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\subsection{Our Models and Results}
We only consider the goal of minimizing the
\emph{average} (or total)
metric distance of all voters from the winning candidate.%
\footnote{Recall that \cite{anshelevich:bhardwaj:postl} and several
follow-up articles studied both the average and median distance.}
Our main result, proved in Section~\ref{sec:general-lower},
is essentially a negative answer to the question of whether any voting
mechanisms can simultaneously have all three desirable properties.
We consider a model in which each voter communicates $b$ bits of
information about her ranking to the mechanism, in a single round.%
\footnote{Analyzing the distortion of multi-round deterministic
mechanisms with limited communication is a very interesting
direction for future work.}
Associated with each $b$-bit string \MSG is a subset \PSet{\MSG} of
rankings.
The \PSet{\MSG} must form a disjoint cover of all possible rankings.
If they did not form a cover, some voters might not have any message
to send, making the mechanism ill-defined.
And if the \PSet{\MSG} were not disjoint,
then it is not clear how a voter with multiple possible messages \MSG
would make the (non-deterministic) choice which one to send;
in particular, this choice could depend on the actual metric
distances, and it might require much more subtle definitions to
place meaningful restrictions on a mechanism to not exploit such
information.
Each voter communicates the (unique) \MSG such that her permutation
is in \PSet{\MSG}.
We require that the same set \PSet{\MSG} is associated with the string \MSG,
regardless of the identity of the voter sending the string.\footnote{%
Our results require this assumption. While studying the power of
mechanisms that allow different voters to use different encodings of
their preferences would be interesting theoretically, voting
mechanisms which treat votes differently \emph{a priori} tend to not
be accepted in practice.}
Under this model, in Section~\ref{sec:general-lower},
we prove the following lower bound:
\begin{theorem} \label{thm:general-lower-intro}
Every one-round deterministic voting mechanism in which each voter
sends only a $b$-bit string to the mechanism has
distortion at least $\frac{2n-4}{b} - 1$.
\end{theorem}
Most mechanisms with limited communication are of a fairly specific
form: voters can communicate only their choices in a (small) set $K$ of
$k$ positions of their ranking,
typically at the top or bottom of their ballots.
(Either giving the candidate for each such position,
or specifying them as a set, as in $k$-approval.)
For such restricted mechanisms, a simpler proof
(in Section~\ref{sec:simple-lower}) gives a lower bound
that is stronger by a factor $\Theta(\log n)$:
\begin{theorem} \label{thm:simple-lower-intro}
Any deterministic one-round social choice rule which receives,
from each voter,
no information about candidates outside positions $K$ in her ranking,
has distortion at least
$\frac{2n-\SetCard{K}}{\SetCard{K}}$.
\end{theorem}
The proof of Theorem~\ref{thm:simple-lower-intro} is significantly
easier and cleaner than the proof of
Theorem~\ref{thm:general-lower-intro},
while still containing some of the key ideas.
Therefore, we present the proof of
Theorem~\ref{thm:simple-lower-intro} \emph{before} that of
Theorem~\ref{thm:general-lower-intro}.
Theorem~\ref{thm:simple-lower-intro} provides a generalization
of Theorem~1 of the recent work
\cite{fain:goel:munagala:prabhu:referee},
which proves linear distortion for the special case when $K$ consists of
the top $k$ positions, for constant $k$.
In fact, \cite{fain:goel:munagala:prabhu:referee} shows these lower
bounds on the \emph{expected squared distortion} of randomized
mechanisms;
this directly implies the same bounds for deterministic mechanisms.
The fact that the lower bound of Theorem~\ref{thm:simple-lower-intro}
is stronger than that of Theorem~\ref{thm:general-lower-intro}
by a factor of $\Theta(\log n)$
is discussed in more detail in Section~\ref{sec:general-lower}.
To see it most immediately,
consider the case $\SetCard{K} = k = \omega(n/\log n), k = o(n)$.
Because $k = o(n)$,
Theorem~\ref{thm:simple-lower-intro} provides a super-constant lower bound
on the distortion.
On the other hand, communicating the positions of $k$ candidates
requires $b = \omega(n)$ bits, so the lower bound
of Theorem~\ref{thm:general-lower-intro} is vacuous.
Closing this $\Theta(\log n)$ gap is an interesting direction for
future work, discussed in Section~\ref{sec:conclusions}.
The reason we consider Theorem~\ref{thm:general-lower-intro} our main
contribution is that it helps us pinpoint the source of high
distortion.
Several recent works have shown lower bounds on the distortion of
different specific classes of social choice rules,
such as score-based rules \cite{anshelevich:bhardwaj:elkind:postl:skowron}
or the above-mentioned top-$k$ ballots \cite{fain:goel:munagala:prabhu:referee}.
Our result implies that regardless of the intricacy of the mechanism,
low communication (within the context studied here) and determinism
are enough to force high distortion.
Communication as a measure of complexity is fairly natural, as
evidenced by the mechanisms typically used in practice for large
numbers of alternatives.
Communication can also be regarded as a proxy for cognitive effort
imposed on the voters, although admittedly, the computation of a
message \MSG in a general $b$-bit bounded mechanism may still require
the voter to first determine her full ranking of all candidates.
\smallskip
The results of Theorems~\ref{thm:general-lower-intro} and
\ref{thm:simple-lower-intro} are lower bounds, raising the question of
how small one can make a mechanism's distortion when communication is
limited.
In Section~\ref{sec:upper}, we address this question, proving the
following theorem.
\begin{theorem} \label{thm:upper-intro}
There is a one-round deterministic social choice rule which,
given only each voter's top $k$ candidates (in order),
selects a candidate with distortion at most
$\frac{79n}{k}$.
\end{theorem}
The deterministic social choice rule of Theorem~\ref{thm:upper-intro}
is a generalization of the Copeland rule to such top-$k$ ballots.
Up to constant factors,\footnote{%
An application of Corollary~5.3 of \cite{DistortionDuality}
gives an upper bound of $\frac{12n}{k}$, which, however,
is still far from matching the lower bound.}
the bounds of Theorems~\ref{thm:simple-lower-intro} and
\ref{thm:upper-intro} match.
Closing the gap between the upper and lower bound is likely difficult,
as even for $k=n$, the best-known lower bound of 3 does not match the
best current upper bound of $2+\sqrt{5} \approx 4.23$ due to
\cite{munagala:wang:improved};
whether there is a deterministic mechanism with metric
distortion 3 is a well-known open question.
Notice also that Theorem~\ref{thm:upper-intro} implies that knowing
each voter's ranking for a constant fraction of candidates is
sufficient to achieve constant distortion,
a fact that may not be a priori obvious.
As we discussed earlier, the main focus in this article is on
\emph{deterministic} mechanisms: as discussed earlier,
the \emph{Random Dictatorship} mechanism
has distortion strictly smaller than 3, achieving small distortion and
low communication simultaneously.%
\footnote{The amount by which it is smaller is of order $1/\SetCard{V}$;
here, $\SetCard{V}$ is the the number of voters, which we consider ``large.''}
\cite{gross:anshelevich:xia:agree} prove a nearly matching lower bound:
they show that every randomized social choice rule in which each voter
only communicates her top $k < n/2$ candidates must have distortion at least
$3-\frac{2}{\Floor{n/k}}$.
However, even for $k=1$, this leaves a gap between the upper
bound of essentially $3$ for Random Dictatorship and the lower bound
of $3-\frac{2}{n}$.
Recently, \cite{fain:goel:munagala:prabhu:referee} shrunk this gap:
they proved that the Random Oligarchy mechanism
--- which samples three voters and outputs a majority of first-place
votes if it exists, and otherwise the choice
of a random voter among the three --- achieves expected distortion
close to $3-\frac{2}{n}$, though there still remains a small gap
between the upper and lower bounds.
As an additional result, in Section~\ref{sec:randomized},
we close this remaining gap:
\begin{theorem} \label{thm:randomized-intro}
There is a simple randomized social choice rule in which each
voter only communicates her first-choice candidate,
and which achieves distortion at most $3-\frac{2}{n}$.
\end{theorem}
\subsubsection*{Nature of Latent Distances}
The optimization objective of the mechanism is expressed in terms of
latent utilities, or more specifically, distances.
A subtle question is whether voters ``know'' their utilities for (or
distances to) candidates, or --- perhaps more philosophically ---
whether these utilities/distances are ``real.''
In general, one attractive feature of the distortion framework is that
it completely obviates the need to address this question:
when a mechanism achieves low distortion, it optimizes robustly over
all possible utility/distance functions consistent with the rankings,
and the question of whether voters could actually quantify the
utilities in a meaningful way is irrelevant.
However, when we focus on the design of mechanisms with low
communication, the question should be addressed explicitly,
as the answer has a strong impact on the design space for mechanisms.
When the mechanism designer has control not only over the
\emph{aggregation} of ballots, but also over the type of information
about voter preferences that is elicited,
this opens the door to designing mechanisms in which agents explicitly
communicate numerical estimates of their utilities for some candidates;
in turn, having such information may allow a mechanism to achieve
lower distortion (as we will see in related work below).
If agents themselves cannot quantify their utilities,
then not only is communication of a ranking imposed by the class of
typically used mechanisms, but it is inherently the only information
about the utilities that agents themselves may have access to.
Which of these two assumptions
(or something between the two along a more fine-grained spectrum)
is more realistic likely depends on the envisioned application.
For example, if software agents vote on a preferred alternative in a
mostly economically motivated setting, then it is very reasonable to
assume that the agents can compute (good approximations of) their
utilities.
On the other hand, when human voters choose between political
candidates, assuming an ability to quantify a metric distance in some
abstract space of political positions is much less realistic.
Thus, we believe that for both assumptions,
there are important and natural settings in which they are justified,
motivating studies of communication-distortion tradeoffs in both types
of scenarios.
\subsection{Related Work}
Communication complexity \cite{kushilevitz:nisan} generally studies
the required communication between multiple parties wishing to jointly
compute an outcome.
Several recent works have studied the communication required
specifically for jointly computing particular economic outcomes,
or --- conversely --- to bound the effects of limited communication on
such economic outcomes.
These include work on auctions and allocations
\cite{alon:nisan:raz:weinstein:limited-interaction,assadi:auctions-interaction,blumrosen:nisan:segal:bounded-communication,blumrosen:feldman:implementation-bounded,dobzinski:nisan:oren:efficiency-interaction},
persuasion \cite{LimitedSignaling},
and general mechanism design
\cite{mookherjee:tsumagari:communication-constraints}.
While the high-level concerns are similar across different domains,
the specific approaches and techniques do not appear to carry over.
The impact of communication more specifically
on social choice rules has been explored before;
see, for instance, \cite{boutilier:rosenschein:incomplete} for an overview.
However, most of the focus in past work has been on the number of bits
that need to be communicated in order to compute the outcome of a
\emph{particular} social choice rule, rather than on proving lower
bounds arising due to limited communication when the social choice rule
is not pre-specified.
A classic paper in this context is by Conitzer and Sandholm
\cite{conitzer:sandholm:vote-elicitation}: they study
\emph{vote elicitation} rules, i.e., protocols by which a mechanism can
interact with voters to determine the winner under a particular voting
rule while not eliciting the full ranking information.
This raises algorithmic questions about whether the information
obtained so far uniquely determines a winner as well as incentive
issues, among others, and a large amount of follow-up literature
(e.g., \cite{ding:lin:voting-partial}) has studied these issues.
Relatedly, Conitzer \cite{conitzer:eliciting-single-peaked} studies
how many comparisons need to be elicited from voters to be able to
reconstruct their complete ranking, and shows that the number is
linear (as opposed to quadratic) when preferences are single-peaked
(on the line).
Several very recent papers have explicitly considered the tradeoff between
communication and distortion in social choice,
both in deterministic and randomized settings.
Perhaps most immediately related is recent work by
Fain et al.~\cite{fain:goel:munagala:prabhu:referee}.
Their focus is on mechanisms with extremely low communication which
achieve low expected squared distortion, a measure somewhere between
expected distortion and deterministic distortion.
They prove that the Random Referee mechanism, which asks two randomly
chosen voters for their top choices, and asks a third voter to choose
between these two choices, achieves constant expected squared
distortion.
Notice that this mechanism elicits different information from
different voters.
Theorem~1 of \cite{fain:goel:munagala:prabhu:referee} shows that this
is unavoidable, in that any mechanism that only obtains top-$k$ lists
(for constant $k$), even from all voters, must have linear expected
squared distortion, implying the same result for the distortion of
deterministic mechanisms.
Our Theorem~\ref{thm:simple-lower-intro} generalizes this result for
deterministic mechanisms to non-constant $k$ and sets other than the
top $k$ positions.
Another very related piece of work is due to
Mandal et al.~\cite{mandal:procaccia:shah:woodruff},
studying the communication-distortion tradeoff in a setting where the
voters have utilities (instead of costs) for the candidates,
and these utilities are only assumed to be non-negative and normalized,
but do not need to satisfy any other properties
(such as being derived from a metric).
The other major modeling difference between our work and
\cite{mandal:procaccia:shah:woodruff} is that they assume that agents
compute their message \MSG to the mechanism directly from their
utility vector, rather than the ranking.
In particular, the mechanism can be designed to allow voters to
express the strength of their preferences, albeit in possibly coarse
form.
This allows for a choice of deterministic/randomized algorithms in two
places: (1) the voters' computation of their message, and
(2) the mechanism's aggregation of the messages into a winner.
\cite{mandal:procaccia:shah:woodruff} give upper and lower bounds for
deterministic and randomized voting rules in this setting.
The positive/algorithmic results in
\cite{mandal:procaccia:shah:woodruff} are obtained primarily by
generalizing an approach of
Benad\`{e} et al.~\cite{BNPS:distortion-rule:application},
asking voters to communicate their top few candidates as well as a
suitably rounded version of their utility for those nominated
candidates.
The bounds are improved in some parameter regimes by having the mechanism
randomly select a subset of candidates and restricting voters to choose
from this subset.
While the results of \cite{mandal:procaccia:shah:woodruff} are clearly
directly related to our work, they are not immediately comparable.
Because the utilities are not derived from metrics, the mechanisms
need to deal with much broader classes of inputs, resulting in
(generally) weaker upper bounds and stronger lower bounds.
On the other hand, the assumption that voters can explicitly quantify
their utilities --- and hence have them elicited by a mechanism ---
gives a mechanism more power than in our setting.
Another related recent piece of work is on approval-based voting,
due to Pierczy\'{n}ski and Skowron \cite{pierczynski:skowron:approval}.
While much of this work focuses on a different notion of distortion
--- analyzing the fraction of voters who \emph{approve} of the winning
candidate in the sense of being ``close enough'' ---
\cite{pierczynski:skowron:approval} also analyzes the (traditional)
distortion of approval-based voting.
Under the type of mechanism that they consider, rather than approving
a given \emph{number} of voters (as in $k$-approval),
voters approve all candidates within a given \emph{distance} of
themselves, i.e., within a ball of given radius around themselves.
This approval radius can be voter-specific or uniform across voters.
In this context, the main result of
\cite{pierczynski:skowron:approval} is to show specific constant
distortion whenever a uniform approval radius ensures that a constant
fraction of voters, bounded away from 0 and 1, have the optimum
candidate within their approval radius.\footnote{%
In particular, when that fraction is between \quarter and \half, the
distortion is at most 3.}
It is of course not clear how a mechanism (or the voters) could
determine such a radius.
Also note that this type of approval-based mechanism does require
voters to quantify their distances,
rather than just interact with their individual ordinal rankings.
Note that Theorem~\ref{thm:upper-intro} can be considered as somewhat
related to this result. It shows that whenever voters communicate
their top $k$ candidates, where $k$ is a constant fraction of the
number of candidates, there is a mechanism with constant distortion.
However, in contrast to the result of \cite{pierczynski:skowron:approval},
not just the identity, but also the ranking of these top $k$
candidates must be communicated; on the other hand, the theorem makes
no assumptions about whether the optimum candidate appears in any of
these top-$k$ rankings.
Low communication complexity of voter preferences is also the focus of
a recent preprint by Bentert and Skowron
\cite{bentert:skowron:few-candidates}.
They study the more ``traditional'' goal of implementing given voting
rules with low communication \cite{boutilier:rosenschein:incomplete},
but are interested in \emph{approximate} implementation of these
rules.
To make approximation meaningful, they focus on score-based rules,
which naturally assign each candidate a score (such as Borda Count,
Plurality, or MiniMax).
Then, the quality of approximation is the ratio between the score of
the winner under full information vs.~the score of the winner under
limited communication.
They focus on mechanisms in which each voter is asked to rank a small
subset of candidates; this subset is either the voter's top $k$
candidates (a deterministic mechanism) or a random subset of $k$
candidates (a randomized mechanism).
Given that the goal in \cite{bentert:skowron:few-candidates} is the
approximate implementation of specific scoring-based voting rules
rather than achieving low distortion, the results are not directly
comparable.
However, the techniques in Section~3.2 of
\cite{bentert:skowron:few-candidates} readily yield a randomized
mechanism with distortion $5+O(\epsilon)$ and very low communication
complexity per voter when the number of voters is sufficiently large.
By asking each voter to compare a uniformly random pair of candidates
(see also \cite{hansen:random-pairs}),
and using the majority of returned votes,
with high probability (by Chernoff and Union Bounds),
one obtains a tournament graph in which each directed edge $(x,y)$
corresponds to at least a $\half-\epsilon$ fraction of voters
preferring $x$ over $y$.
Then, a straightforward modification of the analysis of the distortion
of uncovered set rules in
\cite{anshelevich:bhardwaj:elkind:postl:skowron}
(or a simple application of Corollary~5.3 in \cite{DistortionDuality})
gives a distortion of $5+O(\epsilon)$.
This rule only requires each voter to compute 1 bit in total.
However, different voters are asked to answer different questions,
which is often considered undesirable.
Furthermore, the total communication complexity is $n$ bits, whereas
the Random Dictator mechanism only needs to elicit $\log_2 n$ bits
from one voter.
The recent work of Bentert and Skowron is somewhat related to earlier
work of Filmus and Oren \cite{filmus:oren:top-k-voting}:
they are also interested in the question of when top-$k$ ballots from
voters are sufficient to obtain the correct candidate.
However, \cite{filmus:oren:top-k-voting} study this question under
probabilistic models for the ballots, significantly changing the
nature of the results.
The metric-based distortion view of social choice has proved to be a
very fruitful analysis framework.
In fact, it has been extended beyond social choice to other
optimization problems in which it is natural to assume that a
mechanism only receives ordinal information;
see, e.g., \cite{anshelevich:sekar:blind,anshelevich:ordinal}.
Several modeling assumptions have been proposed that yield lower
distortion than the worst-case bounds of
\cite{anshelevich:bhardwaj:elkind:postl:skowron}.
One such assumption is termed \emph{decisiveness}
\cite{anshelevich:postl:randomized,gross:anshelevich:xia:agree}:
it posits that for every voter, there is a sufficiently clear
first choice among candidates.
When the metric space is sufficiently decisive, significantly stronger
upper bounds on the distortion can be proved.
An alternative approach was proposed in
\cite{OfThePeople,BordaRepresentative}.
The authors assumed that the candidates were ``representative,''
in that they themselves were drawn i.i.d.~uniformly from the set of
voters.
Under this assumption, the authors obtained improved expected
distortion bounds for the case of two candidates
\cite{OfThePeople},
and constant expected distortion for Borda count and
several other position-based scoring rules \cite{BordaRepresentative}.
As mentioned above, the gap between the upper bound of 5
(achieved, e.g., by the Copeland rule)
and the lower bound of 3 has posed an
interesting open question for several years now.
One initial conjecture of \cite{anshelevich:bhardwaj:postl}
was that the Ranked Pairs mechanism might achieve a distortion of 3.
This conjecture was disproved by \cite{goel:krishnaswamy:munagala},
who showed a lower bound of 5 on the distortion of Ranked Pairs.
Very recently, Munagala and Wang \cite{munagala:wang:improved}
have presented a (deterministic) social choice rule with
distortion at most $2+\sqrt{5} \approx 4.23$,
which is the first piece of progress towards closing the gap.
In our and much of the preceding work on metric voting,
the focus is on distortion, while ignoring incentive compatibility.
(Recall the strong impossibility result of
\cite{gibbard:manipulation,satterthwaite:voting}.)
The connection between strategy proofness and distortion in this type
of setting was studied in \cite{feldman:fiat:golomb}.
\section{Introduction} \label{sec:introduction}
\input{introduction}
\section{Preliminaries} \label{sec:preliminaries}
\input{preliminaries}
\section{A Lower Bound for $k$-Entry Social Choice Rules} \label{sec:simple-lower}
\input{simple-lower}
\section{The General Lower Bound} \label{sec:general-lower}
\input{general-lower}
\section{A Near-Matching Upper Bound} \label{sec:upper}
\input{upper}
\section{A Tight Upper Bound for Randomized Algorithms} \label{sec:randomized}
\input{randomized}
\section{Conclusions} \label{sec:conclusions}
\input{conclusions}
\subsubsection*{Acknowledgements}
The author would like to thank
Elliot Anshelevich, Yu Cheng, Shaddin Dughmi, Tyler LaBonte, Jonathan
Libgober, and Sigal Oren for useful conversations and pointers,
and anonymous reviewers for useful feedback.
\bibliographystyle{plain}
\subsection{Voters, Candidates, and Social Choice Rules}
There are $n$ candidates, which we always denote by lowercase letters
at the end of the alphabet.
Sets of candidates are denoted by uppercase letters,
and \ALLCANDS is the set of all candidates.
The preference order (or ranking) of voter $v$ over the candidates
is a bijection $\PERM[v] : \SET{1, \ldots, n} \to \ALLCANDS$,
mapping positions $i$ to the candidate $x = \Perm[v]{i}$ which voter
$v$ ranks in position $i$.
We say that $v$ (strictly) \emph{prefers} $x$ to $y$ iff
$\PermInv[v]{x} < \PermInv[v]{y}$.
When only the ranking, but not the identity, of a voter is relevant,
we will omit the subscript $v$ for legibility.
The set of all voters\footnote{%
We will not need to reference the number of voters explicitly.
In general, we treat the number of voters as ``much larger'' than
the number of candidates, and are only interested in bounds in terms
of the number of candidates.}
is denoted by \ALLVOTERS.
We write \ALLPERMS for the set of all possible rankings
$\PERM: \SET{1, \ldots, n} \to \ALLCANDS$,
and $\PERMMAT = (\Perm[v]{i})_{v \in \ALLVOTERS, i \in \SET{1, \ldots, n}}$
for the rankings of all voters, which we call the \emph{vote profile}.
In the traditional full-information view,
a \emph{social choice rule} (we use the terms
\emph{mechanism} or \emph{voting mechanism} interchangeably)
$\SCRULE : \ALLPERMS^{\ALLVOTERS} \to \ALLCANDS$
is given the rankings of all voters, i.e., \PERMMAT,
and produces as output one \emph{winning} candidate
$\WINNER = \SCRule{\PERMMAT}$.
For most of this article, we are interested only in
\emph{deterministic} social choice rules \SCRULE.
\subsection{Communication-bounded mechanisms}
Our main contribution is to consider communication-bounded social
choice rules.
As in the standard model described above,
we still only consider deterministic \emph{single-round} mechanisms,
i.e., each voter can only send a single message to the mechanism.
However, this message is now also restricted to be at most $b$ bits
long.
This induces $\PARTS = 2^b$ sets
$\PSet{1}, \PSet{2}, \ldots, \PSet{\PARTS}$ of rankings;
when the mechanism receives a message \MSG from voter $v$,
all it learns is that $\PERM[v] \in \PSet{\MSG}$.
As discussed in the introduction,
we assume that the \PSet{\MSG} form a disjoint partition of \ALLPERMS,
i.e., they are pairwise disjoint and cover all rankings:
$\bigcup_{\MSG=1}^{\PARTS} \PSet{\PARTS} = \ALLPERMS$.
The fact that \PARTS is a power of 2 is not relevant anywhere in our
proofs,
so we also consider mechanisms with arbitrary numbers \PARTS of sets.
\begin{definition}[\PARTS-communication bounded social choice rule]
\label{def:communication-bounded-rule}
An \emph{\PARTS-communication bounded social choice rule} consists of
pairwise disjoint sets
$\PSet{1}, \PSet{2}, \ldots, \PSet{\PARTS} \subseteq \ALLPERMS$
with $\bigcup_{\MSG=1}^{\PARTS} \PSet{\PARTS} = \ALLPERMS$,
and a deterministic mapping
$\SCRULE: \SET{1, \ldots, \PARTS}^{\ALLVOTERS} \to \ALLCANDS$.
\end{definition}
Communication-bounded social choice rules that are used in
practice, such as Plurality, Veto, $k$-approval, and combinations
thereof, are of a specific form:
there is a set $K$ of $k$ positions, and voters can communicate the
set of candidates they have in positions in $K$,
possibly with an ordering, but cannot communicate any additional
information about their ranking of candidates in positions outside $K$.
For such mechanisms, we will be able to prove stronger lower bounds on
the distortion, and with a significantly simpler proof.
We define them formally as follows:
\begin{definition} \label{def:k-entry-rule}
A \emph{$k$-entry social choice rule} is an
\PARTS-communication bounded social choice rule with the following
additional restriction on the sets
$\PSet{1}, \PSet{2}, \ldots, \PSet{\PARTS}$:
there exists a set $K \subseteq \SET{1, \ldots, n}$ of at most $k$
positions such that
if $\PERM, \PERMP$ agree for all positions in $K$,
i.e., $\Perm{i} = \PermP{i}$ for all $i \in K$, then
$\PERM \in \PSet{\MSG}$ if and only if $\PERMP \in \PSet{\MSG}$.
\end{definition}
\subsection{Metric Space and Distortion}
The key modeling contribution of the metric-based distortion
\cite{anshelevich:bhardwaj:postl}
objective is to assume that all voters and candidates are
embedded in a pseudo-metric space \DIST.
\Dist{v}{x} denotes the distance between voter $v$ and candidate $x$.
Being a pseudo-metric, it satisfies non-negativity and the triangle
inequality $\Dist{v}{x} \leq \Dist{v}{y} + \Dist{v'}{y} + \Dist{v'}{x}$
for all voters $v, v'$ and candidates $x, y$.
Given our choice of defining the metric only for pairs consisting of a
voter and a candidate, symmetry is not directly relevant.
One can naturally extend the pseudo-metric to pairs of candidates or
pairs of voters, but those distances will never appear in our
mechanisms or proofs.
For our upper bounds, we explicitly allow the distance between
candidates and voters
(and thus also between pairs of candidates or pairs of voters)
to be 0; however, for improved flow, we will still refer to \DIST as
a \emph{metric}.
In our lower-bound constructions, all distances will be strictly
positive; that is, we do not exploit the increased generality for
negative results.
We say that a vote profile \PERMMAT is \emph{consistent} with the
metric \DIST, and write \Legal{\DIST}{\PERMMAT},
if $\Perm[v]{x} < \Perm[v]{y}$ whenever $\Dist{v}{x} < \Dist{v}{y}$.
That is, \PERMMAT is consistent with \DIST iff all
voters rank candidates by non-decreasing distance from themselves.
Notice that in case of ties among distances,
i.e., $\Dist{v}{x} = \Dist{v}{y}$,
several vote profiles are consistent with \DIST.
None of our results depend on any tie breaking assumptions.
The \emph{cost} of candidate $x$ is
$\Cost{x} = \sum_{v} \Dist{v}{x}$,
i.e., the sum of distances of $x$ to all voters.%
\footnote{\cite{anshelevich:bhardwaj:postl} also consider the median
distance as an optimization objective;
here, we only focus on the sum/average objective.}
An \emph{optimum} candidate is any candidate
$\OPT[\DIST] \in \argmin_{x \in \ALLCANDS} \Cost{x}$;
in our analysis, it will not matter which candidate is considered
``the'' optimum candidate in case of ties.
The social choice rule is handicapped by not knowing the metric \DIST,
instead only observing the consistent vote profile \PERMMAT
(or some limited information about it, when communication is restricted).
Due to this handicap, and possibly other suboptimal choices,
it will typically choose candidates with higher cost than
\Cost{\OPT}.
The \emph{distortion} of \SCRULE is the worst-case ratio between the
cost of the candidate chosen by \SCRULE,
and the optimal candidate \OPT[\DIST]
(determined with knowledge of the actual distances \DIST).
Formally,
\[
\Distortion{\SCRULE} \; = \;
\max_{\PERMMAT} \sup_{\DIST: \Legal{\DIST}{\PERMMAT}}
\frac{\Cost{\SCRule{\PERMMAT}}}{\Cost{\OPT[\DIST]}}.
\]
We can think of the distortion in terms of a game between the social
choice rule and an adversary.
First, the adversary chooses the vote profile \PERMMAT.
Then, the social choice rule, knowing only \PERMMAT
(or part of that information, in case of communication restrictions),
chooses a winning candidate \WINNER = \SCRule{\PERMMAT}.
Then, the adversary chooses a metric \DIST consistent with
\PERMMAT that maximizes the ratio between the cost of the
candidate chosen by \SCRULE and the optimum candidate for \DIST.
The goal now is to define a social choice rule \SCRULE
--- under suitable constraints --- that achieves small distortion
\Distortion{\SCRULE}, and to prove lower bounds on all social choice
rules under the given constraints.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,529
|
\section{Intro}
Few debates in the history of physics have shed more ink than that of relationalism versus absolutism. Pragmatists take refuge in absolute structures which provide excellent arenas for performing calculations that can accurately predict the outcomes of laboratory experiments. The epistemologically minded physicist argues for relationalism on the grounds of sound philosophical foundations. Though the debate has evolved considerably since the time of Newton and Leibniz, it is still alive and well today.
To say that Einstein's presentation of General Relativity (GR) has settled the debate, even classically, is wishful thinking. Julian Barbour makes the point bluntly: ``Einstein's approach to Mach's principle has generated much confusion.''\cite{barbour:mach_principle} The confusion that Barbour is referring to (on top of some other important issues) is the distinction between a relational theory and a generally covariant theory. Einstein was led to GR in an attempt to write the laws of gravity in a generally covariant form \cite{einstein:gen_rel}. The idea was to implement Mach's principle by finding generally covariant laws of physics. These motivations are still taught in introductory courses on GR despite the fact that less than a year after Einstein published his theory, Kretschmann \cite{kretschmann:gen_cov} demonstrated that \emph{any} theory with physical content can be written in generally covariant form. For example, one can \emph{Kretchmannize} Newtonian mechanics by substituting every occurrence of $dt$ in the Newtonian action by $\frac{dt}{d\lambda}d\lambda$, where $\lambda$ is some arbitrary parameter. This gives a reparameterization invariant formulation of Newtonian mechanics often called Parameterized Particle Mechanics (PPM) \cite{lanczos:mechanics, kuchar:canqugr, adm:adm_review}. Though this theory is generally covariant with respect to time, it still contains and absolute temporal structure. GR, as a result, can not be called relational because of its general covariance alone. But if general covariance is not sufficient to make GR relational then what makes it relational? Indeed, is GR relational at all?
The answer to this question, according to Barbour \emph{el al.} \cite{barbourbertotti:mach, barbour_el_al:scale_inv_gravity, barbour_el_al:physical_dof}, is very nearly\footnote{In \cite{barbourbertotti:mach, barbour_el_al:scale_inv_gravity}, GR was found not to be Machian with respect to scale transformations. However, in \cite{barbour_el_al:physical_dof}, it was noticed that GR, in CMC gauge, is Machian with respect to \emph{volume preserving} scale transformations improving the situation considerably.} ``yes''. They were able to derive GR on closed spatial topologies using a technique called \emph{best-matching} which was designed to implement temporal and spatial relationalism. The same techniques applied to models of non-relativistic particles, developed in \cite{barbour:scale_inv_particles, barbourbertotti:mach, barbour:timelessness, gergely:geometry_BB1, gergely:geometry_BB2, anderson:triangleland_old}, lead to temporally and spatially relational theories which suffer from the same Problem of Time encountered in GR. The link between the Problem of Time in GR and in these models is well documented in \cite{barbour:eot, kiefer:qu_gr_book, kuchar:time_int_qu_gr, kuchar:canqugr, adm:adm_review, anderson:rel_part_mech_2, kuchar:prob_of_time, kuchar:lect_notes, barbour:timelessness2} and is discussed at some length in \cite{anderson:triangleland_new_2}. From the perspective of Barbour-Bertotti (BB) theory, the only difference between the relational particle models and GR is the choice of the configuration space\footnote{As well as the \emph{local} square root used in the action.}. They are, thus, useful toy models for studying the Problem of Time in GR\footnote{Of course there are certain features of GR that may be important to the Problem of Time, such as the indefiniteness of the kinetic energy and the fact that it is four dimensional, that are not captured by these toy models.}.
In this paper, we focus specifically on temporal relationalism and its implications to the Problem of Time. We justify this in an \apx{spatial_sym} where we show that adding spatial relationalism by best-matching the spatial symmetries of PPM does not change the main results regarding time. We then propose a new method for implementing temporal relationalism, inspired by best-matching, and apply it to PPM. We prove that this method is equivalent to BB's proposal for non-relativistic particles. Using our proposal, we can study the difference between relational and absolute formulations of PPM. We present two main results: 1) PPM can be made temporally relational by applying our procedure which effectively gauges its time translational symmetry, and 2) the absolute structure can be restored by modifying the variation of the relational fields. In this context, it is the different types of variations of these fields, which are controlled by a constraint we call the Mach constraint, that make the theory either temporally relational or absolute. We thus establish a clear distinction between parameterised theories that have hidden backgrounds and those that don't. By adding a Mach constraint to the time translations, we have been successful at separating the issue of Relationalism (ie, background independence), which is characterized by the Mach constraint, from the issue of reparameterization invariance (ie, general covariance), which is characterized by the Hamiltonian constraint. However, this does not address the deeper issue (considered by \cite{barbour_foster:dirac_thm}) of whether the Hamiltonian constraint should be thought of as a generator of gauge transformations, which arises whenever a theory is reparameterization invariant in the canonical variable.
We arrive at our results using Dirac and path integral quantizations. Though these approaches should, in principle, be equivalent, the path integral approach provides a much more detailed and convincing picture of what is going on.
Given the deep relationship between these toy models and GR, our results indicate that it might be possible to determine an absolute formulation of GR. This would be invaluable for either understanding relationalism in GR or understanding how absolute structures may emerge from quantum gravity. Furthermore, the generalization of our results might shed additional light on the results of \cite{brown:gr_time}.
\subsection{Historical Background}
Barbour and Bertotti have provided a technique \cite{barbourbertotti:mach} called \emph{best-matching} for producing a Machian theory. The key idea is that background structures exist in theories when meaning is attached to the absolute position of the system in a gauge orbit generated by a particular symmetry group. Using a procedure mathematically similar to \emph{gauging} a global symmetry\footnote{A precise relation between best-matching and gauge theory has been proposed in \cite{gryb:ym_bm}.}, the dependence of a theory on a background structure can be eliminated. This procedure is known as the \emph{corrected coordinate method} and is the basis of best-matching.\footnote{The idea of the \emph{corrected coordinate method} itself was introduced after \cite{barbourbertotti:mach} and is developed nicely in \cite{barbour:mach_principle, barbour:scale_inv_particles, barbour_el_al:scale_inv_gravity}.}
Barbour and Bertotti \cite{barbourbertotti:mach} used their technique of best-matching on the simplest non-trivial theory they could write down where the 3-metric is diffeomorphism invariant. This led to the Baierlein-Sharp-Wheeler (BSW) action \cite{bsw:bsw_action} for GR, an approach referred to as \emph{geometrodynamics}. It proves that GR, restricted to $\mathbb{R} \times \Sigma^3$ topologies where $\Sigma^3$ is a compact 3 manifold with no boundary, has no background spatial structure and, as a result, is spatially relational. Temporal relationalism is achieved by implementing Jacobi's principle \cite{lanczos:mechanics}: an action principle that uses the weighted average rate of change of the 3 dimensional configuration space variables multiplied by a suitably defined conformal factor to define a metric on configuration space. As a result, according to Barbour's criterion, Einstein succeeded, though not in the way he originally intended, at finding a spatially and temporally relational theory of gravity.\footnote{GR is \emph{not} relational with respect to local scale transformation of the metric and is, hence, not completely Machian by Barbour's criterion. For formulations of geometrodynamics with and without scale invariance see \cite{barbour_el_al:scale_inv_gravity, barbour_el_al:rel_wo_rel, barbour_el_al:physical_dof, barbourbertotti:mach}.}
The stark asymmetry between how temporal and spatial relationalism is achieved is attributed to a conceptual asymmetry between the nature of time and space. However, in the following paper, we show that for non-relativistic particles, Jacobi's principle is equivalent classically and quantum mechanically to applying the corrected coordinate method to the time translational symmetry of PPM. Thus, Jacobi's principle is seen as originating from the temporal generalization of best-matching. We do not claim that this undermines the conceptual beauty of Jacobi's principle nor the unique role played by time due to the quadratic constraints but, rather, we propose it as a mathematical equivalent for deriving temporal relationalism.
The theory we obtain from applying the corrected coordinate method to PPM we will call time-gauged PPM. As we will show, it is equivalent to Jacobi's principle applied to classical mechanics and is temporally relational as a result. PPM on the other hand, is the Kretschmannized version of Newtonian mechanics and, as such, contains a type of ``hidden'' background structure. Hence, we must make a clear distinction between: 1) manifestly absolute theories (like Newtonian mechanics), 2) Kretschmannized theories with absolute structure (like PPM), and 3) truly relational theories (like time-gauged PPM and GR). It is easy to tell (by definition) theories of type 1) from theories of type 2) and 3). However, it is not always clear how to make a distinction between theories of type 2) and and theories of type 3). For instance, GR, in its formulation due to Arnowitt, Deser, and Misner (ADM), has many formal similarities to PPM. In \cite{adm:adm_review}, ADM point out these similarities suggesting that GR might be a theory of type 2) having a hidden background structure. Solving the problem of time would then involve finding this background structure. This was also a view that has long been supported by Kucha$\check{\text{r}}$ \cite{kuchar:canqugr, kuchar:prob_of_time, kuchar:lect_notes}. In light of Barbour and Bertotti's results, it is not surprising that no attempt to find this background structure has been successful. The results of this paper will suggest that the ADM action might have more in common with time-gauged PPM than standard PPM. In light of this confusion, it would be valuable to have a criterion to distinguishing between type 2 and type 3 theories. We propose such a criterion in the context of time-gauged PPM.
In this work, we study closely what happens when the background structure of PPM is eliminated using the corrected coordinate method. The auxiliary fields, which can be thought of as being related to the flat part of the gauge connection,\footnote{See \cite{gryb:ym_bm} for more details.} are not varied in the standard way but in a way where the endpoints are allowed to vary freely. We will find that it is this free endpoint variation that makes the theory relational. If we use a standard fixed endpoint variation instead, the background structure of the theory is restored. That is, when the corrected coordinate method is applied to the global symmetry of this theory, the difference between the relational theory and the absolute theory is the difference between a free and fixed endpoint variation of the auxiliary fields. This result is tied in to the basic structure of the corrected coordinate method and we expect it to hold generally when any global symmetry is gauged in this way. In particular, we expect it hold also in geometrodynamics where the free endpoint variation of the auxiliary fields is known to lead to GR. This raises the question: what would happen in geometrodynamics if we do a fixed endpoint variation of the auxiliary fields instead? This is still an open question though it may be that the answer is related to the results of \cite{brown:gr_time}.
It is a robust result of best-matching and the corrected coordinate method \cite{barbour:scale_inv_particles} that a relational theory will lead to linear momentum constraints. Classically, as is shown in \cite{gryb:ym_bm}, these constraints guarantee that the Noether current associated with the gauged symmetry is zero. Quantum mechanically it means that the wavefunction is a zero eigenvalue eigenstate of the appropriate generalized momentum. For time-gauged PPM, this means that the wavefunction is an energy eigenstate and does not evolve in time. A similar situation arises in geometrodynamics and leads to the problem of time. However, for a reasonable theory of quantum gravity to agree with everyday experience, one requires time to emerge on isolated subsystems of the universe. How can GR ,which is a relational theory, pick up an absolute structure on isolated subsystems? What would this structure look like? In light of the results mentioned above, we can answer this question in PPM. The answer is that, the kernel for time-gauged PPM should vary the auxiliary fields freely at the endpoints. For local subsystems however, this variation needs to reduce to a variation where the endpoints are approximately fixed. This would introduce an absolute structure. This demonstrates the value of having a definite way to distinguish a theory of type 2) from a theory of type 3).
The confusion surrounding relationalism, which started with Einstein, survives today and these foundational issues still divide the quantum gravity community. This is a particularly important observation when one considers that the difficulties encountered in interpreting the Wheeler-DeWitt equation stem, as just explained, from the temporal relationalism of classical GR. The problem of time, in this sense, can be thought of as the quantum mechanical offspring of the relational versus absolute debate.
\subsection{Outline}
The results of this paper will be presented as follows. In \scn{canqu}, we will develop the full canonical formalism and quantization of time-gauged PPM. On our way, we will study Jacobi-Barbour-Bertotti (JBB) theory (a temporally relational theory of particle mechanics) and PPM as these will be, respectively, the relational and absolute equivalents of time-gauged PPM. We will end the section by considering the fixed endpoint variation and how it reintroduces a background structure. In \scn{pathint}, we repeat the analysis of time-gauged PPM with free and fixed endpoint variations but with phase space path integrals. In this context, there is much intuition to be gained about how time disappears and is reintroduced into the quantum theory. Because of this, the path integral may be a powerful tool for studying how time might emerge for isolated subsystems of the universe.
\section{Canonical Quantization} \label{sec:canqu}
\subsection{Review Of Jacobi-Barbour-Bertotti (JBB) Theory}
We begin with a review of JBB theory and its canonical quantization as a simple example of a temporally relational theory of non-relativistic particle mechanics. The purpose of this review is to lay down the ground work for later calculations. The Hamiltonian constraint of this theory leads to a problem of time just as it does in quantum GR. Though the action for this theory was first written down long ago by Jacobi (for a good treatment see \cite{lanczos:mechanics}) it wasn't until relatively recently that Barbour and Bertotti \cite{barbourbertotti:mach} noticed the significance of the theory in regards to relational time and the problem of time.
\subsubsection{Hamiltonian and Dirac Algebra}
The configuration space of JBB theory is $N\times d$ dimensional space (with $N$ the number of particles and $d$ the number of spatial dimensions) of particle positions $q^i_I$, where lower case indices range from 1 to $d$ and upper case indices label particles and range from 1 to $N$. The dynamics of the system is given by trajectories in configuration space parameterized by the arbitrary parameter $\lambda$. The reparameterization invariant action is:
\equa{\label{eq:jbb_action}
S_{\text{JBB}} = \int d\lambda\, 2\sqrt{T(\ensuremath{\dot{q}})}\sqrt{E-V(q)},\quad \text{where}\quad T = \sum_I \frac{1}{2} m_I (\ensuremath{\dot{q}}_I^i)^2.
}
The canonical momenta are given by
\equa{\label{eq:pjbb}
p^I_i = \diby{L_\ensuremath{\text{JBB}}}{\ensuremath{\dot{q}}^i_I} = \sqrt{\frac{E-V}{T}} m_I \ensuremath{\dot{q}}^j_I \eta_{ij}
}
and are easily seen to obey the quadratic identity
\equa{\label{eq:hamcon}
\ensuremath{\mathcal{H}} = V - E + \sum_I \frac{(p^I_i)^2}{2m_I} =0.
}
This quadratic identity suggests that the momenta should be thought of as directional cosines since the length of the momenta are irrelevant pure gauge degrees of freedom. The canonical Hamiltonian $H_{\text{c}} = \sum_I p^I_i q^i_I - L_{\ensuremath{\text{JBB}}}$ is zero. Thus, as in any reparameterization invariant theory, the total Hamiltonian is proportional to the constraints. In this case, (\ref{eq:hamcon}) is the only constraint making it first class. This gives us the total Hamiltonian
\equa{\label{eq:hamjbb}
H_{\text{T}} = N(\lambda) \ensuremath{\mathcal{H}} = N(\lambda) \ensuremath{\left}(V - E + \sum_I\frac{(p^I_i)^2}{2m_I}\ensuremath{\right}).
}
\subsubsection{Equations of Motion}
The fundamental Poisson brackets
\equa{\label{eq:pbjbb}
\pb{q^i_I}{p^J_j} = \delta^J_I \delta^i_j
}
can now be used to work out Hamilton's first and second equations of motion
\begin{align}
\ensuremath{\dot{q}}^i_I = \pb{q^i_I}{H_{\text{T}}} &= N \frac{p^I_j\eta^{ij}}{m_I}, \quad \text{and} \label{eq:h1jbb}\\
\ensuremath{\dot{p}}_i^I = \pb{p^I_i}{H_{\text{T}}} &= -\diby{V}{q^i_I}\label{eq:h2jbb}.
\end{align}
Combining (\ref{eq:h1jbb}) and (\ref{eq:h2jbb}), we immediately recover Newton's laws for the gauge choice $N=1$. However, because we have the constraint (\ref{eq:hamcon}) in addition to Newton's differential equations, this theory only represents Newtonian mechanics constrained to orbits of constant energy $E$.
\subsubsection{Quantum Theory}\label{sec:quantumJBB}
Following Dirac's procedure, we promote the phase space variables to operators and the fundamental Poisson brackets to commutators. Thus we have (in units where $\hbar = 1$):
\equa{
\ensuremath{\left}[ \ensuremath{\hat{q}}^i_I, \ensuremath{\hat{p}}^J_j \ensuremath{\right}] = i\, \delta^J_I \delta^i_j.
}
The operators $\ensuremath{\hat{q}}^i_I$ and $\ensuremath{\hat{p}}^J_j$ act on the wavefunction $\ket{\Psi}$ which also must obey the operator constraint
\equa{
\hat\ensuremath{\mathcal{H}} \ket{\Psi} = 0
}
or
\equa{
\ensuremath{\left}[ V(\ensuremath{\hat{q}}) + \sum_I \frac{(\ensuremath{\hat{p}}^I_i)^2}{2m_I} \ensuremath{\right}] \ket{\Psi} = E \ket{\Psi}.
}
This is just the familiar time independent Schr\"{o}dinger equation requiring that the wavefunction of the system must be an energy eigenstate. The Wheeler-DeWitt equation is completely analogous in this sense which is why JBB theory can be used as a toy model for understanding the problem of time in quantum gravity. In this paper we will show that JBB theory is equivalent quantum mechanically to time-gauged PPM. Before gauging the time translational invariance of PPM, we will quickly review it since we will need many of the derived results later.
\subsection{Parameterized Newtonian Mechanics (PPM)}
\subsubsection{Hamiltonian and Dirac Algebra}
The general idea behind PPM is to treat time as a configuration space variable and solve for it dynamically using an action principle (see \cite{lanczos:mechanics, adm:adm_review, henneaux_teit:quant_gauge} for alternative treatments). The configuration space of PPM is then the \emph{extended configuration space} of the particle positions $q^i_I$ and a variable $q^0$ which, once the equations of motion have been solved for, will represent a Newtonian time. It turns out, as we will see, that $q^0$ is also equivalent to Newtonian time off-shell. The ``dynamics'' are given by trajectories in extended configuration space parameterized by the arbitrary parameter $\lambda$. Here $\lambda$ should be thought of as an arbitrary parameter \emph{not} necessarily related to time. The action is:
\equa{\label{eq:actionpnm}
S_\ensuremath{\text{PPM}} = \int d\lambda\, \ensuremath{\left}[ \frac{T(\ensuremath{\dot{q}})}{\ensuremath{\dot{q}}^0} - \ensuremath{\dot{q}}^0 V(q) \ensuremath{\right}].
}
This action is trivially invariant under translations of the variable $q^0(\lambda) \ensuremath{\rightarrow} q^0(\lambda) + a$. The momenta conjugate to $q^i_I$ are
\equa{
p^I_i = \diby{L_\ensuremath{\text{PPM}}}{\ensuremath{\dot{q}}^i_I} = \frac{1}{\ensuremath{\dot{q}}^0} m_I \ensuremath{\dot{q}}^j_I \eta_{ij}.
}
Comparing this to \eq{pjbb}, we see that the only difference here is that $\ensuremath{\dot{q}}^0$ is now playing the role previously played by the quantity $\sqrt{\frac{E-V}{T}}$. Indeed, in JBB theory $\sqrt{\frac{E-V}{T}}$ is typically identified with the $\lambda$ derivative of the \emph{ephemeris time} $\tau_{\text{EPH}}$ \cite{barbourbertotti:mach}. However, in PPM, $\ensuremath{\dot{q}}^0$ is a configuration space variable while the ephemeris time is \emph{not} in JBB theory. This will have important ramifications for the role played by the different kinds of time variables in the quantum theory.
In extended configuration space there is a momentum conjugate to $q^0$ leading to an extended phase space. This momentum is
\equa{
p_0 = \diby{L_\ensuremath{\text{PPM}}}{\ensuremath{\dot{q}}^0} = - \ensuremath{\left}( \frac{T}{(\ensuremath{\dot{q}}^0)^2} + V \ensuremath{\right}).
}
The momenta obey the identity
\equa{\label{eq:hamconpnm}
\ensuremath{\mathcal{H}} = p_0 + V + \sum_I \frac{(p^I_i)^2}{2m_I} = 0.
}
Again, the canonical Hamiltonian $H_{\text{c}} = \ensuremath{\dot{q}}^0 p_0 + \sum_I p^I_i q^i_I - L_{\ensuremath{\text{PPM}}}$ is zero which means that the total Hamiltonian $H_{\text{T}}$ is proportional to the lone first class constraint (\ref{eq:hamconpnm})
\equa{
H_{\text{T}} = N(\lambda) \ensuremath{\mathcal{H}} = p_0 + V + \sum_I \frac{(p^I_i)^2}{2m_I}.
}
The only (but crucial) difference between this Hamiltonian and the one in (\ref{eq:hamjbb}) for JBB theory is that $p_0$ is a phase space variable instead of a constant $-E$. This will lead to a completely different quantum theory since phase space variables are promoted to operators but constants are not. Because of this, quantum PPM will have an absolute time and be free of a problem of time.
\subsubsection{Equations of Motion}
In addition to the fundamental Poisson brackets $\pb{q^i_I}{p^J_j} = \delta^J_I \delta^i_j$, which are identical to the Poisson brackets (\ref{eq:pbjbb}) of JBB theory, we also have the Poisson brackets
\equa{
\pb{q^0}{p_0} = 1.
}
Using these, we find that the equations of motion for the $\ensuremath{\dot{q}}$'s and the $\ensuremath{\dot{p}}$'s are completely identical to (\ref{eq:h1jbb}) and (\ref{eq:h2jbb}) of JBB theory. However, in PPM we have additionally
\begin{align}
\ensuremath{\dot{q}}^0 = \pb{q^0}{H_{\text{T}}} &= N(\lambda), \quad \text{and} \label{eq:h1pnm} \\
\ensuremath{\dot{p}}_0 = \pb{p_0}{H_{\text{T}}} &= 0. \label{eq:h2pnm}
\end{align}
\eq{h2pnm} expresses conservation of energy while \eq{h1pnm} tells us that picking a function $N(\lambda)$ is tantamount to picking a time gauge. In light of this, the $N=1$ gauge can be thought of as the \emph{Newtonian gauge} as it is the gauge where Newton's laws are manifestly valid.
\subsubsection{Quantum Theory}
Promoting all the phase space variables to operators and Poisson brackets to commutators gives the operator algebra
\equa{
\ensuremath{\left}[ \ensuremath{\hat{q}}^i_I, \ensuremath{\hat{p}}^J_j \ensuremath{\right}] = i\, \delta^J_I \delta^i_j, \quad \text{and} \quad \ensuremath{\left}[ \ensuremath{\hat{q}}^0, \ensuremath{\hat{p}}_0 \ensuremath{\right}] = i.
}
The operators act on the ket $\ket{\Psi}$ which obeys the operator constraint
\equa{
\hat\ensuremath{\mathcal{H}} \ket{\Psi} = \ensuremath{\left}[ V(\ensuremath{\hat{q}}) + \ensuremath{\hat{p}}_0 + \sum_I \frac{(\ensuremath{\hat{p}}^I_i)^2}{2m_I} \ensuremath{\right}] \ket{\Psi} = 0.
}
Associating $\ensuremath{\hat{q}}^0$ with a time operator, this is just the familiar time dependent Schr\"{o}dinger equation. Thus, the quantization of PPM leads to standard quantum theory for non-relativistic particles with no problem of time. The main reason for this is that PPM contains a ``hidden'' background structure that it inherited from standard Newtonian mechanics. It is possible to remove this background structure however by applying a procedure that will be treated in the next section.
\subsection{The Corrected Coordinate Method and PPM}
The corrected coordinate method is a technique developed for best-matching by Barbour and collaborators \cite{barbour:scale_inv_particles, barbour_el_al:scale_inv_gravity} to remove a theory's dependence on some non-physical absolute structure. One can think of it as a way of gauging a global symmetry associated to the absolute structure in question. The link between the corrected coordinate method and standard gauge theory has been explored in \cite{gryb:ym_bm}. For our purposes, the main result that we will need is that the corrected coordinate method will eliminate the dependence of our theory on the non-physical part of the gauge fields associated with a global symmetry. In this case, the global symmetry we are interested in is the time translational invariance of PPM. Gauging this symmetry will reduce the theory to JBB theory.
\subsubsection{The Method}
If the configuration space coordinates contain a physical symmetry under the transformation $q^i_I \ensuremath{\rightarrow} G(\omega^\alpha)^i_j q^j_I$, where $G$ is a matrix representation of the group symmetry and $\omega^\alpha$ are group parameters, then the corrected coordinate method involves replacing the coordinates $q^i_I$ with the \emph{corrected coordinates} $\bar{q}^i_I = G(\omega'^\alpha)^i_j q^j_I$. Here, $\omega'^\alpha$ are dynamical fields called \emph{auxiliary fields}. By construction, they do not contribute to the true degrees of freedom of the system but shift the physical degrees of freedom along gauge orbits. The dynamics of the auxiliary fields is determined by a \emph{free} endpoint variation of the action since the value of the auxiliary fields on the boundary is just as arbitrary as it is in the bulk. In a free endpoint variation we must require the additional constraint
\equa{
\diby{L}{\dot{\omega}'^\alpha} = 0
}
on top of the usual Euler-Lagrange equations. In the Hamiltonian formulation, this is equivalent to imposing the additional free endpoint condition
\equa{\label{eq:machconstraint}
\pi^\alpha = 0,
}
where $\pi^\alpha$ are the momenta conjugate to $\omega'_\alpha$, on top of the usual Hamilton equations of motion. I will call (\ref{eq:machconstraint}) the \emph{Mach constraint}. We will see how the role of the Mach constraint is to eliminate the background structure of a theory leaving a true relational theory behind. In best-matching, the Mach constraint leads to linear momentum constraints which can be solved for to eliminate the auxiliary fields and obtain a theory in terms of the physical degrees of freedom of the system. Here we will apply the corrected coordinate method to gauge the time translational invariance of PPM in an attempt to introduce a relational time.
\subsubsection{Hamiltonian and Dirac Algebra} \label{sec:diracpnm}
We introduce the corrected coordinates
\equa{
\bar{q}^0(\lambda) = G(a(\lambda))q^0 = q^0(\lambda) + a(\lambda)
}
in order to apply the corrected coordinate method to the time translational invariance of PPM. This further extends the configuration space to include the auxiliary field $a(\lambda)$. Next, we substitute $q^0 \ensuremath{\rightarrow} \bar{q}^0$ in the action of (\ref{eq:actionpnm}) giving
\equa{\label{eq:action}
S = \int d\lambda \ensuremath{\left}[ \frac{T}{\ensuremath{\dot{q}}^0 + \ensuremath{\dot{a}}} - (\ensuremath{\dot{q}}^0 + \ensuremath{\dot{a}})V \ensuremath{\right}].
}
The canonical momenta are given by
\begin{align}
p^I_i &= \diby{L}{\ensuremath{\dot{q}}^i_I} = \frac{1}{\ensuremath{\dot{q}}^0 + \ensuremath{\dot{a}}} m_I \ensuremath{\dot{q}}^j_I \eta_{ij} \\
p_0 &= \diby{L}{\ensuremath{\dot{q}}^0} = - \ensuremath{\left}( \frac{T}{(\ensuremath{\dot{q}}^0+ \ensuremath{\dot{a}})^2} + V \ensuremath{\right}). \label{eq:ppm_p0}
\end{align}
Notice that these are the same as the momenta for PPM with the replacement $\ensuremath{\dot{q}}^0 \ensuremath{\rightarrow} \ensuremath{\dot{q}}^0 + \ensuremath{\dot{a}}$. However, we now have an additional momentum conjugate to $a$. It is given by
\equa{
\pi = \diby{L}{\ensuremath{\dot{a}}} = - \ensuremath{\left}( \frac{T}{(\ensuremath{\dot{q}}^0+ \ensuremath{\dot{a}})^2} + V \ensuremath{\right}).\label{eq:ppm_pi}
}
This introduces the new momentum constraint
\equa{
\ensuremath{\mathcal{L}} \equiv p_0 - \pi = 0.
}
We will call this the \emph{linear momentum constraint} in analogy with the diffeomorphism constraint of quantum General Relativity. Because we are doing a free endpoint variation of the $a$'s we must also impose the Mach constraint
\equa{
\pi = 0.
}
Finally, there is also a Hamiltonian constraint
\equa{
\ensuremath{\mathcal{H}} = p_0 + V + \sum_I \frac{(p^I_i)^2}{2m_I} = 0
}
which is identical to the Hamiltonian constraint of PPM. Collecting our results, we have three constraints $\ensuremath{\mathcal{L}} = \pi = \ensuremath{\mathcal{H}} =0$ two linear and one quadratic in the momenta. The constraints that are linear in the momenta are very straightforward and one can solve for them and integrate them out. However, it is very enlightening to keep them around so that we can see how they integrate out explicitly. This will reveal the full structure of the theory and will give us conceptual clues to how time disappears. Also, it will provide a good model for what we should expect to happen in more complicated theories like best-matched particle mechanics or general relativity where the constraints are not as easy to solve for.
The canonical Hamiltonian $H_{\text{c}}$, can once again be shown to be zero. This leaves a total Hamiltonian which is a linear combination of the constraints
\begin{align}
H_{\text{T}} &= N(\lambda) \ensuremath{\mathcal{H}} + L(\lambda) \ensuremath{\mathcal{L}} + M(\lambda) \pi \notag \\
&= N \ensuremath{\left}[ p_0 + V + \sum_I \frac{(p^I_i)^2}{2m_I} \ensuremath{\right}] + L [p_0 - \pi] + M\pi.
\end{align}
We can work out the Dirac algebra by using the fundamental Poisson brackets
\begin{equation}
\begin{array}{ccc}
\pb{q^i_I}{p^J_j} = \delta^J_I \delta^i_j, & \pb{q^0}{p_0} = 1, \text{and} & \pb{a}{\pi} = 1.
\end{array}
\end{equation}
Since the constraints commute with themselves, we are left to work out the Poisson brackets between different constraints. It is a very short calculation to show that
\begin{align}
\pb{\ensuremath{\mathcal{H}}}{\ensuremath{\mathcal{L}}} &=0 \\
\pb{\ensuremath{\mathcal{H}}}{\pi} &=0, \quad\text{and} \\
\pb{\pi}{\ensuremath{\mathcal{L}}} &=0.
\end{align}
Thus, all the constraints are first class.
\subsubsection{Gauge Transformations}
Because $\ensuremath{\mathcal{L}} = 0$ and $\pi = 0$ are ordinary linear momentum constraints they will generate gauge transformations. For discussions of the action of the quadratic constraint of parameterized theories, such as $\ensuremath{\mathcal{H}} = 0$, see \cite{barbour_foster:dirac_thm}. Here we will only be concerned with the gauge transformations generated by ordinary constraints.
We can see the types of gauge transformations generated by the linear constraints by computing the infinitesimal change generated by each of the constraints on the configuration space variables. For $\ensuremath{\mathcal{L}}$ this is
\begin{align}
\delta_\ensuremath{\mathcal{L}} q^i_I & = \epsilon(\lambda) \pb{q^i_I}{q^0 - \pi} =0 \\
\delta_\ensuremath{\mathcal{L}} q^0 & = \epsilon(\lambda) \pb{q^0}{q^0 - \pi} = \epsilon(\lambda) \\
\delta_\ensuremath{\mathcal{L}} a & = \epsilon(\lambda) \pb{a}{q^0 - \pi} =-\epsilon(\lambda).
\end{align}
Hence, $\ensuremath{\mathcal{L}}$ generates the transformation $q^0 \ensuremath{\rightarrow} q^0 + \epsilon$ and $a \ensuremath{\rightarrow} a - \epsilon$ which is clearly an invariance of the action (\ref{eq:action}). Because of its triviality, it has been called the \emph{banal invariance} by Barbour \cite{barbour:scale_inv_particles}. However, by keeping all the constraints we can see that there is yet another even more trivial invariance of the action which I will call the \emph{Machian invariance}. It is generated by the Mach constraint $\pi$
\begin{align}
\delta_\pi q^i_I & = \epsilon(\lambda) \pb{q^i_I}{\pi} =0 \\
\delta_\pi q^0 & = \epsilon(\lambda) \pb{q^0}{\pi} = 0 \\
\delta_\pi a & = \epsilon(\lambda) \pb{a}{\pi} =\epsilon(\lambda).
\end{align}
which is given by the transformation $a \ensuremath{\rightarrow} a + \epsilon$. The reason for the name is that this invariance guarantees that the physical theory is invariant under infinitesimal translations of the $a$'s. This is another way of saying that the $a$'s are arbitrary and therefore cannot affect the physical content of the theory. It is a requirement of a relational theory.
\subsubsection{Equations of Motion}\label{sec:eom}
Using the fundamental Poisson brackets we can easily work out Hamilton's equations of motion for the system. The equations of motion for the spatial variables are unchanged from PPM. They are given by \eq{h1jbb} and \eq{h2jbb}. There are differences however in the equations of motion of the time variables and the auxiliary fields. For the time variables we have
\begin{align}
\ensuremath{\dot{q}}^0 &= \pb{q^0}{H_{\text{T}}} = N + L \label{eq:h1}\\
\ensuremath{\dot{p}}_0 &= \pb{p_0}{H_{\text{T}}} = 0 \label{eq:h2}
\end{align}
We still have conservation of energy from \eq{h2} and an arbitrary $\ensuremath{\dot{q}}^0$ from \eq{h1} but now the Lagrange multiplier of \eq{h1} is different, in general, from that of \eq{h1pnm}. The equations of motion of the auxiliary fields are
\begin{align}
\ensuremath{\dot{a}} &= \pb{a}{H_{\text{T}}} = M - L \label{eq:h1a}\\
\dot{\pi} &= \pb{\pi}{H_{\text{T}}} = 0 \label{eq:h2a}.
\end{align}
\eq{h1a} says that $\ensuremath{\dot{a}}$ is arbitrary. We will use these equations of motion to determine expressions for the gauge fixing functions in the path integral quantization.
In JBB theory, the only variables we have are the $q^i_I$ and the $p_i^I$. They obey the same equations of motion as the $q^i_I$ and the $p_i^I$ of this theory. Furthermore, classically the constraints on the momenta $\ensuremath{\mathcal{L}} = \pi = \ensuremath{\mathcal{H}} =0$ can be reduced to
\equa{
\ensuremath{\left}(V + \sum_I\frac{(p^I_i)^2}{2m_I}\ensuremath{\right}) =0
}
which is identical to the Hamiltonian constraint of \eq{hamcon} with $E=0$. However, the energy can easily be absorbed into the constant part of the potential $V_0$ restoring the equivalence of the constraints. Therefore, as far as the spatial variables are concerned, the two theories are classically equivalent. The only difference is that here we have additional auxiliary fields $a$ which do not affect the physical content of the theory. Their role can be thought of as a way of eliminating $p_0$ from the Hamiltonian constraint.
\subsubsection{Quantum Theory}\label{sec:qupnm}
Promoting the phase space variables to operators and the Poisson brackets to commutators we have
\equa{
\begin{array}{ccc}
\ensuremath{\left}[ \ensuremath{\hat{q}}^i_I, \ensuremath{\hat{p}}^J_j \ensuremath{\right}] = i\, \delta^J_I \delta^i_j, & \ensuremath{\left}[ \ensuremath{\hat{q}}^0, \ensuremath{\hat{p}}_0 \ensuremath{\right}] = i, \quad\text{and} & \ensuremath{\left}[ a, \pi \ensuremath{\right}] = i.
\end{array}
}
The operators act on the wavefunction $\ket{\Psi}$ which must obey the constraints
\equa{
\hat\ensuremath{\mathcal{H}} \ket{\Psi} = \hat\ensuremath{\mathcal{L}} \ket{\Psi} = \hat\pi \ket{\Psi} = 0.
}
That is, the wavefunction must simultaneously satisfy
\begin{align}
\ensuremath{\left}[ V(\ensuremath{\hat{q}}) + \sum_I \frac{(\ensuremath{\hat{p}}^I_i)^2}{2m_I} \ensuremath{\right}] \ket{\Psi} &= -\ensuremath{\hat{p}}_0 \ket{\Psi}, \label{eq:hamconspnmbm}\\
\ensuremath{\hat{p}}_0 \ket{\Psi} &= \hat\pi \ket{\Psi}, \label{eq:lincons}\ensuremath{\quad\text{and}} \\
\hat\pi \ket{\Psi} &= 0.
\end{align}
Combining these leads immediately to the time independent Schr\"{o}dinger equation
\equa{
\ensuremath{\left}[ V(\ensuremath{\hat{q}}) + \sum_I \frac{(\ensuremath{\hat{p}}^I_i)^2}{2m_I} \ensuremath{\right}] \ket{\Psi} = 0
}
with energy $E$ equal to zero. The fact that the energy here appears to be zero is slightly deceiving since the potential $V$ could have a constant term $V_0$ which could be interpreted as the negative of the energy. Thus, this is not necessarily a theory with zero energy. Nevertheless, we have still made the connection quantum mechanically with the quantum version of JBB theory outlined in \scn{quantumJBB}.
\subsubsection{Fixed Endpoint Variation}
We will end this section by solving the interesting problem of using a \emph{fixed} endpoint variation rather than a free endpoint variation when implementing the corrected coordinate method on the time translational invariance of PPM. We will attempt to mirror the discussion of the free endpoint variation of the earlier part of this section.
The phase space and action used in the fixed endpoint variation are identical to those of the free endpoint variation. As a result, the expressions for $p^I_i$, $p_0$, and $\pi$ are identical to the expressions given in \scn{diracpnm}. Furthermore, because the definitions of the momenta are unchanged, they will obey the same identities
\begin{align}
\ensuremath{\mathcal{H}} &= p_0 + V + \sum_I \frac{(p^I_i)^2}{2m_I} = 0, \ensuremath{\quad\text{and}} \\
\ensuremath{\mathcal{L}} &= p_0 -\pi = 0.
\end{align}
However, because we would like to fix the variation at the endpoints we need to drop the Mach constraint. This will lead to the standard canonical analysis of the system. The total Hamiltonian $H_{\text{T}}$ of the system is
\equa{
H_{\text{T}} = N\ensuremath{\mathcal{H}} + L\ensuremath{\mathcal{L}}.
}
Having one less constraint in the system means we have only one non-trivial Poisson bracket, $\pb{\ensuremath{\mathcal{H}}}{\ensuremath{\mathcal{L}}}$, to work out. It is easily seen to be zero. Thus, our two constraints are first class.
Being first class, $\ensuremath{\mathcal{L}}$ will generate gauge transformations. Because the fundamental Poisson brackets are unchanged, $\ensuremath{\mathcal{L}}$ will still generate the banal transformations
\begin{align}
q^0 &\ensuremath{\rightarrow} q^0 + \epsilon, \ensuremath{\quad\text{and}} \\
a &\ensuremath{\rightarrow} a - \epsilon.
\end{align}
and the Hamiltonian constraint $\ensuremath{\mathcal{H}}$ will behave as it did before. However, the absence of a Mach constraint implies that we no longer have a Machian invariance. Thus, if one takes the terminology seriously, we no longer will have a Machian theory. We will now see why one should take this terminology seriously.
If we work out the equations of motion, we find
\begin{align}
\ensuremath{\dot{q}}^i_I &= \pb{q^i_I}{H_{\text{T}}} = N \frac{p^I_j \eta^{ij}}{m_I}, & \ensuremath{\dot{p}}_i^I &= \pb{p_i^I}{H_{\text{T}}} = -\diby{V}{q^i_I}, \\
\ensuremath{\dot{q}}^0 &= \pb{q^0}{H_{\text{T}}} = N + L, & \ensuremath{\dot{p}}_0 &= \pb{p_0}{H_{\text{T}}} = 0, \\
\ensuremath{\dot{a}} &= \pb{a}{H_{\text{T}}} = - L,\ensuremath{\quad\text{and}} & \dot{\pi} &= \pb{\pi}{H_{\text{T}}} = 0.
\end{align}
The gauge freedom provided by the surviving banal invariance allows us to pick a gauge where $L = 0$. In this gauge, it is manifest that this system is classically equivalent to PPM since the equations of motion are identical and the $a$'s are irrelevant variables due to the banal invariance. Since PPM is really just Newton's theory (this can easily be seen by setting $N=1$) it is clear that we now have a theory containing a background absolute structure for time. This is our first indication that the presence of the Mach constraint, or alternatively a free endpoint variation on the auxiliary fields, is the definition of a Machian theory. Its presence provides a clear criterion to distinguish between a Kretchmanized theory, or generally covariant theory, and a true relational theory. We will see now and in the next sections how this criterion is also good at the quantum level.
Let us briefly examine the canonical quantization of the theory first, showing that fixing the endpoints restores an absolute structure to the quantum theory, then provide a much deeper analysis using path integrals that will lead to the same results but will provide us with more intuition. The quantum mechanical operators and commutators are identical to those of \scn{qupnm}. However, now the wavefunction $\ket{\Psi}$ need only obey the two constraints
\begin{align}
\ensuremath{\left}[ V(\ensuremath{\hat{q}}) + \sum_I \frac{(\ensuremath{\hat{p}}^I_i)^2}{2m_I} \ensuremath{\right}] \ket{\Psi} &= -\ensuremath{\hat{p}}_0 \ket{\Psi},\label{eq:hamfixed}\ensuremath{\quad\text{and}} \\
\ensuremath{\hat{p}}_0 \ket{\Psi} &= \hat\pi \ket{\Psi}. \label{eq:linfixed}\\
\end{align}
\eq{linfixed} says that the operators $\ensuremath{\hat{p}}_0$ and $\hat\pi$ are interchangeable. The banal invariance gives us the freedom to make $\hat{q}^0$ and $\hat a$ interchangeable making the auxiliary fields somewhat redundant. We are left with \eq{hamfixed} which is just the time \emph{dependent} Schr\"{o}dinger equation. This makes it clear, that, in the quantum theory, the time dependence has also been restored. We will see that this is true in much more detail when we explore the path integral quantization of this system.
\subsection{Results and Discussions}
We have seen that time-gauged PPM is classically and quantum mechanically equivalent to JBB theory. The role of the auxiliary fields and the Mach constraint is, classically, to remove $p_0$ from the Hamiltonian constraint of PPM. Quantum mechanically, this leads to the time \emph{independent} Schr\"{o}dinger equation rather then its time \emph{dependent} counterpart. We have worked out the Dirac algebra and found the first class constraints. In a Machian, theory there is a Machian invariance associated with the Mach constraint. If this constraint is lifted, the background structure is restored leaving a theory that is no longer Machian despite still being reparameterization invariant. This suggests to us a criterion for distinguishing true relational theories from reparameterization invariant theories. We will now study in more detail the process through which time disappears from the quantum theory during the gauging process by considering the path integral of time-gauged PPM.
\section{Path Integral Quantization}\label{sec:pathint}
To construct the path integral for time-gauged PPM, which, as we have seen, is a gauge theory, we would like to start from the phase space path integral and apply the procedure of Faddeev and Popov for determining the measure to correctly divide out by the gauge volume \cite{faddeev:fp}. Because of the reparameterization invariance of the action there are some subtleties in how to deal with the gauge invariance but many of these details have been worked out in \cite{sg:emer_time_PI}. We will compare the kernel we get to that obtained from JBB theory and will find it to be identical. Furthermore, we will analyze what happens in the path integral language when the auxiliary fields are fixed on the endpoints. Doing this will introduce a new absolute structure into the theory leading to a kernel that is identical to that of PPM. This will provide further evidence that one can switch between a relational theory and a theory with a background structure by switching between free and fixed endpoints respectively. The path integral does not add new conclusions to the results already derived but it provides much deeper insight into what is going on. Furthermore, it provides a powerful mathematical tool for studying the quantum mechanics of the system.
\subsection{Free Endpoint Time-Gauged PPM}\label{sec:freepnm}
In the phase space path integral, we must integrate over the entire phase space which, in this case, constitutes the spatial coordinates $q^i_I$ for each particle, the time coordinate $q^0$, the auxiliary field $a$, and the momenta conjugate to each of these fields. Because this is a gauge theory, the momenta satisfy constraints. In this case, we have the three constraints derived in \scn{diracpnm}\footnote{An alternative starting point for the free endpoint variation of the auxiliary fields is to remove the Mach constraint but integrate over all possible endpoints for $a$. This method leads to the same result as the method outlined in this section but does not illustrate the role of the Mach constraint which is necessary in the canonical approach. See \apx{A} for the details of the derivation without the Mach constraint.}:
\begin{align}
\ensuremath{\mathcal{H}} &= p_0 + V + \sum_I \frac{(p^I_i)^2}{2m_I} = 0, \\
\ensuremath{\mathcal{L}} &= p_0 - \pi = 0, \ensuremath{\quad\text{and}} \\
\pi &= 0.
\end{align}
To satisfy these constraints we add them to the Hamiltonian with Lagrange multipliers $N$, $L$, and $M$. These constraints restrict the system to lie on a hypersurface of three less dimensions on momentum space. To resolve the ambiguity caused by the many to one map between the $\ensuremath{\dot{q}}$'s and the $p$'s, we must further impose gauge fixing constraints for each of the first class constraints. These constraints must uniquely specify $\ensuremath{\dot{q}}$ for a given $p$ and will constrain the system to lie on a hypersurface of 6 less dimensions on phase space (that is, it will be a phase space with 3 less configuration variables). There will be a map from this hypersurface to the true physical phase space of the system that will induce a measure on the hypersurface. This measure must be invariant under canonical transformations and redefinitions of the constraints. The measure satisfying these requirements, proposed by Faddeev and Popov, is the determinant of the Poisson bracket between the first class constraints and their gauge fixing conditions.
In \scn{gaugefix} to \scn{boundarycond} we will set up the technical details necessary to perform the integrations which reduce the kernel to JBB form. Once these are out of the way, in \scn{evalfree} we will focus on how to carry out the integrations paying special attention to the physical processes involved and the disappearance of time.
\subsubsection{Gauge Fixing Conditions}\label{sec:gaugefix}
Inspired by Hamilton's first equations of motion, derived in \scn{eom}, we propose the following natural gauge fixing conditions
\begin{align}
\ensuremath{\mathcal{G}} &= \sum_I \frac{m_I \ensuremath{\dot{q}}^i_I p^I_i}{p_I^2} - f = 0,\\
\ensuremath{\mathcal{F}} &= \ensuremath{\dot{q}}^0 - g = 0, \ensuremath{\quad\text{and}} \\
\ensuremath{\mathcal{P}} &= \ensuremath{\dot{a}} - h = 0,
\end{align}
where $f$, $g$, and $h$ are \emph{arbitrary} functions on phase space. These constraints will specify a unique $\ensuremath{\dot{q}}$ for a given $p$. The arbitrariness of $f$, $g$, and $h$, represents the gauge freedom inherent in the system. Thus, the action in terms of the physical degrees of freedom should not depend on the choice of these functions. To impose these constraints, we add them to the total Hamiltonian of the system with the Lagrange multipliers $\ensuremath{\mathcal{E}}$, $R$, and $S$ respectively.
\subsubsection{Kernel}
The phase space path integral is now an integration over all possible trajectories in phase space for all possible values of $L$, $M$, $N$, $\ensuremath{\mathcal{E}}$, $R$, and $S$ with a measure given by the Faddeev-Popov determinant. The canonical Lagrangian is
\begin{align}
L &= \sum_I \ensuremath{\dot{q}}^i_I p^I_i + \ensuremath{\dot{q}}^0 p_0 + \ensuremath{\dot{a}} \pi - H_{\text{T}} \notag \\
&= \sum_I \ensuremath{\dot{q}}^i_I p^I_i + \ensuremath{\dot{q}}^0 p_0 + \ensuremath{\dot{a}} \pi - N\ensuremath{\mathcal{H}} - L\ensuremath{\mathcal{L}} - M\pi - \ensuremath{\mathcal{E}}\ensuremath{\mathcal{G}} - R\ensuremath{\mathcal{F}} -S\ensuremath{\mathcal{P}}.
\end{align}
where the $\lambda$ derivatives are taken using the definition $\ensuremath{\dot{q}}^i_I(\lambda) = \frac{q^i_I(\lambda + d\lambda) - q^i_I(\lambda)}{d\lambda}$ (and similar formulas for $\ensuremath{\dot{q}}^0$ and $\ensuremath{\dot{a}}$) so that $L$ is a function of phase space. The kernel, $K$, is a function of the differences $Q^i_I \equiv q^i_I(\lambda_{\text{final}}) - q^i_I(\lambda_{\text{initial}})$ and $\tau \equiv q^0(\lambda_{\text{final}}) - q^0(\lambda_{\text{initial}})$ which are the endpoints of the trajectories in extended configuration space. Because we are performing a free endpoint variation for the auxiliary fields, the Mach constraint should eliminate the dependence of the kernel on any boundary data that could be specified for them. We will see explicitly how this happens later. Collecting all this information, we have
\begin{equation}\label{eq:kernelmaster}
K(Q^i_I, \tau) = \int \ensuremath{\mathcal{D}} q^i_I\, \ensuremath{\mathcal{D}} p^I_i\, \ensuremath{\mathcal{D}} q^0\, \ensuremath{\mathcal{D}} p_0\, \ensuremath{\mathcal{D}} a\, \ensuremath{\mathcal{D}} \pi\, \ensuremath{\mathcal{D}} N\, \ensuremath{\mathcal{D}} M\, \ensuremath{\mathcal{D}} L\, \ensuremath{\mathcal{D}} \ensuremath{\mathcal{E}}\, \ensuremath{\mathcal{D}} R\, \ensuremath{\mathcal{D}} S\, \abs{\pb{\chi_a}{\chi_b}} e^{i \int d\lambda\, L}.
\end{equation}
In the above expression, we have collected all the constraints in the variable $\chi_a$ where $a$ and $b$ range over all the constraints.
\subsubsection{Faddeev-Popov Determinant and Partial Gauge Fixing}
To make the Faddeev-Popov determinant easier to compute and deal with we would like to do a partial gauge fixing. In general, the functions $f$, $g$, and $h$ can be arbitrary functions of phase space but this would make the Faddeev-Popov determinant a horrible mess. Instead, we would like to require that the gauge fixing function $f$ be a function only of the spatial coordinates $q^i_I$ and their conjugate momenta $p_i^I$ while the functions $g$ and $h$ be functions of $\lambda$ only. This allows us to simplify the determinant to the expression
\equa{
\abs{\pb{\chi_a}{\chi_b}} = \abs{\pb{\ensuremath{\mathcal{H}}}{\ensuremath{\mathcal{G}}}} = \abs{\pb{V + \sum_I \frac{p_I^2}{2m_I}}{\sum_J m_J \frac{\ensuremath{\dot{q}}^i_J p^J_i}{p_I^2}-f}}.
}
This is still slightly non-trivial but we will only need to write it down formally as we are not interested in solving the full path integral but, rather, just enough of it to be able to compare it to the kernel of JBB theory.
\subsubsection{Boundary Conditions}\label{sec:boundarycond}
To evaluate the path integral we break up each path integration into a discrete product of integrals. Slicing each trajectory into $N$ pieces plus 2 endpoints, we replace the functional dependence on $\lambda$ by a discrete Greek index that can range from 0 to $N$. Integrations over the Lagrange multipliers are normalized to give $\delta$-functions.
For all the configuration space variables $q^i_{I,\alpha}$, $q^0_\alpha$, and $a_\alpha$, the endpoints are fixed by the boundary conditions. This would normally mean that we have $N-1$ integrations instead of the $N$ integrations necessary for the remaining variables (which include the momenta and their Lagrange multipliers). However, for each of the first class constraints we have gauge fixing functions which eliminate one degree of freedom in configuration space. Thus, for these degrees of freedom, we need a different prescription for fixing the boundary conditions. In this case, instead of just dropping an integration and fixing the endpoints to satisfy the boundary conditions, we keep the extra integration then pick gauge fixing functions that guarantee the boundary conditions are satisfied. Because the gauge fixing functions uniquely specify the configuration space variable in question, it is possible to chose them in such a way that the boundary conditions are satisfied. This will amount to keeping an extra integration over the configuration space variable in question then imposing a constraint on the gauge fixing functions.
In our case, we have 3 gauge fixing functions which correspond to the specification of 3 configuration space degrees of freedom. We will now implement the method described above for each of these 3 variables.
\begin{enumerate}
\item The constraint $\ensuremath{\mathcal{F}}$ tells us that the gauge fixing function $g$ determines the configuration space variable $q^0$. The boundary condition for this variable is simply
\equa{\label{eq:q0const}
\sum_{\alpha = 0}^{N-1} \ensuremath{\dot{q}}^0_\alpha \Delta\lambda_\alpha = q^0(\lambda_{\text{final}}) - q^0(\lambda_{\text{initial}}) \equiv \tau_q.
}
where we use the definition $\ensuremath{\dot{q}}^0_\alpha \equiv \frac{q^0_{\alpha+1}-q^0_{\alpha}}{\Delta\lambda_\alpha}$. It is clear from this that choosing $g_\alpha$ such that
\equa{\label{eq:g0const}
\sum_{\alpha = 0}^{N-1} g_\alpha \Delta\lambda_\alpha = \tau_q
}
will guarantee that (\ref{eq:q0const}) is satisfied. The calculations are greatly simplified by noting that $q^0$ is cyclic (that is, it enters the action only through its derivative $\ensuremath{\dot{q}}^0$). This suggests the change of variables with unit determinant $q^0_\alpha \ensuremath{\rightarrow} q^0_{\alpha+1}-q^0_{\alpha} = \ensuremath{\dot{q}}^0_\alpha \Delta\lambda_\alpha$. Because we are allowing all the variables to vary freely, all integrations are still from $-\infty$ to $\infty$.
\item The constraint $\ensuremath{\mathcal{P}}$ tells us that the gauge fixing function $h$ determines the configuration space variable $a$. Imposing boundary conditions on $a$ may seem odd given the fact that we would like to vary $a$ freely at the endpoints. However, the power of the Mach constraint $\pi=0$ is that, even if we try to impose boundary conditions on $a$, the extra gauge freedom introduced by the arbitrary function $h$ should be enough to eliminate the theory's dependence on them. Thus, to test that the Mach constraint is doing its job we will try to impose the conditions
\equa{\label{eq:aconst}
\sum_{\alpha = 0}^{N-1} \ensuremath{\dot{a}}_\alpha \Delta\lambda_\alpha = a(\lambda_{\text{final}}) - a(\lambda_{\text{initial}}) \equiv \tau_a,
}
where we use the definition $\ensuremath{\dot{a}}_\alpha \equiv \frac{a_{\alpha+1}-a_{\alpha}}{\Delta\lambda_\alpha}$, and see if $\tau_a$ disappears from the final theory. It is clear from the above that choosing $h_\alpha$ such that
\equa{\label{eq:h0const}
\sum_{\alpha = 0}^{N-1} h_\alpha \Delta\lambda_\alpha = \tau_a
}
will guarantee that (\ref{eq:aconst}) is satisfied. Again, we will use the variable substitution $a_\alpha \ensuremath{\rightarrow} a_{\alpha+1} - a_\alpha= \ensuremath{\dot{a}}_\alpha \Delta\lambda_\alpha$ to simplify the integrations.
\item The most difficult boundary condition to implement using gauge fixing functions is the one on the $q^i_I$. This condition will not be needed when comparing the kernel of best-matched PPM to that of JBB theory as we won't need to integrate over all the degrees of freedom in order to compare the two theories. For this reason, we will delay this discussion referring the reader to \cite{sg:emer_time_PI} to see how this can be done.
\end{enumerate}
\subsubsection{Evaluating the Path Integral}\label{sec:evalfree}
Our goal is to show that the kernel (\ref{eq:kernelmaster}) is equivalent to the kernel of JBB theory. In order to compare the two theories, we must integrate (\ref{eq:kernelmaster}) over $\ensuremath{\mathcal{D}} q^0$, $\ensuremath{\mathcal{D}} p_0$, $\ensuremath{\mathcal{D}} a$, $\ensuremath{\mathcal{D}} \pi$, $\ensuremath{\mathcal{D}} M$, $\ensuremath{\mathcal{D}} L$, $\ensuremath{\mathcal{D}} R$, and $\ensuremath{\mathcal{D}} S$. The order in which these integrations are done is irrelevant but we will sketch an option that will attempt to highlight the physical processes occurring in the path integrations.
Solving first for $g_0$ using (\ref{eq:g0const}) and $h_0$ using (\ref{eq:h0const}) we will rewrite the kernel of (\ref{eq:kernelmaster}) for completeness specifying the measure exactly using the discretization:
\begin{multline}\label{eq:longkernel}
K(Q^i_I, \tau_q,\tau_a) = \int_{-\infty}^\infty \frac{d^d p_i^{I,0}}{(2\pi)^d} \frac{dN_0}{2\pi} d \ensuremath{\dot{q}}^0_{0} \frac{dR^0}{2\pi} \frac{d p_0^{0}}{2\pi} \frac{dL_0}{2\pi} d \ensuremath{\dot{a}}_{0} \frac{dS^0}{2\pi} \frac{d \pi^{0}}{2\pi} \frac{dM_0}{2\pi} \times \exp \ensuremath{\left}\{ i \ensuremath{\left}( S_0\tau_a + R_0\tau_q \ensuremath{\right}) \ensuremath{\right}\} \\
\times \exp \ensuremath{\left}\{ i \Delta\lambda_0 \ensuremath{\left}[ \sum_I \ensuremath{\dot{q}}^i_{I,0} p^{I,0}_i + \ensuremath{\dot{q}}^0_0 p_0^0 + \ensuremath{\dot{a}}_0 \pi^0 - N_0\ensuremath{\left}( p_0^0 + V^0 + \sum_I \frac{p_{I,0}^2}{2m_I} \ensuremath{\right}) - L_0(p_0^0 - \pi^0) \ensuremath{\right}.\ensuremath{\right}. \\
\ensuremath{\left}. \ensuremath{\left}. - R^0 \ensuremath{\dot{q}}^0_0 - M_0\pi^0 - S^0\ensuremath{\dot{a}}_0 \ensuremath{\right}] \ensuremath{\right}\} \prod_{\alpha = 1}^{N-1} d^d q^i_{I,\alpha} \frac{d\ensuremath{\mathcal{E}}^\alpha}{2\pi} \frac{d^d p_i^{I,\alpha}}{(2\pi)^d} \frac{dN_\alpha}{2\pi} d \ensuremath{\dot{q}}^0_{\alpha} \frac{dR^\alpha}{2\pi} \frac{d p_0^{\alpha}}{2\pi} \frac{dL_\alpha}{2\pi} d \ensuremath{\dot{a}}_{\alpha} \frac{dS^\alpha}{2\pi} \frac{d \pi^{\alpha}}{2\pi} \frac{dM_\alpha}{2\pi} \\
\times \exp \ensuremath{\left}\{ i \Delta\lambda_\alpha \ensuremath{\left}[ \sum_I \ensuremath{\dot{q}}^i_{I,\alpha} p^{I,\alpha}_i + \ensuremath{\dot{q}}^0_\alpha p_0^\alpha + \ensuremath{\dot{a}}_\alpha \pi^\alpha - N_\alpha\ensuremath{\left}( p_0^\alpha + V^\alpha + \sum_I \frac{p_{I,\alpha}^2}{2m_I} \ensuremath{\right}) - \ensuremath{\mathcal{E}}^\alpha \ensuremath{\left}( \sum_I \frac{m_I \ensuremath{\dot{q}}^i_{I,\alpha} p^{I,\alpha}_i}{p_{I,\alpha}^2} - f_\alpha \ensuremath{\right}) \ensuremath{\right}.\ensuremath{\right}. \\
\ensuremath{\left}. \ensuremath{\left}. - L_\alpha(p_0^\alpha - \pi^\alpha) - R^\alpha (\ensuremath{\dot{q}}^0_\alpha - g_\alpha) -R^0 g_\alpha - M_\alpha\pi^\alpha - S^\alpha (\ensuremath{\dot{a}}_\alpha - h_\alpha) - S^0 h_\alpha \ensuremath{\right}] \ensuremath{\right}\}\times \abs{\pb{\ensuremath{\mathcal{H}}}{\ensuremath{\mathcal{G}}}}.
\end{multline}
Here we have made the coordinate transformations $q^0_\alpha \ensuremath{\rightarrow} q^0_{\alpha+1}-q^0_{\alpha} = \ensuremath{\dot{q}}^0_\alpha \Delta\lambda_\alpha$ and $a_\alpha \ensuremath{\rightarrow} a_{\alpha+1} - a_\alpha= \ensuremath{\dot{a}}_\alpha \Delta\lambda_\alpha$ discussed in \scn{boundarycond}. Though this is a long expression it is important to be explicit about the differences between the $\alpha = 0$ terms and the $\alpha = 1,\hdots,N-1$ terms as these differences contain all the information about the boundary conditions which, in turn, contain all the information about time. In the above expression, we insert the Faddeev-Popov determinant only formally since the partial gauge fixing we have used will make it independent of the integrations we are going to perform in order to bring the kernel into JBB form.
We will perform the integrations in two separate steps. The first step will eliminate the kernel's dependence on $\tau_a$ and the arbitrary functions $h_\alpha$. This must be the case since the Mach constraint $\pi = 0$ is supposed to impose free endpoint conditions. That means, if we try to impose an artificial dependence on some boundary data $\tau_a$, the Mach constraint will be able to use the extra gauge freedom to eliminate this dependence. Indeed, the order of the integrations that we will use will highlight this process explicitly. In the second step, we will see how the gauge freedom introduced by the linear momentum constraint $\ensuremath{\mathcal{F}}$ is sufficient to eliminate the kernel's dependence on $\tau_q$. This is how the corrected coordinate method is successful at eliminating time from the quantum mechanical theory. Even though this process occurs in exactly the same manner as the elimination of the $\tau_a$ dependence, we will use an alternate order for doing the integrals that will highlight the cancelation of the $\tau_q$ dependence explicitly.
\begin{description}
\item[Step 1] First impose the Mach constraint by integrating over $dM_\alpha$ for $\alpha = 0, \hdots, N-1$. This will produce the $N$ $\delta$-functions $\delta(\pi^\alpha)$. Integrating over $d\pi^\alpha$ will then impose the Mach constraint
\equa{\pi^\alpha = 0.}
Once we have done this we can use the gauge fixing functions $h_\alpha$ to fix a gauge for the $\ensuremath{\dot{a}}$'s. To do this we will integrate over $dS^\alpha$ for $\alpha = 1, \hdots, N-1$. Note that here $\alpha$ ranges from 1 to $N-1$ only. That is, we will still have an important integration over $dS^0$. Holding off the $dS^0$ integration for now will give the $N-1$ $\delta$-functions $\delta(\ensuremath{\dot{a}}_\alpha - h_\alpha)$ which imply \equa{\ensuremath{\dot{a}}_\alpha = h_\alpha} after integrating over $d\ensuremath{\dot{a}}_\alpha$.
After setting all the $\pi$'s to zero and the $\ensuremath{\dot{a}}$'s to $h$'s, we can complete the final integrations over $dS^0$ and $d\ensuremath{\dot{a}}_0$. The freedom to vary $\ensuremath{\dot{a}}_0$ arbitrarily is unique to the free endpoint variation. This fact is crucial to the disappearance of the $\tau_a$ dependence of the theory. The remaining terms proportional to $S^0$ are
\equa{
i S^0( \tau_a - \Delta\lambda_0 \ensuremath{\dot{a}}_0 - \sum_{\alpha = 1}^{N-1} \Delta\lambda_\alpha h_\alpha )
}
telling us that the $dS^0$ and $d\ensuremath{\dot{a}}_0$ integrations will impose the condition
\equa{
\Delta\lambda_0 \ensuremath{\dot{a}}_0 = \tau_a - \sum_{\alpha = 1}^{N-1} \Delta\lambda_\alpha h_\alpha.
}
which should be solved for $\ensuremath{\dot{a}}_0$. But, since there are no further $\ensuremath{\dot{a}}_0$ terms because of the disappearance of the $\pi$'s due to the Mach constraint, this is just an interesting but irrelevant fact that no longer affects any of the remaining terms in the kernel. We can see that the freedom to vary the endpoints of the $a$'s coupled with gauge freedom of the $h$'s gives us a theory that, at the end of the day, doesn't depend on the $\tau_a$'s.
\item[Step 2] To see how time is eliminated from the quantum theory we could proceed with the same order of integration as before but substitute the Mach constraint with the linear momentum constraint $\ensuremath{\mathcal{F}}$ and the $\ensuremath{\dot{q}}^0$'s for the $\ensuremath{\dot{a}}$'s. This would lead to the irrelevant condition
\equa{
\Delta\lambda_0 \ensuremath{\dot{q}}^0_0 = \tau_q - \sum_{\alpha = 1}^{N-1} \Delta\lambda_\alpha g_\alpha.
}
for\footnote{See \apx{A} Step 2 for more details of this calculation.} $\ensuremath{\dot{q}}^0_0$. However, it is instructive switch the order of the integrations. In this case, we still impose the linear momentum constraint first by integrating over $dL_\alpha$ followed by $dp^\alpha_0$. This should be done for \emph{all} $\alpha$'s and will force the vanishing of the energy: \equa{p^\alpha_0 = 0.} This will kill the $\ensuremath{\dot{q}}^0_\alpha p^\alpha_0$ terms leaving only the $-R^\alpha \ensuremath{\dot{q}}^0_\alpha$ terms proportional to $\ensuremath{\dot{q}}^0_\alpha$. Thus, the linear momentum constraint has left us with only a single $\ensuremath{\dot{q}}^0_\alpha$ term so if we integrate over $\ensuremath{\dot{q}}^0$ \emph{first} then $R^\alpha$ we will get \equa{R^\alpha = 0.} For $\alpha = 0$ this kills the $\tau_q$ term and many of the $g_\alpha$ terms. For $\alpha \neq 0$, this kills the rest of the $g_\alpha$ terms. This method leaves no doubt that the extra gauge freedom destroys the time dependence of the kernel. The previous method, however, gives us better physical intuition for exactly how the gauge freedom accomplishes this.
\end{description}
After these integrations, we are left with the kernel
\begin{multline}
K(Q^i_I, \tau_q,\tau_a) = K_{\ensuremath{\text{JBB}}}(Q^i_I) = \int \ensuremath{\mathcal{D}} q^i_I \ensuremath{\mathcal{D}} \ensuremath{\mathcal{E}} \ensuremath{\mathcal{D}} p^I_i \ensuremath{\mathcal{D}} N \abs{\pb{\ensuremath{\mathcal{H}}}{\ensuremath{\mathcal{G}}}}
\times \exp \ensuremath{\left}\{ i \int d\lambda \ensuremath{\left}[ \sum_I \ensuremath{\dot{q}}^i_I p^{I}_i -\ensuremath{\right}.\ensuremath{\right}. \\
\ensuremath{\left}. \ensuremath{\left}. N \ensuremath{\left}( V + \sum_I \frac{p_{I}^2}{2m_I} \ensuremath{\right}) - \ensuremath{\mathcal{E}} \ensuremath{\left}( \sum_I \frac{m_I \ensuremath{\dot{q}}^i_I p^I_i}{p_I^2} - f \ensuremath{\right}) \ensuremath{\right}] \ensuremath{\right}\}.
\end{multline}
This is identical to the kernel for JBB theory derived in \cite{sg:emer_time_PI} generalized to many particles.
\subsection{Fixed Endpoint Time-Gauged PPM} \label{sec:fixedpnm}
There are two major differences between the \emph{fixed} endpoint variation and the \emph{free} endpoint variation. The first is that we no longer have the Mach constraint. The mathematical result of this is simply to remove two terms from the Lagrangian: the term proportional to the Mach constraint itself and the term proportion to its gauge fixing constraint. The second major difference is in how the boundary conditions are applied. Because we no longer have gauge fixing functions to fix the values of the auxiliary fields we must specify these the old fashioned way on the boundary. For the moment, this will mean that our kernel will depend on $\tau_a$ and $\tau_q$ though we will see later that, in fact, it will only depend on the combination $\tau_a + \tau_q$ which can be clumped into a single time variable $\tau$. It is instructive to go through the same steps as before to see precisely how the fixed endpoint variation will reintroduce a time dependence into the theory.
\subsubsection{Evaluating the Path Integral}
We can impose the boundary conditions the old fashioned way by removing the integral over $\ensuremath{\dot{a}}_0$ and insert instead its value
\equa{\label{eq:a0val}
\Delta\lambda_0 \ensuremath{\dot{a}}_0 = \tau_a - \sum_{\alpha = 1}^{N-1} \lambda_\alpha \ensuremath{\dot{a}}_\alpha
}
in terms of $\ensuremath{\dot{a}}_\alpha$ and $\tau_\alpha$. Inserting this into the action we start with kernel
\begin{multline} \label{eq:fixedkernel}
K(Q^i_I, \tau_q,\tau_a) = \int_{-\infty}^\infty \frac{d^d p_i^{I,0}}{(2\pi)^d} \frac{dN_0}{2\pi} d \ensuremath{\dot{q}}^0_{0} \frac{dR^0}{2\pi} \frac{d p_0^{0}}{2\pi} \frac{dL_0}{2\pi} \frac{d \pi^{0}}{2\pi} \times \exp \ensuremath{\left}\{ i \ensuremath{\left}( \pi_0\tau_a + R_0\tau_q \ensuremath{\right}) \ensuremath{\right}\} \\
\times \exp \ensuremath{\left}\{ i \Delta\lambda_0 \ensuremath{\left}[ \sum_I \ensuremath{\dot{q}}^i_{I,0} p^{I,0}_i + \ensuremath{\dot{q}}^0_0 p_0^0 - N_0\ensuremath{\left}( p_0^0 + V^0 + \sum_I \frac{p_{I,0}^2}{2m_I} \ensuremath{\right}) - L_0(p_0^0 - \pi^0) - R^0 \ensuremath{\dot{q}}^0_0 \ensuremath{\right}] \ensuremath{\right}\} \\ \prod_{\alpha = 1}^{N-1} d^d q^i_{I,\alpha} \frac{d\ensuremath{\mathcal{E}}^\alpha}{2\pi} \frac{d^d p_i^{I,\alpha}}{(2\pi)^d} \frac{dN_\alpha}{2\pi} d \ensuremath{\dot{q}}^0_{\alpha} \frac{dR^\alpha}{2\pi} \frac{d p_0^{\alpha}}{2\pi} \frac{dL_\alpha}{2\pi} d \ensuremath{\dot{a}}_{\alpha} \frac{d \pi^{\alpha}}{2\pi} \\
\times \exp \ensuremath{\left}\{ i \Delta\lambda_\alpha \ensuremath{\left}[ \sum_I \ensuremath{\dot{q}}^i_{I,\alpha} p^{I,\alpha}_i + \ensuremath{\dot{q}}^0_\alpha p_0^\alpha + \ensuremath{\dot{a}}_\alpha (\pi^\alpha - \pi_0) - N_\alpha\ensuremath{\left}( p_0^\alpha + V^\alpha + \sum_I \frac{p_{I,\alpha}^2}{2m_I} \ensuremath{\right}) - \ensuremath{\right}.\ensuremath{\right}. \\
\ensuremath{\left}. \ensuremath{\left}. \ensuremath{\mathcal{E}}^\alpha \ensuremath{\left}( \sum_I \frac{m_I \ensuremath{\dot{q}}^i_{I,\alpha} p^{I,\alpha}_i}{p_{I,\alpha}^2} - f_\alpha \ensuremath{\right}) - L_\alpha(p_0^\alpha - \pi^\alpha) - R^\alpha (\ensuremath{\dot{q}}^0_\alpha - g_\alpha) -R^0 g_\alpha \ensuremath{\right}] \ensuremath{\right}\}\times \abs{\pb{\ensuremath{\mathcal{H}}}{\ensuremath{\mathcal{G}}}}.
\end{multline}
Notable differences between this kernel and the one of \eq{longkernel} include the disappearance of the $M$ and $S$ terms as well as the $d\ensuremath{\dot{a}}_0$ integration. Instead we have a new $\ensuremath{\dot{a}}_\alpha \pi_0$ term and $S_0$ has been replaced by $\pi_0$ in $e^{iS_0\tau_a}\ensuremath{\rightarrow} e^{i\pi_0 \tau_a}$. The next step is to perform the integrations in the same way we did for the free endpoint variation.
\begin{description}
\item[Step 1] We can no longer impose the Mach constraint, of course, as it no longer exists in the problem nor can we fix a gauge for the $a$'s. We can, however, integrate over the remaining variables needed to bring us to the beginning of Step 2. These are just the $\ensuremath{\dot{a}}_\alpha$'s and the $\pi^\alpha$'s. Because we have no $\ensuremath{\dot{a}}_0$ integration, we will just keep the $\pi_0$ integration for now as it will ultimately correspond to an integration over all energies later. Integrating over the $\ensuremath{\dot{a}}_\alpha$'s first will give
\equa{
\pi^\alpha = \pi^0
}
which leaves us with a kernel of the form
\begin{equation}
K(Q^i_I, \tau_q,\tau_a)_{\text{Step 1, fixed}} = \int \frac{d\pi_0}{2\pi} e^{i\pi_0\tau_a} \int \ensuremath{\mathcal{D}}\mu\, I(Q^i_I, \tau_q)_{\text{Step 1, free}}\exp \ensuremath{\left}\{ i \sum_{\alpha = 0}^{N-1} \Delta\lambda_\alpha L_\alpha\pi^0\ensuremath{\right}\}
\end{equation}
where $\ensuremath{\mathcal{D}}\mu$ is short for the integration measure of \eq{kernelmaster} and $I(Q^i_I, \tau_q)_{\text{Step 1, free}}$ is its integrand after Step 1. That is, we can write the result for the kernel after Step 1 of the \emph{fixed} endpoint variation, $K(Q^i_I, \tau_q,\tau_a)_{\text{Step 1, fixed}}$, explicitly as an integral over the integrand after Step 1 of the \emph{free} endpoint variation, $I(Q^i_I, \tau_q)_{\text{Step 1, free}}$ times some simple exponentials. Note that the $\tau_a$ dependence has been restored.
\item[Step 2] In Step 2, we can proceed exactly as before with our results being only slightly different. Imposing the linear momentum constraint by integrating first over $dL_\alpha$ then $\pi_\alpha$ will give
\equa{
p^\alpha_0 = \pi^0.
}
This can be done for all values of $\alpha$ including $0$. It is why the integration over $\pi^0$ will effectively give an integration over an energy instead of the previous case where $p^\alpha$ was just 0.
By integrating over the remaining variables we will see what happens to the $\tau_q$ dependence. First, integrate over $dR^\alpha$ for $\alpha \neq 0$ then integrate over $d\ensuremath{\dot{q}}^0_\alpha$. This will set
\equa{
\ensuremath{\dot{q}}^0_\alpha = g_\alpha.
}
Finally, if we integrate over $dR_0$ then $d\ensuremath{\dot{q}}^0_0$ we get
\equa{
\Delta\lambda_0 \ensuremath{\dot{q}}^0_0 = \tau_q - \sum_{\alpha = 1}^{N-1} \Delta\lambda_\alpha g_\alpha
}
as an expression for $\ensuremath{\dot{q}}^0_0$. This, however, is no longer an irrelevant fact as we still have a remaining $\ensuremath{\dot{q}}^0$ term proportional to $\pi^0$. Inserting this into our remaining integral and noting that the $g_\alpha$ terms cancel, as they should, in the final kernel, we obtain
\begin{multline}
K(Q^i_I, \tau_q,\tau_a) = \int \frac{d\pi^0}{2\pi} e^{\pi^0(\tau_a + \tau_q)} \int \ensuremath{\mathcal{D}} q^i_I \ensuremath{\mathcal{D}} \ensuremath{\mathcal{E}} \ensuremath{\mathcal{D}} p^I_i \ensuremath{\mathcal{D}} N \abs{\pb{\ensuremath{\mathcal{H}}}{\ensuremath{\mathcal{G}}}}\\
\times \exp \ensuremath{\left}\{ i \int d\lambda \ensuremath{\left}[ \sum_I \ensuremath{\dot{q}}^i_I p^{I}_i - N \ensuremath{\left}( \pi^0 + V + \sum_I \frac{p_{I}^2}{2m_I} \ensuremath{\right}) - \ensuremath{\mathcal{E}} \ensuremath{\left}( \sum_I \frac{m_I \ensuremath{\dot{q}}^i_I p^I_i}{p_I^2} - f \ensuremath{\right}) \ensuremath{\right}] \ensuremath{\right}\}.
\end{multline}
Calling $\pi^0 = -E$ and $\tau = \tau_a + \tau_q$ we get
\equa{
K(Q^i_I, \tau)_{\text{fixed}} = \int \frac{dE}{2\pi} e^{iE\tau} K_{\ensuremath{\text{JBB}}}(Q^i_I, E)
}
which is exactly the expression for the kernel of PPM derived in \cite{sg:emer_time_PI}. It is effectively the Fourier transform of the JBB kernel for energy eigenstates evaluated at a particular time $\tau$. Note that $\tau$ is shifted from what we would expect by trying to apply the boundary condition $\tau_q$. One can simply understand this as the result of having an absolute time defined by the absolute orientation of the $a$'s. This kernel is a solution to the time \emph{dependant} Schr\"{o}dinger equation. Hence, we have demonstrated that fixing the endpoints of the variation of the auxiliary fields reinserts a background structure into the theory from which a time dependence emerges.
\end{description}
\section{Results / Conclusions}
\subsection{Summary of Results}
Our first main result is that time-gauged PPM is equivalent classically and quantum mechanically to JBB theory. More specifically, we showed that the background absolute time structure present in PPM can be removed by applying the corrected coordinate method of Barbour to its time translational invariance. Removing this background structure completely removed the time dependence of the quantum theory. In the canonical approach, the disappearance of time was a result of the Mach constraint killing the energy operator and demoting the time dependent Schr\"{o}dinger equation to the time independent Schr\"{o}dinger equation. In the path integral approach, time disappeared because the extra gauge freedom provided by the Mach constraint allowed us to absorb the kernel's dependence on any boundary conditions associated with time into non-physical degrees of freedom. In both cases, we found that introducing the auxiliary fields $a$ and allowing the endpoints to vary freely was sufficient to eliminate the background absolute structure and render the theory temporally relational.
Our second key result concerned returning the absolute structure to time-gauged PPM by altering the corrected coordinate method to fix the endpoints of the auxiliary fields. More precisely, we found that fixing the endpoints of the $a$'s produced a theory that is classically and quantum mechanically equivalent to PPM. In the canonical framework, we can see the background structure emerge as a result of the disappearance of the Mach constraint leaving us with a time dependent Schr\"{o}dinger equation. In the path integral framework, fixing the endpoints means we don't have enough gauge freedom to absorb the time dependence into non-physical degrees of freedom leaving a true physical dependence on the boundary data.
Collecting these results, we obtain a specific criterion for distinguishing a general reparameterization invariant theory from a a truly relational theory: if there is a symmetry in the action, a \emph{free} endpoint variation of the auxiliary fields associated to that symmetry will generate a relational theory while a \emph{fixed} endpoint variation will restore the background structure.
\subsection{Conclusions / Open Questions}
What conclusions can we draw from these results? If we take best-matching or the corrected-coordinate method (which has been shown to be intimately connected to the gauge principle in \cite{gryb:ym_bm}) as a fundamental principle of Nature then we are led immediately to two choices: 1) we keep the endpoints of the auxiliary fields fixed as if they were standard physical variables, or 2) we allow the endpoints to vary freely admitting no knowledge of their absolute orientation. Note that only the second option properly implements Machian ideas while the first option assumes the existence of absolute structures.
According to the analysis in this paper, choosing option 2) would lead to a proper relational theory \emph{even} if the variable we are making relational is time. This fact provides an alternative to using Jacobi's principle for relational time proposed by Barbour. Admittedly, this approach is not as nice conceptually since there is no obvious way to interpret the action principle as a geodesic principle on configuration space. However, we have shown that it is mathematically equivalent to Jacobi's principle not only classically but also quantum mechanically. In \apx{spatial_sym}, we show that these results are not modified by best-matching the spatial symmetries. Thus, as an alternative to Jacobi's principle, it may not be as conceptually pleasing but, owing to the lack of the square root, it is mathematically cleaner. Furthermore, if our results regarding the free endpoint variation extend to geometrodynamics, as we might expect based on the link between GR and our toy model discussed in the introduction, then the ADM action may have less in common with the action of PPM, as has been often suggested \cite{adm:adm_review,kuchar:prob_of_time}, and more in common with the action of time-gauged PPM or, alternatively, the JBB action. If the variation carried out by ADM is equivalent to a free endpoint variation of the auxiliary fields of time gauged geometrodynamics then one should expect this theory to be temporally relational and suffer from a Problem of Time. The fact that all efforts to study the ADM action have led to a Problem of Time would suggest that this is indeed the case.
Alternatively, we can choose option 1). Our analysis in this paper would suggest that such a choice should lead to a theory with a definite background structure. This raises two interesting questions:
First, is there a way to introduce a background structure into GR? This is not a new question and has been raised most notably by Kucha$\check{\text{r}}$ after noting the similarities between the ADM action and the action of PPM \cite{kuchar:lect_notes, kuchar:prob_of_time}. He conjectured that GR might have a hidden background structure and that solving the Problem of Time would involve finding a way to write the theory in terms of this background structure and quantize it. In light of our results, we can understand why this has yet to be achieved\footnote{For alternative explanations of why these attempts may have failed, see \cite{torre:gr_param}.}. If, as in option 2), the variation of the ADM is equivalent to a free endpoint variation then the theory will be relational. On the other hand, if we choose option 1) we might be able to introduce a background structure in GR in the same way we have done for time gauged PPM. Implementing this choice would involve removing the Mach constraint from time gauged geometrodynamics. We are currently exploring this option and its connection to the ADM action. Indeed, it may be that implementing option 1) and option 2) would lead respectively to the time independent and time dependent Wheeler-DeWitt equations considered by York and Brown \cite{brown:gr_time}. It appears that the main difficulty in deriving the theories of York and Brown using the corrected coordinate method on the dynamic 3-geometry analogue of PPM is in motivating the use of a time parameter that is local in space\footnote{The presence of a \emph{local} time variable is the analogue of the \emph{local} square root in the BSW formulation.}. This is the only significant difference, other than the choice of configuration space, between these toy models and GR.
We are now brought to our second question: what would such a background structure mean in GR? Indeed, what does this background structure even mean in the context of time-gauged PPM? At this point, one could only speculate that this structure could be equivalent to the emergent absolute structures that seem to emerge from relational theories on isolated subsystems of the universe. The absolute frame of rest provided by the fixed stars is a good example of how absolute structures can emerge classically from relational theories. In this case, the local absolute theory is very well approximated by the \emph{fixed} endpoint variation of time-gauged PPM (on and off shell) while the global relational theory is given by the \emph{free} endpoint variation\footnote{One could be more rigorous and consider the global theory to be GR of which time gauged PPM would provide the weak gravity, non-relativistic limit.}. For non-relativistic particle mechanics, the emergence of absolute structures on isolated subsystems can be understood classically \cite{barbour:mach_principle} but quantum mechanically we are still lacking a convincing demonstration of this. The path integral techniques developed in this paper and in \cite{sg:emer_time_PI} are ideally suited to studying this problem. In principle, it should be possible to extend these techniques to geometrodynamics in order to study the Problem of Time in quantum GR.
Can we introduce an absolute structure into geometrodynamics, in accordance with \cite{brown:gr_time}, by varying the auxiliary fields of time gauged geometrodynamics while fixing their endpoints? Would the theory we obtain in this way be emergent from GR, a theory which is fully relational, on isolated subsystems of the universe? These are all interesting open questions that can be answered if the results of this work do indeed extend to geometrodynamics.
\section{Acknowledgements}
I would like to warmly thank Julian Barbour for inviting me to the beautiful College Farm, where I learned much about Mach's principle, and for contributing very helpful comments. I would also like to thank Ed Anderson and Brenden Foster for useful comments on the draft as well as Lee Smolin for advice and guidance. Research at the Perimeter Institute is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MEDT. I also acknowledge support from an NSERC Postgraduate Scholarship, Mini-Grant MGA-08-008 from the Foundational Questions Institute (fqxi.org), and from the University of Waterloo.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}
The origin of stars with masses of 10\,M$_\odot$ or more is a
lively debated issue. Only recently approaches to establish an evolutionary
sequence based on observations and theoretical considerations have been presented
\citep{2007prpl.conf..165B,Zinnecker2007}. However, the observational basis
concerning the earliest phases in the formation process of high-mass stars
remains sparse, as many studies target luminous infrared sources revealing
embedded stellar precursors associated with hot as well as high-density gas.
Younger objects that are sometimes addressed as infrared-quiet massive
cores should be best detected using far-infrared and submillimeter dust
continuum surveys as they are assumed to contain large amounts of cold material.
Furthermore, the identification of extinction features against the galactic
mid-infrared background provides a number of candidate objects commonly
referred to as infrared dark clouds (IRDCs).
Only a relatively small number of candidate sources have been investigated in
detail \citep[recently e.g.][]{2007ApJ...656L..85B,2007ApJ...662.1082R,2007A&A...474..883B}
but also results of larger-scale studies suggest that the lifetimes of massive
prestellar cores are very short \citep{2007A&A...476.1243M}.
The thermal emission from cold dust ($< 20$\,K) peaks in the wavelength range
beyond the IRAS 100\,$\mu$m limit. The 170\,$\mu$m ISOPHOT Serendipity
Survey (ISOSS) \citep{1996A&A...315L..64L,1996A&A...315L..71B,2007A&A...466.1205S}
carried out during the ISO mission \citep{1996A&A...315L..27K} provides the first
large-scale survey in this range and is well-suited to search for extremely young
and massive star-forming regions. Selecting compact 170\,$\mu$m sources
(FWHM $<$ 3.5\arcmin)
that are
associated with an IRAS 100\,$\mu$m point source \citep{1988iras....1.....B} and
molecular gas emission, a sample of more than 50 massive candidate sources has
been identified \citep[see][and references therein]{2003PhDT.........3K,2004BaltA..13..407K}.
The large-scale average dust temperatures derived from the flux
ratios at 170 and 100\,$\mu$m are about 18\,K or lower and the resulting
mass-luminosity ratios are M/L\,$\sim$ 0.6\,M$_\odot$/L$_\odot$, implying early
stages in the process of star formation
\citep[cf.][]{2002ApJ...566..931S} .
In order to explore the physical
conditions in this unique sample of cold and massive star-forming regions and to
search for possible high-mass prestellar cores, a multi-wavelength follow-up
survey has been launched. The presence
of very young massive clumps and cores has been confirmed for the regions
ISOSS J20298+3559 \citep{2003A&A...398.1007K}, ISOSS J18364-0221
\citep{2006ApJ...637..380B} and
ISOSS J23053+5953 \citep{2007A&A...474..883B}. In this
paper we report on observations of four more ISOSS star-forming regions
and extend the study of ISOSS J20298+3559.
Our continuum observations with high spatial resolution covering the
submillimeter to near-infrared wavelength range allow us to characterize
the detailed star-forming content. In the submillimeter we trace
the thermal dust emission and identify compact condensations. To constrain their
evolutionary stage we have conducted mid- and far-infrared observations using
\textit{Spitzer}. The
long-wavelength data in the far-infrared and submillimeter is used to estimate
the dust temperatures and total masses of the clumps. The sensitive
mid-infrared data also reveal very young associated stellar objects.
Additional deep near-infrared images provide a good assessment of source
confusion. For the detected young stellar objects
we perform an evolutionary classification from their infrared colors. In the
Discussion section we further characterize a subset using radiative transfer
models.
\begin{table*}
\begin{minipage}[h]{\hsize}
\caption{Observed ISOSS star-forming regions.}
\label{tab_regions}
\centering
\renewcommand{\footnoterule}{} \begin{tabular}{c c c c c c}
\hline\hline
Region & Assoc. IRAS & RA\,(J2000) & DEC\,(J2000)& Distance\footnote{
Kinematic distances obtained using the galactic rotation model of \citet{1993A&A...275...67B}
and identification of kinematically associated molecular cloud complexes \citep{2003PhDT.........3K}.}
& T$_{\rm FIR\,colour}$\footnote{T$_{\rm FIR\,color}$ is derived from the
far-infrared flux ratios at 100\,$\mu$m (IRAS) and 170\,$\mu$m (ISOSS).} \\
ISOSS\dots & point source & & & in kpc & in K \\
\hline
J19357+1950 & 19335+1944 & 19:35:45.9 & +19:50:58 & 4.0 & 17.5\\
J19486+2556 & 19465+2549 & 19:48:36.8 & +25:56:55 & 2.9 & 18.0\\
J20153+3453 & 20134+3444 & 20:15:20.9 & +34:53:53 & 2.0 & 19.0\\
J20298+3559 & 20278+3549 & 20:29:48.3 & +35:59:24 & 1.8 & 16.0\\
J22478+6357 & 22460+6341 & 22:47:54.1 & +63:57:11 & 4.1 & 15.4\\
\hline
\end{tabular}
\end{minipage}
\end{table*}
\section{Observations, Data Reduction and Analysis}
\subsection{Submillimeter continuum emission}
Submillimeter continuum jiggle maps at 450\,$\mu$m and 850\,$\mu$m were obtained with
SCUBA \citep{1999MNRAS.303..659H} at the James Clerk Maxwell Telescope (JCMT) in
July 2001 and May 2003 under good atmospheric transmission conditions
($\tau_{850\,\mu\mathrm{m}} \lesssim 0.2$). Reduction was done using the
ORAC-DR \citep{1999ASPC..172..171J} and SURF \citep{1998ASPC..145..216J} software.
Photometric calibration is based on maps
of Uranus acquired shortly before or after the observations. Further analysis
was done using the MIRIAD \citep{sault-1995-77} software package and followed
the procedure described in \citet{2001ApJS..134..115S}: To account for the deviations
of the JCMT beam from a single Gaussian we used the Uranus maps to construct
symmetric beam models and deconvolve the maps of the target regions. The derived
beam sizes are 7.8 -- 8.8$\arcsec$ at 450\,$\mu$m and 14.7 -- 15.1$\arcsec$ at
850\,$\mu$m. The maps are then restored using a gaussian of 8$\arcsec$ and 14$\arcsec$,
respectively. By fitting
gaussian components to the restored maps we first derive submillimeter continuum
fluxes and deconvolved source sizes at 450\,$\mu$m. We then use
these sizes convolved with the 14$\arcsec$ gaussian
to
extract 850\,$\mu$m fluxes in order to include emission from the same regions only.
The photometric accuracy is estimated to be 30\% at 450\,$\mu$m and 20\% at 850\,$\mu$m and
the pointing uncertainty of the
submillimeter maps is \mbox{2$\arcsec$ rms}.
\subsection{Near-infrared emission}
Near-infrared images in J, H and Ks were taken with the Calar
Alto 3.5\,m telescope using the two prime focus wide field cameras Omega2000
\citep{2003SPIE.4841..343B} and OmegaPrime \citep{1998SPIE.3354..825B}.
Omega2000 features a field of view (FOV) of $15.4\times 15.4\,\mathrm{arcmin^2}$
with a pixel scale of $0.4496\arcsec\,\mathrm{pix^{-1}}$, while the FOV for
OmegaPrime is $6.8 \times 6.8\,\mathrm{arcmin^2}$ with a pixel scale of $\sim
0.4\arcsec\,\mathrm{pix^{-1}}$. The exposure time in the broad band filters was
20 minutes each. The exposures
were dithered on source to allow for sky subtraction. The reduction and photometry
was done using IRAF \citep{1993ASPC...52..173T} and
GAIA\footnote{http://star-www.dur.ac.uk/\~{}pdraper/gaia/gaia.html} and the
photometric calibration for J, H, and Ks is based on the 2MASS point source catalog.
\subsection{Mid- and far-infrared emission}
The \textit{Spitzer} \citep{2004ApJS..154....1W} observations include IRAC
\citep{2004ApJS..154...10F} imaging in all four photometric bands, MIPS
\citep{2004ApJS..154...25R} imaging at 24\,$\mu$m and 70\,$\mu$m and the MIPS
spectral energy distribution (SED) mode.
For the imaging observations the basic flux calibrated data (BCD) of the
\textit{Spitzer Science Center} (SSC) pipeline was used for further data reduction
and analysis. The calibration uncertainties of the data are about 2\% for IRAC
\citep{2005PASP..117..978R}, 4\% for MIPS 24\,$\mu$m \citep{2007PASP..119..994E}
and 10\% for MIPS 70\,$\mu$m \citep{2007PASP..119.1019G}. Cosmetic
corrections and astrometric refinement was performed with the MOPEX software
\citep{2005PASP..117.1113M} and final images were combined using scripts
in IRAF. Aperture photometry and PSF fitting was done with the
aperture corrections given on the SSC
website\footnote{http://ssc.spitzer.caltech.edu}. The MIPS SED
mode calibration is based on a spectrum of $\alpha$ Boo
\citep{2005ApJ...631.1170L} and the measured MIPS 70\,$\mu$m fluxes. The resulting
photometric accuracy is estimated to be 5\% (IRAC), 10\% (MIPS 24), and 20\% (MIPS
70 and SED).
\section{Observational results: Submillimeter emission morphology
and associated mid-infrared sources}
The five ISOSS regions are listed in Table \ref{tab_regions} and displayed in
Figs.~\ref{fig:19357_maps} to \ref{fig:22478_maps}. In the following
we describe their individual submillimeter morphologies and point out the
associated 24\,$\mu$m sources. On the IRAC and near-infrared maps those sources
for which singular counterparts can be identified at shorter wavelengths are
marked.
\subsection{ISOSS J19357+1950}
In this region three submillimeter emission components are resolved at 450\,$\mu$m,
two adjacent parts (SMM1 North and South) and a south-western part (SMM2)
(Fig.~\ref{fig:19357_maps}). They have deconvolved
FWHM sizes of about 19$\arcsec$, 16$\arcsec$ and 20$\arcsec$ which correspond
to $0.31-0.39$\,pc. An arc-shaped
extended emission feature that stretches from South-east to North-west is
visible at 24\,$\mu$m. Two 24\,$\mu$m point sources are detected associated with
SMM1 North, for SMM1 South and SMM2 one and two sources are present respectively.
Both SMM1 North and South are also coincident with emission
at 70\,$\mu$m but are not resolved individually. At the location of SMM2
no distinct 70\,$\mu$m feature is detected.
\begin{figure*}
\centering
\includegraphics[width=18cm]{f1.eps}
\caption{
Observations of the region ISOSS J19357+1950. The left panel
shows the 24\,$\mu$m map and the overlaid contours show the SCUBA 450\,$\mu$m
emission. The mid panel shows a color composite of the IRAC images. The right
panel is
a JHKs color composite using OmegaPrime and Omega2000 observations.
The circles surround the sources investigated in the Analysis section.
} \label{fig:19357_maps}
\end{figure*}
\subsection{ISOSS J19486+2556}
Three submillimeter clumps are detected in this region: They are located along a
chain
from north-east to south-west and termed SMM1 to SMM3 (Fig.~\ref{fig:19486_maps}).
All three appear compact in the
submillimeter: The deconvolved FWHM diameters are about 16$\arcsec$ (SMM1), 10$\arcsec$ (SMM2) and
15$\arcsec$ (SMM3) at 450\,$\mu$m. The corresponding length scales are $0.14-0.22$\,pc.
Both SMM2 and SMM3 are associated with bright 24 and 70\,$\mu$m sources. In the case
of SMM2, several objects are detected at 24\,$\mu$m and the brightest one
coincides with the submillimeter peak as well as with the 70\,$\mu$m source.
For SMM3, one source at 24 and 70\,$\mu$m is located at the submillimeter peak.
In addition, two areas of extended 24 and 70\,$\mu$m emission are found, one
in the south-east of SMM2 and a second one towards the southern rim of SMM3.
The first one is not associated with submillimeter emission. Only a faint source
at 24\,$\mu$m is detected towards the center of SMM1 and it is not
associated with compact 70\,$\mu$m emission.
\begin{figure*}
\centering
\includegraphics[width=18cm]{f2.eps}
\caption{
Observations of the region ISOSS J19486+2556.
Same arrangement and symbols as in Fig.~\ref{fig:19357_maps}.
} \label{fig:19486_maps}
\end{figure*}
\subsection{ISOSS J20153+3453}
This region contains a single submillimeter clump clearly detected in both SCUBA
bands (Fig.~\ref{fig:20153_maps}). The deconvolved 450\,$\mu$m FWHM extension is about
$18\arcsec\times14\arcsec$ i.e. the
projected diameter of the clump is around 0.16\,pc.
In the 24\,$\mu$m band two point-like sources surrounded by
extended emission are detected towards the north-western limb of the submillimeter
clump. The brighter one is located among a cluster of sources
seen at shorter wavelengths. At 70\,$\mu$m none of these are resolved individually,
they blend into a single slightly extended source.
The submillimeter emission peak is offset from the
brighter point source by 11$\arcsec$.
\begin{figure*}
\centering
\includegraphics[width=18cm]{f3.eps}
\caption{
Observations of the region ISOSS J20153+3453.
Same arrangement and symbols as in Fig.~\ref{fig:19357_maps}.
} \label{fig:20153_maps}
\end{figure*}
\subsection{ISOSS J20298+3559}
This region has been studied in detail in \citet{2003A&A...398.1007K}. Four
submillimeter emission peaks are found in this region: Two of them are joined by
extended emission (SMM1 and SMM2) while SMM3 and SMM4 are offset to the west
(Fig.~\ref{fig:20298_maps}).
SMM4 has not been discussed in \citet{2003A&A...398.1007K}. The sizes of the clumps
SMM1 and SMM3 are approximately 0.14 and 0.17\,pc, SMM2 is unresolved ($< 0.07$\,pc)
and SMM4 is more extended: The deconvolved FWHM diameter is
roughly 39$\arcsec$ corresponding to 0.34\,pc. SMM1, SMM2 and SMM4 are associated
with emission
at 70\,$\mu$m. SMM3 was not covered by our 70\,$\mu$m maps. Two 24\,$\mu$m
point sources are detected towards SMM2 and one towards SMM4. SMM1 and SMM3 have
faint counterparts at 24\,$\mu$m.
\begin{figure*}
\centering
\includegraphics[width=18cm]{f4.eps}
\caption{
Observations of the region ISOSS J20298+3559. The left panel
shows the 24\,$\mu$m map and the overlaid contours show the SCUBA 450\,$\mu$m
emission (for SMM3 the 850\,$\mu$m emission is shown). The mid panel shows a
color composite of the IRAC images. The right panel is
a JHKs color composite using OmegaPrime and Omega2000 observations.
The circles surround the sources investigated in the Analysis section.
} \label{fig:20298_maps}
\end{figure*}
\subsection{ISOSS J22478+6357}
The submillimeter emission in this region traces an elongated clump (SMM1) that
is resolved in an eastern and a western component at 450\,$\mu$m
(Fig.~\ref{fig:22478_maps}). The deconvolved 450\,$\mu$m FWHM extensions of these are
$\sim$\,7$\arcsec$ and $\sim$\,12$\arcsec$, corresponding to roughly 0.14
and 0.24\,pc. At 24\,$\mu$m, several point sources are associated. The brightest
are located at the eastern and to the southern limb of the western submillimeter
peaks. The 70\,$\mu$m map reveals emission that coincides with the submillimeter
peaks and with the 24\,$\mu$m sources.
\begin{figure*}
\centering
\includegraphics[width=18cm]{f5.eps}
\caption{Observations of the region ISOSS J22478+6357.
Same arrangement and symbols as in Fig.~\ref{fig:19357_maps}.
} \label{fig:22478_maps}
\end{figure*}
\section{Analysis}
We first present the analysis of the long-wavelength observations
towards the star-forming regions and then address the associated infrared
sources. The extracted fluxes in the far-infrared and submillimeter are listed
in Table~\ref{tab_lw}. In
Figs.~\ref{fig:19357_seds}, \ref{fig:19486_seds}, \ref{fig:20153_sed},
\ref{fig:20298_seds} and \ref{fig:22478_seds}
the spectral energy distributions (SEDs) of the detected clumps are sketched.
\begin{figure*}
\centering
\includegraphics[width=18cm]{f6.eps}
\caption{
Spectral energy distributions of the submillimeter sources in the
region ISOSS J19357+2556.
The diamonds represent photometric data obtained with MIPS (70\,$\mu$m) and
SCUBA (450 and 850\,$\mu$m) and extrapolated data at 93\,$\mu$m for SMM1 South
and SMM2.
Histogram-like bars show the error range of the MIPS SED spectrophotometry
(53-93\,$\mu$m) for SMM1 North.
The lines show modified blackbody fits for $\lambda \ge 93\,\mu$m using
different dust opacities (solid: no ice mantles, dotted: thick ice mantles,
dashed: thin ice mantles).
For SMM1 South and SMM2 two sets of curves are shown for spectral slopes
of 0.3 and 2.0 in $70 \le \lambda \le 93\,\mu$m.
}\label{fig:19357_seds}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=18cm]{f7.eps}
\caption{
Spectral energy distributions of the submillimeter sources in the
region ISOSS J19486+2556.
The diamonds represent photometric data obtained with
SCUBA (450 and 850\,$\mu$m).
Histogram-like bars show the error range of the MIPS SED spectrophotometry
(53-93\,$\mu$m), for SMM1 upper limits are plotted with triangles.
The lines show modified blackbody fits for $\lambda \ge 93\,\mu$m using
different dust opacities (solid: no ice mantles, dotted: thick ice mantles,
dashed: thin ice mantles).
}\label{fig:19486_seds}
\end{figure*}
\begin{figure}
\centering
\resizebox{6.665cm}{!}{\includegraphics{f8.eps}}
\caption{
Spectral energy distribution of the submillimeter source in the
region ISOSS J20153+3453.
The diamonds represent photometric data obtained with
SCUBA (450 and 850\,$\mu$m).
Histogram-like bars show the error range of the MIPS SED spectrophotometry
(53-93\,$\mu$m).
The lines show modified blackbody fits for $\lambda \ge 93\,\mu$m using
different dust opacities (solid: no ice mantles, dotted: thick ice mantles,
dashed: thin ice mantles).
}\label{fig:20153_sed}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=18cm]{f9.eps}
\caption{
Spectral energy distributions of the submillimeter sources in the
region ISOSS J20298+3559.
The diamonds represent photometric data obtained with MIPS (70\,$\mu$m) and
SCUBA (450 and 850\,$\mu$m), extrapolated data at 93\,$\mu$m for SMM1 and
SMM2 and the datapoint at 1.2\,mm for SMM1.
Histogram-like bars show the error range of the MIPS SED spectrophotometry
(53-93\,$\mu$m) for SMM4.
The lines show modified blackbody fits for $\lambda \ge 93\,\mu$m using
different dust opacities (solid: no ice mantles, dotted: thick ice mantles,
dashed: thin ice mantles).
For SMM1 and SMM2 two sets of curves are shown for spectral slopes
of 0.3 and 2.0 in $70 \le \lambda \le 93\,\mu$m.
}\label{fig:20298_seds}
\end{figure*}
\begin{figure}
\centering
\resizebox{6.665cm}{!}{\includegraphics{f10.eps}}
\caption{
Spectral energy distribution of the submillimeter sources in the
region ISOSS J22478+6357.
The diamonds represent photometric data obtained with MIPS (70\,$\mu$m) and
SCUBA (450 and 850\,$\mu$m) and extrapolated data at 93\,$\mu$m for SMM1 West.
Histogram-like bars show the error range of the MIPS SED spectrophotometry
(53-93\,$\mu$m) for SMM1 East.
The lines show modified blackbody fits for $\lambda \ge 93\,\mu$m using
different dust opacities (solid: no ice mantles, dotted: thick ice mantles,
dashed: thin ice mantles).
For SMM1 West two sets of curves are shown for spectral slopes
of 0.3 and 2.0 in $70 \le \lambda \le 93\,\mu$m.
}\label{fig:22478_seds}
\end{figure}
\subsection{Far-infrared spectral slopes}
The examination of the SEDs compiled for the seven detected clumps towards which
MIPS SED observations have been performed shows that the spectral slopes at
wavelengths around 70\,$\mu$m and below do not resemble a single thermal emission
component that would reproduce the submillimeter fluxes. It presumably originates
from warmer dust components or transiently heated very small grains
\citep[cf.][]{2007A&A...474..883B,2005MNRAS.356..810R}. The spectral slopes
(${\rm d}\log(\lambda {\rm F_\lambda}) /{\rm d}\log\lambda$) vary in the range
$0.3 - 2.0$ for 70\,$\mu$m $<\lambda<$ 94\,$\mu$m.
\subsection{Clump dust temperatures and masses}
To characterize the cold component of dust (and gas) that gives rise to the
emission at long wavelengths, we assume that it can be reproduced by isothermal
dust emission. As noted above, the fluxes at wavelengths around 70\,$\mu$m and
below do not conform with this assumption and we therefore use the datapoints at
93, 450 and 850\,$\mu$m to estimate the dust temperature.
In that respect we also assume that the emission at
these wavelengths is optically thin and can be approximated by a modified Planck
spectrum.
We use the dust opacities given in \citet{1994A&A...291..943O} for a MRN distribution
either without ice mantles, with thin and with thick ice mantles
($\kappa_{850{\rm \mu m}}=0.6-1.4\,\rm{cm^2/g}$) corresponding to
emissivity indices of $\beta=1.81-1.85$ at long wavelengths. The dust-to-gas
mass ratio is assumed to be 1/100. In the cases where no MIPS SED measurements
are available, we extrapolate the 70\,$\mu$m flux to 93\,$\mu$m allowing for
spectral slopes of 0.3 and 2.0 (adopting the measured extremes).
We list the resulting
range of dust temperatures and gas masses for each clump in Table~\ref{tab_lw}.
The fitted curves are shown in Fig.~\ref{fig:19357_seds}, \ref{fig:19486_seds},
\ref{fig:20153_sed}, \ref{fig:20298_seds} and \ref{fig:22478_seds}.
\begin{table*}
\begin{minipage}[h]{\hsize}
\caption{Long-wavelength emission and derived properties of the cold component
of the detected clumps.}
\label{tab_lw}
\centering
\renewcommand{\footnoterule}{} \renewcommand{\thempfootnote}{\alph{mpfootnote}}
\renewcommand{\thefootnote}{\alph{footnote}}
\begin{tabular}{c c c c c c c c c c}
\hline\hline
Region & Clump & \multicolumn{4}{c}{Total flux (Jy)} & Size (pc) & Dust temperature (K) & Gas mass (M$_\odot$) \\
ISOSS\dots & & 70\,$\mu$m & 93\,$\mu$m & 450\,$\mu$m & 850\,$\mu$m & & \\
\hline
J19357+1950 & SMM1 N & 3.3 & 6.0 & 2.1 & 0.40 & 0.37 & $16.6-18.7$ & $54-92$\\
& SMM1 S & 0.57 & $0.80-1.3$\footnotemark[1] & 1.0 & 0.21 & 0.31 & $14.1-16.5$ & $34-66$\\
& SMM2 & 0.21 & $0.30-0.47$\footnotemark[1] & 1.9 & 0.35 & 0.39 & $12.0-13.7$ & $85-166$\\
J19486+2556 & SMM1 & $<0.15$ & $<0.26$ & 2.2 & 0.51 & 0.22 & $11.6-12.5$\footnotemark[2] & $69-123$\footnotemark[2]\\
& SMM2 & 2.7 & 4.6 & 2.1 & 0.38 & 0.14 & $16.1-18.1$ & $28-49$\\
& SMM3 & 7.4 & 11 & 4.1 & 0.59 & 0.21 & $16.8-19.0$ & $46-79$\\
J20153+3453 & SMM1 & 13 & 17 & 12 & 2.1 & 0.16 & $15.3-17.0$ & $87-149$\\
J20298+3559 & SMM1 & 1.7 & $2.5-3.7$\footnotemark[1] & 1.3\footnotemark[3] & 0.23\footnotemark[3] & 0.14\footnotemark[3] & $15.8-18.8$\footnotemark[4] & $6-13$\footnotemark[4]\\
& SMM2 & 1.1 & $1.6-2.4$\footnotemark[1] & 0.31\footnotemark[3] & 0.091\footnotemark[3] & $<0.07$\footnotemark[3] & $17.3-21.3$ & $2-3$\\
& SMM4 & 0.41 & 0.81 & 3.9 & 1.0 & 0.34 & $12.1-13.2$ & $46-80$\\
J22478+6357 & SMM1 E & 0.80 & 1.6 & 1.0 & 0.35 & 0.14 & $14.7-16.4$ & $47-81$\\
& SMM1 W & 0.72 & $1.0-1.6$\footnotemark[1] & 1.8 & 0.47 & 0.24 & $13.4-15.5$ & $79-153$\\
\hline
\end{tabular}
\footnotetext[1]{Extrapolated values.}
\footnotetext[2]{Derived using upper limit fluxes.}
\footnotetext[3]{Flux values and sizes from \citep{2003A&A...398.1007K}.}
\footnotetext[4]{Derived from listed fluxes and 0.075\,Jy at 1.2\,mm
\citep{2003A&A...398.1007K}.}
\end{minipage}
\end{table*}
\subsection{Associated infrared sources}
We detect a number of sources at 24\,$\mu$m and shorter wavelengths that are
associated with the submillimeter clumps. To evaluate their nature and
relationship with the submillimeter condensations we select the sources
marked in the mid and right panels of Fig.~\ref{fig:19357_maps} to
\ref{fig:22478_maps} for which we can identify singular counterparts. At
distances of
$2-4$\,kpc the resolution achieved in the Ks band corresponds to $1600-3200$\,AU
and we can only probe the confusion of sources on scales larger than these. For
sources that show no near-infrared counterpart the IRAC resolution at
3.6\,$\mu$m which corresponds to $3000-6000$\,AU limits the search for
multiplicity. In addition we check for optical counterparts in Second Digitized
Sky Survey DSS2-red maps. Sources without optical counterpart are best
candidates to represent embedded objects and therefore the association with the
molecular and dusty clumps is presumably no projection effect.
However, the actual positions of the sources with respect to the clumps
and the foreground extinctions are unknown.
\citet{2006ApJS..167..256R} present a grid of young stellar objects (YSOs)
model SEDs which has been used to analyze mid-infrared sources
detected with \textit{Spitzer} in star-forming regions
\citep{2007ApJ...666..321I,2007ApJ...669..464S}. Utilizing a
Monte-Carlo radiative transfer code, more than 20\,000 2D models were computed
for ten inclinations each.
The different evolutionary phases in the modelled parameter space are
ordered according to the Class system \citep{1994coun.conf..179A} into stages:
Stage 0/I (hereafter: Stage I) are very young objects that are deeply embedded
in an extended envelope with a cavity surrounding an accretion disk, Stage II
contains disk-dominated models and Stage III represents star-dominated systems.
Via a web interface measured fluxes of individual sources can be put into a
model SED fitting routine to constrain source parameters \citep{2007ApJS..169..328R}
and we make use of this below in the Discussion section.
The color-color diagram [3.6]-[5.8] versus [8]-[24]
(Fig.~23 in \citet{2006ApJS..167..256R})
offers a good distinction between the stages. These colors have also been
compiled in \citet{2004ApJS..154..385R} for a number of young stellar objects in
the Elephant Trunk Nebula which have been classified before according to the Class
system \citep[see][and references therein]{2004ApJS..154..385R}. For a
verification of the color assessment we plot their data
and the regions corresponding to the Stages proposed by \citet{2006ApJS..167..256R}
in the upper left panel of Fig.~\ref{fig:irccds}.
The foreground extinction appears to be small for these sources.
The Class I/0 and I sources fall into the corresponding Stage I region
with one exception. The colors of Class I/II and II sources lie in the Stage II region
with one exception. The colors of the debris disk object $\beta$~Pictoris
place it on the border between the Stage II and III regions. These accordances
support the utility of the colors for classification and we apply this method to
our source samples
to get an estimate of their evolutionary status. However, due to the presence of
a significant amount of dust in the regions, an enhanced extinction is expected
for our sources.
In the remaining five panels of Fig.~\ref{fig:irccds} the infrared colors of
the sources marked (and numbered) in Fig.~\ref{fig:19357_maps} to
\ref{fig:22478_maps} are plotted. Sources without optical counterpart in DSS2-red
maps are marked with crosses. Most of the evaluated
sources exhibit colors lying in the Stage II region or in the vicinity of the
Stage I/II transition. Interestingly, a few sources show very red colors that
are consistent with Stage I even if a foreground extinction of
A$_{\rm V} \ge 100$ is assumed. Exhibiting \mbox{[8]-[24] $> 5$}, these sources are
distinct from the sample in the upper left panel.
Two red sources, Source 1 in ISOSS J19486+2556 and Source 6 in ISOSS J22478+6357,
apparently show enhanced 5.8\,$\mu$m emission but still lie within the Stage I
region. Both are very faint in the near-infrared.
Several sources are associated with ``fuzzy'' green features in the IRAC composite
images and/or extended emission features in the near-infrared:
Source 2 in ISOSS J19486+2556, Source 2 in ISOSS J20153+3453, Source 8 in ISOSS
J20298+3559 and also towards the SMM3 peak in ISOSS J19486+2556, where we cannot
identify the appropiate counterparts to the detected 24\,$\mu$m source, and
towards the SMM1 peak in ISOSS J20298+3559, where the 24\,$\mu$m source shows
no counterparts at shorter wavelengths. These features likely trace outflow
activity from young accreting objects which gives rise to shock-excitation
of molecular hydrogen and CO. However, also atomic hydrogen Br\,$\alpha$ line
emission can contribute to the IRAC 4.5\,$\mu$m flux.
The most prominent features in the IRAC composite image of ISOSS J19486+2556 are
two extended red emission features east of SMM2 and south of SMM3 which are also
visible at the MIPS wavelenghts. In the near-infrared very red objects are detected
that are located towards the centers of the features which themselves are not
traced. Extended low-density dust heated to higher temperatures could give rise
to the mid- and far-infrared emission.
In the panel of Fig.~\ref{fig:irccds} corresponding to ISOSS J20298+3559
we also plot the approximate colors of the brightest source in the region,
located western of SMM1. It has been investigated in detail in
\citet{2003A&A...398.1007K} and classified as Herbig B2e star
that is accreting from a disk which is optically thick. The 8 and 24\,$\mu$m
images seem to show non-linear effects due to the brightness of the source which
may affect the derived colors corresponding to Stage II.
\begin{figure*}
\centering
\includegraphics[width=18cm]{f11.eps}
\caption{
IRAC and MIPS infrared color-color diagram.
The regions where most models are Stage I, II and III according to
\citet{2006ApJS..167..256R} are shaded from dark to light grey.
The vectors show an extinction of A$_V$=20 \citep{2005ApJ...619..931I}.
In the upper left panel young stellar objects corresponding to different
evolutionary stages from \citet{2004ApJS..154..385R} are plotted.
In the five remaining panels the mid-infrared sources
associated with submillimeter emission in the ISOSS star-forming regions are
drawn (diamonds, numbering as in Figs.~\ref{fig:19357_maps}--\ref{fig:22478_maps}).
The sources without optical counterpart are marked by crosses.
In the panel corresponding to ISOSS J20298+3559 the approximate colors of
the Herbig B2e star are represented by the triangle.
}\label{fig:irccds}
\end{figure*}
\section{Discussion}
\subsection{The clump population}
Our simple approach to reproduce the long-wavelength emission towards the
sample of submillimeter clumps by one component neglects any effects of the
density and temperature distributions within the clumps and the assumption of
optically thin
emission may be violated in the far-infrared if the (column) density is high
enough. Assuming a density profile that is flat in the innermost region of
100\,AU radius, follows a power-law (${\rm \rho \propto r^{-\alpha}}$
with $\alpha$ = 1.5) in between and is cut-off exponentially at 20\,000\,AU,
an optical depth of unity is reached
at about 100\,$\mu$m for a 100\,M$_\odot$ condensation (dust opacities from
\citet{1984ApJ...285...89D}; Steinacker, private communication). Therefore
we may underestimate the total emission at the shortest wavelengths used
(93\,$\mu$m) and also the mean dust temperatures. Setting
the dust temperature to a canonical value of 20\,K results in a decrease of the
mass by up to a factor of about two for the
coldest temperatures, but would require high optical depths at wavelengths
around 100\,$\mu$m to be consistent with the measured fluxes.
However, the required mass scales similarly to the optical depth.
It seems therefore
unlikely that this effect causes errors in the mass estimation of more than
a factor of two.
The distances to the regions are mainly derived from radial velocities and
carry therefore some uncertainty. In \citet{2003A&A...398.1007K} the distance to
ISOSS J20298+3559 has been constrained by an extinction study to
$1.8 \pm 0.3$\,kpc. We
estimate the other distances to be reliable within $\sim 1$\,kpc which
results in possible mass changes by factors of 1.3 to 1.8.
Another uncertainty factor of the order of two for the mass estimate arises
from the unknown dust
composition, e.g. the dust opacities given by \citet{1994A&A...291..943O} are
about five times higher than those used for the calculation
of the model SED grid \citep{2003ApJ...591.1049W}.
It is obvious from the SED diagrams that the used dust models in combination with
the one-component approach does not fit the data well for all sources: In the
cases of SMM2 in J20298+3559 and SMM1 East in J22478+6357 it seems that
dust models corresponding to a lower value of the emissivity index $\beta$ at
long wavelengths or the introduction of several emission components would
improve the fit. However, the latter approach would increase the number of
variable parameters for the fit above the number of datapoints used. From the
available measurements in the submillimeter range we cannot place constraints on
the dust properties.
The derived dust temperatures (see Table~\ref{tab_lw}) lie between 11.6 and
21.3\,K, approximately in the same range
as the far-infrared color temperatures from ISO and IRAS fluxes of the whole
regions (see Table~\ref{tab_regions}).
In comparison to other studies, the values are consistent with those of cold dust
components derived and assumed for young dense clumps
\citep[e.g.][]{2007ApJ...656L..85B,2005ApJ...634L..57S,2007ApJ...666..982E}.
The estimated masses that reside in these cold dust and gas components range
from 2\,M$_\odot$ up to 166\,M$_\odot$.
Taking the mass estimates at face value, four of the clumps may be massive
enough ($> 100$\,M$_\odot$) to form a high-mass star and an accompanying
cluster, if one assumes that a fraction of 10\% of the total mass ends up in
massive stars \citep{Zinnecker2007}. The star formation efficiency is
observationally not well constrained on the scales of individual clumps.
From the source sizes in Table~\ref{tab_lw} volume-averaged densities
have been derived for our clump sample under the assumption of spherical symmetry.
The resulting densities are ${\rm 4\cdot10^4\,cm^{-3} < n_{H_2} < 10^6\,cm^{-3}}$ with
a median value of ${\rm 2\cdot10^5\,cm^{-3}}$.
Compared to studies of dense condensations found in IRDCs
\citep[][median density ${\rm 3\cdot10^4\,cm^{-3}}$]{2006ApJ...641..389R}
and in Cygnus~X
\citep[][median density ${\rm 2\cdot10^5\,cm^{-3}}$]{2007A&A...476.1243M}
the evaluated clumps seem to be more similar to the latter sample based on a
millimeter continuum survey.
Both the mentioned samples seem to contain candidate objects
representing very early stages of massive star formation
(e.g. massive infrared-quiet cores).
For the sample of \citet{2002ApJ...566..945B} based on IRAS measurements and
containing luminous infrared sources a lower mean density of
${\rm 9\cdot10^3\,cm^{-3}}$ is recalculated in \citet{2007A&A...476.1243M}.
The direct comparison of these densities may be affected by systematically
different derivation methods, but it suggests that the source selection
based on longer wavelengths gives access to presumably less evolved sources.
In the present case the 170\,$\mu$m band measurements constrain the average dust
temperature to lower values than those that can be inferred from the IRAS bands.
\subsection{The nature of the 24\,$\mu$m sources}
The majority of the clumps appear in a stage where 24\,$\mu$m sources are present.
In the following we discuss their ability to generate high-mass star progenitors
and evaluate selected subsamples in more detail.
During their formation high-mass stars develop high luminosities: According to
the approach of \citet{2003ApJ...585..850M} values above
100\,L$_\odot$ are reached already when one solar mass has been accreted.
The corresponding timescale is on the order of several 10$^4$\,years.
Therefore young high-mass stars should stand out as luminous sources.
However, due to the reprocessing of the radiation at short wavelengths by the
surrounding dusty envelopes only a fraction of the luminosity is released
at the infrared wavelengths we probe (here we consider $\lambda \le 24$\,$\mu$m).
If the source is deeply
embedded, we will expect its infrared colors to be very red and the emission at
optical wavelengths to be not detectable. By combining the classification from
the colors with an estimate of the total luminosity including the far-infrared
and submillimeter data we therefore get a handle on the
probability of a source to be a high-mass star precursor: Sources of Stage II
or later should generate high luminosities to be considered candidate massive
young stellar objects.
We consider the sources' locations
in the infrared color-color space as qualitatively describing their evolutionary
stages. From the sample of 24\,$\mu$m sources that has been evaluated in the Analysis
section the majority is estimated to be comparable to Stage II models. As the
majority of the associated clumps does not substantially exceed a
total luminosity of 200\,L$_\odot$, we consider most of these objects as being
rather low-mass YSOs. Their association with the submillimeter clumps suggests
that they have formed inside and that the clumps contain density substructures.
\subsubsection{Very red protostar candidates}
As described above, sources that display very red infrared colors and sources
that are not detected in the shorter-wavelength bands are best candidates to
represent very young objects that may evolve to massive YSOs. Using
Fig.~\ref{fig:irccds} we select Source 5 in ISOSS J19357+1950, Source 1 in ISOSS
J19486+2556 and Source 2 in ISOSS J20153+3453. The remaining sources with
\mbox{[8]-[24] $> 5$} are not associated with massive submillimeter clumps and
probably will not be able to accrete enough material to end up as high-mass
stars. Furthermore, in \citet{2003A&A...398.1007K} the submillimeter source SMM1
in ISOSS J20298+3559 is found to be a candidate Class 0 object. We detect a
24\,$\mu$m source towards this peak which is not seen at shorter wavelengths.
Associated outflow-tracing emission supports the classification as young
protostellar object.
Source 5 in ISOSS J19357+1950 is located towards the massive clump SMM2. We
apply the online model fitter using the
fluxes including upper limits in the H band, at 70, 450 and 850\,$\mu$m.
Source 2 found towards the massive clump ISOSS J20153+3453 SMM1
exhibits very red colors and is associated with a possible outflow.
The online model fitter is fed with the fluxes including upper limits in the
J band, at 70, 450 and 850\,$\mu$m.
For SMM1 in ISOSS J20298+3559 we input the fluxes measured at 24, 70, 450 and
850\,$\mu$m and upper limits in the IRAC bands and in the Ks band.
The range of possible foreground extinction is selected to ${\rm A_V \le 500}$.
The fitting results for these three sources are listed in Table~\ref{tab_ysos1}.
From the inspection of the SED plots we establish a cut-off value of
$\chi^2_{\rm d}$ ($\chi^2$ per datapoint) to select the models that fit the
datapoints within reasonable margins.
We also give the total extinction to the central source by combining the fitted
foreground extinction with the extinction along the line of sight from the
model outer radii inwards.
These parameters are degenerate to some extent and allow models with
different envelope sizes to be fitted via compensating foreground extinctions.
The results are consistent with the interpretation
of the sources as young embedded stellar precursors.
The observed outflow activity signatures accord with high accretion rates for
the latter two sources. In the cases of ISOSS J19357+1950 SMM2 and
ISOSS J20153+3453 SMM1 the estimated large total masses of around
100\,M$_\odot$ residing in cold components of dust and gas would in principle
allow ongoing accretion and the build-up of high-mass stars.
However, due to the multiplicity of
near-infrared sources that are detected in the vicinity of ISOSS 20153+3453 SMM1
the further evolution of this object is not straightforward. Towards
ISOSS J19357+1950 SMM2 there also is a second more evolved source nearby.
In the third case the mass estimate of about 10\,M$_\odot$ for
ISOSS J20298+3559 SMM1 is consistent with the source being an intermediate-mass
star precursor.
The remaining very red candidate Stage I source located towards the clump SMM1
in the region ISOSS J19486+2556 is particularly interesting
because of the low infrared luminosity in comparison to the submillimeter,
making it a candidate for a young and deeply embedded object.
We input the photometric data into the online model fitter including upper
limits in the Ks band, at 70, 450 and 850\,$\mu$m. We choose a cut-off at
$\chi^2_{\rm d} < 49.6$ to select the acceptable fits. The results
for this source are ambiguous: Among the selected models are several belonging
to the Stage I category with central source masses of about 0.1\,M$_\odot$ but
also a majority of Stage II models with central source masses between 3 and
4.5\,M$_\odot$, both with and without envelopes. There is no significant
difference in the fitted total extinction towards the central sources. We
therefore cannot assess and constrain the parameters of this source.
\begin{table*}
\begin{minipage}[h]{\hsize}
\caption{Results of the SED fitting for three very red young stellar objects.}
\label{tab_ysos1}
\centering
\renewcommand{\footnoterule}{} \renewcommand{\thempfootnote}{\alph{mpfootnote}}
\renewcommand{\thefootnote}{\alph{footnote}}
\begin{tabular}{l c c c}
\hline\hline
Region ISOSS\dots & J19357+1950 & J20153+3453 & J20298+3559\\
Source & 5 & 2 & -\\
Associated clump & SMM2 & SMM1 & SMM1\\
\hline
Selected cut-off & $\chi^2_{\rm d} < 14.2$ & $\chi^2_{\rm d} < 9.3$ & $\chi^2_{\rm d}<12.5$\\
Fitted models & Stage I & Stage I & Stage I\\
Total extinction & ${\rm A_V\sim85}$ & ${\rm A_V\sim100}$ & ${\rm A_V > 200}$\\
Central mass (M$_\odot$) & $0.2-2.1$ & $0.7-2.3$ & $0.2-4.6$\\
Envelope accretion & & &\\
\quad rate (M$_\odot$/yr) & $1.6\cdot10^{-6}-8.9\cdot10^{-5}$ & $1.5\cdot10^{-6}-3.4\cdot10^{-4}$ & $4\cdot10^{-6}-7\cdot10^{-4}$\\
\hline
\end{tabular}
\end{minipage}
\end{table*}
\subsubsection{Evolved young stellar objects}
We also detect sources whose SEDs are dominated by emission at mid-infrared
wavelengths which exhibit higher total luminosities and may represent
more massive YSOs. By comparing the spectral energies at 24\,$\mu$m and
70\,$\mu$m we find two candidate sources, located towards SMM4 in
ISOSS J20298+3559 (Source 8) and SMM1 East in ISOSS J22478+6357 (Source 3).
We also use the fluxes from \citet[IRS 6]{2003A&A...398.1007K} for the I, J, H
and Ks bands as well as upper limits at 70, 450 and 850\,$\mu$m for the former
and upper limits in the J band, at 70, 450 and 850\,$\mu$m for the latter source.
The fitting results are compiled in Table~\ref{tab_ysos2}. We consider the first
source to be an embedded proto- or pre-main-sequence
star of intermediate mass as also suggested by \citet{2003A&A...398.1007K}.
It probably has been formed in the SMM4 clump. The results for the second source
are consistent with the source representing an evolved YSO of
at least intermediate mass that is still embedded and presumably has been formed
within the clump SMM1 East.
\begin{table}
\begin{minipage}[h]{\columnwidth}
\caption{Results of the SED fitting for two evolved young stellar objects.}
\label{tab_ysos2}
\centering
\renewcommand{\footnoterule}{} \renewcommand{\thempfootnote}{\alph{mpfootnote}}
\renewcommand{\thefootnote}{\alph{footnote}}
\begin{tabular}{l c c}
\hline\hline
Region ISOSS\dots & J20298+3559 & J22478+6357\\
Source & 8 & 3\\
Associated clump & SMM4 & SMM1 East\\
\hline
Selected cut-off & $\chi^2_{\rm d}<3.3$ & $\chi^2_{\rm d}<2.1$\\
Fitted models & Stage II\footnotemark[1] & Stage II\footnotemark[2]\\
Total extinction & ${\rm 6<A_V<9.4}$ & ${\rm 29<A_V < 41}$\\
Total luminosity (L$_\odot$) & $\sim 150$ & $\sim 2000$\\
Central mass (M$_\odot$) & $3.2-4.9$ & $6-8.5$\\
Preferred inclination & low & -\\
Disk mass (M$_\odot$) & $4\cdot10^{-6}-10^{-1}$ & $10^{-5}-10^{-1}$\\
Accretion rate (M$_\odot$/yr) & $10^{-11}-10^{-6}$ & $5\cdot10^{-11}-2\cdot10^{-5}$\\
System age (yr) & $10^6-10^7$ & $10^6 - 6\cdot10^6$\\
\hline
\end{tabular}
\footnotetext[1]{M$_{\rm disk}$/M$_*\ge7.7\cdot10^{-7}$}
\footnotetext[2]{M$_{\rm disk}$/M$_*\ge9.5\cdot10^{-7}$}
\end{minipage}
\end{table}
\section{Conclusions}
We have analysed multi-wavelength observations of five star-forming regions
that were identified using the ISOPHOT Serendipity Survey at 170\,$\mu$m.
From the discussed results we infer:
\begin{enumerate}
\item We found one to four compact ($\sim 0.2$\,pc) submillimeter
condensations in
every region that represent molecular clumps containing a cold component
of gas and dust. The dust temperature estimates vary between 11.6 and
21.3\,K and accord with the large-scale color temperatures measured in the
far-infrared.
\item The resulting estimated clump masses that reside in these cold components
range from 2 to 166\,M$_\odot$. Four out of twelve clumps may be massive enough
($> 100$\,M$_\odot$) to be promising candidate birthplaces for high-mass stars.
\item We identify multiple associated mid-infrared sources for the majority
of the clumps suggesting that they embody significant density substructures
on smaller scales than probed by the submillimeter observations ($\gtrapprox 0.1$\,pc).
Since the emission of most clumps is dominated by
the cold material we expect thriving star formation from further collapse
of overdensities.
\item Most of the associated sources are considered as low-mass young stellar
objects in an evolutionary state later than Class 0/I partly embedded in the
clumps where they have formed.
\item In the region ISOSS J19357+1950 we find a massive clump of around
$100$\,M$_\odot$. A probably deeply embedded accreting Class 0/I protostar of
$\sim 1$\,M$_\odot$ is detected adjacent to a more evolved young stellar object.
Similarly, in ISOSS J20153+3453 one massive clump of about $100$\,M$_\odot$
is present and a cluster
of sources is detected in the vicinity containing a presumably deeply embedded
accreting protostar with a current mass of $\sim 1.5$\,M$_\odot$. Its
evolutionary state corresponds to Class 0/I and there is evidence
for outflow activity from this object.
\item The source SMM1 in ISOSS J20298+3559 probably represents a young
accreting intermediate-mass star precursor embedded in a molecular clump of
$\sim 10$\,M$_\odot$. Towards SMM4 in the same region we identify an embedded
proto- or pre-main-sequence star of $3.2-4.9$\,M$_\odot$ that likely evolved
from the associated clump of $\sim 60$\,M$_\odot$.
\item In the region ISOSS J22478+6357 a candidate protostar
of $6-8.5$\,M$_\odot$ is found that is embedded in the associated
SMM1 East clump ($\sim 60$\,M$_\odot$).
\item We do not detect stellar precursors that could have current masses
of 10\,M$_\odot$ or more. Therefore we cannot be sure whether such objects
will emerge within the identified clumps. However, the presence of
intermediate-mass proto- or pre-main-sequence star candidates and the
large clump masses indicate that one can expect that also high-mass stars are formed
in these systems.
\item Our study reveals that the search for regions containing large fractions
of cold material in the far-infrared has successfully identified star-forming
regions associated with cold and massive clumps. Compared to sources selected
at shorter wavelengths they may represent less evolved stages of high-mass
star-formation and in that respect substantial impact can be expected from
future (space) missions like Herschel and Planck. Furthermore this study
demonstrates that the detailed star-forming content can only be accessed with a
multi-wavelength approach including sensitive mid- and far-infrared observations.
\end{enumerate}
\begin{acknowledgements}
Based on observations with ISO, an ESA project with instruments funded by ESA
Member States (especially the PI countries: France, Germany, the Netherlands and
the United Kingdom) and with the participation of ISAS and NASA. The ISOSS was
supported by funds from the DLR, Bonn.
Based on observations collected at the
Centro Astron\' omico Hispano Alem\' an (CAHA) at Calar Alto, operated jointly
by the Max Planck Institut f\"ur Astronomie and the Instituto de Astrof\' isica
de Andaluc\' ia (CSIC), and on observations with the
James-Clerk-Maxwell Telescope (JCMT) as well as on
observations made with the Spitzer Space Telescope, which is operated by the Jet
Propulsion Laboratory, California Institute of Technology under a contract with
NASA.
This publication makes use of data products from the Two Micron All Sky Survey,
which is a joint project of the University of Massachusetts and the Infrared
Processing and Analysis Center/California Institute of Technology, funded by the
National Aeronautics and Space Administration and the National Science
Foundation.
The Digitized Sky Surveys were produced at the Space Telescope Science Institute
under U.S. Government grant NAG W-2166.
The Second Palomar Observatory Sky Survey (POSS-II) was made by the California
Institute of Technology with funds from the National Science Foundation, the
National Geographic Society, the Sloan Foundation, the Samuel Oschin Foundation,
and the Eastman Kodak Corporation.
MH thanks J\"urgen Steinacker and Ulrich Klaas (both at MPIA) for their support
and helpful discussions; Kalevi Mattila and Jos\'e Gon\c{c}alves
(University of Helsinki Observatory) for hospitality and fruitful comments.
\end{acknowledgements}
\bibliographystyle{aa}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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{"url":"https:\/\/www.splashlearn.com\/math-vocabulary\/decimals\/decimal","text":"# Decimal \u2013 Definition with Examples\n\nA decimal is a number that consists of a whole and a fractional part. Decimal numbers lie between integers and represent numerical value for quantities that are whole plus some part of a whole.\n\nFor example, in the given image, we have one whole pizza and a half of another pizza. This can be represented in two ways:\n\nFractional form: In fraction form, we can write that there is one and one-half of a pizza. That is $1\\frac{1}{2}$ pizza.\n\nDecimal Form: In decimal form, we will write this as 1.5 pizzas. Here, the dot represents the decimal point and the number before the dot, i.e., \u201c1\u201d represents one whole pizza and the number behind the decimal point represents the half pizza or the fractional part.\n\nYou might have seen decimal numbers like these when you go grocery shopping or on a weighing machine or even a game of baseball!\n\n## What is a Decimal?\n\nWe get decimals when we break a whole into smaller parts. A decimal number then has two components: a whole number part and a fractional part. The decimal place value system for the whole part of a decimal number is the same as the whole number value system. However, we get the fractional part of the decimal number as we move toward the right after the decimal point. The given image shows the decimal place value chart:\n\nNote that as we go from left to right in the decimal place value system, each values is $\\frac{1}{10}$ times smaller than the value to its left.\n\nThe first place after the decimal point is called the \u201ctenths\u201d, which represents a place value of $\\frac{1}{10}$ of the whole or one-tenth of the whole. In decimal form, this fraction is written as \u201c0.1\u201d.\n\nSuch fractions whose denominator is 10 or a positive power of 10 is called a decimal fraction.\n\nThe second place is called the \u201chundredths\u201d, which represents a place value of $\\frac{1}{100}$ of the whole or one-hundredth of the whole. In numerical form, this decimal fraction is written as \u201c0.01\u201d.\n\nAnd the third place is called the \u201cthousandths\u201d, which represents a place value of $\\frac{1}{1000}$ of the whole or one-thousandth of the whole. In numerical form, this decimal fraction is written as \u201c0.001\u201d.\n\nHere\u2019s an example of a decimal number 17.48, in which 17 is the whole number, while 48 is the decimal part.\n\n## How to Read a Decimal\n\nAn informal way to read a decimal is by reading the whole part of the decimal number as you would read any whole number and then reading the decimal dot as \u201cpoint\u201d and then reading each digit of the rational part individually.\n\nFor example, the number 17.48 would be read as \u201cSeventeen point four eight\u201d.\n\nHowever, a more formal way to read decimals is to read the whole part as a whole number, then the decimal dot as \u201cand\u201d and then reading the fractional part altogether but using the place value of the last digit with it.\n\nFor example, take a look at the given number\n\n25.678\n\nHere, the whole part is 25 and the place value of the last digit, 8, is thousandths. So we will read this number as \u201cTwenty-five and six hundred seventy-eight thousandths\u201d.\n\n## Types of Decimals\n\nBased on the number of digits after the decimal point, the decimal numbers can be divided into two categories:\n\nLike decimals: Two decimal numbers are said to be \u201clike\u201d decimals if they have the same number of digits after the decimal point. For example, 6.34 and 2.67 both have two digits after the decimal point so they are Like decimals.\n\nUnlike decimals: Two decimal numbers are said to be \u201cunlike\u201d decimals if they have different number of digits after the decimal point. For example, 5.3 and 6.873 both have different number of digits after the decimal point so they are unlike decimals.\n\n## Solved Examples\n\nExample 1: In which place does the underlined digit lie in 56.782?\nSolution:\n\nThe underlined digit is in the third place after the decimal point. The first place after the decimal point is tenths, the second place is hundredths and the third place is called thousandths. Hence, the undersigned digit is in the thousandths place.\n\nExample 2: What is the place value of the underlined digit in 45.82?\n\nSolution:\n\nThe underlined digit, 8, is in the tenths place. Hence, its place value is 0.8.\n\nExample 3: Give an example of like decimals.\n\nSolution:\n\nTwo decimal numbers are said to be like decimals if they have the same number of digits after the decimal point. Example: 5.99 and 3.65\n\n## Practice Problems\n\n### 1Which of the following pairs of decimals are like decimals?\n\n58.90 and 58.999\n54.22 and 6.32\n83.25 and 83.2\n50 and 60.7\nCorrectIncorrect\nCorrect answer is: 54.22 and 6.32\nBecause these are the only pair of decimal numbers that have the same number of digits after the decimal point.\n\n### 2In what place does the underlined digit in $6.8\\underline{9}3$ lie?\n\nTenths\nHundredths\nOnes\nthousandths\nCorrectIncorrect\nCorrect answer is: Hundredths\nThe underlined digit is in the second place after the decimal point. The first place after the decimal point is tenths, the second place is hundredths.\n\n### 3Which number is in the tenths place in 34.987?\n\n3\n4\n9\n8\nCorrectIncorrect\nCorrect answer is: 9\nThe first place after the decimal point is the tenths place and in the given decimal number, the digit 9 is in the tenths place.\n\n### 4What is the place value of the underlined digit in 124.64$\\underline{8}$?\n\n0.008\n0.08\n0.8\n8\nCorrectIncorrect\nCorrect answer is: 0.008\nThe underlined digit \u201c8\u201d is in the thousandths place. Hence, its place value is 0.008.\n\n## Frequently Asked Questions\n\nA common application of the concept of decimals is in monetary transactions. A dollar is made up of 100 cents, and the prices of objects are written in decimals, such as USD 2.79. Other applications are mainly concerned with the precision required in the field, such as scientific research, sports, jewelry, and vehicle gas meters.\n\nAll integers can be written as decimal numbers. For example, the integer 5 can be written as 5.00. But not all decimal numbers can be written as integers since integers are whole and do not have a fractional part, but decimal numbers have a fractional part and hence can only be written as fractions.\n\nA decimal number is read by saying the whole part of it as a whole number and by individually reading out every number after the decimal point. For example, 27.69 would be read as twenty-seven point six nine.","date":"2022-10-02 00:00:31","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\": 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\": 0.6888794898986816, \"perplexity\": 449.6517757126373}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"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-2022-40\/segments\/1664030336978.73\/warc\/CC-MAIN-20221001230322-20221002020322-00558.warc.gz\"}"}
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{"url":"https:\/\/www.physicsforums.com\/threads\/i-have-no-idea-how-to-do-very-very-basic-proofs-help-guide-me.369671\/","text":"# I have no idea how to do (very very basic) proofs help guide me?\n\n1. Jan 14, 2010\n\n### nachosamurai\n\n1. The problem statement, all variables and given\/known data\n\nLet A and B be any sets.\n\n1: Prove A is the disjoint union of A\\B and A intersect B.\n\n2: Prove A U B is the disjoint union of A\\B, A intersect B, and B\\A.\n\n2. Relevant equations\n\n????\n\n3. The attempt at a solution\n\nI understand most of the basic terminology used. I know disjoint means that no elements in one are in the other; on a Venn diagram they would not be overlapping at all. I know a union includes all elements from either set. I'm guessing a disjoint union just means that they are two disjoint sets?\n\nI understand that these would all be disjoint, by use of Venn diagrams that I drew myself to help get my head around this new, alien type of math problem.\n\nWhat I don't get is how to even begin proving this type of thing. My professor gave us some examples of other proofs, which often begin by supposing X is an element of one of the sets, and working with it. They were on different types of questions, though, so I can't figure out how to translate them to this particular question.\n\nHonestly, I don't really understand how he comes up with his examples on any of this \"discrete\" math.\n\nI'm not asking for you to do this problem for me, but rather to help me learn how to think logically and attack this type of thing myself. How do I even begin to construct a basic proof?\n\nAssistance would be gratefully appreciated.\n\nLast edited: Jan 14, 2010\n2. Jan 14, 2010\n\nIt may be fruitful to think of the equivalency A=B as $$(A\\subset B) \\cap (B\\subset A)$$. That is, show for an element a in A, show that it lies also within B, and for an element b in B, show that it also lies within A. For disjointedness, there are probably many ways to show it, but I think that if I were to solve it I might attack it by contradiction.\n\n3. Jan 14, 2010\n\n### iomtt6076\n\nFirst I just want to make sure I know what you're asking: did you mean to write:\n\n\"Prove $$A\\cup B=(A\\setminus B)\\cup (A\\cap B)\\cup (B\\setminus A)$$?\"\n\n4. Jan 14, 2010\n\n### nachosamurai\n\nOK, I had a slight mistake in the problem. There are two separate but similar problems, and I got confused about which one I was reading half-way through the problem.\n\nI've updated them in the original post, and will repeat it here as well:\n\nLet A and B be any sets.\n\n1: Prove A is the disjoint union of A\\B and A intersect B.\n\n2: Prove A U B is the disjoint union of A\\B, A intersect B, and B\\A.\n\nThe way they are written is identical to that. As I understand it, the \"and\" in these problems does not equal \"union\". Am I correct?\n\nLast edited: Jan 14, 2010\n5. Jan 14, 2010\n\n### iomtt6076\n\nAlright, lets deal with #1:\n\"prove $$A=(A\\setminus B)\\cup (A\\cap B)$$\"\n\nFirst, do you see that $$A\\setminus B=A\\cap B^\\complement$$?\n\n6. Jan 14, 2010\n\n### nachosamurai\n\nWell, I'm still confused why we're turning \"and\" into the union symbol. Is that what we're supposed to do?\n\nAs far as A\\B = A intersect B-compliment, I understand that. Everything in A that's not in B is equal to everything that's in both A and everything that's not in B, and also in A.\n\nBut but but... and = union?\n\n7. Jan 14, 2010\n\n### iomtt6076\n\nThe statement \"union of X and Y\" means $$X\\cup Y$$.\n\nSorry, I don't mean to be difficult, but going back to what you wrote\n\nthat says to me, prove $$A=(A\\setminus B)\\cup (A\\cup B)$$, which isn't true in general. Are you sure it isn't\n\n1: Prove A is the disjoint union of A\\B and A $$\\cap$$ B.\n\n8. Jan 14, 2010\n\n### nachosamurai\n\nah, shoot, yes >.< you're correct, it should be this:\n\n1: Prove A is the disjoint union of A\\B and A intersect B.\n\nThat's what i get for trying to mess around with these symbols, I have no idea how to do it correctly on these boards and didn't even notice I had made a union sign instead of an intersect sign. Going to stick to text for now.\n\n9. Jan 14, 2010\n\n### iomtt6076\n\nOkay, we're on the same page now.\n\nFirst a comment, when you asked if and=union, perhaps you were thinking of if x is in A and B, then $$x\\in A\\cap B$$. But this is different to the 'and' in \"union of X and Y\"\n\nBack to the problem. So you realize $$A\\setminus B=A\\cap B^\\complement$$. We want to show $$A=(A\\setminus B)\\cup (A\\cap B)$$. We can write the right hand side as $$(A\\cap B^\\complement)\\cup (A\\cap B)$$. Does this remind you of some identity?\n\n10. Jan 14, 2010\n\n### nachosamurai\n\nThank you very much, I think I see how I can proceed from here :)\n\nSo basically it's all a matter of re-writing these to be easier to prove, eh?\n\n11. Jan 14, 2010\n\n### iomtt6076\n\nFor these kind of problems, yes. It might also be helpful to draw out the Venn diagrams to visualize what's going on.","date":"2018-03-19 11:42:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.7118447422981262, \"perplexity\": 466.85017016374684}, \"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-2018-13\/segments\/1521257646875.28\/warc\/CC-MAIN-20180319101207-20180319121207-00581.warc.gz\"}"}
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Бори́с Па́влович Асе́ев (1901—1965) — советский конструктор военной техники радиосвязи. Генерал-майор инженерно-технической службы. Лауреат Сталинской премии первой степени.
Биография
Родился 7 (20 ноября) 1901 года в Рязани в семье мелкопоместного дворянина. Рано лишился матери и воспитывался бабушкой.
С 9 лет начал учиться в 1-й Рязанской гимназии, затем в 1-й Московской гимназии. За отличную учебу обучался бесплатно за счет государственного обеспечения, в старших классах занимался репетиторством и совмещал учёбу с работой электромонтера. После гимназии поступил в Московскую школу радиоспециалистов, которую окончил в 1920 году, получив специальность «радиотелеграфист первого разряда».
В 1920 году в Красной Армии. 1920—1922 гг. — командир отделения радиокурсов в Институте связи, лаборант-инженер Учебно-опытного радиотелеграфного дивизиона, в 1922 году смонтировал первый в стране коротковолновый передатчик, использовавшийся в качестве лабораторного стенда для практического обучения слушателей. 1921—1923 гг. инженер и начальник центральной радиостанции радиодивизиона учебной радиобригады, затем начальник батальонной школы 9-го отдельного радиобатальона и преподаватель Повторных курсов командного состава радиочастей Красной Армии в Москве.
В 1927 г. командирован в Ленинград, где занимается созданием кафедры радиопередающих устройств и преподает в Военной академии связи имени С. М. Будённого и в Военно-технической академии им. Ф.Э. Дзержинского (октябрь 1926—август 1934). В 1929 году сдал экстерном экзамены в Ленинградском электротехническом институте им. В. И. Ульянова (Ленина) и получил воинское звание инженер-электрик.
В 1936 году, в составе делегации, совместно с А. Д. Фортушенко командирован в США где участвует в подписании контрактов на поставку в СССР радиотехнического оборудования фирмы RCA (Radio Corporation of America).
НИИТС КА
После создания Научно-исследовательского института техники связи Красной Армии А. Д. Асеев возглавляет один из его отделов, занимает должности инженера-конструктора и главного инженера по технике связи (август 1934 — январь 1935). Организованная Асеевым группа квалифицированных радистов-коротковолновиков разрабатывает и организует производство семейства радиопередатчиков (тип «А») мощностью от 30 до 100 Вт, предназначенных для обеспечения связи советских торговых судов, перевозивших грузы в охваченную войной Испанию. Наиболее удачные из них А‑5/2 и А‑19 широко использовались и на фронтах Великой Отечественной войны. В этот же период, под руководством Асеева, в НИИТС КА была разработана известная радиостанция Север («Омега»).
В 1937 — 1943 гг. Асеев работает над оснащением радиоузлов на трассе Алма-Ата - Ланьчжоу, создаваемых в рамках военной помощи Советского Союза Китаю, во время его борьбы с японской агрессией. C 1941 года возглавляет специальную лабораторию, в которую входят Н. Н. Иванов, М. Г. Марголин, И. Х. Невяжский, Л. М. Финк. Силами лаборатории была разработана аппаратура, позволяющяя настраивать мощный советский передатчик точно на волну немецкой радиовещательной станции и осуществлять контрпропагандистское вещание во время пауз в радиопередачах. 10 апреля 1942 года все участники разработки были удостоены Сталинской премии 1-й степени.
В рамках работ над многоканальными возбудителями мощных радиопередатчиков, Асеевым впервые были разработаны принципы работы и методики расчетов многофазных RC-генераторов.
Наркомат связи
Параллельно с работами в НИИ, в числе 1000 специалистов прикомандированных из Красной Армии в народное хозяйство, Б. П. Асеев занимал должность главного инженера радиоуправления Наркомата связи (октябрь 1937 — июнь 1941). Возглавляемая им группа инженеров, в которую вошли Л. А. Копытин, З. И. Модель, М. С. Нейман, Б. В. Брауде и др. выполнила эскизно-техническое проектирование Куйбышевской радиовещательной станции мощностью 1200 кВт, а затем осуществляло руководство отдельными монтажно-настроечными работами. Строительство станции было закончено в 1943 году в сложных условиях военного времени, за ее создание Б. П. Асеев награжден орденом Трудового Красного Знамени, удостоен ученой степени и получил воинское звание Генерал-майор инженерно-технической службы (23.10.1943).
Работал в составе советских делегации в рамках международных конференций МККР, проходивших в Каире (1938 г.) и Лондоне (1945 г., 1946 г. руководитель), участвовал в разработке Регламента радиосвязи и других международных нормативных документов в этой области.
Участник Великой Отечественной войны. В эти и послевоенные годы (1941—1952) находился на научной и организаторской работе в наркомате — министерстве связи СССР и военном НИИ по технике связи Красной — Советской армии (1941—1952), заместитель начальника этого НИИ по научно-исследовательской части и он же главный инженер.
Умер 13 октября 1965 года в Москве. Похоронен на Донском кладбище.
Научные работы
«Ламповые генераторы и передатчики» (учебный курс 1930 г.)
«Электромагнитные колебания в цепях с распределенными постоянными» (1934 г.). Переиздана с дополнениями в 1936 г. и 1938 г.
«Основы радиотехники» (1947 г.). Рекомендована в качестве учебника для ВУЗов СССР, переиздана в ПНР, Румынии, Китае и ГДР.
Автор разделов «Генератор ламповый» (т. 10, с. 415—417) и «Дуговой генератор» (т. 15, с. 273—274) в Большой Советской энциклопедии.
Награды и премии
два ордена Ленина (21.02.1945, 24.09.1945)
два ордена Красного Знамени (03.11.1944, 1949)
орден Трудового Красного Знамени
орден Красной Звезды (20.01.1943)
орден «Знак Почета»
медали.
Сталинская премия первой степени (1942) — за изобретение электроприбора, имеющего оборонное значение;
Примечания
Ссылки
АСЕЕВ Борис Павлович
Конструкторы СССР
Преподаватели Военной академии связи имени С. М. Будённого
Похороненные на Донском кладбище
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published: true
title: Banana-Walnut Chocolate Chunk Cookies
layout: post
categories: [cookies]
rating: 5
---
### Servings
36
### Ingredients
- 1 cup all-purpose flour
- 1/2 cup whole-wheat flour
- 1 teaspoon coarse salt
- 1/2 teaspoon baking soda
- 3/4 cup (1 1/2 sticks) unsalted butter, softened
- 1/2 cup granulated sugar
- 1/2 cup packed light-brown sugar
- 1 large egg
- 1 1/2 teaspoons pure vanilla extract
- 1/2 cup mashed ripe banana (about 1 large)
- 1 cup old-fashioned rolled oats
- 8 ounces semisweet chocolate, coarsely
- chopped into 1/4-inch chunks
- 1/2 cup coarsely chopped walnuts (about 2 ounces), toasted
### Directions
1. STEP 1
2. Preheat oven to 375 degrees. Whisk together flours, salt,
3. and baking soda in a small bowl; set aside. Put butter and
4. sugars into the bowl of an electric mixer fitted with the
5. paddle attachment; mix on medium speed until pale and
6. fluffy. Reduce speed to low. Add egg and vanilla; mix
7. until combined. Mix in banana. Add flour mixture; mix
8. until just combined. Stir in oats, chocolate chunks, and walnuts.
9. STEP 2
10. Using a 1 1/2-inch ice cream scoop, drop dough
11. onto baking sheets lined with parchment paper,
12. spacing about 2 inches apart. Bake cookies, rotating
13. sheets halfway through, until golden brown and just
14. set, 12 to 13 minutes. Let cool on sheets on wire racks
15. 5 minutes. Transfer cookies to wire racks; let cool completely.
16. Cookies can be stored in airtight containers up to 2 days.
### Source
<a href="http://www.marthastewart.com/338169/banana-walnut-chocolate-chunk-cookies?search_key=chocolate%20chunk%20banana%20cookies" target="new">http://www.marthastewart.com/338169/banana-walnut-chocolate-chunk-cookies?search_key=chocolate%20chunk%20banana%20cookies</a>
|
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\section{Introduction and main results}
\subsection{Introduction}
The focus of this paper is the determination of a second order perturbation of the polyharmonic operator from the Dirichlet to Neumann map. We start with some notations. Let $\Omega\subset\mathbb{R}^n$, $n \geq 3$, be a bounded simply connected domain of $\mathbb{R}^n$ with $C^\infty$ boundary $\Gamma=\partial\Omega$ and let $m \geq 2$. Given a symmetric matrix $A=(A_{jk})$, a vector field $B=(B_j)$ and a potential $q$, we consider the perturbed polyharmonic operator formally given by
\begin{align}\label{1.1}
\mathcal{L}_{A,B,q}(x,D)&= (-\Delta)^m+A(x)D\cdot D+B(x)\cdot D+q(x)\cr
&=(-\Delta)^m+\sum_{j,k=1}^n A_{jk}(x)D_j D_k +\sum_{j=1}^n B_j(x)D_j+q(x),
\end{align}
where $D=i^{-1}\nabla$, $A\in W^{2,\infty}(\Omega,\mathbb{C}^{n^2})$, $B\in W^{1,\infty}(\Omega,\mathbb{C}^n)$ and $q\in L^\infty(\Omega,\mathbb{C})$ are the perturbation coefficients. The operator $\mathcal{L}_{A,B,q}(x,D)$ equipped with domain
\begin{equation}\label{1.2}
\mathcal{D}(\mathcal{L}_{A,B,q})=\{u\in H^{2m}(\Omega)\,:\, \gamma u:=\bigr(u,(-\Delta) u, \ldots, (-\Delta)^{m-1} u\bigr)|_{\Gamma}=0 \},
\end{equation}
is an unbounded closed operator on $L^2(\Omega)$ with purely discrete spectrum, see \cite{GG}.\\
For $m \geq 2$ and $r >0$, we denote by $\mathcal{H}^{m,r}(\Gamma)$ the product space
\begin{align*}
\mathcal{H}^{m,r}(\Gamma)=\prod_{j=0}^{m-1}H^{2m-2j-r}(\Gamma ),
\end{align*}
equipped with norm
\begin{align*}
\|f\|_{ \mathcal{H}^{m,r}(\Gamma)}:=\sum_{j=0}^{m-1}\|f_j\|_{ {H}^{2m-2j-r}(\Gamma)},\quad f=(f_0, \ldots, f_{m-1}) \in \mathcal{H}^{m,r}(\Gamma).
\end{align*}
Assuming $0$ is not an eigenvalue of $\mathcal{L}_{A,B,q}(x,D): \mathcal{D}(\mathcal{L}_{A,B,q}) \rightarrow L^2(\Omega)$ and consider the boundary value problem with Navier boundary conditions
\begin{equation}\label{1.3}
\left\{\begin{array}{ll}
\mathcal{L}_{A,B,q}(x,D)u =0 & \mathrm{in}\, \Omega,\cr
\gamma u =f &\mathrm{on}\, \Gamma,
\end{array}
\right.
\end{equation}
where $f=(f_0, \ldots, f_{m-1})\in \mathcal{H}^{m,\frac{1}{2}}(\Gamma)$. Then \eqref{1.3} has a unique solution $u:=u_f\in H^{2m}(\Omega)$ (see Lemma \ref{L0.1} below) and the boundary measurements are given by the Dirichlet to Neumann map (D-to-N), defined formally by
\begin{equation}\label{1.4}
\Lambda_{A,B,q}: f\mapsto \widetilde{\gamma} u:=(\partial_\nu u,\partial_\nu(-\Delta) u, \ldots, \partial_\nu(-\Delta)^{m-1} u)|_\Gamma,
\end{equation}
where $u\in H^{2m}(\Omega)$ satisfies the boundary value problem (\ref{1.3}). Here $\nu$ denotes the unit outer normal vector to the boundary $\Gamma$ at $x$.
\medskip
Before stating our main result, we recall the following Lemma on the existence and uniqueness of a solution to the problem \eqref{1.3}, the proof is given by Lemma 5.13 in \cite{TH}.
\begin{lemma}\label{L0.1}
Let $A \in W^{2, \infty}(\Omega, \mathbb{C}^{n^2})$, $B \in W^{1, \infty}(\Omega, \mathbb{C}^n)$ and $q \in L^\infty(\Omega, \mathbb{C})$. Suppose that $0$ is not an eigenvalue of $\mathcal{L}_{A,B,q}(x, D)$ and $f \in \mathcal{H}^{m,\frac{1}{2}}(\Gamma)$. Then, there exists a unique solution $u \in H^{2m}(\Omega)$ to \eqref{1.3} satisfying
\begin{align*}
\|u\|_{H^{2m}(\Omega)} \leq C \|f\|_{\mathcal{H}^{m,\frac{1}{2}}(\Gamma)}.
\end{align*}
Furthermore, we have $\widetilde{\gamma} u:=(\partial_\nu u,\partial_\nu(-\Delta) u, \ldots, \partial_\nu(-\Delta)^{m-1} u)|_\Gamma \in \mathcal{H}^{m,\frac{3}{2}}(\Gamma) $ and there exists a constant $C > 0$ such that
\begin{align*}
\|\widetilde{\gamma} u\|_{\mathcal{H}^{m,\frac{3}{2}}(\Gamma)} \leq C \|f\|_{\mathcal{H}^{m,\frac{1}{2}}(\Gamma)}.
\end{align*}
\end{lemma}
As a corollary of above Lemma, the Dirichlet-to-Neumann map $\Lambda_{A,B,q}$ given by \eqref{1.4}
is bounded from $\mathcal{H}^{m,\frac{1}{2}}(\Gamma)$ to
$\mathcal{H}^{m,\frac{3}{2}}(\Gamma)$. We denote by $\|\Lambda_{A,B,q}\|$ its norm.
\medskip
In this paper, we are interested in recovering the coefficients $A$, $B$ and $q$ from the boundary measurements unclosed in the Dirichlet to Neumann map
\begin{equation}\label{1.5}
\Lambda_{A,B,q}: \mathcal{H}^{m,\frac{1}{2}}(\Gamma)\longrightarrow \mathcal{H}^{m,\frac{3}{2}}(\Gamma).
\end{equation}
In practice, the polyharmonic operators $(-\Delta)^m$, $m \geq2$, arise in many areas of physics and geometry, including the study of vibration of beams, the Kirchhoff-Love plate equation in the theory elasticity (for $m=2$), the continuum mechanics of buckling problems and the study of the Paneitz-Branson operator in conformal geometry, for more details we refer to \cite{GGS} and \cite{MVV}.\\
The uniqueness question of determination the lower order perturbations of polyharmonic operator $(-\Delta)^m$, $m \geq2$, was begun by Krupchyk, Lassas and Uhlmann who showed in \cite{K1} the uniqueness of the zeroth and the first order perturbations from the knowledge of D-to-N map. Later, they proved in \cite{K2} that the unique recovery of a first order perturbation of the biharmonic operator, i.e., $m=2$, is possible when the D-to-N map given only on a part of the boundary $\Gamma$, where $\Omega$ is a bounded domain. In case where $\Omega$ is an unbounded domain, we refer the work of Yang \cite{Y}. In \cite{Gk}, Ghosh and Krishnan considered the higher order perturbation of polyharmonic operators with partial data on the boundary, where the coefficients attached to perturbed terms are isotropic. Later, Bhattacharyya and Ghosh established in \cite{B2} the uniqueness of a second order perturbations of a polyharmonic operator $(-\Delta)^m$, $m \geq 2$, from the D-to-N data. More precisely, they showed that, for $m>2$, the unique determination of a $2-$tensor field $A$, a vector field $B$ and a potential $q$ provided $A \in W^{3, \infty}(\Omega)$, $B \in W^{2, \infty}(\Omega)$ and $q \in L^\infty(\Omega)$ with some restrictions on the support of the coefficients and for $m=2$, they assumed that $A$ is an isotropic matrix. In the case of an anisotropic matrix, for $m=2$, they showed in \cite{B1} that the unique determination of second order perturbations of a biharmonic operator is possible from the boundary D-to-N map. Note that in the case of a lower order perturbation up to order $3$ of the biharmonic operator, there is an obstruction to uniqueness, as noted in \cite{B1}. Recently, Bhattacharyya, Krishnan and Sahoo established in \cite{BKS} the unique recovery of a lower order anisotropic tensor perturbations up to order $m$ of a polyharmonic operator from measurements taken on a part of the boundary. All the above mentioned works are stated in the case where all the coefficients of the perturbed polyharmonic operator are sufficiently smooth. In case of lower regularity of the coefficients, we refer to \cite{A1}, \cite{AI}, \cite{KU} and to recent work \cite{BG}.
\smallskip
For the stability question, Choudhury and Krishnan proved in \cite{C1} a logarithmic type stability estimate for the zeroth order perturbation of the biharmonic operator, $(-\Delta)^2$, for the case when the Neumann data is measured on the whole boundary and a log-log type stability estimate for the case when the Neumann data is measured only on slightly more than half of the boundary. Later, Choudhury and Heck established in \cite{CH} a logarithmic type stability estimate for the zeroth order perturbation of the biharmonic operator with partial data when the inaccessible part of the domain is flat. A natural question to ask is whether higher order perturbations of the polyharmonic operator can be stably recovered from the knowledge of D-to-N map. In this work, we show that if we consider a second order perturbation of the polyharmonic operator $(-\Delta)^m$, $m \geq 2$, of the form \eqref{1.1} then the coefficients $(A, B, q)$ depend stably on the bounded map $\Lambda_{A,B,q}$.
\subsection{Main stability results} We here state the main results of this paper concerning conditional logarithmic stability reconstruction of the second order perturbation coefficients $(A, B, q)$ from knowledge of the Dirichlet-to-Neumann map $\Lambda_{A,B,q}$ given by $\eqref{1.5}$.
\smallskip
Let us first indicate the required conditions for admissible coefficients $(A, B, q)$. In what follows, we assume that the second order perturbation $A$ takes the form
\begin{align}\label{cond_A}
A(x)=a(x) \operatorname{id},\quad \textrm{if} ~~m=2.
\end{align}
Let $M >0$ and $\sigma_1 > \frac{n}{2}+3$ be given. We define the class of admissible symmetric tensor $(A_{jk})$, $ \mathcal{A}_{\sigma_1}(M)$, by
\begin{align*}
\mathcal{A}_{\sigma_1}(M)&=\{A \in W^{\operatorname{min}(2m, 6), \infty}(\Omega,\mathbb{C}^{n^2});~~ \|A\|_{H^{\sigma_1}(\Omega)} \leq M,~\textrm{and satisfying \eqref{cond_A}} \},\\
\end{align*}
Given $M >0$ and $\sigma_2 >\frac{n}{2}+1$, we define the class of admissible vector field $B$ and electric potential $q$ respectively by
\begin{align*}
\mathcal{B}_{\sigma_2}(M)&=\{B \in W^{4,\infty}(\Omega,\mathbb{C}^n),~~ \|B\|_{H^{\sigma_2}(\Omega)} \leq M \},
\end{align*}
and
\begin{align*}
\mathcal{Q}(M)&=\{q \in L^{\infty}(\Omega),~~ \|q\|_{L^{\infty}(\Omega)} \leq M \}.
\end{align*}
Let us define, where $\mu >0$, the function $\Phi_\mu$ as follows
$$
\Phi_\mu(t)=\left\{
\begin{array}{lll}
0, & t=0, \\
\abs{\ln t}^{-\mu }+t^{\frac{1}{2}}, & t >0,
\end{array}
\right.
$$
and we denote by $\mathcal{E}^\prime(\Omega)$ the space of all compactly supported distributions in $\Omega$.
\medskip
Our main results state that it is possible to stably determine the symmetric tensor $A$, the vector field $B$ and the potential $q$, given the knowledge of the Dirichlet-to-Neumann map. Precisely, we will prove the following three theorems in sections \ref{section4} and \ref{section5}.
\begin{theorem}
Let $M >0$ and $\sigma_j >0$, $j=1,2,$ as above. Then, there exists a constant $C >0$ and $\mu_1, \kappa_1 \in (0,1)$, such that for any $A^{(j)}\in \mathcal{A}_{\sigma_1}(M) \cap \mathcal{E}^\prime(\Omega)$, $B^{(j)}\in\mathcal{B}_{\sigma_2}(M)\cap \mathcal{E}^\prime(\Omega)$ and $q^{(j)}\in \mathcal{Q}_{}(M)$, $j=1,2$, we have
\begin{align}\label{1.8}
\|A^{(1)}-A^{(2)}\|_{L^{\infty}(\Omega)} \leq C \Phi_{\mu_1}( \|\Lambda_{A^{(1)},B^{(1)},q^{(1)}}-\Lambda_{A^{(2)},B^{(2)},q^{(2)}}\|)^{\kappa_1}.
\end{align}
Here $C$ depends only on $\Omega$, $\sigma_1$, $\sigma_2$, $n$, $m$ and $M$, and $\mu_1$ depends only on $n$ and $m$.
\label{T1.2}
\end{theorem}
\begin{theorem}
Let $M >0$ and $\sigma_j >0$, $j=1,2,$ as above. Then, there exists a constant $C >0$ and $\mu_2, \kappa_2 \in (0,1)$, such that for any $A^{(j)}\in \mathcal{A}_{\sigma_1}(M)\cap \mathcal{E}^\prime(\Omega)$, $B^{(j)}\in\mathcal{B}_{\sigma_2}(M)\cap \mathcal{E}^\prime(\Omega)$ and $q^{(j)}\in \mathcal{Q}_{}(M)$, $j=1,2$, we have
\begin{align}\label{1.9}
\|B^{(1)}-B^{(2)}\|_{L^{\infty}(\Omega)} \leq C \Phi_{\mu_2}( \|\Lambda_{A^{(1)},B^{(1)},q^{(1)}}-\Lambda_{A^{(2)},B^{(2)},q^{(2)}}\|)^{\kappa_2}.
\end{align}
Here $C$ depends only on $\Omega$, $\sigma_1$, $\sigma_2$, $n$, $m$ and $M$, and $\mu_2$ depends only on $n$ and $m$.
\label{T1.3}
\end{theorem}
\begin{theorem}
Let $M >0$ and $\sigma_j >0$, $j=1,2,$ as above. Then, there exists a constant $C >0$ and $\mu_3 \in (0,1)$, such that for any $A^{(j)}\in \mathcal{A}_{\sigma_1}(M)\cap \mathcal{E}^\prime(\Omega)$, $B^{(j)}\in\mathcal{B}_{\sigma_2}(M)\cap \mathcal{E}^\prime(\Omega)$ and $q^{(j)}\in \mathcal{Q}_{}(M)$, $j=1,2$, we have
\begin{align}\label{1.10}
\|q^{(1)}-q^{(2)}\|_{L^{\infty}(\Omega)} \leq C \Phi_{\mu_3}( \|\Lambda_{A^{(1)},B^{(1)},q^{(1)}}-\Lambda_{A^{(2)},B^{(2)},q^{(2)}}\|).
\end{align}
Here $C$ depends only on $\Omega$, $\sigma_1$, $\sigma_2$, $n$, $m$ and $M$, and $\mu_3 \in (0,1)$ depends only on $n$ and $m$.
\label{T1.4}
\end{theorem}
Let us explain the main difficulties and ideas in the proof of
Theorems \ref{T1.2}, \ref{T1.3} and \ref{T1.4}. We start by
realling the qualitative argument due to Bhattacharyya and Ghosh
\cite{B2}. Their starting point
is the following orthogonal identity (see Lemma \ref{L3.1} below)
\begin{equation}
\int_\Omega (A(x)D \cdot D u+B(x)\cdot Du+q(x)u)\overline{u^*}dx=0,\label{*}
\end{equation}
which holds for $u$ and $u^*$ solving $\mathcal{L}_{A^{(2)}, B^{(2)}, q^{(2)}}u=0$ and $\mathcal{L}^*_{A^{(1)}, B^{(1)}, q^{(1)}}u^*=0$ respectively, wherever $\Lambda_{A^{(1)}, B^{(1)},
q^{(1)}}=\Lambda_{A^{(2)}, B^{(2)}, q^{(2)}}$. They, then proceed by constructing of special solutions, called complex geometric optics
solutions (C.G.O- solutions), that are to be used with the integral identity \eqref{*}. We mention that the method of C.G.O- solutions that is used
for proving uniqueness question for higher order elliptic operators
goes back to Sylvester and Uhlmann \cite{SU}. The complex geometric optics solutions to the polyharmonic equation take the special form
\begin{align}\label{**}
u(x,h)=e^{\frac{x \cdot \varrho}{h}}(\alpha_1(x)+h \alpha_2(x)+r(x,h)),
\end{align}
where $\varrho \in \mathbb{C}^n$, $h$ is a small parameter, $\alpha_j$ $j=1,2,$ are complex amplitudes satisfying some transport equations and $r(\cdot, h)$ is remainder term that vanishes when $h$ goes to zero.\medskip\\
Now, any symmetric tensor $A \in H^k(\Omega)$ can be uniquely represented as the sum
\begin{align}\label{***}
A=\widetilde{A} +\nabla_{\text{sym}}V ,
\end{align}
where the covector field $V \in H^{k+1}(\Omega, \mathbb{C}^{n})$ satisfying the boundary condition
\begin{align}\label{***V}
V=0,\quad \textrm{on}~~\Gamma,
\end{align}
and the tensor field $\widetilde{A} \in H^k(\Omega, \mathbb{C}^{n^2})$ satisfies the conditions
\begin{align}\label{***F}
\sum_{j=1}^n \partial_j \widetilde{A}_{jk}=0,\quad \forall k \in \{1, \ldots, n \}.
\end{align}
With the C.G.O-solutions given by \eqref{**} and the integral identity \eqref{*}, they deduce at first, in the case when $m>2$, that $A:=A^{(1)}-A^{(2)}$ is an isotropic matrix, which they fall into the regime of case when $m=2$. Second, by changing the choice of the amplitudes $\alpha_1$ and $\alpha_1^*$, they prove that $B^{(1)}=B^{(2)}$. Hence, the conclusion that $A^{(1)}=A^{(2)}$ and $q^{(1)}=q^{(2)}$ follows easily.\medskip \\
However, in order to obtain a stability estimate for this inverse boundary value problem, the integral identity \eqref{*} transformed to the following integral inequality
\begin{align*}
\abs{\int_\Omega (A D \cdot D u+B \cdot Du+q u)\overline{u^*}dx} \leq \|\Lambda_{A^{(1)}, B^{(1)}, q^{(1)}}-\Lambda_{A^{(2)}, B^{(2)}, q^{(2)}}\| \|u\|_{H^{2m}(\Omega)} \|u^*\|_{H^{2m}(\Omega)}.
\end{align*}
The estimate for the second order perturbation is slightly more complicated. In contrast to the uniqueness result of \cite{B2}, the decomposition \eqref{***} may not be helpful to establish a stability estimate for the 2-tensor field $A$. To remedy this difficulty, we decompose $A$, in the different way, as
\begin{align}
A=A^\prime +\nabla_{\text{sym}}V+\lambda~ id ,
\end{align}
where the function $\lambda \in H^{k}(\Omega, \mathbb{C})$, the covector field $V \in H^{k+1}(\Omega, \mathbb{C}^{n})$ satisfying the boundary condition \eqref{***V} and the tensor field $A^\prime \in H^k(\Omega, \mathbb{C}^{n^2})$ satisfies the conditions \eqref{***F} and
\begin{align}\label{***A}
\operatorname{trace}(A^{\prime})=0.
\end{align}
In fact, using the condition \eqref{***A} we are able to estimate $A^\prime:=A^{\prime (1)}-A^{\prime (2)}$. This technique will be discussed later. After this is established, the stability estimates for the vector fields $V$ and $B$ follow by using the Hodge decomposition. Then by changing the choices of amplitudes , we may estimate the function $\lambda$ and therefore, we conclude the stability estimate for $A$. Finally we will combine the estimates we get for $A$ and $B$ to obtain the stability estimate for the potential $q$. For the case where $m=2$, the stability result would follow similarly except in this case $A=\lambda ~id$.
\medskip
The remainder of this paper is organized as follows. In section $2$ we will build complex geometric optics solutions to the perturbed polyharmonic operator. In section $3$, we derive an integral identity involving the perturbations and we give a specific Hodge decomposition of a symmetric tensor $A$ and a vector field $B$. Sections $4$ and $5$, are concerned to prove the stability estimates respectively in case $m>2$ and case $m=2$. In appendix \ref{appendixA}, we develop the proof of Hodge decomposition of a symmetric tensor.
\section{Carleman estimate and C.G.O- solutions}
The main strategy of the proof of stability estimates on determining the symmetric tensor $A$, the vector field $B$ and the electric potential $q$ from the D-to-N map is the use of complex geometrical solutions to estimate the Fourier transform of the difference of two symmetric tensors $A^{(1)}-A^{(2)}$, the difference of two vector fields $B^{(1)}-B^{(2)}$ and the difference of two potentials $q^{(1)}-q^{(2)}$. The constructed of C.G.O- solutions is based on the use of Carleman estimates. We therefore first outline some known results about Carleman estimate.
\subsection{Local Carleman estimate}
In this section we will first recall the Carleman estimates for semiclassical Lapace operator $(-h^2\Delta)$ and then we use this Carleman estimate to solve an equation involving a conjugated operator of $\mathcal{L}_{A,B,q}(x,D)$.
\smallskip
We start by collecting several known Lemmas and notations. Let $\psi\in C^\infty(\mathbb{R}^n,\mathbb{R})$, consider the conjugated operator with respect the weight function $\psi$
\begin{equation}\label{2.1}
P_\psi(x,hD)=e^{\frac{\psi}{h}}(-h^2\Delta)e^{-\frac{\psi}{h}},\quad h>0.
\end{equation}
Following \cite{KSU}, we say that $\psi$ is a limiting weight function for Laplace operator, if $\nabla \psi \neq 0$ in $\Omega$, and if it satisfies the following Poisson bracket condition
\begin{align*}
\left\{\overline{p}_\psi, p_\psi \right\}(x, \xi)=0\quad \textrm{where}~~p_\psi(x, \xi)=0,\quad x \in \overline{\Omega}, ~\xi \in \mathbb{R}^{n},
\end{align*}
where $p_\psi$ is the semiclassical principal symbol of the opearator $P_\psi(x, hD)$.\\
Let $\omega\in\mathbb{S}^{n-1}$, the limiting weight function $\psi$ can be chosen as
\begin{equation}\label{2.2}
\psi(x)= x\cdot\omega,\quad x\in\mathbb{R}^n.
\end{equation}
In what follows we shall equip the standard Sobolev space $H^s(\mathbb{R}^n)$, $s\in\mathbb{R}$, with the semicalssical norm
\begin{equation}\label{2.3}
\norm{u}_{H_{\mathrm{scl}}^{s}(\mathbb{R}^n)}=\norm{\langle hD\rangle ^su}_{L^2(\mathbb{R}^n)},
\end{equation}
here $\langle \xi\rangle=(1+\abs{\xi}^2)^{1/2}$. Let $\varepsilon >0$, we consider the convexified Carleman weight given by
\begin{align}\label{2.4}
\psi_\varepsilon=\psi+\frac{h}{2\varepsilon}\psi^2.
\end{align}
Therefore, for $0< h < \varepsilon \ll 1$, we have
\begin{align}\label{2.5}
\| \psi_\varepsilon\|_{W^{2, \infty}(\Omega)}= O(1).
\end{align}
We begin with the local Carleman estimate for semiclassical Laplace operator $(-h^2 \Delta)$ which is due to \cite{ST}.\medskip\\
In what follows, ${\,\lesssim\,} $ holds for $\leq$ modulo a multiplicative positive constant independent of $h$.
\begin{lemma}\label{P2.1}
Let $\psi_\varepsilon$ be the convexified Carleman weight function given by (\ref{2.4}). Then for $0 <h \ll \varepsilon$ and $s \in \mathbb{R}$, we have
\begin{equation}\label{2.6}
\frac{h}{\sqrt{\varepsilon}}\norm{v}_{H_{\mathrm{scl}}^{s+2}(\mathbb{R}^n)}{\,\lesssim\,} \norm{P_{\psi_\varepsilon}(x,hD)v}_{H_{\mathrm{scl}}^s(\mathbb{R}^n)},
\end{equation}
for any $v\in C^\infty_0(\Omega)$.
\end{lemma}
Iterating the Carleman estimate (\ref{2.6}) $m$-times, $m\geq2$, we get the following local Carleman estimate for the polyharmonic operator $(-h^2\Delta)^m$, that is
\begin{equation}\label{2.7}
(\frac{h}{\sqrt{\varepsilon}})^m\norm{u}_{H_{\mathrm{scl}}^{s+2m}(\mathbb{R}^n)}{\,\lesssim\,} \norm{e^{\frac{\psi_\varepsilon}{h}}(-h^2\Delta)^me^{-\frac{\psi_\varepsilon}{h}}u}_{H_{\mathrm{scl}}^s(\mathbb{R}^n)},
\end{equation}
for any $u\in C^\infty_0(\Omega)$ and $h, \varepsilon>0$ small enough.
\medskip
Armed with inequality (\ref{2.7}) we shall derive a local Carleman estimate for the perturbed polyharmonic operator $\mathcal{L}_{A,B,q}(x,D)$. Consider the conjugate operator corresponding to the perturbed polyharmonic operator
\begin{align}\label{2.8}
\hspace{-0.7 cm}\mathcal{L}_\psi(x,hD)&=e^{\frac{\psi}{h}}(h^{2m}\mathcal{L}_{A,B,q}(x,D))e^{-\frac{\psi}{h}}\cr
&=e^{\frac{\psi}{h}}(-h^2\Delta)^{m}e^{-\frac{\psi}{h}}+h^{2m-2}e^{\frac{\psi}{h}}A (hD)\cdot (hD) e^{-\frac{\psi}{h}}+h^{2m-1}e^{\frac{\psi}{h}}B\cdot( hD)e^{-\frac{\psi}{h}}+h^{2m}q.
\end{align}
In order to estimate the lower order terms in \eqref{2.8}, we use the following result.
\begin{lemma}\label{L2.2}
Let $t_1$, $t_2 \in \mathbb{R}$ and $V\in H^{t_2}(\Omega)$ with $t_2 > \frac{n}{2}+|t_1|$. Then the estimate
\begin{align}\label{2.9}
\| V u \|_{H_{\mathrm{scl}}^{t_1}(\Omega)} {\,\lesssim\,} \| V \|_{H^{t_2}(\Omega)} \|u\|_{H_{\mathrm{scl}}^{t_1}(\Omega)},
\end{align}
holds for all $u\in C_0^{\infty}(\Omega)$.
\end{lemma}
\begin{proof}
At first, we extend $V$ to $\mathbb{R}^{n}$ and we consider $\tilde{V}\in H^{t_2}(\mathbb{R}^{n})$ with $\tilde{V}=V$, in $\Omega$, and
$$\Vert \tilde{V}\Vert_{H^{t_2}(\mathbb{R}^{n})}{\,\lesssim\,} \,\Vert V\Vert_{H^{t_2}(\Omega)}.$$
Then, we get for $u\in C_0^{\infty}(\Omega)$
$$\mathcal{F}(\tilde{V}u)(\xi)=(\widehat{\tilde{V}}\ast\widehat{u})(\xi)=\int_{\mathbb{R}^{n}}\widehat{\tilde{V}}(\xi-\eta )\widehat{u}(\eta )\,d\eta, $$
which leads directly to
\begin{equation}\label{A.6*}
\langle h \xi\rangle^{t_1}\mathcal{F}(\tilde{V}u)(\xi)=\int_{\mathbb{R}^{n}}\langle h\xi\rangle^{t_1}\langle h \eta \rangle^{-t_1}\,\widehat{\tilde{V}}(\xi-\eta )\langle h \eta \rangle^{t_1}\widehat{u}(\eta )\,d\eta.
\end{equation}
Now, we apply the Peetre's inequality (see \cite{CP} page 90)
\begin{equation}\label{A.7*}
\langle h \xi\rangle^{t_1}\langle h \eta \rangle^{-t_1}\leq 2^{\abs{t_1}/2}\,\langle \xi-\eta \rangle^{\abs{t_1}/2},\quad \forall\,\xi,\, \eta \in \mathbb{R}^{n}.
\end{equation}
Further, we define two functions $w_1$ and $w_2$ by
$$ w_1(\xi)=\langle h\xi\rangle^{t_1}\vert\widehat{u}(\xi)\vert,\;\;w_2(\xi)=\seq{\xi}^{\abs{t_1}}\,\vert\widehat{\tilde{V}}(\xi)\vert,$$
so that we get using (\ref{A.6*}) and (\ref{A.7*}) that
$$\displaystyle\vert \langle h\xi \rangle^{t_1}\mathcal{F}(\tilde{V}u)(\xi)\vert{\,\lesssim\,} \int_{\mathbb{R}^{n}}w_2(\xi-\eta )\,w_1(\eta )\,d\eta=(w_1 \ast w_2)(\xi).$$
Consequently
\begin{align*}
\displaystyle\Vert \tilde{V}u\Vert_{H_{\mathrm{scl}}^{t_1}(\mathbb{R}^n)}&{\,\lesssim\,} \displaystyle\Vert w_1 \ast w_2\Vert_{L^2(\mathbb{R}^{n})}\\
& {\,\lesssim\,} \,\Vert w_1 \Vert_{L^2(\mathbb{R}^{n})}\,\Vert w_2\Vert_{L^1(\mathbb{R}^{n})}\\
&{\,\lesssim\,} \,\Vert u\Vert_{H_{\mathrm{scl}}^{t_1}(\mathbb{R}^n)}\,\Vert V\Vert_{H^{t_2}(\Omega)}\Big(\int_{\mathbb{R}^{n}}\frac{d\xi}{(1+\vert \xi\vert^2)^{t_2-|t_1|}}\Big)^{1/2}.
\end{align*}
Since $t_2 > \frac{n}{2}+|t_1|$ then the last integral converge. This ends the proof.
\end{proof}
Armed with Lemma \ref{L2.2}, we are now in position to derive a Carleman estimate to the perturbed polyharmonic operator, which can be stated as follows.
\begin{lemma}\label{P2.3}
Let $A\in \mathcal{A}_{\sigma_1}(M)$, $B \in \mathcal{B}_{\sigma_2}(M)$ and $q \in \mathcal{Q}_{}(M)$ and let $\psi$ be the limiting Carleman weight function given by (\ref{2.2}) and $-2m\leq s\leq 0$. Then the local Carleman estimate
\begin{equation}\label{2.12}
h^m\norm{u}_{H_{\mathrm{scl}}^{s+2m}(\mathbb{R}^n)}{\,\lesssim\,} \norm{\mathcal{L}_\psi(x,hD)u}_{H_{\mathrm{scl}}^s(\mathbb{R}^n)}\quad m\geq2,
\end{equation}
holds for all $u\in C_0^\infty(\Omega)$ and all $h>0$ small enough.
\end{lemma}
\begin{proof}
To get the Carleman estimate for $\mathcal{L}_\psi(x,hD)$, we add the lower order perturbation to \eqref{2.7}. First we add the zero order perturbation $q$ then for $-2m \leq s \leq 0$, we get
\begin{align}\label{q1}
\|h^{2m} q u\|_{H_{\mathrm{scl}}^s(\mathbb{R}^n)} &\leq h^{2m} \|q\|_{L^\infty(\mathbb{R}^n)} \|u\|_{L^2(\mathbb{R}^n)}
{\,\lesssim\,} h^{2m} \|u\|_{H_{\mathrm{scl}}^{s+2m}(\mathbb{R}^n)}.
\end{align}
Next, we consider the first order perturbation $B$, we compute
\begin{align*}
h^{2m-1}e^{\frac{\psi_\varepsilon}{h}}(B\cdot hD)e^{-\frac{\psi_\varepsilon}{h}}u=h^{2m-1}[i(B\cdot \nabla \psi_\varepsilon)u+ B \cdot (hD)u].
\end{align*}
Moreover, we have $B\cdot(h D)u=h D\cdot (Bu)-h(D\cdot B)u$. Then, by using the fact that $\psi_\varepsilon$ satisfies \eqref{2.5}, we obtain, for $0<h<\varepsilon<1$, that
\begin{align}
\|h^{2m-1}e^{\frac{\psi_\varepsilon}{h}}(B\cdot hD)e^{-\frac{\psi_\varepsilon}{h}}u\|_{H_{\mathrm{scl}}^s(\mathbb{R}^n)}
& {\,\lesssim\,} h^{2m-1} \bigr( \|u\|_{H_{\mathrm{scl}}^{s+2m}(\mathbb{R}^n)}+ \|Bu\|_{H_{\mathrm{scl}}^{s+1}(\mathbb{R}^n)}\bigr).\label{Bu}
\end{align}
Now we estimate the last term in \eqref{Bu}, to do this we distinguish two cases: if $-2m \leq s \leq -1$, we have immediately
\begin{align}\label{bu_1}
\|Bu\|_{H_{\mathrm{scl}}^{s+1}(\mathbb{R}^n)} \leq \|B\|_{L^\infty(\Omega)} \|u\|_{H_{\mathrm{scl}}^{s+2m}(\mathbb{R}^n)},
\end{align}
and if $-1 \leq s \leq 0$, we get from Lemma \ref{L2.2}, for $t_2:=\sigma_2 > \frac{n}{2}+1$ and $t_1:=s+1$,
\begin{align}\label{bu_2}
\|Bu\|_{H_{\mathrm{scl}}^{s+1}(\mathbb{R}^n)} \leq \|B\|_{H^{\sigma_2}(\Omega)} \|u\|_{H_{\mathrm{scl}}^{s+1}(\mathbb{R}^n)}.
\end{align}
Now plugging \eqref{bu_1} and \eqref{bu_2} into \eqref{Bu}, we conclude for $-2m \leq s \leq 0$,
\begin{align}\label{B_}
\|h^{2m-1}e^{\frac{\psi_\varepsilon}{h}}(B\cdot hD)e^{-\frac{\psi_\varepsilon}{h}}u\|_{H_{\mathrm{scl}}^s(\mathbb{R}^n)} {\,\lesssim\,} h^{2m-1} \|u\|_{H_{\mathrm{scl}}^{s+2m}(\mathbb{R}^n)}.
\end{align}
We end up with the term involving the second order perturbation $A$, by a simple computation we have
\begin{align*}
h^{2m-2}e^{\frac{\psi_\varepsilon}{h}}A(hD)\cdot (hD) e^{-\frac{\psi_\varepsilon}{h}}u&=h^{2m-2} A D \psi_\varepsilon \cdot D \psi_\varepsilon u-2h^{2m-2} A D \psi_\varepsilon \cdot (hD) u\cr
&\quad-h^{2m-1} A D \cdot D \psi_\varepsilon u+h^{2m-2} A (hD) \cdot (hD) u .
\end{align*}
Since
$$ A D \psi_\varepsilon \cdot (hD)u=\sum_{j,k=1}^{n}h D_k(A_{jk}(D_j\psi_\varepsilon) u)-\sum_{j,k=1}^{n}h D_k(A_{jk}D_j\psi_\varepsilon )u,
$$
and
\begin{align*}
A (hD) \cdot (hD) u&=\sum_{j,k=1}^{n} h^2 D _{j}D_{k}(A_{jk}u)-h^2 \sum_{j,k=1}^{n} ( D_{j}D_{k}A_{jk})u-2h\sum_{j,k=1}^{n}
(D_{j}A_{jk})(h D_{k}u) \\
&=\sum_{j,k=1}^{n} h^2 D _{j}D_{k}(A_{jk}u)+h^2 \sum_{j,k=1}^{n} ( D_{j}D_{k}A_{jk})u-2h\sum_{j,k=1}^{n}
h D_{k}\bigr(D_{j}(A_{jk})u\bigr).
\end{align*}
Then we obtain, for $0<h<\varepsilon<1$
\begin{align*}
& \| h^{2m-2}e^{\frac{\psi_\varepsilon}{h}}A(hD)\cdot (hD) e^{-\frac{\psi_\varepsilon}{h}}u\|_{H_{\mathrm{scl}}^s(\mathbb{R}^n)} \\&{\,\lesssim\,} h^ {2m-2} \bigr( \| u\|_{H_{\mathrm{scl}}^{s+2m}}+\sum_{j,k=1}^{n} \bigr(\|A_{jk}(D_j\psi_\varepsilon) u\|_{H_{\mathrm{scl}}^{s+2}}+ \|A_{jk} u\|_{H_{\mathrm{scl}}^{s+2}}
+h \|D_{j}(A_{jk}) u\|_{H_{\mathrm{scl}}^{s+2}}\bigr)\bigr).
\end{align*}
Again to estimate the last three terms in the above inequality, we distinguish two cases: if $-2 \leq s \leq 0$ , we obtain from Lemma \ref{L2.2}
\begin{align*}
&\| h^{2m-2}e^{\frac{\psi_\varepsilon}{h}}A(hD)\cdot (hD) e^{-\frac{\psi_\varepsilon}{h}}u\|_{H_{\mathrm{scl}}^s(\mathbb{R}^n)}\cr &{\,\lesssim\,} h^ {2m-2} \Big( \| u\|_{H_{\mathrm{scl}}^{s+2m}(\mathbb{R}^n)}+\sum_{j,k=1}^{n}\bigr( \|A_{jk}D_j\psi_\varepsilon \|_{H^{\sigma_1}(\Omega)}\| u\|_{H_{\mathrm{scl}}^{s+2}(\mathbb{R}^n)}+\|A_{jk} \|_{H^{\sigma_1}(\Omega)}\| u\|_{H_{\mathrm{scl}}^{s+2}(\mathbb{R}^n)}\\&\qquad+\|D_{j}A_{jk} \|_{H^{\sigma_1-1}(\Omega)} \| u\|_{H_{\mathrm{scl}}^{s+2}(\mathbb{R}^n)}\bigr)\Big)
\\&{\,\lesssim\,} h^ {2m-2} \| u\|_{H_{\mathrm{scl}}^{s+2m}(\mathbb{R}^n)},
\end{align*}
with $\sigma_1 > \frac{n}{2}+3$. Now if $-2m \leq s < -2$, we get immediately
\begin{align*}
\| h^{2m-2}e^{\frac{\psi_\varepsilon}{h}}A(hD)\cdot (hD) e^{-\frac{\psi_\varepsilon}{h}}u\|_{H_{\mathrm{scl}}^s(\mathbb{R}^n)} &{\,\lesssim\,}
h^ {2m-2} \| u\|_{H_{\mathrm{scl}}^{s+2m}(\mathbb{R}^n)}.
\end{align*}
Then, we conclude for $-2m \leq s \leq 0$,
\begin{align}\label{A_}
\| h^{2m-2}e^{\frac{\psi_\varepsilon}{h}}A(hD)\cdot (hD) e^{-\frac{\psi_\varepsilon}{h}}u\|_{H_{\mathrm{scl}}^s(\mathbb{R}^n)} &{\,\lesssim\,} h^ {2m-2} \| u\|_{H_{\mathrm{scl}}^{s+2m}(\mathbb{R}^n)}.
\end{align}
Combining \eqref{q1}, \eqref{B_} and \eqref{A_} we get from \eqref{2.7}
\begin{align*}
( \frac{h}{\sqrt{\varepsilon}})^m \| u\|_{H_{\mathrm{scl}}^{s+2m}(\mathbb{R}^n)} {\,\lesssim\,} \| h^{2m}e^{\frac{\psi_\varepsilon}{h}}\mathcal{L}_{A, B, q} e^{-\frac{\psi_\varepsilon}{h}}u\|_{H_{\mathrm{scl}}^{s}(\mathbb{R}^n)}+h^{2m-2}\|u\|_{H_{\mathrm{scl}}^{s+2m}(\mathbb{R}^n)}.
\end{align*}
Choosing now $h < \varepsilon <1$ small enough, we get our desired estimate.
\end{proof}
Let $\mathcal{L}^*_\psi(x,hD)$ the formal adjoint of $\mathcal{L}_\psi(x,hD)$ given by $(\mathcal{L}_\psi u, v )_{L^2(\Omega)}= ( u, \mathcal{L}^*_\psi v )_{L^2(\Omega)},~~u,v \in C_{0}^{\infty }(\Omega )$. Notice that $\mathcal{L}^*_\psi(x,hD)$ has the same form as $\mathcal{L}_\psi(x,hD)$ except $\psi$ is replaced by $-\psi$ and $A,B$ and $q$ are replaced by
\begin{equation}\label{2.19}
A_{jk}^*=\overline{A_{jk}},\quad B_k^*=\overline{B_k}+2\sum_{j=1}^n D_j \overline{A_{jk}},\quad q^*=\overline{q}-i\textrm{div}\overline{B}+\sum_{j,k=1}^n D_k D_j \overline{A_{jk}},\quad j,k=1, \ldots, n.
\end{equation}
Since $A^*$, $B^*$ and $q^*$ lies in the same admissible sets as $A, B$ and $q$, then we have the same local Carleman estimate for $\mathcal{L}^*_\psi(x,hD)$.
\begin{lemma}\label{P2.4}
Let $A\in \mathcal{A}_{\sigma_1}(M)$, $B \in \mathcal{B}_{\sigma_2}(M)$ and $q \in \mathcal{Q}_{}(M)$ and let $\psi$ be the limiting Carleman weight function given by (\ref{2.2}) and $-2m\leq s\leq 0$. Then the local Carleman estimate
\begin{equation}\label{2.20}
h^m\norm{u}_{H_{\mathrm{scl}}^{s+2m}(\mathbb{R}^n)}{\,\lesssim\,} \norm{\mathcal{L}^*_\psi(x,hD)u}_{H_{\mathrm{scl}}^s(\mathbb{R}^n)},
\end{equation}
holds for all $u\in C_0^\infty(\Omega)$, and all $h>0$ small enough.
\end{lemma}
As a consequence of the Carleman estimate (\ref{2.20}) and the Hahn-Banach theorem, we have the following solvability result for $\mathcal{L}_\psi(x,hD)$. The proof is essentielly well-known and we omit it (see \cite{Gk, K2, K1}).
\begin{lemma}\label{P2.5}
Let $A\in \mathcal{A}_{\sigma_1}(M)$, $B \in \mathcal{B}_{\sigma_2}(M)$ and $q \in \mathcal{Q}_{}(M)$ and let $\psi$ be the limiting Carleman weight function given by (\ref{2.2}). Then for any $\mathrm{g}\in L^2(\Omega)$ the equation
\begin{equation}\label{2.21}
\mathcal{L}_\psi(x,hD)r=\mathrm{g}\quad \textrm{in}\,\,\Omega,
\end{equation}
has a solution $r\in H^{2m}(\Omega)$, which satisfying the estimate
\begin{equation}\label{2.22}
h^m\norm{r}_{H_{\mathrm{scl}}^{2m}(\Omega)}{\,\lesssim\,} \norm{\mathrm{g}}_{L^2(\Omega)},
\end{equation}
for all $h>0$ small enough.
\end{lemma}
\subsection{Construction of C.G.O- solutions}
We now give the construction of special solution to the polyharmonic equation $\mathcal{L}_{A,B,q}(x,D)u=0$ in $\Omega$. This proceeds as in \cite{B2} except we precise the constant on the construction of the remainder term in the C.G.O- solutions. The constructed solutions will have the special form
\begin{equation}\label{2.23}
u(x,h)=e^{\frac{\varphi}{h}}(\alpha_1(x)+h \alpha_2(x)+r(x,h)),\quad x\in\Omega,
\end{equation}
where $\varphi(x)=\psi(x)+i\widetilde{\psi}(x)$, $\psi$ and $\widetilde{\psi}$ are the real valued functions given by
\begin{equation}\label{2.24}
\psi(x)=x\cdot\omega,\quad \widetilde{\psi}(x)=x\cdot\tilde{\omega},
\end{equation}
and $\omega,\widetilde{\omega}\in\mathbb{S}^{n-1}$, with $\omega\cdot\tilde{\omega}=0$. Let $\varrho=\omega+i\tilde{\omega}$, the functions $\alpha_1$ and $\alpha_2$ as an amplitude which respectively solve the transport equations
\begin{equation}\label{2.25}
T_{\varrho}(x,D)\alpha_1:=(\varrho\cdot\nabla)^m\alpha_1=0,\quad \textrm{in}\,\,\Omega,\quad\textrm{for}~m \geq 2,
\end{equation}
and
\begin{equation}\label{2.26}
(2\varrho\cdot\nabla)^m\alpha_2=\left\{\begin{matrix}
\bigr(-4\Delta(\varrho \cdot \nabla)+2A\varrho \cdot \nabla +i \varrho \cdot B\bigr) \alpha_1,&\quad \textrm{if}~~m=2, \\
\bigr(-12\Delta(\varrho \cdot \nabla)^{2}-A\varrho \cdot \varrho\bigr)\alpha_1,&\quad \textrm{if}~~m=3,\\
-m\Delta(2\varrho \cdot \nabla)^{m-1}\alpha_1,&\quad \textrm{if}~~ m>3.
\end{matrix}\right.
\end{equation}
Finally, the correction term $r(\cdot, h)\in H^{2m}(\Omega)$.\\
The solution of the transport equation \eqref{2.25} always exists and can be taken in $C^\infty(\overline{\Omega})$. We will present the existence of complex amplitude $\alpha_2$ satisfying \eqref{2.26} in Subsection \ref{Sect3}.
\begin{lemma}\label{L2.6}
Let $A\in \mathcal{A}_{\sigma_1}(M)$, $B \in \mathcal{B}_{\sigma_2}(M)$ and $q \in \mathcal{Q}_{}(M)$ and $\varphi$ as above. Then for any $\alpha_1 \in C^\infty(\overline{\Omega})$ and $\alpha_2\in H^{2m}(\Omega)$ satisfying respectively the transport equations (\ref{2.25}) and \eqref{2.26} and for all $h>0$ small enough, we can construct a solution to $\mathcal{L}_{A,B,q}(x,D)u=0$, $x\in\Omega$, of the form
\begin{equation}\label{2.27}
u(x,h)=e^{\frac{\varphi}{h}}(\alpha_1(x)+h \alpha_2(x)+r(x,h)),\quad x\in\Omega,
\end{equation}
where $\varphi(x)=\psi(x)+i\widetilde{\psi}(x)=x\cdot\varrho$. Moreover the correction term $r(\cdot,h)\in H^{2m}(\Omega)$, satisfying
\begin{equation}\label{2.28}
\norm{r(\cdot,h)}_{H_{\mathrm{scl}}^{2m}(\Omega)}{\,\lesssim\,} h^2 (\|\alpha_1\|_{H^{2m}(\Omega)}+\| \alpha_2\|_{H^{2m}(\Omega)}).
\end{equation}
\end{lemma}
\begin{proof}
Let $\alpha_1\in C^\infty(\overline{\Omega})$ and $\alpha_2\in H^{2m}(\Omega)$ satisfying respectively the transport equations (\ref{2.25}) and \eqref{2.26}. We denote $\mathrm{g}_h\in L^2(\Omega)$ given by
\begin{equation}\label{2.29}
\mathrm{g}_h(x)=-e^{i\frac{\widetilde{\psi}}{h}}\mathcal{L}_{-\varphi}(x,hD)\bigr(\alpha_1(x)+h \alpha_2(x)\bigr),\quad x\in\Omega,
\end{equation}
where $\mathcal{L}_{-\varphi}(x,hD)$ denote the conjugated operator
\begin{equation}\label{2.30}
\mathcal{L}_{-\varphi}(x,hD)=e^{-\frac{\varphi}{h}}h^{2m}\mathcal{L}_{A,B,q}(x,D)e^{\frac{\varphi}{h}}.
\end{equation}
By a simple computation, we get
\begin{align*}
\hspace{-2 cm}-e^{-i\frac{\widetilde{\psi}}{h}}\mathrm{g}_h&= \Big((-h^2\Delta-2h\varrho\cdot\nabla)^m+h^{2m-2}A(hD-i\varrho)\cdot (hD-i\varrho)+h^{2m-1}B\cdot(hD-i\varrho)\cr
&\qquad +h^{2m}q\Big)(\alpha_1+h \alpha_2)\cr
&=\Big((-1)^m \sum_{k=0}^{m}\binom{m}{k}h^{2m-k}(\Delta)^{m-k}(2\varrho\cdot\nabla)^{k} + h^{2m} ( AD\cdot D+B\cdot D+q )\cr
&\quad +h^{2m-1} (-2i A \varrho \cdot D-i \varrho \cdot B )-h^{2m-2} (A \varrho \cdot \varrho ) \Big)(\alpha_1+h \alpha_2)\cr
&= h^{2m}\bigr( \mathcal{L}_{A, B, q}(x, D) (\alpha_1+h \alpha_2)-2i A \varrho \cdot D\alpha_2-i \varrho \cdot B\alpha_2\bigr)\cr
&\quad+(-1)^m \sum_{k=1}^{m}\binom{m}{k}h^{2m-k}(\Delta)^{m-k}(2\varrho\cdot\nabla)^{k}(\alpha_1+h \alpha_2)\cr&\quad+h^{2m-1} \bigr(-2i A \varrho \cdot D\alpha_1-i \varrho \cdot B \alpha_1-A \varrho \cdot \varrho \alpha_2\bigr)-h^{2m-2} A \varrho \cdot \varrho \alpha_1.
\end{align*}
Since $\alpha_1$ satisfying the transport equation \eqref{2.25}, we get
\begin{align}\label{2.31}
-e^{-i\frac{\widetilde{\psi}}{h}}\mathrm{g}_h&= h^{2m}\bigr( \mathcal{L}_{A, B, q}(x, D) (\alpha_1+h \alpha_2)-2i A \varrho \cdot D\alpha_2-i \varrho \cdot B\alpha_2\bigr)\cr
&\quad+(-1)^m \sum_{k=1}^{m-1}\binom{m}{k}h^{2m-k}(\Delta)^{m-k}(2\varrho\cdot\nabla)^{k}(\alpha_1+h \alpha_2)+ h^{m+1}(-1)^m (2\varrho\cdot\nabla)^{m}\alpha_2\cr
&\quad+h^{2m-1} \bigr(-2i A \varrho \cdot D\alpha_1-i \varrho \cdot B \alpha_1-A \varrho \cdot \varrho \alpha_2\bigr)-h^{2m-2} A \varrho \cdot \varrho \alpha_1.
\end{align}
Now we need to show that the terms in right hand side of \eqref{2.31} are terms of order $\mathcal{O}(h^{m+2})$, $m \geq 2$, to do this we distinguish two cases. First, if $m=2$, in that case using the fact that $A$ is isotropic we find out from \eqref{2.31} that
\begin{align*}
-e^{-i\frac{\widetilde{\psi}}{h}}\mathrm{g}_h&= h^{4}\bigr( \mathcal{L}_{A, B, q}(x, D) (\alpha_1+h \alpha_2)-2i A \varrho \cdot D\alpha_2-i \varrho \cdot B\alpha_2+4\Delta (\varrho\cdot\nabla) \alpha_2\bigr)\cr&\quad+ h^{3}\bigr( (2\varrho\cdot\nabla)^{2}\alpha_2-2i A \varrho \cdot D\alpha_1-i \varrho \cdot B \alpha_1+4\Delta (\varrho\cdot\nabla) \alpha_1\bigr).
\end{align*}
In order to get $\|e^{-i\frac{\widetilde{\psi}}{h}}\mathrm{g}_h\|_{L^2(\Omega)}=\mathcal{O}(h^{4})$, we choose the amplitude $\alpha_2 \in H^4(\Omega)$ such that
\begin{align*}
(2\varrho\cdot\nabla)^{2}\alpha_2=2i A \varrho \cdot D\alpha_1+i \varrho \cdot B \alpha_1-4\Delta (\varrho\cdot\nabla) \alpha_1.
\end{align*}
Having chosen $\alpha_2$ in this way, we obtain for $m=2$, the
following equation
\begin{align}\label{2.32}
-e^{-i\frac{\widetilde{\psi}}{h}}\mathrm{g}_h= h^{2m}\bigr( \mathcal{L}_{A, B, q}(x, D) (\alpha_1+h \alpha_2)-2i A \varrho \cdot D\alpha_2-i B\cdot \varrho\alpha_2+4\Delta (\varrho\cdot\nabla) \alpha_2\bigr).
\end{align}
Now, if $m>2$ we get from \eqref{2.31}
\begin{align}\label{2.33}
-e^{-i\frac{\widetilde{\psi}}{h}}\mathrm{g}_h&= h^{2m}\bigr( \mathcal{L}_{A, B, q}(x, D) (\alpha_1+h \alpha_2)-2i A \varrho \cdot D\alpha_2-i \varrho \cdot B\alpha_2\bigr)\cr
&\quad+(-1)^m \sum_{k=1}^{m-2}\binom{m}{k}h^{2m-k}(\Delta)^{m-k}(2\varrho\cdot\nabla)^{k}(\alpha_1+h \alpha_2)\cr&\quad+
h^{m+2}(m(-1)^m \Delta (2\varrho\cdot\nabla)^{m-1}\alpha_2)\cr
&\quad+h^{m+1}\bigr((-1)^m (2\varrho\cdot\nabla)^{m}\alpha_2+(-1)^m m \Delta (2\varrho\cdot\nabla)^{m-1}\alpha_1\bigr)\cr&\quad+h^{2m-1} \bigr(-2i A \varrho \cdot D\alpha_1-i \varrho \cdot B \alpha_1-A \varrho \cdot \varrho \alpha_2\bigr)-h^{2m-2} A \varrho \cdot \varrho \alpha_1.
\end{align}
We observe that for $1 \leq k \leq m-2$, we have $h^{2m-k} \leq h^{m+2}$, then we deduce
\begin{align*}
-e^{-i\frac{\widetilde{\psi}}{h}}\mathrm{g}_h&= \mathcal{O}(h^{m+2}) +h^{m+1}\bigr((-1)^m (2\varrho\cdot\nabla)^{m}\alpha_2+(-1)^m m \Delta (2\varrho\cdot\nabla)^{m-1}\alpha_1\bigr)-h^{2m-2} A \varrho \cdot \varrho \alpha_1.
\end{align*}
Notice that $h^{2m-2}=h^{m+1}$ for $m=3$, and $h^{2m-2} \leq h^{m+2}$ for $m>3$. Then in order to obtain $ -e^{-i\frac{\widetilde{\psi}}{h}}\mathrm{g}_h= \mathcal{O}(h^{m+2})$, we distinguish again two cases : first, if $m=3$, we get from \eqref{2.33}
\begin{align*}
-e^{-i\frac{\widetilde{\psi}}{h}}\mathrm{g}_h&= h^{6}\bigr( \mathcal{L}_{A, B, q}(x, D) (\alpha_1+h \alpha_2)-2i A \varrho \cdot D\alpha_2-i \varrho \cdot B\alpha_2\bigr)\cr&\quad+h^{5}\bigr(- 3 (\Delta)^{2}(2\varrho\cdot\nabla)(\alpha_1+h \alpha_2)
-3 \Delta (2\varrho\cdot\nabla)^{2}\alpha_2-2i A \varrho \cdot D\alpha_1-i \varrho \cdot B \alpha_1-A \varrho \cdot \varrho \alpha_2\bigr)\cr&\quad+h^{4}\bigr(- (2\varrho\cdot\nabla)^{3}\alpha_2-3 \Delta (2\varrho\cdot\nabla)^{2}\alpha_1- A \varrho \cdot \varrho \alpha_1\bigr).
\end{align*}
We choose the amplitude $\alpha_2 \in H^6(\Omega)$ satisfies
\begin{align*}
(2\varrho\cdot\nabla)^{3}\alpha_2=-3 \Delta (2\varrho\cdot\nabla)^{2}\alpha_1- A \varrho \cdot \varrho \alpha_1,\quad \textrm{in}~\Omega.
\end{align*}
Hence, we can immediately conclude that
\begin{align}\label{2.34}
-e^{-i\frac{\widetilde{\psi}}{h}}\mathrm{g}_h&= h^{6}\bigr( \mathcal{L}_{A, B, q}(x, D) (\alpha_1+h \alpha_2)-2i A \varrho \cdot D\alpha_2-i \varrho \cdot B\alpha_2\bigr)\cr&\quad+h^{5}\bigr(- 3 (\Delta)^{2}(2\varrho\cdot\nabla)(\alpha_1+h \alpha_2)
-3 \Delta (2\varrho\cdot\nabla)^{2}\alpha_2\cr &\quad-2i A \varrho \cdot D\alpha_1-i \varrho \cdot B \alpha_1-A\varrho \cdot \varrho \alpha_2\bigr).
\end{align}
Finally, if $m>3$, we choose the amplitude $\alpha_2 \in H^{2m}(\Omega)$ satisfies
\begin{align*}
(2\varrho\cdot\nabla)^{m}\alpha_2=- m \Delta (2\varrho\cdot\nabla)^{m-1}\alpha_1,\quad \textrm{in}~\Omega, \end{align*}
which allows us to obtain from \eqref{2.33}
\begin{align}\label{2.35}
-e^{-i\frac{\widetilde{\psi}}{h}}\mathrm{g}_h&= h^{2m}\bigr( \mathcal{L}_{A, B, q}(x, D) (\alpha_1+h \alpha_2)-2i \varrho \cdot D\alpha_2-i \cdot \varrho\alpha_2\bigr)\cr &\quad+h^{m+2}(m(-1)^m \Delta (2\varrho\cdot\nabla)^{m-1}\alpha_2)+(-1)^m \sum_{k=1}^{m-2}\binom{m}{k}h^{2m-k}(\Delta)^{m-k}(2\varrho\cdot\nabla)^{k}(\alpha_1+h \alpha_2)\cr&\quad+
h^{2m-1} \bigr(-2i A \varrho \cdot D\alpha_1-i \varrho \cdot B \alpha_1-A \varrho \cdot \varrho \alpha_2\bigr)-h^{2m-2} A \varrho \cdot \varrho \alpha_1.
\end{align}
We deduce from \eqref{2.32}, \eqref{2.34} and \eqref{2.35} that
\begin{equation}\label{2.36}
\norm{e^{-i\frac{\widetilde{\psi}}{h}}\mathrm{g}_h}_{L^2(\Omega)}{\,\lesssim\,} h^{m+2}(\|\alpha_1\|_{H^{2m}(\Omega)}+\| \alpha_2\|_{H^{2m}(\Omega)}),\quad m \geq 2.
\end{equation}
So, by Lemma \ref{P2.5}, for $h>0$ sufficiently small, we can conclude that there exists $\widetilde{r}(\cdot,h)\in H^{2m}(\Omega)$ solving
$$
\mathcal{L}_{-\psi}(x,hD)\widetilde{r}(x,h)=\mathrm{g}_h(x)\quad \textrm{in} \,\,\Omega,
$$
such that
\begin{equation*}
h^m\norm{\widetilde{r}(\cdot,h)}_{H_{\mathrm{scl}}^{2m}(\Omega)}{\,\lesssim\,} \norm{\mathrm{g}_h}_{L^2(\Omega)}\leq h^{m+2}(\|\alpha_1\|_{H^{2m}(\Omega)}+\| \alpha_2\|_{H^{2m}(\Omega)}).
\end{equation*}
We denote $r(\cdot,h)=e^{-i\frac{\widetilde{\psi}}{h}}\widetilde{r}(\cdot,h)$. Therefore
\begin{equation*}
\mathcal{L}_{A,B,q}(x,D)(e^{\frac{\varphi}{h}}(\alpha_1(x)+h \alpha_2(x)+ r(x,h))=0,
\end{equation*}
with $r(\cdot, h) \in H^{2m}(\Omega)$, such that
\begin{equation}
\norm{r(\cdot,h)}_{H_{\mathrm{scl}}^{2m}(\Omega)}{\,\lesssim\,} h^2 (\|\alpha_1\|_{H^{2m}(\Omega)}+\| \alpha_2\|_{H^{2m}(\Omega)}).
\end{equation}
The proof is complete.
\end{proof}
Similarly, let consider the following transport equations
\begin{equation}\label{2.38}
T^*_{\varrho}(x,D)\alpha_1^*:=(\overline{\varrho}\cdot\nabla)^m \alpha_1^*=0,\quad \textrm{in}\,\,\Omega,
\end{equation}
and
\begin{equation}\label{2.39}
(2\overline{\varrho}\cdot\nabla)^m\alpha_2^*=\left\{\begin{matrix}
\bigr(4\Delta(\overline{\varrho} \cdot \nabla)-2{A^*}\overline{\varrho} \cdot \nabla -i {B^*} \cdot \varrho\bigr) \alpha_1^*,&\quad \textrm{if}~~m=2, \\
\bigr(-12\Delta(\overline{\varrho} \cdot \nabla)^{2}-{A^*}\overline{\varrho} \cdot \overline{\varrho}\bigr)\alpha_1^*,&\quad \textrm{if}~~m=3,\\
-m\Delta(-2\overline{\varrho} \cdot \nabla)^{m-1}\alpha_1^*,&\quad \textrm{if}~~ m>3.
\end{matrix}\right.
\end{equation}
where $\varrho=\omega+i\tilde{\omega}$ and $A^*$, $B^*$ are given by \eqref{2.19}. We can similarly prove the existence of C.G.O solutions to the adjoint operator $\mathcal{L}^*_{A,B,q}(x,D)$.
\begin{lemma}\label{L2.7}
Let $A\in \mathcal{A}_{\sigma_1}(M)$, $B \in \mathcal{B}_{\sigma_2}(M)$ and $q \in \mathcal{Q}_{}(M)$ and $\varphi$ as above. Then for any $\alpha_1^*\in C^\infty(\overline{\Omega})$ and $\alpha_2^*\in H^{2m}(\Omega)$ satisfying respectively the transport equations (\ref{2.38}) and \eqref{2.39} and all $h>0$ small enough, we can construct a solution to $\mathcal{L}^*_{A,B,q}(x,D)u=0$, $x\in\Omega$, of the form
\begin{equation}\label{2.40}
u^*(x,h)=e^{-\frac{\overline{\varphi}}{h}}(\alpha_1^*(x)+h\alpha_2^*(x)+r^*(x,h)),\quad x\in\Omega,
\end{equation}
where $\varphi(x)=\psi(x)+i\widetilde{\psi}(x)=x\cdot\varrho$. Moreover the correction term $r^*(\cdot,h)\in H^{2m}(\Omega)$, satisfying
\begin{equation}\label{2.41}
\norm{r^*(\cdot,h)}_{H_{\mathrm{scl}}^{2m}(\Omega)}{\,\lesssim\,} h^2(\|\alpha_1^*\|_{H^{2m}(\Omega)}+\| \alpha_2^*\|_{H^{2m}(\Omega)}).
\end{equation}
\end{lemma}
\subsection{Solvability of the transport equation}\label{Sect3}
We will now establish the existence of complex amplitude $\alpha_2$ satisfying \eqref{2.26}. We will follow the same approach used in \cite{IS}. Let $P(D)$ be a differential operator with constant coefficients \begin{align}
P(D)=\sum_{\abs{\alpha}\leq m} a_\alpha D^\alpha,\quad D=-i\nabla,\quad a_\alpha \in \mathbb{C}.
\end{align}
We associate to the operator $P(D)$ its full symbol $p(\xi)$ given by
\begin{align}
p(\xi)=\sum_{\abs{\alpha}\leq m} a_\alpha \xi^\alpha,\quad \xi \in \mathbb{R}^n.
\end{align}
Moreover, we set
\begin{align}\label{p_tilde}
\widetilde{p}(\xi)=\bigr(\sum_{\abs{\alpha}\leq m} \abs{\partial_\xi^\alpha p(\xi)}^2 \bigr)^{\frac{1}{2}},\quad \xi \in \mathbb{R}^n.
\end{align}
Let us first recall the following result due to \cite{IS}, where it is obtained as a consequence of the general theory of \cite{Hor}.
\begin{lemma}\label{L2.8}
Let $P(D) \neq 0$, be a differential operator with constant coefficients. Then for all $k \in \mathbb{N}$, there exists a linear operator $E \in \mathcal{B}(H^k(\Omega))$ such that $P(D)(E(f))=f$, for all $f \in H^k(\Omega)$. Moreover, there exists $C >0$ such that
\begin{align}\label{2.45}
\|E(f)\|_{H^k(\Omega)} \leq C \bigr(\underset{\xi \in \mathbb{R}^n}{\operatorname{sup}} \frac{1}{\widetilde{p}(\xi)} \bigr) \|f\|_{H^k(\Omega)}, \quad \forall f \in H^k(\Omega).
\end{align}
Here $C$ depends only on the order of $P(D)$, $n$ and $\Omega$.
\end{lemma}
For $\varrho=\omega + i \widetilde{\omega} \in \mathbb{S}^{n-1}+i \mathbb{S}^{n-1}$ as above, we consider the differential operator with constant coefficients
\begin{align}
T_{\varrho}(D)=(\varrho\cdot\nabla)^m , \quad \textrm{for}~~m \geq 2.
\end{align}
As a consequence of the above Lemma we have the following result.
\begin{lemma}\label{L2.9}
The operator $ T_{\varrho}(D)$ admits a bounded inverse $ E_{\varrho}$ from $H^k(\Omega)$ to $H^k(\Omega)$, such that, if $f \in H^k(\Omega)$ and $v=E_{\varrho}f \in H^k(\Omega)$, we have
\begin{align}\label{2.47}
\|v\|_{H^k(\Omega)} \leq C \|f\|_{H^k(\Omega)}.
\end{align}
Here $C >0$ is a positive constant depends only on $n$, $m$ and $\Omega$.
\end{lemma}
\begin{proof}
From Lemma \ref{L2.8}, there exists a linear operator $E_{\varrho}\in \mathcal{B}(H^k(\Omega))$ such that
$$T_{\varrho}\circ E_{\varrho}f=f, \quad\textrm{for any } f \in H^k(\Omega).$$
Moreover, since the symbol of the differential operator $T_{\varrho}(D)$ is given by $$p_{\varrho}(\xi)=i^m(\varrho \cdot \xi)^m,\quad \xi \in \mathbb{R}^n.$$
So we have, by \eqref{p_tilde} and H\"older's inequality
$$(\widetilde{p}_{\varrho}(\xi))^2 \geq \sum_{j=1}^n \abs{m! \varrho_j^m}^2 \geq \frac{(m!)^2}{n^{m-1}}\bigr(\sum_{j=1}^n \abs{\varrho_j }^2\bigr)^m \geq \frac{2^m (m!)^2}{n^{m-1}}.$$
Then, from \eqref{2.45} we get
$$ \|E_{\varrho}f\|_{H^k(\Omega)} \leq C \|f\|_{H^k(\Omega)}.$$
This completes the proof of the Lemma.
\end{proof}
We are in the position now to establish the existence of amplitude $\alpha_2$ satisfying \eqref{2.26}.
\begin{lemma}\label{L2.10}
Let $A\in \mathcal{A}_{\sigma_1}(M)$, $B \in \mathcal{B}_{\sigma_2}(M)$ and $\alpha_1 \in C^\infty(\overline{\Omega})$ satisfies \eqref{2.25}. Then, there exists $\alpha_2 \in H^{2m}(\Omega)$ solving the transport equation
\eqref{2.26}.
Moreover, there exists $C>0$ such that we have
\begin{align}\label{2.50}
\|\alpha_2\|_{H^{2m}(\Omega)} \leq C\big( \|\alpha_1\|_{H^{2m}(\Omega)}+\|(\varrho \cdot \nabla)^{m-1} \alpha_1\|_{H^{2m+2}(\Omega)}\big).
\end{align}
Here $C$ is a positive constant depends only on $n$, $m$ and $\Omega$.
\end{lemma}
\begin{remark}
Notice from \eqref{2.26} and Lemma \eqref{L2.9}, that the regularity of the complex
amplitude $\alpha_2$
depends on the regularity of the coefficients $A$ and $B$. This justifies our extra regularity assumption on the coefficients. We hope that the regularity condition imposed on $2-$tensor field can be weakened from $W^{\operatorname{min}(2m, 6), \infty}(\Omega)$ to $W^{2, \infty}(\Omega)$, if we replace $A$ in \eqref{2.26} by its regularization $A_h$ given by
$$A_h=\chi_h \ast A,$$
where $\chi_h(x)=h^{-n\sigma } \chi(h^{-\sigma }x)$ is the usual mollifier and $\sigma \in (0,1/2)$.
\end{remark}
In a similar manner, we can prove that
\begin{lemma}\label{L2.11}
Let $A\in \mathcal{A}_{\sigma_1}(M)$, $B \in \mathcal{B}_{\sigma_2}(M)$ and $\alpha_1^* \in C^\infty(\overline{\Omega})$ satisfies \eqref{2.38}. Then, there exists $\alpha_2^* \in H^{2m}(\Omega)$ solving the transport equation \eqref{2.39}. Moreover, there exists $C>0$ such that we have
\begin{align}\label{2.53}
\|\alpha_2^*\|_{H^{2m}(\Omega)} \leq C \big(\|\alpha_1^*\|_{H^{2m}(\Omega)}+\|(\overline{\varrho} \cdot \nabla)^{m-1} \alpha_1^*\|_{H^{2m+2}(\Omega)}\big).
\end{align}
Here $C$ is a positive constant depends only on $n$, $m$ and $\Omega$.
\end{lemma}
For the sake of simplicity, the norm in right hand side of \eqref{2.50} and \eqref{2.53} is denoted by
\begin{align}
N_{2m, \varrho}(\alpha):=\|\alpha\|_{H^{2m}(\Omega)}+\|(\varrho \cdot \nabla)^{m-1} \alpha\|_{H^{2m+2}(\Omega)}.
\end{align}
\section{Integral estimates and Hodge decomposition}
In this section, we will use the properties of the Dirichlet-to-Neumann map to prove an integral estimate, which relates the difference of coefficients to the difference of the corresponding Dirichlet-to-Neumann map. This integral estimate will be our starting point in proving the stability estimates for the inverse problem under consideration. In the second part of this section we give a specific Hodge decomposition of a symmetric tensor $A$ and a vector field $B$. \smallskip \\
As in the first section we consider a priori constant $M>0$, $\sigma_1 > \frac{n}{2}+3$ and $\sigma_2 > \frac{n}{2}+1$. Also, we consider for $j=1,2$, $A^{(j)} \in \mathcal{A}_{\sigma_1}(M)\cap \mathcal{E}^\prime(\Omega)$, $B^{(j)} \in \mathcal{B}_{\sigma_2}(M)\cap \mathcal{E}^\prime(\Omega)$ and $q^{(j)}\in \mathcal{Q}_{}(M)$ pair of admissible coefficients. We denote
\begin{equation}\label{3.1}
A=A^{(2)}-A^{(1)},\qquad B=B^{(2)}-B^{(1)}\quad\text{and }~~~~ q=q^{(2)}-q^{(1)}.
\end{equation}
We extend $A$, $B$ and $q$ by zero outside
$\Omega$. In the sequel of the next, we denote by $A$, $B$ and $q$ these extensions and by $\mathcal{L}_j(x,D)$, for $j=1,2,$ the operator corresponding to the perturbation $(A^{(j)},B^{(j)},q^{(j)})$, that is
\begin{equation}
\mathcal{L}_{j}(x,D)=\mathcal{L}_{A^{(j)},B^{(j)},q^{(j)}}(x,D),\quad j=1,2.
\end{equation}
For notational convenience, the Dirichlet to Neumann map is denoted by
\begin{equation}
\Lambda^{(j)}=\Lambda_{A^{(j)},B^{(j)},q^{(j)}},\quad j=1,2.
\end{equation}
The key idea in the stability result is to use complex geometric optics solutions $u$ to $\mathcal{L}_2(x, D)u=0$ in $\Omega$ and $u^*$ to $\mathcal{L}_1^*(x, D)u^*=0$ in $\Omega$ and plug them in an integral identity. \smallskip\\
For $A^{(1)}$, $B^{(1)}$ and $q^{(1)}$ as above. In what follows,
we shall need the generalization of Green's formula given in \cite{AS},
\begin{align}\label{3.4}
&\left(\mathcal{L}_{1}(x,D)w , v \right)-\left (w , \mathcal{L}^*_{1}(x,D)v \right)\cr&=\sum_{\ell=1}^{m}\int_\Gamma \left [ \bigr((-\Delta )^{m-\ell}w\bigr)\bigr(\overline{\partial_\nu (-\Delta)^{\ell-1}v}\bigr) - \bigr(\partial_\nu(-\Delta )^{m-\ell}w\bigr)\bigr(\overline{(-\Delta)^{\ell-1}v}\bigr) \right ]ds_x\cr
&:= \left<\gamma^t w , \tilde{\gamma}v\right> -\bigr< \tilde{\gamma}w, \gamma^t v \bigr>,
\end{align}
where $\gamma^t w=((-\Delta)^{m-1} w, \ldots,(-\Delta) w, w)$. Here $\left(\cdot , \cdot \right)$ and $\left<\cdot , \cdot
\right>$ denote the $L^2-$ inner product respectively on $\Omega$
and $\Gamma$, and $ds_x$ stands for the Euclidean surface measure on $\Gamma$.
\smallskip
We start by the following integral identity.
\begin{lemma}\label{L3.1}
Let $A$, $B$ and $q$ given by \eqref{3.1}. For given $f=(f_0, \ldots, f_{m-1}), f^*=(f_0^*, \ldots, f_{m-1}^*)\in \mathcal{H}^{m,\frac{1}{2}}(\Gamma)$, let $u,\,u^*\in H^{2m}(\Omega)$ solutions respectively, to
\begin{equation}
\left\{\begin{array}{ll}
\mathcal{L}_{2}(x,D)u=0 & \textrm{in}\,\Omega,\cr
\gamma u=f & \textrm{on}\,\Gamma,
\end{array}
\right.
;\qquad
\left\{\begin{array}{ll}
\mathcal{L}^*_{1}(x,D)u^*=0 & \textrm{in}\,\Omega,\cr
\gamma u^*=f^* & \textrm{on}\,\Gamma.
\end{array}
\right.
\end{equation}
Then the following identity holds true
\begin{equation}
\int_\Omega (A(x)D \cdot D u+B(x)\cdot Du+q(x)u)\overline{u^*}dx=\bigr<(\Lambda^{(2)}-\Lambda^{(1)})f, (f^*)^t\bigr>,\label{3.6}
\end{equation}
where $(f^*)^t=(f_{m-1}^*, \ldots, f_0^*)$.
\end{lemma}
\begin{proof}
Let $u$ and $u^*$ as in the lemma. Let $u_1\in H^{2m}(\Omega)$ be the unique solution of
\begin{equation}\label{3.7}
\left\{\begin{array}{ll}
\mathcal{L}_{1}(x,D)u_1=0 & \textrm{in}\,\Omega,\\
\gamma u_1=f & \textrm{on}\,\Gamma.
\end{array}
\right.
\end{equation}
Putting $w=u_1-u$, then $w$ solves
\begin{equation}\label{3.8}
\left\{\begin{array}{ll}
\mathcal{L}_{1}(x,D)w=A(x)D\cdot D u+B(x)\cdot Du+q(x)u & \textrm{in}\,\Omega,\cr
\gamma w=0 & \textrm{on}\,\Gamma.
\end{array}
\right.
\end{equation}
Next, we multiply the first equation in (\ref{3.8}) by $\overline{u^*}$ and we then apply the Green's formula \eqref{3.4} to get
\begin{equation}
\int_\Omega (A(x)D\cdot D u+B(x)\cdot Du+q(x)u)\overline{u^*}dx=\int_\Gamma(\Lambda^{(2)}-\Lambda^{(1)})f\cdot\overline{(f^*)^t}ds_x.
\end{equation}
This completes the proof.
\end{proof}
For given $\omega,\widetilde{\omega}\in\mathbb{S}^{n-1}$ with $\omega\cdot\widetilde{\omega}=0$, let $\varrho=\omega+i\widetilde{\omega}$ and let $\varphi(x)=x\cdot\varrho$. For $h>0$ sufficiently small, Lemmas \ref{L2.6} and \ref{L2.7} guarantee the existence of C.G.O- solutions $u\in H^{2m}(\Omega)$ verifying $\mathcal{L}_2(x,D)u=0$ in $\Omega$ and $u^*\in H^{2m}(\Omega)$ verifying $\mathcal{L}_1^*(x,D)u^*=0$ in $\Omega$ and such that
\begin{align}
u(x,h)&=e^{\frac{\varphi}{h}}(\alpha_1(x)+h\alpha_2(x)+r(x,h)),\label{3.10}\\
u^*(x,h)&=e^{-\frac{\overline{\varphi}}{h}}(\alpha_1^*(x)+h \alpha_2^*(x)+r^*(x,h)),\label{3.11}
\end{align}
where $r(\cdot,h)$ and $r^*(\cdot,h)$ satisfy
\begin{align}
\norm{r(\cdot,h)}_{H_{\mathrm{scl}}^{2m}(\Omega)}&{\,\lesssim\,} h^2N_{2m,\varrho}(\alpha_1),\label{3.12} \\ \norm{r^*(\cdot,h)}_{H_{\mathrm{scl}}^{2m}(\Omega)}&{\,\lesssim\,} h^2N_{2m,\overline{\varrho }}(\alpha_1^*),\label{3.12'}
\end{align}
and where $\alpha_1(\cdot,\varrho)$ and $\alpha_1^*(\cdot,\varrho)$ belong to $C^\infty(\overline{\Omega})$ and satisfying the following transport equations respectively
\begin{align}
& (\varrho\cdot\nabla)^m \alpha_1 =0\quad \mathrm{in}\,\,\Omega, \label{3.13}\\
& (\overline{\varrho}\cdot\nabla)^m\alpha_1^* =0\quad \mathrm{in}\,\,\Omega.\label{3.14}
\end{align}
As a consequence of Lemma \ref{L3.1} and the C.G.O- solutions constructed as above, we have the following integral identity.
\begin{lemma}\label{L3.2}
Let $A$, $B$ and $q$ given by \eqref{3.1}. For all $\alpha_1,\,\alpha_1^*\in C^\infty(\overline{\Omega})$ satisfy \eqref{3.13} and \eqref{3.14} respectively, the following identity holds true
\begin{equation}
\int_\Omega \bigr(h^{-1} A \varrho \cdot \varrho \alpha_1+2 A \varrho \cdot \nabla \alpha_1 + i (\varrho \cdot B) \alpha_1\bigr)\overline{\alpha_1^*}dx=\mathcal{I}(h),\label{3.15}
\end{equation}
where
\begin{align}\label{3.16}
| \mathcal{I}(h)|
&{\,\lesssim\,} (e^{C/h} \|\Lambda^{(1)}-\Lambda^{(2)}\|+h+\|A \varrho \cdot \varrho\|_{L^\infty})N_{2m, \varrho}(\alpha_1)N_{2m, \overline{\varrho }}(\alpha_1^*).
\end{align}
Here $C$ is a positive constant independent of $h$.
\end{lemma}
\begin{proof}
Let $u$ be a solution to $\mathcal{L}_2(x, D)u=0 $, in $\Omega$, of the form (\ref{3.10}) and $u^*$ be a solution to $\mathcal{L}_1^*(x, D)u^*=0 $, in $\Omega$, of the form (\ref{3.11}). Thus, we get
\begin{align}
hA D \cdot D u&=e^{\frac{\varphi}{h}}\big(-h^{-1}(A\varrho\cdot\varrho)-2iA\varrho\cdot D+AD\cdot(hD)\big)(\alpha_1+h\alpha_2+r).\label{3.17}
\end{align}
Then the first term in the left hand side of (\ref{3.6}) becomes
\begin{equation}
h\int_\Omega (A D \cdot D u)\overline{u^*}dx=-
\int_\Omega \bigr(h^{-1}(A\varrho\cdot\varrho)\alpha_1 +2i A\varrho\cdot D\alpha_1\bigr)\overline{\alpha_1^*}dx +\mathcal{J}_1(h),\label{3.18}
\end{equation}
where $\mathcal{J}_1$ denotes the integral
\begin{align}
\mathcal{J}_1(h)&=-\int_\Omega \bigr(h^{-1}A\varrho\cdot\varrho(h\alpha_2+r)+2i A \varrho \cdot D (h\alpha_2+r)\bigr) \overline{(\alpha_1^*+h\alpha_2^*+r^*)}dx\cr&\quad-\int_\Omega \bigr( h^{-1}A\varrho\cdot\varrho \alpha_1+2i A \varrho \cdot D \alpha_1 \bigr) \overline{(h\alpha_2^*+r^*)} dx\cr
&\quad +\int_\Omega
(AD\cdot hD(\alpha_1+h\alpha_2+r))\overline{(\alpha_1^*+h\alpha_2^*+r^*)}dx.\label{3.19}
\end{align}
We deduce from the Cauchy-Schwarz inequality and from the inequalities (\ref{3.12}), \eqref{3.12'}, \eqref{2.50} and \eqref{2.53} that
\begin{equation}\label{3.20}
\abs{\mathcal{J}_1(h)}{\,\lesssim\,} (\|A \varrho \cdot \varrho\|_{L^\infty}+h)N_{2m, \varrho}(\alpha_1)N_{2m, \overline{\varrho }}(\alpha_1^*).
\end{equation}
In the same way as above, we can prove that
\begin{align}
h\int_\Omega(B\cdot Du+qu)\overline{u^*}dx &=-i\int_\Omega B\cdot\varrho \alpha_1 \overline{\alpha_1^*}dx + \mathcal{J}_2(h),\label{3.21}
\end{align}
where
\begin{align}\label{3.22}
\mathcal{J}_2(h)&=-i\int_\Omega B\cdot\varrho \bigr(\alpha_1 \overline{(h\alpha_2^*+r^*)} + (h\alpha_2+r)\overline{(\alpha_1^*+h\alpha_2^*+r^*)}\bigr)dx \cr
&\quad+ \int_\Omega \bigr((hB\cdot D+hq)(\alpha_1+h\alpha_2+r)\bigr)\overline{(\alpha_1^*+h\alpha_2^*+r^*)}dx,
\end{align}
and satisfies
\begin{equation}
\abs{\mathcal{J}_2(h)}{\,\lesssim\,} h N_{2m, \varrho}(\alpha_1)N_{2m, \overline{\varrho }}(\alpha_1^*).\label{3.23}
\end{equation}
Collecting (\ref{3.18}) and (\ref{3.21}), we find from (\ref{3.6}) that
\begin{align*}
&\bigr|\int_\Omega \bigr( h^{-1}A\varrho\cdot\varrho \alpha_1+2 A \varrho \cdot \nabla \alpha_1 +iB\cdot\varrho \alpha_1 \bigr) \overline{\alpha_1^*}dx\bigr|\cr &{\,\lesssim\,} h\norm{(\Lambda^{(1)}-\Lambda^{(2)})f}_{\mathcal{H}^{m,\frac{3}{2}}}\norm{f^*}_{\mathcal{H}^{m,\frac{1}{2}}}
+(\|A \varrho \cdot \varrho\|_{L^\infty}+h) N_{2m, \varrho}(\alpha_1)N_{2m, \overline{\varrho }}(\alpha_1^*)\cr
&{\,\lesssim\,} h\norm{\Lambda^{(1)}-\Lambda^{(2)}}\norm{u}_{H^{2m}(\Omega)}\norm{u^*}_{H^{2m}(\Omega)}+ (\|A \varrho \cdot \varrho\|_{L^\infty}+h)N_{2m, \varrho}(\alpha_1)N_{2m, \overline{\varrho }}(\alpha_1^*) \cr
&{\,\lesssim\,} (e^{C/h}\norm{\Lambda^{(1)}-\Lambda^{(2)}}+ h+\|A \varrho \cdot \varrho\|_{L^\infty})N_{2m, \varrho}(\alpha_1)N_{2m, \overline{\varrho }}(\alpha_1^*).
\end{align*}
Hence, the proof of Lemma \ref{L3.2} is completed.
\end{proof}
As a consequence of the above Lemma, we have the following integral identities.
\begin{lemma}\label{L3.3}
Let $A, B$ and $q$ as above. Then, for all $\alpha_1,\,\alpha_1^*\in C^\infty(\overline{\Omega})$ satisfy \eqref{3.13} and \eqref{3.14} respectively, we have
\begin{equation}\label{3.26}
\int_\Omega A \varrho \cdot \varrho \alpha_1\overline{\alpha_1^*}dx=\mathcal{I}_2(h),\quad \textrm{if}~~m>2,
\end{equation}
and
\begin{equation}\label{3.25}
\int_\Omega (2 A \varrho \cdot \nabla \alpha_1 + i \varrho \cdot B \alpha_1)\overline{\alpha_1^*}dx=\mathcal{I}_1(h),\quad \textrm{if}~~m=2,
\end{equation}
where
\begin{align}\label{3.27}
| \mathcal{I}_j(h)| {\,\lesssim\,} (e^{C/h} \|\Lambda^{(1)}-\Lambda^{(2)}\|+h)N_{2m, \varrho}(\alpha_1)N_{2m, \overline{\varrho }}(\alpha_1^*),
\end{align}
for $j=1,2$ and $m \geq 2$. Here $C$ is a positive constant independent of $h$.
\end{lemma}
\begin{proof}
First, we prove \eqref{3.26}. Multiplying the equation \eqref{3.15} by $h$ we can obtain
\begin{align}\label{3.28}
\int_\Omega A \varrho \cdot \varrho \alpha_1\overline{\alpha_1^*}dx=-h\int_\Omega (2 A \varrho \cdot \nabla \alpha_1 + i \varrho \cdot B \alpha_1)\overline{\alpha_1^*}dx+\mathcal{J}_2(h),
\end{align}
where $\mathcal{J}_2(h)$ satisfy \eqref{3.16}. Therefore by using the Cauchy-Schwarz inequality for the first term in the right hand side of \eqref{3.28} we get our desired result \eqref{3.26}.
We move now to prove \eqref{3.27}. Since we assumed that $A$ is isotropic for $m=2$, then, $A \varrho \cdot \varrho=0$. Therefore, using Lemma \ref{L3.2} we get
\begin{align*}
\int_\Omega (2 A \varrho \cdot \nabla \alpha_1 + i \varrho \cdot B \alpha_1)\overline{\alpha_1^*}dx=\mathcal{J}_1(h),
\end{align*}
where
\begin{align*}
| \mathcal{J}_1(h)| {\,\lesssim\,} (e^{C/h} \|\Lambda^{(1)}-\Lambda^{(2)}\|+h)N_{4, {\varrho }}(\alpha_1)N_{4, \overline{\varrho }}(\alpha_1^*).
\end{align*}
This completes the proof.
\end{proof}
We will now give the specific Hodge decomposition of symmetric tensor $A \in H^k(\Omega, \mathbb{C}^{n^2})$ and a vector field $X\in W^{k, p}(\Omega, \mathbb{C}^n)$, with $k \geq 1$ and $p>n$, in the simply connected domain $\Omega \subset \mathbb{R}^n$.
\begin{lemma}\label{L4.1}
Let $k \geq 1$, any $2-$tensor field $A \in H^k(\Omega, \mathbb{C}^{n^2})$ can be uniquely represented as
\begin{align}\label{4.1}
A=A^\prime +\nabla_{\text{sym}}V + \vartheta_A~ id,
\end{align}
where $\vartheta_A \in H^k(\Omega, \mathbb{C})$, the vector field $V
\in H^{k+1}(\Omega, \mathbb{C}^{n})$ satisfies the boundary
condition
\begin{align}\label{4.2}
V=0,\quad \textrm{on}~~ \Gamma,
\end{align}
and the tensor field $A^\prime \in H^k(\Omega, \mathbb{C}^{n^2})$ satisfies the conditions
\begin{align}\label{4.3}
\operatorname{trace}(A^\prime)=0,\quad \sum_{j=1}^n \partial_j A^\prime_{jk}=0,~~ \forall k \in \{1, \ldots, n \}.
\end{align}
Here $\nabla_{\text{sym}}$ is the symmetrized differentiation defined as $$(\nabla_{\text{sym}}V)_{ j,k}=\frac{1}{2}(\partial_{x_j}V_k+\partial_{x_k}V_j).$$
Moreover, the right hand side of \eqref{4.1} depends continuously on $A$ in the following sense :
\begin{align}\label{4.4}
\|A^\prime \|_{H^k(\Omega)} {\,\lesssim\,} \|A\|_{H^k(\Omega)} ,\quad \|V\|_{H^{k+1}(\Omega)} {\,\lesssim\,} \|A\|_{H^k(\Omega)} \quad \textrm{and}~~
\| \vartheta_A \|_{H^k(\Omega)} {\,\lesssim\,} \|A\|_{H^k(\Omega)}.
\end{align}
\end{lemma}
This Lemma is a straightforward adaptation of a Theorem $3.3$ in \cite{SV} and, for shake of completeness, we give the proof in the Appendix \ref{appendixA} .
\begin{lemma}\label{L4.9}
Let $k \in \mathbb{N}$ and $p >n$, any vector field $X \in W^{k, p}(\Omega, \mathbb{C}^n)$ can be represented as
\begin{align}\label{4.80}
X=X^{\prime}+\nabla \vartheta_X,
\end{align}
where $\vartheta_X \in W^{k+1, p}(\Omega, \mathbb{C})$ satisfies the
boundary condition
\begin{align}\label{4.9}
\vartheta_X=0,\quad \textrm{on}~~\Gamma,
\end{align}
and the vector field $X^{\prime} \in W^{k, p}(\Omega, \mathbb{C}^n)$ satisfies the condition
\begin{align}\label{4.100}
\textrm{div}~X^\prime=0 ,\quad \textrm{in} ~~\Omega.
\end{align}
Moreover, the right hand side of \eqref{4.80} depends continuously on
$X$ in the following sense :
\begin{align} \label{4.36}
\|\vartheta_X\|_{W^{k+1, p}(\Omega)} {\,\lesssim\,} \| X\|_{W^{k, p}(\Omega)},\quad \|X^{\prime}\|_{W^{1, p}(\Omega, \mathbb{C}^n)} {\,\lesssim\,} \|\textrm{curl}~X^{\prime}\|_{L^{p}(\Omega)}.
\end{align}
\end{lemma}
The extension by zero outside $\Omega$ of the $2-$tensor field $A^\prime$, the vector fields $V$ and $X^\prime$, and the functions $\vartheta_A$ and $\vartheta_X$ given by the
previous Lemma is still denoted by the same letters.\medskip \\
The idea will be now to use the specific Hodge decomposition given
by Lemmas \ref{L4.1} and \ref{L4.9} to decompose the symmetric tensor $A=A^{(2)}-A^{(1)} \in \mathcal{A}_{\sigma_1}\cap \mathcal{E}^\prime(\Omega)$ and the vector field $ B=B^{(2)}-B^{(1)} \in \mathcal{B}_{\sigma_2}\cap \mathcal{E}^\prime(\Omega)$ and write
\begin{align}
A=A^\prime+\nabla_{\text{sym}}V+\vartheta_A~ id,\label{A}
\end{align}
where $A^\prime \in H^{\sigma_1}(\Omega)$ satisfies \eqref{4.3}, the vector field $V \in H^{\sigma_1+1}(\Omega)$ satisfies \eqref{4.2} and the function $\vartheta_A \in H^{\sigma_1}(\Omega)$. Moreover, we have
\begin{align}\label{A1}
\|A^\prime \|_{H^{\sigma_1}(\Omega)} {\,\lesssim\,} \|A\|_{H^{\sigma_1}(\Omega)} ,\quad \|V\|_{H^{\sigma_1}(\Omega)} {\,\lesssim\,} \|A\|_{H^{\sigma_1}(\Omega)} \quad \textrm{and}~~
\| \vartheta_A \|_{H^{\sigma_1}(\Omega)} {\,\lesssim\,} \|A\|_{H^{\sigma_1}(\Omega)}.
\end{align}
In addition, since $\sigma_1 >\frac{n}{2}+3$~ then by using Sobolev's embedding one can easily see that $A^\prime \in W^{3, \infty}(\Omega, \mathbb{C}^{n^2})$, $V \in W^{4, \infty}(\Omega, \mathbb{C}^n)$ and $\vartheta_A \in W^{3, \infty}(\Omega, \mathbb{C})$.\\
Further, the vector field $B$ can be decomposed by Lemma \ref{L4.9} as
\begin{align}
B=B^\prime+\nabla \vartheta_B,\label{B}
\end{align}
where $B^\prime \in L^{ \infty}(\Omega, \mathbb{C}^{n})$ satisfies \eqref{4.100} and \eqref{4.36}. Hence, the Morrey's inequality yields
\begin{align}\label{B1}
\|B^\prime\|_{L^\infty(\Omega)} {\,\lesssim\,} \|\textrm{curl}~B\|_{L^{\infty}(\Omega)}.
\end{align}
Moreover, the function $\vartheta_B \in W^{1, +\infty}(\Omega, \mathbb{C})$ solving the following boundary value problem
\begin{equation}
\left\{\begin{array}{ll}
\Delta \vartheta_B = \textrm{div}~B & \mathrm{in}\, \Omega,\cr
\vartheta_B =0 &\mathrm{on}\, \Gamma.
\end{array}
\right.
\end{equation}
Therefore, using the theorem on regular solvability of elliptic problem we have $\vartheta_B \in H^{\sigma_2+1}(\Omega)$ and satisfies the following estimate
\begin{align}\label{vartheta_B1}
\| \vartheta_B\|_{H^{\sigma_2}(\Omega)} {\,\lesssim\,} \|B\|_{H^{\sigma_2}(\Omega)}.
\end{align}
\label{section3}
\section{Stability estimates (first case \texorpdfstring{$m>2$}{Lg})} \label{section4}
This section is dedicated to proving the stable determination of
the symmetric tensor $A$, the vector field $B$ and the electric
potential $q$ from the Dirichlet to Neumann map $\Lambda_{A,B,q}$,
in the case $m >2$.
We will use the family of solutions called complex geometric
solutions, constructed in the previous section, to estimate
the Fourier transform of the difference of the coefficients.\\
Consider a priori constant $M>0$, $\sigma_1 > \frac{n}{2}+3$ and $\sigma_2 > \frac{n}{2}+1$. Let $A^{(j)} \in \mathcal{A}_{\sigma_1}(M)\cap \mathcal{E}^\prime(\Omega)$, $B^{(j)} \in \mathcal{B}_{\sigma_2}(M)\cap \mathcal{E}^\prime(\Omega)$ and $q^{(j)}\in \mathcal{Q}_{}(M)$, $j=1,2$, be two sets of coefficients. We denote as section \ref{section3}
\begin{equation}
A=A^{(2)}-A^{(1)},\qquad B=B^{(2)}-B^{(1)}\quad\text{and }~~~~ q=q^{(2)}-q^{(1)}.
\end{equation}
Moreover, without loss of generality, we will assume that
\begin{align}
\|\Lambda^{(1)}-\Lambda^{(2)}\| \ll 1.
\end{align}
In order to prove the stability estimates, we need the following estimate.
\begin{lemma}\label{LA}
Let $k \in \mathbb{N}$ and $n_0 \geq 1$ be given. Let $\eta \in H^\sigma(\Omega)$, such that $\|\eta\|_{H^\sigma(\Omega)} \leq M$, for some
$M>0$ and $ \sigma > \frac{n}{2}+k$, and let $\Lambda >0$. We assume that for any $0 <h \ll 1$ and $\xi \in \mathbb{R}^n$, we have
\begin{align}\label{0}
\abs{\widehat{\widetilde{\eta} } (\xi )} {\,\lesssim\,} \langle \xi \rangle^{n_0}(e^{C_0/h} \Lambda^{\mu_0} +h^{\kappa_0}),
\end{align}
for some $0< \mu_0 \leq 1$, $0< \kappa_0 \leq 1$ and $C_0>0$. Here $\widetilde{\eta}$ denotes the extension of $\eta$ outside $\Omega$ which satisfies $\tilde{\eta} \in H^\sigma(\mathbb{R}^n)$ and $\Vert \tilde{\eta}\Vert_{H^{\sigma}(\mathbb{R}^{n})}{\,\lesssim\,} \,\Vert \eta\Vert_{H^{\sigma}(\Omega)}$. Then there exist $\mu, \kappa \in (0, 1)$ such that
\begin{align}\label{1}
\| \eta \|_{W^{k, \infty}(\Omega)} {\,\lesssim\,} e^{C/h} \Lambda^\mu +h^\kappa,
\end{align}
for any $h>0$ small enough. Here $C$ is positive constant independent of $h$ and $\kappa$ depends only on $n$ and $n_0$. Moreover, for any $n_1 \in \mathbb{N}$ there exist $\ell \gg 1$ and $\kappa_\ell >0$ such that
\begin{align}\label{1*}
h^{-n_1} \| \eta \|_{W^{k, \infty}(\Omega)} {\,\lesssim\,} e^{C/h^\ell} \Lambda^\mu +h^{\kappa_\ell}.
\end{align}
\end{lemma}
\begin{proof}
First, we may bound the $H^{-1}-$norm of $ \eta$. For $R >1$, to be chosen later, we can obtain
from \eqref{0} the following inequality
\begin{align}\label{2}
\| \eta\|_{H^{-1}(\Omega)}^2 &\leq \| \widetilde{\eta}\|_{H^{-1}(\mathbb{R}^n)}^2=\int_{\langle \xi \rangle \leq R} \abs{\widehat{\widetilde{\eta}}(\xi)}^2 \langle \xi \rangle^{-2} d\xi+\int_{\langle \xi \rangle > R} \abs{\widehat{\widetilde{\eta}}(\xi)}^2 \langle \xi \rangle^{-2} d\xi\cr
&{\,\lesssim\,} R^{2n_0+n-2}(e^{{2C_0}/{h}} \Lambda^{2\mu_0} +h^{\kappa_0})+M^2 R^{-2}.
\end{align}
Choosing $R>1$ such that $R^{2n_0+n-2} h^{\kappa_0}=R^{-2}$ that is $R=h^{\frac{- \kappa_0}{n+2n_0}}$ then we deduce from \eqref{2} for $h_0$ sufficiently small that
\begin{align*}
\|\eta\|_{H^{-1}(\Omega)} {\,\lesssim\,} e^{C^\prime/h} \Lambda^{\mu_0} + h^{\frac{\kappa_0}{n+2n_0}},
\end{align*}
for some positive constant $C^\prime$ and all $h < h_0$. Here we used the fact that $h^{-a} \leq e^{C^\prime/h}$, for some $0<a \leq 1$.\\
In order to complete the proof of the theorem, let
$\delta=\frac{1}{2}(\sigma-(\frac{n}{2}+\abs{\alpha}))>0$, where $\alpha \in \mathbb{N}^n$ such that $\abs{\alpha} \leq k$, then using Sobolev's
embedding theorem together with interpolation theorem, we end up getting the following inequality
\begin{align*}
\|D^\alpha \eta\|_{L^{\infty}(\Omega)} & {\,\lesssim\,} \|D^\alpha \eta\|_{H^{n/2+\delta }(\Omega)} \\
& {\,\lesssim\,} \|D^\alpha \eta\|_{H^{-1-\abs{\alpha}}(\Omega)}^{\kappa_1}\|D^\alpha \eta \|_{H^{n/2+2 \delta }(\Omega)}^{ 1-\kappa_1} \\
& {\,\lesssim\,} e^{{ \kappa_1 C^\prime}/{h}} \Lambda^{\mu_0 \kappa_1} + h^{\frac{\kappa_0\kappa_1}{n+2n_0}},
\end{align*}
for some $\kappa_1 \in (0,1)$. This concludes the proof of \eqref{1} with $\mu=\mu_0 \kappa_1$ and $\kappa=\frac{\kappa_0\kappa_1}{n+2n_0}$.\\
We move now to prove \eqref{1*} we observe that \eqref{1} implies in particular
\begin{align*}
\| \eta \|_{W^{k, \infty}(\Omega)} {\,\lesssim\,} e^{C/h^\ell} \Lambda^\mu +h^{\ell\kappa},
\end{align*}
for all $\ell \gg 1$. Choosing $\ell$ such that $\kappa_\ell:=\ell \kappa -n_1 >0$, we obtain
\begin{align*}
h^{-n_1} \| \eta \|_{W^{k, \infty}(\Omega)} {\,\lesssim\,} e^{C/h^\ell} \Lambda^\mu +h^{\kappa_\ell},
\end{align*}
with possibly different constant $C$. This completes the proof.
\end{proof}
As a corollary of Lemma \ref{L3.3} and the decomposition \eqref{A}, we have the following integral identity.
\begin{lemma}\label{L4.2}
Let $A^\prime \in W^{3, +\infty}(\Omega, \mathbb{C}^{n^2})$ and $V
\in W^{4, +\infty}(\Omega, \mathbb{C}^{n})$ given by \eqref{A}. Then, for all $\alpha_1,\,\alpha_1^*\in C^\infty(\overline{\Omega})$ satisfy \eqref{3.13} and \eqref{3.14} respectively, we have
\begin{align}\label{4.5}
\int_\Omega (A^\prime \varrho \cdot \varrho) \alpha_1\overline{\alpha_1^*}dx-\int_\Omega (\varrho \cdot V) \varrho \cdot \nabla (\alpha_1\overline{\alpha_1^*})dx=\mathcal{I}(h),
\end{align}
where
\begin{align}\label{4.6}
| \mathcal{I}(h)| {\,\lesssim\,} (e^{C/h} \|\Lambda^{(1)}-\Lambda^{(2)}\|+h)N_{2m, \varrho}(\alpha_1)N_{2m, \overline{\varrho }}(\alpha_1^*),
\end{align}
for any $h>0$ small enough. Here $C$ is a positive constant independent of $h$.
\end{lemma}
\begin{proof}
Substituting $A$ into the left hand side of \eqref{3.26}, we get
\begin{align}\label{4.7}
\int_\Omega (A^\prime \varrho \cdot \varrho) \alpha_1\overline{\alpha_1^*}dx+\int_\Omega (\nabla_{\text{sym}}V \varrho \cdot \varrho) \alpha_1\overline{\alpha_1^*}dx=\mathcal{I}_2(h),
\end{align}
where $\mathcal{I}_2(h)$ satisfy \eqref{3.27}. Here we used the fact that $\varrho \cdot \varrho=0$.\\
Now, using an integration by part for the second term in the left hand side of \eqref{4.7} and taking into account the condition \eqref{4.2}, we get our desired estimate.
\end{proof}
With the help of the previous lemma, we are now in position to derive an estimate for the $2-$tensor $A^\prime$ as follow.
\begin{lemma}\label{L4.5}
Let $A^\prime$ as above. There exist $C_1>0$ and $\mu_1, \kappa_1 \in (0,1)$, such that we have
\begin{align}\label{4.24}
\|A^\prime\|_{L^{\infty}(\Omega)} {\,\lesssim\,} e^{C_1/h} \|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_1} +h^{\kappa_1},
\end{align}
for any $h >0$ small enough. Here $C_1$ is independent of $h$ and $\kappa_1$ depends only on $n$ and $m$.
\end{lemma}
\begin{proof}
Let $\omega, \widetilde{\omega} \in \mathbb{S}^{n-1}$ and $\xi \in \mathbb{R}^n$ such that $\xi \cdot \omega =\xi \cdot \widetilde{\omega }=\omega \cdot \widetilde{\omega}=0$. Choosing $\alpha_1 (x)=e^{-i x\cdot \xi}$ and $\alpha_1^*(x)=1$,~~for $x \in \Omega$. It is clear that $\alpha_1 $ and $\alpha_1^*$ satisfy, respectively, the transport equations \eqref{3.13} and \eqref{3.14}. Then, by using Lemma \ref{L4.2} and the fact that $\varrho \cdot \xi=0$, we obtain
\begin{align}\label{4.8}
\abs{\int_{\Omega} e^{-i x\cdot \xi} {A^\prime}(x)\varrho \cdot \varrho \mathrm{~d} x} {\,\lesssim\,} \langle\xi \rangle^{2m}\bigr(e^{C/h }\|\Lambda^{(1)}-\Lambda^{(2)}\|+h\bigr).
\end{align}
Now, let us fix $\xi \in \mathbb{R}^n\setminus \{ 0\}$. Following \cite{B2}, we consider the orthonormal basis $\mathscr{B}$ of $\mathbb{R}^n$ as
\begin{align*}
\mathscr{B}=\{ \omega_1, \omega_2, \ldots, \omega_n\} \qquad \textrm{with} \quad \omega_n=\frac{\xi}{\abs{\xi }}.
\end{align*}
The inequality \eqref{4.8} is valid for all $\omega, \widetilde{\omega} \in \mathbb{S}^{n-1}$ such that $\omega \cdot \xi= \widetilde{\omega} \cdot \xi =0$. Then, for $\varrho=\omega_j+i \omega_k$, where $j, k \in \{1, \ldots, n-1\}$, $j \neq k$, we have
\begin{align}\label{4.10}
\abs{\widehat{A^\prime}(\xi) (\omega_j+i \omega_k)\cdot(\omega_j+i\omega_k) } {\,\lesssim\,} \langle\xi \rangle^{2m}\bigr(e^{C/h }\|\Lambda^{(1)}-\Lambda^{(2)}\|+h\bigr).
\end{align}
Replacing $\omega_k$ by $-\omega_k$ in $\mathscr{B}$ and doing the same analysis as before we get
\begin{align}\label{4.11}
\abs{\widehat{A^\prime}(\xi) (\omega_j-i \omega_k)\cdot(\omega_j-i\omega_k) } {\,\lesssim\,} \langle\xi \rangle^{2m}\bigr(e^{C/h }\|\Lambda^{(1)}-\Lambda^{(2)}\|+h\bigr),
\end{align}
for all $j,k\in\{1, \ldots, n-1\}, ~j \neq k.$ Combining \eqref{4.10} and \eqref{4.11} together we obtain
\begin{align}\label{4.12}
| \widehat{A^\prime}(\xi)\omega_j \cdot \omega_j - \widehat{A^\prime}(\xi)\omega_k \cdot \omega_k |{\,\lesssim\,}\langle\xi \rangle^{2m}\bigr(e^{C/h }\|\Lambda^{(1)}-\Lambda^{(2)}\|+h\bigr),
\end{align}
for all $j,k\in\{1, \ldots, n-1\}, ~j \neq k.$ Moreover, their difference gives us
\begin{align}\label{4.13}
| \widehat{A^\prime}(\xi)\omega_j \cdot \omega_k |{\,\lesssim\,}\langle\xi \rangle^{2m}\bigr(e^{C/h }\|\Lambda^{(1)}-\Lambda^{(2)}\|+h\bigr),
\end{align}
for all $ j,k\in\{1, \ldots, n-1\}, ~j \neq k.$ In addition, since $A^\prime$ is
divergence free we get $\sum_{k=1}^{n} \widehat{A^\prime}_{jk}(\xi)\xi_k=0$, for
any $j\in \{1, \ldots, n\}$, then, due to the fact that $A^\prime$ is symmetric we can obtain that
\begin{align}\label{4.14}
\widehat{A^\prime}(\xi)\omega_j \cdot \xi=0, \quad j = {1, \ldots, n}.
\end{align}
Which shows that the vector $\widehat{A^\prime}(\xi)\omega_j$ can be written with respect to orthonormal basis $\mathscr{B}$ as
\begin{align}\label{4.15}
\widehat{A^\prime}(\xi)\omega_j=\sum_{k=1}^{n-1}\alpha_k(\xi)\omega_k\quad j={1, \ldots, n},
\end{align}
where $\alpha_k(\xi) \in \mathbb{C}$. Let us write $\lambda_{j, k}(\xi):=\widehat{A^\prime}(\xi)\omega_j \cdot \omega_k,~\textrm{where}~j, k\in\{1, \ldots, n-1\}.$
Then it follows from \eqref{4.12} and \eqref{4.13} that
\begin{align}
| \lambda_{j, j}(\xi) - \lambda_{k, k}(\xi) |&{\,\lesssim\,} \langle\xi \rangle^{2m}\bigr(e^{C/h }\|\Lambda^{(1)}-\Lambda^{(2)}\|+h\bigr), \label{4.16}
\end{align}
and for $j \neq k$, we have
\begin{align}\label{4.17}
| \lambda_{j, k}(\xi) |&{\,\lesssim\,} \langle\xi \rangle^{2m}\bigr(e^{C/h }\|\Lambda^{(1)}-\Lambda^{(2)}\|+h\bigr).
\end{align}
Let us define the symmetric $n \times n$ matrix $M(\xi)$ given by
\begin{align*}
M(\xi)=\begin{pmatrix}
\lambda_{1,1}(\xi) & \ldots & \lambda_{1,n-1}(\xi) & 0 \\
\vdots & \ddots & \vdots & \vdots \\
\lambda_{n-1,1}(\xi) & \ldots & \lambda_{n-1,n-1}(\xi) & 0\\
0 & \ldots & 0 & 0 \\
\end{pmatrix}.
\end{align*}
Then, we can write $ \widehat{A^\prime}(\xi)={}^t\mathscr{B} M(\xi) \mathscr{B}, $ where $^t \mathscr{B}$ is the transpose of the matrix $\mathscr{B}$. We observe that $M=D+M^\prime$, where $D$ is the diagonal matrix given as $$D(\xi)=\operatorname{Diag}\bigr(\lambda_{1,1}(\xi), \ldots, \lambda_{n-1, n-1}(\xi),0\bigr),$$ and $M^\prime=M-D$. Since $\operatorname{trace}(A^\prime)=0$ then $\operatorname{trace}(D)=0$ which implies, for any $j=1, \ldots,n-1$, that
\begin{align*}
\lambda_{j,j}(\xi)&=-\sum_{\substack{k=1 \\{k \neq j}}}^{n-1} \lambda _{k,k}(\xi)=-\sum_{\substack{k=1 \\{k \neq j}}}^{n-1} \bigr(\lambda_{k,k}-\lambda_{j,j}\bigr)(\xi)-(n-2)\lambda_{j,j}(\xi).
\end{align*}
Then, using \eqref{4.16} we get, for all $j=1, \ldots n-1$
\begin{align}\label{4.18}
| \lambda_{j,j}(\xi)| {\,\lesssim\,} \langle\xi \rangle^{2m}\bigr(e^{C/h }\|\Lambda^{(1)}-\Lambda^{(2)}\|+h\bigr).
\end{align}
On the other hand, it follows from \eqref{4.17} that
\begin{align}\label{4.19}
| M^\prime_{j,k}(\xi)| {\,\lesssim\,} \langle\xi \rangle^{2m}\bigr(e^{C/h }\|\Lambda^{(1)}-\Lambda^{(2)}\|+h\bigr).
\end{align}
Thus, from \eqref{4.18} and \eqref{4.19} one can directly conclude that
\begin{align}\label{4.20}
|\widehat{A^\prime}(\xi) |{\,\lesssim\,} \langle\xi \rangle^{2m}\bigr(e^{C/h }\|\Lambda^{(1)}-\Lambda^{(2)}\|+h\bigr),
\end{align}
here we used the fact that $|\omega_j|=1$, $j={1, \ldots, n}$.\\
However, since $A^\prime$ satisfy the inequality \eqref{A1} then by using Lemma \ref{LA} we obtain our desired estimate.
\end{proof}
The next step is to estimate the vector field $V \in W^{4, \infty}(\Omega, \mathbb{C}^n)$, given by decomposition \eqref{A}. To do this, we use again the Hodge decomposition for a vector fields given by Lemma \ref{L4.9} and we write
\begin{align}
V&=V^\prime+\nabla \vartheta_V,\label{V}
\end{align}
where $V^{\prime} \in W^{3, \infty}(\Omega, \mathbb{C}^n)$ satisfies \eqref{4.100} and
\begin{align}\label{V1}
\|V^\prime\|_{L^{\infty}(\Omega)} {\,\lesssim\,} \|\textrm{curl}~V\|_{L^{\infty}(\Omega)},
\end{align}
Moreover, the function $\vartheta_V \in W^{4, \infty}(\Omega, \mathbb{C})$ satisfies the boundary condition $\vartheta_V=0$, on $\Gamma$, and
\begin{align}\label{vartheta_V1}
\| \vartheta_V\|_{H^{\sigma_1}(\Omega)} {\,\lesssim\,} \|V\|_{H^{\sigma_1}(\Omega)}.
\end{align}
Here $$\textrm{curl}~V=\sum_{j, k=1}^{n} \mathrm{v}_{jk} d x_{j} \wedge d x_{k},$$
with $\mathrm{v}_{jk}(x)=\partial_{x_j}V_k-\partial _{x_k}V_j , ~~j, k=1, \ldots, n$
and $V_{k}(x)=V(x) \cdot e_{k},~ x \in \Omega,$ where $\left(e_{k}\right)_{k}$ is the canonical basis of $\mathbb{R}^{n}$.
\smallskip
The $L^\infty-$norm of the vector field $V^\prime$ can be estimate as follow.
\begin{lemma}\label{L4.7}
Let $V^\prime$ given by \eqref{V}. There exist $C_2 >0$ and $\mu_2, \kappa_2 \in (0, 1)$ such that we have the following estimate
\begin{align}
\|V^\prime\|_{W^{1, \infty}(\Omega)} {\,\lesssim\,} e^{{C_2}/{h}} \|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_2} +h^{\kappa_2},
\label{4.30}
\end{align}
for any $h >0$ small enough. Here $C_2$ is a positive constant independent of $h$ and $\kappa_2 $ depends only on $m$ and $n$.
\end{lemma}
\begin{proof}
First, we start by estimate the Fourier transform of $\textrm{curl}~V$. Let $\xi \in \mathbb{R}^{n}$. We select $\omega, \widetilde{\omega} \in \mathbb{S}^{n-1}$ such that $\xi, \omega$ and $\widetilde{\omega}$ be three mutually orthogonal vectors in $\mathbb{R}^{n}$. Choosing $\alpha_1 (x)=e^{-i x\cdot \xi}(x \cdot \omega)$ and $\alpha_1^*(x)=1$,~~for $x \in \Omega$. It is clear that $\alpha_1 $ and $\alpha_1^*$ satisfy respectively \eqref{3.13} and \eqref{3.14}. Then, by using Lemma \ref{L4.2} and the fact that $\varrho \cdot \xi=0$, we obtain
\begin{align}\label{4.25}
\abs{ \varrho \cdot \widehat{V}(\xi)} {\,\lesssim\,} \langle\xi \rangle^{2m}\bigr( \|A^\prime\|_{L^{\infty}(\Omega)} +e^{C/h }\|\Lambda^{(1)}-\Lambda^{(2)}\|+h\bigr).
\end{align}
In addition Lemma \ref{L4.5} yields
\begin{align}\label{4.27}
\abs{ \varrho \cdot \widehat{V}(\xi)} {\,\lesssim\,} \langle \xi\rangle^{2m} \bigr(e^{C^\prime/h}\|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_1}+h^{\kappa_1} \bigr),
\end{align}
for some constant $C^\prime=\max(C, C_1) > 0$. Replacing $\widetilde{\omega }$ by $-\widetilde{\omega }$ and doing the same analysis as before, we get
\begin{align}\label{4.28}
\abs{ \overline{\varrho} \cdot \widehat{V}(\xi)} {\,\lesssim\,} \langle \xi\rangle^{2m} \bigr(e^{C^\prime/h}\|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_1}+h^{\kappa_1} \bigr).
\end{align}
Combining the inequalities \eqref{4.27} and \eqref{4.28} together, we find that
\begin{align}
\ \abs{ \omega \cdot \widehat{V}(\xi)} {\,\lesssim\,} \langle \xi\rangle^{2m} \bigr(e^{C^\prime/h}\|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_1}+h^{\kappa_1} \bigr).
\label{4.29}
\end{align}
For any fixed $\xi \in \mathbb{R}^n$ with $\xi \cdot \omega=0$. Let us choose $\omega =\frac{\xi_{j} e_{k}-\xi_{k} e_{j}}{|\xi_{j} e_{k}-\xi_{k} e_{j}|}$, for $j \neq k \in \{ 1, \ldots, n\}$, then by multiplying \eqref{4.29} by $\abs{\xi_{j} e_{k}-\xi_{k} e_{j}}$, we obtain
\begin{align}\label{4.26}
\abs{\widehat{\mathrm{v}}_{jk}(\xi)} {\,\lesssim\,} \langle \xi\rangle^{2m+1} \bigr(e^{C^\prime/h}\|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_1}+h^{\kappa_1} \bigr).
\end{align}
Hence, from Lemma \ref{LA} and the inequality \eqref{V1}, there exist $C > 0$ and $\mu, \kappa=\kappa(n, m) \in (0,1)$ such that we have
\begin{align}
\|V^\prime\|_{L^{\infty}(\Omega)} {\,\lesssim\,} \|\textrm{curl}~V\|_{L^{\infty}(\Omega)} {\,\lesssim\,} e^{C/h} \|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu} +h^{\kappa}.\label{4.31'}
\end{align}
In order to complete the proof of the theorem, let $\delta=\frac{1}{2}(\sigma_1-(\frac{n}{2}+1))>0$, then using Sobolev's embedding theorem together with interpolation theorem and Plancherel theorem, we get for $\kappa_0 \in (0,1)$
\begin{align*}
\|DV^\prime\|_{L^{\infty}(\Omega)} & {\,\lesssim\,} \|DV^\prime\|_{H^{n/2+\delta }(\Omega)} \\
& {\,\lesssim\,} \|DV^\prime\|_{H^{-1}(\Omega)}^{\kappa_0}\|DV^\prime\|_{H^{n/2+2 \delta }(\Omega)}^{ 1-\kappa_0} \\
&{\,\lesssim\,} \|V^\prime\|_{L^\infty(\Omega)}^{\kappa_0}.
\end{align*}
This and the inequality \eqref{4.31'} conclude the proof.
\end{proof}
With the help of the above Lemma, we may now prove the $W^{1, \infty}-$norm of $V$.
\begin{lemma}\label{L4.11}
Let $V$ given by \eqref{V}. There exist $C_3>0$ and $\mu_3$, $\kappa_3 \in (0,1)$ such that the following estimate
\begin{align}\label{4.40}
\|V\|_{W^{1, \infty}(\Omega)}&{\,\lesssim\,} e^{C_3/h} \|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_3} +h^{\kappa_3},
\end{align}
holds for any $h >0$ small enough. Here $C_3$ is independent of $h$ and $\kappa_3$ depends on $n$ and $m$.
\end{lemma}
\begin{proof}
In the first step, we will estimate $\widehat{\vartheta}_V$ given by \eqref{V}. Let $\xi \in \mathbb{R}^n$, we choose $\omega, \widetilde{\omega} \in \mathbb{S}^{n-1}$ such that $\xi, \omega$ and $\widetilde{\omega}$ be three mutually orthogonal vectors in $\mathbb{R}^{n}$. By substituting $V$ into the left hand side of \eqref{4.5}, we get for $\alpha_1(x)=e^{-i x \cdot \xi}$
\begin{equation}\label{4.38}
\int_\Omega e^{-i x \cdot \xi} (\varrho \cdot \nabla \vartheta_V) (\varrho \cdot \nabla \overline{\alpha_1^*}) ~dx=-\int_\Omega e^{-i x \cdot \xi} (\varrho \cdot V^{\prime}) (\varrho \cdot \nabla \overline{\alpha_1^*})~dx + \mathcal{I}_3(h),
\end{equation}
where $\mathcal{I}_3$ satisfies
\begin{align}\label{4.39}
| \mathcal{I}_3(h)| {\,\lesssim\,} \langle \xi\rangle^{2m}(\|A^\prime\|_{L^\infty}+e^{C/h} \|\Lambda^{(1)}-\Lambda^{(2)}\|+h)N_{2m, \overline{\varrho }}(\alpha_1^*).
\end{align}
So using integration by parts for the left hand side of the above equality, we obtain
\begin{align*}
- \int_{\Omega} e^{-i x \cdot \xi} \vartheta_V (\varrho \cdot \nabla)^2 \overline{ \alpha_1^*}\mathrm{~d} x =- \int_\Omega e^{-i x \cdot \xi} (\varrho \cdot V^{\prime}) (\varrho \cdot \nabla \overline{\alpha_1^*})~dx + \mathcal{I}_3(h),
\end{align*}
here we used the fact that $ \vartheta_V =0$, on $\Gamma $, and $\omega \cdot \xi=\widetilde{\omega} \cdot \xi=0$. We consider now $\alpha_1^*(x)=-\frac{1}{2}(\omega \cdot x)^2$, $x \in \Omega$, which satisfies \eqref{3.14}.
Thus, one can show that
\begin{align*}
\abs{ \widehat{\vartheta}_V(\xi)}
{\,\lesssim\,} \langle \xi\rangle^{2m}(\|V^\prime \|_{L^{\infty}(\Omega)}+\|A^\prime\|_{L^\infty}+e^{C/h} \|\Lambda^{(1)}-\Lambda^{(2)}\|+h).
\end{align*}
Then from Lemmas \ref{L4.5} and \ref{L4.7}, we obtain
\begin{align*}
\abs{\widehat{\vartheta}_V(\xi)}
{\,\lesssim\,} \langle \xi\rangle^{2m} ( e^{C^\prime/h}\|\Lambda^{(1)}-\Lambda^{(2)}\|^{\theta_1}+h^{\theta_2}),
\end{align*}
where $C^\prime=\max(C, C_1, C_2)$ is a positive constant independent of $h$, $\theta_1 =\min(\mu_1, \mu_2) \in (0,1)$ depends on $n$ and $m$, and $\theta_2=\min(\kappa_1, \kappa_2 ) \in (0,1)$.\\
Moreover, as $\vartheta_V$ verifies \eqref{vartheta_V1} then by applying Lemma \ref{LA}, we get
\begin{align}\label{varth_V}
\|\vartheta_V\|_{W^{2, \infty}(\Omega)}&{\,\lesssim\,} e^{C_0/h} \|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_0} +h^{\kappa_0},
\end{align}
for some positive constant $C_0$ and $\mu_0$, $\kappa_0 \in (0,1)$.\\ Taking into account \eqref{V}, we may conclude from Lemma \ref{L4.7} and \eqref{varth_V} that Lemma \ref{L4.11} is completely proved with $C_3=\max(C_2, C_0)$ which is a positive constant independent of $h$, $\mu_3 =\min(\mu_2, \mu_0)\in (0,1)$ and $\kappa_3=\min(\kappa_2, \kappa_0 )\in (0,1)$ which depends on $n$ and $m$.
\end{proof}
In order to estimate the function $\vartheta_A$ given by \eqref{A}, we start by the following integral identity.
\begin{lemma}\label{L4.12}
Let $B \in \mathcal{B}_{\sigma_2}(M) \cap \mathcal{E}^\prime(\Omega)$ and $\vartheta_A$ given by \eqref{A}. There exist $\ell_1 \gg 1$ and $\mu_4$, $\kappa_4 \in (0,1)$ such that for all $\alpha_1,\,\alpha_1^*\in C^\infty(\overline{\Omega})$ satisfy \eqref{3.13} and \eqref{3.14} respectively, we have
\begin{equation}\label{4.41}
\int_\Omega 2 \vartheta_A (\varrho \cdot \nabla \alpha_1)\overline{\alpha_1^*}dx + i\int_\Omega (\varrho \cdot B) \alpha_1\overline{\alpha_1^*}dx=\mathcal{J}(h),
\end{equation}
where
\begin{align}\label{4.42}
| \mathcal{J}(h)| {\,\lesssim\,} \bigr( e^{C_4/{h^{\ell_1}}} \|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_4}+h^{\kappa_4})N_{2m, \varrho}(\alpha_1)N_{2m, \overline{\varrho }}(\alpha_1^*).
\end{align}
Here $C_4$ is a positive constant independent of $h$ and $\kappa_4, \ell_1$ depend only on $n$ and $m$.
\end{lemma}
\begin{proof}
Substituting $A$ in Lemma \ref{L3.2} we get
\begin{align*}
&\int_\Omega 2 \vartheta_A (\varrho \cdot \nabla \alpha_1)\overline{\alpha_1^*}dx + i\int_\Omega (\varrho \cdot B) \alpha_1\overline{\alpha_1^*}dx\\&=-\int_\Omega \bigr(h^{-1} A^\prime \varrho \cdot \varrho \alpha_1+h^{-1} \nabla_{\text{sym}}V \varrho \cdot \varrho \alpha_1+2 A^\prime \varrho \cdot \nabla \alpha_1+2 \nabla_{\text{sym}}V \varrho \cdot \nabla \alpha_1 \bigr)\overline{\alpha_1^*}dx+\mathcal{I}(h)\cr
&:=\mathcal{J}(h),
\end{align*}
where $\mathcal{I}(h)$ satisfies
\begin{align*}
\abs{ \mathcal{I}(h)}
&{\,\lesssim\,} (e^{C/h} \|\Lambda^{(1)}-\Lambda^{(2)}\|+h+\|A^\prime \|_{L^\infty}+\|V\|_{W^{1, \infty}})N_{2m, \varrho}(\alpha_1)N_{2m, \overline{\varrho }}(\alpha_1^*).
\end{align*}
Then by applying the Cauchy-Schwarz inequality for the first term in the right hand side of the above expression we obtain
\begin{align}\label{J0}
\abs{ \mathcal{J}(h)}
&{\,\lesssim\,} \bigr( h^{-1} (\|A^\prime \|_{L^\infty}+\|V\|_{W^{1, \infty}})+e^{C/h} \|\Lambda^{(1)}-\Lambda^{(2)}\|+h\bigr)N_{2m, \varrho}(\alpha_1)N_{2m, \overline{\varrho }}(\alpha_1^*).
\end{align}
Using now Lemmas \ref{L4.5} and \ref{L4.11}, then with the help of the last part of Lemma \ref{LA}, there exist $\widetilde{\ell}_1, \widetilde{\ell}_4 \gg1$ and $\widetilde{\kappa}_1, \widetilde{\kappa}_4 \in (0,1)$ such that we have
\begin{align*}
\abs{ \mathcal{J}(h)}&{\,\lesssim\,} \bigr(e^{C/{h^{\widetilde{\ell}_1}}} \|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_1} +h^{\widetilde{\kappa}_1}+e^{C/{h^{\widetilde{\ell}_4}}} \|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_3} +h^{\widetilde{\kappa}_4}\bigr)N_{2m, \varrho}(\alpha_1)N_{2m, \overline{\varrho }}(\alpha_1^*).
\end{align*}
This completes the proof with $\ell_1=\max(\widetilde{\ell}_1, \widetilde{\ell}_4)\gg 1$, $\mu_4=\min(\mu_1, \mu_3) \in (0,1)$ and $\kappa_4=\min(\widetilde{\kappa}_1, \widetilde{\kappa}_4) \in (0,1)$.
\end{proof}
\subsection{Stability estimate for the vector field (first case \texorpdfstring{$m>2$}{Lg}) }
We derive in this section a stability estimate for the vector field $B$. First, we will use the Hodge decomposition given by \eqref{B} and the integral identity \eqref{4.41} to estimate the Fourier transform of $\textrm{curl}~ B$. Second, we exploit the boundness of $B$ to prove the stability estimate for $B$ it self.
\smallskip
In what follows, we denote for $B=(B_1, \ldots, B_n)$
\begin{align*}
\mathrm{b}_{jk}=\partial_{x_j}B_k-\partial _{x_k}B_j ,~~ j, k=1, \ldots, n,
\end{align*}
the components of $\textrm{curl}~B$ and $\widehat{\mathrm{b}}_{jk}$ the associated Fourier coefficients.\\
Since the functions $\alpha_1$ and $\alpha_1^*$ are arbitrary solutions of \eqref{3.13} and \eqref{3.14}, respectively, our strategy is to choose suitable solutions $\alpha_1$ and $\alpha_1^*$ such that, in the first step, the first term in the left hand side of \eqref{4.41} becomes zero in order to obtain an estimate of $\textrm{curl}~B$. Second, we return to identity \eqref{4.41} and by choosing again a suitable solutions $\alpha_1$ and $\alpha_1^*$ we may estimate the Fourier transform of $\vartheta_A$ in terms of norm of $B$. We then have the following estimate of $\textrm{curl}~B$.
\begin{lemma}\label{L4.14}
Let $B$ as above. There exist $\ell_2 \gg 1$ and $\mu_5, \kappa_5 \in (0, 1)$ such that we have the following estimate
\begin{align}
\|\textrm{curl}~B\|_{L^{\infty}(\Omega)} {\,\lesssim\,} e^{C_5/{h^{\ell_2}}} \|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_5 } +h^{\kappa_5},
\label{4.44}
\end{align}
for any $h >0$ small enough. Here $C_5$ is a positive constant independent of $h$ and $\kappa_5 $, $\ell_2$ depend only on $n$ and $m$.
\end{lemma}
\begin{proof}
Let $\xi \in \mathbb{R}^{n}$. We select $\omega, \widetilde{\omega} \in \mathbb{S}^{n-1}$ such that $\xi, \omega$ and $\widetilde{\omega}$ be three mutually orthogonal vectors in $\mathbb{R}^{n}$. Let us choose $\alpha_1 (x)=e^{-i x \cdot \xi}$ and $\alpha_1^*(x)=1$, $x \in \Omega$. Then $\alpha_1 $ and $\alpha_1^*$ satisfy respectively \eqref{3.13} and \eqref{3.14}. Using now Lemma \ref{L4.12} and the fact that $\varrho \cdot \xi=0$, we obtain
\begin{align*}
&\abs{\varrho \cdot \widehat{B}(\xi)}{\,\lesssim\,} \langle \xi\rangle^{2m}\bigr(e^{C_4/{h^{\ell_1}}}\norm{\Lambda^{(1)}-\Lambda^{(2)}}^{\mu_4 }+h^{\kappa_4 }).
\end{align*}
Then by doing the same analysis as the first part of the proof of Lemma \ref{L4.7}, we get
\begin{align*}
&\abs{\widehat{\mathrm{b}}_{jk}(\xi)}{\,\lesssim\,} \langle \xi\rangle^{2m+1}\bigr(e^{C_4/{h^{\ell_1}}}\norm{\Lambda^{(1)}-\Lambda^{(2)}}^{\mu_4 }+h^{\kappa_4 }).
\end{align*}
Therefore, by Lemma \ref{LA} there exist $C_5 >0$ and $\mu_5, \kappa_5 \in (0, 1)$ such that we have
\begin{align}
\|\textrm{curl}~B\|_{L^{\infty}(\Omega)} {\,\lesssim\,} e^{C_5/{h^{\ell_1}}} \|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_5 } +h^{\kappa_5}.
\end{align}
This completes the proof.
\end{proof}
The next step is to estimate the $W^{1, \infty}-$norm of $\vartheta_B$.
\begin{lemma}\label{L4.15}
Let $\vartheta_B$ given by \eqref{B}. There exist $\ell_3 \gg 1$ and $\mu_6$, $\kappa_6 \in (0,1)$ such that we have
\begin{align}\label{4.49}
\|\vartheta_B\|_{W^{1, \infty}(\Omega)}&\leq e^{C_6/{h^{\ell_3}}} \|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_6} +h^{\kappa_6},
\end{align}
for any $h >0$ small enough. Here $C_6$ is a positive constant independent of $h$ and $\kappa_6, \ell_3$ depend on $n$ and $m$.
\end{lemma}
\begin{proof}
We start by estimate the Fourier transform of $\vartheta_B$. Let $\xi \in \mathbb{R}^n$, we choose $\omega, \widetilde{\omega} \in \mathbb{S}^{n-1}$ such that $\xi, \omega$ and $\widetilde{\omega}$ be three mutually orthogonal vectors in $\mathbb{R}^{n}$. Substituting $B$ into the left hand side of \eqref{4.41} then we get for $\alpha_1(x)=e^{-i x \cdot \xi}$
\begin{equation}\label{4.50}
\int_\Omega e^{-i x \cdot \xi}(\varrho \cdot \nabla \vartheta_B) \overline{\alpha_1^*} ~dx=-\int_\Omega e^{-i x \cdot \xi} (\varrho \cdot B^{\prime}) \overline{\alpha_1^*}~dx + \mathcal{I}(h),
\end{equation}
where $\mathcal{I}(h)$ satisfies
\begin{align}\label{4.51}
| \mathcal{I}(h)| {\,\lesssim\,} \langle \xi\rangle^{2m} (e^{C_4/h^{\ell_1}} \|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_4}+h^{\kappa_4})N_{2m, \overline{\varrho }}(\alpha_1^*).
\end{align}
So using integration by parts for the left hand side of the above equality, we obtain
\begin{align*}
- \int_{\Omega} e^{-i x \cdot \xi} \vartheta_B (\varrho \cdot \nabla \overline{ \alpha_1^*})\mathrm{~d} x =- \int_{\Omega} e^{-i x \cdot \xi} (\varrho \cdot B^\prime) \overline{\alpha_1^*}\mathrm{~d} x+ \mathcal{I}(h),
\end{align*}
here we used the fact that $\vartheta_B =0$, on $\Gamma $ and $\omega \cdot \xi=\widetilde{\omega} \cdot \xi=0$. Choosing now $\alpha_1^*(x)=-\omega \cdot x$, $x \in \Omega$,
then by using Lemma \ref{L4.14} , we obtain
\begin{align*}
\abs{ \widehat{\vartheta}_B (\xi)}
{\,\lesssim\,} \langle \xi\rangle^{2m} (e^{C/h^\ell} \|\Lambda^{(1)}-\Lambda^{(2)}\|^{\theta_3}+h^{\theta_4}),
\end{align*}
for some positive constant $C$. Here $\ell=\max(\ell_1, \ell_2) \gg 1$, $\theta_3=\min(\mu_4, \mu_5)\in(0,1)$ and $\theta_4=\min(\kappa_4, \kappa_5)$ $\in (0,1)$ depends only on $n$ and $m$.\\
Therefore, since ${\vartheta}_B$ satisfies \eqref{vartheta_B1} then by using Lemma \ref{LA}, we get our desired estimate.
\end{proof}
We are now in position to prove the stability estimate of the first order coefficient $B$ in the case $m >2$.
\begin{proof}[Proof of Theorem \ref{T1.3} (Case $m >2$)]
Using Lemmas \ref{L4.14}, \ref{L4.15} and the estimate \eqref{B1} we get from \eqref{B}
\begin{align}\label{norm_B}
\|B \|_{L^{\infty}(\Omega)}
&{\,\lesssim\,} e^{C/h^\ell} \|\Lambda^{(1)}-\Lambda^{(2)}\|^{\theta_5}+h^{\theta_6},
\end{align}
for some $C >0$. Here $\ell=\max(\ell_2, \ell_3) \gg 1$, $\theta_5=\operatorname{min}(\mu_5, \mu_6) \in (0,1)$ and $\theta_6=\operatorname{min}(\kappa_5, \kappa_6) \in (0,1)$ depends on $n$ and $m$.\\
Let $0 < h_0 <1$, the above statement is valid for all $h < h_0$ and $\|\Lambda^{(1)}-\Lambda^{(2)}\|$ small enough. Then, if $\|\Lambda^{(1)}-\Lambda^{(2)}\|< \varepsilon_0$, such that $-\frac{\theta_5}{2}\ln \varepsilon_0 \geq {C/h_0^\ell}$, the taking $h=\bigr(-\frac{1}{C}\ln (\|\Lambda^{(1)}-\Lambda^{(2)}\|^{\frac{\theta_5}{2}})\bigr)^{\frac{-1}{\ell}}$ in \eqref{norm_B} implies
\begin{align}\label{4.22}
\|B\|_{L^{\infty}(\Omega)} {\,\lesssim\,} \|\Lambda^{(1)}-\Lambda^{(2)}\|^{\frac{\theta_5}{2}}+ \abs{\ln \|\Lambda^{(1)}-\Lambda^{(2)}\|}^\frac{-\theta_6}{\ell}.
\end{align}
Moreover, with this choice of $h$ we have $0 <h < h_0$.\\
We also observe that this type of inequality holds if $\|\Lambda^{(1)}-\Lambda^{(2)}\|\geq \varepsilon_0$. Since in that case we can simply write
\begin{align}
\|B\|_{L^{\infty}(\Omega)} {\,\lesssim\,} M {\,\lesssim\,} \frac{M}{(\sqrt{\varepsilon_0})^{\theta_5}}\|\Lambda^{(1)}-\Lambda^{(2)}\|^{\frac{\theta_5}{2}}.
\end{align}
This concludes the proof of stability estimate of first order coefficient $B$ in the case $m >2$.
\end{proof}
\subsection{Stability estimate for the symmetric tensor (first case \texorpdfstring{$m>2$}{Lg}) }
The aim of this section is to prove the stable determination of the symmetric tensor $A$ in the case $m>2$. To do this we need to estimate the function $\vartheta_A$, given by \eqref{A}.
\begin{lemma}\label{L4.17}
Let $\vartheta_A$ given by \eqref{A}. There exist $\ell_4 \gg 1$ and $\mu_7, \kappa_7 \in (0,1)$ such that we have
\begin{align}\label{4.54}
\|\vartheta_A\|_{L^{\infty}(\Omega)}&\leq e^{C_7/{h^{\ell_4}}} \|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_7} +h^{\kappa_7},
\end{align}
for any $h >0$ small enough. Here $C_7$ is a positive constant independent of $h$ and $\kappa_7, \ell_4$ depend on $n$ and $m$.
\end{lemma}
\begin{proof}
Let $\xi \in \mathbb{R}^n$. We fix $\omega, \widetilde{\omega} \in \mathbb{S}^{n-1}$ such that $\xi$, $\omega$ and $\widetilde{\omega}$ be three mutually orthogonal vectors in $\mathbb{R}^n$. Let us choose $\alpha_1(x)=\omega \cdot x$, $x \in \Omega$ and $\alpha_1^* (x)=e^{i x \cdot \xi}$. It is clear that $\alpha_1 $ and $\alpha_1^*$ satisfy \eqref{3.13} and \eqref{3.14}, respectively. Using now Lemma \ref{L4.12}, we obtain
\begin{align*}
&\abs{\widehat{ \vartheta}_A(\xi) }
{\,\lesssim\,} \langle \xi\rangle^{2m}\bigr(e^{C_4/{h^{\ell_1}}}\norm{\Lambda^{(1)}-\Lambda^{(2)}}^{\mu_4}+h^{\kappa_4}+\|B\|_{L^\infty(\Omega)}\bigr).
\end{align*}
Therefore, \eqref{norm_B} implies
\begin{align*}
&\abs{\widehat{ \vartheta}_A(\xi) }
{\,\lesssim\,} \langle \xi\rangle^{2m}(e^{C^\prime/h^{\ell^\prime}}\norm{\Lambda^{(1)}-\Lambda^{(2)}}^{\theta_7}+h^{\theta_8}),
\end{align*}
for some $C^\prime >0$. Here $\ell^\prime=\max(\ell_1, \ell)$, $\theta_7=\min(\mu_4, \theta_5)$ and $\theta_8=\min(\kappa_4, \theta_6)$.\\
Hence, since $\vartheta_A$ satisfies \eqref{A1} then by using Lemma \ref{LA} we deduce \eqref{4.54}. This ends the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{T1.2} (Case $m >2$)]
By using Lemmas \ref{L4.5}, \ref{L4.11} and \ref{L4.17}, we get from the equality \eqref{A}
\begin{align}\label{normA}
\|A\|_{L^{\infty}(\Omega)}&\leq e^{C/{h^{\ell_4}}} \|\Lambda^{(1)}-\Lambda^{(2)}\|^{\theta_9} +h^{\theta_{10}},
\end{align}
for some positive constant $C$. Here $\theta_9=\min(\mu_1, \mu_3, \mu_7)$ and $\theta_{10}=\min(\kappa_1, \kappa_3, \kappa_7)$ depends only on $n$ and $m$.\\
Choosing now $h=\bigr(-\frac{1}{C}\ln (\|\Lambda^{(1)}-\Lambda^{(2)}\|^{\frac{\theta_9}{2}})\bigr)^{\frac{-1}{\ell_4}}$ and proceeding exactly as the last part of proof of Theorem \ref{T1.3} (Case $m >2$)
we deduce the stability for the second order perturbation in the case $m >2$.
\end{proof}
\subsection{Stability estimate for the electric potential (first case \texorpdfstring{$m>2$}{Lg}) }
We will now establish the stability estimate for the zeroth order term. Let us first estimate the Fourier transform of zeroth-order perturbations as follow.
\begin{lemma}\label{L3.12}
Let $q \in \mathcal{Q}(M)$. For all $\xi \in \mathbb{R}^n$, there exist $\ell_5 \gg 1$ and $\mu_8, \kappa_8 \in (0,1)$ such that the following estimate
\begin{align}\label{3.63}
|\widehat{q}(\xi)| {\,\lesssim\,} \langle \xi\rangle^{2m} \bigr(e^{C_8/{h^{\ell_5}} }\|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_8}+h^{\kappa_8}\bigr),
\end{align}
holds for any $h >0$ small enough. Here $C_8$ is a positive constant independent of $h$ and $\kappa_8, \ell_5$ depend on $n$ and $m$.
\end{lemma}
\begin{proof}
Substituting $u$ and $u^*$ into the left hand side of \eqref{3.6}. Then the third term on the left hand side of \eqref{3.6} becomes
\begin{align*}
\int_{\Omega} q(x) u(x) \overline{u^*}(x)~dx= \int_{\Omega} q(x) \alpha_1(x) \overline{\alpha_1^*}(x)~dx + \mathcal{J}_1(h),
\end{align*}
where
\begin{align}
\mathcal{J}_1(h)= \int_{\Omega} q(x) (h\alpha_2+r) \overline{(\alpha_1^*+h\alpha_2^*+r^*)}~dx+\int_{\Omega} q(x) \alpha_1 \overline{(h\alpha_2^*+r^*)}~dx.
\end{align}
We deduce from the Cauchy-Schwartz inequality and from the inequalities (\ref{3.12}), \eqref{3.12'}, \eqref{2.50} and \eqref{2.53} that
\begin{align}\label{3.61}
\abs{ \mathcal{J}_1(h)}{\,\lesssim\,} hN_{2m, \varrho}(\alpha_1)N_{2m, \overline{\varrho }}(\alpha_1^*).
\end{align}
In the same manner, we get
\begin{align}\label{3.62}
&\int_{\Omega} (AD\cdot D u+ B\cdot Du) \overline{u^*}~dx \cr
&= \int_{\Omega}\bigr((-h^{-2}A\varrho\cdot \varrho-\frac{2i}{h}A\varrho \cdot D+AD\cdot D-\frac{i}{h} \varrho \cdot B +B \cdot D
)(\alpha_1+h\alpha_2+r)\bigr)\overline{(\alpha_1^*+h\alpha_2^*+r^*)}dx\cr
&:=\mathcal{J}_2(h),
\end{align}
where $\mathcal{J}_2(h)$ satisfies
\begin{align*}
\abs{\mathcal{J}_2(h)}{\,\lesssim\,} \bigr(h^{-2}\|A\|_{L^{\infty}}+ h^{-1}\|B\|_{L^{\infty}}\bigr)N_{2m, \varrho}(\alpha_1)N_{2m, \overline{\varrho }}(\alpha_1^*).
\end{align*}
Therefore, using the estimates obtained above and the equality \eqref{3.6}, we deduce
\begin{align}\label{3.59}
\abs{ \int_{\Omega} q(x) \alpha_1(x) \overline{\alpha_1^*}(x)~dx}
&{\,\lesssim\,} \bigr(h^{-2}\|A\|_{L^{\infty}(\Omega)}+h^{-1}\|B\|_{L^{\infty}(\Omega)}\cr &\qquad+e^{C/h }\|\Lambda^{(1)}-\Lambda^{(2)}\| +h\bigr) N_{2m, \varrho}(\alpha_1)N_{2m, \overline{\varrho }}(\alpha_1^*),
\end{align}
Let now $\xi \in \mathbb{R}^n$. We fix $\omega, \widetilde{\omega} \in \mathbb{S}^{n-1}$ such that $\xi$, $\omega$ and $\widetilde{\omega}$ be three mutually orthogonal vectors in $\mathbb{R}^n$. Choosing $\alpha_1(x)=e^{-ix \cdot \xi}$ and $\alpha_1^*(x)=1$ in \eqref{3.59} we get
\begin{align}\label{3.64}
\abs{\widehat{q}(\xi)} {\,\lesssim\,} \langle \xi\rangle^{2m} \bigr(h^{-2}\|A\|_{L^{\infty}(\Omega)}+h^{-1}\|B\|_{L^{\infty}(\Omega)}+e^{C/h }\|\Lambda^{(1)}-\Lambda^{(2)}\|+h\bigr) .
\end{align}
This along with \eqref{norm_B}, \eqref{normA} and \eqref{1*} show that
\begin{align*}
\abs{\widehat{q}(\xi)} &{\,\lesssim\,} \langle \xi\rangle^{2m} \bigr(e^{C_8/{h^{\widetilde{\ell}_4}} }\|\Lambda^{(1)}-\Lambda^{(2)}\|^{\theta_9}+h^{\widetilde{\theta}_{10}}+e^{C_8/{h^{\widetilde{\ell}}} }\|\Lambda^{(1)}-\Lambda^{(2)}\|^{\theta_5}+h^{\widetilde{\theta}_{6}}\bigr),
\end{align*}
for some $C_8 >0$ is a positive constant independent of $h$. Here $\widetilde{\ell}, \widetilde{\ell}_4 \gg 1$ and $\widetilde{\theta}_{6}, \widetilde{\theta}_{10} \in (0,1)$ depend on $n$ and $m$. Then, by choosing $\ell_5=\max(\widetilde{\ell}_4, \widetilde{\ell}) $, $\kappa_8=\min(\widetilde{\theta}_{10}, \widetilde{\theta}_{6})$ and $\mu_8=\min(\theta_5, \theta_9)$ we get our desired estimate.
\end{proof}
With the help of the above Lemma, we may now prove the stability estimate of the potential.
\begin{proof}[Proof of Theorem \ref{T1.4} (Case $m>2$)]
Let $R >1$, to be chosen later, we get from Lemma \ref{L3.12}
\begin{align}
\|q \|_{H^{-1}(\Omega)}^{2} &\leq \|q \|_{H^{-1}\left(\mathbb{R}^{n}\right)}^{2}=\int_{\langle \xi\rangle \leqslant R}\abs{\widehat{q}(\xi)}^{2}\langle \xi\rangle^{-2} d \xi+\int_{\langle \xi\rangle> R}\abs{\widehat{q}(\xi)}^{2}\langle \xi\rangle^{-2} d \xi \cr
&{\,\lesssim\,} R^{n+4m-2}\bigr(e^{{2C_8}/{h^{\ell_5}} }\|\Lambda^{(1)}-\Lambda^{(2)}\|^{2\mu_8}+h^{2\kappa_8}\bigr)+ M^2 R^{-2}.\label{3.650}
\end{align}
Choosing $R=h^\frac{-2\kappa_8}{n+4m}$ then, for $h_0 >0$ sufficiently small we obtain
\begin{align}
\|q\|_{H^{-1}(\Omega)} {\,\lesssim\,} e{^{C^\prime/h^{\ell_5}}}\|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_8}+ h^\frac{2 \kappa_8}{n+4m},\label{3.660}
\end{align}
for some positive constant $C^{'}$ and all $h<h_0$.\\
Let $\varepsilon_0 >0$ such that $- \frac{\mu_8}{2}\ln \varepsilon_0 \geq {C^\prime/h_0^{\ell_5}}$. If $\|\Lambda^{(1)}-\Lambda^{(2)}\|< \varepsilon_0$, we choose $h=\bigr(-\frac{1}{C^{'}}\ln( \|\Lambda^{(1)}-\Lambda^{(2)}\|^{\frac{\mu_8}{2}})\bigr)^{-\frac{1}{\ell_5}}$ in \eqref{3.660}, thus, we obtain $0 < h \leq h_0$ and
\begin{align}
\|q\|_{H^{-1}(\Omega)} {\,\lesssim\,} \|\Lambda^{(1)}-\Lambda^{(2)}\|^{\frac{\mu_8}{2}}+ \abs{\ln \|\Lambda^{(1)}-\Lambda^{(2)}\|}^\frac{-2 \kappa_8}{(n+4m)\ell_5}.
\end{align}
Now if $\|\Lambda^{(1)}-\Lambda^{(2)}\|\geq \varepsilon_0$, we also observe that this type of inequality holds. Indeed, we can simply write
\begin{align}
\|q \|_{H^{-1}(\Omega)} {\,\lesssim\,} M {\,\lesssim\,} \frac{M}{\sqrt{\varepsilon_0}^{\mu_8}}\|\Lambda^{(1)}-\Lambda^{(2)}\|^{\frac{\mu_8}{2}}.
\end{align}
This concludes the proof for stability of zeroth order coefficient, in the case $m
>2$.
