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Q: First [Monday] of the month, Second [Thursday] of the month, etc. Delphi Using Delphi, I need a function to evaluate the current date and see if it's, for example, the Third Sunday of the month, etc.
In pseudocode:
function IsFirst(const CurrentDateTime: TDateTime; const Day: Word): Boolean;
/// Day can be 1-7 (monday to sunday)
begin
Result:= ??
end;
Another function would be needed to calculate the Second, Third, Forth and Last of the month. DateUtils seems to have nothing like that. Any ideas?
A: This can be done using simple math.
Get the DayOfTheWeek and divide the DayOf by seven.
A: This function is what you need:
function IsFirst(const DateTime: TDateTime; const Day: Word): Boolean;
begin
Result := (DayOfTheWeek(DateTime)=Day) and
InRange(DayOfTheMonth(DateTime), 1, 7);
end;
The equivalent function for the second occurrence is:
function IsSecond(const DateTime: TDateTime; const Day: Word): Boolean;
begin
Result := (DayOfTheWeek(DateTime)=Day) and
InRange(DayOfTheMonth(DateTime), 8, 14);
end;
I'm sure you can fill out the details for third, fourth and fifth. You may prefer to write a single general function like this:
function IsNth(const DateTime: TDateTime; const Day: Word;
const N: Integer): Boolean;
var
FirstDayOfWeek, LastDayOfWeek: Integer;
begin
LastDayOfWeek := N*7;
FirstDayOfWeek = LastDayOfWeek-6;
Result := (DayOfTheWeek(DateTime)=Day) and
InRange(DayOfTheMonth(DateTime), FirstDayOfWeek, LastDayOfWeek);
end;
| {
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Melanoplus ascensor är en insektsart som först beskrevs av Scudder, S.H. 1897. Melanoplus ascensor ingår i släktet Melanoplus och familjen gräshoppor. Inga underarter finns listade i Catalogue of Life.
Källor
Gräshoppor
ascensor | {
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Medical offices are drowning under mounds of paper; some practices even have rooms full of paper occupying valuable space that could be used for patient exams that produce revenue. Even those practices using Electronic Health Records are typically still managing some aspects of their practice using paper, whether it be through practice faxes or a variety of other items.
iSALUS offers a truly fresh experience when it comes to paper management. ALL of your paper can be completely eliminated and stored in the iSALUS cloud. | {
"redpajama_set_name": "RedPajamaC4"
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{"url":"https:\/\/math.stackexchange.com\/questions\/3929469\/break-a-stick-into-two-pieces-and-break-the-shorter-one-again-whats-the-pdf-o","text":"Break a stick into two pieces, and break the shorter one again, what's the pdf of the shortest pieces.\n\nBreak a stick of unit length at a uniformly chosen random point. Then take the shorter of the two pieces and break it into two again at a uniformly chosen random point. Let X denote the shortest of the final three pieces. Find the probability density function of X.\n\nI let the shorter one on the first break be x, which should be between (0, $$1\/2$$), so the pdf should be $$f(x)=2, 0 then the conditional probability of $$f(x|y)$$ should be (since it is another uniform distribution) $$f(x|y)=\\frac{1}{y\/2-0}=\\frac{2}{y}$$ by applying conditional probability formular $$f_X(x)=\\int_0^{1\/2}f_{X|Y}(x|y)f_Y{y}dy=\\int_0^{1\/2}\\frac{2}{y}\\times2dy=4ln(y)|^{1\/2}_0$$\n\nI stopped here since it is impossible. Can someone help?\n\nThe last integral should be taken from $$x$$ to $$1\/2$$, since $$f(x \\mid y)$$ is zero when $$x > y$$.\n$$p(y) = 8 - 32 y\\quad {\\rm for}\\ 0\\leq y \\leq 1\/4$$\nand $$0$$ otherwise.\nAfter the first break the short leg is uniformly likely between $$0 \\leftrightarrow 1\/2$$, as shown by the range of the abscissa in the graph.\nAfter this short piece is broken, it longer remaining part is uniformly distributed in $$1\/4 \\leftrightarrow 1\/2$$. The other (shorter) part obeys a triangle distribution between $$0 \\leftrightarrow 1\/4$$. So just write down a triangle distribution that goes between $$0 \\leftrightarrow 1\/4$$.","date":"2021-04-13 17:44:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 15, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9980736374855042, \"perplexity\": 390.6179428726457}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038073437.35\/warc\/CC-MAIN-20210413152520-20210413182520-00099.warc.gz\"}"} | null | null |
Samo Vidovič, slovenski nogometaš, * 24. september 1968, Črna na Koroškem.
Vidovič je nekdanji profesionalni nogometaš, ki je igral na položaju napadalca. V svoji karieri je igral za slovenske klube Ljubljano, Korotan Prevalje, Rudar Velenje, Domžale, Dravograd in Celje ter ob koncu kariere za avstrijske SAK Klagenfurt, DSG Sele/Zell in Eberndorfer AC. Skupno je v prvi slovenski ligi odigral 255 tekem in dosegel 65 golov. Junija 1991 je nastopil na neuradni prijateljski tekmi slovenske reprezentance proti hrvaški.
Sklici
Zunanje povezave
Slovenski nogometaši
Nogometaši NK Ljubljane
Nogometaši NK Korotana
Nogometaši NK Rudarja Velenje
Nogometaši NK Domžal
Nogometaši NK Dravograda
Nogometaši NK Celja
Nogometaši SAK Klagenfurta
Nogometaši DSG Sele/Zell
Nogometaši Eberndorferja AC | {
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\section{Introduction}
\label{sec:introduction}
Charge transfer (CT) in biological molecules attracts recently considerable interest among the physical, chemical, biological and medical communities. It also attracts a broad spectrum of interdisciplinary scientists and engineers. This is because CT constitutes the basis of many biological processes e.g. in various proteins~\cite{Page:2003} including metalloproteins~\cite{GrayWinkler:2010} and enzymes~\cite{Moser:2010} with medical and bioengineering applications \cite{Artes:2014,Kannan:2009}.
CT plays a central role in DNA damage and repair \cite{Dandliker:1997,Rajski:2000,Giese:2006} and it, also, might be an indicator to discriminate between pathogenic and non-pathogenic mutations at an early stage \cite{Shih:2011}.
At least for twenty years, there have been many experimental attempts to recognize the electronic properties of DNA cf. e.g.~\cite{Storm:2001,Porath:2000,Cohen:2005,Yoo:2001,FinkSchoenenberger:1999,Xu:2004}.
Today we know that many factors related to environment (the aqueous solution, the concentration of counterions), extraction process, conducts, purity, base-pair sequence, geometry etc influence CT in DNA. These factors can be categorized into intrinsic and extrinsic. In this work we focus on maybe the most important of the intrinsic factors, i.e. the effect of alternating the base (or base-bair) sequence, which affects the overlaps across the $\pi$-stack. Additionally, we have to discriminate between the words transport (usually implying the use of electrodes), transfer, migration (a transfer over rather long distances). The carriers (electrons or holes) can be inserted via electrodes, generated by UV irradiation or by reduction and oxidation. The nature of charge transport, transfer, migration along the DNA double helix is important for many scientific fields like physics, chemistry, biology, medicine and engineering.
Although unbiased charge transfer in DNA nearly vanishes after 10 to 20 nm~\cite{Simserides:2014, LChMKTS:2015}, DNA still remains a promising candidate as an electronic component in molecular electronics, e.g. as a short molecular wire~\cite{Wohlgamuth:2013}. Favouring geometries and base sequences have still to be explored e.g. incorporation of sequences serving as molecular rectifiers, using non-natural bases or using the triplet acceptor anthraquinone for hole injection~\cite{LewisWasielewski:2013}. Structural fluctuations could be another important factor which influences quantum transport
through DNA molecular wires~\cite{Gutierrez:2010}. The contact of DNA segments with experimentally involved surfaces and interfaces is another research direction e.g. for bio-sensoric applications. For example, charge transfer on the contact of DNA with gold nanoparticles~\cite{Abouzar:2012} and polyelectrolyte multilayers \cite{Poghossian:2013} has been recently investigated.
During the last decade, the scientific literature has been enriched with works studying carrier oscillations within ``molecular'' systems.
Real-Time Time-Dependent Density Functional Theory (RT-TDDFT)~\cite{RT-TDDFT} simulations predicted oscillations ($\approx$ 0.1-10 PHz) within p-nitroaniline and FTC chromophore~\cite{Takimoto:2007}, as well as within zinc porphyrin, green fluorescent protein chromophores and the adenine-thymine base pair~\cite{LopataGovind:2011}.
It was shown that in a simplified single-stranded helix of 101 bases, a collinear uniform electric field induces THz Bloch oscillations~\cite{Malyshev:2009}.
Single and multiple charge transfer within a typical DNA dimer in connection to a bosonic bath has been studied, too~\cite{Tornow:2010}.
Each base pair was approximated by a single site, as in our Tight Binding (TB) approach at the base-pair level denoted in the present article as TB I [cf. Section~\ref{sec:general} and Appendix A]. In the subspace of single charge transfer between base pairs, having initially placed the charge at the donor site and having used a ``typical hopping matrix element'' of 0.2 eV, the authors obtained a period slightly greater than 10 fs. Let us call $T$ the period and $f$ the frequency.
Applying our equation~\cite{Simserides:2014,LKGS:2014} $f= \frac{1}{T} = \frac{\sqrt{(2t)^2 + \Delta^2}}{h}$, with $ t = $ 0.2 eV for the ``typical hopping matrix element'' and difference of the on-site energies $\Delta = 0$ for identical dimers i.e. as in Ref.~\cite{Tornow:2010},
we obtain $T \approx$ 10.34 fs, which agrees splendidly with the dotted line in Fig.~4 of Ref.~\cite{Tornow:2010}.
Recently, we studied B-DNA dimers, trimers and polymers with TB I~\cite{Simserides:2014,LChMKTS:2015,LKGS:2014}.
This approach allowed us to readily determine the spatiotemporal evolution of holes or electrons along a $N$ base-pair DNA segment. With TB I, we have already shown~\cite{Simserides:2014,LKGS:2014} that for all dimers and for trimers made of identical monomers the carrier movement is periodic with frequencies in the mid- and far-infrared i.e. approximately in the THz domain~\cite{ISO20473}. This part of the electromagnetic (EM) spectrum is significant for biological sciences because it can be used to extract complementary to traditional spectroscopic measurements information e.g. on low-frequency bond vibrations, hydrogen bond stretching, bond torsions in liquids and gases etc and because it is relatively non-invasive compared to higher-frequency regions of the EM spectrum~\cite{Yin:2012}. Within TB I, we also showed that, generally, increasing the number of monomers above three, periodicity is lost~\cite{Simserides:2014,LKGS:2014}. Even for the simplest tetramer, the carrier movement is not periodic~\cite{Lambropoulos:2014}.
For periodic cases, we defined~\cite{Simserides:2014,LKGS:2014} the maximum transfer percentage $p$, e.g. the maximum probability to find the carrier at the last monomer having placed it initially at the first monomer and the pure maximum transfer rate $\frac{p}{T}=pf$.
For all cases, either periodic or not, the pure mean transfer rate $k$ (cf.~Eq.~\ref{meantransferrateN(I)} in Appendix A) and the speed $u = kd$, where $d = (N-1) \times $ 3.4 {\AA} is the charge transfer distance, can be used to characterize the system. Within TB I, our analytical calculations and numerical results showed that for dimers $k=2\frac{p}{T}$ and for trimers made of identical monomers $k \approx 1.3108 \frac{p}{T}$.
Using $k$ to evaluate the easiness of charge transfer, we calculated the inverse decay length $\beta$ for exponential fits $k(d)$ and the exponent $\eta$ for power law fits $k(N)$. Studying B-DNA polymers and segments taken from experiments~\cite{Simserides:2014,LChMKTS:2015}, we determined the ranges of values of $\beta$ and $\eta$. Our TB I was used~\cite{Simserides:2014} to reproduce theoretical and experimental findings for various B-DNA segments~\cite{WangLewisSankey:2004,Giese:2001,Murphy:1993,Arkin:1996,Giese:1999}. For monomer-polymers and dimer-polymers, we studied HOMO and LUMO eigenspectra, mean over time probabilities to find the carrier at a particular monomer and mean transfer rates, we illustrated how increasing the number of different parameters involved in TB I, the fall of $k(d)$ or $k(N)$ becomes steeper and we circumscribed the range covered by $\beta$ and $\eta$ ~\cite{LChMKTS:2015}. Finally, both for the time-independent and the time-dependent problem, we analyzed the {\it palindromicity} and the {\it degree of eigenspectrum dependence} of the probabilities to find the carrier at a particular monomer~\cite{LChMKTS:2015}.
As in Refs.~\cite{Simserides:2014,LChMKTS:2015,LKGS:2014,Lambropoulos:2014}, here we call \textit{monomer} a B-DNA base pair and study carrier oscillations in \textit{dimers} and \textit{trimers}. Moreover, for the first time we study \textit{monomers}.
However, here we employ two Tight Binding (TB) approaches:
(I) at the base-pair level, using the on-site energies of the base pairs and the hopping parameters between successive base pairs which is \textit{a wire model}~\cite{CMRR:2007:Chapter} (this was the approach employed in our previous works~\cite{Simserides:2014,LChMKTS:2015,LKGS:2014,Lambropoulos:2014})
and
(II) at the single-base level, using the on-site energies of the bases and the hopping parameters between neighbouring bases, specifically between
(a) two successive bases in the same strand,
(b) complementary bases that define a base pair, and
(c) diagonally located bases of successive base pairs, which is \textit{an extended ladder model} since it also includes the diagonal hoppings (c). Hence, it is more elaborate than the usual ladder model~\cite{CMRR:2007:Chapter}. Extrinsic effects, such as the aqueousness and the presence of counterions can also be taken into account in TB models which then can be renormalized via a decimation procedure to obtain either wire or ladder models, like the ones presented here \cite{Macia:2009}.
Inclusion of the diagonal hoppings is essential e.g. for dimers made of identical monomers with crosswise purines cf. Section 4.
Additionally, TB II allows us to study charge oscillations within monomers, which is not possible for TB I since there a site is a base pair.
A few preliminary results with TB II have been included in Ref.~\cite{PIERS:2015} and in Ref.~\cite{Kaklamanis:2015}.
We assume that isolation of a few consecutive B-DNA base pairs is possible,
e.g. by connecting at the boundaries moieties with very small transfer integrals with the segment of interest.
The TB parameters that we use can be found in Refs.~\cite{Simserides:2014,HKS:2010-2011,MehrezAnantram:2005}.
We solve analytically and numerically, with the eigenvalue method, a system of (I) $N$ or (II) $2N$ coupled differential equations to determine the spatiotemporal evolution of an extra carrier (electron or hole) along a $N$ base-pair DNA segment.
Carriers move either between the HOMOs or between the LUMOs of the relevant sites [(I) base pairs, (II) bases].
A legitimate critique to our theoretical predictions could be relative to a possible comparison with the experiment.
As far as we know, such experiments in so short DNA segments (monomers, dimers, trimers) do not exist.
However, nowadays a variety of experimental techniques can be used to probe CT in biological molecules including optical, electrochemical and scanning probe techniques such as Electrochemical STM and Conductive AFM \cite{Artes:2014}.
Another method is the femtosecond transient absorption spectroscopy, with which one can measure the transfer rates of photoinduced carriers (holes or electrons) in small molecular systems \cite{Berera:2009,Gorczak:2015}. This method has been successfully employed in order to study single and double-stranded DNA oligonucleotides, slightly larger than the systems studied here \cite{CrespoHernandez:2005}.
In our case, we can imagine the isolation of an oligomer by putting at its end moieties with very small hopping integral with the ends of the oligomer under investigation and subsequent use of one of these techniques. The techniques must be in the position to probe oscillations with frequency content in the 0.1 to 1000 THz regime i.e. time scales of 10 ps to 1 fs.
Additionally, our method TB I has already been used to successfully reproduce experimental results with longer DNA segments relative to transfer rates and mean occupation probabilities~\cite{Simserides:2014}.
In this work we show that for monomers TB II predicts periodic carrier oscillations with frequency $f \approx$ 50-550 THz (but with very small transfer percentages), while TB I predicts periodic carrier oscillations with $f \approx$ 0.25-100 THz for dimers and with $f \approx$ 0.5-33 THz for trimers made of identical monomers. In other cases, either with TB I or TB II, oscillations are not strictly periodic, but Fourier analysis shows similar frequency content.
For dimers and trimers TB I and TB II give complementary aspects of the oscillations.
For dimers made of identical monomers we have large carrier transfer and the occupation probability is equally shared between the two monomers which constitute the dimer. For dimers made of identical monomers, if purines are crosswise to purines, interstrand carrier transfer dominates, i.e. we have significant diagonal transfer uncovered by TB II which includes \textit{diagonal hoppings}; if purines are on the same strand, intrastrand carrier transfer dominates. For dimers made of different monomers, TB II basically shows intrastrand carrier transfer (but in small percentage).
In this work we show that THz oscillations in DNA monomers, dimers and trimers exist and we study the frequency content, the maximum transfer percentages, the transfer rates between sites and the mean probabilities to find the carrier at a site. Hence, one could imagine the future built of a source or receiver of EM radiation in the range 0.1 THz to 1000 THz made of tiny DNA segments.
The rest of the article is organized as follows:
In Sec.~\ref{sec:general} we outline our TB I and TB II approaches in general terms skipping technical details, which are presented in Appendix A.
Our results for monomers, dimers and trimers are presented in Sections~\ref{sec:ResultsMonomers}, \ref{sec:ResultsDimers} and \ref{sec:ResultsTrimers}, respectively. In Sec.~\ref{sec:Conclusion} we state our conclusions.
\section{General}
\label{sec:general}
We begin with our notation. By YX we denote two successive base pairs, according to the convention
\begin{eqnarray*}
&\vdots& \\
5' & & 3' \\
\textrm{Y} & - & \textrm{Y}_{\textrm{compl}} \\
\textrm{X} & - & \textrm{X}_{\textrm{compl}} \\
3' & & 5' \\
&\vdots&
\label{bpdimer}
\end{eqnarray*}
for the DNA strands orientation.
We denote by X, X$_{\textrm{compl}}$, Y, Y$_{\textrm{compl}}$ DNA bases, where
X$_{\textrm{compl}}$ (Y$_{\textrm{compl}}$) is the complementary base of X (Y).
In other words, the notation YX means that the bases Y and X of
two successive base pairs are located at the same strand in the direction $5'-3'$.
X-X$_{\textrm{compl}}$ is the one base pair and
Y-Y$_{\textrm{compl}}$ is the other base pair,
separated and twisted by 3.4 {\AA} and $36^{\circ}$, respectively,
relatively to the first base pair, along the growth axis of the nucleotide chain.
For example, the notation GT denotes that one strand contains G and T in the direction $5'-3'$ and the complementary
strand contains C and A in the direction $3'-5'$.
In the sense explained above, we can talk for equivalent dimers i.e.
$\textrm{YX} \equiv \textrm{X}_{\textrm{compl}}\textrm{Y}_{\textrm{compl}}$
and expand the notion of equivalency to $N$-mers.
Furthermore, we suppose that an extra hole or electron inserted in a DNA segment travels through HOMOs or LUMOs, respectively. Hence, for each base pair or {\it monomer},
the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) play a key role.
We utilize two Tight-Binding approximations, the main points of which are explained below. The mathematical details are given in Appendix A. Here we merely describe them in general terms to make the manuscript more accessible for readers from different areas of the rather wide DNA science.
In TB I the carrier is located at a base pair and it can move to the next or to the previous base pair. Hence, this is a \textit{wire model}~\cite{Albuquerque:2014,CMRR:2007}. In TB II the carrier is located at a base and it can move (1) to its complementary base of the same base pair or (2) to the next or the previous base of the same strand or (3) to the diagonally located base of the other strand of the next or the previous base pair in the $5'-5'$ or in the $3'-3'$ direction, respectively. If we ignore (3), this would be a \textit{ladder model}~\cite{Albuquerque:2014,CMRR:2007}. Here, since we also include diagonal hoppings we call it an \textit{extended ladder model}~\cite{Albuquerque:2014,CMRR:2007}.
It will become evident below that in some cases (e.g. in dimers when purines are crosswise to purines) interstrand carrier transfer dominates, in other words we have significant diagonal transfer, which justifies the inclusion of diagonal hoppings in our model TB II. The two TB models are explained schematically in Fig.~\ref{fig:TBITBII}.
\begin{figure} [h!]
\centering
\includegraphics[width=7.5cm]{TBI.eps}
\includegraphics[width=7.5cm]{TBII.eps}
\caption{Our Tight-Binding approaches: TB I (left) and TB II (right).}
\label{fig:TBITBII}
\end{figure}
In Sections \ref{sec:ResultsMonomers}, \ref{sec:ResultsDimers}, and \ref{sec:ResultsTrimers} we show that THz oscillations in DNA monomers, dimers and trimers exist. We study the frequency content of these oscillations; where necessary, we employ Fourier analysis. For periodic cases, we study the maximum carrier transfer percentage $p$ from an initial site to a final site, as well as the relevant maximum transfer rate $pf$ showing not only how fast the transfer is, but also what the maximum carrier transfer percentage is. For all cases, either periodic or not, we use the mean transfer rate $k_{ij}$, which shows not only how fast the transfer from site $i$ to site $j$ is, but also what the mean carrier transfer percentage from site $i$ to site $j$ is (cf. Eq.~\ref{meantransferrateN}).
\section{Monomers}
\label{sec:ResultsMonomers}
TB I cannot be used for charge transfer in monomers, since it considers a monomer as a single site. Hence, we use TB II, supposing that initially we place the carrier at one of the bases. We can prove that an extra hole or electron oscillates between the bases of the two possible monomers (G-C and A-T) with frequency (or period)
\begin{equation}\label{fandT}
f = \frac{1}{T} = \frac{\sqrt{(2t)^2 + \Delta^2}}{h},
\end{equation}
where $t$ is the hopping integral between the complementary bases and $\Delta$ is the energy gap between the on-site energies of the complementary bases.
Our results for A-T and G-C, both for holes and electrons, are shown in Fig.~\ref{fig:monomers}, with parameters from Ref.~\cite{HKS:2010-2011} (``HKS parametrization'') and from Ref.~\cite{MehrezAnantram:2005} (``MA parametrization'').
For HKS parametrization, $f \approx$ 50-200 THz ($T \approx$ 5-20 fs),
for MA parametrization, $f \approx$ 250-550 THz ( $T \approx$ 2-4 fs).
These ranges correspond to wavelength $\lambda \approx$ 545 nm - 6000 nm i.e. from visible to near-infrared and mid-infrared~\cite{ISO20473}.
We can prove that the maximum transfer percentage $p$
[e.g. $\max{(|B_{1}(t)|^2)}$ for initial conditions $A_{1}(0)=1, B_{1}(0)=0$ or vice versa], is given by
\begin{equation}\label{p}
p = \frac{(2t)^2}{(2t)^2 + \Delta^2}.
\end{equation}
We observe that the carrier is not very likely to be transferred between the monomer bases ($p$ is very small in all cases).
The pure maximum transfer rate defined as $pf$ is also here very small in all cases.
The pure mean transfer rate $k$ is also shown.
It can be analytically proven and numerically shown that here $k=2pf$.
$T$, $f$, $p$, $pf$ and $k$ do not depend on which base the carrier is initially placed at.
A snapshot of electron oscillations between G and C in G-C, according to the HKS parametrization, is given in Fig.~\ref{fig:ElectronOscillationsG-C}.
\begin{figure} [h!]
\includegraphics[width=7.8cm]{TandfHKS.eps}
\includegraphics[width=7.8cm]{pandpfkHKS.eps}
\includegraphics[width=7.8cm]{TandfMA.eps}
\includegraphics[width=7.8cm]{pandpfkMA.eps}
\caption{Charge oscillations in A-T and G-C according to the TB single-base approach II:
period $T$, frequency $f$, maximum transfer percentage $p$, pure maximum transfer rate $pf$ and pure mean transfer rate $k$.
1st row: TB parameters from Ref.~\cite{HKS:2010-2011} (HKS parametrization).
2nd row: TB parameters from Ref.~\cite{MehrezAnantram:2005} (MA parametrization).
}
\label{fig:monomers}
\end{figure}
\begin{figure} [h!]
\centering
\includegraphics[width=15cm]{G-C10LUMOOscillations.eps}
\caption{Electron oscillations within G-C, according to TB single-base approach II and HKS parametrization.}
\label{fig:ElectronOscillationsG-C}
\end{figure}
This is a two-level system of given stationary states (the two HOMOs or the two LUMOs) with a ``perturbation'' represented by the hopping integral, which impels an extra carrier to oscillate between these stationary states. Mathematically, the problem is equivalent to a two-level system (e.g. atom) under the influence of an electric field, which impels an electron to oscillate between the two eigenstates [semiclassical approach after Rotating Wave Approximation or the time-dependent problem with a Jaynes-Cummings Hamiltonian in a full quantum mechanical approach].
These problems are well known in the context of quantum optics~\cite{QOL:2016}.
The same applies to the dimer problem within TB I discussed briefly below.
Increasing the number of monomers which make up the DNA segment, either with TB I or TB II, we have to solve generalizations of the above mentioned problem.
In other words, to determine the spatiotemporal evolution of electrons or holes along a $N$ base-pair DNA segment, we have to solve a system of (I) $N$ or (II) $2N$ coupled differential equations. The relevant matrices $A$ are given in Appendix B.
For $N$ base pairs, the system has $N$ states within TB I, or $2N$ states within TB II. Thus, in this work, for TB I we examine systems composed of two (dimers) or three (trimers) states, while for TB II we examine systems composed of two (monomers), four (dimers) or six (trimers) states.
\section{Dimers}
\label{sec:ResultsDimers}
The possible dimers result from permutations with repetition of two out of the four bases taking additionally into account that
at the base located in helix $\sigma=1$ corresponds always its complementary base in helix $\sigma=2$ .
The number of possible permutations with repetition is $PR(4,2) = 4^2 = 16$. However, six of them are equivalent to other six i.e. GG$\equiv$CC, AA$\equiv$TT, AG$\equiv$CT, AC$\equiv$GT, TG$\equiv$CA, TC$\equiv$GA. Hence, the possible dimers are 10.
Recently, with TB I, we proved that the carrier movement in all dimers is strictly periodic~\cite{Simserides:2014,LKGS:2014,Lambropoulos:2014}.
The frequencies (or periods) are given by Eq.~\ref{fandT}, where, now $t$ is the hopping integral between the base pairs and $\Delta$ is the energy gap between the on-site energies of the base pairs.
Using the TB parameters of Ref.~\cite{Simserides:2014}, we found~\cite{Simserides:2014,LKGS:2014,Lambropoulos:2014} $f \approx$ 0.25-100 THz, i.e. $T \approx$ 10-4000 fs, i.e. wavelength $\lambda \approx$ 3-1200 $\mu$m, in other words mainly in the MIR and the FIR range~\cite{ISO20473}.
We found that the maximum transfer percentage $p=1$ for dimers made of identical monomers, but $p<1$ for dimers made of different monomers and that the values of $f, T, p, pf, k$ do not depend on which of the two monomers the carrier is initially placed at (cf. also Eq.~\ref{periodsorfrequencies}).
The HKS parametrization~\cite{HKS:2010-2011} results in the same frequency range, although, the predicted frequencies vary slightly due to the different values of the TB parameters (a summarizing graph is given in Fig.~\ref{fig:dimersHKS}).
\begin{figure} [h!]
\centering
\includegraphics[width=7.5cm]{dimersTandfHKSHOMO.eps}
\includegraphics[width=7.5cm]{dimersppfkHKSHOMO.eps}
\includegraphics[width=7.5cm]{dimersTandfHKSLUMO.eps}
\includegraphics[width=7.5cm]{dimersppfkHKSLUMO.eps}
\caption{Charge oscillations in all possible dimers according to the TB base-pair approach I and the HKS parametrization~\cite{HKS:2010-2011}:
period $T$, frequency $f$, maximum transfer percentage $p$, pure maximum transfer rate $pf$ and pure mean transfer rate $k$. It can be analytically proven and numerically shown that here $k=2pf$.}
\label{fig:dimersHKS}
\end{figure}
Let us now compare TB approaches I and II relatively to the frequency content. Using TB II, one cannot strictly determine periodicity in the carrier movement between the four bases. Hence, $f, T, p, pf$ cannot be defined, but Fourier analysis shows similar frequency content in the THz domain, cf. Appendix C, where we depict the Fourier spectra given by Eq.~\ref{FourierSpectra}, within TB II and HKS parametrization~\cite{HKS:2010-2011}.
Let us start with GG, a dimer made of identical monomers with purines on purines (Fourier spectra in Appendix C, Fig.~\ref{fig:FourierGG}).
If we initially place the hole at A1(G) or A2(G), we obtain the main Fourier amplitude at $f \approx$ 30 THz.
If we initially place the hole at B1 (C) or B2 (C), we obtain the main Fourier amplitude at $f \approx$ 32 THz.
The rest of the frequencies show up with almost negligible amplitudes.
These results are in accordance with TB I and HKS parametrization~\cite{HKS:2010-2011} where for the GG dimer we obtain $f \approx$ 30 THz.
With TB I and the parametrization of Refs.~\cite{Simserides:2014,LKGS:2014} we had obtained $f \approx$ 48 THz~\cite{Simserides:2014,LKGS:2014}.
The amplitudes at the main frequencies are $\approx$ 0.5, in accordance with Eq.~(\ref{FourierSpectra}), expressing the fact that for GG the mean probability to find the hole at a base is, approximately, almost exclusively, equally divided between the base the carrier was initially placed at and the other base of the same strand, cf. also Fig.~\ref{fig:DimersIIHOMO} below.
Let us now continue with GC, a dimer made of identical monomers with crosswise purines (Fourier spectra in Appendix C, Fig.~\ref{fig:FourierGC}).
If we initially place the hole at A1(G) or B2(G), we obtain the main Fourier amplitude at $f \approx$ 0.3 THz.
If we initially place the hole at B1 (C) or A2 (C), we obtain the main Fourier amplitude peak at $f \approx$ 1.6 THz.
The rest of the frequencies show up with almost negligible amplitudes.
These results are in accordance with TB I and HKS parametrization~\cite{HKS:2010-2011} where for the GC dimer we obtain $f \approx$ 0.5 THz.
With TB I and the parametrization of Refs.~\cite{Simserides:2014,LKGS:2014} we had obtained $f \approx$ 4.8 THz~\cite{Simserides:2014,LKGS:2014}.
The amplitudes at the main frequencies are $\approx$ 0.5, in accordance with Eq.~\ref{FourierSpectra}, expressing the fact that for GC the mean probability to find the hole at a base is, approximately, almost exclusively, equally divided between the base the carrier was initially placed at and the diagonally located base at the opposite strand, cf. also Fig.~\ref{fig:DimersIIHOMO} below.
Let us finish with CT, a dimer made of different monomers (Fourier spectra in Appendix C, Fig.~\ref{fig:FourierCT}).
