\) to \(v_0^\infty\) resp.\ \(v_1^\infty\) resp.\ \(v_2^\infty\).
\item
There is a concentration of energy $\hbar>0$, given by the minimal bubbling energy from Definition~\ref{def:bubble energy},
at each $z_j$ in the sense that there is a sequence of radii $r^\nu\to 0$ such that
\begin{align*}
\liminf_{\nu\to\infty} \int_{B_{r^\nu}(z_j)} \tfrac 1 2|\delta\underline{v}^\nu|^2 \;\geq \; \hbar > 0,
\end{align*}
where the energy densities $|\delta\underline{v}^\nu|$ are defined as in \eqref{eq:density}.
\item At least one type of bubble forms near each blow-up point \(z_j=(s_j,t_j)\):
There is a sequence \((w^\nu)\) of (tuples of) maps obtained by rescaling the maps defined on the intersection of the respective domain with \(B_{r^\nu}(z_j)\), which converges in $\cC^\infty_\loc$ to one of the following:
\begin{enumlist}
\item[\bf (S0),(S1),(S2):]
a \(J_\ell^\infty(z_j)\)-holomorphic map \(w_\ell^\infty\colon \mathbb{R}^2 \to M_\ell\) for \(\ell = 0,1,2\), which can be completed to a nonconstant pseudoholomorphic sphere \({\overline{w}_\ell^\infty\colon S^2 \to M_\ell}\);
\item[\bf (D01):]
a \((-J_0^\infty(s_j,0))\times J_1^\infty(s_j,0)\)-holomorphic
map \(w_{01}^\infty\colon \H \to M_0^-\times M_1\) with \(w_{01}^\infty(\partial\H )\subset L_{01}\), which can be extended to a nonconstant pseudoholomorphic disk \(\overline{w}_{01}^\infty\colon (D,\partial D) \to (M_0^-\times M_1,L_{01})\);
\item[\bf (D12):]
a \((-J_1^\infty(s_j,0))\times J_2^\infty(s_j,0)\)-holomorphic map \(w_{12}^\infty\colon \H \to M_1^-\times M_2\) with \(w_{12}^\infty(\partial\H )\subset L_{12}\), which can be extended to a nonconstant pseudoholomorphic disk \(\overline{w}_{12}^\infty\colon (D,\partial D) \to (M_1^-\times M_2,L_{12})\);
\item[\bf (E012):]
a nonconstant
\((J_0^\infty(s_j,0), J_1^\infty(s_j,0), J_2^\infty(s_j,0))\)-holomorphic figure eight bubble between \(L_{01}\) and \(L_{12}\), as in Definition~\ref{def:8};
\item[\bf (D02):]
a nonconstant \((J_0^\infty(s_j,0),J_2^\infty(s_j,0))\)-holomorphic squashed eight bubble with generalized boundary conditions in \(L_{01} \circ L_{12}\), as in Definition~\ref{def:8}.
\end{enumlist}
\end{enumerate}
\end{theorem}
\begin{remark}
If the composition \(L_{01} \circ L_{12}\) is cleanly immersed, then \cite[Thm.\ 2.2]{b:singularity}
guarantees a continuous removal of singularity for figure eight and squashed eight bubbles, in particular for the bubbles produced in cases (E012) and (D02) of Theorem~\ref{thm:rescale}. This allows us to partially
characterize the singular set $S\subset\mathbb{R}$ in (1) at which $\overline v_1$ does not extend continuously, and hence to which $v_0,v_2$
may not extend smoothly: A necessary condition for a bubbling point $z_j$ to lie in $S$ is that a sheet-switching bubble can be found by rescaling near $z_j$, i.e.\ a squashed eight bubble whose boundary arc on \(L_{01} \circ L_{12}\) does not lift to \(L_{01} \times_{M_1} L_{12}\), or a figure eight bubble with $\lim_{s\to-\infty} w^\infty_1(s,-) \neq \lim_{s\to+\infty} w^\infty_1(s,-)$.
However, this is not a sufficient condition, since a tree involving several sheet-switching bubbles at $z_j$ could allow continuous extension of $\overline v_1$ to $z_j$.
\end{remark}
The proof of Theorem~\ref{thm:rescale} will take up the rest of this section.
Our first goal will be to find a subsequence and blow-up points so that (2) and (3) hold together with the following bound on energy densities:
\begin{enumerate}
\item[(0)]
The energy densities \(|\delta\underline{v}^\nu|\) are uniformly bounded away from the bubbling points, that is for each compact subset \(K\subset (-\rho,\rho)^2 {\smallsetminus}\{z_1,\ldots z_N\}\) we have
$
\sup_{\nu\in\mathbb{N}} \; \bigl\| \delta\underline{v}^{\nu} \bigr\|_{L^\infty( K \cap \, (-\rho,\rho)^2 ) }
< \infty $.
\end{enumerate}
Then we will show that (1) follows from Theorem~\ref{thm:nonfoldedstripshrink}.
Suppose that we have already found a subsequence (for convenience again indexed by \(\nu\in\mathbb{N}\)) and some blow-up points \(z_1,\ldots,z_N\in(-\rho,\rho)^2\) such that (2) holds and we have established (3) at each such point.
Now either (0) holds, too, or we can pass to a further subsequence and find another blow-up point \(z_{N+1}=(s_{N+1},t_{N+1})=\lim_{\nu\to\infty}(s^\nu, t^\nu)\in (-\rho,\rho)^2{\smallsetminus}\{z_1,\ldots z_N\}\) such that \(\lim_{\nu \to \infty} |\delta \underline{v}^\nu(s^\nu,t^\nu)| = \infty\).
We can apply the Hofer trick \cite[Lemma 4.3.4]{ms:jh}\footnote{Note that the Hofer trick applies directly to each function $f(x)=|\delta\underline{v}^\nu(x)|$ for $x=(s,t)$, although it is only upper semi-continuous. In the proof, continuity is used only to exclude $f(x_n)\to\infty$ for a convergent sequence $x_n\to x_\infty$. For a bounded upper semi-continuous function $f$, we still have $\limsup f(x_n) \leq f(x_\infty) <\infty$, excluding this divergence.
}
to vary the points \((s^\nu,t^\nu)\) slightly (not changing their limit) and find \(\epsilon^\nu\to 0\) such that we have
\begin{align} \label{eq:vbounds}
\sup_{(s,t)\in B_{\epsilon^\nu}(s^\nu,t^\nu)} |\delta \underline{v}^\nu(s,t)| \leq 2 |\delta \underline{v}^\nu(s^\nu,t^\nu)| =: 2 R^\nu, \qquad
R^\nu \epsilon^\nu \to\infty .
\end{align}
We will essentially rescale by \(R^\nu\) around \((s^\nu,t^\nu)\) to obtain different types of bubbles, depending on where the lines \(\{t= \pm f^\nu(s)\}\) get mapped under the rescaling. We denote by \({\tau_\pm^\nu := R^\nu(\pm f^\nu(s^\nu)-t^\nu)}\) the \(t\)-coordinate of the preimage of the point \((s^\nu, \pm f^\nu(s^\nu))\) under the rescaling \(t\mapsto t^\nu + t/R^\nu\).
After passing to a subsequence, we may assume that \(\tau_\pm^\nu\) converges to \(\tau_\pm^\infty \in \mathbb{R} \cup \{\pm \infty\}\) with \(\tau_-^\infty \leq \tau_+^\infty\).
Then exactly one of the following cases holds:
\begin{enumlist}
\item[\bf (S0)]
$\tau_-^\infty = \tau_+^\infty= \infty$
\item[\bf (S1)]
$\tau_-^\infty = -\infty$ and $\tau_+^\infty = \infty$
\item[\bf (S2)]
$\tau_-^\infty = \tau_+^\infty = -\infty$
\item[\bf (D01)]
$\tau_-^\infty \in \mathbb{R}$ and \(\tau_+^\infty = \infty\)
\item[\bf (D12)]
$\tau_-^\nu = -\infty$ and $\tau_+^\nu \in \mathbb{R}$
\item[\bf (E012)] $\tau_\pm^\infty \in \mathbb{R}$ and \(\tau_-^\infty < \tau_+^\infty\)
\item[\bf (D02)] $\tau_-^\infty = \tau_+^\infty \in \mathbb{R}$
\end{enumlist}
Below, we will for each case specify the rescaled maps and establish their convergence to one of the bubble types in (3) as well as prove the energy concentration in (2).
Thus in all cases we will have proven (2) and (3) for the new blow-up point $z_{N+1}$, and after adding this point we will either have (0) satisfied or be able to find another blow-up point. Since $\hbar>0$ by Lemma~\ref{lem:hbar}, we will find at most $E/\hbar$ such blow-up points in this iteration before (0) holds.
Out of the seven blow-up scenarios just listed, the only case where our rescaling argument will be significantly different from standard rescaling arguments is (D02), in which we will need to appeal to the new analysis of Theorem~\ref{thm:nonfoldedstripshrink}. The rescaling argument in the cases (D01), (D12), (E012) is essentially the same as the standard process of ``bubbling off a disk'', since locally we can fold across the seam to obtain a pseudoholomorphic map to a product manifold.
Before we rescale to obtain bubbles, we record the key properties of the rescaled width function.
\begin{lemma} \label{lem:rescaledwidth}
Given a sequence \((f^\nu)_{\nu\in\mathbb{N}}\) of real-analytic functions shrinking obediently to zero, shifts $s^\nu{\to}s^\infty$, and rescaling factors $\alpha^\nu{\to}\infty$, the rescaled width functions
\(\widetilde f^\nu(s) := \alpha^\nu f^\nu(s^\nu + s/\alpha^\nu)\)
satisfy \(\cC^\infty_\loc(\mathbb{R})\) convergence
$$
\widetilde f^\nu - \widetilde f^\nu(0) \;\underset{\nu\to\infty}{\longrightarrow} \; 0 , \qquad
\widetilde f^\nu / \widetilde f^\nu(0) \;\underset{\nu\to\infty}{\longrightarrow} \; 1.
$$
Moreover, let \(F^\nu\) be the extension of \(f^\nu\) from Definition~\ref{def:obedience}, identify \((s,t)\in \mathbb{R}^2\) with \(z=s+it \in \mathbb{C} \), and set
\begin{align*}
\phi^\nu(s,t) &:= \bigl(s^\nu + s/\alpha^\nu, 2f^\nu\bigl(s^\nu + s/\alpha^\nu\bigr)\, t\bigr) \\
\psi^\nu(z) &:= s^\nu + z/\alpha^\nu - i F^\nu\bigl( s^\nu + z/\alpha^\nu \bigr).
\end{align*}
Then for any \(R>0\) and \(\nu\) sufficiently large, the maps \((\psi^\nu)^{-1} \circ \phi^\nu\) are well defined on \(B_R(0)\). In the special case \(\alpha^\nu := (2f^\nu(s^\nu))^{-1}\), the maps \((\psi^\nu)^{-1} \circ \phi^\nu\) converge \(\cC^\infty_\loc(\mathbb{R}^2,\mathbb{R}^2)\) to \((s,t) \mapsto (s, t+1/2)\).
\end{lemma}
\begin{proof}
The functions \(\widetilde f^\nu(s) - \widetilde f^\nu(0)\) resp.\ \(\widetilde f^\nu(s) / \widetilde f^\nu(0)\) are equal to 0 resp.\ 1 at \(s = 0\), so it suffices to show that \((\widetilde f^\nu(s) - \widetilde f^\nu(0))^{(k)}\) and \((\widetilde f^\nu(s) / \widetilde f^\nu(0))^{(k)}\) converge \(\cC^0_\loc\) to $0$
for every \(k \geq 1\). This convergence follows from the formulas
\begin{align*}
(\widetilde f^\nu - \widetilde f^\nu(0))^{(k)}(s) = (\alpha^\nu)^{1 - k} (f^\nu)^{(k)}(s^\nu + \tfrac s {\alpha^\nu}), \qquad
(\widetilde f^\nu / \widetilde f^\nu(0))^{(k)}(s) = \frac {(\alpha^\nu)^{-k} (f^\nu)^{(k)}(s^\nu + \tfrac s {\alpha^\nu})} {f^\nu(s^\nu)},
\end{align*}
the convergence \(s^\nu \to s^\infty\) and \(\alpha^\nu \to \infty\), and the obedient shrinking \(f^\nu \Rightarrow 0\) in Definition~\ref{def:obedience}.
The domain of $\phi^\nu$ is \([-\alpha^\nu(s^\nu + \rho), -\alpha^\nu(s^\nu - \rho)]\times \mathbb{R}\)
which contains $B_R(0)$ for sufficiently large $\nu$ since $s^\nu\to s^\infty\in (-\rho,\rho)$, $\alpha^\nu\to\infty$, and \(f^\nu \stackrel{\cC^\infty}{\longrightarrow} 0\). The image concentrates at $(s^\nu,0)$, more precisely we have
\begin{align*}
\phi^\nu(B_R(0)) \subset B_{R\delta^\nu}(s^\nu,0) , \qquad \delta^\nu:=\max\bigl\{ (\alpha^\nu)^{-1}, 2 \|f^\nu\|_{\cC^0} \bigr\} \to 0 ,
\end{align*}
since
$\bigl| \phi^\nu(s,t) - (s^\nu, 0) \bigr|^2 \leq s^2 / (\alpha^\nu)^2 + 4 \|f^\nu\|_{\cC^0}^2 t^2
\leq \bigl( \delta^\nu |(s,t)|\bigr)^2$.
Next, we claim that $B_{R\delta^\nu}(s^\nu,0)$ lies in the image of $\psi^\nu$ for sufficiently large $\nu$.
Indeed, given $y\in B_{R\delta^\nu}(s^\nu,0)$, we can solve
\begin{align} \label{eq:banach}
y= \psi^\nu(z)
\end{align}
iff there is a solution
$z \in -\alpha^\nu s^\nu + [-\alpha^\nu\rho,\alpha^\nu\rho]^2$ of \begin{align*}
z = \alpha^\nu\Bigl( y - s^\nu + i F^\nu\bigl( s^\nu + z/\alpha^\nu \bigr)\Bigr)=: H(z).
\end{align*}
The existence of a such a solution follows from Banach's fixed point theorem applied to \(H\). Indeed, \(H\) is a smooth map from \(-\alpha^\nu s^\nu + [-\alpha^\nu \rho, \alpha^\nu\rho]^2\) to itself since \(y \in B_{R\delta^\nu}(s^\nu,0)\) gives
\begin{align*}
\bigl| H(z) + \alpha^\nu s^\nu \bigr| &= \bigl|\alpha^\nu(y + i F^\nu(s^\nu + \tfrac z {\alpha^\nu}))\bigr|
\leq \alpha^\nu ( | s^\nu | + R\delta^\nu + \|F^\nu\|_{\cC^0})
< \alpha^\nu \rho
\end{align*}
for $\nu$ sufficiently large so that \(R\delta^\nu + \|F^\nu\|_{\cC^0} < \rho - |s^\nu| \). The latter holds for large $\nu$ since the left hand side converges to $0$ while $\rho - |s^\nu| \to \rho - |s^\infty| >0$.
Furthermore, \(H\) is a contraction mapping once $\nu$ is large enough so that $\|F^\nu\|_{\cC^1}<1$,
\begin{align*}
\bigl|H(z) - H(w)\bigr| &= \alpha^\nu \bigl| F^\nu\bigl( s^\nu + z/\alpha^\nu \bigr) - F^\nu\bigl( s^\nu + w/\alpha^\nu \bigr) \bigr| \leq \|F^\nu\|_{\cC^1}|z - w|.
\end{align*}
Therefore Banach's fixed point theorem guarantees a (unique) solution \(z \in -\alpha^\nu s^\nu + [-\alpha^\nu\rho,\alpha^\nu\rho]^2\) of \eqref{eq:banach}, which shows that for \(\nu\gg1\), the image of \(\psi^\nu\) contains \(\phi^\nu(B_R(0))\). To show that \((\psi^\nu)^{-1} \circ \phi^\nu\) is a well-defined element of \(\cC^\infty(B_R(0), \mathbb{R}^2)\), it remains to show that \(\psi^\nu\) is injective and has a Jacobian with nonvanishing determinant.
Injectivity again holds once $\|F^\nu\|_{\cC^1}<1$ since
\begin{align*}
\psi^\nu(z) = \psi^\nu(w) &\iff z - w = i\alpha^\nu\Bigl(F\bigl(s^\nu + \tfrac z {\alpha^\nu}\bigr) - F\bigl(s^\nu + \tfrac w {\alpha^\nu}\bigr)\Bigr) \\
&\implies |z - w| \leq \|F^\nu\|_{\cC^1}|z - w|.
\end{align*}
The Jacobian of \(\psi^\nu\) is given by \begin{align} \label{eq:psijacobian}
\mathfrak{J}^\nu(s,t) &:= \on{Jac}(\psi^\nu)(s+it) = (\alpha^\nu)^{-1}\left(\begin{array}{ll}
1 + \partial_s\on{im} F^\nu(s^\nu + \tfrac{s+it}{\alpha^\nu}) & \partial_t\on{im} F^\nu(s^\nu + \tfrac{s+it}{\alpha^\nu}) \\
-\partial_s\on{re} F^\nu(s^\nu + \tfrac{s+it}{\alpha^\nu}) & 1 - \partial_t\on{re} F^\nu(s^\nu + \tfrac{s+it}{\alpha^\nu})
\end{array}\right),
\end{align}
which for \(\nu\gg1\) has nonvanishing determinant since \(F^\nu \stackrel{\cC^\infty}{\longrightarrow} 0\).
This proves that \((\psi^\nu)^{-1} \circ \phi^\nu\) is a well-defined element of \(\cC^\infty(B_R(0), \mathbb{R}^2)\) for \(\nu\gg1\).
We now restrict to the case \(\alpha^\nu := (2f^\nu(s^\nu))^{-1}\). To establish the \(\cC^\infty_\loc(\mathbb{R}^2,\mathbb{R}^2)\)-convergence of \((\psi^\nu)^{-1} \circ \phi^\nu\) to the map \((s,t) \mapsto (s,t+\tfrac 1 2)\), we begin by noting their equality at $(s,t)=(0,-\tfrac 1 2)$,
\begin{align*}
\bigl((\psi^\nu)^{-1}\circ \phi^\nu\bigr)(0,-\tfrac 1 2) = (\psi^\nu)^{-1}(s^\nu,-f^\nu(s^\nu)) = (0,0).
\end{align*}
It remains to show \(\cC^\infty_\loc\)-convergence of the Jacobians
\(\on{Jac}\bigl((\psi^\nu)^{-1} \circ \phi^\nu\bigr) \to \on{Id}\).
Using the inverse of \eqref{eq:psijacobian} and abbreviating $Q^\nu(s,t) := s^\nu + \bigl((\psi^\nu)^{-1}\circ \phi^\nu\bigr)(s,t) /\alpha^\nu$ we have
\begin{align}
& \on{Jac}\bigl((\psi^\nu)^{-1}\circ \phi^\nu\bigr)(s,t)
=
\Bigl(\mathfrak{J}^\nu\bigl((\psi^\nu)^{-1}(\phi^\nu(s,t))\bigr)\Bigr)^{-1} \cdot\on{Jac}(\phi^\nu)(s,t) \nonumber \\
&\hspace{0.35in} =
\frac {\left(\begin{array}{ll}
1 - \partial_t\on{re} F^\nu\circ Q^\nu & -\partial_t\on{im} F^\nu\circ Q^\nu \\
\partial_s\on{re} F^\nu\circ Q^\nu & 1 + \partial_s\on{im} F^\nu\circ Q^\nu
\end{array}\right)\left(\begin{array}{ll}
1 & 0 \\
2(f^\nu)'(s^\nu + s/\alpha^\nu)t & 2\alpha^\nu f^\nu(s^\nu + s/\alpha^\nu)
\end{array}\right)} {(1 + \partial_s\on{im} F^\nu\circ Q^\nu)(1 - \partial_t\on{re} F^\nu\circ Q^\nu) + \partial_t\on{im} F^\nu\circ Q^\nu\partial_s\on{re} F^\nu\circ Q^\nu}.
\label{eq:compjacobian}
\end{align}
The \(\cC^\infty\)-convergence \(F^\nu \to 0\) implies that the first matrix divided by the denominator
converges \(\cC^0_\loc\) to the identity. In fact, this is \(\cC^k_\loc\)-convergence if the derivatives of $Q^\nu$ up to order $k$ are uniformly bounded on compact sets.
In the second matrix we have $2(f^\nu)'(s^\nu + s/\alpha^\nu)t \to 0$ in \(\cC^\infty_\loc\) by the \(\cC^\infty\)-convergence \(f^\nu\to0\) and $(\alpha^\nu)^{-1}\to 0$, and the bottom right entry
$\tfrac {f^\nu(s^\nu + 2f^\nu(s^\nu)s)} {f^\nu(s^\nu)} =\tfrac {\widetilde f^\nu(s)} {\widetilde f^\nu(0)}\to 1$
converges already in \(\cC^\infty_\loc\) by the first statement of the current lemma.
This proves \(\cC^0_\loc\) convergence of the Jacobians and thus \(\cC^1_\loc\)-convergence of the maps \((\psi^\nu)^{-1}\circ\phi^\nu\). Since $Q^\nu$ is given in terms of these maps and $(\alpha^\nu)^{-1}\to 0$, we conclude that its derivatives are uniformly bounded on compact sets, thus the convergence of the Jacobians is in \(\cC^1_\loc\), which implies \(\cC^2_\loc\)-convergence of the maps \((\psi^\nu)^{-1}\circ\phi^\nu\). Iterating this argument proves the claimed \(\cC^\infty_\loc\) convergence.
\end{proof}
\noindent
Continuing with the proof of Theorem~\ref{thm:rescale},
the nontrivial bubbles claimed in (3) are now obtained as follows:
\medskip
\begin{itemlist}
\item[\bf (S1):]
We will obtain a {\bf sphere bubble in \(\mathbf{M_1}\)} by rescaling \begin{align*}
w_1^\nu(s,t):= v^\nu_1(s^\nu + \tfrac s {R^\nu} , t^\nu + \tfrac t {R^\nu} )
\end{align*}
to define maps \begin{align*}
w_1^\nu\colon U_1^\nu := \bigl\{ (s,t) \: | \: -R^\nu(\rho + s^\nu) \leq s \leq R^\nu(\rho - s^\nu), \: -\widetilde f^\nu(s) - R^\nu t^\nu \leq t \leq \widetilde f^\nu(s) - R^\nu t^\nu \bigr\} \;\longrightarrow\; M_1.
\end{align*}
The map \(w_1^\nu\) is pseudoholomorphic with respect to \(\widetilde J_1^\nu(t):=J_1^\nu(s^\nu + s/R^\nu , t^\nu + t/R^\nu)\); due to the convergence \(R^\nu \to \infty\), these almost complex structures converge in
\(\cC^\infty_\loc(\mathbb{R}^2)\) as \(\nu \to \infty\)
to the constant almost complex structure \(\widetilde J_1^\infty:=J_1^\infty(s^\infty,t^\infty)\). By construction, the maps \(w_1^\nu\) satisfy both upper and lower gradient bounds: \begin{gather}
|\delta w_1^\nu(0) | = \tfrac1{R^\nu} |\delta v_1^\nu(s^\nu, t^\nu)| = \tfrac 1 {R^\nu}|\delta \underline v(s^\nu, t^\nu)| = 1 , \label{eq:w1gradbound} \\
\sup_{(s,t)\in B_{R^\nu\epsilon^\nu}(0)} |\delta w_1^\nu(s,t)|
\leq \sup_{(s,t)\in B_{\epsilon^\nu}(s^\nu,t^\nu)} \tfrac1{R^\nu} |\delta \underline{v}^\nu(s,t)|
\leq 2 , \nonumber
\end{gather}
where the second equality in the top line follows for large \(\nu\) from the assumption \(\tau_\pm^\nu \to \pm\infty\). The containment \(s^\nu \to s^\infty \in (-\rho,\rho)\) implies that the left resp.\ right bounds \(R^\nu(\mp \rho - s^\nu)\) of \(U_1^\nu\) have limits \(-\infty\) resp.\ \(\infty\); furthermore, the assumption \(\tau_\pm^\infty = \pm\infty\) and Lemma~\ref{lem:rescaledwidth} implies that the lower resp.\ upper bounds \(-\widetilde f^\nu(s) - R^\nu t^\nu\) resp.\ \(\widetilde f^\nu(s) - R^\nu t^\nu\) of \(U_1^\nu\) converge \(\cC^\infty_\loc\) to \(-\infty\) resp.\ \(\infty\). Hence the maps \(w_1^\nu\) are defined with uniformly bounded differential on balls centered at \(0\) of radii tending to infinity.
Standard compactness for pseudoholomorphic maps (e.g.\ \cite[Appendix B]{ms:jh}\footnote{\label{foot:bounds}
If noncompact symplectic manifolds are involved, then one needs to establish \(\cC^0\)-bounds on the maps before ``standard Gromov compactness'' can be quoted.
}) implies that a subsequence, still denoted by \((w_1^\nu)\), converges \(\cC^\infty_\loc\) to a \(\widetilde J_1^\infty\)-holomorphic map \(w_1^\infty\) defined on \(\mathbb{R}^2\). Its energy is bounded by \(E\), so after removing the singularity (using \cite[Theorem~4.1.2(i)]{ms:jh}) we obtain a \(\widetilde J_1^\infty\)-holomorphic sphere \(\overline w_1^\infty\colon S^2 \to M_1\), which is nonconstant by \eqref{eq:w1gradbound}.
Rescaling invariance of the energy and $\cC^\infty_\loc$-convergence imply energy concentration:
\begin{align*}
\liminf_{\nu\to\infty} \int_{B_{\epsilon^\nu}(s^\nu,t^\nu)} \tfrac 1 2\bigl| \delta\underline{v}^\nu\bigr|^2
\geq \liminf_{\nu\to\infty} \int_{B_{\epsilon^\nu}(s^\nu, t^\nu)} v_1^{\nu\;*} \omega_1 \geq\liminf_{\nu\to\infty} \int_{B_{R^\nu\epsilon^\nu}(0) \cap U_1^\nu} w_1^{\nu\;*} \omega_1
\;\;&\geq\; \int_{\mathbb{R}^2} w_1^{\infty\;*} \omega_1 \\
&\geq\; \hbar_{S^2} > 0.
\end{align*}
\item[\bf (S0,S2):] In complete analogy to (S1), rescaling by \(w_\ell^\nu(s,t) := v_\ell^\nu(s^\nu + s/R^\nu, t^\nu + t/R^\nu)\) yields a nonconstant pseudoholomorphic {\bf sphere in \(\mathbf{M_\ell}\)} and with energy concentration of at least \(\hbar_{S^2}\).
\item[\bf (D01):]
We will obtain a {\bf disk bubble in \(\mathbf{ M_0^-\times M_1}\) with boundary on \(\mathbf{L_{01}}\)} by rescaling
\begin{align*}
w_\ell^\nu(z) := v_\ell^\nu \Bigl( s^\nu + \tfrac s {R^\nu}, -f^\nu\bigl(s^\nu + \tfrac s {R^\nu}\bigr) + \tfrac t {R^\nu} \Bigr)
\end{align*}
to define maps \begin{align*}
w_0^\nu&\colon \bigl\{(s,t) \:|\: -R^\nu(\rho+s^\nu) \leq R^\nu(\rho - s^\nu), \: R^\nu\bigl(-\rho + f^\nu(s^\nu + \tfrac s {R^\nu})\bigr) \leq t \leq 0\bigr\} \;\longrightarrow\; M_0, \\
w_1^\nu&\colon \bigl\{(s,t) \:|\: -R^\nu(\rho+s^\nu) \leq R^\nu(\rho - s^\nu), \: 0 \leq t \leq 2R^\nu f^\nu(s^\nu + \tfrac s {R^\nu})\bigr\} \;\longrightarrow\; M_1.
\end{align*}
In the case of straight seams $t=\pm f^\nu(s) = \pm \delta^\nu$ the rescaled maps easily pair to maps
$w_{01}^{\nu}: (s,t)\mapsto \bigl(w_0^\nu(-s,t),w_1^\nu(s,t)\bigr)\in M_0^-\times M_1$ defined on increasing domains \(B_{r^\nu}(0) \cap \H\) with $r^\nu\to\infty$
in half space, with boundary values in $L_{01}$, which by standard arguments converge and extend to a pseudholomorphic disk.
The squiggly seams require an easier version of the arguments in (E012) to establish convergence \(w_0^\nu \to w_0^\infty\), \(w_1^\nu \to w_1^\infty\) in \(\cC_\loc^\infty(-\H)\) resp.\ \(\cC_\loc^\infty(\H)\) to nonconstant \(J_\ell^\infty(s^\infty,0)\)-holomorphic maps satisfying the Lagrangian seam condition \begin{align*}
(w_0^\infty(s,0), w_1^\infty(s,0)) \in L_{01} \quad \forall s \in \mathbb{R}.
\end{align*}
Then \(w_{01}^\infty(s,t) := (w_0^\infty(s,-t), w_1^\infty(s,t))\colon \H \to M_0^-\times M_1\) is a nonconstant \(\widetilde J_{01}^\infty := (-J_0^\infty(s^\infty,0)) \times J_1^\infty(s^\infty,0)\)-holomorphic map with \(w_{01}^\infty(\partial\H) \subset L_{01}\).
Its energy is bounded by \(E\), so after removing the singularity (using e.g.\ \cite[Theorem~4.1.2(ii)]{ms:jh}) we obtain a nonconstant \(\widetilde J_{01}^\infty\)-holomorphic disk \(\overline w_{01}^\infty\colon D^2 \to M_0^-\times M_1\).
Energy quantization in the case of straight seams is given by
\begin{align*}
\liminf_{\nu\to\infty} \int_{B_{\epsilon^\nu}(s^\nu,t^\nu)} \tfrac 1 2\bigl| \delta\underline{v}^\nu\bigr|^2
\geq \liminf_{\nu\to\infty} \sum_{\ell \in \{0,1\}} \int_{B_{\epsilon^\nu}(s^\nu,t^\nu)} v_\ell^{\nu\;*} \omega_\ell &\geq \liminf_{\nu\to\infty} \int_{B_{r^\nu}(0)\cap\H} w_{01}^{\nu\;*} \bigl( (-\omega_0) \oplus \omega_1 \bigr) \\
&\hspace{0.5in} \geq \int w_{01}^{\infty\;*} \bigl( (-\omega_0) \oplus \omega_1 \bigr) \geq \hbar_{D^2},
\end{align*}
and in the general case just requires more refined choices of domains as in (E012).
\item[\bf (D12):] In complete analogy to (D01), rescaling yields a nonconstant pseudoholomorphic {\bf disk in \break \(\mathbf{ M_1^-\times M_2}\) with boundary on \(\mathbf{ L_{12}}\)} and energy concentration of at least \(\hbar_{D^2}\).
\item[\bf (E012):] We will obtain a {\bf figure eight bubble between \(\mathbf{L_{01}}\) and \(\mathbf{L_{12}}\)} by a rescaling that in the case of straight middle strips of width $2 f^\nu \equiv \delta^\nu \to 0$ amounts to
$w_\ell^\nu(s,t) = v_\ell^\nu \bigl( s^\nu + \delta^\nu s, \delta^\nu t \bigr)$ for $\ell=0,1,2$.
In the general case of squiggly strips, we straighten out the strip by using an $s$-dependent rescaling factor in the $t$ variable:
\begin{align}\label{eq:wrestle}
w_\ell^\nu(s,t) := v_\ell^\nu \bigl(\phi^\nu(s,t) \bigr), \qquad \phi^\nu(s,t) := \Bigl(s^\nu + 2f^\nu(s^\nu)s, 2f^\nu\bigl(s^\nu + 2f^\nu(s^\nu)s\bigr)\, t\Bigr).
\end{align}
Note that $\phi^\nu$ is a diffeomorphism between open subsets of $\mathbb{R}^2$ since $f^\nu>0$, and that it pulls back the seams $\cS_\pm =\bigl\{(x,y)\in\mathbb{R}^2 \,|\, y=\pm f^\nu(x)\bigr\}$ to straight seams $(\phi^\nu)^{-1}(\cS_\pm) = \bigl\{(s,t) \in \mathbb{R}^2\,|\, t=\pm\tfrac 12 \bigr\}$ as in Definition~\ref{def:8} of the figure eight bubble.
Moreover, the rescaled quilt maps $(w_\ell^\nu)_{\ell=0,1,2}$ have the total domain
\begin{align*}
(\phi^\nu)^{-1}\bigl( (-\rho,\rho)^2 \bigr)
= \Bigl\{ (s,t)\in\mathbb{R}^2 \,\Big| \, \tfrac{-\rho-s^\nu}{2f^\nu(s^\nu)} < s < \tfrac{\rho - s^\nu}{2f^\nu(s^\nu)}, \: |t| < \tfrac{\rho}{2f^\nu(s^\nu + 2f^\nu(s^\nu)s)} \Bigr\}
\end{align*}
Since \(s^\nu \to s^\infty \in (-\rho,\rho)\) and \(f^\nu \stackrel{\cC^\infty}{\longrightarrow} 0\), we can find a sequence of radii \(r^\nu \to \infty\) so that these domains contain the balls $B_{r^\nu}(0)$.
In the case of straight seams the maximal radii are $r^\nu = (\rho-|s^\nu|)/\delta^\nu$, but we may also choose smaller radii \(r^\nu \to \infty\) so that in addition $r^\nu\delta^\nu\to 0$. In general we choose \(r^\nu \to \infty\) so that $B_{r^\nu}(0)\subset (\phi^\nu)^{-1}\bigl( (-\rho,\rho)^2 \bigr)$ and
\begin{equation} \label{a}
r^\nu \max_{s \in [-\rho,\rho]} \bigl( f^\nu(s) + (f^\nu)'(s)\bigr) \underset{\nu\to\infty}{\longrightarrow} 0 .
\end{equation}
Finally, we wish to choose $r^\nu\to\infty$ such that in addition the inclusion \(\phi^\nu\bigl(B_{r^\nu}(0)\bigr) \subset B_{\epsilon^\nu}(s^\nu,t^\nu)\) ensures that the gradient bounds \eqref{eq:vbounds} transfer to the rescaled maps.
For that purpose first note that for sufficiently large $\nu\in\mathbb{N}$ from \eqref{a} we also obtain the estimate
\begin{align}\label{c}
\max_{s \in [-r^\nu,r^\nu]} 2f^\nu(s^\nu + 2f^\nu(s^\nu)s) \;\leq\; 3f^\nu(s^\nu) .
\end{align}
Indeed, for \(s \in [-r^\nu, r^\nu]\) and $\nu$ sufficiently large such that $r^\nu \|(f^\nu)'\|_{\cC^0([-\rho,\rho])}\leq \frac 14$ we have
\begin{align*}
f^\nu(s^\nu + 2f^\nu(s^\nu)s) &\leq f^\nu(s^\nu) + \int_0^s 2f^\nu(s^\nu)(f^\nu)'(s^\nu + 2f^\nu(s^\nu)s) \,ds \\
&\leq f^\nu(s^\nu)\bigl(1 + 2r^\nu \|(f^\nu)'\|_{\cC^0([-\rho,\rho])}\bigr)
\;\leq\; \tfrac 32 f^\nu(s^\nu).
\end{align*}
Next, for \((s,t) \in B_{r^\nu}(0)\) we obtain
\begin{align*}
\left|\bigl(s^\nu + 2f^\nu(s^\nu)s, 2f^\nu(s^\nu + 2f^\nu(s^\nu)s)t\bigr) - (s^\nu, t^\nu)\right| &\leq \left|(2f^\nu(s^\nu)s, 2f^\nu(s^\nu + 2f^\nu(s^\nu)s)t)\right| + |(0,t^\nu)| \\
&\leq 3r^\nu f^\nu(s^\nu) + |t^\nu| \\
&= \frac {3 r^\nu(\tau_+^\nu - \tau_-^\nu) + |\tau_+^\nu + \tau_-^\nu|} {2 R^\nu\epsilon^\nu} \; \epsilon^\nu
\end{align*}
from \eqref{c} and the identities
\begin{equation}\label{eq:tau}
f^\nu(s^\nu) = \frac{\tau_+^\nu - \tau_-^\nu}{2 R^\nu},
\qquad
t^\nu = - \frac{\tau_+^\nu + \tau_-^\nu}{2 R^\nu}.
\end{equation}
Thus to obtain the inclusion \(\phi^\nu\bigl(B_{r^\nu}(0)\bigr) \subset B_{\epsilon^\nu}(s^\nu,t^\nu)\) for large $\nu$ it suffices to replace the above $r^\nu$ by a possibly smaller sequence $r^\nu\to\infty$ so that $3 r^\nu(\tau_+^\nu - \tau_-^\nu) + |\tau_+^\nu + \tau_-^\nu| < 2 R^\nu\epsilon^\nu$ for large $\nu$.
This is possible since \(R^\nu\epsilon^\nu \to \infty\) and \(\tau_\pm^\nu \to \tau_\pm^\infty \in \mathbb{R}\).
So from now on, after dropping finitely many terms from the sequence, we are working with a sequence of rescaled quilt maps \eqref{eq:wrestle} together with a sequence \(r^\nu \to \infty\) so that we have the inequalities \eqref{a}, \eqref{c}, and the inclusions
\begin{equation}\label{bd}
\phi^\nu\bigl(B_{r^\nu}(0)\bigr) \subset B_{\epsilon^\nu}(s^\nu,t^\nu) \cap (-\rho,\rho)^2 .
\end{equation}
Thus, restricting the maps from \eqref{eq:wrestle} to the balls $B_{r^\nu}(0)$ of increasing radius amounts to considering the quilt map \(w_\ell^\nu\colon W_\ell^\nu \to M_\ell\) with the domains
\begin{align*}
W_0^\nu := B_{r^\nu}(0) \cap \bigl(\mathbb{R} \times (-\infty, -\tfrac 1 2]\bigr), \:\:\: W_1^\nu := B_{r^\nu}(0) \cap \bigl(\mathbb{R} \times [-\tfrac 1 2, \tfrac 1 2]\bigr), \:\:\: W_2^\nu := B_{r^\nu}(0) \cap \bigl(\mathbb{R} \times [\tfrac 1 2, \infty)\bigr).
\end{align*}
These maps are \((\widetilde J_\ell^\nu, j^\nu)\)-holomorphic, where \(\widetilde J_\ell^\nu = J_\ell^\nu \circ \phi^\nu\) are the almost complex structures on $M_\ell$ with appropriately rescaled domain dependence, and \(j^\nu\) is the complex structure
\begin{align*}
j^\nu(s,t) &:= \bigl(\delta\phi^\nu(s,t)\bigr)^{-1}\circ j_0 \circ \delta\phi^\nu(s,t) \\
&= \left(\begin{array}{ll}
2f^\nu(s^\nu) & 0 \\
4f^\nu(s^\nu)(f^\nu)'(s^\nu + 2f^\nu(s^\nu)s)\,t & 2f^\nu(s^\nu + 2f^\nu(s^\nu)s)
\end{array}\right)^{-1}\left(\begin{array}{ll}
0 & -1 \\
1 & 0
\end{array}\right)\times \\
&\hspace{2in} \times\left(\begin{array}{ll}
2f^\nu(s^\nu) & 0 \\
4f^\nu(s^\nu)(f^\nu)'(s^\nu + 2f^\nu(s^\nu)s)\,t & 2f^\nu(s^\nu + 2f^\nu(s^\nu)s)
\end{array}\right) \\
& = \left(\!\!\begin{array}{ll}
(2f^\nu(s^\nu))^{-1}& 0 \\
- \frac{(f^\nu)'(\cdots)\,t}{f^\nu(s^\nu + \cdots)} & \bigl(2 f^\nu(s^\nu + \cdots)\bigr)^{-1}
\end{array}\!\!\right)\!\!
\left(\!\!\begin{array}{ll}
- 4f^\nu(s^\nu)(f^\nu)'(s^\nu + \cdots)\,t & - 2f^\nu(s^\nu + 2f^\nu(s^\nu)s) \\
2f^\nu(s^\nu) & 0
\end{array}\!\!\right)
\\
&= \left(\begin{array}{cc}
-2t(f^\nu)'(s^\nu + 2f^\nu(s^\nu)s) & -\tfrac {f(s^\nu+2f^\nu(s^\nu)s)} {f^\nu(s^\nu)} \\
\tfrac {f^\nu(s^\nu)\bigl( 4t^2(f^\nu)'(s^\nu+2f^\nu(s^\nu)s)^2 + 1\bigr)} {f^\nu(s^\nu+2f^\nu(s^\nu)s)} & 2t(f^\nu)'(s^\nu+2f^\nu(s^\nu)s)
\end{array}\right) \\
&= \left( \begin{array}{cc}
-\tfrac {t(\widetilde f^\nu)'(s)} {\widetilde f^\nu(0)} & -\tfrac {\widetilde f^\nu(s)} {\widetilde f^\nu(0)} \\
\tfrac {t^2(\widetilde f^\nu)'(s)^2 + \widetilde f^\nu(0)^2} {\widetilde f^\nu(0)\widetilde f^\nu(s)} & \tfrac {t(\widetilde f^\nu)'(s)} {\widetilde f^\nu(0)}
\end{array} \right),
\end{align*}
where we abbreviate
$\widetilde f^\nu(s) := f^\nu(s^\nu + 2f^\nu(s^\nu)s) / 2f^\nu(s^\nu)$.
Note that Lemma~\ref{lem:rescaledwidth} with \(\alpha^\nu := (2f^\nu(s^\nu))^{-1}\) implies \(j^\nu \to i\) in \(\cC^\infty_\loc\),
and the almost complex structures also converge \(\widetilde J_\ell^\nu \to J_\ell^\infty(s^\infty,0)\) in \(\cC^\infty_\loc\)
since $\phi^\nu(s,t) \to 0$ for any fixed $(s,t)$.
Moreover, the maps \(w_\ell^\nu\) satisfy the Lagrangian seam conditions \begin{align*}
(w_0^\nu(s, -\tfrac 1 2 ), w_1^\nu(s, -\tfrac 1 2 )) \in L_{01}, \quad (w_1^\nu(s,\tfrac 1 2), w_2^\nu(s, \tfrac 1 2)) \in L_{12} \qquad \forall \: s \in (-r^\nu, r^\nu).
\end{align*}
The gradient blowup at $(s^\nu,t^\nu)$ in \eqref{eq:vbounds} translates into lower bounds on the gradient \(|\delta w^\nu| := \bigl(|\delta w_0^\nu| + |\delta w_1^\nu| + |\delta w_2^\nu|\bigr)^{1/2}\) at $t=\widetilde t^\nu := \tfrac{t^\nu}{2f^\nu(s^\nu)} = \tfrac {-\tau_+^\nu - \tau_-^\nu} {2(\tau_+^\nu - \tau_-^\nu)} \to \tfrac {-\tau_+^\infty-\tau_-^\infty}{2(\tau_+^\infty-\tau_-^\infty)} \in \mathbb{R}$, since $\phi^\nu(0,\tfrac{t^\nu}{2f^\nu(s^\nu)})=(s^\nu,t^\nu)$ and hence
\begin{align*}
\bigl|\delta \underline w^\nu( 0, \widetilde t^\nu) \bigr|^2
&=
\sum_\ell \bigl|2f^\nu(s^\nu)\partial_sv^\nu_\ell(s^\nu,t^\nu) + 4\widetilde t^\nu f^\nu(s^\nu)(f^\nu)'(s^\nu)\partial_tv^\nu_\ell(s^\nu,t^\nu)\bigr|^2 + \bigl|2f^\nu(s^\nu)\partial_tv_\ell^\nu(s^\nu,t^\nu)\bigr|^2 \nonumber \\
&\geq
3\bigl|\delta\underline v^\nu(s^\nu,t^\nu)\bigr|^2, \quad \nu \gg 0.
\end{align*}
Now by $t^\nu\to t^\infty$, the obedient convergence $f^\nu \Rightarrow 0$ in Definition~\ref{def:obedience}, and \eqref{eq:tau} we obtain a nonzero lower bound for sufficiently large $\nu$,
\begin{equation} \label{eq:E012bounds}
\bigl|\delta \underline w^\nu( 0, \widetilde t^\nu) \bigr|^2
\;\geq \;
2 f^\nu(s^\nu)^2|\delta\underline v^\nu(s^\nu,t^\nu)|^2
\;=\;
\tfrac 1 2 (\tau^\nu_+ - \tau^\nu_-)^2
\;\geq\;
\tfrac 1 4(\tau^\infty_+ - \tau^\infty_-)^2 > 0.
\end{equation}
Next we use \eqref{a}--\eqref{bd} to transfer the upper bound in \eqref{eq:vbounds}
to the gradient of the rescaled maps for sufficiently large $\nu$,
\begin{align}
&\sup_{(s,t) \in B_{r^\nu}(0)} |\delta\underline w^\nu(s,t)|^2 \nonumber\\
&\qquad=
\sup_{(s,t) \in B_{r^\nu}(0)} \sum_\ell
\begin{array}{r}
\bigl| 2f^\nu(s^\nu)\partial_sv_\ell^\nu(\phi^\nu(s,t)) + 4tf^\nu(s^\nu)(f^\nu)'(s^\nu+2f^\nu(s^\nu)s)\partial_tv_\ell^\nu(\phi^\nu(s,t))\bigr|^2 \nonumber\\
+ \phantom{\Big|} \bigl|2f^\nu(s^\nu + 2f^\nu(s^\nu)s)\partial_tv_\ell^\nu(\phi^\nu(s,t))\bigr|^2
\end{array}
\nonumber\\
&\qquad\leq
\Bigl( \bigl( 2f^\nu(s^\nu) + 4 r^\nu f^\nu(s^\nu) \max_{s\in[-\rho,\rho]} |(f^\nu)'(s)| \bigr)^2
+ \bigl(3f^\nu(s^\nu)\bigr)^2
\Bigr)
\sup_{(x,y) \in B_{\epsilon^\nu}(s^\nu,t^\nu)} |\delta\underline v(x,y)|^2 \nonumber\\
&\qquad \leq
18f^\nu(s^\nu)^2
\sup_{(x,y) \in B_{\epsilon^\nu}(s^\nu,t^\nu)} |\delta\underline v(x,y)|^2
\;\leq\;18 (\tau_+^\infty - \tau_-^\infty)^2 .
\label{eq:wbound}
\end{align}
Using these gradient bounds (and the compact boundary conditions in the case of noncompact symplectic manifolds), standard Gromov compactness asserts that after passing to a subsequence, \(w_0^\nu\) resp.\ \(w_1^\nu\) resp.\ \(w_2^\nu\) converge \(\cC^\infty_\loc\) on the interior of \(\mathbb{R}\times(-\infty, -1/2]\) resp.\ \(\mathbb{R} \times [-1/2,1/2]\) resp.\ \(\mathbb{R} \times [1/2,\infty)\).
To obtain convergence up to the seams $\mathbb{R}\times\{\pm 1/2\}$, we will first prove convergence of somewhat differently rescaled maps.
More precisely, to prove convergence of \(w_0^\nu, w_1^\nu\) near the \(L_{01}\)-seam \(\mathbb{R} \times \{-1/2\}\) we consider the maps
$$
u_0^\nu\colon \; U_0^\nu := (-r^\nu, r^\nu) \times (-1/2, 0] \; \to \; M_0,
\qquad
u_1^\nu\colon \; U_1^\nu := (-r^\nu, r^\nu) \times [0, 1/2) \; \to \; M_1
$$
given by rescaling $u_\ell^\nu(s,t) := v_\ell^\nu\bigl(\psi^\nu(s+it)\bigr)$ with the holomorphic map
$$
\psi^\nu(z) := s^\nu + 2f^\nu(s^\nu)z - i F^\nu\bigl( s^\nu + 2f^\nu(s^\nu)z \bigr),
$$
where we identify \((s,t)\in \mathbb{R}^2\) with \(z=s+it \in \mathbb{C} \), and \(F^\nu\) is the extension of \(f^\nu\) from Definition~\ref{def:obedience}.
To see that $\psi^\nu$ is well-defined on \(U_0^\nu \cup U_1^\nu\) for sufficiently large \(\nu\), despite $f^\nu$ resp.\ $F^\nu$ only being defined on \([-\rho,\rho]\) resp.\ \([-\rho,\rho]^2\),
note that \(s^\nu \to s^\infty \in (-\rho,\rho)\) and \(r^\nu f^\nu(s^\nu) \to 0\) by \eqref{a}.
To ensure that \(u_\ell^\nu\) is well-defined for large \(\nu\) and $\ell=0,1$ we moreover need to verify that $\psi^\nu(U_\ell^\nu)$ lies in the domain of $v_\ell^\nu$.
Indeed, firstly we have \(\psi^\nu(U_0^\nu\cup U_1^\nu) \subset (-\rho,\rho)^2\) for large \(\nu\) by \(s^\nu \to s^\infty \in (-\rho,\rho)\), \eqref{a}, and \(F^\nu \stackrel{\cC^\infty}{\longrightarrow} 0\).
Secondly, the bounds required by the seams are
\begin{align}
\on{im} \psi^\nu(s+it) &\leq -f^\nu(\on{re} \psi^\nu(s+it)) \;\quad\forall\: s+it \in U_0^\nu,
\label{eq:U0seambound} \\
\bigl| \on{im} \psi^\nu(s+it) \bigr| &\leq f^\nu(\on{re} \psi^\nu(s+it)) \qquad\forall\: s+it \in U_1^\nu
\label{eq:U1seambound}
\end{align}
for large \(\nu\).
For that purpose we rewrite
\begin{align*}
& \on{im} \psi^\nu(s+it) \pm f^\nu(\on{re} \psi^\nu(s+it)) \\
&=
2tf^\nu(s^\nu) - \on{re} F^\nu\bigl(s^\nu+2f^\nu(s^\nu)(s+it)\bigr) \pm f^\nu\Bigl(s^\nu + 2 f^\nu(s^\nu) s
+ \on{im} F^\nu \bigl(s^\nu+2f^\nu(s^\nu)(s+it)\bigr) \Bigr) \\
&= (2t-1\pm1)f^\nu\bigl(s^\nu+2f^\nu(s^\nu)s\bigr) + E_1^\nu(s,t) - \on{re} E_2^\nu(s,t) \pm E_3^\nu(s,t).
\end{align*}
with
\begin{align*}
E_1^\nu(s,t) &= 2t \bigl( f^\nu(s^\nu) - f^\nu\bigl(s^\nu+2f^\nu(s^\nu)s\bigr) \bigr), \\
E_2^\nu(s,t) &= F^\nu\bigl(s^\nu+2f^\nu(s^\nu)(s+it)\bigr)-F^\nu\bigl(s^\nu+2f^\nu(s^\nu)s\bigr) , \\
E_3^\nu(s,t) &= f^\nu\Bigl(s^\nu+2f^\nu(s^\nu)s +\on{im} F^\nu\bigl(s^\nu+2f^\nu(s^\nu)(s+it)\bigr)\Bigr) - f^\nu\bigl(s^\nu+2f^\nu(s^\nu)s\bigr).
\end{align*}
We can bound \(E_1^\nu,E_2^\nu,E_3^\nu\) for large \(\nu\) and \((s,t)\in U_0^\nu\cup U_1^\nu\subset B_{2r^\nu}(0)\),
using the obedient convergence \(f^\nu \Rightarrow 0\) from Definition~\ref{def:obedience},
\begin{align}
|E_1^\nu(s,t)| &\leq 4t \|(f^\nu)'\|_{\cC^0}f^\nu(s^\nu)s \leq 8 t r^\nu\|(f^\nu)'\|_{\cC^0} C_0 f^\nu\bigl(s^\nu+2f^\nu(s^\nu)s\bigr),
\end{align}
\begin{align}
|E_2^\nu(s,t)| &\leq \|DF^\nu\|_{\cC^0} 2t f^\nu(s^\nu) \leq 2t \|DF^\nu\|_{\cC^0} C_0 f^\nu\bigl(s^\nu+2f^\nu(s^\nu)s\bigr),
\end{align}
\begin{align}
|E_3^\nu(s,t)|
&\leq \|(f^\nu)'\|_{\cC^0}\bigl|\on{im} F^\nu(s^\nu+2f^\nu(s^\nu)(s+it))\bigr| \label{eq:E3bound} \\
&= \|(f^\nu)'\|_{\cC^0}\Bigl|\on{im} \bigl( F^\nu(s^\nu+2f^\nu(s^\nu)(s+it)) - F^\nu(s^\nu+2f^\nu(s^\nu)s) \bigr)\Bigr| \nonumber \\
&\leq \|(f^\nu)'\|_{\cC^0} \|DF^\nu\|_{\cC^0} 2 t f^\nu(s^\nu) \nonumber \\
&\leq 2t \|(f^\nu)'\|_{\cC^0} \|DF^\nu\|_{\cC^0} C_0 f^\nu\bigl(s^\nu+2f^\nu(s^\nu)s\bigr).
\nonumber
\end{align}
Then by \eqref{a} and \(F^\nu \stackrel{\cC^\infty}{\longrightarrow} 0\) we obtain for sufficiently large $\nu$\begin{align*}
\bigl| \on{im} \psi^\nu(s+it) \pm f^\nu(\on{re} \psi^\nu(s+it))
- (2t-1\pm1)f^\nu\bigl(s^\nu+2f^\nu(s^\nu)s\bigr) \bigr|
\leq tf^\nu\bigl(s^\nu+2f^\nu(s^\nu)s\bigr).
\end{align*}
To check \eqref{eq:U0seambound} from this, recall that \(t \in (-1/2,0]\) on $U^\nu_0$ so that
$$
\on{im} \psi^\nu(s+it) + f^\nu(\on{re} \psi^\nu(s+it)) \leq
t f^\nu\bigl(s^\nu+2f^\nu(s^\nu)s\bigr) \leq 0 .
$$
Similarly, on $U^\nu_1$ we have \(t \in [0,1/2)\) so that \eqref{eq:U1seambound} follows from
\begin{align*}
f^\nu(\on{re}\psi^\nu(s+it)) \pm \on{im}\psi^\nu(s+it)
\geq
\bigl(1 - t \pm (2t-1)\bigr) f^\nu\bigl(s^\nu+2f^\nu(s^\nu)s\bigr) \geq 0
\end{align*}
since $f^\nu> 0$, $1-t-2t+1 = 2 - 3t > 0$ and $1-t+2t-1 = t\geq 0$.
Now that \(u_0^\nu, u_1^\nu \) are well-defined, note that the advantage of this rescaling is that the resulting maps are pseudoholomorphic with respect to the standard complex structure \(i\) on their domains (viewed as subsets of $\mathbb{C}$). On the other hand it straightens out only one seam,
\begin{align*}
\psi^\nu \bigl( (-r^\nu,r^\nu)\times\{0\} \bigr)
& =\bigl\{ \bigl(
s^\nu + 2f^\nu(s^\nu)s , - f^\nu\bigl( s^\nu + 2f^\nu(s^\nu)s \bigr)
\bigr) \,\big|\, s\in(-r^\nu,r^\nu) \bigr\} \\
&\subset
\bigl\{(x,y)\in\mathbb{R}^2 \,|\, y=- f^\nu(x)\bigr\}
\;=\;\cS_- ,
\end{align*}
so that we obtain the Lagrangian seam condition
\begin{align*}
(u_0^\nu(s, 0 ), u_1^\nu(s, 0 )) \in L_{01} \qquad \forall \: s \in (-r^\nu, r^\nu),
\end{align*}
but the $L_{12}$-condition would hold on the curved seam $(\psi^\nu)^{-1}(\cS_+)$.
However, we use this rescaling only to prove convergence near the \(L_{01}\)-seam, and to prove convergence of \(w_1^\nu, w_2^\nu\) near the \(L_{12}\)-seam would use the rescaling $z\mapsto s^\nu + 2f^\nu(s^\nu)z + i F^\nu\bigl( s^\nu + 2f^\nu(s^\nu)z \bigr)$.
More precisely, we will below prove $\cC^\infty_\loc$-convergence of $u_\ell^\nu$ near $\mathbb{R}\times\{0\}$ since this yields control of the maps of interest $w_\ell^\nu$ for $\ell=0,1$ near the seam $\mathbb{R}\times\{-\frac 12\}$.
Indeed, $w_\ell^\nu = u_\ell^\nu \circ ((\psi^\nu)^{-1} \circ \phi^\nu)$ is obtained from $u_\ell^\nu$ by composition with the local diffeomorphisms $(\psi^\nu)^{-1} \circ \phi^\nu$ which by Lemma~\ref{lem:rescaledwidth} converge $\cC^\infty_\loc$ to a shift map.
On the other hand, to establish convergence of the $u_\ell^\nu$, we can start from local uniform gradient bounds given by \eqref{eq:wbound} via the reparametrization with $(\psi^\nu)^{-1} \circ \phi^\nu$.
Further, we can work with the ``folded'' maps \(u_{01}^\nu\colon (-r^\nu, r^\nu) \times [0, 1/2) \to M_0^- \times M_1\) given by \(u_{01}^\nu(s,t) := (u_0^\nu(s,-t), u_1^\nu(s,t))\). These satisfy the Lagrangian boundary condition \(u_{01}^\nu(s,0) \in L_{01}\) for \(s \in (-r^\nu,r^\nu)\) and are pseudoholomorphic with respect to
\(K_{01}^\nu(s,t) := \bigl( - J_0^\nu \bigl( \psi^\nu(s-it) \bigr)\bigr) \times J_1^\nu \bigl( \psi^\nu(s+it) \bigr) \),
which converges to
\(K_{01}^\infty := \bigl( -J_0^\infty(s^\infty,0)\bigr) \times J_1^\infty(s^\infty,0)\) in \(\cC^\infty_\loc\).
Now standard compactness for pseudoholomorphic maps implies that after passing to a subsequence, \((u_{01}^\nu)\) converges \(\cC^\infty_\loc\) on \(\mathbb{R}\times[0,1/2)\), and as discussed above this implies \(\cC^\infty_\loc\)-convergence for the corresponding subsequence of \(w_0^\nu, w_1^\nu\) near the \(L_{01}\)-seam \(\mathbb{R} \times \{-1/2\}\).
An analogous argument shows that \(w_1^\nu, w_2^\nu\) converge \(\cC^\infty_\loc\) near \(\mathbb{R} \times \{1/2\}\), so we have now shown that \(w_0^\nu, w_1^\nu, w_2^\nu\) converge \(\cC^\infty_\loc\) everywhere to a \((J_0^\infty(s^\infty,0),J_1^\infty(s^\infty,0),J_2^\infty(s^\infty,0))\)-holomorphic figure eight bubble \(\underline w^\infty\) between \(L_{01}\) and \(L_{12}\).
The lower gradient bound in \eqref{eq:E012bounds} implies that \(\underline w^\infty\) is nonconstant and hence has nonzero energy, hence by Lemma~\ref{lem:hbar} has energy at least \(\hbar_8>0\).
Finally, rescaling invariance and \(\cC^\infty_\loc\)-convergence imply energy concentration of at least \(\hbar_8\) at \((s_{N+1},0)\): \begin{align*}
\liminf_{\nu\to\infty} \int_{B_{\epsilon^\nu}(s^\nu,t^\nu)} \tfrac 1 2\bigl|\delta\underline v^\nu\bigr|^2 \geq \liminf_{\nu\to\infty} \sum_{\ell\in\{0,1,2\}} \int_{U_\ell^\nu} {w_\ell^\nu}^*\omega_\ell \geq \int_{\mathbb{R}^2} \bigl|\delta\underline w^\nu\bigr|^2 \geq \hbar_8>0.
\end{align*}
\item[\bf (D02):]
We will obtain a {\bf
squashed eight bubble in \(\mathbf{ M_{02}}\) with boundary on \(\mathbf{L_{01}\circ L_{12}}\)} by rescaling \begin{align*}
w_\ell^\nu(s, t) := v_\ell^\nu \Bigl( s^\nu + \tfrac s {R^\nu}, \tfrac {\widetilde f^\nu(s)} {\widetilde f^\nu(0)} \tfrac t {R^\nu} \Bigr),
\end{align*}
where we have set \(\widetilde f^\nu(s) := R^\nu f^\nu(s^\nu + s/R^\nu)\), to obtain maps
\begin{align*}
w_0^\nu&\colon
U_0^\nu:=
\bigl\{ (s,t) \: | \: -R^\nu(\rho + s^\nu) \leq s \leq R^\nu(\rho - s^\nu), \: -\tfrac {\rho R^\nu \widetilde f^\nu(0)} {\widetilde f^\nu(s)} \leq t \leq -\widetilde f^\nu(0) \bigr\} \to M_0, \\
w_1^\nu&\colon
U_1^\nu:=\bigl\{ (s,t) \: | \: -R^\nu(\rho + s^\nu) \leq s \leq R^\nu(\rho - s^\nu), \: -\widetilde f^\nu(0) \leq t \leq \widetilde f^\nu(0) \bigr\} \to M_1, \\
w_2^\nu&\colon
U_2^\nu:=\bigl\{ (s,t) \: | \: -R^\nu(\rho + s^\nu) \leq s \leq R^\nu(\rho - s^\nu), \: \widetilde f^\nu(0) \leq t \leq \tfrac {\rho R^\nu \widetilde f^\nu(0)} {\widetilde f^\nu(s)} \bigr\} \to M_2.
\end{align*}
Each \(w_\ell^\nu\) is pseudoholomorphic with respect to \((\widetilde J_\ell^\nu(s,t), j^\nu)\), where the almost complex structure \(\widetilde J_\ell^\nu(s,t) := J_\ell^\nu(s^\nu + s/R^\nu, \widetilde f^\nu(s) t / \widetilde f^\nu(0) R^\nu )\) converges \(\cC^\infty_\loc\) to \(\widetilde J_\ell^\infty := J_\ell^\infty(s_\infty,0)\) and the symmetric complex structure \(j^\nu\) on \(U_0^\nu \cup U_1^\nu \cup U_2^\nu
\subset \mathbb{R}^2\)
\begin{align*}
j^\nu(s,t) := \left( \begin{array}{cc}
-\tfrac {t(\widetilde f^\nu)'(s)} {\widetilde f^\nu(0)} & -\tfrac {\widetilde f^\nu(s)} {\widetilde f^\nu(0)} \\
\tfrac {t^2(\widetilde f^\nu)'(s)^2 + \widetilde f^\nu(0)^2} {\widetilde f^\nu(0)\widetilde f^\nu(s)} & \tfrac {t(\widetilde f^\nu)'(s)} {\widetilde f^\nu(0)}
\end{array} \right)
\end{align*}
converges \(\cC^\infty_\loc\) to the standard complex structure by Lemma~\ref{lem:rescaledwidth}
with \(\alpha^\nu := R^\nu\).
The maps \(w_\ell^\nu\) also satisfy the Lagrangian seam conditions
\begin{gather*}
(w_0^\nu(s, -\widetilde f^\nu(0)), w_1^\nu(s, -\widetilde f^\nu(0))) \in L_{01}, \qquad (w_1^\nu(s, \widetilde f^\nu(0)), w_2^\nu(s, \widetilde f^\nu(0))) \in L_{12}
\end{gather*}
for all \(s\) in \((-R^\nu(\rho + s^\nu), R^\nu(\rho - s^\nu))\). By the \(\cC^\infty_\loc\)-convergence \(\widetilde f^\nu(s) / \widetilde f^\nu(0) \to 1\) proven in Lemma~\ref{lem:rescaledwidth}, we may choose a subsequence and
\(r^\nu \to \infty\) with \(r^\nu \leq R^\nu \epsilon^\nu / 4\) such that we have
\begin{align} \label{eq:D02-tilde-f-bound}
\|\widetilde f^\nu(s) / \widetilde f^\nu(0)\|_{\cC^1((-r^\nu,r^\nu))} \leq 2
\end{align}
for \(\nu\) sufficiently large.
This allows us to translate \eqref{eq:vbounds} into upper and lower bounds on the energy density \(| \delta \underline w^\nu |\) for sufficiently large $\nu$,
\begin{gather*}
| \delta\underline w^\nu(0, R^\nu t^\nu) |
\geq
\tfrac 1 {2R^\nu} | \delta\underline v^\nu(s^\nu, t^\nu) | \geq \tfrac 1 2, \\
\sup_{(s,t) \in B_{r^\nu}(0)} | \delta \underline w^\nu(s,t) | \leq
\sup_{(s,t) \in B_{\epsilon^\nu}(s^\nu,t^\nu)}
\tfrac 4 {R^\nu}| \delta \underline v^\nu(s,t) | \leq 8.
\end{gather*}
Here we estimated
$\tfrac1{R^\nu} | \partial_s v_\ell^\nu|- \tfrac{|(\widetilde f^\nu)'|}{R^\nu\widetilde f^\nu(0)}| \partial_t v_\ell^\nu| \leq
\bigl| \partial_s w_\ell^\nu \bigr| \leq
\left(\tfrac1{R^\nu} + \tfrac{|(\widetilde f^\nu)'|}{R^\nu\widetilde f^\nu(0)}\right) | \delta v_\ell^\nu|$,
used the identity
$\bigl| \partial_t w_\ell^\nu \bigr| =
\tfrac{\widetilde f^\nu}{R^\nu\widetilde f^\nu(0)} | \partial_t v_\ell^\nu|$,
and need to check that $(s,t)\in B_{r^\nu}(0)$ implies
$\bigl( s^\nu + \tfrac s {R^\nu}, \tfrac {\widetilde f^\nu(s)} {\widetilde f^\nu(0)} \tfrac t {R^\nu} \bigr) \in B_{\epsilon^\nu}(s^\nu,t^\nu)$ for sufficiently large $\nu$.
Indeed, \eqref{eq:D02-tilde-f-bound} yields:
\begin{align} \label{eq:D02-containment}
\bigl| \bigl( \tfrac s {R^\nu} \,,\, \tfrac {\widetilde f^\nu(s)} {\widetilde f^\nu(0)} \tfrac t {R^\nu} - t^\nu \bigr)\bigr| / \epsilon^\nu
\;\leq\;
\tfrac {2r^\nu}{R^\nu\epsilon^\nu} + \tfrac{|t^\nu|}{\epsilon^\nu}
\;\leq\;
\tfrac 12 + \tfrac{\tau_+^\nu + \tau_-^\nu}{2 R^\nu \epsilon^\nu} ,
\end{align}
where \(R^\nu \epsilon^\nu \to \infty\) and $\tau_\pm^\nu\to\tau_\pm^\infty \in \mathbb{R}$.
Next we consider the limiting behaviour of the domain \(U_0^\nu \cup U_1^\nu \cup U_2^\nu\).
Its straight boundaries diverge \(-R^\nu(\rho + s^\nu) \to -\infty\) resp.\ \(R^\nu(\rho - s^\nu)\to \infty\) since $s^\nu\to s^\infty \in (-\rho,\rho)$.
The functions \(\pm \rho R^\nu \widetilde f^\nu(0) / \widetilde f^\nu(s)\) of the upper/lower boundary converge \(\cC^\infty_\loc\) to \(\infty\) resp.\ \(-\infty\) by Lemma~\ref{lem:rescaledwidth} with \(\alpha^\nu := R^\nu\) and $R^\nu\to\infty$.
Finally, the straight seams \(\{ t = \pm \widetilde f^\nu(0)\} \) shrink to a single seam $\{t=0\}$ since we have
$ \widetilde f^\nu(0)= \tfrac 1 2(\tau_+^\nu - \tau_-^\nu) \to 0 \).
Now we may apply Theorem~\ref{thm:nonfoldedstripshrink} to this strip shrinking situation to deduce
that after passing to a subsequence, \((w_0^\nu(s, t - \widetilde f^\nu(0))\) resp.\ \((w_2^\nu(s, t + \widetilde f^\nu(0))\) converge in \(\cC^\infty_\loc(-\H)\) resp.\ \(\cC^\infty_\loc(\H)\) to \(\widetilde J_0^\infty\)- resp.\ \(\widetilde J_2^\infty\)-holomorphic maps \(w_0^\infty\) resp.\ \(w_2^\infty\), and that \((w_1^\nu|_{t=0})\) converges in \(\cC^\infty_\loc(\mathbb{R})\) to a smooth map \(w_1^\infty\). Furthermore, at least one of \(w_0^\infty, w_2^\infty\) is nonconstant, and the generalized seam condition \((w_0^\infty(s,0), w_1^\infty(s), w_1^\infty(s), w_2^\infty(s,0)) \in L_{01} \times_{M_1} L_{12}\) is satisfied for \(s \in \mathbb{R}\), so that \((w_0^\infty, w_2^\infty)\) is a nonconstant squashed eight bubble with boundary on \(L_{01} \circ L_{12}\), with energy bounded below by \(\hbar_{L_{01} \circ L_{12}}\).
Finally, rescaling invariance, \(\cC^\infty_\loc\)-convergence, and the containment proven in \eqref{eq:D02-containment} imply energy concentration:
\begin{align*}
\liminf_{\nu \to \infty} \int_{B_{\epsilon^\nu}(s^\nu, t^\nu)} \tfrac 1 2| \delta\underline v^\nu|^2 \geq \liminf_{\nu \to \infty} \sum_{\ell \in \{0,2\}} \int_{B_{\epsilon^\nu}(s^\nu, t^\nu)} v_\ell^{\nu\;*}\omega_\ell &\geq \liminf_{\nu \to \infty} \sum_{\ell \in \{0,2\}}
\int_{B_{r^\nu}(0)}
w_\ell^{\nu\;*}\omega_\ell \\
&\geq \sum_{\ell \in \{0,2\}} \int w_\ell^{\infty\;*}\omega_\ell \geq \hbar_{L_{01} \circ L_{12}}.
\end{align*}
\end{itemlist}
\medskip
\noindent
This ends the construction of a nontrivial bubble in the this last case, (D02), and thus
finishes the iterative construction of a subsequence and blow-up points so that (0), (2), and (3) hold. To establish the $\cC^\infty_\loc$-convergence on the complement of the blow-up points claimed in
(1) we will apply Theorem~\ref{thm:nonfoldedstripshrink} to quilted domains that make up rectangles in \((-\rho,\rho)^2 {\smallsetminus} \{z_1, \ldots, z_N\}\).
Standard elliptic regularity implies that \(v_0^\nu(s, t - f^\nu(s))\) resp.\ \(v_2^\nu(s, t + f^\nu(s))\) converge \(\cC^\infty_\loc\) on the interior of their domains \((-\rho,\rho) \times (-\rho,0] {\smallsetminus} Z\) resp.\ \((-\rho,\rho) \times [0, \rho) {\smallsetminus} Z\). To extend this convergence to the boundary and to establish convergence of \(v_1^\nu(s,0)\), fix a point \((\sigma,0)\) in \((-\rho,\rho) \times \{0\} {\smallsetminus} Z\), and define three maps by rescaling \(v_\ell^\nu\) for $\ell=0,1,2$ and straightening the seams:
\begin{align*}
w_\ell^\nu(s,t) := v_\ell^\nu \bigl( \sigma + s, \tfrac {f^\nu(s + \sigma)} {f^\nu(\sigma)}t \bigr).
\end{align*}
For \(r > 0\) sufficiently small, these maps form a squiggly strip quilt of size \((f^\nu(\sigma), r)\), which is \((\widetilde J_0^\nu, \widetilde J_1^\nu, \widetilde J_2^\nu, j^\nu)\)-holomorphic for \(\widetilde J_\ell^\nu\) and \(j^\nu\) the pulled-back almost complex and complex structures
\begin{align*}
\widetilde J_\ell^\nu(s,t) := J_\ell^\nu( \sigma + s, \tfrac {f^\nu(s + \sigma)} {f^\nu(\sigma)}t), \qquad j^\nu(s,t) := \left(\begin{array}{cc}
-\tfrac {(f^\nu)'(s + \sigma)} {f^\nu(\sigma)}t & -\tfrac {f^\nu(s + \sigma)} {f^\nu(\sigma)} \\
\tfrac { (f^\nu)'(s + \sigma)^2 t^2 + f^\nu(\sigma)^2} {f^\nu(\sigma)f^\nu(s + \sigma)} & \tfrac {(f^\nu)'(s + \sigma)} {f^\nu(\sigma)} t
\end{array}\right).
\end{align*}
The obedient shrinking \(f^\nu \Rightarrow 0\) and the Arz\`{e}la--Ascoli theorem guarantee that after passing to a subsequence\footnote{For those choices of \((f^\nu)\) that arise from natural geometric situations --- e.g.\ from the figure eight bubble --- we expect convergence directly, i.e.\ without passing to a subsequence.},
\(f^\nu(s + \sigma) / f^\nu(\sigma)\) converges in \(\cC^\infty_\loc\); therefore \(\widetilde J^\nu_\ell\) and \(j^\nu\) converge in \(\cC^\infty_\loc\) to almost complex and complex structures \(\widetilde J_\ell^\infty\) and \(j^\infty\). As long as \(r\) was chosen to be small enough, the bound \(\|j^\infty - i \|_{\cC^0} \leq \epsilon\) holds (where \(\epsilon\) is the constant appearing in Thm~\ref{thm:nonfoldedstripshrink}), so Theorem~\ref{thm:nonfoldedstripshrink} implies that \(w_0^\nu(s, t - f^\nu(\sigma))\), \(w_1^\nu(s, 0)\), \(w_2^\nu(s, t + f^\nu(\sigma))\) converge \(\cC^\infty_\loc\) to smooth maps \(w_0^\infty, w_1^\infty, w_2^\infty\) that satisfy a generalized seam condition in \(L_{01} \times_{M_1} L_{12}\). Since \(f^\nu( s + \sigma) / f^\nu(\sigma)\) converges \(\cC^\infty_\loc\), we may conclude that \(v_0^\nu(s, t - f^\nu(\sigma))\), \(v_1^\nu(s,0)\), \(v_2^\nu(s, t + f^\nu(\sigma))\) converge \(\cC^\infty_\loc\) on a neighborhood of \((\sigma,0)\), and the limit maps satisfy a generalized seam condition in \(L_{01} \times_{M_1} L_{12}\). We established convergence away from \((-\rho,\rho) \times \{0\}\) earlier, so we have now proven (1). This finishes the proof of Theorem~\ref{thm:rescale}.
\begin{remark} \label{rmk:noncompact}
The purpose of this remark is to discuss the minimal assumptions which allow one to apply Theorem~\ref{thm:rescale} to symplectic manifolds $M_0, M_1, M_2$ that are not compact.
If the Lagrangian correspondences are compact, then --- unlike the ``bounded geometry'' assumptions in \cite{isom} --- we do not explicitly require uniform bounds on metrics and almost complex structures (which were used in \cite{isom} to show energy concentration in a sequence of pseudoholomorphic maps with unbounded gradient).
Instead we need to ensure convergence of maps which result from rescaling near a blow-up point of the gradients of a sequence of pseudoholomorphic maps. If the rescaled domains contain boundary or seam conditions, then compactness of the Lagrangians implies \(\cC^0\)-bounds so that the rest of our arguments applies in a precompact neighborhood of the Lagrangians.
If the rescaled domains do not contain boundary or seam conditions, or if the Lagrangians in the boundary or seam conditions are noncompact, then \(\cC^0\)-bounds must be obtained a priori from some special properties of the symplectic manifold or Lagrangians.
Note that the \(\cC^0\)-bounds are not merely technical complications --- in general, nontrivial parts of sequences of pseudoholomorphic curves can and will escape to infinity, at best yielding punctures and SFT-type buildings in the limit.
One way to achieve \(\cC^0\)-bounds would be to work with completed Liouville domains and Lagrangians which are cylindrical at infinity, as in Abouzaid--Seidel's definition of the wrapped Fukaya category in \cite{as:wrapped}.
Footnotes \ref{foot:bubblesinstrata} and \ref{foot:bounds} point out the main instances where the specific geometry would have to be considered when dealing with noncompact manifolds. When working with noncompact Lagrangians, one would have to make additional assumptions --- such as ``bounded geometry'' for symplectic manifolds and Lagrangians --- to guarantee uniformity of the elliptic estimates in \cite{b:singularity}.
\end{remark}
\section{Boundary strata and algebraic consequences of strip-shrinking moduli spaces} \label{s:propaganda}
The purpose of this section is to analyze the expected boundary stratification of strip-shrinking moduli spaces and from this predict the algebraic consequences of figure eight bubbling.
While we make an effort to provide convincing arguments for the more surprising features, this part of our exposition will be rather cavalier --- aiming only to explain the rough form of what we expect to be able to make rigorous.
In particular, all Floer cohomology groups will be considered as ungraded and with coefficients in the Novikov field defined over \(\mathbb{Z}_2\), which should be valid as long as sphere bubbling can be avoided. We ultimately expect a fully-fledged graded theory with Novikov coefficients defined over \(\mathbb{Q}\) (resp.\ over \(\mathbb{Z}\) in the absence of sphere bubbling).
\subsection{Boundary stratifications and their algebraic consequences} \label{ss:boundary}
One of the intuitions in the treatment of pseudoholomorphic curve moduli spaces is that sphere bubbling is ``codimension 2'' and disk bubbling is ``codimension 1''. We give a more rigorous statement of this intuition in the polyfold framework and explain its algebraic consequences in Remark~\ref{rmk:codim} below, and will argue that, in a similarly imprecise sense, figure eight bubbling is ``codimension 0'' within the ``zero-width boundary components'' of quilt moduli spaces involving a strip or annulus of varying width.
\begin{remark}[\bf Codimension and algebraic contribution of sphere and disk bubbles] \label{rmk:codim}
In the polyfold setup for pseudoholomorphic curve moduli spaces (whose blueprint is given in \cite{hwz:gw} at the example of Gromov--Witten moduli spaces), the compactified
moduli space is cut out of the ambient polyfold by a (polyfold notion of) Fredholm section, which arises from the Cauchy--Riemann operator. Transversality while preserving compactness can then be achieved by adding a small, compact (possibly multivalued) perturbation, which is supported near the unperturbed moduli space. This equips the perturbed moduli space with the structure of a compact (possibly weighted branched) manifold. For expositions of this theory see e.g.\ \cite{Hofer,hwz:fred2,theguide}.)
An important feature of the ambient space is that there is a sensible notion of ``corner index'' --- a nonnegative integer associated to each point in the polyfold, so that the points of corner index 0 resp.\ resp.\ $\ge 2$ should be thought of as the interior resp.\ smooth part of boundary resp.\ corner stratification.
The transverse perturbation can be chosen compatibly with corner index, so that a Fredholm index 0 section gives rise to a perturbed moduli space lying in the interior of the polyfold, and a Fredholm index \(1\) section gives rise to a perturbed moduli space whose boundary is given by the intersection of the zero set with the smooth (corner index \(1\)) part of the polyfold boundary. The index 0 components of the perturbed moduli space are then typically used to define an algebraic structure, whose algebraic relations arise from the fact that the Fredholm index \(1\) component has nullhomologous boundary corresponding to algebraic compositions of Fredholm index \(0\) contributions. More precisely, the sum over the algebraic contributions of each boundary point is zero, and since these boundary points are given by the zero set of the section restricted to the smooth part of the boundary of the polyfold, the algebraic relations are given by a sum over these boundary strata.
It turns out that interior nodes do not contribute to the corner index, and in particular that curves with sphere bubbles and no other nodes are smooth interior points of the polyfold. This can be understood from the gluing parameters $(R_0,\infty)\times S^1$ used to describe a neighborhood of the node. The corresponding pre-gluing construction provides a local chart for the polyfold, in which the gluing parameters get completed by $\{\infty\}$ to an open disk, which contributes no boundary.
On the other hand, gluing at a boundary node or a breaking is described by a parameter in $(R_0,\infty)$, which gets completed by $\{\infty\}$ to a half-open interval, so that pre-gluing in these cases provides local charts in which parameter $\infty$ indicates a contribution of \(+1\) to the corner index.
Hence each boundary node (e.g.\ from disk bubbling), each trajectory breaking (as in Floer theory), and each extra level of buildings (in SFT) contribute \(1\) to the corner index.
This explains why sphere bubbling does not contribute to algebraic relations of the type discussed here, and instead it is the curves with exactly one boundary node (e.g.\ one disk bubble) or one breaking which contribute to the algebraic relations.
\end{remark}
Following the above remark, we need to analyze the boundary stratification of the polyfolds from which the strip-shrinking moduli spaces are cut out in order to predict the algebraic consequences of figure eight bubbling. For that purpose we describe in the following the {\bf pre-gluing constructions that provide the local charts near figure eight and squashed eight bubbles}:
\begin{itemlist}
\item Gluing a {\bf figure eight} into a pseudoholomorphic quilt has to go along with introducing an extra strip of width $\delta>0$. Since figure eights do not have an $S^1$ symmetry, it then remains to fix the length of neck between the bubble and the quilted Floer trajectory.
However, this gluing parameter in $(R_0,\infty)$ is in fact fixed by the choice of width \(\delta > 0\), as illustrated in Figure~\ref{fig:8gluing}.
Hence, while figure eight bubbles can only appear on the \(\delta=0\) boundary, they do not contribute to the corner index.
This means that a \(\delta=0\) quilted Floer trajectory with any number of figure eight bubbles will still just have corner index $1$. Indeed, the figure eights can only be pre-glued simultaneously since their neck-lengths must all be given by the same strip width $\delta>0$.
\end{itemlist}
\begin{figure}
\centering
\def6in{\columnwidth}
\input{8gluing.pdf_tex}
\caption{A figure eight bubble can be glued to a marked point on a double strip indicated in the left figure. This is done by first fattening the seam in the double strip to a new middle strip of width \(\delta\) and centered puncture at the old marked point, then overlaying a neighborhood of this puncture with a neighborhood of the figure eight singularity in cylindrical coordinates, and finally interpolating between the maps on the new domain. In this construction the neck-length parameter \(R\) is determined by the strip width \(\delta\), as illustrated in the right figure:
Since the seams are not straight in cylindrical coordinates, the relative shift between bubble and triple strip is determined by the positions of the seams having to match.
Note that these figures illustrate the domains of the respective maps, not their images.
}
\label{fig:8gluing}
\end{figure}
\begin{itemlist}
\item The configuration of a {\bf squashed eight bubble} with seam in \(L_{01} \times_{M_1} L_{12}\) attached to a \(\delta=0\) quilted Floer trajectory has corner index $2$ since it can be glued with two independent parameters.
Indeed, the pre-gluing construction is to first widen the seam in
both the base and the bubble to strips of independent widths \(\delta\in [0,1)\) and \(\epsilon \in [0,1)\) (turning the squashed eight into a figure eight in case \(\epsilon>0\)), and to then pre-glue the resulting bubble into the quilt with a gluing parameter \(R\in (R_0,\infty]\).
Here the strip width \(\delta\) is determined by \((R,\epsilon)\) as follows:
Pre-gluing with \(R=\infty, \epsilon>0\) yields a (only approximately holomorphic)
figure eight attached to a middle strip of width \(\delta(\infty,\epsilon)=0\) whereas \(R<\infty, \epsilon=0\) produces a (approximately holomorphic) quilted Floer trajectory with middle strip width \(\delta(R,0)=0\).
Positive strip width \(\delta(R,\epsilon)>0\) is achieved only with \(R<\infty, \epsilon>0\), providing the interior of the chart.
\item
Similarly, a {\bf disk bubble} with boundary in \(L_{01}\) or \(L_{12}\) attached to a \(\delta=0\) quilted Floer trajectory has corner index 2 since the length of the gluing neck is independent of the width \(\delta\ge 0\).
In fact, in our tree setup there would be a constant figure eight between the disk and Floer trajectory, so that the gluing parameter is used to pre-glue the disk into the figure eight, and the width parameter pre-glues the resulting figure eight into the Floer trajectory.
\end{itemlist}
\begin{remark}[\bf Boundary stratification of strip-shrinking moduli spaces] \label{rmk:wtc}
The gluing construction for squashed eights above indicates that the closures of the two top boundary strata given by \(\delta = 0\) quilts with one figure eight bubble (i.e.\ \(R = \infty, \epsilon > 0\)) and by \(\delta = 0\) quilts with no bubbles (i.e.\ \(R < \infty, \epsilon = 0\)) intersect in a corner index 2 stratum consisting of \(\delta=0\) quilts with one squashed eight bubble (i.e.\ \(R = \infty, \epsilon = 0\)).
To see how a sequence of \(\delta=0\) quilts with one figure eight bubble can converge to a \(\delta = 0\) quilt with one squashed eight bubble, note that the moduli space of figure eight bubbles has a boundary stratum in which all energy concentrates at the singularity, so that rescaling yields a squashed eight bubble attached to a constant figure eight.
Note here that different choices of rescaling yield constant figure eights with different values --- namely any seam value of the squashed eight except for its value at infinity. Thus the resulting figure eight here is a true {\bf ``ghost eight"} in the sense that its value would be determined by the choice of a marked point on the squashed eight; however we do not make a specific choice. In fact, a constant figure eight with no further marked point (except at its singularity, where the squashed eight is attached) would not be stable. We will however include these ghost eights as stable figure eight vertices when describing the bubble trees as colored ribbon trees.
Let us compare this to the fact that a disk bubble with boundary on \(L_{01}\) or \(L_{12}\) attached (via a constant figure eight)
to a \(\delta=0\) quilted Floer trajectory lies in the intersection of a stratum of \(\delta>0\) trajectories with disk bubble and a stratum of \(\delta=0\) trajectories with figure eight bubble. In the first stratum the width goes to zero at the corner, whereas in the second stratum the figure eight converges to a constant figure eight with disk bubble.
In this case, however, the constant value of the figure eight is uniquely determined by its attaching point on the quilted Floer trajectory, and the figure eight is stable due to the marked point at which the disk is attached.
\end{remark}
\subsection{Strip shrinking in quilted Floer theory for cleanly-immersed geometric composition} \label{ss:HF}
With this framework in place, we now analyze the boundary stratification of a specific strip-shrinking moduli space, from which we will obtain specific algebraic predictions in Section~\ref{ss:algebra1}.
The isomorphism between quilted Floer homologies \eqref{eq:HFiso} under monotone, embedded composition is proven by applying the cobordism argument in Remark~\ref{rmk:codim}
to a moduli space of quilted Floer trajectories with varying width $\delta\in[0,1]$ of the strip mapping to $M_1$. Here the boundary arises from the strip widths $\delta=0$ and $\delta=1$, since other bubbling or breaking is excluded by the monotonicity assumption.
Recall however that this bubble exclusion fails even in monotone cases as soon as the geometric composition is a multiple cover of a smooth Lagrangian (as in many examples of interest, e.g.\ \cite{w:chekanov}).
In order to obtain a result that allows for general symplectic manifolds and Lagrangians and cleanly-immersed composition \(L_{01} \circ L_{12}\), we need to study the boundary strata of the polyfold which provides an ambient space for a general compactified moduli space of quilted Floer trajectories with varying width.
In addition to breaking and bubbling (of disks, squashed eights, and figure eights), the ends of the interval $[0,1]$ contribute to its corner index.
Based on the previous analysis of gluing parameters, we predict that the
{\bf top boundary strata of the Gromov-compactified strip-shrinking moduli space} (the strata with corner index $1$) are the strata of the following types: \label{Bs}
\begin{itemize}
\item[(B1)]
quilted Floer trajectories for $\delta=1$;
\item[(B2)]
once-broken quilted Floer trajectories for $\delta \in (0,1)$;
\item[(B3)]
quilted Floer trajectories with one disk bubble on a seam for $\delta \in (0,1)$;
\item[(B4)]
quilted Floer trajectories for $\delta=0$ with generalized seam condition;\footnote{
Since the Lagrangian \(L_{01} \circ L_{12}\) is in general just a clean immersion, we require not only that the corresponding seam gets mapped to the composed Lagrangian, but we require this map to lift continuously to \(L_{01} \times_{M_1} L_{12}\), and include the lift as data of the Floer trajectory.
}
\item[(B5)] quilted Floer trajectories for $\delta=0$ with any number of figure eight bubbles.\footnote{
In this case, the generalized seam condition requires a lift to \(L_{01} \times_{M_1} L_{12}\) that is continuous (and hence smooth) on the complement of the bubbling points,
and at each bubbling point is possibly discontinuous in a way that matches with the limits $\lim_{s\to\pm\infty}w_1(s, \cdot)$ of the figure eight.
}
\end{itemize}
Within each such boundary component we may also find curves that include trees of sphere bubbles.
Furthermore, contributions from (B2) and (B3) necessarily involve curves that for fixed $\delta$ are not cut out transversely, i.e.\ these contributions come from a finite set of singular values of $\delta\in (0,1)$.
To argue for our prediction, in particular the necessity of allowing figure eight bubbles in (B5), from a more geometric perspective, let us go through the rather silly example of shrinking the strip in standard Floer theory for a pair of Lagrangians $L_{01}\subset \pt \times M_1$ and $L_{12}\subset M_1^- \times \pt$.
In this case, the boundary component (B1) represents the Floer differential. The boundary components (B2) and (B3) will be empty, since the Cauchy--Riemann operator for each $\delta>0$ is just a rescaling of that for $\delta=1$, and hence all can be made regular simultaneously.
Hence the Floer differential must coincide with the algebraic contributions from (B4--5). Indeed, each Floer trajectory can be viewed as a figure eight bubble by Example~\ref{ex:mickeymouse}, and in this case is attached to the constant Floer trajectory in $\pt\times\pt$.
Broken Floer trajectories are excluded for index reasons.
More evidence for the necessity of (B5) are the Floer homology calculations in \cite{w:chekanov} between Clifford tori and $\mathbb{RP}^n\subset \mathbb{CP}^n$ resp.\ the Chekanov torus in $S^2\times S^2$ using strip shrinking for multiply covered geometric composition, where bubbling can only be excluded for classes of Floer trajectories whose limits are not self-connecting.
Nonzero results for the corresponding entries of the differential from other calculation methods then indirectly show nontrivial figure eight contributions.
At this point, a reader comfortable with evaluation maps into appropriate spaces of chains may skip to the algebraic consequences in Section~\ref{ss:algebra1}.
However, we will construct these algebraic structures from the following more complicated moduli spaces that will simplify both our analytic and algebraic work, and also serve to further solidify our prediction of boundary stratifications.
\subsection{Morse bubble trees arising from strip-shrinking moduli spaces} \label{ss:Morse}
To capture the algebraic effect of figure eight bubbling in terms of Morse chains on $L_{01}\times_{M_1} L_{12}$, we will extend strip-shrinking moduli spaces by allowing Morse flow lines between figure eight, squashed eight, and disk bubbles\footnote{
Here we identify quilted spheres with two patches in \(M_k\) and \(M_{k+1}\) with disks in \(M_k^- \times M_{k+1}\); see footnote~\ref{foot:fold}.
}.
This will be achieved by organizing the tree of bubble vertices and Morse edges into a colored metric ribbon tree (as introduced in \cite[Def.\ 7.1]{mw}) whose root is the base quilt in which the strip was being shrunk.
This approach is analogous to constructing the $A_\infty$-algebra of a single Lagrangian submanifold via trees of disk vertices and Morse edges\footnote{
This has been a partially realized vision in the field for a while. Formally, it follows the $A_\infty$ perturbation lemma \cite[Prop.\ 1.12]{se:bo} for transferring an $A_\infty$-structure on a space of differential chains to the space of Morse chains. A related moduli space setup was proposed in \cite{CL}
but with a different algebraic goal.
}
which we sketch in the following Remark before explaining its generalization to strip-shrinking moduli spaces.
\begin{remark}[\bf Polyfold setup and boundary stratification for trees of disks with Morse edges] \label{rmk:disks}
Consider a single disk bubble attached by a marked point to e.g.\ a Floer trajectory. If we enlarge the moduli space by Morse flow lines between nodal pairs of boundary marked points, then boundary strata with length $0$ flow lines cancel strata with boundary nodes and we obtain a compactified moduli space of metric ribbon trees whose root is the attaching point, vertices are represented by pseudoholomorphic disks (modulo appropriate reparametrizations which we will not discuss), and edges are represented by generalized Morse trajectories (including broken trajectories that compactify the space of finite length Morse trajectories) that are directed toward the root.
In \cite{lw:morsetrees}, assuming the absence of sphere bubbling, this moduli space is described as the zero set of a Fredholm section in an M-polyfold bundle. This section is given by the Cauchy--Riemann operators on each vertex together with the matching conditions for each edge between the endpoints of the Morse trajectory and corresponding marked point evaluations of the disk maps.
The ambient M-polyfold is the space of trees in which vertices are represented by (reparametrization equivalence classes of) not necessarily pseudoholomorphic maps and edges are represented by generalized Morse trajectories.
Nodal configurations with edge length $0$ are interior points of this space by pre-gluing of the nodal disks into a single vertex (this is made rigorous in terms of M-polyfold charts arising from the pre-gluing construction). Hence the boundary stratification of this space is induced by the compactified space of Morse trajectories --- which was given a smooth structure in \cite{w:morse}, with corner index equal to the number of critical points at which a trajectory breaks.
Now the arguments of Remark~\ref{rmk:codim} yield an algebraic structure from counting isolated solutions whose relations are given by summing over the top boundary strata (those with corner index 1).
In this case, adding incoming Morse edges from input critical points yields a curved $A_\infty$-algebra because the top boundary strata --- configurations with exactly one broken trajectory, i.e.\ edge of length $\infty$ --- correspond to the top boundary strata of a space of metric ribbon trees.
The stable trees in the latter realize Stasheff's associahedra, so that the boundary strata yield the $A_\infty$-relations with the exception of terms involving $\mu^1$ or $\mu^0$. These additional terms arise from breaking into two subtrees of which one is unstable with zero or one incoming Morse edge.
Similarly, considering Floer trajectories for pairs of Lagrangians (or quilted Floer trajectories) with several bubble trees (on each boundary component resp.\ seam) yields the relations for the Floer differential coupled with the $A_\infty$-algebras of the Lagrangians.
Sphere bubbling can be included here by extending the ambient space of disk maps and Morse edges to allow for trees of spheres attached to the maps. This introduces the additional complication of isotropy, turning the ambient space into a polyfold (M-polyfolds are a special case with trivial isotropy) and forces the use of multivalued perturbations, thus yielding rational counts.
However, as discussed in Remark~\ref{rmk:codim}, this does not affect the boundary stratification and algebraic consequences.
\end{remark}
Ignoring sphere bubbling as above, we introduce Morse flow lines into strip-shrinking moduli spaces in two stages: First, we allow for Morse edges between disk bubbles and the domain in which they occurred. As in Remark~\ref{rmk:disks} this captures disk bubbling on seams in an algebraic coupling with the curved $A_\infty$-algebras generated by Morse chains on the Lagrangian correspondences that are not involved in the strip shrinking. If such a tree of disks was attached to a seam in the quilt for $\delta>0$, then in the $\delta=0$ limit we represent it by a tree attached to the respective seam of a constant figure eight bubble. We then begin the second stage of extending the strip-shrinking moduli space by introducing Morse flow lines on $L_{01}\times_{M_1} L_{12}$ between the quilt and figure eights. Since a zero width strip with any number of figure eight bubbles is a corner index $1$ boundary point, we need to extend with a single normal parameter, so take all Morse edges of the same length $-\delta$. This extends the strip width parameter from $\delta\in[0,1]$ to \(\delta<0\), and to compactify the resulting extended moduli space we allow for Morse breaking (simultaneously for each figure eight) as $\delta\to\infty$ but also need to take into account that besides disk bubbling (which is dealt with as before) we may have squashed eight bubbles appearing by energy concentration on the base quilt or at the singularity of a figure eight bubble (as in Remark~\ref{rmk:wtc}, with the remaining figure eight being either nonconstant or a ghost eight). We cancel these boundary components again by introducing Morse flow lines on $L_{01}\times_{M_1} L_{12}$, whose length can now vary individually. However, since squashed eights appear between the base quilt and figure eights, the length of these Morse edges has to be accounted for in the condition of figure eights all being at the same ``Morse distance'' from the base quilt. So the bubble hierarchy indicated in Figure~\ref{fig:tree} yields a construction of the extended moduli space (more precisely its part on which $\delta\in [-\infty,0]$): It consists of {\bf Morse bubble trees over quilts of strip width zero} that are organized into colored metric ribbon trees (see \cite[Def.\ 7.1]{mw}) as follows:
\begin{itemlist}
\item
The root is represented by the base quilt in which the strip has been shrunk to width $0$. \label{page-tree}
\item
Other vertices are represented by a pseudoholomorphic disk, squashed eight, figure eight, or ghost eight.
Figure eights and ghost eights are the colored vertices, of which there is exactly one between each leaf and the root.
The squashed eights are exactly the vertices between the root and a colored vertex.