\end{proof}
\section{Stability estimates (second case \texorpdfstring{$m=2$}{Lg})} \label{section5}
The aim of this section is to proving the stable determination of the isotropic tensor $A=a ~id$, the vector field $B$ and the electric potential $q$ from the Dirichlet to Neumann map $\Lambda_{A,B,q}$, in the case $m=2$. The proof follows from similar arguments we used in the previous section.\smallskip\\
We consider for $j=1,2,$ $A^{(j)}=a^{(j)}\operatorname{id} \in \mathcal{A}_{\sigma_1}(M) \cap \mathcal{E}^\prime(\Omega)$, $B^{(j)} \in \mathcal{B}_{\sigma_2}(M) \cap \mathcal{E}^\prime(\Omega)$ and $q^{(j)} \in \mathcal{Q}(M)$ pair of admissible coefficients. We denote
\begin{align}
a=a^{(2)}-a^{(1)},\quad B=B^{(2)}-B^{(1)}\quad\text{and }~~~~ q=q^{(2)}-q^{(1)}.
\end{align}
\subsection{Stability estimate for the vector field (second case \texorpdfstring{$m=2$}{Lg})}
We derive in this section stability estimate for the vector field $B$ in the case $m=2$. As section \ref{section4}, we use the Hodge decomposition for a vector field $B$ and we write
\begin{align}
B&=B^\prime+\nabla \vartheta_B,\label{6.0}
\end{align}
where the vector field $B^{\prime} \in L^{ \infty}(\Omega, \mathbb{C}^n)$ satisfies \eqref{4.100} and
\begin{align}\label{6.1}
\|B^\prime\|_{L^{\infty}(\Omega)} {\,\lesssim\,} \|\textrm{curl}~B\|_{L^{\infty}(\Omega)}.
\end{align}
Moreover, the function $\vartheta_B \in W^{1, \infty}(\Omega, \mathbb{C})$ satisfies
\begin{align}\label{6.2}
\| \vartheta_B\|_{H^{\sigma_2}(\Omega)} {\,\lesssim\,} \|B\|_{H^{\sigma_2}(\Omega)},
\end{align}
here $\sigma_2> \frac{n}{2}+1$.\medskip\\
We start by estimate the $L^{\infty}-$norm of the vector field $B^\prime$.
\begin{lemma}\label{'T3.5}
Let $B^\prime$ given by \eqref{6.0}. Then there exist $\mu_9, \kappa_9 \in (0, 1)$ such that we have the following estimate
\begin{align}
\|B^\prime\|_{L^{\infty}(\Omega)} {\,\lesssim\,} e^{C_9/{h}} \|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_9} +h^{\kappa_9},
\label{'3.27}
\end{align}
for any $h >0$ small enough. Here $C_9$ is a positive constant independent of $h$ and $\kappa_9 $ depends only on $n$.
\end{lemma}
\begin{proof}
Let $\xi \in \mathbb{R}^n$, choosing $\omega, \widetilde{\omega} \in \mathbb{S}^{n-1}$ such that $\xi, \omega$ and $\widetilde{\omega}$ be three mutually orthogonal vectors in $\mathbb{R}^n$. Let us choose $\alpha_1 (x)=e^{-i x \cdot \xi}$ and $\alpha_1^*(x)=1$, $x \in \Omega$ being a solutions of \eqref{3.13} and \eqref{3.14}, respectively. Using now Lemma \ref{L3.3} and the fact that $\varrho \cdot \xi=0$, we obtain
\begin{equation}
\abs{\varrho \cdot \widehat{B}(\xi)}{\,\lesssim\,} \langle \xi\rangle^{4}(e^{C/h}\norm{\Lambda^{(1)}-\Lambda^{(2)}}+h).\label{'3.24}
\end{equation}
Doing the same analysis as the first part of proof of Lemma \ref{L4.7}, we get
\begin{align*}
\|\textrm{curl}~B\|_{L^{\infty}(\Omega)} {\,\lesssim\,} e^{C_9/{h}} \|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_9} +h^{\kappa_9},
\end{align*}
for some positive constant $C_9$ and $\mu_9, \kappa_9 \in (0,1)$. Now, by using \eqref{6.1} we get our desired estimate.
\end{proof}
With the help of above Lemma, we may estimate $\vartheta_B$ given by \eqref{6.0}.
\begin{lemma}
Let $\vartheta_B$ given by \eqref{6.0}. There exist $\mu_{10}, \kappa_{10} \in (0, 1)$ such that we have
\begin{align*}
\|\vartheta_B \|_{W^{1, \infty}(\Omega)} {\,\lesssim\,} e^{C_{10}/{h}} \|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_{10}} +h^{\kappa_{10}}.
\end{align*}
Here $C_{10}$ is a positive constant independent of $h$ and $\kappa_{10} \in (0,1)$ depends only on $n$.
\label{'P3.7}
\end{lemma}
\begin{proof}
Let $\xi \in \mathbb{R}^{n}$. We choose $\omega, \widetilde{\omega} \in \mathbb{S}^{n-1}$ such that $\xi, \omega$ and $\widetilde{\omega}$ be three mutually orthogonal vectors in $\mathbb{R}^{n}$. Substituting $B$ into the left hand side of \eqref{3.25}, we get for $\alpha_1(x)=e^{-i x \cdot \xi}$
\begin{equation}\label{'3.33}
\int_\Omega e^{-i x \cdot \xi} (\varrho \cdot \nabla \vartheta_B) \overline{\alpha_1^*} ~dx=-\int_\Omega e^{-i x \cdot \xi} (\varrho \cdot B^{'} )\overline{\alpha_1^*}~dx + \mathcal{I}(h),
\end{equation}
where $\mathcal{I}$ satisfies
\begin{align}\label{'3.34}
| \mathcal{I}(h)| {\,\lesssim\,} \langle \xi\rangle^4(e^{C/h} \|\Lambda^{(1)}-\Lambda^{(2)}\|+h)N_{4, \overline{\varrho}}(\alpha_1^*).
\end{align}
We apply now an integration by parts for the left hand side of the above equality, we obtain
\begin{align*}
- \int_{\Omega} e^{-i x \cdot \xi} \vartheta_B (\varrho \cdot \nabla \overline{ \alpha_1^*})\mathrm{~d} x =- \int_{\Omega} e^{-i x \cdot \xi}(\varrho \cdot B^\prime) \overline{\alpha_1^*}\mathrm{~d} x+ \mathcal{I}(h),
\end{align*}
here we used the fact that $\vartheta_B =0$, on $\Gamma $ and $\varrho \cdot \xi=0$.\\
Therefore, we obtain for $\alpha_1^*(x)=-\omega \cdot x$, $x \in \Omega$, that
\begin{align*}
\abs{ \widehat{\vartheta}_B(\xi)}
{\,\lesssim\,} \langle \xi\rangle^4 \bigr(\|B^\prime \|_{L^{\infty}(\Omega)}+e^{C/h}\|\Lambda^{(1)}-\Lambda^{(2)}\|+h \bigr).
\end{align*}
This inequality and Lemma \ref{'T3.5} allow us to obtain for some constant $C^\prime >0$ that
\begin{align*}
\abs{ \widehat{\vartheta}_B(\xi)} &{\,\lesssim\,} \langle \xi\rangle^4 (e^{C^\prime/h}\|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_9}+h^{\kappa_9}),
\end{align*}
here $\kappa_9 \in (0,1)$ and $\mu_9=\mu_9(n) \in (0,1)$.\\
In the other hand, since ${\vartheta}_B$ satisfies \eqref{6.2} then by using Lemma \ref{LA} we get our desired estimate.
\end{proof}
By using the above results, we are able to prove the stability estimate for first order coefficient $B$, in the case $m=2$.
\begin{proof}[Proof of Theorem \ref{T1.3} (Case $m=2$)]
By Lemmas \ref{'T3.5} and \ref{'P3.7}, we get from \eqref{6.0}
\begin{align}
\|B \|_{L^{\infty}(\Omega)} & {\,\lesssim\,} \|B^\prime\|_{L^{\infty}(\Omega)}+\|\vartheta_B \|_{W^{1,\infty}(\Omega)}\cr
&{\,\lesssim\,} e^{C^\prime/h} \|\Lambda^{(1)}-\Lambda^{(2)}\|^{\theta_{13}} +h^{\theta_{14}},\label{6.3}
\end{align}
with $C^\prime$ is a positive constant independent of $h$, $\theta_{13}=\operatorname{min}(\mu_9, \mu_{10})$ and $\theta_{14}=\min(\kappa_9, \kappa_{10}) \in (0,1)$ depends only on $n$.\\
Let $\varepsilon_0 \in (0, 1)$ such that $-\ln \varepsilon_0 \geq {C^\prime/h_0}$. Then if $\|\Lambda^{(1)}-\Lambda^{(2)}\|< \varepsilon_0$, choosing $h=\bigr(-\frac{1}{C^\prime}\ln (\|\Lambda^{(1)}-\Lambda^{(2)}\|^{\frac{\theta_{13}}{2}})\bigr)^{-1}$ in \eqref{6.3} and proceeding exactly as the last part of proof of Theorem \ref{T1.3} (Case $m >2$)
we conclude that Theorem \ref{T1.3} is completely proved.
\end{proof}
\subsection{Stability estimate for the isotropic tensor (second case \texorpdfstring{$m=2$}{Lg})}
We derive in this section stability estimate for the isotropic matrix $A=a ~id$. First, we will use Lemma \ref{L3.3} to estimate the Fourier transform of $a=a^{(2)}-a^{(1)}$.
\begin{lemma}\label{'L3.8}
Let $a$ as above. There exist $\mu_{11}, \kappa_{11} \in (0,1)$ such that we have the following estimate
\begin{align*}
\abs{\widehat{a}(\xi)} {\,\lesssim\,} \langle\xi \rangle^4\bigr(e^{C_{11}/{h} }\|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_{11}}+h^{\kappa_{11}}\bigr),
\end{align*}
for any $h > 0$ small enough and for all $\xi \in \mathbb{R}^n$. Here $C_{11}$ is a positive constant independent of $h$ and $\kappa_{11}$ depends only on $n$.
\end{lemma}
\begin{proof}
Let $\xi \in \mathbb{R}^n$. We fix $\omega, \widetilde{\omega} \in \mathbb{S}^{n-1}$ such that $\xi$, $\omega$ and $\widetilde{\omega}$ be three mutually orthogonal vectors in $\mathbb{R}^n$. Choosing $\alpha_1 (x)=(x \cdot \omega) e^{-i x\cdot \xi}$ and $\alpha_1^*(x)=1$,~~for $x \in \Omega$. It is clear that $\alpha_1 $ and $\alpha_1^*$ satisfy respectively \eqref{3.13} and \eqref{3.14}. Then using Lemma \ref{L3.3} we get
\begin{align*}
\abs{\widehat{a}(\xi)}{\,\lesssim\,}\langle \xi\rangle^4\bigr(\|B\|_{L^{\infty}(\Omega)}+ e^{C/h }\|\Lambda^{(1)}-\Lambda^{(2)}\|+h \bigr).
\end{align*}
Now using \eqref{6.3} we get our desired estimate with $C_{11}=\max(C^\prime, C)$, $\mu_{11}=\theta_{13}$ and $\kappa_{11}=\theta_{14}(n)$.
\end{proof}
With the help of the above Lemma, we may now prove the stability estimate of the second-order perturbation, in the case $m=2$.
\begin{proof}[Proof of Theorem \ref{T1.2} (Case $m=2$)]
By Lemma \ref{'L3.8} we have
\begin{align*}
\abs{\widehat{a}(\xi)} {\,\lesssim\,} \langle\xi \rangle^4\bigr(e^{C_{11}/{h} }\|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_{11}}+h^{\kappa_{11}}\bigr).
\end{align*}
Then by using Lemma \ref{LA}, there exist $\theta_{15}, \theta_{16}=\theta_{16}(n) \in (0,1)$ such that we have
\begin{align}\label{normA1}
\|a\|_{L^{\infty}(\Omega)}&\leq e^{C^\prime/h} \|\Lambda^{(1)}-\Lambda^{(2)}\|^{\theta_{15}} +h^{\theta_{16}},
\end{align}
for some constant $C^\prime >0$.\\
Now choosing $h=\bigr(-\frac{1}{C^\prime}\ln (\|\Lambda^{(1)}-\Lambda^{(2)}\|^{\frac{\theta_{15}}{2}})\bigr)^{-1}$ and proceeding exactly as the last part of proof of Theorem \ref{T1.3} (Case $m >2$)
we get our desired estimate. This concludes the proof for stability of second order coefficients from the D-to-N map.
\end{proof}
\subsection{Stability estimate for the electric potential (second case \texorpdfstring{$m=2$}{Lg})}
Now, we establish the stability result for the zeroth order term $q$. Let us first establish the following estimate of the Fourier transform of $q$.
\begin{lemma}
There exist $\ell_6 > 1$ and $\kappa_{12}, \mu_{12} \in (0,1)$ such that we have
\begin{align}\label{'3.40}
|\widehat{q}(\xi)| {\,\lesssim\,} \langle \xi\rangle^4 \left( e^{C_{12}/{h^{\ell_6}} }\|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_{12}}+h^{\kappa_{12}}\right),
\end{align}
for any $h >0$ small enough and for all $\xi \in \mathbb{R}^n$. Here $C_{12}$ is a positive constant independent of $h$ and $\kappa_{12} \in (0,1)$ depends on $n$.
\label{'L3.10}
\end{lemma}
\begin{proof}
Substituting $u$ and $u^*$ into the left hand side of \eqref{3.6}. Then the third term in the left hand side of \eqref{3.6} becomes
\begin{align*}
\int_{\Omega} q(x) u(x) \overline{u^*}(x)~dx= \int_{\Omega} q(x) \alpha_1(x) \overline{\alpha_1^*}(x)~dx + \mathcal{J}_1(h),
\end{align*}
where
\begin{align}
\mathcal{J}_1(h)= \int_{\Omega} q(x) (h\alpha_2+r) \overline{(\alpha_1^*+h\alpha_2^*+r^*)}~dx+\int_{\Omega} q(x) \alpha_1 \overline{(h\alpha_2^*+r^*)}~dx,
\end{align}
then by using the Cauchy-Schwartz inequality together with the inequalities (\ref{3.12}), \eqref{3.12'}, \eqref{2.50} and \eqref{2.53} we deduce
\begin{align}\label{'3.37}
\abs{ \mathcal{J}_1(h)}{\,\lesssim\,} hN_{4, {\varrho}}(\alpha_1)N_{4, \overline{\varrho}}(\alpha_1^*).
\end{align}
Similarly, we have
\begin{align}\label{'3.38}
&\int_{\Omega} (-a\Delta u+ B\cdot Du) \overline{u^*}~dx \cr
&= \int_{\Omega}\big((\frac{-2}{h}a \varrho \cdot \nabla-a \Delta-\frac{i}{h} \varrho \cdot B +B \cdot D
)(\alpha_1+h\alpha_2+r) \big)\overline{(\alpha_1^*+h\alpha_2^*+r^*)}dx\cr&:= \mathcal{J}_2(h),
\end{align}
where $\mathcal{J}_2$ can be estimated as
\begin{align*}
\abs{\mathcal{J}_2(h)}{\,\lesssim\,} h^{-1}(\|a\|_{L^{\infty}}+ \|B\|_{L^{\infty}})N_{4, {\varrho}}(\alpha_1)N_{4, \overline{\varrho}}(\alpha_1^*).
\end{align*}
Combining now the above inequalities, we obtain from \eqref{3.6}
\begin{align}\label{3.82}
\abs{ \int_{\Omega} q(x) \alpha_1(x) \overline{\alpha_1^*}(x)~dx}
&{\,\lesssim\,} (h^{-1}\|B\|_{L^{\infty}(\Omega)}+h^{-1}\|a\|_{L^{\infty}(\Omega)}\cr&\quad+e^{C/h }\|\Lambda^{(1)}-\Lambda^{(2)}\|+h)N_{4, {\varrho}}(\alpha_1) N_{4, \overline{\varrho}}(\alpha_1^*),
\end{align}
Let now $\xi \in \mathbb{R}^n$. We fix $\omega, \widetilde{\omega} \in \mathbb{S}^{n-1}$ such that $\xi$, $\omega$ and $\widetilde{\omega}$ be three mutually orthogonal vectors in $\mathbb{R}^n$. Choosing $\alpha_1(x)=e^{-ix \cdot \xi}$ and $\alpha_1^*(x)=1$ which respectively solve the transport equations \eqref{3.13} and \eqref{3.14}. Now by putting these particular choices of $\alpha_1$ and $\alpha_1^*$ in \eqref{3.82}, we get
\begin{align}\label{'q}
\abs{\widehat{q}(\xi)} {\,\lesssim\,} \langle \xi\rangle^4 \bigr(h^{-1}\|B\|_{L^{\infty}(\Omega)}+h^{-1}\|a\|_{L^{\infty}(\Omega)}+e^{C/h }\|\Lambda^{(1)}-\Lambda^{(2)}\|+h\bigr).
\end{align}
From this and the inequalities \eqref{6.3} and \eqref{normA1}, we can deduce
\begin{align}
\abs{\widehat{q}(\xi)} {\,\lesssim\,} \langle \xi\rangle^4 h^{-1}\bigr(e^{C/h }\|\Lambda^{(1)}-\Lambda^{(2)}\|^{\theta_{17}}+h^{\theta_{18}}\bigr),
\end{align}
for some positive constant $C$. Here $\theta_{17}=\min(\theta_{13},\theta_{15}) \in (0,1)$ and $\theta_{18}=\min(\theta_{14}, \theta_{16}) \in (0, 1)$ depends on $n$.
Using now \eqref{1*} we get our desired estimate.
\end{proof}
With the help of the above Lemma, we may now prove the stability estimate of the potential $q$, in the case $m=2$.
\begin{proof}[Proof of Theorem \ref{T1.4} (Case $m=2$)]
By Lemma \ref{'L3.8} we have
\begin{align*}
|\widehat{q}(\xi)| {\,\lesssim\,} \langle \xi\rangle^4 \left( e^{C_{12}/{h^{\ell_6}} }\|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_{12}}+h^{\kappa_{12}}\right).
\end{align*}
Then for $R >1$, to be chosen later, we obtain
\begin{align}
\|q \|_{H^{-1}(\Omega)}^{2} &\leq \|q \|_{H^{-1}\left(\mathbb{R}^{n}\right)}^{2}=\int_{\langle \xi\rangle \leqslant R}\abs{\widehat{q}(\xi)}^{2}\langle \xi\rangle^{-2} d \xi+\int_{\langle \xi\rangle> R}\abs{\widehat{q}(\xi)}^{2}\langle \xi\rangle^{-2} d \xi \cr
&{\,\lesssim\,} R^{8}\bigr(e^{{2C_{12}}/{h^{\ell_6}} }\|\Lambda^{(1)}-\Lambda^{(2)}\|^{2\mu_{12}}+h^{2\kappa_{12}}\bigr)+ R^{-2}.\label{3.6500}
\end{align}
Choosing $R=h^\frac{-2\kappa_{12}}{10}$ then, for $h_0 >0$ sufficiently small we obtain
\begin{align}
\|q\|_{H^{-1}(\Omega)} {\,\lesssim\,} e{^{C^\prime/h^{\ell_6}}}\|\Lambda^{(1)}-\Lambda^{(2)}\|^{\mu_{12}}+ h^\frac{2 \kappa_{12}}{10},\label{3.6600}
\end{align}
for some positive constant $C^{'}$ and all $h<h_0$.\\
Now choosing $h=\bigr(-\frac{1}{C^\prime}\ln (\|\Lambda^{(1)}-\Lambda^{(2)}\|^{\frac{\mu_{12}}{2}})\bigr)^{\frac{-1}{\ell_6}}$ and proceeding exactly as the last part of proof of Theorem \ref{T1.4} (Case $m >2$) we get our desired estimate. This concludes the proof for stability of zeroth order coefficient.
\end{proof}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,447
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\section{Introduction}
Consider a self-injective basic algebra of finite representation
type over an algebraically closed field. According to Riedtmann's
classification, the stable $AR$-quiver of such an algebra can be
described with the help of an associated tree, which must be
congruent with one of the Dynkin diagrams $A_n, D_n, E_6, E_7$, or
$E_8$ (see \cite{Riedt}). The complete description of the Hochschild
cohomology ring was obtained for an algebras of the types $A_n$ and
$D_n$, see \cite{Erd,Gen&Ka,Ka,Pu} (type $A_n$) and
\cite{Volkov1,Volkov2,Volkov3,Volkov4,Volkov5,Volkov6} (type $D_n$).
Consider algebras of tree class $E_6$. Any algebra of the class
$E_6$ is derived equivalent to the path algebra for some quiver with
relations. Namely, let $\mathcal Q_s$ ($s\in\N$) is the following
quiver:
\begin{figure}[h]
\includegraphics[width=10cm, scale=1]{pust_E6.eps}
\end{figure}
Then any algebra of the class $E_6$ is derived equivalent to one of the two following algebras:
1) $R_s=K\left[\mathcal Q_s\right]/I$, where $K$ is a field, and $I$ is the ideal in the path
algebra $K\left[\mathcal Q_s\right]$ of the quiver $\mathcal Q_s$, generated by
a) all the paths of length $5$;
b) the expressions of the form $\a^3-\b^3$, $\a\g\b$, $\b\g\a$.
2) $R_s^\prime=K\left[\mathcal Q_s\right]/I^\prime$, where $K$ is a field, and $I^\prime$ is the
ideal in the path algebra $K\left[\mathcal Q_s\right]$ of the quiver $\mathcal Q$, generated by
a) all the paths of length $5$;
b) the expressions of the form $\a^3-\b^3$, $\a_{3t}\g_{t-1}\b_{3t-1}$, $\b_{3t}\g_{t-1}\a_{3t-1}$
($1\le t\le s-1$), $\a_0\g_{s-1}\a_{3(s-1)+2}$, $\b_0\g_{s-1}\b_{3(s-1)+2}$.
Henceforth we will often omit indexes in arrows $\a_i$, $\b_i$ and $\g_i$ as long as subscripts are
clear from the context.
The present paper is dedicated to the study of Hochschild cohomology ring structure for algebra
$R_s$. For this algebra we obtain the description of Hochschild cohomology ring structure in terms
of generators and relations. In studies of the structure of cohomology ring we will construct the
bimodule resolution of $R_s$, which could be seen as a whole result.
\section{Statement of the main results}
In what follows, we assume $n=6$.
Let $\HH^t(R)$ is the $t$th group of the Hochschild cohomology ring of $R$ with coefficients in
$R$. Let $\ell$ be the aliquot, and $r$ be the residue of division of $t$ by $11$, $m$ be the
aliquot of division of $r$ by $2$.
Consider the case of $s>1$. To describe Hochschild cohomology ring of algebra $R_s$ we must
introduce the following conditions on an arbitrary degree $t$:
$($1$)$ $r=0$, $\ell\div 2$, $\ell n+m\equiv 0(s)$ or $s=1$;\label{degs}
$($2$)$ $r=0$, $\ell\div 2$, $\myChar=3$, $\ell n+m\equiv 1(s)$ or $s=1$;
$($3$)$ $r=1$, $\ell\div 2$, $\ell n+m\equiv 0(s)$ or $s=1$;
$($4$)$ $r=1$, $\ell\ndiv 2$, $\ell n+m\equiv 0(s)$ or $s=1$;
$($5$)$ $r=2$, $\ell\ndiv 2$, $\ell n+m\equiv 1(s)$ or $s=1$;
$($6$)$ $r=3$, $\ell\div 2$, $\myChar=2$, $\ell n+m\equiv 0(s)$ or $s=1$;
$($7$)$ $r=3$, $\ell\ndiv 2$, $\ell n+m\equiv 0(s)$ or $s=1$;
$($8$)$ $r=4$, $\ell\ndiv 2$, $\ell n+m\equiv 1(s)$ or $s=1$;
$($9$)$ $r=4$, $\ell\ndiv 2$, $\myChar=3$, $\ell n+m\equiv 0(s)$ or $s=1$;
$($10$)$ $r=4$, $\ell\div 2$, $\myChar=2$, $\ell n+m\equiv 1(s)$ or $s=1$;
$($11$)$ $r=5$, $\ell\div 2$, $\myChar=3$, $\ell n+m\equiv 0(s)$ or $s=1$;
$($12$)$ $r=5$, $\ell\ndiv 2$, $\myChar=3$, $\ell n+m\equiv 0(s)$ or $s=1$;
$($13$)$ $r=6$, $\ell\div 2$, $\ell n+m\equiv 0(s)$ or $s=1$;
$($14$)$ $r=6$, $\ell\div 2$, $\myChar=3$, $\ell n+m\equiv 1(s)$ or $s=1$;
$($15$)$ $r=6$, $\ell\ndiv 2$, $\myChar=2$, $\ell n+m\equiv 0(s)$ or $s=1$;
$($16$)$ $r=7$, $\ell\div 2$, $\ell n+m\equiv 0(s)$ or $s=1$;
$($17$)$ $r=7$, $\ell\ndiv 2$, $\myChar=2$, $\ell n+m\equiv 0(s)$ or $s=1$;
$($18$)$ $r=8$, $\ell\div 2$, $\ell n+m\equiv 0(s)$ or $s=1$;
$($19$)$ $r=9$, $\ell\div 2$, $\ell n+m\equiv 0(s)$ or $s=1$;
$($20$)$ $r=9$, $\ell\ndiv 2$, $\ell n+m\equiv 0(s)$ or $s=1$;
$($21$)$ $r=10$, $\ell\ndiv 2$, $\ell n+m\equiv 1(s)$ or $s=1$;
$($22$)$ $r=10$, $\ell\ndiv 2$, $\myChar=3$, $\ell n+m\equiv 0(s)$ or $s=1$.
Let $M=11\frac{2s}{\myNod(2n,s)}$.
\begin{zam}
We will prove in paragraph \ref{sect_res} that the minimal period of bimodule resolution of $R_s$
is $M$.
\end{zam}
Let $\{t_{1, i},\dots,t_{\alpha_i, i} \}$ be a set of all degrees $t$, that satisfy the conditions
of item $i$ from the above list, and such that $0\le t_{j, i}<M$ $(j=1,\dots,\alpha_i)$. Consider
the set $$\mathcal
X=\bigcup_{i=1}^{22}\left\{X^{(i)}_{t_{j,i}}\right\}_{j=1}^{\alpha_i}\cup\{T\},$$ and define a
graduation of polynomial ring $K[\mathcal X]$ such that
\begin{align*}\label{degs2}
&\deg X^{(i)}_{t_{j,i}}=t_{j, i} \:\text{for all} \: i=1,\dots,22 \:\text{and}\: j=1,\dots,\alpha_i;\tag{$\circ$}\\
&\deg T=M.
\end{align*}
\begin{zam}\label{brief_notation}
Hereafter we shall use simplified denotation $X^{(i)}$ instead of $X^{(i)}_{t_{j,i}}$, since lower
indexes are clear from context.
\end{zam}
\begin{obozn}
$$\widetilde X^{(i)}= \begin{cases}X^{(i)},\quad \deg\widetilde
X^{(i)}<\deg T\\TX^{(i)},\quad\text{otherwise.}\end{cases}$$
\end{obozn}
Define a graduate $K$-algebra $\mathcal A=K[\mathcal X]/I$, where $I$ is the ideal generated by
homogeneous elements corresponding to the following relations.
\begin{align*}
&X^{(3)}X^{(1)}=X^{(3)}X^{(2)}=X^{(3)}X^{(3)}=X^{(3)}X^{(5)}=X^{(3)}X^{(8)}=0;\\
&X^{(3)}X^{(9)}=X^{(3)}X^{(10)}=X^{(3)}X^{(11)}=X^{(3)}X^{(12)}=X^{(3)}X^{(14)}=0;\\
&X^{(3)}X^{(16)}=X^{(3)}X^{(17)}=X^{(3)}X^{(19)}=X^{(3)}X^{(21)}=X^{(3)}X^{(22)}=0;\\
&X^{(3)}X^{(4)}=\widetilde X^{(5)};\quad X^{(3)}X^{(6)}=\widetilde
X^{(10)};\\
&X^{(3)}X^{(7)}=2\widetilde X^{(8)};\quad
X^{(3)}X^{(13)}=2\widetilde
X^{(16)};\\
&X^{(3)}X^{(15)}=\widetilde X^{(17)};\quad
X^{(3)}X^{(18)}=\widetilde X^{(19)};\quad X^{(3)}X^{(20)}=\widetilde
X^{(21)}.
\end{align*}
\begin{align*}
X^{(4)}X^{(7)}&=\begin{cases}\widetilde X^{(10)},\quad\myChar
=2,\\0,\quad\text{otherwise};\end{cases}&\text{(r1)}\\
X^{(7)}X^{(7)}&=\begin{cases}s\widetilde X^{(14)},\quad\myChar
=3,\\0,\quad\text{otherwise};\end{cases}&\text{(r2)}\\
X^{(4)}X^{(13)}&=\begin{cases}\widetilde X^{(17)},\quad\myChar
=2,\\0,\quad\text{otherwise};\end{cases}&\text{(r3)}\\
X^{(7)}X^{(18)}&=\begin{cases}-s\widetilde X^{(2)},\quad\myChar
=3,\\0,\quad\text{otherwise};\end{cases}&\text{(r4)}\\
X^{(13)}X^{(20)}&=\begin{cases}\widetilde X^{(10)},\quad\myChar
=2,\\0,\quad\text{otherwise};\end{cases}&\text{(r5)}\\
X^{(18)}X^{(18)}&=\begin{cases}-\widetilde X^{(12)},\quad\myChar
=3,\\0,\quad\text{otherwise};\end{cases}&\text{(r6)}\\
X^{(18)}X^{(20)}&=\begin{cases}-s\widetilde X^{(14)},\quad\myChar
=3,\\0,\quad\text{otherwise}.\end{cases}&\text{(r7)}
\end{align*}
Describe the rest relations as a tables (numbers (r1)--(r7) in tables cells are the number of
relation that defines a multiplication of the following elements).
\setlength{\extrarowheight}{1mm}
\begin{tabular}{c|c|c|c|c|c|c|c|c|c}
&$X^{(1)}$&$X^{(2)}$&$X^{(4)}$&$X^{(6)}$&$X^{(7)}$&$X^{(8)}$&$X^{(9)}$&$X^{(11)}$&$X^{(12)}$\\
\hline
$X^{(1)}$&$\widetilde X^{(1)}$&$\widetilde
X^{(2)}$&$\widetilde X^{(4)}$&$\widetilde X^{(6)}$
&$\widetilde X^{(7)}$&$\widetilde X^{(8)}$&$\widetilde X^{(9)}$&$\widetilde X^{(11)}$&$\widetilde X^{(12)}$ \\
\hline
$X^{(2)}$& &0&0&0&0&0&$\widetilde X^{(8)}$&0&0 \\
\hline
$X^{(4)}$& & & 0&$\widetilde X^{(8)}$&(1)&0&$\widetilde X^{(11)}$&0&$s\widetilde X^{(14)}$ \\
\hline
$X^{(6)}$& & & &0&0&0&0&0&0\\
\hline
$X^{(7)}$& & & & &(2)&0&$s\widetilde X^{(16)}$&0&0 \\
\hline
$X^{(8)}$& & & & & &0&0&0&0\\
\hline
$X^{(9)}$& & & & & & &0&0&$-s\widetilde X^{(19)}$\\
\hline
$X^{(11)}$& & & & & & & &0&$-s\widetilde X^{(21)}$ \\
\end{tabular}
$\quad$
$\quad$
\begin{tabular}{c|c|c|c|c|c|c|c}
&$X^{(13)}$&$X^{(14)}$&$X^{(15)}$&$X^{(16)}$&$X^{(18)}$&$X^{(20)}$&$X^{(22)}$\\
\hline
$X^{(1)}$&$\widetilde X^{(13)}$&$\widetilde X^{(14)}$&$\widetilde X^{(15)}$&$\widetilde X^{(16)}$&$\widetilde X^{(18)}$&$\widetilde X^{(20)}$&$\widetilde X^{(22)}$ \\
\hline
$X^{(2)}$&$\widetilde X^{(14)}$&0&0&0&0&0&$\widetilde X^{(21)}$ \\
\hline
$X^{(4)}$&(3)&0&$\widetilde X^{(16)}$&0&$\widetilde X^{(20)}$&0&0 \\
\hline
$X^{(6)}$&$\widetilde X^{(19)}$&0&$\widetilde X^{(20)}$&0&0&0&0\\
\hline
$X^{(7)}$&$-2\widetilde X^{(20)}$&0&$\widetilde X^{(19)}$&$-\widetilde X^{(21)}$&(4)&0&$-s\widetilde X^{(5)}$ \\
\hline
$X^{(8)}$&$-\widetilde X^{(21)}$&0&0&0&0&0&0\\
\hline
$X^{(9)}$&$-\widetilde X^{(22)}$&$-\widetilde X^{(21)}$&0&0&$s\widetilde X^{(3)}$&$s\widetilde X^{(5)}$&0\\
\hline
$X^{(11)}$&0&0&0&0&$s\widetilde X^{(5)}$&0&0 \\
\end{tabular}
$\quad$
$\quad$
\begin{tabular}{c|c|c|c|c|c|c|c|c}
&$X^{(12)}$&$X^{(13)}$&$X^{(14)}$&$X^{(15)}$&$X^{(16)}$&$X^{(18)}$&$X^{(20)}$&$X^{(22)}$\\
\hline
$X^{(12)}$&0&$-s\widetilde X^{(2)}$&0&0&0&0&0&$s\widetilde X^{(8)}$ \\
\hline
$X^{(13)}$& &$2\widetilde X^{(4)}$&0&$\widetilde X^{(3)}$&$\widetilde X^{(5)}$&$-\widetilde X^{(7)}$&(5)&$\widetilde X^{(11)}$ \\
\hline
$X^{(14)}$& & & 0&0&0&0&0&0 \\
\hline
$X^{(15)}$& & & &$\widetilde X^{(4)}$&0&$\widetilde X^{(6)}$&$\widetilde X^{(8)}$&0\\
\hline
$X^{(16)}$& & & & &0&$-\widetilde X^{(8)}$&0&0 \\
\hline
$X^{(18)}$& & & & & &(6)&(7)&$s\widetilde X^{(16)}$\\
\hline
$X^{(20)}$& & & & & & &0&0\\
\hline
$X^{(22)}$& & & & & & & &0 \\
\end{tabular}
\begin{thm}\label{main_thm}
Let $s>1$, $R=R_s$ is algebra of the type $E_6$. Then the Hochschild cohomology ring $\HH^*(R)$ is
isomorphic to $\mathcal A$ as a graded $K$-algebra.
\end{thm}
Consider the case of $s=1$.
Let us introduce the set $$\mathcal X^\prime=\begin{cases}\mathcal X\cup \left\{X^{(23)}_0,
X^{(24)}_0,
X^{(25)}_0, X^{(26)}_0, X^{(27)}_0, X^{(28)}_0\right\},\quad\myChar\ne 3;\\
\mathcal X\cup \left\{X^{(24)}_0, X^{(25)}_0, X^{(26)}_0, X^{(27)}_0, X^{(28)}_0\right\}, \quad
\myChar=3;\end{cases}$$ and define a graduation of polynomial ring $K[\mathcal X^\prime]$ such that
\begin{align*}
&\deg X^{(i)}_{t_{j,i}}=t_{j, i} \:\text{for all} \: i=1,\dots,22 \:\text{and}\: j=1,\dots,\alpha_i;\\
&\deg T=M \text{ (similar to (\ref{degs2}))};\\&\deg X^{(23)}_0=\deg X^{(24)}_0=\deg
X^{(25)}_0=\deg X^{(26)}_0=\deg X^{(27)}_0=\deg X^{(28)}_0=0.
\end{align*}
Define a graduate $K$-algebra $\mathcal A^\prime=K[\mathcal X^\prime]/I^\prime$, where $I^\prime$
is the ideal generated by homogeneous elements corresponding to the relations described in the case
of $s>1$, and by the following relations:
\begin{align*}
X^{(1)}X^{(i)}=&\begin{cases}\widetilde X^{(i)},\quad
t_1=0;\\0,\quad\text{otherwise},\end{cases}\quad i\in\{23,24,28\};\\
X^{(1)}X^{(i)}=&\begin{cases}\widetilde X^{(i)},\quad
t_1=0;\\\widetilde X^{(2)},\quad t_1>0\text{ and
}\myChar=3;\\0,\quad\text{otherwise},\end{cases}
i\in\{25,26,27\};\\
X^{(9)}X^{(i)}=&\widetilde X^{(8)},\quad i\in\{25,26,27\};\\
X^{(13)}X^{(i)}=&\begin{cases}\widetilde X^{(14)},\quad
\myChar=3;\\0,\quad\text{otherwise},\end{cases} i\in\{25,26,27\};\\
X^{(22)}X^{(i)}=&\widetilde X^{(21)},\quad i\in\{25,26,27\};\\
X^{(j)}X^{(i)}=&0,\quad j\in[2, 28]\setminus\{9,13,22\},\quad
i\in[23, 28],
\end{align*}
where $t_1$ denotes a degree of the element $X^{(1)}$.
\begin{thm}\label{main_thm2}
Let $s=1$, $R=R_1$ is algebra of the type $E_6$. Then the Hochschild cohomology ring $\HH^*(R)$ is
isomorphic to $\mathcal A^\prime$ as a graded $K$-algebra.
\end{thm}
\begin{zam}
From the descriptions of rings $\HH^*(R)$ given in theorems
\ref{main_thm} and \ref{main_thm2} it implies, in particular, that
they are commutative.
\end{zam}
\section{Bimodule resolution}\label{sect_res}
We will construct the minimal projective bimodule resolution of the $R$ in the following form: $$
\dots\longrightarrow Q_3\stackrel{d_2}\longrightarrow Q_2\stackrel{d_1}\longrightarrow
Q_1\stackrel{d_0}\longrightarrow Q_0\stackrel\varepsilon\longrightarrow R\longrightarrow 0
$$
Let $\Lambda$ be an enveloping algebra of algebra $R$. Then $R$--$R$-bimodules can be considered as
left $\Lambda$-modules.
\begin{obozns}$\quad$
(1) Let $e_i,\text{ }i\in \Z_{ns}=\{0, 1,\dots, ns-1\},$ be the idempotents of the algebra
$K\left[\mathcal Q_s\right]$, that correspond to the vertices of the quiver $\mathcal Q_s$.
(2) Denote by $P_{i,j}=R(e_i\otimes e_j)R=\Lambda(e_i\otimes e_j)$, $i,j\in \Z_{ns}$. Note that the
modules $P_{i,j}$, forms the full set of the (pairwise non-isomorphic
by) indecomposable projective $\Lambda$-modules.
(3) For $a\in\Z$, $t\in\N$ we denote the smallest nonnegative deduction of $a$ modulo $t$ with
$(a)_t$ (in particular, $0\le(a)_t\le t-1$).
\end{obozns}
Let $R=R_s$. We introduce an automorphism $\sigma\text{: }R\rightarrow R$, which is mapping as
follows:
$$\sigma(e_i)=\begin{cases}
e_{i+n^2},\quad i\equiv 0,5(n);\\
e_{i+n^2+2},\quad i\equiv 1,2(n);\\
e_{i+n^2-2},\quad i\equiv 3,4(n),
\end{cases}$$
$$\sigma(\g_i)=-\g_{i+n},$$
$$\sigma(\a_i)=\begin{cases}
-\b_{i+3n},\quad i\equiv 0,1(3);\\
\b_{i+3n},\quad i\equiv 2(3),
\end{cases} \sigma(\b_i)=\begin{cases}
-\a_{i+3n},\quad i\equiv 1,2(3);\\
\a_{i+3n},\quad i\equiv 0(3).
\end{cases}$$
Define the helper functions $f\text{: }\Z\times\Z\rightarrow\Z$, $h\text{:
}\Z\times\Z\rightarrow\Z$ and $\lambda\text{: }\Z\rightarrow\Z$, which act in the following way:
$$f(x,y)=\begin{cases}1,\quad x=y;\\0,\quad x\ne y,\end{cases}\quad
h(x,y)=\begin{cases}
1,\quad x\div 2,\text{ }x<y;\\
0,\quad x\ndiv 2,\text{ }x<y;\\
1,\quad x\ndiv 2,\text{ }x\ge y;\\
0,\quad x\div 2,\text{ }x\ge y,
\end{cases}$$
$$\lambda(i)=\begin{cases}i,\quad i\equiv 0,5(\mathrm{mod}\text{ } 6);\\
i+2,\quad i\equiv 1,2(\mathrm{mod}\text{ } 6);\\
i-2,\quad i\equiv 3,4(\mathrm{mod}\text{ } 6).\end{cases}$$
Introduce $Q_r\text{ }(r\le 10)$. Let $m$ be the aliquot of division of $r$ by $2$ for considered
degree $r$. We have
\begin{align*}
Q_{2m}&=\bigoplus_{r=0}^{s-1} Q_{2m,r}^\prime,\quad 0\le m\le n-1,\\
Q_{2m+1}&=\bigoplus_{r=0}^{s-1} Q_{2m+1,r}^\prime,\quad 0\le m\le
n-2,
\end{align*}
where
\begin{multline*}
Q_{2m,r}^\prime=\left(\bigoplus_{i=0}^{f(m,2)}P_{(r+m)n-1+h(m,2)+i,rn}\right)\\
\oplus\bigoplus_{i=0}^{f(m,3)}\left(P_{(r+m)n+1+(m)_3+3i,rn+1}
\oplus P_{\lambda((r+m)n+1+(m)_3+3i),rn+3}\right)\\
\oplus\bigoplus_{i=0}^{f(m,2)}\big(P_{(r+m)n+1+(m+1)_3+3f(m,5)+3i,rn+2}
\oplus P_{\lambda((r+m)n+1+(m+1)_3+3f(m,5)+3i),rn+4}\big)\\
\oplus\left(\bigoplus_{i=0}^{f(m,3)}P_{(r+m+1)n-h(m,3)+i,rn+5}\right),
\end{multline*}
\begin{multline*}
Q_{2m+1,r}^\prime=\left(\bigoplus_{i=0}^{1-f(m,4)}P_{(r+m)n+1+h(m,0)+4f(m,4)+2i,rn}\right)\\
\oplus P_{(r+m+1)n-1+h(m,5)-4f(m,0),rn+1}\oplus P_{\lambda((r+m+1)n-1+h(m,5)-4f(m,0)),rn+3}\\
\oplus P_{(r+m+1)n-1+h(m,0)+4f(m,4),rn+2}\oplus
P_{\lambda((r+m+1)n-1+h(m,0)+4f(m,4),rn+2),rn+4}\\
\oplus\left(\bigoplus_{i=0}^{1-f(m,0)}P_{(r+m+1)n+1+h(m,5)-2f(m,0)+2i,rn+5}\right).\\
\end{multline*}
Now we shall describe differentials $d_r$ for $r\le 10$. Since $Q_i$ are direct sums, their
elements can be concerned as column vectors, hence differentials can be described as matrixes
(which are being multiplied by column vectors from the right). Now let us describe the matrixes of
differentials componentwisely.
\begin{zam}
\textup{Numeration of lines and columns always starts with zero.}
\end{zam}
\begin{obozns}$\quad$
(1) Denote by $w_{i\ra j}$ the way that starts in $i$th vertex and ends in $j$th.
(2) Denote by $w_{i\ra}^{(m)}$ the way that starts in $i$th vertex and has length $m$.
(3) Denote by $w_{\ra i}^{(m)}$ the way that ends in $i$th vertex and has length $m$.
\end{obozns}
Define the helper functions $g\text{: }\Z\rightarrow\Z$ $f_0\text{:
}\Z\times\Z\rightarrow\Z$ and $f_1\text{: }\Z\times\Z\rightarrow\Z$,
which act in the following way:
$$
g(j)=\begin{cases}1,\quad (j)_{2s}<s;\\3\quad\text{otherwise},\end{cases}\quad
f_0(x,y)=\begin{cases}1,\quad x<y;\\0\quad x\ge y,\end{cases}\quad f_1(x,y)=\begin{cases}1,\quad
x<y;\\-1\quad x\ge y.\end{cases}
$$
\centerline{\bf Description of the $d_0$} \centerline{$d_0:Q_1\rightarrow Q_0\text{ -- is an }
(7s\times 6s)\text{ matrix}.$}
If $0\le j<2s$, then
$$(d_0)_{ij}=\begin{cases}
w_{(j+m)n\ra (j+m)n+g(j)}\otimes e_{jn},\quad i=(j)_s;\\
-e_{(j+m)n+g(j)}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
0\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$(d_0)_{ij}=\begin{cases}
w_{(j+m)n+g(j)\ra (j+m)n+g(j)+1}\otimes e_{jn+g(j)},\quad i=j-s;\\
-e_{(j+m)n+g(j)+1}\otimes w_{jn+g(j)\ra jn+g(j)+1},\quad i=j+s;\\
0\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $$(d_0)_{ij}=\begin{cases}
w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
-e_{(j+m)n+5}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=5s+(j)_s;\\
0\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $$(d_0)_{ij}=\begin{cases}
w_{(j+m)n+5\ra (j+m)n+6}\otimes e_{jn+5},\quad i=j-s;\\
-e_{(j+m)n+6}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
0\quad\text{otherwise.}\end{cases}$$
\centerline{\bf Description of the $d_1$} \centerline{$d_1:Q_2\rightarrow Q_1\text{ -- is an }
(6s\times 7s)\text{ matrix}.$}
If $0\le j<s$, then
$$(d_1)_{ij}=\begin{cases}
w_{jn+1+j_1+2f(j_1,2)\ra jn+5}\otimes w_{jn\ra jn+j_1},\quad i=j+2sj_1,\text{ }0\le j_1< 3;\\
-w_{jn+3+j_1\ra jn+5}\otimes w_{jn\ra jn+j_1+2(1-f(j_1,0))},\quad i=j+2sj_1+s,\text{ }0\le j_1< 3;\\
0\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$(d_1)_{ij}=\begin{cases}
w_{\ra (j+1)n+1+g(j+s)}^{(4-j_1)}\otimes w_{jn+g(j+s)\ra}^{(j_1)},\\\quad\quad\quad i=j+2sj_1+s-s(1-f_0(j,2s))f(j_1,2),\text{ }0\le j_1< 3;\\
w_{\ra (j+1)n+1+g(j+s)}^{(1-j_1)}\otimes w_{jn+g(j+s)\ra}^{(j_1+3)},\\\quad\quad\quad i=(j+1)_s+2sj_1+s(1-f_0(j,2s)),\text{ }0\le j_1< 2;\\
0\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$(d_1)_{ij}=\begin{cases}
w_{\ra (j+1)n+g(j)}^{(2)}\otimes e_{jn+g(j+s)+1},\quad i=j+s;\\
w_{\ra (j+1)n+g(j)}^{(1)}\otimes w_{jn+g(j+s)+1\ra}^{(1)},\quad i=(j)_s+6s;\\
e_{(j+1)n+g(j)}\otimes w_{jn+g(j+s)+1\ra}^{(2)},\quad i=(j+1)_s+sf_0(j,4s);\\
0\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$(d_1)_{ij}=\begin{cases}
w_{(j+1)n\ra (j+2)n}\otimes e_{jn+5},\quad i=j+s;\\
w_{(j+1)n+1+j_1+2f(j_1,3)+2f(j_1,2)\ra (j+2)n}\otimes w_{jn+5\ra (j+1)n+j_1+2f(j_1,3)},\\\quad\quad\quad i=(j+1)_s+2sj_1,\text{ }0\le j_1< 4;\\
0\quad\text{otherwise.}\end{cases}$$
\centerline{\bf Description of the $d_2$} \centerline{$d_2:Q_3\rightarrow Q_2\text{ -- is an }
(8s\times 6s)\text{ matrix}.$}
If $0\le j<2s$, then
$$(d_2)_{ij}=\begin{cases}
w_{(j+m-1)n+5\ra (j+m)n+g(j)+1}\otimes e_{jn},\quad i=(j)_s;\\
-f_1(j,s)e_{(j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
f_1(j,s)w_{(j+m)n+g(j)\ra (j+m)n+g(j)+1}\otimes w_{jn\ra jn+1+g(j+s)},\quad i=(j+s)_{2s}+3s;\\
0\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$(d_2)_{ij}=\begin{cases}
w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes e_{jn+g(j)},\quad i=j-s;\\
-w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes w_{jn+g(j)\ra jn+g(j)+1},\quad i=j+s;\\
-f_1(j,3s)e_{(j+m)n+5}\otimes w_{jn+g(j)\ra (j+1)n},\quad i=(j+1)_s;\\
0\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $$(d_2)_{ij}=\begin{cases}
w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
-e_{(j+m+1)n}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=5s+(j)_s;\\
w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn+g(j)+1\ra (j+1)n},\quad i=(j+1)_s,\text{ }j<5s;\\
0\quad\text{otherwise.}\end{cases}$$
If $6s\le j<8s$, then $$(d_2)_{ij}=\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes e_{jn+5},\quad i=5s+(j)_s;\\
-w_{(j+m+1)n-1\ra (j+m+1)n+g(j)}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s,\text{ }j<7s;\\
-e_{(j+m+1)n+g(j)}\otimes w_{jn+5\ra (j+1)n+g(j+s)+1},\quad i=3s+(j+1)_s+sf_0(j,7s);\\
0\quad\text{otherwise.}\end{cases}$$
\centerline{\bf Description of the $d_3$} \centerline{$d_3:Q_4\rightarrow Q_3\text{ -- is an }
(9s\times 8s)\text{ matrix}.$}
If $0\le j<s$, then
$$(d_3)_{ij}=\begin{cases}
w_{(j+m)n+2\ra (j+m)n+5}\otimes e_{jn},\quad i=j;\\
-w_{(j+m)n+4\ra (j+m)n+5}\otimes e_{jn},\quad i=j+s;\\
e_{(j+m)n+5}\otimes w_{jn\ra jn+1},\quad i=j+2s;\\
e_{(j+m)n+5}\otimes w_{jn\ra jn+3},\quad i=j+3s;\\
0\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$(d_3)_{ij}=\begin{cases}
w_{(j+m)n+2+j_1+2(1-f(j_1,0))\ra (j+m+1)n}\otimes w_{jn\ra jn+j_1},\quad i=j-s+2sj_1,\text{ }0\le j_1< 3;\\
-e_{(j+m+1)n}\otimes w_{jn\ra jn+4},\quad i=j+4s;\\
0\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$(d_3)_{ij}=\begin{cases}
w_{\ra (j+m+1)n+g(j+s)}^{(2-j_1)}\otimes w_{jn+g(j)\ra}^{(j_1)},\quad i=2j_1s+j,\text{ }0\le j_1< 2;\\
e_{(j+m+1)n+g(j+s)}\otimes w_{jn+g(j)\ra}^{(2)},\quad i=6s+(j+s)_{2s};\\
0\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $$(d_3)_{ij}=\begin{cases}
w_{\ra (j+m+1)n+g(j)}^{(1-j_1)}\otimes w_{jn+g(j)+1\ra}^{(j_1)},\quad i=j+2sj_1,\text{ }0\le j_1< 2;\\
0\quad\text{otherwise.}\end{cases}$$
If $6s\le j<8s$, then $$(d_3)_{ij}=\begin{cases}
w_{\ra (j+m+1)n+g(j+s)+1}^{(2)}\otimes e_{jn+g(j)+1},\quad i=j-2s;\\
w_{\ra (j+m+1)n+g(j+s)+1}^{(1)}\otimes w_{jn+g(j)+1\ra}^{(1)},\quad i=(j+s)_{2s}+6s;\\
-f_1(j,7s)e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j)+1\ra}^{(2)},\quad i=(j+1)_s+sf_0(j,7s);\\
0\quad\text{otherwise.}\end{cases}$$
If $8s\le j<9s$, then $$(d_3)_{ij}=\begin{cases}
w_{(j+m+1)n+2\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
e_{(j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+1},\quad i=2s+(j+1)_s;\\
w_{(j+m+1)n+1\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j-2s;\\
-w_{(j+m+1)n+3\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j-s;\\
0\quad\text{otherwise.}\end{cases}$$
\centerline{\bf Description of the $d_4$} \centerline{$d_4:Q_5\rightarrow Q_4\text{ -- is an }
(8s\times 9s)\text{ matrix}.$}
If $0\le j<2s$, then
$$(d_4)_{ij}=\begin{cases}
w_{(j+m-1)n+5\ra (j+m)n+1}\otimes e_{jn},\quad i=j,\text{ }j<s;\\
-f_1(j,s)w_{(j+m)n\ra (j+m)n+g(j)}\otimes e_{jn},\quad i=(j)_s+s;\\
-e_{(j+m)n+g(j)}\otimes w_{jn\ra jn+g(j+s)},\quad i=(j+s)_{2s}+2s;\\
e_{(j+m)n+g(j)}\otimes w_{jn\ra jn+1+g(j)},\quad i=j+4s;\\
0\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$(d_4)_{ij}=\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn+1\ra (j+1)n},\quad i=(j+1)_s,\text{ }j<3s;\\
w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes e_{jn+g(j)},\quad i=j;\\
-w_{(j+m)n+g(j+s)+1\ra (j+m+1)n}\otimes w_{jn+g(j)\ra jn+g(j)+1},\quad i=4s+j;\\
-f_1(j,3s)e_{(j+m+1)n}\otimes w_{jn+g(j)\ra (j+1)n},\quad i=(j+1)_s+s;\\
0\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $$(d_4)_{ij}=\begin{cases}
e_{(j+m)n+5}\otimes w_{jn+g(j)+1\ra (j+1)n},\quad i=(j+1)_s,\text{ }j<5s;\\
w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes e_{jn+g(j)+1},\quad i=j;\\
-w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes e_{jn+g(j)+1},\quad i=j+2s;\\
-f_1(j,5s)e_{(j+m)n+5}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=8s+(j)_s;\\
0\quad\text{otherwise.}\end{cases}$$
If $6s\le j<8s$, then $$(d_4)_{ij}=\begin{cases}
-w_{(j+m+1)n-1\ra (j+m+1)n+g(j)+1}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s,\text{ }j<7s;\\
w_{(j+m+1)n-1\ra (j+m+1)n+g(j)+1}\otimes e_{jn+5},\quad i=8s+(j)_s;\\
f_1(j,7s)w_{(j+m+1)n+g(j)\ra (j+m+1)n+g(j)+1}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\\\quad\quad\quad i=2s+(j+1)_s+sf_0(j,7s);\\
-f_1(j,7s)e_{(j+m+1)n+g(j)+1}\otimes w_{jn+5\ra (j+1)n+g(j+s)+1},\\\quad\quad\quad i=6s+(j+1)_s+sf_0(j,7s);\\
0\quad\text{otherwise.}\end{cases}$$
\centerline{\bf Description of the $d_5$} \centerline{$d_5:Q_6\rightarrow Q_5\text{ -- is an }
(9s\times 8s)\text{ matrix}.$}
If $0\le j<s$, then
$$(d_5)_{ij}=\begin{cases}
w_{(j+m)n+1\ra (j+m+1)n}\otimes e_{jn},\quad i=j;\\
w_{(j+m)n+3\ra (j+m+1)n}\otimes e_{jn},\quad i=j+s;\\
(2f(j_1,0)-1)w_{(j+m+1)n-j_1\ra (j+m+1)n}\otimes w_{jn\ra jn+1+j_1},\\\quad\quad\quad i=j+2s(1+j_1),\text{ }0\le j_1< 2;\\
(2f(j_1,0)-1)w_{(j+m+1)n-j_1\ra (j+m+1)n}\otimes w_{jn\ra jn+3+j_1},\\\quad\quad\quad i=j+2s(1+j_1)+s,\text{ }0\le j_1< 2;\\
0\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$(d_5)_{ij}=\begin{cases}
w_{\ra (j+m+1)n+g(j+s)}^{(1)}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
-e_{(j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra}^{(3)},\quad i=(j+1)_s+s(1-f_0(j,2s));\\
0\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$(d_5)_{ij}=\begin{cases}
-w_{\ra (j+m+1)n+g(j)+1}^{(3)}\otimes w_{jn+g(j+s)\ra}^{(1)},\quad i=j+s;\\
w_{\ra (j+m+1)n+g(j)+1}^{(2)}\otimes e_{jn+g(j+s)},\quad i=j-s;\\
w_{\ra (j+m+1)n+g(j)+1}^{(1)}\otimes w_{jn+g(j+s)\ra}^{(3)},\quad i=(j+1)_s+sf_0(j,4s);\\
-f_1(j,4s)e_{(j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra}^{(2)},\quad i=(j)_{2s}+6s;\\
0\quad\text{otherwise.}\end{cases}$$
If $5s\le j<7s$, then $$(d_5)_{ij}=\begin{cases}
f_1(j,6s)e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)+1\ra}^{(1)},\quad i=j+s;\\
w_{\ra (j+m+1)n+g(j+s)+1}^{(3)}\otimes e_{jn+g(j+s)+1},\quad i=j-s;\\
0\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $$(d_5)_{ij}=\begin{cases}
w_{(j+m+1)n+1\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
w_{(j+m+1)n+3\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n},\quad i=s+(j+1)_s;\\
-e_{(j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+2},\quad i=4s+(j+1)_s;\\
-e_{(j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+4},\quad i=5s+(j+1)_s;\\
w_{(j+m+1)n+2\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j-s;\\
-w_{(j+m+1)n+4\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j;\\
0\quad\text{otherwise.}\end{cases}$$
If $8s\le j<9s$, then $$(d_5)_{ij}=\begin{cases}
w_{(j+m+1)n+2\ra (j+m+2)n}\otimes e_{jn+5},\quad i=j-2s;\\
-e_{(j+m+2)n}\otimes w_{jn+5\ra (j+1)n+3},\quad i=3s+(j+1)_s;\\
0\quad\text{otherwise.}\end{cases}$$
\centerline{\bf Description of the $d_6$} \centerline{$d_6:Q_7\rightarrow Q_6\text{ -- is an }
(8s\times 9s)\text{ matrix}.$}
If $0\le j<2s$, then
$$(d_6)_{ij}=\begin{cases}
w_{(j+m)n\ra (j+m)n+g(j)+1}\otimes e_{jn},\quad i=(j)_s;\\
-w_{(j+m)n+g(j)\ra (j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
-e_{(j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j+s)},\quad i=(j+s)_{2s}+3s;\\
e_{(j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+5s;\\
0\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$(d_6)_{ij}=\begin{cases}
w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes e_{jn+g(j)},\quad i=j-s;\\
-w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes e_{jn+g(j)},\quad i=j+s;\\
-w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes w_{jn+g(j)\ra jn+g(j)+1},\quad i=j+3s;\\
e_{(j+m)n+5}\otimes w_{jn+g(j)\ra jn+5},\quad i=(j)_s+7s;\\
0\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $$(d_6)_{ij}=\begin{cases}
w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j+s;\\
-f_1(j,5s)e_{(j+m+1)n}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=(j)_s+8s;\\
e_{(j+m+1)n}\otimes w_{jn+4\ra (j+1)n},\quad i=(j+1)_s,\text{ }j\ge 5s;\\
-w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn+4\ra jn+5},\quad i=j+2s,\text{ }j\ge 5s;\\
0\quad\text{otherwise.}\end{cases}$$
If $6s\le j<8s$, then $$(d_6)_{ij}=\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+1}\otimes e_{jn+5},\quad i=j+s,\text{ }j<7s;\\
-w_{(j+m+1)n\ra (j+m+1)n+1}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s,\text{ }j<7s;\\
-f_1(j,7s)w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes e_{jn+5},\quad i=(j)_s+8s;\\
e_{(j+m+1)n+g(j)}\otimes w_{jn+5\ra (j+1)n+g(j)},\quad i=(j+1)_s+s+s(1-f_0(j,7s));\\
0\quad\text{otherwise.}\end{cases}$$
\centerline{\bf Description of the $d_7$} \centerline{$d_7:Q_8\rightarrow Q_7\text{ -- is an }
(6s\times 8s)\text{ matrix}.$}
If $0\le j<s$, then
$$(d_7)_{ij}=\begin{cases}
w_{(j+m)n+2\ra (j+m)n+5}\otimes e_{jn},\quad i=j;\\
-w_{(j+m)n+4\ra (j+m)n+5}\otimes e_{jn},\quad i=j+s;\\
e_{(j+m)n+5}\otimes w_{jn\ra jn+1},\quad i=j+2s;\\
-e_{(j+m)n+5}\otimes w_{jn\ra jn+3},\quad i=j+3s;\\
0\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$(d_7)_{ij}=\begin{cases}
-e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra}^{(3)},\quad i=(j+1)_s+s(1-f_0(j,2s));\\
w_{\ra (j+m+1)n+g(j+s)+1}^{(3)}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
w_{\ra (j+m+1)n+g(j+s)+1}^{(2)}\otimes w_{jn+g(j+s)\ra}^{(1)},\quad i=j+3s;\\
-w_{\ra (j+m+1)n+g(j+s)+1}^{(1)}\otimes w_{jn+g(j+s)\ra}^{(2)},\quad i=j+5s;\\
0\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$(d_7)_{ij}=\begin{cases}
w_{\ra (j+m+1)n+g(j)}^{(1)}\otimes e_{jn+g(j+s)+1},\quad i=j+s;\\
e_{(j+m+1)n+g(j)}\otimes w_{jn+g(j+s)+1\ra}^{(1)},\quad i=(j)_{2s}+6s;\\
0\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$(d_7)_{ij}=\begin{cases}
w_{(j+m+1)n+4\ra (j+m+2)n}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s+s;\\
-w_{(j+m+1)n+5\ra (j+m+2)n}\otimes w_{jn+5\ra (j+1)n+1},\quad i=(j+1)_s+2s;\\
-e_{(j+m+2)n}\otimes w_{jn+5\ra (j+1)n+2},\quad i=(j+1)_s+4s;\\
-e_{(j+m+2)n}\otimes w_{jn+5\ra (j+1)n+4},\quad i=(j+1)_s+5s;\\
w_{(j+m+1)n+1\ra (j+m+2)n}\otimes e_{jn+5},\quad i=j+s;\\
w_{(j+m+1)n+3\ra (j+m+2)n}\otimes e_{jn+5},\quad i=j+2s;\\
0\quad\text{otherwise.}\end{cases}$$
\centerline{\bf Description of the $d_8$} \centerline{$d_8:Q_9\rightarrow Q_8\text{ -- is an }
(7s\times 6s)\text{ matrix}.$}
If $0\le j<s$, then
$$(d_8)_{ij}=\begin{cases}
w_{(j+m-1)n+5\ra (j+m)n+5}\otimes e_{jn},\quad i=j;\\
-e_{(j+m)n+5}\otimes w_{jn\ra (j+1)n},\quad i=(j+1)_s;\\
-w_{(j+m)n+2\ra (j+m)n+5}\otimes w_{jn\ra jn+1},\quad i=j+s;\\
w_{(j+m)n+4\ra (j+m)n+5}\otimes w_{jn\ra jn+3},\quad i=j+2s;\\
w_{(j+m)n+3\ra (j+m)n+5}\otimes w_{jn\ra jn+2},\quad i=j+3s;\\
-w_{(j+m)n+1\ra (j+m)n+5}\otimes w_{jn\ra jn+4},\quad i=j+4s;\\
0\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$(d_8)_{ij}=\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s,\text{ }j<2s;\\
w_{(j+m)n+g(j+s)+1\ra (j+m+1)n}\otimes e_{jn+g(j+s)},\quad i=j;\\
-w_{(j+m)n+g(j)\ra (j+m+1)n}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+2s;\\
e_{(j+m+1)n}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=(j)_s+5s;\\
0\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$(d_8)_{ij}=\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j+s)+1\ra (j+1)n},\quad i=(j+1)_s,\text{ }j<4s;\\
w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
-e_{(j+m+1)n+g(j)}\otimes w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)+1},\\\quad\quad\quad i=(j+1)_s+3s+s(1-f_0(j,4s));\\
-w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j+s)+1\ra jn+5},\\\quad\quad\quad i=(j)_s+5s;\\
0\quad\text{otherwise.}\end{cases}$$
If $5s\le j<7s$, then $$(d_8)_{ij}=\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+5},\quad i=(j)_s+5s;\\
w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s,\text{ }j\ge 6s;\\
e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\\\quad\quad\quad i=(j+1)_s+s+s(1-f_0(j,6s));\\
w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+5\ra (j+1)n+g(j)+1},\\\quad\quad\quad i=(j+1)_s+3s+sf_0(j,6s);\\
0\quad\text{otherwise.}\end{cases}$$
\centerline{\bf Description of the $d_9$} \centerline{$d_9:Q_{10}\rightarrow Q_9\text{ -- is an }
(6s\times 7s)\text{ matrix}.$}
If $0\le j<s$, then
$$(d_9)_{ij}=\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n}\otimes e_{jn},\quad i=j;\\
e_{(j+m+1)n}\otimes w_{jn\ra jn+1},\quad i=j+s;\\
-e_{(j+m+1)n}\otimes w_{jn\ra jn+3},\quad i=j+2s;\\
0\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$(d_9)_{ij}=\begin{cases}
w_{\ra (j+m+1)n+g(j)}^{(1)}\otimes e_{jn+g(j+s)},\quad i=j;\\
e_{(j+m+1)n+g(j)}\otimes w_{jn+g(j+s)\ra}^{(1)},\quad i=j+2s;\\
0\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$(d_9)_{ij}=\begin{cases}
w_{\ra (j+m+1)n+g(j)+1}^{(1)}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
e_{(j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)+1\ra}^{(1)},\quad i=(j)_{2s}+5s;\\
0\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$(d_9)_{ij}=\begin{cases}
e_{(j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
w_{(j+m+1)n+2\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j;\\
-w_{(j+m+1)n+4\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j+s;\\
0\quad\text{otherwise.}\end{cases}$$
\centerline{\bf Description of the $d_{10}$} \centerline{$d_{10}:Q_{11}\rightarrow Q_{10}\text{ --
is an } (6s\times 6s)\text{ matrix}.$}
If $0\le j<s$, then
$$(d_{10})_{ij}=\begin{cases}
w_{(j+m)n\ra (j+m+1)n}\otimes e_{jn},\quad i=j;\\
-e_{(j+m+1)n}\otimes w_{jn\ra (j+1)n},\quad i=(j+1)_s;\\
-w_{(j+m)n+3\ra (j+m+1)n}\otimes w_{jn\ra jn+1},\quad i=j+s;\\
w_{(j+m)n+1\ra (j+m+1)n}\otimes w_{jn\ra jn+3},\quad i=j+2s;\\
w_{(j+m)n+4\ra (j+m+1)n}\otimes w_{jn\ra jn+2},\quad i=j+3s;\\
-w_{(j+m)n+2\ra (j+m+1)n}\otimes w_{jn\ra jn+4},\quad i=j+4s;\\
w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn\ra jn+5},\quad i=j+5s;\\
0\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$(d_{10})_{ij}=\begin{cases}
f_1(j,2s)w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)},\quad i=j;\\
-e_{(j+m+1)n+g(j)}\otimes w_{jn+g(j+s)\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s+s(1-f_0(j,2s));\\
-w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+2s;\\
-f_1(j,2s)w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=(j)_s+5s;\\
0\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$(d_{10})_{ij}=\begin{cases}
-f_1(j,4s)w_{(j+m+1)n\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n},\quad i=(j+1)_s;\\
w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)+1}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
w_{(j+m+1)n+g(j)\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)},\\\quad\quad\quad i=(j+1)_s+s+s(1-f_0(j,4s));\\
-e_{(j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)+1},\\\quad\quad\quad i=(j+1)_s+3s+s(1-f_0(j,4s));\\
f_1(j,4s)w_{(j+m)n+5\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=(j)_s+5s;\\
0\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$(d_{10})_{ij}=\begin{cases}
-w_{(j+m+1)n\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
w_{(j+m)n+5\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j;\\
w_{(j+m+1)n+3\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+1},\quad i=(j+1)_s+s;\\
-w_{(j+m+1)n+1\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+3},\quad i=(j+1)_s+2s;\\
-w_{(j+m+1)n+4\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+2},\quad i=(j+1)_s+3s;\\
w_{(j+m+1)n+2\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+4},\quad i=(j+1)_s+4s;\\
-e_{(j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+5},\quad i=(j+1)_s+5s;\\
0\quad\text{otherwise.}\end{cases}$$
\begin{thm}\label{resol_thm}
Let $R=R_s$ is algebra of the type $E_6$. Then the minimal projective resolution of the
$\Lambda$-module $R$ is of the form:
\begin{equation}\label{resolv}\tag{$+$} \dots\longrightarrow
Q_3\stackrel{d_2}\longrightarrow Q_2\stackrel{d_1}\longrightarrow
Q_1\stackrel{d_0}\longrightarrow
Q_0\stackrel\varepsilon\longrightarrow R\longrightarrow
0,
\end{equation}
where $\varepsilon$ is the multiplication map $(\varepsilon(a\otimes b)=ab)$; $Q_r\text{ }(r\le
10)$ and $d_r\text{ }(r\le 10)$ were described before; further $Q_{11\ell+r}$, where $\ell\in \N$
and $0\le r\le 10$, is obtained from $Q_r$ by replacing every direct summand $P_{i,j}$ to
$P_{\sigma^\ell(i),j}$ correspondingly $($here $\sigma(i)=j$, if $\sigma(e_i)=e_j)$, and the
differential $d_{11\ell+r}$ is obtained from $d_r$ by act of $\sigma^\ell$ by all left tensor
components of the corresponding matrix.
\end{thm}
To prove that the terms $Q_i$ are of this form we introduce $P_i=Re_i$ is the projective cover of
the simple $R$-modules $S_i$, corresponding to the vertices of the quiver $\mathcal Q_s$. We will
find projective resolutions of the simple $R$-modules $S_i$.
\begin{obozn}
For $R$-module $M$ its $m$th syzygy is denoted by $\Omega^m(M)$.
\end{obozn}
\begin{zam}\label{note_brev}
From here we denote the multiplication homomorphism from the right by an element $w$ by $w$.
\end{zam}
\begin{lem}\label{lem_s0}
The begin of the minimal projective resolution of $S_{rn}$ is of the form
\begin{multline*}
\dots\longrightarrow P_{(r+3)n+5}
\stackrel{\binom{\a}{-\b}}\longrightarrow P_{(r+3)n+2}\oplus
P_{(r+3)n+4}
\stackrel{(\a^2\text{ }\b^2)}\longrightarrow\\
\longrightarrow P_{(r+3)n}
\stackrel{\binom{\g\a^2}{\g\b^2}}\longrightarrow P_{(r+2)n+1}\oplus
P_{(r+2)n+3} \stackrel{\binom{\a\g\phantom{-}0}{-\a\phantom{-}\b}}
\longrightarrow\\\longrightarrow P_{(r+1)n+5}\oplus P_{(r+2)n}
\stackrel{\binom{\phantom{-}\a\phantom{-}\g\a}{-\b\phantom{-}0}}\longrightarrow
P_{(r+1)n+2}\oplus P_{(r+1)n+4} \stackrel{(\a^2\g\text{
}\b^2\g)}\longrightarrow\\\longrightarrow
P_{rn+5}\stackrel{\binom{\a^2}{-\b^2}}\longrightarrow P_{rn+1}\oplus
P_{rn+3} \stackrel{(\a\text{ }\b)}\longrightarrow
P_{rn}\longrightarrow S_{rn}\longrightarrow 0.
\end{multline*}
At that $\Omega^{9}(S_{rn})\simeq S_{(r+4)n+5}$.
\end{lem}
\begin{lem}
The begin of the minimal projective resolution of $S_{rn+1}$ is of the form $$\dots\longrightarrow
P_{rn+2} \stackrel{\a}\longrightarrow P_{rn+1}\longrightarrow S_{rn+1}\longrightarrow 0.$$ At that
$\Omega^{2}(S_{rn+1})\simeq S_{(r+1)n+2}$.
\end{lem}
\begin{lem}
The begin of the minimal projective resolution of $S_{rn+2}$ is of the form
\begin{multline*}
\dots\longrightarrow P_{(r+4)n+3} \stackrel{\b}\longrightarrow
P_{(r+4)n}\stackrel{\g\a}\longrightarrow
P_{(r+3)n+2}\stackrel{\a^2\g}\longrightarrow
P_{(r+2)n+5}\stackrel{\binom{\a^2}{-\b}}\longrightarrow\\\longrightarrow
P_{(r+2)n+1}\oplus P_{(r+2)n+4}\stackrel{(\a\text{
}\b^2)}\longrightarrow
P_{(r+2)n}\stackrel{\g\b^2}\longrightarrow\\
\longrightarrow
P_{(r+1)n+3}\stackrel{\b\g}\longrightarrow
P_{rn+5}\stackrel{\a}\longrightarrow P_{rn+2}\longrightarrow
S_{rn+2}\longrightarrow 0.
\end{multline*}
At that $\Omega^{9}(S_{rn+2})\simeq S_{(r+5)n+3}$.
\end{lem}
\begin{lem}
The begin of the minimal projective resolution of $S_{rn+3}$ is of the form $$\dots\longrightarrow
P_{rn+4} \stackrel{\b}\longrightarrow P_{rn+3}\longrightarrow S_{rn+3}\longrightarrow 0.$$ At that
$\Omega^{2}(S_{rn+3})\simeq S_{(r+1)n+4}$.
\end{lem}
\begin{lem}
The begin of the minimal projective resolution of $S_{rn+4}$ is of the form
\begin{multline*}
\dots\longrightarrow P_{(r+4)n+1} \stackrel{\a}\longrightarrow
P_{(r+4)n}\stackrel{\g\b}\longrightarrow
P_{(r+3)n+4}\stackrel{\b^2\g}\longrightarrow
P_{(r+2)n+5}\stackrel{\binom{\a}{-\b^2}}\longrightarrow\\\longrightarrow
P_{(r+2)n+2}\oplus P_{(r+2)n+3}\stackrel{(\a^2\text{
}\b)}\longrightarrow
P_{(r+2)n}\stackrel{\g\a^2}\longrightarrow\\
\longrightarrow P_{(r+1)n+1}\stackrel{\a\g}\longrightarrow
P_{rn+5}\stackrel{\b}\longrightarrow P_{rn+4}\longrightarrow
S_{rn+4}\longrightarrow 0.
\end{multline*}
At that $\Omega^{9}(S_{rn+4})\simeq S_{(r+5)n+1}$.
\end{lem}
\begin{lem}\label{lem_s5}
The begin of the minimal projective resolution of $S_{rn+5}$ is of the form $$\dots\longrightarrow
P_{(r+1)n} \stackrel{\g}\longrightarrow P_{rn+5}\longrightarrow S_{rn+5}\longrightarrow 0.$$ At
that $\Omega^{2}(S_{rn+5})\simeq S_{(r+2)n}$.
\end{lem}
\begin{proof}
Proofs of the lemmas consist of direct check that given sequences are exact, and it is immediate.
\end{proof}
We shall need the Happel's lemma (see \cite{Ha}), as revised in \cite{Gen&Ka}:
\begin{lem}[Happel]\label{lem_Ha}
Let
$$\dots\rightarrow Q_m\rightarrow Q_{m-1}\rightarrow\dots\rightarrow
Q_1\rightarrow Q_0\rightarrow R\rightarrow 0$$ be the minimal projective resolution of $R$. Then
$$Q_m\cong\bigoplus_{i,j}P_{i,j}^{\dim\mathrm{Ext}^m_R(S_j,S_i)}.$$
\end{lem}
\begin{proof}[Proof of the theorem \ref{resol_thm}]
Descriptions for $Q_i$ immediately follows from lemmas \ref{lem_s0} -- \ref{lem_s5} and Happel's
lemma.
As proved in \cite{VGI}, to prove that sequence \eqref{resolv} is exact in $Q_m$ ($m\le 11$)
it will be sufficient to show that $d_md_{m+1}=0$. It is easy to verify this relation by
a straightforward calculation of matrixes products.
Since the sequence is exact in $Q_{11}$, it follows that
$\Omega^{11}({}_\Lambda R)\simeq {}_1R_{\sigma}$, where
$\Omega^{11}({}_\Lambda R)=\Im d_{10}$ is the 11th syzygy of the
module $R$, and ${}_1R_{\sigma}$ is a twisted bimodule. Hence, an
exactness in $Q_t$ ($t>11$) holds.
\end{proof}
We recall that for $R$-bimodule $M$ the {\it twisted bimodule} is a linear
space $M$, on which left act right acts of the algebra $R$ (denoted by
asterisk) are assigned by the following way:
$$r*m*s = \lambda(r)\cdot m\cdot\mu(s) \text{ for } r,s\in R
\text{ and } m\in M,$$ where $\lambda,\mu$ are some automorphisms of algebra $R$.
Such twisted bimodule we shall denote by ${}_\lambda M_\mu$.
\begin{s}
We have isomorphism $\Omega^{11}({}_\Lambda R)\simeq {}_1R_{\sigma}$.
\end{s}
\begin{pr}
Automorphism $\sigma$ has a finite order, and
$(1)$ if $\myChar=2$, then order of $\sigma$ is equal to $\frac{2s}{\myNod(2n,s)}$;
$(2)$ if $\myChar\ne 2$, then order of $\sigma$ is equal to $\frac{2s}{\myNod(2n,s)}$, if
$\frac{s}{\myNod(n,s)}$ is divisible by $4$, and to $\frac{4s}{\myNod(2n,s)}$ otherwise.
\end{pr}
\begin{pr}
The minimal period of bimodule resolution of $R$ is $11\frac{2s}{\myNod(2n,s)}$.
\end{pr}
\begin{proof}
Since $\Omega^{11}({}_\Lambda R)\simeq {}_1R_{\sigma}$, period is equal to $11a$. There is an
isomorphism $\sigma^a(Q_0)\simeq Q_0$, i. e. $\sigma^a$ must act identically on idempotents. Hence,
if $\myChar=2$, then $a=\deg\sigma$. For the case $\myChar\ne 2$ we just need to show that
$a=\frac{\deg\sigma}{2}$, if $\frac{s}{\myNod(n,s)}$ is not divisible by $4$. Consider the case
$\myChar\ne 2$, $\frac{s}{\myNod(n,s)}$ is not divisible by $4$. Denote by
$b=\frac{\deg\sigma}{2}$. We see that $\sigma^b(e_i)=e_i,$ $\sigma^b(\g_i)=\g_i$,
$$\sigma^b(\a_i)=\begin{cases}
-\a_i,\quad i\equiv 0,2(3);\\
\a_i,\quad i\equiv 1(3),
\end{cases} \sigma^b(\b_i)=\begin{cases}
-\b_i,\quad i\equiv 0,2(3);\\
\b_i,\quad i\equiv 1(3).
\end{cases}$$
Let $x=\sum\limits_{i=0}^{s-1} e_{6i+1}+e_{6i+2}+e_{6i+3}+e_{6i+4}-e_{6i}-e_{6i+5}$. We have:
$x^{-1}=x$, $\sigma^b(w)=xwx^{-1}$, for any path $w$, so $\sigma^b$ is an inner automorphism of
$R$, and $11b$ is a period of bimodule resolution.
\end{proof}
\section{The additive structure of $\HH^*(R)$}
\begin{pr}[Dimensions of homomorphism groups, $s>1$]\label{dim_hom}
Let $s>1$ and $R=R_s$ is algebra of the type $E_6$. Next, $t\in\N\cup\{0\}$,
$\ell$ be the aliquot, and $r$ be the residue of division of $t$ by $11$.
$(1)$ If $r=0$, then $$\dim_K\Hom_\Lambda(Q_t, R)=\begin{cases}
6s,\quad \ell n+m\equiv 0(s)\text{ or }\ell n+m\equiv 1(s),\text{ }\ell\div 2;\\
2s,\quad \ell n+m\equiv 0(s)\text{ or }\ell n+m\equiv 1(s),\text{ }\ell\ndiv 2;\\
0,\quad\text{otherwise.}
\end{cases}$$
$(2)$ If $r=1$, then $$\dim_K\Hom_\Lambda(Q_t, R)=\begin{cases}
7s,\quad \ell n+m\equiv 0(s),\text{ }\ell\div 2;\\
5s,\quad \ell n+m\equiv 0(s),\text{ }\ell\ndiv 2;\\
0,\quad\text{otherwise.}
\end{cases}$$
$(3)$ If $r\in\{2,8\}$, then $$\dim_K\Hom_\Lambda(Q_t, R)=\begin{cases}
3s,\quad \ell n+m\equiv 0(s),\text{ }\ell\div 2;\\
s,\quad \ell n+m\equiv 0(s),\text{ }\ell\ndiv 2;\\
s,\quad \ell n+m\equiv 1(s),\text{ }\ell\div 2;\\
3s,\quad \ell n+m\equiv 1(s),\text{ }\ell\ndiv 2;\\
0,\quad\text{otherwise.}
\end{cases}$$
$(4)$ If $r\in\{3,5,7\}$, then $$\dim_K\Hom_\Lambda(Q_t, R)=\begin{cases}
8s,\quad \ell n+m\equiv 0(s);\\
0,\quad\text{otherwise.}
\end{cases}$$
$(5)$ If $r=4$, then $$\dim_K\Hom_\Lambda(Q_t, R)=\begin{cases}
2s,\quad \ell n+m\equiv 0(s),\text{ }\ell\div 2;\\
6s,\quad \ell n+m\equiv 0(s),\text{ }\ell\ndiv 2;\\
5s,\quad \ell n+m\equiv 1(s),\text{ }\ell\div 2;\\
7s,\quad \ell n+m\equiv 1(s),\text{ }\ell\ndiv 2;\\
0,\quad\text{otherwise.}
\end{cases}$$
$(6)$ If $r=6$, then $$\dim_K\Hom_\Lambda(Q_t, R)=\begin{cases}
7s,\quad \ell n+m\equiv 0(s),\text{ }\ell\div 2;\\
5s,\quad \ell n+m\equiv 0(s),\text{ }\ell\ndiv 2;\\
6s,\quad \ell n+m\equiv 1(s),\text{ }\ell\div 2;\\
2s,\quad \ell n+m\equiv 1(s),\text{ }\ell\ndiv 2;\\
0,\quad\text{otherwise.}
\end{cases}$$
$(7)$ If $r=9$, then $$\dim_K\Hom_\Lambda(Q_t, R)=\begin{cases}
5s,\quad \ell n+m\equiv 0(s),\text{ }\ell\div 2;\\
7s,\quad \ell n+m\equiv 0(s),\text{ }\ell\ndiv 2;\\
0,\quad\text{otherwise.}
\end{cases}$$
$(8)$ If $r=10$, then $$\dim_K\Hom_\Lambda(Q_t, R)=\begin{cases}
2s,\quad \ell n+m\equiv 0(s)\text{ or }\ell n+m\equiv 1(s),\text{ }\ell\div 2;\\
6s,\quad \ell n+m\equiv 0(s)\text{ or }\ell n+m\equiv 1(s),\text{ }\ell\ndiv 2;\\
0,\quad\text{otherwise.}
\end{cases}$$
\end{pr}
\begin{proof}
The dimension $\dim_K\Hom_\Lambda(P_{i,j}, R)$ is equal to the number of linear independent nonzero
paths of the quiver $\mathcal{Q}_s$, leading from $j$th vertex to $i$th, and the proof is to
consider cases $r=0$, $r=1$ etc.
\end{proof}
\begin{pr}[Dimensions of homomorphism groups, $s=1$]
Let $R=R_1$ is algebra of the type $E_6$. Next, $t\in\N\cup\{0\}$,
$\ell$ be the aliquot, and $r$ be the residue of division of $t$ by $11$.
$(1)$ If $r=0$, then $$\dim_K\Hom_\Lambda(Q_t, R)=\begin{cases}
12,\quad \ell\div 2;\\4,\quad \ell\ndiv 2.\end{cases}$$
$(2)$ If $r=1$, then $$\dim_K\Hom_\Lambda(Q_t, R)=\begin{cases}
7,\quad \ell\div 2;\\5,\quad \ell\ndiv 2.\end{cases}$$
$(3)$ If $r\in\{2,8\}$, then $\dim_K\Hom_\Lambda(Q_t, R)=4$.
$(4)$ If $r\in\{3,5,7\}$, then $\dim_K\Hom_\Lambda(Q_t, R)=8$.
$(5)$ If $r=4$, then $$\dim_K\Hom_\Lambda(Q_t, R)=\begin{cases}
7,\quad \ell\div 2;\\13,\quad \ell\ndiv 2.\end{cases}$$
$(6)$ If $r=6$, then $$\dim_K\Hom_\Lambda(Q_t, R)=\begin{cases}
13,\quad \ell\div 2;\\7,\quad \ell\ndiv 2.\end{cases}$$
$(7)$ If $r=9$, then $$\dim_K\Hom_\Lambda(Q_t, R)=\begin{cases}
5,\quad \ell\div 2;\\7,\quad \ell\ndiv 2.\end{cases}$$
$(8)$ If $r=10$, then $$\dim_K\Hom_\Lambda(Q_t, R)=\begin{cases}
4,\quad \ell\div 2;\\12,\quad \ell\ndiv 2.\end{cases}$$
\end{pr}
\begin{proof}
The proof is basically the same as proof of proposition \ref{dim_hom}.
\end{proof}
\begin{pr}[Dimensions of coboundaries groups]\label{dim_im}
Let $R=R_s$ is algebra of the type $E_6$, and let
\begin{equation}\tag{$\times$}\label{ind_resolv} 0\longrightarrow
\Hom_\Lambda(Q_0, R)\stackrel{\delta^0}\longrightarrow \Hom_\Lambda(Q_1,
R)\stackrel{\delta^1}\longrightarrow \Hom_\Lambda(Q_2, R)\stackrel{\delta^2}\longrightarrow\dots
\end{equation} be a complex, obtained from minimal projective
resolution \eqref{resolv} of algebra $R$, by applying functor
$\Hom_\Lambda(-,R)$.
Consider coboundaries groups $\Im\delta^s$ of the complex
\eqref{ind_resolv}. Let $\ell$ be the aliquot, and $r$ be the
residue of division of $t$ by $11$, $m$ be the aliquot of
division of $r$ by $2$. Then$:$
$(1)$ If $r=0$, then $$\dim_K\Im\delta^t=\begin{cases}
6s-1,\quad \ell n+m\equiv 0(s),\text{ }\ell\div 2;\\
2s,\quad \ell n+m\equiv 0(s),\text{ }\ell\ndiv 2;\\
0,\quad\text{otherwise.}
\end{cases}$$
$(2)$ If $r=1$, then $$\dim_K\Im\delta^t=\begin{cases}
s,\quad \ell n+m\equiv 0(s),\text{ }\ell\div 2;\\
3s-1,\quad \ell n+m\equiv 0(s),\text{ }\ell\ndiv 2;\\
0,\quad\text{otherwise.}
\end{cases}$$
$(3)$ If $r=2$, then $$\dim_K\Im\delta^t=\begin{cases}
3s,\quad \ell n+m\equiv 0(s),\text{ }\ell\div 2;\\
s,\quad \ell n+m\equiv 0(s),\text{ }\ell\ndiv 2;\\
0,\quad\text{otherwise.}
\end{cases}$$
$(4)$ If $r=3$, then $$\dim_K\Im\delta^t=\begin{cases}
5s-1,\quad \ell n+m\equiv 0(s),\text{ }\ell\div 2,\text{ }\myChar=2;\\
5s,\quad \ell n+m\equiv 0(s),\text{ }\ell\div 2,\text{ }\myChar\ne 2;\\
7s-1,\quad \ell n+m\equiv 0(s),\text{ }\ell\ndiv 2;\\
0,\quad\text{otherwise.}
\end{cases}$$
$(5)$ If $r=4$, then $$\dim_K\Im\delta^t=\begin{cases}
2s,\quad \ell n+m\equiv 0(s),\text{ }\ell\div 2;\\
6s-1,\quad \ell n+m\equiv 0(s),\text{ }\ell\ndiv 2,\text{ }\myChar=3;\\
6s,\quad \ell n+m\equiv 0(s),\text{ }\ell\ndiv 2,\text{ }\myChar\ne 3;\\
0,\quad\text{otherwise.}
\end{cases}$$
$(6)$ If $r=5$, then $$\dim_K\Im\delta^t=\begin{cases}
6s-1,\quad \ell n+m\equiv 0(s),\text{ }\ell\div 2,\text{ }\myChar=3;\\
6s,\quad \ell n+m\equiv 0(s),\text{ }\ell\div 2,\text{ }\myChar\ne 3;\\
2s,\quad \ell n+m\equiv 0(s),\text{ }\ell\ndiv 2;\\
0,\quad\text{otherwise.}
\end{cases}$$
$(7)$ If $r=6$, then $$\dim_K\Im\delta^t=\begin{cases}
7s-1,\quad \ell n+m\equiv 0(s),\text{ }\ell\div 2;\\
5s-1,\quad \ell n+m\equiv 0(s),\text{ }\ell\ndiv 2,\text{ }\myChar=2;\\
5s,\quad \ell n+m\equiv 0(s),\text{ }\ell\ndiv 2,\text{ }\myChar\ne 2;\\
0,\quad\text{otherwise.}
\end{cases}$$
$(8)$ If $r=7$, then $$\dim_K\Im\delta^t=\begin{cases}
s,\quad \ell n+m\equiv 0(s),\text{ }\ell\div 2;\\
3s,\quad \ell n+m\equiv 0(s),\text{ }\ell\ndiv 2;\\
0,\quad\text{otherwise.}
\end{cases}$$
$(9)$ If $r=8$, then $$\dim_K\Im\delta^t=\begin{cases}
3s-1,\quad \ell n+m\equiv 0(s),\text{ }\ell\div 2;\\
s,\quad \ell n+m\equiv 0(s),\text{ }\ell\ndiv 2;\\
0,\quad\text{otherwise.}
\end{cases}$$
$(10)$ If $r=9$, then $$\dim_K\Im\delta^t=\begin{cases}
2s,\quad \ell n+m\equiv 0(s),\text{ }\ell\div 2;\\
6s-1,\quad \ell n+m\equiv 0(s),\text{ }\ell\ndiv 2;\\
0,\quad\text{otherwise.}
\end{cases}$$
$(11)$ If $r=10$, then $$\dim_K\Im\delta^t=\begin{cases}
2s,\quad \ell n+m\equiv 0(s),\text{ }\ell\div 2;\\
6s-1,\quad \ell n+m\equiv 0(s),\text{ }\ell\ndiv 2,\text{ }\myChar=3;\\
6s,\quad \ell n+m\equiv 0(s),\text{ }\ell\ndiv 2,\text{ }\myChar\ne 3;\\
0,\quad\text{otherwise.}
\end{cases}$$
\end{pr}
\begin{proof}
The proof is technical and consists in constructing the image matrixes from the description of
differential matrixes and the subsequent computations of the ranks of image matrixes.
\end{proof}
\begin{thm}[Additive structure, $s>1$]
Let $s>1$ and $R=R_s$ is algebra of the type $E_6$. Next, $t\in\N\cup\{0\}$,
$\ell$ be the aliquot, and $r$ be the residue of division of $t$ by $11$,
$m$ be the aliquot of division of $r$ by $2$. Then
$\dim_K\HH^t(R)=1$, if one of the following conditions takes place$:$
$(1)$ $r\in \{0,6,7,8\}$, $\ell n+m\equiv 0(s)$, $\ell\div 2$;
$(2)$ $r\in \{0,6\}$, $\ell n+m\equiv 1(s)$, $\ell\div 2$,
$\myChar=3$;
$(3)$ $r\in \{1,9\}$, $\ell n+m\equiv 0(s)$;
$(4)$ $r\in \{2,4,10\}$, $\ell n+m\equiv 1(s)$, $\ell\ndiv 2$;
$(5)$ $r=3$, $\ell n+m\equiv 0(s)$, $\ell\div 2$, $\myChar=2$;
$(6)$ $r=3$, $\ell n+m\equiv 0(s)$, $\ell\ndiv 2$;
$(7)$ $r\in \{4,10\}$, $\ell n+m\equiv 0(s)$, $\ell\ndiv 2$,
$\myChar=3$;
$(8)$ $r=4$, $\ell n+m\equiv 1(s)$, $\ell\div 2$, $\myChar=2$;
$(9)$ $r=5$, $\ell n+m\equiv 0(s)$, $\myChar=3$;
$(10)$ $r\in \{6,7\}$, $\ell n+m\equiv 0(s)$, $\ell\ndiv 2$,
$\myChar=2$.
In other cases $\dim_K\HH^t(R)=0$.
\end{thm}
\begin{proof}
As $\dim_K\HH^t(R)=\dim_K\Ker\delta^t-\dim_K\Im\delta^{t-1}$, and
$\dim_K\Ker\delta^t=\dim_K\Hom_\Lambda(Q_t,R)-\dim_K\Im\delta^t$,
the assertions of theorem easily follows from propositions
\ref{dim_hom} -- \ref{dim_im}.
\end{proof}
\begin{thm}[Additive structure, $s=1$]
Let $R=R_1$ is algebra of the type $E_6$. Next, $t\in\N\cup\{0\}$,
$\ell$ be the aliquot, and $r$ be the residue of division of $t$ by $11$.\\
$($a$)$ $\dim_K\HH^t(R)=7$, if $t=0$.\\
$($b$)$ $\dim_K\HH^t(R)=2$, if one of the following conditions takes place$:$
$(1)$ $r\in \{0,6\}$, $t>0$, $\ell\div 2$, $\myChar=3$;
$(2)$ $r\in \{4,10\}$, $\ell\ndiv 2$, $\myChar=3$.\\ $($c$)$
$\dim_K\HH^t(R)=1$, if one of the following conditions takes
place$:$
$(1)$ $r\in \{0,6\}$, $t>0$, $\ell\div 2$, $\myChar\ne 3$;
$(2)$ $r\in \{1,9\}$;
$(3)$ $r\in \{2,3\}$, $\ell\ndiv 2$;
$(4)$ $r\in \{3,4\}$, $\ell\div 2$, $\myChar=2$;
$(5)$ $r\in \{4,10\}$, $\ell\ndiv 2$, $\myChar\ne 3$;
$(6)$ $r=5$, $\myChar=3$;
$(7)$ $r\in \{6,7\}$, $\ell\ndiv 2$, $\myChar=2$;
$(8)$ $r\in \{7,8\}$, $\ell\div 2$.\\ $($d$)$ In other cases
$\dim_K\HH^t(R)=0$.
\end{thm}
\section{Generators of $\HH^*(R)$}
For $s>1$ introduce the set of generators $Y^{(1)}_t$, $Y^{(2)}_t$,
\dots $Y^{(22)}_t$, such that $\deg Y_t^{(i)}=t$, $0\le t <
11\deg\sigma$ and $t$ satisfies conditions of (i)th item from the
list on page \pageref{degs}. For $s=1$ introduce the set of
generators $Y^{(1)}_t$, $Y^{(2)}_t$, \dots $Y^{(28)}_t$, such that
$\deg Y_t^{(i)}=t$, $0\le t < 11\deg\sigma$ and $t$ satisfies
conditions of (i)th item from the list on page \pageref{degs} for
$i\ge 22$ and $t=0$ if $i>22$. Now let us describe the matrixes of
$Y^{(i)}_t$ componentwisely.
\begin{obozn}
Let us represent the degree $t$ of the generator element in the form
$t=11\ell+r$ ($0\le r\le 10$) and denote by
$\kappa=(-1)^{\lfloor\frac{\ell}{2}\rfloor}$.
\end{obozn}
(1) $Y^{(1)}_t$ is an $(6s\times 6s)$ matrix, whose elements
$y_{ij}$ have the following form:
If $0\le j<s$, then $$y_{ij}=
\begin{cases}
\kappa e_{jn}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$y_{ij}=
\begin{cases}
e_{jn+g(j+s)}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$y_{ij}=
\begin{cases}
e_{jn+g(j+s)+1}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$y_{ij}=
\begin{cases}
\kappa e_{jn+5}\otimes e_{jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
(2) $Y^{(2)}_t$ is an $(6s\times 6s)$ matrix with a single nonzero
element:
$$y_{3s,3s}=w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)+1}.$$
(3) $Y^{(3)}_t$ is an $(7s\times 6s)$ matrix with a single nonzero
element:
$$y_{5s,6s}=\kappa w_{jn+5\ra (j+1)n}\otimes e_{jn+5}.$$
(4) $Y^{(4)}_t$ is an $(7s\times 6s)$ matrix, whose elements
$y_{ij}$ have the following form:
If $0\le j<s$, then $$y_{ij}=
\begin{cases}
-w_{jn\ra jn+g(j+s)}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<5s$, then $y_{ij}=0$.
If $5s\le j<6s$, then $$y_{ij}=
\begin{cases}
\kappa w_{jn+g(j)+1\ra jn+5}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $y_{ij}=0$.
(5) $Y^{(5)}_t$ is an $(6s\times 6s)$ matrix with a single nonzero
element:
$$y_{3s,3s}=w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)}\otimes e_{jn+g(j+s)+1}.$$
(6) $Y^{(6)}_t$ is an $(8s\times 6s)$ matrix, whose elements
$y_{ij}$ have the following form:
If $0\le j<s$, then $$y_{ij}=
\begin{cases}
w_{jn\ra jn+g(j)+1}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$y_{ij}=
\begin{cases}
w_{jn\ra jn+g(j)+1}\otimes e_{jn},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $y_{ij}=0$.
If $4s\le j<5s$, then $$y_{ij}=
\begin{cases}
w_{jn+g(j)+1\ra (j+1)n}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $y_{ij}=0$.
If $6s\le j<7s$, then $$y_{ij}=
\begin{cases}
w_{jn+5\ra (j+1)n+g(j)}\otimes e_{jn+5},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $y_{ij}=0$.
(7) $Y^{(7)}_t$ is an $(8s\times 6s)$ matrix, whose elements
$y_{ij}$ have the following form:
If $0\le j<2s$, then $$y_{ij}=
\begin{cases}
w_{jn\ra jn+g(j+s)+1}\otimes e_{jn},\quad i=(j)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$y_{ij}=
\begin{cases}
-\kappa w_{jn+g(j)\ra jn+5}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $$y_{ij}=
\begin{cases}
-\kappa w_{jn+g(j)+1\ra (j+1)n}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<8s$, then $y_{ij}=0$.
(8) $Y^{(8)}_t$ is an $(9s\times 6s)$ matrix with a single nonzero
element:
$$y_{3s,6s}=w_{jn+g(j)+1\ra (j+1)n+g(j)+1}\otimes e_{jn+g(j)+1}.$$
(9) $Y^{(9)}_t$ is an $(9s\times 6s)$ matrix, whose elements
$y_{ij}$ have the following form:
If $0\le j<s$, then $y_{ij}=0$.
If $s\le j<2s$, then $$y_{ij}=
\begin{cases}
\kappa e_{jn}\otimes e_{jn},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$y_{ij}=
\begin{cases}
e_{jn+g(j)}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $y_{ij}=0$.
If $6s\le j<8s$, then $$y_{ij}=
\begin{cases}
e_{jn+g(j)+1}\otimes e_{jn+g(j)+1},\quad i=j-3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s\le j<9s$, then $$y_{ij}=
\begin{cases}
\kappa e_{jn+5}\otimes e_{jn+5},\quad i=j-3s;\\
0,\quad\text{otherwise.}\end{cases}$$
(10) $Y^{(10)}_t$ is an $(9s\times 6s)$ matrix with a single nonzero
element:
$$y_{5s,8s}=w_{jn+5\ra (j+1)n+5}\otimes e_{jn+5}.$$
(11) $Y^{(11)}_t$ is an $(8s\times 6s)$ matrix, whose elements
$y_{ij}$ have the following form:
If $0\le j<s$, then $$y_{ij}=
\begin{cases}
w_{jn\ra jn+g(j)}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $y_{ij}=0$.
If $2s\le j<3s$, then $$y_{ij}=
\begin{cases}
\kappa w_{jn+g(j)\ra (j+1)n}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $y_{ij}=0$.
If $4s\le j<5s$, then $$y_{ij}=
\begin{cases}
-\kappa w_{jn+g(j)+1\ra jn+5}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $y_{ij}=0$.
If $6s\le j<7s$, then $$y_{ij}=
\begin{cases}
w_{jn+5\ra (j+1)n+g(j)+1}\otimes e_{jn+5},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $y_{ij}=0$.
(12) $Y^{(12)}_t$ is an $(8s\times 6s)$ matrix, whose elements
$y_{ij}$ have the following form:
If $0\le j<s$, then $$y_{ij}=
\begin{cases}
-w_{jn\ra jn+g(j+s)}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<4s$, then $y_{ij}=0$.
If $4s\le j<5s$, then $$y_{ij}=
\begin{cases}
\kappa w_{jn+g(j)+1\ra jn+5}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<8s$, then $y_{ij}=0$.
(13) $Y^{(13)}_t$ is an $(9s\times 6s)$ matrix, whose elements
$y_{ij}$ have the following form:
If $0\le j<s$, then $y_{ij}=0$.
If $s\le j<3s$, then $$y_{ij}=
\begin{cases}
-f_1(j,2s)e_{jn+g(j+s)}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $y_{ij}=0$.
If $5s\le j<7s$, then $$y_{ij}=
\begin{cases}
-f_1(j,6s)e_{jn+g(j+s)+1}\otimes e_{jn+g(j+s)+1},\quad i=j-2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $y_{ij}=0$.
If $8s\le j<9s$, then $$y_{ij}=
\begin{cases}
-\kappa w_{jn+5\ra (j+1)n}\otimes e_{jn+5},\quad i=j-3s;\\
0,\quad\text{otherwise.}\end{cases}$$
(14) $Y^{(14)}_t$ is an $(9s\times 6s)$ matrix with a single nonzero
element:
$$y_{0,0}=\kappa w_{jn\ra (j+1)n}\otimes e_{jn}.$$
(15) $Y^{(15)}_t$ is an $(9s\times 6s)$ matrix, whose elements
$y_{ij}$ have the following form:
If $0\le j<s$, then $$y_{ij}=
\begin{cases}
e_{jn}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $y_{ij}=0$.
If $3s\le j<5s$, then $$y_{ij}=
\begin{cases}
w_{jn+g(j+s)\ra jn+g(j+s)+1}\otimes e_{jn+g(j+s)},\quad i=j-2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<7s$, then $y_{ij}=0$.
If $7s\le j<8s$, then $$y_{ij}=
\begin{cases}
e_{jn+5}\otimes e_{jn+5},\quad i=j-2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s\le j<9s$, then $y_{ij}=0$.
(16) $Y^{(16)}_t$ is an $(8s\times 6s)$ matrix, whose elements
$y_{ij}$ have the following form:
If $j=0$, then $$y_{ij}=
\begin{cases}
w_{jn\ra jn+g(j)+1}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $0<j<2s$, then $y_{ij}=0$.
If $j=2s$, then $$y_{ij}=
\begin{cases}
-\kappa w_{jn+g(j)\ra jn+5}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s<j<8s$, then $y_{ij}=0$.
(17) $Y^{(17)}_t$ is an $(8s\times 6s)$ matrix, whose elements
$y_{ij}$ have the following form:
If $j=0$, then $$y_{ij}=
\begin{cases}
w_{jn\ra jn+g(j+s)+1}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $0<j<2s$, then $y_{ij}=0$.
If $j=2s$, then $$y_{ij}=
\begin{cases}
w_{jn+g(j)\ra jn+5}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s<j<8s$, then $y_{ij}=0$.
(18) $Y^{(18)}_t$ is an $(6s\times 6s)$ matrix, whose elements
$y_{ij}$ have the following form:
If $0\le j<s$, then $y_{ij}=0$.
If $s\le j<3s$, then $$y_{ij}=
\begin{cases}
w_{jn+g(j+s)\ra jn+g(j+s)+1}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $y_{ij}=0$.
If $5s\le j<6s$, then $$y_{ij}=
\begin{cases}
-\kappa w_{jn+5\ra (j+1)n}\otimes e_{jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
(19) $Y^{(19)}_t$ is an $(7s\times 6s)$ matrix, whose elements
$y_{ij}$ have the following form:
If $0\le j<s$, then $y_{ij}=0$.
If $j=s$, then $$y_{ij}=
\begin{cases}
\kappa w_{jn+g(j+s)\ra (j+1)n}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s<j<2s$, then $y_{ij}=0$.
If $j=2s$, then $$y_{ij}=
\begin{cases}
\kappa w_{jn+g(j+s)\ra (j+1)n}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s<j<7s$, then $y_{ij}=0$.
(20) $Y^{(20)}_t$ is an $(7s\times 6s)$ matrix, whose elements
$y_{ij}$ have the following form:
If $0\le j<s$, then $$y_{ij}=
\begin{cases}
\kappa w_{jn\ra jn+5}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$y_{ij}=
\begin{cases}
\kappa w_{jn+g(j+s)\ra (j+1)n}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<3s$, then $y_{ij}=0$.
If $3s\le j<4s$, then $$y_{ij}=
\begin{cases}
-w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $y_{ij}=0$.
If $6s\le j<7s$, then $$y_{ij}=
\begin{cases}
-w_{jn+5\ra (j+1)n+g(j)+1}\otimes e_{jn+5},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
(21) $Y^{(21)}_t$ is an $(6s\times 6s)$ matrix with a single nonzero
element:
$$y_{0,0}=-\kappa w_{jn\ra (j+1)n}\otimes e_{jn}.$$
(22) $Y^{(22)}_t$ is an $(6s\times 6s)$ matrix, whose elements
$y_{ij}$ have the following form:
If $0\le j<s$, then $y_{ij}=0$.
If $s\le j<3s$, then $$y_{ij}=
\begin{cases}
f_1(j,2s)e_{jn+g(j+s)}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$y_{ij}=
\begin{cases}
f_1(j,4s)e_{jn+g(j+s)+1}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $y_{ij}=0$.
(23) $Y^{(23)}_t$ is an $(6s\times 6s)$ matrix with a single nonzero
element:
$$y_{3s,3s}=w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)+1}.$$
(24) $Y^{(24)}_t$ is an $(6s\times 6s)$ matrix with a single nonzero
element:
$$y_{0,0}=w_{jn\ra (j+1)n}\otimes e_{jn}.$$
(25) $Y^{(25)}_t$ is an $(6s\times 6s)$ matrix with a single nonzero
element:
$$y_{s,s}=w_{jn+g(j+s)\ra (j+1)n+g(j+s)}\otimes e_{jn+g(j+s)}.$$
(26) $Y^{(26)}_t$ is an $(6s\times 6s)$ matrix with a single nonzero
element:
$$y_{2s,2s}=-w_{jn+g(j+s)\ra (j+1)n+g(j+s)}\otimes e_{jn+g(j+s)}.$$
(27) $Y^{(27)}_t$ is an $(6s\times 6s)$ matrix with a single
nonzero element:
$$y_{4s,4s}=-w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)+1}.$$
(28) $Y^{(28)}_t$ is an $(6s\times 6s)$ matrix with a single nonzero
element:
$$y_{5s,5s}=w_{jn+5\ra (j+1)n+5}\otimes e_{jn+5}.$$
\section{$\Omega$-shifts of generators of the algebra $\HH^*(R)$}
Let $Q_\bullet\rightarrow R$ be the minimal projective bimodule
resolution of the algebra $R$, constructed in paragraph
\ref{sect_res}. Any $t$-cocycle $f\in\Ker\delta^t$ is lifted
(uniquely up to homotopy) to a chain map of complexes $\{\varphi_i:
Q_{t+i}\rightarrow Q_i\}_{i\ge 0}$. The homomorphism $\varphi_i$ is
called the {\it $i$th translate} of the cocycle $f$ and will be
denoted by $\Omega^i(f)$. For cocycles $f_1\in\Ker\delta^{t_1}$ and
$f_2\in\Ker\delta^{t_2}$ we have
\begin{equation}\tag{$*$}\label{mult_formula}
\cl f_2\cdot \cl f_1=\cl(\Omega^0(f_2)\Omega^{t_2}(f_1)).
\end{equation}
We shall now describe $\Omega$-translates for generators of the
algebra $\HH^*(R)$ and then find multiplications of the generators
using the formula \eqref{mult_formula}.
\begin{obozns}$\quad$
(1) For generator degree $t$ represent it in the form $t=11\ell+r$
($0\le r\le 10$) and denote by
$\kappa=(-1)^{\lfloor\frac{\ell}{2}\rfloor}$.
(2) For translate $\Omega^{t_0}$ represent $t_0$ in the form
$t_0=11\ell_0+r_0$ ($0\le r_0\le 10$) and denote by
$\kappa_0=(-1)^{\ell_0}$.
\end{obozns}
Define the helper functions $z_0\text{:
}\Z\times\Z\times\Z\rightarrow\Z$ and $z_1\text{:
}\Z\times\Z\times\Z\rightarrow\Z$, which act in the following way:
$$z_0(x,\ell,k)=(x-\ell k)_s+(\ell s)_{2s},\quad z_1(x,\ell,k)=(x-\ell k)_s+((\ell+1)s)_{2s}$$
\begin{pr}[Translates for the case 1]
$({\rm I})$ Let $r_0\in\N$, $r_0<11$. $r_0$-translates of the
elements $Y^{(1)}_t$ are described by the following way.
$(1)$ If $r_0=0$, then $\Omega^{0}(Y_t^{(1)})$ is described with
$(6s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m)n}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j+s)}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j+s)+1}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m)n+5}\otimes e_{jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
$(2)$ If $r_0=1$, then $\Omega^{1}(Y_t^{(1)})$ is described with
$(7s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j)}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j)+1}\otimes e_{jn+g(j)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m)n+5}\otimes e_{jn+g(j)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m+1)n}\otimes e_{jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
$(3)$ If $r_0=2$, then $\Omega^{2}(Y_t^{(1)})$ is described with
$(6s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m-1)n+5}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j+s)+1}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j)}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m+1)n}\otimes e_{jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
$(4)$ If $r_0=3$, then $\Omega^{3}(Y_t^{(1)})$ is described with
$(8s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j)+1}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m)n+5}\otimes e_{jn+g(j)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<8s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j)}\otimes e_{jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
$(5)$ If $r_0=4$, then $\Omega^{4}(Y_t^{(1)})$ is described with
$(9s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m-1)n+5}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m)n}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j+s)}\otimes e_{jn+g(j)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j)}\otimes e_{jn+g(j)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<8s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j+s)+1}\otimes e_{jn+g(j)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s\le j<9s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m)n+5}\otimes e_{jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
$(6)$ If $r_0=5$, then $\Omega^{5}(Y_t^{(1)})$ is described with
$(8s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j)}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m+1)n}\otimes e_{jn+g(j)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m)n+5}\otimes e_{jn+g(j)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<8s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j)+1}\otimes e_{jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
$(7)$ If $r_0=6$, then $\Omega^{6}(Y_t^{(1)})$ is described with
$(9s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m)n}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j+s)}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j)+1}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<7s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j+s)+1}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m)n+5}\otimes e_{jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s\le j<9s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m+1)n}\otimes e_{jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
$(8)$ If $r_0=7$, then $\Omega^{7}(Y_t^{(1)})$ is described with
$(8s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j)+1}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m)n+5}\otimes e_{jn+g(j)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<8s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j)}\otimes e_{jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
$(9)$ If $r_0=8$, then $\Omega^{8}(Y_t^{(1)})$ is described with
$(6s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m-1)n+5}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j+s)+1}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j)}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m+1)n}\otimes e_{jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
$(10)$ If $r_0=9$, then $\Omega^{9}(Y_t^{(1)})$ is described with
$(7s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m)n+5}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m+1)n}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j)}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<7s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j+s)+1}\otimes e_{jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
$(11)$ If $r_0=10$, then $\Omega^{10}(Y_t^{(1)})$ is described with
$(6s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m)n}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j)}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j)+1}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m)n+5}\otimes e_{jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
\medskip
$({\rm II})$ Represent an arbitrary $t_0\in\N$ in the form
$t_0=11\ell_0+r_0$, where $0\le r_0\le 10.$ Then
$\Omega^{t_0}(Y_t^{(1)})$ is a $\Omega^{r_0}(Y_t^{(1)})$, whose left
components twisted by $\sigma^{\ell_0}$.
\end{pr}
\begin{pr}[Translates for the case 2]
$({\rm I})$ Let $r_0\in\N$, $r_0<11$. $r_0$-translates of the
elements $Y^{(2)}_t$ are described by the following way.
$(1)$ If $r_0=0$, then $\Omega^{0}(Y_t^{(2)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{3s+z_0(0,\ell_0,n),3s+z_0(0,\ell_0,n)}=w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)+1}.$$
$(2)$ If $r_0=1$, then $\Omega^{1}(Y_t^{(2)})$ is described with
$(7s\times 7s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{2s+z_0(0,\ell_0,n),2s+z_0(0,\ell_0,n)}=w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)+1}\otimes e_{jn+g(j)}.$$
$(3)$ If $r_0=2$, then $\Omega^{2}(Y_t^{(2)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{s+z_0(-1,\ell_0,n),s+z_0(-1,\ell_0,n)}=w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)}.$$
$(4)$ If $r_0=3$, then $\Omega^{3}(Y_t^{(2)})$ is described with
$(8s\times 8s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{z_0(-1,\ell_0,n),z_0(-1,\ell_0,n)}=w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)+1}\otimes e_{jn}.$$
$(5)$ If $r_0=4$, then $\Omega^{4}(Y_t^{(2)})$ is described with
$(9s\times 9s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{6s+z_1(-2,\ell_0,n),6s+z_1(-2,\ell_0,n)}=w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j)+1}.$$
$(6)$ If $r_0=5$, then $\Omega^{5}(Y_t^{(2)})$ is described with
$(8s\times 8s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{6s+z_0(-3,\ell_0,n),6s+z_0(-3,\ell_0,n)}=w_{(j+m+1)n+g(j)+1\ra (j+m+2)n+g(j)+1}\otimes e_{jn+5}.$$
$(7)$ If $r_0=6$, then $\Omega^{6}(Y_t^{(2)})$ is described with
$(9s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<3s+z_1(-3,\ell_0,n)$, then $b_{ij}=0$.
If $j=3s+z_1(-3,\ell_0,n)$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)+1}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s+z_1(-3,\ell_0,n)<j<5s+z_0(-3,\ell_0,n)$, then $b_{ij}=0$.
If $j=5s+z_0(-3,\ell_0,n)$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s+z_0(-3,\ell_0,n)<j<9s$, then $b_{ij}=0$.
$(8)$ If $r_0=7$, then $\Omega^{7}(Y_t^{(2)})$ is described with
$(8s\times 8s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{z_0(-3,\ell_0,n),z_0(-3,\ell_0,n)}=w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)+1}\otimes e_{jn}.$$
$(9)$ If $r_0=8$, then $\Omega^{8}(Y_t^{(2)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{s+z_0(-4,\ell_0,n),s+z_0(-4,\ell_0,n)}=w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)}.$$
$(10)$ If $r_0=9$, then $\Omega^{9}(Y_t^{(2)})$ is described with
$(7s\times 7s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{5s+z_0(-5,\ell_0,n),5s+z_0(-5,\ell_0,n)}=w_{(j+m+1)n+g(j+s)+1\ra (j+m+2)n+g(j+s)+1}\otimes e_{jn+5}.$$
$(11)$ If $r_0=10$, then $\Omega^{10}(Y_t^{(2)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{3s+z_1(-5,\ell_0,n),3s+z_1(-5,\ell_0,n)}=w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)+1}\otimes e_{jn+g(j+s)+1}.$$
\medskip
$({\rm II})$ Represent an arbitrary $t_0\in\N$ in the form
$t_0=11\ell_0+r_0$, where $0\le r_0\le 10.$ Then
$\Omega^{t_0}(Y_t^{(2)})$ is a $\Omega^{r_0}(Y_t^{(2)})$, whose left
components twisted by $\sigma^{\ell_0}$, and coefficients multiplied
by $(-1)^{\ell_0}$.
\end{pr}
\begin{pr}[Translates for the case 3]\label{shifts_3}
$({\rm I})$ Let $r_0\in\N$, $r_0<11$. $r_0$-translates of the
elements $Y^{(3)}_t$ are described by the following way.
$(1)$ If $r_0=0$, then $\Omega^{0}(Y_t^{(3)})$ is described with
$(7s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{5s+(-3\ell_0)_s,6s+(-3\ell_0)_s}=\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes e_{jn+5}.$$
$(2)$ If $r_0=1$, then $\Omega^{1}(Y_t^{(3)})$ is described with
$(6s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s+(-3\ell_0)_s$, then $b_{ij}=0$.
If $j=s+(-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s+(-3\ell_0)_s<j<2s+(-3\ell_0)_s$, then $b_{ij}=0$.
If $j=2s+(-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s+(-3\ell_0)_s<j<3s+(-3\ell_0)_s$, then $b_{ij}=0$.
If $j=3s+(-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)+1},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s+(-3\ell_0)_s<j<4s+(-3\ell_0)_s$, then $b_{ij}=0$.
If $j=4s+(-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)+1},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s+(-3\ell_0)_s<j<5s+(-1-3\ell_0)_s$, then $b_{ij}=0$.
If $j=5s+(-1-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m+2)n}\otimes w_{jn+5\ra (j+1)n+5},\quad i=(j+1)_s+6s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s+(-1-3\ell_0)_s<j<6s$, then $b_{ij}=0$.
$(3)$ If $r_0=2$, then $\Omega^{2}(Y_t^{(3)})$ is described with
$(8s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<(-3\ell_0)_s$, then $b_{ij}=0$.
If $j=(-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
w_{(j+m-1)n+5\ra (j+m)n+g(j)+1}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $(-3\ell_0)_s<j<s+(-3\ell_0)_s$, then $b_{ij}=0$.
If $j=s+(-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
w_{(j+m-1)n+5\ra (j+m)n+g(j)+1}\otimes e_{jn},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s+(-3\ell_0)_s<j<6s+(-1-3\ell_0)_s$, then $b_{ij}=0$.
If $j=6s+(-1-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j)}\otimes w_{jn+5\ra (j+1)n+g(j+s)+1},\quad i=(j+1)_s+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s+(-1-3\ell_0)_s<j<7s+(-1-3\ell_0)_s$, then $b_{ij}=0$.
If $j=7s+(-1-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j)}\otimes w_{jn+5\ra (j+1)n+g(j+s)+1},\quad i=(j+1)_s+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s+(-1-3\ell_0)_s<j<8s$, then $b_{ij}=0$.
$(4)$ If $r_0=3$, then $\Omega^{3}(Y_t^{(3)})$ is described with
$(9s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<6s+(-1-3\ell_0)_s$, then $b_{ij}=0$.
If $j=6s+(-1-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j)+1\ra (j+1)n},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s+(-1-3\ell_0)_s<j<7s+(-1-3\ell_0)_s$, then $b_{ij}=0$.
If $j=7s+(-1-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j)+1\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s+(-1-3\ell_0)_s<j<8s+(-1-3\ell_0)_s$, then $b_{ij}=0$.
If $j=8s+(-1-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n+g(j)\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j-2s;\\
\kappa w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s+(-1-3\ell_0)_s<j<9s$, then $b_{ij}=0$.
$(5)$ If $r_0=4$, then $\Omega^{4}(Y_t^{(3)})$ is described with
$(8s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s+(-1-3\ell_0)_s$, then $b_{ij}=0$.
If $j=2s+(-1-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn+g(j)\ra (j+1)n},\quad i=(j+1)_s;\\
\kappa e_{(j+m+1)n}\otimes w_{jn+g(j)\ra (j+1)n},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s+(-1-3\ell_0)_s<j<3s+(-1-3\ell_0)_s$, then $b_{ij}=0$.
If $j=3s+(-1-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-\kappa e_{(j+m+1)n}\otimes w_{jn+g(j)\ra (j+1)n},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s+(-1-3\ell_0)_s<j<4s+(-1-3\ell_0)_s$, then $b_{ij}=0$.
If $j=4s+(-1-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-\kappa e_{(j+m)n+5}\otimes w_{jn+g(j)+1\ra (j+1)n},\quad i=(j+1)_s;\\
\kappa e_{(j+m)n+5}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s+(-1-3\ell_0)_s<j<5s+(-1-3\ell_0)_s$, then $b_{ij}=0$.
If $j=5s+(-1-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-\kappa e_{(j+m)n+5}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s+(-1-3\ell_0)_s<j<8s$, then $b_{ij}=0$.
$(6)$ If $r_0=5$, then $\Omega^{5}(Y_t^{(3)})$ is described with
$(9s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s+(-1-3\ell_0)_s$, then $b_{ij}=0$.
If $j=s+(-1-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-e_{(j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s+(-1-3\ell_0)_s<j<2s+(-1-3\ell_0)_s$, then $b_{ij}=0$.
If $j=2s+(-1-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-e_{(j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s+(-1-3\ell_0)_s<j<3s+(-1-3\ell_0)_s$, then $b_{ij}=0$.
If $j=3s+(-1-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+1)n\ra (j+m+1)n+g(j)+1}\otimes e_{jn+g(j+s)},\quad i=j-s;\\
w_{(j+m)n+5\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s+(-1-3\ell_0)_s<j<4s+(-1-3\ell_0)_s$, then $b_{ij}=0$.
If $j=4s+(-1-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+1)n\ra (j+m+1)n+g(j)+1}\otimes e_{jn+g(j+s)},\quad i=j-s;\\
w_{(j+m)n+5\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s+(-1-3\ell_0)_s<j<5s+(-1-3\ell_0)_s$, then $b_{ij}=0$.
If $j=5s+(-1-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s+(-1-3\ell_0)_s<j<6s+(-1-3\ell_0)_s$, then $b_{ij}=0$.
If $j=6s+(-1-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s+(-1-3\ell_0)_s<j<9s$, then $b_{ij}=0$.
$(7)$ If $r_0=6$, then $\Omega^{6}(Y_t^{(3)})$ is described with
$(8s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<(-1-3\ell_0)_s$, then $b_{ij}=0$.
If $j=(-1-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n\ra (j+m)n+g(j)+1}\otimes e_{jn},\quad i=j;\\
-w_{(j+m)n+g(j)\ra (j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $(-1-3\ell_0)_s<j<s+(-1-3\ell_0)_s$, then $b_{ij}=0$.
If $j=s+(-1-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n\ra (j+m)n+g(j)+1}\otimes e_{jn},\quad i=j-s;\\
-w_{(j+m)n+g(j)\ra (j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s+(-1-3\ell_0)_s<j<2s+(-1-3\ell_0)_s$, then $b_{ij}=0$.
If $j=2s+(-1-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s+(-1-3\ell_0)_s<j<3s+(-1-3\ell_0)_s$, then $b_{ij}=0$.
If $j=3s+(-1-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s+(-1-3\ell_0)_s<j<8s$, then $b_{ij}=0$.
$(8)$ If $r_0=7$, then $\Omega^{7}(Y_t^{(3)})$ is described with
$(6s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s+(-2-3\ell_0)_s$, then $b_{ij}=0$.
If $j=s+(-2-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s+(-2-3\ell_0)_s<j<2s+(-2-3\ell_0)_s$, then $b_{ij}=0$.
If $j=2s+(-2-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s+(-2-3\ell_0)_s<j<5s+(-2-3\ell_0)_s$, then $b_{ij}=0$.
If $j=5s+(-2-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n+g(j+s)\ra (j+m+2)n}\otimes e_{jn+5},\quad i=j+s;\\
-\kappa w_{(j+m+1)n+g(j)\ra (j+m+2)n}\otimes e_{jn+5},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s+(-2-3\ell_0)_s<j<6s$, then $b_{ij}=0$.
$(9)$ If $r_0=8$, then $\Omega^{8}(Y_t^{(3)})$ is described with
$(7s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<(-2-3\ell_0)_s$, then $b_{ij}=0$.
If $j=(-2-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m)n+5}\otimes w_{jn\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $(-2-3\ell_0)_s<j<s+(-2-3\ell_0)_s$, then $b_{ij}=0$.
If $j=s+(-2-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j+s)+1\ra (j+m+1)n}\otimes e_{jn+g(j+s)},\quad i=j;\\
-\kappa w_{(j+m)n+g(j)\ra (j+m+1)n}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s+(-2-3\ell_0)_s<j<2s+(-2-3\ell_0)_s$, then $b_{ij}=0$.
If $j=2s+(-2-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-\kappa e_{(j+m+1)n}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s+(-2-3\ell_0)_s<j<3s+(-2-3\ell_0)_s$, then $b_{ij}=0$.
If $j=3s+(-2-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s+(-2-3\ell_0)_s<j<4s+(-2-3\ell_0)_s$, then $b_{ij}=0$.
If $j=4s+(-2-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s+(-2-3\ell_0)_s<j<7s$, then $b_{ij}=0$.
$(10)$ If $r_0=9$, then $\Omega^{9}(Y_t^{(3)})$ is described with
$(6s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<(-2-3\ell_0)_s$, then $b_{ij}=0$.
If $j=(-2-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m+1)n}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $(-2-3\ell_0)_s<j<2s+(-2-3\ell_0)_s$, then $b_{ij}=0$.
If $j=2s+(-2-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s+(-2-3\ell_0)_s<j<6s$, then $b_{ij}=0$.
$(11)$ If $r_0=10$, then $\Omega^{10}(Y_t^{(3)})$ is described with
$(6s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<(-2-3\ell_0)_s$, then $b_{ij}=0$.
If $j=(-2-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n\ra (j+m+1)n}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $(-2-3\ell_0)_s<j<s+(-3-3\ell_0)_s$, then $b_{ij}=0$.
If $j=s+(-3-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j)}\otimes w_{jn+g(j+s)\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s+(-3-3\ell_0)_s<j<2s+(-3-3\ell_0)_s$, then $b_{ij}=0$.
If $j=2s+(-3-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j)}\otimes w_{jn+g(j+s)\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s+(-3-3\ell_0)_s<j<3s+(-3-3\ell_0)_s$, then $b_{ij}=0$.
If $j=3s+(-3-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+1)n+g(j)\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s;\\
e_{(j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)+1},\quad i=(j+1)_s+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s+(-3-3\ell_0)_s<j<4s+(-3-3\ell_0)_s$, then $b_{ij}=0$.
If $j=4s+(-3-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+1)n+g(j)\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+2s;\\
e_{(j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)+1},\quad i=(j+1)_s+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s+(-3-3\ell_0)_s<j<5s+(-3-3\ell_0)_s$, then $b_{ij}=0$.
If $j=5s+(-3-3\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n+g(j)\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s;\\
\kappa w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j)},\quad i=(j+1)_s+2s;\\
\kappa w_{(j+m+1)n+g(j)+1\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j+s)+1},\quad i=(j+1)_s+3s;\\
-\kappa w_{(j+m+1)n+g(j+s)+1\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j)+1},\quad i=(j+1)_s+4s;\\
\kappa e_{(j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+5},\quad i=(j+1)_s+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s+(-3-3\ell_0)_s<j<6s$, then $b_{ij}=0$.
\medskip
$({\rm II})$ Represent an arbitrary $t_0\in\N$ in the form
$t_0=11\ell_0+r_0$, where $0\le r_0\le 10.$ Then
$\Omega^{t_0}(Y_t^{(3)})$ is a $\Omega^{r_0}(Y_t^{(3)})$, whose left
components twisted by $\sigma^{\ell_0}$, and coefficients multiplied
by $(-1)^{\ell_0}$.
\end{pr}
\begin{pr}[Translates for the case 4]
$({\rm I})$ Let $r_0\in\N$, $r_0<11$. $r_0$-translates of the
elements $Y^{(4)}_t$ are described by the following way.
$(1)$ If $r_0=0$, then $\Omega^{0}(Y_t^{(4)})$ is described with
$(7s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n\ra (j+m)n+g(j+s)}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<5s$, then $b_{ij}=0$.
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $b_{ij}=0$.
$(2)$ If $r_0=1$, then $\Omega^{1}(Y_t^{(4)})$ is described with
$(6s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes e_{jn},\quad i=j+s;\\
\kappa w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n+g(j)\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $b_{ij}=0$.
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j+s)+1},\quad i=j+s;\\
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n\ra (j+m+2)n}\otimes e_{jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(3)$ If $r_0=2$, then $\Omega^{2}(Y_t^{(4)})$ is described with
$(8s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-e_{(j+m)n+g(j+s)+1}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $b_{ij}=0$.
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $b_{ij}=0$.
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j+s)}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
-e_{(j+m+1)n+g(j+s)}\otimes w_{jn+5\ra (j+1)n+g(j)+1},\quad i=(j+1)_s+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $b_{ij}=0$.
$(4)$ If $r_0=3$, then $\Omega^{3}(Y_t^{(4)})$ is described with
$(9s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $b_{ij}=0$.
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<3s$, then $b_{ij}=0$.
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j)+1},\quad i=j;\\
e_{(j+m+1)n+g(j+s)}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<7s$, then $b_{ij}=0$.
If $7s\le j<8s$, then $$b_{ij}=
\begin{cases}
-e_{(j+m+1)n+g(j)+1}\otimes w_{jn+g(j)+1\ra (j+1)n},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s\le j<9s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n+g(j+s)+1\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(5)$ If $r_0=4$, then $\Omega^{4}(Y_t^{(4)})$ is described with
$(8s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m-1)n+5\ra (j+m)n+g(j+s)}\otimes e_{jn},\quad i=j;\\
e_{(j+m)n+g(j+s)}\otimes w_{jn\ra jn+g(j)},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $b_{ij}=0$.
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes e_{jn+g(j)},\quad i=j;\\
\kappa w_{(j+m)n+g(j+s)+1\ra (j+m+1)n}\otimes w_{jn+g(j)\ra jn+g(j)+1},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $b_{ij}=0$.
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes e_{jn+g(j)+1},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $b_{ij}=0$.
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+5},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $b_{ij}=0$.
$(6)$ If $r_0=5$, then $\Omega^{5}(Y_t^{(4)})$ is described with
$(9s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $b_{ij}=0$.
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j)}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s+s;\\
w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $b_{ij}=0$.
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j)+1}\otimes e_{jn+g(j+s)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $b_{ij}=0$.
If $7s\le j<8s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n+g(j)+1\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s\le j<9s$, then $b_{ij}=0$.
$(7)$ If $r_0=6$, then $\Omega^{6}(Y_t^{(4)})$ is described with
$(8s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j+s)+1}\otimes w_{jn\ra jn+g(j)},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $b_{ij}=0$.
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes e_{jn+g(j)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $b_{ij}=0$.
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
-e_{(j+m+1)n+g(j+s)}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+2s;\\
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)}\otimes e_{jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $b_{ij}=0$.
$(8)$ If $r_0=7$, then $\Omega^{7}(Y_t^{(4)})$ is described with
$(6s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $b_{ij}=0$.
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s+s;\\
-w_{(j+m)n+5\ra (j+m+1)n+g(j)+1}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $b_{ij}=0$.
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j+s)+1},\quad i=j+s;\\
-e_{(j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n+g(j)+1\ra (j+m+2)n}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(9)$ If $r_0=8$, then $\Omega^{8}(Y_t^{(4)})$ is described with
$(7s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m-1)n+5\ra (j+m)n+5}\otimes e_{jn},\quad i=j;\\
\kappa w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $b_{ij}=0$.
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j+s)+1\ra (j+m+1)n}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $b_{ij}=0$.
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j)+1}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
-e_{(j+m+1)n+g(j)+1}\otimes w_{jn+5\ra (j+1)n+g(j)},\quad i=(j+1)_s+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $b_{ij}=0$.
$(10)$ If $r_0=9$, then $\Omega^{9}(Y_t^{(4)})$ is described with
$(6s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $b_{ij}=0$.
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $b_{ij}=0$.
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
-e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $b_{ij}=0$.
$(11)$ If $r_0=10$, then $\Omega^{10}(Y_t^{(4)})$ is described with
$(6s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-\kappa e_{(j+m+1)n}\otimes w_{jn\ra (j+1)n},\quad i=(j+1)_s;\\
\kappa w_{(j+m)n+g(j)\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+2s;\\
-\kappa w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+4s;\\
\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn\ra jn+5},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $b_{ij}=0$.
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $b_{ij}=0$.
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n},\quad i=(j+1)_s;\\
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j)},\quad i=(j+1)_s+2s;\\
-\kappa w_{(j+m+1)n+g(j+s)+1\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j)+1},\quad i=(j+1)_s+4s;\\
\kappa e_{(j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+5},\quad i=(j+1)_s+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
\medskip
$({\rm II})$ Represent an arbitrary $t_0\in\N$ in the form
$t_0=11\ell_0+r_0$, where $0\le r_0\le 10.$ Then
$\Omega^{t_0}(Y_t^{(4)})$ is a $\Omega^{r_0}(Y_t^{(4)})$, whose left
components twisted by $\sigma^{\ell_0}$.
\end{pr}
\begin{pr}[Translates for the case 6]
$({\rm I})$ Let $r_0\in\N$, $r_0<11$. $r_0$-translates of the
elements $Y^{(6)}_t$ are described by the following way.
$(1)$ If $r_0=0$, then $\Omega^{0}(Y_t^{(6)})$ is described with
$(8s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n\ra (j+m)n+g(j)+1}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n\ra (j+m)n+g(j)+1}\otimes e_{jn},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $b_{ij}=0$.
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $b_{ij}=0$.
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes e_{jn+5},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $b_{ij}=0$.
$(2)$ If $r_0=1$, then $\Omega^{1}(Y_t^{(6)})$ is described with
$(9s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $b_{ij}=0$.
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes e_{jn},\quad i=j-s;\\
w_{(j+m)n+g(j+s)+1\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<3s$, then $b_{ij}=0$.
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j)\ra jn+g(j)+1},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $b_{ij}=0$.
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j)+1\ra (j+1)n},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s\le j<9s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j-2s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(3)$ If $r_0=2$, then $\Omega^{2}(Y_t^{(6)})$ is described with
$(8s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
w_{(j+m-1)n+5\ra (j+m)n+g(j)}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $b_{ij}=0$.
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn+g(j)\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes w_{jn+g(j)\ra jn+g(j)+1},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $b_{ij}=0$.
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n+g(j)\ra (j+m+1)n+g(j)+1}\otimes w_{jn+5\ra (j+1)n+g(j+s)+1},\quad i=(j+1)_s+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $b_{ij}=0$.
$(4)$ If $r_0=3$, then $\Omega^{3}(Y_t^{(6)})$ is described with
$(9s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes e_{jn},\quad i=j;\\
w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j)},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $b_{ij}=0$.
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<7s$, then $b_{ij}=0$.
If $7s\le j<8s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n+g(j+s)+1\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
e_{(j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s\le j<9s$, then $b_{ij}=0$.
$(5)$ If $r_0=4$, then $\Omega^{4}(Y_t^{(6)})$ is described with
$(8s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
w_{(j+m-1)n+5\ra (j+m)n+g(j)+1}\otimes e_{jn},\quad i=j;\\
w_{(j+m)n+g(j)\ra (j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+3s;\\
e_{(j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+7s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j)\ra (j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+s;\\
e_{(j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+5}\otimes w_{jn+g(j)\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<8s$, then $b_{ij}=0$.
$(6)$ If $r_0=5$, then $\Omega^{5}(Y_t^{(6)})$ is described with
$(6s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes e_{jn},\quad i=j;\\
w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes e_{jn},\quad i=j+s;\\
e_{(j+m)n+5}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+4s;\\
e_{(j+m)n+5}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<5s$, then $b_{ij}=0$.
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+2)n}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(7)$ If $r_0=6$, then $\Omega^{6}(Y_t^{(6)})$ is described with
$(7s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n\ra (j+m)n+5}\otimes e_{jn},\quad i=j;\\
w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<3s$, then $b_{ij}=0$.
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j+s)+1\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $b_{ij}=0$.
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(8)$ If $r_0=7$, then $\Omega^{7}(Y_t^{(6)})$ is described with
$(6s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes e_{jn},\quad i=j;\\
w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+3s;\\
e_{(j+m+1)n}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+4s;\\
e_{(j+m+1)n}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<6s$, then $b_{ij}=0$.
$(9)$ If $r_0=8$, then $\Omega^{8}(Y_t^{(6)})$ is described with
$(6s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
w_{(j+m)n+g(j)\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $b_{ij}=0$.
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n+g(j)\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)+1},\quad i=(j+1)_s+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n},\quad i=(j+1)_s;\\
w_{(j+m+1)n+g(j)\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)+1},\quad i=(j+1)_s+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n+g(j)+1\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j)},\quad i=(j+1)_s+2s;\\
w_{(j+m+1)n+g(j)\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j+s)+1},\quad i=(j+1)_s+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(10)$ If $r_0=9$, then $\Omega^{9}(Y_t^{(6)})$ is described with
$(7s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $b_{ij}=0$.
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j+s)}\otimes e_{jn},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<5s$, then $b_{ij}=0$.
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+5}\otimes w_{jn+g(j)+1\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n+5\ra (j+m+2)n}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(11)$ If $r_0=10$, then $\Omega^{10}(Y_t^{(6)})$ is described with
$(6s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n\ra (j+m)n+5}\otimes e_{jn},\quad i=j;\\
w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+2s;\\
w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+3s;\\
w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+4s;\\
e_{(j+m)n+5}\otimes w_{jn\ra jn+5},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
w_{(j+m+1)n+g(j)\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
w_{(j+m+1)n+g(j)\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<6s$, then $b_{ij}=0$.
\medskip
$({\rm II})$ Represent an arbitrary $t_0\in\N$ in the form
$t_0=11\ell_0+r_0$, where $0\le r_0\le 10.$ Then
$\Omega^{t_0}(Y_t^{(6)})$ is a $\Omega^{r_0}(Y_t^{(6)})$, whose left
components twisted by $\sigma^{\ell_0}$.
\end{pr}
\begin{pr}[Translates for the case 7]
$({\rm I})$ Let $r_0\in\N$, $r_0<11$. $r_0$-translates of the
elements $Y^{(7)}_t$ are described by the following way.
$(1)$ If $r_0=0$, then $\Omega^{0}(Y_t^{(7)})$ is described with
$(8s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n\ra (j+m)n+g(j+s)+1}\otimes e_{jn},\quad i=(j)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<8s$, then $b_{ij}=0$.
$(2)$ If $r_0=1$, then $\Omega^{1}(Y_t^{(7)})$ is described with
$(9s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes e_{jn},\quad i=j;\\
\kappa w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes e_{jn},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j)\ra (j+m+1)n}\otimes e_{jn},\quad i=j;\\
\kappa w_{(j+m)n+g(j+s)+1\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+s;\\
-\kappa w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j)},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
f_1(j,3s)w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $b_{ij}=0$.
If $6s\le j<8s$, then $$b_{ij}=
\begin{cases}
f_1(j,7s)w_{(j+m)n+5\ra (j+m+1)n+g(j)+1}\otimes e_{jn+g(j)+1},\quad i=j-2s;\\
f_1(j,7s)w_{(j+m+1)n\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=j-s(j_2-6);\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s\le j<9s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n+g(j)\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(3)$ If $r_0=2$, then $\Omega^{2}(Y_t^{(7)})$ is described with
$(8s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $b_{ij}=0$.
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<8s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s(8-j_2);\\
0,\quad\text{otherwise.}\end{cases}$$
$(4)$ If $r_0=3$, then $\Omega^{3}(Y_t^{(7)})$ is described with
$(9s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes e_{jn},\quad i=j;\\
-\kappa w_{(j+m)n+g(j+s)+1\ra (j+m+1)n}\otimes e_{jn},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $b_{ij}=0$.
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s+s(j_2-3);\\
f_1(j,4s)w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<7s$, then $b_{ij}=0$.
If $7s\le j<8s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n+g(j+s)+1\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
\kappa w_{(j+m+1)n+g(j)+1\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s\le j<9s$, then $b_{ij}=0$.
$(5)$ If $r_0=4$, then $\Omega^{4}(Y_t^{(7)})$ is described with
$(8s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $$b_{ij}=
\begin{cases}
f_1(j,s)w_{(j+m-1)n+5\ra (j+m)n+g(j+s)+1}\otimes e_{jn},\quad i=j-sj_2;\\
-f_1(j,s)w_{(j+m)n+g(j+s)\ra (j+m)n+g(j+s)+1}\otimes w_{jn\ra jn+g(j)},\quad i=j+2s;\\
f_1(j,s)e_{(j+m)n+g(j+s)+1}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+6s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes e_{jn+g(j)},\quad i=j;\\
-\kappa w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes w_{jn+g(j)\ra jn+g(j)+1},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<8s$, then $b_{ij}=0$.
$(6)$ If $r_0=5$, then $\Omega^{5}(Y_t^{(7)})$ is described with
$(6s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $b_{ij}=0$.
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
-f_1(j,2s)w_{(j+m+1)n+g(j)\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\\\quad\quad\quad i=(j+1)_s+s(2-j_2);\\
-f_1(j,2s)w_{(j+m+1)n\ra (j+m+1)n+g(j)+1}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $b_{ij}=0$.
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m+2)n}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+2s;\\
\kappa e_{(j+m+2)n}\otimes w_{jn+5\ra (j+1)n+g(j)},\quad i=(j+1)_s+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(7)$ If $r_0=6$, then $\Omega^{6}(Y_t^{(7)})$ is described with
$(7s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
-\kappa w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m+1)n}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
\kappa w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
-f_1(j,4s)e_{(j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s(j_2-2);\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<7s$, then $$b_{ij}=
\begin{cases}
f_1(j,6s)w_{(j+m+1)n+g(j)\ra (j+m+1)n+g(j)+1}\otimes w_{jn+5\ra (j+1)n+g(j)},\quad i=(j+1)_s+s(7-j_2);\\
0,\quad\text{otherwise.}\end{cases}$$
$(8)$ If $r_0=7$, then $\Omega^{7}(Y_t^{(7)})$ is described with
$(6s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $b_{ij}=0$.
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
f_1(j,2s)w_{(j+m)n+5\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
f_1(j,2s)w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+3s;\\
-f_1(j,2s)e_{(j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<6s$, then $b_{ij}=0$.
$(9)$ If $r_0=8$, then $\Omega^{8}(Y_t^{(7)})$ is described with
$(6s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+3s;\\
-\kappa w_{(j+m)n+g(j)\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
-f_1(j,4s)e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s(j_2-2);\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n+g(j+s)+1\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s;\\
-\kappa w_{(j+m+1)n+g(j)+1\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j)},\quad i=(j+1)_s+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(10)$ If $r_0=9$, then $\Omega^{9}(Y_t^{(7)})$ is described with
$(7s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes e_{jn},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j)+1}\otimes w_{jn+g(j)\ra jn+5},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j)+1}\otimes e_{jn+g(j)},\quad i=j-s;\\
-e_{(j+m+1)n+g(j)+1}\otimes w_{jn+g(j)\ra jn+5},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $b_{ij}=0$.
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n+5\ra (j+m+2)n}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(11)$ If $r_0=10$, then $\Omega^{10}(Y_t^{(7)})$ is described with
$(6s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n\ra (j+m)n+5}\otimes e_{jn},\quad i=j;\\
\kappa w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
\kappa w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+2s;\\
-\kappa w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+3s;\\
-\kappa w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+1)n\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $b_{ij}=0$.
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j+s)+1\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n+g(j+s)\ra (j+m+2)n}\otimes w_{jn+5\ra (j+1)n+g(j)},\quad i=(j+1)_s+2s;\\
-\kappa w_{(j+m+1)n+g(j+s)+1\ra (j+m+2)n}\otimes w_{jn+5\ra (j+1)n+g(j)+1},\quad i=(j+1)_s+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
\medskip
$({\rm II})$ Represent an arbitrary $t_0\in\N$ in the form
$t_0=11\ell_0+r_0$, where $0\le r_0\le 10.$ Then
$\Omega^{t_0}(Y_t^{(7)})$ is a $\Omega^{r_0}(Y_t^{(7)})$, whose left
components twisted by $\sigma^{\ell_0}$.
\end{pr}
\begin{pr}[Translates for the case 8]
$({\rm I})$ Let $r_0\in\N$, $r_0<11$. $r_0$-translates of the
elements $Y^{(8)}_t$ are described by the following way.
$(1)$ If $r_0=0$, then $\Omega^{0}(Y_t^{(8)})$ is described with
$(9s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{3s+z_0(0, \ell_0, 2),6s+z_0(0, \ell_0, 2)}=w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)+1}\otimes e_{jn+g(j)+1}.$$
$(2)$ If $r_0=1$, then $\Omega^{1}(Y_t^{(8)})$ is described with
$(8s\times 7s)$-matrix with one nonzero element that is of the
following form{\rm:}
\begin{multline*}
b_{(6s+z_1(-1, \ell_0,
2)+1)_s+2s+(\ell_0s)_{2s},6s+z_1(-1, \ell_0, 2)}=\\
-\kappa_0 w_{(j+m+1)n+g(j+s)+1\ra (j+m+2)n+g(j+s)+1}\otimes w_{jn+5\ra (j+1)n+g(j+s)}.
\end{multline*}
$(3)$ If $r_0=2$, then $\Omega^{2}(Y_t^{(8)})$ is described with
$(9s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{(3s+z_0(-1, \ell_0, 2)+1)_s+s+(\ell_0s)_{2s},3s+z_0(-1, \ell_0, 2)}=e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra (j+1)n+g(j+s)}.$$
$(4)$ If $r_0=3$, then $\Omega^{3}(Y_t^{(8)})$ is described with
$(8s\times 8s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{(2s+z_0(-1, \ell_0, 2)+1)_s+2s+(\ell_0s)_{2s},2s+z_0(-1, \ell_0, 2)}=\kappa\kappa_0 e_{(j+m+1)n+5}\otimes w_{jn+g(j)\ra (j+1)n+g(j)}.$$
$(5)$ If $r_0=4$, then $\Omega^{4}(Y_t^{(8)})$ is described with
$(6s\times 9s)$-matrix with one nonzero element that is of the
following form{\rm:}
\begin{multline*}
b_{(s+z_0(-1, \ell_0, 2)+1)_s+2s+(\ell_0s)_{2s},s+z_0(-1, \ell_0, 2)}=\\
-w_{(j+m+1)n+g(j)\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra (j+1)n+g(j+s)}.
\end{multline*}
$(6)$ If $r_0=5$, then $\Omega^{5}(Y_t^{(8)})$ is described with
$(7s\times 8s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{(s+z_0(-1, \ell_0, 2)+1)_s+2s+(\ell_0s)_{2s},s+z_0(-1, \ell_0, 2)}=\kappa\kappa_0 e_{(j+m+2)n}\otimes w_{jn+g(j+s)\ra (j+1)n+g(j+s)}.$$
$(7)$ If $r_0=6$, then $\Omega^{6}(Y_t^{(8)})$ is described with
$(6s\times 9s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{(s+z_0(-1, \ell_0, 2)+1)_s+s+(\ell_0s)_{2s},s+z_0(-1, \ell_0, 2)}=\\
e_{(j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra (j+1)n+g(j+s)}.$$
$(8)$ If $r_0=7$, then $\Omega^{7}(Y_t^{(8)})$ is described with
$(6s\times 8s)$-matrix with one nonzero element that is of the
following form{\rm:}
\begin{multline*}b_{(s+z_0(-1, \ell_0, 2)+1)_s+2s+(\ell_0s)_{2s},s+z_0(-1, \ell_0, 2)}=\\
-w_{(j+m+1)n+5\ra (j+m+2)n+g(j+s)}\otimes w_{jn+g(j+s)\ra (j+1)n+g(j+s)}.
\end{multline*}
$(9)$ If $r_0=8$, then $\Omega^{8}(Y_t^{(8)})$ is described with
$(7s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{(2s+z_0(-1, \ell_0, 2)+1)_s+s+(\ell_0s)_{2s},2s+z_0(-1, \ell_0, 2)}=-e_{(j+m+1)n+g(j)+1}\otimes w_{jn+g(j)\ra (j+1)n+g(j)}.$$
$(10)$ If $r_0=9$, then $\Omega^{9}(Y_t^{(8)})$ is described with
$(6s\times 7s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{5s+z_0(-1, \ell_0, 2),s+z_0(-1, \ell_0, 2)}=-w_{(j+m+1)n+g(j+s)+1\ra (j+m+2)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra jn+5}.$$
$(11)$ If $r_0=10$, then $\Omega^{10}(Y_t^{(8)})$ is described with
$(8s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{(s+z_0(-1, \ell_0, 2))_{2s}+3s,z_0(-1, \ell_0, 2)}=\kappa_0 w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)+1}\otimes w_{jn\ra jn+g(j+s)+1}.$$
\medskip
$({\rm II})$ Represent an arbitrary $t_0\in\N$ in the form
$t_0=11\ell_0+r_0$, where $0\le r_0\le 10.$ Then
$\Omega^{t_0}(Y_t^{(8)})$ is a $\Omega^{r_0}(Y_t^{(8)})$, whose left
components twisted by $\sigma^{\ell_0}$.
\end{pr}
\begin{pr}[Translates for the case 9]
$({\rm I})$ Let $r_0\in\N$, $r_0<11$. $r_0$-translates of the
elements $Y^{(9)}_t$ are described by the following way.
$(1)$ If $r_0=0$, then $\Omega^{0}(Y_t^{(9)})$ is described with
$(9s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $b_{ij}=0$.
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m)n}\otimes e_{jn},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j)}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $b_{ij}=0$.
If $6s\le j<8s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j)+1}\otimes e_{jn+g(j)+1},\quad i=j-3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s\le j<9s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m)n+5}\otimes e_{jn+5},\quad i=j-3s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(2)$ If $r_0=1$, then $\Omega^{1}(Y_t^{(9)})$ is described with
$(8s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j+s)}\otimes e_{jn},\quad i=(j)_s+f_0(j,s)s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
-\kappa f_1(j,3s)w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes e_{jn+g(j)},\quad i=j;\\
\kappa f_1(j,3s)w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn+g(j)\ra jn+g(j)+1},\quad i=j+2s;\\
\kappa f_1(j,3s)e_{(j+m+1)n}\otimes w_{jn+g(j)\ra jn+5},\quad i=(j)_s+6s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa f_1(j,5s)e_{(j+m)n+5}\otimes e_{jn+g(j)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<8s$, then $$b_{ij}=
\begin{cases}
-f_1(j,7s)w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+5\ra (j+1)n},\\\quad\quad\quad i=(j+1)_s+f_0(j,7s)s;\\
f_1(j,7s)e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+(f_0(j,7s)+2)s;\\
-f_1(j,7s)w_{(j+m+1)n\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+5},\quad i=(j)_s+6s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(3)$ If $r_0=2$, then $\Omega^{2}(Y_t^{(9)})$ is described with
$(9s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m-1)n+5\ra (j+m)n}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
-e_{(j+m)n+g(j)}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j+s)+1}\otimes e_{jn+g(j+s)},\quad i=j-2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<7s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+g(j)\ra (j+m)n+g(j)+1}\otimes e_{jn+g(j+s)+1},\quad i=j-2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $$b_{ij}=
\begin{cases}
-\kappa e_{(j+m)n+5}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s\le j<9s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
\kappa e_{(j+m+1)n}\otimes e_{jn+5},\quad i=j-3s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(4)$ If $r_0=3$, then $\Omega^{3}(Y_t^{(9)})$ is described with
$(8s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $$b_{ij}=
\begin{cases}
-f_1(j,s)e_{(j+m)n+g(j+s)+1}\otimes e_{jn},\quad i=(j)_s+f_0(j,s)s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
\kappa f_1(j,3s)e_{(j+m)n+5}\otimes e_{jn+g(j)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa f_1(j,5s)e_{(j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<8s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j+s)}\otimes e_{jn+5},\quad i=(j)_s+(f_0(j,7s)+6)s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(5)$ If $r_0=4$, then $\Omega^{4}(Y_t^{(9)})$ is described with
$(6s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m-1)n+5}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+g(j)\ra (j+m)n+g(j)+1}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
-e_{(j+m)n+g(j)+1}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j+s)}\otimes e_{jn+g(j+s)+1},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
-\kappa e_{(j+m+1)n}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s+s;\\
-\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes e_{jn+5},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(6)$ If $r_0=5$, then $\Omega^{5}(Y_t^{(9)})$ is described with
$(7s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes e_{jn},\quad i=j;\\
-\kappa w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes e_{jn},\quad i=j+s;\\
-\kappa e_{(j+m)n+5}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+4s;\\
-\kappa e_{(j+m)n+5}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
\kappa f_1(j,2s)e_{(j+m+1)n}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
\kappa f_1(j,2s)w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
-e_{(j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)+1\ra (j+1)n},\quad i=(j+1)_s+(1-f_0(j,4s))s;\\
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j+s)+1},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<7s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+1)n+g(j)\ra (j+m+1)n+g(j)+1}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s+f_0(j,6s)s;\\
-f_1(j,6s)e_{(j+m+1)n+g(j)+1}\otimes e_{jn+5},\quad i=(j)_s+(6+f_0(j,6s))s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(7)$ If $r_0=6$, then $\Omega^{6}(Y_t^{(9)})$ is described with
$(6s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m)n}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j+s)}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j+s)+1}\otimes e_{jn+g(j+s)+1},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m)n+5}\otimes e_{jn+5},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(8)$ If $r_0=7$, then $\Omega^{7}(Y_t^{(9)})$ is described with
$(6s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j+s)+1\ra (j+m+1)n}\otimes e_{jn},\quad i=j+s;\\
\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j)},\quad i=j+2s;\\
-\kappa e_{(j+m+1)n}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+4s;\\
-\kappa e_{(j+m+1)n}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+3s;\\
-e_{(j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
-e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n},\quad i=(j+1)_s+(1-f_0(j,4s))s;\\
-w_{(j+m+1)n\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)+1},\quad i=j+s;\\
w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n+g(j)+1\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s+s;\\
-\kappa e_{(j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+2s;\\
-\kappa w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j+s;\\
-\kappa w_{(j+m+1)n+g(j)\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(9)$ If $r_0=8$, then $\Omega^{8}(Y_t^{(9)})$ is described with
$(7s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j)}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m-1)n+5\ra (j+m)n+g(j)}\otimes e_{jn},\quad i=j-s;\\
e_{(j+m)n+g(j)}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
-e_{(j+m)n+g(j)+1}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m)n+5}\otimes w_{jn+g(j)+1\ra (j+1)n},\quad i=(j+1)_s;\\
\kappa w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
-\kappa e_{(j+m+1)n}\otimes e_{jn+5},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(10)$ If $r_0=9$, then $\Omega^{9}(Y_t^{(9)})$ is described with
$(6s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-\kappa e_{(j+m)n+5}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+1)n\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)},\quad i=j;\\
-e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
-e_{(j+m+1)n+g(j)}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
-\kappa e_{(j+m+2)n}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s;\\
-\kappa w_{(j+m+1)n+g(j+s)+1\ra (j+m+2)n}\otimes e_{jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
$(11)$ If $r_0=10$, then $\Omega^{10}(Y_t^{(9)})$ is described with
$(8s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n\ra (j+m)n+g(j)+1}\otimes e_{jn},\quad i=(j)_s;\\
f_1(j,s)w_{(j+m)n+g(j)\ra (j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j+s)},\quad i=(j)_s+(1+f_0(j,s))s;\\
f_1(j,s)e_{(j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=(j)_s+(3+f_0(j,s))s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes e_{jn+g(j)},\quad i=j-s;\\
\kappa w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes w_{jn+g(j)\ra jn+g(j)+1},\quad i=j+s;\\
\kappa f_1(j,3s)e_{(j+m)n+5}\otimes w_{jn+g(j)\ra jn+5},\quad i=(j)_s+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m+1)n}\otimes w_{jn+g(j)+1\ra (j+1)n},\quad i=(j+1)_s;\\
\kappa w_{(j+m)n+g(j+s)+1\ra (j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j+s)+1\ra (j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
e_{(j+m+1)n+g(j)}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+2s;\\
w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes e_{jn+5},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j)}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
\medskip
$({\rm II})$ Represent an arbitrary $t_0\in\N$ in the form
$t_0=11\ell_0+r_0$, where $0\le r_0\le 10.$ Then
$\Omega^{t_0}(Y_t^{(9)})$ is a $\Omega^{r_0}(Y_t^{(9)})$, whose left
components twisted by $\sigma^{\ell_0}$, and coefficients multiplied
by $(-1)^{\ell_0}$.
\end{pr}
\begin{pr}[Translates for the case 11]
$({\rm I})$ Let $r_0\in\N$, $r_0<11$. $r_0$-translates of the
elements $Y^{(11)}_t$ are described by the following way.
$(1)$ If $r_0=0$, then $\Omega^{0}(Y_t^{(11)})$ is described with
$(8s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n\ra (j+m)n+g(j)}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $b_{ij}=0$.
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j)\ra (j+m+1)n}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $b_{ij}=0$.
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $b_{ij}=0$.
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j)+1}\otimes e_{jn+5},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $b_{ij}=0$.
$(2)$ If $r_0=1$, then $\Omega^{1}(Y_t^{(11)})$ is described with
$(9s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j)\ra (j+m+1)n}\otimes e_{jn},\quad i=j;\\
-\kappa w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j)},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
w_{(j+m)n+5\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+3s;\\
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $b_{ij}=0$.
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n+g(j)\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
-w_{(j+m)n+5\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $b_{ij}=0$.
If $7s\le j<8s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
-\kappa w_{(j+m+1)n+g(j+s)+1\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+2s;\\
\kappa w_{(j+m+1)n\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s\le j<9s$, then $b_{ij}=0$.
$(3)$ If $r_0=2$, then $\Omega^{2}(Y_t^{(11)})$ is described with
$(8s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m-1)n+5\ra (j+m)n+g(j)+1}\otimes e_{jn},\quad i=j;\\
-e_{(j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $b_{ij}=0$.
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes w_{jn+g(j)\ra jn+g(j)+1},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $b_{ij}=0$.
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes e_{jn+5},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $b_{ij}=0$.
$(4)$ If $r_0=3$, then $\Omega^{3}(Y_t^{(11)})$ is described with
$(6s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
-e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $b_{ij}=0$.
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)+1},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n+g(j+s)+1\ra (j+m+2)n}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
\kappa w_{(j+m+1)n+g(j+s)\ra (j+m+2)n}\otimes e_{jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(5)$ If $r_0=4$, then $\Omega^{4}(Y_t^{(11)})$ is described with
$(7s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m-1)n+5\ra (j+m)n+5}\otimes e_{jn},\quad i=j;\\
-\kappa w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+3s;\\
-\kappa w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+4s;\\
-\kappa w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+7s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j)\ra (j+m+1)n}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
-\kappa w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $b_{ij}=0$.
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j+s)+1\ra (j+1)n},\quad i=(j+1)_s+s;\\
-w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)+1},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+5\ra (j+1)n+g(j+s)+1},\quad i=(j+1)_s+4s;\\
-e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+5\ra (j+1)n+g(j)+1},\quad i=(j+1)_s+7s;\\
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+5},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $b_{ij}=0$.
$(6)$ If $r_0=5$, then $\Omega^{5}(Y_t^{(11)})$ is described with
$(6s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $b_{ij}=0$.
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $b_{ij}=0$.
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n+g(j)\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n},\quad i=(j+1)_s;\\
w_{(j+m)n+5\ra (j+m+1)n+g(j)+1}\otimes e_{jn+g(j+s)+1},\quad i=j+s;\\
e_{(j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n+g(j+s)+1\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(7)$ If $r_0=6$, then $\Omega^{6}(Y_t^{(11)})$ is described with
$(6s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+4s;\\
-\kappa w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $b_{ij}=0$.
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $b_{ij}=0$.
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+1)n\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n},\quad i=(j+1)_s;\\
-w_{(j+m)n+5\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n+g(j)\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j)},\quad i=(j+1)_s+2s;\\
-\kappa w_{(j+m+1)n+g(j+s)+1\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j)},\quad i=(j+1)_s+4s;\\
-\kappa w_{(j+m+1)n\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(8)$ If $r_0=7$, then $\Omega^{7}(Y_t^{(11)})$ is described with
$(7s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $b_{ij}=0$.
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)}\otimes w_{jn\ra jn+g(j)},\quad i=j+2s;\\
-w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+3s;\\
-w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<3s$, then $b_{ij}=0$.
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n\ra (j+m+1)n+5}\otimes e_{jn+g(j)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n+g(j+s)+1\ra (j+m+1)n+5}\otimes w_{jn+g(j)+1\ra (j+1)n},\quad i=(j+1)_s;\\
-\kappa w_{(j+m+1)n\ra (j+m+1)n+5}\otimes e_{jn+g(j)+1},\quad i=j;\\
-\kappa w_{(j+m+1)n+g(j)\ra (j+m+1)n+5}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n+g(j)+1\ra (j+m+2)n}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
\kappa w_{(j+m+1)n+g(j)\ra (j+m+2)n}\otimes e_{jn+5},\quad i=j;\\
-\kappa w_{(j+m+1)n+g(j+s)\ra (j+m+2)n}\otimes e_{jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(9)$ If $r_0=8$, then $\Omega^{8}(Y_t^{(11)})$ is described with
$(6s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m)n+5}\otimes w_{jn\ra (j+1)n},\quad i=(j+1)_s;\\
-\kappa w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $b_{ij}=0$.
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)+1\ra (j+1)n},\quad i=(j+1)_s;\\
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $b_{ij}=0$.
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n+g(j+s)+1\ra (j+m+2)n}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s;\\
-\kappa w_{(j+m+1)n+g(j)\ra (j+m+2)n}\otimes w_{jn+5\ra (j+1)n+g(j+s)+1},\quad i=(j+1)_s+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(10)$ If $r_0=9$, then $\Omega^{9}(Y_t^{(11)})$ is described with
$(8s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n\ra (j+m+1)n+5}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
-\kappa e_{(j+m+1)n+5}\otimes w_{jn+g(j)\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n+5\ra (j+m+2)n}\otimes w_{jn+g(j)+1\ra (j+1)n},\quad i=(j+1)_s;\\
\kappa w_{(j+m+1)n+g(j+s)\ra (j+m+2)n}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $b_{ij}=0$.
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n+5\ra (j+m+2)n+g(j+s)}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+2)n\ra (j+m+2)n+g(j+s)}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(11)$ If $r_0=10$, then $\Omega^{10}(Y_t^{(11)})$ is described with
$(9s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n\ra (j+m)n+5}\otimes e_{jn},\quad i=j;\\
\kappa w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
-\kappa w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+2s;\\
-\kappa w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+3s;\\
\kappa w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+4s;\\
-\kappa e_{(j+m)n+5}\otimes w_{jn\ra jn+5},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
-\kappa e_{(j+m+1)n}\otimes w_{jn\ra (j+1)n},\quad i=(j+1)_s;\\
\kappa w_{(j+m)n+g(j)\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j)\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $b_{ij}=0$.
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $b_{ij}=0$.
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j)+1\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<9s$, then $b_{ij}=0$.
\medskip
$({\rm II})$ Represent an arbitrary $t_0\in\N$ in the form
$t_0=11\ell_0+r_0$, where $0\le r_0\le 10.$ Then
$\Omega^{t_0}(Y_t^{(11)})$ is a $\Omega^{r_0}(Y_t^{(11)})$, whose
left components twisted by $\sigma^{\ell_0}$, and coefficients
multiplied by $(-1)^{\ell_0}$.
\end{pr}
\begin{pr}[Translates for the case 12]
$({\rm I})$ Let $r_0\in\N$, $r_0<11$. $r_0$-translates of the
elements $Y^{(12)}_t$ are described by the following way.
$(1)$ If $r_0=0$, then $\Omega^{0}(Y_t^{(12)})$ is described with
$(8s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n\ra (j+m)n+g(j+s)}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<4s$, then $b_{ij}=0$.
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<8s$, then $b_{ij}=0$.
$(2)$ If $r_0=1$, then $\Omega^{1}(Y_t^{(12)})$ is described with
$(9s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes e_{jn},\quad i=j+s;\\
-\kappa w_{(j+m)n+g(j+s)+1\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+3s;\\
\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+4s;\\
-\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j)}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<3s$, then $b_{ij}=0$.
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)},\quad i=j-s;\\
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+s;\\
-w_{(j+m+1)n\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<7s$, then $b_{ij}=0$.
If $7s\le j<8s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
\kappa w_{(j+m+1)n+g(j+s)+1\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s\le j<9s$, then $b_{ij}=0$.
$(3)$ If $r_0=2$, then $\Omega^{2}(Y_t^{(12)})$ is described with
$(8s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j+s)\ra (j+m)n+g(j+s)+1}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m-1)n+5\ra (j+m)n+g(j+s)+1}\otimes e_{jn},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m)n+5}\otimes w_{jn+g(j)\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes w_{jn+g(j)\ra jn+g(j)+1},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $b_{ij}=0$.
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn+g(j)+1\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j+s)}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $b_{ij}=0$.
$(4)$ If $r_0=3$, then $\Omega^{3}(Y_t^{(12)})$ is described with
$(6s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes e_{jn},\quad i=j+s;\\
\kappa e_{(j+m)n+5}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<6s$, then $b_{ij}=0$.
$(5)$ If $r_0=4$, then $\Omega^{4}(Y_t^{(12)})$ is described with
$(7s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-\kappa e_{(j+m)n+5}\otimes w_{jn\ra (j+1)n},\quad i=(j+1)_s;\\
-\kappa w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+3s;\\
\kappa w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+7s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j)\ra (j+m+1)n}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
-\kappa w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<6s$, then $b_{ij}=0$.
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n+g(j)\ra (j+m+1)n+g(j)+1}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+3s;\\
-e_{(j+m+1)n+g(j)+1}\otimes w_{jn+5\ra (j+1)n+g(j+s)+1},\quad i=(j+1)_s+7s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(6)$ If $r_0=5$, then $\Omega^{5}(Y_t^{(12)})$ is described with
$(6s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m+1)n}\otimes w_{jn\ra jn+g(j)},\quad i=j+2s;\\
\kappa e_{(j+m+1)n}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<6s$, then $b_{ij}=0$.
$(7)$ If $r_0=6$, then $\Omega^{6}(Y_t^{(12)})$ is described with
$(6s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n\ra (j+m+1)n}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+j_2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)},\\\quad\quad\quad i=(j+1)_s+(j_2-2)s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
\kappa w_{(j+m)n+5\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(8)$ If $r_0=7$, then $\Omega^{7}(Y_t^{(12)})$ is described with
$(7s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<4s$, then $b_{ij}=0$.
If $4s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n\ra (j+m+1)n+5}\otimes e_{jn+g(j)+1},\quad i=j;\\
-\kappa w_{(j+m+1)n+g(j)\ra (j+m+1)n+5}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $b_{ij}=0$.
$(9)$ If $r_0=8$, then $\Omega^{8}(Y_t^{(12)})$ is described with
$(6s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+3s;\\
-\kappa w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<5s$, then $b_{ij}=0$.
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n+g(j+s)+1\ra (j+m+2)n}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(10)$ If $r_0=9$, then $\Omega^{9}(Y_t^{(12)})$ is described with
$(8s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<4s$, then $b_{ij}=0$.
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
-\kappa e_{(j+m+2)n}\otimes w_{jn+g(j)+1\ra (j+1)n+g(j)},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $b_{ij}=0$.
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+1)n+g(j)+1\ra (j+m+2)n+g(j)}\otimes e_{jn+5},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $b_{ij}=0$.
$(11)$ If $r_0=10$, then $\Omega^{10}(Y_t^{(12)})$ is described with
$(9s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $b_{ij}=0$.
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j+s)}\otimes w_{jn+g(j)\ra (j+1)n+g(j)},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j+s)\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j)},\quad i=j-s;\\
-w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j)\ra jn+g(j)+1},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j)+1\ra (j+1)n},\quad i=(j+1)_s;\\
-w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $b_{ij}=0$.
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j)+1\ra (j+1)n+g(j)},\quad i=(j+1)_s+s;\\
-e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j)+1\ra (j+1)n+g(j)+1},\quad i=(j+1)_s+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j)+1},\quad i=j-3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s\le j<9s$, then $b_{ij}=0$.
\medskip
$({\rm II})$ Represent an arbitrary $t_0\in\N$ in the form
$t_0=11\ell_0+r_0$, where $0\le r_0\le 10.$ Then
$\Omega^{t_0}(Y_t^{(12)})$ is a $\Omega^{r_0}(Y_t^{(12)})$, whose
left components twisted by $\sigma^{\ell_0}$.
\end{pr}
\begin{pr}[Translates for the case 13]
$({\rm I})$ Let $r_0\in\N$, $r_0<11$. $r_0$-translates of the
elements $Y^{(13)}_t$ are described by the following way.
$(1)$ If $r_0=0$, then $\Omega^{0}(Y_t^{(13)})$ is described with
$(9s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $b_{ij}=0$.
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
-f_1(j,2s)e_{(j+m)n+g(j+s)}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $b_{ij}=0$.
If $5s\le j<7s$, then $$b_{ij}=
\begin{cases}
-f_1(j,6s)e_{(j+m)n+g(j+s)+1}\otimes e_{jn+g(j+s)+1},\quad i=j-2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $b_{ij}=0$.
If $8s\le j<9s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes e_{jn+5},\quad i=j-3s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(2)$ If $r_0=1$, then $\Omega^{1}(Y_t^{(13)})$ is described with
$(8s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $$b_{ij}=
\begin{cases}
f_1(j,s)e_{(j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j)},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
-\kappa f_1(j,3s)w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes e_{jn+g(j)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $$b_{ij}=
\begin{cases}
-\kappa f_1(j,5s)w_{(j+m)n+5\ra (j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<8s$, then $$b_{ij}=
\begin{cases}
f_1(j,7s)e_{(j+m+1)n+g(j)}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s+(j_2-6)s;\\
f_1(j,7s)w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes e_{jn+5},\quad i=(j)_s+6s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(3)$ If $r_0=2$, then $\Omega^{2}(Y_t^{(13)})$ is described with
$(6s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $b_{ij}=0$.
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
-f_1(j,2s)e_{(j+m)n+g(j+s)+1}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
-f_1(j,4s)e_{(j+m)n+g(j)}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(4)$ If $r_0=3$, then $\Omega^{3}(Y_t^{(13)})$ is described with
$(7s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes e_{jn},\quad i=j;\\
\kappa w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes e_{jn},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
-\kappa f_1(j,2s)w_{(j+m)n+5\ra (j+m+1)n}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
-f_1(j,4s)w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)+1},\quad i=j+s;\\
-f_1(j,4s)e_{(j+m+1)n+g(j)}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=j+2(1+f_0(j,4s))s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<7s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s+(j_2-5)s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(5)$ If $r_0=4$, then $\Omega^{4}(Y_t^{(13)})$ is described with
$(6s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m-1)n+5\ra (j+m)n}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
-f_1(j,2s)e_{(j+m)n+g(j)}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
-f_1(j,4s)e_{(j+m)n+g(j)+1}\otimes e_{jn+g(j+s)+1},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $b_{ij}=0$.
$(6)$ If $r_0=5$, then $\Omega^{5}(Y_t^{(13)})$ is described with
$(6s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m+1)n}\otimes w_{jn\ra jn+g(j)},\quad i=j+2s;\\
\kappa e_{(j+m+1)n}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
-f_1(j,2s)e_{(j+m+1)n+g(j)}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s+f_0(j,2s)s;\\
-f_1(j,2s)w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
f_1(j,4s)w_{(j+m)n+5\ra (j+m+1)n+g(j)+1}\otimes e_{jn+g(j+s)+1},\quad i=j+s;\\
e_{(j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=j+2(1+f_0(j,4s))s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n+g(j+s)+1\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j+s;\\
\kappa w_{(j+m+1)n+g(j)+1\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(7)$ If $r_0=6$, then $\Omega^{6}(Y_t^{(13)})$ is described with
$(7s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $$b_{ij}=
\begin{cases}
-f_1(j,s)e_{(j+m)n+g(j+s)}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+2f_0(j,s)s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
f_1(j,3s)e_{(j+m)n+g(j+s)+1}\otimes e_{jn+g(j)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes e_{jn+g(j)+1},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
-\kappa e_{(j+m+1)n}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes e_{jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(8)$ If $r_0=7$, then $\Omega^{7}(Y_t^{(13)})$ is described with
$(6s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes e_{jn},\quad i=j;\\
-\kappa w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes e_{jn},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
-f_1(j,2s)e_{(j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s+f_0(j,2s)s;\\
f_1(j,2s)w_{(j+m)n+5\ra (j+m+1)n+g(j)+1}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
-f_1(j,4s)w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j+s)+1},\quad i=j+s;\\
f_1(j,4s)e_{(j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n+g(j)+1\ra (j+m+2)n}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s+s;\\
-\kappa e_{(j+m+2)n}\otimes w_{jn+5\ra (j+1)n+g(j+s)+1},\quad i=(j+1)_s+4s;\\
-\kappa e_{(j+m+2)n}\otimes w_{jn+5\ra (j+1)n+g(j)+1},\quad i=(j+1)_s+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(9)$ If $r_0=8$, then $\Omega^{8}(Y_t^{(13)})$ is described with
$(8s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $$b_{ij}=
\begin{cases}
-f_1(j,s)w_{(j+m-1)n+5\ra (j+m)n+g(j+s)+1}\otimes e_{jn},\quad i=(j)_s;\\
-e_{(j+m)n+g(j+s)+1}\otimes w_{jn\ra jn+g(j+s)},\quad i=(j)_s+(1+f_0(j,s))s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn+g(j)+1\ra (j+1)n},\quad i=(j+1)_s;\\
\kappa w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
-\kappa e_{(j+m+1)n}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
-\kappa e_{(j+m+1)n}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<8s$, then $$b_{ij}=
\begin{cases}
f_1(j,7s)e_{(j+m+1)n+g(j+s)}\otimes w_{jn+5\ra (j+1)n+g(j)+1},\quad i=(j+1)_s+(j_2-3)s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(10)$ If $r_0=9$, then $\Omega^{9}(Y_t^{(13)})$ is described with
$(9s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $b_{ij}=0$.
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes e_{jn},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
-f_1(j,3s)w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $$b_{ij}=
\begin{cases}
-f_1(j,5s)e_{(j+m+1)n+g(j+s)}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<8s$, then $$b_{ij}=
\begin{cases}
f_1(j,7s)e_{(j+m+1)n+g(j)+1}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s\le j<9s$, then $$b_{ij}=
\begin{cases}
-\kappa e_{(j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(11)$ If $r_0=10$, then $\Omega^{10}(Y_t^{(13)})$ is described with
$(8s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n\ra (j+m)n+g(j+s)}\otimes e_{jn},\quad i=(j)_s;\\
-f_1(j,s)e_{(j+m)n+g(j+s)}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes e_{jn+g(j)},\quad i=j-s;\\
-\kappa w_{(j+m)n+g(j+s)+1\ra (j+m+1)n}\otimes w_{jn+g(j)\ra jn+g(j)+1},\quad i=j+s;\\
-\kappa f_1(j,3s)w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn+g(j)\ra jn+5},\quad i=(j)_s+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
\kappa f_1(j,5s)e_{(j+m)n+5}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=(j)_s+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<8s$, then $$b_{ij}=
\begin{cases}
f_1(j,7s)w_{(j+m+1)n\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
-w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+5\ra (j+1)n+g(j)},\quad i=(j+1)_s+(j_2-5)s;\\
e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+5\ra (j+1)n+g(j)+1},\quad i=(j+1)_s+(j_2-3)s;\\
0,\quad\text{otherwise.}\end{cases}$$
\medskip
$({\rm II})$ Represent an arbitrary $t_0\in\N$ in the form
$t_0=11\ell_0+r_0$, where $0\le r_0\le 10.$ Then
$\Omega^{t_0}(Y_t^{(13)})$ is a $\Omega^{r_0}(Y_t^{(13)})$, whose
left components twisted by $\sigma^{\ell_0}$.
\end{pr}
\begin{pr}[Translates for the case 14]
$({\rm I})$ Let $r_0\in\N$, $r_0<11$. $r_0$-translates of the
elements $Y^{(14)}_t$ are described by the following way.
$(1)$ If $r_0=0$, then $\Omega^{0}(Y_t^{(14)})$ is described with
$(9s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{(-3\ell_0)_s,(-3\ell_0)_s}=\kappa w_{(j+m)n\ra (j+m+1)n}\otimes e_{jn}.$$
$(2)$ If $r_0=1$, then $\Omega^{1}(Y_t^{(14)})$ is described with
$(8s\times 7s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{(-3\ell_0)_s,5s+(-1-3\ell_0)_s}=\kappa w_{(j+m+1)n+g(j+s)\ra (j+m+2)n}\otimes w_{jn+g(j)+1\ra (j+1)n}.$$
$(3)$ If $r_0=2$, then $\Omega^{2}(Y_t^{(14)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{4s+(-1-3\ell_0)_s,4s+(-1-3\ell_0)_s}=w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)+1}.$$
$(4)$ If $r_0=3$, then $\Omega^{3}(Y_t^{(14)})$ is described with
$(7s\times 8s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{5s+(-1-3\ell_0)_s,4s+(-2-3\ell_0)_s}=-w_{(j+m+2)n\ra (j+m+2)n+g(j)}\otimes w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)+1}.$$
$(5)$ If $r_0=4$, then $\Omega^{4}(Y_t^{(14)})$ is described with
$(6s\times 9s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{7s+(-1-3\ell_0)_s,4s+(-2-3\ell_0)_s}=-e_{(j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)+1}.$$
$(6)$ If $r_0=5$, then $\Omega^{5}(Y_t^{(14)})$ is described with
$(6s\times 8s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{5s+(-1-3\ell_0)_s,4s+(-2-3\ell_0)_s}=w_{(j+m+1)n+5\ra (j+m+2)n+g(j)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)+1}.$$
$(7)$ If $r_0=6$, then $\Omega^{6}(Y_t^{(14)})$ is described with
$(7s\times 9s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{6s+(-1-3\ell_0)_s,5s+(-2-3\ell_0)_s}=-\kappa w_{(j+m+1)n+g(j)+1\ra (j+m+1)n+5}\otimes w_{jn+g(j)+1\ra (j+1)n+g(j)+1}.$$
$(8)$ If $r_0=7$, then $\Omega^{7}(Y_t^{(14)})$ is described with
$(6s\times 8s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{5s+(-1-3\ell_0)_s,4s+(-2-3\ell_0)_s}=-w_{(j+m+2)n\ra (j+m+2)n+g(j+s)}\otimes w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)+1}.$$
$(9)$ If $r_0=8$, then $\Omega^{8}(Y_t^{(14)})$ is described with
$(8s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{5s+(-2-3\ell_0)_s,5s+(-2-3\ell_0)_s}=\kappa w_{(j+m+1)n\ra (j+m+2)n}\otimes w_{jn+g(j)+1\ra jn+5}.$$
$(10)$ If $r_0=9$, then $\Omega^{9}(Y_t^{(14)})$ is described with
$(9s\times 7s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{s+(-2-3\ell_0)_s,s+(-2-3\ell_0)_s}=-\kappa w_{(j+m+1)n\ra (j+m+2)n}\otimes w_{jn\ra jn+g(j+s)}.$$
$(11)$ If $r_0=10$, then $\Omega^{10}(Y_t^{(14)})$ is described with
$(8s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{(2s+(-3-3\ell_0)_s+1)_s,2s+(-3-3\ell_0)_s}=\kappa w_{(j+m+1)n\ra (j+m+2)n}\otimes w_{jn+g(j)\ra (j+1)n}.$$
\medskip
$({\rm II})$ Represent an arbitrary $t_0\in\N$ in the form
$t_0=11\ell_0+r_0$, where $0\le r_0\le 10.$ Then
$\Omega^{t_0}(Y_t^{(14)})$ is a $\Omega^{r_0}(Y_t^{(14)})$, whose
left components twisted by $\sigma^{\ell_0}$.
\end{pr}
\begin{pr}[Translates for the case 15]
$({\rm I})$ Let $r_0\in\N$, $r_0<11$. $r_0$-translates of the
elements $Y^{(15)}_t$ are described by the following way.
$(1)$ If $r_0=0$, then $\Omega^{0}(Y_t^{(15)})$ is described with
$(9s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $b_{ij}=0$.
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j+s)\ra (j+m)n+g(j+s)+1}\otimes e_{jn+g(j+s)},\quad i=j-2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<7s$, then $b_{ij}=0$.
If $7s\le j<8s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+5}\otimes e_{jn+5},\quad i=j-2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s\le j<9s$, then $b_{ij}=0$.
$(2)$ If $r_0=1$, then $\Omega^{1}(Y_t^{(15)})$ is described with
$(8s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j+s)\ra (j+m)n+g(j+s)+1}\otimes e_{jn},\quad i=(j)_s+f_0(j,s)s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes e_{jn+g(j)},\quad i=j;\\
e_{(j+m)n+5}\otimes w_{jn+g(j)\ra jn+g(j)+1},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $b_{ij}=0$.
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes e_{jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $b_{ij}=0$.
$(3)$ If $r_0=2$, then $\Omega^{2}(Y_t^{(15)})$ is described with
$(6s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
e_{(j+m-1)n+5}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j)\ra (j+m)n+g(j)+1}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $b_{ij}=0$.
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n}\otimes e_{jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
$(4)$ If $r_0=3$, then $\Omega^{3}(Y_t^{(15)})$ is described with
$(7s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes e_{jn},\quad i=j;\\
e_{(j+m)n+5}\otimes w_{jn\ra jn+g(j)},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j+s)+1},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<7s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n+g(j)\ra (j+m+1)n+g(j)+1}\otimes e_{jn+5},\quad i=j+2f_0(j,6s)s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(5)$ If $r_0=4$, then $\Omega^{4}(Y_t^{(15)})$ is described with
$(6s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n}\otimes e_{jn},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $b_{ij}=0$.
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j+s)\ra (j+m)n+g(j+s)+1}\otimes e_{jn+g(j+s)+1},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+5}\otimes e_{jn+5},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(6)$ If $r_0=5$, then $\Omega^{5}(Y_t^{(15)})$ is described with
$(6s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes e_{jn},\quad i=j+s;\\
e_{(j+m+1)n}\otimes w_{jn\ra jn+g(j)},\quad i=j+2s;\\
w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s+(j_2-1)s;\\
w_{(j+m)n+5\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n},\quad i=(j+1)_s+(j_2-3)s;\\
w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)+1},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
e_{(j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j)+1},\quad i=(j+1)_s+5s;\\
w_{(j+m+1)n+g(j+s)+1\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(7)$ If $r_0=6$, then $\Omega^{6}(Y_t^{(15)})$ is described with
$(7s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j)}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n\ra (j+m)n+g(j)}\otimes e_{jn},\quad i=j-s;\\
e_{(j+m)n+g(j)}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $b_{ij}=0$.
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes e_{jn+g(j)+1},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes e_{jn+g(j)+1},\quad i=j+s;\\
e_{(j+m)n+5}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
e_{(j+m+1)n}\otimes e_{jn+5},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(8)$ If $r_0=7$, then $\Omega^{7}(Y_t^{(15)})$ is described with
$(6s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes e_{jn},\quad i=j+s;\\
e_{(j+m)n+5}\otimes w_{jn\ra jn+g(j)},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s+(j_2-1)s;\\
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)+1},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+2)n}\otimes w_{jn+5\ra (j+1)n+g(j+s)+1},\quad i=(j+1)_s+4s;\\
w_{(j+m+1)n+g(j+s)\ra (j+m+2)n}\otimes e_{jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(9)$ If $r_0=8$, then $\Omega^{8}(Y_t^{(15)})$ is described with
$(8s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
w_{(j+m)n+g(j)\ra (j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
w_{(j+m-1)n+5\ra (j+m)n+g(j)+1}\otimes e_{jn},\quad i=j-s;\\
e_{(j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
w_{(j+m)n+g(j)\ra (j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+5}\otimes w_{jn+g(j)\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $b_{ij}=0$.
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
e_{(j+m+1)n}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes e_{jn+5},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $b_{ij}=0$.
$(10)$ If $r_0=9$, then $\Omega^{9}(Y_t^{(15)})$ is described with
$(9s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+5}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n}\otimes w_{jn\ra jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $b_{ij}=0$.
If $6s\le j<8s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j)+1},\quad i=j-3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s\le j<9s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n+g(j)+1\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j-3s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(11)$ If $r_0=10$, then $\Omega^{10}(Y_t^{(15)})$ is described with
$(8s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n\ra (j+m)n+g(j)}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $b_{ij}=0$.
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n}\otimes w_{jn+g(j)\ra (j+1)n},\quad i=(j+1)_s;\\
w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
e_{(j+m)n+5}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n+g(j)\ra (j+m+1)n+g(j)+1}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+2s;\\
w_{(j+m)n+5\ra (j+m+1)n+g(j)+1}\otimes e_{jn+5},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n+g(j)\ra (j+m+1)n+g(j)+1}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
\medskip
$({\rm II})$ Represent an arbitrary $t_0\in\N$ in the form
$t_0=11\ell_0+r_0$, where $0\le r_0\le 10.$ Then
$\Omega^{t_0}(Y_t^{(15)})$ is a $\Omega^{r_0}(Y_t^{(15)})$, whose
left components twisted by $\sigma^{\ell_0}$.
\end{pr}
\begin{pr}[Translates for the case 16]
$({\rm I})$ Let $r_0\in\N$, $r_0<11$. $r_0$-translates of the
elements $Y^{(16)}_t$ are described by the following way.
$(1)$ If $r_0=0$, then $\Omega^{0}(Y_t^{(16)})$ is described with
$(8s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<(-4\ell_0)_s$, then $b_{ij}=0$.
If $j=(-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n\ra (j+m)n+g(j)+1}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $(-4\ell_0)_s<j<2s+(-4\ell_0)_s$, then $b_{ij}=0$.
If $j=2s+(-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s+(-4\ell_0)_s<j<8s$, then $b_{ij}=0$.
$(2)$ If $r_0=1$, then $\Omega^{1}(Y_t^{(16)})$ is described with
$(6s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<(-4\ell_0)_s$, then $b_{ij}=0$.
If $j=(-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $(-4\ell_0)_s<j<s+(-1-4\ell_0)_s$, then $b_{ij}=0$.
If $j=s+(-1-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
-e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s+(-1-4\ell_0)_s<j<5s+(-1-4\ell_0)_s$, then $b_{ij}=0$.
If $j=5s+(-1-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n+g(j+s)\ra (j+m+2)n}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
-\kappa w_{(j+m+1)n\ra (j+m+2)n}\otimes e_{jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s+(-1-4\ell_0)_s<j<6s$, then $b_{ij}=0$.
$(3)$ If $r_0=2$, then $\Omega^{2}(Y_t^{(16)})$ is described with
$(7s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s+(-1-4\ell_0)_s$, then $b_{ij}=0$.
If $j=s+(-1-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j+s)+1\ra (j+m+1)n}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s+(-1-4\ell_0)_s<j<2s+(-1-4\ell_0)_s$, then $b_{ij}=0$.
If $j=2s+(-1-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j)\ra (j+m+1)n}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s+(-1-4\ell_0)_s<j<4s+(-1-4\ell_0)_s$, then $b_{ij}=0$.
If $j=4s+(-1-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s+(-1-4\ell_0)_s<j<7s$, then $b_{ij}=0$.
$(4)$ If $r_0=3$, then $\Omega^{3}(Y_t^{(16)})$ is described with
$(6s\times 8s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{(-1-4\ell_0)_s,(-1-4\ell_0)_s}=\kappa w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes e_{jn}.$$
$(5)$ If $r_0=4$, then $\Omega^{4}(Y_t^{(16)})$ is described with
$(6s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<(-2-4\ell_0)_s$, then $b_{ij}=0$.
If $j=(-2-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-\kappa e_{(j+m+1)n}\otimes w_{jn\ra (j+1)n},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $(-2-4\ell_0)_s<j<2s+(-2-4\ell_0)_s$, then $b_{ij}=0$.
If $j=2s+(-2-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s+(-2-4\ell_0)_s<j<4s+(-2-4\ell_0)_s$, then $b_{ij}=0$.
If $j=4s+(-2-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)+1}\otimes e_{jn+g(j+s)+1},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s+(-2-4\ell_0)_s<j<6s$, then $b_{ij}=0$.
$(6)$ If $r_0=5$, then $\Omega^{5}(Y_t^{(16)})$ is described with
$(7s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<(-2-4\ell_0)_s$, then $b_{ij}=0$.
If $j=(-2-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j+s)}\otimes w_{jn\ra (j+1)n},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $(-2-4\ell_0)_s<j<3s+(-2-4\ell_0)_s$, then $b_{ij}=0$.
If $j=3s+(-2-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s+(-2-4\ell_0)_s<j<7s$, then $b_{ij}=0$.
$(7)$ If $r_0=6$, then $\Omega^{6}(Y_t^{(16)})$ is described with
$(6s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<(-2-4\ell_0)_s$, then $b_{ij}=0$.
If $j=(-2-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $(-2-4\ell_0)_s<j<2s+(-3-4\ell_0)_s$, then $b_{ij}=0$.
If $j=2s+(-3-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s+(-3-4\ell_0)_s<j<6s$, then $b_{ij}=0$.
$(8)$ If $r_0=7$, then $\Omega^{7}(Y_t^{(16)})$ is described with
$(8s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s+(-3-4\ell_0)_s$, then $b_{ij}=0$.
If $j=s+(-3-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s+(-3-4\ell_0)_s<j<3s+(-3-4\ell_0)_s$, then $b_{ij}=0$.
If $j=3s+(-3-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n+g(j)+1\ra (j+m+1)n+5}\otimes w_{jn+g(j)\ra (j+1)n},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s+(-3-4\ell_0)_s<j<6s+(-3-4\ell_0)_s$, then $b_{ij}=0$.
If $j=6s+(-3-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+1)n+g(j+s)\ra (j+m+2)n+g(j+s)}\otimes e_{jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s+(-3-4\ell_0)_s<j<8s$, then $b_{ij}=0$.
$(9)$ If $r_0=8$, then $\Omega^{8}(Y_t^{(16)})$ is described with
$(9s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<(-3-4\ell_0)_s$, then $b_{ij}=0$.
If $j=(-3-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m)n+5}\otimes w_{jn\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $(-3-4\ell_0)_s<j<3s+(-3-4\ell_0)_s$, then $b_{ij}=0$.
If $j=3s+(-3-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s+(-3-4\ell_0)_s<j<4s+(-3-4\ell_0)_s$, then $b_{ij}=0$.
If $j=4s+(-3-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+g(j+s)\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s+(-3-4\ell_0)_s<j<9s$, then $b_{ij}=0$.
$(10)$ If $r_0=9$, then $\Omega^{9}(Y_t^{(16)})$ is described with
$(8s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<(-3-4\ell_0)_s$, then $b_{ij}=0$.
If $j=(-3-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $(-3-4\ell_0)_s<j<6s+(-4-4\ell_0)_s$, then $b_{ij}=0$.
If $j=6s+(-4-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+1)n+5\ra (j+m+2)n+g(j+s)+1}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s+(-4-4\ell_0)_s<j<8s$, then $b_{ij}=0$.
$(11)$ If $r_0=10$, then $\Omega^{10}(Y_t^{(16)})$ is described with
$(9s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s+(-4-4\ell_0)_s$, then $b_{ij}=0$.
If $j=s+(-4-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
-e_{(j+m+1)n+g(j)}\otimes w_{jn+g(j+s)\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s+(-4-4\ell_0)_s<j<5s+(-4-4\ell_0)_s$, then $b_{ij}=0$.
If $j=5s+(-4-4\ell_0)_s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)+1}\otimes e_{jn+g(j+s)+1},\quad i=j-2s;\\
-w_{(j+m)n+5\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s+(-4-4\ell_0)_s<j<9s$, then $b_{ij}=0$.
\medskip
$({\rm II})$ Represent an arbitrary $t_0\in\N$ in the form
$t_0=11\ell_0+r_0$, where $0\le r_0\le 10.$ Then
$\Omega^{t_0}(Y_t^{(16)})$ is a $\Omega^{r_0}(Y_t^{(16)})$, whose
left components twisted by $\sigma^{\ell_0}$, and coefficients
multiplied by $(-1)^{\ell_0}$.
\end{pr}
\begin{pr}[Translates for the case 18]
$({\rm I})$ Let $r_0\in\N$, $r_0<11$. $r_0$-translates of the
elements $Y^{(18)}_t$ are described by the following way.
$(1)$ If $r_0=0$, then $\Omega^{0}(Y_t^{(18)})$ is described with
$(6s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $b_{ij}=0$.
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j+s)\ra (j+m)n+g(j+s)+1}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $b_{ij}=0$.
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes e_{jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
$(2)$ If $r_0=1$, then $\Omega^{1}(Y_t^{(18)})$ is described with
$(7s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes e_{jn},\quad i=j;\\
-\kappa w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes e_{jn},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j+s)+1\ra (j+m+1)n}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)+1},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<7s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s+(j_2-5)s;\\
-w_{(j+m+1)n\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+5},\quad i=j+(6-j_2)s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(3)$ If $r_0=2$, then $\Omega^{2}(Y_t^{(18)})$ is described with
$(6s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m-1)n+5\ra (j+m)n}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $b_{ij}=0$.
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+g(j)\ra (j+m)n+g(j)+1}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $b_{ij}=0$.
$(4)$ If $r_0=3$, then $\Omega^{3}(Y_t^{(18)})$ is described with
$(6s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes e_{jn},\quad i=j;\\
\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j)},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
-f_1(j,4s)e_{(j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n},\quad i=(j+1)_s+(4-j_2)s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n+g(j+s)+1\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
-\kappa e_{(j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(5)$ If $r_0=4$, then $\Omega^{4}(Y_t^{(18)})$ is described with
$(7s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $b_{ij}=0$.
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
w_{(j+m-1)n+5\ra (j+m)n+g(j+s)}\otimes e_{jn},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j+s)\ra (j+m)n+g(j+s)+1}\otimes e_{jn+g(j)},\quad i=j;\\
-e_{(j+m)n+g(j+s)+1}\otimes w_{jn+g(j)\ra jn+g(j)+1},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m)n+5}\otimes w_{jn+g(j)+1\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<7s$, then $b_{ij}=0$.
$(6)$ If $r_0=5$, then $\Omega^{5}(Y_t^{(18)})$ is described with
$(6s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes e_{jn},\quad i=j;\\
-\kappa w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes e_{jn},\quad i=j+s;\\
\kappa e_{(j+m)n+5}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+4s;\\
\kappa e_{(j+m)n+5}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+1)n+g(j)\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s+(2-j_2)s;\\
-w_{(j+m+1)n\ra (j+m+1)n+g(j)+1}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $b_{ij}=0$.
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
-\kappa e_{(j+m+2)n}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(7)$ If $r_0=6$, then $\Omega^{6}(Y_t^{(18)})$ is described with
$(8s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $$b_{ij}=
\begin{cases}
f_1(j,s)w_{(j+m)n\ra (j+m)n+g(j+s)+1}\otimes e_{jn},\quad i=(j)_s;\\
-f_1(j,s)w_{(j+m)n+g(j+s)\ra (j+m)n+g(j+s)+1}\otimes w_{jn\ra jn+g(j+s)},\quad i=(j)_s+(2-j_2)s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
-\kappa f_1(j,3s)w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m+1)n}\otimes w_{jn+g(j)+1\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $b_{ij}=0$.
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
-e_{(j+m+1)n+g(j+s)}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $$b_{ij}=
\begin{cases}
-e_{(j+m+1)n+g(j+s)}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(8)$ If $r_0=7$, then $\Omega^{7}(Y_t^{(18)})$ is described with
$(9s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $b_{ij}=0$.
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes e_{jn},\quad i=j;\\
\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+s;\\
\kappa e_{(j+m+1)n}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+3s;\\
\kappa e_{(j+m+1)n}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j)},\quad i=j;\\
w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j)\ra jn+g(j)+1},\quad i=j+2s;\\
-e_{(j+m+1)n+g(j)}\otimes w_{jn+g(j)\ra jn+5},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<9s$, then $b_{ij}=0$.
$(9)$ If $r_0=8$, then $\Omega^{8}(Y_t^{(18)})$ is described with
$(8s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
w_{(j+m-1)n+5\ra (j+m)n+g(j+s)}\otimes e_{jn},\quad i=j;\\
e_{(j+m)n+g(j+s)}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j+s)}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn+g(j)\ra (j+1)n},\quad i=(j+1)_s;\\
-\kappa w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $b_{ij}=0$.
If $6s\le j<8s$, then $$b_{ij}=
\begin{cases}
-f_1(j,7s)e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+(8-j_2)s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(10)$ If $r_0=9$, then $\Omega^{9}(Y_t^{(18)})$ is described with
$(9s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $b_{ij}=0$.
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)},\quad i=j-2s;\\
-e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<7s$, then $b_{ij}=0$.
If $7s\le j<8s$, then $$b_{ij}=
\begin{cases}
-\kappa e_{(j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s\le j<9s$, then $b_{ij}=0$.
$(11)$ If $r_0=10$, then $\Omega^{10}(Y_t^{(18)})$ is described with
$(8s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $$b_{ij}=
\begin{cases}
f_1(j,s)w_{(j+m)n\ra (j+m)n+g(j+s)+1}\otimes e_{jn},\quad i=(j)_s;\\
-w_{(j+m)n+g(j+s)\ra (j+m)n+g(j+s)+1}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
e_{(j+m)n+g(j+s)+1}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
\kappa f_1(j,3s)w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes e_{jn+g(j)},\quad i=j-s;\\
-\kappa f_1(j,3s)w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes w_{jn+g(j)\ra jn+g(j)+1},\quad i=j+s;\\
-\kappa e_{(j+m)n+5}\otimes w_{jn+g(j)\ra jn+5},\quad i=(j)_s+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<8s$, then $b_{ij}=0$.
\medskip
$({\rm II})$ Represent an arbitrary $t_0\in\N$ in the form
$t_0=11\ell_0+r_0$, where $0\le r_0\le 10.$ Then
$\Omega^{t_0}(Y_t^{(18)})$ is a $\Omega^{r_0}(Y_t^{(18)})$, whose
left components twisted by $\sigma^{\ell_0}$.
\end{pr}
\begin{pr}[Translates for the case 20]
$({\rm I})$ Let $r_0\in\N$, $r_0<11$. $r_0$-translates of the
elements $Y^{(20)}_t$ are described by the following way.
$(1)$ If $r_0=0$, then $\Omega^{0}(Y_t^{(20)})$ is described with
$(7s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n\ra (j+m)n+5}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<3s$, then $b_{ij}=0$.
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $b_{ij}=0$.
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j)+1}\otimes e_{jn+5},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(2)$ If $r_0=1$, then $\Omega^{1}(Y_t^{(20)})$ is described with
$(6s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j)\ra (j+m+1)n}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<3s$, then $b_{ij}=0$.
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)+1},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $b_{ij}=0$.
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(3)$ If $r_0=2$, then $\Omega^{2}(Y_t^{(20)})$ is described with
$(6s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<3s$, then $b_{ij}=0$.
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $b_{ij}=0$.
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n+g(j+s)+1\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(4)$ If $r_0=3$, then $\Omega^{3}(Y_t^{(20)})$ is described with
$(7s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes w_{jn\ra jn+g(j)},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<4s$, then $b_{ij}=0$.
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n+g(j)+1\ra (j+m+1)n+5}\otimes w_{jn+g(j)+1\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<7s$, then $b_{ij}=0$.
$(5)$ If $r_0=4$, then $\Omega^{4}(Y_t^{(20)})$ is described with
$(6s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m-1)n+5\ra (j+m)n+5}\otimes e_{jn},\quad i=j;\\
-\kappa w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+3s;\\
\kappa w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+7s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<6s$, then $b_{ij}=0$.
$(6)$ If $r_0=5$, then $\Omega^{5}(Y_t^{(20)})$ is described with
$(8s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n+g(j)\ra (j+m+1)n+g(j)+1}\otimes w_{jn\ra (j+1)n},\quad i=(j+1)_s;\\
w_{(j+m+1)n\ra (j+m+1)n+g(j)+1}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $b_{ij}=0$.
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+5}\otimes w_{jn+g(j)\ra (j+1)n},\quad i=(j+1)_s;\\
\kappa w_{(j+m+1)n\ra (j+m+1)n+5}\otimes e_{jn+g(j)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $b_{ij}=0$.
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+2)n\ra (j+m+2)n+g(j)}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $b_{ij}=0$.
$(7)$ If $r_0=6$, then $\Omega^{6}(Y_t^{(20)})$ is described with
$(9s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $b_{ij}=0$.
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n\ra (j+m+1)n}\otimes e_{jn},\quad i=j-s;\\
\kappa w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<3s$, then $b_{ij}=0$.
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j)\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<9s$, then $b_{ij}=0$.
$(8)$ If $r_0=7$, then $\Omega^{7}(Y_t^{(20)})$ is described with
$(8s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes w_{jn\ra jn+g(j)},\quad i=j+2s;\\
w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+4s;\\
w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $b_{ij}=0$.
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n+g(j)+1\ra (j+m+2)n}\otimes w_{jn+g(j)\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<8s$, then $b_{ij}=0$.
$(9)$ If $r_0=8$, then $\Omega^{8}(Y_t^{(20)})$ is described with
$(9s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
-\kappa w_{(j+m)n+g(j)\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<7s$, then $b_{ij}=0$.
If $7s\le j<8s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j-2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s\le j<9s$, then $b_{ij}=0$.
$(10)$ If $r_0=9$, then $\Omega^{9}(Y_t^{(20)})$ is described with
$(8s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j)+1}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $b_{ij}=0$.
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n\ra (j+m+1)n+5}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n\ra (j+m+1)n+5}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $b_{ij}=0$.
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n+g(j+s)\ra (j+m+2)n}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
w_{(j+m+1)n+5\ra (j+m+2)n+g(j)}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $b_{ij}=0$.
$(11)$ If $r_0=10$, then $\Omega^{10}(Y_t^{(20)})$ is described with
$(6s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n\ra (j+m)n+5}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<4s$, then $b_{ij}=0$.
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n+g(j)\ra (j+m+2)n}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s;\\
\kappa w_{(j+m+1)n+g(j)+1\ra (j+m+2)n}\otimes w_{jn+5\ra (j+1)n+g(j+s)+1},\quad i=(j+1)_s+3s;\\
-\kappa w_{(j+m+1)n+g(j+s)+1\ra (j+m+2)n}\otimes w_{jn+5\ra (j+1)n+g(j)+1},\quad i=(j+1)_s+4s;\\
\kappa w_{(j+m+1)n+5\ra (j+m+2)n}\otimes w_{jn+5\ra (j+1)n+5},\quad i=(j+1)_s+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
\medskip
$({\rm II})$ Represent an arbitrary $t_0\in\N$ in the form
$t_0=11\ell_0+r_0$, where $0\le r_0\le 10.$ Then
$\Omega^{t_0}(Y_t^{(20)})$ is a $\Omega^{r_0}(Y_t^{(20)})$, whose
left components twisted by $\sigma^{\ell_0}$.
\end{pr}
\begin{pr}[Translates for the case 22]
$({\rm I})$ Let $r_0\in\N$, $r_0<11$. $r_0$-translates of the
elements $Y^{(22)}_t$ are described by the following way.
$(1)$ If $r_0=0$, then $\Omega^{0}(Y_t^{(22)})$ is described with
$(6s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $b_{ij}=0$.
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
f_1(j,2s)e_{(j+m)n+g(j+s)}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
f_1(j,4s)e_{(j+m)n+g(j+s)+1}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $b_{ij}=0$.
$(2)$ If $r_0=1$, then $\Omega^{1}(Y_t^{(22)})$ is described with
$(6s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j)\ra (j+m+1)n}\otimes e_{jn},\quad i=j;\\
-\kappa w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes e_{jn},\quad i=j+s;\\
-\kappa w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j)},\quad i=j+2s;\\
\kappa w_{(j+m)n+g(j+s)+1\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
f_1(j,2s)e_{(j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s+(j_2-1)s;\\
-f_1(j,2s)w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
f_1(j,2s)w_{(j+m)n+5\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+3s;\\
f_1(j,2s)w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=(j)_s+6s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
-f_1(j,4s)w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n},\\\quad\quad\quad i=(j+1)_s+(j_2-3)s;\\
f_1(j,4s)e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+(j_2-1)s;\\
-f_1(j,4s)w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)+1},\quad i=j+s;\\
-f_1(j,4s)w_{(j+m+1)n\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=(j)_s+6s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
-\kappa w_{(j+m+1)n+g(j)\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s+s;\\
-\kappa w_{(j+m+1)n+g(j+s)+1\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+2s;\\
\kappa w_{(j+m+1)n+g(j)+1\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j)},\quad i=(j+1)_s+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(3)$ If $r_0=2$, then $\Omega^{2}(Y_t^{(22)})$ is described with
$(7s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m-1)n+5\ra (j+m)n+g(j)}\otimes e_{jn},\quad i=(j)_s;\\
-f_1(j,s)e_{(j+m)n+g(j)}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=(j)_s+(4-j_2)s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
-f_1(j,3s)e_{(j+m)n+g(j)+1}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa f_1(j,5s)w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $b_{ij}=0$.
$(4)$ If $r_0=3$, then $\Omega^{3}(Y_t^{(22)})$ is described with
$(6s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes e_{jn},\quad i=j;\\
\kappa w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes e_{jn},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s+(j_2-1)s;\\
-f_1(j,2s)w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
-f_1(j,4s)w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)+1},\quad i=j+s;\\
-f_1(j,4s)e_{(j+m+1)n+g(j)}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=(j)_s+(10-j_2)s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n+g(j+s)+1\ra (j+m+2)n}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
\kappa w_{(j+m+1)n+g(j+s)\ra (j+m+2)n}\otimes e_{jn+5},\quad i=j+s;\\
-\kappa w_{(j+m+1)n+g(j)\ra (j+m+2)n}\otimes e_{jn+5},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(5)$ If $r_0=4$, then $\Omega^{4}(Y_t^{(22)})$ is described with
$(8s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $$b_{ij}=
\begin{cases}
w_{(j+m-1)n+5\ra (j+m)n+g(j)+1}\otimes e_{jn},\quad i=(j)_s;\\
-w_{(j+m)n+g(j)\ra (j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j+s)},\quad i=(j)_s+(3-j_2)s;\\
-e_{(j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=(j)_s+(7-j_2)s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
\kappa f_1(j,3s)w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes e_{jn+g(j)},\quad i=j;\\
\kappa f_1(j,3s)w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes w_{jn+g(j)\ra jn+g(j)+1},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $$b_{ij}=
\begin{cases}
-\kappa f_1(j,5s)w_{(j+m)n+g(j)\ra (j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j;\\
-\kappa f_1(j,5s)w_{(j+m)n+g(j+s)+1\ra (j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
-e_{(j+m+1)n+g(j)}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+3s;\\
-w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes e_{jn+5},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j)}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+2s;\\
-w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes e_{jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(6)$ If $r_0=5$, then $\Omega^{5}(Y_t^{(22)})$ is described with
$(9s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $b_{ij}=0$.
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m+1)n}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+s;\\
\kappa e_{(j+m+1)n}\otimes w_{jn\ra jn+g(j)},\quad i=j+2s;\\
-\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+3s;\\
-\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<4s$, then $$b_{ij}=
\begin{cases}
-f_1(j,3s)e_{(j+m+1)n+g(j+s)}\otimes w_{jn+g(j)\ra (j+1)n},\quad i=(j+1)_s+(3-j_2)s;\\
-f_1(j,3s)w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j)},\quad i=j;\\
f_1(j,3s)w_{(j+m)n+5\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j)\ra jn+g(j)+1},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $$b_{ij}=
\begin{cases}
f_1(j,5s)w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<8s$, then $$b_{ij}=
\begin{cases}
-f_1(j,7s)w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j)+1},\quad i=j-2s;\\
e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=(j)_s+(13-j_2)s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s\le j<9s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n+g(j)+1\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j-2s;\\
\kappa w_{(j+m+1)n+g(j+s)+1\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(7)$ If $r_0=6$, then $\Omega^{6}(Y_t^{(22)})$ is described with
$(8s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $$b_{ij}=
\begin{cases}
f_1(j,s)e_{(j+m)n+g(j)}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
-\kappa e_{(j+m+1)n}\otimes w_{jn+g(j)\ra (j+1)n},\quad i=(j+1)_s;\\
\kappa w_{(j+m)n+g(j)\ra (j+m+1)n}\otimes e_{jn+g(j)},\quad i=j-s;\\
-\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn+g(j)\ra jn+5},\quad i=j+5s;\\
\kappa e_{(j+m+1)n}\otimes w_{jn+g(j)\ra jn+5},\quad i=j+6s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j)\ra (j+m+1)n}\otimes e_{jn+g(j)},\quad i=j-s;\\
\kappa w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes w_{jn+g(j)\ra jn+g(j)+1},\quad i=j+3s;\\
-\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn+g(j)\ra jn+5},\quad i=j+4s;\\
-\kappa e_{(j+m+1)n}\otimes w_{jn+g(j)\ra jn+5},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa f_1(j,5s)w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes e_{jn+g(j)+1},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
-e_{(j+m+1)n+g(j)+1}\otimes w_{jn+5\ra (j+1)n+g(j)+1},\quad i=(j+1)_s+5s;\\
-w_{(j+m)n+5\ra (j+m+1)n+g(j)+1}\otimes e_{jn+5},\quad i=j+s;\\
-w_{(j+m+1)n\ra (j+m+1)n+g(j)+1}\otimes e_{jn+5},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j)+1}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+3s;\\
e_{(j+m+1)n+g(j)+1}\otimes w_{jn+5\ra (j+1)n+g(j)+1},\quad i=(j+1)_s+6s;\\
w_{(j+m)n+5\ra (j+m+1)n+g(j)+1}\otimes e_{jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
$(8)$ If $r_0=7$, then $\Omega^{7}(Y_t^{(22)})$ is described with
$(9s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-\kappa e_{(j+m+1)n}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+4s;\\
-\kappa e_{(j+m+1)n}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
-w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+3s;\\
-e_{(j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j+s)},\quad i=j+s;\\
e_{(j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
e_{(j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s+s;\\
w_{(j+m+1)n\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $b_{ij}=0$.
If $5s\le j<7s$, then $$b_{ij}=
\begin{cases}
-e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n},\quad i=(j+1)_s+(j_2-5)s;\\
-f_1(j,6s)w_{(j+m+1)n\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)+1},\quad i=j-s;\\
-w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n+g(j)+1\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s+s;\\
-\kappa w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s\le j<9s$, then $$b_{ij}=
\begin{cases}
\kappa e_{(j+m+2)n}\otimes w_{jn+5\ra (j+1)n+g(j)+1},\quad i=(j+1)_s+4s;\\
-\kappa e_{(j+m+2)n}\otimes w_{jn+5\ra (j+1)n+g(j+s)+1},\quad i=(j+1)_s+5s;\\
\kappa w_{(j+m+1)n+g(j+s)\ra (j+m+2)n}\otimes e_{jn+5},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(9)$ If $r_0=8$, then $\Omega^{8}(Y_t^{(22)})$ is described with
$(8s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
e_{(j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
w_{(j+m)n+g(j)\ra (j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<2s$, then $$b_{ij}=
\begin{cases}
-e_{(j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s\le j<3s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes e_{jn+g(j)},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<4s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes e_{jn+g(j)},\quad i=j-s;\\
-\kappa w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes w_{jn+g(j)\ra jn+g(j)+1},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s\le j<5s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn+g(j)+1\ra (j+1)n},\quad i=(j+1)_s;\\
\kappa w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
\kappa e_{(j+m+1)n}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j-s;\\
-\kappa e_{(j+m+1)n}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s\le j<7s$, then $$b_{ij}=
\begin{cases}
-e_{(j+m+1)n+g(j)}\otimes w_{jn+5\ra (j+1)n+g(j+s)+1},\quad i=(j+1)_s+4s;\\
w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes e_{jn+5},\quad i=j-s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s\le j<8s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
e_{(j+m+1)n+g(j)}\otimes w_{jn+5\ra (j+1)n+g(j+s)+1},\quad i=(j+1)_s+3s;\\
-w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes e_{jn+5},\quad i=j-2s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(10)$ If $r_0=9$, then $\Omega^{9}(Y_t^{(22)})$ is described with
$(6s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $b_{ij}=0$.
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
-f_1(j,2s)w_{(j+m+1)n\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)},\quad i=j;\\
-f_1(j,2s)e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
-f_1(j,4s)e_{(j+m+1)n+g(j)}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $$b_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n+5\ra (j+m+2)n}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
$(11)$ If $r_0=10$, then $\Omega^{10}(Y_t^{(22)})$ is described with
$(7s\times 6s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $$b_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
-\kappa w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+2s;\\
-\kappa w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+3s;\\
-\kappa w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s\le j<3s$, then $$b_{ij}=
\begin{cases}
\kappa f_1(j,2s)w_{(j+m)n+g(j)\ra (j+m+1)n}\otimes e_{jn+g(j+s)},\quad i=j;\\
\kappa f_1(j,2s)w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+2s;\\
\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=(j)_s+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s\le j<5s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j+s)+1\ra (j+1)n},\quad i=(j+1)_s;\\
f_1(j,4s)e_{(j+m+1)n+g(j)}\otimes w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+(j_2-2)s;\\
-f_1(j,4s)w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
-w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=(j)_s+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s\le j<7s$, then $$b_{ij}=
\begin{cases}
-w_{(j+m+1)n\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
-f_1(j,6s)w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+5\ra (j+1)n+g(j)},\\\quad\quad\quad i=(j+1)_s+(7-j_2)s;\\
-f_1(j,6s)e_{(j+m+1)n+g(j+s)+1}\otimes w_{jn+5\ra (j+1)n+g(j)+1},\quad i=(j+1)_s+(9-j_2)s;\\
0,\quad\text{otherwise.}\end{cases}$$
\medskip
$({\rm II})$ Represent an arbitrary $t_0\in\N$ in the form
$t_0=11\ell_0+r_0$, where $0\le r_0\le 10.$ Then
$\Omega^{t_0}(Y_t^{(22)})$ is a $\Omega^{r_0}(Y_t^{(22)})$, whose
left components twisted by $\sigma^{\ell_0}$, and coefficients
multiplied by $(-1)^{\ell_0}$.
\end{pr}
\begin{pr}[Translates for the case 23]
$({\rm I})$ Let $r_0\in\N$, $r_0<11$. $r_0$-translates of the
elements $Y^{(23)}_t$ are described by the following way.
$(1)$ If $r_0=0$, then $\Omega^{0}(Y_t^{(23)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{3s+z_0(0,\ell_0,n),3s+z_0(0,\ell_0,n)}=w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)+1}.$$
$(2)$ If $r_0=1$, then $\Omega^{1}(Y_t^{(23)})$ is described with
$(7s\times 7s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{2s+z_0(0,\ell_0,n),2s+z_0(0,\ell_0,n)}=w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)+1}\otimes e_{jn+g(j)}.$$
$(3)$ If $r_0=2$, then $\Omega^{2}(Y_t^{(23)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{s+z_0(-1,\ell_0,n),s+z_0(-1,\ell_0,n)}=w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)}.$$
$(4)$ If $r_0=3$, then $\Omega^{3}(Y_t^{(23)})$ is described with
$(8s\times 8s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{z_0(-1,\ell_0,n),z_0(-1,\ell_0,n)}=w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)+1}\otimes e_{jn}.$$
$(5)$ If $r_0=4$, then $\Omega^{4}(Y_t^{(23)})$ is described with
$(9s\times 9s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{6s+z_1(-2,\ell_0,n),6s+z_1(-2,\ell_0,n)}=w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j)+1}.$$
$(6)$ If $r_0=5$, then $\Omega^{5}(Y_t^{(23)})$ is described with
$(8s\times 8s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{6s+z_0(-3,\ell_0,n),6s+z_0(-3,\ell_0,n)}=w_{(j+m+1)n+g(j)+1\ra (j+m+2)n+g(j)+1}\otimes e_{jn+5}.$$
$(7)$ If $r_0=6$, then $\Omega^{6}(Y_t^{(23)})$ is described with
$(9s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<3s+z_1(-3,\ell_0,n)$, then $b_{ij}=0$.
If $j=3s+z_1(-3,\ell_0,n)$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)+1}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s+z_1(-3,\ell_0,n)<j<5s+z_0(-3,\ell_0,n)$, then $b_{ij}=0$.
If $j=5s+z_0(-3,\ell_0,n)$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s+z_0(-3,\ell_0,n)<j<9s$, then $b_{ij}=0$.
$(8)$ If $r_0=7$, then $\Omega^{7}(Y_t^{(23)})$ is described with
$(8s\times 8s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{z_0(-3,\ell_0,n),z_0(-3,\ell_0,n)}=w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)+1}\otimes e_{jn}.$$
$(9)$ If $r_0=8$, then $\Omega^{8}(Y_t^{(23)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{s+z_0(-4,\ell_0,n),s+z_0(-4,\ell_0,n)}=w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)}.$$
$(10)$ If $r_0=9$, then $\Omega^{9}(Y_t^{(23)})$ is described with
$(7s\times 7s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{5s+z_0(-5,\ell_0,n),5s+z_0(-5,\ell_0,n)}=w_{(j+m+1)n+g(j+s)+1\ra (j+m+2)n+g(j+s)+1}\otimes e_{jn+5}.$$
$(11)$ If $r_0=10$, then $\Omega^{10}(Y_t^{(23)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{3s+z_1(-5,\ell_0,n),3s+z_1(-5,\ell_0,n)}=w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)+1}\otimes e_{jn+g(j+s)+1}.$$
\medskip
$({\rm II})$ Represent an arbitrary $t_0\in\N$ in the form
$t_0=11\ell_0+r_0$, where $0\le r_0\le 10.$ Then
$\Omega^{t_0}(Y_t^{(23)})$ is a $\Omega^{r_0}(Y_t^{(23)})$, whose
left components twisted by $\sigma^{\ell_0}$, and coefficients
multiplied by $(-1)^{\ell_0}$.
\end{pr}
\begin{pr}[Translates for the case 24]
$({\rm I})$ Let $r_0\in\N$, $r_0<11$. $r_0$-translates of the
elements $Y^{(24)}_t$ are described by the following way.
$(1)$ If $r_0=0$, then $\Omega^{0}(Y_t^{(24)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{0,0}=w_{(j+m)n\ra (j+m+1)n}\otimes e_{jn}.$$
$(2)$ If $r_0=1$, then $\Omega^{1}(Y_t^{(24)})$ is described with
$(7s\times 7s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{6s,6s}=w_{(j+m)n\ra (j+m+1)n}\otimes e_{jn+5}.$$
$(3)$ If $r_0=2$, then $\Omega^{2}(Y_t^{(24)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{5s,5s}=w_{(j+m)n\ra (j+m+1)n}\otimes e_{jn+5}.$$
$(4)$ If $r_0=3$, then $\Omega^{3}(Y_t^{(24)})$ is described with
$(8s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<4s$, then $b_{ij}=0$.
If $j=4s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n\ra (j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s<j<5s$, then $b_{ij}=0$.
If $j=5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n\ra (j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s<j<8s$, then $b_{ij}=0$.
$(5)$ If $r_0=4$, then $\Omega^{4}(Y_t^{(24)})$ is described with
$(9s\times 9s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{s,s}=w_{(j+m)n\ra (j+m+1)n}\otimes e_{jn}.$$
$(6)$ If $r_0=5$, then $\Omega^{5}(Y_t^{(24)})$ is described with
$(8s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $b_{ij}=0$.
If $j=2s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n\ra (j+m+1)n}\otimes e_{jn+g(j)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s<j<3s$, then $b_{ij}=0$.
If $j=3s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n\ra (j+m+1)n}\otimes e_{jn+g(j)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s<j<8s$, then $b_{ij}=0$.
$(7)$ If $r_0=6$, then $\Omega^{6}(Y_t^{(24)})$ is described with
$(9s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $j=0$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n\ra (j+m+1)n}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $0<j<8s$, then $b_{ij}=0$.
If $j=8s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n\ra (j+m+1)n}\otimes e_{jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s<j<9s$, then $b_{ij}=0$.
$(8)$ If $r_0=7$, then $\Omega^{7}(Y_t^{(24)})$ is described with
$(8s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<4s$, then $b_{ij}=0$.
If $j=4s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n\ra (j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s<j<5s$, then $b_{ij}=0$.
If $j=5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n\ra (j+m+1)n}\otimes e_{jn+g(j)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s<j<8s$, then $b_{ij}=0$.
$(9)$ If $r_0=8$, then $\Omega^{8}(Y_t^{(24)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{5s,5s}=w_{(j+m)n\ra (j+m+1)n}\otimes e_{jn+5}.$$
$(10)$ If $r_0=9$, then $\Omega^{9}(Y_t^{(24)})$ is described with
$(7s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<s$, then $b_{ij}=0$.
If $j=s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n\ra (j+m+1)n}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s<j<2s$, then $b_{ij}=0$.
If $j=2s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n\ra (j+m+1)n}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s<j<7s$, then $b_{ij}=0$.
$(11)$ If $r_0=10$, then $\Omega^{10}(Y_t^{(24)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{0,0}=w_{(j+m)n\ra (j+m+1)n}\otimes e_{jn}.$$
\medskip
$({\rm II})$ Represent an arbitrary $t_0\in\N$ in the form
$t_0=11\ell_0+r_0$, where $0\le r_0\le 10.$ Then
$\Omega^{t_0}(Y_t^{(24)})$ is a $\Omega^{r_0}(Y_t^{(24)})$, whose
left components twisted by $\sigma^{\ell_0}$, and coefficients
multiplied by $(-1)^{\ell_0}$.
\end{pr}
\begin{pr}[Translates for the case 25]
$({\rm I})$ Let $r_0\in\N$, $r_0<11$. $r_0$-translates of the
elements $Y^{(25)}_t$ are described by the following way.
$(1)$ If $r_0=0$, then $\Omega^{0}(Y_t^{(25)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{s+z_0(0,\ell_0,n),s+z_0(0,\ell_0,n)}=w_{(j+m)n+g(j+s)\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j+s)}.$$
$(2)$ If $r_0=1$, then $\Omega^{1}(Y_t^{(25)})$ is described with
$(7s\times 7s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{z_0(0,\ell_0,n),z_0(0,\ell_0,n)}=w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes e_{jn}.$$
$(3)$ If $r_0=2$, then $\Omega^{2}(Y_t^{(25)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{3s+z_1(0,\ell_0,n),3s+z_1(0,\ell_0,n)}=w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)+1}.$$
$(4)$ If $r_0=3$, then $\Omega^{3}(Y_t^{(25)})$ is described with
$(8s\times 8s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{6s+z_0(0,\ell_0,n),6s+z_0(0,\ell_0,n)}=w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes e_{jn+5}.$$
$(5)$ If $r_0=4$, then $\Omega^{4}(Y_t^{(25)})$ is described with
$(9s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s+z_1(0,\ell_0,n)$, then $b_{ij}=0$.
If $j=2s+z_1(0,\ell_0,n)$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j+s)\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s+z_1(0,\ell_0,n)<j<4s+z_0(0,\ell_0,n)$, then $b_{ij}=0$.
If $j=4s+z_0(0,\ell_0,n)$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s+z_0(0,\ell_0,n)<j<9s$, then $b_{ij}=0$.
$(6)$ If $r_0=5$, then $\Omega^{5}(Y_t^{(25)})$ is described with
$(8s\times 8s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{z_0(0,\ell_0,n),z_0(0,\ell_0,n)}=w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes e_{jn}.$$
$(7)$ If $r_0=6$, then $\Omega^{6}(Y_t^{(25)})$ is described with
$(9s\times 9s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{s+z_0(0,\ell_0,n),s+z_0(0,\ell_0,n)}=w_{(j+m)n+g(j+s)\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j+s)}.$$
$(8)$ If $r_0=7$, then $\Omega^{7}(Y_t^{(25)})$ is described with
$(8s\times 8s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{6s+z_0(0,\ell_0,n),6s+z_0(0,\ell_0,n)}=w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes e_{jn+5}.$$
$(9)$ If $r_0=8$, then $\Omega^{8}(Y_t^{(25)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{3s+z_1(0,\ell_0,n),3s+z_1(0,\ell_0,n)}=w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)+1}.$$
$(10)$ If $r_0=9$, then $\Omega^{9}(Y_t^{(25)})$ is described with
$(7s\times 7s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{3s+z_1(0,\ell_0,n),3s+z_1(0,\ell_0,n)}=w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)+1}.$$
$(11)$ If $r_0=10$, then $\Omega^{10}(Y_t^{(25)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{s+z_1(0,\ell_0,n),s+z_1(0,\ell_0,n)}=w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)}.$$
\medskip
$({\rm II})$ Represent an arbitrary $t_0\in\N$ in the form
$t_0=11\ell_0+r_0$, where $0\le r_0\le 10.$ Then
$\Omega^{t_0}(Y_t^{(25)})$ is a $\Omega^{r_0}(Y_t^{(25)})$, whose
left components twisted by $\sigma^{\ell_0}$, and coefficients
multiplied by $(-1)^{\ell_0}$.
\end{pr}
\begin{pr}[Translates for the case 26]
$({\rm I})$ Let $r_0\in\N$, $r_0<11$. $r_0$-translates of the
elements $Y^{(26)}_t$ are described by the following way.
$(1)$ If $r_0=0$, then $\Omega^{0}(Y_t^{(26)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{s+z_1(0,\ell_0,n),s+z_1(0,\ell_0,n)}=-w_{(j+m)n+g(j+s)\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j+s)}.$$
$(2)$ If $r_0=1$, then $\Omega^{1}(Y_t^{(26)})$ is described with
$(7s\times 7s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{z_1(0,\ell_0,n),z_1(0,\ell_0,n)}=-w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes e_{jn}.$$
$(3)$ If $r_0=2$, then $\Omega^{2}(Y_t^{(26)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{3s+z_0(0,\ell_0,n),3s+z_0(0,\ell_0,n)}=-w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)+1}.$$
$(4)$ If $r_0=3$, then $\Omega^{3}(Y_t^{(26)})$ is described with
$(8s\times 8s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{6s+z_1(0,\ell_0,n),6s+z_1(0,\ell_0,n)}=-w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes e_{jn+5}.$$
$(5)$ If $r_0=4$, then $\Omega^{4}(Y_t^{(26)})$ is described with
$(9s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s+z_0(0,\ell_0,n)$, then $b_{ij}=0$.
If $j=2s+z_0(0,\ell_0,n)$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+g(j+s)\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s+z_0(0,\ell_0,n)<j<4s+z_1(0,\ell_0,n)$, then $b_{ij}=0$.
If $j=4s+z_1(0,\ell_0,n)$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s+z_1(0,\ell_0,n)<j<9s$, then $b_{ij}=0$.
$(6)$ If $r_0=5$, then $\Omega^{5}(Y_t^{(26)})$ is described with
$(8s\times 8s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{z_1(0,\ell_0,n),z_1(0,\ell_0,n)}=-w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes e_{jn}.$$
$(7)$ If $r_0=6$, then $\Omega^{6}(Y_t^{(26)})$ is described with
$(9s\times 9s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{s+z_1(0,\ell_0,n),s+z_1(0,\ell_0,n)}=-w_{(j+m)n+g(j+s)\ra (j+m+1)n+g(j+s)}\otimes e_{jn+g(j+s)}.$$
$(8)$ If $r_0=7$, then $\Omega^{7}(Y_t^{(26)})$ is described with
$(8s\times 8s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{6s+z_1(0,\ell_0,n),6s+z_1(0,\ell_0,n)}=-w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes e_{jn+5}.$$
$(9)$ If $r_0=8$, then $\Omega^{8}(Y_t^{(26)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{3s+z_0(0,\ell_0,n),3s+z_0(0,\ell_0,n)}=-w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)+1}.$$
$(10)$ If $r_0=9$, then $\Omega^{9}(Y_t^{(26)})$ is described with
$(7s\times 7s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{3s+z_0(0,\ell_0,n),3s+z_0(0,\ell_0,n)}=-w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)+1}.$$
$(11)$ If $r_0=10$, then $\Omega^{10}(Y_t^{(26)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{s+z_0(0,\ell_0,n),s+z_0(0,\ell_0,n)}=-w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes e_{jn+g(j+s)}.$$
\medskip
$({\rm II})$ Represent an arbitrary $t_0\in\N$ in the form
$t_0=11\ell_0+r_0$, where $0\le r_0\le 10.$ Then
$\Omega^{t_0}(Y_t^{(26)})$ is a $\Omega^{r_0}(Y_t^{(26)})$, whose
left components twisted by $\sigma^{\ell_0}$, and coefficients
multiplied by $(-1)^{\ell_0}$.
\end{pr}
\begin{pr}[Translates for the case 27]
$({\rm I})$ Let $r_0\in\N$, $r_0<11$. $r_0$-translates of the
elements $Y^{(27)}_t$ are described by the following way.
$(1)$ If $r_0=0$, then $\Omega^{0}(Y_t^{(27)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{3s+z_1(0,\ell_0,n),3s+z_1(0,\ell_0,n)}=-w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)+1}.$$
$(2)$ If $r_0=1$, then $\Omega^{1}(Y_t^{(27)})$ is described with
$(7s\times 7s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{2s+z_1(0,\ell_0,n),2s+z_1(0,\ell_0,n)}=-w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)+1}\otimes e_{jn+g(j)}.$$
$(3)$ If $r_0=2$, then $\Omega^{2}(Y_t^{(27)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{s+z_1(0,\ell_0,n),s+z_1(0,\ell_0,n)}=-w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)}.$$
$(4)$ If $r_0=3$, then $\Omega^{3}(Y_t^{(27)})$ is described with
$(8s\times 8s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{z_1(0,\ell_0,n),z_1(0,\ell_0,n)}=-w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)+1}\otimes e_{jn}.$$
$(5)$ If $r_0=4$, then $\Omega^{4}(Y_t^{(27)})$ is described with
$(9s\times 9s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{6s+z_0(0,\ell_0,n),6s+z_0(0,\ell_0,n)}=-w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j)+1}.$$
$(6)$ If $r_0=5$, then $\Omega^{5}(Y_t^{(27)})$ is described with
$(8s\times 8s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{6s+z_1(0,\ell_0,n),6s+z_1(0,\ell_0,n)}=-w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)+1}\otimes e_{jn+5}.$$
$(7)$ If $r_0=6$, then $\Omega^{6}(Y_t^{(27)})$ is described with
$(9s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<3s+z_0(0,\ell_0,n)$, then $b_{ij}=0$.
If $j=3s+z_0(0,\ell_0,n)$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)+1}\otimes e_{jn+g(j+s)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s+z_0(0,\ell_0,n)<j<5s+z_1(0,\ell_0,n)$, then $b_{ij}=0$.
If $j=5s+z_1(0,\ell_0,n)$, then $$b_{ij}=
\begin{cases}
-w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s+z_1(0,\ell_0,n)<j<9s$, then $b_{ij}=0$.
$(8)$ If $r_0=7$, then $\Omega^{7}(Y_t^{(27)})$ is described with
$(8s\times 8s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{z_1(0,\ell_0,n),z_1(0,\ell_0,n)}=-w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)+1}\otimes e_{jn}.$$
$(9)$ If $r_0=8$, then $\Omega^{8}(Y_t^{(27)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{s+z_1(0,\ell_0,n),s+z_1(0,\ell_0,n)}=-w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+g(j+s)}.$$
$(10)$ If $r_0=9$, then $\Omega^{9}(Y_t^{(27)})$ is described with
$(7s\times 7s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{5s+z_1(0,\ell_0,n),5s+z_1(0,\ell_0,n)}=-w_{(j+m)n+g(j+s)+1\ra (j+m+1)n+g(j+s)+1}\otimes e_{jn+5}.$$
$(11)$ If $r_0=10$, then $\Omega^{10}(Y_t^{(27)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{3s+z_0(0,\ell_0,n),3s+z_0(0,\ell_0,n)}=-w_{(j+m)n+g(j)+1\ra (j+m+1)n+g(j)+1}\otimes e_{jn+g(j+s)+1}.$$
\medskip
$({\rm II})$ Represent an arbitrary $t_0\in\N$ in the form
$t_0=11\ell_0+r_0$, where $0\le r_0\le 10.$ Then
$\Omega^{t_0}(Y_t^{(27)})$ is a $\Omega^{r_0}(Y_t^{(27)})$, whose
left components twisted by $\sigma^{\ell_0}$, and coefficients
multiplied by $(-1)^{\ell_0}$.
\end{pr}
\begin{pr}[Translates for the case 28]
$({\rm I})$ Let $r_0\in\N$, $r_0<11$. $r_0$-translates of the
elements $Y^{(28)}_t$ are described by the following way.
$(1)$ If $r_0=0$, then $\Omega^{0}(Y_t^{(28)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{5s,5s}=w_{(j+m)n+5\ra (j+m+1)n+5}\otimes e_{jn+5}.$$
$(2)$ If $r_0=1$, then $\Omega^{1}(Y_t^{(28)})$ is described with
$(7s\times 7s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<4s$, then $b_{ij}=0$.
If $j=4s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+5}\otimes e_{jn+g(j)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s<j<5s$, then $b_{ij}=0$.
If $j=5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+5}\otimes e_{jn+g(j)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s<j<7s$, then $b_{ij}=0$.
$(3)$ If $r_0=2$, then $\Omega^{2}(Y_t^{(28)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{0,0}=w_{(j+m)n+5\ra (j+m+1)n+5}\otimes e_{jn}.$$
$(4)$ If $r_0=3$, then $\Omega^{3}(Y_t^{(28)})$ is described with
$(8s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $b_{ij}=0$.
If $j=2s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+5}\otimes e_{jn+g(j)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s<j<3s$, then $b_{ij}=0$.
If $j=3s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+5}\otimes e_{jn+g(j)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s<j<8s$, then $b_{ij}=0$.
$(5)$ If $r_0=4$, then $\Omega^{4}(Y_t^{(28)})$ is described with
$(9s\times 9s)$-matrix with the following elements $b_{ij}${\rm:}
If $j=0$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+5}\otimes e_{jn},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $0<j<8s$, then $b_{ij}=0$.
If $j=8s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+5}\otimes e_{jn+5},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s<j<9s$, then $b_{ij}=0$.
$(6)$ If $r_0=5$, then $\Omega^{5}(Y_t^{(28)})$ is described with
$(8s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<4s$, then $b_{ij}=0$.
If $j=4s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+5}\otimes e_{jn+g(j)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s<j<5s$, then $b_{ij}=0$.
If $j=5s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+5}\otimes e_{jn+g(j)+1},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s<j<8s$, then $b_{ij}=0$.
$(7)$ If $r_0=6$, then $\Omega^{6}(Y_t^{(28)})$ is described with
$(9s\times 9s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{7s,7s}=w_{(j+m)n+5\ra (j+m+1)n+5}\otimes e_{jn+5}.$$
$(8)$ If $r_0=7$, then $\Omega^{7}(Y_t^{(28)})$ is described with
$(8s\times 8s)$-matrix with the following elements $b_{ij}${\rm:}
If $0\le j<2s$, then $b_{ij}=0$.
If $j=2s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+5}\otimes e_{jn+g(j)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s<j<3s$, then $b_{ij}=0$.
If $j=3s$, then $$b_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+5}\otimes e_{jn+g(j)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s<j<8s$, then $b_{ij}=0$.
$(9)$ If $r_0=8$, then $\Omega^{8}(Y_t^{(28)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{0,0}=w_{(j+m)n+5\ra (j+m+1)n+5}\otimes e_{jn}.$$
$(10)$ If $r_0=9$, then $\Omega^{9}(Y_t^{(28)})$ is described with
$(7s\times 7s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{0,0}=w_{(j+m)n+5\ra (j+m+1)n+5}\otimes e_{jn}.$$
$(11)$ If $r_0=10$, then $\Omega^{10}(Y_t^{(28)})$ is described with
$(6s\times 6s)$-matrix with one nonzero element that is of the
following form{\rm:}
$$b_{5s,5s}=w_{(j+m)n+5\ra (j+m+1)n+5}\otimes e_{jn+5}.$$
\medskip
$({\rm II})$ Represent an arbitrary $t_0\in\N$ in the form
$t_0=11\ell_0+r_0$, where $0\le r_0\le 10.$ Then
$\Omega^{t_0}(Y_t^{(28)})$ is a $\Omega^{r_0}(Y_t^{(28)})$, whose
left components twisted by $\sigma^{\ell_0}$, and coefficients
multiplied by $(-1)^{\ell_0}$.
\end{pr}
\begin{proof}
We prove the proposition \ref{shifts_3}, the other propositions are
proved similarly. Show that the following squares are commutative:
$$\begin{CD} Q_{t+t_0} @>{d_{t+t_0-1}}>>Q_{t+t_0-1} \\ @V{f_{t_0}}VV @VV{f_{t_0-1}}V\\ Q_{t_0} @>{d_{t_0-1}}>> Q_{t_0-1}.\end{CD}$$
We shall describe the matrixes of products $d_{t_0-1}f_{t_0}$ and
see that they coincide with $f_{t_0-1}d_{t+t_0-1}$.
$(1)$ If $t_0<11$ and $r_0=0$, then the product $d_{r_0-1}f_{r_0}$
is an $(7s\times 6s)$-matrix $C=(c_{ij})$ with the following
elements $c_{ij}$:
If $0\le j<6s+(-3\ell_0)_s$, then $c_{ij}=0$.
If $j=6s+(-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
-\kappa w_{(j+m)n\ra (j+m+1)n}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
-\kappa w_{(j+m)n+g(j)\ra (j+m+1)n}\otimes w_{jn+5\ra (j+1)n+g(j)},\quad i=(j+1)_s+s;\\
-\kappa w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+2s;\\
-\kappa w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes w_{jn+5\ra (j+1)n+g(j)+1},\quad i=(j+1)_s+3s;\\
-\kappa w_{(j+m)n+g(j+s)+1\ra (j+m+1)n}\otimes w_{jn+5\ra (j+1)n+g(j+s)+1},\quad i=(j+1)_s+4s;\\
-\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn+5\ra (j+1)n+5},\quad i=(j+1)_s+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s+(-3\ell_0)_s<j<7s$, then $c_{ij}=0$.
$(2)$ If $t_0<11$ and $r_0=1$, then the product $d_{r_0-1}f_{r_0}$
is an $(6s\times 6s)$-matrix $C=(c_{ij})$ with the following
elements $c_{ij}$:
If $0\le j<s+(-3\ell_0)_s$, then $c_{ij}=0$.
If $j=s+(-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s+(-3\ell_0)_s<j<2s+(-3\ell_0)_s$, then $c_{ij}=0$.
If $j=2s+(-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s+(-3\ell_0)_s<j<3s+(-3\ell_0)_s$, then $c_{ij}=0$.
If $j=3s+(-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s+(-3\ell_0)_s<j<4s+(-3\ell_0)_s$, then $c_{ij}=0$.
If $j=4s+(-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s+(-3\ell_0)_s<j<5s+(-1-3\ell_0)_s$, then $c_{ij}=0$.
If $j=5s+(-1-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n+5\ra (j+m+2)n}\otimes w_{jn+5\ra (j+1)n+5},\quad i=(j+1)_s+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s+(-1-3\ell_0)_s<j<6s$, then $c_{ij}=0$.
$(3)$ If $t_0<11$ and $r_0=2$, then the product $d_{r_0-1}f_{r_0}$
is an $(8s\times 7s)$-matrix $C=(c_{ij})$ with the following
elements $c_{ij}$:
If $0\le j<(-3\ell_0)_s$, then $c_{ij}=0$.
If $j=(-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
w_{(j+m-1)n+g(j)+1\ra (j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j)},\quad i=j+2s;\\
w_{(j+m-1)n+5\ra (j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+4s;\\
-w_{(j+m-1)n+5\ra (j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $(-3\ell_0)_s<j<s+(-3\ell_0)_s$, then $c_{ij}=0$.
If $j=s+(-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
-w_{(j+m-1)n+g(j)+1\ra (j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j)},\quad i=j+2s;\\
w_{(j+m-1)n+5\ra (j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+3s;\\
-w_{(j+m-1)n+5\ra (j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s+(-3\ell_0)_s<j<6s+(-1-3\ell_0)_s$, then $c_{ij}=0$.
If $j=6s+(-1-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes w_{jn+5\ra (j+1)n+g(j+s)+1},\quad i=(j+1)_s+5s;\\
w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes w_{jn+5\ra (j+1)n+5},\quad i=(j+1)_s+6s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s+(-1-3\ell_0)_s<j<7s+(-1-3\ell_0)_s$, then $c_{ij}=0$.
If $j=7s+(-1-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j)}\otimes w_{jn+5\ra (j+1)n+g(j+s)+1},\quad i=(j+1)_s+4s;\\
w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes w_{jn+5\ra (j+1)n+5},\quad i=(j+1)_s+6s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s+(-1-3\ell_0)_s<j<8s$, then $c_{ij}=0$.
$(4)$ If $t_0<11$ and $r_0=3$, then the product $d_{r_0-1}f_{r_0}$
is an $(9s\times 6s)$-matrix $C=(c_{ij})$ with the following
elements $c_{ij}$:
If $0\le j<6s+(-1-3\ell_0)_s$, then $c_{ij}=0$.
If $j=6s+(-1-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j)+1\ra (j+1)n},\quad i=(j+1)_s;\\
w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j)+1\ra (j+1)n+g(j)+1},\quad i=(j+1)_s+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s+(-1-3\ell_0)_s<j<7s+(-1-3\ell_0)_s$, then $c_{ij}=0$.
If $j=7s+(-1-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j)+1\ra (j+1)n},\quad i=(j+1)_s;\\
w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j)+1\ra (j+1)n+g(j)+1},\quad i=(j+1)_s+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $7s+(-1-3\ell_0)_s<j<8s+(-1-3\ell_0)_s$, then $c_{ij}=0$.
If $j=8s+(-1-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
\kappa w_{(j+m)n+5\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
-\kappa w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j)+1},\quad i=(j+1)_s+3s;\\
\kappa w_{(j+m+1)n+g(j)\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j+s)+1},\quad i=(j+1)_s+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $8s+(-1-3\ell_0)_s<j<9s$, then $c_{ij}=0$.
$(5)$ If $t_0<11$ and $r_0=4$, then the product $d_{r_0-1}f_{r_0}$
is an $(8s\times 8s)$-matrix $C=(c_{ij})$ with the following
elements $c_{ij}$:
If $0\le j<2s+(-1-3\ell_0)_s$, then $c_{ij}=0$.
If $j=2s+(-1-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j+s)+1\ra (j+m+1)n}\otimes w_{jn+g(j)\ra (j+1)n},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s+(-1-3\ell_0)_s<j<3s+(-1-3\ell_0)_s$, then $c_{ij}=0$.
If $j=3s+(-1-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j+s)+1\ra (j+m+1)n}\otimes w_{jn+g(j)\ra (j+1)n},\quad i=(j+1)_s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s+(-1-3\ell_0)_s<j<4s+(-1-3\ell_0)_s$, then $c_{ij}=0$.
If $j=4s+(-1-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes w_{jn+g(j)+1\ra (j+1)n},\quad i=(j+1)_s+s;\\
\kappa w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=j+2s;\\
-\kappa w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s+(-1-3\ell_0)_s<j<5s+(-1-3\ell_0)_s$, then $c_{ij}=0$.
If $j=5s+(-1-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes w_{jn+g(j)+1\ra (j+1)n},\quad i=(j+1)_s;\\
-\kappa w_{(j+m)n+g(j+s)\ra (j+m)n+5}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=j+s;\\
\kappa w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes w_{jn+g(j)+1\ra jn+5},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s+(-1-3\ell_0)_s<j<8s$, then $c_{ij}=0$.
$(6)$ If $t_0<11$ and $r_0=5$, then the product $d_{r_0-1}f_{r_0}$
is an $(9s\times 9s)$-matrix $C=(c_{ij})$ with the following
elements $c_{ij}$:
If $0\le j<s+(-1-3\ell_0)_s$, then $c_{ij}=0$.
If $j=s+(-1-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s+(-1-3\ell_0)_s<j<2s+(-1-3\ell_0)_s$, then $c_{ij}=0$.
If $j=2s+(-1-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
-w_{(j+m+1)n\ra (j+m+1)n+g(j+s)}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s+(-1-3\ell_0)_s<j<3s+(-1-3\ell_0)_s$, then $c_{ij}=0$.
If $j=3s+(-1-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s+s;\\
-w_{(j+m)n+5\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s+(-1-3\ell_0)_s<j<4s+(-1-3\ell_0)_s$, then $c_{ij}=0$.
If $j=4s+(-1-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
-w_{(j+m+1)n\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s+s;\\
w_{(j+m)n+5\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s+(-1-3\ell_0)_s<j<5s+(-1-3\ell_0)_s$, then $c_{ij}=0$.
If $j=5s+(-1-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n},\quad i=(j+1)_s;\\
w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s+(-1-3\ell_0)_s<j<6s+(-1-3\ell_0)_s$, then $c_{ij}=0$.
If $j=6s+(-1-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
-w_{(j+m)n+5\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $6s+(-1-3\ell_0)_s<j<9s$, then $c_{ij}=0$.
$(7)$ If $t_0<11$ and $r_0=6$, then the product $d_{r_0-1}f_{r_0}$
is an $(8s\times 8s)$-matrix $C=(c_{ij})$ with the following
elements $c_{ij}$:
If $0\le j<(-1-3\ell_0)_s$, then $c_{ij}=0$.
If $j=(-1-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
w_{(j+m)n+g(j)\ra (j+m)n+g(j)+1}\otimes w_{jn\ra (j+1)n},\quad i=(j+1)_s;\\
w_{(j+m)n\ra (j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+3s;\\
-w_{(j+m-1)n+5\ra (j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+4s;\\
-w_{(j+m-1)n+5\ra (j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $(-1-3\ell_0)_s<j<s+(-1-3\ell_0)_s$, then $c_{ij}=0$.
If $j=s+(-1-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
w_{(j+m)n+g(j)\ra (j+m)n+g(j)+1}\otimes w_{jn\ra (j+1)n},\quad i=(j+1)_s+s;\\
w_{(j+m)n\ra (j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+s;\\
-w_{(j+m-1)n+5\ra (j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j+s)+1},\quad i=j+3s;\\
-w_{(j+m-1)n+5\ra (j+m)n+g(j)+1}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s+(-1-3\ell_0)_s<j<2s+(-1-3\ell_0)_s$, then $c_{ij}=0$.
If $j=2s+(-1-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes w_{jn+g(j)\ra (j+1)n},\quad i=(j+1)_s;\\
\kappa w_{(j+m)n\ra (j+m)n+5}\otimes e_{jn+g(j)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s+(-1-3\ell_0)_s<j<3s+(-1-3\ell_0)_s$, then $c_{ij}=0$.
If $j=3s+(-1-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j)\ra (j+m)n+5}\otimes w_{jn+g(j)\ra (j+1)n},\quad i=(j+1)_s+s;\\
\kappa w_{(j+m)n\ra (j+m)n+5}\otimes e_{jn+g(j)},\quad i=j;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s+(-1-3\ell_0)_s<j<8s$, then $c_{ij}=0$.
$(8)$ If $t_0<11$ and $r_0=7$, then the product $d_{r_0-1}f_{r_0}$
is an $(6s\times 9s)$-matrix $C=(c_{ij})$ with the following
elements $c_{ij}$:
If $0\le j<s+(-2-3\ell_0)_s$, then $c_{ij}=0$.
If $j=s+(-2-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
-w_{(j+m+1)n\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s+(-2-3\ell_0)_s<j<2s+(-2-3\ell_0)_s$, then $c_{ij}=0$.
If $j=2s+(-2-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
-w_{(j+m+1)n\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
w_{(j+m+1)n+g(j+s)\ra (j+m+1)n+g(j+s)+1}\otimes w_{jn+g(j+s)\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s+(-2-3\ell_0)_s<j<5s+(-2-3\ell_0)_s$, then $c_{ij}=0$.
If $j=5s+(-2-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
\kappa w_{(j+m+1)n\ra (j+m+2)n}\otimes w_{jn+5\ra (j+1)n},\quad i=(j+1)_s;\\
-\kappa w_{(j+m+1)n+g(j+s)\ra (j+m+2)n}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s;\\
-\kappa w_{(j+m+1)n+g(j)\ra (j+m+2)n}\otimes w_{jn+5\ra (j+1)n+g(j)},\quad i=(j+1)_s+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s+(-2-3\ell_0)_s<j<6s$, then $c_{ij}=0$.
$(9)$ If $t_0<11$ and $r_0=8$, then the product $d_{r_0-1}f_{r_0}$
is an $(7s\times 8s)$-matrix $C=(c_{ij})$ with the following
elements $c_{ij}$:
If $0\le j<(-2-3\ell_0)_s$, then $c_{ij}=0$.
If $j=(-2-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
\kappa w_{(j+m)n+g(j)+1\ra (j+m)n+5}\otimes w_{jn\ra (j+1)n},\quad i=(j+1)_s;\\
-\kappa w_{(j+m)n+g(j+s)+1\ra (j+m)n+5}\otimes w_{jn\ra (j+1)n},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $(-2-3\ell_0)_s<j<s+(-2-3\ell_0)_s$, then $c_{ij}=0$.
If $j=s+(-2-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j+s)+1\ra (j+m+1)n}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s;\\
-\kappa w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=j+5s;\\
-\kappa w_{(j+m)n+g(j)\ra (j+m+1)n}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=j+6s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s+(-2-3\ell_0)_s<j<2s+(-2-3\ell_0)_s$, then $c_{ij}=0$.
If $j=2s+(-2-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
-\kappa w_{(j+m)n+g(j+s)+1\ra (j+m+1)n}\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s+s;\\
-\kappa w_{(j+m)n+g(j)\ra (j+m+1)n}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=j+4s;\\
-\kappa w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s+(-2-3\ell_0)_s<j<3s+(-2-3\ell_0)_s$, then $c_{ij}=0$.
If $j=3s+(-2-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=j+4s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s+(-2-3\ell_0)_s<j<4s+(-2-3\ell_0)_s$, then $c_{ij}=0$.
If $j=4s+(-2-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s+(-2-3\ell_0)_s<j<7s$, then $c_{ij}=0$.
$(10)$ If $t_0<11$ and $r_0=9$, then the product $d_{r_0-1}f_{r_0}$
is an $(6s\times 6s)$-matrix $C=(c_{ij})$ with the following
elements $c_{ij}$:
If $0\le j<(-2-3\ell_0)_s$, then $c_{ij}=0$.
If $j=(-2-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
\kappa w_{(j+m)n+5\ra (j+m+1)n}\otimes w_{jn\ra (j+1)n},\quad i=(j+1)_s;\\
\kappa w_{(j+m)n+g(j)+1\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
-\kappa w_{(j+m)n+g(j+s)\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j)+1},\quad i=j+3s;\\
\kappa e_{(j+m+1)n}\otimes w_{jn\ra jn+5},\quad i=j+5s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $(-2-3\ell_0)_s<j<2s+(-2-3\ell_0)_s$, then $c_{ij}=0$.
If $j=2s+(-2-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
w_{(j+m)n+g(j)\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j+s)\ra jn+g(j+s)+1},\quad i=j+2s;\\
-w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j+s)\ra jn+5},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s+(-2-3\ell_0)_s<j<6s$, then $c_{ij}=0$.
$(11)$ If $t_0<11$ and $r_0=10$, then the product $d_{r_0-1}f_{r_0}$
is an $(6s\times 7s)$-matrix $C=(c_{ij})$ with the following
elements $c_{ij}$:
If $0\le j<(-2-3\ell_0)_s$, then $c_{ij}=0$.
If $j=(-2-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
\kappa w_{(j+m)n\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j)},\quad i=j+s;\\
-\kappa w_{(j+m)n\ra (j+m+1)n}\otimes w_{jn\ra jn+g(j+s)},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $(-2-3\ell_0)_s<j<s+(-3-3\ell_0)_s$, then $c_{ij}=0$.
If $j=s+(-3-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j+s)\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $s+(-3-3\ell_0)_s<j<2s+(-3-3\ell_0)_s$, then $c_{ij}=0$.
If $j=2s+(-3-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
w_{(j+m+1)n\ra (j+m+1)n+g(j)}\otimes w_{jn+g(j+s)\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $2s+(-3-3\ell_0)_s<j<3s+(-3-3\ell_0)_s$, then $c_{ij}=0$.
If $j=3s+(-3-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
-w_{(j+m+1)n\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $3s+(-3-3\ell_0)_s<j<4s+(-3-3\ell_0)_s$, then $c_{ij}=0$.
If $j=4s+(-3-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
-w_{(j+m+1)n\ra (j+m+1)n+g(j)+1}\otimes w_{jn+g(j+s)+1\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $4s+(-3-3\ell_0)_s<j<5s+(-3-3\ell_0)_s$, then $c_{ij}=0$.
If $j=5s+(-3-3\ell_0)_s$, then $$c_{ij}=
\begin{cases}
-\kappa w_{(j+m+1)n\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j+s)},\quad i=(j+1)_s+s;\\
\kappa w_{(j+m+1)n\ra (j+m+1)n+5}\otimes w_{jn+5\ra (j+1)n+g(j)},\quad i=(j+1)_s+2s;\\
0,\quad\text{otherwise.}\end{cases}$$
If $5s+(-3-3\ell_0)_s<j<6s$, then $c_{ij}=0$.
$(12)$ If $t_0\ge 11$, then the product matrix $d_{t_0-1}f_{t_0}$ is
a $d_{r_0-1}f_{r_0}$, whose left components twisted by
$\sigma^{\ell_0}$, and coefficients multiplied by $(-1)^{\ell_0}$.
The described matrixes of products $d_{t_0-1}f_{t_0}$ coincide with
$f_{t_0-1}d_{t+t_0-1}$.
\end{proof}
\section{Multiplications in $\HH^*(R)$}
From the descriptions of elements $Y^{(i)}_t$ and
its$\Omega$-translates we can find multiplications of the elements
using the formula \eqref{mult_formula}.
To find the multiplications we need the description of $\sigma^t$
for an arbitrary $t\in\N$. From the description of an automorphism
$\sigma$ we have:
\begin{align*}
\sigma^{2t}(\a_i)&=\begin{cases} (-1)^t\a_{i+6tn},\quad i\equiv
0,2(3);\\\a_{i+6tn},\quad i\equiv 1(3),
\end{cases}
\sigma^{2t}(\b_i)=\begin{cases} (-1)^t\b_{i+6tn},\quad i\equiv
0,2(3);\\\b_{i+6tn},\quad i\equiv 1(3),
\end{cases}\\
\sigma^{2t+1}(\a_i)&=\begin{cases} (-1)^{t+1}\b_{i+3(2t+1)n},\quad
i\equiv 0(3);\\-\b_{i+3(2t+1)n},\quad i\equiv 1(3);\\
(-1)^t\b_{i+3(2t+1)n},\quad i\equiv 2(3),
\end{cases}
\sigma^{2t+1}(\b_i)=\begin{cases} (-1)^t\a_{i+3(2t+1)n},\quad
i\equiv 0(3);\\-\a_{i+3(2t+1)n},\quad i\equiv 1(3);\\
(-1)^{t+1}\a_{i+3(2t+1)n},\quad i\equiv 2(3),
\end{cases}\\
\sigma^t(\g_i)&=(-1)^t\g_{i+tn}.
\end{align*}
We will find a multiplication of elements of the types 4 and 3 for
$s>1$.
Consider two arbitrary elements $Y_{t_4}^{(4)}$ and $Y_{t_3}^{(3)}$.
For its degrees $t_4$ and $t_3$ we have:
\begin{align*}
t_4&=11\ell_4+1,\text{ }\ell_4 n\equiv 0(s),\text{ }\ell_4\ndiv 2;\\
t_3&=11\ell_3+1,\text{ }\ell_3 n\equiv 0(s),\text{ }\ell_3\div 2.
\end{align*}
Let $t=t_4+t_3$; this is the degree of an element
$Y_{t_4}^{(4)}Y_{t_3}^{(3)}$. Then $t=11(\ell_4+\ell_3)+2$. Group of
the degree $t$ has type (5).
Denote by $B=(b_{ij})$ translate matrix of an element
$Y_{t_4}^{(4)}$ by degree $t_3$. This matrix is of the form.
If $0\le j<s$, then
\begin{align*}
b_{ij}&=
\begin{cases}
\kappa \sigma^{\ell_3}(w_{(j+m_4)n+g(j+s)\ra (j+m_4)n+5})\otimes e_{jn},\quad i=j+s;\\
\kappa \sigma^{\ell_3}(w_{(j+m_4)n+g(j+s)+1\ra (j+m_4)n+5})\otimes w_{jn\ra jn+g(j+s)},\quad i=j+3s;\\
0,\quad\text{otherwise;}\end{cases}\\
&=\begin{cases}
\kappa \sigma^{\ell_3}(w_{(j+m_4)n+3\ra (j+m_4)n+5})\otimes e_{jn},\quad i=j+s;\\
\kappa \sigma^{\ell_3}(w_{(j+m_4)n+4\ra (j+m_4)n+5})\otimes w_{jn\ra jn+3},\quad i=j+3s;\\
0,\quad\text{otherwise;}\end{cases}\\
&=\begin{cases}
\kappa \sigma^{\ell_3}(\b_{3(j+m_4)+2}\b_{3(j+m_4)+1})\otimes e_{jn},\quad i=j+s;\\
\kappa \sigma^{\ell_3}(\b_{3(j+m_4)+2})\otimes \b_{3j},\quad i=j+3s;\\
0,\quad\text{otherwise;}\end{cases}\\
&=\begin{cases}
(-1)^{\frac{\ell_4-1}{2}}(-1)^{\frac{\ell_3}{2}} \b_{3(j+\ell_3n)+2}\b_{3(j+\ell_3n)+1}\otimes e_{jn},\quad i=j+s;\\
(-1)^{\frac{\ell_4-1}{2}}(-1)^{\frac{\ell_3}{2}} \b_{3(j+\ell_3n)+2}\otimes \b_{3j},\quad i=j+3s;\\
0,\quad\text{otherwise;}\end{cases}\\
&=\begin{cases}
(-1)^{\frac{\ell_4+\ell_3-1}{2}}\b_{3j+2}\b_{3j+1}\otimes e_{jn},\quad i=j+s;\\
(-1)^{\frac{\ell_4+\ell_3-1}{2}}\b_{3j+2}\otimes \b_{3j},\quad i=j+3s;\\
0,\quad\text{otherwise.}\end{cases}
\end{align*}
If $s\le j<2s$, then
\begin{align*}b_{ij}&=
\begin{cases}
\sigma^{\ell_3}(w_{(j+m_4+1)n+g(j)\ra (j+m_4+1)n+g(j)+1})\otimes w_{jn+g(j+s)\ra (j+1)n},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise;}\end{cases}\\
&=\begin{cases}\sigma^{\ell_3}(w_{(j+m_4+1)n+3\ra (j+m_4+1)n+4})\otimes w_{jn+1\ra (j+1)n},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise;}\end{cases}\\
\end{align*}
\begin{align*}&=\begin{cases}
\sigma^{\ell_3}(\b_{3(j+m_4+1)+1})\otimes \g_j\a_{3j+2}\a_{3j+1},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise;}\end{cases}\\
&=\begin{cases}
\b_{3(j+1)+1}\otimes \g_j\a_{3j+2}\a_{3j+1},\quad i=(j+1)_s+s;\\
0,\quad\text{otherwise.}\end{cases}
\end{align*}
If $2s\le j<4s$, then $b_{ij}=0$.
If $4s\le j<5s$, then
\begin{align*}b_{ij}&=
\begin{cases}
\sigma^{\ell_3}(w_{(j+m_4)n+5\ra (j+m_4+1)n+g(j+s)})\otimes e_{jn+g(j+s)+1},\quad i=j+s;\\
\sigma^{\ell_3}(w_{(j+m_4+1)n\ra (j+m_4+1)n+g(j+s)})\otimes w_{jn+g(j+s)+1\ra jn+5},\quad i=j+2s;\\
0,\quad\text{otherwise;}\end{cases}\\
&=\begin{cases}
\sigma^{\ell_3}(\b_{3(j+m_4+1)}\g_{j+m})\otimes e_{jn+4},\quad i=j+s;\\
\sigma^{\ell_3}(\b_{3(j+m_4+1)})\otimes \b_{3j+2},\quad i=j+2s;\\
0,\quad\text{otherwise;}\end{cases}\\
&=\begin{cases}
(-1)^{\frac{\ell_3}{2}}\b_{3(j+1)}\g_j\otimes e_{jn+4},\quad i=j+s;\\
(-1)^{\frac{\ell_3}{2}}\b_{3(j+1)}\otimes \b_{3j+2},\quad i=j+2s;\\
0,\quad\text{otherwise.}\end{cases}
\end{align*}
If $5s\le j<6s$, then
\begin{align*}b_{ij}&=
\begin{cases}
\kappa \sigma^{\ell_3}(w_{(j+m_4+1)n\ra (j+m_4+2)n})\otimes e_{jn+5},\quad i=j+s;\\
0,\quad\text{otherwise;}\end{cases}\\
&=\begin{cases}
\kappa \sigma^{\ell_3}(\g_{j+m_4+1}\a_{3(j+m_4+1)+2}\a_{3(j+m_4+1)+1}\a_{3(j+m_4+1)})\otimes e_{jn+5},\quad i=j+s;\\
0,\quad\text{otherwise;}\end{cases}\\
&=\begin{cases}
(-1)^{\frac{\ell_4-1}{2}}\g_{j+1}\a_{3(j+1)+2}\a_{3(j+1)+1}\a_{3(j+1)}\otimes e_{jn+5},\quad i=j+s;\\
0,\quad\text{otherwise.}\end{cases}
\end{align*}
Multiply this matrix by an element $Y_{t_3}^{(3)}$, which is
$(7s\times 6s)$-matrix with single nonzero element
$y_{5s,6s}=(-1)^{\frac{\ell_3}{2}}\g_j\otimes e_{jn+5}.$
Multiplication is the matrix $C=(c_{ij})$ with single nonzero
element $c_{5s,4s}=\b_{3(j+1)}\g_j\otimes \b_{3j+2}.$ We must show
that this element coincide with $Y^{(5)}_t$ for degree of type (5).
Element $Y^{(5)}_t$ is matrix with single nonzero element
$y_{3s,3s}=\a_{3(j+1)}\g_j\a_{3j+2}\otimes e_{jn+2}.$ Thus we must
show that
$Y^{(5)}_t-C=\a_{3(j+1)}\g_j\a_{3j+2}-\b_{3(j+1)}\g_j\b_{3j+2}$ is
in $\Im\delta^{t-1}$.
Consider the matrix $D=(d)_{ij}$, which is obtained from
differential matrix $d_{t-1}$ by replacing elements which turns into
$0$ in $\Im\delta^{t-1}$, by $0$. Elements of this matrix are of the
form.
If $0\le j<s$, then
\begin{align*}(d)_{ij}&=\begin{cases}
\sigma^{\ell_3+\ell_4}(w_{jn+1+j_1+2f(j_1,2)\ra jn+5})\otimes w_{jn\ra jn+j_1},\\\quad\quad\quad i=j+2sj_1,\text{ }0\le j_1< 3;\\
-\sigma^{\ell_3+\ell_4}(w_{jn+3+j_1\ra jn+5})\otimes w_{jn\ra jn+j_1+2(1-f(j_1,0))},\\\quad\quad\quad i=j+2sj_1+s,\text{ }0\le j_1< 3;\\
0\quad\text{otherwise;}\end{cases}\\
&=\begin{cases}
(-1)^{\frac{\ell_3+\ell_4+1}{2}}\b_{3j+2}\b_{3j+1}\otimes e_{jn},\quad i=j;\\
e_{jn+5}\otimes \a_{3j+1}\a_{3j},\quad i=j+4s;\\
(-1)^{\frac{\ell_3+\ell_4+1}{2}}\a_{3j+2}\a_{3j+1}\otimes e_{jn},\quad i=j+s;\\
-e_{jn+5}\otimes \b_{3j+1}\b_{3j},\quad i=j+5s;\\
0\quad\text{otherwise.}\end{cases}
\end{align*}
If $s\le j<3s$, then $(d)_{ij}=0$.
If $3s\le j<5s$, then $$(d)_{ij}=\begin{cases}
(-1)^{\frac{\ell_3+\ell_4+1}{2}}\a_{3(j+1)}\g_j\otimes e_{jn+1},\quad i=j+s,\text{ }j<4s;\\
(-1)^{\frac{\ell_3+\ell_4-1}{2}}\b_{3(j+1)}\g_j\otimes e_{jn+4},\quad i=j+s,\text{ }j\ge 4s;\\
(-1)^{\frac{\ell_3+\ell_4-1}{2}}\a_{3(j+1)}\otimes \a_{3j+1},\quad i=(j)_s+6s,\text{ }j<4s;\\
(-1)^{\frac{\ell_3+\ell_4+1}{2}}\b_{3(j+1)}\otimes \b_{3j+1},\quad i=(j)_s+6s,\text{ }j\ge 4s;\\
e_{(j+1)n+g(j)}\otimes w_{jn+g(j+s)+1\ra}^{(2)},\quad i=(j+1)_s+sf_0(j,4s);\\
0\quad\text{otherwise.}\end{cases}$$
If $5s\le j<6s$, then $(d)_{ij}=0$.
$6s$-th row of image matrix multiplied by $(-1)^{\frac{\ell_3+\ell_4-1}{2}}$, coincide with
$Y^{(5)}_t-C$, hence, $Y^{(5)}_t-Y_{t_4}^{(4)}Y_{t_3}^{(3)}\in\Im\delta^{t-1}$, i. e.
$Y^{(5)}_t-Y_{t_4}^{(4)}Y_{t_3}^{(3)}=0$ in the cohomology ring.
Multiplications of other elements, except $Y^{(5)}$, $Y^{(10)}$, $Y^{(17)}$, $Y^{(19)}$ and
$Y^{(21)}$, are similarly considered. To get the whole picture we should prove the following lemma.
\begin{lem}$\text{ }$
$($a$)$ Let $Y^{(5)}$ be an arbitrary element from generators of the
corresponding type. Then there are elements $Y^{(3)}$ and $Y^{(4)}$
such as $Y^{(5)}=Y^{(3)}Y^{(4)}$.
$($b$)$ Let $Y^{(10)}$ be an arbitrary element from generators of
the corresponding type. Then there are elements $Y^{(3)}$ and
$Y^{(6)}$ such as $Y^{(10)}=Y^{(3)}Y^{(6)}$.
$($c$)$ Let $Y^{(17)}$ be an arbitrary element from generators of
the corresponding type. Then there are elements $Y^{(3)}$ and
$Y^{(15)}$ such as $Y^{(17)}=Y^{(3)}Y^{(15)}$.
$($d$)$ Let $Y^{(19)}$ be an arbitrary element from generators of
the corresponding type. Then there are elements $Y^{(3)}$ and
$Y^{(18)}$ such as $Y^{(19)}=Y^{(3)}Y^{(18)}$.
$($e$)$ Let $Y^{(21)}$ be an arbitrary element from generators of
the corresponding type. Then there are elements $Y^{(3)}$ and
$Y^{(20)}$ such as $Y^{(21)}=Y^{(3)}Y^{(20)}$.
\end{lem}
\begin{proof}
The degree 1 has type 3, for all $s$. It only remains to use the
relations for type (3).
\end{proof}
|
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\section{Introduction}
Computational basis states for quantum computing are usually taken from quantized energy levels.
An alternative approach to it is to use oscillator states consisting of multiple quanta (photons or phonons).
In this approach, multiple number states are used to represent each basis state, like logical qubits
for quantum error correction~\cite{Nielsen}.
Thus, the oscillator approach to quantum computing will offer qubits robust to noises or
hardware-efficient quantum error-correcting codes, which are known as
bosonic codes~\cite{Gottesman2001a,Cochrane1999a,Michael2016a,Ofek2016a,Chou2018a,Hu2019a,Fluhmann2019a,Campagne2020a}.
One of such oscillator approaches is to use parametric oscillators.
Their two stable oscillating states can be used for qubits.
There are two types of parametric-oscillator approach:
dissipative and Kerr.
In the dissipative-type approach,
parametrically two-photon driven oscillators with large two-photon loss (larger than single-photon loss)
are used for qubits~\cite{Mirrahimi2014a,Albert2016a}.
Such a two-photon dissipative parametric oscillator becomes a Schr\"{o}dinger cat state
(superposition of two oscillating states) as a steady state~\cite{Mirrahimi2014a,Albert2016a,Milburn},
which has been demonstrated experimentally using superconducting circuits~\cite{Leghtas2015a,Touzard2018a,Lescanne2020a}.
Since this type of qubit is insensitive to bit-flip errors~\cite{Lescanne2020a},
this approach is expected to be useful for fault-tolerant quantum computation~\cite{Guillaud2019a,Chamberland2020a}.
In the Kerr-type approach,
parametrically two-photon driven oscillators with large Kerr nonlinearity~\cite{Milburn1991a,Wielinga1993a},
which we call Kerr-nonlinear parametric oscillators (KPOs)~\cite{Goto2016a,Goto2019a},
are used for qubits.
Low-loss KPOs have recently been realized experimentally using
superconducting circuits~\cite{Wang2019a,Grimm2020a,Yamaji2020a}.
An ideal KPO is lossless (nondissipative), and it can generate a Schr\"{o}dinger cat state from the vacuum state
via quantum adiabatic evolution (quantum bifurcation)~\cite{Cochrane1999a,Goto2016a}.
Moreover, a network of KPOs can solve a combinatorial optimization problem
(ground-state search in the Ising model)
by adiabatic quantum computation~\cite{Farhi2000a,Farhi2001a,Albash2018a} or
quantum annealing~\cite{Kadowaki1998a,Das2008a},
the final state of which is a highly entangled state, a superposition of many-mode coherent states
corresponding to two optimal solutions~\cite{Goto2016a}.
Quantum annealing using KPOs has been developed
in this five years~\cite{Nigg2017a,Puri2017a,Zhao2018a,Onodera2020a,Goto2020a,Kanao2021a}.
The KPOs can also be used for qubits in gate-based quantum computing~\cite{Goto2019a,Goto2016b,Puri2017b,Puri2020a,Kanao2021b,Xu2021a}.
Since the KPO qubit, also known as a Kerr-cat qubit, is robust against bit-flip errors, like the above dissipative-type qubit,
fault-tolerant quantum computation using KPOs has been developed~\cite{Puri2019a,Darmawan2021a}.
The KPO has also offered physically interesting topics, such as
nonclassical traveling-state generation~\cite{Goto2019b,Strandberg2021a},
quantum heating leading to Boltzmann sampling~\cite{Goto2018a},
steady-state entanglement generation~\cite{Manaev2018a,Kewming2020a}, and
phase transition~\cite{Savona2017a,Rota2019a}.
In this work, we investigate nonlinear dynamics of coupled KPOs from the viewpoint of chaos~\cite{Strogatz,Wimberger}.
The KPO, which is nondissipative in an ideal case, is more desirable for the study on chaos
than the two-photon dissipative parametric oscillator
(and also optical parametric oscillators~\cite{Goto2019a,Wang2013a}, another dissipative type),
because dissipation inevitably introduces noises (so-called quantum noises)~\cite{Milburn,Gardiner,Breuer},
and such stochastic noises are undesirable for the study on chaos~\cite{Strogatz}.
Nonlinear dynamics of a KPO have been studied~\cite{Milburn1991a,Hovsepyan2016a}.
To our knowledge, however, the coupled-KPO case has not been explored so far,
though it has been suggested that the studies on chaos in a KPO network would be interesting~\cite{Goto2016a}.
(Chaos in a simplified classical model for the KPO network has also been suggested in the proposal of
a quantum-inspired algorithm called simulated bifurcation~\cite{Goto2019c}, but it has not been investigated in detail.)
Here we treat a system of two nondissipative KPOs with time-independent parameters as the simplest example
sufficient for the study on chaos.
(Similar studies on more KPOs are an interesting next step.)
This paper is organized as follows.
In Sec.~\ref{sec-model}, we introduce the quantum and classical models for the system.
In Sec.~\ref{sec-classical}, we show our results for the classical model,
where the nonintegrability of this model is shown
by the Poincar\'{e} surface of section (SOS)~\cite{Strogatz,Wimberger} and its lower-dimensional version, and
the sensitivity to initial conditions is also shown.
These indicate chaos in the classical model.
In Sec.~\ref{sec-quantum}, we provide our results for the quantum model,
where the SOS and its lower-dimensional version are extended to quantum cases using the Wigner and Husimi functions~\cite{Milburn,Wimberger,Leonhardt},
and the initial-condition sensitivity is also examined using
out-of-time-ordered correlators (OTOCs)~\cite{Hashimoto2017a,Fortes2019a,Akutagawa2020a,Hashimoto2020a,Bhattacharyya2021a}.
Energy-level spacing statistics,
which is a conventional quantum signature of chaos~\cite{Wimberger,Brody1981a,Haller1984a,Berry1987a,Berry1989a},
are also discussed.
Finally, we summarize our results in Sec.~\ref{sec-conclusion}.
\section{Models for two coupled KPOs}
\label{sec-model}
The quantum and classical models for the KPO network have been introduced in Ref.~\citenum{Goto2016a}
and well summarized in Ref.~\citenum{Goto2019a}.
In the following, we provide these models in the case of two KPOs.
\subsection{Quantum model}
The quantum model for two KPOs is given by the following Hamiltonian:
\begin{align}
H &= H_1 + H_2 + H_{\mathrm{I}},
\label{eq-H}
\\
H_i &= \hbar \frac{K}{2} a_i^{\dagger 2} a_i^2 - \hbar \frac{p_i}{2} \! \left( a_i^2 + a_i^{\dagger 2} \right) \!
+ \hbar \Delta a_i^{\dagger} a_i,
\label{eq-Hi}
\\
H_{\mathrm{I}} &= -\hbar \xi_0 \! \left( a_1^{\dagger} a_2 + a_2^{\dagger} a_1 \right),
\label{eq-HI}
\end{align}
where $a_i$ and $p_i$ are the annihilation operator and the parametric pump amplitude, respectively,
for the $i$th KPO,
$K$ is the Kerr coefficient,
$\Delta$ is the detuning of the KPO resonance frequency from half the pump frequency,
$\xi_0$ is the coupling strength between the two KPOs,
and $\hbar$ is the reduced Planck constant.
\subsection{Classical model}
The corresponding classical model is derived by replacing the annihilation operator $a_i$
with a complex amplitude ${\alpha_i = x_i + \mathrm{i} y_i}$
(classical approximation) in the Heisenberg equations of motion for $a_i$~\cite{Goto2016a,Goto2019a}.
Thus we obtain the equations of motion in the classical model:
\begin{align}
&
\frac{\mathrm{d} x_i}{\mathrm{d} t} = \frac{\partial H_{\mathrm{c}}}{\partial y_i}
= \! \left[ K \! \left( x_i^2 + y_i^2 \right) \! + p_i + \Delta \right] \! y_i - \xi_0 y_j,
\label{eq-x}
\\
&
\frac{\mathrm{d} y_i}{\mathrm{d} t} = -\frac{\partial H_{\mathrm{c}}}{\partial x_i}
= - \! \left[ K \! \left( x_i^2 + y_i^2 \right) \! - p_i + \Delta \right] \! x_i + \xi_0 x_j,
\label{eq-y}
\\
&
H_{\mathrm{c}}(\bm{\mathrm{x}},\bm{\mathrm{y}})
=
\! \sum_{i=1,2} \! \left[ \frac{K}{4} \! \left( x_i^2 + y_i^2 \right)^2 \!
- \frac{p_i}{2} \! \left( x_i^2 - y_i^2 \right) \! \right.
\nonumber
\\
&
\qquad \qquad \qquad \quad
\left. + \frac{\Delta}{2} \! \left( x_i^2 + y_i^2 \right) \! \right] \!
-\xi_0 \! \left( x_1 x_2 + y_1 y_2 \right),
\label{eq-Hc}
\end{align}
where ${j \neq i}$.
For convenience, we introduce the potential energy $V_{\mathrm{c}}(\bm{\mathrm{x}})$ defined by the minimum of
$H_{\mathrm{c}}(\bm{\mathrm{x}},\bm{\mathrm{y}})$ with respect to $\bm{\mathrm{y}}$,
which is $H_{\mathrm{c}}(\bm{\mathrm{x}},\bm{0})$ if ${\xi_0 < p_i}$:
\begin{align}
V_{\mathrm{c}}(\bm{\mathrm{x}})
=
\! \sum_{i=1,2} \! \left( \frac{K}{4} x_i^4 - \frac{p_i - \Delta}{2} x_i^2 \right) \! - \xi_0 x_1 x_2.
\label{eq-V}
\end{align}
\subsection{Parameter setting}
In this work, we take the following values for the above parameters:
\begin{align}
\hbar &= 1,
\label{hbar}
\\
K &= 1,
\\
p_1 &= 3,
\\
p_2 &= \pi,
\\
\Delta &= 0,
\label{Delta}
\\
\xi_0 &= 0,~0.3,~\mathrm{or}~1.
\end{align}
Here ${\hbar = K =1}$ means that the units of energy and frequency are $\hbar K$ and $K$, respectively
(thus the unit of time is $K^{-1}$).
The pump amplitudes around 3 leads to mean photon numbers around 3,
because the mean photon numbers for oscillating states are given by ${p_i/K}$ when ${\Delta =0}$~\cite{Goto2016a}.
We choose such small values because we are interested in the dynamics at a few-photon level.
Also, we set $p_2$ to $\pi$ such that $p_2 \simeq p_1$ but $p_2/p_1$ becomes an irrational number,
because then the ratio between the periods of the two KPOs in the decoupled case (${\xi_0=0}$) is irrational,
and the dynamics becomes relatively complex.
The three values of $\xi_0$ correspond to regular (integrable), intermediate, and chaotic (nonintegrable) cases, respectively,
as shown in the next section.
(Note that when ${\xi_0=0}$, that is, the two KPOs are decoupled,
then the Hamiltonian for each KPO is conserved, and the system is integrable by definition~\cite{comment-integrable}.)
The potential energy $V_{\mathrm{c}}(\bm{\mathrm{x}})$
with the above parameters is shown in Fig.~\ref{fig-classical-potential}.
There is a minimum in each quadrant and a maximum at the origin.
Thus, the origin is unstable.
In this work, we investigate the dynamics started around the origin
(around the vacuum state in the quantum case).
\begin{figure}[ht]
\includegraphics[width=7cm]{fig-classical-potential-HR.pdf}
\caption{Potential energy $V_{\mathrm{c}}(\bm{\mathrm{x}})$ in Eq~(\ref{eq-V}).
(a) ${\xi_0=0}$. (b) ${\xi_0=0.3}$. (c) ${\xi_0=1}$.
The other parameters are set as Eqs.~(\ref{hbar})--(\ref{Delta}).}
\label{fig-classical-potential}
\end{figure}
\section{Chaos in the classical model}
\label{sec-classical}
\subsection{Surface of section (SOS)}
We start with nonintegrability of the classical model.
To check nonintegrability,
the Poincar\'e surface of section (SOS)~\cite{Strogatz,Wimberger} is useful, in particular,
for systems with two degrees of freedom (four-dimensional phase space), like the present system.
The SOS in the phase space is defined by the section of the energy surface
${H_{\mathrm{c}}(\bm{\mathrm{x}},\bm{\mathrm{y}})=E}$ ($E$ is a constant) by a plane, e.g., ${y_2=0}$.
In the case of two degrees of freedom (four-dimensional phase space),
the object consisting of the intersection points between the SOS and a trajectory,
which we call the SOS plot, is two-dimensional in general.
If the system is integrable, however, there is another constant of motion~\cite{comment-integrable},
and consequently the SOS plot must be one-dimensional.
Thus we can check whether the system is integrable or not by the dimension of the SOS plot.
\begin{figure}[ht]
\includegraphics[width=8cm]{fig-classical-SOS.pdf}
\caption{SOS plots in the classical model given by Eqs.~(\ref{eq-x})--(\ref{eq-Hc}).
Circles are obtained by plotting ${(x_1,y_1)}$ when trajectories in numerical simulation cross the plane ${y_2=0}$.
(a) ${\xi_0=0}$. (b) ${\xi_0=0.3}$. (c) ${\xi_0=1}$.
The other parameters are set as Eqs.~(\ref{hbar})--(\ref{Delta}).
See Appendix~\ref{appendix-classical-SOS} for details.}
\label{fig-classical-SOS}
\end{figure}
Figure~\ref{fig-classical-SOS} shows the SOS plots in the classical model.
The SOS plot is one-dimensional~\cite{comment-lemniscate} (integrable and regular)
in the decoupled case (${\xi_0=0}$) [Fig.~\ref{fig-classical-SOS}(c)],
two-dimensional (nonintegrable and chaotic)
in the strong-coupling case (${\xi_0=1}$) [Fig.~\ref{fig-classical-SOS}(c)],
and intermediate in the intermediate case (${\xi_0=0.3}$) [Fig.~\ref{fig-classical-SOS}(b)], as expected.
\subsection{Momentum plot at a minimum of potential (MPMP)}
To demonstrate the nonintegrability more clearly,
here we introduce another plot, which we call the momentum plot at a minimum of potential (MPMP).
Instead of the plane ${y_2=0}$ for the SOS,
here we fix the two positions, $x_1$ and $x_2$, at a minimum of the potential $V_{\mathrm{c}}(\bm{\mathrm{x}})$,
and plot the momenta in the $y_1 y_2$ plane.
We focus on a potential minimum, because at such a point, the energetically allowable region of the momenta becomes the largest,
and that will be desirable for visualization.
If the system is integrable, the MPMP must be zero-dimensional (i.e., points), otherwise one-dimensional.
Figure~\ref{fig-classical-MPMP} shows the results of the MPMP, where
we choose the potential minimum in the first quadrant of the $x_1 x_2$ plane.
The MPMP is zero-dimensional for ${\xi_0=0}$ [Fig~\ref{fig-classical-MPMP}(a)],
one-dimensional for ${\xi_0=1}$ [Fig~\ref{fig-classical-MPMP}(c)],
and intermediate for ${\xi_0=0.3}$ [Fig~\ref{fig-classical-MPMP}(b)], as expected.
As we will see in Sec.~\ref{sec-quantum},
the MPMP is particularly useful in the quantum case, in comparison with the SOS plot.
\begin{figure}[hb]
\includegraphics[width=6.4cm]{fig-classical-MPMP.pdf}
\caption{MPMPs in the classical model given by Eqs.~(\ref{eq-x})--(\ref{eq-Hc}).
We choose the potential minimum $(X_1,X_2)$ in the first quadrant of the $x_1 x_2$ plane.
Circles are obtained by plotting ${(x_1,y_1)}$ when trajectories in numerical simulation pass by close to $(X_1,X_2)$.
(a) ${\xi_0=0}$ and ${(X_1,X_2)=(1.73,1.77)}$.
(b) ${\xi_0=0.3}$ and ${(X_1,X_2)=(1.82,1.85)}$.
(c) ${\xi_0=1}$ and ${(X_1,X_2)=(2,2.03)}$.
The other parameters are set as Eqs.~(\ref{hbar})--(\ref{Delta}).
See Appendix~\ref{appendix-classical-MPMP} for details.}
\label{fig-classical-MPMP}
\end{figure}
\subsection{Sensitivity to initial conditions}
Here we also observe the sensitivity to initial conditions in the classical model.
The results are shown in Fig.~\ref{fig-classical-sensitivity}.
The Euclidean distance between two trajectories, $\bm{\mathrm{x}}(t)$ and $\bm{\mathrm{x}}'(t)$,
with a very small deviation in their initial conditions
saturates at a small value in the integrable case (${\xi_0=0}$),
but exponentially grows in the nonintegrable cases (${\xi_0=0.3}$ and 1),
as expected.
(The saturation in the nonintegrable cases comes from the fact that
the energetically allowable regions are finite.)
This together with the above results for SOS and MPMP
indicates chaos in the coupled-KPO system.
\begin{figure}[ht]
\includegraphics[width=6.1cm]{fig-classical-sensitivity.pdf}
\caption{Initial-condition sensitivity in the classical model given by Eqs.~(\ref{eq-x})--(\ref{eq-Hc}).
$\bm{\mathrm{x}}(t)$ and $\bm{\mathrm{x}}'(t)$ are two trajectories
with a very small deviation in their initial conditions.
(a) ${\xi_0=0}$. (b) ${\xi_0=0.3}$. (c) ${\xi_0=1}$.
The other parameters are set as Eqs.~(\ref{hbar})--(\ref{Delta}).
See Appendix~\ref{appendix-classical-sensitivity} for details.}
\label{fig-classical-sensitivity}
\end{figure}
\section{Quantum signatures of chaos}
\label{sec-quantum}
\subsection{Quantum SOS plots}
Here we introduce quantum versions of the SOS plot.
Instead of plotting intersection points in the classical case,
we use the time integral of a quasi-probability distribution, the Wigner function or
the Husimi function (also known as the Q function)~\cite{Milburn,Goto2016a,Wimberger,Leonhardt},
with ${y_2=0}$.
Thus we define the quantum versions of the SOS plot as
\begin{align}
W_{\mathrm{SOS}}(x_1,y_1)
&=
\int_{0}^{T} \! \mathrm{d}t \! \int_{-\infty}^{\infty} \! \mathrm{d}x_2 W(x_1,x_2,y_1,0,t),
\label{eq-WSOS}
\\
Q_{\mathrm{SOS}}(x_1,y_1)
&=
\int_{0}^{T} \! \mathrm{d}t \! \int_{-\infty}^{\infty} \! \mathrm{d}x_2 Q(x_1,x_2,y_1,0,t),
\label{eq-QSOS}
\end{align}
where $W(x_1,x_2,y_1,0,t)$ and $Q(x_1,x_2,y_1,0,t)$ are the Wigner and Husimi functions with ${y_2=0}$
for the state vector, $|\psi (t) \rangle$, at time $t$, and $T$ is the final time in each simulation.
Figure~\ref{fig-quantum-SOS} shows the results of the quantum SOS plots corresponding to the classical ones
in Fig.~\ref{fig-classical-SOS}.
Although the Husimi-type SOS plots shown in Figs.~\ref{fig-quantum-SOS}(a)--\ref{fig-quantum-SOS}(c)
indicate the classical SOS plots in Figs.~\ref{fig-classical-SOS}(a)--\ref{fig-classical-SOS}(c) to some extent,
it is hard to distinguish integrability (one-dimensional) from nonintegrability (two-dimensional)
because of large quantum fluctuations.
(The large fluctuations come from the small mean photon numbers.)
The situation is worse in the Wigner case, as shown in Figs.~\ref{fig-quantum-SOS}(d)--\ref{fig-quantum-SOS}(f),
because of quantum interference.
(Similar results have been reported for a single driven pendulum~\cite{Lee1993a}.).
This is the reason why we have introduced the MPMP in this work.
\begin{widetext}
\begin{figure}[hbt]
\centering
\includegraphics[width=17cm]{fig-quantum-SOS-HR.pdf}
\caption{Quantum SOS plots defined by Eqs.~(\ref{eq-WSOS}) and (\ref{eq-QSOS})
in the quantum model given by Eqs.~(\ref{eq-H})--(\ref{eq-HI}).
Corresponding classical results are shown in Fig.~\ref{fig-classical-SOS}.
(a,d) ${\xi_0=0}$. (b,e) ${\xi_0=0.3}$. (c,f) ${\xi_0=1}$.
The other parameters are set as Eqs.~(\ref{hbar})--(\ref{Delta}).
See Appendix~\ref{appendix-quantum-SOS} for details.}
\label{fig-quantum-SOS}
\end{figure}
\subsection{Quantum MPMPs}
Here we introduce quantum versions of the MPMP as
\begin{align}
W_{\mathrm{MPMP}}(y_1,y_2)
&=
\int_{0}^{T} \! \mathrm{d}t W(X_1,X_2,y_1,y_2,t),
\label{eq-WMPMP}
\\
Q_{\mathrm{MPMP}}(y_1,y_2)
&=
\int_{0}^{T} \! \mathrm{d}t Q(X_1,X_2,y_1,y_2,t),
\label{eq-QMPMP}
\end{align}
where $(X_1,X_2)$ is the position of a minimum of the potential $V_{\mathrm{c}}(\bm{\mathrm{x}})$ in Eq.~(\ref{eq-V}).
Figure~\ref{fig-quantum-MPMP} shows the results of the quantum MPMPs
together with the corresponding classical results
in Fig.~\ref{fig-classical-MPMP}.
The Husimi-type MPMPs shown in Figs.~\ref{fig-quantum-MPMP}(a)--\ref{fig-quantum-MPMP}(c)
clearly indicate the classical MPMPs even with large quantum fluctuations,
which can be regarded as a quantum signature of chaos (nonintegrability).
The Wigner-type MPMPs shown in Figs.~\ref{fig-quantum-MPMP}(d)--\ref{fig-quantum-MPMP}(f)
also indicate the classical MPMPs to some extent
even with quantum interference.
Thus, the MPMP is particularly useful in quantum cases,
in comparison with the SOS plot.
\begin{figure}[b]
\includegraphics[width=14cm]{fig-quantum-MPMP-HR.pdf}
\caption{Quantum MPMPs defined by Eqs.~(\ref{eq-WMPMP}) and (\ref{eq-QMPMP})
in the quantum model given by Eqs.~(\ref{eq-H})--(\ref{eq-HI}).
$(X_1,X_2)$ is set to the position of the potential minimum
in the first quadrant of the $x_1 x_2$ plane.
Circles represent classical results shown in Fig.~\ref{fig-classical-SOS}.
(a,d) ${\xi_0=0}$ and ${(X_1,X_2)=(1.73,1.77)}$.
(b,e) ${\xi_0=0.3}$ and ${(X_1,X_2)=(1.82,1.85)}$.
(c,f) ${\xi_0=1}$ and ${(X_1,X_2)=(2,2.03)}$.
The other parameters are set as Eqs.~(\ref{hbar})--(\ref{Delta}).
See Appendix~\ref{appendix-quantum-MPMP} for details.}
\label{fig-quantum-MPMP}
\end{figure}
\end{widetext}
\subsection{Out-of-time-ordered correlators (OTOCs)}
Here we discuss the sensitivity to initial conditions in the quantum model.
In quantum cases, the initial-condition sensitivity can be evaluated by
the out-of-time-ordered correlators (OTOCs)~\cite{Hashimoto2017a,Fortes2019a,Akutagawa2020a,Hashimoto2020a,Bhattacharyya2021a}
defined by
\begin{align}
C_{i,j}(t)
=
-4\langle \psi (0)| [x_i(t), y_j(0)]^2 |\psi (0) \rangle,
\label{eq-quantum-OTOC}
\end{align}
where $|\psi (0) \rangle$ is an initial state vector,
$[O_1,O_2]=O_1 O_2 - O_2 O_1$ is the commutation relation between two operators,
$x_i(t)$ is the position operator for the $i$th KPO at time $t$ in the Heisenberg representation,
and $y_j(0)$ is the initial momentum operator for the $j$th KPO.
The factor of 4 comes from the definitions of the quadrature amplitudes and their commutation relations:
\begin{align}
x_i(0) = \frac{a_i + a_i^{\dagger}}{2},
\label{eq-x-def}
\\
y_i(0) = \frac{a_i - a_i^{\dagger}}{2\mathrm{i}},
\label{eq-y-def}
\\
[x_i(0),y_i(0)]=\frac{\mathrm{i}}{2}.
\end{align}
The physical meaning of the OTOCs can be extracted by naively replacing the commutator
with the classical Poisson bracket
${(\mathrm{i}/2) \{ x_i(t), y_j(0) \}} = {(\mathrm{i}/2) \partial x_i(t)/\partial x_j(0)}$~\cite{Hashimoto2017a}.
That is, the classical counterpart of $C_{i,j}(t)$, which is denoted by $\tilde{C}_{i,j}(t)$, is given by
\begin{align}
\tilde{C}_{i,j}(t)
=
\left\langle \left( \frac{\partial x_i(t)}{\partial x_j(0)} \right)^2
\right\rangle,
\label{eq-classical-OTOC}
\end{align}
where $\langle \cdot \rangle$ represents the average over trajectories
with different initial conditions related to the quantum initial state.
This classical interpretation suggests that the OTOCs are related to the initial-condition sensitivity.
Figure~\ref{fig-OTOC} shows the results of the OTOCs,
where the solid and dotted lines represent $C_{i,j}(t)$ and $\tilde{C}_{i,j}(t)$, respectively.
First of all, the classical results are in good agreement with the quantum results,
in particular, around the initial time.
This indicates that the above classical interpretation of the OTOCs is valid.
However, unlike the classical initial-condition sensitivity shown in Fig.~\ref{fig-classical-sensitivity},
the OTOCs rapidly increase only around the initial time and soon saturate.
This may be due to quantum fluctuations.
Here it should be noted that the initial rapid increase of the OTOCs does not indicate chaos,
because this can be seen even in the integrable case, as shown in Fig.~\ref{fig-OTOC}(a).
Instead, this naturally occurs when the initial state is around an unstable point (maximum of the potential),
as discussed recently~\cite{Hashimoto2020a,Bhattacharyya2021a}.
On the other hand,
we can find that the oscillation amplitudes of $C_{1,1}$ in the nonintegrable case (${\xi_0=0.3}$ and 1) shown
in Figs.~\ref{fig-OTOC}(b) and \ref{fig-OTOC}(d)
seem smaller than that in the integrable case (${\xi_0=0}$) shown in Fig.~\ref{fig-OTOC}(a).
This difference may be due to more chaotic behavior in the nonintegrable case.
This is another quantum signature of chaos proposed recently~\cite{Fortes2019a}.
\begin{figure}[ht]
\includegraphics[width=6.9cm]{fig-OTOC.pdf}
\caption{OTOCs in the quantum model (solid lines)
and classical counterparts (dotted lines).
(a) ${\xi_0=0}$. (b,c) ${\xi_0=0.3}$. (d,e) ${\xi_0=1}$.
($C_{2,1}$ for ${\xi_0=0}$ is not shown, because
in the decoupled case, $C_{2,1}$ is exactly zero.)
The other parameters are set as Eqs.~(\ref{hbar})--(\ref{Delta}).
See Appendix~\ref{appendix-OTOC} for details.}
\label{fig-OTOC}
\end{figure}
\subsection{Energy-level spacing statistics}
Finally, we check a conventional quantum signature of chaos:
energy-level spacing statistics~\cite{Wimberger,Brody1981a,Haller1984a,Berry1987a,Berry1989a}.
It is known that the energy-level spacing $\Delta_E$ defined by the difference between two neighboring energy levels
obeys the Poisson distribution (${\propto e^{-\beta \Delta_E}}$) in the integrable (regular) case and
the Wigner distribution (${\propto \Delta_E e^{-\beta \Delta_E^2}}$) in the nonintegrable (chaotic) case
($\beta$ is a constant).
This means that the probability for zero spacing decreases as the system becomes more chaotic,
which is due to avoided crossings of energy levels induced by complex interactions in chaotic systems~\cite{Wimberger}.
The two distributions are unified as
${\Delta_E^{\omega} e^{-\beta \Delta_E^{\omega+1}}}$~\cite{Brody1981a,Haller1984a},
where ${\omega=0}$ and 1 correspond to Poisson and Wigner, respectively.
Note that this distribution can be integrated analytically,
which leads to the cumulative level spacing distribution ${\propto (1- e^{-\beta \Delta_E^{\omega+1}})}$.
Figure~\ref{fig-spacing} shows the cumulative distributions of the present quantum model
together with fitting curves using the function form ${A (1- e^{-\beta \Delta_E^{\omega+1}})}$
and the fitting results of the three parameters.
The exponent $\omega$ is larger for stronger coupling and
exceeds 0.5 when ${\xi_0=1}$.
This is another quantum signature of chaos in the coupled-KPO system.
\begin{figure}[ht]
\includegraphics[width=8cm]{fig-spacing.pdf}
\caption{Cumulative energy-level spacing distributions
in the quantum model given by Eqs.~(\ref{eq-H})--(\ref{eq-HI}).
${N(\Delta_E)}$ denotes the number of level spacing equal to or smaller than $\Delta_E$.
Curves and three parameters ($\omega$, $A$, and $\beta$)
are fitting results using the function form ${A (1- e^{-\beta \Delta_E^{\omega+1}})}$.
(a) ${\xi_0=0}$. (b) ${\xi_0=0.3}$. (c) ${\xi_0=1}$.
The other parameters are set as Eqs.~(\ref{hbar})--(\ref{Delta}).
See Appendix~\ref{appendix-spacing} for details.}
\label{fig-spacing}
\end{figure}
\section{Conclusions}
\label{sec-conclusion}
We have investigated the quantum and classical models for two coupled nondissipative KPOs
from the viewpoint of chaos.
Using the Poincar\'{e} surface of section (SOS) plot, the momentum plot at a minimum of potential (MPMP)
(a lower-dimensional version of the SOS plot),
and the initial-condition sensitivity,
we have shown that the classical model with nonzero coupling is nonintegrable and hence exhibits chaotic behavior.
We have also provided quantum signatures of chaos,
using quantum versions of the SOS plot and the MPMP defined with time integrals of
the Wigner and Husimi functions,
out-of-time-ordered correlators (OTOCs), and
energy-level spacing statistics.
We have found that the quantum MPMP can distinguish integrability and nonintegrability clearly
even at a few-photon level (more clearly than the quantum SOS plot),
and also that the long-term behavior, not the initial behavior, of the OTOCs
can be regarded as a quantum signature of chaos.
The present results suggest that the system of coupled KPOs will be useful
not only for quantum computing, but also the study on quantum chaos.
We also expect that such an understanding of the nonlinear dynamics in the KPO network
will be useful for its applications, such as quantum computing and quantum-inspired algorithms~\cite{Goto2019c}.
\section*{Acknowledgments}
This work was supported by JST ERATO (Grant No. JPMJER1601).
\begin{appendix}
\section{SOS plot in the classical model}
\label{appendix-classical-SOS}
To obtain the SOS plots in Fig.~\ref{fig-classical-SOS},
we numerically solve Eqs.~(\ref{eq-x})--(\ref{eq-Hc})
by the fourth-order Runge-Kutta method with a time step of ${\Delta_t=10^{-4}}$
from $t=0$ to 20.
The initial conditions are set as ${x_1(0)=x_2(0)=0}$ and ${y_i(0)=10^{-6} r_i}$,
where $r_1$ and $r_2$ are independent random numbers from the standard normal distribution.
We plot ${(x_1,y_1)}$ when ${y_2 (t) y_2 ({t-\Delta_t}) <0}$,
which means that the trajectory have crossed the plane ${y_2=0}$.
We iterated the simulation 200 times to obtain enough points.
\section{MPMP in the classical model}
\label{appendix-classical-MPMP}
To obtain the MPMPs in Fig.~\ref{fig-classical-MPMP},
we did the same simulation as in the SOS case explained in Appendix~\ref{appendix-classical-SOS}.
We plot ${(y_1,y_2)}$ when ${|\bm{\mathrm{x}} - \bm{\mathrm{X}}|<10^{-3}}$,
where $\bm{\mathrm{X}}$ denotes the position of the potential minimum
in the first quadrant of the $x_1 x_2$ plane.
$\bm{\mathrm{X}}$ is numerically found as
${\bm{\mathrm{X}}=(1.73,1.77)}$ for ${\xi_0=0}$,
${\bm{\mathrm{X}}=(1.82,1.85)}$ for ${\xi_0=0.3}$,
and ${\bm{\mathrm{X}}=(2,2.03)}$ for ${\xi_0=1}$.
We iterated the simulation $10^5$ times to obtain enough points.
\section{Initial-condition sensitivity in the classical model}
\label{appendix-classical-sensitivity}
To obtain Fig.~\ref{fig-classical-sensitivity},
we did the same simulation as above with the following initial conditions (no iteration):
\begin{align}
&
x_1(0)=x_2(0)=x'_2(0)=0,
\label{eq-sensitivity-initial-x}
\\
&
x'_1(0)=10^{-6},
\\
&
y_1(0)=y'_1(0)=0.5\cos (0.65\pi),
\\
&
y_2(0)=y'_2(0)=0.5\sin (0.65\pi).
\label{eq-sensitivity-initial-y}
\end{align}
\section{Quantum SOS plots}
\label{appendix-quantum-SOS}
Here we give the formulations of $W_{\mathrm{SOS}}(x_1,y_1)$ and $Q_{\mathrm{SOS}}(x_1,y_1)$
defined by Eqs.~(\ref{eq-WSOS}) and (\ref{eq-QSOS})
in the photon-number basis.
First of all, the Wigner and Husimi functions are defined as~\cite{Goto2016a,Leonhardt}
\begin{align}
W(\bm{\mathrm{x}},\bm{\mathrm{y}},t)
&=
\left( \frac{2}{\pi} \right)^2 \!
\mathrm{tr} \! \left[ D^{(1)} (2\alpha_1 ) P_1 D^{(2)} (2\alpha_2 ) P_2 {\rho (t)} \right],
\\
Q(\bm{\mathrm{x}},\bm{\mathrm{y}},t)
&=
\frac{1}{\pi^2} {\langle \alpha_1 |} {\langle \alpha_2 |} {\rho (t)} {| \alpha_1 \rangle} {| \alpha_2 \rangle},
\end{align}
where ${\alpha_i = x_i + \mathrm{i} y_i}$,
${D^{(i)} (\alpha ) =e^{\alpha a_i^{\dagger} - \alpha^* a_i}}$ is the so-called displacement operator,
${P_i = e^{\mathrm{i} a_i^{\dagger} a_i}}$ is the parity operator,
${|\alpha_i \rangle = D^{(i)} (\alpha_i ) |0\rangle}$ is a coherent state ($|0\rangle$ is the vacuum state), and
${\rho (t) = |\psi (t) \rangle \langle \psi (t)|}$ is the density operator corresponding to the state vector
$|\psi (t) \rangle$.
In the photon-number basis $\{ |n_1 \rangle |n_2 \rangle \}$,
we obtain~\cite{Goto2016a}
\begin{widetext}
\begin{align}
W(\bm{\mathrm{x}},\bm{\mathrm{y}},t)
&=
\left( \frac{2}{\pi} \right)^2
\sum_{m_1=0}^{\infty} \sum_{m_2=0}^{\infty} \sum_{n_1=0}^{\infty} \sum_{n_2=0}^{\infty}
(-1)^{n_1+n_2}
D^{(1)}_{m_1,n_1} (2\alpha_1 ) D^{(2)}_{m_2,n_2} (2\alpha_2 ) \psi_{n_1,n_2}(t) \psi^*_{m_1,m_2}(t),
\label{eq-W-number}
\\
Q(\bm{\mathrm{x}},\bm{\mathrm{y}},t)
&=
\frac{1}{\pi^2}
\sum_{m_1=0}^{\infty} \sum_{m_2=0}^{\infty} \sum_{n_1=0}^{\infty} \sum_{n_2=0}^{\infty}
{\langle \alpha_1 | n_1 \rangle} {\langle \alpha_2 | n_2 \rangle} \psi_{n_1,n_2}(t) \psi^*_{m_1,m_2}(t)
{\langle m_1 | \alpha_1 \rangle} {\langle m_2 | \alpha_2 \rangle},
\label{eq-Q-number}
\end{align}
with
\begin{align}
&
\psi_{n_1,n_2}(t) = \langle n_1| \langle n_2| \psi (t) \rangle,
\label{eq-psi-number}
\\
&
D^{(i)}_{m_i,n_i} (\alpha)
=
e^{-|\alpha |^2/2} \sqrt{m_i ! n_i !} \sum_{k=0}^{\mathrm{min}(m_i,n_i)}
\! \frac{1}{k!} \frac{\alpha^{m_i - k}}{(m_i-k)!} \frac{(-\alpha^*)^{n_i-k}}{(n_i-k)!},
\label{eq-D-number}
\\
&
{\langle n_i | \alpha_i \rangle} = \frac{\alpha_i^{n_i}}{\sqrt{n_i !}} e^{-|\alpha_i |^2/2}.
\label{eq-alpha-number}
\end{align}
The integrals $\int_{-\infty}^{\infty} \! \mathrm{d}x_2 W(x_1,x_2,y_1,0,t)$ and
$\int_{-\infty}^{\infty} \! \mathrm{d}x_2 Q(x_1,x_2,y_1,0,t)$
required for $W_{\mathrm{SOS}}(x_1,y_1)$ and $Q_{\mathrm{SOS}}(x_1,y_1)$
are obtained by Eqs.~(\ref{eq-W-number})--(\ref{eq-alpha-number}) together with the following formulae:
\begin{align}
&
\int_{-\infty}^{\infty} \! \mathrm{d}x_2 D^{(2)}_{m_2,n_2} (2x_2)
=
\sqrt{\frac{\pi}{2} m_2 ! n_2 !} \sum_{k=0}^{\mathrm{min}(m_2,n_2)}
\! \frac{(-1)^{n_2-k} (m_2 + n_2 -2k -1)!!}{k! (m_2-k)! (n_2-k)!}
\delta_{\mathrm{even}} (m_2+n_2),
\\
&
\int_{-\infty}^{\infty} \! \mathrm{d}x_2 \langle \alpha_2|m_2 \rangle \langle n_2| \alpha_2 \rangle
=
\sqrt{\frac{\pi}{m_2! n_2!}} \frac{(m_2+n_2-1)!!}{2^{(m_2+n_2)/2}} \delta_{\mathrm{even}} (m_2+n_2),
\end{align}
\end{widetext}
where ${\delta_{\mathrm{even}} (n)=1}$ if $n$ is even, otherwise ${\delta_{\mathrm{even}} (n)=0}$.
The numerical results of
$W_{\mathrm{SOS}}(x_1,y_1)$ and $Q_{\mathrm{SOS}}(x_1,y_1)$ shown in Fig.~\ref{fig-quantum-SOS}
are obtained by accumulating the integrals $\int_{-\infty}^{\infty} \! \mathrm{d}x_2 W(x_1,x_2,y_1,0,t)$ and
$\int_{-\infty}^{\infty} \! \mathrm{d}x_2 Q(x_1,x_2,y_1,0,t)$ multiplied by a time step of ${\Delta_t=10^{-3}}$
from ${t=0}$ to ${T=20}$.
Here $\psi_{n_1,n_2} (t)$ necessary for the integrals is obtained by solving the Schr\"{o}dinger equation
with the Hamiltonian in Eq.~(\ref{eq-H}) in the photon-number basis
by the fourth-order Runge-Kutta method with the time step of ${\Delta_t=10^{-3}}$,
the initial state set to the vacuum state, and
the maximum photon number of 30.
\section{Quantum MPMPs}
\label{appendix-quantum-MPMP}
The numerical results of
$W_{\mathrm{MPMP}}(y_1,y_2)$ and $Q_{\mathrm{MPMP}}(y_1,y_2)$ shown in Fig.~\ref{fig-quantum-MPMP}
are obtained by accumulating the Wigner and Husimi functions multiplied by a time step of ${\Delta_t=10^{-3}}$
from ${t=0}$ to ${T=20}$,
where the Wigner and Husimi functions are obtained by using Eqs.~(\ref{eq-W-number})--(\ref{eq-alpha-number})
with $\psi_{n_1,n_2} (t)$ obtained by the same simulation as in Appendix~\ref{appendix-quantum-SOS}.
$(X_1,X_2)$ is numerically found,
as mentioned in Appendix~\ref{appendix-classical-MPMP}.
\section{OTOC}
\label{appendix-OTOC}
The results shown in Fig.~\ref{fig-OTOC} are obtained as follows.
Using ${x(t)=e^{\mathrm{i} Ht} x(0) e^{-\mathrm{i}Ht}}$,
the OTOC defined by Eq.~(\ref{eq-quantum-OTOC}) is
formulated in the eigenenergy basis $\{ |E_k \rangle \}$ as~\cite{Hashimoto2017a}
\begin{widetext}
\begin{align}
&
C_{i,j}(t)=
\sum_{k=0}^{\infty} \sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
\langle \psi (0)|E_k \rangle \langle E_k|[x_i(t),y_j(0)]|E_m \rangle
\langle E_m |[x_i(t),y_j(0)]|E_n \rangle
\langle E_n|\psi (0) \rangle,
\\
&
\langle E_k|[x_i(t),y_j(0)]|E_m \rangle
=
\sum_{l=0}^{\infty}
\left[
e^{\mathrm{i} (E_k - E_l)t} \langle E_k| x_i(0) |E_l \rangle
\langle E_l| y_j(0) |E_m \rangle
-
e^{\mathrm{i} (E_l - E_m)t} \langle E_k| y_j(0) |E_l \rangle
\langle E_l| x_i(0) |E_m \rangle
\right].
\end{align}
\end{widetext}
Thus, we can obtain $C_{i,j}(t)$ by using $\{ |E_k \rangle \}$
obtained by numerically diagonalizing the Hamiltonian in the photon-number basis
with the maximum photon number of 30,
the same as the above simulation,
and using the definitions of $x_i(0)$ and $y_i(0)$ in Eqs.~(\ref{eq-x-def}) and (\ref{eq-y-def}).
For the comparison with the results in Fig.~\ref{fig-classical-sensitivity},
the initial state is set to coherent states
as ${|\psi (0) \rangle = |\alpha^{(0)}_1 \rangle |\alpha^{(0)}_2 \rangle}$,
where ${\alpha^{(0)}_1=x^{(0)}_1 + \mathrm{i} y^{(0)}_1}$
[${x^{(0)}_1=0}$ and ${y^{(0)}_1=0.5 \cos (0.65\pi)}$] and
${\alpha^{(0)}_2=x^{(0)}_2 + \mathrm{i} y^{(0)}_2}$
[${x^{(0)}_2=0}$ and ${y^{(0)}_2=0.5 \sin (0.65\pi)}$],
which correspond to the initial conditions for Fig.~\ref{fig-classical-sensitivity}
in Eqs.~(\ref{eq-sensitivity-initial-x})--(\ref{eq-sensitivity-initial-y}).
The resultant $C_{i,j}(t)$ are shown by the solid lines in Fig.~\ref{fig-OTOC}.
For the classical counterparts,
we calculate two trajectories, $\bm{\mathrm{x}}(t)$ and $\bm{\mathrm{x}}'(t)$,
with the following initial conditions:
\begin{align}
&
x_1(0)=x^{(0)}_1 + \Delta_x r_1,
\label{eq-OTOC-initial-x}
\\
&
x'_1(0)=x_1(0)+\delta_x,
\\
&
x_2(0)=x'_2(0)=x^{(0)}_2 + \Delta_x r_2,
\\
&
y_1(0)=y'_1(0)=y^{(0)}_1 + \Delta_y r_3,
\\
&
y_2(0)=y'_2(0)=y^{(0)}_2 + \Delta_y r_4,
\label{eq-OTOC-initial-y}
\end{align}
where $r_j$ (${j=1, \ldots, 4}$) are independent random numbers from
the standard normal distribution,
${\Delta_x = \Delta_y =0.5}$ correspond to the standard deviations
of the quantum fluctuations of $x_i$ and $y_i$, and
${\delta_x = 0.5}$ is the deviation for the evaluation of the partial derivative
in Eq.~(\ref{eq-classical-OTOC}).
(The large $\delta_x$ comparable to the quantum fluctuations
is used to mimic the saturation of the OTOCs in the quantum model.)
Using the two trajectories, $\tilde{C}_{i,1}(t)$ shown by the dotted lines in Fig.~\ref{fig-OTOC}
are obtained as follows:
\begin{align}
\tilde{C}_{i,1}(t)
=
\left\langle \left( \frac{x'_i(t) - x_i(t)}{\delta_x} \right)^2
\right\rangle,
\end{align}
where the average was taken over $10^4$ iterations.
\section{Energy-level spacing statistics}
\label{appendix-spacing}
The results in Fig.~\ref{fig-spacing} are obtained as follows.
First, we numerically diagonalize the Hamiltonian in Eq.~(\ref{eq-H})
in the photon-number basis with the maximum photon number of 30,
the same as above.
Then, taking the parity invariance of the Hamiltonian into account,
we classify the energy eigenstates into two groups with even and odd total photon numbers.
Here we focus on the even eigenstates and sort the corresponding energies in ascending order.
Thus we obtain the energy-level spacing as the difference between two neighboring energies.
To avoid the effects of the finite photon numbers,
we take 50 spacings from the smallest, which are plotted in Fig.~\ref{fig-spacing}.
The curves in Fig.~\ref{fig-spacing} are obtained by fitting ${A (1- e^{-\beta \Delta_E^{\omega+1}})}$
to the 50 points, where $\omega$, $A$, and $\beta$ are fitting parameters.
\end{appendix}
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{
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Co Marketing Agreement Template co marketing agreement template 2 joint marketing agreement templates pdf free premium templates download. co marketing agreement template 18 marketing template doc excel pdf psd free premium templates ideas. co marketing agreement template sample marketing agreement 7 documents in pdf word templates. Co Marketing Agreement Template co marketing agreement template 7 marketing agreement form samples free sample example format printable. co marketing agreement template 20 marketing agreement template free sample example format free. co marketing agreement template co marketing agreement joint marketing agreement template template. Co Marketing Agreement Template co marketing agreement template marketing agreement gtld world congress printable.
Co Marketing Agreement Template co marketing agreement template 18 marketing template doc excel pdf psd free premium templates ideas. co marketing agreement template sample marketing agreement 7 documents in pdf word templates. co marketing agreement template 7 marketing agreement form samples free sample example format printable. co marketing agreement template 20 marketing agreement template free sample example format free. co marketing agreement template co marketing agreement joint marketing agreement template template.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 970
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San Cristóbal de La Paz era una fortezza in Cile, costruita nel 1621 dal Governatore Cristóbal de la Cerda y Sotomayor. Si trovava a circa cinque o sei chilometri a sud ovest di Yumbel, sulla costa ovest del fiume Claro, a nord della sua confluenza col fiume Laja. Nel 1646 furono costruite una missione e una piccola chiesa, che divennero successivamente un piccolo insediamento con alcuni abitanti. Nel febbraio del 1655 sia la fortezza sia la missione furono distrutti dai Mapuche, e furono ricostruiti nel 1663 sotto il governo temporaneo di Ángel de Peredo. Questo insediamento durò fino all'ascesa dei Mapuche nel 1766, quando fu abbandonato e mai più ricostruito.
Fonti
Francisco Solano Asta-Buruaga y Cienfuegos, Diccionario geográfico de la República de Chile, SEGUNDA EDICIÓN CORREGIDA Y AUMENTADA, NUEVA YORK, D. APPLETON Y COMPAÑÍA. 1899. Pg. 692 San Cristóbal
Diego Barros Arana, Historia general de Chile, Tomo cuarto, Capítulo V, pg.169-170.
Storia del Cile
Fortezze
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\section*{Introduction}
The question of understanding consciousness is in the focus of philosophers and researchers for more than two millennia. Insights range broadly from \emph{``Ignorabimus'' -- ``We will never know.''}\footnote{With this simple statement, Emil du Bois-Reymond concluded his talk on the limits of scientific knowledge about the relation of brain processes and subjective experience at the 45th annual meeting of German naturalists and physicians in 1872.} to mechanistic ideas with the aim to construct artificial consciousness following
Richard Feynman's famous words \emph{``What I cannot create, I do not understand.''}\footnote{Richard Feynman left these words on his blackboard in 1988 at the time of his death as a final message to the world.}.
The major issue that precludes the analysis of consciousness is its subjectivity. Our mind is able to feel and process our own conscious states. By induction, we are also able to ascribe conscious processing to other human beings. However, once we try to imagine to be another species, as Nagel describes in his seminal work ``What is it like to be a Bat?''\cite{nagel1974like}, we immediately fail to follow such experience consciously.
Another significant issue is that we are not able to determine consciousness by means of behavioural observations as Searle demonstrates in his thought experiment \cite{searle1980minds}. Searle describes a room that we cannot enter. One can pass messages written in Chinese to the room and the room returns messages to the outside world. All the messages and questions passed to the room are answered correctly as a Chinese person would. A first conclusion would be that there is somebody in the ``Chinese Room'' who speaks Chinese and answers the questions. However, the person in the room could also have simply access to a large dictionary that contains all possible questions and the respective answers. When we are not able to understand how the information is actually processed, we will never be able to determine whether a system is conscious or not.
In this article, we want to explore these and different thoughts in literature to address the problem of consciousness. We therefore revisit works in philosophy, neuroscience, artificial intelligence, and machine learning. Following the paradigm of cognitive computational neuroscience \cite{kriegeskorte2018cognitive}, we present how the convergence of these fields could potentially also lead to new insights regarding consciousness.
\section*{The Philosophical Perspective}
More than two thousand years ago, Aristotle was convinced that only humans are endowed with a rational soul. All animals, however, live only with the instincts necessary for survival, like biological automata. Along the same line, in the statement ``Cogito ergo sum'' also Descartes realised being self-aware is reserved for human beings. In his view, this insight is fundamental for any philosophical approach \cite{descartes1990meditations}.
Modern philosophy went on to differentiate the problem into an easy and a hard problem. While the ``easy problem'' is to explain its function, dynamics, and structure,
the ``hard problem of consciousness'' \cite{chalmers1995facing} is summarised in the Internet Encyclopedia of Philosophy \cite{Weisberg2020} as:
\emph{``The hard problem of consciousness is the problem of explaining why any physical state is conscious rather than nonconscious. It is the problem of explaining why there is ``something it is like'' for a subject in conscious experience, why conscious mental states ``light up'' and directly appear to the subject.''}
In order to avoid confusion some scientists prefer to speak of ``conscious experience'' or only ``experience'' instead of consciousness \cite{chalmers1995facing}. As already noted, the key problem of deriving models of conscious events is that they can only be perceived subjectively. As such it is difficult to encode such an experience in a way that it can be recreated by others. This gives rise to the so-called ``qualia problem'' \cite{crane2012origins} as we can never be sure, e.g. that the color red consciously looks the same to another person. Extension of this line of thought leads again to Nagel's thought experiment \cite{nagel1974like}.
According to \cite{Weisberg2020}, approaches to tackle the problem from a philosophical point of view are very numerous, but none of them can be considered to be exhaustive:
\begin{itemize}
\item \textbf{Eliminativism} \cite{rey1988question} demonstrates that the mind is fully functional without the experience of consciousness. Being nonfunctional, consciousness can be neglected. \item The view of
\textbf{strong reductionism} proposes that consciousness can be deconstructed into simpler parts and be explained by functional processes. Such considerations gave rise to the global work space theory \cite{newman1993neural,baars1994global,baars1994neurobiological} or integrated information theory \cite{tononi2004information,tononi2008consciousness} in neuroscience. The main critique of this view, is that any mechanistic solution to consciousness that is not fully understood will only mimic true consciousness, i.e., one could construct something that appears conscious that simply isn't as the Chinese Room argument demonstrates \cite{searle1980minds}.
\item \textbf{Mysterianism} proposes that the question of consciousness cannot be tackled with scientific methods. Therefore any investigation is in vain and the explanatory gap cannot be closed \cite{levine2001purple}.
\item In \textbf{Dualism} the problem is tackled as consciousness being metaphysical that is independent of physical substance \cite{descartes1990meditations}. Modern versions of Dualism exist, but virtually all of them require to reject that our world can be fully described by physical principles. Recently, Penrose and Hammeroff tried to close this gap using quantum theory \cite{penrose1994mechanisms,hameroff2014consciousness}. We dedicate a closer description of this view in a later section of this article.
\item Assuming that metaphysical world and physical world simply do not interact does not require to reject physics and gives rise to \textbf{Epiphenomenalism} \cite{campbell1992body}.
\end{itemize}
There are further theories and approaches to address the hard problem of consciousness that we do not want to detail here. To the interested reader, we recommend to study the Internet Encyclopedia of Philosophy \cite{Weisberg2020} as further reading into the topic.
In conclusion, we observe that a major disadvantage of exploring the subject of consciousness by philosophical means is that we are never able to explore the inside of the Chinese Room. Thought alone will not be able to open the black box. Neuroscience, however, offers means to explore the inside by various means of measurement and might offer suitable means to address the problem.
\section*{Consciousness in Neuroscience}
\subsection*{Historical Overview}
In 1924, Hans Berger recorded, for the first time, electrical brain activity using electroencephalography (EEG) \cite{berger1934elektrenkephalogramm}. This breakthrough enabled the investigation of different mental states by means of electrophysiology, e.g. during perception \cite{krauss2018statistical} or during sleep \cite{krauss2018analysis}. The theory of cell assemblies, proposed by Donald Hebb in 1949 \cite{hebb1949organization}, marked the starting point for the scientific investigation of neural networks as the biological basis for perception, cognition, memory, and action. In 1965, Gazzaniga demonstrated that dissecting the corpus callosum which connects the two brain hemispheres with each other results in a split of consciousness \cite{gazzaniga1965observations,gazzaniga2005forty}. Almost ten years later, Weiskrantz et al. discovered a phenomenon for which the term ``blindsight'' has been coined: following lesions in the occipital cortex, humans loose the ability to consciously perceive, but are still able to react to visual stimuli \cite{weiskrantz1974visual,weiskrantz1975blindsight}. In 1983, Libet demonstrated that voluntary acts are preceded by electrophysiological readiness potentials that have their maximum at about $550\,ms$ before the voluntary behavior \cite{libet1983preparation}. He concluded that the role of conscious processing might not be to initiate a specific voluntary act but rather to select and control volitional outcome \cite{libet1985unconscious}.
In contrast to the above mentioned philosophical tradition from Aristotle to Descartes that consciousness is a phenomenon that is exclusively reserved for humans, in contemporary neuroscience most researchers tend to regard consciousness as a gradual phenomenon, which in principle also occurs in animals \cite{boly2013consciousness}, and several main theories of how consciousness emerges have been proposed so far.
\subsection*{Neural Correlates of Consciousness}
Based on Singer's observation that high-frequency oscillatory responses in the feline visual cortex exhibit inter-columnar and inter-hemispheric synchronization which reflects global stimulus properties \cite{gray1989oscillatory,engel1991interhemispheric,singer1993synchronization} and might therefore be the solution for the so called ``binding problem'' \cite{singer1995visual}, Crick and Koch suggested Gamma frequency oscillations to play a key role in the emergence of consciousness \cite{crick1990towards}. Koch further developed this idea and investigated neural correlates of consciousness in humans \cite{tononi2008neural, koch2016neural}. He argued that activity in the primary visual cortex, for instance, is necessary but not sufficient for conscious perception, since activity in areas of extrastriate visual cortex correlates more closely with visual perception, and damage to these areas can selectively impair the ability to perceive particular features of stimuli \cite{rees2002neural}. Furthermore, he discussed the possibility that the timing or synchronization of neural activity might correlate with awareness, rather than simply the overall level of spiking \cite{rees2002neural}. A finding which is supported by recent neuroimaging studies of visual evoked activity in parietal and prefrontal cortex areas \cite{boly2017neural}. Based on these findings, Koch and Crick provided a framework for consciousness, where they proposed a coherent scheme to explain the neural activation of visual consciousness as competing cellular clusters \cite{crick2003framework}. Finally, the concept of neural correlates of consciousness has been further extended to an index of consciousness based on brain complexity \cite{casarotto2016stratification}, which is independent of sensory processing and behavior \cite{casali2013theoretically}, and might be used to quantify consciousness in comatose patients \cite{seth2008measuring}.
\subsection*{Consciousness as a Computational Phenomenon}
Motivated by the aforementioned findings concerning the neural correlates of consciousness, Tononi introduced the concept of integrated information, which according to his ``Integrated Information Theory of Consciousness'' plays a key role in the emergence of consciousness \cite{tononi2004information,tononi2008consciousness}. This theory represents one of two major theories of contemporary research in consciousness. According to this theory, the quality or content of consciousness is identical to the form of the conceptual structure specified by the physical substrates of consciousness, and the quantity or level of consciousness corresponds to its irreducibility, which is defined as integrated information \cite{tononi2016integrated}.
Tegmark generalized Tononi's framework even further from neural-net\-work-based consciousness to arbitrary quantum systems. He proposed that consciousness can be understood as a state of matter with distinctive information processing abilities, which he calls ``perceptronium'', and investigates interesting links to error-correcting codes and condensed matter criticality \cite{tegmark2014consciousness, tegmark2015consciousness}.
Even though, there is large consensus that consciousness can be understood as a computational phenomenon \cite{cleeremans2005computational,seth2009explanatory,reggia2016computational,grossberg2017towards}, there is dissent about which is the appropriate level of granularity of description and modeling \cite{kriegeskorte2018cognitive}. Penrose and Hameroff even proposed that certain features of quantum coherence could explain enigmatic aspects of consciousness, and that consciousness emerges from brain activities linked to fundamental ripples in spacetime geometry. In particular, according to their model of orchestrated objective reduction (Orch OR), they hypothesize that the brain is a kind of quantum computer, performing quantum computations in the microtubeles, which are cylindrical protein lattices of the neurons' cytoskeleton \cite{penrose1994mechanisms,hameroff1996orchestrated,hameroff2001biological}.
However, Tegmark and Koch argue, that the brain can be understood within a purely neurobiological framework, without invoking any quantum-mechanical properties: quantum computations which seek to exploit the parallelism inherent in entanglement, require that the qubits are well isolated from the rest of the system, whereas on the other hand, coupling the system to the external world is necessary for the input, the control, and the output of the computations. Due to the wet and warm nature of the brain, all these operations introduce noise into the computation, which causes decoherence of the quantum states, and thus makes quantum computations impossible. Furthermore, they argue that the molecular machines of the nervous system, such as the pre- and post-synaptic receptors, are so large that they can be treated as classical rather than quantum systems, i.e. that there is nothing fundamentally wrong with the current classical approach to neural network simulations \cite{tegmark2000importance,koch2006quantum,koch2007relation}.
\begin{figure}[tb]
\centering
\includegraphics[width=\linewidth]{global_workspace.pdf}
\caption{The Global Workspace emerges by connecting different brain areas according to Dehaene.}
\label{fig:globalworkspace}
\end{figure}
\subsection*{The Global Workspace Theory of Consciousness}
In the 1990s, Baars introduced the concept of a virtual ``Global Workspace'' that emerges by connecting different brain areas (Figure \ref{fig:globalworkspace}) to describe consciousness \cite{newman1993neural,baars1994global,baars1994neurobiological,baars2007global}. This idea was taken up and further developed by Dehaene \cite{dehaene1998neuronal,dehaene2001towards,dehaene2004neural,sergent2004neural,dehaene2011global,dehaene2014toward}. Today, besides the Integrated Information Theory, the Global Workspace Theory represents the second major theory of consciousness, being intensively discussed in the field of cognitive neuroscience. Based on the implications of this theory, i.e., that consciousness arises from specific types of information-processing computations, which are physically realized by the hardware of the brain, Dehaene argues that a machine endowed with these processing abilities \emph{``would behave as though it were conscious; for instance, it would know that it is seeing something, would express confidence in it,would report it to others, could suffer hallucinations when its monitoring mechanisms break down, and may even experience the same perceptual illusions as humans''} \cite{dehaene2017consciousness}.
\begin{figure}[tb]
\centering
\includegraphics[width=0.7\linewidth]{damasio_model.pdf}
\caption{Simplified view of Damasio's model of consciousness: The protoself processes emotions and sensory input unconsciously. Core consciousness arises from the protoself which allows to put the itself into relation. Projections of emotions give rise to higher-order feelings. With access to memory and extended functions such as language processing the extended consciousness emerges.}
\label{fig:damasiomodel}
\end{figure}
\subsection*{Damasio's Model of Consciousness}
Damasio's model of consciousness was initially published in his popular science book ``The feeling of what happens'' \cite{damasio1999feeling}. Later Damasio also published the central ideas in peer-reviewed scientific literature \cite{damasio2009consciousness}. With the ideas being published first in a popular science book, most publications on consciousness neglect his contributions. However, we believe that his thoughts deserve more attention. Therefore, we want to introduce his ideas quickly in this section.
The main idea in Damasio's model is to relate consciousness to the ability to identify one's self in the world and to be able to put the self in relation with the world. However, a formal definition is more complex and requires the introduction of several concepts first.
He introduces three levels of conscious processing:
\begin{itemize}
\item The fundamental \textbf{protoself} does not possess the ability to recognize itself. It is a mere processing chain that reacts to inputs and stimuli like an automaton, completely non-conscious. As such any animal has a protoself according to this definition. However, also more advanced lifeforms including humans exhibit this kind of self.
\item A second stage of consciousness is the \textbf{core consciousness}. It is able to anticipate reactions in its environment and adapts to them. Furthermore, it is able to recognise itself and its parts in its own image of the world. This enables it to anticipate and to react to the world. However, core consciousness is also volatile and not able to persist for hours to form complex plans.
In contrast to many philosophical approaches, core consciousness does not require to represent representations of the world in words or language. In fact, Damasio believes that progress in understanding conscious processing has been impeded by dependence on words and language.
\item The \textbf{extended consciousness} enables human-like interaction with the world. It builds on top of core consciousness and enables further functions such as access to memory in order to create a autobiographic self. Also being able to process words and language falls into the category extended consciousness and can be interpreted as a form of serialisation of conscious images and states.
\end{itemize}
In Damasio's theory emotion and feelings are fundamental concepts \cite{damasio2001fundamental}. In particular Damasio differentiates emotions from feelings. \textbf{Emotions} are direct signals that indicate a positive or negative state of the (proto-)self. \textbf{Feelings} emerge only in conjunction with images of the world and can be interpreted as a second-order emotion that is derived from the world representation and future possible events in the world. Both are crucial for the emergence of consciousness. Fig.~\ref{fig:damasiomodel} schematically puts the described terms in relation.
After having defined the above concepts, Damasio now goes on to attempt and describe a model of (core) consciousness. In his theory, consciousness does not merely emerge from the ability to identify oneself in the world or an image of the world. For conscious processing, additionally feeling oneself in the sense of desiring to exist is required. Hence, he postulates a feeling, i.e., a derived second-order emotion, between the protoself and its internal representation of the world. Conscious beings as such want to identify oneself in the world and want to exist. From an evolutionary perspective as he argues, this is possibly a mechanism to enforce self-preservation.
In the context of this article, Damasio's theory is interesting for two major reasons. On the one hand, it describes a biologically plausible model of consciousness, as he locates all stages of consciousness to structures in the brain and associates them to the respective function. On the other hand, Damasio describes a mechanistic model that can at least in theory be completely implemented as a computer program.
Hence, we can conclude that neuroscience is able to describe fundamental processes in the brain that give rise to complex phenomena such as consciousness. However, the means of observation in neuroscience are insufficient. Neither EEG nor fMRI have a temporal and spatial resolution that is even close enough to observe what is happening in the brain in-vivo. At this point, the recent massive progress in artificial intelligence and machine learning comes to our attention and will be the focus of our next section.
\section*{Consciousness in Machine Learning and AI}
In artificial intelligence (AI) numerous theories of consciousness exist \cite{sun2007computational, starzyk2010machine}. Implementations often focus on the global work space theory with only limited learning capabilities \cite{franklin1999software}, i.e. most of the consciousness is hard-coded and not trainable \cite{kotov2017computational}. An exception is the theory by van Hateren which relates the consciousness to close to simultaneous forward and backward processing in the brain \cite{van2019theory}. Yet, algorithms that were investigated so far made use of a global work space and mechanistic hard-coded models of consciousness. Following this line, research on minds and consciousness rather focuses on representation than on actual self-awareness \cite{tenenbaum2011grow}.