The main Fourier amplitude is at $f \approx$ 70.75 THz, if the hole is initially placed at C or T, otherwise hole transfer is negligibly small.
Within TB I and the HKS parametrization~\cite{HKS:2010-2011}, for the CT dimer we obtain $f \approx$ 72.5 THz.
With TB I and the parametrization of Refs.~\cite{Simserides:2014,LKGS:2014} we had obtained $f \approx$ 74 THz~\cite{Simserides:2014,LKGS:2014}.
The amplitudes at the main frequencies are $\approx$ 0.25, when the hole is initially placed at C or T and tiny when the hole is initially placed at G or A, in accordance with Eq.~\ref{FourierSpectra}, expressing the fact that for CT the mean probability to find the hole at a base is approximately 0.75 at the base the carrier was initially placed at and approximately 0.25 at the other base of the same strand when these bases are C and T, but, hole transfer is negligibly small when the hole is initially placed at G or A, cf. also Fig.~\ref{fig:DimersIIHOMO} below.
Examples of snapshots of hole oscillations in GG, GC and CT dimers, according to the HKS parametrization and TB II, are given in Fig.~\ref{fig:HoleOscillationsDimers}.
\begin{figure} [h!]
\centering
\includegraphics[width=15cm]{GG1000HOMOoscillations.eps}
\includegraphics[width=15cm]{GC1000HOMOoscillations.eps}
\includegraphics[width=15cm]{CT1000HOMOoscillations.eps}
\caption{ From top to bottom: Hole oscillations within the GG dimer, the GC dimer and the CT dimer, according to the TB single-base approach II and the HKS parametrization. Continuous black (dashed red) lines correspond to the 1st (2nd) base pair.}
\label{fig:HoleOscillationsDimers}
\end{figure}
Hence, generally, both TB approaches, independently of the specific parametrization, predict oscillations in the same THz range. The little differences arise from the specific values used for the on-site energies and the hopping parameters in each parametrization.
TB approaches I and II allow us to determine the mean probability to find the carrier at a site [base pair for I, base for II].
A comparison between the mean probabilities obtained with TB approaches I and II is shown in Fig.~\ref{fig:comparison}, using the HKS parametrization~\cite{HKS:2010-2011}.
Comparing the two TB approaches e.g. looking at Fig.~\ref{fig:comparison}, but also focusing on TB II e.g. looking at Figs.~\ref{fig:DimersIIHOMO}-\ref{fig:DimersIILUMO} below, we reach the following conclusions:
(a) Carrier transfer is large in dimers made of identical monomers: Finally, the probability is equally shared between the two monomers which make up the dimer.
(b) For dimers made of identical monomers, if purines are crosswise to purines, the carrier changes strand (from strand 1 to strand 2 or vice versa), while if purines are on the same strand, the carrier is transferred through the strand it was initially placed at.
(c) For dimers made of different monomers, the carrier is transferred (albeit in small percentage) mainly through the strand it was initially placed at. Hence, carrier transfer is very small in dimers made of different monomers. The carrier basically remains in the base it was initially placed at, while a small percentage passes to the other base of the same strand.
A careful inspection in Figs.~\ref{fig:FourierGG},~\ref{fig:FourierGC} and \ref{fig:FourierCT} shows that the Fourier analysis confirms conclusions (a),(b),(c).
For TB I, in Refs.~\cite{Simserides:2014,LKGS:2014,LChMKTS:2015,Lambropoulos:2014}, we used a somehow different set of TB parameters, the results are similar.
\begin{figure} [h!]
\centering
\includegraphics[width=7.5cm]{MeanProbDimers1H.eps}
\includegraphics[width=7.5cm]{MeanProbDimers2H.eps}
\includegraphics[width=7.5cm]{MeanProbDimers1L.eps}
\includegraphics[width=7.5cm]{MeanProbDimers2L.eps}
\caption{The mean probabilities to find an extra carrier [hole (1st row) or electron (2nd row)] at each site of a DNA dimer,
as determined with
(I) the base-pair TB approach (left column) and
(II) the single-base TB approach (right column).
For TB I, the carrier is initially placed at the 1st monomer, while,
for TB II, it is initially placed at the base of the 1st monomer that belongs to the 1st strand.
We use the HKS parametrization~\cite{HKS:2010-2011}.}
\label{fig:comparison}
\end{figure}
\begin{figure} [h!]
\centering
\includegraphics[width=7.5cm]{Dimers1stHOMO.eps}
\includegraphics[width=7.5cm]{Dimers2ndHOMO.eps}
\includegraphics[width=7.5cm]{Dimers3rdHOMO.eps}
\includegraphics[width=7.5cm]{Dimers4thHOMO.eps}
\caption{The mean probabilities to find an extra hole at each base of a DNA dimer, as determined with TB approach II, the single-base approach.
The hole is initially placed at the 1st base (A1) (top left panel), the 2nd base (B1) (top right panel), the 3rd base (A2) (bottom left panel) or the 4th base (B2) (bottom right panel). We use the HKS parametrization~\cite{HKS:2010-2011}.}
\label{fig:DimersIIHOMO}
\end{figure}
\begin{figure} [h!]
\centering
\includegraphics[width=7.5cm]{Dimers1stLUMO.eps}
\includegraphics[width=7.5cm]{Dimers2ndLUMO.eps}
\includegraphics[width=7.5cm]{Dimers3rdLUMO.eps}
\includegraphics[width=7.5cm]{Dimers4thLUMO.eps}
\caption{The mean probabilities to find an extra electron at each base of a DNA dimer, as determined with TB approach II, the single-base approach.
The electron is initially placed at the 1st base (A1) (top left panel), the 2nd base (B1) (top right panel), the 3rd base (A2) (bottom left panel) or the 4th base (B2) (bottom right panel). We use the HKS parametrization~\cite{HKS:2010-2011}.}
\label{fig:DimersIILUMO}
\end{figure}
Let us now turn our discussion to carrier mean transfer rates $k_{ij}$, within TB II and the HKS parametrization~\cite{HKS:2010-2011}, cf. Fig.~\ref{fig:DimersHOMOLUMOkijall}. The specific values of $k_{ij}$ depend, of course, on the TB hopping parameters used~\cite{Kaklamanis:2015}. Let us call $k_{ij}$ the mean transfer rates of a dimer YX and $k^{\textrm{\scriptsize{compl}}}_{ij}$ the mean transfer rates of the equivalent dimer $\textrm{X}_{\textrm{\scriptsize{compl}}}\textrm{Y}_{\textrm{\scriptsize{compl}}}$. The following two properties hold:
\begin{equation}
k_{ij}=k_{ji},
\label{kij=kji}
\end{equation}
\begin{equation}
k_{ij}=k^{\textrm{\scriptsize{compl}}}_{(5-i)(5-j)}.
\label{k,k' relation for equivalent dimers}
\end{equation}
For dimers made of identical monomers with purine on purine (GG$\equiv$CC, AA$\equiv$TT), hole transfer is almost entirely of intrastrand character i.e. it is along the $5'$-$3'$ or $3'$-$5'$ directions. Moreover, since $k_{13}$ and $k_{24}$ satisfy Eq.~\ref{k,k' relation for equivalent dimers}, in Fig.~\ref{fig:DimersHOMOLUMOkijall} we observe a symmetry in the alternation of colors for the couples of equivalent dimers GG$\equiv$CC and AA$\equiv$TT.
For dimers made of identical monomers with crosswise purines (CG, GC, TA, AT), there is significant diagonal hole transfer, and furthermore, for CG and TA the stronger hole transfer is along the $3'$-$3'$ direction. For dimers made of different monomers (AG$\equiv$CT, AC$\equiv$GT, TG$\equiv$CA, TC$\equiv$GA), hole transfer is almost exclusively of intrastrand character i.e. along the $5'$-$3'$ or $3'$-$5'$ directions; since $k_{13}$ and $k_{24}$ satisfy Eq. \ref{k,k' relation for equivalent dimers}, we observe the same symmetry in the alternation of colors. For the couple TG$\equiv$CA, although $k_{13}$ and $k_{24}$ are the biggest among all other $k_{ij}$, they are very small.
Electron transfer in dimers made of identical monomers with purine on purine (GG$\equiv$CC, AA$\equiv$TT) is qualitatively similar to hole transfer in such dimers. For dimers made of identical monomers with crosswise purines (GC, CG, AT, TA) electron transfer is slightly different than hole transfer in such dimers in the sense that diagonal channels are important but are not, quantitatively, identically important.
Electron transfer in dimers made of different monomers has a significant intrastrand character, but there is also intra-base-pair character in some cases. For the same reasons described above, we observe symmetry in color alternation for $k_{ij}$ of equivalent dimers.
\begin{figure} [h!]
\centering
\includegraphics[width=7.5cm]{kijHOMOidemon.eps}
\includegraphics[width=7.5cm]{kijHOMOdimo.eps}
\includegraphics[width=7.5cm]{kijLUMOidemon.eps}
\includegraphics[width=7.5cm]{kijLUMOdimo.eps}
\caption{Mean transfer rates $k_{ij}$ between bases $i$ and $j$ either for HOMO (hole transfer, 1st row) or for LUMO (electron transfer, 2nd row), for all dimers,
within TB approach II and the HKS parametrization~\cite{HKS:2010-2011}.
The 1st column corresponds to dimers made of identical monomers and
the 2nd row to dimers made of different monomers.}
\label{fig:DimersHOMOLUMOkijall}
\end{figure}
\clearpage
\section{Trimers}\label{sec:ResultsTrimers}
Let us now compare TB I and TB II in trimers made of identical monomers. Using TB I we proved~\cite{Simserides:2014,LKGS:2014} that in trimers made of identical monomers an extra carrier oscillates periodically with
\begin{equation}\label{fandTtrimers}
f = \frac{1}{T} = \frac{\sqrt{t^2 + t'^2}}{h},
\end{equation}
where $t, t'$ are the hopping integrals between the base pairs (when all purines are on the same strand, $t = t'$). With the parametrization of Refs.~\cite{Simserides:2014,LKGS:2014}, we found $f \approx$ 0.5-33 THz ($T \approx$ 30-2000 fs)~\cite{LKGS:2014}; with the parametrization of Ref.~\cite{HKS:2010-2011}, we find $f \approx$ 0.5-21 THz ($T \approx$ 48-2000 fs).
In other words, for trimers made of identical monomers,
the frequency range is narrower than for dimers.
For 0 times crosswise purines, the maximum transfer percentage $p = 1$, while for 1 or 2 times crosswise purines $p < 1$ \cite{Simserides:2014,LKGS:2014}.
TB II in trimers, generally, does not allow one to strictly determine periodicity;
$T, f, p$, and $pf$ cannot be defined.
However, Fourier analysis shows similar frequency content.
(Specific examples are given in Appendix C, where we present Fourier analysis of hole oscillations in two trimers made of identical monomers:
GGG in Fig.~\ref{fig:FourierGGG} and
GCG in Fig.~\ref{fig:FourierGCGandCACandCTC}.)
For hole transfer in GGG, within TB I and the parametrization of Refs.~\cite{Simserides:2014,LKGS:2014}, we found $f \approx $ 34.2 THz;
now with the HKS parametrization~\cite{HKS:2010-2011}, we find $f \approx $ 21.2 THz.
This is in remarkable agreement with the frequencies obtained by Fourier Transform,
within TB II and HKS parametrization~\cite{HKS:2010-2011} shown in Fig.~\ref{fig:FourierGGG}.
Specifically, e.g. for initial placement of the hole at base A1(G), the main frequencies are around 21.2 THz (a double peak) and 42.4 THz (a single peak), while, e.g. for initial placement of the hole at base B1(C), the main frequencies are around 22.5 THz (a double peak) and 45 THz (a single peak).
In other words, for hole transfer in GGG, within the HKS parametrization, the period predicted by TB I agrees with the approximate period predicted by TB II.
The mean probabilities to find an extra carrier at a base and the mean transfer rates in GGG and AAA, within TB II and HKS parametrization~\cite{HKS:2010-2011},
are shown in Fig.~\ref{fig:TrimersGGGAAA}.
We observe that the carrier movement is almost exclusively of intrastrand character,
a fact also evident from the Fourier analysis in Fig.~\ref{fig:FourierGGG}.
Remarkably, with TB II, the probabilities to find the carrier at each base pair are either $\approx$ 0.375, 0.25, 0.375 or $\approx$ 0.25, 0.5, 0.25 depending on the initial placement of the carrier, in agreement with the rules established in Ref.~\cite{LChMKTS:2015} for TB I.
The mean transfer rates $k_{ij}$ confirm the intrastrand character of charge transfer in GGG and AAA.
For hole transfer in GCG, within the HKS parametrization~\cite{HKS:2010-2011},
TB I and TB II give similar results, indicating rather weak transfer.
For example,
placing the hole initially at the first base pair for TB I or placing the hole initially at the first base for TB II, the probability to find the hole at the first base pair is $\approx$ 0.9990 for TB I and 0.9848 for TB II and at the last base pair $\approx$ 0.0008 for TB I and 0.0006 for TB II. This is mirrored in the very small Fourier amplitudes for GCG (Appendix C, Fig.~\ref{fig:FourierGCGandCACandCTC}).
Within TB I, for trimers made of different monomers, carrier movement may be non-periodic~\cite{Simserides:2014,LKGS:2014}. Within TB I, generally, increasing the number of monomers above three, the system becomes more complex and periodicity is lost~\cite{LKGS:2014}; even in the simplest cases, e.g. tetramers made of identical monomers with all the purines on the same strand, there is no periodicity~\cite{Lambropoulos:2014}. Within TB II, as mentioned before, in trimers, generally, periodicity cannot be strictly determined. As a last point, we restrict ourselves in giving (Appendix C, Fig.~\ref{fig:FourierGCGandCACandCTC}) two examples of the frequency content of hole oscillations in trimers made of different monomers, specifically in CAC and CTC.
\begin{figure} [h!]
\centering
\includegraphics[width=7.5cm]{GGGgHLprob.eps}
\includegraphics[width=7.5cm]{AAAgHLprob.eps}
\includegraphics[width=15cm]{kijkjiGGGdAAA.eps}
\caption{GGG and AAA trimers within TB approach II and HKS parametrization~\cite{HKS:2010-2011}.
\textit{Upper panels:}
Mean probabilities to find the carrier at each base after having placed it initially at a particular base.
100000 means that the carrier was initially placed at base A1, 010000 at base B1 etc
(e.g. for GGG the bases are A1(G), B1(C), A2(G), B2(C), A3(G), B3(C)).
The carrier movement is almost exclusively of intrastrand character.
The probabilities to find the carrier at each base pair are either $\approx$ 0.375, 0.25, 0.375 or $\approx$ 0.25, 0.5, 0.25 depending on the initial placement of the carrier, which agrees remarkably with the rules established in Ref.~\cite{LChMKTS:2015} for TB I.
\textit{Lower panel:} The mean transfer rates $k_{ij}$ from base $i$ to base $j$.
The arrows indicate intrastrand transfer from the strand start to the strand end.}
\label{fig:TrimersGGGAAA}
\end{figure}
\section{Conclusion}
\label{sec:Conclusion}
Using two TB approaches,
a wire model called here TB I, where a site coincides with a DNA base pair, and
an extended (i.e. including also diagonal hoppings) ladder model called here TB II,
where a site coincides with a DNA base,
we demonstrated that THz oscillations in DNA monomers, dimers and trimers exist.
We studied various aspects of the effect, e.g.
frequency content,
maximum transfer percentages and transfer rates between sites, and
mean probabilities to find the carrier at a particular site.
We also compared successfully the two TB approaches.
Naturally, TB II allows for greater detail.
For DNA monomers, i.e. for adenine-thymine and guanine-cytosine, with TB II,
we predicted electron or hole oscillations in the range
$f \approx$ 50-550 THz ($T \approx$ 2-20 fs), i.e. $\lambda \approx$ 545 nm - 6 $\mu$m, from visible to near- and mid infrared~\cite{ISO20473}. We found that the maximum transfer percentage $p$ and the pure maximum transfer rate $pf$ between the bases are very small.
For DNA dimers, with TB I, we predicted electron or hole oscillations in the range $f \approx$ 0.25-100 THz ($T \approx $ 10-4000 fs) i.e. $\lambda \approx$ 3-1200 $\mu$m, approximately in the mid- and far-infrared. For dimers made of identical monomers the maximum transfer percentage $p = 1$, but for dimers made of different monomers $p < 1$. With TB II, the carrier oscillations are not strictly periodic but the frequency content is similar to that predicted with TB I. For the mean probabilities to find the carrier at a particular site, the two approaches give coherent, complementary results.
TB II shows that for dimers made of identical monomers, when purines are crosswise to purines, interstrand carrier transfer dominates, i.e. we have significant diagonal transfer, justifying the inclusion of diagonal hoppings in our model, while if purines are on the same strand, intrastrand carrier transfer dominates. For dimers made of different monomers, we carrier transfer is mainly intrastrand character but the transfer percentage is small.
With TB I, for trimers made of identical monomers, the carrier oscillates periodically with $f \approx$ 0.5-33 THz ($T \approx$ 30-2000 fs) if we use the parametrization of Refs.~\cite{Simserides:2014,LKGS:2014}.
With the HKS parametrization~\cite{HKS:2010-2011}, $f \approx$ 0.5-21 THz.
For 0 times crosswise purines $p = 1$, for 1 or 2 times crosswise purines $p < 1$.
With TB II, the carrier oscillations are not strictly periodic but the frequency content is similar to that predicted with TB I. For the mean probabilities to find the carrier at a particular site, the two approaches give coherent, complementary results.
Finally, we would like to mention that increasing the number of monomers,
i.e. constructing an oligomer or a polymer, the frequency spectrum becomes more fragmented and moves towards lower frequencies.
A systematic study of longer segments is beyond the scope of the present manuscript.
It seems that a source or receiver of electromagnetic radiation
made of DNA monomers, dimers or trimers,
with frequencies from fractions of THz to just below PHz,
could be envisaged. \\
\noindent \textbf{* Related Work at} \\
\verb"http://users.uoa.gr/~csimseri/physics_of_nanostructures_and_biomaterials.html" \\
\noindent \textbf{Acknowledgements}
A. Morphis thanks the State Scholarships Foundation-IKY for a Ph.D. research scholarship
via ``IKY Fellowships of Excellence'', Hellenic Republic-Siemens Settlement Agreement.
M. Tassi thanks the State Scholarships Foundation-IKY for a post-doctoral research fellowship
via ``IKY Fellowships of Excellence'', Hellenic Republic-Siemens Settlement Agreement. \\
\clearpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,833 |
{"url":"https:\/\/docs.clamav.net\/manual\/Signatures\/LogicalSignatures.html","text":"# Logical signatures\n\nLogical signatures allow combining of multiple signatures in extended format using logical operators. They can provide both more detailed and flexible pattern matching. The logical sigs are stored inside *.ldb files in the following format:\n\nSignatureName;TargetDescriptionBlock;LogicalExpression;Subsig0;\nSubsig1;Subsig2;...\n\n\nwhere:\n\n\u2022 TargetDescriptionBlock provides information about the engine and target file with comma separated Arg:Val pairs. For args where Val is a range, the minimum and maximum values should be expressed as min-max.\n\n\u2022 LogicalExpression specifies the logical expression describing the relationship between Subsig0...SubsigN. Basis clause: 0,1,...,N decimal indexes are SUB-EXPRESSIONS representing Subsig0, Subsig1,...,SubsigN respectively. Inductive clause: if A and B are SUB-EXPRESSIONS and X, Y are decimal numbers then (A&B), (A|B), A=X, A=X,Y, A>X, A>X,Y, A<X and A<X,Y are SUB-EXPRESSIONS\n\n\u2022 SubsigN is n-th subsignature in extended format possibly preceded with an offset. There can be specified up to 64 subsigs.\n\nKeywords used in TargetDescriptionBlock:\n\n\u2022 Engine:X-Y: Required engine functionality level (range; 0.96).\n\nNote: If the Engine keyword is used, it must be the first keyword in the TargetDescriptionBlock for backwards compatibility. See the FLEVEL reference for details.\n\n\u2022 Target:X: A number specifying the type of the target file: Target Types.\n\n\u2022 FileSize:X-Y: Required file size (range in bytes; 0.96)\n\n\u2022 EntryPoint: Entry point offset (range in bytes; 0.96)\n\n\u2022 NumberOfSections: Required number of sections in executable (range; 0.96)\n\n\u2022 Container:CL_TYPE_*: File type of the container which stores the scanned file.\n\nSpecifying CL_TYPE_ANY matches on root objects only (i.e. the target file is explicitely not in a container). Chances slim that you would want to use CL_TYPE_ANY in a signature, because placing the malicious file in an archive will then prevent it from alerting.\n\nEvery ClamAV file type has the potential to be a container for additional files, although some are more likely than others. When a file is parsed and data in the file is identified to be scanned as a unique type, that parent file becomes a container the moment the embedded content is scanned. For a list of possible CL_TYPEs, refer to the File Types Reference.\n\n\u2022 Intermediates:CL_TYPE_*>CL_TYPE_*: Specify one or more layers of file types containing the scanned file. This is an alternative to using Container.\n\nYou may specify up to 16 layers of file types separated by \u2019>\u2019 in top-down order. Note that the \u2019>\u2019 separator is not needed if you only specify a single container. The last type should be the immediate container containing the malicious file. Unlike with the Container option, CL_TYPE_ANY can be used as a wildcard file type. (expr; 0.100.0)\n\nFor a list of possible CL_TYPEs, refer to the File Types Reference.\n\n\u2022 IconGroup1: Icon group name 1 from .idb signature Required engine functionality (range; 0.96)\n\n\u2022 IconGroup2: Icon group name 2 from .idb signature Required engine functionality (range; 0.96)\n\nModifiers for subexpressions:\n\n\u2022 A=X: If the SUB-EXPRESSION A refers to a single signature then this signature must get matched exactly X times; if it refers to a (logical) block of signatures then this block must generate exactly X matches (with any of its sigs).\n\n\u2022 A=0 specifies negation (signature or block of signatures cannot be matched)\n\n\u2022 A=X,Y: If the SUB-EXPRESSION A refers to a single signature then this signature must be matched exactly X times; if it refers to a (logical) block of signatures then this block must generate X matches and at least Y different signatures must get matched.\n\n\u2022 A>X: If the SUB-EXPRESSION A refers to a single signature then this signature must get matched more than X times; if it refers to a (logical) block of signatures then this block must generate more than X matches (with any of its sigs).\n\n\u2022 A>X,Y: If the SUB-EXPRESSION A refers to a single signature then this signature must get matched more than X times; if it refers to a (logical) block of signatures then this block must generate more than X matches and at least Y different signatures must be matched.\n\n\u2022 A<X: Just like A>Z above with the change of \"more\" to \"less\".\n\nIf the SUB-EXPRESSION A refers to a single signature then this signature must get matched less than X times; if it refers to a (logical) block of signatures then this block must generate less than X matches (with any of its sigs).\n\n\u2022 A<X,Y: Similar to A>X,Y. If the SUB-EXPRESSION A refers to a single signature then this signature must get matched less than X times; if it refers to a (logical) block of signatures then this block must generate less than X matches and at least Y different signatures must be matched.\n\nExamples:\n\nSig1;Target:0;(0&1&2&3)&(4|1);6b6f74656b;616c61;7a6f6c77;7374656\n\nSig2;Target:0;((0|1|2)>5,2)&(3|1);6b6f74656b;616c61;7a6f6c77;737\n46566616e\n\nSig3;Target:0;((0|1|2|3)=2)&(4|1);6b6f74656b;616c61;7a6f6c77;737\n\nSig4;Engine:51-255,Target:1;((0|1)&(2|3))&4;EP+123:33c06834f04100\nf2aef7d14951684cf04100e8110a00;S2+78:22??232c2d252229{-15}6e6573\n(63|64)61706528;S3+50:68efa311c3b9963cb1ee8e586d32aeb9043e;f9c58\ndcf43987e4f519d629b103375;SL+550:6300680065005c0046006900\n\n\n## Subsignature Modifiers\n\nClamAV (clamav-0.99) supports a number of additional subsignature modifiers for logical signatures. This is done by specifying :: followed by a number of characters representing the desired options. Signatures using subsignature modifiers require Engine:81-255 for backwards-compatibility.\n\n\u2022 Case-Insensitive [i]\n\nSpecifying the i modifier causes ClamAV to match all alphabetic hex bytes as case-insensitive. All patterns in ClamAV are case-sensitive by default.\n\n\u2022 Wide [w]\n\nSpecifying the w causes ClamAV to match all hex bytes encoded with two bytes per character. Note this simply interweaves each character with NULL characters and does not truly support UTF-16 characters. Wildcards for \u2019wide\u2019 subsignatures are not treated as wide (i.e. there can be an odd number of intermittent characters). This can be combined with a to search for patterns in both wide and ascii.\n\n\u2022 Fullword [f]\n\nMatch subsignature as a fullword (delimited by non-alphanumeric characters).\n\n\u2022 Ascii [a]\n\nMatch subsignature as ascii characters. This can be combined with w to search for patterns in both ascii and wide.\n\nExamples:\n\n\u2022 Match 'AAAA'(nocase) and 'BBBBBB'(nocase)\nclamav-nocase-A;Engine:81-255,Target:0;0&1;41414141::i;424242424242::i\n\n\u2022 Match 'AAA' and 'hello'(fullword)\nclamav-fullword-A;Engine:81-255,Target:0;0&1;414141;68656c6c6f::f\n\n\u2022 Match 'AAA' and 'hello'(fullword nocase)\nclamav-fullword-B;Engine:81-255,Target:0;0&1;414141;68656c6c6f::fi\n\n\u2022 Match 'AAA' and 'hello'(wide ascii)\nclamav-wide-B2;Engine:81-255,Target:0;0&1;414141;68656c6c6f::wa\n\n\u2022 Match 'AAA' and 'hello'(nocase wide fullword ascii)\nclamav-wide-C0;Engine:81-255,Target:0;0&1;414141;68656c6c6f::iwfa\n\n\n## Special Subsignature Types\n\n### Macro subsignatures\n\nIntroduced in ClamAV 0.96\n\nFormat: ${min-max}MACROID$\n\nMacro subsignatures are used to combine a number of existing extended signatures (.ndb) into a on-the-fly generated alternate string logical signature (.ldb). Signatures using macro subsignatures require Engine:51-255 for backwards-compatibility.\n\nExample:\n\ntest.ldb:\nTestMacro;Engine:51-255,Target:0;0&1;616161;${6-7}12$\n\ntest.ndb:\nD1:0:$12:626262 D2:0:$12:636363\nD3:0:\\$30:626264\n\n\nThe example logical signature TestMacro is functionally equivalent to:\n\nTestMacro;Engine:51-255,Target:0;0;616161{3-4}(626262|636363)\n\n\u2022 MACROID points to a group of signatures; there can be at most 32 macro groups.\n\n\u2022 In the example, MACROID is 12 and both D1 and D2 are members of macro group 12. D3 is a member of separate macro group 30.\n\u2022 {min-max} specifies the offset range at which one of the group signatures should match; the offset range is relative to the starting offset of the preceding subsignature. This means a macro subsignature cannot be the first subsignature.\n\n\u2022 In the example, {min-max} is {6-7} and it is relative to the start of a 616161 match.\n\n### Byte Compare Subsignatures\n\nIntroduced in ClamAV 0.101\n\nFormat: subsigid_trigger(offset#byte_options#comparisons)\n\nByte compare subsignatures can be used to evaluate a numeric value at a given offset from the start of another (matched) subsignature within the same logical signature. These are executed after all other subsignatures within the logical subsignature are fired, with the exception of PCRE subsignatures. They can evaluate offsets only from a single referenced subsignature, and that subsignature must give a valid match for the evaluation to occur.\n\n\u2022 subsigid_trigger is a required field and may refer to any single non-PCRE, non-Byte Compare subsignature within the lsig. The byte compare subsig will evaluate if subsigid_trigger matches. Triggering on multiple subsigs or logic based triggering is not currently supported.\n\n\u2022 offset is a required field that consists of an offset_modifier and a numeric offset (hex or decimal offsets are okay).\n\n\u2022 offset_modifier can be either >> or << where the former denotes a positive offset and the latter denotes a negative offset. The offset is calculated from the start of subsigid_trigger, which allows for byte extraction before the specified match, after the match, and within the match itself.\n\n\u2022 offset must be a positive hex or decimal value. This will be the number of bytes from the start of the referenced subsigid_trigger match within the file buffer to begin the comparison.\n\n\u2022 byte_options are used to specify the numeric type and endianess of the extracted byte sequence in that order as well as the number of bytes to be read. By default ClamAV will attempt to matchup up to the number of byte specified, unless the e (exact) option is specified or the numeric type is b (binary). This field follows the form [h|d|a|i][l|b][e]num_bytes\n\n\u2022 h|d|a|i where h specifies the byte sequence will be in hex, d decimal, a automatic detection of hex or decimal at runtime, and i signifies raw binary data.\n\n\u2022 l|b where l specifies the byte sequence will be in little endian order and b big endian. If decimal d is specified, big-endian is implied and using l will result in a malformed database error.\n\n\u2022 e specifies that ClamAV will only evaluate the comparison if it can extract the exact number of bytes specified. This option is implicitly declared when using the i flag.\n\n\u2022 num_bytes specifies the number of bytes to extract. This can be a hex or decimal value. If i is specified only 1, 2, 4, and 8 are valid options.\n\n\u2022 comparisons are a required field which denotes how to evaluate the extracted byte sequence. Each Byte Compare signature can have one or two comparison_sets separated by a comma. Each comparison_set consists of a Comparison_symbol and a Comparison_value and takes the form Comparison_symbolComparison_value. Thus, comparisons takes the form comparison_set[,comparison_set]\n\n\u2022 Comparison_symbol denotes the type of comparison to be done. The supported comparison symbols are <, >, =.\n\n\u2022 Comparison_value is a required field which must be a numeric hex or decimal value. If all other conditions are met, the byte compare subsig will evalutate the extracted byte sequence against this number based on the provided comparison_symbol.\n\n### PCRE subsignatures\n\nIntroduced in ClamAV 0.99\n\nFormat: Trigger\/PCRE\/[Flags]\n\nPCRE subsignatures are used within a logical signature (.ldb) to specify regex matches that execute once triggered by a conditional based on preceding subsignatures. Signatures using PCRE subsignatures require Engine:81-255 for backwards-compatibility.\n\n\u2022 Trigger is a required field that is a valid LogicalExpression and may refer to any subsignatures that precede this subsignature. Triggers cannot be self-referential and cannot refer to subsequent subsignatures.\n\n\u2022 PCRE is the expression representing the regex to execute. ClamAV identifies the regex string by searching from the beginning of the subsignature for the start-\/ and searching from the end for the end-\/. You may \\-escape any use of \/ within the regex string, but it is not required. For backward compatibility, ; within the expression must be expressed as \\x3B. The regex string cannot be empty and (?UTF\\*) control sequences are not allowed. If debug messages are enabled (i.e. clamscan --debug), then named capture groups are displayed in a post-execution report.