\item
Each edge is labeled by a ``Morse length'' in $[0,\infty]$, and the ``Morse distance'' between each colored vertex and the base quilt (the sum of lengths of edges in between base quilt and figure eight resp.\ ghost eight) is the same. We denote this ``figure eight height'' by $-\delta$ for $\delta\in [-\infty,0]$.
\item
Each edge attached to a disk vertex is represented by a generalized Morse trajectory on $L_{01}$ resp.\ $L_{12}$ of the given length. Each edge between figure eights, squashed eights, and the root is represented by a generalized Morse trajectory on $L_{01}\times_{M_1}L_{12}$ of the given length.
\item
Disks and squashed eights are constant only if the vertex has valence $\ge 3$.
Figure eights are constant only if the vertex has valence $\ge 2$. Ghost eights only appear as colored leaf attached by a ghost edge to a squashed eight vertex of valence $2$. The ghost vertex and ghost edge carry no geometric information other than a length of the ghost edge in $[0,\infty]$.
\item[$\circ$] Later on we generalize these moduli spaces further to obtain algebraic coupling with the Morse chain complexes on \(L_{01}\) and \(L_{12}\). For that purpose we allow ``half-infinite edges'' which terminate at a leaf and are represented by generalized Morse trajectories in the compactifications $\overline{\M}(x_-, L_{ij})$ of unstable manifolds of critical points $x_-$.
\end{itemlist}
Note that constant figure eights are stable from an isotropy point of view when their vertex has valence $\ge 2$. The ghost eight leaves are forced by the fact from Remark~\ref{rmk:wtc} that the moduli space (resp.\ ambient polyfold) of figure eight quilts has a boundary stratum on which all energy concentrates at the singularity, so that rescaling yields a squashed eight bubble that is attached to a constant figure eight quilt, whose value can be varied by changing the rescaling (that's what we denote by ghost eight).
If the figure eight was nonconstant, then the resulting boundary stratum of the moduli space of fixed tree type is cancelled by the boundary stratum in which the Morse edge between a squashed eight and a figure eight vertex is of length $0$.
Our setup including ghost eight leaves is equivalent to requiring squashed eight vertices to have ``Morse distance'' from the base quilt less or equal to the ``figure eight height'', and using the boundary strata resulting from equality to cancel the above mentioned strata in which a figure eight vertex degenerates into a squashed eight.
From here we can note that any breaking of trajectories between squashed eights or figure eights and the main component corresponds to Morse height $-\delta\to\infty$ and thus forces at least one breaking between each figure eight vertex and the root. (Between ghost eights and the root, this can be achieved by a broken Morse trajectory between a squashed eight and the root, or by a ghost edge of length $\infty$.)
If there is just one breaking for each figure eight (and for each ghost eight either the ghost edge is infinite or there is one breaking between the squashed eight and quilt), then the result lies in the top boundary stratum, but any additional breaking of an edge adds to the corner index individually.\footnote{If there are $N$ figure eight vertices, then this effect is analogous to the breaking of finite length Morse trajectories in the $N$-fold Cartesian product of $L_{01}\times_{M_1}L_{12}$ : The first breaking has to happen simultaneously in all components since their lengths are coupled; further breakings are independent since trajectories can be constant in various components.
}
As in Remark~\ref{rmk:disks}, we expect to obtain an ambient polyfold (or M-polyfold if sphere bubbling can be a priori excluded) by replacing the pseudoholomorphic curves and quilts with appropriate spaces of not necessarily pseudoholomorphic maps modulo reparametrization. Making this rigorous will require a precise setup of pre-gluing constructions for squashed eights and figure eights as M-polyfold charts from \cite{b:thesis}, which will also make the predicted boundary stratification rigorous. Further steps in the program are the Fredholm property of the Cauchy--Riemann operator, and formal setups for the construction of gluing-coherent perturbations and orientations.
However, we can already see that the boundary stratification of this polyfold is --- apart from strata of types (B1--4) on page~\pageref{Bs} --- induced by the boundary structure of the compactified Morse trajectory spaces from \cite{w:morse}.
These $\delta=-\infty$ boundary components correspond to the boundary strata of the compactified space of colored metric ribbon trees (given by allowing infinite edge length). Since stable trees of this sort realize Stasheff's multiplihedra \cite[Thms.\ 1.1, 7.6]{mw}, we expect to obtain $A_\infty$-functor relations from this boundary component.
More precisely, we expect to obtain algebraic relations from the {\bf top boundary strata of the compactified strip-shrinking moduli space with Morse bubble trees}, which replace our previous list (B1--5) as follows:
\begin{figure}
\centering
\def6in{\columnwidth}
\input{prediction.pdf_tex}
\caption{A contribution to the differential on \(CF( L_0, (L_{01}\circ L_{12}, \sum_{k,l\geq0} b_{02}^{k|l}), L_2 )\). The two subtrees above the Morse critical points contribute to \(b_{02}^{1|1}\) and \(b_{02}^{0|0}\), respectively.
The dashed lines indicate the level structure of the colored metric ribbon tree. Additional half-infinite edges are labeled with the Morse cochains $b_{01}$ or $b_{12}$, indicating a formal sum of trees whose half-infinite edges are Morse edges starting at the Morse critical points that represent the cochains.
}
\label{fig:prediction}
\end{figure}
\label{BBs}
\begin{enumerate}
\item[(B1')]
quilted Floer trajectories for $\delta=1$ which may include trees of disk bubbles with finite length Morse edges;
\item[(B2')]
once-broken quilted Floer trajectories for $\delta \in (0,1)$ which may include trees of disk bubbles with finite-length Morse edges;
\item[(B2'')]
once-broken quilted Floer trajectories can also appear for $\delta \in (-\infty,0)$, where they consist of Floer trajectories of width $0$ with a colored tree of figure eight height $-\delta$, and may include further trees of disk bubbles with finite-length Morse edges;
\item[(B3')]
quilted Floer trajectories with one disk bubble on a seam for $\delta \in (0,1)$ are canceled as boundary components, but new boundary components contain quilted Floer trajectories for $\delta \in (0,1)$ with trees of disk bubbles, in which one Morse edge is broken once;
\item[(B3'')]
quilted Floer trajectories of width $0$ with a single broken Morse edge can also appear for $\delta \in (-\infty,0)$, i.e.\ in a colored tree of figure eight height $-\delta$ with the broken edge occurring either above the figure eight height or in a tree of disk bubbles on another seam;
\item[(B4')]
quilted Floer trajectories for $\delta=0$ may include trees of disk bubbles with finite length Morse edges attached to seams that are not involved in the shrinking; but when the width goes to zero in the presence of a tree of disk bubbles on $L_{01}$ or $L_{12}$, this is viewed as constant figure eight to which this tree is attached, and is canceled like other strata of type (B5);
\item[(B4'')]
quilted Floer trajectories for $\delta=0$ with squashed eight bubbles are canceled as boundary components, but new boundary components contain quilted Floer trajectories of width $0$ with a colored tree of figure eight height $-\delta=\infty$, all of whose colored vertices are ghost eights with infinite ghost edges;
equivalently, these are quilted Floer trajectories for $\delta=0$ with trees of squashed eights and finite Morse edges attached to the shrunk seam;
\item[(B5')]
quilted Floer trajectories for $\delta=0$ with figure eight bubbles are canceled as boundary component, but new boundary components contain quilted Floer trajectories of width $0$ with a colored tree of figure eight height $-\delta=\infty$, that is between each figure eight or ghost eight and main quilt there is exactly one infinite edge --- either an infinite ghost edge or a once-broken Morse trajectory (of which there is at least one, otherwise this component is listed under (B4'')) --- and all other edges have finite length.
\end{enumerate}
An example of a boundary point of type (B5') is given in Figure~\ref{fig:prediction}, where the dashed lines indicate the level structure of the colored metric ribbon tree resulting from strip shrinking:
In the first level above the root quilt, all vertices are represented by squashed eights, whereas in the third level the vertices are represented by disk bubbles with boundary on $L_{01}$ or $L_{12}$. The second level provides the division since between each leaf and the root there is exactly one vertex represented by a figure eight --- though we did not graphically represent the ghost eight vertices above the two squashed eight leaves on the left subtree.
The figure eight height of this tree is $\infty$, reflected by at least one broken trajectory below each figure eight --- in particular, the graphically unrepresented ghost edges between the squashed eight leaves and ghost figure eight vertices have length $\infty$.
The corner index is $1$ since there is exactly one breaking for each figure eight (resp.\ infinite edge for the ghost eights).
A similar colored tree of corner index $1$ and figure eight height $\infty$ could be obtained by giving the ghost edges above the squashed eight leaves finite lengths, but replacing either both edges below the leaves with a broken trajectory, or having just one broken trajectory at the edge which attaches both leaves to the root.
However, we do not expect algebraic contributions from the latter tree types: Geometrically, this would mean an isolated solution on the boundary of the polyfold containing the moduli space of figure eights with outgoing infinite Morse edge. In terms of our tree setup, such solutions aren't isolated since the length of the finite ghost edge can be varied.
\subsection{Floer homology isomorphism for general cleanly-immersed geometric composition} \label{ss:algebra1}
To generalize the isomorphism between quilted Floer homologies \eqref{eq:HFiso} under monotone, embedded composition to general symplectic manifolds and Lagrangians and cleanly-immersed composition \(L_{01} \circ L_{12}\), we analyzed in Section~\ref{ss:boundary} the boundary strata of the polyfold which provides an ambient space for a general compactified moduli space of quilted Floer trajectories with varying width.
The cobordism argument outlined in Remark~\ref{rmk:codim} then predicts an algebraic identity from summing over the boundary strata (B1--5) on page~\pageref{Bs} resp.\ the refined boundary strata on page~\pageref{BBs}.
We expect the strata of types (B2), (B3) resp.\ (B2'), (B2''), (B3'), (B3'') to appear only at finitely many singular values of strip width $\delta\in (0,1)$ or figure eight height $\delta\in (-\infty,0)$ and to provide a chain homotopy equivalence between two Floer complexes:
The first is in both frameworks defined from the regular strip width $\delta = 1$, with the differential given by counting solutions of type (B1) resp.\ (B1'). In the Morse framework, the second Floer complex is defined from counting regular solutions of types (B4'), (B4''), and (B5'). In the framework of page~\pageref{Bs}, the second complex should arise from solutions of types (B4) and (B5) at $\delta=0$, though it is unclear in what sense the latter might be made regular.
Up to such a chain homotopy equivalence, or assuming there are no singular values in $(-\infty,1)$, we obtain the following identity relating the Floer differential \(\mu^1_{(\delta = 1)}\) arising from strip width $\delta=1$ and the Floer differential \(\mu^{1|0}_{(\delta=0)}\) arising from strip width $\delta=0$ with generalized seam condition in \(L_{01} \times_{M_1} L_{12}\):\footnote{
Note that the moduli spaces with generalized seam condition involve a choice of lift of the seam values to $L_{01}\times_{M_1}L_{12}$. So in case $L_{01}\circ L_{12}$ is a smooth Lagrangian correspondence albeit multiply covered by $L_{01}\times_{M_1}L_{12}$, this Floer complex is generated by lifts of intersection points. The differential \(\mu^{1|0}_{(\delta=0)}\) only counts Floer trajectories with smooth seam lift, whereas the terms \(\mu^{1|k}_{(\delta=0)}\) for $k\ge1$ will allow for jumps in the seam lift.
}
\begin{align} \label{eq:mu1relation}
\mu^1_{(\delta = 1)}(-) = \sum_{k\ge 0} \mu^{1 | k}_{(\delta = 0)} (\, - \, |\, b_{02},\ldots, b_{02}).
\end{align}
In the Morse framework, the moduli spaces defining the differentials \(\mu^1_{(\delta = 1)}\) and \(\mu^{1|0}_{(\delta=0)}\) both allow for trees of disks with finite Morse edges (including trees of squashed eights attached to the seam obtained from strip shrinking). Figure eight bubbling is in both frameworks encoded in the higher operations \(\mu^{1|k}_{(\delta=0)}\) for \(k \geq 1\).
In the framework of (B5), this operation should be defined from quilted Floer trajectories of middle strip width $0$ with $k$ incoming marked points on the seam labeled by the immersion $L_{01}\circ L_{12}$, and \(b_{02}\) should be a chain obtained from a moduli space of figure eight bubbles by evaluation at the singularity.
In the absence of an approach for making the latter rigorous, we will construct the operations\footnote{
When the composition \(L_{01}\circ L_{12}\) is embedded, we expect \(\mu^{1|k}_{(\delta=0)}\) to agree with the \(A_\infty\)-structure map \(CF(L_{01}\circ L_{12}, L_0\times L_2) \otimes CF(L_{01}\circ L_{12},L_{01}\circ L_{12})^{\otimes k} \to CF(L_{01}\circ L_{12},L_0\times L_2)\) on \(\on{Fuk}(M_0^-\times M_2)\).
}
$$
\mu^{1|k}_{(\delta=0)} : \;
CF(L_0, L_{01}\circ L_{12}, L_2) \otimes CM(L_{01}\times_{M_1} L_{12})^{\otimes k} \;\longrightarrow\; CF(L_0, L_{01}\circ L_{12}, L_2)
$$
in the Morse framework from quilted Floer trajectories of strip width $0$ with \(k\) incoming Morse edges (represented by half-infinite trajectories in $L_{01}\times_{M_1} L_{12}$ starting at a Morse critical point) attached (possibly via trees of squashed eights and finite Morse edges) to the middle seam.
Then (B5') indicates that the Morse cochain
\begin{equation}\label{b02}
b_{02} \in CF(L_{01}\circ L_{12}, L_{01}\circ L_{12}) := CM(L_{01}\times_{M_1} L_{12})
\end{equation}
should be defined by counting regular isolated figure eight bubble trees as follows:
\begin{itemlist}
\item
Exactly one vertex is represented by a figure eight, and this vertex lies between every leaf and the root. All vertices between leaves and the figure eight are represented by pseudoholomorphic disks, and all vertices between the figure eight and the root are represented by squashed eights.
\item
Each edge attached to a disk vertex is represented by a finite Morse trajectory on $L_{01}$ resp.\ $L_{12}$, and all other edges are represented by a finite Morse trajectory on $L_{01}\times_{M_1}L_{12}$.
\item
The root vertex is represented by a figure eight or squashed eight with a marked point at the singularity, to which an outgoing Morse edge is attached, i.e.\ a half-infinite trajectory in $L_{01}\times_{M_1} L_{12}$ ending at a Morse critical point.
\item
Disks and squashed eights are constant only if the vertex has valence $\ge 3$.
Figure eights are constant only if the vertex has valence $\ge 2$.
\end{itemlist}
\noindent
Once such operations are defined, \eqref{eq:mu1relation} identifies (up to chain homotopy equivalence) the quilted Floer chain complexes
\begin{align*}
CF\bigl(L_0, L_{01}, L_{12}, L_2 \bigr)
\;\simeq\;
CF\bigl(L_0, (L_{01} \circ L_{12}, b_{02}), L_2 \bigr),
\end{align*}
where the differential on the left hand side is $\mu^1_{(\delta = 1)}$ and the differential on the right hand side is the twisted differential
$\partial_{b_{02}}:= \sum_{k\ge 0} \mu^{1 | k}_{(\delta = 0)} (\, - \, |\, b_{02},\ldots, b_{02})$.
Here the right hand side treats $L_{01} \circ L_{12}$ as an immersion. If this is an embedding then the right hand side is the Floer chain complex of the Lagrangian $L_{01} \circ L_{12}\subset M_0^-\times M_2$ twisted by the Morse cochain $b_{02}$.
An example of a contribution to the twisted Floer differential is Figure~\ref{fig:prediction} without the middle tree.
This result is meaningful if the cyclic Lagrangian correspondence \(L_0, L_{01}, L_{12}, L_2\) is naturally unobstructed in the sense that the differential $\partial =\mu^1_{(\delta = 1)}$ satisfies $\partial^2= 0$. In particular, it asserts that the twisted differential on the right hand side satisfies $\partial_{b_{02}}^2 = 0\).
To understand more intrinsically why the twisted differential squares to zero, we need to go into the $A_\infty$ algebra.
\medskip
\noindent
{\bf Remark on $\mathbf{A_\infty}$
terminology:} {\it In the upcoming sections, we will denote by $\bigl(\mu^d_{01}\bigr)_{d\ge 0}$, $\bigl(\mu^d_{12}\bigr)_{d\ge 0}$, resp.\ $\bigl(\mu^d_{02}\bigr)_{d\ge 0}$ the curved $A_\infty$-algebras associated to $L_{01}$, $L_{12}$, resp.\ $L_{01}\circ L_{12}$, constructed on Morse chain complexes as outlined in Remark~\ref{rmk:disks}. If working with the latter, we will usually assume that $L_{01}\circ L_{12}$ is embedded, though there are extensions to multiply covered and even cleanly immersed cases, as outlined in Remark~\ref{rem:immfuk}.
Moreover, we will call \(b \in CF(L,L)\) a \emph{bounding cochain for the Lagrangian \(L\)} if it satisfies the Maurer--Cartan equation \(\sum_{d\geq 0} \mu^d(b,\ldots,b) = 0\). When a quilted Floer differential is twisted by bounding cochains for each Lagrangian correspondence, it will square to zero. It is however also possible that twisting with more general cochains yields a chain complex.
}
\medskip
\begin{figure} \label{fig:MC}
\centering
\includegraphics[width=\columnwidth]{MC.pdf}
\caption{
The expected boundary strata of a figure eight moduli space explain various algebraic identities in Remark~\ref{rmk:b02}.
If contributions from disk bubbles on $L_{01},L_{12}$ can be excluded
(i.e.\ \(\mu_{01}^0 = \mu_{12}^0 = 0\))
then the element \(b_{02} \in CM(L_{01}\times_{M_1} L_{12})\)
should solve the Maurer--Cartan equation for \(L_{01} \circ L_{12}\).
For monotone, embedded composition, the figure eight bubbles can be excluded to explain the identity $n_{L_{01}} + n_{L_{12}} = n_{L_{01}\circ L_{12}}$ between disk counts.
}
\end{figure}
\begin{remark} \label{rmk:b02}
The vanishing \(\partial_{b_{02}}^2 = 0\) generally follows from the identification of differentials $\partial=\partial_{b_{02}}$ in \eqref{eq:mu1relation} together with the assumption $\partial^2=0$.
In more special cases we expect this to be a consequence of \(b_{02} \in CF(L_{01}\circ L_{12}, L_{01}\circ L_{12})\) being a bounding cochain, i.e.\
satisfying the Maurer--Cartan equation \(\sum_{d=0}^\infty \mu_{02}^d(b_{02},\ldots,b_{02}) = 0\).
(Here we assume \(L_{01} \circ L_{12}\) to be an embedded composition, though this remark should extend to the cleanly-immersed setting.)
This should follow from a cobordism argument illustrated in Figure~\ref{fig:MC}: Consider the 1-dimensional moduli space of figure eight quilts between \(L_{01}\) and \(L_{12}\), with a half-infinite outgoing Morse trajectory on \(L_{01}\times_{M_1}L_{12}\) attached to its singularity.
Extrapolating from the boundary analysis in \S\S\ref{ss:HF}--\ref{ss:Morse}, we expect the 0-dimensional boundary strata to come in two types:
\begin{itemize}
\item Some strata are given by squashed eights with seam in \(L_{01}\times_{M_1} L_{12}\), with one outgoing half-infinite Morse trajectory on \(L_{01}\times_{M_1}L_{12}\) attached to the singularity, and \(d \geq 0\) figure eight bubbles attached to the seam via once-broken Morse trajectories on \(L_{01}\).
The formal sum over the limiting critical points of the outgoing trajectories of such isolated solutions yields \(\mu_{02}^d(b_{02},\ldots,b_{02})\).
\item The remaining strata are given by figure eights with one outgoing half-infinite
Morse trajectory attached to the singularity, and a disk bubble mapping to \((M_k^-\times M_{k+1}, L_{k(k+1)})\) for either $k=0$ or $k=1$ attached to the \(L_{k(k+1)}\)-seam via a once-broken Morse trajectory on \(L_{k(k+1)}\). The formal sum over the limiting critical points of such isolated solutions yields \(C^2(\:|\: \mu_{01}^0)\) resp.\ \(C^2(\mu_{12}^0 \:|\:)\) when \(k = 0\) resp.\ \(k=1\), where \(C^2\) is the curved \(A_\infty\)-bifunctor whose blueprint we sketch in \S\ref{ss:algebra2}.
\end{itemize}
As boundaries of a $1$-dimensional moduli space, the algebraic contributions of these boundary strata should sum to zero. (In fact, this equation is also a formal consequence of the curved \(A_\infty\)-bifunctor relations satisfied by \(C^2\).)
In the special case \(\mu_{01}^0 = \mu_{12}^0 = 0\) (i.e.\ when there are no disk bubbles on $L_{01}$ or $L_{12}$, or their contributions cancel) this yields the expected Maurer--Cartan equation \(\sum_{d\geq 0} \mu_{02}^d(b_{02},\ldots,b_{02}) = 0\).
This also illuminates an identity between disk counts noted in \cite[Remark~2.2.3]{isom}: Working with monotone orientable Lagrangians and embedded composition, one expects both differentials $\partial_{\delta}:=\mu^1_{(\delta>0)}$ and $\partial_{0}:=\mu^1_{(\delta=0)}$
to square to multiples of the identity, $\partial_{\delta}^2 = w_\delta {\rm id}$ resp.\ $\partial_{0}^2 = w_0 {\rm id}$, with
$w_\delta= n_{L_0} + n_{L_{01}} + n_{L_{12}} + n_{L_2}$ resp.\ $w_0= n_{L_0} + n_{L_{01}\circ L_{12}} + n_{L_2}$ given by sums of counts $n_L$ of Maslov index $2$ disks through a generic point on the Lagrangian $L$. Arguing by strip shrinking identifying the differentials, \cite{isom} concluded $w_\delta=w_0$ and hence
$n_{L_{01}} + n_{L_{12}} = n_{L_{01}\circ L_{12}}$.
This identity can now also be seen directly from the above cobordism argument: Monotonicity excludes nonconstant figure eight bubbles, which reduces the boundary strata on the right hand side of Figure~\ref{fig:MC} to the first and last two types, corresponding to $n_{L_{01}\circ L_{12}}$ and $n_{L_{01}}$, $n_{L_{12}}$, respectively.
\end{remark}
Next, we relax the unobstructedness to the assumption that the Lagrangians \(L_0, L_{01}, L_{12}, L_2\) are equipped with cochains
\(\underline{b}=(b_0, b_{01}, b_{12}, b_2)\) so that the twisted differential $\partial_{\underline{b}}$ (which arises from adding marked points labeled with $b_0$, $b_{01}$, $b_{12}$, resp.\ $b_2$ to the $\delta=1$ quilted Floer trajectories) satisfies $\partial_{\underline{b}}^2=0$.
Then we may add these cochains to the previous strip-shrinking moduli space as incoming Morse edges whose starting points represent $b_0, b_{01}, b_{12}$, resp.\ $b_2$ and whose endpoints correspond to marked points on the seams labeled $L_0, L_{01}, L_{12}$, resp.\ $L_2$ anywhere on the quilted Floer trajectory or the attached bubble trees.
Then an analogous cobordism argument yields a chain homotopy equivalence
\begin{align*}
CF\bigl( (L_0,b_0), (L_{01},b_{01}), (L_{12},b_{12}), (L_2,b_2) \bigr)
\; \simeq\;
CF\bigl( (L_0,b_0), \bigl(L_{01}\circ L_{12}, {\textstyle \sum_{k,\ell\geq0}} b_{02}^{k|\ell}\bigr), (L_2,b_2) \bigr),
\end{align*}
where the Morse cochains \(b_{02}^{k|\ell} \in CF(L_{01}\circ L_{12},L_{01}\circ L_{12})\) are obtained by adding incoming Morse edges to the figure eight bubble trees that define \(b_{02}=:b_{02}^{0|0}\).
More precisely, we attach \(k\) incoming Morse edges representing \(b_{12}\in CF(L_{12},L_{12})\) to the \(L_{12}\) seams anywhere on the bubble tree, and we attach \(\ell\) incoming Morse edges representing \(b_{01}\) to the \(L_{01}\) seams.
Figure~\ref{fig:prediction} provides an example of figure eight contributions to the twisted Floer differential for the composed Lagrangian correspondences on the right-hand side of the equivalence.
This demonstrates, as advertised in the introduction, that the isomorphism of quilted Floer homologies \eqref{eq:HFiso} should generalize in a straightforward fashion to the nonmonotone, cleanly-immersed case as isomorphism of quilted Floer homologies with
twisted differentials,
\begin{align*}
HF\bigl( \ldots , (L_{01},b_{01}), (L_{12},b_{12}), \ldots \bigr)
\; \simeq\;
HF\bigl( \ldots , (L_{01}\circ L_{12}, 8(b_{01},b_{12}) ), \ldots \bigr),
\end{align*}
in which the cochain $8(b_{01},b_{12})$ for the composed Lagrangian is obtained from moduli spaces of figure eight bubble trees with inputs $b_{01}$ and $b_{12}$.
In particular, the cochain $8(0,0)=b_{02}$ in \eqref{b02} for vanishing inputs is a generally nonzero count of figure eight bubbles.
\begin{remark}
The cobordism argument in Remark~\ref{rmk:b02} can be adapated to the situation that \(L_{01}, L_{12}\) are equipped with bounding
cochains \(b_{01} \in CF(L_{01}, L_{01})\), \(b_{12} \in CF(L_{12},L_{12})\):
This time, for every \(k, \ell \geq 0\) we consider 1-dimensional moduli spaces of figure eight quilts with one outgoing half-infinite Morse trajectory attached to the singularity, and \(k\) resp.\ \(\ell\) incoming half-infinite Morse trajectories attached to the \(L_{01}\)-seam resp.\ the \(L_{12}\)-seam.
The algebraic contributions of the boundary strata with incoming Morse critical points representing \(b_{01}\) and \(b_{12}\) should sum to zero.
Summing over all \(k, \ell \geq 0\), we obtain the expected equation \begin{align*}
&\sum_{k, \ell \geq 0}
\sum_{
\substack{k_1+\cdots+k_d=k, \\ \ell_1+\cdots+\ell_d=\ell}
}
\mu_{02}^d\bigl(b_{02}^{k_d|\ell_d}, \ldots, b_{02}^{k_1|\ell_1}\bigr) \\
&\hspace{1in} = \sum_{k,\ell\geq 0}\sum_{a+d\leq k} C^2\Bigl(\underbrace{b_{12}, \ldots, b_{12}}_\ell \:|\: \underbrace{b_{01}, \ldots, b_{01}}_{k-a-d}, \mu_{01}^d(b_{01}, \ldots, b_{01}), \underbrace{b_{01}, \ldots, b_{01}}_a\Bigr) \\
&\hspace{1.5in} + \sum_{k,\ell\geq 0}\sum_{a+d\leq \ell} C^2\Bigl(\underbrace{b_{12}, \ldots, b_{12}}_{\ell-a-d},\mu_{12}^d(b_{12},\ldots,b_{12}),\underbrace{b_{12},\ldots,b_{12}}_a \:|\: \underbrace{b_{01}, \ldots, b_{01}}_k\Bigr).
\end{align*}
The right side vanishes, after a reorganization, by the Maurer--Cartan equations for $b_{01}$ and $b_{12}$. Then a reorganization of the left side yields the expected Maurer--Cartan equation for \(L_{01}\circ L_{12}\),
\begin{align*}
\sum_{d\geq 0} \mu_{02}^d\Bigl( {\textstyle\sum_{k,\ell\geq0}} b_{02}^{k|\ell}, \ldots, {\textstyle\sum_{k,\ell\geq0}} b_{02}^{k|\ell}\Bigr) = 0.
\end{align*}
\end{remark}
\subsection{Conjectural categorical framework for geometric composition} \label{ss:algebra2}
\begin{figure}
\centering
\def6in{6in}
\input{functordef.pdf_tex}
\caption{The \(A_\infty\)-bifunctor \(C^2\) will be defined by counting figure eight quilts
with input marked points on the seams as on the left of this figure.
The \(A_\infty\)-functor \(C^2_{L_{01}}\) is then obtained by setting \(M_2:=\pt\), which amounts to eliminating the \(M_2\)-patch and replacing the Lagrangian correspondences \(L_{12}^i\) with Lagrangians
\(L_1^i \subset M_1^-\).
So \(C^2_{L_{01}}\) is defined by counting quilted disks as on the right of this figure.
}
\label{fig:functordef}
\end{figure}
The cochains \(b_{02}^{k|\ell}\) in the previous section can be viewed as the result of a map $CF(L_{12},L_{12})^{\otimes k} \otimes CF(L_{01},L_{01})^{\otimes\ell} \to CF(L_{01}\circ L_{12},L_{01}\circ L_{12})$ applied to $ b_{12}^{\otimes k} \otimes b_{01}^{\otimes \ell}$.
More generally, we propose that the figure eight should not be viewed as an undesirable obstruction, but as a geometric object whose raison d'\^{e}tre is to give rise to maps between Floer cochain groups involving geometric composition.
For that purpose note that the moduli spaces of quilted disks constructed in \cite{mw} to represent Stasheff's multiplihedra can be viewed as configuration spaces of marked points on one of the two seams of the domain of figure eight bubbles. Allowing for marked points on both seams is analogous to the construction of ``biassociahedra'' in \cite{mww}, and will again yield families of marked quilted surfaces parametrized by polyhedra. We can then build moduli spaces of pseudoholomorphic quilts whose domains are given by points in a biassociahedron.
More precisely, we modify the figure eight bubble by placing \(d \geq 0\) input marked points on the \(12\)-seam (between the \(M_1\) patch and the \(M_2\) patch), \(e \geq 0\) input marked points on the \(01\)-seam, and one output marked point at the singular point of the quilt (the left half of Figure~\ref{fig:functordef} is the \(d=1, e=2\) case). The \(12\)-seam is now divided into \(d+1\) segments which we label by Lagrangians \(L_{12}^0, \ldots, L_{12}^d \subset M_1^- \times M_2\). Similarly, we label the segments of the \(01\)-seam by \(L_{01}^0, \ldots, L_{01}^e \subset M_0^- \times M_1\). Given a finite-energy pseudoholomorphic quilt with this domain, and assuming that \(L_{01}^0 \circ L_{12}^0\) and \(L_{01}^e \circ L_{12}^d\) are cleanly immersed and intersect each other cleanly,
it follows\footnote{
\cite{b:singularity} considers figure eight bubbles with no input marked points (i.e.\ with seams mapping to single Lagrangians \(L_{01}, L_{12}\)). However, the proof is local at the singularity, so it also applies to figure eights with marked points.} from
\cite[Removal of Singularity Thm.\ 2.2]{b:singularity} that the limit of the three maps at the output marked point is a generator of \(CF(L_{01}^0 \circ L_{12}^0, L_{01}^e \circ L_{12}^d)\).
The 0-dimensional moduli space of such quilts should therefore define a map
\begin{align}
& C^2(- \:|\: -) \; \colon \;
CF(L_{12}^{d-1}, L_{12}^d) \otimes \cdots \otimes CF(L_{12}^0, L_{12}^0) \nonumber \\
&\hspace{1in}
\otimes CF(L_{01}^{e-1}, L_{01}^e) \otimes \cdots \otimes CF(L_{01}^0, L_{01}^1)
\;\;\longrightarrow\;\; CF(L_{01}^0 \circ L_{12}^0, L_{01}^e \circ L_{12}^d). \label{eq:bifunctor}
\end{align}
\begin{figure}
\centering
\def6in{6in}
\input{functor.pdf_tex}
\caption{
These quilted surfaces represent on the one hand the algebraic expressions in the bifunctor relation of Conjecture~\ref{conj:bifunctor} (with the exception of curvature terms), and on the other hand the expected boundary strata of the 1-dimensional moduli space of figure eight bubbles with one marked point on the \(01\)-seam and two on the \(12\)-seam (with the exception of bubbling that does not involve marked points).
}
\label{fig:bifunctorrel}
\end{figure}
The boundaries of 1-dimensional moduli spaces of such quilts will then give rise to a collection of relations among these maps. These boundary components will arise from several effects. Firstly, the underlying biassociahedron of quilted surfaces has boundary. In the example of Figure~\ref{fig:bifunctorrel}, its boundary strata correspond to the eight algebraic terms not involving $\mu^1$-terms. Secondly, Floer trajectories can break off at any input marked point.
In the example of Figure~\ref{fig:bifunctorrel}, this corresponds to the three algebraic terms in the first row involving pre-composition with $\mu^1_{01}$ or $\mu^1_{12}$. Moreover, energy concentrating at the outgoing marked point (where in cylindrical coordinates two pairs of $01$- and $12$-seams approach each other asymptotically) can be captured geometrically as a Floer trajectory for the composed Lagrangians breaking off. In the example of Figure~\ref{fig:bifunctorrel}, this corresponds to the algebraic term in the bottom left corner involving post-composition with $\mu^1_{02}$. Together, these algebraic terms capture the relations describing an \(A_\infty\)-bifunctor.
Finally, energy can concentrate without marked points being involved, yielding sphere, disk, or figure eight bubbles. Spheres will be interior points of the ambient polyfold, hence do not contribute to the algebraic relation. Disk bubbling can appear on a $01$- or $12$-seam, yielding algebraic terms involving pre-composition with $\mu_{01}^0$ or $\mu_{12}^0$, which reflect curvature of an \(A_\infty\)-algebra associated to a Lagrangian $L_{01}^i$ or $L_{12}^i$.
Since figure eight bubbling does not add to the corner index, we expect additional boundary faces arising from adding any number of figure eight bubbles without marked points to the $02$-seams of these configurations. Algebraically, this will be reflected by \(C^2( \: | \: )\) terms (with $d=e=0$) in any number of entries of \(\mu_{02}^k\), meaning that the \(A_\infty\)-bifunctor itself is curved.
More precisely, we expect for each \(d, e \geq 0\)
and fixed $(y_d,\ldots,y_1), (x_e,\ldots,x_1)$
the following relation:
\begin{align} \label{eq:Aoobifun}
& \sum
C^2\bigl(y_d, \ldots, y_{k+\ell+1}, \mu_{12}^\ell(y_{k+\ell},\ldots, y_{k+1}), y_k, \ldots, y_1 \:|\: x_e, \ldots, x_1\bigr) \:\:\: + \nonumber \\
&\hspace{0.15in} + \: \sum
C^2\bigl(y_d, \ldots, y_1 \:|\: x_e, \ldots, x_{k+\ell+1}, \mu_{01}^\ell(x_{k+\ell}, \ldots, x_{k+1}), x_k, \ldots, x_1\bigr) \nonumber \\
&\hspace{0.3in} = \sum \mu_{02}^n\bigl(
C^2(y_{j_n}, \ldots, y_{j_{n-1}+1} \:|\: x_{i_n}, \ldots, x_{i_{n-1}+ 1}), \ldots,
C^2(y_{j_1}, \ldots, y_{j_0+1} \:|\: x_{i_1}, \ldots, x_{i_0+1}) \bigr).
\end{align}
Here the first sum is over $0\le k \le d, 0\le \ell\le d-k$, the second sum is over $0\le k \le e, 0\le \ell\le e-k$, and the right hand sum is over $n\ge 1$ and order respecting partitions into $n$ (possibly empty) tuples $(x_e,\ldots,x_1)=(x_{i_n},\ldots,x_{i_{n-1}+1}) \cup \ldots \cup (x_{i_1},\ldots,x_{i_0+1})$, and
$(y_d,\ldots,y_1)=(y_{j_n},\ldots,y_{j_{n-1}+1}) \cup \ldots \cup (y_{j_1},\ldots,y_{j_0+1})$.
For example, the $(d,e)=(0,0)$ relation is
\begin{align*}
C^2(\mu_{12}^0 \:|\: ) + C^2( \:|\: \mu_{01}^0) =
{\textstyle \sum_{n=1}^\infty} \; \mu_{02}^n\bigl(C^2( \:|\: ), \ldots, C^2( \:|\: )\bigr).
\end{align*}
The twelve summands not involving curvature terms in the \(d = 1, e = 2\) case, together with their corresponding boundary strata, are shown in Figure~\ref{fig:bifunctorrel}. These relations are exactly what is required of a curved \(A_\infty\)-bifunctor \cite[Def.~8.8]{pretriangulated}, so we are led to the following conjecture.
\begin{conjecture} \label{conj:bifunctor}
Given compact symplectic manifolds \(M_0, M_1, M_2\), there exists a curved \(A_\infty\)-bifunctor \begin{align*}
C^2\colon (\on{Fuk}(M_1^- \times M_2), \on{Fuk}(M_0^- \times M_1)) \to \on{Fuk}(M_0^- \times M_2)
\end{align*}
between Fukaya categories of cleanly-immersed Lagrangians
that sends a pair of Lagrangians \\
\((L_{12}, L_{01})\) with cleanly-immersed composition to \(L_{01} \circ L_{12}\)
and is defined on the morphism level by the maps \(C^2(- \:|\: -)\) in \eqref{eq:bifunctor} in the case that all composed Lagrangians intersect one another cleanly.
\end{conjecture}
\begin{remark}[Immersed Fukaya categories] \label{rem:immfuk}
Conjecture~\ref{conj:bifunctor}
is naturally stated in the immersed setting: Even if the source Fukaya categories were chosen to contain only embedded Lagrangians as objects, the target Fukaya category would still need to contain immersed Lagrangians, since Hamiltonian perturbation of Lagrangian correspondences \(L_{01}, L_{12}\) can always achieve cleanly-immersed but generally not embedded composition.
The Fukaya categories in Conjecture~\ref{conj:bifunctor} will have as objects immersed Lagrangians \(\varphi\colon L \to M\) with clean self-intersections; boundary conditions in \(L\) for a map \(u\colon \Sigma \to M\) on \(\gamma \subset \partial \Sigma\) will require the data of a continuous lift of \(u|_\gamma\) to \(L\).
Cleanly-immersed Lagrangian boundary conditions have been discussed in various settings before, but \cite{bw:bigkahuna} will develop both analysis and algebra from scratch --- the first since we choose to work in the framework of polyfolds, and the second since our version of the immersed Fukaya category requires control of sheet-switching at self-intersection points in terms of cochain labels which encode contributions from sheet switching figure eight bubbles.
\end{remark}
\begin{remark}
Special cases of Conjecture~\ref{conj:bifunctor} will yield \(A_\infty\)-functors similar to the ones constructed in \cite{mww}. The main differences are that \cite{mww} works with \emph{extended} Fukaya categories (whose objects are composable sequences of embedded Lagrangian correspondences) and is necessarily limited to settings (such as monotonicity) in which figure eight bubbling is excluded.
\begin{enumlist}
\item[(i)]
For \(M_2=\pt\) the restriction of \(C^2\) to a fixed unobstructed object \(L_{01}\in \on{Fuk}(M_0^-\times M_1)\) yields a curved \(A_\infty\)-functor \(C^2_{L_{01}}\colon \on{Fuk}(M_1^-) \to \on{Fuk}(M_0^-)\). On the object level, this functor sends \(L_1 \subset M_1^-\) to \(L_{01} \circ L_1 \subset M_0^-\) if this composition is cleanly immersed; on the morphism level, this functor is defined by the maps \(C^2( - \:|\: )\) arising from the moduli spaces of quilted disks with a nonnegative number of marked points on the boundary circle, as on the right of Figure~\ref{fig:functordef}.
\item[(ii)]
For \(M_0=\pt\) the restriction of \(C^2\) to a fixed unobstructed object \(L_{12}\in \on{Fuk}(M_1^-\times M_2)\) yields a curved \(A_\infty\)-functor \(\phantom{.}_{L_{12}}{C^2}\colon \on{Fuk}(M_1) \to \on{Fuk}(M_2)\) that sends \(L_1 \subset M_1\) to \(L_1 \circ L_{12} \subset M_2\) if this composition is cleanly immersed, and is defined by the maps \(C^2( \:|\: - )\) on morphism level.
\item[(iii)]
We expect the special case $C^2_{\Delta_M}$ resp.\ $\!\!\phantom{.}_{\Delta_M}{C^2}$ of both functors to be
the identity functor on $\on{Fuk}(M\times \pt)\cong \on{Fuk}(M)\cong\on{Fuk}(\pt^-\times M)$.
More generally, we will show in \cite{bw:bigkahuna} that \((C^2_{L_{12}^T}, \!\phantom{.}_{L_{12}} C^2)\)
and \((\!\phantom{.}_{L_{12}}{C^2}, C^2_{L_{12}^T})\) form adjoint pairs, where \(L_{12}^T\subset M_2^-\times M_1\) is obtained from \(L_{12}\subset M_1^-\times M_2\) by exchange of factors.
\end{enumlist}
\end{remark}
The constructions we have described should also have flat incarnations, by a general algebraic process. Indeed, given a curved \(A_\infty\)-category \(\cA\), one can form its associated flat \(A_\infty\)-category \(\overline\cA\), an object of which is an object \(X\) of \(\cA\) together with a bounding cochain \(b \in \hom_{\cA}(X,X)\).
A curved \(A_\infty\)-multifunctor \(F\colon (\cA_k,\ldots,\cA_1) \to \mathcal{B}\) induces a flat \(A_\infty\)-multifunctor \(\overline F\colon (\overline \cA_k, \ldots, \overline \cA_1) \to \overline \mathcal{B}\), where \(\overline F\) is defined on the object level by \begin{align*}
\overline F\bigl((X_k, b_k), \ldots, (X_1, b_1)\bigr) := \Bigl( F(X_k,\ldots,X_1), \sum_{\ell_1, \ldots, \ell_k \geq 0} F\Bigl(\underbrace{b_k, \ldots, b_k}_{\ell_k} \:|\: \cdots \:|\: \underbrace{b_1, \ldots, b_1}_{\ell_1}\Bigr) \Bigr).
\end{align*}
Hence \(C^2\), once defined, descends to a flat \(A_\infty\)-bifunctor \begin{align*}
\overline C^2\colon (\overline\on{Fuk}(M_1^- \times M_2), \overline\on{Fuk}(M_0^- \times M_1)) \to \overline\on{Fuk}(M_0^- \times M_2).
\end{align*}
In particular, given \((L_{01}, b_{01}) \in \overline\on{Fuk}(M_0^-\times M_1)\), we can define \(\overline C^2_{(L_{01},b_{01})}\colon\overline\on{Fuk}(M_0) \to \overline\on{Fuk}(M_1)\). Observe that even if \(L_0\) and \(L_{01}\) are unobstructed, \(C^2_{L_{01}}(L_0)=L_0 \circ L_{01}\) may be obstructed, but \(\overline C^2_{(L_{01},0)}\) will specify a bounding cochain for \(L_0\circ L_{01}\) given by a count of figure eight bubbles in this context: quilted disks as on the right of Figure~\ref{fig:functordef} with no input marked points.
\medskip
Thus we hope to have demonstrated that figure eight bubbles are not only analytically necessary and manageable, but also in fact natural algebraic contributions when geometric composition of Lagrangian correspondences is involved.
We end with a brief outlook on how the full embrace of figure eight quilts, and their natural generalization, should yield a symplectic \(A_\infty\) 2-category.
\begin{remark}[Outlook on a proposed \(A_\infty\) 2-category]
A natural way to generalize the notion of figure eight quilt is to allow the seams in the domain to be any positive number \(d\) of circles that intersect tangentially at the south pole of the total domain \(S^2\), to allow any number of input marks on the seams, and to regard the south pole as an output mark. We call such a quilt a ``\((d+1)\)-patch witch ball''. (Then a 3-patch witch ball is the same as a figure eight quilt, and a 2-patch witch ball with label \(M_0\) and \(M_1\) is the same as a pseudoholomorphic disk with boundary marked points mapping to \(M_0^- \times M_1\).)
In future work, we plan to construct moduli spaces of \((d+1)\)-patch witch balls in order to define algebraic structures relating immersed Fukaya categories of different symplectic manifolds. For each \(d\geq1\) we plan to construct a ``map'' \begin{align*}
C^d\colon (\on{Fuk}(M_{d-1}^- \times M_d), \ldots, \on{Fuk}(M_0^-\times M_1)) \to \on{Fuk}(M_0^- \times M_d)
\end{align*}
defined on the object level by sending \((L_{(d-1)d}, \ldots, L_{01})\) to \(L_{(d-1)d} \circ \cdots \circ L_{01}\) (in the case that this composition is cleanly-immersed) and on the morphism level by counting rigid \((d+1)\)-patch witch balls. In particular, \(C^2\) is the conjectural \(A_\infty\)-bifunctor described in \S\ref{ss:algebra2}, and \(C^1\) is the \(A_\infty\)-structure in the immersed Fukaya category \(\on{Fuk}(M_0^-\times M_1)\) (see Remark~\ref{rem:immfuk}).
We expect algebraic identities between the \(C^d\)'s from the boundary stratification of the Deligne--Mumford-type spaces of domains. The moduli spaces of 3-patch witch ball domains are the biassociadra; these are polytopes constructed in \cite{mw} whose local coordinates arise from the relative positions of the marked points. For \(d \geq 3\), an extension of \cite{mw} should yield polytopes parametrizing \((d+1)\)-witch ball domains, whose local coordinates will come from the relative positions of both the marked points and the seams. Preliminary analysis of the moduli spaces of \((d+1)\)-patch witch balls suggests, for each \(d \geq 1\), a relation involving \(C^1, \ldots, C^d\), of which the first is the \(A_\infty\)-equation
\begin{align*}
\sum C^1(x_e, \ldots, x_{k+\ell+1}, C^1(x_{k+\ell}, \ldots, x_{k+1}), x_k, \ldots, x_1) = 0
\end{align*}
and the second is the \(A_\infty\)-bifunctor equation \eqref{eq:Aoobifun}. The resemblance of these relations to the hierarchy of relations satisfied by an \(A_\infty\)-category has led the first author and H.\ Tanaka to formulate in \cite{bt} the notion of an ``\(A_\infty\) 2-category''. Based on this, we plan to construct in \cite{bw:bigkahuna} the \(A_\infty\) 2-category \textsf{Symp}, whose objects are symplectic manifolds, 1-morphisms are Lagrangian correspondences, and 2-morphisms are Floer cochains, and whose structure maps are the \(C^d\)'s that arise from figure eight quilts and their natural generalization to witch balls.
\end{remark}
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{"url":"http:\/\/whatindiadiscovered.blogspot.com\/","text":"## Saturday, 9 September 2017\n\n### Was pi really invented in India?\n\nWikipedia says that pi was invented by William Jones, a mathematician from Wales, in the 1970s. But, little do we know that our own legendary mathematician Aryabhatta had mentioned the value of pi in one of his ancient shloks, written in 455 A.D.\n\nOur ancestors had a very unique way of passing their time in those days. Most of their time was spent in doing yagnas to various demigods and goddesses. One of the peculiar features of the yagnakundas used for doing yagna was their unique shapes. Some yagnakundas were square, some were circle and some semi circle. The area of the yagnakunda, however, was same for all yagnakundas.\n\nNow, a simple mathematical equation, as given below, between the area of square and circle, can give us the radius of a circle whose area is equal to a given square.\n\nBut, how did the ancient Indians build yagnakundas of same area and different shapes? The following shlok written by Aryabhatta in 455 A.D. gives us an idea of how the ancient Indians might have been able to design such intricate structures:\n\n[gaNita pAda, 10]\u00a0Aryabhatiyam (499 CE)","date":"2019-03-19 03:11:34","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8285594582557678, \"perplexity\": 2416.140044011618}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-13\/segments\/1552912201882.11\/warc\/CC-MAIN-20190319012213-20190319034213-00436.warc.gz\"}"}
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Q: Kafka Stream timestamp synchronization in KStream/KTable join with repartitioning I need a solution for a Kstream-Ktable join where records will be out-of-order because of preprocessing in topology. I am using kafka-streams:3.0.1
I have topic1 and topic2, if they are keyed on the same key I get the expected result in KStream-KTable join.
T1, T2,..Tn are timestamps.
topic1 records: [{K1,V1}(T2), {K2,V2}(T4)]
topic2 records: [{K1,A}(T1), {K2,B}(T3)]
join result: [{K1,A}, {K2,B}]
KStream stream = builder.stream("topic1");
KTable table= builder.table("topic2");
stream.leftJoin(table, (v1,v2)->v2)
.to("enrichedTopic");
But, if there is pre-processing in topology which results in repartition, the result will not be the same.
topic1 records: [{K1,V1}(T2), {K2,V2}(T4)]
topic2 records: [{*K1,A}(T1), {*K2,B}(T3)]
KStream stream = builder.stream("topic1");
KTable table= builder.stream("topic2")
.selectKey((k,v)->k.replace("*","") //remove '*' from the key
.repartition()
.toTable();
stream.leftJoin(table, (v1,v2)->v2)
.to("enrichedTopic");
Below are two solutions with their disadvantages. I want to know if there is any better way to achieve this.
Solution 1: Increasing max.task.idle.ms
This gives the right result, but I am afraid that it will slow down the processing if topic2 has slow incoming traffic. As join will wait for topic2 partitions data.
Solution 2: Foreign key join
Not completely sure about the result, as it also involves repartitioning. But, it will also produce an increased number of result if have frequent updates on topic2 for the same key and one record from topic2 is mapped to multiple records of topic1 (topic:topic1 is 1:n).
This will also need a logic to remove the records produced by updates on topic2 as we are only interested in value of topic2 when there is an event on topic1 based on time synchronization.
Please correct if there is anything incorrect in my understanding.
Questions:
*
*Let know me if there is any better approach to solve this.
*Will solution 2 guarantee the correct result if we only consider the first resultent record produced by a foreign key join (in order to ignore the updates from topic2)?
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{
"redpajama_set_name": "RedPajamaStackExchange"
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| 1,768
|
module Codewars.Kata.NthSeries where
import Text.Printf (printf)
seriesSum :: Integer -> String
seriesSum 0 = "0.00"
seriesSum n = printf "%.2f" (sum $ map (1.0/) (take (fromIntegral n) [1, 4 ..]) :: Double)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,447
|
Scenario 6: CPH Failure
The scenario represents the failure of the negotiations about the long-term commitments: countries comply with their less strict pledge announcements until 2030, and stop any other emission reduction after 2020. This scenario represents a sort of threat scenario, illustrating what could happen if no long-term agreement is reached apart from the current Copenhagen Accord.
Trade of carbon permits occurs during the years of the Copenhagen Accord. In TIAM, it is limited to 10% in 2020 and 20% in 2030 of the required emission reduction of each country, including the voluntary emission reductions.
The long-term energy system of the different countries doesn't differ from the energy system of the reference: some effects of short-term mitigation efforts on the energy system remain visible during around 10 years after the end of the climate agreement (for example, higher generation of wind-based electricity to substitute coal-based electricity), but they disappear in the longer term. The measures implemented under the Copenhagen Accord are not deep enough to modify the energy system forever. Such a conclusion could be of course different in case of deeper emission cuts in the first part of the century, permitting for example the penetration of new technologies in a sufficient volume to provoke a decrease of their cost and make them cost-efficient even without any climate constraint.
Despite the return to an energy system similar to the reference, long-term net emissions result to be slightly smaller (5 to 7% smaller) than the emissions of the reference case because of the long-term impacts of forestry measures: by default in TIAM, plantations decided in the first part of the century last during several decades given the lifetime of trees. The reality could be of course different if the countries have no incentive anymore to manage in a sustainable manner their forests.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,471
|
Coroplast is typically used for yard, parking, real estate, and election signs. They are waterproof. If utilizing H-Stakes, place your Rigid Coroplast order with the flutes vertical.
Available in Single Sided or Double Sided.
Also as an option they come with bootleg wire coro stakes for driving into grass and gravel.
Coroplast Yard Signs 12 in x 24 in Just need one? Great! You can order just one coroplast corrugated yard sign on this page.
Coroplast Yard Signs 18 in x 24 in Just need one? Great! You can order just one coroplast corrugated yard sign on this page.
Set of 12 signs for you to use for your service business, yard sale or event.
Set of 18 signs for you to use for your service business, yard sale or event.
Set of 24 signs for you to use for your service business, yard sale or event.
Set of 36 signs for you to use for your service business, yard sale or event.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,897
|
This exclusive Chi Kri sequence was designed by Neil Patel in order to strengthen the mind, the legs, improve co-ordination, focus, and something else really crucial...oh yea, stamina.
Practice at home and enjoy!
Get the full 27 step Chi Kri Warrior Sequence CD with guided instructions here!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,362
|
Q: How to make the div height dynamic How to make the div height dynamic so that it increases the height automatically when data inserted exceed the size of div?
<div class="group"style="background:white;box-shadow: 3px 3px 3px #DDDADA;">
<div class="cadetabox" id="s10">
<div class="capht"><img src="images/user_picture.jpg" width="50" height="50"/></div>
<div class="caphtdtail">
<a href="CADetail.php"><h4>CA Ritesh </h4></a>
<p> learn how to setup video calls learn how to setup video calls learn how to setup video callslearn how to setup video callslearn how to setup video callslearn how to setup video callslearn how to setup video callslearn how to setup video calls Kandivali, Mumbai Kandivali, Mumbai Kandivali, Mumbai Kandivali, Mumbai Kandivali, Mumbai Kandivali, Mumbai </p>
<p>2 yrs Experience</p>
<p style="color: green;">Open Today</p>
</div>
<a href="login-register.php"><span class="likespan"><i class="fa fa-thumbs-o-up" style="font-size: 18px;"></i> 0</span></a>
<a href="#"><span class="span" data-toggle="modal" data-target="#myModal">Call</span></a>
<div class="clearfix"></div>
</div>
</div>
A: You need to use CSS height property to make the div height dynamic.
Use height:auto; Example
And here is a sample of your code.
.caphtdtail {
border: 2px solid;
height: auto;
}
<div class="group" style="background:white;box-shadow: 3px 3px 3px #DDDADA;">
<div class="cadetabox" id="s10">
<div class="capht">
<img src="images/user_picture.jpg" width="50" height="50" />
</div>
<div class="caphtdtail">
<a href="CADetail.php"><h4>CA Ritesh </h4></a>
<p>learn how to setup video calls learn how to setup video calls learn how to setup video callslearn how to setup video callslearn how to setup video callslearn how to setup video callslearn how to setup video callslearn how to setup video calls Kandivali,
Mumbai Kandivali, Mumbai Kandivali, Mumbai Kandivali, Mumbai Kandivali, Mumbai Kandivali, Mumbai</p>
<p>2 yrs Experience</p>
<p style="color: green;">Open Today</p>
</div>
<a href="login-register.php"><span class="likespan"><i class="fa fa-thumbs-o-up" style="font-size: 18px;"></i> 0</span></a>
<a href="#"><span class="span" data-toggle="modal" data-target="#myModal">Call</span></a>
<div class="clearfix"></div>
</div>
</div>
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,512
|
class Performance {
constructor() {
self.BabylonianPerformance = this;
this.reset();
}
reset() {
this.steps = {};
this.lastTime = null;
this.curStep = null;
}
step(stepName) {
// Log old step if there was one
if(this.curStep) {
// Get duration
const duration = this.duration();
// Push current duration
this.steps[this.curStep].push(duration);
}
// Add next step
this.curStep = stepName
if(!(this.curStep in this.steps)) {
this.steps[this.curStep] = [];
}
// Start timer
this.lastTime = this.now();
}
stop() {
// Get duration
const duration = this.duration();
if(!this.curStep) {
return;
}
// Push current duration
this.steps[this.curStep].push(duration);
this.curStep = null;
}
duration() {
return this.now() - this.lastTime;
}
now() {
return performance.now();
}
}
// Only export as Singleton
export default new Performance();
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,012
|
Q: In what order should i put objects to database? I have database in mysql and connection to java with hibernate. I have 4 tables:
Clients 1 to many with Orders
Orders 1 to many with OrdersProducts
OrderProducts many to 1 with Products.
I have clients and products already in database. My program collects products and client from user and I want to add them to database. All I need to do is:
-add Order with ArrayList of OrderProducts
-add all OrderProducts
-update client ArrayList of Orders with new order
-update all Products with ArrayLists of OrderProduts
My question:
Is it matter which objects i put first to database and if so what should be correct sequence?
My entities below: (edited)
@Entity
public class Clients implements Serializable {
@Id
@GeneratedValue
private int idClient;
private String clientName, clientSurname, clientCompany;
@OneToMany(mappedBy = "clients")
private List<Orders> listOfOrders = new ArrayList<Orders>();
}
@Entity
public class Orders implements Serializable {
@Id
@GeneratedValue
private int idOrder;
private double totalAmount;
@ManyToOne
@JoinColumn(name = "idClientFK")
private Clients clients;
@OneToMany(cascade = CascadeType.ALL, mappedBy = "orders")//, fetch = FetchType.EAGER)
private List<OrdersProducts> listOfProductsInOrder = new ArrayList<OrdersProducts>();
}
@Entity
public class OrdersProducts implements Serializable {
@Id
@GeneratedValue
private int idOrderProduct;
private int productAmount;
private float productsValue;
@ManyToOne
@JoinColumn(name = "idOrderFK")
private Orders orders;
@ManyToOne
@JoinColumn(name = "idProductFK")
private Products products;
}
@Entity
public class Products implements Serializable {
@Id
@GeneratedValue
private int idProduct;
private String productName;
private double productValuePerUnit;
private String productUnit;
private int productCount;
private String productCompany;
@OneToMany(mappedBy = "products")
private List<OrdersProducts> listOfOrdersProducts = new ArrayList<OrdersProducts>();
}
A: The sequence to persist a new Order in the database will be:
*
*WHEN USER PUSH NEW ORDER BUTTON: Create a new Orders entity.
*
*Update relation to Clients.
*Add it to listOfOrders list in Clients entity (not necessary to persist an Order).
*WHEN USER ADD A NEW PRODUCT TO THE ORDER: Create an OrdersProducts entity.
*
*Update relations to Orders and Products.
*Add them to listOfProductsInOrder list in Order entity.
*WHEN USER PUSH CHECKOUT ORDER: Persist the Order entity.
But, you have to change:
@OneToMany
by:
@OneToMany(cascade = CascadeType.ALL)
in the Orders entity.
Note: Better give names in singular to the model entities.
EDIT
Ok, I would implement bidirectional relations to increase flexibility:
*
*In your Orders entity you should have a ManytoOne relation to Clients entity.
*In your OrdersProducts you should have a ManytoOne relation to Orders and another to Products.
In general update all the relations before persist the an entity.
For these bidirectional relations the systax would be:
*
*Clients:
@Entity
public class Clients implements Serializable {
...
@OneToMany(fetch = FetchType.EAGER, mappedBy = "idClientFK")
private List<Orders> listOfOrders = new ArrayList<Orders>();
}
*Orders:
@Entity
public class Orders implements Serializable {
...
@OneToMany(fetch = FetchType.EAGER, mappedBy = "idOrderFK")
private List<OrdersProducts> listOfProductsInOrder = new ArrayList<OrdersProducts>();
@JoinColumn (referencedColumnName = "idclient")
@ManyToOne
private Clients idClientFK;
}
*OrdersProducts:
@Entity
public class OrdersProducts implements Serializable {
...
@JoinColumn (referencedColumnName = "idorderproduct")
@ManyToOne
private Products idProductFK;
@JoinColumn (referencedColumnName = "idorder")
@ManyToOne
private Orders idOrderFK;
}
*Products:
@Entity
public class Products implements Serializable {
...
@OneToMany(fetch = FetchType.EAGER, mappedBy = "idProductFK")
private List<OrdersProducts> listOfOrdersProducts = new ArrayList<OrdersProducts>();
}
If you have bidirectional relations you could persist an entire Orders entity or you could persist only one OrdersProducts entity each time. But if you have only unidirectional relations you will have to persist the Clients entity in order to persist one or more new or modified Orders for this client. And navigation across related entities will be restricted to father to children direction.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,399
|
CROWN AGENTS MOVE TO MACEDONIA AND TANSANIA
The well-known British consultants from Crown Agents will be probably advising Macedonia's customs as well. According to publications in the press in Skopje, they may be hired by that country's Government. The entrance of Crown Agents in Bulgaria's Southwestern neighbour is connected with the forthcoming election of Nikola Gruevski for a premier. It was him as a finance minister in the VMRO-DPMNE cabinet in 2002 who began negotiations with Crown Agents for attracting them to the country. However, the deal was frustrated by the socialists coming into power a little while after the start of negotiations.
VMRO's victory during the recent parliamentary elections (July 6, 2006) has opened the road for closing a contract between the former Yugoslav republic and the British consultants.
Meanwhile, Crown Agents ensured themselves an order for consulting the customs in the African state of Tansania. The company's website has published this week an advertisement for hiring customs officers on the black continent.
The British consultants' project in Bulgaria will be completed in end-2006. Meanwhile, Macedonian mass media circulated ads information that since hiring them in November 2001 till now Crown Agents have helped growth of proceeds from imports in Bulgaria to reach 156 per cent.
събота, 22. юли 2006 - 01:00
Lowering heads the old way again
Employers and Unions Demand Termination of Contracts with US Power Plants
The BNB and SANS will jointly check banks against money laundering
Cordovska case – not a single word about crimes!
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,284
|
{"url":"http:\/\/www.nag.com\/numeric\/FL\/nagdoc_fl24\/html\/F06\/f06wbf.html","text":"F06 Chapter Contents\nF06 Chapter Introduction\nNAG Library Manual\n\nNAG Library Routine DocumentF06WBF (DTFSM)\n\nNote:\u00a0 before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.\n\n1\u00a0\u00a0Purpose\n\nF06WBF (DTFSM) performs one of the matrix-matrix operations\n $B\u2190\u03b1A-1B , B\u2190\u03b1A-TB , B\u2190\u03b1BA-1 or B\u2190\u03b1BA-T ,$\nwhere $A$\u00a0is a real triangular matrix stored in Rectangular Full Packed (RFP) format, $B$\u00a0is an $m$\u00a0by $n$\u00a0real matrix, and $\\alpha$\u00a0is a real scalar. ${A}^{-\\mathrm{T}}$\u00a0denotes ${\\left({A}^{\\mathrm{T}}\\right)}^{-1}$\u00a0or equivalently ${\\left({A}^{-1}\\right)}^{\\mathrm{T}}$. The RFP storage format is described in Section 3.3.3 in the F07 Chapter Introduction.\nNo test for singularity or near-singularity of $A$\u00a0is included in this routine. Such tests must be performed before calling this routine.\n\n2\u00a0\u00a0Specification\n\n SUBROUTINE\u00a0F06WBF\u00a0( TRANSR, SIDE, UPLO, TRANS, DIAG, M, N, ALPHA, A, B, LDB)\n INTEGER M, N, LDB REAL\u00a0(KIND=nag_wp) ALPHA, A(*), B(LDB,*) CHARACTER(1) TRANSR, SIDE, UPLO, TRANS, DIAG\nThe routine may be called by its LAPACK name dtfsm.\n\n3\u00a0\u00a0Description\n\nF06WBF (DTFSM) solves (for $X$) a triangular linear system of one of the forms\n $AX=\u03b1B , ATX=\u03b1B , XA=\u03b1B or XAT=\u03b1B ,$\nwhere $A$\u00a0is a real triangular matrix stored in RFP format, $B$, $X$\u00a0are $m$\u00a0by $n$\u00a0real matrices, and $\\alpha$\u00a0is a real scalar.\n\nNone.\n\n5\u00a0\u00a0Parameters\n\n1: \u00a0\u00a0\u2002 TRANSR \u2013 CHARACTER(1)Input\nOn entry: specifies whether the RFP representation of $A$\u00a0is normal or transposed.\n${\\mathbf{TRANSR}}=\\text{'N'}$\nThe matrix $A$\u00a0is stored in normal RFP format.\n${\\mathbf{TRANSR}}=\\text{'T'}$\nThe matrix $A$\u00a0is stored in transposed RFP format.\nConstraint: ${\\mathbf{TRANSR}}=\\text{'N'}$\u00a0or $\\text{'T'}$.\n2: \u00a0\u00a0\u2002 SIDE \u2013 CHARACTER(1)Input\nOn entry: specifies whether $B$\u00a0is operated on from the left or the right, or similarly whether $A$\u00a0(or its transpose) appears to the left or right of the solution matrix in the linear system to be solved.\n${\\mathbf{SIDE}}=\\text{'L'}$\n$B$\u00a0is pre-multiplied from the left. The system to be solved has the form $AX=\\alpha B$\u00a0or ${A}^{\\mathrm{T}}X=\\alpha B$.\n${\\mathbf{SIDE}}=\\text{'R'}$\n$B$\u00a0is post-multiplied from the right. The system to be solved has the form $XA=\\alpha B$\u00a0or $X{A}^{\\mathrm{T}}=\\alpha B$.\nConstraint: ${\\mathbf{SIDE}}=\\text{'L'}$\u00a0or $\\text{'R'}$.\n3: \u00a0\u00a0\u2002 UPLO \u2013 CHARACTER(1)Input\nOn entry: specifies whether $A$\u00a0is upper or lower triangular.\n${\\mathbf{UPLO}}=\\text{'U'}$\n$A$\u00a0is upper triangular.\n${\\mathbf{UPLO}}=\\text{'L'}$\n$A$\u00a0is lower triangular.\nConstraint: ${\\mathbf{UPLO}}=\\text{'U'}$\u00a0or $\\text{'L'}$.\n4: \u00a0\u00a0\u2002 TRANS \u2013 CHARACTER(1)Input\nOn entry: specifies whether the operation involves ${A}^{-1}$\u00a0or ${A}^{-\\mathrm{T}}$, i.e., whether or not $A$\u00a0is transposed in the linear system to be solved.\n${\\mathbf{TRANS}}=\\text{'N'}$\nThe operation involves ${A}^{-1}$, i.e., $A$\u00a0is not transposed.\n${\\mathbf{TRANS}}=\\text{'T'}$\nThe operation involves ${A}^{-\\mathrm{T}}$, i.e., $A$\u00a0is transposed.\nConstraint: ${\\mathbf{TRANS}}=\\text{'N'}$\u00a0or $\\text{'T'}$.\n5: \u00a0\u00a0\u2002 DIAG \u2013 CHARACTER(1)Input\nOn entry: specifies whether $A$\u00a0has nonunit or unit diagonal elements.\n${\\mathbf{DIAG}}=\\text{'N'}$\nThe diagonal elements of $A$\u00a0are stored explicitly.\n${\\mathbf{DIAG}}=\\text{'U'}$\nThe diagonal elements of $A$\u00a0are assumed to be $1$, the corresponding elements of A are not referenced.\nConstraint: ${\\mathbf{DIAG}}=\\text{'N'}$\u00a0or $\\text{'U'}$.\n6: \u00a0\u00a0\u2002 M \u2013 INTEGERInput\nOn entry: $m$, the number of rows of the matrix $B$.\nConstraint: ${\\mathbf{M}}\\ge 0$.\n7: \u00a0\u00a0\u2002 N \u2013 INTEGERInput\nOn entry: $n$, the number of columns of the matrix $B$.\nConstraint: ${\\mathbf{N}}\\ge 0$.\n8: \u00a0\u00a0\u2002 ALPHA \u2013 REAL\u00a0(KIND=nag_wp)Input\nOn entry: the scalar $\\alpha$.\n9: \u00a0\u00a0\u2002 A($*$) \u2013 REAL\u00a0(KIND=nag_wp)\u00a0arrayInput\nNote: the dimension of the array A must be at least $\\mathrm{max}\\phantom{\\rule{0.125em}{0ex}}\\left(1,{\\mathbf{M}}\u00d7\\left({\\mathbf{M}}+1\\right)\/2\\right)$\u00a0if ${\\mathbf{SIDE}}=\\text{'L'}$\u00a0and at least $\\mathrm{max}\\phantom{\\rule{0.125em}{0ex}}\\left(1,{\\mathbf{N}}\u00d7\\left({\\mathbf{N}}+1\\right)\/2\\right)$\u00a0if ${\\mathbf{SIDE}}=\\text{'R'}$.\nOn entry: $A$, the $m$\u00a0by $m$\u00a0triangular matrix $A$\u00a0if ${\\mathbf{SIDE}}=\\text{'L'}$\u00a0or the $n$\u00a0by $n$\u00a0triangular matrix $A$\u00a0if ${\\mathbf{SIDE}}=\\text{'R'}$, stored in RFP format, as described in Section 3.3.3 in the F07 Chapter Introduction.\n10: \u2002 B(LDB,$*$) \u2013 REAL\u00a0(KIND=nag_wp)\u00a0arrayInput\/Output\nNote: the second dimension of the array B must be at least $\\mathrm{max}\\phantom{\\rule{0.125em}{0ex}}\\left(1,{\\mathbf{N}}\\right)$.\nOn entry: the $m$\u00a0by $n$\u00a0matrix $B$.\nIf ${\\mathbf{ALPHA}}=0$, B need not be set.\nOn exit: the updated matrix $B$, or similarly the solution matrix $X$.\n11: \u2002 LDB \u2013 INTEGERInput\nOn entry: the first dimension of the array B as declared in the (sub)program from which F06WBF (DTFSM) is called.\nConstraint: ${\\mathbf{LDB}}\\ge \\mathrm{max}\\phantom{\\rule{0.125em}{0ex}}\\left(1,{\\mathbf{M}}\\right)$.\n\nNone.\n\nNot applicable.\n\nNone.\n\n9\u00a0\u00a0Example\n\nThis example reads in the lower triangular part of a symmetric matrix $A$\u00a0which it converts to RFP format. It also reads in $\\alpha$\u00a0and a $6$\u00a0by $4$\u00a0matrix $B$\u00a0and then performs the matrix-matrix operation $B\u2190\\alpha {A}^{-1}B$.\n\n9.1\u00a0\u00a0Program Text\n\nProgram Text (f06wbfe.f90)\n\n9.2\u00a0\u00a0Program Data\n\nProgram\u00a0Data (f06wbfe.d)\n\n9.3\u00a0\u00a0Program Results\n\nProgram Results (f06wbfe.r)","date":"2015-11-26 03:54:42","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 99, \"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\": 0.9998563528060913, \"perplexity\": 4836.307896771935}, \"config\": {\"markdown_headings\": false, \"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-2015-48\/segments\/1448398446300.49\/warc\/CC-MAIN-20151124205406-00164-ip-10-71-132-137.ec2.internal.warc.gz\"}"}
| null | null |
La rue Pauline-Marie-Jaricot, plus communément appelée rue Pauline-Jaricot, est une rue du quartier de la Sarra située sur la colline de Fourvière dans le arrondissement de Lyon, en France.
Situation et accès
La rue Pauline-Marie-Jaricot débute rue Roger-Radisson, face à l'entrée du parc de la Visitation. Elle se termine place du 158-Régiment-d'Infanterie, juste avant la rue Cardinal-Gerlier.
Plusieurs lignes de transport ont des itinéraires qui passent près de rue Pauline-Marie-Jaricot. Les bus : 55, 90, C14, C20, C21 et les funiculaires : F1, F2 ont un arrêt à proximité.
Son code INSEE est le 69385
Origine du nom
Anciennement rue des Quatre-Vents, c'est par délibération du Conseil municipal du 2 avril 1962 que la rue a pris le nom de Pauline-Marie-Jaricot.
Histoire
Fondatrice de plusieurs œuvres sociales, Pauline Jaricot est née à Lyon le 22 juillet 1799. Elle décède le 9 janvier 1862 à Lyon (5e). La propagation de la foi est son œuvre la plus connue, inspirée par la venue de Pie VII en 1805. Les fonds levés ont permis la création de nombreuses missions. Les papes Léon XIII, Paul VI et Benoît XVI lui ont rendu hommage, Pie XI l'a introduit en béatification, Jean XXIII l'a proclamée vénérable. Enfin, Pauline Jaricot sera béatifiée à Lyon, avec l'accord du pape François, lors d'une cérémonie dirigée par le cardinal Luis Antonio Tagle, prévue le 22 mai 2022.
Notes et références
Voie dans le 5e arrondissement de Lyon
Voie de Lyon se référant à un nom de femme
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 9,596
|
\section{Introduction}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
Peer-to-peer (p2p) activity continues to represent a very significant fraction
of overall Internet traffic, 44\% by one recent account
\cite{DCInfo063008}.
BitTorrent
\cite{BitTorrent,Cohen:2003,Ge:2004,Yang:2004,Qiu:2004,bittor-imc05,Turner:2005}
is a widely deployed p2p
file-sharing network which has recently played a significant
role in the network neutrality debate.
Under BitTorrent, peers join ``swarms" (or ``torrents")
where each swarm corresponds
to a specific data object (file).
The process of finding the peers
in a given swarm to connect to
is typically facilitated through a centralised ``tracker".
Recently, a trackerless BitTorrent client
has been introduced that uses distributed hashing for query
resolution \cite{maymounkov02kademlia}.
For file sharing, a peer is typically uploads
upload pieces (``chunks") of the file to other peers in the
swarm while downloading his/her missing chunks from them.
This chunk swapping constitutes a transaction-by-transaction
incentive for peers to cooperate (i.e., trading rather
than simply download) to disseminate data objects.
Large files may be segmented
into several hundred chunks, all of which the peers of the
corresponding warm must collect,
and in the process disseminate their own chunks
before they can reconstitute the desired file and possibly leave
the file's swarm.
In addition to the framework
in which data objects are segmented into chunks to promote
cooperation through swapping, there is a system whereby the rate
at which chunks are
uploaded is assessed for any given transaction, and peers that
allocate inadequate bandwidth for uploading may be ``choked"
\cite{LZhang07,ITA06}. Choking may also be applied to peers
who, by employing multiple identities (sybils),
abuse BitTorrent's system of allowing newly arrived peers to
a swarm to just download a few chunks (as they clearly cannot
trade what they simply do not as yet possess).
BitTorrent can also rehabilitate peers by (optimistically) unchoking them.
In the following, we do not directly consider upload bandwidth and
related choking issues.
In this paper, we motivate a deterministic epidemiological model
of file dissemination for peer-to-peer file-sharing networks that
employ BitTorrent-like incentives, a generalisation of that
given in \cite{Kesidis:2006b}\footnote{And this paper is a significant
extension of \cite{ICASSP07}.}.
Our model is different from those explored in
\cite{Mass-Voj-05,Yang:2004,Qiu:2004} for BitTorrent,
and we compute different quantities of interest. Our epidemiological
framework, similar to that we used for the spread of multi-stage worms
\cite{Kesidis08}, could also be adapted
for network coding systems.
In \cite{bittor-imc05}, the authors propose a ``fluid" model
of a single torrent/swarm (as we do in the following) and fit it
to (transient) data drawn from aggregate swarms.
The connection to branching
process models \cite{Yang:2004,Ge:2004} is simply
that ours only tracks the number of active peers who possess or
demand the file under consideration, i.e., a single swarm.
Though our model is
significantly simpler than that of prior work,
it is derived directly from an intuitive transaction-by-transaction
Markov process modelling file-dissemination of the p2p network and its
numerical solutions clearly demonstrate the effectiveness of the
aforementioned incentives. A basic assumption in the following is
that peers do not distribute bogus files (or file chunks)
\cite{Walsh:2005:2}.
This paper is organised as follows.
The Markovian model is developed in detail in Section \ref{stoch-sec}.
A proof of its fluid limit is described in
Section \ref{fluid-limit-sec} (and in the Appendix)
including Lyapunov stability and settling time results.
The behaviour of the limiting ODE is studied in Section
\ref{ODEbeh} for specific examples.
In Section \ref{incentives-sec}, we derive conditions under
which the BitTorrent-like system
has improved performance (smaller mean time to completely
acquire the file by a peer) compared to a system of
pure client-server (no chunk-swapping) interactions.
A diffusion approximation is given in \ref{diff-sec}.
The paper concludes with a discussion of open problems in
Section \ref{concl-sec}.
\section{The stochastic model}\label{stoch-sec}
We fix a set $F$ (a file) which is partitioned into $n$
(on the order of hundreds) pieces called chunks.
Consider a large networked ``swarm" of $N$ nodes called peers.
Each peer possesses a certain (possibly empty) subset $A$ of $F$.
As time goes by, this peer interacts with other peers,
the goal being to enlarge his set $A$ until, eventually,
the peer manages to collect all $n$ chunks of $F$.
The interaction between peers can either be a download
or a swap; in both cases, chunks are being copied from
peer to peer and are assumed never lost.
A peer will stay in the network as long as he does not possess all chunks.
After collecting everything, sooner or later a peer departs
or switches off.
By splitting the desired file into many chunks we give incentives
to the peers to remain active in the
swarm for long time during which other peers will
take advantage of their possessions.
\subsection{Possible interactions}
We here describe how two peers, labelled $A, B$, interact.
The following types of interactions are possible:
\begin{enumerate}
\item
{\bf Download:} Peer $A$ downloads a chunk $i$ from $B$. This is possible
only if $A$ is a strict subset of $B$.
If $i \in B$ then, after the downloading
$A$ becomes $A' = A \cup \{i\}$ and but $B$ remains $B$ because it
since it gains nothing from $A$.
Denote this interaction as:
\[
\boxed{
(A \leftarrow B) \rightsquigarrow (A', B)
}
\]
The symbol on the left is supposed to show the type
of interaction and the labels before it,
while the symbol on the right shows the labels after the interaction.
\item
{\bf Swap:} Peer $A$ swaps with peer $B$. In other words, $A$ gets a chunk
$j$ from $B$ and $B$ gets a chunk $i$ from $A$.
It is required that $j$ is not an element of $A$ and $i$ not
an element of $B$.
We denote this interaction by
\[
\boxed{
(A \leftrightarrows B) \rightsquigarrow (A', B')
}
\]
where $A' = A \cup \{j\}$, $B' = B \cup \{i\}$.
We thus need $A \setminus B \neq \varnothing$ and $B \setminus A \neq
\varnothing$.
\end{enumerate}
\subsection{Notation}
The set of all combinations of $n$ chunks, which partition
$F$, is denoted by $\mathscr{P}(F)$, where $|\mathscr{P}(F)|=2^n$ and the empty set is
included.
We write $A \subset B$ (respectively, $A \subsetneq B$) when $A$
is a subset (respectively, strict subset) of $B$.
We (unconventionally) write
\begin{equation*}
\text{ $A \sqsubset A'$ when $A \subset A'$ and $|A'-A|=1$. }
\end{equation*}
If $A \cap B = \varnothing$, we use $A + B$ instead of $A \cup B$;
if $B=\{b\}$ is a singleton, we often write $A+b$ instead of $A + \{b\}$.
If $A \subset B$ we use $B-A$ instead of $B \setminus A$.
We say that \begin{equation*}
\text{ $A$ relates to $B$ (and write $A \sim B$)
when $A \subset B$ or $B \subset A$};
\end{equation*}
if this is not the case, we write $A \not \sim B$.
Note that $A \not \sim B$ if and only if two peers labelled $A$, $B$
can swap chunks.
The space
of functions (vectors)
from $\mathscr{P}(F)$ into $\mathbb{Z}_+$ is denoted by $\mathbb{Z}_+^{\mathscr{P}(F)}$.
The stochastic model will take values in this space.
The deterministic model will evolve in $\mathbb{R}_+^{\mathscr{P}(F)}$.
We let $e_A \in \mathbb{Z}_+^{\mathscr{P}(F)}$ be the vector with
coordinates \[
e_A^B := {\text{\Large $\mathfrak 1$}}(A=B), \quad B \in \mathscr{P}(F).
\]
For $x \in \mathbb{Z}_+^{\mathscr{P}(F)}$ or $\mathbb{R}_+^{\mathscr{P}(F)}$
we let $|x| := \sum_{A \in \mathscr{P}(F)} |x^A|$.
If $\mathscr{A} \subset \mathscr{P}(F)$ then the $\mathscr{A}$-face $\mathbb{R}_+^\mathscr{A}$ of $\mathbb{R}_+^{\mathscr{P}(F)}$ is
defined by $\mathbb{R}_+^\mathscr{A}:=\{x \in \mathbb{R}_+^{\mathscr{P}(F)}:~
\sum_{A \in \mathscr{A}} x^A=0,~ \prod_{B \not \in \mathscr{A}} x^B >0\}$.
\subsection{Defining the rates of individual interactions}
We follow the logic of stochastic modelling of chemical reactions
or epidemics and assume that the chance of a particular interaction
occurring in a short interval of time is proportional to the number
of ways of selecting the peers needed for this interaction
\cite{Kurtz81}.
Accordingly, the interaction rates
{\em must} be given by the formulae described below.
Consider first finding the rate of
a download $A \leftarrow B$, where $A \subsetneq B$,
when the state of the system is $x
\in \mathbb{Z}_+^{\mathscr{P}(F)}$.
There are $x^A$ peers labelled $A$ and $x^B$ labelled $B$.
We can choose them in $x^A x^B$ ways.
Thus the rate of a download $A \leftarrow B$
that results into $A$ getting {\em some} chunk from $B$ should be
proportional to $x^A x^B$.
However, we are interested in the rate
of the {\em specific}
interaction $(A \leftarrow B) \rightsquigarrow (A', B)$,
that turns $A$ into a specific set $A'$ differing from $A$
by one single chunk $(A \sqsubset A')$;
there are $|B -A|$ chunks that $A$ can download from $B$;
the chance that picking one of them is $1/|B-A|$.
Thus we have:
\[
(DR) \quad
\left\{
\text{
\begin{minipage}{12cm}
\em
the rate of the download $(A \leftarrow B) \rightsquigarrow (A', B)$
equals $\displaystyle \beta x^A \frac{x^B}{|B-A|}$,
\\
as long as
$A \sqsubset A' \subset B$,
\end{minipage}
}
\right.
\]
where $\beta > 0$.
Consider next a swap $A \leftrightarrows B$
and assume the state is $x$.
Picking two peers labelled $A$ and $B$ (provided that $A \not \sim B$)
from the population is done in $x^A x^B$ ways.
Thus the rate of a swap $A \leftrightarrows B$
is proportional to $x^A x^B$.
So if we {\em fix} two chunks
$i \in A\setminus B, j \in B\setminus A$
and specify that $A'=A+j, B'=B+i$, then
the chance of picking $i$ from $A\setminus B$ and $j$ from $B\setminus A$
is $1/|A\setminus B| |B\setminus A|$.
Thus,
\[
(SR) \quad \left\{
\text{
\begin{minipage}{12cm}
\em
the rate of the swap $(A \leftrightarrows B) \rightsquigarrow (A', B')$
equals $\displaystyle \gamma
\frac{x^A x^B}{|A \setminus B| |B \setminus A|}$,
\\
a long as
$A \sqsubset A',
\quad
B \sqsubset B',
\quad
A'-A \subset B,
\quad
B'-B \subset A$,
\end{minipage}
}
\right.
\]
where $\gamma > 0$.
\subsection{Deriving the Markov chain rates}
Having defined the rates of each individual interaction
we can easily define rates
$q(x,y)$
of a Markov chain in continuous time and state space $\mathbb{Z}_+^{\mathscr{P}(F)}$
as follows.
Define functions $\lambda_{A,A'}, \mu_{A,B} :
\mathbb{R}^{\mathscr{P}(F)} \to \mathbb{R}$ by:
\begin{subequations} \label{lllmmmm}
\begin{align}
&\lambda_{A,A'}(x)
:=
\left[\beta x^A \sum_{C : C \supset A'}
\frac{x^C}{|C-A|} \right]
~{\text{\Large $\mathfrak 1$}}(A \sqsubset A') \label{lll}
\\
&\mu_{A,B}(x)
:= \gamma
\frac{x^A x^B}{|A \setminus B| |B \setminus A|}
~{\text{\Large $\mathfrak 1$}}(A \not \sim B). \label{mmm}
\end{align}
\end{subequations}
Consider also constants $\delta \ge 0$ and $\alpha^A \ge 0$ for $A \in \mathscr{P}(F)$,
i.e., $\alpha \in\mathbb{R}_+^{\mathscr{P}(F)}$.
The transition rates of the closed conservative Markov chain are given by:
\begin{equation} \label{rates}
q(x,y) :=
\begin{cases}
\lambda_{A,A'}(x),
& \text{ if } y = x-e_A+e_{A'}
\\
\mu_{A,B}(x),
& \text{ if }
\begin{cases}
y = x-e_A-e_B+e_{A'}+e_{B'}
\\
A \sqsubset A', B \sqsubset B', A'-A \subset B, B'-B \subset A,
\end{cases}
\\
\alpha^A
& \text{ if } y = x+e_A
\\
\delta x^F
& \text{ if } y = x-e_F
\\
0, & \text{ for any other value of } y \not = x,
\end{cases}
\end{equation}
where $x$ ranges in $\mathbb{Z}_+^{\mathscr{P}(F)}$.
A little justification of the first two cases is needed:
that $q(x,x-e_A-e_B+e_{A'}+e_{B'}) = \mu_{A,B}(x)$
is straightforward. It corresponds to a swap,
which is only possible when
$A \sqsubset A', B \sqsubset B', A'-A \subset B, B'-B \subset A$.
The swap rate was defined by (SR).
To see that $q(x,x-e_A+e_{A'})=\lambda_{A,A'}(x)$ we observe
that a peer labelled $A$ can change its label to $A' \sqsupset A$
by downloading a chunk from some
set $C$ that contains $A'$, so we sum the rates (DR) over all
these possible individual interactions to obtain the first line
in \eqref{rates}.
We can think of having Poisson process of arrivals of new peers
at rate $|\alpha|$, and that each arriving peer is labelled $A$
with probability $\alpha^A/|\alpha|$.
Peers can depart, by definition, only when they are labelled $F$
and it takes an exponentially distributed amount of time (with mean
$1/\delta$) for a departure to occur. Thus, $q(x, x-e_F)=\delta x^F$.
We shall let $\mathsf Q$ denote the generator of the chain, i.e.\
$\mathsf Q f(x) = \sum_y (f(y)-f(x)) q(x,y)$, when $f$ is an appropriate functional
of the state space.
\begin{definition}
[\bt{x_0}{n} {\alpha} {\beta}{\gamma} {\delta}]
Given $x_0 \in \mathbb{Z}_+^{\mathscr{P}(F)}$ (initial configuration),
$n=|F| \in \mathbb{N}$ (number of chunks),
$\alpha\in \mathbb{R}_+^{\mathscr{P}(F)}$ (arrival rates), $\beta > 0$
(download rate), $\gamma \ge0$ (swap rate), $\delta \ge 0$ (departure rate)
we let \bt{x_0}{n} {\alpha} {\beta}{\gamma} {\delta}
be a Markov chain $(X_t, t \ge 0)$ with
transition rates \eqref{rates} and $X_0=x_0$.
We say that the chain (network) is \underline{open}
if $\alpha^A > 0$ for at least one $A$
and $\delta >0$;
it is \underline{closed} if $\alpha^A=0$ for all $A$;
it is \underline{conservative}
if it is closed and $\delta=0$;
it is \underline{dissipative} if it is closed and $\delta>0$.
\end{definition}
In a conservative network, we have $q(x,y)=0$ if $|y| \neq |x|$ and
so $|X_t| = |X_0|$ for all $t \ge 0$.
Here, the actual state space is the simplex
\[
\{x \in \mathbb{Z}_+^{\mathscr{P}(F)}: |x|=N\},
\]
where $N = |X_0|$.
It is easy to see that
the state $e_F$ is reachable from any other state, but
all rates out of $e_F$ are zero. Hence a conservative network has
$e_F$ as a single absorbing state.
In a dissipative network, we have
$|X_t| \le |X_0|$ for all $t \ge 0$.
Here the state space is
\[
\{x\in \mathbb{Z}_+^{\mathscr{P}(F)}: |x|\le N\},
\]
where $N = |X_0|$.
It can be seen that a dissipative network has many absorbing points.
In an open network, there are no absorbing points. On the other
hand, one may wonder if certain components can escape to infinity.
This is not the case:
\begin{lemma}
If $\alpha^F > 0$ then
the open {\tt BITTORRENT[$x,n,\beta,\gamma,\alpha,\delta$]}
is positive recurrent Markov chain.
\end{lemma}
\proof (sketch)
If $\alpha^F > 0$, $\delta > 0$ the Markov chain is irreducible.
The remainder of the proof is based on a the construction of
a simple Lyapunov function:
\[
V(x) := |x|,
\]
for which it can be shown that there is a bounded set of states $K$
such that
\[
\sup_{x \not \in K} (\mathsf Q V)(x) < 0.
\]
Perhaps the easiest way to
see this is by appealing to the stability of the corresponding ODE limit;
see Theorem \ref{ODEapprox} below and \cite{FK}.
\qed
\subsection{Example: $n=1$}
Let us take the special case where the file consists of a single chunk
$(n=1)$. The state here is $x=(x^\varnothing, x^1:=x^F)$.
The rates are:
\begin{align}
&q\big((x^\varnothing, x^1), (x^\varnothing+1, x^1)\big)
= \alpha^\varnothing \nonumber \\
&q\big((x^\varnothing, x^1), (x^\varnothing, x^1+1)\big)
= \alpha^1 \nonumber \\
&q\big((x^\varnothing, x^1), (x^\varnothing-1, x^1+1)\big)=\beta x^\varnothing x^1 \nonumber \\
&q\big((x^\varnothing, x^1), (x^\varnothing, x^1-1)\big)=\delta x^1.
\label{kmc}
\end{align}
If $\alpha^\varnothing=\alpha^1=0$,
this is the stochastic version of the classical (closed) Kermack-McKendrick
(or susceptible-infective-removed (SIR))
model for a simple epidemic process \cite{Daley-Gani}.
Its absorbing points are states of the form $(x^\varnothing, 0)$.
In epidemiological terminology, $x^1$ is the number of infected
individuals, whereas $x^\varnothing$ is the number of susceptible
ones. Contrary to the epidemiological interpretation, infection
{\em is} desirable, for infection is tantamount to downloading
the file.
\section{Macroscopic description: fluid limit}\label{fluid-limit-sec}
Analysing the Markov chain in its original form is complicated.
We thus resort to a first-order approximation by an ordinary differential
equation (ODE).
Let $v(x)$ be the vector field on $\mathbb{R}_+^{\mathscr{P}(F)}$ with
components $v^A(x)$ defined by
\begin{multline}
\label{vcomp}
\lefteqn{
v^A(x) =
\alpha^A
-x^A \big(\beta \phi_d^A(x) + \gamma \phi^A_s(x) \big)
}
\\
+ \beta
\sum_{B: A \subset B}
\frac{\psi^A_d(x) x^B }{1+|B\setminus A|}
+
\gamma
\sum_{B: A \not \subset B}
\frac{\psi^{A,B}_s(x) x^B }{1+|B\setminus A|}
-\delta x^F {\text{\Large $\mathfrak 1$}}(A=F),
\end{multline}
where
\begin{align}
& \phi_d^A(x) := \sum_{B \supset A} x^B, \quad
\phi_s^A(x) := \sum_{B \not \sim A} x^B
\nonumber \\
& \psi_d^A(x) := \sum_{a \in A} x^{A-a}, \quad
\psi_s^{A,B}(x) := \sum_{a \in A \cap B} x^{A-a}
\label{phipsi}
\end{align}
Consider the differential equation
\begin{equation}
\dot x = v(x) \text{ with initial condition $x_0$.} \label{ODE0}
\end{equation}
Consider the sequence of stochastic models \bt{X_{\text{\tiny $N$},0}}{n}
{N \alpha}{\frac{\beta}{N}}{\frac{\gamma}{N}}{\delta} for
$N \in \mathbb{N}$ and let $X_{\text{\tiny $N$},t}$ be the corresponding jump Markov chain.
\begin{theorem}\label{ODEapprox}
There is a has a unique smooth
(analytic) solution to (\ref{ODE0}), denoted by $x_t$ for $t \ge 0$.
Also, if there is an $x_0 \in \mathbb{R}_+^{\mathscr{P}(F)}$ such that
$X_{\text{\tiny $N$},0}/N \to x_0$
as $N \to \infty$,
then for any $T, \epsilon > 0$,
\[
\lim_{N \to \infty}
P\big(\sup_{0 \le t \le T}|N^{-1} X_{\text{\tiny $N$},t} -x_t| > \epsilon\big)=0.
\]
\end{theorem}
\proof
Let $\mathcal{N}$ be the set of vectors $-e_{F}$, $e_A$,
$-e_A+e_{A'}$, $-e_A-e_B+e_{A'}+e_{B'}$,
where $A, B\in\mathscr{P}(F)$ and
$A \sqsubset A'$, $B \sqsubset B'$.
{From} \eqref{rates}, we have that $q(x,y)=0$ if $y-x \not \in \mathcal{N}$.
Introduce, for each $\zeta \in \mathcal{N}$, a unit rate Poisson process
$\Phi_\zeta$ on the real line, and assume that these Poisson processes
are independent.
Consider the Markov chain $(X_t)$ for the
\bt{X_0}{n}{\alpha}{\beta}{\gamma}{\delta}.
Its rates are of the form
\begin{equation}
\label{Qdef}
q(x,x+\zeta) = Q_\zeta(x), \quad \zeta \in \mathcal{N},
\end{equation}
where $Q_\zeta(x)$ is a polynomial in $2^n$ variables of degree $2$, and
which can be read directly from \eqref{rates}; its coefficients
depend on the parameters $\alpha$, $\beta$, $\gamma$, $\delta$.
We can represent \cite{Kurtz81,Kurtz86} $(X_t)$ as:
\[
X_t=X_0 + \sum_{\zeta \in \mathcal{N}} \zeta
\Phi_\zeta\big(\int_0^t Q_\zeta(X_s) ds\big).
\]
Consider now the Markov chain $\frac{1}{N} X_{\text{\tiny $N$},t}$ corresponding to
to \bt{X_{\text{\tiny $N$},0}}{n}{(N \alpha^{A})}{\beta/N}
{\gamma/N}{\delta}.
The transition rates for $\frac{1}{N} X_{\text{\tiny $N$},t}$ are
\[
q(x/N, (x+\zeta)/N) = N Q_\zeta(x/N), \quad
x \in \mathbb{Z}_+^{\mathscr{P}(F)}, \quad \zeta \in \mathcal{N},
\]
and $0$, otherwise.
Here, $Q_\zeta(x)$ is the polynomial defined through
\eqref{Qdef} and \eqref{rates} and we now assume that its variables
range over the reals.
Therefore, $\frac{1}{N} X_{\text{\tiny $N$},t}$ can be represented as
\[
\frac{1}{N}X_{\text{\tiny $N$},t}=\frac{1}{N}X_{\text{\tiny $N$},0} + \sum_{\zeta \in \mathcal{N}} \zeta
\frac{1}{N}
\Phi_\zeta\bigg(N \int_0^t Q_\zeta(\frac{1}{N}X_{\text{\tiny $N$},s}) ds\bigg).
\]
Define $x_t$ by the (deterministic) integral equation
\begin{equation}
\label{inteq}
x_t = x_0 + \sum_{\zeta \in \mathcal{N}} \zeta
\int_0^t Q_\zeta(x_s) ds
\end{equation}
and assume that it is unique for all $t \ge 0$.
Fix a time horizon $T > 0$ and let
\begin{align*}
B &:= \max_{t \le T} |x_t|,
\\
M_\zeta &:= \max_{|x|\le B} |Q_\zeta(x)|
\\
L_\zeta &:=\sup_{\substack {|x|, |y| \le B \\ x \neq y} }
\frac{|Q_\zeta(x) - Q_\zeta(y)|}{|x-y|}
\\
\tau_\text{\tiny $N$} & :=\inf\{t>0:~ |X_{\text{\tiny $N$},t}|>N B\}.
\end{align*}
We then have:
\begin{multline*}
\Delta_{\text{\tiny $N$},t}:=
\frac{X_{\text{\tiny $N$},t}}{N}-x_t =
\frac{X_{\text{\tiny $N$},0}}{N}-x_0
+ \sum_{\zeta \in \mathcal{N}} \zeta
\bigg[\frac{1}{N} \Phi_\zeta\bigg( N \int_0^t Q_\zeta(X_{\text{\tiny $N$},s}/N) ds \bigg)
- \int_0^t Q_\zeta(X_{\text{\tiny $N$},s}/N)ds \bigg]
\\
+ \sum_{\zeta \in \mathcal{N}} \zeta \int_0^t
\big( Q_\zeta(X_{\text{\tiny $N$},s}/N) - Q_\zeta(x_s) \big) ds
\end{multline*}
Suppose that $t \le T \wedge \tau_\text{\tiny $N$}$. Then,
for all $s \le t$,
\[
|Q_\zeta(X_{\text{\tiny $N$},s}/N) - Q_\zeta(x_s)| \le
L_\zeta |\Delta_{\text{\tiny $N$},s}|.