Although representation will be important to create human-like minds and general intelligence \cite{gershman2015computational,lake2017building, mao2019neurosymbolic}, a key factor to become conscious is the ability to identify a {\it self} in one's environment \cite{dehaene2017consciousness}.
A major drawback of pure mechanistic methods, however, is that the complete knowledge on the model of consciousness is required in order to realise and implement them. As such, in order to develop these models to higher forms such as Damasio's extended consciousness, a complete mechanistic model of the entire brain including all connections is required.
\subsection*{Consciousness in Machine Learning}
A possible solution to this problem is machine learning, as it allows to form and train complex models. The topic of consciousness, however, is neglected in the field to a large extent. On the one hand, this is because of the concerns that the brain and consciousness will never be successfully simulated in a computer system \cite{penrose2001consciousness,hameroff2014consciousness}. On the other hand, consciousness is considered to be an extremely hard problem and current results in AI are still meager \cite{brunette2009review}.
The state-of-the-art in machine learning instead focuses on supervised and unsupervised learning techniques \cite{bishop2006pattern}. Another important research direction is reinforcement learning \cite{sutton1998introduction} that aims at learning of suitable actions for an agent in a given environment. As consciousness is often associated with an embodiment, reinforcement learning is likely to be important for modelling of consciousness.
The earliest work that the authors are aware of attempting to model and create agents that learn their own representation of the world entirely using machine learning date back to the early 1990's. Already in 1990, Schmidhuber proposed a model for dynamic reinforcement learning in reactive environments \cite{schmidhuber1990line} and found evidence for self-awareness in 1991 \cite{schmidhuber1991possibility}. The model follows the idea of a global work space. In particular, future rewards and inputs are predicted using a world model.
Yet, Schmidhuber was missing a theory on how to analyse intelligence and consciousness in this approach. Similar to Tononi \cite{tononi2008consciousness}, Schmidhuber followed the idea of compressed neural representation. Interestingly, compression is also key to inductive reasoning, i.e., learning from few examples which we typically deem as intelligent behaviour.
Solomonoff's Universal Theory of Inductive Inference \cite{solomonoff1964formal} gives a theoretic framework to inductive reasoning. It combines information and compression theory and results in a formalisation of Occam's razor preferring simple models over complex ones, as simple models are more likely from an information theoretic point of view \cite{maguire2016understanding}.
Under Schmidhuber's supervision, Hutter applied Solomonoff's theory to machine learning to form a theory of Universal Artificial Intelligence \cite{hutter2004universal}. In his theory, intelligent behaviour stems from efficient compression of inputs, e.g. from an environment, such that predictions and actions are performed optimally. Again, models capable of describing a global work space play an important role.
Maguire et al. further expand on this concept to extend Solomonoff's and Hutter's theories to also describe consciousness. Following the ideas of Tononi and Koch \cite{rees2002neural} consciousness is understood as data compression, i.e. the optimal integration of information \cite{maguire2016understanding}. The actual consciousness emerges from binding of information and is inherently complex. As such, consciousness can also not be deconstructed into mechanical sub-components, as the decomposition would destroy the sophisticated data compression. Maguire et al. even provide a mathematical proof to demonstrate that consciousness is either integrated and therefore cannot be decomposed or there is an explicit mechanistic way of modelling and describing consciousness \cite{maguire2016understanding}.
Based on the extreme success of deep learning \cite{lecun2015deep}, also several scientists observed similarities in neuroscience and machine learning. In particular, deep learning allows to build complex models that are hard to analyse and interpret at the benefit of making complex predictions. As such both fields are likely to benefit each other in the ability to understand and interpret complex dynamic systems \cite{marblestone2016toward,van2017computational,hassabis2017neuroscience,kriegeskorte2018cognitive,barrett2019analyzing, richards2019deep, savage2019marriage}. In particular, hard-wiring following biological ideas might help to reduce the search space dramatically \cite{zador2019critique}. This is in line with recent theoretical considerations in machine learning as prior knowledge allows to reduce maximal error bounds \cite{maier2019learning}. Both fields can benefit from these ideas as recent discoveries of e.g. successor representation show \cite{stachenfeld2017hippocampus, gershman2018successor, geerts2019probabilistic}. Several scientists believe that extension of this approach to social, cultural, economic, and political sciences will create even more synergy resulting in the field of machine behaviour \cite{rahwan2019machine}.
\begin{figure}[tbp]
\centering
\includegraphics[width=\linewidth]{overview_architectures.pdf}
\caption{Overview on typical architectures in machine learning. Hard-coded paths, i.e. recurrent connections, are indicated by solid arrows, trainable connections by dashed arrows, and training losses are indicated by red double arrows. While archtiectures A-E do not match theories of consciousness, architectures F-H implement theories by Schmidhuber and Damasio.}
\label{fig:architectures}
\end{figure}
\section*{Can Consciousness emerge in Machine Learning Systems?}
After having reviewed philosophy, neuroscience, and the state-of-the-art in AI and machine learning, we can now analyse the most important concepts in the field of machine and deep learning to assess whether they have the potential to create consciousness following one of the previous theories. In particular, we focus on the ability of the system to represent a symbol of self and how this self-awareness is constructed, as all theories of consciousness require at least experiencing the self.
We show a brief overview on important architectures in machine learning in Fig.~\ref{fig:architectures}. We denote hard-wired connections as solid arrows to indicate recurrent modes of specific network parts. Dashed lines indicate trainable connections. They could be implemented as a single feed-forward layer, i.e., a universal function approximator. Without loss of generality, they could also be implemented by other deep feed-forward architectures \cite{maier2019gentle} and could thus be inherently complex. Red double arrows indicate training losses to adjust the trainable weights of the dashed arrows.
Fig.~\ref{fig:architectures} A shows a simple feed-forward architecture that requires external labelled training data $\bf{y}^*$ to adjust its trainable weights given input $\bf{x}$ to produce output $\bf y$. Fig.~\ref{fig:architectures} B shows a similar setup, for the recurrent case. Note that we only indicate a simple recurrent cell here with time-dependent state $\bf h_t$. Without loss of generality, this could also be realised using gated recurrent units \cite{cho2014properties} or long short term memory cells \cite{hochreiter1997long}. Both models fall into the category of supervised learning. Consciousness is not supported by any of the theories presented so far.
In Fig.~\ref{fig:architectures} C, we introduce the concept of ``Emotion'' following Damasio's wording. In machine learning terms, this reflects an additional loss. Now, the system receives an additional input $\bf e$ that is associated to a valence or value. Without loss of generality, we can assume positive entries in $\bf e$ to be associated to desirable states for the system and negative values to undesirable states. As such, training using $\bf e$ falls into the category of reinforcement learning that aims at maximizing future values of $\bf e$. In order to model competing interests and saturation effects, e.g. a full battery does not need to be charged further, we introduce a reference $\bf e^*$ that is able to model such effects. Note that we deem the system to be able to predict the expected future reward $\bf e'$ from its current state $\bf h_t$ following a deep Q learning paradigm \cite{mnih2015human}. Here we use $\bf e'$ and $\bf e^*$ to construct a trainable reinforcement loss, to be able to learn from low-level rewards/emotions $\bf e$. Although being able to learn, the system still needs supervision to train the weights producing output $\bf y$ using a reference. This setup also does not match any theory of consciousness so far.
As self-awareness is a requirement for base consciousness, we deem a world model to be necessary. Such an approach his shown in Fig.~\ref{fig:architectures} D. Given from the produced output $\bf y$, the world model is used to create an expected future input $\bf x'$. In the figure, we chose a recurrent model capturing the state of the world in $\bf w_t$ that is independent of the internal state of the actual agent $\bf h_t$. To gain consciousness, this model misses at least a link from internal to external state and ``emotions'' that would guide future decisions. Adding low-level rewards/emotions results in Fig.~\ref{fig:architectures} E. Again world and self are disconnected inhibiting self-representation and self-discovery. Approaches like this are already being explored for video game control \cite{kaiser2019model}.
With a world-model being present, we are now able to predict future rewards $\bf e''$ that also take into account the state of the world and the chosen action. As such Fig.~\ref{fig:architectures} F is the first one that would implement a trainable version of deep Q learning. Development of consciousness is debatable, as the model does not feature a link between the state of the world and the state of the agent. If we would add a trainable connection from $\bf h_t$ to $\bf w_t$ and vice versa, we would end up with Schmidhuber's Model from 1990 \cite{schmidhuber1990line} for which Schmidhuber found evidence to develop self representation \cite{schmidhuber1991possibility}.
Interestingly, Damasio's descriptions follow a similar line in \cite{damasio1999feeling}. We depict a model implementing Damasio's core consciousness in Fig.~\ref{fig:architectures} G.
As Schmidhuber, Damasio requires a connection from the world model $\bf w_t$ to the body control system $\bf h_t$. However, in his view, consciousness does not emerge by itself. It is enforced by a ``feeling'' that is expressed as a loss in the world of machine learning. As such, the Damasio model of core consciousness requires a loss that aims at the recovery of the image of the self in the world model. If this is implemented as a loss, we are able to express the desire to exist in the world. If implemented merely as trainable weights, we arrive at the theory of integrated information that creates consciousness as maximized compression of the world, the self, and their interaction. Interestingly, these considerations also allow integration of attention \cite{vaswani2017attention} and other concepts of resolving context information in machine learning. Realised in a biological learning framework, e.g. using neuromodulators like Dopamin \cite{russek2017predictive}, the notion of loss and connection will disappear and the models of Damasio, Schmidhuber, Tononi, Koch, and Dehane turn out to be different descriptions of the same principles.
Note that the models of consciousness that we have discussed so far are very basic. They do not concern language, memory, or other complex multimodal forms of processing, planning, induction, or representation. Again, we follow Damasio at this point in Fig.~\ref{fig:architectures} H in which all of these sophisticated processes are mapped into a block ``Memory / Extended Functions''. Note although we omit these extended functions, we are able to integrate them using trainable paths. As such, the model of core consciousness acts as a ``neural operating system'' that is able to update
also higher order functions according to the needs of the environment. With increasing ``extended functions'', the degree of complexity and ``integrated information'' rises measurably as also observed by Casarotto \cite{casarotto2016stratification}.
This brings us back to the original heading of our section: There are clearly theories that enable modelling and implementation of consciousness in the machine. On the one hand, they are mechanistic to the extend that they can be implemented in programming languages and require similar inputs as humans would do. On the other hand, even the simple models in Fig.\ref{fig:architectures} are already arbitrarily complex, as every dashed path in the models could be realised by a deep network. As such also training will be hard. Interestingly, the models follow a bottom-up strategy such that training and development can be performed in analogy to biological development and evolution. The models can be trained and grown to more complex tasks gradually.
\section*{Discussion}
Existence of consciousness in the machine is a hot topic of debate. Even with respect to the simple core consciousness, we observe opinions ranging from ``generally impossible'' \cite{carter2018conscious} through ``plausible'' \cite{dehaene2017consciousness} to ``has already been done'' \cite{schmidhuber1991possibility}. Obviously, all of the suggested models cannot solve the qualia problem or the general problem on how to demonstrate whether a system is truly conscious. All of the emerging systems could merely be mimicking conscious behaviour without being conscious at all (even Fig.~\ref{fig:architectures} A). Yet as already discussed by Schmidhuber \cite{schmidhuber1991possibility}, we would be able to measure correlates of self recognition similar to neural correlates of consciousness in humans \cite{koch2016neural} which could help to understand consciousness in human beings. However, as long as we have not solved how to provide proof of consciousness in human beings, we will also fail to do so in machines as the experience of consciousness is merely subjective.
Koch and Dehaene discussed the theories of global work space and integrated information as being opposed to each other \cite{carter2018conscious}. In the models found in Fig.~\ref{fig:architectures}, we see that both concepts require a strong degree of interconnection. As such, we do not see why both concepts are fundamentally opposing. A global work space does not necessarily have to be encoded in decompressed state. Also, Maguire's view of integrated information \cite{maguire2016understanding} is not necessarily impossible to implement mechanistically, as we are able to use concepts of deep learning to train highly integrated processing networks. In fact, as observed by neuroscience \cite{kriegeskorte2018cognitive}, both approaches might support each other yielding methods to construct and reproduce biological processes in a modular way. This allows the integration of representation \cite{gershman2015computational} and processing theories \cite{sun2007computational} as long as they can be represented in terms of deep learning compatible operations \cite{maier2019learning}.
In all theories that we touched in this article, the notion of self is fundamental. Hence, all presented theories of consciousness require embodiment as basis for consciousness. As such, a body is required for conscious processing. Also the role of emotion and feelings is vital. Without emotion or feelings, the system cannot be trained and thus it can not adopt to new environments and changes of circumstances. In the machine learning inspired models, we assume that a disconnection between environment and self would cause a degradation of the system similar to the one that is observed in human beings in locked-in state \cite{kubler2008brain}. This homeostatsis was also deemed important by Man et al. \cite{man2019homeostasis}.
Similar to the problems identified by Nagel, also the proposed mechanistic machine learning models will not be able to understand ``what it is like'' to be a bat. However, the notion of train-/learnable programs and connections or adapters might offer a solution to explore this in the future. Analogously, one cannot describe to somebody ``what it is like'' to play the piano or to snowboard on expert level unless one really acquires the ability. As such also the qualia problem persists in machine consciousness. However, we are able to investigate the actual configuration of the representation in the artificial neural net offering entirely new mechanisms of insight.
In Damasio's theory, consciousness is effectively created by a training loss that causes the system to ``want'' to be conscious, i.e., "Cogito ergo sum" becomes "Sentio ergo sum". Comparison between trainable connections after \cite{schmidhuber1990line}, attention mechanisms \cite{vaswani2017attention}, and this approach are within the reach of future machine learning models which will create new evidence for the discussion of integrated information and global work spaces. In fact, Schmidhuber has already taken up the work on combination of his early ideas with modern approaches from deep learning \cite{schmidhuber2015learning,schmidhuber2018one}.
With models for extended consciousness, even the notion of the Homunculus \cite{kenny2016homunculus} can be represented by extension of the self with another self pointer. In contrast to common rejection of the Homunculus thought experiment, this recurrent approach can be trained using end-to-end systems comparable to AlphaGo \cite{silver2016mastering}.
Damasio also presents more interesting and important work that is mostly omitted in this article for brevity. In \cite{damasio1999feeling} he also relates structural brain damage to functional loss of cognitive and conscious processing. Also the notion of emotion is crucial in a biological sense and is the driving effect of homeostasis. In \cite{man2019homeostasis}, Damasio already pointed out that this concept will be fundamental for self-regulating robotic approaches.
With the ideas of cognitive computational neuroscience \cite{kriegeskorte2018cognitive} and the approaches detailed above, we will design artificial systems that approach the mechanisms of biological systems in an iterative manner. With the iterations, the artificial systems will increase in complexity and similarity to the biological systems. However, even if we arrive at an artificial system that performs identical computations and reveals identical behaviour as the biological system, we will not be able to deem this system as conscious beyond any doubts. The true challenge in being perceived as conscious will be the acceptance by human beings and society. As Alan Turing already proposed in his imitation game in 1950 to decide whether a machine is intelligent or even conscious \cite{turing1950computing}, the ascription of such by other humans is a critical factor. For this purpose, Turing's Test has already been extended to also account for embodiment \cite{french2012moving}. However, such tests are only necessary, but not sufficient as Gary Marcus pointed out: Rather simple chat bot models are already able to beat Turing's Test in some occasions \cite{vardi2014would}. Hence, requirements for conscious machines will comprise, the similarity to biological conscious processes, the ability to convince human beings, and even the machine itself. As such, we deem it necessary to look as some ethical implications at this point.
\section*{Ethical implications}
Being able to create systems that are indistinguishable from conscious beings that are potentially conscious also raises ethical concerns. First and foremost, in the transformation from core consciousness to extended consciousness, the systems gain the ability to link new program routines. As such systems following such a line of implementation need to be handled with care and should be experimented on in a contained environment. With the right choice of embodiment in a virtual machine or in a robotic body, one should be able to solve such problems.
Of course there are also other ethical concerns, the more we approach human-like behaviour. A first set of robotic laws has been introduced in Asimov's novels \cite{clarke1993asimov}. Even Asimov considered the rules problematic as can be seen from the plot twists in his novels. Aside this, being able to follow the robotic laws requires the robot to understand the concepts of ``humans'', ``harm'', and ``self''. Hence, such beings must be conscious. Therefore, tampering with their memories, emotions, and feelings is also problematic by itself. Being able to copy and reproduce the same body and mind does not lead to further simplification of the issue and implies the problem that we have to agree on ethics and standards of AI soon \cite{jobin2019global}.
\section*{Conclusion}
In this article, we reviewed the state-of-the-art theories on consciousness in philosophy, neuroscience, AI, and machine learning. We find that all three disciplines need to interact to push research in this direction further. Interestingly, basic theories of consciousness can be implemented in computer programs. In particular, deep learning approaches are interesting as they offer the ability to train deep approximators that are not yet well understood to construct mechanistic systems of complex neural and cognitive processes. We reviewed several machine learning architectures and related them to theories of strong reductionism and found that there are neural network architectures from which base consciousness could emerge. Yet, there is still a long way to form human-like extended consciousness.
\section*{Data availability}
All data in this publication are publicly available.
\section*{Code availability}
All software used in this article is publicly available.
\section*{Acknowledgments}
This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation): grant KR5148/2-1 to PK -- project number 436456810, the Emergent Talents Initiative (ETI) of the University Erlangen-Nuremberg (grant 2019/2-Phil-01 to PK). Furthermore, the research leading to these results has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (ERC grant no. 810316).
\section*{Author contributions}
PK and AM contributed equally to this work.
\section*{Ethics declarations}
\subsection*{Competing interests}
The authors declare no competing financial interests.
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Home » National » Civic reception hosted in honour of the President in Thiruvananthapuram
The President, Shri Ram Nath Kovind at the civic reception, at Thiruvanthapuram, in Kerala on October 27, 2017.
Civic reception hosted in honour of the President in Thiruvananthapuram
in National October 28, 2017 0 495 Views
Thiruvanthapuram:28/10/17:The President of India, Shri Ram NathKovind, addressed a civic reception hosted for him at Tagore Theatre, Thiruvananthapuram on October 27, 2017
Speaking on the occasion, the President said that historically Kerala is a land of spiritualism, with a tradition of spiritual leadership and social reform extending from AdiShankaracharya to Sri Narayana Guru to Ayyankali and so many others.
The President said that Kerala's traditions and thinking have been humanistic, people-oriented and democratic. The emphasis on human development and on health-care and education in Kerala has set an example for the rest of the country. In sanitation, Kerala's achievements are praiseworthy. In local self-government and panchayati raj, again Kerala has deepened our democracy.
The President said that not only are the people of Kerala educated and knowledgeable, they are willing to use that education and knowledge to venture out and to help build our nation and to contribute to the well-being and the prosperity of other regions.
The President said that today, Kerala continues to attract the international community. This beautiful state is one of India's leading tourism destinations. He stated that just as outsiders have flocked to Kerala, Kerala's people too have ventured out of their home state and contributed to the world. The Malayalee migrant community is the backbone of the work-force of so many Gulf States. And it faithfully sends back remittances to help in the development of Kerala and of India.
Earlier in the day, the President paid tributes to social reformer Ayyankali by garlanding his statue in Thiruvananthapuram. He also launched the Technocity project and laid the foundation stone for the first government building in Technocity in Thiruvananthapuram. Tomorrow, he will grace the valedictory function of the diamond jubilee celebrations of High Court of Kerala in Kochi before returning to Delhi.
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The Glassware Division manufactures a variety of bottles that can be sold externally (to soft-drink and juice bottlers) or internally to Mouton & Perrier'...\n##### 1point What is [(28.7 x 105) 48.533] 140 calculated to the correct number of significant figures? 5.9*10^4\n1point What is [(28.7 x 105) 48.533] 140 calculated to the correct number of significant figures? 5.9*10^4...\n##### A49-g blcck al coppor @t-13 addeddo JO0 & % waler in 80-& a uminum Cup; Tno = and tho wator have an Initial lemperature 0i 45Part &Fird Iho oqullibrum torpurature at the cup und its contunts. Exprons Vout Gnewor Uelno two slanhlcami Ilauree:SubmltBeqvast Answer\nA49-g blcck al coppor @t-13 addeddo JO0 & % waler in 80-& a uminum Cup; Tno = and tho wator have an Initial lemperature 0i 45 Part & Fird Iho oqullibrum torpurature at the cup und its contunts. Exprons Vout Gnewor Uelno two slanhlcami Ilauree: Submlt Beqvast Answer...\n##### For an electron in a hydrogen atom, how is the value of n of its orbit...\nFor an electron in a hydrogen atom, how is the value of n of its orbit related to its energy?...\n##### 5. cis-[Pt(PEt3)2C12] is normally stable in benzene solution. Upon the addition of a trace amount of...\n5. cis-[Pt(PEt3)2C12] is normally stable in benzene solution. Upon the addition of a trace amount of PPh3, however, it rapidly converts to a mixture of the cis and trans isomers. Draw a mechanism that accounts for this behavior...\n##### From the reaction enthalpies of the following chemical equations:C2H2 (g) + 5\/2 O2 (g) \u00e2\u2020\u2019 2 CO2 (g) + H2O (l) \u00ce\u201dH \u00c2\u00b0 = - 1299.6 kJC (graphite) + O2 (g) \u00e2\u2020\u2019 CO2 (g) \u00ce\u201dH \u00c2\u00b0 = - 393.5 kJH2 (g) + 1\/2 O2 (g) \u00e2\u2020\u2019 H2O (l) \u00ce\u201dH \u00c2\u00b0 = - 285.8 kJIn each of the following statements, indicate whether it is true or false1.- The combustion reaction of ethyne, C2H2, is endothermic.a)tb) f2.-The value of the enthalpy of the reaction: 2 CO2 (g) + H2O (l) \u00e2\u2020\u2019 C2H2 (g) + 5\/2 O2 (g) is \u00ce\u201dH \u00c2\u00b0 = 1299.6 kJ.a)tb) f3.-T\nFrom the reaction enthalpies of the following chemical equations: C2H2 (g) + 5\/2 O2 (g) \u00e2\u2020\u2019 2 CO2 (g) + H2O (l) \u00ce\u201dH \u00c2\u00b0 = - 1299.6 kJ C (graphite) + O2 (g) \u00e2\u2020\u2019 CO2 (g) \u00ce\u201dH \u00c2\u00b0 = - 393.5 kJ H2 (g) + 1\/2 O2 (g) \u00e2\u2020\u2019 H2O (l) \u00ce\u201dH \u00c2\u00b0 = - 285.8 kJ In each of the following statem...\n##### 7.17 and 7.18 Determine the resultant of the parallel force system shown.\n7.17 and 7.18 Determine the resultant of the parallel force system shown....\n##### How do you solve 6\/8=12\/x?\nHow do you solve 6\/8=12\/x?...\n##### How do you simplify the product (3x - 1)(2x + 1) and write it in standard form?\nHow do you simplify the product (3x - 1)(2x + 1) and write it in standard form?...\n##### If the probability of it raining today is 30%, what is theprobability of the complement (NOT raining today).\nIf the probability of it raining today is 30%, what is the probability of the complement (NOT raining today)....\n##### Evaluate the integral by interpreting it in terms of areas In other words, draw a the integral represents, and find the area using picture of the region geometry: [ I6. 9\/drQuestion Help:VideoSubmit Question\nEvaluate the integral by interpreting it in terms of areas In other words, draw a the integral represents, and find the area using picture of the region geometry: [ I6. 9\/dr Question Help: Video Submit Question...","date":"2022-10-05 09:14:53","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.6716412305831909, \"perplexity\": 6425.839477550694}, \"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-2022-40\/segments\/1664030337595.1\/warc\/CC-MAIN-20221005073953-20221005103953-00016.warc.gz\"}"}
| null | null |
using System;
using System.IO;
using System.Security.Cryptography;
using System.Text;
namespace QingFeng.Common.Extensions
{
public static class DecryptHelper
{
public static string Sha512(string text)
{
var sha512 = new SHA512CryptoServiceProvider();
var bytesSha512In = Encoding.Default.GetBytes(text);
var bytesSha512Out = sha512.ComputeHash(bytesSha512In);
var strSha512Out = BitConverter.ToString(bytesSha512Out).Replace("-", "").ToLower();
return strSha512Out;
}
public static string SaltAndHash(string rawString, string salt)
{
var salted = string.Concat(rawString, salt);
return Sha512(salted).ToUpper();
}
public static string Md5(string text)
{
var md5 = new MD5CryptoServiceProvider();
var data = Encoding.Default.GetBytes(text);
var md5Data = md5.ComputeHash(data);
return BitConverter.ToString(md5Data).Replace("-", "").ToLower();
}
public static string Hmacsha1(this string encryptText, string encryptKey)
{
var myHmacsha1 = new HMACSHA1(Encoding.Default.GetBytes(encryptKey));
var rstRes = myHmacsha1.ComputeHash(Encoding.Default.GetBytes(encryptText));
var enText = new StringBuilder();
foreach (var Byte in rstRes)
{
enText.AppendFormat("{0:x2}", Byte);
}
return Convert.ToBase64String(myHmacsha1.Hash);
}
/// <summary>
/// 对称密钥解密
/// </summary>
/// <typeparam name="TAlgorithm"></typeparam>
/// <param name="str"></param>
/// <param name="key"></param>
/// <returns></returns>
public static string Decrypt<TAlgorithm>(this string str, string key)
where TAlgorithm : SymmetricAlgorithm
{
return str.Decrypt<TAlgorithm, UTF8Encoding>(key);
}
/// <summary>
/// 对称密钥解密
/// </summary>
/// <typeparam name="TAlgorithm">对称加密算法</typeparam>
/// <typeparam name="TStringEncoding">字符编码</typeparam>
/// <param name="str"></param>
/// <param name="key"></param>
/// <returns></returns>
public static string Decrypt<TAlgorithm, TStringEncoding>(this string str, string key)
where TAlgorithm : SymmetricAlgorithm
where TStringEncoding : Encoding
{
var s = new MemoryStream(Convert.FromBase64String(str));
var decryptedStream = s.Decrypt<TAlgorithm>(key.ToByteArray());
var bytes = decryptedStream.ToByteArray();
return Activator.CreateInstance<TStringEncoding>().GetString(bytes);
}
public static Stream Decrypt<TAlgorithm>(this Stream stream, byte[] key)
where TAlgorithm : SymmetricAlgorithm
{
var alg = Activator.CreateInstance<TAlgorithm>();
var pdb = new PasswordDeriveBytes(key, null);
alg.Key = pdb.GetBytes(alg.KeySize/8);
alg.GenerateIV();
alg.IV = pdb.GetBytes(alg.IV.Length);
var encryptor = alg.CreateDecryptor();
return new CryptoStream(stream, encryptor, CryptoStreamMode.Read);
}
public static string TripleDesc(string key, string str)
{
var inputByteArray = Encoding.Default.GetBytes(str);
var crypto = new TripleDESCryptoServiceProvider();
var pdb = new PasswordDeriveBytes(key, null);
crypto.Key = pdb.GetBytes(crypto.KeySize/8);
crypto.GenerateIV();
crypto.IV = pdb.GetBytes(crypto.IV.Length);
crypto.CreateEncryptor();
var ms = new MemoryStream();
var cs = new CryptoStream(ms, crypto.CreateEncryptor(), CryptoStreamMode.Write);
cs.Write(inputByteArray, 0, inputByteArray.Length);
cs.FlushFinalBlock();
return Convert.ToBase64String(ms.ToArray());
}
public static byte[] ToByteArray(this Stream stream)
{
var buffer = new byte[1000];
var readLength = 1;
using (var ms = new MemoryStream())
{
while (readLength > 0)
{
readLength = stream.Read(buffer, 0, buffer.Length);
if (readLength > 0)
ms.Write(buffer, 0, readLength);
}
return ms.ToArray();
}
}
public static byte[] ToByteArray(this string str)
{
return str.ToByteArray<UTF8Encoding>();
}
/// <summary>
/// 字符串转换为字节数组
/// </summary>
/// <typeparam name="TEncoding">编码</typeparam>
/// <param name="str"></param>
/// <returns></returns>
public static byte[] ToByteArray<TEncoding>(this string str) where TEncoding : Encoding
{
Encoding enc = Activator.CreateInstance<TEncoding>();
return enc.GetBytes(str);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,175
|
Neuffen is a decorative pendant light, powered by SAMSUNG SMD LEDs and linked with a high-quality MEANWELL driver. It is composed of an extruded aluminum frame and a PMMA diffuser. Moreover, Neuffen is suspended using stainless steel suspension wires to ensure a firm grip.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,384
|
package drawable;
import view.DrawView;
public class DrawCommandGetX implements DrawRequest {
public DrawCommandGetX() {
}
public double get(DrawView view) {
return view.getX();
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,487
|
Centurija (lat. centuria od centum – sto) je osnovna postrojba rimske vojske, broji 70 do 100 vojnika. Javlja se u starom Rimu vjerojatno već tijekom borbe protiv etruščanske prevlasti, kada su i plebejci, krajem 6. stoljeća pr. Kr., bili podvrgnuti vojnoj obvezi.
Rimski kralj Servije Tulije je podijelio rimsku teritoriju na 20 tribusa (plebs podijeljen na rodove) od po četiri centurije, od kojih su tri bile teško a četvrta lako naoružana.
Tijekom rimske republike, centurije – vojno organiziran narod – je birao konzule i na taj način postaju i izborna tijela. Ove dvije funkcije centurija – vojne postrojbe i izborna tijela - se vremenom razilaze. Širenjem teritorije broj tribusa se povećava na 35 a time i broj izbornih centurija koje postaju isključivo administrativne postrojbe, dok se vojna postrojbe preimenuju u legije, kao područne vojne postrojbe, falange – taktičke postrojbe i manipule. 443. pr. Kr. se osniva i prva centurijska skupština.
Na čelu centurije se nalazio centurion (lat. centurio) i po svom socijalnom položaju odgovarao je činu dočasnika, a po funkciji koju je izvršavao zapovjedniku satnije. Tijekom rimske republike probijao se od običnog vojnika (ex caliga), nižih socijalnih slojeva. Imao je dvostruko veću plaću od običnog vojnika a tijekom carstva petostruko. Centurion je bio nositelj rimskog vojnog uređenja. Centurioni nisu bili izbirani samo po hrabrosti, već i po čvrstini i sposobnosti vođenja ljudi. U borbi, obično je zauzimao mjesto na desnoj strani svoje centurije. U doba carstva, bilo je i centuriona iz viteških redova (ex eqiute Romano) koji su bili izbirani iz redova pretorijanaca a podrijetlom Italici. Funkcija centuriona se gubi početkom 6. stoljeća.
Izvori
Vojne postrojbe Starog Rima
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,326
|
Subscribe to Federal Communications Commission
Competition Policy Gets a Top Spot in the White House
By Aaron J. Burstein & Steven A. Augustino on March 8, 2021
Posted in Federal Communications Commission, Federal Trade Commission
Following weeks of speculation about a potential role for Columbia Law Professor Tim Wu in the Biden Administration, the White House announced on March 5 that Wu has been named Special Assistant to the President for Technology and Competition Policy. As an official housed in the National Economic Council ("NEC"), Wu will not directly command staff within federal agencies or set the agencies' enforcement or regulatory agendas. Instead, Wu will most likely focus on coordinating federal agencies' efforts to identify and address competition issues. Given his history, Wu could seek to have particular influence on the Federal Communications Commission ("FCC") and Federal Trade Commission ("FTC") as they shape their Biden Administration agendas.
Continue Reading Competition Policy Gets a Top Spot in the White House
Beginning of a TCPA Clean-Up? FCC Sets Another Robocall Blocking Item for Vote While Addressing Two of Nearly Three Dozen Pending Petitions
By Alysa Z. Hutnik, Laura A. Mazzuchetti & Steven A. Augustino on June 30, 2020
Posted in Federal Communications Commission, Privacy and Information Security, Regulatory Developments
On the same day that the FCC set a call blocking declaratory ruling for vote at its July 2020 Open Meeting, the FCC's Consumer and Governmental Affairs Bureau issued rulings in two long-pending petitions for clarification of the requirements of the Telephone Consumer Protection Act ("TCPA"). Although these clarifications do not address the…
FCC/FTC Stake out Aggressive Robocall Position, Tell Gateway VoIP Providers to Block COVID-19 Robocalls – or Be Blocked Themselves
By Alysa Z. Hutnik, Steven A. Augustino & Chris Laughlin on April 16, 2020
Posted in Enforcement, Federal Communications Commission, Federal Trade Commission, Regulatory Developments, Telemarketing
The FTC and FCC have taken a number of actions to stem unlawful robocalls generally and, during the COVID-19 pandemic, to stem harmful and deceptive calls that seek to exploit the COVID-19 crisis. Even amid the backdrop of their long-standing commitment, the agencies' most recent action stands out as an aggressive new approach to unlawful…
Telemarketing During the Pandemic
By Alysa Z. Hutnik on March 24, 2020
Posted in Advertising Law, Coronavirus, COVID-19 Regulatory, Federal Communications Commission, Privacy Practices, Telemarketing
Over the past few weeks, my colleagues have discussed some of the considerations for marketing around COVID-19, including claim substantiation and price gouging. In the next few posts, we are going to take a deeper dive into a few topics, beginning with telemarketing. Here are some points to keep in mind:
States of Emergency…
The Eleventh Circuit Weighs In On ATDS Definition
By Laura A. Mazzuchetti, Alysa Z. Hutnik & Whitney M. Smith on February 3, 2020
Posted in Advertising Litigation, Federal Communications Commission, Telemarketing
In Glasser v. Hilton Grand Vacations Company, LLC, the Eleventh Circuit addressed a pair of appeals that presented the question of the appropriate definition of an automatic telephone dialing system ("ATDS") as set forth in the Telephone Consumer Protection Act ("TCPA"). In answering that question, the Eleventh Circuit expanded upon the Third Circuit's ruling…
Taking Stock of the TCPA in 2019: What is an "Autodialer"?
By Alysa Z. Hutnik, Laura A. Mazzuchetti, Steven A. Augustino & Paul A. Rosenthal on March 4, 2019
Posted in Consumer Protection, Federal Communications Commission, Privacy and Information Security, Telemarketing
The current and future definition of what qualifies as an automatic telephone dialing system (ATDS or autodialer) remains a hotly debated and evaluated issue for every company placing calls and texts, or designing dialer technology, as well as the litigants and jurists already mired in litigation under the Telephone Consumer Protection Act (TCPA). Last year, the D.C. Circuit struck down the FCC's ATDS definition in ACA International v. FCC, Case No. 15-1211 (D.C. Cir. 2019). Courts since have diverged in approaches on interpreting the ATDS term. See, e.g., prior discussions of Marks and Dominguez. All eyes thus remain fixed on the FCC for clarification.
In this post, we revisit the relevant details of the Court's decision in ACA International, and prior statements of FCC Chairman Ajit Pai concerning the ATDS definition to assess how history may be a guide to how the FCC approaches this issue.
Continue Reading Taking Stock of the TCPA in 2019: What is an "Autodialer"?
Read the Signs: FCC Unleashes Wave of Equipment Marketing Actions Involving LED Signs
By Steven A. Augustino on July 19, 2018
Posted in Federal Communications Commission
As we enter the dog days of summer, the FCC continues to turn up the heat on equipment marketing enforcement. But while million dollar fines for marketing noncompliant devices capture the spotlight, the FCC also quietly issued a number of equipment marketing actions focused on a single type of device: LED signs. In just the last three months, the FCC has settled over ten investigations involving the marketing of LED signs used in digital billboards for commercial and industrial applications without the required authorizations, labeling, or user manual disclosures. Each action involved an entity that either manufactured or sold (or both) LED signs. The agency's recent actions should be a shot across the bow to any retailer of LED signs to ensure that their devices are properly tested and authorized prior to sale. Otherwise, these companies may face significant fines and warehouses of unmarketable devices.
Most consumers might not think that LED signs fall within the FCC's jurisdiction. However, the signs emit radio waves that can interfere with communications services. As a result, the FCC requires most LED signs and other "unintentional" radiators to be tested for compliance with its technical requirements prior to marketing. Importantly, the FCC's rules prohibit the marketing of such devices unless they have been properly authorized, labeled, and carry the required disclosures. Even with the FCC's recent efforts at simplification, the rules regarding equipment marketing are complex, requiring close attention to compliance at every step in the supply chain.…
Continue Reading Read the Signs: FCC Unleashes Wave of Equipment Marketing Actions Involving LED Signs
Podcast: Inside the TCPA – Autodialers
By Alysa Z. Hutnik & Steven A. Augustino on June 22, 2018
Posted in Federal Communications Commission, Telemarketing
Kelley Drye introduces a new Full Spectrum series, "Inside the TCPA," which will offer a deeper focus on TCPA issues and petitions pending before the FCC. Each episode will tackle a single TCPA topic or petition that is in the news or affecting cases around the country. In this inaugural episode, partner Steve Augustino discusses…
FCC Net Neutrality Repeal Published in Federal Register, Triggering Deadlines for Challengers
By Steven A. Augustino & John J. Heitmann on February 27, 2018
On Thursday, February 22, 2018, the Federal Communications Commission (FCC or Commission) published the Restoring Internet Freedom Order (the Order) in the Federal Register.
As we previously discussed, the Order effectively reverses the Commission's 2015 Open Internet Order, reclassifying broadband Internet access service as a lightly regulated Title I "information service" and eliminating…
On the Eve of the FCC's Reclassification of Broadband Services, the FCC and FTC Release Memorandum of Understanding for Oversight of Broadband
By Steven A. Augustino on December 14, 2017
On December 11, 2017, the Federal Communications Commission (FCC) and Federal Trade Commission (FTC) released a draft Memorandum of Understanding (MOU) which will allocate oversight and enforcement authority related to broadband Internet access service (BIAS) between the two agencies. The new MOU was announced three days before the FCC's scheduled vote to reclassify BIAS as an "information service," and is expected to be finalized simultaneously with that vote. The MOU is part of an ongoing effort to address concerns that reversing the current "net neutrality" rules will adversely affect consumers, and provides a guide for Internet service providers (ISPs) and other stakeholders to understand which agency will be taking the lead on oversight and enforcement going forward. However, the extent to which the MOU takes effect will depend upon, among other things, the pending case interpreting section 5 of the FTC Act that is before the Ninth Circuit Court of Appeals.
Continue Reading On the Eve of the FCC's Reclassification of Broadband Services, the FCC and FTC Release Memorandum of Understanding for Oversight of Broadband
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,262
|
Rawlins se poate referi la următorii autori care au denumit cel puțin o specie:
Andrew Rawlins
John E. Rawlins
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,472
|
Agency Zone
Get Food
Food Assistance Guide
Food Assistance Locator
A Fresh Approach Campaign
Dine. Shop. Live.
Current and Upcoming Food Drives
School Pantry
Mobile Pantry
The Emergency Food Assistance Program (TEFAP)
Business and Corporate Partners
Mission – Vision – Values
Foodbanking 101
Food Rescue
Idaho Food Bank Fund
Camille Beckman Foundation offers $40,000 Donation Match
December 27, 2016 / Mark Ellsworth / Donors, News
The Camille Beckman Foundation has generously provided The Idaho Foodbank with a matching donation of $40,000 for the final weeks of 2016.
This is the third year the foundation has come forward with an end-of-year match to help the hungry in Idaho. Camille Beckman is based in Eagle, Idaho and has been a leading producer of cosmetics and personal care products for over 18 years.
Albertsons Gives 1,518 Holiday Meals To Idaho Families In Need
December 19, 2016 / ifb / Donations, News
As Christmas approaches, Albertsons is coming through in the clutch for families in need. The newest shipment of meals is part of Albertsons' "Santa Bucks" campaign, enabling customers to donate at the register by rounding up their purchase or adding $1, $5 or $10 as a donation. The campaign, along with "Turkey Bucks" during November, enables customers and associates to work together to raise the funds needed to provide delicious meals.
A Special Holiday Gift – Paying it Forward
December 16, 2016 / Mark Ellsworth / Stories
The Davis family, who were visiting the Boise Town Square Mall today and ran into a mysterious woman in the parking lot, who handed Mary Alice a Christmas card. She told Mary Alice that she looked like she deserved the gift, and when she opened the card she discovered that it contained a $100 bill. Mary Alice turned to thank the woman, but it was too late – the lady had vanished!
December 5, 2016 / ifb / Nutrition
The Holidays can be a difficult time for people watching their eating habits and trying to stay healthy. Our dietician Alexa shared some helpful and healthful tips for cooking, baking and eating this holiday season.
Holidays Add Strain On Programs
December 5, 2016 / ifb / News
During the holiday season, it can be difficult to reach the people who are in need in Idaho. With Thanksgiving and Christmas come weeks off from school, leading to Backpack and School Pantry organizers working extra hard to make sure the children who need assistance will have the food they need over their extended time off.
Three Generations Volunteer At Foodbank
December 5, 2016 / ifb / News, Stories, Volunteer
This is a letter from Brook Richter, who was one of a family of three generations which came to volunteer at The Idaho Foodbank on the day before Thanksgiving. She comes from a family of caring people with hearts for the mission to solve hunger. We'll let her tell you about her experiences in her own words.
You can help the 1 in 6 children and 1 in 8 adults in Idaho who are facing hunger. Unite with us on the mission for a hunger-free and healthier Idaho.
Sportsmen Against Hunger – Chubbuck
January 25 @ 10:00 am - 4:00 pm MST
C-A-L Ranch Store Chubbuck
Beef Night on the Ice with the Idaho Steelheads
February 1 @ 7:00 pm - 10:00 pm MST
CenturyLink Arena
Organizer: Idaho Beef Council
Main Warehouse and Administrative Offices
3562 South TK Avenue
info@idahofoodbank.org
The Idaho Foodbank has been awarded a four-star rating for 10 consecutive years.
Sign Up for Monthly E-Newsletter
(c) 2020 The Idaho Foodbank
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,750
|
Q: extract multiindex types in pandas I have a dataframe where I set_index() using 3 of the columns. I would like to extract the data type associated with each index. How can I do this efficiently? I don't want to do type(df.index.get_level_values()) since the df is large.
MWE:
import pandas as pd
df = pd.DataFrame({"id": [1,2,1,2], "time": [1, 1, 2, 2], "val": [1,2,3,4]})
df.set_index(keys=["id", "time"], inplace=True)
type(df.index.get_level_values(1))
#pandas.core.indexes.numeric.Int64Index
I would also like to know the type of the actual data that is in the index (ie looking at this i know it's integers but it would be nice to get something like this also:
type(df.index.get_level_values(1).values[0])
#numpy.int64
A: You could use [lev.dtype.type for lev in index.levels]:
import pandas as pd
df = pd.DataFrame({"id": [1,2,1,2], "time": [1, 1, 2, 2], "val": [1,2,3,4]})
df.set_index(keys=["id", "time"], inplace=True)
index = df.index
print([lev.dtype.type for lev in index.levels])
# [<class 'numpy.int64'>, <class 'numpy.int64'>]
# Alternatively, there is the private attribute, `_inferred_type_levels`,
# but this is probably not what you are looking for.
print(index._inferred_type_levels)
# ['integer', 'integer']
index.levels is a FrozenList of 1-dimensional Indexes:
In [172]: list(index.levels)
Out[172]:
[Int64Index([1, 2], dtype='int64', name='id'),
Int64Index([1, 2], dtype='int64', name='time')]
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,354
|
Josh Kopelman writes some extremely effective and highly useful blog posts. This one happens to be about what startup marketing really means. I'm planning a series of venture capital interviews in the near future, I'd be flattered if Josh was one of the participants in that series.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,588
|
\section{Introduction}
\label{sec:intro}
Rapid yet accurate computation of the Voigt function \citep{Armstrong67} is a challenge in many fields of the physical sciences.
Because the convolution integral of a Lorentz and Gauss function does not have an analytical solution, approximations have been discussed in numerous papers.
Whereas many modern ``state-of-the art'' algorithms evaluate the closely related complex error function (also known as complex probability function, plasma dispersion function, or Fadde(y)eva function, cf.\ e.g., \citealp{NIST-handbook,DLMF,Oldham09}) utilizing sophisticated numerical techniques,
``simple'' closed-form expressions still appear to be attractive.
Linear combinations of the Lorentz and/or Gauss functions have been suggested by several authors.
\citet{Flores91} proposed a sum of the Lorentzian and its derivatives and \citet{Melcher77} fitted the Voigt functions by ``generalized Lorentz functions''.
\citet{McLean94} have further developed an approximation originally suggested by \citet{Martin81,Martin81e,Puerta81,Puerta83e} and proposed a
superposition of four Lorentzians \citep[for a recent assessment see][]{Schreier18v}.
Linear combinations of a Lorentz and Gauss function (sometimes called ``pseudo-Voigt'' function and occasionally including a correction term)
have been suggested or used by \citet{Whiting68,Matveev72,Peyre72,Kielkopf73,Wertheim74,Thompson87,Teodorescu94,Ida00} and \citet{Liu01}.
In this note we present an assessment of closed-form expressions for the Voigt function using a combination of Lorentz and Gauss functions.
After a short review of the basic definitions in the next subsection, we describe several combinations using a consistent notation (in chronological order).
The results of our tests using an accurate Voigt function code as reference are presented in Section \ref{sec:results}.
The codes have been implemented in Python and a Scientific Python (\url{scipy.org}) implementation has been used as a reference.
In the final Section \ref{sec:conclusions} we provide a summary and some conclusions.
\section{Theory}
\label{sec:theory}
\subsection{The Voigt function}
\label{sec:voigt}
The Voigt function (normalized to $\sqrt\pi$) is defined by
\begin{equation} \label{vgtFct}
K(x,y) ~=~ {y \over \pi} ~ \int_{-\infty}^\infty {\mathrm{e}^{-t^2} \over (x-t)^2 + y^2} \, \mathrm{d} t
\end{equation}
where $x$ is a measure of the distance to the center peak, and $y$ is essentially the ratio of the Lorentzian and Gaussian width, $y = \sqrt{\ln 2} \gamma_\text{L}/\gamma_\text{G}$.
At the line center $x=0$ the Voigt function can be expressed as the exponentially scaled complementary error function
\begin{align} \label{vgtFct0}
K(0,y) ~=~ \exp(y^2) \bigl( 1 - \erf(y) \bigr) ~=~ \exp(y^2) \erfc(y) ~.
\end{align}
The Voigt function is symmetric, i.e.\ $K(-x,y) = K(x,y)$ and essentially reduces to the Lorentz function for large $|x + \mathrm{i} y|$.
The Voigt function is identical to the real part of the complex function
\begin{align}\label{wDef}
w(z) ~&\equiv~ K(x,y) \,+\, \mathrm{i} L(x,y) ~=~ {\mathrm{i} \over \pi} ~ \int_{-\infty}^\infty ~ {e^{-t^2} \over z-t} ~ \mathrm{d} t \\
~&=~ \exp(-z^2) \erfc(-\mathrm{i} z)
\qquad\hbox{ with }\quad z = x + \mathrm{i} y.
\end{align}
\subsection{The \citet{Whiting68} approximation}
\label{ssec:Whiting}
To our knowledge the first approximation of the Voigt function using a combination of Lorentz and Gauss functions
\begin{equation}
\label{pseudo} K(x,y) ~=~ K(0,y) \left[ \bigr( 1-\eta(y) \bigl) G(x) ~+~ \eta(y) L(x) \right] \\
\end{equation}
with
\begin{align}
\label{lorentz} L(x) ~&=~ \left[ 1 + \left({x \over \xHalf}\right)^2 \right]^{-1} \\
\label{gauss} G(x) ~&=~ \exp{\left( -\ln 2 (x / \xHalf)^2 \right)}
\end{align}
is due to \citet{Whiting68}.
The weight factor is given by the ratio of the widths of the Lorentz and Voigt profiles or
\begin{align}
\label{whitingWeight} \eta ~&=~ y / \xHalf \\
\intertext{with the half width}
\label{whitingWidth} \xHalf ~&=~ \tfrac{1}{2} \left(y+\sqrt{y^2 + 4 \ln 2} \right) ~.
\end{align}
An improved approximation is obtained by adding a correction term (i.e.\ $K(x,y) \longrightarrow K(x,y)+C(x,y)$) that is given by ``kind of Lorentzians and Gaussians'',
\begin{align} \label{whitingCorr}
C(x,y) = 0.016 \, \eta (1-\eta) \, \Biggl[ & \exp{\left(-0.4 (x / \xHalf)^{(9/4)}\right)} \\ \notag
& - {10 \over 10 + (x/\xHalf)^{(9/4)}} \Biggr]
\end{align}
By definition, the approximation \eqref{pseudo} is exact in the center at $x=0$.
Furthermore, both approximations are exact for the limiting cases of pure Lorentz and Gauss functions.
According to the author, ``this approximation matches the Voigt profile within 5 per cent at worst and is generally within 3 per cent or less.''
\subsection{The \citet{Matveev72} approximation}
\label{ssec:Matveev}
The approximation is given by
\begin{equation} \label{matveev}
K(x,y) ~=~ {\sqrt{\ln 2} \over \xHalf} \: \left[ (1 - \eta) G(x) ~+~ {\eta \over \sqrt{\pi\ln 2}} L(x) \right]
\end{equation}
with a correction term
\begin{align}
\label{matveevCorr}
C(x,y) ~=&~ {\eta (1 - \eta) \over \sqrt{\pi \ln 2}} \left({1.5 \over \ln 2} + 1 + \eta \right) \\
&~~~ \times \Biggl[ 0.066 \, \exp{\left(-0.4 \left({x \over \xHalf}\right)^2 \right)} \notag \\
&~~~~~~~~ - {1 \over 40 - 5.5\, \left({x \over \xHalf}\right)^2 + \left(\frac{x}{\xHalf}\right)^4} \Biggr] ~. \notag
\end{align}
The weight $\eta$ is defined as in \eqref{whitingWeight} and the half width is given by a refinement of Whiting's approximation \eqref{whitingWidth}%
\footnote{Note that a prefactor $\gamma_\text{L}$ for the correction term is missing in Matveev's Eq.\ (5) and is correctly inserted in \citet[][after Eq.\ (12)]{Titov97}.
Furthermore, the factor $\eta$ in \eqref{matveevCorr} is not given in \citet[][Eq.\ (12c)]{Titov97}.}
\begin{align} \label{matveevWidth}
\xHalf ~&=~ \tfrac{1}{2} \left(y+\sqrt{y^2 + 4 \ln 2} \right) \\ \notag
&~~~~~~~~ + 0.05 y \left( 1 - {2 y \over y + \sqrt{y^2 + 4 \ln 2}} \right) \\
&=~ \xHalf^\text{W} + 0.05 y \left( 1 - {y \over \xHalf^\text{W}} \right)
\end{align}
Without correction term \eqref{matveevCorr} \citet{Matveev72} reports a ``greatest error of $~\approx 25\%$ at $\eta=0.1$ and $x\approx 3$''.
With correction the maximum error at the line center does not exceed $0.6\%$, and for $x/\xHalf>6$ the error lies within the limits of $1\%$'' for any $y$.
In the intermediate frequency regime ``the error nowhere exceeds $3\%$.''
\subsection{The \citet{Kielkopf73} approximation}
\label{ssec:Kielkopf}
Without correction term this approximation is identical to \eqref{pseudo}, but with the weight and half width defined as
\begin{align}
\label{kielkopfWeight} \eta ~&=~ {y \xHalf \over 1 + y \xHalf} \\
\label{kielkopfWidth} \xHalf ~&=~ \tfrac{1}{2} y \, \left( 1 + k_\text{e} \ln 2 + \sqrt{(1 -k_\text{e} \ln 2)^2 + {4 \ln 2 \over y^2}} \right)
\end{align}
The correction contains the difference of a Gaussian and Lorentzian multiplied with a rational function of $x$ (see Online \ref{app:constants} for numerical values)
\begin{equation} \label{kielkopfCorr}
C(x,y) ~=~ \eta (1-\eta) \bigl( G(x) - L(x) \bigr) {k_1 + k_2 x^2 \over 1 + k_3 x^2 + k_4 x^4} ~.
\end{equation}
This approximation ``is accurate to the order of 0.0001 of the peak value of the function''.
\subsection{The \citet{Thompson87} approximation}
\label{ssec:Thompson}
In contrast to the three approximations discussed so far the weight factors of the \citet{Thompson87} (and \citet{Liu01}, next subsection) approaches are defined by power series of $y$.
The pseudo-Voigt function is written as
\begin{equation} \label{pV_Thompson}
K(x,y) ~=~ {\sqrt{\ln 2} \over \xHalf} \; \left[ (1 - \eta) G(x) + {\eta \over \sqrt{\pi \ln 2}} L(x) \right]
\end{equation}
with the Voigt half width and weight (see Online \ref{app:constants})
\begin{align}
\label{thompsonWidth} \xHalf ~&=~ \Bigl( t_0 + t_1 y + t_2 y^2 + t_3 y^3 + t_4 y^4 + y^5 \Bigr)^{(1/5)} \\
\label{thompsonWeight} \eta ~&=~ \tau_1 {y \over \xHalf} + \tau_2 \left({y \over \xHalf}\right)^2 + \tau_3 \left({y \over \xHalf}\right)^3
\end{align}
According to \citet{Ida00} the maximum deviation of about $1.2\%$ is found at $y / (y + \sqrt{\ln 2}) \approx 0.5$.
\subsection{The \citet{Liu01} approximation}
\label{ssec:Liu}
Introducing a dimensionless parameter $ d = (y-\sqrt{\ln 2}) / (y+\sqrt{\ln 2})$ and approximating the weights (see \qeq{LiuWeights} in the Online Appendix) as
\begin{align}
\label{LiuWeightL} c_\text{L} ~&=~ l_0 + l_1 d + l_2 d^2 + l_3 d^3 \\
\label{LiuWeightG} c_\text{G} ~&=~ g_0 + g_1 d + g_2 d^2 + g_3 d^3
\end{align}
the pseudo-Voigt function is written as
\begin{equation}
K(x,y) ~=~ {c_\text{L} \over \xHalf \sqrt\pi} L(x) ~+~ c_\text{G} {\sqrt{\ln 2} \over \xHalf} \, G(x)
\end{equation}
with the half width given by the \citet{Olivero77} approximation
\begin{equation}
\begin{aligned}
\xHalf(y) ~&=~ \bigl( y + \sqrt{\ln 2} \bigr) \; \bigl( 1 - 0.18121 (1 - d^2) - \beta \sin\pi d \bigr) \\
\beta ~&=~ 0.023665 \exp{(0.6 d)} + 0.00418 \exp{(-1.9 d)}
\end{aligned}
\end{equation}
According to the abstract \citep{Liu01} ``the maximum errors of width, area, and peak \dots\ are 0.01\%, 0.2\%, and 0.55\%, respectively.''
\begin{figure}[h]
\centering\includegraphics[width=\linewidth]{FSchreier_voigt3-figure1.pdf}
\caption{The weight of the Lorentzian: $\eta$ as defined in \eqref{whitingWeight}, \eqref{kielkopfWeight}, \eqref{thompsonWeight} and $c_\text{L}$ defined in \eqref{LiuWeightL}.}
\label{fig_wgt}
\end{figure}
\section{Results}
\label{sec:results}
Ignoring the correction terms Eqs.\ \eqref{whitingCorr}, \eqref{matveevCorr}, and \eqref{kielkopfCorr}, all algorithms use a weighted sum of the Lorentz and Gauss function,
where the Lorentz weight tends to one for $y\gg 1$ and zero for $y\ll 1$.
\qufig{fig_wgt} compares the weights of the Lorentz function for all five approximations.
The weights of \citeauthor{Whiting68} and \citeauthor{Matveev72} are identical by construction, \qeq{whitingWeight}, and the \citeauthor{Thompson87} and \citeauthor{Liu01} weights appear to be very similar (with differences for small and large $y$ only).
For $y \approx 1$ (i.e.\ equal width of the Lorentz and Gauss function) all weights are approximately 0.7.
Note that except for \citeauthor{Liu01} the Gaussian weight is simply given by $1-\eta$.
As discussed in \citet{Schreier11v} and in our previous assessments of simple closed-form approximations of the Voigt function \citep{Schreier16,Schreier17g,Schreier18v},
the range of $y$ values encountered in molecular spectroscopy and atmospheric and astrophysical applications spans many orders of magnitude.
In \qufig{figError} (left) we compare the pseudo-Voigt approximations with reference values for $w(z)$, Eq.\ \eqref{wDef}, obtained with the \texttt{wofz} code
(algorithm originally based on \citet{Poppe90,Poppe90a} and later refined with ideas from \citet{Zaghloul11};
Scientific Python (\url{http://scipy.org}) implementation \texttt{scipy.special.wofz} with at least 13 significant digits according to the documentation).
\begin{figure*}
\centering\includegraphics[width=\textwidth]{FSchreier_voigt3-figure2.pdf}
\caption{The Voigt function (left) and the relative error (right) for $y=10$ (top), $y=1.0$, $y=0.1$, and $y=0.01$ (bottom).
Note the different range of $x$ values in the four rows and the linear $y$ axis of the top left plots.}
\label{figError}
\end{figure*}
The function values shown on the left appear to be in reasonably good agreement with the reference.
However, significant problems show up in the relative errors $|K - K_\text{wofz}| / K_\text{wofz}$ (right side)
and for all approximations the maximum error is larger than 1 percent.
In the line center ($x=0$) the Whiting and Kielkopf approximations are exact by definition (assuming that the exponentially scaled complementary error function $K(0,y) = \erfce(y)$, \qeq{vgtFct0}, is evaluated exactly).
The other approximations have errors of some percent for $y=1$, but smaller errors for smaller and larger $y$.
Note that in the line center the Whiting and Kielkopf corrections vanish, i.e.\ $C(0,y)=0$, in contrast to the Matveev correction.
In the line wings only Matveev's approximation (with and without correction) has relative errors decreasing with increasing $|x|$.
Evaluating the Voigt function for very large $x$ ($x \le 2000$ for $y=10$ and $x \le 100$ for $y=1$) indicates that
for all other approximations the errors become constant for large $x$.
For Kielkopf this asymptotic error is about $10^{-4}$ for $y=10$ and $0.02$ for $y=0.1$ and $0.01$.
For small $y\le 0.1$ Matveev's approximation appears to be superior (with errors less than a few percent), however, for $y>1$ the correction term worsens the accuracy,
and the results shown in the top of \qufig{figError} have been obtained without the correction \eqref{matveevCorr}.
For Kielkopf and Matveev, the largest errors occur for intermediate values of $x$,
whereas the other approximations always fail for large $x$.
\begin{figure*}
\includegraphics[width=\textwidth]{FSchreier_voigt3-figure3.pdf}
\caption{Contour plots of the relative error of the pseudo-Voigt approximations:
In the first three rows the relative errors of the Whiting, Matveev, and Kielkopf approximations without correction terms are shown on the left, and with correction on the right. The number in the title indicates the maximum relative error.}
\label{figContour}
\end{figure*}
The contour plots of relative errors shown in \qufig{figContour} essentially confirm these findings.
Except for the \citet{Liu01} Lorentz-Gauss combination all approximations have at least a small region where the relative accuracy is better than $10^{-4}$
(see \citep{Schreier11v} for a discussion of the $10^{-4}$ criterion).
For Whiting and Kielkopf a relative high accuracy is achieved near the origin for small and large $y$, for Matveev and Thompson only for small $y$.
The maximum relative error indicated in the title of all subplots identifies Kielkopf's code (with correction) as the most accurate approximation.
However, this maximum error is relatively large for modern standards (about six percent in the wings),
and for small $y$ Matveev's code appears to be better.
Despite the significant accuracy problems of all pseudo-Voigt approximations it is nevertheless instructive to test the numerical performance of the codes.
Simple tests within the IPython interpreter \citep{Perez07} indicate that the Kielkopf and Matveev approximations are somewhat slower than the optimized combination of
the \citet{Humlicek82} and \citet{Weideman94} rational approximations \citep{Schreier11v}.
However, evaluation of {HNO$_3$} cross sections in the microwave for a series of pressures and temperatures corresponding to Earth's atmosphere in the $ 0 \,\text{--}\, 120\rm\,km$ altitude
range is about a factor two slower with these two pseudo-Voigt approximations compared to the \citeauthor{Humlicek82}--\citeauthor{Weideman94} combination.
For details see the online appendix \ref{app:speed}.
\section{Summary and Conclusions}
\label{sec:conclusions}
Closed-form expressions for the Voigt function based on combinations of Lorentz- and Gauss-type functions show significant accuracy problems,
with relative errors in the percent range.
Note that in this study we have only considered ``pseudo-Voigt'' approximations based on an analytical, closed-form expression for the weight of the Lorentz and Gauss components
(as a function of $y$, the ratio of the Lorentz and Gauss width).
In several studies pseudo-Voigt approximations have been used for analysis of experimental data, where the weight has been estimated by least squares fitting \citep[e.g.][]{Wertheim74,SanchezBajo97}.
Our conclusions now are therefore similar to those given in \citet{Schreier16,Schreier17g}:
Closed-form expressions as presented here might be desirable for certain applications, but their quality is limited.
In general approximations based on modern state-of-the-art numerical methods, e.g.\ rational approximations as discussed in \citet{Humlicek79, Humlicek82, Weideman94} and \citet{Schreier11v,Schreier18h} are recommended.
\medskip
\section*{Acknowledgments}
Financial support by the Deutsche Forschungsgemeinschaft --- DFG (project SCHR 1125/3-1) is gratefully acknowledged.
\medskip
\begin{small}
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 5,763
|
{"url":"https:\/\/stacks.math.columbia.edu\/tag\/0D6Y","text":"Lemma 21.37.4. Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous and cocontinuous functor of sites. Let $g : \\mathop{\\mathit{Sh}}\\nolimits (\\mathcal{C}) \\to \\mathop{\\mathit{Sh}}\\nolimits (\\mathcal{D})$ be the corresponding morphism of topoi. Let $\\mathcal{O}_\\mathcal {D}$ be a sheaf of rings and let $\\mathcal{I}$ be an injective $\\mathcal{O}_\\mathcal {D}$-module. If $g_!^{Sh} : \\mathop{\\mathit{Sh}}\\nolimits (\\mathcal{C}) \\to \\mathop{\\mathit{Sh}}\\nolimits (\\mathcal{D})$ commutes with fibre products1, then $g^{-1}\\mathcal{I}$ is totally acyclic.\n\nProof. We will use the criterion of Lemma 21.13.5. Condition (1) holds by Lemma 21.37.1. Let $K' \\to K$ be a surjective map of sheaves of sets on $\\mathcal{C}$. Since $g_!^{Sh}$ is a left adjoint, we see that $g_!^{Sh}K' \\to g_!^{Sh}K$ is surjective. Observe that\n\n\\begin{align*} H^0(K' \\times _ K \\ldots \\times _ K K', g^{-1}\\mathcal{I}) & = H^0(g_!^{Sh}(K' \\times _ K \\ldots \\times _ K K'), \\mathcal{I}) \\\\ & = H^0(g_!^{Sh}K' \\times _{g_!^{Sh}K} \\ldots \\times _{g_!^{Sh}K} g_!^{Sh}K', \\mathcal{I}) \\end{align*}\n\nby our assumption on $g_!^{Sh}$. Since $\\mathcal{I}$ is an injective module it is totally acyclic by Lemma 21.14.1 (applied to the identity). Hence we can use the converse of Lemma 21.13.5 to see that the complex\n\n$0 \\to H^0(K, g^{-1}\\mathcal{I}) \\to H^0(K', g^{-1}\\mathcal{I}) \\to H^0(K' \\times _ K K', g^{-1}\\mathcal{I}) \\to \\ldots$\n\nis exact as desired. $\\square$\n\n[1] Holds if $\\mathcal{C}$ has finite connected limits and $u$ commutes with them, see Sites, Lemma 7.21.6.\n\nThere are also:\n\n\u2022 3 comment(s) on Section 21.37: Derived lower shriek\n\nIn your comment you can use Markdown and LaTeX style mathematics (enclose it like $\\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).","date":"2023-03-28 14:22:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 2, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 1.0000026226043701, \"perplexity\": 629.0368559501305}, \"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\/1679296948867.32\/warc\/CC-MAIN-20230328135732-20230328165732-00734.warc.gz\"}"}
| null | null |
{"url":"https:\/\/homework.study.com\/explanation\/how-much-h-2-gas-at-stp-can-be-produced-by-the-reaction-2-na-s-plus-2-h-2o-l-rightarrow-h-2-g-plus-2-naoh-aq-of-3-90-g-of-na-and-excess-water.html","text":"# How much H_2 gas at STP can be produced by the reaction 2 Na(s) + 2 H_2O(l) rightarrow H_2(g) +...\n\n## Question:\n\nHow much {eq}H_2 {\/eq} gas at {eq}STP {\/eq} can be produced by the reaction {eq}2 Na(s) + 2 H_2O(l) \\longrightarrow H_2(g) + 2 NaOH (aq) {\/eq} of {eq}3.90 \\ g {\/eq} of {eq}Na {\/eq} and excess water ?\n\n## Reaction of Metals with Water and Acids\n\n{eq}{\/eq}\n\n1. Reaction of metals with water : Metals react with water to produce metal hydroxides and hydrogen gas. In other words, they reduce the hydrogen in water and bring it to its native state.\n\n$$M + nH_2O \\longrightarrow M(OH)_n + \\frac{1}{2} H_2 \\\\$$\n\nValue of n depends on the valency of metal M.\n\n{eq}{\/eq}\n\n2. Reaction of metals with acids : Metals react with acids to produce salts and hydrogen gas. In other words, they reduce the hydrogen in the acid and bring it to its native state.\n\n$$M + nHA \\longrightarrow MA_n + \\frac{1}{2} H_2 \\\\$$\n\nNote - Metals that are weaker reducing agents than hydrogen ( e.g. Copper, gold, etc) cannot undergo these reactions since they are unable to replace hydrogen from their compounds.\n\n{eq}{\/eq}\n\nBecome a Study.com member to unlock this answer!\n\n{eq}{\/eq}\n\nGiven chemical equation :\n\n$$\\displaystyle 2 Na(s) + 2 H_2O(l) \\longrightarrow H_2(g) + 2 NaOH (aq) \\\\$$\n\nSince water is taken in...","date":"2023-01-31 23:15:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.6505951881408691, \"perplexity\": 6055.014186472631}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764499891.42\/warc\/CC-MAIN-20230131222253-20230201012253-00815.warc.gz\"}"}
| null | null |
/**
*/
#ifndef __ARDUINO_CUBE_EFFECTS_RAIN_CPP__
#define __ARDUINO_CUBE_EFFECTS_RAIN_CPP__ 1
#include <Rain.h>
#include <Arduino.h>
#include <Point.h>
Rain::Rain(Cube *cube, unsigned int iterations, unsigned int iterationDelay, unsigned char minDrops, unsigned char maxDrops)
: Effect(cube, iterations, iterationDelay), minDrops(minDrops), maxDrops(maxDrops) {
}
bool Rain::iterate() {
Point p = Point(0, 0, Cube::SIZE - 1);
unsigned char n = random(minDrops, maxDrops);
for (unsigned char k = 0; k < n; k++) {
p.x = random(0, Cube::SIZE);
p.y = random(0, Cube::SIZE);
cube->turnVoxelOn(&p);
}
delay(iterationDelay);
cube->shiftOnZ(DOWN);
cube->turnPlaneZOff(Cube::SIZE - 1);
return true;
}
#endif /* __ARDUINO_CUBE_EFFECTS_RAIN_CPP__ */
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 2,515
|
\section{Introduction}
\label{sec:Intro}
\IEEEPARstart{W}{ith} the ongoing evolution of mobile devices, the demand for higher data rates in mobile communication systems has been increasing exponentially. According to Wireless World Research Forum's (WWRF) vision for 2020, a mobile traffic growth of 1000 times is expected. International Telecommunication Union (ITU) has opened a programme for this new cellular standard as International Mobile Telecommunication 2020 (IMT-2020) which will be the 5th generation of cellular standards (5G). According to 5G visions of ITU and several communication companies, the services in 5G will require larger data rates, lower latency and higher reliability. All the improvements should be done in a cost effective manner\cite{chin}. In order to satisfy the 1000x data challenge, as stated in \cite{WW5GBE} and \cite{BOCCARDI}, the key technologies are increased bandwidth, increased spectral efficiency and extreme cell densification.
Cell densification is a key enabler for 5G networks, \cite{WW5GBE}, \cite{DAMNJANOVIC}. By shrinking the cell sizes, the spectrum can be reused across on the area and number of users using same resources will decrease which will increase the per user rate. In dense deployments, adding more BSs will also increase the interference levels. In order to overcome this problem, deployment of BSs with lower transmit power is proposed. Low power base stations are named as micro, pico, femto base stations depending on their transmit power. Networks consisting of a mix of these base stations are called Heterogeneous Networks (HetNets).
In HetNets, with the addition of small cells, area spectral efficiency is increased. For the ongoing 3GPP development, there are various scenarios and requirements for the enhancement of small cells \cite{GEN3PP}. Cell range expansion (CRE) is one of the methods in HetNets to offload more users to small cells. It is enabled by through cell biasing and adaptive resource partitioning and is seen as an effective method to balance the load among the nodes in the network and improve overall trunking efficiency. Depending on the bias value, the network can control the number of users (UE) associated with the low-power nodes and therefore control traffic demand at those nodes \cite{DAMNJANOVIC}. A positive bias means that the UE associate itself with small cell as long as the difference of the received power between macro cell and small cell is smaller than this bias value. In this study, we assume a two tier HetNet where there are Macro and Micro base stations and with cell biasing there are three types of UEs: the UEs that are associated with Macro cell are named as Macro UEs, the UEs associated with Micro cell with zero biasing are Direct Micro UEs and the UEs associated with Micro cell with positive biasing are named as Cell Range Expanded (CRE) UEs.
In order to minimize the interference among the users of the system, time/frequency resources should be partitioned carefully in HetNets. For instance, in case when CRE UEs and Macro UEs are being served in same time interval and at the same frequency band, the received signal power of CRE UEs will be lower than interference power coming from Macro cell. Therefore, the resources are needed to be orthogonally shared between CRE and Macro UEs. This can be done by using Almost Blank Subframe (ABS) technique which is a part of Enhanced Intercell Interference Coordination (eICIC) developed by 3GPP working group, \cite{Qiaoyang}. As stated in \cite{Qiaoyang} and \cite{SARABJOT_2}, in ABS Macro BS does not transmit in a portion of the time when CRE UEs are being served, so that CRE UEs do not suffer from the Macro BS interference.
The resource allocation between Direct Micro UEs and Macro UEs can be done by using orthogonal or non-orthogonal deployments. In \cite{ROSENBERG}, it is stated that the frequency band can be shared among Macro and Micro tier users by using one of three different schemes which are Co-channel deployment (CCD), Orthogonal deployment (OD) and Partially Shared deployment (PSD). In CCD all sub-channels can be used by Macro and Micro BSs whereas in OD the sub-channels are shared orthogonally among micro and macro tier users. PSD is a combination of CCD and OD. In PSD, a number of the sub-channels are used by both Macro and Micro BS whereas remaining sub-channels are used only by Macro BS.
In the literature, there are studies regarding how to partition the resources and adjust bias value for HetNets assuming different base station and user distributions, in order to maximize user rates. For instance, in \cite{SARABJOT}, rate distribution of users are found for a two tier HetNet in which, users and base stations are distributed on the area asuming a 2-D Poisson Point Distribution. Using the rate distribution, the authors analyzed the system for different bias and resource partitioning parameters setting $5^{th}$ percentile and median rate as performance indicators.
There are studies to obtain analytical downlink data rate distribution for HetNets and other types of networks. In \cite{GUVENC_1}, a semi-analytical approach depending on computer simulations is utilized to investigate the capacity of a HetNet scenario that is similar to our scenario. Using the semi-analytical approach for capacity distribution, the authors of \cite{GUVENC_1} optimize the HetNet system in terms of fairness and user data rates. Different than \cite{GUVENC_1}, in our study, we follow a fully analytical approach. We obtain an analytical expression for the user rate and do not need to do simulations for further optimization of system parameters. Additionally, in our system the bandwidth is shared among different types of users whereas in \cite{GUVENC_1}, co-channel deployment is assumed.
In many other studies in the literature, e.g., \cite{ANDREWS_1}, \cite{GANTI}, \cite{MUKHERJEE}, \cite{HEATH} the performance of HetNets are modeled using Poisson-Point-Process (PPP) based models since it provides computationally efficient methods for the analytical performance evaluations. However, as stated in \cite{Merwaday2014}, in PPP based models the macro base stations may be very close to each other which is not the case in real life scenarios. In a recent study given in \cite{Guo_Haenggi}, SINR distribution is investigated for a general class of point processes and it has been shown that SINR distribution obtained for a point process is a shifted version of the distribution obtained for the other point process.
In this paper, different than the Hetnet model used in \cite{SARABJOT} and \cite{ANDREWS_1, GANTI, MUKHERJEE, HEATH, Guo_Haenggi}, we model a HetNet for a scenario in which small cells are located at cell edges which is known as Cell-On-Edge (COE) configuration. As stated in \cite{SHAKIR}, the COE configuration has been shown to produce significant gains to operators and mobile customers, including improved cell-edge coverage, increased network capacity to match cell-edge mobile user demands, enhanced end-user experience, and reduced cost of delivering mobile broadband services to such cell-edge mobile users.
According to Next Generation Mobile Networks (NGMN) alliance, in next generation communication systems energy efficiency of the networks is a key factor and it is a central design principle of 5G. Energy efficiency is defined as the number of bits that can be transmitted per Joule of energy, where the energy is computed over the whole network. As stated above 5G should support a 1,000 times traffic increase, but this increase should be done with an energy consumption by the whole network of only half that typically consumed by today's networks. This leads to the requirement of an energy efficiency increase of x2000 \cite{NGMN}. One way to reduce power consumption is bandwidth expansion. In \cite{SE_EE_TOFF_WU}, it is stated that bandwidth expansion enables savings of power consumption of up to 45\% if all cells in the network apply the same bandwidth expansion strategy. In this study, in addition to user rate analysis we have also investigated the variation of Spectral and Energy Efficiency with resource allocation parameters in HetNet model under consideration.
In this study, for the HetNet model with Cell-Edge Located Small Cells, we use a resource allocation scheme which partitions the resources in time and also in frequency domain. We analytically derive the probability distribution of downlink data rates achieved by users in the cell and then verify the proposed analytical model by simulations. We show that the distributions given by the analytical model are highly accurate under different network parameters such as user distribution and bias. By using the analytical rate distribution expression, we optimize the system in terms of $10^{th}$ percentile rate ($R_{10}$). Additionally, by using a similar approach that we have followed to obtain rate distribution, we have also derived the CDF expressions for Spectral Efficiency (SE) and Energy Efficiency (EE). We have also optimized the resource allocation parameters in order to maximize tenth percentile SE ($SE_{10}$) and EE ($EE_{10}$). These results demonstrate the SE and EE trade-off in the studied HetNet model.
The most important contributions of this paper are:
\begin{itemize}
\item Analytical CDF of downlink data rate per user, SE and EE are derived for a HetNet model with COE configuration, for different user distributions and bias values.
\item The analytically derived CDFs have been shown to be very close to CDFs obtained by simulations.
\item Using analytically obtained distributions, optimal resource allocation parameters have been obtained that maximize $R_{10}$, $SE_{10}$ and $EE_{10}$.
\item The results have shown that the optimal values of resource allocation parameters maximizing $R_{10}$ and $EE_{10}$ are close to each other, however these values are not optimal for $SE_{10}$.
\item A single performance metric for jointly maximizing SE and EE has been proposed and the variation of this parameter with HetNet system parameters has also been analyzed. The results have shown the value of resource allocation parameter optimizing joint metric is closer to the value that maximizes SE.
\end{itemize}
The rest of this paper is organized as follows: Section \ref{sec:System_Model} describes our system model for the HetNet in consideration. Section \ref{sec:Analitic} presents the derivation of cumulative distribution function of the rate/user, spectral efficiency and energy efficiency for the network model and resource allocation scheme employed. Section \ref{sec:Results} presents the simulation and analytical results and the paper is concluded in Section \ref{sec:Conclusion}.
\section{System Model }
\label{sec:System_Model}
We consider a heterogeneous network model that consists of Macro and Micro Base Stations (BS) and User Equipments (UE). We assumed the Cell-On-Edge Model in which Micro BSs are located on the edge of a cell that is covered by a Macro BS. The model follows the assumptions, which are in accordance with the 3GPP simulation parameters given in \cite{GEN3PP_2}, are enlisted below;
\begin{itemize}
\item There is 1 Macro BS located in the center of a circular area with a radius of 1km.
\item There are 10 Micro BSs which are located on the ring that is located 0.8 km away from the center. The distance between adjacent Micro BSs are equal.
\item A part of the user equipments (UEs) are located uniformly on all the area.
\item The other part of the UEs are located uniformly within circles that are in the coverage of Micro BSs. The ratio of this part of UEs to all UEs located on all area is equal to $W_{Micro}$.
\end{itemize}
One example topology is shown in Figure \ref{fig:exTop}. By examining Figure \ref{fig:exTop}, the denser distribution of the users around micro base stations can be observed. As $W_{Micro}$ increases, the density around Micro BS will also increase.
\begin{figure}
\includegraphics[scale=0.6]{exTop}
\caption{An example of topology, red circles are Base Station Locations} \label{fig:exTop}
\end{figure}
In the communication system model, we have only considered the down-link communication from the BSs to UEs and assumed that UEs have always something to receive from BSs. The wireless channel between BSs and UEs is modeled by a path loss model for which the received power ($P_{r,i}$ in watts) is related to transmit power ($P_{t,i}$ in watts) of BS with index $i$ as in
\begin{equation}
P_{r,i}= \frac {P_{t,i}} {d_i ^ {\gamma_i} } \label{eqn:P_rec_i}.
\end{equation}
Throughout the paper we use the following convention for BS type to index mapping: the BS with index $i=0$ is the Macro BS and the BSs with index $i>0$ are Micro BSs. In (\ref{eqn:P_rec_i}), $d_i$ is the distance between UE and BS $i$ and $\gamma_i$ is the attenuation parameter for BS $i$ and it is defined by
\begin{equation}
\gamma_i=\begin{cases} \alpha_1, & \mbox{if } i \mbox{=0} \\ \alpha_2, & \mbox{if } i \mbox{$>$0} \end{cases} \label{eqn:alpha_vals}.
\end{equation}
In (\ref{eqn:P_rec_i}), $P_{t,i}$ differs depending on the BS type and it is given as
\begin{equation}
P_{t,i}=\begin{cases} P_1, & \mbox{if } i \mbox{=0} \\ P_2, & \mbox{if } i \mbox{$>$0} \end{cases}. \label{eqn:Pt_vals}
\end{equation}
In this system, each UE calculates its signal power parameter $P_{s,i}$, which is a scaled version of received power $P_{r,i}$ with the bias value of BS $i$ ($B_i$). Depending on the $P_{s,i}$ value, UE associates itself with a BS. The relation between $P_{s,i}$ and $P_{r,i}$ is given by
\begin{equation}
P_{s,i}=P_{r,i} 10^{ \frac {B_i} {10} } \label{eqn:P_s_i}
\end{equation}
where
\begin{equation}
B_i=\begin{cases} 0, & \mbox{if } i \mbox{=0} \\ B, & \mbox{if } i \mbox{$>$0} \end{cases} \label{eqn:B_vals}.
\end{equation}
Each UE is associated with the BS $i$ for which signal power parameter, $P_{s,i}$ is maximum. In the system each UE can be a Macro, Cell Range Extended (CRE) or a Direct Micro UE. The UEs for which $P_{s,i}$ is maximum for $i=0$ are Macro UEs, the UEs for which $P_{s,i}$ is maximum for $i>0$ and $B=0$ are Direct Micro UEs, and the UEs for which $P_{s,i}$ is maximum for $i>0$ and $B>0$ are CRE UEs. Figure \ref{fig:Top_B_15} shows how UEs are associated with BSs for $B=15$ dB.
\begin{figure}
\includegraphics[scale=0.6]{Node_topology_B_15}
\caption{UE Connections for B=15dB} \label{fig:Top_B_15}
\end{figure}
In our system we use a resource allocation scheme which is shown by Figure \ref{fig:case_4}. According to this resource allocation scheme, CRE UEs are served in $\eta$ of time for $0 \leq \eta \leq 1$ whereas Direct Micro UEs and Macro UEs are served in $1-\eta$ amount of time. In addition to partitioning in time domain, we have also employed a partitioning in frequency domain. The whole band is shared orthogonally among adjacent Micro BSs in order to reduce interference coming to CRE UEs from neighboring Micro BSs. Frequency band is orthogonally allocated among Macro and Direct Micro UEs, so that the interference power at Direct Micro UEs and Macro UEs is minimized. In our scheme, $\rho W$ of the total band $W$ will be used by Macro UEs.
\begin{figure}
\includegraphics[scale=0.5]{Macro_Micro_CRE}
\caption{Allocation of time and frequency to UEs} \label{fig:case_4}
\end{figure}
The time/frequency resources that are given to UEs are shared equally among UEs that are connected to the same BS. For example, if a Micro BS has $N_m$ Direct Micro UEs, each user has a total access to a channel having a bandwidth of $(1-\rho)W$ for $\frac{1-\eta}{N_m}$ in one unit of time.
We have also assumed that each UE uses the maximum capacity of the channel assuming Gaussian alphabet is transmitted.
Using the communication model described above, we have investigated parameters such as the cumulative distribution (CDF) of bit rate per UE, spectral and energy efficiency (SE and EE). We have selected $10^{th}$ percentile rate ($R_{10}$), median and tenth percentile Spectral ($SE_{50}, SE_{10}$) and Energy Efficiency ($EE_{50}, EE_{10}$) as Key Parameter Indicators (KPIs). We analytically derived CDF of bit rate per UE using a geometrical approach and verified our analytical results using extensive simulations. A similar approach has also been followed to derive the analytical distributions of SE and EE. Moreover, we have employed the analytical CDF expression of rate, SE and EE in order to select optimal values of resource sharing parameters ($\eta$, $\rho$) for two specific bias values $B$ in order to maximize the KPIs.
\section{Analytical Derivation of Cumulative Distribution of User Rate, Spectral and Energy Efficiency}
\label{sec:Analitic}
In this section, first the analytical formulas for cumulative distribution of capacity/user for the UEs in the heterogeneous network model will be derived by using a geometrical approach. Then, spectral and energy efficiency will be introduced and following a similar procedure the cumulative distributions of these two parameters will also be derived.
As described in Section \ref{sec:System_Model}, there are three types of UEs in the system, which are Macro, Direct Micro and CRE UEs. Each type of these users have different capacity distributions and for given $B$, $\eta$, $\rho$ values a general equation for capacity per user is given as,
\begin{equation}
C_{\zeta}(B, \eta, \rho)=\frac{\eta_\zeta}{N_{BS, \zeta}(B)} W \rho_\zeta log_2(1+ \frac{P_r}{\sigma^2_{N+I, \zeta}}). \label{eqn:Cap_UE}
\end{equation}
In (\ref{eqn:Cap_UE}), $\zeta$ is the index showing type of user. $\eta_\zeta$ is the time sharing parameter for user type $\zeta$. $N_{BS, \zeta}(B)$ is the average number of type $\zeta$ users per base station for bias value $B$, $W$ is the total bandwidth used, $\rho_\zeta$ is the band sharing parameter for user type $\zeta$. $P_r$ is the power of signal received from the associated BS. $\sigma^2_{N+I, \zeta}$ is the variance of $Interference + Noise$ term of user type $\zeta$, where this term is modeled as a Gaussian random variable. $\zeta$, $\eta_\zeta$ and $\rho_\zeta$ are given as,
\begin{equation}
\zeta=\begin{cases} 0, & \mbox{For a macro user}
\\1, & \mbox{For a direct user}
\\ 2, & \mbox{For a CRE user}, \end{cases}
\label{eqn:zeta_types}
\end{equation}
\begin{equation}
\eta_\zeta=\begin{cases} 1-\eta, & \mbox{$\zeta=0, 1$}
\\\eta, & \mbox{$\zeta=2$}, \end{cases}
\label{eqn:eta_types}
\end{equation}
\begin{equation}
\rho_\zeta=\begin{cases} \rho, & \mbox{$\zeta=0$}
\\1-\rho, & \mbox{$\zeta=1$}
\\0.5, & \mbox{$\zeta=2$}. \end{cases}
\label{eqn:rho_types}
\end{equation}
The parameters $\eta$, $\rho$ above, are time and band sharing parameters which are defined as shown in resource allocation scheme illustrated in Fig. \ref{fig:case_4}. In this paper, after obtaining the analytical expression for capacity distribution, we aim to find optimum values of $\eta$ and $\rho$ in order to maximize tenth percentile rate, $R_{10}$.
\subsection{Modeling $Interference+Noise $ Term}
$Interference+Noise $ term is modeled as a zero-mean Gaussian random variable with variance $\sigma^2_{N+I, \zeta}$. Due to the symmetry of the base station locations in heterogeneous network model, $\sigma^2_{N+I, \zeta}$ is assumed to be same for all users of same type. In this section, the variance value for different type of UEs will be presented.
\textbf{Direct Micro UEs:} Sources of interference for Direct Micro UEs are all micro base stations other than the one that is associated with the UE. We have assumed that the total $Interference+Noise$ for these users can be modeled as a Gaussian random variable with variance given by
\begin{equation}
\sigma^2_{N+I, 1}= P_{noise} + \displaystyle \sum _{i=2}^{N_{MICRO}} \frac {P_2} {l_i ^ { \alpha_2 } } \label{eqn:sigma_dir},
\end{equation}
where $l_i$ is the distance between $i^{th}$ Micro BS and $1^{st}$ Micro Base Station for $i=2,3,...N_{MICRO}$.
\textbf{CRE UEs:} Sources of interference for CRE UEs are the micro base stations that use the same portion of the band. Because the band is orthogonally and equally shared by Micro BSs. We have assumed that the total $Interference+Noise$ can be modeled as a Gaussian random variable with variance
\begin{equation}
\sigma^2_{N+I, 2}= P_{noise} + \displaystyle \sum _{i=2n+1}^{N_{MICRO}} \frac {P_2} {l_i ^ { \alpha_2 } } \label{eqn:sigma_CRE},
\end{equation} where $l_i$ is as defined above.
\textbf{Macro UEs:} By inspecting the resource allocation scheme shown in Fig \ref{fig:case_4}, it can be observed that there is no source of interference for Macro UEs, therefore $Interference + Noise$ term depends only on noise, therefore $\sigma^2_{N+I, 0}$ is given by
\begin{equation}
\sigma^2_{N+I, 0}= P_{noise} \label{eqn:sigma_macro}.
\end{equation}
\subsection{Distribution of Received Power}
\label{sec:Dist_general}
With the Gaussian assumption for $Interference + Noise$ term, the distribution of received power, $P_r$, should be obtained first in order to find the distribution of capacity per user which was given by (\ref{eqn:Cap_UE}). Figure \ref{fig:CRE_Contour_15_dB} shows the range extended coverage of Micro BS for $B=15dB$. This is the region for which $P_{r, Micro} 10^{\frac{B}{10}} > P_{r, Macro}$, where $P_{r, Micro}$ is the power received from the closest Micro BS, $P_{r, Macro}$ is the received power from Macro BS and $B$ is the BIAS parameter.
Any point $x, y$ on the contour of this area should satisfy
\begin{figure*}
\centering
\includegraphics[width=17cm,height=9cm]{CRE_x_y}
\caption{CRE region contour for $B=15dB$}
\label{fig:CRE_Contour_15_dB}
\end{figure*}
\begin{equation}
\frac {P_1} {(x^2+y^2) ^ {\frac {\gamma_1} {2} }} = 10^{\frac {B} {10}} \frac {P_2} {((x-800)^2+y^2) ^ {\frac {\gamma_2} {2} } }. \label{eqn:Contour_eqn}
\end{equation}
(\ref{eqn:Contour_eqn}) is numerically solved for various $B$ values and coverage of Micro BS is obtained. In order to simplify the analytical calculations, these coverage regions are approximated by circles centered at points $c=\frac {p_1+p_2} {2} e ^ {j \frac{\pi i}{N_{MICRO}}}, \: i=0,1,2 ...(N_{MICRO}-1) $ and have the radius of $r=\frac {|p_1-p_2|} {2}$. $p_1, p_2$, which are shown in figure \ref{fig:CRE_Contour_15_dB}, are points where the contour intersects the $y=0$ line, and $|\cdot|$ is the magnitude operator. Figures \ref{fig:Coverage_0_dB} and \ref{fig:Coverage_20_dB} show how this approximation is close to the actual coverage region for two different bias values, $B=10dB$ and $B=20dB$. The approximated coverage model of the BSs for same bias values are shown in Figures \ref{fig:AllArea_10_dB} and \ref{fig:AllArea_20_dB}, respectively.
\begin{figure}
\includegraphics[scale=0.5]{coverage0dB}
\caption{Approximated coverage for $B=0dB$}
\label{fig:Coverage_0_dB}
\end{figure}
\begin{figure}
\includegraphics[scale=0.5]{coverage20dB}
\caption{Approximated coverage for $B=20dB$}
\label{fig:Coverage_20_dB}
\end{figure}
\begin{figure}
\includegraphics[width=9.0cm,height=8.5cm]{AllArea_10dB}
\caption{Coverage of different type of BSs for $B=10dB$}
\label{fig:AllArea_10_dB}
\end{figure}
\begin{figure}
\includegraphics[width=9.0cm,height=8.5cm]{AllArea_20dB}
\caption{Coverage of different type of BSs for $B=20dB$}
\label{fig:AllArea_20_dB}
\end{figure}
Using this approximated model for the system, the cumulative distribution of received power ($P_r$) for different type of users will be derived in sections \ref{sec:Pr_Dir_Micro}, \ref{sec:Pr_CRE} and \ref{sec:Pr_Macro}. Using CDFs obtained for $P_r$ for different type of UEs, the capacity per any UE will be derived in section \ref{sec:Cap_User}.
\subsubsection{Distribution of Received Power for Direct Micro Users}
\label{sec:Pr_Dir_Micro}
Received power for a UE is a function of the distance ($D$) between the UE and the BS that is associated with it. Therefore, in order to obtain received power distribution, first we have to obtain CDF of random variable $D$, i.e. $F_D(d)=P(D \le d)$. As shown in Figure \ref{fig:Coverage_0_dB}, direct micro coverage is approximated by a perfect circle with parameters defined in Section \ref{sec:Dist_general}. Using this approximation and keeping uniform user distribution in mind, $F_D(d)$ for a direct micro user can be calculated as
\begin{equation}
P(D \le d) = \frac {S(d)} {S_{DIR}}, \label{eqn:CDF_Micro}.
\end{equation}
In (\ref{eqn:CDF_Micro}), $S(d)$ is the intersection area of the circle centered at Micro BS location with a radius of $d$ ($d \leq R_{max}$) and approximated coverage region for Micro BS (orange colored region in Figure \ref{fig:DIR_Region_plot}). $R_{max}$ is the maximum distance between micro BS location and a Direct Micro UE. $S_{DIR}$ is the area of the approximated direct Micro coverage region (union of orange and green colored regions in Fig. \ref{fig:DIR_Region_plot}). Calculation of the intersection area of two circles is given in Appendix \ref{sec:TwoCircIntersection}.
\begin{figure}
\includegraphics[scale=0.25]{DIR_Region}
\caption{Calculation of Direct Micro Users' distance distribution to serving Micro BS}
\label{fig:DIR_Region_plot}
\end{figure}
Using (\ref{eqn:CDF_Micro}) and the relation between distance and received power that is given by (\ref{eqn:P_rec_i}), the CDF of received power $P_R$ can be expressed as
\begin{equation}
F_{p_r}(P_R) = 1-F_D(\sqrt[-\alpha_2] {\frac{P_R \sigma^2_{N+I, 1}}{P_2}}). \label{eqn:Pr_DirMicro_CDF}
\end{equation}
\subsubsection{Distribution of Received Power for CRE UEs}
\label{sec:Pr_CRE}
The distribution of received power for CRE UEs can be calculated similarly to that of Direct Micro UEs. The approximated coverage model for $B=10dB$ and $B=20dB$ are shown in Figures \ref{fig:CRE_Detail_10_dB}-\ref{fig:CRE_Detail_20_dB}. The range extended coverage of Micro BS is approximated by a perfect circle as defined in Chapter \ref{sec:Dist_general}. Using uniform distribution of UEs, the distribution of the distance between CRE UEs and Micro BS is given by
\begin{equation}
P(D \le d) = \frac {S(d)} {S_{CRE}(B)} \label{eqn:CDF_CRE}.
\end{equation}
In (\ref{eqn:CDF_CRE}), $S(d)$ is the area of the intersection of the circle centered at Micro BS location with a radius $d$ ($d \leq R_{max}$), with approximated CRE region and all region where UEs are uniformly located. This region is colored to orange in Figures \ref{fig:CRE_Detail_10_dB} and \ref{fig:CRE_Detail_20_dB} for different $B$ values. $S_{CRE}(B)$ is the total area of region where CRE users are located for a fixed bias value $B$, geometrically this area can be expressed by the union of orange and green colored regions that are shown in Figures \ref{fig:CRE_Detail_10_dB} and \ref{fig:CRE_Detail_20_dB}.
Using (\ref{eqn:CDF_CRE}) and (\ref{eqn:P_rec_i}), the CDF of received power, $P_R$ can be written as
\begin{equation}
F_{p_r}(P_R) = 1-F_D(\sqrt[-\alpha_2] {\frac{P_R \sigma^2_{N+I, 2}}{P_2}}) \label{eqn:Pr_CRE_CDF}.
\end{equation}
\begin{figure}
\includegraphics[scale=0.3]{CRE_Area_10}
\caption{CRE Region in detail for $B=10dB$}
\label{fig:CRE_Detail_10_dB}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.4]{CRE_Area_20}
\caption{CRE Region in detail for $B=20dB$}
\label{fig:CRE_Detail_20_dB}
\end{figure}
\subsubsection{Distribution of Received Power for Macro UEs}
\label{sec:Pr_Macro}
Distribution of the received power for Macro UEs can be found by calculating the distribution of the distance between Macro UEs and Macro BS. The Macro BS coverage region is modeled as a combination of differently shaped regions as illustrated in Figures \ref{fig:MACRO_Detail_10_dB} and \ref{fig:MACRO_Detail_20_dB} for different $B$ values. The CDF of the distance between Macro UEs and Macro BS is given by
\begin{figure}
\includegraphics[scale=0.35]{MACRO_AREA_10}
\caption{Macro Region in detail for $B=10dB$}
\label{fig:MACRO_Detail_10_dB}
\end{figure}
\begin{figure}
\includegraphics[scale=0.4]{MACRO_AREA_20}
\caption{Macro Region in detail for $B=20dB$}
\label{fig:MACRO_Detail_20_dB}
\end{figure}
\begin{equation}
P(D \le d) = \frac {S(d)} {S_{MACRO}(B)} \label{eqn:CDF_MACRO}.
\end{equation}
In (\ref{eqn:CDF_MACRO}), $S(d)$ is the area of the intersection of the circle centered at Micro BS location with a radius $d$ ($d \leq R_{max}$), with approximated Macro region. This region is colored to orange in Figures \ref{fig:MACRO_Detail_10_dB} and \ref{fig:MACRO_Detail_20_dB}. $S_{MACRO}(B)$ is the total area of the region where Macro users are uniformly located for a given bias value $B$, geometrically it is the union of orange and green colored regions in Figures \ref{fig:MACRO_Detail_10_dB} and \ref{fig:MACRO_Detail_20_dB}.
Using (\ref{eqn:CDF_MACRO}) and (\ref{eqn:P_rec_i}), the CDF of received power $P_r$ is given by
\begin{equation}
F_{p_r}(P_R) = 1-F_D(\sqrt[-\alpha_1] {\frac{P_R \sigma^2_{N+I, 0}}{P_1}}) \label{eqn:Pr_Macro_CDF}.
\end{equation}
\subsection{Distribution of Capacity Per User}
\label{sec:Cap_User}
The relation between the capacity per user $C$ and $P_r$ was given in (\ref{eqn:Cap_UE}). Equations (\ref{eqn:Pr_DirMicro_CDF}), (\ref{eqn:Pr_CRE_CDF}) and (\ref{eqn:Pr_Macro_CDF}) give the CDF's of $P_r$ for each type of UEs. Using these equations and (\ref{eqn:Cap_UE}) the capacity distribution for UE type $\zeta$can be obtained as in (\ref{eqn:C_Zeta}) in terms of the CDF of $P_r$.
\begin{equation}
F_{C}(c|\zeta, B, \eta, \rho) = F_{Pr}(\sigma^2_{N+I, \zeta} (2^{\frac {c N_{BS, \zeta}(B)} {\eta_{\zeta}W\rho_{\zeta}} }-1)), \label{eqn:C_Zeta}
\end{equation}
Using (\ref{eqn:C_Zeta}), the distribution of capacity per any UEin the system can be obtained as given by
\begin{equation}
F_{C}(c|B, \eta, \rho) = \displaystyle \sum _{\zeta=1}^{3} P(\zeta|B) F_{C}(c|\zeta, B, \eta, \rho) \label{eqn:Cap_Any}
\end{equation}
where $P(\zeta|B)$ for a given $B$ value is given by
\begin{equation}
P(\zeta=i|B) = \frac { N_{\zeta}(B) } {N}, \zeta=0,1,2. \label{eqn:P_UE_Type}
\end{equation}
In (\ref{eqn:P_UE_Type}), $N_{\zeta}(B)$ is the average number of UEs of type $\zeta$ for bias value $B$ and $N$ is the total number of UEs in the system. According to system model, a ratio of $1-W_{Micro}$ of all UEs are distributed uniformly to all area, and a ratio of $W_{Micro}$ of UEs are distributed uniformly in Direct Micro coverage area. Using this model $N_{\zeta}(B)$ is given by
\begin{equation}
E N_{ \zeta }(B) =\begin{cases}
N (1-W_{micro}) \frac {S_{MACRO}(B)} {S_{TOT}} , & \mbox{$\zeta=0$} \\
N W_{micro} + N (1-W_{micro}) \frac {S_{DIR}} {S_{TOT}}, & \mbox{$\zeta=1$} \\
N (1-W_{micro}) \frac {S_{CRE}(B)} {S_{TOT}} & \mbox{$\zeta=2$}
\end{cases} \label{eqn:N_zeta}.
\end{equation}
In order to calculate the areas of different type of UEs, calculation of intersection area of three circles should be calculated for large $B$ values. The method used for this calculation is shown in Appendix \ref{sec:ThreeCircIntersection}.
\subsection{Distribution of Spectral and Energy Efficiency}
\label{sec:EE_SE_Section}
Spectral Efficiency (SE) is defined as the experienced data rate of a UE per bandwidth occupied by the UE. SE is expressed by
\begin{equation}
SE_{\zeta}(\eta)=\eta_\zeta log_2(1+ \frac{P_r}{\sigma^2_{N+I, \zeta}}). \label{eqn:SE_UE}
\end{equation}
Energy Efficiency is the rate of UE divided by the total power consumed by the BSs of the system. EE is given by
\begin{equation}
EE_{\zeta}(\eta)=\frac {C_{\zeta}(B, \eta, \rho)}{P_{tot} (\eta)}. \label{eqn:SE_EE}
\end{equation}
The distributions of both SE and EE will be derived by using previously obtained distributions.
\subsubsection{Distribution of Spectral Efficiency}
\label{sec:Dist_SE}
Distribution of $SE$ can be obtained similarly to capacity per UE distribution as found in Section \ref{sec:Analitic}. Using the cumulative distribution of received power $P_r$ (\ref{eqn:C_Zeta}), for user type $\zeta$ and time sharing parameter $\eta$, the distribution of $SE$ can be obtained as given by
\begin{equation}
F_{SE}(s|\zeta, B, \eta) = F_{Pr}(\sigma^2_{N+I, \zeta} (2^{\frac {s} {\eta_{\zeta}} }-1)), \label{eqn:SE_Zeta}
\end{equation}
By using (\ref{eqn:SE_Zeta}), the CDF of $SE$ for any UE in the system can be written as
\begin{equation}
F_{SE}(s|B,\eta) = \displaystyle \sum _{\zeta=1}^{3} P(\zeta|B) F_{SE}(s|\zeta, B, \eta), \label{eqn:SE_Any_UE}
\end{equation}
where $P(\zeta|B)$ is the probability of being type $\zeta$ UE in the system for a bias value of $B$.
\subsubsection{Distribution of Energy Efficiency}
\label{sec:Dist_EE}
In order to obtain EE which is given by (\ref{eqn:SE_EE}), $P_{tot} (\eta)$, total power consumed by BSs should be calculated. Here, we use a Base Station power consumption model which is proposed in \cite{GAUER}. In \cite{GAUER}, the BS power consumption is modeled by a linear power model:
\begin{equation}
P_{in}=\begin{cases} N_{TRX} P_0 + \Delta_p P_{out}, & \mbox{if } \mbox{0 $<$} P_{out} \mbox{$\leq P_{max}$} \\ N_{TRX} P_{sleep}, & \mbox{if } P_{out} \mbox{=0} \end{cases} \label{eqn:P_BS_Cons},
\end{equation}
where, $P_{in}$ is the total power consumed, $N_{TRX}$ is the number of transceivers in BS, $P_0$ is the power consumption at the minimum non-zero output power, $\Delta_p$ is the slope of the load-dependent power consumption, $P_{out}$ is the output power which is limited by $P_{max}$ and $P_{sleep}$ is the sleep mode power consumption of BS. The values of the parameters of Macro and Micro BSs are listed in Table \ref{table:BS_Power_Parameters}.
\begin{table}
\centering
\caption{Base Station Power Consumption Parameters}
\begin{tabular}{|c|c|c|c|c|}
\hline BS Type & $N_{TRX}$ & $P_{0}$ & $\Delta_{p}$ & $P_{sleep}$ \\ \hline
Macro & 6 & 130 & 4.7 & 75 \\ \hline
Micro & 2 & 56 & 2.6 & 39 \\ \hline
\end{tabular}
\label{table:BS_Power_Parameters}
\end{table}
By considering the model given by (\ref{eqn:P_BS_Cons}), the total power consumed by BSs of the system is given by
\begin{equation}
\begin{aligned}
P_{tot}=(1-\eta) P_{in,Macro} (P_{out}=P_{t,Macro}) \\
+ \eta P_{in,Macro} (P_{out}=0) \\
+N_{MICRO} P_{in,Micro} (P_{out}=P_{t,Micro}).
\end{aligned}
\label{eqn:P_tot_BS}
\end{equation}
In (\ref{eqn:P_tot_BS}), $P_{in,Macro} (P_{out}=P_{t,Macro})$ is the total power consumed by Macro BS when the output power is $P_{t,Macro}$ and similarly $P_{in,Micro} (P_{out}=P_{t,Micro})$ is the total power consumed by a Micro BS when the output power is set to be $P_{t,Micro}$.
Using distribution of received power per UE which is given by (\ref{eqn:C_Zeta}), the distribution of $EE$ can be derived as given by
\begin{equation}
F_{EE}(e|\zeta, B, \eta, \rho) = F_{Pr}(\sigma^2_{N+I, \zeta} (2^{\frac {e P_{tot} (\eta) N_{BS, \zeta}(B)} {\eta_{\zeta}W\rho_{\zeta}} }-1)). \label{eqn:EE_Zeta}
\end{equation}
By using (\ref{eqn:EE_Zeta}), the CDF of $EE$ for any UE in the system can be expressed as
\begin{equation}
F_{EE}(e|\eta) = \displaystyle \sum _{\zeta=1}^{3} P(\zeta|B) F_{EE}(e|\zeta,B, \eta, \rho). \label{eqn:EE_Any_UE}
\end{equation}
\section{Numerical Results}
\label{sec:Results}
\begin{figure*}
\centering
\subfloat[$B=10dB$]{\label{subfig:CDF_B10}
\includegraphics[height=7cm]{CDF_B10_w12}}
\subfloat[$B=20dB$]{\label{subfig:CDF_B20}
\includegraphics[height=7cm]{CDF_B20_w12}}
\caption{CDF of Downlink Data Rate per User for $\eta = 0.2$, $\rho=0.5$}
\label{fig:CDF_10_20_dB}
\end{figure*}
\begin{figure*}
\centering
\subfloat[$B=10dB$]{\label{subfig:R10_3D_13_B10}
\includegraphics[height=7cm]{R10_3D_w13_B10}}
\subfloat[$B=20dB$]{\label{subfig:R10_3D_13_B20}
\includegraphics[height=7cm]{R10_3D_w13_B20}}
\caption{Variation of $R_{10}$ as a function of $\rho$ and $\eta$ with $W_{Micro}=1/3$}
\label{fig:R10_3D_W_1_3}
\end{figure*}
\begin{figure*}
\centering
\subfloat[$B=10dB$]{\label{subfig:R10_3D_12_B10}
\includegraphics[height=7cm]{R10_3D_w12_B10}}
\subfloat[$B=20dB$]{\label{subfig:R10_3D_12_B20}
\includegraphics[height=7cm]{R10_3D_w12_B20}}
\caption{Variation of $R_{10}$ as a function of $\rho$ and $\eta$ with $W_{Micro}=1/2$}
\label{fig:R10_3D_W_1_2}
\end{figure*}
\begin{figure*}
\centering
\subfloat[$B=10dB$]{\label{subfig:R10_3D_23_B10}
\includegraphics[height=7cm]{R10_3D_w23_B10}}
\subfloat[$B=20dB$]{\label{subfig:R10_3D_23_B20}
\includegraphics[height=7cm]{R10_3D_w23_B20}}
\caption{Variation of $R_{10}$ as a function of $\rho$ and $\eta$ with $W_{Micro}=2/3$}
\label{fig:R10_3D_W_2_3}
\end{figure*}
In this section, analytical results obtained for cumulative distribution of rate per UE will be compared first with the rate distribution obtained by simulations. The comparisons have been done for different bias ($B$) and resource allocation parameter ($\eta$, $\rho$) values and also for different UE distributions. Then, the analytical rate distribution has been employed in order to optimize system parameters $\eta$, $\rho$ and $B$ for different UE distributions. The optimizations have been done by considering tenth percentile rate $R_{10}$, which is the parameter we have selected as KPI in the system. Optimization of $R_{10}$ has also been done by simulations for comparison aspects. We have investigated the system in terms of Energy and Spectral efficiency. By using the $\rho$ values that maximizes $R_{10}$, we have obtained the variation of tenth percentile and median of EE and SE by both using analytical expression and doing simulations. The analytical and simulation results are obtained with system model parameter values which are listed in Table \ref{table:sim_pars}.
\begin{table}
\centering
\caption{Parameter values}
\begin{tabular}{|c|c|}
\hline
Parameter & Value \\ \hline
$P_{t,Macro}$ & $16$ dB \\ \hline
$P_{t,Micro}$ & $-4$ dB \\ \hline
$P_{noise}$ & $-173$ dBm/Hz \\ \hline
$BS Noise Figure$ & $7$ dB \\ \hline
$W$ & $100$ MHz \\ \hline
$\eta$ & $0 \leq \eta \leq 1$ \\ \hline
$\rho$ & $0 \leq \rho \leq 1$ \\ \hline
$\alpha_{1}$ & $3.5$ \\ \hline
$\alpha_{2}$ & $4$ \\ \hline
$N_{UE}$ & 1000 \\ \hline
$N_{MICRO}$ & 10 \\ \hline
$N_{MACRO}$ & 1 \\ \hline
\end{tabular}
\label{table:sim_pars}
\end{table}
The cumulative probability distribution of rate per UE obtained by analytical formula and simulations are plotted in Figure \ref{fig:CDF_10_20_dB}. The CDFs are obtained for $\eta=0.2$, $\rho=0.5$, $B=10dB$ and $B=20dB$ and also for different UE distributions where $W_{micro}=\frac{1}{3}, \frac{1}{2}, \frac{2}{3}$. The goodness of fit of CDFs obtained by analytical approximation and simulations are compared by using Kolmogorov-Smirnov test \cite{KS_TEST}. Given the analytical distribution, this test shows whether random variables obtained empirically are distributed with the analytical distribution or not for a given level of significance value. In order to test the CDF obtained by analytical approximation, 100 UEs among 1000 UEs are randomly selected and the average level of significance between empirical and analytical distribution is calculated for 400 trials. The average significance values obtained for different UE distributions and $B$ values are given in Table \ref{subtable:KS_R_Significance} for rate distribution. Table \ref{subtable:KS_R_Ratio} shows the ratio of the KS tests passed for a significance value of $0.05$, which is a typical significance value for KS test. Tables \ref{subtable:KS_R_Significance}, \ref{subtable:KS_R_Ratio} and Figure \ref{fig:CDF_10_20_dB} show us that the CDFs obtained by analytical approximation and simulations are very close to each other. Consequently, we have concluded that analytical CDF approximation can be used for optimization of the system in terms of $R_{10}$.
\begin{table}
\centering
\caption{Kolmogorov-Smirnov Test Results for CDF of $R$}
\subfloat[Average Significance Values]{\label{subtable:KS_R_Significance}
\begin{tabular}{|c|c|c|c|}
\hline \backslashbox{$B$(dB)}{$W_{Micro}$} & 1/3 & 1/2 & 2/3 \\ \hline
10 & 0.3320 & 0.4606 & 0.1609 \\ \hline
20 & 0.4149 & 0.2740 & 0.2300 \\ \hline
\end{tabular}}
\subfloat[Pecentage of Kolmogorov-Smirnov Tests Passed for $P_{sig}=0.05$]{\label{subtable:KS_R_Ratio}
\begin{tabular}{|c|c|c|c|}
\hline \backslashbox{$B$(dB)}{$W_{Micro}$} & 1/3 & 1/2 & 2/3 \\ \hline
10 & 0.8783 & 0.9033 & 0.5400 \\ \hline
20 & 0.9117 & 0.7650 & 0.6867 \\ \hline
\end{tabular}}
\label{table:KS_R}
\end{table}
In order to find optimal values of system parameters, we have plotted the variation of $R_{10}$ with respect to $\rho$ and $\eta$ for two different bias values, $B=10dB$ and $B=20dB$ and for 3 different UE distributions. The variation of $R_{10}$ are presented in Figures \ref{fig:R10_3D_W_1_3}, \ref{fig:R10_3D_W_1_2} and \ref{fig:R10_3D_W_2_3}.
Figure \ref{fig:Rate_10thPerc_t3} shows the cross-sections of Figures \ref{fig:R10_3D_W_1_3}, \ref{fig:R10_3D_W_1_2} and \ref{fig:R10_3D_W_2_3} which are plotted for $\rho$ values that maximizes $R_{10}$. Tables \ref{subtable:B10_Res_Comparison} and \ref{subtable:B20_Res_Comparison} show the optimal $\eta$ and $\rho$ values and maximum $R_{10}$ values that are obtained by simulations and using analytical CDF expression for different UE distributions.
\begin{figure*}
\centering
\subfloat[$B=10dB$]{\label{subfig:Rate_10thPerc_B10}
\includegraphics[height=7cm]{R10_B10}}
\subfloat[$B=20dB$]{\label{subfig:Rate_10thPerc_B20}
\includegraphics[height=7cm]{R10_B20}}
\caption{$10^{th}$ Percentile Downlink Data Rate for varying $\eta$ and $W_{Micro}$}
\label{fig:Rate_10thPerc_t3}
\end{figure*}
By examining Figures \ref{fig:Rate_10thPerc_t3} and Tables \ref{subtable:B10_Res_Comparison} and \ref{subtable:B20_Res_Comparison}, it can be observed that analytically obtained results are very close to simulation results. It can also be concluded that the optimal $\rho$ value decreases and optimal $\eta$ value increases with increasing values of $B$. Optimal $\rho$ value also decreases with increasing $W_{micro}$, which says that as number of direct Micro UEs increases, larger portion of the bandwidth should be given to Micro BSs compared to small $W_{micro}$ values.
\begin{table}
\centering
\caption{The Comparison of Optimal Parameter Values and $R_{10}$}
\subfloat[$B=10dB$]{\label{subtable:B10_Res_Comparison}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline $W_{micro}$ & $\eta $ (S) & $\eta $ (A) & $\rho$ (S) & $\rho$ (A) & $R_{10}$ (S) & $R_{10}$ (A) \\ \hline
$1/3$ & 0.17& 0.15&0.79 &0.78 &3.115 & 3.113 \\ \hline
$1/2$ & 0.15& 0.13& 0.68& 0.66 &3.704 & 3.657 \\ \hline
$2/3$ & 0& 0& 0.56& 0.54& 4.768& 4.585 \\ \hline
\end{tabular}}
\subfloat[$B=20dB$]{\label{subtable:B20_Res_Comparison}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline $W_{micro}$ & $\eta $ (S) & $\eta $ (A) & $\rho$ (S) & $\rho$ (A) &$R_{10}$ (S) & $R_{10}$ (A) \\ \hline
$1/3$ &0.43 &0.41 & 0.67& 0.66& 3.433 & 3.416 \\ \hline
$1/2$ &0.36 & 0.35 & 0.53& 0.50& 4.095& 4.011 \\ \hline
$2/3$ & 0.27& 0.25 & 0.38& 0.35 & 4.799 & 4.786 \\ \hline
\end{tabular}}
\label{table:Res_Comparison}
\end{table}
Tables \ref{table:KS_SE} and \ref{table:KS_EE} show the results of KS test when it is applied to the distributions of SE and EE. Examining these results, it can be concluded that the analytical distributions obtained can be used for further optimization of the system in terms of Spectral and Energy Efficiency.
By examining Tables \ref{table:KS_R}, \ref{table:Res_Comparison}, \ref{table:KS_SE} and \ref{table:KS_EE}, it can be concluded that the accuracy of the analytical model decreases with increasing $W_{Micro}$ and $B$ values. The reason behind this is the approximations done to simplify base station coverage models for different type of UEs.
\begin{table}
\centering
\caption{Kolmogorov-Smirnov Test Results for CDF of $SE$}
\subfloat[Average Significance Values]{\label{subtable:KS_SE_Significance}
\begin{tabular}{|c|c|c|c|}
\hline \backslashbox{$B$(dB)}{$W_{Micro}$} & 1/3 & 1/2 & 2/3 \\ \hline
10 & 0.4153 & 0.3296 & 0.2001 \\ \hline
20 & 0.3935 & 0.2968 & 0.2163 \\ \hline
\end{tabular}}
\subfloat[Pecentage of Kolmogorov-Smirnov Tests Passed for $P_{sig}=0.05$]{\label{subtable:KS_SE_Ratio}
\begin{tabular}{|c|c|c|c|}
\hline \backslashbox{$B$(dB)}{$W_{Micro}$} & 1/3 & 1/2 & 2/3 \\ \hline
10 & 0.9233 & 0.8350 & 0.6683 \\ \hline
20 & 0.8967 & 0.7950 & 0.6733 \\ \hline
\end{tabular}}
\label{table:KS_SE}
\end{table}
\begin{table}
\centering
\caption{Kolmogorov-Smirnov Test Results for CDF of $EE$}
\subfloat[Average Significance Values]{\label{subtable:KS_EE_Significance}
\begin{tabular}{|c|c|c|c|}
\hline \backslashbox{$B$(dB)}{$W_{Micro}$} & 1/3 & 1/2 & 2/3 \\ \hline
10 & 0.4055 & 0.3606 & 0.1642 \\ \hline
20 & 0.4144 & 0.2705 & 0.2410 \\ \hline
\end{tabular}}
\subfloat[Pecentage of Kolmogorov-Smirnov Tests Passed for $P_{sig}=0.05$]{\label{subtable:KS_EE_Ratio}
\begin{tabular}{|c|c|c|c|}
\hline \backslashbox{$B$(dB)}{$W_{Micro}$} & 1/3 & 1/2 & 2/3 \\ \hline
10 & 0.9117 & 0.8383 & 0.5400 \\ \hline
20 & 0.9067 & 0.7600 & 0.7000 \\ \hline
\end{tabular}}
\label{table:KS_EE}
\end{table}
Figures \ref{subfig:SE_B10_20}, \ref{subfig:EE_B10_20} and \ref{subfig:theta_B10_20} shows the variation of tenth percentile and median of $SE$, $EE$, $\theta$ with varying $\eta$ and $B$ values, respectively. By inspecting Figures \ref{subfig:SE_B10_20} and \ref{subfig:EE_B10_20}, it can be observed that median SE, and EE decays nearly linearly as $\eta$ increases. This result says us that Macro and Direct Micro UEs constitute more than half of the UEs consequently, as more resources are given to CRE UEs, less data rate, SE and EE are experienced by Macro and Direct Micro UEs. However, this is not the case if $SE_{10}$ and $EE_{10}$ are considered. By observing Figures \ref{subfig:SE_B10_20} and \ref{subfig:EE_B10_20}, it can be concluded that $SE_{10}$ and $EE_{10}$ are concave down functions of $\eta$ and $SE_{10}$, $EE_{10}$ are maximized at different values of $\eta$. When $EE_{10}$ is considered, the $\eta$ value that maximizes $EE_{10}$ is very close to the value that is optimal for $R_{10}$. This is because of the fact that $\eta$ is the parameter that controls $P_{tot}$. However, variation of $R_{10}$ with $\eta$ is much more faster that the variation of $P_{tot}$ with $\eta$. Therefore the variation of $P_{tot}$ with respect to $R_{10}$ can be assumed as constant. If we analyze the system in terms of $SE_{10}$, the situation is different. The $\eta$ value which maximizes $R_{10}$ and $EE_{10}$ does not maximize $SE_{10}$. As $\eta$ increases, the rate of CRE UEs increases however the SE of CRE UEs are still small compared to SE of Macro and Direct Micro UEs. Therefore larger $\eta$ is needed to increase $SE_{10}$. This result also exhibits the SE and EE trade-off in our Heterogeneous Network System model as also shown in Figure \ref{fig:SE10vsEE10_Results}. By inspecting Figures \ref{subfig:SE_B10_20}, \ref{subfig:EE_B10_20} and \ref{subfig:theta_B10_20}, it can also be concluded that the $\eta$ values maximizing $SE_{10}$, $EE_{10}$ and $\theta_{10}$ increase with $B$.
In order to jointly optimize SE and EE, two joint metrics $\theta_{10}$ and $\theta_{50}$ which are defined as in (\ref{eqn:SE_EE_10prod}) and (\ref{eqn:SE_EE_50prod}) are proposed.
\begin{equation}
\theta_{10}=SE_{10} EE_{10} \label {eqn:SE_EE_10prod},
\end{equation}
\begin{equation}
\theta_{50}=SE_{50} EE_{50}, \label {eqn:SE_EE_50prod}
\end{equation}
Figure \ref{subfig:theta_B10_20} shows the variation of $\theta_{10}$ and $\theta_{50}$ with $\eta$. As observed from the figure, it can be concluded that the joint metric $\theta_{10}$ is maximized near the value that is optimal for $SE_{10}$ whereas the joint metric $\theta_{50}$ monotonically decreases with increasing $\eta$. This is due to the fact that $\theta_{50}$ covers the half of the UEs with small $\theta$ values. As $\eta$ increases, the resources given to Macro and Direct Micro UEs decreases, so $\theta_{50}$ decreases. From this observation, it can concluded that $\theta_{50}$ does not give any information about the variation of $\theta$ values of the UEs with smaller $\theta$ values. Therefore $\theta_{10}$ is a better metric if fairness is considered.
\begin{figure*}
\centering
\subfloat[]{\label{subfig:SE_B10_20} \includegraphics[height=10.5cm]{SE_B10_20}}
\subfloat[]{\label{subfig:EE_B10_20} \includegraphics[height=10.5cm]{EE_B10_20}}
\subfloat[]{\label{subfig:theta_B10_20} \includegraphics[height=10.5cm]{Theta_B10_20}}
\caption{Variation of $SE$, $EE$ and $\theta$ as a function of $\eta$ for $\rho=0.5$, $B=10dB$ (top), $B=20dB$ (bottom) and $W_{Micro}=1/2$}
\label{fig:SE_EE_Results}
\end{figure*}
\begin{figure}
\centering
\includegraphics[scale=0.6]{EE10_SE10_1_2}
\caption{$SE_{10}$ vs $EE_{10}$ for $B=10dB, 20dB$ and $W_{Micro}=1/2$}
\label{fig:SE10vsEE10_Results}
\end{figure}
\section{Conclusion}
\label{sec:Conclusion}
In this paper, we have analyzed a Heterogeneous Network with cell-edge located small cells. In the system, there is one Macro and 10 Micro BSs and there are three types of UEs which are Macro, Direct Micro and CRE UEs.
By using a Time/Frequency resource allocation scheme that gives resources to different type of UEs orthogonally, we have obtained the CDFs of user rate, SE and EE by using a geometrical approach. We have compared these CDFs with CDFs obtained by extensive simulations. Our results show that the analytically derived CDFs are very close to the CDFs obtained by simulations. We have selected tenth percentile rate ($R_{10}$), SE ($SE_{10}$) and EE ($EE_{10}$) as Key Parameter Indicators (KPIs) since these parameters inherently consider the fairness of the system and we have used the derived CDF expressions to optimize the system resource allocation parameters $\eta$, $\rho$ to maximize these KPIs. Our results show that the system is optimized around nearly same resource allocation parameters if $R_{10}$ and $EE_{10}$ are considered. However, larger $\eta$ values are needed to maximize $SE_{10}$, where $R_{10}$ and $EE_{10}$ values degrade. This demonstrates Energy Efficiency and Spectral Efficiency trade-off in HetNet system under consideration. When SE and EE are jointly optimized by considering the proposed $\theta$ parameter, we observe that optimum value of $\eta$ is closer to the value of $\eta$ which maximizes SE. This shows that SE is more critical in the SE-EE trade-off.
\appendices
\section{Intersection Area Calculation of Two Circles}
\label{sec:TwoCircIntersection}
Two circles with radius values $R_1$ and $R_2$ ($R_1>R_2$) may be located in four different positions to each other. These cases are shown in Figures \ref{fig:CC_Inters_1}-\ref{fig:CC_Inters_4}. In this section, formulas for the intersection area of these two circles for these four cases will be derived. The different cases in Figures \ref{fig:CC_Inters_1}-\ref{fig:CC_Inters_4} are defined as:
\begin{equation}
Case=\begin{cases}
0, & \mbox{$d > R_1+ R_2$} \\
1, & \mbox{$R_1 < d < R_1+ R_2$} \\
2, & \mbox{$d < R_1 < 2R_2 $} \\
3 & \mbox{$d < R_1, R_1 > 2 R_2$}
\end{cases} \label{eqn:cases_all},
\end{equation}
In (\ref{eqn:cases_all}), $d$ is the distance between centers of the two circles and is defined by
\begin{equation}
d=|C_1-C_2|, \label{eqn:distance_centers}
\end{equation}
where $C_1$ and $C_2$ are the coordinates of centers of the two circles.
(\ref{eqn:S_2_Inters}) gives the intersection area of two circles for different cases: $a$ and $b$ used in (\ref{eqn:S_2_Inters}) are distances between points $C_2,C_3$ and $C_3,C_4$, respectively. Points $C_2$ , $C_3$ and $C_4$ are shown in Figures \ref{fig:CC_Inters_2} and \ref{fig:CC_Inters_3}.
\begin{equation}
S_{intersect}=\begin{cases}
0 , & \mbox{Case 1} \\
cos^{-1}(\frac {a}{R_1})R_1^2-ab\\+cos^{-1}(\frac{d-a}{R_2})R_2^2-b(d-a),& \mbox{Case 2, Case 3} \\
\pi R_2^2, & \mbox{Case 4} \\
\end{cases} \label{eqn:S_2_Inters}
\end{equation}
where
\begin{equation}
a=\frac {-R_2^2+R_1^2+d^2}{2d}, \label{eqn:eqn_for_a}
\end{equation}
\begin{equation}
b=\sqrt{R_1^2-a^2}, \label{eqn:eqn_for_b}
\end{equation}
\begin{figure}
\centering
\includegraphics[scale=0.34]{Intersect_Model_Case1}
\caption{Intersection of two circles, Case-1} \label{fig:CC_Inters_1}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.36]{Intersect_Model_Case2}
\caption{Intersection of two circles, Case-2} \label{fig:CC_Inters_2}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.5]{Intersect_Model_Case3}
\caption{Intersection of two circles, Case-3} \label{fig:CC_Inters_3}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.5]{Intersect_Model_Case4}
\caption{Intersection of two circles, Case-4} \label{fig:CC_Inters_4}
\end{figure}
\section{Intersection Area Calculation of Three Circles}
\label{sec:ThreeCircIntersection}
In order to properly calculate the region where CRE UEs are located, intersection area of three circles should be calculated. According to our system model, the red colored area in Fig. \ref{fig:CC_Inters_3Circ} should be obtained. Using definitions shown in Fig. \ref{fig:CC_Inters_3Circ}, the red area $S_{Int,3}$ can be expressed as
\begin{equation}
S_{Int,3}=2(S_1+S_2)
\label{eqn:Area_Sall}
\end{equation}
where
\begin{figure}
\centering
\includegraphics[scale=0.37]{Intersect_3Circ}
\caption{Intersection Area of 3 Circles} \label{fig:CC_Inters_3Circ}
\end{figure}
\begin{equation}
S_1=\int_{P_{1,x}}^{P_{2,x}} \sqrt{-(x-C_{1,x})^2+R_{CRE}^2}+ C_{1,y} dx
\label{eqn:Area_S1},
\end{equation}
\begin{equation}
S_2=\int_{P_{2,x}}^{P_{3,x}} \sqrt{-(x-C_{2,x})^2+R_{Macro}^2}+ C_{2,y} dx
\label{eqn:Area_S2}
\end{equation}
In (\ref{eqn:Area_S1}), (\ref{eqn:Area_S2}), $P_{1,x}$, $P_{2,x}$, $P_{3,x}$ are the $x$ coordinates of the points $P_1(X,Y)$, $P_2(X,Y)$ and $P_3(X,Y)$, respectively. $P_1(X,Y)$ is the point where two adjacent CRE circle intersect, $P_2(X,Y)$ is the point where Macro circle and CRE circle intersects, $P_3(X,Y)$ is the point where $x$ axes and Macro circle intersects. $C_{1,x}$, $C_{1,y}$, $C_{2,x}$ and $C_{2,y}$ are the $x$ and $y$ coordinates of the centers of Macro and CRE circles respectively. $R_{Macro}$ and $R_{CRE}$ are the radius values of Macro and CRE circles. All these points and variables are shown in Figure \ref{fig:CC_Inters_3Circ}.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,466
|
package com.orpc.example.client;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
import org.springframework.beans.factory.annotation.Autowired;
import org.springframework.web.bind.annotation.PathVariable;
import org.springframework.web.bind.annotation.RequestMapping;
import org.springframework.web.bind.annotation.RestController;
import com.orpc.client.RpcClient;
import com.orpc.example.client.command.UserCommand;
import com.orpc.example.domain.User;
@RestController
@RequestMapping("/user")
public class ClentController {
Logger logger = LoggerFactory.getLogger(this.getClass());
@Autowired
private RpcClient rpcClient;
@RequestMapping("/get/{id}")
public String get(@PathVariable("id") String id) {
User user = new User();
// 成功
for (int i = 0; i < 30; i++) {
UserCommand userCommand = new UserCommand(rpcClient, Long.parseLong(id));
user = userCommand.execute();
logger.info("{}", user);
}
// 失败
for (int i = 0; i < 30; i++) {
UserCommand userCommand = new UserCommand(rpcClient, null);
user = userCommand.execute();
logger.info("{}", user);
}
//
for (int i = 0; i < 10; i++) {
UserCommand userCommand = new UserCommand(rpcClient, Long.parseLong(id));
user = userCommand.execute();
logger.info("{}", user);
}
return "ko...";
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,562
|
{"url":"https:\/\/mathproblems123.wordpress.com\/2015\/12\/13\/print-your-matlab-models-in-3d\/","text":"Home > matlab, Uncategorized > Print your Matlab models in\u00a03D\n\n## Print your Matlab models in\u00a03D\n\nThe emergence of 3D printers opens a whole new level of creation possibilities. Any computer generated model could be materialized as soon as it can be transformed in a language that the 3D printer can use. This is also the case with objects and structures which emerge from various mathematical research topics. Since I\u2019m working on shape optimization problems I have lots of structures that would look nice printed in 3D. Below you can see an example of a 3D model and its physical realization by a 3D printer.\n\nI want to show below\u00a0how can \u00a0you can turn a Matlab coloured patch into a file which can be used by a 3D printer. The first step is to export the Matlab information regarding the position of the points, the face structure and the colours into an obj file format. This is not at all complicated. Vertex information is stored on a line of the form\n\n$\\displaystyle v\\ x\\ y\\ z\\ R\\ G\\ B$\n\nwhere ${v}$ is exactly the character ${v}$, ${x,y,z}$ give the coordinates of the points and ${R,G,B}$ give the colour associated to the point in the RGB format. The face information can be entered in a similar fashion:\n\n$\\displaystyle f\\ t_1\\ t_2\\ t_3$\n\nwhere ${t_1, t_2, t_3}$ are the indices of the points in the corresponding face. Once such an obj file is created, it can be imported in MeshLab (a free mesh editing software). Once you\u2019re in MeshLab you should be able to export the structure into any file format you want, which can be understood by a 3D printer (like STL). Once you have the stl file, you can go on a 3D printing website like Sculpteo and just order your 3D object.","date":"2018-03-21 18:32:55","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\": 0, \"img_math\": 7, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2508295178413391, \"perplexity\": 502.5197036111836}, \"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-2018-13\/segments\/1521257647681.81\/warc\/CC-MAIN-20180321180325-20180321200325-00082.warc.gz\"}"}
| null | null |
Praise for
**THE FIRST BILLION**
"A fast-paced financial thriller... Reich has been compared to John Grisham: He offers the same page-turning David vs. Goliath plots."
— _USA Today_
"Reich does for finance what John Grisham does for the law."
— _San Francisco Chronicle_
"There has been no shortage of writers aspiring to be the John Grisham of Wall Street.... Reich deserves the Grisham mantle... the plot is so suspenseful, the dialogue so believable... you can already envision The First Billion: The Movie."
— _The New York Times_
"Engrossing... destined for a big readership."
— _The Wall Street Journal_
"The high-testosterone worlds of fighter pilots and investment bankers collide in this briskly paced thriller packed with genuine surprises... deliciously plausible, and the short chapters build a DSL-speed momentum."
_—Entertainment Weekly_
"If you want a high-concept espionage, it doesn't get much better than this."
— _Booklist_
"Compelling."
— _Library Journal_
"Gripping."
— _Fort Worth Star-Telegram_
"Suspense crackles.... Reich's talent as a chills-and-thrills storyteller looms large."
— _The Virginian-Pilot_
"Breakneck pace... [Reich's] characters, even the minor ones, are fascinating blends of genius, humor and wickedness.... Reich's new novel can't be beat."
— _Winston-Salem Journal_
"A sure bet... [Reich] is keeping to his lofty standards."
— _Times Record News_ (Wichita Falls, Tex.)
Praise for
**THE RUNNER**
_"The Runner_ is an intriguingly crafted cat-and-mouse hunt.... Move over, Jack Higgins and Robert Ludlum, Reich has grabbed hold of your genre and made it sing."
— _San Francisco Examiner_
"Wonderful... Reich evokes the fascinating world that existed between the hot war and the cold war—he is a master of atmosphere and detail."
—Nelson DeMille, author of _The Lion's Game_
"Irresistible."
— _The Wall Street Journal_
"Keeps pages turning... _The Runner_ confirms all the promise Reich showed in _Numbered Account."_
—Chicago Tribune
"Compelling... well-crafted."
— _Rocky Mountain News_
"Reich is good news for insomniacs who need an excuse to stay up till the wee hours."
— _Daily News_ (New York)
"Compelling... a classic story of good vs. evil... Reich is at his best, and indeed is one of the best, at creating a powerful setting.... [He] masterfully explores one of the oldest philosophical questions: Should good people resort to evil to destroy a greater evil?"
— _Austin American-Statesman_
"[Reich's] intimate knowledge of international intrigue presents the reader with a genuine insider's look at diplomacy, reminiscent of Le Carré."
— _Booklist_
"Keeps the pages turning."
— _Detroit Free Press_
"This is thriller-writing on the grand scale."
— _The Denver Post_
"Reich skillfully keeps us guessing."
— _Chicago Sun-Times_
"Intricately plotted."
— _The Cincinnati Enquirer_
Praise for
**NUMBERED ACCOUNT**
"A smart and sophisticated financial thriller... Wonderfully credible."
— _The New York Times_
"Chilling detail, suspense and intrigue."
— _The Denver Post_
"An excellent debut thriller... Reich has written a gripping tale of murder, money, and immortality."
— _Publishers Weekly_ (starred review)
"The beauty of the book is in its realistic and complex evocation of the small moral dilemmas of corporate man.... Reich's story, along with his insight on how Swiss banks work, makes for a fine, solid book. Big story, big engagement, big entertainment."
— _Newsday_
"A taut thriller."
— _Chicago Tribune_
"While the tension crackles like crisp new banknotes, Reich presents a fascinating view of the super-secret Swiss banking establishment.... _Numbered Account_ is an insider's take readers can bank on."
— _People_ (Page-Turner of the Week)
"Fast-paced... compelling, rich with intrigue and suspense."
— _San Francisco Chronicle_
Fast-paced, hard to put down... Reich knows the workings of the [banking] industry and its capital inside out—and it shows."
— _Business Week_
**A LSO BY CHRISTOPHER REICH**
Numbered Account
The Runner
The Devil's Banker
THE FIRST BILLION
A Dell Book
PUBLISHING HISTORY
Delacorte hardcover edition published September 2002
Dell international mass market edition published February 2003
Dell domestic mass market edition / August 2003
Published by Bantam Dell
A Division of Random House, Inc.
New York, New York
This is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously. Any resemblance to actual persons, living or dead, events, or locales is entirely coincidental.
All rights reserved
Copyright © 2002 by Christopher Reich
Library of Congress Catalog Card Number: 2002022254
No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, or by any information storage and retrieval system, without the written permission of the publisher, except where permitted by law. For information address: Delacorte Press, New York, New York.
Dell is a registered trademark of Random House, Inc., and the colophon is a trademark of Random House, Inc.
eISBN: 978-0-440-33432-3
v3.1_r1
_For my daughters, Katja and Noelle, with love_
# Contents
_Cover_
_Other Books by This Author_
_Title Page_
_Copyright_
_Dedication_
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 20
Chapter 21
Chapter 22
Chapter 23
Chapter 24
Chapter 25
Chapter 26
Chapter 27
Chapter 28
Chapter 29
Chapter 30
Chapter 31
Chapter 32
Chapter 33
Chapter 34
Chapter 35
Chapter 36
Chapter 37
Chapter 38
Chapter 39
Chapter 40
Chapter 41
Chapter 42
Chapter 43
Chapter 44
Chapter 45
Chapter 46
Chapter 47
Chapter 48
Chapter 49
Chapter 50
Chapter 51
Chapter 52
Chapter 53
Chapter 54
Chapter 55
Chapter 56
Chapter 57
Chapter 58
Chapter 59
Chapter 60
Chapter 61
Chapter 62
Chapter 63
Chapter 64
Chapter 65
Chapter 66
Chapter 67
Chapter 68
_Epilogue_
_Acknowledgments_
# 1
#
**_Y OU ARE MILLIONAIRE?"_** she asked.
"Me?" Grafton Byrnes pointed a finger at his chest. "No. I'm afraid not."
"Yes," she insisted, adding a coy smile. "You are millionaire. I can tell. You have nice suit. Beautiful tie. You are confident. It is clear. You are millionaire."
Byrnes unglued his eyes from the leggy blond who'd taken a seat at the bar next to him and looked around the room. The place was called Metelitsa, and it was a restaurant, nightclub, and casino rolled into one, located on the Novy Arbat in the center of Moscow. Red curtains blocked out the summer evening's glare. White tablecloths, smoked mirrors, and croupiers in black ties lent the room a touch of class. But one sniff told Byrnes different: the smoke, the perfume, the heady mix of expensive liquor and easy morals. He could recognize a cathouse by scent alone.
"I'm successful," he said, curtly. "Nothing special."
"You are _very_ successful, I think. Yes, a millionaire." She pronounced the word— _mee-lone-air_ —and her Slavic accent and grave delivery lent the word a patina of its foregone luster. "You would like to buy me drink?"
"Sure," he said, before he could ask himself what he was getting himself into. "What'll you have?"
"Vodka. On rocks with twist of orange."
"Coming right up."
Byrnes was finding it increasingly difficult to keep his eyes off the woman next to him. To call her gorgeous would have been an injustice. She was no more than twenty-one, with white blond hair, satin blue eyes, and the kind of pouty lips that his ex-wife called "bee-stung" and that no amount of collagen injections could reproduce. Her dress was black, short, and tight; her nails were lacquered a rich maroon. But it was her bearing that Byrnes found irresistible: the inquisitive tilt of the head, the brazen posture, the adventurous twinkle to the eyes that seemed to say, "Dare me—I'll try anything." In short, she was every middle-aged divorcé's idea of a fitting companion.
"Bartender!" As Byrnes shifted on his seat to get the barkeep's attention, he inadvertently nudged the man next to him. _"Izvinitye,"_ he said, offering a smile. Excuse me.
The man looked Byrnes up and down, then rose from his stool. He was six four, about two twenty, with a Marine's crew cut and a neck the size of a fire hydrant. He had a buddy next to him who looked like he'd fallen out of the same tree. Byrnes had been warned about guys like this. "Flat tops," they were called. Enforcers for the Russian _mafiya_ , or more politely, point men for the Russian business elite.
_Be careful_ , Byrnes's best friend had told him. _Moscow isn't Paris or Zurich or Rome. It may look like a European city, but it's not. You're in Russia. The whole country is in the shithouse. Two percent of the people are making a fortune and the rest don't have a pot to piss in. It's dangerous over there_.
"Excuse _me,"_ the Russian replied, in decent English. "I hope I not disturb you and pretty lady."
"No," said Byrnes. "My fault. Again, I'm sorry. Let me buy you a drink. We'll call it even."
"No need," said the Russian, with grating politeness. "Have nice evening." He made a show of adjusting his blazer and retook his place. Only a blind man would have missed the nickel-plated revolver nestled beneath his arm—a .357 Colt Python with a pearl handle, if Byrnes wasn't mistaken.
Turning back to the girl, Byrnes found a round of drinks on the counter. Okay, he said to himself, let's start over again. And raising his glass, _"Na Strovye."_
_"Na Strovye."_ She took a sip, then leaned forward and gave him a lingering kiss on the cheek. "My name is Svetlana."
"I'm Graf," he said, knocking back the entire drink. "Good to know you."
"You speak Russian. Why you not tell me so before?"
_"Nemnogo,"_ he said. Just a little. The Air Force would be proud of him for having remembered as much as he did. He also knew how to say, "I am an officer,"
"My serial number is...," and a few choice obscenities.
"I no like Russian men," Svetlana confided in his ear. "So arrogant."
"Me neither," he complained. "So big."
She laughed. "Tell me, Graf, why you are in Moscow?"
"Business," he answered.
_"Beez-ness?_ What do you do?"
Byrnes shrugged, looking away. "Nothing interesting. Just some routine stuff."
His response couldn't have been further from the truth. He'd arrived earlier that afternoon on an emergency visit. All very hush-hush. Forty-eight hours in country to check out the operating equipment of Mercury Broadband, a multinational Internet service and content provider his company was set to bring public in a week's time. Questions had surfaced regarding the firm's Moscow network operations center—namely, whether it owned all the physical assets it claimed to: routers, switches, servers, and the like. He was to find the facility, verify that it contained equipment necessary to provide broadband services to its publicized customer base of two hundred thousand people, and report back.
The IPO, or initial public offering, of shares in the company was valued at two billion dollars, and nothing less than his firm's continued existence depended on what he discovered. A green light meant seventy million dollars in fees, a guarantee of fee-related business from Mercury down the road, and a rescue from impending insolvency.
Shelving the offering meant death—defined either as massive layoffs, the sale of the firm to a larger house, or in the worst case, shuttering up the shop and putting a "Gone Fishing" sign in the window. Permanently.
"And what you do for business?" she asked.
"Investment banking. Stocks. Bonds. Like Wall Street, you know?"
"So, I am right," she announced proudly, dropping a hand onto his leg and allowing it to linger there. "You are millionaire."
"Maybe," he said. "Maybe not. Anyway, it's not polite to talk about money."
"I think you are wrong. Money is sexy," she said, winking. "Aphrodisiac, I think."
He ordered another drink, and when it came he took a greedy sip. He was getting that warm, fuzzy feeling, and liking it. From his perch at the bar, he overlooked a parquet dance floor and a small casino with slot machines and a half dozen gaming tables. A few flat tops had staked out positions at the craps pit. They were dressed to a man in snazzy black suits, open collars, and gold chains. Crisp American greenbacks were exchanged for stacks of blue and silver chips. No one was playing with less than five thousand dollars. Dice tumbled across the green baize tables. Raucous voices lofted across the room, spirited, cajoling, violent. The staccato shouts had a serrated edge and lent the place an aggressive buzz. At five past nine on a Tuesday night, the joint was beginning to jump.
"And why, Graf, you come to Metelitsa?" Svetlana's hand had moved higher on his leg. A single finger danced along the crease of his trousers. "To see me, maybe? See Svetlana?"
She was staring at him, the magnetic blue eyes commanding him nearer. Her lips parted, and he saw a moist band of pink flashing behind the dazzling teeth. He could taste her warm, expectant breath. The scent of her hair, lilac and rosewater, drifted over him... enticing him... seducing him.
"Yes... I mean, no... I mean..." Byrnes didn't know what he wanted to say. He wasn't sure whether it was the vodka or just Svetlana, but suddenly he was decidedly tipsy. He was having trouble focusing, too. Placing a hand on the bar, he stood up unsteadily, bumping once more into the thug next to him.
"Watch it!" barked the linebacker.
_You're in Russia. It's dangerous over there_.
"Sorry, sorry." Byrnes raised his hands defensively. He turned toward Svetlana. "Excuse me. I'll be right back." He mumbled the words "rest room" and "freshen up."
"I help you," she said, resting a hand on his waist. "We go upstairs together. I show you way."
"No, no. I'm all right, really. Where do I go?"
"Up. To right side." She pointed the way, then wrapped her arms around him. "You no leave Svetlana?"
Suddenly, she didn't look so much the unapproachable Russian ice princess as an insecure twenty-year-old frightened she might lose her evening's pay.
"No," he said. "I no leave Svetlana. I come right back." Jesus, now he was even talking like her.
He set off to the rest room, lurching along the bar before recovering his sea legs and guiding himself up the stairs. Inside the john, he turned the tap on full and took turns slapping cold water on his face and taking deep breaths. A minute passed and he began to feel better. That was some vodka he was drinking. Two doubles and he was on his ass. He promised himself he'd have a word with the hotel concierge, tell him he had something different in mind when asking about a place where a gentleman could get a few drinks and some dinner.
Laying both hands on the sink, he took a close look at himself in the mirror. "Come on, kid," he whispered. "Snap out of it."
Staring back was a vital, handsome father of two teenage children gracefully approaching middle age. Strands of silver streaked a generous head of black hair. Fatigue shadowed his flinty eyes. His bold, clefted chin, the brunt of a thousand jokes, evidenced a slight but noticeable sag. Squinting, he wondered what had happened to the gallant airman who had flown his nation's fighters in two armed conflicts, the able pilot who had deadstick-landed a flamed-out F-15 and bailed out over open ocean after he'd lost his hydraulics.
"Still here," tolled a fighting voice deep within him. "Just get lost once in a while."
"You are a huckleberry," he said aloud, angered by his lack of self-restraint. "Your little lady friend probably had your drink spiked. Five'll get you ten her big buddy is waiting downstairs at this very instant to give you his best regards. You came to do a job, not fuck around. Get thyself out of here. Now!"
Five minutes later, Grafton Byrnes left the rest room. His tie was straightened, if a little wet. His jacket was buttoned. His wooziness had faded, replaced by a whopping headache and an ironclad desire to get as far from the premises as possible. Walking to the head of the stairs, he glanced down at the bar. Svetlana was deep in conversation with the two bullies who'd been sitting next to him.
_Idiot!_ he thought. It really was a put-up job.
Spinning on his heel, he headed to the dining room. An illuminated sign along the far wall read "Exit." He snaked through the tables, bumping into diners, slowing only to offer an apology. Reaching the emergency exit, he threw open the door and found himself standing at the top of a fire escape. He put a tentative foot on the rusted landing. The entire structure swayed and groaned. The thing had been built before Stalin had even thought of the words "five-year plan."
_Retreat. Go to plan B_.
But even as he turned to reenter the building, the door slammed shut. There was no handle or doorknob to gain entry.
Byrnes swallowed hard, a bolt of unease creasing his shoulders. He wasn't sure if he was frightened or exhilarated, but a moment later he was attacking the fire escape. Rung by rung, he descended the rickety structure, his steps cautious but not unsure. Six flights of stairs took him down three floors, and when he reached the ground he stood stock still, amazed the thing had actually held together.
He was still dusting the rust off his hands when the emergency exit flung open and his favorite flat top emerged onto the landing, six floors above. _"Allo_ , Graf," the Russian called. "Stop. I want to talk. You owe Tatiana money."
_Tatiana?_ What happened to Svetlana?
It took Byrnes less than a second to decide to get the hell out of there. He might owe Svetlana, or Tatiana, or whatever her real name was, an apology for his sudden departure, but he certainly didn't owe her any money. And even if he did, he didn't want to give it to her pimp. Somehow he didn't peg the guy as a believer in win-win negotiation.
A deep breath and Byrnes was off, running down the alley as fast as his Bally loafers would carry him. He didn't look back to see if the _mafiya_ goon was following him—the angry creaking of the fire escape told him all he needed to know on that account. The sky was a pale blue, softening to azure. A crescent moon hung in the sky. The air smelled of fried potatoes and automobile exhaust. Rounding the corner of Metelitsa, he hightailed it through the parking lot toward the street.
The Novy Arbat had been built in the early sixties as Khrushchev's answer to Manhattan's Fifth Avenue. Four lanes of traffic flowed in either direction, lined by a succession of nondescript offices and run-down apartment buildings, the kind where air conditioners dripped coolant from jury-rigged perches and half the windows were caked with grime. Maybe the Bowery, carped Byrnes, but Fifth? No way.
Reaching the street, he raised a hand in the air.
"Taxi!"
It was a Russian tradition for ordinary drivers to offer their services as taxis in exchange for a few dollars, marks, or francs. In a heartbeat, a red Lada had pulled over and Byrnes was in the passenger seat.
"Hotel Baltschug," he said, then a second later, "No, wait." Digging his hand into his pockets, he found the address of the network operations center he was supposed to visit. If this was Russia, he wanted to get the hell out of it as quickly as possible. He checked the sky again. Plenty of light remained to get his job done. Finish tonight and he could catch the first plane out in the morning. He'd be back in San Francisco at four and in the office by five. Plowing through his E-mails would never be so much fun.
"You know Rudenev Ulitsa?"
_"Rudenev?"_ The driver appeared confused, then it came to him. _"Rudenev! Da. Da."_ He was a small man, near sixty, with a Tatar's eyes and a hairline that started about an inch above his eyebrows. Living proof the Mongols had reached the gates of Moscow.
"Rudenev Ulitsa 99," Byrnes said, yanking a hundred-dollar bill out of his wallet and handing it to the man. "And hurry!"
Five seconds later, the Lada was barreling down the center lane of the Novy Arbat. Byrnes looked over his shoulder out the back window. Late-evening traffic had already closed in around the car. For a moment, he was able to glimpse the parking lot in front of Metelitsa. A long line of cars was pulled up to the valet. Men and women ambled toward the entrance. He saw no sign of his newest friend.
"Rudenev. How long?"
The driver held up a finger. "One hour."
Byrnes sat lower in his seat, catching his breath.
He knew it had been a lousy idea to come to Russia.
# 2
#
**T HE EARLY-MORNING SKY WAS DARK**, a low cloud cover threatening rain as John Gavallan backed his Mercedes 300 SL "Gullwing" from the garage of his home in Pacific Heights and accelerated down Broadway toward his office in the heart of San Francisco's Financial District. It was a short trip: eight minutes in good weather or foul. At 4 a.m., the streets were deserted. The night owls had gone to bed; the early birds were only just beginning to rise. A fat drop of rain plopped onto the windshield, and Gavallan shivered. A week into June and he'd barely seen the sun. He recalled Mark Twain's quote about the coldest winter he'd lived through being the summer he'd spent in San Francisco, and smiled thinly. Normally, the prospect of another dreary day would have soured his mood. Reared in the southernmost nib of the Rio Grande Valley as he was, his blood had been boiled thin by the Texas heat, his soul stone-bleached by the subtropical sun. This morning, though, the stormy skies suited him. What better companion to the acid drizzle corroding the lining of his gut?
Gavallan drove the Mercedes hard, shifting down through the gears, enjoying the engine's finely tuned growl, loving the communion of man and machine. He cracked the window an inch, and a blast of sea air freshened the car. Directly ahead lay the bay, and for a moment he lost himself in its blind expanse, wondering how much time had passed since so much had ridden on a single day's outcome. The answer came immediately. Eleven years and five months. It was the calendar against which he measured his life. There was before the Gulf War and after the Gulf War. And sinking deeper into the black bucket seats, he felt himself strapped inside the cockpit of his F-117 Nighthawk, the turbofan engine rumbling to life beneath him, G suit tight across the waist, hugging his legs and his back. He recalled, too, the shortness of breath beneath the confident smile, the tingling that had taken hold of his stomach as he gave the thumbs-up and taxied onto the runway for takeoff that first night.
A tingling not so different from the one he felt this morning.
Shaking off the memory, Gavallan drove his foot against the accelerator, taking the sports car to seventy miles an hour. The rain hardened and a gust sheeted the windshield with water. Blinded, he downshifted expertly, braking as he crested Russian Hill. "Instrument conditions," he whispered, eyes scanning dials and gauges. A moment later, the wipers cleared the screen. Off to his right loomed the Transamerica Tower, a pale triangular needle framed by a score of steel and concrete skyscrapers. The buildings were dark, except for random bands of light encircling their highest floors. He glanced at the mute forms a moment longer, feeling a kinship with those already at their desks. He'd always thought there was something daredevilish about starting the workday at four in the morning, something not completely sane. It had the whiff of tough duty that had always attracted him, the raised bar of an elite.
At age thirty-eight, John J. Gavallan, or "Jett" as he was known to friends and colleagues, was founder and chief executive of Black Jet Securities, an internationally active investment bank that employed twelve hundred persons in four countries around the globe. Black Jet was a full-service house, offering retail and institutional brokerage, corporate finance advice, and merger and acquisition services. But IPOs had been the ladder it had climbed to prominence. Initial public offerings. The company had made its fortune in the technology boom of the late nineties and, to Gavallan's dismay, it was still suffering a financial hangover from those halcyon days.
Nine years he'd been at it. Up at three, to work by four, finished twelve hours later, fourteen on a busy day. Once, the days had passed with astonishing rapidity. Success was an opiate and mornings bled into evenings in a hazy, frenetic rush. Lately, the clock had assumed a less benign stance. Time meant money, and every month that passed with revenue goals unmet was another inch cut from Black Jet's financial tether.
Dropping a hand to the stereo, Gavallan spun the dial to National Public Radio. The 4 **A.M.** business report was under way, a summary of action on the world's major markets. God, let it be an up day, he thought. In Asia, the Nikkei and Hang Seng Indexes had closed higher, both with solid gains. In Europe, markets were divided, with the London FTSE, or "footsie," strongly ahead and the German DAX and French CAC 40 _("cack quarante")_ lagging only slightly below their highs. But what about New York? He'd been in the business long enough to know there was only one market that really counted. A moment later he had his answer. At seven-oh-five Manhattan time, the futures markets were up sharply, presaging a solid opening in just over two hours.
"Nice!" he said aloud, landing his palm against the varnished oak steering wheel for good measure. It didn't take a genius to know it was best to sell in an up market. But just as quickly his exuberance faded, replaced by a cold apprehension. If all went well, he could celebrate at the end of the day. For now, though, he had to wait. Too many cards remained facedown on the table.
**T HE OFFICES OF BLACK JET SECURITIES** occupied the fortieth and forty-first floors of the Bank of America Tower, a fifty-two-story slab of red carnelian marble not dissimilar to Mies van der Rohe's Seagram Building in New York. The elevator opened, disgorging Gavallan into a brightly lit reception area. Sofas and chairs upholstered in Corinthian leather offset terra-cotta carpeting. A combed birch counter stood to the left, and behind it a seven-foot wall of polished black granite bearing the firm's name in silver matte letters.
_"Six days!"_
Gavallan slowed, turning to meet the source of the words.
"Six days," Bruce Jay Tustin repeated, cresting the interior staircase that led from the trading room on the floor below. "The countdown for Mercury is on. T minus a hundred twenty-two hours. Fuckin' A, bubba!" Tustin was the firm's head of syndicates as well as a member of the executive board. He was forty-five years old, short, and svelte, a bantamweight clad in a Brioni suit. He had a boxer's mug, too—the broad forehead; the flat, broken nose; the sly, determined cast to the eyes.
"How's the book?" Gavallan asked. "Holding strong?" The "book" referred to the nimble piece of software that held all orders and indications of interest for the new issue.
"A few cries in the jungle, but we're working to calm the savages."
Gavallan sensed there was more to it. "Any of the major players backing out?"
"Just one so far. Mutual Advantage in Cincy canceled their order. Said they wanted to put the money into bonds. Doesn't look like anyone else is taking the rumors seriously. The market wants this deal to happen."
"Let's stop it there, Bruce. I don't want a snowball effect. We're standing behind the deal one hundred and ten percent. Keep putting the word out: Mercury is hunky-dory."
Tustin nodded obediently. "You find out who it is bad-mouthing us? Not one of your girlfriends, is it?"
Gavallan shook his head, thinking that someday Tustin's mouth was going to kill him. "Not yet. But we're looking."
"Ah, that's right, I forgot. _She left you_. Hang in there, kid. You're young yet." Tustin clapped Gavallan on the back. Features brightening, he added, "Opening's looking strong, _Jefe_. The market's getting primed for Mercury. Six days. Hoo-yeah!" And pumping his right fist in the air, he spun and bustled down the steps to the trading floor.
"Hoo-yeah," repeated Gavallan, but his parting smile disguised a pressing urge to get to his office. Walking briskly, he shifted his calfskin satchel to his left hand while withdrawing a set of keys from his pocket.
At first glance, he looked more the affluent bachelor than the driven executive. Tall and fit, he'd dressed for the day in his usual outfit: jeans, moccasins, and a faded chambray shirt, throwing on a navy cashmere blazer for good measure. He was finished with uniforms, be they dress blues or three-button worsteds from Savile Row. In the same disobedient spirit, he kept his sandy hair cut long, sure that it brushed his collar and hid the tops of his ears. His face was strong rather than handsome. Creases dimpled weathered cheeks. Wrinkles bracketed eyes hard and gray as agate. His nose was slim and straight, the boldest testament to his Scottish ancestry. His jaw was steadfast, and as usual raised an extra degree, as if he were trying to peer over the horizon. A pillar of the yacht club, you might guess. A regular at the nineteenth hole.
But a second look would give you pause. His gaze was direct, and when not combative, confrontational. His gait was compelling and hinted at tensions simmering within, some urgent, inner purpose. You would never, for example, stop him on the street to ask for directions. It was his hands, though, that gave him away. They were the hands of a brawler, large and callused, the knuckles swollen from long-ago fights. No Ivy Leaguer he, you might say, and take a step back. This one was hewn from rougher stock. This one had required polishing.
Even at this hour, the hallways were abuzz. The day's first conference call originated at four-thirty. Everyone present in the office at that hour—usually about sixty traders, analysts, and brokers—gathered in the company conference room to share earnings announcements, analysts' reports, and street gossip with branches in New York and London. Video cameras, color monitors, and microphones linked the participants, and for thirty minutes they hashed out anything that might boost a particular stock's price or knock it down. Information was the market's universal deity—rational, impartial, and above all merciless—and it was worshiped accordingly.
Inside his office, Gavallan turned on the light. A glance at his watch gave him ten minutes until the conference call began. Not bothering to unbutton his jacket, he sat at his desk and checked his E-mail. Seventy-four new messages had come in since yesterday evening. Hurriedly, his eyes scanned the flat panel screen. The usual brokerage recommendations: Buy Sanmina, hold Microsoft; so-and-so initiating coverage on Nortel. Delete. Delete. Delete. Notes from a few venture capitalists in the Valley. An invitation to a golf tournament in Vegas. "Don't think so," he muttered, hitting the delete key; he'd take his clubs out of storage when the world righted itself. A smattering of messages from his colleagues in the firm. He'd check these later.
"Byrnes, Byrnes, where are you, buddy?" He looked for Grafton Byrnes's handle but didn't see anything. "Damn it," he muttered, rocking in his chair.
He'd hardly slept, expecting his number two to call with an update on the trip to Moscow. At the least, he'd hoped for an E-mail. Finding nothing, he unlocked the top drawer of his desk and located a square slip of paper bearing the initials G.B. and a ten-digit number. He picked up the phone and dialed.
"Hotel Baltschug Kempinski. Good afternoon."
Gavallan snapped to attention. "Yes, good afternoon. I'd like to speak with one of your guests. Mr. Grafton Byrnes."
"One moment."
Where are you, my boy? he wondered, drumming his fingers impatiently on the desk. You're my ace in the hole. Pick up the goddamn phone and tell me everything's all right. Tell me I was a fool to worry and that we can put some champagne and caviar on ice for our European friends.
"Mr. Byrnes is not in the hotel."
"Very good," said Gavallan, though in fact he was curious as to why Byrnes hadn't finished his work yet. Drawing a manila file from his desk, he flipped open the cover. Inside lay the photographs—the reasons for Grafton Byrnes's last-minute trip.
The first showed the façade of a two-story building that could have been a warehouse or a manufacturing plant. A sign above the entry read "Mercury Broadband." The photo was captioned "Moscow Network Operations Mainstation." A second picture purported to show the building's interior: room after room packed with standard telephone switching equipment, circa 1950, gray rectangular dinosaurs sprouting black connector cables like unruly hair.
Founded in 1997, Mercury Broadband was the leading provider of high-speed Internet service in Russia, the Ukraine, Belarus, and the Czech Republic—an area that Gavallan, with his training as a Cold War jet jock, would forever think of as "the communist bloc." Through its network of coaxial cable, fixed wireless, and satellite relays, Mercury Broadband serviced over two million businesses and residential customers and had contracted rights to service an additional twenty-seven million. It also packaged and offered multimedia content and E-commerce in the form of Red Star, a multilingual portal similar to America Online that boasted over seven million subscribers.
But here was the good part: Not only had the company doubled its revenues each of the last three years, it had begun turning a profit as of third-quarter fiscal 2000. In six days, Black Jet Securities would take Mercury Broadband public on the New York Stock Exchange in an IPO set to raise two billion dollars. The seventy million in fees the deal generated was crucial to relieving Black Jet's worsening financial malaise. Every bit as important was the boost to the company's reputation a successful offering would bring. From regional mighty mite to international presence in one fell swoop.
Which brought Gavallan back to the photographs. He had another picture in his manila file, also purporting to show the interior of the Moscow network operations center. This one positively beamed with the latest in Internet hardware—Sun servers, Cisco routers, Lucent switches—and it was this photo he'd showed to his investors.
"I'd like to leave a message," he said. "Please tell him Mr. Gavall—"
"Mr. Byrnes is not in the hotel," the Russian operator interrupted.
"Yes, I heard you. If you don't mind I'd like to leave a—"
"No sir, you do not understand," cut in the operator again. "Mr. Byrnes has checked out."
"That's not possible. He's not due to return to the States until tomorrow. Please check again." And before the operator could protest, he shouted, "Do it!"
"Very well." The "sir" was distinctly missing.
Confused, Gavallan ran his eyes over the computer screen, reconfirming he hadn't received any E-mails from Byrnes. His instructions had been clear: Once Graf picked up something about Mercury— _good or bad_ —he was to let Gavallan know. Immediately.
"Sir? Our records indicate that Mr. Byrnes checked out of the hotel yesterday evening at eleven-thirty."
"Eleven-thirty? You're sure?"
Moscow was eleven hours ahead of San Francisco; 11:30 **P.M.** in the Russian capital meant lunchtime in the office. Byrnes had called in four hours before that, at around eight yesterday morning, to report that he'd arrived safely and would start his investigations the next day. The notion that he'd checked out without spending the night was as unsettling as it was absurd.
"Mr. Byrnes is no longer a guest with us," replied the operator. "If you'd like to speak with our general manager, I'd be happy to connect you."
"No. That won't be necessary."
_"Po Zhausta. Da Svidaniya."_
Gavallan put down the phone and strode to the window. For a long time, he remained still, looking out over the city. Through the rain, he could make out Telegraph Hill, and beyond it the bow lights of a supertanker advancing slowly out to sea. Farther to his left, pale red beacons glimmered atop the cable towers of the Golden Gate Bridge. Staring at the melancholy panorama, he experienced a sudden tremor, a shiver that rustled his spine and caused him to cross his arms and hug himself as if fending off a stern winter's breeze. It was the same yawn of anxiety that had passed over him two days earlier, when on a foggy Monday morning he'd first broached the idea of a trip to Moscow to Grafton Byrnes.
# 3
#
**S O YOU'VE SEEN IT?"** Gavallan had demanded as Grafton Byrnes entered his office.
"Yeah, I've seen it," answered Byrnes with a calm Gavallan did not share. "Not the best PR one of our deals has ever gotten, but not the worst, either."
"I'm not so sure. Timing couldn't be worse, that's for certain."
Byrnes strolled across the room with the easy authority that was his trademark. He was taller by an inch, dressed in a navy crew neck sweater over a white oxford button-down, brown corduroy slacks, and Belgian loafers polished to a spit shine. His face was craggy and lean, with eyes that appraised but never accused, and a smile that forgave all sins.
"Want something to drink? Pellegrino?" Gavallan spun in his chair and opened the compact refrigerator hidden in his credenza. "I've got one of those new lattes in a bottle. How 'bout that?"
Byrnes took up position behind him, peering over his shoulder. "Nothing with caffeine, thanks. I'll take a mineral water. No, no... one without any bubbles."
Gavallan handed him a bottle of Ozarka and selected an ice-cold can of Orange Crush for himself. He considered his teenager's sweet tooth his only vice. Vintage European automobiles, chilled Russian vodka, and Stevie Ray Vaughan playing the blues at excruciating volumes counted as passions, and were thus exempt.
"Skoal, brother," he said, lifting the can of soda pop.
"Skoal, my man."
It was a joke between Texans, "Skoal" being both an informal "Cheers" and the tried-and-true chewing tobacco of their youths.
Gavallan had known Grafton Byrnes his entire adult life. They had met at the Air Force Academy in Colorado Springs, where Byrnes had played regimental commanding officer to Gavallan's plebe. Every time Gavallan mouthed off, it was Byrnes who administered the punishment. A hundred push-ups on the deck. A thousand-yard sprint in shorts and tennis shoes through waist-high drifts of midwinter snow. Two hours of reciting the Uniform Code of Military Justice while doing Roman chairs against the commons room wall. If harsh, the abuse was well-intentioned. It was Byrnes's job to make sure Cadet John J. Gavallan made it through the Zoo, and to that end he tutored him in calculus, instructed him on how to properly hold his knife and fork, and taught him to iron a razor-sharp crease into his trousers.
Retiring from the Air Force a major, Byrnes had followed him to Stanford Business School, then to Black Jet Securities two years after its founding. He was pretty much Gavallan's older brother, and as close a friend as he could ever hope for.
"You know this guy, the Private Eye-PO?" Gavallan asked.
Byrnes shrugged, offering a wry smile. "I do now. Who is he exactly? Or should I say 'what'? Some sort of Internet gadfly?"
"You could say that. Calls himself the Robin Hood of the Valley's pink slip brigade. He spies on the rich to protect the poor."
"The poor being who?" smirked Byrnes. "The laid-off techies who can't afford their Beamer payments?"
"More like the average investor who lost his shirt when tech stocks took a beating."
"Oh, you mean our retail clientele. So he's the bastard responsible for the plunge in our commission revenues. Got it."
Outside, a blanket of fog sprawled across the Bay Area, a pea soup so thick Gavallan had trouble making out the gargoyles on the roof of the Peabody Building a hundred feet away. Rising from his chair, he circled the desk, swiveling the computer monitor 180 degrees so they could both read from the screen. As usual the Private Eye-PO's posting was written in a style somewhere between the Motley Fool and a fifties Hollywood tabloid.
_For weeks now, Wall Street has been in a lather for the $2 billion Mercury Broadband deal being brought to market by Black Jet Securities. Well, kids, your own Private Eye-PO has learned that the offering is fully subscribed, with plenty of savvy investors looking to get in on the action. Caveat emptor. Mercury is not what it appears. My own no less savvy gum-shoes swear to me that Mercury is only a shadow of its trumped-up self, and Red Star, a sheep in AOL's clothing. What do you expect from Black Jet Securities, itself a pretender to the throne? When will Mr. Gavallan learn? Black Jet can never be white-shoe. But, hey, friends, why listen when you can look? After all, isn't seeing believing?_
"You sure you don't know this guy?" asked Byrnes. "This stuff sounds almost personal. He had as much fun knocking you as he did Mercury."
"No one knows him," Gavallan replied testily. "That's his gig. He keeps a bag on his head while he goes around savaging companies. Mercury's not the first company he's skewered."
"I suggest we find him on the double and shut him up."
"I know a guy we can call. Does some work for the government. I'll get on it right away." Sighing, Gavallan turned away from the monitor, massaging the bridge of his nose with his thumb and forefinger. "Every time I read it I feel like I've been socked in the gut. This is not what we need right now."
"No, it's not," Byrnes agreed, "but it's what we got, so we deal with it and move on." His eyes narrowed with concern over a different matter. "You okay, kid? You look a little tired."
"Yeah, yeah, I'm fine. It's just this on top of all the other crap lately..." The words trailed off.
"If it's Manzini who's bothering you, forget it. You had to let his team go. They knew the rules. Around here you eat what you kill. We're not a bulge bracket firm that can rely on our granddaddy's clients to throw us some scraps. GM's not knocking down our door wondering if we might underwrite some debt for them. IBM isn't about to ask us to do a secondary offering. We have to go out and get it."
"Yeah," said Gavallan. "We make money the old-fashioned way— _we earn it."_
"Damn right," said Byrnes emphatically. "Don't beat yourself up over it. They were lucky you kept them on as long as you did. Half those guys were earning a base of three hundred. Look, the Internet vertical was dying. They didn't produce, they got canned. End of story. We're not running a charity here."
A "vertical" was banking jargon for a particular industry segment. The tech sector was divided into E-commerce, web infrastructure, optical equipment, software, and so on. Each vertical was assigned a team of bankers to service businesses operating in that sector. The team consisted of an equity analyst, a few capital markets specialists, the investment bankers who actually drummed up the business, and two or three associates to do the grunt work.
"I'm well aware of that," said Gavallan. "Next time it can be your turn to fire the guy you've been going to Warrior games with for five years. Carroll Manzini's a friend."
But he could see from Byrnes's skeptical expression that he wasn't buying. Byrnes had a more unyielding attitude toward business. You performed or you got cut. That simple. He'd governed by the same draconian principles when Gavallan had served under him at Stealth training in Tonopah, Nevada, the two-thousand-square-mile cut of yucca and scrub known to conspiracy buffs as Area 51. The funny thing was that back then Gavallan had been happy to live by those rules. He was as confident of his own skills as he was disdainful of the saps who didn't make the grade.
Strangely, as chief executive of Black Jet Securities, he was unable to demand of his employees the uncompromising standards he asked of himself. He regretted the most recent firing of twenty-six of his executives and couldn't help but feel in some way responsible for their inability to generate income for the firm. So what if financing activity in the Internet sector had dried up as quickly as a summer squall? That not a single IPO had been done for an Internet play in months? Or that every other bank on the street had slashed their staffs long before?
Frustrated, Gavallan looked around his office. It was large but modest, with tan carpeting, textured ecru wallpaper, and comfortable furniture arranged to promote informal discussions with clients. A floor-to-ceiling window ran the length of the room and gave the office a stagelike feel. The plummeting vista was nothing short of spectacular, and nearing the window more than one client had professed an incipient acrophobia. A second glass wall ran along the interior corridor. When Gavallan was alone at his desk, he made every effort to keep the blinds open, as well as the door. He detested the trappings of authority and wanted everyone at Black Jet to know he was available at all times.
"Maybe you're right," he conceded. "I'm just lousy at that kind of thing. It's easier to hire a man than to kick him out on his tail."
"Oh, but if the world were a fair place," said Byrnes, bowing an imaginary violin.
"Get out of here," said Gavallan. "Come on, cut it out. You look really stupid doing that."
He knew his ideas about an employer's duty were old-fashioned, but he stuck with them nonetheless. His father had worked on the cutting line at Martinez Meats in Harlingen, Texas, for forty years. Forty years hacking the hindquarter off a flayed steer's carcass, eight hours a day, five days a week, in a fluorescent-lit factory that breathed blood and sweated ambition, where temperatures routinely soared to a hundred degrees during the six-month summer. The Martinez family might not splurge on luxuries like air-conditioning and they certainly didn't pay much. (Gus Gavallan's weekly salary of $338 came tucked in a wax-paper envelope delivered Monday mornings at nine o'clock sharp, so that the younger men wouldn't drink their paycheck over the weekend.) But neither did they fire their staff. In those forty years, Martinez Meats never let go a single man or woman except for absence, tardiness, or public inebriation, and his father's devotion to the Martinez family was nearly religious.
Black Jet had barely been in business nine years and Gavallan had already fired, let go, laid off, made redundant—however you wanted to put it—over a hundred men and women, including the latest casualties, Carroll Manzini's tech-team of banking superstars, twenty-six strong. The thought pained him. He wanted to believe that the bond between a man and his employer went beyond business to family. It was a social contract that exchanged loyalty and service for welfare and security. Maybe he was foolish. Maybe at seventeen thousand dollars a year you had a right to that kind of paternalistic relationship. At half a million bucks plus bonus you were on your own.
Byrnes laid a hand on his shoulder and gave it a squeeze. "Toughen up, kid," he said. "Look at you. Your chin's falling into your neck, your ass is dragging, and God knows you need a haircut. And that whining... Christ, you sound like a dooly crying during Hell Week. The Gavallan I knew was a rock. You didn't say a goddamn word that day up at Alamogordo. Not before, during, or after. A fuckin' rock, man."
"Easy to be a rock when you're border trash that doesn't know any better," retorted Gavallan, but already he was smiling, feeling a little better. He was remembering the day in Alamogordo. August 2, 1986. Lead-in-Fighter Training.
**T HE WEATHER HAD BEEN PERFECT,** hot and mostly clear, with only a few thunderheads to keep away from. The two of them were up in a T-38 jet trainer, Byrnes already a combat-tested pilot, the instructor, and Gavallan his student. After an hour of practicing basic fighter maneuvers, the two were heading in for landing, making plans to rendezvous at the O-club for a few beers and a steak after debrief. Then— _Bam!_ —without warning, the jet's turbine engine had exploded, severing the hydraulic main, ripping off a chunk of the tail, and sending the plane into wild, uncontrollable gyrations at four hundred knots. One second they were flying level, the next they were pitching wildly, rolling and yawing, the burnt scrub of New Mexico changing places with the powder blue sky with sickening frequency.
Standing in his office, Gavallan jolted. Sixteen years after the fact, he could hear the whine of the disintegrating engine, the whoosh of the violated air as it battered the jet. Mostly, he recalled the adrenaline rush, the iron fingers grasping his heart and crushing it mercilessly.
"Everything's copacetic," had come Byrnes's voice, calm as a Sunday morning. "Just let me take care of this fire and we'll be jim-dandy to land." And in the same unbothered delivery, he'd begun ticking off the measures to regain control of the plane—depress rudder, bring up left aileron, release the stick to let the nose find its way down.
But strapped into the front seat, Gavallan knew damn well everything was not copacetic. His eyes were glued to the altimeter, watching it tick down from four thousand feet at a hundred feet a second. He could feel the G forces increasing, driving him deeper into his seat, nailing his arms to his side. As he counted the seconds until they augered in, his hands automatically reached for the side of his seat, searching for the ejection handles. But when he found them, he immediately let them go. It was an act of betrayal. Of disbelief. No, it was worse. It was a pilot's cardinal sin: the acknowledgment of his own fallibility.
The altimeter spun merrily counterclockwise, passing eight hundred feet, seven hundred, six.... The plane came out of its death spiral, the nose pointed straight down toward the arid landscape. Gripped with a quiet terror, he waited for the nose to rise. A series of prayers stumbled from his lips. When that failed him, he swore silently. _Come on, you son of a bitch. Come up. Just a little, you mutha, just a little!_
Slowly, the plane righted itself. The nose inched up, the wings leveled to the horizon. And as the ground zipped beneath their wings close enough to slap a long-horn's rump, Byrnes chuckled, as if the whole escapade had been engineered for Gavallan's amusement.
"What'd I tell you, rookie?" he asked.
After landing, the two accomplished their postflight inspection of the debilitated aircraft. A four-by-four-foot section of crumpled metal dangled from the tail, secured by an aluminum thread no wider around than a pencil. Viewing the damage, neither Byrnes nor Gavallan commented. They simply exchanged glances and shrugged their shoulders. That night, "everything's copacetic" entered lore, meaning, of course, just the opposite—that nothing could be more screwed up.
"Okay, okay. I get the message," said Gavallan, walking to his chair and sitting down. "Slap me around a little if I start feeling sorry for myself again."
"Yes sir. You're the boss."
Gavallan eyed Byrnes suspiciously. Sometimes he wasn't so sure. "Look, the pictures of Mercury's network operations center are fakes. I know that company inside and out. The only question is what we're going to do about it."
"You've talked to Kirov?"
"He called me a few minutes ago. He was livid. Said the comments were nonsense. A ploy to drive down the offering price. He hinted it might be political. He wasn't sure, yet."
"Political? Come off it. If there's one thing I can tell you about the Private Eye-PO, it's that he's as American as apple pie. Still glad you crawled into bed with the enemy?"
"Kirov's hardly the enemy. We checked him out backwards and forwards. Even Kroll gave him a clean bill of health. No ties to the _mafiya_ , no indentures to the government, no evidence of corruption or criminal activity. Konstantin Kirov's the first—"
"Stop right there," blurted Byrnes. "I know what you're going to say. He's 'the first truly Western businessman.' The _Financial Times_ said that, right? 'The patron saint of the second Russian perestroika.' Remember, Jett, I read the prospectus, too."
Gavallan shook his head. Byrnes would always be an unrepentant cold warrior. "You know, Graf, you missed our calling. You should start up a new chapter of America Firsters. Bring isolationism back into vogue."
"Okay, okay," said Byrnes, lifting his hands palm up. "He's a wild card, that's all I'm saying."
"Well, he's _our_ wild card, so you better get used to him. If the Mercury IPO goes well, we'll be doing business with Kirov for a decade. We're already talking a secondary offering in a year, and he's asked us to scout some acquisition targets for him. Mercury's a gusher waiting to be tapped, and we're darned lucky they chose us to do the drilling. He asked me if I wanted him to send over his jet to bring me to Moscow. He wants to personally show me the premises. He's worried about how the market's taking it."
"And how _is_ the market taking it?" asked Byrnes. "What's the word from Bruce?"
"Too soon to tell, but this kind of thing is never good. We'll need to engage in some proactive damage control."
"So you believe Mr. Kirov?"
"A hundred percent."
"All right then. Let's look at this closer."
Digging his hands into his pockets, Grafton Byrnes began a slow circuit of the room. "This is an accusation of material fraud. The Private Eye-PO isn't just saying that Mercury isn't up to snuff, he's implying we knew all about it, too, and kept our mouths shut. If those photos are genuine, there's no way Mercury can be doing the business it claims. Two hundred thousand clients in Moscow? Hell, they couldn't service twenty with that stuff. These accusations are tantamount to saying the company's entire P&L is a bunch of garbage. We've got to imagine that most of our customers will either read this or get wind of it and come to the same conclusions themselves. In a few hours, every one of Bruce Jay Tustin's salesmen will be fielding calls asking for us to comment on the Private Eye-PO's claims. Whether we believe Kirov or not, we've got to check on Mercury."
"Agreed."
"And not under his personal auspices, I'm afraid. Tell him you'll pass on the jet. I'll give Silber, Goldi, and Grimm a call instead." Byrnes was talking about the Swiss accounting firm that had performed the due diligence on the deal. "They're in Geneva; it's only a two-hour flight for them. They can have this sorted out by the end of business tomorrow."
"No go," responded Gavallan. "I don't want to bring an outside firm into this. It's too late for that. We can't have anyone thinking we have even the slightest doubts about Mercury, not this far into the quiet period. One of us has to go. Like you said, our head is on the chopping block as much as Mercury's."
_"One of us?"_ Byrnes did not look pleased.
"I'd go if I could, you know that. I've got the dinner on Wednesday."
"Yeah, yeah, I know. Since when did hospitals start honoring border trash as 'Man of the Year'? I'll miss heckling you. I had a few choice tomatoes saved for the occasion." Byrnes collapsed onto the sofa, resting his chin on folded hands. "And how will your friend Kirov feel about this? He's bound to find out."
"He won't like it, but he'll understand," explained Gavallan. "He knows what's involved to get a listing on the Big Board. In the end, he'll thank us for it."
"I hope so. I don't relish getting a guided tour of the Lubyanka."
Rolling his eyes, Gavallan opened the drawer and took out a plane ticket. He'd known all along the actions required of the firm. He'd just wanted Byrnes's opinion on the matter. "Flight goes at one," he said, waving the slim jacket. "Consulate opens at eight. You'll need a visa. If you hurry, you might even have time to get home and pack."
Byrnes picked up the ticket off the desk, opening the sleeve and reading over the flight details. "You're a crafty prick, you know that?"
"What do you expect? I learned from the best."
**R ECALLING THE MOMENT** forty-eight hours earlier, Gavallan caught his reflection in the glass. He was surprised at the man staring back. He looked tired and worn, older than his years. The weight of office, he told himself. The price for making a fortune before the age of forty. And the price for losing it? he wondered. What's that? Do you get some of your youth back? Learn how to take a few days off? Regain the affections of the woman you love?
Gavallan put a stranglehold on his thoughts. Self-pity was a loser's luxury. He heard Byrnes telling him to "toughen up" and felt the wise eyes boring into him.
_Graf, where the hell are you? Give me a call and tell me everything's all right_.
A minute passed as Gavallan considered taking a dozen actions: canvassing the larger hotels in the Russian capital, contacting the U.S. Embassy in Moscow, even calling the Moscow Police directly. All were premature. If Byrnes had checked out of the Baltschug, he had a good reason. It was silly to worry. He'd give his best friend until noon to call or check in, then reassess the situation.
A firm hand rapped on his door. "Morning meeting's about to start, boss."
"Yeah," said Gavallan, without turning. "Be right there."
Returning to his desk, he made a quick check of his agenda. As always, his schedule was packed to bursting. Quarterly earnings review at ten. A powwow to go over acquisition candidates for a new client at eleven. Roundtable with the executive board to discuss new business opportunities at two. And, of course, the black-tie dinner that evening for which he had yet to write a speech.
But even as he catalogued his day's appointments, his thoughts vaulted six thousand miles to the onion domes and cobblestoned streets of a city he'd known forever, but never visited. Moscow.
_Graf_ , he shouted silently. _Talk to me!_
# 4
#
**G RAFTON BYRNES WAS STILL TRYING** to figure out when exactly they had left the city and entered the country. It seemed like only five minutes ago they'd been barreling down the road to Sheremetyevo Airport, the driver busily pointing out Dynamo Stadium, home to Moscow's soccer team, the Ministry of the Interior building built by Stalin, the new Seventh Continent supermarket. Then they'd made a left turn past a car dealership, traveled a ways through a birch forest, and _—bang_!—they were in the Russian countryside. Eight lanes had dwindled to four, and then two, and now they were bouncing down a dirt road smack in the middle of a potato patch that stretched as far as the eye could see in every direction.
Byrnes took out the paper on which he'd written the address of Mercury Broadband's network operations center. "Rudenev Ulitsa?" he asked skeptically, gesturing at the road beneath them.
" _Da_. Rudenev," said his Tatar chauffeur. He blurted a few words in Russian that Byrnes caught as "Long street. Goes to city of Rudenev."
" _Eto Daleko?_ Is it far?"
_"Nyet."_ The man shook his head emphatically. "Very close now."
Byrnes looked at him a second longer, wondering if he might be possessed of some criminal intent. He dismissed the thought out of hand. If the guy wanted to rob him, all he had to do was pull over on any side street and stick a gun in his face. A look over his shoulder confirmed they were not being followed. The road behind them was empty, desperately so. Svetlana's or Tatiana's—or _whatever her name was_ —protectors were no doubt still at Metelitsa, concentrating their efforts on the next unlucky schlemiel. He stared at the setting sun, a dusky orange dome melting into the infinite plain. Russia, he thought, shaking his head. It was like watching a sunset on another planet.
They passed a row of dachas, small brightly painted cottages with steep, angular roofs. He'd always imagined dachas to be quaint, well-constructed cabins that lay hidden in pine glades. Maybe some were. These, however, were slapdash and garish, one plunked down next to the other with not a green tree in sight. The dachas looked uncared-for, as did the gardens and fences that surrounded them. In fact, his one overwhelming impression of Russia so far was of neglect. Offices with shattered windows, roads scarred with potholes, cars rusted beyond belief. He refused to think about the fire escape he'd climbed down an hour ago. He had a feeling the country was running as fast as it could just to stay in the same place. If he'd seen a mule pulling a hay cart, he wouldn't have been surprised. Somewhere back there he hadn't left just Moscow, but the entire twentieth century.
A half mile down the road, a blue strobe flashed urgently. Gripping his hands on the dashboard, Byrnes leaned forward, willing his pilot's eyes to focus. He made out a stubby automobile bestriding the narrow road. The car was white with green doors. The traffic militia, Byrnes groaned inwardly. On his ride in from the airport, he'd noted several similar automobiles parked in the center of tangled intersections. In each case, an olive-smocked policeman had stood nearby paying no mind to the horns blaring around him, doing damned all to right the congested thoroughfares. In a country famous for its corruption, the traffic militia had a reputation second to none. He didn't care to imagine what had brought them this far into the countryside a few minutes before nightfall.
"Shit," spat the driver, clearly sharing his anxiety. Shooting a worried look Byrnes's way, he braked to a halt and produced his papers.
A pug-faced militiaman approached the car. Ducking low, he peered into the windows, looking between Byrnes and the Tatar. The disparity between the two couldn't have been greater: Byrnes in his custom-tailored suit and five-hundred-dollar shoes, the Tatar in worn wool trousers and a frayed red pullover. The militiaman said a few words, then backed away from the car.
"A bad accident ahead. The road is closed," explained the Tatar. "We must go back. But first he wants you to get out and show him your passport."
"I have to get out? How come?" Byrnes didn't know why he was so surprised. In anticipation of the request, he'd already removed his passport and slipped a hundred-dollar bill inside the cover. Preparing a servile smile, he stepped out of the car and walked toward the militiaman.
"Good evening," he said in halting Russian, wanting to show he was one of the good guys.
The militiaman approached slowly, rolling his boots, thumbs tucked into a heavy utility belt. He was a block of a man, more chunky than muscular, heavy around the shoulders and neck. He was dirty. Visibly dirty. Dirt flecked his cheeks. His hair was greasy and uncombed, his mustard uniform dotted with stains. Deliberately, he slid his baton from its holster.
"Passport," he grunted.
Byrnes eyed the baton. Dents and chips and scuffs decorated its length. Losing the smile, he handed over the passport. The baton flicked through the air, so fast as to be a blur, cuffing Byrnes's wrist and sending the passport tumbling to the road. "Hey," he shouted, grabbing at his hand. "Watch it, you sonuva—"
The next blow was faster, if that was possible. Harder, too—a lightning-quick jab to Byrnes's unsuspecting gut. The baton disappeared into his midsection before caroming back a split second later, robbing Byrnes of his belligerence as well as his breath. He fell to a knee, eyes bulging as he prayed for his lungs to start working again.
The militiaman pointed at the hundred-dollar bill lying on the ground. "Yours?" he grunted in English.
"No," coughed Byrnes.
The militiaman motioned for Byrnes to hand it to him. Struggling to his feet, Byrnes picked up the note and his passport and handed them to the policeman.
_"Spaseeba."_ The unkempt cop stared at the passport for a few seconds. "What hotel, please?"
"The Baltschug. In Moscow." From the corner of his eye, Byrnes could see the Tatar, standing at the rear of the car, hands folded in front of him, eyes making a meticulous study of the rocks near his shoes. The militiaman returned to his car, placed a call on his radio, smoked a cigarette, talked a little more on the two-way, then came back. Curling a finger, he motioned for the Tatar to join them. He barked a few words, looking at Byrnes.
"You are not guest at the Baltschug," the Tatar translated. "The hotel does not know you. The officer would like to know where you are staying, please?"
"The Baltschug." Byrnes could not keep the irritation from his voice. "I checked in at four o'clock. Room 335. Look, I have a key." He delved into his pocket for the room key. Not finding it, he tried the other pocket, then his jacket. He remembered the tempting blond leaning close to him, rubbing his leg. "Please tell the officer that he can accompany me back to the hotel. I'll be happy to show him my room. My suitcase, my clothing, everything is there."
But the militiaman was already shaking his head. An amused grin said he'd heard this one a hundred times before. "No," he said in his brusque English before rattling off a few more bursts at the Tatar.
"We must go," said Byrnes's chauffeur worriedly, pulling at his sleeve. "The road is closed. A bad accident farther on."
"Go? Hold on a goddamn minute," cried Byrnes, freeing himself. "The guy still has my passport. I'm not going anywhere." He took a step toward the police officer, his ingrained belief in law and order overruling his common sense. "I'm an American citizen. You have no right to keep my passport. Please, I'd like it back."
"When you check into a hotel, you are to call police," explained the Tatar, scuttling back to the Lada. "They will bring you passport. Now please, we go."
"Ask him how much he wants for it. Here, here's another hundred." The militiaman feinted with the baton, and Byrnes jumped back. "You go," barked the policeman, ignoring the proffered currency. Then slipping the passport into his breast pocket, he ambled back to his beat-up patrol car.
Furious, Byrnes climbed into the Lada. The Tatar started the car, executed a neat three-point turn, then steered them back toward Moscow. Turning in his seat, Byrnes stared behind him. Fading into the distance was the same featureless vista that had played out before him for the past thirty minutes, a rutted, dusty road rolling like a draftsman's straightedge into the horizon. The Tatar began humming a tuneless melody, his breath whistling through chipped teeth. The car bumped along and Byrnes kept staring over his shoulder at the blinking strobes, feeling cheated and unjustly persecuted, asking himself what he might have done differently to effect a better outcome. He had no doubt he'd get his passport back—or that it would cost him another hundred dollars, if not more. He was sure the cop had never called the Baltschug. Of course, there was no accident, but his mind did not allow him to go any further. He waited until he could no longer see the militiaman, then said, "Stop."
The Tatar dashed an annoyed look his way. "We go home now. I take you to hotel. You sleep. I sleep. Okay?"
Byrnes slipped his wallet from his jacket and took out a hundred-dollar bill. "Stop," he repeated. "Please."
The Tatar sighed painfully, as if he knew what Byrnes was going to ask, then slowed the car.
"I must go to Rudenev," Byrnes said. Using his hands, he indicated his desire to make a bell-shaped detour around the militiaman. He was sure the Lada was sturdy enough to handle a few miles through hardscrabble fields. When the Tatar hesitated, Byrnes took out another hundred and pressed both bills into the man's creased palms. Two hundred dollars was probably double his monthly salary. "Please. It's important."
The Tatar stuffed the bills in his pocket and grunted as if Byrnes's request was but the final depredation forced upon him by a world going to the devil. Pulling off the road, he said, "I am Mikhail. Pleased to meet you. You are millionaire, maybe?"
Byrnes shook the callused hand. What was it about this place? "Graf. Likewise."
**T HEY DROVE THROUGH THE FIELDS** for half an hour. The Lada bounced and groaned and rocked, keeping up a steady assault on the Russian potato industry. Never did the needle on the speedometer surpass twenty kilometers per hour. The sky was darkening quickly, and Byrnes thought if they didn't find the network operations center soon, he'd be spending the night in the countryside instead of in his four-hundred-dollar hotel room.
An outcropping of buildings came into shape a kilometer ahead. The silhouettes were low, right-angled, and unimaginative, no different than a strip mall or office park. He thought he could make out a satellite dish.
"Rudenev 99?"
_"Da."_
Byrnes laughed, then clapped his hands and expelled a soft "Hooray!" He knew it was common for satellite downlinks and cable relays to be located at the periphery of metropolitan areas; land was cheaper there and it was easier to lay cable in undeveloped areas. He just hadn't expected to be so far outside the city. Only then did he make out the squadron of small trucks and automobiles parked in front of the buildings. Dark figures scurried like ants back and forth between the vehicles.
As they drew closer, he was able to discern four separate buildings, one at each corner of an intersection. The "ants" were workmen. Some were clad in overalls or jumpsuits, others in denim shorts and T-shirts. To a man, they were busy unloading large rectangular cartons from the trucks and carting them on dollies into the building with the satellite dish on its roof. No one paid the Lada any mind as it climbed onto the road and drew to a labored halt.
With a strong elbow and a few oaths, Byrnes opened the door. "Please wait," he said.
The driver got out of the car and lit a cigarette. Byrnes made a note to ask for his address so he could FedEx him a carton of Marlboros.
Buttoning his jacket, he set off through the throng, intent on making his way into the building. He had only to glance at the cartons being wheeled inside to get a sharp, sick pain in his stomach. Now he knew what Jett had meant when he said he felt as if he'd been socked in the gut. Printed on the boxes were names like Dell, Sun, Alcatel, and Juniper—the brightest lights of the new economy. He walked stiffly, expecting at any moment to be stopped and asked who he was and what he was doing there.
The center of activity was a large warehouse painted a totalitarian gray, windowless and boasting double doors through which a nonstop stream of men filed in and out. Painted on the wall was the Mercury Broadband name and logo. He recognized the building from the picture the Private Eye-PO had posted on the web. No doubt about it: He was in the right place. Taking out his cell phone, he dialed the office. A recorded message informed him the call could not be completed at that time.
"Damn it," he muttered, sliding the phone back into his jacket.
Working to keep his gait slow, his bearing relaxed, Byrnes took up position by the front doors. Fluorescent lights blazed inside. The atmosphere was hushed, as reverential as that of a cathedral. The workmen kept to a long corridor, disappearing into another part of the building. What the hell, he said in a bid to buck himself up. You've come this far, why not go whole hog?
And tightening his tie, he ducked inside Mercury Broadband's Moscow network operations center.
**H IS FIRST IMPRESSION** was that the pictures were wrong.
The operations center was a model of its kind. Rack after rack of servers sat in black metal cages. Video cameras monitored every room. Liebert air conditioners kept the temperature an ideal sixty-five degrees. A corps of technicians manned a sophisticated console keeping tabs on the company's metropolitan operations. Every now and then a red light would flash on a map of the city, indicating a problem at a relay station or outlying node. Immediately, a technician would pick up the phone and attempt to solve the problem.
Byrnes slid from room to room, noticed but unquestioned, his suit and tie and confident posture as good as any E-ZPass. His relief in learning that the Private Eye-PO's pictures were bogus was outweighed only by his desire to know what in the world all the new equipment was being used for. He didn't remember reading any plans for a buildout of this proportion. As unobtrusively as possible, he followed the train of deliverymen through the corridors, passing from the main building to an outstation that had not been visible from the road. Just ahead, a security guard stood in front of a pair of swinging doors. He was holding a clipboard, and as each piece of equipment passed through the doors he checked both the item and the man's name against his list.
Byrnes allowed himself only a moment's hesitation. Then, hurrying his pace, he approached the security guard and handed him his business card. "Good evening," he said in English. "I'm a friend of Mr. Kirov's. He invited me to visit." And before the man could answer, Byrnes thanked him, smiled, and followed the next deliveryman through the doors.
He was standing inside a very large room, one hundred feet long and seventy feet wide. The floor was white. The walls were white. The ceiling was white, and from it hung rafts of fluorescent lights suspended by thin cables. Table after table ran the width of the room. On them was arrayed an army of personal computers: hundreds... no, _thousands_ of PCs arranged one after another in perfect rows. The screens blinked on and off. On and off. He walked closer. One screen read, "Welcome to Red Star. Please enter your password." The computer did as it was asked and the PC logged onto Mercury's signature portal. The welcome screen went blank, replaced a moment later by a familiar web page. Somewhere on the page, he read the greeting "Hello, Sergei Romanov," but a moment later the screen blinked and traveled to another electronic address. The PC continued its peripatetic iterations, bouncing from one site to another for a minute or two, then logging off. A few seconds passed, and it began the same trick again.
Byrnes advanced a few rows and watched another PC perform the same operations, only visiting different websites. He stood mesmerized, floating in a white universe of personal computers, wondering what the hell was going on. He took a few more steps and watched some more.
And then, it hit him.
Taking in the entire room at once, he whispered, "My God. It can't be."
**W HEN HE EMERGED FIVE MINUTES LATER,** his first act was to phone his office. It was near noon in San Francisco. This time the call went through.
"Yeah?" answered a familiar voice.
"Oh, it's you," said Byrnes, a little surprised that Gavallan hadn't answered his private line. "Where's Jett?"
"Not around right now. What's up?"
"Is he close by? It's important I talk to him."
Byrnes caught the sound of an engine revving hard and jogged toward the Lada. A gold Mercedes sedan was flying down the road, leaving a curtain of dust in its wake. No roadblock for him, Byrnes mused; no playing kissy-face with Uncle Vanya of the traffic militia.
"Where are you, Graf?" came the voice in his ear. "You sound a million miles away."
Byrnes tapped his foot nervously. No one but he and Gavallan knew about the excursion to Moscow. "Just get Jett. And hurry."
"Cool down. He's not here. I saw him a while ago, but he may have stepped out."
The Mercedes was a hundred yards away and showed no signs of slowing. Byrnes hesitated, hoping the sedan would pass through the intersection, knowing in his gut it was headed here, and that whoever was inside was looking for him. As the Russian police didn't drive late-model Mercedes that retailed for a hundred grand a pop, he had a feeling he was in for a rougher brand of justice. He looked around. It would be easy to duck back into the building, to hide among the workers. But why? He'd done nothing wrong. As Mercury's banker, he had every right to be here. His visit was unannounced, but not surreptitious. He had every intention of phoning Mr. Kirov once he assembled his findings. The thought of being found cowering inside an empty cardboard box decided the matter. Galvanized, his feet took firm possession of the ground, and he rummaged in his pockets for a business card.
"All right, all right, listen then," he said into the phone. "It's about Mercury. You have to tell Jett everything I'm about to say verbatim. You got that? Verbatim. You won't believe it."
And for the next sixty seconds he rattled off everything he'd seen inside the Moscow network operations center, stopping only when the Mercedes sedan had pulled to a halt ten feet away. "You got that?"
The voice sounded shocked. "Yeah, I got it. It just sounds a little crazy. I mean, that's not even possible, is it?"
But Byrnes didn't answer. By then, the door of the Mercedes had opened and Tatiana, or Svetlana, or whatever the gorgeous pickpocket with the satin blue eyes wanted to call herself, had stepped into the Russian night. In her hand she held her friend's nickel-plated Colt revolver, and she was pointing it at his chest.
_"Allo, Graf."_
# 5
#
**A T PRECISELY 7:15,** Jett Gavallan ducked out of his office, took the elevator to the garage, and jumped behind the wheel of his Mercedes. The rain had stopped and traffic was light as he pulled onto the street two minutes later and accelerated east toward the Embarcadero. He didn't like being away from the office in the mornings; even the most important client meeting left him feeling like a choirboy locked outside of chapel. But today, there was nothing to be done. He had an appointment to keep, one that required discretion and a degree of stealth. One that on no account could be handled inside the walls of Black Jet Securities.
To his left, Chinatown passed in a blur of weeping pagodas and shuttered storefronts. Steering around slower cars, Gavallan made good time, managing to make every light on Pine Street without seeing so much as a hint of yellow. He did not notice the silver Ford Taurus tucked neatly into his lane three cars behind him, following at a discreet distance. If he had, he would have had no reason to be suspicious of it. The FBI was a professional organization. They had made it a point to change their tail car every day since taking up surveillance on Mr. John J. Gavallan ten days earlier.
Gavallan focused his mind on business. He had an earnings review at ten, and to prepare for it he'd spent an hour studying the firm's pro forma revenue estimates for the second quarter. The results were not encouraging. He was beginning to wonder if it wasn't the market that had gone sour, but his own touch.
"Nine years," he whispered to himself, recalling the arduous climb from one-man band to multinational touring act. Today, it felt like ninety.
**G AVALLAN HAD FOUNDED** Black Jet Securities one year out of Stanford Business School. He'd had offers from plenty of blue-chip firms—IBM, Goldman Sachs, Ford—but he'd turned them down. Six years in the Air Force had left him wary of institutional authority. Instead, he rolled up his sleeves, took a job with Sutro & Co., a smallish California investment bank, studied for his registered rep's license, and taught himself the investment business. Twelve months later, he quit his job and hung up his own shingle, taking with him a few of his largest clients and eighty thousand dollars in savings.
For a year or two, he was content to act as a brokerdealer, hiring registered reps and managing clients' money. But south of town, in Silicon Valley, things were happening that quickly caught his eye. Something called the Internet was springing to life, and overnight, companies eager to capitalize on its heralded, but unproven, potential were sprouting up like mushrooms. While their products and strategies differed wildly, they all shared one common trait: a dire, unquenchable need for cash.
It was into this market that Gavallan jumped headfirst in the fall of 1996. He didn't know much about initial public offerings, but that didn't matter. Nor did a lack of pedigree or track record. Jett Gavallan had something none of his rival investment bankers did. Something his business school profs would have labeled a "unique point of differentiation."
During the Gulf War, he had flown twenty-six missions at the controls of an F-117 Nighthawk, the angular black jet known to the world as the Stealth bomber. And there was nothing that a gaggle of laboratory-bred "techies" liked hearing better than what it was like to man the controls of the world's most technologically sophisticated aircraft and drop a laser-guided smart bomb smack on the top of a pack of kaftan-clad, Uncle Sam–jeering camel jockeys. Forget Quake. Forget Doom. Forget Tomb Raider. Here was the real McCoy. A first-person shooter with blood on his hands. And they would be proud to have him battle the wizards of Wall Street to secure funding on their behalf.
For two years, Gavallan traveled between the towns of San Mateo, Menlo Park, and Palo Alto. His first clients were small ones, rinky-dink start-ups happy to raise ten million dollars on the Pacific Stock Exchange. Computer jocks with dirty fingernails who worked in their pajamas. He did eight IPOs the first year. Twenty the second. In time his reputation grew, and with it the quality of his clients and his company's revenues. Black Jet's annual gross rose in a vertiginous spiral. Sixty million dollars, one hundred forty, four hundred. Amazingly, the firm managed to break a billion before the bubble broke and things went to hell.
Since then he'd been fighting to keep his head above water. The company was still making money, just not enough. He was sized for growth, not stasis. Euphemisms like "ramping up," "burn rate," and "top-heavy" and their connotations of boom and bust weren't solely the preserve of Silicon Valley.
He could imagine the earnings review later that morning. Retail brokering was coming back nicely, but nothing like it had been. IPO activity was only just recovering. M&A was off 20 percent the last two years. Only trading was making money, landing on the right side of the latest big leg up. As each managing director reported his or her results, their eyes would creep toward Gavallan. He knew the downcast glances, the uncomfortable silences, the nervous laughter by rote, each person wondering when the ax would fall, and whose head it would be thumping into the wicker basket. God, he hated being the executioner.
"We're not running a charity," Byrnes had said during their last meeting.
Gavallan was all too aware of the fact. Three times in the last year he'd dipped into his savings to fund increases in Black Jet's capital. He'd liquidated his portfolio of stocks, sold off a large chunk of real estate in Montana he'd been planning to build on for his retirement, and cashed out of a promising hedge fund. This morning, he would take the final plunge—a second mortgage on his home. After that... An old adage about tapping a dry well came to mind.
Arriving at the Embarcadero, he was pleasantly surprised to find an empty space in front of the building. He parked hastily, telling himself that the space was an omen of good things to follow. Entering his attorney's office, Gavallan laughed at his desperation. He knew there was no such thing as good luck. Just good timing.
**S PECIAL AGENT **R** **OY** DIGENOVESE,** on temporary assignment to the San Francisco field office of the Federal Bureau of Investigation, double-parked the silver Ford Taurus a safe distance shy of Gavallan. Keeping the engine running, he rolled down the window and lit a cigarette. A glance in the side-view mirror confirmed that DiGenovese was Sicilian in looks as well as name. His hair was black, his eyes the color of midnight wine, his beard pushing up stubble three hours after he'd shaved. He had the brooding, patient gaze of a hunter, and a hundred years ago he might have been found wandering the rugged landscape of southern Sicily clad in chamois pants and a sheepskin vest, a _lupara_ slung over one shoulder, tracking the wolves that regularly ravaged his family's flock. Today, DiGenovese might still be called a hunter, but his prey was decidedly human, and his arsenal more subtle than his ancestor's twelve-gauge shotgun.
Armed with a Juris Doctor and an MBA from New York University, a CPA's credential emblazoned upon his breastplate, Roy DiGenovese was the newest member of the FBI's Joint Russo-American Task Force on Organized Crime. Prior to his studies, he'd spent time in the U.S. Army, earning his Ranger's tab and serving with the 75th Ranger Regiment at Fort Benning, Georgia. Three years into what he hoped to be a lifelong career with the FBI, he was trim and muscular, and possessed of the same killer instinct as the wild-ass teenager who used to rappel out of helicopters in the dead of night, an M16 on his back and a K-bar strapped to his calf.
Setting the cigarette in the ashtray, DiGenovese picked up a scuffed Nikon from the seat beside him and brought it to his eye. The speed-wind whirred nicely as he fired off a dozen stills of Gavallan hauling himself out of the crazy old car with the gullwing doors. Even through the shutter, the man looked tired and in need of a break. It was easy to understand why. Seven days of following Gavallan had convinced DiGenovese he'd made the right decision not to take a job on Wall Street. Twelve hours a day cooped up inside a skyscraper was no way to go through life. The guy's desk might be made of mahogany, but the chain that tied him to it was pig iron, all the same.
As soon as Gavallan disappeared inside the sleek office tower, DiGenovese exchanged the Nikon for a two-way radio. "Zebra two, this is Zebra base, come in."
"Zebra two, roger."
"Maid gone?"
"Two minutes back. On her way to pew number seven at St. Mary's as we speak."
"Good. Tell her to light a candle for us, we who are about to sin."
Gavallan's maid, a middle-aged Guatemalan illegal named Hortensia Estrada, hadn't missed morning mass a single day that week. The service lasted between fifty and sixty minutes, leaving DiGenovese's men plenty of time to do their work.
"You're good to go, Zebra two," said DiGenovese. "Time at target is thirty minutes. I repeat: three-zero minutes. Are we clear?"
"Roger, Zebra base. Three-zero minutes. Walk in the park."
**A RE YOU SURE YOU WANT TO DO THIS?**" Sten Norgren asked, clutching a sheaf of manila folders, legal envelopes, and stray papers to his chest.
"Just give me the documents, Sten. It's not that big a deal."
Hesitantly, Norgren laid the stack of papers on his desk. "It's only for your protection," he pleaded in an injured tone. He was short and barrel-chested, with a florid, cherubic face and curly blond hair. "Isn't it just a wee bit crazy to stuff all your money in one investment?"
"Not if it's your own company," said Gavallan. "Besides, can I tell you a secret?" He motioned the attorney closer. "That stuff about diversification? It's bullshit. Just a ruse to bump up commissions. We can't have our customers buying and holding the same stock for twenty years at a time. We'd be bankrupt by Christmas. Sector rotation, averaging in, market timing—that's the ticket. Churn and burn, Sten, that's the name of the game."
For a moment, Norgren didn't answer, and Gavallan could practically see the lawyer's analytical mind parsing over his statement, deciding whether what he said might actually be true. Then Norgren burst out, "Shut up, you bullshit artist. Sit your butt down in that chair, right now. I've got just the pen to sign your life away—a Mont Blanc that Sherry gave me for Christmas. Lousy thing cost more than my law school diploma."
Gavallan found his way to the chair and sat down. He had always hated lawyers' offices. He had only to set foot inside one for a feeling of imminent bad luck to creep into his neck and shoulders. Norgren's office was no exception, even with the Scandinavian furniture, the credenza packed with photos of blond, smiling kids, and the colorful modern art on the wall.
"Last chance," said Norgren.
"The pen, maestro."
Norgren took a beautiful onyx and gold fountain pen from his pocket and plucked off the cap. "She's all yours."
Recognizing the concern as genuine, Gavallan was flushed with a sudden fondness for the man. These days it was pretty hard to find a lawyer who gave a damn. "Thanks, man, but I know what I'm doing."
"I'm sure you do," said Norgren, a little too quietly.
The firm of Norgren, Piel, and Pine had done the majority of Black Jet's securities work for years: fairness opinions, registration statements for the SEC, legal analyses of all manner of financial instruments. At some point, Gavallan and Sten had become friends. They had dinner once a month and Gavallan took his kids sailing when the westerlies weren't too strong. Norgren had recently earned his pilot's license and was always calling Gavallan to ask his advice on one matter or another, begging him to come up with him for a short flight above the bay. Gavallan always declined politely, without offering an excuse. He hadn't taken the controls of an aircraft since the Gulf.
Gavallan spent a few moments reading through the documents. On top of the pile was a statement from Alameda Trust Corporation, granting him a second mortgage on his home in Pacific Heights in the amount of two million dollars. Beneath it were envelopes containing Gavallan's monthly bank and brokerage statements and the HUD-1 for the recent sale of his property in Montana—all the paperwork Norgren had needed to secure the loan on his behalf.
Gavallan's first action was to endorse the check made out in his name and hand it to Norgren. "You know where to deposit this."
"How much does this bring it to? North of twenty million, if I'm not mistaken?"
"Twenty-five point seven, to be exact," answered Gavallan, meeting his eye. "Don't worry, Sten, I am keeping track."
Black Jet's recent quarterly losses, combined with decreases in the value of securities the firm held for its own account and a certain fifty-million-dollar bridge loan he'd made to a less than investment grade customer, necessitated the capital injections. The Securities and Exchange Commission had strict requirements for firms underwriting new issues, especially ones valued at two billion dollars. Gavallan was not going to lose Mercury on a technicality.
"Why don't you let me make some calls," suggested Norgren. "I know some money center banks who could help out. Your reputation's sterling, Jett. You know that."
"I'm flattered, Sten, but it's hardly a good time. We're due to book our second quarterly loss in a row. We wouldn't get close to the price we deserve."
"I thought business was picking up. Markets are a helluva lot stronger than a year back. I'm sure you could get a great price—two times book at least."
Gavallan wondered if Norgren had already placed a few calls on his behalf. Black Jet's book value was close to four hundred million dollars; two times book gave the firm a value of eight hundred million. Gavallan considered the price, but the prospect of vast riches left him unfazed. And then? he mused. What happens after that? Report to a drone three thousand miles away? Sit on a beach reading paperback novels? Work on his golf game for eleven years in hopes of joining the Senior Tour? Sell? _Never_. The word wasn't in his vocabulary.
"The problem's overhead," said Gavallan. "Business isn't too bad. Revenues might even inch up a little over last year."
"So fire some people, Jett. Come on. Rationalize, downsize, economize."
"Compromise, marginalize, capsize," countered Gavallan, firing off the words like bullets.
"Survive!" shouted Norgren. "Stop being so damned proud and do what everyone else in your place would have done a year ago."
"Proud? I've fired forty men and women so far this year. Is that 'proud' enough for you?"
"It isn't up to me to decide—it's up to the market. It's no sign of failure if you cut back a little, tighten your belt." When Gavallan didn't respond to his prodding, Norgren threw up his arms in frustration. "Whatever you say, Jett. I'm just your lawyer. You pay me to keep you apprised of your best options. Consider yourself informed." And sighing, he bent over Gavallan's shoulder and thrust out a beefy hand to indicate where he should sign. "Here. Here. And here."
Gavallan affixed his violent slash to the documents as indicated. "That it?"
"That's it, my friend." Norgren added his own signature as witness, then gathered up the papers and laid them neatly in the out tray. "First payment isn't for sixty days. After that it's twenty-five grand a month, _every month_. And that's on top of your regular nut. That's a lot for a guy who hasn't taken a salary since Christmas."
Gavallan figured that taking a paycheck was like robbing Peter to pay Paul. He was all too aware of his precarious circumstances. "Not to worry, Sten. I'll have the entire amount paid back by the end of the month."
"Thousand-dollar prepayment penalty, just so you know. Couldn't get them to drop it. Come on, I'll walk you out." When they reached the reception area, Norgren said, "Sorry if I'm being a nervous Nelly. It's just that you're cutting it awfully close this time—I mean putting it all on this deal. Frankly, if you weren't my friend, I'd tell you you were out of your blazing mind. You sure it's going to work out?"
Gavallan smiled slyly as he threw an arm over the attorney's shoulder. "You saying you don't want your Mercury shares? Is that it?"
"Jett, I'm serious. If Mercury goes south—even if you have to shelve it for a few months—you'll be feeling the pain. You and your company. Think about what I said. About cutting. Make it temporary. A three-month vacation."
"Relax, Sten. It's not that big of a deal."
"Better yet, let me make that call. I'm just wondering if it's wise to bet it all on one number. It is a big deal, Jett."
"Nah," said Gavallan, shaking his friend's hand. "Betting it all isn't such a big deal—losing it is. Anyway, didn't you know? The house always wins."
Waving good-bye, Gavallan strode confidently to the elevator. He pressed the button for the ground floor, and as the car descended, his stomach went with it.
_Cutting it close?_ Norgren had no idea. Gavallan was down to three thousand dollars in his checking account, a hundred grand in certificates of deposit, and his prize Mercedes parked out front, its value beyond reckoning. He had a first mortgage of eighteen thousand dollars a month, a second of twenty-five grand kicking in in sixty days, and a quarterly tax payment of two hundred and eighty thousand dollars due on the twenty-first based on a salary he wasn't receiving—and that was before he put one foot out of bed.
Walking to his car, he considered his other obligations. To his three sisters and a widowed mother, all in Texas. To a club of broken and battered men spread around the world whom he'd adopted as his own. To a hospital that this very evening would fete him as its Man of the Year.
"And so, Mercury," he whispered, with a secret hope.
And so, seventy million dollars in fees and a spigot of related business to come down the pike.
And so, a twenty-first-century return to normalcy.
Gavallan started the motor. He had one more stop before returning to the office.
**T HE TEAM OF THREE MEN** and one woman worked quickly, efficiently, and silently. They entered Gavallan's residence through the rear door, disabling the security system, then spreading out through the four-thousand-square-foot home to their assigned target areas. Each knew the house by rote. They had studied architectural drawings of the home as well as an electrical schema of its wiring. They carried the tools of their trade in black web belts hidden beneath striped cotton shirts declaring them employees of Pacific Gas and Electric.
It was a standard "look and listen" job. Two of the men, known in agency lingo as the "ears," planted ultra-high-frequency wireless listening devices in strategic locations throughout the house. Under the dining room table. On top of the refrigerator. Behind the headboard of Gavallan's bed. Each bug had been assigned its own frequency, so that there would be no risk of one transmission interfering with another.
A third man, "the eyes," installed the cameras. They were very small and designed to replace the screws securing the faceplates of standard electrical outlets. Where this proved impractical—in the study, for example, where it was crucial that the lens be granted an unobstructed view of any materials Mr. Gavallan might be reading—he drilled a hole the circumference of a surgical needle into a gilded picture frame and inserted an even smaller model. Afterward, he applied a coat of colored translucent epoxy over the pinhole, making it invisible to the naked eye.
The last member of the team walked straight to Gavallan's private office and installed herself at his desk. She was the only person that morning engaged in a function outside the scope deemed legal by the court order issued the previous day by the Eighth Circuit Court in Washington, D.C. In her belt she carried a set of Czech-made titanium alloy skeleton keys, a dozen picks, and two dummy credit cards. She didn't need any of them. Giving a gentle pull, she discovered the desk to be unlocked. Methodically, she withdrew the papers, set them neatly upon the desk, and photographed them with a digital camera. Once she was finished with the top drawer, she returned the contents to their place and attacked the two larger drawers to her right.
When the team departed twenty-two minutes and fifty-one seconds later, a total of eleven bugs and six wireless cameras had been planted throughout the house. Two hundred twelve photographs of the suspect's most confidential documents waited to be enlarged and scrutinized. Mr. John J. Gavallan, subject of federal warrant SJ-74A001, under investigation in connection with thirty-two counts of international fraud, larceny, and racketeering, could not crap without the FBI knowing exactly how much tissue he used to wipe his ass. Walk in the park.
**R OY DIGENOVESE WAITED** until the Mercedes 300 SL had exited the office car park, then put the Ford in gear and pulled into traffic. He was not particularly worried about losing his mark. Gavallan was a steady driver, fast, aggressive, but safe. He used turn signals and didn't run red lights. A bakery truck pulled away from the curb, momentarily blocking Gavallan's car from view. DiGenovese didn't mind. He knew that when traffic picked up, all he'd have to do would be slide to the left and peek down the road. The white Mercedes, with its slot back and flat roof, would be there as usual, exactly three car lengths ahead of him, sticking out like a sore thumb.
"Zebra base, this is Zebra two, come in."
DiGenovese calmly picked up the walkie-talkie. "Roger, Zebra two."
"Went off like a charm. Target is wired for sound and light. Copy."
"Roger that, Zebra two. Rendezvous at the ranch at 1600. Drive on, Airborne."
DiGenovese put down the walkie-talkie and checked his watch. It was 8:07. In and out in under twenty-three minutes. "Outstanding," he murmured, remembering the long hours he'd put in on the case, the endless calls overseas, the numbing arguments with one after another federal magistrate to obtain his precious search warrants.
Setting up surveillance on Gavallan's residence was the final step in the casting of an all-encompassing electronic net over the suspect. Phone taps had gone into effect last night. Calls in and out of Black Jet, as well as his home, were screened for a succession of keywords and names. Mercury, Moscow, Novastar, Andara, Futura, and Kirov, Baranov, Tustin, and a hundred others. At the first mention of any of them, sophisticated computers at the National Security Agency would track and record the conversations.
Better yet was the second-generation Internet eavesdropping software being installed even as he drove. Nicknamed "Daisy," in deference to the flak brought down on their heads by its predecessor—the ineptly titled "Carnivore" system—the FBI's newest cyber-surveillance tool was housed in a black metal box no larger than a Palm personal assistant and powered by state-of-the-art software developed by the Bureau's in-house programmers. Installed at Gavallan's and Black Jet's wireless and Internet service providers, Daisy monitored every E-mail he or his executives received, their RIM Blackberries, cellular phones, or digital pagers for the list of keywords that DiGenovese and his superiors in D.C. had deemed likely to indicate conversations of a criminal bent.
All Gavallan had to do was breathe one word of his wrongdoing anywhere in his home, office, or car and DiGenovese and his superiors would know it. It was only a matter of time before the man slipped up.
DiGenovese waited a few more seconds, then edged the Ford over to the left, tilting his head to see around the bakery truck. A train of unfamiliar cars clogged the lane in front of him. The white Mercedes was nowhere to be seen. Panicked, DiGenovese craned his neck to the right and left, his eyes darting over every inch of the roiling cityscape. "Fuck," he muttered, chastising himself for his daydreaming. Signaling, he pulled into the fast lane and accelerated. He made it ten yards before a red light stopped him cold. Slamming his hand on the wheel, he swore again, this time loudly. He glanced to his right. There was Gavallan, a hundred yards away, trawling down Hope Street.
DiGenovese leaned on the horn, then jumped into the intersection, cutting off an oncoming taxi. He threw a hand out the window, showing his badge. Horns blared, voices shouted, fists threatened. In fits and spurts, he edged across the cluttered intersection. After what felt like a lifetime, he was barreling up Hope, the Mercedes no longer in sight.
He found Gavallan three blocks away, parked catty-corner to a playground next to St. John's Hospital. The guy was seated in his car, still as a bird. If DiGenovese wasn't mistaken, he was watching a couple of crips playing some early-morning roundball.
"Go figure," DiGenovese whispered. "Go fuckin' figure."
**T HE SCORE WAS 16–8,** with Flint pulling away.
Gavallan sat at a distance watching the two soldiers battle each other on the basketball court, the men rolling this way and that in their graphite low-profile wheelchairs, chasing down rebounds, clearing the ball, making fast breaks. Flint was the quicker of the two and, with his arcing hook, a better shooter. A close look revealed he was missing both legs below the knees and most of his left hand. Jaworski had the better bank shot and was speedier off the mark, but he was going to fat and his stamina was weakening down the stretch. A sliver of shrapnel no bigger than a needle had severed his spinal cord at the twelfth vertebra. He hadn't walked or made love in eleven years.
Gavallan watched another five minutes, until Flint had roundly defeated Jaworski, then started the car and headed back to the office. Passing the hospital's entrance, he felt a jab of shame bow his shoulders. "Man of the Year." The words made him wince. And for the first time, he acknowledged that he might soon have to write a letter explaining why due to financial circumstances, wholly of his own making, he would be unable to meet the terms of his commitment to the hospital.
He drove faster.
He wanted to be back in the office. Byrnes might have called.
# 6
#
**S TANDING ON A GRANITE PEDESTAL** opposite Gavallan's desk was an imposing four-foot statue of a shaman carved from the wood of a Canadian maple by the Haida tribe of the Queen Charlotte Islands, south of Alaska. It was a strange-looking creature, with an abbreviated torso, narrow neck, and large, grotesque head that was all bulging eyes, flattened lips, and flared nostrils.
"The shaman is a mystical and omnipotent medicine man," the dealer in Indian curios had explained to him when he'd first seen the statue three years before. "He knows all, does all, and judges all." Gavallan had locked eyes with the carving and decided at once that he had to have it.
Since then, whenever something unforeseen came up in his life—good or bad, important or trivial—he consulted the shaman. When the markets caught fire or fell in the dumps, when his putts rimmed out or his drives sailed a mile, when his emotional entanglements threatened to suffocate him if his commitment to his business didn't, he consulted the shaman.
The statue didn't offer any answers. He didn't speak in tongues or send telepathic messages. He just looked back, bored, impassive, and generally disdainful of all things human, counseling faith in the grand scheme of things while reminding Gavallan that he wasn't as important a shit as he sometimes got to thinking.
Sinking into his chair, Gavallan gazed imploringly at the shaman. He didn't need any reminders about his human frailties this morning, no rejoinders about hubris, arrogance, or cocksureness. He simply needed its help.
Returning to the office, he'd found no messages waiting from Grafton Byrnes. Nothing on his E-mail or voice mail. No chits left with Emerald, Gavallan's secretary of seven years, to call him back at the Metropol or the National or any of Moscow's better hotels. Nothing. The harried executive in him told him to wait until noon before reacting and to concentrate on other matters. The concerned friend urged him to get on the horn with Konstantin Kirov, tell him of their plans to disprove the Private Eye-PO's accusations, and demand his help in tracking Byrnes down. Respect for his friend's judgment and Gavallan's innate discipline won out. He would wait.
"You take care of my buddy, okay?" he said, holding the shaman's eye.
Opening his satchel, Gavallan withdrew the copies of the documents he'd signed at Norgren's and filed them in his drawer along with the other markers routing his path to perdition. He folded the receipt for the two-million-dollar check in two and slipped it into his pocket. Then he leaned back his chair, kicked his feet up onto the desk, and laughed.
It was not a joyful laugh, nor one with any hint of amusement hidden inside its rolling baritone folds. It was a sad laugh, a mocking laugh, one tinged with doubt, disdain, and wonderment at his own folly. Oh yes, he was cutting it close this time. He was hanging it out there in the wind real far. He'd always been one to enjoy the roll of the dice, to crave the giddiness of a measured risk, but this time he had overextended himself. This time he'd bet on events that he could not control, only witness. This time he'd been plain old stupid, and it was about time he admitted it.
Gavallan felt a wave of reckless anger build inside him, a steady roar expanding in his chest, filling his lungs, and scratching at his throat. If his rage was directed at himself, it was no less explosive for it.
In response, he made himself absolutely still. He slowed his breathing and laid his palms down on his desk as if he were about to stand. But he didn't move, not a muscle. Instead, he closed his eyes and began to count. He'd taught himself this trick years ago, when he was young and wild and given to bouts of unbridled fury. As a teenager he'd gotten into frequent fights. Not the clawing, awkward wrestling bouts of high school rivalries, but knock-down-drag-out, bare-knuckled exchanges with older, stronger men, the winner losing a tooth and the loser going to the hospital for stitches and X rays.
Gavallan didn't know from what spring the violence inside him flowed. His father was distant, but kind; his mother a fixture in the household; his sisters adoringly attentive. He himself was for the most part an obedient, dutiful, and undemanding youngster. Yet there was no doubting the wild streak, the inclination toward anger, the predilection for the nervy, rash act. Twice he was arrested for disorderly conduct. The first instance was when he beat the tar out of a Texas A&M lineman who'd stood up his oldest sister for her senior prom; the second and less valiant occasion occurred when, shit-faced in a Matamoros bar, he picked a fight with the biggest Mexican in the room just to prove he could whip him. He did, but he'd ended up with three broken knuckles, a cracked rib, and an eye swollen to the size of a grapefruit. Only through the benevolence of a local police officer had both acts been expunged from his record.
Aware of this flaw in his character and unwilling to allow it to defeat him, Gavallan had decided to isolate it and raze it from his behavior—or, at the very least, to keep it hidden from public view. Deep down, he knew his anger to be primal and lurking, and impossible to extinguish altogether. But slowly, and with an iron discipline new to him, he'd altered the way he acted.
He had always harbored ambitions, dreams of a life that would take him far away from the twelve-hundred-square-foot cinder-block home where he had grown up sleeping in the same bedroom as his three sisters, away from the unrelenting heat and humidity, from the mosquitoes that preyed on a man from dawn till dusk, from the bleak horizons of his parents' timid expectations.
By the age of fifteen, he knew what he wanted. He wanted to see the world as a pilot in the United States Air Force, and to be an officer and a gentleman in the best sense of the words. He wanted to be honorable, truthful, dependable, and courageous. He wanted to be respected not only for his skills as a pilot but for his integrity and character, and he expected to earn that respect. He wanted a wife and two children, and it was very important to him that he fall truly, madly in love. One day he hoped to wear a general's star on his shoulder.
To others, his dreams appeared fanciful or, worse, illusory. He had no money, no connections, no guidance but his own. But never did he doubt that he would gain his ambitions. He set forth a plan and he did not alter from it. He knew what he had to do. He must work harder than the rest, he must expect unfairness and some degree of intolerance. He must never complain. He must present the world a façade of unrelenting good spirit, equanimity, and drive. Above all, he must harness his rage.
To a large extent, Gavallan succeeded. He tempered his behavior. He fought down his rage and played up his humor. He showed the world what it most liked about itself.
Most of his ambitions were realized, though for a price beyond his reckoning. But deep inside him, the anger still burned, the rage still flickered, and he knew he must be ever watchful. For if he wasn't, one day it would surely rise up and destroy him. In the blink of an eye.
Reaching the count of one hundred, Gavallan exhaled audibly. For now, the anger was gone; the struggle for control won for another day. Happier, he turned and glanced at the pictures on his wall, wanting to share the victory, however minor. There was Gavallan and his father shaking hands on graduation day at the Air Force Academy. The old man looked as stern as ever, paying no mind to the fact that he was wearing his son's dress cap on his head. He never got over his boy's leaving the service, or the less than satisfactory general discharge that had made it official. Until the day he died, he insisted to his friends that his son had left the cockpit over the lack of decent pay.
"Money," sniffed Gavallan. "If only..."
The true cause of his sudden, and not altogether voluntary, separation from the United States Air Force could be found on a ninety-minute videocassette kept shut in the bottom corner of his flight locker alongside his jumpsuit, his flying scarf, and his old Omega Speedmaster. The tape was dated February 25, 1991, and titled _Day 40—Abu Ghurayb Presidential Complex_. It had been made with an infrared camera mounted on the underside of his F-117. The tape was a copy, a pirated bootleg, and his possession of it was a jailable offense. The original was kept in a more secure location, most likely somewhere deep inside the Pentagon where the United States Armed Forces hid its dirty laundry.
Gavallan's eyes dodged his father, only to land on himself. There he was, a twenty-six-year-old superman gussied up for combat, strapped into his G suit, helmet in hand, standing beside the cockpit of his Desert Storm mount, an F-117 he'd christened _Darling Lil_. Look at that smile. Top of the world, eh, kid? The photo had been taken in a hangar at King Khalid Air Force Base in Saudi Arabia. A giant American flag hung from the rafters behind him. Beat that, Tom Cruise!
Another photo showed his mother and three sisters standing at the base of Big Tex, the 150-foot cowboy, at the state fair in Dallas ten years back. Mom, meek and gray, with her haunted smile, the woman who'd gifted him the name of Jett, not out of any premonition of the future, but because of her long-held crush on an unknown actor who'd visited her hometown of Marfa, Texas, one teenage summer, to stand before the cameras as Jett Rink, impetuous wildcatter who struck it rich in the glorious Technicolor Texas epic _Giant_. James Dean did a number on Marfa. Look in the phone book. You'll find a dozen men aged forty and up carrying the ridiculous name of Jett.
Above the photos hung two wooden plaques with attached miniature replicas of an A-10 bomber. Flowery script declared: "Captain John J. Gavallan, USAF, Squadron and Wing Top Gun at Red Flag '89 and '90." Red Flag was the annual competition staged at Nellis Air Force Base outside Las Vegas, where a pilot's proficiency was measured during several days of demanding flight exercises. As always, the mementos triggered a desire to fly, a yearning so strong he could feel it.
_Trade your company, your career, to do it again?_ a skeptical voice demanded.
_Any day_ , he answered.
To be at the stick of a jet was like nothing else in the world. To soar like an eagle and dive like a tern, while enveloped in the sky's royal blue cape. If there was magic in the world, Gavallan had found it in the cockpit of a jet aircraft.
Dismissing his longing, he continued on his nostalgic tour. There was only one place left to visit. Like any sentimental fool, he'd left his heart's graveyard for last.
Opening the bottom drawer of his desk, he rummaged through a dozen photographs, most framed with simple silver settings, a few loose, the dates and places written on the backs. Leaning to his side, he picked up one photo, then the next. With each, he stared into the woman's bold, ebullient green eyes, imagining the touch of her pillowed lips, sighing, smiling, longing, always wishing he could reshape the past. Flipping over the snapshots in turn, he read the inscriptions penned on the back: _Manhattan, Valentine's Day; Chicago, Xmas Eve; Hong Kong, Easter Sunday_. The script was looping and feminine, but never less than purposefully legible. Lingering over the words, he felt happily vulnerable, close to her again.
Cate who was kind but serious. Cate who was shy but sensual. Cate who was painfully honest yet a mystery even to those nearest to her. Cate who never raised her voice but let her eyes argue for her. Cate who declined his proposal of marriage with a single word and no backward glances.
Gavallan brooded for a minute or two, still in a kind of suspended state of disbelief that she'd turned him down. He hadn't seen it coming. Not after two years of dating and six months of living together. One moment, he'd popped the question; the next, she was out of there. Not a sock, stocking, or bobby pin left behind.
_Cate who was gone_.
The last picture in the stack had been taken just a few hours before he'd proposed, and showed the two of them at the rail of Sten Norgren's fifty-foot Wellington as it passed by the Presidio, San Francisco's oldest military installation. Cate's lustrous black hair, sparkling in the mid-morning sun, whipped across her face. Her eyes were partially hidden, but there was no disguising the smile or the quicksilver brilliance of the perfect white teeth. And no mistaking the unalloyed joy behind them.
Bringing the photo closer, he traced a thumb over her features, searching her obscured expression for a hint of what was to come. Looking past the hair into her eyes, checking her smile, he fought to glimpse a trace of discord, a measure of dissembling, some signal of the betrayal that lurked around the corner. He'd been doing the same stupid thing every day for a month, and every day he came away empty. She hadn't given him a clue.
This failure to foresee her actions had left him feeling powerless, the fool. Later, when she'd refused to explain her reasons, or to even speak with him, his emotions had hardened and he'd felt tricked and cheated and vengeful.
A few nights ago, he'd woken in a sweat, trembling, his heart racked with a terrifying anxiety. He hadn't suffered a nightmare. No subconscious spasm that his bet on Mercury would turn sour, no clawing certainty he'd lose everything he'd worked toward since leaving the Air Force, that he might end up penniless and without a means of supporting himself. The fear that stole upon him out of the darkness was deeper and more personal. It was fear sprung from his most desperate insecurity, more a premonition really, a merciless and exacting portrait drawn in black and gray of his life to come.
He saw himself in twenty years. He looked as he did now. He had all his hair, was trim and fit. He knew as you do in the subterfuge of dreams that he still had Black Jet, that he played golf once a week and went sailing on occasion, and that he was as well-off as he would ever need to be. Yet his image was surrounded by a naked aura of despair. Waves of loneliness rose from him like heat from the desert floor. Here was a man who had spent his life wedded to his business, involved in the stark, predictable activity of making money. Here was a drone who embraced repetition and success as a substitute for passion—and who, for all his effort and infinite industry, had no one.
Awake, perched on the edge of his bed in the dead of night, he'd suddenly realized that he had no possibilities without her, that he would never find someone to replace her, that there was no one in the world who could excite him and challenge him and thrill him as she had. No one who would own him so utterly.
Gavallan's phone rang. Bolting forward, he dropped the pictures into the drawer, slid it shut, and picked up the receiver. It was Emerald Chew on his private line.
"Yes, Emerald."
"Sorry to disturb you, but Tony's on his way in. He's very agitated."
_"Agitated?"_ Gavallan dropped his feet to the floor and sat bolt upright. "Did he say what it's ab—"
Just then the door burst open and Antony Llewellyn-Davies, the firm's head of capital markets, rushed into the room.
"Tony, what is it? What's wrong?"
But one look had already told him everything he needed to know.
# 7
#
**H E'S BACK,"** shouted Antony Llewellyn-Davies. "And he's calling our stock a 'scam dog.' Cocky bastard!"
Gavallan rounded his desk, confronting the anxious man in the center of the office. "Who's back?"
"Who do you think? The Private Eye-PO. He's worse than the bloody herp. But this time he's gone too far. It's slander. I swear it is, Jett. Mercury, a scam dog? Never."
Gavallan knew all too well what a scam dog was. Slang used by day traders and Internet stock junkies, it connoted a stock that was at worst a fraud—hence, "scam"—and at best an underperforming or poorly run company—hence, "dog."
"Okay, Tony, let's calm down. Just give it to me from the top."
"If you'll step aside, I believe the expression is 'Better show than tell.' "
Llewellyn-Davies was a tall, thin man with wavy blond hair and a bobbing Adam's apple. An émigré from "the City," as London's financial district was known, he looked every inch an advertisement for the English upper classes. Gray slacks, white shirt, and navy pullover: It was his uniform, and he wore it every day. Add the apple blossom cheeks and the look of childish outrage and he was the old Etonian who'd never grown up. Just thirty-one, he was the youngest member of Black Jet's executive board, and its latest addition.
Gavallan allowed Llewellyn-Davies to pass, and a minute later the men were huddled over the monitor reading the Private Eye-PO's latest salvo at the Mercury Broadband offering.
_Hi, kids! Surprised to hear from me again so soon? Don't be. News this sizzling pops right out of the pan and into your laps. Don't thank me. Thank our sponsors at Black Jet Securities. Last week, we showed you a pretty pic of Mercury Broadband's Moscow network operations facilities. Très déclassé, n'est-ce pas? This week, we go a step further. Your Private Eye-PO has come into possession of documents proving once and for all that Mercury is nothing but a hairy little scam dog with mucho fleas_.
The Private Eye-PO went on to claim that Mercury Broadband had not purchased sufficient Cisco routers to service its two million business and residential customers in Central and Eastern Europe. (Routers formed what was known as the "IP backbone" and were basically sophisticated machines that channeled digital messages to the proper addresses.) As proof, he contrasted a footnoted item in the Mercury offering prospectus with a copy of an internal accounting document from Cisco Systems, the giant manufacturer of Internet operating equipment. Whereas the prospectus claimed that Mercury had purchased over three million dollars' worth of equipment from Cisco in the last year alone (and even listed the products: the 12000 series Gigabit Switch Router, the 7500 series router, and the MC 3810 multiservice access convertor), Cisco's internal "customer revenue summary" showed cumulative sales to Mercury during the period 1999–2002 totaling just $212,000.
The missive ended with an unusually brazen sign-off.
_Shocked, loyal readers? Not as much as the hotshots at Black Jet, I'll bet. Or are they in the know? Here's a quick_ _lesson, Mr. Gavallan: No routers, no customers. No customers, no moolah! Remember, it's never too late to ix-nay the deal, Jett. You've done it before at the last minute—and for less of a reason. Will your pride permit you to do it again? Or is the going price for honesty two billion smackers these days? Hey, all you lowlifes in San Francisco, can you say "due diligence"? Better yet, can you say "class action"? See you in court, Jett!_
"Class action, my ass," spat Gavallan, chewing the inside of his lip, fighting to control the fount of anger welling inside him. "I toured their network operations centers in Kiev, Prague, and St. Petersburg. Their facilities are top of the line. They've got a dozen engineers on payroll in Geneva laying out plans for the new expansion grid. He's got it all wr—" Aware that his words were sounding more like an excuse than an explanation, he cut himself off. "This is bad, Tony."
Llewellyn-Davies crossed his arms, nodding. "Indeed. Do we just ignore this too? I mean we can't, can we? This is the second time in two weeks he's come after us. First Moscow, now this. Hasn't that chum of yours found him yet? The Internet detective, what?"
"No, not yet," said Gavallan, wanting to add that he was hardly a chum. Two days earlier, he'd contacted a man rumored to be the best in the business at what he did—namely, track down thieves and criminals who trafficked hidden inside the web—and provided him with the Private Eye-PO's web address along with instructions that he needed him found within seventy-two hours. "Found" meant a name, address, and telephone number. A price was given: a fifty-thousand-dollar retainer to be wired to an account in the Cayman Islands, and fifty thousand more should the deadline be met.
Llewellyn-Davies printed out the accounting document and handed it over. "Ask me, it looks shoddy enough to be real. Still, these days you can't tell. Any two-bit con artist could run up a copy of Cisco letterhead."
"But why would he?" asked Gavallan, happy to have someone else defending Mercury for once. "Tell me that and I'll tell you if the documents are real or phony."
"Ah! The million-dollar question," declared Llewellyn-Davies. "First answer's obvious: Chap wants to push down demand so he can scoop up some shares himself. Hold 'em or flip 'em, it's all the same. He knows Mercury's a golden goose and he wants to make some dough."
"If it were anyone else, I might agree. But this guy's reputation's too good. He's no pump-and-dump artist. And Mercury's no penny stock. Last few tech IPOs out of the gate he's called to within ten percent of their first day's close. The guy's a sharpshooter."
"The question remains, why is he taking aim at us?" Llewellyn-Davies pursed his lips and put a finger to his chin, and Gavallan noticed that his skin had taken on a peculiar yellowish cast. He couldn't help thinking the man looked even thinner than usual.
"I suggest we call Cisco right away," Llewellyn-Davies went on. "Tell them we're double-checking, engaging in a round of last-minute due diligence." Rising to his feet, he picked up Gavallan's phone and dialed 9 for the main operator. "Let's see if there's anything to this."
An astonished voice bellowed across the room. "Put that fuckin' phone down, Two Names."
Bruce Jay Tustin stormed into Gavallan's office, his cheeks flushed, eyes afire. "What a disaster! I can't believe this son of a bitch. Christ, Jett, what did you do to get this guy so pissed off at us—spill a couple drops on his shoe in the men's room? I mean, this sounds personal. Fuckin' Hatfields and McCoys."
Llewellyn-Davies lowered the phone. "We'll thank you to keep a civil tone, Mr. Tustin."
"Excuse me, your majesty," said Tustin, curtsying before Llewellyn-Davies. "Call Cisco and word will be out before lunch that we're having doubts about our client. I can hear it now. 'The hotshot investment bankers who don't know if the biggest IPO they've ever brought to market is a "scam dog with mucho fleas." ' Jesus, you queens and your dramatics." He put on a terrible English accent and walked mincingly around the room. " _'I suggest we call Cisco right away. Just say we're double-checking, what?'_ What is it, Tony? Those cocktails you take every day starting to include a little gin?"
"Piss off, Bruce. You're a fucking cretin."
"Excuse me for breathing. When did you get such a thin skin?"
"Cool it, you two," said Gavallan. He was in no mood for Tustin's theatrics or his bullying, but he had enough respect for the man to give his counsel a hard listen. Bruce Jay Tustin was the firm's resident historian, its link to the past. He'd come up on Wall Street in the heyday of greenmail, leveraged buyouts, and insider trading, in an era when news of a twenty-billion-dollar merger shook the world and of a man's taking home five hundred million dollars in salary enraged the masses. At one time or another, Bruce Jay Tustin had worked with all the great names—Henry Kravis, Boone Pickens, Carl Icahn—and some of the not so great ones—Ivan Boesky, Martin Siegel, Mike Milken. He was rude, maladroit, and pathologically disrespectful. He was also savvy as hell, and he wasn't shy about shining his Diogenes lamp on Black Jet's own offerings.
"Bruce has a point," said Gavallan. "Best if we keep our lips sealed for the moment."
Llewellyn-Davies replaced the phone in its cradle, but not before whispering a catty, "Fuck you, Bruce."
Gavallan returned his attention to Tustin. "You think it's personal too? What gives you that idea?"
"Hell, I don't know, but what's all that stuff about pride and 'you lowlifes' and canceling a deal at the last minute 'for less of a reason'? Guy sounds like as much of an asshole as me."
"I won't argue with you there." The truth was that Gavallan had also thought the Private Eye-PO's latest remarks sounded personal. Something in his words rang a bell, and though Gavallan couldn't quite put his finger on it, it disturbed him nonetheless to think that somewhere in the past he'd wronged someone badly enough that the person would seek him out and try to return the favor. "You remember anyone we cut off at the last minute?"
"Last minute?" asked Tustin. "Two years ago we canceled the whole goddamn slate overnight. We had eight companies registered that we had to shelve. I think we ended up saving two of 'em. The others went back for more rounds of funding or got snapped up by bigger fish."
"I'll do some checking," volunteered Tony Llewellyn-Davies. "See how many issues we pulled in the last five or six years."
"What's the word on the street, anyway?" Gavallan asked Tustin. "Anyone buying this guy's shtick?"
"Fidelity's already been on the horn with me, and so has Vanguard. I told both of them the guy was full of shit, but Fidelity cut their order. Said they didn't like the Eastern European exposure. If you ask me they're gonna flip and get out."
Gavallan ran a worried finger across his chin. Fidelity was the nation's largest manager of mutual funds. If it backed out, others would too. "Anyone else call?"
"Scudder and Strong," offered Llewellyn-Davies, giving the names of two more influential funds. "Not to worry. They're still buyers, both of them, and looking to build a big position in a couple of their funds. I'm sure we'll hear from others in the course of the morning."
As it so often did, the market was sending Gavallan contradictory signals. Someone believed the Private Eye-PO. Someone else thought he was a raving lunatic. It was a constant game of tug-of-war. Gavallan likened it to guessing which way the wind would turn, and felt the only thing to do was to sniff the air and go on your own instincts. He walked to the window and looked down on the city. The streets were glossy and gray and crowded with automobiles.
As if reading his thoughts, Tustin asked, "You're not thinking of postponing the offering?"
"Might not be an unwise move," mused Llewellyn-Davies. "Give us time to sort everything out. Set this Private Eye-PO chap straight."
"It would be a very unwise move, and we all know it," Gavallan replied sharply. "The question is whether we have a choice. Word hits that Fidelity's cutting their order and others might follow suit. It could kill us."
He watched Tustin and Llewellyn-Davies exchange concerned glances. For all their adversarial banter, the two were close friends. A year earlier, when Llewellyn-Davies had suffered a relapse, Tustin had visited him nightly, bringing books and videos and sometimes sneaking in a plate of the Englishman's beloved five-alarm curry from his favorite Indian restaurant.
"I thought we were safe on that one," said Tustin, his swagger conspicuously absent.
"It's spread out," said Gavallan. "Lehman and Merrill took ten apiece, but it's still a handshake deal. Best case, we're left holding thirty million."
"That's a high price to win some business, Jett, old boy."
"Maybe," said Gavallan.
They were talking about the bridge loan he had floated Mercury to win the deal: a short-term, fifty-million-dollar facility to help the company tie up real estate, purchase much-needed hardware, and lease fiber-optic cable. In good times, bridge loans were a wonderful way to leverage the fees a bank could earn on a transaction. You were out the money maybe ninety days. You charged a juicy premium over prime. And you won the loyalty of your client by showing faith and shouldering some of his risk.
But these weren't good times, and right now the bridge to Mercury was looking to be a damned fool thing. First off, it had eaten up the last of the firm's capital. Second, it had left Black Jet reliant on current earnings to meet its cash flow requirements. Maybe Tustin was right. Maybe it was a high price. But it had been necessary. Crucial even. Black Jet needed the Mercury business and the bridge had won it, allowing the smaller upstart to steal the prestigious offering from under the noses of the big guns in New York.
So far, Gavallan had managed to farm out twenty of the fifty million to a few friendly banks—half of what he'd hoped. If the deal went south, Black Jet would be out thirty million dollars. It would be too late for layoffs. He'd be forced to sell his company to the first interested party at a fire-sale price. _If the deal went south..._
Gavallan swore to himself it would not.
"We can't just sit still," he said, at once disheartened and energized by the latest development. Like all adrenaline junkies, he functioned best in time of crisis. "Our silence will be regarded as an affirmation of the Private Eye-PO's warnings. The pictures were one thing, but Tony's right—he's gone too far this time. It's as if he were building a case against us." _Against me_ , came an unprovoked thought. "Bruce, do me a favor and get Sam and Meg down here, on the double."
Sam Tannenbaum was Black Jet's in-house counsel, Meg Kratzer its head of investment banking.
"Aye, aye," said Tustin, saluting, then turning on a heel and hustling out of the room.
"Stupid git," laughed Llewellyn-Davies, a bit of color returning to his cheeks. "Doesn't even know you're Air Force."
Gavallan laughed too. "You want a coffee? Something to eat while we wait?"
"No thanks. I'm fine as is."
"Sure? I'm thinking of a breakfast burrito. Sausage and egg. Maybe a soda. Didn't teach you to eat like that at Eton, did they?"
"Stone the crows, no. A burrito would probably send me to the heavens." Llewellyn-Davies coughed once, violently. "Pardon," he said, raising a hankie to his mouth.
"You okay, Tony?" Gavallan asked, concerned.
"I'm alive, Jett. That's good enough for me."
"If you need anything..."
"Yes, I know. _Ask."_ Llewellyn-Davies knitted his brow inquisitively. "Not looking for another ticket to the ball tonight, are you? Hoping I might opt out?"
"No," said Gavallan abashedly. "No, no, no."
"Good, because I have every intention of attending. I can't wait to see you mount the dais and make a bloody fool of yourself. You have to pay good money for that kind of entertainment."
"You bastard!" said Gavallan, laughing in earnest for the first time that morning and clapping his friend and colleague on the back. Sometimes it was hard to hide his admiration for Llewellyn-Davies. It had to be damned tough living your life on a leash, he thought, relying on ten different combinations of six different pills—"cocktails," they were called—to be taken six times a day. He remembered the frail, sallow man who'd showed up for the interview seven years earlier, the thousand-yard stare, the unflinching honesty.
"I'm sick," Llewellyn-Davies had said. "You can see that. But I can work. Have to, actually. Can't go out leaving debts behind me. What would my dad say? An accountant, don't you know?"
His resume read like gold. Oxford, Harvard, a year at a bulge bracket firm before being fired for excessive absences. Gavallan had made some calls beforehand. Smart as a whip, came the unanimous response. Polite. Great sense of humor. Clients love him. But, come on, the coughing, the sweating, all those doctor's appointments. How long's he got, anyway? Six months? A year? Who wants to sit next to a fuckin' cadaver all day long? Besides, you never know. Shit may be contagious that way, too.
"The job's for a trader on our Swiss franc book," Gavallan had said. "Pays fifty thousand a year plus a fifteen percent bonus if you don't lose us too much money. If you have to miss work, get someone to cover for you. If you can't find someone to cover for you, call me. Understood?"
Llewellyn-Davies had nodded, his jaw clenched, eyes welling. "Monday is it, then?" he'd asked, wiping his cheek with the back of his hand.
"You kidding?" Gavallan had exclaimed, standing and walking to the door. "You're starting now. Take off that necktie and come with me."
Gavallan looked at Llewellyn-Davies now, wondering if maybe he was remembering the same moment. Seven years later, Tony wasn't simply alive but a vital component of Black Jet Securities and one of Gavallan's most trusted lieutenants. For a few more seconds, neither man spoke, and the silence that invaded the room was soft and comforting.
"Jett, do you think it might be true?" Llewellyn-Davies asked finally, in his mildest brogue. "You think Kirov's having us on, then?"
"Is it true?" For once, Gavallan didn't have an answer. Shrugging, he was unable even to mouth the requisite denials. The answers, he knew, lay elsewhere. In the past. In his judgment. In his greed.
And instead of looking at the delicate features of Antony Llewellyn-Davies, he was meeting the brooding, religious gaze of Konstantin Romanovich Kirov the night they had first met six months before.
# 8
#
**Y OU ARE MR. GAVALLAN,** I think."
"Mr. Kirov. It's an honor."
The two were standing in a plush reception hall on the campus of Stanford University. Kirov had been invited to deliver the annual Grousbeck lecture on foreign affairs. As his subject, he had chosen the current state of the Russian legal system, and his performance had been impressive, an impassioned sixty-minute discourse hitting all the buzzwords a liberal California audience was dying to hear. The need for an independent judiciary, ratification of the nation's highest judges by a legislative authority, freedom of the press, the right to free speech. It was the _Federalist Papers_ dressed up in an Italian blazer, Cartier links, and Lobb shoes, and topped off with an irresistibly cosmopolitan Russian accent. Kirov was still glowing from the standing ovation he'd received.
"Such a joy to speak to an American audience," he said, patting his brow with a folded handkerchief. "If only my countrymen understood the importance of democratic institutions as well as yours. I must always remind myself that you have two hundred years' experience to put your ideas into practice. Russia has a thousand years of an altogether different experience: oppression, tyranny, poverty. In short, the boot." He balled his hands into fists and stamped his foot dramatically on the wooden floor, but his optimistic grin promised Gavallan and the circle of devotees around him that if Konstantin Kirov had anything to say about it, "the boot" would soon be a thing of the past.
"Shall we find someplace private to speak?" Kirov asked Gavallan, and grasping him amicably by the arm, led him from the crowded reception room into a quiet hallway outside. "There, that is better. Now we can talk. Man to man."
Kirov was a slim, compact man with narrow shoulders and an economical step. Leaving the reception area, Gavallan remarked upon the strict posture and bowed head, the carriage of the arms tucked close to his body, and rushed ahead to open the door, as if ushering a clergyman or someone whose life conveyed a sanctity of purpose greater than the never-ending quest for the almighty dollar. This higher calling was also visible in Kirov's face. It was grave and focused, the skin so pale as to be translucent, the eyes dark, deep-set, and menacing as a witches' hollow. His hair was a raven's black. Cropped close to the skull, it accentuated the sharp cheekbones and drawn jowls. But there was fun in him, too. His mouth was puckish, as if ready to smile given the slightest reason. His eyes could surprise you with their playfulness. And he had a fine, boisterous laugh, louder than one would expect from such a small man. Mostly, though, what Gavallan sensed about the man was a monastic self-control, a zealot's singularity of purpose.
"I've heard much about you, Mr. Gavallan," Kirov went on. "Believe me when I say the honor is all mine. Thank you for making the trip to see me. Since you have come, I take it you are familiar with my company, Mercury Broadband?"
"Naturally," said Gavallan, aware he was being patronized. He sure as hell hadn't driven to Palo Alto to brush up on the principles of American democracy. "You've built an exciting platform for the industry. We're all very impressed."
The mandate to take Mercury public was the hottest ticket on the street. All the big boys wanted in. Credit Suisse, Morgan, Goldman. Gavallan considered it a miracle in itself he'd been able to secure an hour of face time with the Russian tycoon.
"Yes, it is time we offer shares in Mercury to the investing public," said Kirov. "Time to show the world Russia is no longer a second-class country. That Russia is not a land _of_ criminals, _for_ criminals, and _by_ criminals. That rights of ownership, once documented, are respected by rule of law."
"I couldn't agree more," said Gavallan, liking the man: the muted confidence, the palpable determination. Of course, if Mercury Broadband had been solely a Russian company, as Kirov was implying, Gavallan wouldn't have touched it with a ten-foot pole. But with operations in Switzerland, the Czech Republic, and Germany—and even an R&D facility literally just up the road in Palo Alto—Mercury Broadband deserved to be called a multinational, and multinationals were exactly the kind of client Gavallan was looking for. "The timing is ideal. The market's hungry for a first-class international operation like Mercury. I'm confident an offering would be greeted favorably."
"I am of the same opinion," said Kirov.
"Mercury's revenues have demonstrated a pattern of consistent growth. You have a solid record of earnings and a sustainable business model. We've taken a close look at the financial statements you so kindly provided, and my colleagues and I believe an offering in the neighborhood of five hundred million dollars is realistic."
"Five hundred million?" Kirov pursed his lips, his expression puckered with uncertainty.
"For ten percent of the company," Gavallan hastened to explain. "We'll go out with fifty million shares priced at ten bucks per, then float another ten percent in a year when the market sees what a great job you're doing and values the company accordingly. We don't want to sell Mercury short before you realize your true worth."
Normally, if a company met its revenue forecasts, it could count on floating additional shares within twelve to twenty-four months at a price significantly superior to the original offering. It was important, therefore, that the client not give away too much of itself at a less than maximal price.
"So you believe Mercury deserves a valuation of five billion dollars?"
"No," said Gavallan. "I'd say ten or fifteen billion, but we'll need time to work the market up to that level." It was crucial he offer Kirov a mildly inflated but marginally realistic value for his company. There were others out there chomping at the bit to get the deal, and he could only guess at how high they had valued Mercury Broadband.
The process of winning the mandate for an IPO was called a "bakeoff" or a "beauty contest," and like all mating rituals it had its own strict rules. Bankers strolled down the runway in their scantiest togs, planted themselves suggestively in the prospective client's lap, and immodestly drew attention to their most lubricious assets—namely, where they ranked in the league tables, the number of IPOs their firm had done in a similar space, and the performance of those stocks six, twelve, and twenty-four months after the offerings.
Next, they turned their attention to the client, whispering tantalizing nothings in his ear about the true market value of the company in play, boasting about the size of the offering—bigger was _always_ better—and giggling, with eager eyes, about how diligently they would support the stock. _Yessir, we'll keep the price up, up, up_. After a drink or two, it was time for the bankers to drop their negligees and show some skin, letting slip that their analyst, invariably an _Institutional Investor_ "first teamer," would start the party off with a bang by issuing a "strong buy" on the stock.
If the client was not yet sufficiently aroused, the banker would trot in the big guns, often the bank's CEO himself, to drive home the firm's overwhelming desire to win the business. With a wantonness that would make even the most jaded harlot blush, the CEO would run his hand through the client's hair, drown him in butterfly kisses, and promise his firmest, longest-lasting, and deepest professional and personal commitment to the stock.
In short, it was a diamond-crusted striptease, and the bank with the nicest tits won.
"I was thinking more in the neighborhood of two billion," Kirov suggested. "We have ambitious plans to expand. When you learn of the full scope of operations, you will be convinced."
"I don't doubt it," conceded Gavallan, not wanting to lose the business before he had it. "Two billion is doable, provided you're willing to part with the extra chunk of your company. I wouldn't advise that at such an early stage."
"Two billion," Kirov repeated, his resolve to be found in the firm set of his jaw, the narrowing of his eyes. "We must have two billion. Now is the time for us to expand. We must strike while the iron is heated."
"Two billion it is. It's big for Nasdaq, but why not."
"I'm afraid that Nasdaq is out of the question," said Kirov, his voice hardly a whisper.
"Oh?" asked Gavallan, knowing that this was how the Russian showed his anger, not with bluster but with discipline, the fist clenching tighter.
"Nasdaq is for new, unproven companies. We are established. We are profitable. A market leader in the East. Perhaps you are not as conversant about our company as you should be. It comes down to a question of face. We, Russians, have a terrible inferiority complex. Several of our nation's larger corporations are already trading on the New York Stock Exchange. We must list Mercury alongside them. It is the New York Stock Exchange or nothing."
Gavallan made the appropriate soothing noises, ego gratification being perhaps the most important job of a chief executive. He'd bring up the listing requirements at a later date— _if there was a later date_. After a promising start, the meeting had embarked on a series of wrong turns. The first order of business was to change the atmosphere. A long, drafty hallway was hardly the place for this conversation.
Gavallan suggested they continue their discussion in the provost's lounge, where they might sit and have a cup of coffee. He had attended business school at Stanford and knew the provost's lounge as the spot where the university president wined and dined the school's more important benefactors. Apparently there was something about oversized club chairs and oil portraits of long-dead scholars that made people free with their checkbooks.
Inside the lounge, the two men sat down, agreeing almost immediately that the chairs were wonderfully comfortable. Settling himself in, Kirov delved into his pocket for a sterling-silver cigarette case. "Sobranie?" Opening it, he offered the case to Gavallan. The cigarettes were long and black, the two-headed Russian eagle stamped boldly above a shiny gold filter. One head faced east, the other west. In Russia, danger had always come from within and without.
"No thank you," said Gavallan. "I don't smoke."
"I know, I know," pleaded Kirov, as he slipped one into his mouth and lit it with a matching silver lighter. "But a man should be allowed one vice." The brows jumped excitedly under the curtain of blue smoke. "After all, we are not saints!"
Kirov drew thoughtfully on the cigarette, inhaling for what seemed an eternity before expelling the smoke in neat flutes through his nose. "I have spoken with a few of your competitors these last days," he said offhandedly. "As you can imagine, a good many are anxious to work with us. Please don't think me rude, but I was hoping you might be able to tell me why I should consider someone outside of New York. Someone so much smaller."
Gavallan made it a point never to discuss his competitors—comparisons conveyed weakness and insecurity. "True, we're a smaller company," he said, launching into a pitch he'd given a thousand times, "but we think our size is one of our advantages. We choose our clients with great care and we like to think they exercise the same scrutiny in choosing us. Our record in the Internet sector is second to none. Of the forty-two companies we've taken public in the last four years, more than fifty percent are trading at significant multiples of their offering price. Not one has gone belly-up. We're selective with whom we work, Mr. Kirov. Black Jet's name on a prospectus has come to indicate a certain quality. We're deeply committed to the companies we offer our investors. The clients for whom we do choose to work receive the complete and dedicated resources of our company."
"So _you_ choose your clients?"
"I prefer to think that we choose each other. Hopefully, bringing Mercury public will be the first step in a long relationship between our two groups."
"So you wish to work with Mercury? You are sure?" Kirov's amused tone indicated he hadn't heard this approach before and just might be buying into it.
"It would be a privilege. And I think I can promise that in the current environment, Black Jet could insure that a Mercury offering would be a home run."
Kirov nodded approvingly. If nothing else, he looked to be enjoying the courtship. He questioned Gavallan about Black Jet's ability to manage so large an offering, its relative inexperience in working with international companies, and its commitment to supporting the stock once it began trading. He asked about Black Jet's analyst, inquiring whether he was on _Institutional Investor's_ first team _(he was, at a salary of four million a year!)_ , and was curious to know if the larger funds would be buyers of the stock, meaning if they would look to build a long-term position in Mercury.
In short, he asked all the right questions. Either he'd been briefed by his chief financial officer or he'd already sat through a dozen of these pitches.
Gavallan addressed each of Kirov's concerns in turn. Knowing he was at a disadvantage to the bulge bracket firms, which could commit a sales force double the size of his own to the IPO and promise a hundred-million-dollar kitty to keep the float active, the stock price above water, he concentrated on Black Jet's strengths: its topflight research team; its position at the vanguard of the new economy; its close ties to the nation's largest mutual funds. In the end, though, it came down to personality. Everybody on the street was offering the same services, more or less. It was a question of whether Kirov liked Miss August or Miss November.
At the close of Gavallan's comments, Kirov placed his hand atop the American's and gave it several pats. "I have received advice from people close to me—people I trust—that you are good man. That your company may soon be very big, very powerful. Like Mercury, I think." Another pat to let him know they were on good terms. "I like you, Mr. Gavallan. You are young. You are ambitious. I sense you are honest, even if you are arrogant." He laughed quietly. _"You choosing your clients_. Very good. I must remember to use that myself one day. But I must have a reason to explain to my own shareholders why I choose your company. We, Russians, like big names. BMW, Gucci, Rolex. We feel we must carry these brands with us to prove our legitimacy. Again our inferiority complex; excuse us. But if I may speak frankly, Black Jet is not yet such a big name."
"You're right, sir. We have only nine years behind us. I hope many more will follow."
"I am sure of it. Absolutely positive," Kirov declared collegially, but the next moment he was wincing, lowering his eyes. The reassuring hand returned to its owner's armrest. "But so much is at risk. It is a critical moment for my country. For so long, we have been held back, our heads pressed beneath the water. Now that we are free, I fear we are terribly greedy. We want to suck in great mouthfuls of this oxygen we call liberty. We claim democracy as ours. We crave progress. Personal progress. Progress measured on the human scale. A phone for every house. Running water. Showers that function. Toilets that flush. Proper medical care. Hospitals stocked with adequate antibiotics, surgical dressings, and sufficient blood. We demand the latest technology.
"You see, technology is our lifeline to the West. We cannot afford to fall farther behind. The Russian people are smart and curious. They are voracious in their hunger for knowledge. We are not a nation of peasants. We are a nation of Ph.D.'s, of scientists, of doctors, and businessmen. Every new PC brought into an EasternEuropean household is a soul saved from our autocraticpast. Every home that logs on to Red Star has a window into the future. And once they see it, they will not let go." Kirov leaned closer, his eyes sparkling with hope. "In the past, weapons and ignorance kept East and Westapart. But the arms race is finished. It is time technology and the quest for knowledge bring us together. The race to advance humankind has begun, and its progress will be measured in computers, not missiles. Over time we will evolve into a single empire, a democratic union of all peoples...." Abruptly, Kirov stopped. He was breathless, and a sheen of perspiration clung to his forehead. His forgotten cigarette had burned down to his fingertips, a two-inch section of ash drooping precariously toward the carpet.
Gavallan found he was breathless, too. Kirov had spoken into his heart. He had addressed all his unsatisfied selves: the conscientious benefactor, the penitent sinner, the advocate of change happiest when striving. He had touched not only his dreams but his desire to dream, which was even more important. In a world scarred with cynicism, Kirov dared to have ideals.
The Russian fixed him with a challenging gaze. "Do you believe, Mr. Gavallan?"
"Yes," said Gavallan, without hesitation. "I do."
Kirov said nothing for a few seconds, his black eyes burning into Gavallan. He had the gift of silence, of dignifying thought for thought's sake. Just then, he noticed the cigarette and rushed to put it out. He smiled, embarrassed, and the evangelist became once more the man. "I'm sorry to say you have put me in a difficult position," he said. "I have very much enjoyed our chat, but I have a late dinner with the president of one of those big names we Russians so like. He has flown in from New York to see me. I think he will promise me the moon if I ask him."
Gavallan sighed as he scooted toward the edge of the chair. Pitch over. Business lost. Next. Despite himself, he acknowledged a jab of disappointment and had to sit straighter to keep his shoulders from sagging. He knew he had had no right to count on winning the business, but he truly believed that Black Jet could do the best job for Kirov.
"Don't let me keep you," he said. "I'll be in the office tomorrow if we might answer any questions for you. If you have a free hour, I'd enjoy showing you around the firm." He rose. "But, Mr. Kirov, I want you to know one thing."
"Yes?"
"I do believe."
Kirov rose from the chair, but a moment later sank back down into its cushioned folds, motioning Gavallan to sit. "I will make you proposition, Mr. Gavallan. We are close to finishing the buildout of our central Russian operations. Kiev, Minsk. These are large cities; maybe a hundred thousand subscribers each. Unfortunately, we need fifty million dollars to complete the construction."
"Fifty million?"
"I am thinking a loan to be repaid from the proceeds of the IPO. It is uncommon?"
"Not at all," said Gavallan, unable to keep the excitement from his voice. Part of him wanted to jump at the chance, another to take a step back. A fifty-million-dollar loan would exhaust Black Jet's resources and leave it perilously exposed to the market's vagaries. It was a tremendous risk. Yet, the fees the deal would bring promised to be tremendous, more than anything Black Jet had ever earned on a single transaction. Add to that the interest on the loan, and of course the prestige... My God, Gavallan said to himself, the prestige alone would do wonders for the company.
He looked at Kirov, doing his best to size him up. The personality contest went both ways. The man was controlling, vain, and at least a little bit of an egomaniac. But his conceit was his strength. How else could he muster the energy, the dedication, the tenacity to build a company like Mercury? Who but the vainest sort of individual would dare talk of aiding his country in such grandiose terms?
Gavallan turned his thoughts to the bigshot flying in from the Big Apple on his big Lear or big Cessna or big Gulfstream. Inside, he smiled. It was a delinquent's smile, an outsider's smile, and it reveled in the pique and fury and disbelief that the overconfident executive would feel when he learned that Black Jet had won the two-billion-dollar mandate to bring Mercury Broadband public. Nothing came easier to a Texas farmboy than spitting in the eye of his betters.
Maybe the Russians weren't the only ones with an inferiority complex.
"Tell you what," said Gavallan. "Cancel that dinner engagement. Let Black Jet take you public and I'll write you a check first thing in the morning for fifty million dollars. Prime plus seven to be repaid out of the proceeds of the IPO." He stuck out his hand.
Konstantin Kirov hesitated, searching Gavallan's eyes. "I can trust you with my baby? It is not just for me, but for my Russia, too."
"Yes, you can trust me."
"Prime plus five and we repay within thirty days."
"No," said Gavallan, tasting the deal, wanting it more than anything, but never so much as to make a poor agreement. "It has to come from the proceeds."
Giving a fateful shrug, Kirov rose laboriously from his chair and grasped Gavallan's hand. "Yes, we shall work together. You are a believer. I see it in your eyes." He laughed richly. "I tell you something. Between us, I never like BMW anyway. But you must promise to call me Konstantin. In Russia, business is family."
Gavallan stood, and though the handshake was awkward and formal, he found himself laughing with his new client, new friend, and new family member, Konstantin Romanovich Kirov.
# 9
#
**T HEY'D MOVED INTO A CONFERENCE ROOM** down the hall. A "working room," they called it, and it was fitted for the late nights and early mornings that claimed so large a part of an investment banker's existence. Besides the glass table and low-backed chairs, there was a refrigerator stocked with Coke, Mountain Dew, Red Bull, and, as if an afterthought in their caffeinated universe, Evian. One cupboard held chips, cookies, and candy bars, and another, rumor had it, fresh fruit—though Gavallan had never seen anyone munching so much as a grape. Next door there was a pantry with a microwave oven, a freezer, and a coffeemaker. A paper plate bearing the remains of Gavallan's sausage and egg burrito sat half in, half out of the trash can. A pall of cigarette smoke hovered below the ceiling. Let mortals worry about ulcers, colitis, and quadruple bypasses. They weren't subject to daily deadlines that could cost a firm tens of millions of dollars and their own paychecks that extra, all-important zero.
Gavallan leaned back in his chair, balancing on its rear legs. He'd already gone over the Private Eye-PO's most recent message and its accusations of misrepresentation and fraud. Reluctantly, he'd let everyone in on Grafton Byrnes's secret visit to Moscow and his failure thus far to report in. He did not, however, feel it necessary to tell them about Byrnes's early checkout.
"Listen, people, our back's up against the wall here," he said. "We need to take a close look at our deal books and see if we can find any holes that correspond to the areas the Private Eye-PO is attacking—namely, the Moscow network operations center and Mercury's hardware purchasing. I don't think we will, but I'm not going to my grave like the captain of the _Titanic_ saying, 'She's unsinkable.' No one's leaving this room until we decide just what the heck we're going to do. _Comprende?"_
His eyes moved from face to face, waiting for someone to pick up the baton. Bruce Jay Tustin, Tony Llewellyn-Davies, Sam Tannenbaum—or "Shirley Temple," as Tustin had christened the blond, ponytailed lawyer—and Meg Kratzer. He was waiting for someone to share his outrage, but outrage, he knew, implied responsibility, and the Mercury deal had been his and his alone from the beginning. Finally, Meg Kratzer chimed in—Meg, for whom silence was an accusation of laziness.
"Look," she said. "We handled all customer and managerial questions in-house. If something weren't kosher about Mercury's Moscow operations, we would have heard about it from one of their customers. Financial, accounting, and operational issues were completed by Silber, Goldi, and Grimm in Geneva. If there were a problem with Mercury's physical plant and inventory, they would have found it—guaranteed! I don't know a bigger tight-ass in the business."
"I do," said Tustin, rolling his eyes and lofting a thumb in Meg's direction.
"I appreciate the compliment, Mr. Tustin," she responded. "It's hard to be more thorough than a Swiss with a microscope and a mandate to inspect. Coming from someone who's such a renowned tight-ass himself, that's very high praise indeed. I will thank you, however, to keep that greasy kid stuff in which you drown your last three remaining hairs off my deal books. I needed a whole bottle of Mr. Clean to get it off last time."
"Very funny," retorted Tustin, above the nervous laughter. "Just so you know, it's _pommade_. That's French, for 'classy.'"
Meg Kratzer circled the table, passing out a thick red three-ring notebook to everyone present. She was a vital, animated woman, short, stocky, and neatly attired in an olive Valentino two-piece. Her red hair was pulled back into a severe bun. Her blue eyes glimmered with healthy determination. At age sixty-three, she was a mother of four, a grandmother of ten, and self-appointed godmother to Jett Gavallan. She'd put in twenty-five years at a well-known securities house, only to be told when she turned fifty that her shelf life had expired. The termination letter called her "irascible, opinionated, and obstinate," and said she was "unable to meet the rapidly evolving dictates of the financial arena."
Gavallan saw those same qualities as forceful, experienced, and demanding, and found her as up-to-the-minute on all matters financial as the most arrogant graduate of Harvard Business School. She was also articulate, responsible, and possessed of a wicked sense of humor.
As the firm's head of investment banking, Meg had supervised the due diligence performed on Mercury. Its being an initial public offering, this involved the systematic deconstruction and analysis of the client company. Balance sheets were audited; bank balances verified; company officers interviewed (and often investigated); clients telephoned and questioned about their relationship with said company; corporate strategies parsed; and physical assets inventoried down to the last pencil and paper clip. It was a strip search really. With rubber glove and all.
Gavallan pulled the deal book closer, glancing at the Mercury name and logo that adorned the cover. The notebook had to weigh five pounds, and inside it was all the information Meg and her team had collected as part of their due diligence on Mercury.
"Let's start with clients," he said, flipping the notebook open. "Section one."
Section one contained single-sheet summaries of over 150 telephone conversations conducted with Mercury's clients in the Czech Republic, Ukraine, Germany, and Russia. Leafing through the pages, he kept a sharp eye out for those customers based in Moscow. He thumbed past the Czech Ministry of Communication, the Kiev Education Committee, Alpha Bank (Minsk branch office), the Dresden Youth League. All declared themselves satisfied with Mercury's product and services. Finally, he arrived in Moscow: the Moscow Municipal Transportation Service, the Moscow State University department of telecommunications, NTV (one of Moscow's larger television networks). Again, all were satisfied. There were more: Romanov Bank, the Greater Russian Health and Casualty Insurance Company, Nezhdanov Construction, Imperial Aluminum Smelting and Manufacturing.
It's bullshit, Gavallan thought, perusing the summaries. Everything the Private Eye-PO had said is patently false. Unadulterated garbage. And again, he wondered who the man could be, why he was trying to savage Mercury, and why he was making the issue so personal, repeatedly mentioning Gavallan's pride.
When they'd finished with section one, Meg directed them to section three, titled "Company Infrastructure," which contained questionnaires filled out by Mercury's management. In an expectant silence, Gavallan and the others read one job description after another, all dictated by the eager and capable executives who worked at Mercury Broadband. Finally, he came upon one provided by a man he knew, Václav Panič, Mercury's CTO—chief technical officer—of European operations, a Czech-born doctor of electrical engineering, formerly a professor of computer science at Brno University.
Gavallan had toured Mercury's Prague office in Panič's company. In his mind, he saw the cool marble floors, the legions of busy workers glued to their workstations, the aisles of servers, routers, and switches housed in trim glass cabinetry. One wall in the office's conference room displayed a map of Mercury's European operations and highlighted its expansion plans. Red fairy lamps depicted network operations centers, white lines denoted the cable or satellite connections, blue lights indicated cities with over twenty thousand subscribers, and green lights showed areas where service was to be offered within twenty-four months. Mercury was driving west to Berlin, south to Budapest, north to the Baltic republics, and east to the oil and mining boomtowns of Siberia. Standing there, Gavallan had felt the company's pulse as surely as if it were his own.
"There's not a scrap in here," said Tony Llewellyn-Davies. "Mercury's as clean as a whistle. Bravo, Meg. Well done, Jett. There's absolutely nothing to worry about, at least nothing we could see."
"That doesn't excuse us if we're wrong," cautioned Gavallan. "Our name's still on the prospectus."
"Not my name, Jett," Bruce Jay Tustin pointed out frankly. "She goes south, you're on your own."
"Thank you, Bruce. You're comforting as usual."
"My pleasure," replied Tustin. "Naturally, I do expect to get your office while you're doing your time in the pokey—oh, excuse me, I mean the men's correctional facility. I've always loved the view."
"Please, Bruce," cut in Tony Llewellyn-Davies, his cheeks pink with anger. "You're being exceptionally rude, even for yourself." He offered Gavallan a look of perfect exasperation, then turned back to Tustin. "You know damned well we agreed I was to get the office."
"No, me," said Meg. "The office is mine. Age before beauty, gents."
Everyone laughed, and the tension in the room was cut by half.
"Thanks, fellas. Thanks, lady," said Gavallan. "I appreciate your efforts. Now if we can finish up, I believe we're scheduled to talk to Silber, Goldi, and Grimm."
Meg Kratzer punched some numbers on the phone. "I've got Jean-Jacques Pillonel, their MD, on conference when we're ready"—"MD" in this case meaning "managing director."
Gavallan reached a hand over the notebook and activated the speakerphone. "Jean-Jacques, it's Jett Gavallan. Good morning."
_"Bonjour, Jett. Ça va?"_
"We've got a minor problem over here. Just a headache, I'm sure. Meg tells me she's gone over it with you. Can you help?"
"Jett, this is nonsense. I read this web page already. Mercury is here in Geneva with us. We spent a week camping in their offices. Certainly there's no question of revenues; we've got the bank statements from UBS and Credit Suisse."
"Jean-Jacques, no one is questioning the revenues. It's a matter of the physical assets." Gavallan leaned over to Meg Kratzer and whispered, "They handled that too, right?" She nodded, and he said into the speakerphone, "Who did the on-site inventory?"
"Mostly, we hired independent specialists," Pillonel replied. "Systems engineers, information technology guys, you know. I supervised the project myself. A favor for my American friends. I know this is a big deal for you."
"Thank you, Jean-Jacques," said Meg, as Gavallan and everyone else at the table rolled their eyes.
"Jett, listen, no worries, my friend. We checked Mercury up and down. We even look in their shorts and count their pubics, you know. Forget this guy on the Net. _Je te dis, ça va_."
Tustin lobbed an arm across the table and punched the mute button. "Ça _va, ça va_. Same thing the fuckin' frogs said about the Maginot Line. _It ees inveencible!_ Look how that turned out."
"He's Swiss, Bruce," Meg pointed out.
Tustin shrugged. "Swiss. French. Whatever. A frog's a frog."
The room tittered nervously and Tustin turned off the mute.
"And Moscow?" asked Gavallan. "Who did you send?"
"I went myself."
"You?" It was odd, not to say completely out of the ordinary, for a senior partner of an internationally prominent accounting firm to hole up in a client's offices and physically inventory its assets. That was a job reserved for "newbies."
"With my associates, of course," Pillonel added quickly. "We have a new office in Moscow, so it was a side trip. Like I say, a favor."
"And you saw all their operations, including the network operations center?"
Suddenly the Swiss adopted a belligerent tone. "Hey, Jett, we put our signature on the offering memorandum. Last time I checked, our name still meant something—or do you pay just _anybody_ two hundred fifty thousand dollars for their help?" The voice regained its diplomatic flavor. "You are worried for nothing. How can Mercury earn so much money without having the equipment to do so? You can't harvest wheat without a thresher—know what I mean? Mercury is doing a hell of a good job, I tell you. Look at their metrics: over four million hits a day. You know I have an order with you to buy a lot of shares."
"And we'll see you get filled," said Gavallan. "Thank you, Jean-Jacques. _Au revoir_."
_"Au revoir, tout le monde."_
For a moment, there was only silence. The sound of pens tapping the table. Legs crossing. Meg Kratzer lit a cigarette and took pains to direct her smoke toward the ceiling.
There it was, Gavallan told himself. The managing director of Europe's largest accounting firm had just confirmed that Mercury's Moscow operations were up and running. Gavallan asked himself why he hadn't called Jean-Jacques Pillonel in the first place. _Because you can only trust your own_ , a cynical voice reminded him. _Because people lie_.
More and more, he was certain the Private Eye-PO had to be someone he knew, someone with a personal ax to grind.
"So, are we back at square one," he asked his colleagues, "or did we just cross the finish line?" Unspoken, but hanging up there near the ceiling with Meg's cigarette smoke and the lingering scent of his half-eaten burrito, were the words "postpone," "shelve," and "cancel."
"Where the hell is Byrnes?" griped Tustin.
"Give him time," said Llewellyn-Davies. "He'll get back to us."
"It's ten o'clock in Moscow. How much time does he need?"
"Relax, Bruce," said Meg. "I'll take Jean-Jacques's word over the Private Eye-PO's anytime. I'm sure Graf will only confirm what we already know."
"Maybe," said Tustin grudgingly. "But I still want to hear from him."
So did Gavallan. Every minute that passed without word from Byrnes fueled his worry over his friend's well-being. Still, he was pleased with the give-and-take of the discussion. If there were any doubts about Mercury, it was best that they surfaced within the confines of the office.
"So, Sam, what's your call?"
"Tough one."
Tannenbaum was the firm's resident bohemian. With his tight jeans, flannel shirt, and flowing blond hair, painstakingly groomed and tied into a ponytail, he looked like a refugee from Big Sur. "We seem to be stuck between believing in ourselves and believing the Private Eye-PO. From what I can gather, Mercury is everything we say it is. You think so. Meg thinks so. Jean-Jacques thinks so. Jupiter Metrix says so. It's a 'go deal.' At the same time, we feel compelled to trust the Private Eye-PO because he's been accurate in the past."
"Jesus, Shirley, you're getting me hard," whined Tustin. "Say what you want to say and let's get on with it."
Tannenbaum shot him a withering look, but refused to be hurried, either by Tustin or by any of the other curious faces staring at him. "Unfortunately, I don't know what to say except that we need to find the Private Eye-PO as quickly as possible and ask him where he's getting his information."
"Only one problem," said Gavallan. "We still don't know who he is."
"Can't we shut him up?" asked Meg. "Slap an injunction on him for false and deprecatory statements? I mean, what he's doing isn't any different from some wiseass issuing a phony earnings warning."
"Sure," said Tannenbaum. "But again, we have to find him first, then we have to get an injunction, and eventually we have to take him to trial. We don't have the time. The balloon is going up in five days."
Gavallan was suddenly restless. Frustration cramped his shoulders and clawed at his neck. Rising from his chair, he walked slowly round the table. All roads kept leading back to the same place. The deal was sound. The Cisco receipts were bullshit. So were the pictures of the Moscow NOC. Some asshole getting his jollies trying to hurt Black Jet or Mercury. It didn't really matter who he was, or why he was doing it. Which left Byrnes. No one knew better than he how important the deal was. Absent his word to the contrary, there was only one way to go.
"Okay, everyone, that's a wrap," Gavallan said. "We all decided on this?" Approaching the table, he extended his hand over its center. "Tony?"
"It's a go, Jett." Llewellyn-Davies laid his hand on top of Gavallan's.
"Bruce?"
"Fuckin' A, bubba. We're going in!" Tustin slapped his hand atop the two others.
"Sam?"
The lawyer looked unsure. "Umm, if you say so. Sure." Another hand joined the pile.
"Meg?"
"Hee-yah!" she shouted, half laughing, throwing her hand on top of the stack. "We're on the road to glory! Two billion or bust!"
Gavallan felt the weight of the four hands on top of his own. For a moment, his eyes passed from one person to the next. Bruce, the congenital loudmouth. Tony, the gutsy survivor. Sam, the reluctant corporate warrior. And Meg, the discarded treasure.
These were more than his friends, more than the closest of colleagues. These were the members of the family he'd chosen for himself. The pillars of the life he had built after his world had crashed in ruins about him. It all came back to people. To teamwork. To mutual accomplishment. He waited a second longer than usual, enjoying the communion of flesh, the union of wills.
"All right then," he said. "We're decided."
Without another word, he pulled his hand from beneath the others and walked out of the conference room.
**B ACK IN HIS OFFICE,** Gavallan stood by the window. Patches of blue peeked through fast-moving clouds. The harbor was alive with mid-morning traffic, tugs and ferries and tankers leaving frothy trails in their wakes. Tired, he pressed a cheek to the glass, enjoying the feel of the cool, slick surface against his skin. "Mercury is solid. Mercury is solid." He repeated the words over and over, a mantra to convince himself and the whole world. But he'd been in the business too long to believe it. Skepticism had become second nature.
Right now only one thing was certain: If what the Private Eye-PO claimed was true and Black Jet Securities went ahead and brought Mercury to market, he, as sole owner of the firm, would be looking at a class action lawsuit of tobaccoesque proportions. Forget recouping the thirty-million-dollar bridge loan. Forget selling the company. Black Jet Securities would be doing a Drexel quicker than he could say "Mike Milken," and he himself would be learning to trade stocks by Touch-Tone phone from the inside of a federal prison.
Returning to his desk, he found the shaman staring at him. He met the squat carving's gaze and stared right back.
"Find him," he ordered the Indian medicine man. "Find him, now!"
# 10
#
**C HILD'S PLAY.**
Jason Vann took a look at the Private Eye-PO's web page and smirked. _An amateur_. He could see it right away. No sidebars. No pull-down menus. No search fields. And certainly no banner advertisements that might earn him a little dough. Just the guy's name written across the top in faggy script, a half dozen hypertexted headlines, and a bunch of charts chronicling the latest goings-on in the exciting world of venture capital financing, tech-related mergers and acquisitions, and initial public offerings.
There were tables showing IPOs coming to market next week, IPOs recently priced, the performance of IPOs just launched, and the year-to-date performances of the Private Eye-PO's personal picks. The symbol for each stock was colored an electric blue, denoting a hyperlink to drive the reader to a related site. Vann double-clicked on a few of the links. As expected, they led to commercial portals that offered free content—Yahoo! Finance, CNBC, Bloomberg. Definitely a one-man show. Best of all, there was an E-mail address at the bottom of the page. PrivateEyePO@hotmail.com. Vann read it, and his smirk took on a decidedly arrogant cast.
This would be the easiest hundred grand he'd ever earned.
The individual whom Jett Gavallan had called "the top man in his field" kept his office in two spartan rooms on the second floor of a modest colonial home in Potomac, Maryland. And the "field" to which Gavallan had been referring was alternately called "cybersleuthing," "systems security," or, if you were a black-hat hacker, "betraying the cause."
If you needed to find someone on the Net quickly—friend or foe, cracker, script kiddie, or gray-haired hacker—Vann was your man. The FBI had called him to discover who had hacked into NORAD and raised the entire United States defense establishment to Defcon 2. Since then, he'd lectured regularly at Quantico. The CIA had paid him handsomely to track down a team of cyberterrorists who had defaced Langley's mainframe. They'd thought so highly of his methods that they'd contracted to keep him on permanent retainer. Five thousand dollars a month so the spooks in Virginia could install a direct line to his home.
And Mr. John Gavallan of San Francisco was paying him a hundred thousand bucks to find out the name and home address of some Net loudmouth calling himself the Private Eye-PO.
_Child's play_.
Vann's offices were small, each room ten by twelve. Windows high on the wall overlooked a green pasture where horses were left to run. Not that Vann spent much time looking. Everything in the world that interested him could be found in this room or the next. Every bare surface was packed with computers and peripheral equipment: PCs, Macs, servers, scanners, printers. At last count he had nine systems up and running, twenty-four seven. He also had some cool _Lord of the Rings_ stuff on the shelves, a Lava lamp he'd gotten for Christmas that he couldn't decide whether was lame or not, and a model of the Eiffel Tower he'd gotten at Paris! Paris! on a trip to Las Vegas last year with his parents.
Scrunching his nose, Vann saddled closer to the monitor. Though not entirely necessary—given the parameters of the assignment—he decided to spend a few minutes studying the Private Eye-PO's web page. He backtracked through a month's worth of the man's weekly columns, basically "rants and raves" about new issues coming to market. Finding the attacks on Mercury Broadband, an IPO managed by Black Jet Securities, he understood why Mr. Gavallan was in such a hurry to find out who had written such mean-spirited words. If it had been his stock the Private Eye-PO was attacking, Vann would have killed the guy.
The first thing Vann did was contact a buddy who worked for Hotmail.com and get him into a private room on IRC, the Internet Relay Chat.
Hotmail.com was a free mail service, and anonymous—that is, you could set up an account there without giving your name, address, phone number, or credit card, any of which would have made it way too easy for someone like Jason Vann to find you. You did, however, have to provide a valid E-mail address to retrieve the password you needed to access the system. Unbeknownst to the lay user, the sign-on page contained an "x field" that recorded the IP address—the "Internet protocol" where the mail was sent.
Vann's contact at Hotmail.com was Ralph Viola, who went by the handle "Stallion."
JV (Jason Vann): My man, I need the 411 on one of your users. Usual terms apply. Here you go: PrivateEye-PO. Whatcha got?
Stallion: Wait a minute while I get the logs.... Okay, got it. Your man's IP=22.154.877.91. Logged on this morning at 7:21 EST. Gaming tonight? We're doing Stalingrad. You can be General von Paulus.
JV: Screw that. Krauts always lose in that one. Too busy, anyway. Who is the ISP?
Stallion: Not so fast, jack. Time to up the scratch. People watching over my shoulder. Five bills'll do the trick.
JV: You're a thief, but since I'm in a hurry, okay. Try it again and I'll brand thee "Highwayman."
Stallion: And thee "Rogue!" The ISP is BlueEarth.com in Palm Beach, Florida. Thanks and aloha, McGarrett!
JV: Aloha!
Since word had gotten out that Vann had joined up with the feds, everyone had started calling him McGarrett. Like Steve McGarrett of _Hawaii Five-O_ , which even the biggest dumb-ass knew was the coolest cop show ever on TV. _"Book him, Danno!"_
He looked down at the name of Private Eye-PO's ISP, or Internet service provider. BlueEarth.com. Every time the Private Eye-PO logged on, his modem was connected to one of BlueEarth's servers, and that server had its own unique and permanent Internet protocol address. Stallion had given him the server address where the Private Eye-PO's mail was last sent and the time of transmission. All Vann had to do was contact BlueEarth.com and find out the IP and corresponding phone number that had logged on to that particular server at 7:21 EST this morning.
_Child's play_.
Vann entered his mail program and pulled up a file containing the names, E-mail addresses, and web handles of people who worked for ISPs. When he'd first gotten hooked on the Net there were maybe a hundred ISPs across the country. Now there were thousands. He guessed BlueEarth was a newcomer, because he couldn't recall ever coming across the name before. No matter; he was sure that somewhere in his files, he'd have something about BlueEarth. Some of the information came from his friends. Some he purchased. Some he procured by more sophisticated means.
Amazingly, the search failed to turn up any associates he might contact at BlueEarth.com, no Ralph "Stallion" Viola he could slip five hundred bucks in exchange for Private Eye-PO's IP and phone number. Vann scratched at his hair, frowning.
Suddenly the screen stuttered, went blank, then colored a sizzling hot pink.
_Reset. Fatal exception at F275A-II/7. 13:52:45_.
Maybe it wouldn't be the easiest hundred thousand he'd ever made.
A long gulp cleaned out the Dew. He tossed the can in the trash and slid back his chair.
It was a lovely day outside: blue sky, a few clouds, temperature closing in on ninety. The Bullises had their Thoroughbreds roaming free in the pasture. He particularly liked the bay gelding and was certain it would have made an excellent charger. If he ever learned to ride, he might ask the Bullises to allow him to take the bay to the jousting tournament at the annual Renaissance Faire in College Park. He toyed with the idea for a few seconds, then discarded it. He'd never be able to find a decent suit of armor. Besides, before that, he'd have to learn how to drive.
Cracking his knuckles, Vann brought his chair close to his PC. It looked like Mr. Gavallan was going to make him earn his money today. Vann didn't like hacking into an ISP, but sometimes a carefully considered violation of an individual's or enterprise's privacy was necessary. If anyone had a problem with it, they could take it up with the FBI. Agent Fox Mulder would be pleased to assist in the matter. And whistling the theme from _The X-Files_ , he began banging code into his computer, working his way, step by laborious step, into BlueEarth.com's innermost sanctum: the customer address files where they guarded the names, phone numbers, and IPs of all their clients.
**T HREE HOURS LATER,** he was still working.
The sun was setting and the small room had grown hot and stuffy, the air as rank and cloying as a high school weight room's. Vann didn't notice. Head bowed, he banged line after line of code into the computer, waiting for the walls to fall. So far, every one of his ploys had failed. He couldn't find a back door. The firewall was impenetrable. And he couldn't keep hacking into the site much longer for fear of being spotted by BlueEarth's security programs.
A voice called from downstairs. "Jason, dinner's ready!"
"Just a second."
Vann tapped at the keys a few moments longer, then threw his hands up. He was beaten and he knew it. "Damn it all!" he muttered, sliding back from his desk and staring at the impotent keyboard.
"Jason!!"
Vann logged off the Net and stalked from his room. There were other ways of finding the Private Eye-PO. It might take a little longer, but in the end, he'd nab him just the same. These "messiah" types were all alike. They craved attention. The anonymous ones were the worst. They couldn't go a day without dropping into some chat room on the web or the IRC to learn what their public thought of them. And next time the Private Eye-PO did that, Jason Vann would be waiting for him. He just hoped it was soon. Vann wanted the fifty-thousand-dollar bonus.
"Coming, Mom," he called.
"And be sure to wash your hands and face."
Vann closed and locked the door behind him. Here he was, thirty-nine years old, and his mother was still telling him to scrub up before dinner. Maybe when he turned forty she would start treating him like an adult.
# 11
#
**G HOSTS IN A FROZEN MIST,** they ran.
Twelve men. Bold apparitions clad in white, doggedly advancing to the same silent cadence, their breath erupting in violent, staccato bursts. Forward. Ever forward. Against the wind. Against the snow. Against themselves.
The cold seeped through their boots, clamping their toes and nipping at their heels with teeth as hungry as a bear trap's. The snow was deep here—two feet, at least—a soupy, devilish mixture of slush and dirt and the spores from the unyielding tundra. And this one week from midsummer's eve. A frantic wind howled around them, clawing at their eyes, scratching their cheeks, slyly slipping beneath the folds of their anoraks and burrowing through their sweaters, their fatigues, and their thermals, biting their skin like ice on fire.
The men's legs were strong, their muscles hard and conditioned, exquisitely calibered pistons willing to carry them over hill and dale hour after hour. Their arms swung by their sides, the dry, rhythmic chafing of the snowsuits sounding like sandpaper scraping velvet. Each man carried a pack, and in that pack a jumble of rocks and stones weighing twenty-five kilograms—fifty-five pounds. They leaned forward as one, their well-toned shoulders and tensed abdomens working in concert to distribute the load. Soon the packs would be filled with a different cargo—timers, fuses, detcord, and plastique, sophisticated devices as far evolved from stones as men were from apes.
The wind died. The icy curtain fell, and for a minute or two the men were permitted a view of the bleached panorama around them. It was a bleak vista, white hills rolling away to the east and west, an endless plain advancing before them. The sky hovered low and gray, a sweeping expanse of nothingness. It was a pale, barren land with no sign of animals, vegetation, or human habitation. Man did not belong here, so far north; his existence counted for nothing. As punishment for their intrusion, the wind picked up so abruptly as to slap the men across the face. They were not welcome here.
Still they ran. Invaders of the Arctic Circle. Five kilometers remained to the halfway point, then back again to base by a different, more difficult route. Another twenty kilometers over uneven, climbing terrain. It was their last training run, a brutal, delirious culmination of four months' preparation. Four months without leave, without a single day's rest, without alcohol, tobacco, or women. Physical conditioning was placed at a premium, but there were mental exercises as well: endless hours mastering English, in particular the American roughneck's slang. Courses in engineering, physics, and the mathematics of high explosives. And, of course, the endless repetition of their tactical objectives. Practicing over and over until every step was memorized and every permutation analyzed, countered, and defeated.
They had been chosen from the best. In other times and other places, similar men had made up the elite forces that had carried names like La Légion Étrangère, the SAS, and the Delta Force. More familiar to them was the Spetsnaz, their own country's vaunted Black Berets.
They were called, simply, Team 7. If the name did not carry the same mystique as those of their illustrious antecedents, it was for good reason: Team 7 did not exist. No record could be found anywhere in the administrative logs of the army, navy, or air force testifying to their founding. No roster listed their names, their ranks, the units from which they had been seconded. When they completed the operation, they would disband and flee to the four corners of the globe, sworn never to speak with one another again.
They were all munitions specialists, five drawn from artillery, four from infantry, and three from underwater demolitions. Explosives were their game, and there were no soldiers anywhere who could better their adeptness with plastique, C-4, or gelignite. They had blown bridges in Kunduz and waterworks in Grozny. They had mined highways in the Sudan and mosques in Eritrea.
It was not, however, their skill under fire that recommended them, but the artist's care with which they practiced their craft. Deft fingers shaped the soft, explosive putty as a sculptor handled his clay, and with the same eye for effect. They could blow out a lock and leave the door standing or bring down a ten-story building with a single charge.
Their target lay thousands of miles away, across the roof of the world. The mission would require speed and stealth, but mostly care and concentration. With the smallest of charges, they would wreak the greatest of damage. Nature would have its revenge on man. And man would fall to his knees in apology. _Never again_ , he would promise. _Never again_.
The shadows moved into the distance, their steps slower, but still confident, a faint humming now dancing from their lips. It was a song they knew well: the anthem of their birthplace. And as their fatigue grew, they hummed louder. They would rebuild their country. They would make it strong once again. Formidable. A force.
A strong wind lashed across the landscape and they were gone, faded to obscurity inside the umbrella of grit and rain and sleet.
Ghosts who had never been.
Soldiers who never were.
A team that did not exist.
# 12
#
**_I N NOMINE PATRIS,_** et Filii, et Spiritus Sancti..."
Konstantin Kirov was dizzy. He had been standing in the front row of the Church of Christ the Savior for two hours, listening with the rapt attention expected of the guest of honor as Archbishop Nikitin, primate of Moscow, droned on and on, giving thanks for Kirov's gift of a fifteenth-century icon by the master Rublev depicting St. Peter slaying the dragon. The icon rested upon the altar. Only fourteen by seven inches, the portrait was a masterpiece of its kind, watercolors and gold leaf applied to a wood canvas, then glazed with albumen. Peter rode astride his stallion, lance carried high. His face was fevered, yet calm, his fear replaced by a trust in the Almighty. A faint halo crowned his head. The dragon, of course, was unseen. Iconography demanded that full attention be given the subject.
Kirov clamped his jaw as the archbishop passed close to him, swinging the censer and scenting the air with pale, acrid smoke. The columns swirled upward toward the cathedral's vaulted ceiling, the vanishing fingers signifying man's prayers lifting unto the Lord. Kirov followed the smoke along its course, viewing the church's interior with a mixture of piety, awe, and disgust. The acres of stained glass, the armies of tortured sculptures, the fabulous array of frescoes and trompe l'oeils awash in gold leaf: It was the Sistine Chapel times ten, without a trace of its grandeur. But what could one expect? Michelangelo had needed seven years for the chapel's ceiling alone; the entire Church of Christ the Savior was constructed in three. Its religiosity was so overwhelming as to be garish, laughable even, thought Kirov. There was not a better example of the contemporary Russian soul to be found in the entire country.
The Church of Christ the Savior was Moscow's latest miracle and the mayor's crowning achievement. Four inferior onion domes crowned each of the cathedral's transepts and surrounded a fifth and dominant dome whose enormous golden gilt swirls were visible across central Moscow—a candle's flame unto the heavens indeed. The church was a larger replica of the original Church of Christ the Savior that had been built on the same site between 1833 and 1883, designed by the architect Konstantin Toms and inaugurated by Czar Alexander II. Stalin in his good graces had torn the church down, melting the gold leaf for the Communist Party's coffers and using the land to erect one of his "Stalin Skyscrapers," atop which he wished to mount a ten-story statue of Lenin. When the land proved sandy and unstable, Stalin shelved the skyscraper and built instead Europe's largest outdoor swimming pool, which he personally christened "the Lido."
"Konstantin Romanovich Kirov, please step forward."
Awoken from his daydream, Kirov placed one foot in front of the other and advanced toward the ornate altar.
"In the name of the holy church, I commend your generosity of heart and spirit, and thank you for the wondrous gift to our diocese." Archbishop Nikitin grasped Kirov's shoulders and bestowed three kisses upon his cheeks, his long, grizzled beard scratching Kirov's face. The mayor followed, placing a bronze medal around his neck. "The city of Moscow is grateful, Konstantin Romanovich," he whispered. "You have done a great service."
"It is my pleasure." The mayor might reek of vodka, but at least he was clean-shaven.
The choir chanted. An organ played. The congregation was dismissed.
In front of the church, Kirov posed for photographs with the archbishop and mayor. It was a happy union of commerce, church, and state. Come morning, the beaming threesome would be on the front page of the city's newspapers.
"Should you need anything, I insist you call me," the mayor said as the crowd broke up. "We must lunch at the Café Pushkin soon. At my table in the library."
Kirov smiled dutifully. "I look forward to it."
The mayor went on talking about his favorite dishes at the tony restaurant, but Kirov only pretended to listen, for a voice in his earpiece had begun speaking. "Excuse me, sir. Rosen here. We have a small problem."
"Yes?" mumbled Kirov, his chin pushed into his chest. The Russian flag decorating his lapel was, in fact, the microphone of his cellular phone.
"Some news on the Net regarding Mercury. This fellow the Private Eye-PO again. You will not be pleased."
"I'll be there at noon," he said.
The mayor eyed him queerly. "I'm sorry, Konstantin Romanovich, but I am not free at noon. Perhaps next week. And if you can get another icon like that, we'd love to have it in the Novodevichy's chapel. Name your price."
**W E MUST FIND HIM,"** Kirov declared. "I want no expense spared."
"It isn't a question of expense, I'm afraid," replied Janusz Rosen. "He leaves us no name, no address."
The two were standing in Kirov's spacious office on the second floor of Mercury Broadband's Moscow headquarters, located in a newly renovated building one block from the Arbat.
"What do you mean, 'no name, no address.' Look here"—Kirov brushed a hand against the monitor displaying the Private Eye-PO's latest attack on Mercury Broadband—"someone is sending us this page, some server at some ISP. He has even given us his E-mail address. Surely we have contacts at Hotmail, if not at Microsoft."
"I've done my best to track him down. He's sharp. He knows how to make himself invisible. If he wishes to remain anonymous, it will be impossible to find him."
"Nothing is impossible." The admission of defeat crouched within the Pole's words angered Kirov. Ten years ago he was lying on a bunk in Lefortovo Prison, Moscow's main military jail, surviving on hardtack and water; today he was on the verge of a deal that would make him a billionaire. "If the mouse won't come to you, offer him some cheese," he said playfully, advancing on the gangly computer scientist. Then the eyes narrowed and the voice dropped a notch. "Find him, Janusz. Or I'll find someone who can. Someone a little hungrier for shares in our nation's most promising public offering. Remind me, will you... are there many U.S. dollar millionaires in Gdansk?"
"No, of course not—I mean yes, I'll do my..." Rosen raised an acquiescent hand, his words drifting off as he scurried down the hallway.
Kirov shut the door quietly and walked in measured paces to his desk. _"Anonymous!"_ he scoffed, shooting the monitor a killing glance. Who would wish himself such a terrible fate?
A hunched, dark man in a houndstooth jacket sat in a chair in the far corner, mumbling angrily into a cellular phone. Kirov ignored him. Picking up the phone, he dialed an internal number. "Boris," he said when a male voice answered. "Bring round the cars. We've a meeting with the prosecutor general himself in half an hour, and a little bird whispered in my ear that it would be wise to be punctual."
Hanging up the phone, he collected a sheaf of papers and shoved them into his briefcase. The papers were unimportant, just something to give the case a little heft.
"So?" asked the swarthy guest. He had mournful black eyes and a swirling salt-and-pepper mustache.
"Nothing more than a 'chat,'" said Kirov, not looking up from his briefcase. "Still, one never knows these days." It was an understatement. Political winds were swirling in violent, unfamiliar patterns; the government a clumsy Hydra, with each head acting independently of the other. One day the boys in the Kremlin were doing their best to promote the affairs of the country's more prominent businessmen, the next they were accusing them of every violation in the penal code, littering included.
"Be careful," ordered the man.
Kirov did his best to smile. "As always."
# 13
#
**W ATER, KONSTANTIN ROMANOVICH?** You look a bit flushed. Something to eat?"
"A sherry would be nice. Perhaps some foie gras."
"I can offer water and a cracker," said Yuri Baranov.
"Thank you, but no." Folding his hands in his lap, Kirov adjusted his immaculate posture and the smile of infinite goodwill that went with it.
For two hours, he had been seated in the same chair listening to Yuri Baranov, the nation's prosecutor general, rant about the sum of one hundred twenty million dollars missing from the coffers of Novastar Airlines. Theft of government property. Illegal exportation of hard currency. Grand larceny. Fraud. Even treason. The accusations went on and on and Kirov was quickly growing tired of them. How many times could a man say he was sorry, but he had no idea what had happened to the money?
"Let us proceed on a new tack," declared Baranov grimly, selecting a document from one of the bottomless stacks that littered his desk. "May I ask if the name Futura Holding conjures any memories?"
"Futura Holding, you say? I'm sorry, but it is not a name to me."
"So I may take it that if you were listed as a director of the company, it would come as a surprise?"
"I am a businessman. I sit on the board of a great many companies. It's difficult to keep track."
Baranov leaned forward in his seat and offered him the document. He was seventy if a day, a gray, stiff man in an ill-fitting suit with yellowing teeth and a well-worn expression of permanent outrage. A poster boy for the old regime, thought Kirov, hating and fearing him in equal measure.
Baranov was known to every Russian over the age of fifty as the man who had tried the arch-spy Oleg Penkovsky, the GRU colonel and war hero who had fed his nation's secrets to JFK and the Americans over an eighteen-month period in 1961 and 1962. Kirov could still remember the fuzzy black-and-white images of Baranov standing on the steps of the Lubyanka calling for Penkovsky to confess his crimes, name his co-conspirators, and publicly apologize to his countrymen if he wished to receive the Rodina's mercy.
_Confess! Collaborate! Apologize! Only then will the Motherland shower her mercy upon you_.
"Do you wish then to deny that you are a director of Futura Holding S.A., domiciled in Lausanne, Switzerland?" Baranov asked.
Kirov shook off the memory and concentrated on the document in his hand. He recognized it immediately. The articles of incorporation for said Futura Holding S.A. The paper was dated March 13th of last year. Kirov was listed as 51 percent shareholder in the company; the purpose of the holding noted as "investments in foreign corporations." "So I am a director of Futura. So what?"
"On March 15th, shares in Novastar were auctioned to the private sector. As the winning bidder, you were permitted to purchase forty-nine percent of the company. A month later, the shares were transferred to Futura in Lausanne, Switzerland."
"That's hardly news. Everybody in the country knows I purchased Novastar. About time someone decided to run one of our national airlines properly. Besides, forty-nine percent is hardly a controlling stake. If I recall, the government owns fifty-one percent."
"A formality. Managerial control of the airline was ceded to the private sector as a precondition to the auction. Therefore Futura is responsible for Novastar's day-to-day operations. The government is a silent partner."
"Apparently no longer."
Baranov continued. "On the seventeenth of March, Novastar management sent a directive to all its foreign sales offices ordering all remittances to be wired to an account in an offshore bank." He picked up a new document and read from it. "I quote, 'All proceeds from advance ticket sales, tour bookings, late fees, and penalties are to be paid into the account of Futura S.A. at the Banque Sino-Suez.' The directive is, in itself, an infraction of our legal code. Revenues accruing to the Russian government are to be transferred to Moscow. I could have you thrown in Lefortovo for that alone. What was the purpose of this measure?"
_Lefortovo_. Stones dripping with damp. Lice-infested beds. Midnight searches of prisoners' cells.
"Ease of accounting. A Swiss firm does all our work."
Baranov dismissed the answer with a sneer. "What concerns me more, however, is that since the time of this directive there has been a shortfall in income of over one hundred million dollars from last year's sales."
"Business has fallen off this year," Kirov explained, his mouth grown parched. "It would help if the government initiated a campaign to bring tourists to the motherland."
For once the pasty lawyer smiled. "Actually, bookings are up fifteen percent over last year."
"Fifteen percent?"
"Fifteen point six to be exact."
Kirov held the attorney's eye, hoping to conceal the tide of unease crashing inside him. First Futura and now mention of Novastar's bookings. Next thing Baranov would say he had the banking records to boot. A word scratched at Kirov's throat, begging to be acknowledged, spoken, screamed. _Spy_. Someone was slipping his organization's most important records out of his offices.
"I don't involve myself in the day-to-day affairs of my companies," he finally said. "I know nothing of the directive, but you have my word it will be discontinued immediately. I'm sure the shortfall in revenues is simply an accounting error."
"One hundred twenty million dollars is more than an accounting error."
"Then the error is surely yours, not mine."
"I think not, Konstantin Romanovich. Don't be surprised to find a government delegation coming to your offices for an early visit one of these days. You know my boys—the ones with ski masks, camouflage utilities, and machine guns. I've been made to understand that you are a demon for order—some might even say obsessive. Who knows what we might find. Perhaps some documents with the Banque Privé de Genève et Lausanne name?"
_The Banque Privé de Genève et Lausanne? How the hell had Baranov come up with that name?_
Kirov colored, but his voice remained calm and modulated. "Is that a threat?"
"One hundred twenty million dollars is missing," said Baranov solemnly. "Return to the state that which it is due and this inquiry will be terminated."
_Confess! Collaborate! Apologize!_ The iron voice echoed across forty years.
"A raid will not be allowed," protested Kirov. "If you wish to launch a formal investigation into my handling of Novastar's affairs, you're welcome to do so. But use the proper channels."
Baranov slammed an open hand on the table. "The Rodina is in a pitiable condition. Our people need money, not justice. The rule of law must take a backseat to economic necessity. We will no longer stand by as you and your like continue to rape the country, as you strip Mother Russia of her wealth to line your own pockets. You oligarchs are jackals, one and all."
"Never have I robbed the Rodina," said Kirov, his voice silk to Baranov's sandpaper. "I do not sell her minerals on the cheap. I do not smuggle her diamonds or gold out of the country. I do not squander her oil. I am a builder. A creator. Look around you. Half the new buildings in this city are mine. Offices. Apartments. Restaurants. I started a television station from nothing and built it into our city's most popular. A thousand rubles says the radio in your car is tuned to my station. It is I who have upgraded our country's phone lines, I who have brought the Internet to our young people and businesses."
"Yes," said Baranov, all outward calm evaporating. "You have constructed buildings, but at twice the true cost. Your offices charge outlandish rent to your own companies. Advertising billings collected by your television station are booked to offshore companies. Income tax—I don't even dare ask what you pay... _or don't_. As for Mercury Broadband and your interest in the upgrading of our nation's infrastructure, it is as suspect as the rest of your operations. Be most assured, Konstantin Romanovich, we are aware of your ambitious plans— _all of them_ —and we will decide which are acceptable."
Kirov was not blind to the threat. He shuddered to think what might happen to the Mercury Broadband IPO should his offices be raided by government troops. The press would be forewarned. Pictures would be broadcast over Russian television by noon and in America by nightfall. The offering would be postponed, or more likely canceled. Two billion dollars gone. And why? Because Kirov had conducted himself according to standard Russian business practice? Because he'd dared to prosper in perilous times?
He blinked, and despite himself his eyelids stuttered. Whatever else might happen, the IPO had to go through. Too many people were relying on its success. He, to build the first great company of the new millennium and to gild his path through the corridors of power. Others, to advance ambitious plans of their own, plans that would restore luster to the country's sword and shield.
Fathoming for the first time the insidious nature of the forces arrayed against him, he shed his mantle of insecurity and donned his fighting gear. If Baranov expected him to roll over and give up, he was sorely mistaken. Kirov had been fighting intimidation his entire life. As a Jew. As an intellectual. And as a businessman.
"Your threats are reprehensible," he declared in a soft, dangerous voice. "But nothing more than I expected from one of Brezhnev's bullyboys. I remind you we live in a democratic society these days. I've even heard a rumor we have rights."
"Thieves have no rights!" Baranov stood, his chair tumbling behind him. "Return to the state that which is its due and the inquiry will disappear. You have my word."
_"Your word?_ Your word is as reliable as the false accusations you've been tossing at me all afternoon." Only his mother's ingrained good manners prevented him from spitting on the floor. Suddenly, he could stand it no longer: the musty room, the weak lightbulbs, the worm-eaten furniture. Any moment, Khrushchev himself would walk through the door and start banging his shoe on the table.
Standing, Kirov buttoned his jacket. "Excuse me," he said politely. "I have a pressing engagement."
Lowering his head, he rushed from the room. There was a spy burrowed inside Mercury, and Konstantin Kirov had to root him out.
# 14
#
**L OOK, MR. GAVALLAN,** it's simply too early to start looking for your friend," said Everett Hudson, a consular officer with the United States Embassy in Moscow. "Twenty-four hours? I don't think they consider a man missing in Russia for a _week_. Until then they just think he's drunk."
Hudson had a squeaky, somewhat unsure voice. A Yalie on his first assignment with the foreign service, guessed Gavallan. Or a baby spy still wet behind the ears. "Mr. Byrnes is not a Russian," he said gravely.
"Of course he isn't," agreed Hudson. "Look, I'll forward the description you gave me to the police, and I'll be more than happy to phone the larger hotels. But I remind you, Moscow is a large city. It covers nine hundred square kilometers and has over ten million inhabitants all included. There's a lot of places to hide."
"Mr. Byrnes isn't hiding. He came to Moscow on extremely urgent business. He is a reliable man. He was due to call me this morning. As he hasn't, I have to assume something..." Gavallan hesitated, searching for the right word. "Well, that something _bad_ has happened to him. He's a former Air Force officer. He's..." Gavallan didn't bother finishing. He had already offered a nutshell explanation of Byrnes's reason for visiting Moscow; it would serve no purpose to offer any further testimonial to his character. "Something's just wrong, okay?"
"Can I be honest with you, Mr. Gavallan?"
"Please." Gavallan took a sip of Coke and set down the can. The clouds had moved on, leaving the sky a pale-washed blue. Whitecaps and a considerable chop attested to a steady offshore breeze. Feeling tired, frustrated, and more than a little pissed off, he kneaded the top of his knuckles while ordering himself not to explode.
"Moscow is kind of a strange city. I've been here four years, and you wouldn't believe the stuff I've seen. What I mean to say is that sometimes people go a little crazy when they get here."
"Crazy?"
"Well, not crazy, but they tend to let go. Especially men. You see, it's kind of a free city these days. After so long under the thumb, the Muscovites have gone a little wild. Let their hair down, if you know what I mean."
"What is your point, Mr. Hudson?"
"Your friend Mr. Byrnes is forty-four years old, correct?"
_We've gone over that_.
"Yes."
"And you mentioned he was divorced?" _We've gone over that, too_.
"Yes."
"Without wanting to sound rude, there's a lot of trouble a forty-four-year-old man can get into over here. If I called the police right this minute and said I was looking for a man like Byrnes, a well-to-do American, first time to Moscow, staying at the Baltschug, missing twenty-four hours, they'd laugh at me. They think every American is in town for one reason and one reason only: to shack up with their women. And they're not half wrong. Why, last week I had a call from the head of human resources for a major accounting firm in New York. She wanted to know if I might be able to explain why so many of her younger managers refused transfers out of Moscow. What was so special about the town that made them so reticent to leave? She said if she knew maybe she could make people stay in their Cleveland office longer."
"If you're trying to insinuate that Mr. Byrnes is off on some drunken jag through Moscow's fleshpots, you're mistaken."
"I'm suggesting no such thing," he said unconvincingly. "I'm just saying relax. Wait a little longer. Honest, Mr. Gavallan. It is too soon to be worried."
"Let me be the judge of that, Mr. Hudson. I've known my friend for a long time and I know when to worry."
"Really?" Hudson's voice grew contemplative. "It's my experience that you never really know anybody. I mean not really. At least not in Moscow. Here anything's possible." Hudson's voice lost its dreamy cast and Gavallan could almost picture him perking up at his desk, sitting straighter, putting on the consular officer's permanent-press smile. "I'll look for your friend—you have my word. Just don't get your hopes up, okay?"
"Thank you, Mr. Hudson. You have my number."
After he hung up the phone, Gavallan spent a moment wondering if what Hudson said was true—about never really knowing anybody. Naw. It was bullshit. If there was one person he did know, it was Grafton Byrnes. Something had to be very wrong for him not to have called by now. Robbery, kidnap, murder. One by one he turned over the possibilities. There was one, however, he had not yet named. It lurked hidden in shadow in the corner of his mind, but he refused to grace it with serious thought.
"Jett," came Emerald's efficient voice on the speakerphone. "I've got Moscow on the line. Mr. Kirov."
It was Gavallan's turn to sit up straighter. Taking a last sip of Coke, he threw the empty can in the trash bin on top of three others—Mountain Dew, A&W Root Beer, and Big Red—then slid back his chair and stood. "I'll take it, thank you." He snapped the receiver to his ear. "Konstantin, you're up late."
"I suppose you know all about this. It's a disgrace, really. Why didn't you call with the news?"
Kirov spoke slowly, his voice so quiet as to be a whisper, and immediately Gavallan sensed the control, the ironfisted discipline, that governed his emotions. Danger, he told himself. But for another moment, he didn't respond. He was unsure whether Kirov was referring to Grafton Byrnes's unannounced visit to Moscow or to the Private Eye-PO's latest broadside.
"I was interested in getting your opinion," Gavallan said noncommittally. "Besides, I thought it could wait until tomorrow morning your time."
"My opinion? What do you think my opinion is? I'm incensed. I am as angry as I have ever been in my life. He really is too much. He's gone too far this time. What I want to know is if anybody out there is stupid enough to believe him."
The Private Eye-PO. Kirov had read the lastest posting on the web.
Gavallan let go his breath, fighting his disappointment. He'd been sure Kirov had called to say that Byrnes had contacted him about his visit to Mercury's Moscow NOC. "Unfortunately, a good many do. Fidelity cut their order this morning. Not a good sign."
"And you? Do you believe it?"
"No, I don't. But I'd like you to tell me I'm right."
"Of course you're right."
"And you've purchased exclusively Cisco routing equipment for your Russian IP backbone?"
"I don't know if we've purchased Cisco exclusively. We buy from Alcatel, Sun, and a dozen others. But we do buy from Cisco, and I can prove it. I'm calling to say that I've asked my chief technical officer in our Geneva office to fax you copies of our purchase receipts from Cisco for the past two years."
"The receipts? Yes, that would be wonderful. Very helpful. Thank you, Konstantin." He swallowed. "Still, if anything is amiss with your platform in Moscow— _anything_ —we can shelve the offering and wait a few months. Demand for Mercury is strong enough that we'll be able to reschedule the issue." The words came hard, tumbling out of his mouth like stones.
"Shelve the offering? Out of the question. We have concrete plans for the money, or have you forgotten what is contained in our prospectus? Shelve the offering? Why ever would you even suggest such a thing? You believe him, is that it? You believe what the Private Eye-PO has said?"
"No, Konstantin, I don't. I want the deal to go through as badly as you. But as a licensed securities dealer, it's my duty to make sure everybody's talking from the same page, that's all."
"And we're paying you very generously for that duty. Moscow is up and running. Everything is a hundred percent operational. Have you got the fax yet?"
Just then, Emerald hustled into the room and laid a sheaf of papers on Gavallan's desk.
"I'm looking at it now for the first time. Give me a minute."
Gavallan's eye passed from one page to the next. The receipts detailed the purchase of over a million dollars worth of various routers and switches. The client was Mercury Broadband Geneva. The manufacturer, Cisco Systems.
All at once, a smile broke out on his face, and he had to work very hard not to burst out laughing. The Private Eye-PO was wrong. He was dead wrong. Someone had fed him a load of malarkey.
"They look good," said Gavallan, as the weight lifted from his shoulders. He read the documents a second time, still not quite believing them. Only one thing bothered him. It was a small detail, but he had spotted it nonetheless. The receipts were dated February 12 of the current year, yet the summary posted by the Private Eye-PO showed sales for the past three years. He dismissed the discrepancy, if it was one. Before his eyes, he had receipts that clearly confirmed Kirov's statement that the Moscow NOC was "up and running."
"They'll make everyone feel a lot better," he said. "I'll post these as a response to the Private Eye-PO on our web page by the end of business today."
"I hope so," said Kirov. "And what about the Private Eye-PO? What do you plan on doing to him? Surely you do not expect us to sit still while our good name is besmirched."
"I have some people on it already. With any luck, we'll have him located by tomorrow, day after at the latest."
"And then? All of us have our part to play to insure Mercury's future. We expect you to take any and all measures to silence this man. Nothing can stand in the way of Mercury Broadband's going public. Nothing."
"And nothing will," said Gavallan. "I'll see to it the Private Eye-PO's mouth is shut— _permanently_ , if I have my way. In the meantime, these receipts refute his accusations nicely. I'd say we're back on track."
"Good," said Kirov. "It's time to put an end to this tomfoolery. There's already been enough snooping."
The line went dead. Hanging up, Gavallan failed to experience the sense of victory, the burst of joy, that Kirov's call and the Cisco receipts should have brought. Instead, a bitter, unsavory taste lingered in his mouth, and he was left with a question.
Exactly what snooping had Kirov been talking about?
**R OY DIGENOVESE STOOD** at the window of a vacant office suite on the forty-first floor of the Peabody Building, peering directly into Jett Gavallan's office seventy feet away. The banker was walking back and forth, one hand to his neck. It was clear he was either very pissed off or very worried about something. "Are you getting a good read now?"
"Yeah, wind's died down so I'm right on target. Hold on." Mills Breitenbach, a tech specialist from the San Fran field office, put a hand to his ear while fiddling with some knobs on a metal device camouflaged to look like a Sony minidisc player. At his feet rested a twelve-inch satellite dish, its cone pointed in Gavallan's direction.
"Hurry up, damn it," said DiGenovese. "Don't want to miss what he's saying."
"Give me a sec. I've got to up the amperage on the beam. Here it comes. Showtime! You're on _Candid Camera."_
Breitenbach punched a button, and Jett Gavallan's voice filled the office. "No, Konstantin, I don't. I want the deal to go through as badly as you. But as a licensed securities dealer, it's my duty to make sure everybody's talking from the same page, that's all."
There was silence as the party on the other end of the phone spoke. DiGenovese noted the exact time. "We'll pick up the other end of this when we get the transcripts from the tap tomorrow," he said to Breitenbach.
Again, Gavallan's voice filled the room, sounding eerily close. "I have some people on it already. With any luck, we'll have him located by tomorrow, day after at the latest."
Breitenbach raised the silver casing to his lips and gave it a kiss. "You are the best, baby!"
The device that allowed the men to listen to a conversation being held seventy feet away through two plates of glass each an inch thick was called a unidirectional lasersat. Shooting a sensitive laser at the window of Gavallan's office, the lasersat read the infinitely subtle vibrations in the glass caused by human speech, then matched the vibrations against a sonic database, or "dictionary," and translated them into distinct words. Measuring the tonal frequency of each syllable, the lasersat was able, to a degree, to re-create the speaker's voice.
"I'll see to it the Private Eye-PO's mouth is shut— _permanently_ , if I have my way," came Gavallan's voice, tinny and emotionless, but recognizable. "In the meantime, these receipts refute his accusations nicely. I'd say we're back on track."
"You getting a load of this?" asked DiGenovese. "These guys are cozier than a pearl and an oyster. Fuckin' Clemenza and Vito Corleone."
Breitenbach smiled and patted the lasersat, a father proud of his baby. "You got what you need?"
"Oh, yeah," said DiGenovese, dark eyes blazing. "More than that. A lot more."
# 15
#
**H E CAME TO.**
The world was as he had left it, a dark, rank confessional, choked with the smoke of a hundred foul Russian cigarettes. He didn't know how long he'd been out—if after the pain had become too much he'd slept, or if it was just a period of nonexistence, where everything inside you kept ticking but your brain shut itself off. His legs burned. The rope that tied him to the chair cut into his calves, restricting circulation. He had that tingly feeling in his toes you get when your feet fall asleep, but they'd been tingling like that all night, and now the tingling had sharpened, so that even though he hadn't stood for hours, his feet screamed as if he were walking across a field of broken glass. His arms were where he'd left them, too, stretched taut in front of him, hands laid flat on a coarse plank, wrists secured by means of leather lanyards strung through the wood. His face throbbed. The right eye had swollen closed. He tried to open his eyelid, but nothing happened. Engine one, shut down and unresponsive.
Boris had left the left eye alone.
Boris from Metelitsa.
Boris, his unblinking Torquemada.
He was seated across the table, his posture rigid, his pale, soulless gaze alert, appraising, mocking, and finally condemning. The gaze never changed. It was the one constant in his swirling, unending nightmare, the hard blue eyes never leaving him even when the pain had become too much and his vision had gone blurry, and the scream had exploded inside him, and mercifully, oh God, yes, mercifully, he'd left the waking world.
Seeing him stir, Boris sat forward. He looked at him sadly and shook his head, as if saying, "One more hard case."
"You call now?"
The voice was as dead as the eyes. It was not a request, nor a plea, nor a command. Slowly he unrolled the chamois leather case containing his tools.
Pliers.
X-Acto knife.
A vial of rubbing alcohol.
A roll of gauze.
A lamp hung above the table, the bulb weak, stuttering. A relentless, pulsating backbeat seeped through the walls, causing the lamp to sway as if they were at sea rocking on an easy swell. Somewhere above him, people were dancing. He thought of his children, children no longer, then pushed their faces from his mind. They did not belong here. He would not tarnish them with this filthy place.
The cone of light swung right, and he looked at the hand splayed on the coarse plank. It was hard not to think of it as another man's hand. The thumb, raw, exposed, slick with blood, and lying next to it the thumbnail, extracted with a backstairs surgeon's precision, broken into two rough-hewn pieces.
At some point, he'd taken a clinical approach to things. An objective view. The pain was his, no mistaking that: the shaft of fire bolting up his arm, the paralyzing scream starting far down in his belly, the cry desperate to escape, discovering the mouth stuffed with a rubber ball and secured with a length of duct tape. Yes, the pain was all his. But as the pliers dug deeper beneath the nail, as the X-Acto knife sliced away layer upon layer of stubborn connecting tissue, as Boris pulled and yanked and twisted, his apathetic, unshakable gaze never wavering, he'd given up the hand.
The beat from above grew louder. The walls quivered with the thud of the bass and he could make out patches of the music. "West End Boys." Boris half sang a few words. _Vest-ent boyz_. He stopped and stared hard.
"You call?"
Grafton Byrnes listened to the music a moment longer, savoring it, knowing it to be the last taste of a sane universe. In the dark hours of his captivity, he had fashioned a plan, but it required patience. And patience meant more pain.
Eyes burning with defiance, he shook his head.
Boris reached for the pliers.
# 16
#
**T HE INVITATION READ:**
_A Midsummer Night's Dream_ ,
_A Fantasy, A Flirtation_
_The St. Jude Children's Hospital's 25th Annual Black_
_and White Charity Ball_
_8 o'clock PM
Governor's Ballroom_,
_The Fairmont Hotel_
Gavallan stepped from the passenger seat of the Range Rover, adjusting his dinner jacket while his date for the evening circled the car to join him. He had just enough time to admire the fairy lamps strung across the portico, the baby ficuses and swirling cypresses dressed with tinsel and crepe to look like Shakespeare's enchanted forest, before Nina Slenczka rushed to link arms with him and guide them up the maroon welcome carpet.
"Remember to smile, hon," she said, her flack's professional grin splitting her ruby red lips. "This one's for the morning papers."
Nina handled all of Black Jet's PR, and to Gavallan's mind the date was strictly business. Not to say he didn't find her attractive. Twenty-nine years old, blond, petite, and lithe, she had dressed for the evening in a skintight black sheath, spaghetti straps, and just enough fabric to cover her nipples and navel, maybe a little more. Yes, she was attractive. Stunning even. But Gavallan wasn't looking.
Gavallan paused in front of the bank of photographers to allow them a few seconds to rejigger their flashes and pop off a few shots.
"Let everyone see those baby blues," Nina said, keeping a tight clutch on his arm, not letting him even think of moving on until the photographers were done. She might be a prig, but she knew her stuff when it came to corporate PR. She was right about the importance of his projecting a confident image, especially when one of his company's issues was under fire.
It was a classic San Francisco evening. An offshore breeze had cleared out the cloud cover, leaving the sky clear, dusted with stars. Across the street from the Fairmont sat the Mark Hopkins Inter-Continental and down the block the Huntington Hotel and the California Club, a gentleman's conclave so stodgy that only ten years ago it had refused entry to a serving mayor due to her sex.
A hundred years ago, Nob Hill had been home to the Big Four: Collis Huntington, Mark Hopkins, Chester Crocker, and Leland Stanford, the railroad and silver barons who'd built California. Setting foot on their stomping grounds, Gavallan never failed to feel bucked up, as if the tycoons had left behind some of their marauding spirit. Tonight was no exception.
Inside the ballroom, he made a beeline for the bar. It proved a long and arduous journey. Every two steps he was accosted by a friend or business acquaintance. Half were eager to congratulate him on the honor to be bestowed that evening, half to learn how the Mercury deal was likely to fare.
"I need a cassette player," he whispered to Nina, after swallowing half of his vodka rocks. "I only need two answers: 'Thank you' and 'Just fine.' I'll say I'm saving my voice for my speech."
"Come on," said Nina, "they're your friends and they're happy for you. You're the star this evening. They have to pay their respects. It's your duty to smile and play the good host."
"And I shall not disappoint," he said gallantly. Despite his distaste for glad-handing and small talk, he recognized that Nina was right, and that of all his duties, civility and good cheer were the ones he could guarantee were met.
Gavallan had been donating to St. Jude's Children's Hospital for eight years, dedicating ever-larger chunks of his salary to the institution and its programs to battle children's cancer, spina bifida, and infantile paralysis. He was quick to point out that he was hardly an ascetic. He had the house in Pacific Heights with the roomfuls of Kreiss furniture and Pratesi bedding. He wore whatever clothes he liked. Music came via the firm of Bang & Olufsen, stereo makers to the King of Denmark; television courtesy of a sleek Sony Plasma screen. He owned two Remington bronzes; some lithos by Branham Rendlen, a local artist he thought was dynamite; and, of course, the Mercedes.
There were other claims on his money. He saw to his mother's needs, helped out with his sisters' occasional purchases—washing machines here, new pickups there, schooling for their kids if they asked. He kept a fair amount in the bank, a little in stocks and bonds. (Or at least he had until he'd stuffed it all into his company.) He had enough to take care of him and his family in comfort should everything go to hell in a handbasket.
The rest he gave away.
The ballroom was filling up quickly. Elegant couples drifted through the carousel of tables, a monochromatic mélange of tuxedos, cocktail dresses, and ball gowns, laughing, chatting, and, to his eye, having a sincerely good time. San Franciscans enjoyed their liquor, and under the influence of a stiff drink or two their voices began to rise and fill the room with a jolly din.
Gavallan ordered another drink, then asked Nina if she wouldn't mind going to their table. Bruce Jay Tustin and Tony Llewellyn-Davies were already seated, Tustin with his wife, Nadia, Two Names with his partner, Giles, another wayward Brit. Meg sat at the adjoining table with her husband of forty years, Harry.
Gavallan greeted his guests with exaggerated bonhomie. He wanted it clear that the day's problems were behind them. Tonight they could relax and let their hair down. "Don't I know you nice folks?" he called, lending his voice a bit of the old Rio Grande twang.
The table stood as one. To Caesar, his due.
"Look who's here," said Bruce Jay Tustin. "And I thought security was supposed to keep the riffraff out. Do you have a ticket, young man?"
Meg sprang from her chair and wrapped her arms around him. "Congratulations, Jett. We're all so proud. You done good."
And then the others were up, shaking his hand, hugging him, treating him like a returning war hero. It was easy to forget that he'd only left them two hours earlier.
"Seriously, Jett, we're honored to share this evening with you," sounded Tony Llewellyn-Davies. "Believe it or not, we care about you deeply." He held Gavallan at arm's length, then proclaimed, "Oh, what the hell. I'll say it for everyone. We love you and we're overjoyed to be here. And that's the last nice word you'll get from any of us this evening." And with that he gave Gavallan a peck on the cheek.
"Here, here," added Giles, a handsome youth in his twenties. The two-carat diamond stud in his ear and eighteen-karat gold Cartier on his wrist hinted that his interest in Tony was more pecuniary than personal. Gavallan hoped his friend wasn't being played for the sap.
"The honor is mine, ladies and gents," he said, touched by the outpouring of affection. "It's rare that you get to work with your friends, and for that I feel both privileged and grateful. Now enough of this smarmy nonsense. Let's sit down and enjoy the evening." Raising his glass, he quoted from Bum Phillips, former coach of the Houston Oilers and honorary "good old boy." "Every man have a drink. Every _good_ man have two!"
"Hoo-yeah!" shouted Tustin, glass raised high.
Gavallan clinked glasses with Tustin and his wife, Two Names, Giles, Meg, Harry, and Nina. He couldn't help but think of the one man who was missing from their ranks. After everyone quieted, he raised his glass again.
"To Grafton Byrnes. Let's pray for his health and safe return."
**I T WAS MIDNIGHT IN POTOMAC,** Maryland. Streets in the leafy suburb were so calm as to be deserted. A warm, gusty evening breeze carried the sweet scent of cut grass and the merry sawing of crickets. On Dumbarton Road, the lights in most houses were dimmed, the occupants asleep. But in the Vann residence, a stuttering spectral light glowed from the second-floor dormer windows.
In his bedroom, Jason Vann dashed from computer to computer, pausing long enough to type in a sentence or two, before moving to the next. Beads of perspiration rolled down his forehead. A hunted look shadowed his drawn face. Round and round he went, enraptured by this game of his creation. A game of cat and mouse. Vann was after the Private Eye-PO. He was trying to lure him into the open, and his bait was praise and scorn and disbelief and any number of the hundred emotions that stock enthusiasts routinely express.
At that moment, he was working five characters on the IRC, the Internet Relay Chat, and they were discussing the Mercury Broadband IPO to be brought to market in five days by Black Jet Securities. Mario was a high school student who was president of his stock club. Julie was a middle-class housewife who grew interested in the market after her husband had lost all of their money. Al was a New York know-it-all, a seasoned investor, and a veteran of many (losing) campaigns. Krystof was a programmer of Polish descent who believed that the stock market was every immigrant's way to riches. Heidi was a computer science teacher in Mamaroneck, New York, who had just invested her first five thousand dollars. And they all lived in a twisted corner of Jason Vann's conniving mind.
Al: The market's gonna gobble up Mercury like a pastrami sandwich. I'm saying double the first day. Think positive.
Krystof: You are sure? I also think it time for big success again.
Heidi: Is it safe?
Mario: I doubled our stock club's fund investing solely in IPOs last year. But be careful. Didn't you see the latest news?
Julie: Where were you when my husband started trading?
Al: The Private Eye-PO don't know his ass from his elbow. He's probably a trader pushing his own stocks, knocking down the others. Caution!
Vann rushed from chair to chair, simulating the voices and thoughts of these five would-be investors. He'd spent three hours online introducing them, getting them into a chat room and allowing them to grow comfortable talking in the open. His job was to create a fictitious universe the Private Eye-PO might stumble upon and wish to join. So far he hadn't had a nibble. He was getting discouraged. It was time to up the ante.
Mario: I disagree. I think he's the only one we can trust. I follow his advice to the letter. If he's a trader, he's a darn good one. Remember what he called Mercury? A scam dog!
Julie: Sounds like you're the Private Eye-PO himself, Mario. Come on, tell us the truth!!
Mario: Ha, ha.
Krystof: Who is this Private Eye-PO? In Poland, you never trust man who does not tell you name. I mean, _his_ name. Excuse me.
Al: No way a company like Black Jet is gonna touch Mercury if it's got problems. No way. Be real. I saw Gavallan on CNBC. The guy's a pro. He was a pilot!