\n\n\u2022 Flags are a series of characters which affect the compilation and execution of PCRE within the PCRE compiler and the ClamAV engine. This field is optional.\n\n\u2022 g [CLAMAV_GLOBAL] specifies to search for ALL matches of PCRE (default is to search for first match). NOTE: INCREASES the time needed to run the PCRE.\n\n\u2022 r [CLAMAV_ROLLING] specifies to use the given offset as the starting location to search for a match as opposed to the only location; applies to subsigs without maxshifts. By default, in order to facilatate normal ClamAV offset behavior, PCREs are auto-anchored (only attempt match on first offset); using the rolling option disables the auto-anchoring.\n\n\u2022 e [CLAMAV_ENCOMPASS] specifies to CONFINE matching between the specified offset and maxshift; applies only when maxshift is specified.\n\nNote: DECREASES time needed to run the PCRE.\n\n\u2022 i [PCRE_CASELESS]\n\n\u2022 s [PCRE_DOTALL]\n\n\u2022 m [PCRE_MULTILINE]\n\n\u2022 x [PCRE_EXTENDED]\n\n\u2022 A [PCRE_ANCHORED]\n\n\u2022 E [PCRE_DOLLAR_ENODNLY]\n\n\u2022 U [PCRE_UNGREEDY]\n\nExamples:\n\nFind.All.ClamAV;Engine:81-255,Target:0;1;6265676c6164697427736e6f7462797465636f6465;0\/clamav\/g\n\nFind.ClamAV.OnlyAt.299;Engine:81-255,Target:0;2;7374756c747a67657473;7063726572656765786c6f6c;299:0&1\/clamav\/\n\nFind.ClamAV.StartAt.300;Engine:81-255,Target:0;3;616c61696e;62756731393238;636c6f736564;300:0&1&2\/clamav\/r\n\nFind.All.Encompassed.ClamAV;Engine:81-255,Target:0;3;7768796172656e2774;796f757573696e67;79617261;200,300:0&1&2\/clamav\/ge\n\nNamed.CapGroup.Pcre;Engine:81-255,Target:0;3;636f75727479617264;616c62756d;74657272696572;50:0&1&2\/variable=(?<nilshell>.{16})end\/gr\n\nFirefox.TreeRange.UseAfterFree;Engine:81-255,Target:0,Engine:81-255;0&1&2;2e766965772e73656c656374696f6e;2e696e76616c696461746553656c656374696f6e;0&1\/\\x2Eview\\x2Eselection.*?\\x2Etree\\s*\\x3D\\s*null.*?\\x2Einvalidate\/smi\n\nFirefox.IDB.UseAfterFree;Engine:81-255,Target:0;0&1;4944424b657952616e6765;0\/^\\x2e(only|lowerBound|upperBound|bound)\\x28.*?\\x29.*?\\x2e(lower|upper|lowerOpen|upperOpen)\/smi\n\nFirefox.boundElements;Engine:81-255,Target:0;0&1&2;6576656e742e6\n\n\n### Image Fuzzy Hash subsignatures\n\nIntroduced in ClamAV 0.105\n\nFormat: fuzzy_img#<hash>#<dist>\n\nFor example if you wanted to match on this image...\n\n...you would make a signature like this:\n\nlogo.png;Engine:150-255,Target:0;0;fuzzy_img#af2ad01ed42993c7#0\n\n\nImage fuzzy hash signatures in 0.105 do not support matching with a hamming distance greater than zero. Support for matching with a hamming distance may be added in a future release. The signatures above explicitly set the hamming distance to 0. But you could also omit it, like this:\n\nlogo.png;Engine:150-255,Target:0;0;fuzzy_img#af2ad01ed42993c7\n\n\nYou can combine the image fuzzy hash subsignature with other logical signature features, like adding additional subsignatures:\n\nlogo.png-2;Engine:150-255,Target:0;0&1;49484452;fuzzy_img#af2ad01ed42993c7\n\n\nor container types:\n\nlogo.png;Engine:150-255,Target:0,Container:CL_TYPE_HTML;0;fuzzy_img#af2ad01ed42993c7\n\n\nClamAV's image fuzzy hash is very close to, but not 100% identical to, the fuzzy hash generated by the Python imagehash package's phash() function. Note that these are only clean-room approximations of the pHash\u2122\ufe0f algorithm. ClamAV's image fuzzy hashes are not expected to match the fuzzy hashes generated using other tools. Some images may match, while others do not.\n\nYou must use ClamAV to generate the fuzzy hash for the most reliable results. A sigtool option does not yet exist to generate a ClamAV image fuzzy hash. So, to generate the image fuzzy hash you can run this command:\n\nclamscan --gen-json --debug \/path\/to\/file\n\n\nThe hash will appear in the JSON above the \"SCAN SUMMARY\" under the object named \"ImageFuzzyHash\".\n\n## Signatures for Version Information (VI) metadata in PE files\n\nStarting with ClamAV 0.96 it is possible to easily match certain information built into PE files (executables and dynamic link libraries). Whenever you lookup the properties of a PE executable file in windows, you are presented with a bunch of details about the file itself.\n\nThese info are stored in a special area of the file resources which goes under the name of VS_VERSION_INFORMATION (or versioninfo for short). It is divided into 2 parts. The first part (which is rather uninteresting) is really a bunch of numbers and flags indicating the product and file version. It was originally intended for use with installers which, after parsing it, should be able to determine whether a certain executable or library are to be upgraded\/overwritten or are already up to date. Suffice to say, this approach never really worked and is generally never used.\n\nThe second block is much more interesting: it is a simple list of key\/value strings, intended for user information and completely ignored by the OS. For example, if you look at ping.exe you can see the company being \"Microsoft Corporation\", the description \"TCP\/IP Ping command\", the internal name \"ping.exe\" and so on... Depending on the OS version, some keys may be given peculiar visibility in the file properties dialog, however they are internally all the same.\n\nTo match a versioninfo key\/value pair, the special file offset anchor VI was introduced. This is similar to the other anchors (like EP and SL) except that, instead of matching the hex pattern against a single offset, it checks it against each and every key\/value pair in the file. The VI token doesn\u2019t need nor accept a +\/- offset like e.g. EP+1. As for the hex signature itself, it\u2019s just the utf16 dump of the key and value. Only the ?? and (aa|bb) wildcards are allowed in the signature. Usually, you don\u2019t need to bother figuring it out: each key\/value pair together with the corresponding VI-based signature is printed by clamscan when the --debug option is given.\n\nFor example clamscan --debug freecell.exe produces:\n\n[...]\nRecognized MS-EXE\/DLL file\nversioninfo_cb: type: 10, name: 1, lang: 410, rva: 9608\ncli_peheader: parsing version info @ rva 9608 (1\/1)\nVersionInfo (d2de): 'CompanyName'='Microsoft Corporation' -\nVI:43006f006d00700061006e0079004e0061006d006500000000004d006900\n630072006f0073006f0066007400200043006f00720070006f0072006100740\n069006f006e000000\nVersionInfo (d32a): 'FileDescription'='Entertainment Pack\nFreeCell Game' - VI:460069006c006500440065007300630072006900700\n0740069006f006e000000000045006e007400650072007400610069006e006d\n0065006e00740020005000610063006b0020004600720065006500430065006\nc006c002000470061006d0065000000\nVersionInfo (d396): 'FileVersion'='5.1.2600.0 (xpclient.010817\n-1148)' - VI:460069006c006500560065007200730069006f006e00000000\n0035002e0031002e0032003600300030002e003000200028007800700063006\nc00690065006e0074002e003000310030003800310037002d00310031003400\n380029000000\nVersionInfo (d3fa): 'InternalName'='freecell' - VI:49006e007400\n650072006e0061006c004e0061006d006500000066007200650065006300650\n06c006c000000\nVersionInfo (d4ba): 'OriginalFilename'='freecell' - VI:4f007200\n6900670069006e0061006c00460069006c0065006e0061006d0065000000660\n0720065006500630065006c006c000000\nVersionInfo (d4f6): 'ProductName'='Sistema operativo Microsoft\nWindows' - VI:500072006f0064007500630074004e0061006d00650000000\n000530069007300740065006d00610020006f00700065007200610074006900\n76006f0020004d006900630072006f0073006f0066007400ae0020005700690\n06e0064006f0077007300ae000000\nVersionInfo (d562): 'ProductVersion'='5.1.2600.0' - VI:50007200\n6f006400750063007400560065007200730069006f006e00000035002e00310\n02e0032003600300030002e0030000000\n[...]\n\n\nAlthough VI-based signatures are intended for use in logical signatures you can test them using ordinary .ndb files. For example:\n\nmy_test_vi_sig:1:VI:paste_your_hex_sig_here\n\n\nFinal note. If you want to decode a VI-based signature into a human readable form you can use:\n\necho hex_string | xxd -r -p | strings -el\n\n\nFor example:\n\necho 460069006c0065004400650073006300720069007000740069006f006e000000000045006e007400650072007400610069006e006d0065006e00740020005000610063006b0020004600720065006500430065006c006c00200047006100\n6d0065000000 | xxd -r -p | strings -el\nFileDescription\nEntertainment Pack FreeCell Game\n\n\n## Icon Signatures for PE files\n\nWhile Icon Signatures are stored in a .idb file, they are a feature of Logical Signatures.\n\nClamAV 0.96 includes an approximate\/fuzzy icon matcher to help detecting malicious executables disguising themselves as innocent looking image files, office documents and the like.\n\nIcon matching is only triggered by Logical Signatures (.ldb) using the special attribute tokens IconGroup1 or IconGroup2. These identify two (optional) groups of icons defined in a .idb database file. The format of the .idb file is:\n\nICONNAME:GROUP1:GROUP2:ICON_HASH\n\n\nwhere:\n\n\u2022 ICON_NAME is a unique string identifier for a specific icon,\n\n\u2022 GROUP1 is a string identifier for the first group of icons (IconGroup1)\n\n\u2022 GROUP2 is a string identifier for the second group of icons (IconGroup2),\n\n\u2022 ICON_HASH is a fuzzy hash of the icon image\n\nThe ICON_HASH field can be obtained from the debug output of libclamav. For example:\n\nLibClamAV debug: ICO SIGNATURE:","date":"2022-07-07 03:47:14","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.37137094140052795, \"perplexity\": 5108.355202694325}, \"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-2022-27\/segments\/1656104683683.99\/warc\/CC-MAIN-20220707033101-20220707063101-00787.warc.gz\"}"} | null | null |
Q: Levin Rule Integrate Power of Oscillating Function I want to numerically integrate a function of the form:
NIntegrate[Exp[I a x] x^-n (1+Exp[I b x])^n,{x,1,10^4},
Method -> {"LevinRule", "LevinFunctions" -> {"TrigRelated"}}]
BTW, Mathematica understands this better using Euler's formula exp = cos + i sin .
The Levin Rule wants a explicit oscillating kernel like Exp[I 1/50 x] . The obvious solution would be to expand the bracket using the binomial theorem and then there would be contributions like Exp[I (a+l b) x] - oscillations with well defined frequency. But then I need to compute N integrals instead of 1 and there will be cancellations.
Is there a smarter way of telling the Levin Rule what to do than the Binomial expansion?
A: The automatic Levin-rule reduction of the integrand picks $e^{iax}$ for the kernel and $e^{-ibx}/x^n$ for the amplitude, which fails to capture the whole oscillatory component. You can rescue the Levin rule manually. See this tutorial for details.
We need to construct a basis $\{w_j\}$, the Levin "kernel," that satisfies a homogeneous linear ODE
$$w_j'(x)=\sum_k A_{j,k}(x) w_k(x) \,.$$
For this problem, we may pick for the kernel the following:
$$w_j(x) = e^{iax}\left(1+e^{ibx}\right)^j\,, \quad j=0,\dots,n \,.$$
A simple calculation (luckily) shows
$$w_j'(x)=i a \, w_j(x)+i b j \left(w_j(x)-w_{j-1}(x)\right) \,,$$
which is programmatically valid for $j=0$ since $w_{j-1}$ is multiplied by $j$.
In the code below, only the kernel levinK[n] and the differential matrix $A$ levinDM[n] are needed to implement the Levin rule in NIntegrate. The others were written for checking the ODE above. While levinDM[] uses levinODE[] and levinVARS[], it could directly compute the matrix.
ClearAll[levinODE, levinICS];
levinODE[n_] :=
Table[w[j]'[x] == I*a*w[j][x] + I*b*j*(w[j][x] - w[j - 1][x]), {j, 0, n}];
levinICS[n_] := Table[w[j][0] == 2^j, {j, 0, n}];
levinVARS[n_] := Array[w, n + 1, 0];
levinK[n_] := Table[Exp[I a x] (1 + Exp[I b x])^j, {j, 0, n}];
levinDM[n_] := CoefficientArrays[
levinODE[n][[All, -1]],
Through[levinVARS[n][x]]
][[2]]; (* Part 2 are the coefficients of the linear part *)
We have to specify the "Amplitude", or NIntegrate will choose $w_0(x)$ for determining it instead of $w_n(x)$.
Block[{n = 4, a = 3, b = 4},
NIntegrate[Exp[I a x] x^-n (1 + Exp[I b x])^n, {x, 1, 10^4},
Method -> {"LevinRule"
, "AdditiveTerm" -> 0
, "Amplitude" -> SparseArray[{{n + 1} -> x^-n}, {n + 1}]
, "Kernel" -> levinK[n]
, "DifferentialMatrix" -> levinDM[n]
, "Points" -> 20 + 10 n (* for a=3, b=4, n up to 8 or higher *)
}]
]
(* -0.01205 + 0.0929632 I *)
The number of collocation "Points" needed depends on how oscillatory the integrand is (a and b) and the dimension of the Levin system (n). I don't know off-hand a formula for it. For a and b on the order of $10^{-3}$, the formula 5 + 5 n works for small n.
A: I thought it was cool how easy it was to do the Levin rule by hand, but there's an easier way. @J.M.'s advice for the OP's previous oscillatory integral applies here, too. A complex integration path can dampen the oscillations, something like {x, 1, 1 + 2I, 10^4 + 2I, 10^4}. It's considerably faster than the Levin rule. Since the integrand is analytic in a simply connected, complex neighborhood of 1 <= x <= 10^4 (for example, Re[x] >= 0), a high-order Gauss rule will be beneficial.
Block[{n = 4, a = 3, b = 4},
NIntegrate[
Exp[I a x] x^-n (1 + Exp[I b x])^n,
{x, 1, 1 + 2 I, 10^4 + 2 I, 10^4},
Method -> {"GaussKronrodRule", "Points" -> 21}]
]
(* -0.01205 + 0.0929632 I *)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,822 |
Q: How to group by string ID in sql server 2008 How do I get :
ID Value
99-07-06-01 1
99-07-06-02 2
99-07-06 3
99-07-08-01 4
99-07-08-02 5
99-07-08 6
to
ID Value
99-07-06 6
99-07-08 15
i.e I wanted to group by the ID [99-07-0x] while summing.
thanks
A: Try this (both MySQL and MS-SQL):
SELECT LEFT(ID, 8) AS nID,
SUM(Value) AS tot
FROM your_table
GROUP BY LEFT(ID, 8)
| {
"redpajama_set_name": "RedPajamaStackExchange"
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1.0.3 Nostalgia – Do we have photo obsessions because we are replicant?
1.0.4 Welcome to China Blogs!
I think I could not live in a world without colors, have you seen the movie Pleasentville? Colors simbolize passion, but for photography, black & white can be a good way for express emotion, sometimes we need colors to express some feeling, others times black & white can be a perfect choice.
Anyway, last December I finally decided how to use an expired 120mm black & white film I found in a small lab, that now doesn't sell films anymore and only prints digital photo. I felt that I had in my hand something precious, 120mm film are rare in Kunming, so I decided to use my 120mm to shoot a becoming rare reality. Kunming city is changing its face and everyday is full of new high buildings, fashion shopping mall and traditional life is getting very rare. People have meal at fast food, or at Chinese Chain restaurants and if they have their meal at home they buy the food at Carrefour, Wallmart, Metro, Trust Mart. The main reason is because a lot of old open markets were destroyed because they were in old urban area. So I took my Lubitel 166+ and move around in an old area that I know it will be destroyed soon: open market, alleys … I wanted to have a piece of this kind of life that tomorrow will gone away.
– Woman watering vegetables with a bottle of Sprite, of course there is no Sprite inside. The bottle cap was drilled to release water when press the bottle.
Nostalgia – Do we have photo obsessions because we are replicant? | {
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What's Really Going On With Demi Lovato And Noah Cyrus?
Kevin Mazur/Getty Images
By Desirée O/April 1, 2021 10:30 am EST
If you're a fan of pop music — or just happen to turn on the radio every once in a while — then you'll likely be aware of the fact that Demi Lovato is the singer behind tunes like "Heart Attack," "Sorry Not Sorry," and "Sober." Noah Cyrus, a.k.a. Miley's little sister, is also in the biz, which is why you can listen to songs that she's released like "July" and "Make Me (Cry)." With both Lovato and Cyrus thrilling listeners with their playlist-worthy music, fans were surely excited when the two decided to team up.
As Cosmopolitan points out, the singers "collaborated together on 'Easy,'" which can be found on Lovato's 2021 Dancing With the Devil... the Art of Starting Over album. Written by Hilary Duff's husband Matthew Koma, "Easy" wasn't originally intended for two people, however, "Lovato decided to make it a collaboration after hearing it," according to Page Six. A source told the outlet, "It's a folksy song about leaving something behind ... It's such a personal album for [Demi] that lays out her journey."
Part of Lovato's journey for this specific album included finding the perfect person to work on the song with her. That turned out to be Cyrus. However, some people apparently believe that there's more going on between the two than just making music together.
Are Demi Lovato and Noah Cyrus dating?
Alberto E. Rodriguez/Getty Images
It seems like 2021 is the year that Demi Lovato decided to put herself out into the world in different ways. For instance, along with releasing a new album, she also came out as pansexual. And she might have started a new relationship as well? That is, if the buzz is to be believed when it comes to her involvement with Noah Cyrus.
After the two stars recorded their song "Easy" together, some were left wondering if the two were actually dating. While, as of this writing, neither Lovato or Cyrus (who may have been in a same-sex relationship with Tana Mongeau in the past) have confirmed or denied the dating rumors, a source told Page Six in March 2021, "They're very close and have been hanging out."
Although the insider "even suggested the pair might have begun a fling," per the outlet, another source apparently thinks that the idea is "weird" and claimed that "they're not dating." They added, "[Lovato and Cyrus] have dinner sometimes, but they're not romantic. They got together for this song that sorta came together a bit last minute, and they hang out." It looks like we'll just have to wait and see if it's indeed a friendship, a romance, or just a professional relationship that has sparked between these two. | {
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#ifndef _GENERATOR_H_
#define _GENERATOR_H_
#include <string.h>
#include <assert.h>
#include <math.h>
#include "ruby.h"
#if HAVE_RUBY_RE_H
#include "ruby/re.h"
#endif
#if HAVE_RE_H
#include "re.h"
#endif
#ifdef HAVE_RUBY_ENCODING_H
#include "ruby/encoding.h"
#define FORCE_UTF8(obj) rb_enc_associate((obj), rb_utf8_encoding())
#else
#define FORCE_UTF8(obj)
#endif
#define option_given_p(opts, key) RTEST(rb_funcall(opts, i_key_p, 1, key))
#ifndef RHASH_SIZE
#define RHASH_SIZE(hsh) (RHASH(hsh)->tbl->num_entries)
#endif
#ifndef RFLOAT_VALUE
#define RFLOAT_VALUE(val) (RFLOAT(val)->value)
#endif
#ifndef RARRAY_PTR
#define RARRAY_PTR(ARRAY) RARRAY(ARRAY)->ptr
#endif
#ifndef RARRAY_LEN
#define RARRAY_LEN(ARRAY) RARRAY(ARRAY)->len
#endif
#ifndef RSTRING_PTR
#define RSTRING_PTR(string) RSTRING(string)->ptr
#endif
#ifndef RSTRING_LEN
#define RSTRING_LEN(string) RSTRING(string)->len
#endif
#define RSTRING_PAIR(string) RSTRING_PTR(string), RSTRING_LEN(string)
/* fbuffer implementation */
typedef struct FBufferStruct {
unsigned int initial_length;
char *ptr;
unsigned int len;
unsigned int capa;
} FBuffer;
#define FBUFFER_INITIAL_LENGTH 4096
#define FBUFFER_PTR(fb) (fb->ptr)
#define FBUFFER_LEN(fb) (fb->len)
#define FBUFFER_CAPA(fb) (fb->capa)
#define FBUFFER_PAIR(fb) FBUFFER_PTR(fb), FBUFFER_LEN(fb)
static char *fstrndup(const char *ptr, int len);
static FBuffer *fbuffer_alloc();
static FBuffer *fbuffer_alloc_with_length(unsigned initial_length);
static void fbuffer_free(FBuffer *fb);
static void fbuffer_free_only_buffer(FBuffer *fb);
static void fbuffer_clear(FBuffer *fb);
static void fbuffer_append(FBuffer *fb, const char *newstr, unsigned int len);
static void fbuffer_append_long(FBuffer *fb, long number);
static void fbuffer_append_char(FBuffer *fb, char newchr);
static FBuffer *fbuffer_dup(FBuffer *fb);
/* unicode defintions */
#define UNI_STRICT_CONVERSION 1
typedef unsigned long UTF32; /* at least 32 bits */
typedef unsigned short UTF16; /* at least 16 bits */
typedef unsigned char UTF8; /* typically 8 bits */
#define UNI_REPLACEMENT_CHAR (UTF32)0x0000FFFD
#define UNI_MAX_BMP (UTF32)0x0000FFFF
#define UNI_MAX_UTF16 (UTF32)0x0010FFFF
#define UNI_MAX_UTF32 (UTF32)0x7FFFFFFF
#define UNI_MAX_LEGAL_UTF32 (UTF32)0x0010FFFF
#define UNI_SUR_HIGH_START (UTF32)0xD800
#define UNI_SUR_HIGH_END (UTF32)0xDBFF
#define UNI_SUR_LOW_START (UTF32)0xDC00
#define UNI_SUR_LOW_END (UTF32)0xDFFF
static const int halfShift = 10; /* used for shifting by 10 bits */
static const UTF32 halfBase = 0x0010000UL;
static const UTF32 halfMask = 0x3FFUL;
static unsigned char isLegalUTF8(const UTF8 *source, int length);
static void unicode_escape(char *buf, UTF16 character);
static void unicode_escape_to_buffer(FBuffer *buffer, char buf[6], UTF16 character);
static void convert_UTF8_to_JSON_ASCII(FBuffer *buffer, VALUE string);
static void convert_UTF8_to_JSON(FBuffer *buffer, VALUE string);
/* ruby api and some helpers */
typedef struct JSON_Generator_StateStruct {
char *indent;
long indent_len;
char *space;
long space_len;
char *space_before;
long space_before_len;
char *object_nl;
long object_nl_len;
char *array_nl;
long array_nl_len;
FBuffer *array_delim;
FBuffer *object_delim;
FBuffer *object_delim2;
long max_nesting;
char allow_nan;
char ascii_only;
} JSON_Generator_State;
#define GET_STATE(self) \
JSON_Generator_State *state; \
Data_Get_Struct(self, JSON_Generator_State, state)
static VALUE mHash_to_json(int argc, VALUE *argv, VALUE self);
static VALUE mArray_to_json(int argc, VALUE *argv, VALUE self);
static VALUE mInteger_to_json(int argc, VALUE *argv, VALUE self);
static VALUE mFloat_to_json(int argc, VALUE *argv, VALUE self);
static VALUE mString_included_s(VALUE self, VALUE modul);
static VALUE mString_to_json(int argc, VALUE *argv, VALUE self);
static VALUE mString_to_json_raw_object(VALUE self);
static VALUE mString_to_json_raw(int argc, VALUE *argv, VALUE self);
static VALUE mString_Extend_json_create(VALUE self, VALUE o);
static VALUE mTrueClass_to_json(int argc, VALUE *argv, VALUE self);
static VALUE mFalseClass_to_json(int argc, VALUE *argv, VALUE self);
static VALUE mNilClass_to_json(int argc, VALUE *argv, VALUE self);
static VALUE mObject_to_json(int argc, VALUE *argv, VALUE self);
static void State_free(JSON_Generator_State *state);
static JSON_Generator_State *State_allocate();
static VALUE cState_s_allocate(VALUE klass);
static VALUE cState_configure(VALUE self, VALUE opts);
static VALUE cState_to_h(VALUE self);
static void generate_json(FBuffer *buffer, VALUE Vstate, JSON_Generator_State *state, VALUE obj, long depth);
static VALUE cState_partial_generate(VALUE self, VALUE obj, VALUE depth);
static VALUE cState_generate(VALUE self, VALUE obj);
static VALUE cState_initialize(int argc, VALUE *argv, VALUE self);
static VALUE cState_from_state_s(VALUE self, VALUE opts);
static VALUE cState_indent(VALUE self);
static VALUE cState_indent_set(VALUE self, VALUE indent);
static VALUE cState_space(VALUE self);
static VALUE cState_space_set(VALUE self, VALUE space);
static VALUE cState_space_before(VALUE self);
static VALUE cState_space_before_set(VALUE self, VALUE space_before);
static VALUE cState_object_nl(VALUE self);
static VALUE cState_object_nl_set(VALUE self, VALUE object_nl);
static VALUE cState_array_nl(VALUE self);
static VALUE cState_array_nl_set(VALUE self, VALUE array_nl);
static VALUE cState_max_nesting(VALUE self);
static VALUE cState_max_nesting_set(VALUE self, VALUE depth);
static VALUE cState_allow_nan_p(VALUE self);
static VALUE cState_ascii_only_p(VALUE self);
#endif
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Aage Valdemar Larsen Roose, född 2 januari 1880 i Kolding, Danmark, död 1970, var en dansk konstnär.
Han var son till spannmålshandlaren Christian Hansen Roose och Lisette Helene Kristine Marie Petersen och från 1907 gift med Wanda Agnes Reichman. Roose studerade vid Det Kongelige Danske Kunstakademi 1899–1901 och bedrev därefter privatstudier för Laurits Tuxen och Peder Severin Krøyer 1901-1905 han reste till Paris 1905 där han studerade vid Académie Colarossi 1905-1906 och under ett flertal studieresor till England, Italien och Sverige. Han kom att bli en Sverigevän och var en period bosatt i Filipstad och efter återflytten till Danmark upprätthöll han kontakten med Sverige genom sommarvistelser i Bohuslän och Stockholms skärgård. Tillsammans med sin fru ställde han ut i Köpenhamn 1910 och ställde därefter ut i ett flertal danska och internationella utställningar. I Sverige ställde han ut separat på Konstsalong Rålambshof i Stockholm 1947, Lunds universitets konstmuseum 1919, Värmlands museum 1950 samt tillsammans med sin fru på Konstnärshuset i Stockholm 1919. Hans konst består av porträtt, figurbilder, djurstudier och landskapsmålningar. Roose finns representerad vid bland annat Nationalmuseum, Moderna museet, Lunds universitets konstmuseum, Värmlands museum och Kunstmuseets kopparstickssamling i Köpenhamn.
Tryckta källor
Svenskt konstnärslexikon del IV sid 511, Allhems Förlag, Malmö.
Noter
Danska konstnärer
Män
Födda 1880
Avlidna 1970
Personer från Kolding
Representerade vid Nationalmuseum
Representerade vid Moderna museet | {
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Ophelia has some worried along eastern Florida coast
JACKSONVILLE BEACH, Fla. – Brisk wind and forceful rain moved Wednesday through this beachside community awaiting Tropical Storm Ophelia, sending tourists scurrying inland and driving surfers out of the churning Atlantic.The tropical depression off the coast strengthened into a tropical storm early Wednesday and could bring high winds and heavy rain to central and northern Florida over the next few days.The Jacksonville area is one of the few spots spared any serious damage from the six hurricanes that have hit the state in the last 13 months."We have to wait and see what (Ophelia) is going to do. The possibilities are endless," said Steve Letro, a meteorologist with the National Weather Service in Jacksonville.Without strong steering currents, it was impossible to say what path Ophelia would take or whether it would strengthen into a hurricane, with winds of at least 74 mph, Letro said.Some computer models had the storm moving farther east and away from the coast, others had it going west, closer to shore, and some showed it heading east and then looping back toward the state.At 5 p.m. EDT, Ophelia, the 15th named storm of the season, was centered 80 miles east-northeast of Cape Canaveral. It had top sustained winds of about 50 mph, up from 40 mph earlier in the day, and was nearly stationary.A tropical storm warning, which means winds exceeding 39 mph are possible within 24 hours, was canceled for areas south of Cocoa Beach but remained in effect north to Flagler Beach.A tropical storm watch was in effect from Flagler Beach to Fernandina Beach. A watch means that tropical storm conditions are possible within 36 hours."This one is going to keep everybody on the edge of their seats for quite some time," Letro said.Ken Lloyd, general manager of a beachside restaurant and bar, was born during Hurricane Dora, which was the last hurricane to hit this beachside community, in 1964."I'm worried about it. They just don't know what it is going to do," Lloyd said.In Volusia County, which suffered damage in three of last year's hurricanes, plans were being made to set up two shelters Thursday."We're just opening them in an extreme abundance of caution," said Dave Byron, a county spokesman.Elsewhere in the Atlantic, Tropical Storm Nate intensified into a hurricane south of Bermuda, and Tropical Storm Maria was upgraded to a hurricane again over the open ocean. Neither was considered a threat to the United States.At 5 p.m., Nate had top sustained winds of 85 mph. It was centered about 200 miles south-southwest of Bermuda and was drifting northeast at 5 mph. It could pass near or just south of Bermuda by Thursday, forecasters said.Forecasters said new observations showed that Maria was again a hurricane with 80 mph winds. It was moving northeast at about 14 mph toward the colder waters of the north Atlantic and was centered about 875 miles east-northeast of Bermuda.Maria and Nate were the fifth and sixth hurricanes of the Atlantic season, which began June 1 and ends Nov. 30. Peak storm activity typically occurs from the end of August through mid-September.Vail, Colorado | {
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{"url":"https:\/\/gamedev.stackexchange.com\/questions\/104874\/glm-z-coordinates-are-flipped","text":"# glm Z coordinates are flipped\n\nI am trying to render a 3D model using OpenGL. For the projection and transformation matrices, I am using glm. I've got my model on the screen and it works just like I intended it to; except one small problem.\n\nI am setting the model's translation matrix as\n\nglm::translate(glm::vec3(0, 0, 4))\n\nto move the model a little bit forward so that I can see it. Since in OpenGL, by default, negative z is out towards the 'camera' and positive z is forward, I expected this to work but it doesn't. It only works if I set it to\n\nglm::translate(glm::vec3(0, 0, -4))\n\nBut this seems weird to me, as I am setting my zNear to 0.01 and zFar to 1000. Is glm's z values flipped or am I doing something wrong here?\n\nHere is my code:\n\nglm::mat4 rotation = glm::mat4(1.0f);\nglm::mat4 translation = glm::translate(glm::vec3(0, 0, -4));\nglm::mat4 scale = glm::mat4(1.0f);\n\nglm::mat4 modelMatrix = translation * rotation * scale;\n\nglm::mat4 projectionMatrix = glm::perspective(70.0f, aspectRatio, 0.01f, 1000.0f);\n\nglm::mat4 transformationMatrix = projectionMatrix * modelMatrix;\n\n\nYou're wrong. OpenGL uses a right-handed coordinate-system where x is right, y is up and z is into the camera, therefore negative z is forwards.\n\u2022 Well thats hard to explain. They are not stored directly, they are used to calculate the projection-matrix. But in some sense you could say that you'd see -0.1 to -1000.0, yes. \u2013\u00a0tkausl Jul 31 '15 at 2:12","date":"2019-11-21 19:22:09","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.6324880719184875, \"perplexity\": 2261.0697764252373}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-47\/segments\/1573496670948.64\/warc\/CC-MAIN-20191121180800-20191121204800-00382.warc.gz\"}"} | null | null |
Oliver Lüttkenhaus (* 28. Juni 1971 in Hamburg) ist ehemaliger deutscher Fußballspieler.