\]
So, if we let
\[
\mathscr{E}_{\text{\tiny $N$},t}:= \frac{X_{\text{\tiny $N$},0}}{N}-x_0
+ \sum_{\zeta \in \mathcal{N}} \zeta
\frac{1}{N}
\bigg[
\Phi_\zeta\bigg( N \int_0^t Q_\zeta(X_{\text{\tiny $N$},s}/N) ds \bigg)
- N \int_0^t Q_\zeta(X_{\text{\tiny $N$},s}/N)ds
\bigg],
\]
we have, by the Gronwall-Bellman lemma, that
\[
|\Delta_{\text{\tiny $N$},t}| \le |\mathscr{E}_{\text{\tiny $N$},t}|
\exp \big( t \sum_{\zeta\in \mathcal{N}} |\zeta| L_\zeta\big),
\quad \text{if }
t \le T\wedge \tau_\text{\tiny $N$}.
\]
Let
\[
\Phi^*_\zeta(t) := \sup_{s \le t} |\Phi_\zeta(s)-s|.
\]
We recall that, as $N \to \infty$,
\begin{equation}
\label{plln}
\frac{1}{N} \Phi^*_\zeta(Nt) \to 0 ~ \text{ a.s.}
\end{equation}
If $s \le t \le \tau_\text{\tiny $N$}$, we have $X_{\text{\tiny $N$},s}/N \le B$
(definition of $\tau_\text{\tiny $N$}$) and so $Q_\zeta(X_{\text{\tiny $N$},s}/N) \le M_\zeta$,
implying that
\[
\sup_{t \le T \wedge \tau_\text{\tiny $N$}}
|\mathscr{E}_{\text{\tiny $N$},t}| \le \big|\frac{X_{\text{\tiny $N$},0}}{N}-x_0\big| +
\sum_{\zeta \in \mathcal{N}} |\zeta| \frac{1}{N} \Phi^*_\zeta(N M_\zeta T)
\]
which converges to zero, a.s., due to \eqref{plln}.
Since
\[
\sup_{t \le T \wedge \tau_\text{\tiny $N$}}
|\Delta_{\text{\tiny $N$},t}|
\le
\sup_{t \le T \wedge \tau_\text{\tiny $N$}}
|\mathscr{E}_{\text{\tiny $N$},t}|
\exp \big( T \sum_{\zeta\in \mathcal{N}} |\zeta| L_\zeta\big),
\]
we have
\[
\sup_{t \le T \wedge \tau_\text{\tiny $N$}}
|\Delta_{\text{\tiny $N$},t}| \to 0, ~ \text{ a.s.}
\]
Now observe that
\begin{align*}
P(\tau_\text{\tiny $N$} \le T) & \le P(\sup_{t \le T \wedge \tau_\text{\tiny $N$}}
|X_{\text{\tiny $N$},t}| > NB)
\\
& \le P(\sup_{t \le T \wedge \tau_\text{\tiny $N$}} |\Delta_{\text{\tiny $N$},t}|
+ \sup_{t \le T \wedge \tau_\text{\tiny $N$}} |x_t| > B) \to 0.
\end{align*}
So we have $\sup_{t \le T} |\Delta_{\text{\tiny $N$},t}| \to 0$ a.s.
To show that $x_t$, defined via \eqref{inteq},
satisfies the ODE $\dot x = v(x)$ with
$v$ given by \eqref{vcomp} is a
matter of straightforward (but tedious) algebra,
see the Appendix.
Uniqueness and analyticity
of the solution of the ODE is immediate from the form of the vector
field (its components are polynomials of degree 2 and hence locally
Lipschitzian).
To show that the trajectories do not explode, we
consider the function
\[
V(x) := \sum_A x^A.
\]
It is a matter of algebra to check that
\[
\langle \nabla V (x), v(x) \rangle
= \sum_A v^A(x) =
\sum_A \alpha^A - \delta x^F
\]
which (since $\delta > 0$)
is negative and bounded away from zero for $x$ outside a bounded
set of $\mathbb{R}_+^{\mathscr{P}(F)}$ containing the origin.
We then apply the Lyapunov criterion for ODEs to conclude
that $x_t$ is defined for all $t \ge 0$ and this justifies the
fact that we could choose an arbitrary time horizon $T$ earlier in
the proof. \qed
\paragraph{\em Comment:}
The quantities defined in \eqref{phipsi},
have physical meanings as follows:
\begin{align*}
\phi_d^A(x) &:= \sum_{B \supset A} x^B =
\text{no.\ of peers from which an $A$-peer can download},
\\
\phi_s^A(x) &:= \sum_{B \not \sim A} x^B =
\text{no.\ of peers an $A$-peer can swap with},
\\
\psi_d^A(x) &:= \sum_{a \in A} x^{A-a} =
\text{no.\ of peers which can assume label $A$ after a download},
\\
\psi_s^{A,B}(x) &:= \sum_{a \in A \cap B} x^{A-a} =
\text{no.\ of peers which can assume label $A$ after a $B$-peer swap}.
\end{align*}
It is helpful to keep these in mind because they aid in
writing down the various parts of $v(x)$, again, see the Appendix.
\section{Behaviour of the limiting ODE}\label{ODEbeh}
Concerning the ODE $\dot x = v(x)$ we consider again three cases,
just as in the stochastic model: an open system ($\alpha^A >0$ for
at least one $A$ and $\delta > 0$), a closed dissipative system
($\alpha^A=0$ for all $A$ and $\delta > 0$), and a closed
conservative system ($\alpha^A=0$ for all $A$ and $\delta=0$).
Qualitatively, the behaviour is different in each case.
In this section, we will try to exemplify this behaviour by means
of examples.
\subsection{The ODE in the absence of BitTorrent incentives}
Absence of BitTorrent incentives means that the file is not
split into chunks, i.e.\ $n=1$.
Thus, the Markov chain is $X_t = (X^\varnothing_t, X^1_t :=X^F_t)$, i.e.\
two-valued $A \in\{\emptyset, 1\}$.
The rates for this case were reported earlier in \eqref{kmc}.
To find the fluid limit, we use \eqref{phipsi} and \eqref{vcomp},
keeping in mind the interpretation of each of the terms in \eqref{phipsi}.
For $A=\emptyset$, we have:
the number of peers from which an $\emptyset$-peer
can download from is $\phi_d^\emptyset(x) = x^1$;
the number of peers that can swap with an $\emptyset$-peer is
$\phi_s^\emptyset(x)=0$;
since no peer can assume value $\emptyset$ after an interaction,
we have $\psi_d^\emptyset(x) = \psi_s^{\emptyset,B}(x) = 0$.
Hence, in the formula for \eqref{vcomp} for $v^\emptyset(x)$
only the first parenthesis survives:
\[
v^\emptyset(x) = \alpha^\varnothing -\beta x^\emptyset x^1.
\]
For $A=1$ we have:
$\phi_d^1(x) = \phi_s^1(x)=0$, $\psi_d^1(x) = x^\emptyset$,
$\psi_s^{1,\emptyset}(x)=0$.
Thus,
\[
v^1(x) = \alpha^1 + \beta x^\emptyset x^1 -\delta x^1.
\]
Here, by simply letting $x:=x^\varnothing$ and $y:=x^1$,
the ODE is
\begin{align*}
\dot x &= \alpha^\varnothing -\beta x y \\
\dot y &= \alpha^1+ \beta xy -\delta y.
\end{align*}
If $\alpha^\varnothing=\alpha^1=\delta=0$ (closed conservative system),
we have $x+y$=constant, say $=1$, and
\[
\dot x = -\beta x(1-x),
\]
the solution of which is the logistic function,
\begin{center}
\begin{tabular}{lcr}
$\displaystyle x_t = \frac{x_0}{x_0+(1-x_0)e^{\beta t}}$:
&
&
\begin{minipage}{7cm}
\epsfig{file=aa.eps,height=3.5cm}
\end{minipage}
\end{tabular}
\end{center}
If $\alpha^\varnothing=\alpha^1=0$, but $\delta >0$ (closed
dissipative system) we have the classical deterministic
SIR epidemic
\begin{align*}
\dot x &= -\beta x y \\
\dot y &= \beta xy -\delta y
\end{align*}
the integral curves of which can be found by solving
\[
\frac{dy}{dx} = \frac{\delta}{\beta x} -1,
\]
whence it follows
\begin{center}
\begin{tabular}{lcr}
$\displaystyle
y = (x_0+y_0) + \frac{\delta}{\beta} \log(x/x_0)-x$:
&
&
\begin{minipage}{7cm}
\epsfig{file=bb.eps,height=5cm}
\end{minipage}
\end{tabular}
\end{center}
Assume $x_0 + y_0 \le 1$.
Notice that the integral curves
are monotonic as long as they start from a point $(x_0, y_0)$
with $x_0 > \delta/\beta$.
They eventually converge to a point of the form $(x^*,0)$
where
\[
(x_0+y_0) + \frac{\delta}{\beta} \log(x^*/x_0)-x^* =0.
\]
In other words, $(x_t, y_t) \to (x^*, 0)$ as $t \to \infty$ and, in fact,
does so exponentially fast.
Notice that if the initial state is in the interior of
the positive orthant then the boundary cannot be reached in finite time.
This is in contrast to the corresponding stochastic model
which can reach the boundary in finite time with positive probability.
In fact, in the open case, it will reach the boundary in finite
time with probability $1$ due to (positive) recurrence.
This remark is generic: it applies to any dimension.
Next, consider the open system case assuming, for simplicity,
that $\alpha^\varnothing = \lambda >0$, but $\alpha^1=0$.
We have
\begin{align*}
\dot x &= \lambda -\beta x y \\
\dot y &= \beta xy -\delta y.
\end{align*}
Since $\delta >0$, the Lyapunov function argument shows that
the system is asymptotically stable. In fact, there is
a unique asymptotic equilibrium which attracts all initial conditions
$(x_0, y_0)$ with $y_0 > 0$.
This unique equilibrium is given by
\[
x^* = \frac{\delta}{\beta}, ~ y^* = \frac{\lambda}{\delta},
\]
as can be seen by setting the right hand side of the ODE equal to zero
\cite{Daley-Gani}.
It should be noticed that the trajectories can be spirals around
$(x^*, y^*)$.
A typical situation
is shown below.
\begin{center}
\epsfig{file=spiral.eps, height=6cm}
\end{center}
In this vector field plot, we took $\beta=3$, $\lambda=5$, $\delta=4$,
so $(x^*, y^*) = (1.33, 1.25)$.
If $y_0=0$ then the trajectory converges to infinity. Indeed,
if no peers initially possess the file and no such peers ever show
up, then the only thing that can happen is an accumulation, at rate
$\lambda$, of ever demanding peers.
The remedy is, clearly, the imposition of $\alpha^1>0$ no matter
how small; then $(x^*, y^*)$ is globally attracting for
all initial states in the closed positive orthant.
In this last example, we find that the eigenvalues of the differential
of the vector field at $(x^*, y^*)$ are complex conjugates
if and only if $\lambda\beta < 4 \delta^2$, and so this is the condition
for the spiralling of trajectories.
\subsection{The ODE for $n=2$ chunks}
Here $A$ can take $4$ values: $\emptyset$, $\{1\}$, $\{2\}$, $\{1,2\}:= F$.
We work out the expression for each
component separately.
Since an $\emptyset$-peer can only download from
$\phi_d^\emptyset(x)=x^1+x^2+x^{12}$
peers\footnote{Where $x^{12}=x^F$ and indexes 1 and 2 represent the chunks that
form a partition of $F$.}, but no peer can download from it or swap with it,
we have $\phi_s^\emptyset(x)=\psi_d^\emptyset(x)=\psi_s^\emptyset(x)=0$, and
so
\[
v^\emptyset(x) = \alpha^\varnothing -\beta x^\emptyset (x^1+x^2+x^{12}).
\]
A $1$-peer can download from $\phi_d^1(x)=x^{12}$ peers;
it can swap with $\phi_s^1(x)=x^2$ peers;
the number of peers that take value $1$ after a download is
$\psi_d^1(x)+x^\emptyset$; and, since no peer can take the
label $1$ after a swap, we have $\psi_s^{1,B}(x)=0$ $\forall B$.
We thus have
\begin{align*}
v^1(x) &= \alpha^1 -x^1(\beta x^{12} + \gamma x^2)
+ \beta \psi_d^1(x) \sum_{B: 1 \in B} \frac{x^B}{1+|B\setminus \{1\}|}
\\
&=
\alpha^1 -x^1(\beta x^{12} + \gamma x^2)
+ \beta x^\emptyset(x^1 + \tfrac{1}{2} x^{12}).
\end{align*}
The expression for $v^2(x)$ is symmetric to that of $v^1(x)$.
A $12$-peer can neither download from or swap with anyone,
so $\phi_d^{12}(x) = \phi_s^{12}(x)=0$;
there are $\psi_d^{12}(x) = x^1+x^2$ peers which can take label $12$
after a download;
since only a $2$-peer can assume label $12$ after a swap, we have
$\psi_s^{12, 1}(x) = x^2$; and, for the same reason,
$\psi_s^{12, 2}(x) = x^1$.
Thus,
\begin{align*}
v^{12}(x) &= \alpha^{12} + \beta \psi_d^{12}(x) x^{12} +
\frac{\gamma \psi_s^{12,1}(x) x^1}{1+|\{1\}\setminus \{1,2\}|}
+
\frac{\gamma \psi_s^{12,2}(x) x^2}{1+|\{2\}\setminus \{1,2\}|}
-\delta x^{12}
\\
&=
\alpha^{12} + \beta (x^1+x^2) x^{12}
+ \gamma (x^2 x^1 + x^1 x^2)
-\delta x^{12}.
\end{align*}
We thus have,
\begin{align*}
&\dot x^\emptyset
=
\alpha^\varnothing -\beta x^\emptyset (x^1+x^2+x^{12})
\\
&\dot x^1
=
\alpha^1 -x^1 (\beta x^{12} +\gamma x^2)
+ \beta x^\emptyset ( x^1 + \tfrac{1}{2}x^{12} )
\\
&\dot x^2
=
\alpha^2 -x^2 (\beta x^{12} +\gamma x^1)
+ \beta x^\emptyset ( x^2 + \tfrac{1}{2}x^{12} )
\\
&\dot x^{12}
=
\alpha^{12} + \beta (x^1+x^2) x^{12} + 2\gamma x^1 x^2 -\delta x^{12}.
\end{align*}
\paragraph{Case 1: closed conservative system.}
Consider the $n=2$ case again, and assume that $\alpha^A =0$ for all $A$,
$\delta=0$ (a closed conservative system). Assume further that
$\gamma=0$. Let
\[
x=x^\varnothing,\quad u=x^1+x^2,\quad w=x^{12}.
\]
We see a reduction in dimension from $4$ to $3$ (owing to
that fact that the sum of the coordinates is constant)
and a further reduction from $3$ to $2$ (owing to the
fact that, for $\gamma=0$, the vector field depends
on $x^1$, $x^2$ through their sum).
We have:
\begin{align*}
\dot x &= -\beta x(u+w) \\
\dot u &= -\beta uw + \beta x(u+w) \\
\dot w &= \beta uw.
\end{align*}
On assuming that
\[
x+u+w=1
\]
and eliminating the variable $w$ we obtain
\begin{align*}
\dot x &= -\beta x(1-x) \\
\dot u &= \beta u^2 - \beta u(1-x) + \beta x(1-x).
\end{align*}
We can qualitatively see the behaviour of this ODE by
looking at the vector field in the $x-u$ plane. A typical
picture is as follows:
\begin{center}
\epsfig{file=logist.eps, height=6cm}
\end{center}
The first is an autonomous equation, encountered earlier.
Its solution is $x_t = x_0/(x_0+(1-x_0)^{\beta t})$.
Letting
\[
f := 1-w = x+u
\]
we have
\[
\dot f = \beta f (1-f) + \beta (1-f) x_t
\]
Assume that $w_0 > 0$, i.e.\ $f_0 < 1$.
The solution of the last ODE is:
\[
f_t = \frac{f_0(1-f_0)^{-1} + x_0 \beta t}
{x_0+(1-x_0)e^{\beta t} + f_0(1-f_0)^{-1} + x_0 \beta t}.
\]
Hence
\[
w_t = \frac{x_0 + (1-x_0) e^{\beta t}}
{x_0 + (1-x_0) e^{\beta t} + x_0 \beta t + (1-w_0)w_0^{-1}}.
\]
We have $\lim_{t \to \infty} w_t =1$,
as expected.
{From} this, we can estimate the time $\tau=\tau(x_0,w_0, \epsilon)$
required for the
system to reach the set $\{(x,u,w):~ w > 1-\epsilon\}$,
where $0< \epsilon <1$ is a given (small) number.
This time can be translated into an estimate for the expected time
required for the original stochastic system to be absorbed by
the state where all peers possess the full file. We shall not attempt
the justification of this statement here.
The time $\tau$ is the solution of the transcendental equation
\[
\frac{1-w_0}{w_0} + x_0 \beta \tau
= \frac{\epsilon}{1-\epsilon}(x_0+(1-x_0) e^{\beta \tau}).
\]
Typical behaviour of this time as a function of
$x_0$ is as follows:
\begin{center}
\epsfig{file=tau.eps, height=6cm}
\end{center}
The three graphs correspond to varying values of $\beta$
ranging from $1$ (top curve) to $5$ (bottom curve) and to
$\epsilon=0.001$, $w_0=0.1$.
\paragraph{Case 2: closed dissipative system.}
Consider again the $n=2$ case with no arrivals, with swap rate $\gamma$ equal
to zero, but with departure rate $\delta > 0$. Using the same
notation as above, we have
\begin{align*}
\dot x &= -\beta x(u+w) \\
\dot u &= -\beta uw + \beta x(u+w) \\
\dot w &= \beta uw-\delta w.
\end{align*}
To substitute out one parameter, change the time
variable to $s=\beta t$, let
$\rho=\delta/\beta$, and write $x'$ for $dx/ds$, so that
\begin{align*}
x' &= -x(u+w) \\
u' &= -uw + x(u+w) \\
w' &= uw-\rho w.
\end{align*}
Assume
\[
x_0+u_0+w_0=1,
\]
so $x_t+u_t+w_t < 1$ for all $t > 0$.
We cannot eliminate the variable $w$ now since there is no
obvious conserved quantity, but we can study the
equilibria of the system.
It seems that the ODE has no obvious analytical solution.
However it can either be integrated numerically or its solution in
terms of power series can be easily found.
Setting the right hand side equal to zero we see that, necessarily,
$w=0$, which leaves us with the possibility $xu=0$.
So points of the form
\[
(x,u,w) = (x,0,0), ~ (0,u,0)
\]
are equilibria.
But not all of them are stable.
For example, any point of the form $(0,u,0)$ with $u>\rho$
is an unstable equilibrium.
Specifically, the differential of the vector field
is given by
\[
Dv =
\begin{pmatrix}
-(u+w) & -x & -x
\\
u+w & x-w & x-u
\\
0 & w & u-\rho
\end{pmatrix}
\]
Evaluated at $(0,u,0)$, it gives
$Dv =
\textstyle
\begin{pmatrix}
0 & 0 & 0
\\
0 & 0 & -u
\\
0 & 0 & u-\rho
\end{pmatrix}$,
and this has eigenvalues $0, -u, u-\rho$.
Evaluated at $(x,0,0)$,
it gives
$Dv=
\textstyle
\begin{pmatrix}
0 & -x & -x
\\
0 & x & x
\\
0 & 0 & \rho
\end{pmatrix}$,
and this has eigenvalues $0, x, -\rho$.
Therefore, the only stable equilibria
are points of the form
\[
(x,u,w) = (0,u,0), \quad 0 \le u < \rho.
\]
In terms of the original variables, the stable equilibria are
\[
(x^\varnothing, x^1, x^2, x^{12}) = (0, x^1, x^2, 0),
\quad 0 \le x^1+x^2 < \rho.
\]
This is as expected: since there is no swapping ($\gamma=0$),
the system eventually settles to a situation where there
are peers with label $1$ and peers with label $2$.
Had $\gamma$ been positive, $x^1$, $x^2$ could not have simultaneously
been positive in equilibrium.
\paragraph{Case 3: Open system.}
Consider the situation as in Case 2, but add arrivals of peers
(known as seeds) possessing the full file.
Choosing variables appropriately, we have
\begin{align*}
x' &= -x(u+w) \\
u' &= -uw + x(u+w) \\
w' &= \lambda+ uw-\rho w.
\end{align*}
In terms of the original variables, we here have $\alpha^\varnothing=
\alpha^1=\alpha^2=0$, $\alpha^{12} = \lambda \beta > 0$.
Here we see that the system eventually settles to the stable
equilibrium
\[
(x,u,w) = (0,0, \lambda/\rho).
\]
We can easily see that the eigenvalues of the differential
of the vector field at the stable equilibrium
are both real and negative: $-\lambda/\rho$ and $-\rho$
(the first one has algebraic multiplicity 1 but geometric multiplicity 2).
Hence there is no possibility of spiralling.
\begin{center}
\epsfig{file=xuw.eps, height=5.5cm}
\end{center}
\subsection{Time to settle}
\label{settle}
Estimating the time for the system to reach an equilibrium
requires cooking up an appropriate Lyapunov function.
The obvious Lyapunov function used earlier gives a crude
lower bound.
Consider a general deterministic system (open or closed),
i.e.\ the differential equation $\dot x = v(x)$ with $v(x)$
given by \eqref{vcomp}. Let $V(x) = |x| = \sum_{A \subset F} x^A$.
Then
\[
\dot V = |\alpha| -\delta x^F,
\]
where $|\alpha| = \sum_{A \subset F} \alpha^A$ is the total arrival rate.
Using the naive inequality $x^F \le V$ we obtain the following:
\begin{corollary}
If the system starts from $x_0$ and if $x_t \to x^*$ then
the time $\tau_r$ required for the trajectory to reach an $r$-ball
centred at $x^*$ satisfies
\[
\tau_r \ge \frac{1}{\delta} \log \frac{|x_0|+|\alpha|}{|x^*|+r}.
\]
\end{corollary}
In the other direction,
consider the last component $v^F(x)$ of the vector field.
Since no seed ($F$-peer) can download or swap, we have
$\phi_d^F(x)=\phi_s^F(x)=0$. So
\begin{align*}
v^F(x) &= \alpha^F
+ \beta \psi_d^F(x) x^F
+ \gamma \sum_{B \subset F} \psi_s^{F,B}(x) x^B - \delta x^F
\\
&= \alpha^F + \beta \sum_{j \in F} x^F x^{F-j}
+ \gamma \sum_{B \subset F} \sum_{i \in B} x^B x^{F-i}
- \delta x^F
\\
&=: \alpha^F + v_+^F(x) - \delta x^F,
\end{align*}
where $v_+^F(x)$ is a quadratic form with positive coefficients.
Assume that the system is closed, so that $\alpha^F=0$ in particular.
Let $\overline {v_+^F}$
be an upper bound on $v_+^F(x)$, i.e., a (good) bound that depends
on the initial state
$x_0$ and the parameters $\beta, \gamma$.
\begin{corollary}
If the system is closed and
$\overline {v_+^F} < \delta$,
then
\[
\tau_r \le \frac{1}{\overline {v_+^F}-\delta} \log\frac{x_0^F}{r}.
\]
\end{corollary}
We conjecture that $v_+^F(x_t)$ decreases along the trajectory $x_t$
as long as $v_+^F(x_0) < \delta$.
If so, the bound above is
$\tfrac{1}{v_+^F(x_0)-\delta} \log\frac{x_0^F}{r}$.
\section{An example of the evaluation of performance improvement in presence of
BitTorrent incentives}\label{incentives-sec}
We address the following question: When is it advantageous to split a file
into chunks? In other words, assuming we fix certain system parameters (e.g.,
arrival rates), will peers acquire the file
faster if the file is split into chunks?
We attempt here to answer the question in a simple case only
by using the deterministic approximation.
Let $\lambda$ be the total peer arrival rate. Let $\beta$ be the download rate.
Assume that only $\varnothing$ peers arrive exogenously.
In the absence of BitTorrent incentives, we have the single-chunk case
\begin{align*}
\dot x^\varnothing &= \lambda - \beta x^\varnothing x^1
\\
\dot x^1 &= \beta x^\varnothing x^1 -\delta x^1.
\end{align*}
The globally attracting stable equilibrium is given by
\[
x^* = (\delta/\beta, ~ \lambda/\delta).
\]
Consider splitting into $n=2$ chunks.
Let $\widetilde x$ be the state of the system.
Suppose that the new parameters are
$\widetilde \lambda = \lambda$, $\widetilde \delta = \delta$, $\widetilde \beta$, $\widetilde \gamma$.
Then
\begin{align*}
&\dot {\widetilde x}^\emptyset
=
\lambda -\widetilde \beta \widetilde x^\emptyset (\widetilde x^1+\widetilde x^2+\widetilde x^{12})
\\
&\dot {\widetilde x}^1
=
-\widetilde x^1 (\widetilde \beta \widetilde x^{12} +\widetilde \gamma \widetilde x^2)
+ \widetilde \beta \widetilde x^\emptyset ( \widetilde x^1 + \tfrac{1}{2}\widetilde x^{12} )
\\
&\dot {\widetilde x}^2
=
-\widetilde x^2 (\widetilde \beta \widetilde x^{12} +\widetilde \gamma \widetilde x^1)
+ \widetilde \beta \widetilde x^\emptyset ( \widetilde x^2 + \tfrac{1}{2}\widetilde x^{12} )
\\
&\dot {\widetilde x}^{12}
=
\widetilde \beta (\widetilde x^1+\widetilde x^2) \widetilde x^{12} + 2\widetilde \gamma \widetilde x^1 \widetilde x^2 -\delta \widetilde x^{12}.
\end{align*}
The new equilibrium is easily found to be
\[
\widetilde x^* = \bigg(\frac{\delta}{\widetilde \beta} \big(\frac{\delta}{\lambda}u+1 \big)^{-1},~
\frac{u}{2},~ \frac{u}{2},~ \frac{\lambda}{\delta}\bigg),
\]
where $u$ is the positive number which solves
\[
q(u)=0,
\]
and
\begin{equation}
\label{qqq}
q(u):= u^2 + \frac{2 \widetilde \beta\lambda}{\widetilde \gamma \delta} u
- \frac{2 \lambda}{\widetilde \gamma}.
\end{equation}
To see this, set the vector field equal to zero and solve for
$\widetilde x^A$, $A\in\{\varnothing, 1, 2, 12:= F\}$.
The simplest way to obtain the solution is by first adding the equations;
this gives
\[
\lambda-\delta \widetilde x^{12} = 0,
\]
whence $\widetilde x^{12} = \lambda/\delta$.
Then add the middle two equations after setting $u=\widetilde x^1 + \widetilde x^2$:
\[
-\widetilde \beta u \widetilde x^{12} -2\widetilde \gamma \widetilde x^1 \widetilde x^2 +
\widetilde \beta x^\varnothing
(u+ \widetilde x^{12}) =0.
\]
Replace $\widetilde x^{12}$ by $\lambda/\delta$ and observe that, due to symmetry,
$\widetilde x^1 = \widetilde x^2 = u/2$. This gives the quadratic equation $q(u)=0$
with $q$ defined by \eqref{qqq}.
Finally, the first equation becomes
\[
\lambda - \widetilde \beta \widetilde x^\varnothing (u + \lambda/\delta)=0,
\]
which is solved for $\widetilde x^\varnothing$ giving:
\[
\widetilde x^{*\varnothing} = \frac{\lambda}{\widetilde \beta}
\big(u+\frac{\lambda}{\delta})^{-1}
= \frac{\delta}{\widetilde \beta} \big(\frac{\delta}{\lambda}u+1 \big)^{-1}
< \frac{\delta}{\widetilde \beta}.
\]
Thus:
\begin{corollary}
If $\widetilde \beta \ge \beta$ then
\[
{\widetilde x}^{*\varnothing}< x^{*\varnothing} .
\]
\end{corollary}
So, by introducing splitting into chunks, we have fewer peers
who have no parts of the file at all.
Using Little's theorem (see below), this can be translated into smaller waiting
time from the time a peer arrives until he gets his first chunk.
Suppose now we are interested in determining how long it
will take for a newly arrived peer to acquire the full file.
On the average, a peer spends time equal to $\lambda^{-1} |x^*|$
before it exits the system. During last part of his sojourn interval
(which is a random variable with mean $1/\delta$), the peer
possess the full file. It thus takes on the average $\lambda^{-1} |x^*|-
\delta^{-1}$ for a peer to acquire the full file. Since
we assume that $\widetilde \lambda = \lambda$, $\widetilde \delta = \delta$,
it suffices to show that
\[
|x^*| > |\widetilde x^*|.
\]
But
\begin{align*}
|x^*| - |\widetilde x^*|
&= \left[\frac{\delta}{\beta}+\frac{\lambda}{\delta}\right]
- \left[\frac{\delta}{\widetilde \beta}
\big(\frac{\delta}{\lambda}u+1 \big)^{-1} + u +\frac{\lambda}{\delta}\right]
\\
&= \frac{\delta}{\beta}
- \frac{\delta}{\widetilde \beta}\big(\frac{\delta}{\lambda}u+1 \big)^{-1}
-u -\frac{\lambda}{\delta}
\\
&= \big(\frac{\delta}{\lambda}u+1 \big)
\left[
\big(\frac{\delta}{\beta}-\frac{\delta}{\widetilde \beta}\big)
+\big(\frac{\delta^2}{\beta \lambda}-1\big) u
-\frac{\delta}{\lambda} u^2
\right]
\end{align*}
recalling $u>0$ solves $q(u)=0$.
So, $|x^*| - |\widetilde x^*| > 0$ if and only if
\begin{equation}
\label{tqqq}
0> \widetilde q(u) := u^2 - \big( \frac{\delta}{\beta}-\frac{\lambda}{\delta} \big) u
- \big( \frac{\lambda}{\beta} - \frac{\lambda}{\widetilde \beta} \big).
\end{equation}
Define $\widetilde u$ as the unique positive number which satisfies
\[
\widetilde q (\widetilde u) =0.
\]
\begin{corollary}\label{incentives-coro}
If $\widetilde \beta \ge \beta$,
a necessary and sufficient condition for $|x^*| > |\widetilde x^*|$
is $u < \widetilde u$.
\end{corollary}
This gives a set of non-vacuous conditions for achieving improvement of
performance by the introduction of BitTorrent incentives.\footnote{The
inequality conditions were mistakenly reversed in the corresponding results of
\cite{ICASSP07}.}
It can be proved that if the parameters
$\beta$, $\delta$, $\widetilde \beta$, $\widetilde \gamma$ are fixed
and if
$\widetilde \beta \ge \beta$
then there exists a value $\lambda_0$ such that for all
$\lambda < \lambda_0$ the inequality $u < \widetilde u$ holds.
To prove this, we observe that
\[
u < \widetilde u \iff q(\widetilde u) > 0
\]
and study the behaviour of $q(\widetilde u)$ as a function of $\lambda$ in
a neighbourhood of zero.
We conjecture that an algebraic condition involving quadratics like
$q$ and $\widetilde q$ is valid for larger values of $n$ also.
To justify the use of deterministic approximation for
estimating performance measures, and, specifically, the use of mean
values, we need to show that as $N \to \infty$,
we can approximate stationary averages in the original stochastic
network by equilibria of the resulting ODE.
It is easy to show that the a.s.\ convergence to the ODE limit
can be translated into convergence of the means, using
a uniform integrability argument. Namely,
\[
\frac{1}{N} E X_{\text{\tiny $N$},t}
\xrightarrow[\text{\tiny $N$} \to \infty]{} x_t \xrightarrow[t \to \infty]{} x^*,
\]
where the second limit concerns the behaviour of the ODE alone.
On the other hand, if we fix $N$ and look at the
asymptotic behaviour of the process $\tfrac{1}{N} X_{\text{\tiny $N$},t}$ as
$t\to \infty$, we have
\[
\frac{1}{N} E X_{\text{\tiny $N$},t}
\xrightarrow[t \to \infty]{}
\frac{1}{N} E \widetilde X_N,
\]
where the law of $\widetilde X_N$ is the stationary distribution of
the chain $(\tfrac{1}{N} X_{\text{\tiny $N$},t})_{t \ge 0}$.
It can be proved that $\tfrac{1}{N} E \widetilde X_N \to
x^*$, as $N \to \infty$.
Arguments for this will be considered in future work.
More detailed estimates on the discrepancy
between the stochastic and deterministic systems can be found in
the recent survey paper \cite{DN08}.
We can also explain the use of $|x^*|$ as a measure of the
sojourn time in the system of a peer, by first using the
approximation outlined above and then appealing to Little's law.
This is as follows.
Consider an open \bt{X_{0}}{n}{\alpha}{\beta}{\gamma}{\delta},
i.e.\ $|\alpha| > 0$, $\delta > 0$.
We know that the Markov chain $(X_t)$ is positive recurrent and has thus
a unique stationary distribution.
It makes sense to assess the performance of the network by looking at
steady-state performance measures, such as the mean time it takes
for an $\varnothing$-peer to become an $F$-peer (a seed).
Consider then the process $(\widetilde X_t, t \in \mathbb{R})$
defined to be a stationary Markov process with time index $\mathbb{R}$
and transition rates as those of $(X_t)$.
The law of the process $(\widetilde X_t, t \in \mathbb{R})$ is unique.
Let $T^A_k, k \in \mathbb{Z}$ be the times at which $A$-peers arrive
(and, say, $T^A_0 \le 0 < T^A_1$, by convention). These are the
points of a stationary Poisson process in $\mathbb{R}$ with rate $\alpha^A$.
Let $W^A_k$ be the sojourn time in the system of a peer
arriving at time $T^A_k$. Since, by assumption, a peer departs only
after it has acquired the full set, the time $W^A_k$ is the sum of the
times it takes for the peer to become a seed plus the time that the
peer hangs out in the system after becoming a seed (the latter is
an exponential time with mean $1/\delta$).
Clearly then, for all $t \in \mathbb{R}$,
\[
\sum_{B \supset A} \widetilde X^B_t = \sum_{k \in \mathbb{Z}}
{\text{\Large $\mathfrak 1$}}(T_k^A \le t < T_k^A + W_k^A).
\]
Using Campbell's formula, we obtain
\begin{equation}
\label{ll}
\sum_{B \supset A} E \widetilde X^B_0
= \alpha^A E^A W^A_0,
\end{equation}
where $E^A$ is expectation with respect to $P^A$--the
Palm probability of $P$ with respect to the point process $(T^A_k, k \in \mathbb{Z})$.
In particular, with $A=\varnothing$, and $\lambda=\alpha^\varnothing$,
we have that
\[
E^\varnothing W_0^\varnothing = \frac{1}{\lambda}
E |\widetilde X_0|,
\]
which can be read as: the mean sojourn time of a $\varnothing$-peer
is, in steady state, equal to the mean number of peers in the
system divided by the rate of arrivals of $\varnothing$-peers.
If $N$ is a parameter of the process as in Theorem \ref{ODEapprox}
then, $\lambda$ being proportional to $N$, we have that
the right hand side converges to something that is proportional
to $|x^*|$, as required.
\section{Diffusion approximation}\label{diff-sec}
Using the functional central limit theorem
for Poisson processes, we can prove, by standard methods, the following:
Again consider the sequence
\bt{X_{\text{\tiny $N$},0}}{n} {N\alpha}{\frac{\beta}{N}}
{\frac{\gamma}{N}}{\delta} for
$N \in \mathbb{N}$, and let $X_{\text{\tiny $N$},t}$ be the corresponding jump Markov chain.
Let $(x_t, t \ge 0)$ be the solution
to the ODE $\dot x = v(x)$ with initial condition $x_0$.
Let
\[
Y_{\text{\tiny $N$},t} := \sqrt{N} (X_{\text{\tiny $N$},t}/N - x_t).
\]
Let $W_\zeta$, $\zeta \in \mathcal{N}$, be i.i.d.\ standard Brownian motions
in $\mathbb{R}$.
Finally, define the (time-inhomogeneous) Gaussian diffusion process $Y$ by
\[
dY_t = \sum_{\zeta \in \mathcal{N}} \zeta \sqrt{Q_\zeta(x_t)} dW_{\zeta,t}
+ Dv(x_t) Y_t dt,
\]
where $Dv(x)$ is the matrix of partial derivatives
of $v(x)$.
\begin{theorem}
\label{diffapprox}
If $\sqrt{N} (X_{\text{\tiny $N$},t}/N-x_0) \to 0$ as $N \to \infty$,
where $x_0 \in \mathbb{R}_+^{\mathscr{P}(F)}$,
then the law of $Y_\text{\tiny $N$}$ (as a sequence of probability
measures in $D[0,\infty)$ with the topology of
uniform convergence on compacta) converges weakly
to the law of $Y$.
\end{theorem}
The proof of this theorem is omitted but the reader is referred to
\cite{Kurtz81} for the relevant arguments.
\section{Final remarks, open problems and future work}\label{concl-sec}
\subsection{Rates of convergence}
We can obtain a computable
rate of convergence of the stochastic model to the ODE
by using a combination of large deviations techniques with
the solution of two optimisation problems. The idea is basically
implicit in the proof of Theorem \ref{ODEapprox} and this is the
reason we wrote the proof explicitly in terms
of the driving Poisson processes $\Phi_\zeta$.
The first problem is so that we obtain
an estimate of the maximum value of $M_\zeta$ of $Q_\zeta(x)$.
In the case of a closed network, this is a quadratic optimisation
problem over the polyhedron $\{|x| \le 1\}$.
The second optimisation problem is for an estimate for $L_\zeta$,
which can be translated to an estimate for the norm of the gradient
$\nabla Q_\zeta(x)$. In the case of a closed
network, we can estimate this by solving
a (large) number of linear programming problems.
Of course, only estimates are needed.
\subsection{Conjectures}
The first one concerns the behaviour of $v_+^F(x_t)$, and was stated
at the end of Section \ref{ODEbeh}.
The second concerning generalisation of Corollary \ref{incentives-coro},
was stated in Section \ref{incentives-sec}.
The third conjecture is more vague: it basically says that we
can evaluate the performance improvement by using a large number
of chunks (say 100), by solving a number of quadratic inequalities.
To this end, it should be remarked that in a deterministic open
network, the unique equilibrium $x^*$ can be found by solving
$n+1$ equations in $n+1$ unknowns, provided that the rate of
arrivals of $A$-peers depends on $A$ through its cardinality alone:
\[
\alpha^A = \lambda_k \text{ if } |A|=k \text{ chunks.}.
\]
Indeed, by symmetry of the vector field, we see that
\[
x^{*A} = x^{*B} \text{ if } |A|=|B|.
\]
So if we define
\[
z^k := \sum_{|A|=k} x^{*A}
\]
we will have
\[
x^{*A} = \binom{n}{k}^{-1} z^k, \text{ if } |A|=k.
\]
Hence if we let
\[
V^k(z^0,\ldots,z^n) := \sum_{|A|=k} v^A(x^*),
\]
where $z$ and $x^*$ are related as above, then
the equations we need to solve are
\[
V^k(z^0,\ldots,z^n) =0, \quad k=0, \ldots, n.
\]
\subsection{A reduction of dimension for a balanced ODE}
If we are interested not only in the equilibria but also in
a more detailed study of the transient behaviour of the ODE, then
we can obtain a rough idea (and bounds) by making the assumption
of full symmetry, i.e.,\ we assume that the arrival rates $\alpha^A$
and the initial states $x_0^A$ depend on $A$ only through $|A|$.
Then the trajectory itself $x^A_t$
depends only on the cardinality of $A$ and so we can
reduce the ODE to an $(n+1)$-dimensional one.
Such a symmetrised ODE can yield more detailed information on the time
to reach a small neighbourhood of the equilibrium point.
(Note that our bounds in \S \ref{settle} are very crude.)
\subsection{Non-Poissonian assumptions}
It may be more reasonable in practise to assume that the
time it takes for a chunk to be downloaded or swapped
is a random variable with a heavy-tailed distribution.
This is not captured by our model. Indeed, the interaction times
are not part of our model at all.
A new, more detailed, model should be worked out.
However, a crude capture of this phenomenon is the replacement of
the Poisson processes $\Phi_\zeta$ by more
general point processes, perhaps with heavy-tailed inter-event times.
As long as these processes obey a functional law of large numbers,
we can (by possibly modifying the scaling parameters) rephrase
Theorem \ref{ODEapprox} and repeat the proof in this more general
case.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,768
|
\section{Introduction\\}
\tpoint{Background and recent history of topology\\}
The history of topology begins with development of algebraic topology in a series of papers published in 1894 and 1895 by Henri Poincar\'{e} (\cite{tophistory}, Preface). This field of mathematics examines properties of geometric objects invariant under continuous, invertible transformations such as stretching or bending, called \textit{homeomorphisms}. These invariant properties include the notions of connectivity and \textit{genus}---informally, the number of ``holes'' in an object---and are irrespective of scale, shape, and any underlying coordinate system. In contrast, classical Euclidean geometry only considers the so-called \textit{rigid transformations} of translation, rotation, and reflection. As such, topology is much less strict in its classification of geometric bodies than is geometry. A classical and well-known example of the generality of topological classification is the equivalence of a coffee mug and a doughnut, as shown in \textbf{Figure \ref{coffee}}. These two objects are certainly not equivalent under the rigid transformations of Euclidean geometry.
\begin{figure}[!h]
\centering
\includegraphics[scale=0.40]{coffee}
\caption{\footnotesize{The transformation of a coffee mug into a doughnut under a continuous, invertible transformation. Intuitively, the transformation is continuous because it does not tear the object, and it is invertible because the transformation can be reversed.}}\label{coffee}
\end{figure}
Though rooted and grown in the realm of pure mathematics for most of its history, topology has recently piqued interest across numerous disciplines, including the biological \cite{bio}, medical \cite{med}, and computational \cite{image} sciences. Advances in these fields have necessitated the development of new methods for data analysis and visualisation, particularly where collected data is of high dimension---that is, measured by a large number of descriptors or independent variables---and difficult to interpret. Carlsson, a major contributor to the development of persistent homology, argues that metric- and coordinate-based analytic methods, specifically outside the field of physics, are often unjustified, subjective, and unnatural, especially where analysis is exploratory and intended to yield preliminary or qualitative results (\cite{topdata}, Section 1). As such, topology, with its classification flexibility and disregard of coordinates and metrics, presents itself as a useful data analytic tool. The application of topology to other areas has given rise to the field of \textit{computational topology}.\\
\tpoint{Persistent homology\\}\label{sec.ph}
Foundational to the methods of computational topology is \textit{persistent homology}. This technique makes use of data sampled from some unknown object, space, or phenomenon to recreate the original object's structure and approximate its topology. This recovery of topological properties has direct application to computer image processing \cite{image}, 3D-modeling, and network analysis \cite{sensor} in physics and computing science. In fields such as biology where data may not necessarily be sampled from a physical object, persistent homology can yield insight into the processes and phenomena underlying and creating the observed data.\\
This topological reconstruction is carried out by building structures, called \textit{simplicial complexes}, using collected data. Although there are numerous ways to construct a simplicial complex, these complexes generally serve to join data points that are deemed sufficiently ``close'' under some pre-specified notion of similarity that is in general not a metric. By observing the \textit{homology}---informally, the connected components, loops, and ``holes''---of a simplicial complex, we approximate the homology of the original, unknown object. Simplicial complexes can be created on any scale, effectively permitting the examination of the original object at any local or global scale or resolution.\\
A fundamental idea underlying persistent homology is that true, global features of the original, unknown object will be present in the simplicial complexees across a wide range of scales. Local features and random noise, on the other hand, will appear only over a limited range of scales. The global significance of a topological feature observed in a simplicial complex, then, is determined by the range of scales over which the feature exists, and is referred to as the feature's \textit{persistence} or \textit{lifetime}. The persistence of all observed features can be represented mathematically by a collection of intervals, and visually by a series of lines, referred to as a \textit{barcode}. Using this barcode, we can make inferences regarding the number of topological features of each dimension present in the original object, referred to as the object's \textit{Betti numbers} or, more precisely put, the dimension of the object's homology groups.\\
The nature of a topological feature is determined by its dimension: zero corresponds to connected components; one to loops, such as those of a circle or torus; two to voids, such as the space enclosed by a sphere; and so on. The number of zeroth-dimensional features are of particular significance in statistics and machine learning, as \textit{clustering}---the grouping of data based on some notion of similarity---is often required in these fields for pattern recognition and general data analysis. Clustering corresponds to the problem of finding connected components in persistent homology. See \textbf{Figure \ref{cluster}} for a visual example of clustering.\\
\tpoint{Linguistic applications\\} \label{linguisticapplications}
As an example to be carried throughout this paper, consider the words of the English language. Each word carries with it a certain related concept, idea, or notion: certain pairs or sets of words may overlap in the ideas associated with them. As a result, a given word may be more closely mentally-associated with one word than another. An example of the large-scale structure that this kind of association can form among words, called a \textit{word association network}, is given in \textbf{Figure \ref{sad}}. This notion of similarity between words, hereafter referred to as \textit{association strength}, allows a word association network to be divided into clusters by persistent homology or other clustering algorithms.\\
A knowledge of how the words of a language cluster together has numerous implications for research and everyday life (\cite{ling1}, Section 6). Such an understanding can suggest new experiments in psychology and psycholinguistics to investigate, for example, how the association of various concepts changes, grows, or degrades during childhood development or with increasing age. In artificial intelligence, the application of word clusters could aid in context recognition for both written and spoken language. Furthermore, electronic dictionaries could be made friendlier to language learners by listing closely-associated words and phrase patterns.\\
\begin{figure}[t]
\begin{subfigure}[H]{0.5\textwidth}
\centering
\includegraphics[scale=0.5]{cluster}
\caption{\footnotesize{An example of data clustering. Each symbol represents a data point; points assigned to the same cluster are represented by the same colour \cite{cluster.fig}.}}\label{cluster}
\end{subfigure}
~
\begin{subfigure}[H]{0.5\textwidth}
\centering
\includegraphics[scale=0.19]{cluster_poor}
\caption{\footnotesize{A small group of words and the strength of the associations between them. The thickness of a line between words is representative of association strength.}}\label{sad}
\end{subfigure}
\caption{}
\end{figure}
\tpoint{Objectives and methods\\}
In this paper, we exposit the theory and method of persistent homology from first principles to the topics of simplicial complex construction, homology groups, Betti numbers, and persistence. We then detail the application of persistent homology to finite sets of data. For ease of visualisation, only data representable in some Euclidean space is considered, and we use the Vietoris-Rips complex construction for its computational efficiency. We note, however, that the methods presented generalise readily to data in any space as well as to other simplicial complex constructions. The theoretical portion of this paper loosely follows select sections from the text by Edelsbrunner (\cite{comptopol}).\\
Additionally, we detail a novel application of persistent homology to linguistics for the purpose of finding clusters of closely-associated English-language words. We compare the clustering abilities of persistent homology against Markov Clustering---an algorithm that has previously been applied to large-scale word association networks (\cite{ling1}, Section 4)---and use the standard graph theoretic modularity index \cite{modularity} to assess the quality of the clusters generated by each method.\\
In addition to the clusters found in the Edinburgh Associative Thesaurus (EAT) \cite{eat} by persistent homology, we present our results for higher-dimensional features. In particular, we include visual examples and offer interpretations of these features in a linguistic context.\\
The data contained in the publicly-available EAT was used in our investigation to compute association strengths between words. The program R, together with the TDA package for topological data analysis \cite{rtda}, as well as van Dongen's Markov Clustering algorithm and code \cite{mcl} were the major tools employed in our analysis. Furthermore, the Pajek Program for Large Network Analysis \cite{pajek} was used to create the visualisations presented in this paper unless otherwise noted.\\
\tpoint{Acknowledgements\\}
We acknowledge the early work of Herbert Edelsbrunner, Afra Zomorodian, Gunnar Carlsson, Robert Ghrist, and Peter Bubenik in developing the field of computational topology. I thank my supervising professor, Giseon Heo, for her guidance throughout this project, and Jisu Kim, among the authors of the R-TDA package, for his advice and technical assistance.\\
\tpoint{Structure of this paper\\}
We present the theory of persistent homology in Sections \ref{simplexsection} through \ref{simplicialhomologysection}. Section \ref{simplexsection} introduces the basic construction of \textit{simplices}, \textit{simplicial complexes}, and \textit{filtrations} on point-cloud data; in particular, we use the \textit{Vietoris-Rips} complex construction. Section \ref{boundscyclessection} develops the \textit{chain}, an algebraic structure on simplicial complexes that underlies persistent homology. We make specific note of a mapping between chains of different dimensions as well as various properties of this map, called the \textit{boundary operator}. Section \ref{simplicialhomologysection} presents \textit{simplicial homology}, the theory of homology groups in the setting of simplicial complexes. Also included is an explicit example demonstrating the calculation of a homology group for a given simplicial complex. In a final theoretical subsection, we briefly define of the \textit{Betti number}, an important numerical summary of a homology group, and present the notion of \textit{barcodes} and \textit{persistence}.\\
Our application of persistent homology to the EAT is detailed in Sections \ref{sectioncomponents} through \ref{interpretationsection}. Section \ref{sectioncomponents} introduces the EAT and defines other topics prerequisite for the proposed analysis, such as the \textit{Markov Clustering algorithm} and the \textit{modularity index} for assessing clustering quality. Clustering results are presented and discussed in Section \ref{resultssection}, with suggestions given for ways to improve the clustering efficacy of persistent homology. Lastly, Section \ref{interpretationsection} briefly examines specific clusters and other topological features found using persistent homology and offers an interpretation of these features in a linguistic context.\\
Following the main body is this paper are two appendices. Appendix Section \ref{appendixproofs} provides algebraic proofs and definitions deemed too technical for the main discussion. Appendix Section \ref{appendiximages} includes additional images of the topological features found in the EAT data using persistent homology.
\section{Simplicial Complexes\label{simplexsection}\\}
In this section, we develop the tools necessary to convert a finite set of points into objects containing information about the topology of the space from which the points were sampled. We build these objects, called simplicial complexes, up from their constituent parts using the Vietoris-Rips complex construction.\\
Our general goal and motivation, as in most statistical investigations, is to elucidate patterns and structure present in a given set of data. This data, when representable in some finite-dimensional Euclidean space, is called a \textit{point-cloud dataset}. In this section, we assume that all points are elements of a fixed, finite-dimensional Euclidean space.\\
\tpoint{Basic simplicial structure\\ \label{basicsimplex}}
We say that a set of points $\{x_0,x_1,\dots,x_k\}$ is \textit{affinely independent} if the set $\{x_i-x_0 \mid 1 \leq i \leq k\}$ is linearly independent. Essentially, affine independence redefines the usual linear independence of vectors by using a fixed, arbitrarily-chosen point of the set as the origin. In the above definition, we use $x_0$ to represent this new origin, although the choice of point in the set is independent of the set's affine independence. It is from sets of affinely independent points that we build basic simplicial structures.\\
\begin{define} \label{simplex}
For some non-negative integer $k$, define the \textit{$k$-simplex} corresponding to a set of $k+1$ affinely independent points $\{x_0,x_1,\dots,x_k\}$ to be the set of all linear combinations of the form $\sum_{i=0}^{k}\lambda_ix_i$, with $\lambda_i$ non-negative for all $i$ and $\sum_{i=0}^{k}\lambda_i=1$. Such a simplex will be denoted $\sigma_{\{x_0,x_1,\dots,x_k\}}$ or $\sigma$ where context is clear.\\
\end{define}
In this setting of Euclidean space, the simplices take on familiar forms: for $k=0,1,2,3$, a $k$-simplex is a point, line, closed triangular region, and solid tetrahedron, respectively, and as shown in \textbf{Figure \ref{eg-simplex}}. More generally, the $k$-simplex corresponding to a set of points is the smallest convex set containing the given points.\\
Note that we require affine independence in the above definition to preclude degenerate simplices from forming, such as the degenerate 2-simplex with all three of its vertices on a single line. Ultimately, affine independence prevents any three points from lying on the same line, any four points from lying in the same plane, and so on.\\
\begin{figure}[!h]
\centering
\includegraphics[scale=0.75]{fig1}
\caption{\footnotesize{$k$-simplices for $k=0,1,2,3$, respectively.}}\label{eg-simplex}
\end{figure}
We can also examine the sub-simplices that make up a given simplex. Following the notation of Definition \ref{simplex}, the notion of a \textit{face} can be introduced.\\
\begin{define} \label{face}
Let $\sigma_A$ be the simplex corresponding to the set $A$ of affinely independent points. We say that $\sigma_B$ is a face of $\sigma_A$ if $B$ is a subset of $A$.\\
\end{define}
For example, the faces of a 3-simplex---that is, a solid tetrahedron---consist precisely of the tetrahedron's four triangular sides, six edges, and four vertices.\\
Now equipped with the notion of simplices and faces, we can create increasingly-complex structures by gluing multiple simplices together.\\
\begin{define}
A \textit{simplicial complex} $X$ is a finite collection of simplices, satisfying the following conditions:
\begin{enumerate}
\item For every simplex $\sigma$ in $X$ and every face $\tau$ of $\sigma$, $\tau$ is also in $X$.
\item For any distinct simplices $\sigma_1$ and $\sigma_2$ in $X$, either $\sigma_1\cap\sigma_2$ is empty or is a face of both $\sigma_1$ and $\sigma_2$.
\end{enumerate}
\end{define}
Informally put, a simplicial complex $X$ contains the faces of all its simplices, and intersects simplices only along entire faces.
\tpoint{Constructions on point-clouds\\} \label{pcconstructions}
By building a simplicial complex from a point-cloud, we can begin to examine the topological properties of the space the data was sampled from. These topological properties include the number of connected components and ``holes'' of any dimension, such as the loops of an $n$-fold torus or the void enclosed by a sphere. While there are numerous ways to construct simplicial complexes on a given point-cloud, we present one method used widely for its computational efficiency.\\
In this subsection, we continue to use the notation of Definition \ref{simplex}.\\
\begin{define}\label{VR-Complex}
Let $P$ be a point-cloud in a Euclidean space equipped with some metric $d(\cdot,\cdot)$. Fix $\varepsilon\geq0$ a non-negative, real number. We construct the \textit{Vietoris-Rips simplicial complex} of radius $\varepsilon$ on $P$, denoted $V_\varepsilon(P)$, according to the following rules:
\begin{enumerate}
\item The 0-simplices of $V_\varepsilon(P)$ are taken to be the points of $P$.
\item Given $x$ and $y$ in $P$, the 1-simplex $\sigma_{\{x,y\}}$ is in $V_\varepsilon(P)$ if and only if $d(x,y)$ is at most $\varepsilon$.
\item If $A$ is a subset of $P$, the simplex $\sigma_A$ is in $V_\varepsilon(P)$ if and only if all faces of $\sigma_A$ are also in $V_\varepsilon(P)$.\\
\end{enumerate}
\end{define}
While $P$ is specified by the given point-cloud, the parameter $\varepsilon$ is free to be chosen arbitrarily. Intuitively, $\varepsilon$ acts as a tuning parameter to adjust the ``coarseness'' of the resulting Vietoris-Rips complex. Here, a pair of points are joined with an edge---that is, a 1-simplex---if and only if those points are within distance $\varepsilon$ of each other. By condition (3), a higher-order simplex is added to the complex only where all faces of the simplex are already present in the complex.\\
Given a point-cloud $P$, we can then construct a family of Vietoris-Rips complexes indexed by a single parameter $\varepsilon$, called a \textit{filtration} of complexes. As $\varepsilon$ increases, new topological features, such as connected components and loops, are created, and existing topological features become connected with other features.\\
Loosely speaking, the range of $\varepsilon$ for which a given topological feature exists is indicative of the feature's significance in the sample space. Features that persist for a wide range of $\varepsilon$ are likely to represent true, global features of the sample space, while those that disappear quickly are likely to be either local features or simply noise in the point-cloud data. We revisit and formally define this notion of feature significance in Section \ref{persistencebarcodes}.\\
\section{Boundaries and Cycles\\} \label{boundscyclessection}
In the following section, we continue to develop the theory of persistent homology by building algebraic structures, called \textit{chains}, on simplicial complexes. In particular, we focus on two kinds of chains central to homology, called \textit{boundaries} and \textit{cycles}, and examine a particular map relating the two, called the \textit{boundary operator}.\\
We make use of a number of well-known group theoretic results presumably present in any introductory-level text. Proofs of elementary claims are included Appendix Section \ref{appendixproofs}, while external references will be made to other works for advanced results outside the scope of this paper. We explicitly reference Goodman's abstract algebra text \cite{goodman} and loosely follow Munkres' algebraic topology text \cite{munkres}.\\
Our general goal and motivation in the following section is, informally, to develop the notion of a loop or cycle in a simplicial complex. The first step in doing so is to formalise the direction, or \textit{orientation}, of such loops. As in previous sections, we consider a single simplex before generalising to simplicial complexes. The following subsection recalls some prerequisite concepts from elementary group theory.\\
\tpoint{Permutations\\ \label{permutationsubsection}}
Recall that a \textit{permutation} of a finite set is a bijection from that set to itself. For example, one permutation of the set $\{1,2,3\}$ is the bijection $\pi_1$ that maps 1 to 3, 3 to 1, and 2 to itself. In other words, $\pi_1$ maps the sequence $(1,2,3)$ to $(3,2,1)$, as shown in \textbf{Figure \ref{pi1}} below. Of particular note are the \textit{transpositions}, that is, permutations that interchange exactly two elements, as in the example just given.
\begin{figure}[h]
$\pi_1: (1,2,3)\xrightarrow{1\leftrightarrow3}(3,2,1)$
\caption{\footnotesize{The permutation $\pi_1$ of the set $\{1,2,3\}$ as introduced above. Note that $\pi_1$ is a transposition because it only switches two elements of the sequence $(1,2,3)$, namely, 1 and 3.}} \label{pi1}
\end{figure}
Define an \textit{ordering} of a finite set $S$ to be an ordered sequence $(x_0,x_1,\dots,x_k)$ of the elements of $S$ in which every element of $S$ appears exactly once. The permutation corresponding to such an ordering is the permutation on $S$ that maps $x_i$ to $x_{i+1}$ for $i=0,1,\dots,k-1$, and maps $x_k$ to $x_0$. In other words,
\begin{equation*}
x_0 \mapsto x_1\ \mapsto \dots x_i \mapsto x_{i+1} \mapsto \dots \mapsto x_n \mapsto x_0,
\end{equation*}
where the arrows represent the mappings of this permutation. Intuitively, the permutation corresponding to an ordering simply ``cycles through'' its elements. For our purposes later on, an ordering will, intuitively-speaking, specify a ``path'' visiting all the vertices of a given simplex.\\
Now let us consider the following example and corresponding \textbf{Figure \ref{123eg}}. Let $\pi_2$ be the permutation of the set $\{1,2,3\}$ that maps 1 to 3, 2 to 1, and 3 to 2. Observe that $\pi_2$ can be viewed as a series of transpositions, first switching 3 with 2, and then switching 1 with 3. Thus we see that $\pi_2$ can be written as the composition of an even number of transpositions.\\
\begin{figure}[h]
$\pi_2: (1,2,3) \xrightarrow{2\leftrightarrow3} (1,3,2) \xrightarrow{1\leftrightarrow3}(3,1,2)$
\caption{\footnotesize{Overall, the above series of transpositions is equivalent to the permutation $\pi_2$ by mapping 1 to 3, 2 to 1, and 3 to 2.\\}}\label{123eg}
\end{figure}
By a well-known result of group theory, this result holds in general: any given permutation of a finite set with at least two elements can be represented as a composition of transpositions. Furthermore, although such a representation is not unique, the number of transpositions used to compose a given permutation will either be invariably even or odd (\cite{goodman}, Section 2.4).\\
In the example of \textbf{Figure \ref{123eg}}, observe that both $(1,2,3)$ and $(3,1,2)$ are orderings of the set $\{1,2,3\}$. As noted above, the permutation $\pi_2$ mapping $(1,2,3)$ to $(3,1,2)$ can be represented by the composition of two transpositions---an even number. Then by the above result, any composition of transpositions mapping $(1,2,3)$ to $(3,1,2)$ must also use an even number of transpositions. We then say that the these two orderings \textit{differ by an even number of transpositions}.\\
In general, we say that two orderings of the same set differ by an even number of transpositions if the permutation mapping one ordering to the other can be written as the composition of an even number of transpositions. Otherwise, we say that the two orderings \textit{differ by an odd number of transpositions}.\\
\tpoint{Oriented simplices\label{orientedsimplicessection}\\}
We next consider orientations of a simplex relative to an ordering of its vertices. Orientation constitutes a subtle yet necessary part of the algebraic structure we will soon impose on simplicial complexes. Here we will assume $\sigma_S$ to be a simplex on $S$ after the notation of Definition \ref{simplex}.\\
\begin{define}\label{orientedsimplex}
For $k$ strictly positive, an \textit{oriented $k$-simplex} is a $k$-simplex $\sigma_S$ together with an ordering of $S$. We say that two orderings of $S$ are of the \textit{same orientation} if and only if the two orderings differ by an even number of transpositions. Furthermore, two oriented $k$-simplices $\sigma_S$ and $\tau_S$ are said to be of the same orientation if their orderings differ by an even number of transpositions.
\end{define}
\begin{nrem}
Recall the result of the previous subsection stating that any permutation on a set of at least two elements can be decomposed into transpositions. Observe that this statement does not hold for singleton sets. Indeed, the only permutation on such a set is the identity map that can be written as the repeated composition of itself any even or odd number of times.\\
For this reason, we define an \textit{oriented 0-simplex} to be a 0-simplex with no orientation.\\
\end{nrem}
We now extend the notion of an oriented simplex to the set of $k$-simplices of a simplicial complex.
\begin{figure}[!h]
\centering
\includegraphics[scale=0.75]{orientedsimplex}
\caption{\footnotesize{Visual representations of oriented $k$-simplices for $k=1,2,3$, respectively. In particular, note the direction of the arrow indicating the simplex's orientation in each case.\\}}\label{oriented}
\end{figure}
\begin{define}
Let $X$ be a simplicial complex with oriented $k$-simplices $\sigma_{i}$, for $i$ in some index set $I_k$. Fix any arbitrary field $\mathbb{F}$. Define a \textit{k-chain} of $X$ over $\mathbb{F}$ to be a formal sum of the oriented $k$-simplices of $X$, denoted $\sum_{i \in I_k}a_i\sigma_{i}$. Here, $a_i$ is an element of $\mathbb{F}$ for all $i$ in $I_k$. We denote by $C_k(X,\mathbb{F})$, or $C_k(X)$ where context is clear, the set of all $k$-chains of $X$ over $\mathbb{F}$.
\end{define}
\begin{nrem}
Informally-speaking, a $k$-chain can be thought of as assigning elements of $\mathbb{F}$ to the oriented $k$-simplices of $X$.
\end{nrem}
\begin{nrem}
Although the final theoretical results of this paper only require $\mathbb{F}$ to have a ring structure, we will restrict our discussion to fields only. This assumption not only appreciably simplifies the development of persistent homology, but also makes our results visually meaningful in the context of simplicial complexes.\\
\end{nrem}
We now further extend $k$-chains by defining an addition operation $\oplus$ on any two $k$-chains.\\
\begin{define} \label{chainop}
Where $+$ is the addition operation of $\mathbb{F}$, define the binary operator $\oplus$ for \textit{$k$-chain addition} via
\begin{align*}
\oplus:C_k(X,\mathbb{F})\times C_k(X,\mathbb{F}) &\rightarrow C_k(X,\mathbb{F})\\
\sum_{i \in I_k}a_i\sigma_{i} \oplus\sum_{i \in I_k}b_i\sigma_{i}&=\sum_{i\in I_k}(a_i+b_i)\sigma_{i}. \numberthis \label{groupaddition}
\end{align*}\\
\end{define}
As a final result of this subsection, we present a crucial property of the set of $k$-chains of a simplicial complex that ultimately makes persistent homology possible.\\
\begin{theorem} \label{abelian}
Fix a non-negative integer $k$, a simplicial complex $X$, and a field $\mathbb{F}$. The set $C_k(X,\mathbb{F})$ of $k$-chains of $X$ over $\mathbb{F}$, together with the $k$-chain addition $\oplus$, forms an Abelian group.\\
\end{theorem}
\begin{nrem}
This proposition can be proven directly by appealing to the definition of an Abelian group. For brevity, we refer the reader to Appendix Section \ref{appendixabelian} for a rigorous proof, but make note of a few important observations here.\\
The required properties of \textit{closure}, \textit{associativity}, the existence of an \textit{identity element}, and the existence of \textit{inverse elements} for $C_k(X,\mathbb{F})$ under $\oplus$ follow readily from the same properties of the additive operation $+$ in $\mathbb{F}$. The \textit{commutativity} of $\oplus$ similarly follows from the commutativity of $+$ in $\mathbb{F}$. In particular, let us consider the existence of inverses and a neutral element in $C_k(X,\mathbb{F})$. Where $0$ denotes the additive neutral element of $+$ in $\mathbb{F}$, observe that $C_k(X,\mathbb{F})$ has additive neutral element $\sum_{i\in I_k}0\sigma_i$.\\
Additionally, an element $\sum_{i\in I_k}a_i\sigma_i$ of $C_k(X,\mathbb{F})$ has additive inverse $\sum_{i\in I_k}(-a_i)\sigma_i$, where $-a_i$ is the additive inverse of $a_i$ in $\mathbb{F}$ under $+$. It is here that the necessity of simplex orientation can be seen: for an oriented simplex $\sigma_S$ on some set of points $S$, we say that the inverse of $\sigma_S$ in $C_k(X,\mathbb{F})$, denoted by $-\sigma_S$, is the same simplex $\sigma_S$ but with reverse orientation. This notion connects our intuition with the algebraic structure of $k$-chains in that to undo the ``loop'' implicit in an ordered $k$-simplex, we simply apply the reverse ``loop''---that is, the same simplex but with a reversed orientation.
\end{nrem}
\tpoint{Boundary operators\\} \label{boundaryoperatorssection}
In the previous subsection, an Abelian group structure was imposed on the set of $k$-simplices of a simplicial complex. We proceed by examining a map between oriented simplices of different dimension, as well as properties of this map fundamental to persistent homology.\\
For simplicity in this subsection, we suppress $\mathbb{F}$ in all notation outside of formal definitions, and will assume $\mathbb{F}$ to be fixed. Furthermore, we use 1 to represent the multiplicative neutral element of $\mathbb{F}$.\\
\begin{notation}
Denote $[x_0,x_1,\dots,x_k]$ to be the oriented $k$-simplex $\sigma_{\{x_0,x_1,\dots,x_k\}}$ with ordering $(x_0,x_1,\dots,x_k)$.
\end{notation}
\begin{notation}
For $j=0,1,\dots,k$, denote $[x_0,\dots,\hat{x}_j,\dots,x_k]$ to be the same oriented simplex but with $\hat{x}_j$ removed, namely, $[x_0,\dots,x_{j-1},x_{j+1}\dots,x_k]$.\\
\end{notation}
\begin{define} \label{boundary}
Define the \textit{boundary} of the oriented $k$-simplex $[x_0,x_1,\dots,x_k]$ to be
\begin{equation*}
\partial_k[x_0,x_1,\dots,x_k] = \sum_{j=0}^k(-1)^j[x_0,\dots,\hat{x}_j\dots,x_k].
\end{equation*}\\
\end{define}
To better illustrate the purpose and intuition of the boundary of a simplex, we present examples involving general $k$-simplices for $k=0,1,2,3$. See \textbf{Figure \ref{boundaryfig}} for a visual representation of the below examples (except for the trivial case where $k=0$). Let $a$, $b$, $c$, and $d$ be arbitrary points. By Definition \ref{boundary}, observe that
\begin{align}
\partial_0[a]&=0,\label{0bound}\\
\partial_1[a,b]&=[b]-[a],\label{1bound}\\
\partial_2[a,b,c]&=[b,c]-[a,c]+[a,b],\label{2bound}\\
\text{and } \partial_3[a,b,c,d]&=[b,c,d]-[a,c,d]+[a,b,d]-[a,b,c].\label{3bound}
\end{align}
\begin{figure}[!h]
\centering
\includegraphics[scale=1]{boundarymap}
\caption{\footnotesize{The boundary of a $k$-simplex for $k=1,2,3$, respectively. Note how the orientation as indicated by the arrows in the latter two diagrams corresponds to the right sides of Equations \ref{2bound} and \ref{3bound}, respectively.\\
Equation \ref{1bound}: a 0-chain remains, the boundary of the original line segment.\label{boundaryfig}\\
Equation \ref{2bound}: a 1-chain remains, forming a loop on the boundary of the original triangular region.\\
Equation \ref{3bound}: a 2-chain remains, forming a loop using the boundary faces of the original tetrahedron.\\}}
\end{figure}
The boundary of a simplex can be developed further through generalisation to the \textit{boundary operator} for $k$-chains, as shown below.\\
\begin{define} \label{boundaryop}
Define the \textit{dimension $k$ boundary operator} on the simplicial complex $X$ via
\begin{align*}
\partial_{k,X}: C_k(X,\mathbb{F}) &\rightarrow C_{k-1}(X,\mathbb{F})\\
\partial_{k,X}\Big(\sum_{i\in I_k}a_i\sigma_i\Big) &= \sum_{i\in I_{k}}a_i\partial_{k-1}(\sigma_i)
\end{align*}
\end{define}
\begin{notation}
As context will make clear whether we are dealing with $\partial_k$ or $\partial_{k,X}$---the boundary of a simplex or of a chain, respectively---we will from now suppress notation and write $\partial_k$ in both cases, for simplicity.\\
\end{notation}
Observe that the boundary operators connect the chain groups of a simplicial complex $X$ by the sequence of maps
\begin{equation} \label{chainmap}
\dots \xrightarrow{\partial_{k+2}}C_{k+1}(X) \xrightarrow{\partial_{k+1}}C_{k}(X) \xrightarrow{\partial_{k}}C_{k-1}(X)\xrightarrow{\partial_{k-1}}\dots \xrightarrow{\partial_{2}}C_{1}(X) \xrightarrow{\partial_{1}}C_{0}(X)\xrightarrow{\partial_0}C_{-1}(X)=\{0\}.
\end{equation}
\begin{nrem}
In order to define $\partial_0$, note that we include $C_{-1}(X)$ as the trivial group $\{0\}$: indeed, the boundary of a 0-simplex is empty, so the boundary operator $\partial_0$ is consistent with the notation introduced thus far.\\
\end{nrem}
As final preparatory work before the formal introduction of simplicial homology groups, we briefly examine properties of the boundary operator and of Equation \ref{chainmap} in the next subsection.\\
\tpoint{Cycles and boundaries\\ \label{cyclesbounds}}
We now focus on properties of the boundary operator previously introduced in Definition \ref{boundaryop}, and in particular, the relationship between the operator's image and kernel. For simplicity, we continue to suppress $\mathbb{F}$ in our notation outside of formal definitions, and assume $\mathbb{F}$ to be fixed.\\
\begin{nlem}\label{homo}
The dimension $k$ boundary operator $\partial_k$ is a homomorphism of groups from $C_k(X)$ to $C_{k-1}(X)$ for all $k\geq1$.\\
\end{nlem}
\begin{proof}
The claim can be proven directly by verifying the definition of a group homomorphism. We will show that the boundary operator $\partial_k$ respects the group operation $\oplus$ of Definition \ref{chainop}. Let $\sum_{i \in I_k}a_i\sigma_i$ and $\sum_{i \in I_k}b_i\sigma_i$ be $k$-chains of a simplicial complex $X$. Observe that\\
\begin{align*}
\partial_k\Big(\sum_{i \in I_k}a_i\sigma_i \oplus\sum_{i \in I_k}b_i\sigma_i\Big) &= \partial_k\Big(\sum_{i \in I_k}(a_i+b_i)\sigma_i\Big) &\text{(by definition of $\oplus$)}\\
&= \sum_{i \in I_k}(a_i+b_i)\partial_k(\sigma_i) &\text{(by definition of $\partial_k$)}\\
&= \sum_{i \in I_k}a_i\partial_k(\sigma_i) \oplus \sum_{i \in I_k}b_i\partial_k(\sigma_i) &\text{(by definition of $\oplus$)}\\
&= \partial_k\Big(\sum_{i \in I_k}a_i\sigma_i\Big) \oplus \partial_k\Big(\sum_{i \in I_k}b_i\sigma_i\Big) &\text{(by definition of $\partial_k$)}\\
\end{align*}
We have shown that the boundary operator $\partial_k$ respects the $C_k(X)$ group operation $\oplus$. Therefore, the boundary operator is a homomorphism of groups.\\
\end{proof}
Recall from group theory that both the \textit{image} and \textit{kernel} of a group homomorphism are themselves groups (\cite{goodman}, Proposition 2.4.12). Then, as a corollary to Lemma \ref{homo}, the image and kernel of the boundary operator are groups, both of which we examine below.\\
\begin{notation}
For non-negative $k$, we denote by $0_k$ the trivial element of the $k$-chain group $C_k(X)$. In other words, we define $0_k=\sum_{i\in I_k}0\sigma_i$. For consistency with Equation \ref{chainmap}, we further denote $0_{-1}$ to be simply 0.\\
\end{notation}
\begin{define} \label{kcycle}
A \textit{$k$-cycle} is a $k$-chain with trivial boundary. More precisely, a $k$-cycle of a simplicial complex $X$ is a $k$-chain $\sum_{i \in I_k}a_i\sigma_{i}$ of $X$ such that
\begin{equation*}
\partial_k\Big(\sum_{i \in I_k}a_i\sigma_{i}\Big)=0_{k-1}.
\end{equation*}
We denote the set of $k$-cycles of a simplicial complex $X$ by $Z_k(X,\mathbb{F})$, or otherwise by $Z_k(X)$ or $Z_k$ where context is clear.\\
\end{define}
It can immediately be seen that the set of $k$-cycles is, by definition, the kernel of the dimension $k$ boundary operator. Therefore, as a corollary to Lemma \ref{homo}, $Z_k(X)$ is a subgroup of $C_k(X)$, for any simplicial complex $X$.\\
\begin{define} \label{kboundary}
A \textit{$k$-boundary} is the boundary of a $(k+1)$-chain. Put precisely, a $k$-chain $\sum_{i \in I_k}b_i\sigma_{i}$ of a simplicial complex $X$ is a $k$-boundary of $X$ if there exists a $(k+1)$-chain $\sum_{i \in I_{k+1}}a_i\sigma_{i}$ in $C_{k+1}(X)$ such that
\begin{equation*}
\partial_{k+1}\Big(\sum_{i \in I_{k+1}}a_i\sigma_{i}\Big)=\sum_{i \in I_k}b_i\sigma_{i}.
\end{equation*}
We will denote the set of $k$-boundaries of $X$ by $B_k(X,\mathbb{F})$ or, where context is clear, simply by $B_k(X)$ or $B_k$.\\
\end{define}
Once again, we see immediately that the set of $k$-boundaries is, by definition, the image of the dimension $(k+1)$ boundary operator. Therefore, as a corollary to Lemma \ref{homo}, $B_k(X)$ is a subgroup of $C_k(X)$ for any simplicial complex $X$.\\
The rest of this subsection will prove a relationship between the set of $k$-cycles and $k$-boundaries.
\begin{nlem} \label{boundofbound}
For any $k\geq0$, the image of a $k$-boundary under the dimension $k$ boundary operator is the trivial $(k-1)$-chain. Equivalently, for any integer $k\geq1$,
\begin{equation*}
\partial_{k-1}\partial_{k}\Big(\sum_{i \in I_{k+1}}a_i\sigma_i\Big)=0_{k-2}.
\end{equation*}\\
\end{nlem}
We will again refer the reader to Appendix Section \ref{appendixbound} for a rigorous proof of this claim, and instead give an example below.\\
Consider the general oriented 3-simplex $[a,b,c,d]$. The image of this simplex under the dimension 3 boundary operator is
\begin{align*}
\partial_3[a,b,c,d] &= [b,c,d]-[a,c,d]+[a,b,d]-[a,b,c].
\end{align*}
Now applying the dimension 2 boundary operator to this result, we observe that
\begin{align*}
\partial_2\big(\partial_3[a,b,c,d]\big) &= \partial_2[b,c,d] -\partial_2[a,c,d] + \partial_2[a,b,d] - \partial_2[a,b,c]\\
&=\big([c,d]-[b,d]+[b,c]\big) - \big([c,d]-[a,d]+[a,c]\big) + \big([b,d]-[a,d]+[a,b]\big) - \big([b,c]-[a,c]+[a,b]\big)\\
&= 0_1,
\end{align*}
Informally put, the boundary of a boundary is trivial.\\
\begin{theorem}
For a simplicial complex $X$, every $k$-boundary of X is a $k$-cycle of $X$.\\
\end{theorem}
\begin{proof}
Using the previously-introduced notation, observe that the given statement is equivalent to $B_k(X)\subset Z_k(X)$. We prove this below.\\
Let $\sum_{i \in I_k}b_i\sigma_i$ be any element of $B_k(X)$. By definition of a $k$-boundary, there exists some $(k+1)$-chain $\sum_{i \in I_{k+1}}a_i\sigma_i$ such that
\begin{equation}
\partial_{k+1}\Big(\sum_{i \in I_{k+1}}a_i\sigma_i\Big)=\sum_{i \in I_k}b_i\sigma_i. \label{subsetpf}
\end{equation}\\
Applying $\partial_k$ to the chosen $k$-boundary, observe that
\begin{align*}
\partial_{k}\Big(\sum_{i \in I_k}b_i\sigma_i\Big) &= \partial_{k}\partial_{k+1}\Big(\sum_{i \in I_{k+1}}a_i\sigma_i\Big) &\text{(by Equation \ref{subsetpf})}\\
&= 0_{k-1}.&\text{(by Lemma \ref{boundofbound})}\\
\end{align*}
Therefore, by definition of a kernel, we have that $\sum_{i \in I_k}b_i\sigma_i$ is an element of $Z_k(X)$, the kernel of $\partial_k$.\\
Since this $k$-chain was chosen arbitrarily from $B_k(X)$, it follows that $B_k(X)\subset Z_k(X)$.\\
\end{proof}
\begin{nrem}
Since we have thus far proven that $B_k(X)$ is itself both a group and a subset of $Z_k(X)$, it follows that $B_k(X)$ is a subgroup of $Z_k(X)$.
\end{nrem}
\section{Simplicial Homology} \label{simplicialhomologysection}
In the previous section, we added a notion of orientation to the simplicial complex introduced in Section \ref{basicsimplex} and defined a group structure on the set of $k$-chains of a simplicial complex. In particular, the results of Section \ref{boundaryoperatorssection} on cycles and boundaries will be fundamental in the development of homology groups in the context of simplicial complexes, called \textit{simplicial homology}.\\
We continue, in this section, to suppress mention of the arbitrary field $\mathbb{F}$ in our notation outside of formal definitions wherever possible. Furthermore, we continue to assume that such an $\mathbb{F}$ is fixed.\\
\tpoint{Homology groups\\ \label{homologygroupssection}}
In this subsection, we fix a simplicial complex $X$ and again denote the set of $k$-chains, $k$-cycles, and $k$-boundaries of $X$ by $C_k$, $Z_k$ and $B_k$, respectively. Furthermore, we use a standard notation to denote operations on a set: where $g$ is an element of a group $G$ with group operation $+$, and where $H$ is a subset of $G$, we define
\begin{equation*}
g+H = \{g+h \mid h\in H\}.\\
\end{equation*}\\
For brevity, we refer the reader to Appendix Section \ref{appendixnormal} for the definition of a normal subgroup and proofs for the related results presented below.\\
We have shown previously in Theorem \ref{abelian} that $C_k$ is an Abelian group. Therefore, it follows immediately that every subgroup of $C_k$ is a normal, Abelian subgroup of $C_k$ (see Appendix Section \ref{appendixnormal}). Furthermore, we proved in Section \ref{cyclesbounds} that both $Z_k$ and $B_k$ are subgroups of $C_k$, and that $B_k$ is a subgroup of $Z_k$.\\
An immediate consequence of these results is a key prerequisite for the development of persistent homology, namely that $B_k$ is a normal subgroup of $Z_k$ (again by Appendix Section \ref{appendixnormal}, since $Z_k$ is Abelian). Consequently, we can now define the \textit{homology group} of a simplicial complex.\\
\begin{define} \label{homologygroup}
The \textit{$k^{\text{th}}$ homology group} $H_k(X,\mathbb{F})$ of a simplicial complex $X$ is the collection of unique \textit{cosets} of $B_k(X,\mathbb{F})$ in $Z_k(X,\mathbb{F})$---that is, the unique equivalence classes of form
\begin{equation*}
z+B_k(X,\mathbb{F}),
\end{equation*}
where $z$ is a $k$-cycle in $Z_k(X,\mathbb{F})$.\\
Equivalently, we write
\begin{equation*}
H_k(X,\mathbb{F}) = \frac{Z_k(X,\mathbb{F})}{B_k(X,\mathbb{F})}
\end{equation*}\\
to mean that $H_k(X,\mathbb{F})$ is the quotient group of $Z_k(X,\mathbb{F})$ modulo $B_k(X,\mathbb{F})$.\\
\end{define}
\begin{notation}
Where context is clear, we will denote $H_k(X,\mathbb{F})$ by $H_k(X)$ or by $H_k$, for simplicity.\\
\end{notation}
\begin{theorem} \label{vectorspace}
The $k^\text{th}$ homology group $H_k(X,\mathbb{F})$ is a vector space over $\mathbb{F}$.\\
\end{theorem}
\begin{nrem}
For brevity, a proof of this claim is omitted from this paper, though the result follows readily once an appropriate vector addition and $\mathbb{F}$-scalar multiplication on $H_k(X,\mathbb{F})$ is defined. In fact, all points of the vector space criteria follow immediately from the status of $C_k(X,\mathbb{F})$ as an Abelian group and $\mathbb{F}$ as a field.\\
The definitions of the above-mentioned vector operations are fairly intuitive and not widely used in this paper, so we refer the reader to Appendix Section \ref{vectorops} for full details.\\
\end{nrem}
Intuitively, the elements of $H_k$ describe the different ``kinds'' of cycles present in a simplicial complex without regard to the complex's boundary elements. This general intuitive understanding of a homology group is formalised in the above definition with the distinct sets of the form $z+B_k$, called cosets. For a fixed $k$-cycle $z_0$, the coset $z_0+B_k$ contains all $k$-cycles of the simplicial complex that differ from $z_0$ only by $k$-boundaries. Thus, if another $k$-cycle $y_0$ differs from $z_0$ by only $k$-boundaries, then $y_0$ is also a member of the coset $z_0+B_k$.\\
We again emphasize that the elements of $H_k$ are the \textit{distinct} equivalence classes of $k$-cycles in the given simplicial complex. Put more rigorously, two $k$-cycles $z_0$ and $y_0$ are in the same coset, or class, if $z_0\oplus(-y_0)$ is a $k$-boundary (where $-y_0$ is the inverse of $y_0$ in $C_k$, as defined in the proof of Proposition \ref{abelian}). In other words, the difference between $z_0$ and $y_0$ is composed only of $k$-boundaries.\\
\tpoint{Homology group example\\} \label{egsection}
To illustrate the concepts developed up to this point, we present a simple yet informative example where we explicitly calculate two homology groups of a given simplicial complex \cite{homoeg}. In this subsection, we use standard notation to represent the kernel and image of a function $f$, namely, $\ker f$ and $\Ima f$, respectively. We also employ the intuitive vector addition and scalar multiplication operations defined for the $k^\text{th}$ homology group, as presented in Appendix Section \ref{vectorops}. Lastly, we use familiar notation from linear algebra to denote the space spanned by a set of chains, as set out below.\\
\begin{notation}
Let $\sigma_i$ be $k$-chains and $a_i$ elements of some fixed field $\mathbb{F}$, for $i=1,2,\dots,n$. Denote by $\spn_\mathbb{F}\{\sigma_1,\sigma_2,\dots,\sigma_n\}$ the set of all $k$-chains of the form $a_1\cdot\sigma_1\oplus a_2\cdot\sigma_2\oplus \dots\oplus a_n\sigma_n$. We refer to $\sigma_1,\sigma_2,\dots,\sigma_n$ as \textit{generators} of $\spn_\mathbb{F}\{\sigma_1,\sigma_2,\dots,\sigma_n\}$.\\
\end{notation}
Considered below is the the simplicial complex $X$, as presented in \textbf{Figure \ref{eg}}.
\begin{figure}[h]
\centering
\includegraphics[scale=1]{eghomology}
\caption{\footnotesize{A simplicial complex with vertices 0,1, 2, 3, and 4 as shown. \label{eg}}}
\end{figure}
We begin by listing the $k$-chain groups of $X$ by taking the span of all $k$-simplices in $X$, for $k=0,1,2$.
\begin{align}
C_0(X) &= \spn\{[0],[1],[2],[3],[4]\}\\
C_1(X) &= \spn\{[0,1],[0,2],[0,3],[0,4],[1,2],[1,3],[2,3],[2,4]\}\\
C_2(X) &= \spn\{[0,1,2],[0,1,3],[0,2,3],[1,2,3]\}
\end{align}\\
Let us first consider the dimension 0 homology group $H_0(X)$. By Definition \ref{homologygroup} of a homology group and by Section \ref{cyclesbounds}, we know that\\
\begin{equation} \label{egH0}
H_0(X) = \frac{Z_0(X)}{B_0(X)} = \frac{\ker \partial_0}{\Ima \partial_1}\\
\end{equation}\\
We will first calculate $\ker\partial_0$. Observe that, for every $0$-simplex [x], we have $\partial_0[x]=0$. Therefore, every 0-simplex is mapped to 0 under the boundary map, and so the set of all 0-simplices is in the kernel of $\partial_0$. In other words,
\begin{equation*}
C_0(X)\subset\ker\partial_0.\\
\end{equation*}
The kernel of $\partial_k$ is necessarily a subset of $C_0(X)$ by the definition of a kernel, so the reverse inclusion also holds. Therefore, we conclude that
\begin{equation} \label{z0}
\ker\partial_0=C_0(X).\\
\end{equation}\\
Let us now move on to determine $\Ima\partial_1$. We can calculate the generators of $\Ima\partial_1$ as image of the generators of $C_0(X)$. In other words,
\begin{align*}
B_0(X) = \Ima\partial_1 &= \partial_1\spn\{[0,1],[0,2],[0,3],[0,4],[1,2],[1,3],[2,3],[2,4]\}\\
&= \spn\{\partial_1[0,1],\, \partial_1[0,2],\, \partial_1[0,3],\, \partial_1[0,4],\, \partial_1[1,2],\, \partial_1[1,3],\, \partial_1[2,3],\, \partial_1[2,4]\}\\
&= \spn\{[1]-[0],\, [2]-[0],\, [3]-[0],\, [4]-[0],\, [2]-[1],\, [3]-[1],\, [3]-[2],\, [4]-[2]\}. \numberthis \label{b0}
\end{align*}\\
Therefore, by substituting Equations \ref{z0} and \ref{b0} into Equation \ref{egH0}, we obtain\\
\begin{equation*}
H_0(X) = \frac{\spn\{[0],[1],[2],[3],[4]\}}{\spn\{[1]-[0],\, [2]-[0],\, [3]-[0],\, [4]-[0],\, [2]-[1],\, [3]-[1],\, [3]-[2],\, [4]-[2]\}}.
\end{equation*}\\
Recall that any two $k$-chains $z_0$ and $y_0$ are considered equivalent in the $k^\text{th}$ homology group if their difference $z_0\oplus(-y_0)$ is a $k$-boundary. In this particular example, one can show that the difference of any two elements of $Z_0$ is a $0$-boundary.\\
For example, consider the 0-cycles $[3]$ and $[4]$, and the $0$-boundaries $[3]-[2]$ and $[4]-[2]$. Note that all of these are generators in the previous equation. Observe that
\begin{align*}
\big([3]-[2]\big) \oplus -\big([4]-[2]\big) &= \big([3]-[2]\big) \oplus \big(-[4]+[2]\big)\\
&= [3] + (1-1)[2] - [4]\\
&= [3] - [4]
\end{align*}
Note that $\big([3]-[2]\big) \oplus -\big([4]-[2]\big)$ is a $0$-boundary since $B_0(X)$ is a group and is closed under 0-chain addition. Therefore, we see that [3] and [4] are equivalent in $H_0(X)$, as the difference between these two cycles is an element of the boundary group.\\
As stated above, this same result holds true in general for every generator of the 0-cycles $Z_0(X)$. From this, it follows that every 0-cycle of $X$ is equivalent to every other 0-cycle of $X$. In other words, $[0]$, $[1]$, $[2]$, $[3]$, and $[4]$ are all elements of the same coset, namely, $[0]+B_0(X)$. Of course, since these cycles are equivalent, we may also represent this coset as $[x]+B_0(X)$, where $x$ is any of 0, 1, 2, 3, or 4. Finally, since $[0]$ is not a 0-boundary itself, we note that this coset is non-trivial---that is, $[0]+B_0(X)$ is not $B_0(X)$.\\
As a result, there is exactly one non-trivial element of the $0^\text{th}$ homology group $H_0(X)$, namely $[0]+B_0(X)$. Pictorially, this result corresponds to the fact that $X$, as shown in \textbf{Figure \ref{eg}}, is composed of exactly one connected component.\\
The $1^\text{st}$ homology group $H_1(X)$ can be calculated in a similar way. Once again, by the definition of a homology group and by previous results, we have that\\
\begin{equation} \label{egH1}
H_1(X) = \frac{Z_1(X)}{B_1(X)} = \frac{\ker\partial_1}{\Ima\partial_2}
\end{equation}\\
Though the work required is somewhat tedious without any additional techniques or methods, one can calculate $\ker\partial_1$ by direct computation to show that
\begin{align*}
\ker\partial_1 = \spn\{&[0,1]+[0,3]-[1,3],\,[0,2]+[2,3]-[0,3],\,[1,2]+[2,3]-[1,3],\,[0,1]+[1,2]-[0,2],\\&[0,2]+[2,4]-[0,4]\} \numberthis \label{z1}
\end{align*}\\
Notably less strenuous is the calculation for $B_1(X)$:
\begin{align*}
B_1(X) &= \Ima\partial_2 = \partial_2\spn\{[0,1,2],[0,1,3],[0,2,3],[1,2,3]\}\numberthis \label{b1}\\
&= \spn\{[0,1]+[1,2]-[0,2],\,[0,1]+[0,3]-[1,3],\,[0,2]+[2,3]-[0,3],\,[1,2]+[2,3]-[1,3]\}.
\end{align*}\\
Now substituting Equations \ref{z1} and \ref{b1} into Equation \ref{egH1}, we obtain, for $H_1(X)$,\\
\begin{align*}
\frac{\spn\{[0,1]+[0,3]-[1,3],[0,2]+[2,3]-[0,3],[1,2]+[2,3]-[1,3],[0,1]+[1,2]-[0,2],[0,2]+[2,4]-[0,4]\}}{\spn\{[0,1]+[1,2]-[0,2],\,[0,1]+[0,3]-[1,3],\,[0,2]+[2,3]-[0,3],\,[1,2]+[2,3]-[1,3]\}}
\end{align*}\\
Observe that the first four generators of $Z_1(X)$ in the numerator are also generators of $B_1(X)$ in the denominator: therefore, any combination of these four elements will necessarily be a boundary! On the other hand, one can see that the fifth generator of $\ker\partial_1$, namely, $[0,2]+[2,4]-[0,4]$ is independent of the boundary elements---this is intuitively clear since none of the boundary generators concern the point labelled as 4.\\
We conclude that $H_1(X)$ has two elements: besides the trivial class $B_1(X)$, we also have the non-trivial class $\big([0,2]+[2,4]-[0,4]\big)+B_1(X)$.\\
Recall from Section \ref{boundaryoperatorssection} that 1-cycles can be visualised as loops, and consider the above result in the context of \textbf{Figure \ref{eg}}. The trivial coset of $H_1(X)$ can be thought of as the class of all loops on \textbf{Figure \ref{eg}} that can be shrunk down to a single point: the loops of this class are exactly those loops that do not make use of the ``arm'' formed by the vertices labelled 2, 4, and 0. On the other hand, the non-trivial coset $\big([0,2]+[2,4]-[0,4]\big)+B_1(X)$ corresponds exactly to those loops that use this extra ``arm'' and hence cannot be reduced to a single point. Refer to \textbf{Figure \ref{egloop}} for a visual example of both cases.
\begin{figure}[h]
\centering
\includegraphics[scale=1]{eghomologyloop}
\caption{\footnotesize{Loops on the simplicial complex of \textbf{Figure \ref{eg}}. The blue loop on the left can be shrunk to a single point, as shown. On the other hand, the red loop on the right cannot be similarly reduced. These loops are examples of elements in the trivial and non-trivial cosets of $H_1(X)$, respectively.}}\label{egloop}
\end{figure}
In short, the result that $H_1(X)$ has exactly one non-trivial element corresponds to the fact that $X$ has exactly one 1-dimensional ``hole''---namely, the loop formed by the vertices labeled 2, 0, and 4.\\
\tpoint{Betti numbers\\ \label{bettinumbers}}
As a final result of this section, we develop a numerical summary of any homology group, called the \textit{Betti number}. The existence of this numerical descriptor stems from the result of Lemma \ref{vectorspace} that $H_k(X,\mathbb{F})$ is itself a vector space over the field $\mathbb{F}$. Consequently, the homology group $H_k(X,\mathbb{F})$ has a well-defined \textit{dimension}---that is, intuitively, the number of non-trivial elements that can be used to generate the homology group.\\
\begin{define}
The \textit{$k^\text{th}$ Betti number} of a simplicial complex $X$, denoted $\beta_k(X)$, is the dimension of the $k^\text{th}$ homology group $H_k(X,\mathbb{F})$ as a vector space over fixed field $\mathbb{F}$. We write
\begin{equation*}
\beta_k(X) = \dim H_k(X,\mathbb{F}),
\end{equation*}
and note in particular that the Betti number $\beta_k(X)$ is independent of the choice $\mathbb{F}$.\\
\end{define}
The Betti numbers of a given simplicial complex $X$ provide an easily-interpretable description of the topology of $X$. As demonstrated by the example in Section \ref{egsection}, the dimension of the $k^\text{th}$-homology group---or equivalently, the number of non-trivial generators---reveals how many holes of dimension $k$ are present in $X$. It is crucial to note that each non-trivial element of a homology group corresponds to a topological feature of $X$, as explained below.\\
In this sense, $\beta_0(X)$ can be interpreted as the number of connected components of $X$, and $\beta_1(X)$ as the number of loops of $X$---or equivalently, the number of 2-dimensional regions enclosed by $X$. Furthermore, $\beta_2$ is the number of \textit{voids}, or enclosed 3-dimensional regions, of $X$.\\
The Betti number can be defined similarly outside the context of simplicial homology. For example, the $k^\textit{th}$ Betti number of a compact manifold $M$, denoted $\beta_k(M)$, is the dimension of the $k^\textit{th}$ homology group of $M$. In other words, $\beta_k(M)$ can be interpreted as the number of connected components, loops, voids, and so on, of the manifold $M$. For intuition, we give a couple examples below of the first three Betti numbers for the sphere and torus.\\
Consider the standard sphere $S$ in three-dimensional Euclidean space. Observe that, since $S$ has a single connected component, $\beta_0(S)=1$. As all loops on $S$ are trivial---that is, since all loops can be shrunk to a single point---we have $\beta_1(S)=0$. Furthermore, since $S$ encloses a single three-dimensional region, we have $\beta_2(S)=1$.\\
As another example, consider the standard torus $T$ in three-dimensional Euclidean space. The torus is composed of a single connected component, so $\beta_0(T)=1$. Also, $T$ has two non-trivial classes of loops, namely, loops around the central ``hole'' of the torus and loops around ``tube'' of the torus: we then have $\beta_1(T)=2$. See \textbf{Figure \ref{torusloops}} for a visualisation of these two classes of loops. Finally, since $T$ encloses one three-dimensional space inside its ``tube'', we have $\beta_2(T)=1$.\\
\begin{figure}[h]
\centering
\includegraphics[scale=0.2]{torusloops}
\caption{\footnotesize{Examples of the two non-trivial classes of loops on a torus, displayed in red and blue \cite{torusloops}.\label{torusloops}}}
\end{figure}
\tpoint{Persistence barcodes\label{persistencebarcodes}\\}
As a final theoretical topic, we will join simplicial homology with the filtrations of simplicial complexes introduced in Section \ref{pcconstructions}. We will continue to use the Vietoris-Rips complex construction of Definition \ref{VR-Complex} for its computational efficiency. Assumed throughout this subsection is the notation of Section \ref{pcconstructions} regarding simplices and Vietoris-Rips simplicial complexes.\\
We begin with a basic property of the a Vietoris-Rips filtration for a fixed point cloud $P$.\\
\begin{nprop}
For two fixed, real, and non-negative $\varepsilon$ and $\varepsilon'$ with $\varepsilon\leq\varepsilon'$, the Vietoris-Rips complex $V_\varepsilon(P)$ is nested inside $V_{\varepsilon'}(P)$. That is, every simplex of $V_\varepsilon(P)$ is also a simplex of $V_{\varepsilon'}(P)$.\\
\end{nprop}
\begin{proof}
The proof of this claim follows immediately from Definition \ref{VR-Complex} of a Vietoris-Rips complex.\\
Suppose $\sigma_S$ is a simplex of $V_\varepsilon(P)$ for some set of points $S$ in $P$. Then by definition, for every pair of points $x$ and $y$ in $S$, it follows that
\begin{equation*}
d(x,y)\leq\varepsilon.