Vann had slid back into Mario's chair when a new name popped onto the screen.
Val: Pros, schmoes. Make up own mind. I buy Mercury and buy big. I have own sources. Nay to Private Eye-PO.
Dismayed, Vann frowned. No way was Val the Private Eye-PO. He sounded like a foreigner. Jumping into Krystof's chair, he tried a ruse.
Krystof [in Polish]: Hello, new friend. Welcome. You are a fellow Pole, perhaps?
Val [in Polish]: From Gdansk. The great Lech Walesa's home. And you?
"Score!" cried Vann aloud, grabbing a Nerf basketball and stuffing it for a quick two points. Then, collapsing back into Krystof's chair, he typed:
Krystof [in Polish]: Kraków. I left in '98.
Vann, whose father's real name was Wladisaw Vanniewski, didn't dare add more. His Polish was rusty; anything more than the basics would expose him as a phony. Anxious to keep the dialogue afloat, he moved to Heidi's chair.
Heidi: A friend of mine is from Warsaw. He made a fortune buying tech stocks. Can they still go up?
There was always at least one total idiot in any chat room.
Val: They can only go up. Mercury will lead way. To heaven!
Boy, thought Vann. He's a real supporter. As he slid back into Al's chair, another name popped onto the screen.
Spade: Hey, kids, you want the inside skinny? Talk to me. Your very own celebrity reporter has come to the rescue. Heidi, dear, listen closely to me if you want the _oop-scay_ on Mercury. All the rest of you neophytes, _am-scray!_
Vann froze in his chair, eyes wide. "Spade" as in Sam Spade. As in the Private Eye-PO. Could it be? Scooting his chair closer to the computer, he felt his heart pounding like a jackhammer inside his chest. The bait had worked. The fish was on the line.
Wiping his forehead, Jason Vann smiled. Now he just had to reel him in.
**T HE FIRST COURSE HAD BEEN CLEARED.** Peter Duchin and his orchestra had begun to play an up-tempo version of "Witchcraft," the vocalist doing a very acceptable Sinatra. Couples flocked from their tables to the dance floor. Deciding he'd done enough penance for one evening, Gavallan turned to Nina and asked if he might have the next dance.
"Sorry, Jett, but I've promised Giles. He's dying to cut the rug."
Gavallan smiled understandingly, though he was a little irked. While same-sex partners might be permitted at society functions, their dancing with each other was still touchy. If Tony or Giles wanted to dance, it had to be with a member of the opposing team. Gavallan thought the whole thing ridiculous. He couldn't care less who did what with whom as long as they were happy. Still, Nina was his date and he wanted to dance. "Try and save one for me, will you?"
"Sure thing, hon."
Gavallan watched the happy couple dodge their way to the dance floor, then stood up and set off in the opposite direction. The path to the bar looked mercifully clear of congestion. If he moved swiftly, he might make it scot-free. Fifteen seconds later he was there, leaning against the oak railing and perusing his choices. Whiskey had been his daddy's drink, but Gavallan preferred vodka. Spotting a familiar bottle with yellow script, he decided on one more of the usual. And why not? It wasn't often you put all your chips on red and gave the wheel a spin. After a day like today, a guy deserved to get hammered. It might even add a few laughs to his speech.
"Hey, chief," he called to the bartender. "Let me have an Absolut Citron."
"How would you like it, sir?"
_"Rocks, no twist,"_ answered a playful feminine voice behind him. _"And pour it heavy."_
Gavallan felt a hand brush his shoulder and turned to face a tall dark-haired woman with glossy bangs that fell shy of amused green eyes.
"That's my line," he said.
"And my drink. You stole it."
She had chosen white for the evening, a simple cotton shift that fell to her knees. Her luxuriant hair had been cut short and barely brushed her shoulders. She wore only a trace of makeup—a dash of eyeliner and a shadow of rouge. She'd never liked coming to these fancy dos. She refused to wear high heels and was shy about her shoulders, complaining they were better suited to a lumberjack than a society maiden. She was his tomboy in waiting. His eyes passed over the swell of her breasts, the planes of her belly, the curve of her hips, remembering.
"Hello, Cate," he said. "You look wonderful."
"I wish I could say the same. You look tired. What happened? Some of your clients beating you up over that last IPO? Trivium, wasn't it?"
"Trillium," he corrected her. "And don't be snippy." Trillium Systems was a maker of enhanced circuit boards whose shares had traded down 50 percent the first week of trading. No one batted a thousand. "Just the usual really. Trying to keep the boat afloat. I'll have to have a word with the shaman to help me out."
"You and your shaman." Cate Magnus's hand went to his cheeks. She leaned closer and checked his eyes. "You okay?"
Suddenly he remembered how overwrought she could become. He used to tease her that she'd been programmed with an extra sensitivity chip. "I'm fine. Nothing that a good night's sleep won't cure."
Cate patted his chest lightly, a sign she'd checked him over and he was in fine fettle. "So is the twenty-million-dollar man ready to entertain the troops? How's the speech? Did you actually write something down or did you plan on winging it?"
Gavallan hadn't given the hospital twenty million dollars outright, but pledged it in annual increments of one million dollars. The third installment was thirty days past due. Not a word had been spoken about the tardy donation.
"You're the writer," he said, sipping at his cocktail. "Me, I just have a couple of drinks and let my silver tongue carry me where it may."
"Silly of me to ask. But be careful, Jett. Too much booze loosens the tongue. You might let a few words slip about all the fires you've been putting out."
"What fires are those?"
"You tell me."
Gavallan registered confusion. "I thought you were a columnist," he complained. "Sounds to me like you're looking for a way to get back on the front page. That why you're here?"
"No," she said. "I slipped by the guards to pay my respects to a pretty neat guy I used to go out with. I think it's great what you've done for the hospital."
"Least I could do, really," he said, searching out her gaze, wanting to stare headlong into her vivid eyes, hoping to find that the connection was still there. But Cate was careful to keep her eyes aloof and darting across the crowd, only briefly engaging his.
"I've been reading that stuff on the web about the deal you've got coming to market," she said. "I hope you're being careful, Jett. I always told you to steer clear of Mercury."
"Come on, let's not start that again."
Cate began to say something, then bit her lip. Offering a noncommittal shrug, she ordered a Stolichnaya straight up, no ice, no chaser. _Her drink_.
Catherine Elizabeth Magnus was a handsome woman, more striking than beautiful. With her angled features, pale complexion, and high cheekbones, she called to mind an exotic strain of royalty. A princess from Liechtenstein, a _Gräfin_ from Pomerania, an Italian _contessa_. Her posture was immaculate, her step light, yet directed. When she walked it was for the audience she'd grown used to long ago. And it was the coupling of patrician bearing with her commoner's unpretentious personality that he found so attractive. It didn't take a genius to figure out why. Cate Magnus was the class Jett Gavallan never had.
She'd worked as a reporter at the _Financial Journal_ for as long as he'd known her, writing a weekly piece for the paper she called "Gold Rush." Every Friday, she filled twelve column inches on the front page of the _Journal_ 's second section with offbeat, funny, and often poignant stories about the ins and outs of surviving in the capitals of the new economy: Silicon Valley, Seattle, Austin, and the few city blocks in Manhattan someone had baptized "Silicon Alley." Her subjects ranged from how the skyrocketing price of real estate was making millionaires out of middle-class home owners to the social etiquette of pink-slip parties to the personal peccadilloes of the new and obscenely rich. The rise and fall of Black Jet Securities would make perfect fodder for her column.
"Speaking of fires, I had an interesting call this afternoon," he said, allowing himself to move a few inches closer to her. "Between you and me, everything the Private Eye-PO has said is bullshit. Complete and utter garbage." He went on to explain about the receipts, his conversation that morning with Jean-Jacques Pillonel, and Konstantin Kirov's personal guarantee that everything was "up and running" in Moscow.
"Kirov himself told you? Well then, I guess you don't have to worry at all."
"Don't start about Kirov. Please, Cate. Not tonight."
"All I said was that you shouldn't trust him. He's an oligarch, for Christ's sake. How do you think he got where he is?"
"He is a businessman, and a damned good one. Neither of us has any idea of the conditions he has to work under over there. I'm not saying he's a saint, but Mercury speaks for itself. It's a gem."
"It sure does."
"What's that supposed to mean?"
"It means he's ruthless and conniving, and maybe even a little more than that. He's a good businessman all right. If that's what you call it."
"Cate!"
Her eyes flashed, and he could feel her straining to rein in her temper. "Okay," she conceded. "You win. Just be careful. Word is you're risking a lot on this deal."
"Whose word is that?"
"Everyone's. No one's. You know how it is. The street's got wind you're putting a lot on the Mercury deal. I just was curious if the rumors are true."
It was Gavallan's turn to shrug. But looking at her, at her lustrous black hair, her keen eyes, her pale, pillowed lips, he had a sudden desire to tell her everything. A need even. Whether she knew it or not, he valued her counsel more than that of any of his colleagues at Black Jet. She was smart. She was well-informed. She was discreet. They'd been together over two years, and though privy to his every insider secret, she'd never once abused his trust.
Cate who was trustworthy.
Cate who was loyal.
Cate who was the most sensuous lover he'd known.
Unable to restrain himself, he ran a hand across her cheek and let it glide through her hair. "I miss you."
"Jett, no," she whispered, her eyes fluttering. It was a plea, a denial, a memory.
"Come on," he said. "Let's dance." And before she could answer, he grabbed her hand and led her to the parquet floor. Continuing its tribute to "Old Blue Eyes," the orchestra launched into "A Foggy Day." Gavallan drew her closer. In seconds, their hands had found familiar places, their bodies secret havens.
"So what do you want to know?" he asked.
Cate looked taken aback. "You're serious?"
"Have I ever kept anything from you?"
"That was when we were... That was before," she said.
_Before_. He hated the word. "You will, however, have to recite the sacred oath."
"Oh, Jett, come on."
"Sorry. You know it's important to me. I am an Eagle Scout, you'll remember. The oath, please."
Cate looked uncertainly to her left and right, then raised her right hand to her shoulder, arranging the fingers in a familiar salute.
_"On my honor I will do my best_
_To do my duty to God and my country_
_and to obey the Scout Law;_
_To help other people at all times;_
_To keep myself physically strong_ ,
_mentally awake, and morally straight."_
Gavallan nodded his approval. "At least I know your time with me was not completely misspent." He cleared his throat. "Anyway, I guess the first thing you should know is that I'm pretty much tapped out. That much of the rumors is true."
And with that he launched into a recitation of the entire day's events: Byrnes's disappearance, the meeting at Sten Norgren's, his taking out the second mortgage, the particulars of his personal and professional liquidity crunch. He left nothing out.
"So, I guess you had a pretty dull day," she said afterward.
Seeing the mischief in her eyes, he laughed. For the first time since he'd woke, he felt as if things might turn out okay.
# 17
#
**T HEY'D DANCED THREE SONGS IN A ROW.** The entree was being served, and suddenly they were the last couple on the floor. Gavallan didn't need to look toward his table to know that Nina was staring daggers into his back. Let her, he thought. I'll take Cate. She can have Giles. Only Tony will be the poorer off.
"So let's get this straight," Cate was saying, "you floated Mercury a fifty-million-dollar bridge loan with no collateral—I mean, other than their stock? Shoot, Jett, I'd be worried, too, about what the Private Eye-PO says."
"Don't be ridiculous," Gavallan countered. "Mercury earned sixty million in profit last year on revenues of three hundred ninety million. No one's disputing that. They couldn't have earned it without the Moscow market. It's one of their biggest."
"I hope you're right, Jett. I really do. Because God forbid that Mercury isn't every inch the company your prospectus says it is, and you bring a fraudulent company public. And in this case I mean 'public' with a capital P. Two billion dollars' worth. Because your life will be over as you know it and everything you hold dear will be taken away from you. Your money. Your company. Everything. The only good news is that you won't have to worry about that second mortgage anymore. You'll have rentfree accommodations for the next seven years or so. Depends on the judge."
Gavallan listened to her assessment, his worry growing because it was the same one he'd made himself. Earlier, he'd told Tustin and Llewellyn-Davies they had to be true to their client. But Cate's skepticism, coupled with his partner's lingering silence, lent him second thoughts, Cisco receipts and Jean-Jacques Pillonel's word notwithstanding.
"A guy I know is tracking down the Private Eye-PO," he said. "Once we find him, I plan on having a heart-to-heart, just him and me, find out why he's going after Mercury before I have a judge slap an injunction on his ass."
"Why do you think he's going after Mercury?" Cate demanded. "Because he has the goods on them."
"Actually, we were looking into the possibility it might be personal, a grudge or something against Black Jet, or maybe even me."
"Oh, come off it. A grudge? Sometimes you really piss me off." The voice had hardly risen, but her eyes had narrowed and a rigid control had taken hold of her body. Dropping her hands, she turned and walked off the dance floor, weaving through the maze of tables to the hallway outside the ballroom. Gavallan knew she meant for him to follow.
She was waiting outside the ballroom, hands on hips, head cocked defiantly.
"Jett, I want you to listen to something I have to say. And I want you to promise me you won't get mad. You sent Graf to Moscow to check on Mercury's operations there and now you can't find him. Gone from the hotel. Not calling back. Whatever. Point is he's disappeared while he was supposed to be looking into Mercury."
"Yeah?"
"And at the same time the Private Eye-PO issues another warning about Mercury. He's never wrong, that guy. You know it and I know it. Accuracy is his hallmark."
"So?"
Cate's eyes widened. "Do I have to connect the dots? Maybe Graf's disappearance isn't a coincidence. Maybe the Private Eye-PO has the goods on Mercury. Maybe Kirov called you to make sure you were still on board."
"That's enough, Cate. Now you're talking like a fool."
"Am I? Think about it, Jett. Just think about it." The challenge hung between them, the ensuing silence warming her concern from professional to personal. Nearing him, she rested a hand on his jacket and neatly brushed a hair from his lapel, so that for a moment, he dared believe she might still love him.
"So what's your advice?" he asked.
"I'll only tell you if you promise to take it."
"Forget it," he said, turning to go back to the party. "I already know what it is. Drop the deal. I'm not going to do it. I _can't."_
"Postpone the offering," she pleaded. "Let me put you in touch with some of our guys in Moscow. Let them look into it. They're hooked into the whole scene."
Gavallan bit his lip, bitter, confused, wanting to say a million things, not daring to say a word. "The offering is going through, Cate. Like I said, Mercury's a gem. I know it, even if you and the Private Eye-PO don't. Now, if you'll excuse me, I have to go give a speech to three hundred of our city's snootiest before they get too sloshed to understand a word I say."
And opening the door, he walked back into the ballroom.
**I N POTOMAC, MARYLAND,** and across the ethereal veins of the Internet, the roundtable between Jason Vann's cast of disgruntled characters and the man calling himself Spade was growing more heated.
Al: Listen to me, kid! You want the inside skinny on Mercury, I'll tell you. You're way off base on this one. My sources tell me Mercury's double the deal you think.
Spade: Whoopee for you! We've all got our sources, honey. And mine is indisputable.
Val: Listen to Al. Where you get silly pictures? I see this and laugh.
Heidi: What picture?
Mario: Go to his website and take a look—www.PrivateEyePO.com. You'll see!
Spade: Thanks, chum. Always nice to know what side your toast is buttered on. As for ye of little faith, the picture cometh straight from the hand of God. Cross my heart and hope to die.
Jason Vann rubbed his hands together, a worried look narrowing his eyes. He was desperate to angle the Private Eye-PO into a private chat room.
Al: If it's "straight from God" you want, come with me, big mouth, and I'll show you something that'll make you close your yap.
Spade: I go everywhere and nowhere. You got the goods, send them to my address at Hotmail.
Al: You want to keep up that winning percentage, you'd be wise to jump my way. You're not the only one with inside info. I've also got some documents from Mercury. And they tell me the opposite of what they tell you.
Val: I come, too. I also know people at Mercury.
Spade: Who? Give me the name, cutie pie. Don't make me beg.
Val: Janusz Rosen. A Pole like myself. He is programmer. Damn good one, too!
Jason Vann stared at the last sentence, wondering who the hell "Val" was, why he was so keen on butting into Mercury's business. If Val was Rosen, then the boys at Mercury were probably running their own gig to track down the Private Eye-PO. Surely, "Spade" knew this.
Al and Spade engaged in a few more volleys, the shadowy Val lurking close by, until by sheer force of will Al broke down Spade's barriers. Immediately, Vann created a private chat room for Al and Spade to enter, then slammed the door closed before Val could sneak in. Once they were inside their cozy, private corner of cyberspace, Spade relented.
Spade: Your 411 better be white hot, chum. Send me the stuff to Ponyfan@earthlink.com, and give your return address. If it's as good as you say, I'll fill you in on the nitty-gritty with Mercury.
Vann jumped out of his chair, roaring. "Gotcha, you big m.f. You are so nailed!" Vann had a dozen buddies at Earthlink. A few calls and he'd have Ponyfan's IP address before he knew it. From there, it would be smooth sailing. By morning, he'd have all the info he needed to earn his fifty-thousand-dollar bonus from Mr. John J. Gavallan: the Private Eye-PO's name, home address, and phone number.
_Child's play!_
**T HE LINE FOR THE VALET CAR PARK** stretched from the curb to the lobby. Gavallan stood near its head, Nina at his side. She'd barely said a word since he'd returned from his extended tête-à-tête with Cate. At least he wouldn't have to worry about how to avoid a good night kiss. Giles was dutifully back with Tony. Meg and her husband, Harry, stood arm in arm, mooning at each other like love-struck teenagers. A cell phone chirped, and every man, woman, and valet froze, listening to hear if it was theirs. Gavallan answered. "Yeah?"
"Jett? That you?"
_"Graf?"_ he asked, the relief spontaneous, bringing a wide smile to his face. "Graf, where the hell are you?" He laughed out loud, thinking it was wonderful. Byrnes was okay. He was safe. The fucking shaman had answered his prayers.
"Where do you think? The heart of the evil empire: Moscow. Back in the USSR."
Gavallan turned his back on the crowd and walked a short distance up the sidewalk. "You were supposed to call this morning, you prick. You had us all worried."
"Sorry. Had to double-check on a few things before I got back to you. Didn't want to give you any information until I knew for sure. Look, I've scoped out Mercury's operations. I made it out to the network operations center. Place is in Timbuktu, I don't mind saying. I've seen their offices in town. It's all like we thought it was. The Private Eye-PO is full of shit. Mercury's up and running."
"So the deal's a go?"
"Green light all the way."
"Fantastic," said Gavallan, controlling his urge to holler. Turning his head, he saw the others locked in a group stare in his direction. He waved a hand and gave a big thumbs-up.
"You there?" asked Byrnes.
"Hell, yes. I'm definitely here."
"I knew you'd be happy. Listen, Jett, everything's copacetic over here. Copy?"
"Yeah, I copy, pard. Thanks for the great news. I'll get that champagne all iced up; you bring back the caviar. Two billion, man. Our biggest fish ever. Can you believe it? Just let me know when you're getting back."
And then the words sunk in and Gavallan held his breath while the hairs on his arms and neck stood on end.
_Everything's copacetic_.
"I'm going to stay the weekend if you don't mind," Byrnes went on. "Moscow's a hell of an interesting place. Thought I might check out some of the sights tomorrow. Saturday, Kirov's invited me out to his summer house in the country. An honest-to-God dacha—can't miss that. By the way, he sends his regards. He's delighted that we decided to take a look for ourselves. Says we're welcome anytime."
"Tell him thank you." It was another man speaking Gavallan's words. "So he wasn't upset when he found out you'd flown over to check out Mercury without letting him know beforehand?"
"I told you he wouldn't be," said Byrnes. "He wanted me to tell you that Mercury must be as transparent as any of its counterparts in the West."
"Did he?"
"Yes, he did. Anyway, I thought I'd fly into New York and meet you for the launch party."
"Sure thing," said Gavallan, searching for words, stumbling. He felt hollow, shaky. A rod of pain, searing and white-hot, fired inside his skull. Wincing, he touched at his forehead. "Um... yeah, sounds good, see you Monday. Oh, and call Emerald and give her your flight details. We'll send a limo to pick you up at JFK. When you see Kirov, ask him if he's free for dinner."
Gavallan waited for a response, but the line was broken and only static answered his words. Besides, it didn't really matter. Grafton Byrnes had told him everything he needed to know.
_Everything's copacetic_.
# 18
#
**G AVALLAN WAS WALKING THE WARD.**
His pace was slow, his steps measured. The click of his heels against the linoleum floor sounded to his anguished ears like the final ticks of a time bomb. With every step, he was tempted to draw a last breath, to squeeze tight his eyelids in anticipation of the blast to come. But what would it destroy? he wondered. What was left that hadn't already been torn apart by his own merciless conscience? What might it damage that hadn't been shredded eleven years ago?
The clock on the wall read 2:15. The room was extremely bright and extremely quiet, a fluorescent universe of hushed sounds. His ear seized each in turn—the rise and fall of a neonatal respirator, the gasp of a fragile patient, the sibilant bleed of oxygen—then he continued his all-night vigil.
He was back at the Zoo, doing tours of the Quad for having missed his second curfew in a month. He was pacing the ready room before his first flight into combat. He was the star witness at his own trial. All that was left to decide was the penalty. The verdict had already been given. Guilty on all counts.
"Everything's copacetic," Byrnes had said, a footnote from their shared history to let Gavallan know he was testifying under duress.
Hardly, mused Gavallan acidly.
A skeleton staff presided at this late hour: a few nurses, orderlies, and cleaners. Through the glass partition, he kept track of a janitor polishing the corridor, his green-clad back bowed and sober, his worn mop eating up miles of hallway with a methodical, unerring rhythm that was a science unto itself.
Gavallan glanced down at the child in his arms, a frail boy swaddled in a sky blue blanket. He'd been awarded the provisional name of Henry, and the name would stick until his mother could come to long enough to provide him a more permanent one. He'd been born one week before, full term, 4 pounds 2 ounces, 14 inches long. To look at him, he was a healthy child. His features were well-formed. Broad nose. Full lips. Dignified chin. His eyes were closed, and a cap of curly black hair crowned his brown skin. But the experienced eye knew differently and ticked off the indicators of the infant's affliction with weary ease. The bluish, trembling lips. The drawn cheeks. The eyes twitching beneath the lids and the head that every minute or so jerked along with them. Ataxic aphasia, they called it, a condition prevalent among children born to crack-addicted mothers.
A tap on the window drew Gavallan's attention.
"Coffee?" asked Rosie Chiu, the head duty nurse, pointing at her own mug. If she was surprised to see a man wearing a dinner jacket beneath his operating gown in the pediatric intensive care ward, she didn't show it. He'd been coming too long for that. Always at night. Always alone.
Gavallan shook his head and said no.
He'd first visited St. Jude's eight years earlier on a Friday evening benefactors' tour. The donors were lectured about the miracle of magnetic resonance imaging, the latest advances in open-heart surgery, and the newest cures in the war against children's leukemia. But it wasn't until Gavallan made it out of the neonatal intensive care unit that he grew angry. His neck grew hot, his suit two sizes too small. Like little Henry, he'd become twitchy all over. He wasn't sure why, but suddenly, he was mad—white-hot, steaming mad. Maybe it was the relentless sunniness of the place—the yellow walls decorated with dancing murals, the cheery nurses, the upbeat smiles—contrasted against the bleak reality of the situation. Even if these kids survived, what did they face? A life lived in medical institutions, state-run homes, or at best foster families. These kids with underdeveloped lungs and diseased eyes, with hair-trigger emotions and chronic aphasia. They had no right to their expectations, he'd railed silently.
But ten minutes later, when Nurse Chiu finished her talk about the hospital's need for volunteers willing to come and walk the infants—to help them grow comfortable with the touch of another human being, to teach them to accept the gaze of another set of eyes, and, yes, just to keep the noisy little gremlins quiet—he'd found himself alone agreeing to return. And he wondered whose expectations he was challenging. His or the kids'?
Gavallan shut his eyes. He couldn't handle another body laid at his feet. Oh no. Byrnes's call had freed him of illusions. Konstantin Kirov was just as Cate had described him—"ruthless and conniving, and maybe even more." This time Gavallan could not look elsewhere for excuses. This time he couldn't fall back on bungled intelligence or fumbled orders. This time it was up to him.
"Don't know if I can handle this one, chief," he whispered to Henry's sleeping brown face. "Think you can give me a hand?"
And he marveled in disbelief at how once upon a time he'd been a warrior.
The whine in his ear built slowly, as it had in the plane itself. A steady high-pitched cry that signaled the powering-up of the aircraft's avionics package. He was going back to the Gulf. To Saudi Arabia. To Iraq. To Desert Storm. To the night the infrared cameras on the underside of _Darling Lil_ recorded the tape that sat even now in his flight locker. The tape titled _Day 40—Abu Ghurayb Presidential Complex_. He was going to his own private little corner of hell, and his familiarity with the territory did little to lessen his terror over the trip.
_"Thunder three-six. Red one. How do you read?"_
_"Roger, Red one. This is Thunder three-six. Ready to copy words. Which way to Wonderland? "_
Gavallan is sitting in the cockpit of _Darling Lil_ , far out on runway two-niner at King Khalid Air Force Base deep in the Saudi Arabian desert. It is 01:15 Continental European Time, the morning of February 25, 1991. Day 40 of Desert Storm. Ground operations had begun twenty-four hours earlier and the vaunted Republican Guard is surrendering en masse. Morale is high. But Gavallan is ever cautious. When will Saddam unleash his biological weapons? Is he waiting until the last minute to launch a nuke at Israel? What exactly is the Iraqi dictator keeping up his sleeve?
Despite the cockpit's airtight seal, the desert air seeps in and surrounds him. It smells of jet fuel and sweat and a million square miles of superheated sand. Gavallan loves the scent. Inside its arid folds, he can taste his country's victory.
_Darling Lil_ is fully loaded for her night's work. Two GBU-27s sit inside the weapons bay. Each a two-thousand-pound package of high explosives capped with a delayed detonation fuse and a laser guidance system to guarantee hand delivery to the target.
"Thunder three-six. You are clear for takeoff."
"Thunder thirty-six copies all. _Salaam Aleik'hum."_
Gavallan wraps the fingers of his left hand lightly around the throttle and guides it forward. For a moment, the plane rocks, as if a boat in a chop, then he releases the brake and the Black Jet begins its shot down the runway. At 180 knots, he rotates the aircraft up and the wheels lift off the ground. He loves this moment, when the aircraft leaves the earth and he feels as if he too has been freed from his temporal moorings. The first climb is brief. At fifty feet, he levels off the aircraft and allows it to build speed to three hundred knots, then pushes up the nose and begins his ascent to his cruising altitude of twenty-four thousand feet.
Outside the cockpit, the sky is cloudy. Few stars are visible. Gavallan's eyes are trained on his instruments: altimeter, flight speed, fuel. Tonight's flight plan is typical of the twenty-two missions he has logged to date. Takeoff to be followed by a rendezvous with a KC-135 to top off the tanks. After the completion of midair refueling, he will cross the Iraqi border and hit two targets, an IOC, or intercept operations center, at Ash Shamiyah, and an SOC, or sector operations center, at Ali Al Salem, one hundred miles to the south. Time to target is two hours forty-seven minutes.
Gavallan runs a hand over his pistol, flight harness, and G suit, fingers probing for the search and rescue map wedged into his leg pocket and the cloth "blood chit" on top of it. The blood chit is to be used in case of forced landing or ejection and carries four "tickets" offering a reward to its holder for helping shepherd the downed airman to safety. The 9mm pistol is in case the ragheads need more convincing.
Suppression of enemy air defenses has been ruled 98 percent successful, but someone has forgotten to inform the Iraqis of the fact. The flak that has greeted Gavallan on his most recent sorties is as hot and heavy as on the first night over Baghdad. Sooner or later, he will be hit. It is a law, not a probability.
He completes refueling without incident. Routine, he says, working to quell his apprehension, feeling restless in the green-glow midnight of the Black Jet's cockpit. He stays on the KC-135's wing for ninety minutes, then "stealths up" and turns east, driving _Darling Lil_ into Iraq. As he kills the primary radio, he glides his thumb over the CD player in his flight suit and hits the play button. Axl Rose screams, "Welcome to the jungle."
Ash Shamiyah goes off without a hitch. A grown man's video game. Bomb armed. Systems check good. Target acquired: a gray rectangle dead center in his infrared display. Bombs away. The long downward ride, his thumb steering death on its unerring path. Thirty seconds later the screen whites out—a desert flower blossoming on his IR display. The IOC is a rectangle no more.
Gavallan pushes the stick left, banking the plane hard into a four G roll. Gut tight, head in a hammerlock, he turns to a heading of 210 degrees, driving _Darling Lil_ to the night's second target. Five minutes later, static tickles his ear. The steely guitars of Guns N' Roses abruptly cut out.
"Thunder three-six. We have a code red, change of target."
Gavallan stiffens. The primary rule of Stealth flight has been broken. Radio contact on the newly installed EML—emergency transmission link.
"Proceed to target designation 'Alpha Golf.' This is a Priority One, Ring One engagement. Do you copy?"
_Priority One. Ring One_.
Unconsciously, Gavallan leans forward, a tiger who has caught scent of his prey. Ring One refers to "command control communications centers," or C3s, the highest-priority target on the modern battlefield. Priority One denotes that the commander of the C3 may be present at the target. In Iraq there is only one man who carries the moniker Priority One, and one of his many palaces is located in Abu Ghurayb—target designation Alpha Golf.
"Thunder three-six. Copy."
"Okay, Tex, this is your chance for the big time. Don't fuck it up."
It is his flight controller, Rob Gettels, and for once Gavallan can't think of a witty response. Suddenly his throat is dry, his stomach jittery. He's a rookie all over again and he's taking the plane up for his first flight. But a second later, the nerves calm, the hand steadies, and the breathing slows. He programs the onboard navigational system and banks the plane north. He is on his way. Priority One. Ring One. The Abu Ghurayb Presidential Complex.
Twenty minutes later, the radio crackles to life.
"Thunder three-six, green light on target Alpha Golf."
"Copy."
Gavallan lowers his seat an inch or two so that he can no longer see out of the cockpit. His world shrinks to the cocoon of instruments surrounding him. The stick between his legs. The throttle and weapons guidance joystick to his left. The infrared display that looks like a six-inch black-and-white television screen. The heads-up display above it.
He is at bombing altitude. A finger toggles the "master-arm" switch. The bomb is primed. Eyes forward on the IR display. Target spotted. A pale stable of buildings silhouetted against the gray desert floor. He has studied the target before, as he has studied all of Hussein's palaces, and he knows the main suite of bedrooms to be in the eastern wing, a slim outcropping from the principal complex of buildings. His middle finger slews the crosshairs back and forth across the palace until he decides he has found the wing. Then, as if a mechanism itself, the thumb locks down. Jett Gavallan does not miss his target. Distance five kilometers. A yellow light flashes. Laser acquisition engaged. Red letters fire on the heads-up display. Target in range. Gavallan hits the "pickle," a red button on top of the stick, and the weapons bay doors open. _Darling Lil_ shudders. Still no ack-ack. No SAMs corkscrewing their way through the night sky. No 57mm shells bursting like flashbulbs on his old Kodak Instamatic. Gavallan does not question. He does not hesitate. He attacks. He is the spearhead of his country's arsenal.
Gavallan depresses the pickle again and the bomb falls from the aircraft. Suddenly lighter, _Darling Lil_ jerks upward, and as his harness strap cuts into his shoulder, he grunts with a secret pleasure. His eyes lock onto the IR screen and the delicate crosshairs positioned over the east wing of the Abu Ghurayb Presidential Complex. All external stimuli disappear. He is in a tunnel. At the far end rests his target. The crosshairs do not move. Thirty seconds to impact. Twenty.
_Too easy_ , a voice whispers. _Where are the SAMs? Where's the flak parade?_ It is the voice that will haunt him for the rest of his life. He sees plumes of exhaust approaching the palace. He counts one, two, three vehicles. Tanks? Jeeps? Trucks? Ours? Theirs? Someone running away? Someone arriving?
Ten seconds.
The crosshairs do not move.
The radio screams. "Thunder three-six. Abort run. Copy?"
The bomb appears on the screen. A dark dot skimming across the ground at an impossible speed. Above the screen, a red light blinks. Fuel warning. Tanks low.
Five seconds.
"I repeat. Abort run. Friendlies in area. We have friendlies on-site."
The words fire in Gavallan's ear as a warning bell sounds in the cockpit. The fuel light is dim. Above it, another light blinks in time to the urgent keening of the warning bell. The Allied Forces Locator. He has engaged friendly forces. His eyes dart between the lights, hesitating. Events blur.
Two seconds.
Only then does the finger dodge right, the crosshairs leave the palace and land in the desert. Or did it go earlier? Before the command? It does not matter. The bomb does not listen. She has been too long on her downward trajectory and it is as if she is too stubborn to alter her course.
"Abort run! Confirm, Thunder three-six!"
One second.
The desert flower blossoms. The screen blanches. A blizzard of white noise. The palace reappears. The east wing is no more, a bonfire of angles fallen in on itself. The heat signatures have disappeared, too, replaced by the blotchy, pulsing quasars that indicate fire.
Ours? Theirs? Coming? Going?
Jett Gavallan does not miss his target.
"Friendlies hit! Friendlies down!" It is Gettels, his operational calm obliterated. "Christ, Tex, I said abort!"
Gavallan blinks his eyes and catapults through space, through time, through the firestorm of his emotions to the present. He is walking. In his sleep, the baby named Henry twitches and is still.
Ten Marines dead. Two in wheelchairs for the rest of their lives. Forward elements of Task Force Ripper, he was to learn later. Scouts who got too far ahead of themselves. Gavallan knows their names to a man. He has sent the families checks for years. But their financial support is skimpy fare for a ravenous conscience. Everywhere he looks he sees pleas for help. Ask me, he begs the unfortunate. Order me. But his appetite for atonement is insatiable. Guilt, he discovers, is a desire, not an emotion. It can be slaked, but never extinguished.
And Saddam? Was he within one hundred miles of the palace? Not likely. News reports showed him touring Baghdad the next day, his beleaguered people showering him with praise.
As for the postscript, well, it went as he had imagined. The immediate transfer out of the theater of operations. The flight stateside. The firm and not so polite request that he resign his commission and never speak of the incident again. More he never learned. Who'd gathered the intelligence? Who gave the order for the raid? Why had the abort command come so late? Was the fuel light faulty? Was the allied locator on the fritz? What did it matter? No amount of rationalizing could scrub the blood off his boots. He had committed the cardinal sin: He had killed his own.
Now, if he didn't watch out, he'd have another name to add to the list. Not a heat signature in desert fatigues, but his best friend in the world. The man who'd stood by his side at weddings, christenings, and funerals. The man he'd worked alongside twelve hours a day, week in, week out, for seven years. The man he'd sailed with to Hawaii, ate steaks with at Alfred's, got drunk with at the Chaya. The only man he knew who gave a good goddamn about John J. Gavallan from Brownsville, Texas.
"Hey, Graf," Gavallan called silently across the miles. "Hang on, bud, I'm coming to get you. Don't ask me how or when, but I'm coming."
_Hundred-hour war_ , the world had called it. _Piece of cake_.
Gavallan looked down at Henry. The boy looked like he was smiling.
Piece of cake, kid.
# 19
#
**_K ER-THUMP!_**
Cate Magnus woke from a sound sleep, stirred by the jarring thud. The noise had come from downstairs. The den, she thought at first, still fuzzy. No, the study, she decided a second later, pinpointing the sound as having come from the room directly beneath her. Sitting up in bed, she trained an ear to the silence. The house was still and part of her wondered if she'd heard anything at all, or if the noise had simply been the slamming of a car door down the block.
It was early morning, and a predawn mist cloaked the bedroom in a grainy light. After a few seconds passed, she was able to make out the ottoman at the foot of the bed and the pile of magazines stacked on top of it. The _Economist, Vogue Italia, Harvard Business Review_ , and, God help her, the _National Enquirer_. Throw them out, she ordered herself. All of them, before they become a fire hazard. Her eyes flitted to the hand-carved walnut desk under the window where she worked on her precious journals, black-speckled notebooks stuffed with daily musings, ideas for the column, personal promises, resolutions and dreams, press clippings of current events, photographs, drawings, and caricatures—a thirty-year-old's running commentary on the world and her place in it.
In the corner stood her rotting, half-drunk armoire, teetering to one side on its bum leg. Beside the armoire rested her easel, her vase and brushes, and the fisherman's bait box that held her oils and acrylics. With the painting I've done lately, I ought to throw those away too, she thought. The guarantee date on her precocious talent had expired ten years ago. But for her treasured possessions, she found no comfort in the familiarity of her surroundings. After a two-year absence, the room remained unfamiliar, foreign, more a hotel room in a distant city than the home for which she'd scrimped and saved for so long.
_Ker-thump!_
The low-pitched noise came again, confident, brazen. Cate could feel the floorboards shiver, as if the house had been punched in the gut. The noise came from the study. Sure of it now, she acknowledged the first intimation of fear. Her stomach knotted itself into a ball and, holding her breath, she sat very, very still. She was not by nature easily frightened, but of late she'd been on edge. She was, she realized, a woman alone in a three-story house in a part of town that might be called "lovingly frayed." Or less generously, "down at its heel."
_The workers!_
It came to her in a shower of relief. At once, her body slackened and her lungs opened for business again. As quickly as her fear had come, it vanished.
For the last twenty days, her home had been a hive of activity as laborers from every guild assembled beneath her roof to help with the pouring of a new concrete slab beneath the existing structure. She'd learned quickly that tradesmen were no respecters of the eight-hour day. Electricians were as likely to show up at seven at night as seven in the morning. Carpenters were happy to stay until you kicked them out.
It's Howie, she told herself, the long-haired foreman who looked as if he couldn't lift a hammer. He's come to check on the job's progress and bum his morning espresso. Caffeine freaks seemed to find one another, and neither Cate nor Howie could start the day without their Lavazza double espressos.
Or maybe it was Gustavo, the drop-dead-gorgeous Basque bricklayer who didn't go a day without asking her for a date. "Meez Magnus, we go deen-er together. You like ke-bab? I show you perfect good time, non?" Ten rejections, and still no sign of giving up.
Three weeks into the project, the cost had skyrocketed from eighteen to thirty thousand dollars, and there was no end in sight. Each day brought a new complication: faulty wiring, rusted pipes, asbestos. _Yes, asbestos!_ Yesterday, she'd learned the hundred-year-old Victorian fixer-upper suffered from a healthy case of softwood termite infestation. Once the slab was completed, the house would have to be tented and fumigated. Cost: seven thousand dollars. Where she'd get the money to pay for it seemed to be no one's concern but hers, and the cause of one hangnail, a persistent headache, and very soon, if she wasn't careful, an ulcer.
Cate had no choice in the matter. The work was obligatory. The building code demanded it, and The Code's will be done. It had all started because of a faulty outlet. First her toaster blew, then her rice steamer. She called in the electrician, who traced the problem to a frayed circuit box beneath the kitchen floorboards. But the circuit box wasn't the real problem, he'd informed her while writing out his bill. The house, it turned out, had been built half on a wooden foundation, half on bare earth. It was a code violation of mythic proportions. By law, he was required to inform the building inspector. She asked him how the house had managed to remain standing through a century of earthquakes, including, if she wasn't mistaken, a couple of doozies in 1906 and in 1989. The electrician didn't know. He only knew that a bare-earth foundation was against code.
_Code!_
She'd learned to hate the word and had reserved a place for it in her personal lexicon alongside "fascist,"
"fibber," and "philanderer," three hall-of-fame baddies.
Even with her name in bold print beneath a weekly column, she barely earned sixty thousand dollars a year. Take away taxes, utilities, car payments, and her mortgage, and she was left with a disposable income of eight hundred dollars a month. Enough for one martini and a cowboy rib eye at Harris's, a couple of movies, a pair of tickets to a Giants game, and maybe a pair of shoes—all depending on how she filled the fridge. Every time she heard a politician say she was "affluent," she wanted to brain him.
_Ka-lunk!_
The sound was louder this time, as if someone had dropped a bowling ball onto her precious stained-pine floor. Cate cocked her head, no longer so confident it was the workers. The problem was the thump itself: The quality of the noise, its pitch and timbre, was unfamiliar. She did not recognize it.
Over the last month she'd become fluent in the buzzes, bangs, and squawks of a construction site. She could rattle off any of a dozen different tasks simply by listening to the frequency of the saw blade or the whine of the drill bit. The thump was not a sledgehammer. It certainly wasn't a pick. No, the sound coming from the study was that of a large object being dropped upon the floor.
_The thump was a stranger, and it scared her_.
Only then did Cate pick up her watch from the night-stand and look at the time. It was 4:06. The streetlights reflecting off the dense fog had lent the sky an eerie luminescence, feigning sunrise and providing a false dawn.
_4:06_.
Cate stared at the dial, anger and fear welling up in her in equal parts. No workman showed up at a job site at 4:06. Even Bob Vila didn't go to _This Old House_ until six-thirty at the earliest! Suddenly, she was wide awake, her senses honed, her radar on full alert. She could smell the oil from the cement mixer parked out front. She could hear the ticking of her watch, the hum of the PC on her desk. The screensaver ran on a loop reading, "John Galt is dead. John Galt is dead." Her capitalist manifesto.
Someone she did not know was inside the house. There was an intruder in her study. _Call the police_. She reached for the phone, but froze halfway there, paralyzed by an older and more wrenching fear. There were worse things than physical peril.
Retrieving her hand, she slid her back against the headboard and waited for a footfall on the landing, for the door to her bedroom to be flung open. For a few moments, the house was silent, and Cate decided it was better for you to go get them than for them to come get you. Gathering her courage, she placed her feet on the ground and stood. For once, she'd make impatience her strong suit. She took one step and stopped, but only for a moment—just long enough to double-check if her sanity was in its proper place, tucked between her aversion to cigarettes and her love of Vermeer—then padded across the room to the bedroom door. The wood planks were cold to the touch and groaned at her meekest step.
_Slowly_ , she ordered herself, concentrating on rolling her feet from heel to toe. You're a Shaolin priest walking on rice paper, she said, quoting from the bible of late-afternoon TV. Calmly, Grasshopper. But to her revved-up ears, she sounded like a newly shoed colt crossing the smithy's floor.
Cracking open the door, she peered to her right and left. The landing was empty, dusted with a sheen of plaster that glowed in the dark like some phosphorescent algae. There were no lights on in the house. Advancing on the staircase, she began to get the motion right, heel to toe, rolling her foot, and her tread fell as delicately as a doe's.
But if her steps were controlled, her mind was running full tilt. She remembered that she hated living alone and cursed herself for moving out of Jett's four-thousand-square-foot home in Pacific Heights. At the same time, she reminded herself she'd had no choice, even though leaving had been the hardest thing she'd ever done.
Continuing her spate of recriminations, she turned to the alarm system—or more specifically, to her practiced nonchalance about turning it on at night. What was the point? With so many workmen traipsing in and out of the house at all hours, it was better to keep an open door. Besides, it was hardly as if there was much to steal: a ten-year-old TV, a few silver candelabra, a stereo she had yet to hook up since her return to singledom.
Her neighborhood on the fringes of Haight-Ashbury wore its poverty like a genteel curse. Rusted VW vans, twice-repainted Olds 98s, SS Camaros with fat racing stripes running across their hoods, lined the curb, their bumper stickers badges of membership to a bygone era. "Drop in, Turn on, Tune out," "Age of Aquarius," and her favorite, "Keep on Truckin'," with the magnificent Crumb icon strolling along flashing the peace sign. On a sunny Saturday afternoon you couldn't pass two houses without hearing Mason Williams's "Classical Gas" or catching the scent of Colombian Gold wafting from an open window.
_But you didn't put in the alarm to protect your possessions_ , a wise voice reminded her. _You installed it to protect_ yourself. _You always knew they would come. You should have known it would be now_.
Laying a hand on the banister, she began her descent. There were fourteen steps to the first floor, the lower six sick with termites. With every step, she craned her neck farther over the rail, curiosity winning over fright as to what or whom she might discover.
_Ka-thunk!_
Cate stopped cold, frozen so still she might have been geologically petrified. Silhouetted against the ivory wall, her figure was slender, well-proportioned, and if ten pounds heavier than she would have liked, the more fit for it. She ran three times a week, made it to Pilates every Saturday morning, and ate enough Cherry Garcia to make it all for naught. She liked to think of herself as strong and capable, but alone in her house at 4 **A.M.** the opinion seemed boastful and ridiculous. Refusing to budge, she asked herself who it could be banging away in her study so contemptuously, who the interested party was who was practically daring her to come down and ask what the hell was going on.
Again she entertained the notion that it was a burglar, but she knew better. Nor could she bring herself to believe it was a rapist, a psychopath, a deviant, even a garden-variety lunatic trying to lure her downstairs to have his way with her. It was none of them. Or anyone else, for that matter, who might have randomly chosen her home to break into on this damp, foggy night.
She knew why there was someone in her house and she knew what they were looking for. She had known for some time that her existence could no longer be accepted with a tolerant grunt or dismissed with a paternal wave. Not with events moving as quickly as they were. It amused her that some people might think her dangerous. Cate Magnus, graduate of the East Coast establishment: Choate, Georgetown, Wharton. She, the failed painter, exiled executive, sucker for beat-up Jeeps and obscure French films. The reporter with a dozen great ideas for books and never the tenacity to complete an outline, the lifelong fugitive from romantic misadventures. Why should anyone be afraid of her? She was someone whose fingers felt more comfortable teasing the keys of a computer than the trigger of a gun.
Cate stared at the pistol in her hand, dull, gray, and bluntly menacing. For the life of her, she could not remember fishing it from the cache on the side of her bed. She noticed, too, that she was wearing her panties and nothing else. Great. Get the gun, but forget your clothes. Show 'em your boobs, then shoot.
_No_ , countered the wise voice again. _You're still fooling yourself. You're a searcher, a collector, a seeker of the truth. You are a woman with a vendetta and the means to exercise it. In fact, you're very dangerous. Never more than now, and you know that, too. As for the gun, don't be coy. You trained five nights a week for a year so that you could hit a nickel at twenty paces. Why did you steal it from your boyfriend's house if not to use it?_
The thud came again. _Ka-thump_.
Suddenly, she knew what they were doing. There were two of them. There were always two. They were trying to get into her safe, the little fireproof model she'd picked up at Home Depot to protect her zip drives and her journals against fire. They were lifting it and dropping it or banging something on top of it in some brutish attempt to pry it open.
Cate reached the first-floor foyer. At the end of the hall, the door to the study was shut, a light burning beneath the crack. She advanced a step, holding the gun in front of her. They really were insolent, she thought, praying anger would fuel her courage.
Something warm and feathery brushed against her leg, and Cate nearly jumped out of her skin. She wanted to scream, but found her heart already lodged in her throat. She looked down and stifled a shrill note of terror.
It was Toby, her gray Angora. Toby, the meowing mauler of Menlo Park, whom she'd threatened to get rid of a hundred times because the damned kitty never shut his mouth. "Shh, Toby." She reached down to pet him, but he was already gone, bounding upstairs to doze in the folds of her duvet. "Coward," she hushed after him.
And straightening her body, she summoned the will to open the study door. _I'm a dangerous woman_ , she thought proudly, taking another step. _I can plug a nickel at twenty paces. I can—_
She didn't hear him coming. Not a footstep or a whisper or even a breath of wind. One second she was alone, the next a large, sweaty hand had clamped itself over her mouth. Cate struggled to turn, to drive an elbow back and into his ribs as she'd learned in self-defense class, but the man was upon her, pulling her into his body, his free hand locking onto her wrist, wrenching the gun loose with one furious twist.
"We're in the library," he said. "We've been waiting for you to join us."
Cate stopped squirming and allowed the man to guide her into her study.
Two men stood by the safe. They'd managed to open it, God knows how. One was perusing her journals, the other tearing through her desk. She knew their type, if not their names. The crew cuts, the aggressive eyes, the pumped-up shoulders and size-twenty necks.
"What are you looking for?" she said when he'd removed his hand.
"You know what," replied the man holding her. "Why are you talking to the police?"
"I'm not." Her fear had vanished, cowering before her mammoth indignation. "You're wasting your time."
"We'll see."
He let her go and spun her around, and for a moment she thought that was it, he was moving to the rough stuff right away. She had no illusions about her ability to guard her secrets. If they beat her, she would talk. Instead, the man brushed by her and devoted himself to a tour of her bookshelves. She remained where she was, quiet, suddenly embarrased by her nudity, covering herself.
After a few minutes, the man gave up his perfunctory search. "Anything?" he asked, turning to his colleagues.
Shrugs were their only response.
He approached Cate, taking her face in his meaty hands and bringing it close to his. He was older, with pitted cheeks, black eyes, and a slit for a mouth. "Keep your mouth closed," he whispered. "Understand?"
When Cate didn't answer, an angry expression contorted his face. "Understand?" he said again, squeezing her cheeks and twisting her jaw.
"Yes," she managed to grunt. "I understand."
A minute later they were gone, leaving the front door open behind them. Cate walked to the door and shut it. As an afterthought, she turned on the alarm. But as she climbed the stairs to her bedroom, a smile of bitter satisfaction played on her lips.
She had them on the run.
# 20
#
**S TOP IT THERE!"** shouted Howell Dodson, deputy assistant director of the FBI, slapping a palm onto his desk. "I want to hear the last part again."
Roy DiGenovese reset the digital recorder, punching the play button when he'd gone back exactly thirty-one seconds. A tinny voice began to speak, the Eastern European accent faintly noticeable.
_"And what about the Private Eye-PO?"_ asked Konstantin Kirov. _"What do you plan on doing to him? Surely you do not expect us to sit still while our good name is besmirched."_
_"I have some people on it already,"_ answered Jett Gavallan. _"With any luck, we'll have him located by tomorrow, day after at the latest."_
_"And then? All of us have our part to play to insure Mercury's future. We expect you to take any and all measures to silence this man. Nothing can stand in the way of Mercury Broadband's going public. Nothing."_
_"And nothing will. I'll see to it the Private Eye-PO's mouth is shut—permanently, if I have my way. In the meantime, these receipts refute his accusations nicely. I'd say we're back on track."_
_"Good,"_ said Kirov. _"It's time to put an end to this tomfoolery. There's already been enough snooping."_
The recording ended, and DiGenovese turned the machine off.
It was eleven-thirty in Washington, D.C., and outside the temperature registered a sweltering ninety-two degrees. From his office on the second floor of the J. Edgar Hoover Building, Howell Dodson, chairman of the Joint Russo-American Task Force on Organized Crime, could see the early lunch crowd making their way to the mall in hopes of staking out shaded benches or dipping their big toes in the Reflecting Pool. It wasn't much of a view. The prime offices were on the opposite side of the building, facing south and offering a panoramic vista of the Capitol, the Washington Monument, and Mr. Thomas Jefferson, fellow Son of Virginia. One day he hoped to gaze out at the Lord of Monticello, but good views required good politicking, and good politicking required a cunning he did not possess.
"What do you say, Roy?" Dodson asked in a slow Williamsburg drawl, his voice the texture of cured tobacco. "Mr. Gavallan talking prudent business practice or did we just hear collusion among conspirators?"
"That depends on Mercury, sir. If the business is legit, I'd say we listened to a bunch of execs who want to stop someone from bad-mouthing their stock. If not, we just tuned in to a group of criminals discussing murder. Me, I opt for the latter. I think we caught some crooks red-handed."
"So the Private Eye-PO is correct? Mercury's nothing but 'a scam dog with mucho fleas'? That what you're saying, Roy?"
"We're getting the same information from our informant in Moscow. Why shouldn't we believe it?"
Dodson couldn't help but chuckle. Three years in the Bureau and Mr. DiGenovese still considered an informant's cant the holy scripture. The boy was a greenhorn. Yes sir. Nothing but a big-city hick. Dodson himself wasn't so much interested in whether what the Private Eye-PO said was correct as in how he came to be in possession of the information. And for that matter, just who in the hell he was. "What's the latest on finding this boy? Mr. Chupik have any luck?"
Lyle Chupik was the Bureau's in-house webhead and the man who'd been charged with tracking down the Private Eye-PO.
"Nothing yet, sir," said DiGenovese. "Says he's close to nabbing him, though."
"Close?" Dodson lifted a thumb beneath his suspenders and let them slap on his chest. "Close don't count but for horseshoes and hand grenades. Isn't that right, Mr. DiGenovese? Mr. Gavallan seems to think he'll have him located today. That leaves us one step behind. And I don't like stomping through another horse's droppings," he whispered, with just a smattering of menace. "Follow?"
"Likewise, sir."
"Good boy. It's time we considered using an outside source. Find me the name of that odd fella does some consulting for us. If I'm not mistaken, he doesn't live too far away. Get him in here this afternoon and put him to work. Here's a dollar. Go buy Mr. Chupik a couple of those chocolate Yoo-Hoos he's so fond of, and tell him better luck next time."
Howell Ames Dodson IV was a Son of the South and ever proud of it. He was tall and lanky, with a shock of brown hair that fell boyishly into devilish blue eyes that teased the world from behind a scholar's half-moon glasses. He favored poplin suits in the summer, worsted gabardine in the fall, and the finest manners all year round. He liked smartly striped shirts, exuberant ties, snazzy cuff links, and pocket squares. He was foppish and a bit of a dandy, and if anyone cared to say a word about it, he'd point them to his unmatched arrest-to-conviction ratio, the commendations he'd received from the President of the United States, and a certain article in the _Washington Post_ he kept tucked away in his desk for just such occasions.
The article described the shooting of four Georgian mafiosi by an unnamed FBI agent in a sting gone sour in the city of Tbilisi late last summer. The article was sketchy in parts. It failed to mention that the agent had shot the men after escaping from their custody or that he'd pulled off the feat fifteen minutes after having two fingers on his left hand severed with a carpet knife.
Sliding the digital recorder toward himself, Dodson listened to the pirated conversation again. "So, Roy," he said when the recording ended. "Think our boy isn't content with a little innocent fraud? That why you asked for this crash meeting? According to you, Mr. Gavallan's joining the big leagues. Premeditated murder is moving up the ladder p.d.q., wouldn't you say?"
"Sir, the Mercury offering is for two billion dollars," answered DiGenovese, leaning across the desk. "Leagues don't get much bigger than that."
"No, son, they do not," said Dodson, rocking in his chair, tapping a pencil on his weathered shipwright's desk, a nineteenth-century antique on loan from the Dodson Family Collection. "Just wish that damned recording didn't make them all sound like robots. Hard to tell if Gavallan's joking or if he's serious."
"Sir, with all due respect, when an associate of a known criminal talks about permanently getting rid of someone, I think that qualifies as serious. Our job is to take a man at his word, not to guess his intentions."
Such fire, mused Dodson, looking at the lean, vital young man seated across the desk. Such drive. His hair was ruffled, his suit wrinkled and in need of a press, but his black eyes were awake and dancing with a mean-spirited ambition. DiGenovese was the kind of agent who wanted to arrest the whole damned world to keep it safe for the police.
"Come now, Roy, we both know that conversation doesn't amount to a hill of beans," he said kindly. "It wouldn't hold a drop in any court in the land. Between you and me, I doubt it would even garner an indictment from so docile a beast as a sitting grand jury. I will grant you one thing, though: It does appear that Mr. Gavallan and Mr. Kirov are closer friends than any of us thought."
Dodson could have added that contrary to DiGenovese's opinion, Kirov was hardly a known criminal, but he didn't want to dampen the boy's enthusiasm. DiGenovese's killer instinct was about all the task force had going for it these days. Truth was, Kirov hadn't ever been charged with a crime, let alone convicted. Not that Dodson didn't think Kirov was dirty. It was just that these days you could label any businessman worth his salt in Russia a suspected criminal. What with all the bribery, extortion, and strong-arming that went on to make the wheels of everyday commerce go round, if you looked closely enough just about anybody was guilty of one infraction or another.
"Now do tell, Roy, what did your team find in Mr. Gavallan's private chambers? Love notes between him and Mr. Kirov? Written promises about how they're going to split the booty? Plans to overthrow the President?"
"No sir," DiGenovese answered without a hint of regret, going on to explain that they hadn't found any documents of an incriminating nature, not with regard to Mercury, Novastar Airlines, or anything else. The bugs were clean too. Only thing they learned was that Gavallan liked to listen to country music. Before going to the ball last night, he'd sat in the bath for half an hour singing along to Bob Wills and the Texas Playboys.
"Bob Wills, eh?" asked Dodson, cleaning his bifocals with a hankie. "At least Mr. Gavallan has himself some taste. Still, it is a shame. Going to all that trouble for nothing. A damned shame indeed." And though his voice displayed no irritation, he was, in fact, hopping mad. Howell Dodson wanted Kirov more than the headstrong Mr. DiGenovese or Mr. Baranov combined. It wasn't ambition but realism that told him the trajectory of his career depended on it.
Konstantin Kirov had popped onto the Bureau's radar half a year back, when Yuri Baranov had launched an investigation into allegations Kirov was embezzling from Novastar Airlines, the country's recently privatized national carrier. Three months into the case, the Russian authorities had managed to slip an informant into Kirov's head office. Since that time, all he'd unearthed were a few documents relating to some shell companies in Switzerland and Kirov's connection with the Dashamirov brothers, a trio of Chechen warlords-cum-businessmen with whom he held interests in some aluminum smelting factories in Perm and a chain of used-car dealerships. As for Novastar, they hadn't managed to find a thing linking Kirov to the missing $125 million, and Dodson had his doubts as to whether the Russian was involved at all—or, to be honest, whether the money was missing in the first place.
The link to Gavallan came as an adjunct to the Novastar inquiry. Baranov's informant had whispered that Mercury Broadband was being used to launder the funds Kirov had skimmed from Novastar. Hence the surveillance on Gavallan. Hence the "Daisy" taps that monitored every E-mail going into and out of Black Jet securities. So far, the Russian stoolie hadn't provided a shred of evidence to back up his claims, and Dodson had taken to wondering if the scuttlebutt on Mercury's Moscow operations center and its failure to purchase adequate routers and switches for its IP backbone weren't just diversions to justify the informant's five-thousand-dollar monthly retainer, all of which came from Howell Dodson's operational budget.
"Sir, I'd like to bring in Gavallan immediately," suggested DiGenovese. "Rustle his feathers a little, question him about his dealings with Kirov."
"The point being?" asked Dodson, with a little pepper. "Only thing you'd get out of him is an invitation to speak with his lawyer. No, son, we'll bring in Gavallan if and when we charge him with a crime. Right now, let's keep the focus on Mr. Kirov, where it belongs."
"But, sir—"
Dodson cut him short with an icy glare. Like every agent who worked for the FBI, he thought twice these days about whom he did and did not arrest. After Whitewater and the special prosecutor's spending forty million dollars of the public's money for little more than a cum-stained dress and a couple of iffy convictions, the government had become more demanding before allowing its lawyers to get involved. These days, the powers that be were asking for a 90 percent probability of conviction before they'd even look at a case. Law enforcement had become a business. Guys like Howell Dodson had to demonstrate a good ROA if they wanted to move up in the ranks, "ROA" meaning "return on _attorneys,"_ not _assets_. And that "return" was convictions.
"Trouble with you, Roy, is that you've got too much piss and vinegar running through your veins. This isn't some Sunday afternoon raid in downtown Mogadishu. We are conducting a sound and systematic investigation into the alleged wrongdoings of some very sophisticated personalities. Time we slow down, examine the evidence."
"Yes sir."
"Well, amen," sang Dodson. "Finally, we agree on something." And he offered his subordinate an approving nod to let him know there were no hard feelings.
Dodson had come to the Bureau late in life, abandoning a promising career as a CPA with an international accounting firm to help balance the scales of justice. Taxes were his bag, but sometime after his thirtieth birthday he'd undergone a conversion. The private sector wasn't for him, he decided. Helping one bigwig after another whittle down their tax exposure brought scant satisfaction. He certainly didn't need the money. The Dodsons were comfortable, thank you very much, Southern planters who'd moved from corn to tobacco to semiconductors without a backward glance. So on a whim, he quit, joined the FBI, and became a thirty-one-year-old neophyte loping over the O-course at Quantico, acing his criminal justice exams, and taking target practice with an H&K 9mm. Time of his life.
As chairman of the Joint Russo-American Task Force on Organized Crime—or the "ratfuckers," as some wiseacre in forensics had nicknamed it—Howell Dodson's mandate was to corral acts of racketeering associated with business endeavors aimed toward the West. In sixty months of operations they'd jailed crooked oil salesmen, murderous rug merchants, and every type of illegal operator in between.
Of late, however, pickings had been lean. Nine months had passed since the last arrest was made, and talk had surfaced about shuttering the task force, assigning its members to more productive areas of the Bureau. Feelers were put out to Dodson about taking a posting to Mexico City as the Bureau's liaison to the _Federales_. It was a lateral move in title, but came with a higher pay-grade salary and a diplomatic allowance. Dodson read it as recompense for his two fingers and wanted none of it. Margaritas, mariachis, and menudo, he summed it up, cringing at the prospect. _No, gracias_.
Mr. John J. Gavallan hadn't been the only man cheering when Kirov entered his life.
"Roy, I want you to humor me," said Dodson, easing back in his chair. "If you're so sure Gavallan's in cahoots with Kirov, start from the get-go and make your case against him. It'll be good for you to polish those argumentative skills. But make it quick. The missus is due in any minute."
Dodson had recently become a father for the second time. At the age of forty-two, he'd been presented with twin baby boys to go along with his sixteen-year-old daughter. Every day at noon, Mrs. Dodson stopped by to leave her boisterous infants with their father while she whipped by Lord & Taylor and Britches of Georgetown to pick up a few household necessities.
"I'll do my best," said DiGenovese, rising from his chair and striding to a bookshelf. Between legal tomes and hefty accounting manuals, room had been cleared for a changing pad, a stack of diapers, and wipes.
"Gavallan's company has hit the skids," he began, pacing slowly, using his hands effectively. "Three years ago, he was on his way to joining the big boys; now he's treading water while guys are passing him left and right. In the last nine months, he's made three infusions of cash into the company to counter quarterly losses and keep his underwriting status with the SEC. Around twenty million and change if I'm not mistaken. The banking records we subpoenaed show he hasn't taken any salary in six months. Bottom line: The guy's hurting and he needs a savior."
"If I might interject. Black Jet was hardly the only company interested in Mercury. All the big-name firms were courting Kirov. Any of them would have jumped at the chance to take his company public."
"And loan him the fifty million to boot?"
"It _is_ a bank's business last time I checked," said Dodson.
DiGenovese grinned madly, the cat who'd swallowed the canary. "Thank you, sir. You just made my case. If anyone would have loaned Kirov the dough, why did Kirov choose Black Jet over so many larger, more prestigious firms—the Merrills and Lehmans of this world? Gavallan's never done a deal in Russia. He's never done an IPO valued at more than a billion dollars. Now, all of a sudden he's taking a Russian company public for two billion. By what stroke of good fortune did Kirov fall into his lap? Let me tell you. Because Gavallan's the only one desperate enough to overlook all of Mercury's shortcomings. Because he and Kirov are thick as thieves in this thing. Because both of them are dying to pull this deal off."
"Dear me, you are drawing a picture of a very cold man. Not exactly the type I'd bet on to donate twenty million to a children's hospital."
"Window dressing," declared DiGenovese. He'd unbuttoned his jacket and was stalking the room like a wolf in his den. "So far he's given two mil, and he's a month late on this year's pledge. Five'll get you ten he never delivers." Abruptly, he stopped his pacing and thrust his hands on Dodson's desk, his peasant's jaw jutting forward. "Mercury's a phony, sir. Kirov's got it dolled up to look like AOL when it's really CompuServe. Gavallan's in bed with him, and together they're going to pull the wool over investors' eyes and pocket the takings. You see what he's pulling down on fees for this deal? Something like seventy million dollars? Seventy _million!"_
"My, my, Roy," drawled Dodson appreciatively. "That would make Mr. Gavallan even more ambitious than you. Isn't that a scary thought? One thing is for certain: Mr. Gavallan's not in this alone. In the first place, Black Jet didn't even do all of the due diligence on this thing. I don't know how many lawyers and accountants and consultants he had signing off on Mercury, but believe me it was a lot. You saying they're part of this, too?"
"Never know."
Dodson nudged his glasses to the end of his nose. "Quite a conspiracy you're cooking up, Roy. Seen any Kennedys flitting around out there in never-never land? Or just Peter Pan and Jett Gavallan?"
Rummaging in his ashtray for a rubber band or two, he wrapped them around his index and middle fingers and, kicking his feet up onto the desk, began to spin the bands forward and back. He liked the bent of the argument, if he wasn't convinced of its veracity. He thought of the transfer to Mexico City, the traffic, the bad water, the horrid food _—enchiladas_ , Heaven forbid—and came to a quick and rational decision that the American angle was just what they needed to drum some new life into the investigation.
A billion-dollar fraud involving a Russian oligarch, a former fighter pilot, and that holy of all holies, the New York Stock Exchange. It was practically treason. He caught himself thinking that the press would have a field day and the man who put Jett Gavallan behind bars would be instantly famous. He stopped himself there. All the infighting, posturing, and backbiting were getting to him. Still, for a last second he couldn't help but imagine that the man who put Gavallan behind bars would have a south-facing office and the promotion to assistant director that came with it.
Rising from his desk, Howell Dodson strode to the window. The harsh glare showed off wrinkles near the eyes normally unseen, a determined cast to the jaw, and a nasty downturn of his pale, fleshy lips. Suddenly, he didn't look so boyish anymore, not every inch the amiable Southern gentleman he pretended to be. Get close enough to any man, he would say, and you could glimpse his true nature. And underneath his easygoing drawl and unflappable smile, Howell Dodson was a nasty sumbitch who did not like to get beaten.
Just then, the cries of his two baby boys exploded from down the hallway. A moment later, a trim, very blond woman bustled through the door, a wailing infant in each arm. Rushing to greet his wife and sons, Dodson softened his expression into a broad smile.
"Hello, Jefferson. Hello, Davis. And how are my two little generals this mornin'?"
# 21
#
**A MOMENTARY LULL HAD DESCENDED** on the trading floor at Black Jet Securities. Phones had stopped ringing—or blinking, as has become their convention—conversation had fallen to a whisper, the shuttle of chairs to and fro between desks had come to a halt. "A rest between rounds," Gavallan liked to call it on good days. Or "the calm before the storm" on bad ones.
Incisex stock had begun trading on Nasdaq—the National Association of Securities Dealers' Automated Quotation System—thirty minutes earlier at 6:55. Ticker symbol CSXI—pronounced "sexy"—Incisex was a pioneer in the field of nanotechnology, the branch of science concerned with building sophisticated engineering devices—engines, motors, valves—on a submicron scale. The company's breakthrough product was a battery-powered valve no bigger than the head of a needle that when surgically implanted into a coronary artery restored proper blood flow to the heart. For men and women suffering from arteriosclerosis or any vasocirculatory problems (and for their cardiologists), it was a godsend. The IPO would net Incisex seventy-five million dollars, funds it would use to move into larger facilities and to upgrade its research and development efforts. Black Jet's fee was the standard 7 percent of the offering.
"Seventeen bid, seventeen and a quarter ask," announced Bruce Jay Tustin from his post behind a dozen color screens and monitors. "We're going up, up, up!"
The issue had been priced at $14 a share and after thirty minutes of trading, demand had lifted the shares 20-odd percent to $17. Checking a screen above Tustin's shoulder, Gavallan saw that buyers outnumbered sellers three to one. It was a far cry from the bonanza days when one new issue after another would double or triple on its first day of trading, but Gavallan wasn't complaining. In a rational world, a first day's gain of 20 percent qualified Incisex stock as "popping" all the same.
"I think we can open the champagne now," he said to an assembly of four men and two women standing to one side of the room. "Mr. Kwok, would you do the honors?"
On cue, Wing Wu Kwok, a newly hatched associate who had accompanied the Incisex brain trust on their two-week road show across America, uncorked a bottle of Moët & Chandon, filled a half dozen flutes, and offered round a silver tray bearing beluga caviar, toast points, and china dishes brimming with chopped egg white and diced onion.
Gavallan accepted a glass of bubbly and raised it high. "To Incisex and that rarest of all marriages—profit and the public good. Cheers!"
There were huzzahs all around. Hugs and handshakes followed the clinking of glasses.
"May the sun be ever in your eyes," Tustin chimed in, "and the wind at your ass."
"Here, here." Gavallan managed a smile, but just barely. Two hours' sleep had left him exhausted and haggard. With Grafton Byrnes's predicament increasingly weighing on him, it was difficult to maintain a cheerful façade. If no one had remarked upon the dark circles beneath his eyes, it was only because he was the boss.
For a few minutes, he mingled with the executives from Incisex, taking refuge from his worry in the cloak of chief executive. He slapped some backs, he partook of the caviar, he extolled his clients' rosy future. But even as half his mind concentrated on projecting a carefree exterior, the other half remained fraught with doubt. Two words from his best friend had turned Cate Magnus's hotly worded suspicions about Kirov and Mercury into Gavallan's worst nightmare.
"Everything's copacetic."
Gavallan had to imagine the rest. Grafton Byrnes was being held against his will somewhere in Russia and would never return. He knew too much. He was sure to be killed, if he wasn't dead already. It was all that simple. That terrible.
Pondering his conclusions, a new thought dawned on Gavallan, one that his trusting mind decided was more frightening than the rest. If Kirov felt so secure that the Mercury offering would come to market that he would risk kidnapping Byrnes, he had to have someone in place at Black Jet to push through the offering despite the chairman's opposition, someone highly enough placed in the company that he might persuade Jett that Byrnes's disappearance was a coincidence, nothing more.
As quickly as it had come, the lull on the trading floor abated. Lights began flashing on the checkerboard consoles that connected Black Jet to over a hundred banks, brokerages, and financial institutions around the world. Voices boomed as traders greeted their clients with news of the strong offering. Casters groaned as the bankers recommenced their daylong jitterbug.
The trading room of Black Jet occupied the entire western length of the fortieth floor. Desks ran perpendicular to floor-to-ceiling windows, twelve carrier decks bisected by a flight tower constructed from the newest in flat-screen monitors. Currencies were to the left of the room, followed by bonds, options, and finally, equities, both domestic and international. Chairs were situated at four-foot intervals and nearly every post was occupied by a man or woman, standing, seated, or in some pose in between. One hundred forty traders in all, and when things heated up, the place took on the frenetic currents of a Middle Eastern bazaar. It was the Casbah gone California, Evian and Odwallah replacing hookahs and hashish.
Gavallan leaned a hand on Tustin's desk, marveling at his ability to goad the price of the stock ever higher. Picking up a receiver, he patched himself into Tustin's call.
"Hey, Brucie, what d'ya got for me on Incisex?" The voice belonged to Frank MacMurray, a trader at Merrill Lynch.
"Her name's 'sexy' and I can give you a block of ten thousand at 18."
"Eighteen? Last bid's 17 **½.** Gimme a break."
"Got ten other johns lined up right behind you, Frankie," Tustin said. "But listen, pal, since you're cute, I'll cut it to 17⅞. Buy or fly."
"Done, and get me ten more at the same price."
"You're filled."
Tustin aimed a finger at another flashing button, this one connecting him with Fidelity Investments, the nation's largest manager of mutual funds. "Yallo, Charlie, what are you looking for?"
Gavallan knew from reading the "book" that Fidelity was a buyer of Incisex. They'd loved the stock's story and planned to build a position in it in one of their biotech funds. Accordingly, they'd given an indication they'd take 10 percent of the issue. No one firm would be allotted a full 10 percent of the offering—in this case over five hundred thousand shares. As it was important that Incisex had a broad and liquid market, Black Jet had a duty to sell shares to a great many customers, some of whom were retail brokerages—Merrill Lynch, Paine Webber, Bear Stearns and the like—that would in turn pass on their allotments to their own clients. To say you wanted 10 percent was equivalent to requesting as much of the new issue as Black Jet might give you. Powerhouses like Fidelity, Strong, Janus, and Vanguard couldn't waste time following small positions in hundreds of stocks. When they committed to a new stock, they expected the issue manager to help them acquire a meaningful stake in the company, somewhere upward of 2 percent of the offering. All through the day, Fidelity would be phoning to buy more shares—especially as the price continued to rise.
"Yo, Brucie, give me everything you got at 18."
Tustin checked his screens for available shares. Many of Black Jet's clients had bought the stock not to buy, but to "flip"—that is, to sell after an hour or two with the expectation of making a small, risk-free profit.
"Got you five grand at 18 and another five at 18 and a teeny," said Tustin. A teeny was a sixteenth of a point. It was Tustin's job to mark up the stock each time he made a sale as a commission to Black Jet. The amount of his markup depended a lot on how good the client was. In the case of Fidelity, one of the firm's best clients, he would slap on a sixteenth at most. "And this just in: a block of twenty thousand at 18 **⅜.** You a buyer?"
"Send 'em over," said Charlie. "We're buying and we're buying big. We're starting to feel good about this baby."
Tustin put down the phone, grinning like a madman. "Five days from now, it'll be Mercury's turn. Two billion dollars. Oh yeah, we're hitting the big time!"
"Yeah, Mercury," said Gavallan, the words stale in his mouth. "Great."
Tustin stared at him oddly. "You okay, Jett? You look kind of like shit. You go out after the ball last night? It was that Nina, I bet. She looked like a goer. Wearing anything less, they'd have arrested her. You always get the sexy ones. But then, you're the boss."
If Tustin was cheeky, it was no more than his usual self. Everyone was in a grand mood since Byrnes had resurfaced; Incisex's successful launch had capped it. Instinct told Gavallan not to reveal his suspicions about Byrnes's situation. He'd explained that Graf was remaining in Moscow for the weekend and would be accompanying Konstantin Kirov to New York come Monday. The words "prisoner" or "hostage" never entered the discussion.
"I'll tell you what Graf's really doing," Tustin went on. "He's shacked up with some Russian babe. I've heard they're lookers over there. Yeah, that's it. Graf's getting himself some commie cooze. Probably got a dozen of them in bed with him."
"Can it, Bruce!" Gavallan barely reined in his outraged voice, infuriated by the insinuation of illicit sex.
But Tustin insisted on going on, his compact figure bouncing up and down in his chair like a jack-in-the-box. "I can see that old fart now. Probably got a club sandwich going, laying there between a blond and a redhead like the filling in an Oreo. Got some pussy in his face and some chick gnawing on his hog. Hoo-yeah! Go, Air Force!"
"I said shut up, Bruce. Now!" Gavallan felt his shoulder tense, his fist bunch up, and he knew that if he didn't leave this second, he'd either pick up Tustin and chuck him across the room or belt him a good one right in the jaw.
"What crawled up _your_ ass and died?" asked Tustin. "Ah, you're jealous, that's it. Maybe Nina _didn't_ take such good care of you. No, no, no. I got it. It's Cate you were after all along. I saw you two, cheek to cheek. You want some of _that_ poontang, that it? I'm a sucker for black bush myself. Drives me cra—"
A cord snapped inside Gavallan and he slugged Tustin, a lightning-fast jab to that oh-so-loud mouth. The trader tumbled into his chair, gasping, raising a hand to his bleeding lip. Thankfully, the Incisex crew had moved down a few aisles and were talking to Mr. Kwok about a listing on foreign exchanges; only the traders in the vicinity saw what happened. For a few seconds, they froze, no one speaking or moving a muscle. Just as quickly, they discounted the act and continued with their work. The expressions on their faces said Tustin had been due a spanking.
"Sorry, man," squealed Tustin, dabbing at his swollen lip. "I was just joking. Really, Jett. No offense, man."
"Damn you, Bruce," whispered Gavallan, sitting down, lowering his head next to Tustin's. "Why can't you just learn to shut up once in a while? Shit. I'm the one who's sorry. I apologize. I was out of line."
And looking into Tustin's pained eyes, he asked himself, _Is it you, Bruce? Has Kirov got his hooks in you?_
Just then, Tustin's private line rang. Gavallan grabbed the phone. "Hello... Yeah, Emerald."
"Jett, I've got a caller who says he has to speak to you right away. He says his name is Jason. He won't give me his last name, but he insists you know him and that it's urgent. Should I send the call down or do you want me to take a message?"
"Tell him I'll be right up. Pass it through to my office." Gavallan handed the phone back to Tustin, a surge of adrenaline making his feet antsy. "Make my good-byes for me. I've got to run.... I'm sorry, man."
Two minutes later, he was upstairs, standing beside his desk. Spotting the shaman, he offered the crude, powerful statue a hopeful nod before picking up the phone.
"Jason, that you?"
"Guess what," said Jason Vann. "Good news. Got a pen handy?"
"Shoot." Gavallan scribbled furiously as Jason Vann rattled off the name, address, phone number, and E-mail of the Private Eye-PO. Gavallan read the name a second time and smirked. "You sure this is the guy?"
"I'm sure that the web page dissing your company originated from his home address. Maybe he's got a kid who's doing it, but I doubt it."
"Why's that?"
"Umm, you're still going to wire me the other fifty thousand dollars, aren't you?"
"Deal's a deal, Jason. I always keep my word."
"Well," said Vann. "It just seems like something this guy might do. You see, I found out a little more about him than you asked. Sometimes I get a little too interested in my work. Occupational hazard."
"Do you now?" Gavallan doubted that Vann knew more about the Private Eye-PO than he did.
"First off, this guy's no dummy. He went to college at M.I.T., then worked for Synertel in Milpitas. He was a big shot. The CTO. But that's not the good stuff. You see, your guy has himself a criminal record. When the company flamed out, he lost it and beat the crap out of the chief executive, before trying to burn down the building. He did nine months in Soledad Medium Security Correctional Facility for Men in California. I guess that explains why he didn't tell anyone his name."
"Guess so," said Gavallan, amazed at all you could find out in the space of twenty-two hours if you knew how and where to look.
"I'm sorry I couldn't get his picture for you," said Jason Vann. "The Department of Motor Vehicles' mainframe has a decent security system. Not that I couldn't have hacked it, but you sounded rushed so I thought I'd stick with the basics."
"No need," said Gavallan. He had a pretty good recollection of what the Private Eye-PO looked like. "Got anything else up your sleeve?"
"Uh, there is one more thing. I hope you don't think me out of place, but I thought I might be able to do you a favor."
"A favor? What do you have in mind?"
"Well, I kind of found out you were in the Air Force and that things didn't go so well for you. You sound like a nice guy—I mean you paid me quicker than anybody else has before—so I just wanted to say that if you ever wanted me to upgrade your discharge, you know, to an honorable one, I can."
"You can?"
"Yeah. Free of charge. Hacking the Pentagon's a piece of cake."
"Good-bye, Jason. I'll wire the remainder of your fee this morning."
Gavallan hung up the phone and turned his attention to the name and address written on the notepaper: _Raymond J. Luca. 1133 Somera Road, Delray Beach, Florida_.
"Ray Luca of Synertel," Gavallan murmured. "Who'd have figured?"
Synertel was a high-flying manufacturer of optical switches that Black Jet had been set to take public for north of five hundred million dollars. Two weeks before the IPO was set to go, the company's primary product was trumped by a competitor, rendering it obsolete before it had even been introduced. Gavallan canceled the IPO on the spot. Three months later, Synertel went bust.
Luca's being the Private Eye-PO explained the pissy note to his warnings. It did not, however, discount the veracity of his statements. Luca might have a bone to pick, but he was telling the truth about Mercury, or at least hinting at it.
Gavallan punched a button on the speakerphone. "Emerald," he began. "Book me a—" He stopped dead, deciding it might be wiser for him to make his own travel arrangements. "Emerald," he started anew. "I've got to run out for a while. Actually, I'm feeling pretty lousy. Forward any calls to me at home. Thanks."
Replacing the receiver, he picked up his jacket and satchel, turned off the lights to his office, and shut the door behind him.
From here on out, Gavallan was on his own.
# 22
#
**W E'RE USING THE SAME GUY,"** announced Roy DiGenovese when he stuck his head into Howell Dodson's office at four-thirty in the afternoon. "Gavallan's paying the same fella we got on contract to the Bureau. Vann. Jason Vann."
Lifting his feet off the desk, Dodson slid his chair forward and afforded DiGenovese his fullest attention. "Do tell, dear boy. I smell progress."
Dodson had been reviewing the casework on Kirov and Mercury, trying to figure out what Gavallan's role in the whole thing was and whether or not it might be wise to alert his friends in the SEC or the Treasury Department about it. It was a thorny issue. The Bureau didn't need any multibillion-dollar lawsuits accusing its very own Howell Ames Dodson IV of maligning, defaming, tarnishing, or slandering a wholly legitimate enterprise. Every request he'd made to Baranov to send some of his investigators over to Mercury's Moscow operations center had been met with deafening silence. The man hadn't lifted so much as a finger. He cared only about Novastar. Mercury was the Americans' problem.
Dodson had the tape from Mercury Broadband USA, the allegations of a paid informant, and that was it. The skeptic inside him refused to follow in DiGenovese's rabid footsteps. When it came to fashioning a winning indictment, they were no better off than they were four weeks ago. Effectively, the decision had been made for him. He didn't dare open his mouth to another federal agency about his concerns over Mercury Broadband. For now, they would remain an in-house matter.
"Vann found the Private Eye-PO," DiGenovese continued, taking a seat opposite Dodson. "His name is Raymond Luca. He's a resident of Delray Beach. M.I.T. grad, and get this... an ex-con."
"And what does Mr. Luca do, pray tell, when he's not playing the Private Eye-PO?"
"No idea. Just got a name and an address. Vann said he could find out more, but he's already run over his hourly commitment and it would run us another few thousand dollars."
"Very well," said Dodson. "Run Mr. Luca's social security number through the IRS, do a thorough credit check on the man, contact M.I.T.'s alumni relations board. Someone can tell us how he earns his daily bread." He shifted in his seat, unsatisfied. "What else did Mr. Vann have to tell us?"
"Nada. Just gave me the same info he gave Gavallan."
"And how much did Mr. Gavallan pay our Mr. Vann?"
"Didn't ask."
"Next time ask," ordered Dodson, wondering if Vann might be holding something back. "And find out where Vann likes his funds wired. I don't take to people double-timing the Bureau—goes against my sense of patriotism. While you're talking to our colleagues at the IRS, why don't you have them take a peek at Mr. Vann's latest 1040s. Might be nice to have some leverage in the future."
DiGenovese had been writing all this down on a notepad he carried in his left hand. Finished, he looked up. "Next flight down to Miami's at seven-fifteen. I booked us two seats."
"Pardon me?"
"You heard Gavallan," DiGenovese said, in a tone as surprised as his superior's. "He wants to permanently shut Luca's mouth."
"And do we have any evidence that Mr. Gavallan's going anywhere near Florida these next few days?"
"Well, no. I mean, not yet. We don't get transcripts of the wiretaps until twenty-four hours after they're picked up. I thought it would be a good idea to have a talk with Luca, let him know that he might be in some danger."
Dodson shot DiGenovese a stern glance as if to say he'd been silly even to think of flying to Florida that evening. In fact, his reluctance to leave so quickly was rooted in his domestic situation. His wife, Clara, was a woman of the times, and would raise holy hell if he popped down to Florida without advance warning. She didn't stand for unannounced departures, late nights at the office, or working more than a half day on weekends unless absolutely necessary—and "necessary" meant that an agent's blood had been spilled.
"Calm down, Roy. If you're so worried about Mr. Luca, give him a call on the telephone. Tell him to lock his front door. I would, on the other hand, enjoy speaking with Mr. Luca about where in God's name he's been getting our confidential information. Book us first thing in the morning."
"You don't think he needs protection?"
"No, Roy, I do not. Now off you go. Book us those seats for tomorrow."
DiGenovese shifted in his chair, and Dodson could see he was using all that Ranger discipline of his to keep from arguing. The Army's fine training won out over DiGenovese's impetuous Sicilian blood, and after a few seconds he complied. "Yes sir. I'll get back to you about the times."
"Good man," said Dodson, beaming. "What's that you always say when things are going well?"
"Drive on, Airborne."
"Yes, yes. Well then, 'Drive on, Airborne.'"
# 23
#
**C ATE MAGNUS HELD THE NOKIA CELL PHONE** close to her ear, clicking the volume higher so she could hear the man's voice over the earsplitting whine of a jacksaw.
"It's just not what we want this week," Jimmy Murphy was saying. "Metrics are so dry. Your readers don't give a fiddler's fart whether Yahoo! gets two million hits a day or two billion. And they care even less what exactly constitutes 'a hit' on a website. This isn't a scientific review here. You're supposed to liven up the rag, not dull it down."
"It's not the methodology I'm interested in, Jimmy," she retorted, pacing the length of her bedroom. "It's the way you can cheat on these things. Use one method and it looks like five hundred users a day are logging onto your site; use another and it's more like five thousand. The whole thing stinks. I mean, who are you supposed to trust?"
"Good question, Cate. Tell you what: Let's leave that question until next month. Give me something lively, something dishy."
Cate lowered the phone from her ear and mouthed a very nasty word in Mr. Jimmy Murphy's general direction. Murphy was the features editor at the _Financial_ _Journal_ , a rail-thin, choleric Kansan who took it as part of his job description to be permanently dissatisfied with his writers' offerings. More and more, he was pushing the column away from the serious fare she favored—namely, an examination of the personal and societal ramifications wrought by a once-in-a-century upheaval in technology—toward dishy, prurient pieces on the lifestyles of the sick and famous. It was partly her mistake. A year ago, she'd written a piece on young women who worked for a certain gentlemen's club in San Mateo that catered to the wild and wildly expensive whims of the valley's glitterati, such as they were. One of the girls she'd interviewed had talked about the habits of one of her regulars, a nationally known Internet exec who liked to do weird things with whipped cream, motherboards, and electrodes on his nipples.
Or there was the time Murphy had sent her to Bangalore, India, to check out the booming matchmaking market for up-and-coming high-tech wizards. It was the Indian women who paid for introductions to men, and the depth of questioning they had to endure approached the ridiculous. "How would you propose to cure your husband's impotence?" "What family remedies can you offer for baldness?" "Would you object to your husband's taking a mistress? Two mistresses?" and her favorite, "What is the proper serving temperature of chicken tika-tika? In Celsius _and_ Fahrenheit, please."
It wasn't lost on her that 90 percent of the _Journal's_ readers were men.
This week's "Gold Rush" dealt with a more serious topic: the internecine warfare going on among competing firms in the field of metrics. "Metrics," as related to the Internet, involved defining precise methodologies to measure usage of the World Wide Web, or more important these days, providing objective information as to exactly how many visitors clicked onto specific websites.
Now that the bloom was off the rose and the new economy was looking a little long in the tooth, metrics had assumed a new importance. Acquisition had replaced IPOs as the prevalent exit strategy for start-ups, and the price a company could demand was directly correlated to the number of hits its website received. Each company in the metrics game claimed to offer the sole, incontrovertible means of measuring a site's popularity. The only hard part for the client was finding the boys who'd put you at the top of the list, and Cate was sure that a little extra vig would better your final score.
"Look, Jimmy," she started again, wincing at the syrupy sound of her voice. "Maybe the piece is a little heavy on the number crunching. Let me talk to rewrite; I'll soften it up, give it a little more color."
Cate was frustrated. She'd finally come up with a story that allowed her to put into practice some of the financial carpentry she'd picked up at Wharton, and no one gave a damn.
"You're not listening to me," carped Murphy. "Where are those personal items we so loved? Remember last year when you followed a Range Rover into and out of a shop six times in three months? We had letters for a year wondering what happened to that lemon—some nut even wanted to buy it. Hey, hey, here's an idea! Hot off the wire. Why not give me something about the house. How does a savvy reporter knee-deep in tech hoopla deal with the down-and-dirty world of home renovation? Give me a thousand words on pouring a new slab. How do they do that, anyway, without having to tear down the house?"
"Noisily," Cate answered, putting a finger to her ear to drown out an eager jackhammer. "Very noisily. Listen, Jimmy, I want you to run my piece as is. Give me metrics this week and I'll give you whatever you want next Friday. Come on, Murph. A favor."
"A favor?" Jimmy Murphy's voice cracked, and she could picture him at his desk, hurriedly figuring the angles. No doubt he was wearing one of his bright red dress shirts with a collar two sizes too big for his scrawny neck. "Deal," he said, finally. "I'll get back to you on a subject. Maybe we can find out what Jim Clark's doing these days. Whatever happened to that boat of his? Maybe you could track it down, go for a sail."
Cate sighed. That was someone else's story. A real writer. Someone who possessed the wherewithal to write a book. "Sure thing, Jimmy. See you."
Collapsing onto her unmade bed, Cate put down the phone while shaking her head. Thank goodness, she'd convinced him to run the column. Time was precious. Even the smallest skirmishes counted as battles. She was mustering her troops, marshaling her evidence for the final assault. Rolling over onto her stomach, she pulled the top sheet off her bed, then the fitted cover. Slipping a hand down the side of the mattress, she found a horizontal indentation, and dug her hand into it until her fingers touched a sheaf of papers. Still there, she confirmed, awarding herself a contented smile. Not the most imaginative of hiding places, but for a girl who'd passed up spy school, not bad.
After replacing the sheets, she made the bed. The room looked better now, friendlier. Her armoire wasn't drunk, just a little tipsy. The desk Jett Gavallan had built for her beamed with memories of their time together. The furniture was a little too "shabby-chic" for her taste, but it would have to do. The furniture, the bedroom, the house, all of it was cover. A mask she'd put on eight years ago.
Her eyes drifted back to the desk, and she thought of Jett. Jett, her erstwhile love. Jett, her weathered Boy Scout. Jett, her pigheaded ex who refused to blink his eyes at the lights of an oncoming train.
Until seeing him last night, she'd thought her loyalties decided upon, her duties sworn. But five minutes in his presence had weakened her resolve. She wondered how much more she could tell him about Mercury before he'd finally accept her words as the truth. How much before she revealed too much about herself.
Rising, Cate turned on the radio and headed to her closet. The raucous jangle of The Clash's "Rock the Casbah" hit her ears, and immediately she felt better. She loved Western music. The hard guitars, the irreverent edge, the joyous mocking of authority.
_Sharif don't like it_
_Rock the Casbah, rock the Casbah!_
She was still shaken from her early-morning visit. That she hadn't been harmed was small consolation, runner-up only to the fact that the men hadn't found what they'd been looking for. Their haphazard rummaging of the house made it clear that no one had any proof she was behind the attacks. They had come to frighten her. They had come to let her know she was being watched and that she could be controlled. They had come to signal that her life as she knew and loved it could come to an end anytime they wanted it to.
They had come to tug at the mask.
Sliding back the door, she chose a pair of faded jeans, a bold blue and white striped dress shirt, and a cowboy's leather belt Jett had given her on a trip to his ranch in Montana. Cate chose her clothing carefully, rarely buying trendy items or accessories that might be out of style the next season. She knew how to read a stitch and checked a garment's cut and the quality of its material before making a purchase. She'd worn enough cheap clothes to know the difference between good and bad. Her only extravagance was a pair of Todd's driving shoes, fire engine red and buffed to a gloss.
Moving to the mirror, she applied her makeup in quick, deft movements. Two strokes apiece for the eyelashes, nothing for the brows—they were too dark as it was, too arched for her liking. A hint of eyeliner. Nothing for the lips. The lips would do on their own, she thought, pressing them together. The lips were her best feature, wide and sensual, full without being grotesque. Yes, she'd keep the lips.
Finished, she took a step back, checking for any sign of the fear she felt bubbling inside of her. Her eyes were clear and registered their usual nonchalance. Her smile was in place, and she was glad to see it still conveyed the promise of mischief, a hint of merriment. She found her face too serious as a whole. The high cheekbones, the narrow nose, the widely spaced eyes—all conspired to lend her a haughty, insolent regard that she felt was the opposite of her true personality.
No, she concluded, giving herself a final looking-over, there was scant sign of fear. And she was cheered by her mastery of her emotions.
Strolling from the bathroom, Cate stopped at the dresser and picked up her handbag. She spent a moment checking the contents—recorder, notepad, digital camera, phone, pager, wallet, hairbrush, Tic Tacs. All present and accounted for.
Just then, her pager buzzed. She picked it up and checked the digital readout. "Urgent information about our mutual friend. Let me know when to send." Excitedly, Cate set down her purse and keyed in a response, then dashed downstairs and stood by the fax machine. A minute later, the phone rang and the fax began to stutter.
The writing on the paper was Cyrillic, the stationery that of the "Prosecutor General of the Russian Republic," but the message was written in English. Dated May 31, the transmission was a copy of a memorandum from Yuri Baranov to "Deputy Assistant Director Howell Dodson of the FBI, Chairman, Joint Russo-American Task Force on Organized Crime."
Cate held her breath, reading the body of the text.
"Pursuant our inquiry re: subject Kirov, Konstantin R., evidence forwarded my offices regarding Novastar Airlines graded sufficient to obtain warrant. Issuing date 7 June. Details of operation to follow. Suggested timetable: Week 23."
_Operation?_ She wondered what they had in mind. Week twenty-three had begun Monday of this week. Damn it, she cursed, why was she always behind the curve?
Cate reread the fax. While there was nothing on the page mentioning Mercury by name, it was a damning document nonetheless. Investors would shy away from an offering for a foreign corporation whose chairman was being investigated on charges of corruption and money laundering by his own government.
Moving to her PC, Cate scanned the document into her hard drive. For all her effort, she was still unsure of the good it would bring. She was sowing doubt, when she needed to be bringing evidence. The article on metrics would help, even if it didn't mention Mercury. More certain was the pain her efforts would cause Jett. He'd lose the deal and his bridge loan to Kirov. He might even have to part with his company. Wouldn't it simply be easier to call Jett up and have a heart-to-heart?
_About what?_ the steely voice inside her demanded. _He's been warned. There's nothing more you can do_.
Cate ignored the voice. One look at Jett Gavallan last night had brought back all her strenuously suppressed feelings. Lowering her eyes, she remembered the touch of his fingers, the defiant glance when she told him to drop the deal, the tide of blood in his eyes. She told herself it wasn't fair for any woman to demand so much of herself.
The hard voice laughed. _Fair? What's fair?_ She only had to call to mind her own past—her struggles, her denials, her battle to rebuild a career from scratch, to carve a new identity for herself—to know that "fair" was not a promise life often kept. But there was more to it than that. There were some things she could never say, no matter how much her heart demanded.
Cate regarded the fax, and her sentiment fled. "Too bad," she whispered, hardening herself to the task. Jett was a big boy. He'd been warned. From here on out he would have to take care of himself. She'd done enough already, even if he didn't know it.
Straightening her back, she accessed her E-mail program and uploaded the fax. After addressing it to her friend in Florida, she hit the send key, confident that he would know how to make proper use of it.
# 24
#
**T HE RAIDING PARTY ASSEMBLED** quietly and with precision. In all they were twenty-two men, divided among three vans and two BMWs from the prosecutor general's office. Crack troops from OMON—the special militia created by Mikhail Gorbachev and now attached to the Ministry of the Interior—the men were dressed in black utilities with matching bullet-proof vests and Kevlar helmets. Nazis for the new millennium. Flash grenades were pinned to their waists and machine pistols dangled from their hands.
The assembly point was Mayakovskya Square, a kilometer from Mercury Broadband's offices. Yuri Baranov moved among the militiamen, offering grunts of encouragement, pats on the back, the occasional grim smile.
"On no account are you to fire a shot," he repeated time and again, until his gruff, tobacco-wearied voice grew sore. "We are all sons of the Rodina, the motherland, even if some of us have lost our way."
He felt old and stiff and spent among such young men. He knew their simmering blood lust, their jacked-up bravado, and it left him uneasy and sad. He'd seen enough suffering in his lifetime to know what those emotions inevitably wrought.
"Move quickly. We must rush the entrance and force the door. We've come to gather evidence—nothing more. Treat the civilians with respect."
On Baranov's signal, the convoy moved out, advancing in tight formation through the serpentine alleys that combed the Moscow cityscape like fissures in a crumbling wall. The prosecutor general rode in the front seat of the lead BMW. His posture was forced, his back barely touching the leather bucket seats. Opulence, even in an automobile, made him uncomfortable. Checking his watch, he leaned forward further, so that his hands clutched the dashboard. The informant had alerted them that Kirov made his banking transfers each day between eleven and twelve o'clock—nine and ten in Switzerland, where the banks had just opened. It was Baranov's goal that warm afternoon to obtain hard-copy proof of Kirov's theft from Novastar Airlines.
One hundred meters from their destination, Baranov turned on the siren. A few seconds later, the sedan screeched to a halt. He jumped out. "Police," he shouted, storming the building's front stairs. "I possess a warrant to search the premises. You are to provide every cooperation."
Jump boots slapped the ground as the troops rushed to his side. Baranov had pulled open the door and taken a step into the building when three hulking men picked him up and carried him back into the street. At once, the thugs were overwhelmed by the onrush of OMON troops and thrown spread-eagle onto the pavement.
Squirming free, Baranov saw a blue metal curtain falling in front of the door. "Quick!" he yelled. "Someone. Inside."
Several of his men struggled to hold down Kirov's security guards, searching them for weapons and giving them a few sharp kicks. The rest were blocked by the confusion at the door. No one could enter the building.
Without making a conscious decision to do so, Baranov charged up the stairs a second time. A single thought galvanized him. He had come for Kirov's banking records, and God help him, he would get them. The barricade was three feet from the ground and falling quickly. Crouching to one knee, then to his hands, he threw himself beneath the metal curtain and tried to crawl inside. The steel curtain struck his back, driving him to the ground.
"Ah," he cried out, feeling old and brittle, hating himself for his weakness. He was half inside, half outside the building, his cheek pressed to a white marble floor. "You will raise the barricade and open the door at once," he called to a team of black-suited bodyguards running at him from across the reception area. "I have a warrant to search the premises."
They were on him in a flash, hands grasping his shoulders, his head, shoving and pushing him back under the curtain. "Out, old man. You have no business here."
"In!" yelled Baranov over his shoulder. "Push me in!"
From beyond the steel curtain, friendly but no less forceful hands took hold of his legs and waist and muscled him forward. He moved an inch one way, then two inches the other. Ferociously angered by such disrespect—for his age, his circumstance, and his office—Baranov gave a mighty grunt and pulled himself forward. The barricade crashed down behind him. He was inside.
"Bring me Kirov," he shouted, climbing to his feet and setting off across the wide reception area. "Tell him he has a visitor!"
**S EATED IN HIS CUSTOMARY CHAIR** in the far corner of Konstantin Kirov's second-floor office, wearing his favorite houndstooth jacket, was a wiry, olive-skinned man with close-cropped black hair, a long, crooked nose, and a black mustache thick enough to sweep the floor. But one did not dwell long on the man's features or his dress. What captured one's attention were the eyes. They were dark and deep-set, twin orbs of unblinking obsidian framed by unusually long, luxuriant lashes. They were the eyes of a zealot.
To meet Aslan Dashamirov's stare was to look into the abyss, to see death and life and know that they were separated only by the razor's edge of his will.
"I understand we have a problem," Dashamirov was saying. "Someone in our organization talking more than he should, being a bit too free with his opinions, taking papers from the workplace that are better left at his desk."
Kirov did not know how Dashamirov had discovered the details of his sit-down with Yuri Baranov the day before, but he knew better than to be surprised. "Yes," he replied. "Some confidential papers have found their way into the prosecutor general's hands. Nothing to worry about in and of itself. What concerns me is how the papers slipped out of the office."
"Any idea who the culprit is?"
"We've narrowed it down to someone in legal or administration. Unfortunately, our staff has doubled in the past year. Don't worry—we'll put our finger on him."
"And is it the same one who has leaked the information regarding Mercury?"
"I certainly hope so."
"And the American?"
"At the dacha. You may have him when he's no longer needed."
Dashamirov bowed his eyes, which was as close as he ever came to saying thank you.
Chechen by birth, a Muscovite by upbringing, Aslan Dashamirov was fifty-two years old, the same age as Konstantin Kirov, and the two had been in business since Kirov had first moved to Moscow—or "the Center," as it was called—from Petersburg. Dashamirov had no pretensions of civility. He was a criminal born and bred, a _Vory v zakone_ —a thief of thieves—a man sworn to conduct his life outside the pale of law and order. Still, he carried a title in the contemporary Russian business world, a position that was acknowledged by none, yet respected by all. Aslan Dashamirov was a _krysha_ —or "roof "—and every businessman engaged in the pursuit of profit somewhere in the Republic kept a man like him on his payroll, whether by choice or not.
A _krysha_ performed a variety of functions. He obtained permits, persuaded politicians, sweet-talked creditors, and harried debtors. He offered protection against racketeers, bargained with corrupt law enforcement officials, secured banking privileges at friendly financial institutions, and helped negotiate the treacherous corridors of the judicial system. His methods were crude but effective, and ranged from bribery and extortion to torture, kidnapping, and murder.
The fee for his services was 15 percent off the top of all Konstantin Kirov's businesses.
"So you're confident the deal will be a success?" he asked.
"Absolutely," declared Kirov. "Absolutely."
"I believed you the first time," said Dashamirov. "Not the second. What is Baranov after?"
"Novastar," volunteered Kirov. "He believes a hundred twenty million is missing from the company's accounts. I told him he was crazy."
"Dollars or rubles?"
"Dollars."
Technically Novastar counted as one of Kirov's private investments. As a long-running enterprise until recently 100 percent controlled by the state, it had never required any of Dashamirov's subtle legerdemain. No scrupulous customs men to brain with a lead pipe. No stubborn inspectors to "bribe" with a blackjack and brass knuckles. No defiant board members to convince with the help of a slender glass mixing rod and a hammer.
"I'm certain Baranov is mistaken about the missing money," Dashamirov said at length. "I know you would never skim a little cream from Novastar without sharing your rewards. We are brothers, nah? Such behavior among kin is unthinkable." He scratched at his mustache, crumpling his brow as if pained. "Still, we cannot allow problems with one business to interfere with another, certainly not at such a delicate moment in our company's history. That is why you hired me. To look after your interests, nah?"
"Why else?" agreed Kirov.
"First we will find our rat," announced Dashamirov. "Then we shall ask him where he got the idea that someone is siphoning a little money from Novastar, and why he wishes to share such silly notions with the government."
At that instant, a siren wailed, the keening so close, so loud, so unexpected, as to make Kirov bunch his shoulders and duck involuntarily. Another siren joined in. Tires screeched. Doors slammed. An entire Army corps was assembling on the pavement beneath his window.
"A raid," Kirov said calmly, remembering Yuri Baranov's veiled threat. And to himself, _He will pay. This will not go unpunished_.
Dashamirov remained immobile as Kirov moved in three directions at once. One hand depressed the internal alarm while the other found the phone. Dialing a number, he strode to the window and looked outside. Two sedans and three vans were parked by the entry. Soldiers were charging up the stairs.
"There's a corridor beneath the building that will take you to the Arbat."
Without a word, Aslan Dashamirov scurried out of the office.
Placing the phone to his ear, Kirov waited for an answer. The number he had dialed connected him to a modern office complex hidden in the forest just north of Moscow, a suburb known as Yasenevo. The sleek gray buildings housed the offices of the FIS, or Foreign Intelligence Service, one of the successors to the KGB, or Committee for State Security. An officious voice answered. _"Da?"_
"Leonid, listen and do not say a word. Yuri Baranov and his men are outside my offices. He's come with his OMON brutes and they're making a show of gaining entry. Send over some of your people immediately, a dozen young men with a little fire in their blood."
Ten years his elder, Major General Leonid Kirov was the ranking officer of FAPSI, the Federal Agency for Government Communication and Information, an offshoot of the former KGB's Eighth Chief Directorate.
"Calm yourself, Konstantin Romanovich. Tell me again what is happening?"
Kirov bit back an epithet, detesting his brother's propensity to give orders and his own to follow them. "It's a business matter," he explained. "The prosecutor general has exhibited more independence than I gave him credit for. All we need is for him to bring in a tank and try to blast his way in. That would make the evening news, don't you think? Where would that leave us?"
The mention of television and its promise of mass and biased dissemination of information sparked in Leonid Kirov a combustible fury. "I imagine that would leave us in the shithouse. Back to Lefortovo for you, retirement on a government pension for me. I don't know which is worse. OMON troops, you say? How many?"
"Twenty, twenty-five. All dolled up in riot gear. If you'd be so kind, Leonid, I would appreciate your doing as I asked. Need I remind you we are five days from immortality? Once the offering is completed, they'll be modeling a bust of you to put in Red Square. Right next to your old boss Andropov and Iron Feliks himself."
Kirov pictured Leonid seated in his brightly lit office, desk immaculate, books and papers aligned at right angles to each other, the large color portrait of the new president hanging in pride of place opposite the door. Leonid would be wearing the navy suit he ironed himself each night, his white dress shirt spotless, silver necktie held in place by the tie clasp Chairman Andropov had awarded him on his twenty-fifth anniversary in the service. His white hair would be brushed and parted just so, his proud chin kept at permanent attention. A single cigarette would be burning in the ashtray, a filthy Belamor Kanal, the brand Stalin had enjoyed, and every minute or two he would allow himself a long, generous puff, then replace it fastidiously.
"Older brother, a response would be welcome."
"Hold the fort," ordered Leonid. "I'll send some men over right away. Whatever you do, keep the press away. It might get messy."
Kirov hung up the phone, only to hear it ring again almost immediately. "Yes."
"Baranov is in the building." It was Boris, and he sounded shaken. "I am sorry, sir. He managed to crawl in under the barricade. What shall I do? He is demanding we raise the barricade and let his deputies enter."
Baranov. Of course he _crawled_ in. The man was a worm. "Do as he asks. Open the door. Give me two minutes, then escort him upstairs."
Flinging down the phone, Kirov fled his office. A minute later he reached the data center. "How long until the files are erased?"
An unshaven tech in a red Adidas T-shirt barked his reply. "Ten minutes, sir."
_Ten minutes. An eternity_. He imagined the documents Baranov would find if he got into the data center before then. The government would see everything. "And we downloaded a backup last night?"
"Yes sir. At 1900 to our data recovery center in Geneva."
"Very good. Go back to your work. Pay the siren no heed."
Continuing down the hall to finance and administration, he found a dozen secretaries and accountants at their desks, diligently stuffing page after page of bank statements, revenue records, and payroll stubs into their shredders with a military efficiency. On the wall a red strobe light flashed in two-second bursts.
"Hurry up," he said. "There, there, you're almost done." Watching them, pride warred with disbelief that one of them might be Baranov's spy.
"Kirov! Where are you?" echoed a familiar voice outside in the hallway. "I have a warrant. I demand you open the doors at once."
"Calm down, Yuri Ivanovich. We have nothing to hide." Closing the door behind him, Konstantin Kirov came face-to-face with the prosecutor general. Behind him stood two of his deputies, breathing hard, pink-cheeked, and Boris. Discreetly, Kirov glanced at his watch. Eight minutes remained until the files were erased. He noticed his jacket jitter ever so slightly with the beating of his heart. "You don't mind if I have a look at the warrant."
"Afterward," said Baranov heatedly. "Move aside. I wish to enter this room."
"No need really. It's only a—"
Brusquely, Baranov and his deputies pushed past Kirov and entered the accounting office. Seeing the men and women shredding documents, Baranov shouted, "Stop. You know who I am. Stop at once. Anyone who does not obey will be placed under arrest."
Several clerks stopped shredding, but most continued. Baranov's cheeks flamed red. "Anyone who does not stop immediately will spend the night in the Lubyanka. With your families. Your children, too."
The shredding ceased at once. Baranov passed from desk to desk, picking up random papers, studying them. He dashed off instructions to one of his deputies, who immediately began gathering all the papers together.
Baranov had found a receipt that interested him. "And what business do you have with the Banque Privé de Genève et Lausanne?" he asked, holding the paper in his hand with a victorious smile.
"A private matter. Nothing to concern so august an office as your own."
"We shall see."
Baranov spent another minute or two examining the shredders, digging his hands into the basket and coming up with wads of slivered paper. "We will take this, too. I know some people who can reconstruct these documents."
"All yours," said Kirov munificently. He was beginning to sweat. He could only pray that the most secret of his documents had already been shredded. Reconstructing them would take a year's time. _A year!_ Anything could happen by then.
"Now, I wish to go to your IT center," said Baranov.
"Do you mind if I ask what it is exactly you want?"
"You know damned well what I want. Now let's go. I believe it's on this floor, just down the corridor."
"If you know your way around so well, I'll allow you to find it yourself." Kirov had no intention of helping Baranov do his job. He had opened the barricade when requested. He had greeted the man cordially. No charges could be brought for obstructing justice. The rest the prosecutor could do on his own. Fuck him!
Baranov left one of his deputies behind in the accounting office and hurried into the long, airy corridor. Kirov followed. A few offices were open, windows raised to let in the warm afternoon breeze. From outside came the sound of car doors slamming, voices shouting, and footsteps entering the building.
_Finally!_
Kirov hastened to a window. A delegation of ten young spies from the FIS had confronted the OMON troops outside. Their leader was a handsome blond man in business attire. His deputies were similarly dressed, but were less handsome and had exchanged neckties in favor of Kalashnikov assault rifles. Shoving broke out between the two groups. One FIS man fell to the ground, pistol-whipped. Then it was the OMON's turn, losing a storm trooper to more conventional means: a well-aimed kick to the balls. Voices rose, then fell.
"Good boy, Leonid," said Kirov softly.
"What is it?" demanded Baranov, bustling alongside.
"See for yourself."
Baranov looked down at the sparking confrontation. "Leave them," he called to his men. "There is to be no fighting. We are all comrades. Let them be." He stormed out of the office, looking this way and that before getting his bearings. He arrived at the entry to the data center as the delegation from Yasenevo poured out from the elevator nearby. Trying the handle, he found it locked. "Konstantin Romanovich, I demand you open the door."
Kirov checked his watch. Fifteen seconds until the files were deleted. He took a breath, rummaging in his pockets for a key. "Ah, here it is." He managed another delay fitting the key into the lock. "There."
Kirov opened the door.
The tech in the red Adidas shirt sat at his desk, studying a manual. "Ah, Mr. Kirov. I have bad news," he said, springing to his feet, his clever eyes taking in Baranov and his deputies. "Terrible, really."
"What?"
"A bug has hit our computers. I'm afraid we have lost all our data."
Baranov stared first at Kirov, then at the technician, and then at Kirov again. Without a word, he turned and left the room.
**K IROV FOUND JANUSZ ROSEN** waiting for him in his office. "Yes, Janusz, what is it?"
"Good news, sir. Great news, even. I found him."
After standing by impotently as Yuri Baranov had carted off two dozen boxes full of Mercury Broadband's financial records, Kirov needed some good news. "Who?"
" 'Who?' " Rosen registered a look of gross disappointment, his glasses falling to the tip of his nose. "Why... _him."_
"Him," of course, was the Private Eye-PO. "About time. What is his name? Where does he live?"
"His name is Raymond J. Luca. An American, naturally. A resident of Delray Beach, Florida. I found him trawling the web early this morning. Another investor invited him into a private chat room and I was able to sneak in."
"Don't look so proud of yourself," said Kirov. "That's what I pay you for, remember?"
**M INUTES LATER,** Kirov stood alone in his office, phone to his ear. He had banished Rosen with a handshake and the promise of more shares in the Mercury IPO. He had told his secretary to hold all calls. The room was silent, a quiet compounded by the absence of sirens and army boots.
"Damn it, girl, answer."
Five rings. Six.
_"Da?_ Allo."
"Tatiana, you don't know how happy I am to hear your voice. I hope you haven't any pressing plans for the evening."
"Konstantin? Is this you? I am tired. I have had a long day. What is it, please?"
Rude, wasn't she? Sometimes he found it hard to believe she was a convent girl. Then again, he hadn't hired her for her good manners.
"Tatiana, I have a trip in mind for you. A junket abroad, actually. Tell me, my little bird, how do you feel about Florida?"
# 25
#
**R AY LUCA WAS IN THE ZONE.**
Perched on the edge of his secondhand office chair inside his four-by-four-foot cubicle on the floor of Cornerstone Trading in downtown Delray Beach, Luca was a model of concentration. All of him—his eyes, his ears, his mind, his square, compact hands with the nicely buffed fingernails, even the downy black hairs on the back of his neck—was dialed into the cascade of information spewing from the twin columns stacked on the desk in front of him.
Ten inches from his all-seeing brown eyes, the wall of color super-VGA displays broadcast a blinking, stuttering, ever-changing array of graphs, bar charts, and streaming price quotations advertising real-time fluctuations of the twenty-seven stocks he was currently following. The setup was called a Level II quotation system, and it allowed him not only to see markets being made in each of these stocks but to directly place a buy or sell an order via an electronic communications network, or ECN. One hour after he'd glued his bottom to the chair, he was finally where he needed to be: deep in "the zone," the Zen-like fusion of focus, mental agility, and intuition necessary to master the godless art of day trading.
It was in this church of unbiased information that Raymond J. Luca, five-foot-five-inch native of Worcester, Massachusetts, and Florida transplant, one-hundred-forty-pound washout from the United States Marine Corps and the Catholic faith, chronic sufferer of duodenal ulcers and incurable myopic, divorced father of three wonderful daughters and Ph.D. from M.I.T., ex–altar boy, ex-tycoon, ex-con, and soon to be ex–day trader, also known as the Private Eye-PO, took his daily communion, a high mass beginning at 9:30 **A.M.** Eastern Daylight Time and ending at 4 in the afternoon, every day of the year save weekends, holidays, and the running of the Flamingo Stakes at Hialeah.
Luca had five open positions at the moment, all buys: Nokia, Solectron, Merck, Juniper, and Amgen. He didn't care what they marketed, manufactured, or sold, who ran them or whether they had a chance in hell of making a decent return over the long run. It didn't matter where they traded—Nasdaq, Amex, or the Big Board—only that they were high-volume stocks that bounced around like a kid on a pogo stick. Volatility was the name of the game.
At the moment he was concentrating on Solectron (symbol SLR), a box maker that after years of double-digit growth and the accompanying rise in share price had suffered a violent tumble to earth. He'd bought eight thousand shares of the stock a few minutes earlier, just after it had made a "double bottom," meaning that twice in the last thirty minutes it had tested its lows and rebounded. Classically, a stock exhibiting this behavior goes on to break through its earlier intraday high. Watching the market makers enter their orders, he noted a couple of things: One, buyers were pouring into the market (also reacting to the double bottom). And two, sellers were few and far between. The stock was set to pop.
Sellers inched into the market, eager to accept the quickly appreciating bids. Luca held on as the stock advanced an eighth, a quarter, a half. An eye flicked to the volume chart and a sixth sense told him the stock was running out of steam. Spotting a bid for eighty round lots, or eight thousand shares, that would lock in his half-point profit, he dashed off an order to sell. Bingo! Four grand in the plus column. In and out in twenty minutes.
"Trade, don't invest." The diligent day trader's motto.
Luca turned his attention to his position in Merck as a gaggle of male voices burst out shouting down the aisle. One raucous laugh stood apart from the others. It was Mazursky—or "the Wizard of Warsaw," as he called himself—and he was crowing about taking down three points on a position inside an hour.
"Thirty grand, baby. Thirty fuggin' large! Oh, yeah! The beers are on me tonight, fellas. And whoever wants to buy me the first shot of Jagermeister will be the recipient of my daily tip. Ooh-yeah!"
Luca shuddered at the Pole's shameless bragging. He didn't need to look to know that Mazursky was doing his victory dance, the revolting little number where he clasped his hands behind his head and rotated his hips and potbelly in ever-widening circles.
Luca felt himself being dragged from the zone, his cerebral connection to the ether evaporating. Annoyed, he leaned even closer to his precious screens, clenching his jaw and grinding his molars in a desperate attempt to lock out the distraction. But it was too late. His connection was severed. He was free-falling back to earth and his place among mortals. Ducking a head outside the cubicle, he saw the regulars crowded around Mazursky's hangout—Krumins, Nevins, Gregorio—all giggling like teenagers.
"Hey, Ray, that goes for you, too," said Mazursky, spotting him and waving him over. "First beer's on me."
Surprised, Luca smiled. It wasn't like Mazursky to count him in. Ray Luca wasn't one of the guys. He didn't share tips on what stocks were about to pop. He didn't discuss his trades or offer advice on how others could make as much money as he did. Part of the reason was that he was naturally a timid person who never did well in groups. People often mistook his shyness for aloofness. Another part was that, well, they were right: He did operate on a different level than these ham-and-eggers did. He was a theorist, an inventor, an evangelist. He was the father of the Synertel fiber-optic switch, a cutting-edge technology that almost _—almost—_ revolutionized the web. If he shared a work space with them it was only a temporary measure, a fluke in the cosmic plane.
Standing, he tucked an errant shirttail into his trousers and ventured a wave. As long as he was out of the zone, why not try to socialize a bit? Truth was it got lonely being a theorist and an inventor. "Hey, Maz," he said. "Beer sounds good. Where you guys heading?"
"What? A word from his highness?" cackled Mazursky. "We serfs are touched."
"Come on, Maz," said Luca. "You guys going to El Torito or what?" Luca felt all eyes on him. Don't look away, he told himself as he jammed both hands into his pockets. Keep your chin up. But already he was fighting for the gray, neutral comfort of the carpet, his chin bobbing up and down, the blinking going haywire. "Umm, what time?"
"I'll be happy to tell you," said Mazursky, "just as soon as you put your stuck-up wop nose up my hairy ass and tell me what I had for dinner last night."
The teenagers burst out laughing and the victory dance began. Round went the hips. Jiggle went the belly. _Ooh-yeah_.
Luca dropped like a stone into his chair, his cheeks afire with humiliation. Instinctually, his eyes began trawling the bank of computer screens, checking stock prices, volume charts, news alerts—anything to lessen the pain of rejection, his shame at wanting to fit in, his anger at himself for not knowing better.
Mazursky, you jerk, he cursed silently. Just you wait. Another month and everything will be different. You'll be begging to buy me a drink, to spend even a minute in the presence of the owner and editor of _The Private Eye-PO_ , the nation's hottest investment newsletter.
And with that he went back to work.
For the past three years, Ray Luca's life had been divided into two halves. Nine to five, he was another "hard-timer" trying to put together a decent grubstake trading the market. It wasn't easy. With alimony claiming six grand a month off his paycheck and child support another three on top of that, he had to make a killing just to keep his head above water. In a good month, he cleared thirty grand. Nine went to his ex-wife, seven to the IRS, and five to settle his penalty to the federal government's Department of Corrections. Living expenses ate up another two grand. Small wonder he was never able to put together a decent capital base.
But every evening he devoted himself to a systematic and thorough dissection of the market for initial public offerings. He educated himself about particular businesses going public. He researched their viability and analyzed their business plans. He compared each upcoming offering against past issues in similar market segments. If the market for IPOs had cooled down, it was to his benefit. Ray Luca was a dyed-in-the-wool contrarian, and he didn't frequent pastures where the grass had been chewed to the roots. Working alone, he was unable to analyze more than two offerings a week. The current market conditions suited him fine. As long as three or four solid new issues hit the street each month, he was on track. His goal was to build a reputation as the nation's foremost prognosticator of IPOs, and on this sunny summer day he could say with equal degrees of modesty and certainty that he had succeeded. Forty thousand hits a day on his website qualified what some might label "hubris" as a mere statement of fact.
Luca sighed, thinking it was a long way from Sand Hill Road in Palo Alto to Cornerstone Trading in Delray Beach. Unlike other casualties of the boom that went bust—the dot-wronged and the dot-bombed—he had no one to blame but himself. He'd been positioned at the right time at the right place with the right technology. Synertel was bulging from two hundred million in VC funding. A white-hot investment bank was set to take the company public. Market capitalization was projected to be eleven billion, leaving Luca's 5 percent stake worth a little more than five hundred million dollars... _and that was before the issue hit the market_.
The bad news had come one week before the IPO was set to begin trading. Luca was in Milwaukee on the fourteenth day of a sixteen-day road trip. He'd just emerged from his thirty-third face-to-face investor meeting, talking up Synertel and its position as a vanguard in Internet transmission technologies. The fund manager evinced interest and promised to put in for 10 percent of the offering. There was talk of Synertel's stock tripling the first day. In Luca's mind, his five-hundred-million stake had already grown to over a billion dollars. His days as a lab rat were near their end, his years of sixteen-hour days, forgone vacations, and forgotten family about to pay off. Ray Luca was as good as a billionaire, and as such qualified to call himself a visionary, a creator, _an evangelist of tomorrow_.
And then it was gone.
Out of nowhere a team from Lucent bettered the speed of Luca's fiber-optic dynametric switch by two gigaseconds. _Two fuckin' gigs!_ Less time than an atom took to circle a molecule, but an eternity in the world of high-speed Internet transmissions.
Black Jet Securities shelved the offering, Jett Gavallan, its CEO, publicly calling for a reevaluation of Synertel's technology. Investor interest evaporated faster than rain in the Mojave. Luca was fired. His wife, doomed to another go-round as a start-up spouse, said "Screw this" and took the girls to live with her mother in Boston. In the space of seventy-two hours, Ray Luca went from billionaire-to-be to bum-in-training. Unemployed, unwanted, and unloved, dismissed by everyone and everything that had meant something to him, he was as instantly obsolete as his very own fiber-optic switch.
Taking in the agro-fluorescent lights, the soda-stained carpeting, and the chest-high cubicles, he wheezed dejectedly. It was time to get out of this jail.
The thought of escape drew his eyes to the battered Samsonite at his feet. Dropping a hand, he unlatched the silver briefcase and gingerly removed the fax he'd received this morning. Simply holding it made his fingers tingle, his stomach swoon. It was his pass to the big time. His golden E-ticket. His invitation to the major leagues. He reread it for the hundredth time, his eyes tripping over the mention of "Prosecutor General," "Joint Russo-American Task Force on Organized Crime," and "FBI." He planned on spending the entire night calling sources in Europe—reporters for the _Financial Times_ , the _Wall Street Journal_ , and the _Washington Post_ —asking if they'd heard anything about Kirov's being arrested or a raid on his offices.
He considered calling Cate Magnus, too. She'd sent him the fax; maybe she could shed some light on what was going on in Russia. He discarded the idea immediately. The rules were clear. Only she was allowed to initiate contact.
"Hi, I'm Cate Magnus," she'd said when he'd picked up the phone at his home on a sultry spring evening hardly four weeks before. "Jerry Brucker at the paper told me it would be worthwhile for us to have a little talk."
"Oh?" He recognized her name, and Brucker was an old pal from M.I.T.
"I've got an interesting piece of news that could do you some good. Mercury Broadband," she whispered. "Take a closer look. I think you'll find something the Private Eye-PO might like to share with his readers."
The next day he'd received an envelope containing the photographs of Mercury's Moscow downlink facility. If her claims sounded sketchy, the Cyrillic letters and twin-headed eagle of the Russian crest stamped on the backs of the photographs did not. A friend had translated the words as "property of the Prosecutor General's office," and Luca had shivered. Next came proof of Mercury's phony purchases from Cisco, then just this morning news of Kirov's impending arrest. If everything Cate Magnus said was true, Mercury wasn't just a scam dog—it was a monumental fraud. An international incident waiting to happen.
Envisioning the Black Jet name on the prospectus, he knew it was meant to be.
"Come on, Jett, just give us some time," he'd pleaded with Gavallan at their last meeting. "Don't cancel the offering. Six months and another round of financing and we'll be in the clear. We'll dust those losers from Lucent."
"Sorry, Ray. I don't think the VC guys would go for it. Six months is a lifetime, you know that. It's tragic. We're all disappointed for you. But unfortunately, this kind of stuff happens."
"Four months," Ray had pleaded, grabbing at Gavallan's sleeve, pawing at him. "I'll double the speed.... Come on, Jett. You gotta believe. Synertel can do it."
"So will Lucent, Ray. It's not the speed. You need a new technology."
_A new technology_. The words had defeated Luca. Four years later, they still did.
Luca put the fax away. He could only hope that when the raid mentioned in the memo took place, he would learn about it. Picking winning stocks, while worthy of admiration, was one thing. Revealing fraud and corruption on an international scale was quite another, and it turned Luca's role from profiteer to patriot. He was defending his country against a new Red Peril. Any aspersions about his past would be bleached clean by the mantle of "Nation's Defender."
On a personal note, it would be Ray Luca's pleasure to cancel Mr. Jett Gavallan's largest IPO. There was a symmetry to the affair that pleased Luca's mathematical mind.
One thing was certain: It would be a helluva way to launch the Private Eye-PO's investment newsletter.
Refocusing his eyes on the collage of screens, Luca felt a new energy plucking him up. He might not ever become a billionaire, but from where he stood in his beat-up Docksiders and floral-print shirt, "millionaire" sounded damned impressive. He'd done the math a thousand times. By multiplying the number of daily hits on his website by the standard browser-to-buyer conversion rate of 2 percent, he'd arrived at the figure of three thousand wise men and women willing to fork over five hundred dollars a year to receive the Private Eye-PO's twice-monthly newsletter. A cool one and a half million in revenues for a start.
Luca felt giddy at the prospect. If nothing else, at least he'd have the money to win visitation rights with his daughters.
It was then that he remembered his sell order for Merck. In the ten minutes he'd been daydreaming, the market had moved against him. Merck was trading at 38½ and falling fast. He sent in his order and was filled at 38½ Instead of making five hundred bucks, he'd lost almost two thousand.
Luca dropped his head into his hands. It was time his luck changed.
# 26
#
**G AVALLAN ARRIVED AT THE RITZ-CARLTON** in Palm Beach a few minutes before midnight. Once in his room, he set down his bags, opened the windows, and stepped onto the balcony. The smell of gardenias and the sound of the sea washing onto the beach greeted him. He always forgot how far south Florida sat, how tropical it could feel. It was hard to believe he was still in the States and not in some island paradise. A second later the first mosquito buzzed his ear and landed on his cheek. So much for paradise. He slapped at it, then went to the bedside phone and checked for messages left at his home. The first was from Tony Llewellyn-Davies.
"Jett, where the hell have you been all day? Thought you were sick in bed, laid up with a summer flu. Anyway, Jett, if you're not in bed now, go there immediately. I've got a piece of bad news. Jack Stuyvesant called from Lehman about the bridge loan to Mercury. Seems his board gave it the thumbs-down. They won't accept the ten-million-dollar tranche to Mercury. Meg told him that Graf had called and said that everything was hunky-dory. She tried to get him to take a smaller piece instead, five million, even three, but Stuyvesant said Lehman wouldn't lend Kirov twenty bucks if it was guaranteed by the full faith and credit of the U.S. government. That's not all, I'm afraid. Barron Bleriaut at Merrill is out, too. Same reasons. At least he was polite about it. Said if we got all the news sorted out about Mercury, he'd be back in. So that's it. Looks like us poor sods are left holding the bag. Fifty million of our best Yankee greenbacks in Mr. Kirov's pocket. 'Course, it will be all to our favor once we get Mercury public, that much more change in _our_ pockets. You might want to call Jack or Barron if you get a chance. A word from the lord of the manor might be in order. Cheers."
Gavallan slumped onto the bed, the phone dangling from his hand. Lehman was out. Merrill was out. Black Jet was left holding the entire fifty-million-dollar bridge loan to Kirov. But maybe it was just as well, he figured. Save an extra lawsuit or two down the road. Running a hand through his hair, Gavallan wasn't sure he could believe the string of bad luck. His right eye twitched, then twitched again, and he realized he'd developed a tic. Maybe this was what it felt like to be shell-shocked.
_Fifty million of our best Yankee greenbacks in Mr. Kirov's pocket_.
That's it, Gavallan said to himself. That's the death knell. He could almost hear the bells pealing.
_Unless somehow he could turn the company..._ No, Gavallan admonished himself, discarding the idea as quickly as it had come. It's foolish to keep hoping.
With great effort, he took off his clothes and climbed under the sheets. Sometime later, he fell asleep.
**F ROM HER SEAT** in the executive jet bound from New York to Miami, Tatiana stared transfixed at the limitless plain of water spreading below her in every direction. She had never seen the ocean, and it made her feel small in a way she never had before. Not forgotten or useless or empty, which was how she felt when she had driven across the endless Russian countryside traveling from her convent school near Novosibirsk to Moscow. But small in a way that left her comfortable and secure, feeling part of something large and wondrous, and maybe even magical.
The ocean, she decided, made her feel happy. It was an odd sensation.
Next to her, Boris Nemov yawned, then looked at his watch. "Eight o'clock. Good. We will land in thirty minutes. Did you get any sleep?"
Tatiana said yes, lying. She was much too agitated to sleep. She could not get Konstantin Kirov's words out of her head. She had never heard him so angry.
"This man is trying to harm us. Not just me, Tatiana, but you, too, and Boris, and everyone in our family at Mercury. He is spreading lies about the company. It is because of him that the American came to Moscow. You know, my sweet bird, that I abhor violence as much as you do, but sometimes..." His voice had trailed off, and she could feel his hurt, his fear, his apprehension.
"Boris will tell you what you must do," he'd gone on. "It will be quick, but messy, and for that I am sorry. Get in. Do the job. Get out. The Americans will think it was one of their own. This type of thing happens every day there. 'Running amok,' they call it."
Tatiana glanced at Boris, who had his nose buried in an American newspaper. "What do you find so amusing in the paper?" she asked.
"Amusing?" Boris cast her a sidelong glance. "Why, nothing. This is the _Wall Street Journal_. Business news. Nothing amusing at all." He began to read the newspaper again, but stopped after a moment, lowering it to his lap. "I am not going to stay with Konstantin Romanovich forever, you know."
"Oh?" Tatiana was surprised at the admission. Herself, she never intended on leaving Kirov. One of his TV crews had found her in a Petersburg brothel, a twelve-year-old runaway doing ten tricks a day. Incensed, Kirov had seen the house shut down and taken her in as his private ward. He gave her lodging, clothing, food. He was kind. (Which meant he'd never tried to sleep with her.) He was important, and she greatly enjoyed being in the employ of someone who commanded so much respect. No, she reassured herself, she would never leave. "What will you do?"
"A few more years and I am going to start my own company," he confided in an excited whisper. "Security, I think. For Westerners doing business in the Rodina. Maybe insurance. Our people will need insurance one day. I am not certain yet." Giving her arm a friendly punch, he smiled. "Maybe we work together. I give you a job."
"Maybe."
"Not what you are doing now. You cannot continue with your work forever. I think you should move into public relations. You are young. You are pretty. How many languages do you have?"
"Four, maybe five, if you count Baku."
"There, you see. If nothing else you can be a translator."
Tatiana smiled, wanting to convey a measure of interest. In truth, the prospect sounded appallingly dull. _Business. Public relations. A translator_. Her world possessed a more pungent vocabulary. _Slut. Thief. Whore_. Words that had been tattooed across her soul long ago. And more recently, _killer_.
She made a show of returning her magazines to her carry-on bag, then leaned back her head and closed her eyes. Enough talk of the future. Of dreams that might never come true. It was time for work. Time to begin steeling her mind to the task ahead.
Killing came easily. All she had to do was imagine a man's body on top of hers, his brow knit in concentration, his mouth open, dripping with lust, his eyes swallowing her whole as if her beauty was his for the taking. She would feel his pounding, taste his sweat. Her vision would grow hazy, the periphery dissolving into a grainy white cloud. Only her target would remain in focus. At the final moment, she would drift outside of herself and watch as another woman pulled the trigger.
Boris had told her it was rage, because she was upset about her time in the convent. She wasn't to blame, he said; anyone who had spent twelve years in a state-run orphanage would feel the same. She recalled the bowls of kasha, twice a day, every day, the haircut every six months, the dull scissors shearing her hair to the scalp, the bar of lye that came next to burn away the lice, taking two layers of skin for good measure.
She remembered the sacred sisters' midnight ministrations. The awkward touches under her gown, the cold raw hands, the bony fingers and ragged nails probing her private places, the sour breath smelling of cabbage and wine and whispering for her to stay quiet, that she was doing God's work, and all the while the chafing of their bristly mounds against her leg, punctuated by the staccato, irreligious grunts.
Tatiana swam through the smells, the sensations, the images, pleased they no longer frightened her or moved her in any way. Yes, she agreed, anyone would feel the same as she. But it was not rage they would feel, or anger. They would simply feel nothing.
Killing was easy if you were not alive.
**G AVALLAN ROSE AT SEVEN.** After a long run on the beach, he showered, then breakfasted on the veranda. The effects of the exercise and the lush surroundings left him feeling restored. Hardly himself, but not the shell who'd crawled into bed the night before. He put in a call to Emerald, explaining he'd be back that night, then left word for Tony or Meg to call him pronto.
At nine sharp, he knocked on the front door of 1133 Somera Road, the residence of Raymond J. Luca. He decided to play it straight from the get-go, explain that he too had learned that something was amiss with Mercury and ask where Luca had gotten his information. But the door never opened. In Gavallan's new world, nothing went as planned.
Returning to his car, he spotted a neighbor walking a pair of toy poodles. He was an older man with gray hair, glasses, and a wary eye behind the welcoming smile. Gavallan asked him if he knew Ray Luca, and if so, where Luca worked.
"You a friend of his?" the man asked.
"You might say that. We were at M.I.T. together." Gavallan thanked his stars for Jason Vann's inquisitiveness.
"Another egghead, eh?" The older man chuckled. "Don't know what I'd do without Ray. Helps me with my taxes. Saves me a couple hundred bucks each year. And the kid won't take a dime. It's not right, I tell him."
"That's Ray. He's a sweetheart. Say, I went by his house, but he's not home. Know where he works?"
Gavallan didn't want to come on like the authorities and made sure not to press too hard. Soon enough, the older man, who'd introduced himself as Ralph O'Mara, gave up the information.
"You can find him at Cornerstone. 714 Atlantic. He's a whiz, that boy. All we talk about is the market."
"Got any recommendations?" Gavallan asked before heading to his car.
"No, just one to stay away from."
Gavallan said good-bye before O'Mara could give him the name. He already knew what it was going to be anyway.
**T HE DELTA AIRLINES 727** inched forward on the runway. Out the window, Howell Dodson counted seven jets lined up in front of him, waiting to take off. Friday morning gridlock at Ronald Reagan National Airport.
"Rush hour—my, my," he said to DiGenovese. "Who'd have thought it? Least we've left the gate. Won't be but fifteen, twenty minutes till we take off. We'll be on the ground by nine, you'll see. Do some of that New York City driving, you can have us in Delray Beach in an hour's time."
Dodson had decided not to alert the Dade field office to their arrival. Protocol demanded that an assistant deputy director be met by the office's ranking agent. He'd have to explain why he was in the area. That meant going into the flimsy case on Kirov and the even flimsier reason for looking up Mr. Raymond Luca. Breathe one word of premeditated murder and someone would suggest setting up surveillance on Luca's house.
No thank you, said Dodson to himself. He didn't care to waste the Bureau's resources on snipe hunts. DiGenovese's hypothesis about Gavallan's murdering ways left him unconvinced.
"Roy," he said, "I think I'm going to avail myself of the free time to catch up on some rest. Twins never did get to sleep last night. Tell you, it's danged tough being a new father at my advanced age." And tucking a pillow under his head, Dodson settled in for a little shut-eye.
DiGenovese sat in the seat next to him, glowering.
**U PON LANDING,** Boris and Tatiana rented a car and the two drove the sixty miles north to Delray Beach. The morning was hot and muggy. The sun sat high in a hazy blue sky. The heat made Boris uncomfortable, and Tatiana wondered if it was too much for him. Every two minutes he had to wipe his brow and take a swig of the bottled water. Tatiana, though, was too taken by her new surroundings to notice the heat. From her first step inside the airport, she was mesmerized. Everything was so clean, the floors waxed a brilliant white and free of cigarette butts, gum wrappers, newspapers. Everyone appeared tanned, fit, and prosperous. And so many smiles. Not a worried brow among them.
They stopped once at a sporting goods store in Fort Lauderdale, where a man was waiting for them in the parking lot. He introduced himself as Andrei and spoke with a Georgian accent. Later Andrei explained he worked with the American branch of the Solnetsevo Brotherhood, the business group that controlled Moscow's northern neighborhoods.
Andrei led them to his car, opened the trunk, and handed Boris a green training bag. Inside was a map of Delray Beach, with instructions on how to find Mr. Raymond Luca and a layout of the building where he worked. He was a "day trader," Boris had explained with some envy, a man who made his living trading the stocks of important companies. Tucked in the bottom of the bag were two 9mm pistols and several boxes of ammunition.
Back in the car, Tatiana took a nail file from her purse and carved an x into the nose of each bullet to make it flatten on impact. Then she fed the bullets into the clip. She enjoyed the crisp click each emitted upon entry. Finished, she used her palm to drive the clip into the pistol.
"I'm sorry, my little bird," Kirov had said, "but on one point we must be clear. There can be no survivors. No witnesses. It is for the best. For your safety and mine."
With the help of Andrei's map and the rental car's onboard navigation system, they found the offices of Cornerstone Trading. Parking the car a block away, Boris told Tatiana to wait while he entered the building and checked if Raymond Luca was in. She watched him cross the street, thinking he did not look so bad dressed like an American in blue jeans, a white button-down shirt, and high-top tennis shoes. It was nice to see him in something other than a black suit.
She was dressed in nearly the same attire, except that her shirt was a blue and white chalk stripe and her tennis shoes were white and dainty.
Boris returned five minutes later.
"He is there. Fourth cubicle to the right."
"What is a 'cubicle'?" Tatiana asked.
"Like a little jail cell. Four walls that rise to your chest and a chair inside. He is seated working at his computer. He wears a baseball cap. Yankees of New York, I think." Though his face was grave, his eyes were bright, overexcited. "You are ready, little sister?"
Tatiana nodded her head. Somewhere back up the road, her tourist's fascination had faded, replaced by a professional's icy detachment. She did not wish to speak. The pistol tucked into her pants, she simply nodded.
"I will be in the alley in back of the building," Boris continued. "Once you enter, you have one hundred twenty seconds. Eight men downstairs. Two upstairs—the managers. Shoot, then move. Shoot, then move. Do you understand?"
Again, Tatiana nodded. Shifting in her seat, she adjusted the bandages that flattened her breasts, then pulled the baseball cap lower on her head. Boris took her hand and kissed it. "Go now."
Tatiana opened the door without a backward glance.
Eight downstairs. Two upstairs. Shoot, then move. Shoot, then move.
One hundred twenty seconds.
Go.
# 27
#
**Y ESTERDAY WAS THE ZONE.** Today was multitasking.
Ray Luca backhanded a glob of ketchup from his mouth and planted his double chili cheeseburger on the only available sliver of free desk. Chewing contentedly, he flicked his eyes from monitor to monitor and screen to screen, from the market being made for Intel to the closed-circuit feed of Thoroughbreds taking their morning run at Hialeah, to the "Money Honey" on CNBC reporting live from the floor of the Exchange and back again. At the same time, he sipped at his coffee, tapped out a series of buy orders, and managed to hum a little ditty.
_Let the good times roll. Yeah baby, let the good times roll_.
The market was up strongly. The sky was as blue as a Tiffany gift box, and on his lap was a completed copy of the Private Eye-PO's latest editorial concerning the Mercury Broadband offering. He particularly liked the title. "Mercury in Mayhem."
Another bite of the double chili cheese, a gulp of coffee, then a moment's glance to reread and edit.
_Private sources report an explosive confrontation Thursday afternoon outside Mercury Broadband's Moscow offices_ _on Kropotkin Ploshad between OMON militia troops led by Russian prosecutor general Yuri Baranov and members of the FIS (read KGB) loyal to Konstantin Kirov. Armed with a search warrant, Baranov had hoped to seize financial records incriminating Kirov in the theft of $125 million from the coffers of Novastar Airlines. Kirov, law-abiding citizen that he is, denied the OMON troops entry, preferring to let his legion of house-trained espiocrats do his talking for him. No doubt he'll call Baranov's visit just another case of political harassment motivated by his advocacy of free speech and a free press_.
The question Luca had yet to answer was what members of the state security apparatus were doing at Kirov's offices and why they had stood to his defense. It was akin to the CIA's defending Ted Turner on American soil.
_Whatever Kirov may say_ , the Private Eye-PO continued, _there can be little doubt, dear hearts, that not only he, but Mercury Broadband as well, is skating on very thin ice. Do tell... if he didn't steal the $125 million, who did? Maybe we should ask Jett Gavallan for the answer? After all, if he's Kirov's banker, who better to point us to the missing loot?_
_Stay tuned, campers, for more news from the Russian Kleptocracy_.
Luca put down the pages, pleased but tired. It had all started just after eleven last night, when Jack Andrew, a correspondent for the _Financial Times_ in Moscow, had called him in a furor to demand how he had known beforehand about the raid on Kirov's offices. Luca dodged the question, instead pounding Andrew for every detail imaginable about the encounter. Afterward, like any solid journalist, he double-checked his source. He phoned his contacts at the _Post_ , the _Wall Street Journal_ , and the _Moscow Times_. All of them said they'd heard whispers about the raid, but as yet could get neither Kirov nor the prosecutor general to confirm or deny.
Adding a few comments here and there, Luca folded up the article and put it back into his briefcase. He'd meant to get it onto his server and uploaded to his web page this morning, but he'd overslept, and his cardinal rule was never to miss an opening. Good thing, too. The market was riding an updraft the likes of which he hadn't seen in a year. Fifteen minutes after the opening the Nasdaq was up 80 points and the Dow up 100.
In a parallel universe, Mazursky and his crew were yelling loud enough to rouse the Miracle Mets. Let 'em, thought Luca. With the news about Kirov, he'd be out of there inside a month. The newsletter would do better than he'd ever imagined. Forget three thousand subscribers. Why not four thousand? Five thousand? Ten, even? Luca would buy a little house and a Boston Whaler he'd had his eye on. He'd arrange a weeklong trip to Disney World for the girls. Maybe, just maybe, he could convince his wife to come back to him.
Enraptured by this rosy vision of the future, he found it difficult to breathe. It could happen, he told himself. It really could. The family back together again. Ray and his four girls. It was all he had ever really wanted.
Minutes passed and the market continued higher, headed straight for the stratosphere. Volume. Tick. S&P futures. All were rocketing up, up, up. One after another he put on a buy, not bothering even to take profits on his earlier positions. At ten o'clock, the Nasdaq was up 150 and the Dow the same. A quick tally showed him ahead twenty-five grand.
Once in a while Luca looked down at the briefcase. Part of him said to close his positions, take his profits, and get home to post his newest article—the sooner the better. But Luca ignored the voice. He wasn't leaving today. Today he was a trader. He could be the Private Eye-PO tomorrow, and for the rest of his life.
**H ELLO, RAY."**
Luca jolted in his chair as if he'd seen a ghost. "Jett Gavallan. What a surprise. What brings you round these parts?"
"I'm sure you can guess. You've been doing some good work—or should I say your sources have. Looks like I was wrong about Mercury."
Luca eyed him warily. "You're going to cancel the deal?"
"Postpone it. The company isn't all bad. Maybe it isn't everything we billed it to be, but there's some decent stuff there. It's Kirov I'm worried about."
"So you heard?" Luca's eyes flashed triumphantly.
"Heard what?"
"Yesterday there was a..." Luca sat back, rubbing at his chin as a mean-spirited grin darkened his features. "Sorry, Jett, you'll have to wait and see."
Gavallan lowered himself onto his haunches so he could look Luca in the eye. "Ray, this isn't about Synertel. I'm sorry about what happened. It was a lousy turn of events. I can imagine it was a letdown."
"A 'letdown,' was it? Is that what you call losing a billion dollars? Having your wife throw you out on the street? Watching your children shy away from you because they're too embarrassed to give you a hug? A 'letdown'?"
"Like I said, I'm sorry it turned out that way. It was a tough break."
"What the hell do you know about 'tough'? You, sitting up there in your luxury penthouse, driving your snazzy car? You bankers are all bloodsuckers. Best friends when times are good, out of there like lightning when things get rough. Payback, Gavallan. This one's on me."
"I did what I had to do. You would have done the same thing if you were in my place. Look at me, Ray. You know it's true. Now, listen, I need your help. I have to know where you got your information about Mercury. I'm trying to work back up the chain, figure out who pulled the wool over our eyes."
Luca laughed, a little wildly. "You're not serious? You don't just expect me to tell you." Shifting his gaze away from Gavallan, he spent a moment tapping an order into his computer. "Tell me, what do I-bankers earn these days? An hourly rate will be fine."
"This is a lot more important than what I earn."
"Two hundred an hour?" Luca cut in. "Or am I out of date? Three hundred? Four?"
"It's not just about Mercury and Black Jet. You're in this too, Ray... or the Private Eye-PO is. We need to talk. You could be in a lot of danger."
"Danger? Ooh, I'm shivering. Can't you see me shaking in my boots?" He tried on another smile, but Gavallan's grim expression stole his mirth. "What kind of danger?" he asked after a moment.
"I'm not sure exactly. But if I can find you, so can Konstantin Kirov. After all the crap you've been spreading on the Net about his company, I don't think he'll be in a charitable mood."
Something in Gavallan's tone reached Luca. The angry cast to his eye softened and the tension left his shoulders. "Okay, okay," muttered Luca. "But I can't leave now. Take a look at the market. I got to make some money."
"Take a break."
"Got too many open positions. Tell you what, though. I'll stop at noon for fifteen minutes. Believe me, that's all we'll need. Meet me next door at Alberto's. We'll have a cup of coffee."
"Deal," said Gavallan, rising to go, happy to get out of the rancid confines. "See you at twelve. Alberto's, right?"
Luca nodded. "And, Jett? Order yourself a drink beforehand. Something strong. You're going to need it."
Leaving the building, Gavallan turned left and headed down the sidewalk to his car. He didn't see the slender young man in the baseball cap enter the building less than a minute after he left.
# 28
#
**L UCA HARDLY HEARD THE FIRST SHOT.**
A door slamming, he thought, keeping his eyes on the screens, but then came the moaning, the fevered imprecations not to shoot, followed by another bang. This time the noise was unmistakable. Achingly loud. Frightening. His ears rang, and then he caught a whiff of smoke and his nose began to burn. Cordite, he thought. Yet for all the sensory data, it came to him slowly. A gun. A very, very big gun.
At first, he thought it had to be Mazursky, some kind of joke he was playing, but a glance down the aisle told him he was wrong on that score. The Wizard of Warsaw lay twenty feet away, his jaw opening and closing like that of a fish out of water, eyes wide open, a pitch-black crater on his forehead starting to leak blood.
And for a split second, Luca thought, Jesus, it took a bullet to shut that loudmouth up.
But by then Krumins was yelling and running toward the front door. Halfway there he seemed to leap out of his shoes and slam against the wall, and when he slid to the ground there was a wide, bloody red swath tracing his path.
Gregorio stood up in his cubicle, and his blond head seemed all at once to vaporize in a cloud of red mist. Nevins crawled past Luca down the aisle. The gun roared, and he went flat and stopped moving, without even a grunt.
"Ray?"
Four feet away stood the shooter. The voice gave her away as a woman and foreign, though it was hard to tell by how she was dressed.
"Ray Luca?" she asked again.
"Yes?" he said, frozen, confused, very, very scared. Kirov, he thought. Kirov sent you. "What do you want?"
But she didn't answer. Striking with the speed of a cobra, she wrapped an arm around his neck, brought him to her chest, and laid the pistol against his temple. Paralyzed, he tried to scream, but the words lodged in his throat.
No, no, it can't be. We're going to Disney World. My wife and daughters, we're going to—
# 29
#
**A LONG ATLANTIC AVENUE** in Delray Beach, traffic slowed to a crawl. Jett Gavallan braked, trying to see ahead and determine what might have caused a traffic jam at eleven-fifteen in the morning. He caught a slew of flashing lights, bright metal, and the rush of uniformed men and women to and fro. A pair of police cruisers, strobes spinning, barred the street a block ahead. An auto accident, he surmised. And a bad one at that.
"Tony, Bruce, I want you both to listen to me," Gavallan was saying into his cell phone. "No more calls to farm out the bridge loan. It's time we show some confidence in the client. If Lehman wants out, fine. Ditto for Merrill. We'll keep all fifty on our books. End of story. I don't want the market to see us sweat."
"It's not a question of seeing us sweat," replied Llewellyn-Davies. "Just simple financial prudence. If I can unload twenty million of our exposure to Kirov, I'm damned well going to."
"No, you're damn well not," barked Gavallan right back.
"He's right, Jett," chimed in Tustin. "Deal goes south, you'll be thanking us, kid."
"And when it goes through you going to fund me the eight hundred grand we passed up?"
_"Youfugginkidddinme?"_ bawled Tustin. "I'm just an employee, bwana."
"Reconsider, Jett," said Llewellyn-Davies. "That's a right decent chunk of risk you're willing to shoulder for eight hundred thousand dollars."
Gavallan shook his head at their tenacity. Not now, fellas; this is not the time. It was imperative everything continue as before, that he not give the slightest hint he was going to scupper the deal before it hit the street, or that he had an inkling that Grafton Byrnes was in a world of trouble.
"The decision has been made," he declared. "No more calls." He hung up.
It was a picture-postcard day, lacy clouds scudding across a pale blue sky, trade winds blowing up from the Caribbean, tangy with sea salt and suntan oil. Close your eyes and you might hear some marimbas and steel drums, catch a scent of jerk pork roasting on the spit. A day to relax, he decided. Play a little golf, take the boat out for a sail, drink a six-pack on the back stoop. A cynical voice laughed at his middle-class musings. In nine years, he'd never taken a day off except when sick. His longest vacation had lasted all of four days, cut short by the minicrash of '98 and the demise of Long Term Capital.
"When you work, work. When you play, play," Graf Byrnes was fond of saying. "But goddamn it, don't think the world is going to stop if you don't show up for work one day. The graveyard is filled with indispensable managers."
Gavallan took the words to heart, deciding that when this thing was over, when he had Graf Byrnes safe and sound back in his office in San Francisco, he'd do some serious playing. A month in Maui. The safari in Kenya he'd promised himself. Maybe he'd charter a yacht, do a little island-hopping near the Bahamas.
_"Alone?"_ a cynical voice asked, and the glow of his dream vacation lost its luster.
"Come on, come on. I'm in a hurry here."
Rapping his palm against the steering wheel, Gavallan urged the column of cars to advance. Yard by yard, the cars edged forward, past the color-coordinated strip malls painted the same gay shade of coral, the casual cafes, the brokerage offices, and the cruise ships offering two-day jaunts to the Bahamas for $99. Delray Beach had the look of a theme park for seniors, with cappuccino and conch fritters replacing cotton candy and corn dogs.
The car in front of him turned onto a side street, offering Gavallan full view of the street ahead. Four patrol cars sat behind the cruisers blocking the road. Parked at odd angles to one another, they looked as if they'd hit a patch of ice and spun to a stop. Two had their noses half to the curb, a third his rear tires on the sidewalk. The last was frozen in the center of his lane, a track of spent rubber thirty feet long attesting to the urgency of his arrival. He sniffed the air. Burnt rubber mixed uneasily with the bloom of summer gardenias and the scent of freshly cut grass.
In the blink of an eye, his curiosity turned to apprehension.
Sliding a knee onto the seat, he lifted himself up and peered over the convertible's windshield. Emergency vehicles jammed the street: three ambulances, rear doors flung open, gurneys absent; a fire truck; a trio of identical navy Crown Vics that screamed federal law enforcement; and bringing up the rear, a TV van, horn blaring, advancing foot by foot. For all the activity, Gavallan had no way of figuring out what exactly had happened. He knew only one thing: This was no auto accident.
A swarm of uniformed men and women buzzed back and forth across the street, running into and out of a building in the center of the block. Two cops carrying spools of yellow and black tape began to walk toward the building, and the words "crime scene" flashed through his head. A gurney emerged from the building and rattled along the sidewalk, shepherded toward an ambulance by three determined paramedics. Their sober pace didn't give Gavallan much hope for the patient. Neither did the woman following them, a middle-aged peroxide blond, hands to her face, sobbing. Another gurney rolled out, this one in a hurry. Above the din, he heard a voice. Strident. Losing its calm. "Move it. We got one alive. I need four units of..."
The words were drowned out by a chopper flying in low overhead, a Bell Ranger hovering a hundred feet in the air. Police? No. More TV.
It was then he recognized the building: the mint green plantation shutters, the barrel tile roof, the Mediterranean arches. Cornerstone Trading.
"All right, sir, let's get a move on," said a tan young traffic cop, patting a hand on the hood of Gavallan's rental car. "Nothing here for you to see. Detour to your right and be on your way."
"Any idea what happened, officer?" Beneath the tourist's smile, Gavallan was aware of his breath coming fast and shallow. He had to fight not to wipe the sweat from his lip.
"Nothing to concern you," answered the policeman. "Just move along. I'm sure you'll be able to read about it tomorrow."
"Looks bad," Gavallan persisted. "Anyone hurt?"
"Move along, buddy. Now!"
Giving a curt wave, Gavallan activated his turn signal and drove the Mustang rental up the block. After finding a place to park two blocks up, Gavallan ran back to the crime scene. By now a sizable crowd had gathered. He threaded his way through the onlookers, stopping on the sidewalk opposite the entry to Cornerstone Trading. He'd hardly had time to gather his breath before a young man standing next to him began to fill him in.
"Guy just lost it, man. Went in and capped his crew, then did himself. Got every one of them. Ten dudes, all dead." He was a handsome Hispanic kid, maybe fifteen, with spiked hair dyed henna, a golden nose stud, and cargo pants cut to the knee. "I heard it, man," he went on. "I work at the Orange Julius next store. It was like this, check it out: _bang, bang, bang, bang_. Shit was loud, and quick, like maybe two seconds between shots."
"You think you ought to tell that to the police?" asked Gavallan.
"The police? Heck, no. I don't need that hassle." Suddenly, the kid jumped back a step, his brown eyes skittish. "You ain't the man, are you?"
"No," said Gavallan. "I ain't the man." He beckoned the boy closer. "You said, 'The guy just lost it.' You know who did it?"
"Nah, man, no one knows. But I know one of the dudes was in there. My man, Ray. 'Fact I made him a burger this morning—his favorite, a double chili cheese with jalapeños. Calls it his 'victory burger.' Dude came in real happy, see, smiling even, and that's something. My man Ray is one serious dude."
A victory burger, Gavallan said to himself, remembering Luca's cocky grin, the mention of having some dirt on Kirov.
"When did it happen?" asked Gavallan.
"When did what happen?"
All at once, Gavallan's patience left him, evaporated under the tropical heat, worn away by the endless string of setbacks, one more trading loss in Black Jet's column, who knew? Grabbing the Hispanic youngster by the arm, he shook him once, hard enough to frighten him. "The shooting," said Gavallan. "The murder. Whatever went on inside of that building."
"Yo, man, chill," the kid said, eyes bugging. "Like an hour ago." He flicked a wrist to check his watch. "Ten, ten-fifteen. Ten-twenty. Round there. We cool now?"
"Yeah, we're cool." Gavallan patted the kid's arm and moved off toward his car. A glance behind told him he'd already been forgotten. The Latino was busy offering his story to the next bystander who'd happened along.
Gavallan wiped the sweat from his forehead.
This was not how the day was supposed to have gone.
**T HE BODIES LAY** where they had fallen. Some sat slumped at their computers, too surprised, too frightened, to have reacted. Others had run, though none had made it more than a few feet from his or her desk. The mess was terrible and overwhelming, gore spackled onto the walls and cubicles in chaotic, Technicolor blotches. Ponds of blood stained the carpet, clotted now, hard as ice. Black Ice.
_Dumdums_ , thought Howell Dodson as he walked slowly down the center aisle of the trading room at Cornerstone Trading. Bullets modified to flatten on impact. Small hole going in; big hole coming out. He passed a victim, his face missing below the hairline, a gaping mask of blood, bone, and gristle.
Despite himself, he gasped. He'd seen men killed, women too. He'd witnessed death many times over in all its inglorious pageantry. He'd sat at a wooden table, arms and legs bound, and watched as the pinky and ring finger of his left hand were severed with a carpet layer's dulled blade. The smell of blood and the scent of fear were familiar companions.
But this was different, he thought, stepping carefully over another corpse. These were the innocent, the unknowing, the unsuspecting. Death didn't belong in these stained, shabby, ordinary corridors.
"Ten bullets, ten bodies," explained Lieutenant Luis Amoro of the Delray Beach Police Department, a beefy Cubano of fifty who looked about two sizes too big for his khaki rayon uniform. "Guy started at the entrance, went seat by seat taking out each of his buddies, then ran upstairs, got the managers. We figure he came back down afterward, looked around, made sure no one was still alive, everything wrapped up nice and neat, then did himself."
"Some shooting." It was the only thing Dodson's normally glib tongue could manage. For all his time on the job, for all the wanton and terrible things he'd seen and experienced, he was having a tough go with this one. The question "Why?" kept jabbing away at his mind, and he had no answer.
Since entering the building, he'd been overwhelmed by a desperate and irrational fear for his sons' welfare. Though the infants were over a thousand miles away in McLean, Virginia, safe in their Talbots sweaters and Eddie Bauer strollers, he wanted nothing more than to hold them in his arms and guarantee their safety. "Christ our savior," he whispered.
Leading the way to the end of the aisle, Amoro knelt beside one of the bodies and pointed to a neat round hole inside the man's hairline by the temple. "We figure he's the doer. Everyone else got theirs from a foot or more, usually in the back of the head."
Dodson eyed the inert form. "Mr. Luca leave any note? Any message for his loved ones?"
"Not a word. Looks like he came in, worked for a little while. Around ten, something must have gotten him pissed. He got up, took out his haymaker, and went about his business." Amoro did a double take. "Hey, how'd you know his name?"
Dodson ignored the question. His eyes were glued to the banks of monitors, the blinking screens of blue and yellow and green. "Wouldn't figure a man to be so upset on such a good day," he said, pointing at the ticker for the Dow Jones Industrial Average. "Market's up three hundred points. I'd say that's cause for celebration. Guess there's just no pleasing some people."
A large, dull gray pistol lay near Luca's outstretched hand.
"A Glock," said Amoro, kneeling down, pointing at the weapon with a pencil. He spoke with a docent's tone, as if the men were touring a museum, not a charnel house. "Serial numbers are filed off, but if you use an acetate wash you can usually bring them back up."
Dodson stooped to get a better look at the weapon. "Where do you suppose Mr. Luca got himself a toy like that?"
"I imagine the same place he got his bullets. We took one out of the wall. He wasn't messing around. These things can penetrate a Kevlar vest. Cop killers, we call 'em. Not a good policy to be on the receiving end of one of these."
Dodson nodded amiably. "I'll take that under advisement, Lieutenant Amoro. Thank you."
"Our boys are checking for prints. We'll do a residue analysis on Luca's hands once we get him to the morgue, just to tie everything up."
"Good idea. Never can be too thorough." Dodson's eyes flitted across the crime scene. While murder was a matter handled by local or state police, the day trading angle and the use of the Internet raised questions of interstate commerce and securities fraud, both crimes squarely in the federal purview. Amoro might know a thing or two about dragging up filed-off serial numbers, but he was far too lax in securing a crime scene.
Laying a hand on the officer's shoulder, Dodson guided him to a quiet corner. "It may interest you to know that Mr. Luca here was the subject of a Daisy tap and a participant in an international investigation involving the Russian _mafiya_. I'm afraid that I'll have to declare this crime scene under federal jurisdiction. I'd like you and your men's fullest cooperation."
Amoro answered with surprising civility. "You want it, it's yours. Worst crime we've had this year is grand theft auto and a rape up on the county line. Between you and me, it's why I transferred out of Miami. It's nice to be able to say that murder's beyond your reach." He added skeptically, "The Russian mafia in Delray Beach? Come on."
"World's a small place," said Dodson. "Now if you'd be so kind, tell your men not to touch a thing. I've called in some of my colleagues from the Miami Dade office. They should be getting here any minute."
He meant the members of the violent crimes unit, sixteen strong. DiGenovese had wanted to alert them yesterday and ask that they put a twenty-four hour watch on Ray Luca. Dodson had said no. The decision would haunt him the rest of his life.
Feeling a tug at his elbow, he turned to see Roy DiGenovese sliding several 8-by-12 photographs from a manila envelope. "Crowd pics from the crime scene an hour after the murders took place," he explained. "Take a look. Second row. Good-looking guy, sunglasses, blond hair."
Dodson slipped his bifocals out of his jacket pocket and looked hard at the face. "Couldn't be," he said. "Must be a resemblance."
"Who else stayed in room 420 of the Ritz-Carlton in Palm Beach last night?"
Dodson was impressed. "My, my, Roy, well done. Seems I taught you well. Anything else up your sleeve?"
"Gavallan got in yesterday night at eleven. He's booked back today at three. American out of Miami. He's driving a Mustang convertible, gold."
"All well and good, Roy. I am a tad curious, however, how Mr. Gavallan slipped past your boys in San Francisco?"
"We were soft," replied DiGenovese unapologetically. "And we were strung too thin. We'd grown used to following him in his car. With two men on duty, it was tough to cover him on foot. Like you said, he must have slipped by."
"Must have. Now let me take another look at these pictures." Dodson brought a photo close to his eyes, shaking his head incredulously. "Come now, Roy, cooking the books with Mr. Kirov is one thing; this is major wetwork. You think he has the _cojones_ for this kind of thing?"
"You heard the tape, sir. Gavallan said if he had his way he'd shut the Private Eye-PO up forever. I don't think it's a coincidence that Gavallan's here. The man has the means, the motive, and the opportunity. I think you taught me that, too."
Dodson didn't believe it was coincidence either, but he couldn't get his arms around pinning Jett Gavallan, a wealthy, law-abiding citizen, a philanthropist, and an ex–Air Force officer, as a mass murderer. You didn't put a square peg in a round hole.
"I'll agree with you that it wasn't poor Mr. Luca here who made such a mess," he said. "My guess is gangland. One of Kirov's American cousins. Let's get on to surveillance in New York. See if any of the shooters in Little Odessa have taken a holiday of late."
"Yes sir. But would you let me bring him in now? Get a B-4 for the records. I'd say we have probable cause."
"All right, Roy, you can bring him in. Have the police issue an APB in the area, put some of our men on his house in San Francisco, get some agents into his office. We want him to know this is for real."
DiGenovese nodded, unable to hide a malicious grin. "I'll take real nice care of him, don't you worry, sir."
"But no arrest warrant until we collect some evidence, and I mean something that will stick in court. His lawyers come charging in now and we'll never get a conviction."
DiGenovese frowned, hanging his shoulders. "What about a fugitive flight alert?"
Again Dodson's instincts told him no. If Gavallan was hanging around the murder scene, he didn't appear to be in any hurry to leave the country or to fear being captured by the police. The acrid scent of burnt powder tickled his nose, making his eyes water. Standing there, feeling his assistant's gaze burrowing into him, appraising him, exhorting him, damning him, Dodson wondered if his hesitancy to act more boldly was really prudence, or just a neatly disguised fear of failing. He forced himself to stare at the bodies, one by one. Each was a member of a family, a loved one who would be missed and mourned and grieved over for years to come. Fathers, brothers, uncles, friends, neighbors. The admission of guilt clutched him by the neck, and he found it difficult to swallow. He tried to argue that he wasn't at fault, that he couldn't have prevented this, but his words rang hollow. He'd let professional hubris and personal comfort interfere with sound police work. He might as well have pulled the trigger himself.
"Put his passport on the watch list," said Dodson. "Get some men to the airport. Send a team of agents to his hotel. And get me his cell number. Guy like that's got to have at least one phone on him at all times."
Excusing himself, he made his way outside and hurried round the corner of the building. There on a neat patch of grass, Howell Dodson fell to his knees and vomited.
_Never again_ , he swore to himself. _Never again_.
**G AVALLAN DROVE THE MUSTANG SLOWLY,** keeping his speed under the limit as he listened to news of the shooting on the radio. The announcer put the final tally at ten dead—eight males, two women. The Latino kid had been right: There were no survivors. The poor joe on the gurney hadn't made it. Police speculated the killer was a disgruntled trader working out of Cornerstone, but had not yet identified him. The announcer spoke of another grim American tragedy. A lonely man. A failed career. A last desperate act.
Gavallan knew better. Ray Luca was the target, even if he'd been made to look like the killer. If Konstantin Kirov hadn't pulled the trigger himself, he was responsible. By now the pattern was clear. Ask a question, risk Kirov's wrath.
He reached the end of Biscayne Boulevard and stopped the car at a red light. Staring out over the placid blue water, he felt a sea change come over him. He was done being the victim. Done feeling guilty. He'd never been well-suited to playing the patsy anyway. A new emotion took hold of him—maybe a whole cocktail of them. Anger. Vengeance. The will to act, not react. He'd come a fair distance in his life, but not so far as to forget his roots, or the struggle he'd waged to get where he was today. He wasn't about to let a smooth-talking Russian take it all away.
The light turned green. A left would take him to his hotel, where he could pick up his belongings. If he hurried, he could make his three o'clock flight home. He gazed up the road, at the seaside hotels and neat bike path. An elderly couple walked hand in hand along the sidewalk.
Gavallan looked to the right. The road offered the same amusements, but led in another direction altogether, to the uncharted places on ancient maps decorated with serpents and dragons.
Gavallan turned right.
# 30
#
**D AMN IT!"** muttered Gavallan as he turned the doorknob and found it locked.
He was standing at the back door of Ray Luca's house, a run-down clapboard cottage with dormer windows, a weather vane, and paint peeling by the bucketload. Bougainvillea, ferns, and frangipani grew untended on three sides of the small home, enough vines and vegetation to qualify the place as a jungle. Frustrated, he took a step back, looking for spots where Luca might have hidden a key. He ran a hand along the door frame; his only rewards a splinter and a dead beetle. A few potted plants dangled from exposed rafters. His fingers probed the moist dirt, again without success. Behind him, a redbrick patio stretched twenty feet in either direction. A hot tub occupied one corner, a rusted hibachi and a flimsy set of lawn chairs the other. He walked to the hibachi and removed the lid. Fired charcoal briquettes dusted the interior. He replaced the lid carefully, his grasp that much tighter because of the sweat rolling down his forearms. The heat and humidity, coupled with his anxiety, made him feel plugged in, electric. He held out his hand and it trembled slightly, not so much with fear as with adrenaline.
He had parked two blocks up the road and walked boldly to Luca's front door, calling out his name to show the world he was a friend. He'd decided that noise was less suspicious than silence, and that an innocent visitor wouldn't think to camouflage his arrival. The neighborhood was sleepy bordering on comatose, with quaint cracker box houses spaced twenty to thirty yards apart and a scarred macadam road shaded by a palm canopy. Though he hadn't seen a soul, he could be sure someone had laid eyes on him. He figured he had fifteen minutes before his window of safety closed. After that he had no idea who might come—police, the FBI, a nosy neighbor.
His anxiety growing as the seconds ticked by, Gavallan returned his gaze to the rear of the house. A watering jar, a can of insecticide, and a terra-cotta pot holding a spade and a trowel sat a few feet from the door. Taking out his handkerchief, he wiped his forehead and dried his palms. _Eeny-meeny-miney-mo_. He chose the watering jar. Wrong again.
The key was under the insecticide.
**I NSIDE THE KITCHEN,** Gavallan stood with his back pressed to the door, listening. He heard the tick of the oven clock, the whir of the ice machine, the deafening static of abandonment. Mostly, though, he heard the draw of his own shallow breathing and the _boom-boomboom_ of blood thumping in his ears.
Satisfied the home was deserted, he made his way through the dining room, past the front door, and into the den, or what his daddy would have called "the parlor." A sky blue La-Z-Boy recliner occupied pride of place, four feet from a big-screen television. Luca hadn't watched TV; he'd bathed in it.
Blinking, Gavallan remembered his father's recliner, an olive velour "EZ-cliner" from Sears, armrests threadbare but spotless after fifteen years. The Captain's Chair, his daddy had called it, though it was strictly for enlisted men. He saw, too, the fifteen-inch black-and-white television, the creatively mangled wire hanger that served as its antenna, and the TV's cinder-block perch, prettied up with a pink pillowcase and a shiny glass jar filled with freshly picked daisies. Cleanliness alone had rescued the Gavallans from poverty.
A curtain fluttered and a faint breath cooled the room, but instead of catching a hint of jasmine and wisteria, he tasted the day-old scent of red beans and rice and the wet, ambition-robbing heat of a Texas summer.
Keep moving, he told himself.
Luca's bedroom lay at the end of a narrow corridor. The queen-size bed was neatly made, colorful stitched pillows strewn over a white bedspread. Poster prints of Monet's water lilies tacked to the wall supplied the culture. Gavallan spotted a few photos of three young girls he presumed to be Luca's daughters—skinny little things with pigtails and overalls, around four, six, and ten. A personal computer sat on a long desk that took up one wall. A screensaver flashed a field of racehorses with the header "254 days until the Flamingo Stakes."
Ray liked the ponies, mused Gavallan. And his "victory burger" with jalapeños.
Six piles of neatly stacked paper were laid out to the left of the computer. Technical charts. Analysts' reports from bulge bracket firms. Typewritten notes. His eye stuck on a page with strangely familiar script. Craning his neck, he looked closer. The header was written in Cyrillic and the body of the text in English. The fax was dated two days earlier, and addressed to Assistant Deputy Director Agent Howell Dodson, Chairman, Joint Russo-American Task Force on Organized Crime.
As he dropped a hand to pick it up, something creaked in another part of the house. It was a distinct sound, high-pitched and whiny, lasting a second or more. It was the kind of noise that made you shiver. A door closing? A footstep?
Ten seconds passed. Fifteen. Gavallan held his breath, his ear tuned to any vibration that might indicate the presence of another. He wasn't feeling so electric anymore. Not so plugged in. Jittery was more like it, the adrenaline long gone. He was breaking and entering into the home of a man shot and killed barely two hours earlier. If the police found him, he could count on a one-way trip to jail with bail an impossibility for days.
The house held its breath and was silent. Using his handkerchief, Gavallan pulled the chair out from under the desk and sat down. He had no intention of leaving any fingerprints. As far as he or anyone else was concerned, he was never here. Picking up the fax, he read about the proposed raid on Kirov's headquarters. A second go-through and he'd memorized the cast's names—Baranov, Skulpin, Dodson of the FBI. He knew the star personally: Kirov, Konstantin R. Replacing the fax on the desk, he recalled an old saw about playing cards: If you can't spot the sucker, it's probably you. A disgusted smile burned his lips.
But if Gavallan thought he'd found his trophy, the souvenir of his secret visit, he was mistaken. A marked-up copy of the newest article for the Private Eye-PO's web page lay crumpled in the trash can by his feet. "Mercury in Mayhem," it was titled, and it offered a blow-by-blow account of Prosecutor General Baranov's failed raid on the offices of Mercury Broadband.
That would have done it, thought Gavallan, reading intently. Word that Kirov was under investigation would have proved the straw that broke the camel's back. And so the victory burger!
"Ah, Ray, you were so close."
Finished reading, he laid the paper to one side. He had no time to digest, just to collect. Still using the handkerchief, he clicked on the mouse and watched as the parade of galloping Thoroughbreds was replaced by a copy of the same article. Closing the file, he thought of burrowing into the computer's directory and deleting it. He decided against it. Mercury was what it was. He'd never planned on abetting a fraud. He wouldn't start by erasing a dead man's last words.
A bedside clock showed the time as 12:08. His window of safety would close in seven minutes. Abruptly, he rose. Collecting the Russian fax, he laid it on top of Luca's last article, then folded the papers in half, as was his habit, script side up. That was when he saw it: ten little numbers printed across the top of the page, indicating the phone number of the sending fax machine. Area code 415 for San Francisco, 472—and he knew the rest by heart.
_Leave_ , a voice told him. _You can be sick outside_.
He had stepped into the corridor outside the bedroom when a door opened and closed. This time there was no mistaking the noise. Footsteps crossed the kitchen floor, squeaking on the checkerboard linoleum. He made out voices. Murmured. Controlled. Guilty.
Gavallan ducked back into the bedroom, eyes desperately seeking a hiding place. Under the bed? Too narrow. Behind the door? Too easy to find. In the closet? He didn't have time to find anything better. The sliding doors were half open. Five steps and he was inside. Edging into the tight space, he moved as far as he could to one side, maneuvering between neatly hung pants and shirts, jostling a golf bag. Laying his fingertips on the sliding doors, he eased them together, leaving a slim crack through which he could see the room.
The man came in first, big as a linebacker, hair cut to a jarhead's exacting specifications—high and tight with plenty of whitewalls showing. Military, Gavallan thought, spotting the caged stance, the disciplined posture. The intruder scoped the room, moving immediately to the computer.
"Tatiana," he called, then issued instructions in what Gavallan took to be Russian.
A young blond girl dashed into the room, her stride as taut as a feline's. A lioness, to be sure. What else would you call a svelte knockout wielding an automatic with a marksman's ease?
"Da, Boris," she answered.
A flash of platinum blond, the wink of gunmetal, and she was gone.
The man named Boris busied himself at Luca's desk, gathering the day trader's papers and shoving them into a plastic duffel he'd produced from his pocket, then sitting down and tapping a blizzard of instructions into Luca's PC. From his hiding place, Gavallan could just about read the windows popping onto the screen, asking Boris if he was sure he wanted to erase the files. A voice inside of him railed and grew frantic. _That's your proof he's destroying. Your evidence that Kirov manipulated the offering from the beginning, that you weren't part of the whole damned scheme_.
Gavallan found the golf clubs. Sliding a hand from the clubhead to the grip, he selected what he thought was a five-iron and deftly withdrew it from the bag. He was no longer thinking, but acting. Rationality had left him when he'd entered the house. Inching the closet door open, he found his vision framed by a fizzing red tide.
_You killed Luca and nine others_.
_You kidnapped Graf Byrnes_.
_You're going to kill him too, if you haven't already_.
Then he was out of the closet, closing the gap between himself and soldier Boris. An eye darted to the door. He could hear Tatiana rummaging through another part of the house. Cocking his wrists, he drew the golf club back, his strength coiling in his arms, his shoulders.
"Hey, Boris."
_"Da?"_
He swung as the man swiveled toward him, involuntarily holding back a fraction as the iron connected. The club struck a glancing blow, toppling Boris from the chair. Gavallan ran to the doorway, ready to deliver a like blow to the girl. Behind him, Boris was already rising, a feral groan escaping his bloodied mouth. No way, muttered Gavallan, retreating a step. Hands slick on the leather Fairway grip, he brought the club back for a second shot. Tatiana appeared in the doorway. Her gun was rising, her laser blue eyes focused on Gavallan's.
_"Nyet_ , Tanya," called Boris, waving her off. He rushed a few words in Russian that Gavallan took as a caution.
Tatiana inched toward the closet. Boris, a hand assaying his bruised jaw, held his ground next to the desk. Gavallan shifted his eyes from one side of the room to the other, from the lithe blond to the hulking thug. He felt tingly and alert and unafraid.
"You, be calm, okay?" said Boris.
"I'm fine. Why don't you two just turn around and leave. This is not your home. You shouldn't be here." His hands tightened on the club. "Just go.... I wouldn't want to hurt you."
"You, hurt us?" Boris wiped at the blood and drool leaking from his mouth. The bastard was smiling.
And then, the telephone rang, an old-fashioned jangle that in the tense silence practically blew the roof right off the house. Boris's eyes shot to the phone. Tanya shifted her head. And in that instant, Gavallan moved. Jumping forward, he drove the iron hard into the soldier's ribs.
"Boris!" screamed the girl as the flat top collapsed to a knee.
Gavallan kept the iron in motion. It rose into the air, then dove in a silver arc, the shaft striking Tanya's hands, sending the pistol pinwheeling across the carpet. The girl registered no disappointment. Planting her feet, she came out swinging. One fist darted at his head, another at his gut. Gavallan sidestepped the blows, and as the girl's momentum carried her by him, he dropped the club and drove an elbow into her back. When she rose from the floor, Gavallan had the automatic in his grasp—a Glock 9mm, he now recognized.
"Freeze," he said, one eye scanning the room for Boris. "Don't move a mus—"
The blow hit him low in the back, a kidney punch delivered with ferocious verve. He wanted to cry, but no sound escaped him. His body was paralyzed. The cords of his neck flexed, his shoulders bowed, his lips bared over screaming teeth. The whole of his being grimaced with a pain it had never known. He collapsed, first to his knees, then to his chest, his arms and hands ignoring his every reflex to cushion his fall.
**H E WASN'T SURE HOW LONG** he was unconscious. A minute. Maybe two. Boris stood by the desk, dumping the last of Ray Luca's papers into his duffel. The computer had been turned off. Tatiana kneeled close by, smelling pleasantly of lilacs and rosewater, the gun once again in her possession. Her head was tilted, and seeing his eyes open, she smiled. "Allo, Mr. Jett."
Hearing Tatiana speak, Boris abandoned his duties. "I'm sorry, sir, but we will kill you now," he said, turning toward Gavallan. "Mr. Kirov, he insists. He says to tell you, it is business only."
"You mean, 'It's only business,'" said Gavallan.
Boris shrugged. "My English is not so good as should be."
Gavallan lifted his head. Watching the blond cock the hammer and level the barrel at his forehead, he felt like a spectator to his own death. He wasn't frightened; he was too groggy for that, too fatigued by pain. He felt only disappointment, a terrible sense of letting Graf Byrnes down, of sentencing his company to an unknown fate, of allowing life to get the better of him.
_"Ray? Ray, you home? What's going on back there?"_ The voice came from inside the house. Boris whispered something to Tanya and she moved toward Gavallan. "Ray? That you?"
Gavallan opened his mouth to cry out, but at the same instant, Tatiana brought the butt of the gun crashing onto his head. The last thought to pass through his mind, even as he drifted into darkness, was that he knew the voice.
_Cate_.
_What the hell are you doing here?_
# 31
#
**G ENERAL KIROV,** some mail."
Major General Leonid Kirov glanced up from his work to see Levchenko, the department's newest probationer, advancing across his office, a small parcel wrapped in brown wax paper in one hand.
"From Belgium," Levchenko announced. He was whey-faced and chubby, more boy than man, and he was wearing the kind of sharp blue Italian suit that passed for a uniform these days among rising members of the service.
"Belgium, eh?" Kirov covered the timetables, bus schedules, and flight itineraries he had been studying, then stood and accepted the package. "What could it be, then? Chocolate? Some Flemish lace?"
He, too, was wearing a blue suit, but its boxy cut, worn serge, and frayed sleeves identified it as a trophy of Soviet tailoring. Still, the creases were razor-sharp and the jacket spotless and wrinkle-free, the result of habit, discipline, and his grandmother's three-kilo iron.
Turning the package over, he checked the franking. The postmark revealed it to have been mailed from Amsterdam the first of May, six weeks earlier. Amsterdam was, of course, in Holland, not Belgium, but he didn't feel like burdening Levchenko with the information. The caliber of probationers being what it was, Kirov supposed he should be grateful the fool hadn't thought Amsterdam in Africa.
"Sign here, General."
As Leonid Kirov scribbled his signature on the clipboard, he could not help but feel bitter and shortchanged. Twenty years earlier, the nation's top graduates had clamored to join the KGB. To say one worked for the _komitet_ gave one a prestige no amount of money could buy. No more. Enterprise, not espionage, had become the career of choice among tomorrow's leaders. Money was what mattered. The crème de la crème of Moscow University and its brethren was not impressed by a starting salary of $150 a month. Waiters at the Marriott Grand Hotel on Tverskaya Ulitsa earned more.
A last look at the deliveries prompted a sigh of disgust. Only two other names were listed on the delivery sheet. One was his own, dated two weeks earlier, signifying receipt of a reconditioned toner cartridge he'd purchased with his own money. Handing back the clipboard, he grunted his thanks. "You may go."
Levchenko gave a flaccid salute and exited the office, slamming the door behind him. Instead of firing off a rebuke, Kirov merely sighed with disgust. Very soon all this would change. Men like Levchenko would be shown the door. Fresh toner cartridges would be found in every laser printer. The Service would cast off its dusty veils and reclaim its proud birthright. And in his new mood of cautious optimism, Leonid Kirov decided the Service wasn't dead. It was just sleeping.
With a few crisp strokes, he gathered the paperwork for his upcoming trip, slipped it into his briefcase, then tucked the briefcase under his desk. Then he patted his breast pocket. The plane ticket was there. Sunday, 11 **A.M.** Novastar Flight 44. Moscow to Perm. A top-secret trip to the Arctic Circle.
Only then did Kirov's eyes return to the glossy brown parcel.
_"Lapis,"_ he whispered. Finally!
Lapis was the work name of an agent he had inserted into Philips, the Dutch electronics behemoth, three years earlier. In early May, Lapis had called in a state of high excitement. He had managed to photograph documents relating to a new eavesdropping technology Philips was developing for the Dutch Intelligence Service. Within Philips, the project was graded "eyes only," and its timely exploitation would allow his department to hack into the Dutch spy service's mainframe and read its take as if it were their own. Six weeks later, the film had arrived. Kirov couldn't help but shake his head. Gone were the days of the diplomatic pouch and emergency couriers. There was no cash in the budget for private jets or even economy-class tickets on KLM. As for commercial courier service, Federal Express had canceled its account two years back on grounds of nonpayment. These days, the Service sent and received its mail through the Russian post, like anyone else.
_Six weeks!_
A gentle shake of the package caused a small hard object to carom inside its folds. It was the film, no question. And despite his dismay, he felt a current of excitement rattle his bones. This was work, he told himself. This was the Service. Running an agent instead of worrying about copiers and toner cartridges.
Leonid Kirov had spent his entire career with the _komitet_. His postings had ranged from Brazil in the sixties to Hong Kong in the seventies, and finally to Washington, D.C., in the last tumultuous years of the regime. His specialty, then as now, was industrial espionage, and in his position as chief of FAPSI he oversaw all espionage measures implemented to advance the country's scientific and technological capabilities.
Outside, a warm sun shone down on the white birch forest that surrounded the office complex. Kirov had always enjoyed the view, finding calm and serenity in the leafy environs. Unfortunately, he could no longer see many of the trees. Dirt an inch thick coated the windows. The window washers had left with Gorbachev. Closing the blinds, he stretched on tiptoes to turn on the electric fan. He would have preferred to open the window, but that was not an option. The "empire at Yasenevo," as some of the intelligence service's detractors called the twin office blocks situated on the outskirts of Moscow, had been constructed in the late 1970s, a prefabricated concrete jigsaw puzzle once a marvel of the Brezhnev era. Soon after its completion, the foundation had mysteriously settled, leaving Kirov's tower "whiff skew," warping the steel superstructure and rendering the windows impossible to open.
Kirov benignly dismissed the shortcomings. He would gladly trade the second-rate power unable to pay its own postage for the fiercesome Soviet State responsible for the frozen windows.
Opening the top drawer, he rummaged for a letter opener. The sound of the tape's being ripped off the wax paper was like a scream in an abandoned church. He upended the package, and a neat black cartridge tumbled onto his desk. Pinching the cartridge between his fingers, he read the ASA number, and below it, written in Lapis's neat script, the actual film speed used in taking the photographs. He scribbled both figures on the corner of the newspaper. Post-its, notepads, and unruled paper were rationed commodities. A moment later he was out of his office, attacking the hallway with the no-nonsense gait of a man half his age.
At seven o'clock on a Friday evening, the building was deserted. Spying had become a nine-to-five job. Walking through the fusty corridors was like touring a ghost town. Doors to many of the offices were open. A glance inside revealed chairs tipped forward onto desks, as per regulations, carpets rolled up, occupants long gone. Some had been let go. Most had fled to the private sector, modern-day defectors.
Four flights of stairs took him to the eighth floor and photo processing. Elevators were out of service over the weekend. Power was supplied by the department's own generators, and the lifts consumed too much electricity. The chief was quick to point out that oil was priced for export and paid for in dollars.
Ah, oil, he mused. In the end, everything always comes back to oil.
He thought of the detailed model of the pump station locked in the old briefing room. He would permit himself a last look while the film was drying.
The lab was open and, like the rest of the building, unoccupied. Kirov flicked on the lights and set to work developing Lapis's film. He was happy to find the necessary chemicals in abundant supply, less so to discover only two pieces of photo paper remaining. He would use one as a proof sheet, the second for any "gems" Lapis might have turned up. There was no use being upset, he decided, reminding himself that a year ago the lab had been out of paper for three months. This was simply the result of democratization—proof positive that unfettered capitalism had no place in modern Russia.
Over the past ten years, the KGB had withered like a rose starved of water. Thirty foreign residences had been closed, staff cut by 80 percent. Typically, a foreign residency could count on a minimum of sixteen officers. Officers were assigned a particular duty, a specific "line" to manage. The PR Line officer was responsible for political, economic, and military affairs. The KR Line officer oversaw counterintelligence. The Line X officer was in charge of collecting scientific intelligence. Other officers took care of signals intelligence, harassed Soviet émigrés in the area, and kept a watchful eye on the local Soviet colony. These days a foreign residency could count itself lucky to have two officers to fulfill all these functions.
Not only had the KGB shrunk, but it had been divided into four separate and self-governing entities. The SBP, or Presidential Security Service, handled the protection of the president. The Border Guards manned the frontiers. The FSB, or Federal Security Service, made up of branches of the _komitet_ that had once repressed internal political dissent, dealt exclusively with domestic police matters. And the FIS, or Foreign Intelligence Service, carried on the job of the First Chief Directorate—namely, the gathering of intelligence designed to further Soviet foreign policy goals and the implementation of a broad range of "active measures," such as disinformation, murder, and the support of international terrorism with the goal of destabilizing the country's enemies.
Kirov could not say with any precision how large the KGB's budget had been in its glory days. Twenty billion dollars? Thirty billion? Fifty? At its height, the KGB and its operatives had numbered in the millions. He knew, however, the size of the _komitet's_ current fiscal operating budget to the penny: $33 million. Less than the combined annual salaries of a Formula 1 race car driver and a top-flight American baseball player.
Kirov bit back a covetous smile. In a matter of hours, the figure would multiply thirtyfold.
**I T HAD BEEN HIS IDEA.**
A way to get the monkey off your back, he'd told Konstantin three months earlier. A way to be free of the state's meddling. The writing was on the wall. The oligarchs were no longer to be tolerated. Look at Gusinsky and Berezovsky and all the others. Forced to trade their assets for their freedom. The favor of the state was capricious, he'd argued. It could be withdrawn as easily as it could be given.
Now it was Konstantin's turn in the hot seat. Everyone knew he'd been stealing from Novastar. Thievery was the oligarchs' acknowledged modus operandi. How long did he think he could keep Baranov at bay?
"What can I do?" Konstantin had asked over lunch at his lavish offices on the Novy Arbat on a squalid March day.
"Same as you've done before. Buy your way out."
"Impossible. Baranov's beyond reproach. Besides, I don't have the money."
"But you will."
"You're talking about Mercury?" Konstantin asked warily. "Impossible. The money's spoken for. We've got to upgrade our systems, build out the infrastructure to handle our future customer load. Routers, switches, servers, firewalls. We're almost there. I'm not the jackal you all think I am. Mercury's for real."
"Of course it is," Leonid soothed. "No one doubts your ambitions or your skills. Selling a piece of your television network to Murdoch was a coup. They still speak of it at the office. Still, younger brother, the offering _is_ for two billion dollars."
"Two billion. Hardly buys you a laptop and a modem these days."
"You're exaggerating. Spend it the right way and two billion could buy you much, much more. You'll have plenty of time to 'upgrade Mercury's infrastructure' later. Right now, I'd be more worried about my freedom. Difficult to upgrade anything from Lefortovo. No DSL there."
Konstantin's hand began to shake. "Is there something you know? Something you're not telling me?" Leonid hesitated for precisely the right amount of time. "Of course not. I'm only talking common sense. You are not invulnerable. A contribution to our well-being—to our _rebirth_ , if you will—could not be ignored."
"And you can guarantee this?" Konstantin pushed away his plate and thrust his monk's head across the table. "How?"
"The Service is not without friends. Some in very high places, I needn't remind you."
"How much?"
"Half."
"Half?" Kirov uttered the word with utter contempt. "Half? You're crazy. And you call me the greedy one."
"The first billion is ours," said Leonid, firmly, as if the decision had already been made. "The second is yours to use as you see fit. Who couldn't call you a patriot?"
"And you could guarantee that my operations remain untouched?"
When Leonid nodded, Konstantin withdrew into himself, eyes glowering at everything and nothing, one hand folded on top of the other in a pose of practiced contemplation. Finally, his head rose and he fixed Leonid with his intense, steadfast gaze.
"It's a deal," he said. "The first billion is yours."
**T WO KEYS EXISTED** to the briefing room. Kirov kept one. The other resided in a certain office in the Kremlin. Unlocking the door, he moved inside and turned on the lights. A halogen spot illuminated an angular white mountain atop a table in the center of the room. Kirov approached reverently, a pilgrim to his shrine. Slowly, with due respect, he removed the sheet, folded it, and laid it on a chair.
As always, the first sight took his breath away. The attention to detail was spectacular. The green and yellow decals with the BP logo; the small diamond-shaped warning signs reading "Danger: Flammable." Every valve turned. The miniature doors really opened. The engineers had taken an industrial complex half a mile long and a quarter of a mile wide and shrunk it down so it fit inside a conference room. It was all there: the oil reservoirs—paint chipped, metal rusting; the power plant; the pump station; the dormitories and administration buildings.
Even the terrain was accurately reproduced, noted Kirov as he circled the table. The target rested on a wide, flat expanse of concrete in the midst of a verdant meadow. Drifts of snow ranged from five to fifty feet in height, depending on the time of year. They'd built a life-size mock-up of it in Severnaya, on the southern rim of the Arctic Circle.
They were there now, training, practicing, awaiting the green light. Team 7 from Department R of the First Directorate. Former Spetsnaz men trained to fight in all weathers. He imagined them clad in white, moving over the rough terrain—white anoraks, white snowsuits, white balaclavas.
Kirov thought of the audacious plan. Soon everything would be different. Seventy-two hours until Mercury went public in New York. Seventy-two hours until the FIS _—oh, fuck it_ , he would call it what it was—until the KGB received a billion dollars into its private account. Seventy-two hours until the planes took off from Severnaya, heading east over the top of the world.
Imagining what was to come, Leonid Kirov shuddered. His brother was right: They would reserve a place for his bust in Red Square, next to Andropov and Iron Feliks. Nothing less would do for the next director of the KGB.
**H E REENTERED THE DARK ROOM** a few minutes later. The timer sounded, and he anxiously moved to the ropes of dangling film to check the negatives. Every frame was a blank, a pearly white slate, overexposed due to heat, low doses of radioactivity... there might be a hundred reasons why. Kirov chucked the worthless film into the trash bin and scowled. He'd had enough of rinsing mercury off his hands.
# 32
#
**G AVALLAN WOKE IN THE BACKSEAT** of a large car. His head was splitting, his mouth bone-dry. With a grunt, he tried to sit up. His back screamed as if gouged by a hundred razor blades. "Shit," he grunted, and fell back.
"Jett, are you all right? Does your head hurt dreadfully? Let me look at you."
Squinting at the bold sun, he made out Cate's form seated behind the wheel. He'd do it, if only to show her. One hand found an armrest, the other the ridge of the rear seat. Teeth gritted, he hauled himself to an upright position.
They were driving north toward Palm Beach along A1A, a two-lane blacktop shaded by gnarled banyans, Norfolk pines, and giant clumps of frangipani. To the right, peeking between the ornate mansions that made up the communities of Gulfstream, Oceanridge, and Manalapan, lay the Atlantic Ocean. To the left were golf courses, more homes, and the intracoastal waterway.
"Jett, who did this to you?" Cate asked, reaching a hand back, laying it to his cheek. "Did you see them?"
Gavallan brushed away her fingers. "You mean you didn't?" Despite her role as savior, she was the enemy. Someone to be distrusted, kept at arm's length.
"I found you alone in the house, lying on the floor. The bedroom window was open. I suppose they left that way."
_"They?_ How did you know there was more than one person?"
"I didn't. They... he... I was just..." She pulled up short, her features crunched into an offended grimace. "I don't suppose thanks are in order."
Gavallan eyed her suspiciously. As usual, she was dressed as if she'd been born to the place: khaki shorts, navy polo shirt, a pair of Ray-Bans hiding her eyes. Two nights ago she'd been the princess of Nob Hill. Today she was a soccer mom. He'd been quick to pick up on her chameleon's gift of adaptability, her ability to look at home in places she'd never set foot in before, to make new acquaintances feel as though they were old friends. She could talk XML with the code pounders from Sun, deliver an address on the future of the Net to an auditorium of grade-schoolers, or bandy about internal rates of return with Meg and Tony, all with equal aplomb. It was her journalist's secret weapon, and when they were dating, he'd often found himself amazed at her social dexterity. Today it made him nervous. He wasn't certain who it was driving the car.
"Thanks." He uttered the words without an ounce of gratitude.
The windows were open, and a stiff, cooling breeze swept through his hair and across his face. Closing his eyes, he inhaled deeply and was reinvigorated by the fresh, salty drafts. The throbbing of his head subsided. The rhythmic stabbing deep inside his belly eased. The pain became bearable. But the deception remained, and he decided it was far worse a companion.
"Stop the car," he said.
"What?"
"I said, 'Stop the car.'"
Cate signaled and guided the car onto the grassy shoulder. Gavallan pushed open the door and lowered himself gingerly to the ground. He had to move, to be free of their faux walnut and Naugahyde confinement. Cate came round and offered a hand, but again, he waved it away.
"Talk, damn it," he said. "Don't just stand there playing nursemaid. Talk to me. What are you doing here? You're in this every bit as deeply as I am—even more, from the looks of things. Your fax number is all over Ray Luca's correspondence. You've been feeding the Private Eye-PO his information. Why, Cate? I want to know what in the world is going on. And then I want to know why you didn't tell me before."
"I wanted to... I was worried... I don't..." She started and stopped a dozen times, groping for a place to begin. Gavallan had never seen her so flustered. All part of the act, he decided.
"Just the truth, Cate. That's all. It's not so hard."
Her features hardened as though she'd been slapped in the face. "If you saw the fax, then you know," she said. "It's about Kirov. He's a criminal—not just a man who cuts a few corners, but a gangster. He's as bad as Al Capone or John Gotti. He's been under investigation by the police for six months now. The Russian prosecutor general and the FBI are all over him. The focus of their inquiries is Novastar Airlines. Kirov took over the company for half of what it was worth and is milking it of every cent, sending its foreign revenues to his private offshore accounts."
"What about Mercury? Is the FBI looking at that too?"
"No one's looking too closely yet, but with Kirov everything's rotten. You've seen the proof. It's hardly a model of propriety."
"You mean the pictures of Mercury's Moscow Operations Center? The Cisco receipts? If the cops aren't concerned about Mercury, why are you trying to pull it down?"
"To get Kirov."
"To get Kirov?" Gavallan smirked, drunk with disbelief. "What the hell does a reporter covering the mating habits of yetis in San Francisco have to do with a Russian billionaire ten thousand miles away? Sick of being a social gadfly? Is that it, Cate? Is this your bid for the big time? Looking for a promotion to hard news? Maybe a Pulitzer? Or is sinking Black Jet what you're after? Dumping me wasn't good enough."
Cate's eyes flared. "You bastard!" She took a step toward Gavallan, raising an opened palm, then stopped, her fury reined in. "You have no idea what you're saying, how your words hurt."
But Gavallan could match neither her emotional nor her physical control. Rushing forward, he pinned her to the car, squaring his face an inch from hers. "Kirov, eh? Bullshit! You don't even know the man. What in the hell could he have done to get you on the warpath?"
"Stop it!"
Gavallan grabbed her by the arms and shook her. "Tell me."
Cate raised a defiant chin, freezing him with her eyes. "He killed a friend."
"Who?" Gavallan fired back with equal vitriol.
"Alexei," she answered, the heat draining from her voice. "He killed Alexei."
"Alexei who?"
"Alexei Kalugin. I loved him."
"Tell me about it." For the moment, he couldn't believe anything she said. _Cate the deceiver_.
"It was so long ago. Another life." She gathered herself for a moment, and when she saw that Gavallan was waiting for her to go on, she drew a deep breath. "His name was Alexei Kalugin. We met at business school. When we graduated, we both took jobs at the K Bank in Moscow. It was our big adventure; our chance to see the world. Alexei started on the trading floor. I worked in international credits, handling the American correspondent banks. After about a month it became clear to both of us that the K Bank wasn't on the up-and-up. Kirov was insisting we grant loans to companies that had no collateral, no creditworthiness whatsoever. It was crazy."
"I'll bet," said Gavallan.
Cate took off her sunglasses and tucked a strand of hair behind her ear. Her motions were clumsy, and he could sense her reticence, her confidence gone AWOL. Vulnerability was a new color for Miss Catherine Elizabeth Magnus, and to his dismay, it rendered her in a flattering light.
"After a couple of weeks, Alexei grew tight with the locals," she went on. "The traders took him under their wing. They treated him as if he were one of their own. Then, it just happened."
"What happened?" asked Gavallan.
"Alexei learned that Kirov and his crew were manipulating the market for aluminum futures. Kirov was buying the stuff from the country's smelters at something like five cents a pound and selling it on the international market at forty-five cents. We're talking major piracy."
"I'd say a markup of nine hundred percent qualifies."
"Alexei showed me what he'd found and I told him he had to go to the police. He didn't want to. He knew it would be dangerous. It was '96, remember. The oligarchs were at war with each other. Anyone who said a bad word about them ended up dead. Every day there were bodies on the street. He just wanted to quit and go back to the States. But I insisted. I held his hand, and together we went to the district attorney, or whatever you call that post in Russian. The next day, Alexei disappeared. We took the Metro to work together. He went to the first floor. I went to the fifth. We had our usual lunch date, but he never showed. They found his body on the banks of the Moskva River a week after that. He had a bullet in his head. His tongue had been cut out. I left the country the same day."
Gavallan kicked at the grass, doing his best to take it all in. He felt aghast and betrayed. Mostly he just felt enraged. Ten people had died this morning, ten precious lives that might have been saved had Cate not withheld her secret history from him. He didn't think it necessary to offer his condolences for one more person he'd never met. Stepping closer, he pointed a finger at her heart. "You worked for Kirov? You knew he's a murderer? Why didn't you tell me?"
Cate shook her head disconsolately. "What's there to say? Yes, I worked for Konstantin Kirov. Yes, I got my boyfriend killed. It's not something I care to remember. Don't be mad, Jett. I told you: It was another life."
"No!" cried Gavallan, slamming his hand against the roof. "It was _our_ life! I told you everything. The best and the worst of it. I gave you _my other life_. What makes you so special you couldn't give me yours?"
"I tried a thousand times. You weren't listening."
"The hell you say. You think if I knew that Kirov killed your boyfriend I'd have gone ahead with the deal? That if the FBI and the Russian government were checking him out, I'd have kept Mercury on the calendar? I'm sorry, ma'am, if you hold so low an opinion of me."
"Don't you be self-righteous with me. The deal's had warning signs on it since day one. You and the rest of the market were so hungry for a winner you never stopped long enough to check them out."
"Bullshit."
"It's true and you know it."
The barb pierced Gavallan, its sting all the sharper because she was right. "You want true?" he railed. "Ray Luca is dead. Nine innocent men and women are dead. None of them will be going home to their families tonight or tomorrow or ever again. All because I've continued pushing Mercury, when you knew I shouldn't have. Oh, and there's something else you ought to know: Graf Byrnes is alive. He called me after you ran out of the ball the other night. He told me the deal was good, that we could go ahead, but he made it clear Kirov had put him up to it. That's where he is right now, I imagine—locked up somewhere in Russia with a gun to his head. For all I know, he could be dead by now. Since you know Kirov so well, honey, why don't you tell me what Graf's chances are."
"Damn you," she shouted, her lips trembling, a solitary tear streaking her cheek. "You've got no right."
"Lady, I have every right. Mercury was my deal. Like it or not, I'm just as responsible as Kirov for those ten people who died today."
"I'm so sorry." The sobs came in huge waves, tremulous currents that racked her shoulders and sent shudders down her spine. Part of Gavallan demanded he comfort her, and almost instinctively, he stepped forward. But, reaching an arm toward her, he caught himself and pulled back. No, he told himself. She deserves this.
"Okay, I should have told you," she said finally. "I see it now. I didn't and I should have and I'm sorry."
"Damn right you should have," he boomed, his anger bursting like a thunderclap around them.
"I said I'm sorry. What more do you want?"
Gavallan said nothing. He felt estranged from her. He decided he'd been right—he didn't know her. Maybe he never had. And that was what hurt most.
"I didn't want to put you at risk," she said, wiping at her tears, fighting to control her breath. "I just wanted to pull down the IPO. I thought if I could stop the Mercury offering, that would be enough to get at Kirov. A man like him only cares about money."
"And Ray Luca was your helper?"
Cate nodded. "A friend at the _Journal_ went to school with him, knew about his playing the Private Eye-PO."
Gavallan turned his back and walked away a few steps. He was working the angles, trying to sift what was left of Mercury from the cinders of Cate's emotional firestorm. He kept revisiting his tour of Mercury's offices in Geneva and Kiev and Prague, seeing room after room of routing equipment, offices humming with motivated employees. Mercury had the vibe of a successful, efficiently run company, and that was something you just couldn't fake. "I saw the fax in Luca's bedroom—the one from the prosecutor general's office. It'd been sent from your home. Where did you get all your information, anyway?"
"One of the detectives who investigated Alexei's murder was part of the task force looking into Kirov's affairs. Detective Skulpin is his name. Vassily Skulpin. We both knew Kirov was behind Alexei's death, but Detective Skulpin could never gather any proof. Over the years we kept in contact, and when Skulpin's task force began to move against Kirov he let me know. Detective Skulpin was the one who told me Kirov had faked the due diligence."
Gavallan winced as if he'd been slapped. "He told you that?"
"He has an informant inside Mercury. The informant said that someone who works for Kirov was covering up its faults, painting a prettier picture than reality allowed. The only proof was the photos. And then the receipts."
Of course Kirov had faked the due diligence. If Luca's claims were true, there was no other way to have slipped it by. _Kirov faked the due diligence_.
"Look," he said. "Let's get to the hotel. I've got to pick up my things. If we hurry we can still make the three o'clock flight back home."
Cate slid behind the wheel and started the engine. They drove in silence for a minute or two, then Gavallan shot her a sidelong glance. "The hotel's just up the road, north side of Manalapan." He brought a hand to his forehead. "Oh, shit, my rental car. I left it a block away from Luca's."
"We'll pick it up later," said Cate. "Right now, let's go get your bags. The Ritz-Carlton, right?"
Gavallan rolled his eyes without humor. "Remind me to have a word with Hortensia about keeping my travel plans quiet," he said, referring to his housekeeper.
"Don't be mad at her, Jett. I called your office to apologize for my behavior at the ball. When they said you were home ill, I spoke to Hortensia. It's not fair to ask her to keep secrets from your friends."
"Yeah. Not like some people I know."
Gavallan's cell phone rang. "Hello." He listened to the man on the other end of the line rant for fifteen seconds, then covered the mouthpiece and shot Cate a sinking glance. "It's Tony. We've got problems."
# 33
#
**J ETT, ARE YOU POSSIBLY IN FLORIDA?"** Tony Llewellyn-Davies was saying. "Bruce, Meg, and I have some unannounced guests who very much would like to speak with you. The gentlemen appear to be from the FBI, and they're asking some very nasty questions about you."
Gavallan's eyes darted to Cate, then back at the road. An hour ago, the news that federal agents had invaded his office would have shocked him. Now, he took it in stride. "Tell your friends they're bang on. Say I came down here to have a word with Ray Luca and find out why he was bad-mouthing our offering. Just be sure to let them know that someone beat me to him."
"I'll relay the message, Jett." A moment passed and Llewellyn-Davies asked if he might put him on the speakerphone. Gavallan said fine. There was another pause and he pictured his friends standing around his desk, the Transamerica Tower and Golden Gate Bridge looming in the background.
"Mr. Gavallan, Special Agent Vernon McNamee of the Federal Bureau of Investigation speaking. Good day, sir."
Against his every reflex, Gavallan found himself saying "Good day" back.
McNamee said, "Sir, we'd like to speak with you about the murder of Mr. Raymond Luca and nine other individuals this morning in Delray Beach, Florida."
"Here I am. Speak."
"We'd prefer to conduct the interview in our offices. We'll be happy to explain everything to you when we meet. The field office nearest to you is in Miami. The federal building on Northwest Second Avenue."
"You want to arrest me for Ray Luca's murder? Is that it?"
"No sir," said McNamee. "I said no such thing. We'd simply like to ask you a few questions. I'm sure it will just be a formality."
"A formality?" Gavallan wondered if the team of FBI agents shaking down his office in San Francisco was also just a formality. "Agent McNamee, let me make something clear. I did not kill Ray Luca. I'll be happy to point you in the right direction, however. The man you are looking for is—" Gavallan stopped himself short. He wanted to say that Konstantin Kirov was the man responsible for Luca's and the others' deaths, and to offer a detailed description of the individuals he believed committed the crime. The first was a six-foot-four-inch male the size of a Sub-Zero refrigerator, approximately thirty-five years of age, blond hair, blue eyes, with a nose that had seen more than a few fistfights. Went by the name of Boris. The other was a woman, platinum hair, blue eyes, maybe nineteen, skinny, and feisty as a cornered bobcat. Tatiana was her name. Russians, both of them, in case McNamee hadn't caught it.
"Do you have a name you'd like to give us?" the FBI agent inquired.
"No, I'm afraid not." For the time being, Gavallan would have to keep his knowledge of Kirov's role in Luca's death, as well as his intention to cancel the Mercury deal, to himself.
"Well, then, sir, it's my duty to inform you that unless you turn yourself into local law enforcement authorities within two hours' time, we will have no option but to issue an arrest warrant on your behalf."
Gavallan drew a breath. Not good. The last place he wanted to be was locked inside a six-by-eight jail cell. "You guys still there? Listen, I want you to get on the horn to Kirov and tell him everything's copacetic with the offering. We're going ahead as planned. Understood?"
"You're sure, Jett?" It was Meg Kratzer. "Maybe it would be wiser to postpone the deal. We can reschedule it six months from now. Put Mercury on the calendar as the first big IPO of the new year."
Gavallan answered for his audience, his script penned by Konstantin Kirov's hand. "No way, Meg. Mercury's a gem. I told you what Graf said. This whole thing with the Private Eye-PO is just a terrible, terrible coincidence. Nothing more. Now, keep your chin up. Come Monday, we'll all be sitting in the Peninsula in New York drinking some bubbly and laughing about the whole thing. Except for Bruce, that is."
"What do you mean, except for me?" Tustin crowed.
"Sorry, Brucie, no children allowed in the bar. We'll be sure to send up some chocolate milk to your room."
Gavallan heard some chuckles and knew he'd won back his team's confidence.
A firm tap on the leg directed his attention to Cate. "What's going on?" she demanded. "What did Bruce say? Are the police looking for you? You didn't mean what you said about Mercury. Go on, now. Tell them what you told me. About Boris and the girl. Tell them who killed Ray."
"Shh," he said to Cate. "Give me a second." Then to McNamee: "Tell you what. You want to talk, get me one of your bosses on the phone. A Mr. Howell Dodson. He runs your task force on Russian organized crime. Name ring a bell? Find him and we can talk till we're blue in the face."
McNamee hesitated, and Gavallan could hear some discussion in the background. After ten seconds, the agent returned. "If you'll give me a minute, I'll patch him in."
"Tell him to call this number." Gavallan rattled off Cate's mobile, hoping he was making it more difficult for anyone to track him down, then hung up. In less time than it took for Cate to fire up her journalist's interrogation, her phone chirped. Gavallan slid it from her bag. "Mr. Dodson, I presume."
"Hello, Mr. Gavallan," replied a smoky Southern voice." I'm sorry to disturb your vacation. Or is it a working holiday like our other famous Texan is so fond of taking?"
"Neither, actually," replied Gavallan flatly. "I came here to speak with Ray Luca. When I learned he was the Private Eye-PO, I wanted to talk to him face-to-face and ask him why he was so intent on discrediting one of our upcoming IPOs."
"That would be Mercury Broadband, would it not?"
"That's correct." Gavallan added, "I take it you're acquainted with Mr. Kirov."
"Not as well as I'd like to be. Perhaps you could introduce us someday.
"I would enjoy meeting you, though, Mr. Gavallan. A little sit-down, just the two of us. How 'bout in an hour's time at your hotel? You're staying at the Ritz-Carlton, I believe. I'm sure you're not too far away."
About a hundred yards if you really want to know, answered Gavallan silently.
Cate had turned the Explorer down a narrow lane leading to the hotel. A pink pastel palace beckoned at the end of a manicured drive. Emerald lawns as smooth as velvet rolled from either side of the road. An imposing portico welcomed guests. Two police cars were parked beneath it, their front doors open. A few uniformed officers mingled with some stiff types whose short haircuts and inviolate posture identified them as members of the law enforcement community.
"Keep driving," Gavallan said coolly, one hand covering the phone. "We're a couple of tourists having a look around. Whatever you do, don't stop. And if they come after us, floor it."
"You're scaring me. What did Dodson say?"
"Just keep driving."
Gavallan froze in his seat, eyes to the fore, phone at his ear. But Cate handled herself as if born to a life of crime. Passing the quartet of police officers, she waved a hand and offered a cool smile, circling the portico at the same steady speed. The officers looked from Cate to Jett to Cate again, somber in their khaki rayon uniforms and Smokey the Bear hats. Tourists didn't rate a second glance, and in a moment the four were talking amongst themselves. There was a fifth man nearby, standing at once among and apart from the police officers. He was a tall, professorial man with neat brown hair and a pair of half-moon bifocals. He was wearing Clarence Darrow's seersucker suit and suede bucks, and he held a phone to his ear.
Howell Dodson. _Had to be_.
A moment later, Cate and Gavallan were through. Gavallan didn't dare look behind him for fear of what he might see. "We clear?" he asked.
Cate's eyes jumped to the rearview mirror and back, and he could see now that her smile was superglued to her teeth and that she was frightened. "We're clear," she said.
"Mr. Gavallan, you still with me?" Dodson was saying.
"I'll take a rain check, if you don't mind," said Gavallan. "For now, why don't you just call off the hounds. Sending your storm troopers into my offices really is a little much."
"I'd say it made the appropriate point. Come now, Mr. Gavallan, let's sit down like a couple of good ole boys and have ourselves a little chat. I'm sure that in no time, we'll have everything all cleared up."
Gavallan chewed on the idea. Dodson was a charming son of a bitch who sounded like he'd be at home as Robert E. Lee's aide-de-camp. The question remained, however, as to whether he would listen to good sense. Gavallan rejected the idea as too risky. Once inside a cell, there would be no way out until Monday morning. Grafton Byrnes could not wait that long.
"Let's just say I know more than I can divulge at the moment," he said. "We can call it a gentlemen's agreement. I'll tell you just as soon as I'm able. Tuesday latest."
Dodson's voice tightened. "You can do better than that. I've got ten bodies that deserve an answer, Mr. Gavallan. Now. Not Tuesday."
Cate patted Gavallan's arm. "Jett!"
"Just a second," Gavallan whispered. Then, "I'm sorry, Mr. Dodson, but that's the best I can do."
"I am trying to be civilized about this. Make no mistake, I have a nasty side. If you choose not to cooperate, I'll slap a warrant on your behind faster than you can say Strom Thurmond and we can conduct our powwow from a federal detention facility instead of a beautiful hotel."
"Believe me, I am sorry. If there were any way I could share with you what I know, I would, sir. For now, I can only say I had nothing to do with Ray Luca's murder. I saw what happened on the news and I'm as shocked by the events as you."
"Two hours, Gavallan. That's what you got to come into our Miami offices. Then we come looking for you. And I mean all of us. The United States government."
"Don't waste your time, Howell. We both know you're looking in the wrong direction. Turn ninety degrees until you're facing due east. Right out over the ocean. That's where you want to go. Catch my drift?"
"Jett!" This time Gavallan could not ignore Cate's plea. "What?" he asked, peeved.
Cate gave her head a slight nudge, behind them. Gavallan eyes fell to the side-view mirror, where a white and blue Palm Beach police cruiser had taken up position on his tail. Behind the car, he could make out a lanky figure beneath the portico charging up the stairs into the hotel.
"Just drive," he said, ending the call.
# 34
#
**F IVE MINUTES LATER,** the police car was still riding their tail.
They were doing the tourist trail, thirty miles an hour along Ocean Boulevard, past Mar-a-Lago, the old Meriwether Post estate Donald Trump purchased in 1990 and renovated to its jazz age glory, past Bethesda-by-the-Sea, the Kennedys' chapel of choice during long-ago winter visits, past the Flagler estate, Worth Avenue, and Green's Pharmacy and Luncheonette. A few billowy clouds hovered low over the ocean—"puffy white fuckers," they'd called them when he was flying.
"Jett, what do I do?" Cate's voice was pitched high, her features frozen in a brittle mask.
"Just keep going," Gavallan advised. "If he hasn't pulled us over yet, he isn't going to."
"I'm not very good at this."
"At what?"
"Running."
"We're not running. Once you see a siren and I tell you to floor it, then we'll be running."
"The police only want to talk with you," she said. "We'll give them the evidence we've gathered about Mercury and tell them the truth."
"I can't do that."
"But you're innocent."
Gavallan gave a quick, bitter laugh. "You know that and I know that. But right now, Howell Dodson isn't looking for the truth. He's looking for a suspect... _any_ suspect." He turned in his seat, wanting to engage her fully. "By eight o'clock tonight, pictures of Cornerstone and the horror of what happened there will be burned into the memories of every man, woman, and child in this country. This is the biggest case the FBI has going. They're not looking for the murderer, they're looking for meat. They need to utter the magic words, 'Suspect in custody.'"
"Dodson said he just wanted to talk," Cate persisted. "You can help them."
"Are you listening to me?" Gavallan retorted. "Haven't you heard a single word that's been said? Dodson threatened to put out an arrest warrant on me. Frankly, I can't say I blame him. You don't need to be Perry Mason to see that I've got 'prime suspect' written all over me." He counted on his fingers. "One: Seventy million dollars in fees that hinge on the successful completion of the Mercury IPO. Absent that, the fifty-million-dollar bridge loan we'll lose if the deal goes south. That's a hundred-twenty-million-dollar swing. Two: Back there in Ray Luca's house, I put my hands all over a snazzy Glock nine-millimeter that for all I know was the murder weapon. And three: I'm here, aren't I? You don't need any more than that for a conviction."
Cutting his gaze to the side-view mirror, he noted that the police car had edged closer, sniffing at their rear like a horny dog. A brown Chrysler hung behind it, and Gavallan wondered for a moment whether he had two cops on his tail. He looked at Cate. She was sitting too straight in her seat. The color had left her cheeks and a sheen of sweat clung to her forehead.
"Just cancel the deal," she said. "Tell the FBI you're pulling Mercury from the market. What more proof do they need than that? Why would you kill Luca if you were going to shutter the IPO?"
"And Graf? What about him? You may not give a good goddamn about what happens to other people, but I do."
Cate started in her seat, turning her head, raising a hand in protest. She stopped halfway there. Mouthing a silent obsenity, she sank back in her seat and locked her gaze straight ahead of them.
"It's like this," Gavallan explained in an even tone, knowing he'd gone too far. "I can't turn myself in, and I can't inform the FBI—or for that matter the SEC, the New York Stock Exchange, or anyone else—that Black Jet is going to cancel the Mercury offering. Kirov has to believe I'm playing ball. He has to think I want the deal to go through as badly as he does. That's why I told Tony to call him and tell him I was standing behind the IPO a hundred percent. That's why I said that stuff about Mercury being a gem and Ray Luca's death a bad coincidence. I'm sending Kirov a message we're on the same team. Maybe it'll keep Graf alive until I can figure out a way to get him home."
"I get it," Cate said. "I'm not sure I like it, but I get it."
"Good," said Gavallan. "Glad to hear you're with the program."
Cate crossed her arms, shooting him a stern glance. "I was always with the program. Now, instead of riding me so hard, why don't you figure out a way to get us off this island."
"I'm working on it. I'm working on it."
Gavallan looked to his left and right, exhaling loudly. He was doing his best to think clearly, to come up with a plan that would get him out from under the FBI's thumb. Sometime during the last two days, his world had been turned upside down, and he was still trying to right it. Graf Byrnes's midnight call, Ray Luca's murder, Cate's miraculous last-second appearance, and a couple of sucker punches to boot—it had all left him feeling as beat-up as a secondhand catcher's mitt.
At two o'clock on a Friday afternoon, eyes glued to the rearview mirror, his stomach in knots that at any moment the police car on his tail would hit the siren and pull him over, Jett Gavallan's emotional reserves had run dry. Grief, hope, worry—all were tapped out, and the only thing he was capable of feeling now was dread. For Graf. For himself and his company. For the whole damned world.
Inclining his head out the window, he caught a glance of himself in the mirror. He looked tired, a lined veteran of too many corporate campaigns. Thirty-eight going on sixty. Yet it wasn't the fatigue that surprised him, but the hunted look in his eyes. He appeared weak. Defeated. Once a warrior, he had been softened by a decade behind a desk, where nerve was a cocktail of figures and formulas, and risk measured in dollars, not lives.
And Graf? a fighting voice asked him. How's he faring right about now? He wouldn't be too thrilled to learn you're feeling a little long in the tooth. Get this through your head: You don't have a choice whether you're tired or not, whether you think you're up to it. Someone else is depending on you. You have an obligation. _A duty_.
The word galvanized him as no other could have, and he remembered a saying that Graf Byrnes had taught him at the Academy, words rich with sacrifice and the blood of history.
_"A man can never do more than his duty. He should never wish to do less."_
They had left the commercial center of Palm Beach and ventured into the northern residential district, where homes lay hidden behind twenty-foot stands of eugenias and gardeners needed cherry pickers to prune the trees. Parked along the curb, battered pickups loaded with lawn mowers and leaf blowers kept company with polished Rolls-Royces whose signature winged hood ornaments had been removed lest they inspire thieving minds. Gavallan wanted to make a U-turn and head for one of the bridges that led to the mainland, but he was fearful any move might be viewed as flight and make the cop want to pull him over.
"Jett!"
The police cruiser had turned on its strobes and hit them twice with its high beams. A moment later, the siren's shrill attack pierced the air.
Gavallan laid a hand on Cate's arm, swiveling in his seat to look over his shoulder. The police officer was waving them to the side. Running was out of the question. Palm Beach was an island. Three bridges linked it to the mainland and there would be a roadblock on every one before they could make it halfway across.
"Pull over," he said. "Up ahead by those hedges."
Cate edged the car to the side of the road, but a few seconds later she still hadn't slowed. He saw her looking at him uncertainly, her lips half moving; then suddenly, she spat out, "Jett, I have a gun in the car."
"What?"
"In the glove compartment. It was for protection. I was afraid of Kirov."
Opening the glove box, he lifted the pistol—a snub-nosed .38 police special—and took out the rental papers. "My God," he said, swallowing hard. "You mean business, don't you." Once the police found the gun, no amount of smooth talking would set them free. "Same goes as before. Pull over. We cooperate. 'Yes sir. No sir.' Whatever you do, don't tell them who I am. There's no way they can have a picture of me by now. We're tourists from California and we'll wing the rest. Somehow, we'll talk our way out of this."
He didn't believe it for a second.
Cate steered the Explorer off the road, braking gently as she brought the car to a halt beneath a cluster of coconut palms. But as her tires sunk into the sandy shoulder, a strange and wonderful thing happened. Instead of following them onto the embankment, the police car pulled into the center of the road and shot past, its V-8 engine growling magnificently. In a moment all that was visible was a pair of taillights flashing back and forth like the blinking eyes of a railroad crossing guard back home in the Rio Grande Valley.
Cate looked at Jett, and he looked right back at her. He was staring into her eyes, marveling at their depth, wondering, as he often had, if he would ever really know her. He continued to her nose, her lips, the swell of her neck.
_I loved you_ , he said to her silently.
A cicada's electric crescendo filled the car. It died down, and then there was only the surf rushing onto the white sand beach and the melancholy drone of a single-engine plane flying high above.
"We're free," she said, in a whisper.
"For now." Gavallan dropped his eyes, uncomfortable with his feelings for her, wanting to trust her, to lower his guard, knowing it wasn't possible. "Let's not press our luck. Let's get off this island. Better yet, let's get out of this state." He looked at his watch. "If Dodson makes good on his offer, the FBI will be checking outgoing flights up and down the coast within the hour; they probably already are. If they know I'm in Florida, we can count on their knowing how I got here and how I planned to go home."
Cate fished in the side compartment for a map. "There's an executive airport in Boca Raton," she said, spreading a multicolor canvas on her lap. "I flew in once with the guys from Redmond to cover one of Microsoft's confabs. It's got a runway long enough for business jets and a few hangars. Think we can charter a plane?"
"'We'? Where do you think _you're_ going?"
"With you."
"But I'm not going home. And I'm not going to be responsible for you."
"No one's asking you to be. I'm thirty, Jett. Last time I checked that qualified as an adult. Correct me if I'm wrong, but wasn't it you who needed looking after about an hour ago?"
Gavallan knew it was more than a question of responsibility. It was a question of trust. Cate had become an unknown commodity. Yes, she had saved his life. Even so, her presence made him antsy, aware that he was in the middle of something bigger than himself, something gray and menacing whose borders he might never discover.
"Look, you've won," he said. "Mercury's not going to come to market. Go home. And thanks. Thanks for saving my butt back there. I mean it. But that's it. This is where it ends."
"And Graf?"
"He's my problem."
"Your problem? You think you can sit there and call me uncaring, brand me with the responsibility of ten people's deaths and expect me just to forget it? I know Grafton Byrnes too. Remember? I'm proud to say that I count him as a friend. You want to be responsible for him? Fine. But you didn't know Ray Luca. And you didn't know Alexei Kalugin. Those two are mine, whether I like it or not. No matter what might happen to Kirov, I have to live with the fact that I was responsible—at least in some way—for getting them killed. You can't just pawn me off. You said it yourself: I'm in this even deeper than you are. Longer, anyway." She spent a moment studying the map. A quizzical expression skirted her features. "By the way, what do you have in mind—I mean if you're not going home, that is? Are you planning on chartering a jet to Moscow, driving up to Kirov's house, banging on his door, and asking him to give you Graf back? Do you have any idea how well-protected a man like Kirov is? He's an oligarch, for Christ's sake. The man has his own private army. The second they know you're in Moscow, they'll whisk you off the streets and stuff you in the same hole where they've put Graf. If they don't just shoot you on sight, that is. Right about now, I'd say you rank number one on Kirov's 'Most Wanted' list."
For a moment, Gavallan didn't answer. He knew well enough that he couldn't just traipse up to Kirov's door and demand his friend's return. In truth, he had no intention of going to Moscow. Securing Graf's return would require a none-too-subtle gambit of barter and blackmail, along with a fair dose of luck. He had only the rudiments of a plan, and they involved his visiting another city on the European continent. Geneva. He needed chips to sit at Kirov's table. What better place was there to get bankrolled than Switzerland?
"If your friend Skulpin's right, Kirov couldn't have faked the due diligence without the help of Silber, Goldi, and Grimm," he said. "They're the ones who visited Kirov's operations. They hired the experts to verify that Mercury's operating platform was up to snuff. They signed off that everything was a hundred percent as advertised. If something was amiss, they'd have to have seen it."
"You told me the other night you'd spoken with Jean-Jacques Pillonel and that he swore the whole thing was good as gold."
"He did."
"Okay then. At least we know where to look."
Gavallan knew the tone of voice too well. Smug, confident, unimpeachable. He couldn't deny her claims on Kirov. On a strictly practical note, it would be safer traveling in her company. The FBI was looking for a lone murderer, not a vacationing couple.
If for Graf's sake alone, he would allow her to come to Geneva with him.
Taking the map from Cate's lap, he spread it across his own. The Boca Raton airport looked to be an hour's drive. His knowledge of private airports taught him they ran the gamut from dirt landing strips with a Coke machine and a gas pump to state-of-the-art facilities equipped to assist their pilots to fly anywhere short of the moon. He was quick to assume that the Boca Raton airport, with its proximity to Palm Beach, Fort Lauderdale, and other monied suburbs of south Florida, ran to the latter variety. On the one hand, it would definitely have several planes available for charter. On the other, it'd be first in line to cooperate with the authorities should questions be asked about flight plans filed that afternoon by a certain investment banker.
Further study revealed several other private airports in the region, but Gavallan liked what Cate had said about a long runway. If they were going to Geneva, they'd require a decent-size jet: a Cessna Citation, an upper-end Lear, a Gulfstream III.
"Boca it is," he said. "Let's get moving. We've got a few stops to make before we get to the airport."
**J ETT GAVALLAN ROLLED ACROSS** the tarmac of the Boca Raton Executive Airport, a bent old man pushed along in a shiny wheelchair by a rather too attractive companion. One stop at the nearest mall had taken care of their requirements. A windbreaker, a broad-visored sun hat, and some dark glasses hid Gavallan's features. The blanket was Cate's idea. Old people got cold, she said, even when the thermometer topped eighty-eight degrees Fahrenheit and humidity was 90 percent plus. The disguise wasn't much, but it might keep the Feds off his trail if they were as eager to find him as they said.
He'd taken other precautions as well. He'd chartered the plane under a fictitious name and paid via E-cash, transferring the fees directly from his bank account to the aircraft leasing company—all before setting foot on airport grounds. He wanted as few people as possible to remember seeing them. In this at least he was successful. Their total time in transit from parking lot to tarmac was ten minutes.
Ahead lay their chariot: a white Gulfstream III with a sporty blue pinstripe running the length of the fuselage. A team of mechanics swarmed around the engines. The pilot and copilot circled the tail, completing their preflight walk-around. A fuel truck lumbered alongside, and a hose was extended to the plane's wing. The sight of the gleaming aircraft did wonders for Gavallan's bruised morale. Airplanes, of every size and vintage, never ceased to thrill him.
"She's a beaut," he said.
"She is," said Cate. "You thinking of getting behind the controls yourself? Give me a show of the Air Force's greatest talent?"
"No," he said coldly. "That part of my life's over. These days I ride just like any other paying customer."
"Maybe someday," suggested Cate.
"Maybe." Gavallan pulled down the brim of his hat to shadow his features.
They'd spent the hour's ride to the airport discussing what to do once they reached Geneva, how to approach Kirov if they were able to extract a confession from Jean-Jacques Pillonel or if by God's grace they got their hands on some material evidence of Silber, Goldi, and Grimm's fraud.
But their conversation hadn't ended there. Sometime during the drive the focus had shifted from freeing Grafton Byrnes to making Kirov pay for his crimes.
"Canceling the Mercury offering might hurt Kirov," Cate had said, "but it's not nearly enough. Not anymore. I want the man to pay. I want him to suffer for the people he's killed."
_And for stealing Black Jet_ , Gavallan added silently.
Canceling the Mercury IPO would deal his company a swift and severe blow. He could forget about the seventy million in fees. He'd have to write off the bridge loan to Kirov, worth another fifty million. And that would be that.
Two choices would be left him. He could embark on a wholesale restructuring of the firm that would require firing a few hundred employees and shuttering his London and Chicago operations. Or he could sell. He and his top executives would pocket large sums, but they would hardly be compensated for the business's true worth. And the prospect of working for another firm left him cold. Were he to leave, his core team of executives would follow, willingly or not. Neither Tony, Bruce, nor Meg fit the mold corporate behemoths demanded these days. Meg was too old. Tony's illness branded him unreliable. And Bruce... well, simply put, Bruce was an asshole. It wouldn't be a week before he'd have called the new managing director a bootlicker or an ass-kisser or God knows what, and that would be the end of Bruce.
"The only way to hurt Kirov is to put him in prison," Cate said. "Rob him of his power, his money, his position."
"Easier said than done," said Gavallan, unable to cloak his pessimism. "He's a Russian citizen. He'll never stand before an American judge to answer for Mercury—if, that is, we can even prove he meddled."
"Oh, he meddled all right. Just like he meddled with Novastar. What we need to do is nail him for stealing the hundred twenty-five million from his own country. Put him in the gulag where he belongs."
"One thing at a time, Cate. I'd say our plates are full as it is."
"I can dream, can't I?"
Cate wheeled the chair to the foot of the stairwell and helped Gavallan board the plane. It wasn't hard to adopt the gait of an older man. His lower back had stiffened and the throbbing in his head had returned with a vengeance. Still, it was impossible to deny the rush of excitement he felt as he entered the fuselage.
"So, you old codger," she said. "Where you headed?"
"Geneva. I hear there are a lot of crooks in those parts. Guess you're coming too?"
Cate stared at him over the top of her sunglasses, but when she spoke the smile had left her voice. "Wouldn't miss it for the world."
# 35
#
**G RAFTON BYRNES ROSE** at the sound of the approaching engine and shuffled to the wall. It was late afternoon. The sky was cloudy, the air growing cooler. He was sick with fever and painfully hungry. The engine meant dinner, if that was what you called a mess tin half filled with weak broth and a few skimpy vegetables. Twice a day, an old, dented truck lumbered into the clearing, delivering the same meal. Twice a day he both cursed and rejoiced. He'd never imagined how famished a man could grow in two days. How terribly, desperately hungry. The stomach did not accept maltreatment complacently. It howled, it stabbed, it cramped.
Glancing up, Byrnes noticed dark clouds gathering overhead. A drop of rain dodged what was left of the roof and caught him on the cheek. Days tended to be warm, but when the sun fell, the temperature plummeted to freezing, the wind picked up, and his teeth chattered like marble on ice. Wiping away the raindrop, he tried to imagine another night lying huddled like an animal in the corner of the shed, toes dug into the dirt, bandaged hands clenched, tucked close to his chest, left with only his trousers and Ascot Chang's finest Egyptian cotton dress shirt to fend off the cold. He began to shiver.
He knew men who'd toughed out eight years in the Hanoi Hilton. He told himself he could stand a couple of days at the Moscow Marriott, or as Konstantin Kirov had eloquently christened the place, "the dacha." Either way, it would be over soon, his freedom granted in one form or another.
He looked down at his bare feet, at the toenails clogged with dirt, at the white, defenseless flesh. "Bastards," he muttered, the shivering growing worse now. "You could have left me my socks."
The shed measured six feet by six feet and had been constructed from the slim, round corpus of birch trees. The walls rose eight feet in height. A padlock secured the door. There were no windows, but by peering through the gaps that separated one log from the next, he had a fine view of the compound. A three-room log cabin with a stone chimney and large picture windows stood a hundred feet to his right. Two smaller structures stood farther away, visible among the towering pines. One was a rotted cabin with a rickety antenna attached to its roof, the other a stone sump house with a redbrick smokestack. In his time at the dacha, Byrnes had yet to see a soul anywhere, save the grizzled man who served as his jailer.
To his left, maybe sixty feet, was another shed like his own: a storage shack, if the shards of coal and wood embedded in the dirt floor were anything to go by. A double fence surrounded the compound, twelve feet high, topped with a run of razor wire. Again he wondered why there were no guards. He stared at the fence. He guessed it was electrified. There was no better guard than twenty thousand volts of raw current.
It would be difficult to get out, Byrnes knew. Difficult, but not impossible. The real question was where he'd go once he was free. He had no money, no shoes. His clothes were tattered and bloody, his face a mess. In his present condition, he could hardly expect to walk back to Moscow.
_Difficult_... but there was a way.
A few rotting signposts stood inside the fences, and Byrnes recognized the place as a military camp of some kind. Though blindfolded during the drive out from Moscow, he'd felt the rise in elevation, especially on the last stretch of road. He could tell by the sun they'd driven north. If he had to guess, he'd say he was in an observation post, something Stalin had built in the paranoid years after the war when the Russians thought every American hiccup presaged a full-scale invasion.
The sound of the approaching motor grew louder. Byrnes's trained ear was quick to notice the smoother, richer growl of the engine. It wasn't the run-down pickup that brought him his meals every day. This was a new-model vehicle with a sturdy V-8. He listened closer. Two trucks, one engine pitched lower than the other.
Pressing his cheek to the coarse wood, he found it suddenly very hard to breathe. He'd warned himself it would happen. It was the natural course of events. He'd signaled Jett the deal was rotten. Jett had canceled the IPO. Kirov had sent his men to make good on his promise.
_Newton's Third Law_ , barked a strict voice from a long-ago classroom. _For every action, there is an equal and opposite reaction_. Or as the modern world had cynically paraphrased it: _No good deed goes unpunished_.
Byrnes stepped away from the wall and brushed the sprinkling of dirt and pine needles from his clothing. He stood a little straighter. This is how they would find him, he decided. With his pride and dignity intact.
A black Chevrolet Suburban pulled into the clearing in front of the main cabin. Doors opened and two of Kirov's troopers got out, dressed in dark suits, shirts open at the collar. Byrnes wondered whether they minted men like that in a factory. Six-feet-something, two hundred pounds of bone and muscle. The first was stocky, with a Marine's crew cut and a Slav's dark scowl. The second, who was taller and had blond hair pulled back into a ponytail, hesitated by the passenger door, then barked out a series of instructions. A moment later, he leaned into the cabin and pulled from it a thin, belligerent man, whom he chucked onto the ground kicking and screaming as if he didn't weigh anything at all. Not finished, the blond giant leaned right back in and came out with a woman, whom he threw over a shoulder and dumped a few steps away, where she lay among the pine needles, silently gathering herself.
Byrnes slid his eyes to the second SUV, of which only the hood was visible. His worry had shifted from himself to the poor wretches fifty feet away. Above the pained whimpering, he heard more voices—economical, cultured, at ease.
Konstantin Kirov appeared, dressed in a charcoal suit, a topcoat tossed over his shoulders in the manner of an Italian aristocrat to ward off the coming rain. Beside him walked a slim, dark-skinned man sporting a traffic cop's mustache and wearing a grimy houndstooth jacket. Byrnes caught the eyes—the steady, soulless gaze—and recognized the type if not the man. He was the muscle.
Kirov and his colleague took up position fifteen yards in front of Byrnes, their backs turned toward him. They stood that way for a minute or so, taut, motionless, two general officers waiting for their troops to pass in review. Another man stumbled into sight, clothes torn, nose bloodied, followed by the big-boned clone who'd shoved him.
Kirov addressed the three unfortunates in a formal voice, and Byrnes was able to pick out a phrase here and there. _"Sorry to have disturbed you." "Over quickly." "Tell the truth. You have nothing to fear."_ And finally, an absurdly polite, _"Spaseeba bolshoi."_ Thank you very much. As if these people hadn't been dragged from their homes or offices and driven to a deserted army outpost two hours outside of Moscow to answer to Kirov for their offenses, real or alleged.
Kirov ambled out of sight, and his partner took over. Immediately, the atmosphere changed, and Byrnes knew the exaggerated politeness had been for show. He had a feeling something terrible was about to happen. It was as if nature knew it, too. The soft breeze had stopped altogether. The birds ceased their incessant chatter. An uneasy stillness reigned.
"You!" shouted Kirov's friend. Byrnes pegged him as an ethnic tribesman, the kind of tough, battle-hardened man you saw on television fighting for his country against the Iraqis or the Slavs or the Russians. From his coloring, Byrnes guessed he was a Chechen.
"Name," he called.
The first man in line said, "Vyasovsky. Rem Vyasovsky."
"You are a thief?"
"No."
"A spy?"
Again, "No."
"You steal papers and give them to the police?"
The man pulled his jacket tight around him. "Of course not," he answered defiantly. "I am a clerk. This is a misunderstanding. If you want my job, you can have it. Fifty dollars a week is not enough for—"
The Chechen advanced three paces and clubbed the man viciously in the head with a ball-peen hammer. The man collapsed without a sound. The woman next to him screamed, and kept on screaming as the Chechen fell to a knee and hit him again and again with the hammer.
"Christ Almighty," murmured Byrnes, something inside him twisting in grief and bewilderment. Somehow he guessed what it was all about, that this was a show for his behalf. Slumping to the ground, he buried his face in the crook of his knees, covering his ears with his hands. Yet, he had to listen. To bear witness. To accord Kirov's victims a last measure of respect.
"Name."
"Ludmilla Kovacs."
"Position?"
"I am a secretary at Mercury Broadband. I work in the finance department for Mr. Kropotkin."
"Do you know Detective Vassily Skulpin?"
"I do not."
"Are you stealing papers from Mercury to give to Prosecutor General Baranov?"
"No." The screams were gone. In their place came crisp emotionless answers. The dialogue went on for some time, and it seemed like the Chechen was pleased with her, that she would not suffer her fellow worker's fate. Then came the horrible thud, the rushed outflow of breath, the slack, undignified thump of the body as it fell to the ground. The blows continued, merciless and mundane, and Byrnes could hear the Chechen's labored, rhythmic breathing above them, greedy, excited, ambitious.
"A ghastly business."
Byrnes jumped at the voice. Looking up, he saw Konstantin Kirov standing at the back of the shed. He was smoking one of his black cigarettes, and he looked pale and unsteady.
"A legal matter," Kirov explained. "Someone has been slipping information out of our offices, giving them to individuals unfriendly to the cause. We're adjudicating the matter in-house."
"Your questioning methods are very efficient."
"They are hardly my methods, but, yes, they are efficient. We can't be certain which of the three stole the information, only that it was one of them. People are so adept at lying these days."
"So you kill them all," said Byrnes without irony. "Clever."
Kirov paid the remark no heed and went on smoking. "Would it surprise you to know that I was once in a position similar to yours? Mr. Dashamirov recruited me in the same manner. More roughly, actually. He put a bullet in my best friend's head, then asked if I wanted the same."
"Is that why I'm here? For recruitment?"
"We're long past recruitment. 'Retirement' might be a more appropriate word."
Again, Byrnes was left to wonder why the deal hadn't been canceled. He was certain Jett had understood his message. He'd heard it in his voice. It came to him that Gavallan had to have a reason not to have canceled the deal, and that he, Grafton Byrnes, might be it. He looked over his shoulder. The woman, Kovacs, lay motionless in the dirt, her blond hair matted with blood. He knew what lay in store, if not today, then soon.
"Doing business in this country's so damned difficult," Kirov complained, dropping his cigarette to the ground, grinding it with the tip of his shoe. "You think I _want_ to be Mr. Dashamirov's partner? I have no choice. What do you think would happen if I gave up? Would Mercury exist? No. Two million legitimate subscribers would lose their connection to the world. Thousands of intelligent men and women would be out of a job. And Russia? What about it? Have you thought what might happen to my country if I threw in the towel just because of Mr. Dashamirov's unsavory methods? Would my country have independent television? Unbiased journalism? The answer is no. It is a question of priorities. Of recognizing what is achievable and doing the necessary to see it through. Of rolling up your sleeves and getting a little dirty in the kitchen."
"Of the greater good?" Byrnes offered.
"Yes, damn it, the greater..." Kirov stopped mid-sentence. His eyes burned with a fervor, an inner fire Byrnes had never seen. More than ever he looked like a crazed monk. "It is too bad you will not see it come to pass."
The whip-crack explosion of three heavy-caliber bullets fired in close succession snapped Byrnes to attention. Glancing over his shoulder, he made out Dashamirov holstering a pistol as he stepped over the corpses. The coup de grace had been administered. Kirov's spy was no longer.
Grafton Byrnes watched Kirov rejoin his partner. After a few words, the two disappeared from sight. An engine fired and one of the vehicles departed. Sickened, Byrnes wondered why he was still alive. The answer came at once. _He still needs you_.
Time passed in strange fits and spurts, and Byrnes knew his fever was worsening. He sat and watched as one after another the corpses were picked up and carried to the stone sump house across the compound. After a time, he heard the muted, regular fall of an ax. Smoke began to course from the chimney. The scent reached him, and he retched.
Sometime later, the second Suburban drove away.
**I T WAS NIGHT** when the van carrying his food arrived. A steady rain pattered the roof, sliding with ease between the irregular birch boughs and making the floor a muddy hell.
Curled into a ball, Byrnes lay in a corner, moaning. As his jailer opened the door, Byrnes moaned louder. "Doctor," he said several times. The jailer set the mess tin on the ground and relocked the padlock with nary a second's hesitation. But Byrnes was sure he'd heard the words, sure he'd noticed him. In the morning when he returned, he would find the prisoner in a similar position. And the next evening, too.
By then, Byrnes would be ready.
# 36
#
**H OWELL DODSON WAS NOT HAPPY** to be in Florida at six o'clock on a Friday evening. His daughter Renee's soft-ball game had begun a half hour ago, and at this very moment he'd hoped to be seated in the bleachers next to his wife, chomping on popcorn, swilling a Coke, and yelling his lungs out for his little girl to belt one over the left field fence. He'd promised her he wouldn't miss the game, and each day this week before he went to work, she'd reminded him of his obligation. _Friday night at seven-thirty, Daddy. It's the league playoffs. You have to come_. In fact, he hadn't just promised to come—he'd sworn it. Cross his heart and hope to die. This was one game the Bureau would not interfere with. And goddamn it, until ten o'clock that morning, he'd had every intention of attending. Until a cold-blooded killer had stormed into Cornerstone Trading in Delray Beach, Florida, and massacred ten innocent people, Howell Dodson would have broken legs to see the game.
"It's all right, Dad," Renee had said when he'd called earlier to tell her he would not be able to make the game. "I know you wanted to come. That's what's important."
"Hit a homer for me, will ya, slugger?"
"Sure thing. I'll try for two even."
Hanging up the phone, Dodson struggled to come to grips with her newfound maturity. When had his little girl grown up on him? When had she become possessed of such poise and understanding? When had she stopped needing him to cheer for her?
Dodson's temporary office was located in a small room in the basement of the Miami-Dade Federal Building. There was a metal desk, a clerk's rolling chair, and a sagging love seat done in transparent plastic slipcovers. The sole artwork came from the U.S. Government Printing Office: a copy of the most recent "Ten Most Wanted" circular.
Standing, Dodson moved to the door, smoothing his blue and white seersucker suit, appraising the knot of his yellow paisley necktie, as if checking that his uniform was presentable for inspection. He looked onto a large, open linoleum floor that might have welcomed the smaller, less prestigious variety of convention. Chiropractors, roofers, or morticians. Desks and chairs were being set up on the double. A man passed carrying a chalkboard. Another labored beneath a half-dozen cases of Coke. Behind him followed a woman with grocery bags full of juice, cookies, and tissues. In an hour or two, the first of the victims' relatives would arrive for questioning. Pinching the bridge of his nose, Dodson sighed. It would be a long and painful night.
From afar, he spotted Roy DiGenovese storming across the floor, dodging a pushcart loaded with potted plants. His eyes were bright, his olive cheeks flushed with excitement. Since Dodson's appointment as director of the Cornerstone investigation, DiGenovese had been more gung ho than usual, almost dangerously so.
"What is it, Roy?" called Dodson. "You look about ready to burst."
"We got Gavallan's prints from the Pentagon. There's a ninety percent probability they match the partials we took from the golf club in Luca's bedroom, as well as the smudges on the closet door. The lab's still comparing them against the prints found at Cornerstone. Nothing yet."
"Lucky for us he's a vet. Always handy to have a suspect's prints on file. Has the Air Force sent us over a copy of his records yet?"
"Due in twenty-four hours."
"Good news." Dodson motioned the younger agent into his office and shut the door behind them. "What about the blood in the house?"
DiGenovese pulled a spiral notepad from his jacket pocket, flipping back a couple of pages. "Gavallan's O-positive. The stuff on the floor is AB-negative."
"How recent?"
"Very. The samples were hardly dry when they collected it. Three hours tops."
"And Luca's blood type?"
"O-positive, too."
"Got sex?"
"Still checking. Preliminary DNA's due by nine."
"What about the acetate test on the murder weapon?"
As Lieutenant Amoro of the Delray Beach P.D. had been so kind to point out, it was nearly impossible to completely erase a weapon's serial number. While the numbers could be filed down so that the human eye could not see them, an overnight's bath in a sodium acetate wash often brought out the latent stamping sufficiently to be identified by an infrared scan.
"Started an hour ago," said DiGenovese. "But I got something better." He was bouncing on his toes, the muscles in his jaw flexing. Yes sir, young Roy was worked up over something.
"Better? Working your magic again, Agent DiGenovese?"
"J45198890," said DiGenovese, reading from his notebook.
"What's that? My tax ID number? The IRS looking for me again?"
"No sir. It's the serial number of Gavallan's Glock."
"The wh—" The words plummeted from Dodson's mouth, a sudden and debilitating disorientation coming over him. "Mr. Gavallan owns a Glock?"
"A Glock 17; a nine millimeter with an extrawide stock to hold seventeen rounds."
Circling the desk, Dodson collapsed in his clerk's chair. "Would I be correct in guessing that's the same model and type of weapon we've got in the lab right now?"
"You would be correct indeed, sir."
Dodson smiled weakly. He knew he was being made fun of, but he didn't feel like sharing in the humor. DiGenovese's revelation didn't just startle him, it forced him to see the case in an entirely new light, to reblock his compass and find a new true north.
Cornerstone had all the hallmarks of a professional job: the marksmanship, the blind entry and exit (meaning no witnesses saw anyone enter or leave the building at the time of the shooting), the speed with which the job was done. The whole thing was too neat. The powder tests on Luca's hand had come back inconclusive. There was residue on his fingers, but not enough for him to have fired the weapon ten times. No one was taking it for a suicide rampage, a day trader run amok. It was bigger than that, a premeditated homicide to be sure.
_But Gavallan?_
Simply put, the man did not fit the profile of a professional killer. The very idea that he possessed the training, the sangfroid, to enter a building and methodically shoot ten innocent people was absurd. A panorama of the bloodshed inside Cornerstone played out in Dodson's mind, and he shuddered. Only a monster could commit that kind of atrocity.
Yet there was Gavallan at the scene of the crime not forty-eight hours after he'd proclaimed himself interested in shutting up the Private Eye-PO forever. And there he was again at Ray Luca's house, leaving behind another man or woman's blood. And now, it turned out he owned a gun identical to that used in the crime.
"Sir, I don't make it my business to try to piece together everything that happened," said DiGenovese, sliding forward on his seat, eyes narrowing. "All I know is that everywhere we look we find Gavallan hiding. He's working in close association with an oligarch, a guy he's gotta know is a gangster. It's a matter of professional life or death if the Mercury deal goes through. I mean, come on, do you think he paid Jason Vann a hundred grand to track down the Private Eye-PO just to talk to him? No sir. Gavallan shelled out that kind of dough because he wanted Luca's mouth closed. And pronto. He wanted the Private Eye-PO dead. Even if he didn't pull the trigger himself, he made it happen. And now he's running."
At some point in his short speech, DiGenovese's tone had shifted from exposition to accusation... and it was Howell Dodson he was accusing.
"That he is, Roy. Full points on that one. And don't you worry, we're going to find him. The question remains whether he's running because he's innocent or because he's guilty."
Dodson had known plenty of innocent men and women who'd refused to cooperate with the authorities for one reason or another. Weak nerves, ingrained distrust of authority, fear of the police, the advice of their friends... the list went on and on. Of course, there was a difference between merely being tight-lipped and going on the lam. The former was not a reliable indicator of guilt. The latter was.
"Roy, I want you on a plane to San Francisco tonight. First thing in the morning, I want you inside Gavallan's house looking for that gun. Don't you worry about breaking and entering—I'll get you a warrant in plenty of time. If you can't get a commercial flight, we'll fire up one of the Bureau's Lears for you. This is key, Roy. You search that house from top to bottom, ya hear? If that gun is there, you find it. Now, shoo! Get on out of here."
**A FTER DIGENOVESE HAD LEFT,** Dodson slid the phone nearer and stared at the Post-it bearing two ten-digit phone numbers that lay next to it. Plucking free the paper, he dialed first one, then the other number. Each time a recorded message informed him the desired subscriber could not be reached. Typical, thought Dodson, for a banker to carry two numbers and answer neither.
Fishing a rubber band out of a side drawer, Dodson began twirling it around his fingers. Forward and back. Forward and back. He asked himself where he would run if he were Gavallan. Home to San Francisco? New York? Overseas? The man had offices in Chicago, London, and Hong Kong, or had he gone to ground in Florida instead? Dodson had the firm impression Gavallan was on the move, that he had an agenda of his own that called for more than eluding the authorities. Whatever it was, Dodson had to give Gavallan one thing: He was a slippery fish.
As of five that afternoon, Dodson and his men had the state of Florida sewn up tight. Partnering with the Florida State Bureau of Investigation, the Coast Guard, municipal police departments, and county sheriffs, Dodson had contacted every airport, harbor, marina, bus station, and train terminal in the state. Faxes were sent to hotels up and down the coast. When agents could not visit a site themselves, a description of the suspect was faxed and a phone call made to get across the urgency of the request.
The phone rang and Dodson answered. "Well, well, Mr. Chupik," he said, recognizing the cocksure voice. Lyle Chupik was the three-hundred-pound, ponytailed, Yoo-Hoo–swilling techie who ran the FBI's computer surveillance lab. "What a surprise."
"I got a track on one of those phone numbers you gave me this afternoon," said Chupik. "A call was placed at two-thirty to a Coastal Aviation in Fort Lauderdale."
"I see you're atoning for your sins."
"If that's atoning, what I'm about to give you ought to send me straight to heaven, and I mean directly to St. Peter. The front of the line."
"Do tell."
"The number we got a nibble on didn't belong to Gavallan."
"It didn't?" Dodson examined the phone numbers. Both had a 415 area code, and the prefixes that followed them were similar. He'd assumed they belonged to Gavallan. "Go on."
"The number belongs to another one of our Daisy taps. A Ms. Catherine Elizabeth Magnus. Ring a bell?"
"I confess I hear a wee tinkling," said Dodson, as a deadly voice inside intoned, _Enter the third murderer_.
"Anyway, that number's connected to a pretty decent phone," Chupik continued. "Kind of a hot rod. It's a WAP device—a wireless assisted protocol. Third-generation equipment. It can send and receive E-mail, as well as download pages from the web. I had the NSA send over the latest Daisy downloads attributed to that phone number. Usually, they sift it for the keywords we give them before sending it over, but I got it raw. This is what I found. At two thirty-two Eastern Daylight Time, the number logged onto a cash transfer site on the Net. Quickpay.com. At two thirty-five, the user ordered sixty-five thousand dollars transferred from an account at the Bank of America in San Francisco to an account at Florida Commerce Bank. The beneficiary was Coastal Aviation."
"And the sender?"
"Drumroll, please.... Mr. John J. Gavallan."
Dodson's stomach tumbled. "Bless your soul, Mr. Chupik. I'll mention your name to St. Peter tonight in my prayers."
"Actually, I'd prefer if you'd mention it to my supervisor. I'm kind of sick of being a GS-15. Time I moved up a notch. You wouldn't want to lose me to the private sector."
"Rather to Satan himself."
**I T TOOK ANOTHER HOUR** for Dodson to put the rest of the pieces together.
While his assistants confirmed that Catherine Magnus had indeed arrived in West Palm Beach that morning—via an American Airlines red-eye, making stops in Las Vegas and Chicago—Dodson contacted Coastal Aviation. They were quick to report that they had, in fact, set up a private charter that afternoon, but neither the names Gavallan nor Magnus appeared on their manifest. The plane in question, a Gulfstream III, was chartered by an elderly man and his nurse. The flight plan called for a leg to Teterboro, New Jersey, then a transcontinental leg to Los Angeles.
"I'm sorry if my knowledge of business jets isn't as up to date as it should be," Dodson had said politely to the desk man at Coastal Aviation. "What is the range of a Gulfstream III?"
"About four thousand miles. But this one's got an extra fuel tank. It can go six thousand easy."
"Pray tell, did the elderly gentleman in question—"
"His name's Dodson, just like you."
Dodson bit back an expletive. He did not abide smart alecks. "Did _Mr. Dodson_ request that the plane be fully fueled?"
"Sure did. Said he was picking up his son in Jersey and didn't want to hang around very long. Funny thing is, he's already half an hour overdue."
"He is?"
"Plane took off at three-fifteen sharp, should have landed at seven latest. This Dodson fella's not a relative of yours, is he?"
"No," said the real Mr. Dodson. "You can rest assured he is not."
**T HE MAP WAS ANCIENT,** circa 1989, a moth-riven relic five feet wide and four feet tall dug up from a closet in the research library on the third floor. Politically, it was obsolete. Myanmar was called Burma. Germany was still two countries. And the Soviet Union was a single rose-colored mass spanning eleven time zones. But Howell Dodson couldn't care less about what belonged to whom, whether Ingushetia was shown as independent or if the Panama Canal was denoted as American territory. All that mattered to his fevered brain was that the map be geographically accurate, and it was.
Leaning over the map, Dodson spread a yardstick in a line from Fort Lauderdale, seeing just how far his six thousand miles would take him. He fanned the yardstick from north to south and east to west, from Alaska to South Africa. Six thousand miles was a long distance, he discovered, and gave a man plenty of places in which to hide.
"By God, he's gone AWOL on us," Dodson whispered to the team of stern, clean-cut agents who had been assigned his acolytes. "Mr. Gavallan's taken a flier on the FBI. I understand if he didn't want to meet me at his hotel. I can see how he wouldn't want to come into our offices right away. I'm not an unreasonable man. But damn it, when a United States citizen flees the country while being sought for questioning in connection with a multiple homicide, that's just wrong. Get me Pierre Dupuis at Interpol. Then get me Yuri Baranov in Russia." Something inside Dodson cracked, and he felt a flash of anger, as white and hot as lightning. "Oh, fuck, get me Crawford at Langley, as well. I suppose they should know about it too." He looked at the eager faces staring at him. "It's time we run along and see the judge upstairs about issuing that arrest warrant."
Howell Dodson would teach Mr. John J. Gavallan not to toy with the United States government.
# 37
#
**F LORIDA HAD DISAPPEARED HOURS AGO,** a tobacco brown smudge swallowed by an azure sea. Distance, darkness, and the pleasant hum of a pressurized cabin relegated Gavallan's worries to another world. Ray Luca wasn't dead. Boris and his blond girlfriend were figments of his imagination. And Howell Dodson and baying hounds were no longer nipping at his heels. Not for the moment, anyway. Flying north by northeast at a speed of 500 knots and an altitude of 42,000 feet, Gavallan's greatest threat lay five feet away, tucked beneath the sheets of a foldout bed.
_Why are you here?_ he asked Cate's sleeping countenance. _Why did you follow me to Florida when you could have phoned just as easily? What else is there you're not telling me about your other life?_ And finally: _Who are you really?_
Rising, he stepped across the cabin and adjusted the powder blue blanket so that it covered her shoulders. Cate stirred, turning onto her side and bringing her knees closer to her chest. A comma of blue black hair fell across a cheek. Her pale, generous lips parted. The cabin lights were dimmed, the door to the cockpit shut. They were in the otherworld of flight, and the sandpaper silence granted her an immunity he was not willing to extend himself.
God, how he wanted to draw back the sheets and crawl into bed next to her, to run his hands up the hard ridges of her back, to slide them around and cup her breast, to kiss that neck, that wonderfully warm and silk-soft neck, to feel her nipple harden beneath his thumb.
_But she doesn't love you anymore. Maybe she never did_.
After years of his not feeling a thing, she'd awoken the dead part of him. She'd made his nerves tingle and his heart dance. She'd made him smile at odd moments. Mostly, she'd given him hope.
And then she'd taken it all away. Like that. In the snap of a finger.
**T HEY'D MET THREE YEARS EARLIER** at an I-bankers' conference one of the big firms had sponsored at the Four Seasons on Maui, this one to chart the Internet's boundless future. It was a lavish shindig. Suites for the big shots, ocean views for everyone else. Unlimited cocktails at the hotel's numerous bars. Breakfast buffets, whale-watching sails, excursions to the neighboring isles of Molokai and Lanai. Thrown in for respectability's sake were a few lectures by industry specialists on topics of burning import, all to be concluded by 11 **A.M.** sharp lest someone miss his or her tee time at Kapalua or the jitney into Lahaina.
The conference wasn't his style: all the glad-handing, everyone so buddy-buddy, patting each other on the back when the day before they'd been vowing to rip out the other guy's guts. It was an exercise in ass-kissing, all expenses paid. Like it or not, though, it was a great way to build the name, to fly the firm's banner where all the big shots could see it.
Gavallan had come to give a talk on the banker's role in preparing start-ups for their IPO. The few stalwarts who caught his 9 **A.M.** speech managed to laugh in the right places, even if it did cause their booze-soaked noggins to ring like the Liberty Bell. Cate was there to deliver a speech on the social ramifications of the Internet, and you can bet not one of the attendees missed her early-morning presentation. She strode to the dais wearing a flowered Hawaiian halter atop a blood red sarong, a white gardenia tucked behind one ear. Her feet were as bare as her midriff. And yes, she'd dared to wear her navel ring.
Today, Gavallan reflected, the outfit would have caused an uproar. Too sexy, too provocative, too disrespectful by half. But this was before the correction. The Nasdaq was making new highs every day. The Dow was puffing like the eighty-year-old geezer it was to keep up. Funding was flowing from venture capitalists like champagne from an excited bottle. This was a celebration of the new economy. A toast to the little engine that could. Graham and Dodd were dead and good riddance to the old blowhards! In short, it was as close to pure bacchanal as Wall Street was ever going to get, in this or any other lifetime.
He'd spotted her by the beach bar the afternoon after she'd given her speech. She'd exchanged her halter and sarong for a black string bikini, and ditched the gardenia in favor of a cycling cap advertising Cinzano. He'd come out of the surf after a mile swim and was still dripping.
"Liked your talk this morning," he'd said, leaning against the bar and asking for a beer. "You're a real believer."
"In the Net, absolutely. In these prices, I'm not so sure. What's your take on things? Is the market really going to keep going up, up, up?"
"For now," he said, seeking out her eyes. "Lot of money on the sidelines waiting to join the parade."
Turning toward him, Cate propped her elbows on the bar and leaned back. "A hundred fifty times earnings is pretty hard to support in the long run, don't you think?"
"Shh!" he said, bringing a finger to his lips. "Trying to upset the apple cart or what?"
"Just saying that reality always catches up to speculation." Cate stole the wedge of lime from Gavallan's beer and bit it between her teeth.
"Not too soon, let's hope. Besides, I didn't hear you mention anything about speculation up there on the podium. 'The Internet is going to radically redefine human existence.' Aren't those your words?"
"Wow. A listener. I'm impressed. You must have been the only guy who wasn't staring at my boobs."
Gavallan choked on his beer, laughing while stumbling back a few steps. "Not necessarily."
"Oh?" Her voice sounded distressed, but her smile confessed her pleasure.
"Just remember, Miss Magnus, when you make money, it's called investing. When you lose it, it's speculation."
"I'll keep that in mind, Mr. Gavallan. At least you don't have to worry as much as the others. You're not a gold digger. Not yet, anyway."
"What's that supposed to mean?"
"It means you've got some common sense." She grinned. "You want the B-school verdict?"
"Why not?"
Cate drew a deep breath. "It means that you alone among your peers have demonstrated prescience and restraint in selecting and bringing to market only those companies whose products not only have a sustainable competitive advantage but whose business models promise long-term profitability." She wagged a finger for him to come closer and, when he did, whispered in his ear. "You know how to separate the pyrite from the gold."
Gavallan backed away, his expression bemused yet appreciative. "Sorry if I'm staring. I didn't know Michael Porter had such a nice ass."
"I pay Professor Porter royalties."
"Can I buy you a drink?"
"Sure. But that means you'll have to take me away from this slum," she said. "At the hotel, everything's comped. I know a decent place in Kahului. A hole in the wall where the windsurfers hang out. You eat meat, don't you? They have great burgers."
Gavallan took the question as an affront to his dignity. "Where I come from in Texas, them's fighting words."
"I know," she winked. "I read the article in _Fortune_. Meet me in the lobby at seven."
**T HEY FEASTED ON CHEESEBURGERS** and mai tais and promised not to say one word about the market. They talked about diving and sailing and designer tequila, consciously steering away from the other's past or anything more frothy than their horoscopes—he was a Scorpio, she a Leo—and their favorite movies—his was _Bridge on the River Kwai_ , hers _Anastasia_. He stuck fifty cents in the jukebox to hear Junior Brown "a-pickin' and a-grinnin'," and she protested, saying that they didn't have any of Pearl Jam's greatest hits. He said that if he hadn't gone into finance, he would have chosen forestry. She lied as adeptly, saying that greeting cards were her secret passion and that journalism just paid the bills.
Before long, they'd broken their promises and she was telling him about her teenage years—high school at Choate, college at Georgetown, business school at Wharton. Her father was in international business, her mother had passed away years ago. He told her about school in Brownsville, about being one of twenty-four Anglos in a graduating class of eight hundred, about thinking he was Mexican until he was six and went home crying to his mother and demanding to know why his hair wasn't black like everyone else's.
Afterward, the two climbed into Gavallan's Jeep for a drive up the Hana Road. She wasn't the only one who knew their way around Maui. Half an hour later, he pulled into a drive-by just past Hoolawa Bridge.
"Come here," he said, running round the Jeep and offering a hand down. "Five minutes to the most beautiful spot on God's green earth."
Cate regarded the trail before them. A dense tropical canopy obscured the path ten yards in. "'And they were never found again,'" she said, shaking her hand free and setting off into the jungle.
The path led up a steep hill, following the course of a tumbling stream. Cate's pace soon slowed and Gavallan took the lead, careful to point out the exposed banyan roots and moss-covered stones that one could trip or stumble on. Though the night was cool, both were soon covered in a light sweat.
"I thought you said five minutes?" Cate asked, stopping and placing her hands on her hips, her breath coming hard.
"Okay, maybe ten. But we're almost there. Fifty yards max." Gavallan brushed back a smattering of low-hanging vines, praying he was on the right trail and that he could find the way back to the Jeep. No sooner had he rounded the next bend than he came upon it: a wide pool fed by a crescent-shaped waterfall that dropped from a cliff twenty feet above. A half moon shone high in the sky, and its reflection was caught in the pool's obsidian calm.
"It's beautiful." Cate stood at the edge of the water, her arms wrapped around herself. "Should we go in?"
"If a little mountain water doesn't scare a city girl, why not?" Gavallan bent low and stuck a hand in the water. _Fur-eezing!_ The stream was fed from the summit of Haleakala, elevation 10,500 feet. Him and his big mouth. "It's great," he said, even as he suppressed a shiver. "Not bad at all."
Cate stepped closer to him, her hands rising to her neck to untie her dress. Suddenly, she stopped and fixed him with a wary grin. "You didn't bring me up here to seduce me, did you? I mean, you don't really think I'd sleep with you on our first date?"
"Of course not... er, um, uh... well, I am an optimist."
"I like optimists," she said, dropping her hand from her dress and running it along his chest. "Tell you what: Think positive and I'll let you buy me dinner tomorrow, too. Deal?"
"Deal."
Then, kicking off a zori, she dipped a toe into the pond. "Not too bad at all. Enjoy yourself."
And before he could ask what she meant, she pushed him into the pond, shorts, T-shirt, Top-Siders, and all.
**G AVALLAN STARED DOWN** at her sleeping form, remembering the moment. _Three years_.
Just then, Cate opened her eyes. "Hello," she said sleepily.
"You never told me why you said no."
"Sorry?"
"You never gave me a reason."
She sat up stiffly. "I didn't know I had to."
"You don't," he said. "But I'm asking you to. It's time we were honest with each other."
Cate threw him a suspicious look. Pulling off the sheets, she rose and walked past him to the bulkhead counter where the first officer had set up an urn of coffee and laid out some cellophane-wrapped sandwiches. "Tuna... chicken salad—what do you feel like?"
Gavallan padded across the cabin, taking up position at her shoulder. "Cate."
She turned to face him. "It just wasn't right."
"What wasn't right? We didn't get along? We weren't okay in bed? We didn't like the same movies? You liked Chinese, I liked Indian? _What_ exactly?"
Cate started to say something, but caught herself. Frowning, she shook her head as if to say, "No, no, I'm not playing this game," then brushed past him.
"Cate, I'm talking to you."
"Excuse me?" she asked, snapping her head. "I don't recall being one of your employees. You can't order me to talk back. Just leave it, okay?"
And shooting him a dismissive glance, she continued to her seat, making a derisive noise on the way, a frustrated exhalation that sounded like a tire bleeding air.
It was the look that did it—the condescending way she had of averting her eyes, of showing him the back of her hand as if warding off an autograph seeker. Until then, he had kept his cool. It had been a difficult day—the hardest he'd known since the war. Neither of them needed another spate of accusations, rebukes, or recriminations. Then she gave him that look, and his calm was a thing of the past. Lost like a candy wrapper out the window of a speeding car. His heart pounded. Blood raged in his ears. Grabbing Cate by the shoulder, he spun her around and looked her straight in the face.
"Stop ducking the question. It's been three months. You think a day goes by without my wondering what happened? What I might have done wrong? I mean, one night you're lying in my arms talking about what's-his-name at the _Journal_ , your editor, and what you're going to write for your next column, the next you're gone. The house is empty. Closets bare. Bathroom deserted. Not a sign you'd ever been there. You even took that chunk of wheatgrass out of the refrigerator. We're not talking a civilized separation here. We're talking a 'scorched earth' retreat. You're damned right I want to know. It's the least I deserve. What happened, Cate? Did you meet someone? Is that it? Just tell me. At least then I'd understand."
"No," she replied coolly. "I didn't meet anyone. It's not that, Jett."
"Then what?"
"It just wasn't going to work. Maybe I could see something you couldn't. It was too painful for me to hang around, so I left."
"That's no answer."
"Oh, Jett, grow up. Stop thinking you're so goddamned special that a girl wouldn't ever dare walk away from you. It didn't work and I left. That's it. Just leave it, okay?"
"No, I won't. I'm sick of leaving things. I'm past closing my eyes and pretending it didn't happen. I need an answer. Do you understand that? _I need it."_ He touched a hand to his chest. "In here. For me."
Cate stared at him for several seconds without answering. He'd surprised her. He could see it. Maybe she didn't want to see how much she'd torn him up, but that was all part of it. He was through hiding his feelings. Abandoning her hostile stance, she leaned forward and put a firm, unemotional hand on his arm. "Jett, we had three good years. Three _great_ years. But they're over. We both have to go on. It's as simple as that."
Gavallan covered her hand with his. "But they're not supposed to be over. We were supposed to be with each other for the rest of our lives."
Her composure left her in stages, like ice slowly melting. She lowered her eyes, and he could see her lip trembling. She began to shake her head. She looked up once, trying to say something. She got out one word—"damn"—and that was it. The tears broke, and after a second she put her head on Gavallan's chest and let them come.
"Just leave it, Jett," she whispered throatily, catching her breath. "Please, just leave it. For me."
Gavallan put his arms around her and hugged her. Okay. He would leave it. For now. _For her_. He hoped that someday she would tell him. But with sadness, he realized it would have to be on her own time, and of her own will.
He helped Cate to her seat, then kneeled and looked out the window. An orange scythe slit the horizon. He checked his watch. It was midnight Eastern Daylight Time, or 6 **A.M.** in Geneva. Their flight plan had taken them northeast from Boca Raton over the Atlantic, past Bermuda, then east toward the European continent where the sun was already rising. In an hour they'd cross the southernmost tip of Ireland, then continue over England and France, entering Swiss airspace from the northwest.
"You think he'll be there?" she asked, eyes glued to the wondrous sight of dawn's approach.
"Pillonel? Yeah, I think so. He's got a place outside of town where he grows his own grapes. Each year he sends over a case of his wine as a Christmas present. Not bad stuff. Anyway, he's always going on about coming out to visit his winery. I figure if it's decent weather, odds are he'll be playing the grand vintner."
"What makes you think he'll talk to us?"
"I can be persuasive when I have to be. Besides, we've got plenty of help. Luca's last letter and that fax to the FBI won't hurt. A guy like Pillonel's got a heck of a lot to lose if he gets caught. He's got to be feeling a little nervous already."
"And you'll play on his guilty conscience?"
"Yeah. And if that doesn't work, I'll beat the living tar out of him."
"Ah, a diplomat."
Gavallan bridled at her dismissive tone. In case she'd forgotten, they'd passed diplomacy a ways back, somewhere after Graf Byrnes had been kidnapped and before Ray Luca had taken a bullet in the head. "Whatever works."
"You sound like Alexei."
"Ah, the mysterious Alexei."
"You're mad I never told you?"
"Shouldn't I be?"
Cate glanced up, her eyes red and swollen. "You can be mad, but don't be unkind. I don't want to cry again for a month."
"I'm sorry."
Cate dropped her eyes to the floor, hiding her hands in the ends of her sweater. "I had to identify his body. Seeing him like that, so damaged, I wanted to die myself. I had urged him to go to the police. I'd hugged him and told him he would be a hero for exposing Kirov. It was my fault. Alexei wasn't a fighter. When he heard me talk about Kirov stealing from his country, breaking the laws that men like him had just made, he adopted my anger as if it were his. He joined my armchair rebellion. It was his way of showing that he loved me."
Still on his knees, Gavallan reached out a hand and touched her cheek. "You can't hold yourself responsible for someone else's actions. Maybe you asked him to go to the police, but he made that decision himself."
"Maybe, but still..." Cate shuddered. "I never realized how bad I might feel. Even now." She reached for his hand, intertwining her fingers with his. "I see now I should have told you. I'm sorry, Jett. Forgive me?"
He nodded, filled with affection for her. Not a sexual yearning, but a stronger, deeper emotion, an encompassing happiness simply that he was there with her.
The cockpit door opened and the pilot stepped into the cabin. "We're an hour out," he said. "Weather looks fine in Geneva—a few clouds, otherwise it should be a sunny day in Switzerland. Mr. Dodson, you have any idea when you'll want us to be ready to take off again? We'd be appreciative if you could give us some idea of our destination ahead of time. We're required to file a flight plan, even if we don't stick to it."
The relationship was strictly business, mercenary all the way. Once they were airborne, Gavallan had bribed him with ten crisp hundred-dollar bills. Ask no questions and he'd tell no tales.
"Be fueled up and ready to go by four. I'll give you a call later this morning to let you know where we're headed."
"That's fine. Couple hours are all we need."
The pilot left. Gavallan took off his watch and reset it for Geneva time. "An hour to go," he said. "Think this bird's got a decent shower?"
Cate pointed to the rear of the aircraft. "Give it a shot. Might as well get your money's worth."
He headed to the shower, but pulled up suddenly, hoping she might be getting out of her seat to join him. "Cate..." he started, but she was still seated, her eyes not on him but glued to the window, staring into the orange dawn.
He could only wonder what she was thinking.
# 38
#
**Y OU ARE HAPPY,** my friend?" asked Aslan Dashamirov.
"Relieved," Konstantin Kirov replied. "I slept better knowing there was no longer a risk of someone slipping our papers to the police. It was a difficult business. I'm glad we've solved the matter."
It was a cold, rainy Saturday morning. The two men walked arm in arm across the muddy field outside of Moscow where Dashamirov had set up one of his used-car lots. A row of crapped-out automobiles ran next to them. Fiats. Ladas. Simcas. None with less than a hundred thousand miles on them, though the odometers showed no more than a quarter of that. Scruffy pennants dangled from a line strung overhead. A ways back, tucked conveniently amongst a copse of baby pines, stood a blue and white striped tent where prices were negotiated and payments made, often in tender as suspect as the cars themselves: televisions, refrigerators, stereos, cigarettes, narcotics, women.
"I'm not so sure," said Dashamirov.
"Oh?"
"No one talked. Not one of them admitted to working with Baranov or with Skulpin. Only the innocent are so brave."
"You didn't give them the chance." Kirov hated himself for playing up to the Chechen. He was a brigand, really, an uneducated hood.
Dashamirov looked at him as if he were a wart on his finger. "I am thinking we did not find the right person."
So that was why his _krysha_ had called the meeting, thought Kirov. He should have known the man wouldn't be so easily put off. Of course, Dashamirov was right. He was always right. This time, though, Kirov had beaten him to the punch.
He'd put his finger on the traitor, a young securities lawyer working in-house on the Mercury deal, and had taken care of the problem himself. Quickly. Neatly. Quietly. A single bullet to the man's brain delivered in the comfort of the traitor's own flat. None of this barbaric business with a hammer. Imagining the fierce blow against the skull, Kirov shivered, a spike of fear running right through him to the pit of his belly.
He stared at Dashamirov. The mustache, the crooked mouth, the eyes at once dead, yet so magnificently alive. The man was a beast. But a smart beast. He was correct in his assumptions. Only the innocent _were_ so brave. The lawyer had spilled his guts after a few threats and a bloody nose. Had Dashamirov pressed him for details about the money missing from Novastar, it would have been Kirov getting the hammer yesterday morning.
_The hammer_.
He ground his teeth.
"What's important," said Kirov, "is that Mercury will go forward without any further problems. For that I have you to thank."
"I was thinking rather about Novastar," said Dashamirov, dropping his arm to his side, quickening his pace as the rain picked up. "The question of the missing funds haunts me, my friend. Where there is one rat, there may be more. Perhaps someone in your organization is stealing the money from the airline. A hundred twenty-five million dollars is too large a sum to take lightly."
"Perhaps," replied Kirov thoughtfully, "though that would be difficult. I alone have signature power over the airline's bank accounts."
"Yes. You are right. Perhaps it would be wise to study the books." He opened his slim, spidery hands in a gesture of conciliation. "If, of course, you do not mind."
It was not a request, and both men knew it. Kirov looked around. A dozen of Dashamirov's clansmen loitered among the cars. _Vor v Zakone_. Thieves of thieves. God knew they were wealthy, but look at them. Standing around in the pouring rain, hair wet, clothing as sodden as the omnipresent cigarettes that dangled from their lips. In four days' time, Dashamirov stood to take home 15 percent of Kirov's billion—a neat $150 million dollars. The next day he would be here, or at one of the other fifty lots he ran in the northern suburbs of Moscow, standing in the rain, drinking filthy coffee, smoking.
"I will speak to my accountant immediately," said Kirov. "He is in Switzerland. It may take some time."
"By all means." The courteous reply was accompanied by a damning smile. "There is no hurry. Have the latest quarterly report for Novastar, as well as the most recent banking statements for our Swiss holding companies, Andara and Futura, in my office by Monday."
"I am in New York Monday," said Kirov, puffing out his chest, trying to muster some authority. "We will price the Mercury offering that afternoon. We can sit down together when I get back in the country on Friday."
"Monday," repeated Dashamirov, less courteously. "By four o'clock. Or else I will begin looking somewhere else for the thief within your company. Somewhere closer to the top."
A bead of sweat broke high on Kirov's back and rolled the length of his spine.
"Monday," he said, knowing it would be impossible.
# 39
#
**T HE JET BANKED HARD TO THE RIGHT** and drifted lower. From her window, Cate stared as the city of Geneva rushed up to greet her, as if she were looking at a postcard from her teenage past. The city looked no different than it had when she'd last seen it, ten years before. The _jet d'eau_ shot a geyser of water two hundred feet into a young blue sky. A flotilla of boats bobbed lazily on the lake's scalloped surface. The prim row of banks and hotels that lined the Quai Guisan nodded a courteous "Welcome back."
Beyond the cityscape, the Saleve rose vertically from a buckle of forest, a brooding granite soldier guarding the town's southern flank. The only Calvinist remaining in a city gone to the devil. But the familiar sights brought forth no haze of nostalgia, neither a wish for the past nor a desire to recall her youth. They promised only trouble. This was her other life. Her secret self. The history she'd sworn to keep hidden. Stealing a glance at Jett, her stomach tightened. In fear. In sorrow. In anticipation. And as the plane touched down, the wheels bouncing once before embracing the runway, she shivered with a premonition of loss. She was certain that everything she'd spent her adult life working toward was about to come undone.
**A WHITE VOLVO** with the orange and blue markings of the airport police waited on the tarmac beside their assigned parking spot. Two policemen, submachine guns tucked under their arms, approached the aircraft.
"Let me handle this," said Cate.
"Be my guest." Gavallan handed her his passport and stepped aside. She didn't know how he could stand there so calmly with a pistol tucked into his waistband.
Customs and immigration were conducted _"sur place."_ The policemen examined their passports. One climbed into the cargo hold to inspect their luggage while the other checked the flight log.
Keeping to English, Cate explained they had nothing to declare and were, in fact, only staying in Geneva for the day. A little sight-seeing. Lunch at the Lion D'Or. A run up to the UN. Would either care to join them? They needed a guide, she said, her itchy nerves fueling the giddy repartee. Someone who knew the language and could provide some local color. Could they tell her where Audrey Hepburn was buried? Wasn't it near Crissier? And didn't Phil Collins live nearby?
Suddenly, the policemen were all smiles. Beneath the blue berets, neither was more than twenty. _"Pheel Collins? Oui, oui, il habite tout près."_ He lives nearby. But neither could come up with the town. As for guides, they were unable to help. _"Désolé, Madame,"_ they replied. They were in the midst of their annual military service and their next scheduled leave was not until the following Friday.
Thirty minutes later, she was driving a rented Mercedes sedan along the highway. Jett sat beside her, a map spread upon his lap. "Keep your eye out for the Aubonne exit," he said. "Looks like it's about twenty klicks down the road. Just up from the lake."
Cate shot him an apprehensive glance, frightened by his retreat into military vernacular. He'd been brooding since they'd crossed over the continent, speaking less and less, avoiding her gaze.
This is the Jett Gavallan I don't know, she mused. The Air Force Academy grad who never whispers a word about his time in uniform. The jet jock who clams up at the first mention of the war he fought. He's going back, she realized. He's suiting up for battle.
"Klicks being what?" she asked. "Kilometers?"
He nodded without looking at her.
"Just don't let me miss the turnoff," she said, though she knew the way to Aubonne as well as to her own home.
"I won't."
Jean-Jacques Pillonel did not live in Aubonne, but in Lussy-sur-Morges, a quaint village situated high on the vine-covered slopes of Lac Leman (she would never call it Lake Geneva) about halfway to Lausanne. She knew the spot only because one of her art teachers had lived there, a man named Luc Caprez with whom at the age of eighteen she'd had her first affair. Luc and his briar pipe, who spoke of the courage to live a dangerous life, dangerous meaning to brave the landscape of your ideals, to pursue your dreams no matter where they led. Luc, who lectured her even while making love.
She kept her foot firmly on the gas, taking the car to 160 kilometers per hour as she passed the exits for Nyon, Gland, and finally, Rolle, where she'd gone to school for four years at Le Rosey. She glimpsed the campus to her left. The schoolhouses were done up as old villas and sat on a plateau cut into the hill. She took in the steep mansard roofs, the limestone façades, and the window boxes heavy with purple and red geraniums.
But it wasn't the sights so much as the smells that lent her a melancholy feeling and sent a current of doubt rustling across her belly. It was the smell of sun-warmed soil carried by an easy lake breeze; of Saturday afternoons trawling the back alleys of Geneva; of Sunday mornings saddling horses at the stable.
It was, she realized, the long-absent smell of her youth.
Cate caught sight of her eyes in the mirror and was frightened at their intensity. When had she adopted the mantle of crusader? she wondered. Had she finally embarked upon the "dangerous life" she'd promised herself she would one day lead? Or was she just tagging along with Jett for the ride?
Until now, she'd been content to fight through others. At the K Bank, she'd transferred her dissatisfaction to Alexei and let him do the dirty work. As a reporter, she hid behind the banner of the paper, relying on its influence and reputation to forward her watered-down causes. In her bid to derail Mercury, she'd recruited Ray Luca to fire her broadsides. As always, she preferred to remain one step removed, a gray eminence sheathed in fear.
But overnight things had changed. The battle had landed on her doorstep with a thud, a personal invitation stained with the blood of innocents. RSVP Konstantin Kirov, Moscow. There was no more escaping, no more hiding behind another.
_This was the dangerous life_.
Yet it was not guilt that had led to her decision. It was you, she said to Gavallan's silent profile, seeing in his strained, concentrated features the determination that had brought him so much success, the confidence that had led him to the brink of disaster, and the generosity of spirit that had captured her heart. I came because of you. Because I can't let you go on with all you don't know. Because your foolish confidence isn't enough to save you. Because I love you and you're all I have left.
As she settled into her seat, Cate's eyes once more found the sparkling asphalt. Grimly, she saw the days ahead playing out. All paths led in the same direction, ended at the same destination. What would happen when he found out? How could she explain? Above all, Jett was an honest man. He detested liars. She was sure she detected a new coolness between them since she'd brought up Alexei. And that was just the tip of the iceberg. How could he ever love a woman whose entire life was a lie? Sooner or later, he would discover the truth. And she would never have a chance to win him back.
"There it is," said Jett. "Aubonne. A thousand meters."
Cate signaled and guided the Mercedes off the highway. "Which way now?" she asked, sliding into the left lane.
"A left under the bridge, then bear to your left again."
_I know_ , she wanted to say. _I used to live here_.
She was struck by a desire to touch him. She reached out a hand, only to pull it right back. Let him go, she told herself silently. He looked at her and she tried to smile. "I'm glad I'm here," she said.
For a moment, Jett's eyes softened, and a question danced beneath his lips. As quickly, it was gone.
"Turn here," he said, spotting a sign with the name of Pillonel's village. "Morges is at the top of this road. Pillonel's house is at 14 Rue de Crecy."
_"Roo-duh-Cray-cee,"_ she repeated, correcting him, her schoolgirl's accent still perfect.
Gavallan eyed her remotely. "You never told me you spoke French."
Cate shook her head, laughing sadly. What the hell? Sooner or later, he was going to find out everything anyway.
# 40
#
**J EAN-JACQUES PILLONEL'S WEEKEND HOME** rested at the end of a short gravel drive, a majestic chalet nestled among the vines with an unobstructed view of the lake. Twin Jags were parked in front of a detached garage. Away to one side sat a barn coupled with two smaller outbuildings. Stacks of crates leaned against one of them, faded pictures of grapes stenciled on the splintered wood. Gavallan figured it must be where he kept the press and bottled the local tipple. All in all, impressive. More the residence of a country squire than the managing director of an accounting firm.
"Jett, but what a surprise," called Pillonel as Gavallan climbed from the car. "And is that Cate I see? You two are together again? _Mais tant mieux_. Come in. Come in. The door is open."
The squire was easy to spot. He stood on the first-floor balcony, clad in khaki work pants and a denim shirt, the nobleman's obligatory sweater tied around his neck. One hand was raised in polite greeting, though Gavallan knew he had to have been wondering who the hell was doing something so decidedly un-Swiss as dropping by without an invitation.
Waving hello, Gavallan allowed Cate to precede him up a groomed path framed by a rose garden in full bloom. She was his calm, the antidote to the rage that had been building in him since they'd landed and that had taken firm grip of his every muscle. Left to himself, he would have run up the path, broken down the door, and wrung Pillonel's neck until he confessed his every last crime, guilty or not.
_Detective Skulpin was right_ , he said to himself. _It had to be you. You handled the on-site inspections. You sounded the all-clear. You toyed with the pictures_.
"Really, I am surprised," Pillonel announced from the head of the stairs. "You are here on vacation? Why didn't you phone me in advance? You're both very naughty."
He was a handsome man, tall, slim, with a bit of the dandy about him. He had a full head of hair that was a shade too black for a fifty-five-year-old and gray eyes that sparkled a little too brightly. He liked to wear ascots when they dined out at night, Gavallan remembered, and he smoked Silk Cuts with an ivory cigarette holder.
"Unfortunately, we're here on business," said Gavallan, climbing the stairs, doing his best to return the hearty handshake. "Mercury."
"Ah. I see," said Pillonel, light as a feather. "The big deal. Cate, may I take your jacket?"
"No thank you," she answered, nearly wincing as he kissed her cheeks in greeting.
"Come along. I was just finishing breakfast." Extending an arm, Pillonel showed them to the balcony. A table littered with croissants, jams, napkins, and a pot of coffee sat near the railing. The lake lay a mile away, a shimmering blue crescent stretching as far as the eye could see in either direction. Beyond it, through a mid-morning haze, rose the snowcapped peaks of the French Haute Savoie. The good life, thought Gavallan.
"Claire will return shortly," said Pillonel. "She's out with the dogs. You remember my wife?"
"Of course," said Gavallan, calling to mind a slightly built, argumentative woman with prematurely gray hair and skin the color of alabaster. He walked to the edge of the balcony and made a show of surveying the surrounding vineyards. "So this is where the Pillonel wine comes from?"
"Yes, the famed Chateau Vauxrien." Pillonel pointed out the boundaries of his estate. "We have only ten hectares. It's a modest parcel, but if the sun shines through September and we don't have too much rain, we can make some good grapes. You would like a glass? I have some open just inside. Last year's vintage. A bit young, but nice. Jett? Cate?"
"No thanks," they both said.
Gavallan turned his back on the vineyard and, crossing his arms, fixed Pillonel with a grave stare. "We've got some major problems with the Mercury deal. I spoke with Graf Byrnes on Wednesday night. He was in Moscow checking out whether the rumors we'd been reading on the Net were true."
"I told you—it's rubbish. Nothing to worry about."
"Graf doesn't agree. He let me know in no uncertain terms that the deal was bad. Unfortunately, circumstances didn't permit him to tell me how bad or what exactly was wrong. Before I cancel it, why don't you tell me what you really know about the company."
"What I _really_ know? Why, we discussed it on the phone the other day. The Private Eye-PO's accusations are ridiculous—frankly, laughable. You can't be serious about canceling the IPO?"
"Oh, you bet I'm serious. The deal's over." Gavallan took a step closer to Pillonel, eyes wandering over every inch of his face, searching out where he kept his guilt hidden. "What do you think Graf could have found, Jean-Jacques? I mean, you promised me on Wednesday everything was hunky-dory. What could it have been? Everything's 'up and running,' right?"
A brisk shake of the head. "I don't know." Swiftly, he added, "Yes, everything is up and running. You said Graf was not able to tell you what was wrong. Why not?"
"I'll tell you in a minute. Let's stay where we are for the time being. The photos? You're certain they're fakes?"
"Positively. They're rubbish. I've seen the facilities myself. You're making much too much of the Private Eye-PO's words. He's a pest. If I were you, I wouldn't even bother."
"Oh, someone bothered, I can tell you that."
He really is a pretty decent actor, Gavallan was thinking. And marveling at the man's practiced deceit, he felt his anger rustle and loosen a notch. A hand dropped to the pocket of his windbreaker. Through the fabric, he let his fingers brush the butt of Cate's pistol. He added, "The Private Eye-PO was killed yesterday. His name was Ray Luca. A gunman entered his workplace and shot him, along with nine other men and women. It was a bloodbath. Didn't you read the papers this morning?"
Pillonel's eyes widened in astonishment. "This is the rampage in Florida I read about. This is the Private Eye-PO? They say a man went crazy. That he killed all his friends, then himself. How horrid."
"He didn't go crazy," said Gavallan flatly. "Take my word for it. It was a professional job."
"You're sure the killer was not Luca? The police sounded like they knew precisely what happened."
"Yes, I'm sure. Who do you think would kill nine innocent people just to get at one man?"
"I have no idea."
"You're lying," said Cate. "You know damn well who might want the Private Eye-PO dead. Who _needed_ him dead. We all know. Ray Luca was a friend. He died with nine innocent men and women because what he said about Mercury was true. You had to know it. You told us yourself you visited the Moscow Operations Center."
"Cate, please, you're mistaken," said Pillonel, retreating, his eyes begging Gavallan for an explanation. _"Je vous en pris...._ Please, Jett, you must have a word with her. I don't know what she is saying.... My God, this is all so crazy."
"You're the one who's mistaken," retorted Cate. "If you think you can jump into bed with Konstantin Kirov and walk away from this untouched, you're a fool. How much is he paying you? A million? Two million? Ten? Or did he promise you shares in the deal? Tell him, Jett. Tell him about Ray Luca. Tell him about Graf."
The mention of money, its hint of bribery and collusion and all things criminal, sparked a radical change in Pillonel. In an instant, his apologetic stance vanished, replaced by one of undisciplined outrage. "That is enough now," he declared, pulling the sweater a little tighter around his neck. "I hope you haven't traveled all the way from the States just to insult me like this, making these fantastical accusations. This is crazy what you say. Really crazy. You are badly mistaken if you think I am involved in some type of illegal affair with Mr. Kirov. I've said it time and time again: Mercury is fine. It's your conduct that is criminal. I'd like you to leave. Now."
But Gavallan did not move. He remained standing at his place near the balcony, stiller than he'd ever been in his life. If he lifted a finger, if he blinked an eye, if he let out his breath, he'd lose control over the animal rage that was clawing at his neck. All too clearly, he imagined himself hitting Pillonel with his fists, pummeling the man until his features were broken, his face a bloody pulp. He felt the gun heavy in his pocket, full of promise. The muscles in his jaw flinched, and a second later the vision passed.
"After six years, Jett, I thought we had a relationship," Pillonel droned on angrily, self-righteously, a man wronged in his own house. "That maybe we were even friends. I see I was wrong. Now, go. Both of you. Take your accusations and make them to the police. Maybe I'll call them myself."
"Friends?" Gavallan asked, cocking his head. "Did I hear you say you thought we were friends?" He advanced on Pillonel. Something inside was stretching, growing taut, moaning like the hull of a submarine down past its depth limits.
Pillonel took another step back, palms raised as if he were calming an angry dog. "Come now, Jett. You stop there or I call the police."
Gavallan grabbed the phone from a side table and thrust it at Pillonel. "Go ahead. Call them. Or do you have the balls?" He threw the phone on the table. Another step. "We know what you've been up to, and it's not what friends do to each other."
Cate said, "Jett, please..."
Gavallan did not remove his eyes from Pillonel. "We know you faked the due diligence reports. Your men scoped out Mercury's assets. Your men signed off on its physical plant and inventory. It couldn't have been anybody else."
"This has gone far enough," said Pillonel, stopping, crossing his arms. "I've had quite enough of your bullying. You will go. Now. I demand it."
But it was Gavallan who had had enough. Later, he wasn't sure what finally made him break: the insistence of Pillonel's denials, the man's elegant ignorance, or just that he was sick of being lied to and didn't know any other way of making Pillonel admit his sins.
Drawing the pistol out of his pocket, he grabbed Pillonel by the collar, yanked him close, and laid the snub-nosed muzzle against his head. "How's that, you fuckin' prick? You want bullying? This is bullying. And I'll tell you something. We aren't going anywhere until you start telling the truth."
"Jett, put it away," pleaded Cate, rushing to his side. "Stop it."
"Don't worry," said Gavallan, cocking the hammer, pressing the barrel harder into Pillonel's forehead. "We're friends. We're just playing. Right, Jean-Jacques? Just palling around?" When Pillonel didn't answer, he said, "Yesterday, two of Kirov's creeps put a bigger gun than this one on my forehead, right there in the same place. Do you know what they said to me, Jean-Jacques? Do you? They said, 'Sorry, Mr. Jett. Mr. Kirov says you have to die. He says it's business only.'"
Gavallan shoved Pillonel across the balcony. The Swiss stumbled over a chair and collapsed on his behind.
"Ten people are dead because of Mercury. ' _For business only.'_ As for Graf, I can only hope he's okay. The reason he couldn't let me know the exact details of what he'd found out about Mercury was because he's with Kirov. A prisoner, I guess, if Kirov hasn't already had him killed. If nothing else, you're going to tell me the truth for him—for Graf Byrnes—so that maybe I might have a chance to get my friend back. Understand?"
Pillonel got to his feet. Righting the upended chair, he brought it to the table and sat down. His tan face had gone gray. _"Mais non,"_ he said. _"Ce n'est pas possible."_
Cate wandered closer. _"Si,"_ she replied. _"C'est bien possible. En fait, c'est la vérité."_ It's the truth.
Gavallan slipped the gun back into his pocket and sat down in a chair next to Pillonel. Just looking at the man made him weary. Accountants had no business being criminals. They lived in a cloistered world of financial reports and P&L statements, of interminable client meetings and rushed lunches. Of clipped fingernails and polished shoes. They had no business consorting with murderers and gangsters.
"Our friend in Moscow is nervous," Gavallan said. "His empire's falling apart. Mercury. Novastar Airlines. So now he's tidying up. Covering his tracks. I'd be scared if I were you. Geneva's a helluva lot closer to Moscow than Florida."
Cate opened her handbag and gave Pillonel the Private Eye-PO's last report, the document titled, "Mercury in Mayhem." When the Swiss executive had read the whole thing, she slipped him Yuri Baranov's fax to the FBI calling for a raid on Kirov's headquarters.
"Call Baranov," Cate suggested. "His number's on the fax. He'll be glad to tell you all about it. His offices have provided us the evidence about Mercury. They have an informant inside the company."
"But this has nothing to do with Mercury," protested Pillonel. "I know nothing about a raid. It is of no concern to me." He made an effort to stand, but Gavallan waved him down. "Sit down. _Now."_
Pillonel shrugged and sat. Affecting a pensive pose, he averted his gaze from his guests. "You know you can see Évian from here?" A tremulous hand pointed to the French side of the lake. "They have a marvelous casino. Right out of the thirties. I go sometimes with Claire. We put on our evening clothes, take the steamer from Ouchy. Maybe we all go, the four of us? Take the waters. Do a little gambling."
When neither Jett nor Cate responded, he shifted in his chair, drawing a breath as he faced his accusers. His color had returned, and he looked remarkably composed. He made a little gesture with his shoulders, a timid shrug that was at once ashamed and contrite. "I'm no murderer. Maybe foolish with the girls. Maybe, I gamble sometimes. But murder? No. That's not me." He sighed. _"Alors_ , how long have you known?"
Gavallan looked down as the anger bled from him. "Since yesterday. Why, Jean-Jacques? What made you do it?"
"Why?" Cate repeated.
Pillonel answered without hesitation. "Money, of course."
Cate shook her head. "You pig."
Pillonel shrugged. Dusting off his shirt, he sipped from his coffee and began to explain.
Seven months ago, Kirov had come to him with a plan to take Mercury public. The thirst for broadband services was unquenchable and Kirov claimed to be in a perfect position to exploit it. Mercury had been growing rapidly for four years. He was already the number two Internet service provider in Russia. Business conditions were stable and the country was increasingly prosperous. It was the time to offer shares. There was only one problem, Kirov confided: Mercury wasn't quite where it should be, the infrastructure not exactly as advertised. Moscow was a problem and so was St. Petersburg. But it was nothing to be concerned about, he promised. The problems would be rectified once Mercury received the infusion of capital an IPO would bring.
"I asked him about his revenues," Pillonel said. " 'How is Mercury making so much money if not through offering broadband services, Internet connectivity?' "
Gavallan raised a hand for him to stop. "What did you know about his revenues?"
"Earlier in the year we'd taken a participation in the German accounting firm that did Mercury's work. When we integrated operations, we took over all their back office operations. We saw the funds coming into Mercury's accounts. In fact, we hold copies of all the financial transfers the company has made over the past three years."
"You're saying you were Mercury's accountants before I farmed out the due diligence to you? That's conflict of interest. You had no right to accept the assignment."
"Of course, you're right," said Pillonel in a dull voice, as if that were the least of his misdeeds. "I asked Kirov where the money was coming from, if not from Mercury. When he just stared at me, saying nothing, looking through me with that charlatan's smile, I knew he had me. We'd been signing off on the books of a thief."
But Gavallan was more interested in something Pillonel had said earlier than in the accountant's belated discovery that Kirov was a thief. "He came to you about the IPO seven months ago?"
"Maybe longer. It was November. I remember, because we were about to take our holiday. Claire and I go every year to the Seychelles. It is beautiful there, and one must get away from the _brouillard_ —you know, the fog."
"How did he know you would be doing the due diligence for us?"
"I've been doing Black Jet's European work for years."
_November_ , repeated Gavallan to himself. But Black Jet hadn't officially won the deal until January.
A few seconds passed. Pillonel offered another of his Gallic shrugs, then rose and said, "Stay here. I'll be right back. I've got something that may interest you." He returned a minute later carrying a raft of notebooks. "Here is the report," he said, handing a green binder to Gavallan. "You'll find the experts' testimony inside. The Moscow station was run-down, but they've fixed it up since. The company's a year behind on its infrastructure. Maybe you burn the papers and close your eyes. Go forward with the offering. The company's really very strong. Kirov just needs time to build up his customer base and modernize his network."
Gavallan read through the notebook, skimming from page to page. It was all there, just as Pillonel had said. Mercury's operations checked out in eight of ten of its major markets. The problems lay in Moscow and St. Petersburg. Mercury had purchased insufficient servers, routers, multiplexers, and the like to handle the number of customers it claimed to have.
As Gavallan absorbed the information, he found himself as impressed with the company as he had been when Kirov first told him about it. Mercury was solid. It possessed excellent market share, capable personnel, and a sound business plan. Maybe the offering wasn't worth two billion dollars, but depending on the true value of its revenues it could be worth eight hundred million, a billion, easy.
"You said you saw the exact flows on money coming into and out of Mercury?"
"Yes. The bank sends us copies of all the account's activity: deposits, transfers, monthly statements."
Gavallan closed the notebooks. At least he'd be able to figure out what Mercury was really worth. He would still cancel the offering; he had to. But that didn't mean his involvement with the company had to end there. There was another way to spin the deal. And imagining the possibilities, Gavallan felt the first glimmerings of hope. For himself. For Black Jet. And for Mercury.
Putting aside the notebooks, he felt a small weight lift from his shoulders. He had his proof that he hadn't been involved in faking the due diligence. Now he would take Pillonel to his offices and recover some of the copies of the funds transfers into and out of Mercury's accounts. If Kirov had done what he suspected, Gavallan would have the chips he needed to sit face-to-face across from the Russian oligarch.
He might just have a chance to win back Byrnes.
"It is enough?"
Looking up, he found Pillonel gazing at him. "Excuse me?"
"It is enough?" the Swiss repeated.
"The report. Yes, it'll do nicely, thank—" Gavallan cut himself short, seeing an unsatisfied look in Pillonel's eye. A moment passed, and he felt his stomach tighten. "You mean there's more?"
"What I've shown you is to protect yourself," said Jean-Jacques Pillonel. "To protect Black Jet. Now I give you something to protect me."
# 41
#
**T HE FAX FROM INTERPOL** arrived on the desk of Detective Sergeant Silvio Panetti of the Geneva Police Department at 9:15 **A.M.** It was a fugitive arrest warrant for an American citizen sought in connection with the murder a day earlier of ten persons in Florida. The FBI had reason to believe he had fled the United States, the fax indicated, and gave the tail number of a private aircraft in which he was said to be traveling. A bold "Urgent" headed the message and it was followed by the instructions that any information was to be forwarded to Assistant Deputy Director Howell Dodson in Washington, D.C., or to the consular officer of the local U.S. embassy.
Panetti yawned and lit his third cigarette of the shift. _Urgent, eh?_ He was impressed. Too often, American law enforcement was interested in tax evaders, money launderers, or other equally bloodthirsty types. Reading the message a second time, his eye tripped over the words "murder" and "ten victims" and "extremely dangerous." A hushed _"Ma foi"_ escaped his mouth. Would someone mind telling him why the fugitive might be headed to Switzerland? And Geneva in particular? The two countries had extradition treaties in place with regard to capital crimes, and lately, no one could argue that Switzerland had been anything but the model of cooperation.
Picking up the fax, he strolled into his boss's office. It was empty, as he'd expected. Saturday was the chief's day for sailing. With this weather, you could bet he was already halfway down the lake to Montreux. Panetti looked up and down the corridor. Seeing no one, he blew a cloud of smoke into the office. A little present for the chief. _Pauvre mec_ had quit smoking the week before and was having a tough go of it. Half the _département_ puffed like chimneys, and the only place in the whole building the chief could get away from the smoke was his own office. Chuckling, Panetti checked that the windows were closed and shut the door behind him, but not before slipping a couple of packets of Gauloise Bleus onto the chief's desk. _Bonne chance, mon lieutenant_.
Returning to his desk, Panetti paused long enough to pick up his lighter, his phone, and his pistol—in that order of importance—then left the office. He wasn't much to look at. Middle-aged, of medium height and medium build, he was one of the Lord's weary travelers. He owned a sad, pouchy face and deep black eyes that guarded a sparkle of mischief. He hadn't shaved this morning, and the two-day stubble combined with yesterday's outfit gave him a shabby charm. Panetti shrugged. At least no one would mistake him for a banker.
Descending the staircase to the parking garage, he called Cointrin to ask for flight operations.
"Claude, I need a favor. Got a list of incoming traffic? Private, not commercial. A jet. Yeah, I'll wait, thanks."
Traffic was light, and he was over the Pont Guisan when he got the answer.
"She's a nice bird," said Claude Metayer, flight operations chief of Geneva International Airport and, to Panetti's everlasting dismay, his brother-in-law.
"You mean the plane is here?" Panetti felt his heart give a rat-a-tat-tat.
"A G-3. Came in an hour ago. Passengers are gone, but if you want to talk to the pilots, I'll tell them you are coming."
"Keep them there," ordered Panetti. "Be there in ten."
"Where are you now?"
"Passing the Hotel President. Why?"
"I'm hungry. Be a pal and get me a brioche. Uh, hold on a sec. And grab a half dozen _pain-au-chocs_ for the boys."
"Eh, Claude?" said Panetti, ramming his foot against the accelerator and throwing the siren onto the roof. "Fuck your _pain-au-chocs."_
**T HERE SHE IS."**
Claude Metayer pointed at a white Gulfstream parked two hundred meters across the tarmac from the control tower. "N278721. That your bird?"
Panetti checked the numbers against those written on the fax. "Yep. That's it. See anyone get out? A man and woman, maybe?"
"No," said Metayer. "But I wasn't looking."
Panetti studied the plane through a set of binoculars. _Mince_ , but she was a beauty. His first thought was "expensive." Whoever owned that plane had to be very wealthy. The words "filthy rich" crossed his mind, and instinctively he sucked in his gut and stood a little straighter. A second later he relaxed. Sometimes he hated being Swiss.
"Where are the pilots?" he asked.
"Downstairs," answered Panetti's brother-in-law. "But go easy. I don't want any blood like last time."
**P ANETTI HAD THE INFORMATION** he needed in sixty seconds. No blood. No threats. Not even a raised voice, thank you very much. The suspect, John J. Gavallan, and his accomplice, Catherine Elizabeth Magnus, had rented a car from Hertz. They were expected back at the plane sometime that afternoon. The pilots had instructions to be refueled and ready to take off at 4 **P.M.** More than that, they said they didn't know, and Panetti believed them. A five-minute stroll took him to the Hertz desk. He flashed his badge and asked for the make, model, and license number of the car the Americans had rented. The answer came immediately. A black Mercedes 420S, Vaud license 276 997 V.
Panetti thanked the employees for their help. He was lighting cigarette number seven of the shift when the manager appeared from his office, waving a fey hand to get his attention.
_"Attendez. Attendez_. Officer, thank goodness you're here."
"Oh?" asked Panetti through a blue haze.
"You are interested in the Americans?"
_"Banh oui."_ Panetti raised a brow, curious as to what the Americans might have done to so disturb this fat old poof.
_"Ils sont terribles, les Amis_. Come, I show you." The manager led Panetti to a bank of phone booths, pointing archly at the third in line. "There. Look. See for yourself."
Panetti sauntered over to the booth. He picked up the receiver and put it to his ear. The dial tone sounded as innocuous as ever. He flicked the coin return. A-OK. "What's wrong?"
_"Non, non, les annuaires,"_ puffed the manager breathlessly. The phone books. And pushing Panetti aside, he pulled open the registry for the canton Vaud. "They stole a page. They ripped it right out. I saw them."
"A page? The whole thing? And you didn't call right away? Next time, I'll have to arrest you for not reporting the incident."
The manager curled his face into a sour smirk. "Very funny."
"Okay. Off you go. Your poodle is waiting."
"I don't own a..." The manager hoomphed, then spun on his heel and hurried back to his office.
When he was out of sight, Panetti sat down on the stool and laid the phone book on his lap. He flipped through the directory several times until he spotted the frayed pennants of the missing page. He had no idea whom Mr. Gavallan might be looking for, but the missing page might indicate where that person—or business, for that matter—might be. Swiss directories were divided alphabetically by city or town, with the locale's name printed on the top outside corner of each page.
Panetti was in luck. The same town was listed at the top of the preceding and succeeding pages.
_Lussy-sur-Morges_.
He had the local police on the line within fifteen seconds. And Mr. Howell Dodson of the FBI a minute after that.
# 42
#
**Y OU'RE SAYING YOU WORK FOR NOVASTAR, TOO?"**
Gavallan asked Jean-Jacques Pillonel on the way to Silber, Goldi, and Grimm's headquarters in downtown Geneva.
"As their accountants, we do all of their bookkeeping," replied Pillonel. "As their _fiduciare_ , we counsel them on setting up offshore accounts, shell companies, the usual song and dance to help our customers avoid paying too much tax."
"And how much is that?" asked Cate from her post in the backseat.
"Why, _any_ , of course," answered Pillonel, who was driving. "When Mr. Kirov purchased Novastar Airlines last year, he came to me to set up a holding company outside of Russia where he could deposit the shares."
"Why would he want to deposit Novastar's shares outside of Russia?" asked Gavallan.
Pillonel smirked, but didn't take his eyes off the road. "You'll see soon enough."
**S ILBER, GOLDI, AND GRIMM'S HEADQUARTERS** were located on the Rue du Rhône, one block from the lake. The newly remodeled building was a symphony of brushed steel and exposed girders. The lines were spare, the profile vibrant and supremely confident. One moment Gavallan thought he was looking at the Beaubourg in Paris; the next, the Hong Kong and Shanghai Bank on Hong Kong Island. Modernism had trumped tradition. Prudence had been declared a four-letter word. So had conservatism, stability, and any other trait that implied the slightest resistance to change.
Once on the third floor, Pillonel guided them along a dim corridor. Stopping in front of an anonymous doorway, he placed an eye to a retinal scanner. The lock disengaged and the door swung open.
"The funny thing is I knew this would happen," he said, allowing Cate and Gavallan to pass and enter the storage room. "I did it anyway, and I'm still not sure why. Foolish, wasn't it?" He looked at Cate. "You wanted to know how much Kirov was paying me? Fifteen million."
"Dollars, I hope."
"No. Francs."
Cate gave him a sad look. "Was it worth it?" Even now, Pillonel's venal nature demanded he think on the answer. _"Alors, non."_
**T HE FIRST ROADBLOCK** was set up one hundred meters north of Silber, Goldi, and Grimm's office at the intersection of Rue du Rhône and Place les Halles. The second was erected fifty meters south, at an intersection not visible from the silver and steel office building. Plainclothes policemen filtered down the busy streets, quietly demanding pedestrians to leave the area, in a few cases forcibly escorting them off the streets. A crisis headquarters was established in the shopping gallery below the Confederation Centre, the office complex that housed the Geneva Stock Exchange. Two armored personnel carriers painted a royal blue arrived. The back doors opened. Twenty-four policemen from the elite _Division D'Intervention Rapide_ , or DIR, of the Geneva Police Department, clad in full battle gear, jumped to the ground, forming into two squads and moving out toward their target. Snipers scrambled up stairwells in adjacent buildings and established shooting platforms with a clear line of sight of Silber, Goldi, and Grimm's lobby.
Watching the activity unfurl around him, Detective Sergeant Silvio Panetti stroked his mustache. _"Mince,"_ he whispered to himself. _"C'est sérieux."_
It had been simple to track down Mr. John J. Gavallan. Lussy-sur-Morges had but two hundred twenty residents. One by one he had read their names to Mr. Howell Dodson of the FBI. Dodson recognized Jean-Jacques Pillonel's name immediately. A team was sent to the man's chalet. Pillonel's wife did not know where her husband had gone. Ten minutes later, a patrol car spotted Gavallan's rental on the Rue du Confédération, a block from Silber, Goldi, and Grimm. The rest Panetti figured out for himself.
A walkie-talkie near him crackled. "In place," said a crisp voice.
_"Entendu,"_ replied Captain Henri L'Hunold, commander of the DIR. "Await my signal."
**S TEPPING INTO THE DOCUMENT STORAGE ROOM,** Jean-Jacques Pillonel took up his tale where he had left off in the car ten minutes earlier.
"As I said, it is part of our job as fiduciaries to keep a permanent record of our customers' accounts. This means keeping copies of the bank confirmations showing all monies that flow into and out of them: every deposit, every wire transfer, every cash withdrawal."
"But you're not a bank yourself?" asked Cate.
"Good Lord, no. But as their accountants we require the confirmations to perform the audits of our customers' accounts. We scan them immediately and transfer them to hard drive. Every month, we download the new confirmations onto our customers' private CDs."
The three were snaking through aisles of chest-high filing cabinets colored a wan yellow. Pillonel was their leader, and he moved like an automaton through the metallic maze, drawing first one CD, then another, his destinations memorized long ago.
"What was Kirov's game?" Gavallan asked. "Didn't he want to pay the tax man his due?"
"Forget the tax man," said Cate. "Kirov didn't even plan on giving the money to Novastar. As far as he was concerned, Novastar's revenues were his, and he made sure they didn't turn up anywhere on the company's ledgers."
"It's a bit more complicated than that," cautioned Pillonel. "Once Kirov won the auction for Novastar, he transferred the company's headquarters from Moscow to Geneva. Moscow was too parochial, he said; an international airline needed an international presence. He asked me to set up a holding company for his forty-nine percent stake in the airline. We were happy to oblige. The company is called Futura. It is domiciled in Lausanne."
"Is Kirov the sole shareholder?" Cate demanded.
"No. There is a second man. His name is Dashamirov. Aslan Dashamirov. You know him?"
Gavallan and Cate said they didn't.
"He is trouble, this man." Pillonel offered a secret smile. "He is Chechen. Not so polished as Mr. Kirov. From the bandit country. Anyway, at the same time as we opened Futura for Mr. Kirov, he asked us to set up a second company, this one offshore in the Dutch Antilles—Curaçao, I believe. That company is named Andara. Now of course we all know why he did this, but I was surprised at his audacity. First, he instructs all of Novastar's foreign offices to transfer their revenues to Futura, instead of to the company's old accounts in Moscow. This means all the money Novastar earns from sales of plane tickets made in Los Angeles or Rio or Hong Kong come to Switzerland."
"I have a feeling we're getting to the good part," said Gavallan, giving Cate a fateful glance.
"If you mean the part that concerns Mercury, you are right," said Pillonel. "From Futura, Kirov would transfer the money into Mercury's accounts here in Geneva. But only at certain times during the year, and just briefly—one day in, the next day out. He timed it so that Mercury's quarterly bank statements showed the effect of the transfer. Usually, the inflows increased Mercury's revenues by around thirty percent."
"Thirty percent? Not kidding around, was he." It was Gavallan's policy to involve himself in the due diligence being done on Black Jet's larger deals, and he remembered poring over Mercury's banking statements, corroborating the balance held at the bank with the sum shown on Mercury's books. In one day, out the next. Clever, but you could only get away with it with the complicity of your accountant.
Then again, fifteen million francs bought a lot of complicity.
Cate said, "So once Mercury booked the funds as revenues, they wired the money back to Futura?"
"Only about ten percent, actually. The rest was always transferred to Andara, the company in Curaçao, for the personal benefit of Mr. Kirov and Mr. Dashamirov."
"That explains why Baranov and the Russian government are so pissed off," Cate said. "The revenues from the foreign rep offices never made it to Moscow. The government privatized Novastar to increase its profitability and bring it up to Western business standards. They expected the fifty-one percent they retained to earn them a decent chunk of hard currency."
Pillonel had completed his rounds of the filing cabinets and was heading toward the back wall, where a long desk divided by partitions into carrels offered a dozen personal computers and printers for everyday use. Next to the desk stood a row of IBM mainframes, their blinking red and green pinlights the only indication they were in service. Sitting down at a carrel, he selected a CD and slipped it into the PC's disc drive. "It's all here. See for yourself."
Gavallan watched from behind Pillonel's shoulder as copies of Novastar's transfers to Futura flashed onto the screen. Two hundred thousand dollars from New York. Three million French francs from Paris. Four hundred thousand deutsche marks from Frankfurt. All the money headed for Switzerland. Pillonel flipped through the transfers, taking the three of them on a paper trail across the globe. Shanghai, Mexico City, Toronto, Chicago, Paris again. Around the world in eighty seconds.
"Like I say, it's all here." Suddenly, Pillonel laughed, a high-pitched, hysterical whinny. "I don't know who is going to be madder—the Swiss because I break the secrecy law, or Kirov because I violated his trust."
Oh, I can tell you the answer to that one, buddy, declaimed Gavallan silently: Kirov by a long shot.
Pillonel switched discs, and a new set of transfers scrolled onto the screen. "Here are the transfers you are most interested in, Jett: the funds injected into Mercury." The amounts were larger, the transfers less frequent. It would be an easy task to back out the amounts Kirov had transferred into Mercury's accounts and arrive at a true reckoning of Mercury's revenues, and thus its market value.
Pillonel switched discs again, and the screen was filled with transfer after transfer out of Mercury and into Andara, Kirov's private strongbox. The sums were staggering. Ten million dollars. Thirty-two million. Six million.
It's the gold seam, thought Gavallan. A hard copy trail showing Konstantin Kirov's meticulously executed efforts to divert Novastar's revenues to his personal account. A how-to manual on stealing from Mother Russia. He found Cate's hand and gave it a squeeze. "I don't suppose Mr. Kirov will be too keen for Baranov to get his hands on these."
"Forget Baranov," said Pillonel acidly. "He's powerless. Kirov will flee the country if any charges are filed against him. He'll set up shop in Marbella with the other Russian expats. They've got a whole little community down there. Like I said, forget Baranov... he's a paper tiger. You want to hurt Kirov, I'll show you something that hurts him."
Pillonel slipped the third compact disc into the e-drive. Once again, the screen was filled with scanned copies of bank transfers. Gavallan leaned closer. It took his middle-aged eyes a few seconds before he could read the names and numbers on the screen. He recognized the account number of Andara, the Curaçao holding company, but the beneficiary was an anonymous numbered account at the Banque Privé de Genève et Lausanne.
"Isn't that your brother's bank?" Gavallan asked. Pierre Pillonel was Jean-Jacques's fraternal twin. One had chosen banking, the other accounting. What more could a Swiss mother desire?
"Yes. Pierre is managing partner for two years now."
Cate put a finger to the screen. "And to whom may I ask does account number 667.984Z belong?"
"Who do you think?" Pillonel scalded her with a reproving glance. "Mr. Kirov, he trusted no one—not even his partner, Mr. Dashamirov. After the Chechen left our meeting, Kirov asked me to open a private account for him here in Switzerland. This man is not content simply to steal from the Russian government—he wants to steal from his partner, too. If I were Kirov, I wouldn't be afraid of the prosecutor general, Mr. Baranov. Baranov can only put him in jail. Me, I am afraid of Mr. Dashamirov. Mr. Dashamirov catches Kirov stealing, he will kill him."
Cate lowered herself to her knees and spun Pillonel in his chair so that he faced her. "You're saying that these transfers show Kirov siphoning off money from Andara to his own private account?"
_"Exactement."_ Suddenly, he stood, forcing his way past her, the compact discs clutched between the fingers of one extended hand. "Take them. Take them all. They're yours. Use them quickly. As I said, I'm not doing this for you—it's for me. I am only safe once Kirov is in jail, or if he is dead. I ask you only one favor. You give me time."
"Time for what?" Gavallan accepted the discs and passed them to Cate, who slipped them into her purse.
"I am not sure yet. If I am a coward, I go to Brazil. Maybe Kirov finds me. Maybe he doesn't. One more man in jail, what does it change? Who's the better off? I've played the game the way I was supposed to. I helped you, my friend. Save your company. Save your friend. I've earned a chance to save myself."
Gavallan realized he didn't have much choice in the matter. Having Pillonel arrested would only alert Kirov to the fact that he was intent on canceling the IPO. He couldn't tell Pillonel to stay home and wait for the police until Tuesday or whenever he was able to find Grafton Byrnes. It boiled down to this: Pillonel was a free man until Gavallan was ready to turn over his evidence to the authorities.
Even then, he couldn't be sure whether the Swiss would arrest him. Though Mercury was technically a Swiss company, the fraud had taken place in conjunction with a listing on the New York Stock Exchange. That was a lot of borders to cross. Borders meant red tape and red tape meant delay.
"Go home," said Gavallan, frustrated. "Go to Brazil. I don't care. But whatever you do, take my advice and keep a low profile. And stay clear of Kirov."
Grabbing one of his arms, Gavallan half pushed Pillonel down the corridor to the elevator. They rode in silence to the lobby, then the elevator opened and Gavallan stepped out. "Cate," he said, looking over his shoulder. "How far to the airport?"
_"Police! Arrêtez!"_
A black-clad figure hit him low in the knees, throwing him to the ground. Gavallan felt the air rush from his lungs, his vision blur, then steady. Iron hands gripped his shoulders, pressing them to the concrete. A knee drilled into his chest. A second later, he was staring into the yawning muzzle of a large-bore pistol.
"Police!" shouted the aggressor. "Do not move!"
# 43
#
**Y OU'RE SURE HE'S HERE?"** Konstantin Kirov asked his brother Leonid as they entered the murky staff auditorium on the ground floor of the Foreign Intelligence Service's headquarters at Yasenevo. The room was at once enormous and stifling. Worn maroon carpeting ran beneath Kirov's feet. Wood-paneled walls hovered over him. The time was 2 **P.M. ,** but imprisoned in the eternal dusk, he had to remind himself that outside the clouds had cleared to usher in a warm summer's afternoon.
"Oh, he's here," replied Leonid. "I spoke with him ten minutes ago. He was upstairs checking on some old friends."
"But there are no cars," Kirov protested. "No sign of his security detail. He's the president, for God's sake. He's not a ghost."
"He's also one of us. He likes to use his tradecraft now and then. Keep himself fit. In practice."
_"Nimble,"_ came a voice from the darkened recesses of the auditorium. _"Like a cat."_ A familiar figure strode onto the stage at the far end of the room. "I can't tell you how advantageous it is being able to get away on occasion. To disappear. It keeps everyone on their toes. Friends. Enemies. Everyone."
The president of the Russian Republic jumped off the stage and advanced on Kirov, fixing him with an odd gaze. He was a slender man with sloped shoulders and a retiring manner. All the same, he demanded the room's focus. There was an unpredictability about him, a hidden strength crouched in his rolling walk, a shy ruthlessness in his eyes. Kirov shook his hand and, from somewhere deep in his Russian blood, obeyed the command to bow his head.
"Seventy-two hours," said the president. "All is in order, I trust?"
"Interest is strong," answered Kirov. "Our bankers report heavy demand for Mercury on all fronts, both institutional and private. A 'bellwether,' some are even calling it."
"And why shouldn't they?" asked the president. "Oil prices remain high. Our GDP is growing at eight percent. Unemployment is falling like a stone and the ruble is stronger than it has been anytime since the new era began. You say demand for Mercury is strong, I say not strong enough."
"I couldn't agree more," said Kirov. "And so does the investing public."
The president ran a hand up and down Kirov's lapel. "I don't want to hear about any of your shenanigans on this one."
"I beg your pardon?" asked Kirov, casting an eye to his brother for backup. Leonid remained silent, his chin dug into his chest.
"I'm talking about Novastar," said the president in a hushed voice. "Not happy with the fortune you're taking out of our aluminum industry, so you're stealing from our airlines, too?"
"A lie," said Kirov. "The airline needs to be restructured, that is all. A few new routes, a little less staff."
"I have your word?"
Kirov nodded, and felt the curse of the damned fall upon him. It took every fiber of his being to keep his eyes locked on the president's. "In fact, I welcome Baranov's investigation."
The president patted Kirov's arm, his brow lifted skeptically. "Don't go too far, Konstantin Romanovich," he whispered. "It's me, Volodya. Remember? The mayor's bagman from Petersburg. If I'm not mistaken, I had the pleasure of ferrying some of your donations to Mayor Sobchak before his untimely passing. You and I know you're robbing Novastar blind. Just keep it quiet. And if you can't, then quiet Baranov." His hand found Kirov's neck, and gave it a squeeze. "Don't worry. You've become much too valuable to your country to put in jail. For the moment, at least."
_Quiet Baranov?_ Had he heard correctly? Kirov mumbled some words, thanking the president.
"You are a good Russian." The president took Kirov's head in his hands and kissed him three times upon the cheek. Releasing him, he walked back toward the stage. "A billion dollars," he said. "Not bad for a new beginning. Do you hear that, comrade Lenin? Or should I say _Mister_ Ulyanov? We've been relegated to stealing scraps from the capitalists' doorstep." Turning his gaze, he stared up at the wall behind him. It was barren, save for the shadow of a familiar profile where a memorial sculpture had once hung. "Without Lenin, who are we? A country of bumbling democrats and corrupt capitalists? A band of impoverished states linked only by the tragedy of our common history?" The president was gathering steam as he spoke. He was giving a speech to convince, even if he was the only one who needed convincing. "We are Russians," he declared. "We did not stop being a superpower when we ceased to be communists. We did not cast off our ideological fetters only to lose our national identity."
If communism didn't work, neither would democracy, Volodya went on. Both were too extreme. He would steer a middle course, but the hand on the tiller would be a firm one. The press would be reined in, the media made an organ of the state once more. As another had said some seventy years before, "the trains would be made to run on time." Some might call it fascism, others benevolent despotism. He saw it differently. Two thousand years of history had made the Russian a serf at heart. He did not simply respect authority—he craved it. And in return for his subjects' obedience, he, Volodya, the fifty-year-old president of Russia, would act as Lord and rebuild their country. He would make sure they ate, see to their education, and care for their sick.
"Most importantly, we will give them something to be proud of," he said. "Nothing less than the country's future is riding on this offering. The state is grateful, Konstantin Romanovich." And here the president's voice turned to ice. "But be sure of one thing: Should anything go wrong, I shall hold you personally responsible. You and you alone."
# 44
#
**T HE CELL WAS TWELVE FEET BY EIGHT,** by Gavallan's measure, curdled cement painted a blinding nautical white floor to ceiling. One wall offered the comforts of a fold-down metal cot—no mattress; no blanket; no pillow—another a stainless steel toilet and matching sink. The door was battleship gray, a solid steel curtain with a rectangular spy hole cut into it. They'd taken his wallet and passport, his belt, his shoes, and his watch. The gun had earned him a kick in the ribs. Cuffed in the backseat of the police car, he'd looked on as a search of the rental car had turned up the authentic due diligence reports Pillonel had cached in his chalet. It went without saying they'd uncovered the compact discs, too. Isolated and alone, Gavallan was back at square one.
Metal groaned, a latch fell, and the viewing slat slid back to reveal a pair of pouchy brown eyes.
"I want to speak to the U.S. Embassy," Gavallan shouted, springing to his feet and rushing the door. "I'm an American citizen. I'd like to know why I am being held."
"Relax," grunted a put-upon voice. "You are thirsty?
Want a Coke? A Fanta?"
"I want to call my embassy. I get a call, don't I?"
"Sure you do. In a couple of days. Perhaps a week."
"A week? You've got to be kidding."
"Next thing you'll be asking for a lawyer."
"Damn straight I want a lawyer," said Gavallan. "Ever heard of innocent until proven guilty?"
An amused chuckle trickled through the slat. "Yeah, but not around here. We suspect someone's guilty, we put him in jail, _then_ we collect the proof. Sometimes it takes a month. Sometimes a year. It depends. I wouldn't worry, _mon ami:_ It's not us who wants you. It's your friends in America. The longest you'll be here is two months. They'll extradite you before then... unless, of course, you fight it. Now sit down and relax. I bring you a Coke, anyway."
"Just give me a phone."
The slat banged shut, and Gavallan slammed his fist against the door. Calm down, he urged himself. No one's going to find you guilty of a murder you didn't commit. Five minutes in front of a judge and you'll be free.
But he wasn't worried about himself so much as Grafton Byrnes. It was the fear of being trapped that rattled him, of being powerless to affect his friend's destiny. It was the stock dream of being chased down a street, your pursuers getting closer and closer while your flailing legs carried you nowhere. It was the terror of the silent scream.
In a little more than sixty hours, Mercury Broadband was set to go public. Kirov would get his two billion dollars. And Grafton Byrnes would have outlived his usefulness.
All Jett Gavallan could do was sit quietly and lament it.
**I T WAS A PERFECT DAY FOR GOLF.** At 5 **P.M.** in Zurich, the sky remained a regal blue, not a cloud to be seen. The temperature had crested at a lovely 75 degrees. The air smelled of pine and grass, and occasionally of the lake a few miles below them. Hay, freshly cut and rolled, sat ready for pickup in the fields nearby.
On the fourteenth green at the Golf & Country Club Zurich, located in the quaint township of Zumikon in the hills above his country's banking capital, Hans-Uli Brunner, Swiss minister of justice, spent a second longer studying the line of his putt. Ten feet for a birdie. Taking a breath, he approached the ball, settling his feet a shoulder's width apart. He looked at the hole, then at the ball, then at the hole again. _A birdie_. On a two-handicap hole, no less, where he already got a stroke. Sink this one and the match was his.
He steadied his head.
He drew back the blade of the putter.
As he stroked the putter toward the ball, an ominous tune chimed from within his golf bag. The first bars of "Beethoven's Fifth." The blade met the ball askew and it sailed three feet past the cup.
"Damn it!"
Stalking to the fringe of the green, he unzipped his bag and answered the call. "Brunner," he said gruffly.
"Is that any way to greet an old friend? And all this time I thought the Swiss were so polite. A nation of innkeepers?"
Brunner looked back toward the pin, where his playing partners were scowling openly at him. "Excuse me," he called, a gloved hand cupped to his mouth. "An emergency."
Though friends of thirty years, the three players did not disguise their displeasure. It was against club rules to carry a cell phone on the golf course, though in Brunner's case, a grudging exception had been made.
Zumikon, as the course was referred to, counted itself the most elite golfing establishment in Switzerland. Accordingly, the rules of golf were worshiped with a sanctity accorded the Ten Commandments. No better proof could be found than the Englishman brought over each April on a seven-month work permit to serve as club manager, normally a retired military man with long golfing experience. Only an Englishman would do. He was their mantle of legitimacy, their direct link to the "ancient cradle of golf."
Brunner hurried a few yards down the fairway until he was out of earshot of his fellow golfers.
"Good afternoon, my dear fellow," he said with a smile, the frustration of his missed putt eons away. He'd recognized the voice immediately, and knew it might promise many good things. "What a pleasant surprise. How have you been?"
"In truth, better, Herr Minister. I'm calling on a matter of some delicacy."
"Go ahead."
And for two minutes, His Honor, Bundesrat Hans-Ulrich Brunner, member in high esteem of the seven-man council that served as Switzerland's executive branch, listened as his "close friend" outlined his problem and how he wanted it resolved.
"Geneva, you say. He's wanted for murder? Yes, yes, I can understand that you want to deal with this on your own. Get him back into your neck of the woods. Good idea. As it happens, I have some close friends in the canton. It will be difficult, but I may be able to arrange things."
"I hope the usual arrangements are acceptable?"
Brunner glanced back at the green. He thought of the missed putt, the heated expressions, the apologies owed. Surely he would have to buy the foursome drinks, maybe even dinner. _A call on the fourteenth green_. They would talk about it for days. "The usual" was hardly adequate.
"It is the weekend," Brunner explained, "and we are talking about Geneva." His apology was pained and heartfelt. _"Alors, la Suisse Romande_. These Calvinists... I'm sorry to say they are notoriously difficult to convince."
"Will a million francs suffice?"
Brunner looked at the three golfers glaring in his direction. One raised his arms as if to say "What the hell is going on?" Brunner waved them onward. He would pick up his ball and return to the clubhouse at once. It was a sin not to finish a round, especially when he had a chance to take them all, and on such a beautiful day... but alas, duty called.
"You're too generous," Brunner responded at once. "Now, as to the account details..."
**I T WAS 8 A.M. PACIFIC DAYLIGHT TIME,** and in San Francisco the fog had returned. It hugged the streets, curling through alleys and climbing the city's steep hills like a fibrous, undulant snake. Approaching the end of Broadway in Pacific Heights, Roy DiGenovese pulled his car into the driveway and killed the engine. He took a moment to finish his double espresso, then wiped his mouth and climbed from the car. He was tired. The flight from Miami had been long and bumpy. A guy six-foot-two just didn't fit in the back of a commercial airliner—at least not in seat 32J he didn't, sandwiched between an Hispanic _Hindenburg_ and the rapper DMX's biggest fan. Maybe someday he'd warrant business-class travel. Maybe someday he'd get to ride in that Lear Mr. Dodson had been going on about. And maybe someday he'd be a Supreme Court justice. DiGenovese laughed at himself. It wasn't so bad being an optimist, he thought. Just keep it real.
Two cars had parked behind him, and their occupants met him on the sidewalk. This morning, they had no need to hide, no call to sneak in the back way. Leading his team of five special agents, DiGenovese knocked on the front door.
An Hispanic woman opened up a few seconds later. "Good morning," she said. She was older, dressed in blue slacks and a 49ers sweatshirt. Her eyes were cautious, scared.
"I'm sorry to bother you so early, ma'am," said DiGenovese, smiling and showing his badge. "We've come to take a look through Mr. Gavallan's belongings. It shouldn't take too long, an hour or two at most. We hold a warrant from a United States Federal Magistrate giving us a right to search the premises. Here's my card. If you'd like, you can call my supervisor. His name is Mr. Dodson. He's at the number written right there on the back."
"Mr. Gavallan, he is okay?"
"He's fine, ma'am."
DiGenovese made it a point to be polite. His mother had spent her working life cleaning homes and offices, and as a child he'd accompanied her on her rounds. He would never forget the dismissive glances, the rude comments, the smug ill will of the moneyed classes.
The woman studied the card for a moment before shrugging and yielding the door. "Okay. You can go."
"Thank you. We'll try to leave things as we found them."
DiGenovese set off through the house, directing his men to take the bigger rooms first: living room, den, guest bedroom, office. He wanted the master bedroom for himself. Gavallan was a former military man. If he kept a gun, odds were it was nearby, either in a night table or a closet.
The house was open and casual, with just the right amount of furniture, not cluttered like the homes of a lot of rich people. The floors were mostly wood, the décor kind of Spanish, giving the place a hacienda-like feel. By the time he reached the bedroom, DiGenovese had decided it was just his style. If, that is, he were to ever become a multimillionaire.
Inside the bedroom, he made straight for the night tables. He pulled out each drawer in turn, finding a few books, a handkerchief, a box of allergy medicine. He moved to the opposite side of the king-size bed. That night table was empty, not even a used Kleenex. Lifting the mattress, he ducked his head and checked for a gun. Nothing.
To the closet. Shelves to the left. A hanging bar to the right. He ruffled through the stacks of shirts and sweaters, at first setting them neatly on the floor and then, growing frustrated, flipping them onto the ground. No bullets. No holster. Nada.
DiGenovese paused, catching sight of himself in the mirror, seeing the furrowed brow, the look of stormy determination. Actually, he didn't want to find the gun. But not finding it drove him crazy just the same. Go figure.