Karriere
Oliver Lüttkenhaus, der damals beim Verbandsligisten VfL 93 Hamburg spielte, nahm im April 1992 beim Bundesligisten Hamburger SV an einem Probetraining teil. Er war Amateurauswahlspieler des Hamburger Fußball-Verbandes. 1993 wechselte er zum HSV. Sein Debüt in der Bundesliga gab er am 8. Spieltag der Saison 1994/95 im Spiel gegen den VfL Bochum. Die Hanseaten gewannen mit 3:1. Insgesamt bestritt er vier Spiele in der Bundesliga. Von 1995 und 1997 stand er für den VfL Wolfsburg in der 2. Bundesliga auf dem Platz.
Weblinks
Einzelnachweise
Fußballspieler (VfL 93 Hamburg)
Fußballspieler (Hamburger SV)
Fußballspieler (VfL Wolfsburg)
Fußballspieler (Eimsbütteler TV)
Deutscher
Geboren 1971
Mann | {
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{"url":"https:\/\/www.physicsforums.com\/threads\/energy-in-b-field-of-infinite-line-current.515668\/","text":"# Energy In B field of infinite line current\n\n1. Jul 20, 2011\n\n### Onufer\n\nMy professor had an example in class the other day where we had an infinite line of current uniformly distributed over a circular wire with radius R. He wanted to look at the energy of the B field produced by this current. He did the normal thing and integrated 1\/2u*B^2 over all space. It obviously diverges since it is an infinite wire. But the interesting thing is that it also diverges if you look the energy per unit length. (it ends up being some constants *(ln infinite - lnR) for the energy of the field outside the wire)\n\nMy intuition (which is often not right) felt that it shouldn't be infinite per unit length. I looked at how Griffiths derived the B^2 energy equation and it looks like it uses the assumption that the surface integral of AXB goes to zero as the radius of the surface goes to infinite...\n\nI have two big questions:\n\n1) which formula for energy of a mag field is more fundamentally right? (less assumptions about the nature of the problem)\n\n(my proff said the 1\/2u*B^2 is most fundamental but it seems like all the proofs i can find do integration by parts and assume the surface integral goes to zero as space goes to infinite to get that equation)\n\n2) When you solve for A you get an integration constant in both parts. (inside and outside the wire) How do you decide what the integration 2nd constant should be?\n\nObviously A needs to be continuous a have a continuous derivative (since there are no surface currents) over all space. The 2 functions I got for A were an S^2 function and an ln(s) function The functions as calculated I had 2 constants and one boundary condition A (must be continuous) so there is another constant that we cant calculate. In electrostatics we often say V=0 at the surface for a problem like this but since A does not have much of a physical meaning. This A constant DOES matter though because of the equation W=.5*(intigral of A*J over all space) (Griffiths 7.31 pg 318) . J is our current density so it is constant and 0 outside of R. But if A has an arbitrary integration constant this integral is completely dependent on that constant which makes no sense since there would be different amounts of energy depending on an arbitrary integration constant.\n\nAlso I may as well show my equations for A and B:\n\nu=mu not\nI= total current\ns is the radius for cylindrical coordinates\n\nInside:\nA =-u*I*s^2\/(4*pi*R^2)+C1 in the Z direction (assuming current is flowing up)\nB = u*s*I\/(2*pi*R^2) in the phi direction\n\nOutside:\nA = -u*I*ln(s)\/(2*pi)+C2 in the z direction\nB = u*I\/(2*pi*s) in the phi direction\n\nI hope that was reasonably clear.....\nThanks in advance for the help!\n\n2. Jul 20, 2011\n\n### Onufer\n\nBtw all Griffiths equations mentioned and the derivation of B^2\/(2u) were on page 318\n\n3. Jul 20, 2011\n\n### nnnm4\n\nThe formula that keeps the surface integral in correct in the general case for any magnetic field of any form. The formula that neglects it is only true if the field goes to zero at the boundaries (infinity). Which one is more fundamental? I don't know really what that would entail. I will say though that the latter formula is much nicer and is *true in any physically realizable problem in electrodynamics*.\n\n4. Jul 21, 2011\n\n### Onufer\n\nCool. That's what i thought too but wasn't sure. Applying that volume and surface integral equation i get:\n\nW = 1\/(2u) *(integral B^2 over inside volume + integral B^2 over outside volume - surface integral of AxB)\n\nI'm gonna divide everything by z since i was interested in the energy per unit length.\n\nthe first volume integral ends up being (after dividing by z)\n\nu^2*I^2\/(8*pi)\nIts kinda cool that it has no dependence on the radius of the wire...\n\nFor volume outside B^2 integral i will integrate from R the outside radius to some arbitrary s value which will be denoted by s'\nThis gives\n\nu^2*I^2\/(2*pi)(ln(s') - ln(R))\n\nThe last surface area integral I will do at the radius s' the enclosing the previous volume integral. It is important to note the top and bottom of this cylindrical area contributes zero since BxA points radially while da points up thus there dot product is zero. For all other spots the BxA lines up perfectly with da allowing to simply multiply their magnitudes while observing the sign. If A is negative (which it will be if we go out far enough) AxB points outward so the integral will be positive but there is a negative sign in front of it...\n\n-u^2*I^2*ln(s)\/(2*pi)+C2*I\/2\n\nnotice that the first part here EXACTLY cancels out the first part of the outside volume integral. (the part that was diverging when we tried to integrate B^2 over all space). The bounds of the integral are irrelevant as long as we are outside the wire.\n\nSumming these integrals gets\n\nW\/z (energy per unit length)= u*I^2\/(16*pi) - u*I^2*ln(R)\/(4*pi) + C2*I\/2\n\nHere is why I am interested in the integration constant of A... The energy seems oddly to depend on it. I am hoping there is a justifiable boundry condition on C2 that it would be a function of R and do some canceling with the middle term to get a reasonable answer (the problem goes crazy if you have a small radius)\n\nIs there a boundary condition I am not thinking or have I taken the math too far out of the realm of physics... I have learned a lot from playing around with this problem but i would really like it to work out to something meaningful....\n\nIf anyone has any thoughts I would really appreciate it.\n\n5. Jul 21, 2011\n\n### Dickfore\n\nIf you are worried about these kinds of infinities, please calculate the total energy of the electrostatic field of a point charge.\n\n6. Jul 21, 2011\n\n### Onufer\n\nI would say its zero because it takes zero energy to move a point charge in from infinite since its field does not effect itself and we generally consider the energy of a charge distribution to be the energy required to assemble the system. probably wrong though.\n\nI'm not uncomfortable with infinities, its just that for a similar problem in electrostatics, with an infinite line of uniformly bound charge, i don't think the energy of that system goes to infinite per unit length. Thus why would this problem?\n\nIn the electrostatic problem we would generally make the assumption that V = 0 a the surface allowing us to define the integral. I was just wondering if there is a similar condition we can use for the magnetic vector potential.","date":"2018-10-21 05:35:30","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9164966940879822, \"perplexity\": 482.0878321544777}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-43\/segments\/1539583513760.4\/warc\/CC-MAIN-20181021052235-20181021073735-00086.warc.gz\"}"} | null | null |
Q: CSS - Active link change text I am trying to create an accordion link (that expands the accordion)
I would like the text change depending on whether the link is active or not.. for example:
DEFAULT : "View all Comments"
ACTIVE : "Hide Comments"
I know I can use CSS to style the content colours, style, etc., but wondering if there is a way I can change the actual text in the link that is dependent on the link being active or not.
A:
.accordion-item::before { /* default text of the accordion item */
content : "View all comment";
}
.accordion-item:focus:before { /* triggered when the accordion item is clicked/focused */
content : "Hide Comments";
}
<a href="#" class="accordion-item"></a>
Here is a sample of how to do this, assuming you will have several link that stands for each accordion items.
Note that you could have used the new HTML data-* attribute to kind of automatize the process in case you are generating the HTML before being interpreted through language such as Php or JavaScript like following :
A: You can use pseudo element like this
<a class="toggleText"></a>
CSS:
.toggleText::after
{
content:"View all Comments";
}
.toggleText:active::after
{
content:"Hide Comments";
}
psuedo elements might not supported on old browsers you wish you want to support.
check support here:http://caniuse.com/#search=%3A%3Aafter
and for content : http://caniuse.com/#search=CSS%20Generated%20content%20for%20pseudo-elements
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,270 |
From beneath the spinning dough, Rosario winks at me, his disc elastic and expanding in a whir of fingertips. Last December, UNESCO awarded pizza making in Naples World Heritage status and I'm beginning to appreciate why.
Not that the heritage committee would recognise anything about my attempt at twirling dough. I stretch it with one hand, then flip it across to the other, spinning it slightly when I toss it back. Slap, slap, slap, flop; it folds in half. My pizzaiolo tells me it takes passion to make pizza — I have that of a limp anchovy.
Naples is where the Margherita was invented in 1889 and so, fed up of the tasteless, bloating pizzas I make at home, I'm at a Pizza Experience class at Pizzeria al 22 in Naples led by pizzaiolos qualified in teaching this centuries-old art. The class begins calmly. Giovanni, one of my two maestros, shows me how to make the dough, swirling the flour and other dry ingredients into a bowl of water and squashing out lumps. I then work at kneading the mixture. On the wall are ageing photos of Giovanni with his grandfather, who started this pizzeria in 1935 and who he began helping after school. "A pizzeria is a very happy place to work," he tells me, as his staff dance to the radio in the kitchen.
A magician with a well-floured bench, Giovanni's next trick involves separating the dough into individual pies by stuffing a portion of it into his partly closed fist. As he tightens his fingers, the excess dough balloons out above his thumb. He gives this a gentle twist and — pop — is left holding the smoothest, most perfectly formed sphere of dough. I practise while the wood-fired oven fizzes and crackles behind us. "It's relaxing, right?" Giovanni smiles. "In the pizzeria you can live a long life."
Rosario, my other maestro, then lets me have a go at stretching and spinning one of these balls. Inevitably, my effort collapses in a puff of flour, but it's not a disaster. We squish it back into shape and dress it with fresh ingredients — tomatoes from Vesuvius, hard cheese, mozzarella, basil, oil. Finally, we slide my creation on to a long-armed peel (shovel-like tool) and shimmy it into position alongside the blazing logs.
The spell cast by the gentle rhythms of the past couple of hours is broken. Strictly, Neapolitan pizzas should only spend 60-90 seconds in the oven. No one speaks as we watch the cheese bubble. I lean towards it and almost get my head whacked by the end of Rosario's peel. "Look out! The baker has right of way!" he laughs as he shunts the shovel quickly in and out again.
Mere seconds later, the room is filled with the aromas of tomato, basil and sweet malty bread, and I'm presented with a Margherita that I would genuinely pay money to eat. I fold a hot, oozing corner into my mouth and smile, a little oil dribbling down my chin. If I can recreate this at home there's a chance I'll never eat anything else again.
Why is pizza so popular across the world?
Because it's easy to reproduce. First, it was a very cheap dish. It was the food of the people, not so much of the poor. Today, however, it has become a food that can satisfy even the finest, most gourmet palates. There has been a great evolution of pizza, especially in terms of quality.
Your pizza was voted the world's best in the Where to Eat Pizza guide — what is special about it?
Part of our work is scientific. My bread base is very digestible because I use a blend of my own 00 flour that I came to after lots of research. I also try and create different perceptions for customers, such as with my dessert pizza, for which I use apricot jam and add powdered olives that have been dehydrated in the oven.
You champion local producers and traditional Neapolitan pizzas that only use specific local ingredients. Why?
About 70% of my menu uses products from my territory. With the farmers and workers here we've created a micro-economy. I want to show what's great about the Alto Casertano area, with onions from Alife, local oil, the Caiazzana olive, black pork from Caserta, Romano cheese and local mozzarella.
Why do you still prepare dough the traditional way, by hand?
We have five people in my pizzeria dedicated to kneading dough. It's all about sensory perception. By touch, they can understand the temperature, the malleability, the gluten level, the elasticity. Some people use machines for kneading but they are an imposed force. They don't listen to the dough. A machine doesn't have the same experience as my hands. The dough is alive. By feeling it they know exactly what needs to be done. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,360 |
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'Family Affairs' interview with Sir Nicholas Wall
Date:12 AUG 2010
Full interview first published in FLBA's [2010] Family Affairs 49, summer edn. Extracts reproduced with kind permission.
Sir Nicholas Wall became President of the Family Division earlier this year despite a well-reported hiccough from the then Lord Chancellor, Jack Straw. It was a popular appointment among the profession, who felt they had got their man.
Sir Nicholas was educated at Dulwich College and won a scholarship to Trinity College, Cambridge, where he read law and became president of the Union. Called to the Bar by Grays Inn in 1969, he enjoyed a successful family practice in Mitre Court (now Hare Court) and rapidly made his way up the judicial ladder, being appointed to the High Court bench in 1993 at the age of 48. He was appointed a Lord Justice in 2004.
Known for an independence of mind and compassionate judgments, the new President made a rallying speech to Cumberland Lodge delegates this year, reminding us how far the Family Bar had come during his own professional lifetime, how good the best advocacy among Family counsel can be, and pledging to support the FLBA to the best of his ability during the choppy weather ahead.
The President spoke to Philip Cayford.
Congratulations on your appointment, which you will know was an extremely popular one.
"Thank you for your congratulations on my appointment. I am pleased to think that it was popular, and am determined to do my best for a beleaguered family justice system in very difficult times."
What, generally, do you see as the biggest problems facing the administration of Family Law within England and Wales today?
"The biggest problems are, I think, financial - but allied to the whole question of resources is the argument about how we go about doing things. Can we go on having long, adversarial private law disputes about contact? Should we be more economical in our use of experts in both private and public law cases, for example? I do not want to see family law turned into the preserve of the affluent. Take just one example; vulnerable people, who often lack the financial resources to litigate, need the court's protection from exploitation by their partners. Only family court orders can provide that protection."
How do you think our relations are likely to be with the new coalition Government? How much notice will (or can) the new Government take of Family Law problems (including the ongoing resources issue?)
"Judges work with governments, whatever their political persuasion. Everyone is being asked to make massive cuts. The Family Justice Review has survived the election and the impression I have gained is that the new government is well aware of the problems besetting the Family Justice System and is anxious to address them. On the other hand, the government must recognise that family law is important. People care most about their personal relationships, their children and their money. Family law touches everyone at some point in their lives, usually when people are most vulnerable. Any government would ignore this at its peril.
"I have met the Minister with responsibility for Cafcass (Tim Loughton) and have corresponded with Jonathan Djanogly. I am looking forward to a constructive dialogue, and intend to be robust when necessary."
At a recent Bar conference on the future of the Bar it seemed to be accepted that many legal aid practitioners were now not earning the same as plumbers but significantly less. What effect do you think the existing (and likely future) legal aid cuts are having on the Family Bar?
"The legal aid cuts are very serious. Family Lawyers represent some of the most disadvantaged people in society. As I say, I am very anxious that the Bar should not become the preserve of the affluent. Furthermore, competent representation in family justice is very important. Family law requires skills not needed in other jurisdictions. Specialisation is crucial."
What are you able to say about the ongoing problems within Cafcass?
"Sadly, I cannot say that as much progress has been made on this as I would like. I have made it clear that I am not going to renew the interim guidance and I am considering closely the discontinuance of the 'duty guardian' scheme, which it seems, has not succeeded. It is also clear to me that the government is likely to invest heavily in the Family Justice Review, where Cafcass is high on the agenda. I therefore anticipate that there will not be a great deal of immediate change, and that the positive aspects of the Interim Guidance will need to be built upon. Given the published timetable of the FJR, we should be able to re-assess the position in the autumn, with a view to a root and branch revision, following final recommendations, in 2011. So watch this space."
In light of a) your comments in Re D, b) the Washington Declaration c) the new research from Dr Marilyn Freeman and Professor Parkinson and now d) the comments of Mostyn J in Re AR, where are we now on international child relocation? Is it still a question of finding a rich or tenacious (or both) litigant to push the right case to the Supreme Court before any effective review of Payne v Payne can take place?
"As I said recently in Re D [2010] EWCA Civ 50 (which I am delighted to see that you have all read) there is a perfectly respectable argument for the proposition that Payne v Payne places too great an emphasis on the wishes and feelings of the relocating parent, and ignores or relegates the harm done to children by a permanent breach of the relationship which children have with the left behind parent. However, all relocation cases are (1) very difficult; and (2) highly fact specific. Re D was plainly not the case upon which to base a re-appraisal of Payne. Furthermore, as I also made clear in Re D, we operate a doctrine of precedent and it will be either for the government to change the law or for the Supreme Court to reconsider the issue in a suitable case. I do not think that a litigant would necessarily have to be either rich or tenacious to get to the Supreme Court, but a finely balanced case is likely to turn on the trial judge's exercise of discretion, with which it may be difficult to interfere."
The Law Commission is considering pre-nuptial agreements, but is there any other pressing issue that you believe it should look at? Do you believe, for example that s25 itself needs fundamental reconsideration?
"I am in favour of no fault divorce, and support cohabitants being given the right to make financial claims against each other. I suspect that the Law Commission has to have a long fuse. It may have sometimes to wait 10 or 20 years before a government catches up with it."
What were your early legal influences and why did you come to the Bar?
"I came to the law in order to earn my living. Law was very well taught at university and all the teachers seemed to regret being academics and not in practice. I still have difficulty understanding academic comment on some of my decisions! I came to the bar - as opposed to becoming a solicitor because I did not like the idea of instructing somebody else and then watching them make a pig's breakfast of a case. I would rather do it myself."
Is there any one person who has particularly inspired you during your life and if so, whom?
"The lawyer for whom I had the greatest respect was Roger Ormrod, although I hope that it is easier to appear in front of me."
What are your extra-curricular interests? (I know you used to be an active Refresher!)
"I collect, bind and restore books. I would happily spend the rest of my life doing so. I would also like to research Elizabethan cryptography, if time permits after I retire. It is kind of you to recall that I was a geriatric Refresher, who can claim to have played in the last game of the barristers against their clerks at the Oval."
What might you have done had you not come to the Bar?
"I would have gone into publishing or the British Library."
What would you like your legacy to be (in due course?)
"If I can leave behind a family justice system which is properly valued by outsiders, and which is rightly recognised as the equal to the other branches of the justice system I will be content. I greatly value the dedication and genuine altruism of all who work in the family justice system, and I hate to see them unjustly criticised by the ignorant or the tendentious. I have been kept going by the ideas which the system has generated, and I hope to have an inclusive Presidency which embraces change and throws up interesting and radical ideas."
Would you recommend the Family Bar today to a member of your family or to a bright young graduate looking for a career?
"Yes, although I came into it by accident. But be aware. The young family lawyer will have many frustrations, and will not grow rich. But he or she will have immense job satisfaction, and will have the real privilege of seeing humanity at both its best and its worst."
Interview with Sir Mark Potter
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"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction}
The dual superconductor picture proposed long ago \cite{dualsuper} is believed to be a promising mechanics for quark confinement.
For this mechanism to work, however, magnetic monopoles and their condensation are indispensable to cause the dual Meissner effect leading to the linear potential between quark and antiquark, namely,
area law of the Wilson loop average.
The Abelian projection method proposed by 't Hooft \cite{tHooft81} can be used to introduce such magnetic monopoles into the pure Yang-Mills theory even without matter fields.
Indeed, numerical evidences supporting the dual superconductor picture resulting from such magnetic monopoles have been accumulated since 1990 in pure SU(2) Yang-Mills theory \cite{SY90,SNW94,AS99}.
However, {\it the Abelian projection method explicitly breaks both the local gauge symmetry and the global color symmetry} by partial gauge fixing from an original non-Abelian gauge group $G=SU(N)$ to the maximal torus subgroup, $H=U(1)^{N-1}$.
Moreover, the Abelian dominance \cite{SY90} and magnetic monopole dominance \cite{SNW94} were observed only in a special class of gauges, e.g., the maximally Abelian (MA) gauge and Laplacian Abelian (LA) gauge, realizing the idea of Abelian projection.
For $G=SU(2)$, we have already succeeded to settle the issue of gauge (in)dependence by {\it introducing a gauge-invariant magnetic monopole in a gauge independent way}, based on another method: a non-Abelian Stokes theorem for the Wilson loop operator \cite{DP89,Kondo98b} and a new reformulation of Yang-Mills theory rewritten in terms of new field variables \cite{KMS06,KMS05,Kondo06} and \cite{KKMSSI05,IKKMSS06,SKKMSI07}, elaborating the technique proposed by Cho \cite{Cho80} and Duan and Ge \cite{DG79} independently, and later readdressed by Faddeev and Niemi \cite{FN99}.
For $G=SU(N)$, $N \ge 3$, there are no inevitable reasons why degrees of freedom associated with the maximal torus subgroup should be most dominant for quark confinement.
In this case, the problem is not settled yet.
In this talk,
we give a theoretical framework for describing {\it non-Abelian dual superconductivity} in $D$-dimensional $SU(N)$ Yang-Mills theory, which should be compared with the conventional Abelian $U(1)^{N-1}$ dual superconductivity in $SU(N)$ Yang-Mills theory, hypothesized by Abelian projection.
We demonstrate that {\it an effective low-energy description for quarks in the fundamental representation} (abbreviated to rep. hereafter) {\it can be given by a set of non-Abelian restricted field variables} and that {\it non-Abelian $U(N-1)$ magnetic monopoles} in the sense of Goddard--Nuyts--Olive--Weinberg \cite{nAmm} {\it are the most dominant topological configurations for quark confinement} as conjectured in \cite{KT99,Kondo99Lattice99}.
\section{Wilson loop and gauge-inv. magnetic monopole}
A version of a non-Abelian Stokes theorem (NAST) for the Wilson loop operator originally invented by Diakonov and Petrov \cite{DP89} for $G=SU(2)$ was proved to hold \cite{Kondo98b} and was extended to $G=SU(N)$ \cite{KT99,Kondo08} in a unified way \cite{Kondo08} as a path-integral rep. by making use of a coherent state for the Lie group.
For the Lie algebra $su(N)$-valued Yang-Mills field $\mathscr{A}_\mu(x)=\mathscr{A}_\mu^A(x) T_A$ with $su(N)$ generators $T_A$ ($A=1, \cdots, N^2-1$),
the NAST enables one to rewrite a non-Abelian Wilson loop operator \begin{align}
W_C[\mathscr{A}]
:=& {\rm tr} \left[ \mathscr{P} \exp \left\{ ig_{\rm YM} \oint_{C} dx^\mu \mathscr{A}_\mu(x) \right\} \right]/{\rm tr}({\bf 1})
,
\end{align}
into the surface-integral form:
\begin{equation}
W_C[\mathscr{A}] = \int d\mu_{\Sigma}(g) \exp \left[ ig_{\rm YM} \int_{\Sigma: \partial \Sigma=C} F \right] ,
\end{equation}
where $d\mu_{\Sigma}(g):=\prod_{x \in \Sigma} d\mu(g_{x})$, with an invariant measure $d\mu$ on $G$ normalized as $\int d\mu(g_{x})=1$, $g_{x}$ is an element of a gauge group $G$ (more precisely, rep. $D_R(g_{x})$ of $G$),
the two-form $F:=dA=\frac12 F_{\mu\nu}(x) dx^\mu \wedge dx^\nu$ is defined from the one-form $A := A_\mu(x) dx^\mu$,
$
A_\mu(x) = {\rm tr}\{ \rho[ g_{x}^\dagger \mathscr{A}_\mu(x) g_{x} + ig_{\rm YM}^{-1} g_{x}^\dagger \partial_\mu g_{x} ] \} ,
$
by
\begin{align}
F_{\mu\nu}(x) &= \sqrt{2(N-1)/N} [\mathscr{G}_{\mu\nu} (x)
+ ig_{\rm YM}^{-1} {\rm tr} \{ \rho g_{x}^\dagger [\partial_\mu, \partial_\nu] g_{x} \} ],
\end{align}
with the field strength $\mathscr{G}_{\mu\nu}$ defined by
\begin{align}
\mathscr{G}_{\mu\nu} (x)
&:= \partial_\mu {\rm tr} \{ \mathbf{n}(x) \mathscr{A}_\nu(x) \} - \partial_\nu {\rm tr} \{ \mathbf{n}(x) \mathscr{A}_\mu(x) \}
\nonumber\\&
+ \frac{2(N-1)}{N} ig_{\rm YM}^{-1} {\rm tr} \{ \mathbf{n}(x) [\partial_\mu \mathbf{n}(x), \partial_\nu \mathbf{n}(x) ] \}
,
\end{align}
and a normalized traceless field $\mathbf{n}(x)$ called the color field
\begin{equation}
\mathbf{n}(x) := \sqrt{N/[2(N-1)]} g_{x} \left[ \rho - \bm{1}/{\rm tr}(\bm{1}) \right] g_{x}^\dagger .
\end{equation}
Here $\rho$ is defined as $\rho := | \Lambda \rangle \langle \Lambda |$ using a reference state (highest or lowest weight state of the rep.) $| \Lambda \rangle$ making a rep. of the Wilson loop we consider.
Note that ${\rm tr}(\rho) = \langle \Lambda | \Lambda \rangle = 1$ follows from the normalization of $| \Lambda \rangle$.
Finally, the Wilson loop operator in the fundamental rep. of $SU(N)$ reads \cite{Kondo08}
\begin{align}
& W_C[\mathscr{A}]
= \int d\mu_{\Sigma}(g) \exp \left\{ ig_{\rm YM} (k, \Xi_{\Sigma}) + ig_{\rm YM} (j, N_{\Sigma}) \right\} ,
\label{NAST-SUN}
\nonumber\\
& k:= \delta *f = *df, \quad j:= \delta f ,
\quad
f:= \sqrt{2(N-1)/N} \mathscr{G} ,
\nonumber\\
& \Xi_{\Sigma} := * d\Theta_{\Sigma} \Delta^{-1} = \delta *\Theta_{\Sigma} \Delta^{-1} , \
N_{\Sigma} := \delta \Theta_{\Sigma} \Delta^{-1} ,
\end{align}
where two conserved currents, ``magnetic-monopole current'' $k$ and ``electric current'' $j$, are introduced,
$\Delta:=d\delta+\delta d$ is the $D$-dimensional Laplacian, and $\Theta$ is an antisymmetric tensor of rank two called the vorticity tensor:
$
\Theta^{\mu\nu}_{\Sigma}(x)
:= \int_{\Sigma} d^2S^{\mu\nu}(x(\sigma)) \delta^D(x-x(\sigma)) ,
$
which has the support on the surface $\Sigma$ (with the surface element $dS^{\mu\nu}(x(\sigma))$) whose boundary is the loop $C$.
Incidentally, the last part $ig_{\rm YM}^{-1} {\rm tr} \{ \rho g_{x}^\dagger [\partial_\mu, \partial_\nu] g_{x} \}$ in $F$ corresponds to the Dirac string \cite{Kondo97,Kondo98a}, which is not gauge invariant and does not contribute to the Wilson loop in the end.
For $SU(3)$ in the fundamental rep., the lowest-weight state $\langle \Lambda |=(0,0,1)$ leads to
\begin{equation}
\mathbf{n}(x) = g_{x} (\lambda_8/2) g_{x}^\dagger \in SU(3)/[SU(2) \times U(1)] \simeq CP^2 ,
\end{equation}
with the Gell-Mann matrix $\lambda_8:={\rm diag.}(1,1,-2)/\sqrt{3}$,
while for $SU(2)$, $\langle \Lambda |=(0,1)$ yields
\begin{equation}
\mathbf{n}(x) = g_{x} (\sigma_3/2) g_{x}^\dagger \in SU(2)/U(1) \simeq S^2 \simeq CP^1 ,
\end{equation}
with the Pauli matrix $\sigma_3:={\rm diag.}(1,-1)$.
The existence of magnetic monopole can be seen by a nontrivial Homotopy class of the map $\mathbf{n}$ from $S^2$ to the target space of the color field $\mathbf{n}$ \cite{KT99}:
For $SU(3)$,
\begin{align}
& \pi_2(SU(3)/[SU(2) \times U(1)])=\pi_1(SU(2) \times U(1))
\nonumber\\&
=\pi_1(U(1))=\mathbb{Z} ,
\end{align}
while for $SU(2)$
\begin{equation}
\pi_2(SU(2)/U(1))=\pi_1(U(1))=\mathbb{Z} .
\end{equation}
For $SU(3)$, the magnetic charge of the non-Abelian magnetic monopole obeys the quantization condition \cite{Kondo08}:
\begin{equation}
Q_m := \int d^3x k^0 = 2\pi \sqrt{3} g_{\rm YM}^{-1} n , \ n \in \mathbb{Z} .
\end{equation}
The NAST shows that {\it the $SU(3)$ Wilson loop operator in the fundamental rep. detects the inherent $U(2)$ magnetic monopole which is $SU(3)$ gauge invariant}.
The rep. can be classified by its {\it stability group} $\tilde H$ of $G$ \cite{KT99,Kondo08}.
For the fundamental rep. of $SU(3)$, the stability group is $U(2)$.
Therefore, the non-Abelian $U(2) \simeq SU(2) \times U(1)$ magnetic monopole follows from $\tilde H=SU(2)_{1,2,3} \times U(1)_{8}$, while the Abelian $U(1) \times U(1)$ magnetic monopole comes from $\tilde H=U(1)_{3} \times U(1)_{8}$.
The adjoint rep. belongs to the latter case.
The former case occurs only when the weight vector of the rep. is orthogonal to some of root vectors.
The fundamental rep. is indeed this case.
For $SU(2)$, such a difference does not exist and $U(1)$ magnetic monopoles appear, since $\tilde H$ is always $U(1)$ for any rep..
For $SU(3)$, our result is different from Abelian projection: two independent $U(1)$ magnetic monopoles appear for any rep., since
\begin{align}
& \pi_2(SU(3)/U(1) \times U(1))=\pi_1(U(1) \times U(1)) =\mathbb{Z}^2 .
\end{align}
\section{Numerical simulations}
The SU(3) Yang-Mills theory can be reformulated in the continuum and on a lattice using new variables.
For $SU(3)$, two options are possible, maximal for $\tilde H=U(1)^2$ \cite{Cho80c,FN99a} and minimal for $\tilde H=U(2)$ \cite{KSM08}.