\end{equation*}
By assumption, $\varepsilon\leq\varepsilon'$, and so also,
\begin{equation*}
d(x,y)\leq\varepsilon'.
\end{equation*}\\
Therefore, by definition of the Vietoris-Rips complex, it follows that $\sigma_S$ is a simplex of $V_{\varepsilon'}(P)$. We have then proven the desired result, namely that $V_\varepsilon(P)$ is a subset of $V_{\varepsilon'}(P)$.\\
\end{proof}
We can form a \textit{chain of nested simplicial complexes} using the Vietoris-Rips construction by varying $\varepsilon$. Indeed, given a sequence of increasing $\varepsilon_i$, where $\varepsilon_1<\varepsilon_2<\dots\varepsilon_n<\varepsilon_{n+1}<\dots$ we have\\
\begin{equation} \label{vrstream}
V_{\varepsilon_1}(P)\subset V_{\varepsilon_2}(P)\subset\dots\subset V_{\varepsilon_n}(P)\subset V_{\varepsilon_{n+1}}(P) \subset\dots,
\end{equation}\\
As a result, we can define maps between the homology groups of these complexes, namely\\
\begin{equation} \label{homologychain}
H_k\big(V_{\varepsilon_1}(P)\big)\xrightarrow{\varphi_1} H_k\big(V_{\varepsilon_2}(P)\big)\xrightarrow{\varphi_2} \dots \xrightarrow{\varphi_{n-1}} H_k\big(V_{\varepsilon_n}(P)\big)\xrightarrow{\varphi_n} H_k\big(V_{\varepsilon_{n+1}}(P)\big)\rightarrow\dots,
\end{equation}\\
where $k\geq2$. For the rest of this subsection, we will denote $H_k\big(V_{\varepsilon_n}(P)\big)$ by $H_k^{\varepsilon_n}$, and the map from $H_k^{\varepsilon_n}$ to $H_k^{\varepsilon_{n+1}}$ by $\varphi_n$, for ease of notation. While the specific maps $\varphi_n$ are not significant for our purposes, the effect of these maps on the above homology groups is certainly of note. Consider the map $\varphi_n$, and suppose that $z_0$ and $y_0$ are distinct elements of $H_k^{\varepsilon_{n}}$. Note that elements of $H_k^{\varepsilon_n}$ are always mapped forward to $H_k^{\varepsilon_{n+1}}$ in Equation \ref{homologychain} by $\varphi_n$. However, $\varphi_n$ is not necessarily surjective---that is, there may be some elements of $H_k^{\varepsilon_{n+1}}$ that are not the image of any element of $H_k^{\varepsilon_{n}}$ under $\varphi_n$. Such elements are said to be \textit{born at time} $\varepsilon_n$.\\
Observe that $\varphi_n$ will map $z_0$ and $y_0$ to elements of $H_k^{\varepsilon_{n+1}}$ that may either be distinct or identical. Where $\varphi_n(z_0)$ and $\varphi_n(y_0)$ are distinct, we say that $z_0$ has \textit{persisted} from $\varepsilon_n$ to $\varepsilon_{n+1}$, and similarly so for $y_0$. On the other hand, if $\varphi_n(z_0)$ is equal to $\varphi_n(y_0)$, we say that one of $z_0$ or $y_0$ has \textit{died}. By convention that will soon become apparent, we choose the element of the pair that was born last to be the one to die at $\varepsilon_{n+1}$. Thus, if $z_0$ was born before $y_0$, we say that $y_0$ \textit{dies at time} $\varepsilon_{n+1}$.\\
At this point, we can develop a simple yet intuitive visual representation of the birth and death times of all homological elements appearing in Equation \ref{homologychain}.\\
\begin{define} \label{barcodedef}
Fix some $k\geq0$, and let $Y_k$ be the set of all $k^{\text{th}}$ homology group elements of Equation \ref{homologychain} at the time they are born. Consider, for each element $y$ in $Y_k$, the real, half-open interval $[b_y,d_y)$, where $b_y$ and $d_y$ are the birth and death times of $y$, respectively. Define the \textit{$k$-barcode} corresponding to the filtration given in Equation \ref{vrstream} by the collection of intervals
\begin{equation*}
\{[b_y,d_y)\mid y\in Y_k\}
\end{equation*}\\
\end{define}
Although beyond the scope of this paper, it can be shown that the homological features appearing in a nested filtration of complexes---for example, the object given in Equation \ref{homologychain}---is ismorphic to the collection of the filtration's corresponding $k$-barcodes for all $k\geq0$. In other words, the birth and death times of all features in a filtration uniquely determines the filtration's corresponding barcode, and vice-versa. We can now, easily and without algebraic notation, represent the homology of a filtration of simplicial complexes as an intuitive series of intervals!
\section{Components of the Proposed Clustering Analysis \label{sectioncomponents}\\}
Now that the theory of persistent homology and its application to simplicial complex filtrations has been developed in the previous sections, we begin the second major portion of this paper. Our focus now shifts to the application of persistent homology to real-world data---in particular, we examine a sizable dataset from the field of linguistics.\\
This section serves as an introduction to the major components relevant to the data analysis that was carried out. In the following subsections, we briefly discuss the \textit{Edinburgh Associative Thesaurus} dataset, the \textit{modularity index} for assessing the quality of a clustering method, the \textit{Markov Clustering} algorithm, and the particular persistent homology techniques applied in this study.\\
\tpoint{Edinburgh Associative Thesaurus\\ \label{eatsubsection}}
The Edinburgh Associative Thesaurus (EAT) is a large dataset containing information on mental associations made between words of the English language \cite{eat}. As discussed in Section \ref{linguisticapplications}, an individual will associate various ideas, concepts, and notions with a given word. As expected, these associations will vary from person to person based on culture, personal experience, worldview, or any number of factors that shape how an individual thinks. For example, one person may associate the word GERMAN most strongly with the word FRENCH, whereas another may associate GERMAN with KRAUT, as shown in \textbf{Figure \ref{french}}. The set of word associations form, for each person, a network between words of the English language, called a \textit{word association network}.\\
\begin{figure}[h]
\centering
\includegraphics[scale=0.15]{french}
\caption{\footnotesize{A small portion of a word association network, highlighting relationships made with the words FRENCH and GERMAN. In this diagram, edge thickness is proportional to the association strength, also labeled numerically on each edge.}\label{french}}
\end{figure}
Differences and similarities in word association networks between persons are of particular interest to researchers, again as discussed in Section \ref{linguisticapplications}. The EAT, for example, has previously been used to find and classify semantic and psychological links between words, as well as to maximize advertising efficacy by making use of common associations.\\
The EAT database was constructed using 8,400 \textit{stimulus words}. Each of these stimulus words was presented on paper to approximately 100 different subjects. Each subject was prompted to write down, as quickly as possible, the first word that came to mind after viewing the stimulus.\\
The data comprising the EAT contains all stimulus words, all responses, and the number of times that a response was given for each stimulus. Based on this data, we assigned a numerical index to each word and calculated the proportion of occurrence for each ordered pair of words. For example, if 25 out of 100 people presented with the word CAT responded with DOG, then the proportion of occurrence of DOG after seeing CAT is 0.25. As the proportion of occurrence is not necessarily symmetric, we took the \textit{association strength} between two words to be the maximum proportion of occurrence between them. Continuing the previous example, if the proportion of occurrence of CAT after seeing DOG is 0.40, we take the strength of the association between CAT and DOG to be 0.4---that is, the maximum of 0.25 and 0.4.\\
In total, the data used in our analysis included 305,134 associations between 23,181 unique words.\\
\tpoint{Modularity index\label{modularitysection}\\}
As one of this project's main objectives is to compare the clustering abilities of persistent homology to other methods, it is essential that we have a scale to measure the performance of each technique. To this end, we make use of the \textit{modularity index} $Q$ for weighted graphs \cite{modularity}---that is, graphs for which a weight has been assigned to each edge. As the formal development of modularity is outside the scope of this paper, we instead provide only a definition and intuitive explanation of this measure of clustering performance.\\
The modularity index $Q$ is a numerical value between -1 and 1 that describes how well a given graph has been partitioned into clusters. Loosely-speaking, modularity measures the difference in density between the connections within clusters and the connections between clusters. A set of clusters that more effectively separates the vertices of a graph will have a higher modularity: we thus seek to maximize $Q$ to obtain the best possible clustering of a graph.\\
In the below definition of modularity, we assume the following notation. Let the vertices of a graph be labelled according to some index set $V$, and let $\omega_{i,j}$ represent the weight of the edge between the vertices labelled $i$ and $j$. Note that $\omega_{i,j}$ is taken to be 0 if no such edge exists. Let $M$ be the sum of all edge weights in the graph, and $k_i$ the sum of the weights of all edges attached to the vertex labelled $i$. Lastly, define $\delta(i,j)$ to be the function that equals 1 when the vertices labelled $i$ and $j$ have been assigned to the same cluster, and 0 otherwise.\\
\begin{define}
Using the notation above, given a weighted graph and a partitioning of its vertices into clusters, define the modularity index $Q$ of this clustering to be
\begin{equation*}
Q=\frac{1}{M}\sum_{i,j\in V}\big[\omega_{i,j}-\frac{k_ik_j}{M}\big]\delta(i,j)
\end{equation*}
\end{define}
Essentially, the sum above only considers pairs of vertices $i$ and $j$ in the same cluster. Intuitively, we then see that $Q$ is increased by $\omega_{i,j}$, the weight of edges within a cluster, and decreased by $\frac{k_ik_j}{M}$, a measure of the complexity of the graph around the vertices $i$ or $j$. In the context of machine learning, this definition of modularity is essentially the fundamental problem of balancing the interpretability of a model with its complexity.\\
\tpoint{Markov Clustering algorithm\\}
\textit{Markov Clustering} (MCL) is an algorithm developed by van Dongen for separating a graph or network into clusters---that is, partitioning the vertices of a graph into non-overlapping subsets containing vertices that are similar in some way \cite{mcl}. In general, a cluster of a graph is characterised by a higher proportion of edges within the cluster then outside the cluster. As discussed in section \ref{linguisticapplications}, the problem of clustering is relevant to the fields of image analysis, machine learning, general pattern recognition in computing science, and bioinformatics.\\
Most clustering methods and algorithms, however, become computationally infeasible for increasingly large datasets---that is, these algorithms are not scalable. The MCL algorithm, on the other hand, is presented as a computationally efficient and scalable means of extracting clusters from even very large networks. Although MCL has been used widely in the field of bioinformatics, the algorithm has been applied previously to linguistics in the creation of a dictionary of French synonyms (\cite{ling1}, Section 4.1) and a study of word clustering in the Japanese language (\cite{ling1}, Section 4.2).\\
Based on its precedent use in linguistic analysis, we chose to use MCL in this study as a performance benchmark for persistent homology. Although a detailed exposition of MCL is outside the scope of this paper, we present here a brief, intuitive description of this algorithm.\\
The scalability of MCL to large graphs stems from the algorithm's use of random walk simulations on the graph being considered. More specifically, the idea underlying MCL is that, by randomly traveling along the edges of a graph, one is more likely at any point to stay within a single cluster than one is to exit the cluster. Based on this idea, MCL alternates between periods of simulating long and short random walks. These periods are respectively referred to as the \textit{expansion} and \textit{inflation stages} of the algorithm. Longer random walks are more likely to travel between clusters, thus allowing potential clusters to expand and include more vertices. Shorter random walks, on the other hand, are more likely to stay within a cluster---this serves to remove weak elements of a potential cluster and strengthen the connection between vertices that are strongly similar.
In our application of MCL, the words of the EAT are interpreted as vertices of a graph. Furthermore, for any pair of words, the weight of the edge connecting them is taken to be their association strength.\\
MCL is dependent on a choice of \textit{inflation parameter} that determines the ``strength'' of the inflation stage of the algorithm. In our study, we performed MCL on the EAT database for a wide range of inflation parameter values. We then calculated the modularity $Q$ of the clustering created by each iteration of the algorithm. The results of this test can be found in Section \ref{mclresults}.\\
\tpoint{Clustering with complexes and persistent homology\label{phtests}\\}
The main focus of this study is persistent homology's ability to find clusters and higher-dimensional topological features such as loops and voids in large datasets. In this subsection, we describe specifically how persistent homology was used to extract clusters and other topological features from the EAT. We will assume the notation used in Section \ref{pcconstructions} pertaining to complex constructions.\\
In order to apply the Vietoris-Rips construction to the EAT, we generalise Definition \ref{VR-Complex}. Since our dataset $P$---that is, the words in the EAT---cannot be placed meaningfully in Euclidean space, we instead take the metric $d(x,y)$ to be one minus the association strength between the two words $x$ and $y$. In other words, $d$ becomes a measure of dissimilarity. Note that this transformation of association strength to dissimilarity is necessary to ensure that pairs of words with high association strength have a short ``distance'' between them. After this modification, though we do not have a true metric and cannot properly visualise the complexes created, we are still able to construct a Vietoris-Rips filtration.\\
To illustrate, consider the example previously given in Section \ref{eatsubsection} using the words CAT and DOG. We will denote, as above, the words in the EAT dataset by $P$. We previously supposed the association strength between CAT and DOG to be 0.4: therefore, the ``distance'', or dissimilarity, between these words is
\begin{equation*}
d(\text{CAT},\text{DOG})=1-0.4=0.6.
\end{equation*}
Therefore, the 1-simplex built from the ``points'' CAT and DOG will be present in the complex $V_{0.7}(P)$, but not in $V_{0.5}(P)$.\\
Using the R-TDA package for topological data analysis, we constructed a filtration of Vietoris-Rips complexes on the EAT and determined the $k$-barcodes, for $k=0,1,2$, corresponding to this filtration. Recall from Section \ref{persistencebarcodes} that the barcode is equivalent to the persistent homology of the filtration: both contain information about the birth and death times of each topological feature that appears.\\
We first considered maximizing cluster modularity over the set of all Vietoris-Rips complexes created. In subsequent sections, we refer to this method as \textit{simple clustering by similarity}. Ultimately, this is a naive method that clusters together pairs of words with similarity above a specified threshold parameter---in other words, we simply take connected components as they appear in a single Vietoris-Rips complex. Note that this method only uses properties of the Vietoris-Rips complex, and not of persistent homology. We performed such clustering over a large number of threshold values in order to maximize the cluster modularity $Q$.\\
In contrast, the next method, referred to as \textit{clustering by persistence}, does make use of the persistent homology of the constructed Vietoris-Rips filtration. Here we consider the \textit{persistence}---that is, the difference between the birth and death time---of each 0-dimensional homological feature. We fix a persistence threshold parameter and cluster two words $x$ and $y$ together if and only if the 1-simplex connecting $x$ and $y$ has a lifetime greater than the specified threshold.\\
We note that, at the time of this study, the R-TDA package did not have the functionalities necessary to perform the above analysis. Although we developed code to extract clusters and topological features of arbitrary dimension from a homology of the Vietoris-Rips filtration, such code will be made available in a future publication and is not presented here.\\
\section{Clustering Results\\ \label{resultssection}}
This section presents results for each of the clustering tests introduced in Section \ref{sectioncomponents} and compares these methods using the modularity index as defined in Section \ref{modularitysection}. Furthermore, we discuss a modification of the clustering by persistence method to increase persistent homology's clustering effectiveness relative to MCL.\\
\tpoint{Markov Clustering results \label{mclresults}\\}
We applied the Markov Clustering algorithm to partition the 23,181 words of the EAT dataset into groups of closely-associated words. Due to the dependence of MCL on a choice of inflation parameter, we iterated the algorithm 241 times using a range of inflation parameter values between 1.20 and 6.00. For each iteration, we calculated the modularity of the clustering produced and looked to maximize this quantity over the inflation parameter values tested. \textbf{Figure \ref{mcl_Q}} presents a plot of the modularity value calculated for each MCL iteration against the inflation parameter used.\\
\begin{figure}[h]
\centering
\includegraphics[scale=0.6]{mcl_modularity}
\caption{\footnotesize{A plot showing the relationship between MCL's inflation parameter value and the quality of the clusters produced by the algorithm for the EAT dataset.}\label{mcl_Q}}
\end{figure}
As shown in \textbf{Figure \ref{mcl_Q}}, the maximum modularity value attained by MCL is 0.3996, occurring at an inflation parameter value of 1.28. This particular iteration produced 319 distinct word clusters.\\
\tpoint{Simple clustering by similarity results\\}
In this test, we used individual Vietoris-Rips complexes to cluster the words of the EAT dataset, as set out in Section \ref{phtests}. Similar to the MCL algorithm, the Vietoris-Rips complex construction is dependent on a parameter $\varepsilon$, adhering the notation of \ref{pcconstructions}. As such, we looked to maximize modularity over the 31 values of $\varepsilon$ chosen. A plot of the results is presented in \textbf{Figure \ref{ph_time_Q}}.\\
\begin{figure}[h]
\centering
\includegraphics[scale=0.6]{ph_time_modularity}
\caption{\footnotesize{A plot showing the relationship between Vietoris-Rips parameter $\varepsilon$, called the \textit{filtration parameter}, and modularity index in the simple clustering by similarity method.}\label{ph_time_Q}}
\end{figure}
Modularity was at a maximum of 0.1345 for a value 0.6421 of the filtration parameter $\varepsilon$. Furthermore, the number of clusters produced at optimal modularity was found to be 21,523. Observe that the optimal modularity value produced by this method is considerably lower than that of the MCL algorithm.\\
\tpoint{Clustering by persistence results\label{phresults}\\}
In our final test, we used the persistent homology of a filtration of Vietoris-Rips complexes to cluster the words of the EAT. As described in Section \ref{phtests}, our method of clustering was once again dependent on a \textit{persistence threshold} parameter. Recall that, for this method, we cluster two words together if and only if the 0-simplex connecting the two words has a lifetime greater than the chosen persistence threshold. As in previous subsections, we maximized modularity over 33 threshold values: a plot of the results is presented in \textbf{Figure \ref{ph_pers_Q}}.\\
\begin{figure}[h]
\centering
\includegraphics[scale=0.6]{ph_pers_modularity}
\caption{\footnotesize{A plot showing the relationship between persistence threshold and modularity index in the clustering by persistence method.}\label{ph_pers_Q}}
\end{figure}
Observe that the maximum modularity value attained by this method is 0.2146, occurring at a persistence threshold of 0.209. At this point, 18,882 distinct clusters were present. Although this maximum modularity value is higher than that of the simple clustering by similarity method, it is still lower than the maximum modularity attained by the MCL algorithm.\\
\tpoint{Discussion\\}
In this subsection, we compare the results of each of the previous methods, and focus specifically on simple clustering by similarity and clustering by persistence. Discussed is a modification that may potentially increase persistent homology's efficacy as a clustering method, relative to MCL.\\
As noted in Section \ref{phtests}, the simple clustering by similarity method is naive in that it only considers the connected components of a single simplicial complex in a Vietoris-Rips filtration. This results in a clustering that groups any given vertex according to solely the strength of its direct connections to other vertices. Furthermore, the modularity values for this method suffer from a problem that largely motivates persistent homology in the first place---noise in the data. Short-lived features, or in this setting, connected components that are created but quickly join with a larger component in the Vietoris-Rips filtration, are counted the same as components with a long lifetime. The failure of the simple clustering method to address this issue fragments the generated clusters into smaller pieces and yields a lower overall modularity.\\
For this reason, clustering by persistence performs notably better than simple clustering by similarity. The former considers the overall significance of each connected component in the entire filtration and removes those components with lifetimes deemed insignificant. However, the method still suffers from a similar weakness in that it doesn't look beyond a vertex's direct connection to its neighbours, ultimately fragmenting potential clusters into small pieces. This effect is clearly seen in the high number of clusters generated by these two methods, as shown in previous subsections. MCL seems to do better in this regard by considering the probability that a random walk on the vertices of a graph stays within a given cluster, rather than considering just each individual connection between vertices.\\
For these reasons, a modification to persistent homology that would likely improve the method's clustering ability for the EAT dataset is to construct simplicial complexes based on some measure of \textit{vertex density} rather than just the similarity between two words. One such (simplistic) density measure could, for example, consider the vertices adjacent to the immediate neighbours of a given vertex. Loosely-speaking, the proportion of those vertices that are still immediately adjacent to the given vertex would function as a measure of how dense a dataset is at the given vertex. The use of such a method would require a different formulation of persistent homology, so we do not discuss it further in this paper.\\
\section{Linguistic Interpretation of Results\\} \label{interpretationsection}
In this final section, we present some specific results of our analysis of the EAT dataset with persistent homology. In particular, we give examples of clusters, loops, and voids, as well as potential interpretations of each of these features in a linguistic context.\\
To extract the clusters displayed in this section, we disregarded all edges with a lifetime less than approximately 0.209---the value of the persistence threshold maximizing the modularity index, as found in Section \ref{phresults}. During our analysis, we noted that higher-dimensional features such as loops and voids have significantly shorter lifespans than do clusters: as such, we selected the loops and voids presented here from those found to be most persistent---that is, those having the longest lifetime in the Vietoris-Rips filtration. Features were then extracted from the R-TDA results using the code described in Section \ref{phtests}.\\
Each of the images displayed in this section were created using the Pajek program for large network analysis and visualisation \cite{pajek}. We make particular note that the placement of each word in a diagram has no effect on the interpretation of the network structures formed. Rather, we used the Kamada-Kawai and Fruchterman-Reingold graph-drawing algorithms \cite{graphdraw} to arrange the words in an visually-pleasing and interpretable way. In each diagram, the thickness of an edge connecting two words is proportional to the association strength of the word pair, also indicated numerically.\\
Similar images of the features found in the EAT are presented in Appendix Section \ref{appendiximages}.
\tpoint{Clusters\\}
Clusters are the simplest and most interpretable feature in the EAT data. Our results agree with common intuition in that words tend to associate closely with others sharing a similar underlying idea. However, we observed it was not uncommon for the words of a given cluster to be connected to a few central, yet thematically unrelated words. This sort of structure appears in \textbf{Figure \ref{cluster_pupil}} with the words EYE and SCHOOL, and in \textbf{Figure \ref{cluster_death}} with DEATH.\\
While most clusters seem to be based largely on a common theme, a number of clusters span multiple ideas and concepts. Of particular interest are these intra-cluster connections between distinct ideas. The cluster in \textbf{Figure \ref{cluster_pupil}} is a representative example, where two clearly distinct themes are present: eyesight and school. Connecting these two ideas is PUPIL, a word which may be taken as both a synonym to the word student or an anatomical part of the eye. We observe that, in this case, the formation of a cluster linking these two concepts is due to PUPIL's multiple meanings in the English language.\\
\begin{figure}[h]
\centering
\includegraphics[scale=0.2]{cluster_pupil}
\caption{\footnotesize{A word cluster displaying two separate themes: eyesight and school. The connection between them is mediated by PUPIL, due to this word's multiple meanings in the English language.}\label{cluster_pupil}}
\end{figure}
\textbf{Figure \ref{cluster_death}} displays the same phenomenon. In contrast to \textbf{Figure \ref{cluster_pupil}}, however, the connection between the two themes of thought and death through CONCEPTION is not due entirely to the structure of the English language. First, CONCEPTION is related to IDEA, perhaps through a mental association with the word CONCEPT or the way in which ideas are ``born'' in an individual's mind. Second, CONCEPTION may be taken in a biological sense as a synonym to BIRTH.\\
\begin{figure}[h]
\centering
\includegraphics[scale=0.2]{cluster_death}
\caption{\footnotesize{A cluster whose members group around DEATH and IDEA. The word CONCEPTION connects the themes present in the graph.}\label{cluster_death}}
\end{figure}
\tpoint{Loops\\}
The example loop presented in this subsection may be easily seen by following word associations of higher strength---that is, the thicker edges---in the diagram. Thinner edges represent weak associations that were disregarded in our clustering method, as described at the beginning of this section.\\
In a linguistic context, a loop in the EAT data can be interpreted as a chain of closely-associated words linking one word back to itself. For example, in \textbf{Figure \ref{loop_cat}}, a certain ``train of thought'' connecting the words of the loop can be seen. We make particular note that, as in clusters, the words of a loop need not share a common theme.\\
\begin{figure}[h]
\centering
\includegraphics[scale=0.25]{loop_cat}
\caption{\footnotesize{A small loop found in the EAT data. The sequence of words composing the loop form a ``train of thought'' between them. For example, we see that CAR and DOG are connected by ROVER, both a kind of vehicle and a common dog name.}\label{loop_cat}}
\end{figure}
\tpoint {Voids\\}
Lastly, we present an example of a void found in the EAT data, and compare these features to loops. Voids are topologically equivalent to spheres: in general, they enclose some three-dimensional space. In contrast, loops, as shown in the previous section, enclose two-dimensional spaces.\\
Both loops and voids are composed of links between strongly-associated words. A loop, however, is restricted in the sense that, at any particular word, there are only two directions in which the loop can proceed. A void, on the other hand, due to its higher-dimensional nature, is not restricted in this way. This ``freedom'' is illustrated in \textbf{Figure \ref{void_math}}, particularly in how EQUATION is linked to all of ALGEBRA, MATHS, MATHEMATICS, and SUM---all of the other words of the void, in this case. For this reason, the words of a void are more closely associated with each other than the words of a loop.\\
We found that the words of a void generally share exactly one common theme. In \textbf{Figure \ref{void_math}}, this theme is clearly mathematics. In contrast, the loop of \textbf{Figure \ref{loop_cat}} contained words related to both animals and transportation.\\
\begin{figure}[h]
\centering
\includegraphics[scale=0.2]{void_math}
\caption{\footnotesize{A visual representation of a void found in the EAT data. Note in particular the high level of connectivity between words of the void and their central theme of mathematics. As the words here were visualised in three-dimensional space, we indicate vertices further in the background with grey labels.}\label{void_math}}
\end{figure}
\newpage
\section{Appendix: Algebraic Proofs and Definitions \label{appendixproofs}}
This appendix contains supplemental materials and proofs deemed too technical for the main body of Sections \ref{boundscyclessection} and \ref{simplicialhomologysection} of this paper. We assume the notations introduced in those sections.
\tpoint{Proof of Proposition \ref{abelian}\\} \label{appendixabelian}
\textbf{Proposition.} \textit{Fix a non-negative integer $k$, a simplicial complex $X$, and a field $\mathbb{F}$. The set $C_k(X,\mathbb{F})$ of $k$-chains of $X$ over $\mathbb{F}$, together with simplex addition $\oplus$, forms an Abelian group.}\\
\begin{proof}
We will verify the standard group criteria and show that $\oplus$ is commutative. Let $\underset{i\in I_k}{\sum}a_i\sigma_i$ and $\underset{i\in I_k}{\sum}b_i\sigma_i$ be arbitrary elements of $C_k(X,\mathbb{F})$.\\
\textit{$\bullet$ Closure under $\oplus$:}\\
Observe that
\begin{equation*}
\underset{i\in I_k}{\sum}a_i\sigma_i \oplus \underset{i\in I_k}{\sum}b_i\sigma_i = \underset{i\in I_k}{\sum}(a_i+b_i)\sigma_i.
\end{equation*}
Since $\mathbb{F}$ is closed under $+$ as a field, it follows that $(a_i+b_i)$ is in $\mathbb{F}$ for all $i$ in $I_k$. Therefore, $\underset{i\in I_k}{\sum}(a_i+b_i)\sigma_i$ is an element of $C_k(X,\mathbb{F})$, proving that $C_k(X,\mathbb{F})$ is closed under $\oplus$.\\
\textit{$\bullet$ Commutativity of $\oplus$:}\\
Observe that
\begin{align*}
\underset{i\in I_k}{\sum}a_i\sigma_i \oplus \underset{i\in I_k}{\sum}b_i\sigma_i &= \underset{i\in I_k}{\sum}(a_i+b_i)\sigma_i &\text{(by definition of $\oplus$)}\\
&=\underset{i\in I_k}{\sum}(b_i+a_i)\sigma_i &\text{(by commutativity of $+$ in $\mathbb{F}$)} \\
&=\underset{i\in I_k}{\sum}b_i\sigma_i \oplus \underset{i\in I_k}{\sum}a_i\sigma_i &\text{(by definition of $\oplus$)}
\end{align*}
Therefore $\oplus$ is commutative in $C_k(X,\mathbb{F})$.\\
\textit{$\bullet$ Identity element:}\\
Let 0 denote the neutral additive element of $\mathbb{F}$ under $+$. We will prove that $\underset{i\in I_k}{\sum}0\sigma_i$ is additive neutral for $\oplus$ in $C_k(X,\mathbb{F})$. Observe that\\
\begin{align*}
\underset{i\in I_k}{\sum}a_i\sigma_i \oplus \underset{i\in I_k}{\sum}0\sigma_i = \underset{i\in I_k}{\sum}(a_i+0)\sigma_i = \underset{i\in I_k}{\sum}a_i\sigma_i
\end{align*}
since 0 is additive neutral in $\mathbb{F}$. Additionally, by the commutativity of $\oplus$ proven above,
\begin{equation*}
\underset{i\in I_k}{\sum}a_i\sigma_i \oplus \underset{i\in I_k}{\sum}0\sigma_i = \underset{i\in I_k}{\sum}0\sigma_i \oplus \underset{i\in I_k}{\sum}a_i\sigma_i = \underset{i\in I_k}{\sum}a_i\sigma_i.
\end{equation*}
Therefore, $\underset{i\in I_k}{\sum}0\sigma_i$ is additive neutral for $\oplus$ in $C_k(X,\mathbb{F})$.\\
\textit{$\bullet$ Inverse elements:}\\
Given any $\underset{i\in I_k}{\sum}a_i\sigma_i$ in $C_k(X,\mathbb{F})$, consider $\underset{i\in I_k}{\sum}(-a_i)\sigma_i$, where $(-a_i)$ denotes the additive inverse of $a_i$ in $\mathbb{F}$. Observe that
\begin{align*}
\underset{i\in I_k}{\sum}a_i\sigma_i \oplus \underset{i\in I_k}{\sum}(-a_i)\sigma_i &= \underset{i\in I_k}{\sum}(a_i+-a_i)\sigma_i=\underset{i\in I_k}{\sum}0\sigma_i
\end{align*}
since $-a_i$ is the additive inverse of $a_i$ under $+$.\\
Therefore, $\underset{i\in I_k}{\sum}(-a_i)\sigma_i$ is the additive inverse of $\underset{i\in I_k}{\sum}a_i\sigma_i$ under $\oplus$.\\\\
Therefore, by the above criteria, $C_k(X,\mathbb{F})$ with group operation $\oplus$ forms an Abelian group.\\
\end{proof}
\tpoint{Proof of Lemma \ref{boundofbound}} \label{appendixbound}\\
\textit{Lemma.} For any $k\geq0$, the image of a $k$-boundary under the dimension $k$ boundary operator is the trivial $(k-1)$-chain. Equivalently, for any integer $k\geq1$
\begin{equation*}
\partial_{k-1}\partial_{k}\Big(\sum_{i \in I_{k}}a_i\sigma_i\Big)=0_{k-2}.
\end{equation*}\\
\begin{proof}
Fix $k\geq1$ as supposed, and let $[x_0,x_1,\dots,x_k]$ be any oriented $k$-simplex. Observe that
\begin{align*}
\partial_{k-1}\partial_{k}[x_0,x_1,..,x_k] &= \partial_{k-1}\sum_{i=0}^k(-1)^i[x_0,..,\hat{x}_i,..,x_k]\\
&= \sum_{i=0}^k(-1)^i\partial_{k-1}[x_0,..,\hat{x}_i,..,x_k]\\
&= \sum_{i=0}^k\Big[\sum_{j=0}^{i-1}(-1)^j(-1)^i[x_0,..,\hat{x}_j,..,\hat{x}_i,..,x_k] + \sum_{j=i+1}^{k}(-1)^{j-1}(-1)^i[x_0,..,\hat{x}_i,..,\hat{x}_j,..x_k]\Big]\\
&= \sum_{i=0}^k\sum_{j=0}^{i-1}(-1)^j(-1)^i[x_0,..,\hat{x}_j,..,\hat{x}_i,..,x_k] + \sum_{i=0}^k\sum_{j=i+1}^{k}(-1)^{j-1}(-1)^i[x_0,..,\hat{x}_i,..,\hat{x}_j,..x_k]\\
&= \sum_{0\leq j<i\leq k}(-1)^j(-1)^i[x_0,..,\hat{x}_j,..,\hat{x}_i,..,x_k] - \sum_{0\leq i< j\leq k}(-1)^{j}(-1)^i[x_0,..,\hat{x}_i,..,\hat{x}_j,..x_k]\\
&= \sum_{0\leq j<i\leq k}(-1)^j(-1)^i[x_0,..,\hat{x}_j,..,\hat{x}_i,..,x_k] - \sum_{0\leq j< i\leq k}(-1)^{i}(-1)^j[x_0,..,\hat{x}_j,..,\hat{x}_i,..x_k]\\
&= 0_{k-2}.\\
\end{align*}
Therefore, for all $k\geq1$, we have $\partial_{k-1}\partial_{k}[x_0,x_1,..,x_k]=0$. Extending this result to $k$-chains, we have
\begin{equation*}
\partial_{k-1}\partial_{k}\Big(\sum_{i \in I_{k}}a_i\sigma_i\Big)=\sum_{i \in I_{k}}a_i\partial_{k-1}\partial_{k}\sigma_i= 0_{k-2},
\end{equation*}
Thus proving the desired result.\\
\end{proof}
\tpoint{Normal subgroups and related results \label{appendixnormal}\\}
In this subsection, we recall the definition of a \textit{normal subgroup} and further prove some basic results required in Section \ref{homologygroupssection}.\\
Assume throughout that $G$ is a group, where the group operation on elements $g_1$ and $g_2$ of $G$ is denoted by $g_1g_2$. Furthermore, let the inverse of $g_1$ under the group operation be denoted by $g_1^{-1}$. Lastly, we suppose that $H$ is a subgroup of $G$.\\
\textit{Definition.} We say that $H$ is a normal subgroup of $G$ if, for every element $g$ of $G$ and every element $h$ of $H$, the element $ghg^{-1}$ is in $H$. Equivalently, we say that $H$ is invariant under conjugation by $G$.\\
\textit{Lemma.} If $G$ is an Abelian group, then every subgroup $H$ of $G$ is a normal subgroup of $G$.\\
\begin{proof}
Suppose $g$ and $h$ are arbitrary elements of $G$ and $H$, and suppose that $G$ is an Abelian group. We will verify that $H$ is a normal subgroup of $G$ by appealing to the definition of a normal subgroup. Observe that
\begin{align*}
ghg^{-1}&=gg^{-1}h &\text{(since $G$ is Abelian)}\\
&=h\in H
\end{align*}
Therefore, $ghg^{-1}$ is in $H$, and it follows by definition that $H$ is a normal subgroup of $G$.
\end{proof}
\tpoint{Vector operations on homology groups \label{vectorops}}\\
In this section, we define a vector addition $\oplus_H$ and scalar multiplication $\cdot$ appropriate for the homology group $H_k(X,\mathbb{F})$ as a vector space.\\
\begin{define}
Let $\oplus_H$ be a binary operation on $H_k(X,\mathbb{F})$ defined via
\begin{align*}
\oplus_H: H_k(X,\mathbb{F})\times H_k(X,\mathbb{F})&\rightarrow H_k(X,\mathbb{F})\\
\big(z_0+B_k(X)\big) \oplus_H \big(y_0+B_k(X)\big) &= (z_0\oplus y_0)+B_k(X),
\end{align*}
where we recall that $\oplus$ is the additive operation of the Abelian group $C_k(X,\mathbb{F})$ presented in Definition \ref{chainop}.\\
\end{define}
\begin{notation}
We will suppress notation and write $\oplus_H$ as $\oplus$, as context will make clear whether we are dealing with the sum of homology group elements or of chains.\\
\end{notation}
\begin{nrem}
The above addition operation follows the standard group theoretic formulation for the addition of cosets.\\
\end{nrem}
\begin{define}
Let $\cdot$ be the $\mathbb{F}$-scalar multiplication defined via\\
\begin{align*}
\cdot: \mathbb{F}\times H_k(X,\mathbb{F})&\rightarrow H_k(X,\mathbb{F})\\
c\cdot\big(\sum_{i \in I_k}a_i\sigma_i + B_k(X)\big)&= \sum_{i \in I_k}(ca_i)\sigma_i+B_k(X),
\end{align*}
where $ca_i$ denotes the result of the multiplicative operation of $\mathbb{F}$ on $c$ and $a_i$.\\
\end{define}
\section{Appendix: Additional Images \label{appendiximages}}
\begin{figure}[h]
\centering
\includegraphics[scale=0.25]{cluster_food}
\caption{\footnotesize{Highly-centralised clustering around FOOD.}}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[scale=0.25]{cluster_play}
\caption{\footnotesize{A number of themes present in a single cluster.}}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[scale=0.25]{cluster_bastard}
\caption{\footnotesize{An example of intricate structure within a cluster.}}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[scale=0.25]{loop_blood}
\caption{\footnotesize{A large loop with weak connections to other loops.}}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[scale=0.25]{loop_napalm}
\caption{\footnotesize{Another large loop}}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[scale=0.2]{void_forever}
\caption{\footnotesize{The most persistent void found in the EAT dataset.}}
\end{figure}
\clearpage
\begin{bibsection}
\begin{biblist}
\bib{tophistory}{book}{
TITLE = {History of topology},
EDITOR = {James, I. M.},
PUBLISHER = {North-Holland, Amsterdam},
YEAR = {1999},
PAGES = {x+1056},
ISBN = {0-444-82375-1},
}
\bib{bio}{article}{
Author = {Kovacev-Nikolic\ Violeta and Bubenik\, Peter\, and Nikolić\, Dragan\, and Heo\, Giseon},
Journal = {Statistical Applications in Genetics \& Molecular Biology},
Number = {1},
Pages = {19 - 38},
Title = {Using persistent homology and dynamical distances to analyze protein binding.},
Volume = {15},
Year = {2016}
}
\bib{med}{article}{
Author = {Mata, Gadea, and Morales, Miguel, and Romero, Ana, and Rubio, Julio},
ISSN = {0167-8655},
Journal = {Pattern Recognition Letters},
Pages = {55},
Title = {Zigzag persistent homology for processing neuronal images.},
Year = {2015},
}
\bib{image}{article}{
Author = {Shengxiang, Xia},
ISSN = {20081898},
Journal = {Journal of Nonlinear Sciences \& Applications (JNSA)},
Number = {1},
Pages = {126 - 138},
Title = {A topological analysis of high-contrast patches in natural images.},
Volume = {9},
Year = {2016},
}
\bib{topdata}{article}{
Author = {Carlsson, Gunnar},
ISSN = {02730979},
Journal = {Bulletin (New Series) of the American Mathematical Society},
Number = {2},
Pages = {255 - 308},
Title = {Topology and Data.},
Volume = {46},
Year = {2009},
}
\bib{sensor}{article}{
Author = {Gamble, Jennifer, and Chintakunta, Harish, and Krim, Hamid},
ISSN = {0165-1684},
Journal = {Signal Processing},
Pages = {1 - 18},
Title = {Coordinate-free quantification of coverage in dynamic sensor networks.},
Volume = {114},
URL = {http://login.ezproxy.library.ualberta.ca/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=edselp&AN=S0165168415000791&site=eds-live&scope=site},
Year = {2015},
}
\bib{eat}{misc}{
title = {EAT: Edinburgh Associative Thesaurus},
year = {2015},
url = {http://www.eat.rl.ac.uk/},
}
\bib{cluster.fig}{misc}{
Author={Chire}
Title={SLINK-Gaussian-data.svg, https://commons.wikimedia.org/w/index.php?curid=17087089}
url={https://commons.wikimedia.org/w/index.php?curid=17087089}
}
\bib{ling1}{article}{
Author = {Joyce, Terry, and Miyake, Maki},
ISSN = {9783540781585},
Journal = {Large-scale Knowledge Resources. Construction \& Application},
Pages = {116},
Title = {Capturing the Structures in Association Knowledge: Application of Network Analyses to Large-Scale Databases of Japanese Word Associations.},
Year = {2008},
}
\bib{rtda}{manual}{
title = {TDA: Statistical Tools for Topological Data Analysis},
author = {Brittany T. Fasy and Jisu Kim and Fabrizio Lecci and Clement Maria and Vincent Rouvreau. The included GUDHI is authored by Clement Maria and Dionysus by Dmitriy Morozov and PHAT by Ulrich Bauer and Michael Kerber and Jan Reininghaus.},
year = {2015},
note = {R package version 1.4.1},
url = {https://CRAN.R-project.org/package=TDA},
}
\bib{mcl}{misc}{
title = {MCL - a cluster algorithm for graphs},
year = {2015},
url = {http://micans.org/mcl/},
}
\bib{ghrist}{article}{
AUTHOR = {Ghrist, Robert},
TITLE = {Barcodes: the persistent topology of data},
JOURNAL = {Bull. Amer. Math. Soc. (N.S.)},
VOLUME = {45},
YEAR = {2008},
NUMBER = {1},
PAGES = {61--75},
ISSN = {0273-0979},
DOI = {10.1090/S0273-0979-07-01191-3},
}
\bib{thesis}{article}{
author={Kovacev-Nikolic, Violeta},
title={Persistent homology in analysis of point-cloud data},
date={2012},
}
\bib{pajek}{misc}{
title = {Program Package Pajek / PajekXXL},
year = {2016},
url = {http://mrvar.fdv.uni-lj.si/pajek/},
}
\bib{munkres}{book}{
Author = {Munkres, James R.},
ISBN = {0201045869},
Publisher = {Menlo Park, Calif. : Addison-Wesley, c1984.},
Title = {Elements of algebraic topology.},
Year = {1984},
}
\bib{goodman}{book}{
Author = {Goodman, Frederick M.},
ISBN = {0130673420},
Publisher = {Upper Saddle River, NJ : Prentice Hall, c2003.},
Title = {Algebra : abstract and concrete : stressing symmetry.},
URL = {http://login.ezproxy.library.ualberta.ca/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=cat03710a&AN=alb.3989528&site=eds-live&scope=site},
Year = {2003},
}
\bib{homoeg}{misc}{
title = {Homology Theory --- A Primer | Math $\cap$ Programming},
year = {2016},
url = {http://jeremykun.com/2013/04/03/homology-theory-a-primer/},
}
\bib{mph}{article}{
Author = {Carlsson, Gunnar, and Zomorodian, Afra},
ISSN = {0179-5376},
Journal = {Discrete and Computational Geometry},
Keywords = {Universities and colleges -- Analysis, Computer science -- Analysis},
Number = {1},
Pages = {71},
Title = {The Theory of Multidimensional Persistence.},
Year = {2009},
}
\bib{graphdraw}{misc}{
title = {Force-directed graph drawing --- Wikipedia, The Free Encyclopedia},
year = {2016},
url = {https://en.wikipedia.org/wiki/Force-directed_graph_drawing},
}
\bib{torusloops}{misc}{
author = {Wootonjames},
title = {ToricCodeTorus - Toric code - Wikipedia, the free encyclopedia},
year = {2010},
url = {https://en.wikipedia.org/wiki/Toric_code\#/media/File:ToricCodeTorus.png},
}
\bib{comptopol}{book}{
AUTHOR = {Edelsbrunner, Herbert and Harer, John L.},
TITLE = {Computational topology},
NOTE = {An introduction},
PUBLISHER = {American Mathematical Society, Providence, RI},
YEAR = {2010},
PAGES = {xii+241},
ISBN = {978-0-8218-4925-5},
MRCLASS = {00-02 (05C10 52-02 55-02 57-02 65D18 68U05)},
MRNUMBER = {2572029},
MRREVIEWER = {Andrzej Kozlowski}
}
\end{biblist}
\end{bibsection}
\end{document}
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{"url":"https:\/\/datawarrior.wordpress.com\/2017\/06\/01\/sammon-embedding-with-tensorflow\/","text":"Embedding algorithms, especially word-embedding algorithms, have been one of the recurrent themes of this blog. Word2Vec has been mentioned in a few entries (see this); LDA2Vec has been covered (see this); the mathematical principle of GloVe has been elaborated (see this); I haven\u2019t even covered Facebook\u2019s fasttext; and I have not explained the widely used t-SNE and Kohonen\u2019s map (self-organizing map, SOM).\n\nI have also described the algorithm of Sammon Embedding, (see this) which attempts to capture the likeliness of pairwise Euclidean distances, and I implemented it using Theano. This blog entry is about its implementation in Tensorflow as a demonstration.\n\nLet\u2019s recall the formalism of Sammon Embedding, as outlined in the previous entry:\n\nAssume there are high dimensional data described by\u00a0$d$-dimensional vectors,\u00a0$X_i$\u00a0where\u00a0$i=1, 2, \\ldots, N$. And they will be mapped into vectors\u00a0$Y_i$, with dimensions 2 or 3. Denote the distances to be\u00a0$d_{ij}^{*} = \\sqrt{| X_i - X_j|^2}$\u00a0and\u00a0$d_{ij} = \\sqrt{| Y_i - Y_j|^2}$. In this problem,\u00a0$Y_i$\u00a0are the variables to be learned. The cost function to minimize is\n\n$E = \\frac{1}{c} \\sum_{i,\n\nwhere\u00a0$c = \\sum_{i.\n\nUnlike in previous entry and original paper, I am going to optimize it using first-order gradient optimizer. If you are not familiar with Tensorflow, take a look at some online articles, for example, \u201cTensorflow demystified.\u201d This demonstration can be found in this Jupyter Notebook in Github.\n\nFirst of all, import all the libraries required:\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport tensorflow as tf\n\n\nLike previously, we want to use the points clustered around at the four nodes of a tetrahedron as an illustration, which is expected to give equidistant clusters. We sample points around them, as shown:\n\ntetrahedron_points = [np.array([0., 0., 0.]), np.array([1., 0., 0.]), np.array([np.cos(np.pi\/3), np.sin(np.pi\/3), 0.]), np.array([0.5, 0.5\/np.sqrt(3), np.sqrt(2.\/3.)])]\n\nsampled_points = np.concatenate([np.random.multivariate_normal(point, np.eye(3)*0.0001, 10) for point in tetrahedron_points])\n\ninit_points = np.concatenate([np.random.multivariate_normal(point[:2], np.eye(2)*0.0001, 10) for point in tetrahedron_points])\n\n\nRetrieve the number of points,\u00a0N, and the resulting dimension,\u00a0d:\n\nN = sampled_points.shape[0]\nd = sampled_points.shape[1]\n\n\nOne of the most challenging technical difficulties is to calculate the pairwise distance. Inspired by this StackOverflow thread and Travis Hoppe\u2019s entry on Thomson\u2019s problem, we know it can be computed. Assuming Einstein\u2019s convention of summation over repeated indices, given vectors $a_{ik}$, the distance matrix is:\n\n$D_{ij} = (a_{ik}-a_{jk}) (a_{ik} - a_{jk})^T = a_{ik} a_{ik} + a_{jk} a_{jk} - 2 a_{ik} a_{jk}$,\n\nwhere the first and last terms are simply the norms of the vectors. After computing the matrix, we will flatten it to vectors, for technical reasons omitted to avoid gradient overflow:\n\nX = tf.placeholder('float')\nXshape = tf.shape(X)\n\nsqX = tf.reduce_sum(X*X, 1)\nsqX = tf.reshape(sqX, [-1, 1])\nsqDX = sqX - 2*tf.matmul(X, tf.transpose(X)) + tf.transpose(sqX)\nsqDXarray = tf.stack([sqDX[i, j] for i in range(N) for j in range(i+1, N)])\nDXarray = tf.sqrt(sqDXarray)\n\nY = tf.Variable(init_points, dtype='float')\nsqY = tf.reduce_sum(Y*Y, 1)\nsqY = tf.reshape(sqY, [-1, 1])\nsqDY = sqY - 2*tf.matmul(Y, tf.transpose(Y)) + tf.transpose(sqY)\nsqDYarray = tf.stack([sqDY[i, j] for i in range(N) for j in range(i+1, N)])\nDYarray = tf.sqrt(sqDYarray)\n\n\nAnd DXarray and DYarray are the vectorized pairwise distances. Then we defined the cost function according to the definition:\n\nZ = tf.reduce_sum(DXarray)*0.5\nnumerator = tf.reduce_sum(tf.divide(tf.square(DXarray-DYarray), DXarray))*0.5\ncost = tf.divide(numerator, Z)\n\n\nupdate_rule = tf.assign(Y, Y-0.01*grad_cost\/lapl_cost)\ninit = tf.global_variables_initializer()\n\n\nThe last line initializes all variables in the Tensorflow session when it is run. Then start a Tensorflow session, and initialize all variables globally:\n\nsess = tf.Session()\nsess.run(init)\n\n\nThen run the algorithm:\n\nnbsteps = 1000\nc = sess.run(cost, feed_dict={X: sampled_points})\nprint \"epoch: \", -1, \" cost = \", c\nfor i in range(nbsteps):\nsess.run(train, feed_dict={X: sampled_points})\nc = sess.run(cost, feed_dict={X: sampled_points})\nprint \"epoch: \", i, \" cost =\n\n\nThen extract the points and close the Tensorflow session:\n\ncalculated_Y = sess.run(Y, feed_dict={X: sampled_points})\nsess.close()\n\n\nPlot it using matplotlib:\n\nembed1, embed2 = calculated_Y.transpose()\nplt.plot(embed1, embed2, 'ro')\n\n\nThis gives, as expected,\n\nThis code for Sammon Embedding has been incorporated into the Python package mogu, which is a collection of numerical routines. You can install it, and call:\n\nfrom mogu.embed import sammon_embedding\ncalculated_Y = sammon_embedding(sampled_points, init_points)\n\n\n\u2022 Kwan-Yuet Ho, \u201cSammon Embedding,\u201d\u00a0Everything About Data Analytics, WordPress (2016). [WordPress]\n\u2022 Kwan-yuet Ho, \u201cWord Embedding Algorithms,\u201d\u00a0Everything about Data Analytics, WordPress\u00a0(2016). [WordPress]\n\u2022 Kwan-yuet Ho, \u201cToying with Word2Vec,\u201d\u00a0Everything about Data Analytics, WordPress. (2015) [WordPress]\n\u2022 Kwan-yuet Ho, \u201cLDA2Vec: a hybrid of LDA and Word2Vec,\u201d\u00a0Everything about Data Analytics, WordPress. (2016) [WordPress]\n\u2022 John W. Sammon, Jr., \u201cA Nonlinear Mapping for Data Structure Analysis,\u201d IEEE Transactions on Computers 18, 401-409 (1969).\n\u2022 Wikipedia: Sammon Mapping. [Wikipedia]\n\u2022 Github repository: stephenhky\/SammonEmbedding. [Github]\n\u2022 Laurens van der Maaten,\u00a0Geoffrey Hinton, \u201cVisualizing Data using t-SNE,\u201d\u00a0Journal of Machine Learning 1, 1-48 (2008). [PDF]\n\u2022 Teuvo Kohonen, \u201cSelf-Organizing Maps,\u201d Springer (2000). [Amazon]\n\u2022 GloVe: Global Vectors for Word Representation. [StanfordNLP]","date":"2018-12-13 08:42:56","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\": 11, \"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\": 0.566981852054596, \"perplexity\": 6582.430841586686}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-51\/segments\/1544376824601.32\/warc\/CC-MAIN-20181213080138-20181213101638-00304.warc.gz\"}"}
| null | null |
Ingria este o regiune istorică situată în Rusia actuală, pe malul Golfului Finic, între Sudul Lacului Ladoga și fluviul Narva.
Istorie
Evul Mediu
În timpul vikingilor, Ingria era un cap de pod pe drumul comercial al varegilor care se duceau în Estul Europei.
Suedezii îi puseseră numele de Ingermanland vechiului teritoriu care aparținea prinților de Novgorod, de la numele fiicei regelui Suediei Olof Skötkonung, care se numea Ingigerd Olofsdotter. La căsătoria sa cu Iaroslav I cel Înțelept, în 1019, ea a primit acest ținut ca dar de căsătorie.
Note
Vezi și
Regiunea Leningrad
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,888
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Call for Applications – Master of Science (MSc) in Immunology (2023) from the Department of Immunology and Molecular Medicine, Faculty of Medical Sciences, University of Sri Jayewardenepura
Course Fee: LKR 1,200,000/=
The program was designed in partnership with SungKyunKwan University, the Republic of Korea as a part of the Leading University Project for International Cooperation which aims to develop educational and scientific research in the fields of Immunology and Molecular Medicine.
Application Closing Date – 15th February 2023
University Website Link – www.sjp.ac.lk/ | www.graduate.sjp.ac.lk
Masters in Environment Management (MEM) 2023/24 – University of Colombo
Diploma in Management 2023/2024 – Uva Wellassa University
Certificate Course in Tourism 2023 – University of Sri Jayewardenepura
Master Degree Programmes (2022) Colombo Intake – PGIA, University of Peradeniya
June 7, 2022 December 26, 2022
Study Programmes (Courses) in Demography 2023 – University of Colombo
Diploma in Management 2021/2022 – Uva Wellassa University (UWU)
Bachelor of Arts (B.A) External Degree Programme (2022) – University of Jaffna
Diploma in Scientific Tea Manufacturing and Quality Management 2022 (2023) – University of Ruhuna
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 7,662
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\section{Introduction and statement of the results}
Positive trigonometric sums play an important role in Harmonic
Analysis, Orthogonal Polynomials, Approximation Theory and many
other branches of mathematics. In this note we discuss two
trigonometric inequalities which appear in the book of R. Askey
\cite{RA1975}. The first one is inequality (1.29) in \cite{RA1975},
which reads as
\begin{equation}\label{e1.1}
\frac{\sin (n-1)\theta}{(n-1)\sin \theta}-\frac{\sin
(n+1)\theta}{(n+1)\sin \theta}\leq\frac{4n}{n^2-1}\Big[1-\frac{\sin
n\theta}{n\sin \theta}\Big]\,,\qquad 0\leq\theta\leq\pi
\end{equation}
(actually, (1.29) appears in \cite{RA1975} as a strict inequality
and under the assumption $0<\theta<\pi$). It was proved by M. S.
Robertson \cite{MR1945} while studying the coefficients of univalent
functions. The second one was proved by Askey and Gasper
\cite{AG1976} and reads as
\begin{equation}\label{e1.2}
\frac{\sin (n-1)\theta}{(n-1)\sin \theta}-\frac{\sin
(n+1)\theta}{(n+1)\sin \theta}\leq
\frac{(3+\cos\theta)n}{n^2-1}\Big[1-\frac{\sin n\theta}{n\sin
\theta}\Big]\,,\qquad 0\leq\theta\leq\pi\,.
\end{equation}
This is inequality (8.17) in \cite{RA1975}, and as Askey wrote, it
is sharper than \eqref{e1.1}. For more information on positive
trigonometric sums and positive finite linear combinations of
classical orthogonal polynomials we refer to \cite{RA1975, DM1998,
DM2001} and the references therein.
We find it more convenient to reformulate inequalities \eqref{e1.1}
and \eqref{e1.2} in terms of the Chebyshev polynomials of the first
kind. Let us recall that the $m$-th Chebyshev polynomial of the
first kind $T_m$, $m\in \mathbb{N}_0$, is defined by
$$
T_m(x)=\cos m\theta,\qquad x=\cos\theta\in [-1,1],\ \ \theta\in
[0,\pi],
$$
and its derivative is
$$
T_m^{\prime}(x)=m\,\frac{\sin m\theta}{\sin \theta},\qquad
x=\cos\theta\,.
$$
Using
\begin{eqnarray*}
&&\frac{\sin (n-1)\theta}{\sin\theta}=\frac{\sin
n\theta\cos\theta-\cos
n\theta\sin\theta}{\sin\theta}=\frac{xT_n^{\prime}(x)}{n}-T_n(x)\,,\\
&&\frac{\sin (n+1)\theta}{\sin\theta}=\frac{\sin
n\theta\cos\theta+\cos
n\theta\sin\theta}{\sin\theta}=\frac{xT_n^{\prime}(x)}{n}+T_n(x)\,,
\end{eqnarray*}
we find that inequality \eqref{e1.1} of Robertson is equivalent to
the inequality
\begin{equation}\label{e1.3}
f_1(x):=T_n(x)+2-\frac{x+2}{n^2}\,T_n^{\prime}(x)\geq 0\,,\qquad
n\geq 2,\ \ x\in [-1,1]\,,
\end{equation}
while the Askey-Gasper inequality \eqref{e1.2} is equivalent to the
inequality
\begin{equation}\label{e1.4}
f_2(x):=T_n(x)+\frac{x+3}{2}-\frac{3(x+1)}{2n^2}\,T_n^{\prime}(x)
\geq 0\,,\qquad n\geq 2,\ \ x\in [-1,1]\,.
\end{equation}
Since $\max_{x\in [-1,1]}T_n^{\prime}(x)=T_n^{\prime}(1)=n^2$, we
have
$$
f_1(x)-f_2(x)=\frac{1-x}{2n^2}\,\big(n^2-T_n^{\prime}(x)\big) \geq
0,\qquad x\in [-1,1]\,,
$$
hence inequality \eqref{e1.3} is a consequence of inequality
\eqref{e1.4}. Thus we naturally arrive at the following
\begin{problem}\label{p1}
Find the largest constant $a=a(n)\geq 0$, $\,n\geq 2$, such that
$$
g_n(a;x):=T_n(x)+2-\frac{x+2}{n^2}\,T_n^{\prime}(x)-a\,\frac{1-x}{n^2}\,
\big(n^2-T_n^{\prime}(x)\big)\geq 0,\qquad x\in [-1,1]\,.
$$
\end{problem}
The cases $n=2,\;3$ are trivial: from $\,T_2(x)=2x^2-1\,$ and
$\,T_3(x)=4x^3-3x\,$ one finds
$$
g_2(a;x)=(1-a)(1-x)^2\,,\qquad
g_3(a;x)=\frac{4}{3}(2-a)(1-x)^2(1+x)\,,
$$
whence $a(2)=1$ and $a(3)=2$.
We assume henceforth that $n\geq 4$. By the Askey--Gasper inequality
\eqref{e1.2}, $g_n(1/2,x)=f_2(x)\geq 0$, $x\in [-1,1]$, and
therefore $a(n)\geq 1/2$. We show below that $a(n)$ cannot be
essentially larger than $1/2$. Indeed, let
$$
x_k=\cos \frac{k\pi}{n}, \qquad k=1,\ldots,n-1,
$$
be the zeros of $T_n^{\prime}$. Then $\,T_n(x_k)=(-1)^{k}\,$ and
$$
g_n(a;x_k)=2+(-1)^k-a(1-x_k),\qquad k=1,\ldots,n-1\,.
$$
The condition that $\,g_n(a;x_{n-1})\geq 0$ in the case of even $n$
and $\,g_n(a;x_{n-2})\geq 0$ in the case of odd $n$ implies
respectively
$$
a\leq\frac{1}{1-x_{n-1}}=\frac{1}{1+x_{1}}
=\frac{1}{1+\cos\frac{\pi}{n}},\qquad n - \text{ even,}
$$
and
$$
a\leq\frac{1}{1-x_{n-2}}=\frac{1}{1+x_{2}}
=\frac{1}{1+\cos\frac{2\pi}{n}},\qquad n - \text{ odd.}
$$
Since both upper bounds for $a$ tend to $1/2$ as $n$ grows, it
follows that $a=1/2$ is the best possible (the largest) absolute
constant, ensuring that $g_n(a;x)\geq 0$ for every $x\in [-1,1]$ and
all $n\in \mathbb{N}$, $n\geq 2$. In this sense, the Askey-Gasper
inequality \eqref{e1.4} is the best possible.
It turns out that the upper bounds for $a(n)$ found above actually
provide the solution to Problem~\ref{p1}. Specifically, we prove the
following statement.
\begin{theorem}\label{t1}
Let $n\in \mathbb{N}$, $n\geq 4$. Then
\begin{equation}\label{e1.5}
T_n(x)+2-\frac{x+2}{n^2}\,T_n^{\prime}(x)-a(n)\,\frac{1-x}{n^2}\,
\big(n^2-T_n^{\prime}(x)\big)\geq 0\,,\quad x\in [-1,1]\,,
\end{equation}
where
$$
a(n)=\begin{cases}\ds{\frac{1}{1+\cos\frac{\pi}{n}}},& \text{ if }\
n\ \text{ is even},\vspace*{1ex}\\
\ds{\frac{1}{1+\cos\frac{2\pi}{n}}},&\text{ if }\ n \ \text{ is
odd}.
\end{cases}
$$
The constant $a(n)$ is the best possible in the sense that
\eqref{e1.5} fails for any larger constant. The equality in
\eqref{e1.5} is attained only at $x=1$ and $x=-\cos\frac{\pi}{n}$ if
$n$ is even, and at $x=\pm 1$ and $x=-\cos\frac{2\pi}{n}$ if $n$ is
odd.
\end{theorem}
\begin{figure}[htp]
\centering
\includegraphics[scale=0.65,clip]{Fig1.pdf} \hspace*{0.5cm}
\includegraphics[scale=0.65,clip]{Fig2.pdf}
{\small \caption{The graphs of
$T_n(x)+2-\frac{x+2}{n^2}\,T_n^{\prime}(x)-a(n)\,\frac{1-x}{n^2}\,
\big(n^2-T_n^{\prime}(x)\big)$ for $n=12$ (left) and $n=13$
(right).}} \label{fig1}
\end{figure}
The typical behavior of the function in \eqref{e1.5} in the cases of
even and odd $n$ is shown in Figure~1. The graphs suggest that in
the interval $[0,1]$ this function could be non-negative for a
larger constant $a$ than the one specified in Theorem~\ref{t1}. We
show that this is indeed the case by proving that the
non-negativeness in $[0,1]$ persists with $a=1$.
\begin{theorem}\label{t2}
Let $n\in \mathbb{N}$, $n\geq 3$. Then
\begin{equation}\label{e1.6}
T_n(x)+x+1-\frac{2x+1}{n^2}\,T_n^{\prime}(x)\geq 0\,,\quad x\in
[0,1]\,,
\end{equation}
or, equivalently,
\begin{equation}\label{e1.7}
T_n^{\prime}(1)-T_n^{\prime}(x)\geq
(1-x)\,T_n^{\prime\prime}(x)\,,\qquad x\in [0,1]\,.
\end{equation}
The equality in \eqref{e1.6}-\eqref{e1.7} occurs only for $x=1$ and
if $n\equiv 2$ $(\!\!\!\mod 4)$, for $x=0$.
\end{theorem}
For $n=0,\,1,\,2,$ \eqref{e1.7} becomes an identity. Inequality
\eqref{e1.7} provides an interesting complement to the finite
increment formula $T_n^{\prime}(1)-T_n^{\prime}(x)=
(1-x)\,T_n^{\prime\prime}(\xi)$, $\xi\in (x,1)$. Moreover,
\eqref{e1.7} implies a similar property of the ultraspherical
polynomials $P_n^{(\lambda)}$, $\lambda\geq 1$.
\begin{corollary}\label{c1}
For every ultraspherical polynomial $P_n^{(\lambda)}$, $\lambda\geq 1$,
there holds
$$
P_n^{(\lambda)}(1)-P_n^{(\lambda)}(x)\geq
\frac{d}{dx}\,\big\{P_n^{(\lambda)}(x)\big\}\,(1-x)\,,\qquad x\in
[0,1]\,.
$$
\end{corollary}
Corollary~\ref{c1} easily follows from
$T_n^{\prime}=n\,P_{n-1}^{(1)}$ and the fact that if $\mu\geq \lambda$,
then $P_n^{(\mu)}$ is represented as a linear combination of
$\{P_m^{(\lambda)}\}_{m=0}^{n}$ with non-negative coefficients. There
are examples showing that Corollary~\ref{c1} is not true if $\lambda<1$,
the case $\lambda=0$ is particularly easy to verify. A challenging
problem is to characterize all pairs of parameters $(\alpha,\begin{equation})$
ensuring similar inequality for the Jacobi polynomials
$P_n^{(\alpha,\begin{equation})}$.
\medskip
The paper is organized as follows. In the next section we propose a
short elementary proof of the Askey-Gasper inequality \eqref{e1.4}
(and thereby of the Robertson inequality \eqref{e1.3}). In Section~3
we present a proof of Theorem~\ref{t2}. The proof of
Theorem~\ref{t1} is given in Section~4.
\section{Proof of the Askey--Gasper inequality (1.4)}
\setcounter{equation}{0} We prove the following statement:
\begin{proposition}\label{pp1}
Let $n\in \mathbb{N}$, $n\geq 2$. Then
\begin{equation}\label{e2.1}
f_2(x)=T_n(x)+\frac{x+3}{2}-\frac{3(x+1)}{2n^2}\,T_n^{\prime}(x)\geq
0,\qquad x\in [-1,1].
\end{equation}
The equality in \eqref{e2.1} is attained only at $x=1$ and if $n$ is
odd, at $x=-1$.
\end{proposition}
\medskip\noindent{\bf Proof.} From $\,T_2(x)=2x^2-1\,$ and $\,T_3(x)=4x^3-3x\,$ we find
\begin{eqnarray*}
&& f_2(x)=\frac{1}{2}\,(1-x)^2,\ \ n=2,\\
&& f_2(x)=2(1-x)^2(1+x),\ \ n=3,
\end{eqnarray*}
hence Proposition~\ref{pp1} is true for $\,n=2,\,3$, and we assume
henceforth $n\geq 4$.
We shall use in this and in the next section the differential
equation satisfied by $T_n$,
\begin{equation}\label{e2.2}
(1-x^2)\,T_n^{\prime\prime}(x)-x\,T_n^{\prime}(x)+n^2\,T_n(x)=0\,,
\end{equation}
as well as the identity
\begin{equation}\label{e2.3}
n^2\,\big[T_n(x)\big]^2+(1-x^2)\,\big[T_n^{\prime}(x)\big]^2=n^2\,.
\end{equation}
Denote by $\tau$ the largest zero of $T_n^{\prime\prime}$. \medskip
\emph{Case 1: $x\in (\tau,1]$}. Using \eqref{e2.2}, we rewrite $f_2$
in the form
$$
f_2(x)=\frac{x+3}{2n^2}\,\big(T_n^{\prime}(1)-T_n^{\prime}(x)\big)
-\frac{1}{n^2}\,(1-x^2)T_n^{\prime\prime}(x)\,.
$$
We observe that in this case the inequality $f_2(x)\geq 0$ is
equivalent to
\begin{equation}\label{e2.4}
\frac{x+3}{2(x+1)}\,\frac{1}{1-x}\,\int_{x}^{1}T_n^{\prime\prime}(u)\,du
\geq T_n^{\prime\prime}(x)\,,\qquad x\in (\tau,1]\,.
\end{equation}
Since $T_n^{\prime\prime}$ is positive and monotonically increasing
in $(\tau,1]$,
$$
\frac{1}{1-x}\,\int_{x}^{1}T_n^{\prime\prime}(u)\,du\geq
T_n^{\prime\prime}(x)
$$
and \eqref{e2.4} is a consequence of the inequality
$$
\frac{x+3}{2(x+1)}\geq 1\,,
$$
which is obviously true for $x\in (\tau,1]$. Notice in the last two
inequalities the equality holds only when $x=1$. Hence, $f_2(x)\geq
0$ for $x\in (\tau,1]$, and the equality is attained only for $x=1$.
\medskip
\emph{Case 2: $x\in [-1,\tau]$}. In this case we rewrite inequality
$f_2(x)\geq 0$ in the form
\begin{equation}\label{e2.5}
1+T_n(x)+\frac{1+x}{2}\geq\frac{3(1+x)}{2n^2}\,T_n^{\prime}(x),
\qquad x\in [-1,\tau]\,.
\end{equation}
The left-hand side of \eqref{e2.5} is non-negative for $x\in
[-1,\tau]$. On the other hand, the right-hand side of \eqref{e2.5}
is non-positive for $x\in [\cos\frac{2\pi}{n},\cos\frac{\pi}{n}]$ as
$T_n^{\prime}(x)\leq 0$ therein. Since $\tau\in
(\cos\frac{2\pi}{n},\cos\frac{\pi}{n})$, \eqref{e2.5} will be proved
if we show that
$$
1+T_n(x)+\frac{1+x}{2}\geq\frac{3(1+x)}{2n^2}\,T_n^{\prime}(x),
\qquad x\in \big[-1,\cos\frac{2\pi}{n}\big]\,,
$$
and it suffices to prove the inequality
\begin{equation}\label{e2.6}
\Big(1+T_n(x)+\frac{1+x}{2}\Big)^2
\geq\frac{9(1+x)^2}{4n^4}\,\big[T_n^{\prime}(x)\big]^2, \qquad x\in
\big[-1,\cos\frac{2\pi}{n}\big]\,.
\end{equation}
We estimate the left-hand side of \eqref{e2.6} by the arithmetic
mean - geometric mean inequality:
\begin{equation}\label{e2.7}
\Big(1+T_n(x)+\frac{1+x}{2}\Big)^2\geq 2(1+x)(1+T_n(x))\,,
\end{equation}
and apply identity \eqref{e2.3} to express
$\big[T_n^{\prime}(x)\big]^2$ in the right-hand side of
\eqref{e2.6},
$$
\big[T_n^{\prime}(x)\big]^2=\frac{n^2(1-T_n(x))(1+T_n(x))}{1-x^2}\,.
$$
It follows from \eqref{e2.7} that \eqref{e2.6} will hold for $x\in
\big[-1,\cos\frac{2\pi}{n}\big]$ if
\begin{equation}\label{e2.8}
2(1+x)(1+T_n(x))\geq\frac{9(1+x)}{4n^2(1-x)}\,(1+T_n(x))(1-T_n(x))\,.
\end{equation}
Since $(1+x)(1+T_n(x))\geq 0$ and $1-T_n(x)\leq 2$, the above
inequality will be certainly true if
$$
n^2(1-x)\geq \frac{9}{4}\,,\qquad x\in
\big[-1,\cos\frac{2\pi}{n}\big]\,.
$$
To see that the above inequality is true, we make use of
$\sin\alpha>\frac{2}{\pi}\,\alpha$, $\alpha\in (0,\pi/2)$. For $x\in
\big[-1,\cos\frac{2\pi}{n}\big]$,
$$
n^2(1-x)\geq
n^2\Big(1-\cos\frac{2\pi}{n}\Big)=2n^2\sin^2\frac{\pi}{n}>
2n^2\,\Big(\frac{2}{\pi}\,\frac{\pi}{n}\Big)^2=8>\frac{9}{4}\,.
$$
Thus, the inequality $f_2(x)\geq 0$ is proved in Case 2, too, and it
remains to check when the equality is attained. Tracing backward our
proof, we see that the equality in \eqref{e2.8} holds only if either
$x=-1$ or $T_n(x)+1=0$, while the equality in \eqref{e2.7} holds
only if $T_n(x)+1=(1+x)/2$. Both conditions imply that $x=-1$ and
$T_n(-1)=-1$, and the latter holds if and only if $n$ is odd. \hfill $\Box$
\section{Proof of Theorem 2}
\setcounter{equation}{0}
Let us set
\begin{eqnarray*}
\varphi_n(x)&:=&T_n(x)+x+1-\frac{2x+1}{n^2}\,T_n^{\prime}(x)\,,\\
\psi_n(x)&:=&T_n^{\prime}(1)-T_n^{\prime}(x)-(1-x)\,T_n^{\prime\prime}(x)\,.
\end{eqnarray*}
From $T_n^{\prime}(1)=n^2$ and the differential equation
\eqref{e2.2} we have
\begin{equation*}
\begin{split}
\varphi_n(x)&=\frac{x+1}{n^2}\,\big(n^2-T_n^{\prime}(x)\big)
-\frac{1}{n^2}\,\big(x\,T_n^{\prime}(x)-n^2\,T_n(x)\big)\\
&=\frac{x+1}{n^2}\,\big(T_n^{\prime}(1)-T_n^{\prime}(x)\big)-\frac{1-x^2}{n^2}
\,T_n^{\prime\prime}(x)\\
&=\frac{x+1}{n^2}\,\psi_n(x)\,,
\end{split}
\end{equation*}
therefore inequalities \eqref{e1.6} and \eqref{e1.7}, i.e.,
$\varphi_n(x)\geq 0$ and $\psi_n(x)\geq 0$, $x\in [0,1]$, are
equivalent.
From
$$
T_n(x)=\cos n\theta,\ \ T_n^{\prime}(x)=n\,\frac{\sin
n\theta}{\sin\theta}\,,\qquad x=\cos\theta\,,\ \ \theta\in
[0,\pi]\,,
$$
we find
$$
\varphi_n(0)=\begin{cases} 2,& n\equiv 0\ (\!\!\!\!\mod 4)\\
1-1/n,& n\equiv 1\ (\!\!\!\!\mod 4)\\
0,& n\equiv 2\ (\!\!\!\!\mod 4)\\
1+1/n,& n\equiv 3\ (\!\!\!\!\mod 4)\,,
\end{cases}
$$
hence $\varphi_n(0)\geq 0$, with the equality holding only if
$n=4k+2$, and we may assume further $x\in (0,1]$.
If $\tau>0$ is the largest zero of $T_n^{\prime\prime}$, then, by
the same argument as in \emph{Case~1} in the preceding section, we
obtain
$$
\psi_n(x)=(1-x)\Big\{\frac{1}{1-x}\int_{x}^{1}T_n^{\prime\prime}(u)\,du-
T_n^{\prime\prime}(x)\Big\}\geq 0,\qquad x\in [\tau,1]\,,
$$
with the equality holding only for $x=1$, so we may restrict our
consideration to the case $x\in (0,\tau)$. Furthermore,
$T_n^{\prime}(x)\leq 0$ for $x\in
\big[\cos\frac{2\pi}{n},\cos\frac{\pi}{n}\big]$, and then obviously
$\varphi_n(x)>0$ for $x\in
\big[\cos\frac{2\pi}{n},\cos\frac{\pi}{n}\big]$. Since $\tau$ lies
in this interval, it remains to prove either of the inequalities
$\varphi_n(x)>0$ and $\psi_n(x)>0$ when $x\in
(0,\cos\frac{2\pi}{n})$. There is nothing to prove if $n=4$, so we
assume $n\geq 5$. It suffices to show that $\varphi_n(t)>0$ (or
$\psi_n(t)>0$) for every critical point $t$ of $\psi_n$ (i.e. zero
of $\psi_n^{\prime}$) in the interval $(0,\cos\frac{2\pi}{n})$.
Since
$$
\psi_n^{\prime}(x)=(x-1)T_n^{\prime\prime\prime}(x)\,,
$$
the critical points of $\psi_n$ in $(0,\cos\frac{2\pi}{n})$ are
zeros of $T_n^{\prime\prime\prime}$.
Let $t\in (0,\cos\frac{2\pi}{n})$ be a zero of
$T_n^{\prime\prime\prime}$. From
$$
\cos\frac{2\pi}{n}=1-2\,\sin^2\frac{\pi}{n}
<1-2\,\Big(\frac{2}{\pi}\,\frac{\pi}{n}\Big)^2=1-\frac{8}{n^2},
$$
we conclude that
\begin{equation}\label{e3.1}
0<t<1-\frac{8}{n^2}\,.
\end{equation}
Since $y=T_n^{\prime}(x)$ satisfies the differential equation
$$
(1-x^2)y^{\prime\prime}-3x\,y^{\prime}+(n^2-1)\,y=0
$$
(this can be seen e.g., by differentiating \eqref{e2.2}), and
$y^{\prime\prime}(t)=0$, we have
$$
T_n^{\prime\prime}(t)=\frac{n^2-1}{3t}\,T_n^{\prime}(t)\,.
$$
Now, \eqref{e2.2} with $x=t$ yields
$$
(1-t^2)\,\frac{n^2-1}{3t}\,T_n^{\prime}(t)-t\,T_n^{\prime}(t)+n^2\,T_n(t)=0\,,
$$
whence
\begin{equation}\label{e3.2}
\frac{T_n^{\prime}(t)}{n^2}=-\frac{3t\,T_n(t)}{n^2-1-(n^2+2)t^2}\,.
\end{equation}
Let us point out that \eqref{e3.1} implies $n^2-1-(n^2+2)t^2>0$,
since
$$
t^2<t<1-\frac{8}{n^2}<1-\frac{3}{n^2+2}=\frac{n^2-1}{n^2+2}\,.
$$
Replacing $T_n^{\prime}(t)/n^2$ with the right-hand side of
\eqref{e3.2} in $\varphi_n(t)$, we obtain
\begin{equation*}
\begin{split}
\varphi_n(t)&=t+1+\Big(1+\frac{3t(2t+1)}{n^2-1-(n^2+2)t^2}\Big)\,T_n(t)\\
&=(t+1)\Big\{1+\frac{n^2-1-(n^2-4)t}{n^2-1-(n^2+2)t^2}\,T_n(t)\Big\}\\
&\geq (t+1)\,\Big\{1-\frac{n^2-1-(n^2-4)t}{n^2-1-(n^2+2)t^2}\Big\}
\end{split}
\end{equation*}
(we have used that the factor in front of $T_n(t)$ in the curly
brackets is positive and $T_n(t)\geq -1$). Hence, to prove
$\varphi_n(t)>0$ it suffices to show that
$$
1-\frac{n^2-1-(n^2-4)t}{n^2-1-(n^2+2)t^2}>0\,,
$$
which is equivalent to
$$
\frac{t\,\big[n^2-4-(n^2+2)t\big]}{n^2-1-(n^2+2)t^2}>0\,.
$$
This inequality is true, since the numerator in the left-hand side
is positive. Indeed, from \eqref{e3.1} we have
$$
t<1-\frac{8}{n^2}<1-\frac{6}{n^2+2}=\frac{n^2-4}{n^2+2}.
$$
The proof of Theorem~\ref{t2} is complete.\hfill $\Box$
\section{Proof of Theorem 1}
\setcounter{equation}{0} Recall that
$$
f_1(x)=T_n(x)+2-\frac{x+2}{n^2}\,T_n^{\prime}(x),
$$
and set
$$
f_3(x):=\frac{1}{n^2}\,(1-x)(n^2-T_n^{\prime}(x)),
$$
$$
F_a(x):=(1+a)f_1(x)-f_3(x)\,.
$$
Clearly, Theorem~\ref{t1} is equivalent to the following statement:
\begin{theorem}\label{t3}
Let $n\in \mathbb{N}$, $n\geq 4$ and
\begin{equation}\label{e4.1}
a=\begin{cases}\cos\frac{\pi}{n},& \text{ if }\
n\ \text{ is even},\vspace*{1ex}\\
\cos\frac{2\pi}{n},&\text{ if }\ n \ \text{ is odd}.
\end{cases}
\end{equation}
Then
\begin{equation}\label{e4.2}
F_a(x)\geq 0\,,\qquad x\in [-1,1]\,.
\end{equation}
The equality in \eqref{e4.2} occurs only for $x=1$ and
$x=-\cos\frac{\pi}{n}$ if $n$ is even, and for $x=\pm 1$ and
$x=-\cos\frac{2\pi}{n}$ if $n$ is odd. For every constant $a$,
smaller than the one specified in \eqref{e4.1}, inequality
\eqref{e4.2} fails to hold.
\end{theorem}
\medskip\noindent{\bf Proof.} According to Robertson's inequality, $f_1(x)\geq 0$ in
$[-1,1]$, therefore if $a_1>a_2$, then $F_{a_1}(x)\geq F_{a_2}(x)$
for every $x\in [-1,1]$. In particular, for every $a>0$,
$$
F_a(x)\geq F_0(x)=T_n(x)+x+1-\frac{2x+1}{n^2}\,T_n^{\prime}(x)
=\varphi_n(x)\,,\qquad x\in [-1,1]\,.
$$
In view of Theorem~\ref{t2}, $\varphi_n(x)\geq 0$ for every $x\in
[0,1]$, with the equality holding only for $x=1$ and if $n=4k+2$,
for $x=0$. Therefore, with $a$ as given in \eqref{e4.1}, we have
$F_a(x)\geq \varphi_n(x)>0$ for every $x\in (0,1)$, and it remains
to prove inequality \eqref{e4.2} and clarify the cases of equality
only when $x\in [-1,0]$.
Let us denote by
$$
x_k=\cos\frac{k\pi}{n},\qquad k=0,\ldots,n\,,
$$
the zeros of $(1-x^2)T_n^{\prime}(x)$, and let $H(f;x)$ be the
Hermite interpolating polynomial, with interpolation nodes
$x_0,x_1,x_1,x_2,x_2,\ldots,x_{n-1},x_{n-1}, x_n$, for a
differentiable function $f$, i.e., $H(f;x)$ is determined by the
conditions
\begin{eqnarray*}
&&H(f;x_k)=f(x_k),\quad k=0,1,\ldots,n,\\
&&H^{\prime}(f;x_k)=f^{\prime}(x_k), \quad k=1,\ldots,n-1\,.
\end{eqnarray*}
A straightforward calculation shows that
\begin{equation}\label{e4.3}
\begin{split}
H(f;x)=&\frac{\big[T_n^{\prime}(x)\big]^2}{2n^4}\,\big[(1+x)\,f(x_0)
+(1-x)\,f(x_n)\big]\\
&+(1-x^2)\,\sum_{k=1}^{n-1} \frac{\ell_k^2(x)}{(1-x_k^2)^2}\,
\mathcal{L}_k(f;x)\,,
\end{split}
\end{equation}
where, for $k=1,\ldots, n-1$, $\ell_k$ are the Lagrange basis
polynomials for interpolation at the zeros of $T_n^{\prime}$,
$$
\ell_k(x)=\frac{T_n^{\prime}(x)}{(x-x_k)T_n^{\prime\prime}(x_k)},
$$
and
\begin{equation}\label{e4.4}
\mathcal{L}_k(f;x):=(1-x_k\,x)f(x_k)
+(1-x_k^2)(x-x_k)f^{\prime}(x_k)\,.
\end{equation}
It follows from the uniqueness of the Hermite interpolation
polynomial that $H(f;\cdot)\equiv f(\cdot)$ whenever $f$ is a
polynomial of degree at most $2n-1$, in particular,
$H(f_i;\cdot)\equiv f_i(\cdot)$ for $i=1,\,3$ and
$H(F_a;\cdot)\equiv F_a(\cdot)$\,.
From $T_n(x_k)=(-1)^k$, $0\leq k\leq n$, and $T_n^{\prime}(x_k)=0$,
$1\leq k\leq n-1$, $T_n^{\prime}(x_0)=n^2$,
$T_n^{\prime}(x_n)=(-1)^n\,n^2$, we find
\begin{equation}\label{e4.5}
\begin{split}
& f_1(x_0)=f_3(x_0)=0,\\
&f_1(x_k)=2+(-1)^k,\ \ f_3(x_k)=1-x_k,\quad 1\leq k\leq n-1;\\
&f_1(x_n)=f_3(x_n)=2(1+(-1)^n)\,.
\end{split}
\end{equation}
In particular, \eqref{e4.5} yields
$$
F_a(x_0)=0\,,\quad F_a(x_n)=2a\big[1+(-1)^n\big]\,,
$$
in agreement with the claim of Theorem~\ref{t3} that $F_a(x)$
vanishes at $x_0$ and if $n$ is odd, at $x_{n}$; in addition,
$F_a(x_n)>0$ if $n$ is even and $a>0$. Moreover, from
$F_a(\cdot)\equiv H(F_a;\cdot)$, \eqref{e4.3} and \eqref{e4.4} we
infer
\begin{equation}\label{e4.6}
F_a(x)=\frac{a\big[1\!+\!(\!-\!1)^{n}\big]}{n^4}\,
(1\!-\!x)\big[T_n^{\prime}(x)\big]^2
\!+\!(1\!-\!x^2)\,\sum_{k=1}^{n-1}\frac{\ell_k^2(x)}{(1\!-\!x_k^2)^2}
\,\mathcal{L}_k(F_a;x)\,.
\end{equation}
In order to find $\mathcal{L}_k(F_a;x)$, $1\leq k\leq n-1$\,, we
firstly evaluate $\mathcal{L}_k(f_i;x)$, $i=1,\,3$\,.
Making use of the differential equation \eqref{e2.2} and
$$
f_1^{\prime}(x)=\Big(1-\frac{1}{n^2}\Big)\,T_n^{\prime}(x)
-\frac{x+2}{n^2}\,T_n^{\prime\prime}(x), \quad
f_3^{\prime}(x)=\frac{1}{n^2}\,
\big[T_n^{\prime}(x)-(1-x)\,T_n^{\prime\prime}(x)\big]-1\,,
$$
we find
\begin{equation}\label{e4.7}
f_1^{\prime}(x_k)=(-1)^{k}\,\frac{x_k+2}{1-x_k^2}\,,\quad
f_3^{\prime}(x_k)=\frac{(-1)^{k}}{1+x_k}-1\,,\qquad
k=1,\ldots,n-1\,.
\end{equation}
From \eqref{e4.4}, \eqref{e4.5} and \eqref{e4.7} we obtain
$$
\mathcal{L}_k(f_1;x)=(1-x_k\,x)\big(2+(-1)^k\big)
+(-1)^{k}(x_k+2)(x-x_k)\,,
$$
or, equivalently,
\begin{equation}\label{e4.8}
\mathcal{L}_k(f_1;x)=
\begin{cases}(1-x_k)(3+2x+x_k),& k \text{ - even},
\vspace*{1ex}\\
(1+x_k)(1+x_k-2x),& k \text{ - odd}.
\end{cases}
\end{equation}
Likewise, we get
$$
\mathcal{L}_k(f_3;x)=(1-x_k\,x)(1-x_k)
+(x-x_k)\,\big[(-1)^{k}(1-x_k)-1+x_k^2\big]\,,
$$
which simplifies to
\begin{equation}\label{e4.9}
\mathcal{L}_k(f_3;x)=
\begin{cases}(1-x_k)(1+x_k^2-2x_k\,x),& k \text{ - even},
\vspace*{1ex}\\
(1-x_k^2)(1+x_k-2x),& k \text{ - odd}.
\end{cases}
\end{equation}
Now $\mathcal{L}_k(F_a;x)=(1+a)\,\mathcal{L}_k(f_1;x)
-\mathcal{L}_k(f_3;x)$, \eqref{e4.8} and \eqref{e4.9} yield
\begin{proposition}\label{pp2.5}
For $1\leq k\leq n-1$, we have
$$
\mathcal{L}_k(F_a;x)=
\begin{cases}
(1-x_k)\big[2(1+x)(1+a+x_k)+(a-x_k)(1+x_k)\big],& k \text{ - even},
\vspace*{1ex}\\
(a+x_k)(1+x_k)(1+x_k-2x),& k \text{ - odd}.
\end{cases}
$$
\end{proposition}
Let us note that \eqref{e4.6} and Proposition~\ref{pp2.5} hold for
an arbitrary constant $a$. We show below that, with $a$ as given in
\eqref{e4.1}, $\mathcal{L}_k(F_a;x)>0$ for $x\in (-1,0]$, except for
a specific index $k$. When $k$ is even, we prove even more.
\begin{proposition}\label{pp3}
If $n\ge 4$ is even, $k$ is even, $1<k< n-1$, and $a=x_1$, then
$$
\mathcal{L}_k(F_a;x)>0,\qquad -1\leq x\leq 1\,.
$$
\end{proposition}
\medskip\noindent{\bf Proof.} In this case $a-x_k=x_1-x_k>0$, and it follows from
Proposition~\ref{pp2.5} that
$$
\mathcal{L}_k(F_a;x)> 2(1+x)(1-x_k)(1+a+x_k)\geq 0,\qquad -1\leq
x\leq 1\,.
$$
\hfill $\Box$
\begin{proposition}\label{pp4}
If $n\geq 4$, $k$ is odd, $1\leq k\leq n-1$, and $a$ is given by
\eqref{e4.1}, then
\begin{equation*}
\mathcal{L}_k(F_a;x)\begin{cases} >0, & -1<x\leq 0,\quad 1\leq
k<n-2\,, \vspace*{1ex}\\
\equiv 0,& k=n-2,\ \ n\ \text{ - odd},\vspace*{1ex}\\
\equiv 0,& k=n-1,\ \ n\ \text{ - even}\,.
\end{cases}
\end{equation*}
\end{proposition}
\medskip\noindent{\bf Proof.} If $n$ is even, then $a=x_1=-x_{n-1}$, thus $a+x_{n-1}=0$ and
$\mathcal{L}_{n-1}(F_a;x)\equiv 0$ in view of
Proposition~\ref{pp2.5}. If $n$ is odd, then $a=x_2=-x_{n-2}$ and
$a+x_{n-2}=0$, which by Proposition~\ref{pp2.5} implies
$\mathcal{L}_{n-2}(F_a;x)\equiv 0$. Finally, if $k<n-2$ is odd, then
$a+x_k>a+x_{n-2}\geq x_2+x_{n-2}=0$, and from
Proposition~\ref{pp2.5} we infer
$$
\mathcal{L}_k(F_a;x)=(a+x_k)(1+x_k)(1+x_k-2x)>0, \qquad -1\leq x<
\frac{1+x_{n-3}}{2}.
$$
Since $(1+x_{n-3})/2>0$, it follows that $\mathcal{L}_k(F_a;x)>0$
for every $x\in [-1,0]$. \hfill $\Box$
\medskip
We are ready to accomplish the proof of the claim of
Theorem~\ref{t3} in the case $x\in [-1,0]$.\smallskip
Assume first that $n\geq 4$ is even. If $a=x_1$, then \eqref{e4.6}
and Proposition~\ref{pp4} imply
\begin{equation}\label{e4.10}
F_{x_1}(x)=\frac{2x_1}{n^4}\,(1-x)\big[T_n^{\prime}(x)\big]^2 +
(1-x^2)\,\sum_{k=1}^{n-2}\frac{\ell_k^2(x)}{(1-x_k^2)^2}
\,\mathcal{L}_k(F_a;x)\,.
\end{equation}
Since $\ell_k(x_{n-1})=0$ for $1\leq k\leq n-2$ and
$T_n^{\prime}(x_{n-1})=0$, it follows that
$F_{x_1}(x_{n-1})=F_{x_1}(-x_1)=0$. If, on the other hand, $x\in
(-1,0]$ and $x\ne x_{n-1}$, then all the summands in the sum in the
right-hand side of \eqref{e4.10} are non-negative, by virtue of
Propositions~\ref{pp3} and \ref{pp4}, with at least one of them
strictly positive, therefore $F_{x_1}(x)>0$ in this case. If
$a<x_1$, then from \eqref{e4.6} and Proposition~\ref{pp2.5} we
obtain
\begin{equation*}
\begin{split}
F_a(x_{n-1})&=(1-x_{n-1}^2)\,\sum_{k=1}^{n-1}
\frac{\ell_k^2(x_{n-1})}{(1-x_k^2)^2} \,\mathcal{L}_k(F_a;x_{n-1})\\
&=\frac{\mathcal{L}_{n-1}(F_a;x_{n-1})}{1-x_{n-1}^2}
=a+x_{n-1}<x_1+x_{n-1}=0\,,
\end{split}
\end{equation*}
showing that the inequality $F_a(x)\geq 0$ fails to hold for
$x=x_{n-1}$.\smallskip
Now assume that $n\geq 5$ is odd. If $a=x_2$, then \eqref{e4.6} and
Proposition~\ref{pp4} imply
\begin{equation}\label{e4.11}
F_{x_2}(x)=
(1\!-\!x^2)\,\Big[\sum_{k=1}^{n-3}\frac{\ell_k^2(x)}{(1-x_k^2)^2}
\,\mathcal{L}_k(F_a;x)\!+\!\frac{\ell_{n-1}^2(x)}{(1-x_{n-1}^2)^2}
\,\mathcal{L}_{n-1}(F_a;x)\Big]\,.
\end{equation}
Since $\ell_k(x_{n-2})=0$ for $k\ne n-2$, we have
$F_{x_2}(x_{n-2})=0$. If $x\in (-1,0]$, $x\ne x_{n-2}$, then, in
view of Propositions~\ref{pp3} and \ref{pp4}, all the summands in
the brackets in \eqref{e4.11} are non-negative, and at least one of
them is strictly positive, therefore $F_{x_2}(x)>0$. Finally, if
$a<x_2$, then from \eqref{e4.6} and Proposition~\ref{pp2.5} we find
\begin{equation*}
\begin{split}
F_a(x_{n-2})&=(1-x_{n-2}^2)\,\sum_{k=1}^{n-1}
\frac{\ell_k^2(x_{n-2})}{(1-x_k^2)^2} \,\mathcal{L}_k(F_a;x_{n-1})\\
&=\frac{\mathcal{L}_{n-2}(F_a;x_{n-2})}{1-x_{n-2}^2}
=a+x_{n-2}<x_2+x_{n-2}=0\,,
\end{split}
\end{equation*}
i.e., $F_a(x_{n-2})<0$ if $a<x_2$.
The proof of Theorem~\ref{t1} is complete.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 850
|
webeasties
Immune Response from Start to Finish: Part 2
By kbonham on November 2, 2010.
[I've been hooked on the immune system since I was a kid and my dad showed me electron micrographs of macrophages eating bacteria in Scientific American. Now that I'm in graduate school studying immunology, and macrophages in particular, my dad asked if I could give a play-by-play of an immune response. Here you go Dad:]
Part 2: T-cells, B-cells and adaptive immunity
If you've ever had the flu (and I mean for real influenza, not some sissy man-flu), you know how much it sucks. But don't blame the virus. Many of the most unpleasant symptoms - extreme fatigue, snot-filled sinuses, and high fever - are all a result of your immune system trying to kill that nasty infection (ok, I guess you can blame the virus). In part 1, I described how the innate immune system usually blocks bugs from getting in, or kills them quite rapidly. That fever is a result of all the cytokines released by the macrophages and other immune cells (higher body temperatures are thought to speed up the immune response or make the environment less hospitable to the pathogens). That snot is mostly comprised of mucous secreted by the inflamed tissues of the nose, and dead neutrophils that swarmed in kamikazi-style to gobble up whatever bacteria or virus they could find. That fatigue is an attempt to conserve energy that might be needed to fight the infection. Most of the time, the innate immune system does a pretty good job on its own. But if you get to the point where you can't breathe through your nose, it's a struggle just to sit up in bed and you could fry an egg on your stomach, your innate immune system just isn't enough. That's where the adaptive immune system comes in.
Dendritic cells: the tissue's messengers
Most tissues in your body contain specialized cells called dendritic cells (DC's) which form the bridge between the innate and the adaptive immune system. DC's are constantly sampling their immediate environment through a process called macropinocytosis - which means literally "big drinking." They are constantly enveloping fluid and molecules with their cell membranes and drawing it in to process and digest it. Most of the time, the DC doesn't see anything particularly surprising - mostly debris from dead or dying cells, secreted hormones and other signaling molecules, and extracellular matrix proteins. But during an infection, the DC will see a PAMP (that's pathogen-associated molecular pattern that I talked about last time), or get a signal from cytokines released by another immune cell (like a macrophage) that saw one. In either case, the DC becomes activated, and stops macropinocytosis, essentially preserving a snap-shot of the local molecular environment.
Lymphatics are the immune system's highway
The next stage of the immune response occurs when the DC picks up stakes and heads for a lymph node. The lymphatic system is not something most people think of, but it's essential for the immune system. It's almost like a parallel circulatory system, but it carries tissue fluid around the body instead of blood. It also doesn't have a pump like the heart, but is instead moved around by normal body movement (this is why people confined to beds need to be continually moved or massaged, otherwise fluid pools in the extremities). Anyway, DC's travel along the lymphatics until they find the nearest lymph node. This is where the magic happens.