He moved into the bathroom.
Drawers. Nil. Medicine cabinet. Nil. Beneath the sink. Nil.
_"Roy!"_
The call came from Gavallan's office. DiGenovese hurried to the oak-paneled study, collaring his excitement. "What do you got?"
"Check it out," said Rosemary Duffy. She was a short, stocky woman, thirty, with cherubic cheeks and sparkling blue eyes. "Gavallan's holster. Minus the piece."
DiGenovese rushed forward and examined the leather. It was creased and worn from long years of cradling a pistol. He rubbed a finger inside it, and it came away oily. "What do you think? A long time since the pistol's been removed?"
Duffy smelled the holster. "A week. A month. Hard to tell."
Within minutes the study became a charnel house of wild, barely disciplined activity. Books were pulled off the shelves. Pillows pulled from the sofa and gutted. The stereo yanked from its tethers. This time it was DiGenovese who got lucky. Pulling a well-thumbed copy of the Bible from the shelves, he spotted a hidden compartment in the wall. "Rosie," he called. "Get over here. Do your stuff."
Within a minute, Duffy had opened the compartment. Reaching in her hand, she came out with a cardboard carton six inches long, three inches wide and three inches high. The word "Remington" was neatly printed on each side of the box.
DiGenovese opened the carton of 9mm shells.
Half the shells were missing.
"Sonuvabitch!"
**H OWELL DODSON PUT DOWN THE PHONE.** He felt lightheaded, bewildered, and ashamed. How could he have been so wrong about someone? Why hadn't he listened more closely to Roy DiGenovese's warnings earlier? Why, even after the murders in Delray Beach, had he been so slow to warm to Gavallan as the prime suspect?
A holster with no gun, DiGenovese had told him.
A half-empty box of bullets.
And now this.
Dodson stared at the manila envelope that had arrived a few minutes earlier stamped "Department of the Air Force: Confidential" and the sheaf of papers that comprised Captain John J. Gavallan's service record lying neatly on the desk beside it. Pushing his bifocals onto the bridge of his nose, he began to read the papers again. Once was not enough. His conscience was as obdurate as his investigative instinct and it demanded he be presented with the error of his ways a second time.
He stopped a few pages in, his index finger frozen halfway down. The entry was innocuous enough: "Summer Semester 1985 / USAF SOC / Grade: Pass." And below it, in capital letters, signifying a commendation: "HONOR GRADUATE."
Translated, the entry stated that during the summer between his junior and senior year at the Air Force Academy, Jett Gavallan had attended the Air Force equivalent of Army Ranger training—the Special Operations Air Command course—and graduated at the top of his class.
When Dodson asked DiGenovese about the Air Force commandos, his assistant whistled long and low. "They're hard-asses, sir. Mostly trained for rescue ops, but rescue ops in hot situations. Lot of gunplay, hand-to-hand combat, that kind of thing. Mean muthas, if you get my drift. Best thing I could say is I'd let them back me up any day. They're pros."
A little probing got Dodson the following: Special Operations Air Commandos were trained to scuba dive and parachute, to support themselves off the land for periods of up to three months, and to master land navigation and map reading. That wasn't all. They were also taught to be experts in small arms and had to qualify as sharpshooters with an M16.
Jett Gavallan wasn't just a pilot. He'd trained as a commando. To use sophisticated weapons. To kill with his hands.
Gavallan was their man, plain and simple.
Dodson read a little further. Even with the glasses, he had to squint to make out the letters. Though he tried to focus on the words, all he could see were bodies. Bodies pitched onto their desks. Bodies strewn across the floor. Bodies slumped in the corner. A tear slid down Dodson's cheek and fell to the paper.
Removing his bifocals, Howell Dodson rubbed at his eyes.
It was time he got a new prescription.
# 45
#
**I N THE CLEARING,** the pickup's engine grumbled, then died.
Grafton Byrnes lay in the corner of the shed, curled into a fetal position, his face half bathed in mud. A steady rain fell. His clothes were sopping wet, as if he had just emerged from a swimming pool. His hair was matted and dripping. The sky was darkening, choked with clouds. He had no idea what time it was, only that it was evening.
_A little longer_ , he told himself. _You're almost there_.
An eerie wind whistled through the pines as rain blew through the cracks in the wall, peppering him like sand on a windy day at the beach. He was cold. He shivered in waves, violent spasms that racked his body, the tremors beginning in his lower back, then traveling up his spine with icy, muscular fingers that wrapped themselves around his ribs and squeezed mercilessly, provoking terrible, wrenching grunts.
The truck's door opened and closed. Byrnes clamped his jaw. By force of will, he stopped shivering. He lay still. Absolutely still.
Boots trudged through the mud, slurping and sucking. Keys jangled. Metal scratched metal and the padlock to the shed opened.
Byrnes gripped the stone close to his chest, the stabbing of his wounded thumb stoking his resolve to act. This was his chance. He was sick and getting sicker. His throat was raw, and he had begun coughing. He was starved and feverish. Another night in the open and he'd be too weak to stand, let alone escape.
A boot landed near his head. The mess tin holding his ration of tepid soup dropped into the mud, spilling half its contents. He made no move toward it. That morning, like the night before, he'd played the dying wretch, murmuring "Doctor" over and over again. Now he was silent. He sensed his jailer's presence, could smell the pig shit on his boots. He urged him closer. He wanted to feel his breath, to look into his eyes. Then he would strike.
The jailer hawked and spat on Byrnes's back, then he muttered a word and laughed.
The boots moved away. One step. Two.
_No!_ screamed Byrnes in private torment. _You cannot leave_. He gripped the stone harder. It was blunt and heavy. Trying to dig his way under the wall, he'd found it beneath six inches of topsoil and clay. Great treasures had been more easily won.
The jailer stopped, and Byrnes heard his breathing, the jagged wheeze of a lifelong smoker. He sensed the man's indecision. There came a new sound—the rustle of clothing—followed by a distinctive two-tiered _click_. The rain seemed to amplify it, and Byrnes knew it was a firing pin being cocked. He clenched his body, willing himself not to move.
_Lie still. Lie absolutely still_.
The gun fired, a deafening explosion inside the shed. The bullet impacted the ground an inch from Byrnes's eyes, blasting him with mud and stone.
_Lie still_.
Seconds passed.
The boots approached and prodded his ribs. First gently. Then less gently. Byrnes scrunched his face, biting back the pain. A labored groan as the jailer knelt on his haunches and slid his hands beneath the prisoner. Another grunt as he turned him over.
Byrnes opened his eyes. And in the moment before he smashed the rock against the Russian's cheek, he met his jailer's gaze.
"Bastard, go spit on someone else."
_"Chto?"_
The rock crushed the man's face, toppling him to the earth, leaving him sitting upright, stunned and immobile. A jagged gash on his cheek leaked blood.
Rushing to his feet, Byrnes brought the stone above his head. He was slow and awkward, and by the time he'd clamored to his feet, the jailer was up too, a mean, dumb grin on his face. A hand fell to his belt, and dropping his gaze from Byrnes, he searched for his pistol. Byrnes charged, ramming the Russian with his head, driving him against the wall. It was then he knew that his jailer was drunk. It wasn't the smell so much as the man's general lassitude, the confused coordination.
Throwing his left arm high and pinioning the man's neck, Byrnes scrabbled for the pistol, his infected thumb screaming at every contact. "Stop it," he yelled, retreating a second later, the pistol held in his right hand. He was irate, crazed, divinely pissed off. "You think you can lock a man up, barely feed him, leave him to die slow? Do you? Answer me!"
The Russian was leering crazily, teetering on his feet. He wasn't drunk—he was absolutely shit-faced. Three sheets to the fuckin' wind. "You ready? _Eh, Amerikanski?"_
"Don't," said Byrnes, his anger seeping from him. _"Nyet_. You stay there."
Muttering, the Russian took a step forward, spreading his arms as if entering the wrestlers' circle. "Come. You want fight?"
"Stay there."
The pistol was an old .22 long barrel. A peashooter. The cylinder held six slugs. Holding it proved difficult, but Byrnes managed by using both his hands, the palm of his left hand pressing the butt firmly into his right. "Stay right there," he said again. He had no desire to kill a man.
Then everything happened quickly, but in distinct steps, so that afterward Byrnes was able to dissect them in minute detail.
The Russian leaped forward, growling like a bear. Byrnes fired the pistol into his gut. A meek geyser of blood spouted forth, then died. The Russian swatted at it as if it were a fly, nothing more, and kept coming. Byrnes raised the gun. At a distance of two feet and closing, he fired into the man's chest. It was a bull's-eye. The jailer collapsed at the knees and fell face forward to the ground without uttering so much as a whisper.
Byrnes looked down at the body, the acrid scent of the spent cordite sickening his stomach. His ears rang from the shots, dizzying him. "Stupid fool," he said, half out loud, kicking the corpse lightly.
Kneeling, he turned the Russian over and began unbuttoning his coat. He started at the neck and worked his way down, helping the buttons through the eyelets with his index fingers, not daring to let his thumbs do the work. Even so, the pain was nearly too much. Several times, he drew his hands away and swore viciously.
Trouble arrived with the third button. It was stuck. He tried everything to get it undone but it would not advance through the eyelet. "Sonuvabitch," he said, taking a deep breath, looking toward the door. He needed the jacket. He needed something dry, something warm. Oh Jesus, he needed it.
"Slowly," he urged himself.
Moving closer to the body, he leaned over the Russian's chest. There was surprisingly little blood and the coat was not as dirty as he'd expected. With iron discipline he commanded his fingers to move. His left index and middle fingers carefully spread the eyelet wide. With his right index finger, he maneuvered the drab gray button through it. A smile creased his face. "Gotcha!"
_"Nyet!"_ screamed the Russian, sitting up, wrapping his hands around Byrnes's neck, squeezing with all his might, sharp uncut nails digging into his flesh. _"Nyet, Amerikanski."_
In a moment, the jailer was on top of him, straddling his chest, the man's weight full on his neck, strangling him. Byrnes fought at the hands, but could not grip them. The gun. Where was the gun? Byrnes groped around in the dirt. He was oblivious to the pain, to the daggers flaying his arms. Then he had it. Grasping the barrel, he bought the handle in a wide arc and struck the Russian across the bridge of the nose. Once. Twice. Blood gushed from both nostrils, but still the hands kept their grip, still those mad, leering eyes bored into him.
Byrnes felt the life ebbing from him, his vision dimming. Lowering the gun to the dirt, he turned it quickly and took hold of it by the grip. With a single fluid motion, he brought it up, laid the barrel against the jailer's temple, and pulled the trigger. Gunpowder exploded and a spigot of blood blew out the opposite side of the jailer's head. The death grip on Byrnes's neck lessened. The light went out in the Russian's eyes. Slumping, he collapsed on top of Byrnes, stone dead.
**T HE ENGINE RUMBLED ROUGHLY** while the heater blasted him like a wind from hell. Behind the wheel of the pickup, Grafton Byrnes sat staring at the fence. The sliding ten-foot gates granting one entry and exit to Konstantin Kirov's "dacha" were closed. Next to him on the seat was a remote-control device with a nine-digit keypad. He picked it up, held it in his right hand, using the fingers of his left to peck out a couple of tries. It was hopeless. He didn't even know how many digits the code required. Three? Four? Five?
"Fuckin' useless," he muttered, dropping it on the seat.
Byrnes was wearing his jailer's jacket, as well as his socks and boots. The gun was back in the shed with the dead Russian. It turned out it was loaded with five bullets, not six, and between them, they'd fired them all. He had drunk his soup and found a chunk of bread in the pickup. He was alive and relatively well and had a few hundred rubles, a pocketknife, and a pack of cigarettes to get him to the U.S. Embassy in Moscow.
If, that is, he could get through the double fences.
He stared at them awhile longer, wondering what twenty thousand volts would do to a car. If he drove over the metal, would it short out? Would the rubber tires ground the charge? Or would the touch of the fender conduct the electricity through the chassis and fry him like an egg on a griddle?
There was only one way to find out.
Byrnes put the truck into reverse and backed up about a hundred feet. Finding neutral, he gunned the engine a few times. He was a hot rod driver on a Saturday night. "Big Daddy" Don Garlits waiting for the green light. He imagined the Christmas tree counting down. The lights blinking red, red, red, and finally green.
Ramming the gearshift into first, Byrnes floored the truck. He passed the main cabin, the radio shack, the crematorium. And as he hit the fence, he loosed a savage howl.
Metal buckled, wire bent and moaned, the engine roared, and then he was clear, hurtling down the rutted dirt road at sixty kilometers an hour.
It was only then that Byrnes looked at the fuel gauge.
The needle hovered on empty.
# 46
#
**G AVALLAN WATCHED THE LAKE SLIDE BY,** a moss green mirror shattered into myriad shards by the sun's piercing rays. It was eight o'clock in the evening. After twenty-seven hours in custody, he'd been released with hardly a word, escorted from the rear of the police station, and ordered into the backseat of an unmarked Audi. Every time he asked a question the plainclothes officer next to him would mutter _"Ça va,"_ and give him a smile like he was the dumbest fuck on the planet Earth.
"Where are you taking me?"
_"Ça va."_
"Where is Miss Magnus?"
_"Ça va."_
"Is Mr. Pillonel in jail?" _Or was the rat ever taken there in the first place?_
_"Ça va."_
They played stop and go through a succession of traffic signals, turning left on the Guisan Bridge and crossing over the lake. Angry gray clouds spilled over the mountains on the French side a few miles up, gathering low above the surface and advancing toward them. A flash of lightning exploded from the sky. They were in for a gully washer.
The car slowed and came to a halt at the center of the bridge. Reflexively, he dropped a hand to the door and let his fingers toy with the handle. He had no illusions about his status. He might have been relieved of his cuffs, but he was hardly a free man. The car's doors were locked, the windows rolled up. One glance at his taciturn companion with the sinewy forearms assured Gavallan he was still a prisoner. The only question was where he was headed.
With a jerk, the car took off, zero to fifty in five seconds flat. The storm clouds were moving quickly toward them, a sheet of black rain dicing the water. The driver continued along the Rue du Mont Blanc, past the tourist shops selling cuckoo clocks, Swatches, and chocolate bars, veering left through a tunnel that took them under and around the train station. A sign ahead showed Annecy and Lyons to the left, Lausanne, Montreux, and Genève Aeroport to the right. The Audi shunted right.
Two minutes later they were out of the city, accelerating down an open stretch of highway. Green fields stretched to their left and right. Bales of hay sat rolled and wrapped in opaque plastic, ready for pickup and transport to the farmer's loft. The driver lowered his window an inch. Immediately the rich, loamy scent of ground under cultivation flooded the car. He shook loose a cigarette and, half turning, offered it to Gavallan.
"Smoke?"
"No thanks."
A whistling roar built in the air around them and suddenly an MD-11 passed directly over their heads, its pale metal belly close enough to touch. Strobing yellow landing lights beckoned to Gavallan's right, and beyond them the crenellated façade of the landing terminal.
The airport.
He was going home.
He didn't like the idea, but there was no use fighting it.
It wasn't until the car passed through a sentry gate and drove onto the tarmac that he started questioning the mechanics of his release. Didn't extradition require weeks, if not months, of legal wrangling? Shouldn't he have been asked if he wished to fight the order? If he hadn't been charged, by what authority were the Swiss loading him onto a plane to send him back to America? And why the hell were they letting him climb back aboard the chartered G-3?
He could see the plane crouched on the apron a few hundred yards away, landing lights on, turbines spinning lazily, an iridescent stream of exhaust escaping the engines. He had to wonder who was waiting at the other end. Dodson and his crew from the Joint Russo-American Task Force? Or would representatives of the Florida police comprise his handpicked welcoming committee? And why was he being smuggled out of the country like a plague bacillus?
Another Audi was parked next to the plane. He saw a door open and Cate's figure emerge. She seemed to hesitate, not wanting to board the plane. Two policemen bracketed her and began walking her to the aircraft. It was then that Gavallan sat up straighter, his nose pressed against the window. The plane was too big. It had too many windows. It wasn't a G-3 but a G-5; no mistaking it. The detailing was different too. A red pinstripe that hadn't been there before ran the length of the fuselage just below the windows. It wasn't the chartered jet at all.
And then he spotted the flag painted high on the tail, and he shivered.
The white, blue, and red tricolor of Russia.
**H E CAUGHT UP TO CATE** as she was about to mount the stairs.
"You okay? Did they keep you locked up this whole time?"
Cate lifted her shoulders, giving a fatigued nod. Her eyes were red, her hair being blown about her by a whipping wind.
Two familiar faces waited at the top of the stairs. Boris and Tatiana. A few hours behind in their forty-million-dollar jet, but no less vigilant.
"Hello, Mr. Jett," said Boris, as if they were old acquaintances from the club. His jaw was blue, swollen like a grapefruit, but his eyes said "No hard feelings." "You come now. We hurry. Storm will be here fast."
Gavallan glanced behind him. The Swiss police had formed themselves into a phalanx, and their stolid expressions said there was no going back. Offering Cate his hand, he guided her up the stairs. She mounted the first step, then stopped. Turning, she grabbed his shoulders and kissed him. "Tell me you'll understand."
Gavallan searched her eyes for an explanation, but saw only confusion and hurt. "What?"
Fighting the wind, Cate drew back the hair from her face and wiped away a tear. She opened her mouth to speak, then shook her head as if the thought were not worth mentioning. With a silken touch, her hand slipped from his. As quickly, she was herself again. The eyes cleared, the jaw firmed.
She mounted the steps rapidly, nodding perfunctorily to Boris as she entered the cabin. Over the wind, Gavallan was only just able to hear what he said to her.
"Good evening, Miss Kirov. Your father sends his regards."
# 47
#
**T HE TWELVE MEMBERS OF TEAM 7** crouched low on the riverbank, knees dug into the sandy moraine, watching, waiting. Fifty yards away, inside the compound, a man left the administration building and headed toward the pump house. He walked slowly, taking time to stretch and light a cigarette.
"Mark?" whispered Team Leader Abel. Each member of Team 7 was known only by his operational name. Personal details were not to be shared.
"Mullen. Jonathan D. Shift supervisor," responded Baker, his second in command. He did not add that Mullen was thirty-four years of age, an engineer who had received his degree at Purdue University in the state of Indiana. They had long ago memorized the faces and vital statistics of the crew who worked here. Mullen was easy. He never went without a Yankees windbreaker.
The American stopped a few feet from the pump house, flicked his cigarette to the ground, then opened the door and disappeared from sight.
To a man, Team 7 focused their eyes on the industrial landscape that lay beyond the fence, a dull metallic carcass sprawled beneath the half light of the midnight sun. Pump Station 2 of the Trans-Alaska Pipeline, or TAPS, lay at the foot of the Endicott Range on the border of the Arctic National Refuge. Its job was to guarantee that crude oil flowed smoothly through a sixty-five-mile section of pipe along the environmentally sensitive south fork of the Koyukuk River. The pipeline began two hundred miles to the north at Prudhoe Bay and cut south in a zigzagging pattern to Valdez, the southernmost Alaskan port that remained free of ice year-round. There, the oil was loaded through one of four primary pumping berths onto the giant supertankers that carried it to points south in America, Europe, and Asia. Over a million barrels of oil arrived at Valdez each day, and at any time some nine million barrels filled the length of the pipeline.
Pump Station 2 was built on a flat rectangle of land five hundred yards long and two hundred yards wide that had been razed from surrounding grasslands and forest. Three oil storage reservoirs stood in a row on the western side of the station, mint green lozenges two stories high and a hundred feet in diameter capable of holding 420,000 barrels of oil. Due to the breakdown of two of the Valdez Marine Terminal's four pumping berths, the reservoirs were topped out.
In the center of the facility, a power plant had been constructed capable of generating four megawatts of electricity daily. It took fuel to make energy and energy to move fuel. The power plant stood gleaming in the dusky night, an elaborate steel Tinkertoy with blue and red lights blinking from catwalks and stairways and metal mesh terraces.
Dormitories, administrative offices, and the pumping station itself occupied the grounds on the eastern side of the compound. Staffing ran between ten and twenty-five persons, depending on whether maintenance was being performed. Current manpower stood at eleven. The exclusively male complement worked twelve-hour shifts, seven days on, seven off. In five minutes, at the designated strike time of 2 A.M., a skeleton staff was set to be on duty: a foreman and a technician. Others were asleep, catching up on some precious rack time before suiting up for their grueling shifts at 6 a.m. Six days into their shift, the lot could be counted on to be tired, irascible, and unobservant.
The pipeline entered the complex from the north, a giant stainless steel tube forty-eight inches in diameter lifted three feet above the ground by a series of vertical support members, or VSMs, located every sixty feet. From afar, the pipeline looked as if it had been built yesterday. But Team 7 knew different.
The Trans-Alaska Pipeline was a disaster waiting to happen. Defective berthing pumps at one end. Rusted and corroding pipeline in between. Hazardous drilling practices on the North Slope. Over 50 percent of all shutoff valves—valves strategically placed to isolate sections of the pipe and minimize the volume of spills—were inoperative. The earthquake monitoring system designed to cut flow through the pipeline no longer functioned. A year earlier, a temblor measuring 5.7 on the Richter scale had shaken the residents of Central Alaska out of their beds. The monitor hadn't given a peep. Oil had continued to flow as normal. Not a single valve closed automatically, not one pump station shut itself down. It was a miracle the pipeline had not snapped clean in two.
Completed in 1977, the TAPS was an aging, brittle dinosaur, one slip from an ecological disaster of heartrending proportions.
Team 7 had arrived to give it the push.
Ghosts no longer, the members of Team 7 had exchanged their anoraks, fatigues, and combat boots for the casual attire favored by American blue-collar workers. They wore blue jeans and corduroys, denim jackets and parkas, work boots and baseball caps. In place of rank, they boasted the insignia of western apparel: North Face, Nike, and Levi's. The uniform of the enemy.
They had buried their parachutes, jumpsuits, and altimeters two miles away in holes four feet deep, now filled and covered with stones and moss and the natural vegetation of the region. Each wore a compact backpack no different from one a college student might be likely to carry. In it they hefted eight 125-gram sticks of C-4 explosives, a length of det cord, three electronic fuses, and a model TA9 remote detonator no larger than a transistor radio. All traces of the C-4 would theoretically vanish in the blaze following the explosion. If, however, investigators were to discover a trace of the plastique and to analyze its chemical signature, they would learn it belonged to a shipment stolen from an American armory two years earlier.
No one carried a weapon. Ghosts did not leave behind corpses.
From somewhere in the wilderness, a foghorn sounded. One bleat, rude and ominous, then silence.
The members of Team 7 scattered.
They were divided into three squads of four persons each, designated, in American military vernacular, as "Alpha,"
"Bravo," and "Charlie." Alpha and Bravo Squads climbed from the protective cornice of the riverbank and ran at a crouch to the fence surrounding the enclosure. The fence stood only six feet high. It was designed to keep animals away, not to deter intruders. Coldfoot was the nearest town and it was seventy miles away. Springing over the fence, they landed softly on the balls of their feet, eyes peeled for oil workers.
Alpha Squad moved to the right, toward the giant reservoirs filled to capacity with North Slope crude, oil from the mammoth field at Prudhoe Bay. Skirting the rear of the reservoirs, they kept out of sight of the supervisor's office (located some two hundred feet across an open concrete field) until they reached the fat, white intake pipes that fed oil into the tanks. Team Leader Abel slung his pack to the ground and removed two sticks of the green C-4, several fuses, and a length of det cord. He gave Baker one stick. One stick he kept for himself.
Immediately, Baker began to roll the stick between his palms to soften the putty. As the C-4 grew malleable, he broke the explosive in two, affixing a slim strip to joints in the pipe that had recently been welded together.
At the same time, Abel ran up the metal staircase attached to the side of the reservoir. He stopped halfway to the top where a blunt valve extended from the side of the wall. The valve allowed for the manual release of oil from the reservoir. After softening the explosives, he fashioned a long tubular section and wrapped it around the valve. With his fingers, he worked the putty into the crease at the base of the valve, as if stanching a leak. Plastique was a forgiving mistress, he thought as he pressed the putty against the cold metal; hit it with a hammer, burn it, shoot it even, and still it would not ignite.
Between his fingers, he held an electronic fuse, two inches in length, one half inch in diameter. From his pocket, he withdrew the det cord and plugged it into the electronic timer. Next he stabbed the det cord deep inside the putty. Det cord was simply a thin plastic cord filled with PETN, a fast-burning explosive. With a glance over the stairs, he snapped his fingers and dropped the cord to the ground where Baker picked it up, similarly attached it to the electronic fuse, and inserted it into the C-4.
From the corner of his eye, Abel spotted the other two members of his squad doing a like job on the next reservoir in line. He checked his watch. They were ahead of schedule.
Bravo Squad had split in two. Two men were now at the north end of the complex, lying on their backs beneath the pipeline itself. They worked quickly and efficiently, molding the plastique to the joints of the pipe, where one forty-foot section was welded to another. Det cord was produced, electronic fuses primed and inserted.
The other two men of Bravo Squad moved to the pump station itself. Sliding against the wall, they lifted their eyes over the windowsill and glanced inside. They saw no one. As expected, the staffers on duty were huddled inside the supervisory shed, where they would remain unless an equipment failure or breakdown summoned them to one part or another of the compound.
Turning the corner of the building, they opened the door and entered. Inside, they moved to the control panel, a wall of dials and gauges, none younger than twenty years old. Screwdrivers were produced. Wirecutters. Needle-nose pliers and a miniature battery. Their work required five minutes' time. The sensitive gauges that comprised the leak detection system and monitored the pressure of oil flowing through the pipeline had all been "adjusted." Even when all oil had ceased coursing through Pump Station 2, it would relay flow as "normal" to the other ten stations up and down the line.
A half mile north of Pump Station 2, Charlie Squad swarmed on top of and around a remote gate valve. The valve looked like the conning tower of a submarine. A red pennant flew from its uppermost walkway, crackling in the wind. Ninety-five such valves were placed up and down the length of the Trans-Alaska Pipeline, eighty-six of them remote-controlled to close in event of a rupture or spill. Plastique was carefully formed to the joists and the undercarriage of the 78,000-pound valve. The charge used was minimal, enough to rupture the pipeline cleanly without igniting the oil inside.
Tasks accomplished, Alpha, Bravo, and Charlie Squads met up at their assembly point, one hundred yards from the periphery of Pump Station 2. No one spoke a word. All took a knee as Abel activated the TA9 transmitter. Three white pinlights came to life, indicating that the electronic fuses were primed and a signal established. Moving his thumb to the ignition switch, Abel paused and, in the second before he depressed the button, imagined the horror of what he was about to unleash.
The charges placed on the reservoirs would simulate a "sparking" incident that occurred when rusting, corroded pipes brushed against each other. The oil would ignite. The reservoirs would blow. The ensuing explosion would shoot hundreds of thousands of gallons of flaming oil hundred of yards in every direction, scorching the sensitive landscape, fouling the air, and incinerating the crew of Pump Station 2. Seventy miles distant from the nearest habitation, the explosion would go unnoticed until the next day when Pump Station 2 did not respond to its routine morning calls.
The charges placed a half mile north of the station would rupture the pipe and allow the crude oil to flow freely onto the Alaskan plain. Oil would spill at the rate of forty thousand barrels per hour. As each barrel held forty-seven gallons of oil, nearly a million gallons of North Slope crude would foul the pristine meadows of the Arctic National Refuge each hour. The oil would form first a pond, then a lake, and soon it would spread into a black viscous ocean. The oil would seep into the ground and foul the water table. It would leak into the streams and the nearby Yukon River. Entire colonies of steelhead trout and chinook, chum, and coho salmon would be destroyed, their pristine habitats forever fouled.
As the oil spread across the rolling plain, it would take with it rookeries of Canada geese. It would tar the nests of the sandhill crane. It would permanently spoil thousands of acres of feeding area for caribou, elk, moose, and Roosevelt elk. By the time the spill had been stopped, somewhere between three and seven million gallons of oil would have blackened the Alaskan landscape.
Abel pressed the button once, firmly. Clouds of green smoke burst from the oil reservoirs and, farther away, from the remote gate valve. But there were no explosions, no fireworks, no hellish cataclysm to light the early-morning sky.
The Klaxon sounded again, this time longer, a full three seconds.
The only explosion, if indeed it was one, came from the sky, where a hundred feet above the ground, rafts of fluorescent lights flickered to life. The lights hung from the ceiling of an enormous hangar, eight hundred yards by a thousand.
Alaska had come to Severnaya high on the Siberian Plain.
A digital clock hanging from the observation tower at the far side of the hangar stood frozen at 8:23:51. The soldiers cheered, if briefly. On this last dry run, they had bettered their time by twenty-two seconds.
Their cheer died down, replaced by a grim determination, a silent resolve. One man after the other met his comrades' eyes. The time for training was past. After four months, the operation was at hand.
Clapping one another on the back, they moved off at a jog to return to their barracks. It was time to write the letter. In a month or two, their parents, girlfriends, loved ones (none were married or had children), would receive the short note explaining simply that Jan, or Ivan, or Sergei had decided to leave the country to seek a new life outside of Russia. He didn't know where or how long he might be gone, only that his absence would be a long one and that they should move on with their lives without him.
One meal remained, one night's sleep. Tomorrow, they would board planes to take them east across the top of the world.
To their destiny.
To America.
# 48
#
**T HEY TOOK OFF INTO THE STORM,** the last plane out before the clouds enveloped the airport, and Gavallan wondered if the pilot had disobeyed the control tower and said, "To hell with it, I'm taking off whether you like it or not." The sky was black, absolutely black, the plane jolting up and down and every which way with sudden, violent tremors.
"I want to talk to Cate," he said to Boris. "Excuse me, I mean Miss Kirov. Your boss's daughter."
The two men were seated facing each other at the rear of the roomy cabin. Cate was up front with the sofas and conference tables, Tatiana her assigned guardian.
"Sorry, Mr. Jett. You are not to talk to her." Sweat coursed from his forehead and his complexion had gone sallow. "Right now, you stay in seat."
"Just give me five minutes," Gavallan persisted, undoing his safety belt, standing. "It's important. I'll be right back."
Despite his sickly mien, Boris was up in a flash, thrusting an open palm against Gavallan's chest. "You sit. Understand? You talk to Kirova when you get to Moscow. Okay?"
Gavallan knocked away the offending hand. "Yeah, I understand."
Sitting down, he refastened his seat belt. Boris waited a moment, glowering above him. The plane hit an air pocket, fell for a second, then pancaked, shoving Boris into his seat. His hands scrambled for his seat belt. His mouth was open, breath coming fast and hard.
"You _should_ be scared, buddy," Gavallan whispered.
He knew he should be scared, too, but right now anger was kicking fear's ass in the emotional war raging inside him. Leaning his head to the right, he caught sight of Cate, seated forward in a separate grouping of sofa and lounge chairs closer to the cockpit. Even now, she looked as if she had things under control. Eyes closed, hands laid calmly on the armrests, head back, she looked as though she was taking a nap. He knew she had to be frightened to death. Why didn't she just show it like anybody else?
Suddenly, it was painful even to look at her.
He stared out the window. The wings were torquing something awful. The pilot had flown them directly into the maw of a thunderstorm. Either he was one crazy mother or he was under instructions to get his new passengers to Moscow as quickly as possible. Either way, he was reckless—the pilot's cardinal sin—and Gavallan hated him for it.
A bolt of lightning struck the aircraft, a hellishly bright flashbulb that bathed the cabin in pure, electric luminescence. Then came the thunder, a rollicking, tumultuous clap that seemed to explode inside the cabin itself. The plane rolled into a thirty-degree bank, the nose going down, down, down. Skeins of Saint Elmo's fire flitted around the bulkhead, a freakish blue and white light emanating from every piece of exposed metal. The port engine whined furiously, the turbine seeking purchase somewhere in the maelstrom of conflicting air currents. The fuselage shuddered as if God had taken the plane in his hand and was shaking it to within an inch of its life.
Gavallan looked around. Soldier Boris's eyes were closed, his chest pumping up and down, hyperventilating. Fore, Tanya had gone whiter than the dead. Her diamond blue eyes were wider than they'd ever been, the cords of her neck stretched to breaking. Her mouth was parted, and over the rattle and hum he could hear her moaning. Anytime now, he figured, she'd either break out into hysterics or throw up all over herself.
He caught Cate's eye. She was scared all right, and despite his distrust of her, his unremitting fury that she had deceived him not once but time and time again, he wanted to be next to her.
The shaking worsened. The starboard overhead luggage bin fell open. A handheld fire extinguisher tore loose from its clasps and crashed onto Boris's head. Oxygen masks dangled from the ceiling. In the galley, plates tumbled from their shelves, shattering. A chaotic choreography danced to the nerve-jangling accompaniment of Tatiana's grating scream.
Then, just as suddenly, there was calm. The plane righted itself. The nose came up and they resumed a steady climb. The engines purred. Sunlight flooded the cabin.
Unbuckling himself, Gavallan crossed to the Russian. Boris was shaken, and a gash on his forehead was seeping blood. Bastard, thought Gavallan, too bad it didn't break your neck. Finding his handkerchief, he pressed it to the cut. "Keep pressure on it."
_"Spaseeba,"_ said the Russian, removing the compress, seeing the blood and swearing. "You want to talk, you go now," he said, jutting a thumb over his shoulder toward Cate. "Maybe you don't have so much chance later. I take Tatiana to the bathroom. Clean her up. Go. I owe you favor."
Gavallan waited until Boris passed him, an arm around Tatiana's shoulder en route to the lavatory, then walked fore and took a seat facing Cate. He wanted to make light of the bumpy ride, to offer her his pilot's confident smile and say, "That was nothing," but the words caught in his throat. He'd left his store of niceties back on the tarmac, along with his willful naïveté. One question needed to be asked.
"Did he know about us?"
Cate looked at him for a moment, not saying anything, her flashing eyes boring into him with unsettling frankness. "Who? Father?" She gave a tired laugh. "Yes, Jett, he knew."
Gavallan glanced out the window. They had climbed above the clouds and were soaring across an azure sea. Sporadic lightning flashed below in a downy gray quilt, smothered eruptions that reminded him of distant gunfire.
"Well, that explains a lot," he said. "You both had me going, I'll say that. Jett, the consummate dealmaker. Mr. Big Shot wangling Mercury away from Goldman and Merrill and every other big swinging dick on the street. Hell, those suckers didn't have a chance. At least I know how Pillonel learned that Black Jet was getting the deal a month before I did."
"What do you mean?"
"Didn't you hear him this morning? Your father recruited him in November to do his dirty work. You know, to fake the due diligence and say that Mercury was more than the sum of its parts. The funny thing is, Black Jet didn't win the deal until January. Remember? You refused to toast the occasion. I drank the entire bottle of DP myself."
"Yes, I remember."
"I paid your father fifty million dollars of my firm's money to win a deal he had every intention of giving me anyway. This is enormous, Cate. I handed a man fifty million bucks to give me the royal screwing of the century. I sank my company for no reason whatsoever."
"Jett, don't do this to yourself."
"And you knew the whole time that it was rotten. The story just gets better and better."
"My father was involved. It couldn't be legitimate. It's that simple." Her tone was apologetic, conciliatory. "I tried saying everything I could to put you off the deal: 'Kirov's a crook.' 'You can't trust an oligarch.' I reminded you he'd gone bankrupt twice before."
"Yeah, yeah, yeah," said Gavallan. "We've already had this conversation."
"What else did you want me to say?"
"How about the truth?"
"I already told you. If you'd done your job, you would never have touched the deal to begin with."
"If you'd told me he was your father, if you'd told me about what happened to Alexei, I would have pulled the plug in a New York minute." He looked at the floor for a moment, then back at Cate. "Why?" he asked again.
She hesitated, her emotions close to the surface. "I couldn't. I just couldn't."
"Of course you could! Ten people, Cate. Ten people are dead. Graf... the company..." He shook his head, and then the anger, the frustration, the deception, grew too much for him to bear. Balling his hand into a fist, he pounded on the armrest once, twice, three times, with all his might. "He's my friend. My best friend. He's got kids. He doesn't deserve this."
"I didn't know what would happen," Cate shot back. "None of us did. You can't blame me. You have no right, no right at all. You don't know what I've been through, why I'm even here."
"Then tell me. But this time, I'd appreciate the truth, _Miss Kirov."_
Cate sat straighter, and when she spoke the apology that had cracked her voice had fled. Anger, disdain, conviction, seeped in, bonding the fissures. "Five years ago, I swore that Konstantin Kirov would never be a part of my life again. I vowed to myself that my father would never touch me again in any way. I moved back to the States. I changed my name. I found a job as a journalist. I built myself a new life from scratch. I became Cate Magnus and I stopped being Konstantin Kirov's daughter. I tried to pretend my father no longer existed, but it was impossible. For me, he will always exist, his birthright like a disease." She took a breath. "Did you know I skated, Jett? That I was an alternate to the Russian Olympic team in 1988 when I was only fifteen? The day I left Moscow, I quit. Did you know that my favorite writer is Chekhov? Or that I adore Tchaikovsky? That I cry every time I hear the Violin Concerto in D Minor? Since coming back to the States, I haven't read a page of Chekhov or listened to a single piece of Tchaikovsky. I can't, because _he_ gave me those things. He gave me his love of literature, of art, of music, and I will have nothing to do with him. Nothing! It's like having dirt all over your body that you can't get off. No matter how much I wash, how hard I scrub, I can't clean his blood out of my veins or his name from my soul. Inside, I will always be Katya Kirov. And I will always hate being her. At least on the outside I can be someone I like. Someone other people might like, too."
"You could have told me. I would have understood."
"I don't want you to understand! That's the whole point." Cate squirmed in her seat, and he could sense the frustration that was consuming her. "For me, he does not exist. Or do you think I should have given up everything I'd built, all I had become, to help you avoid a bad business deal?" She stopped, staring hard into his eyes. "Besides, Jett, I did tell you. You just weren't listening."
"I didn't listen? To what?" And then it hit him. He exhaled grimly, stunned. "You said no because he was your father."
Cate nodded. "When I saw that no matter what I said you wouldn't back away from the deal, I had no choice. If we stayed together, I knew it was inevitable you'd find out the truth, my secret history. I couldn't allow that. No matter how happy we might have been together"—she grabbed Jett's hands and squeezed them lovingly—"I would have been terrified of that day. I can see now that you would have understood... that it's me who's the problem... but I don't care. Even now, I despise you seeing me as his daughter. I hate you knowing. I'm not like him, Jett. Not at all."
"Of course you're not," said Gavallan after a moment.
But he was unable to bring himself to sit next to her.
**S O, IS CATE YOUR REAL NAME?"** he asked. The door to the lavatory was open and he could see Boris wiping a washcloth across Tatiana's face. "I mean, if your last name's Kirov, maybe the rest is different, too."
"Actually, it's Ekaterina Konstantinovna Elisabeth. My mother was a quarter English. Her grandmother married an English soldier who'd come to fight alongside the Whites in 1920."
"Where'd you come up with Magnus?" But even as he asked, the answer came to him. "Oh, I get it. 'Magnus' as in great... as in 'Catherine the Great.' Clever."
A modest shrug. "I had to come up with something."
All you had to do was look and you'd have known, Gavallan scolded himself. The high cheekbones, the Slavic eyes. It was all in front of you the whole time. He remembered how their conversations had always turned awkward when he'd made even the slightest mention of her father, the moderately successful international trader. Never a picture. Never a word.
"And what you said about Kirov—er... your father—it's true?"
"You mean about killing Alexei? Yes. It's true. Pretty awful, huh?"
"It's beyond that."
"All in a day's work for Mr. Kirov," she said, her jaw riding high, eyes to the fore, the soldier bearing up under her ungodly burden. He could tell she was fighting to keep it together, doing whatever jig or two-step she danced to prevent all those jagged edges rustling around inside her from ripping her to bits.
"What hurt most was the betrayal," she went on, the hurt ripe in her voice eight years later. "Learning that your father wasn't the man he'd built himself up to be. He meant everything to me. Mommy was dead. I had no brothers or sisters. He was the world."
"I can imagine."
"Did you know that originally he was a curator at the Hermitage? Icons were his specialty. He was one of the world's leading authorities on religious subjects. When the winters grew cold and the heating in our apartment building gave out, we'd spend whole weekends inside the museum just to keep warm. He would take me through the workshops below the palace and show me how the paintings were renovated—so much paint, so much albumen, so much shellac. You should have heard him preach. 'Art was honest. Art was untainted. Art was the truth. Everything we could be, if only we tried.' This was in '85 or '86. 'Perestroika' was the word of the day. Glasnost was in full bloom. Suddenly, it was okay to admit how worm-eaten the regime was. Art was his way of proving that even in a lousy world, light still shines. Or at least that's what he had me believe. All the while he was smuggling icons from the museum's stock out of the country, building up a fortune on the side."
"What about Choate? What about growing up in Connecticut?"
"Don't worry, Jett, I'm not a total phony. I'm still a Choatie. My father had me thinking that one of his rich American friends was paying my tuition. When he was arrested and the checks suddenly stopped coming, I was able to convince the headmaster to let me finish up my classes and graduate. One semester without tuition, he could let slide. He couldn't kick out the valedictorian, could he?"
"I guess not," said Gavallan.
"Anyway, soon Kirov was back in business. No more skulking through dark alleys. Now he could conduct his affairs in the open. The K Bank, he called it. Finally, he was the businessman he'd always aspired to be. Everything aboveboard. On the straight and narrow. I forgave him. Worse, I believed in him again. 'Katya, we are making Russia great again!' he would say. 'Come join me. Work at my side.' You know how persuasive he can be."
Gavallan nodded. Yes, he knew. He had believed Kirov too. Every word.
"I took a plane to Moscow the same day I finished my exams at Wharton," she continued. "I couldn't wait to get to work. To help make Russia great again. To rebuild my country. The Rodina, we call it. The motherland. And then..."
Behind them the lavatory opened, and Cate clipped her words. The sound of running water mixed with weary sobs drifted into the cabin. Checking over his shoulder, he saw Boris's muscled shoulders easing into the gangway. Cate tapped his knee, and he said, "What?"
When he turned back, he saw that she'd opened her purse and was handing him her pink compact. "What should I do with these?" she asked, a thumb flicking her makeup kit open. Tucked inside were the minidiscs Pillonel had given them from Silber, Goldi, and Grimm.
"Jesus, you still have those?"
Cate nodded eagerly, her eyes darting over his shoulder. "Take them. Quickly."
Gavallan recalled the painstakingly correct and intimate strip search to which he'd been subjected in Geneva. He'd assumed Cate, as a fellow prisoner, had suffered like treatment. "No. They're better with you," he said, glancing over his shoulder. "If anything happens, get them to Dodson."
"But—"
"Cate. Keep them. Use them if you get a chance." He held her eyes, signaling he had no illusions about what awaited him when they landed.
Rising, he headed aft, loitering in the cramped gangway long enough to allow her to conceal the financial records that were their only proof against Konstantin Kirov and the key to the salvation of Black Jet Securities.
# 49
#
**W HAT DO YOU MEAN** he's not in your booking facility?" Howell Dodson demanded, the phone to his ear. He was very angry. His cheeks had points of red in them, and he jabbed at his distant interlocutor with the arm of his bifocals. "You only got him yesterday. Would you be so kind as to tell me what goes on in the Swiss penal system between Saturday night and Sunday afternoon?"
"He was released on order from the government," responded the unnamed party who had fielded Dodson's call. "I am sorry."
_"Released? To whom? When?_ I'm the government who wants him. Do you mean to tell me some other country has issued a warrant for Gavallan's arrest?"
_"Non, non_. You misunderstand," the polite French-accented voice chirped. "Our government ordered his release. The _Swiss_ government, Monsieur Dodson."
Dodson chewed on his eyeglasses, fighting a rearguard action against fury, guilt, and incredulity. Gavallan was gone? It couldn't be. Lord help him, it just couldn't be. He looked toward the matching strollers parked in a corner of his office. The boys were having their morning nap, bless their souls, while their mother attended a Baptist service in Georgetown. Outside, a cloudy sky promised rain. At nine-thirty on a Sunday morning, the streets of the nation's capital were asleep.
"Who signed for his release?" Dodson asked, in a calmer voice to avoid disturbing his two dozing generals.
_"Un instant, je vous en prie_. One moment."
Waiting, Dodson walked across the room and gazed down at Jefferson and Davis bundled up in their powder blue blankets. It was hard not to lean over and give each a kiss on the cheek. Gone barely two days and he had missed them like the dickens.
Learning that Gavallan had been detained and incarcerated by the Swiss gendarmes, Dodson had returned to Washington the night before. It had turned out Gavallan was their man after all. He owned a gun similar to that used in the Cornerstone shooting. The gun was missing—ergo, he had taken it with him. He'd received training as an elite commando. And of course, he had every reason to want Luca dead. Though as yet circumstantial, the evidence was overwhelming.
In Geneva, the slippery voice returned to the phone. "A lawyer named Merlotti signed for Mr. Gavallan."
"And he's with the government?" Dodson asked.
_"Non, non_. You misunderstand. He's a private citizen, of course. A prominent attorney, actually."
"But you said Mr. Gavallan was released to the government."
_"Non, non_. You misunderstand," the man said again in his singsong voice. "I say that the government permitted Mr. Gavallan to be released to Mr. Merlotti."
"And for whom does Mr. Merlotti work?"
"That I do not know."
Of course not, Dodson grumbled inwardly. No doubt it would constitute a violation of your canons of secrecy, confidentiality, and inbred chicanery. "I'm sorry, sir, but I didn't get your name yet?"
"LeClerc. Georges LeClerc."
"Well, Mr. LeClerc," Dodson said, "if I cannot speak with Mr. Gavallan, would you be so kind as to connect me with your own Detective Sergeant Panetti?"
"That is not possible. Sergeant Panetti is on holiday."
"Will he be back tomorrow?"
_"Non, non_. You misunderstand. He is on _summer_ holiday. He will return in three weeks."
If Howell Dodson "misunderstood" one more time, he vowed to himself, he was going to catch the next plane to Geneva and beat LeClerc over the head with the phone until he _under_ stood that the FBI meant business. Then the words sunk in.
"Three weeks!" Dodson shouted, losing his cool, then checking his voice and darting a glance at the twins. Jefferson stirred and began to cry. "You've got to be—"
The light went on in his head, and he stopped arguing. It was a put-up job. LeClerc was running interference for some very powerful, very nasty shit who'd pulled some strings high up in the Swiss government to have Jett Gavallan released. Some VVIP who did not want anyone knowing his identity.
"And that's it?" Dodson picked up the wailing infant and held him to his shoulder. Patting his boy on the back, bouncing lightly as he walked the room, he wondered if this officious Swiss prick actually expected the United States Federal Bureau of Investigation, the goddamned finest law enforcement outfit on Earth, to give up searching for a fugitive wanted for capital murder as easy as that. The mere suggestion infuriated him.
"I'm afraid we cannot help. Mr. Gavallan is no longer in the country."
"Isn't he?" asked Dodson. For once they were getting somewhere. "Has he returned to America?"
"I'm afraid I cannot say where he has gone."
_Of course not_. "Just one last thing," said Dodson, as Davis began to stir in his carriage. "The girl who was with him? Miss Magnus? Where is she?"
"They leave together," LeClerc replied promptly, eager to be free of his responsibilities to international justice.
"That's what I thought," said Dodson. _"Au revoir."_
_Asshole_ , he added silently, in a most ungentlemanly tone.
**T HEY CANNOT DO THAT,"** declared Roy DiGenovese unequivocally. "If a suspect is detained on the basis of an international warrant, he may be released only to the custody of the government that issued the warrant, and then only if he's waived his right to fight extradition. It's a mistake. Has to be. He must have been transferred to a different jail, maybe to a federal prison. There's one near Bern. It makes sense. He'd be nearer our embassy."
DiGenovese had rushed back from San Francisco, arriving at six that morning. Still glowing from his triumph, he was dressed in a sports shirt and blazer, his black hair neatly combed. Dutifully, he held young Jefferson in his arms, cradling him back and forth.
"That's what I aim to find out," Dodson stated. "I'm as appalled as you are."
The phone rang and he picked it up. It was the international operator with the private number of a Mr. Silvio Panetti. Jotting down the phone number on his blotter, he thanked the operator, then called Panetti.
The detective answered on the third ring. Dodson introduced himself and asked what in the world was going on with Gavallan.
"Business as usual," answered Panetti, sounding half in the bag. "We pick up your Mr. Gavallan Saturday afternoon. A formal arrest on the Interpol mandate cannot be filed until Monday. Through their contacts, Mr. Gavallan's lawyers were able to secure his release before the charges were ever filed. Officially, Gavallan was never apprehended. It is a triumph of technicalities."
Dodson thought it was a crock of shit, pardon his French, and he planned on filing a formal complaint. "Any idea where he went, Detective Sergeant?"
"You know, Monsieur Dodson, I am on vacation," Panetti protested perfunctorily. "I am not supposed to discuss official police business. On the other hand, I was not planning on taking this vacation, so what the hell, I tell you. After leaving the station, Gavallan drive directly to the _aeroport_. Please understand, I did not see him, not with my eyes, but my friend say he climb on a private jet."
"Was Miss Magnus with him?"
"Yes. She go, too."
"Any destination?"
_"Je ne sais pas_. I don't know and I don't ask. I am already too much involved, I think. I am smart, Mr. Dodson, not brave. You want to know where Gavallan go? You find out for yourself."
"Surely you can phone the airport...."
"Surely you can, too. _Au revoir, Monsieur. Bonne chance."_
Dodson hung up the phone.
"Where is he?" asked DiGenovese. "Is he headed back this way?"
"Gone," said Dodson, taking Jefferson from his assistant and laying him on his shoulder. "Vanished into the night."
# 50
#
**T HE PLANE TOUCHED DOWN** at Moscow Sheremetyevo Airport just after midnight. A light rain fell, collecting into greasy puddles on the tarmac. The air smelled stubbornly of smoke and exhaust. Deplaning, they were led to a convoy of black Chevrolet Suburbans. A corps of rugged, sloe-eyed men in navy tracksuits lined the path. One waved his hand at Gavallan, pointing the way to an opened door. A funeral cortege, thought Gavallan as he slid into the backseat. A day or two and the same cars will be taking me to the cemetery.
The ride into the city took forty minutes. Cate sat up front, sandwiched between the driver and Boris. Tatiana slouched next to Gavallan, sullen and listless after the flight. The driver turned on the radio and a mishmash of wailing voices, discordant guitars, and arrhythmic tambourines filled the car. Top 40 from the Muslim republics to the south, Gavallan thought. It was brash, unsettling, and foreign, and it made him feel alone and abandoned. Stretching an arm forward, he found Cate's shoulder. A moment later, she took his hand, intertwining her fingers with his.
For a time they drove dead straight along a quiet four-lane highway. Billboards advertising a variety of products kept them company. Samsung. Volvo. Fisherman's Friend. Cate asked the driver a question and he answered politely, as if she were a guest, not a prisoner. She'd picked up two languages in a day: French, now Russian. Waiting for her translation, he thought, _This is the real Cate, and I don't know her at all_.
"We're going to my father's clubhouse in Sparrow Hills," she said, turning and meeting his eye. "Across the river where all the _nomenklatura_ used to live. Brezhnev, Chernenko, Andropov."
"Just like Pacific Heights, huh?" Gavallan said icily.
They were in the city now, and it looked like the other parts of Russia he'd visited, only larger, more impressive, more desolate. The highway had been swallowed by a boulevard eight lanes across and they continued without regard for traffic signals. Green meant "go"; red meant "go faster." The grand avenues craved a dignified audience—skyscrapers of steel and glass, noble town houses, even a decent minimall. Instead, they were awarded stooped stone apartments and crumbling office buildings weeping soot and grime, all wedged together, all the same height, all devoid of personality. And then Gavallan remembered why: Personalities were allowed only inside the Kremlin. Or maybe these days in Sparrow Hills.
Suddenly everyone was sitting straighter, stiffly even. The driver turned off the music. Cate's shoulders left the seat. Even Tanya lifted her head from the glass to look. The motorcade descended a long slope, and ahead he could make out a bridge and, running beneath it, the choppy, evanescent surface of a broad, fast-moving river. To his left, the night sky softened, lit with a warm chiffon glow.
And then he saw it too. Bathed in the arc of a hundred discreet spotlights, a tall, curving fortress wall ran the length of the riverbank. The wall was painted an imperial yellow, with stone battlements rising every fifty feet, and behind it, silhouetted against the blue black sky, soared the swirling onion domes and proud towers that housed the seat of the Russian government. The Kremlin.
He was in the heart of Mother Russia, and to his eye, it still looked every bit the evil empire.
**J ETT, MY FRIEND.** How nice to see you again."
"Cut the bullshit, Konstantin," said Gavallan, walking past the man, ignoring the outstretched hand, the offer to play it as if the events of the past five days were nothing more than a difference of opinion. "We're not friends now. We never were."
"I suppose we weren't," replied Kirov. He looked fatigued. His pallor was funereal, his eyes pouchy and rimmed with wine black circles. "Come and sit. I'll be brief, then we can go to bed."
"I'd rather stand."
The two men faced each other in a stark, glacial space the size of an emperor's ballroom. The floor was a sea of pale travertine, the walls painted a glossy white. A sleek Italian couch and matching chair, both an incongruous orange, sat in the center of the room, a low-slung coffee table showing too much chrome between them. The only other furniture was an antique cocktail cabinet miles away at the far end of the room. If they seemed alone, it was an illusion. A brace of security guards stood outside the door, ready to enter at a moment's notice.
Cate had been shown to a study across the foyer. "I haven't seen my father in six years," she'd said. "I'll happily wait a few more minutes."
"A drink?" asked Kirov. "I heard you had a rough flight in. Something to calm your stomach? Cognac? Brandy? A Fernet, perhaps?" He strode to the liquor cabinet and poured two snifters of brandy from a cut-glass decanter. Even at one in the morning, he was his usual elegant self, dressed in a tailored navy suit and solid maroon tie.
"No," said Gavallan. "I want to talk to Graf Byrnes."
"I'm afraid that isn't possible. He's spending a few days at my dacha in the country. It's quite remote. No electricity. No phones. But don't worry: I'll make sure you two see each other tomorrow."
"That won't do. I want to speak to him now. You and I have nothing to discuss until I know he's alive and well."
"Oh, he's alive. You have my word. As for 'well,' that's a different matter altogether. I'd like to say his condition rests squarely upon you. What you do. What you don't do."
"News flash, Kirov: Mercury isn't going near the market until either Graf or I say so. Without our go-ahead, the deal will be pulled. Enough controversy has surrounded it already. My disappearance will be the last straw."
"Will it?" Kirov sneered, lifting the snifter to his lips and taking a generous draft. "There seems to be some concern that you've gone a little crazy. Hitting Mr. Tustin on the trading floor. Flying to Florida without alerting your staff. Fleeing the FBI. I have it on good authority that the offering will go forward as planned without your go-ahead."
"Whose authority is that?"
"Now, now, Jett. You don't expect me to show you all my cards, do you? Suffice it to say it's someone who can run the show perfectly well in your absence. Besides, you shouldn't be too angry if your friends decide not to follow your orders."
Seething, Gavallan circled the grouping of furniture. Who did Kirov have his hooks into? Bruce? Tony? Meg? Had the words not come from Kirov's mouth, Gavallan never would have thought it possible. Despite his fury, his heart beat slowly. His hands were cool and dry. His vision had sharpened. It had been eleven years since he'd felt this way. It was his calm in the face of a coming storm. "Battle-bright," they called it.
"And just what do you think is going to happen down the road?" he asked. "Mercury won't last two weeks once it goes public. You'll have analysts crawling over your operations like flies on shit. They're a tough group—nosy, ambitious, eager to make their reputation at your expense. They'll suss out the company's problems in no time."
"I'm not worried. With proceeds from the offering, we'll quickly shore up any remaining operational deficiencies."
"The money Mercury receives from the offering is slated for acquisitions that will insure you meet your forecast growth rate. That's cash to move forward, not to come up to speed. Miss one quarter's estimates and the stock will fall into the cellar. Miss two and it's all over. The price will dip below a dollar and you'll be delisted from the Exchange."
"I can assure you we have no intention of missing our estimates," said Kirov. "As per your own instructions, we have a few surprises in the pipeline. 'Unexpected' good news that will increase our earnings and allow us to beat our own optimistic expectations. What did you call it, Jett? 'Sandbagging'?"
"Sandbagging" was a common enough practice, a simple trick designed to goose the price of new issues six months out. The idea was to keep a little good news in your back pocket: a juicy contract about to be signed, word of another cable route about to be granted, a new and unforeseen use for a company's proprietary technology—anything that would augment your revenue stream and boost your earnings. Six months down the road, when the time came to issue your first earnings report, you peeled away the blinds and announced that "due to the dramatic customer response" to your new software or router or "fill-in-the-blank," your earnings had beat forecasted estimates by a nickel. The stock jumped 10 percent and everyone was smiling. Bankers. Customers. The investing public.
"Sandbagging's one thing," retorted Gavallan. "Lying about your customers and your revenues is another. What are you going to say about your problems with Novastar? Having the prosecutor general riding your tail doesn't quite fit with your investment scenario. It's my experience that investors prefer to see CEOs of newly listed companies in the boardroom, not in jail."
Kirov laughed softly, but his irritation was beginning to show. He was blinking incessantly, his fingers appraising the knot of his tie. "I agree that jail isn't part of our 'investment scenario.' If you're talking about Mr. Luca's article, I read it, too. 'Mercury in Mayhem,' I believe it was titled. A shame no one else will have the pleasure. Boris is very thorough. He promises me he erased the story from Mr. Luca's computer and that he confiscated every copy in the apartment."
"Wrong again," said Gavallan. "Even Boris couldn't stop Luca from E-mailing the article to his friends before he was killed. It's a matter of time until it turns up on the Net."
"So what?" spat Kirov. "One more rumor floated by a dead lunatic. One more piece of jetsam drifting over the ether. The public will pay it no mind. As for Yuri Baranov, I don't think he's going to be holding office much longer. I have it from a reliable source that the president is dissatisfied with his performance. Let me be the first to proclaim the investigation into Novastar Airlines closed."
Gavallan stared into Kirov's eyes, catching a glint of real malice. He wasn't sure what Kirov was hinting at—Baranov's impending firing or his murder. He knew only that he was dealing with a killer, a man utterly without morals for whom murder was a legitimate business tool.
"I think there's been a little misunderstanding between us," he said, walking up to the oligarch, standing close to him to emphasize the difference in their heights, in the beams of their shoulders. "I'm the guy's got you by the short and curlies, not the other way around."
"Is that right?" Kirov kept his eyes locked on Gavallan's, neither man giving an inch.
"Before I visited Silber, Goldi, and Grimm's offices this morning, I took a few precautions to cover my ass, just in case something like this happened. You see, I'm pretty thorough, too. First thing I did was make a copy of Pillonel's original due diligence report and send it to my lawyer. We spoke, and I filled him in on everything that's gone on over the past couple of days. I told him that if I didn't get in touch by Tuesday morning at the opening, he should contact the stock exchange and the SEC's enforcement division. I gave him instructions to hand over the real due diligence report and to inform them that Black Jet was pulling the Mercury IPO."
"You're lying."
"Am I?" Gavallan picked up the brandy and downed it in a gulp. Fuck it. He needed a drink even if the poison came from a scoundrel like Kirov. "Pillonel was a big help too. Sang like a canary, Jean-Jacques did, right into my attorney's tape recorder. I wouldn't say the confession was entirely of his own free will, but so what—it'll do in the short run."
"You're lying." Kirov broke off his stare and retreated behind the sofa. "You didn't have time to make a copy."
"We had plenty of time."
"No, no. It's not possible. It simply isn't." The words were high-pitched, almost hysterical. Kirov's mouth twitched and his eyes furrowed in thought. "Why should you have bothered taping a confession? Was it not your intention to turn Pillonel over to the police? No. No. You're lying." And as he reasoned through Gavallan's actions, his voice calmed, the steady confidence returning. "You couldn't have known you were being followed. You had every intention of flying back to the States with your precious evidence. Maybe even with Pillonel. There was no reason to take precautions at that point. I wouldn't have. You're lying. I know it."
Gavallan shook his head, his iron gaze letting Kirov know he was not. Putting down the snifter, he pointed a blunt finger at his host, his jailer, his willing executioner. "Here's the deal: Tomorrow morning, you will wire me the fifty million dollars you borrowed from Black Jet. With interest. Graf, Cate, and I climb on board a commercial airliner and fly back to the States. And you will issue a statement that due to unforeseen market conditions, you've decided to postpone the offering to a later date." Gavallan thought about Ray Luca and the others at Cornerstone, enraged that no one would ever be brought to trial for the crimes. "Believe me, you're getting off easy."
Kirov's eyes seemed to bulge from their sockets, to expand with boiling hatred. "So now you're issuing ultimatums? Look around you—you're hardly in the right place. If you like ultimatums, however, I'll be happy to give you one of my own: The Mercury offering will go through. It will be a bigger success than any of us dares imagine. We shall earn our two billion and then some. And you, dear friend, will help see to it. Do you know why? Do you? Because if you don't, Mr. Grafton Byrnes will die. Slowly. Terribly. Very, very painfully. And you will be on hand to watch it."
"Fuck you, Konstantin. You've got the wrong guy. I don't respond well to extortion."
Kirov laughed, an ugly derisory snort. "We'll see very shortly what you do or do not respond to. Personally I think your story about Pillonel is utter shit. But not to worry: One way or another we'll ferret out the truth. Either Jean-Jacques Pillonel will tell me or you will." He smiled invitingly. "I guarantee it."
# 51
#
**T HIS WAS WHERE ALL PATHS LED.**
To Russia.
To Moscow.
To her father.
Cate waited alone in the wood-paneled den off the entry hall. The lights were dim, and the room smelled of new carpet and worn leather. Through the heating vents, murmurs of a violent conversation drifted to her ears. Jett and her father were arguing, and it made her afraid. She'd spent her last teenage years here. Something about the Edwardian house seemed to goad its inhabitants into perfectly dreadful behavior. She used to lie with her ear to the floor, listening to every word of her parents' fights, wincing, crying, silently ordering them to stop and make up.
_The past_.
Everywhere she looked it was crowding in on her, suffocating her with nightmares and obligations.
Moving to the window, she drew a curtain and peeked outside. If she lifted her eyes, she could make out the top floors of Moscow State University, towering above a stand of trees. Well past midnight, the building's lights were ablaze. Built in the late 1940s as one of seven
"Stalin Skyscrapers" meant to showcase Soviet prowess in architecture and engineering, the university was ever the brilliant trophy. The stern spires and bold, conformist tower were masterpieces of their kind and stirred in her pangs of nostalgia so strong as to be painful. It was not the first time this evening she'd been overcome with sentiment.
Passing St. Basil's, the Novodevichy Monastery, the Kremlin, even the most mundane of office buildings, she'd found her throat choked with emotion. These were the landmarks not only of the city but of a childhood she'd willed dead and buried, and each in turn provoked a cascade of memories. Cate and her mother pausing for a tea in one of the unsmiling cafes that dotted the upper levels of the GUM department store. Cate skating for the first time on an impromptu ice rink in the courtyard of their apartment building, the result of a broken main that had spewed water into the air for two weeks running. A reverent Cate, barely thirteen, passing through Lenin's tomb for the first time, frightened for the life of her to stare down at the great man's embalmed face, her teacher stopping her and forcing her to look, berating her in the sacrosanct hall to open her eyes and gaze upon the motherland's savior. She'd obeyed and fainted straightaway.
But the stirring went deeper than nostalgia. It went to her heart. To her blood. It was her history awakening inside her. The past beckoning her to return. She was no longer Catherine Elizabeth Magnus, but _Ekaterina Konstantinovna Elisabeth Kirova_ , a Russian woman born in Leningrad to a Catholic mother and a Jewish father thirty years ago. There was nothing her devotion to the West could do about it. Nothing her love for Ayn Rand or her addiction to Bruce Springsteen could do to rectify the error of her birth. All were accessories she'd acquired to paper over her true colors. Garments designed to deceive, to camouflage, to lie. The intended victim, of course, being none other than Katya Kirov herself.
Too wound up to sit, she dropped the curtain and made a tour of the room. The walls were covered with photographs, cartoons, framed articles, and here and there a diploma or honorary citation. Their common link was Konstantin Kirov. There was her father with Boris Yeltsin. Her father with Gorbachev. A photo with Bush the Elder. Oh, how he loved mingling with the big names, if only so he might position himself as champion of the free media. If, that is, one's definition of "free media" meant using your television stations, your newspapers, your radio networks, to trumpet your own pet causes. If "free media" meant decrying taxes on aluminum production in order to favor your smelters in Krasnoyarsk. Or savaging the academic who had issued a report claiming that oligarchs exerted a drag on the economy equal to two percentage points of GNP. If so, then Kirov was your man.
Cate stared at her nails and stupidly wished she'd had a manicure before coming. She felt dirtied by her time in a jail cell. Catching a glimpse of her reflection, she flicked a strand of hair from her face, then rushed to her purse to apply some lipstick, only to throw the makeup back inside before she'd finished. Why did she give a damn about pleasing her father? She hated him and everything he stood for. He was a thief, a plunderer, a murderer. The epithets grew stale on her tongue, and pausing for breath, she was left with her original question: Why did she give a damn?
Unhappy sitting, she returned to the window and looked outside. A stream of headlights rolled up and down Kutuzovsky Prospekt. Marshal Mikhail Kutuzov, hero of Borodino, who had defeated Napoleon not on the battlefield but off it, by withdrawing his troops from Moscow and burning the city in his wake. There was something about his methods, something about sacrificing one's children for personal glory, be it a nation's or a businessman's, that rang a bell with her.
And taking a breath, she found the answer to her question. It had been lying in front of her for days, months, years even. She gave a damn simply because he was her father. Her blood. And she would never be free of her ties to him.
Worse still, he had defeated her once again. For all her actions to halt Mercury, her promises to avenge Alexei's death, her desire to help Jett, she'd come up short. She still had no way to punish her father for his sins. She was ever the little girl powerless in her father's presence. And she hated herself because of it.
**H ELLO, FATHER.** It's been a long ti—"
Konstantin Kirov crossed the study in three quick steps, slapping her hard across the face before she could finish her words. "Shut up, whore."
Cate fell back onto the couch. Her hand dabbed at her mouth and came away red with blood. She struggled for something to say, but the onrush of emotions, hot and angry and prideful, cluttered her throat, leaving her defenseless and speechless.
Kirov gazed down at her, shaking his head. He looked older, smaller, ascetic even, but he had the same energy, the same conviction.
"How dare you even look me in the eye?" he went on. "Look away. Look at the ground. Out the window. Just don't set your eyes on me." He stalked to the window, threw back the curtains, then turned on her again. "Here is my darling daughter returned from America with her new name and new boyfriend. Have you any idea the shame you bring to my house? The disgust I feel showing you to the men who work for me? I brought you into this world. I cared for you in difficult times. I gave you an education worthy of a princess. And how do you repay it? First by sending your weak-spined boyfriend to the police with some ludicrous accusations that I was fixing the market for aluminum. I'll never forget that boy. That Kalugin. He lasted five minutes before spilling his guts, sobbing that you put him up to it. You should thank me for relieving you of his company. It was a favor, believe me."
Aghast, Cate stared at her father. He was no longer just a corrupt businessman, no longer merely a killer even. He'd become a monster. Inhuman. A beast. "Stop it," she said, her voice a whisper.
But Kirov went right on, trampling over her words as he had always trampled over her wishes, her desires, her opinions. "And now," he said, "after I allow you to make a new start, you dare to use all your resources to destroy the greatest professional achievement of my life. You conspire with the prosecutor general's office, you feed that sick-minded day trader rumors, you turn my partner against me—"
"Stop it!" she shouted. "Stop your lying! You can lie to Jett. You can lie to Baranov, to your adoring public. But you will not lie to me. I am your daughter, though the word scalds my tongue. With me you will speak the truth." Cate stood and pushed her way past him.
"The truth?" Kirov spun, following her, his expression saying he found her suggestion murderously amusing. "Oh, it's the truth you want, is it? You are a big girl now. A grown woman. I suppose I can tell you the truth. The truth is simple: We are building a new country. We are raising a phoenix from the ashes. What you may consider extreme is in fact mundane."
"I'm all for building a new country," she said through tears. "But legitimately."
"Legitimately?" Kirov jumped on the term. "The word is not in the Russian vocabulary. How can there be legitimacy when no one knows how to define it? You think everything must be done the American way. It is easy for them. They draw upon a tradition of common law dating back a thousand years. A thousand years ago Moscow was a swamp. Huns, Goths, Tatars... we had them all at one time or another, riding pell-mell across our territories. Law was whoever had the faster horse, the sharper sword. 'Kleptocracy' is hardly a recent term. Only this time it's the businessmen doing the heavy lifting, not the government. Have you any idea what it took to bring Mercury this far? What it costs to bribe the Czech communications minister? The going rate to secure cable construction permits in Kiev? Do you? So what if we're not up to Western standards of transparency? We're starting from so far back it's a miracle we've gotten this far. If we'd kept to the letter of the law, Mercury would consist of two cans and a string. Be reasonable, my love. We are only asking for a chance."
"But you cheat. You lie. You kill. Ten people, Father. Why? Just to disguise the murder of one?"
"What are the lives of ten people to insure the prosperity, the education, the livelihood, of thousands? I would have killed a hundred if necessary. A thousand, if the Rodina demanded it."
"Another lie. You didn't kill Ray Luca and the others for the Rodina. You killed them to help yourself. To take Mercury public. To steal your billion dollars and make yourself rich."
Kirov approached her slowly, reaching out and taking her face in his hands. "But, Katya, don't you see? I had no other choice. As Mercury goes, so goes the country. I _am_ the Rodina."
Cate grasped her father's wrists and took his hands from her face. She felt sickened, her soul nauseated. "No," she said. "You are not the Rodina. You are one man. You are greedy and desperate and you will fail. Oh, Father, you will fail. You cannot build a country on evil. If anyone should know it, it is we Russians. Hasn't our history taught you anything?"
"Yes," he said, suddenly thoughtful, sliding his hands into his pockets, pursing his lips. "It has taught me that perhaps we weren't ruthless enough. I, for one, will not repeat the mistake."
"You won't succeed. We won't allow you to. Not I, and not Jett."
Kirov laughed softly. "The defiant ones. A pity, really."
Cate looked at her father, wondering for the thousandth time how she could share his blood, carry his genes. "I'm the one who is ashamed. I am not your daughter. Not anymore."
Kirov's smile disappeared, and an ugly resolve settled about his face. "Be thankful you are, Katya. Be thankful you are."
His eyes said the rest.
_Or you would be dead, too_.
# 52
#
**T WO VEHICLES APPROACHED** from across the valley, their xenon headlights cutting an electric blue swath before them.
_Christ, they're driving like hell_ , thought Grafton Byrnes, squinting at the harsh beams. As they neared, he yanked at the wheel, steering the truck onto the shoulder of the narrow road. The cars flew past in a flash, but a flash was all the time Byrnes needed to recognize them. Twin black Suburbans. Outland sentinels from the Kirov fleet.
Byrnes rammed his foot impotently against the accelerator. The engine did not respond. The pickup continued its downhill run, the gearshift parked in neutral, speedometer showing seventy kilometers per hour. He had run out of gas two miles from the dacha. Somehow he'd babied the truck to the edge of the slow grade that led from the hilltop observation post to the sweeping flatlands below. He'd been coasting for a while now. It was hard to tell how long. Five minutes. Maybe ten. He checked the rearview. The taillights were already specks, Satan's fireflies receding into the distance. They would be back. And when they came they wouldn't be coasting at a leisurely fifty miles an hour. They'd be hauling ass at a hundred easy, looking for the truck they'd passed five minutes earlier.
Despite his anxiety, a wave of exhaustion swept over him, and Byrnes gripped the wheel more tightly. His vision blurred. His jaw fell to his chest. Just as quickly the exhaustion passed, the band of cold sweat dampening his forehead its only reminder. He took a breath, steadying himself. Had he really expected to get away? He was feverish and half starved. His body was struggling to fight off the infection raging in his hands. He couldn't touch the steering wheel without wincing. How could he have thought himself in any condition to make it to Moscow?
Because he had been trained as an officer and an officer's duty was to escape.
Because Jett Gavallan would have done the same damned thing.
Because there was no other choice.
The slope began to flatten. The needle on the speedometer eased to the left—65... 60... 55. The rain had stopped. A half moon played hide-and-seek behind fast-moving clouds, its slow-blinking light casting a silver shadow across an endless vista of waist-high grass. Desperately, Byrnes scanned the horizon, looking for some sign of a village, a service station, an all-night 7-Eleven, where he could pop in, buy a coffee, and make a lifesaving call to the embassy in Moscow. The plain was infinite and dark, sheaths of grass waving back and forth in a whispering wind.
The speed continued to bleed—45... 40... 35. Byrnes guided the pickup off the road, letting it cut a path across the grass for a few hundred yards, hoping he might find a gully, a hollow, where he could hide the truck. No such luck. The truck hobbled to an arthritic halt on flat ground. Byrnes got out and looked back. He was close enough to the road to see the pavement. The roof of the pickup shimmered in the moonlight. Where to go? Where to hide? He didn't think about his chances. He had none.
He began to run. South. Toward Moscow.
The ground was hard and even. The grass fell effortlessly before him. He crossed back over the highway, hoping to confuse the pursuers he knew would soon come. His step acquired an ugly, pounding rhythm. Once he'd routinely run three miles in eighteen minutes. He'd cranked out a hundred sit-ups in a hundred twenty seconds and dropped from the bar after twenty-two pull-ups. Once he'd eaten nails for breakfast, spat fire, and drove his country's hottest jets.
_Once..._
Byrnes laughed bitterly at himself. He was forty-four years old. He drank a half bottle of wine every night with dinner. In the twenty-odd years since he'd graduated from college, he'd added thirty pounds to his runner's frame. The last time he'd run any kind of distance was a year ago on vacation in Hawaii with his fifteen-year-old boy, Jeff. After a lousy half a mile, old Dad had veered off the white sand beach and ditched at sea, crashing his well-marbled bulk into the delightful ocean water.
Byrnes thought of Jeff now, and of his daughter, Kirsten. He saw their faces in front of him. He ran to them. He ran to the warm saltwater oasis. His breath came hard. He was sweating, really sweating, beads of perspiration rolling off his forehead, stinging his eyes. The boots were small, tight in the toe. A blister was coming up on the heel. Another hour and his feet would be bleeding.
Still, he ran.
He ran because he was scared. Scared of going back to the dacha, scared of being caught, scared of what they would do to him. He didn't have the strength to go through another session with Boris.
"No," he whimpered aloud at the thought, fear beginning to grip him.
Mostly, he was scared he might betray his friend and the company they'd built together.
And for a minute his steps lengthened, his gait quickened, and he swore that he would not allow himself to be used by Kirov.
He thought of the pistol, of the cylinder that held five bullets instead of six. It was an old rancher's trick. You always left the barrel that was in the firing position empty. That way there were never any accidents. To advance the cylinder, you had to pull the trigger.
He wanted the gun.
He wanted the bullet. One bullet.
Mr. Kipling knew what to do in such an instance. Mr. Kipling, every soldier's favorite.
_When you're wounded and left on Afghanistan's plains_ ,
_And the women come out to cut up what remains_ ,
_Jest roll to your rifle and blow out your brains_
_An' go to your Gawd like a soldier_.
Panting, he recited the quatrain aloud. Again and again. Until he had no more breath left to talk with. His stride slowed. His legs grew heavy. His chest burned.
_An' go to your Gawd like a soldier_.
He heard the roar of a motor and looked behind him. Xenon beams swept over the grass; the murderous engine growled. He ran harder, dodging to the left, shooting quick glances over his shoulder.
"No," he said aloud, sucking in short, dry breaths. "God, no."
The lights dodged left, too.
Byrnes ran.
# 53
#
**H E HIT ME.** Six years and not even a hello. Just a slap across the face."
Cate walked into the bedroom, a hand to her mouth. She looked gray, pale, her eyes drifting here and there. Gavallan was at her side in an instant. Taking hold of her hand, he pulled it from her mouth and examined the wound. A nasty cut marred her lower lip. It had stopped bleeding, but without a stitch might open again. Closing the door behind her, he ventured a quick look into the hallway. A shadow sunk back into the doorway of the next room. One of Kirov's security boys. So far he'd counted nine of them patrolling the corridors.
"Come in," he said, leading her to the bathroom. "Let's get that cleaned up."
"Kind of you, Mr. Gavallan. It's not often a disloyal, disgraceful slut gets any TLC, especially at two o'clock in the morning."
He moistened a washcloth and dabbed at her lip. He had no words for her, no way to assuage her tortured feelings. Abruptly, she pushed him away and stormed into the bedroom.
"I'm leaving," she said. "Damned if he can keep me here." She spotted her travel bag and scooped it up. "After all, I'm a traitor to his blood. An unrealistic dreamer who's getting back at her father for simply protecting his own interests. He shouldn't want anything to do with me." She reached the door and turned the knob. Locked. She tried again and again, finally slamming her fist against the wood-grained panels. "Let me out," she cried. "I'm going home. _My real home_. My name isn't Kirov. It's Magnus. Do you hear? I'm an American now."
Gavallan laid his hands on her shoulders, turning her slowly, taking her in his arms. "Sit down. Have a glass of water. It's going to be all right."
"No, it's not. It's not going to be all right. He's going to kill us. Like he killed Luca. Like he killed Alexei. Like he kills anyone who's in his way."
"No, Cate, he's not going to kill us. He just wants to frighten us a little."
She turned, staring at the walls, knowing as well as Gavallan that the room was wired for sound, and probably for pictures, too. "You win, Daddy," she said. "I'm scared. I'm scared as hell."
Gavallan got her to the bed and gave her some water. After a few minutes she recovered her calm. Her eyes cleared and her breathing eased. "Shit, that hurts," she said, touching her lip. "The little prick."
She caught Gavallan's eye, and they laughed. After a minute he walked to the television and turned it on. He flicked through the channels looking for something loud or raucous enough to allow them to talk or at least whisper freely. He stopped at Channel 33, a smile flitting across his face. A basketball game was under way, Lakers versus the Knicks. Game three of the finals. Turning up the volume, he retook his place on the bed next to Cate. "Tell me what your father had to say."
"He's rebuilding the country and we're stopping him. Mercury's his greatest professional achievement and we're letting a few minor details sour our view of the whole enterprise. We don't see the big picture. I'm the criminal, not him. I'm the one guilty of treason. Of harming the Rodina. He's gone insane, Jett. I swear it. _'L'état, c'est moi.'_ He practically uttered the words himself."
"What about tomorrow? Do you know where he's taking us?"
"No. He didn't say. We didn't end the conversation on an up note. He implied I should be glad not to be in Ray Luca's shoes with all I've done. What about Graf? Oh, Jett, I'd forgotten him for a moment. How is he?"
"Alive, from what I gather. More than that your father didn't say, except that we'll be seeing Graf tomorrow."
"Thank God," said Cate. "What else did you say to him? I hope you didn't threaten him."
"Only with the truth." Gavallan nodded subtly for her to go along. "I told him about Pillonel's confession and that my attorney in the States will turn over the due diligence reports if he doesn't hear from us. I told him I wanted the fifty million back from the bridge loan and that we had all better be on a plane to the States tomorrow. Graf included."
Kobe Bryant swished a three-point bomb and the crowd at the Staples Center went crazy.
Cate cast her head to one side. "Did he agree?"
Gavallan heard the hope in her voice. "No. He's convinced I'm lying. Says he'll find out for himself tomorrow whether I'm telling the truth."
"What does that mean?" Cate looked away, and when her eyes returned to him they carried the dreadful intent of her father's words. "No, Jett, he can't. You've got to—"
"Shh." Gavallan nodded reassuringly. "I'll be all right. He still needs me. I have the feeling he can't walk away from this deal."
"So do I," said Cate. "There's more to this than just Mercury's continued success as a company. It's much more than a mere business matter—Mercury's grown larger than just an initial public offering."
Gavallan rose and walked to the window, drawing back the curtain and peeking outside. The view gave onto an interior courtyard where two of Kirov's Suburbans were parked. A trio of guards were busy giving the SUVs the chauffeur's professional polish, leaning their butts against the chassis, talking furtively, and smoking cigarettes. Each cradled an Uzi beneath his arm. Gavallan tried to open the window but found it locked. The frame had been nailed to the sill.
Releasing the curtain, he took a second look at his carpeted prison cell. The room was large and luxuriously decorated in shades of brown and ochre, with a wooden four-poster bed, a sofa, a desk, a wet bar, a plasma-screen television hanging from one wall, and what looked like an authentic Matisse hanging from another. Welcome to the Stalag Four Seasons.
"Looks like we're here for the duration."
"The duration?"
"Of the night." He refused to think about the next day, about Kirov's keen desire to find out exactly what he did or did not know about the Russian's operations.
"I know you wanted to get me alone," said Cate, "but isn't this a bit much?"
"Hey, you know what we Boy Scouts say: 'Take it where you can get it.'"
"Very romantic."
Gavallan slid his arm around her waist and drew her near him so that their shoulders were rubbing against each other. Turning toward her, he lifted her jaw with the tip of his finger. He looked at her eyes, serious, compassionate, and defiant, and the faint circles beneath them; at her cheeks wiped clean of blush; at her broken lip stern, uncompromising, slightly aquiver. "You don't look so bad all rough-and-tumble, Miss Magnus."
"It's Kirov. Better get used to it."
"Okay, Miss Kirov. But not for long."
"Is that a promise?"
He answered with a kiss, gentle as a candle's breath.
"Oww," she moaned, smiling. She stood. "Stay here a second." Opening her night bag, she went round the room and covered the Matisse with a skirt, the mirror with a pair of pants, and the triptych of Moscow by night with her blouse. "I don't care if my father hears me," she said, "but I'll be damned if I let him see me."
**H E ALWAYS BEGAN WITH HER SHOULDERS.** The skin there was a shade darker, more luminous, an intimation of her mysterious self. Gently, he pulled her shirt away and kissed her, breathing deeply to get the scent of her, enjoying the firm response of her flesh, feeling the muscles quiver at his touch. He kissed her neck, the cusp of her jaw, and then, unable to wait any longer, he lay her down on the bed. She threw her arms above her head and narrowed her eyes. It was a temporary capitulation, a tactical maneuver to lure him helter-skelter into her ambush. She moaned, and he could feel himself falling into her, a boundless, head-over-heels plunge into a warm, velvety abyss.
Somewhere within him he found the power to stop, if only for a second. He raised himself on an elbow to look at her. He saw not just her beauty but the sum of her self staring back at him: her strength, her courage, her will. Her humor, her obstinance, her frailty, her fear. She met his gaze, and her frank ardor roused in him a heady sensation, a cocktail equal parts respect, desire, honor, and lust that he had come to recognize as love.
"Jett."
Her voice was husky, ripe, unfulfilled. Raising a hand to the back of his head, she ran her lithe fingers through his hair and pulled him to her.
He surrendered.
**I N ANOTHER BEDROOM,** in a less surveilled wing of the house, Konstantin Kirov lay awake, unable to sleep. Through a drizzly haze he was visited by a revolving medley of faces—Baranov, Volodya, Leonid, Dashamirov—each taking a turn to lambaste and curse and threaten him. Scariest of all was the father of modern Russia himself, Lenin, all too alive, rising from his dank tomb and waving an angry fist at him. "Mercury must go through!" he shouted as if addressing a band of discontented dock-workers in Petersburg. But instead of bread and peace, he was extolling the benefits of free market economics, of unfettered capitalism. "The offering is essential for the well-being of the nation. The president demands it. Your brother demands it. The future of the Rodina depends on it. On you, Konstantin Romanovich. _On you."_
Sitting up, he pushed back his sheets and rubbed his eyes. He didn't know why he was worrying so. He had Gavallan. He had Byrnes. The Private Eye-PO was no more. True, he had a few loose ends to tie up, but soon those would be eliminated as well. He'd tracked Jean-Jacques Pillonel and his wife to a hotel near the Zurich airport where they were spending the night awaiting a 9 **A.M.** departure to Mahé in the Seychelles. With a sly smile, Kirov silently advised all bettors not to wager on the Pillonels making the flight. Seats 2A and 2B would remain unoccupied, their occupants last-minute no-shows.
And then there was Baranov. Yuri Ivanovich Baranov, the prosecutor general who didn't know when enough was enough. In the morning, Kirov would have a word with him, too, and that would be another problem taken care of once and for all. Mercury would go through exactly as everyone demanded, Lenin included!
Instead of lamenting his fate, Kirov urged himself to celebrate it.
One thing still bothered him: Katya. His beloved and unloving Katya. Sadly, he recalled the sting of his hand across her cheek. I'm sorry, my love, he apologized, seeing the blood curling from her lip, her eyes wide with shock and pain and fury.
Oh, Ekaterina Konstantinovna, why can you not understand your father? Why can you not see the sacrifices that must be made to insure our people's welfare? And our family's? Is it wrong to desire a nice station in this life? To earn enough to provide a few luxuries to brighten our short days? Can you not see that I am a visionary, a leader, and, as will be evident in a few short hours, a patriot, too?
Floundering for an answer, Kirov scowled, then rose from the bed. Crossing the room, he sat down in front of a bank of small video monitors, twelve in all, discreetly hidden behind a false wall of books. His daughter's room was dark. She had covered several of the cameras, but not those embedded in the crown molding. Playing with the controls, he was able to zoom in on the bed. Faintly, he made out her sleeping form, and next to her, Gavallan. It really was a pity about their not marrying. He could have used an investment banker in the family. He had little hope of Katya—or _Cate_ , as she called herself these days—falling for the next director of Black Jet Securities.
Turning up the volume, he heard only steady breathing.
"Sleep, Katya, sleep," he whispered, kissing a finger and touching it to the monitor.
Kirov returned to bed and soon fell into an uneasy slumber. The dream came as he knew it would, the walls closing in on him, the ceiling falling toward the bed. He could smell the damp, taste the rot of centuries. Somewhere deep inside a voice promised him he would never go free.
_Lefortovo_.
**G AVALLAN ROSE FROM THE BED** and padded to the bathroom. Darkness his cloak, he found the sink, lowered himself to a knee, and set to work. The first screw came off easily, the second cost his fingertips a layer of skin. Careful to make as little noise as possible, he jostled free the capton—a slim rectangular piece of metal that controlled the vertical motion of the drain—and laid it beside him. So much for the grip. Now he needed a blade. His hands ran from the U-shaped PVC drainage pipe to the smaller bore fishnet cables that supplied the water. A long slim rod, smooth and round as a screwdriver, ran between them, a bolt attaching it on either end. Only brute strength would free it. Sliding himself farther under the sink, Gavallan fastened his hand around the rod, counted to three, and yanked it furiously downward. The rod broke off cleanly, with hardly a snap.
Suddenly, he smiled. There was a time when his parents would have been glad if he'd said he wanted to be a plumber, or a carpenter, or just about anything else that would have stopped him from walking around town with his fists in front of him looking for a brawl to get into. With a bolt of clarity, he remembered how he felt in those days. The wild yearnings that would well up inside him, the unheroic desire to slug another man in the face—always someone bigger, someone imposing—to see the blood gush from his nose, maybe even hear the crunch of bone. For the life of him, he'd never understood why he was such a mean little bastard.
Now, these twenty-five years later, he had the answer. Divinity. God, nature, the force—whatever you wanted to call it—had provided him with some early on-the-job training for what was to come later in life.
_For what was to come tomorrow_.
Gathering the rod and capton, he slid from beneath the sink. A length of curtain wire would bind the two together; padding from beneath the carpet would serve as a grip.
He only needed something to sharpen the rod into a killing blade.
# 54
#
**_Y OU WANT KIROV_**, _I can help. Meet me at Pushkinskaya Metro, southwest exit, at seven o'clock. And make sure to bring a briefcase. You won't believe the shit I have on him."_
A coarse laugh, and the call ended.
Yuri Baranov, prosecutor general of the Russian Republic, put down the phone. Eyes rimmed with sleep, he checked his watch. It was six o'clock. Through the curtains, a hazy sun filtered in. It took him a few moments to clear the cobwebs from his head and evaluate whether the call was legitimate or a crank. Since the investigation into Novastar had begun, his office had been inundated with complaints against Konstantin Kirov. Everything from an employee's griping about her unfair dismissal to anonymous promises to obtain Novastar's offshore banking records. Baranov thought the call a ten-to-one shot, but decided to go anyway.
Rising, he ducked beneath the clothesline that bisected his one-room apartment, picking off a shirt, some clean underwear, and a pair of socks, then shuffled to the window. There was a carton of milk on the sill, along with a jar of pickles, some plums, and a plate of smoked herring left over from last night's dinner. He owned a refrigerator, but it was broken and he couldn't afford to repair it, never mind the electricity to run it. Opening the window, he brought the food inside and performed a hurried ballet, dressing and eating at the same time. A strip of herring while he buttoned his shirt. A plum while he threaded his belt. A last sip of milk as he knotted his tie.
Four days after seizing some eight hundred fifty-three pages of documents from Kirov's headquarters, his investigators had yet to find the evidence they needed to link Kirov to the millions of dollars stolen from Novastar Airlines. Oh, they'd dug up false receipts, double billings to clients, all manner of petty schemes to launder money and avoid paying income taxes. The practices were illegal. The state would file suit. But they'd come across no smoking gun that Baranov could set before a magistrate. The few documents he had found from the Banque Privé de Genève et Lausanne had led nowhere. The Swiss bank would not even confirm that Kirov was the holder of the numbered account.
Finished dressing, he considered taking some of the precautions that had become second nature to any government official working to put a crimp in an oligarch's style. He thought about calling his deputy, Ivanov, and asking him to come along. No, he decided; Ivanov deserved to eat breakfast with his family. Better to request a police escort. Baranov dismissed that idea, too. The police would never show up on time, even if they had a car parked in Pushkin Square. Besides, he wasn't so old that he couldn't meet an informant on his own. He was hardly meeting a gang of thugs in a dark alley at midnight. This was Pushkin Square. Early on a Monday morning there would be throngs of passersby.
Dressed in yesterday's trousers, his scuffed briefcase strangely light in his hand, he headed down the stairs and walked the fifty meters to the subway. The morning air was crisp and clean, not yet fouled by the legions of automobiles that had taken Moscow hostage these last years. Street signs advertised the latest American films. One showed four grotesquely obese Negroes seated on a couch, smiling like idiots. Baranov had no doubt but that the picture was an unquestionable masterwork, something Eisenstein himself might have directed. Giant billboards demanded he drink Coke and enjoy it. Part of him bristled at this relentless onslaught of Western imperialism, this secret invasion of the Rodina that was occurring can by can, frame by frame, ad by ad.
_Relax, Yuri_ , he told himself in a voice that belonged to the new millennium. _Let the people enjoy themselves. Life is hard enough as it is. Besides, Coke beats the hell out of Baikal any day_.
He arrived at Mayakovskaya station at six forty-five. Descending the escalator to the Circle line, he ran his impromptu caller's words over and over in his mind. _You want Kirov, I can help_ , the man had said. Baranov tried to put a face to the voice. Was it an older man or a younger one? A Muscovite or someone from Petersburg? He decided the voice was familiar. Was it someone in his own office? Or someone they'd interrogated from Kirov's? A Mercury insider, perhaps? Vexed at his inability to come up with an answer, he caught himself breathing harder and gnashing his teeth.
He had forgotten just how much he hated Konstantin Kirov.
**J EAN-JACQUES PILLONEL** was having a terrible dream.
He saw himself from afar, a tired, bent man dressed in prisoner's garb, gray dungarees, a matching work shirt, his feet carrying the heavy boots one saw on the rougher sort of motorcyclist. The man, who was at once him and not him, was marching in a circle around a dusty yard. There were no walls, but a voice told him he was in prison and that he was not free to go anywhere else. He continued his rounds, but with each circuit his steps grew heavier, his body denser, his mass harder to move. He began to sweat. He was not frightened by his plight as a prisoner so much as by the impending impossibility of mere locomotion. He realized that his burden was not one of extraneous weight but of conscience, and that he would never be rid of this load. A current of anxiety seized him, threatening to paralyze his every muscle.
The scene shifted and he was looking in the mirror at this man who was and was not himself. He was gaunt, poorly shaven. His eyes were lost, forlorn. This isn't right, he was telling the familiar visage in the mirror. The reward for honesty must be greater, the relief more fulfilling, certainly longer lasting. The anxiety grew stronger, arcing up his spine, bowing his shoulders. Sensing he had no more time, he raised a fist and drove it into the mirror. The looking glass shattered. Everywhere shards of green and silver glass fell to the floor.
Struggling to the surface of consciousness, he felt a rustling in the bed next to him. A kick in the legs. He heard a shout, but it was muffled, distant.
_"Claire? "_
He opened his eyes.
His wife of thirty-two years stood across the room, held in the arms of a black-clad intruder. He had her by the neck, one hand over her mouth, the other pinning a knife to her throat.
"Claire!" he yelled, sitting up. A half second later a coarse, powerful hand cupped his mouth and forced him back down onto his bed.
"Silence!" The voice belonged to a stocky figure clad entirely in black. Black trousers. Black sweater. A black stocking snubbing the nose, rendering the lips flat, grotesque. The intruder wore plastic gloves and in one of his hands he held the knife. It was a monster, the blade twelve inches long, partly serrated, curling upward to a hungry tip.
"You've been a naughty boy," he said in accented French. "You don't know how to keep secrets."
_"Non,"_ Pillonel argued. "I can. I can."
The flattened lips drew back into a smile. "We shall see, Monsieur Pillonel."
**T HE SUBWAY PULLED INTO PUSHKIN SQUARE** at six fifty-seven. The timing was perfect, thought Yuri Baranov while riding the wooden escalators up to the mezzanine level. And as he entered the tunnel that passed beneath Tverskaya Ulitsa to the Metro's southwest exit, his gait assumed a triumphant rhythm. Something told him this was the real thing. That Kirov's goose was finally cooked. His step faltered only once, when he wondered whether the informant might wish some quid pro quo. Immunity for his own crimes, perhaps, which Baranov could grant. Or money, which he couldn't. Marching past the babushkas hawking their flowers and the Chechens their pirated videos, he decided he wanted Kirov so badly he'd be tempted to dish out a little of his own savings if it might help secure the villain's conviction.
A humble table stood at the end of the tunnel, covered with an embroidered muslin cloth and decorated with twenty or so candles of varying colors and heights, all burning. The candles served as a memorial to the innocent victims killed at the spot a few years back by a Chechen guerrilla's bomb. Some had whispered it was a ploy by the president to drum up support for the never-ending war against the insurgent republic. Baranov didn't believe a word of it. Volodya was an honorable man. Who else would give him free rein to pull in thieves like Kirov?
It was with a subdued smile that Yuri Baranov mounted the stairs to the southwest exit of the Pushkinskaya Metro station. He did not notice the phalanx of young, crewcut males who quickly erected a chain of sawhorses to block the tunnel behind him. Nor did he pay attention to the scaffolding at the head of the stairs, or the seesaw pounding of a jackhammer nearby. Construction was an omnipresent hazard in modern Moscow and the century-old subway stations were in constant need of repair.
The first shot took him high in the leg. He hadn't heard a thing and had it not been for the spout of blood that erupted from his pant leg, he would have thought it a bee sting at worst. One hand grasped the railing for support, while the other fell to his thigh. "This is absurd," he heard himself saying, and then somewhat irrationally, "It's Monday morning, for Christ's sake," as if murder were not a state-approved way to begin the workweek. His eyes darted around, but he saw nothing. A sense of desperation seized him. Frantically, he tried to continue up the stairs. He took one step and fell to the pavement, writhing in pain.
"Get up, Baranov. It's unseemly for government officials to grovel. Especially honest ones."
_It was the voice from the telephone_. The voice he couldn't quite place. Only now, he knew exactly to whom it belonged. Grimacing, Baranov lifted his head and squinted to make out the figure at the top of the stairs. "You," he said.
"Who else?"
Konstantin Kirov stood in a black suit with a black tie, hands on his hips, offering a gaze as morbid as his attire. "I have a message from the president. He asked me to deliver it personally." Kirov snapped his fingers, and someone tossed him a large rifle. A Kalashnikov. With a halting, unsteady motion, Kirov cleared the chamber and brought the weapon to his shoulder. The gun looked ridiculously large in the small man's hands.
"He said, 'Be quiet,' " Kirov finished.
Baranov raised himself to his feet. He felt neither fear nor lament, but a pervasive contempt for this pitiable excuse for a human being.
"Liar!" he shouted.
A hail of bullets riddled his body in time to the jackhammer's renewed assault.
**T ELL ME THE TRUTH,"** said Konstantin Kirov.
"Yes, I promise."
"What did he want?"
Pillonel hesitated, and the knife dug in. "Mercury," he said. "They knew I had faked the due diligence. They wanted proof."
"And you gave it to them. Without so much as a call to a lawyer or the local police, you gave it to them."
"They knew," said Pillonel. "They already knew, goddamn it. Gavallan said he was going to the SEC with or without my help. He was going to report me to the Swiss authorities." The intruder had tied his hands and feet to the bedposts with elastic cord and was kneeling beside the bed. In one hand, the man held the knife delicately, as if ready to fillet a fish, the point inserted meanly between Pillonel's ribs. In the other, he had a cell phone, which he pressed to Pillonel's ear. Pillonel had an urge to explain everything at once. "Gavallan had a gun. He put it to my head. I thought he would kill me. I had no choice. Of course I gave them the real books."
"I can understand your anxiety at being confronted with your misdeeds. But why did you take them to your offices?"
"Gavallan demanded I show him Mercury's exact financial condition—how much money the company had really been earning, its revenues, its expenses, its profits."
"And you showed him. How kind of you to be so helpful." The voice was more ominous because of its even tone, the complete absence of aggression, irony, or anger. "Did you ever once consider telling him he was mistaken, to leave you alone?"
"I couldn't. I told you, he had a gun. He said you had killed the man on the Internet, that you would kill me next."
"I never knew you for such a gullible sort." Kirov laughed, then resumed his unhurried interrogation. "And after Mercury, what did you show them? Did Gavallan have any idea he was so close to the crown jewels?"
"Nothing. I gave them nothing."
"Novastar?"
"It did not come up."
"Not even a mention? What about Futura and Andara? Baranov knew well enough about them. Didn't Miss Magnus have any questions about them? You didn't show them the holding company's banking records?"
Pillonel lay still, the lie poised above him like the blade of a guillotine. "I'm no fool. The records would take me down too."
"If you gave them Mercury, you were already going down. If I were you, I might have taken the opportunity to win over the authorities, show them the error of my ways, maybe even try to offer up something to protect myself. I'm sorry I must be so thorough in this matter, but I'm sure you can understand that it is of the utmost importance I learn exactly what materials you gave Mr. Gavallan and Miss Magnus."
Pillonel looked at his wife, his eyes begging her forgiveness. "I gave them nothing," he whimpered. "Only Mercury. Novastar did not come up."
"Ah, Jean-Jacques, you are a poor liar. Calm yourself now. You have nothing to worry about. I have them both with me—Cate and Mr. Gavallan. No more harm can be done. You don't have to worry. I think you know what will happen if you decide to go to the authorities."
"Yes, absolutely. Not a word."
"Now tell me the truth and you'll be on your way to Mahé before you know it. What evidence did you give Gavallan?"
_Mahé. Sanctuary. A new life_.
Pillonel grasped at the words, seeking solace and safety. His hands came away scratched and empty. _Kirov was also a poor liar_. "Nothing."
"Good. I'm happy for it. As for the confession, you know that they don't hold up in court when made under duress. Don't be too hard on yourself. I wouldn't be surprised if Gavallan's lawyer throws the thing away."
"What confession is that?" he blurted.
Pillonel heard Kirov murmur something like "I knew it " under his breath. Then he heard a harsher "Damn him," and he realized he'd said something wrong. Something very, very wrong.
"Well," scoffed Kirov, "at least this conversation wasn't a _total_ waste of time. Give the phone to Sergei."
Sergei took back the phone and after a moment hung up.
"Well?" said Pillonel, eyes paralyzed with hope.
"Good news and bad news. The bad news is you're both to die. The good news is you go first." And even as the words left his mouth, he slid the razor-sharp blade between Pillonel's ribs, puncturing his heart and killing him instantly.
# 55
#
**W HEN WILL YOU PUT SOME FURNITURE** in this place?" asked Leonid Kirov, throwing open the door to his younger brother's study. "Every time I walk in I'm sure I've come to the wrong address. A museum or a mausoleum, I don't know which."
"I need space to think, Leonid. To imagine. To dream." Konstantin Kirov crossed the floor with a statesmanlike gait, extending a hand in welcome. "It is from rooms like this that our country will be reborn."
He was in an exuberant mood. Baranov was dead. Pillonel, too, but not before exposing Gavallan as one more paper tiger, his ruse about the taped confession a last, desperate ploy. All obstacles had vanished. Only time separated Konstantin Romanovich Kirov from reaping his billion-dollar reward.
He'd decided he'd had enough of Dashamirov, too. Fifteen percent was too much to dole out for a little protection now and then. Besides, he had a new _krysha:_ the _komitet_. A few words to Leonid's colleagues in domestic security and the vile Chechen would be a memory. A billion dollars bought that kind of service.
"Come sit down. Have some breakfast. Not often we get a chance to catch up on things, just the two of us."
Leonid took his place at a table that had been set up for the two of them. Fastidiously attaching his napkin to his collar, spreading it across his chest, he appraised the bounteous meal. Broiled kippers, poached eggs, sausages, melon, bacon, and hashed brown potatoes. A grunt signaled his satisfaction. Lifting his knife and fork, he met his brother's eyes. "It's all over the radio this morning. You can't change the station without hearing it. A return to the days of yore. The gangsters are back. Nothing like a little fear to keep the naysayers in line. Well done. The president is pleased."
"Honesty was his only vice," said Kirov. "He was admirable in his way. Just outdated. Obsolete."
"Baranov?" scoffed Leonid. "He was a pain in the ass. Always has been. Even during the old regime, we called him 'our conscience.' That was not a compliment, I can promise you. God, but you made it bloody enough. How many times did you shoot him?"
"A full clip. I thought he was worth it."
"What do you mean, _you_ thought? Don't tell me you got your hands dirty, younger brother?"
"I discovered I had a rather emotional attachment to the prosecutor general. I decided he merited my personal attentions. A hell of a way to relieve some stress, I can tell you that."
Leonid said nothing, but there was no denying the look of admiration. Younger brother had finally done something worthwhile. "Witnesses?"
"A few. We took their names."
"Give them to me. We don't want any trouble."
Kirov shivered, for the first time feeling the power of the state in his hands. No longer was he beholden to the likes of Baranov or Dashamirov. From this day forward, Konstantin Kirov was a partner of the state. An equal of Mother Russia.
_He_ was _the Rodina_.
"And you?" Kirov asked. "All goes well? Where are you going with those boots? Perm?"
"Severnaya, if you want to know."
"Severnaya? Good God, that's the Arctic Circle. What gives you reason to go up there?"
Leonid gave a look at his boots. It was a proud look, Kirov noticed. A look of deep satisfaction. "Oil, if you must know."
"Have we discovered a new field? Wonderful news." Immediately, Kirov began to scheme how he could get in on things—leasing drilling equipment, securing a contract for the construction of the new pipeline, arranging a turnkey operation; there were a hundred ways to make a fortune when one was the first to learn of such news.
"Not exactly, younger brother. There is a new field, but it is not ours. These days it's not a question of too little oil, but too much. The world is drowning in the stuff. If OPEC ever opens the spigots we'll be back at fourteen dollars a barrel and that will be the end of us. If our country is to continue growing, oil prices must remain high. Twenty-seven dollars a barrel at least. Only then can we earn enough to keep our GDP growing at eight percent a year. Continue at this rate and in ten years we'll be a superpower again. One decade. It's not really so long, is it?"
"Not long at all. Then why the trip to Severnaya? It's awfully far to travel if there's no oil there."
"An exercise in prevention, younger brother. While we may wish for higher prices, others abhor the idea. One in particular has taken to the notion of self-sufficiency. Unfortunately, they have the resources. It would be devastating to our country should they exploit them. We must see to it they do not consider the option." Leonid finished chewing a bite of sausage, then asked offhandedly, "Speaking of America, you do have Mr. Gavallan here, don't you?"
Kirov felt himself jolt, his stomach rebel.
"Don't look so surprised," Leonid continued. "Just because the _komitet_ 's stinking bankrupt doesn't mean we don't do our job. Is he here or out at the field observation post with the other one? Excuse me, I mean your 'dacha.'"
"Mr. Gavallan is here. He'll be joining his colleague at the dacha."
"And Katya?"
"As well."
Leonid set down his cutlery, pulling the napkin from his neck and wiping his mouth clean with one stroke. His plate was spotless. "They are dangerous. Either of them can compromise the operation."
Kirov wanted to disagree. Never would he allow Cate or Gavallan to interfere with Mercury. Then, he realized Leonid wasn't talking only about Mercury. He was talking about Severnaya, the preemptive exercise he had cooking on the cusp of the Arctic Circle. Somehow the two had become hopelessly intertwined.
"Gavallan, of course," he added, a bit uncertainly. "I had no intention of continuing our working relationship. But Katya... Naturally, she'll remain in Moscow under my supervision."
"Cut the crap, Konstantin. You know what has to be done." He leaned across the table, his square gray head looming foremost in Kirov's vision. "No one can compromise the _komitet_ , younger brother. Our name may have changed, but our principles haven't. I'm sorry, but that's that. After all, this is the second time the little missy has tried to put you away. You should be happy to have an excuse to be rid of her."
"Come now, Leonid, let's be realistic. Gavallan is one thing, but family... Katya is my only daughter. She's strong-willed, of course, but nothing more—"
"No buts, younger brother. Remember where you live. The only family you have is the state." Leonid stood, buttoning his jacket. "So I can tell him you'll take care of matters? Clean things up? We don't like to leave a mess. That hasn't changed either."
Kirov swallowed hard, the taste of his bile acidic, repellent. He felt tricked, massively deceived. A victim. "Yes. Tell the president to have no worries."
"He'll be most grateful. Good luck, and remember, you are representing the country. The president will be watching on television. Oh, I almost forgot." Leonid reached into his jacket and handed his brother a small blue velvet box.
Opening it, Kirov saw a colonel's polished golden oak leaves. "What's this?"
"Message from the president. You work for us now."
**S HE HEARD IT ALL.** Not every word, but snippets here and there. Enough to piece the conversation together. Enough to grow as frightened as she'd ever been in her life.
"He's going to kill us," she repeated silently, as if repetition would make the certainty less ghastly. In her panic, she reverted to her journalist's guise. There's a word for it, she told herself. When a father kills his child... there's a word for it. But her distress was such that she couldn't remember what it was. Plain old "murder" fit the bill, and that was bad enough.
Kneeling inside the den, Cate kept her head tilted toward the heating vents. She had come downstairs ten minutes earlier, Boris her escort. Her father wished to speak with her, she'd been informed. Alone. But as Boris locked her in, she caught the back of her uncle Leonid charging into the living room. He was unmistakable. The blue suit. The stiff shoulders. The iron gray hair.
Her father and uncle had been estranged during her childhood. Curious as to what common bond had brought them together, she'd pressed her ear to the grate. Listening, she had forced herself not to cry out at the tales of barbarity bandied about by the two men.
The doors to the den opened.
"He is ready to see you," said Boris, motioning to follow him across the foyer.
"Of course."
It was moving day in Sparrow Hills. At nine o'clock, the clubhouse was a picture of commotion. The twin front doors stood open wide, the muscular growl of a supercharged V-8 flooding the entry. The snout of a black SUV pulled into view. Car doors opened and slammed. Boots slapped the pavement. A steady stream of her father's bullies entered and exited the house, at least half sporting Uzis slung over their shoulders. Luggage was brought downstairs. Another Suburban arrived.
At last, her father emerged from the living room.
"Good morning, then," he said, with an affable smile. "I apologize for my behavior last night. I was distraught. I hope at least that you slept well."
It was an act. A murderous masquerade. "Fine. And you? Sleep of the innocent?"
"Always," he replied in his soft, deathly courteous tone. "I wanted to have a last word with you before you set off."
"I thought we covered everything last night."
Her father stepped closer, patting her arms understandingly. "Katya, there's so much you don't know. So much I want to explain to you. I'm sending you with Jett to my dacha for a few days. When I return from New York, we will sit and talk. I'm not the ogre you think. I will listen to what you have to—"
"What is there to talk about? Mercury is a lie, but you're going ahead with the deal anyway. You hold your daughter as if she were a prisoner." She shook off his hands. "We have nothing to talk about. Not now. Not ever."
Kirov retreated a step, a blithe smile on his lips. "I can see you're upset. It is understandable. When I return, we can speak again. If you'll excuse me, I must hurry. The pricing is set for four p.m. this afternoon in Manhattan. Bye-bye, Katya."
She fixed him with an unloving stare. "Don't you mean _adieu_ , Father?"
# 56
#
**T HE GLOVES WERE OFF,** the last semblance of civility fading as quickly as the Moscow skyline behind them. They rode in separate cars, Gavallan in the lead vehicle with Boris and two guards, Cate bringing up the rear with Tatiana and another two guards of her own. A glance over his shoulder earned him a twisted smile and a view of an Uzi pointed directly at his back, a taut finger laid across the trigger.
They lumbered across the Moskva River, then joined the Outer Ring Road, leaving the city along the path they'd taken the night before. Instead of turning off at Sheremetyevo, they continued north toward St. Petersburg. After that he was lost. The road markers were in Cyrillic and he couldn't decipher a word. The highway narrowed to two lanes and all signs of the city tapered off. Potato fields spread to their left and right, bordered by elevated dirt berms—half levee, half road. Occasionally, he caught sign of a town away in the distance and wondered how, without any marked exits, one was supposed to reach them. Birch forests came and went as if moved en bloc.
Gavallan shifted in his seat, laying an arm across the backrest. It was hard to sit still. Tucked into the waistband of his undershorts was the shank he'd fashioned the night before. He had no idea how he'd use it, or even if he'd be given a chance. Pitted against an Uzi with a full clip, a handmade dagger didn't amount to much. Whatever happened, he wouldn't go easy.
**H ER NAME WAS KATYA,** once again, and as she drove, a gothic fantasy played in her head. She was the Czarina en route to Ekaterinburg. Anastasia, of course, on her last journey. Her fate was sealed, but she was too proud to acknowledge it. How many nights until the brigade of toughs stormed the lodge and forced her to the cellar? How long until her father's eager band of revolutionaries signed their name to her short history?
The first intimations of disaster came at 11:06 by the digital clock on the dashboard. The driver left the highway at an exit marked "Svertloe" and took up a new course on a single-lane macadam road leading intrepidly across a meadow-grass plain. Once the preserve of boyars, or nobles, and the wealthy bourgeoisie, dachas tended to be rustic cottages located in pine forests or near lakes or mountains. Most served as weekend retreats and could be found within thirty miles of the city. But one look at this stale landscape told her that no right-thinking man would build a dacha within a hundred miles of this place.
The road began a steady climb uphill toward a pine forest. The macadam quit, replaced by hard-packed dirt. She glimpsed silver. Straining her eyes, she made out a fence. She leaned forward, knowing it was her destination. One fence became two, each ten feet high and topped with curls of barbed wire. The gate, though, was in ruins, bent and mangled, lying to one side. They entered the compound, and she looked around. There were a few log cabins, nothing quaint or rustic about them. _The dacha_ , indeed. One more of her father's sick jokes. The car pulled up in front of the largest building. She saw the windows and gasped. They were decorated with stout iron bars placed three inches apart.
This was where all roads led.
To Russia.
To her father.
To her death.
**G AVALLAN SPOTTED THE RUINED FENCE** and knew it was Graf. He was alive. He had escaped. He had crashed through the fence. Right now he was in Moscow alerting the embassy. It was a matter of time before they sent out their delegates in the company of the Russian militia. His blood stirred and he grew giddy with a desperate joy.
Then he saw the battered truck parked behind the main building, and his spirits crashed to earth. The pickup's fender was dented, the windshield cracked. Whoever had driven through the fence hadn't gotten far.
The SUV lumbered to a halt in front of a large cabin. Gavallan spotted the bars and knew he would have to act fast. Once inside, they'd be locked up and then he'd have no chance for surprise. He imagined that the day's agenda called for interrogation and torture, followed sometime in the afternoon by death. Call it the Russian trinity. He'd have to hit someone before he got locked up. He swallowed hard, steeling himself to the task. He'd never killed anyone, not with his hands. He was a pilot. Tell him to drop a couple bombs from twenty thousand feet and he was your man. Ask him to shove a three-inch blade into a man's belly and he'd say, "No thanks, that's the next guy's job." Except today there wasn't a next guy. Today there was him and Cate and five Russian thugs with at least two Uzis and a couple of handguns between them. He looked at the driver and at Boris. Who would be first? It didn't matter so long as he had one of the machine guns. That's what he needed. From then on out it would be a crapshoot.
"We are arrived," said Boris.
Gavallan descended slowly, pushing his stomach out to keep pressure on the shank, make sure it remained inside his waistband. The air was dry and dusty, hinting of resin and mint. He looked around, his eyes making a desperate survey of the compound. Besides the main building, there were three smaller cabins, shacks, really. Two stood to his left, fifty yards away. A third was closer, more a shed, constructed from pale birch wood. Gavallan thought he saw something move inside it. He looked closer. He could see the fingers of two hands extended through gaps in the wall, grasping the wood.
_Graf_.
His heart beat with a violent resolve.
The second Suburban pulled into the clearing and stopped. Tatiana jumped from the car, and a moment later Cate appeared. Behind them, Boris's cronies had formed a small welcoming committee. The Uzis were out, and not just for show.
Gavallan walked over to Cate. "It's gonna be okay," he said, taking her hand.
"No, Jett," she said. "It's not."
# 57
#
**T HEY STOOD IN THE CLEARING** in front of the cabin waiting for Boris to open the door, a vacation party anxious to get into their summer rental. Gavallan held Cate's hand, every bit as much for his comfort as hers. "You okay?" he asked.
Cate nodded, shifting her head toward him. "We have to talk."
The blunt nose of an Uzi jabbed Gavallan's back before he could reply. "Quiet. No speak."
"Take it easy, bud," said Gavallan. Irritated, he turned to face his newly appointed guardian angel, all two hundred and forty pounds of him. He wanted to shove the guy, gun or no gun. "We're not going anywhere. Give us a break."
"Fuck you." The guard had white blond hair done in a burr cut, dull blue eyes, and pitted cheeks that had fought a losing battle against teenage acne. He feinted with the Uzi and Gavallan jumped back, drawing a bored chortle from the spectators.
The drivers lolled against the doors of their Suburbans, arms crossed, smoking and chatting up Tatiana, who was dressed like a California teenager in Levi's, cowboy boots, and a black tank top. Her shoulder holster and pearl-handled .357 Magnum were strictly adult fare, though, and christened her the flat tops' dream date. She responded to their catcalls desultorily, her voice flat, her eyes glued to the cabin, to Gavallan and Cate.
She was a pro, Gavallan decided. She was trouble.
Shifting his eyes around the clearing, he took in the trees that stood stiff as sentries, the furrowed track that had brought them here, the twin fences, and the ruined gate. The entry to a storm cellar could be seen a ways off, next to a depleted woodpile. In the same direction were two smaller cabins, one with an antenna, the other a crude smokestack. But Gavallan's interest was first and foremost on the shed. He took a step toward it, pointing. "Is Mr. Byrnes in there?"
No one answered.
"Boris, is Mr. Byrnes in there?" The Uzi stabbed his back and Gavallan spun rapidly, knocking it away. "Hit me again with that thing and I'll ram it sideways up your ass."
Finished unlocking the cabin, Boris hurried back toward Gavallan. "Why you no shut up? We ask you once, twice. Still, you talk."
"You can't just—"
Boris fired a fist into his jaw, knocking Gavallan to a knee. "Shut up. _Ponimayu?"_
"Jett!" Cate jumped to his side and Boris picked her up kicking and struggling and carried her back a step or two. Setting her down, he rattled off a barrage of words at her. Cate relaxed again. She stood rock-still, her eyes glued to Boris. She was playing the obedient schoolgirl, and Gavallan was glad for it.
_A little longer, my girl. Play along a little longer_.
Slowly, Gavallan found his way to his legs. He hadn't been hit like that in a long time. It didn't hurt so much as make him want to give Boris one right back. Brushing the pine needles off his pants, he checked for the butt of his shank. It was still in place. I owe you one, buddy, he promised himself, meeting Boris's eye. Payback. And it's coming sooner than you think.
"My father forbids you to talk," Cate explained a moment later. "To me or to anyone. Graf is in the shed. He says if you want the same punishment as him, all you have to do is keep speaking."
_"Ponimayu?"_ Boris repeated, firing two fingers into his chest. "You understand now?"
"Loud and clear."
Boris jumped onto the porch and waved his arm for them to follow. "Inside."
The Uzi nipped at Gavallan's back and he took a step forward, bending to help Cate with her bag. "I'll get it," he said. He needed the bag every bit as much as the shank that was cutting into his waist. The bag was his decoy. A prop to buy him time.
"Thanks," she whispered, her smile a present.
Gavallan crossed the threshold and looked around. The floor was wooden, swept clean and covered with a sisal throw rug. Four battered desk chairs were scattered about the place. A trestle table took up one wall. On it was a propane-fueled heating ring, a few dishes, and a tray of cutlery. A portable Honda generator sat in a corner, along with a space heater and two jerry cans he presumed were filled with gasoline. A pile of dirty magazines littered another corner. Man's fundamental needs had been reduced to heat, food, and jerking off.
"Nice place," said Gavallan. "Tell me, is it a time-share or do you own it outright?"
"You will only stay a few days," said Boris.
"We shouldn't be staying here at all. You know your boss is in trouble. Come on, Boris, it's time to call it quits. Let's all get back into the cars and go back to Moscow. I'll buy you a drink at the Kempinski."
Gavallan waited for him to say "Shut up," to throw another punch. But this time Boris merely laughed. "You think I should quit? And do what?"
"You've got a good head for the market. Use it. With your knowledge, I bet you could find a job as a broker in no time."
"With you? With Black Jet?"
"Why not? It's better than staying with Kirov. Where do you want to start? San Francisco? New York? Let's get Mr. Byrnes and head back to town."
"New York, eh?" Boris hummed a few bars of "On Broadway." _Un Brod-vey_. Abruptly, his gaze darkened. "Mr. Kirov is not in trouble. You are in trouble, Mr. Jett. Go with Ivan. He show you to your room."
"Boris, listen to me—"
"Shut up, Mr. Jett."
All trace of the Russian's former good nature had vanished. Gavallan knew why: He was steeling himself for the job ahead. Putting on his armor. As Ivan led the way down the hall, Gavallan grabbed Cate's hand. "Hang in there," he said.
The first room offered a cot, a table, and a wooden bucket. The second was less accommodating. A peek inside revealed a sturdy wooden chair with broad, flat armrests and a stiff back bolted to a concrete floor. He'd seen chairs like it before, but usually they had straps for your arms and legs and came with a metal bowl and a few electrodes to clamp on your freshly shaven head. The floor was stained black and sloped toward a drain in its center.
"Jett... oh, Jesus, no." Cate's gait faltered, and Gavallan rushed to support her. "Go," he said, propelling her forward. Sensing he had a moment, he put his mouth to her ear. "Hit the floor when I tell you."
"What?" Cate asked, brow knitted.
Seeing Ivan's eyes on them, Gavallan backed off and didn't answer.
Ivan opened the door to the room at the far end of the corridor. "Come," he said, motioning them closer.
Cate ventured a look behind her and Gavallan nodded for her to go on, his eyes gifting her with the confidence he was lacking. She stepped into the room and, moving to the left, disappeared from Gavallan's sight. A last check over his shoulder showed Boris hovering near the front door, distracted, barking instructions to Tatiana and her suitors.
There were two cots placed against opposite walls with a window in between them. Cate stood to his left, arms crossed over her chest. She was nervous, her sea green eyes flicking this way and that.
"Which one is mine?" Gavallan asked, pointing at the beds. His body had gone rigid; his hands itched for action. His jaw still tingled from Boris's punch, and fighting blood stirred inside him. Ivan stood in front of him, the Uzi pushed back to his side, his forearm resting on top of it.
"Ex-cuze me, I no—" he began to answer, his fractured English bringing an ugly grin to his lips.
But by then Gavallan was already moving.
Shoving Cate's overnight bag into Ivan's stomach, he drove the white-haired Russian into the far wall. While one hand blocked the Uzi's rise, the other dropped the bag and freed the shank from his pants. With curt, vicious thrusts, he rammed the blade into Ivan's neck, once, twice, then brought his arm around in a windmill and stabbed the Russian in the back. His actions were savage, feral, unthinking. Ivan fought to push his attacker away, to bring up the Uzi, but his efforts were divided, unfocused. Hugging him close, Gavallan shoved home the shank. The Russian's back arched in spasm. His fingers left Gavallan and grasped at his ruined throat, but the only sound he could produce was the clotted cough of a man choking to death on his own blood. His body shuddered, then was still.
_"Ivan!"_
Boris's strident voice echoed through the cabin as his footsteps pounded down the hallway. Gavallan freed the submachine gun from Ivan's shoulder and let the corpse fall to the floor. "Down," he yelled to Cate as he darted to the doorway and his thumb kicked off the safety. He ducked a head into the corridor and a chunk of wood exploded from the door frame, accompanied by the ear-numbing blast of a large-bore handgun.
Blindly, Gavallan stuck the Uzi into the corridor and fired. Three short bursts. Left. Right. Then left again. He could hear the bullets strike Boris, three fastballs thudding into a catcher's mitt. His steps slowed violently and the Russian collapsed to the floor.
Gavallan peered into the hall. Boris was on his stomach, one hand patting the ground as if he were a wrestler signaling his surrender. The pistol lay a few inches away. Gavallan fired a quick burst and Boris's skull disintegrated, freckling the walls with gore.
"The others are coming," Cate shouted. "Hurry!"
"Get the gun and stay here," Gavallan instructed her.
With a leap, he cleared Boris and made for the open front door. Running, he glanced out the window. The two drivers were rushing the cabin. Tatiana was nowhere to be seen. Stopping short, he fired through the glass in a wide arc. His goal wasn't to kill but to halt Kirov's soldiers' advance. Both men dived headlong to the ground and, as if trained for this exact scenario, began crawling in different directions. The nearer sought refuge in the lee of the landing. The other skidded backward on his hands and knees toward the automobiles.
_You can only get one_ , a voice whispered in Gavallan's head.
Steadying himself, he took aim and fired. A short burst, five bullets max. The black suit approaching the cabin stopped moving. Gavallan fired again. Filaments from the man's jacket flittered into the air where the bullets struck.
"Cate," he yelled, "get on your hands and knees and crawl to me."
Gavallan had slammed the front door and was running from window to window, scouring the woods for sign of Tatiana's platinum hair, her blue jeans running among the trees. He didn't see her anywhere. Fire broke out from the front of the house. Bullets thudded into the cabin, then found the windows. Glass shattered and tinkled to the floor, sending him tumbling to the floor. Lifting his head above the windowsill, he saw their driver firing his Uzi over the Suburban's hood. It's a feint, Gavallan decided. He's keeping us pinned down for the girl. For Tatiana.
"Take the Uzi," he said to Cate, trading her the machine gun for Boris's .44 automag. "If he tries to leave the car, fire." He showed her how to hold the gun at arm's length and helped fashion her finger around the trigger. "Just short bursts. Fire; let go. Fire; let go. You don't have many bullets left."
Cate accepted the weapon, tried to get a feel of its heft. "Short bursts," she said, her eyes keen.
"Yeah, and keep looking every now and again. He may try to rush you."
"And you?"
Gavallan had remembered the woodpile twenty five feet from the cabin and the boarded-up entry to the storm cellar next to it. He'd already located the stairs to the cabin's cellar. The only question was whether there was a passageway leading between the two. Given the severity of Russian winters, he was counting on it. "I've got to check on something. I'll be right back."
Mindful that speed was a factor, he moved off before she could protest. The automag leading the charge, he crashed down the stairs to the cellar. The room was dank and dark. He scurried along the walls, his hand checking the concrete for a door. He found nothing. He took a step backward, puzzled, and a hollow thud greeted his footfall. He was standing on a trapdoor.
Falling to his knee, he slipped two fingers into the rusty pull ring and yanked open the door. Stairs led to an abyss. Slowly, he descended them, one by one, and when he reached the bottom, he stopped. The room was pitch dark. He waved a hand in front of his face. Nothing. He listened. Nothing.
_But what did you expect to hear over the hobnailed beating of your own heart?_ a voice chided him.
Hurry, he commanded himself. Cate is alone. And then, more frighteningly: You could be wrong. Tatiana may know another way into the house.
Groping the wall, he set out, holding the gun in front of him as he would have a flashlight. He calculated that twenty paces would take him to the storm cellar. Water dripped from the ceiling. Instinctively, he lowered his head. Something damp and sticky danced across his face. Grimacing, he swiped it away.
Ten paces.
_"Jett! Come here! Now!"_
Gavallan spun his head in the direction of her voice. He retreated a step. It was the driver. He'd gotten bored and was mounting his own lonely charge. Just then, the door to the storm cellar opened and sunshine flooded the passage. Gavallan froze, squinting to adjust to the light. A black cowboy boot landed on the stairs forty feet in front of him.
"Jett!" Cate's voice came again.
Gavallan slid backward, his head turning one way, then the other. On the stairwell, the boots became blue jeans, and the blue jeans were joined by a pale hand holding the pearl-handled .357 Magnum. Gavallan dug his feet into the dirt floor. There was no going back. Bringing his left hand up to the grip of the .44 automag, he assumed the Stableford stance: left foot forward, right arm extended, left hand supporting his shooting wrist. He waited until he saw her face—the diamond blue eyes, the pouting lips. "Stop," he yelled.
Tatiana's only reaction was to raise the gun as quickly as she could. Gavallan hesitated, but only for a fraction of a second.
Then he fired three times.
**H E FOUND CATE** standing in the center of the front room. "I killed him," she said.
"So I see." The driver had, after all, decided to mount a charge—a very ill-advised one. His body lay twisted and prone a few feet from the Suburban. "Good shot."
Cate shrugged, laying the Uzi on the table with a professional's ease.
"Sure you never fired one of those before?" he asked.
"I never said that."
"I just assumed..."
Cate gave a crisp shake of the head. "Don't assume too much. Remember, you didn't even know my real name until yesterday."
Gavallan knew she meant it as a joke, but he could not laugh. He was upset, jittery, waiting for the adrenaline to run down, for the electric colors to fade. "Come on. There's someone here who's very anxious to see us."
"Oh, Jesus, I almost for—" Cate bolted out the door, jumping off the porch and making toward the shed. "Graf!" she called. "We're coming, Graf!"
# 58
#
**M IND EXPLAINING THIS?"**
Gavallan rested on a knee next to Grafton Byrnes, fingering the frayed bullet hole in his friend's jacket.
Pale, unshaven, dark circles denting his eyes, Byrnes sat on the bare earth outside the shed, legs spread, sipping from a cup of water. His lower lip was cracked and swollen. A minute earlier, he'd smiled to show Cate and Gavallan the incisor he'd lost after he'd been recaptured the night before and returned to the camp.
"All you need to know is I wasn't wearing it when it happened," he said.
"I hope the guy that was got what he deserved."
Byrnes looked away, his voice as distant as his gaze. "Oh yeah."
"All right then," said Gavallan, seeking to rouse the fighting spirit in Byrnes. He knew their freedom was an illusion, a temporary gift that might be yanked away at any time. It was a long trek to the border and he needed Byrnes at his side, not lagging behind.
Gavallan's eyes kept coming back to his friend's hands. The bandages covering his thumbs were torn, the gauze stained black with dirt and blood. His palms were colored rust, dried blood tattooing the flesh. "You okay, pard?"
Byrnes caught his glance. "Six months," he said, raising his right hand, turning it over in the sunlight. "That's how long I've heard nails take to grow back. Tell you one thing. I'm not ever getting another fucking manicure in my life."
"Amen to that," said Gavallan, patting him on the shoulder. He knew he could never appreciate the barbarity his friend had suffered. A glance at the bandages, at the wounded eyes, told him enough.
A breeze came up, rustling the trees, scattering pine needles across the dirt, and freighting the air with the scent of turned earth, loam, and, somewhere distant, burning leaves. It was a melancholy scent, and Gavallan was overcome with sorrow and sadness and a sense of failed responsibility.
"You ready?" he asked, getting to his feet. "Time to saddle up."
"I thought you'd never ask."
Byrnes stood shakily, throwing an arm to Gavallan's shoulder for support. He took a few steps toward the clearing to better see the shot-up cabin, the bullet-riddled Suburban, the corpses sprawled pell-mell in the dirt. He stopped. Turning, he fixed Gavallan with a stunned, disquieted gaze, almost as if looking through him. Then he rushed forward and wrapped his arms around his friend, hugging him tightly. "Thank you," he said, pushing his cheek into Gavallan's hair, and Gavallan knew he was crying. "Thank you for coming to get me."
Gavallan returned the hug. He tried to say, "Anytime—that's what brothers do for each other," but something was blocking his throat and he couldn't trust himself to speak.
**T HE SECOND SUBURBAN** had survived the shoot-out intact. Not a dent in its black armor, nor a streak of dirt marring the high-gloss finish. Gavallan and Byrnes walked toward it, Cate following a step behind.
"Why didn't you just cancel the deal after I left you the message?" Byrnes asked.
"What message was that?"
"About the network operations center."
"It's a wreck. We know that. Just like the Private Eye-PO said."
"No," protested Byrnes, stopping short, waiting for Cate and Gavallan to face him. "It's not a wreck at all. On the contrary. That's what I called to tell you. It's a state-of-the-art facility. The NOC is Kirov's beard. Don't you see? It's his disguise. It's what fooled us."
"Fooled us?" asked Gavallan. "How?"
Byrnes described the vast room filled with row upon row of personal computers logging on and off Red Star, Mercury's wholly owned and operated Internet portal. "There were a thousand in there, maybe two thousand. I couldn't count them all. Each logs onto Red Star, then visits a site or two—Amazon, Expedia, the high-traffic sites. Some make a purchase, then they log off. A minute later, they dial Red Star back up again. Over and over, ad infinitum. All running off some master program."
"Metrics," explained Cate, pushing a comma of hair off her forehead. "Has to be."
"I was thinking the same thing," said Byrnes.
"You knew?" Gavallan demanded.
"God, no. But it makes sense. I just wrote about the same kind of shenanigans for the paper. You know... how websites use metrics to manipulate the tally of monthly visitors. It's a gag to fool the firms that measure Red Star's traffic. Make them think Mercury has more customers than it really does. Jett, when you were doing your due diligence on Mercury, didn't you talk to a metrics firm to validate Kirov's claims about Red Star's size?"
"Jupiter in San Jose. Their report tallied perfectly with Mercury's figures. Two hundred thousand subscribers in Moscow alone."
"Of course it did," said Cate. "He knew Jupiter or someone like them would be called in to check how many hits Red Star got every day. He couldn't risk there being a discrepancy. He needed two hundred thousand subscribers to justify his sky-high revenues, and two hundred thousand he got. Only his customers weren't customers at all. They were straw men, or maybe I should say 'straw machines.'" Cate took a breath. "Don't you see? It's a twenty-first-century Potemkin village."
"You're saying he set up shop out here and created a cybercommunity of Red Star fanatics?" asked Gavallan.
Cate nodded disgustedly. "Kirov had it worked out to a fault so you wouldn't question how rapidly the company's revenues were increasing. He knew from the beginning the kind of revenues Mercury had to post to max out its IPO. He could get the money easy. He stole it from Novastar. The subscribers were the hard part. That's what required the creative thinking."
"My God," muttered Gavallan, shaken. "He played us like a fiddle."
"More like a Stradivarius," said Cate. "But his performance is over. And there will be no encore, thank you very much."
Grafton Byrnes signaled his incomprehension. "Hold on, I'm missing something here. What's Novastar Airlines got to do with this?"
Cate explained to him about her dealings with Ray Luca and what had happened in Delray Beach, about the trip to Geneva and Jean-Jacques Pillonel's complicity with Konstantin Kirov to hide transfers from Novastar Airlines to Mercury Broadband and then to Kirov's personal accounts.
"But what put you onto Kirov's case in the first place?"
"Don't ask," said Gavallan, and Cate elbowed him.
"Actually, he's my father," she answered.
Byrnes's eyes registered shock. "You said 'father.' You don't mean...?"
Cate nodded.
"Can't say I see a resemblance."
"Thank God for that." She went on with her explanation: "I don't think we'll ever learn who Detective Skulpin's informant was, but whoever it was that had the guts to go up against my father, I'd like to thank him."
"I think you can forget about that," said Byrnes reticently. "On Friday, Kirov—er, your father—showed up here with a nasty piece of work named Dashamirov. They had three employees of Mercury with them. Dashamirov went to work on them...." The words trailed off. "Anyway, you can figure it out."
Cate Magnus shut her eyes, and a chill seemed to pass through her. "I'm sorry, Graf. I'm sorry about my father. About everything that's happened to you."
"Don't be," Byrnes said. "You didn't have a damned thing to do with this. You're a good egg—I can't imagine the guts it must have taken to come back and face him. The hardest thing a kid can do is step outside the shadow of a parent, especially a father. And then if he happens to be a rogue like Kirov, well..." Byrnes shook his head, then leaned forward and kissed her on the cheek. "Thank you for coming, too."
Cate shrugged forlornly. "Tell me I'm forgiven?"
Byrnes brought her to his chest. "You're forgiven, kid. Big time."
**T HE SPEEDOMETER ROSE STEADILY.** 180... 190... 200 kilometers per hour. Hands clutched to the steering wheel, Gavallan kept his heel hard against the accelerator and sent the Suburban hurtling across the green Russian plains. They'd left the dacha an hour ago and were headed back to Moscow.
A cell phone lying on the front seat between Cate and Gavallan chirped. She picked it up and read the digital readout. "Him again."
For the last thirty minutes the phones they'd taken off Boris, Tanya, and the two drivers had rung more and more frequently. The digital readout indicated that it was the same caller every time—no doubt Kirov calling from his private jet, eager to know how the interrogation of "Mr. Jett" was proceeding.
"Jett, we've got to answer. He'll know something's wrong if we don't."
"No," said Gavallan. "Not yet he won't. When you're forty thousand feet up it's a crapshoot if your call will go through. Besides, what are you going to say—'Hi, Dad. Having a great time. Wish you were here'?"
"He's right," said Grafton Byrnes. "It'll buy us some time."
Cate cut off the call. "Have it your way."
"Look, he's still at least four hours outside of New York," said Gavallan. "Believe me, he'll put it off to atmospherics. Now get on with your story. How can you be so sure you heard right?"
"I was there. Right next to the study. Everyone was going every which way. The front door was open. I got almost every word." Cate pinched her voice and added her father's nasal timbre. "'I thought he deserved my personal attention. I gave him the full clip.' Animal," she added angrily, pounding the sideboard with her fist.
"And your uncle Leonid said the president was pleased?"
"It sounded as if Father was doing him a favor. Like the president wanted Baranov out of the way too."
"Of course he did," said Byrnes from his post in the backseat. "The president made his career as a spy. He's just looking out for his cronies who are still in the trade. It's the old boy network, Volga style. If Kirov's promised him some money from the offering, you can bet the president will do what he can to help him."
"'An exercise in prevention,' Uncle Leonid said," Cate informed them. "Something to keep oil prices high and stop America from developing its own resources."
"What do you think it is?" wondered Gavallan aloud. "The only major resources we have are in Texas and Alaska, and I'd scratch off Texas from the git-go—most of those are old wells with a only few good years left in them. Alaska's our treasure trove. If we ever get around to developing it."
Byrnes laughed bitterly. "Hell, I can think of a dozen ways to stop us from opening the land up there to drilling. All Kirov has to do is hire himself a few good lobbyists. That'll tie up Congress for a couple years right there."
Cate didn't share in the humor. "But Leonid was going to Siberia. They're going to do something!"
"Prevention, huh?" said Gavallan. "Only way to prevent us from exploiting our reserves is to keep us from drilling in the Arctic National Refuge. I mean, what other new resources do we want to exploit? Sons of bitches. If they try anything to ruin that land..."
Gavallan didn't know if he should laugh, cry, or scream bloody murder. He shouldn't have worried about the bush-league charges of defrauding his investors. Dodson's accusations of murder didn't amount to anything. No, he'd really hit the jackpot this time. He'd moved up to the big time—the bulge bracket all the way. Black Jet Securities was underwriting the KGB in its efforts to economically sabotage the United States, however they intended to do it. He had set his company on a line to commit a crime that was tantamount to treason. Willingly or not, he was abetting his nation's oldest, and still its most formidable, enemy. A country that until recently spied on its citizens as a matter of course, that tortured, imprisoned, and executed men and women without trial or benefit of counsel, that believed human freedoms were secondary to the will of the state. A country that even now was on the slippery slope to fascism.
Cate handed Gavallan the cell phone. "Call your office, Jett. Tell them they've got to cancel the offering."
It was 4 A.M. in San Fran. The office was just coming to life. A voice answered, "Black Jet," and Gavallan hung up. "Graf," he said urgently, looking over his shoulder, "when did you leave me that message?"
"Same day I got into Moscow. I got spooked by Tatiana at a dinner club and decided to check out the NOC for myself, then and there. I was sure you'd gotten it."
"Well, I didn't." Gavallan paused, thinking of Kirov's spy. He recalled the first intimations in San Francisco that someone had to be slipping Kirov information, then the Russian's gloating confirmation last night that he'd lured one of Gavallan's lieutenants to his side. "Who took the call?"
Byrnes fixed him with a cynical glance. "Who's always loitering around your office the last six months waiting to have an urgent word? Who'd we catch looking in your drawers before Memorial Day? Who's the one attending all of Mercury's due diligence meetings when they never had before?"
"Jesus," said Gavallan as a face came to mind. Family. One of the inner circle. A small part of him died, and he swore revenge. "Never said a word."
"Fucking ingrate," murmured Byrnes.
"Call back," Cate implored. "Cancel the offering. Tell them all—Bruce, Tony, Meg. Call the SEC, too. And the stock exchange. If you won't, I will."
"And then what?" asked Gavallan, throwing the cell phone onto the seat between them. "What happens to Kirov after we cancel the offering? You think that's going to put an end to him? Hell, it won't even put a crimp in his style."
He could see the events of the following days un-spooling like clips from the evening news. Kirov being detained in Manhattan, then handed over to the Russian authorities. Kirov being set free as Russian prosecutors bemoaned a lack of hard evidence. Kirov appearing triumphant a year later, trumpeting his latest highflyer. There would follow an IPO in Paris or Frankfurt. A private placement in London. The world was full of believers. Gavallan knew it for a fact, firsthand.
"We have the Novastar evidence," said Cate. "The proof he stole from the country. That ought to land him in jail."
"And we're going to keep it," declared Gavallan. "We're going to use it for ourselves."
"But we have to give it to the prosecutor general," Cate protested.
"Baranov's dead with the president's consent," Gavallan said in disgust. "If his successor has any sense, he'll give your father and Novastar Airlines a wide berth."
Cate shook her head, fashioning an answer, but the words died on her tongue.
"Remember what you said to me back in Florida when we were boarding the plane?" Gavallan asked. "You said canceling the offering wasn't enough. You said that you wanted your father to pay for Ray Luca, for the others at Cornerstone, for Alexei and Graf. Well, now you can add the three that Graf saw killed too. And the others to come."
Byrnes leaned forward to be nearer to Gavallan and Cate. "What are you saying, Jett? That you're not going to cancel the deal?"
"Of course we'll cancel it. We have to. Just not now."
"But when? Look around you, buddy. We're a hundred miles from Moscow. It's three o'clock in the afternoon. I hope you don't plan on delivering the message in person. Given what you told me about Kirov and his family relationship with the KGB, I don't think it's going to be a wise idea to line up at the Aeroflot counter and purchase three first-class tickets to New York—if, that is, there's even a flight leaving tonight."
"I've got until nine-thirty tomorrow morning New York time."
"You're pushing it, Jett. This is way outside the envelope."
_The envelope?_ They'd broken through the envelope days ago. All he wanted was a return to earth. A chance to get back to where he was before all the madness had begun.
Cate laid her hand on Gavallan's, and when she spoke her voice had acquired the edge of dangerous dissatisfaction that he himself felt. "What do you have in mind?"
Gavallan looked at her, and saw she was game. "Plenty."
# 59
#
**_J UST BECAUSE THE KOMITET_** **_WAS BANKRUPT_** did not mean they stopped doing their job....
The car was a black four-door Chaika, property of the FSB, the division of the directorate concerned with internal security. The binoculars had been lifted from Directorate 6, the Border Guard, but the men seated sternly behind the dashboard, Lieutenant Dmitri Mnuchin and Major Oleg Orlov, were from FAPSI of the Eighth Chief Directorate, and as such, Major General Leonid Kirov's own.
Mnuchin and Orlov were old hands at this sort of thing—the sitting and waiting, the long idle hours, the marathon sessions of chai and chewing gum. You would not know it, however, from their looks. Both were lean, athletic, and possessed of an alert, aggressive gaze. Both spent their free time in the gym and on the soccer field. They were the new breed: the smart young men who would reinvigorate the _komitet_.
From their vantage point three miles to the west of Army Forward Observation Post 18—recently ceded to Konstantin Kirov and renamed, according to secret transcripts of Kirov's conversations "the dacha"—Mnuchin and Orlov had an unobstructed view of the wooded hilltop. Their assignment was to maintain Level 1 surveillance on Kirov's men—that is, to keep track of their whereabouts, but not to worry about their specific activities. It was an undemanding job, nothing like their usual work involving the installation and monitoring of sensitive eavesdropping apparatus. Both held doctorates from Moscow State in electrical engineering. Today all that was required were a pair of binoculars and a logbook to note the time and nature of their targets' movements.
"A hundred rubles he doesn't do it," Mnuchin said, a loving hand appraising the stubble of his new crew cut.
"You're on. Konstantin Romanovich is every bit as cold as the General. If he were here, I wouldn't be surprised if he did the job himself."
"Never. No man can kill his own daughter. Frankly, I think he's sick. I would have told the General to fuck off."
"The hell you say," Orlov said with a smirk. "You would cut your dick off with a butter knife if General Kirov told you to."
Shrugging his agreement, Mnuchin picked up the binoculars. "Anything for Mother Russia." A moment later, his posture stiffened and the grin dropped from his face. "They're leaving."
"Already? Impossible. They've been there hardly thirty minutes." Orlov picked up the logbook and noted the time: 12:47. Laying the journal by his side, he drew on his seat belt, taking care that it did not interfere with the pistol he wore beneath his left arm, and checked that the mirrors were adjusted properly.
"False alarm," called out Mnuchin. "Only one vehicle."
"You get the signal?"
"Not yet."
The _komitet_ had its own man inside Kirov's organization. He had promised to signal when the executions had been carried out: Two flashes of his high beams would mean that the American and Kirov's daughter were dead. The Suburban rushed past, its midnight-tinted windows making it difficult to get a clear look into the interior.
"Give the plates to dispatch," said Mnuchin, settling back into his seat. "If they want, they can assign a team."
Orlov called in the license plates and advised central dispatch of the events. The report would be forwarded to their superior officer, who would either contact General Kirov with the news or make a decision for himself. Either way, it meant another few hours of sitting in the car. "You think we should call up there? See what's going on?"
Mnuchin trained his binoculars on the dacha. All he could see were the broken fence and the tail end of the second Suburban. "Why? We wouldn't want to interrupt their fun."
**T HE CELL PHONE RANG AGAIN.**
Cate checked her watch. It was nearly four o'clock. They were driving south on the M4 motorway, nearing the Moscow city limits. For miles they would see no one, then traffic would come to a halt as they came upon a convoy of ten or twelve broken-down trucks, tailpipes spewing exhaust, tires wobbling precariously, lumbering down the center of the road. Jett would steer the Suburban onto the shoulder, negotiate the borderland of waist-deep potholes and basketball-size rocks, until once past the trucks he could reclaim his position on the pavement.
"Leave it," said Gavallan.
Cate stared at the phone as if it were a bomb. She knew her father. She knew his impatience. He was not a man who allowed "atmospherics" to stand in his way. "No," she said brusquely, surprised at the force of her reply. "I won't."
And before Jett could make a move, she picked up the phone and put it to her ear.
_"Da."_ It was another woman's voice, rougher, more unpolished than her own. If it didn't sound exactly like Tatiana, it didn't sound like Katya Kirov either.
"Give me Boris," ordered her father.
"He is busy," Cate responded.
"Is Gavallan talking?"
"Not yet."
"Tell Boris to hurry up."
"Sure."
"And my daughter..."
"What about her?" Cate stared out the window, willing her soul to become as desolate as the passing countryside.
"Please make it as painless as possible. Surprise her if you can. It is better if she does not know it is coming. As her father, it would please me. It is the least I can do."
"You are too kind."
A long silence followed. As Jett stared daggers at her, Cate wondered if she had gone too far, if she'd tipped her hand. Then her father's voice came back, as focused and self-centered as before. "Have Boris call me as soon as he's done. I've been having a terrible time getting through. The pilot says it's the aurora borealis acting up this time of year. If there is a problem, have him try me at my hotel. He has the number."
Cate hung up.
"What did he say?" Gavallan asked.
Cate met his eyes. "He wants Boris to call him when we're dead."
**M OSCOW.**
Rush hour in the Center. Ten minutes inside the city limits and Gavallan decided it was every third world hellhole he'd ever known. Jakarta. Bangkok. São Paulo. Traffic was snarled. Militiamen stood impotent amid the blaring horns and packed metal, smoking cigarettes. The pollution was choking and oppressive. Inside the narrow urban canyons, the sky was bleached a puke yellow, a swirling sea of grit, garbage, and carbon monoxide. The heat was oppressive. Combined with the noxious smells, the jangling din, the stop-and-go traffic, it left Gavallan off balance and wary.
"There's the embassy," said Cate, pointing ahead of them at a large traditional yellow and cream colored building on the right-hand side of the road. "That's the main building there. But the consular offices are around the corner."
"Where do I park?"
"You don't. Just pull over."
A tall concrete wall painted white surrounded the complex. Entrance was gained through a reinforced gate guarded by two Marine sentries and untold plainclothes security guards. Spotting the Stars and Stripes flying behind the wall, Gavallan pulled into the right lane and cut his speed.
"You ready, pard?" he asked, catching Byrnes's eyes in the rearview mirror. "When I stop, you skedaddle. Don't let anyone stop you from getting inside those four walls. They touch you, scream bloody murder."
"Don't worry about me. That there is sovereign territory of the United States of America. I'm as good as home."
Gavallan shifted his gaze to the side-view mirror and the gray Chaika sedan that had been on his tail, precisely three cars behind him, for the last thirty minutes. He looked at the two men inside the car—dark suits, dark glasses, short haircuts, chilling caricatures of the once and future totalitarian state. He looked back at Byrnes, not betraying a thing.
"Yeah, well, don't get too comfortable. I want you out of there tomorrow morning."
"Swissair flight 1915 to Geneva," Byrnes recited. "Departs at nine-fifteen; arrives ten-fifteen local time."
They'd gone over the formalities several times already. Byrnes was to ask to see Everett Hudson, the consular officer with whom Gavallan had spoken when he was in San Francisco. He was to explain that he had been kidnapped and to ask for immediate medical attention. Any requests to have him speak with the local police were to be politely but vigorously turned down. The embassy would supply clothing and a place to sleep.
"If they give you any trouble about issuing a new passport overnight, tell them to call the senator." Gavallan figured his contributions to the winning side had been hefty enough to guarantee him at least one favor. Besides, the senator was a former mayor of San Francisco. It was the least she could do for one of the city's own.
Byrnes leaned against the door, his fingers gripping the release. "You're sure about what you're doing?"
"Yeah, I'm sure. It's the only way." Then a lick of pragmatism tempered his confidence and he added in a subdued voice, "You might want to have a word with the defense attaché. Give him some advance warning. I'll have the other side warmed up and waiting."
"Just keep it low and slow. Even if we are all buddy-buddy now, remember, you're not flying the friendly skies of United. And watch out near the Polish border—they scramble on a heartbeat these days."
"You know, some people might think you're still my CO."
Byrnes didn't smile. His eyes did not flicker. "Good luck."
Gavallan stopped the car directly in front of the embassy, but only for a second. "Go. Get the hell out of here."
The passenger door opened and Byrnes was gone, running across the sidewalk to the Marine security guards. Gavallan accelerated. In his rearview mirror, he watched his close friend pass into the compound and disappear from view. It was only then that he voiced his newest suspicions to Cate. "Bad news."
"Oh?"
Discreetly, he poked a thumb behind him. "We've got company."
# 60
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**M R. KIROV,** it is an honor to welcome you to New York and to Black Jet Securities," boomed Bruce Jay Tustin as he greeted Konstantin Kirov outside the main entrance of 11 Madison Avenue.
"The honor is mine," said Kirov, climbing from the limousine. Shaking Tustin's hand, he glanced up at the building, a noble façade of steel and glass. "It's a privilege to be here."
"If you don't mind, let's get upstairs. We're in a bit of a hurry. We've got a lot of people waiting for the big event."
"Do I have time to button my jacket?" He was always taken aback by the American ability to be overly polite and unbearably rude at the same time. He followed Tustin through the swinging doors and into the lobby, where Tustin pinned a badge on his jacket and shepherded him past the security desk.
At three-thirty in the afternoon, the lobby was pleasantly busy. A steady stream of men and women churned past Kirov. White, black, mulatto, Asian, Hispanic—as many ethnic mixes as in the former Soviet Union. There was an eagerness to their faces, an alacrity to their step, a forthrightness in their demeanor, that both amazed and frightened him. Such confidence in the world. Such faith that the system would not disappoint. He was sure every one of them held a valid claim on dreams of expensive cars and luxury apartments and vacations in Paris. No doubt they already possessed color televisions, PCs, cellular telephones, digital cameras, Japanese stereos, and closets full of fine clothing, most of which they never wore. They owned refrigerators choked with fresh vegetables, eggs, milk, cheese, leftover Chinese food, soda, and foreign mineral water. Still, they ate out twice a week. They had bank accounts and ATM cards and Swiss watches and cable TV. Many owned automobiles. In short, they had everything. And look at them. Hungry as wolves for more. Bravo!
Kirov was a student of the American brand of greed, a fan of the excess that capitalism bred. He had always been curious as to how the old barons of the Kremlin, all dead and buried (and, he hoped, rotting in hell), could have believed that dogma and political creed could suffocate the competitive drive of the human soul, could stifle man's innate desire to exploit his talents to the best of his abilities, to toil and be compensated accordingly. What hubris! What arrogance! What barbarity!
_I am the first of the new breed_ , Kirov told himself with the same ambitious cynicism he read painted on the faces around him. _I am a pioneer sent to show my countrymen the path to success. To midwife Russia's transition to a modern economy_.
A few bold Americans had lit the way a century earlier. Men who had overseen the growth of the railroads, the introduction of oil, the mass manufacture of steel. Some called them "robber barons," but Kirov thought differently. They were men of vision, builders, creators, founders of a new empire. The riches they amassed were small compensation for the legacy they left behind.
He was no different. Bold? Yes. Aggressive? Always. Immoral? Unethical? Unscrupulous? Let the next generation judge. He was a modern-day Gould. A twenty-first-century Vanderbilt, if not quite a Rockefeller.
They entered a waiting elevator and Tustin pressed the button for the twelfth floor. "Tired, sir?"
Kirov breathed deeply, suddenly feeling quite at home. "On the contrary, invigorated."
He looked closely at Tustin, standing with his arms behind his back like a latter-day Napoleon. The banker was dressed in a bold gray pinstripe, a pink broadcloth shirt, and a blaring red necktie that could be heard back in Petersburg. His hair was slicked back with enough pomade to fill a lake. There was a slight bruise on his lip where Gavallan had punched him, but Kirov decided not to mention it. He was playing it by the book, pretending to be a client just like any other.
"Any word from Mr. Gavallan?" he asked.
"None, but I'm sure he'll check in shortly."
"I'm sure he will too. Still, it is disturbing."
Tustin simply lowered his eyes, and Kirov thought, _Here is a man who cares less for Gavallan than I_. "I see the markets are up today," he said.
"The Dow's up one twenty, the Nasdaq about the same. Sentiment is very positive lately. Maybe you're bringing us luck. After all, you brought us some blue skies. For the last couple of days, the city has had nothing but rain."
"You know the old saying. 'When angels travel, the heavens smile.' And what about the pricing?"
"I'm sure you'll be pleasantly surprised. There are a couple of formalities we like to engage in before we make an official announcement. We've got a conference room reserved. Like I said, a few people will be joining us."
"Very good," said Kirov, keeping the smile pasted to his face. Inside, however, he was worried. _Formalities? What formalities could remain at this eleventh hour?_
The doors slid back and Tustin requested that Kirov follow him. They walked past the elevator bank and onto the trading floor, threading their way through aisle upon aisle of men and women seated in front of a myriad screens. And as they walked, something strange and marvelous happened. The room grew hushed. The incessant chatter died down. At first Kirov heard one pair of hands begin clapping, then another. He looked around, eager to spot the source of the applause, wondering in his vain yet insecure mind if it was mocking or adulatory. The next thing he knew every person in the room had risen and was pounding his or her hands together. Respectfully. Enthusiastically. Lovingly. Every living soul on the trading floor of Black Jet Securities was saluting his arrival.
Slowing his gait, Kirov raised a hand to acknowledge the applause. He selected an expression of imperious solemnity to greet the masses. He was Alexander riding into Macedon. Caesar returning to Rome. Chuikov arriving in Red Square after taking Berlin.
"It is too much, really," he said, bowing to speak into Tustin's ear.
"Nonsense."
And then Kirov heard the music and he stopped walking altogether. The strains of "The International," the majestic Russian national anthem, played from hidden loudspeakers. The applause died and all eyes fell on him. Kirov was stunned, and for a few seconds he didn't know what expression to choose. The music grew louder, and his skin shivered with goose bumps. Emotion plucked at his eyes, and Kirov was damned if he wasn't crying, this man born to peasant stock, this servant of free speech, this disciple of technology. This son of Russia.
Tustin patted his shoulder, nodding as if to say it was all right to shed a tear, that his pride was well-deserved, and for a moment Kirov loved him, too, as he loved everyone else in the room. This handsome, well-attired, overtly intelligent assemblage of financial professionals.
The anthem came to an end and the applause again started up, but only briefly. Kirov offered the victor's smile expected of him, gave a final wave, then followed Tustin to a conference room that took up a corner of the floor. Twenty or thirty people were milling about the glassed-in room, drinking champagne, munching on canapés, and making small talk.
"Janusz, Václav, Ed, hello. So glad you could make it." One by one he greeted his underlings from Mercury, then the others who had shepherded the Mercury offering through the offering process. Lawyers, bankers, accountants. And there was old man Silber himself—gray, bent, and exceedingly ugly, a Swiss gnome indeed. Kirov shook his hand. Apparently, the dinosaur hadn't yet gotten the word about the fate of his in-house tout, Pillonel.
"Welcome to Black Jet," said Antony Llewellyn-Davies, tapping him on the shoulder and handing him a glass of champagne. "We're delighted you were able to make it on time. One never knows with those small jets."
"What is small about a G-5?"
"Oh, nothing, I just..."
"Thank you." Kirov accepted the champagne, averting his gaze. The Englishman always left him feeling nervous and inadequate, with his soft eyes and snobby manner.
A spoon clinked a glass and the room fell silent. Bruce Jay Tustin cleared his throat, and those around him stepped back to clear a small space. "Ladies and gentlemen, if I might have your attention, please. It's time for us to conduct some important business...."
**D ON'T LOOK BEHIND YOU,"** Gavallan instructed Cate, laying a hand on her leg. "They've been there since we got into the city. Maybe before, but I didn't pick them up."
"How can you be sure?"
"I got the first two numbers of their license plates. I'm sure."
"It could be routine," said Cate. "The traffic militia getting ready to shake us down for a little bribe."
Gavallan eyed her doubtfully. "We both know better than that."
"But why didn't they stop Graf?"
"I can't say. Probably they didn't have orders to. All I know is that we stick out like a sore thumb in this car. We've got to ditch it in a hurry."
They had crossed the river and were driving south on Kutuzovsky Prospekt, a broad boulevard eight lanes across. Traffic was heavy, but moving. Stone apartment buildings five stories high, each a block long, lined the street. Gavallan maneuvered the large SUV into the center lane, checking the rearview mirror. A few seconds later, the Chaika followed, a hearse amid a carpet of colorful Fiats, Fords, and Opels.
_They're obvious about it, that's for sure_ , thought Gavallan.
"You know where we are?" he asked.
"Of course."
"It's time to abandon ship. Find us a good place around here for us to get away from those goons."
"Ahead is a factory district. There are a lot of side streets, alleys really, that separate the different warehouses and manufacturing plants. It used to be kind of run-down. You wouldn't want to go there at night, I'll tell you that."
"Sounds good."
"You really want to just leave the car?"
"They won't be expecting us to. It'll give us a head start at least."
Gavallan kept the Suburban in the center lane, pointing out to Cate their best possible path. Approaching the next stoplight, he slowed to insure he would be the last car across as it turned red. The light turned from green to yellow. He waited, watching the cars nose in aggressively from his left. The light turned red. At the last instant, he gunned the engine, making it through the intersection amid a barrage of horns and obscene gestures as a wave of cars closed off the street and left the Chaika behind him, marooned.
He drove twenty yards farther and then, blocked by the grid of automobiles in front of him, stopped. "Get out."
He and Cate opened their doors and ran across the three lanes of traffic. Reaching the sidewalk, Gavallan glanced behind him. "Holy shit."
Heads were popping out of several of the cars stuck in traffic ahead of them. Two men appeared from a yellow Fiat. Another two from a white Simca. A lone man from a Mercedes. All left their vehicles and began threading through the gridlock toward them. Swallowing hard, Gavallan looked back. The goons from the Chaika were out too, rushing through the intersection as if fording a stream, brandishing pistols for cars to stop.
"Move! Move! Move!" Gavallan yelled.
Cate led the way, running up the sidewalk to the first side street and dashing right. Fifty yards up she crossed the pavement, took another left, then ducked into an alley that ran between two apartment buildings. Her strides were long, her arms pumping, her eyes aimed to the fore. Gavallan stayed at her heel, daring a glance behind them every ten or fifteen steps. He counted seven men running after them. They looked to be bunched in groups: three a hundred yards back, another three seventy yards away, and a lone man fifty yards and closing.
Coming to the end of the alley, Cate darted to the right. They were confronted by two crumbling roads that led at odd angles toward low, decrepit wooden warehouses set in fields of uncut grass. Cate continued to the right. They passed through the field, Gavallan stumbling in a pothole and catching sight of the lone runner, nearer now, a gun in his right hand.
"We've got to get off the road," he panted, catching up to Cate. "There's one guy back there we're not going to shake."
Cate nodded, her lips drawn taut. At the far side of the warehouse, they came to another street. Apartments on both sides. All of them newer, almost modern—the prefab monstrosities the press used to mock: paper-thin walls, plumbing that leaked from the ceiling like rain, air currents that rushed between the crevices that separated one unit from the other. They found another alley. Cate ducked left and after ten steps halted.
"What?" asked Gavallan.
"Come on. Hustle." She was already crawling through an open window into a ground-floor apartment. Gavallan followed, slamming the window behind him, ripping the curtains closed. He was in a bedroom. It was neat. A nicely made-up single bed covered with a red top sheet. Posters of Los Angeles and Mexico City on the walls. A crib. A dresser with mirrored drawers.
Into the hallway. A shout. Gavallan found Cate in the front room, speaking feverishly to a young dark-haired woman cradling a baby on her lap. The woman stared at Gavallan with intense, frightened eyes. Smells of soup and burnt toast. Another instant and they were out the front door, walking briskly down a dim corridor.
Up the stairs. One flight. Two. Gavallan followed, too winded to ask any questions, happy to have someone else take the lead. After four stories, they reached the rooftop. The door was locked. Gavallan stepped past Cate, raised his leg, and kicked viciously at the handle. Wood splintered. The door flung open, rebounding on its hinges. Sunlight flooded the stairwell.
Cate ran to the edge of the roof and peeked her head over. Raising her arm, she signaled Gavallan back. He dropped to a crouch and eased himself toward the parapet. The seven men were gathered in the street. Arms gesticulated wildly. Raised voices drifted up to them. Then there was a screech of tires. A silver sedan rounded the corner, shuddering to a halt, disgorging four men.
"We can't wait here," said Gavallan, mopping the sweat from his eyes. "They're mustering an army down there."
Cate backed away from the precipice. Setting her hands on her hips, she looked first left, then right. "These apartments are built one next to the other. We can work our way along the roof. At the end of the block, we'll go downstairs and come out on the next street over."
They jogged across the rooftops, easily jumping the gaps between buildings, until they'd reached the end of the street. Lowering himself to his belly, Gavallan ventured a glance below. The men, now eleven in number, stood a hundred yards away, still congregated in the center of the street. An automobile approached from the other direction and made the mistake of honking at them. Immediately, one of the men broke off from the group and pounded savagely on the intruder's hood. A head came out the window. Words were exchanged. Several more of the secret policemen approached. In a moment, they had the driver out of the car and on the ground, and began kicking him.
"Now's our chance," said Cate. "Let's get down to the street."
"But we don't have a car."
"Don't worry," she answered, already moving toward the stairwell. "I'll get us one."
**I T'S BEEN A WHILE** since we've had an occasion to use this room," Bruce Jay Tustin began. "There's no need to mention that it's been a rough year, but boy, _it's been a_ _rough year!_ I guess it was natural, then, for the Mercury Broadband offering to pose some problems of its own. It wasn't the easiest deal to put together, but it's a testament to our professionals and to Mercury's solid management team that we were able to stay focused and surmount those obstacles, so that we're able to stand here among one another today."
"Here, here," murmured the assembly.
Tustin affected a modest stance, his pugilist's chin tucked into his collar. "Let me say that I'm not the one who should be giving this speech. That privilege belongs to another man, someone who for very grave reasons cannot be here today. For those of you who just flew in, I'd like to say that I don't know any more about Jett's whereabouts or his status than you do. I think it best that we offer him our prayers and keep the faith. I'm sure everything will turn out for the best."
Silence reigned as John J. Gavallan, the firm's founder, majority shareholder, and guiding spirit, was sent their prayers. But only for five seconds—then the voices began to swell again. Standing at once among and apart from the assembly, Kirov felt a violent tick in his brain. Enough of the preliminaries. It was time to get to the main event. What had they priced the damn security at?
Finally, Tustin clinked his glass one more time.
"They say 'All's well that ends well,' " he intoned. "And, ladies and gentlemen, I stand before you this evening with news that the Mercury Broadband deal will end very well indeed!" Pulling a note card from his jacket, he slipped on a pair of bifocals. "I don't need these, but I hear they make me look sexy," he said, to a chorus of groans. Then he read: "After a three-week road show that took our executives from Shanghai to Stockholm, from Pittsburgh to Peoria, and after a total of seventy-four investor meetings, I am happy to offer the following comments: The Mercury order book stands at forty times oversubscribed. We have an unprecedented thirty ten-percent orders. And on one-to-one meetings, we scored a cumulative hit ratio of ninety-two percent."
Translated, Tustin's words meant that they had orders for forty times as many shares as they would allocate. Thirty of their clients had asked to take as much of the offering as Black Jet would give them. And 92 percent of the firms with whom Mercury executives had met to pitch the offering had put in orders. By any measure, it was an extraordinary success.
So much for the Private Eye-PO, scoffed Kirov silently. So much for Baranov and Gavallan and even Katya. There would be no mourning any of them. They had brought their fates upon themselves. No one ever said empire building was without pain.
Tustin continued over the sustained hollering and applause. "I guess there's only one piece of information left to give you guys. For that, let me turn the floor over to Tony." He walked over to Llewellyn-Davies and gave him a big bear hug. "Two Names, you done good."
"But seriously, folks, we have had some difficulties with Mercury," Llewellyn-Davies declared as his smile faded and his cheeks grew taut. "Like it or not, though, the time has come for us to put a price on this thing. So here goes. Based on the market's appetite for Mercury stock and using some valuation models of businesses in similar spaces, we've finally come up with something." He shot Meg Kratzer a glance. "This is going out on the hoot and holler, isn't it?"
Meg held up the speaker box. "You're going out live, Tony."
"Great," he said. "Super. So anyway, where was I? Oh yeah, _pricing_. Ladies and gentlemen... Mr. Kirov... tomorrow morning at nine-thirty, shares of Mercury Broadband—ticker symbol MBB—will be issued at thirty dollars a share. _Three dollars above our highest estimate!"_
Llewellyn-Davies crossed the room and placed himself in front of Kirov.
"Mr. Kirov," he said formally, as if asking him to swear in court. "As chairman and majority shareholder of Mercury Broadband, do you accept the price?"
Kirov had already done the math. Thirty dollars a share brought the total offering to 2.2 billion dollars. Deducting Leonid's share and the underwriting expenses, he would still pocket over a billion dollars. And that was just for the 33 percent of the company that was being offered to the public. Were he to value a hundred percent of the shares, Mercury had a theoretical worth of nearly seven billion dollars.
"Thank you, Mr. Llewellyn-Davies, Mr. Tustin," he said. "On behalf of all my employees and colleagues at Mercury, I accept."
Applause erupted. Whistles and catcalls.
And taking a sip of champagne, Kirov thought, _Screw Vanderbilt. Fuck Mr. Gould. I'm a Rockefeller now_.
# 61
#
**S ORRY, SORRY.** It is too late. We are closed today. You go home to Moscow. Come here tomorrow."
He was tall and mustachioed and the name tag on his washed-out flight suit read "Grushkin, Colonel Pyotr R." His English was outstanding, if not his grammar. Bending to check a register on his desk, he scratched at his generous crop of iron gray hair and said, "No, come Wednesday instead. Tomorrow, I am booked. Mr. Hamada from Tokyo."
Gavallan and Cate were standing inside the cluttered operations office of the Grushkin Flight Academy, formerly known as Hulskvoe Air Force Base. The room smelled of sweat, cottonseed oil, and the lingering exhaust of high-octane jet fuel. One step inside had turned Gavallan's stomach to water. He was back where he'd never wanted to be again in his life.
Through the open door behind them, they could see the blue Toyota Cressida that Cate had flagged down to bring them here, its driver counting his $120 fare, and behind him, parked not ten feet away, a Mig-25 Foxbat dressed for combat in khaki camouflage war paint. With its swept-back wings, boxy fuselage, and sharp, angular nose, the Mig recalled the old F-111 Starfighter, only bigger, heavier, and, from what he'd been taught, slower to turn. A few airmen tended to the bird, throwing chocks under its wheels, climbing a ladder to the cockpit to check on the instruments, leading a hose out for refueling.
Guards had left Hulskvoe ten years ago, when budgetary constraints had shuttered the base along with seventy-one of its brethren across the Russian landmass. Since then it had been put to more profitable uses. Budding aviators, flight enthusiasts, and any other individuals interested in piloting some of the world's most sophisticated fighter aircraft came to Hulskvoe to attend any of the day- or weeklong courses that were offered. Prices began at $2,000 a day and went from there.
"We're not interested in going for a ride," said Cate. "Not exactly."
"No?" asked Grushkin playfully. "Who are you? Media, I suppose? You want free ride in my plane and you promise to show my school on television? Look, I need the press, but flights aren't free. Fuel, upkeep." He rubbed his fingers together to show how expensive it was to care for a state-of-the-art fighter. "Listen, we make deal. I give you discount. Fifty percent off. A thousand cash. Dollars, not lira, eh? But you don't get to take home a flight suit."
"I think you've got things a little mixed up," said Cate. "We're not press and we don't want you to give us a ride in your plane."
"No?" Grushkin's manner turned from solicitous to suspicious on a dime.
"We'd like to make you an offer," she said.
"An offer?" Grushkin stepped around the desk, arms crossed over his chest. "What is it exactly you want?"
Gavallan told him, and Grushkin laughed boisterously. "You got to be kidding."
Gavallan pulled out his wallet and laid his American Express Platinum Card on the table. "On the contrary. I've never been more serious. How does a million sound? Dollars, not lira."
**T HE OFFICES OF AMERICAN EXPRESS** Travel Related Services–European Division occupied the top four floors of a Victorian building on the Bahnhofstrasse, one block from the Zurich main station. From his window, Benno Notzli, chief of Centurion and Platinum Card Services, had a pleasant view of Johannes Pestalozzi's statue and the manicured square in which it stood. Pestalozzi, as all Swiss children were taught, was the sixteenth-century schoolteacher recognized as the father of modern pedagogy, and the statue showed him merrily helping a child to walk. A McDonald's restaurant bordered the south side of the park, the luxury department store Globus the west side. The time was 6:49, and Notzli had paused in packing his briefcase for his 7 P.M. departure to listen to a band of Peruvian musicians who'd taken up station below his window. He didn't particularly care for Peruvians or any of the ambling bands of musicians who turned up across Switzerland during the summer like fleas on a dog. To begin with, they were impecunious. Secondly, they were foreigners. Lastly, they were not clients of American Express. He did, however, enjoy their haunting mountain melodies. Especially those played with the pan flute.
The phone on his desk began to ring. Seeing it was his private line, he hurried to answer. "Notzli."
_"Herr Direktor_ , we have a rather interesting call from Russia. You'd better have a look at the file. I'm sending it up immediately."
"Not again." Notzli sat down with a thump, giving his briefcase a longing glance. So much for a timely departure. _Russians!_ He was well-acquainted with the country and its newly affluent citizens. Every weekend another group of Russian businessmen accompanied by their wives, mistresses, nannies, and children flitted their way up and down the Bahnhofstrasse, buying everything that wasn't nailed down. Fifty thousand francs at Bucherer. A hundred thousand at Chanel. Twenty thousand at Bally. Rolexes, furs, diamonds, ostrich shoes, cashmere topcoats, and couture, couture, couture. Shopping sprees of orgiastic dimension.
Notzli knew that most of the merchandise went to government officials flown to Zurich for the weekend to pocket "soft payments" from their counterparts in the private sector for services rendered—past, present, and future. Not that it was his business. It was Notzli's job to review the client's credit and make spot decisions authorizing or denying such purchases.
"What is it?" he asked.
"An odd request from an airport. The Grushkin Flight Academy."
"An airport? Just give me the customer and the amount."
"Mr. John J. Gavallan. An American. The amount is one million dollars."
_"One million dollars!"_ Notzli coughed, coming to attention in his chair.
By now the purchase request and client record was flashing on his monitor. The record showed the client's complete credit history, his average monthly expenditures, days payable, and most recent purchases. It also listed the client's estimated personal net worth, his annual income, and any known assets. Finally, it assigned the entire package a letter grade denoting the client's overall creditworthiness.
Last year, Mr. Gavallan had spent $214,987.15. He paid his bills promptly, averaging fifteen days and his stated annual income was $3.5 million. His overall grade was an A+.
Mr. Gavallan was the real thing.
"Do you have the customer on the line?" Notzli asked.
"Yes sir, I'll transfer him immediately."
Adjusting his tie and smoothing his hair, Notzli introduced himself, then gave his title. "So, Mr. Gavallan, sir, I understand you would like to make a rather large purchase. Please bear in mind, it is necessary for us to take some precautions. I hope you don't mind my asking a few questions to verify your identity."
"Not at all. Shoot."
Notzli asked for Mr. Gavallan's social security number, his date of birth, and his mother's maiden name. Gavallan replied correctly. Then Notzli asked for the small four-digit number printed on the right-hand side of the card. Again, Gavallan supplied the correct response.
"I hope you don't find my questions too intrusive. It's just that your request is coming from an odd location. Normally, significant purchase requests come from jewelry stores, art galleries, even auction houses. You, sir, are at an airport in the region south of Moscow."
"That's right," said Gavallan. "The town is called Hulskvoe, if you're interested."
"May I be so bold as to inquire, sir, what you wish to purchase for one million dollars?"
"A plane. A Mig-25 Foxbat. I'm a pilot myself, and I thought it would be neat to have one to tool around with on weekends."
"Is that right?" Notzli didn't know a Mig Foxbat from a jumbo jet. He was a train man, himself. Antique miniatures. Double-A gauge. "And you're certain this aircraft is worth one million dollars?"
"Actually, it's worth a lot more than that. Production price is around twenty-eight million a copy, but they're having a fire sale."
"You're serious?"
"Yes, I'm serious. I must have this plane."
Benno Notzli stared at the screen, evaluating the man's impeccable credit history and the reasonable voice on the other end of the phone. It was his job to see to it his clients were satisfied, that they were able to purchase the baubles, bangles, trinkets, and, well... _planes_ that they simply "must have." One look at the annual salary and credit grade made the decision a snap. If the man wanted to fork over a million dollars for a Mig-25 Foxbat, he could be Notzli's guest. AmEx would be happy to pocket its customary 2 percent fee on the transaction.
"There should be no problem, Mr. Gavallan. I'll be happy to authorize the purchase."
"Thank you, Mr. Notzli."
"And fly safely."
"I intend to," said Gavallan.
All in all, a most pleasant man, decided Notzli, already halfway out the door. If he hurried, he just might make the 7:13.
**C ATE MAGNUS TOOK A SEAT** at Colonel Pyotr Grushkin's desk. Pulling the phone toward her, she dialed information and asked for the number of the headquarters of the Federal Bureau of Investigation in Washington, D.C. The mere act made her jumpy. The thought of asking a Russian operator for the phone number of the Main Adversary's vaunted internal police was hard to fathom.
Waiting, she watched Jett and Grushkin walk around the Mig, Grushkin pointing out the flaps and ailerons beneath the wing, stooping to inspect the landing gear. Jett looked nervous—fidgeting, nodding frequently, wringing his hands, then brushing them off. Well, she thought, that makes two of us.
The operator returned with the number. She hung up and dialed. It took her two disconnections and a string of "Would you please hold"s before she was connected with her intended party.
"This is Dodson."
"Mr. Dodson, this is Catherine Magnus. I'm sure you know who I am."
"Yes, Miss Magnus. I hope you don't mind my saying I'm a bit surprised to hear from you. How can I be of service?"
"How can you be of service?" If she snapped at him, it was because she was still incensed at his role in her predicament. Were it not for Dodson, she would be safely in the States as she spoke. There would be no question of Mercury's opening for trading tomorrow morning and she could still look at herself in the mirror. "I'll tell you how. First, you can revoke the warrant for Jett Gavallan's arrest. He didn't kill Ray Luca. I was there too—I mean in Florida. Yes, he was looking for Luca, but not to kill him. He wanted to know why Luca was trying to spoil the Mercury Broadband IPO Mr. Gavallan's company was underwriting. Unfortunately, he got there late—we both did, actually. The same people who killed Mr. Luca nearly killed Jett."
"Miss Magnus—"
"If you want to know where to find Luca's killers, I'll be happy to tell you. Drive north from Moscow on the Petersburg road. Take a turnoff for a place called Svertloe and go east another—"
"Miss Magnus, please—"
"You'll find them near a dirty cabin in a small pine forest. They're dead, I'm afraid. We had to kill them. Do you understand, Mr. Dodson? We had to do your job for you!"
_"Miss Magnus, please calm yourself. If you'd like my cooperation, you'll need to compose yourself. Please, ma'am."_
But Cate had no more words. She was crying, her breath coming in great big drafts, as if she'd been drowning and needed air. She'd killed someone. She'd ended a life. It didn't matter that the man was trying to kill her. Even now, after everything, she could not summon any enmity toward him. She saw him dodging round the nose of the Suburban, running at the house, his eyes so ambitious, focused, blazing with mission. She had aimed the gun and pulled the trigger and he had fallen dead without uttering so much as a whimper. She could feel her finger tight against the trigger, the gentle, even pleasant bucking of the gun, the dull fireworks as the casings ejected and tinkled onto the cabin floor. The bullets struck him in the chest, in a neat diagonal from spleen to shoulder, and down he went. She was expecting more drama, more blood, a shout, the acknowledgment of his wounds... something to punctuate the loss of a life. But he just fell and stopped moving and his eyes were still open and that was it.
"It was Kirov," she said, gathering herself. "He sent two of his killers to do the job. Check the flights in and out of Florida. You must have the tail number of his plane somewhere. Look for a late Thursday or early Friday arrival and a Friday evening departure." Cate mentioned Boris and Tatiana and offered descriptions of them.
"Konstantin Kirov? You mean Mr. Gavallan's partner?"
"No, I mean Konstantin Kirov, the man that tried to kill us and is hoping to defraud the investing public out of two billion dollars."
"Let me get this straight. Are you saying that Jett Gavallan does not want the Mercury deal to happen?"
"Of course he doesn't want it to come to market. What Ray Luca was saying about Mercury was true, more or less. Jett looked into it and discovered some serious accounting discrepancies. He would never represent a company that wasn't exactly as advertised. Contrary to your screwed-up line of thought, he is not a dishonest man."
Dodson cleared his throat. "I appreciate the information, Miss Magnus. You can be sure we'll look into it. But if you'd like any cooperation from our side, I'm afraid you'll have to come back to the United States. I take it you are in Moscow now?"
"South of it. Hulskvoe. It's a former Red Air Force base." Drumming her nails on the desktop, she managed to slow her breathing and get a grip on herself. "Actually, Mr. Dodson, _I_ want to help _you."_
"You do?"
"Yes. That is, if you're still interested in jailing Konstantin Kirov for skimming two hundred million dollars from Novastar Airlines?"
"Oh yes, ma'am, we're still very interested in Mr. Kirov. But I think you're mistaken on your figures. Kirov stole a hundred twenty-five million from Novastar."
"No, Mr. Dodson, it's you who are mistaken. I have in my possession Novastar's banking records for the past three years. Every transfer into and out of the company. They're all there. I also have the complete banking history of a company called Andara, and one called Futura. I even have a couple of numbered accounts nobody's ever heard of. I guarantee you, it's enough evidence to see Konstantin Kirov convicted in any court in the world."
"And you're willing to turn this over to the government?"
"I am."
A palm muffled the mouthpiece and Cate could hear Dodson's heated voice summoning someone named Roy. Waiting, she watched Jett climb into the Mig's cockpit and Grushkin take his place next to him. Jett looked more comfortable now, and she found her own nerves settling too. Then she reminded herself that in a little while she would have to take Grushkin's place, and her hard-won repose vanished. Suddenly, the Mig looked very big and very dangerous.
"Miss Magnus, you've piqued my interest," she heard Dodson's voice say. "What is it you want?"
"Just a little help getting home."
"Oh?"
Cate outlined Jett's plan for the next twenty-four hours and how the FBI could help.
"Anything else?" Dodson asked. "Dinner with the President? An audience with the Pope?"
"No, thank you," Cate replied, all business. "That's all." Her sense of humor had deserted her sometime back, probably in a dusky pine clearing in the plains north of Moscow. "Is that a yes?"
It took Dodson a long time to answer.
**S HE HAD ONE CALL YET TO MAKE.** As usual, she'd saved the hardest for last. Half a dozen times already, she'd picked up the handset only to slam it right back down. Grushkin had brought her a flight suit and draped it over the door. A helmet with a dark sunshield sat on the desk in front of her, and she could see her reflection in it. She asked herself who she really was, Cate Magnus or Katya Kirov. And who, after all was said and done, she would choose to remain. The answers came more easily than she expected. As Jett said, there was only one direction: straight ahead.
Picking up the phone, she dialed the nine-digit number that she recognized as belonging to the north side of Moscow. It was a hard part of town, and the voice that answered the phone matched it perfectly. _"Da?"_
Catherine Elizabeth Magnus did not hang up.
**Y OU READY?"** Gavallan asked Cate.
"Yeah," she said, then more certainly, "Yes, I'm ready. Jesus, Jett, what am I supposed to say—hee-hah, let's git? I'm scared, that's what I am. Are you?"
Glancing to his right, he caught sight of her beneath the Perspex bubble next to him. Wearing the oversized helmet, she looked thin and vulnerable. He could see that she was trying to smile and having a hard time of it. Shifting his eyes to the fore, he gazed down the slim strip of asphalt rolling to the horizon. He waited for his heart to beat faster, for the prickly fingers to scratch at the back of his neck, but his heart was calm, and so was his psyche. In the final analysis, he was just flying a plane. Besides, it wasn't takeoffs that frightened him. It was what he'd find up there.
"Am I what?" he asked, a half second later.
"Are you ready?"
"Absolutely," he said, fingering the throttle, inching it ever so slightly forward. Immediately, the engine roared. The aircraft begin to rumble. "Let's go to Germany."
**C OLONEL PYOTR GRUSHKIN WATCHED** his beloved Mig taxi to the end of the runway, turn slowly, then barrel down the asphalt and take off over the golden fields of wheat swaying in the warm evening breeze. Wings sweeping back toward the fuselage, the aircraft climbed higher and higher into the azure sky. The American rocked the fighter port and starboard, a gentleman's good-bye, and Grushkin's heart went with it.
When the Mig was barely a speck in the sky, he walked into his office and made a phone call.
"Jerzy, this is Pyotr. Listen, I have a student taking the jet out for a long run toward the border. Nothing to worry about—just a training exercise. But in case anything funny happens, maybe you could keep your eyes closed."
"What do you mean, 'keep my eyes closed'?" Jerzy asked.
"Take a short break. Forget you saw anything. If any tough guys ask, say everything's quiet as the grave."
"This is a serious matter you are talking about, Colonel. A question of the motherland's security."
"I think it is more a question of a thousand American dollars, _nyet?"_ There was a pause, and Grushkin pictured his old crew chief seated at his obsolete radar array, a cigarette burning between his fingers, a tepid cup of coffee on his desk. "Please, Jerzy. A favor."
"It is a very quiet evening. I would be surprised if anything of interest appeared on our screens. Good-bye, Pyotr."
When Grushkin returned to the hangar, he was confronted by a pageant of disappointed faces. He stared back, then slowly allowed a broad, shit-eating grin to crack his stoic face.
"Hey, don't look so glum, you dirty bastards," he shouted. "Somebody break out the vodka! We're fucking millionaires!"
# 62
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**T HE SAFE COURSE** was to keep the plane low, respect a two-hundred-foot ceiling, bleed the speed to five hundred miles per hour, well under supersonic, and take the Mig for a sunset cruise over the rooftops of Eastern Europe. A check of the instruments showed what Gavallan thought of the safe course. Speed: 650 knots. Altitude: 30,000 feet and climbing. Screw the safe course. It was long gone anyway. He'd thrown safety to the wind when he'd busted into Ray Luca's home in Delray Beach Friday afternoon. No, he decided, he'd chucked it earlier than that. He even had the date: January 10, somewhere around three o'clock, when after a boozy lunch at Alfred's in the financial district, he'd signed Konstantin Kirov as a client and pledged Black Jet Securities' every effort to make the Mercury offering a grand slam.
Rolling his shoulders, Gavallan tried to get comfortable in the scooped-out seat. One hand fought the stick. He was holding it too rigidly, nudging the aircraft left every few seconds to compensate for a slight oversteer. The other hand rested on the throttle like a leaden weight, keeping his airspeed steady.
A click of his thumb activated the intercom. "How ya doin'?"
Cate sat beside him in her own self-enclosed turret, his airsick RIO, or radar intercept officer, in her sky blue flight suit and pearl white helmet. "Alive," she whispered. "Just barely."
"We're about eleven hundred miles out," he said. "Another two hours and we'll be on friendly soil."
"Just hurry, Jett."
Cate had greeted the initial rush of speed with an exhilarated "Wow!" and then, a few seconds later, as they'd slowed dramatically, a less enthusiastic "Uh-oh." She'd used two of Grushkin's doggy bags, and Gavallan didn't think there was anything left in her tummy for a third.
"I am," he said. "You can count on it."
Gavallan released his thumb and turned his eyes back to the bank of instruments. He'd expected it to be easier than this. He'd expected it all to come right back, as if sliding into the cockpit after an eleven-year break were the same thing as slipping on an old jacket and finding that it still fit. Instead, the seat felt tight on his bottom. The cockpit was much too small, the stick unresponsive. It wasn't a question of whether he could still fly. He could. The Mig was not especially challenging in that regard. The cockpit configuration was similar to that of the A-10 he'd piloted prior to going into the Stealth program. Aircraft design dictated that form follow function and the throttle, stick, and navigation systems were all in similar places. The gauges and the heads-up display, or HUD, with their Cyrillic lettering might be difficult to read and the airspeed indicator was in kilometers, not knots per hour, but when it came down to it, the Mig was just another jet. All the same, he was flying poorly, stiffly, with no grace, no feel for the aircraft. Even the familiar tightness of the G suit around his thighs and across his stomach, the shoulder harness's stiff bite, failed to comfort him.
Relax, he told himself. You were born to do this. _Born to fly_.
The words set him on a slingshot journey back through time in which he reviewed his every accomplishment as a pilot. Baghdad. Tonopah. Colorado Springs. The images shot past his mind's eye with increasing speed, faster and faster, one on top of the other, blurred, ill-focused, until just as quickly they froze and he saw himself at age fifteen, lying on the hood of his father's Chevy on a hot summer night in Texas. The car was a hot rod, a fire engine red '68 Camaro with a 454 engine, twin chrome exhausts, and a white racing stripe painted down the hood. After spending all afternoon washing and waxing it, he'd driven twenty miles outside of town and parked in the middle of the open plain where alone in the gathering dusk, he could watch the jets from Beeville Air Station, fifty miles to the north, screech across the sky. He would lie there for an hour, looking up at their gleaming silver bodies, listening to their engines shake the very pillars of the sky, dreaming upon the white contrails they left behind. He was born to fly. It had come to him with a certainty that was raw and cold and frightening. Shivering in the ninety-nine-degree dusk, he'd known he belonged up there.
_So, fly_ , he told himself now. _Relax and fly, goddamn it_.
He gazed at the countryside below. The sun had fallen below the horizon, and its waning rays burnt the Earth's cusp a flaming ochre. The sky above was dark and supple and inviting.
Gavallan's eyes fell to the radar array, a square black screen six inches by six inches located on the instrument console. The screen was dark except for his own orange blip and a flashing triangle that was a passenger jet ninety miles to the north. He'd been flying for an hour, and so far he had detected no sign of Russian air patrols. Either Grushkin was a man of his word or Russian air defenses were perilously lax.
Checking his coordinates on the heads-up satellite navigation system, he put the plane into a seventy-degree roll and brought his heading to west-southwest. Doing some quick math, he figured he'd put the bird down at Ramstein Air Force Base outside of Frankfurt at around 10 P.M. local time. From then on, they'd be living on the good graces of others.
Five minutes passed. Gavallan checked his coordinates against a map on his knee and decided he was somewhere just south of Kraców, Poland, safely out of Russian airspace. "We're going to start looking mighty suspicious to our flyboys anytime now," he said to Cate. "Time to call ahead and give the boys in blue our arrival time." He checked his radio log and dialed in Ramstein Air Force Base, home to the 86th Airlift Wing. As he keyed the mike a second time, a steady howl sounded outside his earphones. At the same time, a red square blinked on his console. _Fire_. Starboard engine. His eyes kicked right. The gauge showing the exhaust gas temperature was maxed out, full in the red. He pulled the handle to activate the fire extinguisher and cut fuel flow to the engine. At the same time, he cut back on the throttle, shut down the engine, and put the plane into a steep dive. A check over his shoulder revealed nothing. But the gauge didn't lie.
The plane shuddered, as if hit from the side.
"Jett!"
"Hold on, sweetheart, just a little problem."
"What is it?"
Gavallan's heart was racing; a lump lodged high in his throat. The stick was bucking in his hand. He jerked it to the right, but there was no response. A high-pitched buzz saw whined in his earphones. He was losing control of the aircraft.
_This isn't my plane_ , he protested silently. I haven't trained in a Mig. A second check over his shoulder showed flames licking the wing. Immediately he hit the auxiliary extinguisher, and a gust of white puffed from beneath the wing. The flames flickered, then disappeared.
And then the world turned upside down on him. The Mig rolled over and went nose down, spinning in a slow roll.
"Jett, help us. Stop this. Oh, God... no, no!"
Gavallan looked at Cate, her eyes wide with terror, her helmet pinned to the canopy.
A voice inside him whispered, _You were born to fly. So, relax and fly_.
"Just a little glitch," he said, in the voice Grafton Byrnes had taught him that hot and sunny day in Alamagordo. "Not to worry."
Still inverted, he pulled back on the stick, depressed the ailerons to stop the spin, and pulled the nose through. Gently he goosed the port engine. The single turbine hummed confidently. It was working. The plane was responding to his touch. He was guiding the aircraft instead of allowing it to guide him. A well of confidence grew in his chest, warm and reassuring. It was the pilot's bravado coming back. The certainty he could do anything, if by sheer force of will alone.
And there, as he plummeted toward the earth at four hundred miles an hour, a dam burst in his mind. A clarity of thought, of memory, of action, came to him that he had not possessed for years.
_Priority One. Ring One_.
The words struck him like a lightning bolt.
The attack on Abu Ghurayb. Saddam's Presidential palace.
He saw himself in the cockpit of the F-117— _no, damn it, he is there..._ the stick between his legs, the joystick to his left, the infrared display screens. _He is there_. Inside _Darling Lil_ , ten thousand feet above the Iraqi desert.
_He is at bombing altitude. A finger toggles a switch. Bomb armed. Eyes forward on the IR display. Target spotted. A_ _stable of buildings silhouetted against the gray desert floor. His finger slews the crosshairs back and forth across the palace until he decides he has found the wing. Then, as if a mechanism itself, the thumb locks down. A yellow light flashes. Laser acquisition engaged. Red lights fire on the heads-up display. Target in range. Gavallan hits the pickle and the weapons bay door opens_. Darling Lil _shudders. He depresses the pickle again and the bomb falls from the aircraft. He feels the aircraft jerk upward, as if freed from its moorings_.
_As the bomb falls, his eyes lock onto the IR screen and the delicate crosshairs positioned over the east wing of the Presidential palace. All external stimuli disappear. He is in a tunnel. At the far end rests his target. Thumb locked. The crosshairs do not move_.
_"Thunder three-six. Red Leader One. Copy?"_
_The bomb appears on the screen. A lethal black dot skimming across the ground at an impossible speed. A red light blinks. A fuel warning. Tanks low. Gavallan pays it no mind. It will wait_.
_"Roger Red One. Come in."_
_"Friendlies in the area. We have friendlies on-site. Abort run. I repeat: Abort run."_
_At the sound of the word "friendlies," Gavallan's finger is already moving, skewing the crosshairs away from the palace, guiding the "smart bomb" away from the American troops_.
_On the console, a second light blinks—yellow, urgent. It is the Allied Forces Locator warning him he has engaged friendly forces_.
_"Abort run! Confirm, Thunder three-six!"_
_But the pilot's instincts have beaten the verbal command by a second, maybe two. An eternity in the electronic world that can be translated into two hundred fifty feet of fall time_.
_Gavallan keeps his thumb pressed to the right, ordering the bomb to follow his instructions. But the bomb does not listen. She has been on her downward trajectory too long and it is as if she is too stubborn to alter her course_.
_The desert flower blossoms. The IR screen blanches. A blizzard of white noise. The palace reappears. The east wing is no more, a bonfire of angles fallen in on itself. The heat signatures have disappeared, too, replaced by the blotchy, pulsing quasars that indicate fire_.
Inside the Mig, Gavallan lets the images fade away. He has seen enough. In an instant, the past has vanished. But it is a different past than the one he has known. A different reality than the one he has lived with these eleven years. No longer will he question his response, second-guess his reflexes. He knows now that he did everything he could, more even, to prevent the bomb from injuring American Marines. Governed by his instincts, he ordered the bomb off its course even before he himself had fully received the command. If his actions were not sufficient to save the lives of ten men, to prevent two others from being robbed of their ability to live full and decent lives, they were still all he could demand of himself. He was an accessory, yes, and for that he would always feel horror and revulsion. But he would no longer feel the guilt, the shame, the dishonor, no longer believe that it was his own poor reactions that had caused those tragic events.
He would never be free of that night, but he was no longer its prisoner.
Slowly, the nose righted itself and the wings found the horizon. The plane shuddered again and was still. They were gliding on a lake of ice.
"Just a little engine problem," he said to Cate. "All taken care of. Sit tight. I'll have us down in a jif."
"Hurry, Jett... thank you... but hurry."
"Roger that."
Bringing the airspeed down to 250 miles an hour, Gavallan let go a long breath. The Mig flew straight on its course, a black eagle skipping across the European sky.
**R AMSTEIN TOWER,** this is United States Air Force Captain John Gavallan, retired. Serial number 276-99-7200. I've got a Russian Mig under my butt that I'd like to put down at your place. You should have word about our arrival. Copy?"
"Copy, Captain Gavallan. Sorry, but we have no word of your status. You are negative for a landing. Please exit secure airspace immediately." There was a pause, and the communications link crackled with white noise. A new voice sounded in Gavallan's earphones. "Captain Gavallan, this is Major Tompkins. You are roger for a landing. Please proceed to vector two seven four, descend to fifteen thousand feet. Welcome back to the Air Force."
"Roger that," said Gavallan. Same old. Same old.
**A T 10:07 LOCAL TIME,** Gavallan brought the Mig to a perfect three-point touchdown on runway two-niner at Ramstein Air Force Base, thirty miles south of Frankfurt, Germany. A jeep waited at the end of the runway, blue siren flashing, to guide them to their parking spot. Gavallan followed at a distance, keeping his ground speed to a minimum. Finding his spot, he killed the engines. Airmen dashed beneath the Mig and threw blocks under his tires. Gavallan waited until they reappeared, flashing him the "thumbs-up," before opening his canopy and unbuckling his seat harness.
The twin, rounded hooks of a flight ladder coupled onto the fuselage and, reluctantly, he climbed out of the cockpit. He stopped at the bottom rung, not wanting his foot to touch the ground. The crackle of avionics still echoed in his ear. The "by the seat of your pants" rush that came with flying a jet lingered inside him like a melancholy phantom. For a few seconds he listened to the cry of the turbine engines winding down and sniffed at the burnt rubber and let the wind brush his cheek. Technically, he owned the plane, but he had no plans to fly it again. Jets belonged to his past, and he knew well enough not to look back.
Jumping to the ground, he jogged around the nose of the aircraft to help Cate out of the cockpit. "Never again," she said. "And you did that for a living?"
"It's not so bad once you get the hang of it."
A major in neatly pressed blues approached. "Captain Gavallan? I'm Calvin Tompkins, executive officer in charge of field security. Welcome to Ramstein."
Gavallan accepted the outstretched hand. "This is Miss Magnus."
"Evening, ma'am," Tompkins said, offering a crisp nod of the head. "I understand you two are headed stateside."
"We need some transportation. The Mig's got a lousy range—fifteen hundred miles max."
"If you'll follow me, I'm sure we can accommodate you. We've got a Lear fueling up as we speak, courtesy of Mr. Howell Dodson of the FBI. I'm afraid it doesn't have such wonderful range either. You'll have to stop in Shannon, Ireland, to refuel, but it'll have you to New York by morning. We had you scheduled for ten forty-five, but I'm afraid we've hit a bit of a glitch."
"A glitch?" asked Cate, her voice taut.
"Just a solenoid that needs replacing," said Tompkins. "Should have it changed out any sec."
Gavallan knew his luck had been too good. "So what's the new departure time?"
"Right now, we're looking at a midnight ETD."
"Midnight?"
"And you shouldn't have to dally in Shannon long. An hour tops."
Gavallan scratched the back of his neck, rejiggering his math. Takeoff at midnight. Hit Shannon by two-thirty. Takeoff from Ireland at three-thirty. Setting the whole operation to New York time, they'd land at JFK around six o'clock. Enough time should everything go according to schedule.
"Just one question, Captain Gavallan."
"Yeah?"
Tompkins pointed to the Mig behind them. "What exactly do you want us to do with your plane?"
# 63
#
**I T WAS PAST MIDNIGHT,** and in room 818 of the Peninsula Hotel in New York City, Konstantin Kirov was sleeping. The telephone rang. Instantly, he was awake, knocking back the sheets, fumbling for the handset. _"Da?_ Kirov."
"Wake up, younger brother. Trouble."
"What do you mean? I thought you were in Siberia."
"I am. But I had a few of my men keep tabs on the dacha. Gavallan has escaped. He took Katya and the other American with him."
"Impossible," said Kirov, sitting up, grabbing at his wristwatch, squinting to read the time. "I assigned my best man to look after them. There were four guards with him."
"All dead," said Leonid. "We found five bodies including Tatiana and, I imagine, your 'best man.' From what we pieced together, Gavallan had a dagger of sorts and used it to kill one of the guards and take his weapon. From there it's anybody's guess."
Kirov tried to imagine Boris and Tatiana and the others dead. A quick rage ignited inside him. He knew why Leonid was watching the dacha. He had posted his men there to make sure Kirov did not spare his daughter's life. "If you were watching, why the hell did you let them drive away?"
"An oversight on our part." There was a pause. "We were able to track Gavallan to Moscow," said Leonid finally. "I'm sorry to say we were unable to keep in contact with him afterward."
"You lost him?"
"Regrettably," said Leonid. "Have you heard anything from your contact at Black Jet?"
"Not a word. I finished dinner with them an hour ago. The deal is going ahead as planned. As far as they are concerned Gavallan is missing in action. Some think he may be involved with the murders in Miami. Others don't dare to think anything. The deal is simply too important for their company."
"Most probably he is still in Moscow with your daughter. Nonetheless, you may see fit to take precautions."
"Precautions?"
"To eliminate any threats should they become localized. After all, Gavallan holds no concrete proof to stop the deal, does he?"
"Concrete? No. But from what I understand he doesn't need any. A call to the right parties will suffice."
"Perhaps we can assume Mr. Gavallan has decided to join with our side in this matter. From everything you've told me, he needs the deal as much as you."
"And if he does not?"
"There is no going back, Konstantin Romanovich," came Leonid's icy response. "Neither for you nor I. We will not embarrass the president. We will not disappoint the state. We will have our money."
Leonid hung up.
Rubbing a hand over his face, Kirov wondered what else could go wrong. He knew he should be worried, but his sheer lack of options left him emboldened instead. He told himself that if Gavallan had wanted to cancel the deal he would have done it already. There had to be a reason he hadn't contacted his partners, and that reason was that he wanted the deal to go through. He wanted his seventy million in fees. He wanted to keep control of his company. Kirov had always pegged him as a greedy one. Smooth, yes, silky smooth, but greedy, too. He was, after all, a banker.
_There was no going back_.
Repeating Leonid's words, Kirov felt a steely resolve firm up inside him. Rising, he crossed to the desk and retrieved his electronic address book from his briefcase. He found the name he needed quickly. He dialed a Manhattan number and a Russian voice answered.
"This is Kirov," he said. "Get your boss on the line. Now."
Gavallan might be in Russia, but Kirov was not going to take any chances. If he could get away from Boris, he might be capable of any number of things. The American was more resourceful than he had anticipated.
A familiar Russian voice came on the telephone and Kirov explained what he wanted. After haggling a few minutes they settled on a price. Satisfied, Kirov hung up, then punched the console for a new line. The hotel operator answered immediately.
"Room 544," he said.
The phone rang three times, four. Finally, a groggy voice answered. "Yes?"
"Some news concerning Mr. Gavallan. It seems he is no longer with my people in Moscow. Are you sure you haven't had any word from him?"
"Lord no. Not a whisper. You're certain he's gone?"
"Still in Russia, no doubt, but out of my control."
"Damn it, Konstantin..."
"Shut up. I'm calling to tell you to be prepared, that's all. The offering will go through. Do you understand?"
"Yes."
Hanging up the phone, Kirov turned off the lights and went back to bed. It wouldn't do to look haggard on the most important day of his life. Sleep came easier than expected. It helped immensely to know that when he visited the New York Stock Exchange in the morning, he would have plenty of friends with him.
**G AVALLAN PACED THE TARMAC** at Shannon International Airport, tired, frustrated, and impatient. Salt and brine from the ocean laced the air, giving the predawn sky a welcome bite. He told himself he should be asleep in the plane like Cate, gathering his energy for the coming day. Lord knew, he was tired. But he was too keyed up to sleep.
Delays. Delays.
They had landed at two o'clock local time to top off their tanks before crossing the Atlantic. Three hours later, they were still there. A bulb in the starboard fuel gauge had burned out and the pilot had refused to take off until it had been replaced. Gavallan had tried to bribe him, but such was military operating procedure that the pilot would not consider the proposal for all the money in the world. The future tottered on the availability of a lousy ten-cent part. Gavallan wanted to scream.
A mist was building over the grass that bordered the runways. Soon it would turn to fog and the airport would be socked in. He looked up briefly, catching the blinking lights of another plane flying high overhead. He couldn't know it, but inside the plane a short, wiry man slept, a blanket pulled to his neck. He was traveling to America for the first time. In fact, it was the first time he had ever traveled anywhere outside of his country. A matter of some importance had forced a hasty and unplanned departure. A business arrangement that needed squaring.
In his sleep, he was dreaming of the old country. Of the rough mountains where he had grown up. Of the rocky soil and rushing streams. Of the impoverished villages and the indomitable people who inhabited them. Some called it the "bandit country," and in truth it was a land that robbed its people of much. But out of nature's cruelty, they had learned to rely on themselves. To count on one another. In these mountains, a man's word was his most valuable asset. He gave it sparingly and with his fullest commitment. While nature was capricious, man had an obligation to be steadfast. To break one's word, then, was to break with his fellow man. Nature could not be punished for its whimsy, but a man could. And the punishment would be awful.
The man dreamt of such punishment.
In his sleep, he smiled.
Gavallan lowered his eyes from the sky. The twin beams of an airport jeep cut through the light fog, advancing rapidly on him. It was the pilot, and as he passed he held up a small cardboard box for his passenger's inspection. "Five minutes and we're out of here."
Finally, thought Gavallan, jogging toward the plane.
# 64
#
**G RAFTON BYRNES PASSED THROUGH** the revolving doors of the Banque Privé de Genève et Lausanne on the Quai Guisan in Geneva at precisely 10 **A.M.** Tuesday morning. Announcing himself to the receptionist, he was shown to a conference room on the fourth floor. The picture window offered a splendid view of Lake Geneva. Byrnes ticked off the sights, running left to right. The Wilson House, where the League of Nations had first met in 1919; the enormous gray stone monuments that housed the European seat of the United Nations; and farther on, past copses of oak trees and manicured lawns, the building where GATT, the General Agreement on Tariffs and Trade, was overseen.
There was a soft knock on the door, and a hunched, portly figure clutching a pad of paper in one hand and a cup of coffee in the other hurried into the room. "Hello, Mr. Byrnes. I am Pierre Pillonel. Welcome to our bank." He stared at his visitor through thick, owlish spectacles. His hair was mussed and his cheeks flushed and red-veined. If his demeanor was timid, his voice was anything but—a rolling, confident baritone that a politician would kill for. Setting down the paper and coffee, he extended a hand, pulling it back at the last moment. "Excuse me, I see you are injured."
"It's nothing," said Byrnes, turning his hand this way and that to show he was in no way hindered. "A mishap with my car. I find I'm getting clumsier with age. Thank you for seeing me on such short notice."
"A friend of my brother's is a friend of mine. Excuse me if I'm not quite myself. I'm still reeling from the news."
"I'm afraid I don't..." It was then that Byrnes noted the beleaguered cast to Pillonel's eyes. They were red and puffy. His nose was runny, his cheeks not flushed, but inflamed.
"You have not heard? Jean-Jacques is dead. He was in Zurich on his way to a short vacation. A robber surprised him and Claire in their hotel. They were both killed. It's terrible. I shudder." The baritone cracked and a tear rolled down Pillonel's cheek. He tried to keep a brave front, but a moment later a sob racked his chest, his stern mouth quivered, and he began to cry in earnest. "I'm sorry," he said, wiping at his eyes. "I don't know why I came to work. My wife told me to stay home. She said I was a fool to come."
"My condolences," said Byrnes, without sympathy. He wasn't surprised Pillonel was dead. The news hadn't hurried his pulse a beat. If anything, he experienced a brief and satisfying surge of justice done, even if it was cruel on his part. Jean-Jacques Pillonel was as responsible for his ruined thumbs as Boris. He deserved partial credit for the deaths in Florida, and if things didn't turn around quickly, they could stick him with the dismantling of Black Jet Securities, too.
Cautiously returning his gaze to his host, Byrnes caught a passing glimpse of his own reflection in the window. Dressed in a charcoal Brooks Brothers suit, hair neatly combed, thumbs discreetly bandaged, he actually looked presentable. A short discussion with the embassy's legal attaché, a man Byrnes pegged as the local CIA resident, had produced a diplomatic passport, an interest-free loan in the amount of a thousand dollars, and a ticket to Geneva the next morning with an onward connection to New York (including an armed escort onto the plane). A hot meal, a soft bed, and ten hours' sleep had done the rest. Moscow, Boris, and the dacha were quickly fading into a corner of his memory he hoped to rarely visit.
"There, I am better," Pierre Pillonel said after a minute, taking a last swipe at his nose. "Please excuse me."
The two men sat at a lacquered maple conference table, taking their time to unbutton their jackets and nap their slacks, uncap their pens, and take a sip of the mineral water that had been poured for them prior to their arrival.
"So?" said Pillonel, a false, professional smile pulling at his cheeks. "How may I be of assistance to you?"
"As you may know, Black Jet Securities is set to take Mercury Broadband public later today on the New York Stock Exchange," Byrnes began. "It's a large deal. A two-billion-dollar equity offering."
"I've read about it. Should I be asking to buy some shares?"
"I'm afraid that wouldn't be such a good idea."
_"Non?_ Why not?"
"Sadly, we've come into possession of evidence showing that Mercury is not exactly the company we sold our investors. Konstantin Kirov, Mercury's chairman, has been siphoning large sums of money from another of his investments, Novastar Airlines, and using the funds to inflate Mercury's balance sheet."
"When you say a large sum, you mean how much exactly?"
"Hundreds of millions of dollars."
_"Dieu,"_ Pillonel said under his breath.
Byrnes nodded in agreement. At least they were talking the same language. "Naturally we're canceling the offering. This morning before the opening bell, we'll announce that the IPO has been shelved indefinitely. It will be an embarrassment to Black Jet and a setback to Mercury Broadband, which we feel is still a vibrant, attractive company. We're quite upset at the development. As Mercury's bankers, we feel we should have spotted the problem earlier. If we'd chosen our partners more wisely this wouldn't have happened."
Byrnes let the words hang there, checking for a response from Pillonel—a sympathetic shrug, a world-weary sigh, an admission that "Yes, this could happen to any of us"—but the Swiss banker remained unmoved, his gaze not giving away a thing.
"Black Jet Securities has an obligation to shelter Mercury from Kirov's misdeeds," Byrnes continued. "We want to do everything possible to insure that Mercury's future as a viable enterprise does not suffer because of its chairman's bad behavior. I like to think the Russian government has a right to the money stolen from Novastar."
The mention of money lit a fire behind Pillonel's eyes. Abruptly, he sat straighter, lifting his chin from his neck. "But of course you are right. One cannot condone such behavior. These oligarchs are too much. They think the entire country is their own private fief. They steal a little from here, a little from there. Their conduct is deplorable." He took a sip of water and shrugged fatalistically. "But how do you hope to convince Mr. Kirov to give back the money?"
"I don't. He's a crook and a murderer. He'd never give it back. But I can convince you."
_"Me?"_
Byrnes delved into his jacket pocket for a translucent envelope and flipped the minidisc onto the table. "Jean-Jacques was working in cahoots with Mr. Kirov to help Mercury defraud Black Jet and the investing public. They cooked the books together and Kirov paid Jean-Jacques to falsify the due diligence Silber, Goldi, and Grimm performed on Mercury. When Mr. Gavallan presented Jean-Jacques with the evidence this past Saturday, your brother broke down and revealed what he'd done. Somehow Konstantin Kirov got word of his duplicity. Your brother wasn't going on vacation. He was getting the hell out of the country. You don't really think Jean-Jacques was killed by a thief, do you?"
Byrnes stared at Pierre Pillonel. It was hard to believe he and Jean-Jacques were twins. One was the model of continental sophistication, the other its opposite. "If you look at the disc, you'll find that Kirov transferred the money he stole from Novastar to your bank."
"To the Banque Privé?" Pillonel slid the disc back to Byrnes. "I'm sure I wouldn't know. I am not his account manager. Many of our clients hold numbered accounts. I don't have to tell you of our secrecy requirements."
Lies. Lies. Everywhere lies, rued Byrnes. Since when had dishonesty become the currency of discretion? He waited a moment, taking a deep breath. He felt depressed. Deeply and achingly depressed. Leaning across the table, he whispered, "Cut the bullshit, Pierre. You know Konstantin Kirov is a client of yours. Your brother sent him to you nine months ago to open an account and I wouldn't doubt it if the three of you went out and broke bread together and told each other how you were going to screw the world."
Pillonel shook his head and lifted a finger. His mouth even moved, but he couldn't bring himself to protest.
"You are a partner at the bank, correct?"
"Yes," said Pillonel. "Managing partner, in fact."
"And as such you are liable for the firm's debts and grievances, _non?"_
"It is a private bank," said Pillonel. "I am a partner. Therefore I am liable. It is the law."
"Then let me make this clear," Byrnes went on, his voice as cold and hard as a diamond. "If you don't wire every cent of the money Konstantin Kirov stole from Novastar Airlines back to the airline itself, I will make sure that you are shown to have been involved in Kirov's scheme from the very beginning. Whether you really were or not, I don't know and I don't care. But if you don't cooperate, I will do my best to link your brother's fraudulent behavior with your own and tie all three of you together into one great big daisy chain. Family being family, and twins being especially close..." Byrnes shook his head, letting the threat of a public trial, the front-page articles, the two-minute reports on the evening news finish the sentence for him. "Don't answer now. Check the disc. It's all there."
Without another word, Pillonel left the conference room. Byrnes stood and looked out at the lake, calm and glassy, promising a hundred summer idylls. He was wondering where Gavallan was, if he'd made it to New York, and even then, if he could pull off his plan. Or more precisely, if Cate would allow him to.
And after that? Byrnes asked himself. What are you going to do? Go back to work? Sit back down at your desk as if the last seven days hadn't happened? He wasn't sure. He knew he wanted to see his kids. He thought about making amends with his wife and chucked the idea posthaste. That part of his life, at least, was over. He decided Pierre Pillonel hadn't been so wrong to venture to his office while in mourning for his brother. There comes a point in life when your work and your self—your own idea of who you really are—grow so intertwined as to be inseparable. Byrnes realized he'd reached that point a long time ago. When you spend twelve hours a day, day in and day out for seven years, you pretty much become the job. _And so, where to?_ Home, thought Byrnes. To San Francisco. To Black Jet. If Jett could succeed in saving the company, he wanted to be there at his side to help.
Five minutes later, Pillonel returned, accompanied by a dour, rail-thin man whom he introduced as Monsieur Buffet, the bank's in-house counsel. The attorney shook Byrnes's hand once, as if sealing a bargain. He had dark, depthless eyes, and as he spoke they remained drilled on Byrnes. "You realize that the bank abhors criminal behavior in every shape and form. That we do not as a matter of highest principle deal with persons of anything but the most sterling character. And that we knew nothing— _I repeat, nothing_ —about Mr. Kirov's activities vis-à-vis Novastar Airlines."
"Yes, I realize all that," said Byrnes. See no evil, hear no evil, speak no evil.
"And should the bank agree to your request, that should in no way be construed as demonstrating either our knowledge of or our complicity in Mr. Kirov's affairs."
Again, Byrnes nodded.
"A terrible business," said Pierre Pillonel, waving his attorney into a far chair. "Black days. So hard to know who to trust, who not to."
"I can imagine."
"Naturally, we are prepared at this instant to wire the funds to the account you mention... or to _any other account_ you may wish for us to help you set up." Pillonel paused, but only for the shortest of moments. "A numbered account with our affiliate in the Bahamas, perhaps?"
Byrnes kept a mirthless smile to himself. What did the French say? _Plus ça change, plus c'est la même chose_. "No thank you. Novastar's account at the Moscow Narodny Bank will be fine." He handed Pillonel a piece of stationery bearing the account numbers. "By three-thirty today, gentlemen."
# 65
#
**I T WAS THE QUIET TIME.**
The time for reflection. The time to put your personal thoughts in order, separate the good from the bad and take a measure of your life. The time to settle things. The last free moments before the operation went tactical, because once it went tactical and you were doing what you'd trained these last four months to be doing, the only things you thought about were the mission, your part in it, and maybe, if you had the courage, whether you'd get out of it on the other end alive.
The members of Team 7 sat at the edge of the landing strip, using parachutes for seats, twelve castaways eating their rations of Pop-Tarts, Fritos, and protein bars, drinking their Gatorades and Diet Cokes. They were Americans, all of them—the baseball caps and work boots, the insouciant smiles, the two-day beards. Or so you'd swear until looking closer. And then, as you examined each one by one, you would shake your head. Here, the cheekbones too high, the eyes vaguely Asiatic. There, the blond hair a shade too blond. This one's gaze too dark, mirroring a fatalism bred over centuries. That one's face too gaunt, hunted, fearful.
They were born of the East. Mother Russia's children.
A stiff wind snapped at the waist-high grass that bordered the strip. Behind them, the Bering Sea lapped at a beach even more desolate than the deserted airfield. The water was calm and glassy, a dark, dark green that went on forever. If you stood on your tiptoes and the air was clear enough, which it wasn't so late in the evening, and you had the right frame of mind, the proper imagination, you might just see the Alaskan coast forty miles away.
But none of the men looked. No one stood. It was the quiet time.
It had been a long journey to the abandoned airfield on the very edge of the Chukchi Peninsula. Seventeen hours without sleep and the mission had not yet begun. From Severnaya they had traveled to Nordvik by a rusting Tupolev transport, and from Nordvik to Anadyr by a snazzy Air Force Ilyushin. The last hundred miles had been traveled in the rear of a Kam truck that smelled as if it had been routinely used to haul sheep to the slaughterhouse. Each leg of the mission was cut off from the next. Compartmentalized. No one asked where they came from or where they were going.
They were spirits.
Ghosts that never were.
A team that did not exist.
Somewhere in the wind danced the drone of a faraway engine. The team rose to their feet and looked to the sky. The drone grew into a silhouette and the silhouette into a silver form. A minute passed and the Beechcraft 18 came into sight. It was a vintage 1960s floatplane that had earned its stripes ferrying fishermen to and from the Canadian wilds. Its new incarnation called for a more hazardous duty, and the oversized radial engines had been souped up accordingly. Pontoon floats grew from the bottom of the plane, and as the Beech hovered low over the airfield they looked like twin torpedoes, primed and ready to drop. Wheels bobbed from the floats, and the plane struck the landing strip with a military finesse.
Barely had it stopped before the commandos had pulled themselves aboard. Webbing had replaced seats in the stripped-down fuselage. Blankets would do for heating. The men took their places, throwing their chutes on the floor between their feet. Their packs, and the sensitive cargo they contained, they held in their laps.
The Beechcraft turned and roared down the runway, lifting gracefully into the gray-tinged sky. The forecast was good, notwithstanding the gusting northerlies. This high in the latitudes, the wind was a constant, and if not your friend, an enemy to be made peace with.
Inside the fuselage, the men checked their equipment a final time, then closed their eyes. They did not sleep. They rehearsed. They concentrated. They willed themselves to their highest level.
The quiet time was over.
# 66
#
**I N NEW YORK CITY,** on this third Tuesday in June, the sun rose at 5:24. The dawn promised a flawless day. Wisps of cumulonimbus raked a hazy blue sky. A freshening breeze kept the temperature in the low sixties, dousing Wall Street with the honest, vital scent of the East River. Outside the New York Stock Exchange workers draped an enormous banner emblazoned with Mercury Broadband's logo across the building's proud Doric columns. Measuring fifty feet by thirty-five, the banner was decorated with a stylized drawing of Mercury's helmet—the disclike headplate garlanded with two lightning bolts—and the company name, painted gold against a royal blue background.
Inside the building, television crews set up for what promised to be a hectic day. Twelve networks had constructed production facilities on the mezzanine level ringing the Exchange's principal trading floor. Making the circuit, one passed cramped, brightly lit ministudios for CNN, CNBC, the BBC, Deutsch Fernsehen, Nippon Television.... Journalists could be glimpsed applying their makeup, brushing their hair, and practicing their "good morning smiles."
By 7 A.M., the first reports were going out live to audiences around the world. The talk today centered on one subject: the Mercury Broadband IPO. What would be the first day pop? Would the stock keep its head? Was Mercury an exception to a moribund market or the pioneer of a long-awaited rally in technology stocks?
**K ONSTANTIN KIROV ROSE AT SEVEN-FIFTEEN,** showered, shaved, and dressed in a sober gray suit and maroon tie. Despite last night's warnings, he'd slept remarkably well. What will be, will be, he told himself. He'd taken every precaution. He was convinced that once the stock began trading, no one would have the nerve to stop it. If Gavallan were going to make a move, he would have done it long before now. What was the American saying? "No news is good news."
Giving himself a final once-over in the mirror, he asked himself if he was being too confident, too cocksure. Up came his hand with a last spritz of cologne. No, he decided, just realistic.
Picking up his briefcase, Kirov left his suite and took the elevator to the first floor, where he was joined for breakfast in the main dining room by Václav Panič, the CTO of Mercury's European operations, and Janusz Rosen. The bankers were absent, no doubt putting in an appearance at the office before making their promised rendezvous at the Broad Street entrance to the stock exchange at nine o'clock. Kirov ordered a large breakfast, then picked at it. His appetite had deserted him.
At eight-thirty, he and his colleagues decamped to a black stretch limousine berthed in front of the hotel. Kirov settled into the backseat for the drive downtown. The chauffeur announced that due to congestion on the FDR Drive, they would be taking the West Side Highway. Traffic was moderate and they made good time, passing the Javits Center, the USS _Intrepid_ —a mothballed aircraft carrier used for various charity functions—and the reconstructed World Financial Center.
The limousine turned onto Broad Street, and through the windows Kirov stared at an imposing neoclassical building at the far end of the street. A steep flight of stairs led to the building, and even he could recognize the statue of George Washington at the top of the steps. The chauffeur explained that the building was Federal Hall, the seat of the United States government from 1776 to 1791. Across from Federal Hall stood the old headquarters of J. P. Morgan & Co., from whose offices the legendary financier had built his empire and dictated the course of the American economy.
To Kirov's left rose the New York Stock Exchange itself. It could have been a temple on Mount Athos, so perfect was its architecture: the soaring Doric columns, the broad plinth, the bas-relief sculpture running lengthwise beneath the roof.
The limousine pulled to a stop. Kirov got out of the car without waiting for the door to be opened. Staring up at the Mercury Broadband banner that hung in front of the fabled Exchange, he gasped.
My God, he thought, I've done it.
**T HE WHEELS OF THE LEARJET** touched down at John F. Kennedy International Airport at 8:47 A.M. Eastern Daylight Time. The eight-passenger aircraft performed an abbreviated rollout, braking sharply and making a quick starboard turn off the runway. The doors to the flight deck opened, the engines revved, and the plane began an easy ride to its parking slot. Unbuckling his safety belt, Gavallan leaned forward, rocking slightly. Through the cockpit windscreen, he watched the impressive girth of a China Airlines jumbo jet cross their path. Inexplicably, the plane came to a halt directly in front of them.
"What's keeping the guy?" Gavallan shouted to the flight deck.
"Waiting for an inbound jet. It'll just be a couple of minutes."
"A couple minutes?" Gavallan wiped a hand across his face, looking to Cate for reassurance. Her only response was to bite her lip and go back to patting her foot nervously.
After an eternity—three minutes by his watch—the Lear arrived at its designated parking slot. The engine died and the plane rocked forward as the brakes were applied and stopped. Rushing to the door, Gavallan leaned hard on the exit lever. The door opened inward, sunlight flooded the cabin, and he went down the stairwell.
A small entourage waited. Three agents of the federal government left the comforts of their four-wheel mount and hurried to the plane. Gavallan recognized the tall, lanky man with the shock of brown hair, the seersucker suit, and the pair of bifocals perched on his forehead as Dodson. Four days earlier he'd seen him talking on the phone beneath the portico of the Ritz-Carlton.
"Mr. Gavallan, Howell Dodson. It's a pleasure, sir," the FBI man said, extending a hand. "Nice flight?" But if his voice was politeness itself, his posture was stiff, his face a mask of tension.
"We're here, that's what counts."
"Miss Magnus, I presume." Dodson gave her his hand and with a cock of the head shepherded them toward the waiting car. "We've got a helicopter standing by to ferry us to Manhattan."
"Tell me the rotors are turning," said Gavallan.
"The rotors are turning, Mr. Gavallan," said Dodson. "Are you sure we can't call ahead? Pull in Kirov as soon as he shows up? We do have resources available."
"No, thank you. That's not part of the deal." This was something Gavallan had to do himself. The FBI was there in a supporting role only, even if the Bureau didn't know it yet. Reaching the sedan, Gavallan tried to open the door, only to find Dodson's hand placed firmly against the window. "Just a second there. You can see that I've kept up my end of the bargain. I wouldn't want to go any further without seeing some good faith from your side."
"You don't trust us?" asked Cate, stepping forward.
"I'm not in the trust business." The smile was gone, the eyes direct, demanding.
Opening her purse, Cate drew out her pink compact, clicked it open, and handed Dodson a slightly dusted minidisc. "I'm not sure what program was used to store the information on the disc. You'll have to do your best with it."
"All that counts is that the data's there. Three years' of banking records, correct?"
"Oh, it's there all right," said Cate. "And then some."
"Thank you kindly." Dodson handed the disc to a fat, unattractive young man chafing in a catalogue-ordered blue twill suit. "Here you are, Mr. Chupik. I don't mean to rush you, but you have eight minutes to let me know what's on this disc."
"Piece of cake," said Chupik, sliding into the front seat and feeding the disc into his laptop computer. "I'll do it in five."
**T HE JUMP LIGHT BURNED RED.**
The members of Team 7 stood as one, affixing their static lines to the jump cable. Team Leader Abel shuffled forward through the bare fuselage and opened the main cabin door. With a mighty rush, a chill midnight wind swept through the airplane. The biting cold stung his cheeks and brought tears to his eyes. Grasping either side of the door, he looked outside. A pine forest rushed beneath them, a dense lush carpet close enough to touch. They'd been crossing it for thirty minutes and still it ran on, measurelessly.
Stepping back, Abel checked his watch and signaled "Five" with his fingers.
All eyes were on him, yet no one responded. There was no need. All tactical contingencies had been dissected, analyzed, solved, and solved again. The time for words had passed. The time for deeds had arrived.
The Beechcraft 18 began a slow ascent. The altimeter rose from 250 feet to 300, then 350, the magnificent radial engines sawing the air with demonic fervor. Several modifications had been made to prepare the plane for its current purpose. All passenger seats had been stripped, all carpet and insulation torn out until the interior cabin was an aluminum and iron husk. Auxiliary fuel tanks were installed in the rear of the fuselage, gifting the plane with a two-thousand-mile range. A sophisticated satellite navigation system had been installed to insure that the men located their target. And unbeknownst to all—even the pilot—a remote-controlled detonation system was attached to the starboard fuel tank: three pounds of plastique governed by a long-distance radio signal.
The Beechcraft leveled off at 400 feet. The pilot slowed the aircraft's speed to 250 knots. From this height and at this speed, the soldiers of Team 7 would jump. It was a standard LALO jump: low altitude, low opening. Once outside the aircraft, they would fall fifty feet before the static line deployed their chutes. Five seconds later they would impact the ground at three times the usual landing velocity.
The forest vanished with a silent white clap. The tundra ran before them, a pale wilderness advancing to the edge of the world.
And then he saw it. Pump Station 2. A necklace of orange lights glimmering far on the horizon. A wisp of smoke rose from the power plant. No homing signal could have been better. Despite his training, Team Leader Abel's throat swelled and grew tight.
He raised three fingers.
**P ASSING THROUGH THE DOORS** at 18 Broad Street, Gavallan received his visitor's badge, walked through the metal detector, then slid through the turnstiles that governed admittance to the Exchange. He'd been on the floor a dozen times over the years, yet he never entered the building without getting a certain buzz in the hollow of his stomach. It was no different this morning, except that coiled among his normal feelings of awe and respect was the unmistakable frisson of danger.
Dodson followed him closely, showing his badge, and Roy DiGenovese entered next. Mr. Chupik had stayed in the car. He'd needed only three minutes to open Pillonel's files. Scrolling page by page, transfer by transfer, deposit by deposit, through Novastar's banking history, Dodson had looked on with a reverent gaze, saying the same words over and over again: "Well, ain't that sweet."
"Miss Magnus doesn't care to join us?" Dodson asked once the three men had assembled in the small foyer just inside the entryway.
"I think she'd prefer to wait outside. She's seen enough." Gavallan didn't add that Kirov was her father, or that she had plenty to do on her own outside the building. Some things the FBI didn't need to know.
"A rough few days, Mr. Gavallan?"
"You can say that."
"I know you had wished to speak with Mr. Kirov alone. Fine by us. Still, I'm sure you'll be happy to know we've taken some steps to see that Mr. Kirov does not flee the premises. If you'll just follow me for a moment."
Dodson led the way down a short corridor, stopping at an unmarked door and knocking once. An African-American agent wearing a navy windbreaker with the yellow letters FBI stenciled on its breast poked his head out the door and said, "Kirov's here. We got him on the closed circuit. He's just leaving the specialist's booth. Did you get what you wanted?"
Dodson grinned while patting the man's shoulder. "You have no idea, Agent Haynes." The grin disappeared, and Dodson found his no-nonsense self. "Our operation is a go. Alert building security that we will be making an arrest. It might be wise to trade your wind-breakers for some trading jackets. And bring along a few of your men. Calm, brisk, and orderly, Agent Haynes. Am I clear? We keep our weapons concealed at all times."
As the agents conferred, Gavallan peeked into the waiting room. Eight men and women dressed in the same navy windbreakers stood around drinking coffee, shooting the shit, and checking the pumps on their street-sweeper shotguns. It was the FBI's Tuesday morning coffee klatch.
"They're going to stay in here, right?" he asked.
"Strictly backup. I'm sure we won't have the slightest need for them."
"All right then," said Gavallan. "Let's go."
# 67
#
**C ATE WAITED IN FRONT** of the visitors' entrance to the Exchange, pacing back and forth, craving a cigarette, though she'd never smoked in her life. The morning air was cool and invigorating, the sidewalk bathed in the shadow of the surrounding skyscrapers. Still, she was sweating. Every minute or so, she checked her watch. _Where was he?_
She searched the parade of faces, men and women walking purposefully up and down the street. Businessmen in three-piece suits, tourists in shorts and T-shirts, artists carrying sketchbooks and easels. At the corner of Wall Street and Broad, street vendors were selling black-and-white photos of Manhattan, magazines, financial texts. The pavement pulsated with the vibrant human cargo. Hugging her arms around herself, Cate wondered if she was doing the right thing. She knew very well the consequences of her actions. Once taken, there would be no going back.
"He'll serve two or three years, tops. And there's no guarantee of that," Pillonel had scoffed in the archives of Silber, Goldi, and Grimm's headquarters. "Besides, it's not the government he should be afraid of, it's his partner."
She thought of the nasty little dacha north of Moscow, the crude torture chamber with its floor stained black by blood. She remembered Alexei and Ray Luca. She forced herself to imagine the countless others who had suffered or died at Kirov's hands, and the countless more who would surely follow. The blood ties to her father, frayed and fragile, unraveled yet further and finally broke, taking with them her doubt. Someone had to stop her father. At last, she had a way.
A tented canopy had been erected on the sidewalk. Beneath it, two long tables were stacked high with caps and T-shirts bearing the Mercury logo. Handsome young men and women were giving the merchandise to passersby, along with brochures describing the company. Cate looked on, disgusted. It was a fraud, a farce, a fairy tale with a very unhappy ending.
She stopped her pacing long enough to check her watch and compare the time against that of the clock on Federal Hall. Both read 9:20. Her heart raced. Where was he?
"Ekaterina Kirova?"
_"Da?"_ Cate spun. A wiry, dark-haired man attired in a neat houndstooth jacket stood in front of her. She'd never met him before, but she knew him intimately: the soulless eyes, the distrustful smile, the shadow of a beard pushing up an hour after shaving. "Dangerous," Pillonel had said of her father's partner. His _krysha_. "From the bandit country."
"You have something for me?" he asked.
Retrieving the compact from her purse, she removed the last disc and told him what he would find. "Hurry," she said.
But in contrast to her anxious demeanor, the Chechen was all too relaxed. He held the disc between his fingers, examining it this way and that as if deciding whether or not to purchase an expensive piece of jewelry. "No need. Everything is already taken care of."
"What will you do?"
The man from the bandit country met her gaze, and she felt a chill pass through her. Saying nothing, he slipped the disc into his pocket, bowed ever so slightly, and walked off.
**T HE PARTY OF THREE HAD GROWN TO SIX.** Dodson and Gavallan led the way. DiGenovese, Haynes, and the muscle came behind. Haynes and his two agents had donned the shapeless jackets favored by specialists on the floor. Strung out along the corridor that ran parallel to the floor, weaving in and out of the milling throngs of traders, brokers, and specialists, the group managed to avoid looking like the war party it was.
Dodson pulled up at one of the double doors leading onto the floor. "All right, Mr. Gavallan. Here we are. You heard Agent Haynes. Kirov just left the specialist's booth and is on his way up to the podium. Lead on. And remember—calm, brisk, and orderly. We find Kirov and we take him into custody."
The New York Stock Exchange was divided into four trading rooms: the Main Room, the Garage, the Blue Room, and 30 Broad Street. There was no hierarchy among them. The Exchange's seventeen trading posts, scattered across the floor like giant bumpers on a billiard table, were divided evenly between them. Wide passageways lead from one room to the next. But when people thought of the Big Board, it was the Main Room they envisaged. It was here that trading was inaugurated from an elevated podium every morning at nine-thirty, and here that was halted every afternoon at four.
Gavallan led the way into the Main Room. It was large and airy as a convention hall, two hundred by two hundred feet. The ceiling stood several stories above a century-old plank floor. American flags of every size and shape dominated the décor, sprouting from every trading post and hanging on every wall. Brokers' booths ringed the floor's perimeter. Ninety percent of orders to buy and sell shares traveled electronically through the "superdot" computer system directly to the specialists' booths, where they were automatically mated, buyer with seller, at an agreed upon price. This 90 percent, however, accounted for only half the share volume that traded each day. The remaining 10 percent of trades accounted for the other 50 percent of the volume, and these large, or "block," trades required the human attention of both broker and specialist.
Lowering his shoulder, Gavallan nudged his way through a knot of brokers talking last night's hoops and walked onto the floor of the New York Stock Exchange. Keeping a driven pace, he wound his way across the floor, passing the trading posts where IBM, 3M, Freddie Mac, and AIG were traded. The posts bristled with television monitors, flat screen displays, computer keyboards. Eleven minutes from the opening—9:18:25, by the digital clocks hanging high on every wall—each was surrounded by clumps of specialists balancing their orders prior to the start of trading. It was difficult to see more than fifteen feet ahead.
Gavallan reached the post that housed the electronic offices of Spalding, Havelock, and Ellis, the specialist firm assigned to trade Mercury's stock. The booth was a hive of activity. Twenty or thirty brokers crowded around Deak Spalding, the firm's top trader, shouting to be heard. It was a scene that played out whenever there was strong demand for a stock, or strong pressure to sell it.
Gavallan glanced toward the podium. A Mercury Broadband banner was draped across the balcony below it. Another larger one hung on the wall behind it, just below the gargantuan American flag that daily paid tribute to the United States of America and the free market it fostered.
"Well, look who's here," said Deak Spalding. "The devil himself, back from the dead. Hey, guy, how are you? I had old man Grasso himself here not two minutes ago, with your buddy Kirov and some of your troops. Gonna be a big opening. Gotta love it."
Spalding was a broad, florid man with an Irishman's ruddy nose and gift for gab. A pink carnation adorned his lapel.
"Doing good, Deak, thanks. Which way'd he—?" A soft hand fell on Gavallan's shoulder and he spun to see to whom it belonged. "Hello, Tony."
"Jett. You're back. Thank God, you're all right."
"You weren't expecting me?"
"Frankly, none of us were," said Tony Llewellyn-Davies. "Not a word from you since Friday. The FBI saying you're a murderer. We didn't know where you'd gone or what you'd been up to."
He was dressed nattily in a double-breasted blue blazer with his requisite gray flannel slacks and club-striped tie. His cheeks were flushed, his blue eyes excited.
"I find that a little hard to believe," said Gavallan. "You if anybody should have been able to tell them. After all, if you're such good friends with Konstantin Kirov you ought to have known."
Llewellyn-Davies bit back his surprise, his Adam's apple bobbing visibly. "We're hardly 'friends.' I'm sure I hardly know him any better than you do."
"Cut the crap, Tony. I've spoken to Graf. He told me about the call... the one you conveniently forgot to relay to me. You knew firsthand Mercury was rotten a week ago. Actually, I guess you knew it a long time before that. Anyway, it stops here. We're pulling the plug on the deal. It's over. I just want to have a quick word with Kirov before I let everyone else know."
"Jett, no... you're mistaken. You're talking nonsense. Really, you are."
"How could you? We built something. We did it together. Seven years, Tony. Christ, you're on the board as it is. What was it? More money? A spot at the top? What he offer you?"
Looking at his associate, Gavallan felt betrayed, ashamed, and naive. Part of him still thought it couldn't be. Not Tony, of all people.
"I don't know. Respect. A chance." Llewellyn-Davies sobbed, a single pathetic cry, and lowered his head. "I'm sorry, Jett. Give me a minute to explain. Not here—come into the booth. It's already embarrassing enough as it is." He tried to smile, and a tear ran down his cheek. "The floor doesn't need to see a pooftah having a good cry."
"I haven't got the time. Tell it to your next employer."
Llewellyn-Davies grabbed at Gavallan's sleeve. "No, Jett. Please. I can make it right. You've got to believe me. Don't be a stupid git. It's just me... _Tony_. Come on."
The official clock read 9:20:51. Gavallan found Dodson and asked him to stay right where he was and, no matter what, to prevent Spalding from initiating trading in the stock. "Give me two minutes. I'll be right back."
"Two minutes, Mr. Gavallan. Then we get Mr. Kirov ourselves."
But Gavallan was already moving, and Dodson's words were drowned by a chorus of babbling voices. Gavallan and Llewellyn-Davies walked the short distance to the Black Jet Securities booth. Curious faces greeted them along the way, along with cries of "Jett, great to see you,"
"Hey, boss," and "We got a kicker today!"
Llewellyn-Davies opened the door to the manager's office and showed Gavallan in.
It was more a shoe box than a place of business. Two desks pushed against each other crowded one wall. Next to them stood a waist-high server, a monitor, and a printer. There was a refrigerator and a microwave oven, a Bridge data monitor, and another desk covered by telephones. The walls were papered with notices from the Exchange. Like any other essentially blue-collar workplace, there were the obligatory topless photos. Tastelessly, someone had glued a picture of Meg Kratzer's face onto the torso of a black woman with enormous breasts. A second door led to the corridor outside the floor.
"Out, both of you," Llewellyn-Davies said to a pair of clerks. "On the double."
Gavallan nodded at them and they left.
Llewellyn-Davies shut the door, then turned, leaning his back against it. "What a mess, eh?"
"You've got a minute, Tony. Get going."
"Oh, fuck a minute. Come to your senses. Seventy million dollars. The firm's future, for Christ's sake. Let it go."
"It's done, Tony. The deal's canceled."
Llewellyn-Davies stared at him, his pinched, patrician features clamped into a mask of hate. "I'm sorry, Jett, but that's out of the question. Too much work. Too much sweat." The tears had vanished. His eyes were clear, burning with an inner purpose, a rage that Gavallan had never seen in him before. "We need this. You, me, all of us. It's our bloody savior. Can't have you taking us all down as a matter of pride or principle. I don't want to hear about rules. Sod all the rules. Made to be broken, what?"
"Mercury's revenues are a sham. Kirov's going to jail. The FBI's got information tying him to the theft of a couple hundred million dollars from one of the companies he controls. The Russian government is all over him. Now come on. Let's go outside and talk to Deak Spalding."
"Kirov assured me he's remedied the shortfalls in infrastructure. It's only a question of months until his revenues are up to snuff. It's time to close an eye. For everyone's good."
What was he trying to do? Gavallan wondered. Intimidate him? Threaten him? Did Llewellyn-Davies actually for a moment think he might change his mind? Gavallan stepped closer to the man he'd been so godawful stupid to trust. "Move, Tony. I have to go."
"Afraid not, chum."
It was then that Gavallan saw the gun. It was a strange gray pistol with a silencer. Plastic, he thought. The bullets would be too. No metal detector in the world could have sniffed it out.
"Some fancy hardware, Tony. A present from Kirov?"
"You damn fool, Jett," said Llewellyn-Davies, shaking his head, his voice tightening. "Don't you see, it's your fault. All of this. Mercury's a gem, just like you said. We've got to see it to market."
"Out of the way." Gavallan stepped forward, and the Englishman fired a round into the floor.
"Christ," shouted Gavallan, freezing, raising a hand. "Have you lost your mind? Put it down."
Llewellyn-Davies held the gun out in front of him, grasping the butt with both hands to control the palsied shaking. "Sorry, Jett. No can do. It's not that I'm not grateful for everything you've done for me. I am, believe me. It's just that it's time I did something for myself. Think ahead. What do you think happens to me if the deal goes sour? Do you think we don't all know how strung-out the firm is? How long do you think the new owners of Black Jet will keep me on? One look at my health records and they'll pack me off with a nice little check and a pat on the back. 'One less liability.' 'Start with a clean sheet.' All that utter crap. I won't have it. I've worked too bloody hard for too bloody long to start over again somewhere else—Christ, if there's someone else who'll even have me."
"It's over, Tony. We'll all make out okay. Put away the gun. What are you going to do? Shoot me? Here, in the Exchange? And then what? The FBI's right outside. Where are you going to run?"
"Yes, I bloody well am going to shoot you. Don't have much choice, do I?"
Someone banged on the door to the office. "Hey, open up. Jett, you in there?" There was no mistaking Bruce Tustin's obnoxious voice. "Gavallan, you there? I saw you crossing the floor. You can hide from your girlfriends, but not from your uncle Bruce... Jett?"
Gavallan nodded toward the door. "Your move, Tony."
Llewellyn-Davies extended his arm, eyes wincing, head turning slightly away. A moment later, his hand dropped. He began crying. "Oh, damn it all. Damn you..."
Gavallan walked up to his former friend, gently prying the gun from his hand. "Go on now. Get out of here. I never want to see you again."
# 68
#
**K ONSTANTIN KIROV MOUNTED THE STAIRS** to the balcony slowly, a valedictory climb to his new orbit high in the capitalist universe. Reaching the top, he crossed the narrow landing. There was room for fifteen people, maybe a few more. Advancing on the podium, he let his eyes wander over the trading floor. He had expected to play to an audience, but the preoccupied traders were going about their business as if he were not there. One by one his colleagues joined him, and he greeted each with a firm handshake.
The clock directly across the room read 9:28:45. The swell of voices rose as Richard Grasso, president of the Exchange, showed Kirov how to ring the bell, jocularly begging him to wait until the appointed moment. Kirov only half listened. His eyes were scouring the floor for sign of Antony Llewellyn-Davies, the sly Englishman who three months before had agreed to be his spy inside of Black Jet Securities. Minutes ago, Llewellyn-Davies had rushed off, worried he'd seen Gavallan. Kirov was left to wonder whether in fact he had, and if so, whether the Englishman had done as he'd been told.
A crew from Russian Channel One gathered on the floor below, camera pointed in his direction, a red light indicating film was rolling. Reflexively, Kirov stood a little straighter. He was aware that at that very instant his image was being broadcast across the Russian continent. To Moscow. To Leningrad. To Kiev and Minsk. To Odessa, Alma-Ata, Ulan Bator, and Vladivostok. Across eleven time zones, the picture of Konstantin Kirov, Russia's "first Western businessman," the "patron saint of the second Russian perestroika," was gazing down upon the country's citizens. He forgot about Gavallan and Llewellyn-Davies. His heart fluttered madly.
Grasso nudged his shoulder. "Thirty seconds, Mr. Kirov."
The clock read 9:29:30.
Meg Kratzer rubbed his back. "Congratulations," she said. "We're all so happy for you. Just thrilled."
Kirov mouthed a thank-you, wishing he could have arranged for a prettier woman to be at his side.
_"Kirov!"_
The voice came from below. Nervously, he looked to the left and right. _"Kirov!"_
Good Christ, it was Gavallan. He had climbed on top of the trading post nearest the podium and was shouting at _him_.
"The offering is canceled. Mercury's over. The specialists are closing their books. The FBI is in the building. Come down right now. We want to talk to you."
Richard Grasso looked appalled. "Jett, mind telling me what is going on here?"
"Just hold on to Kirov. Keep him there. We're coming to arrest him."
"Yes, yes, of course." Grasso nodded his head vigorously, but when he checked over his shoulder all he saw was Kirov's narrow shoulders retreating down the stairs.
**I T HAD BEEN A STRESSFUL DAY** for the president.
New uprisings in Grozny threatened the fragile Chechen peace. A group of demonstrators from Greenpeace had camped in front of St. Basil's protesting the country's use of mammals, dolphins in particular, as instruments of war. And an independent newspaper in the south had uncovered decade-old evidence of a bribe he'd carried for Mayor Sobchak back in his days in Leningrad. The travails of politics. Sometimes he didn't think it worth it.
Pouring himself a glass of mineral water, Volodya settled into his chair and turned on the television. Quickly, he found Channel One. The screen filled with the picture of Konstantin Kirov standing on the podium of the New York Stock Exchange. Finally, some good news. He didn't care for the man, but as a representative of Russian business he was acceptable. His English was colloquial and flawless, his dress impeccable. And there was no doubting the man's resourcefulness. Given the proper training, he might have made a decent spy.
The president turned up the volume. An American stock analyst was calling for Mercury stock to rise dramatically the first day, touting the inauguration of Russia into the club of Western nations. Henceforward, the commentator intoned, one could expect a flood of Russian multinationals to be quoted on the world's major exchanges.
The president smiled.
He looked closer. There was a commotion brewing. Konstantin Kirov's face had taken on a decidedly worried cast, and he was looking this way and that. The president leaned forward, eyes glued to the television. The camera panned lower, focusing on a wild man who had climbed atop one of the trading posts on the floor of the Exchange. The commentator stopped speaking, and one could hear with astonishing clarity what the man was shouting. "Kirov. The offering is canceled. Mercury's over." And then, to the president's horror, "The FBI's in the building."
The camera panned back up and Kirov could be seen fleeing the balcony, leaving his colleagues and advisers questioning one another.
Lifting the remote control, the president turned off the television. He felt sick to his stomach. Kirov had despoiled his country's reputation in front of millions of viewers. Tomorrow, the story would be front-page news. One more Russian thief. Another doomed enterprise. Worse, the man had failed the Service. There would be no money. No money at all.
The president reached for a phone. One fiasco he might be able to explain away; two would reek of conspiracy. There could be no more embarrassments, not even the hint of intrigue. His budding relations with America and the economic favors they promised were too valuable to risk.
His assistant answered, and Volodya roared, "Find me Major General Kirov. Immediately!"
**K ONSTANTIN KIROV RUSHED DOWN THE STAIRS** from the podium, eager to be free of the building. To be free of the city. Of the whole damned country. Four of his men were waiting on the ground floor. They were new faces, dark, sullen, part of the New York crew he'd summoned the night before.
"Get me to the car," he said. "Yours, not mine. A bit of trouble. We must move quickly."
"Follow me," answered one of the men, his accent southern, unfriendly.
Kirov eyed the man, not liking his swarthy features, his dead eyes. But what choice did he have? They set off down the hallway at a dignified clip. Off the floor, the building was quiet and well-lit, and for a few seconds Kirov maintained the illusion that he would be able to waltz scot-free from the building. He soothed himself with the notion that he could still salvage Mercury. He would put his own money into the firm. He would upgrade the infrastructure. He would create the company he had sold to all of Wall Street. If he didn't take the company public today, who cared? He would be back in six months or a year with something even better. Forget Black Jet. Forget Gavallan. He would go to the big boys this time. Bulge bracket only. Salomon. First Boston. Lehman. They'd fight over themselves for the deal.
Fifty feet ahead, twin sets of brass-framed double doors led to the street. A black sedan lolled at the curb, its back door opened. Kirov saw daylight and thought, _Freedom_.
Then he heard the strident voice coming from behind him.
"Mr. Kirov, this is the FBI. Please stop where you are. You are under arrest, sir."
Turning, he saw a tall brown-haired man in a summer suit walking toward him, his gun drawn and hanging at his side. Gavallan was next to him. Two more men whom Kirov took to be law enforcement agents followed close behind. "You're under arrest, Mr. Kirov. Lie down on the floor, sir. Tell your men to do the same."
"Come on, Konstantin," said Gavallan. "Do as you're told. Don't make this tougher than it has to be."
Kirov looked back toward the exit. At the end of the hallway, a pair of the Exchange's security guards, clad in dove gray uniforms, their hands drifting toward their holsters, walked slowly, uneasily, toward him and his bodyguards. Passersby hugged the walls, sensing trouble.
Kirov took another look at Gavallan, then darted toward the exit. At the same time, his bodyguards moved in the opposite direction. They had no guns. They made no move to appear menacing. They simply walked rapidly toward the federal agents, obstructing their line of sight.
Passing the gray-clad security guards, Kirov murmured to his men, "Hold them here. I just need a minute."
Both men, soldiers belonging to the New York side of the Solnetsevo Brotherhood, nodded and took up position in the center of the hallway.
Kirov ran, not daring to look behind, as if he were being chased by the ghosts of his own conscience. He heard the sounds of a scuffle, Gavallan's voice calling after him. Strangely, he sounded more perfunctory than upset. The life seemed to have gone out of the man. Funny—he hadn't pegged Gavallan as a quitter. Passing through one door, then the next, Kirov emerged on the sidewalk. Twenty feet away a car door stood open, and a man inside was gesturing for him to hurry. He caught the words "Hurry, damn you. Run!" Kirov slowed only to lower his head and threw himself into the backseat.
"Thank God," he whispered, his cheek touching the cool black upholstery. "Get me out of here. Fast!"
**O NE MOMENT THE BEECHCRAFT** was flying straight on its course, its speed a comfortable 250 knots, altitude 400 feet. It had lined up perfectly on its inbound azimuth. The landing site, a circle of knee-high heather sprouting from the snow, was visible. The pilot had opened the cockpit door. Leaning out of his seat, he offered a thumbs-up to the valiant warriors. "Godspeed," he said, though with the tumult of the air invading the fuselage and the propeller engines buzzing so close it was doubtful anyone heard him.
The next moment the plane was no longer there.
Three pounds of plastique ignited the four hundred gallons of jet fuel in the starboard wing, which in turn ignited the auxiliary tanks housed at the rear of the fuselage and then the fuel tanks in the port wing. Expanding at 7,800 meters per second, an enormous, wickedly powerful fireball engulfed the plane. Joint tore from joint, bolt from superstructure. In one-hundredth of a second, the elemental explosion shattered the plane and everyone inside of it into ten thousand pieces, showering the pristine Alaskan tundra with a black and silver rain.
Some attributed the melted tire and grotesquely twisted propeller that landed squarely in the infield of Pump Station 2's summer baseball diamond to a practical joke played by some local miners. Others offered no explanation at all, content to merely scratch their heads. No planes had been reported in the area. The explosion was heard only faintly and seen by no one. Alaska was nothing if not mysterious.
In Severnaya, Leonid Kirov removed his hand from the transmitter. He had tried and he had failed. There would be no bust in Red Square. No promotion waiting upon his return. The president had made his disappointment abundantly clear. The penalty for failure was as severe as the reward for success was generous.
Such it had been in Russia, and such it would always be.
His hand fell to his jacket, hanging on the chair behind him. His fingers probed the jacket's pocket. It was there, as he knew it would be. He felt the cool metal, the smooth expanse of the grip, the curled menace of the trigger. Slowly, he drew the pistol out and laid it on the table. He lit a cigarette, but the smoke tasted harsh, unwelcome.
Standing, he put on his jacket and straightened his tie. He spent a moment adjusting the tie clasp, his gift from Andropov, then drew himself to attention. And raising the pistol, he was careful to keep his chin raised just so, his eyes to the fore. The gun touched his temple, and as he pulled the trigger he made sure to lean his head sideways into the barrel.
**S ETTLED INTO THE TOWN CAR'S BACKSEAT,** Konstantin Kirov expelled a sigh of relief. He was hardly home free, but with a little luck, he'd make it to Teterboro and be airborne and en route to his private hideaway in the Exumas before the authorities could track him down. A man did not make it to his position in life without taking a few precautions, without setting aside a few dollars for a rainy day or establishing a place to keep his head down if the waters grew too rough. He'd lie low for a few years, cultivate his relations with the country's entrepreneurs, work on his memoirs. A return to Moscow was out of the question, at least until a new president took office. As for Mercury, that too would be put on hold. His plan to bring the company public had dissolved the moment he'd heard the words "FBI" and "under arrest."
Looking over his shoulder, he caught sight of Gavallan running down the stairs of the Exchange, pulling up in the middle of the street, arms raised high in exasperation.
_"Yeb vas,"_ he muttered. Fuck you.
He truly hoped he'd never see the man again in his life.
Suddenly, he was very thirsty. "Do you have something to drink? Some water, perhaps? Perrier? Evian?"
Two men sat in the front.
"Of course," said the one in the passenger seat. He turned and looked at Kirov. "Anything for my partner," said Aslan Dashamirov.
"But—why—how?" Kirov choked on his own confusion.
"You've been a naughty boy, Konstantin Romanovich," said Dashamirov, waving a slim silver disc between his fingers. "Have you never heard of honor among thieves?"
Kirov threw a hand to the door, fingers clawing for the release. He would make a deal with the FBI. He would show them the inner workings of the Russian underworld. He would forfeit his entire fortune.
With a sturdy thump, the doors locked, and Dashamirov laughed.
Konstantin Kirov cast a last look behind him. Katya had joined Gavallan, and the two stood in the center of Wall Street. He thought he saw his daughter raise a hand and wave, but he couldn't be sure. Tears had blurred his vision.
# Epilogue
#
**T HE GAVEL SLAMMED WITH FINALITY** and a short, exultant cry went up from the executives gathered on the podium. Jett Gavallan shook hands with the Russian president, and then it was everyone else's turn, Meg's and Bruce's and Graf's. Each received the same firm grip, the same swift shake, the same sober nod. The president turned to Cate and kissed her on the cheek three times in the Russian custom. He had been learning English, and Gavallan overheard a few words.
"We are grateful to you both for saving our airline. I only hope the public will treat it as fairly."
"I'm certain it will," answered Cate graciously.
Novastar Airlines had begun the day trading on the New York Stock Exchange at $14 a share and had closed at $15.25. As thanks for returning to Novastar the money that Kirov had stolen, the president had awarded Gavallan the mandate to bring the company public a year later. Black Jet Securities had brought the $500 million offering to market at the upper end of its price range. A first day's jump of nearly 10 percent wasn't too bad for a Russian company, all things considered.
The president clapped a hand on Gavallan's arm. "Now we must talk about our aluminum industry. It is not in good condition. When can you come again to Moscow?"
"Not for a while, I'm afraid. This is our last trip until the big day. Cate can't fly much longer and I don't want to be away when the moment arrives."
"A boy or a girl?"
Gavallan looked at Cate. Her cheeks wore a slight flush, but at seven months pregnant, she'd never looked more beautiful. "It will be a surprise," he said. "But Mr. Byrnes will be happy to travel to Moscow—say in a week? He has some business with another company we're helping to sell."
"Mercury, yes?" asked the president.
"Yes," said Gavallan. "Mercury's being purchased by Bluephone, an Anglo-French telecom company."
"What is the price?"
"One billion."
"Rubles or dollars?"
Gavallan smiled. They both knew the answer to that one.
Cate wrapped an arm through his and gave him a squeeze. Actually, if you added the 50 percent stake in Novastar Cate had inherited from her father and her 85 percent ownership of Mercury, they would be nigh on billionaires. But they had decided not to keep the money, feeling that it didn't really belong to them. The shares in Novastar and her proceeds from the Mercury sale were to be placed in a philanthropic foundation Cate would chair.
With a final handshake, the president left with his entourage. Graf Byrnes headed down the stairs a moment later, with Bruce Jay Tustin and Meg Kratzer in tow. Gavallan stood at the podium, looking over the paper-strewn floor, the blinking monitors, the bold American flags. Ten minutes after the end of trading, the floor of the New York Stock Exchange was quiet, though not deserted. Traders had returned to their posts to tally their books. Brokers were on the phone with their head offices. Over a billion shares had exchanged hands. The cogs of capitalism never stopped turning, Gavallan mused.
Slipping his hand into his wife's, their fingers intertwining, he walked with her down the stairs and across the floor. "See you at seven," he said. "You thinking dinner out?"
"How 'bout room service?"
"You got it."
They walked outside the building. A fierce summer sun cut through the latticework of skyscrapers, warming their cheeks. Ahead, Graf Byrnes was climbing into the rear of a limousine that would take them to Black Jet's midtown offices. "You coming?" he shouted.
"Be right there."
Gavallan kissed his wife on the cheek. "Seven o'clock," he said. "It's a date." Then he brought her close and whispered, "Hey, we did it."
Cate didn't answer. He saw a memory dance in her eyes, a tear well up, then die.
# Acknowledgments
I acknowledge with gratitude the help of Andrea O'Connell, Wyc Grousbeck, Richard Pops, Henrique M. L. Gregorio, and Barron Emile Eyraud, who gave willingly of their time and made the calls that set the ball rolling. In San Francisco, Mitch Whiteford, Michael Graham, David Golden, and Cristina Morgan showed me the inside of the tech banking world. In New York, I owe a debt of thanks to Jeffrey Zorek, Richard Cunningham, Paul Meeks, David Ballard, Kevin Keys, Christine Walton, and Derek Reisfield. Murray Teitelbaum shepherded me around the New York Stock Exchange and had an answer to every question. In Moscow, Alexander Poudov was a guide _par excellence_. Andrew Jack of the _Financial Times_ gave me a cup of hot tea and steered me through the treacherous alleys of the Russian oligarchy. As always, I can't thank my wife, Sue, enough for her patience and interest in my work. Bill Massey, my brilliant editor at Bantam Dell, hounded me tirelessly and the book is the better for it. Thank you, Bill. My thanks also to Martin Fletcher at Headline in London for his support and unwavering good taste. Irwyn Applebaum and Nita Taublib oversaw every aspect of the work from beginning to end. It is a privilege to work with such talented and energetic professionals. I am lucky to have one of the finest literary agents in the business and his colleagues working on my behalf. My heartfelt appreciation goes out to Richard Pine, Sarah Piel, and Lori Andiman at Arthur Pine Associates.
Lastly, I would like to thank my brother, Bill, who is always there with a kind word, solid advice, and a ready ear. You're one in a billion.
If you enjoyed
**THE FIRST BILLION,**
get ready for the next pulse-pounding suspense novel from
**Christopher Reich...**
**THE
DEVIL'S
BANKER**
COMING FROM DELACORTE PRESS
**IN SEPTEMBER** 2003!
# The Devil's Banker
On sale September 2003
# 1
#
**I T IS DIFFICULT TO WALK** casually with five hundred thousand dollars taped to your belly. More difficult still when any of the men brushing past you would gladly slit your throat were they to suspect the king's ransom you carried.
The man who had chosen the warrior's name Abu Sayeed snaked through the alleys of the smugglers' bazaar, careful to check his impatient step. He was close now, but he could not hurry. To hurry invited attention. And attention meant trouble he could not afford.
Around him, shopkeepers leaned in open doorways, smoking cigarettes and sipping cups of tea. He could sense their eyes upon him as they studied his bearing, gauging its strength, deciding whether he was a predator or prey. Instinctively, he stood straighter and thrust his chin forward. But all the while he kept his pace relaxed, his face slack, even as the claws dug into him.
The money was divided into fifty packets, each containing ten thousand dollars, each wrapped and waterproofed in transparent plastic. The packets had sharp, cruel corners that chafed and cut his flesh. He had been traveling for thirty-six hours. His chest and back were flayed as if scored by a cat-o'-nine-tails. Only by thinking of the operation was he able to continue. The prospect of the infidels' death invigorated him with the strength of the Pharaoh's army.
At four **P.M. ,** the summer sun was at its fiercest. Dust devils arose on the dusty road, swirled lazily, then spun themselves out. After a brief siesta, the bazaar was rousing itself to life. Beneath fluorescent lights, shelves sagged with cartons of Dunhill cigarettes, Toshiba laptops, and Paco Rabanne cologne, all brought overland from Afghanistan to avoid duty and tax. Other windows displayed less mundane goods: Kalashnikov rifles, Colt pistols, and Claymore mines. Hashish, heroin, even human chattel could be found at the right address. If there was a free market on Earth, mused Sayeed, it was here on the western outskirts of Peshawar, the gateway to the Khyber Pass.
Stopping to purchase a cube of diced sugar cane, he cast his gaze behind him. His depthless black eyes scoured the street, checking for the misplaced face, the averted gaze, the anxious dawdler. So close, he must keep his senses keen. He did not believe that the crusaders knew his identity. Still, he must be cautious. Members of the American Special Forces infested Peshawar as lice infest a beast. Most were easy to spot, with their Oakley sunglasses, Casio watches, and desert boots. A few even dared enter the bazaar, where foreigners were not welcome and Pakistani law held no sway.
The thought of the Americans brought a contemptuous smile to his lips. Soon they would learn that they could not run. The fire was coming. It would burn them in their heartland. It would scald them from within.
And, for a moment, the claws loosened their grip. The pain subsided, and he basked in the glow of destruction.
Satisfied his trail was clean, Sayeed spat out the sinewy cane and crossed the narrow road. To look at, he was no different from any of the thousands of souls who eked out an existence trafficking the porous border that separated Pakistan from Afghanistan. His _shalwar kameez_ , the baggy shirt and trousers that made up the local dress, was filthy and stiff with dried sweat; his black headdress smothered with red alkali dust. His beard belonged to the most fervent of believers, as did the AK-47 he carried slung over a shoulder and the bejeweled dagger strapped to his calf.
But Sayeed was not Pakistani, nor was he a Pashtun from the southern provinces of Afghanistan or an Uzbek from the north. Born Michael Christian Montgomery in London, England, Sayeed was the bastard offspring of a cancerous British officer and a teenage Egyptian whore. His father had died while he was a boy, leaving him a polished accent and not much more. Unable to care for him, his mother returned to Cairo and gave him over to the _madrassas_ , the religious schools that gifted him with an Islamic education. His childhood was brutish and short. It was a natural progression to the camps where he learned the creed of the gun, memorized the verse of violence, and worshiped at the altar of rebellion. And from there to the killing fields of Palestine, Chechnya, and Serbia.
At twenty, the Scorpion found him.
At twenty-one, Michael Christian Montgomery ceased to exist. It was Abu Mohammed Sayeed who swore the oath, accepted the mark, and joined _Hijira_.
Skirting a convoy of carts piled high with Korean fabrics, Tibetan rugs, and Panasonic televisions still in their factory packaging, he reached the Tikram mosque. The doors were open, and inside the shadowy hall, a few men lay on prayer rugs, prostrate in worship. His eyes returned to the street. Scanning the intersection ahead, he felt a new pain lash his back. This time, however, it was not the jagged belt that provoked his discomfort. It was fear. He could not see the store. Somehow, he had taken a wrong turn. He was lost.
Frantically, Sayeed turned his head this way and that. It could not be. He was at the Tikram mosque. He had seen the photographs. He had studied the maps. Despair washed over him. Others were waiting. The countdown had begun. Seven days. The thought of failure turned his bowels to water.
Terrified, he wandered into the street. A horn blared in his ear, loud, very loud, but from another universe altogether. Sayeed jumped back a step and a jitney lumbered past, passengers hanging from the doors, clinging to the luggage rack. In its wake, a cloud of rank exhaust choked the already oppressive air. He could not go on. He could not go back. Truly, he was damned.
The exhaust dissipated and he saw it. The gold letters emblazoned on a black field. **BHATIA'S GOLD AND PRECIOUS JEWELRY**. His despair vanished. In its place came joy. The light of a thousand suns.
"Inshallah, God is great," he whispered, a bolt of piety swelling his heart.
Guards stood on either side of the doorway, Kalashnikovs to their chest, fingers tickling the trigger guard. Sayeed passed them without a glance. They were not there to protect jewelry, but cash, primarily U.S. dollars, and gold ingots. Bhatia's reputation as a jeweler might be suspect, but his trustworthiness as a _hawaladar_ , or money broker, was unquestioned. Faisan Bhatia had long served the local smuggling community as its agent of choice. He was the only broker in the region able to handle the large sums that Abu Sayeed required.
In Arabic, _hawala_ means _to change_. And in Hindu, _trust_. Put simply, it was the _hawala_ broker's job to affect transfers of cash from one city to another. Some of his clients were traders eager to repatriate their earnings after selling their haul in the bazaar. Others, simple folk wishing to send money home to loved ones in Karachi, Delhi, or Dubai. Both groups shared a distrust of the bureaucracy and paperwork demanded by the country's less-than-solvent banks. For them, _hawala_ was a welcome alternative. A system built on trust, hidden from intrusive eyes. A system that had been in place when Arab traders plied the Silk Road hundreds of years ago.
Bhatia, a fat Indian with a streak of gray in his hair, stood imperiously behind the counter. As Sayeed approached, he eyed the customer's caked clothing and unwashed face with undisguised contempt.
"I would like to make a transfer," Abu Sayeed whispered when he was close enough to taste the man's breath. "It is a matter of some urgency."
The Indian did not move.
"The Scorpion sent me."
Faisan Bhatia's eyes flickered, but only for an instant. "Come this way."
# 2
#
**I T WAS THE MOST GODAWFUL** frightening place she had ever been. Some parts of Jakarta came close. Jakarta with its garish slums, oppressive pollution, and packs of teenage muggers giggling with hostile intent. Macao had a few dark corners where you didn't dare venture. And everyone knew about Rio, the gorgeous bad boys on motorbikes, streaking past with their razors at the ready. But here... the unremitting heat, the hostile stares, and, worst of all, the _burqa_ draped over her head and shoulders, baking her like a ruddy Christmas goose... this topped it all.
Her name was Sarah Churchill, operational designation: EMERALD, and through her black gauze veil, she watched the target advance across the intersection. She could see that he was in distress, trying not to limp, compensating by standing too straight and puffing out his chest. Two days she'd been tracking him, up and down the mountain passes, a distance of sixty miles. She was hurting, too, but she'd be damned if she'd show it. Her feet were raw and blistered in their leather sandals, her legs fatigued beyond measure. A little while ago, her lower lip had cracked and she could feel a trickle of blood, salty and strangely reassuring, on her tongue.
A trio of Indian women clad in red and orange saris scurried across her path, and she mimicked their gait. The "second-class shuffle," she called it—head bent, shoulders hunched, eyes fixed to the ground like a dog that's been beaten too much.
Drawing in her shoulders, Sarah made herself shrink beneath the full-length garment. Her horizons seemed to shrink before her and she bridled at her training. _Blend in with your environment:_ the first rule of tradecraft taught at Fort Monckton, where all good little English boys and girls go to learn to be spies. Ever the prize pupil, she kept her back hunched and continued to hug the inside of the street.
She was too tall. That was the problem. You didn't see many Pakistani women who stood five feet nine inches in stocking feet. Her height came from her father, a six foot four inch Welshman. Her hair, a raven's black and cut to her shoulders, was her mother's gift, as were her sierra brown eyes.
Her attitude, though, was all her own, and not subject to amendment or revision. She was plucky, outspoken, and a sucker for a smile. Five years ago at IONEC, the Intelligence Officer's New Entrant's Course, she'd set the women's mark for the fifty-mile hike, but when, at her graduation ceremony, her instructors called her their toughest recruit, she'd broken down and cried like a baby.
Her earpiece crackled with static. "Primary still visual?"
"He's gone into the store," she whispered. "Bhatia's Gold and Jewelry." She spelled the name slowly, enunciating each letter just as Matron had taught her at Roedean. "It's the bloody _hawala_ , all right. Time to call in the reinforcements."
"Give us a GPS read."
"Coming up." She found the global positioning device on her belt and hit the locate/transmit button. Within a second, the stationary satellites that comprised the Central Intelligence Agency's proprietary GPS had established her exact latitude and longitude to within six inches of where she actually stood, and her altitude above sea level to within four. She'd been transmitting her location every hundred meters since she'd entered the bazaar. Taken together, the coordinates constituted a route marker for the cavalry, or in more dire circumstances, a path to get her the hell out of Dodge.
"EMERALD, you are mapped. An A-team is moving in to clean up. ETA is twelve minutes."
_"Twelve minutes?_ He could be out the back and halfway back to Pesh by then. Dammit, tell them to hurry."
Located on the western outskirts of Peshawar, the smugglers' bazaar encompassed an area as big as the City of London, with half again as many alleys, roads, and lanes. Few of the roads were marked, if they even possessed a name. There were certainly no addresses.
The bazaar had sprung up as an informal "gray market," trading in goods stolen across the Afghani border. Carts had given way to shacks, and now most of the stores were housed in sturdy concrete bungalows. A patchwork of dubious signs advertised the wares. Marks and Spencer. Maytag. Pringle of Scotland. Sony. And her absolute favorite: Sacks Fifth Avenue. Though wholly within Pakistan's borders, the bazaar was treated as its own autonomous region. Crime was rampant. Thieves, pickpockets, and worse roamed freely, practicing their trade on the weak and unsuspecting. It was up to the victim to catch the criminal. Once he did, the punishment was up to him too. If there was any rule at all, it was the harsh tribal custom of the Pathan tribesmen who made it their home.
"Maintain visual," snapped the voice.
"How's the picture?" she asked. "Getting what you need?"
"Reception's a little fuzzy. Keep still for a second. I need to reset the color balance."
Sarah held still, staring out at the bustling street. It was hard to imagine that seven thousand miles away a technician was deciding whether the picture was too red or too green. The Sony microdigital camera embedded in her sunglasses was a gift from the boys in Langley. She liked to think of it as a "welcome to our side of the pond" present given upon her secondment from MI6. The Yanks always had the neater toys. The camera's images ran to a transmitter in her belt that relayed both audio and visual signals to a spot station nearby. The spot station, in turn, sent the signals on to Langley. The boys at Langley had also given her a machine pistol, three spare clips of ammo, and a tab of cyanide tucked inside a neat little compartment where her wisdom tooth used to be.
"Give us a slow scan left to right."
Sarah turned her head as directed, the camera capturing the same exotic imagery as her eyes: the mosque and its beautifully carved doors, the merchant stringing fresh offal in his front window, the gunsmith tooling a rifle barrel on the sidewalk, and finally, Bhatia's Gold and Jewelry, where she could make out a tall, lean figure standing at the far counter. Abu Muhammed Sayeed, nicknamed OMAR, for operational purposes.
But they couldn't get the smell. The acrid whiff of long-tended fires; the spiced scent of lamb on a spit; the eye-watering odor of men who toiled and sweated in the one-hundred-degree heat and had not bathed in weeks.
"Close enough? Or would you gents like me to stick my head inside the store and say hello?"
"Negative. Just give me a walk-by. Nice and brisk. We can slow down the pictures on this side."
Sarah crossed the street, dodging a howling Vespa, doing her best to keep to a walk. She was sure that somewhere in the Koran there was a _hadith_ banning "righteous women" from running, just as the holy lessons banned them from everything else except catering to the whims and desires of "righteous man folk."
Stepping onto the raised walkway, she continued past the jewelry store, letting her gaze fall on the array of gold chains in the window. The doorway gaped beside her. Two guards with AK's stood at attention inside. Surveillance cameras stared down from the corners. A portly Indian was talking to Sayeed. There were no other customers.
"Confirmed. OMAR on premises. Looks like he's got some muscle in there" came the voice in her earpiece. "Keep it moving."
For a quarter second longer she watched, then continued her promenade. At that very moment, however, there was a flurry of movement inside the store, and she stopped. It was a clumsy, jerky halt, a dead giveaway. And there she stood for one second... two... a perfect silhouette frozen in the doorway.
"He's going in the back," Sarah whispered. "I mean the two are going together. So is one of the guards. Where are the bully boys?" she asked, desperation crowding inside her.
"ETA nine minutes. Do not jeopardize your cover. Proceed to the Tikram mosque and continue surveillance from there."
"Nine min—" Her rigid training cut short her protest. In her mind, however, she howled with frustration.
At the end of the walkway, Sarah stepped off the curb and stopped. The courtyard to her right was filled with automobile tires. Hundreds and hundreds of brand-new tires, stacked neatly upon one another, row after row, rising thirty feet in the air. Turning, she peered across the intersection toward the mosque. It would be safer to watch from there. An A-team was inbound. She knew what that meant: bullets and lots of them. Abu Sayeed was not the type to turn himself over to the authorities and say, "Okay, Officer. I'll come along quietly."
"EMERALD, this is Ranger." A new voice sounded in her ear, calm, authoritative. _Ranger_. The DO himself. The Director of Operations. "Go on into the store. Take a look around."
_"Go in?"_
"We wouldn't want him to sneak out on us, would we? Not before the party starts. It's a jewelry store," he went on. "Have a look at a necklace. Buy whatever you like. Call it my treat. You can put it on my expense account."
Her mouth had grown dry, her tongue parched and swollen.
"I don't think they take American Express," she answered blithely, knowing that the banter was to relax her, to deceive her about the peril of his command. And make no mistake, it _was_ an order. He was asking her to flit by her lonesome into a shop with the biggest underworld financier on the northwest frontier, and a hardened terrorist associated with a group so secret, so rife with all manner of rumor, that no one even dared whisper its name—if it even had one—because until now no one had wanted to acknowledge its existence. One supreme evil commander was enough for the world these days.
Across the street a fierce-looking man was staring daggers at her. He wore a black headdress and a black _dish-dasha_ and his beard hadn't been cut in a decade. The poster boy for Islamic Ignorance refused to avert his gaze, lips trembling, eyes afire, his entire being a vessel of hate. Through the veil, she met his accusing glare, and from his obduracy, his anger, his bewildering disrespect of the superior sex, she drew the courage she herself lacked.
"Roger last," she said. "I think I'll have a look at some of Bhatia's tat."
"Good girl," said Ranger. And Sarah thought that if he ever called her that again, she'd slug him in the jaw. But by then it didn't matter. She was moving, not thinking. She dodged the curtain of sparks sent up from the gunsmith's forge. She grimaced in her private netherworld as she passed the coils of lamb intestines dangling from the butcher's hook. And then she was inside the store, admiring Mr. Bhatia's mediocre wares as if they were the Crown Jewels.
**T HE MONEY SAT IN A PILE** on a table in Bhatia's private office. With a barber's straight-edged razor, the Indian opened each packet, then handed the bills to an associate to count. When he was finished, he grunted. "Five hundred thousand dollars, as you claimed."
"The Scorpion does not lie," said Abu Sayeed. A bountiful rain had doubled the poppy harvest. One ton of raw opium was Allah's gift to Hijira: his benediction upon the holocaust to come.
"It is not easy to move such a sum," said Bhatia. "How quickly do you need it?"
"Immediately."
"Today?"
"Now."
Bhatia's grave features registered concern. "Where is the money to be sent?"
"Paris."
"Hmm." Bhatia's eyes narrowed, and he mumbled a few words to himself, shaking his head. Sayeed knew it a ruse, the Indian figuring how large a fee he might get away with. "It can be done. However, the cost for such a transaction is two percent."
"One percent."
"Impossible. No one keeps such cash on the premises. A bank will have to be involved. There will be borrowing costs. Overnight at least. Maybe longer. It cannot be avoided. And of course the risk. One and a half."
Sayeed disliked negotiation, but in some cases, it was necessary. Five thousand dollars was a small fee to insure swift delivery of money to Paris. Small, indeed, compared to the damage it would wreak. "One," he repeated. He had his orders. "The Scorpion will show his appreciation."
"How?"
Abu Mohammed Sayeed clamped his hand over the Indian's. "He will not sting you."
**F OUR HUNDRED AND SEVENTY MILES** above the Indian Ocean, an Intruder Geosynchronous SIGINT (signals intelligence) satellite tasked by the National Reconnaissance Agency to monitor mobile communications in the Pakistan–Northern India–Afghanistan triangle responded to an emergency override command. In the freezing infinity of space, guidance boosters fired for a half second. Rectangular electromagnetic-phased array panels minutely altered their attitude. In an instant, the satellite's field of surveillance or "footprint" shifted forty miles to the north and twenty-two miles to the east, and centered on codename EMERALD's last relayed GPS coordinates.
Several minutes later, the satellite intercepted an open-air cellular transmission based in Peshawar with a respondent in Paris. Along with two thousand three hundred and twenty-nine other calls it had concurrently captured, the satellite's transponders relayed the signal to a ground-based listening station at Diego Garcia, maintained by the U.S. Air Force's 20th Space Satellite Group. In real time, the listening station directed the signals to the National Security Agency in Fort Meade, Maryland, where the conversation was analyzed by a team of parallel-linked IBM supercomputers for any of a thousand "keywords" in one hundred languages and dialects. In .025 of a second, the supercomputer determined the call was of "strategic or military" value, coded it "urgent," and forwarded a digital copy of the conversation to an analyst at the Central Intelligence Agency's headquarters in Langley, Virginia.
The analyst, realizing he was in possession of "real time intelligence," or information of an immediate strategic concern, phoned the Deputy Director of Operations and requested a crash meeting.
"Third Floor. CTCC," said Admiral Owen Glendenning. "Get up here on the double and bring me a copy of the call."
**"S O," TRUMPETED FAISAN BHATIA,** reentering the office after a fifteen-minute absence. "Everything is arranged. The money can be picked up at Royal Joailliers. It is located at the Place Vendôme in Paris. Do you wish their address?"
"Of course." Abu Sayeed smiled secretly. The Scorpion had informed him that Bhatia would use Royal Joailliers. Royal called itself an "haute joaillier," meaning that nothing in its satin-lined showcases sold for less than ten thousand dollars. The cartels were their best clients—Colombians, Mexicans, Russians—and it was their practice to keep unconscionable sums of cash on the premises. When Sayeed had written down the address, Bhatia inquired if he would like to provide him with the recipient's name. "That is not necessary," said Sayeed.
"Very well. The recipient must use a password to identify himself. In this case, a dollar bill will do nicely." Bhatia slid a worn U.S. banknote across the table. "You will take it with you. As soon as possible, I advise you to transmit the serial numbers on the lower left-hand side of the bill to the recipient. When he presents himself to Royal Joailliers, he must give them the identical numbers in sequence. Only then will he be given the money. There can be no mistakes. It is agreed?"
Sayeed knew the rules of _hawala_ well. The Scorpion had made use of the informal banking system for years to funnel funds to his operatives. "It is agreed," he said.
"May I offer the use of my telephone?"
"I have my own."
"Very good, then. You will join me for something to eat. If I may say, you look rather done in." Bhatia clapped his hands, barking an order to an unseen consort. A moment later, his wife entered carrying a tray with two porcelain cups and a china teapot. A younger woman followed, bearing a goat's head upon a silver platter. In the cloying ninety-degree heat, flies swarmed the tray, attacking the staring, gelatinous eyes.
"Please," said Bhatia, extending a hand toward the Pakistani delicacy.
But Sayeed was not interested in food. Glancing at the monitor that broadcast the interior of Bhatia's showroom, he watched as a woman clad in a full-length _burqa_ examined a tray of jewelry. She had been there the entire time he had been with Bhatia. The picture grew fuzzy, as if losing reception, then snapped back into focus. A tinge of unease soured his stomach. The clock read 4:45. It would be 12:45 in Paris. He wanted to leave. He wanted to make the call. His brother would be waiting.
Without a further word, he stood.
Bhatia set his cup of tea on the table and sprang to his feet. "It was an honor to serve the Scorpion," he intoned gravely, leading the way to the beaded curtains. "You will pass on my regards?"
"The monitor," said Sayeed, lifting his finger toward the screen, "it is a closed circuit system?"
"No," answered Bhatia proudly. "Wireless. New from Japan."
Sayeed stalked from the office without another word.
# 3
#
**A DMIRAL OWEN GLENDENNING** stood at the rear of the Counterterrorism Command Center on the sixth floor of the Agency's headquarters in Langley, Virginia, digesting the latest information. It was too soon to hope, yet he could neither stifle nor ignore the first flush of optimism that reddened the back of his neck and had him curling his toes. Keep on him a little longer, girl, and we're there. Just a little longer.
Projected onto a seven-foot screen, a live feed from Pakistan offered him Sarah Churchill's point of view as she examined a selection of gold chains. She raised her head and Glendenning was faced with a close-up of a frantic jewelry salesman blabbing the usual nonsense about high quality and best price. A simultaneous English translation ran across the bottom of the screen.
A second screen broadcast the footprint of the Central Intelligence Agency's spy satellites on a political map of the globe. Each satellite's footprint was indicated by a shadowed area. Some shadows remained stationary; others crept across the map with the turning of the Earth.
The seal of the CIA highlighted against a navy blue background lit up a third screen, currently unused.
At four A.M., the Counterterrorism Command Center was fully staffed and humming. Three rows of analysts occupied the front position of the auditorium-sized command room. All enjoyed brand-new workstations, the latest flat-panel displays, and state-of-the-art ergonomic chairs that had cost twelve hundred dollars a pop. Glendenning thought it looked like a scene from a movie—the way Hollywood dreamed the espionage community operated. It had been a long time since the Company had access to funds like this, but with the war on terrorism running full bore, the spigots were open wide.
Glendenning was a small, slightly built man with thinning gray hair and an unremarkable face. Dressed in his usual uniform of khaki trousers, white short-sleeved button-down shirt complete with pen protector, and Rockport deck shoes, he looked more like a NASA engineer than a seasoned spymaster. He saved his dress whites and the six rows of ribbons he had accumulated during thirty years of service for his appearances before the permanent Senate subcommittee on intelligence. Briefings that had once been little more than secret check-writing ceremonies had lately taken an adversarial turn. "Where were the results Glendenning had promised?" the more daring senators demanded. A few hundred million dollars in confiscated accounts was fine and dandy, but what about the terrorists behind it? Warm bodies, not frozen assets, were the order of the day.
Picking up a phone, he dialed the Foreign Terrorist Asset Tracking Center (FTAT) two floors below. "Get me Halsey."
Strictly speaking, FTAT was a Treasury operation. Treasury funded it. Treasury supervised it. But when the scope of the investigation into worldwide terrorist financing had become clear, all involved had decided to move FTAT's operations to Langley.
There was a time not too far back that the very idea of the CIA contacting Treasury to share information was practically a jailable offense. There was law enforcement and there was intelligence and never the twain shall meet. But the events of September 11, 2001, had changed all that. With the passage of the Patriot Act, communication between the United States' varied and multiple law enforcement and intelligence agencies was not only permitted, it was encouraged. The old concept of "stovepiping," or keeping information inside the particular agency, or, as was the case with the FBI, inside the individual department that had discovered it, was thrown out the door. Concerns about infractions on civil liberties and personal privacy were quickly dismissed. The threat beyond the country's borders took preeminence, and was, Glendenning knew, far greater than anyone could be told.
"This is Halsey," answered a deep, gravelly voice.
"Don't you have a home either?"
Allan Halsey, chief of the Foreign Terrorist Asset Tracking Center, gave a shallow laugh. "Not according to my wife."
"We nabbed the call," said Glendenning. "The money's being moved as we speak. Come on up and we'll run it from here."
"How much?"
"We're guessing five hundred grand or five million. Either way it's the real deal."
"I don't like it," said Halsey. "Risky to move so much."
"I couldn't agree more. Something's about to pop. Who's running your team in Paris?"
"Chapel, Adam."
"Don't know him. A new guy?"
"Treasury pulled him in after the WTC."
"Military?"
"God no," said Halsey. "A Quant jock all the way. Kid was on the fast track at Price Waterhouse. Partner at twenty-nine. National audit manager."
"Sounds like a real killer," said Glendenning. "Ought to have 'em quaking in their boots."
"Come on now, Glen. He's the new kind of soldier. You know, brains instead of brawn. Different war we're fighting this time."
"That's what they tell me. In the end, though, you still have to shoot the bad guys."
"Don't worry about Adam," said Halsey, his voice quieter, more confident, as if vouchsafing a secret. "He can hold his own."
|
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{"url":"https:\/\/zenodo.org\/record\/3941403\/export\/csl","text":"Journal article Open Access\n\n# Implicit and Explicit Regularization for Optical Flow Estimation\n\nKarageorgos, Konstantinos; Dimou, Anastasios; Alvarez, Federico; Daras, Petros\n\n### Citation Style Language JSON Export\n\n{\n\"DOI\": \"10.3390\/s20143855\",\n\"language\": \"eng\",\n\"author\": [\n{\n\"family\": \"Karageorgos, Konstantinos\"\n},\n{\n\"family\": \"Dimou, Anastasios\"\n},\n{\n\"family\": \"Alvarez, Federico\"\n},\n{\n\"family\": \"Daras, Petros\"\n}\n],\n\"issued\": {\n\"date-parts\": [\n[\n2020,\n7,\n10\n]\n]\n},\n\"abstract\": \"<p>In this paper, two novel and practical regularizing methods are proposed to improve existing neural network architectures for monocular optical flow estimation. The proposed methods aim to alleviate deficiencies of current methods, such as flow leakage across objects and motion consistency within rigid objects, by exploiting contextual information. More specifically, the first regularization method utilizes semantic information during the training process to explicitly regularize the produced optical flow field. The novelty of this method lies in the use of semantic segmentation masks to teach the network to implicitly identify the semantic edges of an object and better reason on the local motion flow. A novel loss function is introduced that takes into account the objects’ boundaries as derived from the semantic segmentation mask to selectively penalize motion inconsistency within an object. The method is architecture agnostic and can be integrated into any neural network without modifying or adding complexity at inference. The second regularization method adds spatial awareness to the input data of the network in order to improve training stability and efficiency. The coordinates of each pixel are used as an additional feature, breaking the invariance properties of the neural network architecture. The additional features are shown to implicitly regularize the optical flow estimation enforcing a consistent flow, while improving both the performance and the convergence time. Finally, the combination of both regularization methods further improves the performance of existing cutting edge architectures in a complementary way, both quantitatively and qualitatively, on popular flow estimation benchmark datasets.<\/p>\",\n\"title\": \"Implicit and Explicit Regularization for Optical Flow Estimation\",\n\"type\": \"article-journal\",\n\"id\": \"3941403\"\n}\n65\n47\nviews","date":"2021-09-19 08:30:18","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\": 0.2732301652431488, \"perplexity\": 2284.7556118188268}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780056752.16\/warc\/CC-MAIN-20210919065755-20210919095755-00200.warc.gz\"}"}
| null | null |
The Ultimate Surfer is an American television surfing competition show that aired on ABC from August 23 to September 21, 2021. In March 2022, the series was canceled after one season.
Format
The Ultimate Surfer, an eight-episode series, gathers some of the world's best amateur surfers to live and train together as they battle it out on the consistent and perfect waves created at the "Surf Ranch" in Lemoore, California. The contestants compete in team and individual challenges, focusing on specific elements of surfing, such as, barrels, cutbacks, turns, and front and back-side surfing where they will compete for the highest score by the judges.
Weekly eliminations will leave two men and two women as finalists. They can either form alliances or rivalries by sending their fellow surfers into a "surf-off", where they must surf at nighttime to get the best score to move on in the competition.
The winner's prize is $100,000 and an opportunity to compete on the WSL Championship Tour, the pinnacle of professional surfing, as well as earn the title of "The Ultimate Surfer".
Production
On November 13, 2019, it was announced that ABC had ordered the series, with Kelly Slater as special correspondent. Craig Piligian, Erik Logan and Dana White served as executive producers. On April 7, 2021, it was announced that the series would premiere on August 16, 2021, later being rescheduled to August 23, 2021. On May 6, 2021, it was announced that Jesse Palmer would host the show, with Erin Coscarelli and Joe Turpel serving as commentators.
On March 30, 2022, it was announced that the series has been cancelled after one season.
Surfers
The surfers were revealed on May 6, 2021.
Contestant progress
Key
The contestant won the competition.
The contestant came in second place in the competition.
The contestant won the surfing competition with their partner.
The contestant was nominated to surf in the surf off.
The contestant was eliminated in the surf off.
The contestant won their way back into the competition.
The contestant failed to return back into the competition.
Episodes
Reception
Notes
See also
The Ultimate Fighter
References
External links
2020s American reality television series
2021 American television series debuts
2021 American television series endings
American Broadcasting Company original programming
English-language television shows
Sports entertainment
Surfing competitions
Surfing mass media
Television shows filmed in California
World Surf League
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\section{Introduction}
One of the most interesting properties of a system of bosons is that under
certain conditions it is possible to have a phase transition at a critical
value of the temperature in which a macroscopic fraction of the bosons can condense into the
ground state. This was first predicted over 70 years ago for the ideal
nonrelativistic Bose gas \cite{bose24,einsteinpreus24} and is nowadays
well known to happen if the spatial dimension $D\geq 3$. (See
\cite{landaulifshitz69} for the case $D=3$ and \cite{may64} for general $D$.) The analogous phenomenon for ideal relativistic gases has also been
studied and the criterion on the spatial dimension has been found to be the
same as that for nonrelativistic gases \cite{kapusta81,haberweldon81,haberweldon82a}.
Of course it is of great interest to see what influence different types of
interactions have on the condensation. In nonrelativistic theories in
certain models it is known what happens if interactions are included
\cite{huang93}. In relativistic theories a detailed study of $\lambda
\phi^4$ theory at finite temperature and density has been given in flat
\cite{bensonbernsteindodelson91,bernsteindodelson91} as well as in curved spacetime
\cite{kirstentoms95}.
Rather than examine self-interacting fields, or the interactions among
different quantum fields, a simpler problem is to study what happens
for a quantum field under external conditions, where by external conditions we
mean for example gravitational or electromagnetic background fields or
simply boundary conditions imposed on the field. Several different
situations have already been considered. Let us only mention the analysis
in the presence of boundaries and static gravitational fields
\cite{toms92,toms93,alta,singh,park,cog,dowken,bytvan,kkthesis}
and in the presence of (mainly constant) magnetic
fields
\cite{schafroth51,schafroth55,may59,may65,toms95,toms95a,daicicfrankelkowalenko94,daicicfrankelgailiskowalenko94,elmforsperssonskagerstam93,elmforsliljenbergperssonskagerstam95,daicic,rojas}.
One of the main questions in all these kinds of considerations is, whether
or not Bose-Einstein condensation can occur. In the above literature
rather involved analysis has been done in order to arrive at the conclusion
whether or not Bose-Einstein condensation can occur. Recently, the authors
gave a very simple criterion to decide whether a system might condense or not
\cite{physlett}. Here, we plan to give more details of the calculations and to
apply the criterion to a large class of examples. This will include the
examples treated in the above mentioned
literature as well as some new cases. Furthermore we will show how the
criterion might be used to decide if a system reveals the Meissner-Ochsenfeld
effect or not.
It is worth emphasizing precisely what is meant by BEC since different definitions are possible. For the free Bose gas in three or more spatial dimensions the specific heat has a non-smooth behaviour at a critical temperature which signals a phase transition. In terms of the quantum field theory the phase transition may be interpreted as symmetry breaking with the scalar field developing a non-zero expectation value at the critical temperature. Associated with this phase transition is a sudden growth in the occupancy of the ground state. In the present paper we will adopt this as our definition of BEC. It is possible to have a build-up of particles in the ground state without a phase transition occurring. In this case there is no unique way to identify a temperature associated with the build-up of particles in the ground state. For the particular case of a constant magnetic field in three spatial dimensions it was shown by Rojas \cite{rojas} that although there was no phase transition there could still be a build-up of particles in the ground state which could be interpreted as BEC. A similar situation occurs for bosons confined by a harmonic oscillator potential \cite{KirstenTomsPRA}.
\section{Criterion for the occurrence of Bose-Ein\-stein condensation}
\subsection{Relativistic quantum field theory}
Let us consider a complex relativistic scalar field which may interact with
background electromagnetic or gravitational fields but which is otherwise
free. We restrict our attention to an ultrastatic spacetime
of the form ${\cal M} ={\mathbb R}\times \Sigma$ with metric
\begin{equation}
ds^2=dt^2 -g_{ij} (x) dx^idx^j \label{2.1}
\end{equation}
and
with the field
obeying any boundary conditions in the spatial directions.
The action functional for the complex field $\Phi$ will be chosen to be
\begin{eqnarray}
S&=&\int dt \int\limits_{\Sigma}d\sigma_x\,\, \left\{
(D^{\mu} \Phi )^{\dagger} (D_{\mu} \Phi )-m^2 \Phi^{\dagger} \Phi
-U_0 (x)\right.\nonumber\\
&&\quad\quad\quad\left. -U_1 (x) \Phi^{\dagger} \Phi \right\},\label{2.2}
\end{eqnarray}
$d\sigma_x $ is the invariant volume element on the manifold $\Sigma$.
In order to have a scalar field exclusively under external conditions,
the functions $U_0 (x)$ and $U_1 (x)$ may depend on the background
gravitational or electromagnetic fields, but are independent of the scalar
field. They are also assumed to be independent of the time. The kinetic term is the usual gauge-covariant derivative
\begin{equation}
D_{\mu} \Phi =\partial _{\mu} \Phi -ie A_{\mu} \Phi .\label{2.3}
\end{equation}
In an ultrastatic spacetime, finite temperature and density are easily
incorporated in the Euclidean time formalism
\cite{bernard74,kapusta89,landsmanvanweert87}. It has been shown, that the
partition function ${\cal Z}$ may be represented in the form
(see for example \cite{toms94})
\begin{equation}
{\cal Z} =\int [d\Phi ][d\Phi^{\dagger}] \exp \left\{-\tilde S\right\},
\label{2.4}
\end{equation}
where
\begin{eqnarray}
\tilde S &=& \int\limits_0^{\beta}d \tau \int\limits_{\Sigma}d\sigma_x\,\,
\Big\{\left[\dot{\Phi}^{\dagger} +ie(A_0-i\mu)\Phi^{\dagger}\right]\nonumber\\
&&\quad\quad\times\left[\dot{\Phi}-ie(A_0-i\mu)\Phi\right]
+|{\mathbf D} \Phi |^2\nonumber\\
&& +(m^2+U_1 (x) ) \Phi^{\dagger} \Phi +U_0 (x)
+J\Phi +J^{\dagger} \Phi^{\dagger}\Big\}.\label{2.5}
\end{eqnarray}
In order to decide whether or not Bose-Einstein condensation will occur we need
the charge in the excited states. It may be obtained directly from the effective
action. Eliminating the dependence on the sources $J$ and $J^{\dagger}$ by
\begin{eqnarray}
\bar{\Phi} &=& \frac{\delta W}{\delta J}\left.\right|_{\mu,J^{\dagger}}\nonumber\\
\bar{\Phi}^{\dagger} &=& \frac{\delta W}{\delta J^{\dagger}}
\left.\right|_{\mu,J},\label{2.6}
\end{eqnarray}
with the background fields $\bar{\Phi}$, $\bar{\Phi}^{\dagger}$, the effective action is
defined by
\begin{eqnarray}
\Gamma [\mu,\bar{\Phi} ,\bar{\Phi}^{\dagger} ]& =&W [\mu , J, J^{\dagger}]\nonumber\\
&& -
\int\limits_0^{\beta} d\tau \int\limits_{\Sigma}d\sigma_x\,\, [J\bar{\Phi} +J^{\dagger} \bar{\Phi}^{\dagger} ]
\label{2.7}
\end{eqnarray}
with
\begin{equation}
W[\mu , J, J^{\dagger}] =-\ln {\cal Z} [\mu , J, J^{\dagger}].\label{2.8}
\end{equation}
It is known, that a minimization of the effective action is equivalent to
a minimization of the Helmholtz free energy \cite{toms95}.
In terms
of the effective action the charge is then defined by
\begin{equation}
Q=-\frac 1 {\beta} \frac{\partial\Gamma }{\partial \mu} \left.
\right|_{\bar{\Phi}, \bar{\Phi}^{\dagger} }.\label{2.9}
\end{equation}
We may now proceed with the analysis of the effective action. It is
straightforward to show that
\begin{equation}
\Gamma [\mu, \bar{\Phi}, \bar{\Phi}^{\dagger} ] =\tilde S [\bar{\Phi}, \bar{\Phi}^{\dagger} ] +\frac 1 2 \ln \det
(l^2 \tilde S _{,ij} ),\label{2.10}
\end{equation}
where we used DeWitt's condensed notation \cite{dewitt65}. $l$ is
an arbitrary unit of length introduced to keep the argument of the logarithm in (\ref{2.10}) dimensionless. Specializing to the case of a background static magnetic field only, and
choosing the gauge
\begin{equation}
A_0 =0, \qquad {\mathbf\nabla}\cdot{\mathbf A }=0,\label{2.11}
\end{equation}
we have
\begin{equation}
\frac 1 2 \ln\det (l^2 S_{,ij}) =\Gamma_++\Gamma_-,\label{2.11a}
\end{equation}
where
\begin{eqnarray}
\Gamma_{\pm} &=&\frac 1 2 \ln\det\left\{l^2\left[
-\left(\frac{\partial}{\partial \tau} \mp e\mu\right)^2
- {\mathbf D} ^2\right.\right.\nonumber\\
&&\quad\quad\quad\left.\left. +m^2 +U_1 (x) \right]\right\}.\label{2.12}
\end{eqnarray}
For the calculation of (\ref{2.12}) we will use the zeta function scheme
\cite{hawking77,critchleydowker76}.
In this scheme one defines
\begin{equation}
\Gamma_{\pm} =-\frac 1 2 \zeta_{\pm} ' (0) +\frac 1 2 \zeta_{\pm} (0)
\ln l^2 \label{2.13}
\end{equation}
with the
generalized zeta functions
\begin{equation}
\zeta_{\pm} (s) =\sum_{j=-\infty}^{\infty}\sum_N (\lambda_{jN} ^{\pm} )^{-s}
.\label{2.14}
\end{equation}
Here, $\lambda _{jN}^{\pm}$ are the eigenvalues of the fluctuation
operator (see (\ref{2.12})) given by
\begin{equation}
\lambda_{jN}^{\pm} =\left(\frac{2\pi }{\beta} j \pm ie\mu\right)^2
+\sigma_N,\label{2.15}
\end{equation}
with the eigenvalues $\sigma_N$ of the spatial part of the Klein-Gordon
operator,
\begin{equation}
(-{\mathbf D}^2 +U_1 (x)+m^2 )f_N (x) =\sigma_N f_N (x) ,\label{2.16}
\end{equation}
with a complete orthonormal set of functions $f_N (x)$. It is easily seen that
$\Gamma_+=\Gamma_-$, so that we skip the index $\pm$ in the notation and
consider $\zeta_+$.
Now all the definitions and preparation to explain our criterion for the
appearance of Bose-Einstein condensation have been given. For the moment we do not
want to consider any specific situation but want to assume a quite
general structure of the eigenvalues $\lambda_{jN}$ or, more specifically,
of the energy $E_N^2 =\sigma_N $. As we will see, to decide if
Bose-Einstein condensation appears, the only relevant information on $E_N$
is about its continuous part. Let us assume that $\sigma_N$ splits into the
sum of a discrete part $\sigma_{\mathbf p}^d$ (${\mathbf p}$ is just a set of labels for the
discrete part of the spectrum), and a continuous part which we can deal with by imposing box normalization. The box will be taken to have sides $L_1,...,L_q$,
for some $q$ and thus we write
\begin{equation}
E_N^2 =\sum_{i=1}^q \left(\frac{2\pi}{L_i}\right)^2 l_i^2 +\sigma_{\mathbf p}^d\;.\label{2.17}
\end{equation}
In the limit $L_i\to\infty$ the starting point for the analysis of $\zeta (s)$
is
\begin{eqnarray}
\zeta (s )&=&\frac{V_q\beta}{(4\pi)^{\frac{q+1} 2}}
\frac 1 {\Gamma (s)} \sum_{l=-\infty}^{\infty} \sum_{\mathbf p}
\int\limits_0^{\infty}dt\,\, t^{s-1-\frac q 2}\nonumber\\
&&\times \exp\left\{\left(\frac{2\pi il}{\beta}
-\mu\right)^2
t -\sigma_{\mathbf p}^d t\right\}.\label{2.18}
\end{eqnarray}
Doing a resummation in $l$, this is equivalent to
\begin{eqnarray}
\zeta (s )&=&\frac{V_q\beta}{(4\pi)^{\frac{q+1} 2}}
\frac{\Gamma\left( s-\frac{q+1} 2\right)}
{\Gamma (s)} \sum_{l=-\infty}^{\infty} \sum_{\mathbf p} \nonumber\\
& &\times \int\limits_0^{\infty}dt\,\, t^{s-1-\frac{ q+1} 2} e^{-\frac{\beta^2}{4t} l^2
-\sigma_{\mathbf p}^d t +\beta\mu l}\nonumber\\
&=&\frac{V_q\beta}{(4\pi)^{\frac{q+1} 2}}\frac{\Gamma\left( s-\frac{q+1} 2 \right)}
{\Gamma (s)}\zeta_{d,\Sigma} \left( s-\frac{q+1} 2\right)\nonumber\\
& &+\frac{V_q\beta}{(4\pi)^{\frac{q+1} 2}} \frac 2 {\Gamma (s)} \sum_{l=1}^{\infty}
\sum_{\mathbf p} \left(\frac 2 {\beta l} \sqrt{\sigma_{\mathbf p}^d}\right)^{\frac{q+1} 2 -s}\nonumber\\
&&\times K_{\frac{q+1} 2 -s} (\beta l \sqrt{\sigma_{\mathbf p}^d}) \left(e^{\beta\mu l}+e^{-\beta\mu l}\right)\;,\label{2.19}
\end{eqnarray}
where we introduced the zeta-function of the discrete part of the
spatial section,
\begin{equation}
\zeta_{d,\Sigma} (s) =\sum_{\mathbf p}(\sigma_{\mathbf p}^d)^{-s}.\label{2.20}
\end{equation}
The condition for Bose-Einstein condensation to occur is that $\mu$ must
reach a critical value $\mu_C$ set by the lowest eigenvalue in the spectrum,
\begin{equation}
\mu_C^2 =\sigma_{\mathbf 0}^d =E_0^2 .\label{2.21}
\end{equation}
If the charge $Q$ in the excited states, Eq.~(\ref{2.9}), remains bounded
as $\mu\to\mu_C$, then Bose-Einstein condensation occurs, because for the
total charge large enough, it is not possible to accommodate it all in the
excited states. If $Q$ is not bounded as $\mu\to\mu_C$, then any amount of
the total charge can reside in the excited states and Bose-Einstein condensation
will not occur. We therefore need to look at the behaviour of $(\partial
/\partial \mu) \zeta (0)$ and $(\partial/\partial\mu )\zeta' (0) $
as $\mu\to\mu_C$.
Using the definition (\ref{2.9}) and representation (\ref{2.19}) for
$\zeta (s)$, one immediately finds
\begin{eqnarray}
Q&=&V_q\frac 2 {(4\pi)^{\frac{q+1} 2}} \sum_{l=1}^{\infty} \sum_{\mathbf p} \beta l \left(\frac 2
{\beta l} \sqrt{\sigma_{\mathbf p}^d}\right)^{\frac{q+1} 2}
K_{\frac{q+1} 2} (\beta l \sqrt{\sigma_{\mathbf p}^d})\nonumber\\
& &\hspace{2cm}\times \left(e^{l\beta \mu }-e^{-l\beta\mu}\right).\label{2.22}
\end{eqnarray}
The convergence of the sums is defined through the behaviour of the
MacDonald functions for large arguments \cite{gradshteynryzhik65}
\begin{equation}
K_{\nu} (z) \sim \sqrt{\frac{\pi}{2z}}e^{-z} \left\{ 1+{\cal O} (z^{-1})
\right\}.\label{2.23}
\end{equation}
It is clear that in the limit $\mu\to\mu_C$ all but the
contributions coming from ${\mathbf p}={\mathbf 0}$ are finite. Thus in the leading approximation,
neglecting finite pieces, we write
\begin{equation}
Q(\mu\to\mu_C) =d_0V_q\left(\frac{\sqrt{\sigma_{\mathbf 0}^d}}{2\pi\beta}\right)^{q/2}
\sum_{l=1}^{\infty}l^{-q/2} e^{-\beta l (\sqrt{\sigma_{\mathbf 0}^d}-\mu )}.
\label{2.24}
\end{equation}
We have allowed for the possible degeneracy of the ground state by introducing the degeneracy factor $d_0$. It is seen clearly that in the limit $\mu\to\mu_C=\sqrt{
\sigma_{\mathbf 0}^d}$ the charge remains
finite for $q\geq 3$; thus for $q\geq 3$ Bose-Einstein condensation occurs,
and for $q\leq 2$ Bose-Einstein condensation does not occur.
The detailed behaviour for $\mu\to\mu_C$ is most clearly extracted using the
Mellin-Barnes integral representation of the exponential in (\ref{2.24}),
\begin{equation}
e^{-v} =\frac 1 {2\pi i} \int\limits_{c-i\infty}^{c+i\infty} d\alpha
\,\,\Gamma(\alpha ) v^{-\alpha},\label{2.25}
\end{equation}
with $\Re v >0$ and $c\in {\mathbb R}$, $c>0$. Using this in Eq.~(\ref{2.24}),
one finds
\begin{eqnarray}
Q(\mu\to\mu_C) &=&\frac{d_0V_q}{2\pi i} \left(
\frac{\sqrt{\sigma_{\mathbf 0}^d}}{2\pi\beta}\right)^{\frac q 2}
\int\limits_{c-i\infty}^{c+i\infty}d\alpha\,\,
\Gamma (\alpha )\nonumber\\
&&\times (\sqrt{\sigma_{\mathbf 0}^d}-\mu )^{-\alpha}\beta^{-\alpha}
\zeta_R \left(\alpha +\frac q 2\right)\;,\label{2.26}
\end{eqnarray}
where, in order to allow for interchanging the sum and the integral one has to
impose $\alpha > 1-q/2$. $\zeta_R(s)$ is the Riemann $\zeta$-function, which is analytic in $s$ except at $s=1$ where it has a simple pole with residue 1. Closing the contour to the left of the rightmost pole
then gives the following leading behaviour~:\\
\\
$q=0$, pole of order one at $\alpha =1$,
\begin{equation}
Q(\mu\to\mu_C ) =\frac{d_0}{\beta (\mu_C-\mu)},\label{2.27}
\end{equation}
$q=1$, pole of order one at $\alpha =1/2$,
\begin{equation}
Q(\mu\to\mu_C) =\frac{d_0V_1}{\sqrt{2} \beta} \left(\frac{\mu_C}{\mu_C-\mu}\right)
^{1/2}, \label{2.28}
\end{equation}
$q=2$, pole of order two at $\alpha =0$,
\begin{equation}
Q(\mu\to\mu_C) =-\frac{d_0V_2}{2\pi\beta} \mu_C\ln\beta (\mu_C-\mu).\label{2.29}
\end{equation}
As mentioned, for $q\geq 3$ no divergent contribution results. The restriction
$q\geq 3$ includes a large number of previously known results, often established
by long and detailed calculations, as special cases. These and some
new examples are summarized in Sec.~3.
\subsection{Nonrelativistic quantum field theory}
We will consider a nonrelativistic field theory described by the complex Schr\"{o}dinger field $\Phi$ whose action functional is
\begin{eqnarray}
S&=&\int dt\int_{\Sigma}d\sigma_x\Big\lbrace\frac{i}{2}\left(\Phi^\dagger \dot{\Phi}-\dot{\Phi}^\dagger\Phi\right)-\frac{1}{2m}|{\mathbf D}\Phi|^2 \nonumber\\
&&\quad\quad-U_1({\mathbf x})\Phi^\dagger\Phi\Big\rbrace\;.\label{2.2.1}
\end{eqnarray}
$d\sigma_x$ is the invariant volume element on the $D$-dimensional Riemannian manifold $\Sigma$. We will consider $\Sigma$ to be compact. (In the case $\Sigma={\mathbb R}^D$ we will impose box normalization with the infinite box limit taken.) $U_1({\mathbf x})$ is an arbitrary time independent potential.
As before,
${\mathbf D}={\mathbf \nabla}-ie{\mathbf A}$ is the gauge covariant derivative, with ${\mathbf A}$ a vector potential describing a background magnetic field
and $|{\mathbf D}\Phi|^2$ is short
for $g^{ij}D_i\Phi^\dagger D_j\Phi$ where $g_{ij}$ is the Riemannian metric on $\Sigma$.
The theory described by (\ref{2.2.1}) has the local gauge invariance
\begin{eqnarray}
\Phi&\rightarrow&e^{ie\theta}\Phi\;,\nonumber\\
{\mathbf A}&\rightarrow&{\mathbf A}+{\mathbf\nabla}\theta\;,\nonumber\\
\end{eqnarray}
which gives rise to a conserved current, and a conserved charge which is given by
\begin{equation}
Q=\int_{\Sigma}d\sigma_x|\Phi|^2\;.\label{2.2.2}
\end{equation}
We will deal with the charge rather than the particle number to mirror the relativistic case considered above as closely as possible. The conserved charge is dealt with by introducing a Lagrange multiplier $\mu$, which is the chemical potential. After a rotation to imaginary time $\tau$, the partition function is again expressed in the form (\ref{2.4}), with
\begin{eqnarray}
\tilde{S}\lbrack\bar{\Phi},\bar{\Phi}^\dagger\rbrack&=&\int_{0}^{\beta}d\tau\int_{\Sigma}d\sigma_x\Bigg\lbrace\frac{1}{2}\left(\Phi^\dagger\frac{\partial}{\partial\tau}\Phi-\frac{\partial}{\partial\tau}\Phi^\dagger\Phi\right) \nonumber\\
&&\quad\quad+\frac{1}{2m}|{\mathbf D}\Phi|^2+U_1({\mathbf x})|\Phi|^2\nonumber\\
&&\quad\quad-e\mu|\Phi|^2+J\Phi+J^\dagger\Phi^\dagger\Bigg\rbrace\;.
\label{2.2.3}
\end{eqnarray}
The field $\Phi$ has been coupled to a complex source $J$ as in (\ref{2.5}). The effective action can now be defined as in Sec.~2.1. In place of (\ref{2.10}) we find
\begin{eqnarray}
\Gamma\lbrack\mu,\bar{\Phi},\bar{\Phi}^\dagger\rbrack&=&
\tilde{S}\lbrack\bar{\Phi},\bar{\Phi}^\dagger\rbrack
+\ln\det l\Big\lbrack
\frac{\partial}{\partial\tau}-e\mu\nonumber\\
&&\quad-\frac{1}{2m}{\mathbf D}^2+
U_1({\mathbf x})\Big\rbrack\;,\label{2.2.4}
\end{eqnarray}
where $\bar{\Phi}$ is the background Schr\"{o}dinger field. The second term in (\ref{2.2.4}) has arisen from performing the functional integral in (\ref{2.4}) over the Schr\"{o}dinger field.
In order to regularize the determinant which appears in (\ref{2.2.4}) we will again use the $\zeta$-function method. This time we will let $f_N({\mathbf x})$ denote the eigenfunctions of $\displaystyle{-\frac{1}{2m}{\mathbf D}^2+U_1({\mathbf x})}$~:
\begin{equation}
\left\lbrack-\frac{1}{2m}{\mathbf D}^2+U_1({\mathbf x})\right\rbrack f_N({\mathbf x})=\sigma_Nf_N({\mathbf x})\;.\label{2.2.5}
\end{equation}
(These $f_N({\mathbf x})$ differ only by a trivial rescaling from the $f_N({\mathbf x})$ used in Sec.~2.1.) The $f_N({\mathbf x})$ are seen to be stationary state solutions to the Schr\"{o}dinger equation for whatever boundary conditions are imposed. The eigenvalues $\sigma_N$ are the energy levels for the first quantized system. The set $\lbrace f_N({\mathbf x})\rbrace$ is assumed to be complete and orthonormal.
Because the functional integral in (\ref{2.4}) extends over all fields periodic in the imaginary time coordinate $\tau$ with period $\beta=T^{-1}$, the eigenvalues of the operator appearing in (\ref{2.2.4}) are
\begin{equation}
\lambda_{jN}=2\pi ijT+\sigma_N-e\mu\;,\label{2.2.6}
\end{equation}
where $j=0,\pm1,\pm2,\ldots\;$. The generalized $\zeta$-function is given by (see \ref{2.14})
\begin{equation}
\zeta(s)=\sum_{j=-\infty}^{\infty}\sum_N(\lambda_{jN})^{-s}\;,\label{2.2.7}
\end{equation}
and we define
\FL
\[
\ln\det l\left\lbrack
\frac{\partial}{\partial\tau}-e\mu-\frac{1}{2m}{\mathbf D}^2+
U_1({\mathbf x})\right\rbrack
\]
\FR
\begin{equation}=-\zeta'(0)+\zeta(0)\ln l\;.\label{2.2.8}
\end{equation}
Using the summation formula in \cite{TomsPRB}, it is easy to show that
\begin{equation}
\zeta(s)=\sum_N(\sigma_N-e\mu)^{-s}+\zeta_T(s)\;,\label{2.2.9}
\end{equation}
where we have defined
\begin{equation}
\zeta_T(s)=\frac{T^{-s}}{\Gamma(s)}\sum_N\sum_{n=1}^{\infty}
\frac{e^{-n\beta(\sigma_N-e\mu)}}{n^{1-s}}\;.\label{2.2.10}
\end{equation}
The first term in (\ref{2.2.9}) has no explicit temperature dependence, and corresponds to the zero-point energy contribution to the effective action. We will ignore it in what follows. It disappears if normal ordering of the Hamiltonian is adopted.
The term in $\zeta_T(s)$, which we may call the thermal $\zeta$-function, is easily shown to have the following values~:
\begin{eqnarray}
\zeta_T(0)&=&0\;,\label{2.2.11}\\
\zeta_T'(0)&=&-\sum_N\ln\left\lbrack1-e^{-\beta(\sigma_N-e\mu)}\right\rbrack\;.\label{2.2.12}
\end{eqnarray}
These last two results show that the second term in (\ref{2.2.4}) has a simple relation to the thermodynamic potential $\Omega$ defined by
\begin{equation}
\Omega=T\sum_N\ln\left\lbrack1-e^{-\beta(\sigma_N-e\mu)}\right\rbrack\;. \label{2.2.13}
\end{equation}
(We could have adopted $\Omega$ as our starting point; however, our criterion is most simply expressed using the generalized $\zeta$-function, and in addition we wished to parallel the relativistic calculation.)
We will again assume that $\sigma_N$ splits up into the sum of a discrete part $\sigma_{\mathbf p}^d$, and a continuous part which is dealt with by box normalization as in (\ref{2.17}) with $E_N^2$ replaced by $\sigma_N$. In the large box limit, the thermal $\zeta$-function becomes
\begin{equation}
\zeta_T(s)=\frac{V_q}{(4\pi)^{q/2}}\frac{T^{q/2-s}}{\Gamma(s)}
\sum_{\mathbf p}\sum_{n=1}^{\infty}\frac{e^{-n\beta(\sigma_{\mathbf p}^d - e\mu)}}{n^{1+q/2-s}}\;.\label{2.2.14}
\end{equation}
As in the relativistic case, the lowest mode $\sigma_{\mathbf 0}^d$ plays the crucial role in determining whether or not Bose-Einstein condensation can occur. We therefore split the sum over $\mathbf p$ in (\ref{2.2.14}) by separating off the lowest mode for special treatment. The critical value of the chemical potential is given by
\begin{equation}
e\mu_C=\sigma_0=\sigma_{\mathbf 0}^d\;,\label{2.2.15}
\end{equation}
and we wish to study the behaviour of the charge as $\mu\rightarrow\mu_C$. If we write
\begin{equation}
\zeta_T(s)=\zeta_T^{(0)}(s)+\zeta_T^{(\ne0)}(s)\;,\label{2.2.16}
\end{equation}
where
\begin{equation}
\zeta_T^{(0)}(s)=\frac{d_0V_q}{(4\pi)^{q/2}}\frac{T^{q/2-s}}{\Gamma(s)}
\sum_{n=1}^{\infty}\frac{e^{-n\beta e(\mu_C- \mu)}}{n^{1+q/2-s}}\label{2.2.17}
\end{equation}
is the lowest mode contribution (with degeneracy $d_0$), and $\zeta_T^{(\ne0)}(s)$ is given by (\ref{2.2.14}) with the sum over $\mathbf p$ restricted to non-zero values, it is easy to see that because the argument of the exponential in (\ref{2.2.14}) will always be negative, even for $\mu=\mu_C$, $\zeta_T^{(\ne0)\prime}(0)$ and $\displaystyle{\frac{\partial}{\partial\mu}\zeta_T^{(\ne0)\prime}(0)}$ remain finite. Bose-Einstein condensation is therefore determined from a knowledge of $\zeta_T^{(0)}(s)$.
From (\ref{2.2.17}) we have $\zeta^{(0)}(0)=0$ and
\begin{equation}
\zeta_T^{(0)\prime}(0)=\frac{d_0V_q}{(4\pi)^{q/2}}T^{q/2-s}
\sum_{n=1}^{\infty}\frac{e^{-n\beta e(\mu_C- \mu)}}{n^{1+q/2}}
\;.\label{2.2.18}
\end{equation}
The charge is given by
\begin{equation}
Q=-T\frac{\partial}{\partial\mu}\Gamma=T\frac{\partial}{\partial\mu}
\zeta_T^{(0)\prime}(0)+\cdots\;,\label{2.2.19}
\end{equation}
where terms which remain finite as $\mu\rightarrow\mu_C$ have been dropped. Using (\ref{2.2.18}) we have
\begin{equation}
Q(\mu\rightarrow\mu_C)\simeq ed_0V_q\left(\frac{T}{4\pi}\right)^{q/2}\sum_{n=1}^{\infty}
\frac{e^{-n\beta e(\mu_C- \mu)}}{n^{q/2}}
\;.\label{2.2.20}
\end{equation}
At $\mu=\mu_C$ the sum appearing in this expression for the total charge diverges for $q\le2$. This gives us the necessary and sufficient condition for Bose-Einstein condensation to occur. For $q\ge3$, Bose-Einstein condensation does not occur.
We can obtain more detailed information on how $Q$ diverges as $\mu\rightarrow\mu_C$ as before by making use of (\ref{2.25}). It is easily seen that (\ref{2.2.20}) becomes
\begin{eqnarray}
Q(\mu\rightarrow\mu_C)&=&\frac{ed_0V_q}{2\pi i}\left(\frac{T}{4\pi}\right)^{q/2}
\int_{c-i\infty}^{c+i\infty}d\alpha\Gamma(\alpha)\nonumber\\
&&\times\left(\frac{T}{\mu_C-\mu}\right)^\alpha\zeta_R(\alpha+q/2)
\;.\label{2.2.21}
\end{eqnarray}
This result may be used to show that
\begin{eqnarray}
Q(\mu\rightarrow\mu_C)&\simeq&\frac{eTd_0}{\mu_C-\mu}\quad (q=0);\label{2.2.22}\\
Q(\mu\rightarrow\mu_C)&\simeq&\frac{1}{2}ed_0V_1T(\mu_C-\mu)^{-1/2}\quad (q=1);\label{2.2.23}\\
Q(\mu\rightarrow\mu_C)&\simeq&\frac{eTd_0V_2}{4\pi}\ln\left(\frac{T}{\mu_C-\mu}\right)\quad (q=2).\label{2.2.24}
\end{eqnarray}
Only the leading part of $Q$ which diverges as $\mu\rightarrow\mu_C$ has been shown in these expressions. The way in which $Q$ diverges as $\mu\rightarrow\mu_C$ is seen to be the same for relativistic and nonrelativistic systems.
\section{Application of the criterion to examples}
In this section we will consider several examples of an ideal Bose gas
under possible external conditions and we will see that the conclusion
whether or not Bose-Einstein condensation might occur can be drawn very
easily using our criterion of section 2.
Let us start with the free ideal Bose gas in a $(D+1)$-dimensional Min\-kow\-ski
space-time. Then the eigenvalues of the Klein-Gordon operator (\ref{2.16})
are
\begin{equation}
E_{\vec k}^2 =\vec k ^2 +m^2,\qquad \vec k \in {\mathbb R} ^D.\label{4.1}
\end{equation}
In the non-relativistic case we have
\begin{equation}
E_{\vec k}=\frac{1}{2m}{\vec k}^2,\qquad \vec k \in
{\mathbb R}^D\;.\label{4.1nr}
\end{equation}
In either case we see that $q=D$, and conclude that BEC can only occur for $D\ge3$. This agrees with conclusions of Refs.~\cite{haberweldon81,haberweldon82a,kapusta81} for the relativistic gas, and Refs.~\cite{may59,Ziff} for the non-relativistic gas.
There are many possibilities of restricting the Minkowski space
by imposing boundary conditions in one or more directions. For example
imposing Dirichlet boundary conditions in one direction,
$f_N (x_i=0) =f_N (x_i=L)$, the energy
eigenvalues eq.~(\ref{2.17}) are
\begin{equation}
E_{n,\vec k}^2 = \vec k^2 +\left(\frac{\pi n}{L}\right)^2 +m^2,
\,\,\vec k \in{\mathbb R} ^{D-1},\,\, n\in{\mathbb N} ,\label{4.2}
\end{equation}
for the relativistic field, and
\begin{equation}
E_{n,\vec k}=\frac{1}{2m}\left\lbrace{\vec k}^2+
\left(\frac{\pi n}{L}\right)^2\right\rbrace,\,\,\vec k \in{\mathbb R} ^{D-1},\,\, n\in{\mathbb N} ,\label{4.2nr}
\end{equation}
for the non-relativistic field. In this case we find $q=D-1$ and therefore BEC is only expected for $D\ge4$ (corresponding to $q\ge3$). If we impose Dirichlet boundary conditions in $p$ of the spatial dimensions then $q=D-p$ and BEC would only be expected for $D\ge3+p$. The choice of Dirichlet boundary conditions is not important here, and periodic or Neumann boundary conditions would lead to the same result. The restriction $D\ge3+p$ holds for any choice of boundary conditions which results in a discrete spectrum in $p$ spatial dimensions.
Another important case where the energy spectrum contains a discrete part is found when there is a constant external magnetic field present. Starting with a single
component constant magnetic field $B$ one encounters the well known Landau
levels giving
\begin{equation}
E_{n,\vec k _{\bot}}^2
=(2n+1)eB+\vec k _{\bot}^2+m^2,\,\,n\in{\mathbb N}_0,\,\,
\vec k _{\bot} \in{\mathbb R} ^{D-2},\label{4.3}
\end{equation}
for the relativistic field, and
\begin{equation}
E_{n,\vec k_{\bot}}=\frac{1}{2m}\left\lbrace(2n+1)eB+{\vec k}_{\bot}^2\right\rbrace\,\,\label{4.3nr}
\end{equation}
in the nonrelativistic case. In both cases the eigenvalues are degenerate with degeneracy $(SeB)/(2\pi)$ with the (formal) volume $S$ of
${\mathbb R} ^2$. The single component magnetic field results in $q=D-2$, and condensation can occur only for $D\geq 5$. This was shown for the relativistic field in \cite{daicicfrankelkowalenko94,daicicfrankelgailiskowalenko94} and for the non-relativistic field in \cite{may65}. In particular BEC is absent for $D=3$ \cite{schafroth55}.
In the general case, a magnetic field in $D$ spatial dimensions is characterized by $\delta$ independent components, where $D=2\delta$ or $2\delta+1$ depending upon whether $D$ is even or odd \cite{toms95a,toms94}. It is straightforward to show that
\begin{equation}
E_{n,\vec k _{\bot}}^2=\sum_{j=1}^l (2n_j +1) eB_j +\vec k_{\bot}^2+m^2
,\label{4.4}
\end{equation}
for the relativistic field, and
\begin{equation}
E_{n,\vec k_{\bot}}=\frac{1}{2m}\left\lbrace
\sum_{j=1}^l (2n_j +1) eB_j +\vec k_{\bot}^2\right\rbrace
,\label{4.4nr}
\end{equation}
for the non-relativistic field. In both cases $n_j\in{\mathbb Z}_0$ and $\vec k _{\bot} \in{\mathbb R} ^{D-2l}$. The degeneracy is $\prod_{j=1}^{p}eB_jL_{2j-1}L_{2j}/(2\pi)$ where $B_j\ j=1,\ldots,p$ represent the independent field strengths and $p\le\delta$. ($L_j$ is the length of the box in the $j^{\rm th}$ direction with $L_j\rightarrow\infty$ assumed.) In either case, $q=D-2p$, so that every component of the magnetic field reduces the effective dimension by 2. The criterion for BEC is $D\ge3+2p$ in agreement with the explicit calculations in Refs.~\cite{toms95a,toms94}.
Another example is to consider a field on the space ${\mathbb S}^{p}\times{\mathbb R}^{D-p}$ where ${\mathbb S}^{p}$ is the $p$-dimensional sphere. In this case we have
\begin{equation}
E_{n,\vec k}^2=n(n+p-1)a^{-2}+{\vec k}^2+\xi a^{-2}+m^2\,\label{einrel}
\end{equation}
for the relativistic field, and
\begin{equation}
E_{n,\vec k}=\frac{1}{2m}\left\lbrack n(n+p-1)a^{-2}+\xi a^{-2}+\vec k^2\right\rbrack\;,\label{einnr}
\end{equation}
for the non-relativistic field. $a$ represents the radius of the sphere, and $\xi$ is a dimensionless coupling constant which is related to a possible coupling of the scalar field to the scalar curvature. We have $q=D-p$ here, and BEC requires $D\ge3+p$. In the case of greatest physical interest we have $p=D=3$ which corresponds to the Einstein static universe. BEC does not occur in this case as shown recently in Ref.~\cite{smithtoms}.
In the examples we have discussed so far the energy spectrum has been explicitly known so that a direct evaluation of the thermodynamics has been possible. Our next example is to consider a case where the spectrum is not explicitly
known; namely a single component constant magnetic field in a cylindrical
box of radius $R$, where the field satisfies Dirichlet boundary
conditions \cite{toms94}.
The eigenvalues for this problem are given in implicit form only and the
equation reads
\begin{equation}
M\left( \frac n 2 -\frac 1 4 \gamma , n+1; \frac 1 2 eBR^2\right) =0
\label{4.5}
\end{equation}
with $n\in{\mathbb N}$, $\gamma =(2/eB)[E_N-eB(n+1)-\vec k _{\bot}^2 ]$,
$\vec k _{\bot}^2 \in{\mathbb R} ^{D-2}$,
and $M(a,b;c)$ the confluent hypergeometric function
\cite{gradshteynryzhik65}. The zeroes of (\ref{4.5}) are a discrete
set of values for $\gamma$; thus $q=D-2$ and condensation occurs
for $D\geq 5$, a criterion derived without any problem. The exact analysis of the problem would be impossible due to the lack of an explicit form for the zeros of the confluent hypergeometric function.
A similar situation to the last example occurs if we consider a gas confined by an arbitrarily shaped cavity of dimension $p$
in a space of dimension $D$. It is impossible to write down any explicit form for the energy eigenvalues. If the confinement is achieved by imposing Dirichlet boundary conditions say, then the portion of the energy spectrum coming from the cavity would be discrete and $q=D-p$. A more complete analysis of this problem will be given elsewhere \cite{KKDJTinprep}.
As our final example we consider a non-relativistic gas in a harmonic oscillator potential. If we assume that
\begin{equation}
U({\mathbf x})=\frac{1}{2}m\sum_{j=1}^{p}\omega_j^2x_j^2\;,\label{pot}
\end{equation}
with $p\le D$, then
\begin{equation}
E_{n,\vec k}=\frac{1}{2m}\left\lbrace\vec k^2+\sum_{j=1}^{p}(n_j+\frac{1}{2})\omega_j\right\rbrace\;,\ \ \vec k\in{\mathbb R}^{D-p}\;,\label{spect}
\end{equation}
are the energy levels. We have $q=D-p$ here so that $D\ge3+p$ is
required for BEC. In the case of greatest physical interest $D=3$
it can be seen that BEC does not occur. At this stage it is worth repeating the comment in the last paragraph of Sec.~1. We are interpreting BEC in the sense of symmetry breaking with an associated phase transition, such as that which occurs for the free Bose gas with $D=3$. This does not occur unless $D\ge3+p$. We are not saying that there cannot be a build-up of particles in the
ground state unassociated with a phase transition as the temperature is lowered.
This concludes the list of examples for the relativistic and non-relativistic Bose gas. As we
have seen, known examples and in addition new examples can be dealt
with very easily.
\section{Appearance of the Meissner-Ochsenfeld effect for a general ho\-mo\-ge\-ne\-ous
mag\-net\-ic field}
\subsection{Relativistic scalar field in an external magnetic field}
Let us consider the situation where a general homogeneous magnetic field is
present. The energy eigenvalues then read
\begin{equation}
E_{N}^2 =\sum_{j=1}^p (2n_j+1) \nu_j +\sum_{j=2p+1}^D
\left(\frac{2\pi l_j}{L_j}\right)^2 +m^2 \label{3.1}
\end{equation}
with the limit $L_j =L \to \infty$ taken and $\nu_j =eB_j$. $p$ is the number of non-zero components of the magnetic field as discussed in the last section. The degeneracy
for each Landau level is $(L^2/(2\pi ))^p\prod_{j=1}^p \nu_j $. (This degeneracy corresponds to $d_0$ introduced in Sec.~2.) Thus for the
zeta function we have
\begin{eqnarray}
\zeta (s) &=& \frac{L^D}{2^{D-p}\pi^{D/2}\Gamma (s)}\prod_{j=1}^p \nu_j
\sum_{l=-\infty}^{\infty} \sum_{n_1,...,n_p=0}^{\infty} \times\label{3.2}\\
& &\int\limits_0^{\infty} dt\,\, t^{s-1+p-D/2} e^{\left(\frac{2\pi il}{\beta}
-\mu\right)^2 t -\sum_{j=1}^p (2n_j +1) \nu_j t -m^2 t}.\nonumber
\end{eqnarray}
For the magnetization
\begin{equation}
M_i =-\frac 1 {L^D\beta} \frac{\partial}{\partial B_i } \Gamma \label{3.3}
\end{equation}
we need $(\partial /\partial B_i) \zeta (s)$. (The magnetization and the choice of units is described in the Appendix.)
Using the same kind of steps as for the derivation of
(\ref{2.19}), we find
\begin{eqnarray}
\frac{\partial}{\partial B_i} \zeta (s)
&=&\frac{L^D\beta}{2^{D+1-p}\pi^{\frac{D+1} 2} \Gamma (s)}
\left(\prod_{j=1}^p\nu_j\right)\nonumber\\
&&\quad\times\left\{ -e \Gamma
\left(s-\frac{q-1} 2 \right)\tilde{\zeta} _{d,\Sigma}\left(
\frac{q-1} 2 -s\right)\right.\nonumber\\
& &\quad\left. +\frac 1 {B_i} \Gamma\left( s-\frac{q+1} 2\right)
\zeta_{d,\Sigma} \left(\frac{q+1} 2 -s\right)\right\}\nonumber\\
& &+\frac{L^D\beta}{2^{\frac{D+1} 2 +s} \pi^{\frac{D+1} 2}\Gamma (s)}
\prod_{j=1}^p\nu_j \sum_{l=1} ^{\infty} \sum_{n_1,...,n_p=0}^{\infty} e^{\beta \mu l}\nonumber\\
&&\quad\times \left(\frac{\sqrt{\sigma_{\mathbf p}^d}}{\beta l}\right)^{\frac{q-1} 2 -s}\Bigg\{-e(2n_i+1)\nonumber\\
&&\quad\quad\times K_
{\frac{q-1} 2 -s } (\beta l \sqrt{\sigma_{\mathbf p}^d})\nonumber\\
&&\quad+\frac 2 {B_i} \left(\frac{\sqrt{\sigma_p^d}}{\beta l}\right)
K_{\frac{q+ 1} 2 -s } (\beta l \sqrt{\sigma_{\mathbf p}^d})\Bigg\}
\nonumber\\
& &+(\mu\to-\mu),\label{3.4}
\end{eqnarray}
where we defined
\begin{equation}
\tilde{\zeta} _{d,\Sigma} (s) =\sum_{n_1,...,n_p=0}^{\infty} (2n_i+1)(\sigma_{\mathbf p}^d)^{-s},\label{3.5}
\end{equation}
and abbreviated to $(\mu\to-\mu)$ the terms obtained from replacing $\mu$ in the second set of terms in (\ref{3.4}) with $-\mu$. In order to decide whether or not the Meissner-Ochsenfeld effect takes place we can consider the limit $B_i \to 0$ and see if a nonvanishing
magnetization results. This is equivalent to the generalization of the
Meissner-Ochsenfeld effect from $D=3$ dimensions
\cite{elmforsliljenbergperssonskagerstam95} to arbitrary dimension $D$. When $B_i\rightarrow0$ this removes one of the discrete degenerate Landau levels and replaces it with two additional continuous labels. (See eq.~(4.1).) The effect of this is to increase $q$ by 2 to $q_{eff}=q+2$. Our general analysis showed that the gas could only condense for $q\ge3$. Thus if $q_{eff}\ge3$, or $q=1,2$ there may be interesting behaviour when a magnetic field is present even though strict BEC cannot occur. The case $q=1$ in the limit $B_i\rightarrow0$ can be compared with the gas for $B_i=0$ with $q=3$; the case $q=2$ in the limit $B_i\rightarrow0$ can be compared with the gas for $B_i=0$ with $q=4$. In either case we wish to study the behaviour of the magnetization as $\mu$ becomes close to the critical value $\mu_c$. As argued in our criterion for BEC, the leading term comes from the smallest energy eigenvalue and it reads
\begin{eqnarray}
M_i (B_i \to 0)&=&\frac{e\beta^{-q/2}}{2^{\frac{D+2} 2 } \pi^{D/2}}
\left(\prod_{j=1}^p\nu_j \right) \mu ^{\frac{q-2} 2}\nonumber\\
&&\quad\times \sum_{l=1}^{\infty} l^{-q/2}
e^{-\beta l (\mu_C-\mu)},\label{3.6}
\end{eqnarray}
which is very similar to the behaviour of the charge, Eq.~(\ref{2.24}).