In our reformulation, all the new variables $\mathscr{C}_\mu$, $\mathscr{X}_\mu$ and $\mathbf{n}$ are obtained
from $\mathscr{A}_\mu$:
\begin{equation}
\mathscr{A}_\mu^A \Longrightarrow (\mathbf{n}^\beta, \mathscr{C}_\nu^k, \mathscr{X}_\nu^b)
,
\end{equation}
once the color field $\mathbf{n}$ is determined by solving the reduction condition:
\begin{equation}
\bm\chi[\mathscr{A},\mathbf{n}]
:=[ \mathbf{n} , D^\mu[\mathscr{A}]D_\mu[\mathscr{A}]\mathbf{n} ]
= 0
,
\label{eq:diff-red}
\end{equation}
On a four-dimensional Euclidean lattice, gauge field configurations $\{ U_{x,\mu} \}$ are generated by using the standard Wilson action and pseudo heat-bath method.
For a given $\{ U_{x,\mu} \}$, color field $\{ \bm{n}_{x} \}$ are determined by imposing a lattice version of reduction condition. Then new variables are introduced by using the lattice version of change of variables \cite{lattice-f}.
\begin{figure}[h]
\includegraphics[height=4.0cm,width=7.0cm]{potential-L24b600-31.ps}
\vspace{-0.6cm}
\caption{$SU(3)$ quark-antiquark potential: (from above to below)
full potential $V_f(r)$, restricted part $V_a(r)$ and magnetic--monopole part $V_m(r)$ at $\beta=6.0$ on $24^4$ ($\epsilon$: lattice spacing).}
\label{fig:quark-potential}
\end{figure}
Fig. \ref{fig:quark-potential} shows the full $SU(3)$ quark-antiquark potential $V(r)$ obtained from the $SU(3)$ Wilson loop average $\langle W_C[\mathscr{A}] \rangle$, the restricted part $V_a(r)$ from the $\mathscr{V}$ Wilson loop average $\langle W_C[\mathscr{V}] \rangle$, and magnetic--monopole part $V_m(r)$ from $\langle e^{ ig_{\rm YM} (k, \Xi_{\Sigma}) } \rangle$. They are gauge invariant quantities by construction. These results exhibit infrared $\mathscr{V}$ dominance in the string tension (85--90\%) and non-Abelian $U(2)$ magnetic monopole dominance in the string tension (75\%) in the gauge independent way.
\section{Conclusion}
We have shown:
(i) The $SU(N)$ Wilson loop operator can be rewritten in terms of a pair of gauge-invariant magnetic-monopole current $k$ ($(D-3)$-form) and the associated geometric object defined from the Wilson surface $\Sigma$ bounding the Wilson loop $C$, and another pair of an electric current $j$ (one-form independently of $D$) and the associated topological object, which follows from a non-Abelian Stokes theorem for the Wilson loop operator \cite{Kondo08}.
(ii) The $SU(N)$ Yang-Mills theory can be reformulated in terms of new field variables obtained by change of variables from the original Yang-Mills gauge field $\mathscr{A}_\mu^A(x)$ \cite{KSM08}, so that it gives an optimal description for the non-Abelian magnetic monopole defined from the $SU(N)$ Wilson loop operator in the fundamental rep. of quarks.
(iii) A lattice version of the reformulated Yang-Mills theory can be constructed \cite{lattice-f}.
Numerical simulations of the lattice $SU(3)$ Yang-Mills theory give numerical evidences that the restricted field variables become dominant in the infrared for correlation functions and the string tension ({\it infrared restricted non-Abelian dominance}) and that the $U(2)$ magnetic monopole gives a most dominant contribution to the string tension obtained from $SU(3)$ Wilson loop average ({\it non-Abelian magnetic monopole dominance}).
See \cite{KSSK10} for more informations.
\begin{theacknowledgments}
This work is supported by Grant-in-Aid for Scientific Research (C) 21540256 from Japan Society for the Promotion of Science
(JSPS).
\end{theacknowledgments}
\bibliographystyle{aipproc}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 274 |
static void send_index_page(struct mg_connection *conn) {
const char *data;
int data_len;
char var_name[100], file_name[100];
mg_printf_data(conn, "%s",
"<html><body>Upload example."
"<form method=\"POST\" action=\"/handle_post_request\" "
" enctype=\"multipart/form-data\">"
"<input type=\"file\" name=\"file\" /> <br/>"
"<input type=\"submit\" value=\"Upload\" />"
"</form>");
if (mg_parse_multipart(conn->content, conn->content_len,
var_name, sizeof(var_name),
file_name, sizeof(file_name),
&data, &data_len) > 0) {
mg_printf_data(conn, "%s", "Uploaded file:<pre>");
mg_send_data(conn, data, data_len);
mg_printf_data(conn, "%s", "/pre>");
}
mg_printf_data(conn, "%s", "</body></html>");
}
static int ev_handler(struct mg_connection *conn, enum mg_event ev) {
if (ev == MG_REQUEST) {
send_index_page(conn);
return MG_TRUE;
} else if (ev == MG_AUTH) {
return MG_TRUE;
} else {
return MG_FALSE;
}
}
int main(void) {
struct mg_server *server;
// Create and configure the server
server = mg_create_server(NULL, ev_handler);
mg_set_option(server, "listening_port", "8080");
// Serve request. Hit Ctrl-C to terminate the program
printf("Starting on port %s\n", mg_get_option(server, "listening_port"));
for (;;) {
mg_poll_server(server, 1000);
}
// Cleanup, and free server instance
mg_destroy_server(&server);
return 0;
}
| {
"redpajama_set_name": "RedPajamaGithub"
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\section{Introduction}\label{sec:introduction}}
\IEEEPARstart{S}{ensor} Networks and Wireless Sensor Networks (WSN) are two main components involved in the development of the
Internet of Things (IoT). Security and privacy handling for Sensor Networks present new issues due to specific constraints. Low resources on computation, hardware functionalities and energy consumption in WSNs. We can divide research work into two categories: security and privacy for the data being sent over the network on one side, and node authentication and trust management on the other side.
Both have been actively explored the last ten years, and some solutions have been brought by researchers. However, from our knowledge none of these works propose a complete model for both content access, security, privacy and trust management. In this paper, we focus on addressing authentication and trust management issues.
\subsection{Overview}
During our researches, we have separated existing work into two distinct research areas. The first one is authentication and trust management issues in decentralized networks and WSN. Then we consider ongoing work on blockchains and their applications.
\subsubsection{Authentication and trust for decentralized networks}
We can find a lot of different approaches for authentication in WSN and the IoT in the literature. As outlined by Medaglia {\textit et al.}\cite{Medaglia2010}, WSNs have specific security constraints on node authentication to ensure data validity and confidentiality.
Trust management is tied to authentication mechanisms, as a the mean to identify the trustee and the truster. We take previous work on trust evaluation in distributed networks by Sun {\textit et al.}\cite{sun2006trust}, as a reference on issues concerning trust in decentralized networks for our work.
\subsubsection{Blockchain as a secured data structure}
Recent work by Zyskind et al.\cite{Zyskind2015} shows the interest of the blockchain as a personal data management platform focused on privacy. They outlined how the blockchain helps leveraging user control over data in the context of social networks and big data. Foutiou et al.\cite{Fotiou2016} describe a decentralized name based security system using blockchains to secure contents access in Information-Centric Networking based architectures. These approaches prove usability of the blockchain as a secure decentralized data structure for new applications, but none has been used to provide node authentication and trust management in Wireless Sensor Networks (WSN) and in the Internet of Things (IoT).
\subsection{Our Contribution}
We propose a model based on blockchain data structure used to store decentralized authentication and node trust informations. This model is evolutive, adaptative and ensure reliability over time.
\subsection{Organization}
We first explain briefly the blockchain data structure as presented in Bitcoin. Then we present issues in decentralized node authentication and trust management for WSN. The last part of this paper describes our model of a blockchain based solution for authentication and trust management which provide a solution to overcome decentralized networks issues.
\section{The blockchain technology}
In 2008, a person or group of persons known under the name of Satoshi Nakamoto published a paper\cite{Nakamoto2008} dealing with a new decentralized peer-to-peer electronic cash system. This paper introduces the blockchain as a new data structure to store financial transactions, as well as an associate protocol to ensure the validity of the blockchain in the network.
\subsection{Data structure}
In his paper, Nakamoto describes the blockchain as a database modeled by a linear sequence of blocks, each one containing cryptographic hashes corresponding to the previous and current block to ensure continuity and immutability. Bitcoin uses the blockchain to store financial transactions and contracts.
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{blockchain-bitcoin-link-blocks}
\caption{Bitcoin block chaining mechanism. The Merkle root of all transactions is included in the block header and then used as input for the next block in the chain.}
\label{fig:Bitcoin-block-chaining}
\end{figure}
The chaining method used in Bitcoin (Figure \ref{fig:Bitcoin-block-chaining}) ensures the immutability by using the hash of the previous header block hash in the current block. The header includes the root hash of the Merkle tree of all transactions in the block. This way transactions cannot be changed without changing the root Merkle hash and then invalidating the block. Due to the way the blockchain is built, fork chains can append with different valid blocks storing different transactions. The Bitcoin protocol resolves this issue by selecting the longest blockchain as the correct one. Note that due to this choice, even after being included in a valid block, transactions can be considered valid only after a subsequent block has been calculated and successfully included in the blockchain by the majority of the network \cite{eyal2016bitcoin}.
\subsection{Secure distributed storage based on blockchains}
We consider the blockchain data structure outside of its application in Bitcoin, as a generic decentralized secured data storage structure. It is possible to use any data payloads other than transactions as parts of the block. The block is then divided in two parts, (a) the block constants and header and (b) the data payloads, as shown in Figure \ref{fig:generic-block-parts}.
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{generic-block-parts}
\caption{Blocks used as generic storage. We use structured payloads and replace transactions by payloads in the Merkle root hash computation.}
\label{fig:generic-block-parts}
\end{figure}
A single modification in one payload of a block will change its Merkle root hash value, and then invalidate it. This solution thus provides secure and reliable storage distributed among all peers in the network. Note that this implies that the complete blockchain and all datas linked to it must be duplicate on all peers. The size of data payloads will influence both the hash calculation algorithm and bandwidth used to maintain the blockchain. Due to bandwidth restriction in the platform we use, we choose to limit the total size of a block to 5 MB to ensure we have enough storage for important security and trust informations without overloading the network with blockchain control data.
\section{Authentication and Trust in WSN}
Authentication and trust can be seen as two parts of the same problem \cite{solhaug2007trust}. Authentication allows us to be sure to who we are dealing with, trust giving us insights of how we can rely on and dealing with a potential risk on an action. If we consider the presence of a master authority in charge of authentication and trust management, we can easily ensure a good security and privacy level in the network. However, this has a major drawback, the master authority becomes the central part of the network security and thus the critical point of vulnerability in the network. In decentralized and ad-hoc networks, this approach is impossible, because we don't have a node which can assure to be connected at every moment of the network life.
\subsection{Our Framework}
To ensure proper organization and content management in decentralized networks, we use a common content model based on Service Oriented Architecture\cite{erl2005service} adapted for our application and compatible with CoAP protocol\cite{shelby2014constrained}. This approach allows us to design a RESTful model for interaction with the internet, and his currently outlined at one promising approach to organize sensor networks\cite{de2011service}.
\subsubsection{Network services model}
Before further introspection on our blockchain based model, we must define the network model we use. Wireless Sensor Networks can be well described as decentralized networks composed of resource constrained nodes based on embedded devices. We choose to model the network as an undirected graph \(G=(V,E)\), each vertex describing a node in the network, and each edge links two nodes within transmission range from each other. Then we associate abilities and services to nodes, providing resources on the network. In this model we define two entities formalized as a set of characteristics vectors.
\begin{itemize}
\item Network Node (NN) defines a vector of Node Properties (NP) and another of Node Abilities (NA)
\end{itemize}
\begin{equation}
NN = \begin{pmatrix}
NP = \begin{bmatrix}
name & energy & cpu \\
\end{bmatrix} \\
NA = \begin{bmatrix}
camera & storage \\
\end{bmatrix}
\end{pmatrix}
\end{equation}%
\begin{itemize}
\item Available Services (AS) defines an Abilities Dependencies (AD) vector, a Resources Dependencies (RD) vector and a Resources Provider (RP) vector
\end{itemize}
\begin{equation}
AS = \begin{pmatrix}
AD = \begin{bmatrix}
camera & storage \\
\end{bmatrix} \\
RD = \begin{bmatrix}
\\
\end{bmatrix} \\
RP = \begin{bmatrix}
videostream & videorecording \\
\end{bmatrix}
\end{pmatrix}
\end{equation}%
Each node stores services in a Service Registry (SR). Nodes having the storage ability can store services they cannot deploy to ensure reuse of these services in the future on other nodes.
In the next section, we refer to our service model and related abbreviations to describe our solution providing authentication and trust management mechanisms for decentralized networks.
\section{Blockchain Authentication and Trust Module (BATM)}
Public Key Infrastructure (PKI) is a major component to resolve authentication in networks. In 1991, Zimmerman introduce a new concept named web of trust for his Pretty Good Privacy (PGP) encryption program~\cite{zimmermann1995official}, which was then standardized by the IETF under the OpenPGP name. Current version of the standard is described in RFC 4880\cite{callas2007rfc}. OpenPGP use PKI to provide three main functionalities.
\begin{itemize}
\item Confidentiality with Encryption
\item Authentication via Digital Signature
\item Web of Trust via identity validation from peers
\end{itemize}
BATM proposes a new way to achieve these goals using the blockchain as the database to store public keys, digital signature and peer informations, allowing each component of the network to validate informations about every other node in the network.
This section explains the global design of BATM in regard to three aspects. First, we focus on authentication, public keys, block mining and their mutual influence. Then we explain principles and particularities of the block exchange protocol and associate rules. Finally, we describe how the combination of authentication and protocol rules allows to define a trust management model.
\subsection{BATM authentication}
BATM associates cryptographic keys with each NN and AS in the network. We use the idea contained in the PGP model of a master key to identify a NN or AS among its lifespan. This key is only used to generate secondary keys for encryption and digital signature. As in most PKI, private keys are the main component of the system, and so key management is particularly critical. An attacker can easily spoof NN identity if he retrieves its keys. In this regard, implementations will need to be careful in the choice of the keyring to store private keys, but we won't address this issue in this paper.
\subsubsection{BATM block mining}
We assimilate each data payload as an event providing informations about the status of a NN and its cryptographic informations. At authentication, a node submits a credential payload containing its master public key along with secondary keys. We ensure key renewal to mitigate attacks known for guessing keys by using key validity timeouts.
As network security and privacy relies on informations contained in the blockchain, our design forbid to add blocks uniquely by resolving the problem and satisfy header hash requirement. More precisely, only authenticated nodes can mine new blocks, and only if they haven't issued a payload to be included in the block. To fullfill these requirements, miners must choose which payloads to include in the block they try to resolve.
To be valid, a block must both resolves the problem and contains a valid Miner Approval (MA) payload generated by the Miner, illustrated by the algorithm in Figure \ref{alg:block-validation}. This kind of payload contain a digital signature of a random value contained in the previous block MA payload, and must correspond to a successfully authenticated node.
\begin{figure}
\begin{algorithmic}[1]
\Require currentblock previousblock
\Ensure block validity
\If{ not(HashCurrentBlock resolves problem)}
\State return false
\EndIf
\If{ not(MinerApproval payload valid)}
\State return false
\EndIf
\If{ CurrentBlock has event payload for miner NN}
\State return false
\EndIf
\If{ not(all payloads in block valid)}
\State return false
\EndIf
\State return true
\end{algorithmic}
\caption{Block validity check algorithm.}
\label{alg:block-validation}
\end{figure}
\subsubsection{BATM data payloads}
When a NN or AS requests to join the network for the first time, it issues a specific Credential Payload (CP) to all NNs. A CP contains public keys needed to operate in the network. Authentication request is approved when an authenticated NN includes the CP in a valid block.
Credential status of the NN / AS can be subsequently updated by renew payload and revoke payload. Note that when revoking his credential, a NN / AS must provide a new credential payload to remain authenticated in the network. Miners will try to include revoke payload and new credential payload in the same block to ensure continuity of node status in the network.
If we allow submission of payloads without further verification, every node could be allowed to propose payloads in the network. To overcome this issue, payloads use a system of signed hash digests. Every payload must have a hash digest signed by payload issuer as its last entry. This way, our payload verification algorithm can easily check the validity of the data. Note that revoke payload use the master key to sign data hash, whereas other payloads use the current signature subkey.
BATM uses 6 different payload types as follows.
\begin{itemize}
\item MA (Miner Approval)
\item NN and AS Payloads
\begin{itemize}
\item Credentials
\item Renew
\item Blame
\item Ban
\item Revoke
\end{itemize}
\end{itemize}
We provide a detailed description of data contained in BATM payloads in Figure \ref{fig:batm-data-payload}. Note that Blame and Ban payloads are specific payloads used in BATM trust management model.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{batm-data-payloads}
\caption{Data payloads available in BATM. All blocks must have a Miner Approval payload to be valid, to verify which node has allowed the authentication entry.}
\label{fig:batm-data-payload}
\end{figure}
We showed how our model of a PKI using the blockchain ensure reliable storage for cryptographic material, and how we use it to perform NN authentication on the network. We then propose a trust management model using informations contained in the blockchain.
\subsection{BATM trust management}
The first need for a definition of trust originated from social studies to characterize relations between people in the society. In this context, we consider relevant to use Gambetta's definition of trust\cite{Gambetta1990} as an assumption on the level of subjective probability about how a particular agent will perform an action from a subjective point of view. Note that we understand the term of subjective probability as a reputation level applied to the realization correctness of a subsequent action in the future.
This interpretation of trust implies that the reputation level associated with an agent must vary over time to match the actual realization of the action. Good behaviour must be rewarded, and bad behaviour must be punished to maintain accurate prediction on the realization of actions.
\subsubsection{Knowledge based trust for BATM}
The BATM module includes a trust model called Human-like Knowledge based Trust (HKT), based on human like behaviour to maintain a reputation level for each node. HKT is a compromise between a mutual surveillance by all nodes on the network and the presence of a trust center.
We use the payloads contained in the blockchain as an indication of each node behaviour on the network over time. This way, we ensure a node cannot fool others by tampering data or pretending to be someone else. Thus we ensure reliability of trust evaluation without the need of a trust center. Following development will be targeted at NN trust evaluation, but same principles apply to AS, with the particularity that AS reputation level is echoed on each node in the network, thus modifying reputation level on each node using it.\\
For each payload type, HKT defines events and associates them reputation factors. We note \(C_{evt}\) the reputation factor for the event, and \(T_{evt}\) the time the event occured.
To make the NN reputation evolve naturally over time, each event reputation factor must be weighted by a function evolving in time since the event occurs. As we want to decreasing contribution of a particular event to the NN reputation level over time, we need to use a continuous decreasing function such as \(e^{-x}\).
During it first authentication, a NN has no passed action to compute a reliable trust value. Thus we choose to grant a base trust value to all nodes when a trusted node gives them access to the network by including their credentials in the blockchain.
For the simulations, we used the following values for event reputation factors.
\begin{itemize}
\item \(C_{approval} = 1\)
\item \(C_{auth} = 8\)
\item \(C_{renew} = 2\)
\item \(C_{blame} = -8\)
\item \(C_{ban} = -16\)
\end{itemize}
We have estimated the following formulas to determine the reputation of node over time.\\
\begin{equation}
\forall evt \in (N,Blk(t)): C_{N,t} = \sum C_{evt}
\end{equation
\vspace{-.2cm}
\begin{equation}
Reputation(N,t_{now}) = C_{auth} + \sum_{t=t_{first}}^{t=t_{now}} C_{N,t} * e^{\frac{-(t_{now}-t)}{256}}
\end{equation}%
In this formula, \(t_{first}\) corresponds to the first block in the blockchain after node has authenticated. \(C_{t}\) is used as the global coefficient for all events concerning the node at \(t\) (the sum of all \(C_{evt}\) at \(t\)).
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{trust-simul-3-nodes-modif.png}
\caption{Reputation level simulation over 500 hours with 3 NN. After 2 blames, a ban is declared on the node and becomes effective for a 84 hours.}
\label{fig:trust-simul-3-nodes}
\end{figure}
\subsubsection{Trust evaluation}
As we defined it earlier, we consider trust as a probability level that an action will be performed correctly by a NN. In this context, we perform trust evaluation by comparing the current reputation level of a NN to trust him doing certain actions in the network. Currently, we only defined blockchain related actions and associate them with a trust level \(A_{evt}\). This level quantifies the minimum reputation level for a node to be trusted to fullfill subsequent event \(evt\). The trust level is weighted by the number of authenticated NN noted \(N_{auth}\), in order to be less restrictive on actions in a small network and more on huge networks. This behaviour allows NN on the network to work properly and becoming trust defaultive in small network, and then raise the trust level required when more nodes are available.
BATM defines the following trust levels, as equivalent for trust events \(C_{evt}\) described earlier.
\begin{itemize}
\item \(A_{auth}\)
\item \(A_{ban}\)
\item \(A_{blame}\)
\item \(A_{approval}\)
\end{itemize}
We provide following formula to compute values of \(A_{act}\) over time, with \(A_{app}\) being an application factor, allowing applications to be more or less restrictive on actions.
\begin{equation}
A_{evt} = C_{evt} + \frac{A_{app} * (N_{auth} - 1)}{ C_{evt}}
\end{equation}
Note that reputation level simulation in Figure \ref{fig:trust-simul-3-nodes} does not consider minimum trust level required for a NN to fullfill an action.
We showed how BATM with HKT provide a powerful solution to authenticate nodes and evaluate trust in decentralized networks. The system can be made instable by malicious authenticated nodes overloading the network and submitting lots of valid payloads for inclusion in the blockchain. To overcome this issue, we define specific rules for payload submission in the network to improve stability of the system over time.
\subsection{BATM payload rules}
To avoid abuse from NN which can overload the network with payloads to be validated, we introduce specific rules on the payload exchange protocol for BATM. We consider two type of rules : timers, key validity timeouts and event reputation factors described earlier. Timers are limitation in time used to discard payloads and blocks submitted by NN overloading the network.
A set of timers defines the minimum amount of time expected between two payloads of the same type. BATM currently uses 3 timers as follows.
\begin{itemize}
\item \(T_{renew}\) is the usual time between two key renewal by a node. If needed, a node is authorized to renew its keys at \(T_{renew} / 2\). Here we ensure there are at most 2 renews in a \(T_{renew}\) for a given NN.
\item \(T_{blame}\) is the minimum time between two blames on a NN given by the same blamer NN.
\item \(T_{banrecover}\) is the time during which a banned NN will be forbidden to mine subsequent block as punishment for a bad behaviour in the network.
\end{itemize}
These timers imply that key validity timeouts must respect the following rules for BATM to work properly.
\begin{itemize}
\item \(T_{subkey}\) is the timeout for subkeys before renewal. It must be greater than \(T_{renew}\), but less than \(50 * T_{renew}\) to be overcome issue that an attacker may be able to guess the key from data collected in the network.
\item \(T_{masterkey}\) is the timeout for the master key. In our model, it should be greater than \(10 * T_{subkey}\) and no more than \(50 * T_{subkey}\) to protect it from key guessing attacks.
\end{itemize}
As indication, simulation results showed in Figure \ref{fig:trust-simul-3-nodes} used following timer values, in hours.
\begin{itemize}
\item \(T_{renew} = 168\)
\item \(T_{blame} = 42\)
\item \(T_{banrecover} = 84\)
\end{itemize}
As these rules can be defined to different values regarding the application using BATM, we use the first block in the blockchain to store values to be used. Thresholds will be defined in the future to overcome a problem with a malicious initial NN, and what we called the origin block problem.
\subsubsection{Origin block problem}
At startup, the network contains no authenticated node to realize BATM authentication and trust evaluation, and the blockchain is empty. This means we need a method to forge the first block. We choose to let any node craft this special block from its own parameters. In fact, the main problem is to ensure proper operation in the beginning of the network life, then BATM will adapt itself to events occuring in the network. If the first NN is malicious, it will be banned by others node early in the network.
Since the first block contains all mutable values used in BATM, a possible attack will be the inclusion of specific values which will tend the system to misbehave. To counter this threat, future work will provide a formula to estimate the probability of BATM instability from these values.
\section{Future work}
We presented the concept and model of BATM, with early results on reputation evaluation over time. The next step is to evaluate each part of BATM completely and the global model. The model itself will be improved depending on the results, with the objective of more adaptative algorithms taking AS and NN characteristics as defined in our model.
\subsection{Trust model}
BATM with HKT provide a simple way to manage trust in decentralized networks. More researches on HKT performance must be conducted, and the model itself may evolve to consider more parameters in trust and reputation evaluation. We think about considering NN and AS abilities in account for specific actions, and enhance the reputation calculations. For example, a blamer reputation could influence the reputation factor of its blame, and we may introduce a time coefficient to raise trust on the overall time presence of the NN or AS in the network. Another possibility is to raise resilience over DoS attacks by requiring blames from different NN or AS before banning one.
In this paper, we consider self-organizing networks with no constraints on which NN may ask to authenticate. We also let possible a derivative model, using a network master key to allow blockchain supervision and eliminate the first block problem. A possibility may be to allow specific network, for example vendor-specific networks, where NN can provide a proof of membership by prior network master key signature.
\subsection{Real world testing}
If simulation results fullfill our expectations, BATM will be included in Multicast Services for Linux (MSL), an implementation our SOA network model. Note that MSL is at development stage for now with now release date. Moreover, MSL is intended to be used as a real world proof of concept for our overall design including SOA model and BATM with HKT.
\section{Conclusion}
This paper proposes a new application for the blockchain as a secured decentralized storage for cryptographic keys as well as trust informations in the context of autonomous Wireless Sensor Networks. The Blockchain Authentication and Trust Module and its Human-like Knowledge based Trust model shows how to use to immutability of the blockchain to provide solutions to high problematics in the field of decentralized ad-hoc networks. More precisely, we show how it is possible to build a complete solution providing authentication mechanisms as well as trust evaluation in a self-organized and evolutive network.
\subsubsection*{Resources}
The Service Oriented model is currently under development into the Multicast Services for Linux (MSL) framework. MSL is a free software and will be publicly available at https://bullekeup.github.io/MSL, under the AGPL license. BATM will be available in MSL in the next months. HKT simulation files for GNU Octave and MATLAB are available by mail on request at axel.moinet@u-bourgogne.fr.
\ifCLASSOPTIONcompsoc
\section*{Acknowledgments}
\else
\section*{Acknowledgment}
\fi
The authors would like to thank the Burgundy Region and the FEDER european fund for funding this research work.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\enlargethispage{.5cm}
\bibliographystyle{IEEEtran}
| {
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} | 3,942 |
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<h2 id="treebank-statistics-ud_latin-ittb-features-proper">Treebank Statistics: UD_Latin-ITTB: Features: <code class="language-plaintext highlighter-rouge">Proper</code></h2>
<p>This feature is language-specific.
It occurs with 1 different values: <code class="language-plaintext highlighter-rouge">Yes</code>.</p>
<p>1107 tokens (0%) have a non-empty value of <code class="language-plaintext highlighter-rouge">Proper</code>.
82 types (0%) occur at least once with a non-empty value of <code class="language-plaintext highlighter-rouge">Proper</code>.
61 lemmas (1%) occur at least once with a non-empty value of <code class="language-plaintext highlighter-rouge">Proper</code>.
The feature is used with 3 part-of-speech tags: <tt><a href="la_ittb-pos-ADJ.html">ADJ</a></tt> (494; 0% instances), <tt><a href="la_ittb-pos-PROPN.html">PROPN</a></tt> (314; 0% instances), <tt><a href="la_ittb-pos-NOUN.html">NOUN</a></tt> (299; 0% instances).</p>
<h3 id="adj"><code class="language-plaintext highlighter-rouge">ADJ</code></h3>
<p>494 <tt><a href="la_ittb-pos-ADJ.html">ADJ</a></tt> tokens (2% of all <code class="language-plaintext highlighter-rouge">ADJ</code> tokens) have a non-empty value of <code class="language-plaintext highlighter-rouge">Proper</code>.</p>
<p>The most frequent other feature values with which <code class="language-plaintext highlighter-rouge">ADJ</code> and <code class="language-plaintext highlighter-rouge">Proper</code> co-occurred: <tt><a href="la_ittb-feat-Case.html">Case</a></tt><tt>=EMPTY</tt> (494; 100%), <tt><a href="la_ittb-feat-Degree.html">Degree</a></tt><tt>=EMPTY</tt> (494; 100%), <tt><a href="la_ittb-feat-Gender.html">Gender</a></tt><tt>=EMPTY</tt> (494; 100%), <tt><a href="la_ittb-feat-InflClass.html">InflClass</a></tt><tt>=EMPTY</tt> (494; 100%), <tt><a href="la_ittb-feat-Number.html">Number</a></tt><tt>=EMPTY</tt> (494; 100%).</p>
<p><code class="language-plaintext highlighter-rouge">ADJ</code> tokens may have the following values of <code class="language-plaintext highlighter-rouge">Proper</code>:</p>
<ul>
<li><code class="language-plaintext highlighter-rouge">Yes</code> (494; 100% of non-empty <code class="language-plaintext highlighter-rouge">Proper</code>): <em>cor., rom., metaph., philipp., hebr., physic., eccli., metaphys., ephes., phys.</em></li>
</ul>
<p><code class="language-plaintext highlighter-rouge">Proper</code> seems to be <strong>lexical feature</strong> of <code class="language-plaintext highlighter-rouge">ADJ</code>. 100% lemmas (17) occur only with one value of <code class="language-plaintext highlighter-rouge">Proper</code>.</p>
<h3 id="propn"><code class="language-plaintext highlighter-rouge">PROPN</code></h3>
<p>314 <tt><a href="la_ittb-pos-PROPN.html">PROPN</a></tt> tokens (5% of all <code class="language-plaintext highlighter-rouge">PROPN</code> tokens) have a non-empty value of <code class="language-plaintext highlighter-rouge">Proper</code>.</p>
<p>The most frequent other feature values with which <code class="language-plaintext highlighter-rouge">PROPN</code> and <code class="language-plaintext highlighter-rouge">Proper</code> co-occurred: <tt><a href="la_ittb-feat-Case.html">Case</a></tt><tt>=EMPTY</tt> (314; 100%), <tt><a href="la_ittb-feat-Gender.html">Gender</a></tt><tt>=EMPTY</tt> (314; 100%), <tt><a href="la_ittb-feat-InflClass.html">InflClass</a></tt><tt>=EMPTY</tt> (314; 100%), <tt><a href="la_ittb-feat-Number.html">Number</a></tt><tt>=EMPTY</tt> (314; 100%).</p>
<p><code class="language-plaintext highlighter-rouge">PROPN</code> tokens may have the following values of <code class="language-plaintext highlighter-rouge">Proper</code>:</p>
<ul>
<li><code class="language-plaintext highlighter-rouge">Yes</code> (314; 100% of non-empty <code class="language-plaintext highlighter-rouge">Proper</code>): <em>ioan., matth., tim., ierem., luc., iac., petr., ier., malach., tit.</em></li>
</ul>
<p><code class="language-plaintext highlighter-rouge">Proper</code> seems to be <strong>lexical feature</strong> of <code class="language-plaintext highlighter-rouge">PROPN</code>. 100% lemmas (14) occur only with one value of <code class="language-plaintext highlighter-rouge">Proper</code>.</p>
<h3 id="noun"><code class="language-plaintext highlighter-rouge">NOUN</code></h3>
<p>299 <tt><a href="la_ittb-pos-NOUN.html">NOUN</a></tt> tokens (0% of all <code class="language-plaintext highlighter-rouge">NOUN</code> tokens) have a non-empty value of <code class="language-plaintext highlighter-rouge">Proper</code>.</p>
<p>The most frequent other feature values with which <code class="language-plaintext highlighter-rouge">NOUN</code> and <code class="language-plaintext highlighter-rouge">Proper</code> co-occurred: <tt><a href="la_ittb-feat-Case.html">Case</a></tt><tt>=EMPTY</tt> (298; 100%), <tt><a href="la_ittb-feat-Gender.html">Gender</a></tt><tt>=EMPTY</tt> (298; 100%), <tt><a href="la_ittb-feat-InflClass.html">InflClass</a></tt><tt>=EMPTY</tt> (298; 100%), <tt><a href="la_ittb-feat-Number.html">Number</a></tt><tt>=EMPTY</tt> (298; 100%).</p>
<p><code class="language-plaintext highlighter-rouge">NOUN</code> tokens may have the following values of <code class="language-plaintext highlighter-rouge">Proper</code>:</p>
<ul>
<li><code class="language-plaintext highlighter-rouge">Yes</code> (299; 100% of non-empty <code class="language-plaintext highlighter-rouge">Proper</code>): <em>gen., sap., trin., prouerb., act., nom., apoc., deut., exod., prou.</em></li>
</ul>
<p><code class="language-plaintext highlighter-rouge">Proper</code> seems to be <strong>lexical feature</strong> of <code class="language-plaintext highlighter-rouge">NOUN</code>. 100% lemmas (30) occur only with one value of <code class="language-plaintext highlighter-rouge">Proper</code>.</p>
<h2 id="relations-with-agreement-in-proper">Relations with Agreement in <code class="language-plaintext highlighter-rouge">Proper</code></h2>
<p>The 10 most frequent relations where parent and child node agree in <code class="language-plaintext highlighter-rouge">Proper</code>:
<tt>ADJ –[<tt><a href="la_ittb-dep-nmod.html">nmod</a></tt>]–> NOUN</tt> (23; 96%),
<tt>PROPN –[<tt><a href="la_ittb-dep-conj.html">conj</a></tt>]–> PROPN</tt> (2; 100%).</p>
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"redpajama_set_name": "RedPajamaGithub"
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Emily Jean "Emma" Stone (Scottsdale, Arizona, 6 de novembre de 1988) és una actriu estatunidenca. Va obtenir l'Oscar a millor actriu (2017) pel seu paper a la comèdia musical La La Land.