T-cells and B-cells
Lymph nodes are the home of specialized cells called T-cells and B-cells. T- and B-cells undergo a unique process in which their DNA at a particular location is cut up and scrambled to generate a receptor that is completely unique, and can be almost infinitely diverse. As a result, T-cell receptors (TCR) and B-cell receptors (BCR) are capable of recognizing just about anything, but each individual cell has a unique receptor that is incredibly specific. In other words, in contrast to a macrophage, which might recognize all bacteria in a very general way, an individual T-cell might only recognize a single strain of a single species of bacteria, while another T-cell would be blind to that strain, but recognize another related strain. When the dendritic cell from an infected tissue arrives at the lymph node, carrying with it the bits of microbe (called antigens) that it found when it was activated, most T-cells and B-cells will ignore it. Because of the huge diversity in their receptors, the chance that any given T-cell or B-cell will recognize something from a particular bug is quite small.
But there are a lot of these cells living in the lymph node, and if even one is able to see that DC and its cargo, it goes nuts. Combining signals that it receives through its own receptor, as well and signals given off by the dendritic cell, the T-cell or B-cell will start to proliferate, cloning itself (and the unique receptor it has) thousands or hundreds of thousands of times. As they proliferate, B-cells start to randomly modify their receptor, and undergo a mini process of natural selection. Most of the modifications are useless or even destructive, and cause the B-cell to decrease in its ability to recognize the antigen. When this happens, the B-cell can't receive the activating signals any more and will die off. As with evolution, however, some of these changes will be beneficial, and increase the B-cell's recognition. These lucky B-cells will get stronger signals, and will proliferate more. Eventually, B-cells will turn their receptor (that's bound to the surface of the cell) into a secreted form called an antibody, which they produce in huge numbers. All these antibodies will head to the site of infection and patrol the blood stream, glomming onto the pathogen wherever it's found, neutralizing them (in the case of viruses and toxins) or making it easier for macrophages to see and eat them (in the case of larger things like bacteria). Antibodies can also cause complement fixation, which punches holes in bacteria and causes them to explode.
Antibodies are great for stuff that's happening outside of your cells, but many infections (including ALL virus infections) happen inside cells where antibodies can't get to. That's where T-cells come in (there are lots of different T-cells that do lots of different things, but I'm only going to focus on "killer" T-cells). Cells are constantly presenting bits of the proteins they express on their cell surface for T-cells to look at. Under normal conditions, T-cells will never see anything other than normal, self proteins, and nothing will happen. But in the case of a virus infection, for instance, viral proteins will also be presented. When a T-cell gets activated in the lymph node, it will head into the blood stream and then migrate to the site of the infection. When it gets there, any cells that are expressing foreign proteins that the T-cell recognizes will trigger the T-cell to release signals that force the infected cell to commit suicide. Any pathogens that survive will hopefully get mopped up by antibodies.
This whole process, from infection, to T- and B-cell activation, to clearance of the pathogen and resolution of the immune response usually takes 1-2 weeks. If you have a cold, you may only have symptoms for a couple days but that's only the part that you notice. In reality, your immune system spends a lot of time working behind the scenes to keep you free and clear. In the next section, I'll talk about immunological memory, vaccines, and the pesky problem of persistent pathogens.
Immune response from start to finish series
Part 1: Invasion and detection: Innate immunity
Part 2: T-cells, B-cells and adaptive immunity (current)
Part 3: Immune Memory
Implications of the Immune Response
[NOTE: Because this post is now a magnet for spam, commenting has been closed. If you want to leave a comment, please send it to me via e-mail: webeastiesblog (at) gmail.com]
[I've been hooked on the immune system since I was a kid and my dad showed me electron micrographs of macrophages eating bacteria in Scientific American. Now that I'm in graduate school studying immunology, and macrophages in particular, my dad asked if I could give a play-by-play of an immune…
A Bitter Sweet Nobel - Beutler, Janeway, and the Dawn of Innate Immunity
Monday's announcement for the Nobel Prize in physiology or medicine should have been a happy occasion for my lab. On the one hand, it was given for early discoveries in the field of innate immunity - my field! On the other hand, it was given to a scientist that many* feel is undeserving of the…
Meet the new white cell, same as the old white cell
The field of immunology has a few quirks. I'm sure this is no different than other fields of study, but one of the most puzzling (and sometimes infuriating) of these quirks is an obsession with categorizing different types of cells. Case in point, a recent paper in Nature Immunology: A semi-…
By Sherry Austin (not verified) on 02 Nov 2010 #permalink
This is an awesome blog. I happened to stumble upon it and I am very impressed. I can't wait to read the rest of your work and plan to check out the other SINT items you listed.
By Mage (not verified) on 05 Nov 2010 #permalink
We, Beasties Sporulates
A little over 4 years ago, I joined up with three friends from grad school and launched a brand new science blog, "We, Beasties!" The name was meant to be a play on a phrase from Paul de Kruif's somewhat tongue-in-cheak translation of the first-ever microbiologist Antonie von Leeuwenhoek's term "…
Anti-science is not a state of mind
Can you be skeptical about GM but believe in climate change? So asks Alice Bell in The Guardian. The answer is of course, "Yes," but you can also be a fundamentalist Christian while believing in evolution and being a great scientist, so being able to hold two things in your brain at the same time…
Living Factories: How Scientists Engineer Microbes to Make Useful Molecules
A couple of weeks ago, I gave a talk for Tonight, I'm presenting at the Science In The News (SITN) Spring lecture series. If you're in the vicinity of Boston, you can come watch at 7pm in Pfizer Auditorium, located in the Mallinckrodt Chemistry Lab, 12 Oxford Street, Cambridge MA 02138. If you can…
UCS response to my piece in SciAm
On Thursday, I had a post published on Scientific American's guest Blog about claims that genetically modified food crops could contain allergens. In it, I am critical of the Union of Concerned Scientists (a science advocacy and policy organization), for what I read as misplaced opposition to…
Science, Racism and Political Correctness
Two weeks ago, the Heritage Foundation (a conservative think-tank) released a position paper based largely on the academic research of one Jason Richwine. The conclusion (roughly paraphrased): Hispanic people have lower IQ's than white people, so an overly permissive immigration policy will drag…
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,954
|
Scene::Scene(UINT width, UINT height, std::wstring name)
: DXSample(width, height, name)
{
}
void Scene::onInit()
{
this->udpateTitle();
auto instanceData = this->makeInstanceData();
{//ƒOƒ‰ƒtƒBƒbƒNƒXƒpƒCƒvƒ‰ƒCƒ"'̏‰Šú‰»
std::vector<char> byteCode;
createShader(this->mpVertexShader.GetAddressOf(), this->mpDevice.Get(), "VertexShader.cso", &byteCode);
//"ü—̓ŒƒCƒAƒEƒg'̍쐬
std::array<D3D11_INPUT_ELEMENT_DESC, 1> elements = { {
{ "POSITION", 0, DXGI_FORMAT_R32G32B32_FLOAT, 0, 0, D3D11_INPUT_PER_VERTEX_DATA, 0 },
} };
auto hr = this->mpDevice->CreateInputLayout(elements.data(), static_cast<UINT>(elements.size()), byteCode.data(), byteCode.size(), this->mpInputLayout.GetAddressOf());
if (FAILED(hr)) {
throw std::runtime_error(""ü—̓ŒƒCƒAƒEƒg'̍쐬'ÉŽ¸"s");
}
createShader(this->mpVSDummy.GetAddressOf(), this->mpDevice.Get(), "VSDummy.cso", &byteCode);
//ƒWƒIƒƒgƒŠƒVƒF[ƒ_
createShader(this->mpGeometryShader.GetAddressOf(), this->mpDevice.Get(), "GeometryShader.cso", &byteCode);
createShader(this->mpGeometryShader2.GetAddressOf(), this->mpDevice.Get(), "GeometryShader2.cso", &byteCode);
//ƒsƒNƒZƒ‹ƒVƒF[ƒ_
createShader(this->mpPixelShader.GetAddressOf(), this->mpDevice.Get(), "PixelShader.cso", &byteCode);
}
{//IA—p'̃oƒbƒtƒ@ì¬
std::array<Vertex, 3> data = { {
{ { 0.0f, 0.5f, 0 } },
{ { 0.5f, -0.5f, 0 } },
{ { -0.5f, -0.5f, 0 } },
} };
CreateIABuffer(this->mpVertexBuffer.GetAddressOf(), this->mpDevice.Get(), static_cast<UINT>(data.size()), data.data(), D3D11_BIND_VERTEX_BUFFER);
std::array<uint16_t, 6> indices = { {
0, 1, 2, 3, 4, 5
} };
CreateIABuffer(this->mpIndexBuffer.GetAddressOf(), this->mpDevice.Get(), static_cast<UINT>(indices.size()), indices.data(), D3D11_BIND_INDEX_BUFFER);
}
}
void Scene::onUpdate()
{
}
void Scene::onKeyUp(UINT8 key)
{
if (key == 'Z') {
this->mMode = static_cast<decltype(this->mMode)>((this->mMode + 1) % eMODE_COUNT);
this->udpateTitle();
}
}
void Scene::onRender()
{
//GPU'É•K—v'ȃf[ƒ^'ðÝ'è'·'é
//"ü—̓AƒZƒ"ƒuƒ‰ƒXƒe[ƒW
std::array<ID3D11Buffer*, 1> ppBufs = { {
this->mpVertexBuffer.Get(),
} };
std::array<UINT, 1> strides = { { sizeof(Vertex) } };
std::array<UINT, 1> offsets = { { 0 } };
this->mpImmediateContext->IASetVertexBuffers(0, static_cast<UINT>(ppBufs.size()), ppBufs.data(), strides.data(), offsets.data());
this->mpImmediateContext->IASetIndexBuffer(this->mpIndexBuffer.Get(), DXGI_FORMAT_R16_UINT, 0);
this->mpImmediateContext->IASetPrimitiveTopology(D3D11_PRIMITIVE_TOPOLOGY_POINTLIST);
this->mpImmediateContext->IASetInputLayout(this->mpInputLayout.Get());
//'¸"_ƒVƒF[ƒ_
this->mpImmediateContext->VSSetShader(this->mpVertexShader.Get(), nullptr, 0);
//ƒsƒNƒZƒ‹ƒVƒF[ƒ_
this->mpImmediateContext->PSSetShader(this->mpPixelShader.Get(), nullptr, 0);
switch (this->mMode) {
case eMODE_POINT_TO_TRIANGLE:
this->mpImmediateContext->GSSetShader(this->mpGeometryShader.Get(), nullptr, 0);
this->mpImmediateContext->Draw(3, 0);
break;
case eMODE_NO_INPUT_TO_TRIANGLE:
this->mpImmediateContext->IASetInputLayout(nullptr);
this->mpImmediateContext->IASetVertexBuffers(0, 0, nullptr, nullptr, nullptr);
this->mpImmediateContext->VSSetShader(this->mpVSDummy.Get(), nullptr, 0);
this->mpImmediateContext->GSSetShader(this->mpGeometryShader2.Get(), nullptr, 0);
this->mpImmediateContext->Draw(1, 0);
break;
default:
assert(false);
}
}
void Scene::onDestroy()
{
}
void Scene::udpateTitle()
{
std::wstring title = L"";
switch (this->mMode) {
case eMODE_POINT_TO_TRIANGLE: title += L"eMODE_POINT_TO_TRIANGLE"; break;
case eMODE_NO_INPUT_TO_TRIANGLE: title += L"eMODE_NO_INPUT_TO_TRIANGLE"; break;
default:
assert(false && "–¢ŽÀ'•");
}
this->setCustomWindowText(title.c_str());
}
std::vector<Scene::InstancedParam> Scene::makeInstanceData()const
{
int i = 0;
float work = 0.f;
auto calPos = [&i, &work, this]() {
const float start = -0.5f;
const float interval = 1.f / this->M_INSTANCED_COUNT;
auto r = start + work;
work += interval;
return DirectX::SimpleMath::Vector3(r, 0, ((i++ % 2) == 0) ? 0 : 0.5f);
};
std::vector<InstancedParam> data;
data.resize(this->M_INSTANCED_COUNT);
data.at(0).offset = calPos();
data.at(0).color = DirectX::SimpleMath::Vector4(1, 0, 0, 1);
data.at(1).offset = calPos();
data.at(1).color = DirectX::SimpleMath::Vector4(0, 1, 0, 1);
data.at(2).offset = calPos();
data.at(2).color = DirectX::SimpleMath::Vector4(0, 0, 1, 1);
data.at(3).offset = calPos();
data.at(3).color = DirectX::SimpleMath::Vector4(1, 1, 0, 1);
data.at(4).offset = calPos();
data.at(4).color = DirectX::SimpleMath::Vector4(0, 1, 1, 1);
return data;
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,489
|
Whether you recognize the term or not—you've all seen and many of you have created gated content. That term refers to putting something on your website or landing page that people want and asking them for information in exchange for that information.
In most cases, you are asking people for their email address and allowing them to pass through the "gate" to a hidden web page where they can download the information that you're offering. An alternative is that you would email them the information once you have their email address.
Every field is another barrier you are asking the person to knock down to get to your content. You are asking them to work harder and risk their anonymity with every field. They assume, for example, that if they include a phone number—you're going to call. They have to decide if what you're offering is worth that intrusion.
It makes sense that you would want to build a database of people who genuinely have some interest in your company or offerings. And on the surface, it makes sense that you would reduce the number of fields so more people actually finish the task and get the item you are offering in exchange.
But there's a weird inverse relationship in this kind of marketing. As the number of people who complete the form increases, the amount of information you have on them and the more you can confidently say they are a qualified lead diminishes. Why? More hurdles to leap mean you are in essence, testing the audience to see how badly they want what you're offering.
Remember, the more you ask them in the form, the fewer completions you will get. But the fewer things you ask, the less you know about your leads.
So when considering whether or not you should create gated content, the first question you need to ask yourself is "why are we doing this?" If you are looking for qualified sales leads then you should actually use more fields. You will get a smaller group of people who actually complete the form and access your information, but you'll know they really want it. You will also have gathered enough information about them to get a sense of how strong a lead they are.
On the flip side, if you are just trying to build up your database so you can keep marketing to the audience, then reduce the number of fields to increase participation. But you have to accept that many of the people, especially if you just require an email address and nothing more, may not have much actual interest in your product or service.
The length of your form not only reflects the value of what you're offering but it also reflects the genuine interest of the prospect. A shorter form will get you a larger database that you can market to down the road. But there will be a lot of tire kickers on that list. A longer form that tells you more about the prospect and what they're interested in. The additional questions will reduce the size of the database but will increase the likelihood of a genuine potential sale being among them.
So as always with marketing—start with why.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 735
|
\section{Introduction}
Most classical Cepheids and RR Lyrae are single-mode periodic and radial pulsators. For many years, the only exception known from this simple understanding was a long-term lightcurve modulation, discovered by and named after \citet{Blazhko1907}, found among some RR Lyrae. With better quality of photometry, it was found that various modes of multi-mode pulsations exist within these variables. For an introduction to these oscillations, see \citet{Moskalik2014}. In brief, there are several relevant classes:
First, there are F+1O (fundamental and first overtone) double-mode Cepheids \citep{Osterhoff1957a,Osterhoff1957b} with periods ratios $P_{2}/P_{1}$=0.694 -- 0.746. Several hundreds of these are known within our galaxy and the Magellanic Clouds (e.g. \citet{Sos2008b,Soszynski2010a,Soszynski2011b,Soszynski2012,Marquette2009,Smolec2010}), and in M31 \citep{Poleski2013a} and M33 \citep{Beaulieu2006}.
Second, many Cepheids are known with a period ratio of $P_{2}/P_{1}$=0.80 -- 0.802, known as 1O+2O (first and second overtone) double-mode Cepheids. Since the discovery by \citet{Mantegazza1992}, about 500 such stars have been found in the Magellanic Clouds \citep{Sos2008b,Soszynski2010a,Soszynski2012,Marquette2009} and only 19 within our galaxy \citep{Smolec2010,Soszynski2011b}. This implies a lower occurrence rate in our galaxy, perhaps due to the higher metallicity within our galaxy.
Third, there are F+1O double-mode RR Lyrae stars (RRd). Since the first discovery by \citet{Jerzykiewicz1977}, ~2,000 RRd stars have been found in the Magellanic Clouds
\citep{Soszynski2009,Soszynski2010b,Soszynski2012} as well as in our galaxy and in many globular clusters \citep{Wils2010,Poleski2013b}. This class generally clusters around a period ratio $P_{2}/P_{1}$=0.742 -- 0.748, with a few exceptions of stars with high metallicity, and a ratio as low as 0.726 \citep{Soszynski2011a}.
Fourth, a very few examples of triple mode pulsators among Cepheids and RR Lyrae have been found \citep{Sos2008a}. The most frequent sub-class are F+2O RRd Lyrae-type stars. Nine such stars are known, with period ratios $P_{3}/P_{1}$ = 0.582 -- 0.593 (see \citet{Moskalik2013} and references therein).
Fifth, with precise time-series photometry such as OGLE and Kepler, low-amplitude (a few percent) secondary pulsations have been found, commonly believed to originate from non-radial modes. These modulations occur among Cepheids and RR Lyrae in the range $P_{2}/P_{1}$=0.60 -- 0.64 (\citet{Moskalik2013,Moskalik2014} and references therein).
Differences of period ratios within one group are attributed to differences in metallicity. As metals in stars are present in various compositions and amounts, period ratios are not believed to exist in discrete steps, but in continuous distributions \citep{Buchler2007}. In this paper, we argue that there might be one sub-class of F+1O Cepheids and RR Lyrae that show period ratios in small, discrete steps, very close to $P_{2}/P_{1}=1/\sqrt{2}$. One might wonder, and rightly so, how stars should be able to calculate a square root, obey the calculations and adjust their pulsations accordingly. Let us elaborate.
\section{Motivation and method}
\subsection{Motivation}
Simple dynamical non-linear models can reproduce the frequency content and the asymmetry of variable star light curves well \citep{Hippke2014}. These models generally involve irrational numbers, usually the \textit{golden ratio}, the most irrational number (as it has the slowest continued fraction expansion convergence of any irrational number) and $\sqrt{2}$, sometimes coined \textit{Pythagoras' constant}, the first number to be proven irrational. This has inspired us to search for double-mode variable stars with the same period ratio.
\subsection{Search method}
We have used the largest available variable star catalogs, namely OGLE-II and OGLE-III, EROS \citep{Afonso1999}, and MACHO \citep{Alcock1995}. At the time of writing, OGLE-IV and the Catalina survey have not yet published their data. These surveys are publicly available and publish all found and significant period ratios in their catalogs. Figure~\ref{fig:petersen}) shows all OGLE data. From a literature review, we have collected data for multiperiodic stars in M31 and M33. We have also retrieved the timeseries photometry for all candidate stars and recalculated the frequencies.
\section{Results}
\subsection{Search results}
We have found no star with a period ratio equaling the golden ratio (see Figure~\ref{fig:petersen}). This number splits two populations of RR Lyrae stars, as shown in a kernel density estimate (Figure~\ref{fig:kernelgolden}). We have chosen the kernel density estimate \citep{Rosenblatt1956,Parzen1962} instead of a histogram to avoid bin size decisions due to the low number of stars. If real, this split could possibly indicate some sort of watershed in convergence towards and from this irrational period ratio. However, the split is not exact, and it has been argued \citep{Dziembowski2012} that the period ratio clustering among these RR Lyrae is caused by an f-mode instability with high angular degree of \textit{l}=42, 46, 52. The location of these stars above and below the golden ratio could therefore be a simple coincidence, and further modeling will be required to decide this question. We have searched for a fine structure in the stars in this region, but found nothing.
\begin{figure}
\resizebox{\hsize}{!}
{\includegraphics{f1.pdf}}
\caption{Petersen diagram of main pulsation period versus period ratio for all OGLE-III stars in this range. Crosses and dots represent Cepheids and RR Lyrae, respectively. The skewed lines indicate stability limits for the inclosed Cepheid population \citep{Buchler2007}. The four diamond stars labeled with their catalog designations are closest to the period ratio of $1/\sqrt{2}$ (upper horizontal line). The lower horizontal line represents the golden ratio.}
\label{fig:petersen}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}
{\includegraphics{f7.pdf}}
\caption{Kernel density estimate for the OGLE stars betwen 0.595 and 0.650 with the golden ratio shown as the dotted vertical line.}
\label{fig:kernelgolden}
\end{figure}
We then focused our search on $1/\sqrt{2}$, the region of F/1O Cepheids and RR Lyrae. From an extensive literature review, we have only three papers listing stars in the vicinity. The first summarizes the Cepheids in our galaxy (see \cite{Wils2004} and references therein). We have obtained their original raw data \citep{Welch2004}, which is of fair to low quality -- in one case (DZ CMa) only 28 data points were retrieved, which does not allow for a stringent determination of the secondary pulsation. While the uncertainties in the period ratio might be large in these data, one could assume that they cancel out in a distribution graph. In Figure~\ref{fig:kernel}, we have plotted a kernel density estimate for these 20 F/1O Cepheids, and it seems that their center is indeed $1/\sqrt{2}$. This would mean that there is a class of Cepheids (and possibly RR Lyrae) which clusters at this period ratio, and is more abundant in our own galaxy. This is possibly an effect of higher metallicity within our galaxy, as compared to the Magellanic Clouds.
\begin{figure}
\resizebox{\hsize}{!}
{\includegraphics{f4.pdf}}
\caption{Solid line: kernel density estimate for the 20 stars towards the galactic bulge \citep{Welch2004}. These stars seem to center around $1/\sqrt{2}$, shown by the vertical dotted line. Dashed line: kernel density estimate for the 17 stars in M31 \citep{Lee2013}. These have the same center, but seem to spare the exact value. When using alternative kernels (e.g. gaussian instead of epanechnikov), the effect can also be seen in the galactic bulge stars.}
\label{fig:kernel}
\end{figure}
The second paper presents 5 Cepheids in M33 \citep{Beaulieu2006}, but also suffers from a low number of only 33 measurements. The third paper shows 17 Cepheids in M31 \citep{Lee2013}, and one star with a period ratio of 0.706 caught our attention. Closer examination, however, reveals that this ratio could be as low as 0.699, when using the additional I-band photometry, bringing it further away from $1/\sqrt{2}$. We have thus excluded these data from Figure~\ref{fig:petersen}, but also show the kernel density in Figure~\ref{fig:kernel}. It seems that these also cluster around $1/\sqrt{2}$, however spare the exact number.
The next step was the analysis of the few hundred Cepheids and RR Lyrae from the OGLE catalogs. At first glance, our search for stars with a period ratio of $\sqrt{2}$ has also delivered no hit, meaning that no star possesses this exact ratio, within its errors. However, we have discovered a few stars that come very close and have analyzed them in more detail. These come from the OGLE catalogs, as MACHO and EROS have no stars in the vicinity with period ratios lower than 0.712.
\subsection{Closest candidates}
We have found one RRd Lyrae and four Cepheids (of F/1O type) in the very closest vicinity of the period ratio $P_{2}/P_{1}$ = $1/\sqrt{2}$. Their main pulsation period $P_{1}$ spans a wide range of 0.3 days for the shortest RR Lyrae pulsator and 7.9 days for the longest Cepheid period (see Table~\ref{tab:table1}. When plotting $P_{1}$ versus the period ratio, $P_{2}/P{1}$ (see Figure~\ref{fig:spacing}), there is apparently a fine structure present. We define the deviation from (half) square root two as $D_{X}=(P_{2}/P_{1}) - 1/\sqrt{2}$. The values for $D_{X}$ range from $+0.000391(2)$ (BLG-RRLYR-09117, $0.06\%$) to $-0.001959(5)$ (BLG-T2CEP-209, $0.3\%$). When using LMC-CEP-1717 as a reference ($1D_{1}$), we find BLG-RRLYR-09117 having $-1.004 D_{1}$, LMC-CEP-1082 as $-0.991 D_{1}$, EY Car as $-0.965 D_{1}$ and BLG-T2CEP-209 showing $+5.026 D_{1}$.
\begin{figure}
\resizebox{\hsize}{!}
{\includegraphics{f3.pdf}}
\caption{Spacings of the stars with period ratios close to $1/\sqrt{2}$. Symbol sizes approximate uncertainties. Dotted horizontal lines in the upper half of the figure indicate integer multiples of the suggested spacing $D_{1} \approx 0.000390$.}
\label{fig:spacing}
\end{figure}
While the proximity of the period ratios to $\sqrt{2}$ seems solid, since the errors are small compared to the spacing, we must be clear that the fine structure is at best suggestive at this time. In the following, we will analyze error estimates and potential data glitches.
\begin{table*}
\caption{Candidate stars\label{tab:table1}}
\begin{tabular}{lcccrl}
\hline
Designation & $P_{1}$ (days)&$P_{2}$ (days)&$R=P_{2}/P_{1}$&$D_{x}=R-1/\sqrt{2}$&Remark \\
\hline
LMC-CEP-1717\tablefootmark{a} & 3.32153 & 2.34738 & 0.7067169 & $-0.000389(3)$ & reference = $1D_{1}$\\
BLG-RRLYR-09117\tablefootmark{a} & 0.30968 & 0.21910 & 0.7074980 & $+0.000391(2)$ & $-1.004 D_{1}$\\
LMC-CEP-1082\tablefootmark{a} & 7.86600 & 5.56514 & 0.7074930 & $+0.000386$ & $-0.991 D_{1}$, see text\\
BLG-T2CEP-209\tablefootmark{b} & 1.18128 & 0.83298 & 0.7051480 & $-0.001959(5)$ & $+5.026 D_{1}$\\
EY Car\tablefootmark{c} & 2.87602 & 2.03481 & 0.7075108 & $+0.000404(8)$ & $-0.965 D_{1}$\\
\hline
\end{tabular}
\tablefoottext{a}{OGLE II+III+IV data}
\tablefoottext{b}{OGLE III data}
\tablefoottext{c}{5 sources, see text}
\end{table*}
\section{Discussion}
\subsection{Data quality}
As with any data, the available photometry has limited precision, which results in statistical uncertainties. To get an impression of the data, we refer to Figure~\ref{fig:fold} for an exemplary phase fold of BLG-RRLYR-09117. Figure~\ref{fig:09117kernel} shows the accompanying Lomb-Scargle periodigrams for $P_{1}$ and $P_{2}$.
For the frequency analysis using Fourier Transforms, \textsc{Period04} \citep{Breger2004} offers a calculation based on the error matrix of a least-squares calculation, as well as a Monte Carlo Simulation. Using LMC-CEP-1082 as an example, both methods typically estimate uncertainties for $P_{1}$ at $6\times10^{-7}$ and for $P_{2}$ at $6\times10^{-6}$. When performing phase folds with such shifted frequencies, one can already visually see the breakdown of the phase fold as increased scatter. This means the estimated uncertainties are probably on the right order.
There is another potential way to derive uncertainties: directly from the period ratios. As described, we have several stars whose deviations are almost identical. For the closest match, BLG-RRLYR-09117 with the reference LMC-CEP-1717, we find $-1.004 D_{1}$. In case these period ratios were in fact identical, the measurement errors would be \textit{a posteriori} on the order of $1\times10^{-7}$.
Of course this assumes that the fine structure hypothesized is real.
\begin{figure*}
\centering
\includegraphics[width=0.49\textwidth]{f5a.pdf}
\includegraphics[width=0.49\textwidth]{f5b.pdf}
\caption{Phase fold for BLG-RRLYR-09117 to $P_{1}$=0.30968d (left) and $P_{1}$=0.21910d (right). The amplitude for $P_{1}$ is $\approx$ 0.45mag, for $P_{2}$ lower, $\approx 0.2$ mag. Error bars are not plotted, as the spread of points gives a much clearer indication of the true error of the 10,305 data points.\label{fig:fold}}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.49\textwidth]{f6a.pdf}
\includegraphics[width=0.49\textwidth]{f6b.pdf}
\caption{Lomb-Scargle periodograms for BLG-RRLYR-09117 focusing on $P_{1}$ (left) and $P_{2}$ (right, without pre-whitening). After removing the primary pulsation, $P_{2}$ is the most significant period. Note different vertical axis.\label{fig:09117kernel}}
\end{figure*}
\subsection{Potential errors}
The easiest (erroneous) explanation for discrete spacings could be the unavoidable quantization in spectral analysis algorithms. There are two main causes for this. The first is based on the simple fact that the algorithms use data types and -structures with a finite bit size, e.g. floating-points in 64-bits. The second comes from limited computing resources, requiring the use of a finite step size in a frequency search. We have analyzed both of these issues in detail.
For the data structures, it can easily be seen that time (e.g. MJD=1000.12345) and flux data (e.g. 10.12345 mag) fit into 64-bits floating points with ease. However, one does not always know which data types the algorithm of the chosen application uses. Also, the required calculations include mathematical functions (mainly sines and cosines) which the application might use via an external system call, and might be rounded there, introducing external error sources. To resolve the data structure issue, we have derived the OGLE photometry as raw data, and merged and reprocessed these using different applications and different methods (\textsc{PERIOD04}, NASA Periodogram Service, and the commercial application \textit{Origin}). We have also calculated all periods separately for OGLE II and OGLE III and compared the values to the OGLE II and III catalog references and found all values to be identical. For the merged data, our results are in between the separate OGLE II and OGLE III values for each dataset. This comes from the additional datapoints and longer time baseline which increase the precision of the measurements. All the results from different algorithms are identical within their errors (with the exception of LMC-Cep-1082, as will be discussed below).
One could still assume a potential data type issue being present in all applications we used. To check this, we have re-written a Lomb-Scargle algorithm in the completely open-source language Free Pascal, using the Lazarus environment. This has the advantage that the complete math unit including sines and cosines can be debugged and modified. By using smaller data types in the subunits (or overloading them with smaller data types), one can even introduce artificial rounding artefacts. With adequate use, we get the exact same results as described above, concluding that today's spectral analysis software does not produce discrete artefacts in period searches.
The second easy cause for such discretization errors could originate from the step size in a frequency search. We chose the same approach as above, with the advantage that the step rate can readily be adjusted in all applications, and the effect can immediately be seen. Using Lomb-Scargle, one can oversample indefinitely without penalty in precision, only at the expense of computing time. Using adequate step rates (e.g. $1\times10^{-8}$), we find that this cannot be causing discrete steps.
Unfortunately, it is impossible to prove that no other errors exist, and one might argue that the data itself contains a glitch. As all OGLE data are processed the same way, it might be that periodic read-out times of the CCD, or other observational requirements introduce spurious frequencies. To exclude this, one needs data from a completely different source. This has proven difficult, but with an extensive literature search, we have found sufficient data for just one other star in the same period ratio range, EY Car. This Cepheid has been monitored over six decades. We found data from five sources, and treated these equally. We have merged the data, removed obvious outliers (caused by e.g. cosmic ray hits) and subtracted the average zero point. \citet{Mitchell1964} published 23 photoelectric observations using UBV-filters, taken in 1953. We have rejected 3 of these as outliers and thus kept 20. Later, \citet{Pike1979} published 52 observations (2 rejected) taken in 1978. From \citet{Mantegazza1992} we gathered 30 good CCD observations, and 111 observations (1 rejected) from \citet{Berdnikov1995}, taken between 1980 and 1995. Finally, we found 718 (44 rejected) data points from the ASAS-3 survey \citep{Pojmanski2005} between 2003 and 2009. Our period search then included the same checks explained above. The result with Fourier and Lomb-Scargle is $D_{X}$ = +0.000404(8), that is $-0.965 D_{1}$. EY Car has slightly larger estimated errors, but seems to show the discrete step as well.
The sample is still small, and there might be data for one or more other stars in the archives. We encourage any search and analysis.
\subsection{Stability of pulsations and ratios over time}
It is well known that Cepheids and RR Lyrae show period variations over individual cycles \citep{Hippke2014}, or over longer times \citep{Pietrukowicz2001,Poleski2008,Jurcsik2012}. One might therefore argue that the period ratios are a transient phenomenon, vanishing over time. Indeed, all pulsations end one (far) day. With respect to EY Car, we would argue that this generally happens on time scales larger than a few decades.
We have checked the stability of the OGLE data as well, and find them stable over these 18 years (1996 to 2014). However, we had some trouble when calculating the values for LMC-CEP-1082. This ``new FU/FO double-mode pulsator (...) has the longest periods among currently known variables of this class'' \citep{Moskalik2008}\footnote{This reference links to the arXiv preprint of the paper. In the published version, this section was ommited.}, and due to the positive period-luminance relation among Cepheids, this star is particularly bright.
When calculating the secondary pulsation period and the ratio for the OGLE II, III and IV\footnote{After completing the first draft of this paper, we have received the pre-release OGLE-IV data and recalculated all frequencies. The results remained unchanged, the uncertainties decreased. We thank Igor Soszynski for his help.} datasets separately, we got results that were significantly different. We considered three possible causes for this: (1) systematic data errors, (2) deficiencies in the Fourier Transform (FT), and (3) a frequency shift over time.
As a first step in our analysis, we have calculated Lomb-Scargle periodograms for all datasets and periods, as well as a moving window-function of various lengths, to determine possible time shifts. Our analysis shows that there is considerable jitter over time in both periods and the resulting ratio. It is however unclear whether this stems from instrumental errors or from a physical cause. We also note that LS and FT give significantly different results for the secondary pulsation. The result is also very sensitive to the exact pre-whitening frequency. The derived period ratio deviations are 0.000386 (LS) and 0.000566 (FT). We note that the LS result ($-0.991 D_{1}$) is very close to the proposed discrete spacing, and is considered to be the better method for data with large gaps.
\subsection{Occurrence rate}
With only one RRd Lyrae star (among $\approx$2,000 known) showing a period ratio very close to $1/\sqrt{2}$, the occurrence rate is probably smaller than 1:1,000. Within the F/1O Cepheid group, we found 4 candidates (among $\approx$500 known), so that the occurrence rate is about 1:100. This estimate suffers from the small sample size, but will be enlarged in the near fututure with new data from time series surveys (e.g. ASAS, OGLE, Catalina)
\subsection{Just a coincidence?}
We have asked the obvious question: how large is the probability that the postulated fine structure is just a coincidence? The sample of stars within $5D_{x}\approx 0.02$ to $1/\sqrt{2}$ consists of only 5 stars, and all of these seem to show the spacing. As the candidate stars are the only ones so close to $P_{2}/P_{1}$ = $1/\sqrt{2}$, we cannot use statistical sample distribution tests. However, we can use the deviations from the fine structure together with the number of candidates. For the OGLE stars, we see a deviation to $D_{x}$ within 1\% (see Table~\ref{tab:table1}). The probability that 4 out of 4 stars fall into this bin is then $2\times1\%^{n-1}$, therefore $0.02^{4-1}$, or 1:125,000. When including the lower data quality star EY Car, with an error of $\approx 3.5\%$ to $D_{x}$, we find the probability as $0.07^{5-1}$, that is 1:41,649.
Admittedly, there is an infinite number of (less and less relevant) irrational numbers. Therefore, there will also be an infinite number of irrational numbers that show a discrete spacing with $n$ stars.
\subsection{Other factors causing unusual period ratios?}
We have considered other factors causing a close connection to an irrational number, and/or the fine structure. In the range $P_{2}/P_{1}$ between 0.65 and 0.71, only one star was found to be discussed in detail in the literature. In a recent paper, the RRab Lyrae \textit{V1127 Aquilae} was studied with data from the CoRoT space telescope \citep{Chadid2010}. A second significant period ($P_{2}/P_{1}=0.6965$) was found and several explanations for this were discussed. As this ratio is relatively close to the ones we discuss here, we review these possibilities.
One explanation would be a simple blend of \textit{two} stars, either as binaries or just by optical proximity. In dense regions such as the Magellanic Clouds or the Galactic Bulge, this seems not unlikely. However, we argue against this being present in these few stars for two reasons: First, linear combinations of the two frequencies have been found in all candidate stars. Second, it is improbable that all such blends would yield a certain period ratio. The pulsation periods in such variable stars are distributed over a wide range (0.2 days to over 10 days for RR Lyrae and Cepheids, and longer or shorter for other types of variables), which would result in various period ratios not clustering around 0.707. The same is true for the possibility of tidal effects in binary systems, with five of our selected having $P_{1} < 0.3$d. The rotation period seems to be equally unsuited for an explanation. The authors of the above mentioned paper argue that from measured line-widths \citep{Peterson1996}, an upper limit of the rotational velocity of 10-100d can be obtained. The stars discussed herein have shorter $P_{1}$ and $P_{2}$ than 10d.
The remaining explanations are double mode behavior or multi-periodicity, generally assumed to be due to radial and non-radial modes.
\subsection{Linear combinations of frequencies}
When performing a Fourier analysis of these double-mode pulsators, linear combinations of the frequencies are always detected. The strongest are:
\begin{eqnarray}
f_{3}=f_{1}+f_{2} \label{1} \\
f_{4}=2f_{1}+f_{2} \label{2}
\end{eqnarray}
In the following, we shall neglect all other possible, and usually weaker combinations. It is interesting to note that a frequency ratio of $P_{2}/P_{1} = 1/\sqrt{2}$ (or $f_{2}/f_{1} = \sqrt{2}$) requires these linear combinations to possess the following connection:
\begin{equation}
\frac{f_{4}}{f_{2}} = \frac{f_{3}}{f_{1}} \label{3} \\
\end{equation}
This can be shown by inserting (\ref{1}) and (\ref{2}) into (\ref{3}) which yields
\begin{equation}
\frac{2f_{1}+f_{2}}{f_{1}+f_{2}} = \frac{f_{2}}{f_{1}} \\ \label{4}
\end{equation}
which has the trivial solution:
\begin{equation}
\frac{f_{2}}{f_{1}} = \sqrt{2}
\end{equation}
This finding offers no explanation, but a purely phenomenological suggestion on how the $\sqrt{2}$ ratio arises. This finding might allow the improvement of existing linear pulsation models by adding an additional restriction.
\section{Conclusion}
We have found a small set of four Cepheids and one RR Lyrae with period ratios $P_{2}/P_{1} \approx 1/\sqrt{2}$. It is remarkable that these cover the whole range of fundamental periods from short (BLG-RRLYR-09117, $P_{1}=0.31$ days) to the longest known double-mode pulsator (LMC-CEP-1082, $P_{1}=7.87$ days), despite the strong negative connection of period ratios to main pulsation period. It is also remarkable that this is found both among RR Lyrae and Cepheids. We attribute this to the non-linear dynamics in these stars, and look forward to further modeling in this area.
We found the slight deviations from this special number to be present in integer multiples of the discrete real number $D_{1} \approx 0.000390(2)$. Common pulsation theory has no basis for the occurrence of such a fine structure. It would therefore be convenient to regard these findings as erroneous, or ``in contemporary vernacular, the astronomical community never lets facts stand in the way of a good idea'' \citep{Preston2014}. We believe that our treatment and findings are sound, and encourage the collection and analysis of further data to shed more light on this interesting phenomenon.
\begin{acknowledgements}
AZ's work is supported by the National Science Foundation under grant no. PHY07-57035 and partially by the Humboldt Foundation.
\end{acknowledgements}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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El Canto del Loco (The Singing of the Madman in English) was a Spanish pop rock band, although its members recognize that some of their songs are more in the style of power pop. The name of the band comes from the song El Canto del Gallo by Radio Futura.
The group sometimes known through the abbreviation ECDL was created in 1994 by Dani Martín and Ivan Ganchegui (who left the group in 2002) although the final line up would not come together until years later. Influenced mainly by other Spanish groups from the 1980s and with five studio albums, El Canto del Loco has gone on to sell more than a million albums, making them one of the most successful bands on the Spanish music scene in recent years.
The group announced their separation in 2010 as the members of the group wanted to pursue solo careers.
ECDL received three nominations for the MTV Europe Music Awards in the category of "Best Spanish Artist", which they won on two occasions, and they were awarded two Premio Ondas as "Best Live Act" (2004) and "Best Spanish Artist or Group" (2005).
History of the group
Origins
El Canto del Loco began in 1994 when Dani Martín was studying at the Cristina Rota School of Dramatic Arts. There, he met Iván Ganchegui, a guitarist, and their shared musical tastes resulted in the formation of the band. Both were fans of Radio Futura and in particular of their song El canto del gallo (The Song of the Rooster), which led them to name the group El Canto del Loco. At this time the group also included a female drummer, a bassist and another guitarist. However, the latter soon left the group because of other commitments, and the day before their first concert David Otero, Dani's cousin, joined the group as a substitute.
The drummer and bassist also decided to leave the group not long after. The band's original drummer was replaced by Jandro Velázquez, an electrician, who was the son of friends of Dani's parents and who he first met in a flamenco competition. The position of bassist was filled by Chema Ruiz who was studying physiotherapy at the same university as David and who was a friend of one of his friends. With this line-up the band started to meet in a warehouse in the Algete area of Madrid in order to rehearse and their friends would act as critics.
Beginnings and first albums
A year later the group recorded a demo record, which they sent to different record companies. However, it was only when Dani met the record producer Pedro del Moral that they got their first opportunity. Moral listened to the demo and took it to the Ariola record company (currently Sony BMG), where Paco Martín, who had discovered groups such as Radio Futura and Hombres G, heard it. The group then received a call from Martín, who proposed a trial concert along with two other groups, with the objective that the record company would sign one of them. Despite the fact that the concert was not very good El Canto del Loco were eventually signed to the record label.
The group's first studio album went on sale on 16 June 2000, produced by Alejo Stivel, the former singer with the Spanish group Tequila. Before this album went on sale it was suggested that the group change their name, with other suggestions including "Superratones" (Super Rats), "Los móviles" (The Mobiles) and "La dulce sonrisa de Lulú" (Lulú's Sweet Smile). However, the group decided to keep their name and for this reason the first album was also called El Canto del Loco.
The arrival of Nigel Walker as producer produced a change in direction for the group and their second album, called A contracorriente (Against the Flow), was released on 1 March 2002, which had a more mature style. In the same year El Canto del Loco also received a nomination at the MTV Europe Music Awards for "Best Spanish Artist", although the group Amaral actually won the award.
Estados de ánimo (2003-2004)
Following the voluntary exit of Iván Ganchegui from the band, the rest of the group decided to take a break. However, a few days later David and Dani had both written several songs on their own and it was decided to record them using the four remaining members of the band. After a delay due to the busy work schedule of Nigel Walker, who was working on the new album by La Oreja de Van Gogh, the group recorded their third studio album which they called Estados de ánimo (States of Mind) and which went on sale on 26 May 2003.
In August of the same year their song Pasión (Passion), from the El canto del loco album, was included on the soundtrack of the film La fiesta (The Party), directed by Carlos Villaverde and Manu Sanabria, which is famous for being one of the cheapest Spanish films ever made (on a budget of 6,000 euros). In October, the group collaborated in the recording of the album Tony Aguilar y amigos (Tony Aguilar and friends), organized by the Los 40 principales disc jockey Tony Aguilar. In collaboration with a number of other singers, El Canto del Loco played on the song Latido urbano (Urban Beat), which was the single taken from the album and which went on sale in November of that year. The profits from the record went to children's oncology hospitals run by the Asociación Española contra el Cáncer Spanish Cancer Association. They also recorded another of the songs on the album along with Aguilar himself, called Casi un Universo (Almost a Universe).
At the end of the year the group won the "Best Spanish Artist" award at the MTV Europe Music Awards beating artists such as Alejandro Sanz and La Oreja de Van Gogh.
In January 2004, they were commissioned to record a new version of the theme tune for the television series 7 vidas (7 lives), which had previously been interpreted by the singer Raimundo Amador. In the summer of that year they participated in a tribute album for Radio Futura, called Arde la calle (Burn the Street!) on which they interpreted the song Escuela de calor (School of Heat).
Zapatillas (2005-2007)
David Otero visited the island of Phi Phi (Thailand) in March 2005, where he saw the damage caused by the 2004 Indian Ocean earthquake and tsunami. Along with other artists he decided to found an organisation called Kuarkx with the objective of collecting money for those affected by the disaster. Otero, with the help of his cousin Dani, wrote the song Despiértame (Wake me up!) and made it available over the Internet where it could be downloaded for 1.15 euros. When El Canto del Loco were recording their new album they decided to include the song and dedicate a proportion of the album's profits to the same cause. Their new album, which was called Zapatillas (Sneakers), was released on 21 June.
The group released their first album outside Spain in September of the same year. It was aimed at the United States and Latin America and was called 12 estados de ánimo (12 States of Mind), it is a compilation album containing the best songs from the three albums they had so far released in Spain. That same month the group was nominated for the third time in the MTV Europe Music Awards category for "Best Spanish Artist". The group gained second prize in the event celebrated in November in Lisbon.
In July 2006, while they were touring with Hombres G, they released a collection of their concerts in indoor venues. The collection was called Pequeños grandes directos (Great Little Gigs) and included recordings of performances in the Sala Caracol in Madrid (22-11-2002); Sala Bikini in Barcelona (30-12-03) and Sala Oasis in Zaragoza. The collection was released in a limited edition of 50,000 copies.
After finishing the Zapatillas tour and before Dani Martín started filming Yo soy la Juani (I am the Juani) directed by Bigas Luna rumours started to circulate regarding the possible break up of the group. The group denied these rumours and simply stated that after a long break they would return with new songs.
The group also participated in the recording of a compilation album in aid of a Telethon held by TV3. The album, which went on sale on 10 December 2006, was sold for 9 euros with the proceeds going to charities working with people suffering from chronic pain. It sold in conjunction with a number of Catalan newspapers (Avui, El Punt, La Vanguardia and El Periódico de Catalunya). The song chosen by the group was "Puede ser" (It Can Be), from their second studio album A Contracorriente, but with the difference that it was sung in Catalan under the title Pot ser. The song was later included on the compilation album Arriba el telón (Raise the curtain).
In 2007, Dani and David, along with the manager of El Canto del Loco, Carlos Vázquez, created the record label El Manicomio (Madhouse) Records, which was supported by the multinational Sony BMG and which released the debut album of the group Sin Rumbo (Without Direction).
Personas (2008-2009)
The recording of the group's next album started in October 2007 and lasted until February 2008. The album went on sale on 1 April under the name Personas (People). On release of the album El Canto del Loco also announced a tour to start at the end of 2009, where they would be supported by the group Sin Rumbo and the singer Lucas Masciano.
On 12 June 2008 Jandro announced that he would leave the group citing personal reasons. Despite this the group confirmed that the tour would not be affected and would take place with a new drummer, Carlos Gamón, who had previously played with the group Amaral and the singer Najwa Nimri.
The group played in the Rock in Rio festival on 28 June that year. This was the first time that the festival had been held in Spain. They received poor critical reviews although the performance was popular with their fans.
De personas a personas 2008
The album "De personas a personas" (From people to people) was released at the end of 2008. It was a special limited edition of their "Personas" album with a new format (at 30x30cm the packaging was the same size as a vinyl album). However, the album was more than a rerelease of "Personas" as in addition to the 13 songs from the original album there were also six new songs and a DVD containing a lot of unpublished material.
Radio La Colifata presenta a El Canto del Loco 2009
In 2009 El Canto del Loco released two albums. The first was 'Radio La Colifata presenta: El Canto del Loco' which contained 19 of the group's hits recorded live in Buenos Aires as well as a previously unpublished song called Quiero aprender de ti (I Want to Learn From You) and also a DVD with new videos.
Por mi y por todos mis compañeros
In addition ECDL also released Por mi y por todos mis compañeros (For Me and All My Mates) in 2009. It was an ambitious project that contained 11 standards of Spanish music all of which were recorded, arranged and produced by the band. It included songs from Smash, Los Piratas, Quique González, Enrique Urquijo, Los Ronaldos and Joan Manuel Serrat among others. The release also included a DVD called Y por mí el primero (And For Me the First), which contained a recording of the band's rehearsals.
Members
Dani Martín: vocals
David Otero: guitars and backing vocals
Chema Ruiz: bass
Jandro Velázquez: drums (2000–2008)
Iván Ganchegui: guitar (2000-2002)
Influences
The group's main musical influence has been groups such as Radio Futura, of who they are known fans, Los Ronaldos, Los Rodríguez and Hombres G, who they are also good friends with. They have also been influenced to a lesser degree by groups such as Alejo Stivel (a member of this band was their first producer), Nacha Pop, Nachote Popeye and Duncan Dhu.
The members of the group also recognize that some of the songs on their album Estados de ánimo were also influenced by more recent bands such as Estopa (Tow), La Cabra Mecánica (The Mechanical Goat), M Clan.
Collaborations
El Canto De Loco collaborated with Mexican singer, Natalia Lafourcade in the song Contigo, which became popular in Mexico.
El Canto De Loco has also collaborated with Spanish band, Sueño De Morfeo
Discography
Studio albums
Live albums
Compilation albums
Reissue album
DVD
Awards and nominations
References
External links
Official El Canto del Loco web page
Official web page for El Manicomio Records
Spanish rock music groups
Spanish pop rock music groups
Rock en Español music groups
Musical groups established in 2000
Sony Music Latin artists
MTV Europe Music Award winners
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
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Q: how to get or reflect the name of the bash function which is called? i did not yet found a solution to this. Anyone a hint?
i sometimes write bash functions in my shell scripts and i love to have my scripts being verbose, not just for debugging. so sometimes i would like to display the "name" of a called bash function as a "variable" in my scripts.
what i did sometimes is setting just a regular variable containing the function name. like this:
test ()
{
funcName=test
echo "function running..."
echo "\$0 is : $0"
echo "function name is : $funcName"
}
but that is kinda stupid.
Is there something better?
A: Sometimes it's enough to read man bash:
FUNCNAME
An array variable containing the names of all shell
functions currently in the execution call stack. The
element with index 0 is the name of any
currently-executing shell function. The bottom-most
element (the one with the highest index) is "main".
This variable exists only when a shell function is
executing. Assignments to FUNC- NAME have no effect and
return an error status. If FUNCNAME is unset, it loses
its special properties, even if it is subsequently
reset.
Example usage:
#!/usr/bin/env bash
func()
{
echo I am inside "$FUNCNAME"
}
foo()
{
echo I am inside "$FUNCNAME"
}
func
foo
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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package org.batfish.datamodel;
import static org.hamcrest.Matchers.equalTo;
import static org.hamcrest.Matchers.notNullValue;
import static org.hamcrest.Matchers.nullValue;
import static org.junit.Assert.assertEquals;
import static org.junit.Assert.assertNull;
import static org.junit.Assert.assertThat;
import com.google.common.collect.ImmutableList;
import com.google.common.collect.ImmutableMap;
import com.google.common.testing.EqualsTester;
import java.util.Collections;
import java.util.Map;
import org.batfish.common.util.BatfishObjectMapper;
import org.batfish.vendor.VendorStructureId;
import org.junit.Test;
/** Tests of {@link AclAclLine} */
public class AclAclLineTest {
@Test
public void testExplicitActions() {
Ip ip1234 = Ip.parse("1.2.3.4");
Ip ip2345 = Ip.parse("2.3.4.5");
ExprAclLine block1234 =
ExprAclLine.rejectingHeaderSpace(
HeaderSpace.builder().setSrcIps(ip1234.toIpSpace()).build());
ExprAclLine allow2345 =
ExprAclLine.acceptingHeaderSpace(
HeaderSpace.builder().setSrcIps(ip2345.toIpSpace()).build());
IpAccessList acl =
IpAccessList.builder()
.setName("acl")
.setLines(ImmutableList.of(block1234, allow2345))
.build();
IpAccessList testAcl =
IpAccessList.builder()
.setName("aclThenDeny")
.setLines(ImmutableList.of(new AclAclLine("aclAclLine", acl.getName())))
.build();
Map<String, IpAccessList> acls =
ImmutableMap.of(acl.getName(), acl, testAcl.getName(), testAcl);
Flow.Builder fb =
Flow.builder().setIpProtocol(IpProtocol.OSPF).setDstIp(Ip.ZERO).setIngressNode("node");
{
// The testACL should explicitly permit the flow on some line, since it was permitted by acl.
FilterResult r2345 =
testAcl.filter(fb.setSrcIp(ip2345).build(), "eth", acls, Collections.emptyMap());
assertThat(r2345.getAction(), equalTo(LineAction.PERMIT));
assertThat(r2345.getMatchLine(), notNullValue()); // did not fall off end
}
{
// The testACL should explicitly reject the flow on some line, since it was rejected by acl.
FilterResult r1234 =
testAcl.filter(fb.setSrcIp(ip1234).build(), "eth", acls, Collections.emptyMap());
assertThat(r1234.getAction(), equalTo(LineAction.DENY));
assertThat(r1234.getMatchLine(), notNullValue()); // did not fall off end
}
{
// The testACL should reject the flow by falling off the end, since it was not explicitly
// handled by acl.
Ip ip3456 = Ip.parse("3.4.5.6");
FilterResult r3456 =
testAcl.filter(fb.setSrcIp(ip3456).build(), "eth", acls, Collections.emptyMap());
assertThat(r3456.getAction(), equalTo(LineAction.DENY));
assertThat(r3456.getMatchLine(), nullValue()); // signifies fell off the end
}
}
@Test
public void testDefaultTraceElement() {
assertNull(new AclAclLine("n", "a").getTraceElement());
}
@Test
public void testEquals() {
TraceElement traceElement1 = TraceElement.builder().add("a").build();
TraceElement traceElement2 = TraceElement.builder().add("b").build();
VendorStructureId vsId1 = new VendorStructureId("a", "b", "c");
VendorStructureId vsId2 = null;
new EqualsTester()
.addEqualityGroup(
new AclAclLine("name1", "acl1", traceElement1, vsId1),
new AclAclLine("name1", "acl1", traceElement1, vsId1))
.addEqualityGroup(new AclAclLine("name2", "acl1", traceElement1, vsId1))
.addEqualityGroup(new AclAclLine("name1", "acl2", traceElement1, vsId1))
.addEqualityGroup(new AclAclLine("name1", "acl1", traceElement2, vsId1))
.addEqualityGroup(new AclAclLine("name1", "acl1", traceElement1, vsId2))
.addEqualityGroup(new Object())
.testEquals();
}
@Test
public void testJsonSerialization() {
{
AclAclLine aclAclLine = new AclAclLine("lineName", "aclName");
AclAclLine clone = (AclAclLine) BatfishObjectMapper.clone(aclAclLine, AclLine.class);
assertEquals(aclAclLine, clone);
}
{
AclAclLine aclAclLine =
new AclAclLine(
"lineName",
"aclName",
TraceElement.builder().add("a").build(),
new VendorStructureId("a", "b", "c"));
AclAclLine clone = (AclAclLine) BatfishObjectMapper.clone(aclAclLine, AclLine.class);
assertEquals(aclAclLine, clone);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,438
|
Si discute se il nome che gli viene ascritto, Robert, sia reale o meno, anche perché già Wace è di per sé un nome.
Biografia
Studiò a Parigi e divenne canonico di Bayeux negli ultimi anni di vita.
Opere
Wace è conosciuto soprattutto per i suoi tre capolavori:
La Vie des Saints (Vita dei Santi), opera in versi, che includono le vite di santa Margherita e san Nicola
Le Roman de Brut o Brut d'Angleterre (circa 1155, dedicato a Eleonora d'Aquitania)
Le Roman de Rou, epopea sui duchi di Normandia.
Roman de Brut
Il Roman de Brut (circa 1155) è basato sulla Historia Regum Britanniae di Goffredo di Monmouth: quest'opera non può essere considerata storica in senso moderno, sebbene Wace distingua spesso tra ciò che sa e ciò che non sa o ciò che non è in grado di trovare. Wace infatti narra la fondazione della Britannia da parte di Bruto di Troia e arriva fino a dove termina la leggendaria storia della Britannia di Goffredo di Monmouth. Wace fu anche il primo a parlare della Tavola Rotonda e a chiamare Excalibur la spada di re Artù, sebbene nel complesso egli aggiunga solo pochi dettagli all'opera di Goffredo. Il Roman de Brut fu a sua volta la base per il Bruto di Layamon, un poema allitterato in inglese medio, e per la Cronaca di Piers Langtoft.
Roman de Rou
Il Roman de Rou, secondo Layamon, fu commissionato a Wace da re Enrico II e un'ampia parte è dedicata a Guglielmo il Conquistatore e alla conquista normanna dell'Inghilterra. In quest'opera Wace si riferisce anche a tradizioni orali provenienti dalla sua famiglia. Inoltre, menziona l'apparizione della cometa di Halley e traccia un breve racconto autobiografico della sua vita (III, 5299-5318 - citato più sopra).
Bibliografia
Opere originali
WACE, Roman de Brut, edito da I. Arnold, 2 vol., Parigi, 1938-1940.
WACE, Roman de Rou, edito da J. Holden, 3 vol., Parigi, 1970-1973.
WACE, Vie de Sainte Marguerite, edito da J. Aristide, Parigi, 1879.
Studi
I. ARNOLD - M. PELAN, La partie arthurienne du Roman de Brut, Parigi, 1962
J. WEISS, Wace's Roman de Brut. A History of the British. Text and Translation, Exeter, 2006
C. FOULON, "Wace" in Arthurian Literature in the Middle Ages, Roger Sherman Loomis (ed.), Clarendon Press: Università di Oxford, 1959, ISBN 0-19-811588-1
C. BRATU, « Je, auteur de ce livre »: L'affirmation de soi chez les historiens, de l'Antiquité à la fin du Moyen Âge. Later Medieval Europe Series (vol. 20). Leiden: Brill, 2019 ISBN 978-90-04-39807-8.
C. BRATU, "Translatio, autorité et affirmation de soi chez Gaimar, Wace et Benoît de Sainte-Maure." The Medieval Chronicle 8 (2013): 135-164.
Voci correlate
Letteratura anglonormanna
Conquista normanna dell'Inghilterra
Medioevo inglese
Anglo-normanni
Re Artù
Tavola Rotonda
Excalibur
Historia Regum Britanniae
Goffredo di Monmouth
Altri progetti
Collegamenti esterni
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 291
|
\section{The effect of post-selection
\label{subsec: post-selection}
}
In this section, we discuss the effect of post-selection to mitigate errors in QSCI.
To be specific, we consider the post-selection technique introduced in Sec.~\ref{subsec:ground-state}, which exploits the conservation of particle number and spin, targeting the bit-flip noise.
In the following, we consider to measure an $N$-qubit computational basis state.
We assume that the state is prepared without the effects of noise but each bit of the measurement result is flipped with error probability $p$. It is equivalent to the situation where the bit-flip noise is introduced to each qubit independently after the input state is generated. The probability that the $N$-bit string describing the state is measured correctly is $(1-p)^N$.
\subsection{Jordan-Wigner mapping}
Let us now assume that we consider an electronic Hamiltonian converted by the Jordan-Wigner mapping. In this case, the number of 1's in the $N$-bit string, which we denote by $n_1$, corresponds to the number of electrons in the system
and is
sometimes known prior to the calculation for a ground state or an excited state.
One can thus perform the post-selection for a measurement outcome that excludes resulting bit strings with the number of 1's not equal to $n_1$.
Although one may still get incorrect results, the probability is reduced.
More concretely, the probability to get a result with correct $n_1$ is
\begin{equation}
(1-p)^N+n_0n_1p^2(1-p)^{N-2}+O(p^4),
\end{equation}
where we define $n_0:=N-n_1$.
After the post-selection, the probability to get the correct result is thus
\begin{align}
&\dfrac{(1-p)^N}{(1-p)^N+n_0n_1p^2(1-p)^{N-2}+O(p^4)}\\
&=\dfrac{1}{1+n_0n_1p^2/(1-p)^2+O(p^4)}\\
&=\dfrac{1}{1+n_0n_1p^2+O(p^3)}\\
&=1-n_0n_1p^2+O(p^3).
\end{align}
The error rate of getting an incorrect result is reduced from $1-(1-p)^N\sim pN$ to $n_0n_1p^2$ with the ratio being
\begin{equation}
\dfrac{n_0n_1p^2}{pN}=\left(\dfrac{n_0}{N}\right)\left(\dfrac{n_1}{N}\right)Np \sim O(pN),
\end{equation}
which is less than one in a sensible situation $Np \ll 1$, where the original success probability $(1-p)^N$ is not too small.
Although we here considered a computational basis state as the input state,
we expect that
the post-selection similarly works
for a general input state that is a superposition of computational basis states with some fixed $n_1$.
Note that, if one also knows the total spin $S_z$ of electrons
prior to the calculation, one can count the number of 1's separately for up- and down-spin electrons, and make the post-selection more efficient.
\subsection{Other mappings}
For most of the other fermion-qubit mappings, it is not expected that the reduction of the error probability from $O(p)$ to $O(p^2)$ happens. For example, in the parity mapping~\cite{bravyi2002fermionic,seeley2012bravyi} and the Bravyi-Kitaev mapping~\cite{bravyi2002fermionic}, the states $\ket{01}$ and $\ket{11}$ are connected by just one bit flip, but both of them are one-electron states. This bit flip, which cannot be detected by the post-selection, occurs with probability $O(p)$, and thus the error rate after the post-selection is still $O(p)$. The same is true for a Hamiltonian with a reduced number of qubits using symmetries, where there is always a bit flip that does not change the total number of electrons.
\section{Details of the algorithms}
In this section, we present several detailed discussions on the QSCI algorithms.
\subsection{Choice of $\beta_i$ parameters and variational inequalities in sequential diagonalization scheme}
\label{subsec:details-sequential}
Here we discuss the sequential diagonalization scheme, introduced in Sec.~\ref{sssec:method-sequantial}, on how to choose the $\beta_i$ parameters and a potential violation of the variational inequality, following the discussion in Ref.~\cite{higgott2019variational}.
Suppose $k$ low-lying eigenstates of $\hat{H}$, $\ket{E_0}, \cdots, \ket{E_{k-1}}$, are known exactly.
Then, the effective Hamiltonian to find the $k$-th eigenstate can be exactly constructed as
\begin{align}
\hat{H}^{(k)\prime} =\hat{H}+
\sum_{i=0}^{k-1}\beta_i \ket{E_i}\bra{E_i}.
\end{align}
This can be formally expressed as
\begin{align}
\hat{H}^{(k)\prime} =
\sum_{i=0}^{k-1}(E_i + \beta_i) \ket{E_i}\bra{E_i} + \sum_{i\geq k} E_i \ket{E_i}\bra{E_i},
\end{align}
where $E_i$ represents the $i$-th eigenvalue of $\hat{H}$ in this appendix.
For $\beta_i > E_k - E_i$ ($i=0,\cdots, k-1$), the following inequality holds for an arbitrary $\ket{\psi}$ with $\bra{\psi}\ket{\psi}=1$:
\begin{align}
\bra{\psi} \hat{H}^{(k)\prime} \ket{\psi} \geq E_k,
\end{align}
where the equality holds if and only if $\ket{\psi}=\ket{E_k}$ up to a phase factor.
In the language of the eigenvalue problem of Eq.~\eqref{eq:eigenvalue-eq}, this implies $E_{R_k}^{(k)\prime}\geq E_k$, where $E_{R_k}^{(k)\prime}$ is the smallest eigenvalue of $\bm{H}_{R_k}^{(k)\prime}$, the subspace matrix for $\hat{H}^{(k)\prime}$, defined in the same way as Eq.~\eqref{eq:sequential-matrix}.
In practice, the condition $\beta_i > E_k - E_i$ can be utilized if one has prior knowledge on the energy spectrum, e.g., based on variational quantum algorithms.
But even without such knowledge, one may still rely on the stronger condition of $\beta_i > 2\sum_j \abs{c_j}$~\cite{higgott2019variational}, which is written in terms of the coefficients $c_j$ of the qubit Hamiltonian $\hat{H}=\sum_j c_j P_j$, expressed by the Pauli strings $P_j$.
In reality, the effective Hamiltonian cannot be exactly constructed as the $k$ low-lying eigenstates would be obtained only approximately and, hence, the inequality $E_{R_k}^{(k)}\geq E_k$ is not guaranteed.
For instance, in the problem to find the first excited state,
the effective Hamiltonian $\hat{H}^{(1)}$ is constructed with $|\psi_{\rm out}^{(0)}\rangle$, the output state for the ground state obtained by the preceding step in sequential diagonalization.
Unless the output state perfectly overlaps with the true ground state $\ket{E_0}$, or $|\langle\psi_{\rm out}^{(0)}| E_0\rangle|=1$,
there is no guarantee that $\ev{\hat{H}^{(1)}}{\psi}$ is bounded by the exact eigenvalue $E_1$.
Instead, $\min_\psi \ev{\hat{H}^{(1)}}{\psi}$ is only bounded as~\cite{higgott2019variational}:
\begin{align}
E_1 - O((E_1-E_0)\epsilon_0) \leq \min_\psi \ev{\hat{H}^{(1)}}{\psi} \leq E_1 +\beta_0\epsilon_0,
\end{align}
where $\epsilon_0 = 1 - |\langle\psi_{\rm out}^{(0)}| E_0\rangle|^2$ and $\bra{\psi}\ket{\psi}=1$.
A concrete example for breaching the variational inequality is given as follows.
Consider a system with the unique ground state, i.e., $E_0 < E_1$. Suppose that one has a poor output state $|\psi_{\rm out}^{(0)}\rangle$ that is orthogonal to the true ground state $\ket{E_0}$, i.e., $\epsilon_0=1$.
Then, $\bra{\psi}\hat{H}^{(1)} \ket{\psi} \geq E_0$ for any positive $\beta_0$, where the equality holds if and only if $\ket{\psi}=\ket{E_0}$ up to a phase factor.
This means that the variational inequality $E_{R_1}^{(1)}\geq E_1 (> E_0)$ is violated at least in the limit where the subspace $\mc{S}_{R_1}^{(1)}$ is enlarged to cover the part of the Fock space necessary to express $\ket{E_0}$.
Such a subspace can be constructed, e.g., if the input state is chosen to be $|\psi_{\rm in}^{(1)}\rangle = \ket{E_0}$ with a sufficiently large $R_1$.
\subsection{An optimal shot allocation for evaluating expectation values of multiple observables in conventional method}
\label{subsec:appendix-scaling-multiple-operators}
In this subsection, we describe the details of the numerical estimation of computational cost in Sec.~\ref{subsec:scaling} for evaluating multiple operators. In the numerical simulation, we considered a situation where we want to calculate the expectation values of the nuclear gradient $\left\{\pdv{\hat{H}}{x_i} \mid i=1,\dots,3N_{\text{atom}} \right\}$ and the nuclear Hessian $\left\{\pdv{\hat{H}}{x_i}{x_j} \mid i,j=1,\dots,3N_{\text{atom}} \right\}$ along with the Hamiltonian $\hat{H}(\left\{x_i\right\})$, where $x_i$ are the nuclear coordinates and $N_\mr{atom}$ is the number of atoms in the molecule.
The most naive way of doing it would be to calculate each expectation value completely separately. This is, though, too naive to be considered as the optimal strategy; all the observables are linear combinations of operators $a^\dagger_i a_j$ and $a^\dagger_i a_j a^\dagger_k a_l$ in the fermionic basis, and the expectation values of these operators can be reused among the operators.
Before discussing the optimal strategy for evaluating expectation values of multiple observables, let us review the one for a single observable, following the discussion in Ref.~\cite{rubin2018application}.
Consider a quantum state $\ket{\psi}$ and the expectation value of an operator $\hat{O}$ which can be written as a sum of operators $\hat{O}_l$:
\begin{equation}
\hat{O}=\sum_{l=1}^{L}\hat{O}_l.
\end{equation}
Each term $\hat{O}_l$ can be either a Pauli string or a sum of Pauli strings that commute with each other, which admits the projective measurement on eigenvalues of each $\hat{O}_l$.
We denote the variance of each term $\hat{O}_l$ per one shot by $\sigma_l^2:=\text{Var}(\hat{O}_l) := \ev{\hat{O}_l^2}{\psi}-\ev{\hat{O}_l}{\psi}^2$. By measuring each term $\hat{O}_l$ with a number of shots $M_l$, the observed expectation value has the variance $\sum_l \sigma_l^2/M_l$. Employing the method of Lagrange multiplier with the Lagrangian
\begin{equation}
\mathcal{L}=\sum_l M_l +\lambda \left( \sum_l \dfrac{\sigma_l^2}{M_l} -\epsilon^2\right),
\end{equation}
one can get the optimal allocation of the number of shots with the total variance of the expectation value fixed to $\epsilon^2$,
which is
\begin{equation}
M_l\propto \sigma_l.
\end{equation}
In general, $\sigma_l$ is not exactly known a priori, so one may use $\sigma_l$ for Haar random states to get a reasonable strategy. One may also try to improve the strategy by dividing the shot budget for one evaluation of an expectation value into several iterations: one can simply evaluate the expectation value with a mildly optimized strategy in the first iteration, and then, in the rest of the iterations, one can adjust the strategy by calculating $\sigma_l$ by using the expectation values obtained in the previous iterations.
Generalizing the above discussion, let us consider a situation where one calculates the expectation values of a set of operators $\left\{\hat{O}^{(i)} \mid i=1,\dots,n\right\}$.
We assume that $\hat{O}^{(i)}$ is decomposed as
\begin{equation}
\hat{O}^{(i)}=\sum_{l=1}^{L} \hat{O}^{(i)}_l,
\end{equation}
where all of $\left\{\hat{O}^{(i)}_l\mid i=1,\dots,n\right\}$ are simultaneously measurable for each $l$, i.e., $[\hat{O}^{(i)}_l, \hat{O}^{(j)}_l] = 0$ for any $i,j$.
In our numerical simulation, the grouping was done by firstly taking the sum of all the observables $\hat{O}^{(i)}$ with each Pauli string with negative coefficient multiplied by $-1$ to make it positive. Then the greedy qubit-wise grouping of Refs.~\cite{mcclean2016theory, crawford2021efficient} was used.
Our aim here is to find a good strategy to estimate the expectation values of all the operators $O^{(i)}$ with statistical error less than $\epsilon$.
Note that one can always rescale the observables so that the required precision is the same for all observables even when one requires different precision for different operators.
To get an analytical solution, we choose the following Lagrangian with slightly modified constraint,
\begin{equation}
\mathcal{L}=\sum_l M_l +\lambda \left(\sum_l \sum_i \left(\dfrac{{\sigma_l ^{(i)}}^2}{M_l}\right)-\epsilon_{\text{tot}}\right),
\end{equation}
where $\epsilon_{\text{tot}}$ can be $N\times \epsilon$ but it turns out that the choice of $\epsilon_{\text{tot}}$ does not affect the final result.
By solving the extremal condition of this Lagrangian, one can get the best shot allocation that minimizes the total number of shots, while keeping the sum of the variances of all the operators less than $\epsilon_{\text{tot}}$. The result implies that
\begin{equation}
M_l\propto \sqrt{\sum_i {\sigma_l^{(i)}}^2}.
\end{equation}
By estimating the variance of each operator $\hat{O}^{(i)}$ with this shot allocation, and by adjusting the total number of shots so that the statistical error of each operator is $\epsilon$ at worst, one can obtain the total number of shots $\sum_l M_l$ with desired precision for all the operators. This may not be the optimal shot allocation to achieve the statistical error $\epsilon$ for each operator as we are minimizing the total variance rather than the maximum value of the variances, but this will give a reasonable strategy that is analytically available.
There is one comment to make on the evaluation of nuclear Hessians.
In the following, the expectation value is always taken by an ansatz state $\ket{\psi(\bm{\theta}(\left\{x_i\right\}))}$ parametrized by the ansatz parameters $\bm{\theta}(\left\{x_i\right\})$; we assume that $\bm{\theta}(\left\{x_i\right\})$ is optimized so that $\ket{\psi(\bm{\theta}(\left\{x_i\right\}))}$ has the minimum energy within the ansatz for each $\left\{x_i\right\}$. We denote the energy expectation value of the state by $E(\left\{x_i\right\})$. In the case of the nuclear gradient,
\begin{equation}
\pdv{E(\left\{x_i\right\})}{x_i}=\expval{\pdv{\hat{H}(\left\{x_i\right\})}{x_i}}
\end{equation}
holds thanks to the Hellmann-Feynman theorem, and it suffices to compute the right-hand side to obtain the nuclear gradient of the energy.
In the case of the nuclear Hessian of the energy~\cite{mitarai2020theory}, on the other hand, it is in general necessary to evaluate the contribution of derivatives acting on the state as well as on the Hamiltonian operator,
\begin{equation}
\pdv{E(\left\{x_i\right\})}{x_i}{x_j}=\expval{\pdv{\hat{H}(\left\{x_i\right\})}{x_i}{x_j}}+\dots
\end{equation}
In our numerical simulation, we ignored the contribution of the derivatives acting on the state for simplicity.
In the case of QWC, this contribution requires additional quantum resources.
On the other hand, in the case of QSCI, one can generate and diagonalize the Hamiltonians at small finite distance $x_i\to x_i\pm \delta$ to get the derivatives of the state within the same selected subspace of the Fock space with no additional quantum resources. If we take this contribution into account properly, the advantage of QSCI will increase.
It should also be noted that, although we evaluate $O(N_{\text{atom}}^2)$ observables in the numerical simulations for the hydrogen chain in the main text, due to the rich geometrical symmetry of the molecules, many of the observables are zero as an operator.
It is likely that, for more generic molecules, the crossing-point of QSCI and QWC comes at a larger number of qubits.
\section{Details of numerical simulations and experiments}
\label{sec:appendix-details-of-sim-and-exp}
In this section, we explain details of the numerical simulations and the experiment on quantum hardware in the main text.
For all the molecules examined in this study, the second-quantized electronic Hamiltonian under the Born-Oppenheimer approximation is generated by OpenFermion~\cite{mcclean2020openfermion} interfaced with PySCF~\cite{sun2018pyscf} using the Hartree-Fock orbitals with the STO-3G minimal basis set, unless otherwise stated. The electronic Hamiltonians are mapped to qubit ones by the Jordan-Wigner transformation. The molecular geometries used in our study are shown in Table~\ref{tab: geometries}. Stable geometries for diatomic molecules are taken from CCCBDB database~\cite{johnson2022nist} and Ref.~\cite{wang2016relativistic}, while those for the other molecules are taken from PubChem~\cite{kim2023pubchem}, except for the hydrogen chains which are not in their stable geometries.
We list the details specific to each of simulations and experiment in the following.
\begin{table*}[]
\caption{Geometries of molecules. ``$(\mr{X}, (x,y,z))$" denotes three dimensional coordinates $x,y,z$ of an atom X in units of \AA.
\label{tab: geometries}
}
\begin{tabular}{c|p{12cm}}
\hline \hline
Molecule & Geometry \\ \hline
\ce{H2O} & (O, (0, 0, 0)), (H, (0.2774, 0.8929, 0.2544)), (H, (0.6068, -0.2383, -0.7169)) \\
\ce{H}$_n$ ($n=4,6,8,10,12)$ & (H, (0, 0, 0)), (H, (0, 0, 1.0), \dots, (H, (0, 0, $n \times 1.0$)) \\
\ce{LiH} & (Li, (0, 0, 0)), (H, (0, 0, 1.595))\\
\ce{N2} & (N, (0, 0, 0)), (N, (0, 0, 1.1))\\
\ce{O2} & (O, (0, 0, 0)), (O, (0, 0, 1.2))\\
\ce{F2} & (F, (0, 0, 0)), (F, (0, 0, 1.4))\\
\ce{Cl2} & (Cl, (0, 0, 0)), (Cl, (0, 0, 2.0))\\
\ce{HCl} & (H, (0, 0, 0)), (Cl, (0, 0, 1.3))\\
\ce{CO} & (C, (0, 0, 0)), (O, (0, 0, 1.1))\\
\ce{Cr2} & (Cr, (0, 0, 0)), (Cr, (0, 0, 1.6))\\
\ce{Benzene} &(C, (-1.2131, -0.6884, 0)), (C, (-1.2028, 0.7064, 0.0001)), (C, (-0.0103, -1.3948, 0)), (C, (0.0104, 1.3948, -0.0001)), (C, (1.2028, -0.7063, 0)), (C, (1.2131, 0.6884, 0)), (H, (-2.1577, -1.2244, 0)), (H, (-2.1393, 1.2564, 0.0001)), (H, (-0.0184, -2.4809, -0.0001)), (H, (0.0184, 2.4808, 0)), (H, (2.1394, -1.2563, 0.0001)), (H, (2.1577, 1.2245, 0))\\
\ce{Naphthalene} &(C, (0, -0.7076, 0)), (C, (0, 0.7076, 0.0001)), (C, (1.225, -1.3944, 0.0001)), (C, (1.225, 1.3944, 0)), (C, (-1.225, -1.3943, 0)), (C, (-1.225, 1.3943, 0)), (C, (2.4327, -0.6958, 0)), (C, (2.4327, 0.6959, -0.0001)), (C, (-2.4327, -0.6958, -0.0001)), (C, (-2.4327, 0.6958, 0)), (H, (1.2489, -2.4822, 0.0001)), (H, (1.2489, 2.4821, -0.0001)), (H, (-1.2489, -2.4822, -0.0001)), (H, (-1.249, 2.4821, 0.0001)), (H, (3.3733, -1.239, -0.0001)), (H, (3.3732, 1.2391, -0.0001)), (H, (-3.3733, -1.239, -0.0001)), (H, (-3.3732, 1.239, 0))\\
\ce{Anthracene} &(C, (-1.225, 0.706, 0.0001)), (C, (-1.2251, -0.7061, 0.0001)), (C, (1.2251, 0.7061, 0.0002)), (C, (1.2251, -0.7061, 0.0001)), (C, (0, 1.3937, 0.0001)), (C, (0, -1.3938, 0)), (C, (-2.4504, 1.393, -0.0001)), (C, (-2.4505, -1.393, 0)), (C, (2.4505, 1.3929, 0)), (C, (2.4505, -1.3929, 0)), (C, (-3.6587, 0.6956, -0.0001)), (C, (-3.6588, -0.6955, -0.0001)), (C, (3.6587, 0.6956, -0.0002)), (C, (3.6587, -0.6956, -0.0002)), (H, (0, 2.4838, 0)), (H, (0, -2.4839, -0.0001)), (H, (-2.4742, 2.4808, -0.0001)), (H, (-2.4744, -2.4809, 0)), (H, (2.4742, 2.4808, 0)), (H, (2.4743, -2.4808, 0)), (H, (-4.5989, 1.2394, -0.0003)), (H, (-4.5991, -1.2391, -0.0002)), (H, (4.5989, 1.2393, -0.0003)), (H, (4.5989, -1.2393, -0.0004))\\
\ce{Tetracene} & (C, (0, 0.7045, -0.0002)), (C, (0, -0.7046, -0.0001)), (C, (-2.451, 0.7058, 0)), (C, (-2.4511, -0.7058, 0.0002)), (C, (2.4511, 0.7057, 0.0001)), (C, (2.4511, -0.7058, -0.0001)), (C, (1.2254, 1.3923, -0.0001)), (C, (1.2254, -1.3924, -0.0003)), (C, (-1.2254, 1.3923, -0.0002)), (C, (-1.2255, -1.3923, 0.0002)), (C, (-3.6764, 1.3928, -0.0001)), (C, (-3.6764, -1.3929, 0.0002)), (C, (3.6764, 1.3929, 0.0003)), (C, (3.6765, -1.3929, -0.0001)), (C, (-4.8846, 0.6957, -0.0001)), (C, (-4.8847, -0.6955, 0.0001)), (C, (4.8846, 0.6957, 0.0004)), (C, (4.8847, -0.6956, -0.0001)), (H, (1.2253, 2.4825, -0.0001)), (H, (1.2254, -2.4825, -0.0003)), (H, (-1.2254, 2.4824, -0.0003)), (H, (-1.2255, -2.4824, 0.0003)), (H, (-3.6999, 2.4807, -0.0002)), (H, (-3.7001, -2.4808, 0.0003)), (H, (3.6999, 2.4807, 0.0004)), (H, (3.7001, -2.4807, -0.0003)), (H, (-5.8248, 1.2393, -0.0002)), (H, (-5.8249, -1.2392, 0.0002)), (H, (5.8248, 1.2394, 0.0005)), (H, (5.8249, -1.2392, -0.0002))\\
\hline \hline
\end{tabular}
\end{table*}
\subsection{Noiseless simulation for ground state}
\label{subsec:setup-noiseless-vqe}
\begin{figure}[h!]
\includegraphics[width=.45\textwidth]{figures/rsp-ansatz.pdf}
\caption{Real-valued symmetry-preserving ansatz with $n$ qubits and depth $d$.}
\label{fig:rsp-ansatz}
\end{figure}
In Sec.~\ref{subsec:ground-state-simulation-with-noiseless-vqe}, the \ce{H2O} molecule with six active electrons and five active orbitals, is chosen to find the ground state by QSCI. In the VQE calculation for preparing the input states, the BFGS optimizer is employed through the scientific library SciPy~\cite{virtanen2020scipy}, and the real-valued symmetry-preserving ansatz~\cite{ibe2022calculating} is used to construct parametric quantum circuits with depth 10 (Fig.~\ref{fig:rsp-ansatz}).
The initial state of the ansatz circuits is set to be the Hartree-Fock state, and the initial parameters in the optimization are randomly chosen.
\subsection{Noiseless simulations for excited states}
\label{subsec:setup-noiseless-vqd}
In Sec.~\ref{subsec:simulation-excited-h2o}, QSCI is demonstrated for the same \ce{H2O} molecule but to find excited states.
To prepare the input states, the VQD calculations are performed in the same setup as the previous VQE calculation,
but with the penalty terms~\cite{mcclean2016theory,ryabinkin2018constrained,kuroiwa2021penalty} added to the Hamiltonian for constraining the resulting states to have $S_z=0$ and $N_e=6$; specifically, the following operator (in atomic units) is added to the Hamiltonian
\begin{equation}
3.0 (\hat{S}_z)^2 + 3.0 (\hat{N}_e-6)^2,
\end{equation}
where $\hat{S}_z$ is the operator for the total electron spin in $z$-direction, and $\hat{N}_e$ for the particle number operator of electrons in the active space.
Furthermore, the overlap terms to constrain the state to be orthogonal to lower energy eigenstates~\cite{higgott2019variational} are added with coefficients of unity (in Hartree).
For the sequential diagonalization scheme of QSCI, the coefficients $\beta_i$ for ensuring orthogonality are also set to unity.
\begin{figure}[h!]
\includegraphics[width=.25\textwidth]{figures/ryansatz.pdf}
\caption{Ry ansatz with 8 qubits. All the rotational gates have independent parameters. The depth is set to be 8 in our experiment.}
\label{fig:ryansatz}
\end{figure}
\subsection{Noisy simulation and experiment}
\label{ssec:setup-experiment}
For the noisy simulation and experiment in Sec.~\ref{sec:noisy-simulation-experiment}, the input states are prepared by noiseless VQE simulations. The VQE calculations are performed with the BFGS optimizer and Ry ansatz (Fig.~\ref{fig:ryansatz}) with depth 8. Other details are described in the main text.
\section{Supplemental numerical results}
In this section, we provide additional numerical results to supplement the contents in Sec.~\ref{sec:numerical}.
\subsection{Scaling of computational costs with various molecules}
\label{ssec:more-results-scaling}
Figure~\ref{fig:all-scaling} shows the scaling of the classical and quantum computational costs, discussed in Sec.~\ref{subsec:scaling}, for different types of molecules. Here, we test three kinds of molecules: hydrogen chains, diatomic molecules, and aromatic molecules. The data for hydrogen chains are exactly the same as in the main text. Diatomic molecules are \ce{N2}, \ce{O2}, \ce{F2}, \ce{Cl2}, \ce{HCl}, \ce{CO}, and \ce{Cr2} with cc-pVQZ basis, and we tested them with various active spaces just as in the main text for \ce{Cr2}.
To test with larger molecules, four aromatic molecules are chosen: benzene, naphthalene, anthracene, and tetracene.
The Hamiltonian is generated by using the Hartree-Fock orbitals with STO-3G basis.
The active space of $n$ orbitals and $n$ electrons with varying $n$ was employed for the diatomic and aromatic molecules.
The geometries of these tested molecules are summarized in Table~\ref{tab: geometries}.
As can be seen in Fig.~\ref{fig:all-scaling}, hydrogen chains with various numbers of atoms show the worst scalings, while \ce{Cr2} is one of the least expensive systems among others.
\begin{figure*}
\begin{minipage}{0.9\textwidth}
\subfloat[][$R$ required for energy error $\epsilon=\SI{0.001}{Hartree}$.]{
\includegraphics[width=.45\textwidth]{figures/scaling/qubit-to-R-all-0.001.pdf}
}
\subfloat[][$1/\abs{c_R}^2$ for energy error $\epsilon=\SI{0.001}{Hartree}$.]{
\centering
\includegraphics[width=.45\textwidth]{figures/scaling/qubit-to-cR-all-0.001.pdf}
}
\subfloat[][$R$ required for energy error $\epsilon=\SI{0.01}{Hartree}$.]{
\centering
\includegraphics[width=.45\textwidth]{figures/scaling/qubit-to-R-all-0.01.pdf}
}
\subfloat[][$1/\abs{c_R}^2$ for energy error $\epsilon=\SI{0.01}{Hartree}$.]{
\includegraphics[width=.45\textwidth]{figures/scaling/qubit-to-cR-all-0.01.pdf}
}
\subfloat[][$R$ required for energy error $\epsilon=\SI{0.1}{Hartree}$.]{
\includegraphics[width=.45\textwidth]{figures/scaling/qubit-to-R-all-0.1.pdf}
}
\subfloat[][$1/\abs{c_R}^2$ for energy error $\epsilon=\SI{0.1}{Hartree}$.]{
\includegraphics[width=.45\textwidth]{figures/scaling/qubit-to-cR-all-0.1.pdf}
}
\end{minipage}
\caption{Estimated $R$ and $1/\abs{c_R}^2$ for various molecules with the same setup as Figs. \ref{fig:scaling-a} and \ref{fig:scaling-b}. The number of qubits is varied by changing the number of atoms for hydrogen chains, and by changing the active space for the other molecules.}
\label{fig:all-scaling}
\end{figure*}
\subsection{Sampling simulations with various molecules}
\label{ssec:appendix-sampling}
In this subsection, we present similar results as Fig.~\ref{fig:conventional} but with various other molecules. The results, shown in Fig.~\ref{fig:all-sampling}, show the same features as \ce{H6} in the main text, such as the small standard deviation for QSCI and $1/\abs{c_R}^2$ giving an accurate estimation of the number of shots for given accuracy $\epsilon$. One can also see that the standard deviation is almost constant for hydrogen chains with various numbers of atoms, while the absolute error is highly dependent on the number of atoms. Comparing the three 12-qubit systems, it can be seen that the difference between the standard deviation and the absolute error depends on the system.
\begin{figure*}
\begin{minipage}{0.9\textwidth}
\subfloat[][\ce{H4} (8 qubits)]{
\includegraphics[width=.45\textwidth]{figures/sampling/H4_diff.pdf}
}
\subfloat[][\ce{H6} (12 qubits)]{
\includegraphics[width=.45\textwidth]{figures/sampling/H6_diff.pdf}
}
\subfloat[][\ce{H8} (16 qubits)]{
\includegraphics[width=.45\textwidth]{figures/sampling/H8_diff.pdf}
}
\subfloat[][\ce{H10} (20 qubits)]{
\includegraphics[width=.45\textwidth]{figures/sampling/H10_diff.pdf}
}
\subfloat[][\ce{H2O} (12 qubits)]{
\includegraphics[width=.45\textwidth]{figures/sampling/H2O_diff.pdf}
}
\subfloat[][\ce{LiH} (12 qubits)]{
\includegraphics[width=.45\textwidth]{figures/sampling/LiH_diff.pdf}
}
\end{minipage}
\caption{Sampling simulation with various molecules. See Fig.~\ref{fig:conventional} and the main text for details.}
\label{fig:all-sampling}
\end{figure*}
\subsection{Accuracy of expectation values of observables other than the Hamiltonian in QSCI}
\label{ssec:appendix-multiple-observable-accuracy}
\begin{figure}[h!]
\begin{minipage}{0.45\textwidth}
\subfloat[][$\epsilon=0.001$]{
\includegraphics[width=\textwidth]{figures/multiple_obs/multiple_obs_qsci_0.001.pdf}}
\subfloat[][$\epsilon=0.01$]{
\includegraphics[width=\textwidth]{figures/multiple_obs/multiple_obs_qsci_0.01.pdf}}
\subfloat[][$\epsilon=0.1$]{
\includegraphics[width=\textwidth]{figures/multiple_obs/multiple_obs_qsci_0.1.pdf}}
\end{minipage}
\caption{Histograms of absolute errors of nuclear gradients and Hessians for \ce{H4}, \ce{H6}, and \ce{H8} molecules in QSCI, compared to the exact, CASCI value. Each plot corresponds to QSCI calculations with an energy error $\epsilon$. Absolute errors are in units of Hartree, Hartree/\AA, or $\text{Hartree}/\text{\AA}^2$, depending on the observables. }
\label{fig:appendix-multiple-observables}
\end{figure}
Here, we examine the accuracy of the expectation values of observables other than the Hamiltonian, estimated for the output state obtained by QSCI calculation.
Figure~\ref{fig:appendix-multiple-observables} shows the histograms for absolute errors of the expectation values for the gradient and Hessian, where the absolute error for an observable $\hat{O}$ is defined by
\begin{equation}
\abs{\ev{\hat{O}}{{\psi_{\rm out}}}-\ev{\hat{O}}{\psi_\text{exact}}}.
\end{equation}
Here, $\ket{{\psi_{\rm out}}}$ is the output state of QSCI calculation with the idealized sampling from the exact ground state with $R$ given in Fig.~\ref{fig:scaling-a} for each error tolerance $\epsilon$ for energy, and $\ket{\psi_\text{exact}}$ is the exact ground state.
The observables $\hat{O}$ are set to be the nuclear gradient $\pdv{\hat{H}}{x_i}$ $(i=1,\dots,3N_\text{atom})$ and the Hessian $\pdv{\hat{H}}{x_i}{x_j}$ $(i,j=1,\dots,3N_\text{atom})$, where $N_\text{atom}$ is the number of atoms in the molecule and $x_i$ are coordinates of the nuclei.
The absolute error is shown in the unit of Hartree, Hartree/\AA, or $\text{Hartree}/\text{\AA}^2$, depending on the observables. Although there are some observables (i.e., components of the gradient or Hessian) whose expectation values exhibit larger absolute errors than that of the energy, the expectation values of the majority of the observables have similar accuracy as the energy.
\subsection{Bond length dependence}
\label{ssec:appendix-bond-length}
\begin{figure}[h!]
\includegraphics[width=.45\textwidth]{figures/bond-to-R-with-PEC.pdf}
\caption{Estimated $R$ and $1/\abs{c_R}^2$ for \ce{H2O} molecule (14 qubits) with various bond lengths. Potential energy curves are also shown for reference. The estimation method of $R$ and $1/\abs{c_R}^2$ are the same as in Figs. \ref{fig:scaling-a} and \ref{fig:scaling-b}.}
\label{fig:pec}
\end{figure}
The Hartree-Fock calculation is known to perform better for a stable geometry of a molecule than for the dissociation limit, so it is worth studying if QSCI also performs worse in the dissociation limit. Figure~\ref{fig:pec} shows the result of the same numerical analysis as Fig.~\ref{fig:scaling-a}, but for various bond lengths of \ce{H2O} molecules. The Hamiltonian is generated by the Hartree-Fock orbitals using STO-3G basis without specifying the active space, and is of 14-qubit after the Jordan-Wigner mapping. The bond lengths of two H-O bonds are taken to be equal, and the H-O-H angle is fixed to \ang{104.45}. The result implies that, although there is some dependency on the bond length for larger $\epsilon$, the dependency disappears for smaller $\epsilon$. It can be expected from the result that the potential energy surface calculated by QSCI has a relatively constant accuracy, at least compared to the Hartree-Fock result, when the error tolerance is not very large.
\subsection{Comparison to ASCI}
\label{ssec:comparison-to-asci}
Here, we investigate if there is
a possibility that QSCI outperforms the state-of-the-art selected CI methods
by taking ASCI for illustration.
ASCI is a
selected CI method
solely based on classical computation, which adaptively searches for the optimal subspace of the Fock space for the diagonalization.
In Fig.~\ref{fig:asci}, we compare QSCI with ASCI, for which we follow the description in Ref.~\cite{tubman2020modern}.
Here, we use the QSCI method with the idealized sampling from the ground state obtained by the exact diagonalization (full-CI) calculation.
The target molecule is the linear hydrogen chain \ce{H10}
with the equal separation of 1.0 \AA.
The basis set is STO-3G and the Hamiltonian with the Hartree-Fock orbitals is mapped to the 20-qubit one by the Jordan-Wigner mapping.
For ASCI, in addition to the parameter $R$ (called as $N_{tdets}$ in Ref.~\cite{tubman2020modern}), there are two additional parameters: they are denoted by $\epsilon$ and $N_{cdets}$ in that paper, and are denoted by $\delta$ and $R_{\text{core}}$, respectively, in the following. The parameters $\delta$ and $R_{\text{core}}$ determine the size of the search space for the iterative search for the new determinants, while the cost for the generation and diagonalization of the Hamiltonian, which is common for both ASCI and QSCI, are determined solely by $R$. We fixed $\delta=\SI{0.05}{Hartree}$ and $r:=R/R_{\text{core}}=10$ or $20$ for ASCI, and run QSCI and ASCI calculations with various $R$.
While in the case of $r=10$ the two methods perform similarly, QSCI performs better for $r=20$, where less computational cost is required for searching for a better set of configurations in ASCI. The result shows that, depending on the hyperparameters for ASCI, there is a possibility that QSCI performs better, at least in the case of the idealized sampling from the exact ground state.
\begin{figure}[h!]
\includegraphics[width=.45\textwidth]{figures/ASCI_H10_d1.0_epsH0.05.pdf}
\caption{Comparison of ASCI and QSCI for \ce{H10} molecule (20 qubits). The ASCI and QSCI energies as the difference to the exact CASCI energy are plotted against the size $R$ of the selected subspace. The parameter $r$ determines the size of the search space for ASCI.}
\label{fig:asci}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
In this work, we proposed QSCI, a class of hybrid quantum-classical algorithms, to find low-lying eigenvalues and eigenstates of a many-electron Hamiltonian.
Taking rough approximations of such eigenstates as input, QSCI selects important electron configurations to represent the eigenstates by sampling the input states on quantum computers, and then classically diagonalizes the Hamiltonian in the subspace(s) spanned by the selected configurations to yield better approximations for the eigenstates and their energies.
QSCI is robust against noise and statistical fluctuation, as quantum computation is used only to define the subspaces. A quantum speed-up potentially arises in that sampling a quantum state is, in general, classically intractable.
We verified the algorithms for ground and excited states of small molecules by numerical simulations and experiment, where the latter was conducted on the quantum device with the 8-qubit quantum circuits.
We discussed potential utility of QSCI in various aspects: for instance, taking a state obtained by VQE as the input state, QSCI can be used to refine the VQE result, which may not be accurate enough due to statistical fluctuation, physical noise, and poor optimization;
QSCI can be used as a technique for eigenstate tomography, which enables estimation of a variety of observables with no additional quantum computational cost.
We also argued that QSCI is potentially feasible to tackle challenging molecules such as the chromium dimer by exploiting quantum devices with several tens of qubits, assisted by a high-performance classical computing resource for diagonalization.
\begin{acknowledgements}
The authors thank Amazon Web Services for supporting this work through their Amazon Braket service.
A part of this work is supported by JST PRESTO JPMJPR2019 and JPMJPR191A, MEXT Quantum Leap Flagship Program (MEXTQLEAP) Grant No. JPMXS0118067394 and JPMXS0120319794, and JST COI-NEXT program Grant No. JPMJPF2014.
\end{acknowledgements}
\section{Discussion}
\label{sec:discussion}
In this section, we discuss various aspects of QSCI. We start from its classical and quantum computational costs in Sec.~\ref{ssec:discussion-computational-cost}, and then discuss its benefits for refining VQE results in Sec.~\ref{ssec:discussion-qsci-refine-vqe}. In Sec.~\ref{subsec:state-preparation}, several ideas for preparing input states are introduced. The aspect of QSCI as a selected CI is discussed in Sec.~\ref{ssec:discussion-qsci-as-selected-ci}, and
ideas for future directions are finally introduced.
\subsection{Computational costs}
\label{ssec:discussion-computational-cost}
Here classical and quantum computational costs are examined.
In QSCI, classical computing is used for generating the truncated Hamiltonian matrix $\bm{H}_R$ and diagonalizing it.