We see that for $q\geq 3$ the sum is convergent, even for $\mu =\mu_C$, and
thus $M_i (B_i \to 0) =0$ for that case. However, for $q=1,2,$ we find
using once more (\ref{2.25}),\\
\\
$q=1$
\begin{equation}
M_i (B_i \to 0) =\frac{eT}{2^{\frac{D+2} 2 } \pi^{\frac{D-1} 2}}
\left(\prod_{j=1}^p\nu_j\right)\frac 1 {\sqrt{\mu_C} \sqrt{\mu_C-\mu}},\label{3.7}
\end{equation}
$q=2$
\begin{equation}
M_i (B_i \to 0) =-\frac{eT}{2^{\frac{D+2} 2 } \pi^{\frac{D} 2}}
\left(\prod_{j=1}^p\nu_j\right)\ln (\mu_C-\mu ).\label{3.8}
\end{equation}
For $q=0$ in the limit $B_i \to 0$ the effective dimension gets increased to $q=2$ and
thus no condensation can occur. Thus $\mu < \mu_C$, the sums are convergent
and though $M_i (B_i \to 0) =0$ in that case in the given approximation.
The magnetization laws, Eqs.~(\ref{3.7}) and (\ref{3.8}), strongly
depend on how $\mu\to\mu_C$ for $B_i\to 0$. It is clear, that the
charge in the continuum above the lowest Landau level will condense into
the ground state in the limit $B_i \to 0$. Let us use the notation
$Q_{gr}(\mu\to\mu_C)$ for that charge. This is however exactly the charge
given in Eqs.~(\ref{2.28}) and (\ref{2.29}), and the limit
$M_i (B_i \to 0)/Q_{gr} (\mu\to\mu_C)$ is seen to be finite. If we introduce the charge density $\rho_{gr}=Q_{gr}/L^D$, it reads for $q=1,2$,
\begin{equation}
\frac{M_i (B_i \to 0)}{\rho_{gr} (\mu\to\mu_C)} =-\frac e {2\mu_C}.\label{3.9}
\end{equation}
Using the universal knowledge of the charge $Q_{gr}$ in the ground state
as a function of the critical temperature \cite{toms92,toms93},
\begin{equation}
Q_{gr} =Q \left(1-\left(\frac T {T_c}\right)^ {D-1} \right)\label{3.10}
\end{equation}
with the total charge $Q$, and total charge density $\rho=Q/L^D$, we arrive at
\begin{equation}
M_i (B_i \to 0) =-\frac e {2\mu_C} \rho
\left(1-\left(\frac T {T_c}\right)^ {D-1} \right),
\label{3.11}
\end{equation}
with $\mu_C =\sqrt{m^2+\sum_{j=1,j\neq i}^p \nu_j}$ as the magnetization law
for $q=1,2$.
For $q=1$ in three spatial dimensions ($D=3$) the result reduces to the one of
ref.~\cite{elmforsliljenbergperssonskagerstam95}, and we agree with their
conclusions on the Meissner-Ochsenfeld effect.
\subsection{Non-relativistic scalar field}
The energy levels for a $p$-component constant magnetic field are
\begin{equation}
E_{n,l}=\sum_{j=1}^{p}(2n_j+1)\frac{eB_j}{2m}+\frac{1}{2m}
\sum_{j=2p+1}^{D}\left(\frac{2\pi l_j}{L_j}\right)^2\;,
\end{equation}
where $n_j=0,1,\ldots$, and $L_j=L\rightarrow\infty$ as before. The degeneracy is $L^{2p}\prod_{j=1}^{p}eB_j/(2\pi)$. The effective action is determined as described in Sec.~2.2, and the magnetization found as in (4.3). We can again argue that $q=1,2$ are the interesting values as $B_i\rightarrow0$ since in this limit we obtain $q_{eff}=q+2=3,4$ for which BEC is possible.
The lowest state contribution is of greatest interest in this limit and we find
\begin{eqnarray}
M_i(B_i\rightarrow0)&\simeq&-\frac{e}{2m}
\left(\frac{m}{2\pi\beta}\right)^{q/2}
\left(\frac{eB_j}{2\pi}\right)\nonumber\\
&&\quad\times\sum_{n=1}^{\infty}
\frac{e^{-n\beta e(\mu_c-\mu)}}{n^{q/2}}\;,\label{star}
\end{eqnarray}
as the leading contribution. (Note that $\mu_c=\frac{1}{2m}\sum_{j=1}^{p}eB_j$ here.) Apart from a numerical prefactor, this is the same as in the relativistic case in (4.6). For $q\ge3$ the sum converges and we find $M_i\rightarrow0$ as $B_i\rightarrow0$ even if $\mu=\mu_c$. The sum occurring in (\ref{star}) is the same as that occurring in the ground state charge (2.51).. If we write $\rho_{gr}=Q(\mu\simeq\mu_c)/L^D$ as the ground state charge density we find
\begin{equation}
M_i(B_i\rightarrow0)\simeq-\frac{\rho_{gr}}{2m}\;.
\end{equation}
This is essentially the same as the result found by Schafroth \cite{schafroth55} for $D=3$.
\section{Conclusions}
In this article we considered the general setting of a quantum field,
relativistic or nonrelativistic, without self-interactions under general
external conditions. We tried to answer the question whether or not the system
might undergo a Bose-Einstein condensation, where by Bose-Einstein condensation we mean a phase transition associated with a build-up of particles in the ground state. Our main result is that the crucial feature governing Bose-Einstein condensation is the dimension
$q$ associated with the continuous part of the eigenvalue spectrum of the
Hamiltonian for nonrelativistic systems or the spatial part of the Klein-Gordon
operator for relativistic systems. In either case Bose-Einstein condensation
can only occur if $q\geq 3$. Having this criterion at hand, many of previous results
were obtained easily in section 3. In addition some new applications were given. The criterion was shown by studying the lowest eigenvalue of a differential
operator. This relates to the idea of the effective infrared dimensions
and finite size effects studied earlier \cite{Hu}.
Furthermore, we applied our ideas to the appearance of the Meissner-Ochsenfeld
effect for a general homogeneous magnetic field. With the help of our
criterion a discussion of this effect is greatly simplified and special
cases are easily recovered
\cite{schafroth55,elmforsliljenbergperssonskagerstam95,daicic}.
The extension of our method to study Bose-Einstein condensation in theories with
self-interaction
\cite{bensonbernsteindodelson91,bernsteindodelson91} is of obvious interest.
Recently we have shown how this
problem can be tackled very efficiently using $\zeta$-function methods
\cite{kirstentoms95}. It is very likely that the method described in the present paper
can be extended to interacting field theory. In particular it is possible
to define an effective field theory describing the lowest modes
as in ref. \cite{Hu}.
\section*{Acknowledgments}
The work of KK is supported by the DFG under contract number Bo 1112/4-1.
|
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O Forte de Miranda localizava-se no Presídio de Miranda, à margem direita do rio M'boteteí (atual rio Miranda), afluente da margem esquerda do rio Paraguai, hoje cidade de Miranda, no estado de Mato Grosso do Sul, no Brasil.
História
O forte remonta a um Presídio (colônia militar) fundado em 1797 pelo governador e capitão-general da capitania de Mato Grosso, Caetano Pinto de Miranda Montenegro (17??-1804) (SOUZA, 1885:139), povoado com indígenas de várias etnias, principalmente Terena (SILVA, 2001). Para a sua defesa, foi erguido um reduto de planta poligonal quadrangular, com um redente ao centro de cada face. O seu plano mostra os edifícios de serviço distribuídos no interior terrapleno, ao abrigo das muralhas, erguidas com estacas de madeira e terra apiloada, com um fosso defendendo o perímetro externo. Este Presídio estava sob o comando do Tenente de Dragões Francisco Rodrigues do Prado, quando por determinação de Ricardo Franco de Almeida Serra, as suas forças contra-atacam o Fortim de San José de Apa, assegurando, na Guerra de 1801, em definitivo para Portugal a fronteira sul do Mato Grosso (MELLO, 1963:28).
Neste Presídio funcionou, de fevereiro de 1799 a fevereiro de 1801 pelo menos, um Armazém Real (Armazém Real do Presídio de Miranda), um depósito de armas, munições, fardamentos, ferramentas, alimentos, equipamentos náuticos e tudo o mais necessário ao uso das forças militares da Coroa Portuguesa ou mesmo das suas repartições civis.
SOUZA (1895), menciona que a exploração do major Luiz d'Allincourt em 1826 encontrou essa fortificação aberta e arruinada (op. cit., p. 139).
De acordo com TAUNAY (s.d.), existiu neste povoado "…sito a 210 quilômetros a NO de Nioac" (op. cit., p. 41) "(…) sobre o local de velha fortificação, outrora bem construído quartel, então muito deteriorado pelo fogo (…)" (op. cit., p. 33), incendiado, como a vila, pelas forças invasoras paraguaias no contexto da Guerra da Tríplice Aliança (1864-1870).
Não confundir o Presídio oitocentista que originou a vila de Miranda com a Colônia Militar de Miranda, erguida pelo Império também na região, à época da Guerra da Tríplice Aliança (1864-1870) também envolvida no episódio da Retirada da Laguna, mas localizada 80 quilômetros SSO de Nioac (TAUNAY, s.d.:41).
Bibliografia
BARRETO, Aníbal (Cel.). Fortificações no Brasil (Resumo Histórico). Rio de Janeiro: Biblioteca do Exército Editora, 1958. 368p.
FERRAZ, Antônio Leôncio Pereira. Memória sobre as Fortificações de Mato Grosso (Separata da Revista do Instituto Histórico e Geográfico do Brasil). Rio de Janeiro: Imprensa Nacional, 1930.
GALLO, José Roberto (Arq.). Fortificações de Mato Grosso do Sul. Campo Grande: 8º DR/IPHAN/FNPM/MinC Escritório Técnico/MS, mar. 1986.
GARRIDO, Carlos Miguez. Fortificações do Brasil. Separata do Vol. III dos Subsídios para a História Marítima do Brasil. Rio de Janeiro: Imprensa Naval, 1940.
LEVERGER, Augusto (Almte.). Apontamentos para o Diccionário Chorografico da Província do Mato Grosso. RIHGB. Rio de Janeiro: Tomo XLVII, Partes I e II, 1884. p. 307-504.
SILVA, Jovam Vilela da. A lógica portuguesa na ocupação urbana do território mato-grossense. História & Perspectivas. Uberlândia: nº 24, jan.-jun. 2001.
SOUSA, Augusto Fausto de. Fortificações no Brazil. RIHGB. Rio de Janeiro: Tomo XLVIII, Parte II, 1885. p. 5-140.
TAUNAY, Alfredo d'Escragnolle. A Retirada da Laguna. s.l.: Edições Melhoramentos, s.d..
Ver também
Lista de fortificações no Mato Grosso do Sul
Ligações externas
Miranda
Miranda (Mato Grosso do Sul)
Fundações no Brasil em 1797
|
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Harry Reid Calls Trump "a Sexual Predator Who Lost the Popular Vote"
Read his no-holds-barred attack on the president-elect.
Patrick Caldwell
News EditorBio | Follow
Gary Cameron/ZUMA
Sen. Harry Reid, on his way out as Senate minority leader, released a blistering statement condemning President-elect Donald Trump Friday morning. "The election of Donald Trump has emboldened the forces of hate and bigotry in America," said Reid. Reid attacked Trump for inciting fear among a wide swath of Americans whom he insulted during his presidential campaign, including African Americans, Latinos, Muslims, and the LGBT community.
Reid, the Democrats leader in the Senate since 2005, didn't run for reelection this year and will be leaving Congress when his term expires in January. The election of Catherine Cortez Masto to replace Reid was one of the lone bits of bright news for Democrats this week. But Reid struck a dire note in his statement on Trump's election, warning that the next president should not be normalized as just a regular politician now, and that the fear among these minority groups is "entirely rational, because Donald Trump has talked openly about doing terrible things to them."
"If this is going to be a time of healing," Reid says, "we must first put the responsibility for healing where it belongs: at the feet of Donald Trump, a sexual predator who lost the popular vote and fueled his campaign with bigotry and hate. Winning the electoral college does not absolve Trump of the grave sins he committed against millions of Americans. Donald Trump may not possess the capacity to assuage those fears, but he owes it to this nation to try."
Read Reid's full statement below:
"I have personally been on the ballot in Nevada for 26 elections and I have never seen anything like the reaction to the election completed last Tuesday. The election of Donald Trump has emboldened the forces of hate and bigotry in America.
"White nationalists, Vladimir Putin and ISIS are celebrating Donald Trump's victory, while innocent, law-abiding Americans are wracked with fear—especially African Americans, Hispanic Americans, Muslim Americans, LGBT Americans and Asian Americans. Watching white nationalists celebrate while innocent Americans cry tears of fear does not feel like America.
"I have heard more stories in the past 48 hours of Americans living in fear of their own government and their fellow Americans than I can remember hearing in five decades in politics. Hispanic Americans who fear their families will be torn apart, African Americans being heckled on the street, Muslim Americans afraid to wear a headscarf, gay and lesbian couples having slurs hurled at them and feeling afraid to walk down the street holding hands. American children waking up in the middle of the night crying, terrified that Trump will take their parents away. Young girls unable to understand why a man who brags about sexually assaulting women has been elected president.
"I have a large family. I have one daughter and twelve granddaughters. The texts, emails and phone calls I have received from them have been filled with fear – fear for themselves, fear for their Hispanic and African American friends, for their Muslim and Jewish friends, for their LBGT friends, for their Asian friends. I've felt their tears and I've felt their fear.
"We as a nation must find a way to move forward without consigning those who Trump has threatened to the shadows. Their fear is entirely rational, because Donald Trump has talked openly about doing terrible things to them. Every news piece that breathlessly obsesses over inauguration preparations compounds their fear by normalizing a man who has threatened to tear families apart, who has bragged about sexually assaulting women and who has directed crowds of thousands to intimidate reporters and assault African Americans. Their fear is legitimate and we must refuse to let it fall through the cracks between the fluff pieces.
"If this is going to be a time of healing, we must first put the responsibility for healing where it belongs: at the feet of Donald Trump, a sexual predator who lost the popular vote and fueled his campaign with bigotry and hate. Winning the electoral college does not absolve Trump of the grave sins he committed against millions of Americans. Donald Trump may not possess the capacity to assuage those fears, but he owes it to this nation to try.
"If Trump wants to roll back the tide of hate he unleashed, he has a tremendous amount of work to do and he must begin immediately."
Trump Election Protests Continue to Erupt Across the Country
|
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Q: Multiple line plotting in one graph - how to automatically adjust the x- and y-axis to find all graphs Following from my previous question here
R: Select vector (numeric) from data frame, sample n=10 subsets of size i=5 and i= 10 within vector and calculate mean for each of these samples
I want to plot the distribution of the resulting 2 vectors. I am using epdfPlot to create one density and then the second one adding it to the plot, however, I have the problem that the xlim and ylim are adjusted for graph I plot first but not for the graph(s) added.
epdfPlot(as.numeric(unlist(vector1)),
curve.fill = TRUE,
curve.fill.col = rgb(1, 0, 0, alpha = 0.5),
epdf.col = "red",
)
epdfPlot(as.numeric(unlist(IRR_mean_diversified)),
curve.fill = TRUE,
curve.fill.col = rgb(0, 0, 1, alpha = 0.5),
epdf.col = "blue",
add = TRUE)
I tried with min/max function of the values, however, that truncates the graphs.
Thank you!
|
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| 7,750
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Q: Access to page timeline with depreciated offline_access I develop flash websites that feature posts from a Facebook page. Since I'm not accessing a visitor's account, I don't want them to go through the oauth process. I only want one feed in Json form, but that feed is a page and not a specific user. I've gotten around this by writing a script that I visit to grant an offline_access token. I use that token to access the one feed I need. On page load, I use the graph API Json URL to get the feed and parse the data in flash.
Now that offline_access is going away, I'm trying to find the best way to access the feed in raw json form that will work in flash.
I have read this page (http://developers.facebook.com/roadmap/offline-access-removal/), and can't find a scenario that helps me as all flows require a user to access an app of some kind. Does this change mean that I can no longer access a page feed without the process being transparent to the visitor? If need be I can be asked to be an admin of the pages I need access to, if that helps.
A: That document outlines all your options -
*
*If acting as the page itself you can get a token which won't expire - Scenario 5 in the document
*If acting on behalf as a user you need them to come back at least every 60 days – Scenario 2 or 3 in the document
|
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| 1,501
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A Lua implementation of some of the Python itertools library.
This is being created as a learning excercise in Lua. Therefore it is subject to breaking changes without notice.
## Dependencies
Tests are written using [busted](http://olivinelabs.com/busted/)
## Usage
Add itertools.lua to you project directory and require it.
```lua
local itertools = require 'itertools'
```
## Testing
Once [busted](http://olivinelabs.com/busted/) is installed, simply run ```$ busted``` from the project directory.
## License
MIT License
See LICENSE for details.
|
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| 9,197
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{"url":"https:\/\/www.physicsforums.com\/threads\/effective-potential.305214\/","text":"# Effective Potential\n\n1. Apr 5, 2009\n\n### roeb\n\n1. The problem statement, all variables and given\/known data\n\nI am trying to do problem 5, I seem to be having a hard time with the algebra.\nhttp:\/\/img24.imageshack.us\/img24\/4224\/landau.th.png [Broken]\n\n2. Relevant equations\n\n3. The attempt at a solution\n\nTo find maximum value of the effective potential we just do: dU\/dr = 0.\nI get (which I have verified to be correct)\n$$r_m^{n-2} = \\frac{n*\\alpha }{m \\rho^2 v^2}$$\n\nI rearrange that into\n$$r_m = (\\frac{n*\\alpha }{m \\rho^2 v^2})^{1\/(n-2)}$$\n\nPlugging that into the Ueff equation for r, I get (and simplifying a bit, let x = n\/(n-2) ).\n\n$$\\frac{m^x v^{2x} \\rho^{2x}}{2 n^{2x\/n} \\alpha^{2x\/n}} - \\frac{\\alpha m^x v^{2x} \\rho^{2x}}{n^x \\alpha^x}$$\n\nAt this point I kind of get stuck, I can do the following, but it never seems to turn out 'right'.\n\n$$\\frac{m^x v^{2x} \\rho^{2x}}{\\alpha^x n^x} ( \\frac{1}{2 \\alpha^{2\/n} n^{2\/n}} - a)$$\n\nAnyone know what I am doing wrong? it must be something simple because I am getting very close to the correct answer, it's just those terms on the right don't seem to become 'nice'.\n\nLast edited by a moderator: May 4, 2017\n2. Apr 5, 2009\n\n### roeb\n\nWhoops, I seem to have figured it out, apparently it's n^(-1) not n^2\/n","date":"2017-11-22 23:10:48","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.8035032153129578, \"perplexity\": 772.3055983981805}, \"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-2017-47\/segments\/1510934806676.57\/warc\/CC-MAIN-20171122213945-20171122233945-00748.warc.gz\"}"}
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Q: How can we handle threshold limit of SharePoint? SharePoint is configured with a default threshold limit of 5000 items in a list or library. This means that if the items in a list or library exceed that number, any views, grouping and queries will fail, So there is any work-around/solution to handle this issue of SharePoint by API? and Why SharePoint set such a rule?
A: SharePoint list view threshold limitation is used to make sure SharePoint high Performance and stable.
You could use Indexed Column in CAML Query or Rest API to handle this limitation:
SharePoint List View Threshold: Your FAQs Answered
A: Normally we dont need huge amount of data on a single page. It will be use less for a user as he cant see all.
But we can use as workaround to fetch data in chunks and append array of data in loop. Check the below blog. i hope it may help you.
https://abdulazizfarooqi.wordpress.com/2016/03/20/sharepoint-defeat-5000-threshold/
Another way is to use index column as mentioned already, use more filters in your query to get the less amount of data which may be useful for end user rather just bring all at once.
In SPFX we can use react component paging to fetch data as per page using rest api.
|
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"redpajama_set_name": "RedPajamaStackExchange"
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As a shop, the staff dedicate themselves to offer the best service to their clients while maintaining very competitive prices. At Royal Pet Store you will find all major brands of pet food and also a large selection of pet toys, care products and daily needs.
Need more info? Don't see a brand in this list just ask if it is in stock.
At Royal Pet Store, one can find various Tropical and cold water fish, Fish food, Medications and Vitamins. One can also find Fish tanks and Aquariums, Aquarium decorations, Gravel Substrates, Stands and Furniture. For the upkeep of your aquarium they have Aquarium equipment available such as Testing equipment, Maintenance equipment and pond supplies too. For the upkeep of water they also sell Filtration, Water Conditioners and Additives.
At Royal Pet Store, the staff can help in finding the right Bird Accessories, Cages, Carriers and Furniture. One can also find Food, Health Care products such as Vitamins and Supplements.
At Royal Pet Store, you can find Cat Accessories, beds, Food and Health care products. One can also find Cat Litter, Repellents and training is provided, Stain & odour Control products.
Royal Pet Store have various dog accessories and beds, dog food, Health care products, Stain and Odour Control Products are also available. Training and Behaviour is also provided by Royal Pet Store.
Royal Pet Store also have various reptiles such as turtles, tortoises, snakes, geckos, frogs, crabs and much more. They also have accessories, equipment, stands and furniture. At Royal Petstore one can also find Food and health care products.
At Royal Pet store, one can also find ferrets, guinea pigs, hamsters and rabbits and much more. They also have Bedding, Litter, Food, Health care products and much more.
Pet Grooming Services at Royal Pet Store are also available for Birds, Cats, Dogs and Small animals.
Royal pet store focuses mainly on ensuring you have all the supplies you need for your pets, whilst they do stock a variety of pets and the list above is simply a quick sample of the type of pets they stock, they can also point you in the right direction if they do not have the type of pet you are looking for in stock at the time.
|
{
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| 5,953
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SkyscraperPage Forum > Global Projects & Construction > Proposals
NEW YORK | 247 Cherry Street | 1,013 FT | 77 FLOORS
Posted Jun 12, 2019, 12:35 PM
ψ Sailor of the Mind ψ
Location: Riverview Estates Fairway (PA)
What kind of judge is that? Aren't judges suppose to go with the rule of law and especially pertaining to land, the codes and conducts. These towers are perfectly legal. This is turning into a mockery. A sham, and a waste of time. These hacks have nothing better to do than to stifle positive growth. All of these affordable units at risk, yet they rather the area remain a taller version of the Queensbridge Projects.
The LES towers will improve the area, not degrade it.
NEW YORK | Mid/Low Rise Compilation and Updates ✰✰✰✰✰
NEW YORK CITY | Highrise Compilation and Directory 2021 Version ✪✪✪✪✪
YIMBY Forums 🗽
Find all posts by chris08876
Posted Jun 12, 2019, 1:22 PM
citybooster
Originally Posted by chris08876
Totally agree, and just hope five years from now we will be seeing construction at least some of these sites and really bring life to an area that can use it.
Find all posts by citybooster
NYguy
New Yorker for life
Location: Borough of Jersey
Originally Posted by yankeesfan1000
Then what's the point of zoning...?
What kind of judge is that? Aren't judges suppose to go with the rule of law and especially pertaining to land, the codes and conducts. These towers are perfectly legal. This is turning into a mockery. A sham, and a waste of time.
Exactly. They not only want to go and rezone specific areas to block development, they want to ignore existing zoning as well. That's why ultimately, the judges ruling is a temporary one, and would be reversed if it were a final one. But in the end, a stall tactic is what the NIMBYs want, as it would give them more time to try to rezone the area.
NEW YORK heals.
"Office buildings are our factories – whether for tech, creative or traditional industries we must continue to grow our modern factories to create new jobs," said United States Senator Chuck Schumer.
Find all posts by NYguy
Posted Jun 13, 2019, 10:31 AM
giantSwan
Location: Northeast, United States
Originally Posted by NYer34
Once again, New York Shitty rears its head!
We demolish beautiful buildings (Hotel Pennsylvania, 417 Park, 5-9 East 51st, 16 East 52nd, any number of entire blocks on the UES) ...
... for trash ("Facebook Tower" (just wait till Congress breaks it up!), 417 Park, Tower Fifth) ...
... While proposals to build on some of the city's ugliest, least historically significant, vacant, pedestrian-unfriendly lots (247 Cherry and its neighbors) ...
... Get derailed.
Find all posts by giantSwan
Sure, sure....
https://www.crainsnewyork.com/letter...ht-its-process
Boro prez on tower plan: It's not just the height—it's the process
Nowhere does the editorial "Tower foes' disregard for zoning laws is the height of irresponsibility" (June 11) mention that the siting of three out-of-scale, predominantly luxury developments is in the midst of the Two Bridges large-scale residential development (LSRD) area, which was created pursuant to a special permit. This glaring omission would lead readers to believe that the proposed developments were as-of-right, which is completely untrue.
The special permit that created the LSRD contained findings and site plans which called for a low-to-moderate-rise neighborhood of low-to-moderate-income residents–and for distribution and placement of buildings which "will not unduly increase the bulk of buildings, density of population or intensity of use in any block, to the detriment of the occupants of buildings in the block or nearby blocks…" That's why the tallest building constructed since the LSRD was created is 27 stories.
The current site plan for the Two Bridges LSRD—based on the findings—clearly conforms to the planned character of the current neighborhood. The City cannot replace that plan with a new one without a new public review process—let alone one which inserts four mostly-luxury skyscrapers (one development has two towers), triples the number of residents, and more than doubles the height of surrounding buildings in one fell swoop. ULURP is required.
The "details" that lead inescapably to this conclusion were nowhere to be found in your editorial, which was big on alarm and short of substance.
Posted Jun 29, 2019, 3:43 AM
\Its time that these anti-development hacks get out of office. City needs change. Might be time to try something new.
A city with a housing crisis should not have local politicians against housing and affordable units.
If you provide schools, they complain. If you provide 100's of affordable units, they complain. If you offer public improvement plans, they complain. Yah just can't win.
There is nothing wrong with the process, but instead, its the people calling the shots. NYC has been doing quite well way before these hacks came into office and power, and we can't stifle decades of economic growth, housing additions all of a sudden. The current city administration needs to go, and I hope the voters try something new.
Under Bloomberg, the city was great. Under DeBlasio, in the beginning, it was okay... but now... now it seems like a mockery, with a mayor that is to busy striving to get less than 5% of the vote, which he will get, if not less. He's pulling a Chris Christie, and abandoning the city for his own selfish interests.
Posted Jul 18, 2019, 2:22 PM
matt19215
The NIMBY's are at it again
https://therealdeal.com/2019/07/18/c...into-jeopardy/
An appellate court has ruled that the owner of a long-term ground lease at the site of a controversial Two Bridges development should have a say in how the land is used.
The owner, Little Cherry LLC, has 25 years left on a ground lease at the 235 Cherry Street site, according to The City. They have so far refused to give consent to the development, which calls for constructing a 1,000-foot tower above the one-story building on their site, as well as a senior residence next door.
In 2018, the city granted approval to JDS Development, L+M Development, CIM Group and Starrett Corporation to build rental towers at 247 Cherry Street, 260 South Street and 259 Clinton Street.
However the ruling may cause problems for the development, which has faced a string of earlier challenges. In May, residents filed a petition against the city, the City Planning Commission, the Department of Buildings and the Department of City Planning, alleging the approval of the project was illegal. The City Council also filed a petition objecting to the approval.
JDS spokesperson, Marci Clark, told The City that the company would "continue to defend the case vigorously," and that "the case nor the ruling will impact the development in any meaningful way."
However Little Cherry's attorney, Raymond Hannigan, saw the ruling differently, describing it as "a serious decision putting a roadblock in the way of Mr. Stern's development."
"He can't build his megaproject without getting our client's consent," Hannigan said. "And we're refusing to give our consent."
Find all posts by matt19215
Location: Brooklyn, NYC/Polanco, DF
Ugh, the endless delays from the pandering politicans and their sycophants.
This is why it takes 20 years and 3x the cost of other cities to get anything built.
Find all posts by Crawford
Zapatan
DENNAB
Location: NA - Europe
Can they actually do anything besides bitch and complain? If not then it's all good.
Find all posts by Zapatan
Originally Posted by Zapatan
That's what they do, they bitch because thats their nitche but to all of us its like April fools
and it may come across as rude, but what do you expect from a bunch of fools...
who know nothing about what makes cities cool. And so they drool like a bunch of ghouls over old school zoning that has been in place since they were in pre school.
... And so they ridicule to give existence to their pitiful lives over developments that would provide benefits to those who need affordable housing as means of medicine.
Posted Jul 25, 2019, 3:57 AM
https://citylimits.org/2019/07/24/tw...suit-rezoning/
LES Groups Try to Revive Rezoning that De Blasio Admin. Rejected
By Sadef Ali Kully
As Two Bridges community groups and the City Council prepare to reconvene in Manhattan Supreme Court next week to face the city and developers over the contentious proposal for four waterfront towers, some community organizations have filed a rezoning application with the Department of City Planning.
This month, community groups Tenants United Fighting for the Lower East Side (TUFF-LES), CAAAV: Organizing Asian Communities and Good Old Lower East Side (GOLES) filed a rezoning application with DCP reviving a portion of an older community rezoning plan for Chinatown and Lower East Side.
"It was a strategy we developed two and a half years ago. It's based on the Chinatown Working Group. So it's kind of exciting to plan and see it come to fruition this way. It has all been very deliberate. This is not by luck," said Melanie Wang, Chinatown Tenant Union Lead Organizer for CAAAV.
The rezoning proposal for Two Bridges takes a portion of the CWG rezoning plan applying to the Lower East Side Waterfront, called subdistrict D. The proposed rezoning would limit heights to 350 feet for new development and it would require up to 55 percent guaranteed on-site permanently affordable housing. Hotels, big box stores, and clubs would require a special permit with the requirement for public review. And it would also require community facilities such as schools, supermarkets or nonprofit recreation centers.
The land use application must be submitted and then certified by the Department of City Planning. The City Planning Commission must approve the application and then the application goes to the City Council for a final vote. According to DCP, there is no difference between public and private rezoning applications: Both must complete a land use application, undertake the appropriate environmental review, and follow the ULURP process.
"[City Planning staff] have accepted our pre-application statement. They declared it complete. Now they have assigned staff from their environmental unit to work with us on the environmental review component. But we were the ones drafting it and their job is to give us feedback," says Paula Segal, attorney with the Equitable Neighborhoods Practice at the Community Development Project, who is working with the community groups on the rezoning application and also represents the TUFF-LES group lawsuit against the proposed Two Bridges development.
CAAAV, TUFF-LES and GOLES are now looking to engage the community further and update residents about the community plan while they pursue lawsuits against the proposed Two Bridges development.
On August 2nd the parties will meet again in Manhattan Supreme Court to either hear the judge's decision or have the case adjourned.
Their lawsuit concerns a joint application for a minor modification filed by four developers would allow three new mixed-use high-rise buildings, a 1,008-foot rental tower at 247 Cherry Street by JDS Development Group; a 798-foot dual-tower project at 260 South Street by L+M Development Partners and CIM Group; and a 730-foot building at 259 Clinton Street by Starrett Corporation.
According to environmental review documents, the four towers would bring in 11,000 square feet for retail and over 2,700 new residential units to the area; 25 percent of those units will be affordable. Two hundred of those 690 affordable units would be set aside for seniors (although the details of how affordability will be measured has not been shared).
Earlier this month, a state appellate court panel of judges ruled that developers of the proposed 1,008-foot tower at 235-247 Cherry Street must obtain tenant Little Cherry LLC's consent in order to proceed with their development project. It was another win for opponents.
"Defendants failed to establish, as a matter of law, that plaintiff is not a party in interest whose consent is required for the zoning lot merger. A ground lease tenant has an interest in a tract of land akin to the fee owner. Plaintiff in this case identified multiple adverse effects of the zoning lot merger that plaintiff be adversely affected," the judges ruled, according to court records.
In an emailed statement, Ray Hannigan, attorney at Herrick's Litigation Department, who represents Little Cherry, said the ruling means the developers "must obtain consent from Little Cherry of the developer's plans, which Little Cherry does not intend to give."
The Best Forumer
this should be a good thiing.
The suburbs are second-rate. Cookie-cutter houses, treeless yards, mediocre schools, and more crime than you think. Do your family a favor and move closer to the city.
Find all posts by The Best Forumer
Posted Aug 3, 2019, 1:54 AM
https://www.nytimes.com/2019/08/02/n...velopment.html
Plan for Supertall Tower Looming Over Lower East Side Is Halted, for Now
The ruling was an unusual setback for developers planning luxury skyscrapers in neighborhoods dotted with low-rise building
By Matthew Haag
The justice, Arthur F. Engoron of State Supreme Court, overruled a city agency's approval of the project in 2016, ordering the developers to essentially start over and go through the city's lengthy and arduous public review process.
The justice's searing opinion echoed what many opponents have said about similar super-tall buildings in Manhattan: They overwhelm the neighborhood, disrupt the character of the area and displace less wealthy residents.
"The irreparable harm here is twofold," Justice Engoron wrote in his opinion. "First, a community will be drastically altered without having had its proper say. Second, and arguably more important, allowing this project to proceed without the City Council's imprimatur would distort the City's carefully crafted system of checks and balances."
The ruling does not outright kill Two Bridges project. The public review, however, will delay construction, require developers to seek additional community input and make the buildings' future less certain.
The City Council, whose members filed the lawsuit against the project along with the Manhattan borough president, Gale A. Brewer, will have the final say about Two Bridges at the end of the public review period.
"We're very grateful that the State Supreme Court agreed and that the community, the Borough President, and the City Council will have an opportunity to provide real input and help shape the future of this neighborhood," the council speaker, Corey Johnson, said in a statement.
The developers — JDS Development Group, Starrett Development and a joint venture between CIM Group and L+M Development Partners — said they would appeal the ruling.
"These projects were lawfully approved and met all legal requirements," James Yolles, a spokesman for the group, said in a statement. "They were proposed after years of community consultation, public review and environmental analysis, and in compliance with zoning that's been in place for more than 30 years."
The Two Bridges development would also include upgrades to a public-housing complex and a local subway station. The city's Law Department, which defended the city in the lawsuit, said it was "considering the city's legal options."
Posted Aug 3, 2019, 7:21 PM
Such a shame, NIMBYs
urbanview
Location: Kampala
Don't like the activist judge, but not too sad because these buildings are not very attractive overall IMO.
Last edited by urbanview; Aug 4, 2019 at 9:55 AM.
Find all posts by urbanview
I wonder if this is going to cause these towers to be halted until the next cycle (The LES Towers)
The Lower East Side has the Largest Surplus of Unsold Luxury Condos, Thanks to One Manhattan Square
The city currently has a problem unloading the luxury condos flooding the market.
The New York Times last week published a piece about the glut of unsold apartments in the luxury stratosphere. That, despite the bonkers pace of construction in the high-end market, supply is outstripping demand. The data, culled from a StreetEasy audit, found that roughly 25-percent of the more than roughly 16,000 new luxury units built since 2013 are still available.
Put another way, one in four are unsold.
According to the report, Manhattan had the largest surplus, with over 2,400 of unsold luxury units, roughly 60-percent. But it is the Lower East Side that leads the city in fewest new condos sold, thanks solely to Extell's giant obelisk blocking the Manhattan Bridge – One Manhattan Square. The 80-story skyscraper, which forever ruined the skyline, hasn't been able to shift its 815-unit stock. To date, only 173 condos have closed, or about 21-percent.
As the Streeteasy study notes, "with a median price of $1.1 million citywide, and more than $2.3 million within Manhattan, these new condos remain out of reach of most New York home buyers."
https://www.boweryboogie.com/2019/09...hattan-square/
Highly unlikely, as these are rental towers. The rental market is booming, the condo market is slow.
This one also has a large component of affordable housing, be that as it is.
Zerton
One Manhattan Square. The 80-story skyscraper, which forever ruined the skyline...
Ugh, that would be the Verizon Building. One Manhattan Square is really nice.
If all others accepted the lie which the Party imposed, if all records told the same tale, then the lie passed into history and became truth. -Orwell
Find all posts by Zerton
Camstonisland
I think it's location combined with its height is quite jarring on the skyline, greatly affecting views of the Manhattan Bridge and the lower east side as a whole. Verizon, on the other hand, is in with the cluster, if ugly and at the edge. Until the other towers get built along the river with it, it stands out apart from the skyline.
Find all posts by Camstonisland
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,091
|
This website requires JavaScript to be fully operational
Please turn JavaScript on
Sunday Cinema | Muscle Shoals
Date: Sunday 12 March 2023
A documentary that celebrates Rick Hall, the founder of FAME Studios in Rated PG- Coarse language | Muscle Shoals, Alabama, and the signature sound he developed in songs such as "I'll Take You There", "Brown Sugar", and "When a Man Loves a Woman".
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Tourism Management Platform for an Activity Operator by IBIS Technology, Ltd.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,924
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Sydney L. Moss Ltd., Escape from the Dusty World: Chinese Paintings and Literati Works of Art, London, 1999, no. 73.
The carving style on this cup would appear to be closely related to that of You Kan (also known as Zhi Sheng). You Kan was a prolific carver, particularly skilled at the representation of flora and fauna. According to J. Chapman in The Art of Rhinoceros Horn Carving in China, London, 1999, p. 139, it is possible that he may also have been known as You Bei (rhinoceros horn cup You), a resident of Wuxi who was summoned to work in the Imperial ateliers by the Kangxi emperor.
A similarly carved example bearing You Kan's signature was sold Christie's, Hong Kong, 1 November 2004, lot 951 (Fig.1). Particularly similar is the carving of the leaves and the manner in which the rocky base extends over the rim into the mouth of the cup just above the handle.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,326
|
package org.usfirst.frc.team2084.CMonster2015;
import org.usfirst.frc.team2084.CMonster2015.commands.ClearFaultsCommand;
import org.usfirst.frc.team2084.CMonster2015.commands.CloseToteGateCommand;
import org.usfirst.frc.team2084.CMonster2015.commands.DriveHeadingCommand;
import org.usfirst.frc.team2084.CMonster2015.commands.DriveToLocationCommand;
import org.usfirst.frc.team2084.CMonster2015.commands.ExtendToteEjectorCommand;
import org.usfirst.frc.team2084.CMonster2015.commands.FallbackDriveCommandGroup;
import org.usfirst.frc.team2084.CMonster2015.commands.LowPowerRaiseToteLifterCommand;
import org.usfirst.frc.team2084.CMonster2015.commands.LowerContainerHookCommand;
import org.usfirst.frc.team2084.CMonster2015.commands.LowerToteLifterCommand;
import org.usfirst.frc.team2084.CMonster2015.commands.NormalDriveCommandGroup;
import org.usfirst.frc.team2084.CMonster2015.commands.OpenToteGateCommand;
import org.usfirst.frc.team2084.CMonster2015.commands.RaiseContainerHookCommand;
import org.usfirst.frc.team2084.CMonster2015.commands.RaiseToteLifterCommand;
import org.usfirst.frc.team2084.CMonster2015.commands.ResetGyroCommand;
import org.usfirst.frc.team2084.CMonster2015.commands.ResetHeadingSetpointCommand;
import org.usfirst.frc.team2084.CMonster2015.commands.RetractToteEjectorCommand;
import org.usfirst.frc.team2084.CMonster2015.commands.RotateToCommand;
import org.usfirst.frc.team2084.CMonster2015.commands.SetEncodersEnabledCommand;
import org.usfirst.frc.team2084.CMonster2015.commands.SetGyroEnabledCommand;
import org.usfirst.frc.team2084.CMonster2015.commands.SetHeadingCommand;
import edu.wpi.first.wpilibj.Joystick;
import edu.wpi.first.wpilibj.buttons.JoystickButton;
import edu.wpi.first.wpilibj.smartdashboard.SmartDashboard;
/**
* This class is the glue that binds the controls on the physical operator
* interface to the commands and command groups that allow control of the robot.
*/
public class OI {
// BEGIN AUTOGENERATED CODE, SOURCE=ROBOTBUILDER ID=DECLARATIONS
public JoystickButton normalDriveButton;
public JoystickButton fallbackDriveButton;
public JoystickButton resetGyroButton;
public JoystickButton extendToteEjectorButton;
public JoystickButton retractToteEjectorButton;
public Joystick driveJoystick;
public JoystickButton raiseToteLifterButton;
public JoystickButton lowerToteLifterButton;
public JoystickButton raiseContainerHookButton;
public JoystickButton lowerContainerHookButton;
public JoystickButton closeToteGateButton;
public JoystickButton openToteGateButton;
public Joystick secondaryJoystick;
// END AUTOGENERATED CODE, SOURCE=ROBOTBUILDER ID=DECLARATIONS
public OI() {
// BEGIN AUTOGENERATED CODE, SOURCE=ROBOTBUILDER ID=CONSTRUCTORS
secondaryJoystick = new Joystick(1);
openToteGateButton = new JoystickButton(secondaryJoystick, 2);
openToteGateButton.whenPressed(new OpenToteGateCommand());
closeToteGateButton = new JoystickButton(secondaryJoystick, 4);
closeToteGateButton.whenPressed(new CloseToteGateCommand());
lowerContainerHookButton = new JoystickButton(secondaryJoystick, 8);
lowerContainerHookButton.whileHeld(new LowerContainerHookCommand());
raiseContainerHookButton = new JoystickButton(secondaryJoystick, 6);
raiseContainerHookButton.whileHeld(new RaiseContainerHookCommand());
lowerToteLifterButton = new JoystickButton(secondaryJoystick, 7);
lowerToteLifterButton.whenPressed(new LowerToteLifterCommand());
raiseToteLifterButton = new JoystickButton(secondaryJoystick, 5);
raiseToteLifterButton.whenPressed(new RaiseToteLifterCommand());
driveJoystick = new Joystick(0);
retractToteEjectorButton = new JoystickButton(driveJoystick, 5);
retractToteEjectorButton.whileHeld(new RetractToteEjectorCommand());
extendToteEjectorButton = new JoystickButton(driveJoystick, 6);
extendToteEjectorButton.whileHeld(new ExtendToteEjectorCommand());
resetGyroButton = new JoystickButton(driveJoystick, 12);
resetGyroButton.whenPressed(new ResetGyroCommand());
fallbackDriveButton = new JoystickButton(driveJoystick, 10);
fallbackDriveButton.whenPressed(new FallbackDriveCommandGroup());
normalDriveButton = new JoystickButton(driveJoystick, 9);
normalDriveButton.whenPressed(new NormalDriveCommandGroup());
// SmartDashboard Buttons
SmartDashboard.putData("Rotate To Command", new RotateToCommand());
SmartDashboard.putData("Drive Heading Command", new DriveHeadingCommand());
SmartDashboard.putData("Drive To Location Command", new DriveToLocationCommand());
SmartDashboard.putData("Low Power Raise Tote Lifter Command",
new LowPowerRaiseToteLifterCommand());
SmartDashboard.putData("Clear Faults Command", new ClearFaultsCommand());
SmartDashboard.putData("Set Gyro Enabled Command", new SetGyroEnabledCommand());
SmartDashboard.putData("Set Encoders Enabled Command", new SetEncodersEnabledCommand());
SmartDashboard.putData("Reset Gyro Command", new ResetGyroCommand());
SmartDashboard.putData("Set Heading Command", new SetHeadingCommand());
SmartDashboard.putData("Reset Heading Setpoint Command", new ResetHeadingSetpointCommand());
// END AUTOGENERATED CODE, SOURCE=ROBOTBUILDER ID=CONSTRUCTORS
}
// BEGIN AUTOGENERATED CODE, SOURCE=ROBOTBUILDER ID=FUNCTIONS
public Joystick getDriveJoystick() {
return driveJoystick;
}
public Joystick getSecondaryJoystick() {
return secondaryJoystick;
}
// END AUTOGENERATED CODE, SOURCE=ROBOTBUILDER ID=FUNCTIONS
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,149
|
\section{Introduction} \label{sec:intro}
Solar flares are the most energetic events in the solar system. Over a typical duration of $\sim$(10 - 1000)~s, they can release up to $10^{32}$~erg of energy --- stored in stressed active region magnetic fields --- into directed mass motions, heating, and acceleration of supra-thermal charged particles, including electrons, protons and heavier ions \citep{kontar_et_al}. Solar flares, together with Coronal Mass Ejections (CMEs), are the main drivers of space weather at Earth and can sometimes even significantly affect Earth- and space-based technology systems like power grids, flight navigation, satellite communications. Predicting solar flares requires, first of all, the determination of parameters such as properties of sunspot groups or of the coronal magnetic field configuration, that are thought to be important for the understanding of fundamental processes in solar plasma physics. Second, at a more technological level, these parameters are used as input values for algorithms that realize predictions providing, for example (but not exclusively), a binary flare/no-flare outcome \citep{bletal12,wh04,gaetal02}.
Most recent flare prediction algorithms belong to the machine learning framework \citep{boco15,coqa09,lietal07,yuetal09,yuetal10}. In this setting, data properties utilized for prediction are named {\em{features}}. In the case of {\em{supervised}} learning, a set of historical data is at disposal where features are tagged by means of {\em{labels}} representing the observation outcome, and the prediction task consists in determining the label associated to the incoming features' set. On the other hand, {\em{unsupervised}} methods do not use any training set and data are clustered in different groups according to similarity criteria involving data features.
A crucial aspect of flare prediction, characterized by notable physical implications, is to provide hints on which data features mostly correlate with the labels. In statistical learning theory this practice is known as {\em{feature selection}} although applications often refer to it as {\em{feature importance}}, which better points out the fact that at the end of the process features are ranked according to their importance in the prediction task (in the following we will use the two terms equivalently). Feature selection can be realized by using advanced implementations of standard neural network approaches \citep{oletal04,ga91} or by means of regularization methods. This second approach aims to optimize a functional made of two terms: the discrepancy term measures the distance between prediction and data in the training set, while the penalty term imposes a constraint on the number of features that significantly contribute to the prediction itself. Two examples of regularization methods for feature selection are LASSO \citep{tibshirani_regression_1996} and $l1$-penalized logit ($l1$-logit in the following) \citep{wu_genome-wide_2009}. Both approaches utilize an $l1$-norm penalty term to reduce the complexity; on the other hand, LASSO measures the discrepancy assuming that the noise on the data is Gaussian, while $l1$-logit relies on a maximum likelihood procedure in which the probability function to maximize is the binomial distribution. In the framework of flare prediction, each one of these two methods presents a specific limitation. In fact, LASSO is intrinsically a regression method and therefore it is not naturally appropriate for applications like flare prediction that may require a binary yes/no response. On the other hand, $l1$-logit is a classification method, but it predicts the binary condition by applying a fixed threshold on the flare occurrence probability, i.e., the applied threshold is the same whatever the dataset used for training is.
The present paper introduces a novel approach to flare prediction with feature importance whose aim is to overtake both previous limitations. The perspective of such approach is hybrid and rather general: first, a regularization method for regression is applied to the training set with an $l1$ penalty term that promotes sparsity, thus realizing feature importance (more specifically, this regularization step reconstructs the vector of weights, with which each feature contributes to the prediction in the training set). Then, the set of the real values obtained by multiplying the weights times the feature values in the training set is automatically clustered in two classes by means of a clustering technique. Clustering is an unsupervised learning approach that organizes a set of samples into meaningful clusters based on data similarity. Data partition is obtained through the minimization of a cost function involving distances between data and cluster prototypes. Optimal partitions are obtained through iterative optimization: starting from a random initial partition, samples are moved from one cluster to another until no further improvement in the cost function optimization is noticed. Therefore, in the second step of the hybrid approach, clustering performs an automatic thresholding, which depends on the historical set used for the training phase and is, therefore, intrinsically data dependent. The resulting algorithm presents several advantages with respect to standard one-step approaches: it selects the most significant features, since, in the first step, it relies on a regularization technique that promotes sparsity; it is a classification method, since at the end it produces two clusters, each one corresponding to a specific outcome of the prediction; it performs classification in a flexible, data-adaptive way, which makes it significantly efficient in providing good performances with respect to standard skill scores. The hybrid approach in this paper utilized LASSO in the regularization step and Fuzzy C-means \citep{bezdek_pattern_1981} to cluster the LASSO outcome, although other feature selection and clustering algorithms can be applied.
In order to corroborate the effectiveness of this hybrid approach we utilized a set of data from the National Oceanic and Atmospheric Administration (NOAA) Space Weather Prediction Center (SWPC) and compared our results with the ones provided by $l1$-logit, as far as both the classification and feature importance abilities are concerned, and with some of the most used machine learning approaches in flare forecasting, as far as just the prediction effectiveness is concerned.
The plan of the paper is as follows. Section 2 illustrates the kind of data, prediction algorithms will deal with. Section 3 introduces our hybrid approach for flare prediction with feature importance. Section 4 applies the hybrid approach to the set of SWPC data described in Section 2 and compares its performances to the ones obtained by $l1$-logit and by other machine learning methods. Our conclusions are offered in Section 5.
\section{Categorical data}
Solar Active Regions (ARs) are classified according to magnetic field complexity indicators. For example, ARs tracked by the National Oceanic and Atmospheric Administration (NOAA) Space Weather Prediction Center (SWPC) are typically classified by using the following $5$ indicators (features): the area, the McIntosh indices \citep{mcintosh_classification_1990}, and the Mount Wilson index \citep{hale_magnetic_1919}. The area index is computed in fractions (millionths) of a solar hemisphere. The McIntosh scheme uses white light emissions to represent sunspot structure and is composed by three independent variables: the {\em{Zurich class}} $Z$ of leading/trailing spot size and separation, which may assume $7$ categorical values; the {\em{penumbral class}} $p$ of primary spot regularity, which may assume $6$ categorical values; the {\em{compactness class}} $c$ of internal spot distribution, which may assume $4$ categorical values. Finally, the Mount Wilson scheme groups sunspots into classes based on the complexity of magnetic flux distribution in associated active regions, according to rules set by the Mount Wilson Observatory in California; this feature may assume $8$ categorical data.
In order to apply machine learning algorithms, either supervised or unsupervised, we need to transform the categorical information contained in the above sunspot classifications (specifically, the McIntosh and Mount Wilson indices) into numerical data. This can be done by either transforming the categorical variables into {\em{dummy variables}} \citep{hardy_regression_1993} or computing occurrence frequencies in a historical database. In this paper we used this second approach, which preserves the dimension of the space where to perform the data analysis. Specifically, we have considered the SWPC database covering the December 1988 to June 1996 time range and we have computed the frequency with which a sunspot classified by a specific value of a fixed indicator produces a flare greater than a given class. Anyhow, we have verified that the use of the dummy variables does not improve the effectiveness of the prediction for all methods considered in this paper. On the other hand, we are also aware that the use of frequencies requires the availability of a labelled dataset whose content, in principle, may increase while new data are at disposal.
More formally, and focusing on the specific case of the value $A$ for the Zurich class in the McIntosh classification, we denoted by $N_{Z=A}^{(C1\rightarrow C9)}$, $N_{Z=A}^{(M1\rightarrow M9)}$, and $N_{Z=A}^{(\ge X1)}$ the occurrences of flaring events of class $C$, $M$ and $X$, respectively, and computed the frequencies associated to flaring events of class greater or equal to a specific class as
\begin{equation}\label{eq:fc1}
f_{Z=A}^{(\ge C1)} = \frac{N_{Z=A}^{(C1\rightarrow C9)} + N_{Z=A}^{(M1\rightarrow M9)} + N_{Z=A}^{(\ge X1)}}{\#~A~occurrences}
\end{equation}
(with the corresponding no-flare-event frequency defined as $f_{Z=A}^{(no flare)}:= 1 - f_{Z=A}^{(\ge C1)}$);
\begin{equation}\label{eq:fm1}
f_{Z=A}^{(\ge M1)} = \frac{N_{Z=A}^{(M1\rightarrow M9)} + N_{Z=A}^{(\ge X1)}}{\#~A~occurrences}
\end{equation}
(with the corresponding no-flare-event frequency defined as $f_{Z=A}^{(no flare)}:= 1 - f_{Z=A}^{(\ge M1)}$);
\begin{equation}\label{eq:fx1}
f_{Z=A}^{(\ge X1)} = \frac{N_{Z=A}^{(\ge X1)}}{\#~A~occurrences}
\end{equation}
(with the corresponding no-flare-event frequency defined as $f_{Z=A}^{(no flare)}:= 1 - f_{Z=A}^{(\ge X1)}$). Similar formulas can be written for each one of the other categorical predictors.
We finally notice that the same dataset used for computing these frequencies has been used as training set for the supervised machine learning algorithms utilized in the following. On the other hand, the database of SWPC indicators covering the time range between August 1996 and December 2010 has been used as test set for both the supervised and unsupervised machine learning methods.
\section{The hybrid approach}\label{pred_alg}
We denote with $X$ the matrix with dimension $N$ (number of active regions) $\times$ $F$ (number of features) whose columns contain the feature values for each specific active region in the training set; $\beta$ is the $F \times 1$ vector containing the $F$ model parameters to determine and $y$ is the $N \times 1$ data vector used in the training set and made of $0$ and $1$ values. $l1$-logit has been designed 'ad hoc' to perform classification with feature importance \citep{wu_genome-wide_2009}. This is a constrained maximum-likelihood method that allows the estimation of the model parameters while best-fitting the data. The logit parameter estimation method solves the minimum problem
\begin{equation}\label{logit-3}
{\hat{\beta}} = \arg\min_{\beta} [-\sum_{i=1}^N \log (1+e^{-y_i(X_i \cdot \beta +c)})]~,
\end{equation}
i.e., it searches for the maximum likelihood of the model parameter vector $\beta$ when one assumes that each component $y_i$ of the vector $y$ is the realization of a random variable described by the Bernoulli distribution. In equation (\ref{logit-3}) $X_i$ is the $i$-th row of matrix $X$ and $c$ is a positive constant. In order to realize feature selection, $l1$-logit adds the condition that $\| \beta \|_1$ is small, which mathematically points out the few parameters that most significantly contribute to the classification. When a new active region $x$ is at disposal ($x$ is a vector of $F$ components), then ${\hat{y}}=\sum_{k=1}^F x_k {\hat{\beta}}_k$ is computed and its sign denotes the outcome of the prediction. This implies that the classification threshold here is fixed and equal to zero independently of the dataset used for training.
We now introduce an approach to flare prediction with feature selection which, differently than $l1$-logit, is hybrid and data-dependent. The first step of this two-step approach utilizes LASSO \citep{tibshirani_regression_1996} to perform feature selection. Specifically, we look for the solution of the minimum problem
\begin{equation}\label{hybrid-1}
{\hat{\beta}} = \arg\min_{\beta} (\|y - X \beta \|_2^2 + \lambda \|\beta \|_1)~,
\end{equation}
where the regularization parameter $\lambda$ is optimized by means of a Cross Validation procedure \citep{stetal74}. Then, in the second step, we apply a clustering method for partitioning the output of ${\hat{y}}=X{\hat{\beta}}$. In a classical clustering approach like Hard C-Means (HCM) \citep{jaetal99}, each sample may belong to a unique cluster, while in a fuzzy clustering formulation a different degree of membership is assigned to each sample with respect to each cluster, which implies a much higher flexibility in accounting for data characteristics. Therefore, in the second step of our hybrid approach, we used Fuzzy C-Means (FCM) \citep{bezdek_pattern_1981}, which is the fuzzy extension of HCM. In this framework, the FCM functional is given by
\begin{equation}\label{FCM-1}
J_m({\hat{y}},{\hat{z}},U) = \sum_{k=1}^N \sum_{j=1}^C (u_{jk})^m d_{jk}^{2}~,
\end{equation}
where ${\hat{z}}=\{{\hat{z}}_j | {\hat{z}}_j \in \mathbb{R}~,~j=1,\ldots,C\}$ is the set of the $C$ centroids of the clusters, the component $u_{jk} \in [0,1]$ of the $C \times N$ matrix $U$ represents the membership of the $k$-th sample to the $j$-th cluster, $d_{jk}$ is the distance between the $j$-th centroid and the $k$-th sample, and $m$ is the fuzzifier parameter. The FCM optimization problem is the one to (iteratively) determine the components of the matrix $U$ and of the vectors ${\hat{z}}$ given the components of the vector ${\hat{y}}$.
\section{Application to SWPC data and analysis of results}
In this Section we have compared the performances of $l1$-logit and our hybrid approach during the analysis of the SWPC test set covering the time range between August 1996 and December 2010 (the cardinality of such set is $22222$); for both methods we used the data collected between December 1988 and June 1996 as training set (the cardinality of this second set is $17600$). Further, we have also analyzed the same test set by means of other four classical machine learning methods: the (unsupervised) clustering HCM and FCM algorithms, a standard Multi Layer Perceptron (MLP) \citep{ruetal86} and a Support Vector Machine (SVM) \citep{cortes_support-vector_1995}. For the latest two methods, which are supervised, we used the same training set as in the case of $l1$-logit and the hybrid method. All these prediction algorithms have been applied to predict flares with class above $C1$ and $M1$, respectively. From now on, for sake of brevity, we will indicate with $\geq C1$ and $\geq M1$ all flares with class above $C1$ and $M1$, respectively. We have not considered flares with class above $X1$ since they are rare in this dataset (less than $1\%$ in the training set and around $0.5 \%$ in the test set).
By means of the frequency matching process described in Section 2, each sample is transformed into a $5$-dimensional vector. Note that the first four components range from $0$ to $1$, while the fifth one, i.e., the sunspot area, goes from $0$ up to $10^2$. Since the differences between component variances can affect the flare prediction performances, a standardization step preceded the application of the machine learning algorithms. We also note that frequency matching must be performed for each case of interest, i.e, separately for the $\ge C1$- and $\ge M1$-flare predictions; therefore, for both the training set and the test set, we have constructed two subsets: the first subset, indicated with $\#1$, is constructed using the frequencies of flares of class at least C1 (i.e., by applying (\ref{eq:fc1}) and analogous); the second subset, indicated with $\#2$, to the frequencies of flares of class at least M1 (i.e., by applying (\ref{eq:fm1}) and analogous).
As explained in the previous section, the main advantage of the hybrid approach is in the fact that the way it partitions the set of LASSO outcomes is driven by the input data. This is clearly described in Figure \ref{fig:fig0}, showing how FCM automatically identifies the probability threshold. It is interesting to note that this threshold depends on the flare class under consideration and in any case is different than the fixed value provided by $l1$-logit, which splits the regressions values at $0.5$.
The threshold value determines the prediction, whose performance can be measured by means of specific scores. Many skill scores can be found in literature for the assessment of flare prediction performances \citep{bletal12}. All these scores are linked to the forecast contingency tables made up of four elements:
\begin{itemize}
\item The number of flares predicted and observed (true positives, TP).
\item The number of flares not predicted but observed (false negatives, FN).
\item The number of flares predicted but not observed (false positives, FP).
\item The number of flares not predicted and not observed (true negatives, TN).
\end{itemize}
We have validated the six flare prediction algorithms by means of the following skill scores defined in terms of the above elements. Specifically, the probability of detection
\begin{equation}\label{pod}
POD = \frac{TP}{TP+FN}~;
\end{equation}
the accuracy
\begin{equation}\label{acc}
ACC = \frac{TP + TN}{TP + TN + FP + FN}~;
\end{equation}
the false alarm ratio
\begin{equation}\label{far}
FAR = \frac{FP}{TP + FP}~.
\end{equation}
These scores range from $0$ to $1$ and best predictions correspond to small FAR values and high values for the other scores. We also utilized two scores with values ranging from $-1$ to $1$: the Heidke skill score
\begin{equation}\label{hss}
HSS = \frac{2 \cdot (TP \cdot TN - FN \cdot FP)}{(TP + FN) \cdot (FN + TN) + (TP + FP) \cdot (FP + TN)}~;
\end{equation}
and the true skill statistics
\begin{equation}\label{tss}
TSS = \frac{TP}{TP+FN} - \frac{FP}{FP + TN}~.
\end{equation}
Also in this case good prediction performances correspond to high values of the scores. Figure \ref{fig:f1} and Figure \ref{fig:f2} present the values of all five skill scores for the $\geq C1$ flare prediction and the $\geq M1$ flare prediction, respectively. Moreover, Table 1 and Table 2 provide the results of the feature selection processes preformed by $l1$-logit and the hybrid technique. Specifically, the tables contain the weights $\beta$ with which the sunspot area, the McIntosh indices, and the Mount Wilson index contribute to the flare prediction process for the two methods.
\startlongtable
\begin{deluxetable}{cccccc}
\tablecaption{Feature importance in $\geq C$ class flare prediction computed from the training set. For each method, the values correspond to the weights associated to the features divided by the sum of the weights, i.e. $\bar \beta_k := \hat \beta_k / \sum_{j=1}^F \hat \beta_j $.}
\label{tab:C-flare}
\tablehead{
& \colhead{MtWilson} & \colhead{McIntosh $Z$} & \colhead{McIntosh $p$} & \colhead{McIntosh $c$} & \colhead{Area} \\
}
\startdata
hybrid & $0.198$ & $0.268$ & $0.222$ & $0.164$ & $0.147$ \\
$l1$-logit & $0.189$ & $0.332$ & $0.219$ & $0.154$ & $0.104$
\enddata
\end{deluxetable}
\startlongtable
\begin{deluxetable}{cccccc}
\tablecaption{Feature importance in $\geq M$ class flare prediction computed from the training set. See caption of Table 1 for the meaning of table entries.}
\label{tab:M-flare}
\tablehead{
& \colhead{MtWilson} & \colhead{McIntosh $Z$} & \colhead{McIntosh $p$} & \colhead{McIntosh $c$} & \colhead{Area} \\
}
\startdata
hybrid & $0.281$ & $0.117$ & $0.047$ & $0.146$ & $0.407$ \\
$l1$-logit & $0.181$ & $0.264$ & $0.217$ & $0.142$ & $0.194$
\enddata
\end{deluxetable}
\begin{figure}[ht!]
\subfigure[]
{
\includegraphics[width=0.49\textwidth]{hybrid_flare_no_flare_50_bins_distribution_C_class.pdf}
}
\subfigure[]
{
\includegraphics[width=0.49\textwidth]{hybrid_flare_no_flare_50_bins_distribution_M_class.pdf}
}
\caption{(a) $\geq C$ class flare prediction. Split of the Lasso regression output by means of the Fuzzy C-means algorithm.
The x-axis shows the values of the regression outcomes provided by the cross validated Lasso algorithm.
Blue and green colors represent the two clusters identified by the Fuzzy C-means algorithm.
Blue (resp. green) cluster is the set of all the events for which the hybrid method returns a no-flare (resp. flare) prediction. (b) The same as in (a) but for $\geq M$ class flares.}
\label{fig:fig0}
\end{figure}
\begin{figure}[ht!]
\plotone{skill_comparison_hmi_C_class.pdf}
\caption{Comparison of performance between the six flare prediction algorithms in terms of skill scores.
The bar plots represent the skill score values obtained by applying each method to the test set for the prediction of $ \geq C1$ flares.}
\label{fig:f1}
\end{figure}
\begin{figure}[ht!]
\plotone{skill_comparison_hmi_M_class.pdf}
\caption{The same as in Figure \ref{fig:f1} but for the prediction of $ \geq M1$ flares.}
\label{fig:f2}
\end{figure}
\section{Discussion and conclusions}
This paper introduces a novel approach to flare prediction, which utilizes indices associated to ARs data and which is also able to automatically indicate the ones, among such features, that mostly contribute to prediction. The approach is intrinsically hybrid, in the sense that it is based on the combination of the ability of regularization to perform feature selection with the ability of clustering to classify in a data-adaptive fashion. In the present implementation we have used LASSO in the feature selection step and FCM in the clustering step. In fact, LASSO guarantees a notable degree of generality in regularization while FCM guarantees a notable degree of flexibility in data adaptation. Anyhow, we have tested the hybrid approach using different combinations of feature selection and clustering methods involving $l1$-logit and HCM: the results of both feature importance ranking and prediction were comparable.
We validated the approach against a NOAA SWPC dataset and by comparing the results with the ones provided by $l1$-logit and other standard machine learning flare prediction algorithms. This comparison showed that the hybrid approach outperforms $l1$-logit in the case of HSS and TSS, that are often considered \citep{bletal12} the most reliable skill scores in the game (for example, ACC tends to reach its maximum when the threshold is $0.5$, which is not fully appropriate in the case of unfrequent events such as M and X class flares). This is particularly true for TSS and for the prediction of flares belonging to class M or higher. More in general, the hybrid method predicts with a performance rate which is very similar to the one of the other two unsupervised clustering algorithms, while, coherently, $l1$-logit works similarly to the other two supervised regularization methods. The higher forecasting effectiveness of the hybrid approach with respect to $l1$-logit is due to the fact that it performs classification with a thresholding procedure which is data adaptive, while $l1$-logit utilizes a fixed negative/positive threshold. We note that the threshold in $l1$-logit could be tuned heuristically, searching `a posteriori' for the values that provide the maximum for TSS and HSS and that the advantage of fuzzy clustering is that it realizes such search `a priori' and in an automatic way.
The hybrid approach and $l1$-logit can be compared also as far as their feature selection power is concerned. Table 1 clearly shows that, in forecasting $\geq$ C1 flares, the two methods indicate the same features as the ones that mostly contribute to the prediction. Results are different when predicting $\geq$ M1 flares, since LASSO gives the highest emphasis to the AR area, while $l1$-logit points out more significantly two of the three McIntosh indices as mostly significant. A clarification of this contradictory outcome shall be obtained by means of a systematic application of these two methods against either several SWPC datasets or features extracted from SDO/HMI images; this activity is part of the tasks currently addressed by the H2020 project FLARECAST, which will provide a technological platform for the testing of flare prediction algorithms and for the validation of the forecasting and feature selection results.
\acknowledgments
The authors have been supported by the H2020 grant Flare Likelihood And Region Eruption foreCASTing (FLARECAST), project number 640216. The authors kindly thank Prof. Shaun Bloomfield for providing the SWPC data and Dr. Annalisa Perasso for useful discussion.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,108
|
Q: Change http 403 to 401 on failed custom spring security expression i have written a custom Expression root for my @PreAuthorize annotations .
The logic itself works fine. However the application returns a 403, but i need to return a 401.
public class JwtConfigurer extends SecurityConfigurerAdapter<DefaultSecurityFilterChain, HttpSecurity> {
@Override
public void configure(HttpSecurity http) throws Exception {
http.addFilterBefore(new OwnTokenFilter(), UsernamePasswordAuthenticationFilter.class)
.exceptionHandling()
.authenticationEntryPoint(new HttpStatusEntryPoint(UNAUTHORIZED));
}
}
The OwnTokenFilter extracts a jwt token and provides it to the SecurityContext. My expectation was, that if the authorization fails, an UNAUTHORIZED was returned, but it is simply ignored. I am using Spring Boot 2.1.x
My expression root looks like
public class ExpressionRoot extends SecurityExpressionRoot implements MethodSecurityExpressionOperations {...
public boolean hasRoleOneOf(final String ... expectedRoles) {
...
return roleMatched? true : false;
}
Thank you
A: Found a solution
import org.springframework.http.ResponseEntity;
import org.springframework.security.access.AccessDeniedException;
import org.springframework.web.bind.annotation.ControllerAdvice;
import org.springframework.web.bind.annotation.ExceptionHandler;
import org.springframework.web.bind.annotation.ResponseBody;
import static org.springframework.http.HttpStatus.UNAUTHORIZED;
@ControllerAdvice
public class ExceptionAdvice {
@ExceptionHandler(AccessDeniedException.class)
@ResponseBody
public ResponseEntity<String> handleControllerException(AccessDeniedException ex) {
return new ResponseEntity<>(ex.getMessage(), UNAUTHORIZED);
}
}
Not sure if it is the best one :)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,248
|
\section{Introduction}
An interesting framework to study the behaviour of QCD scattering amplitudes
in the limit of large centre--of--mass energies $\sqrt{s}$ and fixed momentum
transfer $\sqrt{-t}$ is the Balitsky--Fadin--Kuraev--Lipatov (BFKL)
formalism~\cite{LLBFKL}.
The BFKL framework can be applied to processes characterised by a colour
octet exchange as well as colour singlet exchange (diffractive processes).
The case of $t=0$ colour singlet scattering corresponds to forward
scattering, whereas the case of $t\not=0$ is called non--forward scattering.
The optical theorem relates the amplitude for forward diffractive scattering
to the amplitude for a colour octet exchange, which will be exploited in the
check of some of the results derived in this paper. If the momentum transfer
is perturbative, i.e. $-t \gg \Lambda_{\rm QCD}^2$, it is possible to use the
non--forward BFKL equation to study high--$t$ diffraction in the high energy
limit, which is characterised by a final state with two systems with
transverse momentum $-t$ and far apart in rapidity.
The colour singlet exchange in the non--forward case results in a
rapidity gap in jet activity. The non--forward BFKL equation thus provides
a useful theoretical framework to study diffractive physics from first
principles in QCD.
The differential cross--section for diffractive scattering,
\begin{eqnarray}
\frac{\mathrm{d}\sigma}{\mathrm{d}t} &=& \frac{|A(s,t)|^2}{16\pi s^2},
\label{eq:xsec}
\end{eqnarray}
can be described within the BFKL framework by a factorised scattering
amplitude of the form
\begin{eqnarray}
\frac{|A(s,t)|}{s} &=& \left| \int {\mathrm{d}}^2 \mathbf{k}_a\, \int {\mathrm{d}}^2 \mathbf{k}_b
\, \Phi_{\mathrm{A}} (\mathbf{k}_a,\mathbf{q}) \, \Phi_{\mathrm{B}} (\mathbf{k}_b,\mathbf{q}) \,
\frac{f \left({\mathbf{k}}_a,{\mathbf{k}}_b,{\mathbf{q}},{\mathrm{Y}} \right)}
{({\mathbf{k}}_a-{\mathbf{q}})^2 {\mathbf{k}}_b^2} \right|,
\end{eqnarray}
where $\mathrm{Y}$ is the rapidity separation of the scattered probes, $\Phi_{\rm
A}({\bf k}_a,{\bf q})$ and $\Phi_{\rm B}({\bf k}_b,{\bf q})$ are the
process--dependent impact factors and the four--point gluon Green's function,
$f({\bf k}_a,{\bf k}_b,{\bf q},{\rm Y})$, is universal. For illustration, in
Fig.~\ref{fig:BFKLladder} a typical dominant contribution to this colour
singlet exchange in the high energy limit is shown. As explained below in
Sec.~\ref{Separation} the non--forward BFKL equation describes the evolution
of $f({\bf k}_a,{\bf k}_b,{\bf q},{\rm Y})$ as a function of the rapidity
separation Y. This formalism can be applied, for example, to the study of the
diffractive production of vector mesons in photon--proton collisions at HERA
for large center--of--mass energies and transverse momentum squared
$|t|>\Lambda_{\rm QCD}^2$. In this kinematical region the Leading Logarithm
(LL) terms $\left(\alpha_s \ln{s/|t|}\right)^n$ generated in the perturbative
series must be resummed. Non--perturbative contributions are included in the
proton parton densities and in the meson light--cone wave function present in
the corresponding impact factor. For the $\rho,~\phi$ and $J/\psi$ mesons the
transverse momentum spectrum and spin density matrix elements have recently
been measured at HERA~\cite{HERA}. From the
theoretical side, there has been an intense activity in the study of these
processes, see e.g.~Ref.~\cite{Largettheory}. The case of photon dissociating
to a photon has also been studied in
Ref.~\cite{photonphoton}. Another example of the application
of this approach is the description of events with interjet rapidity gaps in
photon--hadron and hadron--hadron
collisions~\cite{hadronhadron}.
\begin{figure}[tbp]
\begin{fmffile}{qqladder}
\begin{fmfgraph*}(150,150)
\fmfset{arrow_len}{3mm} \fmfset{arrow_ang}{15} \fmfpen{1pt}
\fmfstraight
\fmfleft{pi1,pi2} \fmfright{po1,po2}
\fmf{fermion}{pi1,vup1,vup2,po1} \fmf{fermion}{pi2,vul1,vul2,po2}
\fmffreeze \fmf{phantom}{vul1,vl14,vl13,vl12,vl11,vup1}
\fmffreeze \fmf{phantom}{vup2,vl21,vl22,vl23,vl24,vul2} \fmffreeze
\fmf{gluon}{vl21,vl11}
\fmf{gluon}{vl22,vl12}
\fmf{gluon}{vl23,vl13}
\fmf{gluon}{vl24,vl14}
\fmf{gluon,label=$\bf{k}_a\downarrow$}{vul1,vl14}
\fmf{gluon}{vl14,vl13}
\fmf{gluon}{vl13,vl12}
\fmf{gluon}{vl12,vl11}
\fmf{gluon,label=$\bf{k}_b\downarrow$}{vl11,vup1}
\fmf{gluon,label=$\uparrow \bf{k}_b-\bf{q}$}{vup2,vl21}
\fmf{gluon}{vl21,vl22}
\fmf{gluon}{vl22,vl23}
\fmf{gluon}{vl23,vl24}
\fmf{gluon,label=$\uparrow \bf{k}_a-\bf{q}$}{vl24,vul2}
\end{fmfgraph*}
\end{fmffile}\centering
\caption{A Feynman diagram contributing to the LL approximation for
quark--quark scattering with colour singlet exchange.}
\label{fig:BFKLladder}
\end{figure}
The original analytic solution to the non--forward LL BFKL
equation~\cite{Lipatov:1985uk}
proved to be significantly more complicated than its forward counterpart. In
order to exhibit the conformal invariance that proved vital in solving the
forward BFKL equation, it is necessary to perform a Fourier transform from
transverse momentum space to impact parameter representation. The non--forward
BFKL equation at next--to--leading logarithmic (NLL) accuracy
($\alpha_s \left(\alpha_s \ln{s/|t|}\right)^n$ terms) will be
significantly more complicated, with the conformal invariance being
explicitly broken by the running of the coupling (as it happens in the
NLL forward case~\cite{NLLpapers}). It is therefore important
to investigate new strategies to find the solution to this equation.
In this paper we present a new approach to obtaining the solution to the
non--forward BFKL equation to LL accuracy. The method of solution is based on
the separation of different contributions to the BFKL kernel by a phase space
slice. Once this separation is performed it is possible to apply an iterative
approach similar to the one presented in
Ref.~\cite{Iterative} for the case of the
BFKL equation describing a colour octet exchange. This approach has recently
been generalised~\cite{Solution} to solve this BFKL
equation in the NLL approximation both in QCD
and $N\!=\!4$ supersymmetric Yang--Mills theory~\cite{Andersen:2004uj}
(reviews can be found in~\cite{Reviews}).
In the present work we concentrate on extending this iterative approach to
the solution of the BFKL equation describing diffractive processes. The
presented method of solving the BFKL equation directly in transverse momentum
space has the benefit that it allows for a direct inspection of all involved
momenta. In particular it is possible to study the diffusion of the transverse
scales along the evolution in rapidity.
The same method of solution can be applied to the non--forward BFKL equation
in the NLL approximation. The corresponding analysis will be performed when
the calculation of the non--forward NLL kernel is completed~\cite{NFNLL}.
The structure of this paper is as follows. In section~\ref{Separation} the LL
BFKL equation in the non--forward case is presented and the phase space
slicing regularisation procedure is performed. In section~\ref{Solution} the
non--forward LL BFKL equation is solved by iterating the kernel, obtaining in
this way an explicit expression for the four--point gluon Green's function.
In section~\ref{Analysis} the behaviour of the four--point gluon Green's
function is analysed for different values of the momentum transfer. A study
of the diffusion properties is performed of not just the four--point gluon
Green's function, but also, in section~\ref{sec:analysis-toy-cross}, for
cross--sections calculated with toy impact factors. The conclusions
are presented at the end.
\section{The Non--Forward LL BFKL Equation}
\label{Separation}
The LL BFKL equation describing colour singlet exchange with a non--zero
momentum transfer squared, $- t = {\bf q}^2$, was originally calculated in
Ref.~\cite{LLBFKL}. The starting point in the present work will be Eq.~(4.16)
of Ref.~\cite{Forshaw:1997dc} where the integral equation for the Mellin
transform in rapidity of the four--point gluon Green's function, $f_\omega
\left({\bf k}_a,{\bf k}_b, {\bf q}\right)$, is given by
\begin{eqnarray}
\omega f_\omega \left({\bf k}_a,{\bf k}_b, {\bf q}\right) &=&
\delta^{(2)} \left({\bf k}_a-{\bf k}_b\right)
+\frac{{\bar \alpha}_s}{2 \pi} \int d^2{\bf k'}
\left[\frac{-{\bf q}^2}{\left({\bf k}'-{\bf q}\right)^2 {\bf k}_a^2}
f_\omega \left({\bf k}',{\bf k}_b, {\bf q}\right) \right. \nonumber\\
&&\hspace{-3.4cm}+
\frac{1}{\left({\bf k}'-{\bf k}_a\right)^2}
\left(f_\omega \left({\bf k}',{\bf k}_b, {\bf q}\right) -\frac{{\bf k}_a^2}{{{\bf k}'}^2+\left({\bf k}_a-{\bf k}'\right)^2}
f_\omega \left({\bf k}_a,{\bf k}_b, {\bf q}\right) \right)\nonumber\\
&&\hspace{-3.4cm}\left.+\frac{1}{\left({\bf k}'-{\bf k}_a\right)^2}
\left(
\frac{\left({\bf k}_a-{\bf q}\right)^2{{\bf k}'}^2}{\left({\bf k}'-{\bf q}\right)^2 {\bf k}_a^2}
f_\omega \left({\bf k}',{\bf k}_b, {\bf q}\right)
-\frac{\left({\bf k}_a-{\bf q}\right)^2}{\left({\bf k}'-{\bf q}\right)^2
+\left({\bf k}_a-{\bf k}'\right)^2}
f_\omega \left({\bf k}_a,{\bf k}_b, {\bf q}\right)\right)\right],
\end{eqnarray}
where $\mathbf{k}_a$ and $\mathbf{k}_b$ describe the two--dimensional
transverse momenta of the exchanged gluons in the $t$--channel (see Fig.~\ref{fig:BFKLladder}) and we define
${\bar \alpha}_s \equiv \alpha_s N_c / \pi$. The driving term $\delta^{(2)}
\left({\bf k}_a-{\bf k}_b\right)$ corresponds to a simple two gluon exchange.
It is more convenient to rearrange the terms in this expression
and, as the integration variable, to use the transverse momenta of the
$s$--channel gluons, ${\bf k} = {\bf k}' - {\bf k}_a$, i.e.
\begin{eqnarray}
\omega f_\omega \left({\bf k}_a,{\bf k}_b, {\bf q}\right) &=& \delta^{(2)} \left({\bf k}_a-{\bf k}_b\right) \nonumber\\
&&\hspace{-4cm}+\frac{{\bar \alpha}_s}{2 \pi} \int d^2{\bf k}
\left\{ \left[ \frac{1}{{\bf k}^2}
\left(1 + \frac{\left({\bf k}_a-{\bf q}\right)^2\left({\bf k}+{\bf k}_a\right)^2}{\left({\bf k}+{\bf k}_a -{\bf q}\right)^2 {\bf k}_a^2} \right)
-\frac{{\bf q}^2}{\left({\bf k}+{\bf k}_a-{\bf q}\right)^2 {\bf k}_a^2} \right]
f_\omega \left({\bf k}+{\bf k}_a,{\bf k}_b, {\bf q}\right) \right.\nonumber\\
&&\hspace{-1.2cm}\left.-\frac{1}{{\bf k}^2}
\left(\frac{{\bf k}_a^2}{\left({\bf k}+{\bf k}_a\right)^2+{\bf k}^2}
+\frac{\left({\bf k}_a-{\bf q}\right)^2}{\left({\bf k}+{\bf k}_a -{\bf q}\right)^2 + {\bf k}^2} \right)
f_\omega \left({\bf k}_a,{\bf k}_b, {\bf q}\right)\right\}.
\end{eqnarray}
At this stage it is possible to separate the exchange terms form the
s--channel contributions by introducing a phase space
slicing parameter $\lambda$ (at NLL it is useful to use dimensional
regularisation to show the cancellation of infrared divergences, at LL this
is not needed). It is also convenient to use the following approximation
\begin{eqnarray}
\label{approximation}
f_\omega \left({\bf k}+{\bf k}_a,{\bf k}_b, {\bf q}\right) &=&
f_\omega \left({\bf k}+{\bf k}_a,{\bf k}_b, {\bf q}\right)
\left(\theta\left({\bf k}^2 - \lambda^2\right)+
\theta\left(\lambda^2-{\bf k}^2\right)\right) \nonumber\\
&& \hspace{-1cm}
\simeq f_\omega \left({\bf k}+{\bf k}_a,{\bf k}_b, {\bf q}\right)
\theta\left({\bf k}^2 - \lambda^2\right) +
f_\omega \left({\bf k}_a,{\bf k}_b, {\bf q}\right)
\theta\left(\lambda^2-{\bf k}^2\right).
\end{eqnarray}
In all the results presented in this paper we have made sure this
approximation is valid by checking that the four--point Green's function is
insensitive to the value of the slicing parameter for small values of
$\lambda$.
Therefore, the non--forward BFKL equation for the Green's function can
be written in a very simple form:
\begin{eqnarray}
\label{SimpleEqn}
\left(\omega -\omega_0 \left({\bf k}_a, {\bf q}, \lambda \right)
\right) f_\omega \left({\bf k}_a,{\bf k}_b, {\bf q}\right) ~=~
\delta^{(2)} \left({\bf k}_a-{\bf k}_b\right) \nonumber\\
&&\hspace{-5cm}+ \int \frac{d^2{\bf k}}{\pi {\bf k}^2}
\theta \left({\bf k}^2-\lambda^2\right)
\xi \left({\bf k}_a,{\bf k},{\bf q}\right)
f_\omega \left({\bf k}+{\bf k}_a,{\bf k}_b, {\bf q}\right).
\end{eqnarray}
In order to write the equation in this way the following notation has been
introduced:
\begin{eqnarray}
\xi\left({\bf k}_a,{\bf k},{\bf q}\right) &=&
\frac{{\bar \alpha}_s}{2}
\left( 1 + \frac{\left({\bf k}_a-{\bf q}\right)^2\left({\bf k}+{\bf k}_a\right)^2-{\bf q}^2 {\bf k}^2}{\left({\bf k}+{\bf k}_a -{\bf q}\right)^2 {\bf k}_a^2}
\right),
\end{eqnarray}
and
\begin{eqnarray}
\omega_0 \left({\bf k}_a, {\bf q},\lambda\right) &=&
\frac{{\bar \alpha}_s}{2 \pi} \int
\frac{d^2{\bf k}}{{\bf k}^2} \left[\theta \left(\lambda^2-{\bf k}^2\right)
\left(1 + \frac{\left({\bf k}_a-{\bf q}\right)^2\left({\bf k}+{\bf k}_a\right)^2-{\bf q}^2 {\bf k}^2}{\left({\bf k}+{\bf k}_a -{\bf q}\right)^2 {\bf k}_a^2}
\right) \right.\nonumber\\
&-&\left.\frac{{\bf k}_a^2}{\left({\bf k}+{\bf k}_a\right)^2+{\bf k}^2}
-\frac{\left({\bf k}_a-{\bf q}\right)^2}{\left({\bf k}+{\bf k}_a -{\bf q}\right)^2 + {\bf k}^2} \right].
\end{eqnarray}
The latter expression corresponds to the Regge trajectory in our
regularisation. In the case of $\mathbf{q}={\bf 0}$, the trajectory for the
colour--octet exchange should be obtained, and indeed we find
\begin{eqnarray}
\label{trajforw}
\omega_0 \left({\bf k}_a, {\bf 0},\lambda\right) &=&
\frac{{\bar \alpha}_s}{\pi} \int
\frac{d^2{\bf k}}{{\bf k}^2} \left[\theta \left(\lambda^2-{\bf k}^2\right)-
\frac{{\bf k}_a^2}{\left({\bf k}+{\bf k}_a\right)^2+{\bf k}^2}\right]
~=~ - {\bar \alpha}_s \ln{\frac{{\bf k}_a^2}{\lambda^2}}.
\end{eqnarray}
Moreover, the whole solution to the non--forward BFKL equation has the correct
limit for $q\to0$ as found in Ref.~\cite{Iterative}.
The expression for the non--forward LL Regge trajectory can be simplified if
we make use of the forward limit in Eq.~(\ref{trajforw}), i.e.
\begin{eqnarray}
\omega_0 \left({\bf k}_a, {\bf q},\lambda\right) &=&
\frac{1}{2} \, \left( \omega_0 \left({\bf k}_a, {\bf 0},\lambda\right)+
\omega_0 \left({\bf k}_a-{\bf q}, {\bf 0},\lambda\right) \right) \nonumber \\
&+&\frac{{\bar \alpha}_s}{2 \pi} \int
\frac{d^2{\bf k}}{{\bf k}^2} \, \theta \left(\lambda^2-{\bf k}^2\right)
\left(\frac{\left({\bf k}_a-{\bf q}\right)^2\left({\bf k}+{\bf k}_a\right)^2-{\bf q}^2 {\bf k}^2}{\left({\bf k}+{\bf k}_a -{\bf q}\right)^2 {\bf k}_a^2}-1
\right).
\end{eqnarray}
The last integral is negligible when $\lambda$ is small, and
therefore the non--forward trajectory takes the simple form
\begin{eqnarray}
\label{eq:trajappr}
\omega_0 \left({\bf k}_a, {\bf q},\lambda\right) &\simeq&
-\frac{{\bar \alpha}_s}{2} \left(\ln{\frac{{\bf k}_a^2}{\lambda^2}}
+\ln{\frac{\left({\bf k}_a-{\bf q}\right)^2}{\lambda^2}}\right).
\end{eqnarray}
With these conventions we proceed in the next section to iterate
Eq.~(\ref{SimpleEqn}) to find the solution for the non--forward
four--point gluon Green's function.
\section{Solution to the Equation}
\label{Solution}
The non--forward LL BFKL Green's function is the solution to the integral
equation
\begin{eqnarray}
f_\omega \left({\bf k}_a,{\bf k}_b, {\bf q}\right) ~=~
\frac{1}{\omega -\omega_0 \left({\bf k}_a, {\bf q}, \lambda \right)}
\left\{\delta^{(2)} \left({\bf k}_a-{\bf k}_b\right) \right.\nonumber\\
&&\hspace{-6cm}\left.+ \int \frac{d^2{\bf k}}{\pi {\bf k}^2}
\theta \left({\bf k}^2-\lambda^2\right)
\xi \left({\bf k}_a,{\bf k},{\bf q}\right)
f_\omega \left({\bf k}+{\bf k}_a,{\bf k}_b, {\bf q}\right) \right\}.
\label{one_over_w}
\end{eqnarray}
for $\lambda\to0$. If this expression is iterated, the Green's function can
be expressed in terms of a kernel per iteration acting on the initial
condition with a series of poles in the $\omega$ complex plane:
\begin{eqnarray}
\label{iterating}
f_{\omega} \left({\bf k}_a ,{\bf k}_b,{\bf q}\right) &=&
\frac{\delta^{(2)} \left({\bf k}_a - {\bf k}_b \right)}{\omega - {\omega}_0 \left({\bf k}_a,{\bf q},\lambda\right)} \nonumber\\
&&\hspace{-1cm}+ \int \frac{d^2 {\bf k}_1}{\pi {\bf k}_1^2}
\frac{\theta\left({\bf k}_1^2-\lambda^2\right)\, \xi\left({\bf k}_a,{\bf k}_1,{\bf q}\right)}{\omega - {\omega}_0 \left({\bf k}_a,{\bf q},\lambda\right)}
\frac{\delta^{(2)} \left({\bf k}_a +{\bf k}_1 - {\bf k}_b\right)}
{\omega - {\omega}_0 \left({\bf k}_a+{\bf k}_1,{\bf q},\lambda\right)} \nonumber\\
&&\hspace{-1cm}+ \int \frac{d^2 {\bf k}_1}{\pi {\bf k}_1^2}
\int \frac{d^2 {\bf k}_2}{\pi {\bf k}_2^2}
\frac{\theta\left({\bf k}_1^2-\lambda^2\right)\, \xi\left({\bf k}_a,{\bf k}_1,{\bf q}\right)}{\omega - {\omega}_0 \left({\bf k}_a,{\bf q},\lambda\right)}
\frac{\theta\left({\bf k}_2^2-\lambda^2\right)\, \xi\left({\bf k}_a+{\bf k}_1,{\bf k}_2,{\bf q}\right)}{\omega - {\omega}_0 \left({\bf k}_a+{\bf k}_1,{\bf q},\lambda\right)}\nonumber\\
&&\hspace{1cm}\times\frac{\delta^{(2)} \left({\bf k}_a +{\bf k}_1 +{\bf k}_2- {\bf k}_b\right)}{\omega - {\omega}_0 \left({\bf k}_a+{\bf k}_1+{\bf k}_2,{\bf q},\lambda\right)} \nonumber\\
&&\hspace{-1cm}+ \cdots
\end{eqnarray}
Each action of the kernel corresponds to an interaction between the reggeised
gluons exchanged in the $t$--channel, building up, in this way, the LL BFKL
ladder. The poles are integrated over when going from the $\omega$--plane to
rapidity space using the Mellin transform
\begin{eqnarray}
f \left({\bf k}_a,{\bf k}_b, {\bf q}, {\rm Y}\right)
&=& \frac{1}{2 \pi i}
\int_{a-i \infty}^{a+i \infty} d\omega ~ e^{\omega {\rm Y}} f_{\omega}
\left({\bf k}_a ,{\bf k}_b, {\bf q}\right),
\end{eqnarray}
where Y is the rapidity span of the BFKL ladder.
This integration can be performed to finally obtain the solution of the
non--forward LL BFKL equation for the four--point Green's function
as (with $y_0 \equiv$ Y)
\begin{eqnarray}
\label{eq:nonf_soln}
&&\hspace{-1cm}f({\bf k}_a ,{\bf k}_b, {\bf q}, {\rm Y})
= \left(
\frac{\lambda^2}{{\bf k}_a^2}\frac{\lambda^2}{\left({\bf k}_a-{\bf q}\right)^2}\right)^{\frac{\bar{\alpha}_s}{2} \, {\rm Y}}
\left\{ \frac{}{} \delta^{(2)} ({\bf k}_a - {\bf k}_b) \right. \\
&+& \sum_{n=1}^{\infty} \prod_{i=1}^{n} \int d^2{\bf k}_i
\frac{\theta\left({\bf k}_i^2- \lambda^2\right)}{\pi {\bf k}_i^2} \,
\xi \left({\bf k}_a+\sum_{l=1}^{i-1} {\bf k}_l,{\bf k}_i,{\bf q}\right)
\nonumber \\
&\times& \left. \int_0^{y_{i-1}} d y_i
\left(\frac{\left({\bf k}_a+\sum_{l=1}^{i-1} {\bf k}_l\right)^2}
{\left({\bf k}_a+\sum_{l=1}^i {\bf k}_l\right)^2}
\frac{\left({\bf k}_a+\sum_{l=1}^{i-1} {\bf k}_l -{\bf q}\right)^2}
{\left({\bf k}_a+\sum_{l=1}^i {\bf k}_l-{\bf q}\right)^2}\right)
^{\frac{\bar{\alpha}_s}{2} \, y_i} \delta^{(2)} \left(\sum_{l=1}^{n}{\bf k}_l
+ {\bf k}_a - {\bf k}_b \right) \right\}. \nonumber
\end{eqnarray}
This solution has the correct forward limit of Ref.~\cite{Iterative}
when the momentum transfer tends to zero. We would like to stress again
that this method of solution is directly
applicable to the non--forward BFKL equation also at NLL accuracy, with the
difference that the introduction of the phase space slice is more conveniently
performed in the language of dimensional regularisation.
\section{Analysis of the Gluon Green's Function}
\label{Analysis}
To study the dependence of the non--forward LL BFKL four--point gluon Green's
function on the transverse momentum scales we choose to integrate over all
external angles and define the quantity
\begin{eqnarray}
{\bar f} \left(\left|{\bf k}_a\right|,\left|{\bf k}_b\right|,\left|{\bf q}\right|, {\rm Y}\right) &=& \int_0^{2 \pi} d \theta_{qa} \int_0^{2 \pi} d \theta_{qb} \, f \left({\bf k}_a, {\bf k}_b, {\bf q}, {\rm Y}\right),
\end{eqnarray}
where $\theta_{qi}$ is the angle between the vectors ${\bf k}_i$ and ${\bf q}$.
As a first analysis, in Fig.~\ref{2} the value of $\left|{\bf k}_b\right|$ is
fixed to 5 GeV and the dependence on $\left|{\bf k}_a\right|$ is studied.
When the value of the modulus of both momenta coincides, the angular
integrated Green's function shows a $\delta$--functional behaviour corresponding to the
two gluon exchange limit. This dependence is caused by the driving term of
the integral equation whose influence is stronger for lower energies. When
Y is increased from 1 in Fig.~\ref{2} to ${\rm Y}=3$ in Fig.~\ref{3} the
influence of the driving term diminishes as a consequence of a larger number
of effective rungs in the BFKL ladder. It is interesting to note that the
influence of the momentum transfer $q$ is larger in regions of low scales
of $k_a$. The general trend is that the four--gluon Green's function
decreases with increasing $q$. Comparing Fig.~\ref{2} with Fig.~\ref{3} it
can be seen that this effect is increasing with rapidity.
\begin{figure}
\centerline{\epsfig{file=qScale_plot_1.eps,width=12cm,angle=0}}
\caption{The angular integrated Green's function for fixed
$\left|{\bf k}_b\right|=5$ GeV as
a function of $\left|{\bf k}_a\right|$ for different values of the momentum
transfer. The rapidity span is low, Y = 1.}
\label{2}
\end{figure}
\begin{figure}
\centerline{\epsfig{file=qScale_plot_3.eps,width=12cm,angle=0}}
\caption{The angular integrated Green's function for fixed
$\left|{\bf k}_b\right|=5$ GeV as
a function of $\left|{\bf k}_a\right|$ for different values of the momentum
transfer $q$. The rapidity span is Y = 3.}
\label{3}
\end{figure}
Let us now proceed to the study of the diffusion of the transverse scales in
the BFKL ladder with increasing rapidity span. We choose in this analysis to
concentrate on the evolution of the transverse scale along the left hand side
of the ladder depicted on Fig.~\ref{fig:BFKLladder}, but, as previously
mentioned, one of the benefits of having the solution to the non--forward
BFKL equation expressed in terms of explicit phase space integrals as in
Eq.~(\ref{eq:nonf_soln}) is that it is possible to study any of the momenta
in the ladder.
Diffusion is normally studied in terms of the mean of the transverse momentum
along the ladder, with the width of the distribution indicated by the
standard deviation (see
e.g.~Ref.~\cite{Forshaw:1997dc,DIF}). This choice
is useful, since it allows analytic studies. However, a plot of the mean
internal transverse momentum plus/minus the standard deviation as a function
of the rapidity along the BFKL ladder fails, by construction, to display the
different behaviour of diffusion to low and high scales. Given the
possibilities offered by the solution written in terms of explicit phase
space integrals, we therefore choose to study diffusion in terms of the
average value $\langle\tau\rangle$ of $\tau=\ln((k_a+\sum
k_i)^2/\mathrm{GeV^2})$ as a function of the rapidity ${\rm Y}'$ along the
ladder.
Specifically, for a given value of $k_a, k_b$, and Y we solve the
non--forward BFKL equation according to Eq.~(\ref{eq:nonf_soln}) by a
MonteCarlo integration method. For each configuration point in $n-$momenta
phase space $\{k_i,y_i\}$ we can trace the evolution of $\tau$ along the ladder
and, at the same time, calculate the weight of this configuration to the total
solution. In this way it is possible to calculate both the average value of
$\tau$ along the ladder, $\langle\tau\rangle{\scriptstyle( {\rm Y}')}$, and the
quantities
\begin{align}
\begin{split}
\sigma^2_1{\scriptstyle( {\rm Y}')}&= \frac{\displaystyle 2\int_{<\tau>{\scriptstyle( {\rm Y}')}}^\infty d\tau (\tau-\langle\tau\rangle{\scriptstyle( {\rm Y}')})^2 \bar{f}(k_a,k_b,q,{\rm Y})}{\displaystyle\int_0^\infty d\tau \bar{f}(k_a,k_b,q,{\rm Y})},\\
\sigma^2_2{\scriptstyle( {\rm Y}')}&=\frac{\displaystyle 2\int_0^{<\tau>{\scriptstyle( {\rm Y}')}} d\tau
(\tau-\langle\tau\rangle{\scriptstyle( {\rm Y}')})^2 \bar{f}(k_a,k_b,q,{\rm Y})}{\displaystyle\int_0^\infty d\tau \bar{f}(k_a,k_b,q,{\rm Y})}.
\end{split}
\end{align}
For a gluon Green's function $f(\mathbf{k}_a,\mathbf{k}_b,\mathbf{q},{\rm Y})$ that
is symmetric in $\tau$, the lines of $\langle\tau\rangle{\scriptstyle( {\rm Y}')}$,
$\langle\tau\rangle{\scriptstyle( {\rm Y}')}+\sigma_1{\scriptstyle( {\rm Y}')}$, and
$\langle\tau\rangle{\scriptstyle( {\rm Y}')}-\sigma_2{\scriptstyle( {\rm Y}')}$ would
reproduce the plot of the mean plus/minus the standard deviation. This is
true for $q=0$~GeV, as shown in Fig.~\ref{fig:diff_q}. Here we have plotted
the above mentioned three lines for $k_a=5$~GeV, $k_b=4$~GeV, $q=0$~GeV,
$\alpha_s=0.23$, and ${\rm Y}=1,2,3$. ${\rm Y}'$ is rescaled to lie between 0 and 1, so as to plot all the three cases on the same figure.
\begin{figure}
\begin{center}
\epsfig{file=cigar_q_0.eps,width=7.5cm,angle=0}
\epsfig{file=cigar_q_2.eps,width=7.5cm,angle=0}
\end{center}
\caption{The diffusion properties of the gluon Green's function in terms of
the lines formed by $\langle\tau\rangle$, $\langle\tau\rangle+\sigma_1$,
and $\langle\tau\rangle-\sigma_2$ along the BFKL ladder. Shown for
$k_a=5$~GeV, $k_b=4$~GeV, $q=0$~GeV (left) and $q=2$~GeV (right), for
rapidity spans of ${\rm Y}=1,2,3$.}
\label{fig:diff_q}
\end{figure}
However, for $q>0$~GeV the distribution of $\tau$ is no longer symmetric, as
can be seen on the right hand side plot in Fig.~\ref{fig:diff_q}, where we
have plotted the same quantities for $q=2$~GeV. It is apparent that there is
less diffusion to smaller scales, the average is unchanged, and the diffusion
to scales larger than the average is almost unchanged compared to the case
$q=0$~GeV.
It is also apparent that for the set of parameters investigated here, there
is no significant diffusion into regions where the coupling is expected to
become unperturbatively large. As it is well known, the influence of softer
scales is larger at higher energies, but in this work we confirm the fact
that the diffusion into the infrared is drastically reduced when there is
some momentum transfer, acting, in this way, as an efficient infrared
cut--off. It will be interesting to investigate if this picture holds at NLL.
\section{Analysis of a Toy Cross--Section}
\label{sec:analysis-toy-cross}
Let us now turn to the study of diffractive cross--sections. To this end we
need to define some suitable impact factors. We choose a generic example from
Ref.\cite{Forshaw:1997dc} modelling the photo--production of a vector meson:
\begin{eqnarray}
\Phi_{\rm A}(\mathbf{k}_a,\mathbf{q})=\alpha_s h^2 \int\mathrm{d}\rho \,
\mathrm{d}\tau \rho \,
(1-\rho)\left[\frac 1 {\left(\mathbf{q}^2 \rho^2 \tau(1-\tau)+m^2\right)}
- \frac 1
{\left((\mathbf{k}_a-\rho \, \mathbf{q})^2\tau(1-\tau)+m^2 \right)}\right],
\end{eqnarray}
and similarly for $\Phi_{\rm B}$. Here, $h$ is a normalisation constant which
we choose arbitrarily such that $\alpha_s h^2=1$. This choice obviously means
that the normalisation of the toy cross--section reported in this paper is
completely arbitrary. We have plotted this impact factor divided by $k_a^2$ in
Fig.~\ref{fig:imp_fac} for $q=0$~GeV and the mass of the meson $m=3.1$~GeV.
\begin{figure}[tbp]
\centering
\epsfig{file=imp_fac.eps,width=10cm}
\caption{The impact factor $\Phi_A(k_a)/k_a^2$ for $q=0$~GeV and $m=3.1$~GeV.}
\label{fig:imp_fac}
\end{figure}
One further complication arises compared to studies of the four--point gluon
Green's function due to the integration over $k_a$ and $k_b$ when calculating
the differential cross--section as in Eq.~(\ref{eq:xsec}). The
approximations of Eqs.~(\ref{approximation}) and~(\ref{eq:trajappr}) are valid
only when $\lambda\ll k_a$. When $k_a$ is integrated over it is therefore
not possible to have $\lambda$ fixed. This situation is similar to the one
encountered in Ref.\cite{Andersen:2004nm}. This problem can be solved in
several different ways. In the present analysis we choose to always have
$\lambda<20 \, k_a$ and $\lambda\le1$~GeV. We have checked that the results
here presented do not depend on these choices.
On Fig.~\ref{fig:xsecvsdeltay} we have plotted the resulting cross--section
of Eq.~(\ref{eq:xsec}) as a function of the rapidity span of the BFKL
ladder for $q=0,1,2$~GeV, $\alpha_s=0.2$ and $m=3.1$~GeV. The exponential
rise of the cross--section as a function of rapidity is evident for all $q$.
In the previous section we showed how the Green's function diminishes as
the momentum transfer rises, this translates here into smaller cross--sections.
\begin{figure}
\centerline{\epsfig{file=xsec_Dy_q.eps,width=10cm}}
\caption{The toy cross--section as a function of the rapidity separation of the
produced vector mesons ($m=3.1$~GeV), each produced with transverse momentum $q=0,1,2$~GeV.}
\label{fig:xsecvsdeltay}
\end{figure}
In Fig.~\ref{fig:xsecvsq} we have plotted the $q-$dependence of the
cross--section for rapidity spans of ${\rm Y}=0,1,2,5$. We see an exponential
fall--off with $q$ for all Y. Furthermore, we observe that
$\mathrm{d}\sigma/\mathrm{d}t$ at $t=0$~$\mathrm{GeV}^2$ increases with Y.
The curves in Fig.~\ref{fig:xsecvsq} are presented on a linear (right) and
a logarithmic (left) scale.
\begin{figure}
\begin{center}
\epsfig{file=log_xsec_q.eps,width=7.5cm}
\epsfig{file=xsec_q.eps,width=7.5cm}
\end{center}
\caption{The toy cross--section as a function of the transverse momentum of
the produced vector mesons, for rapidity separations of ${\rm Y}=0,1,2,5$ units
of rapidity.}
\label{fig:xsecvsq}
\end{figure}
These results could have also been obtained using an analytic approach to
solving the non--forward BFKL equation at LL accuracy. Nevertheless it is when
discussing diffusion properties that the method presented in this paper has
advantages. For the analysis of diffusion in cross--sections, a second issue
arises compared to the study of diffusion of the Green's function. The
perturbative scale is now set by both the meson mass $m$ and the momentum
transfer squared $-t$, while the scales $k_a$ and $k_b$ are no longer fixed. It
therefore becomes interesting to study not only the distribution of the
average transverse scale $\langle\tau\rangle$ along the ladder, but also at
the ends of it, where it will describe the average scale of the transverse
momentum connecting the impact factors to the gluon Green's function.
In Fig.~\ref{fig:xsec_diff} we have plotted the distribution of the average
momentum scale in a similar way to the one used in Fig.~\ref{fig:diff_q} for
the Green's function. It is interesting first to note that the spread along
the chain is not significantly larger than that at the ends of the chain (for
the rapidity spans considered here). Secondly, it is comforting to see that
the typical scales remain perturbative for all values of the momentum
transfer $q$, for the chosen value of the mass of the vector meson
($m=3.1$~GeV). The average value of $\tau$ for $q=0$~GeV
($\langle\tau\rangle\approx3.9$) corresponds to a scale of the internal momenta of
the BFKL exchange of roughly 7~GeV. The logarithmic scale of momenta on
Fig.~\ref{fig:diff_q} and Fig.~\ref{fig:xsec_diff} emphasises the region of
soft momenta. It might therefore be helpful to study the effect of the
increase in scattering momenta on the internal scales directly: an increase
of $q$ from 0~GeV to 2~GeV leads to and increase in the upper (UV) lines
corresponding to an increase in the internal momenta of roughly 2.4~GeV, while
the increase in the lower (IR) lines corresponds to an increase of roughly
1.7~GeV.
\begin{figure}
\begin{center}
\epsfig{file=xsec_diff_Y_2.eps,width=7.5cm,angle=0}
\epsfig{file=xsec_diff_Y_3.eps,width=7.5cm,angle=0}
\end{center}
\caption{The range of relevant internal transverse momentum scales in terms
of $\langle\tau\rangle, \langle\tau\rangle+\sigma_1,
\langle\tau\rangle-\sigma_2$ for the diffractive cross-- section for
$\alpha_s=0.2$, $m=3.1$~GeV, ${\rm Y}=2,3$, and $q=0,1,2$~GeV. ${\rm Y}'$ is
the rescaled (to unity) rapidity along the BFKL ladder.}
\label{fig:xsec_diff}
\end{figure}
\section{Conclusions}
We have presented a new solution to the non--forward BFKL equation at leading
logarithmic accuracy that allows a study of diffusion properties directly in
momentum space. We have investigated the behaviour of the gluon Green's
function as a function of transverse scales, including a study of the
IR/UV diffusion. Then we extended this study to the analysis of a
toy cross--section.
The presented framework is very efficient for solving BFKL evolution
equations and the solution allows immediate insight into the momentum
configurations of the evolution. We hope to extend the iterative method for
solving the non--forward BFKL equation to next--to--leading logarithmic
accuracy, once the appropriate integral kernel is calculated.
\subsubsection*{Acknowledgements}
\label{sec:acknowledgements}
We acknowledge useful discussions with Jochen
Bartels, Victor Fadin, Jeff Forshaw, Lev Lipatov and Leszek Motyka. ASV
would like to thank the Cavendish Laboratory at the University of Cambridge
and both authors wish to thank the CERN Theory Division for hospitality. JRA
acknowledges the support of PPARC (postdoctoral fellowship PPA/P/S/2003/00281).
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,085
|
\section{Introduction}\label{intro}
Astrometric measurements have revealed that a subset of the Milky Way (MW) satellite galaxies coherently orbit their host galaxy within a spatially thin plane (`thin' describes systems with minor-to-major axis ratios of $c/a \lesssim 0.3$, and `coherent' indicates that a majority of satellites share the same orbital direction) \citep[e.g.,][]{LyndenBell1976,Kroupa2005,Pawlowski2012a}.
Recently, precise proper motions from $\textit{Gaia}$ Data Release 2 have affirmed an even tighter orbital alignment of MW satellites than previously measured \citep{Fritz2018,Pawlowski2020}.
Similar structures have also been observed around Andromeda (M31) \citep{Ibata2013,Conn2013} and Centaurus A \citep{Muller2018}.
However, the spatial and kinematic coherence of satellite planes beyond the Local Group (LG) is less certain because of projection effects, distance uncertainties, and the inaccessibility of proper motions.
Even at the relatively close distance of M31, currently only two of its satellites have measured proper motions \citep{Sohn2020}, making it difficult to determine true 3D orbital alignment of the entire satellite population.
The cosmological significance of these satellite planes remains a topic of ongoing investigation, largely because of a lack of consensus on the incidence of planarity in both simulations and observations.
Studies using dark matter-only (DMO) simulations have often yielded conflicting interpretations of how rare satellite planes are in the standard cosmological model of cold dark matter with a cosmological constant ($\Lambda\mathrm{CDM}$).
Most analyses of DMO simulations find such configurations to be rare, highly significant, and therefore possibly in conflict with $\Lambda\mathrm{CDM}$ \citep[e.g.][]{Metz2008,Pawlowski2014,Buck2016}.
However, DMO simulations combined with semi-analytic models of galaxy formation suggest that planes might be more common \citep{Libeskind2009,Cautun2015}, but this is not a universal result \citep{Pawlowski2014b,Ibata2014a}.
Results from baryonic simulations have varied too, often relying on a much smaller sample of host-satellite systems compared to what is available from DMO simulations.
Some baryonic simulations show evidence for a more natural presence of satellite planes in the universe \citep[e.g.][]{Libeskind2007,Sawala2016}.
While other baryonic results show that satellite planes can be uncommon, but find conflicting evidence for whether planes can be explained by anisotropic satellite accretion along filamentary structures \citep{Ahmed2017,Shao2018,Shao2019}.
Beyond just checking for the presence and significance of satellite planes in simulations, several authors have also explored what may cause planes to form, with mixed results.
Though one might expect the host halo to affect satellite planes,
\citealt{Pawlowski2014} found no connection between planes and host halo properties.
Some authors have argued either for \citep{Libeskind2011} or against \citep{Pawlowski2012b} the preferential infall of satellites along cosmic filaments as a causal factor in the formation of satellite planes.
\citealt{Li2008} proposed the accretion of satellites in small groups as an explanation of correlated orbits, and \citealt{Wetzel2015a} showed that $25-50$ per cent of satellite dwarf galaxies in MW-mass hosts today previously were part of a group.
\citealt{Metz2007} has even speculated that satellite planes arise naturally from the creation of tidal dwarf galaxies in fly-bys or mergers of larger galaxies.
Several authors have investigated the orbital stability of LG satellite planes.
Recently, \citealt{Riley2020} showed that globular clusters and stellar streams around the MW do not seem to be members of the satellite plane, suggesting that plane members may be recently accreted or in a particularly stable orbital configuration.
\citealt{Pawlowski2017} noted that integrating present-day satellite orbits either forward or backward in time typically leads to the disintegration of the plane, especially when sampling measurement uncertainties on satellite galaxy positions and velocities.
\citealt{Shaya2013} took a different approach and, by searching the dynamical parameter space of Local Volume satellites, found past trajectories that could possibly lead to the observed satellite planes.
Many previous attempts to investigate satellite planes have relied on simulations that may not resolve the dynamical evolution of ``classical'' ($M_*\geq10^5\mathrm{\,\mathrm{M}_{\odot}}$) dwarf galaxies, or that do not include baryonic physics.
Insufficient resolution can lead to artificial satellite destruction \citep[e.g.][]{Carlberg1994,vanKampen1995,Moore1996,Klypin1999a,vanKampen2000,Diemand2007,Wetzel2010,vandenBosch2018}.
This may introduce a bias in satellite plane metrics if the destruction is spatially varying (such as near the host disk), and because earlier infalling satellites are preferentially destroyed, leading to an age bias that correlates with satellite orbit today \citep{Wetzel2015a}.
If baryonic effects act to create or destroy planes of satellites, then dark matter-only simulations may not be able to wholly capture the theoretical picture of satellite plane formation.
The central disk in baryonic simulations tidally destroys satellites, altering their radial profile at small distances from the host \citep[e.g.,][]{DOnghia2010,Sawala2017,GK2017b,Nadler2018,Kelley2018,RodriguezWimberly2019,Samuel2020}.
This leads the surviving satellites to have more tangentially biased orbits \citep{GK2017b,GK2019a}, but these effects do not necessarily imply an effect on planarity.
In addition, \citealt{Ahmed2017} found that the members of satellite planes in baryonic versus DMO simulations of the same host halo can be different, suggesting that baryonic effects may alter halo occupation in unexpected ways and hence affect satellite planes.
\citealt{GK2019a} also noted that satellites in baryonic simulations of LG-like pairs do not necessarily trace the most massive subhalos in DMO runs of the same systems.
Outside of the MW, the satellite plane around M31 is somewhat more ambiguous.
Taken as a whole, M31's satellites do not appear to be particularly planar, but a subset of 15 satellites lie within a significantly spatially thin plane and most of those are kinematically aligned, based on line-of-sight velocities \citep{Conn2013,Ibata2013}.
Many works have focused in on this particular subset, but it is important to understand the overall satellite distribution, because there are no clear evolutionary differences between M31 plane members and non-members \citep{Collins2015}.
Satellite planes outside of the LG are more difficult to robustly characterize because of projection effects and larger distance uncertainties.
Studies using the Sloan Digital Sky Survey (SDSS) database have revealed that while there is evidence for spatial flattening of satellites \citep[e.g.,][]{Brainerd2005}, their kinematic distribution is unlikely to indicate a coherently orbiting satellite plane \citep{Phillips2015}.
Furthermore, the Satellites Around Galactic analogues (SAGA) survey \citep{Geha2017}, which aims to study satellites of $\sim100$ MW analogues in the nearby Universe, has found little evidence for coherently orbiting satellite planes \citep{Mao2020}.
In this paper, we seek to understand if the FIRE-2 simulations contain satellite planes similar to those found in the Local Group, whether those satellite planes are long-lived or transient, and if the presence of satellite planes correlates with host or satellite properties.
We leave comparisons to systems outside of the LG for future work.
We organize this paper as follows: in Section~\ref{sims} we describe our simulations and satellite selection criteria, in Section~\ref{observations} we describe the 3D positions and velocities of Local Group satellites used, in Section~\ref{methods} we describe the plane metrics we apply to simulations and observations, in Section~\ref{results} we present our results of planarity in simulations compared to observations, and in Section~\ref{discussion} we discuss our conclusions and their implications for observed satellite planes.
\section{Simulations}
\label{sims}
The zoom-in simulations we use in this work reproduce the mass functions, radial distributions, and star formation histories of classical ($M_*\geq10^5\mathrm{\,\mathrm{M}_{\odot}}$) dwarf galaxies around MW/M31-like hosts \citep{Wetzel2016,GK2019a,GK2019b,Samuel2020}.
We use two suites of cosmological zoom-in hydrodynamic simulations from the Feedback In Realistic Environments (FIRE) project\footnote{\url{https://fire.northwestern.edu/}}.
Latte is currently a suite of 7 isolated MW/M31-mass galaxies with halo masses M$_{\rm 200m} = 1 - 2 \times 10^{12}\, \mathrm{M}_{\odot}$\footnote{`200m' indicates a measurement relative to 200 times the mean matter density of the Universe} introduced in \citealt{Wetzel2016}.
We selected the Latte halos for zoom-in re-simulation from a periodic volume dark matter simulation box of side length 85.5 Mpc.
We selected two of the Latte halos (m12r and m12w) to host an LMC-mass subhalo at $z=0$ within their initial DMO simulations, though after re-simulation with baryonic physics the orbital phase of these subhalos changes and they are no longer near pericenter \citep{Samuel2020}.
Latte gas and star particles have initial masses of $7070\, \mathrm{M}_{\odot}$, but at $z = 0$ a typical star particle has mass $\approx 5000\, \mathrm{M}_{\odot}$ because of stellar mass loss.
Dark matter particles have a mass resolution of m$_{\rm dm} = 3.5 \times 10^4\, \mathrm{M}_{\odot}$.
The gravitational softenings (comoving at $z > 9$ and physical at $z < 9$) of dark matter and stars particles are fixed: $\epsilon_{\rm dm} = 40$ pc and $\epsilon_{\rm star} = 4$ pc (Plummer equivalent).
The gas softening is fully adaptive, matched to the hydrodynamic resolution, and the minimum gas resolution (inter-element spacing) and softening length reached in Latte is $\approx 1$ pc.
We also use an additional simulation of an isolated MW/M31-mass galaxy (m12z), simulated at higher mass resolution (m$_{\rm baryon,ini} = 4200\, \mathrm{M}_{\odot}$).
The second suite of simulations we use is ``ELVIS on FIRE''.
This suite consists of three simulations, containing two MW/M31-mass galaxies each, wherein the main halos were selected to mimic the relative separation and velocity of the MW-M31 pair in the LG \citep{GK2014,GK2019a,GK2019b}.
ELVIS on FIRE has $\approx 2 \times$ better mass resolution than Latte: the Romeo \& Juliet and Romulus \& Remus simulations have m$_{\rm baryon,ini} = 3500\, \mathrm{M}_{\odot}$ and the Thelma \& Louise simulation has m$_{\rm baryon,ini} = 4000\, \mathrm{M}_{\odot}$.
We ran all simulations with the upgraded FIRE-2 implementations of fluid dynamics, star formation, and stellar feedback \citep{Hopkins2018}.
FIRE uses a Lagrangian meshless finite-mass (MFM) hydrodynamics code, \textsc{GIZMO} \citep{Hopkins2015}.
\textsc{GIZMO} enables adaptive hydrodynamic gas particle smoothing depending on the density of particles while still conserving mass, energy, and momentum to machine accuracy.
Gravitational forces are solved using an upgraded version of the $N$-body \textsc{GADGET-3} Tree-PM solver \citep{Springel2005}.
The FIRE-2 methodology includes detailed subgrid models for gas physics, star formation, and stellar feedback.
Gas models used include: a metallicity-dependent treatment of radiative heating and cooling over $10-10^{10}$ K \citep{Hopkins2018}, a cosmic ultraviolet background with early HI reionization ($z_{\rm reion}\sim10$) \citep{FaucherGiguere2009}, and turbulent metal diffusion \citep{Hopkins2016,Su2017,Escala2018}.
We allow gas that is self-gravitating, Jeans-unstable, cold (T $<10^4$ K), dense ($n>1000$ cm$^{-3}$), and molecular (following \citealt{Krumholz2011}) to form stars.
Star particles represent individual stellar populations under the assumption of a Kroupa stellar initial mass function \citep{Kroupa2001}.
Once formed, star particles evolve according to stellar population models from \textsc{STARBURST99} v7.0 \citep{Leitherer1999}.
We model several stellar feedback processes including core-collapse and Type Ia supernovae, continuous stellar mass loss, photoionization, photoelectric heating, and radiation pressure.
For all simulations, we generate cosmological zoom-in initial conditions at $z = 99$ using the \textsc{MUSIC} code \citep{Hahn2011}, and we save 600 snapshots from $z = $ 99 to 0, with typical spacing of $\lesssim$25 Myr.
All simulations assume flat $\Lambda\mathrm{CDM}$ cosmologies, with slightly different parameters across the full suite: $h = 0.68 - 0.71$, $\Omega_\Lambda = 0.69 - 0.734$, $\Omega_m = 0.266 - 0.31$, $\Omega_b = 0.0455 - 0.048$, $\sigma_8 = 0.801 - 0.82$, and $n_{\rm s} = 0.961 - 0.97$, broadly consistent with \citealt{PlanckCollaboration2018}.
\subsection{Halo finder}\label{halo_finder_subsection}
We use the \textsc{ROCKSTAR} 6D halo finder \citep{Behroozi2013a} to identify dark matter halos and subhalos in our simulations.
We include a halo in the catalog if its bound mass fraction is $> 0.4$ and if it contains at least 30 dark matter particles within a radius that encloses 200 times the mean matter density, R$_{\rm 200m}$.
We generate a halo catalog for each of the 600 snapshots of each simulation, using only dark matter particles.
The subhalos that we use in this work (within 300 kpc of their host) are uncontaminated by low-resolution dark matter particles.
We then construct merger trees using \textsc{CONSISTENT-TREES} \citep{Behroozi2013b}.
We describe our post-processing method for assigning star particles to (sub)halos further in \citealt{Samuel2020}.
First, we identify all star particles within 0.8 R$_{\rm halo}$ (out to a maximum 30 kpc) of a halo as members of that halo.
Then, we further clean the member star particle sample by selecting those (1) that are within 1.5 times the radius enclosing 90 per cent of the mass of member star particles (R$_{90}$) from both the center-of-mass position of member stars and the dark matter halo center, and (2) with velocities less than twice the velocity dispersion of member star particles ($\sigma_{\rm vel}$) with respect to the center-of-mass velocity of member stars.
We iterate through steps (1) and (2) until the total mass of member star particles ($\mathrm{M}_{\mystar}$) converges to within 1 per cent.
Finally, we save halos for analysis that contain at least 6 star particles and that have an average stellar density $> 300\, \mathrm{M}_{\odot} \, \mathrm{kpc}^{-3}$.
We performed this post-processing and the remainder of our analysis using the \texttt{GizmoAnalysis} and \texttt{HaloAnalysis} software packages \citep{WetzelHaloAnalysis2020,WetzelGizmoAnalysis2020}.
\subsection{Satellite selection}
Throughout this paper we refer to the central MW/M31-mass galaxies in our simulations as hosts, and their surrounding population of dwarf galaxies within 300 kpc as satellites.
Our host galaxies have stellar masses in the range $\mathrm{M}_{\mystar}\sim10^{10-11}\, \mathrm{M}_{\odot}$ and dark matter halos in the mass range $\mathrm{M}_\mathrm{200m}=0.9-1.7 \times10^{12}\, \mathrm{M}_{\odot}$.
The eight Latte+m12z simulations contain a single isolated host per simulation.
Each of the three ELVIS on FIRE simulations contains two hosts in a LG-like pair, surrounded by their own distinct satellite populations.
Thus, we use a total of 14 host-satellite systems to study satellite planes in this work.
Our fiducial redshift range is $z=0-0.2$ (114 snapshots), giving us a time baseline of $\sim2.4$ Gyr over which to examine the presence of satellite planes at late times in our simulations.
We present our results treating each snapshot as an independent realization and stacking snapshots across hosts.
This allows us to mitigate the time-variability and host-to-host scatter in the satellite distribution at small distances from the host, and achieve robust comparisons of simulations and observations.
We also consider a longer time window ($z=0-0.5$, 219 snapshots, $\sim5.1$ Gyr) in Section~\ref{time_subsection} in order to examine the lifetimes of planar structures and the coincidence of spatial thinness and kinematic coherence in our simulations.
We consider two ways to select simulated satellite galaxies for comparison to the MW.
Our primary method is to select a fixed number of satellites around each host, by choosing the 14 satellites with highest stellar mass from our simulations, to match the number of observed MW satellites that have $M_*\geq10^5\mathrm{\,\mathrm{M}_{\odot}}$.
We also choose the 15 most massive satellites around hosts for our comparison to M31 (see Section~\ref{m31_subsection} for more details).
Satellites with $M_*\geq10^5\mathrm{\,\mathrm{M}_{\odot}}$ contain $\geq20$ star particles and have peak halo masses of $M_\mathrm{peak}\geq8\times10^8\mathrm{\,\mathrm{M}_{\odot}}$ ($\gtrsim 2.3\times10^4$ dark matter particles prior to infall).
Satellite galaxies with $M_*\geq10^5\mathrm{\,\mathrm{M}_{\odot}}$ are also nearly complete in observations \citep[e.g.][]{Koposov2007,Tollerud2008,Walsh2009,Tollerud2014,Martin2016}, so we choose this as our nominal stellar mass limit to select satellites around the MW and M31.
As an example, at $z=0$, the satellite with the lowest stellar mass in our fixed-number satellite selection criteria has $\mathrm{M}_{\mystar}= 5.6\times10^4\, \mathrm{M}_{\odot}$ ($11$ star particles), which is enough to at least indicate the presence of a true satellite, given that it also satisfies the criteria outlined in Section~\ref{halo_finder_subsection}.
We also consider a stellar mass threshold selection method in Section~\ref{selection_effects_subsection} whereby we require satellites to have $M_*\geq10^5\mathrm{\,\mathrm{M}_{\odot}}$ and maintain the same distance cutoff ($d_\mathrm{host}\leq300\mathrm{\,kpc}$).
This selection means that the number of satellites considered around all hosts varies from 10 to 31 in the redshift range $z=0-0.2$.
See \citealt{Samuel2020} for more details on the radial distributions and resolution of simulated satellites meeting our criteria, and completeness estimates in the Local Group.
See \citealt{GK2019a,GK2019b} for how the stellar mass, velocity dispersion, dynamical mass, and star-formation histories of satellite dwarf galaxies in our simulations all broadly agree with MW and M31 observations, making these simulations compelling to use to examine planarity.
\section{Observations}\label{observations}
We consider all known MW satellite galaxies with $M_*\geq10^5\mathrm{\,\mathrm{M}_{\odot}}$ and $d_\mathrm{host}\leq300\mathrm{\,kpc}$, based on the satellite stellar masses and galactocentric distances listed in Table A1 of \citealt{GK2019a}.
While we are not confident that our halo finder is able to correctly identify analogues of the Sagittarius dwarf spheroidal (SgrI) galaxy, given its significant tidal interactions, we include it in our observational sample, because it is a historical member of the MW's satellite plane.
Excluding SgrI from the MW satellite galaxy sample does not significantly change the resulting spread in the MW's plane metrics, and therefore we achieve essentially the same results in our comparisons to simulations regardless of this choice.
For each observed satellite, we take the sky coordinates and heliocentric distances with uncertainties from \citealt{McConnachie2012}.
Furthermore, we include Crater 2 and Antlia 2, which meet our stellar mass and distance criteria as described in \citealt{Samuel2020}, and use the positions and uncertainties from their discovery papers \citep{Torrealba2016,Torrealba2018}.
This brings the total number of MW satellites that we consider in this study to 14.
We consider effects of observational incompleteness from the Galactic disk in Section~\ref{obs_completeness}.
We use proper motions from $\textit{Gaia}$ Data Release 2 as presented in \citealt{Fritz2018}.
We use the larger of the statistical or systematic uncertainties on $\textit{Gaia}$ proper motions, which typically is the systematic uncertainties.
We take line-of-sight heliocentric velocities ($v_\mathrm{los}$) for MW satellites and their uncertainties from \citealt{Pawlowski2020} and \citealt{Fritz2018}, where available.
To supplement this, we use the proper motions and $v_\mathrm{los}$ for the Magellanic Clouds presented in \citealt{Kallivayalil2013}, and Antlia 2's kinematics come from its discovery paper \citep{Torrealba2018}.
In our analysis of the MW satellite plane, we first sample the heliocentric distances and velocities (and proper motions) 1000 times assuming Gaussian distributions on the uncertainties.
We then convert these values to a Cartesian galactocentric coordinate system using Astropy \citep{astropy:2013,astropy:2018}.
We measure planarity on the resulting satellite phase space coordinates in the same way we describe for simulated satellites in Section~\ref{methods}.
We take a different approach to sample M31's satellites.
We impose the same stellar mass limit of $M_*\geq10^5\mathrm{\,\mathrm{M}_{\odot}}$ and 3D distance limit of $d_\mathrm{host}\leq300\mathrm{\,kpc}$, but we additionally require that the projected distance from M31 listed in \citealt{McConnachie2012} adhere to $d_\mathrm{host, proj}\leq150\mathrm{\,kpc}$, because M31's satellite population is most complete within this range from the Pan-Andromeda Archaeological Survey \citep[PAndAS,][]{McConnachie2009} coverage.
We sample 1000 line-of-sight distances for each satellite, using the posterior distributions published in \citealt{Conn2012} where available, and elsewhere assuming Gaussian distributions on distance uncertainties \citep{McConnachie2012,Martin2013a}.
We assume that M32 and NGC205 have the same posterior distance distribution as M31 itself because they are too close to M31 to reliably determine their line-of-sight distances.
The double-peaked posteriors of AndIX and AndXXVII cause the actual number of satellites within $d_\mathrm{host}\leq300\mathrm{\,kpc}$ of M31 in each sample to range from 14 to 16, but this is unlikely to cause significant differences in our analysis.
We take the line-of-sight velocities for M31 satellites from \citealt{McConnachie2012,Tollerud2012,Collins2013}, and we use them for the 2D kinematic coherence metric described in Section~\ref{methods}.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{plots/plane_metrics_diagram_3panel.png}
\vspace{-7 mm}
\caption{
Diagram showing each plane metric that we use, as measured on the 3D positions and velocities of 14 MW satellites ($M_*\geq10^5\mathrm{\,\mathrm{M}_{\odot}}$ and $d_\mathrm{host}\leq300\mathrm{\,kpc}$), shown in order of decreasing stellar mass.
All planes are centered on the MW and observational uncertainties are neglected here for visual clarity.
RMS height ($\Delta_\mathrm{h}$, left) is the root-mean-square distance of satellites from the satellite midplane.
Axis ratio (middle) is the ratio of the minor-to-major axes ($c/a$) from the moment of inertia tensor of satellite positions.
The ellipse shown has the same minor-to-major axis ratio as the MW's satellites.
Orbital pole dispersion ($\Delta_\mathrm{orb}$, right) is the root-mean-square angle in the range $[0^\circ, 360^\circ]$ of the angular momentum unit vectors of satellites around their average direction.
We show each metric in the same projection, to illustrate that the MW's satellite plane is kinematically coherent \textit{within} a spatially thin plane.
}
\label{fig:plane_metrics_diagram}
\end{figure*}
\section{Methods}\label{methods}
Figure~\ref{fig:plane_metrics_diagram} is a visual demonstration of how we measure planarity using two spatial metrics and one kinematic metric.
We show these metrics as measured on the MW's 14 satellites with $M_*\geq10^5\mathrm{\,\mathrm{M}_{\odot}}$ and $d_\mathrm{host}\leq300\mathrm{\,kpc}$.
For clarity we do not show the effects of observational uncertainties here, which have the largest effect on kinematic coherence, but we do include them in our analysis.
Our planarity metric definitions are based on and consistent with those from e.g., \citealt{Cautun2015,Pawlowski2015,Pawlowski2020}.
We require all planes to pass through the center of the host galaxy.
Below, we describe in detail each metric and how we calculated it at each simulation snapshot.
\subsection{Spatial metrics of planarity}\label{spatial_subsection}
We measure the spatial coherence of satellite galaxies in two ways: root-mean-square (RMS) height ($\Delta_{\rm h}$) and minor-to-major axis ratio ($c/a$).
The RMS height of a satellite distribution characterizes the vertical spread of satellites above and below a plane using the RMS component of satellites' 3D positions along the direction normal to a plane according to Equation~\ref{eq1}.
This can be thought of as the thickness or height of the plane.
We randomly generate $10^4$ planes centered on the host galaxy and quote the minimum value amongst these iterations.
\begin{equation}\label{eq1}
\Delta_{\rm h}=\sqrt{\frac{\sum_{\rm{i}=1}^{\rm N_{sat}}(\uvec{n}_{\perp}\cdot\vec{x}_{\rm i})^2}{\rm N_{sat}}}
\end{equation}
We also use the minor-to-major axis ratio ($c/a$) of the satellite spatial distribution to characterize satellite planes with a dimensionless metric.
This is the ratio of the square root of the eigenvalues of the inertia tensor corresponding to the minor ($c$) and major ($a$) axes.
We define a modified moment of inertia tensor treating satellites as unit point masses, weighting each one equally regardless of its stellar or halo mass, so it is a purely geometrical measure of the satellite distribution.
The elements of the 3D inertia tensor are given by Equation~\ref{eq2}.
\begin{equation}\label{eq2}
I_{ij} = \sum_{k = 1}^{\mathrm{N}_\mathrm{sat}}\sum_{\alpha = 1}^{3} \delta_{ij}r_{\alpha,k}^2 - r_{\alpha i,k}r_{\alpha j,k}
\end{equation}
We explored a third metric of spatial planarity, enclosing angle, motivated by the desire to mitigate effects of radially concentrated satellite distributions on planarity measurements.
We define enclosing angle as the smallest angle that encompasses the population of satellites, as measured off of the `midplane' of the satellite plane.
Similar to the galactocentric latitude ($b_c$) used in Section~\ref{selection_effects_subsection}, the coordinate origin is placed at the center of the host galaxy.
Enclosing angle ranges from 0 to 180 degrees by definition, where a measured angle of near 180 degrees indicates an isotropic distribution of satellites.
Similar to the method used for RMS height, in practice we randomly orient planes centered on the host galaxy from which to measure enclosing angle, and find the minimum angle from these iterations.
We found that this metric was significantly noisier over time compared to the other spatial metrics, and often selected a different plane orientation from RMS height and axis ratio, so we do not use it in our final analysis.
\subsection{Kinematic metrics of planarity}\label{kinematic_subsection}
We consider both 3D and 2D measures of orbital kinematic coherence of satellite populations to compare against observed 3D velocities of satellites in the MW, and line-of-sight velocities ($v_\mathrm{los}$) of satellites around M31.
The 3D metric we use is orbital pole dispersion ($\Delta_\mathrm{orb}$), which describes the alignment of satellite orbital angular momenta relative to the average satellite orbital angular momentum for the entire satellite population.
We are not taking into account the magnitude of satellite orbital velocities, so orbital pole dispersion is a measure of purely directional coherence in satellite orbits.
The orbital pole dispersion is defined as the RMS angular distance of the satellites' orbital angular momentum vectors with respect to the population's average orbital angular momentum direction, given by Equation~\ref{eq3}.
A system with all satellite orbital angular momenta aligned will have $\Delta_\mathrm{orb}=0^\circ$, while a random, isotropic distribution of satellite velocities has $\Delta_\mathrm{orb}\sim180^\circ$
\begin{equation}\label{eq3}
\Delta_{\rm orb}=\sqrt{\frac{\sum_{i=1}^{\rm N_{sat}}[\arccos(\uvec{n}_{\rm orb,avg}\cdot\uvec{n}_{\rm orb,i})]^2}{\rm N_{sat}}}
\end{equation}
To investigate 2D orbital kinematic coherence around M31 we examine whether satellites share the same `sense of orbital direction' around their host galaxy.
We measure this by computing the maximum fraction ($f^{\rm max}_{\vlos}$) of satellites with opposing (approaching or receding) $v_\mathrm{los}$ on the left and right `sides' of a satellite distribution.
A fraction close to unity indicates a highly coherent system, and a fraction of 0.5 represents a purely isotropic system.
We compute this fraction along $10^4$ randomly generated lines of sight.
\subsection{Statistically isotropic realizations of satellite positions and velocities}\label{isotropic_subsection}
To compare the `true' satellite planes (as measured at each snapshot) across different simulations, we quantify the likelihood of measuring thinner or more kinematically coherent planes in a statistically isotropic distribution of satellites.
This is a more general characterization of planarity, independent of the actual values measured for observed systems, that can also address whether satellite planes are statistically significant.
We generate isotropic realizations of satellite positions by randomly generating $10^4$ polar and azimuthal angles for each satellite, keeping their radial distance from the host fixed, following \citealt{Cautun2015}.
For isotropic kinematic distributions, we generate random unit velocities (using a similar prescription as for the randomization of angular coordinates) while also randomizing the angular spatial coordinates of each satellite.
We then take measurements of each plane metric for each of the $10^4$ realizations.
We quantify the significance of a planar alignment by quoting the fraction ($f_{\rm iso}$) of isotropic realizations with smaller values of plane metrics than the true value at each snapshot.
In effect this is the conditional probability of finding a more planar distribution of satellites among the isotropic realizations.
A fraction $f_{\rm iso}\leq0.5$ indicates that the true satellite distribution is more planar than a statistically isotropic distribution of satellites, and we define $f_{\rm iso}\leq0.05$ to mean the true satellite distribution is significantly planar.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{plots/kde_disk_sm_14_z02_3panel.png}
\vspace{-6 mm}
\caption{
Planarity of simulated satellite galaxies ($\mathrm{N}_\mathrm{sat}=14$ and $d_\mathrm{host}\leq300\mathrm{\,kpc}$) around MW/M31-mass hosts compared to the MW's satellite plane.
We model incompleteness in the simulations by excluding any satellites that lie within $\pm12^\circ$ of the plane of the host galaxy's stellar disk.
We generate KDEs (purple) using 114 snapshots over $z=0-0.2$ for each of the 14 simulated hosts, and the solid vertical colored lines are the distribution medians.
We show MW observations (black) for 14 satellites with 68 (95) per cent spread from observational uncertainties.
The number in the top right of each panel is the per cent of snapshots that lie at or below the MW upper 68 per cent limit.
For all metrics we consider, we find some ($1-5$ per cent) snapshots that are at least as planar as the MW, though they are rare.
}
\label{fig:mw_completeness_planes}
\end{figure*}
\section{Results}\label{results}
\subsection{Comparisons of simulations and the Local Group}
As we showed in \citealt{Samuel2020}, the simulations are a reasonable match to the radial distribution of satellites in the LG as a function of both distance from the host and stellar mass of the satellite.
This provided an important first benchmark of just the 1D radial positions of satellites in our simulation.
We now seek to leverage the full 3D positions and velocities of satellites in our simulations (and around the MW) to characterize satellite planes.
We compare our simulations to observations of LG satellites, leaving comparisons to systems such as other MW/M31 analogues and Centaurus A for future work.
In this section, we make physically rigorous comparisons using mock observations that include disk completeness corrections.
In subsequent sections we further explore selection effects on measured satellite planes and possible physical origins of satellite planes.
\subsubsection{MW-like planes}\label{mw_subsection}
We select the 14 most massive satellites in $\mathrm{M}_{\mystar}$ within $d_\mathrm{host}\leq300\mathrm{\,kpc}$ to compare planarity in simulations and the 14 MW satellites in our observational sample.
Furthermore, we apply a simple completeness correction for seeing through the MW's disk by first excluding all satellites that lie within a galactocentric latitude of $|b_c|\leq12^\circ$ from the host's galactic disk \citep{Pawlowski2018}, and then choosing the 14 most massive satellites from the remaining population.
See Section~\ref{selection_effects_subsection} for an investigation of how disk incompleteness affects planarity metrics.
Figure~\ref{fig:mw_completeness_planes} shows plane metrics for simulated satellites stacking over 114 snapshots spanning $z=0-0.2$, compared to the MW satellite plane.
Spatial plane metrics for the MW are tightly constrained by well-measured 3D positions of MW satellites.
The MW's satellite plane is thinner and more kinematically coherent than most of our simulated satellite systems.
We define MW-like planes as those with plane metrics at or below the one sigma upper limit on the MW's corresponding distribution.
Notably, the MW's plane is significantly spatially flattened compared to the average simulation when measured by RMS height and axis ratio.
While MW-like spatial planes are rare in our simulations, we do identify satellite populations that are as thin as the MW's plane in $1-2$ per cent of our full sample.
We compute each plane metric independently, but we discuss instances of satellite planes that are simultaneously both thin and kinematically coherent in Section~\ref{time_subsection}.
The occurrence of thin planes in $1-2$ per cent of snapshots holds over both our fiducial time baseline of $z=0-0.2\approx2.4$ Gyr (114 snapshots per host, 1,596 snapshots in total) and also over the longer interval $z=0-0.5\approx5.1$ Gyr (219 snapshots per host, 3,066 snapshots in total), an indication of the robustness of the measurement.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{plots/hist_m31_nsat15iter_z02_3panel.png}
\vspace{-6 mm}
\caption{Planarity of the 15 simulated satellite galaxies with the highest stellar mass within $d_\mathrm{host, proj}\leq150\mathrm{\,kpc}$ of each MW/M31-mass host (consistent with completeness in PAndAS).
We generate KDEs (green) using 114 snapshots over $z=0-0.2$ for each of the 14 simulated hosts.
The solid vertical colored lines are the medians of each distribution.
We show the M31 data (black) for 16 satellites with 68 (95) per cent spread in plane metrics from line-of-sight (LOS) distance uncertainties.
The number in the top of each panel is the per cent of snapshots that are at least as planar as the M31 upper 68 per cent limits.
Selected in this general way, the simulations are about as planar as M31.
The average RMS height (left) of simulations is somewhat thicker than M31's satellite population as a whole, but M31 is still within $\sim1\sigma$ of the simulation peak.
Typical simulation axis ratios (center) are even more similar to M31's satellites.
In the right panel, more planar snapshots are shown to the right of the M31 value.
LOS velocity uncertainties are too small to broaden the M31 velocity coherence measurement
M31's satellites are slightly more kinematically coherent than most simulations, but only by $\sim1\sigma$, consistent with the spatial planarity comparisons.
}
\label{fig:m31_planes}
\end{figure*}
The uncertainties in 3D velocities of MW satellites are much larger than the uncertainties on their 3D positions, and this leads to a much wider spread in orbital pole dispersion of the MW compared to the spatial metrics.
However, the MW's satellites still have highly correlated orbits relative to the simulations, with only 5 per cent of the simulations having a plane at least as kinematically coherent as the MW's upper one sigma limit during $z=0-0.2$.
The fraction of the full sample containing these planes actually increases to 8 per cent when measured over $z=0-0.5$, likely from the correlated infall of satellites in groups or along filaments at earlier times.
The spread in the MW's orbital pole dispersion is large compared to the spatial metrics, so we also provide the fraction of the simulation sample lying at or below the median MW value, 0.3 per cent.
There are even a few (5) snapshots that extend below the MW distribution.
The MW's satellite kinematics, while rare, do not appear to be extreme outliers compared to our simulations.
This broadly agrees with \citealt{Pawlowski2020}, who found that $\sim2-3$ per cent of hosts at $z=0$ in the IllustrisTNG simulations \citep{Pillepich2018,Nelson2019} have satellites as orbitally aligned as the MW.
However, the comparison between our work and theirs is not one-to-one: they vary the number of satellites included in plane calculations ($\mathrm{N}_\mathrm{sat}=3-11$) in both simulations and observations in order to account for the ``look elsewhere'' effect (the spurious detection of high significance events from searching a large parameter space), but they find that their conclusions do not vary for any number of plane members greater than three.
The IllustrisTNG simulations they use allow them to analyze a larger number of hosts, in part because they choose to include dark subhalos as satellites in order to maximize their sample size of hosts with at least 11 satellites.
The larger host sample size comes at the cost of resolution though, with $m_{\rm DM} = 7.5\times10^6 \, \rm{M}_{\odot}$, $m_{\rm baryon} = 1.4\times10^6 \, \rm{M}_{\odot}$, and $\epsilon_{\rm DM, *} = 0.74$ kpc.
In contrast, our planarity metrics are predicated on matching the number of observed satellites ($\mathrm{N}_\mathrm{sat}=14$) and we only have 14 hosts.
Instead, we leverage our time resolution to increase our sample size given that our planes are often transient features (see Section~\ref{time_subsection}).
Our simulations also have order-of-magnitude higher resolution, which may allow planes of satellites to survive that would be disrupted in lower resolution simulations. This is evidenced by their broad agreement with the MW and M31 in their radial distributions down to $\sim50$ kpc \citep{Samuel2020}.
Our measured plane metrics should be considered upper limits on absolute planarity at each snapshot.
If we instead varied $\mathrm{N}_\mathrm{sat}=3-14$, to test for the look-elsewhere effect, we might find even thinner or more coherent planes.
Likewise, our quoted fractions of MW-like planes are upper limits on the incidence of MW-like planarity, as this can only be diminished by accounting for the look-elsewhere effect.
Because we are always choosing a larger number of plane members ($\mathrm{N}_\mathrm{sat}=14$) than used by \citealt{Pawlowski2020}, which yields larger values of plane metrics in our case, we instead compare the fractions of our samples that are MW-like for consistency.
As a more rigorous test, we examine the instances of planarity for which simulations are simultaneously spatially thin and kinematically coherent.
We do not find any such simultaneously thin and coherent instances during $z=0-0.2$ in the simulations.
However, looking further back in time to $z=0.5$, we find 10 snapshots that are simultaneously as thin and kinematically coherent as the MW are today.
This amounts to 0.3 per cent of the total sample of snapshots over $z=0-0.5$.
This level of simultaneous spatial and kinematic planarity agrees with \citealt{Pawlowski2020}, who find that thin and coherent MW-like planes occur in $<0.1$ per cent of IllustrisTNG hosts, than when we examine individual plane metrics.
Notably, the instances of simultaneous planarity in our simulations occur in 2 out of 14 hosts (m12b and m12z).
In both cases, the simultaneous spatial and kinematic planarity occurs around the time of the first pericentric passage of a massive ($\mathrm{M}_{\mystar}\geq10^8\, \mathrm{M}_{\odot}$) satellite galaxy.
The massive satellite that passes near m12b meets our criteria for being an LMC analog.
We explore the influence of LMC-like companions further in Section~\ref{lmc_subsection}.
We do not see a significant difference in planarity between satellites of isolated hosts and satellites of hosts in LG-like pairs.
Both the medians and ranges of plane metrics for each host type are essentially the same, so we do not further separate our results by host type.
In Section~\ref{isotropic_subsection}, when we compare true satellites distributions to statistically isotropic distributions, the paired and isolated hosts do not appear systematically different from each other either.
This is consistent with results from \citealt{Pawlowski2019}, who reported no significant differences in planarity between dark matter-only simulations of isolated MW-mass halos and paired LG-like halos in the ELVIS simulations \citep{GK2014}.
\subsubsection{M31-like planes}\label{m31_subsection}
For comparison to M31's satellites, we mimic the completeness of PAndAS in our simulations.
We first select all simulated satellites within $d_\mathrm{host}\leq300\mathrm{\,kpc}$.
Then, we randomly choose a line of sight from which to observe the simulation, and we select only the satellites that then fall within a (2D) projected radius of 150 kpc from the host galaxy.
We choose the 15 satellites with greatest stellar mass that fall within our mock PAndAS-like projection, to match the number of M31 satellites in our observational sample.
We repeat this process along $10^3$ random lines of sight.
In order to meet the 15 satellite criteria, we do not impose a lower limit on the stellar mass of satellites.
At $z=0$, the lowest mass satellite included in this sample has $\mathrm{M}_{\mystar} \approx 1.8 \times 10^4\, \mathrm{M}_{\odot}$.
While most simulations easily meet the 15 satellite criteria, there are a few hosts that have fewer than 15 luminous satellites within the mock survey area.
At $z=0$, four of the isolated hosts have fewer than 15 satellites selected (as few as 9 satellites) for some lines of sight, so we exclude those snapshots.
All simulations meet the satellite quota along most lines of sight, and in particular the hosts in LG-like pairs never suffer from this issue.
The results that we achieve with this satellite number selection method are essentially the same as for a stellar mass selection method ($M_*\geq10^5\mathrm{\,\mathrm{M}_{\odot}}$).
We use the full 3D phase space coordinates of these satellites to calculate spatial plane metrics, because the 3D spatial coordinates of each satellite within the coverage of PAndAS are well known.
We calculate planarity metrics along each of $10^3$ lines of sight at each snapshot over $z=0-0.2$ for each simulated host.
Figure~\ref{fig:m31_planes} shows that when considering the 15 most massive satellites, M31-like planes are common in our simulations.
In particular, the axis ratios of simulated satellite systems are typically as planar as the full sample of M31 satellites.
More than 10 per cent of simulations are more planar than M31 for RMS height, so M31 is slightly thinner than our average simulation, but still lies within $\sim1\sigma$ of the simulation median.
Furthermore, throughout $z=0-0.2$ the simulations have many instances of satellite configurations that are simultaneously as spatially thin and kinematically coherent as M31's satellites are under our selection criteria.
Radial (line-of-sight) velocities are currently the only kinematic information available for all of M31's satellites that we consider, so we cannot compute the 3D orbital pole dispersion of them as we did for the MW's satellites.
We quantify kinematic coherence of satellites using $f^{\rm max}_{\vlos}$, where a larger fraction indicates greater kinematic coherence (see Section~\ref{kinematic_subsection} for details).
As Figure~\ref{fig:m31_planes} shows, 14 per cent of simulations are more kinematically coherent than M31's satellites, though this is still within about $1\sigma$ of the simulation median.
None of our simulations have all satellites sharing the same sense of orbital direction.
\citealt{Buck2016} have pointed out that a 2D metric like $f^{\rm max}_{\vlos}$ likely overestimates the true 3D kinematic coherence, so we may be overestimating the kinematic coherence in both our simulated and observed samples.
The velocity coherence plot (right panel) is shown as a histogram because the underlying distribution is essentially discretely binned.
Because each satellite population contains 15 satellites, the fraction of satellites sharing coherent velocities varies from 0.53 to 1.0 in steps of $\sim0.07$ (see Section~\ref{methods} for calculation details).
We find that the M31 satellite population as a whole is not significantly more planar than our simulations.
This agrees with \citealt{Conn2013} who found that M31's overall satellite population is consistent with a statistically isotropic distribution of satellites, though the 15 most-planar of its satellites lie within an exceptionally thin (12 kpc) plane.
While \citealt{Buck2015} use a different plane fitting method different from ours (a fixed-height plane), they also recover many instances of satellite planes as thin as the most-planar subset of M31 satellites.
We stress that our comparison to observations is not predicated on selecting the most planar subset of satellites in either simulations or observations.
This is because we prioritize a wholistic view of the planarity of the satellite population as a whole, rather than highly planar subsets of those satellites.
Other than having coherent LOS velocities, which do not unambiguously indicate orbital coherence, the member satellites of M31's plane are not significantly different from non-members, suggesting that they do not have different formation mechanisms or evolutionary histories \citep{Collins2015}.
In addition, sampling the satellite distributions to calculate plane metrics is computationally expensive (see Section~\ref{observations}), and this is made more difficult by finding optimal planes for all satellite combinations.
We defer such an investigation to future work.
For the rest of this work, we do not investigate M31-like planes further.
Instead, we examine MW-like planes, given that completeness is more certain out to the virial radius, and precise 3D velocities of MW satellites are available.
The availability of 3D velocities of MW satellites provides a more realistic metric of kinematic coherence.
\begin{figure*}
\begin{multicols}{2}
\includegraphics[width=0.45\textwidth]{plots/mw_plane_lifetimes_sm_14_disk_z05_3metric.png}\par
\includegraphics[width=0.45\textwidth]{plots/generic_plane_lifetimes_sm_14_z05_3metric.png}\par
\end{multicols}
\vspace{-6 mm}
\caption{
Satellite plane lifetimes measured over $z=0-0.5$ (219 snapshots per host, $\sim25$ Myr spacing) for the 14 satellites with the greatest $\mathrm{M}_{\mystar}$ within $d_\mathrm{host}\leq300\mathrm{\,kpc}$.
We define lifetimes independently for each plane metric.
\textit{Left:} MW-like planes are those that have plane metrics at or below the MW upper one sigma limits.
We have applied the same completeness correction for seeing through the disk as in Section~\ref{mw_subsection}.
Such planes are rare and short-lived, with most lasting $<0.5$ Gyr and none surviving for longer than $\sim$1 Gyr.
Two out of the three instances of MW-like planes lasting $>$500 Myr occur in hosts that experience a pericenter passage of an LMC-like satellite.
\textit{Right:} Generic planes are any flattened or kinematically coherent systems whose plane metrics fall below the lower 68 percent limits of our simulations shown in red in Figure~\ref{fig:combined_selections_planes}.
Generic planes are also typically short-lived and many last for only a single snapshot.
Half of the hosts have an instance of a generic plane that last $>1$ Gyr, and two of those experience an LMC-like pericenter passage.
While some generic planes live for a few Gyr, those planes are not typically simultaneously spatially thin and kinematically coherent.}
\label{fig:plane_lifetimes}
\end{figure*}
\subsection{Statistical significance and lifetimes of planes}
\subsubsection{Statistical significance of planes}\label{statistical_subsection}
We now move from absolute metrics of planarity to a more general investigation of planarity, that is not predicated on MW or M31 observations.
We characterize the statistical significance of satellite planes in our simulations by randomizing the positions and velocities of satellites in order to form a statistically isotropic distribution as a control sample (see Section~\ref{isotropic_subsection} for how we set this up).
By generating $10^4$ isotropic iterations and acquiring plane metrics from them, we create a bank of plane metrics that one might expect to measure if the distribution is statistically isotropic.
This isotropic bank is used to compute plane significance by calculating the fraction ($f_{\rm iso}$) of isotropic iterations that are \textit{more} planar than the true measured value at each snapshot.
In effect, this provides an estimate of the probability of finding a thinner or more coherent plane in a random distribution of satellites.
Small fractions ($f_{\rm iso}<0.05$) indicate a rare plane with high significance, while larger fractions ($f_{\rm iso}\geq0.5$) show that the measured plane is consistent with an isotropic distribution of satellites.
We distinguish between two different measures of plane statistical significance: conditional probability and marginalized probability (following \citealt{Cautun2015}).
Marginalized probability refers to the significance of a system's planarity relative to an ensemble of planarity measurements on that system where the number of satellites considered is allowed to vary from the minimum number of points needed to define a plane (3) to some maximum
We concentrate our analysis on conditional probability, because it represents the significance of a system's planarity given a certain set of constraints (such as completeness or total number of satellites).
We calculate the significance of planes on simulations across $z=0-0.2$, and on the observed positions and velocities of MW satellites.
Again, for the simulations, we remove satellites obscured by the host disk at $|b_c|\leq12^\circ$.
This is the same selection that we used in Figure~\ref{fig:mw_completeness_planes}.
By these simple metrics, and without correcting for selection or the look-elsewhere effect, the MW's plane is highly significant relative to a statistically isotropic distribution.
Less than one per cent of the MW's isotropic realizations of its satellites have a thinner plane ($f_{\rm iso}=0.003$ for RMS height or axis ratio), or a more kinematically coherent plane ($f_{\rm iso}=0.005$ for orbital pole dispersion).
In comparison, many of our simulation snapshots have median $f_{\rm iso}\gtrsim0.5$, indicating that they are broadly consistent with and have no meaningful degree of planarity relative to a statistically isotropic distribution of satellites.
See Appendix~\ref{isotropic_appendix} for a visual representation of $f_{\rm iso}$ for each host during $z=0-0.2$.
About half of both the isolated and paired hosts have median $f_{\rm iso}<0.5$, and this similarity indicates that the paired host environment does not significantly enhance the statistical significance of satellite planes.
About half of the hosts have $\sim5-10$ per cent of their snapshots with $f_{\rm iso}<0.05$, indicating significant spatial planes for these particular snapshots.
Only 3 out of the 14 hosts have significant kinematic coherence relative to a statistically isotropic distribution of satellite velocities, consistent with previous studies \citep[e.g.][]{Metz2008,Pawlowski2014,Ahmed2017,Pawlowski2020}.
Notably, none of the hosts have satellites that are simultaneously highly spatially significant ($f_{\rm iso}<0.05$) and highly kinematically significant relative to a statistically isotropic distribution at any snapshot during $z=0-0.2$.
In general, hosts that with small ($<0.25$) median $f_{\rm iso}$ for spatial planarity metrics do not have correspondingly small $f_{\rm iso}$ for kinematic coherence (orbital pole dispersion), and vice versa.
While our simulations contain instances of planes that are simultaneously as spatially thin and kinematically coherent at the MW in an absolute sense (by directly comparing plane metrics), the planes found in our simulations are not as significant relative to a statistically isotropic distribution.
\subsubsection{Lifetimes of planes}\label{time_subsection}
Thus far we have focused our analysis on the spatial and kinematic coherence of satellite galaxies in our simulations over $z=0-0.2$ ($\sim$2.4 Gyr).
In this section, we seek to understand if the satellite planes we find are long-lived and stable, or merely transient configurations, across $z=0-0.5$ ($\sim5.1$ Gyr, 219 snapshots).
This longer time baseline allows us to examine the time evolution of satellite plane structures as satellites make multiple orbits around their host.
A satellite in the inner regions of its host's halo may complete an orbit in under 1 Gyr.
Satellites in the most outer regions of the host halo take $\sim$3-4 Gyr to undergo a complete orbit.
We consider a plane to be ``long-lived'' if it persists for $\geq1$ Gyr, lasting for at least one satellite orbital timescale in the inner halo.
We deem any planar configurations lasting $<1$ Gyr to be ``short-lived'', and we consider those lasting $<500$ Myr to be ``transient'' alignments that do not indicate coherence amongst satellite orbits because they are so short.
We examine the distribution of plane lifetimes over $z=0-0.5$ separately for MW-like planes and generically flattened satellite systems.
We define MW-like planes as those with plane metric values at or below the upper 68 per cent limits on MW values: RMS height $\leq28$ kpc, axis ratio $\leq0.24$, or orbital pole dispersion $\leq67^\circ$.
``Generically'' flattened means having plane metric values: RMS height $\leq54.5$ kpc, axis ratio $\leq0.45$, or orbital pole dispersion $\leq74^\circ$, defined by the lower 68 per cent limit on simulation plane metrics during $z=0-0.2$).
We measure plane lifetimes ($\Delta t_{\rm plane}$) as the amount of time that a system spends consecutively at or below these plane metric thresholds.
Whether a satellite system is planar for only a single snapshot ($\sim25$ Myr) or many consecutive snapshots, we count it as a single instance of planarity.
Figure~\ref{fig:plane_lifetimes} shows that for both MW-like and generic planes, most planar instances are transient alignments and many last for just one snapshot ($\Delta t_{\rm plane} < 25\, \rm{Myr}$).
There are 348 snapshots with MW-like planes in our simulations across all hosts over $z=0-0.5$ (219 snapshots per host, 3,066 in total) and amongst all three 3D plane metrics.
Out of the total 89 separate instances of MW-like planes, most (56) are in kinematic coherence and only one of them lasts for $\gtrsim1 \text{Gyr}$.
This only occurs for one host, m12b, which also happens to experience a close passage of an LMC-like satellite during that time, that we discuss further in Section~\ref{lmc_subsection}.
There are 1,796 snapshots and 177 separate instances of generically flattened planes in our simulations.
Almost all generic planes last $<1$ Gyr, with only a small fraction ($<10$ per cent) of separate instances extending up to 3 Gyr.
One host, m12f, has a generic kinematic plane lasting 3 Gyr and also experiences an LMC-like passage during this time.
We conclude that satellite planes in our simulations, regardless of exact plane metric, are typically transient alignments that do not indicate a long-lived orbiting satellite structure, though the presence of LMC-like satellites can lead to longer-lived planes.
We also examine our simulations for instances of satellite configurations that are simultaneously spatially thin and kinematically coherent.
We use the same plane metric thresholds as above to look at how often a satellite system meets the kinematic threshold and at least one of the spatial thresholds at the same snapshot.
We do not find any instances of simultaneously thin and coherent MW-like planes over $z=0-0.2$ using either our fiducial selection method ($\mathrm{N}_\mathrm{sat}=14$ and $d_\mathrm{host}\leq300\mathrm{\,kpc}$) or combining that with a completeness correction due to seeing through the host's galactic disk.
However, there are several instances of coincident thinness and coherence over $z=0-0.5$, especially when we apply a completeness correction for seeing through the host disk.
In particular, m12b and m12r have up to 13 snapshots ($\sim325$ Myr) of simultaneous spatial and kinematic planarity over $z=0-0.5$.
We also consider the coincidence of generic planes, and find that m12b, m12r, and m12f all have snapshots with simultaneous spatial and kinematic planes even without implementing a completeness correction for the host disk.
Both m12f and m12b have LMC satellite analogues during this time, as we discuss in Section~\ref{lmc_subsection}.
Interestingly, none of our hosts in LG-like pairs exhibit simultaneous planarity, reinforcing the result that LG-like host environments are not more likely to have satellite planes.
\citealt{Shao2019} looked at plane lifetimes in the EAGLE simulations.
They considered both a different sample size ($\mathrm{N}_\mathrm{sat}=11$) and a longer time baseline ($z\approx0-2\approx10.5$ Gyr).
This leads them to identify thinner planes in an absolute sense, because fewer satellites create a thinner plane.
This time window may also catch some MW-like hosts as they are still being formed by mergers of smaller galaxies and before they have been able to form most of their stellar mass \citep[e.g.][]{Santistevan2020}.
However, they too found that most instances of MW-like spatially thin planes were short-lived ($<1$ Gyr), but some systems remain orbitally coherent for upwards of 4 Gyr.
Though we do not find such long-lived kinematic planes in our sample, this generally agrees with our findings.
\subsection{Selection effects on measured planarity}
\subsubsection{Observational incompleteness from the host disk}\label{obs_completeness}
In our analysis of MW-like planarity thus far, we have applied a fixed obscuration correction for seeing through the host disk, masking out everything that lies within $|b_c|\leq12^\circ$ (where $c$ indicates a galactocentric coordinate system).
We now analyze how the relative incidence of MW-like planes changes as a function of how much of the sky is obscured by the host's disk.
We vary the region obscured by the galactic disks of simulated hosts from $b_c=0^\circ$ (completely unobscured) to $|b_c|\leq45^\circ$ (majority obscured) in increments of $\Delta|b_c|=3^\circ$.
For each obscured region we select the 14 most massive satellites in $\mathrm{M}_{\mystar}$ within $d_\mathrm{host}\leq300\mathrm{\,kpc}$ of a host to use in the plane sample.
We define the relative incidence of MW-like planes as follows: we compute the fraction of snapshots with MW-like planes for each obscured region, and normalize it to the `true' (unobscured, $|b_c|=0^\circ$) fraction of snapshots with MW-like planes.
We repeat this process for each plane metric individually.
However, for $|b_c|\geq30^\circ$ there are typically fewer than 14 luminous satellites in the unobscured region and near $|b_c|\sim40^\circ$ there are only about 10 satellites available on average, so we cannot draw strong conclusions about completeness effects in those limits.
Figure~\ref{fig:host_disk} shows the incidence of MW-like planes, measured independently for each metric, as a function of disk obscuration angle.
We find that such incompleteness artificially boosts the fraction of snapshots with MW-like spatial planes for any value of $|b_c| > 0$.
In particular, near the fiducial obscuration we adopt for MW-like planes in previous sections ($|b_c|=12^\circ$), the incidence of MW-like planes is increased by about an order of magnitude.
For $|b_c|\lesssim40^\circ$, disk obscuration has a much smaller and opposite effect on kinematic planarity compared to spatial planarity; MW-like kinematic planes tend to be somewhat washed out by incompleteness.
Near our fiducial obscuration for the MW, the relative incidence of MW kinematic planes is about 0.77.
As expected, disk obscuration has the largest effect on planarity when $|b_c|\sim45^\circ$, where so much of the sky is obscured that any detected satellites would appear to be in a plane purely due to incompleteness.
Our results show that observational incompleteness from the host disk can have a strong effect on measured spatial planarity.
If the MW's satellite population is incomplete from seeing through the Galactic disk at our fiducial level, then MW observations may be overestimating the spatial planarity of MW satellites by a factor of $\sim10-20$.
To a much lesser degree, MW observations may underestimate the kinematic coherence of satellites by a factor of $\sim1-2$.
Because this incompleteness may bias our analysis of the underlying causes of satellite planes, we only use a host disk correction when comparing directly to MW observations in Sections~\ref{mw_subsection} -- ~\ref{time_subsection}.
For the remainder of this paper, we do not include a host disk correction.
\begin{figure}
\includegraphics[width=\columnwidth]{plots/mwlike_vs_disk_obscuration_z02_sm14_norm.png}
\vspace{-5 mm}
\caption{
Effects of disk incompleteness on measured planarity.
We define the relative incidence of MW planes as the fraction of snapshots during $z=0-0.2$ with MW-like planes normalized to the true or unobscured fraction ($|b_c|=0^\circ$).
We select the 14 most massive satellites in $\mathrm{M}_{\mystar}$ within $d_\mathrm{host}\leq300\mathrm{\,kpc}$ of each host, but for $|b_c|\geq30^\circ$ there are usually fewer than 14 luminous satellites available.
The horizontal line represents consistency with the unobscured fraction.
The arrow shows the fiducial obscuration that we adopt for MW-like planes, $|b_c|=12^\circ$.
Spatial planarity ($c/a \leq 0.24$, $\Delta_\mathrm{h}\leq 28$ kpc) is much more affected by host disk obscuration than kinematic planarity.
Spatial planarity jumps an order of magnitude between $|b_c|=0^\circ$ and $|b_c|\sim10^\circ$.
Kinematic planarity ($\Delta_\mathrm{orb}\leq67^\circ$) is slightly diminished by host disk obscuration.
At $|b_c|=12^\circ$, we are $8.5-18.5\times$ more likely to measure a MW-like spatial plane and $1.3\times$ less likely to measure a MW-like kinematic plane.
As expected, when nearly half of the sky is obscured spatial planarity is highly likely to be measured.}
\label{fig:host_disk}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{plots/kde_selection_z02_3panel.png}
\vspace{-6 mm}
\caption{
Planarity of simulated satellite galaxies ($d_\mathrm{host}\leq300\mathrm{\,kpc}$) selected using a fixed number method versus a stellar mass threshold.
Note that we do not include a correction for completeness due to seeing through the host galactic disk here.
We generate KDEs using 114 snapshots over $z=0-0.2$ for each of the 14 simulated hosts.
The vertical lines are the medians for each distribution.
The red distributions are the 14 most massive satellites in stellar mass, while the blue distributions are all satellite galaxies with $M_*\geq10^5\mathrm{\,\mathrm{M}_{\odot}}$.
Thin and coherent planes are rare in the simulations using these particular selections and time baseline, but using the number selection for satellites yields lower (more planar) metrics because the stellar mass selection allows for many more satellites to be included ($\mathrm{N}_\mathrm{sat}=10-31$).}
\label{fig:combined_selections_planes}
\end{figure*}
\subsubsection{Method of selecting simulated satellites}\label{selection_effects_subsection}
We also explore how using a fixed number selection for satellites compared to using a stellar mass threshold affects planarity measurements.
Our primary method of satellite selection throughout this work is to choose the 14 most massive satellites by rank-ordering them in stellar mass, because the number of satellites in a sample strongly correlates with the measured planarity \citep[e.g.][]{Pawlowski2019}.
In terms of observational completeness and resolution in simulations, another way to select satellites may be to impose a simple stellar mass threshold.
So we test our fixed number selection against a stellar mass threshold method: $M_*\geq10^5\mathrm{\,\mathrm{M}_{\odot}}$ and $d_\mathrm{host}\leq300\mathrm{\,kpc}$.
However, this leads to a range of numbers of satellites selected around each host, which makes it difficult to compare plane metrics across simulations and observations.
The total number of satellites with $M_*\geq10^5\mathrm{\,\mathrm{M}_{\odot}}$ and $d_\mathrm{host}\leq300\mathrm{\,kpc}$ per host varies from 10-31 during $z=0-0.2$ in our simulations.
Figure~\ref{fig:combined_selections_planes} shows that planes with $\mathrm{N}_\mathrm{sat}=14$ tend to be both thinner and more kinematically coherent than planes with $M_*\geq10^5\mathrm{\,\mathrm{M}_{\odot}}$, because while some $M_*\geq10^5\mathrm{\,\mathrm{M}_{\odot}}$ satellite populations have $\mathrm{N}_\mathrm{sat}<14$, more actually have $\mathrm{N}_\mathrm{sat}>14$.
One consequence is that when using the $\mathrm{M}_{\mystar}$ selection the simulations never reach the MW's RMS height (27 kpc) during $z=0-0.2$, but the fixed number selection does.
The small bump in the $\mathrm{N}_\mathrm{sat}=14$ orbital pole dispersion distribution is from a single host, m12f, during the snapshots following a close passage of an LMC-like satellite.
We discuss effects of such an LMC-like companion further in the following section.
This selection exercise highlights an important aspect of the satellite plane problem: many of the conclusions drawn about the nature of satellite planes are sensitive to satellite selection method, likely because of underlying sensitivity to the number of satellites in the sample.
Had we used the stellar mass threshold as our fiducial selection method in previous sections, we would have found more evidence for tension between simulations and observations, but deciding whether that tension is cosmologically significant is hampered by the sensitivity of plane metrics to both incompleteness and sample selection.
\subsection{Exploring physical explanations of planes}
\subsubsection{Influence of an LMC-like satellite}\label{lmc_subsection}
The presence of a massive satellite galaxy near pericenter, like the Large Magellanic Cloud (LMC), has been suggested as a possible explanation for the dynamical origin of the MW's satellite plane \citep{Li2008,DOnghia2008}, from the accretion of multiple satellites in a group with the LMC \citep[e.g.,][]{Wetzel2015b,Deason2015,Jethwa2016,Sales2017,Jahn2019}.
We seek to determine whether or not the presence of an LMC-like companion has an effect on the planarity in simulations.
We compare planarity metrics measured on systems experiencing an LMC-like passage to those without an LMC-like passage.
We identify pericentric passages of four LMC-mass analogues in our simulations based on the following selection criteria:
\begin{enumerate}[(i)]
\item $t_{\rm peri} > 7.5$ Gyr ($z < 0.7$)
\item M$_{\rm sub,peak} > 4\times10^{10}\, \mathrm{M}_{\odot}$ and $\mathrm{M}_{\mystar} > 10^{9}\, \mathrm{M}_{\odot}$
\item $d_{\rm{peri}} < 50$ kpc
\item The satellite is at its first pericentric passage.
\end{enumerate}
This broad time window allows us to capture a larger number LMC-like passages, which tend to be rare as we have defined them.
The minimum mass is consistent with measurements of the LMC's mass \citep{Saha2010}, and the maximum pericenter distance reflects the measured distance and orbit of the LMC \citep{Freedman2001,Besla2007,Kallivayalil2013}.
Table~\ref{tab:LMC-table} lists the four hosts in our simulations with LMC satellite analogues that meet these criteria, all of which are from simulations of isolated MW-like hosts rather than paired/LG-like hosts.
We emphasize that these satellites are not the only sufficiently massive satellites in the simulations, but that they are the only instances that satisfy all our LMC analogue criteria simultaneously.
To compare planarity during LMC-like passages and otherwise, we first select all snapshots within $\pm5$ snapshots (a time window of $\sim250$ Myr) of the LMC-like pericenter passage in each of the four simulations containing an LMC analogue.
This gives us a total of 44 snapshots that we classify as occurring close enough to an LMC analogue pericenter to exhibit any dynamical effects of group infall.
We compare plane metrics from those snapshots to plane metrics measured on all other simulations (excluding the four hosts with LMC analogues) up to the earliest snapshot included in the LMC sample ($z\sim0-0.7$, 247 snapshots per host).
We apply our fiducial disk obscuration correction, masking out all satellites within $|b_c|\leq12^\circ$ of the hosts' galactic disks in our simulations.
To calculate plane metrics we select $\mathrm{N}_\mathrm{sat}=14$ of the most massive satellites ranked by stellar mass.
\begin{table*}
\centering
\caption{
Properties of the LMC satellite analogues at their first pericentric passage about their MW/M31-mass host in our FIRE-2 simulations. We select satellites with M$_{\rm sub,peak} > 4 \times 10^{10}\, \mathrm{M}_{\odot}$ and $\mathrm{M}_{\mystar} > 10^{9}\, \mathrm{M}_{\odot}$ that have their first pericenter after 7.5 Gyr ($z < 0.7$) and within 50 kpc of their host.
}
\begin{tabular}{lllllll}
\hline
Host &
M$_{\rm sub,bound}$ [$10^{10}\, \mathrm{M}_{\odot}$] &
M$_{\rm sub,peak}$ [$10^{11}\, \mathrm{M}_{\odot}$] &
$\mathrm{M}_{\mystar}$ [$10^{9}\, \mathrm{M}_{\odot}$] &
$t_{\rm peri}$ [Gyr] &
$z_{\rm peri}$ &
$d_{\rm peri}$ [kpc] \\
\hline
m12b & 12.0 & 2.1 & 7.1 & 8.8 & 0.49 & 38 \\
m12c & 5.1 & 1.6 & 1.2 & 12.9 & 0.07 & 18 \\
m12f & 6.0 & 1.5 & 2.6 & 10.8 & 0.26 & 36 \\
m12w & 4.9 & 0.8 & 1.3 & 8.0 & 0.59 & 8 \\
\hline
\end{tabular}
\label{tab:LMC-table}
\end{table*}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{plots/kde_lmc_nsat14_mwpercent.jpg}
\vspace{-7 mm}
\caption{
Planarity of satellites of hosts experiencing a first pericenter passage of an LMC satellite analogue (red) compared to all other hosts without an LMC-like passage (blue).
We rank order satellites by stellar mass and choose the 14 most massive around each host within $d_\mathrm{host}\leq300\mathrm{\,kpc}$.
We select snapshots within $\pm125$ Myr of LMC-like passages that occur during $z=0-0.7$.
Only 4 hosts have such LMC-like passages (see Table~\ref{tab:LMC-table}).
Vertical colored lines are the medians of the simulation distributions.
MW planarity values are the vertical black lines and shaded regions.
We apply a disk obscuration correction and omit satellites within $|b_c|\leq12^\circ$.
LMC passages push towards $\sim20$ per cent lower plane metric medians and smaller ranges of spatial planarity metrics.
MW-like planes are $\sim2-3$ times more likely to be measured during an LMC-like passage.
}
\label{fig:lmc_passages}
\end{figure*}
Figure~\ref{fig:lmc_passages} summarizes our results for the planarity of satellites during an LMC analogue pericenter passage compared to all other satellite systems during $z\sim0-0.7$.
In general, the presence of an LMC analogue leads to thinner and more kinematically coherent satellite planes on average.
The presence of an LMC analogue shrinks the range of spatial plane metric values and slightly shifts them towards smaller (thinner) values.
In particular, the range of axis ratios is much smaller in the presence of an LMC analogue.
The right panel of Figure~\ref{fig:lmc_passages} also shows that the presence of an LMC analogue is correlated with more of the simulation distribution having tighter orbital alignment of satellites.
For all three metrics, we are $\sim2-3$ times more likely to measure a MW-like plane during an LMC pericentric passage compared ot the general simulation sample.
Thus, the presence of the LMC on first infall may contribute significantly to the thin and kinematically coherent satellite plane around the MW.
We find that the main reason for enhanced planarity in systems with LMC analogues is that the LMC analogues bring several satellites with them that are counted in the plane metric calculations, and because it is only at first pericenter there has not been enough time for the LMC and its satellites to dissociate from each other \citep[e.g.,][]{Deason2015}.
The four LMC analogues each bring in $2-4$ satellites with $M_*\geq10^5\mathrm{\,\mathrm{M}_{\odot}}$, consistent with the results presented in \citealt{Jahn2019} for both likely satellites of the LMC and FIRE-2 simulation predictions for satellites of LMC-mass hosts.
Of the $2-4$ LMC analogue satellites, $1-3$ of them are counted toward the $\mathrm{N}_\mathrm{sat}=14$ satellites in the plane sample.
The host with the most planar configuration that we find (m12b), which also has instances of simultaneous spatial and kinematic planarity, brings in four satellites with $M_*\geq10^5\mathrm{\,\mathrm{M}_{\odot}}$ and three of these (plus the LMC analogue itself) are counted in the plane calculations.
This means that the LMC analogue and its satellites account for $\sim30$ per cent of the plane sample, so spatial and kinematic coherence of the LMC subgroup can easily drive the measured plane metrics to lower values.
\citealt{Shao2018} examined whether anisotropic accretion or group accretion could explain the formation of satellite planes.
They ultimately found that most massive satellites were singly accreted, and that anisotropic accretion rather than group accretion correlated more with planarity.
In light of this we test for whether planarity correlates in general with group accretion and average infall times.
We find that overall most of the satellites in our sample were either singly accreted or accreted as groups of two.
We do not find a strong correlation between such group accretion and planarity in our full sample.
So we stress that our key result is that only a sufficiently massive LMC-like satellite near first pericenter shows a clear sign of enhancing planarity.
Two hosts with LMC analogues (m12f and m12b) experience extended periods of planarity ranging from $0.7-3$ Gyr (Section~\ref{time_subsection}).
We conclude that LMC-like satellites contribute significantly to satellite planes with moderate lifetimes ($\sim1-2$ Gyr), but that they are unlikely to have a permanent effect on the satellite distribution on longer timescales.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{plots/dmo_mpeak14_z02_3panel.png}
\vspace{-6 mm}
\caption{
Planarity in baryonic versus dark matter-only (DMO) simulations.
We compare the 14 most massive baryonic satellites and subhalos to DMO subhalos within $d_\mathrm{host}\leq300\mathrm{\,kpc}$.
We rank order subhalos by $\mathrm{M}_\mathrm{peak}$ and satellites by $\mathrm{M}_{\mystar}$.
We generate KDEs using 114 snapshots per host over $z=0-0.2$ for each of the hosts and the vertical colored lines are the medians of each distribution.
All three samples show similar planarity, but a few systems have baryonic satellites with greater kinematic coherence.
We conclude that there are no significant differences in planarity of baryonic versus DMO simulations.
}
\label{fig:dmo}
\end{figure*}
\subsubsection{Baryonic versus dark matter-only simulations}\label{dmo_subsection}
We also ran all of our simulations without baryonic physics, except for one of the isolated hosts, m12z.
We compare planarity of these dark matter-only (DMO) simulations to our baryonic simulations in order to investigate potential baryonic effects on satellite planes, given that many previous studies of planes have used DMO simulations.
We ran the DMO simulations with the same number of DM particles and the same gravitational force softening.
We compare planes in our baryonic simulations to planes in their DMO counterparts by selecting luminous satellites and dark matter subhalos within $d_\mathrm{host}\leq300\mathrm{\,kpc}$.
We choose the 14 most massive object from each sample by rank ordering satellite galaxies by $\mathrm{M}_{\mystar}$ and subhalos using $\mathrm{M}_\mathrm{peak}$.
Figure~\ref{fig:dmo} shows the distributions of planarity for satellite galaxies and subhalos, both selecting the top 14 subhalos by $\mathrm{M}_\mathrm{peak}$ and the top satellite galaxies by $\mathrm{M}_{\mystar}$, which are not identical samples because of scatter in the $\mathrm{M}_{\mystar} - \mathrm{M}_\mathrm{peak}$ relation.
The satellite galaxy distributions (red) are identical to those in Figure~\ref{fig:combined_selections_planes}.
While the three distributions have slightly different shapes, they have almost the same ranges and medians.
Using a rank ordering selection, the planarity of DMO subhalos is essentially identical to that of baryonic satellites.
We find that this general result is robust with respect to rank ordering subhalos by different properties such as $\mathrm{M}_\mathrm{halo}$, $\mathrm{V}_\mathrm{peak}$, and $\mathrm{V}_\mathrm{circ}$.
The one exception is a small population of baryonic satellites that extend to lower orbital dispersion values due to the passage of an LMC-like satellite (see Section~\ref{lmc_subsection}).
These results are surprising in light of the differences in the radial distributions of satellites and subhalos in our simulations, wherein DMO subhalos are more radially concentrated around their host than luminous satellites \citep{Samuel2020}.
One might expect to find thinner planes in DMO simulations because subhalos reside spatially closer to the host halo.
Alternatively, one might expect baryonic simulations to show greater planarity, given that the surviving population is biased to more tangential orbits \citep[e.g.][]{GK2017b}.
\citealt{Ahmed2017} observed both a difference in the significance and satellite membership of their planes in baryonic versus DMO simulations of the same four systems.
Satellite membership here refers to whether the satellites contributing to planes belong to the same subhalos in baryonic and DMO runs of the same systems.
While we do not explicitly consider differences in satellite membership, we do find that our DMO spatial planes are also typically more significant relative to a statistically isotropic distribution of satellites than their baryonic counterparts, with most having $P<0.5$ during $z=0-0.2$.
However, the significance of DMO kinematic planes is on par with the baryonic simulations.
So while we find that the significance of spatial planes may be slightly overestimated in DMO simulations relative to baryonic simulations, the absolute planarity is not much different from that in baryonic simulations.
If, instead, we select subhalos at a fixed value of $M_\mathrm{peak}\geq8\times10^8\mathrm{\,\mathrm{M}_{\odot}}$, DMO simulations typically have many more subhalos meeting this criteria.
This difference in number of subhalos in the plane sample reduces planarity in DMO simulations, because planes with more members are general less planar \citep{Pawlowski2019}.
\begin{table}
\centering
\caption{
Correlations between planarity in simulations and properties of the host and radial distribution of satellites.
We select satellite galaxies by rank ordering them by stellar mass at each snapshot and choosing the 14 most massive.
We measure host halo properties using only dark matter.
We quote the correlation coefficients ($r$) and $p$-values given by the Spearman correlation test.
For brevity, we only show correlations with $p<0.1$, though we note that only $p\lesssim0.01$ indicates a significant correlation in our sample.}
\label{tab:corr-table}
\begin{tabular}{llll}
\hline
Planarity metric & Host/system property & $r$ & $p$-value \\
\hline
RMS height & Host halo concentration & 0.49 & 0.07 \\
& Host $\rm{M}_{*}$ & 0.55 & 0.04 \\
& Host halo axis ratio (c/a) & 0.60 & 0.02 \\
& Host $\rm{M}_{*}/\rm{M}_{\rm{halo}}$ & 0.54 & 0.04 \\
& $R_{50}$ & 0.68 & 0.01 \\
\hline
Axis ratio & Host halo axis ratio (c/a) & 0.52 & 0.06 \\
\hline
Orbital dispersion & $R_{90}/R_{10}$ & 0.59 & 0.03 \\
& $R_{90}/R_{50}$ & 0.58 & 0.03 \\
\hline
\end{tabular}
\end{table}
\subsubsection{Correlations between plane metrics and host-satellite system properties}\label{correlations_subsection}
Finally, we explore relationships between satellite planarity and host and satellite system properties, but we find few correlations.
We quantify correlation using the Spearman correlation coefficient ($r$) and $p$-value, applied to the median value of each plane metric and host property over $z=0-0.2$ for all hosts.
None of the correlations that we found are particularly strong, as all have $r<0.7$.
We summarize correlations in Table~\ref{tab:corr-table}, where we only show correlations with $p<0.1$ for brevity.
Only correlations with $p\lesssim0.01$ indicate a statistically significant correlation in our sample, and there is only one correlation meeting this criteria.
We considered four host properties: stellar mass, dark matter halo mass ($\mathrm{M}_\mathrm{200m}$), stellar-to-total mass ratio, and halo concentration.
Both of the spatial metrics correlate with the host halo axis ratio, such that more triaxial host halos are more likely to have thinner satellite planes.
RMS height is also correlated with host stellar mass, whereby more massive host disks may act to disrupt rather than promote thin planes.
Orbital dispersion does not correlate significantly with any of the host properties.
While there is some evidence for spatial planarity correlating with host halo axis ratio, the correlations are not strong ($0.5<r<0.6$).
We conclude that it is unlikely that host properties drive the formation of satellite planes because we do not find strong and consistent correlations between planarity and host properties.
We also explored the alignment of planes with respect to both the host galaxy disk and the host halo minor axis, but we found no conclusive correlations among our sample.
Given the polar satellite plane around the MW, and the more aligned plane around M31, our results support that we expect no consistent correlation with the disk.
This results agrees with \citealt{Pawlowski2019}, who found that satellite plane metrics did not correlate with host properties like halo concentration or halo formation time in dark matter only simulations.
We also test for correlations between planarity and the radial distribution of satellites.
The strongest correlation that we find ($r=0.68$ and $p=0.01$) exists between RMS height and $R_{50}$, the radius enclosing 50 per cent of the satellites.
This correlation may arise from more satellites being near pericenter, rather than actually being flattened into a thin plane, because RMS height is a dimensional quantity (unlike dimensionless axis ratio), so we would expect it to correlate with satellite distances.
We also examine planarity as a function of $R_{90}/R_{50}$ and $R_{90}/R_{10}$, where $R_{90}/R_{50}$ is the ratio of the distance from the host that encloses 90 per cent of the satellite population to the distance from the host that encloses 50 per cent of the satellite population, and $R_{90}/R_{10}$ is similarly defined.
These ratios describe the radial concentration of the satellites around their host, and they are the only metrics that significantly correlate with orbital dispersion.
In both cases, more concentrated satellite systems are correlated with less kinematically coherent planes.
\section{Summary and Discussion}\label{discussion}
We explored the incidence and origin of planes of satellite galaxies in the FIRE-2 simulations, using satellites around 14 MW/M31-mass galaxies over $z=0-0.2$.
We compared to and provided context for satellite planes in the Local Group, including all satellites with $M_*\geq10^5\mathrm{\,\mathrm{M}_{\odot}}$ around the MW and within the PAndAS survey of M31.
We summarize our main results as follows.
\subsection{Rareness Of Planes}
\begin{itemize}
\item MW-like planes exist in our simulations, but they are relatively rare among our randomly selected $\sim10^{12}\,M_{\odot}$ halos at $z=0-0.2$: planes at least as thin or coherent as the MW's plane occur in $\sim1-5$ per cent of all snapshots, and planes as thin and coherent according to \textit{spatial and kinematic metrics simultaneously} occur in $\sim 0.3$ per cent of snapshots.
\item \textit{However}, if we select halos that feature a LMC-mass satellite analogue near its first pericentric passage, then the frequency of MW-like or thinner planes dramatically increases to $7-16$ per cent, with $\sim 5\%$ at least as thin as the MW plane by spatial and kinematic metrics simultaneously.
\item If we consider M31's satellite population as a whole, the planarity of satellites around M31 is common in our simulations. By every spatial or kinematic (or simultaneous) measure we consider, M31's satellites lie within $\sim1\sigma$ of the median of randomly selected halos of similar mass that we simulated.
\item Most of our simulations are not significantly planar relative to a statistically isotropic distribution of satellites.
\end{itemize}
\subsection{Physical Origins of Planes}
\begin{itemize}
\item Most MW-like thin satellite planes are transient and last $<1$\,Gyr in our simulations. However, the presence of an LMC satellite analogue near pericenter produces longer lifetimes of $\sim 0.7-3\,$Gyr. More generically flattened satellite systems survive for up to $\sim2-3$\,Gyr, even without requiring a massive satellite like an LMC analogue.
\item We do not find significant differences in planarity of satellites around hosts in Local Group-like pairs versus isolated hosts.
\item Dark matter-only (DMO) simulations show no significant differences in planarity compared to their baryonic-simulation counterparts, when selecting a fixed number of satellites in each sample.
\item Correlations between plane thickness and other satellite population properties (radial concentration) or host properties (mass, concentration, size, axis ratio) are generally modest or weak. Plane thickness is generally larger for more radially extended satellite distributions, as expected. The one property that strongly correlates with the presence of spatially thin and kinematically coherent planes is the presence of an LMC analogue near first pericentric passage.
\end{itemize}
\subsection{Observational and Selection Effects}
\begin{itemize}
\item Plane metrics can be sensitive to the satellite selection method in simulations and observations. Selecting just the 14 satellite galaxies with the highest stellar mass in the simulation produces thinner planes compared to selecting \textit{all} satellites with $M_{\ast}>10^{5}\,M_{\odot}$, because the latter tends to select more satellites, which produces thicker planes.
\item Incompleteness from the inability to see through the host galaxy disk (as in the MW) can increase the probability of measuring MW-like spatial planes by as much as a factor $\sim 10$. This bias is opposite in sign but much smaller for kinematic planes.
\item We have \textit{not} corrected in any of our analysis for any `look-elsewhere' effects, including the choice to look for `planes', the choice of definition of `plane', sample selection, number of satellites, etc. These corrections only would \textit{decrease} the statistical significance of the observed planes, as outliers from simulations.
\end{itemize}
\subsection{Discussion}
While only 1-2 percent of snapshots for all 14 hosts during z=0-0.2 contain planes at least as thin as the MW's, we do not interpret this as a tension with $\Lambda\mathrm{CDM}$ cosmology.
Instead, we identify the mere presence of MW-like planes in the simulations as evidence that cosmological simulation indeed can form thin planes of satellites, as long as they have adequate mass and spatial resolution.
We find that planes are much more common in the presence of LMC analogues, as suggested by \citealt{Li2008,DOnghia2008}, which provides evidence that future work should prioritize comparing the MW against simulations with an LMC analog.
Considering the entire M31 satellite population, M31-like satellite planes are common in our simulations, and combined with the fact that our simulations are only marginally more planar than a statistically isotropic distribution of satellites, this may indicate that M31's satellites as a whole are not significantly planar.
The lack of strong correlations between planarity and other properties of the host-satellite systems leaves us with few physical explanations for the MW's highly coherent satellite plane.
Our most promising result points to the presence of the LMC near first pericenter as a likely primary driver of planarity.
If our simulations are representative of the MW, then the observed MW plane is likely to be a temporary effect that will wash out in subsequent orbits of the LMC \citep{Deason2015}.
We have deliberately approached our analysis of satellite planes as agnostically as we can.
In choosing a fixed number of satellites for our nominal selection method, we have tried to both show the clearest comparisons between our simulations and LG observations, as well as mitigate the confounding effects of correlations between $\mathrm{N}_\mathrm{sat}$ and planarity.
Further studies of the most-planar subsamples of simulated satellites, as examined in \citealt{Pawlowski2013} and extended in \citealt{SantosSantos2020}, may yield more insight into the nature of satellite planes.
We defer an analysis of satellite sub-samples to future work.
We also have not yet considered a comparison to satellite systems outside of the LG.
There is evidence for satellite planes outside of the LG around Centaurus A \citep{Muller2018}, and recent studies have examined planarity around hosts in SDSS \citep{Ibata2014b,Brainerd2020} and the SAGA survey \citep{Mao2020}.
Connecting LG hosts to a statistical sample of similar hosts will be crucial in evaluating the significance of planar alignments and the validity of proposed formation mechanisms, demonstrating the need for large surveys with e.g., the Nancy Grace Roman Space Telescope, which promises to significantly augment the observational sample of MW analogues.
LG galaxies are also aligned with large scale structure, along a local sheet, which is not captured in our simulations and may play a part in the formation of satellite planes \citep{Neuzil2020}.
Simulations that can accurately reproduce this large scale structure may offer new insight into satellite planes \citep{Libeskind2020}.
\section*{Acknowledgements}
We thank Marcel Pawlowski for insightful comments and discussion that improved this manuscript.
This research made use of Astropy,\footnote{http://www.astropy.org} a community-developed core Python package for Astronomy \citep{astropy:2013, astropy:2018}, the IPython package \citep{ipython}, NumPy \citep{numpy}, SciPy \citep{scipy}, Numba \citep{numba}, and matplotlib, a Python library for publication quality graphics \citep{matplotlib}.
JS, AW, and SC received support from NASA through ATP grants 80NSSC18K1097 and 80NSSC20K0513; HST grants GO-14734, AR-15057, AR-15809, and GO-15902 from the Space Telescope Science Institute (STScI), which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555; the Heising-Simons Foundation; and a Hellman Fellowship.
We performed this work in part at the Aspen Center for Physics, supported by NSF grant PHY-1607611, and at the KITP, supported NSF grant PHY-1748958.
PFH was provided by an Alfred P. Sloan Research Fellowship, NSF grant \#1715847 and CAREER grant \#1455342, and NASA grants NNX15AT06G, JPL 1589742, 17-ATP17-0214.
MBK acknowledges support from NSF grant AST-1517226 and CAREER grant AST-1752913 and from NASA grants NNX17AG29G and HST-AR-13888, HST-AR-13896, HST-AR-14282, HST-AR-14554, HST-AR-15006, HST-GO-12914, and HST-GO-14191 from STScI.
CAFG was supported by NSF through grants AST-1517491, AST-1715216, and CAREER award AST-1652522, by NASA through grant 17-ATP17-0067, and by a Cottrell Scholar Award from the Research Corporation for Science Advancement.
We ran simulations using the Extreme Science and Engineering Discovery Environment (XSEDE) supported by NSF grant ACI-1548562, Blue Waters via allocation PRAC NSF.1713353 supported by the NSF, and NASA's HEC Program through the NAS Division at Ames Research Center.
\section*{Data Availability}
Full simulation snapshots at $z = 0$ are available for m12i, m12f, and m12m at \url{ananke.hub.yt}.
The publicly available software packages used to analyze these data are availble at: \url{https://bitbucket.org/awetzel/gizmo\_analysis}, \url{https://bitbucket.org/awetzel/halo\_analysis}, and \url{https://bitbucket.org/awetzel/utilities}.
\bibliographystyle{mnras}
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{"url":"https:\/\/academy.vertabelo.com\/course\/ms-sql-window-functions\/final-quiz\/final-quiz\/question-5","text":"Kickstart 2020 with new opportunities! - hours only!Up to 80% off on all courses and bundles.-Close\nIntroduction\nThe Final Quiz\n8. Question 5\nYou\u2019ve Completed the Windows Function Course!\n\n## Instruction\n\nThat's the right answer! Question 5 awaits.\n\n## Exercise\n\nFor each treatment, show the following information: TreatmentDate, DoctorId, Category, Name, Score and the average score of treatments that are in the same category and are included in the following window frame: two previous rows, the current row, and the three following rows, in terms of the treatment date. Name the column AvgScore.","date":"2020-02-19 17:41:41","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.20827452838420868, \"perplexity\": 6055.801495110024}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875144165.4\/warc\/CC-MAIN-20200219153707-20200219183707-00341.warc.gz\"}"}
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{"url":"https:\/\/www.reddit.com\/r\/math\/comments\/bopw2i\/why_is_math_something_that_could_be_interesting\/","text":"Press J to jump to the feed. Press question mark to learn the rest of the keyboard shortcuts\n3\nPosted by5 days ago\n\n## Why is math something that could be interesting?\n\nI could understand completely well why someone might pursue a career in a scientific field: biology, chemistry, physics, medicine, etc., but I've always wondered why anyone would choose a field like math. Throughout my life, like many others probably, math has just been a boring subject where you repeat formulaic steps over and over again. Even when I got to calculus, I found finding derivatives and integrals satisfying, but... not something I would want to do my whole life.\n\nI understand that I probably glossed over the whole field of math just a bit too much, but that's unfortunately the scope under which I view math in as it was taught to me in school.\n\nSo... how could you make me interested in math?\n\n54% Upvoted\nSort by\nlevel 1\n57 points \u00b7 5 days ago\n\n\"music is just a boring subject where you practice scales over and over again\"\n\nlevel 1\n35 points \u00b7 5 days ago \u00b7 edited 5 days ago\n\nThroughout my life, like many others probably, math has just been a boring subject where you repeat formulaic steps over and over again.\n\nThen you haven't actually done any mathematics.\n\nMathematics as a discipline is about finding proofs, not doing calculations. Mathematicians don't answer questions such as \"What is 13 squared subtract 13?\", they answer questions such as \"Is it necessarily the case that the square of any whole number, subtract the original number, is even?\". The former you can literally bash into a calculator and get the answer, but the latter actually requires thought to answer. (Sure, you could check a lot of numbers, but you'd never be able to check ALL of them. And even if you think the answer is obviously \"yes\", how do you KNOW that your answer is correct?)\n\nOr, have another example of a question that a mathematician might try to answer: \"If you cut two opposite corners out of a chessboard, can you cover the remaining 62 squares with 31 dominoes (each domino covering two adjacent squares)?\" Seriously, try it out yourself. (When you come to the conclusion that it's impossible, ask yourself \"OK, but how do I know that it is in fact impossible and that there isn't some sneaky configuration of dominoes that I've missed?\".)\n\nOf course, these two questions I've posed are fairly low-level mathematics problems, which have already been solved. Some example of problems that mathematicians haven't yet solved:\n\n\u2022 Does there exist a cuboid, for which all edges, all face diagonals, and the space diagonal are whole number lengths?\n\n\u2022 What is the shortest route, travelling along roads, that visits these seven cities? How quickly can a solution be found?\n\n\u2022 Given a photo of a knot in a loop of string, can you determine quickly if the knot can be untangled?\n\n\u2022 Given any loop in the plane that doesn't intersect itself, are there necessarily four points on the loop that form the vertices of a square?\n\n\u2022 What is the largest possible sofa you can move around a corridor?\n\nThe key thing is, these problems are fun and interesting. That's why mathematicians like to work on mathematics.\n\nlevel 1\nAlgebra12 points \u00b7 5 days ago\n\nThroughout my life, like many others probably, math has just been a boring subject where you repeat formulaic steps over and over again.\n\nIt's sad that this has been your experience of \"mathematics\". What you are describing has no mathematical content whatever. Mathematics is a completely different subject.\n\nlevel 2\nApplied Math4 points \u00b7 4 days ago\n\nYes but it's not OP's fault (at least not completely). High school maths, in most countries, is exclusively computational.\n\nCalculate this or solve that but it rarely explains why. Even a 5 year old could do derivatives if you showed them a table of common derivatives and explained them the product and chain rule. Simply substitute this expression when you see that other expression and do this thing when you find a product. Would that mean they understood derivatives? Certainly not, but that's not too dissimilar to what high school students go through.\n\nUnderstanding what maths is all about is something that students who are passionate about maths do on their own, and it works for us, but it's wrong. Teachers are there for a reason and they should help all students understand what maths is, there is no good excuse that justifies students doing it on their own.\n\nI know that it isn't maths teachers' fault, they have to prepare students for tests without indulging in proofs\/explanations because the current educational system is flawed.\n\nlevel 1\n12 points \u00b7 5 days ago\n\nUm, having spent my life doing math I can say that if not for the teaching aspect of it I would quite literally never do anything involving formulas (let alone formulaic steps) nor deal with any derivatives nor (Riemann) integrals.\n\nMath is about \"finding patterns\" in the general sense but better: math is the art of putting rigorous proof to the apparent patterns we all see. It's taught horribly so I don't blame you at all for your post, but there is a reason that those of us who do it find it so beautiful.\n\nlevel 1\n26 points \u00b7 5 days ago\n\nMath is about constructing arguments regarding abstract objects, it's not just calculating specific quantities ad nauseam. There are people who enjoy this. At this level there is nothing formulaic about it.\n\nlevel 1\n6 points \u00b7 4 days ago\n\nFor some perspective, if calculus is as far as you have gotten then all the math you have ever seen (aside from the notation used to present it) is essentially over 350 years old. I don't think your impressions of other subjects are based on how they were understood that long ago. Even if you took physics only at the high school level, you have still probably heard about 20th century developments like relativity and quantum mechanics.\n\nIf you want to get an impression of how far math has progressed beyond what you think of as math, take a look at the list of topics in current mathematical research areas in the 47 page file here: https:\/\/mathscinet.ams.org\/msc\/pdfs\/classifications2010.pdf. There are topics there that were known 350 years ago, such as derivatives and prime numbers, but modern math for the most part has nothing to do with what you learned in math classes.\n\nThe standard website where people post current research papers in math is the arXiv: see https:\/\/arxiv.org\/archive\/math. Going back 350 years ago, most of the subject areas listed there (excluding \"general mathematics\" and \"history and overview\") did not exist at all aside from some elementary examples.\n\nlevel 2\n1 point \u00b7 4 days ago\n\nEven if calculus is as far as someone gets that's still a lot of possible math. I think the major issue is not that the math seen in school is before the explosion in our subject but that only the most elementary aspects of the subject is seen. For example if a bit of Euclidean geometry is taught at all then it's the mostly boring bits. There are beautiful parts which are not show like Euclids construction of the pentagon inscribed in a circle (in book 4 I think). It is a superb piece of math that holds its own in comparison to modern math.\n\nlevel 1\n8 points \u00b7 5 days ago\n\nGalileo said it himself \"Mathematics is the language in which God has written the universe.\" Science is nothing without math to explain it, that's why it's so important. Using math you can predict the motion of an object, to explaining the electrical processes in your neurons that arise to thoughts. Even predicting the stock market can be as simple as a inch long equation (in a nutshell). Math even breaks down to the simplest forms of nature, Maxwell's equations explain how something as fundamental as light works. The odds of a sports team winning can be predicted with math too. Theres only a handful of things in nature that are truly unpredictable, and everything else can be explained mathematically.\n\nlevel 2\nPDE3 points \u00b7 5 days ago\n\nA similar quote to yours is\n\nMathematics is the poetry of rational thought\n\nWhich I believe was Einstein. And I think this captures both the interest and ubiquitousness of mathematics. It is an elegant language of formulating arguments at the end of the day. Almost anything can be \"mathematised\", I've seen talks on waves crashing, fish swarming black holes forming, electrons interacting, origami, juggling, knitting everything. As long as you're willing to commit an idea to a precise language it can be everywhere. And there's a beauty in this unified way of approaching problems.\n\nlevel 1\n7 points \u00b7 5 days ago\n\nbecause it's like a giant puzzle and puzzles are fun\n\nlevel 1\n2 points \u00b7 5 days ago\n\nThis isn't really a direct answer to your question, but something that I think people don't mention enough.\n\nThere are things that make mathematics very interesting (people have mentioned many things in the comments) but in general, mathematicians don't find every mathematical object or idea interesting. This should not be surprising - consider whatever field you find interesting. I'm sure you aren't completely enraptured by every single detail. For example, someone who studies philosophy might find moral philosophy to be very interesting, but might find aesthetics to be rather \"boring\". Likewise, the process of learning philosophy (or physics, or art, or biology, etc) necessarily will require you to learn certain topics which will be a bit \"less than interesting.\" But the motivation to learn them comes from the desire to understand the bigger idea that requires that knowledge.\n\nWhat's a bit unfortunate is that people become scared away from mathematics while they're learning the more \"boring\" things, and they never get to see the things that are more interesting, or more deep in some sense.\n\nAnd in case you were looking for a more direct answer, I would say look into chaos theory - it's basically the study of systems which are \"really complicated\" and evolved in really strange ways over time. If you look around online you can find some very simple explanations of chaotic systems that don't require any mathematics to understand. The most straightforward example is weather - small disturbances now could cause very large changes in the future (hence the so-called \"butterfly effect\").\n\nlevel 1\n2 points \u00b7 5 days ago \u00b7 edited 4 days ago\n\nMathematicians don't calculate derivatives and integrals all day. What I'm about to say is an oversimplification, but roughly speaking, engineers and scientists calculate integrals and derivatives, while mathematicians are the ones who build fields like calculus from the ground up and make sure they're on a solid logical foundation. Boradly speaking, the type of questions mathematicians are interested in are the questions that concern the structure of the mathematical objects and universes they choose to explore.\n\nFor instance, here's a totally artificial mind-numbingly boring \"math\" problem that you might get in a calculus class: Calculate the integral of of the function f(x) = e2x + sinxcosx\n\nHere's an actual math problem that interested mathematicians back in the day: Precisely which real-valued functions are integrable? How do you prove it? To see the solution to this problem, you would usually need to take an analysis course (it's now a standard theorem), but most people never take a \"real\" math course before giving up on math altogether.\n\nAnd there's a lot, lot more to math than simply answering these types of well-defined problems. For instance, a mathematician might take a familiar object, and ask if they can work with a much broader set of objects of which the original object is a special case. They try to say things about these objects that apply in as many cases as possible. Then they try to generalize these new general objects even further, so that they all fall under some very broad umbrella, losing some structure in the process but gaining the power to make increasingly more general statements that apply in more and more cases, and which allow you to eventually prove theorems that allow you to glean very deep and all-encompassing insights. And on the other side of the coin, a mathematician might take a very broad class of object and narrow their attention to a particular case that seems to exhibit all sorts of interesting behaviors that are absent for typical objects in the class, and try to see what hidden treasures they can dig up and how those insights can be applied in some other situation. Then they might again try to generalize so as to identify another class of objects with useful characteristics of which the special object of interest is a special case, perhaps a new class that succeeds in retaining the interesting structure that they've identified in that object. Or maybe they would try to find a generalized description of the behavior of that original object so that they're still talking about the particular object, but they're finding a wider umbrella to describe its behavior. So math is about finding balance between stuff like that, and also much, much more that I can't even begin to touch upon in one comment.\n\nSome other things, right off the top of my head: Mathematicians love definitions that formalize, in a satisfactory way, intuitive notions in different contexts like \"essentially the same\" or \"natural\" or \"bigger\/smaller\" or \"minimal\/maximal\" or \"holes\" or \"sufficiently\/arbitrarily close\" or \"association\" or \"messy,\" etc. Mathematicians tend to want proofs to follow from the most illuminating insight possible, rather than from a bunch of opague computations or clever tricks. Mathematicians constantly strive to reduce unfamiliar concepts to things they have already become familiar with, and this is essentially how all problems get solved. All of these issues are navigated with an aesthetic sense characteristic of creative human beings. But again, this barely covers a tiny portion of what mathematics is.\n\nThese remarks probably won't make too much sense until you've studied a lot of math and you've actually seen a lot of it in action. Calculus class almost always does a terrible job of showing you any of what I described.\n\nlevel 2\nApplied Math1 point \u00b7 4 days ago\n\nbut most people never take a \"real\" math course before giving up on math altogether.\n\nThat's probably the biggest issue with maths education. Introducing high school students to proofs as soon as possible is frowned upon but it makes everything much easier in my opinion.\n\nI sometimes help high school students with their maths homework and I always give them an outline of a proof because it usually helps them overcome the initial feeling of \"this is true because the teacher said so\" and understand that \"this is true because I can see it must be true\".\n\nStudents can choose to get through proofs by rote learning or they can choose to work through proofs understanding them. With the help of their teachers they'll hopefully find that understanding proofs is the easiest path even though it sounds harder.\n\nlevel 1\n2 points \u00b7 4 days ago\n\nThinking that math is about numbers and equations is like thinking English is about letters and words and Physics is about forces. Those are the \"tools\" (can't think of a better word) of those fields but they aren't the end goal or the purpose of those fields\n\nlevel 1\n2 points \u00b7 3 days ago\n\nPhysics would not exist as it is today wothout math, but math would exist just by itself fine.\n\nlevel 1\n1 point \u00b7 5 days ago\n\nI have long been fascinated by mathematics - despite my remarkable lack of understanding. I'm convinced that there is a great deal about the Universe that will remain unknowable until and unless I achieve that understanding. Still trying; I recently taught myself about logarithms and the trigonometric functions.\n\nWhen I was in university, my approach could best be described by the metaphor 'when the sage points at the moon, the fool looks at the finger'. Most of my mathematical education consisted of looking really hard at the finger and completely ignoring the moon. Truth be told, it wasn't until I was well into my thirties before I even realized that there was a moon.\n\nTo be frank, from my perspective maths education in the United States looks a lot like art or music education - it's a sieve to identify the students who are worth teaching and shunt everyone else in a different direction. Can't grasp the calculus the way we're teaching it? The Humanities building is over that way.\n\nlevel 2\nAlgebra4 points \u00b7 5 days ago\n\nwhen the sage points at the moon, the fool looks at the finger'.\n\nThat's a good one. Thanks.\n\nlevel 1\n1 point \u00b7 4 days ago\n\nAhem.\n\nMathematics is very boring for some time but given enough time it will then turn mysterious for some more time until finally it yields moments of marvel few and far between an indefinite stretch of madness...\n\nadjusts the tin foil hats on my 47 copies of dummit & foote\n\nlevel 1\n1 point \u00b7 3 days ago\n\nPhysics would not exist as it is today wothout math, but math would exist just by itself fine.\n\nlevel 1\n1 point \u00b7 5 days ago\n\nI found finding derivatives and integrals satisfying, but... not something I would want to do my whole life.\n\nI'd imagine most mathematicians agree with you. Thankfully, that's not what research mathematics is about!\n\nHave you ever taken a proof-based or proof-heavy class, such as an \"introduction to proofs\"? Or, at least, a logic or set theory class? The motivations behind those classes are much closer to what \"real\" math is about (or at least pure math) - although their scope is dwarfed by the depth and complexity of modern mathematics.\n\nThere's nothing formulaic about higher-level math. It's hard to capture exactly what mathematics research is to a layman, but let me give an example from number theory:\n\nGoldbach's Conjecture: Every even number greater than 2 can be written as the sum of two prime numbers. For example, 6 = 3+3; 8 = 5+3; 22 = 17 + 5.\n\nGoldbach's is an open problem. We've ran a bunch of calculations with computers and we haven't found an exception yet - but we don't truly know that it works for all even numbers > 2. We have no proof. It's impossible to test an infinite amount of numbers, after all! So, is it true? Well, we don't know - but some very talented people are actively working on it, and many suspect it is true.\n\nThese are the sorts of problems (pure) mathematicians tackle. Of course, most problems are far more complex or high-level than that, but it still captures the \"essence\" of mathematics research: posing problems and solving them via logical connections. You'll probably notice that this problem isn't really computational, and it definitely can't be solved by some rote procedure.\n\nSo, how do we solve those problems? Why, with logical reasoning and proof techniques. For a fairly elementary example, consider this proof that there's an infinite amount of prime numbers.\n\nMathematics would be quite dull if it was just applying formulas over and over again. Fortunately, that's not mathematics - that's engineering! (I jest, of course, but reducing mathematics to just formulaic steps is like reducing engineering to just calculations.)\n\nlevel 1\n-6 points \u00b7 5 days ago(0 children)\nlevel 2\nApplied Math3 points \u00b7 4 days ago\n\nYeah you're right, Applied Maths is a lie and it doesn't exist.\n\nlevel 3\n1 point \u00b7 4 days ago\n\nProof?\n\nlevel 4\nApplied Math5 points \u00b7 4 days ago\n\nSuppose Applied Maths professors existed, they would at least reply to students' emails.\n\nMy professors never do and that implies they don't exist.\n\nlevel 5\n1 point \u00b7 3 days ago\n\nBecause they think they are God\n\nContinue browsing in r\/math\nCommunity Details\n\n802k\n\nMembers\n\n485\n\nOnline\n\nWelcome to r\/math!\n\nThis subreddit is for discussion of mathematical links and questions. Please read the FAQ and the rules below before posting.\n\nIf you're asking for help understanding something mathematical, post in the Simple Question thread or \/r\/learnmath. This includes reference requests - also see our lists of recommended books and free online resources. Here is a more recent thread with book recommendations.\n\nIf you are asking for a calculation to be made, please post to \/r\/askmath or \/r\/learnmath.\n\nIf you are asking for advice on choosing classes or career prospects, please post in the stickied Career & Education Questions thread.\n\nPlease be polite and civil when commenting, and always follow reddiquette.\n\nr\/math Rules\n1.\nNo homework problems\n2.\nStay on-topic\n3.\nBe excellent to each other\n4.\nNo low-effort image posts\n5.\nNo career or education related questions\nRecurring Threads & Resources\n\nEverything about X - every Wednesday\n\nWhat Are You Working On? - posted Mondays\n\nCareer and Education Q&A - Every other Thursday\n\nSimple Questions - Posted Fridays\n\nClick here to chat with us on IRC!\n\nUsing LaTeX\n\nTo view LaTeX on reddit, install one of the following:\n\nMathJax userscript (install Greasemonkey or Tampermonkey first)\n\n[; e^{\\pi i} + 1 = 0 ;]\n\nPost the equation above like this:\n\n[; e^{\\pi i}+1=0 ;]\n\nUseful Symbols\n\nBasic Math Symbols\n\n\u2260 \u00b1 \u2213 \u00f7 \u00d7 \u2219 \u2013 \u221a \u2030 \u2297 \u2295 \u2296 \u2298 \u2299 \u2264 \u2265 \u2266 \u2267 \u2268 \u2269 \u227a \u227b \u227c \u227d \u228f \u2290 \u2291 \u2292 \u00b2 \u00b3 \u00b0\n\nGeometry Symbols\n\n\u2220 \u221f \u00b0 \u2245 ~ \u2016 \u27c2 \u2adb\n\nAlgebra Symbols\n\n\u2261 \u225c \u2248 \u221d \u221e \u226a \u226b \u230a\u230b \u2308\u2309 \u2218\u220f \u2210 \u2211 \u22c0 \u22c1 \u22c2 \u22c3 \u2a00 \u2a01 \u2a02 \ud835\udd95 \ud835\udd96 \ud835\udd97\n\nSet Theory Symbols\n\n\u2205 \u2216 \u2201 \u21a6 \u21a3 \u2229 \u222a \u2286 \u2282 \u2284 \u228a \u2287 \u2283 \u2285 \u228b \u2296 \u2208 \u2209 \u220b \u220c \u2115 \u2124 \u211a \u211d \u2102 \u2135 \u2136 \u2137 \u2138 \ud835\udcdf\n\nLogic Symbols\n\n\u00ac \u2228 \u2227 \u2295 \u2192 \u2190 \u21d2 \u21d0 \u2194 \u21d4 \u2200 \u2203 \u2204 \u2234 \u2235 \u22a4 \u22a5 \u22a2 \u22a8 \u2ae4 \u22a3\n\nCalculus and Analysis Symbols\n\n\u222b \u222c \u222d \u222e \u222f \u2230 \u2207 \u2206 \u03b4 \u2202 \u2131 \u2112 \u2113\n\nGreek Letters\n\n\ud835\udee2\ud835\udefc \ud835\udee3\ud835\udefd \ud835\udee4\ud835\udefe \ud835\udee5\ud835\udeff \ud835\udee6\ud835\udf00\ud835\udf16 \ud835\udee7\ud835\udf01 \ud835\udee8\ud835\udf02 \ud835\udee9\ud835\udf03\ud835\udf17 \ud835\udeea\ud835\udf04 \ud835\udeeb\ud835\udf05 \ud835\udeec\ud835\udf06 \ud835\udeed\ud835\udf07 \ud835\udeee\ud835\udf08 \ud835\udeef\ud835\udf09 \ud835\udef0\ud835\udf0a \ud835\udef1\ud835\udf0b \ud835\udef2\ud835\udf0c \ud835\udef4\ud835\udf0e \ud835\udef5\ud835\udf0f \ud835\udef6\ud835\udf10 \ud835\udef7\ud835\udf19\ud835\udf11 \ud835\udef8\ud835\udf12 \ud835\udef9\ud835\udf13 \ud835\udefa\ud835\udf14\n\nOther math subreddits\nr\/learnmath\n\n118k members\n\nr\/mathbooks\n\n10.0k members\n\nr\/matheducation\n\n14.9k members\n\nr\/CasualMath\n\n9.2k members\n\nr\/puremathematics\n\n10.8k members\n\nr\/mathpics\n\n18.4k members\n\nr\/mathriddles\n\n8.7k members\n\nRelated subreddits\nr\/Mathematica\n\n4.8k members\n\nr\/matlab\n\n20.4k members\n\nr\/sagemath\n\n774 members\n\nr\/actuary\n\n14.0k members\n\nr\/algorithms\n\n59.4k members\n\nr\/compsci\n\n527k members\n\nr\/statistics\n\n76.4k members\n\nModerators\nu\/yesmanapple\nGeometry\/Topology\nu\/inherentlyawesome\nAlgebraic Topology\nu\/amdpox\nGeometric Analysis\nu\/AngelTC\nAlgebraic Geometry\nu\/dogdiarrhea\nDynamical Systems","date":"2019-05-20 05:08:26","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\": 0.4606190323829651, \"perplexity\": 1351.422940690705}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-22\/segments\/1558232255562.23\/warc\/CC-MAIN-20190520041753-20190520063753-00535.warc.gz\"}"}
| null | null |
package com.mnxfst.stream.pipeline.element.script;
import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.net.URL;
import java.util.ArrayList;
import java.util.List;
import javax.script.ScriptEngine;
import javax.script.ScriptEngineManager;
import org.apache.commons.lang3.StringUtils;
import com.mnxfst.stream.message.StreamEventMessage;
import com.mnxfst.stream.pipeline.PipelineElement;
import com.mnxfst.stream.pipeline.config.PipelineElementConfiguration;
import com.mnxfst.stream.pipeline.message.PipelineElementSetupFailedMessage;
/**
* Implements a script evaluator which extracts the {@link StreamEventMessage#getEvent() event content}
* from the message and applies the configured script. The evaluator may modify the message content
* and returns the identifier of the next pipeline element which must receive the message
* @author mnxfst
* @since 04.03.2014
*
*/
public class ScriptEvaluatorPipelineElement extends PipelineElement {
public static final int ERROR_CODE_EVENT_CONTENT_MISSING = 1;
/** configuration option holding the name of the script engine to use */
public static final String CONFIG_SCRIPT_ENGINE_NAME = "script.engine.name";
/** prefix to configuration option holding the scripts to be used for initialization - script.code.init.0 ... n */
public static final String CONFIG_SCRIPT_INIT_CODE_PREFIX = "script.code.init.";
/** configuration option holding the script to be executed for an event */
public static final String CONFIG_SCRIPT_EVAL_CODE = "script.code.eval";
/** configuration option holding the variable where the script expects the input */
public static final String CONFIG_SCRIPT_INPUT_VARIABLE = "script.var.input";
/** configuration option holding the variable where the script writes the identifier of the next pipeline element to */
public static final String CONFIG_SCRIPT_OUTPUT_NEXT_ELEMENT_VARIABLE = "script.var.output.nextelement";
private ScriptEngine scriptEngine;
private List<String> initScripts = new ArrayList<>();
private String evalScript = null;
private String scriptInputVariable = null;
private String scriptOutputNextElementVariable = null;
public ScriptEvaluatorPipelineElement(PipelineElementConfiguration pipelineElementConfiguration) {
super(pipelineElementConfiguration);
}
/**
* @see akka.actor.UntypedActor#preStart()
*/
public void preStart() throws Exception {
// initialize the script engine
ScriptEngineManager factory = new ScriptEngineManager();
try {
this.scriptEngine = factory.getEngineByName(getStringProperty(CONFIG_SCRIPT_ENGINE_NAME));
} catch(Exception e) {
context().parent().tell(new PipelineElementSetupFailedMessage(
getPipelineElementConfiguration().getPipelineId(), getPipelineElementConfiguration().getElementId(),
PipelineElementSetupFailedMessage.GENERAL, e.getMessage()), getSelf());
return;
}
// iterate from zero to max, read out init code snippets and interrupt if an empty one occurs
for(int i = 0; i < Integer.MAX_VALUE; i++) {
String initScript = getStringProperty(CONFIG_SCRIPT_INIT_CODE_PREFIX + i);
if(StringUtils.isNotBlank(initScript)) {
this.initScripts.add(loadScript(initScript.toString()));
} else {
break;
}
}
// fetch the script to be applied for each message
String scriptUrl = getStringProperty(CONFIG_SCRIPT_EVAL_CODE);
// this.evalScript = getStringProperty(CONFIG_SCRIPT_EVAL_CODE);
if(StringUtils.isBlank(scriptUrl)) {
context().parent().tell(new PipelineElementSetupFailedMessage(
getPipelineElementConfiguration().getPipelineId(), getPipelineElementConfiguration().getElementId(),
PipelineElementSetupFailedMessage.GENERAL, "Required script code missing"), getSelf());
return;
}
this.evalScript = loadScript(scriptUrl);
// retrieve the name of the variable where the script expects the input
this.scriptInputVariable = getStringProperty(CONFIG_SCRIPT_INPUT_VARIABLE);
if(StringUtils.isBlank(this.scriptInputVariable)) {
context().parent().tell(new PipelineElementSetupFailedMessage(
getPipelineElementConfiguration().getPipelineId(), getPipelineElementConfiguration().getElementId(),
PipelineElementSetupFailedMessage.GENERAL, "Required input variable missing"), getSelf());
return;
}
// retrieve the name of the variable where the script writes the identifier of the next pipeline element to
this.scriptOutputNextElementVariable = getStringProperty(CONFIG_SCRIPT_OUTPUT_NEXT_ELEMENT_VARIABLE);
if(StringUtils.isBlank(this.scriptOutputNextElementVariable)) {
context().parent().tell(new PipelineElementSetupFailedMessage(
getPipelineElementConfiguration().getPipelineId(), getPipelineElementConfiguration().getElementId(),
PipelineElementSetupFailedMessage.GENERAL, "Required output (next element) variable missing"), getSelf());
return;
}
// if the set of init scripts is not empty, provide them to the script engine
if(!initScripts.isEmpty()) {
for(String script : initScripts) {
this.scriptEngine.eval(script);
}
}
}
/**
* @see com.mnxfst.stream.pipeline.PipelineElement#processEvent(com.mnxfst.stream.message.StreamEventMessage)
*/
protected void processEvent(StreamEventMessage message) throws Exception {
if(message != null) {
if(StringUtils.isBlank(message.getEvent())) {
// TODO what to do?
reportError(ERROR_CODE_EVENT_CONTENT_MISSING, "Required event content missing");
return;
}
// provide message to script engine
this.scriptEngine.put(scriptInputVariable, (String)message.getEvent());
this.scriptEngine.eval(this.evalScript);
// fetch the content from the input variable as it may have been modified ... if the script sets it
// to null, it will be ignored
String modifiedEventContent = (String)this.scriptEngine.get(scriptInputVariable);
if(StringUtils.isNotBlank(modifiedEventContent))
message.setEvent(modifiedEventContent);
// fetch the next element identifier
String nextElementId = (String)this.scriptEngine.get(scriptOutputNextElementVariable);
if(StringUtils.isNotBlank(nextElementId)) {
forwardMessage(message, nextElementId, true);
}
}
}
/**
* Load scrtip from url
* @param url
* @return
*/
protected String loadScript(final String url) throws IOException {
StringBuffer scriptContent = new StringBuffer();
URL initScriptUrl = new URL(url);
BufferedReader reader = new BufferedReader(new InputStreamReader(initScriptUrl.openStream()));
int c = -1;
while((c = reader.read()) != -1) {
scriptContent.append((char)c);
}
return scriptContent.toString();
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,219
|
477 Employers SideBar Navigation, use key tab or arrows keys to navigate through the menu. Expand collapse using space or enter key.
The OMERS Supplemental Plan for Police, Firefighters and Paramedics (the "Supplemental Plan") offers optional benefits for members of the police sector, firefighters and paramedics. As an OMERS employer, you may offer a benefit to a specific class or classes of employees. Supplemental benefits will "top up" the pension that a member already earns in the OMERS Primary Pension Plan (the "Primary Plan").
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{
"redpajama_set_name": "RedPajamaC4"
}
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Investors, though, remain empty-handed. Although the effort to recoup funds isn't over, court records suggest investors are unlikely to get much back.
"I haven't seen a dime and I'm not sure I ever will," said John Edlund, a Sun City Roseville resident who invested $50,000 with the Redding firm.
Edlund purchased a financial instrument called a "viatical." The concept was perfectly legitimate – a life insurance policy converted into an investment. Viaticals, also known as life settlements, arose 30 years ago, when dying AIDS patients began selling their life insurance policies to raise cash. They've become a big business, with Wall Street financial firms investing in them.
But authorities said the owners of Redding-based Secure Investment Services lied to investors about the risks involved and improperly used their money to prop up a rapidly crumbling business. The losses to investors will run into the millions.
The toll has been more than financial. Wolok's business partner in Michigan, a man named David Goldenberg, committed suicide two months after he and the other defendants were indicted in 2007. The principal of Secure Investment Services, 76-year-old Donald Neuhaus of Redding, died of natural causes soon after.
In Roseville, the case has turned neighbors into enemies. Several of the defrauded investors were Sun City residents who were lured into the scheme by one of their own, a sales agent and fellow resident named Robert Koppel. After pleading guilty to securities fraud, Koppel was sentenced Oct. 25 to three years' probation. He returned to a frosty reception from some of his Sun City neighbors.
Prosecutor Lee Bickley, an assistant U.S. attorney, said Koppel "at least at some point" believed the investment program was for real. But she said the evidence showed that eventually Koppel realized the investment plan was fraudulent and he knowingly misled investors. Court records show that Koppel sold more than $2 million worth of viaticals.
It's not surprising that a community like Sun City was fertile ground for the Redding company; officials said many of the investors were in their 50s or older. Experts say senior citizens, because of their advanced age, tend not to be squeamish about investing in something like another person's death benefit.
What's more, the investment was sold as perfectly safe. Koppel made presentations to Sun City's investment club under the title, "Double Digit Returns With No Market Risk," according to Edlund.
Authorities said Secure Investment's investors were told that all the policyholders had been examined by a doctor, who provided forecasts of how soon they would die. Funds would be set aside, in escrow accounts, to keep up premium payments on the policies so they wouldn't lapse and become worthless. Finally, a bonding company from Michigan would pay investors their guaranteed returns if the policyholders lived longer than expected.
The doctor was a fake. Most of the policyholders lived longer than expected. The bonding company, run by Wolok and Goldenberg, was a sham.
The entire scheme unraveled. Secure Investment needed more cash to keep those policies active and began using dollars from new investors to prop up the old policies belonging to earlier investors. Because of that, federal investigators labeled Secure Investment a Ponzi scheme.
There wasn't much left of Secure Investment when the authorities got a court order seizing its assets in August 2007. Despite taking in millions from investors, the Redding company had just $440,000 in cash.
A court-appointed receiver was brought in to convert the life insurance policies into cash in order to repay investors. It's proved to be daunting.
The receiver, a Dallas attorney named Michael Quilling, said in a recent court filing that he's been able to generate about $8 million, mostly in insurance proceeds, after some policyholders died. But he's spent nearly $10 million on premiums to keep policies from lapsing and is nearly out of cash.
In short, the receiver has run into the same problems as Secure Investment – the premiums keep coming due, and the policyholders haven't died quickly enough.
"Unfortunately, there's no science to any of this," said John Yun, a lawyer with the Securities and Exchange Commission who has worked on the case.
Quilling has found an investment firm willing to pay $5 million for the remaining life policies on the Redding firm's books, according to a court filing. But most of the proceeds would go to repay $4.3 million he borrowed from a bank to pay premiums.
SEC officials declined to predict how much money investors can expect. "The game's not completely over yet, so we'll see," said Michael Dicke, associate regional director at the SEC's office in San Francisco. Quilling couldn't be reached for comment.
As part of his criminal sentence, Wolok was ordered to make restitution totaling $17 million, but investors shouldn't count on that. "From our review of his bank records and what we know of this person, he doesn't have the capacity," Bickley said.
Besides Wolok and Koppel, four others have pleaded guilty in the case. They include Neuhaus' daughter, Kimberly Snowden of Redding; and three former sales agents, Robert and Barbara Eberle and Clifford Palm. All are awaiting sentencing.
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{
"redpajama_set_name": "RedPajamaC4"
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\section{Introduction}
{\em Introduction.}---
The standard model (SM) of elementary particle physics still holds a mysterious puzzle in its matter contents.
{\em Who ordered three copies of quarks and leptons in the SM?}
As I.\,I.\,Rabi famously quipped for the muon {\em ``Who ordered that?''}, up-type quarks (up $u$, charm $c$, top $t$), down-type quarks (down $d$, strange $s$, bottom $b$), charged leptons (electron $e$, muon $\mu$, tauon $\tau$) and neutrinos (electron neutrino $\nu_e$, muon neutrino $\nu_\mu$, tau neutrino $\nu_\tau$) carry the same quantum charges and are distinguished by their Yukawa couplings to the Higgs boson, namely their masses.
After the SM was proposed, this simple and profound mystery has been one of great interests in the SM over a long period of history.
Nevertheless, a satisfactory explanation for an origin of three-generation structures has not been naturally given from the viewpoint of four dimensional (4D) field theory.
Among attempts to reveal an origin of triply multiple copies of the SM fermions, a promising hypothesis is that there exist compactified extra spacial dimensions in addition to 4D that we live in, although our world apparently looks four dimensional.
In higher dimensional theories, fields are expanded by Kaluza--Klein (KK) expansions, which are to decompose extra dimensional parts of higher dimensional fields into a complete set spanned by the KK mode wavefunctions.
A novel proposal by Bachas is based on quantized magnetic fluxes in toroidal compactification \cite{Bachas:1995ik}, where the magnetic fluxes provide a degeneracy of the lowest KK-expanded wavefunctions.
The degeneracy of the lowest KK modes should be regarded as the family replication of matters after dimensional reduction.
After the Bachas's proposal, Ibanez et al. pointed out that the degenerated mode functions can be analytically expressed by an elliptic function and discussed their convergence properties \cite{Cremades:2004wa}.
However, such higher dimensional theories contain dimensionful coupling constants and are non-renormalizable.
Thus, these theories possess less predictability in general.
In 2001, Arkani-Hamed et al. proposed a splendid ultraviolet (UV) completion, called {\em (de)construction} of an extra dimension \cite{ArkaniHamed:2001ca}, where a fifth dimension can be effectively established by the multiplicity of 4D renormalizable gauge theories.
In the paper, the authors utilized the knowledge of lattice gauge theories to interpret the non-renormalizable higher dimensional gauge theories by renormalizable ones.
As the results, (de)constructed higher dimensional theories acquire a predictability without loss of several essential properties.
In this Letter, we apply the (de)construction mechanism to the flux compactification where there exists a (nontrivial) topological index associated with the index theorem, and construct renormalizable, asymptotically free, 4D local gauge theories that dynamically generate the fifth and sixth dimensions with magnetic fluxes.
A main subject of this study is to establish the UV completion of magnetized toroidal compactifications and formulate latticized gauge theories where the index theorem \cite{Atiyah:1963zz} is applicable.
We find that two dimensional Moose diagram is insufficient due to the Nielsen--Ninomiya theorem \cite{Nielsen:1981hk} with the index theorem for zero-modes where the index of the Dirac operator in the latticized space is inevitably zero.
Thus, it is remarkable that it is necessary to treat three dimensional (3D) Moose diagram to correctly regulate the theory from (de)constructing points of view.
\vspace{5pt}
{\em Multiple zero-modes with fluxes.}---
Here, we briefly review an original theory proposed in Ref.\,\cite{Cremades:2004wa}.
We consider a six dimensional (6D) gauge theory that has magnetic fluxes (or magnetic monopoles) in toroidal compactification $T^2$.
It contains nonvanishing flux background $b = \int_{T^2} F$ of the field strength $F=(ib/2) dz \wedge d \bar z$, which is provided by a vector potential
\begin{gather}
A^{(b)}(z) = \frac{b}2 {\rm Im} \, (\bar z dz),
\label{bgd}
\end{gather}
where two Cartesian coordinates of the fifth and sixth directions $y_5$ and $y_6$ are expressed by $z \equiv (y_5 + i y_6)/2 \pi R$.
The single-valuedness of the 6D action under contractible loops, e.g., $z \to z+1 \to z+1+i \to z + i \to z$, demands the Dirac charge quantization,
\begin{gather}
\frac{qb}{2\pi} = m \in \mathbb{Z}.
\end{gather}
We perform the KK decomposition of the 6D Weyl spinor $\Psi$ and scalar $\Phi$ by a complete set as
\begin{align}
\Psi(x^\mu, z) &=\sum_n \chi_n(x^\mu) \otimes \psi_n(z),\\
\Phi(x^\mu, z) &= \sum_n \varphi_n(x^\mu) \otimes \phi_n(z).
\end{align}
for $n=0,1,2, \cdots$, where $\chi_n$ denotes 4D Weyl spinors.
Here, the KK-decomposed wavefunctions in the extra dimensional parts $\psi_n$ and $\phi_n$ are chosen to be eigenstates of the covariant derivative $D = 2 \partial/\partial \bar z + \pi m z$ and the Laplace operator $\Delta = \{D^\dag, D\}/2$ for extra dimensions as $i \slashed{D} \psi_n = m_n \psi_n$ and $\Delta \phi_n = m_n^2 \phi_n$, respectively, as defined in Ref.\,\cite{Cremades:2004wa}.
In the Letter, we adopt a unit $2 \pi R =1$ for the compactification radius $R$ and set a $U(1)$ charge as $q=1$.
On the flux background \eqref{bgd} for $m > 0$, the lowest KK-decomposed modes $\psi_0$ are $m$-multiply degenerated as
\begin{gather}
\psi^j_{+, 0}(z) = {\cal N} e^{\pi i m z {\rm Im} \, z} \, \vartheta
\begin{bmatrix}
j/m\\[5pt]
0
\end{bmatrix}
(mz, mi), \label{eq:eigenvalue_CIM_eq}
\end{gather}
with $j=0,1, \cdots, m-1$ and the Jacobi theta function
\begin{gather}
\vartheta
\begin{bmatrix}
\alpha\\[5pt]
\beta
\end{bmatrix}
(\nu, \tau) = \sum_{\ell \in \mathbb{Z}} e^{\pi i (\alpha+\ell)^2 \tau + 2\pi i(\alpha + \ell)(\nu +\beta)},
\end{gather}
where $\alpha$ and $\beta$ are real parameters, and $\nu$ and $\tau$ take complex values with ${\rm Im} \, \tau>0$.
On another hand, $\psi_{-, 0}$ possesses no normalizable zero-mode wavefunction, where we decompose the two dimensional (2D) spinor as $\psi_0 = (\psi_{+, 0}, \psi_{-, 0})^T$ carrying the 2D chiralities.
Also, ${\cal N}$ denotes a normalization factor, given as ${\cal N} = (2m)^{1/4}$.
Note that the lack of $\psi_{-, 0}$ implies that the chiral spectra are realizable in the low energy effective theory.
The zero-modes \eqref{eq:eigenvalue_CIM_eq} are localized at different regions on the torus $T^2$ and their schematic shapes are Gaussian-like.
The important point is that the degeneracy of the zero-modes corresponds to the family replication after dimensional reduction.
Although mode functions of the scalar are the same as those of the spinor, an exception remains in the KK mass spectra.
Indeed, it is straightforwardly found that the KK mass spectra of the spinor are calculated as
\begin{gather}
m^2_n = 4 \pi m n. \label{eq:KK_mass}
\end{gather}
(The KK mass spectra of the scalar are given as $m^2_n = 4 \pi m (n+1/2)$.)
Although they are not necessary in this Letter, the concrete wavefunctions of excited KK modes $\psi_n$ and $\phi_n$ for $n\geq 1$ are analytically calculated, as discussed in Ref.\,\cite{Hamada:2012wj}.
\vspace{5pt}
{\em (De)Construction of magnetized dimensions.}---
\begin{figure}[t]
\centering
\includegraphics*[width=5cm]{moose.eps}
\caption{A schematic picture of three dimensionally extended Moose diagram (or discretized toroidal extra directions plus another direction associated with the domain-wall fermion), which is necessary to utilize the index theorem. Link functions $Q_M(\bar{y})$ with $\bar{y}=(y_5,y_6,y_7)$ and $M=5,6,7$, correspond to gauge fields in the lattice gauge theory.}
\label{fig:6d_lat}
\end{figure}
In order to (de)construct magnetized toroidal dimensions, we start from the Wilson fermion on the 3D ``Moose'' diagram in Fig.\,\ref{fig:6d_lat} or namely discretized three dimensions.
The parts associated with the seventh dimension of fields satisfy the Dirichlet boundary condition.
The Lagrangian of the 4D Weyl spinor in three Cartesian coordinates reads:
\begin{widetext}
\begin{align}
{\cal L}
&= \frac{1}{2}\sum_{\bar{y}} \bar{\eta}_{\bar{y}} i \left[\sum_{M=5,6,7} \Gamma_M\big(Q_{M}(\bar{y}) \eta_{\bar{y}+\hat{M}} - Q^\dagger_{M}(\bar{y}-\hat{M} ) \eta_{\bar{y}-\hat{M}} \big) \right] - M_0 \sum_{\bar{y}} \bar{\eta}_{\bar{y}} \eta_{\bar{y}}\notag\\
& \quad\qquad - \frac{1}{2}\sum_{\bar{y}} \bar{\eta}_{\bar{y}} \left[\sum_{M=5,6,7} \big(Q_{M}(\bar{y}) \eta_{\bar{y}+\hat{M}}
+ Q^\dagger_{M}(\bar{y}-\hat{M} ) \eta_{\bar{y}-\hat{M}} -2 \bar{\eta}_{\bar{y}} \eta_{\bar{y}}\big) \right],
\end{align}
\end{widetext}
where $\bar{y} \equiv (y_5, y_6, y_7)$ is a discretized coordinate for the 3D Moose diagram which is defined as $y_M \in \{1/N_M, 2/N_M, \cdots, 1\} \,\, (M=5,6,7)$, $\hat{M}$ is a unit vector for the $M$-th direction and $\Gamma_M$ is the gamma matrices for extra three dimensions.
As is the case with the lattice gauge theory, $\eta_{\bar{y}} = {\eta}_{\bar{y}}(x)$ and $\bar{\eta}_{\bar{y}} = \bar{\eta}_{\bar{y}}(x)$ are regarded as independent fermion fields each other in the 4D Minkowski spacetime, and $M_0$ is a real positive parameter of order one.
Since we are interested in the low energy spectrum of the Dirac operator, thus we extract the low lying modes of the Dirac operator as the same as the work in Ref.\,\cite{Tomiya:2016jwr}.
To be precise, after integrating out only the seventh direction of the latticized fermion fields with a trivial link function $Q_7(\bar{y})=1$ and a Pauli--Villars infrared regulator field, chiral fermions are realized in the residual 2D, similarly to a derivation of the overlap-Dirac operator in the context of the 4D lattice gauge theory.
For practical reasons, we employ the M\"obius domain-wall Dirac operator rather than the standard domain-wall Dirac operator.
This is due to the fact that the M\"obius domain-wall Dirac operator can realize more appropriate chiral symmetry \cite{Brower:2005qw} only through a slight extension of the seventh direction in comparison with the standard one, e.g., Ref.\,\cite{Fukaya:2004kp}.
Also, it is expected that the M\"obius domain-wall fermion with $N_7$ lattice points becomes asymptotically equivalent to the standard one with $2N_7$ lattice points \cite{Brower:2012vk}.
The Hermitian domain-wall Dirac operator $H_\text{DW}$ \cite{Kaplan:2009yg} with the corresponding Pauli--Villars infrared regulator filed is given in Refs.\,\cite{Brower:2005qw, Brower:2012vk} as
\begin{gather}
\label{eq:def_dw_eff}
H_\text{DW}=\frac{1}{2}\text{sgn}_\text{rat}(H_M),
\end{gather}
where the matrix sign function $\text{sgn}_\text{rat}(H_M)$ by use of the rational approximation is defined as
\begin{gather}
\label{eq:polar}
\text{sgn}_\text{rat}(H_M)=\frac{1-(T(H_M))^{N_7}}{1+( T(H_M))^{N_7}},
\end{gather}
with the transfer matrix $T(H_M)=(1-H_M)/(1+H_M)$.
The kernel operator (the Hermitian M\"obius-Dirac operator) $H_M$ is given as
\begin{gather}
\label{eq:dw_kernel}
H_M=\Gamma_7\frac{2 D_W}{2+D_W},
\end{gather}
where $\Gamma_7$ is a chirality operator, namely the Pauli matrix $\sigma_3$, and $D_W$ is the Wilson--Dirac operator with the mass $-M_0$.
Here, $0 < M_0 < 2$ is required to realize a correct pole structure, and we set $M_0 = 1$.
We adopt the same choice of parameters as those reported in Ref.\,\cite{Hashimoto:2014gta}.
Note that if we start from the overlap-Dirac operator, which is realized in the limit of $N_7 \to \infty$ and it has an exact sign function, the theory becomes nonlocal and phenomenologically unnatural.
To introduce magnetic flux background, we assume that link functions $Q_M(y_5, y_6)$ in the toroidal directions acquire the following expectation values:
\begin{align}
Q_5(y_5,y_6) &=
\begin{cases}
1 & \quad (y_5 \neq 1),\\
\exp[-i b y_6] & \quad (y_5= 1),\\
\end{cases}
\\
Q_6(y_5, y_6) &= \exp[i b y_5],\label{eq:def_mag_u}
\end{align}
where $y_M \in \{1/N_M,2/N_M, \cdots,1 \} \,\, (M=5, 6)$, and $b = 2\pi m$ is required from the consistency of single particle wavefunction \cite{AlHashimi:2008hr}.
\vspace{5pt}
{\em (De)Constructed zero-mode wavefunctions.}---
In order to obtain zero-mode wavefunctions, the main task is to solve an eigenvalue problem,
\begin{align}
H_\text{DW} \psi_n^{(D)} (y) = \lambda_n \psi_n^{(D)} (y),
\label{eq:eigenvalue_eq}
\end{align}
where $y \equiv (y_5,y_6)$.
Eigenvalues and eigenvectors in the eigenvalue problem correspond to the KK-decomposed mass spectra and mode functions.
To realize Gaussian-localized zero-modes, it is necessary to select appropriate bases out of eigenvectors in Eq.\,\eqref{eq:eigenvalue_eq} such that all of kinetic terms are canonically normalized.
This is because, on the lattice, there is no reason to be eigenstates of a covariant translational operator $\tilde{Y}$ defined in Ref.\,\cite{Abe:2014noa}.
Note that the zero-mode wavefunctions in Ref.\,\cite{Cremades:2004wa} are all eigenstates of the operator $\tilde Y$.
In our practical calculation, we take the number of lattice points in the seventh direction is sixteen, i.e., $N_7 = 16$, and set $N_5=N_6=30$, $20$ and $10$ for comparison.
Also, we fix the number of magnetic fluxes as $m=3$.
Our code is implemented by Python 3.4 and Numpy with Cython from the scratch, and the calculation is performed in our laptop and desktop computers.
By the setup previously shown, it is possible to realize localization profiles of the KK wavefunctions as well as several lower modes of the KK mass spectra $m_n^2=4 \pi m n$ \eqref{eq:KK_mass} as shown in Fig.\,\ref{fig:kkmass}.
It is remarkable that negative chirality modes never appear in the lowest eigenvalues, as recognized in the continuum theory, and thus the (de)construction mechanism also lead to chirality projection via the presence of nonvanishing fluxes (monopoles).
Note that the degeneracy of each of KK levels is controlled by $N_7$, while the mass squared ratios of the KK spectra are determined by $N_5$ and $N_6$.
The deviation between the continuum and (de)constructed theories can be recognized, depending on the numbers of $N_5$ and $N_6$.
Next, we depict the (de)constructed zero-mode wavefunctions $\psi^{(D), i}_0 \,\, (i=0, 1, 2)$ in Fig.\,\ref{fig:zero-mode-wavefunctions-ma3}, where one can easily find satisfactory agreements between each of zero-mode wavefunctions for Eq.\,\eqref{eq:eigenvalue_eq} and those in the continuum theory \eqref{eq:eigenvalue_CIM_eq}.
It is also found that the scalar components are discretely realized in a similar manner.
Thus, we conclude that the magnetized extra dimensions can be (de)constructed.
\begin{figure}[tb]
\centering
\includegraphics*[width=0.375\textwidth]{KKspectra.eps}
\caption{The KK spectra in the continuum (black cross symbols) and (de)constructed (colored dots) theories, where the square (rhombus and circle) symbols denote those of $N_5=N_6=30$ ($20$ and $10$), respectively.}
\label{fig:kkmass}
\end{figure}
\begin{figure*}[tb]
\centering
\includegraphics[width=0.3\textwidth]{figdec0.eps}\hspace{0.033\textwidth}
\includegraphics[width=0.3\textwidth]{figdec1.eps}\hspace{0.033\textwidth}
\includegraphics[width=0.3\textwidth]{figdec2.eps}
\caption{Zero-mode wavefunctions for Eq.\,\eqref{eq:eigenvalue_eq} (orange dots) and those in the continuum theory \eqref{eq:eigenvalue_CIM_eq} (shaded curved surfaces), where we set $N_5=N_6=30$.}
\label{fig:zero-mode-wavefunctions-ma3}
\end{figure*}
\vspace{5pt}
{\em Conclusion and discussion.}---
In this Letter, we have reconstructed the toroidal compactification with magnetic fluxes on the basis of an idea by Arkani-Hamed et al., {\em (de)construction}.
As a main result, we have concretely established the renormalizable, asymptotically free, four dimensional local gauge theories where family replications, e.g., three-generation structures of the SM fermions, originate from nonvanishing magnetic fluxes in the toroidal compactification.
Because the toroidal compactification with fluxes possesses nontrivial topological index, it turned out that the (de)construction of such a situation is required to utilize the three dimensional Moose diagram with the Wilson fermion, namely the domain-wall fermion.
In this case, although the total topological index is zero, contributions from the zero-moodes and the heaviest KK modes (i.e., doubler modes) cancel out each other.
As long as we focus on the lower modes, nontrival topological index is effectively realized for the 6D theory.
In addition, the localization profiles of multiple zero-mode wavefunctions in the continuum theory have been realized by the multiplicity of the latticized gauge theories.
Although it is not mentioned in the main text, we can reproduce effective coupling constants, especially (three-point) Yukawa coupling constants, after dimensional reduction.
If we focus on the lowest modes among the KK-decomposed modes, the 6D fields are expressed as
$\Psi(x^\mu, z) = \sum_j \chi^j_{0}(x^\mu) \psi^j_{+, 0}(z)$ and $\Phi(x^\mu, z) = \sum_j \varphi^j_{0}(x^\mu) \phi^j_{+, 0}(z)$, respectively.
Then, integrating out 6D Yukawa interaction terms, e.g., the interaction among quarks and the Higgs boson in the SM, along two extra dimensions is expressed as
\begin{align}
{\cal L}_{\rm Yukawa} &\propto \int_{T^2} d^2z \, \Psi(x^\mu, z) \bigl( \Psi(x^\mu, z) \bigr)^\dag \Phi(x^\mu, z) \notag\\
&= \left(\int_{T^2} d^2z \, \psi^i_{+, 0}(z) \bigl( \psi^j_{+,0}(z) \bigr)^\dag \phi^k_{0}(z) \right) \notag\\
& \hspace{60pt} \times \chi^i_{0} (x^\mu) \bigl( \chi^j_{0}(x^\mu) \bigr)^\dag \varphi_{0}^k(x^\mu).
\end{align}
Thus, the effective Yukawa couplings are calculated by overlap integrations of three kinds of the zero-mode wavefunctions \cite{Cremades:2004wa}:
\begin{gather}
y_{ijk} = \int_{T^2} d^2z \, \psi_{+, 0}^i(z) \bigl( \psi_{+, 0}^j(z) \bigr)^\dag \phi_0^k(z),
\end{gather}
up to an overall factor.
Since the zero-mode wavefunctions succeeded to be (de)constructed, the corresponding Yukawa coupling constants should be calculated as follows:
\begin{gather}
y^{(D)}_{ijk} = \sum_{y} \psi^{(D), i}(y) \bigl( \psi^{(D), j} (y) \bigr)^* \phi^{(D), k}(y).
\end{gather}
Detailed analyses of the (de)constructed Yukawa couplings are left for our future work.
In particular, corrections from small values of $N_5, N_6$ and $N_7$ to Yukawa couplings are characteristic features of our setup.
Those might be detected in the future experiments \cite{Dawson:2013bba}.
As a related topic, the lattice quantum chromodynamics with magnetic fluxes is actively investigated in the context of the relativistic heavy ion collision at LHC and RHIC.
Elliptic flows of the quark gluon plasma generate the hugest magnetic field in the world, and thus researches of the equation of states and phase transition with external magnetic fluxes have attracted much attentions \cite{DElia:2010abb, Bali:2011qj, Endrodi:2015oba}.
A crucial difference between such researches and ours appears in a coupling to gluons, which makes the system under consideration more intricate, and for example they derive chiral symmetry breaking.
For this reason, our setup is not related to such researches directly.
However, our research in this Letter provides a great insight of the role of chiral symmetry on the lattice in the presence of the external magnetic fluxes.
It should be mentioned to a relation between our model and an effective Hamiltonian of 3D topological insulators (TI) \cite{Zhang:2009zzf, Araki:2013qva}.
As introduced previously, our Lagrangian is the Wilson fermion in the 3D, and is coincident with the effective Hamiltonian of 3D TI.
Recently, surface states of 3D TI with the external magnetic field have been observed in an experiment \cite{Xu:2014}.
It would indicate that the physics of 3D TI is the same as the extra dimensional theory which can predict family replications of elementary particles.
\vspace{5pt}
{\em Acknowledgment.}---
The authors would like to thank Hiroyuki Abe, Yutaka Hosotani, Tetsuya Onogi, Hooman Davoudiasl, Taku Izubuchi, Hidenori Fukaya and Issaku Kanamori for helpful comments.
Y.T. would like to sincerely thank Masafumi Kurachi for valuable discussions at each stage of this research.
Y.T. is supported in part by Grants-in-Aid for Scientific Research No.\,16J04612 from the Ministry of Education, Culture, Sports, Science and Technology in Japan.
A.T. is supported by NSFC under Grant No.\,11535012.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Innovation translates into antidepressant effectiveness.
Goodwin GM.
Unipolar depression is a major cause of disability in developed societies. There is significant unmet need because existing treatments are neither as effective nor as free of adverse effects as we would hope. Agomelatine is a new antidepressant with a novel profile of pharmacological action. Its efficacy in major depression has been reported in short-term placebo-controlled trials and in direct head-to-head comparison with existing products. The pooling of data from the agomelatine short-term studies allows analysis of sub-groups within the study populations. There is a trend for a larger effect size to be seen with more severely ill patients as a consequence of placebo responses falling. Efficacy is also reported in the maintenance of effect seen after clinical response in a recently completed relapse prevention study, analysed initially after 6 months. What is unusual, certainly compared with the selective serotonin reuptake inhibitors (SSRIs) and venlafaxine, is the clear absence of an excess of early relapses soon after agomelatine withdrawal. This parallels the absence of an immediate withdrawal effect seen in the pattern of subjective symptoms in another study. It underlines the distinct and novel pharmacological properties of the product, and has evident clinical advantages for the patient. Agomelatine is well tolerated in other respects; only occasional dizziness emerges at significant rates higher than with placebo in the controlled data.
J Psychopharmacol
Acetamides, Animals, Antidepressive Agents, Circadian Rhythm, Depressive Disorder, Humans, Receptor, Melatonin, MT1, Receptor, Melatonin, MT2, Recurrence, Serotonin 5-HT2 Receptor Antagonists, Serotonin Antagonists, Severity of Illness Index, Time Factors, Treatment Outcome
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,884
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Marième Badiane, née le à Brest (Finistère), est une joueuse française de basket-ball.
Biographie
Née de parents basketteurs Abdoulaye Badiane, ex-Étendard de Brest, et Cathy Silinski, ancienne pivot de l'Étoile Saint-Laurent Brest, elle commence le sport par la danse avant d'être rattrapée par le basket-ball et d'être sélectionnée pour rejoindre trois ans le Centre fédéral :
Elle signe son premier contrat avec Roche Vendée. . Elle n'y reste qu'un an et rejoint une autre équipe de Ligue 2, Reims. Bien que courtisée durant l'été par des clubs belge et de LFB, elle choisit de rester une nouvelle saison à Reims pour y terminer sa licence de biologie avant d'entamer une carrière exclusivement sportive. Sur la lancée d'une bonne saison en Champagne (12,1 pts et 8,1 rebonds de moyenne), elle signe en en LFB pour l'USO Mondeville.
Ses et lors de la sixième journée 2016-2017 sont importants dans la victoire à l'extérieur face à Basket Landes, alors seule équipe invaincue. Elle récidive avec et le lors d'une victoire face au Hainaut. Son mouvement préféré est le reverse.
Fin , elle annonce s'engager pour trois saisons avec Lyon ASVEL, club récemment alorsrepris en mains par Tony Parker. Puis dès , elle renouvelle son engagement à Lyon pour la période 2020-2023. Elle fait l'impasse sur la saison LFb 2020-2021 pour cause de maternité.
Équipe nationale
Elle connait sa première sélection internationale avec les U16 et remporte une médaille de bronze au championnat d'Europe 2010 avec des statistiques personnelles de 6,7 points à 51,0 % et 6,3 rebonds, qui font d'elle la troisième marqueuse de l'équipe derrière Olivia Époupa et Valériane Ayayi. Deux ans plus tard avec les U18, la France gagne la médaille d'or, Marième Badiane alignant 6,6 points, 9,0 rebonds et 1,8 passe décisive. Invitée à citer ses qualités, elle cite : .
À l'été 2013, elle remporte avec sa future coéquipière de Reims Pauline Lithard,la médaille d'argent du Championnat du monde avec l'Équipe de France de basket-ball féminin des 19 ans et moins pour des statistiques personnelles de 4,4 points et 8,8 rebonds. L'été suivante, elle remporte le championnat d'Europe U20 face à l'Espagne () dans un tournoi et une finale dans lesquelles elle a pris une part majeure (8,1 points à 49,0 % d'adresse et 8,9 rebondsavec notamment deux paniers à trois points et un contre décisifs en fin de rencontre.
Elle est retenue début 2016 dans la pré-sélection pour les qualifications du championnat d'Europe 2017 : . Cependant, elle se blesse au quadriceps lors de la dernière journée de championnat avant le rassemblement, ce qui la contraint au forfait.
Clubs
Palmarès
Sélection nationale
Seniors
Médaille d'or des Jeux de la Francophonie 2017
Médaille d'argent Championnat d'Europe 2019 à Belgrade (Serbie)
Jeunes
Médaille d'or au Championnat d'Europe 2014 des 20 ans et moins
Médaillée d'argent au Championnat du monde de basket-ball féminin des moins de 19 ans 2013
Médaille d'or au Championnat d'Europe 2012 des 18 ans et moins
Médaille de bronze au Championnat d'Europe 2010 des 16 ans et moins
En club
Championne de France : 2018-19
Match des Champions LFB : 2019
Finaliste du Championnat de France : 2021-22
Notes et références
Liens externes
Joueuse française de basket-ball
Joueuse du Centre fédéral
Joueuse de Roche-Vendée
Joueuse de Champagne Basket
Joueuse de l'USO Mondeville
Joueuse de basket-ball de Lyon ASVEL
Joueuse internationale française de basket-ball
Naissance en novembre 1994
Naissance à Brest
Sportif lié à Brest
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 4,918
|
Q: Editing a Select Menu | Discord.js I'm trying to get all the text channel in the guild and list them into a Select Menu and I'm getting some trouble.
Here is my code :
const { PermissionOverwrites, Permissions, Collection, MessageEmbed, MessageActionRow, MessageSelectMenu } = require("discord.js");
const dotenv = require("dotenv");
dotenv.config();
module.exports = {
data: {
name: `select-raid-options`
},
async execute (interaction) {
let guild = interaction.client.guilds.cache.get(process.env.GUILD_ID);
switch (interaction.values[0]) {
case 'raid_channel':
let raidChannel = guild.channels.cache.find(channel => channel.name === 'raid');
raidChannel.send('Raid');
break;
case 'select_channel':
// Create the Embed Message
const channelSelectorMessage = new MessageEmbed()
.setColor('#e4e8eb')
.setTitle('Veuillez sélectionnez le channel où vous souhaitez publier votre raid.')
.setTimestamp()
// Create the Select Menu
let channelSelectorSelect = new MessageSelectMenu()
.setCustomId('select-raid-channel')
.setPlaceholder('Nothing selected')
.addOptions([
{
label: `Annuler`,
description: 'Annulez la publication de votre raid.',
value: 'cancel',
},
]);
// Get all channels in the guild
guild.channels.cache.forEach(channel => {
// Check if the channel is not the raid channel
if (channel.name !== 'raid') {
// add channel to the select menu
channelSelectorSelect.addOptions([
{
label: `${channel.name}`, // Option label
description: `${channel.name}`, // Option description
value: `${channel.id}`, // Option value
},
]);
}
})
console.log(channelSelectorSelect); // Debug to see the select menu
interaction.user.send({embeds: [channelSelectorMessage], components: [channelSelectorSelect]}); // Send the message with the select menu to the user
}
await interaction.reply({ content:'Information envoyée !', ephemeral: true });
}
}
The console.log of the channelSelectorSelect return me this :
MessageSelectMenu {
type: 'SELECT_MENU',
customId: 'select-raid-channel',
placeholder: 'Nothing selected',
minValues: null,
maxValues: null,
options: [
{
label: 'Annuler',
value: 'cancel',
description: 'Annulez la publication de votre raid.',
emoji: null,
default: false
},
{
label: '- ',
value: '953674264408653825',
description: '- ',
emoji: null,
default: false
},
{
label: '̀',
value: '953674264408653826',
description: '̀',
emoji: null,
default: false
},
{
label: '',
value: '953674264408653827',
description: '',
emoji: null,
default: false
},
{
label: ' - ',
value: '954015397336735807',
description: ' - ',
emoji: null,
default: false
},
{
label: '',
value: '954015876892475433',
description: '',
emoji: null,
default: false
},
{
label: '́',
value: '954015934081794128',
description: '́',
emoji: null,
default: false
},
{
label: '',
value: '954016142844895302',
description: '',
emoji: null,
default: false
},
{
label: '',
value: '954016204966727680',
description: '',
emoji: null,
default: false
},
{
label: ' - ',
value: '954020253757739018',
description: ' - ',
emoji: null,
default: false
},
{
label: '3wa',
value: '963018402132267048',
description: '3wa',
emoji: null,
default: false
},
{
label: 'tickets',
value: '963018417936420895',
description: 'tickets',
emoji: null,
default: false
},
{
label: 'Degré Zero',
value: '973526523200016384',
description: 'Degré Zero',
emoji: null,
default: false
},
{
label: 'test-commands',
value: '973528693714595860',
description: 'test-commands',
emoji: null,
default: false
}
],
disabled: false
}
And when I send the message with the selector, i get this error :
DiscordAPIError: Invalid Form Body
components[0]: The specified component type is invalid in this context
If somebody have an idea to how to fix it ?
Thank you
A: The other answer is correct, I figured I'd put the code since that page doesn't demonstrate how to dynamically add options.
// Create the Message Action row
let channelSelectorSelect = new MessageActionRow()
.addComponents(
new MessageSelectMenu()
.setCustomId('select-raid-channel')
.setPlaceholder('Nothing selected')
.addOptions([{
label: `Annuler`,
description: 'Annulez la publication de votre raid.',
value: 'cancel',
}]);
)
guild.channels.cache.forEach(channel => {
if (channel.name !== 'raid') {
// add channel to the select menu - the below line was changed
channelSelectorSelect.components[0].addOptions([{
label: `${channel.name}`,
description: `${channel.name}`,
value: `${channel.id}`,
}]);
}
})
A: You need to add the select menu to an action row. If you don't know how visit this page in the Discord.js Guide
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,045
|
Tues. #2 is HERE. Please follow us on Tumblr or Twitter, subscribe to us on YouTube or like us on Facebook. Don't miss The Wrestling Nerdcast, Nickolye's 205 Live Recap after 7 PM ET and Glove Up or Shut Up at 8! With the cheap plugs out of the way here's Wednesday's latest news.
The full Cody Rhodes vs. Nick Aldis rematch from the NWA 70th Anniversary show has been released for free on YouTube in its entirety -- and we have it available for you to watch below!
Independent wrestler Kara Noia was the "fan" on WWE Raw this week.
Via social media Joey Mercury's attorney announced his client's charges were dismissed: "While I am pleased that his cases have been dismissed based on his actual innocence (not a legal technicality), I want you to know that the fact that he was arrested and charged in two cases in which he was never involved in is an embarrassment to our criminal justice system. The actions of the Winter Park, Florida Police Department represents the sloppiest and most incompetent police work anywhere. Mr. Birch never should have been charged with either of those crimes. A review of the evidence which was on videotape makes it absolutely clear that the perpetrator of the crimes was not Adam Birch. Furthermore, the fact that Adam Birch was not in the United States when the crimes were committed illustrates the absurdity and grave injustice in charging and causing the arrest of Mr. Birch. While we are grateful that the Orange County State's Attorney's office ultimately realized that Mr. Birch was absolutely innocent, it took them too long to do so. Mr. Birch has suffered irreparable harm to his excellent reputation all because of disgraceful police work. He was arrested in Chicago and subjected to inhumane treatment while he was incarcerated in Chicago and during his transport to Florida from Illinois, before he could post a bond and be released. On a personal note, I have found Adam Birch to be an absolute gentleman in all of my dealings with him."
Also via social media Matt Hardy says he's not done yet: "My last @WWE match hasn't happened yet, so this particular cycle shall be #BROKEN.. But I'm sure the DELIGHTFUL cycle of Hardys & Hennigs doing battle shall continue in the future."
New WWE videos including (but not limited to) last night's SmackDown Live highlights are below.
Courtesy WWE.com here's the updated TLC card for December 16th.
^ Braun Strowman earns WWE Universal Championship match if he wins; Baron Corbin becomes permanent Raw GM if he wins, but loses the position if he loses.
According to the article at ksdk.com/article/news/local/sources-xfl-is-bringing-pro-football-back-to-st-louis-in-2020/63-618061504 St. Louis is the first city to officially get a XFL team in 2020.
205 Live results courtesy of WrestlingInc.com are below - don't scroll down if you don't want to know.
* Noam Dar defeated Mike Kanellis.
* Hideo Itami defeated a local enhancement talent. Ariya Daivari came out after the match and attacked the jobber, apparently to show up Itami.
* Cedric Alexander and Mustafa Ali defeated Tony Nese and WWE Cruiserweight Champion Buddy Murphy.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,990
|
De Kerkdijk is een eeuwenoude dijk uit 1325. Het verbindt Doornspijk met de oude Zuiderzeekust, wat nu het Veluwemeer is.
De naam van de Kerkdijk is ontstaan door de Sint Ludgeruskerk die eeuwenlang op de plek stond waar tot 1825 het oorspronkelijke kerkdorp Doornspijk lag, aan de Zuiderzee. In de 14e eeuw werd de dijk ook wel Wessinghedike genoemd, omdat bewoners van de buurt Wessinge in Doornspijk daar naar de kerk gingen. De mensen die op de hoger gelegen gronden van de Veluwe woonden en die in Doornspijk naar de kerk wilden, liepen eeuwenlang te voet over de kerkdijk, Kerkepad op zondag naar de kerkdienst van Ludgeruskerk te gaan.
In februari 1825 brak er een storm uit, Doornspijk lag toen nog aan de Zuiderzee en kreeg te maken met een overstroming, de kerk uit de 12e eeuw overspoelde met de omliggende boerderijen. De Kerk en de boerderijen werden zwaar beschadigd. Een blikseminslag op 18 oktober zorgde ervoor dat de resten door brand werden verwoest. De Sint Ludgeruskerk en het kerkdorp Doornspijk verdwenen voorgoed.
Overlevenden van de ramp trokken naar de hoger gelegen zandruggen van de Veluwe.
Op de oude plek waar de oude kerk gestaan heeft, is de plattegrond van de fundamenten nog te zien.
Het huidige dorp Doornspijk ontstond in de loop van de 19e eeuw rondom de nieuwe kerk die in 1829 weer helemaal opnieuw opgebouwd was.
Het lag veel verder van de zee op de hogere gronden van de Veluwe, helemaal veilig was het nog niet. De grote overstroming van 1916 bewijst dat. De stormvloed van 1916 was de aanleiding voor het afsluiten van de Zuiderzee. Het gevaar van overstromingen verdween pas door de bouw van de Afsluitdijk in 1932.
De Kerkdijk uit 1325 ligt er nog, is nu een onderdeel van een van de Klompenpaden routes het Thornspiccerpad is een wandelroute van 21 km. Links van de dijk ligt de Goorkolk wat een blijvende herinnering is aan de dijkdoorbraak van 1881. De Kolk ontstond door de kracht van het water wat ook 'Wiel' genoemd wordt.
Dijk in Gelderland
Geografie van Elburg
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,122
|
Q: Java weird behaviour. .jar can't be opened Whenever I try to double-click a .jar file, it gives me a message saying that it can't locate or load the main class of the file. However, if I try to open the .jar file in the command line like this:
java -jar myfile.jar
Then, it runs. Is it something wrong the the system variables or something? I'm having a lot of trouble with java and uninstalling and reinstalling it again doesn't seem to solve it.
Thanks in advance!
EXTRA INFO: I'm running on Win7 64-bit, with both JDK 7 Update 9 (64-bit) and Java 7 Update 9 (64-bit). It also used to run .jar files just by double-clicking before, I don't know what happened.
A: Check that these two configurations are in place:
C:\>assoc .jar
.jar=jarfile
C:\>ftype jarfile
jarfile="C:\Program Files (x86)\Java\jre6\bin\javaw.exe" -jar "%1" %*
With the latter pointing to the correct location of your javaw.exe. It should work then.
A: Double clicking on the jar in Win7 won't pass the -jar flag to the java program, so nothing will happen.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,566
|
{"url":"https:\/\/lemon.cs.elte.hu\/trac\/lemon\/changeset\/722\/lemon-1.2\/lemon\/maps.h","text":"Changeset 722:b52189c479fb in lemon-1.2 for lemon\/maps.h\n\nIgnore:\nTimestamp:\n08\/02\/09 17:22:43 (12 years ago)\nBranch:\ndefault\nPhase:\npublic\nMessage:\n\nDoc improvements for several graph maps (#302)\n\nFile:\n1 edited\n\nUnmodified\nAdded\nRemoved\n\u2022 lemon\/maps.h\n\n r721 \/\/\/ but data written to it is not required (i.e. it will be sent to \/\/\/ \/dev\/null). \/\/\/ It conforms the \\ref concepts::ReadWriteMap \"ReadWriteMap\" concept. \/\/\/ It conforms to the \\ref concepts::ReadWriteMap \"ReadWriteMap\" concept. \/\/\/ \/\/\/ \\sa ConstMap \/\/\/ \/\/\/ In other aspects it is equivalent to \\c NullMap. \/\/\/ So it conforms the \\ref concepts::ReadWriteMap \"ReadWriteMap\" \/\/\/ So it conforms to the \\ref concepts::ReadWriteMap \"ReadWriteMap\" \/\/\/ concept, but it absorbs the data written to it. \/\/\/ \/\/\/ \/\/\/ In other aspects it is equivalent to \\c NullMap. \/\/\/ So it conforms the \\ref concepts::ReadWriteMap \"ReadWriteMap\" \/\/\/ So it conforms to the \\ref concepts::ReadWriteMap \"ReadWriteMap\" \/\/\/ concept, but it absorbs the data written to it. \/\/\/ \/\/\/ It can be used with some data structures, for example \/\/\/ \\c UnionFind, \\c BinHeap, when the used items are small \/\/\/ integers. This map conforms the \\ref concepts::ReferenceMap \/\/\/ integers. This map conforms to the \\ref concepts::ReferenceMap \/\/\/ \"ReferenceMap\" concept. \/\/\/ \/\/\/ stored actually. This value can be different from the default \/\/\/ contructed value (i.e. \\c %Value()). \/\/\/ This type conforms the \\ref concepts::ReferenceMap \"ReferenceMap\" \/\/\/ This type conforms to the \\ref concepts::ReferenceMap \"ReferenceMap\" \/\/\/ concept. \/\/\/ \/\/\/ The \\c Key type of it is inherited from \\c M and the \\c Value \/\/\/ type is \\c V. \/\/\/ This type conforms the \\ref concepts::ReadMap \"ReadMap\" concept. \/\/\/ This type conforms to the \\ref concepts::ReadMap \"ReadMap\" concept. \/\/\/ \/\/\/ The simplest way of using this map is through the convertMap() public: \/\/\/ \\brief This class represents the inverse of its owner (IdMap). \/\/\/ \/\/\/ This class represents the inverse of its owner (IdMap). \/\/\/ \\brief The inverse map type of IdMap. \/\/\/ \/\/\/ The inverse map type of IdMap. The subscript operator gives back \/\/\/ an item by its id. \/\/\/ This type conforms to the \\ref concepts::ReadMap \"ReadMap\" concept. \/\/\/ \\see inverse() class InverseMap { explicit InverseMap(const IdMap& map) : _graph(map._graph) {} \/\/\/ \\brief Gives back the given item from its id. \/\/\/ \\brief Gives back an item by its id. \/\/\/ \/\/\/ Gives back the given item from its id. \/\/\/ Gives back an item by its id. Item operator[](int id) const { return _graph->fromId(id, Item());} \/\/\/ It wraps a standard graph map (\\c NodeMap, \\c ArcMap or \\c EdgeMap) \/\/\/ and if a key is set to a new value, then stores it in the inverse map. \/\/\/ The values of the map can be accessed \/\/\/ with stl compatible forward iterator. \/\/\/ The graph items can be accessed by their values either using \/\/\/ \\c InverseMap or \\c operator()(), and the values of the map can be \/\/\/ accessed with an STL compatible forward iterator (\\c ValueIterator). \/\/\/ \/\/\/ This map is intended to be used when all associated values are \/\/\/ different (the map is actually invertable) or there are only a few \/\/\/ items with the same value. \/\/\/ Otherwise consider to use \\c IterableValueMap, which is more \/\/\/ suitable and more efficient for such cases. It provides iterators \/\/\/ to traverse the items with the same associated value, however \/\/\/ it does not have \\c InverseMap. \/\/\/ \/\/\/ This type is not reference map, so it cannot be modified with \/\/\/ \\brief Forward iterator for values. \/\/\/ \/\/\/ This iterator is an stl compatible forward \/\/\/ This iterator is an STL compatible forward \/\/\/ iterator on the values of the map. The values can \/\/\/ be accessed in the [beginValue, endValue) range. public: \/\/\/ Constructor ValueIterator() {} \/\/\/ \\e ValueIterator& operator++() { ++it; return *this; } \/\/\/ \\e ValueIterator operator++(int) { ValueIterator tmp(*this); } \/\/\/ \\e const Value& operator*() const { return it->first; } \/\/\/ \\e const Value* operator->() const { return &(it->first); } \/\/\/ \\e bool operator==(ValueIterator jt) const { return it == jt.it; } \/\/\/ \\e bool operator!=(ValueIterator jt) const { return it != jt.it; } \/\/\/ \\brief Returns an iterator to the first value. \/\/\/ \/\/\/ Returns an stl compatible iterator to the \/\/\/ Returns an STL compatible iterator to the \/\/\/ first value of the map. The values of the \/\/\/ map can be accessed in the [beginValue, endValue) \/\/\/ \\brief Returns an iterator after the last value. \/\/\/ \/\/\/ Returns an stl compatible iterator after the \/\/\/ Returns an STL compatible iterator after the \/\/\/ last value of the map. The values of the \/\/\/ map can be accessed in the [beginValue, endValue) public: \/\/\/ \\brief The inverse map type. \/\/\/ \/\/\/ The inverse of this map. The subscript operator of the map \/\/\/ gives back the item that was last assigned to the value. \/\/\/ \\brief The inverse map type of CrossRefMap. \/\/\/ \/\/\/ The inverse map type of CrossRefMap. The subscript operator gives \/\/\/ back an item by its value. \/\/\/ This type conforms to the \\ref concepts::ReadMap \"ReadMap\" concept. \/\/\/ \\see inverse() class InverseMap { public: }; \/\/\/ \\brief It gives back the read-only inverse map. \/\/\/ \/\/\/ It gives back the read-only inverse map. \/\/\/ \\brief Gives back the inverse of the map. \/\/\/ \/\/\/ Gives back the inverse of the CrossRefMap. InverseMap inverse() const { return InverseMap(*this); } \/\/\/ \\brief Gives back the \\e RangeId of the item \/\/\/ \/\/\/ Gives back the \\e RangeId of the item. \/\/\/ \\brief Gives back the \\e range \\e id of the item \/\/\/ \/\/\/ Gives back the \\e range \\e id of the item. int operator[](const Item& item) const { return Map::operator[](item); } \/\/\/ \\brief Gives back the item belonging to a \\e RangeId \/\/\/ \/\/\/ Gives back the item belonging to a \\e RangeId. \/\/\/ \\brief Gives back the item belonging to a \\e range \\e id \/\/\/ \/\/\/ Gives back the item belonging to the given \\e range \\e id. Item operator()(int id) const { return _inv_map[id]; \/\/\/ \\brief The inverse map type of RangeIdMap. \/\/\/ \/\/\/ The inverse map type of RangeIdMap. \/\/\/ The inverse map type of RangeIdMap. The subscript operator gives \/\/\/ back an item by its \\e range \\e id. \/\/\/ This type conforms to the \\ref concepts::ReadMap \"ReadMap\" concept. class InverseMap { public: \/\/\/ \/\/\/ Subscript operator. It gives back the item \/\/\/ that the descriptor currently belongs to. \/\/\/ that the given \\e range \\e id currently belongs to. Value operator[](const Key& key) const { return _inverted(key); \/\/\/ \\brief Gives back the inverse of the map. \/\/\/ \/\/\/ Gives back the inverse of the map. \/\/\/ Gives back the inverse of the RangeIdMap. const InverseMap inverse() const { return InverseMap(*this); \/\/\/ \\c bool value for graph items (\\c Node, \\c Arc or \\c Edge). \/\/\/ For both \\c true and \\c false values it is possible to iterate on \/\/\/ the keys. \/\/\/ the keys\u00a0mapped to the value. \/\/\/ \/\/\/ This type is a reference map, so it can be modified with the \/\/\/ subscription operator. \/\/\/ subscript operator. \/\/\/ \/\/\/ \\tparam GR The graph type. \/\/\/ mapped to the value. \/\/\/ \/\/\/ This map is intended to be used with small integer values, for which \/\/\/ it is efficient, and supports iteration only for non-negative values. \/\/\/ If you need large values and\/or iteration for negative integers, \/\/\/ consider to use \\ref IterableValueMap instead. \/\/\/ \/\/\/ This type is a reference map, so it can be modified with the \/\/\/ subscription operator. \/\/\/ subscript operator. \/\/\/ \/\/\/ \\note The size of the data structure depends on the largest \/\/\/ \\brief Dynamic iterable map for comparable values. \/\/\/ \/\/\/ This class provides a special graph map type which can store an \/\/\/ This class provides a special graph map type which can store a \/\/\/ comparable value for graph items (\\c Node, \\c Arc or \\c Edge). \/\/\/ For each value it is possible to iterate on the keys mapped to \/\/\/ the value. \/\/\/ \/\/\/ The map stores for each value a linked list with \/\/\/ the items which mapped to the value, and the values are stored \/\/\/ in balanced binary tree. The values of the map can be accessed \/\/\/ with stl compatible forward iterator. \/\/\/ the value (\\c ItemIt), and the values of the map can be accessed \/\/\/ with an STL compatible forward iterator (\\c ValueIterator). \/\/\/ The map stores a linked list for each value, which contains \/\/\/ the items mapped to the value, and the used values are stored \/\/\/ in balanced binary tree (\\c std::map). \/\/\/ \/\/\/ \\ref IterableBoolMap and \\ref IterableIntMap are similar classes \/\/\/ specialized for \\c bool and \\c int values, respectively. \/\/\/ \/\/\/ This type is not reference map, so it cannot be modified with \/\/\/ the subscription operator. \/\/\/ the subscript operator. \/\/\/ \/\/\/ \\tparam GR The graph type. \/\/\/ \\brief Forward iterator for values. \/\/\/ \/\/\/ This iterator is an stl compatible forward \/\/\/ This iterator is an STL compatible forward \/\/\/ iterator on the values of the map. The values can \/\/\/ be accessed in the [beginValue, endValue) range. public: \/\/\/ Constructor ValueIterator() {} \/\/\/ \\e ValueIterator& operator++() { ++it; return *this; } \/\/\/ \\e ValueIterator operator++(int) { ValueIterator tmp(*this); } \/\/\/ \\e const Value& operator*() const { return it->first; } \/\/\/ \\e const Value* operator->() const { return &(it->first); } \/\/\/ \\e bool operator==(ValueIterator jt) const { return it == jt.it; } \/\/\/ \\e bool operator!=(ValueIterator jt) const { return it != jt.it; } \/\/\/ \\brief Returns an iterator to the first value. \/\/\/ \/\/\/ Returns an stl compatible iterator to the \/\/\/ Returns an STL compatible iterator to the \/\/\/ first value of the map. The values of the \/\/\/ map can be accessed in the [beginValue, endValue) \/\/\/ \\brief Returns an iterator after the last value. \/\/\/ \/\/\/ Returns an stl compatible iterator after the \/\/\/ Returns an STL compatible iterator after the \/\/\/ last value of the map. 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Explorarea este activitatea de căutare sau călătorie cu scopul de a găsi în regiuni necunoscute, inclusiv spațiu cosmic, petrol, gaz, cărbune, apă, peșteri, sau informații. Explorarea implică și căutarea prin informațiile din diferite epoci cu scopul de a găsi lucruri neștiute până în acel moment.
Termenul este utilizat și pentru a descrie primele incursiuni ale oamenilor unei anumite culturi în arealul geografic sau cultural al altei culturi. Deși explorarea a existat încă de la începuturile ființelor umane, apogeul a fost atins în timpul Perioada Marilor descoperiri, când navigatorii europeni au călătorit în jurul lumii pentru a descoperi noi teritorii și culturi.
În cercetarea științifică, explorarea este unul dintre scopurile studiului (celelalte doua fiind descrierea și explicația). Explorarea este încercarea de a dezvolta înțelegerea grosieră inițială a unui anumit fenomen.
Exploratori celebri
Pytheas (380 – c. 310 BC) - explorator grec. Primul care a înconjurat Marea Britanie și care a explorat Germania. A ajuns în Thule, despre care se crede că sunt Insulele Shetland sau Islanda.
Erik cel Roșu (950 - 1003) - explorator Viking norvegian. După ce a fost alungat din Islanda, a navigat spre Groenlanda unde s-a și stabilit.
Leif Eriksson (980 - 1020) - explorator norvegian. Despre el se crede că este primul european care a ajuns în America de Nord.
Friar Julian (sec. al XIII-lea) - explorator maghiar Dominican friar.
Marco Polo (1254 - 1324) - explorator italian .
Ibn Battuta (1304 - 1377) - explorator Berber.
Zheng He (1371 - 1433) - explorator chinez.
John Cabot (c. 1450 - 1499) - explorator italian. A descoperit Newfoundland pe care l-a revendicat pentru Regatul Angliei.
Bartolomeu Diaz (c. 1450 - 1500) - explorator portughez. A navigat din Portugalia și a ajuns la Capul Bunei Speranțe.
Christopher Columbus (1451 - 1506) - explorator italian. A navigat în 1492 și a descoperit "Lumea Nouă" a Americilor.
Afonso de Albuquerque (1453 - 1515) - (Afonso cel Mare) navigator și colonizator portughez, întemeietorul puterii portugheze în India.
Amerigo Vespucci (c. 1454 - 1512) - explorator italian. A navigat în 1499 și 1502. A explorat coasta estică a Americii de Sud.
Juan Ponce de León (c. 1460 - 1521) - explorator spaniol. A explorat Florida în încercarea sa de a găsi Fântâna Tinereții.
Piri Reis (c. 1465/1470 – 1554/1555) - explorator otoman.
Pedro Álvares Cabral (c. 1467 - c. 1520) - explorator portughez, recunoscut ca primul european care a găsit calea maritimă spre Brazilia.
Vasco da Gama (c. 1469 - 1524) - explorator portughez. A navigat din Portugalia până în India ocolind Capul Bunei Speranțe.
Vasco Núñez de Balboa (c. 1475 - 1519) - explorator spaniol. Primul european care a traversat ismul Panama și care să vadă oceanul Pacific de pe țărmurile Americii.
Francisco Pizarro (c. 1475 - 1541) - explorator spaniol. Cuceritor al Imperiului Incaș.
Juan Sebastián Elcano (1476 - 1526) - explorator spaniol. A încheiat primul înconjor al Pământului într-o singură expediție după ce căpitanul său, Magellan, a fost omorât.
Sebastian Cabot (1476 sau 1477 - 1557) - navigator. Efectuează explorări prin America Centrală și de Sud.
Ferdinand Magellan (1480 - 1521) - explorator portughez. A inițiat prima expediție de înconjor al Pământului. A trecut prin Strâmtoarea Magellan și a numit oceanul Pacific. A murit în Filippine după ce le-a revendicat pentru Spania.
Giovanni da Verrazzano (c. 1485 - 1528) - explorator italian. A explorat coasta nord estică a Americii, din South Carolina până în Newfoundland.
Hernán Cortés (1485 - 1545) - explorator spaniol. A cucerit imperiul Aztec pentru Spania.
Jacques Cartier (1491 – 1557) - explorator francez. A descoperit Canada.
Hernando de Soto (c. 1496 - 1542) - explorator spaniol. A explorat Florida, și a descoperit Râul Mississippi.
Francisco Vásquez de Coronado (c. 1510 - 1554) - explorator spaniol. A căutat Cele Șapte orașe ale Aurului și a descoperit Marele Canion.
Sir Francis Drake (c. 1540 - 1596) - explorator englez. Primul englez care a făcut înconjurul Pământului și care a supraviețuit.
Pedro Sarmiento de Gamboa (1532 - 1592) explorator spaniol al Pacificului.
Alvaro de Mendaña de Neyra (1541-1596) explorator spaniol al Pacificului.
Willem Barents (ca.1550 - 1597) - navigator și explorator olandez, celebru pentru explorările arctice pentru găsirea "Pasajului de Nord-Est".
William Adams (1564 - 1620) - navigator englez. Explorează zona arctică iar mai târziu se stabilește în Japonia, fiind unul dintre primii europeni care cunoaște această țară.
Pedro Fernandes de Queirós )1565-1614) navigator portughez. A explorat Pacificul în serviciul Coroanei Spaniole.
Luis Váez de Torres (c. 1565-? ) navigator spaniol sau portughez. A explorat Pacificul în serviciul Coroanei Spaniole.
Henry Hudson (1570 - 1611) - explorator englez. A explorat mare parte din Atlanticul de Nord, inclusiv Labrador, coasta Groenlanda, și Hudson Bay. Se presupune că a murit în 1611, omorât de propriul echipaj.
Antão de Andrade (1580 - 1634) - misionar iezuit portughez. Primul european care ajunge în mod sigur în Tibet.
William Baffin (1584 - 1622) - navigator și explorator arctic englez
Cristóbal de Acuña (1597 - 1676?) - misionar iezuit spaniol. A explorat America de Sud.
Vitus Bering ([1681]] - 1741) - explorator danez. A explorat Estul Îndepărtat Siberian și Alaska revendicând-o pentru Rusia.
James Cook (1728 - 1779) - Căpitan de navă englez. A explorat mare parte din Pacific inclusiv Noua Zeelandă, Australia și Hawaii.
Louis Antoine de Bougainville (1729 - 1811) - navigator francez. Expediții în Oceania.
Jean François La Pérouse (1741–1788) ofițer și explorator francez a cărui expediție a dispărut în Oceania.
Alessandro Malaspina (1754-1810) - explorator italian. A explorat Pacificul și coasta de vest a Americii de Nord pentru Coroana Spaniolă.
William Bligh (1754 - 1817) - navigator și om politic englez. A explorat Tahiti și Indiile de Vest.
Alexander MacKenzie (1764-1820) explorator scoțian-canadian, care în 1789, căutând Pasajul de Nord-vest, a urmat cursul râului numit acum după el, până în Oceanul Arctic, și în 1793 a traversat Munții Stâncoși, ajungând la Pacific în 1793, învingându-i pe Lewis și Clark cu 12 ani.
Alexander von Humboldt (1769 - 1859) - explorator și om de știință german a cărui muncă a dus la întemeierea biogeografiei.
Căpitanul Meriwether Lewis (1774 - 1809) - explorator American care a condus expediția lui Lewis și Clark în Louisiana Purchase și nord-vestul Pacific în 1804-1806.
Charles Wilkes (3 aprilie 1798 – 8 februarie 1877) - ofițer naval american naval care a comandat expediția de explorare a Statelor Unite ale Americii
David Livingstone (1813 – 1873) - misionar și explorator scoțian în Africa Centrală. Este primul european care a văzut Cascada Victoria, pe care a numit-o în cinstea reginei Victoria.
Sir Samuel White Baker (1821 - 1893) - călător și explorator englez. Explorează zona Nilului și Extremul Orient.
Henry Morton Stanley (1841 – 1904) - jurnalist și explorator din Țara Galilor în Africa Centrală celebru prin căutarea lui David Livingstone, și prin fraza pe care a rostit-o când l-a găsit: "Dr. Livingstone presupun?"
Fridtjof Nansen (1861 - 1930) - explorator, om de știință și diplomat norvegian. Primul om care să înconjoare Groenlanda în 1888 și care a navigat în Oceanul Arctic cu Fram în 1893-1896, în încercarea sa de atinge Polul Nord împreună cu Hjalmar Johansen.
Otto Sverdrup (1854 - 1930) - explorator norvegian. S-a alăturat lui Fridtjof Nansen străbătând Groenlanda în 1888, și fiind căpitanul misiunii arctice Fram dintre 1893-1896, precum și celei de-a doua expediție Fram dintre 1898-1902. A întocmit harta părții cea mai de nord a Canadei în 1898-1902.
Roald Amundsen (1872 - 1928) - explorator norvegian. A condus prima expediție care a reușit în Antarctica între anii 1910 și 1912. El a reușit de asemenea să fie prima persoană care să traverseze North West Passage. Considerat cel mai mare explorator al tuturor timpurilor.
Robert Bartlett (1875 - 1946) - Căpitan din Newfoundland. A condus peste 40 expediții arctice, mai multe decât oricine înainte și după el. Este primul care a navigat la nord de latitudinea de 88° N.
Knud Rasmussen (1879 - 1933) - explorator polar și antropolog din Groenlanda. Rasmussen a fost primul care să traverseze Northwest Passage prin intermediul săniilor trase de câini.
Ahmed Pasha Hassanein (1889 - 1946) - explorator, diplomat egiptean, unul dintre cei doi câștigători ne-europeni ai medaliei de aur ai Royal Geographical Society în 1924, King's chamberlain, participant la Olimpiada din 1924, fotograf, autor și descoperitorul lui Jebel Uweinat, autorul cărții "The Lost Oases", scrisă în trei limbi.
Colonel Noel Andrew Croft (1906 - 1998) -a deținut recordul pentru cea mai lungă expediție arctică făcută în anii 1930 pentru aproximativ 60 de ani.
Sir Edmund Percival Hillary (n 20 iulie 1919-11 ianuarie 2008) - explorator din Noua Zeelandă, care împreună cu Tenzing Norgay, a reușit să cucerească Muntele Everest.
Neil Armstrong (n 5 august 1930- 25 august 2012) - astronaut american - primul om care a pus piciorul pe Luna la 20 iulie 1969.
Iuri Gagarin (9 martie 1934 – 27 martie 1968) - cosmonaut sovietic care la 12 aprilie 1961 a devenit primul om care a călătorit în spațiu cosmic și primul om care a orbitat în jurul Pământului.
Reinhold Messner (n 17 septembrie 1944) - explorator italian, primul om care să cucerească toate cele 14 vârfuri mai înalte de 8 000 de metri.
Frank Cole (1954 - 2000) - explorator canadian, producător de film. Este primul Nord American care să traverseze deșertul Sahara singur pe o cămilă în 1990. A fost omorât de bandiți în timpul celei de-a doua traversări din anul 2000.
Exploration by area
Exploration of the Pacific Ocean
Exploration of the Pacific Northwest
Exploration of the Americas
European exploration of Arabia
European exploration of Asia
European exploration of Africa
European exploration of Australia
European exploration of North America
European exploration of South America
Exploration of the High Alps
Exploration of Mercury
Exploration of Venus
Moon Landings and Exploration of the Moon
Exploration of Mars
Friar Marcos
Vezi și
Explorator
Listă de exploratori
Legături externe
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How Running For Our Lives May Have Made Humans Smarter
By Joel N. Shurkin 2013-01-28T22:55:01Z
This image shows a 3-D reconstruction of a mouse brain based on magnetic resonance imaging (MRI). The forebrain is seen in green, the midbrain in yellow and the cerebellum in orange.
(Image: © Garland Lab, UC Riverside)
(ISNS) -- Could athletic prowess be linked to the size of our brains? Some new research suggests that exercise-loving mice have larger midbrains then their more mellow counterparts.
Scientists now think that the ability to run far and fast helped us evolve both physically and mentally. For evidence, look to the common house mouse.
Scientists at the University of California, Riverside have taken house mice, bred for both their propensity and abilities on treadmill wheels, and found they had a larger section of their brain called the midbrain and other parts of the brain outside the cerebellum -- the motor control center -- than plain house mice. But the overall size of their brains did not vary significantly.
Whether the mice evolved larger midbrains because they exercised, or exercised because they had bigger brains, remains to be determined.
In the classic comedy sketch, "The 2,000-Year-Old Man," Mel Brooks, playing the title character, said the main form of transportation long ago "was mostly fear."
"You'd hear an animal growling and you'd go two miles in a minute," he said. That was probably true, at least in spirit.
Theodore Garland and his team didn't go that far back. They bred dozens of generations of mice for athleticism and studied them for almost 20 years. Garland said that he uses mice instead of the conventional laboratory rats because rats are more expensive and take longer to turn over generations.
The relationship between brains and body is especially important in mammals, which have a greater brain size relative to body size compared with most other organisms. With increased brain size comes a greater complexity in behavior and thought.
In 2004, Daniel Lieberman of Harvard University and Dennis Bramble at the University of Utah suggested that one of the reasons humans survived and evolved is that they learned to run faster and further. That allowed them, like Mel Brooks' character, to get away from predators, and they also could walk farther so they could track down animals and bring food home.
Those who could run faster and walk farther reproduced more than those who could not, they wrote, so humans evolved endurance. Our legs grew longer, toes shortened; we lost hair, and gained more complex middle ears for balance. This transformation changed the whole neurobiology of the human species, according to David Raichlen of the University of Arizona.
Human brains grew in size, and we became smarter. Whether one is related directly to the other is not known, and is the reason for the kind of research Garland's team does.
Another thing Garland and his team were trying to test is a theory called mosaic evolution. As animals evolve, do certain areas of their bodies change independently of what happens to the rest of the body, or does the whole body generally evolve simultaneously? Garland wanted to see whether parts of the mouse brain evolved independently of the rest of the brain.
They used four strains of athletic mice and four strains of more indolent mice. The athletic mice were given access to exercise wheels and by all accounts had a fabulous time.
"They love going on the wheels and excel in both motivation and abilities," said Garland, whose work was published in The Journal of Experimental Biology.
The brains of the mice then were dissected and studied. Garland said that the midbrain of the mice was studied using an MRI device because they were too small to dissect. The difference between the growth of the midbrain relative to the overall size of the brain seems to support the mosaic evolution concept. For whatever reason, the non-cerebellum part grew faster than the brain as a whole.
The midbrain, which accounts for about 10 percent of a mouse's brain, affects the senses, motor control and alertness. The midbrain also is the site of reward centers, an important factor for this latest research. When mammals encounter something pleasurable -- food, sex, music -- the reward center sends out chemical signals of pleasure. Presumably, the larger the proportion of the midbrain, the greater the ability to experience that pleasure.
Raichlen, who calls Garland's work "an amazing step forward," said this new "data show you can select a behavioral trait and see a change in the brain."
Having this happen in the midbrain reinforces pleasurable behavior; in this case, it would be running.
Garland's next experiment, however, is to see whether it is the exercise that is increasing the midbrain in mice instead of the opposite. He thinks using MRIs and questionnaires with living humans could test to see if the same things are true of human joggers.
Joel Shurkin is a freelance writer based in Baltimore. He is the author of nine books on science and the history of science, and has taught science journalism at Stanford University, UC Santa Cruz and the University of Alaska Fairbanks.
Inside Science News Service is supported by the American Institute of Physics.
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Martin Yelling's Long Run Home
Martin Yelling | Long Run Home | Julias House | Macmillan Cancer Support | Southmead Hospital Charity
Martin Yelling has completed Ironmans (including Kona), he's represented his country, he's run Comrades, he runs marathons. What's left to truly test him? 630 miles. 115,000 feet of ascent (like climbing Everest 4 times). 21 days of running. That should do it!
Martin is running the South West Coastal Path, the UK's longest national trail. He's running from Minehead in North Somerset back to his home in Poole in Dorset (hence the name of the challenge - Long Run Home) via, Devon, Cornwall and Land's End. He's going to be averaging 30 miles a day, and if that doesn't sound tough enough, factor in some huge hills and very tough terrain. This is a very, very difficult challenge. He sets off on July 11th.
So why is Martin doing it? To quote him: 'I've been fortunate enough to have travelled a bit with my sporting career and that of my much faster wife. We've been privileged to have run in some awesome and inspiring places all over the world. Mountains, rivers, forests and vistas to admire for miles. Yet never have I felt more at home than when running along the South West Coast Path. It feels like my spiritual running home. It's a coastal landscape with an ever changing feel and face. With one breath its beauty is startling. With another its harshness is stark. Its rugged and iconic cliff tops, hard edges, steep steps, windswept beaches and seductive coves. The history, heritage, geology and of the path is dramatic, exciting, and iconic.'
Martin is also raising money for 3 charities that are close to his heart: Julia's House in Poole (for which Martin's wife Liz is a patron), Macmillan Cancer Support and the Frenchay Hospital in Bristol (part of the Southmead Hospital Charity). He's hoping to raise £10,000 so please support him here if you can.
As well as donating to his charities, you can also support Martin by running with him on a stage of the route (details here). You don't have to run the whole 30 miles with him. He'd appreciate any support along the way. It's a very long run home…
www.longrunhome.co.uk
Skid Row Marathon Pip Haylett went along to a special screening of Skid Row Marathon
Record breaking weekend at Bounemouth Ma...Race report: Bournemouth Marathon Festival (BMF) - October 4-5, 2014
Join Barry Thompson's ASICS Greater Ma...Charity feature: Organisers are encouraging runners to join Barry's Project 15...
FREE 12 week marathon schedule from Mart...In his role as ambassador for the Human Race Pace Series, international running ...
Skid Row Marathon - a must-see filmSpecial screening of this very special film
Best foot forward for the new Sheffield ...Still time to enter this new event on 30th October
Charities given boost as Asda Foundation...Crowds turn out in force for the 7th Hull 10k
Asda Foundation Burnley 10k hands charit...Runners and spectators out in force to support Burnley 10k
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The embankment holding the railway line built in 1858 runs beside the site of The Castle, and the two arches built to support it changed the whole look of the village.
The Castle Inn was built in 1853, just before the advent of the railway in 1859. The Old White Hart Ale House, which previously occupied a nearby site, was the haunt of members of the Hawkhurst Gang of Smugglers.
In 1748 the landlady Mrs Payne welcomed the gang and helped to send two customs officers on the path to their murders.
Over in Poole, smuggler John Diamond started off the horrific events by storming the Customs House to retrieve "his" Contraband Tea impounded there. After the raid the gang were seen in Fordingbridge. A shoe maker, Daniel Chater on hearing of the two hundred pounds reward for information regarding Diamond informed on him and he was duly arrested.
uniform did not stand a chance. They were lashed to a horse, then dragged up Bowes Hill and through the country to Rake, where William Galley was buried in a shallow grave.
Two days later, Chater was thrown down a well at Ladyholt Park. lt was only after sometime that an anonymous letter was received and seven members of the gang were apprehended and hung in Chichester in 1749. From then on violence slowly decreased.
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Censored
Censored
DISTRACTION AND DIVERSION INSIDE CHINA'S GREAT FIREWALL
_Margaret E. Roberts_
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyright © 2018 by Princeton University Press
Published by Princeton University Press,
41 William Street, Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press,
6 Oxford Street, Woodstock, Oxfordshire, OX20 1TR
press.princeton.edu
All Rights Reserved
ISBN 978-0-691-17886-8
Library of Congress Control Number 2017962808
British Library Cataloging-in-Publication Data is available
This book has been composed in Minion Pro
Printed on acid-free paper ∞
Typeset by Nova Techset Pvt Ltd, Bangalore, India
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
For my parents,
Don and Barbara Roberts
CONTENTS
_Acknowledgments_ | ix
---|---
1 Introduction |
1.1 The Puzzle of Porous Censorship |
1.2 Distraction and Diversion |
1.3 Implications and Challenges to Conventional Wisdom |
1.4 The Plan of the Book |
2 A Theory of Censorship |
2.1 Why Do Governments Censor? |
2.2 Citizens Are Rationally Ignorant |
2.3 Traditional Media Care about Story Costs |
2.4 Citizens Exchange Low-Cost Information Through Social Media |
2.5 What Is Censorship? |
2.6 The Mechanisms of Censorship |
2.7 Fear |
2.8 Friction |
2.9 Flooding |
2.10 Conclusion |
3 Censorship in China |
3.1 Modern History of Information Control in China |
3.2 Censorship of the Chinese Internet |
4 Reactions to Experience with Censorship |
4.1 China's Targeted Censorship Strategy |
4.2 The Costs of Observable Censorship |
4.3 Matched Comparison of Censored and Uncensored Social Media Users |
4.4 An Experimental Study of Consumers of Social Media |
4.5 Conclusion |
5 The Powerful Influence of Information Friction |
5.1 The Effects of Content Filtering on the Spread of Information |
5.2 Structural Frictions and the Great Firewall |
5.3 When Does Friction Fail? |
5.4 Conclusion |
6 Information Flooding: Coordination as Censorship |
6.1 What Effect Can Propaganda Have in the Digital Age? |
6.2 Flooding in China |
6.3 Detection of Information Flooding in Newspapers and Online Media |
6.4 The Influence of Flooding on the Spread of Information |
6.5 Conclusion |
7 Implications for a Digital World |
7.1 Why Porous Censorship Matters |
7.2 Authoritarian Resilience |
7.3 Implications for Free Speech in Democracies |
7.4 A Call for Future Research |
8 Appendix |
8.1 Description of the China Urban Governance Survey |
8.2 Words Related to Censorship, Mutual Information |
8.3 Tibet Self-Immolations Negative Binomial Model |
_Works Cited_ |
_Index_ |
ACKNOWLEDGMENTS
Writing a book is not something done in solitude; rather, it is reliant on the insights, conversations, support, and feedback of an entire community. I am indebted to many for their help throughout this process. The origins of this book are in my conversations with Gary King and Jennifer Pan, with whom I originally began working on censorship and who encouraged me to pursue the idea beyond what is censored to how censorship impacts people. Their ideas and feedback are embedded in this work, and I am thankful to them for being wonderful mentors, collaborators, and friends. Similarly influential was Brandon Stewart, whose work appears within this manuscript, and who through a long friendship and collaboration has strongly impacted how I think both about data analysis and social science.
This book grew out of my dissertation that began at Harvard. I owe a great debt to my many colleagues in graduate school whose ideas, criticisms, and friendship enriched my time at Harvard and vastly improved my work. Not only Brandon and Jen, but also the rest of my reading group—Shelby Grossman, Chiara Superti, Amanda Pinkston, and Vanessa Williamson—read many versions of my work and provided invaluable feedback. My dissertation committee—Gary King, Jeff Frieden, Elizabeth Perry, and Dustin Tingley—provided me with incredible feedback and support throughout my time at Harvard. They provided outstanding examples of scholars and teachers, and I learned a tremendous amount from them.
When I moved to University of California, San Diego, I was immediately welcomed into a warm community of scholars who took me under their wing. I could not be more indebted to Susan Shirk and Lei Guang and the 21st Century China Center for providing feedback, mentorship, and funding while I was finishing the manuscript. My other colleagues in Political Science, School of Global Policy and Strategy, and at the San Diego Supercomputer, in particular Claire Adida, Jude Blanchette, Jesse Driscoll, Amarnath Gupta, James Fowler, Stephan Haggard, Seth Hill, Ruixue Jia, David Lake, Kai Lin, Megumi Naoi, Barry Naughton, Simeon Nichter, Sam Popkin, Christina Schneider, Branislav Slantechev, Victor Shih, David Wiens, and Yiqing Xu, were generous with their time and provided me invaluable feedback on my work. Thanks to Justin Grimmer, Tim Groeling, Xiao Qiang, and Jeremy Wallace for traveling to San Diego and providing detailed feedback on the manuscript for my book conference. Thanks to Will Hobbs who worked with me on censorship and who helped me think more deeply about how sudden information changes can impact censorship outcomes.
I offer thanks to those people who supported me during my fieldwork in China. In particular, Kai Lukoff and Maoliang Ye provided incredible support and valuable conversations during my time in Beijing. A special thanks to Tianguang Meng for his tremendous insights and support. Peter Volberding and Jeff Javed were the best fieldwork roommates anyone could ask for. Thanks to Frances Chen, Yichen Guan, Yingjie Fan, Amy Jiang, Adam Jin, Heather Liu, Junjie Liu, Fei Meng, Hongmiao Wang, and LuShuang Xu for their assistance at different points in the process. Thanks also to those who helped me who prefer to remain anonymous.
Many others provided valuable feedback throughout the process. Other than those I have already listed, Alexei Abrahams, Pablo Barbará, Chris Cairns, Any Catalinac, Joan Cho, Jeff Colgan, Greg Distelhorst, Ruben Enikolopov, Roya Ensafi, Nick Feamster, Scott Gehlbach, Anita Gohdes, Judy Goldstein, Sheena Greitens, Andy Hall, Jingkai He, Nancy Hearst, Yue Hou, Haifeng Huang, Kyle Jaros, Ashley Jester, Iain Johnston, Aaron Kaufman, Jeehye Kim, Patrick Lam, Peter Lorentzen, Chris Lucas, Andrew MacDonald, Dan Mattingly, Gwyneth McClendon, Blake Miller, James Ashley Morrison, Rich Nielsen, Maria Petrova, Maggie Peters, Jakub Redlicki, Jake Shapiro, Daniel Smith, Jim Snyder, Arthur Spirling, David Steinberg, Zachary Steinert-Threlkeld, Daniel Treisman, Alex Storer, Brian Tsay, Sam Tsoi, Disy Trinh, Rory Truex, Josh Tucker, Felicity Vabulas, Jessica Weiss, Vanessa Williamson, Yuhua Wang, Jason Wu, and Yin Yuan. For those who I have forgotten to list, it is not out of lack of appreciation but rather out of my own forgetfulness.
This book would not have been possible without the financial support of generous research institutions. I thank 21st Century China Center, the Carnegie Corporation of New York, the Policy Design and Evaluation Lab, National Science Foundation (grant 1500086), DARPA (contract W31P4Q-13-C-0055/983-3), Hellman Fellows Fund, Institute for Quantitative Social Science, and the Weatherhead Center for International Affairs for financial support. I thank the Ashford family for their generous funding and personal support throughout my time at Harvard through the Ashford Fellowship.
Thanks to Eric Crahan and those at Princeton University Press for their support throughout the process. The book was much improved by anonymous reviewers who read this manuscript and provided feedback. Thanks also to Madeleine Adams for reading and improving the manuscript.
A final thanks to my wonderful family for entertaining and improving my ideas and reminding me to take breaks. First and foremost, thanks to my husband David for constantly challenging me to communicate my ideas more clearly, providing me unconditional love and support, and for being an outstanding example of a scholar. This book would not have been possible without him. Thanks to my siblings Emily, Heather, and Allen for reading the book many times, entertaining me with food, jokes, and playlists, and reminding me to speak to a broader audience. Thanks to my first teachers and mentors, my parents Don and Barbara, for three decades and counting of thought-provoking conversation. I have always tried to emulate their thirst for knowledge and commitment to truth. I dedicate this book to them.
Censored
CHAPTER ONE
Introduction
China has four million websites, with nearly 700 million Internet users, 1.2 billion mobile phone users, 600 million WeChat and Weibo users, and generates 30 billion pieces of information every day. It is not possible to apply censorship to this enormous amount of data. Thus censorship is not the correct word choice. But no censorship does not mean no management.
—Lu Wei, Former Director, State Internet Information Office, China, December 20151
**1.1 THE PUZZLE OF POROUS CENSORSHIP**
As more people around the world gain access to the Internet, government censorship seems an increasingly futile exercise. Traditional conceptions of censorship that could completely control information, such as watertight bans on access, prepublication review, or government-enforced prohibitions on content, seem silly when you consider that every second millions of Internet users around the world are sending one another instant messages, participating in online forums, and tweeting to hundreds of thousands of followers. Even the world's most famous censors recognize this reality. As the former "gatekeeper of the Chinese Internet" Lu Wei stresses in the epigraph to this chapter, the thirty billion pieces of information generated each day by Chinese citizens quite simply cannot be censored.
Yet recognizing the impossibility of complete control of online discourse has not kept authoritarian regimes from spending billions of dollars trying. On the face of it, authoritarian efforts of information control seem halfhearted. Even censorship in one of the most sophisticated censorship regimes in the world—China—could be seen as faltering attempts at "information management." For the most part, these efforts at censorship are porous—frequently circumvented by savvy Internet users, accidentally evaded by citizens wasting time on the web, and rarely enforced with punishment.2
Indeed, most censorship methods implemented by the Chinese government act not as a ban but as a _tax_ on information, forcing users to pay money or spend more time if they want to access the censored material. For example, when the government "kicked out" Google from China in 2010, it did so simply by throttling the search engine so it loaded only 75 percent of the time.3 If you wanted to use Google, you just had to be a bit more patient. The Great Firewall, China's most notorious censorship invention that blocks a variety of foreign websites from Chinese users, can be circumvented by savvy Internet users by downloading a Virtual Private Network (VPN). Social media users in China circumvent keyword censoring of social media posts by substituting similar words that go undetected for words that the government blocks, making content easy to find if you spend more time searching.4 Newspapers are often instructed by censors to put stories on the back pages of the newspaper, where access is just a few more flips of the page away.5
Porous censorship is not unique to China or even to the modern time period. Instead of shutting off the whole Internet, Iran has been known to simply throttle it and make it slower during elections.6 The Russian government uses armies of online bots and commentators to flood opposition hashtags and make it more difficult, but not impossible, for people to find information on protests or opposition leaders.7 Even before the Internet, in the late nineteenth century, British censors banned translations of French literature they considered obscene, but allowed untranslated versions to circulate freely, allowing unlimited access to those willing to expend the effort to read them in French.8 In East Germany during the cold war, the government decided against enforcing restrictions on satellite dishes that enabled citizens to watch West German television, effectively allowing East Germans who were interested enough to find a way to buy a satellite dish to have access to it.9
Why do governments attempt to control information when these controls are easily circumvented? Conventional wisdom posits that these porous censorship strategies are futile for governments as citizens learn quickly to circumvent censorship that is not complete or enforced. Many have stressed that information, which is often called "non-excludable" because it is easily shared, is difficult to control once it has become known to a portion of the public, as it can spread quickly.10 "Information wants to be free," originally coined by Stewart Brand, captures the idea that information technology makes information easy to copy and thus difficult to control.11 More puzzling is that many governments have the capacity to enforce censorship more forcefully, but choose not to do so. Periodic VPN crackdowns indicate that China could make the Firewall less permeable, but much of the time the government chooses not to.12 The government could implement draconian punishments for those who evade censorship, creating strong disincentives for circumvention, but most circumvention is not even illegal. Using censorship that taxes, rather than prohibits, information in China—and in other countries around the world—seems to be a design choice, not an operational flaw—but why?
**1.2 DISTRACTION AND DIVERSION**
In this book, I shed light on the puzzle of porous censorship by showing that even easily circumventable censorship has an important impact on information access for the typical person in most circumstances, and, for this very reason, is strategically useful for authoritarian regimes. Many censorship methods require citizens to spend more time or money accessing information that the government would like to slow down. Only a minority of citizens who are interested enough in the information and have the education and resources to pay the costs of evasion are motivated and equipped enough to circumvent censorship. For the majority of citizens, who are less interested in politics and are not willing to spend significant time becoming informed,13 small costs of access and government distractions can divert citizens to information that is less dangerous to the regime. Even though it is possible to access most information, as normal citizens get lost in the cacophony of information available to them, their consumption of information is highly influenced by the costs of obtaining it. I argue that there are massively different implications for the spread of political information of having certain information completely free and easy to obtain as compared to being available but slightly more difficult to access.
Part of the inconsistency between conventional wisdom about censorship and the reality of censorship results from the lack of conceptual clarity about the mechanisms by which censorship affects the public's consumption of information. We lack a theory of censorship. I provide a typology of the three ways in which censorship can affect individuals. What most people think of when they think of censorship is _fear_ —threats of punishment, such as losing a job, prison, or worse—which may deter citizens from spreading or accessing information. Fear works by prohibiting particular information and through this inducing self-censorship. But the threat of punishment must be observable to be credible—those who are not aware of punishment cannot be deterred by it. Although fear is a more complete form of censorship because it can be enforced, fear is problematic for authoritarian regimes because it can cause backlash, draw attention to censored information, and create information-gathering problems for governments. Fear is more difficult to use in the digital age because prohibitions on information are difficult for governments to enforce when information is easily copied.
The other two less well-known censorship mechanisms I introduce— _friction_ and _flooding_ —have proven themselves more useful in the age of the Internet. Friction—increasing the costs, either in time or money, of access or spread of information—diverts citizens' attention by imposing barriers to information access. A slow webpage, a book removed from a library, reordered search results, or a blocked website can all be used to increase the costs of access to information. Friction is often circumventable—it can be evaded simply by sustaining these costs. However, it does not have to be observable in order to work and therefore can more easily be explained away or go unnoticed. Friction's counterpart, _flooding_ , is information coordinated as distraction, propaganda, or confusion, such as astroturfing, online propaganda, or government-mandated newspaper articles. Flooding competes with information that authoritarian governments would like to hide by diluting it and distracting from it. As with the friction mechanism, while flooding can be discounted or avoided, flooding requires the consumer to take time and effort to separate out good information from bad information.
I offer a wide range of empirical evidence—from online experiments to nationally representative surveys, datasets of millions of geo-located social media posts, and leaked propaganda archives—to show that friction and flooding effectively divert and distract most people away from censored information. Even though a minority of people will pay the costs to circumvent censorship, friction and flooding are useful to governments because they separate those who are willing to pay the cost of evasion from those who are not, enabling the government to target repression toward the most influential media producers while avoiding widespread repressive policies. I focus my empirical evidence on the citizen production and consumption of information on the Chinese Internet. China is a nearly ideal case for testing how each mechanism of censorship affects citizens' consumption of information and political behavior because the Chinese government implements a wide variety of censorship tactics, which function through each of the three censorship mechanisms. Furthermore, China's censorship system has become the model for many authoritarian regimes: evidence exists that others are trying to emulate it.14 A better understanding of how the Chinese censorship system works will allow us to predict the future impacts of information control across a wide range of authoritarian regimes.
Censorship is difficult to study empirically because it is often intended to go undetected. Recently, entire subfields in computer science have emerged dedicated to detecting censorship because governments are not typically forthcoming with their tactics.15 In this book, I move beyond what is censored to take up the challenging task of measuring individuals' reactions to censorship _while_ they are being subjected to it. Using large social media datasets, measures of the spread of online information, online experiments, and surveys, I answer the questions: How do individuals react when observing censorship? How does Internet users' behavior change when particular pieces of information are more difficult to access? Are Internet users who come across distracting online propaganda likely to spread and share it? The evidence I present shows that although many people are resistant to censorship when they notice and observe it, they are very affected by it when they are inconvenienced by it, do not notice it, or can explain it away.
My findings of how censorship influences individuals may explain why we see so many regimes using porous censorship strategies even though these methods are easy to thwart. Although many would see the fact that a minority of capable citizens can route around censorship as detrimental to the regime's censorship efforts, I argue that circumventible censorship can be useful to authoritarian regimes precisely because it has different effects on different segments of the population. Porous censorship drives a wedge between the elite and the masses. The savvy members of the elite easily circumvent censorship, discount propaganda, read blocked information, and enter into banned social networks. By contrast, friction and flooding prey on the rest of the public's short attention spans, busy schedules, and general lack of interest in politics, nudging them toward an information environment that is disconnected from their more well-educated, well-to-do, and politically sophisticated counterparts. By separating the elite from the masses, the government prevents coordination of the core and the periphery, known to be an essential component in successful collective action.16 Although a portion of savvy and politically concerned citizens may be willing to pay the costs imposed by friction and flooding, less interested individuals often are not, making wider discontent among the broader population significantly less likely and reducing the accountability of political entities.
The strategy of porous censorship allows the government to avoid widespread use of observable repression, which is well known to spark popular backlash.17 Autocrats face significant trade-offs when making citizens fearful of speaking out. Highly constraining forms of censorship that operate through deterrence must be observable to their targets; otherwise deterrence cannot work. As I will show using social media data, surveys, and online experiments, when censorship is observable, political entities call attention to the information they would like to make off-limits. The observation of censorship intended to create deterrence can instead create opportunities for push-back, signal government weakness, and create increased interest in the off-limits topic. Repression that deters citizens from speaking out also creates information and surveillance problems for the government, as governments often rely on input from the media and population to identify local corruption and on information in the public sphere to identify new pockets of dissent.18
Incomplete censorship, by contrast, is more easily masked by political entities, giving the government the cover of plausible deniability.19 Flooding can front as concerned citizens who are voluntarily writing pro-government content online or are spontaneously gathering in a pro-government parade, and friction can front as technological errors or algorithmic quirks, which ordinary citizens may not be aware of or may explain away. If a link on the Internet redirects to an error page, it is difficult to tell whether the page is down or the government has blocked it. If a book is missing from a library shelf, is it lost, not ordered, or removed by the government? If a social media post does not appear in a news feed, is it because the algorithm predicts you might not be interested in it, or because of government manipulation?20 Because information is widespread and has many substitutes, small impediments to reading information and even silly distractions can significantly affect users' consumption of political information.
The strategy of porous censorship does, however, have an Achilles' heel. Although for most citizens most of the time, small impediments to accessing information and government-encouraged distractions can divert them to more benign information, there are cases when the typical citizen will take the time to seek out restricted information and evade censorship. I show that in periods of crisis, such as the 2015 Tianjin explosion, citizens are more likely to spend time seeking out methods of accessing restricted information. Similarly, when censorship is imposed suddenly and disrupts habits, such as the case of the Instagram block during the 2014 Hong Kong protests, citizens are more likely to find ways to continue consuming information and entertainment to which they are accustomed.21 Thus, the strategy of porous censorship can be counterproductive and dangerous to the regime when it uses this censorship too decisively during times it needs censorship most. If information were to disrupt the Chinese political system, it would be during a period when the majority of people were willing to pay the price imposed by censorship to collectively inform themselves.
**1.3 IMPLICATIONS AND CHALLENGES TO CONVENTIONAL WISDOM**
The findings I present in this book challenge many conventional notions of censorship and have implications for research in digital politics, the politics of repression, and political communication.
_**Censorship Is More Than Fear**_
First, this book speaks to the strategies that modern autocracies use to prevent large-scale dissent. Many scholars have puzzled over the resilience of some authoritarian regimes.22 Some argue that the resilience of autocracies is due in part to successful repression; that autocrats have survived by forcefully extinguishing opposition groups.23 Others have maintained that autocrats are successful in part by creating institutions that are better able to share power with the opposition and respond to citizens' concerns.24 Still others have credited authoritarian resilience to brainwashing or enforced symbolism, through cultlike nationalism, religion, or ideology.25
In this book I demonstrate that autocrats have methods outside of direct repression, accommodation, or brainwashing to maintain power, even in the modern era. Autocrats have a large toolbox available to them to nudge citizens away from activist circles, dangerous information, and focal points that could facilitate coordination.26 These methods are not forceful, do not accommodate, and are often not meant to directly persuade. Instead, they create small inconveniences that reroute users to information and social networks that are more palatable to the regime, decreasing the mobilization capacity for opposition, often without citizens being aware of it. Although less forceful than repression or brainwashing, these methods are surprisingly effective in changing the behavior of the vast majority of citizens who are too busy to engage deeply in politics.
_**Censorship Is Customized**_
Second, this book speaks to a long-standing question of whether and how governments can control social media in the information age. Many scholars believed that the Internet, which expanded the number of citizens involved in public discourse, would force governments to become more accountable to citizens because of the speed with which large numbers of citizens could participate in everyday public debate.27 Yet the failure of the Internet to create the expected accountability in some authoritarian regimes led other scholars to argue that this new technology in fact played into the hands of the autocrats.28 Some of these writers hypothesized that the Internet had not reached its political potential because of extreme self-censorship and fear.29 Others discerned that the Internet created opportunities for authorities to use sophisticated hidden technologies that could manipulate citizens without their consent or being aware of it.30
The findings in this book cut a middle path between these arguments by showing that Internet censorship has very different impacts on different types of individuals, which allows governments to use these differential effects strategically to maximize censorship's impact while minimizing its costs. The findings in this book suggest that the low probability of the government following through on punishment for millions of Internet users who engage daily in off-limits discussion has diminished the government's ability to enforce self-censorship on those engaged in public discourse. Self-censorship, by itself, does not "purify" the Internet in many authoritarian regimes as some have suggested, and online criticism of autocrats is commonplace.31 For the majority of citizens, this book provides evidence that political entities have a wide range of effective tools available to them to interfere with the Internet without citizens being aware of it or motivated enough to circumvent it.32 However, these tools work not because they are sophisticated enough to prevent access to information, but precisely because they have holes: they can affect the majority of the public's information-seeking behavior simply by inconveniencing them, without interfering so much to cause widespread public backlash. Small costs of access, not draconian punishments or sophisticated manipulation, can have huge effects on the behavior of the majority.
Because censorship affects different segments of the population differently, its impact is more than simply hidden manipulation and instead is a story of customized repression. The fact that the majority are affected by diversion and distractions allows regimes the flexibility to selectively target punishment for speech toward journalists, activists, and other high-profile elites. Because friction and flooding are not effective for highly capable and motivated individuals, autocrats use targeted fear to contain the spread of information at elite levels.33 Just as the Internet has enabled more micro-targeting of information and advertising toward particular individuals, the evidence I present suggests that censorship as well is becoming increasingly customized to individual behavior and capabilities.
Despite the cunning of the Chinese censorship system, I highlight the ways in which the censorship system can be undermined in particular periods. I show that the regime is more constrained in making censorship porous during crises when individuals are motivated to seek out information. The more that citizens are willing to overcome friction, the less able the government is to use censorship methods other than fear. This puts the government in a difficult situation, as direct repression will frequently cause backlash. Although the government will try to ramp up all forms of censorship during periods of crisis, these are also the periods that are most likely to force government accountability and concessions.
_**More Media Does Not Always Lead to Better Information**_
I caution against a rosy economic model of information where more producers of information will always lead to better information outcomes. Some scholars have posited that as the number of producers of information and media outlets increases, the government's influence over the media will decrease because governments will have a more difficult time forcing media to keep silent.34 One implication is that the digital age, where there are many more producers of information, will lead to a less biased news media.35
However, these models only consider coercion of media and media capture as methods of censorship and do not consider the impacts that governments have on the _distribution_ of information. The results in this book show that even if media that contains better information exists, if government can create frictions on the distribution of information through censorship, then this media will not reach most of the public.36 Governments that have direct control over information distribution can use friction to de-prioritize media that they find to be objectionable. Even if articles on the Internet contain good information, if they are buried in a search engine by government censorship, very few people will access it.
Moreover, even if governments do not directly control the distribution of information, they can use the fact that anyone can enter into the Internet discourse to flood the information environment with their own version of events. By hiring paid commentators or distributing online propaganda, governments can crowd out information that they find objectionable, undermine the credibility of competing media, and distract citizens from events that reflect badly on them. Counterintuitively, the ability for anyone to produce media can result in the production of less reliable information because some governments and entities will have incentives and resources to produce and spread unreliable information en masse.
_**A Broader Definition of Censorship Has Implications for Democracies**_
Last, because this book is about censorship that does not always function through fear, it has broader implications for censorship outside of authoritarian systems. Democracies generally have laws that prevent them from directly repressing free speech—they cannot use fear-based methods of censorship. However, democratic governments have vast powers to affect the costs of access to information by producing legislation that regulates information such as the availability of data, the transparency of the government, and the functioning of the Internet. The findings in this book suggest that even small impediments to access imposed by any regime can have significant political effects, and therefore that manipulation of information in democracies can also have a widespread impact on the public's political knowledge.
As I will discuss in the conclusion, recent events in democracies highlight the importance of a broader definition of censorship. Evidence that taxes on the accessibility of information can have large political impacts37 suggests that society should be concerned with the extent that a few Internet companies and Internet service providers have primary control over the speed and convenience with which information can be accessed. If too few individuals, companies, or politicians wield significant power to make certain political information easy to access while making other information more difficult (for example through fast lanes on the Internet or reordering search results) in an effort to advance their own interests, this could have political impacts in democracies similar to the impacts of search filtering and firewalls in autocracies. Similarly, as traditional media have been decimated by competition from the Internet, small costs of access to data imposed by federal or local government may have an impact on content reported to the public in the traditional press. The broader definition of censorship I provide in this book emphasizes the importance of institutionalizing and facilitating government transparency and competition between information distributors as well as producers in democracies so that what information is provided, at what speed and when, to the press and public is not completely the result of political motivations and strategy.
Citizens in democracies recently have been shown to be susceptible to flooding as well. Distractions and misinformation spread online by cheap Internet commentators or automated bots increase the burden on the public to separate the signal from the noise, and many confuse good and bad information.38 Denial of service attacks that flood the websites of media, Internet companies, nongovernmental organizations, and government with too much traffic so that they become unavailable have the power to silence information channels selectively.39 As soon as these strategies are used for political purposes, they become political censorship. Although much censorship research has focused on the Internet in autocracies, more research needs to be done to study how censorship extends to democratic environments on the Internet as these undoubtedly have important political impacts.
**1.4 THE PLAN OF THE BOOK**
I proceed by offering a theory of the strategic interplay between government censorship and citizens' consumption and production of information. First, I describe the incentives of the government—why it would choose to censor and the costs it might incur from censorship. Next, I develop a model of how both citizens and the media interact with information. Using this model, I elucidate the three main ways in which censorship can influence the media and the public—fear, friction, and flooding. I then provide examples of each of these mechanisms in various communication media, and describe when each will have more or less impact on the spread of information. Fear, which is censorship based on deterrence, is by nature very constraining but must be observable in order to have an impact. Fear has to be credible in order to create deterrence; otherwise, it may instead draw attention to authoritarian weakness or create backlash. Therefore, it is discreetly targeted toward the most capable and motivated individuals. Friction, which imposes small taxes on information access, and flooding, which creates distractions, by contrast, do not need to be obviously driven by political entities to have an impact on information consumption and dissemination. Friction and flooding are more porous but less observable to the public than censorship using fear, and therefore are more effective with an impatient or uninterested public.
Chapter 3 provides an overview of the modern history of censorship in China and outlines the institutional structure and methods of censorship in China today. It describes how the Chinese censorship system has evolved from a model that was designed to micromanage every citizen's consumption and production of information to one that relies on porous censorship. It provides an overview of the main methods by which the Chinese government censors the Internet and the bureaucratic system that implements this censorship. Practically, it describes why China provides a good empirical test for the impact of porous censorship.
Chapter 4 explores how citizens react when they observe censorship online in China. Although many scholars have suggested that fear and self-censorship are the main forms of control of the Chinese Internet, I show that typical Internet users do not act afraid after experiencing online censorship and instead are angered by observing it. Using a matched pair study of users who forward the same social media post, but where one experiences censorship and the other does not, I study how experience with censorship affects the writings of Internet users. I find that, all else being equal, those who have experienced censorship persist in writing about the censored topic and are more likely to complain about censorship, even as they become increasingly targeted with censorship. I then survey Internet users about how they would feel if they experienced censorship. I find that Internet users, particularly those who report having experienced censorship, are much more likely to report being unfazed or angry about censorship than fearful or worried. Last, using online experiments, I randomly assign users in a lab experiment to come across a censored webpage. I find that the observation of censorship creates more, not less, interest in the censored topic and also decreases support for government censorship policies. I explore how the Chinese government, likely aware that experience with censorship can undermine its reputation, adopts a two-pronged censorship strategy targeting high-profile users with fear-based censorship while attempting to make online censorship efforts less observable to the public.
Chapter 5 demonstrates that small, less observable frictions on information have a powerful influence on the online behavior of Chinese citizens. First, I analyze the spread of information about 120 self-immolation events in Tibet through social media in China. I find that the best predictor of the number of social media posts that accompany a self-immolation event is whether the event occurs on the weekend, when the censors are slower to censor, suggesting that the speed of censorship has important implications for the spread of information in China. Next, I estimate the effect of the Great Firewall on the behavior of citizens in China. Using surveys and direct measures of those evading censorship through data from the social media platform Twitter, I find that those who evade the Firewall are technologically savvy, well-educated, high-income Internet users in China who have high levels of political efficacy. I find that the Firewall pulls this political elite away from their potential followers. I show that newly blocked websites have precipitous declines in usage directly following their block, showing how small impediments to access have an immediate impact on traffic from typical Chinese users. But I find that friction has an Achilles' heel, and is more commonly circumvented during crises and moments of sudden implementation.40
In chapter 6, I demonstrate that flooding in both online and traditional news media in China coordinates messages to distract the public from sensitive events. Using plagiarism detection software and leaked archives from the Chinese government to identify instances of flooding both online and in traditional news media, I show that the government uses propaganda to distract with coverage of the mundane details of Party meetings or with encouraging quotes and positive thoughts directed at the public.41 Using estimates of search results for reposting of propaganda articles around the web, I show that for the most part this strategy is effective—highly coordinated propaganda used by the Chinese government is more likely than articles that are less coordinated to be re-shared in both the domestic and international social media spheres.
Chapter 7 concludes with a discussion of the implications of my findings for politics in both democracies and autocracies as information technology and social media become more central components of political communication. I lay out specific directions for future research in the area of censorship and discuss censorship's potential for long-term political impacts on domestic and international politics.
1 "美记者质疑中国" 网络审查 "鲁炜:内容审查用词不当" December 9, 2015. Available at: <http://news.china.com/domestic/945/20151209/20903585.html>.
2 Yang (2009 _a_ , pg. 2) describes many of the ways in which Chinese netizens circumvent Internet control and calls government control over the Internet "only partly effective." Xiao (2011) similarly emphasizes how Internet controls in China are easily evaded.
3 Millward, Steven, "Google+ Not Actually Blocked in China, Just Being Slowly Throttled," _Tech in Asia_ , June 30, 2011. Available at: <https://www.techinasia.com/google-plus-china>.
4 Branigan, Tania, "How China's internet generation broke the silence," _Guardian_ , March 24, 2010. Available at: <https://www.theguardian.com/world/2010/mar/24/china-internet-generation-censorship>; Hiruncharoenvate, Lin and Gilbert (2015).
5 "Ministry of Truth: Personal Wealth, Income Gap," _China Digital Times_ , February 6, 2013. Available at: <https://chinadigitaltimes.net/2013/02/ministry-of-truth-personal-wealth-income-gap/>.
6 See Aryan, Aryan and Halderman (2013, pg. 5) and Esfandiari, Golnaz "Iran Admits Throttling Internet to 'Preserve Calm' During Election," _Radio Free Europe_ , June 26, 2013. Available at: <http://www.rferl.org/a/iran-Internet-disruptions-election/25028696.html>.
7 Goncharov, Maxim, "The Dark Side of Social Media," _TrendLabs Security Intelligence Blog_ , December 7, 2011. Available at: <http://blog.trendmicro.com/trendlabs-security-intelligence/the-dark-side-of-social-media/>.
8 Reynolds (2014, pg. 188).
9 Kern and Hainmueller (2009, pg. 394–395).
10 Taubman (1998, pg. 266) stresses that the decentralized nature of the Internet means no censorship methods are foolproof. Yang (2009 _b_ , pg. 30) contends that online activism is powerful because it can be more easily multiplied. Esarey and Xiao (2011) show that digital media has more critical content than newspapers in China.
11 Barlow, John Perry, "The Economy of Ideas," _Wired_ , March 1, 1994. Available at: <https://www.wired.com/1994/03/economy-ideas/>.
12 "China Cracks Down on VPNs During Political Meetings," _Wall Street Journal_ , <http://blogs.wsj.com/chinarealtime/2016/03/10/china-cracks-down-on-vpns-during-political-meetings/>.
13 Many scholars in political communication have shown that most people are not willing to spend time informing themselves about politics. For example, Sniderman, Tetlock and Brody (1991) show that voters rely on heuristics to make political judgements, Popkin (1994) explores how voters use information shortcuts to make choices, Conover and Feldman (1984) develop a theory of how people have ideology under low information, and Hamilton (2004, pg. 11) explains how media consumers can be rationally ignorant.
14 See Diamond (2015, pg. 151), and Soldatov, Andrei and Irna Borogan, "Putin brings China's Great Firewall to Russia in cybersecurity pact," _Guardian_ , November 29, 2016, <https://www.theguardian.com/world/2016/nov/29/putin-china-internet-great-firewall-russia-cybersecurity-pact>. As a result, scholars have advocated for more research on the Chinese censorship system; see Shorey and Howard (2016).
15 For an overview of the challenges measuring censorship see Burnett and Feamster (2013).
16 Barberá et al. (2015) show that the periphery is critical to the success of protests, Steinert-Threlkeld (2017) shows that the periphery can even instigate successful protests, and Chenoweth and Stephan (2011, pg. 39–40) show that total numbers and recruitment are a strong predictor of successful protest movements.
17 Dickson (2016, pg. 7).
18 Egorov, Guriev and Sonin (2009); Liebman (2005); Lorentzen (2014); Shirk (2011, pg. 19); Stockmann (2012, pg. 140); Qin, Strömberg and Wu (2017).
19 Stockmann (2012) makes a similar argument about the traditional media in China, arguing that the commercialization of the media provides cover for government propaganda. The concept of plausible deniability has also been used widely in the literature on repression, for example, Conrad and Moore (2010, pg. 461) argue that plausible deniability of torture allows the state to shift the blame.
20 See Knockel, Ruan and Crete-Nishihata (2017) for an example of how censorship is used surreptitiously in the Chinese social media platform WeChat.
21 Hobbs and Roberts (2016).
22 Nathan (2003); Anderson (2006); Gilley (2003).
23 Davenport (2007, pg. 7) describes the "Law of Coercive Responsiveness," that autocrats respond to opposition movements with force. Brownlee (2007, pg. 33) argues that autocrats have been able to repress opposition groups to consolidate power.
24 Wintrobe (1990, pg. 851) and Wintrobe (1998) stresses the patronage and public services dictators can provide as a substitute for repression. Dickson (2016); He and Warren (2011); and Lorentzen (2013) elaborate on how the Chinese government creates channels to respond to citizens' concerns. Magaloni (2008); Bueno De Mesquita et al. (2003); and Boix and Svolik (2013) describe how dictators create power-sharing institutions to prevent overthrow by other elite.
25 See Wedeen (1999).
26 Note that this is the same "nudge" logic with a darker take as that used in the behavioral economics literature; see Thaler and Sunstein (2009).
27 Ferdinand (2000, pg. 5), Lynch (2011), Bellin (2012, pg. 138), Diamond (2010, pg. 70).
28 Morozov (2011), MacKinnon (2012), Kalathil and Boas (2010), Rød and Weidmann (2015), Steele and Stein (2002).
29 Kalathil and Boas (2010, pg. 26), Wacker (2003, pg. 88).
30 MacKinnon (2012, pg. 6), Morozov (2011, pg. 97).
31 Zhang, Yuxin, "China: Self-Censorship Displaces Western Threats," _Diplomat_ , March 3, 2015. <http://thediplomat.com/2015/03/china-self-censorship-displaces-western-threats/>.
32 This finding provides support for some of the arguments in MacKinnon (2012) and Morozov (2011).
33 This finding is more in line with arguments made in studies that emphasize the impact of fear in controlling the spread of information. Kalathil and Boas (2010); Wacker (2003).
34 See Besley and Prat (2006, pg. 4), Gentzkow, Glaeser and Goldin (2006, pg. 189).
35 Edmond (2013, pg. 1441).
36 Edmond (2013, pg. 1442) allows for the possibility that governments can invest in "large-scale fixed investments for information control" online that may allow them to control the Internet despite the decentralized nature of the Internet.
37 Byrnes, Nanette, "How the Bot-y Politic Influenced This Election," November 8, 2016. Available at: _MIT Technology Review_ <https://www.technologyreview.com/s/602817/how-the-bot-y-politic-influenced-this-election/>. Epstein and Robertson (2015), Bond et al. (2012).