Biografia
Stone va néixer a Scottsdale a l'estat d'Arizona, filla de Jeff i Krista Stone, té un germà petit: Spencer. Va ser membre del Teatre de la Joventut de la Vall durant la infància, un teatre regional a Phoenix, Arizona, on va aparèixer en la seva primera representació teatral, The wind in the Willows amb onze anys. Stone va anar a la Sequoya Elementary School i després al Cocopah Middle School per estudiar sisè grau. Va viure durant dos anys al col·legi, moment en què aparegué en 16 produccions a la Vall de Teatre Juvenil, incloent-hi: Winnie the Pooh de Nadal de cua, La princesa i el pèsol, La Ventafocs, The Wiz, Titanic, Honk!, La sireneta, Schoolhouse Rock Live!, Alícia al país de les meravelles i José i el Technicolor Dreamcoat Amazing.
Emma va anar al Xavier College Preparatory com a estudiant de primer any per un semestre. En acabar l'escola es traslladà amb la seva mare a Los Angeles amb quinze anys. Començà a la televisió i aconseguí un paper principal en una sèrie dramàtica i d'acció de la cadena Fox Drive el 2007. Aquell any va debutar al cinema amb la pel·lícula Supersortits amb el paper de Jules. Després va participar en les pel·lícules The Rocker (2008), The House Bunny (2008), Paper Man (2009) i Ghosts of Girlfriends Past (2009). També va ser protagonista a Rumors i mentides (2010), paper que va fer que estigués nominada a un Globus d'Or a la millor actriu musical o còmica alhora que va donar veu en l'adaptació cinematogràfica del còmic Marmaduke (2010).
El 2011 va rodar Amor, boig i estúpid, al costat de Steve Carell, Ryan Gosling i Julianne Moore. Va ser nominada per l'Oscar a la millor actriu de repartiment pel seu paper de Sam, una addicte a les drogues en recuperació en el film Birdman (2014), al costat de Michael Keaton i dirigida pel mexicà Alejandro González Iñárritu. Ha protagonitzat dues cintes dirigides per Woody Allen, al costat de Colin Firth a Màgia a la llum de la lluna (2014) i acompanyada de Joaquin Phoenix a Irrational man (2015). A la comèdia musical La La Land (2016), dirigida per Damien Chazelle, amb la seva interpretació de Mia, una aspirant a actriu al costat del pianista interpretat per Ryan Gosling va obtenir l'Oscar a la millor actriu (2017), Globus d'Or a la millor actriu musical o còmica i el BAFTA a la millor actriu, entre d'altres premis.
Filmografia
Referències
Actors de cinema d'Arizona
Actors de sèries de televisió estatunidencs
Guanyadores del premi Oscar a la millor actriu
Actors de doblatge estatunidencs
Actors d'Arizona
Guanyadors del Globus d'Or
Guanyadores del premi BAFTA a la millor actriu | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 311 |
Franz Lechthaler (* 3. August 1890 in Volxheim; † 29. Juni 1967 in Eschwege) war ein deutscher Polizeioffizier und Täter des Holocaust.
Leben
Lechthaler war der Sohn eines Maurermeisters. Er besuchte die Volksschule in Volxheim und machte eine Malerlehre. 1911 wurde er Soldat und im Ersten Weltkrieg an der Ostfront eingesetzt. Im Zuge der Verkleinerung des Heeres wurde er 1920 im Range eines Wachtmeisters entlassen, auch weil er sich 1918 im besetzten Kiew in den Soldatenrat seiner Schwadron hatte wählen lassen. Lechthaler bewarb sich für die Offizierslaufbahn im Polizeidienst und wurde 1921 in Erfurt zum Polizeileutnant befördert. Er war in Aachen, Bochum und Wiesbaden stationiert. In Wiesbaden heiratete er und bekam eine Tochter. Nach der Machtergreifung 1933 bewarb er sich um eine Aufnahme in die NSDAP. Er wurde aber nicht aufgenommen, weil er sich seinerzeit in Kiew den Revolutionären angeschlossen hatte und sich gegenüber seiner Polizeieinheit bei der Befehlsausgabe anlässlich der Reichspräsidentenwahl 1932 für Hindenburg und gegen Hitler ausgesprochen hatte. 1938 wurde er in Königsberg zum Major im Polizeidienst befördert und erhielt nun das Mitgliedsbuch der NSDAP rückwirkend zum 1. Mai 1933.
Lechthaler wurde im Dezember 1939 zum Kommandeur des nach Kriegsausbruch aufgestellten Reserve-Polizei-Bataillons 11 ernannt, das zu dem Zeitpunkt in Pultusk im besetzten Polen stationiert war. Das Bataillon bestand zeitweise aus vier Kompanien, die an verschiedenen Aktionen gegen die Zivilbevölkerung in Polen, Litauen und Weißrussland beteiligt waren. Dabei wurden Intelligenzler und Juden ausgeraubt und ermordet. Lechthaler baute seit August 1941 in Kaunas eine litauische Hilfstruppe für die Ordnungspolizei auf und setzte Teile des Bataillons bei der Bewachung des Zwangsghettos in Kaunas ein. Der Befehlsweg von Lechthaler zu denjenigen seiner Polizisten, die im Herbst 1941 an Massenmorden der Einsatzgruppe 3 in Kaunas beteiligt waren, konnte im späteren Gerichtsprozess gegen Lechthaler nicht beweiskräftig nachvollzogen werden.
Im Oktober 1941 wurde Lechthaler mit 326 Polizisten in zwei Kompanien und 457 litauischen Hilfstruppen nach Minsk verlegt, wo er die 707. Infanteriedivision bei Sicherungsmaßnahmen unterstützte. Die Truppe begann dort unverzüglich mit der von ihr sogenannten "Aktion Judenrein". Am 14. Oktober wurden in Smolewitsch etwa 1400 jüdische Einwohner erschossen. Am 26. Oktober 1941 erhielt Lechthaler vom Chef des Stabes (Ia) der 707. Infanteriedivision Fritz Wedig von der Osten den Befehl, alle Juden in Sluzk zu erschießen. Lechthaler war selbst vor Ort und gab die entsprechenden Befehle an seine Kompaniechefs weiter, dabei befahl er auch die Ermordung der Kinder. Lechthalers Kompanien führten auch andernorts in Weißrussland Massenerschießungen von Juden durch; so wurden in Kletsk zwischen 1800 und 3500 Juden ermordet. Unter dem Befehl eines Offiziers des Polizeibataillons räumten zwei Kompanien der Lechthaler unterstellten Litauer das Zivilgefangenenlager in Minsk und ermordeten dabei 1675 "Kommunisten".
Lechthaler war im zweiten Halbjahr 1942 Taktiklehrer an der im besetzten Pelplin eingerichteten Polizeischule. Er besuchte in Dresden-Hellerau einen Regimentsführerlehrgang und wurde im September 1942 zum Oberstleutnant der Schutzpolizei befördert. Als Regimentsführer des Polizeiregiments 17 leitete er Einsätze gegen Partisanen in Polen und Jugoslawien. Bei Kriegsende wurde er nach eigenen Angaben von einem SS- und Polizei-Gericht wegen Befehlsverschleppung zu drei Monaten Gefängnis verurteilt, wegen der deutschen Kapitulation im Mai 1945 sei die Strafe aber nicht mehr vollstreckt worden.
Im Juni 1945 wurde Lechthaler aus britischer Kriegsgefangenschaft entlassen und zog nach Eschwege, der Heimatstadt seiner Ehefrau. Er arbeitete dort als Buchhalter. Bei der Entnazifizierung wurde er 1947 als "entlastet" eingestuft. Seit dem Nürnberger Prozess gegen die Hauptkriegsverbrecher 1945/46 war die Beteiligung von Einheiten des von Lechthaler geführten Polizeibataillons an dem Massaker an Juden im Oktober 1941 im weißrussischen Sluzk öffentlich bekannt, da ein Bericht des deutschen Gebietskommissars von Sluzk, Heinrich Carl, zum Beweismaterial genommen worden war und 1948 als Teil der Prozessakten des IMT veröffentlicht wurde. Gegen Lechthaler und einen der damaligen Kompaniechefs, Willy Papenkort, wurde Ende der 1950er Jahre ermittelt, und Lechthaler kam am 28. April 1960, zwei Monate vor Fristablauf wegen Verjährung, in Untersuchungshaft.
Während Lechthaler vom Landgericht Kassel wegen "Beihilfe zum Totschlag" zu dreieinhalb Jahren Zuchthaus verurteilt wurde, erhielt Papenkort einen Freispruch mangels Beweisen. Nachdem der Bundesgerichtshof das Urteil aufgehoben hatte, wurde Lechthaler im Revisionsverfahren wegen Beihilfe zum Totschlag von "mindestens 15 Kindern" zu zwei Jahren Gefängnis verurteilt. Papenkort, der sich vor Gericht, ebenso wie Lechthaler, auf Befehlsnotstand berufen hatte, wurde entsprechend der seinerzeitigen Rechtsprechung in der Bundesrepublik Deutschland erneut freigesprochen.
Literatur
"LG Kassel 9. Januar 1963". In: Justiz und NS-Verbrechen. Sammlung deutscher Strafurteile wegen nationalsozialistischer Tötungsverbrechen 1945–1966, Bd. XVIII, hrsg. von Irene Sagel-Grande, H. H. Fuchs und C. F. Rüter. University Press, Amsterdam 1978, Nr. 546, S. 779–849.
Wolfgang Curilla: Der Judenmord in Polen und die Deutsche Ordnungspolizei 1939–1945. Ferdinand Schöningh, Paderborn 2011, ISBN 978-3-506-77043-1.
Wolfgang Curilla: Die deutsche Ordnungspolizei und der Holocaust im Baltikum und in Weissrussland, 1941–1944. F. Schöningh, Paderborn 2006, ISBN 3-506-71787-1.
Stefan Klemp: Nicht ermittelt. Polizeibataillone und die Nachkriegsjustiz. Ein Handbuch. Klartext, Essen 2005, ISBN 3-89861-381-X.
Peter Lieb: Täter aus Überzeugung? Oberst Carl von Andrian und die Judenmorde der 707. Infanteriedivision 1941/42, in: Vierteljahrshefte für Zeitgeschichte 50 (2002), Heft 4, S. 523–557 (PDF im Heftarchiv, 6,7 MB)
Christian Gerlach: Kalkulierte Morde. Die deutsche Wirtschafts- und Vernichtungspolitik in Weißrußland 1941 bis 1944. Hamburger Edition, Hamburg 1999.
Einzelnachweise
Polizist der Ordnungspolizei
Täter des Holocaust
Person (deutsche Besetzung Polens 1939–1945)
Person (deutsche Besetzung Weißrusslands 1941–1944)
Person (deutsche Besetzung Litauens 1941–1945)
Person (deutsche Besetzung Jugoslawiens 1941–1945)
Person (deutsche Besetzung Italiens 1943–1945)
Person im Ersten Weltkrieg (Deutsches Reich)
Verurteilte Person (NS-Kriegsverbrechen)
NSDAP-Mitglied
Deutscher
Geboren 1890
Gestorben 1967
Mann | {
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Q: Are basic HTML Canvas drawing/animation frameworks (or at least large portions of a framework), such as processing.js still necessary? Allow me to explain, so that this doesn't just get marked as an "opinion-based" question.
I'm learning processing.js right now, and I can't help but notice many of the similarities in functionality with what already exists in the Canvas API of Vanilla-JS. Perhaps writing a set of large-scale animations is much more complicated in plain old Canvas than it is in processing?
I'm asking this because, as I continue to learn more about the vanilla APIs, I'm seeing a lot of new functionality added in JS over the years that is starting to (VERY SLOWLY) make certain aspects of popular frameworks, no longer necessary (jQuery being a great example). I'm curious as to whether or not this is the case with Canvas and processing.js as well.
Personally, I'm trying to determine whether or not I should still be spending a lot of time in processing.js (I'm not asking you to make that decision for me though, but I just want some information that can help me decide what's best for me).
A: Stackoverflow allows specific non-coding questions about programming tools-like ProcessingJS, but your question seems likely to be closed as too broad.
Even so, here are my thoughts...
Native Canvas versus ProcessingJS
*
*Html5 canvas was born with a rich set of possibilities rivaling Photoshop itself. However, native canvas is a relatively low-level tool where you must handle structuring, eventing, serialization and animation with your own code.
*ProcessingJS adds structure, eventing, serialization, animation & many (amazing!) mathematical functions to native canvas. IMHO, ProcessingJS is a higher-level tool that's well worth learning.
Extending native canvas into a higher level tool instead of a low-level tool
*
*With about 500 lines of javascript, you can add a reusable framework to native canvas that adds these features in within a higher level structure: eventing (including drag/drop, scaling, rotating, hit testing, etc), serialization / deserialization.
*With about 100 more lines you can add a reusable framework to native canvas that does animation with easing.
*Even though native canvas was born with most of the capabilities needed to present even complex content, a PathObject is sorely needed in native canvas. The PathObject would serialize paths to make them reusable. With about 50 lines you can create a reusable PathObject.
Here's a fairly useless recommendation :-p
Try to use the right tool for the job (yeah, not specifically helpful).
*
*Learning native canvas alone will let you do, maybe 70% of pixel display tasks.
*Coding the extensions (above) will get you to 90%.
*Using a tool like ProcessingJS will get you to 98%.
*Yes, there are always about 2% edge cases where you either "can't get there" or must reduce your design requirements to accommodate coding limitations.
A slightly more specific recommendation
Since ProcessingJS merely extends native canvas, IMHO it's well worthwhile to take a few days and learn native canvas. This knowledge will let you determine the right tool for the job.
| {
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One might go into a review like this one wondering how many words will pass before Joy Division is brought up. In this case, the answer is 16. Many are too quick to classify Interpol as mimics and lose out on discovering that little more than an allusion is being made. The music made by both bands explores the vast space between black and white and produces something pained, deftly penetrating, and beautiful. Save for a couple vocal tics, that's where the obvious parallels end. The other fleeting comparisons one can one whip up when talking about Interpol are several — roughly the same amount that can be conjured when talking about any other guitar/drums/vocals band formed since the '90s. So, sure enough, one could play the similarity game with this record all day and bring up a pile of bands. It could be a detrimental thing to do, especially when this record is so spellbinding and doesn't deserve to be mottled with such bilge. However, this record is a special case; slaying the albatross this band has been unfairly strangled by is urgent and key. Let's: there's another Manchester band at the heart of "Say Hello to the Angels," but that heart is bookended by a beginning and end that approaches the agitated squall of Fugazi; the torchy, elegiac "Leif Erikson" plays out like a missing scene from the Afghan Whigs' Gentlemen; the upper-register refrain near the close of "Obstacle 1" channels Shudder to Think. This record is no fun at all, the tension is rarely resolved, and — oh no! — it isn't exactly revolutionary, though some new shades of gray have been discovered. But you shouldn't allow your perception to be fogged by such considerations when someone has just done it for you and, most importantly, when all this brilliance is waiting to overwhelm you. | {
"redpajama_set_name": "RedPajamaC4"
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CC=g++
CC_OSX=clang
PROTOBUF_DIR=protobuf-2.5.0
PROTOBUF_SRC=$(PROTOBUF_DIR)/src
PROTOBUF_LIB=$(PROTOBUF_DIR)/src/.libs/libprotobuf.a
INC=-I$(PROTOBUF_SRC)
CFLAGS=-m32 -O3 -Wall -fpermissive --std=c++0x
LDFLAGS=-m32 -pthread
OS=$(shell uname -s)
ifneq "$(OS)" "Darwin"
LDFLAGS += -static
else
CC=$(CC_OSX)
LDFLAGS += -lc++
endif
default: demoinfogo
demoinfogo: src/demofile.o src/demofiledump.o src/demoinfogo.o src/demofilebitbuf.o src/demofilepropdecode.o src/generated_proto/steammessages.pb.cc src/generated_proto/cstrike15_gcmessages.pb.cc src/generated_proto/netmessages_public.pb.cc src/generated_proto/cstrike15_usermessages_public.pb.cc
$(CC) $(INC) $^ $(PROTOBUF_LIB) $(LDFLAGS) -o $@
src/demofile.o: src/demofile.cpp
$(CC) $(CFLAGS) $(INC) -c $< -o $@
src/demofiledump.o: src/demofiledump.cpp src/generated_proto/steammessages.pb.cc src/generated_proto/cstrike15_gcmessages.pb.cc src/generated_proto/netmessages_public.pb.cc src/generated_proto/cstrike15_usermessages_public.pb.cc
$(CC) $(CFLAGS) $(INC) -c $< -o $@
src/demoinfogo.o: src/demoinfogo.cpp
$(CC) $(CFLAGS) $(INC) -c $< -o $@
src/demofilebitbuf.o: src/demofilebitbuf.cpp
$(CC) $(CFLAGS) $(INC) -c $< -o $@
src/demofilepropdecode.o: src/demofilepropdecode.cpp
$(CC) $(CFLAGS) $(INC) -c $< -o $@
src/generated_proto/steammessages.pb.cc: src/steammessages.proto
mkdir -p src/generated_proto
$(PROTOBUF_SRC)/protoc --proto_path=./src --proto_path=$(PROTOBUF_SRC) --cpp_out=./src/generated_proto $<
src/generated_proto/cstrike15_gcmessages.pb.cc: src/cstrike15_gcmessages.proto
mkdir -p src/generated_proto
$(PROTOBUF_SRC)/protoc --proto_path=./src --proto_path=$(PROTOBUF_SRC) --cpp_out=./src/generated_proto $<
src/generated_proto/netmessages_public.pb.cc: src/netmessages_public.proto
mkdir -p src/generated_proto
$(PROTOBUF_SRC)/protoc --proto_path=./src --proto_path=$(PROTOBUF_SRC) --cpp_out=./src/generated_proto $<
src/generated_proto/cstrike15_usermessages_public.pb.cc: src/cstrike15_usermessages_public.proto
mkdir -p src/generated_proto
$(PROTOBUF_SRC)/protoc --proto_path=./src --proto_path=$(PROTOBUF_SRC) --cpp_out=./src/generated_proto $<
.PHONY: clean get_protobuf protobuf image container
clean:
rm -rf src/*.o src/generated_proto/*.pb.* demoinfogo
get_protobuf:
curl -L https://github.com/google/protobuf/releases/download/v2.5.0/protobuf-2.5.0.tar.gz | tar xvz
protobuf:
cd protobuf-2.5.0 ; ./configure CFLAGS="-O3 -m32 -DNDEBUG" CXXFLAGS="-m32 -DNDEBUG" LDFLAGS=-m32 ; make
image:
docker build -t demoinfogo:latest .
container:
docker run -it --rm --name demoinfogo -v $(PWD):/demoinfogo demoinfogo:latest
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,104 |
\section{Introduction}
A surprising, but well-replicated, result in the study of deep neural network (DNN) optimization is that it is often possible to significantly ``prune'' the number of parameters after training with little effect on performance \citep{jan89, moz89a, moz89b, lecun89, kar90, han15, bla20}. Because DNNs have become increasingly large and require considerable storage, memory bandwidth, and computational resources \citep{han15}, finding a reduced form is desirable and has become an area of major research. In addition to the practical considerations, DNN pruning has become of theoretical interest, as the search for subnetworks that can be pruned with minimal effect on performance led to new insight on the success of large DNNs, via the development of the Lottery Ticket Hypothesis (LTH) \citep{fra19, fra20_linearmode}. A considerable body of literature has been devoted to expanding, interrogating, and critiquing the LTH, furthering the understanding of sparse DNNs.
While many different methods exist for choosing which parameters to prune, including ones that take into account the gradient and the Hessian of the loss function \citep{lecun89, has92, yu2018, lee2019, lee2020}, the simple approach of pruning based on a global threshold of parameters' magnitudes has been found to be robust and competitive with more complicated, state-of-the-art methods \citep{bla20}. In addition, early work on the LTH found that magnitude pruning could be successfully performed early during training (although, iteratively pruning based on magnitude was found to yield considerably better results) \citep{fra19}. Subsequent research has related the point in training time when iterative magnitude pruning can be successfully applied to a phenomenon known as ``linear mode connectivity'' \citep{fra20_linearmode}. However, why magnitude pruning does not work before the emergence of linear mode connectivity, is not well understood. Therefore, there is a need to develop new, theoretically driven tools for studying magnitude pruning, especially pre-convergence \citep{fra21_pruninginitialization}.
DNN training can be viewed as a discrete dynamical system, evolving the associated parameters along a trajectory. Despite the fact that this perspective is familiar (indeed, it is often taught in introductions to gradient descent), the complex dependence of the trajectory on the choice of optimization algorithm, architecture, activation functions, and data has kept dynamical systems theory from providing much insight into the behavior of DNNs. However, the recent development of Koopman operator theory \citep{Koopman1931, Koopman1932, Mezic2005, Budisic2012}, a rigorous, data-driven framework for studying nonlinear dynamical systems, has been successfully applied to DNNs and machine learning, more generally \citep{die20, dogra2020, tano2020, mano2020, mohr2021, naiman2021}.
This recent success motivated us to ask whether we could leverage Koopman operator theory to provide insight into the pruning of DNNs. We first made use of Koopman methods to define a new class of pruning algorithms. One method in this class, which we refer to as Koopman magnitude pruning, is shown to be equivalent to magnitude pruning at convergence. Early during training, we again find that the two methods can be equivalent. The dynamical systems perspective afforded by Koopman pruning allows us to gain insight into the parameter dynamics present when the equivalence holds and when it does not. We find that the breaking of the equivalence occurs at a similar time that non-trivial structure emerges within the DNN. Finally, we extend our work beyond magnitude based pruning, showing that gradient pruning is equivalent to another form of Koopman pruning. Thus, the Koopman framework unifies magnitude and gradient pruning algorithms. Our contributions are the following:
\begin{itemize}
\item A new class of theoretically motivated pruning methods;
\item New insight on how training dynamics may impact the success (or lack thereof) of magnitude pruning pre-convergence;
\item A unifying framework for magnitude and gradient based pruning.
\end{itemize}
\section{Koopman Operator Theory Based Pruning}
Koopman operator theory is a dynamical systems theory that lifts the nonlinear dynamics present in a finite dimensional state space, to an infinite dimensional function space, where the dynamics are linear \citep{Koopman1931, Koopman1932, Mezic2005}. While gaining an infinite number of dimensions would appear to make the problem intractable, numerical methods have been developed over the past two decades to find good finite dimensional approximations of the relevant spectral objects \citep{mezic2004, Mezic2005, rowley2009, schmid2010, tu2014, williams2015, mezic2020}. Given the powerful and interpretable tools that exist for studying linear systems, Koopman operator theory has found success in a number of applied settings. Below we discuss the basics of Koopman mode decomposition, a key element of Koopman operator theory, and use it to motivate a new class of pruning algorithms. We broadly refer to these methods as \textit{Koopman based pruning}. For more details on Koopman operator theory, we refer the interested reader to \citet{Budisic2012}.
\subsection{Koopman Mode Decomposition}
The central object of interest in Koopman operator theory is the Koopman operator, $\textbf{U}$, an infinite dimensional linear operator that describes the time evolution of observables (i.e. functions of the underlying state space variables) that live in the function space $\mathcal{F}$. That is, after $t > 0$ amount of time, which can be continuous or discrete, the value of the observable $g \in \mathcal{F}$, which can be a scalar or a vector valued function, is given by
\begin{equation}
\label{eq:KO}
\textbf{U}^t g(p) = g \left[ \textbf{T}^t(p) \right].
\end{equation}
Here $\textbf{T}$ is the nonlinear dynamical map evolving the system and $p$ is the initial condition or location in state space. In the case of training a DNN, $\textbf{T}$ is determined by the choice of optimization algorithm and its associated hyperparameters (e.g. learning rate), architecture, activation functions, and order in which the training data is presented. If the optimizer is (a variant of) stochastic gradient descent, then each epoch will have its own $\textbf{T}$, since the dynamic map depends on the ordering of the training data (see Sec. 2.3). The DNN parameter values that are set at initialization, $\theta(0)$, are the associated initial condition [i.e. $p = \theta(0)$]. The observable, $g$, could be chosen to be the identity function or some other function that might be relevant to a given DNN, such as its loss or gradient. However, note that Eq. 1 must hold for all $g \in \mathcal{F}$. For the remainder of the paper, it will be assumed that the state space being considered is of finite dimension, and that $\mathcal{F}$ is the space of square-integrable functions.
The action of the Koopman operator on the observable $g$ can be decomposed as
\begin{equation}
\label{eq:KMD}
\textbf{U} g(p) = \sum_{i = 0}^R \lambda_i \phi_i(p) \textbf{v}_i,
\end{equation}
where the $\phi_i$ are eigenfunctions of $\textbf{U}$, with $\lambda_i \in \mathbb{C}$ as their eigenvalues and $\textbf{v}_i$ as their Koopman modes \citep{Mezic2005}. The decomposition of Eq. \ref{eq:KMD} is called the Koopman mode decomposition.
The Koopman mode decomposition is powerful because, for a discrete dynamical system, the value of $g$ at $t \in \mathbb{N}$ time steps in the future is given by
\begin{equation}
\label{eq:KMD time}
g\left[ \textbf{T}^t(p) \right] = \textbf{U}^t g(p) = \sum_{i = 0}^R \lambda_i^t \phi_i(p) \textbf{v}_i.
\end{equation}
From Eq. \ref{eq:KMD time}, we see that the dynamics of the system in the directions $\textbf{v}_i$, scaled by $\phi_i(p)$, are given by the magnitude of the corresponding $\lambda_i$. Assuming that $|\lambda_i| \leq 1$ for all $i$, finding the long time behavior of $g$ amounts to considering only the $\phi_i(p)\textbf{v}_i$ whose $\lambda_i \approx 1$.
Given that the Koopman operator is linear, and it is known that even simple DNNs can have parameters that exhibit nonlinear dynamics \citep{saxe2014exactsolutions}, it may seem that applying Koopman methods to the training of DNNs is an over simplification. However, the Hartman-Grobman Theorem, a foundational result in dynamical systems theory, guarantees local neighborhoods around hyperbolic fixed points where the linearization captures the dynamical properties of the full, nonlinear system (i.e. the two are conjugate) \citep{wiggins2003nonlinear}. Additional work by \citet{lan2013} extended the neighborhood where this conjugacy holds to the entirety of the basin. Therefore, as long as the DNN is in a basin of attraction of some fixed point, we can assume that the Koopman mode decomposition will provide an accurate representation to the true, nonlinear dynamics.
There exist a number of different methods for computing the Koopman mode decomposition. While they differ in their assumptions on the dynamics of the system, they are all data-driven and require multiple snapshots of some $g$. In the case of DNN training, we can take $g$ to be the identity function, and collect $\tau + 1$ column vectors of the values the DNN parameters, $\theta$, each taken at a different training step. That is, to compute the Koopman mode decomposition, the data matrix $\textbf{D} = \left[\theta(0), \theta(1),..., \theta(\tau) \right] \in \mathbb{R}^{N \times (\tau+1)}$ must be constructed. For more details on the method used in this paper to compute the Koopman modes, and its computational demands, see Appendix \ref{appendix:KMD}.
\subsection{Koopman Based Pruning}
Any combination of DNN optimizer, architecture, activation function, and ordering of training data has an associated Koopman operator, $\textbf{U}$, which governs the flow of DNN parameters during training. This Koopman operator, in turn, has various directions along which the flow exponentially grows and shrinks (i.e. the $\phi_i[\theta(0)]\textbf{v}_i$).
Previous work has shown that, over a non-trivial window of training, approximations of the Koopman operator can well capture weight and bias dynamics \citep{dogra2020,tano2020}, suggesting that the $\phi_i[\theta(0)]\textbf{v}_i$ may be useful for pruning. If we order the modes by the real part of their eigenvalues, i.e. $\text{re}(\lambda_1) > \text{re}(\lambda_2)>...>\text{re}(\lambda_R)$, and assume that $\lambda_1 = 1$ (a reasonable assumption in the case of stable dynamical systems, which we find corroborated in our experiments below), then $\phi_1[\theta(0)]\textbf{v}_1$ corresponds to the predicted fixed point of training. That is, the first mode identifies the parameter values that the DNN would take after training reached convergence. All additional modes would identify parameters that experienced increasingly larger exponential decays.