Exploiting the Slater-Condon rules, one can generate the sparse matrix $\bm{H}_R$
efficiently in both $R$ and the number of orbitals (see, e.g., Ref.~\cite{tubman2020modern} for details). For diagonalizing $\bm{H}_R$, one can employ algorithms
to diagonalize a sparse matrix, such as the Lanczos method or the Davidson method.
The generation and diagonalization of the Hamiltonian matrix are common procedures in the selected CI methods, and it is reported~\cite{garniron2019quantum} that $R\simeq \SI{5e7}{}$ of Slater determinants are manageable when a state-of-the-art high-performance computing resource is available, even for the method that repeats the Hamiltonian generation and diagonalization.
In our method, such a repetition is not needed, and thus the computational cost should be smaller.
As already discussed in Sec.~\ref{subsec:scaling},
Fig.~\ref{fig:scaling-a} suggests that, for some challenging molecules of $\sim$50 qubits, the QSCI calculation is feasible in terms of the classical cost by the current state-of-the-art classical computing, while meeting the accuracy requirement of $\epsilon \lesssim \SI{0.001}{Hartree}$.
Note such a system size would be beyond the reach of the exact diagonalization.
The quantum computational time is $t_Q=N_{\text{shot}}\times t_{\text{prepare}}$,
where $N_{\text{shot}}$ is the number of shots for the sampling, i.e., the repetitions of the input-state preparation and measurement, and $t_{\text{prepare}}$ is the time needed for a single shot.
Note that the total computational time can be reduced if
multiple quantum computers are available, since the sampling procedures are completely parallelizable. $t_{\text{prepare}}$ highly depends on the type of quantum device to be used and the way to prepare the input state. For example, the Sycamore processor used in the Google's quantum supremacy experiment~\cite{arute2019quantum} can achieve $N_{\text{shot}}=\SI{1e6}{}$ in 200 seconds for a quantum circuit with 53 qubits and 20 repetitions of entangling operations, which corresponds to $N_{\text{shot}}\sim\SI{4e8}{}$ in a day.
Hence, Fig.~\ref{fig:scaling-b} implies that the sampling cost is affordable for \ce{Cr2} with several tens of qubits, while it may be challenging at the moment to achieve $\epsilon=\SI{0.001}{Hartree}$ for a hydrogen chain with, say, 50 qubits.
We remark that the sampling cost can be significantly reduced if one can
prepare a state $\ket{\Delta\psi}$ that is orthogonal to a classically tractable state $\ket{\psi_c}$ such that, for some complex numbers $\alpha$ and $\beta$,
$\ket{\psi_{\text{GS}}}=\alpha \ket{\psi_c} + \beta \ket{\Delta\psi}$
approximates the ground state, and can sample from $\ket{\Delta\psi}$ on a quantum computer. The state $\ket{\psi_c}$ can be the Hartree-Fock state or more intricate states such as the CISD state. For example, if $\abs{\alpha}^2=0.9$, then the sampling cost for a given precision can be reduced by a factor of ten. On the other hand, $\ket{\Delta\psi}$ can be prepared, e.g., by the method of Ref.~\cite{radin2021classically}.
\subsection{
Use of QSCI to refine VQE results
}
\label{ssec:discussion-qsci-refine-vqe}
QSCI can be viewed as a post-processing technique for VQE and its variants, when they are used to prepare the input states.
Our methods have the following advantages:
\begin{description}
\item[Error reduction]
By virtue of the classical diagonalization of a Hamiltonian matrix generated classically, the proposed methods can refine the VQE results,
as demonstrated in Sec.~\ref{sec:numerical} and Sec.~\ref{sec:noisy-simulation-experiment}.
Although results of noiseless VQE simulations are used to prepare the input states in our numerical and experimental studies, our results suggest that QSCI is also effective to refine \textit{dirty} VQE results subject to the statistical and physical errors.
Figure~\ref{fig:vqe-experiment} also shows the effectiveness of the post-selection: the rate of the readout error, which is one of the major sources of physical errors, can be reduced from $O(p)$ to $O(p^2)$ with the Jordan-Wigner mapping, as discussed in Appendix~\ref{subsec: post-selection}.
Note that QSCI does not require extra gate operations for the measurement, unlike expectation-value estimations in VQE.
As already shown in Fig.~\ref{fig:noiseless-vqe}, our method is also effective to improve the quality of the input state even in the absence of physical and statistical errors. This feature may enable one to use ansatzes with shallower circuits, or to reduce the number of optimization steps in VQE, by employing QSCI to improve the final result.
\item[Reliability] Our method is free of errors in the sense that the resulting ground-state energy is exact within the subspace spanned by the quantum-selected configurations.
This means that the obtained energy is a definite upper bound for the exact ground-state energy, which is not the case in conventional VQE because of physical and statistical errors, as discussed in Sec.~\ref{subsec:ground-state}.
This is advantageous for comparing the QSCI result obtained on noisy quantum devices with the results of classical variational methods such as CISD or density matrix renormalization group (DMRG)~\cite{white1992,white1993,Schollwock2005}: the variational nature of these methods guarantees that the method that gives the lowest energy is the most accurate one.
Similar variational inequalities hold for excited states in the single diagonalization scheme of QSCI, while there is no such guarantee in the sequential diagonalization scheme. Although the latter appeared to be more accurate in our numerical simulation, the former is of great use if one is interested in giving rigorous upper bounds on excited-state energies.
\item[Handiness]
As one has the classical representations of the eigenstates as output,
one can compute the expectation values of a large class of observables with no additional quantum computation.
Our method becomes more valuable
when more observables are to be evaluated, as exemplified in Fig.~\ref{fig:scaling-b}. Moreover, one can also analyze the classical vectors themselves, which may be useful to study the significance of each Slater determinant.
\end{description}
\subsection{
Use of QSCI with more general input states
}
\label{subsec:state-preparation}
As discussed in the previous sections, input states for ground state can be prepared by VQE, and those for excited states by its variants, but the proposed methods are applicable to more general input states.
Our method can in principle be applied to any kind of input states
that can be prepared and sampled on a quantum computer.
We give an incomplete list of possible preparation schemes for input states in the following:
the adiabatic state preparation~\cite{farhi2000quantum, aspuru2005simulated}, the imaginary time evolution~\cite{williams2004probabilistic, terashima2005nonunitary, mcardle2019variational,mao2022measurementbased}, classically-boosted VQE~\cite{radin2021classically}, classically-optimized shallow ansatz circuits~\cite{okada2022identification}, unitary coupled-cluster ansatz circuits with classically-optimized parameters~\cite{mcclean2016theory, romero2018strategies, kuroiwa2023clifford+, hirsbrunner2023beyond}, and parametrized states classically optimized by Clifford circuits~\cite{mitarai2022quadratic, ravi2022cafqa}.
Note that the performance of QSCI depends on the quality of the input state and also on the form of the exact eigenstate. For example, if the exact eigenstate is the equal superposition of all the computational basis states, then our algorithm will not perform well.
The algorithm can also be useful for a Hamiltonian that has an exactly-known ground state. For example, one can calculate an exact ground-state energy of a system that is solvable with the Bethe ansatz, but there are quantities, such as a class of correlation functions, that cannot be computed efficiently~\cite{verstraete2009quantum}. Our method provides
the classical representation of an approximate eigenstate,
which means that one can evaluate various physical quantities without additional quantum resource, as we already discussed for states prepared by VQE. The preparation of the Bethe ansatz states on quantum computers is addressed in Refs.~\cite{van2021preparing, sopena2022algebraic}.
Moreover, although we proposed the method as a hybrid quantum-classical algorithm, one can apply the method to input states that can be sampled efficiently on classical computers.
This is shortly discussed in Sec.~\ref{ssec:outlook}.
\subsection{
QSCI as selected CI
}
\label{ssec:discussion-qsci-as-selected-ci}
As selected CI methods, the novelty of QSCI comes simply from how to define the subspace on which we construct the subspace Hamiltonian. Quantum computers are used to sample important configurations from the input state, and there is a quantum speed-up when the input state is hard to sample classically. In selected CI methods, the subspace of the Fock space for the diagonalization is either fixed by the method, e.g., CISD, or adaptively chosen according to the algorithm. We have shown experimentally that CISD performs worse even when compared to the QSCI result on the current NISQ device (Fig.~\ref{fig:vqe-experiment}).
One of the most advanced methods for sampling dynamically important bases is the adaptive sampling configuration interaction (ASCI) algorithm developed by Tubman and co-workers~\cite{tubman2016deterministic,tubman2020modern}. The idea of systematically selecting important bases based on perturbation theory was developed about 50 years ago~\cite{bender1969pr,whitten1969jcp,huron1973jcp,buenker1974tca,buenker1975tca}, and a selection scheme based on Monte Carlo methods was proposed in the 1990s~\cite{greer1995jcp,greer1998jcpss}. However, systematically selected CI was not widely used in quantum chemistry calculations for many years. Recently, it has undergone rapid development and is now becoming applicable to large-scale quantum chemical simulationst~\cite{evangelista2014jcp,holmes2016jctc,schriber2016jcp,holmes2016jctc2,tubman2016deterministic,ohtsuka2017jcp,schriber2017jctc,sharma2017semistochastic,chakraborty2018ijqc,coe2018jctc,coe2019jctc,abraham202jctc,tubman2020modern,zhang2020jctc,zhang2021jctc,chilkuri2021jcc,chilkuri2021jctc,goings2021jctc,pineda2021jctc,jeong2021jctc,coe2023jctc,seth2023jctc}. Indeed, Tubman \textit{et al.} showed that ASCI is capable of handling 34 electrons in 152 spatial orbitals~\cite{tubman2020modern}.
ASCI has hyperparameters that define the size of the search space to adaptively select the configurations, and we will see in Appendix~\ref{ssec:comparison-to-asci} that, with some set of hyperparameters, QSCI can perform better than ASCI.
\subsection{Outlook}
\label{ssec:outlook}
QSCI is applicable to diverse systems, and has many directions for generalizations.
\begin{itemize}
\item It would be possible to consider a hybrid of the proposed method and another adaptive selected CI method, such as ASCI, by combining the configurations suggested by QSCI with those of the other method. In this way, one could improve the results of the state-of-the-art selected CI methods by using quantum computers.
\item QSCI is essentially a selected CI where the configurations are randomly selected according to a probability distribution $p(x) = \abs{\braket{x}{\psi_{\mathrm{in}}}}^2$. A classical counterpart of this approach is called Monte-Carlo configuration interaction (MCCI)~\cite{greer1995jcp,greer1998jcpss}. MCCI does not seem to have been extensively studied since the first proposal in 1995, and the use of a more sophisticated (classical) probability distribution for MCCI is yet to be explored. It would be an interesting future work to use a classically tractable $p(x)$ for MCCI and compare/combine it with QSCI. For example, some of the tensor network states, such as the matrix product states (MPS) and the multi-scale entanglement renormalization ansatz (MERA) states, can be efficiently sampled on classical computers~\cite{ferris2012perfect}.
\item Our method, compared to the conventional VQE, has an advantage that it can evaluate physical observables classically with no additional quantum computational cost. One may leverage this feature by using QSCI for the geometry optimization problem of a molecule, or a molecular dynamics calculation. In those applications, one may skip the sampling for some iterations, and continue to run with the same state subspace defined by the $R$ electron configurations, thereby reducing the quantum computational cost further.
\end{itemize}
We remark that the performance of QSCI depends highly on the quality of the input state.
It would be great if there is a way to start from an input state with modest quality, and then improve the quality of the input state by an iterative use of QSCI.
\section{Benchmark of QSCI with noisy simulation and experiment}
\label{sec:noisy-simulation-experiment}
\begin{figure*}
\begin{minipage}{\textwidth}
\subfloat[][Noisy simulator w/o post-selection]{
\includegraphics[width=.45\textwidth]{figures/ionq-experiment/manylines_exp-sampling-noisysim-nops.pdf}
}
\subfloat[][Noisy simulator w/ post-selection]{
\includegraphics[width=.45\textwidth]{figures/ionq-experiment/manylines_exp-sampling-noisysim-ps.pdf}
}
\subfloat[][IonQ device w/o post-selection]{
\includegraphics[width=.45\textwidth]{figures/ionq-experiment/manylines_exp-sampling-ionq-nops.pdf}
}
\subfloat[][IonQ device w/ post-selection]{
\includegraphics[width=.45\textwidth]{figures/ionq-experiment/manylines_exp-sampling-ionq-ps.pdf}}
\end{minipage}
\caption{
QSCI results for the ground state of the linear hydrogen chain $\ce{H4}$ on 8 qubits by the noisy simulator [(a), (b)] and by the IonQ device [(c), (d)], with and without post-selection, compared with the conventional method of quantum-expectation value estimation and CISD, which uses 27 Slater determinants. The resulting energies are plotted in Hartree as deviations from the one obtained by the exact diagonalization (CASCI).
Lines specified by ``VQE'' show the exact energy value of the parametrized state at each iteration, and the states at four selected iterations are used as the input states for the QSCI calculations, shown by ``QSCI'' with four different values of $R$.
The conventional method uses the QWC grouping in the energy estimation.
The markers on the solid lines show the average value of some trials while each line without markers shows the result of one of the trials. The number of trials is ten for both QSCI and the conventional method on the noisy simulator, one for the conventional method and five for QSCI on the IonQ device.
}
\label{fig:vqe-experiment}
\end{figure*}
In this section, we describe the result of the experiment for the ground state of the hydrogen chain \ce{H4} (8 qubits), conducted on
the IonQ 11-qubit device through Amazon Braket service, along with the result of noisy sampling simulation using Qulacs with the identical setup. We first run a VQE calculation of a linear \ce{H4} molecule
with bond lengths \SI{1.0}{\AA} on a noiseless state-vector simulator. We use the STO-3G basis set without freezing any orbitals, and thus the problem Hamiltonian is 8-qubit. The so-called Ry ansatz with depth 8 is employed for the VQE calculation. See Appendix~\ref{ssec:setup-experiment} for details, including the circuit diagram of the ansatz. Then, we perform QSCI calculations on the quantum hardware and the noisy simulator using four sets of parameters at four distinct iterations of the VQE calculation. We use 10,000 shots for each sampling, and the most frequent $R$ configurations are selected to define the subspace, with and without the post-selection.
The post-selection of the sampling result is performed using the number of electrons $N_e=4$ and the spin $S_z=0$.
For noisy simulation, to simulate the physical noises on the device, single-qubit depolarizing noise is added after each gate and bit-flip noises are added at the end of the circuit to mimic the measurement error. The level of each type of the noise is determined by the single-qubit and two-qubit gate fidelities, and the measurement fidelity of the actual device: 99.61\%, 96.868\%, and 99.824\%, respectively\footnote{More precisely, the error rate of the single-qubit depolarizing noise for each single-qubit gate is set to $p_1$, where $p_1$ is the single-qubit gate infidelity. For the two-qubit gates, single-qubit depolarizing noise is applied to each of the two qubits with probability $1-\sqrt{1-p_2}$ for a two-qubit gate infidelity $p_2$. The bit-flip noise is applied to each qubit with probability $p_{\text{ro}}$, the measurement infidelity.}.
For comparison, on the quantum device and the noisy simulator, the calculation of the expectation value of the energy using a conventional method is performed with 10,000 shots. The QWC grouping and the shot allocation optimized for Haar random states are employed.
Error mitigation techniques, which may improve the result at the cost of additional quantum resources, are not employed in this study.
The results are presented in Fig.~\ref{fig:vqe-experiment}. By comparing the results from the noisy simulator and the quantum device, one can see that they have a reasonable agreement, although the result from the quantum device seems to be more affected by the errors. Moreover, it is clear that the post-selection is powerful in both simulation and experiment. It is particularly worth noting that even on the physical device, some of the QSCI calculations with $R=27$ do outperform the result of CISD, which also diagonalizes the subspace Hamiltonian with 27 Slater determinants, and achieve the chemical accuracy on the 8-qubit system.
Some minor comments are in order: firstly, at the earlier iterations, the number of sampled (and post-selected) configurations was sometimes less than the given $R$, because the state is concentrated in some computational basis states. In that case, we only used the sampled configurations for the QSCI calculation; secondly, CASCI result, i.e., the exact diagonalization result, corresponds to\footnote{The number of Slater determinants which have the required particle number and $S_z$ is $\binom{4}{2}\cdot \binom{4}{2}=36$.} $R=36$; this number may seem to be comparable to $R=27$, but it is still a non-trivial task to choose 27 configurations out of 36 possibilities.
\section{Introduction}
Recent years have seen a rapid development of quantum computers towards their practical use. Although current quantum devices are prone to errors due to physical noise, ways to achieve \textit{quantum advantage} over classical computations have been explored experimentally~\cite{arute2019quantum, zhong2020quantum, madsen2022quantum},
and such noisy intermediate-scale quantum (NISQ) devices are believed to become useful in the near future~\cite{preskill2018quantum}.
Quantum chemistry is at the top of the list of such useful applications (see, e.g., Refs.~\cite{cao2019quantum, mcardle2020quantum, cerezo2021variational, bharti2022noisy, tilly2022variational}): for instance, energy eigenvalues of a molecular Hamiltonian can be calculated by quantum algorithms developed for NISQ devices, where the most notable is the variational quantum eigensolver (VQE)~\cite{peruzzo2014variational} to find the ground-state energy.
However, VQE faces several challenges to be overcome for practical use.
The major obstacle comes from errors caused by statistical fluctuation and physical noise inherent in the noisy devices. Suppressing the statistical error to a practically acceptable level needs a prohibitively large number of samples~\cite{gonthier2020identifying}, and error mitigation techniques~\cite{viola1999dynamical, temme2017error, li2017efficient,endo2018practical,koczor2021exponential,huggins2021virtual,mcardle2019error,bonet2018low,maciejewski2020mitigation,endo2021hybrid} for reducing physical noise require even more samples to compensate the additional statistical error they introduce~\cite{wang2021can,takagi2022fundamental, tsubouchi2022universal,takagi2022universal}.
In particular, the effect of the errors can spoil the \textit{variational} nature of VQE: that is, the energy estimated by quantum devices is not guaranteed to give an upper bound on the exact ground-state energy.
This is problematic because lowering the resulting energy of VQE does not necessarily mean approaching to the exact ground state.
Besides, there are other challenges for VQE such as the barren plateau problem, which can interrupt the optimization~\cite{mcclean2018barren}.
In this paper, we propose a class of hybrid quantum-classical methods, which we call quantum-selected configuration interaction (QSCI), to find low-lying eigenvalues and eigenstates of a many-electron Hamiltonian.\footnote{We focus on applications to quantum chemistry in this paper. However, the proposed methods can be applied to a variety of many-body Hamiltonians, including many-electron and spin problems in condensed matter physics.} QSCI is noise resilient
and, in principle, free of costly optimization of parametrized quantum circuits.
In particular, QSCI sets rigorous upper bounds on the ground-state energy\footnote{QSCI can also set rigorous upper bounds on the excited-state energies, depending on its algorithmic implementation.} even under the effect of physical and statistical errors.
Here we outline a version of QSCI for finding a ground state:
suppose that an approximate ground state, which we call an \textit{input state} in this paper, can be prepared on a quantum computer; one then repeats a measurement of the state to identify the computational basis states, or electron configurations, that are important to express the ground state~\cite{kohda2022quantum};
one then diagonalizes, on classical computers, the truncated Hamiltonian matrix in the subspace spanned by the identified configurations to obtain the smallest eigenvalue and eigenvector.
The resulting eigenvalue approximates the ground-state energy.
The diagonalization is classically tractable unless the number of selected configurations is exponentially large in the system size.
The algorithm can be extended to find excited states by enlarging the subspace or by repeating the procedure for each energy eigenstate.
Since the matrix elements of the Hamiltonian in the computational basis can be exactly calculated on classical computers, the diagonalization results in an energy that gives a definite upper bound on the exact ground-state energy regardless of the quality of the subspace spanned by the identified configurations; the quality only affects how tight the bound is. The states need to be measured only in the computational basis, and thus no additional gate operation is required for the measurement. In the presence of symmetries with conserved quantities such as the particle number, the post-selection of the computational basis states in the sampling outcome allows one to mitigate the bit-flip errors. We experimentally demonstrate the effectiveness of the post-selection in this paper.
The algorithm may take any quantum states as the input states, if they roughly approximate the desired eigenstates and can be prepared on quantum devices. Such input states can be prepared, e.g., by parametrized quantum circuits moderately optimized via
VQE and its variants~\cite{tilly2022variational}, and other preparation schemes are discussed in Sec.~\ref{subsec:state-preparation}. Sampling from such quantum states can be hard for classical computers~\cite{arute2019quantum}, and thereby providing a potential quantum speed-up in QSCI.
QSCI can also be advantageous as a technique for \textit{eigenstate} tomography in that it can (classically) estimate the expectation values of a variety of observables at no additional quantum cost: as we already have the classical representation of the state, one can efficiently compute the expectation values using that representation.
Unlike QSCI, other efficient tomography techniques such as classical shadows~\cite{huang2020, zhao2021}, neural network tomography~\cite{Torlai_2018}, and tensor network tomography~\cite{Cramer_2010} do not exploit the fact that the states of our interest are eigenstates of some problem Hamiltonian.
As the name suggests, QSCI can be viewed as a configuration interaction (CI), where the many-body basis set is determined by quantum computers via sampling of an input state.
There are established techniques~\cite{helgaker2014molecular} that choose fixed basis sets.
A common approach in electronic structure theory is to select only one- and two-particle excitations from a reference wavefunction. When the reference wavefunction is chosen to be Hartree-Fock, the resulting method is known as CI with singles and doubles (CISD). If the reference wavefunction is a correlated wavefunction beyond the mean-field approximation, the method is called multi-reference CISD (MR-CISD). In the context of quantum computing, MR-CISD has sometimes been called as quantum subspace expansion (QSE)~\cite{mcclean2017hybrid,takeshita2020increasing,urbanek2020jctc}.
Another approach is the adaptive selection of a suitable basis set for a target system. In quantum chemistry, a systematic selection of important bases has a long history~\cite{bender1969pr,whitten1969jcp,huron1973jcp,buenker1974tca,buenker1975tca,nakatsuji1983cluster,cimiraglia1987jcc,harrison1991jcp,greer1995jcp,greer1998jcpss}. And there are recently active studies along with such a systematic selected CI~\cite{evangelista2014jcp,holmes2016jctc,schriber2016jcp,holmes2016jctc2,tubman2016deterministic,ohtsuka2017jcp,schriber2017jctc,sharma2017semistochastic,chakraborty2018ijqc,coe2018jctc,coe2019jctc,abraham202jctc,tubman2020modern,zhang2020jctc,zhang2021jctc,chilkuri2021jcc,chilkuri2021jctc,goings2021jctc,pineda2021jctc,jeong2021jctc,coe2023jctc,seth2023jctc}. Thanks to such developments, systematic selected CIs are now gradually being considered as a promising approach for large-scale quantum chemical simulations. The likely reason for this revival is that selected CI is an algorithm that can be adapted to current classical computer architectures with sufficient memory. QSCI may be seen as a new systematic selected CI that utilizes quantum computers.
Our methods are capable of selecting electron configurations which are necessary to describe the eigenstates to some accuracy but are missed in the conventional methods with a fixed basis set. Note that our methods call the diagonalization procedure at most only once for each eigenstate, while the adaptive methods iteratively repeat the diagonalization to search for a configuration to be added in the basis set; our methods require much less classical computational time compared to those adaptive methods.
The classical diagonalization is already utilized in various hybrid quantum-classical algorithms to find energy eigenstates. Most notable is QSE, which spans the subspace by states built upon the reference VQE state,
and is widely used for various applications, e.g., excited state calculations~\cite{mcclean2017hybrid}, band structure calculations~\cite{yoshioka2022variational}, and noise reduction~\cite{mcclean2017hybrid,bonet2018low, takeshita2020increasing,mcclean2020decoding,yoshioka2022generalized, epperly2022theory}.
More generally, one can span the subspace by various methods~\cite{huggins2020non,motta2020determining, parrish2019quantumfilter, stair2020multireference,parrish2019quantum, seki2021quantum,baek2022say, kirby2022exact}, which are sometimes collectively called as the quantum subspace diagonalization. In those methods, however, the matrix elements of the subspace Hamiltonian are calculated on quantum computers, and thus are subject to the physical and statistical errors. There is a proposal~\cite{radin2021classically} where some of the matrix elements are classically calculated, but the method still requires some matrix elements which are efficiently computable only by quantum computers for a possible quantum speed-up. In QSCI, on the other hand, all the matrix elements are classically computed, giving up the use of more complex and physically-motivated states as basis states that define the subspace.
The rest of the paper is organized as follows. The proposed methods are introduced in Sec.~\ref{sec:methods}, and numerically tested in Sec.~\ref{sec:numerical}. A demonstration on a quantum device is presented in Sec.~\ref{sec:noisy-simulation-experiment}, along with a noisy simulation as a preparatory study. We discuss aspects of the proposed methods in Sec.~\ref{sec:discussion}, and finally conclude in Sec.~\ref{sec:conclusion}. Details of the algorithms, numerical simulations and experiment, as well as supplemental numerical results are given in the appendices.
\section{Methods}
\label{sec:methods}
In this section, we present the methods of QSCI.
Two ways of implementation are introduced: single diagonalization scheme in Sec.~\ref{sssec:method-single} and sequential diagonalization scheme in Sec.~\ref{sssec:method-sequantial}.
They are designed for finding multiple energy eigenstates, and reduce to the same simplified method when used for finding the ground state alone.
After introducing necessary ingredients, we begin with the algorithm specific to finding the ground state, which is simple and illustrative, and then proceed to the two methods which can also find excited states.
\subsection{
Preliminary
}
We consider electronic structure problems of molecules in the second-quantization formalism with the Born-Oppenheimer approximation.
A Hamiltonian and wave functions for electrons, in this setup, can be mapped onto $N_q$ qubits such that the Slater determinants\footnote{Instead, linear combinations of Slater determinants such as configuration state functions may be mapped to the computational basis states. QSCI can work with such a mapping, if the matrix elements of the Hamiltonian in the computational basis can be efficiently computed by classical computation.} for the Hartree-Fock state and its excitations are associated with the computational basis states $\ket{x}$, where $x\in{\{0, 1\}^{N_q}}$ is an $N_q$-bit string (see, e.g., Ref.~\cite{cao2019quantum,mcardle2020quantum}).
In the Jordan-Wigner mapping, which we adopt in the numerical study, $N_q$ corresponds to the number of spin orbitals, and ``1'' or ``0'' represents whether each spin orbital is occupied or not.
The methods can work with other mapping schemes such as the Bravyi-Kitaev mapping~\cite{bravyi2002fermionic}, although the fermion-qubit correspondence is less intuitive and the error mitigation (discussed later) is less effective.
We denote the qubit Hamiltonian by $\hat{H}$.
A linear combination of all the computational basis states,
\begin{align}
\ket{\psi}
=\sum_{x\in{\{0, 1\}^{N_q}}} \alpha_x \ket{x},
\label{eq:general_state}
\end{align}
encompasses the full-CI wave function.
Note that for a fixed number of electrons only a subset of the computational basis states is needed.
In the full-CI method, sets of the CI coefficients \{$\alpha_x$\} that correspond to energy eigenstates are found by diagonalizing the Hamiltonian in the full Fock space.
The method is costly due to the combinatorial growth of the Fock-space dimension as the number of spin-orbitals increases.
For reducing the computational cost, there exist various classical approaches which truncate the Fock space and approximate the sum in Eq.~\eqref{eq:general_state} using a fixed or adaptively selected basis set, as mentioned in the previous section.
In line with these efforts, but from a different viewpoint, we propose methods which harness quantum computers to identify important computational basis states, or electron configurations, for truncating the Fock space.
\subsection{QSCI for ground state
\label{subsec:ground-state}
}
We now describe the explicit algorithms.
We begin with the algorithm for finding the lowest eigenvalue and the corresponding eigenstate (ground state) of an electronic Hamiltonian $\hat{H}$ on $N_q$ qubits.
For simplicity, we assume the ground state is unique.
When the degeneracy exists, the algorithms given in the next subsection, which is aimed at finding multiple eigenstates, can be straightforwardly applied.
Indeed, the algorithm introduced in this subsection is a special case of each of the two algorithms in the next subsection.
Let $\ket{{\psi_{\rm in}}}$ be an input state, which roughly approximates the ground state, and suppose $\ket{{\psi_{\rm in}}}$ can be prepared by a quantum circuit with $N_q$ qubits.
Then, one prepares the input state on a quantum computer and measures the state in the computational basis, which results in an outcome bit string $x\in{\{0, 1\}^{N_q}}$.
Repeating such a sampling procedure (or shot) for $N_{\rm shot}$ times, one counts how many times each $x$ appears.
Based on the total sampling result, the most frequent $R$ computational basis states are selected to define the set
\begin{align}
\mc{S}_R = \{ \ket{x} | x\in{\{0, 1\}^{N_q}}, R~{\rm most~frequent} \},
\label{eq:set_GS}
\end{align}
where $R$ is a positive integer manually determined.
This is to truncate the Fock space.
One may in principle include all the computational basis states appeared in the measurements, while choosing an appropriately small $R$ can reduce the computational cost for diagonalization.
One then solves the eigenvalue problem in the subspace spanned by $\mathcal{S}_R$:
\begin{align}
\bm{H}_R\bm{c} = E_R\bm{c},
\end{align}
where $\bm{H}_R$ is the $R\times R$ Hermitian matrix defined by
\begin{align}
(\bm{H}_R)_{xy}= \mel{x}{\hat{H}}{y}~{\rm for}~\ket{x}, \ket{y} \in \mathcal{S}_R,
\end{align}
and $\bm{c}$ is an eigenvector with eigenvalue $E_R$, satisfying $\bm{c}^\dagger \bm{c}=1$.
This step of the algorithm proceeds via classical computations: calculations of the matrix elements $\mel{x}{\hat{H}}{y}$ and the diagonalization of $\bm{H}_R$.
The former calculations can be efficiently done by some classical method, e.g., by the Slater-Condon rules in the fermionic basis.
The latter diagonalization is performed to obtain the smallest eigenvalue $E_R$ and the eigenvector $\bm{c}$, which are output of the algorithm.
See Sec.~\ref{ssec:discussion-computational-cost} for further discussion on costs of these classical computations.
Here, $E_R$ approximates the exact ground-state energy of $\hat{H}$, while $\bm{c}$ approximately gives the (normalized) CI coefficients, or the vector representation of the ground state, respectively.
The corresponding quantum state, which we call the \textit{output state}, is constructed as
\begin{align}
\ket{{\psi_{\rm out}}}=\sum_{\ket{x}\in \mathcal{S}_R} c_x \ket{x},
\label{eq:output-state-gs}
\end{align}
where $c_x$ is an element of the eigenvector $\bm{c}$.
The output state $\ket{{\psi_{\rm out}}}$ approximates the true ground state of $\hat{H}$.
We remark that one does not need to realize the output state on quantum computers. Retaining the eigenvector $\bm{c}$ as classical data is enough for the application explained below.
The output state can be used to estimate the expectation values of observables other than the Hamiltonian for the ground state, solely based on classical computations.
Specifically, suppose that an observable in question is represented by a qubit operator $\hat{O}$. If the matrix elements $\mel{x}{\hat{O}}{y}$ can be efficiently computed on classical computers, so does the expectation value $\ev{\hat{O}}{{\psi_{\rm out}}}$, which is expected to give an approximation to the expectation value for the true ground state.
In particular, if $\hat{O}$ can be expressed as a linear combination of $\text{poly}(N_q)$ Pauli strings,
which is the case in many physical quantities, its expectation value can be efficiently computed on classical computers.
Comments are in order for technical details.
We identify the set $\mathcal{S}_R$ to span the subspace by sampling the input state.
In this way, we expect that important computational basis states, or Slater determinants, to describe the ground state wave function can be selected.
This is because in the sampling procedure a bit string $x$ occurs with the probability $\abs{\bra{x}\ket{{\psi_{\rm in}}}}^2$, while $\bra{x}\ket{{\psi_{\rm in}}}$ gives the CI coefficient of the corresponding Slater determinant in the input wave function $\ket{{\psi_{\rm in}}}$.\footnote{See, e.g., Eq.~\eqref{eq:general_state}. There, the CI coefficients can be expressed as $\alpha_x = \bra{x}\ket{\psi}$.}
Indeed, $\mathcal{S}_R$ gives the $R$ Slater determinants with the largest coefficients $\abs{\bra{x}\ket{{\psi_{\rm in}}}}$ in $\ket{{\psi_{\rm in}}}$, under the ideal situation where physical noise can be ignored and the sampling is performed with an infinite number of shots.
In passing, we sometimes adopt a method equivalent to this ideal situation to define $\mc{S}_R$ in the numerical study: that is, we just pick up the $R$ Slater determinants with the largest absolute values of the CI coefficients in the input state, instead of performing actual sampling procedures.
We call this method as the \textit{idealized sampling} in this paper.
Note that we assume the input state roughly approximates the true ground state.
This is just to ensure the two states share the important computational basis states, and there is no need for a precise agreement between the CI coefficients of the two states.
Such an input state can be prepared, e.g., by a parametrized quantum circuit moderately optimized via VQE.
In Sec.~\ref{subsec:state-preparation}, we discuss methods to prepare the input state, including non-VQE based ways.
The set $\mathcal{S}_R$ is defined in Eq.~\eqref{eq:set_GS} by specifying $R$, the number of the computational basis states retained in the subspace.
But this is not the unique choice.
For instance, one may define the set by taking all the computational basis states in the measurement outcome, as already mentioned.
Or, one may instead set a threshold on the rate of occurrence $f_x$ for an outcome $x$ in the total sampling result, and then define an alternative set $\mathcal{S}_\epsilon = \{ \ket{x} | f_x \geq \epsilon \}$ with a threshold parameter $\epsilon$,
For a proof-of-principle demonstration, we adopt Eq.~\eqref{eq:set_GS} to define the subspace for diagonalizing the Hamiltonian in the rest of the paper.
In reality, physical noise and statistical fluctuation, the latter due to a finite number of shots, cannot be ignored, causing some errors in the output.
However, the effect is only indirect and the method is robust against those errors: that is, the errors can degrade the quality of the selected subspace by missing important configurations or by picking up irrelevant configurations in the sampling procedures, but the lowest eigenvalue and eigenvectors are exact within the subspace.
The latter point, the exactness within the subspace, results from the use of diagonalization for the matrix $\bm{H}_R$, whose elements are exactly computed.
Consequently, the obtained energy $E_R$ sets an upper bound on $E_{\rm exact}$, the true ground-state energy of $\hat{H}$:
\begin{align}
E_{\rm exact}\leq E_R.
\label{eq:variational-inequality}
\end{align}
Note that this variational inequality holds even under statistical fluctuation and physical noise.
The situation is in contrast with VQE, where such an inequality is not guaranteed as the energy is directly measured on quantum computers and hence is susceptible to the errors.\footnote{In VQE, physical noise in the state preparation can hardly lead to the violation of the variational inequality, but it may be possible that an error during the measurement procedure causes it. The use of error mitigation techniques can also lead to the breakdown of the inequality.
}
It is also worth mentioning that, for a given sampling outcome, increasing $R$, the subspace size, always leads to a better approximation of the ground-state energy: $E_{\rm exact} \leq E_{R_a} \leq E_{R_b}$ for $R_a > R_b$.
This can be used to see if the calculation converges.
On the other hand, smaller $R$ can reduce the classical computational cost.
Such a trade-off between the accuracy and cost is discussed in Sec.~\ref{subsec:scaling}.
The algorithm finds the lowest energy state in the subspace $\mc{S}_R$, which gives an approximation to the ground state in the {\it full} Fock space.
When there exists symmetry in the Hamiltonian, there are associated conserved quantities, e.g., the total electron number $N_e$ (or the charge of molecule) and the $z$-component of total electron spin $S_z$.
Given this, one may wish to find the lowest energy state in a specific symmetry sector. In such a case, the method can be similarly applied but by relying on the subspace with fixed conserved quantities. For $N_e$ and $S_z$, this can be easily achieved as follows since each computational basis state corresponds to a Slater determinant with definite $N_e$ and $S_z$: one prepares an input state with the desired values of $(N_e, S_z)$, for which the sampling results in configurations each with the desired $(N_e, S_z)$; or, if such an input state cannot be prepared, one may post-select the sampling outcome, where one discards an outcome $x\in{\{0, 1\}^{N_q}}$ if it conflicts with the desired $(N_e,S_z)$. It is worth noting that the variational inequality~\eqref{eq:variational-inequality} still holds in each sector of Fock space specified by $(N_e, S_z)$.
\begin{figure*}
\includegraphics[width=\textwidth]{figures/schematic/qsci-for-ground-state.pdf}
\caption{Schematic description of the QSCI algorithm for finding the ground state. When selecting the configurations, one may post-select the configurations by using conserved quantities such as the electron number or spin $S_z$ to mitigate the errors.}
\label{fig:algorithm-gs}
\end{figure*}
Physical noise can cause a contamination of symmetry sectors: for an input state with fixed $(N_e, S_z)$, sampling on a noisy device can result in electron configurations with unwanted values of $(N_e, S_z)$, due to the bit-flip noise\footnote{Note that an error that corresponds to a phase-flip error occurring at the end of a circuit does not affect the probability distribution $\abs{\bra{x}\ket{{\psi_{\rm in}}}}^2$ and hence the sampling outcome.} or readout error.
Nevertheless, one can mitigate such errors by post-selecting the sampling outcome according to the conserved quantities, as described above.
One then diagonalizes the Hamiltonian in the post-selected subspace.
We find that the post-selection is particularly effective to mitigate the readout error in the Jordan-Wigner mapping, while it is also applicable to other fermion-qubit mapping schemes (see Appendix~\ref{subsec: post-selection} for discussions).
The algorithm is schematically summarized in Fig.~\ref{fig:algorithm-gs}.
\subsection{
QSCI for multiple energy eigenstates
\label{subsec:excited-states}
}
\begin{figure*}
\begin{minipage}{\textwidth}
\subfloat[][Single diagonalization]{
\includegraphics[width=\textwidth]{figures/schematic/qsci-for-excited-state-single.pdf}}
\subfloat[][Sequential diagonalization]{
\includegraphics[width=\textwidth]{figures/schematic/qsci-for-excited-state-multi.pdf}}
\end{minipage}
\caption{Schematic descriptions of the QSCI algorithms for finding the ground state and the first excited state: (a) single diagonalization scheme, and (b) sequential diagonalization scheme. In both panels, $|\psi_{\rm in}^{(0)}\rangle$ ($|\psi_{\rm in}^{(1)}\rangle$) is the input state for the ground (first excited) state. In the panel (b), the overlap term is constructed from the preobtained output state $|\psi_{\rm out}^{(0)}\rangle$ to define the effective Hamiltonian for the first excited state, $\hat{H}^{(1)} =\hat{H}+\beta_0|\psi_{\rm out}^{(0)}\rangle\langle \psi_{\rm out}^{(0)}|$.}
\label{fig:algorithm-es}
\end{figure*}
We now extend the algorithm to find multiple energy eigenstates, including low-lying excited states.
For this, we note that the previous algorithm can output multiple eigenvectors, which can be taken to approximate excited states as well as the ground state.
Yet, the quality of the approximation would not be satisfactory for the excited states, as the subspace is tailored for the ground state.
Hence, we introduce extra input states to construct subspace(s) which can capture the excited states.
In the following, we present two distinct algorithms to find multiple energy eigenstates and energies, schematically shown in Fig.~\ref{fig:algorithm-es}.
The first algorithm, which we call the single diagonalization scheme, constructs a common subspace for both ground and excited states of interest, and performs the diagonalization in the subspace to simultaneously obtain all the desired eigenstates and energies.
On the other hand, the second algorithm, dubbed as the sequential diagonalization scheme, constructs multiple subspaces, each tailored for each energy eigenstate, and sequentially diagonalizes the Hamiltonian in each subspace.
Both of the algorithms contain the algorithm specific to the ground state, introduced in the preceding subsection, as a special case.
\subsubsection{
Single diagonalization scheme
\label{sssec:method-single}
}
Here we describe the single diagonalization scheme.
Suppose one seeks for $N_s$ low-lying eigenstates of $\hat{H}$, which consist of the ground state(s) and subsequent excited states.
In this case, one prepares multiple input states $|\psi_{\rm in}^{(i)}\rangle$ ($i=0,1,\cdots,N_{\rm in}-1$), which correspond to the low-lying energy eigenstates.
Here, we allow $N_{\rm in}\leq N_s$, although the natural choice would be $N_{\rm in}=N_s$.
For each of the input states, one repeats the sampling procedure as in the previous subsection.
One then obtains the set of important configurations $\mc{S}_{R_i}^{(i)}$, formed by most frequent $R_i$ bit strings in the total sampling outcome for the $i$-th input state.
Combining all the sets $\mc{S}_{R_i}^{(i)}$, one constructs the common subspace\footnote{Note that the set $\mc{S}_R$ defined here agrees with the definition~\eqref{eq:set_GS} in the preceding subsection when $N_{\rm in}=1$.}:
\begin{align}
\mc{S}_R =
\mc{S}_{R_0}^{(0)} \cup \mc{S}_{R_1}^{(1)} \cup \cdots \cup \mc{S}_{R_{N_{\rm in}-1}}^{(N_{\rm in}-1)}.
\label{eq:set_single-step}
\end{align}
In this case, the parameters $R_i$ may be eigenstate dependent, while $R$ is the number of the elements in the common subspace $\mc{S}_R$.
$R\geq N_s$ is required to yield at least $N_s$ eigenvectors in the diagonalization procedure shortly explained.
One may treat all $R_i$ as free parameters, which determine $R$ in turn.
Or, one may first choose a value for $R$ and, then, decide each $R_i$ following some strategy.
There are various ways for the latter strategy, depending on the purpose of using the algorithm.
For example, if one prioritizes the ground state in terms of accuracy, a possible choice would be $R_0=R$ and $R_i=0$ for $i\neq 0$, albeit extreme.
Or, if one wishes to treat all the input states on equal footing, each of $R_i$ can be chosen as equal as possible.\footnote{One can make each of $R_i$ as equal as possible by the following cycle of procedures, starting from an empty set $\mathcal{S}_R$, for a given $R$: in the first cycle, for each of the $N_\text{in}$ input states, the most frequent bit string is selected from the sampling outcome and then added to $\mathcal{S}_R$;
this procedure is executed from the 0-th input state to $(N_{\rm in}-1)$-th input state, where one skips the state if the selected bit string already exists in $\mathcal{S}_R$;
in the second cycle, the second frequent bit string is added to $\mathcal{S}_R$ for each input state according to the same rule;
one repeats such a cycle until $\mathcal{S}_R$ is filled with $R$ distinct bit strings.
Suppose such procedures finished after completing $R'$ cycles.
Then, one can ensure that at least $R'$ most frequent bit strings for each input state are included in $\mathcal{S}_R$.
This implies $R'$ or $R'+1$ most important configurations in each input state are included in the common subspace~\eqref{eq:set_single-step}, in the ideal situation where statistical fluctuation and physical noise can be ignored.}
With the common subspace $\mc{S}_R$ constructed, one then diagonalizes the Hamiltonian in $\mc{S}_R$ as in the previous subsection: one constructs the $R\times R$ Hermitian matrix $\bm{H}_R$, solves the eigenvalue equation $\bm{H}_R\bm{c} = E_R\bm{c}$, and then picks up $N_s$ low-lying eigenvectors and eigenvalues, $(\bm{c}^{(0)}, E_R^{(0)}), (\bm{c}^{(1)}, E_R^{(1)}), \cdots, (\bm{c}^{(N_s-1)}, E_R^{(N_s-1)})$, where $\bm{c}^{(i)\dagger} \bm{c}^{(j)}=\delta_{ij}$.
Here, $E_R^{(i)}$ ($E_R^{(0)}$) approximates the true energy of the $i$-th excited state (ground state), when the ground state is unique, for instance.
The corresponding output states can be constructed as
\begin{align}
|\psi_{\rm out}^{(i)}\rangle =\sum_{\ket{x}\in \mathcal{S}_R} c_x^{(i)} \ket{x},
\label{eq:output-state_single-step}
\end{align}
for $i=0,1,\cdots, N_s-1$.
Note that the algorithm in the previous subsection is a special case of the single diagonalization scheme with a single input state ($N_{\rm in}=1$).
In this method, one can apply the same error mitigation technique by the post-selection as described in the previous subsection.
The variational inequality now holds for each of energy eigenstates by Cauchy's interlace theorem~\cite{helgaker2014molecular} (see also Refs.~\cite{hylleraas1930numerical, macdonald1933successive}):
\begin{align}
E^{(i)}_{\rm exact} \leq E_R^{(i)},
\label{eq:variational-inequality-single}
\end{align}
for $i=0,1,\cdots, N_s-1$, where $E^{(i)}_{\rm exact}$ is the $i$-th eigenvalue (in ascending order) by the exact diagonalization.
We remark that QSE~\cite{mcclean2017hybrid} and multistate-contracted VQE (MCVQE)~\cite{parrish2019quantum}, which also rely on the subspace diagonalization to obtain excited states, need to measure matrix elements, while the current method exactly calculates the matrix elements.
Hence, we expect our method to be more noise-robust with the guarantee of the variational inequality.
\subsubsection{
Sequential diagonalization scheme
}
\label{sssec:method-sequantial}
We now give another scheme of QSCI to find excited states.
The sequential diagonalization finds the ground state(s) and subsequent excited states by sequential diagonalization procedures of the Hamiltonian $\hat{H}$ in distinct subspaces.
The algorithm is similar to the variational quantum deflation (VQD)~\cite{higgott2019variational}, a variant of VQE for excited states.
Suppose one seeks for the $k$-th excited state\footnote{We implicitly assume the ground state is unique for ease of illustration. One can straightforwardly translate the description here to cases of degenerate ground (and possibly excited) states.} given that preceding $(k-1)$ excited states and ground state are already obtained by this method with the output states $|\psi_{\rm out}^{(i)}\rangle$ ($i=0,1,\cdots, k-1$).
As in the previous methods, one repeats the preparation and measurement of the input state $|\psi_{\rm in}^{(k)}\rangle$ to obtain the set of important configurations:
\begin{align}
\mc{S}_{R_k}^{(k)} = \{ \ket{x} | x\in{\{0, 1\}^{N_q}}, R_k~{\rm most~frequent} \}.
\label{eq:set_multi-step}
\end{align}
One then has to find the lowest energy state of $\hat{H}$ in this subspace, under the restriction that this state is orthogonal to the states already found, $|\psi_{\rm out}^{(i)}\rangle$ ($i=0,1,\cdots, k-1$).
This can be achieved by diagonalizing the following effective Hamiltonian\footnote{This is not the unique choice of the effective Hamiltonian. For instance, the orthogonality can be imposed without introducing extra parameters though the implementation would be less suitable for NISQ devices~\cite{lee2018generalized}.} in the subspace spanned by $\mc{S}_{R_k}^{(k)}$:
\begin{align}
\hat{H}^{(k)} =\hat{H}+
\sum_{i=0}^{k-1}\beta_i
|\psi_{\rm out}^{(i)}\rangle
\langle \psi_{\rm out}^{(i)} |,
\label{eq:sequential-Heff}
\end{align}
where $\beta_i$ are real parameters, which need to be sufficiently large for ensuring the orthogonality.
The additional terms correspond to the overlap terms in VQD.
This is equivalent to solving the eigenvalue equation
\begin{align}
\bm{H}_{R_k}^{(k)}\bm{c}^{(k)} = E_{R_k}^{(k)}\bm{c}^{(k)},
\label{eq:eigenvalue-eq}
\end{align}
and then pick up the smallest eigenvalue $E_{R_k}^{(k)}$ and eigenvector $\bm{c}^{(k)}$, normalized by $\bm{c}^{(k)\dagger} \bm{c}^{(k)}=1$.
Here, $\bm{H}_{R_k}^{(k)}$ is the $R_k \times R_k$ Hermitian matrix defined by
\begin{align}
(\bm{H}_{R_k}^{(k)})_{xy}= \mel{x}{\hat{H}^{(k)}}{y}~{\rm for}~\ket{x}, \ket{y} \in \mathcal{S}^{(k)}_{R_k},
\label{eq:sequential-matrix}
\end{align}
whose matrix elements can be efficiently calculated by classical computations based on the expression
\begin{align}
(\bm{H}_R^{(k)})_{xy}
= \mel{x}{\hat{H}}{y} +\sum_{i=0}^{k-1}\beta_i c_x^{(i)}c_y^{(i)*}.
\end{align}
One then constructs the output state
\begin{align}
|\psi_{\rm out}^{(k)}\rangle =\sum_{\ket{x}\in \mathcal{S}_{R_k}^{(k)}} c_x^{(k)} \ket{x},
\label{eq:output-state_multi-step}
\end{align}
which approximates the $k$-th excited state.
Note that the expressions are specific to the $k$-th excited state.
In order to find entire (low-lying) spectrum, one has to repeat the above procedure sequentially, starting from $k=0$, the ground state, which can be found by the ground-state algorithm already explained.
This is similar to VQD, but the QSCI method does not require extra circuits to calculate the overlap terms.
The coefficients $\beta_i$ can be chosen in the same manner as VQD.
We want the smallest eigenvalue of $\bm{H}_{R_k}^{(k)}$ to approximate $E^{(k)}_{\rm exact}$, the $k$-th eigenvalue of $\hat{H}$.
Following the discussion in Ref.~\cite{higgott2019variational}, it suffices to choose $\beta_i > E^{(k)}_{\rm exact}-E^{(i)}_{\rm exact}$ for $i=0,\cdots, k-1$;
or, one may apply the looser condition of $\beta_i > 2\sum_j \abs{c_j}$, where $c_j$ are coefficients in the qubit Hamiltonian $\hat{H}=\sum_j c_j P_j$, expressed by the Pauli strings $P_j$ (see Appendix~\ref{subsec:details-sequential} for details).
In practice, the condition $\beta_i > E^{(k)}_{\rm exact} - E^{(i)}_{\rm exact}$ can be utilized if one has prior knowledge on the energy spectrum, e.g., based on variational quantum algorithms.
Even if such information is not available, one may still rely on the looser condition $\beta_i > 2\sum_j \abs{c_j}$.
Note that in the sequential diagonalization scheme, the variational inequality like Eq.~\eqref{eq:variational-inequality-single} is not guaranteed due to the inexactness of the effective Hamiltonian, i.e., as Eq.~\eqref{eq:sequential-Heff} would be constructed only by approximate eigenstates in practice (see Appendix~\ref{subsec:details-sequential} for further discussion).
\section{Benchmark of QSCI with noiseless simulations
\label{sec:numerical}
}
In this section, we test various aspects of QSCI for small molecules by noiseless numerical simulations, where the effects of physical noise are not included. In Secs.~\ref{subsec:ground-state-simulation-with-noiseless-vqe} and \ref{subsec:simulation-excited-h2o}, QSCI calculations are performed for ground states and excited states, using input states prepared by VQE and VQD~\cite{higgott2019variational}, a variant of VQE for excited states. Then the scalability of QSCI is examined in Sec.~\ref{subsec:scaling}, and finally the effect of the statistical error in QSCI is studied in Sec.~\ref{ssec:sampling-simulation}. For the numerical simulations, a quantum-circuit simulation library Qulacs~\cite{suzuki2021qulacs} is used with the help of QURI Parts~\cite{quri_parts}, a library for developing quantum algorithms. The simulations in Sec.~\ref{ssec:sampling-simulation} are carried out by the sampling simulator which takes into account the statistical error, while all the other simulations are performed by the state-vector simulator, where the expectation values are exactly calculated without errors.
For each simulation and experiment in this paper, the molecular Hamiltonian is first prepared as the second-quantized electronic Hamiltonian using the Born-Oppenheimer approximation with Hartree-Fock orbitals using the STO-3G basis unless otherwise stated, and converted to the qubit one by the Jordan-Wigner mapping. Active spaces are explicitly specified when employed, otherwise the full-space Hamiltonians are used.
The electronic Hamiltonians are generated by OpenFermion~\cite{mcclean2020openfermion} interfaced with PySCF~\cite{sun2018pyscf}. The molecular geometries and other details are shown in Appendix~\ref{sec:appendix-details-of-sim-and-exp}. Stable geometries are chosen for all the molecules except for the hydrogen chains, and a potential impact of unstable geometry is briefly analyzed in Appendix~\ref{ssec:appendix-bond-length}.
\subsection{QSCI for ground state}
\label{subsec:ground-state-simulation-with-noiseless-vqe}
\begin{figure}
\includegraphics[width=.45\textwidth]{figures/History-H2O-CAS65-log_20221025.pdf}
\caption{
The result of QSCI, the proposed method, for the ground state of \ce{H2O} molecule by noiseless simulation, shown with optimization history of VQE, which is used to prepare the input states of QSCI.
Each of the resulting energies is shown as the difference to the CASCI result $E_{\rm exact}$ in Hartree. The dash-dotted line shows the result by the state-vector simulation of VQE. The lines specified by the parameter $R$ show the results of QSCI, $E_R - E_{\rm exact}$, for the given value of $R$, using the parametrized state at each iteration of VQE as the input state. The parameter $R$ determines the classical computational cost for QSCI, as explained in the main text.}
\label{fig:noiseless-vqe}
\end{figure}
We first show the result of numerical simulation for ground state with input states prepared by noiseless VQE. We choose \ce{H2O} molecule with five active spatial orbitals and six active electrons as our problem, which leads to a 10-qubit Hamiltonian after the Jordan-Wigner mapping.
In the VQE calculation, the parametrized quantum circuit is constructed by the real-valued symmetry-preserving ansatz~\cite{gard2020efficient, ibe2022calculating} with the depth 10, and is optimized by the Broyden–Fletcher–Goldfarb–Shanno (BFGS) optimizer in the scientific library SciPy~\cite{virtanen2020scipy}.
See Appendix~\ref{subsec:setup-noiseless-vqe} for details.
The QSCI calculation is performed for each iteration of the VQE optimization: given the values of ansatz parameters obtained at the iteration, the input state is prepared by the parametrized quantum circuit with those values assigned; then, the QSCI calculation with the idealized sampling introduced in Sec.~\ref{subsec:ground-state} is performed to estimate the ground-state energy $E_R$ for a given $R$, the number of configurations in the subspace $\mc{S}_R$.
This calculation is repeated for all the iterations of VQE with different values of $R$.
The effect of the uncertainty due to the finiteness of the number of shots is addressed later in Secs.~\ref{ssec:sampling-simulation}, \ref{sec:noisy-simulation-experiment}, and Appendix~\ref{ssec:appendix-sampling}.
In Fig.~\ref{fig:noiseless-vqe}, the result is shown along with the optimization history of VQE: $E_R - E_{\rm exact}$ is plotted (in Hartree) for each optimization step of VQE, where $E_{\rm exact}$ is the ground-state energy obtained by the exact diagonalization in the active space, called the complete active space configuration interaction (CASCI).
The energies obtained by VQE are shown in the same way.
Comparing the results at the last iteration in the plot, one can see that QSCI gives a lower energy than VQE for $R\gtrsim 16$.
This shows that the method is able to improve the results of VQE even in the noiseless setting,
where the effect of error mitigation is not present.
We emphasize that, as discussed in Sec.~\ref{subsec:ground-state}, a lower energy by QSCI means that the energy is closer to the exact ground-state energy, which is manifested in the plot where $E_R - E_{\rm exact}$ is always positive.
It is notable that we can already achieve the chemical accuracy\footnote{In this paper, we define the chemical accuracy
by $\SI{1}{kcal/mol} \simeq \SI{1.6e-3}{Hartree}$ for the deviation of the calculated energy from the one obtained by the exact diagonalization of the Hamiltonian.
} of $\SI{1.6e-3}{Hartree}$ with $R\sim 16$ while the CASCI in this case uses 100 determinants to express the ground state.\footnote{
For the active space restriction of five active orbitals and six active electrons with $S_z=0$, the number of the Slater determinants
is $\binom{5}{3}\cdot \binom{5}{3}=100$.
If one does not know
the number of electrons and $S_z$ of the ground state before the calculation, then one would need to deal with the full Hamiltonian in the Fock space of $2^{10}=1024$ dimensions.}
A similar tendency is observed for iterations of $\gtrsim 200$. Note, in this case, the VQE results already achieve the chemical accuracy.
On the other hand, for intermediate iterations of 70--200, the VQE results do not reach the chemical accuracy, while QSCI can improve them to meet the chemical accuracy if $R \gtrsim 16$.
This suggests that an intermediate result of VQE, which is not seeing convergence in the optimization yet, is already useful as an input state of QSCI,
and that one can reduce the number of optimization steps for VQE by employing QSCI as a post-processing.
We note that the QSCI results do not monotonically decrease,
as a QSCI calculation for an input state with a lower energy does not necessarily result in a lower output energy.
\subsection{QSCI for excited states}
\label{subsec:simulation-excited-h2o}
\begin{figure}
\begin{minipage}{0.5\textwidth}
\subfloat[][$T_1$ state]{
\includegraphics[width=\textwidth]{figures/excited_H2O_state1.pdf}}
\subfloat[][$S_1$ state]{
\includegraphics[width=\textwidth]{figures/excited_H2O_state2.pdf}}
\end{minipage}
\caption[]{
Same as Fig.~\ref{fig:noiseless-vqe} but for the first ($T_1$) and second ($S_1$) excited states of \ce{H2O} with $S_z = 0$, along with optimization histories of VQD, shown by dotted lines, for input-state preparation; the energy differences are plotted by the absolute values.
For the QSCI calculation of the $T_1$ ($S_1$) state, the input state(s) corresponding to the lower energy states, i.e., $S_0$ ($S_0$ and $T_1$) state(s), are prepared by converged sets of parameters of VQD.
QSCI results are shown for the sequential diagonalization and the single diagonalization with two types of configuration selection, as described in the main text. The QSCI calculations with sequential diagonalization are done with $R_i=16$ for $i=0,1,2$, while the value of $R$ is set to $R=16$ for single diagonalization.}
\label{fig:noiseless-vqd}
\end{figure}
We next show the results of noiseless simulations for excited states of \ce{H2O} using the two distinct implementations of QSCI presented in Sec.~\ref{subsec:excited-states}, namely the single diagonalization and sequential diagonalization schemes, which take the input states for excited states as well as the ground state.
The numerical setup is similar to the previous subsection, with some differences explained below.
In the input-state preparation, we employ VQE for the ground state and VQD for excited states, each with the same ansatz and optimizer as in the previous subsection;
we use the same 10-qubit Hamiltonian, but with the overlap terms and penalty terms~\cite{mcclean2016theory,ryabinkin2018constrained,kuroiwa2021penalty} in VQD, for orthogonality between the eigenstates and for symmetry restrictions (charge neutrality for the molecule and $S_z=0$ for the total electron spin) on the excited states.
Under the same symmetry restrictions, the first excited state is a triplet state ($T_1$) and the second excited state is a singlet state ($S_1$), according to the exact diagonalization.
The VQD calculation for $T_1$ requires information of the ground state ($S_0$) to generate the overlap term, for which the ansatz state is used with the converged parameters in VQE.
A similar procedure is applied for $S_1$, but with the extra overlap term for $T_1$ added.
With the input states for $S_0$, $T_1$ and $S_1$, we perform QSCI calculations to find $T_1$ and $S_1$, where the idealized sampling is used.
See Appendix~\ref{subsec:setup-noiseless-vqd} for details.
The results are shown in Fig.~\ref{fig:noiseless-vqd} along with the optimization history of VQD, in the similar way as Fig.~\ref{fig:noiseless-vqe}, but for $|E_R^{(i)} - E_{\rm exact}|$ ($i=1$ for $T_1$ and $i=2$ for $S_1$).
At each iteration, three types of QSCI calculations are performed: sequential, single-ground-state, and single-mixed. Sequential diagonalization uses the $T_1$ ($S_1$) state at the iteration, and one (two) lower energy state(s) at their final iterations as input states. Two single diagonalization methods use different input states: ``single-ground-state'' uses the ground state prepared by the converged VQE calculation, and that is why they are constant in the plot; on the other hand, ``single-mixed'' uses the two (three) states as input states, and selects $R$ configurations so that each of the two (three) states contributes as equally as possible, as explained in Sec.~\ref{sssec:method-single}.
Note that $R$ is the dimension of the common subspace $\mc{S}_R$ in Eq.~\eqref{eq:set_single-step}.
The coefficient(s) $\beta_0$ (and $\beta_1$ for $S_1$ state) of the overlap term(s) for orthogonality is set to $\beta_0=\beta_1=\SI{1}{Hartree}$, which is sufficiently larger than the energy gaps between the states in question.
For sequential diagonalization, the values of $R_i$ are fixed to $R_i=16$ for $i=0,1$ and 2, corresponding to $S_0$, $T_1$, and $S_1$ states, respectively;
for single diagonalization, the value of $R$ is set to $R=16$,
so that the sizes of the subspace Hamiltonian matrices to be diagonalized are the same among all the setups.
Comparing the three QSCI results for excited states, the sequential diagonalization performs the best
except for the initial steps of iterations where the quality of the input state is significantly low.
Moreover, the sequential diagonalization outperforms the VQD calculation, even with a moderate value of $R_i=16$.
For some larger $R$, the single diagonalization is also expected to improve and eventually outperform the VQD result at the same iteration as it can achieve the same representability as the sequential one\footnote{To show this explicitly, assume $N_s=2$ for simplicity. The single diagonalization with the subspace $\mc{S}_{R}=\mc{S}_{R_0}^{(0}\cup\mc{S}_{R_1}^{(1)}$, where the subspaces on the right-hand side denote those of the sequential diagonalization, have at least the same representability as the sequential diagonalization calculation with $\mc{S}_{R_1}^{(1)}$.}.
Although the sequential diagonalization seems to be better in terms of performance, it should be noted that there is no guarantee for the variational inequality in the sequential diagonalization. The inequality for excited states holds in the single diagonalization, as explained in Sec.~\ref{subsec:excited-states}.
\subsection{Scaling of computational costs}
\label{subsec:scaling}
\begin{figure}
\begin{minipage}{0.45\textwidth}
\subfloat[][\ce{Cr2}]{
\includegraphics[width=\textwidth]{figures/scaling/qubit-to-R-Cr2.pdf}
}
\subfloat[][Hydrogen chain]{
\includegraphics[width=\textwidth]{figures/scaling/qubit-to-R-Hchain.pdf}
}
\end{minipage}
\caption{Estimated $R$ required for a given energy error $\epsilon$. Results with (a) expanding active spaces (\ce{Cr2}) or (b) various numbers of atoms (hydrogen chain) are shown by markers, along with the linear fit of each plot.}
\label{fig:scaling-a}
\end{figure}
We now investigate the scalability of the proposed method by estimating the classical and quantum computational costs
to calculate the ground states for molecular Hamiltonians of various sizes. More concretely, we estimate the minimum value for $R$ and the required number of shots $N_{\rm shot}$ to obtain the ground-state energy within an error $\epsilon$ for those Hamiltonians.
For this sake, we employ the chromium dimer \ce{Cr2} with various active spaces
and the linear hydrogen chains with different numbers of atoms. Both \ce{Cr2} and hydrogen chains are known to be challenging molecules in quantum chemistry (see, e.g., Refs.~\cite{larsson2022chromium, motta2020ground} and references therein), while the hydrogen chains are also expected to show a clear scaling in the number of atoms. For \ce{Cr2}, the cc-pVQZ basis set is used with $n$ active orbitals and $n$ active electrons with $n=2,4,\dots,12$;
the Jordan-Wigner mapping produces $4,8,\cdots,24$-qubit Hamiltonians, respectively. For the linear hydrogen chains, we consider
$4,6,\cdots,12$ hydrogen atoms equally separated by a distance \SI{1.0}{\AA}; we use the STO-3G basis set without specifying the active space, corresponding to full-space Hamiltonians of $8,12,\cdots, 24$-qubit after the Jordan-Wigner mapping, respectively.
For each setup, the exact ground state of the Hamiltonian is prepared as the input state, and the QSCI calculation is performed by the idealized sampling introduced in Sec.~\ref{subsec:ground-state}, which picks up the $R$ Slater determinants with the largest absolute values of CI coefficients in the input-state wavefunction.
Then, for a given accuracy $\epsilon$, the minimal $R$ that satisfies $\abs{E_R -E_{\rm exact}} \leq \epsilon$ is determined, where $E_R$ is the energy obtained by QSCI with the $R$ configurations and $E_{\rm exact}$ by the exact diagonalization. In Fig.~\ref{fig:scaling-a}, the results are plotted for each molecule by varying the number of qubits, for $\epsilon=0.1, 0.01$ and $0.001$~Hartree; they are extrapolated by fitting (shown by lines) to discuss the feasibility for larger system sizes.
As detailed in Sec.~\ref{ssec:discussion-computational-cost},
we infer that the diagonalization with $R\simeq \SI{5e7}{}$ configurations is achievable by the current state-of-the-art classical computing according to the reports~\cite{stampfuss2003improved, garniron2019quantum}.
The result for \ce{Cr2} suggests that $R$ is expected to be manageable even when we require $\epsilon=\SI{0.001}{Hartree}$ for a system larger than 50 qubits, where the exact diagonalization in the whole Fock space, i.e., CASCI, is challenging for classical computers. In the case of the hydrogen chains, on the other hand, the exponential growth of $R$
is more clearly observed, and it may become hard to achieve $\epsilon=\SI{0.001}{Hartree}$ for a system much larger than 50 qubits due to the limitation of classical computing. Note that the two scalings have slightly different meanings: the active space is enlarged for \ce{Cr2} while fixing the molecule, i.e., the system size, while the system size itself is enlarged for the hydrogen chains. The results may suggest that our method is more suited to a localized system with many electrons involved, rather than a spatially extended system. For similar studies and results for several diatomic and aromatic molecules, see Appendix~\ref{ssec:more-results-scaling}.
\begin{figure}
\begin{minipage}{0.5\textwidth}
\subfloat[][\ce{Cr2}]{
\includegraphics[width=.9\textwidth]{figures/scaling/qubit-to-cR-Cr2.pdf}
}
\subfloat[][Hydrogen chain]{
\includegraphics[width=\textwidth]{figures/scaling/qubit-to-cR-H10-include-0.01.pdf}
}
\end{minipage}
\caption{Estimated number of shots for a given energy error $\epsilon$. For QSCI, the number of shots are approximated by $1/\abs{c_R}^2$, where $R$ for each $\epsilon$ is obtained in Fig.~\ref{fig:scaling-a} and $c_R$ is the CI coefficient of the input state with $R$-th largest absolute value. The reasoning for this approximation is explained in the main text.
For comparison,
the results of the conventional expectation-value estimation with QWC grouping are plotted for each $\epsilon$, fitted by a logarithmic function in the plot: the required numbers of shots for evaluating the energy are shown by dashed curves; for hydrogen chains, the ones for evaluating the gradients and Hessians in addition to the energy are shown by dash-dotted curves; the precision of gradients and Hessians are set to be $\epsilon/\text{\AA}$ and $\epsilon/\text{\AA}^2$, respectively.}
\label{fig:scaling-b}
\end{figure}
We next estimate the number of shots for sampling required
to achieve an error of $\epsilon$ by using the value of $1/\abs{c_R}^2$ for each setup (Fig.~\ref{fig:scaling-b}). Here, $c_R$ is the CI coefficient that has the $R$-th largest absolute value in the input state, where $R$ is taken to be the values shown in Fig.~\ref{fig:scaling-a}. When the state is sampled $1/\abs{c_R}^2$ times, the probability of obtaining the $R$-th most significant configuration is $O(1)$, and in that sense, $1/\abs{c_R}^2$
gives a rough estimator for the number of shots required to sample $R$ most significant configurations.
We see in the next section, especially in Fig.~\ref{fig:conventional},
that this gives a ballpark estimate of the required number of shots for a given accuracy.
For comparison, the total number of shots required in a conventional expectation-value estimation
is also estimated. More precisely, we analytically estimate the number of shots for which the standard deviation of the expectation-value estimations equals $\epsilon$ for the exact ground state (see, e.g., Ref.~\cite{kohda2022quantum}). In the conventional methods, the expectation value of the Hamiltonian, which is expressed as a linear combination of Pauli strings, is estimated by directly measuring the quantum state in the basis of the Pauli strings multiple times and taking the average of the measurement outcome. To reduce the number of measurements, we employ the qubit-wise commuting (QWC) grouping~\cite{mcclean2016theory} with the sorted insertion algorithm~\cite{crawford2021efficient}.
The total shot is distributed to each of the groups
with the shot allocation optimized for the exact ground state\footnote{This shot allocation may not be possible in practice without prior knowledge of the exact ground state, but
this estimation gives the lower-bound on the required number of total shots among possible shot allocation strategies, for a given error tolerance with the given grouping method and the state.
}~\cite{wecker2015progress, rubin2018application}.
Note that, although there are methods that are capable of reducing the number of measurements better than QWC, they require more gate operations for measurements than QWC does; QWC requires a layer of single-qubit rotations after the state preparation, which is minimal for methods that measure the Pauli strings directly, while QSCI requires no gate operation. Most of the other grouping methods are thus expected to be more vulnerable to noise, and QWC is chosen for a fair comparison in this study.
Figure~\ref{fig:scaling-b} shows the values of $1/\abs{c_R}^2$ in QSCI
for various numbers of qubits, along with the estimated numbers of shots in QWC. For the hydrogen chains, the results of QWC are fitted by a function $a(N_q)^b$ with parameters $a$ and $b$ as they are expected to be polynomial in the number of qubits\footnote{More precisely, the fit was performed by a function
\begin{equation}
\log (N_{\text{shot}}(N_q))= B\log(N_q)+A,
\end{equation}
where $A$ and $B$ are the free parameters.
Similarly in Fig.~\ref{fig:scaling-a}, a linear function $c N_q+d$ was used to fit the data for $\log(R)$, rather than $D\times 2^{C N_q}$ for $R$.
},
while the scaling of QSCI is unclear and fitting is not performed.
In the case of \ce{Cr2}, the number of shots for the proposed method seems to be consistently smaller than that of QWC, while the advantage of QSCI, in terms of reducing the effect of statistical fluctuation, is more non-trivial in hydrogen chains with the numbers of qubits $N_q\gtrsim 30$.
The more operators are evaluated with the same output state, the more advantageous QSCI becomes; as we already noted in the previous section, QSCI does not require any additional quantum computation to evaluate additional observables, because QSCI
outputs the classical vector representation of the state, and the expectation values are evaluated classically.
On the other hand, in the conventional methods, quantum computational cost becomes more expensive, e.g., to measure additional Pauli strings introduced by the extra operators.
To exploit this feature, we explore a scenario where the nuclear gradient and Hessian are evaluated along with the energy in the case of the hydrogen chains.
For the shot allocation in the QWC grouping, we developed a method that is optimized for measuring multiple operators at once and is used in the simulation; see Appendix~\ref{subsec:appendix-scaling-multiple-operators} for details.
The result, shown also in Fig.~\ref{fig:scaling-b} (b), implies that such a scenario makes QSCI much more advantageous\footnote{It is numerically shown in Appendix~\ref{ssec:appendix-multiple-observable-accuracy} that the accuracy of the gradients and Hessians in QSCI are of the same order as $\epsilon$ when expressed in the units of Hartree$/\mathrm{\AA}$ and Hartree$/\mathrm{\AA}^2$, respectively.}, as the number of shots for QWC significantly increases.
QSCI generally outperforms QWC in terms of the sampling cost within the range of the system size that we studied.
Although the scaling of QSCI seems to be worse than that of QWC in hydrogen chains, we should emphasize here that, even if QWC outperforms QSCI at, say, 50 qubits, it does not mean that QSCI is not useful for Hamiltonians with more than 50 qubits: QSCI has various features, such as error mitigation and the explicit representation of the output state, over the conventional methods, in addition to the reduction of the number of shots. The result should be interpreted as an implication that QSCI can be advantageous in moderately smaller but still classically-challenging systems, even when we only consider the effect of reduction of the number of shots.
\subsection{Sampling simulation}
\label{ssec:sampling-simulation}
For assessing the effect of the statistical error during the sampling in QSCI, sampling simulation with different numbers of shots is performed. The result for a linear \ce{H6} molecule (12 qubits) is shown in Fig.~\ref{fig:conventional}, and results for other molecules are in Appendix~\ref{ssec:appendix-sampling}. For this simulation, the exact ground state is used as the input state, and we include all the configurations obtained in the sampling into the basis set $\mathcal{S}_R$ and we do not specify $R$ beforehand.
For comparison, we also performed a conventional sampling estimation for the exact ground state with QWC grouping and a shot allocation optimized for Haar random states.
For both QSCI and QWC, we perform 10 trials of sampling simulation for each number of shots, and the average of the absolute differences to the exact ground-state energy is plotted along with the standard deviation of the 10 trials. The absolute differences to the exact value are much smaller in QSCI compared to the conventional sampling with QWC grouping.
It is worth noting that the standard deviation of QSCI energy is smaller than its average difference, while those of QWC sampling are roughly equal. Energy values obtained by QSCI are biased estimators for the exact expectation values even when using the exact ground states as input states. Thus, the absolute difference can roughly be calculated as a sum of the intrinsic bias existing in QSCI and the standard deviation which comes from statistical fluctuation of the subspace $\mathcal{S}_R$. In QWC, on the other hand, the statistical error is the only source of error. One can say that the QSCI result is much less affected by the statistical error compared to the conventional method.
Furthermore, as one can see in Fig.~\ref{fig:conventional}, $1/\abs{c_R}^2$ calculated in the previous simulation gives a relatively accurate estimation of the total shots that gives an average error close to $\epsilon$. Thus the plots in Fig.~\ref{fig:scaling-b} for both QWC and QSCI give reasonable estimations of the number of shots with expected average error $\epsilon$, and the comparison is fair in this sense.
\begin{figure}
\includegraphics[width=.45\textwidth]{figures/sampling/H6_diff.pdf}
\caption{Energy error results for both QSCI and conventional QWC in sampling simulations. For each set of 10 trials for each method, the standard deviation and the average of the absolute error to the exact value obtained by exact diagonalization are shown. QSCI energy error using $1/\abs{c_R}^2$ shots obtained in Fig.~\ref{fig:scaling-b} for $\epsilon=0.1,0.01,0.001$ Ha are also plotted for reference. The horizontal line indicates the chemical accuracy, \SI{1.6}{mHa}.}
\label{fig:conventional}
\end{figure}
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 1,068
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Q: Issue while building the kernel for Wilink8 module I am trying to integrate the Wilink8 WiFi module with the ZedBoard(Zync series SoC from Xilinx) and when i tried to compile mine linux- Kernel to create the uImage, I am getting the following error:
In file included from drivers/net/wireless/ti/wlcore/main.c:43:0:
drivers/net/wireless/ti/wlcore/version.h:1:39: error: macro "__TIMESTAMP__" might prevent reproducible builds [-Werror=date-time]static const char *wlcore_timestamp = __TIMESTAMP__;
drivers/net/wireless/ti/wlcore/main.c:5911:2: warning: initialization from incompatible pointer type
.sched_scan_stop = wl1271_op_sched_scan_stop,
drivers/net/wireless/ti/wlcore/main.c:5911:2: warning: (near initialization for 'wl1271_ops.sched_scan_stop')
cc1: some warnings being treated as errors
make[5]: *** [drivers/net/wireless/ti/wlcore/main.o] Error 1
make[4]: *** [drivers/net/wireless/ti/wlcore] Error 2
make[3]: *** [drivers/net/wireless/ti] Error 2
make[2]: *** [drivers/net/wireless] Error 2
make[1]: *** [drivers/net] Error 2
make: *** [drivers] Error 2
Can you help me to rectify this.
A: You may disable treating this warning as error as desribed in this post or remove line containing timestamp macro from driver source code.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 500
|
{"url":"https:\/\/dataspace.princeton.edu\/handle\/88435\/dsp01vh53wz90q?mode=full","text":"Please use this identifier to cite or link to this item: http:\/\/arks.princeton.edu\/ark:\/88435\/dsp01vh53wz90q\nDC FieldValueLanguage\ndc.contributor.authorSwaminathan, Mohan\ndc.contributor.otherMathematics Department\ndc.date.accessioned2022-06-16T20:33:27Z-\ndc.date.available2022-06-16T20:33:27Z-\ndc.date.created2022-01-01\ndc.date.issued2022\ndc.identifier.urihttp:\/\/arks.princeton.edu\/ark:\/88435\/dsp01vh53wz90q-\ndc.description.abstractIn this thesis, we present new results on three aspects of moduli spaces of pseudoholomorphic curves: smoothness, compactness and bifurcations.In the first part of this thesis, dealing with smoothness, we give a functorial construction of a so-called relative smooth structure on the moduli spaces of solutions to the (perturbed) pseudo-holomorphic curve equation. In the second part of this thesis, dealing with compactness, we prove a quantitative version of Gromov\u2019s compactness theorem for closed pseudo-holomorphic curves of genus 0 in a symplectic manifold. In the third part of this thesis, dealing with bifurcations, we study moduli spaces of embedded pseudo-holomorphic curves in a Calabi\u2013Yau 3-fold. Performing a careful bifurcation analysis of these moduli spaces in generic 1-parameter families leads, in some cases, to the construction of an integer valued invariant of Calabi\u2013Yau 3-folds which counts embedded curves with suitably defined integer weights. This part is joint with Shaoyun Bai.\ndc.format.mimetypeapplication\/pdf\ndc.language.isoen\ndc.publisherPrinceton, NJ : Princeton University\ndc.relation.isformatofThe Mudd Manuscript Library retains one bound copy of each dissertation. Search for these copies in the library's main catalog: <a href=http:\/\/catalog.princeton.edu>catalog.princeton.edu<\/a>\ndc.subjectGromov-Witten\ndc.subjectHolomorphic curve\ndc.subjectSymplectic topology\ndc.subject.classificationMathematics\ndc.subject.classificationTheoretical mathematics\ndc.titleNew results in the analysis of pseudo-holomorphic curves","date":"2023-03-23 05:53:55","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\": 0.5343203544616699, \"perplexity\": 916.0588129272251}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296944996.49\/warc\/CC-MAIN-20230323034459-20230323064459-00176.warc.gz\"}"}
| null | null |
The process of a Order Utah short sale can be extremely difficult to understand and potentially upsetting to homeowners. It can be overwhelming to understand the requirements, restrictions and myriad of documents. This is why selecting an agent that truly knows how to execute a short sale in Utah is so important.
The short sale process is full of intricacies that a smart and experienced real estate agent will be able to navigate successfully. There are so many differences between a normal home sale and a short sale that a seasoned realtor is required. Additionally, an experienced Utah short sale agent can speed the process along as fast as possible so you can move forward.
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Utah short sale experts make every effort to ensure that a deficiency order is not issued to the seller. In other words, the seller will be able to walk away without owing any additional money on the property. A best-case scenario would works like viagra include the agent being able to secure a small moving expense stipend to help the seller find new housing.
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An experienced agent will clearly explain the tax and credit implications of a short sale to the seller. The agent will be aware of the details of the Buy Mortgage Debt Relief Act of 2007 and will share applicable information to the seller (the provisions of this act are still valid).
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,522
|
Q: Rotate triangle ABC around the origin How do you get from triangle ABC (blue) to triangle ABC (red)
The instructions are to rotate it $270$ degrees
I am trying to help a friend but forgot how to do this? Is it using a formula?
A: $\triangle ABC$ $ \rightarrow$ $\triangle A'B'C'.$
$x,y$ $\rightarrow$ $ x',y'$, a rotation about the origin by $\alpha =90°$, clockwise:
1)$x' = x\cos(\alpha) +y\sin(\alpha)$.
2)$y' = -x\sin(\alpha) + y\cos(\alpha)$,
$\alpha = 90°.$
$A(-2,3)$ $ \rightarrow$ $A'(3,2)$.
$B(-2,0)$ $\rightarrow$ $B'(0,2).$
$C(-4,2)$ $\rightarrow$ $C'(2,4).$
Can you derive 1),2) from scratch?
A: Think of the vertices as being complex numbers.
*
*$A=-2+3\,i$
*$B=-2+0\,i$
*$C=-4+2\,i$
Each point may be rotated $270^\circ$ or $\frac{3\pi}{2}$ radians about the pole $0$ by multiplying by
$$e^{\frac{3\pi}{2}i}=\cos\left(\frac{3\pi}{2}\right)+i\,\sin\left(\frac{3\pi}{2}\right)=-i$$
This gives
*
*$A^\prime=3+2\,i$
*$B^\prime=2\,i$
*$C^\prime=2+4\,i$
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,233
|
{"url":"https:\/\/en.wikipedia.org\/wiki\/Quil_(instruction_set_architecture)","text":"# Quil (instruction set architecture)\n\nQuil is a quantum instruction set architecture that first introduced a shared quantum\/classical memory model. It was introduced by Robert Smith, Michael Curtis, and William Zeng in A Practical Quantum Instruction Set Architecture.[1] Many quantum algorithms (including quantum teleportation, quantum error correction, simulation,[2][3] and optimization algorithms[4]) require a shared memory architecture. Quil is being developed for the superconducting quantum processors developed by Rigetti Computing through the Forest quantum programming API.[5][6] A Python library called pyQuil was introduced to develop Quil programs with higher level constructs. A Quil backend is also supported by other quantum programming environments.[7][8]\n\n## Underlying Quantum Abstract Machine\n\nIn the paper presented by Smith, Curtis and Zeng, Quil specifies the instruction set for a Quantum Abstract Machine (QAM,) akin to a Turing machine, yet more practical for accomplishing \"real-world\" tasks.[1] The state of the QAM can be represented as a 6-tuple ${\\displaystyle (|\\Psi \\rangle ,C,G,G',P,\\kappa )}$ where:\n\n\u2022 ${\\displaystyle |\\Psi \\rangle }$ is the (quantum) state of a fixed but arbitrary number of qubits ${\\displaystyle N_{q}}$ indexed using a 0-based indexing.\n\u2022 ${\\displaystyle C}$ is a classical memory of a number ${\\displaystyle N_{c}}$ of classical bits indexed using a 0-based indexing.\n\u2022 ${\\displaystyle G}$ a fixed but arbitrary list of static gates (quantum gates that do not depend on parameters like the Hadamard gate.)\n\u2022 ${\\displaystyle G'}$ a fixed but arbitrary list of parametric gates (gates that depend on a number of complex parameters like the phase shift gate that requires an angle parameter to be completely defined.)\n\u2022 ${\\displaystyle P}$ a sequence of Quil instructions to be executed, representing the program. The length of ${\\displaystyle P}$ is denoted by ${\\displaystyle |P|}$.\n\u2022 ${\\displaystyle \\kappa }$ an integer program counter pointing to the next instruction to be executed. ${\\displaystyle \\kappa }$ always starts at 0 (pointing to the ${\\displaystyle 0^{th}}$ instruction) and ends at ${\\displaystyle |P|}$ indicating program halting (note that the last instruction has the index ${\\displaystyle |P|-1}$.) The program counter is incremented after every instruction, except for special control flow instructions (conditional and unconditional jumps, and the special HALT instruction that halts the program by setting ${\\displaystyle \\kappa }$ to ${\\displaystyle |P|}$.\n\nThe sematics of the QAM are defined using tensor products of Hilbert spaces and the linear maps between them.[1]\n\n## Features\n\nQuil has support for defining possibly parametrized gates in matrix form (the language does not include a way to verify that the matrices are unitary, which is a necessary condition for the physical realizability of the defined gate) and their application on qubits. The language also supports macro-like definitions of possibly parametrized quantum circuits and their expansion, qubit measurement and recording of the outcome in classical memory, synchronization with classical computers with the WAIT instruction which pauses the execution of a Quil program until a classical program has ended its execution, conditional and unconditional branching, pragma support, as well as inclusion of files for use as libraries (a standard set of gates is provided as one of the libraries.)\n\n## Rigetti QVM\n\nRigetti Computing developed a Quantum Virtual Machine in Common Lisp that simulates the defined Quantum Abstract Machine on a classical computer and is capable of the parsing and execution of Quil programs with possibly remote execution via HTTP.[9]\n\n## Example\n\nThe following example demonstrates the classical control flow needed to do quantum teleportation of the qubit in register 2 to register 1[10][11]:\n\n# Declare classical memory\nDECLARE ro BIT[2]\n# Create Bell Pair\nH 0\nCNOT 0 1\n# Teleport\nCNOT 2 0\nH 2\nMEASURE 2 ro[0]\nMEASURE 0 ro[1]\n# Classically communicate measurements\nJUMP-UNLESS @SKIP ro[1]\nX 1\nLABEL @SKIP\nJUMP-UNLESS @END ro[0]\nZ 1\nLABEL @END\n\n\nExamples of the implementations of the quantum fourier transform and the variational quantum Eigensolver are given in the paper.\n\n## References\n\n1. ^ a b c Smith, Robert S.; Curtis, Michael J.; Zeng, William J. (2016-08-10). \"A Practical Quantum Instruction Set Architecture\". arXiv:1608.03355 [quant-ph].\n2. ^ McClean, Jarrod R.; Romero, Jonathan; Babbush, Ryan; Aspuru-Guzik, Al\u00e1n (2016-02-04). \"The theory of variational hybrid quantum-classical algorithms\". New Journal of Physics. 18 (2): 023023. arXiv:1509.04279. Bibcode:2016NJPh...18b3023M. doi:10.1088\/1367-2630\/18\/2\/023023. ISSN\u00a01367-2630.\n3. ^ Rubin, Nicholas C. (2016-10-21). \"A Hybrid Classical\/Quantum Approach for Large-Scale Studies of Quantum Systems with Density Matrix Embedding Theory\". arXiv:1610.06910 [quant-ph].\n4. ^ Farhi, Edward; Goldstone, Jeffrey; Gutmann, Sam (2014-11-14). \"A Quantum Approximate Optimization Algorithm\". arXiv:1411.4028 [quant-ph].\n5. ^ \"Rigetti Launches Full-Stack Quantum Computing Service and Quantum IC Fab\". IEEE Spectrum: Technology, Engineering, and Science News. Retrieved 2017-07-06.\n6. ^ \"Rigetti Quietly Releases Beta of Forest Platform for Quantum Programming in the Cloud | Quantum Computing Report\". quantumcomputingreport.com. Retrieved 2017-07-06.\n7. ^ \"XACC Rigetti Accelerator\". ornl-qci.github.io. Retrieved 2017-07-06.\n8. ^ Doiron, Nick (2017-03-07), jsquil: Quantum computer instructions for JavaScript developers, retrieved 2017-07-06\n9. ^ The @rigetti high-performance quantum virtual machine.: rigetti\/qvm, Rigetti Computing, 2019-04-26, retrieved 2019-04-28\n10. ^ Nielsen, Michael A.; Chuang, Isaac L. (2000). Quantum Computation and Quantum Information. Cambridge University Press. p.\u00a027. ISBN\u00a0978-0-521-63503-5.\n11. ^ Computing, Rigetti (28 May 2019). \"pyQuil Documentation\" (PDF). pyQuil Documentaion. Retrieved 6 June 2019.","date":"2019-09-20 18:48:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 17, \"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\": 0.8182157278060913, \"perplexity\": 5822.678645606148}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-39\/segments\/1568514574058.75\/warc\/CC-MAIN-20190920175834-20190920201834-00304.warc.gz\"}"}
| null | null |
package com.intellij.spellchecker.compress;
import junit.framework.TestCase;
public class UnitBitSetTest extends TestCase {
public void testUnitValue() {
int bitsPerUnit = 256;
for (int i = 0; i < bitsPerUnit - 1; i++) {
UnitBitSet bs = new UnitBitSet(new byte[2], new Alphabet());
bs.setUnitValue(0, i);
assertEquals(i, bs.getUnitValue(0));
assertEquals(0, bs.getUnitValue(1));
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 135
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Diversity Works
A Twitter List by StanfordVPGE
Abe (2019) -- SGS Film Festival
July 21, 2021 - 12:00pm
CELEBRATING DIVERSITY at STANFORD
Diversity has been a core value at Stanford since the university's founding in 1891. That year, Stanford opened its doors to men and women from diverse religious, national, and racial backgrounds in its quest to train useful citizens.
The diversity of Stanford's first class reflects our commitment to provide opportunities for advancement to any deserving student. We value the rich perspectives, skills, and ideas people from varied backgrounds bring to the Stanford community.
ABOUT STANFORD
Stanford Profiles
Research Centers A - Z
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,853
|
Die Synkanzerogenese bezeichnet eine Verstärkung der Tumorauslösung durch gleichzeitiges oder aufeinanderfolgendes Einwirken mindestens zweier krebserregender Stoffe. Dies ist insbesondere dadurch möglich, dass beide Stoffe im gleichen Organ einen Tumor auslösen können.
Beispiel
Das wiederkehrende Einwirken von Tabakrauch und hochprozentigem Alkohol erhöht das Risiko für das Auftreten eines Zungengrundkarzinoms im Vergleich zur alleinigen Einwirkung nur eines Stoffes.
Vergleiche
Kokanzerogenese bezeichnet den Effekt, dass bestimmte selbst nicht krebserregende Stoffe (= Kokanzerogene oder Promotoren) den krebsauslösenden Effekt von Karzinogenen verstärken.
Onkologie
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,429
|
hills If Air makes you feel like you're in the clouds, this self-titled album will make you feel cosmic. Favorite track: Lost.
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Lachsior Astronoid harness the original energy of the Big Bang. It is overwhelming in it's power and aim! Transcendental odes. Favorite track: I Dream in Lines.
First pressing of the "Astronoid" CD in digipack. Printed on 300gsm stock with a 6-panel panorama layout.
Includes unlimited streaming of Astronoid via the free Bandcamp app, plus high-quality download in MP3, FLAC and more.
All songs and lyrics written by Brett Boland.
Mixed by Brett Boland & Daniel Schwartz.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 5,466
|
Bálint Hóman (1885-1951), historien et homme politique hongrois
Hans Linthorst Homan (1903-1986), homme politique et diplomate néerlandais
Korie Homan (née en 1986), joueuse de tennis néerlandaise
Rachel Homan (née en 1989), curleuse canadienne
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 3,647
|
\subsection{Constraint tightening of robust OCP}
Recall that with SLS, it suffices to consider the uncertain system $\xx = Z\hat{\sA} \xx + Z \hat{\sB} \uu + \bfSigma \tildeww$ for robust control since under constraint~\eqref{eq:over_approx_constr}, the additive filtered disturbances $\bfSigma \tildeww$ can realize all possible values of the lumped uncertainties $\eta_t = \Delta_A x_t + \Delta_B u_t + w_t$ at each time $t$. Compared with the actual uncertain perturbation $\eta_t$, the over-approximating additive disturbance $\bfSigma \tildeww$ has a simple structure with $\lVert \tilde{w}_t \rVert_\infty \leq 1$ which makes constraint tightening of the robust OCP~\eqref{eq:robustOCP} straightforward.
Let us take the state constraint tightening of $x_t \in \mathcal{X}$ as an example. The state constraint is a polyhedral set $\mathcal{X} = \lbrace x \in \mathbb{R}^{n_x} \mid F_x x \leq b_x \rbrace$. Denote the number of linear constraints in defining $\mathcal{X}$ as $n_\mathcal{X}$ and $\facet(\mathcal{X}) = \{(F_x(:,i), b_x(i)) \mid i = 1, \cdots, n_\mathcal{X} \}$ as the set of all linear constraint parameters of $\mathcal{X}$. Then, an arbitrary state constraint is represented by $f^\top x \leq b, (f, b) \in\facet(\mathcal{X})$.
For the dynamical system~\eqref{eq:surrogate_dynamics} with filtered disturbance $\bfSigma \tildeww$, the affine constraint~\eqref{eq:tilde_affine_constr} parameterizes all achievable closed-loop system responses $(\tildePhix,\tildePhiu)$ such that $\xx = \tildePhix \tildeww , \uu = \tildePhiu \tildeww$ under a linear state feedback controller $\uu = \KK \xx$~\cite[Theorem 1]{chen2021level}. Plugging in $x_t = \tilde{\Phi}_x^{t,t}x_0 + \sum_{i=1}^t \tilde{\Phi}_x^{t,t-i} \tilde{w}_{i-1}$, the tightening of the state constraint is given by
\begin{equation} \label{eq:state_constr_tightening}
\begin{aligned}
&f^\top \widetilde{\Phi}_x^{t,t} x_0 + \sum_{i=1}^t \lVert f^\top \widetilde{\Phi}_x^{t,t-i} \rVert_1 \leq b, \\
&\forall (f, b) \in \text{facet}(\mathcal{X}), \quad t = 0, \cdots, T-1
\end{aligned}
\end{equation}
which follows from a direct application of the H\"older's inequality and the fact $\lVert \tilde{w}_t \rVert_\infty \leq 1$. Similarly, we can tighten the terminal constraint as
\begin{equation} \label{eq:terminal_constr_tightening}
f^\top \widetilde{\Phi}_x^{T,T} x_0 + \sum_{i=1}^T \lVert f^\top \widetilde{\Phi}_x^{T,T-i} \rVert_1 \leq b, \ \forall (f, b) \in \text{facet}(\mathcal{X}_T),
\end{equation}
and tighten the control input constraints as
\begin{equation}\label{eq:input_constr_tightening}
\begin{aligned}
&f^\top \widetilde{\Phi}_u^{t,t} x_0 + \sum_{i=1}^t \lVert f^\top \widetilde{\Phi}_u^{t,t-i} \rVert_1 \leq b, \\
&\forall (f, b) \in \text{facet}(\mathcal{U}), \quad t = 0, \cdots, T-1.
\end{aligned}
\end{equation}
\subsection{Convex tightening of the robust OCP}
Finally, we can synthesize a robust state feedback controller $\KK$ as shown in the following theorem.
\begin{theorem} \label{thm:convex}
Consider the convex quadratic program
\begin{equation} \label{eq:convex_inner_approx}
\begin{aligned}
\underset{\tildePhix, \tildePhiu, \bfSigma}{\textrm{minimize}} & \quad \Big \lVert \begin{bmatrix}
\mathbf{Q}^{1/2} & \\ & \mathbf{R}^{1/2}
\end{bmatrix} \begin{bmatrix}
\tildePhix(:,0) \\ \tildePhiu(:,0)
\end{bmatrix}x_0 \Big \rVert_2^2 \\
\textrm{subject to} & \quad \textrm{affine constraint~\eqref{eq:tilde_affine_constr}} \\
& \quad \textrm{uncertainty over-approximation constraint~\eqref{eq:over_approx_constr}} \\
& \quad \textrm{tightened constraints~\eqref{eq:state_constr_tightening}, \eqref{eq:terminal_constr_tightening}, \eqref{eq:input_constr_tightening}} \\
&\quad x_0 = x(k)
\end{aligned}
\end{equation}
where $\tildePhix \in \mathcal{L}_{TV}^{T, n_x \times n_x}$, $\tildePhiu \in \mathcal{L}_{TV}^{T, n_u \times n_x}$, $\bfSigma \in \mathcal{L}_{TV}^{T, n_x \times n_x}$ is parameterized in Section~\ref{sec:filter_parameterization}, and $\mathbf{Q} = \blkdiag(Q, \cdots, Q, Q_T)$, $\mathbf{R} =\blkdiag(R, \cdots, R)$. For any feasible solution $(\tildePhix, \tildePhiu, \bfSigma)$ of the above problem~\eqref{eq:convex_inner_approx}, the linear state feedback controller $\KK= \tildePhiu \tildePhix^{-1}$ is feasible for the robust OCP~\eqref{eq:robustOCP} with polytopic model uncertainty.
\end{theorem}
The proof of Theorem~\ref{thm:convex} directly follows from our derivation of the constraints in~\eqref{eq:convex_inner_approx} from the previous sections. Note that since each entry of $d_t$ is lower bounded by $\sigma_w >0$ from~\eqref{eq:over_approx_constr}, any feasible $\bfSigma$ of problem~\eqref{eq:convex_inner_approx} is invertible. Some remarks about formulation~\eqref{eq:convex_inner_approx} are given as follows.
\begin{remark}
The objective function in~\eqref{eq:convex_inner_approx} is a quadratic nominal function for the system $\xx = Z \hat{\sA} \xx + Z \hat{\sB} \uu + \bfSigma \tildeww$ considering surrogate disturbances where $\tildePhix(:,0) x_0$ and $\tildePhiu(:,0)x_0$ model the nominal predicted states and control inputs, respectively. It aims to stabilize the closed-loop system to the origin, and it equals to the nominal cost considered in~\eqref{eq:nominalcost} when the sub-diagonal blocks in the filter $\bfSigma$ are restricted to be zero in which case $\tildePhix(:,0) = \Phix(:,0)$ and $\tildePhiu(:,0) = \Phiu(:,0)$ following from~\eqref{eq:change_var}.