38 Nyhan and Reifler (2010), Ratkiewicz et al. (2011, pg. 301–302).
39 Woolf, Nicky, "DoS attack that disrupted internet was largest of its kind in history, experts say," _Guardian_ , October 26, 2016. Available at: <https://www.theguardian.com/technology/2016/oct/26/ddos-attack-dyn-mirai-botnet>.
40 This draws on work with William Hobbs; see Hobbs and Roberts (2016).
41 This draws on work with Brandon Stewart, Jennifer Pan, and Gary King; see Roberts and Stewart (2016); King, Pan and Roberts (2017).
CHAPTER TWO
A Theory of Censorship
**2.1 WHY DO GOVERNMENTS CENSOR?**
Just as political entities have incentives to promote particular types of information to their constituents,1 they also have reasons to control, slow down, or prevent citizens from consuming other types of information. Governments are entrusted by the public to carry out particular responsibilities—to act in the public's interest, or at least in accordance with a selection of the public's demands.2 Information is dangerous to political entities as they require support from at least some part of the citizenry, and information that reveals that they are not fulfilling their role can negatively affect their survival. In democracies, negative information can result in fewer votes, fewer political contributions, or even the forced resignation of politicians.3 In autocracies, information can persuade citizens to discontinue their support for the regime, undermine regime policies in everyday life, or persuade citizens to go out and protest.4 For political parties and interest groups, damaging information can galvanize citizens to discontinue political and financial contributions to a particular cause.5
Outside of accountability, information can also be dangerous to political entities insofar as information can act as a tool to facilitate coordination and protests that can threaten political entities' survival. At the most basic level, information can facilitate the logistics of collective action by communicating where and when protests will take place.6 Information can also indicate shared discontent among citizens that may embolden individuals to take action against the regime.7 Because large-scale protest can threaten a regime's survival, slowing the spread of these "signals of discontent" and logistical planning of protests can prevent or slow large-scale unrest.
In their ideal world, political entities would like no one to know dangerous political information that could reveal their lack of accountability to the public, decrease their political support or financial standing, or facilitate collective action to overthrow them. Conveniently, they have many tools to slow the spread of information. First, political entities often know more about their own performance than the public does.8 They can selectively reveal or hide information to avoid accountability for their own financial and political gain.9 Second, governments and other similarly powerful organizations have significant power over the infrastructure of the flow of information—from telecommunications to laws that govern speech—which allows them to control what and how information is shared between citizens. By changing laws, infrastructure, or incentives for media, governments can wield influence over the information that the public consumes, taking pre-emptive action against dissent.10
However, even though governments have interest in and power over constraining speech, there are costs associated with reducing transparency and implementing censorship. These trade-offs between the benefits and costs of repression and censorship are often referred to as "the dictator's dilemma."11 One form of the dictator's dilemma is when the government would like to enforce constraints on public speech but repression could backfire against the government.12 If the population indeed wants to hold an authority accountable, then the observation of censorship itself may be enough to undermine the political entity. Censorship could be seen as a signal that the political entity has something to hide and is not in fact acting as an agent for citizens. This could incentivize citizens to seek out information that the authority is trying to conceal. Francisco (2005), among others, has found that consistent repression can backfire against the regime by creating a more violent opposition or signaling opportunities for discontent.13
Another form of the "dictator's dilemma" is that even if the dictator would like to censor, by censoring the autocrat has more difficulty collecting precious information about the public's view of the government. Fear of punishment scares the public into silence and this creates long-term information collection problems for governments, which have interests in identifying and solving problems of governance that could undermine their legitimacy. As Wintrobe (1998, pg. 20) puts it, dictators cannot know "whether the population genuinely worships them or worships them because they command such worship." Political entities in general, and autocracies in particular, have few methods to gather information about how citizens feel about their performance.14 Significantly contracting the horizontal flow of information between citizens may harm the vertical flow of information from citizens to the government, potentially obscuring fixable political problems and preventing the government from addressing them before they become too significant to overcome.15 Greater transparency facilitates central government monitoring of local officials, ensuring that localities are carrying out central directives and not mistreating citizens.16 Allowing citizens to express grievances online also allows governments to predict and prevent the organization of protests.17 If citizens and officials are too scared to report problems, the government may face unexpected widespread public dissatisfaction that could lead to revolution.18
What should perhaps be considered a third "dictator's dilemma" is that censorship can have economic consequences that are costly for authoritarian governments that retain legitimacy from economic growth. Communications technologies facilitate markets, create greater efficiencies, lead to innovation, and attract foreign direct investment.19 Censorship is expensive—government enforcement or oversight of the media can be a drag on firms and requires government infrastructure. For example, governments that require social media companies to hire censors impose extra burdens on the development of these firms.20 Economic stagnation and crises can contribute to the instability of governments.21 Censorship can exacerbate crises by slowing the spread of information that protects citizens.22 When censorship contributes to crises and economic stagnation, it can have disastrous long-term political costs for governments.
This book focuses on the ways in which governments balance the overwhelming incentive to keep certain information away from the public against these significant costs of suppressing information. Although the strategies that governments use to suppress information vary by time period and target, I find that increasingly in a digital age political entities balance these competing incentives by making information they would rather the public not know more difficult to find without creating direct punishments for spreading or accessing it. By "taxing" particular types of information while still allowing access to this information, authorities slow information's spread while avoiding many of the costs of repression. I provide evidence of this by showing that small costs of access to information are enough to prevent the majority of people who are not very interested in politics from accessing it. Even though these forms of censorship are porous and sometimes are not even considered to be censorship, they can prevent political authorities from being held accountable and, because they do not seem repressive, are also less likely to create a backlash. These techniques enable governments to selectively target their most powerful, repressive tools of censorship toward influential individuals and media, rather than typical citizens, and create the perception of free flow of information while still retaining effective control.
In order to understand how censorship affects the spread of information and the strategic interaction between citizens, the media, and the government, we first have to describe the incentives and costs citizens and media encounter when consuming and sharing information. In this chapter, I review our current understanding of how typical citizens consume information and propose a framework for thinking about how the media, individuals, and political groups collect, synthesize, and spread information. Then, I present a working definition of censorship. Based on how information is shared and what censorship is, I describe three mechanisms— _fear, friction_ , and _flooding_ —through which censorship impedes expression of and access to information and the strategic interaction between citizens and governments in the context of each mechanism. Throughout this discussion, I explain how these mechanisms are strategically used by governments to maximize the impact while reducing the costs of censorship.
**2.2 CITIZENS ARE RATIONALLY IGNORANT**
In order to understand how censorship works to influence the spread of information and the accountability of political entities, we must first have a basic model of how people consume information. By describing how individuals go about deciding what information to consume, we can then pinpoint how censorship methods will influence the likelihood that they read, watch, or listen to a given piece of information. Through decades of research, political scientists have already laid the empirical and theoretical groundwork for understanding citizens' consumption of information in both democratic and autocratic environments. I draw on their expertise here to provide a basic framework of when and why citizens consume political information and draw out how this might vary across contexts.
The public's basic problem is they have far more information than they could possibly consume in the time that they have—they are overwhelmed with available information and have only limited attention to focus on particular pieces of information.23 How do consumers of information make the decision of what articles to read and what programs to watch from what sources, if they do not have time to consume everything? Citizens rely on the expected costs and benefits of the information to make this decision.24 Downs (1957) describes useful information as that which aids citizens in consumption, production, or political participation. For example, information is useful if people can use it to make decisions about what car to buy, how to do their jobs better, or who will best represent them. Information in these cases is more likely to be useful the more likely it is to be true, and therefore citizens consume information from sources they trust and from sources that they believe share their views.
Consumers also benefit from the entertainment value of information.25 They enjoy sharing and hearing information from friends in social situations or in gossip. Those who look for information that they enjoy might select sources that focus on the drama of the events.26 By the same logic, consumers avoid or resist information that creates cognitive dissonance, or conflicts with their long-held beliefs or experiences.27 Such information makes consumers uncomfortable, and therefore they derive less benefit from consuming it than information that accords with their current beliefs.
Even though they would like to consume information that benefits them, consumers have difficulty evaluating what information is useful or entertaining. In the best case, information is an _experience good_ , where citizens know the value of information only after they consume it.28 For example, a person can verify that a telephone number on a website indeed reached their representative or can verify that consuming a piece of information was entertaining after reading it, but cannot tell before. But in many cases, information is worse—a _credence good_ —where even after consuming information, citizens do not know its veracity.29 For example, after reading a rumor about a politician we may still not know whether the information is true. Therefore, people often decide to consume information if they receive signals that it may be important or if it is from sources that they trust. If everyone is talking about an article or news program, a citizen might be more likely to read or watch it. The placement of a news article within a publication or the decision of a news organization to include a story may signal to a citizen that the story is valuable. Citizens therefore rely extensively on the media and on political elites to set the agenda for the information that they consume and to provide them with trustworthy information.30
Consumers of information are likely to seek out information if its benefits outweigh its costs of access. Consumers of information are influenced by the monetary cost of information and the amount of time they have to spend consuming it. Some information is almost inescapable, such as information blared on a megaphone or billboards on the street; this information is difficult for citizens to avoid consuming, regardless of its benefits. However, for information that takes energy to seek out and consume, changes in the cost of information influence consumption. The amount a consumer is willing to pay in terms of time and money depends on the benefits they can expect to get from the information. A trading company might be willing to pay for a subscription to a high-cost periodical about markets, whereas an individual trading small amounts of money in the stock market may not. A person affected by an oil spill may check out a library book or spend time searching on the Internet about the potential health consequences, whereas a citizen far away from the spill may not.
Some citizens naturally benefit more from spending time informing themselves than others. What communication scholars have called the "political elite"—a small subsection of people who are well educated, are interested in politics, and have political connections—spend more time thinking about and informing themselves about politics because doing so is consistent with their own self-interest.31 However, even these individuals who may have monetary interest in or impact on politics because of their jobs, connections, or investments are not able to read _all_ information and have to select information to read. Although they might spend more time overall consuming information, elites are also affected by the expected benefits and costs of information and consume information with a higher expected benefit and lower expected cost.
Outside of the elite, most people have very few substantial incentives to be informed about politics.32 Since citizens have a low probability of being pivotal in political situations like voting or protest, the utility of being informed to make better decisions on how to participate is also low. Citizens are for the most part _rationally ignorant_ and spend little time investing in information about politics because doing so is unlikely to benefit them. In general, scholars have found that typical consumers of information are very poorly informed and consume little political information.33 Zaller (1992, pg. 18) describes consumption of political information as having a low mean with a high variance. At best, typical citizens are motivated to be informed about the happenings inside their own society so that when information appears that pertains to their well-being,34 they can react in a way that maximizes their utility. Whether because of too little time, belief that they have little control over their political situation, or distractions from entertainment, there is substantial evidence that most citizens in even developed democracies are not at all informed about their political situation and have very little intention to spend the effort to inform themselves.35
In autocracies, scholars have speculated that citizens spend even less time consuming political information because they have few opportunities to participate outside of stateorganized political organizations, few incentives to seek alternative viewpoints, and reasons to avoid cognitive and moral dissonance.36 Those living in autocratic environments also typically are exposed to fewer alternative viewpoints.37 When citizens in autocracies are exposed to political information, there is substantial evidence that they are likely to accept the views expressed in the mainstream media.38 Even if they do not agree with the information, if citizens feel that they cannot publicly oppose the government, many will seek out information that confirms the government viewpoint to avoid hearing negative information about politics that are difficult to change. Of course, there may be periods when citizens are more likely to seek out political information. Evidence suggests that when there is greater uncertainty about the political situation, such as during crises or government transition, citizens in autocracies are more likely to seek out information and may be more affected by media.39
Using this evidence, the theory in this book is based on a model that citizens, both in democracies and to a greater extent in autocracies, are typically willing to pay only small costs to inform themselves of their political situation. For the most part, citizens consume information that is easy to access, confirms their beliefs, and is from sources that they have reasons to trust. Citizens have a highly _elastic_ demand for political information: small increases in the cost of political information will strongly decrease the probability that a citizen consumes it. Citizens will spend time and money searching for and understanding political information only if they receive signals that knowing that information will be immediately pertinent to their own lives, if they happen upon it as a by-product of information they are consuming for other reasons,40 or when they are in a crisis situation that creates incentives to gather and search to information.
**2.3 TRADITIONAL MEDIA CARE ABOUT STORY COSTS**
If citizens rely on low-cost information from sources that they trust, undoubtedly the traditional media plays a large role in what citizens consume. But how do free and commercialized media decide what to cover, when they are not politically constrained?41 There are infinitely many possible stories the media could cover on any given day. How do the media decide what information to collect, present, and write up for their readers?
Like citizens, the media face a cost-benefit trade-off in the stories that they cover. The financial benefit to the media from reporting on a story is based primarily on how many readers they can attract to their publication. The more subscriptions, clicks, or views a media outlet receives, the more money it can receive from advertisers.42 As such, traditional media will typically create stories that pander to a particular group of tastes or a particular type of person.43 Among their target audience, media stories will direct content to their marginal consumer, rather than necessarily their loyal base, in order to attract the greatest numbers of people. Commercialized media will also be particularly interested in attracting consumers that advertisers would like to target, those who are easily swayed by advertisements or who purchase goods for their households.44
Because citizens' trust will be important in attracting an audience to news stories, the media will be very concerned with their own credibility. A media outlet that is seen as providing false information or missing important stories in its purview will be less frequented than one that is thought to be more reliable.45 Traditional media, therefore, would like to develop a track record for stories that the public, and in particular the marginal consumer, believes are important and of interest.
However, media face the same trade-offs consumers do in that there are costs associated with writing stories. Some stories, particularly those that involve investigation, may involve months of document review, travel, and interviews. Other stories could involve very low costs, such as reporting what a politician stated in a press release, the outcome of a baseball game, or even reprinting a story another media outlet produced. If these two types of stories are of equal interest to the audience, the media outlet will typically report the low-cost story rather than that with high fixed costs because the expected net benefit of low-cost stories is larger.46
At times, the costs and benefits of a story will be uncertain, which will affect the probability that the media will pursue it. The media might easily assess their audiences' interest in the outcome of, say, a baseball game, but journalists cannot always tell immediately which stories will be useful to spend time and money investigating before the information has actually been extracted and analyzed. Investigative journalism, like research, has many dead ends and is a "precarious profession."47 Media must rely on signals of importance and projected costs to know what stories to pursue. If information is too difficult to access, an investigative story will not be successful no matter how potentially interesting it is. Risk-averse media will be more likely to pursue stories that have a higher certainty of success. To reduce costs, the media frequently rely on data collection and analysis by third parties, such as the government. In these cases, a story's success will depend on the degree of transparency of these parties. Data availability will affect the media's costs of production of a story and eventually citizens' cost of access to the story.48
**2.4 CITIZENS EXCHANGE LOW-COST INFORMATION THROUGH SOCIAL MEDIA**
Increasingly, as information has become more accessible to the public directly through the Internet and the traditional media have lost resources from online competition, information has bypassed the media completely, and information can flow directly between individuals. Social media may present unfolding events and accounts of individuals, such as real-time accounts of earthquakes or protests, directly to the public, without the filter of the traditional media.49 Sometimes those who are more interested in politics or more informed than the average individual will take it upon themselves to gather information directly, synthesize stories, and distribute them to the public through social media, even though they are not members of the traditional media. Other times, the wider public themselves share information and accounts with one another without aggregation or filter.50 In these cases, information and stories become known and widely shared without passing through the media.
However, despite the power of social media to share direct accounts of events, most information shared on social media is still stories produced by the traditional media, and therefore traditional media continue to have a powerful influence over what consumers read and share, even in an online environment.51 The information that passes directly between individuals is also more likely to be low cost and most accessible, such as opinion or entertainment, because the public has relatively little time and few tools to gather and synthesize information.52 As a result, it sometimes can be lower-quality information—or even false information—since the public is not invested in its own credibility or reputation, like the media.53 Even so, this pathway for information to bypass the media cannot be discounted and is dangerous for authoritarian governments.54 Whereas before information could be controlled simply by controlling the media, with social media such control is much less straightforward.
Figure 2.1: Pyramid that represents the collection and distribution of information.
Figure 2.1 presents a diagram that summarizes the collection and distribution of information in a pyramid. The bottom of the pyramid represents all potential stories, information disaggregated in data, events, individuals, and documents that could be synthesized, analyzed, and presented to the public. The bottom level of the pyramid is too large for any one person to consume given a limited attention span. The second tier of the pyramid represents the stories the media synthesize for the public. Which stories filter from the bottom to the second tier will depend on how costly they are for the media to synthesize and how lucrative the media expect them to be for their business model. The top of the pyramid are the stories that are consumed by the public, a subset of those synthesized by the media. Each level of the pyramid is a subset of the level before, filtered by the public's demand for information, their trust in the media sources that present the stories, and the cost of access to information. Social media, which are represented in the diagram by an arrow from the bottom and middle tiers to the top tier of the pyramid, allow events to be presented without the filter of the media and provide another avenue for stories already produced by the media to be shared among individuals.
**2.5 WHAT IS CENSORSHIP?**
For the reasons discussed before, political groups may have an interest in affecting the flow of information between levels of the pyramid. Formally, I define censorship as the _restriction_ of the public expression of or public access to information by _authority_ when the information is thought to have the capacity to undermine the authority by making it accountable to the public.55 Central to this definition are the actions that could be restricted by censorship. In this book, I consider two types of actions that are restricted by censorship: expression of information and access to information.56
Expression refers to the ability of a person or the media to share information unimpeded in the public sphere, whether through writing or public speech. People may be restricted from expressing themselves because they are prohibited by the law, or they are fearful to do so because they have been intimidated. They may also have had the tools of expression withheld from them; for example, they may not have access to the Internet or may purposely have been kept illiterate.57 Infringements on media and individual expression can occur at any level of the information pyramid. Those with primary information may be unable to express their views in interviews to the media, preventing the flow of information from the bottom to the middle level. Or citizens and the media may be unable to print or share an already prepared story, preventing information from spreading from the middle to the top level of the pyramid.
Expression is different than access to information, which is the ability to consume information. Barriers imposed on access to information also can occur at any level of the pyramid. The media may be restricted from accessing certain types of data or particular people for interviews. Authorities may prevent the media and individuals from accessing data or analyses that they have collected and generated. Books may have been written, but may be difficult to access if they are banned from libraries or schools. Websites may exist, but they may be blocked by firewalls or removed from the indices of search engines.
Freedom of access to information is often as treated secondary to freedom of expression;58 however, this is like arguing that the chicken is more important than the egg: access is necessary for expression, since the expression of information often relies on it first being accessed or collected. Similarly, expression is necessary for access—it is difficult to access unexpressed information. The two are inextricably linked, and both constrain the movement of information up the pyramid. However, as I will explain later, barriers to access are often less observable than restrictions on expression, and therefore can be more easily used by political entities when direct repression is too costly.
_2.5.1 How Is Information Restricted?_
Now that we have specified what is restricted, what do we mean by restriction itself? Unlike popular illustrations of censorship, like a journalist with tape over their mouth or a TV producer with a hand over the red button on live television, restriction does not need to be complete to be considered censorship. Restriction is imposing any type of cost on expression or access to information, be it large or small, as long as this restriction is conducted purposefully to decrease the accountability of the authority. Much of censorship is incomplete or porous. Censorship also does not necessarily have to involve laws or punishment. One of the most-cited examples of censorship in China is the Great Firewall, which is by no means complete censorship as it can easily be circumvented by a Virtual Private Network (VPN).59 However, the costs of doing so are not insignificant, as finding a VPN takes time and often costs money. Censorship lies on a continuum of costliness; and, as I will demonstrate later, even small restrictions on freedom of access and expression can have significant impacts on the spread of information and may sometimes be more effective than more repressive forms of censorship because they are less likely to draw attention to government censorship efforts.
_**2.5.2 Who Is "Authority"?**_
Although the main focus of this book is on censorship conducted by governments, the government is not the only political entity that can censor. Most of the literature on censorship has referred to the government as the primary perpetrator, primarily because the government is often the entity with the most power to control freedom of expression and freedom of access to information and is the primary agent accountable to the public. By shaping laws, the government can make particular types of expression illegal, and because the government often has control of telecommunications infrastructure and government data, it can easily influence the costs of access to information. Facing elections or political opponents, governments purport to represent their constituents and rely on public opinion for the maintenance of their own power, and therefore have strong incentives to manipulate the spread of information.
However, there are other political entities that the public wishes to hold accountable and that have power over the costs of expression of and access to information. Many organized groups have sufficient resources to collect or discourage the collection of information.60 For example, bureaucracies are often the primary source of data on their own performance, and, unless legally mandated, they sometimes opt not to collect this information, collect it in a way that obscures their performance, or simply make it unavailable to the public.61 Interest groups, with a political objective and the money that underlies it, can effectively manipulate information. These groups collect data on the topic that they are concerned about and make this data either easily available or difficult to access for reporters, citizens, and the media. They can commission studies that often turn out to support their cause. These groups may have the power to lobby the government to ban particular types of research or publication of information.62
Companies can influence access to information and are at times held accountable to the public. In many cases, keeping information secret is imperative to the functioning of a company, be it for property rights protection or protection of the company's business plan from competitors. However, there are aspects of company activity where they are accountable to the public, for example, in the realm of product safety or environmental and labor regulation. Companies are also increasingly handling local public services, being hired to carry out foreign military operations, or taking on the role of government.63 To the extent that companies nominally acting in the interest of the public conceal information in order to reduce their accountability to the public, their activity would be classified, under this definition, as censorship.64
**2.6 THE MECHANISMS OF CENSORSHIP**
I now turn to how government censorship affects which stories come to the attention of the media and the public. Instead of enumerating the many different ways in which governments censor across different media and technologies, I create a typology of censorship restrictions based on the _mechanism_ through which censorship slows the flow of information.65 The technology used to censor and the information medium that is targeted by censorship both change over time. However, the ways in which authorities limit the freedom of expression and access to information by changing the incentives of individuals and the media are general to all of these technologies, time periods, and media. By defining censorship by its mechanism, we understand it in its most general sense and can more easily understand its new incarnations in the future.
Each of the mechanisms of censorship I lay out below influences the cost-benefit calculation for the media and citizens to express or access information and therefore filters information between the levels of the pyramid. However, these mechanisms have different levels of observability to the public and thus vary in their probability of creating political backlash, economic costs, or information-gathering problems for the authority. By analyzing how each mechanism affects its target and the costs it has for its implementer, I elucidate the strategic interactions among governments, citizens, and the media.
The first way that censorship operates—fear—affects the flow of information by _deterring_ the media or individuals from distributing, analyzing, collecting, or consuming certain types of information. Fear creates the awareness of consequences of facilitating the flow of information. It makes expression or access to information more costly because of its punitive consequences. It is explicitly consequential, but very observable, and therefore has a greater potential to backfire and create information-gathering problems for the authority.
Friction, the second type of censorship, acts like a tax on information by directly increasing the costs of distribution of and access to information, _diverting_ the media and individuals away from censored information. If information is simply more costly to collect, analyze, or distribute, even if there are no punitive costs of accessing or distributing that information, individuals and the media will be less likely to come across it or distribute it. Individuals may not even be aware that the information is purposely made costly to access. Simply by becoming frustrated accessing it or by being entirely unaware that it can be accessed, the public will be less likely to pay attention to it. Friction is not explicitly consequential and is often less observable than fear and therefore is less likely to create direct backlash among citizens. However, friction is often more porous than fear and can be circumvented, particularly during periods of crisis or heightened political awareness.
Flooding, the last type of censorship, vastly decreases the costs of particular information in order to increase the relative costs of competing information. Flooding can influence the media by presenting them with cheap, prepackaged, easy-to-publish information. Or it can influence the public, which has too much information to consume in too little time, pushing particular types of information to the top of the pyramid in order to de-emphasize others. Flooding creates _distractions_ that require individuals to spend more energy to sift through the available information. Flooding is even less observable than friction because it affects the costs of information indirectly. As a result, it has the least potential for backlash from citizens because even if citizens realize a particular piece of information is being promoted by the government, they may not be sure what it is meant to compete with or obscure. However, flooding is the most porous form of censorship as it does not interfere directly with the information it competes with.
Fear, friction, and flooding are exhaustive categories of censorship in that they can be used to describe all forms of censorship, but they are not necessarily mutually exclusive. Censorship can act through two or more mechanisms simultaneously. For example, the government putting an activist in jail could create both friction for journalists hoping to write about stories for which the activist is a source and fear that the government is serious about cracking down on stories related to those that the activist has knowledge about. Similarly, the removal of a genre of book from the library will create both friction for citizens at the library and flooding if the books are replaced by others that provide a distraction or interest citizens in political propaganda.
For the remainder of this chapter, I will discuss each of these mechanisms of information manipulation in turn. Using examples from different countries, a variety of political entities, and across time periods, I will develop an intuition of how each mechanism differs from the others and its various incarnations across time and space, including the current age of the Internet. I explain when and where these different mechanisms will be more or less successful at stopping the spread of information, suggesting when authorities will decide to use them for the purpose of censorship.
**2.7 FEAR**
Raif Badawi, a Saudi Arabian blogger, was arrested on June 17, 2012. Saudi police accused him of insulting Islam through the Internet, and of apostasy, or the abandonment of Islam. Badawi's primary crime was that he had created the website Liberal Saudi Network, which questioned aspects of Islam and sought freer discussion of politics in Saudi Arabia. On May 7, 2014, Badawi was sentenced to ten years in prison and one thousand lashes. On January 9, 2015, the first fifty lashes were administered, and they were scheduled to be repeated every Friday for twenty weeks.66 Badawi was not new to this type of treatment by the Saudi regime. In 2008, he had been questioned for apostasy and subsequently released.67
Badawi's arrest and sentence is only one of many examples of the first way in which censorship operates: fear. Whereas Badawi was not deterred from writing, as others in his situation might have been, the punitive action and public floggings that resulted from his writing have one purpose—to illustrate the punishment that accompanies particular types of speech and deter others who may be considering similar types of activities. Censorship through fear functions by dissuasion—by prohibiting the expression of or access to information and articulating its punishment so that citizens are discouraged from doing so. Censorship through fear is based fundamentally on the _awareness_ of the punishment that can be expected if the collection, production, or consumption of particular types of information is carried out.
_**2.7.1 Legal Deterrence**_
Perhaps the most observable way fear is produced is through censorship laws. Governments create laws that prohibit particular types of expression or consumption of information and then publicize these laws so that citizens and the media are aware of the punishment that will befall them if they commit the crimes associated with the laws. Many authoritarian governments have laws that prevent public speech on particular topics; for example, Saudi Arabia outlaws insulting Islam through the Internet. Russia recently outlawed spreading false information on the Internet and requires the author to prove the information's veracity rather than the government to disprove it.68 Similarly, Iran has Internet censorship laws that prohibit information that is "immoral" or endangers national security.69 The Chinese government has a Law on Guarding State Secrets, which prevents the publication or spread of any "state secrets," the definition of which is sufficiently vague to cover the discussion of many different types of information.70 In totalitarian societies, like North Korea, fear controls the lives of individuals who must be extremely careful not to even insinuate political narratives outside of the government line.71
Journalists and other traditional producers of media are subject to laws that could cost them their jobs or lead to their arrest. Reporters Without Borders estimates that 826 journalists around the world were formally arrested in 2013.72 Governments can control media organizations and journalists by requiring them to obtain licenses or press cards or by controlling personnel appointments directly. Academics can be dismissed from their jobs when they interfere too much with politics,73 and countries can reject visas of international journalists and academics who do not follow publication guidelines.74 Governments may anticipate periods when elites and media might cause trouble, arresting them in anticipation of their speaking out.75
Even in democracies, expression is limited by laws that threaten punishment. Defamation laws exist in most countries, though they vary considerably in the extent to which they are used. South Korea has recently come under criticism for using criminal defamation laws to target journalists. In November 2014, then President Park Geun-hye of South Korea filed a criminal lawsuit against six journalists for reporting on a leaked government document.76 Special interests, too, have been successful in a variety of democracies in advocating for lawmakers to pass laws that limit expression on particular issues. In a handful of states in the United States, agriculture lobbies have passed anti-whistleblower bills, often referred to as "aggag" bills, that make documenting anything inside an animal facility with the purpose to commit defamation illegal.77 Other recent examples of criminal defamation laws in democracies include those in Indonesia, where social media users and journalists have spent time in jail for alleging political corruption or questioning the existence of God.78
_**2.7.2 Intimidation**_
Fear not only originates in legal deterrence, or the threat of state punitive action, but can also take the form of extra-legal intimidation or threats, where government actors or other authorities can dissuade citizens or journalists from consuming, expressing, or collecting particular types of information. These types of threats are less observable to the public than censorship laws, which are often explicitly laid out in government documents and publicly enforced. Intimidation can be more specifically targeted toward particular people or entities, often toward public figures, journalists, and academics who could potentially influence the opinions of large groups of people.
In China, local government officials are known to intimidate journalists and social media users who, while typically operating within the law, sometimes uncover and print information that reflects badly on of local officials. Although at times the central government actively encourages investigative journalism, local officials find investigative work particularly threatening because it brings their failings to the attention of their central government bosses.79 As such, local government officials employ intimidation tactics, extra-legally assaulting or kidnapping journalists who try to expose local corruption, land grabs, or local riots.80
In many semi-authoritarian countries, democratic institutions do not allow censorship to be explicitly codified into law, and therefore government officials often take it upon themselves to deter activists and journalists from reporting on particular issues. Mikhail Beketov, a Russian journalist, advocated for the resignation of the government in the Russian city Khimki. His car was blown up and his dog beaten to death shortly after his story was published. Continuing to write, Beketov was beaten by thugs so badly in 2008 that he was confined to a wheelchair and died from his injuries in 2013. An investigation of the beating was suspended "for lack of evidence."81
Reporters Without Borders estimates that in 2013, 71 journalists were killed, 87 kidnapped, and 2,160 threatened or physically attacked.82 Along with countries that are notorious for threatening journalists, such as Syria, democracies such as Mexico and Brazil have significant numbers of missing or murdered journalists, as well as countries that have seen large protest events, like Turkey and the Ukraine. In India, eight journalists were killed in 2013, and more were threatened by police. In short, even in countries where censorship laws may not exist, intimidation still can be a tool for inducing fear-based self-censorship.
_**2.7.3 Reward**_
Fear tactics can come with carrots, not just sticks. A government may facilitate the promotion of a journalist who refuses to say negative things about the government or may pay off a media outlet for keeping particular information secret. An employer may promote employees who do not speak out on particular political positions. In this book, I will consider such rewards a form of fear because the implied opposite of reward is sanction. If failing to criticize the government as a journalist means you will be promoted, the implication is that you will not be promoted if you criticize the government. Like threats, rewards must be observable to work—citizens must know there is a payoff to be incentivized to limit their speech. In this case, the fear is in failing to receive the reward.
**_2.7.4 When Does Fear Affect Access and Expression?_**
In order for fear-based methods of censorship to deter the production and consumption of information and affect the flow of information up the pyramid, citizens and the media must (1) be aware of the consequences of consuming or producing information, and (2) believe that these consequences will be administered. The threat or law in conjunction with the probability of punishment must also be significant enough to outweigh the expected benefit of the production, dissemination, or consumption of information.
Although it may seem obvious that for deterrence to work one has to know about a threat, this point is central to understanding the differences among the three censorship mechanisms. The requirement that citizens be aware of a threat of punishment means that governments must either target individuals whom they would like to deter—which can be time consuming and costly—or alert the public at large about the threat by making laws or "examples" well known to the public. That fear works by making punishment observable distinguishes it from friction and flooding, which can affect citizens without their being aware of government interference.
Second, it must be credible that the threat will be enforced. Enforcement of laws and the carrying out of threats is expensive, and it may not always be optimal for authorities to enforce.83 Citizens may try to evaluate the willingness of the authority to follow through with punishment by evaluating how costly the information is to the authority. The more the authority's survival is threatened by the information, the more the media and citizens might be concerned that it will be worthwhile to the authority to punish those who spread it. Therefore, citizens may think that punishment might be more likely the more damaging the information they intend to spread is. If others, who they believe are in many ways similar to themselves, have been punished for similar crimes, citizens may believe that they also are susceptible to punishment.
Citizens and the media may also try to determine whether the authority has the capacity to punish, or what the costs to the authority are for punishment. First, authorities have direct costs of punishing. For example, if many citizens and journalists in the country are violating the censorship law, the state's capacity to punish everyone violating the law is less than in a society where only one person has violated it. Citizens and the media may feel more comfortable speaking out if many others are doing so.
Authorities may also pay external costs for punishments that citizens and the media will take into account when evaluating their capacity to punish. In the case of the blogger Raif Badawi in Saudi Arabia, the government incurred international costs from outraged Western governments in terms of diplomacy and international press coverage.84 Some types of censorship may not be politically viable internally, and could cause outrage and public backlash. Threats of punishment for dissemination of information will be less credible when the costs of censorship are higher for the political authority.
In other cases, it might be clear that authorities will incur costs if they _do not_ punish citizens and the media for violating censorship laws. In these cases, authorities' censorship laws are more credible. For example, in countries where substantial consensus exists over censorship laws, such as child pornography laws in the United States or religious blasphemy laws in religiously devout countries, the government would face public outcry if enforcement were lax. In other cases, where the government has promised extensive enforcement of censorship laws, failing to follow through on enforcement could hurt its future credibility and reputation.
If the threat is credible, fear-based methods can have an enormous impact. If fear-based methods can intimidate all people who know a piece of information, they can effectively keep information secret and reduce trust and coordination among citizens.85 Ambiguity of fear-based approaches can silence third parties who are not even on the radar of the government, simply because these third parties do not know whether they could be targeted or what would happen to them if they were.86
However, governments face many trade-offs when implementing fear-based methods of censorship. Restricting speech may make it more difficult for governments to understand the opinions of the public, which has in the past led to the surprise demise of regimes.87 Speech can reveal corruption of local officials or bureaucrats, which can be beneficial to central leaders and the effective functioning of governments.88 Limitations on freedom of speech can also limit scientific discussion or the spread of information about disasters, which can have damaging consequences for the economy and for the safety of the population.
Further, when the threat is not credible, fear tactics are dangerous methods of censorship for governments. Because awareness of the threat is necessary for its efficacy, the paradox of fear-based censorship is that these methods create awareness of censorship. Because information is an experience or credence good, citizens assess whether or not to consume it based on signals of its importance. Therefore, by articulating which types of information are off-limits, governments bring attention to the issues that could undermine them, a phenomenon popularly known as the "Streisand effect" for Barbra Streisand, who sued the California Coastal Records Project for posting pictures online of her coastal mansion.89 Instead of protecting her privacy, her suit drew more attention to the photos, which were widely circulated online after news spread that she was suing the project.
Censorship may act as a signal to citizens or the media that the information is important and in this way might motivate them to go looking for or write more about the censored topic than they would have had they not experienced intimidation. Just as Kepler was assured that having his book banned would only "make it read more attentively"90 and as books that are censored in China sometimes receive more attention than those that are uncensored,91 prohibited material where punishment is not credible might be more interesting to producers and consumers of media because it signals importance. Fear may alert citizens to subject matter of importance, which may make them more aware of the issue subsequently.92
The government may also pay a reputational cost of censorship if the censorship law itself is unpopular. If the individual believes that the law or threat is unjust or that censorship signals weakness, then support for the authority may decrease among the censored. This could particularly be true if censorship itself signals that the political authority is not secure enough to withstand the spread of sensitive information. Since authorities will often target those who disagree with them in the first place, signaling such weakness with threats that are not credible could be very dangerous, since it essentially alerts the enemy to the authority's lack of capacity.
_2.7.5 The Internet Has Made Fear-Based Censorship More Costly for Governments_
The costs to governments of fear-based methods of censorship are more severe in the information age, as there has been an increase in the number of producers of information in the public domain.93 Whereas before, authorities only had to target the traditional media to influence what information became widely known to the public, because social media has introduced the possibility that the public find and broadcast stories without the traditional media, fear would need to be credibly targeted toward the widespread public to be similarly effective. The more people writing critiques of the government online, the more costly it becomes to punish them all, in terms of both the physical and reputational costs of repression. Credibly threatening punishment for millions of online users who are simultaneously sharing information with one another is—even by the most sophisticated censors' accounts—currently impossible.94
The "Streisand effect" is also exaggerated in the age of information. Overwhelmed with information on the web, the challenge of the consumer in the information age is knowing what to read. Awareness of censorship without credible threats of punishment signals which topics might be more interesting to seek out and write about. Whereas a person may not find information about a particular corrupt official if they simply type in "corruption" in a search engine, if they realize they have been censored talking about a particular official, then they will know that this information is somehow important to the government. If these writers are not deterred by government threats, they may be incentivized to continue searching and possibly continue writing on the topic, backfiring against fearinduced methods of censorship.
Moreover, the Internet is a more useful conduit of information for governments when individuals are not fearful about speaking out. Social media does not only provide a forum for individuals to speak to each other, it also serves as a way for authoritarian regimes to monitor public opinion.95 If individuals are fearful of expressing themselves online, the value of online information for understanding citizen concerns is lessened.
Authorities take into account the credibility of their threats to citizens, their reputation, the possibility for backfire, and censorship's effects on the government's future access to information about the population when deciding the circumstances under which they should use fear-based methods of censorship. If censorship laws are very popular, rarely violated, and quickly enforced, fear-based methods of censorship will be credible and will have their intended effect. But when censorship laws are unpopular and threats of punishment target large numbers of people who are already violating censorship laws, the threat of enforcement is less credible and instead reminds the population of unpopular policies. In these cases, authorities may focus on targeting fear-based censorship toward those who are most likely to find and synthesize damaging information: the political elite and the media.
**2.8 FRICTION**
On January 12, 2010, Google publicly alleged that it had been hacked by Chinese sources. In response, it threatened to stop obeying laws required of foreign businesses operating in China to censor particular content from its search engine. The resulting conflict between Google and the Chinese government came to a head in March 2010, when Google began redirecting its mainland users to google.com.hk, Google's unfiltered Hong Kong search site.96
Reacting to Google's refusal to filter content, the Chinese government began blocking the unfiltered Hong Kong site. In addition, throughout mainland China, the government throttled access to Google services, allowing access to Gmail and Google's social media site Google Plus for only about 75 percent of requests.97 China Unicom announced that it had removed Google search from its search platform, further restricting its accessibility for Chinese users.98 Mainland users could access Google, but doing so required more work and more patience. For years, traffic to the search giant creeped along slowly, until Google was finally completely blocked in 2014.99
Google spends millions of dollars a year making search faster because research has found that faster search means more users. Google market research shows that slowing search by one second creates approximately a one percent drop in search volume. A video stalling can be aggravating enough that 80 percent of Internet users will become frustrated and leave Google's YouTube.100 Consistent with this research, when China began slowing Google's services in China, Google lost a large portion of the Chinese market. Since 2010, Google's market share in China has declined precipitously. In 2010, Google accounted for over 10 percent of page views among search engines in China, but its share had dropped to less than 2 percent by 2013.101
Despite its failure to comply with Chinese censorship, Google was never illegal to Chinese users. Throttling and later blocking Google with the Great Firewall is a form of friction, which affects the accessibility of information. Friction does not create consequences for successful access; there is no downside for an individual for having accessed that information. Instead, it taxes information by imposing costs on the process of accessing information—frustrating individuals by requiring more time, money, or resources, or by simply preventing them from coming across the information by reducing the probability that it will be found. Friction can impose small or large taxes on information; it can impose very small costs in terms of time and money or can make information very difficult to find. As I show in subsequent chapters, for those with little time, patience, or resources, even small frictions—not to mention costly frictions—can have significant effects on whether citizens consume particular types of information or whether the media will collect particular types of data.
Friction affects access to and expression of information by making certain facts, data, social media posts, or news articles difficult to obtain. In doing so, it reduces the likelihood that a particular viewpoint will be communicated to the public or a particular fact revealed. Just like sales taxes reduce the number of products a person can buy on a limited budget, friction imposes constraints on what can be written within the short period of a news cycle because the media and citizens have a finite amount of time to gather and synthesize information.
As such, friction reprioritizes consumption and production of information by affecting its price. As discussed earlier, both citizens and the media are affected by the cost of information when deciding what to read and what stories to undertake. As I will discuss in more detail later in this section, this is particularly true in the digital world, where media competition is fierce and citizens are overwhelmed with the amount of information they could potentially consume. In this type of environment, the costs and inconvenience of access will be one of the most important factors in prioritizing the consumption and production of information.
Of course, there are certain types of information that by their nature have lower costs of access than other types of information. For example, readers are more likely to consume news that is printed in their native language. Collecting data is easier for some types of measurements than others: a day with severe air pollution is easier to observe than knowing that your food or water is contaminated with trace amounts of lead. Even though such "natural" frictions can have large implications for politics, to the extent that these costs of access are not manipulated by an interested party these frictions are not knowingly orchestrated and thus are not a form of censorship.
However, to the extent that convenience can by manipulated by interested authorities, friction is an important form of censorship. Costs of access do impose restrictions on public access and expression of information and have surprisingly strong effects on the public's and media's consumption of information. Even though the public is not prohibited from sharing or accessing information that is affected by friction, the time and money it takes to do so affects the likelihood that this information will spread.
Friction is also distinct from fear because it does not have to be observable to be effective. Like taxes, citizens will be aware of some frictions; for example, they may run across an error page that was obviously taken down by the government, or be told that an online search cannot be run because of government regulations. However, like value-added taxes of inputs to a product or sales tax that is priced into a product, many frictions will be hidden from the public. The citizen will simply experience more difficulty in accessing or spreading information, without knowing or understanding that access to that information is affected by the government.
Friction and fear are not mutually exclusive and can overlap. Which mechanism stops or throttles the flow of information may also be person-specific. For example, when a person comes across a webpage that has clearly been removed by the government, they might experience both friction, because the information is more difficult to find, and fear, if the removal of the website is paired with a threat of punishment for continuing to search for that information. However, another person who is more oblivious to signals of censorship or less easily frightened of the government may just experience friction and not fear.
Relatedly, fear-based censorship can create downstream friction. Fear imposed on the media might make stories about sensitive topics more difficult to find for the public, even if the public is not aware of punishment or does not experience fear themselves. Importantly, however, friction does not require fear and is often imposed without any fear. Slower Internet, throttled websites, or purposefully uncollected data are all common examples of friction that do not require fear but still affect the flow of information. As we will see later in this chapter, friction is often used as a substitute for fear when fear is too costly for the authority.
_**2.8.1 Friction Imposed on Distribution of Information**_
The most direct form of friction is cost applied to communication between the media and the public—between the middle and top levels of the information pyramid. These frictions require individuals to spend extra time or resources consuming or sharing news, social media, books, or opinion pieces that have already been written. Costs on the distribution of information are distinct from frictions between the lower levels of the information pyramid in collecting primary data or information that could then be analyzed and later printed in news, social media, books, or other forms of media. I will first discuss direct frictions on the dissemination of information, and in the next section explore frictions on the initial collection of information.
Perhaps the most infamous type of friction is the Great Firewall, China's sophisticated mechanism for blocking foreign websites that the government finds objectionable, including eventually Google. For people who are physically in China, websites such as Voice of America News (voanews.com), Facebook (facebook.com), and Twitter (twitter.com) return an error page instead of the website itself.102
Although the Firewall is technologically sophisticated, it is relatively easy to circumvent. Users simply have to download a Virtual Private Network (VPN), which allows them to log onto a computer outside of China and access the blocked site indirectly through that computer. Depending on the technological capabilities of the person, the Firewall is a large friction or very small friction imposed on foreign information. Simply knowing that the Firewall exists and can be jumped is a first step to overcoming the friction imposed by the Firewall, and not every person in China realizes that the Firewall exists. Even if a person knows that the Firewall can be jumped, using a VPN imposes small costs on a user's paycheck and time. VPNs can sometimes cost money—typically only a few dollars a month, but such costs would be prohibitive for many Chinese citizens. They also are invariably slower than using a computer directly, with costs of logging in and waiting for two computers to load a website. Sometimes, too, VPNs are shut down by the Chinese government; in these cases, users have to spend time finding an alternative VPN before they can jump the Firewall. As I will show in later chapters, for the typical consumer uninterested in politics, these small costs can be sufficient to significantly affect their behavior.
China is not the only country that filters access to particular sites—countries all over the world block content from reaching IP addresses within their borders. Both Saudi Arabia and Iran have sophisticated blocking systems that filter websites deemed immoral or with political content the government finds objectionable. Other countries block particular websites; for example, Turkey has blocked YouTube and Pakistan has blocked Google's platform Blogger.103 Other countries use temporary measures to block websites, such as denial of service attacks to shut down websites during particularly sensitive periods or protest events.104
When authorities do not want to use a Firewall to filter information, they can impose friction by simply slowing information transfer to make it more annoying for citizens to access. Internet blackouts can be strategically timed to prevent reporting of repression or protests.105 Countries can also throttle websites, making them slower to load or unreliable. These types of methods are intended to aggravate the user, while not outright blocking the site. Iran has been known not to black out the Internet but simply make it slower during periods of unrest.106 The recent debate in the United States about net neutrality fundamentally concerns the same issues of speed and friction. Proposals supported by telecom companies would make fast "lanes" on the Internet for content providers that agreed to pay more. Essentially, this would make the relative speed of some information slower than others. For impatient consumers waiting for websites to load, marginally faster websites would gain a larger audience and therefore more revenue and influence. If the selection of which content could move through fast lanes were political, this could act like online political censorship in more authoritarian environments.
Another example of online friction is search filtering and keyword blocking, where Internet search engines reorder the search results or disallow particular searches for political purposes. In China, search engines such as Google and Baidu were originally required to block particular searches that include sensitive people or events. Social media sites such as Sina Weibo were required to do the same when users were searching for social media posts related to sensitive people or events. However, recently, Chinese websites have moved past simply prohibiting some searches and instead search services reorder search results in a way that obscures the omission of certain results to the user.107 By burying sensitive webpages in the depths of the results, these search filtering methods of censorship create friction for particular types of information, and users may be completely unaware that this information is being de-prioritized. Similar concerns have been launched against U.S. social media companies that control the prioritization of information to users. While unverified, Google and Facebook have been accused of prioritizing particular types of news for users for political purposes.108
While there are many examples of frictions online, friction is not new or unique to the Internet age. In print publications, governments also try to de-emphasize certain types of information so fewer people come across them.109 In China, government information agencies often mandate that particular pieces of news be published on the back pages of the newspaper, or below the fold, so as to reduce the total number of people who will come across that piece of information. Leaked directives show explicit directions for placement within newspapers.110 These government directives also mandate particular titles that are intended to be less sensational and attract less attention than newspapers might print otherwise.
Even authorities without direct access to media infrastructure, for example companies, politicians, and interest groups, can use the news cycle to make it more difficult for consumers to come across bad news. "Bad news Fridays," as media advisors in the field frequently refer to it, is when groups announce their bad news at or after 5:00 PM on Friday, when most journalists have gone home and most citizens are not paying attention to the news.111 Because citizens may have stopped paying attention to news for the weekend, and the story is unlikely to run three days later on Monday, they are less likely to come across this negative piece of information.
Controversies over the material within libraries or in textbooks also reflects conflict over friction, as student access to this public information is nearly costless, whereas outside information takes initiative and resources for students to procure. These controversies are widespread, including controversies over the portrayal of historical wars,112 the portrayal of race and class,113 or ideology and nationalism,114 and occur in countries around the world.
_**2.8.2 Friction on the Collection of Primary Information**_
In the previous section, I described instances of information friction where access to media or information that had already been written and collected was throttled or made more costly. In these cases, friction throttled information traveling to the top level of the pyramid. In this section, I describe when the data necessary to write a story, book, or social media post is difficult to access, or when information is censored at the bottom of the pyramid. This is a deeper form of friction because it occurs at the initial stages of story discovery. Like a value-added tax on an input to a product that shows up in the price of the good, but is invisible to the consumer, this friction on collection of primary information can often be more invisible because it is further removed from the public.
Data collection has a natural cost—certain types of information are more difficult to collect than others, even without political manipulation. For example, it is less costly to go outside and observe the weather where you are than to collect information about the weather in an unoccupied part of Antarctica, thousands of miles away. It is less time-consuming to conduct an interview in your native tongue than in a foreign language. However, these frictions can be exacerbated by authorities (the government, political parties, interest groups, etc.) who have an interest in making particular types of information even more difficult to collect.
In this section, I will outline three different types of friction-based censorship that apply to the collection of data. First, authorities can throttle access to information they have already collected, making the information they collect more difficult for secondary sources to gain access to. Second, authorities may simply refrain from collecting particular types of information, even information that would be natural for them to collect. Last, for journalists, academics, or citizens interested in collecting particular types of data, authorities can throttle access to the collection of data, even if they themselves do not collect it.
**2.8.2.1 Friction on Data Access**
In the first case, groups and governments can slow access to information they already have, making it more difficult for citizens and journalists to collect information necessary to complete an article, social media post, or study on their topic of interest. In the United States and many democracies, Freedom of Information Act (FOIA) requests are meant to reduce friction in citizens' and the media's access to information. However, companies and interest groups lobby for exceptions from FOIA requests. For example, in 2003, data collected by the Department of Transportation about safety defects in cars was made exempt from FOIA requests obstensibly to prevent one auto manufacturer from gaining an advantage over another. In another example in the United States, the Tiahrt Amendment, passed in 2003, made private a government database that traced guns used in crimes back to dealers. Making this information impossible to access has made lawsuits against gun dealers and research about gun violence more difficult to conduct, as other sources of information must be used.115
Even without banning information, government agencies can make information harder to access under FOIA laws. When a bureaucracy does not want to release information, it can quote exorbitant prices for FOIA requests for copying and sending the information to citizens, the only aspect of a FOIA request a bureaucracy is allowed to charge for. The ACLU, for example, has been quoted hundreds of thousands of dollars for access to police records.116 Journalists have been quoted hundreds of thousands of dollars for FOIA requests to access information about FBI contracts with defense companies.117 Such exorbitant costs have motivated a few states to put caps on the amount that can be charged for access to information under FOIA laws.
In authoritarian environments, costs of access can be even more arbitrary. When a public disaster strikes in China, local government officials are quick to control the sources of information, ensuring government officials follow victims' families to control their interactions with the press, or placing victims and their families in a hotel or a place where officials can control press access.118 Although the Chinese government has become increasingly transparent in providing data about governance online, these data are often subject to scrutiny before they are posted, making it impossible to know their veracity. Local governments in China are known to collect inaccurate GDP and environmental data, inflating numbers to make it seem as if their jurisdictions are growing faster, and with less pollution, than they are in actuality.119 Such manipulation of government statistics makes estimating true GDP numbers in China difficult for investors, the media, and researchers.120 Archives in China are not consistently open to academics and are often subject to gatekeepers or dependent on the sensitivity of the time period. The government will sometimes allow researchers more access to historical information that supports their current political agenda.121
**2.8.2.2 Failure to Collect Data**
Authorities can fail to collect information, thereby making it very difficult for the media to access. The logic behind the failure to collect data is often obscure—does the authority simply not have the resources for collection, or are they purposefully trying to hide information? For example, in the aftermath of the Ferguson police shooting in 2014, where an unarmed man was shot and killed by the police, the media attempted to print statistics reflecting the number of Americans killed by police each year. In fact, police do not collect these data, and the Justice Department only has statistics on "justifiable" police killings, which rely on voluntary statistics from police departments, where "justifiable" may be defined differently across departments. Data on unjustifiable police killings are not collected.122 Scholars have found that discrepancies between sources of the number of police killings is due to the failure of police to report the data or the misclassification of police killings as homicides that were not due to the police.123 Whether these failures to report are a result of error or a result of incentives is subject to speculation.
Autocracies are also notorious for refraining from collecting or for mis-collecting information that makes access to data more difficult for journalists, citizens, and researchers. For example, in Beijing, until 2012 the Chinese government consistently refused to collect data on fine particulate matter (PM 2.5), a particular type of air pollutant that is thought to have damaging effects on health. While the U.S. embassy posted PM 2.5 data for Beijing on its blocked Twitter feed, the Chinese government posted pollution data without PM 2.5 included.124 Similarly, China has reported enormous progress in fighting crime over the past years after announcing a large anti-crime campaign. Yet the crime statistics reported from China are somewhat unbelievable—over two-fifths of counties reported "cracking" 100 percent of murders.125 Many assume that local police simply omitted cases they thought were too difficult to solve.
**2.8.2.3 Friction Imposed on Data Collection**
When data are not collected by authorities, journalists and citizens may make an effort to collect the data themselves. Data collection by the media and citizens can be expensive and time consuming even without intervention by third parties, but authorities will sometimes try to make the collection of data itself more costly for anyone attempting it. For example, in China, lead poisoning is one of the major threats to public safety and in particular children's safety. At one point, estimates suggested that up to one-third of Chinese children could be affected by high levels of lead in their blood.126 Interviews conducted by Human Rights Watch showed that political entities within China have made data collection on the severity of the problem extremely difficult. Hospitals, particularly in rural areas in Anhui, Hunan, Yunnan, and Shaanxi provinces, have repeatedly refused to give lead tests to parents and children.127 Local hospitals have given misinformation about the health consequences of lead poisoning and the recommended treatment.128 Although people are aware that lead poisoning is a problem, such lack of information collection has lessened widespread pressure on local governments, as local residents have found it difficult to organize because verifying that they are affected by the problem is difficult.129
Making data collection more difficult is not unique to the authoritarian context. For example, in war zones, the U.S. government has been accused of impeding journalists' access to particular sites, allowing journalists only in "approved" areas of the war zone. In the Persian Gulf War, journalists were escorted to parts of the war zone based on U.S. military decision making. The U.S. military made it difficult to access other types of information about the war outside of the perspectives of these "press pools," meaning that the U.S. media largely saw the war from one perspective. Those who were found violating press pool guidelines were removed from the area, which meant that journalists could sometimes not collect information needed to corroborate the U.S. government's account of the war.130
While we have only touched on a few examples for illustration in this section, authorities have myriad methods for imposing anything from small to significant costs on data collection. This censorship often does not involve punishment but can still have significant effects on the spread of information. Further, these frictions are often difficult for the public and media to observe—the lack of information is often less noticeable than its existence. It will typically not be apparent to the public that a story is not covered because of data unavailability. The few times it is made clear to the public that this information cannot be accessed, often the underlying intent is also opaque: are FOIA requests expensive because of politically motivated manipulation of information or because of true administrative costs? Does China not collect PM 2.5 data because they lack the technological capabilities or because they hope to mask the true levels of pollution? Unlike fear, where censorship must be explicit to deter, the intent behind friction is less apparent and can be explained away by authorities and observers, offering the cover of plausible deniability.
_**2.8.3 When Does Friction Affect Access and Expression?**_
Friction-based methods of censorship are more easily implemented by the authority the more the authority has control over information generation at the lower levels of the information pyramid. Information that can be collected only by the authority, either because they alone have access to the information or because only they have the expertise to collect it, can easily be made more costly: the authority can simply fail to collect it, or if they do decide to collect it, they can make sure it is unavailable to the public. Control over information also enables the authority to provide reasons for information restriction. For example, if the government restricts where the military can provide journalists access to war zones, it may provide safety reasons for keeping journalists out of particular locations. Since authorities can control access at the data-collection stage, these types of actions are also more difficult for the public to detect, as the public will not always notice the _lack_ of a story in the media. In contrast, when the authority does not control sources of information, such as events that happen quickly and are highly observable to large numbers of people, friction-based methods of censorship will be difficult to use. Information about events like environmental crises, large accidents that may implicate the government, or natural disasters that occur in large population areas will spread quickly, before information friction can be imposed. In these cases, high levels of censorship or the lack of reporting may be indications that the government is ignoring or repressing the issue, which could backfire and cause more, not less, consternation among the public.131
Once information has already been collected or is already known to some number of people, control over the information environment through friction requires political power or access to the media infrastructure and distribution of media.132 The more control the government has over Internet content providers, newspapers, or television stations, the more successful it will be at creating friction. The more tools that the government has to control search engines or slow the delivery of particular types of information, the more the government will be able to impose friction on already created media articles. Governments of smaller countries, where many citizens rely on foreign news sources or Western Internet content providers like Twitter and Facebook, will have fewer options for imposing friction because they do not have direct jurisdiction over the outside media content.133 Similarly, governments in areas where television or newspaper broadcasts span international borders will have trouble controlling information from outside.
Even if information is not particularly well known initially, it may be difficult for the authority to create friction if another organized entity has a strong incentive to seek out the information and publish it. For issues where there is a stronger civil society, more factionalism, or other types of political competition, it will be more difficult for authorities to hide information since it will likely be collected, published, and promoted by the competing entity.134 Such competition also makes friction harder to justify—for example, the Chinese government had trouble justifying publishing pollution numbers that did not include levels of PM 2.5 because the U.S. embassy in Beijing was publishing these numbers on its own Twitter feed.
Citizens and the media will be more susceptible to friction if the cost added by censorship to the information is enough to offset the benefits of consuming or disseminating information. Importantly, how citizens and the media are affected by friction will depend on their _elasticity of demand for information_ ; in other words, how much their behavior changes in reaction to small changes in the price of information. In general, the more elastic citizens' and the media's demand for information—the more affected they are by changes in the price of information—the more effective friction-based methods will be since citizens and the media will be more likely to substitute lower-cost information for that which has become higher-cost. As a result, friction will have a greater impact the less citizens and the media are willing to seek out the political information.
Because of the importance of elasticity of demand for information, friction will have a stronger impact when it is invisible—if the friction induced by the authority becomes known to its target, the citizen or media may take it as a signal that the information is important and their demand for this information might become more inelastic. If the observation of censorship creates more inelastic demand for information, more people will be willing to overcome costs to find the information because it was censored, creating the "Streisand effect" discussed earlier.135 Relatedly, friction-based methods are more effective when censorship can be obscured or blamed on another cause. Plausible deniability will muddle the signal of the importance of information and citizens may explain friction away or ignore it. Like in contexts of direct repression, "plausible deniability" on the part of the government might be more credible when friction is implemented by an agent—a private company or a local level of government—rather than the central government itself.136
Citizens' and the media's demand for information may become more inelastic when external events make them attune to politics or when censorship disrupts their habits. During political crises, the public may sense threat and therefore be more likely to overcome barriers to seek out information. As I'll show in chapter 5, when an explosion shook Tianjin during August of 2015, many more people were willing to search out information across the Great Firewall. Similar research has shown that during revolutions and political crises, citizens may be more willing to seek out media to inform themselves of ongoing events.137 Similarly, in democracies, heightened attention that results from crises or protests can draw attention to the limits on government data, or underreporting of events within the media.138
Similar to the economic observation that addictive goods are less price-sensitive, friction that suddenly disrupts habits will be less effective because people are generally more willing to spend time and money to continue habitual behavior. If friction gets in the way of something a user needs—for work or for entertainment, for example—they may be more willing to pay the costs of censorship evasion. As I'll discuss more in chapter 5, when the very addictive social media website Instagram was blocked in China, millions of people evaded the Firewall to access Instagram because they were accustomed to doing so,139 and were subsequently exposed to censored political information.
Those who learn to overcome frictions during political crises or in reaction to habit disruption may have non-linear effects on the information environment. Once a user has learned to overcome friction, they may more easily be able to do it again in the future, or apply it to other types of information. For example, when someone learns to evade the Great Firewall, they may more easily jump it again in the future. As more people learn how to evade censorship, they may be more likely to share this information between their friends. Friction will therefore be less effective if it is broadly applied to both political information and entertainment or other types of information that citizens are highly motivated to access; in such cases, people may learn how to overcome frictions in order to access entertainment and once they do, they will be able to apply it to other forms of information, such as political information.140
Of course, friction—like all forms of censorship—can have economic and political consequences that are detrimental for authorities. Blocking websites that could potentially be useful to economic growth (like Google, GitHub, or Dropbox) is like imposing a tariff on valuable inputs for local Internet companies. However, because taxes on information are more easily disguised than a prohibition that has to be communicated and enforced, censorship by friction is less likely to have the same backfire effects as fear. Friction will also not make citizens fearful of speaking their mind and therefore will allow the government to monitor public opinion. For these reasons, friction solves many of the dilemmas inherent in the imposition of censorship through fear.
_**2.8.4 The Impact of Friction Has Strengthened in the Information Age**_
The Internet has decreased the absolute per unit cost of information substantially. According to Internet Live Stats, there are more than one trillion individual webpages on the Internet.141 With a known URL and a good Internet connection, information from all over the world can be accessed within seconds. As a result, many scholars thought that censorship in the age of the Internet would be futile—in President Clinton's words, "like nailing jello to a wall."142 Easily copied or added to the trillions of websites online, many suspected that information simply could not be controlled.
However, despite the overall decrease in the cost of information, the relative cost of competing information is still relevant for consumers of information. Because information is easily substituted and citizens have a limited amount of time, the relative costs of information drive consumption patterns. Having more near-equivalent copies of information on the web increases the substitutability of information because if one piece of information entails costs, consumers are likely to be able to find something similar enough more cheaply elsewhere. Many recent studies have documented consumers' extreme impatience on the Internet. Google research has shown that even small, micro-second delays in searches can significantly decrease user search.143 Krishnan and Sitaraman (2013) find that with each one second delay in video start-up, 5.8 percent of users switch to something else. Viewers who experience slow video equal to one percent of the duration of the video watch five percent less of the video than those with no delay. Mark, Voida and Cardello (2012) find that users on average switch windows on a computer 37.1 times an hour, with time staying on one window on average just over one minute. Yeykelis, Cummings and Reeves (2014) find that sympathetic arousal increases right around switching between tasks and windows on the computer, suggesting physiological reasons for impatience and distraction on computers. As a result, small changes in search algorithms have been shown to drastically shift consumption of websites and news.144 As the elasticity of demand for information increases because of the Internet, censorship methods that rely on the friction mechanism become more potent.
Unlike fear, which is more costly as more people become producers of information, the price of friction is scalable for increased numbers of Internet users. Search filtering one website many people use, for example, takes only one manipulation to affect all consumers. Removing a post from the Internet affects all people who would potentially look for that post. A dataset that is removed from a government website affects access for the entire public.
Friction, particularly in the digital age, facilitates plausible deniability and is less transparent than fear, which must be observed in order to function. Because a search on the Internet is ordered by algorithms and Internet errors frequently occur without censorship, even if the user comes across evidence left behind by friction, it can be difficult to know whether a technical issue or an authority purposely manipulating the web was the cause. Algorithms for search are protected intellectual property and difficult for the public to understand—an Internet user could not tell you whether the search results were ordered in a particular way because of an algorithm acting in the interest of the public, or because of an algorithm censoring in the interest of the authority.145 If a user clicks on a link that is blocked by the Great Firewall, it is difficult to know whether that page is down because of the government or because the website itself is undergoing maintenance. Indeed, many websites in China "go down for maintenance" around the anniversary of the 1989 Tiananmen Square protests.146 A link may be broken because it has been moved or because it has been intentionally broken by authorities. The intricacies and complicated technology of the Internet disguise friction-based censorship.
The costless nature of the Internet has caused traditional journalism to be more susceptible to government frictions. Because of the "24-hour" nature of the news cycle, the fast pace of the Internet, and the fleeting nature of stories, costly investigation has become less profitable. In the United States, full-time professional staff at newspapers dropped 27 percent between 2000 and 2010, due to rapid declines in advertising revenue.147 In particular, resources for investigative reporting and international reporting have been cut, and newspapers are more reluctant to allow their short-staffed journalists covering daily news to invest in high-cost stories.148 These costs and uncertainties involved in story creation are more important in an era where competition between media outlets is more fierce and subscriptions to papers are being substituted with access to the Internet. Economic models of media show that advertising revenue is one of the most important components of independent press, as when the media has more advertising revenues, it is more difficult for the government to capture.149 Traditional media outlets are closing their doors, reducing resources for investigative journalism, and relying more on information from political entities, opinion articles, or the syndication of news, which are less costly to produce.150
Of course, traditional media have now been augmented by online news. In particular, "citizen journalists," who spend their own time researching and reporting news, have received significant amounts of attention in popular and academic presses.151 However, citizen journalists typically have more stringent monetary constraints on reporting than traditional journalists. Without an institution funding their salaries and the costs of investigation, citizens have fewer resources to invest in their stories. They also typically have less training to help them determine which stories might be successful. Citizen journalists will typically be more affected by the costs of a story in terms of time and money. Social media written by citizen journalists are more likely to carry re-shares of stories or opinions, which have lower costs of production, than the news media.152 The decline of the traditional media and rise of citizen journalists have therefore made the information distributed and shared online more susceptible to frictions imposed by governments.
**2.9 FLOODING**
The last type of censorship is flooding, the coordinated production of information by an authority with the intent of competing with or distracting from information the authority would rather consumers not access. Flooding occurs when groups systematically create information and disseminate it at low cost, to make it convenient for the media to print on a large scale or easy for the public to access. Although it seems counterintuitive that the production of information could be considered a form of censorship, by making a particular piece of information very easy to access, flooding raises the relative cost of information from alternative sources and therefore can have the same impact as censorship. Contrary to common wisdom that more information is always better, flooding with irrelevant or less valuable information reduces the amount of time that citizens can spend consuming more valuable information and taxes good information by requiring more time to separate good information from bad information.
Flooding and friction are tightly paired: flooding causes friction by making information contemporaneous to the flooding effort more costly to access. However, the mechanism of increasing the costs is different: flooding increases the relative cost of information indirectly through competing information, whereas friction directly increases the costs of information, raising the absolute costs of that information. As I will show throughout this book, flooding and friction are often used together as authorities try to slow the spread of one type of information and promote another. Like friction, flooding is not always mutually exclusive of fear—the observation of propaganda can create fear and cause people to self-censor.153 However, fear is not necessary for the flooding mechanism to work—citizens may be completely unaware that flooding is happening and still be affected by it. In fact, flooding is often used by authorities when fear and friction are too costly or could create backlash.
Like friction, there are natural types of flooding that affect citizens' consumption of information. Reporting on the Olympics, for example, has been shown to compete with reporting on natural disasters—areas affected by disasters that occur during the Olympics receive less aid because people are distracted by the games.154 Although such distractions have important political and economic consequences, I do not consider distractions like the Olympics to be flooding-based censorship, insofar as they are not designed to decrease the accountability of political entities. However, to the extent that distractions can be coordinated by political organizations, political entities themselves can have a similar impact on citizens' consumption of information, an effort I consider to be a form of censorship.
As discussed in more detail at the end of this chapter, flooding is an increasingly convenient form of censorship for authorities because now that much of the media is digitized it is relatively inexpensive to produce distracting information in large quantities. Producing superfluous information is often less costly to the authority than trying to find a way to throttle information that is already produced or to intimidate individuals so much that they refuse to speak out. In countries where information is relatively free already and citizens are less likely to be intimidated, flooding is an increasingly attractive option for political entities interested in controlling information. Flooding is also easier to use to affect information systems across borders, where authorities have less control over legal punishment, intimidation, and infrastructure that could allow them to create fear or impose friction.
In this section, I consider two main types of flooding. In the first case, the authority produces information that it then disseminates directly to the public at the top of the information pyramid. In the second case, the authority provides prepackaged information to the media at the middle level of the pyramid, with the intent of reducing the media's costs for a story, thus encouraging the media to print the story the authority designed in exclusion of others. In each case, the flooding may not include content directly related to the information the flooding is trying to suppress, but rather might take the form of entertainment or distraction that might be more interesting to the consumer. Much like advertising, flooding always occurs in a competitive context, attempting either to distract from or to overwhelm another party's version of events.
_**2.9.1 Flooding Directly to the Public**_
The first type of flooding competes directly with information already available to the public. This type of flooding operates at the dissemination stage, where the flooded material competes for attention with information being disseminated by media or individuals. Many times, the information that is being used to compete is not attributed to the government, even though it is government-coordinated.
Flooding directed at the public can be used for persuasion, confusion, or distraction. In each case, the purpose of flooding is to crowd out alternative viewpoints or perspectives. Sometimes flooding can take the form of traditional propaganda; the government can use its control over the media to create an omnipresent perspective on an issue for which the government has come under criticism. A classic example in China is the TV program _Xinwen Lianbo_ , which airs nightly across China and represents the viewpoint of the central government. This news show stands in direct competition with Western viewpoints that cast China in a chaotic light, in which the chaos stems from autocracy. Instead, the oft-repeated Chinese propaganda viewpoint is that chaos stems from democracy, as Chinese reporters often focus on protests or political conflict in democracies. The TV program is shown on all local TV stations every night during prime time, crowding out alternative prime time TV.155
Traditional propaganda has been augmented by an online environment where paid commentators are cheap and can spread information widely. Authorities all over the world have designed flooding strategies to compete with online information that reflects poorly on them. On Twitter, governments have amassed "Twitter armies," coordinating the promotion of their version of events, increasing the ratio of Tweets that reflect well on their own perspective, and thus increasing the probability that consumers of information will run into their version of events. The Turkish, Israeli, and Palestinian governments all hire or enlist people to promote their ideas on Twitter. Such flooding strategies have resulted in "Twitter wars," where each government produces an onslaught of its version of events.156 Even political parties in the United States are beginning to employ online armies that defend their candidate.157 Web browsing data show that during the 2016 presidential election there were millions of shares of false news stories, which may have contributed to significant confusion about the candidates.158
Flooding can be used to undermine competing sources, either by directly disparaging them or by confusing the public into discounting the traditional press. Authoritarian governments around the world are known to widely disparage the Western press, using their own control over media to caution readers about bias in independent media.159 Contradicting facts or persuading citizens to question reality, sometimes known as "gas-lighting," can also be the purpose of propaganda in order to confuse or distort citizens' perceptions of reality and thereby undermine the free press.160 Most recently, the Russian government has been accused by the U.S. government of spreading misleading news online about the candidates in the 2016 election, undermining facts reported in the free press in an attempt to influence the election.161 These efforts tax the public's attention, either misleading voters, or requiring them to spend more time to separate out which information is credible from that which is false.
Governments also use online armies for the second purpose of flooding—distraction—hiring thousands of people to post distracting information during sensitive events, what analysts have called "third-generation" Internet controls.162 During protests surrounding Russian elections in 2011, pro-Kremlin Twitter users allegedly flooded anti-Kremlin hashtags with slogans and meaningless tweets, making it more difficult for anti-Kremlin users to find useful information and coordinate action.163 Similar tactics have allegedly been used in democracies—the Mexican government, for example, has been accused of trying to thwart protests with bots that flood Twitter hashtags with meaningless punctuation.164 As I will discuss more in chapter 6, China's "Fifty Cent Party," social media users who post at the government's direction, often write positive social media posts to shift attention from negative events.165
Flooding is not a phenomenon specific to the Internet; it also appears in traditional and popular media. During sensitive periods, Chinese propaganda authorities mandate that newspapers write "positive news" to make negative events less salient.166 Russian authorities are also known to employ this tactic—the Kremlin has been know to suggest that news and television should focus on positive stories.167
_**2.9.2 Flooding Directed at the Media**_
The second type of flooding occurs when the authority directs the information to the media. By collecting data, analyzing it, and presenting these results to the media in an easily reportable format, the authority can encourage the media to report on a particular story. The media then may present this story to the public using the prepackaged version to reduce media costs. The public, however, may not recognize that the source of the information is the government itself, but instead view the news as independent.
Politicians, companies, and interest groups sometimes spend money to support scientific research that can bolster their product or cause. Research findings generated by authorities are then often picked up by the media to present a "balanced" view of the issue, but often do not disclose their funders directly.168 Infamously, Philip Morris sponsored research claiming that cigarettes were not damaging to health. According to prosecutors, the company created the Tobacco Industry Research Committee (TIRC) in 1954 "to refute, undermine, and neutralize information coming from the objective scientific and medical community."169 In 1998, tobacco companies sponsored the Center for Indoor Air Research (CIAR) to counter research on the detrimental effects of second-hand smoke. Confidential documents released as part of a 1998 settlement show that Philip Morris used the 244 studies produced by CIAR from 1989 to 1999 to fight public perceptions that second-hand smoke causes cancer.170 Although Philip Morris is one particularly pernicious example, research funded by special interests and companies to balance scientific consensus is very common, with examples ranging from head injuries in the NFL to skepticism on climate change.171
Authorities can also use press releases to try to drive media coverage.172 Grimmer (2013, pg. 129–130) finds that local papers will often use large amounts of language from U.S. legislators' press releases directly in news articles. He argues that press releases are purposely written in the style of news stories, acting as a subsidy for newspapers because journalists do not even have to rewrite them in order to print. Similarly, beginning with the Clinton administration and accelerating under the Bush administration, dozens of U.S. bureaucracies regularly released video press releases that supported particular government policies. These press releases were directly aired on local news programs without acknowledging that the government itself had produced them.173
Countries can use their own news agencies to push their soft power abroad. China's _Xinhua_ news agency's worldwide expansion reflects a strategy to promote China's view of political events in foreign papers, as _Xinhua_ spins stories quite differently than Western media.174 In many countries with a small domestic news presence, _Xinhua_ is frequently syndicated.175 By making _Xinhua_ stories deliberately cheap, China promotes its perspective on world events, increasing its international audience.
_**2.9.3 When Does Flooding Affect Access and Expression?**_
In order for the flooding mechanism to be effective, the media and citizens need to be likely to consume the low-cost information produced by the authority instead of the information with which it intends to compete. Like friction, flooding-based methods of censorship will be more effective when the media and citizens have a higher elasticity of demand for information, or when the cost of information is a primary determinant of whether the media uses the information or citizens consume it. When citizens and the media are very affected by the cost of information, low-cost information produced and disseminated by the authority will be readily consumed and citizens will be unwilling to sort through the distracting information to find its alternative.
When propaganda is too obvious, it can generate a "boomerang effect," where propaganda is discredited by the public.176 Flooding will therefore be more effective if citizens are not aware that it is generated by the authority. The authority could achieve this in one of two ways. First, it could build trust in its own sources of media and discredit alternative sources of media so that citizens are more likely to turn to government sources for information. Chinese state-run news sources, for example, proactively discredit Western media, though they still struggle with maintaining credibility in the eyes of the public.177 Alternatively, it could hide the fact that it is manipulating the information environment, using media or individuals who look as though they are not related to the authority to spread the information. Individuals hired to post pro-government content on social media are examples of authority-funded information that is supposed to appear spontaneous. The less that individuals and the media are aware that the authority is behind the information, the less they will be able to discredit it.
However, unlike friction, it is less imperative for authorities that their efforts to flood remain invisible since the public typically has fewer objections to the production of information than its suppression. Further, awareness of flooding does not usually bring attention directly to the information that the authority is trying to hide, and therefore does not undermine the authority as much as awareness of efforts of friction or fear. In this way, flooding should be a less risky strategy than friction or fear because awareness of these methods will not be as likely to produce backlash.
_**2.9.4 Flooding Has Become Cheaper with the Internet**_
Most authorities can participate in flooding if they have sufficient funds to do so—authorities may not have control over information they are trying to hide, but almost always have some type of information that they can promote.178 The cost of creating information with which to flood will depend on the medium through which the authority intends to promote the information. The cost of flooding to the authority has decreased substantially in the digital age—Twitterers working for the Russian government, for example, use opposition hashtags to make it more difficult for the opposition to coordinate. The content of these tweets does not matter as long as they use the correct hashtag, and therefore such messages are very low cost. This type of flooding can even be implemented by bots rather than humans.
Similarly, flooding is easier to disguise in the age of the Internet than before the information age. In the past, personal opinions were not frequently voiced in the public domain—newspapers distributed information and officials and community leaders could make speeches or write op-eds, but typical citizens did not announce their positions. Today, however, individuals share descriptions of even mundane aspects of their lives and personal views on issues. These individuals do not necessarily have a public reputation to protect. The prevalence of online opinion makes it easier to disguise those who are paid by the government as individuals who genuinely hold these views.
The ability for individuals to bypass the media to share stories directly with the public also can allow for flooding to be more effective. Whereas the media decides which stories to include by making editorial decisions, social media is prioritized to consumers through algorithms. Authorities interested in using flooding can reverse-engineer and take advantage of these algorithms and coordinate to prioritize their information to consumers. If propagandists can find ways for users to engage with information through "clickbait" headlines or by disguising themselves as news media, they may be more likely to filter to the top of a social media algorithm.179
In other more traditional media, flooding may be more expensive. If authorities hope to spread their message in ways that look like news articles in traditional media, they need to distribute press releases that newspapers are likely to pick up and print. If these messages are persuasive, not just distracting, then the autocrat might need to collect data or conduct research to make the flooded messages convincing. Flooding via television requires either some control of the media network or large amounts of money for paid programming and advertising. Alternatively, some authorities may host high-profile events or news conferences to attract attention, and these events themselves can be expensive.
Flooding is the least draconian mechanism of censorship and it is rarely illegal even in democracies, and therefore it is the least likely to cause backlash. However, like friction and fear, it can have long-term costs for governments. If citizens discover government flooding efforts, it may undermine the credibility of the information environment, making the spread of misinformation and rumors more likely. As with censorship by friction, authorities may also trick themselves with their own propaganda, assuring themselves that they have more support than they actually do by measuring the balance of online opinion.
Because flooding is the least objectionable to the public of all the mechanisms, it will be used relatively more than friction and fear in societies with strict information freedom laws and intense competition between groups, for example, in democracies. Flooding may be used to complement or substitute for methods of fear and friction in cases where the authority does not have complete control over information, such as during sensitive periods or sudden high-profile events that could endanger the existence of the authority. In periods of crisis, when friction breaks down because individuals are motivated to seek out information, governments may turn to flooding as the public may seek out information of an unfolding situation and come across government-flooded media.180 However, these may also be moments when the public is willing to spend more time ascertaining the credibility of the source, and therefore flooding, like friction, may be relatively less effective during crises than during time periods where public demand for information is more elastic.
**2.10CONCLUSION**
Fear, friction, and flooding—the mechanisms of censorship—are not specific to information-communication technologies or media. Over time, regime types, and issue areas, we have seen instances of each of these mechanisms of information manipulation in this chapter. However, the blend of strategies will vary across government structures, depend on the threats the governments face, and change in new information environments. As the structure and flow of information changes due to new technologies, the way that citizens and the media react to censorship and the costs censorship has for authorities change. In particular, the nature of the trade-off between using fear or punitive methods to create censorship and more porous methods based on friction or flooding will depend on the goals of the state and the costs that censorship inflicts on governments.
The logic of fear, friction, and flooding all point to a strategy of porous censorship in the digital age. In particular, the Internet has made fear more costly for regimes and friction and flooding relatively cheap. As more people participate online, repression must be credible to a larger number of people in order to enforce self-censorship. Yet because information is easily duplicated, substituted, and replaced, small variations in the cost of access in the form of friction can have large impacts on what the majority of Internet users consume. Similarly, the Internet has made flooding relatively cheap as government propaganda efforts online can be more easily automated and masked.
1 Mayhew (1974) and Bernays (1923) describe how political parties and corporations advertise to their constituents and consumers.
2 Even in autocracies, leaders have to work in some of the public's interest. Shirk (1993, pg. 107) describes how constituent preferences are aggregated in authoritarian regimes; Bueno De Mesquita et al. (2003, pg. 8) develops the idea of the selectorate within autocracies.
3 Enikolopov, Petrova and Zhuravskaya (2011).
4 Much of the literature on collective action explores this. For example, Tilly (1978, pg. 8) talks about the importance of beliefs in mobilization; O'Brien and Li (2006, pg. 38–39) explore how perceptions and information about state policy can generate rightful resistance in rural China.
5 Friedman (1999, pg. 24) describes how the media and information affects consumer boycott success; Popkin (1994, pg. 27) shows how information and the media influences how voters think about elections and the government.
6 Little (2016) argues that information can hold autocrats accountable either by spreading negative information or by facilitating collective action. Enikolopov, Makarin and Petrova (2016) show that social media primarily played a coordination role in protests in Russia. See also González-Bailón et al. (2011).
7 Lohmann (1994, pg. 42), Kuran (1989, pg. 42), Chwe (2001, pg. 7), Lewis (2008, pg. 52).
8 Stiglitz (2002, pg. 461).
9 Rose-Ackerman (1978).
10 Ritter and Conrad (2016, pg. 85), Guriev and Treisman (2015, pg. 6).
11 See Wintrobe (1998, pg. 20), Francisco (2005); in China see Dickson (2016).
12 Francisco (2005, pg. 58–59).
13 For other work on backfire against repression, see Lichbach (1987), O'Brien and Deng (2015).
14 Charron and Lapuente (2011, pg. 399); Egorov, Guriev and Sonin (2009); Liebman (2005); Lorentzen (2014); Stockmann (2012, pg. 140).
15 Lorentzen (2015).
16 Shirk (2011, pg. 19).
17 Distelhorst and Hou (2017).
18 Kuran (1997, Chapter 4); Chen and Xu (2017 _b_ ).
19 Shirk (2011, pg. 1), Choi (2003).
20 The American Chamber of Commerce's recent survey of U.S. firms operating in China shows that 71% of firms report that censorship hurts their business. "2016 China Business Climate Survey Report." The American Chamber of Commerce in the People's Republic of China (2016). <https://www.amchamchina.org/policy-advocacy/business-climate-survey/2016-business-climate-survey>. Domestic firms are hurt by censorship also. See Tate, Ryan, "Costs of Censorship Haunt 'Chinese Twitter' IPO," _Wired_ , April 17, 2014. <https://www.wired.com/2014/04/weibo-ipo-cost-of-oppression/> and Beech, Hannah, "China's Great Firewall is Harming Innovation, Scholars Say," _Time_ , June 2, 2016. <http://time.com/4354665/china-great-firewall-innovation-online-censorship/>.
21 O'Donnell (1973); Gasiorowski (1995).
22 Alexievich (2006, pg. 211–212); see also "Information Control and Self-Censorship in the PRC and the Spread of SARS," Congressional-Executive Commission on China White Paper, <https://www.cecc.gov/publications/issue-papers/information-control-and-self-censorship-in-the-prc-and-the-spread-of-sars>.
23 Sniderman, Tetlock and Brody (1991); Conover and Feldman (1984); Popkin (1994); Hamilton (2004).
24 Lupia and McCubbins (1998, pg. 25–26), Hamilton (2004, pg. 8).
25 Downs (1957, pg. 214–216).
26 Prior (2005, pg. 577), Baum (2002, pg. 91).
27 See Festinger (1957, pg. 30) on cognitive dissonance and avoidance of information; see Nyhan and Reifler (2010, pg. 307) on the potential for corrections to backfire and increase misperceptions.
28 Hamilton (2004, pg. 9).
29 See Darby and Karni (1973, pg. 68–69) for a discussion of credence qualities of goods.
30 McCombs and Shaw (1972); Zaller (1992, pg. 48) discusses how accessibility of information influences public opinion; Iyengar (1990, pg. 4) discusses how accessibility is influenced by the media.
31 Zaller (1992, pg. 6) defines the political elite as those who "devote themselves full time to some aspect of politics or political affairs"; Converse (1964) shows that elites think about and interact with political information differently than the mass public.
32 Hamilton (2004, pg. 11).
33 Downs (1957, pg. 259), Angus et al. (1960, pg. 180–181), Converse (1964, pg. 34), Graber (1988, pg. 105).
34 RePass (1971, pg. 391), Krosnick (1990, pg. 66–67)
35 See also Delli Carpini and Keeter (1996, pg. 62–105), Caplan (2007, pg. 94–113), Lewis-Beck (2008, pg. 161–180).
36 See Festinger (1957, pg. 30) for a discussion of cognitive dissonance; relatedly Frey (1986) discusses selective exposure to information. Kuran (1998, pg. 158–161) discusses how moral dissonance can lead to rationalization.
37 Stockmann (2012, pg. 41) finds that Chinese people seek information that is consistent with their beliefs. Stockmann (2012, Chapter 8) finds that Chinese citizens will select Chinese newspapers to read that they find more reputable, but will not invest substantial time in seeking out Western sources that are less readily available.
38 Geddes and Zaller (1989, pg. 327–341).
39 This comes from the theory of _media dependency_ —that crises force citizens to rely on mass media, see; Ball-Rokeach and DeFleur (1976); Loveless (2008).
40 For example, their own entertainment; see Baum (2003).
41 Although the media is not free in many countries around the world, in this section I focus on the media's incentives when it is free and commercialized and turn to government constraints on the media later in the chapter.
42 Hamilton (2011, pg. 277–288) outlines five incentives of media—advertising, subscription, persuasion, nonprofit, and expression. Here we consider the first two, though later in the chapter the other three are discussed.
43 Mullainathan and Shleifer (2005); Gentzkow and Shapiro (2006) describe how audience preferences affect the content of news.
44 See Hamilton (2004) for a detailed discussion of media bias that arises from the market. Of course, the media may also be motivated by what they see as their journalistic contribution or their philanthropic contribution, even if the mass public may not be most interested in these types of stories; see Zaller (1999); Hamilton (2011). High-quality stories that challenge the status quo might win awards or improve the journalist's reputation in the eyes of other journalist professionals. Though even with journalists' personal preferences for impact, successful journalists also must attract large audiences, and therefore to a certain extent journalists are subject to mass preferences and cannot solely follow their own definition of impact; see Zaller (1999).
45 Whereas untrustworthy sources are avoided by much of the public, "objective" reporting is not always valued by the marginal consumer and will not always be in the interest of the news outlet, as consumers may value a particular perspective. An outlet's slant can provide branding that distinguishes one news outlet from another; see Hamilton (2004, Chapter 2).
46 For more discussion of how marginal costs influence the content of reporting, see Hamilton (2011), Petrova (2012), Besley and Prat (2006), Gentzkow, Glaeser and Goldin (2006).
47 Mollenhoff (1981, pg. 4).
48 Hamilton (2005, Chapter 2).
49 Shirky (2008).
50 Tufekci and Wilson (2012).
51 Mitchell et al., "The Modern News Consumer," _Pew Research_ , <http://www.journalism.org/2016/07/07/pathways-to-news/>.
52 Carpenter (2008, pg. 539).
53 See Del Vicario et al. (2016) for a discussion of the spread of online misinformation.
54 Tufekci and Wilson (2012).
55 The skeleton of this definition is from Lasswell (1930).
56 These actions come from the freedoms discussed in Ingram (2000, Chapter 1), which includes another form that we don't consider here: freedom of communication. I consider freedom of communication as subsumed by freedom of access and expression.
57 Milner (2006); Weidmann et al. (2016).
58 Ingram (2000, Chapter 1).
59 Using a VPN is not penalized in China at the time of writing; however, there has been some discussion of making them illegal; see "Chongqing regulations may penalize VPN users," _Global Times_ , March 28, 2017. <http://www.globaltimes.cn/content/1040014.shtml>.
60 Typically groups that have solved the "collective action problem," as in Olson (2009).
61 Gormley and Balla (2013, Chapter 1).
62 For more discussion of interest groups and censorship, see Petrova (2012), Sobbrio (2011), Alston, Libecap and Mueller (2010).
63 Calland (2007), Avant (2005, pg. 60).
64 See Soley (2002) for a more complete exploration of corporate censorship. Crabtree, Fariss, and Kern (2015) explore private censorship in Russia. Lam (2017) explores private censorship in Hong Kong.
65 For an overview of online censorship technologies, see Deibert et al. (2008), Deibert et al. (2010), and Deibert et al. (2011), Freedom House's "Freedom the Net" project provide reports of censorship across time and countries.
66 "Saudi blogger Badawi 'flogged for Islam insult'," _BBC_ , January 9, 2015. <http://www.bbc.com/news/world-middle-east-30744693>.
67 "Saudi Arabia: Website Editor Facing Death Penalty," _Human Rights Watch_ , December 22, 2012. <https://www.hrw.org/news/2012/12/22/saudi-arabia-website-editor-facing-death-penalty>.
68 Stone, Jeff, "Russian Internet Censorship, Social Media Crackdown Makes It Easy for Putin to Stay Popular," _International Business Times_ , August 6, 2014, <http://www.ibtimes.com/russian-internet-censorship-social-media-crackdown-make-it-easy-putin-stay-popular-1651078>.
69 Fassihi, Farnaz, "Iran's Censors Tighten Grip," _Wall Street Journal_ , March 16, 2012, <http://www.wsj.com/articles/SB10001424052702303717304577279381130395906>.
70 "1989 Law on Guarding State Secrets" _Congressional-Executive Commission on China_ , <http://www.cecc.gov/resources/legal-provisions/1989-law-on-guarding-state-secrets-chinese-and-english-text>; Xu, Beina and Eleanor Albert, "Media Censorship in China," _Council on Foreign Relations_ , February 17, 2017, <http://www.cfr.org/china/media-censorship-china/p11515>.
71 See Demick (2010, pg. 51–56) for an example derived from interviews with North Korean defectors.
72 "71 Journalists Were Killed in 2013," _Reporters Without Borders_ , January 25, 2016, <https://rsf.org/en/news/71-journalists-were-killed-2013>.
73 Buckley, Chris, "Outspoken Chinese Professor Says He Was Dismissed," _New York Times_ , October 19, 2013, <http://www.nytimes.com/2013/10/20/world/asia/xia-yeliang-an-outspoken-chinese-professor-says-he-has-been-dismissed.html?_r=0>.
74 "Spiked in China? A ChinaFile Conversation," _China File_ , November 12, 2013, <http://www.chinafile.com/conversation/spiked-china>.
75 Truex (2016).
76 Haggard and You (2015); "South Korea: Stop Using Criminal Defamation Laws," _Human Rights Watch_ , December 14, 2014, <http://www.hrw.org/news/2014/12/14/south-korea-stop-using-criminal-defamation-laws>.
77 Oppel, Richard, "Taping of Farm Cruelty is Becoming the Crime," _The New York Times_ , April 6, 2013, <http://www.nytimes.com/2013/04/07/us/taping-of-farm-cruelty-is-becoming-the-crime.html>.
78 Cochrane, Joe, "Defamation Law Reminds Critics of Indonesia's Past," _New York Times_ , September 22, 2014, <http://www.nytimes.com/2014/09/23/world/asia/23indonesia.html>.
79 Tong (2011, pg. 49–79).
80 "14 Police Suspended After Beating Up Journalists," _Global Times_ , January 27, 2015, <http://english.sina.com/china/2015/0126/777177.html>; Iritani, Evelyn, "Beating Death of Journalist Spurs Inquiry," _Los Angeles Times_ , January 25, 2007, <http://articles.latimes.com/2007/jan/25/world/fg-chideath25>.
81 Barry, Ellen, "Journalist in Russia, Badly Beaten in 2008, Dies," _New York Times_ , April 8, 2013, <http://www.nytimes.com/2013/04/09/world/europe/mikhail-beketov-russian-journalist-beaten-in-2008-dies.html?_r=0>.
82 "71 Journalists Were Killed in 2013," _Reporters Without Borders_ , January 25, 2016, <https://rsf.org/en/news/71-journalists-were-killed-2013>.
83 See Becker (1968) for a discussion of the economics of crime enforcement.
84 Black, Ian, "Global Outrage at Saudi Arabia as Jailed Blogger Receives Public Flogging," _The Guardian_ , January 11, 2015, <https://www.theguardian.com/world/2015/jan/11/flogging-global-outrage-saudi-arabia-silent>.
85 Friedrich and Brzezinski (1965, pg. 169–170); Young (2016) shows evidence of how fear can impact the probability of dissent in Zimbabwe. Pearce and Kendzior (2012) estimate the impact of fear in Azerbaijan.
86 Stern and Hassid (2012), Bahry and Silver (1987, pg. 1068).
87 Kuran (1989).
88 Tong (2011, Chapter 1), Tong and Sparks (2009), Lorentzen (2013); Liebman (2005).
89 "What is the Streisand Effect?" _Economist_ , April 16, 2013, <http://www.economist.com/blogs/economist-explains/2013/04/economist-explains-what-streisand-effect>.
90 Eisenstein (1983, pg. 254).
91 Wong, Chun Han, and Olivia Geng, "Book Ban Rumors Boost Authors in China," _Wall Street Journal_ , October 13, 2014, <http://blogs.wsj.com/chinarealtime/2014/10/13/rumors-of-book-ban-boosts-authors-in-china/>.
92 A general model for this reaction is explained in Marcus, Neuman and MacKuen (2000), where emotions like fear alert citizens of something new and important, inspiring them to gather more information about the subject.
93 See Mossberger, Tolbert, and McNeal (2007) and Conroy, Feezell, and Guerrero (2012) for a description of how the Internet has influenced political participation.
94 "美记者质疑中国" 网络审查 "鲁炜:内容审查用词不当" December 9, 2015. Available at: <http://news.china.com/domestic/945/20151209/20903585.html>. Note that as surveillance technology improves, the crediblility of fear-based methods of censorship may increase. Yet, the other costs of censorship, including backlash, information gathering, and economic costs will still impact censorship decisions.
95 Lorentzen (2015); Meng, Pan and Yang (2014); Chen, Pan and Xu (2015); Chen and Xu (2017 _a_ ).
96 In a talk at Google headquarters, Google founder Sergey Brin claimed that the motivation behind Google's actions in China were based on his own experiences with information control growing up in the Soviet Union. Gustin, Sam, "Google's Sergey Brin Leads Charge Against Chinese Web Censorship," _Daily Finance_ , March 24, 2010, <http://www.dailyfinance.com/2010/03/24/googles-sergey-brin-leads-charge-against-chinese-web-censorship/>.
97 Millward, Steven, "Google+ Not Actually Blocked in China, Just Being Slowly Throttled," _Tech in Asia_ , June 30, 2011. Available at: <https://www.techinasia.com/google-plus-china>.
98 Hille, Kathrin, and Justine Lau, "China Unicom to Drop Google Search on Phones Using Android," _Financial Times_ , March 25, 2010, <http://www.ft.com/cms/s/0/22975096-37b0-11df-88c6-00144feabdc0.html>.
99 Levin, Dan, "China Escalating Attack on Google," _New York Times_ , June 2, 2014, <https://www.nytimes.com/2014/06/03/business/chinas-battle-against-google-heats-up.html>
100 Hoelzle, Urs, "The Google Gospel of Speed," _Google Think Insights_ , January 2012, <http://www.thinkwithgoogle.com/articles/the-google-gospel-of-speed-urs-hoelzle.html>.
101 Millward, Steven, "Baidu down, Qihoo up, Google dead: 2013 was a year of drama for China's search engines," _Tech in Asia_ , January 6, 2014, <https://www.techinasia.com/how-baidu-qihoo-google-performed-in-china-in-2013>.
102 For a list of websites currently blocked by the Great Firewall from China, visit GreatFire.org.
103 Rosen, Jeffrey, "Google's Gatekeepers," _New York Times_ , November 28, 2008, <http://www.nytimes.com/2008/11/30/magazine/30google-t.html>.
104 "State Blamed in LiveJournal Attack," _Moscow Times_ , April 5, 2011, <https://themoscowtimes.com/news/state-blamed-in-livejournal-attack-6116>.
105 Gohdes (2015).
106 Aryan, Aryan and Halderman (2013, pg. 5).
107 Knockel, Ruan and Crete-Nishihata (2017).
108 Bump, Philip, "Did Facebook Bury Conservative News? Ex-staffers Say Yes." _Washington Post_ , May 9, 2016, <https://www.washingtonpost.com/news/the-fix/wp/2016/05/09/former-facebook-staff-say-conservative-news-was-buried-raising-questions-about-its-political-influence/>. Shultz, David, "Could Google Influence the Presidential Election?" _Science Magazine_ , October 25, 2016. <http://www.sciencemag.org/news/2016/10/could-google-influence-presidential-election>.
109 See Gang and Bandurski (2011, pg. 56) for how the CCP tries to "grab the megaphone," or set the agenda, within traditional media.
110 "Ministry of Truth: Personal Wealth, Income Gap," _China Digital Times_ , February 6, 2013, <https://chinadigitaltimes.net/2013/02/ministry-of-truth-personal-wealth-income-gap/>.
111 Patell andWolfson (1982).
112 Fackler, Martin, "U.S. Textbook Skews History, Prime Minister of Japan Says," _New York Times_ , January 29, 2015, <http://www.nytimes.com/2015/01/30/world/asia/japans-premier-disputes-us-textbooks-portrayal-of-comfort-women.html>.
113 Turner, Cory, "The Great U.S. History Battle," _National Public Radio_ , February 24, 2015, <http://www.npr.org/sections/ed/2015/02/24/388443955/the-great-u-s-history-battle>.
114 Chen, Te-Ping, "Protest Over 'Brainwashing' Schools," _Wall Street Journal_ , September 2, 2012, <https://blogs.wsj.com/chinarealtime/2012/09/02/thousands-protest-hong-kongs-moral-and-national-education-push/>.
115 Grimaldi, James V., and Sari Horwitz, "Industry Pressure Hides Gun Traces, Protects Dealers from Public Scrutiny," _Washington Post_ , October 24, 2010, <http://www.washingtonpost.com/wp-dyn/content/article/2010/10/23/AR2010102302996.html>. Gup (2008, pg. 12).
116 Gup (2008, pg. 57); Masnick, Mike, "Michigan State Police Say It Will Cost $545k to Discover What Info It's Copying Off Mobile Phones During Traffic Stops," _TechDirt_ , April 20, 2011, <https://www.techdirt.com/blog/wireless/articles/20110420/01070213969/michigan-state-police-say-itll-cost-545k-to-discover-what-info-its-copying-off-mobile-phones-during-traffic-stops.shtml>.
117 Sampson, Zack, "Want to see the work Booz Allen did for the FBI? Get ready to fork over enough to buy a house," September 30, 2015, <https://www.muckrock.com/news/archives/2013/sep/30/want-see-work-booz-allen-did-fbi-get-ready-fork-ov/>.
118 McDonell, Steven, "Yangtze Ferry Disaster: Chinese authorities start righting capsized ship; death toll rises to 75," _ABC_ , June 4, 2015, <http://www.abc.net.au/news/2015-06-04/bad-weather-interrupts-work-to-find-china-ferry-survivors/6522012>.
119 Wallace (2016); Ghanem and Zhang (2014); Buckley, Chris, "China Burns Much More Coal Than Reported, Complicating Climate Talks," _New York Times_ , November 3, 2015, <http://www.nytimes.com/2015/11/04/world/asia/china-burns-much-more-coal-than-reported-complicating-climate-talks.html?_r=0>.
120 Bradsher, Keith, "Chinese Data Mask Depth of Slowdown, Executives Say," _New York Times_ , June 22, 2012, <http://www.nytimes.com/2012/06/23/business/global/chinese-data-said-to-be-manipulated-understating-its-slowdown.html?pagewanted=all>.
121 Cunningham, Maura, "Denying Historians: China's Archives Increasingly Off-Bounds," _Wall Street Journal_ , August 19, 2014, <https://blogs.wsj.com/chinarealtime/2014/08/19/denying-historians-chinas-archives-increasingly-off-bounds/>.
122 Fischer-Baum, Reuben, "Nobody Knows How Many Americans The Police Kill Each Year," _Five Thirty Eight_ , August 19, 2014, <http://fivethirtyeight.com/features/how-many-americans-the-police-kill-each-year/>.
123 Loftin et al. (2003).
124 Wong, Edward, "On Scale of 0 to 500, Beijing's Air Quality Tops 'Crazy Bad' at 755," _New York Times_ , January 12, 2013, <http://www.nytimes.com/2013/01/13/science/earth/beijing-air-pollution-off-the-charts.html>.
125 "Murder mysteries," _Economist_ , April 6, 2013, <http://www.economist.com/news/china/21575767-official-figures-showing-sharp-drop-chinas-murder-rate-are-misleading-murder-mysteries>.
126 LaFraniere, Sharon, "Lead Poisoning in China: The Hidden Scourge," _New York Times_ , June 15, 2011, <http://www.nytimes.com/2011/06/15/world/asia/15lead.html?pagewanted=all>; "My Children Have Been Poisoned: A Public Health Crisis in Four Chinese Provinces," _Human Rights Watch_ June 2011, <http://www.hrw.org/sites/default/files/reports/china0611WebInside_0_0.pdf>.
127 Cohen and Amon (2011), Human Rights Watch, pg. 25–27.
128 Human Rights Watch, pg. 25–27.
129 Human Rights Watch, pg. 21–22.
130 Boydston (1992); Apple, R.W. Jr, "War in the Gulf: The Press; Correspondents Protest Pool System," _New York Times_ , February 12, 1991, <http://www.nytimes.com/1991/02/12/us/war-in-the-gulf-the-press-correspondents-protest-pool-system.html>.
131 Jansen and Martin (2003).
132 This point is made by Edmond (2013) who shows that the more centralized control is over media, the easier it is to control.
133 Pan (2016).
134 One of Wolfsfeld (2011, pg. 23–25) five principles of political communication is that when "authorities lose control over the political environment, they lose control of the news." This is because once political control is lost, more competing sources are able to leak and report information that the media can use to contradict the authority.
135 See Jansen and Martin (2015) for a discussion of how authorities seek to mitigate censorship backlash.
136 See Mitchell, Carey, and Butler (2014) for a discussion of plausible deniability of human rights violations.
137 Ball-Rokeach and DeFleur (1976), Loveless (2008).
138 Fischer-Baum, Reuben, "Nobody Knows How Many Americans The Police Kill Each Year," _Five Thirty Eight_ , August 19, 2014, <http://fivethirtyeight.com/features/how-many-americans-the-police-kill-each-year/>.
139 Hobbs and Roberts (2016).
140 This is similar to the argument made in Baum (2003), when entertainment and news are paired they may be more easily consumed by citizens. This is also true in a censored information environment.
141 "Total Number of Websites," _Internet Live Stats_ , accessed May 2, 2017, <http://www.internetlivestats.com/total-number-of-websites/>.
142 "A Giant Cage," _Economist_ , April 6, 2013, <http://www.economist.com/news/special-report/21574628-internet-was-expected-help-democratise-china-instead-it-has-enabled>.
143 Brutlag (2009).
144 Athey and Mobius (2012).
145 Tufekci (2014).
146 "Websites to 'Close' for China's 'Internet Maintenance Day'," _Register_ , June 4, 2013, <http://www.theregister.co.uk/2013/06/04/chinas_internet_maintenance_day_shutters_sites/>.
147 Edmonds et al. (2012).
148 Nichols and McChesney (2009).
149 Gehlbach and Sonin (2014); Petrova (2011).
150 Nichols and McChesney (2009).
151 Bulkley, Kate, "The Rise of Citizen Journalism," _Guardian_ , June 10, 2012, <https://www.theguardian.com/media/2012/jun/11/rise-of-citizen-journalism>; Tufekci and Wilson (2012).
152 Carpenter (2008), Nichols and McChesney (2009).
153 See Huang (2015) for a discussion of the signaling function of propaganda.
154 Eisensee and Strömberg (2007).
155 Bandurski, David, "China Announces 'Newsy' Changes for CCTV's Official Nightly News Broadcast," _China Media Project_ , February 4, 2008, <http://cmp.hku.hk/2008/02/04/china-announces-newsy-makeover-for-cctvs-official-nightly-news-broadcast/>.
156 Zeitzoff (2017).
157 Halper, Evan, "Be Nice to Hillary Clinton Online—or Risk a Confrontation with her Super PAC," _LA Times_ , May 9, 2016, <http://www.latimes.com/politics/la-na-clinton-digital-trolling-20160506-snap-htmlstory.html>; Markoff, John, "Automated Pro-Trump Bots Overwhelmed Pro-Clinton Messages, Researchers Say," _New York Times_ , November 17, 2016, <https://www.nytimes.com/2016/11/18/technology/automated-pro-trump-bots-overwhelmed-pro-clinton-messages-researchers-say.html>.
158 Allcott and Gentzkow (2017).
159 Allen-Ebrahimian, Bethany, "How China Won the War Against Western Media," _Foreign Policy_ , March 4, 2016, <http://foreignpolicy.com/2016/03/04/china-won-war-western-media-censorship-propaganda-communist-party/>.
160 Pomerantsev (2014) describes this phenomenon in Russia.
161 Flegenheimer, Matt and Scott Shane, "Countering Trump, Bipartisan Voices Strongly Affirm Findings on Russian Hacking," _New York Times_ , January 5, 2017, <https://www.nytimes.com/2017/01/05/us/politics/taking-aim-at-trump-leaders-strongly-affirm-findings-on-russian-hacking.html>.
162 Deibert et al. (2010, pgs. 6–7).
163 Goncharov, Maxim, "The Dark Side of Social Media," _TrendLabs Security Intelligence Blog_ , December 7, 2011, <http://blog.trendmicro.com/trendlabs-security-intelligence/the-dark-side-of-social-media/>.
164 Suárez-Serrato et al. (2016).
165 King, Pan and Roberts (2017), "China's Paid Trolls: Meet the 50-cent party," _New Statesman_ , <http://www.newstatesman.com/politics/politics/2012/10/china's-paid-trolls-meet-50-cent-party>.
166 Brady (2008, pg. 95), Stockmann (2012, pg. 82); Stockmann and Gallagher (2011).
167 Pomerantsev (2014).
168 Shapiro (2016).
169 "Complaint for Injunctions, Mandatory Injunctions, Damages, Restitution, Disgorgement, Penalties, and Other Relief," _UCSF Truth Tobacco Industry Documents_ , June 5, 1996, <http://www.library.ucsf.edu/sites/all/files/ucsf_assets/wacomplaint.pdf>.
170 Muggli et al. (2001).
171 Kain (2009, pg. 700), Stone (2011, pg. 398).
172 Cook (1989, pg. 108–109).
173 Barstow, David, and Robin Stein, "Under Bush, a New Age of Prepackaged TV News," _The New York Times_ , March 13, 2005, <http://www.nytimes.com/2005/03/13/politics/under-bush-a-new-age-of-prepackaged-tv-news.html>.
174 See Roberts, Stewart and Airoldi (2016) for a quantitative analysis of media slant between Xinhua and Western news sources.
175 Shambaugh (2013, pg. 230).
176 Jansen and Martin (2003).
177 Gang and Bandurski (2011).
178 Gunitsky (2015) provides a useful overview of how regimes can take advantage of the Internet using online propaganda.
179 Roberts, Hannah, "Google made changes to its search algorithm that unintentionally made it vulnerable to the spread of fake news, sources say," _Business Insider_ , December 10, 2016, <http://www.businessinsider.com/google-algorithm-change-fake-news-rankbrain-2016-12>.
180 Baum and Groeling (2010) show that the government has an informational advantage at the beginning of a crisis, when less information about the crisis is out, but the public is still attentive to the event.
CHAPTER THREE
Censorship in China
The mechanisms of fear, friction, and flooding introduced in the last chapter illustrate the ways in which censorship can affect citizens and the media. The impact of each of these mechanisms on the spread of information varies across contexts. Some political entities, like totalitarian regimes, have opted for fear-based forms of censorship when power is concentrated in the hands of the government and awareness of control does not thwart government objectives. In other periods, authoritarian regimes have opted for friction- and flooding-based censorship that slows, but does not completely prevent, access to information, in order to reduce the probability of backlash or allow for more economic growth and internationalization. The constraints political entities face, the goals of the government, and the technological environment affect the capability of authorities to use each mechanism of censorship and the ways in which citizens will react to censorship.
In this chapter, I describe how the theory of censorship applies to modern China. First, I follow information control in China from the Mao era to the present. I describe how the Chinese state has adapted its censorship strategy to the changing goals of the state and the technological environment to balance information control with economic development, information collection, and fear of popular backlash. I follow these tradeoffs through the Cultural Revolution, reform and opening, to the turning point of the Tiananmen Square protests and the rise of the Internet. I delineate the current institutional censorship structure and strategy in China and describe how the censorship system's ability to control information consumption while minimizing the perception of control enables it to effectively prioritize information for the vast majority of the population while minimizing the economic and political side effects of censorship.
**3.1 MODERN HISTORY OF INFORMATION CONTROL IN CHINA**
In this section I describe how the Chinese government's information control apparatus has evolved since the Mao era. The modern history of China not only provides context for the current period but also illustrates the political trade-offs between censorship mechanisms. In particular, highly observable methods of censorship based primarily on fear in the Maoist era have shifted toward less complete mechanisms of censorship based on friction and flooding after reform in an effort to contain the negative side effects of fear-based censorship on the economy, government information gathering, and political backlash. Even after the Tiananmen crisis, a turning point in information control strategy in China, the government opted to strengthen methods of censorship that were less intrusive to the average citizen rather than revert to censorship based on overt control, an indication of how overt censorship methods conflicted with its goals of economic growth and internationalization. The government's emphasis on friction- and flooding-based methods of censorship has only accelerated with the advent of the Internet, and the government has developed institutions that create porous censorship, relying on citizens' high elasticity of demand for information instead of their self-discipline.
_3.1.1 Censorship under Mao (1949–1976)_
Under Mao Zedong, the Chinese government exercised extensive authority in all areas of citizens' lives.1 The Party viewed information control as a central component of political control, and Party dogma, ideology, and doctrine pervaded every part of daily routine. Propaganda teams were located in workplaces and schools to carry out work and education in the spirit of Party ideology and to implement mass mobilization campaigns.2 Ordinary citizens were regularly encouraged to engage in self-criticism—publicly admitting and promising to rectify "backward" thoughts.
Under Mao, the introduction of "thought work" (思想工作) into aspects of everyday life meant that fear played a primary role in controlling the spread of information, as each citizen was aware of political control over speech and fearful of the consequences of stepping over the line. Although the severity of punishment for transgressions varied throughout the Maoist era, everyday speech could land citizens in jail or worse—criticizing your cat (in Chinese, a word that sounds like Mao) or giving your children unpatriotic names could be considered criminal.3 Drastic punishments for both formal and informal speech and a system that encouraged citizens to report their closest friends and family members to authorities led to an environment of extreme self-censorship.4 Fear transformed personal relationships between individuals from "friendship" to "comradeship," where private information could not be shared even between friends as betrayal of friendship was encouraged and commonplace.5
Newspapers, which became entirely state-run soon after Mao took power, were seen as a mouthpiece of the state rather than a mechanism of government oversight.6 The Chinese Communist Party (CCP) adopted a Leninist view of the media: newspapers should speak for the Party. By promoting Party ideology, and encouraging mass mobilization and positive thoughts, Leninist principles encourage the media to simultaneously educate and speak for the working class. Media—including radio, newspapers, and television—were run by central, provincial, and municipal governments directly, and were guided by the Central Propaganda Department of the CCP.
During this period, China, which was closed off from the Western world in an information environment completely controlled by the state, had arguably among the most "complete" control of information a country could muster, akin to today's North Korea. Citizens were aware of political control of all of their actions in their work, personal, and public lives. Cut off from private or non-state media and with relatively little interaction with citizens from other countries, they had little choice but to obediently follow propaganda controlled by Party media for fear of stark repercussions.
But even with ideological uniformity and totalitarian control based on repression, the Communist Party and the Chinese people paid a high price for highly observable forms of censorship that controlled citizens through brainwashing and deterrence. First, citizens' and officials' awareness of political control stifled the government's ability to gather information on the performance of policies, contributing to severe problems of economic planning and governance. The Great Leap Forward, in which around thirty million people died of starvation in the late 1950s, has been partially attributed to local officials' fear of reporting actual levels of grain production to the center, leading them to report inflated numbers.7 Even after the Great Leap Forward, the inability of the Chinese bureaucracy to extract true economic reports from local officials and citizens led to greater economic instability and failed economic policies and plans.8
Such extensive control also imposed explicit constraints on economic growth. Large amounts of trade with other countries was not possible without loosening restrictions on the exchange of information with foreigners. Innovation and entrepreneurship require risk-taking, creativity, and access to the latest technology, all difficult under high levels of fear that encourage risk-aversion. Millions of people were affixed with class labels that made them second-class citizens or were imprisoned in Chinese gulags that prevented them from participating in the economy. Often those who were persecuted had high levels of education and skills that the Chinese economy desperately needed.9 The planned economy in concert with high levels of fear stifled economic productivity and kept the vast majority of Chinese citizens in poverty.
Finally, even in a totalitarian society with little contact with the outside world, government ideological control over the everyday lives of citizens decreased the government's legitimacy and sowed the seeds of popular discontent. Mao's goal of ideological purity led him to encourage the Cultural Revolution, a decade-long period of chaos in China based on the premise of weeding out ideologically incorrect portions of society, which in the process killed millions of people and completely disrupted social order. The chaos of the Cultural Revolution, combined with resentment toward the extreme ideological left in the Chinese political system that had spawned it, created openings for dissent. In 1974, a poster written in Guangzhou under a pseudonym called explicitly for reform. Similar protests followed—during the first Tiananmen Incident in 1976, thousands of people turned out to protest the ideological left, and several years later, in the Democracy movement in 1978 and 1979, protesters explicitly called for democracy and human rights, including free speech.10
_3.1.2 Censorship Reform before 1989_
When Deng Xiaoping gained power in 1978, he initiated policies of reform and opening that were in part a reaction to the intense dissatisfaction of Chinese citizens with the Cultural Revolution and prying hand of the government in their personal affairs. One of the hallmarks of Deng's transition to a market economy, which began in 1978, was the government's retreat from the private lives of citizens and from control of the media. Important leaders within Deng's government realized the trade-offs between individual control and entrepreneurship, creativity, and competition required by the market and decreased government emphasis on ideological correctness of typical citizens in China.11 In the late 1970s and early 1980s, the CCP rehabilitated those who had been political victims during the Cultural Revolution, removing class labels and releasing political prisoners, a process that enabled more than twenty million additional people to participate in the economy, many of whom had high levels of education.12 As Gold (1985) describes, the "omnipresent fear" that had been common in the Mao era lessened and personal relationships again became primarily private and economic. Citizens began to criticize the government and express dissatisfaction, privately at first, but later more publicly.
The government not only retreated from the private lives of individuals to stimulate the economy and address dissatisfaction, but also loosened its control over the media in order to reduce its own economic burden in the information industry.13 Media—which under Mao had been publicly funded—imposed large operational costs on the government. As other aspects of the Chinese economy privatized, the government began to commercialize the media to lessen its strain on government resources. The commercialization of news allowed the news media to respond to citizens' demands for entertainment and economic, international, and political news, which proved to be extremely lucrative for Chinese media companies. The lessened control also allowed Chinese media to compete with the new onslaught of international information that began to pour in as international trade and interactions increased, and Chinese media companies were able to innovate to retain market share in an increasingly competitive information environment.14
The 1980s also witnessed an increasing decentralization of the economy from the central Party planning system to the localities.15 As the government began to decentralize its control, it began to rely on the media to ensure that local officials were acting in the interests of the Party. A watchdog media could help keep local businesses, officials, and even local courts in check.16 Investigative journalism on local corruption was first encouraged by members of the central government in the 1980s, a trend that accelerated in the 1990s. While investigative journalism serves citizens by exposing corrupt politicians or lax economic practices, it also serves the state by exposing the defective aspects of its own system. Freer media in a decentralized state, it has been argued, can serve the government's own interest as much as it can serve the interests of citizens.17
Although restrictions on the press were significantly relaxed, the CCP did not completely loosen constraints on speech or the media during the 1980s, and there was significant political conflict within the Party over how free the media should be immediately following reform and opening. High-level members of Deng's government Hu Yaobang and Zhao Ziyang supported greater freedom of the press, while more conservative members such as Deng Liqun thought that more emphasis should be placed on government-controlled thought work.18 Conflict between these two groups within the Party and Deng's vacillation between them were hallmarks of CCP politics during this period. For example, Deng Liqun was head of the Central Propaganda Department from 1981 to 1985, but was removed because of his conservative stances for the more liberal Zhu Houze, who was later replaced for being too open to reform.19 This oscillation between stances on media reflected a larger internal dispute over the extent to which the Party should play a role in the press.
However, the CCP did take significant steps toward relaxing control over the flow of information in the 1980s to loosen enforcement over speech, particularly relative to the Maoist era. By 1982, the Chinese constitution began to guarantee free speech and expression for all Chinese citizens, including freedom of the press, assembly, and demonstration.20 Commercialization of Chinese newspapers began in 1979 with the first advertisement and gradually the press began making more profit from sales of advertising and less from government subsidies.21 Radio and television, which had previously been controlled by the central and provincial levels of government, expanded rapidly to local levels of government and was also commercialized.22
By the late 1980s, prominent citizens and officials were calling for even more expanded versions of free speech. Even prominent government figures, such as Hu Jiwei, former editor of the government newspaper the _People's Daily_ , called publicly for more freedom of speech, emphasizing how freedom of speech could promote political stability by revealing citizens' grievances rather than keeping them hidden from the government.23 Hu was in the midst of drafting a press law to clarify the Party's role in the press when the student protests were sparked by the death of Hu Yaobang in April 1989. These pro-democracy protests centered in Beijing's Tiananmen Square spread all over China, culminating in an internal CCP crisis and a large-scale violent crackdown on protesters on June 4, 1989, that was condemned internationally.
The June 4 crisis marked a turning point in government strategy with respect to the media and the press. Whether or not a freer media had in fact contributed to the Tiananmen protests, there was widespread consensus among Party elites after the crackdown that the loosening of media restrictions had aggravated the student demonstrations. In particular, during the months of the protests, reformers within the Party had allowed and even encouraged newspapers to discuss the protests.24 In the immediate aftermath of the crackdown on the protesters and clearing of the square on June 4, 1989, censorship ramped up quickly. A large-scale crackdown on journalists, activists, and academics reintroduced widespread fear into the private lives of influential individuals, particularly among those who had been involved in the protest events. Government officials were ordered to return to the model of the media serving the Party and expressing enthusiasm for government policies.25
_**3.1.3 Post-Tiananmen: Control Minimizing the Perception of Control**_
One might think that after an event as consequential as the 1989 protests in Tiananmen Square, the government might return indefinitely to tight control and thought work that had existed under Mao, as many leaders thought that loosened control had culminated in a direct threat to the regime. Yet a return to complete restriction of information and pervasive fear to control private and public communication between citizens was also not consistent with continued expansion and internationalization of the market economy on which the regime sought to base its legitimacy. Although the belief among government officials that free media had contributed to unrest prevented the CCP from returning to the extent of press freedom before Tiananmen Square, Deng did not return to the version of pre-reform information control that relied on fear-based control of individuals' everyday lives and instead quickly reversed the post-Tiananmen crackdown on speech.
Instead, government policy evolved toward a censorship strategy that attempted to minimize the perception of information control among ordinary citizens while still playing a central role in prioritizing information for the public. In essence, the government strengthened mechanisms of friction and flooding while for the most part staying out of the private lives of citizens. A few years after the Tiananmen crisis, the CCP returned to an apparent loosening of control, and commercialization of the media resumed in the mid-1990s.26 After Deng's famous "Southern Tour" in 1992, meant to reemphasize the economy, broader discussions and criticisms of the state were again allowed, even publicly and even about democracy.27
Even though the government did not revert to Maoist-era censorship, the government tightened its grip on the media, officials, journalists, and technology in a way that allowed targeted control: by managing the gatekeepers of information, the government could de-prioritize information unfavorable to itself and expand its own production of information to compete with independent sources. Even though the media had already undergone significant commercialization, the government strengthened institutional control over the media.28 First, the CCP created stricter licensing requirements to control the types of organizations that could report news. They also required that journalists reapply for press cards, which required training in government ideology.29 Despite extensive commercialization that created the perception among readers that news was driven by demand rather than supply, the government retained control over the existence, content, and personnel decisions of newspapers throughout the country, allowing the government to effectively, if not always explicitly, control publishing.30
Deng also strengthened control over Party propaganda and strategies of flooding. In 1990, one of the Party's leading news agencies, _Xinhua_ , was close to bankrupt.31 The government proactively changed its propaganda strategies after Tiananmen Square, adapting Western theories of advertising and persuasion, and linking thought work with entertainment to make it more easily consumed by the public.32 After Tiananmen, the CCP decided to instruct newspapers to follow _Xinhua_ 's lead on important events and international news, much as they had done with the _People's Daily_ during the 1960s.33 In the 1990s, the Party also renewed its emphasis on "patriotic education" in schools around the country, ensuring that the government's interpretations of events were the first interpretations of politics that students learned.34
**3.2CENSORSHIP OF THE CHINESE INTERNET**
During the period following the Tiananmen crackdown, China witnessed the arrival of the web in 1995, which complicated the government's ability to control the gatekeepers of information as channels of information transitioned from a "one to many" model, where a few media companies transferred information to many people, to a "many to many" model where everyday people could contribute to media online and easily share news and opinions with each other.35 Had the government been worried about complete control over the information environment, we would expect it to try to slow the expansion of the Internet within the country. Instead, China actively pursued it. The Chinese government aggressively expanded Internet access throughout the country and encouraged online enterprises, as the CCP saw these as linked to economic growth and development.36
Yet as it was pursuing greater connectivity, the government simultaneously developed methods of online information control that would allow it to channel information online. In 1994, the government issued regulations for the Internet, concurrent with the Internet's arrival in China, stipulating that the Internet could not be used to harm the interest of the state.37 Immediately, the state began developing laws and technology that allowed it more control over information online, including filtering, registration of online websites, and capabilities for government surveillance.
Descendants of the post-1989 period, the institutions that now implement information control in China for both news media and the Internet are aimed at targeting large-scale media platforms and important producers of information in both traditional and online media to make it more difficult for the average consumer to come across information that the Chinese government finds objectionable. The CCP also retains control over key information channels in order to have the capacity to generate and spread favorable content to citizens. The CCP's direct control over these information providers allows them the flexibility to make censorship restrictions more difficult to penetrate during particular periods and to loosen constraints during others. This censorship system is essentially a taxation system of information on the Internet, allowing the government to have it two ways: by making information _possible_ to access, those who care enough (such as entrepreneurs, academics, or those with international business connections) will easily bypass controls and find the information they need, but for the masses, the impatience that accompanies surfing the web makes the controls effective even though they are porous.
The Party's primary avenue for influencing information control is through the CCP Propaganda Department (中共中央宣传部), hereafter CCPPD. The CCPPD is the main institution that monitors and devises strategies both for what content should be censored and for what types of content should be disseminated across all information media in China, including the Internet, mobile phones, print publications, radio, television, art, and education materials used in schools or vocational training.38 The CCPPD can issue directives to gatekeepers in any of these media, from TV producers to Internet content providers to those in charge of education, to either censor or spread particular types of information.
The CCPPD is the center of the Chinese information control apparatus, but it delegates to a variety of smaller institutions in the state branch of the government responsibility to carry out censorship and propaganda in different media in China.39 The State Council Information Office (国务院信息办公室; SCIO) and General Administration of Press and Publication (新闻出版总署) are responsible for published media, including the licensing of publishers and Internet publishers, monitoring news and foreign journalists, and banning and pre-screening books. The Ministry of Culture (文化部) and the Ministry of Education (教育部) are responsible for regulation of the arts and education, respectively. The Ministry of Industry and Information Technology (工业和信息化部) is responsible for regulating the information technology industry. In 2011, the State Internet Information Office was established specifically to regulate content on the Internet.40 The CCP also has institutions in charge of punishing those who violate information laws. The Ministry of Public Security (公安部), the institution that overseas the police, enforces censorship laws from violations of the Internet laws to publishing licensing laws. The Ministry of State Security (国家安全部) is in charge of intelligence gathering, which includes Internet surveillance that may be used to enforce information control laws.41
In 2013, president Xi Jinping upgraded the State Internet Information Office to create a new, separate administration for regulating Internet content and cyberspace, called the Cyberspace Administration of China (锢家互联网信息办公室; CAC), run by the Central Cybersecurity and Informatization Leading Small Group (中央网络安全和信息化领导小组) and personally chaired by Xi Jinping. Xi, allegedly worried that there were too many bureaucracies in control of regulating the Internet,42 formed the CAC to streamline Internet control.43 Even though the CAC duplicates some of the responsibilities of existing ministries for information control, the new administration also placed regulation of the Internet and the State Internet Information Office directly under Xi's control.44 The CAC sought to more strictly enforce censorship online, including shutting down websites that do not comply with censorship regulations, and increasing the prevalence of the government's perspective online by digitizing propaganda. The creation of the CAC shows the importance the Xi administration has placed on managing content on the Internet.
The institutions described above use a variety of laws and regulations to control information in their respective purviews. In China, these laws tend to be relatively ambiguous to give the state maximal flexibility in their enforcement. Censorship laws disallow a wide range of political discourse, including anything that "harms the interest of the nation," "spreads rumors or disturbs social order," "insults or defames third parties," or "jeopardizes the nation's unity."45 Although, due to widespread discussion of protest events and criticism of the government online, the government cannot possibly (and likely would not want to) arrest all those who violate a generous interpretation of this law, these institutions keep a close watch particularly on high-profile journalists, activists, and bloggers, developing relationships with these key players to control content and arresting those they view as dangerous. These activities are facilitated by surveillance tools that require users to register for social media sites with their real names and require Internet providers to keep records of users' activities.46 Since Xi Jinping became president in 2012, additional laws and regulations have been written to prevent "hacking and Internet-based terrorism."47
The government keeps a much closer watch on the media infrastructure itself than on typical citizens. The propaganda department issues directives to the traditional media ordering them either not to report on content or to promote particular types of content. Online news portals are not allowed to post news that is not from state news outlets unless they themselves have a journalistic license from the state.48 The CCPPD issues censorship directives to social media companies to filter content, ordering them to delete individual posts that are about particular topics, most frequently topics related to collective action, activists, pornography, or criticism of censorship.49 Some websites automatically filter content by keywords, preventing individuals from posting anything with a sensitive term before the post is reviewed by censors. Online search providers, such as Baidu and Yahoo, omit particular websites from being listed when a user searches for sensitive words. This process, known as search filtering, also occurs within social media firms, such as Sina Weibo, which omits references to sensitive trending topics and disallows users from searching for media posts that contain particular terms.50
Not only can the government order traditional media to print particular articles and stories, but it also retains flooding power on the Internet. The Chinese government allegedly hires thousands of online commentators to write pseudonymously at its direction.51 This so-called Fifty Cent Party, described more in chapter 6, is an army of paid Internet commentators who work at the instruction of the government to influence public opinion during sensitive periods. In other work, we have shown that these propagandists are largely instructed to promote positive feelings, patriotism, and a positive outlook on governance and are unleashed during particularly sensitive periods as a form of distraction.52 This is largely in line with President Xi's own statements that public opinion guidance online should promote positive thinking and "positive energy."53 Other scholars have posited that they are also sometimes instructed to defame activists or counter government criticism.54
In the case of international websites, where the government does not have the jurisdiction to directly control the media, the Great Firewall,55 officially referred to by the Chinese government as the "Golden Shield Project" (金盾工程), blocks particular websites from IP addresses within China, preventing Chinese citizens from accessing websites the government deems objectionable.56 The Great Firewall is sophisticated enough to censor particular pages or images from being accessed in China and also includes surveillance capabilities by tracking requests from computers inside China to foreign websites.57 It can also throttle websites, making them slower, instead of censoring them outright.
Because the government focuses control on gatekeepers of information, rather than individuals, from the perspective of an ordinary citizen in China the information control system poses few explicit constraints. Street stands sell tens of different types of newspapers, all competing for attention by pandering to the consumer. It is not obvious what stories have been redacted from the newspapers and which the editors have been ordered to print. Internet access is widely available in China, with a flourishing social media environment where even vitriolic criticisms of the Chinese government are common. As I will show in chapter 4, many Chinese citizens are not scared by censorship. Censorship does not interfere with most citizens' daily lives or perceived access to information. In fact, many citizens do not even know that certain types of censorship exist.
For those who are aware of censorship and are motivated to circumvent it, censorship poses an inconvenience rather than a complete constraint on their freedom. Even though some foreign websites are blocked, they can be accessed with a VPN. Foreign newspapers and banned books can be bought in underground bookstores or in Hong Kong. Social media posts can be written with slightly different keywords to evade censors who filter sensitive terms.58 For the well-educated, well-to-do elite, censorship is annoying but rarely makes information impossible to obtain. The porous nature of censorship allows those who really care about finding the information to access it, while effectively prioritizing information for those who are less interested.
That information _can_ be accessed sidesteps many of the trade-offs that were present in the Maoist era. The perception that information control in China is porous means that many citizens believe that they consume relevant information and this in part prevents widespread backlash that more repressive forms of censorship could create. Pervasive criticism of the government online and in the international and domestic media also provides the central government with access to information about its own performance. The technical ability to access information imposes fewer constraints on the economy than complete inaccessibility of information would, as companies and students can evade censorship to access information that is central to their education, business, or research.
Yet, the system provides effective control of the typical person. Typical citizens in China can access almost any information that they want to access, if they can find the time and the resources. But because they are uninterested in politics, busy, and often unaware of the existence of alternative information, most people in China do not go out of their way to find information that is difficult to gain access to. Instead, they consume information that is relatively easy to find, often that which is prioritized by the information control infrastructure of the government. The flexibility of the censorship system drives a wedge between technologically savvy, politically interested individuals who easily circumvent censorship restrictions and those who are more affected by them.
This is not to say that the Chinese government has perfected information control, or that its current strategy is without its own trade-offs and risks for the government. First, any type of censorship creates a drag on the economy. Many companies operating in China have reported that information friction like the Great Firewall has severe negative impacts on their businesses because it limits access to technologies.59 Chinese websites are burdened with hiring censors, which makes them less competitive in international business. For average citizens, even though censorship is less observable than it might be under censorship that employs high levels of fear, it is not without footprints. When citizens run into traces of censorship, it can decrease the legitimacy of the government regime and cause backlash, as I will show empirically in the next chapter. In this way, the government censorship regime can undermine its own stated purpose.
For the government, the strategy of porous censorship also entails risks precisely because it is possible to sidestep. Porous information control relies on citizens to be indifferent enough to be satisfied with the information the government prioritizes. Certain types of events—such as financial crises, natural disasters, or government scandals—may create sufficient incentives for citizens to search out and find information that they otherwise might not come across. Moments when enough citizens are motivated enough to learn how to outsmart government media control are those when the information management strategy comes under the most pressure.
However, the Chinese government has come a long way in managing information while avoiding many of its costs. Comparing the information management strategy under Mao, which micro-managed every aspect of citizens' lives, to today information management is less visible and less costly. Yet even while minimizing the perception of control, the government is able to wield significant influence over which information citizens will come across. I turn to measuring how censorship influences the spread of information in the subsequent chapters.
1 Walder (1988, Chapter 1) describes patterns of authority under Mao.
2 See Shambaugh (2007) for a summary of propaganda under Mao and Unger (1982) for a detailed description of education under Mao.
3 Link (2002).
4 Shirk (1982, pg. 130–135).
5 Vogel (1965).
6 Zhao (1998 _b_ , pg. 19).
7 Li and Yang (2005).
8 Huang (1994).
9 Naughton (1996, pg. 89–91).
10 Teiwes and Sun (2004); Brodsgaard (1981).
11 Gold (1985).
12 Naughton (1996, pg. 89–91).
13 Lynch (1999, Chapter 2).
14 Stockmann (2012, pg. 50–59).
15 Naughton (1996, Chapter 3).
16 Liebman (2011).
17 Zhou (2000).
18 Brady (2008, pg. 40).
19 Brady (2008, pg. 40–41).
20 Zhao (1998 _b_ , pg. 44).
21 Shirk (2011, pg. 1–9).
22 Zhao (1998 _b_ , pg. 55).
23 Zhao (1998 _b_ , pg. 36). "没有新闻自由就没有真正的安定," 世界经济导报, http://www.64memo.com/disp.aspx?Id=9075&k=%E5%A4%A9%E5%AE%89%E9%97%A8.
24 Gang and Bandurski (2011, pg. 71–72), Brady (2008, pg. 42).
25 Zhao (1998 _b_ , pg. 45–46).
26 Zhao (1998 _b_ , pg. 47–50).
27 Ding (2001, pg. 33).
28 Brady (2009).
29 Stockmann (2012, pg. 60).
30 Stockmann (2012, pg. 52).
31 Brady (2008, pg. 113).
32 Brady (2008, pg. 73).
33 Brady (2008, pg. 113).
34 Zhao (1998 _a_ ).
35 National Committee for Cadre Training Materials (2011, pg. 5).
36 Hong (2014, pg. 11).
37 "中华人民共和国计算机信息系统安全保护条例," state Council, http://www.lawinfochina.com/display.aspx?lib=law&id=12136&CGid=.
38 Shambaugh (2007).
39 Because there are many institutions with authority over various aspects of the media, in practice this control can be redundant and fragmented. See Yang (2013) for discussion.
40 Buckley, Chris, "China Sets Up Agency to Tighten Grip on Internet," _Reuters_ , May 4, 2011, <http://www.reuters.com/article/us-china-internet-idUSTRE7436SA20110504>.
41 See Wang and Minzner (2015) for a description of the recent history of China's security apparatus.
42 "习近平谈关于加快完善互联网管理领导体制,"人民网, _People's Daily Online_ , November 15, 2013, <http://politics.people.com.cn/ท/2013/1115/c1001-23559689.html>.
43 Alshabah, Nabli, "Information Control 2.0: The Cyberspace Administration of China Tames the Internet," _Merics China Monitor_ , September 16, 2016, <http://www.merics.org/en/merics-analysis/analyseschina-monitor/information-control-20.html#c15313>.
44 Lam (2015).
45 "Falling Short: Appendix II: Media Law in China," _Committee to Protect Journalists_ <https://cpj.org/reports/2008/06/12ii-2.php>.
46 Deibert et al. (2010, pg. 465).
47 Lam (2015), see: <http://www.npc.gov.cn/npc/xinwen/2016-11/07/content_2001605.htm>, <http://www.gov.cn/jrzg/2012-12/28/content_2301231.htm>.
48 Wang and Faris (2008).
49 See MacKinnon (2009); King, Pan and Roberts (2013, 2014); Cairns and Carlson (2016); Miller (2017); Knockel, Ruan and Crete-Nishihata (2017) for a discussion of what is censored.
50 For a discussion of search filtering and keyword blocking see: Ng (2013); Knockel et al. (2015).
51 Han (2015).
52 King, Pan and Roberts (2017).
53 Bandurski, David, "The CCP's Positive Energy obsession," _China Media Project_ , December 15, 2015, <http://cmp.hku.hk/2015/12/15/chinas-obsession-with-positive-energy/>.
54 Han (2015); Bandurski (2008); Miller (2016).
55 The Great Firewall is under the purview of the Ministry of Public Security, but the implementation of the Firewall is overseen in conjunction with the Ministry of Industry and Information Technology. The CCPPD, public security, and other government agencies provide directives to these ministries with instructions on which sites should be blocked; see Tai (2014).
56 For a list of websites blocked in China, visit greatfire.org.
57 Tai (2014, pg. 68–69).
58 MacKinnon (2008); Hiruncharoenvate, Lin and Gilbert (2015); King, Lam and Roberts (2017).
59 Zimmerman, James, "Censorship in China Also Blocks Business Growth," _Wall Street Journal_ , May 17, 2016, <http://www.wsj.com/articles/censorship-in-china-also-blocks-business-growth-1463504866>.
CHAPTER FOUR
Reactions to Experience with Censorship
On December 20, 2010, Fang Binxing, the architect of the Great Firewall, opened an account on Sina Weibo. Within minutes, Sina Weibo users began following and commenting on Fang's wall, deriding the censorship mastermind and encouraging users to "surround and watch" (围观), or subject him to public scrutiny.1 Despite high-speed censorship of their comments, these users got their point across, with comments such as: "Old dog why don't you die?" (老狗何不去死?), "Destined to be nailed to the history of disgrace" (注定被钉在历史的耻辱上), and "SB" (short for a Chinese curse word). Ridiculed, Fang deleted his account and the comments with it within three hours.2
The history of the Internet in China is littered with examples of simmering public resentment against censorship boiling over publicly. In January 2013, journalists from the newspaper _Southern Weekly_ took to the streets to protest what they saw as excessive censorship of a New Year's editorial. The journalists posted their concerns with censorship of the newspaper on their Sina Weibo accounts, prompting petitions advocating more freedom of speech that were circulated and signed by scholars, students, lawyers, and intellectuals.3 In another example in 2013, the Chinese government began blocking GitHub.com, an American-owned website that allows computer programmers to share code. Programmers around China protested the block online, led by former China Google executive Kai-Fu Lee, whose Weibo post concerning the block was forwarded 80,000 times: "GitHub is the preferred tool for programmers to learn and connect with the rest of the world. Blocking GitHub is unjustifiable, and will only derail the nation's programmers from the world, while bringing about a loss in competitiveness and insight."4
These examples are only a few among many in which awareness of censorship in China has inspired widespread backlash against the government. Censorship is costly to authoritarian regimes because it can create anger and reduce trust among the broader population, create economic inefficiencies, and complicate government efforts to collect information from the public. These costs have been exacerbated by the information age because as more people participate online, more people have firsthand experiences with censorship. Media training manuals provided to government officials in China emphasize that one of the main differences between online media and traditional media is that anyone can participate: "The capacity of individuals and societal organizations to broadcast has strengthened, the number of people broadcasting information has increased... everyone now has a microphone." The expanded number of people in the public sphere means that the structure of communication "has changed from 'few to many' to 'many to many,'" which means that a "small, mundane problem can quickly escalate into a political problem."5 Autocrats have historically retained strict control over the few in traditional media, but controlling the many in the digital age is more difficult.
Perhaps due to evidence of self-censorship of journalists, activists, and academics,6 the academic literature has suggested that self-censorship may be the primary form of government control over the ordinary user of the Internet.7 Fear, unknown threats, and arrests of typical citizens might persuade citizens to restrain their speech online. The implication is that if the government can be more threatening and make censorship more constraining, it could increase control of information, even in an online environment.
However, these common conceptions of the functioning of censorship in China have overlooked the risks that widespread repression entails for the government in the information age. Deterrence must be observable to work and the expansion of people involved in public discourse has reduced the credibility of government threats. Instead, potential backlash against censorship can create unrest that reduces the legitimacy of the regime. Thus, increases in observable online constraints for everyday users can counterintuitively _decrease_ rather than increase control of information.
Aware of these costs, the government pursues two different censorship strategies: one for typical Internet users and another for activists, public opinion leaders, and journalists. For typical Internet users, the government uses the strategy of porous censorship to walk the fine line of controlling information while preventing censorship from backfiring. The goal of this strategy is to divert information with friction and through distractions in the form of flooding. This form of control generally does not make the information off-limits—typically it is still _possible_ to access. For individuals who are intent on accessing information, porous censorship allows them to do so, limiting backlash and allowing the government to plausibly deny interference. For those who are indifferent, the logic of porous censorship relies on citizens' busy schedules, relative indifference to politics, and numerous alternative sources of information and entertainment in Chinese media, wagering that, for the most part, citizens are not willing to go out of their way to gather information about politics.
The government reserves more traditional fear-based censorship strategy to target journalists, public opinion leaders, and activists. Government training manuals show that the government realizes that traditional media and a few public opinion leaders can control the online agenda in China. During public crises, the government advises focusing control over the news media and key online social media stories.8 This, in turn, creates a secondary friction and flooding effect on the average Internet user by decreasing the prevalence of objectionable information and steering the conversation away from topics that are less desirable for the government without creating general awareness of censorship. This two-pronged strategy of censorship drives a wedge between key public opinion leaders and the public. Because these different groups experience different censorship tactics, the public is more likely to believe that key public opinion leaders are similarly unconstrained in posting and accessing media online.
In this chapter, I first describe China's two-pronged strategy of censorship, in particular how typical Internet users are less likely to be targeted with fear-based censorship than journalists, activists, and key opinion leaders. Using surveys, online experiments, and a unique set of social media datasets, I show that, despite government efforts to signal to the public the consequences of spreading sensitive information online, fear-based methods of censorship do not deter much of the large online population in China, which is accustomed to regularly reading and discussing political information. Signals that particular information is off-limits do not persuade online users to avoid the topic. In fact, for ordinary citizens who consume and produce political information online in China, experience with censorship and awareness of censorship negatively affects their opinion of the state and may even make them more likely to read and write about topics that are viewed by the state as more sensitive, as they are alerted to topics the Chinese government deems dangerous.
**4.1 CHINA'S TARGETED CENSORSHIP STRATEGY**
As Internet use has expanded in China, the Chinese government has made efforts to minimize its potentially destabilizing political impact. Laws that govern Internet activity specify that a wide variety of information is not allowed to be written or re-shared on the Internet, including information that "harms the interest of the nation," "spreads rumors or disturbs social order," "insults or defames third parties," or "jeopardizes the nation's unity."9 These regulations are sufficiently ambiguous that they give the state considerable leeway on the types of people they can punish for online behavior.
These laws apply to all Internet users, but journalists, activists, academics, and public opinion leaders are more likely to be punished than typical users of the Internet, even if they write similar information. In the recent crackdown on online rumors, many of those arrested were high-profile Internet users who hold disproportionate sway over the online community or were already involved in offline activities the CCP would consider subversive. Some Internet laws even state that information that is re-shared many times is more likely to be considered criminal by the government, thereby targeting higher-profile social media users whose posts are more likely to be re-shared.10 In one recent prominent example, Charles Xue, a Chinese businessman who was famous for his liberal commentary on social media and had more than ten million social media followers, was arrested in August 2013 and detained for almost eight months. Three weeks before his arrest, Xue and other prominent microbloggers attended a meeting where they were warned about the "social responsibilities of Internet celebrities," which included spreading positive messages on Weibo instead of negative messages. The government accompanies arrests with efforts to discredit these high-profile individuals—Charles Xue was arrested for and later confessed publicly to soliciting prostitutes.11
The government targets high-profile journalists, academics, and social media users in a way that obscures fear-based censorship to the rest of the online community. Individuals who have been subject to government censorship report experience similar to that of Xue's of being "invited to tea" by government officials, where they are sometimes asked for information, told to write or not write about certain topics, and offered threats or rewards for particular types of behavior. These private, targeted conversations are likely more influential because they are more credibly threatening—if the government is willing to sit down to tea with you, they not only care sufficiently to do something about your behavior, but can track you down. Because these conversations are private, they can also be concealed from a larger public who might object to such repression. Although even previously detained microbloggers sometimes show surprising persistence in continuing to write about topics that are off-limits,12 we would expect that targeted individuals will be more likely to self-censor than those who do not feel singled out by the government.
However, for most people who violate these broad Internet laws in China, there is no punishment at all—thousands of social media users daily write content that could be considered in violation of these laws, and the vast majority of them are not punished. Fear for the typical Internet user is the knowledge that such enforcement _could_ happen at any time, though for all practical purposes it is extremely unlikely. The Chinese government's own training manuals on how to control public crises provide evidence that it strategically uses targeted fear-based censorship of the media to control public crises on social media, rather than attempting to micromanage typical Internet users. Aware that they cannot control all social media users, the government warns against ignoring or disrespecting average citizens' opinions and encourages authorities to avoid ignoring the issue.13 Focusing on controlling traditional media and public opinion leaders, who can set the agenda, rather than on controlling typical social media users, can lead the conversation away from what is off-limits, but also will avoid incurring the large costs of widespread censorship documented in this chapter.
In recent years, China has sought to strengthen the credibility of the enforcement of these censorship laws for Internet users. First, to improve surveillance of Internet users, the government has begun to require that users provide official identification when they register for social media accounts. In 2012, Sina Weibo, China's largest microblogging platform (similar to Twitter), began to require real name registration for users.14 In 2015, real name registration became required on all Chinese social media sites.15 Real name registration in combination with more invasive surveillance technologies make government efforts to hold Internet users accountable for their online actions more credible, since users can more easily be tracked and identified. These types of surveillance may increase in the future as the government is experimenting with more detailed tracking of users and online credit scores.16 Better surveillance could allow the government to more credibly use fear on China's large Internet population.
Second, the government has increased the frequency of arrests of social media users. In 2013, under the new President Xi Jinping, the Ministry of Public Security conducted a campaign cracking down on online "rumors" and other information deemed illegal in China. Although many of the targets were famous microbloggers as described above, the crackdown also involved some typical Internet users who were thought to be spreading misinformation.17 Exact estimates of the number of people arrested are unknown, but many suspect that during the crackdown thousands of people were arrested for their online writing.18
However, even with real name registration and increasing numbers of arrests, enforcement against all users who violate China's broad Internet laws is difficult for the government, and this difficulty creates protections for Internet users on the Chinese web who push the limits of censorship. Because millions of users share political information with one another every second on the Chinese Internet, users are—at least for now—shielded by the masses from being the target of the government's censure, even if the government has perfect information about what each individual is writing, reading, and sharing online. Later in this chapter, I describe the behavior of typical Chinese Internet users when they experience censorship, either when they are consuming online content, or when they are posting. I find that signals of censorship do not deter the production or consumption of information, as one might expect if Internet users were indeed frightened by Chinese Internet repression. Instead, such experience undermines the Chinese government's online censorship laws and can inspire more criticism and discussion of sensitive issues online in China. If the Chinese government were to engage in widespread fear tactics online in the future, it would also have to deal with the very real possibility of widespread backlash.
**4.2 THE COSTS OF OBSERVABLE CENSORSHIP**
In this section, I show that users who experience censorship are more likely to be angered or intrigued by the experience than to be fearful of government reprimands. In the empirical tests below, I study instances where Internet users run into observable censorship on the web. Measuring individual reactions to experience with censorship is difficult, as it requires simultaneously measuring censorship and observing how citizens respond. First, I study pairs of similar social media users who shared identical posts, but where one was censored and the other was not. I find that experiencing censorship does not cause users to avoid the censored topic; if anything, they interact more with it—writing more about the topic and complaining about government censorship policies. I also provide survey evidence that corresponds to this behavioral evidence: social media users do not report being fearful after having a social media post removed by the censors, but instead report anger or indifference.
Next, using a lab experiment conducted in China, I study what happens when consumers of social media observe censorship online. I find that when consumers click on a link that redirects to a censored error page, they seek out more information about the topic and are subsequently less likely to support government censorship policies.
The following studies do not provide evidence that self-censorship in China does not exist. Certainly some individuals fear to write about particular topics online because of the risk of repercussions. What the findings in this chapter do show is that when censorship is obvious to citizens, they may push back against the government. The results suggest that draconian methods of censorship could negatively affect government legitimacy. They also explain why the government primarily targets gatekeepers of information, such as journalists, activists, and users with many followers, but for the most part has avoided directly threatening the typical user with fear-based methods of censorship.
**4.3 MATCHED COMPARISON OF CENSORED AND UNCENSORED SOCIAL MEDIA USERS**
How does experience with censorship affect social media users' perceptions of the government? One way that social media users may experience censorship is when their own social media posts are deleted by government censors. Government directives issued daily to social media sites indicate what types of topics should be deleted from social media websites. Censors at the social media company then delete individual posts in accordance with the directive. A user who experienced censorship by having their post deleted receives a signal of the kinds of topics that the government currently considers too sensitive for the Internet.
If social media users believe that they could have their account blocked, be the subject of a police visit, or at worse jailed if they continue writing on that topic, then they might take the signal of censorship as an indication that they should avoid that topic in future writings. Experience with censorship could cause a chilling effect if the user perceives censorship as a wider signal of a government crackdown. If, however, social media users do not feel that punishment of continued writing on the censored topic is likely, censorship could be a signal of government weakness, of what the government fears, or that the government objects to social media users' opinions, and it may instead motivate the social media user to write more on the topic.
_4.3.1 Research Design and Social Media Data_
We cannot simply compare users who were censored to users who were not censored to understand the impact of censorship. Users who are censored write about different topics and have different opinions than users who are not censored, and therefore we would likely be measuring the differences between those groups of people rather than the impact of censorship. A perfectly scientific approach to test how users respond to experience with censorship would be to conduct an experiment that randomly assigns censorship to a set of individuals to see how they react. A randomized treatment and control group would ensure that, on average, there were no other differences between those who were censored and those who were not censored that might explain the response to censorship. However, randomly administering censorship on the Internet for research purposes without the consent of users would be unethical to implement.
With observational data, however, we can approximate such an experiment. I find instances where two social media users write identical or nearly identical posts, where one was censored and one was not.19 I ensure that they are similar in all other observable respects, including how frequently they post and how frequently they are censored. Using this "matched pair design" allows me to compare the subsequent writings of the matched pair to estimate the impact of censorship.
To find matched social media users, I use a dataset of Weibo posts that was collected and made available by Fu, Chan and Chau (2013). Fu, Chan and Chau (2013) created a list of Weibo users with more than one thousand followers using the Sina Weibo User Search Application Programming Interface (API), and then followed these users throughout 2012. Their project, Weiboscope,20 provides data for 14,387,628 unique users during this period.
The Weiboscope project collected the microblogs from each of the users' timelines in almost real time—before the censors had a chance to remove them—and also revisited each users' previous posts at least once a day, and frequently more than once a day, to record whether the post had been censored. If the post was removed, the authors documented the last time the message was seen before it was removed.21
The Weiboscope data provides an almost ideal dataset to test netizen reactions to censorship because many users were followed over a relatively long time. The approximate date and time of censorship is known, revealing the approximate time that users were "treated" with censorship.
To find matched users, I first preprocess the entire dataset by removing all non-textual data from the microblogs, including emoticons and user names. After preprocessing, I find all pairs of posts with identical text, but with different censorship statuses. I require that matches have more than fifteen characters to ensure that two identical posts do not have different meanings because of their context, such as posts that only include short context-dependent phrases such as "reposting Weibo" (转发微博). To further ensure that the posts were written in the same context, I require that the matched posts were posted on the same day.
The removal of social media posts by the censors happens very quickly in China, as reflected in Weiboscope data. The data indicate that 14 percent of the censored posts were not seen after they were first collected; that is, they were deleted before the automated scraper had time to return to them. Half of the censored posts were last seen only a half day after they were first posted and then were removed from the web. More than 80 percent of censored posts were last seen less than two days after they were written. However, for a few posts, censorship occurs significantly after the posts were written. Since I want to study the reaction of Weibo users to censorship and Weibo users are more likely to notice censorship the more quickly it happens, I remove all matches where the censored post had not yet been censored more than two days after posting.
I ensure that the matched users wrote posts with similar sensitivities in the past. I calculate the censorship rate for each user before they wrote the matched post. Using matching,22 I remove pairs where the overall historical censorship rate and censorship rate in the most recent ten-day time period are very different for the two users. This ensures that users in my final group of matched posts will both have experienced similar amounts of censorship overall and will have similar recent experiences with censorship.
In addition to censorship history, I ensure that other attributes of the paired users are similar. Since verified users, who are typically more famous and have more followers, may be more salient to censors than users who are not verified, I only consider matches where matched users have the same verification status. The Weiboscope data also indicate whether the Weibo post contains an image: I match on the inclusion or exclusion of an image in the matched post.
With these restrictions, I find 174 matched posts, or 87 pairs of posts, one censored and one not censored, written by similar users. Matches appear in each month of 2012. Within each match, the users wrote identical posts (usually both reposted the same post) on the same day, have similar censorship histories, and have identical verification statuses. The matched posts discuss topics that we would expect to be censored during this time period, including posts about activists and human rights lawyers, posts describing the corruption of top leaders, posts describing land demolitions and subsequent protests, posts regarding protests in Hong Kong, posts mentioning the leaked online sex video showing official Lei Zhengfu, and many posts that talk about the removal of then Chongqing Party Secretary Bo Xilai from the CCP. Some matched posts are also complaints about censorship, ranging from complaints about censorship of investigative journalism, to complaints about the deletion of microblogs, to complaints about the censorship of scenes of nudity in the screening of the film _Titanic_ in 3-D.
_4.3.2 Do Weibo Users Persist after Censorship?_
Using these 174 matches, we are interested in analyzing how the censored user's behavior after being censored differed from the behavior of the similar user who posted the same content but was not censored. I test their reaction to censorship using four metrics: (1) How similar are their subsequent posts to the censored post? (2) How likely are they to use sensitive words after censorship? (3) How likely are they to complain about censorship after being censored? (4) How likely are they to be censored after the matched post?
Similarity of Posts to Censored Post
First, I test whether the users persist in talking about the censored topic. Do censored users take government censorship as a signal that they should avoid a topic, declining to write further about that topic and self-censoring? Or do they take government censorship as a signal of the topic's importance and persist in writing about the topic more than their uncensored counterparts?
Because the posts from the matched pairs cover a wide range of topics, we cannot measure simply whether the censored group or uncensored group talks more about one particular topic after censorship. What we want to measure is how _similar_ the posts they write after censorship are to the matched post. If the censored group self-censors, we would expect them to avoid the topic of the particular post of theirs that was censored. If they rebel, however, we would expect them to continue to write about it.
To measure similarity between the text of the censored post and the subsequent posts, I use a measure of string similarity that estimates the number of overlapping pairs of characters between each of the users' subsequent posts and the matched posts.23 String similarity of 1 means strings are identical, string similarity of 0 means they have no overlapping sets of two consecutive characters. The more overlapping characters the two strings have, the higher their similarity.
I chose this method because it is very simple and transparent, and it corresponds with words in Chinese, which are typically two characters long. It also is correlated with censorship—having a higher string similarity with the censored post increases the likelihood the post is missing. For each user in the matched dataset, I measure string similarity between the matched posts and each post that the user wrote during the period from ten days before to ten days after the match. Figure 4.1 plots average string similarity for matched censored and uncensored users by time before and after the post. Although the censored group does talk slightly more similarly to the matched post than the control before censorship, this, difference increases after censorship. The censored group writes more similarly to the matched post than the uncensored group does after censorship, even accounting for the small differences between the two groups before.24 Being censored, at first glance, seems to inspire as much or more writing _similar_ to the censored topic rather than a trend away from that topic.
Use of Sensitive Words
String similarity is a useful metric in this context because short posts such as microblogs that display high levels of similarity are also typically quite similar to each other in terms of topical content. However, it could be that the similarity that we are measuring is not related to the topic of the matched post, but rather the similarity is because of the ancillary words within the post. To ensure that we are measuring similarity of the _sensitive_ content of the matched post, I estimate the words that most predict censorship in the matched posts by taking the one hundred words that are most related to the matched posts in comparison to a sample of uncensored posts written by the users.25 These words are highly predictive of censorship and include such words as "punishment," "representatives," "miscarriage of justice," "stir up chaos," and "overthrow." For each post the users wrote in the ten days before and after censorship, I measure the number of times the user mentions a word within this list and divide by the post length to standardize across posts.
Figure 4.1: String similarity to matched post (censored users - uncensored users). Censored users are as or more likely to write posts similar to the matched post after censorship than uncensored users.
Figure 4.2: Sensitive words in posts, normalized (censored users - uncensored users). Censored users are as or more likely to use sensitive words after censorship than uncensored users.
Figure 4.2 plots the average proportion of each post that is one of these one hundred words. Whereas on average the censored and uncensored groups use the words similarly before the matched post, the censored group is as or more likely to use these words after censorship, even though they had received a signal that these words are off-limits.26
Complaints about Censorship
Does the censored user complain more about censorship than the uncensored user? To study this, I sampled three thousand social media posts from the users on the second, third, and fourth days after the matched post. For each post, I recorded whether the post contained a complaint about censorship.
Censored users were twice as likely to complain about censorship after the matched posts than uncensored users—1 in 62 of the posts of the censored users complained explicitly about censorship, whereas only 1 in 100 of the posts written by uncensored users complained about censorship. This provides evidence that the censored users did indeed notice the censorship, as they talk about that experience in their subsequent posts. It also indicates that they feel more rather than less empowered to object to censorship directly to their censors after experiencing censorship.
Censorship Rate after Matched Post
Even though the censored and uncensored groups have identical censorship rates before the matched posts, the censored group was more likely to be censored in comparison to the uncensored group after writing the matched post. Figure 4.3 shows the before and after missingness of the posts. Although part of this effect may be due to the topical persistence of the censored group, these differences in censorship after matching may be too stark to be completely explained by the fact that the censored group tends to continue talking about the topic more than the uncensored group. I expect that the differences in censorship rates are partly due to increased attention by the censors, who may be flagging users after censoring them. This makes the results even more striking, since users are not only persisting after being censored, but are persisting in talking about the same topic in the face of increased scrutiny by the censors.27 It also suggests that users who have been censored in the past subsequently become more targeted by the censors, providing evidence that censorship focuses on those who the censors believe are more likely to disobey the rules.
Figure 4.3: Censorship rates (censored users - uncensored users). Censored users are more likely to be censored after censorship than uncensored users.
_4.3.3 Case Studies "Weibo is Democracy!"_
Why would the censored group persist in writing about the censored topic when they have just received a signal that the topic is off-limits and they are under heightened scrutiny by the censors? A closer look at a few censored individuals provides some insight into the thought process of the censored users. Take the match between User Zhang and User Liu.28 User Zhang and User Liu both shared identical posts on the same day, "voting" for a decrease in censorship of the Internet. User Zhang was, ironically, censored, while User Liu was not. Whereas User Zhang posted about the Internet only four times in the ten days before the post, he has twelve posts in the ten days after censorship that mention the Internet, six of which occur the day after he was censored.
Why would User Zhang be so relentless? User Zhang opposes censorship so strongly that he will do everything to defy the censors. He sees censorship as an indication that the government is trying to cover up corruption—he sees censorship as the direct result of corrupt officials. In the few days after being censored, User Zhang shares:
"//@经济抠门男: 这是贪官们怕微博把他们都给爆料了吧,想挡住大伙 的嘴, 徒劳"
"It's because the corrupted officials are worried that Weibo will spill all their (negative) secrets, so they try to shut everyone up, it's useless."
"网络反腐 今天我们贩了,但明天我们一定会迎来胜利. 都贪汗,你 不贪,你就没办法在官场混!"
"Online anti-corruption, although we lost today, but we will have the victory tomorrow. Everyone is corrupted, if you don't, you can't survive in the government."
User Zhang's match, User Liu, also frequently writes about corruption and the rule of law in China. User Liu also opposes censorship, having written an identical post to User Zhang "voting" for a decrease in censorship on the Internet. However, User Liu is not reminded of how much he hates censorship because, unlike User Zhang, his complaint was not censored. User Liu therefore writes fewer posts about the Internet in the period directly following the match.
In another example, User Zhu is censored when he posts the following text about Lei Zhengfu, the Party official who was caught on video with an eighteen-year-old woman, suggesting that users watch the pornographic video:
"【重庆霄政富同志,请辟谣!】 资深调查记者举报,重庆市北砖区委 书记雷政富 (正厅级), 与重庆市开县赵家镇18岁女青年赵红霞发生不正 当男女关系后,又动用权力将赵红霞抓捕,试图封口. 由于本人水平有 限, 无法识别图片真伪,群众的眼睛是雪亮的,诚邀广大网友一起鉴定!"
"Chongqing Comrade Lei Zhengfu, please tell me it isn't so! A senior investigative reporter reported that the Chongqing Beibei District Secretary Lei Zhengfu (at the department level), had an improper relationship with an Chongqing city, Kaixian county 18 year old woman Zhao Hongxia and used his power to arrest Zhao Hongxia to try to seal it. One person's ability is limited, can't validate the authenticity of the picture, everyone's eyes are good, we invite all the Internet users to try together!"
Despite being censored, User Zhu writes more about the scandal after censorship than before, outpacing his uncensored match who shared the identical post, User Li. Why is User Zhu so persistent? In User Zhu's own post, written after he was censored:
"【微博就是民主! 】 微博 (Twitter中国版本) 的出现,彻底颠覆了中 国的舆论格局,每个人都能发出声音,瞬间无限放大,传统媒体全跟在微 搏屁股后面做文章. 微博本身具有 "自净" 功能谣言必将被真相戳破,别 封杀微博,赋予网友分自由,政府不用费心政改,媒体也不用呼吁新闰自 由,只是,纪委要下岗了"
"Weibo is democracy! The appearance of Weibo (the Chinese version of Twitter) completely overturns the public opinion structure in China, everyone can express their opinion, and it will escalate instantly and infinitely, all the traditional media are following Weibo to make stories. Weibo itself has a 'self-cleansing' function, myths will be busted by the truth, so don't block Weibo, give the Internet users enough freedom, the government does not have to worry about revising the law, and the media don't have to appeal for freedom of information, the Commission for Discipline Inspection can step down."
In essence, the act of censorship has been interpreted by this user as weakness—social media expresses truth and to censor social media is to hide the truth. In the ten days before the matched post, User Zhu's censorship rate was zero percent, the same as his match User Li. However, in the ten days after the matched post, User Zhu's censorship rate skyrocketed to 33 percent, while User Li's increased only to 4 percent. Despite being flagged by the censors, User Zhu persists, continuing to talk about the political topics he believes are important and criticizing censorship itself.
_4.3.4 Survey of Internet Users in China_
Why does experience with censorship embolden social media users and cause them to complain about their repression, despite increased scrutiny from government censors and indications that their writings are off-limits? Clearly, for the producers of online media studied in this section, the small increase in the probability of government reprisal was outweighed by countervailing forces that cause Internet users to persist in speaking about the topic that initially interested them and in some cases to write more about politically sensitive information in China.
To verify that the results presented here generalize to a broader population, following Dickson (2016, pg. 71–72), I included a question in a representative survey of urban residents in China conducted in the summer of 2015, asking respondents who were Internet users whether they had experienced censorship in the form of deletion of a social media post, removal of an account, not being able to post, or not being able to search for a term.29 Eleven percent of the respondents admitted to having experienced one of these forms of censorship. Respondents were then asked: "Whether or not you have experienced [these forms of censorship], if you had experienced them, how would you feel?" Respondents were then given a set of emotions, from which they could choose none, one, or more than one.
The survey results presented in table 4.1 indicate that censorship does not inspire much fear. Very few of the respondents indicated that they would be worried or fearful after experiencing censorship—only 5 percent of all Internet users and 7 percent of users who admitted to having experienced censorship said they would be worried or fearful. The largest proportion of users said that they wouldn't care, 35 percent of all Internet users and 41 percent of those who had experienced censorship. More than being fearful, users reported that they would be angry—23 percent of all users and 36 percent of users who had experienced censorship said they would be angry or extremely angry. This suggests that social media users could be pushing the limits of censorship because they disagree with it and hope to undermine it. Interestingly, the survey evidence indicates that the distribution of respondents' expected emotions among those who have experienced censorship and those who have not are similar, suggesting that the results in this study are more broadly applicable to the population of users who do not regularly post sensitive material online.
Table 4.1: Emotional reactions to censorship, urban survey in China. | All users | Users with censorship experience
---|---|---
Wouldn't care
Angry or extremely angry
Sad
Worried or fearful
Proud or would brag to friends
Happy
Didn't answer | 0.35
0.23
0.12
0.05
0.03
0.00
0.34 | 0.41
0.36
0.38
0.07
0.10
0.00
0.05
**4.4 AN EXPERIMENTAL STUDY OF CONSUMERS OF SOCIAL MEDIA**
Last, I study how online consumers of information react when they come across censored information. Although censorship in China is less obvious to consumers of social media than it is to producers, occasionally a consumer will come across a censored page. One way observation of censorship occurs is through a link, when a page that the person is visiting links to a page that has been censored. When the consumer clicks the link, the user will be taken to an error page generated by the Internet content provider instead of the page with the original content. An example error page is displayed in figure 4.4.
Figure 4.4: Error page indicating that a post has been removed, from sina.com.cn.
Another way consumers of social media can observe censorship is within a forum or a discussion thread. When posts are censored within a forum, the content of the individual comment is removed, but the rest of the conversation still exists. Therefore, the user will see a removed post where the original post once stood. This can also occur on Sina Weibo, where sometimes when an individual post is removed, the rest of the thread remains uncensored.
Does the observation of censorship influence the likelihood that sensitive topics will be read? When consumers come across a censored page, they can often guess the topic of the censored post because of the context of the censorship. For example, if the user clicked on a link to a censored page, the title of the link will often give an indication of what the original post was about. Within a forum, consumers of information will see the discussion surrounding the missing post, and therefore may be able to guess the topic of the post's content, even though it is missing.
The observation of censorship may be a signal of what information that the government views as "in bounds" and thereby may affect the behavior of individuals. Many scholars have posited that authoritarian governments signal to citizens what they should be consuming and talking about and that these signals affect citizen behavior. For example, Singapore uses "out-of-bounds markers" to indicate what should and what should not be discussed.30 Brady (2009) posits that a similar information control strategy is used in China, where "frames" or information norms show by example what should and should not be discussed in public. A netizen might observe censorship and infer Party guidelines, following the Party's lead in what information she should be consuming online. If the public does take the signal of censorship as an indication of what it should be consuming, we would expect this to have an important impact on the spread of information.
If consumers of information sometimes happen upon indications of censorship in social media, how do they react? If the consumer is primarily acting on fear or on the realization of norms produced by the censored post, interaction with a censored post will cause the reader to avoid further interaction with the topic. On the other hand, if censorship creates countervailing signals that undermine government legitimacy or enhance readers' curiosity, it might instead pique readers' interest, drawing them to more posts about the same topic and undermining the legitimacy of the government. These basic questions motivate an additional test: an experiment on the influence of awareness of censorship on consumers of social media posts in China.
_Experimental Design_
I conducted an experiment to study how consumers of social media posts react to censorship with two universities in China students at over the summer of 2013. Students were given a computer, either a laptop in a coffee shop or a computer in a lab setting, and were given access to the Internet. They were provided with a list of social media posts in a blog aggregator that I designed for the study. They were told to read whichever posts interested them, and told that they would be asked a few questions about the blogs they read after five minutes of reading, but they were not expected to cover all blogs or all the blog topics. The subjects were aware that their actions online were being watched, as the description of the process indicated that their behavior online was being recorded, simulating an environment of surveillance.
The social media posts covered four different topics about ongoing events in China: (1) protests over the construction of paraexlyne (PX) plants in Yunnan, (2) protests in Hong Kong against the Chinese government, (3) a scandal alleging that a Chinese Communist Party official's son, Li Tianyi, was involved in the gang rape of a woman, and (4) blogs speculating that the Chinese economy would soon descend into an economic crisis similar to the one that had occurred in the United States. Because this study was conducted within China, more sensitive topics could not be used, but pre-testing of the experiment suggested that these were all topics Chinese citizens expect to be sensitive and could be censored. All blogs within the study existed online and had not been censored at the time of the study.
Students were asked to read blogs that interested them and then told that they would be asked a few questions. They could judge which social media posts might interest them because the title of the post was displayed on the main page of the aggregator, and each title contained information that would allow them to determine the post's topic. It was clear that they could not cover all topics or read all posts during the time allotted.
For the treated group, the first post the reader clicked on in a randomly selected topic would not link to the blog itself, but instead to the error page associated with the blog's Internet content provider, indicating censorship. The student would then return to the blog aggregator and choose another post to read. For the control group, all links on the main page would direct to the full content of the post, and not to an error page.
The question of interest was: How does the experience with censorship influence the topic the reader selects next? Is the reader less likely to pick a post on that topic because she has received a signal from the government that that topic is off-limits? To measure this, I installed technology within the website to track the behavior of each individual. I could therefore observe when an individual was treated with a censored link and which link she decided to click on after encountering the censored page.
A comparison between the group that encountered censorship when they clicked on their initial topic of interest and those who did not encounter censorship on that topic is the causal effect of the awareness of censorship on the reader's consumption decisions. If the censored page created mainly fear and anxiety, we would expect participants to avoid the censored topic in the post they selected subsequently. If the censored page inspired curiosity or indignation, however, we would expect participants to be equally or even more likely to click on the censored topic. Finally, for those who were treated with censorship at some point during the course of their time on the blog aggregator, we can study how this affected their opinions on the validity of government censorship to explore how the observation of censorship affects government legitimacy.
_Results: Participants Compensate for Censorship_
The vast majority of the subjects recognized the error page as censorship. The last question in the survey following the experiment presented a screenshot of the censorship error page to the participants and asked them what they thought this error page indicated. An overwhelming 84 percent of the respondents said that the page was due to purposeful deletion of posts. Treatment was also successfully randomized—there was no difference between treated and control among other observed demographic covariates.
Censorship did not dissuade people from reading more about the same topic—instead it made people more interested in the topic. To estimate how the observation of censorship influenced readers in the most rigorous way possible, I selected the first post each participant had clicked on as the treatment post. This helps control for initial interest in the post, as presumably readers initially pick the topic that they are most interested in. The dependent variable of interest is whether the second post the participant clicked on discussed the same topic, or a different topic than the one that was initially deleted.
As shown in figure 4.5, readers who first clicked on a censored post were more likely to click on a post within the same topic next than those who first clicked on an uncensored post. This indicates that readers were not deterred by censorship, for if they were they would be less likely to click on the same topic having observed censorship. In fact, readers seem to compensate for the fact that they were censored by clicking on more posts about the topic overall than uncensored readers. I estimated on average how many total _uncensored_ posts a person who had come across censorship initially would read in comparison to the person who initially did not come across censorship. The result was indistinguishable from zero: on average, a person who came across censorship initially would click enough subsequent posts to read between one and two uncensored posts about the topic by the end of the time period, about the same number as those who came across an uncensored post originally. This indicates that when readers are aware of censorship, they are willing and able to compensate for it.
Figure 4.5: Effect of censored social media post on the probability of clicking on the same topic subsequently. The treated group (top) is more likely to click on a post of the same topic than the control group (middle). The difference in means is at the bottom of the plot.
Experience with censorship decreased treated users' agreement with government censorship policies. After the experiment, subjects were asked about whether they thought that particular categories of information should be regulated on the Internet, including online discussion, games, uncivilized language, ads, rumors, pornography, spam, violence, and false information. As shown in figure 4.6, when asked about how much regulation of the Internet there should be across these nine categories, subjects who observed censorship indicated that they thought the Internet should be less regulated across the board. This difference in opinion suggests that experience with censorship, instead of deterring social media users from seeking out information, may instead undermine the legitimacy of the government's information laws.
Figure 4.6: Effect of observing censored social media post on support for Internet regulation. Treated units (black) are on average less likely to support regulation across the board than control units (gray). A t-test of the overall difference between treated and control has a p-value of 0.08.
The implication of these results is that awareness of censorship on the part of consumers of social media posts does not dissuade these consumers from reading more about a topic. Instead, observing censorship interested readers in learning more about the topic and created disillusionment with government censorship laws. Inducing awareness of censorship in this case backfired to undermine government policy.
The findings presented in this chapter could explain why the Chinese government is constrained in implementing more draconian policies when censoring the Internet. If there were no cost to censorship, then the government might censor anything about itself that could be seen as unseemly. However, because censorship is costly for the government, it refrains from censoring too broadly and attempts to hide its censorship footprints online. In earlier work, my coauthors and I found that the Chinese government allows much criticism to go uncensored online, and instead focuses censorship on collective action—in other words, censoring the information that is a direct threat to the regime.31 Internet content providers in China recently have tried to make censorship less observable by concealing error pages to the authors of censored blogs and simply reordering search results rather than throwing errors when search filtering, perhaps to lessen the backfire effect that censorship creates.
The evidence presented here suggests that government censorship policies are complicated by the fact that Internet users can compensate for censorship when they observe it. Topics the Chinese government would like to see removed online are constantly changing and developing as politics, international events, and protest events unfold. The government cannot simply signal what is off limits to online users because Internet users will not automatically avoid these topics and "purify" the information environment. Observable censorship instead can create more interest in the topic and undermine the reputation of the government.
**4.5 CONCLUSION**
In this chapter, I showed that signals of what topics are off limits do not induce typical Internet users to avoid topics. Using a matched pair design of microbloggers, evidence from a survey, and online experiments, I show that Internet users seem to read and share as much or more about topics that they believe to be censored. Further, the evidence suggests that experience with censorship may in fact undermine the government's reputation, inspiring social media users to complain about censorship, reevaluate their opinions on government censorship laws, and experience feelings of anger.
The evidence presented here is consistent with the incentives of online users in high information environments. In these situations, observable censorship attracts citizens to information, as it signals importance. Because so many people participate online, netizens are largely protected from punishment. Therefore, signals of off-limits information embolden users and reduce government legitimacy rather than inspiring widespread self-censorship.
If fear is not what is preventing typical Internet users from consuming off-limits information, then what control does the government have over the information environment? In the next chapter, I introduce more online data to show that when Internet users are not explicitly aware of what is off-limits, they are highly affected by the cost of access to information. Friction and inconvenience have a more significant impact on the typical Internet user in China than deterrence does.
1 "关注就是力量围观改变中国," 南方周末, January 14, 2010, <http://www.infzm.com/content/40097>.
2 "Netizens Force Fang Binxing (方滨兴), Father of the GFW, Off of Sina Microblog," _China Digital Times_ , December 20, 2010, <http://chinadigitaltimes.net/2010/12/netizens-force-fan-binxing-father-of-the-gfw-off-of-sina-microblog/>.
3 "Special Feature: The 'Southern Weekly' Controversy," _Freedom House_ , January 18, 2013, <https://freedomhouse.org/cmb/2013_southern_weekly>.
4 Kan, Michael, "GitHub Unblocked in China after Former Google Head Slams Its Censorship," _Computer World_ , January 23, 2013, <http://www.computerworld.com/article/2493478/internet/github-unblocked-in-china-after-former-google-head-slams-its-censorship.html>.
5 National Committee for Cadre Training Materials (2011, 5–6).
6 For discussions of self-censorship of journalists, activists, and academics see Stern and Hassid (2012); Link (2002); Lee and Lin (2006).
7 Kalathil and Boas (2010, pg. 26), Wacker (2003, pg. 88).
8 National Committee for Cadre Training Materials (2011).
9 "Falling Short: Appendix II: Media Law in China," _Committee to Protect Journalists_ , <https://cpj.org/reports/2008/06/12ii-2.php>.
10 "New Rules Create Online Rumor Straitjacket," _Xinhua_ , September 9, 2013, <http://news.xinhuanet.com/english/china/2013-09/09/c_125354622.htm>.
11 Feng, Wang, "Outspoken Chinese American Investor Charles Xue Detained in Beijing 'Prostitution Bust'," _South China Morning Post_ , August 25, 2013, <http://www.scmp.com/news/china-insider/article/1299448/outspoken-chinese-american-investor-Charles-xue-detained-beijing>.
12 For example, Wang Gongquan, a microblogger whose account with more than one million followers was shut in 2012 and who himself was arrested for four months at the end of 2013, reactivated his account right after returning from prison, posting about his time in jail and attracting over 10,000 followers in one day before it was shut down (wertime, David, "Wang Gongquan: The mysterious return of the microblogger," _Sydney Morning Herald_ , January 31, 2014, <http://www.smh.com.au/world/wang-gongquan-the-mysterious-return-of-the-microblogger-20140131-hvamg.html>).
13 National Committee for Cadre Training Materials (2011).
14 Fu, Chan and chau (2013).
15 Chin, Josh, "China is Requiring People to Register Real Names For Some Internet Services," _Wall Street Journal_ , February 4, 2015, <https://www.wsj.com/articles/china-to-enforce-real-name-registration-for-internet-users-1423033973>.
16 Chin, Josh, and Gillian Wong, "China's New Tool for Social Control: A Credit Rating for Everything," _Wall Street Journal_ , November 28, 2016, <https://www.wsj.com/articles/chinas-new-tool-for-social-control-a-credit-rating-for-everything-1480351590>.
17 ""打击网络谣言" 台前幕后," 南方周末, September 5, 2013, <http://www.infzm.com/content/93974>.
18 Wee, Sui-Lee, "Chinese police arrest 15,000 for Internet crimes," _Reuters_ , August 18, 2015, <http://www.reuters.com/article/us-china-Internet-idUSKCN0QNlA520150818>.
19 Miller (2017) shows that censors may miss posts because they are overwhelmed or deliberately thwarting censorship directives.
20 <http://weiboscope.jmsc.hku.hk/datazip/>.
21 They also documented the error message related to the removed post. From the authors' own experiments, "Permission Denied" indicates that the post had been censored, whereas "Weibo does not exist" usually indicates government censorship but could also mean that the post had been deleted by the poster. While the team anonymizes the identity of the user, they include a subset of information about the user, including whether the user was "verified." Verified users on Weibo are typically those whose identity has been verified by the online platform and are typically the most prominent or famous users, who have more followers.
22 Iacus, King and Porro (2009).
23 I use the string kernel similarity measure from the _kernlab_ package in R.
24 The difference-in-difference estimate of treatment after censorship on text similarity is positive and significant. Including user fixed effects and other controls also produces a positive estimate.
25 This is measured by estimating the words with highest mutual information in the matched posts in comparison to a random sample of uncensored posts from the same time period; see Manning, Raghavan and Schütze (2008). A list of these words are included in the appendix.
26 The difference-in-difference estimate of treatment after censorship on sensitive word use is positive and significant. Including user fixed effects and other controls also produces a positive estimate.
27 The difference-in-difference estimate of treatment after censorship on missing posts is positive and significant. It is robust to including user fixed effects and other controls.
28 Names are pseudonyms; Weiboscope does not provide user names.
29 More information about the survey methodology is provided in the appendix.
30 Benner, Tom, "Singapore's new generation wants a kinder, chiller country," _Global Post_ , August 12, 2015, <https://www.pri.org/stories/2015-08-12/singapores-new-generation-wants-kinder-chiller-country>.
31 King, Pan, and Roberts (2013, 2014).
CHAPTER FIVE
The Powerful Influence of Information Friction
In May 2011, I had been following news about a local protest in Inner Mongolia, where an ethnic Mongol herdsman had been killed by a Han Chinese truck driver in a dispute. In the following days increasingly large numbers of local Mongols began protesting outside of government buildings, culminating in sufficiently large-scale protests that the Chinese government imposed martial law. These protests were the largest that Inner Mongolia had experienced in twenty years.1
A few months later, I arrived in Beijing for the summer. At dinner with a friend, discussing ethnic divisions in China, I brought up the Inner Mongolia protest event. My friend struggled to recollect the event, saying that she had not heard of it. A few minutes later, she remembered that indeed a friend of hers had mentioned something about it, but when she looked for information online, she could not find any. Her assumption had been that the protest itself could not have been that important if she had not received word.
It was difficult by design for my friend to have learned of the protest event. Bloggers who posted information about the protest online had their posts quickly removed from the Internet by censors.2 Local media were not reporting on the event, so news of the protest was reported mainly by foreign sources many of which had been blocked by the Great Firewall. Even for the media, information was difficult to come by, as reporting on the protests on the ground had been banned, and the local Internet had been shut off by the government.
Of course, information about the protest was not _impossible_ to find on the Internet. I had been following the news from Boston, and even in China, the simple use of a Virtual Private Network and some knowledge of which keywords to search for had uncovered hundreds of news stories about the protests. But my friend, a well-to-do, politically interested, tech-savvy woman, was busy and Inner Mongolia is several hundreds of miles away. After a cursory search that turned up nothing, she had thought that the news was either unimportant or non-existent.
A few years later, in an interview in 2015, I asked a woman in China (henceforth referred to pseudonymously as Lina) to describe how she used social media. A young professional who had studied abroad in the United States, she used Chinese social media platforms like WeChat and Weibo and also frequently jumped the Firewall to connect with her friends in the United States on Facebook. Although for the most part she read news in Chinese from government newspapers like _People's Daily_ and from more commercial Chinese newspapers like _Southern Weekend_ , she also made time when she could to read the _New York Times_ and the BBC's website.
Lina, unlike the average citizen, was very invested in politics and followed political events closely. She was involved in multiple organizations that advocated for gender equality and was an opinionated feminist. Because of her feminist activism, I asked her whether she had heard of the five female activists who had been arrested earlier that year in China, including in Beijing, for their involvement in organizing a series of events meant to combat sexual harassment.3 The arrests of these five women were covered extensively in the foreign press and had drawn international outcry—articles about the activists had appeared in the _New York Times_ and on the BBC. Multiple foreign governments had publicly called for their release. Within China, activists had organized social media accounts to provide updates on their imprisonment, and lawyers and students had petitioned the Chinese government for their release. But posts about their detention were highly censored and the Chinese news media were prohibited from reporting on it. Lina, who participated in multiple feminist social media groups and made an effort to read Western news, still had not heard about their imprisonment.
In interviews, I kept encountering examples like these—where people living in China exhibited surprising ignorance about Chinese domestic events that had made headlines in the international press. People I interviewed had not heard that the imprisoned Chinese activist Liu Xiaobo had won the Nobel Peace Prize. They had not heard about major labor protests that had shut down factories or bombings of local government offices. Although this ignorance was widespread among Chinese citizens who had not traveled abroad, did not jump the Firewall, and rarely used the Internet, it was also surprisingly common among tech-savvy, globally traveled, well-educated Chinese citizens. Despite the _possibility_ of accessing this information, without newspapers, television, and social media blaring these headlines, they were much less likely to come across these stories.
As Internet penetration has expanded, scholars have posited that the masses would be the beneficiaries of this new technology, at the expense of the powerful.4 The Internet creates transparency, providing minute-to-minute news on actions of governments, politicians, companies, and interest groups. The Internet allows citizens to communicate with one another instantly, creating a new forum for civil society that can spread signals of discontent and organize action against the government, before bumbling censors delete this information.5 According to these scholars, the Internet's ability to thwart government- and interest-group-led efforts of censorship makes it impossible for anything to be kept secret.6
However, as these interviews and the following empirical tests suggest, the perception that the Internet makes government control over information obsolete is fundamentally flawed. Although the Internet has made a lot of information _possible_ to find, much information is still disaggregated or difficult for the public to access. Market research provides evidence that Internet users are for the most part lazy consumers of information, and consume only the most accessible information.7 Governments and interest groups, which are the main actors gathering and providing access to information, determine what information consumers are most likely to read.
Porous censorship is surprisingly effective in the age of the Internet. The fact that information about these domestic events is _possible_ to access creates the perception that information is free-flowing, and suggests that difficulty of access is due to lack of interest rather than government manipulation. However, the possibility of access does not mean that citizens will consume information. Even though censorship is easily circumvented, citizens often do not have the patience to circumvent it. Citizens are also often unaware of censorship, and therefore cannot counteract it. Although some citizens will take the time and spend resources to seek out information that is difficult to access, costs of access will have larger effects on the average Internet user, who has fewer resources and little to gain from circumventing censorship.
In this chapter, I study the effect of two of these friction-based censorship methods on the spread of information on the Chinese Internet. I show that these methods of censorship have significant effects on the prevalence of information online and the regularity with which users access off-limits information. First, I study content filtering, the removal of social media posts online. Previous studies have estimated that anywhere between 1 percent and 10 percent of social media posts are removed by censors on Chinese social media sites.8 Whereas in the previous chapter I studied the direct effect of the removal of social media posts on the users whose posts were noticeably deleted by the government, in this chapter I will study the indirect effect of content filtering, or how the removal of posts influences the vast majority of users who do not come across the error page but simply do not have access to the post. For these users, content filtering simply makes information about an event difficult to find—netizens do not know that information has disappeared; instead they simply have trouble finding it or do not run across it. I show that small perturbations in the timing of content filtering influences the spread of information about self-immolation events in Tibet. When censors are a bit slower, news about self-immolations spreads to more people on Sina Weibo than when the censors are a bit faster.
Second, I study the effect of the Great Firewall of China on the frequency with which Chinese citizens access foreign information. Outside of removal of social media posts, the Great Firewall of China is perhaps the most obvious example of online information friction in China. The Great Firewall blocks access from Chinese IP addresses to a list of foreign websites the government deems objectionable. Although the Firewall can be circumvented by logging into a foreign computer through a Virtual Private Network, which allows users to access the Internet through a third-party computer, seeking out VPNs and paying a small fee for using them increases the cost of information that can be accessed only by circumventing the Firewall.
Using nationally representative survey data and geo-located data on Chinese users of Twitter, I show that very few people in China evade the Great Firewall, despite the fact that during the time period of the survey, it was relatively easy to do so. I show that the types of users who are willing to circumvent the Great Firewall are more likely to reside in large cities, be technologically savvy, be interested in politics, and discuss sensitive political events than typical Internet users in China. I find that when a new website is blocked, its popularity in China is significantly reduced. In doing so, friction creates a small but effective wall between the general public, who are less interested in politics, and the well-educated and disillusioned wealthy class, decreasing the likelihood of anti-government mobilization.
**5.1 THE EFFECTS OF CONTENT FILTERING ON THE SPREAD OF INFORMATION**
This section will focus on one of the Chinese government's censorship methods: content filtering, which is the selective removal of social media posts online in China. While the Great Firewall blocks foreign websites, the vast majority of social media posts in China are written on the platforms of Chinese-owned Internet content providers (ICPs). As mentioned in chapter 3, the Chinese government devolves responsibility for content filtering of social media posts to each of these individual ICPs. When a user posts something that is objectionable to the government online, the website is responsible for removing this material from the Internet. If government censors find too much objectionable material on a website, they have the authority to shut the entire website down. Under threat of extinction, ICPs employ thousands of censors who remove content based on directives from government agencies.
Large-scale studies of content filtering in the past have shown that the government focuses the efforts of censors on removing posts related to protest events or those who could organize protests. Although censorship in China can be a relatively disaggregated process, King, Pan, and Roberts (2013, 2014) show that, surprisingly, the government largely does not target criticism of government policies, but rather removes all posts related to collective action events, activists, criticism of censorship, and pornography regardless of their support or criticism of the government. Consistent with this theory, Bamman, O'Connor and Smith (2012) show that censorship focuses on social media posts that are geo-located in more restive areas, like Tibet. The primary aim of government censorship seems to be to stop information flow from protest areas to other parts of China, or to prevent people from knowing about protest events. Since large-scale protest is known to be one of the main threats to the Chinese regime,9 success for the Chinese censorship program is preventing the spread of information about protests in order to reduce their scale.
Despite extensive content filtering, if users were motivated and willing to invest time in finding information about protests, they could overcome information friction to find such information. First, information is often published online before it is removed by Internet companies. There usually exists a lag of several hours to a day before content is removed from the Internet. Therefore, even the most objectionable material will spend a period of time online and will be available to the public before it is removed.
Second, Internet content providers will occasionally miss posts related to an event. Automated methods of content filtering are not sophisticated or very successful, and therefore much of content filtering is done by hand. Since censors cannot read every post on the Chinese Internet, they may miss a fraction of objectionable material. Netizens who want to discuss a particular event may also find ways to trick the censors, either by finding websites that are less carefully watched (talking about politics on a dating website, for example), or by finding phrasing that makes posts about the topic difficult for the censors to find.
Last, if the event is reported in the foreign press, Internet users could access the information by jumping the Great Firewall using a VPN. Even if Chinese Internet content providers remove information about these events, Twitter, Facebook, or other blocked social media will often contain information describing the events. These social media companies do not filter content at the instruction of the Chinese government, and therefore, as long as a user has the know-how and money to access a VPN, they can access the entire unfiltered foreign web.
However, despite the possibility of accessing information in the face of content filtering, the slightly increased costs of information due to content filtering reduce the probability that netizens will come across information about protests, and therefore have a significant influence on the number of people who know about an event. This is primarily because content filtering reduces the prevalence of information on the Chinese Internet, meaning that fewer people encounter this information while searching the web. Content filtering particularly focuses on social media users who have large numbers of friends,10 targeting those who are in positions to spread information about the sensitive event to the largest audience.
Content filtering not only decreases the accessibility of information; it also can disaggregate accounts of an event, reducing the salience of the event and increasing uncertainty about what exactly occurred. Users cannot tell whether the lack of information online about a particular event is due to government censorship or to general lack of interest in the event. If no one seems to be talking about protests in Inner Mongolia, for example, even people who know about them may assume that the protests do not have widespread support. They may assume that the protest was started by radicals, and that people they follow online, their "friends," do not support these people. Unless they subsequently spend time searching and aggregating multiple accounts of the event, netizens may not realize the importance of an event even if they simply happen upon one post related to it.
All of these small costs add up to create a multiplying influence on the inaccessibility and disaggregation of information about the protest. The more content filtering, the fewer people happen upon the information and the fewer new postings occur online. The fewer the new postings, the fewer people know about the event, and those who do are more likely to think the event has fewer followers than they would if there were more discussion online. The fewer people who know about the event, the fewer people know there is any reason to spend time searching for the information. The smaller the number of social media posts that are related to the event, the fewer people out protesting, and the smaller effect the protest has on governance.
**Tibetan Self-immolation Protests**
Between March 2011 and July 2013, 120 Tibetans self-immolated within China, the majority of whom died. Although the exact reasons for the self-immolations are unknown and probably vary, writings by self-immolators call for Tibetan independence, greater Tibetan autonomy, or the return of the Dalai Lama, all policy stances that the Chinese government opposes.
These protest events represented a major political problem for the Chinese government, as this spate of self-immolations followed large-scale protests in 2008, where thousands of young people protested in Tibet, some waving the Tibetan flag.11 The self-immolations themselves were also sometimes followed by larger-scale protests. Since peaceful relations between minority groups and fighting independence movements are central to Chinese national security policy, the immolations were a direct challenge to the Chinese government. Self-immolations in other countries also have a history of causing political upheaval; for example, Mohamed Bouazizi, a Tunisian businessman, is credited with sparking the Arab Spring in 2010 after self-immolating in political protest.12
To discourage such protest events, the government responded rapidly to the self-immolation events, removing the self-immolator as quickly as possible and increasing police presence to prevent protests in the aftermath. Police punished villagers and families of Tibetans who self-immolated to discourage future events. Monasteries were often surrounded with police forces since many of the self-immolators were Buddhist monks.13
News of Tibetan self-immolations was uniformly and quickly censored on social media websites.14 Unlike most political discussions, where one thread will generate thousands of re-shares in China, threads spreading news of Tibetan self-immolators are quickly cut off. Since information about self-immolations is scarce, social media accounts of self-immolations are often uncertain, for example, "I heard there was a self-immolation today in Tibet. Is it true?" News about specific self-immolation events is an example of a topic that is uniformly censored by all social media websites in China, and therefore fear of writing and reading about self-immolation events should be constant across events.
**More Discussion of Self-immolations on the Weekends When Censorship Is Lower**
To study how censorship influences the spread of news surrounding self-immolation events, I collected a random sample of social media postings from Sina Weibo, BBS websites, and Sina blogs related to self-immolations between March 2011 and July 2013 before the Chinese government was able to censor them.15 Discussion about self-immolations on social media in China naturally clusters around self-immolation events. In this section, I define "bursts" of social media posts about an event as the spike in volume of social media discussion at the time of the event. Social media is characterized by bursts of activity,16 but some events receive more attention than others. In this context, since I obtain posts _before_ censorship, I am estimating how many posts were _written_ about the event within my sample. However, depending on how quickly censorship occurs, many of these posts were _available_ to readers only for a number of hours.
Of course, even after posts about the event are censored, it is possible for netizens to find information about any of these immolation events online. First, some posts are never removed—they are missed by the censors. Second, netizens could jump the Great Firewall and read any number of foreign websites that report on these events. Censorship in this case is _not_ an information blackout by any means. Instead, the degree and quickness of censorship will determine the effort a netizen would have to expend to find information about the event. As a result, this form of censorship functions through friction, a continuous variable that indicates the degree of difficulty in finding a piece of information, not a dichotomous variable that indicates complete availability or total unavailability.
Despite relatively high censorship across self-immolation events, some self-immolations receive more attention from social media users than others—some have longer bursts, or more discussion about the event. Figure 5.1 shows the variation in bursts within my sample across the 120 self-immolation events between 2011 and 2013. Some events receive barely any attention at all, while others have a large amount of social media discussion associated with the event.
Why would some immolations receive more attention than others? It could be that the nature of the event was such that certain events received more attention from the public than others. The age of the immolator is usually something people note when discussing self-immolation events, with younger immolators often discussed with more grief than older immolators. Monks who self-immolate might have a larger network of followers, leading to more attention about the event. Self-immolation events that appear in clusters might build on one another, generating more attention.
However, in an environment of high censorship, the fact that some immolators receive barely any online attention at all and that the spread of information about self-immolations is overall so stifled could be explained by variation in friction caused by the control of information. If differences in burst lengths between self-immolation events were due to variation in censorship, we would expect that social media posts that were online for a longer period would also have more time to be re-shared by others. The faster the censorship, the fewer people would know about the event, and the smaller the information available associated with it. The quicker that censors react to an event, the less online discussion about that event.
Figure 5.1: Number of social media posts after each of 120 self-immolation events between 2011 and 2013; sample from Sina Weibo, BBS, and Sina blogs.
I do not have real-time data on censorship of self-immolation events because the infrastructure required to detect real-time censorship over such a long period is prohibitively large. However, real-time analyses of censorship over short time periods can be conducted, and a few authors have uncovered the regular schedule of censors. Using real-time data collected by King, Pan and Roberts (2013), which was collected around the same time that the posts within my sample were written, I find that censorship is low on the weekends relative to weekdays.17 Typically, censorship occurs within one day of posting: a post written Friday is most likely to be censored on Friday. However, a post written on Friday is second most likely to be censored on Monday, not on Saturday or Sunday. This suggests that fewer censors are working on the weekends than on weekdays.
Self-immolations, however, can happen on any day of the week, and do. An analysis of all self-immolation events over the past two years shows that self-immolations are no more likely to happen on one day of the week than another. Since the act of self-immolation is so drastic, there is likely very little strategy involved in the particular day of the week chosen.
If information friction were effective in stopping the spread of information, we might expect that bursts related to self-immolations would be longer on the weekend, given that it might take longer for censors to locate and delete these postings and therefore provide a longer period of time for others to read and repost these social media posts. As shown in figure 5.2, bursts associated with self-immolations on the weekends are significantly longer than those associated with self-immolations that occur on a weekday.18
Of course, the length of the social media burst could be due to other variables besides censorship. Monks might be more likely to self-immolate on a weekend, and also are more likely to gain a larger following. The age of the self-immolator could be related to both the timing and the following. In order to control for these variables, I collected data about the specific circumstances of each self-immolation event, including the age, whether the self-immolator was a monk, and the time since the last self-immolation to capture any clustering effects. The results are robust to these controls, and I show a full model controlling for these characteristics in the appendix.
Figure 5.2: Weekend self-immolations (top) have more discussion than weekday self-immolations (bottom).
To verify that this effect was not a result of people simply writing more posts on the weekend, I downloaded a random sample of blogs from the same source mentioning Tibet during the same time period. These posts are largely not sensitive; most talk about economic opportunities in Tibet, traveling in Tibet, or Tibetan culture. Surprisingly, people are significantly _less likely_ to write about Tibet on the weekend than on a weekday.
The fact that the speed of content filtering influences the number of posts about a self-immolation event indicates that small costs of access to information, such as the timing of censorship, influence the spread of information about protest events throughout China. Even in this case, where fear and self-censorship should be constant across immolation events, the timing of censorship dictated by censors' schedules is correlated with the number of people who wrote about the event.
**5.2 STRUCTURAL FRICTIONS AND THE GREAT FIREWALL**
It could be that short-term frictions like the content filtering described in the previous section have short-term effects on the spread of information, but in the long run people can learn to overcome these frictions. Indeed, many scholars who have maintained that censorship cannot work in the age of information have argued that eventually users will discover the information because censorship is porous and they will learn to circumvent it.19 Censorship in China has frequently been described as a "cat and mouse game," where citizens adapt to censorship technologies, which forces the government to change these technologies in an effort to prevent this circumvention. Perhaps in a repeated game, netizens would learn to gain access to information that was off-limits.
In this section, I will show that, on the contrary, structural frictions can have persistent effects on the majority of the population, even over the long term. I focus on how a form of censorship that has persisted over years, the Great Firewall, affects the information citizens access in China. Even though the costs of evading the Firewall are relatively low and the technology to do so is available, not many people in China regularly jump the Firewall. Those who regularly jump the Firewall are exceptional in that they typically have more resources, more technical capabilities, and an unusual interest in politics, and therefore have lower costs of access and greater expected benefits of evasion. Thus the Great Firewall structurally separates activist issue publics in China from the Chinese public at large.
In this chapter I use two methods of measuring who in China "jumps" or evades the Firewall and who does not. First, I use a survey of urban users in China in which respondents were asked about their Internet behavior, including whether or not they circumvented censorship. Second, I directly observe Twitter users who are using a VPN to tweet from China. Twitter has been blocked from Chinese IP addresses since 2009. As such, it is difficult to know when a user on Twitter is from China, as their IP address is routed through a third computer so they typically cannot be traced to China. Although users in China must use a VPN to post on the blocked site, if they are using mobile phones or other geo-located devices, their location is recorded and sent to the Twitter API. Thus, many users record their location in China even if they are using a VPN. Using this information, I sample users who have tweeted at some point from China to estimate the effect of the Great Firewall on Twitter use and the differences in conversations among people in China who are evading censorship and those who are using Chinese alternatives to Twitter, like Sina Weibo.
_**5.2.1 "Jumping" the Great Firewall**_
The Great Firewall of China blocks particular foreign websites from Chinese IP addresses. These websites include a wide range of content that the Chinese government deems objectionable, from foreign news websites that regularly report on sensitive events in China, like the _New York Times_ , to pornographic websites, to companies that compete with Chinese Internet companies and refuse to censor content at the direction of the Chinese government, for example, Twitter, Facebook, and Google.
"Jumping" the Great Firewall in China at the time of writing is not penalized and is not difficult, but it does involve time and money. First, a citizen in China must use the Internet in order for it to be possible to jump the Great Firewall. China has around 649 million Internet users, which is about 48 percent of the Chinese population. The vast majority of Internet users (about 85 percent) access the Internet from their mobile phone.20 In the survey of urban residents conducted in China in 2015, 60 percent of respondents reported having used the Internet, a larger proportion because those sampled were urban residents, who are more likely to have access to Internet infrastructure.21 Eighty-six percent report that one of the ways that they access the Internet is through their mobile device.
If a person in China uses the Internet, the first step to evading the Firewall is to actually realize that the Firewall exists and that evading censorship is in fact possible. Among Internet users, 48 percent did not know what evasion of the Firewall meant when asked whether they had jumped the Firewall. Ruling out those who do not use the Internet and those who do not know that evasion is possible, this leaves us with only 30 percent of the total sample who both use the Internet and recognize that the Great Firewall restrictions can be circumvented.
Next, a user has to find a Virtual Private Network that would allow them to evade the Firewall. As VPNs are periodically blocked by the Chinese government, this entails locating and downloading a VPN that has not yet been blocked from within China. To use a VPN, users sometimes have to pay small sums of money, typically a few dollars a month. Users also must be patient, since VPN access to the Internet also can be quite slow, as traffic is routed through a third-party computer first before reaching the user. VPNs are also regularly blocked and thwarted, and therefore "jumping" the Firewall sometimes requires several working VPNs. Even though these barriers are by no means insurmountable, of the people who used the Internet and knew that evasion of the Firewall was possible, only 16 percent stated that they had used a VPN to jump the Firewall. That means that, out of the entire sample, only 5 percent of urban residents reported that they had jumped the Firewall.
Why don't users who know that evasion is possible jump the Firewall? The survey included a question to ask users who knew that evasion was possible but reported not having evaded censorship why they did not jump the Firewall. Only 2 percent of these users said that they did not jump the Firewall because they were fearful and only 9 percent of the users said that they did not jump the Firewall because of legal concerns. The vast majority of users said they did not jump the Firewall because they didn't have a reason to (45 percent), they did not know how (15 percent), or it was too bothersome (14 percent). Small frictions that users have no reason to overcome, rather than fear or deterrence, seem to keep people from accessing information blocked by the Chinese government.
_5.2.2 Who Evades Censorship? Evidence from the Survey_
Who evades the Firewall and how are these netizens different from the rest of the population? In this section, I describe the correlates of Firewall evasion among Internet users in China. I find that citizens who jump the Firewall tend to be highly educated, concerned about politics, and have sufficient skills to evade censorship. Consistent with the survey respondents' own accounts of why they do not jump the Firewall, those who jump the Great Firewall have relatively more skills and resources at their disposal and can overcome the technical difficulties of acquiring a VPN. Those who jump the Firewall also have more reasons to evade censorship: they are generally more interested in international politics, participate more in politics both online and offline, and overall have a lower opinion of the government than those who do not use VPNs.
**5.2.2.1 Those Who Jump the Great Firewall Are Younger and Have More Education and Resources**
Most significantly, citizens who jump the Firewall tend to be much younger than those who do not. Survey respondents who grew up during the age of the Internet are significantly more likely to jump the Firewall than those who entered their twenties before the Internet was introduced to China. Figure 5.3 shows the propensity to use the Internet, know about the Firewall, and evade the Firewall by age. Chinese citizens in their fifties, sixties, and seventies largely do not use the Internet and do not know much about the Great Firewall. There is a significant increase in those who jump and know about the Great Firewall for those younger than 35: approximately 10–20 percent of those younger than 35 know about and evade the Firewall. Respondents around the age of 35 were younger than 18 when the first Internet cafés were appearing in major cities in China and therefore are the beginning of the generation who grew up with the Internet.22
Figure 5.3: Internet use, knowledge of the Great Firewall, and evasion of censorship by age.
In addition to being younger, those who jump the Firewall have attained far more skills through education than those who do not evade censorship. Seventy-five percent of those who jump the Firewall are either in college or have a college degree. In comparison, among those who do not evade censorship, only 25 percent have a college degree. Controlling for age, having a college degree means that a user is 10 percentage points more likely to jump the Great Firewall.
Technical capability is not the only thing needed to jump the Firewall. Access to VPNs cost a small amount of money. Individuals who jump the Firewall have an average family income of 141,800 _yuan_ per year, or $22,839.23 Those who know what the Firewall is but do not jump it have an average family income of 87,100 _yuan_ per year, or about $14,029. Those who do not use the Internet, in comparison, have an average income of 57,685 _yuan_ per year, or $9,291. Those who jump the Great Firewall are also much less likely to be migrant workers in China—only 23 percent of those who jump the Great Firewall have a household registration in a rural area outside of the area, whereas 33 percent of all respondents had household registrations in rural areas and have migrated to the urban area where they were interviewed. Indeed, the Firewall exacerbates what scholars have called the digital divide by creating barriers to information that those with greater wealth and resources can more easily circumvent.24
Evaders of censorship are also more likely to be networked with foreigners, which would give them both more technological capacity to evade the Firewall and more reason to do so. Twenty-five percent of those who jump the Great Firewall say they can understand English, as compared with only 6 percent of all survey respondents.25 Twelve percent of those who jump the Great Firewall work for a foreign-owned enterprise or foreign-based venture, compared to only 2 percent of all survey respondents. Forty-eight percent of those who jump the Great Firewall have been abroad,26 compared with 17 percent of all respondents. Figure 5.4 shows a map of the proportion of Internet users in each province who indicated that they use a VPN. Unsurprisingly, the highest rates of censorship evasion are in large East Coast provinces with more connections to the outside world.
Figure 5.4: Proportion of internet users who evade censorship, by province. White indicates no data available.
**5.2.2.2 VPN Users: More Knowledgable about Politics with Less Trust in the Government**
As described in the previous section, VPN users are typically better educated, have a higher income, and have more foreign connections than their counterparts who do not go to the trouble to evade censorship. Not only by their social standing, but also by their high levels of political knowledge, perceived efficacy, and participation, VPN users are part of the wealthy and educated class in China. However, VPN users on the whole have less trust in the government than users who do not jump the Firewall. Perhaps because they are very interested and participate in politics but do not trust the government, these citizens seek out information and social networks that the government blocks.
Those who evade the Firewall are more knowledgeable about politics than their counterparts. Forty percent of VPN users say they are interested in politics and 60 percent say they follow international politics, whereas only 28 percent of all respondents say that they are interested in politics and 30 percent say they follow international politics in the full sample. VPN users score higher than any other group on a political knowledge test within the survey—they are more likely to correctly name the Chinese president, U.S. president, and Japanese prime minister than those who do not use VPNs.
VPN users are also substantially more politically active and have high self-perceived political efficacy. They are more likely than any other group to have participated in political meetings, to have expressed their views to the government leadership, to have expressed their views through the media, to have participated in political protests, and to have petitioned the government.27 Those who evade the Firewall also report much higher levels of political efficacy than their counterparts. When asked, "When you encounter unfair treatment, is your ability to solve the problem higher or lower compared with people you know?" VPN users are 11 percentage points more likely than those who do not evade the Firewall to report that they are much more or relatively more influential than people they know. These results hold up even when controlling for age, employment status, and college education, suggesting that VPN users see themselves as very capable in political situations, even in comparison to others within their social class.
Despite high levels of participation and political efficacy, VPN users are more distrustful of the government than even their well-to-do counterparts. Those who evade the Firewall believe the government is more corrupt than those who do not, even controlling for employment status, college education, and age. Figure 5.5 shows VPN users' and non-VPN users' answers to how vulnerable different levels of government are to corruption. VPN users, shown in the dotted line, consistently think that all types of government officials asked about in the survey are more corrupt than do their fellow citizens who do not jump the Firewall.
Figure 5.5: Evasion of the Great Firewall and perceptions of corruption in government.
Naturally, VPN users are much stronger advocates of free speech and are more opposed to censorship than their fellow citizens who do not jump the Firewall. Seventy-two percent of VPN users in the survey disagreed with the statement, "The government should regulate the whole Internet," while only 60 percent of those who know about the Firewall but do not jump it disagreed with the statement, and only 57 percent of those who do not know about the Firewall disagreed with the statement. Eighty-five percent of VPN users agreed with the statement, "Internet governance should not violate individual freedom of expression," whereas only 77 percent of those who know about the Firewall but do not jump it agreed, and only 74 percent of those who did not know about the Firewall but were Internet users agreed.
Overall, the types of citizens in China who use VPNs are potentially threatening to the government. This set of people who jump the Firewall are well-endowed in terms of financial standing, education, technological sophistication, political and international connections, and political interest and knowledge. They frequently participate politically and know how to get things done, but overall they have less faith in government than their non-censorship-evading counterparts.
However, these users represent a very small fraction of the Chinese public. Conveniently, from the CCP's perspective, the Great Firewall separates this small group of political and well-educated users from the larger public simply by user selection, without resorting to repression or force. Because the broader public is less sophisticated and less politically interested, they will not take the time to enter into digital conversation across the Firewall.28 Thus, the Firewall succeeds in creating a porous but effective barrier between a skeptical class and the public they would have to connect with to have a bigger political impact.
_**5.2.3 Observations of VPN Users Using Geo-location**_
Of course, surveys rely on user-reported findings of evasion. These responses could contain measurement error due to social desirability bias, individuals rushing through the survey, or individuals misinterpreting questions. To retrieve a behavioral measure of evasion of the Great Firewall, in this section I identify Chinese users of the social media website Twitter, which is blocked in China. Although IP addresses that could locate individuals to China are masked by VPNs, the geographical location of the user is sometimes recorded by Twitter, particularly if the user is using a mobile phone or wants to publicly reveal their location by "checking in" at their location. In these cases, Twitter records the latitude and longitude of the user's Tweet and makes this information available for researchers through its API. If users geo-locate on Twitter to China, they must be using a VPN since Twitter is blocked from China.
To identify VPN users in China on Twitter, I use a random sample of approximately one-third of all geo-located tweets from China and Hong Kong (which is not affected by the Great Firewall) during September 2014 downloaded from the Twitter API. Geo-located tweets themselves are only a small subset of all Tweets; scholars have estimated that geo-located tweets are approximately 2–3 percent of the whole Twitter sample.29 Therefore, the sample represents approximately .66 percent to 1 percent of total Twitter users from China and Hong Kong. My sample from the Twitter API returned on average 1,690 unique users per day tweeting from mainland China and on average 905 unique users per day tweeting from Hong Kong. As the sample is approximately 1 percent of all Twitter users, a cursory estimate suggests that 169,000 unique Twitter users post each day from mainland China and 90,500 unique users post per day from Hong Kong. Given that there are about 5,388,354 Internet users in Hong Kong, this suggests that approximately 1.7 percent of Internet users in Hong Kong post on Twitter every day.30 In comparison, in mainland China with about 675,131,785 Internet users, only .02 percent of Internet users post on Twitter every day.31 Assuming Hong Kong provides an approximation for how much Chinese users _would use_ Twitter _without_ government censorship restrictions, this comparison suggests that China has about 1–2 percent of the Twitter users it would have without the Firewall restrictions. The Firewall, although easy to evade, is extremely effective at keeping Chinese users away from off-limits sites, even those that are popular in politically, culturally, and linguistically similar areas.
What types of users tend to jump the Firewall? To explore how Chinese Twitter users differ from typical social media users in China, we sampled geo-located Sina Weibo posts from the Beijing area in September 2014 and compared them with geo-located Twitter posts from the same area and time period.32 After removing common Chinese words, figure 5.6 shows the fifty most frequently used terms on Twitter in China with their translations, and figure 5.7 shows the fifty most commonly used words on Sina Weibo with their translations. Consistent with the survey data, the most commonly used words on Twitter in China reflect the differences in the two user populations. Twitter users are more likely to use political words like "peace," "freedom," "country," and even to mention activists like Ilham Tohti, a Uyghur economist who was sentenced to life in prison in September 2014 for calling attention to repression in Xinjiang.33 Chinese Twitter users are also likely to use words about technology such as "Endomondoendorphins," a hashtag associated with the sports tracking application Endomondo. Chinese Twitter users—even those signed up in Chinese—are also much more likely to use English words, indicating that Chinese users of Twitter are more educated and internationalized. Weibo users, on the other hand, are starkly apolitical, most commonly discussing feelings and mood, or documenting the events of the day with words like "today," "tomorrow," "time," and "goodnight." They also are likely to discuss celebrities—one of the most popular words during this period is Wei Chen (魏晨), a popular Chinese pop star and singer.
Figure 5.6: Fifty most common words after stopword removal, Chinese Twitter. Left panel, original words; right panel, English translation. Words are scaled in proportion to their frequency.
Figure 5.7: Fifty most common words after stopword removal, Sina Weibo. Left panel, original words; right panel, English translation. Words are scaled in proportion to their frequency.
Figure 5.8: Fifty most common words after stopword removal, Hong Kong Twitter. Left panel, original; right panel, English translation. Words are scaled in proportion to their frequency.
Of course, these differences could be due partly to platform. Perhaps Twitter is simply a more political platform than Weibo and so Chinese citizens sign on to Twitter to discuss politics and use Weibo to chat with friends apolitically. However, mainland China users are conspicuously more political than their Chinese counterparts on Twitter who are not affected by the Firewall. Figure 5.8 shows the fifty most commonly used words on Twitter during the same period in Hong Kong. Instead of discussing politics, Hong Kong Twitter users are more similar to Sina Weibo users, discussing music, love, feelings, and celebrities. This suggests that the Firewall itself creates a barrier only particular types of users are willing to cross—users who tend to be highly educated, political, and technologically savvy. The Great Firewall creates two mostly disconnected social media communities—separating users who are dedicated to political activism away from the less engaged public.
_5.2.4 Government Crackdowns on Websites_
Despite the fact that the censorship imposed by the Great Firewall is incomplete and can be easily circumvented with a Virtual Private Network, small costs have large effects on which websites Chinese citizens are likely to access. In the last section, I showed that only a small fraction of the Chinese population regularly uses a VPN to evade the Firewall. In this section, I show that small costs of access also have dynamic effects on the types of information that Chinese users access. To do this, I show that when websites are blocked by the Great Firewall, fewer Chinese users access these sites. I focus on three cases of website blocks in China: (1) Google, which was slowly throttled but eventually completely blocked at the beginning of June 2014; (2) Wikipedia, which has been blocked intermittently by the Chinese government, but was completely blocked on May 19, 2015; and (3) Instagram, which was suddenly blocked during Hong Kong democracy protests on September 29, 2014. In each case, I use an interrupted time series analysis of website visits to understand how censorship influences Chinese traffic to the site.
Given the evidence provided in chapter 4 that observations of censorship backfire, it may seem contradictory that when a website is blocked in China it does not create more interest in the website. However, many users in China do not know that the Firewall exists and do not realize that the webpage is down due to censorship. Further, the government has developed ways to make censorship more ambiguous, throttling websites to make them slower instead of, or before, outright blocking them. I show at the end of this chapter that when Great Firewall blocks are more sudden and more obvious, more users download VPNs in an effort to jump the wall. The evidence provided here suggests that the more porous censorship is, the more easily it is disguised, and the more effective it can be at diverting access.
5.2.4.1 Case 1: Google
As I described earlier, Google entered into a conflict with the Chinese government in 2010, when it alleged that Chinese hackers had breached its servers. In March 2010, Google began redirecting traffic from the mainland to its Hong Kong website, which does not abide by mainland China censorship policies. Instead of outright blocking Google with the Great Firewall, the Chinese government began throttling the search engine, so it connected only part of the time. Finally, in June 2014, before the twenty-fifth anniversary of Tiananmen Square, the Chinese government blocked Google services outright.34
Even though Google was not completely blocked in China immediately after the dispute, the throttling of Google took an immediate toll on the number of Google users in China. Figure 5.9 shows the fraction of worldwide traffic from Chinese IP addresses to Google.com over the 2009 to 2015 period.35 Initially after the alleged hack, Google received more traffic from China, perhaps as a result of increasing news about Google in China. However, immediately after the redirect, when the Chinese government began throttling Google, traffic decreased precipitously. When Google was finally completely blocked in 2014, traffic from mainland China had already declined from an average of 10 percent of Google's traffic to 5 percent of its traffic. After the complete block, Chinese traffic made up less than 1 percent of the world's traffic to Google.com.
Figure 5.9: Proportion of Google traffic originating in China, 2010–2015.
The Google case shows that the Chinese censors can control the popularity of websites by throttling them. Simply making a website slower frustrates users, giving them an incentive to switch websites. Such censorship also obscures the reason for the throttling. Was Google slow, or was this government censorship? In this way, porous censorship can diminish the backlash effects described in the previous chapter.
5.2.4.2 Case 2: Wikipedia
Like Google, Wikipedia has long had a fraught relationship with the Chinese government. Created in 2001 and first blocked in 2004, particular pages of Wikipedia such as descriptions of the Tiananmen Square protests have long been blocked. But the entire Wikipedia website, too, has occasionally been made unaccessible from Chinese IP addresses.36
Figure 5.10: Chinese language Wikipedia page views, May 2015. All Wikipedia pages were blocked from mainland China beginning on May 19, 2015.
The entire Wikipedia site was again subject to the Great Firewall block on May 19, 2015.37 To estimate the influence of the block on access to Wikipedia, I use data on the number of page views of Chinese Wikipedia (zh.wikipedia.org) during May 2015.38 For each day, I summed the total number of page views on all Chinese-language Wikipedia pages. Figure 5.10 shows a sharp drop in the number of page views of Chinese Wikipedia pages, which occurred precisely on the day of the block—page views decreased from 11 million total page views per day to approximately 8 million.
From Wikipedia page views, we can't tell how many of the 11 million daily page views originated in China itself and which were accessed by users outside of mainland China—many of the page views may originate in Hong Kong, Taiwan, or from Chinese speakers living in other countries. If most of the 11 million page views per day were from mainland China the Wikipedia block decreased the number of mainland page views by 30 percent. However, it is likely that users of Chinese Wikipedia are more likely to originate from areas other than mainland China, which has its own Wikipedia-like website called Baidu Baike. Therefore, we expect that the block decreased the number of mainland users by substantially more than 30 percent. Regardless, the stark decrease in page views driven by the Wikipedia block reflects the amount of power the Chinese government has to affect the popularity of Chinese-language websites around the globe simply through friction.
5.2.4.3 Case 3: Instagram
Under very different circumstances, another widely popular social media website—Instagram—was also blocked in 2014. On September 26, 2014, pro-democracy protests broke out in Hong Kong over reforms to Hong Kong's electoral system initiated by the mainland Chinese government. Thousands of people took to the streets in what later became known as the "umbrella revolution." On September 29, 2014, the Chinese government blocked the social media website Instagram from mainland Chinese IP addresses, due to increased popularity among Hong Kong protesters.39
To study how the Instagram block influenced mainland users' access to Instagram, my coauthor and I sampled geo-located Instagram posts from across mainland China during September and October 2014. To do this, we used a geographical grid of China and randomly sampled locations in this grid. For each randomly sampled location, we obtained all posts from the immediate area of the sample for the period, which we estimate reflects .25 percent of all Instagram posts.40 The Instagram block halved the number of unique users accessing Instagram geo-locating to China overnight; only 53 percent of users continued to use Instagram after the block. Like Wikipedia, the Instagram block shows the powerful impact of filtering on traffic to popular social media websites.41
**5.3 WHEN DOES FRICTION FAIL?**
In the last section, I showed that by throttling and blocking websites, the Chinese government wields extensive power over the number of mainland Chinese users who frequent a website. Despite the ability to circumvent censorship, small costs of evasion generally decrease the number of people accessing the newly blocked website. However, this does not mean that Chinese Internet users are completely passive, at the whim of government censors. As I discussed in the previous chapter, Chinese users express substantial contempt for censorship, and make efforts to evade censorship when they are aware of it. In this chapter, I showed that particularly political and wealthy users with lower levels of trust in the government are likely to acquire Virtual Private Networks to evade censorship.
When will increases in costs of access change the behavior of citizens and prevent them from accessing a website? Users implicitly engage in a cost/benefit analysis when deciding whether to spend the time to evade censorship. They will be more likely to spend the time and money to access blocked information when (1) blocked information is valuable to them, and (2) when censorship is applied suddenly, disrupting their short-term habits and raising awareness of censorship itself. In this section, I show how the circumstances of the block can have very different effects on user evasion behavior.
The value of blocked information depends partially on how easily information can be substituted by alternatives within China. The Chinese government has actively encouraged homegrown versions of foreign websites, and these homegrown versions are more easily controlled by censors than their foreign counterparts.42 If the functionality of a foreign website can be easily substituted by an unblocked Chinese site, users may be unlikely to spend the time and resources to evade censorship.
Even if there are substitutes to the blocked website, the more suddenly a website is blocked, the more disruptive it will be to the habits of users, giving them incentives to seek out evasion technology to access the censored website rather than substitute with a Chinese version. If a citizen uses Gmail, a sudden block of Google may disrupt e-mail conversations or short-term projects, causing users to seek out a VPN to continue their short-term conversations. Slow, incomplete blocks, on the other hand, allow users to finish short-term projects with difficulty without seeking out a VPN, but eventually frustrate users enough so that they switch to the mainland Chinese substitute.
The three blocked websites discussed in the previous section—Google, Wikipedia, and Instagram—provide stark contrasts in the value and suddenness of each of their blocks. Instagram, an extremely popular photo site, does not have a direct Chinese analog and was blocked suddenly by the Chinese government during the Hong Kong protests, disrupting millions of users' photo sharing overnight. Google and Wikipedia, on the other hand, both have mainland Chinese competitors—Google's analog in China is Baidu.com and Wikipedia's analog in China is Baidu Baike. Further, Google and Wikipedia have long been throttled by the Chinese government so not being able to access these websites was by no means sudden or unexpected.
Figure 5.11: iPhone download rank in China of VPN Express, 2014–2015.
These three blocks have very different implications for the number of people who sought to evade the Firewall because of the block. Figure 5.11 shows the download rank of the iPhone application VPN Express in mainland China on the days of each of the Google, Wikipedia, and Instagram blocks.43 VPN Express increased in popularity only slightly after the Google block and was not more popular after the Wikipedia block than it was before—the block did not increase the popularity of evasion software. In contrast, the day of the Instagram block, the rank of VPN Express skyrocketed from the 1,229th most downloaded application in China to the sixth most downloaded application in China overnight. The Instagram block encouraged new users to download Virtual Private Networks, whereas the Google and Wikipedia blocks had few immediate effects.
The sharp increase in acquisition of Virtual Private Networks after the Instagram block highlights the difficulties the Chinese government faces in censoring suddenly during crises. Unlike Google and Wikipedia, where censorship was initiated from longstanding conflict between the companies and the Chinese government, censorship of Instagram was motivated by a protest event that the Chinese government was worried would affect political opinion of citizens in the mainland. Because of its quickly evolving nature, the government may have thought that it did not have time to slowly throttle Instagram, as it had done with Google and Wikipedia, before outright blocking it. Even though the government decreased citizens' overall access to Instagram, the Instagram block inspired more censorship evasion that facilitated user access to long-blocked websites such as Twitter and Facebook. Hobbs and Roberts (2016) show a spike in Twitter and Facebook downloads and user signups on the day of the Instagram block.
In general, crisis events complicate the government's ability to effectively use friction. Crises enhance citizen awareness of their political situation, which may make them more likely to spend the time and money necessary to find information, no matter how costly. As suggested by evidence in the previous chapter, sudden censorship that might accompany political crises may also alert citizens to the information that the government is trying to keep quiet, giving them incentives to seek out information that is not immediately available to them. Sudden censorship also disrupts habits, giving users incentives to seek out newly blocked informations.44 Evidence from geo-located Instagram users during the Instagram block shows that users who continued to use Instagram through a VPN had significantly more likes and posts previous to the block than users who did not continue to use Instagram by evading the wall. It could be that the Instagram users downloaded a VPN simply to continue their habits, and in doing so were exposed to information already blocked by the Firewall.
Outside of VPN downloads before and after the sudden block of Instagram, the survey conducted in China in the summer of 2015 also suggests the Chinese citizens seek out censored information during crises. On August 12, 2015—which happened to fall in the middle of the survey of urban residents in China—explosions rocked the city of Tianjin, China, causing hundreds of deaths and injuries in apartment buildings nearby. Netizens uploaded to the web videos of the explosion—which was caused by a storage facility with overheated chemicals and could be seen from miles away—and the fire that spread from the explosion site to neighboring buildings.45
The enumerators of the 2015 survey happened to be between two waves of surveys in Tianjin when the Tianjin explosion took place. One of the starkest differences between the two waves of respondents in Tianjin was their propensity to have jumped the Firewall. For each day that the enumerators were in the field, figure 5.12 shows the proportion of those interviewed who jumped the Firewall. On the days before the explosion, only 5 percent of Internet users said they had jumped the Firewall. On the days following the explosion, almost 30 percent of Internet users admitted to having evaded censorship. Although we cannot be certain that these effects were directly due to the explosion, having been interviewed after the explosion in Tianjin predicts a 22 percentage point increase in the probability of jumping the Firewall, controlling for a battery of demographic and political characteristics. This evidence suggests that crises in which people have physical and financial incentives to seek out information can undermine the effectiveness of friction-based censorship.
Figure 5.12: Proportion of Internet users who say they jump the Firewall, before and after the Tianjin explosion on August 12, 2015. Size of points reflects the sample size on each day.
**5.4 CONCLUSION**
In sum, this chapter showed that small barriers to information access have important effects on the information consumption patterns of individuals in China. As I found first in the case of self-immolation events in Tibet, events that correspond to slightly quicker censorship are less likely to spread on social media than those that correspond with downtimes for the censors. Slight changes in the speed of censorship affect how much information is available to people in China about important political events within the country. Sudden censorship of websites starkly decreases how much they are visited by Chinese citizens. These blocks are particularly effective when censorship is porous—when governments throttle, but do not suddenly block, websites.
Even frictions that do not change over time have persistent effects on the online behavior of Chinese citizens. Very few people in China are knowledgeable and motivated enough to circumvent censorship from the Firewall, despite the fact that circumvention tools for the Great Firewall have existed for a decade. A back-of-the-envelope comparison to use of Twitter in Hong Kong indicates that Twitter use in mainland China is about 1–2 percent of what it would be were Twitter not blocked. This number is consistent with the results of our survey, which suggest that only 5 percent of urban respondents had jumped the Firewall. Those who do not jump the Firewall report that they have no reason to, that they do not know how, or that jumping the Firewall is too bothersome.
Those who take the time to circumvent the Firewall are from the upper class: are interested in politics, are highly educated, are networked with foreigners, and have high political efficacy and high incomes. While on blocked websites, they discuss sensitive topics like activists, the government, and human rights. The Great Firewall, therefore, separates the political discussion of the political elite from the rest of the public, strangling the potential for collective action by decreasing the following of those in the elite who are skeptical of the government.46
However, the very porous nature of censorship also means that there are limits on how much it can influence citizens. When there are few substitutes for blocked information, when information is blocked suddenly, or during crises when individuals have incentives to seek out off-limits information, citizens will be more likely to spend time and money thwarting the costs of access that the government has imposed on information. During these periods, the government will try to manage the conversation by dominating it, rather than by stalling it. In the next chapter I turn to the more indirect but increasingly popular method of censorship that functions through flooding. I show that the Chinese government uses flooding to distract from or compete with sensitive information in China, particularly during sensitive periods when friction may be less effective.
1 "China's Inner Mongolia 'Under Heavy Security'," _BBC_ , May 30, 2011, <http://www.bbc.com/news/world-asia-pacific-13592514>.
2 Evidence of censorship of this event is shown in King, Pan and Roberts (2013).
3 Branigan, Tania, "Five Chinese Feminists Held over International Women's Day Plans," _Guardian_ , March 12, 2015, <https://www.theguardian.com/world/2015/mar/12/five-chinese-feminists-held-international-womens-day>.
4 Ferdinand (2000, pg. 5), Lynch (2011), Bellin (2012, pg. 138), Diamond (2010, pg. 70).
5 Yang (2009 _b_ , pg. 30).
6 Taubman (1998, pg. 266).
7 Hoelzle (January 2012).
8 Fu, Chan and Chau (2013); King, Pan and Roberts (2013).
9 See Chen (2012) and Cai (2010) for a discussion of protests in China.
10 Zhu et al. (2013).
11 Greve (2013).
12 Lotan et al. (2011).
13 Greve (2013).
14 The experiment in King, Pan, and Roberts (2014) shows the high censorship rate of social media posts related to self-immolations.
15 I sampled by requiring the social media post to contain the words self-immolation (自焚)and Tibetan (藏)from the social media analytics company Crimson Hexagon.
16 Ratkiewicz et al. (2010).
17 The Weiboscope data provided by Fu, Chan, and Chau (2013) also show lower censorship on weekends in comparison to weekdays.
18 Self-immolations that happen on Friday are considered weekend immolations because they often happen later in the day, and therefore discussion of these events often does not occur until the weekend.
19 Yang (2009 _b_ , pg. 30).
20 "CNNIC "发布第35 次《中国互联网络发展状况统计报告》," Cyberspace Administration of China, February 3, 2015, <http://www.cac.gov.cn/2015-02/03/c_1114237273.htm>.
21 More information about the survey methodology is provided in the appendix.
22 FlorCruz, Jaime A., and Lucrezia Seu, "From Snail Mail to 4G, China Celebrates 20 Years of Internet Connectivity," _CNN_ , April 23, 2014, <http://www.cnn.com/2014/04/23/world/asia/china-Internet-20th-anniversary/>.
23 Using an exchange rate from July 2015 http://www.x-rates.com/average/?from=USD&to=CNY&amount=1&year=2015 of 6.208627 _yuan_ per U.S. dollar.
24 Norris (2001); Schlozman, Verba and Brady (2010) show how the Internet serves the wealthy already; censorship exacerbates this.
25 Questions about English ability were only asked to those with some college education, otherwise English ability is assumed to be zero.
26 "Abroad" here includes Hong Kong and Macao.
27 Interestingly, VPN users are less likely to have voted in elections than other users.
28 These findings are consistent with Chen and Yang (2017) who find that being given a VPN does not incentivize censorship evasion without attitional incentives to jump the Firewall.
29 Leetaru et al. (2013).
30 Internet Live Stats, 2014 numbers. <http://www.internetlivestats.com/internet-users/china-hong-kong-sar/>.
31 Internet Live Stats, 2014 numbers. <http://www.internetlivestats.com/internet-users/china/>.
32 I remove posts by users who had indicated when they signed up for Twitter that their default language was a language other than Chinese to try to remove tourists and ex-pats from the sample. See Hobbs and Roberts (2016) for more details.
33 "China jails prominent Uighur academic Ilham Tohti for life," _BBC_ , September 23, 2014, <http://www.bbc.com/news/world-asia-29321701>.
34 Levin, Dan, "China Escalating Attack on Google," _The New York Times_ , June 2, 2014, <https://www.nytimes.com/2014/06/03/business/chinas-battle-against-google-heats-up.html?_r=0>.
35 Data from Google's Transparency Report: https://www.google.com/transparencyreport/traffic/explorer/?r=CN&l=WEBSEARCH&csd=1235354784827&ced=1471030200000.
36 Pan, Philip P, "Reference Tool on Web Finds Fans, Censors," _Washington Post_ , February 20, 2006, <http://www.washingtonpost.com/wp-dyn/content/article/2006/02/19/AR2006021901335.html>.
37 Fox-Brewster, Thomas, "Wikipedia Disturbed Over Fresh China Censorship," _Forbes_ , May 22, 2015, <http://www.forbes.com/sites/thomasbrewster/2015/05/22/wikipedia-disturbed-over-fresh-china-censorship/#6d046845f842>.
38 Page view data at <http://stats.grok.se/>.
39 "Instagram Appears Blocked in China," _BBC_ , September 29, 2014, <http://www.bbc.com/news/technology-29409533>.
40 If geo-located Instagram posts are 1 percent of all posts and the sample covered approximately 25 percent of all residences in China, then the sample reflects approximately .25 percent of all posts.
41 For more detail on the study of the Instagram block in China, see Hobbs and Roberts (2016).
42 Pan (2016).
43 Information about download ranks obtained from App Annie, appannie.com.
44 This is related to the "cute cat" theory of censorship proposed by Zuckerman (2014) where censorship that is blunt enough to include entertainment can create more backlash.
45 "China Explosions: What We Know about What Happened in Tianjin," _BBC_ , August 17, 2015, <http://www.bbc.com/news/world-asia-china-33844084>.
46 Barberá et al. (2015), Steinert-Threlkeld (2017), Chenoweth and Stephan (2011, pg. 39–40).
CHAPTER SIX
Information Flooding: Coordination as Censorship
In the afternoon of August 3, 2014, a 6.5-magnitude earthquake hit Yunnan province in China. The earthquake killed hundreds and injured thousands of people, destroying thousands of homes in the process. School buildings toppled and trapped children, reminiscent of the 2008 Sichuan earthquake, which killed 70,000 people, when the government was heavily criticized for shoddy construction of government buildings. Emergency workers rushed to the scene to try to rescue survivors.1
Eight hours after the earthquake struck, the Chinese official media began posting coordinated stories—not about the earthquake, but about controversial Internet personality Guo Meimei. Guo had reached Internet celebrity status three years earlier, in 2011, when she repeatedly posted pictures of herself dressed in expensive clothing and in front of expensive cars on Sina Weibo, attributing her lavish lifestyle to her job at the Red Cross in China.2 Although Guo did not work at the Red Cross, her boyfriend, Wang Jun, was on the board of the Red Cross Bo-ai Asset Management Ltd., a company that coordinated charity events for the Red Cross. The expensive items that Guo had posed with on social media in 2011 were allegedly gifts from Wang. Attracting millions of commentators on social media, the 2011 Guo Meimei scandal highlighted issues with corruption of charities in China, and donations to the Red Cross plummeted.
By 2014, when the earthquake hit, the Guo Meimei scandal was old news, long forgotten by the fast pace of the Internet. On July 10, 2014, Chinese officials had arrested Guo on allegations of gambling on the World Cup.3 Then, out of the blue, on midnight of August 4, 2014, _Xinhua_ posted a long, detailed account of a confession made by Guo Meimei that included admissions of gambling and engaging in prostitution.4 On the same day, many other major media outlets followed suit, including major media outlets such as CCTV,5 the _Global Times_ ,6 _Caijing_ ,7 _Southern Weekend_ ,8 _Beijing Daily_ ,9 and _Nanjing Daily_.10
The overwhelming number of newspapers sensationalizing the Guo Meimei confession on August 4 seemed too coincidental to be uncoordinated. Indeed, the _China Digital Times_ received a government leak on August 4 that directed websites to "prominently display Xinhua and CCTV coverage of Guo Meimei, and to actively organize and direct commentary."11 The state propaganda apparatus seemed to be actively trying to engage the public with a story about a celebrity turned criminal and prostitute.
Why engage the public on Guo Meimei on that particular date? Many netizens and foreign media alleged that the Chinese government directed the coordination of news as a distraction from the Yunnan earthquake, which had the potential to reveal failings of the government's earthquake preparedness. Netizens noted that the Weibo account of the _People's Daily_ posted a dozen times about Guo Meimei _before_ reporting on the Yunnan earthquake—the major news story of the day—on the morning of August 4.12 The focus on Guo, who had enraptured Chinese netizens years earlier, may have been a ploy to distract netizens from an unraveling crisis in southwest China.13 In response, the Chinese Red Cross posted on its Weibo, "Rescue teams are working through the night, and time is of the essence.... So please, forget Guo Meimei."14
In the previous chapter, I showed that small costs that inconvenience online users can have important effects on citizens' online behavior, the spread of information about political events in China, and the potential for coordination between highly educated, internationalized citizens and the public. In this chapter, I focus on a different form of porous censorship, _flooding_ , or the promotion of information, which changes the relative costs of access by making competing information cheaper and off-limits information relatively more expensive.
The difference between friction and flooding can be explained by an analogy to taxes on something the government would rather people not buy, like gas-guzzling cars. If the government wanted fuel inefficient cars to circulate less within the economy, then it could either tax these cars directly, increasing the cost and therefore decreasing demand, or subsidize fuel efficient cars, which would increase the relative cost of fuel inefficiency and decrease demand. Similarly, the government can affect the cost of information by making the information harder to access (friction), or by promoting competing or distracting information (flooding). If successful, flooding-based censorship should have very similar effects to friction on the spread of information and the behavior of individuals.
The actual content of flooding can take a variety of different forms. As I show in the rest of the chapter, like the case of Guo Meimei, flooding in China often occurs when the government promotes information completely unrelated to a negative event the government would rather not be salient in the minds of the public. I will show a number of cases of flooding where government-flooded information is meant to distract. In other cases, governments use flooding to downplay or to control the narrative of an event by mandating the promotion of their own views on the event that compete with alternative views. This might be more likely to happen when the government has already lost control of a narrative of an event so that they cannot simply ignore it. Last, flooding sometimes has a more long-term goal, not in response to one event directly but with the intention to shape citizens' perspectives on politics. In these cases, flooding might be used to promote the government's overarching narrative about world events or encourage citizens to have a positive outlook on their own life. This latter type of flooding may not be in response to one event in particular, but may be meant to shape citizen perspectives on politics and their reactions to subsequent crises.
In addition to affecting the relative costs of information, flooding also acts as a less observable form of censorship because it does not bring attention to the information the political entity is trying to hide. Information flooding occurs when a group or government promotes its viewpoint by ensuring that a particular piece of information or a particular perspective is repeated from many different sources in the news media or social media. Because of the repetition, this information is highly accessible and may be virtually impossible for citizens to avoid. Citizens are likely to come across the information and are likely to share it with others, and the more citizens spend time consuming the information, the more it will distract them from other ongoing events.
As I noted in the previous chapter, friction is less effective for governments during crises, as citizens may be more willing to go out of their way to seek out censored information and information may have already spread to large networks of people, making it difficult to control. When the government has less control over information it would like to stall, the government will often resort to spreading alternative versions of events or distractions to de-emphasize the negative information, even when they cannot prevent citizens from accessing it. Flooding can also be an attempt to gain control of the narrative of a quickly escalating story, rather than preventing citizens from knowing it in the first place. Flooding might also be relatively more effective when the public is searching for new information, or early in an unfolding crisis.15
Unlike friction, which can be measured by observing post removal or website blocks, flooding is more difficult to measure because it is often disguised as the typical spread of information. To empirically test how information flooding strategies implemented by the Chinese government influence the spread of information online, I describe coordinated efforts by the Chinese government to publish information in traditional and online media. Drawing on previous work,16 I show that coordination of information is designed as a censorship strategy by the Chinese government, to provide news stories and viewpoints that overwhelm existing news stories or online information. I identify instances of propaganda in Chinese newspapers using plagiarism detection software and identify propaganda posts by using leaked e-mail archives from Chinese online government commentators.17 I find that these flooding efforts reverberate in both the domestic and the international blogosphere, suggesting that the Chinese government is in fact effective in distracting from alternative news sources and promoting its own version of events.
In the next section, I review information flooding as a method of censorship and highlight how it differs from previous theories of propaganda. In the following section, I discuss the use of flooding strategies by the Chinese government. I then describe the data I use to identify the Chinese government's flooding efforts. In the last section, I estimate how information flooding influences the spread of information online.
**6.1 WHAT EFFECT CAN PROPAGANDA HAVE IN THE DIGITAL AGE?**
Using similar logic to scholars who maintain that censorship is impossible in a digital age, many scholars have argued that propaganda, or the promotion of information, is also outdated in the age of the Internet.18 These researchers maintain that propaganda can be effective only when the state can control the agenda, or when the number of sources of information is constrained. As the number of media sources has proliferated with the advent of the Internet, consumers of information have more choices over the sources of information. In these environments, consumers of information select out of biased state media sources and into sources with more "reliable" information.19
A few authors oppose this view, arguing that although state media is recognizable, propaganda is a signal of government power and therefore is closely followed by citizens.20 Propaganda, in these authors' views, creates norms that citizens are trained to follow.21 Even if this propaganda is unbelievable, by inducing participation in propaganda, the state can create rituals and standards that encourage compliance.22 In particular, in China, the government emphasizes propaganda to promote cultural governance, appealing to citizens' emotions to prevent protests and keep them in line with government policy.23
In this chapter, I show that propaganda can also be effective as a form of censorship in the online environment by influencing the relative costs of information. Political entities coordinate propaganda by repeating information from multiple sources so that it is low cost to citizens. Because such repetition increases the availability of information to citizens, the population becomes more likely to consume propaganda, regardless of whether they can identify the source. The "flooding" of information in the news media and blogosphere by political entities works not so much to signal power as to prioritize the consumption of government-produced news over news produced by other groups or by citizens themselves. Propaganda is effective because political entities have the resources to make it easy to access and low cost, and, for impatient online users, low-cost stories are more likely to be read and to reverberate throughout the blogosphere.
Coordination of propaganda has long been used by governments and other organized groups to promote information. China's 1977 Propaganda Directive explicitly directs the Propaganda Department to coordinate stories among the news media in order to "promote the CCP's current line."24 More recently, governments and mobilized interests around the world have organized "Internet armies" to flood the blogosphere at the same moment. Notoriously, the Chinese "Fifty Cent Party" allegedly pays Chinese netizens to post at the direction of the government. Although the Chinese have been criticized for this strategy, other governments have adopted similar strategies, including Israel, where representatives have been recruited to post on blogs that are "anti-Zionist,"25 and recently Turkey, which has a 6,000-member social media team to write pro-government posts.26
Not only the central government, but also individual Chinese government officials and companies use coordinated flooding strategies to bolster support. These groups pay public relations companies or unemployed citizens to post positive accounts of them online, like companies in the United States that will pay for positive reviews of their company on the web. Such actions have created scandals in China from time to time, because these local officials and companies regularly pay newspapers for articles that reflect positively on them.27
If governments, companies, and politicians were using such messages only to signal their own strength, they would want to take credit for these online messages. However, governments typically try to cover up the fact that they pay people to write online propaganda. Instead, they prefer that it appear as if "everyone" is writing pro-government comments or reporting a news story the government finds favorable or distracting. Part of the strategy of information flooding is issuing propaganda from many different sources, so as to disguise the fact that the information originated with the government.
Disguising the source of information flooding provides an added benefit to the government in that it has less potential to sully government credibility. Even though state news media already promote government versions of events, too much blatant government cheerleading may make the government media appear more biased. Except in obvious cases of coordination, like the Guo Meimei case described in the introduction, where netizens noticed the coordinated stories, most citizens may not realize that their local paper or the social media forums they visit are infiltrated with government propaganda. Incomplete control of the information environment allows the government to hide its own influence in the media by mixing with normal users.
By creating a multiplier effect in the news media and online, information flooding if successful can be worth the investment even though it is porous and cannot require readers to pay attention. The more sources a government or interest group can pay to cover their story from their perspective, the more other news groups and other social media users pick up the story and share it with others. Flooding begets more flooding, and, if effectively done, this domino effect of information dissemination can be exponential. What began as a propaganda message can seem like an online event created by citizens as more and more people read and share the story.
Flooding creates friction for stories that are less desirable to the government. In interviews with Fifty Cent Party members, artist-dissident Ai Weiwei reveals that Fifty Cent Party members are often instructed to distract from current stories that are less desirable to the government.28 This account is consistent with evidence from my work studying a leaked e-mail archive of online government-paid commentators that I describe later in this chapter.29 Flooding of entertainment and "soft news" stories like the Guo Meimei confession brings these stories to the forefront at the expense of stories that could shed a negative light on the government. If citizens are distracted by the accessibility of flooded stories, they are less likely to read other stories.
**6.2 FLOODING IN CHINA**
Coordination of information to produce such flooding is key to the information strategies of the Chinese propaganda system. Like many organized groups, the Chinese government is in the perfect position to coordinate because it has the resources and infrastructure to do so. First, the institution of propaganda in China is built in a way that makes coordination easy. The Propaganda Department is one of the most extensive bureaucracies within the Chinese Communist Party, infiltrating every level of government.30 It is managed and led directly from the top levels of the CCP.
From the very top of the Party, messages are coordinated throughout the news media within China, through every medium of news, including television, print, and radio.31 The government controls the personnel in every major media organization within China and requires each journalist to be government-certified. For day-to-day monitoring of content, the government issues propaganda directives to editors, who then decide what to include in the newspaper.32 Post-publication monitoring is conducted by retired propaganda officials who make sure that newspapers are following the issued guidelines.33
The extent of newspaper coordination within China has waxed and waned throughout recent Chinese history. During the Maoist period and Cultural Revolution, articles within the _People's Daily_ coordinated news around the country—smaller newspapers would reprint _People's Daily_ articles when instructed.34 With reform and opening after 1979, the coordination of news within China was significantly loosened and the Central Propaganda Department was weakened.35 In the lead-up to the Tiananmen Square pro-democracy movement in 1989, newspapers were less coordinated, and several Chinese newspapers became well known as critics of the Party.
The events of 1989 caused a complete reversal in the CCP's strategy toward propaganda and coordination. After the crisis in Tiananmen Square, the government decided to strengthen its grip on propaganda. For example, in 1990, one of the Party's leading news agencies, _Xinhua_ , was close to bankrupt. However, the government decided to use _Xinhua_ as the coordinating agency following 1989, instructing newspapers to follow _Xinhua_ 's lead on important events and international news, much as they had done with the _People's Daily_ during the 1960s.36 _Xinhua_ is now one of the most profitable news agencies in the country because it leads the coordination of news.
Coordination of government media has now extended beyond traditional media to the blogosphere. As mentioned earlier, paid government commentators, or the "Fifty Cent Party," promote government-sanctioned news online. In addition, the government is known to contact high-profile social media users and important online opinion leaders before important events in order to coordinate political messages among highly followed social media users in China.37
**6.3 DETECTION OF INFORMATION FLOODING IN NEWSPAPERS AND ONLINE MEDIA**
In this section, I will show that coordination of information across government newspapers and online is often used for the purpose of censorship. In particular, coordination of information is used to distract from or prevent the dissemination of other types of information that the government would rather the public not see. I will show that the Chinese government uses newspaper and online propaganda not only to persuade, but also to throttle access to other forms of information. In the next sections, I will provide evidence that such coordination does indeed influence the spread of information online.
To show that flooding is used for censorship, I must first identify propaganda. This is a difficult task because the spread of propaganda in China is a clandestine operation. For the most part, the government would rather that citizens not be able to distinguish propaganda from typical social media posts or the regular commercial news media. Stockmann (2012) describes how the commercialization of print media in China has allowed for propaganda to blend in with news, which may make it more believable and distracting for citizens, who may discount information that they believe originated with the government.38 The government's online propaganda program is also a secretive enterprise where online propagandists attempt to blend into the normal social media environment. In this section, I use leaked propaganda directives from the government and leaked e-mail archives linked to online propagandists to reverse engineer general propaganda trends. I show that the instructions and trends in propaganda are consistent with an information flooding strategy intended to distract from alternative viewpoints or events that could negatively affect public opinion toward the government.
_6.3.1 Identification of Propaganda in Newspapers_
To identify newspaper propaganda in China, my coauthor and I combined leaked propaganda directives published online by the _China Digital Times_39 with detection of coordination in a large collection of government newspapers.40 Since the Chinese government uses propaganda directives to facilitate coordination, we reverse engineered propaganda by identifying days when all newspapers publish the same or nearly identical articles. We used the leaked propaganda directives to validate this measure and describe in more detail the kinds of instructions provided by the government to promote coordination. We found that large numbers of coordinated newspaper articles appear during sensitive political meetings and around sensitive political scandals, seemingly to distract or downplay the events.
To find coordinated newspaper articles, we collected every newspaper article from twenty-five provincial and city newspapers in China for the year 2012. The newspapers were scraped from each newspaper's "digital" website. These sites are different from the online news sites in that they contain only digital copies of the printed newspaper and do not include online advertisements. Although some Chinese newspapers' websites differ from their printed papers, the articles on the digital website reflect the content of the printed newspaper exactly. We recorded the date on which each article was published, the full text, and the page number of each of the articles. In total, the dataset contains 111,789 articles during the year 2012.
To identify moments of coordination, we grouped the articles by the date on which they were written. For each day, we compared each pair of articles written on that date to look for overlap between articles using open-source plagiarism detection software called Copyfind.41 Copyfind works by identifying overlap between phrases of a specified length within the document pair. It then estimates the percentage of overlap in these phrases between the two documents, allowing for slight imperfections between phrase matches. High levels of phrase overlap indicate plagiarism, or in this case article coordination.
Of course, some forms of coordination between articles occur naturally, without directives from higher levels of government. Some newspapers may reprint _Xinhua_ or the _People's Daily_ not at the direction of the government, but to save on cost, a process called syndication. However, it is unlikely that coordination of identical stories across the vast majority of papers would occur naturally simply because of syndication. Only seldom will editors of the majority of papers decide to report on the same story. In the cases where the story was important enough that all papers would want to print an article covering it, typically a subset of the papers will devote resources to printing their own version of the event. Therefore, syndication that occurs naturally, not by design, should occur across small subsets of papers, and almost never across every paper. If an identical story were printed across every paper, there is a high chance that coordination was designed by the Propaganda Department.
We found that reprints of newspaper articles across papers corresponded with this expectation. Among groups of article reprints, more than 50 percent of articles had a total of only two newspapers that printed overlapping stories. More than 95 percent of overlapping articles were featured in fewer than ten newspapers, or fewer than half of the papers. Less than 1 percent of overlapping articles had more than fifteen newspapers that were coordinated. Printing of identical news stories among large numbers of newspapers is a rare phenomenon.
To validate that the coordination among large groups of articles is by government design rather than by chance, we looked for leaked propaganda directives that correspond to the coordinated articles identified by the algorithm. The _China Digital Times_ (CDT) contains a collection of both propaganda and censorship directives collected from journalists in China. These directives are posted on the CDT's website <http://chinadigitaltimes.net>.42 If some of the highly coordinated days also correspond to leaked directives, this will validate that our algorithm is picking up moments of government-induced coordination.
Many of the coordinated events we found within the newspapers had corresponding leaked propaganda directives. In one example, a leaked propaganda directive on November 18, 2012, indicated that all media should emphasize a _Xinhua_ article that urges readers to study and implement the 18th Party Congress's collective learning ideology.43 In the newspaper corpus we collected, nineteen of the twenty-five newspapers printed a version of this article that had over 70 percent phrase overlap with others that had printed the same article, and all of the newspapers for which we could collect page numbers printed the article on their first page.
In another example, in March 2012, a propaganda directive indicated that newspapers should report the _Xinhua_ version of a meeting between Wen Jiabao and reporters.44 Nineteen out of twenty-three newspapers printed that day included the same version of this story. On November 28, 2012, a leaked propaganda directive indicated that newspapers should follow _Xinhua_ in reporting on the death of Luo Yang, the main architect of the J-15 Chinese fighter jet.45 Seventeen of twenty-three newspapers that printed that day had the same reprinted version of the story. The correspondence between these leaked directives and the coordinated articles with reprints across many papers provides strong evidence that highly coordinated newspapers are often the result of government efforts in propaganda.
6.3.1.1 Coordination: Sensitive Time Periods, on Sensitive Issues
Now that we have developed a measure of propaganda across provincial and city newspapers, we can identify when coordination is used by the government, to better understand the purpose of newspaper propaganda. Does coordination happen at regular intervals? Or during particular time periods? When does the Party decide it needs to control the information environment?
Figure 6.1 maps the number of coordinated events over time, where at least 70 percent of newspapers printed the same article. The largest numbers of coordinated articles occurred during Party meetings, particularly the extremely sensitive period of the power transition between Hu Jintao and Xi Jinping in November 2012. During this time period, there was extremely high coordination among papers—at the highest point, four separate articles were coordinated on one day across almost all papers. The coordinated articles that are printed during the meeting are about the proceedings of the events and Party ideology. By forcing all papers to print the same version of the event, the Party prevents alternative interpretations of the most high-level Party meetings and spreads the Party's perspective on its own governance.
The prevalence of propaganda during this period also aligns with journalists' own account of Party media control around important meetings. Journalists reported being required to avoid any negative news or commentary for the entire month of November 2012.46 Further, many of the articles coordinated during this period were printed on the front pages of the papers. The Party's interpretation of the meetings and news about the meetings were by design the first thing newspaper readers saw during the November transition of power from Hu Jintao to Xi Jinping, de-emphasizing other stories that could detract from the Party's spotlight or call into question the Party's power.
Figure 6.1: Number of times that articles are coordinated across more than 70% of papers by day, 2012. Highest levels of coordination appear during national meetings.
The other main cluster of coordinated news articles outside of the March and November meetings was in August 2012. This corresponds to an extremely sensitive event within the Party—the trial of the wife of the Party Secretary of Chongqing Bo Xilai, Gu Kailai, who was given a deferred death sentence in August 2012 for murdering a British businessman. The cluster of propaganda around this time may have occurred as a distraction from this sensitive event. Although not all of the articles discuss the event directly, some allude to it, urging Party members to understand the full meaning of "socialism with Chinese characteristics," and encouraging Party discipline.
Coordination seems to be not only a tool to distract from sensitive events within the Chinese government but also a means of control of newspapers so that alternative views of these events cannot gain traction in the media. Articles relating to the sentencing of Wang Lijun and Gu Kailai in 2012 are frequently coordinated across papers and appear in the leaked propaganda directives, but the message is to de-emphasize the event. The purpose of coordination in these circumstances is to prevent the newspapers from providing alternative accounts of the incident or sensationalizing it, thereby explicitly censoring the editorial leeway of the newspapers.
Similarly, discussion of policies on the environment, corruption, food safety, real estate prices, and relations with Japan are also sometimes coordinated, again perhaps to prevent sensationalist versions of these events. Leaked propaganda directives associated with these events tend to instruct that they should be de-emphasized to the back pages of the newspaper and not "hyped."47 Coordination, in this case, is used not to distract, as it is during Party meetings or sensitive periods, but to control the number of alternative stories about the policy implementation or sensitive event.
Overall, in this section, I showed that newspaper propaganda is often used as a tool of censorship by filling the front pages of the newspapers during sensitive periods and controlling reporting and placement of stories within newspapers on sensitive topics. Rather than always being used to persuade or cajole the reader, as we often think of propaganda, the flooding strategy of the Chinese government is to affect the likelihood that readers come across particular articles or accounts of events. In the next section, I look to see whether these same strategies also appear in the Chinese government's approach to online propaganda.
_6.3.2 Identification of Online Propaganda_
To identify the online flooding strategy employed by the Chinese government, I turn to leaked e-mail archives from a local propaganda department. These archives, leaked by blogger "Xiaolan,"48 provide several years of e-mail correspondence between a local propaganda department in Zhanggong county in Jiangsu province and government officials who had been tasked with posting online propaganda. Many of the e-mails include posts that internet commentators nicknamed the "Fifty Cent Party" made at the direction of the local propaganda department. Although we have no way of knowing whether the e-mail archive is complete, the leaked e-mails give us a window into the instructions and propaganda posts that were coordinated by government entities over this period.
In order to identify the online propaganda, a team of research assistants went through each of the 2,000 e-mails in the archive to extract the details and reports of the online propaganda posts. In total, the research assistants identified 43,757 online propaganda posts on around 2,000 accounts over a two-year time period.49 A full description of the post collection process and full analysis of the posts are included in King, Pan and Roberts (2017).
6.3.2.1 China's Fifty Cent Party: Highly Coordinated Cheerleading
If propaganda were meant to persuade, online propaganda posts should address the political questions that are highly contested within the blogosphere. Current conceptions of online propaganda in China posit that the Fifty Cent Party is primarily tasked with countering anti-government rhetoric online. Social media users are accused of being Fifty Cent Party members when they defend government positions in heated online debates about policy, or when they attack those with anti-government views. In large part, scholars and pundits have viewed Fifty Cent Party members as attackers aimed at denouncing or undermining pro-West, anti-China opinion.50 Fifty Cent Party members, for the most part, have been seen in the same light as traditional propaganda—as intending to persuade rather than to censor.
For the most part, however, the leaked online e-mail archive containing Fifty Cent Party posts does not suggest that the purpose of Fifty Cent Party posts is to take on critics of the Chinese government, or persuade people to support Chinese government policy. Very few of the thousands of posts in the archive were _argumentative_ in nature or were aimed at defending the government against attackers or attacking its critics. Instead, the vast majority of Fifty Cent Party posts seem to be designed to distract from political arguments happening on the Internet. Like coordination among newspapers, the coordination of online propaganda serve as censorship or distraction, rather than for persuasion.
Instead of attacking, the largest portion of Fifty Cent Party posts in the leaked e-mail archive were aimed at _cheerleading_ for citizens and China—patriotism, encouragement or motivation of citizens, inspirational quotes or slogans, gratefulness, or celebrations of historical figures, China, or cultural events. Many of the posts were not even political in nature. For example, many remember heroes who sacrificed for China: in ite "是你们抛头颅洒热 血换来了我们今天的幸福生活,向你们致敬! 你们永远活在 我们心中!" "We salute you who shed your blood in exchange for our happy life today! You will always live in our hearts!" Others encourage citizens to keep trying to achieve their goals: "幸福不易须珍惜,明日振兴催奋进。" "Happiness doesn't come easily, so treasure it; tomorrow reenergize and advance with courage."
In order to measure the target of propaganda, we divided a random sample of 200 leaked Fifty Cent Party posts into five different categories. If propaganda posts were meant to persuade, the posts should fall into either (1) argumentative praise or criticism, including praise or criticism of the government that takes a position vis-à-vis another viewpoint; or (2) taunting of foreign countries, including comparisons of China to other countries with the sentiment that China is better, or insults toward other countries. These first two categories are how most pundits had described the purpose of the Fifty Cent Party in the past. If posts were meant to distract or change the subject, posts would fall into (3) non-argumentative praise or suggestions, including praise of current government officials, programs, or policies, which is not responding to an alternative viewpoint; (4) cheerleading for China, including patriotism, encouragement or motivation of citizens, inspirational quotes and slogans, thankfulness, gratefulness, inspiration or gratefulness for historical figures, or cultural references and celebrations; or (5) factual reporting, including descriptions of current government programs, projects, events, or initiatives, without praise or criticism.
We found that the majority of the leaked Fifty Cent posts, 85 percent, fell into the cheerleading category. The second most prevalent type of post was non-argumentative praise or criticism (11 percent). The last type of post we found within the sample were posts that contained factual reporting about government programs or policies (4 percent). In the random sample, we found no examples of taunting of foreign countries or argumentative praise or criticism. In King, Pan and Roberts (2017) we use automated methods to extend this analysis to the rest of the post and to accounts that we predict to be associated with the Fifty Cent Party in other provinces. We find similar results; overwhelmingly, the posts we identify cheerlead and distract, rather than argue online.
The observation that most of the posts seem to be intended to make people feel good about their lives—and not to draw attention to anti-government threads on the Internet—is consistent with recent indications from Chinese propaganda officials that propagandists attempt to promote "positivity." The Chinese Communist Party has recently focused on encouraging art, TV shows, social media posts, and music to focus on creating "正能量," or "positive energy," to distract from increasingly negative commercial news.51 Although sources do not directly lay out the reasoning behind this positive energy strategy, reorienting the public toward the positive instead of trying to counter negative criticism is a way to distract the public from negative online discourse without drawing more attention to it.
Not only do the leaked e-mails show that online flooding in China is aimed at generating positive sentiment rather than countering criticism, the timing of the posts themselves provide indications that online propaganda is used during sensitive periods and to distract from highly sensitive events. Figure 6.2 shows a timeline of the posts retrieved from the e-mails. Although the leaked archive gives us only a small view of the online propaganda system in China, the online propaganda effort like the coordination of articles in the newspapers is quite "bursty," or focused within particular periods. The two major bursts within the time period are focused on promoting Chinese patriotic festivals, including Qingming festival, which is a traditional Chinese cultural holiday to celebrate ancestors, and Martyr's Day, a new day created by the CCP to celebrate military heroes. A third burst is focused on the promotion of Xi Jinping's slogan the "Chinese Dream," which was a major focus of propaganda during the spring of 2013. A group of propaganda posts is also clustered around the Third Plenum of the 18th Party Congress, an important Party meeting in November 2013, and another group of posts comments on a local government development initiative.
However, outside of these cultural, slogan, and meeting bursts, there are two large bursts of propaganda posts—one at the end of June and beginning of July 2013, and another in May 2014—which do not have an obvious purpose. These bursts contain typical cheerleading posts, making it difficult to determine the reason the posts were concentrated within such a short period.
A closer look at the e-mails associated with the first burst reveals that this concentrated set of positive propaganda may have been aimed at distracting from riots against the government in Xinjiang, which occurred on June 26, 2013, and killed 27 people.52 The posts at the beginning of the burst are associated with an e-mail from the Zhanggong propaganda department written to a blind copied group of individuals only a few days after the riots with examples of online propaganda condemning terrorism in Xinjiang, promoting national unity, and promoting harmony between nationalities. One hour later, the Zhanggong propaganda department reported to the higher-level city of Ganzhou that its team had posted hundreds of microblogs promoting positivity, the Chinese Dream, tolerance, diversity, and revitalization. The timing of the e-mails suggests that the propaganda posts were a follow-up to the original posts that condemned terrorism in Xinjiang. It also suggests that when online propaganda is meant to react to a crisis only a small number of propaganda posts actually address the event directly. Instead, most posts are focused on distracting from the negative event with "positive energy."
Figure 6.2: Timeline of Fifty Cent Party posts in leaked e-mail archive, reproduced from King, Pan and Roberts (2017).
The second burst of posts does not have an e-mail trail like the first burst, but its timing is also associated with an event in Xinjiang. On April 30, 2014, a knife attack and bombing in the Urumqi railway station killed three people and injured dozens immediately following a visit by Chinese President Xi Jinping.53 In the days that followed, thousands of posts appear in the leaked dataset that cover a wide range of topics, including the people's livelihood and good governance. Although there is no direct evidence in the e-mail archive that the burst is a response to the bombing, the parallels between this burst and the one in June 2013 provide suggestive evidence that this burst, too, was created with the purpose of distraction.
**6.4 THE INFLUENCE OF FLOODING ON THE SPREAD OF INFORMATION**
Having described the logic behind flooding, I now show that government coordination of information has an important influence on the prevalence of information that appears within social media. Either because government efforts to spread information are high in volume or because netizens unwittingly pick up flooded information and reshare it, increases in coordination are associated with significant increases in the spread of government-sanctioned information online. As the ratio of government-initiated to citizen-initiated information online increases, citizens will be more likely to come across government propaganda relative to alternative viewpoints and will be more likely to share them.
To show this, I estimate the impact of both the coordination of newspaper articles and paid online commentators on the web. I find that the _particular wording_ of the coordinated newspaper article spreads throughout both the domestic and also, more surprisingly, the international blogosphere, showing that information flooding strategies have a multiplicative effect on the spread of information. Next, I show that online discussions of Qingming festivals in recent years are many times more likely to reflect the government's framing of the festival, evidence that online propaganda indeed influences the tone of the conversation.
_6.4.1 How Does Newspaper Coordination Influence the Spread of Information?_
How does the coordination among newspapers in China influence the spread of information? If information is sufficiently coordinated, are others more likely to reprint these stories? Do social media users and other commentators pick up the same language used in the coordinated newspaper articles?
To answer these questions, I estimate whether newspaper articles that were more coordinated were more likely to appear on non-news sites and within individual social media posts. To do this, I take consecutive word strings from more coordinated articles to see if they appear more frequently on the web than those where there is less coordination among newspapers. For each unique cluster size of coordinated newspaper articles ranging from two to twenty, I sample 100 coordinated articles.54 For each of these 1,290 articles, I sample five 30-character strings randomly from the text of the article. I then use the Google API to search each of these 6,450 strings on Google and record the number of search results returned.
I obtain three different search result metrics. First, I count the number of search results Google returned overall. I also count specifically the number of search results returned on sina.com.cn,55 the most popular blogging site in China. Last, I count the number of search results returned on blogspot.com, the most popular blogging site within the United States, which is blocked by the Great Firewall in China.56
I measure the relationship between coordination and the number of search results, both within and outside of China. Figure 6.3 shows the number of coordinated papers on the x-axis, and the log of the search results on the y-axis. There is a very strong correlation between the degree of coordination across newspapers and number of search results containing the strings of the coordinated articles.
It could be that these results are simply a reflection of coordination by government propaganda workers who could repost the same wording as in the provincial newspapers at the direction of the Chinese government. However, within the leaked Fifty Cent posts, we did not see examples of the Fifty Cent Party reposting copies of traditional media stories on the web. We would also not expect the Fifty Cent Party to write social media posts on websites outside of the Great Firewall, such as blogspot.com, which are less likely to be frequented by the average Chinese citizen. The evidence suggests that the link between coordination and social media mentions is not simply through other coordinated measures initiated by the CCP. Instead, the multiplicative relationship between coordination and search results likely indicates that the propaganda is being reprinted by regular Chinese citizens and also by citizens abroad.
_6.4.2 How Do Online Commentators Influence the Framing of a Holiday?_
How do government online commentators influence the tenor and framing of online conversations? In this section, I study how the tenor of the online conversation about the Qingming festival in China has changed as the government's effort to affect online conversations through paid commentators has increased in recent years.
Figure 6.3: Relationship between average coordination and number of Google search results for all websites, sina.com.cn, and blogspot.com.
The Qingming festival, also known as the Tomb Sweeping Festival or the holiday of Pure Brightness, occurs every spring and is traditionally a time when Chinese pay respect to their ancestors by visiting their graves and presenting them with offerings. The holiday has its origins in ancient China, but was largely disallowed in the Maoist era. After reform, the Qingming festival has returned to China and is now widely observed throughout the country.57
Qingming is problematic for the Chinese government because these rituals have frequently turned political. The Tiananmen Incident in 1976 originated in a gathering to remember former Premier of China Zhou Enlai on Qingming, but turned into a protest against government officials. The holiday brings attention to graves of those who were killed during sensitive political events, like the 1989 Tiananmen Square protest. In recent years, before the Qingming festival, in an effort to prevent instability, government police have detained outspoken family members of those killed on June 4, 1989.58
As Qingming has become more popular in recent years, the Chinese government has made an effort to take back the festival for its own purposes. In particular, the CCP has tried to link Qingming to the recognition of Communist martyrs by showing officials on television visiting the graves of famous revolutionary heroes.59 This framing is consistent with an increasing emphasis on the CCP as a nationalist, unifying force by emphasizing the sacrifices the party has made to unify and provide stability and prosperity for the country and distracting from the more contentious history of the Qingming holiday.
The "martyr" framing of Qingming is reflected in the leaked Fifty Cent Party posts discussed earlier in this chapter. The largest volume spike in the leaked archive occurred on the Qingming festival (figure 6.2). Many of these posts link Qingming with revolutionary heroes. The burst contains posts such as: "Qingming is the day where we relive the fire of the revolution and we commune with the martyrs." ("清明是重温革命烽火,与先烈对话的日子"), "Mourn the martyrs, the great men who gave their lives for the birth of new China. Because of them we now have a happy life! Because of them we now have international status! Because of them the Chinese people can finally once again stand proudly among the nations of the world." ("深切悼念那些为新中国诞生而献身的英烈和伟人们,因为 有了他们我们才有现在的幸福生活! 因为他们,我们才有现在 的国际地位! 因为他们,中华民族才又一次傲立世界民族之林.")
Can the CCP be successful in changing the tenor of the tomb sweeping festival with its coordinated framing of the festival? To study how the online conversation has changed as the government has increased its focus on online propaganda,60 I gathered a random sample of social media posts on baidu.com, sina.com.cn, and sohu.com that mentioned the word "Qingming" during April 1–April 5 of each of the years 2012–2016. In each year, I counted the proportion of posts that mentioned the word "martyr" (either "先烈" or "烈士") during that period.61
The results are plotted in figure 6.4. Since Xi Jinping took office, there has been a remarkable six-fold increase in discussion of martyrs online associated with the Qingming festival. Although the word "martyr" was used in less than 0.5 percent of posts that mentioned "Qingming" on these websites in 2012, in 2014–2016 almost 3 percent of all posts that mention "Qingming" also mention "martyr." The increase in prevalence of posts equating Qingming with martyr's day reflects a shift in the way that online information portrays the holiday and its purpose to the public. It also shows that the government can wield significant influence over the tenor of the online discussion through flooding.
Figure 6.4: Proportion of posts about Qingming that mention martyrs, April 1–5, 2012–2016.
**6.5 CONCLUSION**
Information flooding is the least identifiable form of censorship of all the mechanisms described in this book. Particularly with the expansion of the Internet, the government can hide its identity and post online propaganda pretending to be unrelated to the government. Coordinated efforts to spread information online reverberate throughout social media because citizens are more likely to come across them and share them. Such coordination can distract from ongoing events that might be unfavorable to the government and can de-prioritize other news and perspectives.
We might expect that coordinated government propaganda efforts would be meant to persuade or cajole support from citizens on topics that citizens criticize the government about. However, the evidence presented in this chapter indicates that governments would rather not use propaganda to draw attention to any information that could shed a negative light on their performance. Instead, governments use coordinated information to draw attention away from negative events toward more positive news or their own overarching narrative, or to create positive feelings about the government among citizens. This type of flooding is even more difficult to detect, and dilutes the information environment to decrease the proportion of information that reflects badly on the government.
1 Jacobs, Andrew, "Earthquake Kills Hundreds In Southwest China," _New York Times_ , August 3, 2014, <http://www.nytimes.com/2014/08/04/world/asia/deadly-earthquake-in-southwest-china.html>
2 Hong, Haolan, and Jaime FlorCruz, "Red Cross China in Celebrity Crisis," _CNN_ , July 6, 2011, <http://www.cnn.com/2011/WORLD/asiapcf/07/06/china.redcross/>.
3 "警方透露郭美美赌球犯罪细节: 8人团伙境外网站开户," _iFeng_ , <http://ent.ifeng.com/a/20140710/40171148_0.shtml>.
4 "从炫富到涉赌, 她为何堕入犯罪深渊? 一一 郭美美涉嫌赌博犯罪 罪被刑拘的背后," 新华, <http://news.xinhuanet.com/legal/2014-08/04/c_1111914547.htm>.
5 "郭美美设赌局牟利数十万" 商演 "为名从事性交易," _CCTV_ , <http://news.cntv.cn/special/video/guomeimei/index.shtml>.
6 "郭美美案件引外媒关注:郭美美栽了 公权力莫再栽," _Global Times_ , <http://world.huanqiu.com/exclusive/2014-08/5095684.html>.
7 "央视播出郭美美画面: 素颜戴眼镜 戴着手铐下楼(图)," _Caijing_ , <http://politics.caijing.com.cn/20140804/3641759.shtml>.
8 ""炫富女" 郭美美涉赌被刑拘 想对红会说声对不起," _Southern Weekend_ , <http://www.infzm.com/content/102888>.
9 "王军谈包养郭美美:她是我一生的噩梦," _Beijing Daily_ , <http://news.sina.com.cn/c/2014-08-04/023930624032.shtml>.
10 "郭美美: 23年素描: 20岁时因红会炫富事件走红," _Nanjing Daily_ , <http://news.sina.com.cn/c/2014-08-04/014130623848.shtml>.
11 "Minitrue: Guo Meimei's Confession," _China Digital Times_ , August 4, 2014, <http://chinadigitaltimes.net/2014/08/minitrue-guo-meimeis-confession/>.
12 Yuen, Lotus, "The Bizarre Fixation on a 23-Year-Old Woman," _ChinaFile_ , August 6, 2014, <https://www.chinafile.com/reporting-opinion/media/bizarre-fixation-23-year-old-woman>.
13 Boehler, Patrick, and Cece Zhang, 'I Like to show off': Chinese Celebrity Guo Meimiei Confesses to Prostitution, Gambling Charges on State TV," _South China Morning Post_ , August 4, 2014, <http://www.scmp.com/news/china-insider/article/1566142/i-show-guo-meimei-confesses-all-charges-cctv-broadcast>; "Heavy Media Takedown of Guo Meimei Angers Chinese Netizens," _Offbeat China_ , August 4, 2014, <http://offbeatchina.com/heavy-media-takedown-of-guo-meimei-angers-chinese-netizens>.
14 Larson, Christina, "Stated Confession of Alleged Call Girl Guo Meimei Distracts From a Charity's Earthquake Relief Efforts," _Bloomberg_ , August 5, 2014, <http://www.bloomberg.com/news/articles/2014-08-05/guo-meimei-falsely-claimed-to-be-a-big-shot-at-chinas-red-cross>.
15 Baum and Groeling (2010) have found that the public is more manipulable in democracies at the beginning of a conflict, when they have accumulated less information.
16 Roberts and Stewart (2016).
17 King, Pan and Roberts (2017).
18 Lynch (1999, pg. 3–4), Lieberthal (1995).
19 See Stockmann (2012, Chapter 8).
20 Huang (2015).
21 Brady (2008, pg. 134).
22 Wedeen (1999).
23 Perry (2013).
24 Brady (2008, pg. 15).
25 Liphshiz, Cnaan, "Israel Recruits 'Army of Bloggers' to Combat Anti-zionist Web Sites," _Haaretz_ , January 19, 2009, <http://www.haaretz.com/print-edition/news/israel-recruits-army-of-blogers-to-combat-anti-zionist-web-sites-1.268393>.
26 Albayrak, Alya and Joe Parkinson, "Turkey's Government Forms 6,000-Member Social Media Team Volunteers to Promote Ruling Party's Perspective in Sphere Dominated by Protestors," and _Wall Street Journal_ , September 16, 2013, <http://online.wsj.com/news/articles/turkeys-government-forms-6000member-social-media-team-1379351399>.
27 Chen, Wang, Shanshan Wang, Zhongyuan Ren, and Yishi Zhu, "Dirty Business for China's Internet Scrubbers, _CNBC_ , February 19, 2013, <http://www.cnbc.com/id/100472398>.
28 "An Insider's Account of the '50 Cent Party,'" _Freedom House China Media Bulletin_ , May 12, 2011, <http://www.freedomhouse.org/article/china-media-bulletin-issue-no-22/#2>.
29 King, Pan, and Roberts (2017).
30 Lieberthal (1995, pg. 194–199)
31 In this chapter, I focus explicitly on coordination within the print news and online media in China.
32 Brady (2008, pg. 19).
33 Brady (2008, pg. 22).
34 Yu (1964, pg. 110–121).
35 Brady (2008, pg. 40).
36 Brady (2008, pg. 113).
37 This strategy is described in case studies in propaganda documents, such as National Academy for Propaganda Cadres (2011).
38 See Stockmann and Gallagher (2011) for a description of how this phenomenon has impacted perceptions of Chinese legal policy.
39 <http://chinadigitaltimes.net/>.
40 Roberts and Stewart (2016).
41 Copyfind software available at: <http://plagiarism.bloomfieldmedia.com/z-wordpress/software/copyfind/>.
42 The vast majority of the leaked directives the CDT collects are censorship directives, detailing what the newspapers should not print rather than what they should print. Some of the CDT directives are a combination of censorship and propaganda directives, suggesting that if papers would like to write on a topic, they can only use the _Xinhua_ or _People's Daily_ version of events.
43 "【真理部】 中央政治局集体学习," _China Digital Times_ , <http://chinadigitaltimes.net/chinese/2012/11/%E4%B8%AD%E5%AE%A3%E9%83%A8%EF%BC%9A%E4%B8%AD%E5%A4%AE%E6%94%BF%E6%B2%BB%E5%B1%80%E9%9B%86%E4%BD%93%E5%AD%A6%E4%B9%A0/>
44 "【真理部】 温总理记者见面会," _China Digital Times_ , <http://chinadigitaltimes.net/chinese/2012/03/%E4%B8%AD%E5%AE%A3%E9%83%A8%EF%BC%9A-%E6%B8%A9%E6%80%BB%E7%90%86%E8%AE%B0%E8%80%85%E8%A7%81%E9%9D%A2%E4%BC%9A/>
45 "【真理部】 罗阳逝世," _China Digital Times_ , <http://chinadigitaltimes.net/chinese/2012/11/%E4%B8%AD%E5%AE%A3%E9%83%A8%EF%BC%9A%E7%BD%97%E9%98%B3%E9%80%9D%E4%B8%96/>
46 Duggan, Jennifer, "China Internet Censored for Party Congress," _Al Jazeera_ , November 17, 2012, www.aljazeera.com/indepth/features/2012/11/20121115105540550384.html.
47 "Directives from the Ministry of Truth: Food Safety," _China Digital Times_ , August 17, 2012, <http://chinadigitaltimes.net/2012/08/directives-from-the-ministry-of-truth-food-safety/>.
48 <https://xiaolan.me/50-cent-party-jxgzzg.html>.
49 The archive's authenticity was verified by locating posts from the archive online.
50 Bandurski, David, "China's Guerrilla War for the Web," September 24, 2008, <https://blogs.law.harvard.edu/guorui/2008/09/24/chinas-guerrilla-war-for-the-web/>; Lam, Oiwan, "When China Briefly Unblocked Facebook, Trolls Rushed In," _Hong Kong Free Press_ , November 26, 2015, <https://www.hongkongfp.com/2015/11/26/when-china-briefly-unblocked-facebook-trolls-rushed-in/>.
51 "正义网: 时代需要" 周小平们 "的网络正能量," 正义网 <http://www.jcrb.com/opinion/zywy/201410/t20141023_1443348.html>, "广电总局: 鼓励拍摄 正能量电视剧," 北京商报 <http://it.sohu.com/20120904/n352242285.shtml>, "媒 体要准确把握舆论导向 传播社会能量," 华商网 <http://ehsb.hsw.cn/shtml/hsb/20141111/191042.shtml>, "鲁炜: 让网络空间清朗起来的 "六个具体目标", <http://news.ifeng.com/mainland/special/luwei/content-4/detail_2013_10/30/30805526_0.shtml>
52 Forsythe, Michael, "Xinjiang Violence Leaves 27 Dead after Clash with Police," _Bloomberg_ , June 26, 2013, <http://www.bloomberg.com/news/articles/2013-06-26/xinjiang-violence-leaves-27-dead-after-attack-on-police-stations>.
53 "Deadly China Blast at Xinjiang Railway Station," _BBC_ , April 30, 2014, <http://www.bbc.com/news/world-asia-china-27225308>.
54 For some coordinated cluster sizes, there were not 100 unique instances in my dataset. For these clusters, I took all of the coordinated articles.
55 Using site: sina.com.cn within the search results.
56 I used the Google API to do this, and close examination indicates that the number of returned search results is fairly accurate. The number of search results when you simply search Google from a desktop is often very inaccurate, which is why using the API (application programming interface) is important.
57 See Johnson (2016) for a discussion of the history of the Qingming festival and its current treatment today in China.
58 Laris, Michael, "Tiananmen's Edgy Proximity," _Washington Post_ , April 6, 1999, <https://www.washingtonpost.com/archive/politics/1999/04/06/tiananmens-edgy-proximity/bcb50b9f-9f7b-4bb6-9b22-d7badbded429/>.
59 Johnson (2016).
60 Bandurski, David, "A 'Year of Innovation' for Internet Controls," _China Media Project_ , January 7, 2016, <http://cmp.hku.hk/2016/01/07/39575/>.
61 Posts were sampled from the online social media analytics company Crimson Hexagon.
CHAPTER SEVEN
Implications for a Digital World
When I was in China in 2006, coming across blocked websites was less common than it is today. Some Wikipedia pages were blocked and some links in the _New York Times_ website would return an error, but otherwise, I could access information I would normally use with no extra effort. I posted pictures on Facebook, I used Google. Twitter had not started yet, but I instant messaged through Skype. The paradigm of incomplete censorship had not yet been widely applied—the Great Firewall did not affect many websites and did not yet create major differences in the accessibility of online news between people living in the United States and those in China.
Each subsequent time I visited China, the government had added friction to information from the Western world. Google and Gmail can no longer be accessed without a VPN. Facebook, Twitter, Instagram, the _New York Times_ , YouTube—so many of the Internet content providers from which Internet users around the world receive their news—are no longer available in China without evading censorship. When traveling to China, I have become accustomed to using a VPN to do simple things like keep up with e-mail, collaborate with coauthors on Google Drive or Dropbox, or wish a friend happy birthday on social media.
Despite being _able_ to access websites from the United States, doing so was frustrating and time consuming. I have found myself increasingly turning to the vibrant and ever-expanding world of Chinese social media. I signed up for WeChat, Weibo, QQ, and 126 e-mail because these platforms were more convenient. These companies are some of the most exciting technology companies in the world, with innovative platforms that attract audiences in their own right. However, the politically inspired division between the networks, information, and online communities in China and in the rest of the world worries me. Although the difference between the Chinese and English languages naturally creates its own barrier and imposes friction on sharing information, understanding, and friendships across the Pacific, the seclusion of Chinese social media users from the rest of the world on the Internet as a result of censorship creates an environment that could exacerbate misunderstandings, nationalism, and differences in perceptions between the population of one of the most important and powerful countries and the rest of the world. Despite many pundits' characterization of porous censorship in China as "futile,"1 this book provides evidence that typical users are affected enormously by these small costs of information without always realizing it, providing credence to worries of polarization not only within China but also between China and the rest of the world.
**7.1 WHY POROUS CENSORSHIP MATTERS**
This book has shown the surprising effects of porous censorship, or small but circumventable taxes on information, on what citizens read, share with others, and think about politics. The perception that the Internet allows us the possibility to access so much information reassures us that governments and political entities cannot completely hide information that could keep them accountable. But the impatience and indifference to politics inherent in the busy schedules of typical citizens mean that ease of access has strong effects on what information the majority of the public consumes and shares. With the decline of the traditional media, increases in hidden forms of censorship combined with the ability of governments to engage in online political propaganda have created an environment where political entities have a surprising amount of control over what information is easiest for their constituents to come across. Censorship that functions through friction and flooding can exacerbate divisions and reduce political entities' accountability even when they do not explicitly make any information off-limits.
In this book, I have described the ways in which citizens can be affected by censorship. Fear affects citizens through deterrence, by making individuals frightened to share or access information. Fear is effective only when it involves a credible threat, when governments are able to credibly commit to punishing individuals for speech. Fear without credibility can have unintended negative consequences for governments, by creating a public backlash against censorship or by creating incentives for citizens to conceal information the government would find valuable. Friction influences the spread of information by making it more difficult to access, and similarly flooding affects citizens by introducing information that distracts from the information the political entity would rather keep off-limits. Friction and flooding are more effective when citizens' elasticity of information is high, or when information has more substitutes. Users are even more affected when they do not know that censorship is happening because they do not have the awareness necessary to counteract it. Search filters, throttling, and paid online propaganda commentators can be explained away by a search engine's algorithm, too much Internet traffic, or a slew of concerned citizens. Friction and flooding, although technically easy for citizens to circumvent, divert and distract the public, particularly when their real purpose and potential consequences go unnoticed.
The influence of fear, friction, and flooding on individuals' consumption of information has clear implications for government strategies of censorship in the digital age. Porous censorship is useful to authoritarian regimes precisely because only some individuals circumvent it. Fear, which is difficult to use to credibly target millions of Internet users, can be primarily used against high-profile individuals and the traditional media who could influence large numbers of people. For the typical Internet user, censorship based on friction and flooding is used to nudge individuals away from conversations, focal points, and networks the government prefers they would not be involved in, separating potential support from the activist core.
This strategy of porous censorship mitigates many of the costs that fear-based censorship creates for authoritarian governments. Fear comes with the potential for backlash against censorship or, if applied too broadly, may limit the ability of governments to collect true information from citizens and discourage complaints that keep local government officials in check. Porous censorship, on the other hand, frustrates the vast majority of citizens from accessing information the government deems dangerous, while not making any information explicitly off-limits and allowing online consumers to feel as if all information is possible to access.
Using online experiments, large social media datasets, datasets of newspapers, leaked archives of propaganda, and a nationally representative survey, I provided evidence that supports this theory of censorship in the empirical sections of this book, showing that the vast majority of Chinese Internet users' consumption and production of information is not affected by fear and deterrence. As I showed in chapter 4, when citizens observe censorship, they become _more_ interested in the information, seeking out information that is related to it and continuing to write about off-limits topics. Because online users are looking for signals of topics' importance, censored information draws attention to topics the government would rather not be discussed.
In contrast, in chapters 5 and , I show that censorship that taxes information has large effects on the typical individual's production and consumption of information. Internet users do not circumvent the Firewall because they do not know it exists, find it bothersome, or have no reason to. Small variations in barriers to access, like blocked websites, censors' schedules, or the timing of coordinated propaganda have large impacts on the amount of information about a topic online. Even though many people report being angered by censorship, when it goes unnoticed it can have surprisingly large impacts.
However, because porous censorship can be circumvented, it has vulnerabilities. During crises, or in periods when censorship suddenly disrupts their habits, even typical citizens are likely to search out information regardless of the costs. As I have highlighted throughout this book, moments of crisis and instability may make it difficult for authoritarian regimes to control access to and spread of information because citizens' demand for information becomes more inelastic. For authoritarian regimes that adopt these strategies, I expect that unexpected, sudden crises that draw large-scale public attention—like the Tianjin explosions described in chapter 5—will likely be the moments when information will be less affected by censorship and governments are more likely to be held accountable.
**7.2 AUTHORITARIAN RESILIENCE**
The findings in this book speak to a growing literature that puzzles over the resilience of authoritarian governments in the face of the third wave of democratization and the expansion of the Internet.2 The evidence suggests that the resilience of authoritarian regimes is due not only to repression and responsiveness but also to other tools used to slow down coordination of collective action. The friction- and flooding-based methods of censorship described in this book do not employ force, make information impossible to access, or respond to citizens' concerns. Instead, they nudge citizens away from activist circles and alternative viewpoints that could facilitate collective action that is dangerous for the regime.
The evidence presented in this book suggests that authoritarian governments may adopt this strategy of porous censorship in part because citizens themselves are strategic consumers of information. Consumers are faced with a problem of information overload and are therefore seeking signals of information importance. Awareness of censorship can draw consumers _toward_ rather than _away_ from information. Repression and responsiveness address issues directly, drawing attention to the issue the government would like to ignore. Throttling or distraction, in contrast, distract and divert citizens to other topics.
In comparison to direct repression, porous censorship bolsters authoritarian resilience by manipulating citizens' incentives so that they choose, rather than are forced, to engage in the desired behavior. As Aldous Huxley, author of _Brave New World_ , writes in a forward to the second edition, "A really efficient totalitarian state would be one in which the all-powerful executive of political bosses and their army of managers control a population of slaves who do not have to be coerced." Rather than making books unavailable or information impossible to access, censorship taxes manipulate incentives so that most citizens _choose_ to consume information palatable to the government. It provides distractions and alternative arguments that citizens select into because they are more readily available and more widely circulated among the population.
The contrast this book draws between repression and porous censorship can be applied to other areas of authoritarian control and could predict ways in which new technologies could be used in the future. For example, experts now worry that as improvements in surveillance technology make online behavior easy to track, governments will increase their ability to track dissidents. Many governments around the world are expanding their surveillance programs, though the details of these programs are shrouded in secrecy. The Chinese government is experimenting with a social credit score, which could include information about what users write on the Internet and could have implications for the ability to borrow money, get a job, or be granted a visa.3 Although some say that such a credit system would provide a much-needed measure for lenders to evaluate potential borrowers,4 others say the idea stretches too far into the political behavior of individuals and instead could act as a mechanism to punish individuals for engaging in political conversation online.5
A framework centered on repression would expect autocrats to use surveillance technologies to track individuals online and make punishment more credible. Indeed, authoritarian governments have used these technologies to infect dissidents' computers and phones.6 Certainly, surveillance technology will make fear-based censorship easier for autocrats, and this is something that the wider academic and policy community should be greatly concerned about.
But less acknowledged is that the enormous data-collection programs that accompany surveillance could be used to personalize friction and flooding to a wide range of users. Surveillance could make friction and flooding more powerful by allowing for targeting of individuals with information that makes them less likely to come across information that damages the regime. Just as Google targets advertisements to users, governments could use information gathered from surveillance to target distracting information to users who might have recently read or searched for sensitive topics. In this way, governments could use surveillance to exert influence over users without even revealing that they are interfering, much less resorting to coercion.
As I have shown in this book, governments are already taking advantage of online information to target censorship within the population. But as the Internet becomes more customized and more data about consumers becomes available, so will censorship be customized to the individual rather than be applied in a blanketed way to the whole population. Such targeted censorship will increase its plausible deniability, as citizens will experience censorship differently. It may also further increase the digital divide between wealthy and marginalized populations,7 as those with less resources and education are less likely to be able to recognize and circumvent censorship when they encounter it.
**7.3 IMPLICATIONS FOR FREE SPEECH IN DEMOCRACIES**
The evidence and theory presented in this book have broader implications than just for information provision in autocracies. Traditionally, scholars have drawn a stark contrast between freedom of speech in autocracies and democracies by focusing on _freedom of expression_ , or an individual's liberty to express themselves in the public sphere.8 This perspective is based on the traditional structure of the media, where very few people expressed their views to the masses. Free speech theorists before the digital revolution worried that certain people would be excluded from broadcast media.9 Autocracies banned particular voices from entering into the media, whereas democracies allowed more freedom in who could take the public stage.
However, the introduction of digital media has made the contrast between democracies and autocracies less stark. As more people have begun to participate in the public sphere, the bandwidth for public speech has widened. As Balkin (2004) notes, the contest over free speech is not so much anymore _whether_ someone can take the public stage, but instead which voices will rise to the top and which will be lost in the cacophony. Censorship consists not only of preventing individuals from speaking but also of determining how their speech is prioritized and presented to the public, the contest over the "code" and structure of the Internet.10
In all societies, political entities have enormous amounts of power in how information is organized in the public sphere. Corporations wield significant power over the algorithms behind online search and there is evidence that the order of search results has electoral influences.11 Bureaucracies have a significant say in what data they collect and how available they make it. Interest groups and political parties invest large amounts of money in flooding the Internet with their own interpretation of events, drowning out the concerns of skeptics. As the financial belt of traditional journalism tightens, political groups have greater sway over affecting the content of articles by feeding journalists stories or refusing to respond to inquiries for interviews or data.
If the prioritization of information for political purposes has the impact of censorship, as this book suggests, then in democracies we have to rethink how we can protect free speech in a world of information overload. Perhaps the algorithms behind online search should be more transparent to the public so that we can have open debates about what voices get shuffled to the surface and what voices are buried. Perhaps we should concern ourselves more with competition between online search, social media, and Internet service providers to ensure that consumers are always provided a variety of perspectives. Maybe government data transparency should be standardized across bureaucracies to avoid selective, incentive-based transparency.12 As democratic environments have shown that they too are susceptible to friction and flooding,13 this book's broader perspective on censorship in the digital age demands that we at least ask questions about how we view freedom of information in democracies in the age of the Internet and what policies should be implemented to protect it.
**7.4 A CALL FOR FUTURE RESEARCH**
There is still much we do not understand about the political implications of censorship in authoritarian environments and the direction that these efforts are headed. As authoritarian governments develop sophisticated techniques to prioritize information for their citizens through porous censorship, we as researchers must keep up with our own study of the political impacts of censorship to help understand and inform others of the potential political results of these efforts. From an academic perspective, understanding the future of authoritarianism will require understanding how effectively these governments can control the agenda. From a policy perspective, understanding how citizens are influenced by media will allow us to better predict political outcomes and develop technologies that promote freedom of information.
Scholars should increasingly seek to find ways to study government censorship and propaganda efforts so that these strategies can be uncovered and made public. Here I describe two directions researchers might take to explore the implications of the findings in this book: better understanding the long-term economic, educational, and ideological impacts of censorship, and extending the theory to other countries and theories of repression.
_**7.4.1 Long-term Economic and Ideological Impacts of Censorship**_
While this book focused on the short-term impacts of censorship on access to information, future research should focus on enumerating the long-term impacts of censorship, including its economic, educational, and international implications. Despite the surprising impact of porous censorship, which I have highlighted, the findings presented here also suggest that censorship is accompanied with important costs to the Chinese economy, even if it is easily bypassed. Friction and flooding in China act like tariffs and subsidies of information, skewing the market for information and creating inefficiencies. Like all international trade protections, some Chinese companies benefit from these market distortions. Sina Weibo, Renren, and Baidu are all protected from international competition because their competitors Twitter, Facebook, and Google are blocked by the Great Firewall. Arguably, censorship has created space for some local Chinese companies to innovate and prosper without foreign competition.
For many technology companies in China, however, the frictions associated with censorship impose large costs on innovation. Start-ups working on developing new technology in China must use VPNs to access important coding tools such as those provided by Google. Social media companies must employ tens, sometimes hundreds, of censors to ensure that their content is in line with government guidelines. These taxes undoubtedly slow innovation in China and retard the growth of the technology industry. Enumerating the economic costs of censorship on the Chinese economy will help us better understand its impact on international trade and economic growth in China and the types of companies that stand to win and lose from the censorship apparatus.
Beyond the economy, the costs of censorship are also dramatically felt in the area of education. As the Chinese school system is one of the primary conduits for information, the government can control accessibility of information by controlling educational materials. Banning textbooks14 and censoring academics15 may make foreign information less accessible and easier to control, but it also throttles information from the world's leading scholars from entering the Chinese school system. Budding computer scientists in China start with a penalty, as they must use a VPN to use some of the world's leading software. Frustrated with government censorship, academics are likely to look for positions outside of China.16 A better understanding of the long-term impacts of censorship on human resources in China will be important to understand China's transition to a high-skilled economy.
The long-term ideological effects of censorship are perhaps the most important. Censorship creates disparities in access to information across political classes in China, exacerbating income disparities and potentially sowing the seeds of long-term political conflict. The political elite, tech-savvy, internationally educated Chinese citizens jump the Firewall easily, while those in rural areas are more affected by censorship because they are less aware of it and do not have the tools to circumvent it. Censorship is regressive in that it exacerbates the digital divide between classes. Perhaps more problematic, it creates an information chasm between the poor and the political elite that could turn into political or ideological conflict in the future.17
Though the digital era has made large strides toward bridging gaps between countries and cultures, censorship in China throttles cultural exchange between China and the West. As the Chinese government increasingly manipulates the media to prioritize its own version of international events, it also ties its hands in how it can resolve international conflicts.18 The bifurcation of media consumption between Chinese and Americans may slow the formation of common ground and common knowledge that may be essential to cooperation between East and West in the future. Future research seeking to understand how media bifurcation affects citizens' support of international cooperation is essential to understanding how censorship will affect the U.S.–China relationship.
_**7.4.2 Fear, Friction, and Flooding around The World**_
While the empirical sections of this book focused on the Chinese case, more work needs to be done to test the impacts of fear, friction, and flooding in other authoritarian environments. This research is increasingly imperative as evidence suggests that China's model of censorship is being exported to other regimes. How do other authoritarian systems of censorship differ and converge with the Chinese model? As the theory in this book suggests, the particular combination of fear, friction, and flooding will depend on the political structure, level of economic development, and technological capabilities of the regime. A better understanding of the predictors of the various incarnations of censorship would allow us to anticipate its implementation across political environments.
Similarly, are the reactions to the observation of censorship similar across authoritarian regimes? This book suggests that citizens' elasticity of demand for information varies by their interests, education, and information environment. Are there some countries and contexts where porous censorship is more or less effective? Are there political environments that tend to be more prone to backlash? Exploring the impact of censorship comparatively would allow us to understand how fear, friction, and flooding travel to other cases and environments.
1 August, Oliver, "The Great Firewall: China's Misguided—and Futile—Attempt to Control What Happens Online," _Wired_ , October 23, 2007, <http://www.wired.com/2007/10/ff-chinafirewall/>.
2 Nathan (2003); Anderson (2006); Gilley (2003).
3 Hamilton Gillian, "China's Social Credit Score Is Doomed to Fail," _Financial Times_ , November 16, 2015, <http://blogs.ft.com/beyond-brics/2015/11/16/chinas-social-credit-score-system-is-doomed-to-fail/>.
4 Tsang, Tim, "How New Credit Scores Might Help Bridge China's Financial Inclusion Gap," _Center for Financial Inclusion Blog_ , June 6, 2016, <https://cfi-blog.org/2016/06/06/how-new-credit-scores-might-help-bridge-chinas-credit-gap/>.
5 Chin, Josh, and Gillian Wong, "China's New Tool for Social Control: A Credit Rating for Everything," _Wall Street Journal_ , November 28, 2016, <https://www.wsj.com/articles/chinas-new-tool-for-social-control-a-credit-rating-for-everything-1480351590>.
6 Marczak, Bill, and John Scott-Railton, "The Million Dollar Dissident: NSO Group's iPhone Zero-Days Used against a UAE Human Rights Defender," _Citizen Lab_ , August 24, 2016, <https://citizenlab.org/2016/08/million-dollar-dissident-iphone-zero-day-nso-group-uae/>.
7 Norris (2001).
8 Scanlon (1972).
9 Balkin (2004, pg. 6).
10 Lessig (1999, Chapter 7); Stockmann and Luo (2017).
11 Epstein and Robertson (2015).
12 Bertot, Jaeger and Grimes (2010).
13 Byrnes, Nanette, "How the Bot-y Politic Influenced This Election," _MIT Technology Review_ , November 8, 2016, <https://www.technologyreview.com/s/602817/how-the-bot-y-politic-influenced-this-election/>.
14 Chen, Andrea, and Zhuang Pinghui, "Chinese Universities Ordered to Ban Textbooks That Promote Western Values," _South China Morning Post_ , January 30, 2015, <http://www.scmp.com/news/china/article/1695524/chinese-universities-instructed-ban-textbooks-promote-western-values>.
15 Phillips, Tom, "It's Getting Worse: China's Liberal Academics Fear Growing Censorship," _Guardian_ , August 6, 2015, <https://www.theguardian.com/world/2015/aug/06/china-xi-jinping-crackdown-liberal-academics-minor-cultural-revolution>.
16 Phillips, Tom, and Ed Pilkington, "No Country for Academics: Chinese Crackdown Forces Intellectuals Abroad," _Guardian_ , May 24, 2016, <https://www.theguardian.com/world/2016/may/24/academics-china-crackdown-forces-intellectuals-abroad>.
17 See Pan and Xu (forthcoming) and Wu and Meng (2016) for more discussion of potential ideological conflict in China.
18 Weiss (2014).
CHAPTER EIGHT
Appendix
**8.1 DESCRIPTION OF THE CHINA URBAN GOVERNANCE SURVEY**
The 2015 China Urban Governance Survey was carried out by the Research Center on Data and Governance at Tsinghua University during the summer of 2015. Urban residents above the age of 18 were sampled via GPS Assisted Area Sampling (Landry and Shen, 2005) to ensure that both residents and migrants were represented within the sample. The survey sampled 50 county-level cities and urban districts in 24 provinces, excluding Inner Mongolia, Jilin, Guizhou, Tibet, Qinghai, Ningxia, and Xinjiang. The total number of respondents surveyed was 3,513, among 5,526 sampled–a response rate of 63.6 percent.
**8.2 WORDS RELATED TO CENSORSHIP, MUTUAL INFORMATION**
This table of words are those that have high mutual information and higher probability of appearing in a censored than an uncensored Weibo post written by matched users. These are the words used to understand how sensitive words appear before and after censorship in the matched Weibo analysis in chapter 4. | Word related to censorship | English translation
---|---|---
1 | 代表们 | representatives
2 | 以示 | to show
3 | 国资委 | State-owned Assets Supervision Commission
4 | 不作为 | inaction; dereliction of duty
5 | 党外 | outside of the party
6 | 反对党 | opposition
7 | 喜马拉雅山 | Himalayas
8 | 张牙舞爪 | to bear fangs and brandish claw
9 | 纠纷 | dispute
10 | 长成 | grow up to be
11 | 受到 | be subjected to
12 | 呼吁 | advocate for
13 | 亮出 | reveal
14 | 毕竟 | after all
15 | 危害 | harm
16 | 人大 | National People's Congress
17 | 处罚 | punishment
18 | 好不容易 | with great difficulty
19 | 特首 | Chief Executive of Hong Kong or Macao
20 | 请求 | request
21 | 集团 | (corporate) group
22 | 代言人 | spokesperson
23 | 冤案 | miscarriage of justice
24 | 惩罚 | punishment
25 | 纠正 | correct
26 | 老百姓 | ordinary people
27 | 宣称 | claim
28 | 白天 | daytime
29 | 表叔 | uncle who wears a watch
30 | 不去 | don't go
31 | 伪劣 | counterfeit or shoddy products
32 | 何刪删 | deleted
33 | 作画 | paint
34 | 使用者 | user
35 | 借的 | borrowed
36 | 值得一看 | worth a look
37 | 农友 | fellow peasant
38 | 出生入死 | willing to risk life and limb
39 | 出车 | dispatch a vehicle
40 | 利刀 | sharp knife
41 | 嘿嘿嘿 | heh heh heh (evil laughter)
42 | 学术界 | academia
43 | 居所 | residence
44 | 屈才 | waste talent
45 | 屋前 | in front of house
46 | 张思之 | Zhang Sizhi (rights lawyer)
47 | 弹雨 | hail of bullets
48 | 心弦 | heartstrings
49 | 恩情 | kindness
50 | 我校 | our school
51 | 拨动 | stir up
52 | 接轨 | adapt to
53 | 敌杀 | poison
54 | 新闻界 | press
55 | 枪淋 | hail of bullets
56 | 梭镖 | spear
57 | 水流 | water stream
58 | 海淀 | Haidian
59 | 特殊性 | particularity
60 | 监视 | surveillance
61 | 相望 | gaze at each other
62 | 知心话 | Caring Talk (program on
| |
Radio Free Asia)
63 | 裸身 | naked body
64 | 隔山 | separated by mountains
65 | 隔水 | separated by waters
66 | 鼓噪 | stir up chaos
67 | 控制 | control
68 | 推翻 | overthrow
69 | 贺卫方 | He Weifang (professor and activist)
70 | 举报 | report
71 | 重庆 | Chongqing
72 | 最高 | highest/supreme
73 | 无辜 | innocent
74 | 泰坦尼克号 | Titanic
75 | rose | rose
76 | 二十多 | over 20
77 | 借给 | lend
78 | 公学 | public school
79 | 军哥 | military brother
80 | 出息 | achievement
81 | 刪减 | censor/deduct
82 | 利剑 | sharp sword
83 | 到瓜瓜 | to Guagua (refers to B. Guagua)
84 | 十亿 | billion
85 | 又是 | also is
86 | 双规 | detained and investigated
87 | 哈罗 | Harrow school
88 | 围困 | besiege
89 | 子孙们 | offspring
90 | 忐忑 | nervous
91 | 扩大化 | escalation
92 | 新词 | new words
93 | 朝阳区 | Chaoyang district
94 | 永不忘 | never forget
95 | 派驻 | dispatch
96 | 淹死 | drown
97 | 珍重 | treasure
98 | 看后 | after viewing
99 | 答复 | reply
100 | 行船 | sailing
**8.3 TIBET SELF-IMMOLATIONS NEGATIVE BINOMIAL MODEL**
To supplement the analysis in chapter 5, I model the length of the social media bursts associated with self-immolation protests in Tibet using a negative binomial regression, where the length of the burst is the dependent variable and whether or not the self-immolation occurred on a weekend is the main independent variable of interest. Shown in table 8.1, controlling for characteristics of the self-immolator, I find that whether the event was on a weekend is still a significant predictor of the length of the burst.
Table 8.1: Negative binomial Model showing the impact of a self-immolation happening on a weekend on the length of the social media burst associated with that immolation event. | _Dependent variable: Length of social media burst_
---|---
|
numafterdate
|
(1) | (2)
Weekend | 0.692*** | 0.764***
|
(0.187) | (0.189)
Age
| |
−0.002
| |
(0.010)
Monk
| |
0.290
| |
(0.192)
Died
| |
0.321
| |
(0.396)
Constant | 1.282*** | 0.865*
|
(0.124) | (0.487)
Observations | 120 | 111
Log likelihood | −320.273 | −294.755
_θ_ | 1.228*** (0.194) | 1.321*** (0.219)
Akaike inf. crit. | 644.546 | 599.509
_Note_ : *p<0.1; **p<0.05; ***p<0.01
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INDEX
Page numbers in _italics_ indicate graphs and charts.
access to/expression of information: defining censorship as restriction of, –; fear affecting, –; flooding affecting, –; friction affecting, , –
ACLU (American Civil Liberties Union), –
age of Firewall jumpers, –, __
Ai Weiwei,
American Chamber of Commerce, 2016 China Business Climate Survey Report, 25n20
American Civil Liberties Union (ACLU), –
authoritarian regimes: future study needs, ; porous censorship in China as model for, ; resilience of, –, –; restricting access to or expression of information, , –. _See also_ costs of censorship to authorities and society
Badawi, Raif, –,
Baidu Baike, , , , ,
Balkin, Jack M.,
Bamman, David,
Barberá, Pablo, 8n16
Baum, Matthew A., 76n140, 91n180
BBC website, ,
BBS websites,
beatings and murders of journalists, –
_Beijing Daily_ ,
Beketov, Mikhail, –
Bernays, Edward L., 21n1
bifurcation of media by censorship, –
Blogger,
blogspot.com,
Bo Xilai, , –
Boix, Carles, 11n24
Brady, Anne-Marie,
Brand, Stewart,
Brazil, missing/murdered journalists in,
Brin, Sergey, 56n96
Brody, Richard A., 5n13
broken links,
Brownlee, Jason, 11n23
Bueno De Mesquita, Bruce, 11n24, 21n2
bureaucracies, censorship by, ,
bursts of activity: of propaganda posts, _i219_ , –, __ ; on social media, , , , , _–_
Bush, George W.,
CAC (Cyberspace Administration of China), –
_Caijing_ ,
California Coastal Records Project,
Cardello, Armand,
CCP. _See_ Chinese Communist Party
CCPPD (Chinese Communist Party Propaganda Department), , , , , 109n55, , –,
CCTV,
_CDT (China Digital Times)_ , , –
censorship. _See_ porous censorship in China; theory/model of censorship
Center for Indoor Air Research (CIAR),
Central Cybersecurity and Informatization Leading Small Group,
Chan, Chung-hong,
Chau, Marie,
Chenoweth, Erica, 8n16
China, censorship in. _See_ porous censorship in China
_China Digital Times (CDT)_ , , –
China Unicom,
China Urban Governance Survey, , –, _, , _, –,
Chinese Communist Party (CCP): flooding and, , –, –, , , ; in historical overview of censorship in China, –, , –, , , , ; Martyr's Day, , __ ; Qingming festival, efforts to reframe, –, __ ; reactions to experience of censorship and, ,
Chinese Communist Party Propaganda Department (CCPPD), , , , , 109n55, , –,
CIAR (Center for Indoor Air Research),
"citizen journalists," –
clickbait,
Clinton, William J., ,
cognitive dissonance, consumer avoidance of, –
collective action and power of information, 21n4,
company/corporate censorship,
Conover, Pamela Johnston, 5n13
content filtering, , –; defined, ; effectiveness of, –; by ICPs, –; porous nature of, –; of Tibetan self-immolation protests, –, _, _; types of content filtered,
coordination of information, as censorship strategy. _See_ flooding
Copyfind,
corporate/company censorship, , –
cost/benefit analysis by consumers: elites, costs versus benefits of obtaining information for, –; rational ignorance model of costs and benefits of information consumption, –; social media, exchange of low-cost information through, –, __ ; website blockages, different effects of, –, _, _
costs of censorship to authorities and society, –; economic costs, –, , –, –, –; educational costs, –; fear, costs of use of, –, , ; fear, Internet increasing costs of using, –; flooding, costs to government of, ; ideological/political costs, –, –; negative reactions to censorship, –; observed versus unobserved, –, –, –; rational ignorance model of costs and benefits of information consumption, –, –,
costs to media: cost-benefit trade-off of stories covered, –; Internet costlessness and susceptibility to friction, ; social media, exchange of low-cost information through, –, __
credence goods,
crime statistics in China,
crises: censorship contributing to, ; flooding in, ; friction in, –, __ ; public willingness to overcome censorship barriers during, , , , , , , , –, , , –, __ , ; sudden implementation of censorship measures, backlash against, , , ,
Cultural Revolution, , , ,
Cummings, James J.,
Cyberspace Administration of China (CAC), –
data: access to, friction on, –; collection of, friction on, –; failure to collect, –
Davenport, Christian, 11n23
defamation laws,
defining censorship, –
democracies: crises drawing attention to government data or reporting limitations in, ; defamation laws in, ; flooding, susceptibility to, , ; implications of porous censorship for, –, –
Democracy Movement (1978–79) in China,
demographics of Firewall jumpers, , –, _, , _,
Deng Liqun, –
Deng Xiaoping, , ,
denial of service attacks, ,
deterrent mechanism, fear as, . _See also_ fear
Dickson, Bruce, 11n24,
dictator's dilemma, –
distraction mechanism, flooding as, . _See also_ flooding
distribution of information, government impacts on, , –
diverting mechanism, friction as, . _See also_ friction
Downs, Anthony,
Dropbox, ,
East Germany, porous censorship in,
economic costs of censorship, –, , –, –, –
economic crisis, Chinese rumors of,
Edmond, Chris, 15n36, 72n132
education and technical capability of Firewall jumpers, __ , –
educational costs of censorship, –
elasticity of demand for information, –, ,
elites: costs versus benefits of obtaining information for, –; defined, 29n31; demographics of Firewall jumpers, , –, _, , _, , ; government targeting of, –; porous censorship separating elites from masses, , , , ; selective use of fear against, –, , ; targeted censorship strategy, –; two-pronged approach to elites and masses to minimize negative reaction, –
Enikolopov, Ruben, 22n6
error pages, as evidence of censorship, , , , , –25n21, –, __ , –, , ,
Esarey, Ashley, 4n10
experience goods,
expression of/access to information. _See_ access to/expression of information
Facebook, , , , , , , ,
failure to collect data, –
fake news,
Fang Binxing,
fear, –; access to/expression of information affected by, –; costs of use of, –, , ; credibility of enforcement threat, –, ; as deterrent mechanism, ; elite levels, selective use at, –, , ; experience of censorship not leading to, –, __ , ; flooding and, –, , ; friction and, –, –, , , ; implications for digital world, –; Internet increasing costs of using, –; intimidation, extra-legal, –; of legal deterrence, –; in Maoist era China, , , ; as mechanism of censorship, –, , ; observability required for use of, , , , , ; other authoritarian environments, importance of studying, ; real name registration requirements for social media and, ; rewards as form of, ; in theory/model of censorship,
Feldman, Stanley, 5n13
feminist activists protesting sexual harassment, –
Fifty Cent Party, , , , , , –, __ , ,
fine particulate matter (PM 2.5), Chinese failure to collect data on, , ,
firewalls, as form of friction, –. _See also_ Great Firewall
flooding, , –, –; ability to detect, ; access to/expression of information affected by, –; costs to government of, ; in crises, ; defined, , –; democracies susceptible to, , ; different forms of, –; direct to public, –; as distraction mechanism, ; distribution of media and, ; effectiveness of, , –; elites and the masses, driving wedge between, , ; fear and, –, , ; Fifty Cent Party, activities of, , , , , , –, __ , , ; friction and, –, , , , , ; historical introduction into China's censorship policy, ; implications for digital world, –, ; Internet and cost-effectiveness of, , –; measurement and testing of, –; as mechanism of censorship, –, , ; media-directed, –; in modern China, , –; natural forms of, ; newspaper coordination and spread of information, –, __ ; newspapers, propaganda in, –; observability, difficulty of, , –, ; observability interfering with effectiveness of, , –; online propaganda, –; other authoritarian environments, importance of studying, ; propaganda as, –, –; Qingming festival and, , __ , , –, __ ; resistance to, ; at sensitive time periods, –, __ ; spread of information, influence on, –, _, _; in theory/model of censorship, ; during Yunnan earthquake (2014), –
FOIA (Freedom of Information Act) requests, –,
foreign connections of Firewall jumpers,
foreign information, effect of Great Firewall on ability to access, , –
Freedom House, "Freedom the Net" project, 41n65
Freedom of Information Act (FOIA) requests, –,
Frey, Dieter, 31n36
friction, –, –, –; ability to detect, ; access to/expression of information affected by, , –; on collection of primary information, –; in crises, –, __ ; crises affecting viability of, –, __ ; on data access, –; on data collection, –; defined, ; on distribution of information, , –; as diverting mechanism, ; effectiveness of, –, , , , –; elasticity of demand for information and, –; elites and the masses, driving wedge between, , , ; failure to collect data as, –; failures of, –, _, _; fear and, –, –, , , ; firewalls, as form of, –; flooding and, –, , , , , ; Google blockage as form of, –; historical introduction into China's censorship policy, ; implications for digital world, –, ; increasing levels of, –; Internet strengthening impact of, –; as mechanism of censorship, –, –, ; natural sources of, ; observability, effectiveness not requiring, , ; other authoritarian environments, importance of studying, ; in print publications, –; resistance to, , , , –; search filtering and keyword blocking, –; slowed information transfer, ; as tax on information, , –, ; in theory/model of censorship, ; Tibet, information about self-immolation events in, , . _See also_ content filtering; Great Firewall
Friedman, Monroe, 22n5
Fu, King-wa,
future research needs, –
gas-lighting,
GDP (gross domestic product) numbers in China, –
General Administration of Press and Publication,
geographic distribution of Firewall jumpers, –, __
geo-location, VPN users using, –
GitHub, ,
_Global Times_ ,
Gmail, , ,
Gold, Thomas B.,
Golden Shield Project. _See_ Great Firewall
Google, , –, , , , , , , , –, __ , –, __ , ,
Google API, –
Google Drive,
Google Plus,
Great Firewall (Golden Shield Project), –; age of Firewall jumpers, –, __ ; amount of Internet and ability to jump, , __ ; China Urban Governance Survey on evasion of, , –, _, , _, –, ; economic protection of Chinese companies by, –; education and technical capability of Firewall jumpers, __ , –; effectiveness of, –; financial resources of Firewall jumpers, –; foreign connections of Firewall jumpers, ; foreign information, effect on ability to access, , –; geographic distribution of jumpers of, –, __ ; geo-location, VPN users using, –; in historical overview of Chinese censorship, –, ; "jumping" or circumvention of, , , , , –, , , –; political knowledge and trust in government of jumpers of, –, __ ; social media response to architect of, ; theory/model of censorship and, , , –, , , ; Twitter users in China, , –, _, _; VPNs and VPN users, , , , , , , , , , , –, __ , , , –, , ; website blockages, –, _, , , _, , –
Great Leap Forward,
Grimmer, Justin,
Groeling, Tim, 91n180
gross domestic product (GDP) numbers in China, –
Gu Kailai, –
Gunitsky, Seva, 89n178
Guo Meimei, –, ,
Hamilton, James, 5n13, 32n42, 33n44
He, Baogang, 11n24
historical overview of censorship in China, , –; Internet era (1995–), porous censorship in, –; Maoist era (1949–1976), –, ; post-Tiananmen soft controls (1989–1995), –; protests and reform (1976–1989), , –
Hobbs, William,
Hong Kong pro-democracy protests (umbrella revolution; 2014), , , , , –
Hu Jintao,
Hu Jiwei, –
Hu Yaobang, –,
Human Rights Watch,
Huxley, Aldous, _Brave New World_ (1932),
ICPs (Internet content providers), content filtering by, –
implications of censorship for digital world, , –; authoritarian resilience, –; democracies, impact of porous censorship on, –, –; effectiveness of porous censorship strategy, –; fear, –; flooding, –, ; friction, –, ; future research needs, –
India, missing/murdered journalists in,
Inner Mongolia, protests in, –,
Instagram, , , –, –, __ ,
Internet: China, number/percentage of users in, ; fear, increasing costs of using, –; flooding, affecting cost-effectiveness of, , –; friction, strengthening impact of, –; government targeting of elite use of, –; porous censorship in China in Internet era (1995-), –; State Internet Information Office, , . _See also_ Great Firewall; media; social media; _specific sites_
Internet content providers (ICPs), content filtering by, –
Internet Live Stats,
intimidation, extra-legal, –
Iran, censorship in,
Israel, flooding in, –,
Iyengar, Shanto, 28n30
J-15 Chinese fighter jet,
Kepler, Johannes,
keyword blocking and search filtering, –, , , , _–_
King, Gary, , , ,
Krishnan, S. Shunmuga,
Kuran, Timur, 31n36
Law of Coercive Responsiveness, 11n23
Law on Guarding State Secrets,
lead poisoning in China, –
Lee, Kai-Fu,
legal deterrence: fear and, –; in modern China, –, ; targeted use of, –
Lei Zhengfu, ,
Leninism, –
Li, Lianjiang, 21n4
Li Tianyi,
Liberal Saudi Network,
links, broken,
Little, Andrew T., 22n6
Liu Xiaobo,
long-term impact of censorship, –
Lorentzen, Peter L., 11n24
Lu Wei,
Luo Yang,
MacKuen, Michael, 53n92
Magaloni, Beatriz, 11n24
"maintenance" slowdowns or closedowns,
Makarin, Alexey, 22n6
Maoist era censorship (1949–1976), –,
Marcus, George E., 53n92
Mark, Gloria,
"martyr" framing of Qingming festival, –, __
Martyr's Day, , __
matched comparison of censored and uncensored social media users, –; case study of observability, –; complaints about censorship, ; methodology and data, –; persistence after censorship, –; rates of censorship after matched posts, –, __ ; sensitive words, use of, –, __ , , _–_; similarity of new posts to previously censored posts, –, __ ; survey of emotional reactions to censorship, –, __
Mayhew, David, 21n1
media: audience preferences and coverage by, –; bifurcation by censorship, –; cost-benefit trade-off of stories covered, –; distribution of information, government impacts on, , –; expanded number of people with access to, ; flooding directed at, –; friction as used by, –; government access to infrastructure and control of, –, ; Internet costlessness and susceptibility to friction, ; in Internet era (1995–), –, ; Maoist era view of, –; more media not equal to better information, , –; positive news requirements, ; in post-Maoist China, –, –; post-Tiananmen Square (1989–1995), –; social media information exchange and continuing influence of, –, _. See also_ costs to media; social media; _specific sites and organs_
media dependency, 31n39
Mexico: flooding in, ; missing/murdered journalists in,
Miller, Blake, 125n19
Ministry of Culture,
Ministry of Education,
Ministry of Industry and Information Technology, , 109n55
Ministry of Public Security, , 109n55,
Ministry of State Security,
model of censorship. _See_ theory/model of censorship
murders and beatings of journalists, –
_Nanjing Daily_ ,
net neutrality,
Neumann, W. Russell, 53n92
_New York Times_ , , , ,
news cycles,
newspapers: coordination and spread of information, –, __ ; propaganda in, –; U.S., newspaper staff reductions in, . _See also specificnewspapers by name_
North Korea, censorship in, ,
Nyhan, Brendan, 28n27
O'Brien, Kevin J., 21n4
observability: of censorship in modern China, ; consumer responses to evidence of censored information, –, _, , _; deterrent measures requiring, ; differing reactions to observed versus unobserved censorship, –, –, –, , ; fear requiring, , , , , ; flooding, difficulty of observing, ; flooding affected by, , –, ; friction not requiring, , ; matched censored and uncensored social media users, case study of, –; more interest in topic, observable censorship creating, , , , , –; porous censorship allowing avoidance of, –, –, ,
O'Connor, Brendan,
Olympics, reporting on,
126 e-mail,
Pakistan, blocked websites in,
Palestinian authority, flooding by, –
Pan, Jennifer, , , ,
paraexlyne (PX) plants,
Park Geun-hye,
patriotic education,
_People's Daily_ , , , , , , , , 204n42
Persian Gulf War, –
Petrova, Maria, 22n6
Philip Morris-sponsored smoking research,
plagiarism detection software, , ,
plausible deniability, , , , ,
PM 2.5 (fine particulate matter), Chinese failure to collect data on, , ,
political elites. _See_ elites
political knowledge and trust in government of Firewall jumpers, –, __
Popkin, Samuel, 5n13, 22n5
porous censorship in China, –; authoritarian resilience and, –, –; concept of, –; customizability and differential effects, –; distribution of information, government impacts on, ; effectiveness of, –, –; fear selectively used against elites, –, , ; historical overview, ( _see also_ historical overview of censorship in China); implications of, , – ( _see also_ implications of censorship for digital world); increasing levels of, –; mechanisms of, – ( _see also_ fear; flooding; friction); as model for other authoritarian regimes, ; in modern Internet era (1995–), –; more media not equal to better information, , –; observable oppression, allowing avoidance of, –, –, –, , ; observed versus unobserved censorship, differing reactions to, –; reactions to, –, – ( _see also_ reactions to experience of censorship); reasons for using, –, –; separating elites from masses, , , , ; as tax on information, , , , –, ; theory/model of, – ( _see also_ theory/model of censorship); weaknesses of,
positive news requirements,
press. _See_ media
press releases, as flooding,
propaganda: bursts of activity in propaganda posts, _i219_ , –, __ ; CCPPD (Chinese Communist Party Propaganda Department), , , , , 109n55, , –, ; as flooding, –, – ( _see also_ flooding); in newspapers, –; online, –; Zhanggong local propaganda department, , –
Pure Brightness holiday (Tomb Sweeping or Qingming festival), , __ , , –, __
PX (paraexlyne) plants,
Qingming festival (Pure Brightness or Tomb Sweeping festival), , __ , , –, __
rational ignorance model of information consumption, –, –,
reactions to experience of censorship, –, –; compensating for censorship, –, _, _; consumer responses to evidence of censored information, –, _, , _; costs to authority, negative reactions as, –; emotional responses to being censored, survey of, –, __ ; government targeting of elite use of Internet, –; information production and consumption, signals of censorship not deterring, ; in modern Internet-era China (1995–), –; more interest in topic, observable censorship creating, , , , , –; observed versus unobserved censorship, –, –, –, ; two-pronged approach to elites and masses to minimize, –. _See also_ matched comparison of censored and uncensored social media users
real name registration, –
Red Cross in China, –
Reeves, Byron,
Reifler, Jason, 28n27
Renren,
Reporters Without Borders, ,
rewards, as form of fear,
Roberts, Margaret E., , , , ,
Russia: censorship in, , , –, 56n96, , ; U.S. elections, alleged interference in, –
Saudi Arabia, censorship in, –, ,
SCIO (State Council Information Office),
search filtering and keyword blocking, –, , , , _–_
selectorate, 21n2
self-censorship, –, ,
self-immolation events in Tibet. _See_ Tibetan self-immolation protests
sensitive terms: related to censorship and mutual information, –, __ , , _–_; Twitter and Sina Weibo common terms compared, –, _, _
Shirk, Susan L., 21n2
Sichuan earthquake (2008),
Sina blogs,
Sina Weibo, , , , , , , , , , , , , , , , , , ,
sina.com.cn, ,
Singapore, censorship in,
Sitaraman, Ramesh K.,
Skype,
Smith, Noah,
smoking research, tobacco company-sponsored,
Sniderman, Paul, 5n13
social credit score,
social media: bursts of activity on, ; censorship in China in Internet era (1995–), –; consumer responses to evidence of censored information, –, _, , _; crackdown on bloggers and spreaders of "rumors," –, ; economic costs of censorship to, ; exchange of low-cost information through, –, __ ; official identification required to register for, –; prioritization of information by, . _See also_ Internet; matched comparison of censored and uncensored social media users
sohu.com,
South Korea, defamation laws in,
Southern Tour (1992),
_Southern Weekend_ , ,
_Southern Weekly_ ,
special interest groups, censorship by, , , –,
State Council Information Office (SCIO),
State Internet Information Office, ,
Steinert-Threlkeld, Zachary C., 8n16
Stephan, Maria J., 8n16
Stockmann, Daniela, 31n37,
Streisand effect, , ,
string similarity, , __
sudden implementation of censorship measures, backlash against, , , ,
surveillance technology, –
Svolik, Milan W., 11n24
syndication, –
Syria, censorship in,
Taubman, Geoffry, 4n10
tax on information, porous censorship as, , , , –,
technical capability and education of Firewall jumpers, __ , –
terminology. _See_ sensitive terms
Tetlock, Philip E., 5n13
theory/model of censorship, –; advantages of porous censorship, –, –; audience preferences and media coverage, –; authorities restricting access to or expression of information, , –; costs of censorship for authorities and society, –; defining censorship, –; expression of/access to information, –; mechanisms of censorship, –, ( _see also_ fear; flooding; friction); media cost-benefit trade-off of stories covered, –; rational ignorance model of costs and benefits of information consumption, –; reasons why governments censor, –; restriction of expression of or access to information, , ; social media, exchange of low-cost information through, –, __
thought work, ,
Tiahrt Amendment (U.S.),
Tiananmen Square protests: 1976, , , , ; 1989, , , , ,
Tianjin explosion (2015), , , –, __
Tibetan self-immolation protests, –; content filtering of, –, _, _; friction's impact on spread of information about, , ; negative binomial model, _–_; weekend versus weekday discussion of, –, _, _
Tilly, Charles, 21n4
TIRC (Tobacco Industry Research Committee),
_Titanic_ (3-D version; movie),
tobacco company-sponsored smoking research,
Tobacco Industry Research Committee (TIRC),
Tohti, Ilham,
Tomb Sweeping Festival (Qingming or Pure Brightness festival), , __ , , –, __
traditional media. _See_ media
trust in government and political knowledge of Firewall jumpers, –, __
Turkey: blocked websites in, ; flooding in, –, ; missing/murdered journalists in,
Twitter, , , , –, , , , , , –, _, _, , , ,
Ukraine, missing/murdered journalists in,
umbrella revolution (Hong Kong pro-democracy protests; 2014), , , , , –
United Kingdom, porous censorship of "obscene" French literature (late 19th century) in,
United States: agricultural anti-whistleblower statutes in, ; child pornography laws in, ; Ferguson police shooting (2014) and failure to collect data on police shootings, –; FOIA requests, –, ; media-directed flooding in, –; net neutrality in, ; newspaper staff reductions in, ; prioritization of information by social media companies in, ; Russia's alleged interference in U.S. elections, –; smoking research, tobacco company-sponsored, ; war zones, journalist access to, –,
Virtual Private Networks (VPNs), , , , , , , , , , , –, __ , , , –, ,
Voice of America News,
Voida, Stephen,
VPNs (Virtual Private Networks), , , , , , , , , , , –, __ , , , –, ,
Wang Gongquan, 119n12
Wang Jun,
Wang Lijun,
Warren, Mark E., 11n24
website blockages, –, _, , , _, , –. _See also specificwebsites_
WeChat, , 10n20, ,
Wei Chen,
Weibo. _See_ Sina Weibo
Weiboscope project, –, 133n28, 160n17
whistleblowers, statutes against,
Wikipedia, –, __ , , –, __ ,
Wintrobe, Ronald, 11n24,
Wolfsfeld, Gadi, 73n134
words. _See_ sensitive terms
Xi Jinping, –, , , , , , ,
Xiao, Qiang, 2n2, 4n10
_Xinhua_ , , , , , , 204n42,
Xinjiang: railway station knife attack and bombing (2014), –; riots in (2013), , –,
_Xinwen Lianbo_ (TV program),
Xue, Charles,
Yahoo,
Yang, Guobin, 2n2, 4n10
Yeykelis, Leo,
Young, Lauren E., 52n85
YouTube, , ,
Yunnan earthquake (2014), ,
Zaller, John R., 28n30, 29n31,
Zhanggong local propaganda department, , –
Zhao Hongxia,
Zhao Ziyang, –
Zhu Houze,
Zimbabwe, censorship in, 52n85
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{
"redpajama_set_name": "RedPajamaBook"
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Pedal (senlat. pedalis "som har med foten att göra", av lat. pes, genitiv pedis "fot") är klaviatur som sköts med fötterna. Orgeln har en hel klaviatur.
Musikinstrument
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{
"redpajama_set_name": "RedPajamaWikipedia"
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Tag: Law and Order
Posted on November 10, 2011 April 5, 2019
A Tale of Two Authors
Compare how two authors deal with book reviews that they believe to be defamatory.
First, Chris McGrath, author of "The Attempted Murder of God: Hidden Science You Really Need to Know" took blogger Vaughan Jones to the High Court over a review that Jones posted on the Amazon website, of all places. The judgement on whether this case can proceed is expected today.
Historian Niall Ferguson was similarly upset by a negative review. His book Civilisation was eviscerated by Pankaj Mishra in the London Review of Books (a much more credible and prominent platform than Amazon's product review pages). Ferguson felt he had been defamed as a racist. However, in contrast to Chris McGrath, Ferguson chose a different forum to express his grievance and demand satisfaction – the letters page.
This approach – fighting words with more words – is precisely the kind of counter-speech I advocated in my 'Way of The Blogs' piece for the Guardian a couple of years ago. It offers a form of redress to the aggrieved person, while avoiding censorship, and it is also much cheaper. I think it is a much classier way of dealing with critics, than hauling them down to the Royal Courts of Justice.
Posted on August 26, 2011 April 5, 2019
Putting the Power of Censorship in the Hands of the Mob
English Defence League / Unite Against Fascism protest, by Matthew Wilkinson on Flickr
Here's a post first published earlier today on Labour List (a new venue for me). I hope there will be comments to which I can respond in a follow-up post.
The riots seem to have brought out the worst in our politicians. You would think our political class would be well aware of the perils of knee-jerk responses and short term expediency, but apparently not. First, a few Conservative MPs (the Prime Minister among them) have called for social networks to be interfered with in times of crisis – an astonishingly cynical and hypocritical idea, given our condemnation of the Iranian and Egyptian regimes when they did the same thing.
Not to be outdone, a group of Labour politicians have now put opportunism and short-term thinking above the principles of good democracy. The leaders of thirteen London Boroughs, together with John Biggs AM and MPs Rushanara Ali and Jim Fitzpatrick, called for a proposed EDL march in Tower Hamlets to be banned on account of the cost of policing, which they say "would simply be too great".
The potential cost of policing the march wass half a million pounds, which is be no small sum to remove from London's clean-up effort. But the costs of banning the EDL march will be much higher in the long term. It will fuel resentment among those wishing to march, and award them the status of 'free speech martyrs' that they crave, but do not deserve. Their warped view of immigration and their fantastical idea of what constitutes 'true' British culture will remain unchallenged once again. This will only lead to more tension and conflict that the police will have to spend time and resources to contain.
Citing costs as a reason to deny political or artistic expression is a classic argument used by despots abroad to suppress internal opposition. Of course, there is no comparison between our democracy and their tyrannies… but that's an argument that carries zero weight when you're campaigning for human rights in those places. Cameron's suggestion that we censor social media, and the Labour call for the banning of this EDL event, will hamstring the fight for free expression elsewhere: "You do it, so why shouldn't we?"
Worse, this excuse also puts the power of censorship into the hands of the mob. For example, in 2004, a small and unrepresentative group of youths were able to stop performances of Behzti at the Birmingham Rep Theatre (which they found offensive), by threatening to cause chaos that the police were unable to stop, on grounds of cost. Six years later, another theatre had to fight tooth-and-nail to ensure that the police would guarantee the safety of performers in another play by the same playwright. If this precedent persists, then we give extremists like the EDL, the BNP, or Islam4UK an ongoing permit to shut down any gathering they disagree with. Already we've seen local councils bullied into withdrawing Moonfleece, a play that challenges far-right extremism… because those same extremists threatened 'trouble'! Arguments that seek to ban the EDL, however well-intentioned, slide inexorably into the banning of others, and eventually, banning everyone.
When the riots erupted across our cities earlier this month, we rightly saw them as a threat to our way of life. We demanded the police throw all their resources at the problem, regardless of the cost in these austere times. The right to freedom of expression must be protected by the police with equal vigour, and it's odd that our London councillors have forgotten this.
To argue that the EDL must be allowed their right to march is only the beginning of the discussion. Those who advocate the right to free expression have a moral obligation to challenge those who preach hate and division. No one is arguing that an EDL march will not exacerbate tensions in Tower Hamlets, but these can be diffused without trampling on the right to association and assembly. This is where we need leadership, from those very same elected Labour representatives who signed the letter in the Guardian on Monday. I met and campaigned with Rushanara Ali and Jim Fitzpatrick when I lived in Tower Hamlets – They are both deeply respected in their constituencies. They, together with the Mayor of London and the Metropolitan Police, have both the wit and the standing to co-ordinate and lead a peaceful response to the EDL. Why did they not playing a central role in the Unite Against Fascism counter-protest? So far it has only garnered support from the unions and the mosques.
It is down to our politicians to present the contrast between the thuggery of the far-right, and the vibrancy of multicultural inner-city life, all while respecting free speech. Granted, this is not as simple as just banning the march. But we elect our Members of Parliament and Councillors to take on these difficult tasks, not to engage in easy, knee-jerk letter-writing. Time for Labour to lead.
Police prepare for an EDL march in Leicester. Photo by robotswanking on Flickr
Pædos, Prisoners, and Cameron's Attack on Human Rights
First they came for the prisoners.
A few weeks ago, MPs voted to ignore the European Court of Human Rights. The court in Strasbourg had said that a blanket ban on prisoners voting was incompatible with human rights law, and that the British government should rectify this. Following a debate in the House of Commons, Parliament thumbed its nose at the Court, as MPs voted 234 to 22 to keep a full ban on prisoners. Our Prime Minister put blatant populism above politics, declaring that "giving prisoners the vote makes me sick" (even if that means paying £143 million in compensation from the barren public purse).
Then they came for the paedophiles.
This week, we heard that those convicted of sex offences might not have to stay on the sex-offenders register for life. Last year, the Supreme Court ruled that those included on the register should be able to appeal against permanent inclusion on the list, and on Tuesday it rejected a Home Office appeal against the ruling. The Government now has to formulate a policy based on this decision. At PMQs, David Cameron called the situation "appalling".
There are clear similarities between these two stories. Both present issues where what might be considered the popular and common-sense approach is over-ruled by judges, forcing the Government to do something counter-intuitive. Both stories will inspire tabloid frothing at judge-made law. And in both cases, there are actually good and sober reasons why the judges ruled as they did, and why we should support their decisions. In the case of prisoners voting, such a change could catalyze the reform of prisons into places that offer better rehabilitation for convicts. Moreover, if a person will be released within the lifetime of a parliament, why shouldn't they have a say on who will be representing them once they're out? Similar arguments exist for sex offenders: In cases where a prisoner has been rehabilitated, coming off the sex offenders register might help reintegration.