If $\phi_1[\theta(0)]\textbf{v}_1$ is accurate in its approximation of the fixed point, then pruning parameters based on their magnitude in $\phi_1[\theta(0)]\textbf{v}_1$ could prove to be beneficial. We refer to this as \textit{Koopman magnitude pruning} (KMP) to denote its similarity to global magnitude pruning (GMP), a popular and robust approach to pruning \citep{bla20}. We make this connection explicit in Section 3.
Alternatively, the exponentially decaying modes, i.e. $\phi_2[\theta(0)]\textbf{v}_2$, $\phi_3[\theta(0)]\textbf{v}_3$..., might be expected to convey information about which parameters are most coherently dynamic. We refer to the strategy that prunes parameters based on their magnitude in these modes as \textit{Koopman gradient pruning} (KGP) to denote its similarity to gradient pruning. We make this connection explicit in Section 5.
These algorithms, KMP and KGP, are two of many possible realizations of pruning based on the Koopman mode decomposition. For example, KMP can be easily extended to layer magnitude pruning, by splitting $\phi_1[\theta(0)]\textbf{v}_1$ into subvectors and pruning each layer based on thresholds computed from each subvector. Algorithm \ref{algo:Koopman pruning} provides a general form for Koopman based pruning. Note that we follow standard practice in the pruning literature and denote a DNN by $f(x; \theta)$, where $x$ is its input and $\theta$ is its $N$ parameters. The goal of pruning is to find a mask, $m \in \{0, 1\}^N$, such that the accuracy of $f(x; \theta \odot m)$ is comparable to the original DNN. Here $\odot$ denotes element-wise multiplication. To determine which parameters to prune (i.e. which $m_i = 0$), a scoring function, $s$, is used. In the case of GMP, $s(\theta_i) = |\theta_i|$. The DNN is compressed an amount $c$ by keeping only the $\frac{100}{c}\%$ of parameters with the largest scores. If $c = 4$, then only the parameters with scoring function values that are in the top 25\% are kept.
\begin{algorithm}[t]
\caption{General form of Koopman based pruning.}\label{algo:Koopman pruning}
\begin{enumerate}
\item Construct the data matrix $\textbf{D} = \left[\theta(0), \theta(1),..., \theta(\tau) \right]$ from $\tau + 1$ snapshots of the parameter values $\theta$ during training.
\item Compute the Koopman mode decomposition from \textbf{D} to obtain $R$ Koopman triplets, $(\lambda_i, \phi_i, \textbf{v}_i)$.
\item Use a scoring function, $s$, that is a function of $(\lambda_i, \phi_i, \textbf{v}_i)$, to create a pruning mask, $m$, that compresses the network by an amount $c$. In the case of KMP, $s(\theta, \lambda_i, \phi_i, \textbf{v}_i) = |\phi_1[\theta(0)]\textbf{v}_1|$, assuming that $\lambda_1 = 1$.
\item Create the pruned DNN, $f(x; \theta \odot m)$.
\end{enumerate}
\end{algorithm}
\subsection{Koopman Operator's Dependence on Order of Training Data}
As noted above, the dynamical map involved in training a DNN, $\textbf{T}$, is dependent on the choice of optimizer, architecture, activation function, and order in which the training data is presented. Given that the action of the Koopman operator is related to $\textbf{T}$ via Eq. \ref{eq:KO}, the Koopman mode decomposition must also be dependent on these quantities. In the experiments that are performed in the remainder of this paper, optimizer, activation function, and architecture will be fixed ahead of time. However, because we made use of SGD, the ordering of the training data, referred to in some literature as ``SGD noise'' \citep{fra20_linearmode}, will vary epoch to epoch.
A consequence of this is that, at each epoch, the dynamical system being considered will change, as the mapping from $\theta(t)$ to $\theta(t+1)$ will depend on which subset of training data is shown at training iteration $t$. Note that it will also depend on the batch size (i.e. what fraction of the training data is shown before updating $\theta$), however this will also be fixed in the experiments we perform. While Koopman operator theory has been extended to random and non-autonomous dynamical systems \citep{macesic2018nonauto, crnjaric2020randomkoop}, we can view the training over the course of an epoch as a deterministic process, by fixing ahead of time the ordering of the training data. For the remainder of this paper, we will be computing the Koopman mode decomposition for each epoch by using only the parameter evolution data from that epoch.
The impact that SGD noise has on a DNN has been found to depend on the amount of training \citep{fra20_linearmode}. Near initialization, two copies of the same DNN trained on different orderings of the data lead to convergence on different local minima, whose linear path sees a ``hill'' (i.e. increase) of testing error. This suggests that they are in different parts of the loss landscape. However, with increased training, SGD noise brings the two copies of the DNN to the same ``valley'', where their linear path has no significant rise in testing error. This phenomenon is called linear mode connectivity \citep{fra20_linearmode}, and it is known to be related to the success of being able to find winning tickets in the Lottery Ticket Hypothesis framework. However, the cause of its emergence, and whether it is related to other dynamical phenomena, are not currently well known.
\section{Equivalence of Global and Koopman Magnitude Pruning in the Long Training Limit ($t \rightarrow \infty$)}
We start by examining how global and Koopman magnitude pruning compare in the long-training time limit. We consider a DNN, initialized with parameters $\theta(0)$, optimized via SGD on training data $x$, with the order of the $i^{\text{th}}$ epoch's data denoted by $x_i$. As the number of epochs goes to $\infty$, the DNN will converge to $\theta^*$, a local minimum in the loss landscape. This requires that $|| \theta(t) - \theta^*||_2 \rightarrow 0$, as $t \rightarrow \infty$. Therefore, for $\epsilon \in \mathbb{R}^+$, there exists $t_\epsilon \in \mathbb{N}$, such that $|| \theta(t) - \theta^*||_2 < \epsilon$, for all $t > t_\epsilon$. Here, $|| \cdot ||_2$ is the $l^2$ norm. Note that this additionally implies that at $t_\epsilon$ the DNN is in the basin of attraction of $\theta^*$.
Previous work on approximating the Koopman operator has found that if a system is sufficiently near a fixed point (and if the trajectory is sufficiently ``nice''), then the computed Koopman mode decomposition should well approximate the true dynamics. That is, for $\epsilon \in \mathbb{R}^+$, there exists $t_\epsilon^{\text{Koop}} \in \mathbb{N}$, such that $||\tilde{\theta}^*- \theta^*||_2 < \epsilon$, for all $t > t_\epsilon^{\text{Koop}}$. Here, $\tilde{\theta}^* = \phi_1[\theta(t_\epsilon^\text{Koop})]\textbf{v}_1$ is the computed Koopman mode corresponding to $\lambda = 1$.
Assuming that a DNN has trained for at least $\max(t_\epsilon, t_\epsilon^\text{Koop})$ training steps, then
\begin{equation}
|| \tilde{\theta}^* - \theta(t)||_2 < 2\epsilon,
\end{equation}
If $\epsilon$ is sufficiently small, then $m(|\tilde{\theta}^*|, c) \approx m(|\theta(t)|, c)$, and the two methods are thus equivalent.
To test whether this result holds in practice, we used ShrinkBench, a recent open source toolkit for standardizing pruning results and avoiding bad practices that have been identified in the pruning literature \citep{bla20}. The results of pruning a custom ShrinkBench convolutional DNN called ``MnistNet'' \citep{bla20}, pre-trained on MNIST \footnote{https://github.com/JJGO/shrinkbench-models/tree/master/mnist}, and ResNet-20, pre-trained on CIFAR-10 \footnote{https://github.com/JJGO/shrinkbench-models/tree/master/cifar10}, are presented in Fig. \ref{fig:KP GMP eq}. Each DNN was trained from a point near convergence for a single epoch, so that the parameter trajectories could be captured. All hyperparameters of the DNNs were the same as the off-the-shelf implementation of ShrinkBench, except that we allowed for pruning of the classifier layer. Training experiments were repeated independently three times, each with a different random seed. Our code has been made publicly available \footnote{https://github.com/william-redman/Koopman\_pruning}.
\begin{figure}
\centering
\includegraphics[width = 0.75\textwidth]{KMP_GMP_eq.png}
\caption{\textbf{Equivalence of Koopman and global magnitude pruning at convergence.} Left, compression vs. accuracy of MnistNet trained on MNIST. Right, compression vs. accuracy of ResNet-20 trained on CIFAR-10. Dashed lines are mean accuracy immediately after pruning and solid lines are mean accuracy after a single epoch of refining. Error bars are standard deviation.}
\label{fig:KP GMP eq}
\end{figure}
We found that, for both MnistNet and ResNet-20, KMP and GMP produced nearly identical results, both immediately after pruning and after one epoch of refinement (Fig. \ref{fig:KP GMP eq}). These results support the claim developed above that KMP and GMP are equivalent in the long training time limit.
\section{Koopman Magnitude Pruning Pre-Convergence}
Motivated by the equivalence of Koopman and global magnitude pruning in the long training time limit, we next examined how the two methods compared before training converged. In this regime, we might expect KMP to outperform GMP, since computing the Koopman mode decomposition requires multiple snapshots of the training data. This provides KMP with considerably more dynamic information than GMP, which is based on a single snapshot.
As previous work has found that ResNet-20 trained on CIFAR-10 does not achieve linear mode connectivity until at least 5 epochs of training \citep{fra20_linearmode}, we examined pruning performance at epoch 1 and epoch 20 to capture the effect that linear mode connectivity had. We find that, despite the additional information, KMP performs similarly to GMP at both epochs (Fig. \ref{fig:early training}, right column).
\begin{figure}
\centering
\includegraphics[width = 0.75\textwidth]{Early_training.png}
\caption{\textbf{Koopman and global magnitude pruning pre-convergence.} Left column, compression vs. accuracy of MnistNet trained on MNIST. Right column, compression vs. accuracy of ResNet-20 trained on CIFAR-10. Top row, results of pruning DNN after 1 epoch of training. Bottom row, results of pruning DNN after 20 epochs of training. In both cases, only data from epoch 1 and epoch 20, respectively, are used to compute the Koopman mode decomposition. Dashed lines are mean accuracy immediately after pruning and solid lines are mean accuracy after a single epoch of refining. Error bars are standard deviation.}
\label{fig:early training}
\end{figure}
There are two possible explanations for this similarity in performance. The first is that KMP and GMP prune the same parameters, making them equivalent methods (even before convergence). The second is that KMP and GMP prune distinct parameters, but those that remain result in a subnetwork with similar accuracy. To test these two hypotheses, we compare the overlap between the parameters not pruned by GMP and KMP at different compressions and different epochs (see Appendix \ref{appendix:overlap} for more details on this metric). The right column of Fig. \ref{fig:mask overlap} shows that the two methods largely prune the same parameters. From epochs 1-10, the mean overlap is roughly $>95\%$, even for a compression of 64. At epoch 20, there is a decrease in mean overlap and an increase in standard deviation. This suggests that the two methods are equivalent at the earliest epochs of training, with the equivalence breaking at some subsequent epoch.
\begin{figure}
\centering
\includegraphics[width = 0.75\textwidth]{mask_overlap.png}
\caption{\textbf{Overlap between Koopman and global magnitude pruning masks.} Left, overlap of surviving parameters found via KMP and GMP on MnistNet training on MNIST. Right, overlap of surviving parameters found via KMP and GMP on ResNet-20 trained on CIFAR-10. Error bars are minimum and maximum of three seeds.}
\label{fig:mask overlap}
\end{figure}
Because KMP makes use of dynamical systems theory, the equivalence of KMP and GMP during (at least) the first 10 epochs has a strong implication on the trajectory in parameter space that is induced by training: at the end of each epoch, the largest parameters of the DNN must be the \textit{same} ones that are expected to be the largest at convergence, if the order of training data were to remain the same. In this sense, the expected identity of the largest parameters are ``static''. The overlap between masks found using KMP from increasingly distant epochs shows a decrease with time (Appendix \ref{appendix:overlap}, Fig. \ref{fig:mask overlap time}), suggesting that SGD slowly re-orders the identity of the largest parameters epoch-to-epoch, making these initial largest parameters inaccurate.
By epoch 20, KMP and GMP stop being equivalent and, on average, start pruning a larger number of different parameters. This implies that, even if the training data was ordered the exact same way for all the remaining epochs, the identity of (some of) the largest parameters are expected to evolve. To contrast this with the ``static'' dynamics found during epochs 1--10, we call this regime ``active''. We speculated that this change from ``static'' to ``active'' might be caused by emerging structure in the parameter evolution. To test this, we pruned ResNet-20 at epochs 1 and 20 using layer shuffle pruning (LSP). This method preserves the amount of compression in each layer, but randomly chooses which parameters to remove. Previous work has found that many state-of-the-art methods designed to prune at initialization do not do better than LSP \citep{fra21_pruninginitialization}. We find that after only a single epoch of training, both GMP and KMP do no better, and in fact, do worse, than LSP for nearly all compressions tested. However, by epoch 20, both GMP and KMP were better than LSP for compression values of less than 16. These results support the idea that non-trivial structure is emerging in the DNN at a similar time that KMP and GMP stop being equivalent. When this structure is not present (e.g. at epoch 1), the two methods prune the same parameters.
We also examined the ability of KMP and GMP to prune MnistNet, trained on MNIST, after epochs 1 and 20. While we found similar results to ResNet-20 (Figs. \ref{fig:early training} and \ref{fig:mask overlap}, left column), there are subtle differences in the behavior of the overlap between pruning masks. For MnistNet, the overlap is smallest at epochs 1 and 5, suggesting that the two methods are not equivalent at the earliest part of training (Fig. \ref{fig:mask overlap}, left). That this is different to what we found for ResNet-20 may be due to the fact that the two models are at different phases of training at equivalent epochs. Previous work has shown that, shortly after initialization, linear mode connectivity emerges in LeNet, trained on MNIST \citep{fra20_linearmode}. This does not happen for ResNet-20, trained on CIFAR-10, until at least epoch 5. Additionally, we find that, for MnistNet, KMP and GMP outperform LSP for compression above 8 as early as epoch 1. These MnistNet results further support the fact that, as structure beyond what LSP can find emerges, the dynamics associated with the identity of the largest parameters goes from being ``static'' to ``active''.
Finding what is driving this ``static'' to ``active'' transition, and whether it is related to the emergence of linear mode connectivity and/or other noted transitions during training \citep{gur18, achille2018critical, kalimeris2019sgd}, will be the focus of future work.
\section{``Near'' Equivalence of Global and Koopman Gradient Pruning}
Thus far, we have focused only on magnitude based pruning. In this section, we extend our work to another method that has found success and is of general interest: gradient based pruning \citep{yu2018, lee2019, lee2020, bla20}. In practice, this way of pruning is accomplished by scoring based on the gradient of the loss function, $\nabla_\theta L$, as propagated through the network. For instance, global gradient pruning (GGP) scores using the magnitude of the gradient times the parameter values, $s(\theta) = \big|\nabla_\theta L \cdot \theta(t) \big|$ \citep{bla20}. Given that the gradient is applied to update the parameters of the DNN, $\nabla_\theta L$ is directly related to the parameters' dynamics.
From the Koopman mode decomposition, we have that the parameters at training step $t$, $\theta(t)$, are given by
\begin{equation}
\label{eq:Koopman mode decomp}
\theta(t) = \tilde{\theta}^* + \sum_{i = 2}^R \lambda_i^t \tilde{\theta}_i,
\end{equation}
where the $\tilde{\theta}_i$ are the scaled Koopman modes (i.e. $\tilde{\theta}_i = \phi_i(p) \textbf{v}_i$). As before, we use $\tilde{\theta}^*$ to denote the scaled Koopman mode that has $\lambda = 1$. Importantly, Eq. \ref{eq:Koopman mode decomp} tells us that the future state of the parameters is given by the sum of $\tilde{\theta}_i$ times their decaying eigenvalues. While there are $R-1$ modes in this sum, those with large norms (i.e. $|\tilde{\theta}_i | \gg 0$) are expected to make the most substantial contribution. For simplicity, we consider only the largest modes that also have real, positive eigenvalues. These modes have easily interpretable dynamics (i.e. exponential decay) and are directly related to $\nabla_\theta L$. Pruning using the magnitude of these modes is what we call Koopman gradient pruning (KGP).
The Koopman spectrum of a ResNet-20 trained on CIFAR-10 for 20 epochs is shown in Fig. \ref{fig:KGP GGP equiv}. In addition to the presence of an eigenvalue close to $1$, corresponding to $\tilde{\theta}^*$, there is also an eigenvalue that satisfies the above criteria. We refer to its corresponding mode as $\tilde{\theta}_2$.
\begin{figure}
\centering
\includegraphics[width = 0.75\textwidth]{KGP_JGP_equiv.png}
\caption{\textbf{Koopman gradient pruning.} Left, real and imaginary parts of the computed Koopman eigenvalues. Red dots denote modes that had a norm of greater than $0.85$. Here, $\lambda_1 = 1.000$ and $\lambda_2 = 0.075$. Right, compression vs. accuracy of ResNet-20 trained for 20 epochs on CIFAR-10. Dashed lines are mean accuracy immediately after pruning and solid lines are mean accuracy after a single epoch of refining. Error bars are standard deviation.}
\label{fig:KGP GGP equiv}
\end{figure}
As expected from the discussion above, pruning ResNet-20 after 20 epochs of training using KGP is very similar to pruning using the scoring function $s(\theta) = |\nabla_\theta L|$, a method we refer to as just gradient pruning (JGP) (Fig. \ref{fig:KGP GGP equiv}, right). We note that we only expect the two methods to be ``nearly'' equivalent (and not truly equivalent, as in the case of KMP and GMP in the long training time limit) because, KGP finds exponential decaying modes from dynamic data, whereas JGP is the average gradient computed over a batch of input data \citep{bla20}.
These results highlight the fact that, while gradient and magnitude based pruning methods have seemingly little to do with each other, they are in fact, related in the Koopman operator framework. Whereas magnitude pruning is, at certain phases of training, equivalent to pruning using $|\tilde{\theta}^*|$, gradient pruning is equivalent to pruning using the exponentially decaying modes.
\section{Discussion}
Inspired by the widespread success Koopman operator theory has found in a number of applied fields, and its recent application to problems in machine learning, we extended Koopman tools to study deep neural network pruning \citep{jan89, moz89a, moz89b, lecun89, kar90, han15, bla20}. By making use of the Koopman mode decomposition, we defined a new class of pruning algorithms (Algorithm \ref{algo:Koopman pruning}) that are theoretically motivated and data-driven in implementation. We found that both global magnitude and gradient pruning \citep{bla20} have equivalent (or nearly equivalent) methods at certain regimes of training in the Koopman framework (Figs. \ref{fig:KP GMP eq}, \ref{fig:KGP GGP equiv}). In this way, Koopman pruning provides a unified perspective on these ``disparate'' methods.
Applying KMP to DNNs in the early part of training revealed that it could be equivalent to GMP (Fig. \ref{fig:mask overlap}). From a dynamical systems perspective, this equivalence implies that the largest parameters, at the end of each epoch, are the same ones that are expected to remain the largest, if training continued using the same ordering of the data. We refer to this behavior as "static". Given that we found the identity of the largest parameters evolved with training (Appendix \ref{appendix:overlap}, Fig. \ref{fig:mask overlap time}), pruning too early may confine the sparsified DNN to a subspace where it stays in the static regime and therefore is unable find a good local minimum. This is additionally supported by the fact that layer shuffle pruning (LSP) outperforms GMP during this part of training, suggesting that the structure present in the GMP mask is more ``harmful'' than a randomly structured mask (Fig. \ref{fig:early training}).
However, at a certain subsequent point in training, the masks produced by GMP and KMP stop overlapping as completely (Fig. \ref{fig:mask overlap}). This suggests that there is some transition in the dynamics, with the largest parameters becoming ``active''. As this transition seems to occur at a similar time that GMP starts to outperform LSP, and at a similar time that linear mode connectivity has been reported to emerge \citep{fra20_linearmode}, we speculate that it is related to the sparsified DNN still being able to evolve and find a good local minima in the subspace it is restricted to.
By making use of ShrinkBench, our results can be easily replicated and avoid some of the bad practices that have been recently identified as being pervasive in the DNN pruning literature \citep{bla20}. Our custom implementations are integrated into the ShrinkBench framework, allowing for additional investigation into Koopman pruning.
While dynamical systems theory is a natural language with which to frame DNN optimization, the complex dependencies on optimizer, architecture, activation function, and training data have historically kept efforts in this direction to a minimum. This has led to a reliance on heuristic methods, such as iterative magnitude pruning, whose basis of success is still not clear \citep{elesedy2021}. Other groups have attempted to more principally examine DNN behavior by studying spectral objects, such as the spectrum of the Hessian matrix \citep{gur18} and the spectrum of the weight matrices \citep{martin2021}. This work has been influential and led to new insight, however the connection between these metrics and the (future) dynamics of DNN training is not direct. An exception to this is the recent work on using inertial manifold theory to study the Lottery Ticket Hypothesis, which is similar in spirit to our Koopman based pruning methods \citep{zhang2021validating}.
Given the data-driven tools that have been developed for Koopman operator theory \citep{mezic2004, Mezic2005, rowley2009, tu2014, williams2015, mezic2020} and the rich theoretical literature that connects Koopman objects to the state space geometry \citep{Mezic2005, arbabi2017ergodic, mauroy2013iso, mezic2020spectrum}, recent work has focused on bring Koopman operator theory to the study of DNNs \citep{die20, dogra2020, tano2020, mano2020, mohr2021, naiman2021}. This work has shown that Koopman methods can:
\begin{itemize}
\item Approximate the parameter dynamics during training and lead to speed-ups in wall clock time when substituted for standard optimizers \citep{dogra2020, tano2020};
\item Predict the number of layers necessary for a hierarchical support vector machine to achieve a given level of performance \citep{mano2020, mohr2021};
\item Shed light on the behavior of sequence neural models \citep{naiman2021}.
\end{itemize}
Our current work further illustrates the power that Koopman operator theory can bring. In particular, we found that the Koopman mode decomposition encodes dynamical information about training that could be leveraged to prune DNNs, enables a framework that unifies disparate pruning algorithms, and provides insight on how training dynamics impact the success of magnitude pruning. To the best of our knowledge, these could not have been done using any other linearization method (e.g. principal component analysis). We believe that continued exchange between the Koopman and machine learning communities will lead to further advances.
\subsubsection*{Acknowledgments}
W.T.R. was partially supported by a UC Chancellor's Fellowship. W.T.R., M.F., R.M and I.M were partially supported by the Air Force Office of Scientific Research project FA9550-17-C-0012. I. G. K. was partially supported by U.S. DOE (ASCR) and DARPA ATLAS programs.
\section*{Reproducibility Statement}
Forethought to ensure reproducibility was taken in both the design and description of our methods and results. We made use of ShrinkBench, a recent open-source framework for developing and testing new pruning methods \citep{bla20}, to implement all our experiments. This not only allowed us to report comparisons between our methods and existing methods across a number of different compressions, but will also enable easy sharing of our algorithms, as they are integrated into the ShrinkBench framework. Additionally, we have provided general pseudo-code for implementing Koopman based pruning (Algorithm \ref{algo:Koopman pruning}) and a description of the method we used to compute the Koopman mode decomposition (Appendix \ref{appendix:KMD}), to make it straightforward for others to understand and implement.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,447 |
\section{Introduction}
\label{Sec-1:introduction}
Generalized parton distributions (or GPDs) have become an essential tool
in the study of nucleon structure in QCD; see
Refs.~\cite{Goeke:2001tz,Diehl:2003ny,Belitsky:2005qn,Boffi:2007yc} for a
review. They parametrize the nucleon matrix elements of quark and gluon
light-ray operators at non-zero momentum transfer and unify the concepts of
parton density and elastic form factor. As such they provide a comprehensive
description of the nucleon's quark and gluon single-particle structure and its
spin-flavor dependence. At twist-2 level the nucleon's quark structure is
described by 4 chiral-even (quark helicity-conserving) and 4 chiral-odd (quark
helicity-flipping) GPDs; the number corresponds to that of independent
quark-nucleon helicity amplitudes \cite{Diehl:2001pm}. The chiral-even GPDs
reduce to the usual unpolarized and helicity--polarized quark parton
distribution functions (PDFs) in the limit of zero momentum transfer. These GPDs
appear in the collinear QCD factorization of amplitudes of hard exclusive
processes such as deeply virtual Compton scattering
\cite{Ji:1996nm,Radyushkin:1997ki,Belitsky:2001ns} and exclusive meson
production with longitudinal photon polarization \cite{Collins:1996fb} and can
be accessed experimentally in this way. The chiral-odd GPDs reduce to the quark
transversity PDFs in the limit of zero momentum transfer. Relating these GPDs to
hard exclusive processes has proved to be challenging. The chiral-odd GPDs
decouple from single vector meson production at leading twist in all orders in
perturbative QCD due to the chirality requirements for massless fermions
\cite{Collins:1999un}. It has been argued that the chiral-odd GPDs could be
probed in diffractive electroproduction of two mesons with large invariant mass
(rapidity gap) \cite{Ivanov:2002jj,Enberg:2006he,Beiyad:2010cxa}, but the
proposed kinematics is difficult to access and no data are presently available.
Recent theoretical work suggests that hard exclusive electroproduction of
pseudoscalar mesons ($\pi^0, \eta, \pi^+$) may be described by a hard scattering
mechanism involving the twist-2 chiral-odd nucleon GPDs and the twist-3
chiral-odd meson distribution amplitude \cite{Ahmad:2008hp,Goloskokov:2009ia
Goloskokov:2011rd,Goldstein:2012az}; see Ref.~\cite{Kroll:2016aop} for a
summary. A large chiral-odd distribution amplitude is induced by the dynamical
breaking of chiral symmetry in QCD, and its normalization can be determined
model-independently in terms of the chiral condensate
\cite{Goloskokov:2009ia,Goloskokov:2011rd}. While the mechanism is formally
power-suppressed and no strict factorization has been established at this level,
the pseudoscalar production amplitudes have been calculated in a modified hard
scattering approach, which implements suppression of large--size $q\bar q$
configurations in the meson through the QCD Sudakov form factor
\cite{Goloskokov:2009ia,Goloskokov:2011rd}. The results agree well with the
$\pi^0$ and $\eta$ electroproduction data from the JLab CLAS experiment at 6 GeV
incident energy, regarding both the absolute cross sections and the dominance of
transverse photon amplitudes ($L/T$ ratio) inferred from the
azimuthal--angle--dependent response functions
\cite{Bedlinskiy:2012be,Kubarovsky:2016yaa}. A tentative spin-flavor separation
of the chiral-odd GPDs has been performed by combining data in $\pi^0$ and
$\eta$ electroproduction using the different sensitivity of the two channels
\cite{Kubarovsky:2016yaa}. Further dedicated experiments in pseudoscalar meson
electroproduction are planned with the JLab 12 GeV Upgrade.
In order to interpret the pseudoscalar meson production data and assess the
potential of this method, it is necessary to gain more insight into the
properties of the chiral-odd GPDs from other sources.
Contrary to the chiral-even GPDs, in the chiral-odd case
neither the zero-momentum transfer limit of the GPDs (transversity PDFs) nor the
local operator limit of the GPD (form factor of local tensor operator)
correspond to structures that are easily measurable, so that little useful
information can be obtained in this way. The transversity PDFs can be extracted
from polarization observables in semi-inclusive deep-inelastic scattering,
and in principle also from dilepton production in polarized proton-proton
collisions, but the methods have large theoretical
and experimental uncertainties; see
Refs.~\cite{Bacchetta:2012ty,Anselmino:2013vqa,Kang:2014zza} and references
therein. The form factors of local tensor operators, which constrain the lowest
$x$--moment of the chiral-odd GPDs, have been calculated in lattice QCD
\cite{Gockeler:2006zu} and in various dynamical models of nucleon structure;
see Refs.~\cite{Barone:2001sp,Burkardt:2015qoa} for a review.
The $x$--dependent chiral-odd GPDs have been studied in quark bound--state
models of nucleon structure
\cite{Pasquini:2005dk,Burkardt:2007xm,Chakrabarti:2008mw,
Kumar:2015yta,Chakrabarti:2015ama}.
Besides these
estimates not much is known about the properties of the chiral-odd GPDs.
The limit of a large number of colors in QCD (large--$N_c$ limit) provides
a powerful model-independent method for studying the spin-flavor structure
of nucleon matrix elements \cite{'tHooft:1973jz,Witten:1979kh,Coleman:1980mx}.
The conceptual basis and practical implementation of this approach have been
described extensively in the literature, see Ref.~\cite{Jenkins:1998wy} for
a review. In the large--$N_c$ limit QCD becomes semi-classical, and baryons
can be described by mean field solutions in terms of meson fields
\cite{Witten:1979kh}. While the dynamics remains complex and cannot be solved
exactly, and the form of the mean field solution is not known, qualitative
insights can be obtained by exploiting known symmetry properties of the mean
field \cite{Witten:1983tx,Balachandran:1982cb}.
The resulting scaling relations for baryon mass splittings, meson-baryon
coupling constants, electromagnetic and axial form factors, and other
observables are generally in good agreement with observations
\cite{Jenkins:1998wy,Dashen:1993jt,Jenkins:1993zu,Jenkins:1995td,Dashen:1993as}.
In matrix elements of quark bilinear operators (vector or axial vector
currents, tensor operators) the large--$N_c$ limit identifies leading and
subleading spin--flavor components and implies a parametric hierarchy in
nucleon structure. The approach can be extended to parton densities
\cite{Diakonov:1996sr,Efremov:2000ar}, where it suggests a large flavor
asymmetry of the polarized antiquark distribution
$\Delta\bar u - \Delta\bar d$, as supported by the recent RHIC $W^\mp$
production data \cite{Aggarwal:2010vc,Adare:2010xa}. The $N_c$--scaling of
chiral--odd quark distributions (transversity PDFs) was considered in
Refs.~\cite{Pobylitsa:1996rs,Schweitzer:2001sr,Pobylitsa:2003ty},
and that of local chiral-odd operators (tensor charges) in
Refs.~\cite{Kim:1995bq,Ledwig:2010tu}, in the context
of calculations in the chiral-quark soliton model of the large--$N_c$ nucleon.
A general method for the $1/N_c$ expansion of GPDs was described
in Ref.~\cite{Goeke:2001tz} and applied to chiral-even GPDs.
In this article we study the spin-flavor structure of the nucleon's
chiral--odd GPDs in the large--$N_c$ limit and discuss its implications.
We derive the $N_c$--scaling of the chiral--odd GPDs using the method of
Ref.~\cite{Goeke:2001tz} and observe interesting differences between the
chiral--even and chiral--odd cases. We show that the findings can be
explained as the result of natural $N_c$--scaling of the nucleon--quark
helicity amplitudes associated with the chiral-odd GPDs \cite{Diehl:2001pm,Diehl:2000xz}.
The spin-flavor structure obtained in the large--$N_c$ limit is found
to be consistent with that observed in an analysis of the JLab CLAS $\pi^0$ and $\eta$
hard exclusive electroproduction data, assuming that these processes are
mediated by twist--3 chiral-odd meson distribution amplitudes.