\end{remark}
\begin{remark}
One important feature of our method is that the robust controller synthesis is decomposed into two steps: uncertainty set over-approximation by a surrogate additive disturbance and constraint tightening. This framework enjoys flexibility to adapt to different uncertainty assumptions. For example, when the actual additive disturbances $w_t$ are bounded in terms of $2$-norm, i.e., $\lVert w_t \rVert_2 \leq \sigma_w$, it is a natural choice to assume our virtual disturbances $\tilde{w}_t$ are bounded by $\lVert \tilde{w}_t \rVert_2 \leq 1$ in~\eqref{eq:unit_norm}. A similar robust OCP formulation to~\eqref{eq:convex_inner_approx} can be derived following the same process but using $\lVert \cdot \rVert_2$ instead. Another main difference will be that the diagonal matrices $\Sigma^{t, 0}$ of the filter have to be parameterized as $\Sigma^{t, 0} = \sigma_{t-1} I$ in this case since $\lVert \cdot \rVert_2$ does not allow processing the constraint~\eqref{eq:constr_0_transformed} in a row-wise manner.
\end{remark}
\subsection{Finite-horizon system level synthesis}
To apply SLS, as the first step, we stack all relevant state, control input, and uncertainty variables over horizon $T$ as
\begin{equation}\label{eq:signal_def}
\begin{aligned}
&\xx = [x_0^\top \ \cdots \ x_T^\top]^\top, \quad \uu = [u_0^\top \ \cdots \ u_T^\top]^\top, \\
&\bfeta = [x_0^\top \ \eta_0^\top \ \cdots \ \eta_{T-1}^\top]^\top, \quad \ww = [x_0^\top \ w_0^\top \ \cdots \ w_{T-1}^\top]^\top.
\end{aligned}
\end{equation}
Note that the initial state $x_0$ is set as the first component in $\bfeta$ and $\ww$, and $x_0$ can be interpreted as a known disturbance from the origin in this case. The vectors in~\eqref{eq:signal_def} can be interpreted as finite horizon signals. The parameterization of the LTV state feedback controller $\KK \in \mathcal{L}_{TV}^{T, n_u \times n_x}$ is represented by the block-lower-triangular matrix~\eqref{eq:BLT} with entries $K^{t, t-i}$. The controller $\KK$ can be interpreted as an LTV linear operator and the state feedback controller is given by $\uu = \KK \xx$. Similarly, we stack the dynamics matrices and uncertainty matrices as
\begin{equation} \label{eq:matrices_stack}
\begin{aligned}
&\mathbf{\hat{A}} = \blkdiag(\hat{A}, \cdots, \hat{A}), \quad \mathbf{\hat{B}} = \blkdiag(\hat{B}, \cdots, \hat{B}), \\
&\mathbf{\Delta}_A = \blkdiag(\Delta_A, \cdots, \Delta_A),\! \mathbf{\Delta}_B = \blkdiag(\Delta_B, \cdots, \Delta_B).
\end{aligned}
\end{equation}
With the above defined compact notations, the open-loop dynamics of the system can be equivalently written as
\begin{align}
&\xx = Z \hat{\sA}\xx + Z\hat{\sB} \uu + \bfeta
\end{align}
and the closed-loop dynamics under $\uu = \KK \xx$ follows as
\begin{equation} \label{eq:cl_dyn}
\xx = Z (\hat{\sA} + \hat{\sB} \KK) \xx + \bfeta
\end{equation}
where $Z$ is a block down-shifting operator with the first block sub-diagonal filled with identity matrices and zeros everywhere else. Note that the lumped uncertainty $\bfeta$ also depends on $\KK$ and $\xx$ as will be shown in the next subsection.
From~\eqref{eq:cl_dyn}, the mapping from the lumped uncertainty to the state and control input under the feedback controller is given by
\begin{equation} \label{eq:explicit_map}
\begin{bmatrix}
\xx \\ \uu
\end{bmatrix} = \begin{bmatrix}
(I - Z (\hat{\sA} + \hat{\sB} \KK))^{-1} \\
\KK (I - Z (\hat{\sA} + \hat{\sB} \KK))^{-1}
\end{bmatrix} \bfeta.
\end{equation}
From the block-down-shifting operator $Z$, we know that the matrix inverse in~\eqref{eq:explicit_map} exists and has a block-lower-triangular structure. The maps from $\bfeta$ to $(\xx, \uu)$ in~\eqref{eq:explicit_map} are called \emph{system responses}, and we introduce maps $\Phix \in \mathcal{L}_{TV}^{T, n_x \times n_x}$, $\Phiu \in \mathcal{L}_{TV}^{T, n_u \times n_x}$ following~\cite{anderson2019system} such that
\begin{equation} \label{eq:system_response}
\begin{bmatrix}
\xx \\ \uu
\end{bmatrix} = \begin{bmatrix}
\Phix \\
\Phiu
\end{bmatrix} \bfeta.
\end{equation}
The relationship between the state feedback controller $\KK$ and the system responses $(\Phix, \Phiu)$ is given by~\eqref{eq:explicit_map}. The following theorem allows us to equivalently transform design of the feedback controller $\KK$ into design of the system responses $(\Phix, \Phiu)$ without explicitly using the nonlinear map~\eqref{eq:explicit_map}.
\begin{theorem}~\cite[Theorem 2.1]{anderson2019system}
\label{thm:SLS}
Over the horizon $ t = 0, 1, \cdots, T$, for the system dynamics~\eqref{eq:system_dyn} with the block-lower-triangular state feedback control law $\KK \in \mathcal{L}_{TV}^{T, n_u \times n_x}$ defining the control action as $\uu = \KK \xx$, we have:
\begin{enumerate}
\item The affine subspace defined by
\begin{equation} \label{eq:affine_constr}
\begin{bmatrix} I - Z \hat{\sA} & -Z \hat{\sB} \end{bmatrix}
\begin{bmatrix} \Phix \\ \Phiu \end{bmatrix} = I, \ \Phix, \Phiu \in \mathcal{L}_{TV}
\end{equation}
parameterizes all possible system responses~\eqref{eq:system_response}.
\item For any block-lower-triangular matrices $\lbrace \Phix, \Phiu \rbrace \in \mathcal{L}_{TV}$ satisfying~\eqref{eq:affine_constr}, the controller $\KK = \Phiu \Phix^{-1} \in \mathcal{L}_{TV}$ achieves the desired responses~\eqref{eq:system_response}.
\end{enumerate}
\end{theorem}
Theorem~\ref{thm:SLS} shows the equivalence between system responses and linear state feedback controllers through the affine constraint~\eqref{eq:affine_constr}, with which optimization problems originally on $\KK$ can be transformed into one on $(\Phix, \Phiu)$. Such transformation may result in a convex problem in $(\Phix, \Phiu)$ while the original one is not, and provides a direct description on the effects of the uncertainty $\bfeta$ on the states and control inputs. More importantly, as shown in this paper, such transformation can reveal additional structured properties of the problem in the space of system responses that can be exploited for high-performance controller design.
\subsection{Dynamics of lumped uncertainty}
By the definition of lumped uncertainty~\eqref{eq:system_dyn} and the compact notations from~\eqref{eq:signal_def}, \eqref{eq:matrices_stack}, we have
\begin{equation}
\bfeta = Z \begin{bmatrix}
\DDelta_A & \DDelta_B
\end{bmatrix} \begin{bmatrix}
\xx \\ \uu
\end{bmatrix} + \ww.
\end{equation}
Then, the system responses~\eqref{eq:system_response} allows an explicit characterization of the dynamics of $\eta_t$ under the controller $\KK$ as
\begin{equation} \label{eq:eta_dynamics}
\bfeta = Z \begin{bmatrix}
\DDelta_A & \DDelta_B
\end{bmatrix} \begin{bmatrix}
\Phix \\ \Phiu
\end{bmatrix} \bfeta + \ww
\end{equation}
which can be equivalently decomposed into the following equations~\footnote{We adopt the convention that when $t=0$, the summation terms in~\eqref{eq:decomposed_dynamics} vanish. }
\begin{equation}\label{eq:decomposed_dynamics}
\begin{aligned}
\eta_t & = \Delta_A (\Phi_x^{t, t} x_0 + \sum_{i=1}^{t} \Phi_x^{t, t-i} \eta_{i-1} ) + \\
& \quad \Delta_B (\Phi_u^{t, t} x_0 + \sum_{i=1}^{t} \Phi_u^{t, t-i} \eta_{i-1} ) + w_t
\end{aligned}
\end{equation}
for $t = 0, \cdots, T-1$. The dynamics of the lumped uncertainty~\eqref{eq:eta_dynamics} exactly describes how the values of $\eta_t$ are jointly and uniquely determined by the uncertainty parameters $(\DDelta_A, \DDelta_B, \ww)$ and the closed-loop system responses $(\Phix, \Phiu)$, which implicitly encode a linear state feedback controller $\KK= \Phiu \Phix^{-1}$. The uniqueness can be seen from~\eqref{eq:decomposed_dynamics} that $\eta_t$ is only dependent on $\eta_i$ with $i \leq t-1$. Therefore, once the controller $(\Phix, \Phiu)$ and the uncertainty parameters $(\DDelta_A, \DDelta_B, \ww)$ are fixed, the values of $\eta_t$ for $0 \leq t \leq T-1$ are correspondingly decided. When $(\Delta_A, \Delta_B)$ and $w_t$ are unknown, the values of $\eta_t$ also become uncertain and we denote by $\mathcal{R}(\bfeta; \{\Phix, \Phiu \})$ the set of all possible values of $\{\eta_t\}_{t=0}^{T-1}$ under the uncertainty assumption in Section~\ref{sec:formulation} for a given controller defined by $\{\Phix, \Phiu \}$.
Despite being exact, the lumped uncertainty value set $\mathcal{R}(\bfeta;\{\Phix, \Phiu\})$ generated by~\eqref{eq:eta_dynamics} is too complex to use for solving the robust OCP~\eqref{eq:robustOCP}. In the next section, we over-approximate $\mathcal{R}(\bfeta;\{\Phix, \Phiu\})$ by an uncertainty set with simpler structure that is amenable for solving the robust OCP~\eqref{eq:robustOCP}.
\section{Introduction}
\input{introduction}
\section{Problem formulation}
\input{formulation}
\section{Characterization of effects of uncertainty}
\input{lumped_uncertainty}
\section{Uncertainty set over-approximation}
\input{over_approximation}
\section{Convex formulation of robust OCP}
\input{convex_OCP}
\section{Simulation}
\input{simulation}
\section{Conclusion}
\input{conclusion}
\bibliographystyle{ieeetr}
\subsection{Parameterization of the filter}
\label{sec:filter_parameterization}
In~\cite{chen2021level}, the filter $\bfSigma$ is parameterized as a block diagonal matrix $\bfSigma = \blkdiag(I, \sigma_0 I, \cdots, \sigma_{T-1} I)$ with which $\mathcal{R}(\bfSigma \tildeww)$ represents a Cartesian product of $\ell_\infty$ norm balls with radii $\sigma_t > 0$. Using this parameterization, we bound the lumped uncertainty $\eta_t$ by an $\ell_\infty$ ball of varying radius $\sigma_t$ at each time $t$. This over-approximation method has demonstrated outstanding performance in robust MPC of systems subject to norm-bounded model uncertainty~\cite{chen2021level}. To handle the polytopic model uncertainty considered in this paper and further reduce the conservatism of robust MPC, we propose a novel method for over-approximating the uncertainty set $\mathcal{R}(\bfeta;\{\Phix, \Phiu\})$ using a more flexible parameterization of the filter $\bfSigma$.
\begin{figure}
\centering
\includegraphics[width = 0.8\columnwidth]{sigma_param.png}
\caption{Blue: value set of the lumped uncertainty $\eta_t$. Green: over-approximation by the filtered disturbance signal $\sigma_t \tilde{w}_t$ under block diagonal parameterization of the filter $\bfSigma = \blkdiag(I, \Sigma^{1,0}, \cdots, \Sigma^{T-1, 0})$ with $\Sigma^{t,0} = \sigma_{t-1} I$ (left) or $\Sigma^{t,0} = \diag(d_{t-1})$ (right). }
\label{fig:ball_bound}
\end{figure}
Specifically, we consider the full block-lower-triangular parameterization of $\bfSigma \in \mathcal{L}_{TV}^{T, n_x \times n_x}$ where the sub-diagonal blocks $\Sigma^{t, t-i}$ are non-zero. The only restriction on $\bfSigma$ is that the matrix blocks $\Sigma^{t, 0}$ on the diagonal of $\bfSigma$ are diagonal matrices with positive entries, i.e., $\Sigma^{t, 0} = \diag(d_{t-1})$ for $1 \leq t \leq T$ where $d_{t-1} \in \mathbb{R}^{n_x}$ satisfies $d_{t-1} > 0$. Additionally, we require $\Sigma^{0,0}= I$ such that the first component of $\bfSigma \tildeww$ is still the initial state $x_0$. By construction, $\bfSigma$ is invertible. Such parameterization contains the block-diagonal parameterization used in~\cite{chen2021level} as a special case, and is therefore less conservative. Indeed, when the sub-diagonal matrix blocks of $\bfSigma$ are enforced zero, $\mathcal{R}(\bfSigma \tildeww)$ represents a sequence of hyperrectangles instead of norm-balls, and is able to bound the values of $\eta_t$ in a tighter way as shown in Figure~\ref{fig:ball_bound}.
\subsection{Formulation of the over-approximation problem}
The uncertainty set over-approximation problem can be stated as finding $(\Phix, \Phiu, \bfSigma)$ such that
\begin{equation}\label{eq:set_over_approx}
\mathcal{R}(\bfeta;\{\Phix, \Phiu\}) \subseteq \mathcal{R}(\bfSigma \tildeww)
\end{equation}
for all $(\Delta_A, \Delta_B) \in \mathcal{P}$ and $w_t \in \mathcal{W}$. This is equivalent to the following problem:
\begin{problem}[Uncertainty set over-approximation]
\label{prob:over_approx}
Find a controller $(\Phix, \Phiu)$ and a filter $\bfSigma$ such that
\begin{equation} \label{eq:robust_equalities}
Z \begin{bmatrix}
\DDelta_A & \DDelta_B
\end{bmatrix} \begin{bmatrix}
\Phix \\ \Phiu
\end{bmatrix} \bfSigma \tildeww + \ww = \bfSigma \tildeww
\end{equation}
has a solution $\tildeww^* \in \mathcal{W}_{\tildeww}$ for all possible realizations of the uncertainty $(\DDelta_A, \DDelta_B, \ww)$.
\end{problem}
To see the equivalence, note that for any given realization of $(\DDelta_A, \DDelta_B, \ww)$, the value of the lumped uncertainty is uniquely determined by~\eqref{eq:eta_dynamics} which we denote as $\bfeta^*$. Since the filter $\bfSigma$ is invertible, $\tildeww^* = \bfSigma^{-1} \bfeta^*$ is the unique solution to the system of equations~\eqref{eq:robust_equalities}. Then~\eqref{eq:set_over_approx} holds if and only if $\tildeww^* \in \mathcal{W}_{\tildeww}$ for all possible $(\DDelta_A, \DDelta_B, \ww)$.
\subsubsection{Change of variables}
The bilinear terms $\Phix \bfSigma$ and $\Phiu \bfSigma$ in~\eqref{eq:robust_equalities} make it challenging to solve Problem~\ref{prob:over_approx} since they lead to a non-convex optimization problem. To resolve this issue, we apply the change of variables
\begin{equation} \label{eq:change_var}
\tildePhix = \Phix \bfSigma, \quad \tildePhiu = \Phiu \bfSigma
\end{equation}
where $\tildePhix, \tildePhiu \in \mathcal{L}_{TV}$ can be interpreted as system responses mapping $\tildeww$ to $(\xx, \uu)$ for the system $\xx = Z\hat{\sA} \xx + Z \hat{\sB} \uu + \bfSigma \tildeww$ in closed-loop with $\uu = \KK \xx$. According to \cite[Theorem 1]{chen2021level}, all achievable closed-loop system responses $(\tildePhix, \tildePhiu)$ are parameterized by
\begin{equation} \label{eq:tilde_affine_constr}
\begin{bmatrix} I - Z \hat{\sA} & -Z \hat{\sB} \end{bmatrix}
\begin{bmatrix} \tildePhix \\ \tildePhiu \end{bmatrix} = \bfSigma,
\end{equation}
which is jointly affine in $(\tildePhix, \tildePhiu, \bfSigma)$. Under the affine constraint~\eqref{eq:tilde_affine_constr}, Problem~\ref{prob:over_approx} is equivalently transformed into finding $(\tildePhix, \tildePhiu, \bfSigma)$ such that
\begin{equation}\label{eq:robust_feasibility}
Z \begin{bmatrix}
\DDelta_A & \DDelta_B
\end{bmatrix} \begin{bmatrix}
\tildePhix \\ \tildePhiu
\end{bmatrix} \tildeww + \ww = \bfSigma \tildeww,\ \tildeww \in \mathcal{W}_{\tildeww}
\end{equation}
is feasible for all $(\Delta_A, \Delta_B) \in \mathcal{P}$ and $w_t\in \mathcal{W}$. The invertibility of $\bfSigma$ guarantees the equivalence between searching $(\Phix, \Phiu, \bfSigma)$ under constraint~\eqref{eq:affine_constr} and searching $(\tildePhix, \tildePhiu, \bfSigma)$ under constraint~\eqref{eq:tilde_affine_constr} (see~\cite[Remark 1]{chen2021level} for more details).
\subsection{Convex over-approximation constraints}
We now present convex sufficient conditions on $(\tildePhix, \tildePhiu, \bfSigma)$ such that the robust feasibility of~\eqref{eq:robust_feasibility} is guaranteed. The block-down-shifting operator $Z$ in~\eqref{eq:robust_feasibility} makes it possible to decompose and analyze the equality constraints for $t = 0, \cdots, T-1$ sequentially.
\subsubsection{Case $t=0$}
For bounding $\eta_0$ at $t = 0$, we have the following constraint from~\eqref{eq:robust_feasibility}:
\begin{equation} \label{eq:constr_0}
\Delta_A x_0 + \Delta_B \tilde{\Phi}_u^{0,0} x_0 + w_0 = \Sigma^{1,1} x_0 + \Sigma^{1,0} \tilde{w}_0
\end{equation}
where we plug in $\tilde{\Phi}_x^{0,0} = I$. The above constraint is robustly feasible with a solution $\lVert \tilde{w}_0 \rVert_\infty \leq 1$ if and only if
\begin{equation} \label{eq:constr_0_transformed}
\lVert {\Sigma^{1,0}}^{-1} (\Delta_A x_0 + \Delta_B \tilde{\Phi}_u^{0,0} x_0 - \Sigma^{1,1} x_0 + w_0) \Vert_\infty \leq 1
\end{equation}
for all $(\Delta_A, \Delta_B) \in \mathcal{P}$ and $\lVert w_0 \rVert_\infty\leq \sigma_w$. Constraint~\eqref{eq:constr_0_transformed} is non-convex in the design parameter $\Sigma^{1,0}$, but under our diagonal matrix parameterization $\Sigma^{1,0} = \diag(d_0)$ (see Section~\ref{sec:filter_parameterization}) it can be rewritten as
\begin{equation} \label{eq:constr_0_convex}
\lVert e_i^\top(\Delta_A x_0 + \Delta_B \tilde{\Phi}_u^{0,0} x_0 - \Sigma^{1,1} x_0 + w_0) \rVert_1 \leq d_{0, i}
\end{equation}
for $i = 1, \cdots, n_x$ where $d_{1,i}$ denotes the $i$-th entry of $d_1$ and $e_i$ is the $i$-th {standard basis}. The change of norm type in~\eqref{eq:constr_0_convex} is due to the fact the left-hand side (LHS) of~\eqref{eq:constr_0_convex} is now a vector, and therefore the matrix $\infty$-induced norm reduces to the $\ell_1$ norm. Then, the following constraints
\begin{equation}\label{eq:constr_0_convex_design}
\begin{aligned}
&\lVert e_i^\top (\Delta_A x_0 + \Delta_B \tilde{\Phi}_u^{0,0} x_0 - \Sigma^{1,1} x_0 ) \Vert_1 + \sigma_w \leq d_{0,i}, \\
& \forall (\Delta_A, \Delta_B) \in \vertex(\mathcal{P}), \quad i = 1, \cdots, n_x
\end{aligned}
\end{equation}
guarantee~\eqref{eq:constr_0_convex} is robustly feasible, where $\vertex(\mathcal{P}) = \{ (\Delta_{A,j}, \Delta_{B,j}), j = 1, \cdots, M\}$ denotes the set of vertices of the uncertainty set $\mathcal{P}$ defined in~\eqref{eq:polytopic_uncertainty}. The robust feasibility guarantee follows from the triangle inequality of $\lVert \cdot \rVert_1$, the norm bound $\sigma_w$ on $w_0$, and the fact that the LHS of~\eqref{eq:constr_0_convex} is convex in $(\Delta_A, \Delta_B)$, and that the maximum of a convex function over a convex polytope is achieved at the polytope vertices~\cite{boyd2004convex}.
Now constraint~\eqref{eq:constr_0_convex_design} is convex in the design parameters $\tilde{\Phi}_u^{0,0}, \Sigma^{1,1}$ and $d_1$. It guarantees that for all possible realizations of $(\Delta_A, \Delta_B, w_0)$ and the generated lumped uncertainty $\eta_0$ generated, we can always find $\tilde{w}_0^*$ such that $\eta_0 = \Sigma^{1,1} x_0 + \Sigma^{1,0} \tilde{w}_0^*$ with $\lVert \tilde{w}_0^* \rVert_\infty \leq 1$. We fix $\tilde{w}_0^*$ as the $\tilde{w}_0$-component of the solution $\tildeww^*$ to~\eqref{eq:robust_feasibility}.
\subsubsection{Case: $t = 1$}
To bound $\eta_1$, similarly we first write down the relevant equality constraints from~\eqref{eq:robust_feasibility} as
\begin{equation}\label{eq:constr_1}
\begin{aligned}
&\Delta_A(\tilde{\Phi}_x^{1,1} x_0 + \tilde{\Phi}_x^{1,0} \tilde{w}_0^*) + \Delta_B(\tilde{\Phi}_u^{1,1} x_0 + \tilde{\Phi}_u^{1,0} \tilde{w}_0^*) + w_1 \\
& = \Sigma^{2,2} x_0 + \Sigma^{2,1} \tilde{w}_0^* + \Sigma^{2,0} \tilde{w}_1
\end{aligned}
\end{equation}
where $\tilde{w}_0^*$ is the solution from the previous step and captures the effects of $\eta_0$ on future perturbations $\eta_t$ for $t \geq 1$. Using the diagonal matrix parameterization of $\Sigma^{2,0} = \diag(d_1)$, following the same steps as in the case $t = 0$ and grouping the terms in~\eqref{eq:constr_1} by $(x_0, \tilde{w}_0^*, \tilde{w}_1)$, we have that if the inequalities
\begin{equation}\label{eq:constr_1_w}
\begin{aligned}
&\lVert e_i^\top(\Delta_A\tilde{\Phi}_x^{1,1} + \Delta_B \tilde{\Phi}_u^{1,1} - \Sigma^{2,2})x_0 \rVert_1 + \\
&\lVert e_i^\top(\Delta_A\tilde{\Phi}_x^{1,0} + \Delta_B \tilde{\Phi}_u^{1,0} - \Sigma^{2,1})\tilde{w}_0^* \rVert_1 + \lVert e_i^\top w_1 \rVert_1 \leq d_{1,i}
\end{aligned}
\end{equation}
for $1 \leq i \leq n_x$ hold robustly, then the robust feasibility of~\eqref{eq:constr_1} is guaranteed. However, the exact value of $\tilde{w}_0^*$ is unknown to us since it depends on $(\Delta_A, \Delta_B, w_0)$. To address this issue, we treat $\tilde{w}_0^*$ as uncertainty satisfying $\lVert \tilde{w}_0^* \rVert_\infty \leq 1$ and further tighten the constraint~\eqref{eq:constr_1_w} as
\begin{equation}\label{eq:constr_1_convex_design}
\begin{aligned}
&\lVert e_i^\top(\Delta_A\tilde{\Phi}_x^{1,1} + \Delta_B \tilde{\Phi}_u^{1,1} - \Sigma^{2,2})x_0 \rVert_1 + \\
&\lVert e_i^\top(\Delta_A\tilde{\Phi}_x^{1,0} + \Delta_B \tilde{\Phi}_u^{1,0} - \Sigma^{2,1})\rVert_1 + \sigma_w \leq d_{1,i}, \\
& \forall (\Delta_A, \Delta_B) \in \vertex(\mathcal{P}), \quad i = 1, \cdots, n_x
\end{aligned}
\end{equation}
by applying the H\"older's inequality on $\lVert a^\top \tilde{w}_0^* \Vert_1 \leq \lVert a \rVert_1 \lVert \tilde{w}_0^* \rVert_\infty \leq \lVert a \rVert_1$ and $\lVert e_i^\top w_1 \rVert_1 \leq \lVert e_i \rVert_1 \lVert w_1 \rVert_\infty \leq \sigma_w$.
\subsubsection{General case}
We repeat this process from $t = 0$ to $t = T-1$ to obtain a set of convex constraints on $(\tildePhix, \tildePhiu, \bfSigma)$:
\begin{equation}\label{eq:over_approx_constr}
\begin{aligned}
& \lVert e_i^\top (\Delta_A \tilde{\Phi}_x^{t,t} + \Delta_B \tilde{\Phi}_u^{t,t}- \Sigma^{t+1,t+1})x_0 \rVert_1 + \sigma_w +\\
& \sum_{i=1}^t \lVert e_i^\top (\Delta_A \tilde{\Phi}_x^{t,t-i} + \Delta_B \tilde{\Phi}_u^{t,t-i}- \Sigma^{t+1,t+1-i})\rVert_1 \leq d_{t, i}, \\
& \forall (\Delta_A, \Delta_B) \in \vertex(\mathcal{P}), i = 1, \cdots, n_x, t = 0, \cdots, T-1.
\end{aligned}
\end{equation}
Any feasible solution $(\tildePhix, \tildePhiu, \bfSigma)$ to constraints~\eqref{eq:over_approx_constr} guarantees the existence of a virtual disturbance signal $\tildeww^* \in \mathcal{W}_{\tildeww}$ such that $\bfeta = \bfSigma \tildeww^*$ for all possible realization of uncertainties. In other words, $\mathcal{R}(\bfeta;\{\Phix, \Phiu\}) \subseteq \mathcal{R}(\bfSigma \tildeww)$ holds, and it suffices to consider the uncertain dynamical system $\xx = Z\hat{\sA} \xx + Z \hat{\sB} \uu + \bfSigma \tildeww$ with the surrogate additive disturbance $\bfSigma \tildeww$ for solving the robust OCP~\eqref{eq:robustOCP}. The synthesized controller $\KK = \tildePhiu \tildePhix^{-1}$ enjoys robustness guarantees for the original uncertain dynamical system~\eqref{eq:dyn}. We present the details of the robust OCP solutions in the next section.
\subsection{Varying uncertainty parameters}
We evaluate the conservatism of our proposed method on a 2-dimensional system drawn from~\cite{bujarbaruah2021simple}. The system nominal dynamics and problem constraints are given as
\begin{equation} \label{eq:example}
\begin{aligned}
&\hat{A} = \begin{bmatrix}
1 & 0.15 \\ 0.1 & 1
\end{bmatrix}, \quad \hat{B} = \begin{bmatrix}
0.1 \\ 1.1
\end{bmatrix}, \\
&\mathcal{X} = \Big \{ x \in \mathbb{R}^2 \vert \begin{bmatrix}
-8 \\ -8
\end{bmatrix} \leq x \leq \begin{bmatrix}
8 \\8
\end{bmatrix} \Big \}, \\
& \mathcal{U} = \{ u \in \mathbb{R} \vert -4 \leq u \leq 4\}.
\end{aligned}
\end{equation}
Then we consider the following polytopic model uncertainty:
\begin{equation} \label{eq:uncertainty_form}
\begin{aligned}
&\Delta_A \in \conv \Big \{ \begin{bmatrix}
\epsilon_A & 0 \\ 0 & 0
\end{bmatrix} , \begin{bmatrix}
-\epsilon_A & 0 \\ 0 & 0
\end{bmatrix} \Big \}, \\
& \Delta_B \in \conv\Big \{ \begin{bmatrix}
0 \\ \epsilon_B
\end{bmatrix}, \begin{bmatrix}
0 \\ -\epsilon_B
\end{bmatrix} \Big \}
\end{aligned}
\end{equation}
and the norm bounded additive disturbances $\lVert w_t \rVert_\infty \leq \sigma_w$ for robust MPC. The uncertainty parameters $\epsilon_A, \epsilon_B$ and $\sigma_w$ are to be specified. The cost weights are chosen as $Q = 10I, R = 1, Q_T = 10I$.
We compare the conservatism of our proposed method, which is denoted as polytopic SLS MPC, and tube MPC~\cite{langson2004robust} for varying values of $(\epsilon_A, \epsilon_B, \sigma_w)$. For each fixed $(\epsilon_A, \epsilon_B, \sigma_w)$, we first apply an iterative algorithm~\cite{grieder2003robust} to find the maximal robust control invariant set and use it as the terminal set $\mathcal{X}_T$. By construction, $\mathcal{X}_T$ gives the largest feasible domain for any robust MPC algorithm and can be achieved by the exact yet computationally expensive dynamic programming approach~\cite[Chapter 15]{borrelli2017predictive}. We do a $15 \times 15$ uniform grid search of initial conditions $x_0$ over $\mathcal{X}_T$ and solve both polytopic SLS MPC and tube MPC with the sampled $x_0$ and horizon $T = 10$. The coverage of the feasible domain of each MPC method is given by the ratio of sampled feasible initial conditions, and a coverage close to $1$ indicates minimal conservatism of the method even compared with the maximal robust control invariant set.
For tube MPC, we select the tube cross section as the disturbance invariant set for the nominal dynamics with an LQR controller (see~\cite{chen2020robust} for details). We report the coverages of polytopic SLS MPC and tube MPC in Figure~\ref{fig:coverage}. In Figure~\ref{fig:coverage_eps_A} we fix $\epsilon_B = \sigma_w = 0.1$ and vary $\epsilon_A$ from $0.05$ to $0.4$ with step size $0.05$, while in Figure~\ref{fig:coverage_w} we fix $\epsilon_A = \epsilon_B = 0.1$ and vary $\sigma_w$ from $0.1$ to $0.7$ with step size $0.1$. Figure~\ref{fig:coverage} suggests that polytopic SLS MPC not only outperforms tube MPC in conservatism, but also achieves a feasible domain close to the maximal robust control invariant set while being polynomially solvable through~\eqref{eq:convex_inner_approx}. The average solver time of the robust OCP is $0.074$ seconds for polytopic SLS MPC and $0.540$ seconds for tube MPC in this example.
\begin{figure}
\centering
\begin{subfigure}{0.8 \columnwidth}
\includegraphics[width = \textwidth]{LCSS_conservatism_eps_A_15.pdf}
\caption{Coverage of feasible regions for varying $\epsilon_A$ with $\epsilon_B = 0.1, \sigma_w = 0.1$}
\label{fig:coverage_eps_A}
\end{subfigure}
\hfill
\begin{subfigure}{0.8 \columnwidth}
\includegraphics[width = \textwidth]{LCSS_conservatism_sigma_w.pdf}
\caption{Coverage of feasible regions for varying $\sigma_w$ with $\epsilon_A = 0.1, \epsilon_B = 0.1$.}
\label{fig:coverage_w}
\end{subfigure}
\caption{The coverages of the feasible domain (ratio of feasible sampled initial states) of polytopic SLS MPC and tube MPC with varying $\epsilon_A$ (top) or $\sigma_w$ (bottom). Polytopic SLS MPC significantly outperforms tube MPC in conservatism in the face of large uncertainties, and has a feasible domain similar to the maximal robust control invariant set (i.e., coverage $\approx 1$).}
\label{fig:coverage}
\end{figure}
\subsection{Randomly generated examples}
We now compare the coverages of polytopic SLS MPC and tube MPC on $50$ randomly generated $2$-dimensional systems. The entries of the nominal dynamics $\hat{A} \in \mathbb{R}^{2 \times 2}$ are sampled independently and uniformly from the interval $[-2, 2]$. The spectral radius of $\hat{A}$ is then scaled to a uniformly sampled number from the interval $[0.5, 2.5]$. The entries of $\hat{B} \in \mathbb{R}^{2 \times 1}$ are uniformly and independently sampled from $[-1, 1]$. The state and control input constraints follow from~\eqref{eq:example}, and the model uncertainty has the form~\eqref{eq:uncertainty_form}. No terminal constraint is applied. The uncertainty parameters are fixed as $\epsilon_A = 0.2, \epsilon_B = 0.1, \sigma_w = 0.1$ and the MPC horizon is chosen as $T = 10$.
Different from the previous example, this time we do not use the maximal robust control invariant set as the reference for each randomly generated problem instance. Instead, we sample initial conditions uniformly from the state space $\mathcal{X}$. Figure~\ref{fig:random_conservatism} shows that polytopic SLS MPC can achieve significant improvement in conservatism compared with tube MPC on randomly generated problem instances. The average solver time of the robust OCP for is $0.056$ seconds for polytopic SLS MPC and $0.128$ seconds for tube MPC.
\begin{figure}
\centering
\includegraphics[width = 0.8 \columnwidth]{LCSS_random_conservatism.pdf}
\caption{The coverages of the feasible domain of polytopic SLS MPC and tube MPC on randomly generated problem instances, arranged in an ascending order according to that of polytopic SLS MPC. Different from Figure~\ref{fig:coverage}, the initial states are sampled from the state space $\mathcal{X}$ rather than the maximal robust control invariant set.}
\label{fig:random_conservatism}
\end{figure}
|
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{"url":"https:\/\/proofassistants.stackexchange.com\/questions\/470\/type-checking-undecidable-in-extensional-type-theory\/479","text":"# Type Checking Undecidable in Extensional Type Theory\n\nWhat is the difference between intensional vs extensional type theories and how come the type checking is undecidable for extensional type theory? Also, how does it affect the expressiveness of theorem provers?\n\nExtensional type theory is characterized by the reflection rule, which says that if the identity type $${\\rm Id}(a,b)$$ is inhabited, then $$a\\equiv b$$ ($$a$$ and $$b$$ are judgmentally equal). It is called extensional type theory because this means that the judgmental equality coincides with the identity type, and the latter is extensional (or, at least, more extensional than the judgmental equality would be in the absence of the reflection rule --- just how extensional it is depends on whether you have principles like function extensionality and univalence). Intensional type theory is so-called because its judgmental equality is intensional, whereas its identity types can be even more extensional than those in extensional type theory (because the reflection rule is incompatible with the \"strongest extensionality principle\", namely univalence).\nIn a dependent type theory, type-checking is complicated because of the conversion rule that if $$a:A$$ and $$A\\equiv B$$ then $$a:B$$. This essentially requires that a type-checking algorithm must include an algorithm for checking judgmental equality. When combined with the reflection rule, this means a type-checking algorithm would have to include an algorithm for checking inhabitation of a type (namely the identity type). But inhabitation of types can be used to encode the truth of arbitrary mathematical statements, so (e.g. by the halting problem) it is impossible to have a terminating algorithm for checking inhabitation of types, and hence impossible to have a terminating type-checking algorithm for extensional type theory.","date":"2022-07-02 18:16:31","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 7, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8582831621170044, \"perplexity\": 427.35638974232745}, \"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-2022-27\/segments\/1656104189587.61\/warc\/CC-MAIN-20220702162147-20220702192147-00061.warc.gz\"}"}
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{"url":"https:\/\/pwntestprep.com\/wp\/tag\/ratios\/","text":"## Posts tagged with: ratios\n\nHere\u2019s a problem from Khan Academy\u2019s SAT practice section. Please explain this one for me.\n\nA marine aquarium has a small tank and a large tank, each containing only red and blue fish. In each tank, the ratio of red fish to blue fish is 3 to 4. The ratio of fish in the large tank to fish in the small tank is 46 to 5. What is the ratio of blue fish in the small tank to red fish in the large tank?\n\nA ratio of 3 red to 4 blue means the total must be a multiple of 7, so it might help you to first multiply the totals in each tank by 7. The 46:5 ratio for large tank to small tank becomes 322:35.\u00a0Let\u2019s just pretend those are the actual numbers of fish in the tank.\n\nIn the small tank, 4\/7 of the 35 fish in the tank are blue, so 20 fish are blue.\n\nIn the large tank, 3\/7 of the 322 fish are red, so 138 fish are red.\n\nTherefore, the ratio of blue fish in the small tank to red fish in the big tank is 20:138, which simplifies to 10:69.\n\nNote that while multiplying everything by 7 was helpful (in my opinion) in making the math more intuitive, you don\u2019t\u00a0need to. You can simply enter the following in your calculator and ask it to convert to a fraction:\n\nIf it\u2019s falling at a constant rate during the period, then we can conclude that it fell (3.4 mb\/hr)(5 hr) = 17 mb.\n\nNow we just need to convert mb to mm Hg, using the given scale.\n\nSolve that for x\u00a0and you get roughly 12.8.\n\nfrom Tumblr https:\/\/ift.tt\/2w1cbUe\n\nA bag contains mangoes that are either green or yellow. The ratio of green mangoes to yellow mangoes in the bag is 3 to 5. When two green mangoes are removed and ten yellow mangoes are added, the ratio becomes 2 to 5. How many green mangoes were originally in the bag?\n\nSay that the original number of green mangoes is\u00a0g and the original number of yellow mangoes is\u00a0y. So you know that .\n\nWhen two green mangoes are removed, and 10 yellow mangoes are added, the ratio becomes .\n\nNow we have two equations and two variables\u2014we can solve for g to get the answer we seek!\n\nManipulate the first equation to set it equal to\u00a0y:\n\nNow substitute into the other equation:\n\nYou can check your work on this easily enough. If there were originally 18 green mangoes and the ratio of green to yellow was 3 to 5, then there were originally 30 yellow mangoes. What happens if you take away 2 green mangoes and add 10 yellow mangoes? You end up with 16 green and 40 yellow, which simplifies to a ratio of 2 to 5, just like the question said.\n\nOne last note here. You didn\u2019t provide answer choices for this question so I assume there weren\u2019t any. However, if there\u00a0were answer choices here, this would be an ideal question to backsolve\u2014essentially to \u201ccheck your work\u201d like we just did in the last paragraph without doing the work!\n\nCan you do test 4 section 4 number 35 please?\n\nThis is a very SAT-ish question: the math is pretty trivial, but conceptually the question is still a bit tricky.\n\nThere are a couple things you need to have nailed down to get this one right.\n\nFirst, you need to recognize that they\u2019re asking you for a ratio of two dynamic pressures, so they\u2019re asking you for a ratio of two\u00a0q\u2018s. Let\u2019s call them and . We\u2019ll say is the one that corresponds to a velocity of v and\u00a0 is the one that corresponds to a velocity of\u00a01.5v.\n\nSecond, you need to make sure you\u2019re providing the ratio that\u2019s asked for: the\u00a0q of the faster fluid to the\u00a0q of the slower fluid. Which fluid is faster\u2014the one with velocity\u00a0v or the one with velocity 1.5v? 1.5v is always going to be a larger number than v, so that\u2019s the faster fluid. Therefore, we need to calculate the ratio of\u00a0 to .\n\nHow do you do Test 2 Section 3 Number 18?\n\nThe key here is to recognize that you\u2019re dealing with similar triangles (pro tip: similar triangle questions often take this \u201chourglass\u201d form). The two angles at point B are vertical, so they must be congruent. And because segments AE and CD are parallel, you\u2019ve got alternate interior angles for the rest. When all the angles in two triangles are congruent, those triangles are similar.\n\n(It might be easier to see the alternate interior angles if you extend the lines\u2026expand the image on the right.)\n\nAnyway, now that you\u2019ve established that these are similar triangles, you just need to use ratios to solve. If and [latexBD=5[\/latex], then each set of corresponding sides will be in the same ratio. So we can solve for\u00a0BC thusly:\n\nBut wait\u2014you\u2019re not quite done! The question asks for\u00a0CE, not\u00a0BC!\n\nCEBCBE = 4 + 8 = 12\n\nThere.\u00a0Now you\u2019re done.\n\nWhats the best and fastest way of doing Practise test 1, section 4, question 23?\n\nWould it be calculating the ratio of all the options and then comparing it to the Human Resources ratio?\n\nThanks!\n\nYeah, that\u2019s the best and fastest way, although you can save yourself some calculator keystrokes by rounding. This isn\u2019t just a time saver: the more keystrokes you make, the more likely you are to make an error! Rounding to the nearest million (it\u2019s already in thousands) will work just fine. So, my calculations would look like this:\n\nHuman resources 2007 to 2010:\n\nAgriculture\/natural resources 2007 to 2010:\n\nEducation 2007 to 2010:\n\nHighways and transportation 2007 to 2010:\n\nPublic safety 2007 to 2010:\n\nThere you have it: the Human resources and Education budgets have the closest ratios.\n\nTest 6 #20 from the no calculator section\n\nThe circle, you\u2019re told, has a radius of 1, so that means its circumference is . The arc between A and B has a length of . How much of the circumference is that?\n\nFor this one, let\u2019s plug in. Let\u2019s say Observer A observes an intensity of 16, so\u00a0I = 16. Observer B would then observe an intensity of 1, so his\u00a0I = 1.\n\nBoth observers are observing the same ratio signal, so the value of\u00a0P will be the same for both. Because it won\u2019t change, we can plug in for that, too. Let\u2019s say\u00a0P = 100. (You can really pick anything for this, but 100 is a good choice because you\u2019ll end up taking its square root.)\n\nNow solve both equations for\u00a0r. I\u2019m going to use and to keep track of which distance is which.\n\nThe question asks for the ratio . We can calculate that!\n\nThat frankenfraction is not as gnarly as it looks. Just remember that dividing by a fraction is the same as multiplying by its reciprocal, and simplify:\n\nThe circle graph above shows the percent of 4th graders at an elementary school who have the indicated numbers of pets in their homes. If 68 of the 4th graders have at least one pet, how many have exactly two pets?\n\n(A) 16\n(B)17\n(C)20\n(D)33\n(E)34\n\n\u201cAt least one\u201d means one, two, or more than two\u2014anything but zero pets. So that\u2019s 30 + 20 + 35 \u00a0= 85 percent of the students.\n\nIf 68 students represent 85 percent of the class, how many students represent 20%? We can do a ratio:\n\nAt a certain camp, the counselor-to-camper ratio is 2 to 9. If the camp has 18 counselors, how many campers does it have?\n\nIn this case, all we need to do is set up our ratio and be careful to keep our units straight:\n\nCross multiply and solve:\n\nSo there are 81 campers at the camp.\n\nA new high-tech transportation system is to be built connecting a city at sea level and a suburb 3,500 feet above sea level. The maximum allowable grade is 3 percent, which means that the track for the new system can ascend no more than 3 feet for every 100 feet of horizontal length. What is the minimum whole number of feet of track needed for the new system?\n\nA) 113,167 ft\nB) 116,615 ft\nC) 116,615 ft\nD) 116,667 ft\nE) 120,167 ft\n\nHow many times does this track need to go up 3 feet?\n\n3500\/3 = 1166.666\u2026\n\nEvery time it goes up 3 feet, it needs to have 100 feet of horizontal length, so multiply that by 100:\n\n100 \u00d7 1166.666\u2026 = 116,667\n\nYou need to be registered and logged in to take this quiz. Log in or Register\n\n## How\u2019s everyone else doing on this quiz?\n\n10% got 5 right\n17% got 4 right\n15% got 3 right\n26% got 2 right\n31% got 0 or 1 right\n\n Source: Married to the Sea.\n\nThis started out as an Old MacDonald\u2019s farm question.\u00a0No wait,\u00a0I thought to myself,\u00a0not depressing enough.\n\nThe prize this week: You\u2019ll get the satisfaction of knowing that you probably solved this problem in less time than I spent staring at my computer screen trying to come up with a clever prize for this week. I swear I used to be more creative.\n\nIn Wendell\u2019s house, the ratio of unopened credit card offers to out-of-date phone books is 9 to 5. The ratio of magazines to crushed loose cigarettes is 25 to 7, and the ratio of McDonald\u2019s Happy Meal toys to rotting, half-eaten pizzas is 3 to 2. There are 6 used-up batteries lying around for each broken VCR. The ratio of crushed loose cigarettes to McDonald\u2019s Happy Meal toys is 5 to 8, and the ratio of used-up batteries to out-of-date phone books is 5 to 7. There are 30 magazines for each 4 broken VCRs. If there were 108 unopened credit card offers, how many rotting, half-eaten pizzas would Wendell have in his house?\n\nPlease note that, as usual for the weekend challenge, I\u2019m taking\u00a0a concept that SAT has been known to test, and extending it to an extreme to which the SAT would not go (not only in subject matter, but in scope as well). All of this is to say that these weekend challenges are meant to be fun for you, not to freak you out if you can\u2019t get them. They\u2019re usually a degree or two harder than what the SAT will throw your way.\nPut your answers in the comments. I\u2019ll post a solution Monday (probably late in the day). Good luck!UPDATE: Nice work, Chris. And special thanks to Catherine from kitchen table math, who pointed out that my original wording (\u201cIf there are\u2026how many does\u2026\u201d) made non-integer quantities (fractional batteries??) quite unsavory. I\u2019ve changed the wording of the last sentence a bit now for the benefit of people who find this in the archives and want to torture themselves with this question later.\n\nSolution below the cut.\n\nThe SAT would never throw such a complex question at you, but the solution I advocate is one that might help you on the harder ratio questions the SAT will\u00a0toss your way. Remember that on ratio questions, units are paramount. When you\u2019re presented with ratios of more than two things and asked to suss out the relationship between just two of those things, the best and most elegant solution is to line up all the fractions and multiply them together, eliminating unwanted units along the way. Let me show you what I mean:\n\nWhat we\u2019re given:\u00a0a bunch of ratios relating together the following things (in order of appearance):\n\ncredit card offers (CCO)\nphone books (PBK)\nmagazines (MGZ)\ncigarettes (CIG)\nHappy Meal toys (HMT)\nhalf-eaten pizzas (HEP)\nbatteries (BAT)\nVCRs (same, duh.)\n\nWhat we want: the ratio of unopened credit card offers (CCO) to rotting, half-eaten pizzas (HEP). Once we have that ratio, then\u00a0we\u2019ll deal with the 108 CCO.\n\nHow we get there:\u00a0Start by listing the ratios that contain the units you want. Make sure to put CCO on top, and HEP on bottom.\n\nSimplify\u2026\n\nWe need to get rid of PBK and HMT, so let\u2019s find some ratios we can use to do so, one at a time.\n\nSimplify\u2026\n\nAnd then we just keep going. This might get monotonous (it\u2019s a challenge question), but it\u2019s really just the same procedure over and over again until the desired result.\n\nSimplify\u2026\n\nRepeat:\n\nSimplify\u2026\n\nRepeat:\n\nSimplify\u2026\n\nRepeat:\n\nGasp\u2026Deep breath\u2026Simplify\u2026\n\nThat wasn\u2019t so bad now, was it?\n\nNow set up a proportion to see how many HEP will exist if there are 108 CCO:\nNote: Although it will get gnarly with things like fractional batteries, if you\u2019re good with your calculator you can also do this by plugging in 108 for CCO and solving consecutive ratios until you end at 16 HEP. I\u2019m not going to spell that way out here. For some more realistic SAT-style questions that can be solved similarly, look here.\n Source.\n\nSo here\u2019s the thing with ratios and proportions on the SAT: they\u2019re really easy. No, seriously, where are you going? Come back! They\u2019re easy, I swear. All you have to do is keep very close track of your units, and you\u2019ll be good to go. That means when you set up a proportion, actually write the units next to each number. Make sure you\u2019ve got the same units corresponding to each other before you solve, and you\u2019re home free. Pass Go, collect your \\$200, and spend it all on Lik-M-Aid Fun Dip. So uh\u2026let\u2019s try one?\n\n1. A certain farm has only cows and chickens as livestock. The ratio of cows to chickens is 2 to 7. If there are 63 livestock animals on the farm, how many cows are there?\n\n(A) 13\n(B) 14\n(C) 16\n(D) 18\n(E) 49\n\nThe SAT writers would love for you to set up a simple proportion here and solve:\n\n$\\large&space;\\inline&space;\\dpi{300}&space;\\fn_cm&space;\\frac{2}{7}=\\frac{x}{63}$\n\nHooray! = 18! That\u2019s answer choice (D)! NOT SO FAST, SPANKY. You just solved for a terrifying hybrid beast, the COWNIMALKEN. Let\u2019s look at that fraction more carefully, with the units included:\n\n$\\large&space;\\inline&space;\\dpi{300}&space;\\fn_cm&space;\\frac{2\\:cows}{7\\:chickens}=\\frac{x\\:cows}{63\\:animals}$\n\nSo when you casually multiplied by 63 and solved, you solved for a unit that won\u2019t do you any good: the [(cow)(animal)]\/(chicken), or COWNIMALKEN. That\u2019s terrifying. Nature never intended it to be so. Not only are you unwisely playing God, but you\u2019re getting an easy question wrong. Before we can solve this bad boy, we need to make sure our units line up on both sides of the equal sign. So let\u2019s change the denominator on the left to match the one on the right. Get rid of \u201c7 chickens\u201d and replace it with \u201c9 animals.\u201d Get it? Because cows count as animals, if there are 2 cows for every 7 chickens, that means there are 2 cows for every 9 animals.\n\n$\\large&space;\\inline&space;\\dpi{300}&space;\\fn_cm&space;\\frac{2\\:cows}{9\\:animals}=\\frac{x\\:cows}{63\\:animals}$\n\nNow, we can solve:\u00a0x = 14 cows. That\u2019s choice (B). See how the units cancel out nicely when you\u2019ve properly set up a ratio question? That should make the hairs on your neck stand on end.\n\n##### Look out for this tricky crap, too:\n\nBut but but! There\u2019s one more thing you need to watch out for. Sometimes they\u2019ll give you units that aren\u2019t quite as easily converted. Like so:\n\n1. The ratio of students to teachers at a certain school is 28 to 3. The ratio of teachers to cafeteria workers is 9 to 2. What is the ratio of cafeteria workers to students?\n\n(A) 1 to 42\n(B) 2 to 28\n(C) 3 to 37\n(D) 9 to 56\n(E) 3 to 14\n\nHere, we have a few options. It\u2019s not too hard to find a number of teachers that will work with both ratios, so I\u2019ll leave it to you to figure out how to solve it that way if you prefer. Instead let me point out that there\u2019s a pretty elegant solution here that comes from simply multiplying the two ratios together, essentially solving for the expression we\u2019re looking for. Peep the skillz:\n\n$\\large&space;\\inline&space;\\dpi{300}&space;\\fn_cm&space;\\frac{28\\:&space;students}{3\\:&space;teachers}\\times&space;\\frac{9\\:teachers}{2\\:cafeteria\\:workers}$\n\nWhat happens to the teachers? They cancel! So multiply, and simplify:\n\n$\\large&space;\\inline&space;\\dpi{300}&space;\\fn_cm&space;=\\frac{252\\:students}{6\\:cafeteria\\:workers}=\\frac{42\\:students}{1\\:cafeteria\\:workers}$\n\nSince the question asked for the ratio of cafeteria workers to students, just flip it and you\u2019re done! 1 cafeteria worker to 42 students. That\u2019s choice (A). Ah-mazing.","date":"2019-07-21 08:49:25","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 5, \"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\": 0.5711043477058411, \"perplexity\": 1017.4256773388264}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-30\/segments\/1563195526940.0\/warc\/CC-MAIN-20190721082354-20190721104354-00210.warc.gz\"}"}
| null | null |
\section{Introduction}
In this paper we consider a \textit{family of perturbation functions}
\begin{equation*}
F_{u}:X\times Y_{u}\rightarrow \mathbb{R}_{\infty }:=\mathbb{R}\cup
\{+\infty \},\text{ with }u\in U,
\end{equation*
and where $X$ and $Y_{u},$ $u\in U,$ are given locally convex Hausdorff
topological vector spaces (briefly, lcHtvs), the index set $U$ is called the
\textit{uncertainty set }of the family\textit{, }$X$ is its \textit{decision
space}, and each $Y_{u}$ is a \textit{parameter space}. Note that our model
includes a parameter space $Y_{u},$ depending on $u\in U,$ which is a
novelty with respect to the \textquotedblleft classical" robust duality
scheme (see \cite{LJL11} and references therein, where a unique parameter
space $Y$ is considered), allowing us to cover a wider range of applications
including uncertain optimization problems under linear perturbations of the
objective function. The significance of our approach is illustrated along
the paper by relevant cases extracted from deterministic optimization with
linear perturbations, uncertain optimization without linear perturbations,
uncertain conic optimization and infinite optimization.\textbf{\ }The
antecedents of the paper are described in the paragraphs devoted to the
first two cases in Section 2.
We associate with each family $\left\{ F_{u}:u\in U\right\} $\ of
perturbation functions corresponding optimization problems whose definitions
involve continuous linear functionals on the decision and the parameter
spaces. We denote by $0_{X},$ ${0}_{_{X}}^{\ast },$ $0_{u},$ and
0_{u}^{\ast },$\textit{\ }the null vectors of $X,$ its topological dual
X^{\ast },$ $Y_{u},$ and its topological dual $Y_{u}^{\ast },$ respectively.
The optimal value of a minimization (maximization, respectively) problem
\mathrm{(}$P$\mathrm{)}$ is denoted by $\inf \mathrm{(}$P$\mathrm{)}$ ($\sup
\mathrm{(}$P$\mathrm{)}$); in particular, we write $\min \mathrm{(}$P
\mathrm{)}$ ($\max \mathrm{(}$P$\mathrm{)}$) whenever the optimal value of
\mathrm{(}$P$\mathrm{)}$ is attained. We adopt the usual convention that
\inf \mathrm{(P)=+\infty }$ ($\sup \mathrm{(}$P$\mathrm{)=-\infty }$)\textbf
\ }when the problem\textbf{\ }$\mathrm{(P)}$ has no feasible solution
\textbf{\ }The associated optimization problems are the following:
\begin{itemize}
\item \textit{Linearly perturbed uncertain problems}: for each $(u,x^{\ast
})\in U\times X^{\ast },
\begin{equation*}
\mathrm{(}\text{\textrm{P}}_{u}\mathrm{)}{_{{x^{\ast }}}:}\quad \quad
\inf_{x\in X}\left\{ {{F_{u}}(x,{0_{u}})-\left\langle {x^{\ast },x
\right\rangle }\right\} .
\end{equation*}
\item \textit{Robust counterpart of }$\left\{ \mathrm{(}P_{u}\mathrm{)}{_{
x^{\ast }}}}\right\} _{u\in U}:$
\begin{equation*}
{\mathrm{(RP)}_{{x^{\ast }}}:}\quad \quad \inf_{x\in X}\left\{ {\mathop
{\sup }\limits_{u\in U}{F_{u}}(x,{0_{u}})-\left\langle {x^{\ast },x
\right\rangle }\right\} .
\end{equation*}
\end{itemize}
Denoting by $F_{u}^{\ast }:X^{\ast }\times Y_{u}^{\ast }\rightarrow
\overline{\mathbb{R}}$, where $\overline{\mathbb{R}}:=\mathbb{R\cup \{\pm
\infty \}}$, the \textit{Fenchel conjugate} of $F_{u}$, namely,
\begin{equation*}
F_{u}^{\ast }(x^{\ast },y_{u}^{\ast }):=\sup\limits_{(x,y_{u})\in X\times
Y_{u}}\Big\{\langle x^{\ast },x\rangle +\langle y_{u}^{\ast },y_{u}\rangle
-F_{u}(x,y_{u})\Big\},\quad (x^{\ast },y_{u}^{\ast })\in X^{\ast }\times
Y_{u}^{\ast },
\end{equation*
we now introduce the corresponding dual problems:
\begin{itemize}
\item \textit{Perturbational dual of }${(\text{\textrm{P}}_{u})_{{x^{\ast }}
}:$
\begin{equation*}
\mathrm{(}\text{\textrm{D}}_{u}\mathrm{)}{_{{x^{\ast }}}:}\quad \quad
\mathop {\sup }\limits_{y_{u}^{\ast }\in Y_{u}^{\ast }}-F_{u}^{\ast }(
x^{\ast }},y_{u}^{\ast }).
\end{equation*
Obviously
\begin{equation*}
\sup \mathrm{(}\text{\textrm{D}}_{u}\mathrm{)}_{x^{\ast }}\leq \inf \mathrm{
}\text{\textrm{P}}_{u}\mathrm{)}_{x^{\ast }}\leq \inf \mathrm{(}\text
\textrm{RP}}\mathrm{)}_{x^{\ast }},\forall u\in U.
\end{equation*}
\item \textit{Optimistic dual of} $\mathrm{(}$\textrm{RP}$\mathrm{)
_{x^{\ast }}:$
\begin{equation*}
\mathrm{(}\text{\textrm{ODP}}\mathrm{)}_{x^{\ast }}:\quad \quad
\sup_{(u,y_{u}^{\ast })\in \Delta }-F_{u}^{\ast }(x^{\ast },y_{u}^{\ast }),
\end{equation*
where
$\Delta :=\left\{ {\left( {u,y_{u}^{\ast }}\right) :u\in U,\ {y^{\ast }}\in
Y_{u}^{\ast }}\right\} $ is the disjoint union of the spaces ${Y_{u}^{\ast }}
$.
We have
\begin{equation*}
\sup {(\mathrm{ODP})_{{x^{\ast }}}}=\mathop {\sup }\limits_{u\in U}{\left( {
\mathrm{D}_{u}}}\right) _{{x^{\ast }}}}\leq \inf {\mathrm{(RP)}_{{x^{\ast }
}.}
\end{equation*}
\end{itemize}
We are interested in the following desirable robust duality properties:
$\bullet $ \textit{Robust duality} is said to hold at $x^{\ast }$ if $\inf
\mathrm{(RP)}_{{x^{\ast }}}}=\sup {(\mathrm{ODP})_{{x^{\ast }}}}$,
$\bullet $ \textit{Strong robust duality} at $x^{\ast }$ means $\inf
\mathrm{(RP)}_{{x^{\ast }}}}=\max {(\mathrm{ODP})_{{x^{\ast }}}}$,
$\bullet $ \textit{Reverse strong robust duality }at $x^{\ast }$ means $\min
{\mathrm{(RP)}_{{x^{\ast }}}}=\sup {(\mathrm{ODP})_{{x^{\ast }}}}$,
$\bullet $ \textit{Min-max robust duality} at $x^{\ast }$ means $\min
\mathrm{(RP)}_{{x^{\ast }}}}=\max {(\mathrm{ODP})_{{x^{\ast }}}}$.
Each of the above desirable properties is said to be \textit{stable} when it
holds for any $x^{\ast }\in X^{\ast }$. The main results of this paper
characterize these properties in terms of formulas involving the
\varepsilon $-minimizers and $\varepsilon $-subdifferentials of the
objective function of the robust counterpart problem \textrm{(RP)}
_{0_{X}^{\ast }}$, namely, the functio
\begin{equation*}
p:=\sup\limits_{u\in U}F_{u}(\cdot ,0_{u}).
\end{equation*}
Theorem \ref{thm31} characterizes robust duality at a given point $x^{\ast
}\in X^{\ast }$ as a formula for the inverse mapping of the $\varepsilon
-subdifferential at $x^{\ast }$ without any convexity assumption. The same
is done in Theorem \ref{thm41} to characterize strong robust duality. In the
case when a primal optimal solution does exist we give a formula for the
exact minimizers of $p-x^{\ast }$ to characterize dual strong (resp.
min-max) robust duality at $x^{\ast }$, see Theorem \ref{thm51} (resp.
Theorem \ref{thm52}). We show that stable robust duality gives rise\textbf{\
}to a formula for the $\varepsilon $-subdifferential of $p$ (Theorem \re
{thm61}, see also Theorem \ref{thm31}). The same is done for stable strong
robust duality (Theorem \ref{lem71}). A formula for the exact
subdifferential of $p$ is provided in relation with robust duality at
appropriate points (Theorem \ref{thm81}). The most simple possible formula
for the exact subdifferential of $p$ (the so-called \textit{Basic Robust
Qualification condition}) is studied in detail in Theorem \ref{thm82}. All
the results from Sections 1-8 are specified for the two extreme cases (the
case with no uncertainty and the one in absence of linear perturbations),
namely, Cases 1 and 2 in Section 2 (for the sake of brevity, we do not give
the specifications for Cases 3 and 4). It is worth noticing the generality
of the mentioned results (as they do not require any assumption on the
involved functions) and the absolute self containment of their proofs. The
use of convexity in the data will be addressed in a forthcoming paper.
\section{Special Cases and Applications}
In this section we make explicit the meaning of the robust duality of the
general model introduced in Section 1, composed by a family of perturbation
functions together with its corresponding optimization problems. We are
doing this by exploring the extreme case with no uncertainty, the extreme
case in absence of perturbations, and two other significant situations. In
all these cases, we propose \textit{ad hoc} families of perturbation
functions allowing to apply the duality results to given optimization
problems, either turning back to variants of well-known formulas for
conjugate functions or proposing\textbf{\ }new ones.
\noindent Let us recall the robust duality formula, $\inf {\mathrm{(RP)}_{
x^{\ast }}}=}\sup \mathrm{(}\mathrm{ODP}\mathrm{)}_{x^{\ast }},$ i.e.,
\begin{equation}
\;\;\mathop {\inf }\limits_{x\in X}\mathop {\sup }\limits_{u\in U}\left\{ {
F_{u}}\left( {x,{0_{u}}}\right) -\left\langle {{x^{\ast }},x}\right\rangle
\right\} =\mathop {\sup }\limits_{\left( {u,y_{^{u}}^{\ast }}\right) \in
\Delta }-{F_{u}^{\ast }}\left( {{x^{\ast }},{y_{u}^{\ast }}}\right) .
\label{eq1}
\end{equation
We firstly study the two extreme cases: the case with no uncertainty and the
one with no perturbations.
\textbf{Case 1. The case with no uncertainty:} Deterministic optimization
with linear perturbations deals with parametric problems of the form:
\begin{equation*}
\mathrm{(P)}_{{x^{\ast }}}:\quad \quad {\mathop {\inf }\limits_{x\in X}
\left\{ {{f}(x)-\left\langle {{x^{\ast }},x}\right\rangle }\right\} {,}
\end{equation*
where $f:X\rightarrow \mathbb{R}_{\infty }$ (i.e., $f\in (\mathbb{R}_{\infty
})^{X}$) is the \textit{nominal objective function} and the \textit{paramete
} is ${x^{\ast }\in X}^{\ast }.$ Taking a singleton uncertainty set
U=\left\{ u_{0}\right\} ,$\ $Y_{u_{0}}=Y$ and $F_{u_{0}}=F\ $such that $
F\left( {x,{0_{Y}}}\right) ={f}(x)}$ for all $x\in X,$ \eqref{eq1} reads
\begin{equation}
\mathop {\inf }\limits_{x\in X}\left\{ {F\left( {x,{0_{Y}}}\right)
-\left\langle {{x^{\ast }},x}\right\rangle }\right\} =\mathop {\sup
\limits_{{y^{\ast }}\in {Y^{\ast }}}-{F^{\ast }}\left( {{x^{\ast }},{y^{\ast
}}}\right) , \label{11}
\end{equation
which is \textit{the fundamental perturbational duality formula} \cit
{Bot-book1}, \cite{Rock74}, \cite{Za02}. Stable and strong robust duality
theorems are given in \cite{BuJeWu06} (see also \cite{DGLV17} and \cite{JL09}
for infinite optimization problems).
\textbf{Case 2. The case with no linear perturbations:} Uncertain
optimization without linear perturbations deals with families of problems of
the form
\begin{equation*}
\mathrm{(P):}\quad \quad \left\{ {\mathop {\inf }\limits_{x\in X}{f_{u}}(x)
\right\} _{u\in U},
\end{equation*
where $f_{u}\in (\mathbb{R}_{\infty })^{X},$ $u\in U.$ Taking $F\ $such that
$F_{u}(x,0_{u})=f_{u}(x)$ for all $u\in U,$ the problem $\mathrm{(}$\textrm{
}$_{u}\mathrm{)}{_{{0}_{_{X}}^{\ast }}}$ represents here the scenario of
\mathrm{(P)}$ corresponding to $u\in U,$\ while ${\mathrm{(RP)}_{{0
_{_{X}}^{\ast }}}$ is the \textit{robust} \textit{counterpart} (also called
\textit{pessimistic} or \textit{minmax counterpart} in the robust
optimization literature) of $\mathrm{(P),}$ namely,
\begin{equation*}
\text{\textrm{(RP)}}\quad \quad \inf_{x\in X}\mathop {\sup }\limits_{u\in U}
{f_{u}}(x)}.
\end{equation*
\newline
For instance, if $F_{u}(x,y_{u})=f_{u}(x),\ $for all $y_{u}\in Y_{u},$ and
\operatorname{dom}f_{u}\neq \emptyset $, we have
\begin{equation}
F_{u}^{\ast }\left( {x,y_{u}^{\ast }}\right) =\left\{
\begin{array}{ll}
{f_{u}^{\ast }\left( {x^{\ast }}\right) ,} & \mathrm{if}\;\;{y_{u}^{\ast }
=0_{u}^{\ast }, \\
{+\infty ,} & \mathrm{if}\;\;y_{u}^{\ast }\neq 0_{u}^{\ast }
\end{array
\right. \label{12}
\end{equation
Then \eqref{eq1} writes
\begin{equation}
{\left( {\mathop {\sup }\limits_{u\in U}{f_{u}}}\right) ^{\ast }}({x^{\ast }
)=\mathop {\inf }\limits_{u\in U}f_{u}^{\ast }({x^{\ast }}), \label{13}
\end{equation
which amounts, for $x^{\ast }={0}_{_{X}}^{\ast },$ to the $\inf -\sup
\textit{\ duality in robust optimization,}\ also called \textit{robust
infimum} (recall that any constrained optimization problem can be reduced to
an unconstrained one by summing up the indicator function of the feasible
set to the objective function):
\begin{equation*}
\inf_{x\in X}\sup_{u\in U}f_{u}(x)=\sup_{u\in U}\inf_{x\in X}f_{u}(x).
\end{equation*
Robust duality theorems without linear perturbations are given in \cit
{WSS15} for a special class of uncertain non-convex optimization problems
while \cite{DGLV17} provides robust strong duality theorems for uncertain
convex optimization problems which are expressed in terms of the closedness
of suitable sets regarding the vertical axis of $X^{\ast }\times \mathbb{R}.$
\textbf{Case 3. Conic optimization problem with uncertain constraints:}
\textit{Consider the uncertain problem}
\begin{equation*}
\mathrm{(P):}\ \ \ \ \ \ \ \ \left\{ {\inf_{x\in X}f(x)\;}\;\text{\textrm
s.t.}}{\text{ }\;{H_{u}}(x)\in -{S_{u}}}\right\} _{u\in U},
\end{equation*
where, for each $u\in U$, $S_{u}$ is an ordering convex cone in ${Y}_{u},$
H_{u}\colon X\rightarrow {Y}_{u}$, and $f\in (\mathbb{R}_{\infty })^{X}$.
Denote by ${S_{u}^{+}:=}\left\{ {y_{u}^{\ast }\in Y}_{u}^{\ast
}:\left\langle {y_{u}^{\ast },y_{u}}\right\rangle \geq 0,\forall {y_{u}\in
S_{u}\right\} $ the dual cone of $S_{u}$\textbf{.}
Problems of this type arise, for instance, in the production planning of
firms producing $n$\ commodities from uncertain amounts of resources by
means of technologies which depend on the available resources (e.g., the
technology differs when the energy is supplied by either fuel gas or a
liquid fuel). The problem associated with each parameter $u\in U$\ consists
of maximizing the cash-flow $c\left( x_{1},...,x_{n}\right) $\ of the total
production, with $x_{i}$\ denoting the production level of the $i$-th
commodity, $i=1,..,n.$\ The decision vector $x=\left( x_{1},...,x_{n}\right)
$\ must satisfy a linear inequality system $A_{u}x\leq b_{u},$\ where the
matrix of technical coefficients $A_{u}$\ is $m_{u}\times n$\ and $b_{u}\in
\mathbb{R}^{m_{u}},$\ for some $m_{u}\in \mathbb{N}.\ $Denoting by $\mathrm{
}_{\mathbb{R}_{+}^{n}}$ the indicator function of $\mathbb{R}_{+}^{n}$
(i.e., $\mathrm{i}_{\mathbb{R}_{+}^{n}}(x)=0,$ when $x\in \mathbb{R
_{+}^{n}, $ and $\mathrm{i}_{\mathbb{R}_{+}^{n}}(x)=+\infty ,$ otherwise),
the uncertain production planning problem can be formulated a
\begin{equation*}
\mathrm{(P):}\ \ \ \ \ \ \ \ \left\{ {\inf_{x\in \mathbb{R}^{n}}f(x)=-c(x)+
\mathrm{i}_{\mathbb{R}_{+}^{n}}{(x)\;}\;\text{\textrm{s.t.}}{\text{
\;A_{u}x-b_{u}\in -}\mathbb{R}_{+}^{m_{u}}\right\} _{u\in U},
\end{equation*
with the space $Y_{u}=\mathbb{R}^{m_{u}}$ depending on the uncertain
parameter $u.$
For each $u\in U$, define the perturbation function
\begin{equation*}
{F_{u}}(x,{y_{u}})=\left\{
\begin{array}{ll}
f{(x),} & {\mathrm{if}\;{H_{u}}(x)+{y_{u}}\in -{S_{u}}}, \\
{+\infty ,} & {\mathrm{else}}
\end{array
}\right.
\end{equation*
On the one hand, $\mathrm{(RP)}_{0_{X}^{\ast }}$ collapses to the robust
counterpart of $\mathrm{(P)}$\ in the sense of robust conic optimization
with uncertain constraints:
\begin{equation*}
\mathrm{(RP):}\;\;\;\;\;\inf_{x\in X}f(x)\;\;\text{\textrm{s.t.}}\ \ \;{H_{u
}(x)\in -{S_{u}},\;\;\forall u\in U.
\end{equation*
On the other hand, it is easy to check that
\begin{equation*}
F_{u}^{\ast }({x^{\ast }},{y}_{u}^{\ast })=\left\{
\begin{array}{ll}
{{{\left( {f+y_{u}^{\ast }\circ {H_{u}}}\right) }^{\ast }}({x^{\ast }}),} &
\mathrm{if}\;y_{u}^{\ast }\in S_{u}^{+},} \\
{\ +\infty ,} & {\mathrm{else,}
\end{array
}\right.
\end{equation*
$\mathrm{(ODP)}_{0_{X}^{\ast }}$ is nothing else than the optimistic dual in
the sense of uncertain conic optimization:
\begin{equation*}
\mathrm{(ODP):}\quad \quad \sup\limits_{u\in U,{y}_{u}^{\ast }\in S_{u}^{+}
\mathop {\inf }\limits_{x\in X}\left\{ {f(x)+\left\langle {y_{u}^{\ast },
H_{u}}(x)}\right\rangle }\right\}
\end{equation*
(a special case when $Y_{u}=Y$, $S_{u}=S$ for all $u\in U$ is studied in
\cite[page 1097]{DMVV} and \cite{LJL11}). Thus,
$\bullet$ \textit{Robust duality holds at $0_{X}^{\ast }$} means that $\inf
\mathrm{(RP)}=\sup \mathrm{(}\mathrm{ODP}), $\
$\bullet $ \textit{Strong robust duality holds at $0_{X}^{\ast }$} means
that
\begin{equation*}
\inf \left\{ {f(x):{H_{u}}(x)\in -{S_{u}},\forall u\in U}\right\}
=\max\limits_{\QATOP{u\in U\hfill }{{y_{u}^{\ast }}\in S_{u}^{+}\hfill
}\inf\limits_{x\in X}\left\{ {f(x)+\left\langle {y_{u}^{\ast },{H_{u}}(x)
\right\rangle }\right\} .
\end{equation*
Conditions for having such an equality are provided in \cite[Theorem 6.3
{DMVV}, \cite[Corollaries 5, 6]{DL17}, for the particular case $Y_{u}=Y$ for
all $u\in U$.
\textit{Strong robust duality and uncertain Farkas lemma:} We focus again on
the case where $Y_{u}=Y$ and $S_{u}=S$ for all $u\in U$. For a given $r\in
\mathbb{R}$, let us\textbf{\ }consider the following statements:
\begin{itemize}
\item[(i)] $H_{u}(x)\in -S,\;\forall u\in U\quad \Longrightarrow \quad
f(x)\geq r$,
\item[(ii)] $\exists u\in U,\exists {y_{u}^{\ast }}\in S^{+}$ such that
f(x)+\left\langle {y_{u}^{\ast }},H_{u}(x)\right\rangle \geq r,\;\forall
x\in X.$
\end{itemize}
\noindent Then, it is true that the Strong robust duality holds at
0_{X}^{\ast }$ if and only if $\left[ (i)\Longleftrightarrow (ii)\right] $
for each $r\in \mathbb{R},$ which can be seen as an uncertain Farkas lemma.
For details\textbf{\ }see \cite[Theorem 3.2]{DMVV} (also \cite[Corollary 5
and Theorem 1]{DL17}).\textbf{\ }
It is worth noticing that when return to problem $\mathrm{(P)}$, a given
robust feasible solution $\overline{x}$\ is a minimizer if and only if $f
\overline{x})\leq f(x)$ for any robust feasible solution $x.$ So, a robust
(uncertain) Farkas lemma (with $r=f(\bar{x})$) will lead automatically to an
optimality test for $\mathrm{(P).}$ Robust conic optimization problems are
studied in \cite{BS06} and \cite{Vera17}.
\textbf{\textbf{Case 4. Discretizing infinite optimization problems:}} Let
f\in (\mathbb{R}_{\infty })^{X}$ and $g_{t}\in \mathbb{R}^{X}\;$for all
\;t\in T$ (a possibly infinite index set). Consider the set $U$ of nonempty
finite subsets of $T,$ interpreted as admissible perturbations of $T,$ and
the parametric optimization problem
\begin{equation*}
\mathrm{(P):}\quad \left\{ \inf_{x\in X}f(x)\;\mathrm{s.t.}\text{
g_{t}(x)\leq 0,\text{ }\forall t\in S\right\} _{S\in U}.
\end{equation*
Consider the parameter space $Y_{s}:={\mathbb{R}^{S}}$ (depending on $S$)
and the perturbation function ${F_{S}}:X\times {\mathbb{R}^{S}}\rightarrow
\mathbb{R}_{\infty }$ such that, for any $x\in X$ and ${{{{\mu :=}({\mu _{s}
)}_{s\in S}\in }\mathbb{R}^{S},}$
\begin{equation*}
{F_{S}}\left( {x,{\mu }}\right) =\left\{
\begin{array}{ll}
{f(x),} & {\mathrm{if}\;{g_{s}}(x)\leq -{\mu _{s}},\;\forall s\in S,} \\
{+\infty ,} & {\mathrm{else}}
\end{array
}\right.
\end{equation*
We now interpret the problems associated with the family of function
perturbations $\left\{ {F_{S}:S\in U}\right\} .$ One has $Y_{s}^{\ast }=
\mathbb{R}^{S}}$ and
\begin{equation*}
F_{S}^{\ast }\left( {{x^{\ast }},{\lambda }}\right) =\left\{
\begin{array}{ll}
{{\left( {f+\sum\limits_{s\in S}{{\lambda _{s}g_{s}}}}\right) ^{\ast }}(
x^{\ast }}),} & {\mathrm{if}\;{\lambda }\in \mathbb{R}_{+}^{S},} \\
{+\infty ,} & {\mathrm{else}}
\end{array
}\right.
\end{equation*
The robust counterpart at $0_{X}^{\ast },$
\begin{equation*}
\mathrm{(RP)}_{0_{X}^{\ast }}:\quad \inf f(x)\;\;\mathrm{s.t.}\ \;\;{g_{t}
(x)\leq 0\text{\ for all }t\in T,
\end{equation*
is a general infinite optimization problem while the optimistic dual at
0_{X}^{\ast }$ is
\begin{equation*}
\mathrm{(ODP)}_{0_{X}^{\ast }}:\quad \sup\limits_{S\in U,{{\lambda }\in
\mathbb{R}_{+}^{S}}}\left\{ \mathop {\inf }\limits_{x\in X}\left(
f(x)+\sum\limits_{s\in S}{{\lambda _{s}g_{s}}(x)}}\right) \right\} ,
\end{equation*
or, equivalently, the Lagrange dual of $\mathrm{(RP)}_{0_{X}^{\ast }},$
i.e.,
\begin{equation*}
\mathrm{(ODP)}_{0_{{X}^{\ast }}}:\quad \quad \sup\limits_{{\lambda }\in
\mathbb{R}_{+}^{(T)}}\left\{ \mathop {\inf }\limits_{x\in X}\left(
f(x)+\sum\limits_{t\in T}{{\lambda _{t}g_{t}}(x)}}\right) \right\} ,
\end{equation*
where, for each $\lambda =\left( {\lambda _{t}}\right) _{t\in T}\in \mathbb{
}_{+}^{(T)}$ (the subspace of $\mathbb{R}^{T}$ formed by the functions
\lambda $ whose support, supp$\lambda :=\left\{ t\in T:{{\lambda _{t}\neq 0}
\right\} ,$ is finite),
\begin{equation*}
\sum\limits_{t\in T}{{\lambda _{t}g_{t}}(x):}=\left\{
\begin{array}{ll}
\sum\limits_{t\in \text{supp}\lambda }{{\lambda _{t}g_{t}}(x),} & {\mathrm{i
}\;\lambda \neq 0,} \\
0, & \mathrm{if}\;\lambda =0
\end{array
\right.
\end{equation*
Following \cite[Section 8.3]{GL98}, we say that $\mathrm{(RP)}_{0_{X}^{\ast
}}$ is \textit{discretizable} if there exists a sequence $\left(
S_{r}\right) _{r\in \mathbb{N}}\subset U$ such that
\begin{equation}
\inf \mathrm{(RP)}_{0_{X}^{\ast }}=\lim_{r}\inf \left\{ f(x):{g_{t}}(x)\leq
0,\ \forall t\in S_{r}\right\} , \label{2.10}
\end{equation
and it is \textit{reducible }if there exists $S\in U$ such that
\begin{equation*}
\inf \mathrm{(RP)}_{0_{X}^{\ast }}=\inf \left\{ f(x):{g_{t}}(x)\leq 0,\
\forall t\in S\right\} .
\end{equation*
Obviously, $\inf \mathrm{(RP)}_{0_{X}^{\ast }}=-\infty $ entails that
\mathrm{(RP)}_{0_{X}^{\ast }}$\ is reducible which, in turn, implies that
\mathrm{(RP)}_{0_{X}^{\ast }}$ is discretizable.
Discretizable and reducible problems are important in practice. Indeed, on
the one hand, discretization methods generate sequences $\left( S_{r}\right)
_{r\in \mathbb{N}}\subset U$ satisfying \ref{2.10} when $\mathrm{(RP)
_{0_{X}^{\ast }}$ is discretizable; discretization methods for linear and
nonlinear semi-infinite programs have been reviewed in \cite[Subsection 2.3
{GL17} and \cite{LS07}, while a hard infinite optimization problem has been
recently solved via discretization in \cite{LR17}. On the other hand,
replacing the robust counterpart (a hard semi-infinite program when the
uncertainty set is infinite) of a given uncertainty optimization problem,
when it is reducible, by a finite subproblem allows many times to get the
desired tractable reformulation (see e.g., \cite{BEN09} and \cite{BJL13}).
\begin{example}[Discretizing linear infinite optimization problems]
\label{Example1}Consider the problems introduced in Case 4 above, with
f(\cdot ):=\left\langle c^{\ast },\cdot \right\rangle $ and ${g_{t}
(x):=\left\langle a_{t}^{\ast },\cdot \right\rangle -b_{t},$ where $c^{\ast
},a_{t}^{\ast }\in X^{\ast }$ and $b_{t}\in \mathbb{R},$ for all $t\in T.$
Then, $\mathrm{(RP)}_{0_{X}^{\ast }}$ collapses to the linear infinite
programming proble
\begin{equation*}
\mathrm{(RP)}_{0_{X}^{\ast }}:\;\;\;\inf \quad \left\langle c^{\ast
},x\right\rangle \;\;\mathrm{s.t.}\ \ \left\langle a_{t}^{\ast
},x\right\rangle \leq b_{t},\ \forall t\in T,
\end{equation*
whose feasible set we denote by $A.$ So, $\inf \mathrm{(RP)}_{0_{X}^{\ast
}}=\inf_{x\in X}\left\{ \left\langle c^{\ast },x\right\rangle +i_{A}\left(
x\right) \right\} .\;$ We assume that $A\neq \emptyset .$ \newline
Given $S\in U$ and ${{{\mu ,\lambda }\in }}\mathbb{R}^{S},$
\begin{equation}
{F_{S}}\left( {x,{\mu }}\right) =\left\{
\begin{array}{ll}
\left\langle c^{\ast },x\right\rangle {,} & {\mathrm{if}\;\ \left\langle
a_{s}^{\ast },x\right\rangle \leq b_{s}-{\mu _{s}},\;\forall s\in S,} \\
{+\infty ,} & {\mathrm{else,}
\end{array
}\right. \label{2.11}
\end{equation
and
\begin{equation}
F_{S}^{\ast }\left( {{x^{\ast }},{\lambda }}\right) =\left\{
\begin{array}{ll}
{{{\sum\limits_{s\in S}{{\lambda _{s}b_{s}}}}},} & {\mathrm{if}\;{{
\sum\limits_{s\in S}{{\lambda _{s}a_{s}^{\ast }}}}}}}=x^{\ast }{{-c^{\ast }}
\text{ }\mathrm{and}\text{ }{{\lambda _{s}\geq 0\,,\ }\forall }s\in S, \\
{+\infty ,} & {\mathrm{else.}
\end{array
}\right. \label{2.12}
\end{equation
Hence, $\mathrm{(ODP)}_{0_{X}^{\ast }}$ collapses to the so-called \textit
Haar dual problem \cite{GLV14}} of $\mathrm{(RP)}_{0_{X}^{\ast }},$
\begin{equation*}
\mathrm{(ODP)}_{0_{X}^{\ast }}:\quad \sup \left\{ -{{{\sum\limits_{t\in
\operatorname{supp}{\lambda }}{{\lambda _{t}b_{t}:-{{{\sum\limits_{t\in \operatorname
supp}{\lambda }}}}\lambda _{t}a_{t}^{\ast }=}}}}c^{\ast },}}\text{ }{\lambda
}\in \mathbb{R}_{+}^{(T)}\right\} ,
\end{equation*
i.e.,
\begin{equation}
\sup \mathrm{(ODP)}_{0_{X}^{\ast }}=-\inf_{S\in U,{{{\lambda }\in }}\mathbb{
}_{+}^{S}}\left\{ {{{\sum\limits_{s\in S}{{\lambda _{s}b_{s}:
\sum\limits_{s\in S}{{\lambda _{s}a_{s}^{\ast }=}}}}-c^{\ast }}}\right\} .
\label{2.13}
\end{equation
From (\ref{2.13}), if $\inf \mathrm{(RP)}_{0_{X}^{\ast }}=\max \mathrm{(ODP)
_{0_{X}^{\ast }}\in \mathbb{R},$ then there exist $S\in U$ and ${{{\lambda
\in }}\mathbb{R}_{+}^{S}$ such that
\begin{equation}
{{{\sum\limits_{s\in S}{{\lambda _{s}}}}}}\left( {{{{{{a_{s}^{\ast },}b_{s}}
}}}\right) {=-}\left( c^{\ast },\inf \mathrm{(RP)}_{0_{X}^{\ast }}\right) .
\label{2.14}
\end{equation
Let $A_{S}:=\left\{ x\in X:{\ \left\langle a_{s}^{\ast },x\right\rangle \leq
b_{s},\;\forall s\in S}\right\} .$ Given $x\in A_{S},$ from (\ref{2.14}),
\begin{equation*}
0\geq {{{\sum\limits_{s\in S}{{\lambda _{s}}}}}}\left( {\left\langle
a_{s}^{\ast },x\right\rangle -{{{{b_{s}}}}}}\right) {=-}\left\langle c^{\ast
},x\right\rangle +\inf \mathrm{(RP)}_{0_{X}^{\ast }}.
\end{equation*
Since
\begin{equation*}
\inf \mathrm{(RP)}_{0_{X}^{\ast }}\leq \left\langle c^{\ast },x\right\rangle
,\forall x\in A_{S},
\end{equation*
\begin{equation}
\inf \mathrm{(RP)}_{0_{X}^{\ast }}=\inf \left\{ \left\langle c^{\ast
},x\right\rangle :{\left\langle a_{s}^{\ast },x\right\rangle \leq
b_{s},\;\forall s\in S}\right\} , \label{2.15}
\end{equation
so that $\mathrm{(RP)}_{0_{X}^{\ast }}$ is reducible. Conversely, if (\re
{2.15}) holds with $\inf \mathrm{(RP)}_{0_{X}^{\ast }}\in \mathbb{R}$ and
\operatorname{cone}\left\{ \left( {{{{{{a_{t}^{\ast },}b_{t}}}}}}\right) :t\in
T\right\} +\mathbb{R}_{+}\left( 0_{X}^{\ast },1\right) $ is weak$^{\ast }
\textbf{-}closed, since $\inf \mathrm{(RP)}_{0_{X}^{\ast }}\leq \left\langle
c^{\ast },x\right\rangle $ is consequence of $\left\{ {\left\langle
a_{s}^{\ast },x\right\rangle \leq b_{s},\;\forall s\in S}\right\} ,$\ by the
non-homogeneous Farkas lemma in lcHtvs \cite{Chu66} and the closedness
assumption, there exist ${{{\lambda }\in }}\mathbb{R}_{+}^{S}$ and $\mu {\in
}\mathbb{R}_{+}$ such that
\begin{equation*}
{-}\left( c^{\ast },\inf \mathrm{(RP)}_{0_{X}^{\ast }}\right) ={{
\sum\limits_{s\in S}{{\lambda _{s}}}}}}\left( {{{{{{a_{s}^{\ast },}b_{s}}}}}
\right) {+}\mu \left( 0_{X}^{\ast },1\right) ,
\end{equation*
which implies that $\mu =0$ and $\inf \mathrm{(RP)}_{0_{X}^{\ast }}=\max
\mathrm{(ODP)}_{0_{X}^{\ast }}{.}$ The closedness assumption holds when $X$
is finite dimensional (guaranteeing that any finitely generated convex cone
in $X^{\ast }\times \mathbb{R}$ is closed). So, as proved in \cite[Theorem
8.3]{GL98}, a linear semi-infinite program $\mathrm{(RP)}_{0_{X}^{\ast }}$
is reducible if and only if (\ref{2.15}) holds if and only if $\inf \mathrm
(RP)}_{0_{X}^{\ast }}=\max \mathrm{(ODP)}_{0_{X}^{\ast }}.$\newline
We now assume that $\inf \mathrm{(RP)}_{0_{X}^{\ast }}=\sup \mathrm{(ODP)
_{0_{X}^{\ast }}\in \mathbb{R}.$ By (\ref{2.13}), there exist sequences
\left( S_{r}\right) _{r\in \mathbb{N}}\subset U$ and $\ \left( {\lambda
_{r}\right) _{r\in \mathbb{N}},$ with ${\lambda }^{r}{\in }\mathbb{R
_{+}^{S_{r}}$ for all $r\in \mathbb{N},$\ such tha
\begin{equation*}
\lim_{r}\inf_{{\lambda }^{r}{\in }\mathbb{R}_{+}^{S_{r}}}\left\{ {{
\sum\limits_{s\in S_{r}}{{\lambda _{s}^{r}b_{s}:}\sum\limits_{s\in S_{r}}{
\lambda _{s}^{r}a_{s}^{\ast }=}}}}-c^{\ast }}}\right\} =-\sup \mathrm{(ODP)
_{0_{X}^{\ast }}.
\end{equation*
Denote $v_{r}:=-{{{\sum\limits_{s\in S_{r}}{{\lambda _{s}^{r}b_{s}.}}}}}$
Then,
\begin{equation}
{{{\sum\limits_{s\in S_{r}}{{\lambda _{s}}}}}}\left( {{{{{{a_{s}^{\ast },
b_{s}}}}}}\right) {=-}\left( c^{\ast },v_{r}\right) , \label{2.16}
\end{equation
with $\lim_{r}v_{r}=\inf \mathrm{(RP)}_{0_{X}^{\ast }}.$ Let $A_{r}:=\left\{
x\in X:{\ \left\langle a_{s}^{\ast },x\right\rangle \leq b_{s},\;\forall
s\in S}_{r}\right\} ,$ $r\in \mathbb{N}.$ Given $x\in A_{r},$ from (\re
{2.16}),
\begin{equation*}
0\geq {{{\sum\limits_{s\in S_{r}}{{\lambda _{s}^{r}}}}}}\left( {\left\langle
a_{s}^{\ast },x\right\rangle -{{{{b_{s}}}}}}\right) ={-}\left\langle c^{\ast
},x\right\rangle +v_{r}.
\end{equation*
Since $v_{r}\leq \left\langle c^{\ast },x\right\rangle $ for all $x\in
A_{r}, $
\begin{equation*}
v_{r}\leq \inf \left\{ \left\langle c^{\ast },x\right\rangle :{\left\langle
a_{s}^{\ast },x\right\rangle \leq b_{s},\;\forall s\in S}_{r}\right\} \leq
\inf \mathrm{(RP)}_{0_{X}^{\ast }}.
\end{equation*
Thus,
\begin{equation*}
\lim_{r}\inf \left\{ \left\langle c^{\ast },x\right\rangle :{\left\langle
a_{s}^{\ast },x\right\rangle \leq b_{s},\;\forall s\in S}_{r}\right\} =\inf
\mathrm{(RP)}_{0_{X}^{\ast }},
\end{equation*
i.e., $\mathrm{(RP)}_{0_{X}^{\ast }}$ is discretizable. Once again, the
converse is true in linear semi-infinite programming \cite[Corollary 8.2.1
{GL98}, but not in linear infinite programming.
\end{example}
\section{Robust Conjugate Duality}
We now turn back to the general perturbation function ${F_{u}}\colon X\times
{Y_{u}}\rightarrow \mathbb{R}_{\infty },$ $u\in U$ and let $\Delta :=\left\{
(u,y_{u}^{\ast }):u\in U,y_{u}^{\ast }\in Y_{u}^{\ast }\right\} $ be the
disjoint union of the spaces $Y_{u}^{\ast }$. Recall that
\begin{equation}
{\mathrm{(RP)}_{{x^{\ast }}}:}\quad \inf_{x\in X}\left\{ {\mathop {\sup
\limits_{u\in U}{F_{u}}\left( {x,{0_{u}}}\right) -\left\langle {{x^{\ast }},
}\right\rangle }\right\} , \label{RPx-star}
\end{equation
\begin{equation}
\mathrm{(ODP)}{_{{x^{\ast }}}:}\quad \ \ \sup\limits_{(u,\mathbf{y
_{u}^{\ast })\in \Delta }-F_{u}^{\ast }\left( {{x^{\ast }},y_{u}^{\ast }
\right) . \label{ODPx-star}
\end{equation}
Define $p\in \overline{\mathbb{R}}^{X}$ and $q\in \overline{\mathbb{R}
^{X^\ast}$ such that
\begin{equation}
p:=\mathop {\sup }\limits_{u\in U}{F_{u}}(\cdot ,{0_{u}})\ \text{\ \ \ and \
\ \ }q:=\mathop {\inf }\limits_{\left( {u,y_{u}^{\ast }}\right) \in \Delta
}\;F_{u}^{\ast }(\cdot ,y_{u}^{\ast }). \label{pq}
\end{equation
One then has
\begin{equation}
\left\{
\begin{array}{l}
{{p^{\ast }}({x^{\ast }})=-\inf {{\mathrm{(RP)}}_{{x^{\ast }}}},\quad q(
x^{\ast }})=-\sup \mathrm{(}{{\mathrm{ODP)}}_{{x^{\ast }}}\bigskip }} \\
{{q^{\ast }}=\mathop {\sup }\limits_{\left( {u,y_{u}^{\ast }}\right) \in
\Delta }{{\left( {F_{u}^{\ast }\left( \cdot {,y_{u}^{\ast }}\right) }\right)
}^{\ast }}=\mathop {\sup }\limits_{u\in U}F_{u}^{\ast \ast }\left( \cdot {,
0_{u}}}\right) \leq p}
\end{array
}\right. \label{30}
\end{equation
and hence,
\begin{itemize}
\item \textit{Weak robust duality} always holds:
\begin{equation}
p^{\ast }(x^{\ast })\leq q^{\ast \ast }(x^{\ast })\leq q(x^{\ast }),\text{
for all }x^{\ast }\in X^{\ast }. \label{WD}
\end{equation}
\item \textit{Robust duality at }$x^{\ast }$ means:
\begin{equation}
p^{\ast }(x^{\ast })=q(x^{\ast }). \label{SD}
\end{equation}
\end{itemize}
Robust duality at $x^{\ast }$ also holds when either $p^{\ast }(x^{\ast
})=+\infty $ or $q(x^{\ast })=-\infty .$
As an illustration, consider Case 4 with linear data, as in Example \re
{Example1}. Then, $p\left( x\right) =\left\langle c^{\ast },x\right\rangle
\mathrm{i}_{A}\left( x\right) ,$ $\operatorname{dom}p=A,$ and so
\begin{equation*}
p^{\ast }\left( 0_{X}^{\ast }\right) =\sup_{x\in \mathbb{R}^{n}}\left(
-p\left( x\right) \right) =-\inf_{x\in \mathbb{R}^{n}}\left\{ \left\langle
c^{\ast },x\right\rangle +\mathrm{i}_{A}\left( x\right) \right\} =-\inf
\mathrm{(RP)}_{0_{X}^{\ast }}.
\end{equation*
Similarly, from (\ref{2.12}),
\begin{equation*}
q\left( {{x^{\ast }}}\right) =\inf_{S\in U,{{{\lambda }\in }}\mathbb{R
^{S}}\left\{ {{{\sum\limits_{s\in S}{{\lambda _{s}b_{s}:}\sum\limits_{s\in S
{{\lambda _{s}a_{s}^{\ast }=}}}}x^{\ast }-c^{\ast }}}\right\} ,
\end{equation*
$\operatorname{dom}q=c^{\ast }+\operatorname{cone}\left\{ a_{t}^{\ast }:t\in T\right\}
$ and
\begin{equation}
q\left( 0_{X}^{\ast }\right) =\inf_{S\in U,{{{\lambda }\in }}\mathbb{R
_{+}^{S}}\left\{ {{{\sum\limits_{s\in S}{{\lambda _{s}b_{s}:
\sum\limits_{s\in S}{{\lambda _{s}a_{s}^{\ast }=}}}}-c^{\ast }}}\right\}
=-\sup \mathrm{(ODP)}_{0_{X}^{\ast }}. \label{36}
\end{equation}
\subsection{Basic lemmas}
Let us introduce the necessary notations. Given a lcHtvs $Z$, an extended
real-valued function $h\in \overline{\mathbb{R}}^{Z}$, and $\varepsilon \in
\mathbb{R}_{+}$, the set of $\varepsilon $-minimizers of $h$ is defined by
\begin{equation*}
\varepsilon -\text{argmin }h:=\left\{
\begin{array}{ll}
\{z\in Z\,:\,h(z)\leq \inf_{Z}h+\varepsilon \}, & \mathrm{if
\;\;\inf\limits_{Z}h\in \mathbb{R}, \\
\emptyset , & \mathrm{if}\;\;\inf\limits_{Z}h\not\in \mathbb{R}
\end{array
\right.
\end{equation*
or, equivalently,
\begin{equation*}
\varepsilon -\text{argmin }h=\{z\in h^{-1}(\mathbb{R})\,:\,h(z)\leq
\inf_{Z}h+\varepsilon \}.
\end{equation*
Note that $\varepsilon -$argmin $h\neq \emptyset $ when $\inf_{Z}h\in
\mathbb{R}$ and $\varepsilon >0.$ Various calculus rules involving
\varepsilon -$argmin have been given in \cite{Volle95}.