It is crucial to remember that in both cases, all the courts did was rule against an absolutist approach: No 'blanket' ban on prisoners' votes; and sex offenders have the right to appeal, not an absolute right to come off the register. The best comparisons for these issues are with parole or bail – you have the right to apply for it, but you might not get it. It is left to magistrates and judges to decide, depending on the actual circumstances.
So there may well be good reasons why extending the rights of some pretty unpleasant people might improve the whole of society… but it is for the penal reform groups to advance that argument. My concern is with how both these stories have been discussed by politicians – The Prime Minister in particular. With his bully-pupit, he has set a terrible example, placing the blame with the judiciary. His comments are clearly designed to undermine the European Court, the Convention on Human Rights and its manifestation in British law, the Human Rights Act (HRA). David Cameron and his allies have never been comfortable with that document, and these outbursts are designed to soften MPs and the public into agreeing to a watered-down Bill of Rights that will make our standing as citizens more tenuous.
Everyone remembers Pastor Martin Neimöller's famous poem, which begins "First they came for the Communists" and ends with the narrator alone, with no-one left to speak in his defence. The moral should be clear: If you don't stand up for the human rights of others, then eventually you will lose your own rights; stand up for the rights of others, and you protect yourself. But while we remember the poem, I think we fail to relate it to the present day. Neimöller's victims, the Jews, the Trade Unionists, and the Communists, are all inoffensive and mainstream today, so we assume we are far away from the oppression described. But what we forget is that during Neimöller's lifetime, all these groups were among the most vilified: the rhetorical equivalent of paedophiles and prisoners today.
What the Prime Minister seems to forget, is that Human Rights laws are designed to protect the most hated in our society, not least because these people are always amongst the most vulnerable too. They are supposed to frustrate our gut reaction. They are meant to be inconvenient. That the Courts' rulings have caused outrage is actually a feature of our democracy, and not a bug. Kudos to the 22 MPs who recognised that, and shame on the Prime Minister. By undermining the principle of human rights, he undermines us all.
This was crossposted over at LiberalConspiracy.org in a more succint form. It got a fairly good response in the comments, although Tyler makes a good point:
Voting is not a human right. As is so often confused by so many on the liberal left, it is a CIVIL right. It is thus conferred on people by the laws of the land. It is granted to an individual by citizenship, and is not unalienable or transferrable, unlike free speech etc.
If it were a human right there would be no real reason why children shouldn't have the vote, for example…
As such, this argument that voting is some form of human right is simply the wrong one.
Mea culpa, but the central points remain intact.
Liberty, Whatever the Cost [Updated]
"There is not enough poxes for your houses" says Jay Rosen to the pundits discussing #Tucson. Well, here's an astonishing quote from a non-pundit which goes places no politician dares to tread:
This shouldn't happen in this country, or anywhere else, but in a free society, we're going to be subject to people like this. I prefer this to the alternative.
That was spoken by John Green, the father of Christina Green, 9-year-old girl killed at the shootings on Saturday. His statement eloquently explains the tough trade-off between liberty and security. He acknowledges the limits of Government, and that ackowledges that horrible things will happen in a free society, and explicitly says that this is a preferable state of affairs. It is a difficult case to make at the best of times (I have tried on a few occasions, regarding cannabis, ID cards, and other civil liberties). For Mr Green to say it at the depth of his grief is truly courageous.
Compare this to Nick Clegg and David Cameron, who seem to want to have it both ways. If you want to argue for more civil liberties, I think you must acknowledge that the mythological state of absolute security does not exist, that there can be negative consequences to liberty… and that we should all be comfortable with that.
When I read this quote I instinctively assumed it was referring to the idea of liberty in general, and did not think too much about the particular tyoe of liberty that Green was advocating. However, a colleague points out that he can only be referring to gun-control (or lack thereof in the American system). And as many others have been arguing these past few days, liberty and the unfettered 'right to bear arms' do not necessarily go hand in hand. Indeed, surely the whole point of consituting a state is to get away from all that! So it is worth adding a line here to emphasise that I do not share Mr Green's views on gun control, and am relieved that we do not have that sort of 'liberty' here in the UK. There's no point in whitewashing my original post though – I think it best to leave my excesses and embarrasments for all to read.
Having said that, I think my central point remains. Mr Green acknowledges that his ideology has negative aspects, and he embraces them anyway.
Posted on October 27, 2010 April 4, 2019
Geeks on the March
… and the April, and the May.
The latest fundraising project for the Libel Reform Campaign is the Geek Calendar. The video below features a number of eminent scientists and science journalists explaining why the libel laws are so terrible, why science and medicine are particularly threatened, and therefore, why they agreed to feature in the calendar.
The Geek Calendar project is, I think, a fantastic example of a good idea that has been very well executed, with the help of new technologies. (To add a disclaimer lest the reader thinks I am sucking my own trumpet, the project was not managed by me – though as part of the Libel Reform Campaign I did get to watch the team in action at all stages.) The above video is a classic example of how a little forward thinking creates a significant amount of added value. The 'geeks' (including celebrities such as Jonathan Ross) were already being photographed – so why not do a quick interview while you're there?
The Geek Calendar team have also been using behind the scenes imagery to build momentum for the project. At the other end of the production line, there have been several opportunities for us to spread the word and seed the #GeekCalendar hashtag via social networking sites – when the shop went 'live' for pre-orders; at the launch party last week; and when the calendars arrived through people's letterboxes.
It also helps to have a strong constituency for the message and product. As Nick Cohen pointed out in April, it is clear that one reason that the Libel Reform campaign has been so successful in lobbying the government (both the Labour administration, and the post-election Coalition) is that there exists a community of technologically savvy, but also very motivated and passionate geeks, to drive the message forward. Earlier this year, Christina Odone labelled this group "the Lib Dem Spooky Posse of Internet Pests" after a forestorm of tweeting against her during a spat with former MP Dr Evan Harris. Over at the New Statesman blog, David Allen Green gives a little more insight into the 'Skeptics' movement. These people would hate to be compared to the religious Right in the USA… but in their dedication to their cause, and their belief that their engagement can actually cause change, I percieve more than a passing similarity.
Posted on July 1, 2010 April 4, 2019
Anti-Bribes
Kabul artist Aman Mojadidi dressed up in a policeman's uniform, set-up his own check-point, and began offering bribes to passing motorists. The stunt was a protest against the high-levels of corruption in the city:
"On behalf of the city of Kabul and the Kabul police, if you have paid a bribe or 'tip' to someone in the past, I apologize," the officer says in Dari to the disbelieving driver. "Please take 100 Afghanis," or about $2.
Mojadidi wanted to draw attention to the pervasive misuse of power in Afghanistan and to see how Afghan drivers would react when he apologized on behalf of the widely scorned police force.
H/T @RohanJay (whom fans of media freedoms should follow). The stunt reminded me of the story earlier this year about the Zero Rupee note, an innovation by 5th Pillar designed to combat bribe culture in India. From the CommGap report:
Fed up with requests for bribes and equipped with a zero rupee note, the old lady handed the note to the official. He was stunned. Remarkably, the official stood up from his seat, offered her a chair, offered her tea and gave her the title she had been seeking for the last year and a half to obtain without success.
The problem of bribe-culture of course begins when public officials are paid too little in the first place. One hopes that these high-profile, amusing-yet-persuasive interventions inspire the politicians of those countries to address the underlying issues, if they can. Charter Cities are one way of guaranteeing standards of pay and public standards, though I recoil at the colonialist mindset such projects seem to promote. Are there more internationalist, left-wing versions of the underlying idea, I wonder?
The Big Libel Gig
Last night, the Libel Reform Campaign staged 'The Big Libel Gig', an evening of comedy, science and politics. Scientists Simon Singh and Brian Cox joined doctors Ben Goldacre (author of 'Bad Science') and Peter Wilmshurst. Politicians Evan Harris (Lib Dem), Peter Bottomley (Con) and Paul Farrelly (Lab) also took a turn, alongside the proper comedians: Robin Ince, Marcus Brigstocke, Ed Byrne, Shappi Khorsandi and Dara O'Briain.
Some of my photos from backstage and in the wings are online at Flickr:
In parliament, the campaign reached a tipping point – the majority of eligible MPs have now signed Early Day Motion 423 which calls for reform.
Unfortunately, the libel laws are still being used to suppress discussion in the public interest. Professor Francisco Lacerda is a Swedish academic who has been threatened with a libel suit by an Israeli lie detector manufacturer. He visited London last week, to highlight how England's libel laws prevent him from publishing research about technology being used by the DWP in England. Millions of pounds of public money has been spent on this technology.
Posted on December 23, 2009 April 4, 2019
Answering the McCann Question
Mark Pack asked me to write a guest piece for the Liberal Democrat Voice on Libel Reform. It was a good opportunity to dig a little deeper into the argument for reform, and rebutt one of the most common objections to making changes.
The clamour for a change to our pernicious libel laws grows louder every day. In November, Index on Censorship and English PEN published Free Speech is Not For Sale, a report into the state of libel in England & Wales, and the bizarre phenomenon of libel tourism. Impressed by this report, Jack Straw announced the creation of a working group to deliver reform. Lib Dem peer Lord Lester announced on the BBC Radio 4 PM programme he will begin drafting a libel bill, and MPs have begun to sign EDM 423 (tabled by Dr Evan Harris) which demands a libel overhaul. High profile cases like the recent battle between Trafigura and the BBC, and the suing of cardiologist Peter Wilmshurst, have shown the general public what a blight on free speech our libel laws have become, and a petition for change is nearing ten thousand signatures (do sign it if you haven't already).
Not everyone is convinced by the need for reform, however. Some people resist the need for change, and it is up to campaigners to win the argument. Since the Bournemouth conference made a brave manifesto commitment to libel reform in September, 'campaigners' now includes every Liberal Democrat party activist. Liberal Democrat Voice is the ideal place to explore the sticking points of this debate a little more deeply. Continue reading "Answering the McCann Question"
The Execution of Gary Glitter
I watched The Execution of Gary Glitter tonight. Just as executions have a morbid fascination, dramas about executions, like Dead Man Walking or The Green Mile seem to have that same fascination (although perhaps one degree removed).
I think the death penalty is a valid subject for Channel 4, a public service broadcaster. Though it is not a live debate here, it is a real and divisive issue for our cultural cousins in the USA. The hanging of paedophiles is an oft repeated thought experiment, whenever a Huntley or a Vanessa George is arrested, and it is sufficiently discussed in the UK for pollsters to regularly ask the public's opinion on the issue. According to the programme, 54% of British adults support its reintroduction.
The device of using Gary Glitter felt like exactly that, & hopelessly crass. If we executed people in the UK they'd be poor & unknown. (@leylandrichard on Twitter)
There's no doubt that the choice of Glitter as the anti-hero was was a fantastic marketing ploy. He is, shall we say, the most culturally significant bogeyman we have. However, this also gave the narrative extra depth, because his rock-star past allowed the programme makers to pass commentary on popular culture. The Daily Mirror headlines for a Glitter trial felt real, and the MP3 remix sending Gary Glitter back to No.1 (on downloads) on the day of his execution was an obvious slam dunk. It is an uncomfortable thought, but I think he is the protagonist many writers would have chosen. The device cannot simply be marked down as the product of pure cynicism.
Continue reading "The Execution of Gary Glitter"
Posted on September 17, 2009 April 4, 2019
Anatomy of Injustice
I've just attended the launch of the CPJ report Anatomy of Injustice: The Unsolved Killings of Journalists in Russia. Index on Censorship hosted a debate as part of the Free Word Festival.
Manana Aslamazyan, Jo Glanville, Nina Ognianova and Richard Sambrook discuss the report. Photo by englishpen on twitter
A culture of impunity has sprung up in Russia. The murderer of Anna Politkovskaya has not been brought to justice, and the authorities are under no pressure to take investigations to their conclusion. For the panel, the blame for this climate of indifference lies in a large part with the Russian media. According to Manana Aslamazyan, there is no culture of solidarity amongst Russian journalists. They fall into three categories:
A sizable group of cynics, who are content to game the system and support the regime;
A larger group of under-trained, provincial journalists, who live in fear of reprocussions and do nothing to upset the status quo;
A small group of "mad" campaigning journalists, who persist in holding power to account.
It is this group which is being murdered. "An entire granch of journalism has been taken out" said Richard Sambrook, Director of BBC gobal news. Investigative journalism has been effectively killed off in Russia.
It therefore falls to the Western journalists to keep Russia from sliding further into a deadly authoriarianism, and to support their beleagred Russian colleagues. Foreign media can be a thorn in the side of the Russian authorities, says Aslamazyan, even 'name-and-shame' those in the domestic media community who are complicit in corruption and failure to accurately report. By leading the way, Western journalists can embolden their Russian counterparts. Indeed, said Oleg Panfilov (director of the Centre for Journalism in Extreme Situations), Russian journalists often ask foreign correspondents in Moscow to cover a trial on their behalf. A report in the Financial Times of London is worth more than dozens of domestic reports.
Panfilov's mentioning of the FT dove-tailed neatly with a comment by the author of the report, Nina Ognianova, who suggested that campaigners should focus on "shared interests" that the West has with Russia, rather than the rejected notion of "shared values". If the Russian government, and even the Russian public, are not outraged by the killing of journalists, then perhaps a campaign that aims for the wallet, rather than the heartstrings, might have more effect. Business journalists, lead by (say) the Financial Times, should place more emphasis on how the decline of investigative journalism leads to corruption… which stunts the economy and ensures fewer returns on investment. When the Russian elite realises that its own business interests are being irrevocably damaged by this culture of impunity, then perhaps they may be motivated to stop it.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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Q: Thinstation Citrix ICA PNAgent Configuration Help I have an old Thinstation build that connects directly to a Citrix server via ICA. I have a new Citrix farm that has an SSL relay out front, and I can't seem to figure out how I connect to this new farm using the PNAgent module.
Can anyone point me in the right direction?
A: From my research it doesn't appear that this is possible with Thinstation. I have, instead, built a custom SLAX distro which works via cd boot and starts the ica client.
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{
"redpajama_set_name": "RedPajamaStackExchange"
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Q: General question about convolution and Fourier I'm working on a system that includes a bunch of elements and I arrived at the following general expression for the output:
$$\mathcal{F}\left\{ T\cdot\left(\left[T\cdot\mathcal{F}\left\{ E_{1}\right\} \right]* h\right)\right\} $$
where \$ mathcal{F}\$ is the Fourier transform of whatever is inside of it.
From the way it looks, it's begging for me to employ the convolution theorem, and so I did:
$$\mathcal{F}\left\{ T\cdot\left(\left[T\cdot\mathcal{F}\left\{ E_{1}\right\} \right]* h\right)\right\} =\mathcal{F}\left\{ T\right\} *\left(\mathcal{F}\left\{ \left[T\cdot\mathcal{F}\left\{ E_{1}\right\} \right]* h\right\} \right)=$$ $$\mathcal{F}\left\{ T\right\} *\left[\mathcal{F}\left\{ T\cdot\mathcal{F}\left\{ E_{1}\right\} \right\} \cdot\mathcal{F}\left\{ h\right\} \right]$$
$$=\mathcal{F}\left\{ T\right\} *\left[\left(\mathcal{F}\left\{ T\right\} *\mathcal{F}\left\{ \mathcal{F}\left\{ E_{1}\right\} \right\} \right)\cdot\mathcal{F}\left\{ h\right\} \right]$$
Assuming I did these steps right, I'm not sure how to carry on from here? can I make it simpler?
Thanks in advance and hope you're all safe and healthy!
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{
"redpajama_set_name": "RedPajamaStackExchange"
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A hellenizmus vagy hellenisztikus korszak rendkívül összetett fogalom, melynek értelme és használata az idők folyamán sokszor változott. A "hellenizmus" alapvetően a görög kultúra elemeinek más kultúrák elemeivel keveredését jelenti, amely egyszerre a görög műveltség terjedése, és ennek révén új, a korábbitól különböző kultúrák megjelenése. A hellenizmus legjellemzőbb tulajdonsága az eklektika.
Politikai értelemben az ókori történelem egyik időszakát képviselő hellenisztikus civilizációt, kulturális téren az ókor művészetének egyik korstílusát, a hellenisztikus művészetet jelöli. Időhatárai pontosan nem meghatározhatóak. Kezdetét általában III. Alexandrosz makedón király uralkodásához kötik. Politikai értelemben akkor ért véget, amikor Kr. e. 30-ban az utolsó hellenisztikus monarchiát, a Ptolemaida Egyiptomot is bekebelezte a Római Birodalom. A kulturális értelemben vett hellenizmus azonban a birodalom görög nyelvű területein továbbra is jelen volt, és csak a kereszténység szorította ki végleg az i. sz. 4-5. században, sőt Egyiptomban az ősi egyiptomi és a görög civilizáció keveredését jelentő hellenizmus egészen az arab hódításig meghatározta a mindennapokat és bizonyos elemei a 10. századig éltek.
A "hellenizmus", mint történeti fogalom gyakorlatilag közmegegyezésen alapul, nincsenek kézzelfogható idő- és térbeli korlátai. Felső időhatára általában az actiumi csata (i. e. 31). Addigra azonban a hellenisztikus kultúrák vagy Parthia (Mezopotámiától Kelet-Iránig), vagy Róma (Kisázsia és Közel-Kelet) kulturális hegemóniájában olvadtak fel, Egyiptomban pedig az ősi kultúra idomította magához a hellénséget. I. e. 31-ben a hellenisztikus kultúra már sehol sem létezett, III. Alexandrosz halálakor még sehol sem – bizonyos mértékig kivéve Egyiptomot, Boszporoszt, valamint magát Rómát, ahol azonban a hellenizmus jóval korábbi eredetű, és inkább kapcsolható a korai görög gyarmatosításhoz, mint a későihez.
Korai hellenizmus
A korai görög gyarmatosítás folyamata az i. e. 7. századra a mediterráneumban már erős hellenizációt okozott, amely nem azonos ugyan a "hellenizmus" korával, de civilizációs következményei hasonlóak ahhoz. Ezért néha "hellenisztikus" jelzővel látják el a görög gyarmatvárosok különböző kultúráit is.
A görögség ez idő tájt hasonló diaszpórákat alkotott, mint korábban az asszírok, későbben pedig a zsidók. A korai görög gyarmatosítás következményei Egyiptomban és az etruszk-latin területeken váltak a leglátványosabbá, hatásai végigkövethetők az ókor folyamán. Hasonlóan a görög gyarmatvárosok görögségének hatása látható Küréné környékén és a későbbi Boszporoszi Királyságban.
A görög kultúra exportja először szinte mindenhol a képzőművészetek terén jelentkezett, úgy a szobrászatban, mint a grafikus ábrázolási módokban. Ezenkívül a görög mintájú természeti istenek kultuszai terjedtek – hol egybeolvadva a régiekkel, hol szinte változatlanul átvéve azokat. Egyiptomban sajátos, csak erre a területre jellemző kultuszok alakultak ki, melyek a helyi totemisztikus eredetű, illetve fétis-jellegű istenségek kultuszainak részleges megváltozásával járt. Ilyenek például Szerápisz, Mnevisz és Ápisz, melyek valamennyien termékenység-jelképek. A görögök is átvettek bizonyos vallási elemeket, mint például Harpokratész, a gyógyító, és Ízisz, az anyaistennő.
A Közel-Keleten és Kisázsiában valamennyi egyiptomi-görög szinkretista isten rendkívüli népszerűségnek örvendett, és nagy szerepük volt további kultuszok beolvasztásában, melyekből végül is a misztériumvallások nőtték ki magukat. A hellenisztikus misztérium-vallások tulajdonképpen a korai görög gyarmatosítás egyenes következményei (Adonisz, Kübelé, Mithrász, stb.), de ezekhez több évszázad szükségeltetett.
A korai görög gyarmatosítás sehol sem hozott nagy kiterjedésű politikai hegemóniát, azaz jelentős területszerzést. Az államisággal még nem rendelkező helyeken az anyavárosokhoz hasonló önálló poliszok jöttek létre, városállamok sorozata, melyek azonban nem ismerték el az anyaváros főségét, de helyi államokat sem alakítottak ki. A dél-itáliai, fekete-tengeri és észak-afrikai görög kolóniák ugyanolyan jellegűek voltak, mint balkáni rokonaik. Az egyes birodalmakban – mint például Egyiptomban – létesülő görög diaszpórák mindig valamilyen engedély birtokában alakultak, ami egyben a város politikai önállóságának feladását is jelentette.
A folyamat ugyanolyan jellegű volt, mint például a föníciai kolonizálás, vagy a zsidók diaszpóra-jelensége. Független, önálló gyarmatvárosok alakultak, melyek csak etnikumukban tartoztak Görögországhoz. E települések bekeretezték a Földközi- és Fekete-tenger partvidékét, de az anyaváros számára jelentős területszerzéssel nem jártak. A "körüllakó" (perioikosz) őslakók ritkán asszimilálódtak, néha kultúra-csere történt. A "nagy hellén gyarmatosítás" ráadásul csak egy mellékterméke a görög polisz fejlődésének, hiszen a behatárolt területtel és eltartóképességgel, ezenkívül rendkívül csekély politikai toleranciával rendelkező poliszok kénytelenségből küldték a hellén világ peremterületeire a politikailag kellemetlen, és a népességszaporulat miatt már el nem tartható polgáraikat. A gyarmatváros gyakran kapcsolatban maradt anyavárosával, ugyanakkor a "barbárokkal" folytatott kereskedelem közvetítőivé vált.
Klasszikus hellenizmus
III. Alexandrosztól a görög gyarmatosítás új formája indult meg. Kulturális megjelenése a keleti és nyugati civilizáció elemeinek összekeveredése.
A hellenizmus korában már nem alapítottak új településeket, hanem a görögök az agresszív hódítás után a meglévő régiekben telepedtek le, elvegyülve a helyi lakossággal, ami jelentős kulturális összeolvadást hozott. A hellenizmus korának városait az őslakosok és kiszolgált katonák lakták. Nagy területek kerültek közvetlenül görög befolyás alá, mind politikai, mind gazdasági síkon is. Azonban a hellén világ nem volt felkészülve egy világbirodalomra – még a monarchikus Makedónia sem –, ezért fordult Alexandrosz a keleti despotátusok formuláihoz. Ez pedig hiba volt, mert a két hagyomány összekapcsolódása után egyik fél sem lehetett maradéktalanul elégedett, főképpen nem a görög-makedón veteránok.
A hellenizmus korának kezdete sem a művészetekben, sem a tudományokban nem hozott törést, irányváltást, bár később a változás, alakulás folyamatos volt. Nem lehet meghatározni egy olyan pillanatot, amikortól például a filozófia hellenisztikussá válik, nincs olyan filozófus, akit a hellenizmus elsőjeként meg lehetne említeni. Még Epikurosz sem ilyen. A platóni Akadémosz folyamatosan változó irányultságú volt, a vezető személyétől függően. Az arisztotelészi Peripatosz már Alexandrosz előtt is "hellenisztikus" jellegű volt, a Sztoa Poikilé tanai későbbiek, Diogenész és Antiszthenész pedig korábbiak. Alexandrosz hódításai csak politikailag hoztak újat, hiszen a diadokhosz államok egészen Egyiptom bukásáig fontos szerepet játszottak, a hellenizmust mégis leginkább művészeti és filozófiai síkon emlegetjük.
Ahogy a hellenizmust időben nem lehet pontosan behatárolni, a fogalomnak tartalmát sem, hogy mit lehet egyáltalán hellenisztikusnak tekinteni. Hiszen ha valamennyi kultúrát ide soroljuk, amelyben görög elemek jelentek meg, akkor az etruszkoktól kezdve a késő római korig mindent fel lehet sorolni. Ezenkívül nincs két olyan terület, ahol az újabb görög gyarmatosítás akárcsak hasonló eredménnyel járt volna. Az Alexandrosz halála után széteső, és egyre inkább aprózódó birodalom valamennyi utódállamában eltérő politikai, kulturális és gazdasági viszonyok alakultak ki.
Általánosítható hellenisztikus stílus egyáltalán nincsen, éppen a rendkívüli sokféleség és az atipikus elemek a jellemzőek. A helyi hagyományok mindenütt szívósan fennmaradtak, a művészetekben, a filozófiában különböző "iskolák" jöttek létre, melyek helyre jellemzőek, nem időre. A jogalkotás és törvényhozás a görög intézmények mellett a helyi szokásjogokat is figyelembe vette. Még az egyes országoknak sem volt egységes nyelvük, még akkor sem, ha néhány nyelv, mint például az arameus nagy területen elterjedt. Az egyetlen közös nyelv a hellének által koinénak nevezett középgörög köznyelv, amelynek eredeti szókincse sok jövevényszóval bővült. A koiné a legtöbb görög nyelvjárást kiszorította. A "hellén" megjelölés ekkor már nem "görögöt" takart, hanem a diadokhosz-államok arisztokráciáinak valamennyi tagját, tekintet nélkül az etnikumra, míg a köznép – ismét etnikai megkülönböztetés nélkül – a "népek". A. A. Long a hellenizmus meghatározására is éppen elegendőnek tartja a koiné nyelv kialakulását.
Hellenisztikus filozófia
A hellenizmus kulturális termékei sajátságos, egyedi, atipikus képet mutatnak. A klasszikus antik filozófia minden vonatkozásában a polisz ideológiáját tükrözte, bármelyik irányzatról legyen is szó (Arisztotelészig), a hellenisztikus filozófia egyfajta individualisztikus metamorfózison ment át. De ez nem kapcsolható Alexandrosz birodalmához, hiszen például a irányzat elsőjeként elkönyvelt Diogenész korábbi annál. Ez a képzőművészetben a szélsőséges realizmus képében jelent meg, és mint az amarna-művészet párhuzama is mutatja, valamiféle hanyatlás látható kifejeződése. Míg korábban nem volt jellemző a fatalizmus, ekkortól nagy szerepet kapott a Sors, melynek kizárólag a Bölcs ember nincs alárendelve. Ez nem más, mint a keleti misztériumvallások és az antik filozófia egyesítése. Jellegzetes alakulatai a hellenizmusnak a sztoikus, a cinikus, a peripatetikus, epikureus és szkeptikus rendszerek. A kifejezetten platonista irányzatok visszaszorultak, már csak azért is, mert az Akadémosz ligetében egyre inkább csak matematikát oktattak.
A sztoikusok a 'logoszt', a világmindenség általános rendező elvét keresték, elsőjük Kitioni Zénón, akinek irányát Kleanthész és Khrüszipposz foglalták rendszerbe. Világnézetük panteisztikus (csakúgy, mint a vallási kérdésekben lanyha érdeklődést mutató peripatetikusoké), ezzel összefüggésben az ember "világpolgárságát" hirdették (koszmopolitész). Ez utóbbi kifejezetten a makedón világbirodalom hatása. A cinikusok Diogenész és Antiszthenész követői, akik az egyszerű népnek beszéltek. A cinikus filozófia még nem volt cinikus, a szó mai, pejoratív értelmében. A szkepticizmus fő képviselői Metrodosz és Pürrhón, akik szerint az élet nyugalmának elérése csak a megismerés lehetetlenségének belátásával lehetséges. Kialakultak a szaktudományok, melyeknek első kidolgozói az arisztotelészi peripatetikusok voltak. Alexandriában a filológiával kísérleteztek, elsősorban Homérosz műveinek elemzésével.
A hellenizmus filozófiai és társadalmi áramlatai nagymértékben elősegítették a keleti jellegű istenek tiszteletének terjedését, a misztériumvallások kialakulását, és mágikus szemléletmód előtérbe kerülését. Elsőrendű táptalaja volt a kereszténységnek.
Az úgynevezett "hellenizmus" tehát nem más, mint a görög-makedón telepesekkel érkező hellén kultúra asszimilálódása, idomulása a helyi formákhoz. A lakosság különböző etnikumainak elkerülhetetlen keveredése a hellén elemek kezdeti túlsúlyát és a helyben talált formáktól való elkülönültségét átváltotta hellenizmussá, amelynek további fejlődése is eltérő lett területenként. Az egy, ami megegyezik, az a hellén elemek fokozatos felszívódása, eltűnése. Ahogy a diadókhosz-államok fennhatósága megszűnt egy-egy volt birodalom-részben, a hellének minden maradék elkülönülése, elkülöníthetősége is megszűnt, csakúgy, mint a további görög etnikum beáramlása.
Kapcsolódó szócikkek
Hellenisztikus nyelvek
Külső hivatkozások
Hegyi Dolores – Kertész István – Németh György – Sarkady János: Görög történelem a kezdetektől Kr. e. 30-ig. Budapest, Osiris, 2005.
A.A. Long: Hellenisztikus filozófia, Osiris Kiadó, 1998.
A művészet története – az antik világ, Corvina Kiadó, 1986.
Ókor
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"redpajama_set_name": "RedPajamaWikipedia"
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[](http://badge.fury.io/js/ember-railio-grid) [](https://travis-ci.org/Fabriquartz/ember-railio-grid) [](http://emberobserver.com/addons/ember-railio-grid) [](https://codeclimate.com/github/Fabriquartz/ember-railio-grid)
An Ember addon to display a list of Objects, and be able to filter, sort and paginate this list. With build-in selection and be able to give actions for the selected objects.
## Install
In your application's directory:
```sh
$ ember install ember-railio-grid
```
## Basic usage
The grid can be used with an existing array of objects, or can be connected to the store to get the objects from the store.
In your Handlebars templates when using an existing list:
```handlebars
{{data-grid content=list
properties=listProperties}}
```
When you want to get the content from the store, keep in mind you have to pass the store in your current project to the data-grid.
```handlebars
{{data-grid modelName="modelName"
store=store
properties=listProperties}}
```
#### Properties
The properties are the cols you want to show and their extra information like formatting and styling.
It should be an array with objects for each property. At least it should contain key and label:
```js
listProperties: [
{
key: 'name',
label: 'First name'
},
{
key: 'age',
label: 'Age'
}
]
```
It is possible to use multiple properties in just one cell by giving an array of keys. By default, without giving a custom format function, both values are shown seperated with a comma. See the example below for using your own format function. The format function gets the properties in the same order as you specify it in the key list.
```js
listProperties: [
{
key: ['firstName', 'lastName'],
label: 'Name',
format: function(first, last) {
return `${last}, ${first}`;
}
}
]
```
If you want your cells to have some styling, you could add it to the property. The following styling options can be used inside the style object:
- width *(in em)*
- horizontalAlign
- verticalAlign
- backgroundColor
- fontFamily
- fontWeight
- italic *(boolean)*
- fontColor
- borderWidth *(in px)*
- borderColor
- borderStyle
```js
listProperties: [
{
key: 'name',
label: 'Name',
style: {
width: 30,
horizontalAlign: 'center',
verticalAlign: 'top',
backgroundColor: '#EFEFEF',
fontFamily: 'Monospace',
fontWeight: '300',
italic: true,
fontColor: '#777',
borderWidth: 2,
borderColor: 'grey',
borderStyle: 'dashed'
}
}
]
```
Instead of giving a fixed value, you could use a function to style the cell depending on it's value:
```js
listProperties: [
{
key: 'total',
label: 'Total',
style: {
backgroundColor: function(total) {
if (total > 1000) { return 'green'; }
if (total < 0) { return 'red'; }
return '#EFEFEF';
}
}
},
{
key: ['in', 'out'],
label: 'diff',
format: function(incoming, outgoing) {
return incoming - outgoing;
},
style: {
fontColor: function(incoming, outgoing) {
if (incoming > outgoing) { return 'green'; }
if (incoming < outgoing) { return 'red'; }
return 'black';
}
}
}
]
```
#### Paginating
The grid can be used to paginate the content. For a given content, it will use the build-in filterer. When using the api for getting the content, the api should handle the paginating. For now, there is just one paginator, but there will be more available in the future (or you could write your own). You are able to show the paginator on top, bottom, or both. Without a page size, the grid will show all content.
```handlebars
{{data-grid content=list
properties=listProperties
topPaginator="page-picker-paginator"
bottomPaginator="page-picker-paginator"
pageSize=20}}
```
#### Filtering
You could optional add a filtering bar, where you can filter the content by each col. For a given content, it will use the build-in filterer. When using the api for getting the content, the api should handle the filtering.
```handlebars
{{data-grid content=list
properties=listProperties
filterEnabled=true}}
```
Possible filter options:
- equals
- contains
- is greater than
- is greater than or equal to
- is lower than
- is lower than or equal to
- starts with
- ends with
#### Predefined filters
When you use the API data manger to load the data, you can add some predefined
filters. These filters will be added to the request, just like the filters from
the optional filter bar would.
```js
filters: [
{
filter: 'gt', propertyPath: 'price', value: 50
}
]
```
And can be passed to the grid like:
```handlebars
{{data-grid content=list
properties=listProperties
filters=filters}}
```
#### Sorting
The grid has a build-in sorting, or can be sorted from the store when using a model. Each property can be used to sort by clicking on it's header. The order of adding the sortings, will be the order for importancy. So the first added sorting will be the most important, even if it is changed from ASC to DESC. Only removing it and re-adding will change its importancy.
#### Actions
The grid has a build-in selection mechanism. You will be able to select specific rows when you have passed some actions (select-boxes appear when at least one action is passed). After selecting, you can use the passed actions for the selected items. The actions need to be a list like:
```js
actionList: [
{
label: 'something',
action: function(items) {
// do something for each selected item
}
},
{
label: 'open',
action: function(items) {
// open the selected items
}
}
]
```
And can be passed to the grid like:
```handlebars
{{data-grid content=list
properties=listProperties
actionList=actionList}}
```
##### doubleClick action
You could also pass a doubleClickAction, which will be called after double-clicking a row. It passes the object of that row to the function.
```js
editObject(object) {
// do something with this object
}
```
And can be passed to the grid like:
```handlebars
{{data-grid content=list
properties=listProperties
doubleClickAction=editObject}}
```
#### Selection
The grid has a build-in selection mechanism. Instead of that, you could use your own. You need to pass an array with the selected items, and three functions to select and deselect the items:
- `toggleItem` to select / deselect a single item: gets the clicked item
- `selectPage` to select the current page: gets an array of the items on the current page
- `clearSelection` to clear your selection
You could pass it like:
```handlebars
{{data-grid content=list
properties=listProperties
selection=selection
toggleItem=(action "selectItem")
selectPage=(action "selectCurrentPage")
clearSelection=(action "clearSelect")}}
```
If you want to use the build-in selection mechanism, but without using an actionList, you can pass the following property:
- `selectionEnabled`
You can use the selection outside the addon by passing an empty array by the following name. Use this at your own risk.
- `_selection`
#### Component Injection
You are able to inject a component into the grid, which will display that component instead of the value of that field.
The value, object and propertyPath will be passed into the injected component by default, to use this function you have these two options:
- `component` is the name of the component you want to inject.
- `componentProperties` are the properties you want to pass along with it.
```js
listProperties: [
{
key: 'name',
label: 'First name',
component: 'foo-bar',
componentProperties: ['foo', 'bar', 100]
}
]
```
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}
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\section{Introduction}
One of the approaches to the analysis of complex and expensive to evaluate computational models is surrogate modeling. Surrogate modeling methods allow to build a cheap-to-evaluate model that preserves some properties of the initial computational model.
One of the widely used surrogate modeling methods is Polynomial Chaos Expansion (PCE)~\cite{PolChao}. This method allows to model the response of the original system as a polynomial expansion over some functional basis of orthogonal polynomials. PCE shows impressive results when applied to the system in which inputs are sampled from some probability distribution~\cite{Chkifa2017,Hampton2015,Jakeman2017,Peng2016}. PCE is broadly used as a powerful tool in Uncertainty Quantification~\cite{1703.05312,Guo2018}. In order to build the PCE of some computational model, it is needed to compute the coefficients of PCE. This can be done with the least-squares method. However, the accuracy of the surrogate model that is obtained with the least-squares depends on the so-called~\emph{experimental design} (hereinafter ED). ED is a set of samples~--- points that are taken from the domain of computational model of interest according to some rule. So, the problem of proper selection of ED arises.
To solve this problem, sampling methods of design of experiments (DoE)~\cite{DoE} are widely used. So-called space-filling designs such as Latin hypercube sampling (LHS) or Sobol' sequence sampling are extensively applied.
One of the classes of DoE methods is the class of optimal design methods~\cite{Diaz2018,OptDesign}. The methods from this class make it possible to sample such ED that is optimal with respect to some criterion (\textit{e.g.} A-optimality, C-optimality, D-optimality, S-optimality, \textit{etc})~\cite{ACDopt}. The main advantage of the optimal design sampling over the other DoE sampling methods is that for the construction of an accurate surrogate model much smaller number of runs of the initial computational model is required.
In this paper, we propose a new method for sampling of D-optimal~\cite{OptDesign} ED by direct gradient-based optimization of the objective:
\begin{equation}
\det(A ^T A)\rightarrow\max,
\label{eq:objective}
\end{equation}
where $A$ is a model matrix that consists of values of all the basis polynomials evaluated at corresponding ED.
We derive an analytical expression for the gradient of \Cref{eq:objective} and test the proposed sampling method on the ordinary least-squares polynomial approximation of the multivariate function. In the context of this testing, an accuracy comparison with LHS, Sobol' sequence sampling, and Maxvol-based sampling~\cite{Goreinov2010,rect_maxvol} is carried out. Also, Lebesgue constant growth~\cite{2014arXiv1407.3291G,Matthias} is investigated.
\section*{Related work} In the paper~\cite{Simpson2001}, a review of statistical techniques for building an approximation of expensive computational codes is conducted. The methods of interest are the design of experiments, response surface methodology, neural networks, kriging.
This review describes an application of metamodeling techniques in engineering design and the issues with application of statistical methods in deterministic computer experiments.
The scope of the paper~\cite{Giunta2003} covers applications in computational engineering design studies of special DoE techniques. These techniques are designed for deterministic computer simulations and include Hammersley sequence sampling, LHS, and orthogonal array sampling. Also, pseudo-Monte Carlo sampling and quasi-Monte Carlo sampling were included into consideration.
In the paper~\cite{Goel2008}, two criteria of experimental design are considered. The first criterion allows to reduce the effect of noise during the surrogate construction while the second criterion helps to reduce bias errors. It is stated that a good sampling method should fulfill both criteria at the same time.
In this paper, multiple criteria for the assessment of widely-used experimental design methods (such as LHS and D-optimal sampling methods) are used. It is demonstrated that the majority of the sampling methods fulfill only one of the criteria but not the other.
\section{Problem statement}
\label{sec:problem}
Let us consider a computational model describing a certain system (for example, physical) $f (\vec x),$ where $\vec x = (\vector x_d)^T \in \mathcal{X}\subset\mathbb{R}^d$ is the column-vector of the input variables, $y\in\mathbb{R}$ is the output variable, and $\mathcal{X}$ is the set of admissible vectors $\vec x.$
We consider the model $f(\vec x)$ as a black box: we assume that it can be represented in the form of an expansion over a certain basis of orthogonal polynomials:
\begin{equation*}
f(\vec{x}) = \sum\limits_{j\in\mathbb{N}_0}c_j\Psi_j(\vec{x}),\;\; \vec{x}\in\mathcal{X}
\end{equation*}
where $\Psi_j(\vec x)$ is a multivariate polynomial, $c_j$ are expansion coefficients, and $\mathbb{N}_0\equiv\mathbb{N}\cup\{0\}$ is an extended set of natural numbers.
An element of a $d-$dimensional polynomial basis is defined as the tensor product of univariate polynomials:
$$\Psi_{\vec\alpha}(\vec x) = \prod\limits_{i = 1}^d\psi_{\alpha_i}(x_i), \;\; \vec\alpha\in\mathbb{N}^{d}_0, $$
where $\alpha_i$ is the degree of the univariate polynomial.
By choosing the set of multi-indices $\vec\alpha\in\mathcal{A}\subset\mathbb{N}^{d}_0 $ for some rule, we obtain a polynomial expansion of our model of interest:
\begin{equation}\label{eq:1}
f(\vec{x})\approx\widetilde f(\vec{x}) = \sum\limits_{\vec\alpha\in\mathcal{A}}\tilde{c}_{\vec\alpha}\Psi_{\vec\alpha}(\vec x).
\end{equation}
Our goal is by evaluating the function of interest $f(\vec{x})$ at points from its domain to recover the coefficients of the expansion \cref{eq:1}.
In order to truncate the number of terms in the expansion, we will choose the set $\mathcal{A}$ as follows~\cite{Truncation}:
$$\mathcal{A} = \left\{\vec\alpha\in\mathbb{N}^{d}_0\colon\|\vec\alpha\|_q = \sum\limits_{i = 1}^d\alpha_i^q\leq p^q\right\},$$
where $p$ is the total degree of the polynomial, and $q\in(0,1].$
It is easy to see that the cardinality of the set $|\mathcal{A}| $ is decreasing with decreasing of $q$. This truncation scheme allows to decrease the number of terms in polynomial expansion while keeping the same total degree $p$.
In this paper we consider the case $q = 1,$ \emph{i.\,e.} $\|\vec\alpha\|_1 = \sum\limits_{i = 1}^d\alpha_i\leq p.$
The cardinality of the set $|\mathcal{A}|$ in this case (or, what is the same, the number of terms of a polynomial expansion) is
\begin{pmatrix}d + p\\ p\end{pmatrix}.$
Let us define as an \emph{experimental design} (ED) the following matrix $X:$
\begin{equation*}
X = [\vector \vec x_n]^T\in\mathbb{R}^{n\times d},\;\; n\geq|\mathcal{A}|.
\end{equation*}
The model (Vandermonde-like) matrix $A\in\mathbb{R}^{n\times|\mathcal{A}|} $ is defined as:
\begin{equation}
A_{ij} = \prod\limits_{k = 1}^d\psi_{\vec\alpha_j^{(k)}}(
x_i^{(k)}
),
\label{eq:vandermonde}
\end{equation}
where $\vec\alpha_j^{(k)}$ is the $k$-th component of the multi-index $\vec\alpha_j\in\mathcal{A} = \{\vec\alpha_1,\vec\alpha_2,\ldots,\vec\alpha_{|\mathcal{A}|}\},$ and $x_i^{(k)}$ is the $k$-th component of the $i$-th point $\vec{x}_i$ of input. It is worth noting that all the elements of $\mathcal{A}$ are ordered arbitrary and fixed.
The coefficients in \Cref{eq:1} can be found as a solution to the ordinary least-squares minimization problem:
\begin{equation}\label{eq:LSM}
\tilde{\vec{c}} = A^+\mathcal{Y} = (A^TA)^{-1}A^T\mathcal{Y},
\end{equation}
where $\tilde{\vec{c}} = (\tilde{c}_{\vec\alpha_1},\tilde{c}_{\vec\alpha_2},\ldots,\tilde{c}_{\vec\alpha_{|\mathcal{A}|}})^T$ is a column-vector of coefficients of polynomial expansion, and $\mathcal{Y} = (\vector y_n)^T$,
$y_i=f(\vec{x}_i)$ is a column-vector of model responses at ED matrix $X$.
We will call a matrix $\widetilde{X}\in\mathbb{R}^{n\times d}$ an \emph{optimal} ED if the following D-optimality criterion holds for it.
\begin{definition}\label{def:dopt}
$\widetilde{X}\in\mathbb{R}^{n\times d} $ is an D-optimal ED if the following criterion is satisfied:
$$\widetilde{X}:\det B(\widetilde{X}) = \max\limits_{X\in\mathbb{R}^{n\times d}}\det B(X),$$
\end{definition}
where $B(X) = A^T(X)\cdot A(X)$ is a symmetric non-negative definite matrix.
\section{Objective function and its gradient}
\label{sec:GD}
To solve the problem posed in \Cref{sec:problem}, we will use the gradient descent method.
Since we use a D-optimality criterion, it is quite a natural way to optimize the following objective function:
$$W(X) = \det B(X),$$
where matrix $B$ is non-negative definite.
Thus, the problem of finding a D-optimal experimental design $\widetilde{X}$ can be written as follows:
$$\widetilde{X} = \argmax\limits_{X\in\mathbb{R}^{n\times|\mathcal{A}|}} W(X)= \argmax\limits_{X\in\mathbb{R}^{n\times|\mathcal{A}|}} \det B(X) = \argmax\limits_{X\in\mathbb{R}^{n\times|\mathcal{A}|}} \det A(X)^TA(X),$$
for a fixed set $\mathcal{A}$ (\textit{i.e.} for a fixed total degree $p$).
For the problem posed in this way, the standard approach is to replace the original problem with the equivalent one~\cite{log_det}:
\begin{equation}\label{problem}
\widetilde{X} = \argmin\limits_{X\in\mathbb{R}^{n\times|\mathcal{A}|}} \widehat{W}(X) = \argmin\limits_{X\in\mathbb{R}^{n\times|\mathcal{A}|}} \log\det B^{-1}(X) = \argmin\limits_{X\in\mathbb{R}^{n\times|\mathcal{A}|}}( -\log\det B(X)).
\end{equation}
It is worth noting that since the function $\widehat{W}(X)$ is differentiable, we can use gradient descent to find the minimum.
\subsection{Analytical calculation of gradient}
We define $G =\nabla_{X}\widehat{W}(X)\in\mathbb{R}^{n\times d}$ as the gradient matrix of the function $\widehat{W}(X).$ The gradient matrix element is
$$G_{ij} = \dfrac{\partial\widehat{W}(X)}{\partial \vec{x}_i^{(j)}}.$$
Having defined the gradient matrix in such a way, we can obtain an analytical expression for finding its elements $G_{ij}.$
First of all, let us consider a one-dimensional case ($d = 1$). In such a case, the experimental design $X$ is presented as a column-vector of $n$ one-dimensional points
$X = (\vector x_n)^T\in\mathbb{R}^n$.\\
Corresponding gradient matrix $G = \left[\dfrac{\partial\widehat{W}(X)}{\partial x_1},\ \ldots,\
\dfrac{\partial\widehat{W}(X)}{\partial x_n}\right]^T\in\mathbb{R}^{n\times{1}}.$
The matrix $A(X)\in\mathbb{R}^{n\times(p+1)}$ will have the following form:
\[A(X) =
\begin{pmatrix}
\psi_0(x_1) & \psi_1(x_1) & \hdots & \psi_p(x_1) \\
\psi_0(x_2) & \psi_1(x_2) & \hdots & \psi_p(x_2) \\
\vdots & \vdots & \ddots & \vdots \\
\psi_0(x_n) & \psi_1(x_n) & \hdots & \psi_p(x_n)
\end{pmatrix},\]
where $p$ is the total polynomial degree.
Then the matrix $B(X)\in\mathbb{R}^{(p+1)\times(p+1)}$, $B(X)= A(X)^{T}\cdot A(X)$ is symmetric and is represented as follows:
\[B(X) =
\begin {pmatrix}
\sum\limits_{m = 1}^{n}\psi_0^2(x_m) & \sum\limits_{m = 1}^{n}\psi_0 (x_m)\psi_1(x_m) & \hdots & \sum\limits_{m = 1}^{n}\psi_0(x_m)\psi_p(x_m) \\
\sum\limits_{m = 1}^{n}\psi_1(x_m)\psi_0(x_m) & \sum\limits_{m = 1}^{n}\psi_1^2(x_m) & \hdots & \sum\limits_{m = 1}^{n}\psi_1(x_m)\psi_p (x_m) \\
\vdots & \vdots & \ddots & \vdots \\
\sum\limits_{m = 1}^{n}\psi_n(x_m)\psi_0(x_m) & \sum\limits_{m = 1}^{n}\psi_n(x_m)\psi_1(x_m) & \hdots & \sum\limits_{m = 1}^{n}\psi_n^2(x_m)
\end{pmatrix},\]
where $ B = \sum\limits_{m = 1}^{n}A_{mi}A_{mj} = \sum\limits_{m = 1}^{n}\psi_i(x_m)\psi_j(x_m).$
Let us formulate the Lemma that allows us to calculate the gradient of the objective.
\begin{lemma}\label{lemma:one}
For the one-dimensional case $(d=1)$, $k$-th $(k\in\{1,\,2\, \ldots,\, n\})$ component of the gradient matrix $G$ is equal to:
$$ G_k = \dfrac{\partial\widehat{W}(X)}{\partial x_k} = -\sum\limits_{ij}(B^{-1}(X))_{ji}\cdot\left[\dfrac{\partial\psi_i(x_k)}{\partial x_k}\cdot\psi_j(x_k) + \psi_i(x_k)\cdot\dfrac{\partial\psi_j(x_k)}{\partial x_k}\right].$$
\end{lemma}
Now, we generalize the result obtained in \cref{lemma:one} to the multidimensional case when $d > 1$. Let us also recall that $\vec\alpha_j^{(k)}$ is the $k$-th component of the multi-index $\vec\alpha_j\in\mathcal{A}.$
\begin {theorem*}\label{th:one}
The element of the matrix $G$ for $d > 1$ is expressed as follows:
\begin{multline*}
G_{kl} = -\sum\limits_{ij}(B^{-1}(X))_{ji}\cdot\left[\dfrac{\partial \psi_{\vec\alpha_i^{(k)}}({x}_k^{(l)})}{\partial {x}_k^{(l)}}\cdot\psi_{\vec\alpha_j^{(k)}}({x}_k^{(l)}) + \psi_{\vec\alpha_i^{(k)}}({x}_k^{(l)})\cdot\dfrac{\partial \psi_{\vec\alpha_j^{(k)}}({x}_k^{(l)})}{\partial {x}_k^{(l)}}\right] \\
\times\prod\limits_{\substack{q=0\\(q\neq{k})}}^{d-1}\psi_{\vec\alpha_i^{(q)}}({x}_k^{(q)})\cdot\psi_{\vec\alpha_j^{(q)}}({x}_k^{(q)}).
\end{multline*}
\end{theorem*}
The proofs of \Cref{lemma:one} and \Cref{th:one} are in \cref{sec:appendix}.
\cref{th:one} allows to calculate the gradient of the objective function $\widehat{W}(X).$ This gradient, in turn, can be used in any of the algorithms of gradient descent.
\subsection{Block-coordinate gradient descent heuristic}
We can compute ED faster by replacing the full gradient with another descent direction. At each iteration of the gradient descent algorithm, we change only the coordinates corresponding to one $d$-dimensional point.
We construct the gradient matrix $G$ at the step $k$ of gradient descent as follows. Using \cref{th:one}, we compute $G(X^{(k)})$, and then we choose a row $l$ of the matrix $G(X^{(k)})$ such that the following condition holds:
\begin{equation}\label{eq:line}
l = \argmax_i\sum\limits_{j = 1}^{d}|G_{ij}(X^{(k)})|.
\end{equation}
Now we change the matrix $G(X^{(k)})$ by setting the elements of the remaining $n-1$ rows equal to zero:
$$
G(X^{(k)})
:=
\begin{pmatrix}
0 & 0 & \hdots & 0\\
G_{l1}(X^{(k)}) & G_{l2}(X^{(k)}) & \hdots & G_{ld}(X^{(k)})\\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \hdots & 0
\end{pmatrix}.
$$
The matrix $G(X^{(k)})$ obtained in such a way is the descent direction that we use.
Let us consider the calculation of the gradient matrix element $G$ at the step $k + 1$ of the gradient descent in more detail.
According to \cref{th:one}:
$$G_{ml}(X^{(k+1)}) = -\sum\limits_{ij}(B^{-1}(X^{(k+1)}))_{ji}\cdot\dfrac{\partial B_{ij}(X^{(k+1)})}{\partial X^{(k+1)}_{ml}}.$$
Since the matrices at the $k$-th and at the $(k + 1)$-th steps differ in exactly one row $l$ (see \Cref{eq:line}), we can simplify the calculation of the inverse matrix $B^{-1}(X^{(k+1)})$ with the use of previously computed $B^{-1}(X^{(k)}).$
Inspired by ideas of maxvol~\cite{Goreinov2010}, we can do this by using Sherman-Morrison-Woodbury formula~\cite{SWM}:
\begin{equation}\label{eq:SWM}
B^{-1}(X^{(k+1)}) = B^{-1}(X^{(k)}) - B^{-1}(X^{(k)})U[I_{2\times 2}+VB^{-1}(X^{(k)})U]^{-1}VB^{-1}(X^{(k)}),
\end{equation}
where $U\in\mathbb{R}^{|\mathcal{A}|\times 2}$ and $V\in\mathbb{R}^{2\times |\mathcal{A}|}$:
$$U
=
\begin{pmatrix}
A_{l1}(X^{(k+1)}) & A_{l1}(X^{(k)}) \\
A_{l2}(X^{(k+1)}) & A_{l2}(X^{(k)}) \\
\vdots & \vdots \\
A_{l|\mathcal{A}|}(X^{(k+1)}) & A_{l|\mathcal{A}|}(X^{(k)})
\end{pmatrix},
$$
$$V
=
\begin{pmatrix}
A_{l1}(X^{(k+1)}) & A_{l2}(X^{(k+1)}) & \hdots & A_{l|\mathcal{A}|}(X^{(k+1)}) \\
-A_{l1}(X^{(k)}) & -A_{l2}(X^{(k)}) & \hdots & -A_{l|\mathcal{A}|}(X^{(k)})
\end{pmatrix}.
$$
All the steps described above are summarized in \cref{alg:block_coordinate}.
Thus, for each computation of the gradient matrix $G$ it is necessary to calculate~\cref{eq:SWM} that requires $\mathcal{O}(|\mathcal{A}|^2)$ operations in comparison with the calculation of the full-size matrix $B^{-1}$ that takes $\mathcal{O}(|\mathcal{A}|^3)$ operations.
\begin{algorithm}
\caption{Block-coordinate gradient descent}
\label{alg:block_coordinate}
\begin{algorithmic}[1]
\STATE{Initialize $X^{(0)}\in\mathcal{D}\subset\mathbb{R}^{n\times|\mathcal{A}|}$}
\STATE{Calculate $B^{-1}(X^{(0)})$}
\STATE{Using \cref{th:one}, calculate $G(X^{(0)})$}
\STATE{Find a row with the largest $L_1$-norm $l := \argmax_i\sum\limits_{j = 1}^{d}|G_{ij}(X^{(0)})|$.}
\STATE{$G_{ij}(X^{(0)}) := 0$ for $\forall i\neq l,j$}
\STATE{$X^{(1)}\leftarrow\ $GRADIENT DESCENT($X^{(0)}, G(X^{(0)})$)}
\STATE{Initialize $k := 1$}
\WHILE{NOT CONVERGED}
\STATE{$B^{-1}(X^{(k)}) := B^{-1}(X^{(k-1)})\left(I - U[I_{2\times 2}+VB^{-1}(X^{(k-1)})U]^{-1}VB^{-1}(X^{(k-1)})\right)$}
\STATE{Using \cref{th:one}, calculate $G(X^{(k)})$}
\STATE{Find a row with the largest $L_1$-norm $l := \argmax_i\sum\limits_{j = 1}^{d}|G_{ij}(X^{(k)})|$.}
\STATE{$G_{ij}(X^{(k)}) := 0$ for $\forall i\neq l,j$}
\STATE{$X^{(k+1)}\leftarrow\ $GRADIENT DESCENT($X^{(k)}, G(X^{(k)})$)}
\STATE{Update $k := k + 1$}
\ENDWHILE
\RETURN $X^{(k)}$
\end{algorithmic}
\end{algorithm}
\section{Numerical experiments}
\label{sec:numerical}
\subsection{Setting}
In this section, we assess the efficiency of the proposed sampling method (denoted as \textit{GD sampling}) by conducting a comparative study with other sampling methods in terms of approximation accuracy and Lebesque constant growth.
In \Cref{sec:error}, comparison of the accuracy of the least-squares polynomial approximation build on the sampled points is carried out. The proposed sampling method is tested on four analytical models with varying complexity and input dimensionality. They include three two-dimensional analytical functions namely:
\begin{itemize}
\item Rosenbrock function,
\item Sine-cosine product function (denoted as \emph{sincos}),
\item Gaussian function.
\end{itemize}
Also, the proposed sampling method is tested on the Piston simulation function that is effectively seven-dimensional. Since all of the testing models are analytical (consequently, cheap to evaluate), the appropriate relative approximation error on a test set is then utilized to evaluate the accuracy of the resulting polynomial expansions.
Despite the fact that the model matrix $A$ can be constructed over different polynomial bases (\textit{e.g.} Legendre polynomials or Hermite polynomials), in the numerical experiments below we are considering Chebyshev polynomials as the basis functions without\- loss of generality:
$$
\begin{aligned}
\psi_i(x) &= T_i(x),\\
T_{i+1}(x) &= 2x\cdot T_i(x) - T_{i-1}(x),
\end{aligned}
$$
with $ T_1(x) = x$, $T_0(x) = 1$.
Since we consider sampling methods that have stochastic nature, the corresponding EDs that are chosen by these sampling methods are affected. In order to take this fact into the account, we run each analysis 50 times. The aim of repetitions is to assess the effect of stochastic variations.
We consider the following sampling techniques:
\begin{itemize}
\item LHS~\cite{McKay1979},
\item Sobol' sequence~\cite{Sobol1967},
\item Maxvol-based~\cite{Goreinov2010,rect_maxvol}.
\end{itemize}
Examples of ED obtained with the mentioned above sampling techniques for the two-dimensional case are shown at \cref{pic:ed_sample}.
\begin{figure}[ht]
\centering{
\begin{subfigure}[t]{0.32\linewidth}
\centering{\includegraphics[width=1\linewidth]{pics/lhs_40.pdf}}\caption{LHS}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.32\linewidth}
\centering{\includegraphics[width=1\linewidth]{pics/sobol_40.pdf}}\caption{Sobol' sequence.}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.32\linewidth}
\centering{\includegraphics[width=1\linewidth]{pics/maxvol_40.pdf}}\caption{Maxvol sampling.}
\end{subfigure}}\caption{\label{pic:ed_sample} ED of the size 40 obtained with different sampling methods.}
\end{figure}
\subsection{Error of approximation}\label{sec:error}
As an accuracy measure of the least-squares polynomial approximation built on ED obtained with each of the sampling techniques, relative error in the infinity norm is utilized:
$$\delta_\infty = \frac{\|f - \tilde{f}\|_\infty}{\|f\|_\infty},$$
where $$\|f\|_\infty\equiv\max\limits_{\vec x\in\mathcal{D}}|f(\vec x)|,$$
for some test set of points
$\mathcal{D}\subset\mathbb{R}^{d}$.
For all of the experiments below, $\delta_\infty$ is calculated on the test set~$\mathcal D$ of the size $N_{\text{test}}=|\mathcal D|= 10^6$. As a numerical optimizer for GD sampling, we use BFGS method from \texttt{scipy.optimize}.
\subsubsection{Rosenbrock function}
First of all, we will approximate with a polynomial expansion a well-known two-dimensional Rosenbrock function:
\begin{equation}
f(x,y) = \left(1-x\right)^2 + 100\left(y-x^2\right)^2.
\label{eq:rosenbrock}
\end{equation}
Recall that $l$ is the number of terms in polynomial expansion (in one-dimensional case, it is a total degree minus one of such an expansion), and $n$ is the number of points that make up the ED matrix $X$. In the experiments below, we consider the case $n=l$ when the number of points equals the number of term in polynomial expansion (in such a case, model matrix $A$ is square).
The performance of different sampling methods is compared in terms of infinity norm of the relative error on the test set for the varying size of experimental designs (\cref{pic:4}). Each analysis is repeated 50 times in order to estimate statistical uncertainty.
\begin{figure}[htb]
\centering{
\begin{subfigure}[t]{0.4\linewidth}
\centering{\includegraphics[width=1\linewidth]{pics/rosen.pdf}}\caption{Plot of Rosenbrock function.}
\end{subfigure}
\vfill
\begin{subfigure}[t]{0.49\linewidth}
\centering{\includegraphics[width=1\linewidth]{pics/rosen_col-10_col-20.pdf}}\caption{}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.49\linewidth}
\centering{\includegraphics[width=1\linewidth]{pics/rosen_col-10_col-20_short.pdf}}\caption{}
\end{subfigure}}
\caption{\label{pic:4}\textbf{(a)}: Plot of Rosenbrock function on the domain of interest: $[-1,1]\times[-1,1]$. \textbf{(b)}: Test error for different number of terms (from $l = 10$ to $l = 20$) in polynomial expansion. The box-plots are obtained from 50 repetitions. \textbf{(c)}: The same as \textbf{(b)} but focused on two best sampling methods.}
\end{figure}
One can observe that ED based on LHS and Sobol' sequences show a poor performance compared to the GD sampling and Maxvol sampling. At the same time GD sampling and Maxvol sampling have the same performance. A drop in the accuracy from $l=13$ to $l=14$ for all the sampling methods is connected with the increase of the total degree of the polynomial expansion~$\widetilde f(\vec x)$, since the total degree of \cref{eq:rosenbrock} is 4, we get exact representation.
\subsubsection{Sincos function}
Now let us consider another two-dimensional function on a square $[-1,1]\times[-1,1]$ that we have denoted as \emph{sincos}:
$$
f(x,y) = \sin\left(\frac{x^2}{2} - \frac{y^2}{4} + 3\right)\cdot\cos\left(2x+1-e^y\right).
$$
As in the case of Rosenbrock function, for sincos function we compute the approximation error on $N_{\text{test}}=10^6$ test points for ED sizes in the range from $l = 30$ to $l = 48$ (\cref{pic:5}). On \cref{pic:5} the trend of decreasing approximation error with the increase of ED size for D-optimal sampling methods can be seen.
\begin{figure}[htb]
\centering{
\begin{subfigure}[t]{0.37\textwidth}
\centering{\includegraphics[width=1\textwidth]{pics/sincos.pdf}}\caption{Plot of sincos function.}
\end{subfigure}
\vfill
\begin{subfigure}[t]{0.49\textwidth}
\centering{\includegraphics[width=1\textwidth]{pics/sincos_col-30_col-48.pdf}}\caption{}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.49\textwidth}
\centering{\includegraphics[width=1\textwidth]{pics/sincos_col-30_col-48_short.pdf}}\caption{}
\end{subfigure}
}\caption{\label{pic:5}\textbf{(a)}: Plot of sincos function on the domain of interest: $[-1,1]\times[-1,1]$. \textbf{(b)}: Evolution of box-plots of the test error for different number of terms in polynomial expansion. The box-plots are obtained from 50 repetitions. \textbf{(c)}: The same as \textbf{(b)} but focused on two best sampling methods.}
\end{figure}
\subsubsection{Gaussian function}
The final test on two-dimensional functions is a Gaussian function on the box domain $[-1,1]\times[-1,1]$:
$$f(x,y) = 2e^{-\frac{7}{2}\left(x^2+y^2\right)}.$$
On \cref{pic:6} one can see results similar to \cref{pic:5}: D-optimal sampling methods perform much better than LHS and Sobol' sequence sampling, and approximation error decreases with the increase in ED size.
\begin{figure}[htb]
\centering{
\begin{subfigure}[t]{0.37\linewidth}
\centering{\includegraphics[width=1\linewidth]{pics/gauss.pdf}}\caption{Plot of Gaussian function.}
\end{subfigure}
\vfill
\begin{subfigure}[t]{0.49\linewidth}
\centering{\includegraphics[width=1\linewidth]{pics/gauss_col-44_col-64.pdf}}\caption{}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.49\linewidth}
\centering{\includegraphics[width=1\linewidth]{pics/gauss_col-44_col-64_short.pdf}}\caption{}
\end{subfigure}}\caption{\label{pic:6}\textbf{(a)}: Plot of Gaussian function on the domain of interest: $[-1,1]\times[-1,1]$. \textbf{(b)}: Evolution of box-plots of the test error for different number of terms in polynomial expansion. The box-plots are obtained from 50 repetitions. \textbf{(c)}: The same as \textbf{(b)} but focused on two best sampling methods.}
\end{figure}
D-optimal design sampling methods consistently outperform other sampling methods. Moreover, such methods generally behave in a more stable way resulting in smaller variability between repetitions. Especially, this property becomes more important as the size of ED becomes larger.
Let us consider the case when more points are sampled than the number of terms in polynomial expansion (so-called \emph{oversampling}).
So, we complement the results for the approximation of Gaussian function (\cref{pic:6}) by considering two cases of oversampling: when the number of sampled points is 1.1 times more than the number of terms in polynomial expansion (\cref{pic:7:a}), and the case when we sample 2.5 times more points than the number of terms in corresponding expansion (\cref{pic:7:b}). As we can see on \cref{pic:7}, oversampling allows to improve the approximation (especially when compared to \cref{pic:6}) mainly for LHS and Sobol' sequence sampling. With the increase of the oversampling factor (from~1.1 to~2.5) Sobol' sequence sampling and GD sampling show more stable performance.
\begin{figure}[ht]
\centering{
\begin{subfigure}[t]{0.49\linewidth}
\centering{\includegraphics[width=1\linewidth]{pics/factor_1_1_gauss_col-44_col-64.pdf}}\caption{\label{pic:7:a}}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.49\linewidth}
\centering{\includegraphics[width=1\linewidth]{pics/factor_2_5_gauss_col-44_col-64.pdf}}\caption{\label{pic:7:b}}
\end{subfigure}}
\caption{\label{pic:7}Accuracy of approximation for different sizes of ED: \textbf{(a)} the number of sampled points is 1.1 times more than the number of terms (\textit{i.e.,} $n = \lceil1.1\cdot l\rceil)$, and \textbf{(b)} the number of sampled points is 2.5 times more than the number of terms in polynomial expansion (\textit{i.e.,} $n = \lceil2.5\cdot l\rceil)$.}
\end{figure}
Summarizing the results of the tests on two-dimensional functions, it can be stated that oversampling allows to significantly improve approximation accuracy for LHS and Sobol' sequence sampling, while practically has no effect on D-optimal sampling methods.
It means that GD sampling or Maxvol sampling can be effectively used in the case of the tight budget on the number of runs of the complex model of interest.
At the same time, it can be also noticed that oversampling provides more stable performance for GD sampling and Sobol' sequence sampling methods.
\subsubsection{Piston simulation function}
In order to apply the proposed sampling method to high-dimensional surrogate modeling, we will consider a Piston simulation function~\cite{Kenett2013}. This function has a seven-dimensional input (see \cref{tbl:piston_descr}) and an one-dimensional output that effectively models the time (in seconds) that takes piston to complete one cycle within a cylinder. The cycle time is determined by a composition of functions:
$$C(\vec{x}) = 2\pi\sqrt{\frac{M}{k+S^2\cdot\frac{P_0V_0}{T_0}\cdot\frac{T_a}{V^2}}},$$
$$\text{where}\ V = \frac{S}{2k}\left(\sqrt{A^2+4k\frac{P_0V_0}{T_0}T_a}-A\right),$$
$$A = P_0S+19.62M-\frac{kV_0}{S}.$$
\begin{table} [ht]
\centering{
\begin{tabular}{|p{1.3cm}||p{3.7cm}|p{2.4cm}|p{1cm}|}
\hline
Variable & Name & Range & Units \\
\hline
$M$ & Piston weight & [30, 60] & $kg$\\
$S$ & Piston surface area & [0.005, 0.020] & $m^2$\\
$V_0$ & Initial gas volume & [0.002, 0.010] & $m^3$\\
$k$ & Spring coefficient & [1000, 5000] & $N/m$\\
$P_0$ & Atmospheric pressure & [90000, 110000] & $N/m^2$\\
$T_a$ & Ambient temperature & [290, 296] & $K$\\
$T_0$ & Filling gas temperature & [340, 360] & $K$\\
\hline
\end{tabular}\caption{\label{tbl:piston_descr}Description of Piston simulation variables.}
}
\end{table}
In the \cref{tbl:piston}, an approximation error (a median value over 30 runs) for two different sizes ($l=1750$ and $l=1850$) of ED and various number of sampling points~$n$ can be found.
\begin{table}[htb]
\centering
\begin{tabular}{|p{3cm}||p{1.4cm}|p{1.4cm}|p{1.4cm}|p{1.4cm}|p{1.4cm}|}
\hline
\multicolumn{6}{|c|}{Number of expansion terms, $l = 1750$} \\
\hline
Number of samples & $n = 1750$ & $n = 1760$ & $n = 1770$ & $n = 1780$ & $n = 1790$\\
\hline
$\delta_{\text{GD}}$ & \textbf{0.0702} & \textbf{0.0593} & \textbf{0.0577} & \textbf{0.0486} & \textbf{0.0477}\\
$\delta_{\text{Sobol}}$ & 0.5161 & 0.4676 & 0.3283 & 0.2656 & 0.2420\\
$\delta_{\text{LHS}}$ & 0.3963 & 0.2494 & 0.3556 & 0.3481 & 0.2126 \\
$\delta_{\text{Maxvol}}$ & 0.0928 & 0.0601 & 0.0526 & 0.0427 & 0.0400\\
\hline
\end{tabular}
\medskip
\begin{tabular}{|p{3cm}||p{1.4cm}|p{1.4cm}|p{1.4cm}|p{1.4cm}|p{1.4cm}|}
\hline
\multicolumn{6}{|c|}{Number of expansion terms, $l = 1850$} \\
\hline
Number of samples & $n = 1850$ & $n = 1860$ & $n = 1870$ & $n = 1880$ & $n = 1890$\\
\hline
$\delta_{\text{GD}}$ & \textbf{0.0511} & \textbf{0.0492} & \textbf{0.0401} & \textbf{0.0388} & \textbf{0.0352}\\
$\delta_{\text{Sobol}}$ & 0.5925 & 0.5987 & 0.3521 & 0.2506 & 0.2517\\
$\delta_{\text{LHS}}$ & 0.2357 & 0.6181 & 0.6009 & 0.3658 & 0.3169 \\
$\delta_{\text{Maxvol}}$ & 0.0548 & 0.0489 & 0.0488 & 0.0462 & 0.0382\\
\hline
\end{tabular}
\caption{\label{tbl:piston} Accuracy of the least-squares polynomial approximation of Piston simulation function.}
\end{table}
From \cref{tbl:piston}, we can see that D-optimal sampling methods are the best. Also, it can be noted that oversampling allows to reduce the approximation error for the ED sampled with Sobol' sequence method.
Overall, it should be noted that in spite of similar performance of Maxvol sampling and GD sampling, the latter is more flexible as it is able to sample arbitrary points from the domain of interest whereas Maxvol sampling is limited with a discrete set of initial points.
\subsection{Lebesgue constant estimation}
In this Subsection we conduct a numerical estimation for the Lebesgue constant.
Let $\mathsf P(X)\colon C\rightarrow C$ be a projector on the span of the selected basis,
based on the experimental design matrix~$X$:
\begin{equation*}
(\mathsf P(X)f)(\vec x)=\widetilde f(\vec x),
\end{equation*}
where~$\widetilde f$ is given by~\eqref{eq:1}
with coefficients defined in~\eqref{eq:LSM}.
By definition, Lebesgue constant~$\Lambda(X)$ is the $\infty$-norm of the operator~$\mathsf P(X)$:
\begin{equation*}
\Lambda(X)=\sup_{\normi|f|=1}\normi|\mathsf P(X)f|.
\end{equation*}
Let $f^*\in\Imm\mathsf P(X)$ be the best polynomial approximation of $f$
in $\infty$-norm,
so that $\normi|f^*-f|$ reaches minimum.
Then,
\begin{multline*}
\normi|f-\mathsf P(X)f|
\leq \normi|f-f^*| + \normi|f^*-\mathsf P(X)f|
=\normi|f-f^*| + \normi|\mathsf P(X)(f^*-f)|\\
\leq \normi|f-f^*| + \Lambda(X)\normi|f^*-f|
=(1+\Lambda(X))\normi|f-f^*|.
\end{multline*}
Thus, the Lebesgue constant can be utilized as an estimation of the approximation error obtained with our method in comparison with the best polynomial approximation of the same degree.
We will numerically estimate the Lebesgue constant using the same technique as for the estimation of the approximation error. Namely, we will take the maximum over the fixed set~$\mathcal D$ of points from the domain of interest:
\begin{equation*}
\Lambda(X)\approx
\Lambda^{\mathcal D}(X)=\max_{\vec x\in{\mathcal D}}\sup_{\normi|f|=1}\abs|(\mathsf P(X)f)(\vec x)|.
\end{equation*}
To take the supremum over $f$, we apply the following trick.
Let us write the expansion~\eqref{eq:1} as a scalar product
of
vector of basis functions $\Psi(\vec x)=\{\Psi_{\alpha_1}(\vec x),\,\Psi_{\alpha_2}(\vec x),\,\ldots\}$ and
vector of coefficients~$c$ (see \Cref{eq:LSM}).
So, we obtain
\begin{equation*}
(\mathsf P(X)f)(\vec x)=\Psi(\vec x)\cdot (A^TA)^{-1}A^T\mathcal Y.
\end{equation*}
Taking the supremum with respect to the function~$f$ is equivalent to taking the supremum over all vectors of $\mathcal Y$ such that $\norm|\mathcal Y|_1=1$, which in turn coincides with the first norm of the corresponding vector:
\begin{equation*}
\Lambda^{\mathcal D}(X)=\max_{\vec x\in{\mathcal D}}\norm|\Psi(\vec x)\cdot (A^TA)^{-1}A^T|_1.
\end{equation*}
Estimation of the Lebesgue constant growth for ED obtained with particular sampling technique w.r.t. size of ED allows us to implicitly estimate accuracy of the least-squares polynomial approximation built using this ED.
At first, let us consider a one-dimensional case. It effectively means that the number of terms in polynomial expansion is equal to the total polynomial degree plus one: $l = p+1$. We use a test set of the size $N_{\text{test}}=10^6$ on the interval $[-1,1].$ We calculate the estimate of the Lebesgue constant $\Lambda_l$ for the points sampled by all the sampling techniques with respect to the number of such points in the range from $p=1$ to $p=9$. Also, we plot values of the Lebesgue constant for the Chebyshev roots as a quasi-optimal reference (\cref{pic:8}).
As it can be seen from \cref{pic:8:a}, D-optimal sampling methods show much slower Lebesgue constant growth compared to LHS and Sobol' sequence. On the more detailed \cref{pic:8:b} one can find that the Lebesgue constant estimates for D-optimal sampling methods shows comparable growth with the Chebyshev nodes.
\begin{figure}[htb]
\centering{
\begin{subfigure}[t]{0.49\linewidth}
\centering{\includegraphics[width=\linewidth]{pics/lebeg_1D.pdf}}
\caption{\label{pic:8:a}}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.49\linewidth}
\centering{\includegraphics[width=\linewidth]{pics/lebeg_1D_short.pdf}}
\caption{\label{pic:8:b}}
\end{subfigure}}
\caption{\label{pic:8}\textbf{(a)}: Lebesgue constant estimation for the different ED sizes (\textit{i.e.} total polynomial degrees). \textbf{(b)}: The same data as \textbf{(a)} but exclusive LHS and Sobol' sequence.}
\end{figure}
We can also estimate the Lebesgue constant for the two-dimensional case. Results are shown on \cref{pic:9} where the size of ED is varying from 10 to 70 points. Since all sampling methods have a stochastic nature, for each ED size $l$ the main model was run 50 times and results were organized in a corresponding box-plot.
From \cref{pic:9}, it can be seen that, as expected, in the two-dimensional case D-optimal sampling techniques perform much better than LHS and Sobol' sequence that is in consistence with the corresponding results on accuracy of the approximation (\cref{pic:4} -- \cref{pic:6}).
\begin{figure}[htb]
\centering{\centering{\includegraphics[width=0.9\linewidth]{pics/lebeg_2D_col-10_col-70_v3.pdf}}}\caption{\label{pic:9}Lebesgue constant estimation for the ED size varying from 10 to 70.}
\end{figure}
\subsection{Sampling from non-rectangular domain}
In this subsection, we demonstrate an ability of the proposed method to sample points not only from rectangular domains but also from domains of arbitrary shape. The results showed on \cref{pic:10} were obtained with the use of \texttt{IPOP} optimizer~\cite{Wchter2005} for three two-dimensional domains with various shapes.
As it can be seen from \cref{pic:10}, GD sampling shows quite a nice coverage of non-rectangular domains.
\begin{figure}[htb]
\centering{
\begin{subfigure}[t]{0.32\linewidth}
\centering{\includegraphics[width=1\linewidth]{pics/ipopt_circle.png}}\caption{Cirle-shape domain.}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.32\linewidth}
\centering{\includegraphics[width=1\linewidth]{pics/ipopt_cheese.png}}\caption{Three quarters of a circle.}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.32\linewidth}
\centering{\includegraphics[width=1\linewidth]{pics/ipopt_diamond.png}}\caption{Diamond-shape.}
\end{subfigure}}\caption{\label{pic:10}EDs sampled from three non-rectangular two-dimensional domains for $n = 50$ and total degree $p = 5$.}
\end{figure}
\section{Conclusions}
\label{sec:conclusion}
In this work, a new sampling method for finding a D-optimal experimental design is proposed. The developed method is based on the gradient descent algorithm and allows to minimize the log-det functional that determines the model matrix of corresponding experimental design.
The proposed sampling method was applied to the problem of the least-squares polynomial approximation of multivariate functions. Its efficiency was demonstrated by numerical comparison with the other sampling methods in the task of the multivariate function approximation. Numerical estimations on the Lebesgue constant growth were obtained and demonstrated quite a slow growth for the proposed sampling method.
In the future, we plan to modify the proposed sampling method in order to make it adaptive and test it on the weighted least-squares polynomial approximation.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,741
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Q: NSButton not displaying alt image when disabled I recently noticed that my app, when running on recent OSes (>10.10 I think) was not displaying button images correctly.
Seems that any time a button was set to disabled, the button drew as a solid grey block instead of the Alt Image which certainly used to draw fine.
I see a few mentions of this on the inter webs but never an answer. Is this an known issue with recent Mac OS?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,568
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\section{Introduction}
Said briefly, Dunn semantics is a modelling of formal languages where truth and falsity are the only truth values but they are not always related functionally to formulas. In Dunn semantics, there are four admissible interpretations for a formula: it can be just true, just false, neither true nor false and both true and false. Dunn semantics is especially associated to the logic \textbf{FDE}, which is important because its generality has proved to be fruitful as a basis for developing further logics. There are three well-known logics that can be obtained as extensions of \textbf{FDE}, that is, by eliminating one of the admissible interpretations in the most general Dunn semantics. One of those logics is (strong) Kleene logic, \textbf{K$_{3}$}, which leaves the both true and false interpretation out; another is González-Asenjo/Priest's \textbf{LP}, which leaves the neither true nor false interpretation out. The third one is classical logic, which leaves out the two that are left out in \textbf{K$_{3}$} and \textbf{LP}.
But those are not the only logics that can be studied as variations on \textbf{FDE}. David Nelson \cite{Nelson1949} clearly distinguished between truth and falsity constructive conditions for the connectives, very much in the spirit of Dunn semantics for \textbf{FDE} to be made explicit below. It was precisely working with Nelson's logic \textbf{N4}\footnote{See \cite{KamideWansing2012} for an overview of this logic.} that Wansing \cite{Wansing2005} obtained his connexive logic \textbf{C}. He changed the falsity condition for $A\rightarrow B$ in \textbf{N4} and the resulting logic was no longer subclassical, but \emph{contra-classical}. This means that, without enriching the language, it validates arguments that are not valid in classical logic.\footnote{And, because of the Post-completeness of classical logic, this means that the resulting logic must be either trivial ---i.e. validates every argument---, which is not, or come with the invalidity of some classically valid arguments. In fact, the resulting logic is \emph{hyper-connexive}, because besides validating the core connexive schemas, namely
\noindent
$\sim \! (A\rightarrow\sim \! A)$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Aristotle's Thesis
\noindent
$\sim \! (\sim \! A\rightarrow A)$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Variant of Aristotle's Thesis
\noindent
$(A \rightarrow B) \rightarrow \sim \! (A \rightarrow \sim \! B)$\ \ \ \ \ \ Boethius' Thesis
\noindent
$(A \rightarrow \sim \! B) \rightarrow \sim \! (A \rightarrow B)$\ \ \ \ \ \ Variant of Boethius' Thesis
\noindent
it also validates the converses of Boethius' Theses.}
After Wansing, Omori \cite{Omori2016} used the same idea, modifying the falsity condition for the conditional on top of \textbf{LFI1} to get another connexive logic, \textbf{dLP}. After that, he has shown (cf. \cite{Omori20XX}) that a number of well-known and new paraconsistent and relevant logics can be obtained also by modifying appropriately the falsity condition for some connectives while leaving the \textbf{FDE}-like truth and falsity conditions for the remaining ones, in most cases even negation, fixed. More recently, Wansing and Unterhuber \cite{WansingUnterhuber2019} modified the falsity condition of Chellas' basic conditional logic \textbf{CK} and they obtained a (weakly) connexive logic. Such a general approach to logics ---viz. starting with Dunn-like evaluations for \textbf{FDE} and then modify them to obtain different logics--- has been called the `Bochum Plan'.
But Dunn semantics is not only a tool for crafting new logics, but also to provide new understandings of already existing ones. For example, Omori and Wansing \cite{OmoriWansing2019} have put forward a systematization of connexive logics based on certain controlled modifications in the conditional's truth and falsity conditions, showing that, in general, modifying the truth condition has led only to \emph{weak connexivity} (Boethius' Theses hold only in rule form, if at all), whereas modifying the falsity condition has led to \emph{hyper-connexivity} (i.e. not only do Boethius' Theses hold, but also their converses). More recently, Estrada-González and Cano-Jorge \cite{EstradaCano2021} showed how a Dunn semantics can help to address some of the objections raised against Reichenbach's three-valued logic.
With this background, the aim of this paper is twofold. First, to show that many contra-classical logics can be presented as variants of \textbf{FDE}, obtained by modifying at least one of the truth or falsity conditions of some connective. Whereas this way of presenting contra-classical logics is not new, my contribution here is that the diet of examples will be enriched and systematized: examples will be given for all the possible modifications in the given language; when possible, taken from existing literature. Second, to give a precise explanation of how and why the contra-classicality is obtained. The presentation using Dunn semantics provides a clear understanding of the source of contra-classicality, namely, connectives that have either the classical truth or the classical falsity condition of another connective. This requires a fine-grained analysis of the sorts of modifications that can be made to an evaluation condition, and I provide such analysis here.
The structure of the remaining of the paper is as follows. In Section 2, I revisit, at tutorial speed, the basics of Dunn semantics for \textbf{FDE}. In Section 3, I show how to treat systematically some contra-classical logics through modifications of the truth and falsity conditions for the connectives in \textbf{FDE}. This procedure opens at least two problems. One is to explain where the contra-classicality comes from; the other is to explain whether the modified connectives are still the intended connectives and why. In Section 4, I will argue that the systematization using Dunn semantics provides a clear understanding of the source of contra-classicality, namely, connectives that have either the classical truth or the classical falsity condition of another connective. The second problem is left open and it will be tackled in a separate work.
\section{Dunn semantics and FDE}
Consider a language $L$ consisting of formulas built, in the usual way, from propositional variables with the connectives $\{\sim, \wedge, \vee, \rightarrow\}$.\footnote{The treatment of additional standard connectives, like 0-ary connectives ---$\top$, $\bot$---, modal connectives ---at least the alethic ones, $\square$, $\lozenge$--- and the usual quantifiers ---$\forall x$, $\exists x$---, is left for a future work.} I will use the first capital letters of the Latin alphabet, `$A$', `$B$', `$C$'\ldots as variables ranging over arbitrary formulas.
A key feature of Dunn semantics is that, to achieve full generality with respect to the relations between formulas and truth values, the predicates ``is true'' and ``is false'' should not be understood \emph{functionally}, that is, \emph{being true} does not imply \emph{not being false} ---nor vice versa---, and \emph{being false} does not imply \emph{not being true} ---nor vice versa---. More formally, a \emph{Dunn model} for a formal language $L$ is a relation $\sigma$ between propositional variables and values $1$ (\emph{truth}) and $0$ (\emph{falsity}), that can be extended to cover all formulas. Said otherwise, a formula can be related to the truth values, via an assignment $\sigma$, in one of the following four ways:
\begin{itemize}
\item $A$ is true but not false, represented `$1\in\sigma(A)$ and $0\notin\sigma(A)$'; more briefly, $\sigma(A)=\{1\}$
\item $A$ is true but also false, represented `$1\in\sigma(A)$ and $0\in\sigma(A)$'; more briefly, $\sigma(A)=\{1, 0\}$
\item $A$ is neither true nor false, represented `$1\notin\sigma(A)$ and $0\notin\sigma(A)$'; more briefly, $\sigma(A)=\{ \ \}$
\item $A$ is false but not true, represented `$0\in\sigma(A)$ and $1\notin\sigma(A)$'; more briefly, $\sigma(A)=\{0\}$
\end{itemize}
Now, let $\Gamma$ be a set of formulas of a logic \textbf{L}. $A$ is a \emph{logical consequence} of $\Gamma$ in \textbf{L}, $\Gamma\models_{\textbf{\tiny{L}}}A$, if and only if, for every evaluation $\sigma$, $1\in\sigma(A)$ if $1\in\sigma(B)$ for every $B\in\Gamma$. $A$ is a \emph{logical truth} in \textbf{L} if and only if $\Gamma\models_{\textbf{\tiny{L}}}A$ and $\Gamma =\varnothing$. An argument is \emph{invalid in} \textbf{L} if an only if there is an evaluation in which the premises are true, i.e. $1\in\sigma(B)$ for every $B\in\Gamma$, but the conclusion is not, i.e. $1\notin\sigma(A)$.\footnote{For simplicity, these definitions will be adapted for the other logics in this paper, just with the respective changes in language.}
\textbf{FDE} is a logic that can be presented as the result of evaluating formulas and arguments, built in the usual way from a countable set of propositional variables and the connectives $\{\sim, \wedge, \vee, \rightarrow\}$, according to the following assignments, where $A$ and $B$ stand for any formula:
\begin{itemize}
\item $\sim \! A$ is true iff $A$ is false; $\sim \! A$ is false iff $A$ is true
\item $A\wedge B$ is true iff $A$ is true and $B$ is true; $A\wedge B$ is false iff $A$ is false or $B$ is false
\item $A\vee B$ is true iff $A$ is true or $B$ is true; \ \ $A\vee B$ is false iff $A$ is false and $B$ is false
\item $A\rightarrow B$ is true iff $A$ is false or $B$ is true; $A\rightarrow B$ is false iff $A$ is true and $B$ is false
\end{itemize}
Using a Dunn model, the evaluation of formulas in \textbf{FDE} is defined recursively as follows:
\medskip
\noindent
Either $1\in\sigma(p)$ or $1\notin\sigma(p)$, and either $0\in\sigma(p)$ or $0\notin\sigma(p)$
\noindent
$1\in\sigma(\sim \! A)$ iff $0\in\sigma(A)$
\noindent
$0\in\sigma(\sim \! A)$ iff $1\in\sigma(A)$
\noindent
$1\in\sigma(A\wedge B)$ iff $1\in\sigma(A)$ and $1\in\sigma(B)$
\noindent
$0\in\sigma(A\wedge B)$ iff either $0\in\sigma(A)$ or $0\in\sigma(B)$
\noindent
$1\in\sigma(A\vee B)$ iff either $1\in\sigma(A)$ or $1\in\sigma(B)$
\noindent
$0\in\sigma(A\vee B)$ iff $0\in\sigma(A)$ and $0\in\sigma(B)$
\noindent
$1\in\sigma(A\rightarrow B)$ iff $0\in\sigma(A)$ or $1\in\sigma(B)$
\noindent
$0\in\sigma(A\rightarrow B)$ iff $1\in\sigma(A)$ and $0\in\sigma(B)$
\medskip
\noindent
Although $A\rightarrow B$ can be defined as $\sim \! A\vee B$, considering it explicitly right from the start with a separate sign will greatly simplify the exposition. A \emph{biconditional}, $A\leftrightarrow B$, can be defined as $(A\rightarrow B)\wedge(B\rightarrow A)$.
The above model-theoretic semantics for \textbf{FDE} can be represented in a tabular way as follows:
\begin{center}
\begin{tabular}{c|c}
$\sim \! A$ & $A$ \\
\hline
$\{0\}$ & $\{1\}$ \\
$\{1, 0\}$ & $\{1, 0\}$ \\
$\{ \ \}$ & $\{ \ \}$ \\
$\{1\}$ & $\{0\}$ \\
\end{tabular}
\hfil
\begin{tabular}{c|cccc}
$A\wedge B$ & $\{1\}$ & $\{1, 0\}$ & $\{ \ \}$ & $\{0\}$\\
\hline
$\{1\}$ & $\{1\}$ & $\{1, 0\}$ & $\{ \ \}$ & $\{0\}$\\
$\{1, 0\}$ & $\{1, 0\}$ & $\{1, 0\}$ & $\{0\}$ & $\{0\}$\\
$\{ \ \}$ & $\{ \ \}$ & $\{0\}$ & $\{ \ \}$ & $\{0\}$\\
$\{0\}$ & $\{0\}$ & $\{0\}$ & $\{0\}$ & $\{0\}$\\
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{c|cccc}
$A\vee B$ & $\{1\}$ & $\{1, 0\}$ & $\{ \ \}$ & $\{0\}$\\
\hline
$\{1\}$ & $\{1\}$ & $\{1\}$ & $\{1\}$ & $\{1\}$\\
$\{1, 0\}$ & $\{1\}$ & $\{1, 0\}$ & $\{1\}$ & $\{1, 0\}$\\
$\{ \ \}$ & $\{1\}$ & $\{1\}$ & $\{ \ \}$ & $\{ \ \}$\\
$\{0\}$ & $\{1\}$ & $\{1, 0\}$ & $\{ \ \}$ & $\{0\}$\\
\end{tabular}
\hfil
\begin{tabular}{c|cccc}
$A\rightarrow B$ & $\{1\}$ & $\{1, 0\}$ & $\{ \ \}$ & $\{0\}$\\
\hline
$\{1\}$ & $\{1\}$ & $\{1, 0\}$ & $\{ \ \}$ & $\{0\}$\\
$\{1, 0\}$ & $\{1\}$ & $\{1, 0\}$ & $\{1\}$ & $\{1, 0\}$\\
$\{ \ \}$ & $\{1\}$ & $\{1\}$ & $\{ \ \}$ & $\{ \ \}$\\
$\{0\}$ & $\{1\}$ & $\{1\}$ & $\{1\}$ & $\{1\}$\\
\end{tabular}
\end{center}
\noindent
Note that $\{1\}$, $\{0\}$, $\{1, 0\}$ and $\{ \ \}$ are not truth values, but assignments or evaluations; more precisely, collections of truth values.\footnote{One might say that they are \emph{generalized truth values}. (See for example \cite{ShramkoWansing2011}.) Let that be. It is still the case that they are not truth values like 1 and 0.} So, under this presentation, it would be rather wrong to call \textbf{FDE} `a four-valued logic'.
Now it can be seen more clearly how three well-known logics, (strong) Kleene logic, \textbf{K$_{3}$}, González-Asenjo/Priest's \textbf{LP} and classical logic can be obtained as extensions of \textbf{FDE}, as I mentioned in the Introduction. \textbf{K$_{3}$} is obtained by ignoring the evaluation $\{1, 0\}$; \textbf{LP} is obtained by ignoring the evaluation $\{ \ \}$; by ignoring those two evaluations at once, one obtains classical logic. Let us move now to the less familiar cases.
\section{Dunn semantics for contra-classical logics}
The evaluation conditions for the binary connectives above have the following general shape
\medskip
\noindent
$1\in\sigma(A\copyright B)$ iff $1\in\sigma(A)$ \emph{copyright} $1\in\sigma(B)$
\medskip
\noindent
$0\in\sigma(A\copyright B)$ iff $v_{i}\in\sigma(A)$ \emph{connective} $0\in\sigma(B)$
\medskip
\noindent
where $v_{i}\in\{1, 0\}$, `\emph{copyright}' stands for a metalinguistic counterpart of $\copyright$ and `\emph{connective}' stands for a metalinguistic counterpart of some other connective, in general distinct from $\copyright$.\footnote{Note that I assume classical reasoning at the meta-theoretical level ---not because I think it is unavoidable but to make things simpler--- and take ``$A$ is false or $B$ is true'' as equivalent to ``If $A$ is true then $B$ is true''. Thus, the truth condition for $A\rightarrow B$ also fits the general shape of truth conditions.} Even more generally, an evaluation condition, regardless of whether it is a truth or falsity condition, has the following general shape:
\begin{center}
$v_{i}\in\sigma(A\copyright B)$ iff $v_{j}\in\sigma(A)$ \emph{relation} $v_{k}\in\sigma(B)$
\end{center}
\noindent
i.e. a value ---truth or falsity-- is assigned to a formula iff there is a certain relation between the assignments of the components.
With this in mind, the changes in an evaluation condition might be of one among the following kinds (the list is not meant to be exhaustive):
\begin{itemize}
\item C1. The \emph{value} assigned to at least one of the components is changed ---ex. gr. from $1\in\sigma(A)$ to $0\in\sigma(A)$---;
\item C2. At least one \emph{kind of assignment} is changed ---ex. gr. from $1\in\sigma(A)$ to $1\notin\sigma(A)$---;
\item C3. The \emph{relation between the assignments} is changed ---ex. gr. from ``$1\in\sigma(A)$ and $0\in\sigma(B)$'' to ``if $1\in\sigma(A)$ then $0\in\sigma(B)$''---;
\item C4. \emph{Extra conditions are added} ---ex. gr. from ``$1\notin\sigma(A)$ or $0\in\sigma(B)$'' to ``$1\notin\sigma(A)$ or $0\in\sigma(B)$, and $0\notin\sigma(B)$ or $0\in\sigma(A)$''---;
\item C5. A \emph{mixture} of the above ---ex. gr. from ``$1\in\sigma(A)$ and $0\in\sigma(B)$'' to ``$1\notin\sigma(A)$ or $0\in\sigma(B)$''---.
\end{itemize}
There are special cases of C5, namely mixing C1 and C2, that will be called `tweakings', as they are not so radical changes. All the other changes will be called `modifications'.\footnote{This might go against some uses in the literature, as `tweaking' is used for any kind of change. I try to put some order in the terminology associated to the changes made to the evaluation conditions.} A \emph{Dunn atom} is an expression of the form $v_{i}\in\sigma(A)$ or $v_{j}\notin\sigma(A)$, with $v_{i}, v_{j}\in\{1, 0\}$. Let $v_{i}\in\sigma(A)$ (resp. $v_{j}\notin\sigma(A)$) be a Dunn atom. I will say that $v_{j}\notin\sigma(A)$ (resp. $v_{i}\in\sigma(A)$), with $v_{i}, v_{j}\in\{1, 0\}$ and $v_{i}\neq v_{j}$, is its \emph{Boolean counterpart}. (And I will assume that the relation of being a Boolean counterpart is symmetric.) For instance, the following cases ---horizontal-wise--- are Boolean counterparts of each other:
\begin{center}
\begin{multicols}{2}
$1\in \sigma(\sim A)$
$0\notin \sigma(\sim A)$
\end{multicols}
\end{center}
\begin{center}
\begin{multicols}{2}
$0\in \sigma(A \wedge B)$
$1\notin \sigma(A\wedge B)$
\end{multicols}
\end{center}
\begin{center}
\begin{multicols}{2}
$0\notin \sigma(A \vee B)$
$1\in \sigma(A\vee B)$
\end{multicols}
\end{center}
A \emph{tweaking} is, then, a modification in the evaluation conditions of a connective by changing at least one of its Dunn atoms by their Boolean counterparts in the right-hand side of the `iff'. As an illustration of tweakings, consider negation as evaluated in \textbf{FDE} in the upper left corner and three connectives obtained by changing at least part of its evaluation conditions:
\begin{multicols}{2}
\begin{center}
$1\in \sigma(\sim \! A)$ iff $0\in \sigma(A)$
$0\in \sigma(\sim \! A)$ iff $1\in \sigma(A)$
\end{center}
\begin{center}
$1\in \sigma(\sim_{1} \! A)$ iff $0\in \sigma(A)$
$0\in \sigma(\sim_{1} \! A)$ iff $0\notin \sigma(A)$
\end{center}
\begin{center}
$1\in \sigma(\sim_{2} \! A)$ iff $1\notin \sigma(A)$
$0\in \sigma(\sim_{2} \! A)$ iff $1\in \sigma(A)$
\end{center}
\begin{center}
$1\in \sigma(\sim_{3} \! A)$ iff $1\notin \sigma(A)$
$0\in \sigma(\sim_{3} \! A)$ iff $0\notin \sigma(A)$
\end{center}
\end{multicols}
In fact, $\sim_{3}$ is an old acquaintance: it is \emph{Boolean} negation. Having it available will make presentation easier, and hence I will use a special sign for it: $\neg A$. Another connective that can be added to the language to make presentations easier is the \emph{material} conditional, $A\supset B$, evaluated as follows:
\noindent
$1\in\sigma(A\supset B)$ iff $1\notin\sigma(A)$ or $1\in\sigma(B)$
\noindent
$0\in\sigma(A\supset B)$ iff $1\in\sigma(A)$ and $0\in\sigma(B)$
\noindent
Note that the material conditional can be regarded in its turn as a tweaking on the evaluation conditions for \textbf{FDE}'s conditional.
The truth tables for both Boolean negation and material conditional are as follows:
\begin{center}
\begin{tabular}{c|c}
$\neg A$ & $A$ \\
\hline
$\{0\}$ & $\{1\}$ \\
$\{ \ \}$ & $\{1, 0\}$ \\
$\{1, 0\}$ & $\{ \ \}$ \\
$\{1\}$ & $\{0\}$ \\
\end{tabular}
\hfil
\begin{tabular}{c|cccc}
$A\supset B$ & $\{1\}$ & $\{1, 0\}$ & $\{ \ \}$ & $\{0\}$\\
\hline
$\{1\}$ & $\{1\}$ & $\{1, 0\}$ & $\{ \ \}$ & $\{0\}$\\
$\{1, 0\}$ & $\{1\}$ & $\{1, 0\}$ & $\{ \ \}$ & $\{0\}$\\
$\{ \ \}$ & $\{1\}$ & $\{1\}$ & $\{1\}$ & $\{1\}$\\
$\{0\}$ & $\{1\}$ & $\{1\}$ & $\{1\}$ & $\{1\}$\\
\end{tabular}
\end{center}
\medskip
\noindent
A \emph{material biconditional}, $A\equiv B$, can be defined as $(A\supset B)\wedge(B\supset A)$.
\medskip
Let me begin now with the examples of contra-classical logics using the machinery from above.
\paragraph{Examples: Modifying the evaluation conditions for negation.}
Let all the evaluation conditions fixed, except the truth condition for negation,
\noindent
$1\in\sigma(\sim \! A)$ iff $0\in\sigma(A)$
\noindent
and replace this condition by the following condition
\noindent
$1\in\sigma(\sim \! A)$ iff $0\notin\sigma(A)$
\noindent
which validates $A\vee\sim\sim \! A$ but not $A\vee\sim \! A$. Paul Ruet's \cite{Ruet1996} introduced this connective with the notation $\circlearrowright$, but I will write it `$\sim_{R}$'. Its truth table would be as follows:
\begin{center}
\begin{tabular}{c|c}
$\sim_{R} \! A$ & $A$ \\
\hline
$\{1, 0\}$ & $\{1\}$ \\
$\{0\}$ & $\{1, 0\}$ \\
$\{1\}$ & $\{ \ \}$ \\
$\{ \ \}$ & $\{0\}$ \\
\end{tabular}
\end{center}
\noindent
Note that the negation so modified is a demi-negation in the sense of Humberstone \cite{Humberstone1995}, since $\sigma(\sim_{R}\sim_{R}~A)$ $=\sigma(\neg A)$, for all $\sigma$, as it can be easily verified.
Now, let all the evaluation conditions fixed, except the falsity condition for negation,
\noindent
$0\in\sigma(\sim \! A)$ iff $1\in\sigma(A)$
\noindent
and replace this condition by the following condition
\noindent
$0\in\sigma(\sim \! A)$ iff $1\notin\sigma(A)$
\noindent
then one obtains Kamide's \cite{Kamide2017} demi-negation of his logic \textbf{CP}, extensively studied in \cite{OmoriWansing2018}, and that I will write as `$\sim_{K}$'. Its truth table would be as follows:
\begin{center}
\begin{tabular}{c|c}
$\sim_{K} \! A$ & $A$ \\
\hline
$\{ \ \}$ & $\{1\}$ \\
$\{1\}$ & $\{1, 0\}$ \\
$\{0\}$ & $\{ \ \}$ \\
$\{1, 0\}$ & $\{0\}$ \\
\end{tabular}
\end{center}
\noindent
As it is highlighted in \cite{OmoriWansing2018}, \textbf{CP} validates both $\sim_{K} \! (A\wedge\sim_{K} \! \sim_{K} \! A)$ and $\sim_{K} \! \sim_{K} \! (A\wedge\sim_{K} \! \sim_{K} \! A)$ ---which makes it negation-inconsistent---. Note that $\sim_{K}$ is a demi-negation too, since $\sigma(\sim_{K}\sim_{K} A)=\sigma(\neg A)$, for all $\sigma$, yet in general $\sigma(\sim_{R} \! A)\neq\sigma(\sim_{K} \! A)$.
\paragraph{Examples: Modifying the evaluation conditions for conjunction.} Let all the evaluation conditions fixed, except the truth condition for conjunction,
\noindent
$1\in\sigma(A\wedge B)$ iff $1\in\sigma(A)$ and $1\in\sigma(B)$
\noindent
and replace this condition by the following condition
\noindent
$1\in\sigma(A\wedge B)$ iff $1\in\sigma(A)$ or $1\in\sigma(B)$
\noindent
This is a version of \texttt{tonk}: it has the truth condition of disjunction and the falsity condition of conjunction. Defined like that, \texttt{tonk} turned out to be inexpressible in classical logic, but it is in an \textbf{FDE}-like setting. This is its truth table:
\begin{center}
\begin{tabular}{c|cccc}
$A\wedge_{t} B$ & $\{1\}$ & $\{1, 0\}$ & $\{ \ \}$ & $\{0\}$\\
\hline
$\{1\}$ & $\{1\}$ & $\{1, 0\}$ & $\{1\}$ & $\{1, 0\}$\\
$\{1, 0\}$ & $\{1, 0\}$ & $\{1, 0\}$ & $\{1, 0\}$ & $\{1, 0\}$\\
$\{ \ \}$ & $\{1\}$ & $\{1, 0\}$ & $\{ \ \}$ & $\{0\}$\\
$\{0\}$ & $\{1, 0\}$ & $\{1, 0\}$ & $\{0\}$ & $\{0\}$\\
\end{tabular}
\end{center}
\noindent
The resulting logic would be contra-classical, since $\sim \! (A\wedge_{t}\sim \! A)\equiv(A\wedge_{t}\sim \! A)$.\footnote{This also shows that certain classically equivalent definitions of connectives are not so in non-classical contexts. For example, the \texttt{tonk} $\wedge_{t}$ cannot be defined as a connective satisfying both the introduction rules for disjunction and the elimination rules for conjunction, since $A\wedge_{t}B\not\models B$ when $\sigma(A)=\{1\}$ and $\sigma(B)=\{ \ \}$.}
Now let all the evaluation conditions fixed, except the falsity condition for conjunction,
\noindent
$0\in\sigma(A\wedge B)$ iff $0\in\sigma(A)$ or $0\in\sigma(B)$
\noindent
and replace this condition by the following condition
\noindent
$0\in\sigma(A\wedge B)$ iff $0\in\sigma(A)$ and $0\in\sigma(B)$
\noindent
Then one obtains Arieli and Avron's \cite{ArieliAvron1996} \emph{informational meet}, $\otimes$, of their logic \textbf{BL}$_{\supset}$. Following our previous convention, I will write `$\wedge_{AA}$' instead of `$\otimes$'. Its truth table is as follows:
\begin{center}
\begin{tabular}{c|cccc}
$A\wedge_{AA} B$ & $\{1\}$ & $\{1, 0\}$ & $\{ \ \}$ & $\{0\}$\\
\hline
$\{1\}$ & $\{1\}$ & $\{1\}$ & $\{ \ \}$ & $\{ \ \}$\\
$\{1, 0\}$ & $\{1\}$ & $\{1, 0\}$ & $\{ \ \}$ & $\{0\}$\\
$\{ \ \}$ & $\{ \ \}$ & $\{ \ \}$ & $\{ \ \}$ & $\{ \ \}$\\
$\{0\}$ & $\{ \ \}$ & $\{0\}$ & $\{ \ \}$ & $\{0\}$\\
\end{tabular}
\end{center}
\noindent
\textbf{BL}$_{\supset}$ is a contra-classical logic because $\sim \! (A\wedge_{AA} B) \equiv (\sim \! A \wedge_{AA} \sim \! B)$ holds in it.
\paragraph{Examples: Modifying the evaluation conditions for disjunction.} Let all the evaluation conditions fixed, except the truth condition for disjunction,
\noindent
$1\in\sigma(A\vee B)$ iff $1\in\sigma(A)$ or $1\in\sigma(B)$
\noindent
and replace this condition by the following condition
\noindent
$1\in\sigma(A\vee B)$ iff $1\in\sigma(A)$ and $1\in\sigma(B)$
\noindent
This is dual to the \texttt{tonk} presented above; its truth table is as follows:
\begin{center}
\begin{tabular}{c|cccc}
$A\vee_{t} B$ & $\{1\}$ & $\{1, 0\}$ & $\{ \ \}$ & $\{0\}$\\
\hline
$\{1\}$ & $\{1\}$ & $\{1\}$ & $\{ \ \}$ & $\{ \ \}$\\
$\{1, 0\}$ & $\{1\}$ & $\{1, 0\}$ & $\{ \ \}$ & $\{0\}$\\
$\{ \ \}$ & $\{ \ \}$ & $\{ \ \}$ & $\{ \ \}$ & $\{ \ \}$\\
$\{0\}$ & $\{ \ \}$ & $\{0\}$ & $\{ \ \}$ & $\{0\}$\\
\end{tabular}
\end{center}
\noindent
It is the only modified connective for which I have not found a previous appearance in the logic literature, so I will not say anything else about it. And maybe this is so with good reason, as the table is the same as that for informational meet.\footnote{The reader might wonder about other modifications in the truth condition, not exactly the one given above. If they have a suggestion already studied in the literature, I would greatly appreciate it.}
Now let all the evaluation conditions fixed, except the falsity condition for disjunction,
\noindent
$0\in\sigma(A\vee B)$ iff $0\in\sigma(A)$ and $0\in\sigma(B)$
\noindent
and replace this condition by the following condition
\noindent
$0\in\sigma(A\vee B)$ iff $0\in\sigma(A)$ or $0\in\sigma(B)$
\noindent
Then one obtains Arieli and Avron's \cite{ArieliAvron1996} \emph{informational join}, $\oplus$, of their logic \textbf{BL}$_{\supset}$. Again, following our previous convention, I will write `$\vee_{AA}$' instead of `$\oplus$'. The table is as follows:
\begin{center}
\begin{tabular}{c|c c c c}
$A\vee_{AA} B$ & \{1\}& \{1,0\}& $\{ \ \}$& \{0\} \\
\hline
\{1\} & \{1\} & \{1,0\} & \{1\}& \{1,0\}\\
\{1,0\}& \{1,0\}& \{1,0\} &\{1,0\}&\{1,0\}\\
$\{ \ \}$& \{1\}& \{1,0\} &$\{ \ \}$&\{0\}\\
\{0\}& \{1,0\}& \{1,0\} &\{0\}&\{0\}
\end{tabular}
\end{center}
\noindent
Contra-classicality enters \textbf{BL}$_{\supset}$ not only through $\wedge_{AA}$, but also with this connective since $\sim \! (A\vee_{AA} B) \equiv (\sim \! A \vee_{AA} \sim \! B)$ holds in it. With this one can also show that \textbf{BL}$_{\supset}$ is negation-inconsistent, since it validates both $((A\supset A)\vee_{AA} \sim \! (A\supset A))$ and $\sim \! ((A\supset A)\vee_{AA} \sim \! (A\supset A))$.
\paragraph{Examples: Modifying the evaluation conditions for the conditional.} Let all the evaluation conditions fixed, except the truth condition for the conditional,
\noindent
$1\in\sigma(A\rightarrow B)$ iff if $0\in\sigma(A)$ or $1\in\sigma(B)$
\noindent
and replace it by the following condition
\noindent
$1\in\sigma(A\rightarrow B)$ iff $1\in\sigma(A)$ and $1\in\sigma(B)$
\noindent
then one obtains a logic with the four valued generalization of a connective that has been studied several times in the recent history of logic although in Kleene-like three valued logics: it is Reichenbach's \cite{Reichenbach1935} and \cite{Reichenbach1944} \emph{quasi-implication}, de Finetti's \cite{deFinetti1936} conditional (or, if not identical, at least very similar to it), and Blamey's \cite{Blamey1986} \emph{transplication}. This is its truth table:
\begin{center}
\begin{tabular}{c|cccc}
$A\rightarrow_{DF} B$ & $\{1\}$ & $\{1, 0\}$ & $\{ \ \}$ & $\{0\}$\\
\hline
$\{1\}$ & $\{1\}$ & $\{1, 0\}$ & $\{ \ \}$ & $\{0\}$\\
$\{1, 0\}$ & $\{1\}$ & $\{1, 0\}$ & $\{ \ \}$ & $\{0\}$\\
$\{ \ \}$ & $\{ \ \}$ & $\{ \ \}$ & $\{ \ \}$ & $\{ \ \}$\\
$\{0\}$ & $\{ \ \}$ & $\{ \ \}$ & $\{ \ \}$ & $\{ \ \}$\\
\end{tabular}
\end{center}
\noindent
It validates, among other things, $(A\rightarrow_{DF} B)\models A$.\footnote{See \cite{EgreRossiSprenger2020} for a recent comprehensive study of such connective in three-valued settings.}
Now consider an expansion of \textbf{FDE} with the material conditional and let all the evaluation conditions fixed, except the falsity condition for the material conditional,
\noindent
$0\in\sigma(A\supset B)$ iff $1\in\sigma(A)$ and $0\in\sigma(B)$
\noindent
and replace it by the following condition
\noindent
$0\in\sigma(A\supset B)$ iff $1\notin\sigma(A)$ or $0\in\sigma(B)$
\noindent
then one obtains \textbf{MC}, `material connexive logic', introduced in \cite{Wansing2020}. The truth table for such conditional is the following one:
\begin{center}
\begin{tabular}{c|cccc}
$A\rightarrow_{W} B$ & $\{1\}$ & $\{1, 0\}$ & $\{ \ \}$ & $\{0\}$\\
\hline
$\{1\}$ & $\{1\}$ & $\{1, 0\}$ & $\{ \ \}$ & $\{0\}$\\
$\{1, 0\}$ & $\{1\}$ & $\{1, 0\}$ & $\{ \ \}$ & $\{0\}$\\
$\{ \ \}$ & $\{1, 0\}$ & $\{1, 0\}$ & $\{1, 0\}$ & $\{1, 0\}$\\
$\{0\}$ & $\{1, 0\}$ & $\{1, 0\}$ & $\{1, 0\}$ & $\{1, 0\}$\\
\end{tabular}
\end{center}
\noindent
Connexive logics including such a conditional typically validate both Boethius' Thesis and its converse, i.e. \emph{Wansing's Thesis} $\sim \! (A\rightarrow_{W} B)\leftrightarrow_{W} (A\rightarrow_{W} \sim \! B)$.\footnote{Richard Sylvan \cite{Sylvan1989BG4} dubbed `hyper-connexivism' the thesis that, in addition to Aristotle's and Boethius' Theses, the converses of the latter also hold.} They are also negation-inconsistent; as witnesses, take $(A\wedge \sim \! A)\rightarrow_{W} A$ and $\sim((A\wedge \sim \! A)\rightarrow_{W} A)$.\footnote{A connexive variant of the more general version of Belnap-Dunn logic, including the negation $\neg$, was studied in \cite{Omori2016} under the name `\textbf{dBD}'.}
\paragraph{Example: Modifying several evaluation conditions at the same time.} If one modifies at least two falsity conditions, for example, that for the conditional as in the connexive logic above, that for disjunction as in \textbf{BL}$_{\supset}$, and the falsity condition for conjunction so that one gets
\noindent
$0\in\sigma(A\wedge B)$ iff either $1\in\sigma(A)$ and $0\in\sigma(B)$, or $0\in\sigma(A)$ and $1\in\sigma(B)$
\noindent
then one obtains Francez's \cite{Francez2019a} \textbf{PCON}, which inherits the contra-classical features of the logics on which it is based.\footnote{`\textbf{PCON}' stands for `poly-connexive logic'. It is \emph{multi-contra-classical} in the sense that it validates contra-classical theses for more than two connectives.} (Although, as the peculiar falsity condition for conjunction witnesses, this logic was motivated by considerations independent from of those by Avron and Arieli.)
I have not found in the literature a contra-classical logic that can be described as a variant of \textbf{FDE} in which the truth conditions for two or more connectives are modified. I would greatly appreciate suggestions on this regard.
\section{The source of contra-classicality}
At this point, at least two questions can be asked:
\noindent
(Q1) What is the connection between contra-classicality and the evaluation conditions such that certain modifications in the latter produce the former?
\noindent
(Q2) Are the modified connectives still the intended connectives?
\noindent
Perhaps the second is the more pressing one. It is far from clear that, say, Kamide's demi-negation is still a negation, or whether the de Finetti connective is still a conditional, to mention just two examples of the difficulty. Nonetheless, I will leave it for another occasion. Tackling at least the first one is already an important contribution.\footnote{Nonetheless, see again \cite{OmoriWansing2018} for the question whether Kamide's connective is a negation, and \cite{EgreRossiSprenger2020} and \cite{EstradaRamirez202X} for discussion about the conditionality of the de Finetti conditional.}
As we have seen above, contra-classicality is not restricted to modifying just one of the evaluation conditions, either the truth or the falsity condition; contra-classical logics can be obtained by modifying either of them. As the history of connexive logic witnesses, most contra-classical logics have been obtained by modifying the truth conditions alone. In the case of connexive logics, it has been the truth condition for the conditional (combined with Boolean negation rather than de Morgan negation); see for example the connexive logics so obtained in \cite{Angell1962}, \cite{McCall1966}, \cite{Pizzi1977}, \cite{Pizzi1991} and \cite{Priest1999}. And a recent wave of connexive logics, after \cite{Wansing2005} ---such as \cite{Omori2016} and \cite{WansingUnterhuber2019}--- modify the falsity condition (using the de Morgan negation).\footnote{In all fairness, connexive logics have been obtained by other means than model-theoretically. See for example \cite{RahmanRuckert2001} or \cite{McCall2014}, for proof theoretic-based connexive logics. In fact, as a referee correctly points out, this sort of presentation of logics would allow for further extensions of the Bochum Plan, in considering systematic and controlled modifications in the proof-theoretic machinery.}
Moreover, not any change in the evaluation conditions, not even a large number of them, produces contra-classical logics. For example, if one tweaks the falsity conditions of all of $\sim$, $\wedge$, $\vee$ and $\rightarrow$ as follows
\noindent
$0\in(\sim \! A)$ iff $0\notin\sigma(A)$
\noindent
$0\in\sigma(A\wedge B)$ iff either $1\notin\sigma(A)$ or $1\notin\sigma(B)$
\noindent
$0\in\sigma(A\vee B)$ iff $1\notin\sigma(A)$ and $1\notin\sigma(B)$
\noindent
$0\in\sigma(A\rightarrow B)$ iff $0\notin\sigma(A)$ and $1\notin\sigma(B)$
\noindent
one obtains a four-valued generalization of Sette's \textbf{P}$^1$, which is not contra-classical.\footnote{It would be worth comparing this generalization with those studied in \cite{Omori2017}. This is left for another work.}
What kind of modifications does the job then? Although it remains to be properly proved, the examples strongly suggest that only those modifications that make one of the evaluation conditions classically equivalent to the corresponding classical evaluation condition for some other connective deliver contra-classicality. In the examples on Section 3,
\begin{itemize}
\item the modified truth condition for negation is (classically) that of identity;
\item the modified falsity condition for negation is (classically) that of identity;
\item the modified truth condition for conjunction is (classically) that of disjunction;
\item the modified falsity condition for conjunction is (classically) that of disjunction;
\item the modified truth condition for disjunction is (classically) that of conjunction;
\item the modified falsity condition for disjunction is (classically) that of conjunction;
\item the modified truth condition for the conditional is (classically) that of conjunction;
\item the modified falsity condition for the conditional is (classically) that of conjunction;
\end{itemize}
It is in this case of modified evaluation conditions where contra-classicality enters the scene: the modified evaluation condition endows the connective with properties of some other connective, and hence validates things that it does not validate in classical logic.
\section{Conclusions}
In this paper, by using Dunn semantics I gave systematic changes in the evaluation conditions for negation, conjunction, disjunction and the conditional, and relate most of them with already existing contra-classical logics. This means that those contra-classical logics can be regarded as variants of \textbf{FDE}, obtained by modifying the evaluations conditions for certain connectives. Then I argued that such systematization provides a clear understanding of the source of contra-classicality, namely, connectives that have either the classical truth or the classical falsity condition of another connective.
A pressing question remains open at this point: are the modified connectives still the intended connectives? Why? This is, as I have said, left for future work. But there are at least two more paths to follow after this investigation. First, other standard connectives, like the 0-ary connectives or constants, some modal connectives or the usual quantifiers, can be given Dunn semantics. Finding examples of contra-classical logics involving those other connectives might be instructive as well to test the explanatory power that the Dunn semantics seems to possess. Second, the entailment relation was assumed to be Tarskian. But it has recently been argued ---see \cite{EgreRossiSprenger2020}, \cite{EstradaRamirez202X}--- that certain connectives, like transplication, are closer to its intended connective when the entailment relation is not Tarskian, in particular, when it is non-transitive. This suggests that the underlying notion of entailment would be allowed to vary as well, not only the truth and falsity conditions, which would make the space of contra-classical logics even richer. This idea also deserves a systematic exploration.
\bibliographystyle{eptcs}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Nine people survived a helicopter crash in Kamchatka — regional Health Ministry
Information on the rest of the tourists and crew members is being investigated, acting Health Minister of the region Marina Volkova said
PETROPAVLOVSK-KAMCHATSKY, August 12. /TASS/. Nine people survived the crash of an Mi-8 helicopter in the Kronotsky Nature Reserve in Kamchatka, acting Health Minister of the region Marina Volkova told TASS.
"According to our information, nine people survived. Information on the rest of the tourists and crew members is being investigated," she said.
On August 12, 2021, at 9:50 am (00:50 Moscow time), rescuers received information about a hard landing of the Mi-8 helicopter of the Vityaz-Aero company in the area of the Kuril Lake of the Kronotsky Reserve.
A source at emergency services said the helicopter sank in the lake. There were 16 people on board: 13 passengers and three crew members. It is known that the crashed helicopter was inspected before departure and had no technical problems, the crew did not report any flight problems, and the landing was carried out in fog.
The investigating authorities opened a criminal case under the article "Violation of air transport safety rules resulting in the death of two or more persons" after the Mi-8 crash with tourists on board in Kamchatka. A hotline has been launched for relatives of passengers and members of the helicopter crew.
The Vityaz-Aero airline has been operating since 2009, its fleet consists of more than 20 Mi-8 helicopters of the MTV, AMT, T and PS and Mi-2 modifications. It operates flights to remote areas of Kamchatka, including passenger transportation of tourists to hard-to-reach areas.
Emirates flight makes emergency landing at St. Petersburg's Pulkovo
It was reported that the plane had landed safely
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,453
|
\section{Introduction}\label{introduction}
Globalization encourages the use of the intellectual property (IP)-based system-on-chip (SoC) designs and embedded microcontrollers. Especially, their applicability as the low-edge edge computing devices in the Internet-of-Things (IoT) scenarios, and low-end embedded computing nodes in the cyber-physical systems (CPS, like the industrial control systems), is widespread \cite{mohan2013s3a,levshun2019design,wang2018model,giakoumis2018chaos,ratasich2019roadmap,shafique2018intelligent,moura2019cyber}. However, this trend increases the chances of malicious hardware design intrusions, known as Hardware Trojans (HTs), into the modern-day SoCs \cite{tehranipoor2010survey,subramani2018hardware}. HTs can lead to several unwanted payloads, i.e., information leakage, change in the timing characteristics, malfunctioning and denial-of-service (DoS)~\cite{tehranipoor2010survey,subramani2018hardware}. The effects can be catastrophic, such as system failure and leakage of secret encryption keys (e.g., failure of an ice-detection module in the P-8A Poseidon~\cite{villasenor2013hidden} and chip insertion attack in 2018~\cite{bloomberg_security}, making it imperative to develop effective HT detection techniques.\par
\begin{figure}[!t]
\centering
\includegraphics[width=1\linewidth]{Figs/Threat_Model.pdf}
\caption{Left side, all possible security vulnerabilities in IC Supply Chain. Right side, targeted attack model.}
\label{fig:threat_model}
\end{figure}
Some of the contemporary HT detection techniques utilize the circuit timing behavior \cite{zarrinchian2017latch,plusquellic2018detecting,nandhini2018delay,fang2018prefetch,xue2018hardware}, power consumption \cite{lodhi2017power,hoque2017golden,lodhi2016self,zhang2018data}, current or electromagnetic signals based \textbf{golden signatures} to detect anomalous electrical behavior \cite{gbade2014signature,lodhi2014hardware}. However, \textit{in case of the third-party-IP based designs, it is nearly impossible to extract the golden signatures}. Moreover, in most of the 3PIP-based SoCs, the facility to integrate the IPs can be trusted (as, shown in Figure~\ref{fig:threat_model}). On the other hand, it is difficult to \textit{guarantee} that the IP vendors can be trusted. To address this issue, various IP analysis-based approaches have been proposed to get the golden behavior \cite{zhang2014detrust,waksman2013fanci,zhang2015veritrust,haider2014hatch,ngo2015hardware,zareen2018detecting,cui2018hardware,liu2019hardware,cui2018hardware} but these techniques inherently pose the following limitations:\par
\begin{enumerate}
\item \textit{Accuracy of estimated golden behavior:} Due to limited access to IPs, they use different estimation algorithms to extract the golden model \cite{he2017hardware}. However, due to measurement inaccuracies and behavioral estimation, they cannot guarantee the accuracy of the golden model.
\item In the case of reverse engineering, sensors for golden data extraction \textit{cannot encompass all the possible input conditions} for larger ICs because of the inherent data loss during the quantization in analog-to-digital conversion \cite{pelgrom2017nyquist}.
\end{enumerate}
These limitations raise a key research question: \textit{How can signature-based HT detection techniques effectively work if an intruder (at foundry) exploits the intricacies of estimation algorithms to insert Trojans that remain dormant during the testing stages \cite{bhunia2014hardware}?} For example, in the SoC design, some hard or firm IPs may hide Trojans depending on the aging of the chip \cite{becker2013stealthy}, and they only get activated during run-time \cite{hasan2015tenacious,mossa2017hardware,mossa2017self}.\par
To address this issue, several run-time detection approaches have been developed to monitor a SoC for its entire operational lifetime, providing an important last-line of defense \cite{forte2013temperature,bao2015temperature,zhao2015applying,lodhi2017power,bao2016reverse,lodhi2016self,iwase2015detection}. However, most of these techniques are based on side-channel analysis (SCA), which require \textbf{precise calibration} for differentiating between the process variations and the intrusion behavior. They also rely on the premise that triggering of payload results in a substantially higher current flow. To avoid the complex requirements of the SCA-based run-time HT detection techniques, alternative behavior can be used to sniff the abnormalities during run-time. One of the most prominent ones is \textit{the communication behavior} because, in real-world scenarios, modules are connected via communication channels and therefore, most of the intrusions have an impact on the communication behavior without affecting the communication protocols.
\subsection{Motivational Analysis}
To validate the aforementioned observation, we analyze the communication behavior of an MC8051 microcontroller for the Gaussian and exponentially distributed input data, in the presence of multiple trust-hub Trojan benchmarks, i.e., MC8051-T200, T300, T400, T500, T600, T700, and T800 \cite{trust-HUB}. Our analysis in Figure~\ref{fig:com_behavior} shows that all the benchmarks have some effects on the communication patterns. For example, MC8051-T600 slightly changes the communication behavior in case of the exponential distribution. However, in the case of Gaussian distribution, it significantly changes the communication behavior of the microcontroller. In short, this analysis shows that \textit{communication behavior can be monitored to identify abnormalities during run-time without estimating the functional behavior of a particular module}.
\begin{figure}[!t]
\centering
\includegraphics[width=1\linewidth]{Figs/Motivational_Analysis_PSL.pdf}
\caption{Experimental analysis to study the effects of Trust-Hub HT benchmarks (i.e., MC8051-T200, T300, T400, T500, T600, T700, and T800) on the communication behavior of MC8051 in an SoC (it consists of four single-core MC8051 IPs that are linked with each other using AMBA 2.0 bus). Note, all these experiments are performed using two different input data distributions, i.e., Gaussian and Exponential distributions. This analysis shows that all the implemented HT benchmarks have a detectable impact on communication patterns of MC8051, which can be observed by analyzing the change in the shape of the communication distribution in MC8051. }
\label{fig:com_behavior}
\end{figure}
This observation leads to a research challenge about obtaining the golden statistical model in the presence of untrusted IPs. To address this challenge, either we assume that Trojans remain dormant during the design stage, fabrication stage, and testing stage, or we assume that at least on the IPs is trusted. However, in real-world applications, Trojans may be active during the design stage, fabrication stage, or testing stage. Therefore, in this paper, we assume that at least one of the IPs is trusted, and the defender deploys the defense inside the trusted IP or in the IP integration platform. Based on this assumption, we postulate a hypothesis, \textit{``information leakage using such communication channels, which are linked with a trusted IP can have a detectable impact on the statistical behavior of the trusted IP communication''.}
\begin{figure}[!t]
\centering
\includegraphics[width=1\linewidth]{Figs/Motivational_Analysis_PSL_2.pdf}
\caption{Experimental analysis to study the untrusted IPs on the communication of the trusted IPs. The SoC used in this analysis consists of four single-core MC8051 IPs that are linked with each other, and other IPs like RS232, vga\_lcd, ethernet, using AMBA 2.0 bus. Note, all these experiments are performed using two different input data distributions, i.e., Gaussian and Exponential distributions, and different Trust-Hub HT benchmarks (e.g., AES-T100, ethernetMAC10GE-T100, vga\_lcd-T100, RS232-T1000, memctrl-T100) with different payloads like information leakage and blocking/jamming the communication. In this analysis, only one MC8051 IP is trusted, and one HT benchmark is active at a time. This analysis shows that all the implemented HT benchmarks have a detectable impact on communication patterns of the trusted MC8051 IP, which can be observed by analyzing the change in the shape of the communication distribution of the trusted MC8051.}
\label{fig:com_behavior_2}
\end{figure}
To illustrate this hypothesis, we perform a motivational case study about four single-core MC8051 interacting with multiple IPs, i.e., vga\_lcd, AES, Ethernet, RS232, memory controller, in an SoC, as shown in Figure~\ref{fig:com_behavior_2}. In this case study, we assume that only one single-core MC8051 is trusted, and the rest of the IPs are untrusted. For a comprehensive analysis, we implemented the multiple Trojans, i.e., AES-T100, ethernetMAC10GE-T100, vga\_lcd-T100, RS232-T1000, memctrl-T100, with different payloads like information leakage and blocking/jamming the communication. To study the minimum impact of the payload of the Trojan, we also assume that only one Trojan is active at a time. By analyzing the results in Figure~\ref{fig:com_behavior_2}, we made the following observations:
\begin{itemize}
\item Communication jamming/blocking caused by un-trusted IP has a significant impact on the statistical properties of the trusted IP communication. For example, in the case of S3 and S5 of Figure~\ref{fig:com_behavior_2}, when Trojan gets the trigger in vga\_lcd or memctrl, it initiates the blank display (in the case of vga\_lcd-T100) or enables the premature sleep signal (in the case of memctrl-T100). Hence, it blocks communication with trusted MC8051.
\item Constant information leakage through untrusted IP has a significant impact on the communication traffic, which eventually changes the statistical properties of the communication of the trusted IP. For example, in the case of S1 of Figure~\ref{fig:com_behavior_2}, the communication traffic increases after the Trojan (AES-T100) gets the trigger. Hence, it changes the statistical properties of the communication of the trusted MC8051.
\item If the control signals or internal protocol signals are compromised, then it also has a relatively less impact on communication with trusted IP. For example, in the case of S2 and S4 of Figure~\ref{fig:com_behavior_2}, the triggered Trojans sometimes increase the communication traffic and sometimes decreases the communication traffic depending upon the payload. Hence, it can also change some of the statistical properties of the communication of the trusted MC8051.
\end{itemize}
The experimental analysis in Fig.~\ref{fig:com_behavior} shows that even the simple communication between MC8051 and UART maintains a distinct communication pattern. Therefore, we can safely conclude that it is possible to extract distinct communication patterns in both the high-end and low-end microcontrollers. These communication patterns can be used to define an appropriate statistical model that represents the communication activities in both low-end and high-end microprocessors.
\subsection{Associated Research Challenges:}
Based on these observations, we conclude that run-time monitoring of the communication behavior of the trusted IPs can be used to detect the abnormal activities of un-trusted IPs. This observation is only valid if the untrusted IPs are linked with trusted IP via active communication channels. However, run-time monitoring of the communication behavior of the trusted IPs poses the following research challenges:
\begin{enumerate}
\item \textbf{Modeling of communication behavior:} Unlike the side channel parameters, typical mathematical modeling techniques cannot be used to model communication behavior because of its probabilistic nature. One of the solutions is to use sophisticated statistical modeling of communication behavior, but it comes with huge overhead. Therefore, the main questions that need to be addressed are:
\begin{itemize}
\item How to statistically model the communication behavior of the trusted IP that can be used for run-time monitoring with minimum overhead?
\item What is the minimum number of observations required to learn/extract a well-defined communication pattern?
\item How to translate the extracted communication patterns to the respective statistical model?
\item How to store and use the golden statistical model for run-time analysis?
\end{itemize}
\item \textbf{Extracting the golden communication behavior:} In run-time monitoring, the fundamental challenge is to extract the golden behavior during the design time that can effectively be used for HT detection.
\end{enumerate}
\subsection{Our Novel Contributions}
To address above mentioned challenges, we propose a novel methodology that leverages the statistical traffic modeling of communication channels (\textit{SIMCom}, Section \ref{proposed-methodology}) in the SoC to sniff the possible anomalies in 3PIP units \footnote{The archive version of original submission is also available online: F. Khalid, et al. "SIMCom: Statistical Sniffing of Inter-Module Communications for Run-time Hardware Trojan Detection." arXiv preprint arXiv:1901.07299 (2018).}. The proposed methodology consists of the following three key steps, shown in Figure~\ref{fig:Novel}. \par
\begin{enumerate}
\item \textbf{Statistical modeling of communication behavior (Section \ref{TM-8051_model}):} To extract the golden communication behavior that can be used during the run-time for HT detection. We assume that in an SoC at least one of the IPs is trusted, and by analyzing the communication of the trusted IP, we can extract the requires golden behavior. Therefore, We propose to statistically model the normal communication traffic of the trusted IP, during the design phase of the SoC. This behavior is extracted by obtaining a statistical traffic model, which is characterized based on the following observations:
\begin{enumerate}
\item How often the packets are injected in the communication channel?
\item On average, how far does each packet travel?
\item What portion of the total traffic has been injected in the communication channel by each module?
\end{enumerate}
\item To identify the appropriate statistical model, we propose to choose the Hurst exponent because of the following reasons:
\begin{itemize}
\item It is very sensitive to the distribution of the time series.
\item It requires very few observations to estimate the Hurst exponent. For example, the minimum observations required for Hurst exponent is approximately 240.
\end{itemize}
\item \textbf{Run-time monitoring:} To design the run-time monitoring setup, we propose to translate the statistical model of communication behavior into their corresponding property specification language (PSL) assertions (see Sections \ref{PSL} and \ref{PSL_run} ). These assertions are embedded into the RTL description of SoC along with the traffic monitoring units. During the run-time or testing time, these assertions are verified based on the run-time values of the statistical parameters, i.e., Hurst exponent, hop probability, standard deviation.
\end{enumerate}
\begin{figure}[!t]
\centering
\includegraphics[width=1\linewidth]{Figs/Novel_Contributions.pdf}
\caption{Design, test and run-time flow of our methodology for statistical modeling of communication of the trusted IPs (SIMCom). The highlighted box represents novel contributions. The evaluation/testing is done on an MC8051-based SoC and a LEON3-based SoC.}
\label{fig:Novel}
\end{figure}
\normalcolor
To evaluate the effectiveness of SIMCom, we implemented three SoCs. First, SoC consists of four single-core MC8051 that are linked with universal asynchronous receiver/transmitter (UART) modules via and AMBA 2.0 bus. Second SoC consists of four single-core MC8051 IPs that are linked with each other and other IPs like RS232, vga\_lcd, ethernet, using AMBA 2.0 bus. Third SoC consists of four single-core LEON3 IPs that are linked with each other and other IPs like RS232, vga\_lcd, ethernet, using AMBA 2.0 bus. The experimental results show that with the help of multiple parameters, SIMCom is able to detect all the implemented Trojan benchmarks \cite{trust-HUB} with less than 1\% area and power overhead.
\subsection{Paper organization}
The rest of the paper is organized as follows: Section~\ref{pre_liminaries} provides the provide a brief overview of statistical modeling of communication behavior using Hurst exponent, the standard deviation of the injection distribution and the hop distribution. Section~\ref{proposed-methodology} elaborates on the proposed methodology based on the statistical traffic modeling of SoC communication network for run-time hardware Trojan detection. Section~\ref{sec:HW} provides the details about implementations of the hardware modules that compute statistical parameters during run time. For illustration, Section~\ref{case-study} present the detailed analysis of three SoCs, i.e., SoC1 consisting of four single-core MC8051 with UART modules, SoC2 consisting of four single-core MC8051 linked with each other and AES, ethernet, memctrl, BasicRSA, RS23s modules, and SoC3 consisting of four single-core LEON3 connected with each other and AES, ethernet, memctrl, BasicRSA, RS23s modules. Section \ref{comparison} discusses the detection approaches, attributes, pros and cons, and a brief comparison w.r.t HT detection accuracy of the proposed approach and the state-of-the-art techniques. it Finally, Section \ref{conclusion} concludes the paper.
\section{Preliminaries}
\label{pre_liminaries}
Before proceeding to the details of the technical contributions, in this section, we provide a brief overview statistical modeling of communication behavior.
\subsection{Statistical Traffic Modeling}\label{Statistical-Traffic-Modeling}
In order to extract the statistical communication behavior of the SoC, which can be used in turn to sniff any anomaly, we utilized the statistical model for SoC traffic provided in \cite{soteriou2006statistical}. The model is composed of the following three key statistical parameters: \par
\begin{enumerate}[leftmargin=*]
\item \textit{\textbf{Hurst Exponent ($H$):}} Also known as the ``index of dependence'' or `` the index of long-range dependence'', it quantifies the relative tendency of a time series either to regress strongly to the mean or to cluster in a direction \cite{kleinow2002testing}. Typically H value ranges from 0 to 1. A value from 0.5 to 1 indicates that a high value in the series will probably be followed by another high value. A value from 0 to 0.5 indicates that it is more likely that a high value will be followed by a low value. A value of H equal to 0.5 indicates a completely uncorrelated series. The Hurst exponent can be estimated by using the following equation:
\begin{equation}
H = \dfrac{\log_{10}{\left( E\left[\dfrac{R(s)}{S(n)}\right] \right) }}{a_2 \times \log_{10} n}
\label {eq:eq1}
\end{equation}
Where:\par
$R(s)$: Magnitude range of the time series\par
$S(n)$: Average peak magnitude of the time series\par
$n$: Total number of observations\par
$a_2$: Positive constant\par
In the traffic modeling of an SoC, the ``\textit{H}'' exponent corresponds to the magnitude of the packet injection peaks and their time arrival injection patterns.
\item \textit{\textbf{Spatial Hop Distribution ($P$):}} It is determined by three factors: (1) the communication pattern of an application, (2) the power consumption of each communicating module and (3) the number of communicating modules required to complete an application. It is possible that the communication pattern and power consumption are fixed for a particular application, but the number of communicating modules, for an application, can vary and affect the hop counts. Therefore, Soteriou et. al. \cite{soteriou2006statistical} proposed a parameter, called \textit{acceptance probability ($p$)} of the packet, over an average hop count as a metric for equivalent spatial traffic distribution. For a total $N$ number of modules, if an application is mapped on $n$ modules, with $n \leq N$, then the acceptance probability is defined as $p = \dfrac{1}{n-1}$. In reality, a packet may only have a limited number of receivers $r$, $n \geq N$, then the acceptance probability is $p = \dfrac{1}{r-1}$. The hop distance also depends upon the communication topology and thus the spatial hop distribution ($P_{h>d}$) for a particular $p$ can be defined as:\par
\begin{equation}
P_{h>d} = (1- p)^{s(d)}
\label {eq:eq2}
\end{equation}
Where, if $s$ represents a source module, then $n(d)$, $s(d)$ and $d$ represent, the total number of modules whose hop distance (Manhattan) is $d$ from $s$, number of modules with hop distance $1$ to $d$, and the maximum hop distance, respectively. The values for $n(d)$ and $s(d)$ for different communication topologies are given in Table \ref{tab:tab1}.
\item \textit{\textbf{Probability Distribution of Spatial Injection ($\sigma$):}} It represents the standard deviation of the probability distribution at which a module injects the packets into the communication network of an SoC. In this work, we used the Gaussian probability distribution.
\end{enumerate}
\begin{table}[H]
\centering
\caption{$n(d)$ and $s(d)$ for different SoC communication topologies \cite{soteriou2006statistical}}
\label{tab:tab1}
\resizebox{0.7\textwidth}{!}{
\begin{tabular}{|c|c|c|}
\hline
\textbf{Topology} & \textbf{$n(d)$} & \textbf{$s(d)$} \\ \hline
Ring & $2$ & $2d$ \\ \hline
2D Mesh & $4d$ & $2d \times (d+1)$ \\ \hline
3D Mesh & $(8\sum_{a=1}^{d} (d-a)) + 4d + 2$ & $\sum_{a=1}^{d} 4a \times (2d-1a+1) + 2d$ \\ \hline
\end{tabular}}
\vspace{0.1in}
\end{table}
\section{SIMCom: Statistical Sniffing of Communication Behavior} \label{proposed-methodology}
This section explains our proposed methodology for runtime hardware Trojan detection based on the traffic modeling of SoC communication.
Using an appropriate threat model is one of the foremost steps in developing any methodology for detecting intrusions in the domain of hardware security. In this paper, we assume that the 3PIP vendors are not trusted, and the SoC integration is performed in a trusted facility \cite{xiao2016hardware}. Moreover, we also assume that at least one 3PIP has to be trusted, which can be built in-house, such that its VHDL is available to insert checks\footnote{Note, If none of the 3PIPs are trusted, then finding Trojan is an extremely hard problem as no hardware access is known, and no golden behavior is known.}.
SIMCom consists of two major phases, as depicted by the dotted rectangles in Figure~\ref{fig:methodology}.
\textbf{During the design time}, SIMCom requires the following steps to design the runtime monitors:
\begin{enumerate}
\item First, it extracts the communication behavior of the trusted IP and statistically models the extracted communication behavior of the trusted IP. Note, the traffic modeling is done under the premise that at least one of the IP is trusted. The statistical modeling of the trusted IP can be used to detect HTs. The statistical communication behavior of the trusted IP in an SoC is obtained using the following statistical parameters.
\begin{enumerate}
\item The input packets are generated with respect to the standard spatial injection distribution, e.g., Gaussian distribution.
\item Next, the \textit{Hurst exponent} is computed for each communication channel using the R/S method.
\item Hop distribution is computed using the total number of available communication channels and an active communication channel.
\end{enumerate}
\item Then, it uses the statistical model to define the corresponding property specification language assertions. These assertions are inserted into the RTL or Verilog/VHDL of the trusted IP.
\item The runtime verification of the statistical parameters-based PSL assertion requires the hardware modules that compute the above-mentioned statistical parameters, i.e., the Hurst exponent ($H$), probability of hop distribution ($P$) and standard deviation of input injection distribution ($\sigma$). Therefore, during the design time, the designer designs these hardware modules and integrated with critical communication channels.
\end{enumerate}
\begin{figure*}[!t]
\centering
\includegraphics[width=1\textwidth]{Figs/SIMCom.pdf}
\caption{SIMCom: Statistical sniffing of inter-module communication for runtime HT detection}
\label{fig:methodology}
\end{figure*}
\textbf{During the run-time}, the values of the statistical parameters computed from the corresponding hardware modules are used to verify the associated PSL assertions. Note, for secure communication, and all the PSL assertions should be verified. If one of the assertions fails the verification then the communication channel is considered as intruded. After the verification failures, any of the suitable state-of-the-art recovery mechanisms can be applied, i.e., re-routing the communication, backup communication paths, backup components that are associated with critical computations or communication~\cite{cui2014high}.
\section{Hardware implementation}\label{sec:HW}
To extract the statistical parameters during the runtime for verification of the PSL assertions, we design the hardware modules for computing the Hurst exponent, hop distribution, and standard deviation.
\begin{enumerate}
\item \textit{Hurst Exponent:} To compute the estimation, we used one of the most commonly used methods, i.e., the R/S method. The reason behind choosing this method is that it requires less number of observations to estimate the Hurst exponent.
\begin{algorithm}[!t]{
\footnotesize
\caption{Hurst Exponent Estimation}
\label{algo:hurst1}
\begin{algorithmic}[1]
\Statex \textbf{Input:}
\State $n$: Number of the observations = 512
\State $X[0:n-1]$ : number of communication packets per observation
\State Generate three computation blocks using $X[0:n-1]$
\Statex \textbf{Computation Block 1:}
\State Compute the mean and the standard deviation of $X[0:n-1]$
\State Compute the mean centered series: $h[0:n-1]= X[0:n-1]-M$
\State Compute cumulative deviation by summing up the mean centred values:
\Statex $Y[0:n-1]=\sum_{i=0}^{0}h[i], \sum_{i=0}^{1}h[i], ..., \sum_{i=0}^{n-1}h[i]$
\State Compute the Range (R) of $Y[0:n-1]$
\State Compute $R/S$
\State Compute $E(R/S)_0$
\Statex \textbf{Computation Block 2:}
\State Compute the mean (M) and standard deviation (S) of $X[0:(n/2)-1]$ and $X[n/2:n-1]$
\State Compute the mean centered series:
\Statex $h_1[0:(n/2)-1]= X[0:(n/2)-1]-M$ and $h_2[n/2:n-1]= X[n/2:n-1]-M$
\State Compute cumulative deviation by summing up the mean centred values:
\Statex $Y[0:(n/2)-1]=\sum_{i=0}^{0}h_1[i], \sum_{i=0}^{1}h_1[i], ..., \sum_{i=0}^{(n/2)-1}h_1[i]$
\Statex $Y[n/2:n-1]=\sum_{i=n/2}^{(n/2)}h_2[i], \sum_{i=n/2}^{(n/2)+1}h_2[i], ..., \sum_{i=n/2}^{n-1}h_2[i]$
\State Compute the Range (R) of $Y[0:(n/2)-1]$ and $Y[n/2:n-1]$
\State Compute $(R/S)_1$ and $(R/S)_2$
\State Compute $E(R/S)_1 = \frac{(R/S)_1 + (R/S)_2}{2} $
\Statex \textbf{Computation Block 3:}
\State Compute the mean (M) and standard deviation (S) of $X[0:(n/4)-1]$, $X[n/4:(n/2)-1]$, $X[n/2:(3n/4)-1]$ and $X[3n/4:n-1]$
\State Compute the mean centered series:
\Statex $h_1[0:(n/4)-1]= X[0:(n/4)-1]-M$, $h_2[n/4:(n/2)-1]= X[n/4:(n/2)-1]-M$, $h_3[n/2:(3n/4)-1]= X[n/2:(3n/4)-1]-M$ and $h_4[3n/4:n-1]= X[3n/4:n-1]-M$
\State Compute cumulative deviation by summing up the mean centred values:
\Statex $Y[0:(n/4)-1]=\sum_{i=0}^{0}h_1[i], \sum_{i=0}^{1}h_1[i], ..., \sum_{i=0}^{(n/4)-1}h_1[i]$
\Statex $Y[n/4:(n/2)-1]=\sum_{i=n/4}^{n/4}h_2[i], \sum_{i=n/4}^{(n/4)+1}h_2[i], ..., \sum_{i=n/4}^{(n/2)-1}h_2[i]$
\Statex $Y[n/2:(3n/4)-1]=\sum_{i=n/2}^{n/2}h_3[i], \sum_{i=n/2}^{(n/2)+1}h_3[i], ..., \sum_{i=n/2}^{(3n/4)-1}h_3[i]$
\Statex $Y[3n/4:n-1]=\sum_{i=3n/4}^{3n/4}h_4[i], \sum_{i=3n/4}^{(3n/4)+1}h_4[i], ..., \sum_{i=3n/4}^{n-1}h_4[i]$
\State Compute the Range (R) of $Y[0:(n/4)-1]$, $Y[n/4:(n/2)-1]$, $Y[n/2:(3n/4)-1]$ and $Y[3n/4:n-1]$
\State Compute $(R/S)_1$, $(R/S)_2$, $(R/S)_3$ and $(R/S)_4$
\State Compute $E(R/S)_2 = \frac{(R/S)_1 + (R/S)_2 + (R/S)_3 + (R/S)_4}{4} $
\Statex \textbf{Hurst Exponent:}
\State Compute $E(R/S) = \frac{E(R/S)_0 + E(R/S)_1 + E(R/S)_2}{3} $
\State Compute Hurst exponent $H = 0.37 \times log{E(R/S)}$
\end{algorithmic}}
\end{algorithm}
\begin{figure}[!t]
\centering
\includegraphics[width=1\textwidth]{Figs/HW_Implementation.pdf}
\caption{Data flow of the hardware module to compute the Hurst exponent. In this paper, we set the number of observations to 512. The reason behind choosing this value is that for fast convergence of Hurst exponent, the number of observations should be greater than 300. Note, To estimate the Hurst exponent, we used one of the most commonly used methods, i.e., the R/S method. The reason behind choosing this method is that it requires fewer number of observations to estimate the Hurst exponent. We implemented state-of-the-art modified non-restoring algorithm~\cite{sutikno2012simplified} for computing the square root function in standard deviation.}
\label{fig:HW}
\end{figure}
\begin{itemize}
\item After establishing a communication channel, it observes and stores the number of the packets per clock cycle, as denoted by $X$ in Algorithm~\ref{algo:hurst1}.
\item To estimate the Hurst exponent, it is important to take the average value of R/S values using the different data distribution obtained from the same data. Therefore, we propose to use three computational blocks, as shown in Algorithm~\ref{algo:hurst1} and Figure~\ref{fig:HW}. The data flow and hardware modules in each computational block are the same but the sizes of hardware modules are different. The computational block 1 uses the complete data for estimating the R/S value. The computational block 2 divides the data into two equal parts. Then this block estimates the respective R/S values using each half data and takes the average to estimate the R/S value. Similarly, the computational block 3 repeats the same procedure, but it divides the data into four equal parts.
\item In each computational block, the first step is to compute the mean value (M) and standard deviation (S) of the input data series $X$. The hardware modules to compute the mean value consists of ``n'' 64-bit adders, one 9-bit shifter and one output register, as shown in Figure~\ref{fig:HW}. The hardware module for computing the standard deviation consists of ``n'' 64-bit subtractors, ``n'' 64-bit multipliers, one 9-bit shifter, one module to compute the square root and one output register. Note, we implemented state-of-the-art modified non-restoring algorithm~\cite{sutikno2012simplified} for computing the square root function in standard deviation.
\item After computing the mean and standard deviation, each computational block computes the mean-centered data series ($H$) by subtracting the mean value from each input data series ($X$), as shown in Algorithm~\ref{algo:hurst1}. Then, it computes the cumulative deviation ($Y$) by summing up the mean-centered data series ($H$) and computes the magnitude range ($R$) of the cumulative deviation ($Y$). Finally, it computes the R/S using a 64-bit divider. Note, there is no extra hardware for the mean-centered series and each computational block uses the values from the standard deviation module. The hardware module of computing $Y$ consists of one 64-bit accumulator, as shown in Figure~\ref{fig:HW}, and the hardware module computing $R$ consists of two comparators, two multiplexers and one 64-bit subtractor.
\item Finally, it computes the R/S by taking the average of R/S values computed from each computational block. Finally, the average R/S value is used to compute the Hurst exponent using $H = 0.37\times log_{10} (average value of R/S)$, as shown in Algorithm~\ref{algo:hurst1} and Figure~\ref{fig:HW}.
\end{itemize}
\item \textit{Standard Deviation}
We have not used a separate block for computing the standard deviation. We use the intermediate output from the standard deviation block of Hurst exponent.
\item \textit{Hop Probability }
In order to compute the hop probability, SIMCom computes the number of active channels by counting the the acknowledgment signals during the channel establishment. Note, this parameter is effective in the case of denial-of-service, jamming or communication blocking.
\end{enumerate}
\section{Case Studies}\label{case-study}
To illustrate the scalability of the SIMCom, we implemented the three SoCs, as shown in Fig\ref{fig:exp_setup}.
\begin{enumerate}
\item \textbf{SoC1} consists of four single-core MC8051 with UART modules.
\item \textbf{SoC2} consists of four single-core MC8051 linked with each other and AES, ethernet, memctrl, BasicRSA, RS23s modules.
\item \textbf{SoC3} consists of four single-core LEON3 connected with each other and AES, ethernet, memctrl, BasicRSA, RS23s modules.
\end{enumerate}
All the SoC1 are tested on the Trust-Hub HT benchmarks~\cite{trust-HUB}. Note for SoC1, and the input data distribution is Gaussian and exponential. The workloads used for LEON3 in SoC3 are 64-bit encrypted multiplication, subtraction, addition, and division with randomly distributed input data. The input data is encrypted using AES. Results outputted using VGA display, ethernet, and RS232. For MC8051 in SoC2, we use the same set of applications with 32-bit. For area and power overhead analysis, we synthesized the SoCs using Cadence Genus (Encounter) tool with the TSMC 65nm library.
\subsection{SoC1: Four MC8051 IPs are connected with UART IPs}
In this section, we illustrate the effectiveness of SIMCom by applying it on the SoC with AMBA bus connecting an MC8051 microcontroller and a UART module, as shown in Figure \ref{fig:exp_setup}.
This experimental setup consists of four MC8051 Master modules, which can initiate the communication with the eight slave modules along with two UART modules. The main motivation of choosing the MC8051 microcontroller as our case study is the availability of its open-source Trojan benchmarks at the trusthub.org \cite{trust-HUB}. We first obtained the communication behavior of the \textit{un-intruded MC8051} while communicating with the UART modules. We used this behavior to obtain the corresponding PSL assertions and embed them into the SoC. The key assumption of the experimental setup is that \textit{Though all instances of the MC8051 can have the Trojan but Trojan in only one IP module is triggered at a particular time considering a unique sequence happening at run time with a very low probability of triggering (almost zero)}.
\begin{figure}[!t]
\centering
\includegraphics[width=1\linewidth]{Figs/exp_setup2.pdf}
\caption{Experimental Setup and a brief overview of all the implemented SoCs.}
\label{fig:exp_setup}
\end{figure}
\subsection{Pre-Market Test Stage Traffic Model of MC8051} \label{TM-8051_model}
For obtaining this behavior, we implemented the MC8051 microcontroller in VHDL and synthesized it for the Spartan 6 (xc6slx45)\footnote{Since the overall SoC fits into this small FPGA, there was no need to use a bigger FPGA chip. We target a low-cost SoC, for which Spartan 6 suffices the need.} FPGA. The communication behavior is extracted by measuring the number of packets for the specific duration and computing the required traffic model parameters, $H$, $P$ and $\sigma$. This phase of our methodology is divided into the following steps:
\begin{enumerate}
\item The first step is to extract the communication model. We developed the model by injecting the packets using the Gaussian distribution, as shown in Figure \ref{fig:exp_setup}. This probability distribution not only reflects the actual traffic very well, but it also incorporates the environmental and process variations \cite{kreutz2001communication}. In our experimental setup, all the master modules, i.e., MC8051s, inject packets into the SoC. The value of $\sigma$ of MC8051 for the first 10,000 clock cycles is kept as 18.844 and maintained over the 100,000 clock cycles, as shown in Figure~\ref{fig:results_time}.
\item Afterwards, we need to identify the number of modules that a particular master can access through the bus. In this case study, we assume that every MC8051 can communicate with all the slave modules ($n = 8$). However, for some specific instruction sets, MC8051 does not communicate with all the slave modules. For example, the instructions that require the UART communication module only. In our setup, there are only \textit{``two''} slaves that are using the UART communication modules. Therefore, the acceptance probability ($p$) for this case study must be greater than 0.125 as shown in Figure \ref{fig:results_time}. Its average value over the 100,000 clock cycles is indicated in column ``$P$'' under ``T000 (Without Trojan)'' of Table \ref{tab:tab2}. \par
\begin{algorithm}[!t]{
\footnotesize
\caption{Hurst Exponent Estimation in SoC consists of MC8051 and UART}
\label{algo:hurst}
\begin{algorithmic}[1]
\Statex \textbf{Input:}
\Statex \tab $b_{size}$: Minimum Number of Clock Cycles (Bin Size)
\Statex \tab $max_{clk}$: Maximum Number of Clock Cycles
\Statex \tab $np$: Number of packet per bin; $np[0:99]$: Number of packet per bin
\Statex \tab $H$: Hurst Exponent; $H[0:99]$: Hurst parameter array
\Statex \textbf{Initialize:}
\Statex \tab $bin_{size}$: 99; $max_{size}$: 99; $n_{packets}$: 0; $i$: 0; $j$: 0;$k$: 0;
\While{$k \leq 10$}
\While{$i \leq max_{clk}$}
\While{$j \leq b_{size}$}
\State $j = j + 1\ @\ posedge\ clock$; $np = np + 1 \ @\ each\ packet$
\EndWhile
\State $np[i] = np $; $i = i + 1$
\If {$np[i] \geq np[i-1]$}
\State $np_{max} = np[i]$ //Maximum number of packets per Bin
\Else
\State $np_{max} = np[i-1]$
\EndIf
\If {$np[i] \leq np[i-1]$}
\State $np_{min} = np[i]$ //Minimum number of packets per Bin
\Else
\State $np_{max} = np[i-1]$
\EndIf
\State $np_{total} = np[i] + np[i+1]$; $j = 0$
\EndWhile
\State $np_{average} = np_{max}/i$
\State $H[k] = \log_{10}(np_{max} - np_{min})/(np_{average}\times\log_{10} (max_{clk}. b_{size}))$
\State $i = 0$; $H_{total} = H[k] + H[k-1]$
\EndWhile
\State $H = H_{total}/k $
\end{algorithmic}}
\end{algorithm}
\item The last step in obtaining the communication model is to estimate the Hurst exponent for MC8051. For this, we have developed Algorithm \ref{algo:hurst}, which counts the number of packets for a specific duration and estimates the Hurst exponent ($H$). In Algorithm \ref{algo:hurst}, a counter ($np$) is used to count the number of packets for a specific bin size. For the considered study, its value is 100 clock cycles. The counting is done 100 times to find the maximum ($np_{max}$) and minimum ($np_{min}$) number of packets in 100 clock cycles. However, for a larger number of clock cycles, the memory requirement for storing these parameters is very high. Therefore, to reduce the memory requirement, we divided the whole duration into 100 equal parts. For example, in Algorithm \ref{algo:hurst}, the total time duration is 10,000 clock cycles, which is divided into 100 smaller time duration part. For each duration, the number of packets is calculated, which is stored in a variable array $np[0:99]$ to compute its overall range ($np_{max} - np_{min}$) and average ($\dfrac{np_0 + np_1 + .. + np_{99}}{100}$). Finally, the Hurst exponent is computed by averaging out the obtained values. We repeated these steps 10 times, i.e., data of the 10,000 clock cycles data is stored in one array and then the whole process is repeated 9 times to get an overall $H$ exponent for 100,000 clock cycles. This value is tabulated as 0.692 in the column ``$H$'' under ``T000'' (without Trojan) of Table \ref{tab:tab2}. \par
\end{enumerate}
\subsection{PSL Assertions}\label{PSL}
We analyzed the extracted pre-market test stage communication model of the AMBA bus, and translated it into the following PSL assertions based on the traffic model parameters, $H$, $P$ and $\sigma$:
\begin{enumerate}
\item \textbf{P Assertion:} It states that, over a given number of clock cycles, the acceptance probability ($P$) for a particular module must be equal to the value of acceptance probability ($P$) to the obtained communication behavior:
\begin{align*}
\texttt{\textbf{always} (P[1:100K] == P\_design)} \tag{I}\label{prop:1}
\end{align*}
Moreover, the acceptance probability ($P$) for a particular architecture must be within the prescribed limits. The maximum and the minimum values of $P$ are obtained as ($P_{max} = \dfrac{1}{n-1}$) and ($P_{min} =\dfrac{1}{r-1}$), respectively, where $n$ and $r$ represent the number of modules that each module can communicate with, and the number of modules that a particular instruction set/application need to communicate with, respectively. This is elaborated in Assertion \ref{prop:2}:
\begin{align*}
\texttt{\textbf{always} (P[1:100K] <= Pmax\ \&\& P[1:100K] >= Pmin)} \tag{II}\label{prop:2}
\end{align*}
\item \textbf{$\sigma$ Assertion:} This assertion states that over a given number of clock cycles, the value of $\sigma$ for a particular module must be equal to the value of $\sigma$ from the pre-market test stage communication behavior:
\begin{align*}
\texttt{ \textbf{always} (sigma[1:100K]== sigma\_design)} \tag{III}\label{prop:3}
\end{align*}
\item \textbf{H Assertion:} This assertion ensures that over a give number of clock cycles, the Hurst exponent ($H$) for a particular module must be equal to value of $H$ obtained at the pre-market test stage:
\begin{align*}
\displaystyle \texttt{ \textbf{always} (H[1:100K]== H\_design)} \tag{IV}\label{prop:4}
\end{align*}
\end{enumerate}
In state-of-the-art HT detection techniques for SoC and microprocessors, the process variation margin is considered as approximately 10\%, however, this margin depends upon the technology parameters~\cite{wang2008detecting,brasser2018special}. Therefore, to incorporate the environmental changes and process variation, we proposed to introduce a 10\% variation cap on the $H$, $P$, $\sigma$ based on the analysis presented in~\cite{gil2012studying,korak2014effects}.
\begin{table}[!t]
\centering
\caption{Effects of the trust-hub Trojan benchmarks on different modules of MC8051 and corresponding statistical modeling. $\times$ and $\checkmark$ shows that Trojan either affect or does not affect a particular functionality.}
\label{tab:tab2}
\resizebox{0.7\textwidth}{!}{
\begin{tabular}{|c|c|c|c|c|}
\hline
\textbf{Trojans} & \textbf{\begin{tabular}[c]{@{}c@{}}UART\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}Jump \\ Disabling\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}ALU \\ Operations\end{tabular}} & \textbf{($H$,$\sigma$,$P$ )} \\ \hline
\textbf{T000} & -- & -- & -- & (0.694,18.844, 0.252)\\ \hline
\textbf{T200} & -- & -- & -- & --\\ \hline
\textbf{T300} & $\checkmark$ & $\times$ & $\checkmark$ & (0,0,0)\\ \hline
\textbf{T400} & $\times$ & $\times$ & $\checkmark$ & (0.704,14.617, 0.252)\\ \hline
\textbf{T500} & $\checkmark$ & $\times$ & $\checkmark$ & (0.701,15.019, 0.252)\\ \hline
\textbf{T600} & $\checkmark$ & $\checkmark$ & $\checkmark$ & (0.644,14.472, 0.252) \\ \hline
\textbf{T700} & $\checkmark$ & $\times$ & $\times$ & (0.672,14.728, 0.252)\\ \hline
\textbf{T800} & $\checkmark$ & $\times$ & $\times$ & (0.639,14.476, 0.252) \\ \hline
\end{tabular}}
\end{table}
\subsection{Run-time Detection and Validation}\label{PSL_run}
The last step of the proposed methodology is to embed the translated assertions along with the $H$, $P$ and $\sigma$ estimation blocks into the SoC. To illustrate the proposed methodology, we validated it with the MC8051 benchmarks intrusions, which are available online at trust-hub.org \cite{trust-HUB}. These intrusions affect different types of communication behavior. Some of them affect the UART communication, other affect the jump based instruction and some intrusions affect the ALU operations, as given in Table \ref{tab:tab2}. Each row in Table \ref{tab:tab2} represents the benchmark Trojans, and the columns describe their effects on different operations. We validated the proposed methodology for the intrusions T300 to T800, because these are all the intrusions that can directly or indirectly affect the communication of MC8051 among the set of the benchmark Trojans. To evaluate the consistency checking of SIMCom, we analyzed the runitme and average statistical behavior for the experimental setup.
\subsubsection{Run-time Impact Analysis}
To elaborate on the practicality of SIMCom, we have computed the statistical communication behavior of the case study for 100,000 clock cycles, as shown in Figure \ref{fig:results_time}. The figure depicts the communication behavior with respect to $H$, $\sigma$ and $P$. This run-time analysis exhibits the following key observations:\par
\begin{enumerate}
\item \textbf{Hurst Exponent:} The Trojan benchmarks exhibit a significant impact on the Hurst Exponent depending on the input data distribution (i.e., Gaussian or Exponential). For instance, in Figure \ref{fig:results_time}, if the input data distribution is\textit{ Gaussian}, then a few of the implemented Trojan benchmarks (i.e., MS8051-T400, T500, and T700) have a significant impact on the Hurst exponent (see: label 1). However, if the input data distribution is \textit{exponential}, then almost all the HTs exhibit a significant impact on the Hurst exponent (see: label 2).
\item \textbf{Probability of Hop Distribution:} All the implemented Trojans exhibit no impact on the probability of hop distribution, except the MC8051-T300 because when it is triggered, it halts a communication channel, as shown in Figure \ref{fig:results_time} (see: labels 3 and 4).
\item \textbf{Standard Deviation of Injection Distribution:} All the implemented Trojans deviate from the original values because all of them affect the input data injection, as shown in Figure \ref{fig:results_time}. For instance, MS8051-T500 and T700 replace valid data with intruded data. However, MC8051-T400 disables \textit{interrupt handling}, T600 \textit{interrupts the jump} and T800 \textit{manipulates the stack} pointer, which indirectly disrupts the input data or respective control modules.
\end{enumerate}
The values of $H$ and $\sigma$ for MC8051-T300 are ``0'' because upon triggering T300 blocks the UART communication altogether, and \textit{hence no data traffic flows at all}, resulting in the corresponding '0' values for \textit{H} and sigma. \par
\begin{figure}[!t]
\centering
\includegraphics[width=1\linewidth]{Figs/PSL_Results_time.pdf}
\caption{Runtime Impact Analysis of implemented HTs (i.e., MC8051-T200, T300, T400, T500, T600, T700, T800) on the statistical model ($H$, $P$, $\sigma$) of the implemented case study for 100,000 clock cycles and the values are averaged out for 10,000 clock cycle duration. Labels 1, 2, 3 and 4 of this figure is described in Section \ref{TM-8051_model}.}
\label{fig:results_time}
\vspace{0.1in}
\end{figure}
\subsubsection{Average Impact Analysis}
In order to elaborate the consistency of SIMCom, we have analyzed the average communication behavior of the MC8051-based UART communication network, as shown in Figure \ref{fig:results_avg}, which depicts the communication behavior with respect to $H$, $\sigma$ and $P$. This analysis exhibits the following key observations:\par
\begin{enumerate}
\item \textbf{Hurst Exponent:} The average behavior of the proposed statistical model shows that, on the average, all the implemented Trojan benchmarks have a significant impact on the $H$ irrespective of the input data distribution, as shown in Figure \ref{fig:results_avg}. However, the value of $H$ for MC8051-T300 is ``0''. Hence, this can be considered as the weakest Trojan for the communication-aware HT detection (see: label A and D). Though it shows that the run-time impact of implemented Trojans is more in case of the exponential distribution, in average analysis, this impact is less as compared to the impact on the Gaussian distribution. The reason behind this behavior is that the data rate decreases exponentially over any period of time in the exponential distribution.
\item \textbf{Probability of Hop Distribution:} Even in the average-case analysis, all the implemented Trojans exhibit no impact on the probability of hop distribution, except for the MC8051-T300 as shown in Figure \ref{fig:results_avg} (see: labels D and E). The reason behind this behavior is the blockage of the communication of a channel when this Trojan benchmark gets an activation trigger. Hence this parameter can be used for detecting such (T300) kinds of Trojans, which are otherwise difficult to be detected.
\item \textbf{Standard Deviation of Injection Distribution:} All the implemented Trojans deviate from the original values because all of them affect the input data injection, except the MC8051-T300, as shown in Figure~\ref{fig:results_avg} (see: labels C and F). For instance, MS8051-T500 and T700 replace valid data with the intruded data. However, the MC8051-T400 disables \textit{interrupt handling}, the T600 \textit{interrupts the jump} and the T800 \textit{manipulates the stack} pointer, which indirectly disrupts the input data or respective control modules.
\end{enumerate}
\begin{figure}[!t]
\centering
\includegraphics[width=1\linewidth]{Figs/PSL_Results.pdf}
\caption{Average Impact Analysis of implemented Trojans (i.e., MC8051-T200, T300, T400, T500, T600, T700, T800) statistical model of 8051 based UART communication network for 100000 clock cycles. Note: T000 represents the pre-market test stage communication behavior of the MC8051 based experimental case study with no Trojans inserted.}
\label{fig:results_avg}
\vspace{0.1in}
\end{figure}
\subsection{Experimental Analysis for SoC2 and SoC3}
We analyze timing behavior of the communication pattern, false positives, false negatives and HT detection accuracy in SoC2 and SoC3.
\begin{itemize}
\item \textbf{Timing behavior of H, P, and S:} Figures~\ref{fig:result_MC8051}~and~\ref{fig:result_LEON3} show the timing behavior of Hurst exponent, standard deviation, and hop probability for three communication channels, i.e., MC8051/LEON3 with AES, BasicRSA, and Ethernet, in the presence of four HT benchmarks, i.e., AES-T100, AES-T200, BasicRSA-T100, and EthernetMAC10GE-T600. This analysis shows the significant change in H and S. Note, only the communication blocking affects the hop probability. Therefore, by analyzing all these parameters, SIMCom detects HTs and its payload type. For example, if it affects the P, then the payload is either denial-of-service, communication blocking or jamming.
\item \textbf{HT Detection Accuracy:} Figures~\ref{fig:FP_FN_8051}~and~\ref{fig:FP_FN_LEON3} show the false positives, false negatives and HT detection accuracy of the different configurations of SIMCom, i.e., SIMCom with only H, SIMCom with only P, SIMCom with only S, and SIMCom with H, P and S. The analysis shows that HT detection accuracy of SIMCom with H, P and S is approximate, 99\%. Note, SIMCom with the only P detects only those Trojans that blocks the communication channels like EthernetMAC10GE-T600.
\end{itemize}
\begin{figure}[!t]
\centering
\includegraphics[width=1\linewidth]{Figs/Results_LEON3.pdf}
\caption{Experimental results to show the effect of HT benchmarks, i.e., AES-T100, AES-T200, BasicRSA-T100 and EthernetMAC10GE-T600, on communication between \textbf{MC8051} and AES, ethernet and basic RAS module respectively. Note, in this analysis the number of the observations (n) is 512.}
\label{fig:result_MC8051}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=1\linewidth]{Figs/FP_FN_MC8051.pdf}
\caption{False positive, False-negative and HT detection accuracy of the SIMCom and the state-of-the-art run-time detection technique~\cite{ngo2015hardware}, in the presence of different HT benchmarks, on \textbf{MC8051-based SoC}. Note, these analyses are based on the 10000 observations, where the total number of HT activations is 1000. In these experiments, the overall accuracy is computed as $Accuracy = \frac{TP+TN}{TP+TN+FP+FN}$, where TP, TN, FP, and FN represent true-positives, true-negatives, false-positives, and false-negatives, respectively.}
\label{fig:FP_FN_8051}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=1\linewidth]{Figs/Results_LEON3.pdf}
\caption{Experimental results to show the effect of HT benchmarks, i.e., AES-T100, AES-T200, BasicRSA-T200 and EthernetMAC10GE-T600, on communication between \textbf{LEON3} and AES, ethernet and basic RAS module respectively. Note, in this analysis the number of the observations (n) is 512.}
\label{fig:result_LEON3}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=1\linewidth]{Figs/FP_FN_LEON3.pdf}
\caption{False positive, False-negative and HT detection accuracy of the SIMCom and the state-of-the-art run-time detection technique~\cite{ngo2015hardware}, in the presence of different HT benchmarks, on \textbf{LEON3-based SoC}. Note, these analyses are based on the 10000 clock cycles, where the total number of HT activations is 1000. In these experiments, the overall accuracy is computed as $Accuracy = \frac{TP+TN}{TP+TN+FP+FN}$, where TP, TN, FP, and FN represent true-positives, true-negatives, false-positives, and false-negatives, respectively.}
\label{fig:FP_FN_LEON3}
\end{figure}
\subsection{Overhead Analysis}
For area and power overhead analysis, we synthesized the SoCs using Cadence Genus (Encounter) tool with the TSMC 65nm library. The overhead analysis presented in Figure~\ref{fig:overhead} shows the following observations:
\begin{enumerate}
\item The area and power overhead associated with SIMCom in the SoCs with the same number of communication channels relatively decreases with the increase in the complexity of the modules. For example, in SoC3, the area and power overhead is less than 1\%.
\item If the number of communication channels increases, the overhead of the SIMCom also increases. Therefore, we propose to use the state-of-the-art methodology for distributing the runtime monitors in the SoC~\cite{khalid2018runtime}. This distribution technique proposes to three topologies, i.e., a global monitor, region-based distribution, and channel-based monitors, that can be selected based on the power and area budget.
\begin{itemize}
\item In global monitor topology, a single monitor is used to handle all the communication channels. This monitor randomly or systematically selects the communication channel and performs the PSL assertion-based verification. This topology exhibits the minimum area and power overhead but it decreases the HT detection accuracy.
\item In channel-based topology, a monitor is attached to each communication channel. This topology exhibits the maximum HT detection accuracy but the area and power overheads are huge as compare to the global monitor.
\item In region-based topology, the SoC is divided into such small regions such that each region has the same number of communication channels. Then for each region a global monitor is inserted. This topology exhibits better HT detection accuracy as compared to global monitor, and better area and power overhead as compared to the channel-based topology.
\end{itemize}
It is important to note that based on the criticality of the SoC regions, different topologies can be used for a different portion of the SoC. For example, the global monitors can be used in the regions which are not vulnerable to security attacks, and the channel-based topology can be used in the most vulnerable region.
\end{enumerate}
\begin{figure}[!t]
\centering
\includegraphics[width=1\linewidth]{Figs/Overhead_Analysis2.pdf}
\caption{Area and power overhead analysis of implemented SoCs.}
\label{fig:overhead}
\end{figure}
\section{Comparative Discussion with state-of-the-art Techniques} \label{comparison}
The run-time property checking technique can be used to detect the HTs at the hardware or software level. The fundamental issue with this technique is to extract the properties that provide maximum coverage. To address this issue, researchers have used the specification or functionality as property. For example, Ngo et al.~\cite{ngo2015hardware} extract the properties from the functionality and specification of the IC to design hardware property checkers (HPC). Similarly, Bloom et al.~\cite{bloom2009support} proposed to check the liveliness property at the OS level. However, both of these techniques are applicable to functional Trojans and cannot detect the information leakage Trojans because they do not affect the functionality. Therefore, to detect such Trojans, we propose to exploit the statistical behavior of the communication for extracting the properties which can be used to detect functional and information leakage Trojans.
Table \ref{tab:tab4} provides a comparison of SIMCom with the state-of-the-art techniques that are relevant. The comparative analysis is based on five parameters. \textit{Overhead} provides the count of the extra component requirement for run-tim HT detection. \textit{Detection Approaches}, describes how a particular technique is detecting or protecting the intrusions that have any effect on SoC communication. \textit{Attributes}, indicates the key parameters used by a technique to protect or detect the intrusions. The final two comparison parameters describe the \textit{Pros and Cons} of a particular technique. The proposed technique is found to be better than other state-of-the-art techniques in the following ways:
\begin{table}[!t]
\centering
\caption{Comparative discussion with the state-of-the-art Techniques}
\label{tab:tab4}
\resizebox{1\textwidth}{!}{
\begin{tabular}{|c|c|c|c|l|l|}
\hline
\textbf{Techniques} & \begin{tabular}[c]{@{}c@{}}\textbf{Run-time}\\ \textbf{Overhead}\end{tabular} & \begin{tabular}[c]{@{}c@{}}\textbf{Detection}\\ \textbf{Approaches}\end{tabular} & \textbf{Attributes} & \multicolumn{1}{c|}{\textbf{Pros}} & \multicolumn{1}{c|}{\textbf{Cons}} \\ \hline
\begin{tabular}[c]{@{}c@{}}\textbf{Kim et al.} \\ \cite{kim2011system} \end{tabular} & \begin{tabular}[c]{@{}c@{}} MUX based \\Wrappers,\\ Registers \end{tabular} & \begin{tabular}[c]{@{}c@{}}Protecting \\ Wrappers, \\ Hidden \\ Signals \end{tabular} & \begin{tabular}[c]{@{}c@{}}Master ID,\\ Restricted \\Addressing,\\ Ready and \\Grant Signals \end{tabular} & \begin{tabular}[c]{@{}l@{}}1. Low On-chip area \\ overhead \\ 2. Applicable to the SoC \\bus intrusions\end{tabular} & \begin{tabular}[c]{@{}l@{}}1. If the intrusion uses the \\ communication network\\ without affecting protocol,\\ then it fails. \end{tabular} \\ \hline
\begin{tabular}[c]{@{}c@{}}\textbf{Kulkarni et al.} \\ \cite{kulkarni2016adaptive,kulkarni2016svm} \end{tabular} & \begin{tabular}[c]{@{}c@{}} Trained\\ ML Model \end{tabular} & \begin{tabular}[c]{@{}c@{}}ML Tools \\i.e., k-NN,\\ MBW, SVM\end{tabular} & \begin{tabular}[c]{@{}c@{}}Source and\\ Destination \\Core, Packet\\ Transfer Path,\\ Distance\end{tabular} & \begin{tabular}[c]{@{}l@{}}1. Low Latency \end{tabular} & \begin{tabular}[c]{@{}l@{}}1. Cannot detect Trojans\\ if the IPs are intruded.\\ 2. Only applicable if \\intrusion affects the \\communication protocols \end{tabular} \\ \hline
\begin{tabular}[c]{@{}c@{}}\textbf{Ngo et al.} \\ \cite{ngo2015hardware} \end{tabular} & \begin{tabular}[c]{@{}c@{}} Assertions \end{tabular} & \begin{tabular}[c]{@{}c@{}}PSL\\ Assertions\end{tabular} & \begin{tabular}[c]{@{}c@{}}Functional \\Specifications\end{tabular} & \begin{tabular}[c]{@{}l@{}}1. Low area and power\\ overhead \end{tabular} & \begin{tabular}[c]{@{}l@{}}1. Cannot detect Trojans\\ that steals the information.\end{tabular} \\ \hline
\begin{tabular}[c]{@{}c@{}}\textbf{ SIMCom} \end{tabular} & \begin{tabular}[c]{@{}c@{}} $H$, $P$ and $\sigma$\\ computing\\ blocks, \\ Four 3.2KB \\ Memories \end{tabular} & \begin{tabular}[c]{@{}c@{}}PSL \\Assertions\end{tabular} & \begin{tabular}[c]{@{}c@{}}Hurst exponent,\\ Hop Distribution,\\ Standard Deviation \\of Injection \\ Distribution\end{tabular} & \begin{tabular}[c]{@{}l@{}}1. Detects the Trojans in \\3PIP modules that use \\SoC communication\\ network \\ 2. It also detects Trojans,\\ which do not affect the \\protocol directly. \end{tabular} & \begin{tabular}[c]{@{}l@{}}1. Only applicable if\\ intrusion affects the \\SoC communication\end{tabular} \\ \hline
\end{tabular}}
\end{table}
\begin{enumerate}
\item The proposed technique does not require any wrappers and hidden control signals, like the LOCK signal, for protecting against the Trojans, as some other techniques require, e.g., \cite{kim2011system}.
\item Unlike \cite{kulkarni2016adaptive,kulkarni2016svm}, our technique does not presume that Trojan payload activation provides a very high change in the communication channel or any change in communication protocols. Moreover, it can also detect the Trojans, which have indirect effects on communication channels.
\item The proposed technique can be used for any known bus system by adding the $H$, $P$, and $\sigma$ measurement blocks and PSL assertions obtained from the communication behavior of the pre-market test stage.
\item SIMCom is a hardware property checking (HPC) based HT detection technique at the hardware-level like proposed by Ngo et al.~\cite{ngo2015hardware}. However, unlike the existing HPC-based technique~\cite{ngo2015hardware} is based on the functional properties of the protocol. Therefore, it is unable to detect the Trojans which leak the information without affecting the communication protocols. On the other hand, SIMCom detects the functional Trojans as well as the information leakage Trojans. Figures~\ref{fig:FP_FN_8051} and~\ref{fig:FP_FN_LEON3} show that HT detection accuracy of the existing HPC-based technique~\cite{ngo2015hardware} for most of the AES Trojans is 0 because these Trojans leak the information without changing the functionality. The detailed hardware implementation of SIMCom is given in Figure~\ref{fig:HW}.
\end{enumerate}
\section{Conclusions}\label{conclusion}
This paper utilizes a statistical traffic modeling of SoC for sniffing HTs during run-time. The proposed methodology consists of two major steps: The first step is to extract the pre-market test stage traffic model by computing three parameters, i.e., Hurst exponent ($H$), spatial hop distribution ($P$) and spatial injection distribution ($\sigma$). Then, this extracted behavior is translated to its corresponding PSL assertions, which are embedded into the given SoC along with the blocks, which compute the traffic modeling parameters. For illustration purpose, for three SoCs, i.e., SoC1 ( four single-core MC8051 and UART modules), SoC2 (four single-core MC8051, AES, ethernet, memctrl, BasicRSA, RS232 modules), and SoC3 (four single-core LEON3 connected with each other and AES, ethernet, memctrl, BasicRSA, RS23s modules microcontrollers). For validation purposes, we used all available benchmarks on Trust-Hub. The experimental results show that the proposed methodology is able to detect all intrusions with less than 1\% area and power overhead. To the best of our knowledge, this is the first time that a statistical sniffing of SoC's communication traffic based approach is utilized to detect hardware Trojans in 3PIP modules.
\section*{Acknowledgment}
This work is supported in parts by the Austrian Research Promotion Agency (FFG) and the Austrian Federal Ministry for Transport, Innovation, and Technology (BMVIT) under the "ICT of the Future" project, IoT4CPS: Trustworthy IoT for Cyber-Physical Systems.
\section*{References}
\bibliographystyle{elsarticle-num}
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{
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\section{Introduction}
\label{sec:introduction}
Recent studies have shown that a considerable fraction of massive
galaxies at $1.5 < z < 2.5$ have quiescent stellar populations
(e.g.,\citealt{labbe2005}; \citealt{kriek2006};
\citealt{williams2009}). Among the most massive galaxies ($M_{*} >
10^{11}$ \msun ) approximately 40\% are no longer forming stars
(e.g.,\citealt{whitaker2011}; \citealt{brammer2011}). Surprisingly,
these massive quiescent galaxies have been found to be extremely
compact (e.g.,\citealt{daddi2005}; \citealt{trujillo2006};
\citealt{vandokkum2008}; \citealt{franx2008}; \citealt{vanderwel2008};
and numerous others), compared to their likely present-day
counterparts.
Searches for ultra-dense low-redshift counterparts by
\citet{trujillo2009} and \citet{taylor2010a} found only a handful of
compact sources at $z\sim0$, which have relatively young stellar
populations (\citealt{trujillo2009}; \citealt{ferremateu2012}). The
dearth of massive, old compact objects at low redshift implies that
massive galaxies must have undergone severe structural evolution in
size.
Errors in the size estimates have been invoked as a possible
explanation for the compactness of massive high-redshift galaxies.
Initial concerns that the size may have been underestimated, due to an
envelope of low surface brightness light, have been addressed with
deep Hubble Space Telescope Wide Field Camera 3 (HST WFC3) imaging
(\citealt{szomoru2010}; \citeyear{szomoru2012}), and by stacking
results (e.g \citealt{vanderwel2008}; \citealt{cassata2010};
\citealt{vandokkum2008}, \citeyear{vandokkum2010}). The light could
also be more concentrated due to the presence of active galactic
nuclei (AGNs) in these galaxies. However, spectra of subsamples of
these galaxies have shown that the light is dominated by evolved
stellar populations, not AGNs (\citeauthor{kriek2006}
\citeyear{kriek2006}, \citeyear{kriek2009a}; \citealt{vandesande2011};
\citealt{onodera2012}).
The question of whether stellar masses are accurate out to $z\sim 2$
remains, however, a serious concern: an overestimate in stellar mass
would bring the galaxies closer to the $z\sim0$ mass-size relation. To
date, basically all (stellar) masses have been derived by fitting the
spectral energy distributions (SEDs). This method suffers from many
systematic uncertainties in stellar population synthesis (SPS) models
(e.g \citealt{conroy2009}; \citealt{muzzin2009}) and is essentially
untested at $z > 1.5$.
\begin{deluxetable*}{c c c c c c c c c c c c}
\tabletypesize{\scriptsize}
\tablewidth{0pt}
\tablecaption{Photometric Properties}
\tablehead{
\colhead{Catalog} & \colhead{ID} & \colhead{$J_{\rm aper}$} & \colhead{$H_{\rm aper}$} & \colhead{$K_{\rm aper}$} & \colhead{$K_{\rm tot}$} & \colhead{$(U-V)_{\rm~rf}$} & \colhead{$(V-J)_{\rm~rf}$} & \colhead{24 $\mu \rm{m}$} & \colhead{SFR$_{24\mu m}$} \\
& & & & & & & & \colhead{ $[\mu\rm{Jy}]$} & \colhead{\msun\ yr$^{-1}$} }\\
\startdata
NMBS-COS & 7447 & 21.09 & 20.72 & 20.63 & 19.64 & 1.20 & 0.31 & $\lesssim 18$ & $\lesssim 13$\\
NMBS-COS &18265 & 22.67 & 20.85 & 20.61 & 19.62 & 1.72 & 0.85 & $\lesssim 18$ & $\lesssim 15$\\
NMBS-COS & 7865 & 22.75 & 21.51 & 21.07 & 20.02 & 1.89 & 0.90 & $\lesssim 18$ & $\lesssim 19$\\
UDS &19627 & 21.40 & 20.91 & 20.65 & 20.19 & 1.37 & 0.71 & $\lesssim 30$ & $\lesssim 29$\\
UDS &29410 & 20.59 & 20.18 & 19.81 & 19.36 & 1.62 & 0.96 & $232 \pm 15$ & $241\pm16$ \\
\enddata
\tablecomments{Aperture and total magnitudes for our targets. Aperture
magnitudes have been measured in fixed 1.5 arcsec diameter aperture
for targets in NMBS-I, while the targets in UDS have 1.75 arcsec
diameter apertures. Rest-frame colors have been derived from the
spectra in Johnson $U$, $V$, and $2MASS J$ filters. For the 24 $\mu$m
fluxes we provide 3-$\sigma$ upper-limits of 18 $\mu$Jy for the
galaxies in NMBS-COSMOS \citep{whitaker2012}, and 30 $\mu$Jy for
UDS-19627 \citep{toft2012}, as these galaxies are not detected with
MIPS. UDS-29410 has a strong MIPS detection, which is likely due to an
obscured AGN.}
\label{tab:phot_sample}
\end{deluxetable*}
Direct stellar kinematic mass measurements, which do not suffer from
these uncertainties, can be derived by measuring the galaxy's velocity
dispersion and the shape and extent of its luminosity profile,
i.e.,the S\'ersic $n$ parameter and effective radius. In particular,
for low-redshift galaxies in the Sloan Digital Sky Survey (SDSS),
\citet{taylor2010b} showed that stellar mass is a very good predictor
of dynamical mass, but only when non-homology of luminosity profile is
properly accounted for using a S\'ersic-dependent virial factor
(e.g.,\citealt{cappellari2006}). Although dynamical measurements of
massive galaxies are common at low redshift, spectroscopic studies
become much more difficult at higher redshift as the bulk of the
light, and stellar absorption features used to measure kinematics,
shift redward into the near-infrared (NIR) (e.g.,\citealt{kriek2009a};
\citealt{vandokkum2009a}).
New technology such as the new red arm of the LRIS spectrograph at
Keck (working beyond 1$\micron$) makes it possible to measure velocity
dispersions up to $z\sim1.5$ (\citealt{newman2010},
\citealt{bezanson2012}). Deep NIR spectroscopy is, however, required
to push stellar kinematic studies to even higher redshift. From a
$\sim$29\,hr spectrum of an ultra-compact galaxy at $z=2.2$ obtained
with Gemini Near-IR Spectrograph \citep{kriek2009a},
\citet{vandokkum2009a} found a high, though uncertain, velocity
dispersion of $\sigma =
510^{+165}_{-95}\,$km$~$s$^{-1}$. \citet{onodera2012} used the MOIRCS
on the Subaru telescope to observe the rest-frame optical spectrum of
a less-compact, passive, ultra-massive galaxy at $z=1.82$, but the low
spectral resolution and signal-to-noise ratio (S/N) severely limited the accuracy
of their velocity dispersion: $\sigma = 270 {\pm
105}\,$km$~$s$^{-1}$. X-Shooter (\citealt{dodorico2006};
\citealt{vernet2011}), the new ultraviolet (UV) to NIR spectrograph at
the Very Large Telescope (VLT), can provide the required S/N and
resolution. The capabilities of X-Shooter for this kind of
measurements were demonstrated in \citet{vandesande2011}, who found
$294 \pm 51\,$km$~$s$^{-1}$ for a massive quiescent galaxy at
z=1.8. \citet{toft2012} also use X-Shooter and present a dynamical
measurement of a galaxy at redshift $z=2.04$ with similar
results. Taken all together, these results indicate that the dynamical
and stellar masses are consistent with $z\sim0$. With the small
number of measurements beyond $z>1.5$, however, the sample is still
too small to draw any firm conclusions on whether the stellar masses
are truly reliable.
Here we present a sample of five massive quiescent galaxies with high
signal-to-noise (S/N), medium-resolution, UV to NIR spectra at
$1.4<z<2.1$ observed with X-Shooter on the VLT. The main goal of this
paper will be to test if the stellar mass measurements at high
redshift are reliable.
The paper is organized as follows. In \textsection \ref{sec:data} we
present our sample of high-redshift galaxies, the photometric and
spectroscopic data, and describe our data reduction. In \textsection
\ref{sec:properties} we determine structural properties and stellar
populations, and derive stellar and dynamical masses. We complement
our results with stellar kinematic results from other studies at low
and high redshift in \textsection \ref{sec:compilation}. In
\textsection \ref{sec:mass_comparison} we compare our dynamical to the
stellar masses. In \textsection \ref{sec:evolution} we study the
structural evolution of high-redshift quiescent massive galaxies. In
\textsection \ref{sec:inside_out} we compare our results with previous
measurements and hydrodynamical simulations. Finally, in \textsection
\ref{sec:conclusion} we summarize our results and
conclusions. Throughout the paper we assume a $\Lambda$CDM cosmology
with $\Omega_\mathrm{m}$=0.3, $\Omega_{\Lambda}=0.7$, and $H_{0}=70$
km s$^{-1}$ Mpc$^{-1}$. All broadband data are given in the AB-based
photometric system.
\begin{figure*}[ht] \epsscale{1.1}
\plotone{fig1.eps}
\caption{Comparison of our spectroscopic sample to the full population
at similar redshift. Symbol size of the squares represent the density
of galaxies from the NMBS-I and UDS at $1.4<z<2.1$ with mass
$>10^{10.5} M_{\odot}$. (a) Rest-frame U-V and V-J
colors. Color coding is based upon the sSFR derived from SED fitting,
red colors indicates low sSFR (quiescent), and blue colors indicate high
sSFR(star-forming). Galaxies in the top left region, as marked by the
black line, all have low sSFR rates. This region is therefore often
used to select quiescent galaxies at high redshift
\citep{williams2009}. All but one of our galaxies fall within this
region, but their sSFR indicate that they are all have quiescent
stellar populations. The vertical dotted line discriminates between
young post-starburst like (left) vs. old quiescent (right) as
indicated by \citet{whitaker2012}. The strong Balmer absorption lines
spectroscopically confirm the young ages of this
sample. (b) Rest-frame U-V vs. stellar mass. At
fixed mass, we find that most of our galaxies have similar colors to
the entire population, except for NMBS-COS7447 and UDS-19627 on the
blue side. (c) $H$-band aperture magnitude vs.
stellar mass. It is clear that our sample was selected on magnitude,
and at fixed mass they are among the brightest galaxies, consistent
with their post-starburst nature.
}
\label{fig:selection_effects_sp}
\end{figure*}
\begin{deluxetable*}{c c c c c c c c c l}[!ht]
\tabletypesize{\scriptsize}
\tablewidth{0pt}
\tablecaption{Targets and Observations}
\tablehead{
\colhead{Catalog} & \colhead{ID} & \colhead{R.A.} & \colhead{Dec.} & \colhead{Exp. Time} & \colhead{Slit Size NIR} & \colhead{S/N $J$} & \colhead{S/N $H$}& \colhead{S/N $_{{4020<\lambda \AA<7000}}$} & \colhead{Telluric Standard Star} \\
& & & & \colhead{[min]} & \colhead{[arcsec]} & \colhead{[\AA$^{-1}$]} & \colhead{[\AA$^{-1}$]} & \colhead{[\AA$^{-1}$]} & \colhead{$Hipparcos~ID$} }\\
\startdata
\vspace{0.15cm}
NMBS-COS & 7447 &10:00: 6.96 & 2:17:33.77 & 120 & 0''.9 & 4.98 & 8.48 & 6.31 & 050307, 000349 \\
\vspace{0.15cm}
NMBS-COS &18265 &10:00:40.83 & 2:28:52.15 & 90 & 0''.9 & 3.37 & 6.99 & 4.18 & 050684, 000349 \\
NMBS-COS & 7865 &10:00:17.73 & 2:17:52.75 & 434 & 0''.9 & 1.64 & 5.86 & 4.12 & 049704, 057126, 046054, \\
\vspace{0.05cm}
& & & & & & & & & 040217, 059987 \\
UDS &19627 & 2:18:17.06 &-5:21:38.83 & 300 &0''.6, 0''.9 & 3.80 & 7.94 & 5.90 & 012377, 114656, 008352, \\
\vspace{0.05cm}
& & & & & & & & & 000328, 015389 \\
UDS &29410 & 2:17:51.22 &-5:16:21.84 & 120 & 0''.9 & 3.75 & 7.09 & 4.35 & 012377
\enddata
\tablecomments{R.A. and Dec. are given in the J2000 coordinate
system. Exposure times are given for the NIR arm, the UVB and VIS arms
had slightly shorter exposure times due to the longer read out. Except
for UDS19627, all targets were observed with a 0.9" slit in the
NIR. S/N ratios have been determined from comparing the residual of
the velocity dispersion fit to the flux, and are given for the $J-$
and $H-$band as well as for the region in which we determine the
velocity dispersion. Last column gives the Tellurics Standard Stars
from the Hipparcos catalog that were observed before and after each
target.}
\label{tab:sample_cat}
\end{deluxetable*}
\begin{figure*}
\epsscale{1.1}
\plotone{fig2.eps}
\caption{Color Images of our five spectroscopic targets. Except for
UDS-19627, all galaxies have available HST-WFC3 Imaging. For each
target we show the composite color image on the left side, the best
S\'ersic model from GALFIT and the residual after we subtract the
best-fitting model from the original image on the right side. The
lower panel shows the intrinsic surface brightness profile with all
available bands. Different colors show the different filters, as
indicated on the bottom right. Vertical dashed lines show the
effective radii for each profile, while the dotted lines shows the
FWHM/2 of the PSF. We find color gradients, such that the redder
bands have smaller effective radii, for all galaxies but
NMBS-COS7447. For that case, the sizes are similar within the errors,
but this could be caused by the extra flux of the red arc-like feature
in the southeast.
}
\label{fig:sizes}
\end{figure*}
\begin{figure*}
\epsscale{1.1}
\plotone{fig3.eps}
\caption{UV to NIR X-Shooter spectra in combination with medium- and
broad-band data (blue diamonds). The binned spectra ($\sim10\AA$) are
shown in black, together with the best-fitting BC03 $\tau$-model as
shown in red. Gray areas indicate regions with strong atmospheric
absorption. The UVB spectrum is missing for UDS-29410, due to an
instrument problem during the observations. The good agreement between
the BC03 models and the spectroscopic data over this large wavelength
range is astonishing. Both NMBS-COS7865 and UDS-19627 show a small
deviation from the best-fitting model around $1\micron$, which is
caused by the absence of good telluric calibrators. From stellar
population synthesis modeling, we find a variety of ages that range
from 0.5-2 Gyr. We find no emission lines, and other signs of star
formation, and with little to no dust (see Section \ref{subsec:sps}).}
\label{fig:spectra_page_1}
\end{figure*}
\begin{figure*}
\epsscale{1.1}
\plotone{fig4.eps}
\caption{Rest-frame optical part of the spectrum focused on the Balmer
break. As in Figure \ref{fig:spectra_page_1}, the X-Shooter spectrum
is shown in black, but this time in higher resolution ($\sim 4\AA$
observed, or $\sim100$ km s$^{-1}$ rest-frame). The most prominent
absorption and emission features are indicated by the blue dashed
lines. The clear detection of absorption lines enables us to measure
stellar velocity dispersions. We use pPXF to fit the best-fitting BC03
$\tau$ model to the spectrum and find velocity dispersions that range
from 275-435 km s$^{-1}$. The convolved best-fit BC03 template is
shown in red.}
\label{fig:spectra_page_2}
\end{figure*}
\begin{figure*}
\epsscale{1.1}
\plotone{fig5.eps}
\caption{Rest-frame optical part of the spectrum focused on $Mgb, Na
D$, and $H\alpha$. As in Figure \ref{fig:spectra_page_2}, the
X-Shooter spectrum is shown in black, with high resolution of $\sim
4\AA$ observed, or $\sim100$ km s$^{-1}$ rest-frame. The most
prominent absorption and emission features are indicated by the blue
dashed lines. The convolved best-fit BC03 template is shown in red.}
\label{fig:spectra_page_3}
\end{figure*}
\section{Data}
\label{sec:data}
\subsection{Target Selection}
\label{sec:target_selection}
The galaxies in this paper are drawn from the \mbox{NMBS-I}
\citep{whitaker2010} and the UKIDSS-UDS \citep{williams2009}. They
were selected to be bright in the $H$-band, and to have $z >1.4$, in
order to obtain sufficient S/N. The SED
from the broadband and medium-band photometry was required to indicate
that they have quiescent stellar populations, and the rest-frame
optical imaging could not show signs of large disturbance due to
e.g.,mergers. We note that NMBS-COS7447 was presented in
\citet{vandesande2011}, and UDS-19627 was presented in
\citet{toft2012}. All data for both galaxies have been re-analyzed
according to the following procedure for consistency. Our selection
had no priors on mass or size, but could be biased in either one of
these parameters. Full information on the photometric properties of the targets is
listed in Table \ref{tab:phot_sample}.
To investigate possible biases, we compare our targets to a sample of
galaxies with mass $>10^{10.5} M_{\odot}$ at $1.4<z<2.1$ from the
NMBS-I and the UDS. Rest-frame $U-V$ and $V-J$ colors are commonly
used to distinguish between star-forming and quiescent galaxies at
this redshift (e.g.,\citealt{williams2009}). Figure
\ref{fig:selection_effects_sp}$a$ shows the UVJ-diagram for all
galaxies at redshifts between $1.4<z<2.1$ with mass $>10^{10.5}
M_{\odot}$, together with the sample presented here. The sizes of the
squares indicate the density of galaxies. For our targets, the
rest-frame colors have been measured from the spectra, while for the
full sample rest-frame colors are based on the broadband and medium-band
data. As demonstrated by \citet{williams2009}, non-star-forming
galaxies can be identified using a color selection indicated by the
black lines. Within this selection region, our targets fall in the
area occupied by young, quiescent galaxies \citep{whitaker2012}. The
median specific star formation rates (sSFR), as indicated by the
different colors, are in good agreement with the full high-redshift
sample at the same place in the UVJ diagram. For their mass, however,
NMBS-COS7447 and UDS-19627 have slightly bluer colors as compared to
the full sample (Figure \ref{fig:selection_effects_sp}$b$). At fixed
mass, the targets are among the brightest galaxies, except for
NMBS-I-7865 (\ref{fig:selection_effects_sp}$c$). This may not come as
a surprise as they are among the youngest quiescent galaxies, and thus
have relatively low $M/L$.
\subsection{Imaging}
\label{subsec:imaging}
Four different imaging data sets are used to measure the surface
brightness profiles of our galaxies, as summarized below. (1) All our
targets in the NMBS-I COSMOS field were observed with HST-WFC3
$H_{160}$ as part of the program HST-GO-12167 (PI: Franx). Each target
was observed for one orbit (2611 sec), using a four point dither
pattern, with half pixel offsets. Reduction of the data was done in a
similar way as to the reduction described in \citet{bouwens2010}, but
without sigma-clipping in order to avoid masking the centers of
stars. The drizzled images have a pixel scale of $0''.06$, with a full
width at half-maximum (FWHM) of the point-spread function (PSF) of
$\sim0''.16$. (2) Our NMBS-I targets are complemented with HST-ACS
$I_{814}$ imaging from COSMOS (v2.0, \citealt{koekemoer2007};
\citealt{massey2010}), which has a $0''.03$ pixel scale and PSF-FWHM
of $\sim0''.11$. (3) For UDS-29410 we make use of the HST-ACS F814W,
HST-WFC3 $J_{125}$ and $H_{160}$ from UDS-CANDELS
(\citealt{grogin2011}; \citealt{koekemoer2011}). These data have the
same properties as the data described in (1) and (2). (4) For
UDS-19627 we use ground based data from UKIDSS-UDS,
(\citealt{lawrence2007}; \citealt{warren2007}) Data Release 8 in the
J, H, and K-band, as no as HST data is available. Imaging in all three
bands were drizzled to a pixel scale $0''.134$, and the FWHM of the PSF
is $0''.7$ in the K-band.
\subsection{Spectroscopic Observations}
Observations were performed with X-Shooter on the VLT UT2
(\citealt{dodorico2006}; \citealt{vernet2011}). X-Shooter is a second
generation instrument on the VLT that consists of three arms: UVB, VIS,
and NIR. The wavelength coverage ranges from 3000 to 24800 $\AA$ in
one single exposure. The galaxies were observed in both visitor and
service mode, and the observations were carried out between 2010 January
and 2011 March (Programs: Fynbo 084.A-0303(D), Van de Sande
084.A-1082(A), Franx 085.A-0962(A), Toft 086.B-0955(A)). Full
information on the targets and observations is listed in Table
\ref{tab:sample_cat}. All observations had clear sky conditions and
an average seeing of $0''.$. A $0''.9$ slit was used in the NIR, except
for the 1st hour of UDS-19627 where a $0.''6$ slit was used. For the
$0''.9$ slit, this resulted in a spectral resolution of 5100 at
1.4$\mu$m. Observing blocks were split into exposures of 10-15
minutes each with an $ABA'B'$ on-source dither pattern. For most
targets, a telluric standard of type B8V-B9V was observed before and
after our primary target, in order to create a telluric absorption
spectrum at the same airmass as the observation of our target.
\subsection{Spectroscopic Reduction}
Data from the three arms of X-Shooter must be analyzed separately and
then combined to cover the full range from the UV to NIR. In the NIR we
identified bad pixels in the following way. The data were corrected
for dark current, flatfielded, and sky subtracted using the average of
the preceding and subsequent frames. The ESO pipeline (ver. 1.3.7;
\citealt {goldoni2006}) was used to derive a wavelength solution for
all orders. The orders were then straightened using integer pixel
shifts to retain the pixels affected by cosmic-rays and bad
pixels. Additional sky subtraction was done on the rectified orders,
by subtracting the median in the spatial direction. Cosmic rays and
bad pixels were identified by LA-Cosmic \citep{vandokkum2001}, and a
bad pixel mask was created.
Further 3$\sigma$ clipping was done on the different exposures,
corrected for dithers, to identify any remaining outliers. The bad
pixel masks of different orders were combined into a single file and
then transformed back to the raw frame for each exposure. Masks will
follow the same rectification and wavelength calibration steps as the
science frames.
Next the flatfielded and sky subtracted observations were rectified
and wavelength calibrated, only this time we used interpolation when
rectifying the different orders. Again, additional sky subtraction was
done. Per order, all exposures were combined, with exclusion of bad
pixels and those contaminated with cosmic rays present in the mask
file.
The telluric spectra were reduced in the same way as the science
frames. We constructed a response spectrum from the telluric stars in
combination with a stellar model for a B8V/B9V star from a blackbody
curve and models from \citet{munari2005}. Residuals from Balmer
absorption features in the spectrum of the tellurics were removed by
interpolation. All the orders of the science observations were
corrected for instrumental response and atmospheric absorption by
dividing by the response spectrum.
The different orders were then combined, and in regions of overlap
weighted using the S/N of the galaxy spectrum. A noise spectrum was
created by measuring the noise in the spatial direction below and
above the galaxy. If the regions exceeded an acceptable noise limit,
from contamination by OH lines or due to low atmospheric transmission,
this spatial region was discarded for further use. The two-dimensional (2D) spectra were
visually inspected for emission lines, but none were found. A 1D
spectrum was extracted by adding all lines (along the wavelength
direction), with flux greater than 0.1 times the flux in the central
row, using optimal weighting.
Absolute flux calibration was performed by scaling the spectrum to the
available photometric data. The scaling was derived for each
individual filter that fully covered the spectrum. For our targets in
NMBS-I we used the following filters: $J, J2, J3, H, H1, H2, and Ks$,
while for the targets in the UDS we only used $J$ and $H$. We then
used an error-weighted average obtained from the broadband magnitudes,
and scaled the whole spectrum using this single value. After scaling,
no color residuals were found, and no further flux corrections were
applied to the spectrum.
\begin{deluxetable*}{c c c c c c c c c c c}[!hb]
\tablecaption{Stellar Population Synthesis Properties}
\tablehead{
\colhead{Catalog} & \colhead{ID} & \colhead{$z_{phot}$} & \colhead{$z_{spec}$} & \colhead{$\log \tau$} & \colhead{$\log Age$} & \colhead{$Z$ } & \colhead{$A_{V}$} & \colhead{$\log M_{*}$} & \colhead{$\log$ SFR} & \colhead{$\log$ sSFR}\\
& & & & \colhead{(yr)} & \colhead{(yr)} & & \colhead{(mag)} & \colhead{($M_{\odot}$)} & \colhead{($M_{\odot}\mathrm{yr}^{-1}$)} & \colhead{($\mathrm{yr}^{-1}$)} \\ }
\startdata
NMBS-COS & 7447 & 1.71$\pm$ 0.03 & 1.800 & 7.80 & 8.74 & 0.020 & 0.00 & 11.27 & -0.08 & -11.35 \\
NMBS-COS &18265 & 1.60$\pm$ 0.03 & 1.583 & 7.00 & 8.96 & 0.020 & 0.45 & 11.42 & -99.00 & -99.00 \\
NMBS-COS & 7865 & 2.02$\pm$ 0.05 & 2.091 & 7.20 & 9.41 & 0.008 & 0.05 & 11.68 & -99.00 & -99.00 \\
UDS &19627 & 1.94$\pm$ 0.06 & 2.036 & 7.90 & 8.74 & 0.050 & 0.20 & 11.24 & 0.56 & -10.68 \\
UDS &29410 & 1.44$\pm$ 0.02 & 1.456 & 7.90 & 8.82 & 0.050 & 0.35 & 11.29 & -99.00 & -11.28 \\
\enddata
\tablecomments{Derived stellar population synthesis properties from
FAST. We use stellar templates from \citealt{bruzual2003}, with an
exponentially declining star formation history with timescale $\tau$,
together with a \citet{chabrier2003} IMF, and the \citet{calzetti2000}
reddening law. No errors are provided, as the $68\%$ confidence all
fall within one grid point. The real errors are dominated by
systematic uncertainties.}
\label{tab:stellar_prop}
\end{deluxetable*}
A low resolution spectrum was constructed by binning the 2D spectrum
in wavelength direction. Using a bi-weight mean, 20 good pixels,
i.e.,not affected by skylines or strong atmospheric absorption, were
combined. The 1D spectrum was extracted from this binned 2D spectrum
in a similar fashion as the high resolution spectrum \textit{(see
Figure \ref{fig:spectra_page_1} - \ref{fig:spectra_page_3} )}.
For the UVB and VIS arms we used the ESO reduction pipeline (version
1.3.7, \citealt {goldoni2006}) to correct for the bias, flatfield, and
dark current, and to derive the wavelength solution. The science
frames were also rectified using the pipeline, but thereafter, treated
in exactly the same way as the rectified 2D spectra of the NIR arm, as
described above.
\section{ Structural Properties and Stellar populations}
\label{sec:properties}
\subsection{Stellar Population Properties}
\label{subsec:sps}
We estimate the stellar population properties by fitting the
low-resolution ($\sim10\AA$ in the observed frame) spectrum in the Visual
and NIR in combination with the broadband and medium-band photometry with
SPS models. We exclude the UVB part of the spectrum due to the lower
S/N and the extensive high S/N broadband photometry in this
wavelength region. Stellar templates from \citeauthor{bruzual2003} (\citeyear{bruzual2003}, BC03)
are used, with an exponentially declining star formation history (SFH) with
timescale $\tau$, together with a \citet{chabrier2003} initial mass
function (IMF), and the \citet{calzetti2000} reddening law.
Using the FAST code \citep{kriek2009a}, we fit a full grid in age, dust
content, star formation timescale, and metallicity. We adopt a grid
for $\tau$ between 10 Myr and 1 Gyr in steps of 0.1 dex. The age range
can vary between 0.1 Gyr and 10 Gyr, but the maximum age is
constrained to the age of the universe at that particular redshift. We
note, however, that this constraint has no impact on our results as
the galaxies are young. Step size in age is set as high as the BC03
templates allow, typically 0.01 dex. Metallicity can vary between
$Z=0.004$ (subsolar), $Z=0.08$, $Z=0.02$(solar), and $Z=0.05$
(supersolar). Furthermore, we allow dust attenuation to range from 0
to 2 mag with step size of 0.05. The redshift used here is from
the best-fitting velocity dispersion (see Section
\ref{subsec:dispersions}). Results are summarized in Table
\ref{tab:stellar_prop}.
Due to our discrete grid and the high quality data, and also because
metallicity and age are limited by the BC03 models, our $68\%$
confidence levels are all within one grid point. Our formal errors
are therefore mostly zero, and not shown in Table
\ref{tab:stellar_prop}. This does not reflect the true uncertainties,
which are dominated by the choice of SPS models, IMF, SFH, and extinction
law (see e.g.,\citealt{conroy2009}; \citealt{muzzin2009}).
The low sSFR confirms the quiescent nature of the galaxies in our
sample, and they match well with the sSFR of the general population in
the same region of the UVJ diagram (Figure
\ref{fig:selection_effects_sp}$a$). We find a range of metallicities,
with the oldest galaxy having the lowest metallicity. However, due to
the strong degeneracy between age and metallicity, we do not believe
this result to be significant. Overall, the dust content in our
galaxies is low.
We find very similar results for NMBS-C7447 as compared to
\citet{vandesande2011}, and the small differences can be explained by
the newer reduction. For UDS-19627 we find a slightly lower mass as
compared to \citet{toft2012}, which is likely due to the lower dust
fraction that we find, i.e $A_v=0.2$ vs. $A_v=0.77$ from
\citet{toft2012}.
The galaxies in our sample are not detected at 24 $\mu$m, leading to a
3-$\sigma$ upper-limit of 18 \ $\mu$Jy for the galaxies in
NMBS-COSMOS, and 30\ $\mu$Jy for UDS-19627 (see
\citealt{whitaker2012}; \citealt{toft2012}). UDS-29410 has a strong
detection at 24 $\mu$m of $232 \pm 15 \mu Jy$. From these upper
limits we calculate the dust-enshrouded SFRs that are listed in Table
\ref{tab:sample_cat}. We find a high SFR for UDS-29410, but we find no
other signs for this high SFR. That is, we find no emission lines and
the best fitting SPS model indicates a low SFR. Therefore, we think
that the strong 24 $\mu$m detection is likely due to an obscured AGN.
\begin{deluxetable*}{c c c c c c c c c c c c c c c c c c c}[!th]
\tabletypesize{\scriptsize}
\tablewidth{0pt}
\tablecaption{Compilation of Masses and Structural Parameters for High-Redshift Galaxies}
\tablehead{
\colhead{Reference$^\mathrm{a}$} & \colhead{ID} & \colhead{$z_{spec}$} & \colhead{$r_e$} & \colhead{$n_{sersic}$} & \colhead{${b/a}$} & \colhead{$\sigma_{e}$} & \colhead{$\sigma_{e} / \sigma_{ap}$} & \colhead{$\beta(n)$} & \colhead{$\log M_{dyn}$} & \colhead{$\log M_{*,corr}$} & \colhead{Filter}}\\
\startdata
0 & 7447 & 1.800 & 1.75 $\pm$ 0.21 & 5.27 $\pm$ 0.23 & 0.71 $\pm$0.02 & 287$^{+ 55}_{- 52}$ & 1.048 & 5.16 & 11.24$^{+0.13}_{-0.12}$ & 11.22 & $H_{\rm F160W}$ \\
0 & 18265 & 1.583 & 0.97 $\pm$ 0.12 & 2.97 $\pm$ 0.06 & 0.26 $\pm$0.02 & 400$^{+ 78}_{- 66}$ & 1.065 & 6.61 & 11.38$^{+0.13}_{-0.11}$ & 11.32 & $H_{\rm F160W}$ \\
0 & 7865 & 2.091 & 2.65 $\pm$ 0.33 & 4.82 $\pm$ 0.15 & 0.83 $\pm$0.02 & 446$^{+ 54}_{- 59}$ & 1.031 & 5.42 & 11.82$^{+0.09}_{-0.09}$ & 11.64 & $H_{\rm F160W}$ \\
0 & 19627 & 2.036 & 1.32 $\pm$ 0.17 & 3.61 $\pm$ 0.73 & 0.48 $\pm$0.06 & 304$^{+ 43}_{- 39}$ & 1.059 & 6.18 & 11.24$^{+0.10}_{-0.09}$ & 11.20 & $K$ \\
0 & 29410 & 1.456 & 1.83 $\pm$ 0.23 & 2.59 $\pm$ 0.03 & 0.54 $\pm$0.02 & 371$^{+114}_{- 90}$ & 1.045 & 6.88 & 11.61$^{+0.19}_{-0.15}$ & 11.24 & $H_{\rm F160W}$ \\
\nodata & \nodata & \nodata & \nodata & \nodata & \nodata & \nodata & \nodata & \nodata & \nodata & \nodata \nodata \\
\enddata
\tablecomments{This Table will be published in its entirety in the
electronic edition of ApJ, and can also be downloaded from
\url{http://www.strw.leidenuniv.nl/~sande/data/}. A portion is shown
here for guidance regarding its form and content.
Spectroscopic redshifts $z_{spec}$ are obtained from the velocity
dispersion fit as described in Section
\ref{subsec:dispersions}. Structural parameters $r_e$, $n_{sersic}$,
and $b/a$ are derived using GALFIT on available imaging, as described
in Section \ref{subsec:sizes}. $\sigma_e$ is the velocity dispersion
within a circular aperture of size $r_e$ from Section
\ref{subsec:dispersions}, and $\sigma_{e} / \sigma_{ap}$ is the
aperture correction we apply to the observed velocity dispersion as
described in Appendix \ref{sec:app_aper_corr}. From Equation
\ref{eq:kn} we calculated $\beta (n)$, and dynamical masses are
derived using Equation \ref{eq:mdyn}. Stellar masses as given here are
corrected to account for the difference between the catalog magnitude
and our measured magnitude. The filter in which the structural
parameters are measured is given in the last column.
}
\tablenotetext{a}{References: 0) this work
1) \citet{bezanson2012};
2) \citet{vandokkum2009a};
3) \citet{onodera2012};
4) \citet{cappellari2009};
5) \citet{newman2010};
6) \citet{vanderwel2008} and \citet{blakeslee2006};
7) \citet{toft2012}.}
\label{tab:sample_results}
\end{deluxetable*}
\subsection{Velocity Dispersions}
\label{subsec:dispersions}
The clear detection of absorption lines in our spectra, together with
the medium resolution of X-Shooter, enable the measurement of accurate
stellar velocity dispersions. We use the unbinned spectra in
combination with the Penalized Pixel-Fitting (pPXF) method by
\citet{cappellari2004} and our best-fitting BC03 models as templates.
Spectra were resampled onto a logarithmic wavelength scale without
using interpolation, but with masking of the bad pixels. The effect of
template mismatch was reduced by simultaneously fitting the template
with a $\sim$17-order Legendre Polynomial. Our results depend only
slightly on the choice of the order of the polynomial (Appendix
\ref{sec:app_disp_test}). Together with the measured velocity
dispersion, the fit also gives us the line-of-sight velocity, and thus
$z_{\rm spec}$.
We also look at dependence of the velocity dispersion on the template
choice. In particular for the younger galaxies in our sample that
show a clear signature of A-type stars, we find a dependence of the
measured velocity dispersion as a function of template age. A more
stable fit is obtained when restricting the wavelength range to $4020
\AA < \lambda <7000 \AA$, which excludes the Balmer break region (see
also Appendix \ref{sec:app_disp_test}).
The errors on the velocity dispersion were determined in the following
way. We subtracted the best-fit model from the spectrum. Residuals are
shuffled in wavelength space and added to the best-fit template. We
then determined the velocity dispersion of 500 of these simulated
spectra. Our quoted error is the standard deviation of the resulting
distribution of the measured velocity dispersions. When we include the
Balmer break region in the fit, the formal random error decreases, but
the derived dispersion becomes more dependent on the chosen stellar
template. In total we have three high-quality measurements, and two
with medium quality. We note that if we exclude the two galaxies with
medium-quality measurements from our sample, our main conclusions
would not change.
The velocity dispersion found here for NMBS-C7447 agrees well with the
results from \citet{vandesande2011}. For UDS-19627 we find a slightly
lower value as compared to \citet{toft2012}. However, they use a
different method for constructing the template for the velocity
dispersion fit. When we fit the spectrum of UDS-19627 in the same way
as was described in \citet{toft2012}, we find a similar answer as
theirs.
All dispersions are corrected for the instrumental resolution ($\sigma$=23
km s$^{-1}$) and the spectral resolution of the templates ($\sigma$=85
km s$^{-1}$). Furthermore, we apply an aperture correction to our
measurements as if they were observed within a circular aperture
radius of $r_e$. In addition to the traditional correction for the
radial dependence of velocity dispersion
(e.g.,\citealt{cappellari2006}), we account for the effects of the
non-circular aperture, seeing and optimal extraction of the 1-D
spectrum. The aperture corrections are small with a median of 4.8$\%$
(See Appendix \ref{sec:app_aper_corr}). The final dispersions and
corresponding uncertainties are given in Table
\ref{tab:sample_results}.
\subsection{Surface Brightness Profiles}
\label{subsec:sizes}
Radial profiles are measured for all galaxies on all available imaging
as described in Section \ref{subsec:imaging}. Galaxies are fitted by
2D S\'ersic radial surface brightness profiles
\citep{sersic1968}, using GALFIT (ver. 3.0.2;
\citealt{peng2010}). Relatively large cutouts of $25''\times 25''$
were provided to GALFIT to ensure an accurate measurement of the
background, which was a free parameter in the fit. All neighboring
sources were masked using a segmentation map obtained with SExtractor
\citep{bertin1996}. In the case of UDS-19627, the close neighbor was
fitted simultaneously. Bright unsaturated field stars were used for
the PSF convolution. All parameters, including the sky, were left free
for GALFIT to determine.
Even though galaxies at low redshift are well-fitted by single
S\'ersic profiles (e.g.,\citealt{kormendy2009}), this does not
necessarily have to be true for galaxies at $z\sim2$. Therefore, we
correct for missing flux using the method described in
\citet{szomoru2010}. We find very small deviation in residual-corrected
effective radii, with a median absolute deviation of
3.4$\%$. Color images and measured profiles are shown in Figure
\ref{fig:sizes}.
We repeated the measurements using a variety of PSF stars
($N\sim25$). We find an absolute median deviation in the half-light
radius of $\sim 3\%$ for HST-WFC3, $\sim 3.5\%$ for HST-ACS, and $\sim
10\%$ for the ground-based UDS-UKIDSS data, due to variations in the
PSF. The largest source of uncertainty in the measured profiles is,
however, caused by the error in the sky background estimate. Even
though these galaxies are among the brightest at this redshift, using
the wrong sky value can result in large errors for both $r_e$ and
$n$. We determine the error in the sky background estimate by
measuring the variations of the residual flux in the profile between 5
and 15 arcsec. For sizes derived from HST-ACS, the absolute median
deviation in the effective radius due to the uncertainty in background
is $\sim 13\%$, and for HST-WFC3 $\sim 12\%$. Due to the deeper
ground-based UDS-UKIDSS data, the uncertainty for UDS-19627 due to the
sky is $\sim 8\%$. All of our results are summarized in Table
\ref{tab:sample_results}.
We note that we find a smaller size and larger $n$ for UDS-19627 as
compared to \citet{toft2012}, which cannot be explained within the
quoted errors. We have compared our results with the size measurements
from \citet{williams2009} and R. J. Williams (2012, private communication),
who also use UDS-UKIDSS data for measuring structural
parameters. They too find a smaller size in the $K$ band of $r_e =
1.63\,$kpc, with a similar axis ratio of $q = 0.53$, while keeping the
S\'ersic index fixed to $n=4$. Furthermore, we compare the size of
UDS-29410 obtained from the ground-based UDS-UKIDSS data, to the size
from HST-WFC3 to test how reliable the ground-based data are for
measuring structural parameters. From the ground-based UDS $H$ band we
find $r_e = 1.97 \pm 0.11$ kpc, and $n=2.47 \pm 0.22$ for UDS-29410
which is consistent with the measurements using the HST-WFC3 data
within our $1-\sigma$ errors. From these two independent results, we
are confident that our size measurement for UDS-19627 is correct.
In what follows, we will use the mean
effective radius and S\'ersic $n$ from the band which is closest to
rest-frame optical $r'$. The effective radii reported here are circularized, $r_e = \sqrt{ab}$.
\subsection{Dynamical Masses}
\label{subsec:mdyn}
Combining the size and velocity dispersion measurements we are now
able to estimate dynamical masses using the following expression:
\begin{equation}
M_{\rm dyn}=\frac{\beta(n)~ \sigma_{e}^2~r_e }{G}.
\label{eq:mdyn}
\end{equation}
Here $\beta(n)$ is an analytic expression as a function of the
S\'ersic index, as described by \citet{cappellari2006}:
\begin{equation}
\beta(n) = 8.87 - 0.831n + 0.0241n^2.
\label{eq:kn}
\end{equation}
This is computed from theoretical predictions for $\beta$ from
spherical isotropic models described by the S\'ersic profile, for
different values of $n$, and integrated to one $r_e$
(cf. \citealt{bertin2002}). The use of a S\'ersic-dependent virial
constant $\beta(n)$ gives a better correspondence between $M_{dyn}$
and $M_{*}$ for galaxies in the SDSS \citep{taylor2010b}. This does
require however, that the total stellar masses are also derived using
the luminosity of the derived S\'ersic profile. Thus we correct our
total stellar mass, as derived from the total magnitude as given in
the catalogs (measured with Sextractor), to the total magnitude from
the S\'ersic fit. We note that the values for $\beta$ that we find
are all close to 5, a value often used in the literature
(e.g.,\citealt{cappellari2006}). Our dynamical masses and corrected
stellar masses are given in Table \ref{tab:sample_results}.
\section{Compilation of Kinematic Studies}
\label{sec:compilation}
In order to study the structural evolution of quiescent galaxies,
we combine the results from different kinematic studies at
various redshifts. Where possible, we apply similar corrections as
described above.
\begin{figure}
\epsscale{1.1}
\plotone{fig6.eps}
\caption{Comparison of the stellar mass vs. the
dynamical mass. Gray squares are non-star-forming galaxies from
the SDSS. Different symbols are from a compilation of high-redshift
galaxies as described in Section \ref{sec:sample_highz}. The dashed
line is for equal dynamical and stellar mass. Low-redshift galaxies
are all below the line, as is expected given the contribution of dark
matter. All our high-redshift galaxies have dynamical masses that are
close to the stellar mass. This suggests that the stellar
mass measurements at high-redshift are robust for passive galaxies.}
\label{fig:mdyn_mstar}
\end{figure}
\begin{figure*}
\epsscale{1.1}
\plotone{fig7.eps}
\caption{(a) Stellar mass divided by the dynamical
mass vs. redshift. Galaxies below the line have dynamical mass
greater than the stellar mass, above the line is the non-physical
regime. For SDSS galaxies with stellar mass $>10^{11} M_{\odot}$, we
find that they have a median $M_*/M_{dyn}$ of 0.59. Up to redshift
$z=1.5$ we find a similar slightly lower median value ($\sim0.5$), but
it rapidly increases at $z>1.5$ with median a of $M_{*}/M_{dyn} =
0.9$. The solid line is the best-fit $M_{*}/M_{dyn}\propto
(1+z)^{0.17\pm0.11}$. We caution that this result might be biased due
to the selection effects as explained in Section \ref{sec:properties},
and relatively large measurement errors. (b) Stellar
mass divided by the dynamical mass vs. the evolution in the
effective radius at fixed dynamical mass. Galaxies that have small
effective radii at fixed dynamical mass also show higher ratios of
$M_{*} / M_{dyn}$, although there is a significant scatter. The solid
line is the best-fit $M_{*} / M_{dyn} \propto \left(r_e(z) \, / \,
r_e(z\sim0.1)\, \right)^{0.16\pm0.10}$.}
\label{fig:redshift_delta_mass}
\end{figure*}
\subsection{Low-Redshift Sample}
\label{sec:sample_sdss}
At low redshift we select galaxies from the SDSS DR7. Stellar masses are based on
MPA\footnote{http://www.mpa-garching.mpg.de/SDSS/DR7/} fits to the
photometry following the method of \citet{kauffmann2003}, and
\citet{salim2007}. Star formation rates (SFRs) are based on
\citet{brinchmann2004}. Structural parameters are from the NYU
Value-Added Galaxy Catalog (NYU-VAGC, \citealt{blanton2005}). For all
galaxies, velocity dispersions were aperture corrected as described in
Section \ref{subsec:dispersions}, and stellar masses are calculated
with a \citet{chabrier2003} IMF. We furthermore correct the stellar
masses using the total magnitude from the best S\'ersic fit. All
dynamical masses were derived using Equation \ref{eq:mdyn}. For making
an accurate comparison between low- and high-redshift galaxies, we
only select non-star-forming galaxies, i.e sSFR $ < 0.3/t_H$ (see
\citealt{williams2009}), where $t_H$ is the age of the universe at the
given redshift.
\subsection{Intermediate- and High-redshift Sample}
\label{sec:sample_highz}
Our high redshift sample consists of a collection of both optical and
NIR spectroscopic studies of individual
galaxies. \citet{vanderwel2008} present a sample of quiescent galaxies
at $z\sim1$, which itself is a compilation of three studies in the
following fields: Chandra Deep Field South (CDF-S;
\citealt{vanderwel2004}; \citeyear{vanderwel2005}), the Hubble Deep
Field North (HDF-N; \citealt{treu2005a}; \citeyear{treu2005b}), and
cluster galaxies in MS 1054-0321 at $z=0.831$ \citep{wuyts2004}. We
derive stellar masses for this sample by running the stellar
population code FAST on available catalogs, i.e., FIREWORKS
\citep{wuyts2008} for the CDS-S, \citet{skelton2012} for the HDF-N,
and FIRES \citep{forsterschreiber2006} for MS 1054-0321. For CDF-S and
HDF-N the stellar masses are corrected using the total magnitude from
the best $n=4$ fit to be consistent with the structural parameters
from \citet{vanderwel2008}. For MS 1054-0321, we use structural
parameters and stellar mass corrections based on the results by
\citet{blakeslee2006}, who fit S\'ersic profiles with $n$ as a free
parameter. We note that \citet{martinez2011} also study a sample of
four $z\sim1$ galaxies, but find dynamical masses that are
significantly lower than their stellar masses, in contrast to the
result by \citet{vanderwel2008}.
Other high-redshift results included here are from \citet{newman2010}
and \citet{bezanson2012}, who use the upgraded red-arm of LRIS on Keck
to obtain UV rest-frame spectra of galaxies at $z \sim1.3$ and $z \sim
1.5$ respectively. Velocity dispersions for two galaxies at $z=1.41$
are presented by \citet{cappellari2009}, and have been observed with
VLT-FORS2 (see also \citealt{cenarro2009}). Using NIR spectrographs,
\citeauthor{onodera2012} (\citeyear{onodera2012}, Subaru-MOIRCS) and
\citeauthor{vandokkum2009a} (\citeyear{vandokkum2009a}, GNIRS)
obtained velocity dispersions for two galaxies at $z=1.82$ and
$z=2.186$. Similar to the current study, \citet{toft2012} study
UDS-19627 using VLT X-Shooter. Dynamical masses were derived using to
Equation \ref{eq:mdyn}. Note that for the studies of
\citet{cappellari2009}, \citet{onodera2012}, \citet{vandokkum2009a},
and \citet{toft2012} no stellar mass corrections were applied due to
the absence of the necessary information. All structural and kinematic
properties of our high-redshift sample are listed in Table
\ref{tab:sample_results}.
\section{Are stellar masses reliable?}
\label{sec:mass_comparison}
The main goal of this paper is to see whether the stellar masses at
$z\sim2$ are reliable. Here we compare our stellar masses, as derived
from the spectra and photometry, to our dynamical masses, which are
derived from effective radii and stellar velocity dispersions (Figure
\ref{fig:mdyn_mstar}). Gray squares represent the density of
non-star-forming, low-redshift galaxies from the SDSS as described in
Section \ref{sec:sample_sdss}. Other symbols are the high-redshift
studies as described in Section \ref{sec:sample_highz}. The one-to-one
relation for $M_{dyn}$ and $M_*$ is indicated by the dashed line. Note
that the region above the line is nonphysical with stellar masses
being higher than the dynamical mass.
Most $z>1.5$ galaxies in this sample are very massive, in the range
$11.2 <$ log M$_{dyn}/$M$_{\odot} < 11.8$. At all redshifts, stellar
and dynamical masses are tightly correlated and dynamical mass, which
includes baryonic and dark matter, is on average higher than stellar
mass. Thus, we infer that the stellar masses of our galaxies are
broadly correct, and that the apparent size evolution of massive
galaxies in photometric studies cannot be explained by errors in the
photometric masses (see also \citealt{vanderwel2008}).
Figure \ref{fig:redshift_delta_mass}a shows the ratio of the stellar
and dynamical mass as function of redshift for all galaxies with
stellar mass $>10^{11} M_{\odot}$. We see that the average ratio at
low-redshift for massive galaxies is a factor of 0.59 with a scatter
of 0.12 dex. We note that For MS 1054-0321, the ratio of the stellar
to dynamical mass are slightly higher as compared to low redshift
galaxies. Up to redshift $z\sim1.5$ we find a similar value
($\sim0.5$) with similar scatter, but at higher redshift, the ratio
seems to decline. For galaxies at $z>1.5$ we find a median ratio of
$M_{*} / M_{dyn} = 0.9$. We quantify the evolution in this ratio by
fitting the relation:
\begin{equation} M_{*} / M_{dyn} \propto (1+z)^{\alpha}.
\label{eq:delta_mass_redshift}
\end{equation}
We use a linear least-squares fit in log-log space using the function
$MPFIT$ \citep{markwardt2009}, which takes the errors on each
individual data point into account. We find a best-fitting value of $
\alpha = 0.17 \pm 0.11$, which is shown as the solid black line in
Figure \ref{fig:redshift_delta_mass}a. The uncertainty is derived from
1000 bootstrap simulations, where we draw data points randomly from
the sample. The quoted error is the standard deviation from the
resulting distribution of points. Even though the fit is statistically
significant at the $1- \sigma$ level, due to the relatively large
measurements errors as compared to low redshift, and the possible
selection bias of the high-redshift samples, we are cautious to draw
any strong conclusions from this result.
\begin{figure*}
\epsscale{1.15}
\plotone{fig8.eps}
\caption{Structural comparison of our spectroscopic sample to the full
quiescent galaxy population. (a) Effective radii
vs. mass for low- and high-redshift galaxies. Gray squares are
non-star-forming $z\sim0$ galaxies from the SDSS, with the dashed line
the best-fit to Equation \ref{eq:mstarre}. High-redshift quiescent
galaxies ($1.4 < z< 2.1$) from CANDELS (\citealt{grogin2011};
\citealt{koekemoer2011}) are shown as small green circles, together
with our spectroscopic sample shown as red circles, and the sample by
\citet{bezanson2012} shown as cyan circles. At fixed mass, our
high-redshift galaxies have smaller effective radii, similar to what
has been found by many studies. (b) Evolution in
effective radius at fixed stellar mass, thus corrected for the $M_* -
r_e$ relation in panel (a). Big green squares are the median
effective radii in bins of redshift for the CANDELS data. The solid
line is the best-fit $r_e \propto (1+z)^{-1.02\pm0.05}$. At similar
redshift, we find that our sample and that of \citet{bezanson2012} are
mostly below this fit, indicating that our samples our biased towards
smaller effective radii. (c) Similar to the
panel (b), but now divided by $(1+z)^{-1.02}$ for a better comparison of
our spectroscopic sample to the CANDELS data. When comparing the
median of our sample to the binned median of other quiescent galaxies
at similar redshift, we find smaller effective radii by a factor
$\sim1.28$. This might be explained by our selection which is based on
aperture magnitude, which tends to be biased towards smaller galaxies.
}
\label{fig:selection_effects_mre}
\end{figure*}
It is tempting to speculate that the evolution in $M_*/M_{dyn}$ might
have been caused by a decrease in the dark matter fraction as a
function of redshift. For galaxies growing in size over time, the
dark matter fraction within $r_e$ will also increase. As the dark
matter profile is less steep than the stellar mass profile, the dark
matter to stellar mass fraction increases with radius, in a similar
fashion as shown here (e.g.,\citealt{hopkins2009a}). If so, this could
also indicate that the IMF at high-redshift is very similar to the IMF
at low-redshift.
Figure \ref{fig:redshift_delta_mass}b shows $M_{*} / M_{dyn}$ vs.
the evolution of the effective radius at fixed dynamical mass (see
Section \ref{sec:size_evolution} and Figure
\ref{fig:redshift_size}). Although there is significant scatter, we do
find the galaxies with high $M_{*} / M_{dyn}$ also tend to have
smallest size at fixed dynamical mass. Galaxies that are closest to
the present-day mass-size relation (dashed vertical line) show lower
ratios of stellar to dynamical mass.
We test this claim by using the the following equation:
\begin{equation}
M_{*} / M_{dyn} \propto \left(r_e(z) \, / \, r_e(z\sim0.1)\, \right)^{\alpha}.
\label{eq:delta_mass_re}
\end{equation}
We find $\alpha = -0.16 \pm 0.10$, where the error is determined in a
similar way as described for Equation \ref{eq:delta_mass_redshift} using the bootstrap method
Furthermore, we use the Spearman's rank test on the intermediate- and
high-redshift data. This confirms that there is an anti-correlation
with a probability of 96$\%$. The best-fitting Spearman's rank
correlation coefficient is $-0.28 \pm 0.08$. Even though we find a
weak anti-correlation, this agrees with the idea that the decreasing
ratio of $M_{*} /M_{dyn}$ with time might be correlated to the size
growth of massive galaxies.
\section{Structural evolution of quiescent galaxies}
\label{sec:evolution}
In this section, we will re-examine the structural evolution
of massive quiescent galaxies but now using dynamical measurements.
\subsection{Bias towards Compact Galaxies}
\label{sec:bias}
As noted in Section \ref{sec:target_selection}, this sample is biased
towards young quiescent galaxies. Therefore, we will first investigate
whether our sample and that of \citet{bezanson2012} are biased in size
as compared to other high-redshift galaxies. We gathered structural
properties of galaxies from two studies that use CANDELS data in the
UDS and GOODS-South fields (\citealt{patel2012};
\citealt{szomoru2012}). We compare to a subsample of these galaxies
that are determined to be quiescent from their rest-frame U-V and V-J
colors (see e.g.,Figure \ref{fig:selection_effects_sp}a). When
comparing the effective radii vs. the stellar mass in Figure
\ref{fig:selection_effects_mre}$a$, we find that our galaxies (red
circles) and those of \citeauthor{bezanson2012} (\citeyear{bezanson2012}, cyan circles) are in
general more compact as compared to the high-redshift CANDELS galaxies
(small green circles).
For low-redshift galaxies we parametrize the mass-size relation by:
\begin{equation}
r_e=r_c \left( \frac{M_{\rm *}}{10^{11}M_{\odot}}\right)^b
\label{eq:mstarre}
\end{equation}
(\citealt{shen2003}; \citealt{vanderwel2008}). Using a linear
least-squares fit in log-log space, we find best fitting values of
$r_c=4.32~$kpc and $b=0.62$. This is slightly different from the fit
by \citet{shen2003} who find $r_c=4.16~$kpc and $b=0.56$. The
difference may be explained by different selection criteria, and their
use of an older release version of SDSS. Figure
\ref{fig:selection_effects_mre}$b$ shows the evolution in effective
radius, by comparing galaxies with similar mass at different
redshifts. Using both the SDSS and the CANDELS data, we examine the
amount of evolution in size by fitting the following relation:
\begin{equation}
r_e \propto (1+z)^{\alpha},
\label{eq:delta_re_redshift}
\end{equation}
We find $\alpha = -1.02 \pm 0.05$ (linear fit in log-log space). Our
spectroscopic targets and those of \citet{bezanson2012} are mostly
below this best-fit relation, being smaller by a factor of $\sim1.28$
as compared to median in redshift bins (big green squares). This is
especially clear from Figure \ref{fig:selection_effects_mre}$c$, where
we correct for the evolution in size.
This bias might be explained by the method our targets are
selected. As our selection is based on the magnitude within a fixed
aperture of $1''.5$, instead of the total magnitude, we create a bias
towards compact galaxies. For galaxies with similar total magnitudes,
the smaller galaxies will be brighter within a photometric aperture,
and thus make it into our sample. In what follows, we correct for
this bias by increasing our sizes and those of \citet{bezanson2012} by
a factor of $1.28$, and decreasing the velocity dispersion by a factor
of $\sqrt{1.28}$.
\begin{figure*} \epsscale{1.15} \plotone{fig9.eps}
\caption{Redshift evolution of the effective radius of passive
galaxies. (a) Size vs. dynamical mass. The dashed line
is the best-fit derived using Equation \ref{eq:mdre} for
non-star-forming SDSS galaxies. At fixed dynamical mass, we find that
$z\sim2$ galaxies (red circles) are smaller by a factor $\sim3$
compared to low-redshift galaxies. (b) Evolution of
the effective radius at fixed dynamical mass vs. redshift. The
solid line is the best-fit $r_e \propto (1+z)^{-0.97\pm0.1}$, similar
to what has been found by other stellar kinematic studies at
high-redshift. Similar to Figure \ref{fig:selection_effects_mre}, our
sample is below the best-fit line to the entire high-redshift
sample. (c) Evolution of the effective radius using
an evolving mass function at constant number density. We now compare
galaxies at high-redshift, to more massive galaxies at low-redshift,
assuming the mass evolves as $\Delta \log M/M_{\odot} \sim
0.15z$. This time, we find an even stronger evolution, with $r_e
\propto (1+z)^{-1.16\pm0.1}$. }
\label{fig:redshift_size}
\end{figure*}
\subsection{Evolution in Size}
\label{sec:size_evolution}
In Figure \ref{fig:redshift_size}a we plot effective radius vs.
dynamical mass. Symbols are the same as in Figure
\ref{fig:redshift_delta_mass}. For the low-redshift galaxies we
parametrize the mass-size relation according to the following equation
\begin{equation}
r_e=r_c \left( \frac{M_{\rm dyn}}{10^{11}M_{\odot}}\right)^b,
\label{eq:mdre}
\end{equation}
and find $r_c=3.23~$kpc and $b=0.56$ (dashed line). This is in good
agreement with $b=0.56$ and $r_c=3.26~$kpc as found by
\citet{vanderwel2008}. At fixed dynamical mass, we see that all our
galaxies have smaller effective radii as compared to
low-redshift. This finding is further illustrated in Figure
\ref{fig:redshift_size}b, where we compare the effective radii at
fixed dynamical mass to the mass-size relation at $z\sim0$. The solid
line is the best-fit as described by equation
\ref{eq:delta_re_redshift}, with $\alpha=-0.97\pm0.1$. This result is
in agreement to with what has been found in previous kinematical studies
(\citealt{vanderwel2008}; \citealt{newman2010}). The scatter between
different studies is considerable, with the work by
\citet{vandokkum2009a} having the largest size difference while that by
\citet{onodera2012} having the smallest. Our sample falls in between
these two extremes, i.e.,we find smaller sizes as compared to
\citet{onodera2012}, but larger effective radii than
\citet{vandokkum2009a}.
Instead of comparing galaxies sizes at fixed dynamical mass, we will
now take into account that galaxies do grow in mass
(e.g.,\citealt{patel2012}). In \citet{vandokkum2010} they find that,
for a sample selected at a constant number density, the stellar mass
evolves as
\begin{equation} log M_{n}/M_{\odot} = 11.45-0.15z
\label{eq:mass_evolution}.
\end{equation}
The number density on which this result is based, $n = 2 \times
10^{-4}$Mpc$^{-3}$, corresponds to an average mass of $\log
M_*/M_{\odot} \sim 11.2$ at $z\sim 2$, similar to our sample. Assuming
that the mass evolves as $\Delta \log M/M_{\odot} \sim 0.15z$, we will
compare effective radii for galaxies at different redshifts. For
example, a galaxy with $\log M_{dyn} /M_{\odot}= 11$ at $z\sim 2$ will
be compared with a $z\sim 0$ galaxy with $\log M_{dyn}/M_{\odot} =
11.3 $.
However, the evolution in stellar mass is determined for a complete
sample of both star forming and quiescent galaxies, while in this
paper we only look at quiescent galaxies. Therefore, we assess whether
the evolution in size at constant cumulative number density is
different for the quiescent population as compared to the full
population. This was already done for galaxies in CANDELS by
\citet{patel2012} at $n_{cum} = 1.4\times10^{-4} \rm{Mpc}^{-3}$, which
corresponds to a median mass of $\log M_*/ M_{\odot} \sim 10.9$ at
$z\sim1.8$. The sample studied here, however, has a median mass of
$\log M_*/M_{\odot} \sim 11.2$ at $z\sim1.8$, which corresponds to
$n_{cum} = 2.5\times10^{-5} \rm{Mpc}^{-3}$. Thus, we repeat the
analysis by \citet{patel2012}, but now using our CANDELS sample
(Section \ref{sec:bias}) at this lower constant cumulative number
density. Our results are similar to \citet{patel2012}, i.e.,the
quiescent population has smaller effective radii as compared to full
population, but the difference in size between the quiescent and
star-forming population is slightly smaller at $z<1.8$ as compared to
\citet{patel2012}. The difference is due to the fact that at our lower
constant cumulative number density, we find a higher fraction of
quiescent galaxies. Therefore, the effective radii of the full
population will be closer to the effective radii of the quiescent
population, as compared to \citet{patel2012}. To correct for this
difference in size, we increase the effective radii of all the
quiescent galaxies in our combined sample by the size difference of
the quiescent population as compared to the full. We correct each
galaxy individually by finding the correction factor at this
particular redshift. Below $z<1.8$, the median correction factor is
$\sim 1.05$, while at $z\sim2.1$ the correction factor is $\sim 1.6$.
Figure \ref{fig:redshift_size}c shows the evolution in size at fixed
number density as a function of redshift. Not surprisingly, the
evolution in effective radii is more extreme, as we are now comparing
$z\sim 2$ galaxies to more massive, and therefore bigger galaxies at
$z\sim0$. Using Equation \ref{eq:delta_re_redshift}, we find that
$\alpha = -1.16\pm0.1$ provides the best fit. In conclusion, assuming
that galaxies evolve in both mass and size, we find that the effective
radii have to grow by a factor $\sim 4$ from $z\sim 2$ to the present
day.
\begin{figure*} \epsscale{1.15}
\plotone{fig10.eps}
\caption{Redshift evolution in stellar velocity dispersion within one
effective radius ($\sigma_e$). (a) $\sigma_e$ vs.
dynamical mass. At fixed dynamical mass, we find that our galaxies
have higher velocity dispersion as compared to low-redshift
galaxies. (b) Evolution of velocity dispersions at
fixed dynamical mass vs. redshift. We find the $\sigma_e \propto
(1+z)^{0.49\pm0.08}$, i.e at fixed dynamical mass the velocity
dispersion decreases by a factor of $\sim1.7$ from $z\sim2$ to the
present day. (c) Evolution of the velocity
dispersion for an evolving mass function at constant number
density. Opposite to the evolution in size, we now find a milder
evolution of $\sigma_e \propto (1+z)^{0.31\pm0.08}$.}
\label{fig:redshift_sigma}
\end{figure*}
\subsection{Evolution in Velocity Dispersion}
\label{sec:sigma_evolution}
In Figure \ref{fig:redshift_sigma}a we compare the stellar velocity
dispersion within one $r_e$ vs. the dynamical mass for both low-
and high-redshift samples. The dashed line is the parametrization of
the $\sigma_e - M_{dyn}$ relation for low$-z$ galaxies using the
following equation:
\begin{equation} \sigma_e=\sigma_c \left( \frac{M_{\rm
dyn}}{10^{11}M_{\odot}}\right)^b.
\label{eq:mdsig}
\end{equation} We find that $\sigma_c=148.9~$km s$^{-1}$ and $b=0.24$.
Our high-redshift sample is clearly offset from low-redshift galaxies
in the SDSS, i.e.,at fixed mass they have higher velocity
dispersions. Comparison of the velocity dispersion at fixed dynamical
mass, as seen in \ref{fig:redshift_sigma}b, shows a clear evolution in
$\sigma_e$, such that velocity dispersion decreases over time. From
this figure the increase in accuracy for the velocity dispersion
measurements of this study, as compared to other studies at similar
redshift, is also clearly noticeable. Again we use the following
simple relation to quantify the amount of evolution:
\begin{equation} \sigma_e \propto (1+z)^{\alpha},
\label{eq:delta_sigma_redshift}
\end{equation}
and find that $\alpha = 0.49\pm0.08$. From $z\sim2$ to $z\sim0$ the
stellar velocity dispersions decrease by a factor $\sim 1.7$. Again,
we note that we apply a correction to the velocity dispersions in our
sample, in order to correct for the bias towards more compact galaxies
(section \ref{sec:bias}).
If we now compare low- and high-redshift galaxies using an evolving
mass function as described above, we find that the velocity dispersion
decreases less with cosmic time than when compared at fixed dynamical
mass. Here, we also take into account that quiescent galaxies at
constant cumulative number density are smaller as compared to the full
sample, and therefore they have higher velocity dispersions. Similar
to as described above, we therefore corrected the velocity dispersions of
all galaxies in our combined sample. At $z<1.8$, velocity dispersions
on average decrease by a factor of $\sim \sqrt{1.05}$, while at
$z\sim2.1$ they decrease by a factor of $\sim \sqrt{1.6}$. If we use
equation \ref{eq:delta_sigma_redshift}, we find that
$\alpha=0.31\pm0.08$. In other words, the velocity dispersion within
one $r_e$ decreases by a factor $\sim 1.4$ from $z\sim2$ to
present-day.
\begin{figure*}
\epsscale{1.15}
\plotone{fig11.eps}
\caption{Redshift evolution of the central and effective densities, as
calculated according to Equations \ref{eq:rho1} and \ref{eq:rhoe}. Top
row shows the results for the effective mass density, while the bottom
row shows the mass density within 1 kpc. \textit{Left panels:} the
density within $r_e$ vs. dynamical mass (a). We find
that low-redshift galaxies show a large scatter at fixed dynamical
mass. Still, our high-redshift galaxies have higher densities within
$r_e$ at fixed mass. In contrast to $\rho_e$, the density within 1 kpc
vs. dynamical mass (d) shows very little scatter, and
we find only a small difference between low- and high-redshift
galaxies. \textit{Middle panels:} evolution in the density at fixed
dynamical mass vs. redshift. We find a strong evolution for the
effective density (b) with $\rho_e \propto
(1+z)^{2.86\pm0.15}$, or a decrease by a factor of $\sim2$ from
high-redshift to $z\sim0$. For the central density, however, we only
find that $\rho_{1 kpc} \propto (1+z)^{0.74\pm0.15}$, or a decrease of
$\sim 2.3$. \textit{Right panels:} evolution of the density with an
evolving mass at constant number density. Similar to the
effective radii, we find a stronger evolution for the effective
density, i.e.,$\rho_{e} \propto (1+z)^{2.93\pm0.15}$, while the central
density requires very little to no evolution ($\rho_{1 kpc} \propto
(1+z)^{0.42\pm0.15}$).}
\label{fig:redshift_rho}
\end{figure*}
\subsection{Evolution in Mass Density}
\label{sec:density_evolution}
Next, we will focus on the central and effective mass densities using a
similar approach as described in \citet{saracco2012}. In short, using
the intrinsic S\'ersic profile we can calculate the fraction of the
luminosity that is within 1 kpc as compared to the total
luminosity. For a S\'ersic profile this ratio is given by
\citep{ciotti1991}:
\begin{equation} \frac{L_{1kpc}}{L_{tot}}
=\frac{\gamma(2n,x)}{\Gamma(2n)}.
\label{eq:lum_ratio}
\end{equation}
Here, $\Gamma (2n)$ is the complete gamma function, $\gamma (2n,x)$
the incomplete gamma function, $x=b_n (r_{1kpc} / r_{e})^{1/n}$, with
$b_n = 1.9992n-0.3271$. Using this ratio we can now calculate the
dynamical mass within one 1 kpc and within $r_e$ from the total mass:
\begin{equation} {M_{1kpc}} =\frac{L_{1kpc}}{L_{tot}} M_{dyn}.
\label{eq:mass1}
\end{equation}
Here we assume that the dynamical mass profile follows the light
profile, and furthermore that the mass-to-light ratio of the galaxy is
radially constant. The detection of small color gradients in our
galaxies indicates, however, that this is not the case, but the effect
on the derived densities is small (\citealt{saracco2012}; see also
\citealt{szomoru2012}). Finally, the densities are calculated as
follows:
\begin{equation} {\rho_{1kpc}} = \frac{M_{1kpc} }{4/3 \pi r_{1kpc}^3},
\label{eq:rho1}
\end{equation}
and
\begin{equation} {\rho_{e}} = \frac{0.5 M_{dyn}}{4/3 \pi r_e^3 }.
\label{eq:rhoe}
\end{equation}
As for $r_e$ and $\sigma_e$, we now compare the density as a function
of dynamical mass (see Figure \ref{fig:redshift_rho}). The top row
shows the results for the mass density within one effective radius,
while the bottom row compares the central density within 1 kpc. The
first thing to notice is the large scatter for low-redshift galaxies
when looking at $\rho_e$ vs. $M_{dyn}$, while $\rho_{1kpc}$ vs.
$M_{dyn}$ shows a tight relation. The density-mass relation can be
parametrized by:
\begin{equation} \rho=\rho_c \left( \frac{M_{\rm
dyn}}{10^{11}M_{\odot}}\right)^b.
\label{eq:mrho}
\end{equation}
For the density within $r_e$ we find $\rho_{c,e} = 4.7\times10^8
$M${_\odot} $kpc$^{-3}$ and $b_e=-0.68\pm 0.15$, and for the central
density within 1 kpc $\rho_{c,1kpc} = 6.6\times10^9 $M${_\odot}
$kpc$^{-3}$ and $b_{1kpc}=0.56$.
When we compare the galaxies in our high-redshift sample to galaxies in
the SDSS, we find that they have higher densities within
$r_e$. Comparison at equal dynamical mass shows that the effective
densities are higher by a factor of $\sim50$ (Figure
\ref{fig:redshift_rho}b) for our sample. The same comparison, but now
for the central density within 1 kpc, reveals only mild evolution,
approximately a factor of $\sim3$ from $z\sim2$ to the present. When fitting
\begin{equation} \rho \propto (1+z)^{\alpha},
\label{eq:delta_rho_redshift}
\end{equation} we find that $\alpha_e = 2.86\pm0.15$, while
$\alpha_{1kpc} = 0.74\pm0.15$.
Instead of comparing galaxies at fixed mass, we again take into
consideration that galaxies evolve in mass when comparing low- and
high-redshift galaxies. Again, we correct for the fact that quiescent
galaxies at constant cumulative number density are smaller as compared
to the full sample. This time, we find that $\rho_e$ evolves even
faster as compared to the equal mass comparison $\alpha_e =
2.93\pm0.15$. The density within 1 kpc, however, requires a decrease
less than a factor of $\sim2$, with $\alpha_{1kpc} = 0.42\pm0.15$,
from $z\sim2$ to the present
\section{Discussion}
\label{sec:inside_out}
In the previous section we have found that in order for the
high-redshift galaxies in our sample to evolve into typical
present-day early-type galaxies, strong structural evolution is
required. Effective radii need to increase, and the velocity
dispersion within $r_e$ has to decrease. The density within the
effective radius has to decrease by more than an order of
magnitude. However, the central density can remain almost the same,
consistent with inside-out growth.
The dominant physical mechanism for this structural evolution is still
a subject of ongoing debate. Size growth dominated by major mergers
seems to be unlikely as it would increase the masses too much, which
would make extremely massive galaxies too common in the local
universe. As the mass and size increase at approximately at the same
rate in major mergers, the galaxies would also remain too compact for
their mass. Minor merging could offer a solution to the problem, as it
can grow a galaxy in effective radius ($r_e$) steeper than $r_e
\propto M_*$ (\citealt{villumsen1983}; \citealt{naab2009};
\citealt{bezanson2009}; \citealt{hopkins2009b} ). In this scenario,
the observed compact high-redshift galaxies may simply be the cores of
local massive early-type galaxies, which grow inside-out by accreting
(smaller) galaxies, and thus assemble a significant part of their mass
at later times (\citealt{vanderwel2009}; \citealt{oser2010}). In this
section we will examine whether dry minor merging agrees with our
findings.
From a simple estimate, based on the virial theorem,
\citet{bezanson2009} predict how the effective radii changes if a
massive galaxy undergoes a series of minor mergers. With only eight
1:10 mergers, the effective radii can grow by a factor of $\sim5$
while only having the mass increase by a factor of $\sim~2$. This is
also described by \citet{naab2009}, who state that if an initial
system undergoes a mass increase by a factor of 2 due to accretion of
very small systems, then the final radius of the system is four times
larger, the velocity dispersion is reduced by a factor of 2, and the
density is reduced by a factor of 32. This analytic prediction is
confirmed by their hydrodynamical cosmological simulation and
consistent with the observational size evolution as presented here.
Using hydrodynamic simulations of galaxy mergers,
\citet{hopkins2009a}, also find evidence for size evolution. When they
compare the effective radii of quiescent galaxies at fixed mass, they
find an evolution in size of $r_e \propto (1+z)^{-0.48}$ for galaxies
with $\log M_*/M_{\odot} = 11$, which is weaker than found by this
study and many others. \citet{oser2012} find a size evolution in their
hydro simulation, which is much stronger: $r_e \propto (1+z)^{-1.44}$,
on the high side of current observational results.
\citet{oser2012} find a similar evolution in velocity dispersion of
$\sigma \propto(1+z)^{0.44}$, to that found in this work. In contrast,
\citet{hopkins2009a}, predict that high-redshift quiescent galaxies
have roughly the same or at most a factor $\sim 1.25$ larger velocity
dispersions.
Evolution of the density is also discussed in both
\citet{bezanson2009} and \citet{hopkins2009b}. Based on photometric
data, both studies find that while the density within one effective
radius is higher at high-redshift, the central density of
high-redshift galaxies is very similar to local massive
ellipticals. From hydro simulations, \citet{naab2009} show that the
central density within 1kpc decreases by a factor of 1.5 from $z = 2$
to $z = 0$, caused by dynamical friction from the surviving cores of
the infalling systems. Similarly, \citet{oser2012} show that the central
density evolves only weakly, while the density within $r_e$ decreases
rapidly by more than an order of magnitude, in good agreement with
what we find here. From a study of central galaxies in three
$\sim10^{13}\,$\msun galaxy groups, simulated at high resolution in
cosmological hydrodynamical simulations, \citet{feldmann2010} come to
the same conclusion. They find that the effective density of these
galaxies decreases by 1-2 orders of magnitude between $z=1.5$ and
$z=0$, while the density within 2 kpc stays roughly constant.
This is in contrast with the findings of \citet{saracco2012}, who find
no evidence for higher mass densities within one effective radius when
comparing their $z\sim1.5$ galaxies to low-redshift cluster
galaxies. Furthermore, the large scatter that they observed in the
effective density and the apparent evolution, is simply due a peculiar
analytic feature in the S\'ersic profile.
In Figure \ref{fig:redshift_delta_mass}a we show that the ratio of
$M_{*}/M_{dyn}$ appears to have evolved from $z\sim2$ to $z\sim0$. As
compared to SDSS galaxies with $\log M_*/M_{\odot} >11$, we find that
the median $M_{*}/M_{dyn}$ is higher by $50\%$ at $z>1.5$, and that
$M_{*}/M_{dyn} \propto (1+z)^{0.17\pm 0.011}$. However, this result
is uncertain due to the selection effects inherent in this sample and
large measurement errors in both masses. We note that this effect is
predicted by simulations; as the effective radius of a galaxy grows,
the dark matter fraction within $r_e$ will also
increase. \citet{hopkins2009a} predicts evolution by a factor of
$\sim1.25$ for galaxies with $\log M_*/M_{\odot} = 11$, with the
effect increasing with stellar mass. \citet{Hilz2012} also find a
strong evolution in the dark matter fraction in their
hydro-simulation, and predict that quiescent galaxies at $z\sim2$ have
lower dark matter fractions ($\gtrsim 80\%$). They mention that it is
mainly driven by the strong size increase, which therefore probes a
larger region that is dominated by dark matter.
\section{summary and conclusion}
\label{sec:conclusion}
In this paper, we present deep UV-NIR spectroscopy of five massive
($>10^{11}$ \msun) galaxies at $z\sim2$, using X-Shooter on the
VLT. These spectra enable us to measure stellar velocity dispersions
with higher accuracy than done before at this redshift: we triple the
sample of $z>1.5$ galaxies with well constrained ($\delta\sigma<100$
km s$^{-1}$) velocity dispersion measurements. We find that the
stellar velocity dispersions are high (290-450 km s$^{-1})$ compared
to equal-mass galaxies in the SDSS.
We combine these kinematic results with size measurements using GALFIT
on HST-WFC3 $H_{160}$ and UDS K-band imaging, and use these
measurements to derive dynamical masses. Stellar masses are obtained
from SPS modeling on the VIS-NIR spectra in
combination with the available broadband and medium-band data. The
SPS-modeling shows that our galaxies have ages ranging from 0.5 to 2 Gyr,
and show no signs of on-going star formation.
We find good correspondence between the dynamical and stellar masses,
with the dynamical mass being higher by \mbox{$\sim15\%$}. Our results
suggest that stellar mass measurements for quiescent galaxies at
high-redshift are robust.
We complement our results with stellar kinematic results from other
studies at low and high redshift to study the structural evolution of
massive quiescent galaxies. At fixed dynamical mass, we find that the
effective radius increases by a factor of $\sim2.8$, while the
velocity dispersion decreases by a factor of $\sim1.7$ from $z\sim2$
to the present day. Furthermore, we study how the mass density within
$r_e$ and 1 kpc evolves with time. We find a strong decrease of the
mass density within one effective radius (factor of $\sim21$), while
it only decreases mildly within 1 kpc (factor of $\sim2.3$). Instead
of comparing galaxies at fixed dynamical mass, we also use an evolving
mass limit as defined by fixed number density. By accounting for
concurrent mass growth in our comparison of high- and low-redshift
galaxy populations, we find an even stronger evolution in galaxy sizes
(factor of $\sim4$). We find that velocity dispersion decreases less
dramatically with time, differing by only a factor of $\sim1.4$
between $z\sim2$ and $z\sim0$. Finally, for the mass density within
$r_e$, we find a stronger evolution, but interestingly, the mass
density within 1 kpc is consistent with no evolution. This finding
implies that massive quiescent galaxies grow inside out.
We examine if our results are compatible with the current idea of inside-out growth
through dry minor mergers. Our findings are qualitatively consistent with predictions
from hydrodynamical simulations which show similar evolution in size, velocity dispersion, and
mass density within one effective radius.
Finally, we find that even though the stellar masses are consistent
with the dynamical masses, the ratio of $M_{*}/M_{dyn}$ may slightly
decrease with time. This, too, is predicted by minor merging
simulations, which show that the size growth due to minor merging will
also change the fraction of dark matter as compared to the stellar
mass within an effective radius. This is due to the fact that the
dark matter profile is less steep than the stellar mass profile, and
thus the dark matter to stellar mass fraction increases with radius.
Despite the vastly improved accuracy of our derived dynamical masses
and stellar population parameters, the broader inferences of our study
are still limited by the small number of high-redshift galaxies with
such information. We have shown that our sample is biased towards
younger galaxies, compared to a stellar mass limited sample at
$z\sim2$, with smaller effective radii as compared to the full
population of quiescent galaxies at $z\sim2$. Only with a larger unbiased
sample of massive quiescent galaxies at high redshift can we start to
comprehend the final phase that massive galaxies go through in
becoming today's ellipticals.\\
\acknowledgments{
We thank the anonymous referee for the constructive comments which
improved the quality and readability of the paper. We also thank
Daniel Szomoru for providing his residual-correct code and the galaxy
sizes for GOODS-S CANDELS; Andrew Newman for providing the corrected
stellar masses; Shannon Patel for the structural parameters of
galaxies in UDS CANDELS; and Ivo Labb\'e for helpful comments
regarding the SFRs from 24 $\micron$flux. It is a pleasure to
acknowledge the contribution to this work by the NMBS
collaboration. We also thank Adam Muzzin for useful
discussions.\\ This research was supported by grants from the
Netherlands Foundation for Research (NWO), the Leids Kerkhoven-Bosscha
Fonds. Support for the program HST-GO-12167.1 was provided by NASA through
a grant from the Space Telescope Science Institute.\\ This work is
based on observations taken by the CANDELS Multi-Cycle Treasury
Program with the NASA/ESA HST, which is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract
NAS5-26555.\\ This publication also makes use of the Sloan Digital Sky
Survey (SDSS). Funding for the creation and distribution of the SDSS
Archive has been provided by the Alfred P. Sloan Foundation, the
Participating Institutions, the National Aeronautics and Space
Administration, the National Science Foundation, the U.S. Department
of Energy, the Japanese Monbukagakusho, and the Max Planck
Society. The SDSS Web site is http://www.sdss.org/. The SDSS
Participating Institutions are the University of Chicago, Fermilab,
the Institute for Advanced Study, the Japan Participation Group, Johns
Hopkins University, the Max Planck Institut fur Astronomie, the Max
Planck Institut fur Astrophysik, New Mexico State University,
Princeton University, the United States Naval Observatory, and the
University of Washington.}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,898
|
De Provinciale Statenverkiezingen 1916 waren Nederlandse verkiezingen die in juni 1916 werden gehouden voor de leden van Provinciale Staten in elf provincies. In Gelderland, Groningen en Noord-Brabant werden de verkiezingen gehouden op 14 juni, in Friesland, Overijssel, Utrecht en Zuid-Holland op 15 juni, in Drenthe, Limburg en Zeeland op 16 juni en in Noord-Holland op 19 juni.
Deze verkiezingen waren de laatste verkiezingen voor Provinciale Staten waarbij het districtenstelsel van toepassing was. Door een wijziging van de Grondwet in 1917 werd vanaf de verkiezingen in 1919 het stelsel van evenredige vertegenwoordiging gehanteerd.
Uitslagen
Landelijk overzicht
Uitslagen per provincie naar partij
Eerste Kamerverkiezingen
De leden van Provinciale Staten kozen op 11 juli 1916 bij Eerste Kamerverkiezingen in tien kiesgroepen 17 nieuwe leden van de Eerste Kamer.
De leden van Provinciale Staten kozen op 14 juni 1917 in elf kiesgroepen naar provincie een geheel nieuwe Eerste Kamer.
Provinciale Statenverkiezingen zijn daarmee behalve provinciaal ook nationaal van belang.
Nederlandse politiek in 1916
1916
Nederland
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,730
|
package com.kyxw007.kftpserver.ftp.command.handlers;
import com.kyxw007.kftpserver.ftp.controller.FtpContext;
import org.apache.log4j.Logger;
import java.io.IOException;
import java.io.OutputStream;
import java.io.PrintWriter;
/**
* Created by kyxw007 on 15/5/2.
*/
public class PwdCommandHandler implements CommandHandler {
private FtpContext ftpContext;
private static Logger logger = Logger.getLogger("KFTP:");
public PwdCommandHandler(FtpContext ftpContext) {
this.ftpContext = ftpContext;
}
@Override
public void execute(String cmd) {
try {
OutputStream os = ftpContext.getSocketClient().getOutputStream();
PrintWriter pw = new PrintWriter(os);
pw.println("257 \""+ftpContext.getPdir()+"\" is current directory.");
logger.info("(" + ftpContext.getUsername() + ") (" + ftpContext.getClientIp() + ")> 257 \"" + ftpContext.getPdir() + "\" is current directory.");
pw.flush();
} catch (IOException e) {
e.printStackTrace();
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,191
|
using System.Collections.Generic;
using RolfMichelsen.Dragon.DragonTools.IO.Disk;
using Xunit;
namespace RolfMichelsen.Dragon.DragonTools.test
{
public class MemoryDiskTest
{
private MemoryDisk disk = null;
private const int Heads = 1;
private const int Tracks = 2;
private const int Sectors = 2;
private const int SectorSize = 10;
public MemoryDiskTest()
{
disk = new MemoryDisk(Heads, Tracks, Sectors, SectorSize);
}
[Fact]
public void SectorEnumeratorNotInitialized()
{
var enumerator = ((IEnumerable<ISector>) disk).GetEnumerator();
var sector = enumerator.Current;
Assert.Null(sector);
}
[Fact]
public void SectorEnumeratorReturnsAllSectors()
{
var enumerator = ((IEnumerable<ISector>) disk).GetEnumerator();
int sectorCount = 0;
while (enumerator.MoveNext())
{
sectorCount++;
}
Assert.Equal(Heads*Tracks*Sectors, sectorCount);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,219
|
{"url":"https:\/\/zbmath.org\/?q=an:0977.57042","text":"zbMATH \u2014 the first resource for mathematics\n\nNon-abelian cohomology of abelian Anosov actions. (English) Zbl\u00a00977.57042\nThe articles of A. N. Livshits [Math. Notes 10, 758-763 (1971); translation from Mat. Zametki 10, 555-564 (1971; Zbl 0227.58006); and Math. USSR, Izv. 6, 1278-1301 (1972); translation from Izv. Akad. Nauk SSSR., Ser. Mat. 36, 1296-1320 (1972; Zbl 0252.58007)] started the study of cocycles over Anosov diffeomorphisms (i.e., actions of the group $$\\mathbb{Z}$$ of integers) and flows (i.e., actions of the group $$\\mathbb{Z}$$ of real numbers). An important conjecture asserts that tori, nilmanifolds, and infranilmanifolds are the only ones supporting Anosov diffeomorphisms. The articles of Livshits have created a lot of research and the article under review includes a survey of related results and then develops a new technique for calculating the first cohomology of certain classes of actions of higher-rank abelian groups $$\\mathbb{Z}^k$$ and $$\\mathbb{R}^k$$ $$(k\\geq 2)$$ with values in a linear Lie group. The main result of the article asserts that the corresponding cohomology trivializes. The methods applied to get the result have geometric nature and are not using global information about the action based on harmonic analysis. As the authors note, the methods can be developed to apply to cocycles with values in certain infinite dimensional groups and to rigidity problems.\n\nMSC:\n 57S25 Groups acting on specific manifolds 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)\nFull Text:","date":"2021-01-28 09:41:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.30544760823249817, \"perplexity\": 822.8420380901009}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-04\/segments\/1610704839214.97\/warc\/CC-MAIN-20210128071759-20210128101759-00386.warc.gz\"}"}
| null | null |
<?php
namespace Meinhof\Templating\Twig\Extension;
use Meinhof\Helper\UrlHelperInterface;
use Meinhof\Export\ExportEvent;
class UrlExtension extends \Twig_Extension
{
protected $helper;
protected $relativeRoot;
protected $parameters;
public function __construct(UrlHelperInterface $helper)
{
$this->helper = $helper;
}
public function onExport(ExportEvent $event)
{
$this->relativeRoot = $event->getRelativeRoot();
$this->parameters = $event->getParameters();
}
public function getName()
{
return 'url';
}
public function getFunctions()
{
return array(
'url' => new \Twig_Function_Method($this, "getUrl")
);
}
public function getUrl($obj, array $params=array(), $absolute=false)
{
if (!$obj) {
return "";
}
if (is_array($this->parameters)) {
$params = array_merge($this->parameters, $params);
}
if (is_object($obj)) {
$obj = $this->helper->getUrl($obj, $params);
}
if (is_string($obj)) {
return $absolute ? $obj : $this->relativeRoot.$obj ;
}
throw new \InvalidArgumentException("Could not get the url.");
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,235
|
\section{Introduction}
An egocentric camera captures rich and varied information of how the wearer interacts with their environment.
The challenge for the visual understanding of this information is currently significant and not only incited by the enormous variety of such interactions but also by limitations in the available visual descriptors, e.g.~those rooted in motion or appearance.
Supervised learning from labelled examples is used to alleviate some of these ambiguities.
Egocentric datasets~\cite{Lee12,fathi2013modeling,spriggs2009temporal,Damen2014a} and interaction recognition methods~\cite{fathi2013modeling,Pirsiavash12,Fathi2012,McCandless13}
differ in the features used and classification techniques adopted, yet they all assume a semantically distinct set of \textit{pre-selected} verbs or verb-noun combinations for supervision.
When free annotations are available - unbounded choice of verbs or verb-nouns - from audio scripts~\cite{Alayrac15learning} or textual annotations~\cite{Damen2014a}, a single label is selected to represent each interaction using a majority vote. Less frequent annotations are treated as outliers, though they typically represent a meaningful and correct annotation.
For example, lifting an object from a workspace could be described as \textit{pick-up}, \textit{lift}, \textit{take} or \textit{grab}; all valid labels.
Note that assuming multiple \textit{valid} labels is different from the problem of Ambiguous Label Learning, \cite{chen2015matrix,hullermeier2006learning}, where the aim is to find a single valid label from a mixed set of related and unrelated labels.
Egocentric video offers a unique insight into object interactions in particular.
The camera is ideally positioned to capture objects being used and, equally interesting, the different ways in which the same object is used.
One interaction (e.g.~\textit{open}) applies to a wide variety of objects, and each video can be labelled by multiple valid labels (e.g. \textit{open door} vs \textit{push door}).
In this context, recognition cannot be simplified as a one-vs-all classification task. Capturing the semantic relationships between annotations and the visual ambiguities between accompanying video segments can better represent the space of possible interactions.
Figure~\ref{fig:motivational} shows a graphical abstract of our work.
Given a dataset of egocentric object interactions with free annotations, we
contribute four diversions from previous attempts:
(i)~We treat all free annotations as valid, correct labellings,
(ii)~A graph that combines semantic relationships with visual similarities is built, inspired by previous work on object class categories in single images~\cite{fang2012measuring}~(Sec.~\ref{sec:methodEmbed}),
(iii)~A~test video is embedded into the previously learnt semantic-visual graph and the probability distribution over its possible annotations is estimated (Sec.~\ref{sec:methodClass}) and
(iv)~When verb meanings are available, we discover semantic relationships between annotations using WordNet~(Sec.~\ref{sec:methodAH}).
We test semantic embedding (SEMBED) on three public egocentric datasets \cite{Damen2014a,spriggs2009temporal,Fathi2012}. We show that as the number of verb annotations and their semantic ambiguities increase, SEMBED outperforms classification approaches.
We also show that
incorporating higher level semantic relationships, such as the hyponymy relationship, improves the results.
Note that while we focus on \textit{egocentric object interaction recognition} as a rich domain of semantic and visual ambiguities, some of the arguments can apply to action recognition in general.
\begin{figure}[!t]
\centering
\includegraphics[width=1.0\textwidth]{./introExample.png}
\caption{Given a dataset of free annotations, with potentially ambiguous semantic labelling~(left),
we propose to deviate from the one-vs-all classical approach~(middle) and instead build a graph that encapsulates semantic relationships and visual similarities in the training set~(right). Recognition then amounts to embedding an unlabelled video (denoted by `??') into the graph and estimating the probability distribution over potential labels.}
\label{fig:motivational}
\end{figure}
\section{Embedding Object Interactions - Prior Work}
\label{sec:Ego_RW}
To the best of our knowledge, embedding for egocentric action recognition has not been attempted previously. We first review works on recognising egocentric
object interactions, then review works which incorporate semantic knowledge for recognition tasks.
\noindent \textbf{Egocentric Object Interaction Recognition:} \hspace{4pt}
Egocentric action recognition works range from self-motion~\cite{Kitani11} (e.g. walk, cycle) to high-level activities (e.g.~\cite{spriggs2009temporal,kuehne2015towards,lade2010task,Bleser15,Sudeep10}).
On the task of object interaction recognition, approaches vary in whether they use hand-centred features~\cite{ishihararecognizing,kumarfly}, object-specific features~\cite{fathi2013modeling,Damen2014a,McCandless13,ren11} or a combination~\cite{Lee12,li2015delving}.
Ishihara~\emph{et~al}~\cite{ishihararecognizing} use dense trajectories in addition to global hand shape features and apply a linear SVM to determine the action class. Kumar~\emph{et~al}~\cite{kumarfly} sample and describe superpixel regions around the hand. Their method allows for hand detectors to be trained spontaneously with the user performing the action.
Object-specific features are better suited for recognising verb-noun actions (e.g. \textit{pick-cup vs pick-plate}) rather than a general \textit{picking} action.
In Damen \emph{et~al}~\cite{Damen2014a}, spatio-temporal interest points have been used to discover object interactions in an unsupervised manner.
The works of Fathi~\emph{et~al}~\cite{fathi2013modeling,Fathi2012,li2015delving,fathi2011learning} have tested features including gaze, colour, texture and shape for verb-noun action classification.
Of these, \cite{fathi2013modeling} specifically discusses the change in the object state as a useful feature to recognise object interactions.
Though attempting video summarisation primarily, Ghosh~\emph{et~al}~\cite{Lee12} introduces a collection of features that could be used to classify object-interactions such as distance from the hand, saliency, objectness represented using a spatio-temporal pyramid to detect change.
These features were proven useful for segmenting object-interactions from a lengthy video, but have not been tested for action classification \textit{per se}.
On several publicly available datasets, Li~\emph{et~al}~\cite{li2015delving} compare motion, object, head motion and gaze information along with a linear SVM for object interaction classification.
Their results prove that Improved Dense Trajectories (IDT) proposed by~\cite{Wang2011} outperform other motion features.
With the emergence of highly-discriminative appearance-based features, pre-trained Convolutional Neural Networks (CNN) on ImageNet have also been tested.
In \cite{Moghimi_2014_CVPR_Workshops}, CNN is evaluated for distinguishing manipulation from non-manipulation actions on an RGB-D egocentric dataset.
Ryoo~\emph{et~al}~\cite{Ryoo_2015_CVPR} combine CNN with IDT along with novel time series pooling for dog-centric manipulation and non-manipulation actions.
More recently, fine-tuned multi-stream CNN approaches have achieved state of the art results on egocentric datasets~\cite{Ma16,Singh16}, though are tuned on each dataset independently.
Based on~\cite{li2015delving,Ryoo_2015_CVPR} conclusions, in this work we report results on IDT as a state-of-the-art motion feature and pre-trained CNN features a state-of-the-art appearance feature. Testing tuned CNNs is left for future work.
\vspace*{8pt}
\noindent \textbf{Semantic Embedding for Object and Action Recognition:} \hspace{4pt}
Using linguistic semantic knowledge for Computer Vision tasks, including action recognition, has been fuelled by the accessibility of text or audio descriptions from online sources.
One such dataset which made this possible was gathered from {Y}ou{T}ube videos \cite{chen2011collecting} with free annotations. The dataset includes a variety of real-world scenarios, though not limited to egocentric or object-interactions.
For each video, multiple annotators were asked to describe the video.
Both \cite{Motwani12,Guadarrama13} use this dataset for action recognition.
In Motwani and Mooney~\cite{Motwani12}, the most frequently annotated verb for each video is used, and verbs are grouped into classes using semantic similarity measures, extracted from the {W}ord{N}et hierarchy as well as information corpuses.
Videos are described by HoG and HoF features around spatio-temporal interest points.
Guadarrama \emph{et~al}~\cite{Guadarrama13} find subject, object and verb triplets in an attempt to automatically annotate the action. They create a separate semantic hierarchy for each, formulated by co-occurrences of words within the free annotations and use Spearman's rank to find the distances between clusters.
Semantic links are used to generate specific, rather than general, annotations and a classifier is trained for each leaf node within the hierarchies.
Their method allows zero-shot action annotation by trading-off specificity and semantic similarity.
While combining semantics, both works use majority voting to limit the description per class to a single verb.
Another recent {Y}ou{T}ube dataset was collected of users performing tasks while narrating their actions \cite{Alayrac15learning}.
Labels are extracted from audio descriptions using automatic speech recognition.
Verb labels are then used to align videos using a {W}ord{N}et similarity measure as well as visual similarity (HoF and CNN) to find the sequence of actions in a task.
Semantics have also been used for object recognition with images.
Jin \emph{et~al} \cite{jin2005image} use {W}ord{N}et to remove noisy labels from images which have multiple labels.
Similarly, Ordonez \emph{et~al}~\cite{ordonez2015predicting} use {W}ord{N}et to find the most frequently-used object labels amongst multiple annotations.
We build our work on Fang and Torresani \cite{fang2012measuring}, where images are embedded in a semantic-visual graph.
In~\cite{fang2012measuring}, images are clustered depending on the semantic relationships between the labels and edges of the graph are weighted with the visual similarity. They use {I}mage{N}et as the database for training, and benefit from the fact that images within {I}mage{N}et are organised according to the {W}ord{N}et hierarchy.
We differ from~\cite{fang2012measuring} in how we add visual links to the semantic graphs as will be explained next.
\section{Semantic Embedding of Egocentric Action Videos}
\label{sec:method}
We next, in Sec.~\ref{sec:methodEmbed}, explain how we build a semantic-visual graph (SVG) that encodes label and visual ambiguities in the training set.
In Sec.~\ref{sec:methodClass}, we detail how videos with an unknown class are embedded in SVG, and how the probability distribution over their annotations is estimated.
Finally, in Sec.~\ref{sec:methodAH} we explore further semantic relationships when verb meanings are annotated.
\subsection{Learning the Semantic-Visual Graph}
\label{sec:methodEmbed}
The Semantic-Visual Graph (SVG) is a representation of the training videos, with three sources of information encoded. First, videos that are semantically linked, e.g. have the same label, are linked in SVG. Second, nodes that are visually similar, yet semantically distinct, should also be linked as these indicate visual ambiguities.
Third, edge weights correspond to the normalised visual similarity, over neighbouring nodes, using a visual descriptor and a defined distance measure.
In this section we explain how SVG$_u$, an undirected graph, is constructed, then normalised to achieve the directed graph SVG.
SVG$_u$ is an undirected graph, where one node $x_i \in \textrm{SVG}_u$ corresponds to one training video.
Assume AX($x_i$, $x_j$) is a binary function that checks whether two video labels are semantically related.
Initially, AX($x_i$, $x_j$) is \textit{true} when both videos are annotated by the exact same verb.
This assumption is revisited in Sec.~\ref{sec:methodAH}.
Edges in SVG$_u$ are created between nodes with a semantic relationship:
\begin{equation}
x_i \frown x_j \in \textrm{SVG}_u \iff AX(x_i, x_j) = true
\end{equation}
\noindent The undirected edge $x_i \frown x_j \in \textrm{SVG}_u$ is assigned a weight $w_{x_i \frown x_j} = D_v (x_i, x_j)$ where $D_v$ is a distance measure defined over the visual descriptor chosen.
Assume $rank (D_v(x_i, x_j))$ is a function that returns the relative position of the distance measure amongst all the remaining pairs of videos such that,
\begin{equation}
\resizebox{1.0\textwidth}{!}{$rank(D_v(x_i, x_j)) = n \iff D_v(x_i, x_j) = min_n(D_v(x_k, x_l)) \quad \forall x_k, x_l \in \textrm{SVG}_u \quad and \quad AX(x_k, x_l) \ne true$}
\end{equation}
\noindent and $min_n$ is the $n^{th}$ minimum element in the list. In addition, assume \\$rank_i (D_v(x_i, x_j))$ is a function that returns the relative position of $D_v(x_i,x_j)$ amongst all nodes not connected to $x_i$ such that,
\begin{equation}
\resizebox{1.0\textwidth}{!}{$rank_i(D_v(x_i, x_j)) = n \iff D_v(x_i, x_j) = min_n(D_v(x_i, x_l)) \quad \forall x_l \in \textrm{SVG}_u \quad and \quad AX(x_i, x_l) \ne true$}
\end{equation}
\noindent Further links are added to SVG$_u$ to encode visual ambiguities such that,
\begin{equation}
x_i \frown x_j \in \textrm{SVG}_u \iff rank(D_v(x_i, x_j)) \le m \quad or \quad rank_i(D_v(x_i, x_j)) = 1
\end{equation}
\noindent where $m$ is the number of visual connections in SVG$_u$ that correspond to the top $m$ visually similar and semantically dissimilar nodes in SVG$_u$.
We differ from~\cite{fang2012measuring} in that we ensure each node is connected to its top visually similar but semantically distinct node.
The undirected graph SVG$_u$ is then converted to a directed graph by replacing each edge with two directed edges.
\begin{equation}
x_i \frown x_j \in \textrm{SVG}_u \Rightarrow \{x_i \rightarrow x_j, x_j \rightarrow x_i\} \in \textrm{SVG}
\end{equation}
\noindent The weights of directed edges are initially the same as the weights for their undirected counterparts however they
are normalised to define the probability of traversing from video $x_i$ to $x_j$,
\begin{equation}
\label{eq:travProb}
P(x_i \rightarrow x_j)=\frac{1/w_{x_i\rightarrow x_j}}{\sum\limits_k{1/w_{x_i \rightarrow x_k}}} \quad \quad \forall x_i \rightarrow x_k \in \textrm{SVG}
\end{equation}
\noindent The reciprocal of the weights is taken so that the most visually similar path will have the highest probability.
\begin{figure}[t]
\centering
\includegraphics[width=1.0\textwidth]{./embeddingExample.png}
\label{eq:prob}
\caption{The Semantic-Visual Graph (SVG) is built for training data, with semantic links (green) and visual links (blue) between videos. Given a test video $x$, two nearest neighbours are found (yellow) and a Markov Walk of 2 steps (step1-red and step2-orange) finds the probability distribution over potential labellings. Ref. supplementary material for animation.}
\label{fig:graphEx}
\end{figure}
\subsection{Embedding in Semantic-Visual Graph}
\label{sec:methodClass}
Given a test video, $x$, we first embed the video into SVG then use the Markov Walk (MW) method from \cite{fang2012measuring} to determine $Class\,(x)$.
To embed $x$, we begin by finding the set $\mathcal{R}$ which contains the $z$ closest neighbours to $x$ based on visual distance, such that
\begin{equation}
\mathcal{R} = \{x_i \in \textrm{SVG} \mid rank(D_v(x,x_j)) \le z\}
\end{equation}
\noindent We embed $x$ into SVG by adding directed edges connecting $x$ to nodes in\\ \mbox{$\mathcal{R}$: $x \rightarrow x_i \quad \forall x_i \in \mathcal{R}$}
with normalised weights $P(x \rightarrow x_i)$.
Following the embedding, MW attempts to traverse the nodes in the directed graph to estimate the probability of $Class(x)$. Given the Markovian assumption and a predefined number of steps $t$, we calculate the probability distribution of reaching a node $x_i$ as follows
\begin{equation}
P(x_{i+t} \mid x) = \prod_{x_i \in R} \Bigl( P(x \rightarrow x_i) \prod_{j = 1}^t P(x_{i+j-1} \rightarrow x_{i+j}) \Bigr)
\end{equation}
\noindent To perform MW efficiently, we construct the vector q such that
\begin{equation}
q(i) =
\begin{cases}
P(x \rightarrow x_i) &x_i \in \mathcal{R} \\
0 &\mbox{otherwise}
\end{cases}
\label{eq:matQ}
\end{equation}
\noindent We also construct a matrix $A$ such that $A(i,j)=P(x_i \rightarrow x_j)$ (Eq.~\ref{eq:travProb}), note that this matrix is asymmetrical as nodes have a different set of neighbours in SVG.
Accordingly, $P(x_{i+t} \mid x) = q^T A^{t}$
where $q^T$ is the transpose of $q$ and $t$ is the number of steps in MW.
We can then accumulate $P(Class(x))$ for every unique annotation $ax \in AX$ as follows
\begin{equation}
P(Class(x) = ax) = \sum_{AX(x_{i+t}, ax) = true} P(x_{i+t} \mid x)
\end{equation}
\noindent We then select $\arg\max_{Class(x)}P(Class(x))$ as the semantic label of $x$.
Figure~\ref{fig:graphEx} shows an example of SVG and video embedding. In the figure, given two nearest neighbours $z = 2$ and two steps in MW $t = 2$, the probability distribution over possible labellings is calculated.
\subsection{Semantic Relationships: Synsets and Hyponyms}
In Sec.~\ref{sec:methodEmbed}, videos are considered semantically linked only when the annotated verbs are the same.
SVG then enables handling ambiguities via incorporating visual similarity links in the graph.
However, further semantic relationships, such as synonymy and hyponymy relationships, can be exploited between annotations.
In linguistics, two words are \textit{synonyms} if they have the same meaning, and the set of all synonyms is a \textit{synset}.
Moreover, two words are described as a \textit{hyponym} and a \textit{hypernym} respectively if the first is a more specific instance of the second.
The terms originate from the Greek word $hyp\acute{o}$ and $hyp\acute{e}r$ - \textit{under} and \textit{over}.
Synonymy and hyponymy relationships are encoded in lexical databases. WordNet (v3.1, 2012) is a commonly-used lexical database that
is based on six semantic relations~\cite{miller1995wordnet}.
In the WordNet verb hierarchy, verbs are first separated into their various meanings by the notation $\langle word \rangle.v.\langle s\rangle$ where $s \ge 1$ is the number of disjoint meanings.
The meanings are then arranged in hierarchies that encapsulate semantic relationships.
To benefit from such hierarchies, verbs should be annotated with their meanings.
We annotate~\cite{Damen2014a} using verb meanings, and Fig.~\ref{fig:annotationEx} shows how such annotations of the same action can be synonyms and hyponyms, as annotators chose different or more specific action descriptions.
\label{sec:methodAH}
\begin{figure}[b!]
\centering
\includegraphics[width=1.0\textwidth]{./annotationExamples.pdf}
\caption{
Five free annotations for two sequences from the BEOID dataset~\cite{Damen2014a}, and the respective semantic relationships between the annotations from WordNet~\cite{miller1995wordnet}. In the hierarchy, each parent-child relationship represents a hypernym-hyponym pair. The dotted circle encapsulates a synonymy relationship. The start and end times of the actions are also shown. For placing a cup on a mat (left), synonyms $put.v.1$ and $place.v.1$ were chosen by annotators. $put\_down.v.1$, a hyponym of $put.v.1$ was also used. For washing a cup (right), the verbs $wash.v.3$, $wash\; up.v.3$ and $rinse.v.1$ were chosen. $rinse.v.1$ is a hyponym of $wash.v.3$.}
\label{fig:annotationEx}
\end{figure}
Given annotated meanings, we define the term action synsets (AS) to indicate that annotations are linked by a synonymy relationship solely, and the term action hyponym (AH) to indicate that annotations are linked by both the synonymy or the hyponymy relationships.
For comparison, we define the term action meaning (AM) where annotations are linked only when the annotation matches exactly.
We use the general term AX where $AX \in \{$AM, AS, AH$\}$ is one of the the possible types of semantic relationships tested.
\section{Datasets, Experiments and Results}
\label{sec:datasets}
We selected three publicly available datasets that primarily focus on object interactions from egocentric videos~\cite{de2008guide,Fathi2012,Damen2014a} (Figure \ref{fig:datasetInfo}).
\noindent \textbf{Verb annotations:} We exploited the annotations provided by the authors to split the CMU and GTEA+ sequences into object-interaction segments. For CMU, object-interaction annotations are only provided for the activity of \textit{making brownies}.
Annotators chose from 12 disjoint verbs to ground-truth segments.
In GTEA+ annotators chose from verb-noun pairing to ground-truth, e.g. \textit{cut\_{}cucumber} versus \textit{divide\_{}bun} and similarly \textit{squeeze\_{}ketchup} versus \textit{compress\_{}bun}.
When removing the nouns, verbs could be used interchangeably but free annotations were not available to annotators.
\begin{figure}[t]
\centering
\begin{minipage}[b]{0.8\textwidth}
\resizebox{\textwidth}{!}{
\centering
\begin{tabular}{|l|c|c|c|c|c|}
\hline
\textbf{Name} & \textbf{Users} & \textbf{Seq.} & \textbf{OI Seg.s} & \textbf{Used OI Seg.s} & \textbf{Semantic Verbs} \\ \hline
\textcolor{cmuColour}{CMU} ~\cite{de2008guide} & 5 & 35 & 516 & 406 & 12 (33.8, 30.5) \\ \hline
\textcolor{gteaColour}{GTEA+} ~\cite{Fathi2012}& 13& 30 & 3371 & 1000$^{\star}$ & 25 (40.0, 75.5) \\ \hline
\textcolor{beoidColour}{BEOID} ~\cite{Damen2014a} & 3-5 & 58 & 1488 & 1225 & 75 (16.3 34.2) \\ \hline
\end{tabular}
}
\centering
\vspace*{6pt}
\includegraphics[width=0.7\textwidth]{./videoLengthDistribution.pdf}
\end{minipage}
\caption{Dataset details (top) and video length distributions (bottom). Number of users, segments, Object-Interaction (OI) segments and used segments in the results (length $<$ 40s) are detailed. We report the number of annotated verbs along with $\mu$ and $\sigma$ for the number of segments per verb. $^{\star}$: Due to the size of GTEA+ we sampled 1000 videos randomly. Ref. supplementary material for frequencies of verb annotations per dataset.}
\label{fig:datasetInfo}
\end{figure}
While BEOID contains a variety of activities and locations, ranging from a desktop to operating a gym machine, it does not provide action-level annotations so we annotated BEOID using free annotations\footnote{Annotations can be found at: \url{http://www.cs.bris.ac.uk/~damen/BEOID/}}, allowing annotators to split video sequences into object-interaction segments in addition to choosing the verb.
We recruited 20 native English speakers.
These annotators were given a free textbox to label each segment with the verb that best described the seen interaction \textit{in their opinion}. Once
a verb has been chosen, the annotators were given the set of potential meanings extracted from WordNet for the chosen verb. Again, they were asked to select the meaning that, \textit{in their opinion}, best suited the segment.
Multiple annotators (8-10) were asked to label each task to intentionally introduce variability in the choice of verbs and start-end times of object interaction segments.
\noindent \textbf{Motion and Appearance Features:} We test two state-of-the-art feature descriptors to represent both the motion and the appearance of the videos.
These are the Improved Dense Trajectories~(IDT)~\cite{Wang2013} and Overfeat Convolutional Neural Networks pre-trained for ImageNet classes~(CNN)~\cite{sermanet-iclr-14}.
For CNN features, we take every 5th frame from 30fps video, starting always from the first frame, and rescale to 320x240 pixels.
\noindent \textbf{Encodings:} We test two encodings, using Bag of Words (BoW) \cite{csurka2004visual} and Fisher Vectors (FV)~\cite{Sanchez2013} with Euclidean distance. For IDT, when creating the BoW and FV representations, we use a 25\% random sample from every video to model the Gaussians for efficiency.
We vary the number of Gaussians ($\gamma_{fv}$) and the size of the codebook ($\gamma_{bow}$) in reported results.
\noindent \textbf{Classification:}
In all results, leave-one-person-out cross validation has been used. Namely, when testing a video containing one person performing an action, all other videos captured from the same person are excluded from the training set.
For SVM results, as the tested datasets contain an imbalance in the distribution of instances per class, we weight the classes by the term $w(c) = 1/prior(c)^\lambda$
where $\lambda \in [0,1]$ is the exponent that best fits the distribution of segments per verb for a given dataset (ref supplementary material).
\begin{table}[t]
\centering
\caption{As the number of verbs increases from \textcolor{cmuColour}{12} to \textcolor{beoidColour}{75}, the best performance changes from SVM to SEMBED. Results are obtained with $\gamma_{fv} = 10$ and $\gamma_{bow} = 256$, $k=$\{\textcolor{cmuColour}{3},\textcolor{gteaColour}{5},\textcolor{beoidColour}{5}\}, $m$ = 240, $z$=\{\textcolor{cmuColour}{2},\textcolor{gteaColour}{6},\textcolor{beoidColour}{4}\}, $t$=\{\textcolor{cmuColour}{20},\textcolor{gteaColour}{20},\textcolor{beoidColour}{8}\} for CNN and $z$=\{\textcolor{cmuColour}{4},\textcolor{gteaColour}{5},\textcolor{beoidColour}{14}\}, $t$=\{\textcolor{cmuColour}{4},\textcolor{gteaColour}{20},\textcolor{beoidColour}{10}\} for IDT. For completion, state-of-the-art results on verb-noun classes are reported under `Other Works' thus are not directly comparable to our verb only results.}
\resizebox{\textwidth}{!}{%
\begin{tabular}{|l|C{1.0cm}|C{1.3cm}|C{2.0cm}||C{1.0cm}|C{1.3cm}|C{2.0cm}||C{1.0cm}|C{1.3cm}|C{2.0cm}||C{1.0cm}|C{1.3cm}|C{2.0cm}||C{1.0cm}|C{2.6cm}|}
\cline{1-13}
\textit{FEATURES} & \multicolumn{6}{c||}{\textbf{CNN}} & \multicolumn{6}{c||}{\textbf{IDT}} &\multicolumn{2}{c}{}\\ \cline{1-13}
\textit{ENCODING} & \multicolumn{3}{c||}{\textbf{FV}} & \multicolumn{3}{c||}{\textbf{BOW}} & \multicolumn{3}{c||}{\textbf{FV}}& \multicolumn{3}{c||}{\textbf{BOW}} &\multicolumn{2}{c}{} \\ \hline
\textit{METHOD} & \textbf{\footnotesize{SVM}} & \textbf{\footnotesize{K-NN}} & \textbf{\footnotesize{SEMBED}} & \textbf{\footnotesize{SVM}} & \textbf{\footnotesize{K-NN}} & \textbf{\footnotesize{SEMBED}} & \textbf{\footnotesize{SVM}} & \textbf{\footnotesize{K-NN}} & \textbf{\footnotesize{SEMBED}} & \textbf{\footnotesize{SVM}} & \textbf{\footnotesize{K-NN}} & \textbf{\footnotesize{SEMBED}} & \textbf{Verbs} &\textbf{Other Works} \\ \hline
\textcolor{cmuColour}{\textbf{CMU}}~\cite{de2008guide} & 58.6 & 46.6 & 46.3 & 55.9 & 43.3 & 52.0 & \textcolor{cmuColour}{\textbf{69.4}} & 58.1 & 57.4 & 55.9 & 57.6 & 61.6 &\textcolor{cmuColour}{\textbf{12}} & 48.6 \cite{spriggs2009temporal}, 73.4 \cite{taralova2011source} \\ \hline
\textcolor{gteaColour}{\textbf{GTEA+}}\cite{Fathi2012} & 15.6 & 30.0 & 31.0 & 25.1 & 33.5 & 33.6 & \textcolor{gteaColour}{\textbf{43.6}} & \textcolor{gteaColour}{\textbf{43.4}} & 42.1 & 27.8 & 34.5 & 40.3 & \textcolor{gteaColour}{\textbf{25}} &60.5 \cite{li2015delving}, 65.1~\cite{Ma16} \\ \hline
\textcolor{beoidColour}{\textbf{BEOID}}~\cite{Damen2014a} & 20.9 & 34.4 & 37.5 & 15.2 & 19.1 & 19.6 & 38.7 & 36.0 & 37.4
& 34.8 & 39.6 & \textcolor{beoidColour}{\textbf{45.0
}} & \textcolor{beoidColour}{\textbf{75}} &- \\ \hline
\end{tabular}%
}
\label{table:resultsTableAV}
\end{table}
\begin{figure}[!h]
\centering
\includegraphics[width=1.0\textwidth]{./newGamma.pdf}
\caption{Results as $\gamma_{fv}$ and $\gamma{bow}$ vary for CMU, GTEA+, BEOID. Results were shown with $k=5$, $m = 240$, $z=10$, $t=10$. Similar performance is seen for other parameters.}
\label{fig:gammaFigure}
\end{figure}
\noindent\textbf{Results on annotated verbs:} \hspace{4pt}
\label{sec:ResultsAV}
Table~\ref{table:resultsTableAV} compares the three datasets for every $\langle$features, encoding, classifier$\rangle$ combination.
The following conclusions can be made: (i) for all datasets, motion features~(IDT) outperform appearance features~(CNN) when classifying verbs without considering the object used. (ii) for CMU and GTEA+, we produce comparable results to published results using motion information on the same datasets. These are reported under `Other Works' but are not directly comparable as published works tend to report on verb-noun classes.
(iii) For the three datasets with varying number of verbs, as the number of verbs increases (\textcolor{cmuColour}{12} $\rightarrow$ \textcolor{beoidColour}{75}) with an increase in semantic ambiguity, SEMBED outperforms standard classifiers (SVM and K-NN). While the table shows the best results for encoding, Fig.~\ref{fig:gammaFigure} reports comparative results as $\gamma$ is changed - $\gamma_{fv}=10$ generally led to higher accuracies on all datasets, compared to $\gamma_{bow} = 256$.
We test the sensitivity of SEMBED to its key parameters $z$ and $t$ and report results in
Fig.~\ref{fig:sembedZandT} showing the accuracy over various features for BEOID and across the three datasets for IDT-BOW (Ref. supplementary material for all combinations).
As noted, $z$ and $t$ behave differently for the various appearance and motion descriptors as well as for different encodings.
Generally, SEMBED is more sensitive to the choice of $z$ than $t$.
This is because the Markovian Walk (MW) is unable to represent the probability distribution over labels unless the starting positions are representative of the visual ambiguity.
Figure~\ref{fig:sembedZandT} also shows that MW isn't too helpful for CMU (as $t$ increases, accuracy decreases) because it
has visually distinctive verb classes.
On all datasets, SEMBED is resilient to changing $m$ values; the results are comparable on $180 \le m \le 400$.
\begin{figure}[b]
\includegraphics[width=1\textwidth]{./sembedZandTEvaluation.png}
\caption{Evaluation of SEMBED sensitivity to $z$ and $t$ parameters with $m = 240$.}
\label{fig:sembedZandT}
\end{figure}
\noindent\textbf{Results on annotated verbs and meanings:} \hspace{4pt}
As mentioned earlier, we also annotate BEOID with verb-meaning ground-truth. This resulted in 108 $\langle word \rangle.v.\langle s\rangle$ annotations for the 1225 segments in the dataset.
Note the increase in the number of classes from 75 when using verbs only to 108 when using verb-meaning ground-truth.
This increase is due to two reasons - one \textit{helpful}, another \textit{problematic}.
For example, it is \textit{helpful} when annotators choose between $hold.v.1$: \textit{``keep in a certain state, position''} and $hold.v.2$: \textit{``hold in one's hand''}.
Annotators would then use the first for when a button is pressed and the second for when an object is grasped.
However, frequently, WordNet meanings can appear ambiguous resulting in \textit{problematic} cases, especially in the context of egocentric actions. An example of this is the action of turning a tap on so water would flow. Annotators used $turn.v.1$: \textit{``change orientation or direction''} and $turn.v.4$: \textit{``cause to move around or rotate''} interchangeably.
In WordNet though, $turn.v.1$ and $turn.v.4$ are not semantically related, introducing unwanted ambiguity affecting the ground-truth labels. While we accept that WordNet may not be the best method to incorporate meaning, we report results as semantic links are incorporated.
\begin{table}[t]
\centering
\caption{As synonymy (AS) and then hyponymy (AH) semantic relationships are incorporated, accuracy increases for all features on the BEOID dataset. $\gamma_{fv} = 10$, $\gamma_{bow} = 256$, $m$ = 240, \{AM,AS,AH\}: $z_{CNN}$=\{3,3,2\}, $t_{CNN}$=\{20,20,14\}, $z_{IDT}$=\{6,10,13\}, $t_{IDT}$=\{20,20,2\}.}
\resizebox{\textwidth}{!}{%
\begin{tabular}{|l|C{1.0cm}|C{1.3cm}|C{2.0cm}||C{1.0cm}|C{1.3cm}|C{2.0cm}||C{1.0cm}|C{1.3cm}|C{2.0cm}||C{1.0cm}|C{1.3cm}|C{2.0cm}||C{1.5cm}|}
\cline{1-13}
\textit{FEATURES} & \multicolumn{6}{c||}{\textbf{CNN}} & \multicolumn{6}{c||}{\textbf{IDT}} \\ \cline{1-13} \textit{ENCODING} & \multicolumn{3}{c||}{\textbf{FV}} & \multicolumn{3}{c||}{\textbf{BOW}} & \multicolumn{3}{c||}{\textbf{FV}} & \multicolumn{3}{c||}{\textbf{BOW}} \\ \hline
\textit{METHOD} & \textbf{\footnotesize{SVM}} & \textbf{\footnotesize{K-NN}} & \textbf{\footnotesize{SEMBED}} & \textbf{\footnotesize{SVM}} & \textbf{\footnotesize{K-NN}} & \textbf{\footnotesize{SEMBED}} & \textbf{\footnotesize{SVM}} & \textbf{\footnotesize{K-NN}} & \textbf{\footnotesize{SEMBED}} & \textbf{\footnotesize{SVM}} & \textbf{\footnotesize{K-NN}} & \textbf{\footnotesize{SEMBED}} & \textbf{Classes} \\ \hline
AM & 13.2 & 24.6 & 26.2 & 12.1 & 7.8 & 11.7 & 25.9 & 28.5 & 32.2 & 26.1 & 31.6 & \textbf{38.2} &108\\ \hline
AS & 17.9 & 25.6 & 27.1 & 12.7 & 8.1 & 12.7 & 29.8 & 30.4 & 33.5 & 29.6 & 33.6 & \textbf{40.6} &102\\ \hline
AH & 18.1 & 25.0 & 26.9 & 12.2 & 7.4 & 16.3 & 36.2 & 33.1 & 34.5 & 29.1 & 35.2 & \textbf{41.9} &84\\ \hline
\end{tabular}%
}
\label{table:resultsTableASH}
\end{table}
\begin{figure}[t]
\label{fig:sembedExample}
\includegraphics[width=1.0\textwidth]{./3Graphs.png}
\includegraphics[width=1.0\textwidth]{./ex1_modified.png}
\caption{SVG for three semantic levels on BEOID (top). Example using AH (bottom), SVM and K-NN produce incorrect results. The Markov walk of SEMBED allows the video to be correctly classified.}
\label{fig:ex_sembed}
\end{figure}
We test the three types of semantic relationships $AX = \{$AM,AS,AH$\}$.
Histograms of all classes for the various semantic relationships are included in the supplementary material.
Table~\ref{table:resultsTableASH}
shows that embedding consistently improved performance as synsets and then hypernyms are grouped. Results also demonstrate the advantages of introducing semantic links between videos. Additionally, IDT continues to outperform CNN.
Figure~\ref{fig:ex_sembed} shows one example of SEMBED in action when using meanings and AH semantic links\footnote{Video with results available at: \url{youtube}}.
It should be noted that the best performance of SEMBED on meanings is inferior to using verbs only. This is due to the difficulty in assigning meanings to verbs as previously noted.
Approaches to address meaning ambiguities are left for future work.
\section{Conclusion and Future Directions}
The paper proposes embedding an egocentric action video in a semantic-visual graph to estimate the probability distribution over potentially ambiguous labels.
SEMBED profits from semantic knowledge to capture interchangeable labels for the same action, along with similarities in visual descriptors.
While showing clear potential, outperforming classification approaches on a challenging dataset, results merely evaluate the $\arg\max$ label when compared to ground-truth.
Further analysis of the probability distribution will be targeted next.
Other approaches to identify semantically related object-interaction labels from, for example, other lexical sources, overlapping annotations or object labels will also be attempted.
SEMBED's ability to scale to other object interactions and more discriminative visual descriptors will also be tested.
\bibliographystyle{splncs03}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,520
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"""
Tests for the Tattler plugin.
"""
import unittest
import tattler
from .fake_module import fake_object, fake_function, fake_factory
class TestTattler(unittest.TestCase):
"""Tests for tattler."""
def setUp(self):
self.tattler = tattler.Tattler()
self.tattler._paths = [
'tests.test_tattler.fake_function',
'tests.test_tattler.fake_object.method',
'tests.fake_module.FakeClass.method',
]
self.tattler.init()
def tearDown(self):
self.tattler.destroy()
def test_tattle(self):
"""Test calling functions we're tattling on."""
self.tattler.start()
with self.assertRaises(tattler.TattleTale):
fake_function()
with self.assertRaises(tattler.TattleTale):
fake_object.method()
self.tattler.stop()
# Should not raise anything
fake_function()
fake_object.method()
def test_tattle_not_started(self):
"""Test initializing the plugin but not starting it."""
# Should not raise anything
fake_function()
fake_object.method()
def test_tattle_method(self):
"""Test that a method receives the correct value of self."""
another_fake = fake_factory()
result = another_fake.method()
assert result is another_fake
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,689
|
<HTML>
<BODY BGCOLOR="white">
<PRE>
<FONT color="green">001</FONT> <a name="line.16"></a>
<FONT color="green">017</FONT> <a name="line.17"></a>
<FONT color="green">018</FONT> package org.apache.commons.math3.stat.clustering;<a name="line.18"></a>
<FONT color="green">019</FONT> <a name="line.19"></a>
<FONT color="green">020</FONT> import java.util.Collection;<a name="line.20"></a>
<FONT color="green">021</FONT> <a name="line.21"></a>
<FONT color="green">022</FONT> /**<a name="line.22"></a>
<FONT color="green">023</FONT> * Interface for points that can be clustered together.<a name="line.23"></a>
<FONT color="green">024</FONT> * @param <T> the type of point that can be clustered<a name="line.24"></a>
<FONT color="green">025</FONT> * @version $Id: Clusterable.java 1416643 2012-12-03 19:37:14Z tn $<a name="line.25"></a>
<FONT color="green">026</FONT> * @since 2.0<a name="line.26"></a>
<FONT color="green">027</FONT> */<a name="line.27"></a>
<FONT color="green">028</FONT> public interface Clusterable<T> {<a name="line.28"></a>
<FONT color="green">029</FONT> <a name="line.29"></a>
<FONT color="green">030</FONT> /**<a name="line.30"></a>
<FONT color="green">031</FONT> * Returns the distance from the given point.<a name="line.31"></a>
<FONT color="green">032</FONT> *<a name="line.32"></a>
<FONT color="green">033</FONT> * @param p the point to compute the distance from<a name="line.33"></a>
<FONT color="green">034</FONT> * @return the distance from the given point<a name="line.34"></a>
<FONT color="green">035</FONT> */<a name="line.35"></a>
<FONT color="green">036</FONT> double distanceFrom(T p);<a name="line.36"></a>
<FONT color="green">037</FONT> <a name="line.37"></a>
<FONT color="green">038</FONT> /**<a name="line.38"></a>
<FONT color="green">039</FONT> * Returns the centroid of the given Collection of points.<a name="line.39"></a>
<FONT color="green">040</FONT> *<a name="line.40"></a>
<FONT color="green">041</FONT> * @param p the Collection of points to compute the centroid of<a name="line.41"></a>
<FONT color="green">042</FONT> * @return the centroid of the given Collection of Points<a name="line.42"></a>
<FONT color="green">043</FONT> */<a name="line.43"></a>
<FONT color="green">044</FONT> T centroidOf(Collection<T> p);<a name="line.44"></a>
<FONT color="green">045</FONT> <a name="line.45"></a>
<FONT color="green">046</FONT> }<a name="line.46"></a>
</PRE>
</BODY>
</HTML>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,863
|
Q: Can't convert object of type java.lang.String to type - Firebase Error its firebase Error I cant get it please Help me to solve and tell me the reason why its happens.
sorry For english and try to solve it
E/AndroidRuntime: FATAL EXCEPTION: main
Process: com.example.kunalgharate.weedon, PID: 16793
com.google.firebase.database.DatabaseException: Can't convert object of type java.lang.String to type com.example.kunalgharate.weedon.Friends
at com.google.android.gms.internal.zg.zzb(Unknown Source)
at com.google.android.gms.internal.zg.zza(Unknown Source)
at com.google.firebase.database.DataSnapshot.getValue(Unknown Source)
at com.firebase.ui.database.ClassSnapshotParser.parseSnapshot(ClassSnapshotParser.java:22)
at com.firebase.ui.database.ObservableSnapshotArray.getObject(ObservableSnapshotArray.java:160)
at com.firebase.ui.database.CachingObservableSnapshotArray.getObject(CachingObservableSnapshotArray.java:40)
at com.firebase.ui.database.FirebaseRecyclerAdapter.getItem(FirebaseRecyclerAdapter.java:180)
at com.firebase.ui.database.FirebaseRecyclerAdapter.onBindViewHolder(FirebaseRecyclerAdapter.java:217)
at android.support.v7.widget.RecyclerView$Adapter.onBindViewHolder(RecyclerView.java:6482)
at android.support.v7.widget.RecyclerView$Adapter.bindViewHolder(RecyclerView.java:6515)
at android.support.v7.widget.RecyclerView$Recycler.tryBindViewHolderByDeadline(RecyclerView.java:5458)
at android.support.v7.widget.RecyclerView$Recycler.tryGetViewHolderForPositionByDeadline(RecyclerView.java:5724)
at android.support.v7.widget.RecyclerView$Recycler.getViewForPosition(RecyclerView.java:5563)
at android.support.v7.widget.RecyclerView$Recycler.getViewForPosition(RecyclerView.java:5559)
at android.support.v7.widget.LinearLayoutManager$LayoutState.next(LinearLayoutManager.java:2229)
at android.support.v7.widget.LinearLayoutManager.layoutChunk(LinearLayoutManager.java:1556)
at android.support.v7.widget.LinearLayoutManager.fill(LinearLayoutManager.java:1516)
at android.support.v7.widget.LinearLayoutManager.onLayoutChildren(LinearLayoutManager.java:608)
at android.support.v7.widget.RecyclerView.dispatchLayoutStep2(RecyclerView.java:3693)
at android.support.v7.widget.RecyclerView.dispatchLayout(RecyclerView.java:3410)
at android.support.v7.widget.RecyclerView.consumePendingUpdateOperations(RecyclerView.java:1710)
at android.support.v7.widget.RecyclerView$1.run(RecyclerView.java:346)
at android.view.Choreographer$CallbackRecord.run(Choreographer.java:871)
at android.view.Choreographer.doCallbacks(Choreographer.java:683)
at android.view.Choreographer.doFrame(Choreographer.java:616)
at android.view.Choreographer$FrameDisplayEventReceiver.run(Choreographer.java:857)
at android.os.Handler.handleCallback(Handler.java:751)
at android.os.Handler.dispatchMessage(Handler.java:95)
at android.os.Looper.loop(Looper.java:154)
at android.app.ActivityThread.main(ActivityThread.java:6123)
at java.lang.reflect.Method.invoke(Native Method)
at com.android.internal.os.ZygoteInit$MethodAndArgsCaller.run(ZygoteInit.java:867)
at com.android.internal.os.ZygoteInit.main(ZygoteInit.java:757)
Application terminated.
Activity Main
import android.content.Context;
import android.content.Intent;
import android.os.Build;
import android.support.design.widget.TabLayout;
import android.support.v4.view.ViewPager;
import android.support.v7.app.AppCompatActivity;
import android.os.Bundle;
import android.support.v7.widget.LinearLayoutManager;
import android.support.v7.widget.RecyclerView;
import android.support.v7.widget.Toolbar;
import android.util.Log;
import android.view.Menu;
import android.view.MenuItem;
import android.view.View;
import android.widget.TextView;
import android.widget.Toast;
import com.firebase.ui.database.FirebaseRecyclerAdapter;
import com.google.firebase.auth.FirebaseAuth;
import com.google.firebase.database.DatabaseReference;
import com.google.firebase.database.FirebaseDatabase;
import com.squareup.picasso.Picasso;
import de.hdodenhof.circleimageview.CircleImageView;
public class MainActivity extends AppCompatActivity {
private FirebaseAuth mAuth;
RecyclerView mSubsList;
DatabaseReference mFriendsDatabase;
DatabaseReference mUsersDatabase;
private String mCurrent_user_id;
@Override
protected void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.activity_main);
Toolbar toolbar = findViewById(R.id.main_page_toolbar);
setSupportActionBar(toolbar);
getSupportActionBar().setTitle("WeedON");
mAuth = FirebaseAuth.getInstance();
mCurrent_user_id = mAuth.getCurrentUser().getUid();
mSubsList =findViewById(R.id.subscribes_list);
mSubsList.setHasFixedSize(true);
mSubsList.setLayoutManager(new LinearLayoutManager(this));
mFriendsDatabase = FirebaseDatabase.getInstance().getReference().child("subscribers").child(mCurrent_user_id);
mFriendsDatabase.keepSynced(true);
mUsersDatabase = FirebaseDatabase.getInstance().getReference().child("Users");
mUsersDatabase.keepSynced(true);
}
private void signOut() {
mAuth.signOut();
}
@Override
protected void onStart() {
super.onStart();
FirebaseRecyclerAdapter<Friends,MainViewHolder> firebaseRecyclerAdapter = new FirebaseRecyclerAdapter<Friends,MainViewHolder>(
Friends.class,
R.layout.single_service_layout,
MainViewHolder.class,
mFriendsDatabase
)
{
@Override
protected void populateViewHolder(MainViewHolder viewHolder, Friends model, int position) {
}
};
mSubsList.setAdapter(firebaseRecyclerAdapter);
}
@Override
public boolean onCreateOptionsMenu(Menu menu) {
super.onCreateOptionsMenu(menu);
getMenuInflater().inflate(R.menu.main_menu, menu);
return true;
}
@Override
public boolean onOptionsItemSelected(MenuItem item) {
super.onOptionsItemSelected(item);
if(item.getItemId() == R.id.main_logout_btn){
// mUserRef.child("online").setValue(ServerValue.TIMESTAMP);
signOut();
Intent intent = new Intent(this, LoginActivity.class);
startActivity(intent);
finish();
// sendToStart();
}
if(item.getItemId() == R.id.main_settings_btn){
Intent settingsIntent = new Intent(MainActivity.this, SettingsActivity.class);
startActivity(settingsIntent);
}
if(item.getItemId() == R.id.main_all_btn){
Intent settingsIntent = new Intent(MainActivity.this, AllServicesActivity.class);
startActivity(settingsIntent);
}
return true;
}
public static class MainViewHolder extends RecyclerView.ViewHolder{
View mView;
public MainViewHolder(View itemView) {
super(itemView);
mView =itemView;
}
}
}
Database Screenshot
enter image description here
its Not showing layout Error or code Error what should i do ?
A: By looking at your database I can see that your mFriendsDatabase DatabseReference points to some 'String' Values and you are telling your firebaseRecyclerAdapter that those are of type 'Friends,that is why the exception is thrown.Most probably your are doing a mistake while saving the value to database,instead of saving a Friends object you are saving some String value.But if that's not the case you can change your firebaseRecyclerAdapter to this...
FirebaseRecyclerAdapter<String,MainViewHolder> firebaseRecyclerAdapter = new FirebaseRecyclerAdapter<String,MainViewHolder>(
String.class,
R.layout.single_service_layout,
MainViewHolder.class,
mFriendsDatabase
)
{
@Override
protected void populateViewHolder(MainViewHolder viewHolder, String mString, int position) {
Log.d("TAG",mString);
}
};
mSubsList.setAdapter(firebaseRecyclerAdapter);
A: To display those messages, please use the following code:
DatabaseReference rootRef = FirebaseDatabase.getInstance().getReference();
DatabaseReference uidRef = rootRef.child("subscribers").child(mCurrent_user_id);
ValueEventListener eventListener = new ValueEventListener() {
@Override
public void onDataChange(DataSnapshot dataSnapshot) {
for(DataSnapshot ds : dataSnapshot.getChildren()) {
String message = ds.getValue(String.class);
Log.d("TAG", message);
}
}
@Override
public void onCancelled(DatabaseError databaseError) {}
};
uidRef.addListenerForSingleValueEvent(eventListener);
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,262
|
Q: Issue in receiving emails I am doing my project in mvc4 using c#. I have a contact page i my website. My need is that i have to receive messages to my email id from other id's, when clicking the Send button.I use the following code
public void ReceiveMail(string name,string email,string message)
{
MailMessage msg = new MailMessage();
HttpContext ctx = HttpContext.Current;
msg.To.Add(new MailAddress("MyEmailId"));
msg.From = new MailAddress(email);
msg.Subject =name + "send a message";
msg.Priority = MailPriority.High;
msg.Body = message;
SmtpClient SmtpServer = new SmtpClient("smtp.gmail.com");// i am confused what to write here
SmtpServer.Send(msg);
}
It shows the error
The SMTP server requires a secure connection or the client was not authenticated.
The server response was: 5.7.0 Must issue a STARTTLS command first.
at4sm42219747pbc.30 - gsmtp
I don't know from which server i got the mail. Then how can i solve this issue . Please help me
A: Sending emails with Gmail requires some additional settings. At first, port number should be 587 (instead of default 25). At second, Gmail requires secure connection. And of course you should provide valid credentials.
All in all, initialization of SmtpClient should look like this:
SmtpClient SmtpServer = new SmtpClient("smtp.gmail.com", 587);
SmtpServer.EnableSsl = true;
SmtpServer.Credentials = new NetworkCredential("username@gmail.com", "password");
A: as the error says, a STARTTLS command should be used first. Thas means gmail only accepts mail via secure connection. In this answer enableSsl was set to true. As the documentation from microsoft says, the SmtpClient class has such an property too. Furthermore you should leave your credentials in the smptClient. I think gmail only accepts mail from authenticate users. I think the whole problem is solved here.
A: You need to use NetworkCredential to login into Gmail SMTP server. Error is very apparent.
SmtpClient smtp = new SmtpClient("smtp.gmail.com", 587);
smtp.EnableSsl = true;
smtp.UseDefaultCredentials = false;
smtp.Credentials = new NetworkCredential("your-email", "your-password");
A: Have you tried:
smtpServer.Host = "smtp.gmail.com";
smtpServer.Port = 587;
smtpServer.Credentials =
new NetworkCredential("SenderGmailUserName", "SenderPassword");
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,730
|
Home Gym /
With our comprehensive fitness apps reviews, finding the best health and workout apps is easier than ever. Most of us are using our smartphones to do just about everything these days. We manage almost every aspect of our lives in a tiny device that fits in our pockets and goes everywhere with us. It's hard to imagine how we ever lived without them.
We use smartphones to keep our schedules on track, connect with friends, and Google anything and everything on a whim. In these days of COVID-19, many people have even substituted their personal trainer and regular gym visits with their phones. How is this possible, and does it work?
In short, yes, it can, but you need to choose the right app that best fits your health and fitness goals - no easy task when there are over a quarter of a million to choose from, and the list is growing.
Rather than wade through a seemingly endless stream of fitness apps, we've found 7 of the best of the bunch to help lighten your load and make your choice more straightforward.
#1. Couch-to-5K Running App for Beginners
Couch-to-5K is one of the best health and fitness apps for beginners. It is targeted at the beginning runner who may not have the confidence or knowledge needed to build their fitness level to run 5K comfortably.
You get three 30-minute workouts per week to prepare yourself for a 5K (3.1 miles) run in as little as nine weeks.
The app tracks your time and distance, with a virtual coach providing verbal cues to spur you on.
Run Faster (Couch-To-5K, 10K, Half-Marathon and Marathon Running) With Chinese Massage Points - FREE Natural Acupressure Trainer
Run Faster without Medications using Traditional Chinese Massage Points; Easily find the right Massage Points with simple Full HD Videoclips and Photos in the App
#2. Zwift for Treadmills and Cyclists
If your schedule means you do most of your training inside long after the sun sets or before it rises, but you miss being outside with all the other fitness enthusiasts, then Zwift might be just the thing.
Zwift is an online virtual training ground where you can run around a course surrounded by other users running or riding with you in a virtual world. Once you have downloaded the app and joined up, you then pair it up with a compatible treadmill and cast the image to your tablet or TV.
Zwift used to be exclusive to cyclists with compatible smart-trainer but has since been made available to the running community. You can train with friends, join a group, or test your endurance in an online race. Zwift can be incredibly motivating if your enthusiasm is a little lacking.
This is one of the best health and fitness apps for cyclists as well as those who workout on treadmills.
#3. RockMyRun Seamlessly Fusing Fitness and Music
Music fuels the workout of many fitness enthusiasts who rely on the upbeat tempo of their favorite workout tunes to stir up the adrenaline for that last big push to the finish line.
RockMyRun understands what music lovers need to keep them going. You can use RockMyRun to match the tempo of your workout - perfect for interval training. Pump up your jam when you need to push yourself harder or slow it down for a steadier pace. The faster the runner goes, the quicker the tempo will be, with the app choosing from a wide selection of genres.
Syncs perfectly with an Apple Watch
Apple Watch Series 3 (GPS, 38mm) - Space Gray Aluminum Case with Black Sport Band
GPS; Retina display; Swimproof; Optical heart sensor; Stores music, podcasts, and audiobooks
#4. BetterMe for Walking and Weightloss
If your exercise style is more laid back and geared towards long-term weight loss, then BetterMe will fit right in. The app gives you choices in what you want to achieve, including to sleep better, lose weight, or get fit. When you first fire up the app, it will ask you a series of questions about your goals.
Once the questionnaire is complete, the app will use your information to tailor-make a personalized weight loss program with exercises and meal plans. You will also gain access to a water tracker, progress charts, motivational notifications, and articles from leading nutrition and fitness experts.
#5. Muscle Booster Workout Planner for Strength Training
Muscle Booster's claim to fame is as a replacement for your trainer. A smart training algorithm helps you create the perfect exercise regime for building lean muscle based on your goals: weight loss, muscle gain, or being more active.
You will receive weekly updates and gain access to a library of more than 200 exercises. The exercises focus on muscle groups like the chest, arms, back, belly, and legs, so it might be possible to set the app up to skip leg day, but we don't recommend it.
The app algorithm uses your data to create a workout plan that includes strength training, cardio, and recovery periods to maximize your results.
#6. MindBody: Fitness, Salon and Spa for Mental and Physical Fitness
We all feel the effects of stress on occasion and could do with a healthy release when it does. MINDBODY: Fitness, Salon, and Spa isn't a fitness app in the strictest sense of the term. What it does do is help you find the best value fitness in your local area.
If you fancy a bit of yoga, then fire up the app to find the best value classes in town or the latest special deals. Alternatively, if you're looking for a beginner work out to get ready for summer on the beach, then the app will show you where. There are over 52,000 businesses listed in the app, covering everything from yoga to Pilates and HIIT to kickboxing.
Whatever type of fitness you are into or if you are looking for something different, then MINDBODYFITNESS can show you how to get it locally and at the best price. The app also contains locations of top-rated spas in your area where you can relax and pamper yourself after a tough week. It also allows you to connect your fitness tracker to track calories burned and more.
MINDBODY: Fitness, Salon & Spa
Search any location worldwide to discover health, wellness, and beauty businesses; Sign up for classes, reserve a spot on the waitlist, and book or request appointments
#7. Fitbit Fitness Tracker
Fitbit has become somewhat of a household name over the years, with its comprehensive range of fitness trackers held in high regard by fitness enthusiasts worldwide. The Fitbit app dives a lot deeper than just counting steps, as it's grown into a fully featured fitness app that tracks everything from calories burned to sleep patterns.
The Fitbit app works best when paired with a Fitbit device, but it's not necessary. You will record a more in-depth analysis of your activity if you do have one, though. There is a free version and a premium service that grants you access to workouts and recipes from fitness gurus like Ayesha Curry, data-driven advice, workout plans, and premium challenges, to name just a few benefits of a monthly subscription.
For other wearable technology selections, click here.
Fitbit Charge 4 Fitness and Activity Tracker with Built-in GPS, Heart Rate, Sleep & Swim Tracking, Rosewood/Rosewood, One Size (S &L Bands Included)
Whatever your fitness goals, there is an app that can help you get there. This list is by no means a comprehensive one, as there are thousands of fitness apps on the Google Play and Apple App stores.
However, the list above details just a few of the top-rated apps that we think are worth your time. Most have free trials available, so you can check out their features before you pay.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,955
|
Марі́я Оле́гівна (? — після 1146) — руська княгиня з роду Ольговичів, династії Рюриковичів. Донька чернігівського князя Олега Святославича. Була посватана за Скарбимира, палатина польського князя Болеслава Кривоустого. Але 1117 року він повстав проти сюзерена, у зв'язку з чим Марія вийшла заміж за іншого палатина — Петра Властовича. 1146 року, після покарання чоловіка, повернулася з великим ескортом на свою батьківщину.
Примітки
Джерела та література
Ольговичі
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,664
|
ArmyStudyGuide.com - A FREE Online and Audio Army Board Study Guide for U.S. Army Promotion Boards and Soldier / NCO Boards.
Quiz yourself with prep-questions for the Army Soldier and NCO Boards and SGT / SSG Promotion Boards.
Take the Army Study Guide with you wherever you go by downloading our free app to your iPhone. Quiz yourself on more than 1,000 questions, covering 38 topics-and be well on your way to preparing for the US Army Promotion Boards and Soldier/NCO Boards.
Plug in data from the DA Form 705 (Army Physical Fitness Test Scorecard) and get your results for the APFT.
Have questions about your Army benefits? Send them on to our experts or browse for the answers to similar questions here.
schools where you can advance your education and in turn, your career opportunities.
Are You Ready for a Civilian Career?
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,638
|
"Jurassic Park: High Heels Edition" Puts Everyone (Dinos Too) in Stilettos…And It's Everything
XVP Comedy has created the viral Jurassic Park video to end all viral Jurassic Park videos. Jurassic World star Bryce Dallas Howard got a lot of attention (both good and bad) for doing every scene of her dino disaster blockbuster in high heels. So what if EVERYONE had high heels…even the raptors? If this doesn't make you laugh, check your pulse.
Share this news story on Facebook!
Kardashian Imitator In Coma After Butt Implant Explosion At Local Gym
NBC 4 in San Antonio is reporting that Serena Beuford, 27, is in a coma after suffering major injuries at her local gym. She was filming a video for Instagram while doing squats and her butt implants ruptured.
"…she heard a loud pop. Soon after, she fell to the floor screaming in agony… saying that her butt was gone.
According to Beuford's sister Jackie, Serena had visited an unlicensed clinic to get a 64-inch bottom.
She said her sister wanted to become famous on Instagram. That clinic is now under investigation."
Doctors say the damage affects her colon and lower intestines, and that it will take several major procedures to undo the damage done.
Butt implants aren't a good idea (do you really want to be a Kardashian that badly?), but if you must undergo plastic surgery, make sure it's with a reputable and certified physician.
J.K. Rowling Announces New "Harry Potter and the Cursed Child" for 2016
In a stream of late night tweets (for American fans at least), celebrated author and philanthropist J.K. Rowling has confirmed that a new Harry Potter story will premiere in 2016…but it won't be on film.
I'm also very excited to confirm today that a new play called Harry Potter and the #CursedChild will be opening in London next year.
— J.K. Rowling (@jk_rowling) June 26, 2015
It will tell a new story, which is the result of a collaboration between writer Jack Thorne, director John Tiffany and myself. #CursedChild — J.K. Rowling (@jk_rowling) June 26, 2015
I don't want to say too much more, because I don't want to spoil what I know will be a real treat for fans. #CursedChild
However, I can say that it is not a prequel! #CursedChild — J.K. Rowling (@jk_rowling) June 26, 2015
To answer one inevitable (and reasonable!) question – why isn't #CursedChild a new novel? – I am confident that when audiences see the play
they will agree that it was the only proper medium for the story. #CursedChild — J.K. Rowling (@jk_rowling) June 26, 2015
I've had countless offers to extend Harry's story over the years, but Jack, John and Sonia Friedman are a dream team! #CursedChild
It has been a huge pleasure to share with them (and soon, with you!) this untold part of Harry's story. #CursedChild — J.K. Rowling (@jk_rowling) June 26, 2015
The new play will no doubt break box office records in what will be a banner year for Potter fans. This new play featuring Harry himself, plus the new Fantastic Beasts and Where to Find Them film trilogy starring Eddie Redmayne, and the opening of the Wizarding World of Harry Potter at Universal Studios Hollywood. It's a great time to be a wizard!
As the play will be a huge success in London, we hope to see it replicated on Broadway and on tour in the United States as soon as 2017.
Share this awesome news on Facebook!
Bad Lip Reading Tackles "The Avengers" and We All Win
The Redneck Avengers are here to save 'murica. So many LOL's!
Share this hilarious video on Facebook!
Poke Your Screen for this Whole Video – It's Mesmerizing
Start the video and put it in full-screen mode – then follow the directions (hint: you'll need to put your finger in the middle of the screen). Then enjoy 3 and a half minutes of bliss. Namie Amuro's got a catchy little beat going here too!
Share this amazing video on Facebook!
What This Guy Can Make Out of Bananas Is Unreal!
Master Disney Animator's New Short Will Take Your Breath Away
A Cappella Cover of "Chandelier" Puts Pitch Perfect To Shame
One Guy, Over a Dozen Disney and Pixar Voices: "Let It Go"…Again
DMCA Copyright Policy | Privacy Policy
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 567
|
La nacionalidad del buque es el vínculo jurídico que une a un determinado buque con el Estado al que pertenece y cuyo pabellón nacional o bandera de popa enarbola.
En tiempos pasados, la nacionalidad del buque era un elemento subjetivo de gran importancia porque le permitía gozar de la protección de las leyes del Estado de su abanderamiento mientras se encontrase en aguas internacionales.
Modernamente, con la proliferación de los pabellones o banderas de conveniencia, la nacionalidad del buque ha ido perdiendo su vigencia.
Derecho internacional
En el Derecho internacional público, el otorgamiento de la nacionalidad del buque se regula por la Convención de las Naciones Unidas sobre el Derecho del Mar, hecha en Montego Bay el 10 de diciembre de 1982, cuyo artículo 91 establece lo siguiente: «1. Cada Estado establecerá los requisitos necesarios para conceder su nacionalidad a los buques, para su inscripción en un registro en su territorio y para que tengan el derecho de enarbolar su pabellón. Los buques poseerán la nacionalidad del Estado cuyo pabellón estén autorizados a enarbolar. Ha de existir una relación auténtica entre el Estado y el buque. 2. Cada Estado expedirá los documentos pertinentes a los buques a que haya concedido el derecho a enarbolar su pabellón».
Régimen jurídico español
En el Derecho español, la regulación la encontramos en la Ley 14/2014, de 24 de julio, de Navegación Marítima y en el Real Decreto 1027/1989, de 28 de julio, sobre abanderamiento, matriculación de buques y registro marítimo.
El artículo 88 de la citada Ley 14/2014, de 24 de julio, de Navegación Marítima, establece lo siguiente: «El abanderamiento es el acto que otorga el derecho a enarbolar el pabellón español. Todos los buques matriculados en el Registro de Buques y Empresas Navieras estarán abanderados en España. Las condiciones para la concesión del abanderamiento se rigen por lo dispuesto en la normativa de marina mercante». Y de acuerdo con el artículo 90: «Los buques debidamente matriculados y abanderados en España tendrán, a todos los efectos, la nacionalidad española».
Por otro lado, el artículo 14 del citado Real Decreto 1027/1989 establece lo siguiente: «Se entiende por abanderamiento de un buque el acto administrativo por el cual y tras la tramitación, prevista en este Real Decreto, se autoriza a que el buque arbole el pabellón nacional». Y de conformidad con su artículo 2: «Para estar amparados por la legislación española, acogidos a los derechos que ésta concede y arbolar la bandera española, los buques, embarcaciones y artefactos navales deberán estar matriculados en uno de los Registros de Matrícula de Buques de las Jefaturas Provinciales de Marina Mercante. Cada buque, embarcación o artefacto naval sólo podrá estar matriculado en uno de los Registros enunciados en el párrafo anterior».
Véase también
Organización Marítima Internacional
Pabellón de conveniencia
Referencias
Derecho del mar
Derecho marítimo
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,948
|
Subsets and Splits
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