The $\varepsilon -$subdifferential of $h$ at a point $a\in Z$ is the set
(see, for instance, \cite{HMSV95})
\begin{eqnarray*}
\partial ^{\varepsilon }h(a) &:=&\left\{
\begin{array}{ll}
\{z^{\ast }\in Z^{\ast }\,:\,h(z)\geq h(a)+\langle z^{\ast },z-a\rangle
-\varepsilon ,\forall z\in Z\}, & \mathrm{if}\;\;h(a)\in \mathbb{R}, \\
\emptyset , & \mathrm{if}\;\;h(a)\not\in \mathbb{R
\end{array
\right. \\
&=&\Big\{z^{\ast }\in (h^{\ast })^{-1}(\mathbb{R})\,:\,h^{\ast }(z^{\ast
})+h(a)\leq \langle z^{\ast },a\rangle +\varepsilon \Big\}.
\end{eqnarray*}
It can be checked that if $h\in \overline{\mathbb{R}}^{X}$ is convex and
h(a)\in \mathbb{R}$, then $\partial ^{\varepsilon }h(a)\neq \emptyset $ for
all $\varepsilon >0$ if and only if $h$ is lower semi-continuous at $a$.
The inverse of the set-valued mapping $\partial ^{\varepsilon
}h:Z\rightrightarrows Z^{\ast }$ is denoted by $M^{\varepsilon }h:Z^{\ast
}\rightrightarrows Z.$ For each $(\varepsilon ,z^{\ast })\in \mathbb{R
_{+}\times Z^{\ast }$, we have
\textbf{\ }
\begin{equation*}
\Big(\partial ^{\varepsilon }h\Big)^{-1}(z^{\ast })=\Big(M^{\varepsilon }
\Big)(z^{\ast })=\varepsilon -\text{argmin }(h-z^{\ast }).
\end{equation*
Denoting by $\partial ^{\varepsilon }h^{\ast }(z^{\ast })$ the $\varepsilon
-subdifferential of $h^{\ast }$ at $z^{\ast }\in Z^{\ast }$, namely,
\begin{equation*}
\partial ^{\varepsilon }h^{\ast }(z^{\ast })=\Big\{z\in (h^{\ast \ast
})^{-1}(\mathbb{R})\,:\,h^{\ast \ast }(z)+h^{\ast }(z^{\ast })\leq \langle
z^{\ast },z\rangle +\varepsilon \Big\},
\end{equation*
where $h^{\ast \ast }(z):=\sup\limits_{z^{\ast }\in Z^{\ast }}\{\langle
z^{\ast },z\rangle -h^{\ast }(z^{\ast })\}$ is the biconjugate of $h$, we
have
\begin{equation*}
(M^{\varepsilon }h)(z^{\ast })\subset (\partial ^{\varepsilon }h^{\ast }\big
(z^{\ast }),\ \forall (\varepsilon ,z^{\ast })\in \mathbb{R}_{+}\times
Z^{\ast },
\end{equation*
with equality if and only if $h=h^{\ast \ast }.$
For each $\varepsilon \in \mathbb{R}_{+}$, we consider the set-valued
mapping $S^{\varepsilon }:X^{\ast }\rightrightarrows X$ as follows:
\begin{equation*}
\begin{array}{ll}
S^{\varepsilon }(x^{\ast }) & :=\left\{ x\in p^{-1}(\mathbb{R
)\,:\,p(x)-\langle x^{\ast },x\rangle \leq -q(x^{\ast })+\varepsilon
\right\}
\end{array}
\label{J-ep}
\end{equation*}
If $q(x^{\ast })=-\infty $, then $S^{\varepsilon }(x^{\ast })=p^{-1}(\mathbb
R}).$ If $q(x^{\ast })=+\infty ,$ then $S^{\varepsilon }(x^{\ast
})=\emptyset .$
Since $p^{\ast }\leq q$, it is clear that
\begin{equation}
S^{\varepsilon }(x^{\ast })\subset (M^{\varepsilon }p)(x^{\ast }),\ \forall
\varepsilon \geq 0,\ \forall x^{\ast }\in X^{\ast }. \label{31}
\end{equation}
\begin{lemma}
\label{lem31} Assume that $\operatorname{dom} p\neq \emptyset $. Then, for each
x^{\ast }\in X^{\ast } $, the next statements are equivalent:
$\mathrm{(i)}$ \textit{Robust duality holds at }$x^{\ast }$\thinspace $,$
i.e., $p^{\ast }(x^{\ast })=q(x^{\ast })$,
$\mathrm{(ii)} $ $\left( M^{\varepsilon }p\right) (x^{\ast })=S^{\varepsilon
}(x^{\ast }),\quad \forall \varepsilon \geq 0$,
$\mathrm{(iii)}$ $\exists \bar{\varepsilon}>0:\left( M^{\varepsilon
}p\right) (x^{\ast })=S^{\varepsilon }(x^{\ast }),\quad \forall \varepsilon
\in ]0,\bar{\varepsilon}[$.
\end{lemma}
\noindent \textit{Proof.} $[\mathrm{(i)}\Rightarrow \mathrm{(ii)}]$ By
definitio
\begin{eqnarray*}
\left( M^{\varepsilon }p\right) (x^{\ast }) &=&\varepsilon -\operatorname{argmin
(p-x^{\ast }) \\
&=&\left\{ x\in p^{-1}(\mathbb{R})\,:\,p(x)-\langle x^{\ast },x\rangle \leq
-p^{\ast }(x^{\ast })+\varepsilon \right\} .
\end{eqnarray*
By $\mathrm{(i)}$ we thus have\ $\left( M^{\varepsilon }p\right) (x^{\ast
})=S^{\varepsilon }(x^{\ast }).$
$[\mathrm{(ii)}\Rightarrow \mathrm{(iii)}]$ It is obviously true.
$[\mathrm{(iii)}\Rightarrow \mathrm{(i)}]$ Since $p^{\ast }(x^{\ast })\leq
q(x^{\ast })$, $\mathrm{(i)}$ holds if $p^{\ast }(x^{\ast })=+\infty .$
Moreover, since $\operatorname{dom}p\neq \emptyset $, one has $p^{\ast }(x^{\ast
})\neq -\infty .$ Let now $p^{\ast }(x^{\ast })\in \mathbb{R}$. In order to
get a contradiction, assume that $p^{\ast }(x^{\ast })\not=q(x^{\ast })$.
Then $p^{\ast }(x^{\ast })<q(x^{\ast })$ and there exists $\varepsilon \in
]0,\bar{\varepsilon}[$ such that $p^{\ast }(x^{\ast })+\varepsilon
<q(x^{\ast })$. Since $\inf_{x\in X}\left\{ p(x)-\left\langle x^{\ast
},x\right\rangle \right\} =-p^{\ast }(x^{\ast })\in \mathbb{R}$ and
\varepsilon >0,$ we have $\varepsilon -\operatorname{argmin}(p-x^{\ast })\neq
\emptyset .$ Let us pick $x\in (M^{\varepsilon }p)(x^{\ast })=\varepsilon
\operatorname{argmin}(p-x^{\ast }).$ By $\mathrm{(iii)}$, we have $x\in
S^{\varepsilon }(x^{\ast })$ and
\begin{equation*}
-p^{\ast }(x^{\ast })\leq p(x)-\langle x^{\ast },x\rangle \leq -q(x^{\ast
})+\varepsilon ,
\end{equation*
which contradicts $p^{\ast }(x^{\ast })+\varepsilon <q(x^{\ast })$.\qed
For each $\varepsilon \in \mathbb{R}_{+}$, let us introduce now the the
following set-valued mapping $J^{\varepsilon }:U\rightrightarrows X$:
\begin{equation} \label{J-ep}
J^{\varepsilon }(u):=\left\{ x\in p^{-1}(\mathbb{R})\,:\,p(x)\leq
F_{u}(x,0_{u})+\varepsilon \right\} ,
\end{equation
with the aim of making explicit the set $S^{\varepsilon }(x^{\ast })$. To
this purpose, given $\varepsilon _{1},\varepsilon _{2}\in \mathbb{R}_{+}$,
u\in U$, and $y_{u}^{\ast }\in Y_{u}^{\ast }$, let us introduce the
set-valued mapping $A_{(u,y_{u}^{\ast })}^{(\varepsilon _{1},\varepsilon
_{2})}:X^{\ast }\rightrightarrows X$ such that
\begin{equation*}
A_{(u,y_{u}^{\ast })}^{(\varepsilon _{1},\varepsilon _{2})}(x^{\ast }):
\Big\{x\in J^{\varepsilon _{1}}(u)\,:\,(x,0_{u})\in (M^{\varepsilon
_{2}}F_{u})(x^{\ast },y_{u}^{\ast })\Big\}.
\end{equation*}
\begin{lemma}
\label{lem32} For each $x^{\ast }\in X^{\ast }$, $\varepsilon
_{1},\varepsilon _{2}\in \mathbb{R}_{+}$, $u\in U$, and $y_{u}^{\ast }\in
Y_{u}^{\ast }$, one has
\begin{equation*}
A_{(u,y_{u}^{\ast })}^{(\varepsilon _{1},\varepsilon _{2})}(x^{\ast })\
\subset \ S^{\varepsilon _{1}+\varepsilon _{2}}(x^{\ast }).
\end{equation*}
\end{lemma}
\noindent \textit{Proof.} Let $x\in J^{\varepsilon _{1}}(u)$ be such that
(x,0_{u})\in (M^{\varepsilon _{2}}F_{u})(x^{\ast },y_{u}^{\ast }).$ Then we
have\textbf{\ }$F_{u}^{\ast }(x^{\ast },y_{u}^{\ast })\in \mathbb{R}$ an
\textbf{\ }$F_{u}(x,0_{u})\in \mathbb{R}$. Moreove
\begin{equation*}
F_{u}(x,0_{u})+\varepsilon _{1}\geq p(x)\geq F_{u}(x,0_{u})\in \mathbb{R
\text{,}
\end{equation*
implying $p(x)\in \mathbb{R}$ and, by (\ref{30}),
\begin{eqnarray*}
p(x)-\langle x^{\ast },x\rangle &\leq &F_{u}(x,0_{u})-\langle x^{\ast
},x\rangle +\varepsilon _{1}\leq -F_{u}^{\ast }(x^{\ast },y_{u}^{\ast
})+\varepsilon _{1}+\varepsilon _{2} \\
&\leq &-q(x^{\ast })+\varepsilon _{1}+\varepsilon _{2},
\end{eqnarray*
that means $x\in S^{\varepsilon _{1}+\varepsilon _{2}}(x^{\ast })$ . \qed
\begin{lemma}
\label{lem33} Assume that
\begin{equation}
\operatorname{dom}F_{u}\neq \emptyset ,\text{ }\forall u\in U. \label{32}
\end{equation
Then, for each $x^{\ast }\in X^{\ast },\varepsilon \in \mathbb{R}_{+},\eta
>0 $, one has
\begin{equation*}
{S^{\varepsilon }}({x^{\ast }})\ \ \subset \ \ \bigcup\limits_{\QATOP{u\in
U\hfill }{y_{u}^{\ast }\in Y_{u}^{\ast }\hfill }}\bigcup\limits_{\QATOP
\scriptstyle{\varepsilon _{1}}+{\varepsilon _{2}}=\varepsilon +\eta \hfill }
\scriptstyle{\varepsilon _{1}}\geq 0,\;{\varepsilon _{2}}\geq 0\hfill
}A_{(u,y_{u}^{\ast })}^{(\varepsilon _{1},\varepsilon _{2})}(x^{\ast }).
\end{equation*}
\end{lemma}
\noindent \textit{Proof.} Let $x\in p^{-1}(\mathbb{R})$ be such that $x\in
S^{\varepsilon }(x^{\ast })$, i.e.,
\begin{equation*}
p(x)-\langle x^{\ast },x\rangle \leq -q(x^{\ast })+\varepsilon .
\end{equation*
We then have, for any $\eta >0$,
\begin{equation*}
q(x^{\ast })<\langle x^{\ast },x\rangle -p(x)+\varepsilon +\eta
\end{equation*
and, by definition of $q$ and $p,$ there exist $u\in U$, $y_{u}^{\ast }\in
Y_{u}^{\ast }$ such that
\begin{equation}
F_{u}^{\ast }(x^{\ast },y_{u}^{\ast })\leq \langle x^{\ast },x\rangle
-p(x)+\varepsilon +\eta \leq \langle x^{\ast },x\rangle
-F_{u}(x,0_{u})+\varepsilon +\eta . \label{33}
\end{equation
Since $p(x)\in \mathbb{R}$, $F_{u}^{\ast }(x^{\ast },y_{u}^{\ast })\neq
+\infty $. In fact, by \eqref{32}, $F_{u}^{\ast }(x^{\ast },y_{u}^{\ast
})\in \mathbb{R}$. Similarly, $F_{u}(x,0_{u})\in \mathbb{R}$. Setting
\begin{equation*}
\alpha _{1}:=p(x)-F_{u}(x,0_{u}),\ \alpha _{2}:=F_{u}^{\ast }(x^{\ast
},y_{u}^{\ast })+F_{u}(x,0_{u})-\langle x^{\ast },x\rangle ,
\end{equation*
we get $\alpha _{1}\in \mathbb{R}_{+}$, $\alpha _{2}\in \mathbb{R}.$
Actually {$\alpha _{2}\geq 0$ since, by definition of conjugate, }
\begin{equation*}
F_{u}^{\ast }(x^{\ast },y_{u}^{\ast })=\sup_{z\in X,y_{u}\in Y_{u}}\left\{
\langle x^{\ast },z\rangle +\langle y_{u}^{\ast },y_{u}\rangle
-F_{u}(z,y_{u})\right\} ,
\end{equation*
i.e., if $z=x$ and $y_{u}=0_{u},
\begin{equation*}
F_{u}^{\ast }(x^{\ast },y_{u}^{\ast })\geq \langle x^{\ast },x\rangle
-F_{u}(x,0_{u}),
\end{equation*
so that
\begin{equation*}
F_{u}^{\ast }(x^{\ast },y_{u}^{\ast })+F_{u}(x,0_{u})-\langle x^{\ast
},x\rangle \geq 0.
\end{equation*
Then, by \eqref{33}, $0\leq \alpha _{1}+\alpha _{2}\leq \varepsilon +\eta $.
Consequently, there exist $\varepsilon _{1},\varepsilon _{2}\in \mathbb{R
_{+}$ such that $\alpha _{1}\leq \varepsilon _{1}$, $\alpha _{2}\leq
\varepsilon _{2}$, $\varepsilon _{1}+\varepsilon _{2}=\varepsilon +\eta $.
Now $\alpha _{1}\leq \varepsilon _{1}$ means that $x\in J^{\varepsilon
_{1}}(u)$ and $\alpha _{2}\leq \varepsilon _{2}$ means that $(x,0_{u})\in
(M^{\varepsilon _{2}}F_{u})(x^{\ast },y_{u}^{\ast })$, and we have $x\in
A_{(u,y_{u}^{\ast })}^{(\varepsilon _{1},\varepsilon _{2})}(x^{\ast })$.
\qed
\medskip
For each $x^{\ast }\in X^{\ast }$, $\varepsilon \in \mathbb{R}_{+}$, let us
define
\begin{eqnarray*}
\mathcal{A}^{\varepsilon }(x^{\ast }):= &&\bigcap\limits_{\eta
>0}\bigcup\limits_{\QATOP{u\in U\hfill }{y_{u}^{\ast }\in Y_{u}^{\ast
}\hfill }}\bigcup\limits_{\QATOP{\scriptstyle{\varepsilon _{1}}+{\varepsilon
_{2}}=\varepsilon +\eta \hfill }{\scriptstyle{\varepsilon _{1}}\geq 0,\;
\varepsilon _{2}}\geq 0\hfill }}A_{(u,y_{u}^{\ast })}^{(\varepsilon
_{1},\varepsilon _{2})}(x^{\ast }) \\
&=&\bigcap_{\eta >0}\bigcup\limits_{\QATOP{u\in U\hfill }{y_{u}^{\ast }\in
Y_{u}^{\ast }\hfill }}{{\bigcup\limits_{\QATOP{\scriptstyle{\varepsilon _{1}
+{\varepsilon _{2}}=\varepsilon +\eta \hfill }{\scriptstyle{\varepsilon _{1}
\geq 0,\;{\varepsilon _{2}}\geq 0\hfill }}}}\Big\{x\in J^{\varepsilon
_{1}}(u)\,:\,(x,0_{u})\in (M^{\varepsilon _{2}}F_{u})(x^{\ast },y_{u}^{\ast
})\Big\}
\end{eqnarray*}
\subsection{Robust duality}
We now can state the main result on characterizations of the robust
conjugate duality.
\begin{theorem}[Robust duality]
\label{thm31} Assume that $\operatorname{dom} p\neq \emptyset $. Then for each
x^{\ast }\in X^{\ast } $, the next statements are equivalent:
$\mathrm{(i)}$ $\inf {\mathrm{(RP)}_{{x^{\ast }}}}=\sup {(\mathrm{ODP})_{
x^{\ast }}}}$,
$\mathrm{( ii)} $ $\left( M^{\varepsilon }p\right) (x^{\ast })=\mathcal{A
^{\varepsilon }(x^{\ast }),\quad \forall \varepsilon \geq 0$,
$\mathrm{( iii)} $ $\exists \bar{\varepsilon}>0: \ \left( M^{\varepsilon
}p\right) (x^{\ast })=\mathcal{A}^{\varepsilon }(x^{\ast }),\quad \forall
\varepsilon \in ]0,\bar{\varepsilon}[$.
\end{theorem}
\noindent \textit{Proof.} We firstly claim that if $\operatorname{dom} p\neq
\emptyset $ then for each $x^* \in X^\ast$, $\varepsilon \in \mathbb{R}_+$,
it holds:
\begin{equation} \label{eqlem34}
S^\varepsilon (x^*) \ = \ \mathcal{A}^\varepsilon (x^*).
\end{equation}
In deed, as $\operatorname{dom} p\neq \emptyset $, \eqref{32} holds. It then
follows from Lemma \ref{lem33}, $S^\varepsilon (x^*) \ \subset \ \mathcal{A
^\varepsilon (x^*)$. On the other hand, for each $\eta > 0$, one has, by
Lemma \ref{lem32},
\begin{equation*}
\bigcup\limits_{\QATOP{u\in U\hfill }{y_{u}^{\ast }\in Y_{u}^{\ast }\hfill
} \bigcup\limits_{\QATOP{\scriptstyle{\varepsilon _{1}}+{\varepsilon _{2}
=\varepsilon +\eta \hfill }{\scriptstyle{\varepsilon _{1}}\geq 0,\;
\varepsilon _{2}}\geq 0\hfill }} A^{(\varepsilon_1, \varepsilon_2)}_{(u,
y_u^*)}(x^*) \ \subset \ S^{\varepsilon + \eta} (x^*).
\end{equation*}
Taking the intersection over all $\eta > 0$ we get
\begin{equation*}
\mathcal{A}^\varepsilon (x^*) \subset \bigcap\limits_{\eta> 0}
S^{\varepsilon + \eta} (x^*) = S^\varepsilon (x^*),
\end{equation*}
and \eqref{eqlem34} follows. Taking into account the fact that (i) means
p^{\ast }(x^{\ast })=q(x^{\ast })$, the conclusions now follows from
\eqref{eqlem34} and Lemma \ref{lem31}. \qed
\medskip
For the deterministic optimization problem with linear perturbations (i.e.,
non-uncertain case where $U$ is a singleton), the next result is a direct
consequence of Theorem \ref{thm31}.
\begin{corollary}[Robust duality for Case 1]
\label{corol31} Let $F\colon X\times Y\rightarrow \mathbb{R}_{\infty }$ be
such that $\operatorname{dom}F(\cdot ,0_{Y})\neq \emptyset $. Then, for each
x^{\ast }\in X^{\ast },$ the fundamental duality formula (\ref{11}) holds,
i.e.,
\begin{equation*}
\mathop {\inf }\limits_{x\in X}\left\{ {F(x,{0_{Y}})-\left\langle {{x^{\ast
},x}\right\rangle }\right\} =\mathop {\sup }\limits_{y\in {Y^{\ast }
}-F^{\ast }(x^{\ast },y^{\ast }),
\end{equation*
if and only any of the (equivalent) conditions (ii) or (iii) in Theorem \re
{thm31}\ holds, where
\begin{equation}
\mathcal{A}^{\varepsilon }(x^{\ast })=\bigcap\limits_{\eta
>0}\bigcup\limits_{y^{\ast }\in Y^{\ast }}\Big\{x\in X\,:\,\left(
x,0_{Y}\right) \in \left( M^{\varepsilon +\eta }F\right) (x^{\ast },y^{\ast
})\Big\}. \label{A-ep1}
\end{equation}
\end{corollary}
\noindent \textit{Proof.} Let $F_{u}=F:X\times Y\rightarrow \mathbb{R
_{\infty },\;p=F(\cdot ,0_{Y})$. In this case, one has,
\begin{equation*}
J^{\varepsilon }(u)=\left\{ x\in X\,:\,F(x,0_{Y})\in \mathbb{R}\right\} ,\
\forall \varepsilon \geq 0,
\end{equation*
and $\mathcal{A}^{\varepsilon }(x^{\ast })$ will take the form \eqref{A-ep1
. The conclusion follows from Theorem \ref{thm31}. \qed
For uncertain optimization problem without linear perturbations, the
following result is a consequence of Theorem \ref{thm31}.
\begin{corollary}[Robust duality for Case 2]
\label{corol32} Let $(f_{u})_{u\in U}\subset \mathbb{R}_{\infty }^{X}$ be a
family of extended real-valued functions, $p=\sup_{u\in U}f_{u}$ be such
that $\operatorname{dom}p\neq \emptyset $. Then, for each $x^{\ast }\in X^{\ast
}, $ the $\inf -\sup $ duality in robust optimization (\ref{13}) holds,
i.e.,
\begin{equation*}
{\left( {\mathop {\sup }\limits_{u\in U}{f_{u}}}\right) ^{\ast }}({x^{\ast }
)=\mathop {\inf }\limits_{u\in U}f_{u}^{\ast }({x^{\ast }}),
\end{equation*
if and only any of the (equivalent) conditions (ii) or (iii) in Theorem \re
{thm31}\ holds, where
\begin{equation}
{{\mathcal{A}^{\varepsilon }}({x^{\ast }})=\bigcap\limits_{\eta >0}
\;\bigcup\limits_{u\in U}{\bigcup\limits_{\QATOP{\scriptstyle{\varepsilon
_{1}}+{\varepsilon _{2}}=\varepsilon +\eta \hfill }{\scriptstyle{\varepsilon
_{1}}\geq 0,\;{\varepsilon _{2}}\geq 0\hfill }}}}}\left\{ {{{{{J^{
\varepsilon _{1}}}}(u)\cap ({M^{{\varepsilon _{2}}}}{f_{u}})({x^{\ast }})}}}
\right\} , \label{A-ep-xstar}
\end{equation
wit
\begin{equation*}
{{{{{J^{{\varepsilon _{1}}}}(u)=\{x\in p}}}}^{-1}(\mathbb{R}):\ f_{u}(x)\geq
p(x)-\varepsilon _{1}\}.
\end{equation*}
\end{corollary}
\noindent \textit{Proof.} Let $F_{u}(x,y_{u})=f_{u}(x),$ for all $u\in U$
and let $p=\mathop{\sup }\limits_{u\in U}{f_{u}}$. Then, by \eqref{J-ep},
\begin{equation}
{J^{\varepsilon }}(u)=\left\{ x\in p^{-1}(\mathbb{R})\,:\,f_{u}(x)\geq
p(x)-\varepsilon \right\} ,\ \forall \varepsilon \geq 0. \label{34}
\end{equation
Moreover, recalling (\ref{12}),\ for each $u\in U$ such that $\operatorname{dom
f_{u}\neq \emptyset $, $(x^{\ast },y_{u}^{\ast })\in X^{\ast }\times
Y_{u}^{\ast }$, and $\varepsilon \geq 0$,
\begin{equation}
{\left( {{M^{\varepsilon }}{F_{u}}}\right) \left( {{x^{\ast }},y_{u}^{\ast }
\right) =\left\{
\begin{array}{ll}
{\left( {{M^{\varepsilon }}{f_{u}}}\right) \left( {x^{\ast }}\right) ,} &
\mathrm{if}\ \ y_{u}^{\ast }=0_{u}^{\ast },} \\
\emptyset , & \text{else}
\end{array
\;}\right. } \label{35}
\end{equation
Finally, for each $(x^{\ast },\varepsilon )\in X^{\ast }\times \mathbb{R
_{+} $, $\mathcal{A}^{\varepsilon }({x^{\ast }})$ takes the form as in
\eqref{A-ep-xstar}. The conclusion now follows from Theorem \ref{thm31}.
\qed
\section{Strong Robust Duality}
We retain the notations in Section 3 and consider the robust problem
\mathrm{(RP)}_{x^{\ast }}$ and its robust dual problem $\mathrm{(ODP)
_{x^{\ast }}$ given in \eqref{RPx-star} and \eqref{ODPx-star}, respectively.
Let $p$ and $q$ be the functions defined by \eqref{pq} and recall the
relations in \eqref{30}, that is,
\begin{equation*}
\left\{
\begin{array}{l}
{{p^{\ast }}({x^{\ast }})=-\inf {{\mathrm{(RP)}}_{{x^{\ast }}}},\quad q(
x^{\ast }})=-\sup \mathrm{(}{{\mathrm{ODP)}}_{{x^{\ast }}}\medskip }} \\
{{q^{\ast }}=\mathop {\sup }\limits_{\left( {u,y_{u}^{\ast }}\right) \in
\Delta }{{\left( {F_{u}^{\ast }\left( \cdot {,y_{u}^{\ast }}\right) }\right)
}^{\ast }}=\mathop {\sup }\limits_{u\in U}F_{u}^{\ast \ast }\left( \cdot {,
0_{u}}}\right) \leq p}
\end{array
}\right.
\end{equation*}
In this section we will establish characterizations of \textit{strong robust
duality at $x^{\ast }$}. Recall that the strong robust duality holds at $x^*$
means that $\inf {\mathrm{(RP)}_{{x^{\ast }}}}=\max {(\mathrm{ODP})_{
x^{\ast }}}}$, which is the same as:
\begin{equation*}
\exists (u,y_{u}^{\ast })\in \Delta :p^{\ast }(x^{\ast })=F_{u}^{\ast
}(x^{\ast },y_{u}^{\ast }).
\end{equation*}
For this, we need a technical lemma, but firstly, given $x^* \in X^\ast$, $\
u\in U$, $y_{u}^{\ast }\in Y_{u}^{\ast }$, and $\varepsilon \geq 0$, let us
introduce the set
\begin{equation*}
B_{(u,y_{u}^{\ast })}^{\varepsilon }({x^{\ast }})= \bigcup\limits_{\QATOP
\scriptstyle{\varepsilon _{1}}+{\varepsilon _{2}} =\varepsilon \hfill }
\scriptstyle{\varepsilon _{1}}\geqslant 0,{\varepsilon _{2}}\geqslant
0\hfill }} A^{(\varepsilon_1, \varepsilon_2)}_{(u, y_u^*)}(x^*)
=\bigcup\limits_{\QATOP{\scriptstyle{\varepsilon _{1}}+{\varepsilon _{2}
=\varepsilon \hfill }{\scriptstyle{\varepsilon _{1}}\geqslant 0,{\varepsilon
_{2}}\geqslant 0\hfill }}\Big\{x\in J^{\varepsilon _1}(u)\, :\, (x,0_{u})
\in ( M^{ \varepsilon _{2}} F_{u}) (x^\ast ,y_u^\ast )\Big\}.
\end{equation*}
\begin{lemma}
\label{lem41} Assume that $\operatorname{dom}F_{u}\neq \emptyset $, for all $u\in
U,$ holds and let $x^{\ast }\in X^{\ast }$ be such that
\begin{equation*}
q(x^{\ast })=\min\limits_{\begin{subarray}{l} u \in U \\ y_u^* \in Y_u^*
\end{subarray}}F_{u}^{\ast }(x^{\ast },y_{u}^{\ast }).
\end{equation*
Then there exist $u\in U$, $y_{u}^{\ast }\in Y_{u}^{\ast }$ such that
\begin{equation*}
S^{\varepsilon }(x^{\ast })=B_{(u,y_{u}^{\ast })}^{\varepsilon }(x^{\ast
}),\ \forall \varepsilon \geq 0.
\end{equation*}
\end{lemma}
\noindent \textit{Proof.} By Lemma \ref{lem32} we have $B_{(u,y_{u}^{\ast
})}^{\varepsilon }(x^{\ast })\subset S^{\varepsilon }(x^{\ast })$.
Conversely, let $x\in S^{\varepsilon }(x^{\ast })$. By the exactness of $q$
at $x^{\ast }$, there exist $u\in U$ and $y_{u}^{\ast }\in Y_{u}^{\ast }$
such that
\begin{equation*}
p(x)-\langle x^{\ast },x\rangle \leq -F_{u}^{\ast }(x^{\ast },y_{u}^{\ast
})+\varepsilon .
\end{equation*
Since $p(x)\in \mathbb{R}$ and $\operatorname{dom}F_{u}\neq \emptyset $, for all
u\in U,$ we have $F_{u}^{\ast }(x^{\ast },y_{u}^{\ast })\in \mathbb{R}$,
F_{u}(x,0_{u})\in \mathbb{R}$,
\begin{equation*}
\Big(p(x)-F_{u}(x,0_{u})\Big)+\Big(F_{u}(x,0_{u})+F_{u}^{\ast }(x^{\ast
},y_{u}^{\ast })-\langle x^{\ast },x\rangle \Big)\leq \varepsilon .
\end{equation*
Consequently, there exist $\varepsilon _{1}\geq 0,\varepsilon _{2}\geq 0$
such that $\varepsilon _{1}+\varepsilon _{2}=\varepsilon ,$
\begin{equation*}
p(x)-F_{u}(x,0_{u})\leq \varepsilon _{1}\ \text{and\ }F_{u}(x,0_{u})+F_{u}^
\ast }(x^{\ast },y_{u}^{\ast })-\langle x^{\ast },x\rangle \leq \varepsilon
_{2},
\end{equation*
that is, $x\in J^{\varepsilon _{1}}(u)\ \text{and }(x,0_{u})\in
(M^{\varepsilon _{2}}F_{u})(x^{\ast },y_{u}^{\ast }) $. Thus, $x\in
A_{(u,y_{u}^{\ast })}^{(\varepsilon _{1},\varepsilon _{2})}(x^{\ast
})\subset B_{(u,y_{u}^{\ast })}^{\varepsilon }(x^{\ast })$, since
\varepsilon _{1}+\varepsilon _{2}=\varepsilon .$ \qed
\begin{theorem}[Strong robust duality]
\label{thm41} Assume that $\operatorname{dom}p\neq \emptyset $ and let $x^{\ast
}\in X^{\ast }$. The next statements are equivalent:
$\mathrm{(i)} $ \ $\inf {\mathrm{(RP)}_{{x^{\ast }}}}=\max {(\mathrm{ODP})_{
x^{\ast }}}}$,
$\mathrm{(ii)}$ $\exists u\in U,\ \exists y_{u}^{\ast }\in Y_{u}^{\ast }:$
\left( M^{\varepsilon }p\right) (x^{\ast })=B_{(u,y_{u}^{\ast
})}^{\varepsilon }(x^{\ast }),\forall \varepsilon \geq 0$,
$\mathrm{(iii)} $ $\exists \bar{\varepsilon}>0,\ \exists u\in U,\exists
y_{u}^{\ast }\in Y_{u}^{\ast }:$ $\left( M^{\varepsilon }p\right) (x^{\ast
})=B_{(u,y_{u}^{\ast })}^{\varepsilon }(x^{\ast }),\forall \varepsilon \in
]0,\bar{\varepsilon}[$.
\end{theorem}
\textit{Proof.} Observe firstly that (i) means that
\begin{equation*}
p^{\ast }(x^{\ast })=q(x^{\ast })=\min\limits_{\begin{subarray}{l} u \in U
\\ y_u^* \in Y_u^* \end{subarray}}F_{u}^{\ast }(x^{\ast },y_{u}^{\ast }).
\end{equation*
As $\operatorname{dom}p\neq \emptyset $, \eqref{32} holds, and then by Lemmas \re
{lem31} and \ref{lem41}, $\mathrm{(i)}$ implies the remaining conditions,
which are equivalent to each other, and also that $\mathrm{(iii)}$ implies
p^{\ast }(x^{\ast })=q(x^{\ast })$.
We now prove that $\mathrm{(iii)}$ implies $q(x^{\ast })=F_{u}^{\ast
}(x^{\ast },y_{u}^{\ast })$. Assume by contradiction that there exists
\varepsilon >0$ such that $q(x^{\ast })+\varepsilon <F_{u}^{\ast }(x^{\ast
},y_{u}^{\ast })$, and without loss of generality one can take $\varepsilon
\in \left] 0,\bar{\varepsilon}\right[ ,$ where $\bar{\varepsilon}>0$
appeared in (iii). Then, by $\mathrm{(iii)}$, $\left( M^{\varepsilon
}p\right) (x^{\ast })=B_{(u,y_{u}^{\ast })}^{\varepsilon }(x^{\ast }).$
Pick $x\in \left( M^{\varepsilon }p\right) (x^{\ast })=B_{(u,y_{u}^{\ast
})}^{\varepsilon }(x^{\ast }).$ Then, there are $\varepsilon _{1}\geq
0,\varepsilon _{2}\geq 0$, $\varepsilon _{1}+\varepsilon _{2}=\varepsilon $
and $x\in J^{\varepsilon _{1}}(u)$ and $(x,0_{u})\in (M^{\varepsilon
_{2}}F_{u})(x^{\ast },y_{u}^{\ast })$. In other words,
\begin{eqnarray}
&&p(x)\leq F_{u}(x,0_{u})+\varepsilon _{1}, \label{eq41a} \\
&&F^{\ast }((x^{\ast },y_{u}^{\ast })+F_{u}(x,0_{u})\leq \langle x^{\ast
},x\rangle +\varepsilon _{2}. \label{eq41b}
\end{eqnarray
It now follows from \eqref{eq41a}-\eqref{eq41b} that
\begin{eqnarray*}
p^{\ast }\left( x^{\ast }\right) &\geq &\langle x^{\ast },x\rangle -p\left(
x\right) \geq\langle x^{\ast },x\rangle -F_{u}(x,0_{u})-\varepsilon _{1} \\
&\geq &\langle x^{\ast },x\rangle +F_{u}^{\ast }(x^{\ast },y_{u}^{\ast
})-\langle x^{\ast },x\rangle -\varepsilon _{2}-\varepsilon _{1}=F_{u}^{\ast
}(x^{\ast },y_{u}^{\ast })-\varepsilon >q(x^{\ast }),
\end{eqnarray*
which contradicts the fact that $p^{\ast }(x^{\ast })=q(x^{\ast })$. \qed
In deterministic optimization with linear perturbations we get the next
consequence from Theorem \ref{thm41}.
\begin{corollary}[Strong robust duality for Case 1]
\label{corol41} Let $F:X\times Y\rightarrow \mathbb{R}_{\infty }$,
p=F(\cdot ,0_{Y})$, and assume that $\operatorname{dom}p\neq \emptyset $. Then,
for each $x^{\ast }\in X^{\ast }$, {\ the strong duality for $\mathrm{(P)
_{x^{\ast }}$ in Case 1 holds at $x^{\ast }$, i.e.,
\begin{equation*}
\mathop {\inf }\limits_{x\in X}\left\{ {F\left( {x,{0_{Y}}}\right)
-\left\langle {{x^{\ast }},x}\right\rangle }\right\} =\mathop {\max
\limits_{{y^{\ast }}\in {Y^{\ast }}}-{F^{\ast }}\left( {{x^{\ast }},{y^{\ast
}}}\right) ,
\end{equation*
}if and only if one of the (equivalent) conditions (ii) or (iii) in Theorem
\ref{thm41} holds with $B_{(u,y_{u}^{\ast })}^{\varepsilon }(x^{\ast })$
being replaced by
\begin{equation}
B_{y^{\ast }}^{\varepsilon }(x^{\ast }):=\big\{x\in X\,:\,(x,0_{Y})\in
(M^{\varepsilon }F)(x^{\ast },y^{\ast })\big\}. \label{By-star}
\end{equation}
\end{corollary}
\textit{Proof.} It is worth observing that we are in the non-uncertainty
case (i.e., $U$ is a singleton), and the set $B_{(u,y_{u}^{\ast
})}^{\varepsilon }(x^{\ast })$ writes as in \eqref{By-star} for each
(x^{\ast },y^{\ast })\in X^{\ast }\times Y^{\ast }$, $\varepsilon \geq 0$.
The conclusion follows from Theorem \ref{thm41}. \qed
In the non-perturbation case, Theorem \ref{thm41} gives rise to
\begin{corollary}[Strong robust duality for Case 2]
\label{corol42} Let $(f_{u})_{u\in U}\subset \mathbb{R}_{\infty }^{X}$,
x^{\ast }\in X^{\ast }$, and $p=\sup\limits_{u\in U}f_{u}$ such that
\operatorname{dom}p\neq \emptyset $. Then, the robust duality formula
\begin{equation*}
\Big(\sup\limits_{u\in U}f_{u}\Big)^{\ast }(x^{\ast })=\min\limits_{u\in
U}f_{u}^{\ast }(x^{\ast })
\end{equation*
holds if and only if one of the (equivalent) conditions (ii) or (iii) in
Theorem \ref{thm41} holds with $B_{(u,y_{u}^{\ast })}^{\varepsilon }(x^{\ast
})$ being replaced by
\begin{equation}
B_{u}^{\varepsilon }(x^{\ast }):=\bigcup\limits_{\QATOP{\scriptstyle
\varepsilon _{1}}+{\varepsilon _{2}}=\varepsilon \hfill }{\scriptstyle
\varepsilon _{1}}\geqslant 0,{\varepsilon _{2}}\geqslant 0\hfill }}\Big
J^{\varepsilon _{1}}(u)\cap (M^{\varepsilon _{2}}f_{u})(x^{\ast })\Big).
\label{41}
\end{equation}
\end{corollary}
\textit{Proof.} Let $F_{u}(x,y_{u})=f_{u}(x)$, $p=\mathop{\sup
\limits_{u\in U}{f_{u}}$, and, from \eqref{34} and \eqref{35} (see the proof
of Corollary \ref{corol32}),
\begin{equation*}
B_{(u,y_{u}^{\ast })}^{\varepsilon }({x^{\ast }})=\left\{
\begin{array}{ll}
\bigcup\limits_{\QATOP{\scriptstyle{\varepsilon _{1}}+{\varepsilon _{2}
=\varepsilon \hfill }{\scriptstyle{\varepsilon _{1}}\geqslant 0,{\varepsilon
_{2}}\geqslant 0\hfill }}\Big( J^{\varepsilon _{1}}(u)\cap (M^{\varepsilon
_{2}}f_{u})(x^{\ast })\Big) , & \mathrm{if}\ \ y_{u}^{\ast }=0_{u}^{\ast },
\\
\ \ \ \ \ \emptyset , & \text{else}
\end{array
\right.
\end{equation*}
which in our situation, collapses to the set $B_{u}^{\varepsilon }(x^{\ast
}) $ defined by \eqref{41}. The conclusion now follows from Theorem \re
{thm41}. \qed
\section{Reverse strong and min-max robust duality}
Given $F_{u}:X\times Y_{u}\rightarrow (\mathbb{R}_{\infty })^{X}$ for each
u\in U$, $p=\sup\limits_{u\in U}F_{u}(\cdot ,0_{u})$, and $x^{\ast }\in
X^{\ast }$, we assume in this section that the problem $\mathrm{(RP)
_{x^{\ast }}$ is finite-valued and admits an optimal solution or, in other
words, that $\mathrm{argmin}(p-x^{\ast })=(M^{0}p)(x^{\ast })\neq \emptyset
. For convenience, we set
\begin{equation*}
(Mp)(x^{\ast }):=(M^{0}p)(x^{\ast }),\ \ S(x^{\ast }):=S^{0}(x^{\ast }),\
\mathrm{and}
\end{equation*
\begin{equation}
\mathcal{A}(x^{\ast }):=\mathcal{A}^{0}(x^{\ast })=\bigcap_{\eta
>0}\bigcup\limits_{\QATOP{u\in U\hfill }{y_{u}^{\ast }\in Y_{u}^{\ast
}\hfill }}{{\bigcup\limits_{\QATOP{\scriptstyle{\varepsilon _{1}}+
\varepsilon _{2}}=\eta \hfill }{\scriptstyle{\varepsilon _{1}}\geq 0,\;
\varepsilon _{2}}\geq 0\hfill }}}}\Big\{x\in J^{\varepsilon
_{1}}(u)\,:\,(x,0_{u})\in (M^{\varepsilon _{2}}F_{u})(x^{\ast },y_{u}^{\ast
})\Big\}. \label{51}
\end{equation}
\begin{theorem}[Reverse strong robust duality]
\label{thm51} Let $x^{\ast }\in X^{\ast }$ be such that $(Mp)(x^{\ast })\neq
\emptyset $ and let $\mathcal{A}(x^*)$ be as in \eqref{51}. The next
statements are equivalent:
$\mathrm{(i)}$ $\min {\mathrm{(RP)}_{{x^{\ast }}}}=\sup {(\mathrm{ODP})_{
x^{\ast }}}}$,
$\mathrm{(ii)} $ $(Mp)(x^{\ast }) = \mathcal{A} (x^{\ast })$.
\end{theorem}
\noindent \textit{Proof.} \ Since $(Mp)(x^{\ast })\neq \emptyset $,
\operatorname{dom}p\neq \emptyset $. It follows from Theorem \ref{thm31} that
\left[ {\mathrm{(i)}\Longrightarrow \mathrm{(ii)}}\right] $. For the
converse, let us pick $x\in (Mp)(x^{\ast })$. Then by (ii), for each $\eta
>0 $ there exist $u\in U$, $y_{u}^{\ast }\in Y_{u}^{\ast }$, $\varepsilon
_{1}\geq 0,\varepsilon _{2}\geq 0$ such that $\varepsilon _{1}+\varepsilon
_{2}=\eta $, $x\in J^{\varepsilon _{1}}(u)$, $(x,0_{u})\in (M^{\varepsilon
_{2}}F_{u})(x^{\ast },y_{u}^{\ast })$ and we have
\begin{eqnarray*}
q(x^{\ast }) &\leq &F_{u}^{\ast }(x^{\ast },y_{u}^{\ast })\leq \langle
x^{\ast },x\rangle -F_{u}(x,0_{u})+\varepsilon _{2} \\
&\leq &\langle x^{\ast },x\rangle -p(x)+\varepsilon _{1}+\varepsilon
_{2}\leq p^{\ast }(x^{\ast })+\eta .
\end{eqnarray*
Since $\eta >0$ is arbitrary we get $q(x^{\ast })\leq p^{\ast }(x^{\ast })$,
which, together with the weak duality (see (\ref{WD})), yields $q(x^{\ast })
= \langle x^{\ast },x\rangle -p(x) = p^{\ast }(x^{\ast }) $, i.e., (i) holds
and we are done. \qed
In the deterministic case we obtain from Theorem \ref{thm51}:
\begin{corollary}[Reverse strong robust duality for Case 1]
\label{corol51} Let $F:X\times Y\rightarrow \mathbb{R}_{\infty }$, $x^{\ast
}\in X^{\ast }$, $p=F(\cdot ,0_{Y})$, and
\begin{equation*}
\mathcal{A}(x^{\ast })=\bigcap_{\eta >0}\bigcup\limits_{y^{\ast }\in Y^{\ast
}}\Big\{x\in X\,:\,(x,0_{Y})\in (M^{\eta }F)(x^{\ast },y^{\ast })\Big\}.
\end{equation*
Assume that $(Mp)(x^{\ast })\neq \emptyset $. Then the next statements are
equivalent:
$\mathrm{(i)}$ $\mathop {\min }\limits_{x\in X}\left\{ {F\left( {x,{0_{Y}}
\right) -\left\langle {{x^{\ast }},x}\right\rangle }\right\} =\mathop {\sup
\limits_{{y^{\ast }}\in {Y^{\ast }}}-{F^{\ast }}\left( {{x^{\ast }},{y^{\ast
}}}\right)$,
$\mathrm{(ii)} $ $(Mp)(x^{\ast }) = \mathcal{A} (x^{\ast })$.
\end{corollary}
\begin{corollary}[Reverse strong robust duality for Case 2]
\label{corol53} Let $(f_{u})_{u\in U}\subset \mathbb{R}_{\infty }^{X}$,
p=\sup\limits_{u\in U}f_{u}$, $x^{\ast }\in X^{\ast }$, and
\begin{equation*}
\mathcal{A}(x^{\ast }):=\bigcap_{\eta >0}\bigcup\limits_{u\in
U}\bigcup\limits_{\QATOP{\scriptstyle{\varepsilon _{1}}+{\varepsilon _{2}
=\eta \hfill }{\scriptstyle{\varepsilon _{1}}\geq 0,\;{\varepsilon _{2}}\geq
0\hfill }}\Big(J^{\varepsilon _{1}}(u)\cap (M^{\varepsilon
_{2}}f_{u})(x^{\ast })\Big),
\end{equation*
where
\begin{equation*}
J^{\varepsilon _{1}}(u)=\Big\{x\in p^{-1}(\mathbb{R})\,:\,f_{u}(x)\geq
p(x)-\varepsilon _{1}\Big\}.
\end{equation*
Assume that $(Mp)(x^{\ast })\neq \emptyset $. Then the next statements are
equivalent:
$\mathrm{(i)}$ $\Big(\sup\limits_{u\in U}f_{u}\Big)^{\ast }(x^{\ast
})=\inf\limits_{u\in U}f_{u}^{\ast }(x^{\ast }),$\ with attainment at the
first member,
$\mathrm{(ii)} $ $(Mp)(x^{\ast }) = \mathcal{A} (x^{\ast })$.
\end{corollary}
Now, for each $u\in U$, $y_{u}^{\ast }\in Y_{u}^{\ast }$, $x^{\ast }\in
X^{\ast },$ we se
\begin{equation*}
J(u):=J^{0}(u)=\Big\{x\in p^{-1}(\mathbb{R})\,:\,F_{u}(x,0_{u})=p(x)\Big\},
\end{equation*
\begin{equation*}
(MF_{u})(x^{\ast },y_{u}^{\ast }):=(M^{0}F_{u})(x^{\ast },y_{u}^{\ast })
\mathrm{argmin}\Big(F_{u}-\langle x^{\ast },\cdot \rangle -\langle
y_{u}^{\ast },\cdot \rangle \Big),
\end{equation*
and
\begin{equation}
B_{(u,y_{u}^{\ast })}(x^{\ast }):=B_{(u,y_{u}^{\ast })}^{0}(x^{\ast })=\Big\
x\in J(u)\,:\,(x,0_{u})\in (MF_{u})(x^{\ast },y_{u}^{\ast })\Big\}.
\label{52}
\end{equation}
\begin{theorem}[Min-max robust duality]
\label{thm52} Let $x^{\ast }\in X^{\ast }$ be such that $(Mp)(x^{\ast })\neq
\emptyset $. The next statements are equivalent:
$\mathrm{(i)}$ {\ $\min \mathrm{(RP)}_{x^{\ast }}=\max \mathrm{(ODP)
_{x^{\ast }}$,}
$\mathrm{(ii)}$ $\exists u\in U,$ $\exists y_{u}^{\ast }\in Y_{u}^{\ast }:$
\ (Mp)(x^{\ast })=B_{(u,y_{u}^{\ast })}(x^{\ast })$,
where $B_{(u,y_{u}^{\ast })}(x^{\ast })$ is the set defined in \eqref{52}.
\end{theorem}
\noindent \textit{Proof.} By Theorem \ref{thm41} we know that $[\mathrm{(i)
\Longrightarrow \mathrm{(ii)}]$. We now prove that $[\mathrm{(ii)
\Longrightarrow \mathrm{(i)}]$. Pick $x\in (Mp)(x^{\ast })$ which is
non-empty by assumption. Then by $\mathrm{(ii)}$, $x\in B_{(u,y_{u}^{\ast
})}(x^{\ast })$, which yields $x\in J(u)$ and $(x,0_{u})\in (MF_{u})(x^{\ast
},y_{u}^{\ast })$. Hence,
\begin{eqnarray*}
q(x^{\ast }) &\leq &F_{u}^{\ast }(x^{\ast },y_{u}^{\ast })\leq \langle
x^{\ast },x\rangle -F_{u}(x,0_{u}) \\
&\leq &\langle x^{\ast },x\rangle -p(x)\leq p^{\ast }(x^{\ast })\leq
q(x^{\ast }),
\end{eqnarray*
which means that $q(x^{\ast })=F_{u}^{\ast }(x^{\ast },y_{u}^{\ast
})=\langle x^{\ast },x\rangle -p(x)=p^{\ast }(x^{\ast })$ and $\mathrm{(i)}$
follows. \qed
\begin{corollary}[Min-max robust duality for Case 1]
\label{corol52} Let $F:X\times Y\rightarrow \mathbb{R}_{\infty }$, $x^{\ast
}\in X^{\ast }$, $p=F(\cdot ,0_{Y})$, and for each $y^{\ast }\in Y^{\ast }$,
\begin{equation*}
B_{y^{\ast }}(x^{\ast }):=\Big\{x\in X\,:\,(x,0_{Y})\in (MF)(x^{\ast
},y^{\ast })\Big\}.
\end{equation*
Assume that $(Mp)(x^{\ast })\neq \emptyset $. The next statements are
equivalent:
$\mathrm{(i)} $ {$\mathop {\min }\limits_{x\in X}\left\{ {F\left( {x,{0_{Y}}
\right) -\left\langle {{x^{\ast }},x}\right\rangle }\right\} =
\mathop {\max
}\limits_{{y^{\ast }}\in {Y^{\ast }}}-{F^{\ast }}\left( {{x^{\ast }},
y^{\ast }}}\right)$, }
$\mathrm{(ii)} $ $\exists y^{\ast }\in Y^{\ast }$:\, $\ (Mp)(x^{\ast }) =
B_{y^{\ast }}(x^{\ast })$.
\end{corollary}
\begin{corollary}[Min-max robust duality for Case 2]
\label{corol54} Let $(f_{u})_{u\in U}\subset \mathbb{R}_{\infty }^{X}$,
p=\sup\limits_{u\in U}f_{u}$, $x^{\ast }\in X^{\ast }$, and and for each
u\in U$,
\begin{equation*}
B_{u}(x^{\ast }):=J(u)\cap (Mf_{u})(x^{\ast }),
\end{equation*
where $J(u)=\{x\in p^{-1}(\mathbb{R})\,:\,f_{u}(x)=p(x)\}$. Assume that
(Mp)(x^{\ast })\neq \emptyset $. Then the next statements are equivalent:
$\mathrm{(i)} $ $\Big(\sup\limits_{u \in U} f_u\Big)^\ast (x^*)
=\min\limits_{u \in U} f_u^\ast (x^*)$, with attainment at the first member,
$\mathrm{(ii)} $ $\exists u \in U$:\, $\ (Mp)(x^{\ast }) = B_u(x^{\ast })$.
\end{corollary}
\section{Stable robust duality}
Let us first recall some notations. Given ${F_{u}}:X\times {Y_{u}
\rightarrow \mathbb{R}_{\infty }$, $u\in U,\ p=\mathop {\sup }\limits_{u\in
U}{F_{u}}(\cdot ,{0_{u}})$ and $q=\mathop {\inf }\limits_{\QATOP
\scriptstyle
u\in U\hfill }{\scriptstyle y_{u}^{\ast }\in Y_{u}^{\ast }\hfill
}F_{u}^{\ast }(\cdot ,y_{u}^{\ast })$. Remember that $p^{\ast }(x^{\ast
})\leq q(x^{\ast })$ for each $x^{\ast }\in X^{\ast }$. \textit{Stable
robust duality} means that $\inf {\mathrm{(RP)}_{{x^{\ast }}}}=\sup {
\mathrm{ODP})_{{x^{\ast }}}}$ for all $x^{\ast }\in X^{\ast }$, or
equivalently,
\begin{equation*}
p^{\ast }(x^{\ast })=q(x^{\ast }),\ \forall x^{\ast }\in X^{\ast }.
\end{equation*
Theorem \ref{thm31} says that, if $\operatorname{dom}p\neq \emptyset ,$ then
stable robust duality holds if and only if for each $\varepsilon \geq 0$ the
set-valued mappings $M^{\varepsilon }p,\ \mathcal{A}^\varepsilon:X^{\ast
}\rightrightarrows X$ coincide, where, for each $x^{\ast }\in X^{\ast }$,
\begin{eqnarray*}
(M^{\varepsilon }p)(x^{\ast }):= &&\varepsilon -\mathrm{argmin}(p-x^{\ast }),
\\
\mathcal{A}^{\varepsilon }(x^{\ast }):= &&\bigcap_{\eta >0}\bigcup\limits_
\QATOP{u\in U\hfill }{y_{u}^{\ast }\in Y_{u}^{\ast }\hfill }}{
\bigcup\limits_{\QATOP{\scriptstyle{\varepsilon _{1}}+{\varepsilon _{2}
=\varepsilon +\eta \hfill }{\scriptstyle{\varepsilon _{1}}\geq 0,\;
\varepsilon _{2}}\geq 0\hfill }}}}\Big\{x\in J^{\varepsilon
_{1}}(u)\,:\,(x,0_{u})\in (M^{\varepsilon _{2}}F_{u})(x^{\ast },y_{u}^{\ast
})\Big\}.
\end{eqnarray*
Consequently, stable robust duality holds if and only if for each
\varepsilon \geq 0$, the inverse set-valued mappings
\begin{equation*}
(M^{\varepsilon }p)^{-1},\ (\mathcal{A}^{\varepsilon
})^{-1}:X\rightrightarrows X^{\ast },
\end{equation*
coincide. Recall that $(M^{\varepsilon }p)^{-1}$ is nothing but the
\varepsilon $-subdifferential of $p$ at $x$.
Let us now make explicit $(\mathcal{A}^{\varepsilon })^{-1}$. To this end we
need to introduce for each $\varepsilon \geq 0$ the ($\varepsilon $-active
indexes) set-valued mapping $I^{\varepsilon }:X\rightrightarrows U$ with
\begin{equation} \label{I-ep-x}
\ I^{\varepsilon }(x)=\left\{
\begin{array}{ll}
\Big\{u\in U\,:\,F_{u}(x,0_{u})\geq p(x)-\varepsilon \Big\}, & \mathrm{if
\quad p(x)\in \mathbb{R}, \\
\ \ \ \emptyset , & \mathrm{if}\quad p(x)\not\in \mathbb{R}
\end{array
\right.
\end{equation
We observe that $I^{\varepsilon }$ is nothing but the inverse of the
set-valued mapping $J^{\varepsilon }:U\rightrightarrows X$ defined in
\eqref{J-ep}.
\begin{lemma}
\label{lem61} For each $(\varepsilon ,x)\in \mathbb{R}_{+}\times X$ one has
\begin{equation*}
{(\mathcal{A}^{\varepsilon })^{-1}}(x)=\bigcap\limits_{\eta >0}
\bigcup\limits_{\QATOP{\scriptstyle{\varepsilon _{1}}+{\varepsilon _{2}
=\varepsilon +\eta \hfill }{\scriptstyle{\varepsilon _{1}}\geqslant 0,
\varepsilon _{1}}\geqslant 0\hfill }}{\bigcup\limits_{u\in {I^{\varepsilon
_{1}}}(x)}\mathrm{proj}_{X^{\ast }}^{u}{\partial ^{{\varepsilon _{2}}}}{F_{u
}(x,{0_{u}})}},
\end{equation*
where ${{\mathrm{proj}_{X^{\ast }}^{u}:}}X^{\ast }\times Y_{u}^{\ast
}\longrightarrow X^{\ast }$ is the projection mapping $\mathrm{proj
_{X^{\ast }}^{u}(x^{\ast },y_{u}^{\ast })=x^{\ast }.$
\end{lemma}
\noindent \textit{Proof.} Let $(\varepsilon ,x,x^{\ast })\in \mathbb{R
_{+}\times X\times X^{\ast }$. One has
\begin{eqnarray*}
x^{\ast }\in (\mathcal{A}^{\varepsilon })^{-1}(x) &\Leftrightarrow & x\in
\mathcal{A}^{\varepsilon }(x^{\ast }) \\
&\Leftrightarrow &\left\{
\begin{array}{l}
\forall \eta >0,\exists u\in U,\exists y_{u}^{\ast }\in Y_{u}^{\ast
},\exists (\varepsilon _{1},\varepsilon _{2})\in \mathbb{R}_{+}^{2}\text{
\mathrm{such\ that} \\
\varepsilon _{1}+\varepsilon _{2}=\varepsilon +\eta ,\text{ }x\in
J^{\varepsilon _{1}}(u)\ \mathrm{and}\ (x,0_{u})\in (M^{\varepsilon
_{2}}F_{u})(x^{\ast },y_{u}^{\ast }
\end{array
\right. \\
&\Leftrightarrow & \left\{
\begin{array}{l}
\forall \eta >0,\exists u\in U,\exists y_{u}^{\ast }\in Y_{u}^{\ast
},\exists (\varepsilon _{1},\varepsilon _{2})\in \mathbb{R}_{+}^{2}\text{
\mathrm{such\ that\ } \\
\varepsilon _{1}+\varepsilon _{2}=\varepsilon +\eta ,\text{ }u\in
I^{\varepsilon _{1}}(x),\ \mathrm{and}\ (x^{\ast },y_{u}^{\ast })\in \Big
\partial ^{\varepsilon _{2}}F_{u}\Big)(x,0_{u}
\end{array
\right. \\
&\Leftrightarrow &\left\{
\begin{array}{l}
\forall \eta >0,\exists u\in U,\exists (\varepsilon _{1},\varepsilon
_{2})\in \mathbb{R}_{+}^{2}\text{ }\mathrm{\ such\ that\ } \\
\mathrm{\ }\varepsilon _{1}+\varepsilon _{2}=\varepsilon +\eta ,\text{ }u\in
I^{\varepsilon _{1}}(x),\ \mathrm{and}\ x^{\ast }\in \mathrm{proj}_{X^{\ast
}}^{u}\Big(\partial ^{\varepsilon _{2}}F_{u}\Big)(x,0_{u}
\end{array
\right. \\
& \Leftrightarrow & x^{\ast} \in \bigcap\limits_{\eta>0} \bigcup\limits_{\QATOP{\scriptstyle{\varepsilon_{1}}+{\varepsilon _{2}}=\varepsilon +\eta
\hfill }{\scriptstyle{\varepsilon {1}}\geqslant 0, \varepsilon_{1} \geqslant 0\hfill}} \bigcup\limits_{u\in {I^{\varepsilon_{1}}}(x)} \mathrm{proj}_{X^{\ast}}^{u}(\partial^{\varepsilon_{2}} F_{u})(x,0_{u}).
\end{eqnarray*}
\qed
\medskip
Now, for each $(\varepsilon, x) \in \mathbb{R}_+ \times X$, let us set
\begin{equation} \label{61}
C^\varepsilon (x) := \bigcap\limits_{\eta >0} \bigcup\limits_{\QATOP
\scriptstyle{\varepsilon _{1}}+{\varepsilon _{2}}=\varepsilon +\eta \hfill }
\scriptstyle{\varepsilon _{1}}\geqslant 0,{\varepsilon _{1}}\geqslant
0\hfill }}\bigcup\limits_{u\in I^{\varepsilon _{1}}(x)}\mathrm{proj}^u
_{X^{\ast }}(\partial ^{\varepsilon _{2}}F_{u})(x,{0_{u}}).
\end{equation}
Applying Theorem \ref{thm31} and Lemma \ref{lem61} we obtain:
\begin{theorem}[Stable robust duality]
\label{thm61} Assume that $\operatorname{dom}p\neq \emptyset $. The next
statements are equivalent:
$\mathrm{(i)} $ $\inf {\mathrm{(RP)}_{{x^{\ast }}}}=\sup {(\mathrm{ODP})_{
x^{\ast }}}}$ for all $x^* \in X^\ast$,
$\mathrm{(ii)} $ $\partial^\varepsilon p(x) = C^\varepsilon (x), \ \forall
(\varepsilon, x) \in \mathbb{R}_+ \times X$,
$\mathrm{(iii)} $ $\exists \bar\varepsilon > 0$: \ $\partial^\varepsilon
p(x) = C^\varepsilon (x), \ \forall (\varepsilon, x) \in ]0,
\bar\varepsilon[ \times X$.
\end{theorem}
\begin{corollary}[Stable robust duality for Case 1]
\label{corol61} Let $F:X\times Y\rightarrow \mathbb{R}_{\infty }$ be such
that $\operatorname{dom}F(\cdot ,0_{Y})\neq \emptyset $. Let ${{\mathrm{proj
_{X^{\ast }}:}}X^{\ast }\times Y^{\ast }\longrightarrow X^{\ast }$ be the
projection mapping $\mathrm{proj}_{X^{\ast }}(x^{\ast },y^{\ast })=x^{\ast
}. $ Then, the next statements are equivalent:
$\mathrm{(i)} $ $\inf\limits_{x \in X} \Big\{ F(x, 0_Y) - \langle x^*,
x\rangle \Big\} = \sup\limits_{y^* \in Y^\ast} - F^\ast (x^*, y^*), \
\forall x^* \in X^\ast$,
$\mathrm{(ii)}$ $(\partial ^{\varepsilon }p)(x)=\bigcap_{\eta >0}\mathrm{pro
}_{X^{\ast }}(\partial ^{{\varepsilon }}{F})(x,{0_{Y}}),\ \forall
(\varepsilon ,x)\in \mathbb{R}_{+}\times X$,
$\mathrm{(iii)}$ $\exists \bar{\varepsilon}>0$: \ $(\partial ^{\varepsilon
}p)(x)=\bigcap_{\eta >0}\mathrm{proj}_{X^{\ast }}(\partial ^{{\varepsilon }}
F})(x,{0_{Y}}),\ \forall (\varepsilon ,x)\in ]0,\bar{\varepsilon}[\times X$.
\end{corollary}
\textit{Proof.} Let $U=\{u_{0}\}$ and $F=F_{u_{0}}:X\times Y\rightarrow
\mathbb{R}_{\infty }$, $Y=Y_{u_{0}}$, $p=F(\cdot ,0_{Y})$. Then for each
(\varepsilon ,x)\in \mathbb{R}_{+}\times X$,
\begin{equation*}
I^{\varepsilon }(x)=\left\{
\begin{array}{ll}
\{u_{0}\},\ \ & \mathrm{if}\quad p(x)\in \mathbb{R}, \\
\emptyset , & \mathrm{if}\quad p(x)\not\in \mathbb{R}
\end{array
\right.
\end{equation*
an
\begin{equation}
\bigcup\limits_{\QATOP{\scriptstyle{\varepsilon _{1}}+{\varepsilon _{2}
=\varepsilon \hfill }{\scriptstyle{\varepsilon _{1}}\geqslant 0,{\varepsilon
_{1}}\geqslant 0\hfill }}\bigcup\limits_{u\in I^{\varepsilon _{1}}(x)
\mathrm{proj}_{X^{\ast }}^{u}(\partial ^{\varepsilon _{2}}F_{u})(x,{0_{u}})
\mathrm{proj}_{X^{\ast }}\Big(\partial ^{\varepsilon }F\Big)(x,0_{Y}).
\label{62}
\end{equation
The conclusion now follows from \eqref{61}-\eqref{62} and Theorem \ref{thm61
. \qed
\begin{remark}
Condition $\mathrm{(ii)}$ in Corollary \ref{corol61} was quoted in \cite
Theorem 4.3]{Grad16} for all $(\varepsilon ,x)\in ]0,+\infty \lbrack \times
\mathbb{R},$ which is equivalent.
\end{remark}
\begin{corollary}[Stable robust duality for Case 2]
\label{corol62} Let $(f_{u})_{u\in U}\subset \mathbb{R}_{\infty }^{X},$
p=\sup\limits_{u\in U}f_{u}$ and assume that $\operatorname{dom}p\neq \emptyset
. The next statements are equivalent:
$\mathrm{(i)} $ $\Big(\sup\limits_{u \in U} f_u\Big)^\ast (x^*)
=\inf\limits_{u \in U} f_u^\ast (x^*)$, \ $\forall x^* \in X^\ast$,
$\mathrm{(ii)} $ $(\partial^\varepsilon p)(x) = C^\varepsilon (x), \ \forall
(\varepsilon, x) \in \mathbb{R}_+ \times X$,
$\mathrm{(iii)}$ $\exists \bar{\varepsilon}>0$: \ $(\partial ^{\varepsilon
}p)(x)=C^{\varepsilon }(x),\ \forall (\varepsilon ,x)\in ]0,\bar{\varepsilon
[\times X$, \newline
where $C^{\varepsilon }(x)$ is the set
\begin{equation}
C^{\varepsilon }(x)=\bigcap\limits_{\eta >0}\bigcup\limits_{\QATOP
\scriptstyle{\varepsilon _{1}}+{\varepsilon _{2}}=\varepsilon +\eta \hfill }
\scriptstyle{\varepsilon _{1}}\geqslant 0,{\varepsilon _{1}}\geqslant
0\hfill }}\bigcup\limits_{u\in I^{\varepsilon _{1}}(x)}(\partial
^{\varepsilon _{2}}f_{u})(x),\ \ \forall (\varepsilon ,x)\in \mathbb{R
_{+}\times X. \label{63}
\end{equation}
\end{corollary}
\textit{Proof.} Let $F_{u}:X\times Y_{u}\rightarrow \mathbb{R}_{\infty }\
be such that $F_{u}(x,y_{u})=f_{u}(x)$ for all $u\in U$. Then for any
(\varepsilon ,x)\in \mathbb{R}_{+}\times X$,
\begin{equation*}
I^{\varepsilon }(x)=\left\{
\begin{array}{ll}
\Big\{u\in U\,:\,f_{u}(x)\geq p(x)-\varepsilon \Big\}, & \mathrm{if}\quad
p(x)\in \mathbb{R}, \\
\ \ \emptyset , & \mathrm{if}\quad p(x)\not\in \mathbb{R}
\end{array
\right.
\end{equation*
\begin{equation*}
(\partial ^{\varepsilon }F_{u})(x,0_{u})=(\partial ^{\varepsilon
}f_{u})(x)\times \{0_{u}^{\ast }\},\ \ \ \forall (u,\varepsilon ,x)\in
U\times \mathbb{R}_{+}\times X,
\end{equation*
and $C^{\varepsilon }(x)$ reads as in \eqref{63}. The conclusion now follows
from Theorem \ref{thm61}. \qed
\section{Stable strong robust duality}
We retain all the notations used in the Sections 3-6. Given $(\varepsilon,
u) \in \mathbb{R}_+ \times U$ and $y_u^* \in Y_U^\ast$ we have introduced in
Section 4 the set-valued mapping $B_{(u,y_{u}^{\ast })}^{\varepsilon } :
X^\ast \rightrightarrows X$ defined by
\begin{equation*}
B_{(u,y_{u}^{\ast })}^{\varepsilon }({x^{\ast }}) =\bigcup\limits_{\QATOP
\scriptstyle{\varepsilon _{1}}+{\varepsilon _{2}}=\varepsilon \hfill }
\scriptstyle{\varepsilon _{1}}\geqslant 0,{\varepsilon _{2}}\geqslant
0\hfill }}\Big\{x\in J^{\varepsilon _1}(u)\, :\, (x,0_{u}) \in ( M^{
\varepsilon _{2}} F_{u}) (x^\ast ,y_u^*)\Big\}.
\end{equation*
Let us now define $B^\varepsilon : X^\ast \rightrightarrows X$ by setting
\begin{equation*}
B^\varepsilon (x^*) := \bigcup\limits_{\QATOP{u \in U}{y_u^* \in Y_u^\ast}}
B_{(u,y_{u}^{\ast })}^{\varepsilon }(x^*) , \ \ \forall x^* \in X^\ast.
\end{equation*}
\begin{lemma}
\label{lem71} For each $(\varepsilon, x) \in \mathbb{R}_+ \times X$ we have
\begin{equation*}
(B^\varepsilon)^{-1} (x) = \bigcup\limits_{\QATOP{\scriptstyle{\varepsilon
_{1}}+{\varepsilon _{2}}=\varepsilon \hfill }{\scriptstyle{\varepsilon _{1}
\geqslant 0,{\varepsilon _{2}}\geqslant 0\hfill }} \bigcup\limits_{u\in
I^{\varepsilon _{1}}(x)}\mathrm{proj}^u_{X^{\ast }}(\partial ^{\varepsilon
_{2}}F_{u})(x,{0_{u}}).
\end{equation*}
\end{lemma}
\noindent \textit{Proof.} $x^* \in (B^\varepsilon)^{-1} (x)$ means that
there exist $u \in U$, $y_u^* \in Y_u^\ast$ $\varepsilon_1 \geq 0$,
\varepsilon_2 \geq 0$, such that $\varepsilon_1 + \varepsilon_2 =
\varepsilon $, $x \in J^{\varepsilon_1} (u) $, and $(x, 0_u) \in
(M^{\varepsilon_2}F_u)(x^*, y_u^*)$, or, equivalently, $u \in
I^{\varepsilon_1} (x)$, and $(x^*, y_u^*) \in
(\partial^{\varepsilon_2}F_u)(x, 0_u)$. In other words, there exist $u \in U
, $y_u^* \in Y_u^\ast$ such that $x \in B^\varepsilon_{(u, y_u^*)} (x^*)$,
that is $x \in B^\varepsilon (x^*)$. \qed
For each $\varepsilon \geq 0$ let us introduce the set-valued mapping
D^{\varepsilon }:=(B^{\varepsilon })^{-1}$. Now Lemma \ref{lem71} writes:
\begin{equation}
D^{\varepsilon }(x)\ =\ \bigcup\limits_{\QATOP{\scriptstyle{\varepsilon _{1}
+{\varepsilon _{2}}=\varepsilon \hfill }{\scriptstyle{\varepsilon _{1}
\geqslant 0,{\varepsilon _{2}}\geqslant 0\hfill }}\bigcup\limits_{u\in
I^{\varepsilon _{1}}(x)}\mathrm{proj}_{X^{\ast }}^{u}(\partial ^{\varepsilon
_{2}}F_{u})(x,{0_{u}}),\ \forall (\varepsilon ,x)\in \mathbb{R}_{+}\times X.
\label{71}
\end{equation
Note that
\begin{equation}
C^{\varepsilon }(x)=\bigcap\limits_{\eta >0}D^{\varepsilon +\eta }(x),\
\forall (\varepsilon ,x)\in \mathbb{R}_{+}\times X, \label{eq72}
\end{equation
and that $D^{\varepsilon }(x)=\emptyset $ whenever $p(x)\not\in \mathbb{R}$.
We now provide a characterization of stable strong robust duality in terms
of $\varepsilon$-subdifferential formulas.
\begin{theorem}[Stable strong robust duality]
\label{thm71} Assume that $\operatorname{dom} p \ne \emptyset$, and let
D^\varepsilon$ as in \eqref{71}. The next statements are equivalent:
$\mathrm{(i)} $ $\inf {\mathrm{(RP)}_{{x^{\ast }}}}=\max {(\mathrm{ODP})_{
x^{\ast }}}} = \max\limits_{\QATOP{u \in U\hfill }{y_u^* \in Y_u^\ast\hfill
} - F_u^\ast (x^*, y_u^*),\ \ \forall x^* \in X^\ast$,
{$\mathrm{(ii)} $} $\partial^\varepsilon p(x) = D^\varepsilon (x), \ \forall
(\varepsilon, x) \in \mathbb{R}_+ \times X$.
\end{theorem}
\noindent \textit{Proof.} $[\mathrm{(i)} \Longrightarrow \mathrm{(ii)}]$ Let
$x^* \in \partial^\varepsilon p(x)$. Then $x \in (M^\varepsilon p)(x^*)$.
Since strong robust duality holds at $x^*$, Theorem \ref{thm41} says that
there exist $u \in U$, $y_u^* \in Y_u^\ast$ such that $x \in
B^\varepsilon_{(u, y_u^*)}(x^*) \subset B^\varepsilon(x^*)$, and finally
x^* \in D^\varepsilon (x)$ by Lemma \ref{lem71}. Thus $\partial^\varepsilon
p(x) \subset D^\varepsilon (x)$.
{\ Now, let $x^{\ast }\in D^{\varepsilon }(x)$. By Lemma \ref{lem71} we have
$x\in B^{\varepsilon }(x^{\ast })$ and there exist $u\in U$, $y_{u}^{\ast
}\in Y_{u}^{\ast }$ such that $x\in B_{(u,y_{u}^{\ast })}^{\varepsilon
}(x^{\ast })$. By Lemma \ref{lem32} and the definition of $B_{(u,y_{u}^{\ast
})}^{\varepsilon }(x^{\ast })$ we have $x\in S^{\varepsilon }(x^{\ast })$,
and, by \eqref{31}, $x\in (M^{\varepsilon }p)(x^{\ast })$ which means that
x^{\ast }\in \partial ^{\varepsilon }p(x)$, and hence, $D^{\varepsilon
}(x)\subset \partial ^{\varepsilon }p(x)$. Thus (ii) follows.}
{\ $[\mathrm{(ii)}\Longrightarrow \mathrm{(i)}]$} If $p^{\ast }(x^{\ast
})=+\infty $ then $q(x^{\ast })=+\infty $ and one has $p^{\ast }(x^{\ast
})=F_{u}^{\ast }(x^{\ast },y_{u}^{\ast })=+\infty $ for all $u\in U$,
y_{u}^{\ast }\in Y_{u}^{\ast }$, and (i) holds. Assume that $p^{\ast
}(x^{\ast })\in \mathbb{R}$ and pick $x\in p^{-1}(\mathbb{R})$ which is
non-empty as $\operatorname{dom}p\neq \emptyset $ and $p^{\ast }(x^{\ast })\in
\mathbb{R}$. Let $\varepsilon :=p(x)+p^{\ast }(x^{\ast })-\langle x^{\ast
},x\rangle $. Then $\varepsilon \geq 0$ and we have $x^{\ast }\in \partial
^{\varepsilon }p(x)$. By (ii) $x\in D^{\varepsilon }(x)$ and hence, there
exist $\varepsilon _{1}\geq 0,\varepsilon _{2}\geq 0$, $u\in U$, and
y_{u}^{\ast }\in Y_{u}^{\ast }$ such that $\varepsilon _{1}+\varepsilon
_{2}=\varepsilon $, $u\in I^{\varepsilon _{1}}(x)$, $(x^{\ast },y_{u}^{\ast
})\in (\partial ^{\varepsilon _{2}}F_{u})(x,0_{u})$. We have
\begin{eqnarray*}
q(x^{\ast }) &\leq &F_{u}^{\ast }(x^{\ast },y_{u}^{\ast })\leq \langle
x^{\ast },x\rangle -F_{u}(x,0_{u})+\varepsilon _{2} \\
&\leq &\langle x^{\ast },x\rangle -p(x)+\varepsilon _{1}+\varepsilon
_{2}=p^{\ast }(x^{\ast })\ \ \ \ \ \ \mathrm{(by\ definition\ of\
\varepsilon ) \\
&\leq &q(x^{\ast }),
\end{eqnarray*
and finally, $q(x^{\ast })=F_{u}^{\ast }(x^{\ast },y_{u}^{\ast })=p^{\ast
}(x^{\ast })$, which is (i).\qed
Next, as usual, we give two consequences of Theorem \ref{thm71} for the
non-uncertainty and non-parametric cases.
\begin{corollary}[Stable strong duality for Case 1]
\label{corol71} Let $F:X\times Y\rightarrow \mathbb{R}_{\infty }$,
p=F(\cdot ,0_{Y})$, $\operatorname{dom}p\neq \emptyset $. The next statements are
equivalent:
$\mathrm{(i)} $ $\inf\limits_{x \in X} \Big\{ F(x, 0_Y) - \langle x^*,
x\rangle\Big\} = \max\limits_{ y^* \in Y^ \ast} - F^\ast (x^*, y^*), \
\forall x^* \in X^\ast$,
$\mathrm{(ii)}$\ \ $\partial ^{\varepsilon }p(x)=\mathrm{proj}_{X^{\ast
}}(\partial ^{\varepsilon }F)(x,{0_{y}}),\ \forall (\varepsilon ,x)\in
\mathbb{R}_{+}\times X.$
\end{corollary}
\textit{Proof.} This is the non-uncertainty case (i.e., the uncertainty set
is a singleton) of the general problem $\mathrm{(RP)}_{x^{\ast }}$, with
U=\{u_{0}\}$ and $F_{u_{0}}=F:X\times Y\rightarrow \mathbb{R}_{\infty }$. We
have from \eqref{62},
\begin{equation}
D^{\varepsilon }(x)=\mathrm{proj}_{X^{\ast }}(\partial ^{\varepsilon }F)(x,
0_{Y}}),\ \forall (\varepsilon ,x)\in \mathbb{R}_{+}\times X. \label{72}
\end{equation
The conclusion now follows from Theorem \ref{thm71}. \qed
\begin{corollary}[Stable strong duality for Case 2]
\label{corol72} Let $(f_{u})_{u\in U}\subset \mathbb{R}_{\infty }^{X}$,
p=\sup\limits_{u\in U}f_{u}$, and $\operatorname{dom}p\neq \emptyset $. The next
statements are equivalent:
$\mathrm{(i)} $ $(\sup\limits_{u \in U} f_u)^\ast(x^*) = \min\limits_{ u\in
U} f_u^\ast (x^*), \ \forall x^* \in X^\ast$,
$\mathrm{(ii)}$\ \ $\partial ^{\varepsilon }p(x)=D^{\varepsilon }(x),\
\forall (\varepsilon ,x)\in \mathbb{R}_{+}\times X$, where
\begin{equation}
D^{\varepsilon }(x)=\bigcup\limits_{\QATOP{\scriptstyle{\varepsilon _{1}}+
\varepsilon _{2}}=\varepsilon \hfill }{\scriptstyle{\varepsilon _{1}
\geqslant 0,{\varepsilon _{2}}\geqslant 0\hfill }}\bigcup\limits_{u\in
I^{\varepsilon _{1}}(x)}(\partial ^{\varepsilon _{2}}f_{u})(x),\ \forall
(\varepsilon ,x)\in \mathbb{R}_{+}\times X, \label{73}
\end{equation
and
\begin{equation*}
I^{\varepsilon }(x)=\left\{
\begin{array}{ll}
\Big\{u\in U\,:\,f_{u}(x)\geq p(x)-\varepsilon \Big\} & \mathrm{if}\quad
p(x)\in \mathbb{R}, \\
\ \ \emptyset & \mathrm{if}\quad p(x)\not\in \mathbb{R}\text{.
\end{array
\right.
\end{equation*}
\end{corollary}
\textit{Proof.} In this non-parametric situation, let
F_{u}(x,y_{u})=f_{u}(x)$. It is easy to see that in this case, the set
D^{\varepsilon }(x)$ can be expressed as in\ \eqref{73}, and the conclusion
follows from Theorem \ref{thm71}. \qed
\section{Exact subdifferential formulas: Robust Basic Qualification conditio
}
Given ${F_{u}}:X\times {Y_{u}}\rightarrow \mathbb{R}_{\infty },\ u\in U$, as
usual, we let $p=\mathop {\sup }\limits_{u\in U}{F_{u}}(\cdot ,{0_{u}})$,
q:=\mathop {\inf }\limits_{\left( {u,y_{u}^{\ast }}\right) \in \Delta
}\;F_{u}^{\ast }(\cdot ,y_{u}^{\ast })$. Again, we consider the robust
problem $\mathrm{(RP)}_{x^{\ast }}$ and its robust dual problem $\mathrm
(ODP)}_{x^{\ast }}$ given in \eqref{RPx-star} and \eqref{ODPx-star},
respectively. Note that the reverse strong robust duality holds at $x^{\ast
} $ means that, for some $\bar{x}\in X$, it holds:
\begin{equation}
-p^{\ast }(x^{\ast })=\min {\mathrm{(RP)}_{{x^{\ast }}}}=\sup_{u\in U}F_{u}
\bar{x},0_{u})-\langle x^{\ast },\bar{x}\rangle =p(\bar{x})-\langle x^{\ast
},\bar{x}\rangle =\sup {(\mathrm{ODP})_{{x^{\ast }}}}=-q(x^{\ast }).
\label{strongrobustduality}
\end{equation}
Now, let us set, for each $x\in X$,
\begin{eqnarray}
D(x):= &&D^{0}(x)=\bigcup\limits_{u\in I(x)}\mathrm{proj}^u_{X^{\ast
}}(\partial F_{u})(x,{0_{u}}), \label{81} \\
C(x):= &&C^{0}(x)=\bigcap\limits_{\eta >0}\bigcup\limits_{\QATOP{\scriptstyl
{\varepsilon _{1}}+{\varepsilon _{2}}=\eta \hfill }{\scriptstyle{\varepsilon
_{1}}\geqslant 0,{\varepsilon _{2}}\geqslant 0\hfill }}\bigcup\limits_{u\in
I^{\varepsilon _{1}}(x)}\mathrm{proj}^u_{X^{\ast }}(\partial ^{\varepsilon
_{2}}F_{u})(x,{0_{u}}), \label{82}
\end{eqnarray
where $I^{\varepsilon _{1}}(x)$ is defined as in \eqref{I-ep-x} and
\begin{equation}
I(x):=\left\{
\begin{array}{ll}
\left\{ u\in U:F_{u}(x,0_{u})=p(x)\right\} , & \mathrm{if}\quad p(x)\in
\mathbb{R}, \\
\emptyset , & \mathrm{if}\quad p(x)\not\in \mathbb{R}
\end{array
\right. \label{I-x}
\end{equation}
\begin{lemma}
\label{lem81} For each $x \in X$, it holds
\begin{equation*}
D(x) \subset C(x) \subset \partial p (x).
\end{equation*}
\end{lemma}
\noindent \textit{Proof.} The first inclusion is easy to check. Now let $x^*
\in C(x)$. For each $\eta > 0$ there exist $(\varepsilon_1, \varepsilon_2 )
\in \mathbb{R}_+^2$, $u \in I^{\varepsilon_1}(x)$, and $y_u^* \in Y_u^\ast$
such that $\varepsilon_1 + \varepsilon_2 = \eta$ and $(x^*, y_u^*) \in
(\partial^{\varepsilon_2} F_u)(x, 0_u)$. We then have $F_u^\ast (x^*, y_u^*)
+ F_u(x, 0_u) - \langle x^*, x\rangle \leq \varepsilon_2$, $p(x) \leq F_u(x,
0_u) + \varepsilon_1$ (as $u \in I^{\varepsilon_1}(x)$), and $p^\ast (x^*)
\leq q(x^*) \leq F^\ast_u(x^*, y^*_u)$. Consequently,
\begin{equation*}
p^\ast (x^*) + p(x) - \langle x^*, x\rangle \leq F_u^\ast (x^*, y_u^*) +
F_u(x, 0_u) + \varepsilon_1- \langle x^*, x\rangle \leq\varepsilon_1 +
\varepsilon_2 = \eta .
\end{equation*}
Since $\eta > 0$ is arbitrary we get $p^\ast (x^*) + p(x) -\langle x^*,
x\rangle \leq 0$, which means that $x^* \in \partial p (x)$. The proof is
complete. \qed
\begin{theorem}
\label{thm81} Let $x \in p^{-1}(\mathbb{R})$ and $C(x)$ be as in \eqref{82}.
The next statements are equivalent:
$\mathrm{(i)} $\ \ $\partial p (x) = C(x)$,
$\mathrm{(ii)} $\ Reverse strong robust duality holds at each $x^* \in
\partial p (x) $,
$\mathrm{(iii)} $ Robust duality holds at each $x^* \in \partial p (x)$.
\end{theorem}
\noindent \textit{Proof.} $[ \mathrm{(i)} \Longrightarrow \mathrm{(ii)}]$
Let $x^* \in \partial p (x)$. We have $x^* \in C(x) = (\mathcal{A})^{-1} (x)$
(see Lemma \ref{lem61} with $\varepsilon = 0$). Then $x \in \mathcal{A}
(x^*)= S(x^*)$ (see \eqref{eqlem34} with $\varepsilon = 0$), and therefore,
\begin{equation*}
- p^\ast (x^*) \leq p(x) - \langle x^*, x\rangle \leq - q (x^*) \leq -
p^\ast(x^*),
\end{equation*}
\begin{equation*}
- p^\ast (x^*) = \min\limits_{z \in X} \{ p(z) - \langle x^*, z\rangle \} =
p(x) - \langle x^*, x\rangle = -q(x^*) ,
\end{equation*}
that means that reverse strong robust duality holds at $x^*$ (see
\eqref{strongrobustduality}).
$[ \mathrm{(ii)} \Longrightarrow \mathrm{(iii)}]$ is obvious.
$[ \mathrm{(iii)} \Longrightarrow \mathrm{(i)}]$ By Lemma \ref{lem81} it
suffices to check that the inclusion $`` \subset"$ holds. Let $x^* \in
\partial p (x)$. We have $x \in (Mp)(x^*)$. Since robust duality holds at
x^*$, Theorem \ref{thm31} (with $\varepsilon = 0$) says that $x \in \mathcal
A} (x^*) $. Thus, $x^* \in \mathcal{A} ^{-1} (x)$, and, by Lemma \ref{lem61
, $x^* \in C(x)$. \qed
In the deterministic and the non-parametric cases, we get the next results
from Theorem \ref{thm81}.
\begin{corollary}
\label{corol81} Let $F:X\times Y\rightarrow \mathbb{R}_{\infty }$,
p=F(\cdot ,0_{Y})$, and $x\in p^{-1}(\mathbb{R})$. The next statements are
equivalent:
$\mathrm{(i)} $\ \ \ $\partial p (x) = \bigcap\limits_{\eta > 0 } \mathrm
proj}_{X^\ast} (\partial^\eta F)(x, 0_Y)$,
$\mathrm{(ii)} $\ \ \ $\min\limits_{z \in X} \Big\{ F(z, 0_Y) - \langle x^*,
x \rangle \Big\} = \sup\limits_{y^* \in Y^\ast} - F^\ast (x^*, y^*) , \
\forall x^* \in \partial p (x)$,
$\mathrm{(iii)} $ \ $\inf\limits_{z \in X} \Big\{ F(z, 0_Y) - \langle x^*, x
\rangle \big\} = \sup\limits_{y^* \in Y^\ast} - F^\ast (x^*, y^*) , \
\forall x^* \in \partial p (x)$.
\end{corollary}
\textit{Proof.} Let $F_{u}=F:X\times Y\rightarrow \mathbb{R}_{\infty }$ and
p=F(\cdot ,0_{Y})$. We then have
\begin{equation*}
C(x)=\bigcap\limits_{\eta >0}\mathrm{proj}_{X^{\ast }}(\partial ^{\eta
}F)(x,0_{Y}),\ \forall x\in X,
\end{equation*
(see Corollary \ref{corol61}) and the conclusion follows directly from
Theorem \ref{thm81}. \qed
\begin{corollary}
\label{corol82} Let $(f_{u})_{u\in U}\subset \mathbb{R}_{\infty }^{X}$,
p=\sup\limits_{u\in U}f_{u}$, $x\in p^{-1}(\mathbb{R})$. The next statements
are equivalent:
$\mathrm{(i)} $\ \ $\partial \left( \sup\limits_{u \in U} f_u\right) (x) =
C(x)$,
$\mathrm{(ii)} $\ \ $\max\limits_{z \in X} \Big\{ \langle x^*, z \rangle -
p(z) \Big\} = \inf\limits_{u \in U} f_u^\ast (x^*), \ \forall x^* \in
\partial p (x)$,
$\mathrm{(iii)} $ \ $\left( \sup\limits_{u \in U} f_u\right) ^\ast (x^*) =
\inf\limits_{u \in U } f_u^\ast (x^*), \ \forall x^* \in \partial p (x)$,
where
\begin{equation}
C(x)=\bigcap\limits_{\eta >0}\bigcup\limits_{\QATOP{\scriptstyle{\varepsilon
_{1}}+{\varepsilon _{2}}=\eta \hfill }{\scriptstyle{\varepsilon _{1}
\geqslant 0,{\varepsilon _{2}}\geqslant 0\hfill }}\bigcup\limits_{u\in
I^{\varepsilon _{1}}(x)}\mathrm{proj}^u_{X^{\ast }}(\partial ^{\varepsilon
_{2}}f_{u})(x),\forall x\in X. \label{83}
\end{equation}
\end{corollary}
\textit{Proof.} Let $F_u(x, y_u) = f_u(x)$. Then it is easy to see that in
this case, $C(x) $ can be expressed as in \eqref{83}. The conclusion now
follows from Theorem \ref{thm81}. \qed
\medskip
Let us come back to the general case and consider the most simple
subdifferential formula one can expect for the robust objective function
p=\sup\limits_{u\in U}F_{u}(\cdot ,0_{u})$:
\begin{equation}
\partial p(x)=\displaystyle\bigcup_{u\in I(x)}\mathrm{proj}^u_{X^{\ast
}}\left( \partial F_{u}\right) (x,0_{u}), \label{84}
\end{equation
where the set of active indexes at $x$, $I(x)$, is defined by \eqref{I-x}.
In Case 3 \ we have
\begin{equation*}
p(x)=\left\{
\begin{array}{ll}
f(x), & \mathrm{if}\quad H_{u}(x)\in -S_{u},\forall u\in U, \\
+\infty , & \mathrm{else}
\end{array
\right.
\end{equation*
$I(x)=U$ for each $x\in p^{-1}(\mathbb{R})$, and \eqref{84} writes
\begin{equation*}
\partial p(x)=\bigcup\limits_{\QATOP{u\in U,\ z_{u}^{\ast }\in S_{u}^{+}}
\langle z_{u}^{\ast },H_{u}(x)\rangle =0}}\partial (f+z_{u}^{\ast }\circ
H_{u})(x),
\end{equation*
which has been called \textit{Basic Robust Subdifferential Condition} (BRSC)
in \cite{BJL13} (see \cite[page 307]{HU-Lem1} for the deterministic case).
More generally, let us introduce the following terminology:
\begin{definition}
Given $F_{u}:X\times Y_{u}\rightarrow \mathbb{R}_{\infty }$ for each $u\in U
, and $p=\sup\limits_{u\in U}F_{u}(\cdot ,0_{u})$, we will say that \textbf
Basic Robust Subdifferential Condition holds at a point } $x\in p^{-1}
\mathbb{R})$ if \eqref{84} is satisfied, that is $\partial p(x)=D(x)$.
\end{definition}
Recall that, in Example \ref{Example1}, $p\left( x\right) =\left\langle
c^{\ast },x\right\rangle +\mathrm{i}_{A}\left( x\right) ,$ where $A=p^{-1}
\mathbb{R})$ is the feasible set of the linear system. So, given $x\in A,$
\partial p(x)$ is the sum of $c^{\ast }$ with the normal cone of $A$ at $x,$
i.e., Basic Robust Subdifferential Condition (at $x$) asserts that such a
cone can be expressed in some prescribed way.
\begin{theorem}
\label{thm82} Let $x\in p^{-1}(\mathbb{R})$. The next statements are
equivalent:
$\mathrm{(i)}$\ \ Basic Robust Subdifferential Condition holds at $x$,
$\mathrm{(ii)}$\ Min-max robust duality holds at each $x^{\ast }\in \partial
p(x)$,
$\mathrm{(iii)}$ Strong robust duality holds at each $x^{\ast }\in \partial
p(x)$.
\end{theorem}
\noindent \textit{Proof.} $[\mathrm{(i)} \Longrightarrow \mathrm{(ii)}]$ Let
$x^* \in \partial p (x)$. We have $x^* \in D(x)$ and by \eqref{81} there
exist $u \in I(x)$ (i.e., $p(x) = F_u(x, 0_u)$), $y_u^*\in Y_u^\ast$, such
that $(x^*, y_u^*) \in \partial F_u)(x, 0_u)$. Then,
\begin{eqnarray*}
p^\ast (x^*) &\geq& \langle x^*, x\rangle - p(x) = \langle x^*, x \rangle -
F_u(x, 0_u) = F_u^\ast (x^*, y_u^*) \\
&\geq& q(x^*) \geq p^*(x^*) .
\end{eqnarray*}
It follows that
\begin{equation*}
\max\limits_{z \in X} \{ \langle x^*, z\rangle - p(z) \} = \langle x^*,
x\rangle - p(x) = F_u^* (x^*, y_u^*) = q(x^*),
\end{equation*}
and min-max robust duality holds at $x^*$.
$[\mathrm{(ii)} \Longrightarrow \mathrm{(iii)}]$ is obvious.
$[\mathrm{(iii)} \Longrightarrow \mathrm{(i)}]$ By Lemma \ref{lem81}, it
suffices to check that $\partial p (x) \subset D(x)$. Let $x^* \in \partial
p (x)$. We have $x \in (M p) (x^*)$. Since strong robust duality holds at
x^*$, Theorem \ref{thm41} says that there exist $u \in U$, $y_u^* \in
Y_u^\ast$ such that $x \in B^0_{(u, y_u^*)} (x^*)$, that means (see
\eqref{52})
\begin{equation*}
(x, 0_u) \in (MF_u)((x^*, y_u^*) , \ (x^*, y_u^*) \in \partial F_u)(x, 0_u),
\end{equation*}
and by \eqref{81}, $x^* \in D(x)$. \qed
As usual, Theorem \ref{thm82} gives us corresponding results for the two
extreme cases: non-uncertainty and non-perturbation cases.
\begin{corollary}
\label{corol83} Let $F:X\times Y\rightarrow \mathbb{R}_{\infty }$,
p=F(\cdot ,0_{Y})$, and $x\in p^{-1}(\mathbb{R})$. The next statements are
equivalent:
$\mathrm{(i)} $\ \ $\partial p (x) = \mathrm{proj}_{X^\ast} (\partial F) (x,
0_Y)$,
$\mathrm{(ii)} $\ \ $\max\limits_{z \in X} \Big\{ \langle x^*, z \rangle -
F(z, 0_Y) \Big\} = \min\limits_{y^*\in Y^\ast} F^\ast (x^*, y^*), \ \forall
x^* \in \partial p (x)$,
$\mathrm{(iii)} $ \ $p ^\ast (x^*) = \min\limits_{y^* \in Y^\ast } F^\ast
(x^*, y^*), \ \forall x^* \in \partial p (x)$.
\end{corollary}
\textit{Proof.} In this case we have, by \eqref{72}, $D(x)=\mathrm{proj
_{X^{\ast }}(\partial F)(x,0_{Y}) $ and the conclusion is a direct
consequence of Theorem \ref{thm82}. \qed
\begin{corollary}
\label{corol84} Let $(f_{u})_{u\in U}\subset \mathbb{R}_{\infty }^{X}$,
p=\sup\limits_{u\in U}f_{u}$, $x\in p^{-1}(\mathbb{R})$. The next statements
are equivalent:
$\mathrm{(i)} $ \ $\partial p (x) = \bigcup\limits_{u \in I(x)} \partial f_u
(x) $,
$\mathrm{(ii)} $\ \ $\max\limits_{z \in X} \Big\{ \langle x^*, z \rangle -
p(z) \Big\} = \min\limits_{u \in U} f_u^\ast (x^*), \ \forall x^* \in
\partial p (x)$,
$\mathrm{(iii)} $ \ $(\sup\limits_{u \in U} f_u) ^\ast (x^*) =
\min\limits_{y^* \in Y^\ast } f_u^\ast (x^*), \ \forall x^* \in \partial p
(x)$.
\end{corollary}
\textit{Proof.} In this non-parametric case, let $F_u(x, y_u) = f_u(x)$, $p
= \sup\limits_{u \in U} f_u$. We have
\begin{equation*}
D(x) = \bigcup_{u \in I(x)} \partial f_u (x), \ I(x) = \{ u \in U\, :\,
f_u(x) = p(x) \in \mathbb{R}\}
\end{equation*}
and the Theorem \ref{thm82} applies. \qed
\textbf{Acknowledgements} This research was supported by the National
Foundation for Science \& Technology Development (NAFOSTED) from Vietnam,
Project 101.01-2015.27 \textit{Generalizations of Farkas lemma with
applications to optimization}, by the Ministry of Economy and
Competitiveness of Spain and the European Regional Development Fund (ERDF)
of the European Commission, Project MTM2014-59179-C2-1-P, and by the
Australian Research Council, Project DP160100854. \textbf{\ }
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,910
|
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