The present study generalizes previous results in the $1/N_c$ expansion of
chiral-even GPDs \cite{Goeke:2001tz}, quark transversity distributions
\cite{Pobylitsa:1996rs,Schweitzer:2001sr,Pobylitsa:2003ty}, and matrix
elements of local chiral-odd operators \cite{Kim:1995bq} and uses the formal
apparatus developed in these earlier works. The description of the apparatus
and explicit quotation of chiral-even results is intended only to make the
present article readable. An intuitive and independent derivation of the
$N_c$--scaling of the chiral-odd GPDs based on nucleon--quark helicity amplitudes
was given in Ref.~\cite{Schweitzer:2016vkq}.
\section{Chiral-odd GPDs}
\label{Sec-2:GPDs}
GPDs parametrize the non-forward nucleon matrix elements
of QCD light--ray operators of the general form
\cite{Goeke:2001tz,Diehl:2003ny,Belitsky:2005qn,Boffi:2007yc}
\be\label{Eq:generic-matrix-element}
{\cal M } (\Gamma) =
P^+\int\frac{\di z^-}{2\pi}\,e^{ixP^+z^-}
\la N, p^\prime |\bar{\psi} (-z/2)\;\Gamma\,
\psi (z/2)\,|N, p \ra \, |_{z^+=0,\;\bm{z}_T=0} ,
\ee
where $P \equiv \frac{1}{2}(p' + p)$ is the average nucleon 4--momentum,
$z$ is a light-like distance, and the 4--vectors are described by their
light-cone components
$z^\pm = (z^0 \pm z^3)/\sqrt{2}, \, \bm{z}_T=(z^1,z^2),$ etc. The light-ray
operator generally contains a gauge link along the light-like path defined by
$z$, which we do not indicate for brevity.
$\Gamma$ denotes a generic matrix in spinor indices and defines the spin
structure of the operator. In the chiral-even case
the relevant spinor matrices are $\Gamma=\gamma^+$ and $\gamma^+\gamma_5$,
and the matrix elements are parametrized as
\ba\label{Eq:def-chiral-even-GPD-1}
{\cal M}(\gamma^+)
&=&
\bar{u}' \biggl[
\gamma^+\;H
+\frac{i\sigma^{+j}\Delta_j}{2M_N}\;E\biggr] u \,,\\
\label{Eq:def-chiral-even-GPD-2}
{\cal M}(\gamma^+\gamma_5)
&=&
\bar{u}^\prime \biggl[
\gamma^+\gamma_5\;\widetilde{H}
+\frac{\gamma_5\Delta^+}{2M_N}\widetilde{E}\biggr] u .
\ea
In the chiral-odd case the spinor matrix is $\Gamma = i\sigma^{+j}\, (j = 1, 2)$,
and the matrix element is parametrized as \cite{Diehl:2001pm}
\ba\label{Eq:def-chiral-odd-GPD}
{\cal M}(i\sigma^{+j})
&=&
\bar{u}\biggl[
i\sigma^{+j}\;H_T
+\frac{P^+\Delta^j-\Delta^+P^j}{M_N^2}\;\widetilde{H}_T\nonumber\\
&&
+\frac{\gamma^+\Delta^j-\Delta^+\gamma^j}{2M_N}\;E_T
+\frac{\gamma^+P^j-P^+\gamma^j}{M_N}\;\widetilde{E}_T\biggr]
u .
\ea
Here $u \equiv u (p, \lambda)$ and $u' \equiv u(p', \lambda')$ are the
bispinors of the initial and final nucleon (the choice of polarization states
will be specified later) and $\Delta \equiv p' - p$ is the 4--momentum transfer.
The GPDs $H=H(x,\xi,t)$, etc., are functions of the average
quark plus momentum fraction $x$, the plus momentum transfer to the quark,
$\xi = -\Delta^+/(2 P^+) = (p-p^\prime)^+/(p+p^\prime)^+$, and
the invariant momentum transfer to the nucleon, $t = \Delta^2$.
For brevity we do not indicate the dependence of the GPDs on the normalization
scale of the QCD operator.
The quark fields in Eq.~(\ref{Eq:generic-matrix-element}) carry flavor indices (suppressed
for brevity), and the correlator is generally a matrix in flavor space. The usual
flavor-diagonal GPDs are obtained with the operator
\be
H^f \;\; \leftrightarrow \;\; \bar\psi_f \ldots \psi_f
\hspace{2em} (f = u, d) ,
\label{flavor}
\ee
where the matrix element refers to the proton state. Alternatively one may consider the
isoscalar and isovector combinations $u\pm d$ of operators and GPDs. In the following
we shall specify the flavor and isospin structure of the matrix element as needed.
The chirally-even and odd GPDs satisfy certain symmetry relations in $\xi$,
resulting from time reversal invariance,
\be\label{Eq:xi-dependence}
{\rm GPD}(x,-\xi,t) = \begin{cases}
+\,{\rm GPD}(x,\xi,t)&{\rm for}\;\;{\rm GPD}=
H,\; \widetilde{H},\; E,\;\widetilde{E},\;
H_T,\;\widetilde{H}_T,\;E_T ,\\
-\,{\rm GPD}(x,\xi,t)&{\rm for}\;\;{\rm GPD}=\widetilde{E}_T .\end{cases}
\ee
Integration over the variable $x$ reduces the light-ray operators in
Eq.~(\ref{Eq:generic-matrix-element}) to local operators. In the chiral-even
case these are the vector and axial vector currents, so that the $x$--integral
(or first moments) of the GPDs coincide with the electromagnetic and axial
form factors of the nucleon \cite{Ji:1996nm}. In the chiral-odd case the
local operator is the tensor operator $\bar{\psi}(0) i\sigma^{\mu\nu}\psi (0)$,
and the first moments of the GPDs are
\be
\label{Eq:form-factors}
\int_{-1}^1\di x\, \{ H_T, \; \widetilde{H}_T, \; E_T, \; \widetilde{E}_T \}
(x,\xi,t)
\;\; = \;\;
\{ H_T(t), \; \widetilde{H}_T(t), \; E_T(t), \; 0 \} ,
\ee
where $H_T(t), \widetilde{H}_T (t)$ and $E_T (t)$ are the nucleon's tensor
form factors [with the same flavor structure as the GPDs, cf.~Eq.~(\ref{flavor})].
The vanishing of the first moment of $\widetilde{E}_T$ is a consequence of
the antisymmetry in $\xi$, Eq.~(\ref{Eq:xi-dependence}); its higher moments
are non-zero. The higher $x$--moments of chirally-even and odd GPDs are
polynomials in $\xi$ (generalized form factors).
In the limit of zero momentum transfer (forward limit) the chiral-even GPDs
$H$ and $\widetilde H$ reduce, respectively, to the unpolarized and
helicity PDFs, $H (x, \xi = 0, t = 0) = f_1$ and
$\widetilde H (x, \xi = 0, t = 0) = g_1$. The GPDs $E_q$ and
$\widetilde E_q$ are also non-zero in the forward limit but do not reduce to
any known PDFs, as these GPDs correspond to nucleon helicity-flip components
of the matrix element (see Sec.~\ref{sec:helicity}). The chiral-odd GPD $H_T$
reduces in the forward limit to the transversity PDF,
\begin{equation}
H_T (x, \xi = 0, t = 0) \;\; = \;\; h_1 (x) .
\label{H_T_forward}
\end{equation}
Its first moment is known as the nucleon's tensor charge. Because the local
tensor operator is not a conserved current, the tensor charge is
scale--dependent and cannot directly be related to low--energy properties
of the nucleon. The forward limit of
the chiral-odd GPDs $\widetilde{H}_T$ and $E_T$
is not related to any known PDFs,
while $\widetilde{E}_T$ vanishes in the forward limit again
due to its antisymmetry in $\xi$, Eq.~(\ref{Eq:xi-dependence}).
Other aspects of the GPDs, such as their partonic interpretation,
are described in
Refs.~\cite{Goeke:2001tz,Diehl:2003ny,Belitsky:2005qn,Boffi:2007yc}.
In Eqs.~(\ref{Eq:def-chiral-even-GPD-1})--(\ref{Eq:def-chiral-odd-GPD}) the GPDs
appear as invariant amplitudes, arising from a particular decomposition of the matrix
elements into bilinear forms between nucleon spinors. In applications to exclusive
pseudoscalar meson production processes it is natural to introduce the
combination
\be
\bar{E}_T \; \equiv \; E_T + 2\widetilde{H}_T
\label{E_T_bar_def}
\ee
of the chiral-odd GPDs, which corresponds to a different invariant decomposition
of the matrix element Eq.~(\ref{Eq:def-chiral-odd-GPD}) \cite{Goloskokov:2011rd}.
An alternative representation
of the GPDs as nucleon-quark helicity amplitudes will
be described in Sec.~\ref{sec:helicity}.
\section{Chiral--odd GPDs in large--$N_c$ limit}
\label{Sec-3:large-Nc}
In the large--$N_c$ the nucleon mass scales as $M_N \sim N_c$, while the nucleon
size remains stable, $\sim N_c^0$. The $1/N_c$ expansion of GPDs is performed
in a class of frames where the initial and final nucleon move with 3--momenta
$p^k, \,p^{\prime k} \sim N_c^0 \; (k = 1, 2, 3)$ and have energies
$p^0, \, p^{\prime 0} = M_N + O(1/N_c)$, which implies an energy and momentum
transfer $\Delta^0 \sim N_c^{-1}, \, \Delta^k \sim N_c^0$, and thus
\begin{equation}
\Delta^i\; \sim \; N_c^0 \hspace{2em} (i = 1, 2), \hspace{2em}
\xi \; \sim \; N_c^{-1}, \hspace{2em}
|t| \; \sim \; N_c^0 .
\end{equation}
In the partonic variable $x$ one considers the parametric region
\begin{equation}
x \; \sim \; N_c^{-1} ,
\label{nc_scaling_x}
\end{equation}
corresponding to non-exceptional longitudinal momenta of the quarks
and antiquarks relative to the slowly moving nucleon,
$x M_N \sim (\textrm{nucleon size})^{-1} \sim N_c^0$. Likewise, it is assumed that the
normalization scale of the light-ray operator is $\sim N_c^0$,
so that the typical quark transverse momenta are $\sim N_c^0$.
Equation~(\ref{nc_scaling_x}) corresponds to the intuitive picture of
a nucleon consisting of $N_c$ ``valence'' quarks and a ``sea'' of $O(N_c)$
quark--antiquark pairs, with each quark/antiquark carrying on average a
fraction $\sim 1/N_c$ of the nucleon's light-cone momentum. Altogether, the
$N_c$--scaling relations for GPDs can then be expressed in the form
\begin{equation}
{\rm GPD}(x, \xi, t) \;\; \sim \;\;
N_c^k \; \times \; \textrm{function}(N_c x, N_c\xi, t) ,
\label{nc_scaling_gpd_generic}
\end{equation}
where the scaling exponent $k$ depends on the GPD in question and the isospin
component ($u + d, u - d$) and can be established on general grounds,
while the scaling function on the
right-hand-side is stable in the large--$N_c$ limit and can only be
determined in specific dynamical models.
A practical method for performing the $1/N_c$ expansion of baryon matrix elements of
quark bilinear operators was given in Refs.~\cite{Pobylitsa:2000tt,Goeke:2001tz},
using collective quantization of an abstract mean-field solution with known
symmetry properties. One considers a generic correlator of the form
\be
\la B^\prime, \bm{p}^\prime |\overline{\psi}_{\alpha' f'}(x')\psi_{\alpha f}(x)|
B, \bm{p} \ra,
\label{corr}
\ee
where the quark fields are at space-time points $x$ and $x'$, and $(\alpha, \alpha')$
and $(f, f')$ are the Dirac spinor and flavor indices. We restrict ourselves to the $SU(2)$
flavor sector and assume exact isospin symmetry. The baryon states are characterized by
their momenta $\bm{p}$ and $\bm{p}'$, and spin--isospin quantum numbers
$B \equiv \{S, S_3, T, T_3 \}$ and $B' \equiv \{S', S_3', T', T_3' \}$, and normalized such that
\be\label{Eq-R:norm}
\la B^\prime, \bm{p}^\prime| B, \bm{p} \ra
= 2p^0(2\pi)^3\delta^{(3)}(\bm{p}^\prime - \bm{p})
\,\delta_{B^{ }B^\prime}\;, \;\;\;
\,\delta_{B^{ }B^\prime} \equiv
\delta_{S^{ }S^\prime}\,\delta_{S_3^{ }S_3^\prime} \,
\delta_{T^{ }T^\prime}\,\delta_{T_3^{ }T_3^\prime} .
\ee
For simplicity we do not specify the color indices of the quark fields in Eq.~(\ref{corr})
and do not indicate the gauge link (in the case of GPDs the gauge link can be eliminated
by choosing the light-cone gauge; in a more general case it can be included explicitly
by an appropriate redefinition of the quark fields \cite{Pobylitsa:2003ty}).
One evaluates the correlator Eq.~(\ref{corr}) starting with the expectation value of the
bilinear operator in the localized mean field characterizing the large--$N_c$ baryon (``soliton''),
centered at the origin,
\be
\langle \overline{\psi}_{\alpha' f'}(x')\psi_{\alpha f}(x) \rangle
\;\; = \;\; {\cal F}(x^{\prime 0}-x^0,\bm{x}',\bm{x})_{\alpha f; \alpha' f'} .
\label{vacuum}
\ee
While the specific form of the function $F$ depends on dynamics and can only be determined
in models, its symmetry properties in the large--$N_c$ limit can be established on general
grounds. In leading order of $1/N_c$ the baryon mean field is static (time-independent),
so that the correlator depends only on the relative time $x^{\prime 0} - x^0$. Most importantly,
the mean field intertwines spatial and isospin degrees of freedom (``hedgehog symmetry'')
\cite{Witten:1983tx}, so that a rotation in flavor space by an $SU(2)$ matrix $R$ and a
simultaneous spatial rotation with a rotation matrix $O(R)$ and spin rotation $S(R)$
leave the correlator Eq.~(\ref{vacuum}) invariant,
\be\label{Eq-R:hedgehog-symmetry}
S(R)_{\alpha\beta}\, R_{fg}\,
{\cal F}(x^{\prime 0}-x^0,O(R)\bm{x}',O(R)\bm{x})
_{\beta g, \beta', g'} \,
R^{-1}_{g'f'}\, S(R^{-1})_{\beta'\alpha'}
\; = \; {\cal F}(x^{\prime 0}-x^0,\bm{x}',\bm{x})_{\alpha f; \alpha'f'}
\, ,
\ee
where
\be
O^{ji}(R) \;\; \equiv \;\; \tfrac{1}{2} \, {\rm tr} [R^{-1} \tau^j R \tau^i]
\hspace{2em} (i, j = 1, 2, 3) ,
\label{O_def}
\ee
and $O(R)\bm{x}'$ and $O(R)\bm{x}$ denote the rotation of the 3--vectors
$\bm{x}'$ and $\bm{x}$ with the matrix $O(R)$.
The mean field breaks translational and rotational/isorotational invariance and does
not correspond to states of definite momentum and spin/isospin quantum numbers.
The matrix element between nucleon states of definite momentum and spin/isospin is
obtained by quantizing the collective motion in coordinate and isospin space and
projecting on states with appropriate quantum numbers. In this way one obtains a
representation of the baryon matrix element Eq.~(\ref{corr}) in the form
\cite{Witten:1979kh,Pobylitsa:2000tt}
\ba
\la B^\prime, \bm{p}^\prime |\overline{\psi}_{\alpha' f'}(x')\psi_{\alpha f}(x)|
B, \bm{p} \ra
&=& 2M_B N_c \; \int\di R\;\phi^\ast_{B^\prime}(R)\,\phi_{B^{ }}(R) \;
\int\di^3X\,e^{i(\bm{p}' - \bm{p})\cdot\bm{X}} \nonumber\\
&&\times \; R_{fg}\,
{\cal F}(x^{\prime 0} - x^0,\bm{x}'-\bm{X},\bm{x}-\bm{X})
_{\alpha g; \alpha' g'}
\, (R^{-1})_{g'f'}\, + \dots ,
\label{Eq-R:correlator-mean-field}
\ea
where the dots indicate subleading terms in $1/N_c$ and $M_B\sim N_c$ is the baryon mass
(note that $M_{B'} = M_B$ in leading order of $1/N_c$).
The integral over the position of the center of the mean field, $\bm{X}$, with wave
functions $\exp (-i\bm{p}\bm{X})$ and $\exp (i\bm{p}'\bm{X})$ projects the
correlator Eq.~(\ref{vacuum}) on baryon states with momenta $\bm{p}$ and $\bm{p}'$.
The integral over the flavor rotation $R$ with rotational wave functions
$\phi_{B^{ }}(R)$ and $\phi^\ast_{B^\prime}(R)$ projects on baryon states
with spin/isospin quantum numbers $B$ and $B'$. The hedgehog symmetry of the
mean field [cf.~Eq.(\ref{Eq-R:hedgehog-symmetry})] implies that the baryon states
occur in representations with equal spin and isospin, $S = T$, and the rotational
wave functions are given by the Wigner finite rotation matrices as \cite{Goeke:2001tz}
\be
\phi_B (R) \; \equiv \;
\phi_{S_3T_3}^{S=T}(R) \; = \;
\sqrt{2S+1}\;(-1)^{T_{ }+T_3} D_{-T_3,S_3}^{S=T}(R) .
\ee
Using Eq.~(\ref{Eq-R:correlator-mean-field}) one can evaluate the matrix element of
any bilinear quark operator in leading non-vanishing order of the $1/N_c$ expansion.
The hedgehog symmetry of the mean field, Eq.(\ref{Eq-R:hedgehog-symmetry}), restricts
the spin-isospin structures emerging from the rotational integral and determines the
$N_c$--scaling of the spin-flavor components of the matrix element. These relations
depend on the specific form of the operator considered.
The chiral even GPDs were evaluated in this way in Ref.~\cite{Goeke:2001tz}. Here
the operators are the light ray operators of Eq.~(\ref{Eq:generic-matrix-element})
with the chirally-even
spinor matrices $\Gamma = \gamma^+$ and $\Gamma = \gamma^+ \gamma^5$,
cf.~Eq.~(\ref{Eq:def-chiral-even-GPD-1}). It is instructive to perform the
integration over collective coordinates in Eq.~(\ref{Eq-R:correlator-mean-field})
in two steps. In the first step one considers the correlator integrated over the
coordinate $\bm{X}$ but not yet over rotations; i.e., projected on momentum states but not yet
on spin/isospin states. This correlator describes the GPDs of a large--$N_c$
baryon that has not yet been projected on spin-isopsin states (``soliton GPDs'').
In the chiral-even case it was found to be of the form \cite{Goeke:2001tz}
\be
M_N \int\frac{\di z^-}{2\pi}\,e^{ixP^+z^-}
\la {\rm sol}, \bm{p}^\prime|\bar{\psi}_{f'}(-z/2)\,
\left\{
\begin{array}{c}
\gamma^+ \\[3.5ex]
\gamma^+ \gamma_5
\end{array} \right\}
\psi_{f}(z/2)\,|{\rm sol}, \bm{p} \ra\biggl|_{z^+=0,\bm{z}_T=0}
=
\left\{
\begin{array}{l}
\displaystyle
\delta_{f'\!f} \, H_{\rm sol}
\; - \; \frac{i\varepsilon^{3jk}\Delta^j}{2M_N} \; D^k_{f'\!f} \,
E_{\rm sol}
\\[2.5ex]
\displaystyle
D^3_{f'\!f} \, \widetilde{H}_{\rm sol}
\; - \; \frac{\Delta^3\Delta^j}{(2 M_N)^2}
\, D^j_{f'\!f} \, \widetilde{E}_{\rm sol}
\end{array} \right\} ,
\label{gpd_sol_even}
\ee
where $H_{\rm sol}, E_{\rm sol}, \widetilde{H}_{\rm sol}$ and
$\widetilde{H}_{\rm sol}$ are functions of $x, \xi$ and $t$, and
[cf.~Eq.~(\ref{O_def})]
\ba
D^i_{f'\!f} \;\; \equiv \;\; D^i_{f'\!f} (R) \;\; \equiv \;\;
{\tfrac{1}{2}} (\tau^j)_{f'\!f} \; O^{ji} (R) .
\label{D_ff_def}
\ea
This expression embodies the hedgehog symmetry expressed by
Eq.~(\ref{Eq-R:hedgehog-symmetry}): the
flavor-singlet structure $\propto \delta_{f'\!f}$ is independent of the rotation matrix
$R$ defining the orientation of the soliton, while the flavor-nonsinglet structures
$\propto (\tau^j)_{f'\!f}$ are accompanied by rotation matrices and coupled with the
spatial directions defined by the light-ray operator ($z$--direction) and the momentum transfer
$\bm{\Delta}$. In the second step one then projects the soliton matrix element
on spin-isospin states by performing the integral over rotations, using
\be
\int\di R\;\phi_{S_3^\prime T_3^\prime}^{\ast S^\prime =T^\prime = 1/2}(R)\,
\; \phi_{S_3^{ }T_3^{ }}^{S^{ }=T^{ }=1/2}(R)
\;
\left\{ \begin{array}{c}
1
\\[2ex]
O^{ji}(R)
\end{array}
\right\}
\;\; = \;\;
\left\{ \begin{array}{c}
\delta_{S_3' S^{}_3} \,
\delta_{T_3' T^{}_3}
\\[2ex]
-\,\frac{1}{3} \, (\sigma^i)_{S_3^\prime S^{}_3} \, (\tau^j)_{T_3' T^{}_3}
\end{array} \right\} .
\ee
The resulting nucleon matrix elements can be expressed in a transparent form by introducing
a shorthand matrix notation for the nucleon spin components,
\be
\sigma^0 \; \equiv \;
\sigma^0 (S_3', S_3) \; \equiv \; \delta_{S_3', S_3}, \hspace{2em}
\sigma^i \; \equiv \;
\sigma^i (S_3', S_3) \; \equiv \; (\sigma^i)_{S_3', S_3} ,
\label{shorthand}
\ee
and correspondingly for the quark flavor components,
\be
\tau^0 \; \equiv \; \tau^0 (f', f) \; \equiv \; \delta_{f'\!f},
\hspace{2em}
\tau^j \; \equiv \; \tau^j (f', f) \equiv (\tau^j)_{f'\!f} .
\ee
In this notation the nucleon matrix elements become (we consider the
proton with $T_3' = T_3 = 1/2$)
\ba\label{Eq:XXa}
{\cal M}(\gamma^+)
&=& \sigma^0 \, \tau^0 \, H_{\rm sol}
+ \frac{i (\bm{\Delta}\times
\bm{\sigma})^3 \, \tau^3}{3 (2M_N)}\,E_{\rm sol} ,\\
\label{Eq:XXb}
{\cal M}(\gamma^+\gamma_5)
&=& -\frac{\sigma^3 \, \tau^3}{3} \, \widetilde{H}_{\rm sol}
+ \frac{\Delta^3\, \bm{\Delta}\cdot\bm{\sigma} \, \tau^3}{3 (2M_N)^2}
\widetilde{E}_{\rm sol} .
\ea
Equations~(\ref{Eq:XXa}) and (\ref{Eq:XXb}) express the spin--flavor symmetry
characteristic of the large--$N_c$ limit:
the spin--singlet matrix element is also a flavor--singlet, and the spin---nonsinglet
one is a flavor--nonsinglet. They also allow one to determine the explicit $N_c$--scaling
of the chiral--even GPDs. The $N_c$--scaling of the soliton GPDs
in Eq.~(\ref{gpd_sol_even}) follows from the fact that the spatial size of the
mean field is $\sim N_c^0$, and from the kinematic prefactors emerging from the
$1/N_c$--expansion of the spin structures in the matrix element, and is given by
[cf.~Eq.~(\ref{nc_scaling_gpd_generic})]
\be
\{ H_{\rm sol}, \, E_{\rm sol}, \, \widetilde{H}_{\rm sol}, \, \widetilde{E}_{\rm sol} \}
(x, \xi, t) \;\; \sim \;\; \{ N_c^2, \, N_c^3, \, N_c^2, \, N_c^4\}
\; \times \; \textrm{function}(N_c x, N_c\xi, t) .
\ee
The $N_c$--scaling of the leading flavor components of the chiral--even nucleon
GPDs is thus obtained as \cite{Goeke:2001tz}
\be
\{ H^{u+d}, \, E^{u-d}, \, \widetilde{H}^{u-d}, \, \widetilde{E}^{u-d} \}
(x, \xi, t) \;\; \sim \;\; \{ N_c^2, \, N_c^3, \, N_c^2, \, N_c^4\}
\; \times \; \textrm{function}(N_c x, N_c\xi, t) .
\label{chiral_even_leading}
\ee
The respective opposite flavor combinations are suppressed by one order in $1/N_c$, i.e.,
$H^{u-d} \sim N_c$, etc.
We now apply this method to the chiral-odd GPDs and derive their $N_c$--scaling.
The calculations are performed in complete analogy to the chiral--even case
described above \cite{Goeke:2001tz}. Using the specific decomposition of the chiral--odd
correlator Eq.~(\ref{Eq:def-chiral-odd-GPD})
and performing the $1/N_c$ expansion of the components,
we obtain the chiral--odd soliton GPDs as [cf.~Eqs.~(\ref{gpd_sol_even})
and (\ref{D_ff_def})]
\ba
&& M_N\int\frac{\di z^-}{2\pi}\,e^{ixP^+z^-} \nonumber
\la {\rm sol}, \bm{p}^\prime |\bar{\psi}_{f'}(-z/2)\,i\sigma^{+j}
\psi_{f}(z/2)\,|{\rm sol}, \bm{p}\ra\biggl|_{z^+=0,\bm{z}_T=0} \\
&& \; = \;
\frac{\Delta^j}{2M_N}
\,\delta_{f'\!f} \,\bar{E}_{T,\,\rm sol}
\; + \; i\varepsilon^{3jk} \,D_{f'\!f}^k \, H_{T,\,\rm sol}
\; + \; \frac{i\varepsilon^{jkl}\Delta^k}{2 M_N} \, D_{f'\!f}^l \,
\widetilde{E}_{T,\,\rm sol} .
\label{gpd_sol_odd}
\ea
The hedgehog symmetry is again manifest in the structure of the right-hand side.
Notice that the large--$N_c$ matrix element has only three independent structures,
and that the GPDs $E_T$ and $\widetilde{H}_T$ appear only in the combination
$\bar{E}_T$, Eq.~(\ref{E_T_bar_def}).
Projecting on nucleon states ($T_3' = T_3 = 1/2$) we obtain, in the matrix notation of
Eqs.~(\ref{Eq:XXa},~\ref{Eq:XXb}) ($\bm{e}_3$ denotes the unit vector in the 3--direction)
\ba\label{Eq:XXc}
{\cal M}(i\sigma^{+j})
\; &=& \; \sigma^0\, \tau^0 \,
\frac{\Delta^j}{2M_N}\, \bar{E}_{T,\,\rm sol}
\; + \; \frac{(\bm{e}_3 \times\bm{\sigma})^j \, \tau^3}{3}\,H_{T,\, \rm sol}
\; - \; \frac{(\bm{\Delta}\times\bm{\sigma})^j \, \tau^3}{3 (2 M_N)}\,
\widetilde{E}_{T, \rm sol} \hspace{2em} (j = 1, 2).
\ea
The result again expresses the spin-flavor symmetry characteristic of the large--$N_c$
limit. The $N_c$--scaling of the chiral-odd soliton GPDs is found to be
\be
\{ \bar{E}_{T, {\rm sol}}, \, H_{T, {\rm sol}}, \, \widetilde{E}_{T, {\rm sol}} \} (x, \xi, t)
\;\; \sim \;\; \{ N_c^3, \, N_c^2, \, N_c^3 \}
\; \times \; \textrm{function}(N_c x, N_c\xi, t) .
\ee
We can thus identify the leading flavor components of the chiral-odd nucleon
GPDs and determine their $N_c$--scaling,
\be
\label{Eq:scaling-HT}
\{ \bar{E}_T^{u + d}, \, H_T^{u-d}, \, \widetilde{E}_T^{u-d} \} (x, \xi, t)
\;\; \sim \;\; \{ N_c^3, \, N_c^2, \, N_c^3 \}
\; \times \; \textrm{function}(N_c x, N_c\xi, t) .
\ee
The respective other flavor
components are suppressed by at least one power of $1/N_c$,
\be
\label{Eq:scaling-HT-subleading}
\{ \bar{E}_T^{u-d}, \, H_T^{u+d}, \, \widetilde{E}_T^{u+d} \} (x, \xi, t)
\;\; \sim \;\; \{ N_c^2, \, N_c, \, N_c^2 \}
\; \times \; \textrm{function}(N_c x, N_c\xi, t) .
\ee
These results confirm our earlier, intuitive
derivation of the $N_c$--scaling using helicity amplitudes \cite{Schweitzer:2016vkq}.
The large--$N_c$ limit exposes an interesting difference between the chiral--even
and chiral-odd quark correlation functions in the nucleon, regarding the number
of independent nucleon spin structure components, as described by the matrices
$\sigma^0$ and $\sigma^i \, (i = 1, 2, 3)$. It can be exhibited by projecting the
spin matrices $\sigma^i \, (i = 1, 2, 3)$
on the orthogonal 3--vectors $\bm{e}_3$ (the direction
defined by the light-ray operator), $\bm{\Delta}_T \equiv
\bm{\Delta} - (\bm{e}_3 \cdot \bm{\Delta}) \bm{e}_3$ (the component of $\bm{\Delta}$
orthogonal to $\bm{e}_3$), and
\be
\bm{n}_T \;\;\ \equiv \;\; \bm{e}_3 \times \bm{\Delta}
\ee
(the normal vector of the plane defined by $\bm{e}_3$ and $\bm{\Delta}_T$, or
the complement of $\bm{\Delta}_T$ in the transverse plane). In the chiral-even correlators
Eqs.~(\ref{Eq:XXa}) and (\ref{Eq:XXb}) one finds that all spin structures
\be
\sigma^0, \;\; \bm{e}_3 \cdot \bm{\sigma}, \;\; \bm{\Delta}_T \cdot \bm{\sigma}, \;\;
\bm{n}_T \cdot \bm{\sigma}
\label{spin_structure_even}
\ee
are non-zero and occur with four independent coefficient functions. In the
chiral-odd correlators Eq.~(\ref{Eq:XXc}), however, the transverse nucleon
spin structures $\bm{\Delta}_T \cdot \bm{\sigma}$ and
$\bm{n}_T \cdot \bm{\sigma}$ occur only in the combination
\be
\bm{e}_3 \times \bm{\sigma} ,
\label{spin_structure_odd}
\ee
which {\em does not depend on the direction of the transverse
momentum} $\bm{\Delta}_T$, and there are only three independent coefficient
functions. One sees that the reason why there are only
three independent chiral-odd GPDs is that the
large--$N_c$ nucleon does not correlate the direction of the transverse
quark spin (as defined by the light-ray operator with $\sigma^{+j}$) with that of the
transverse nucleon spin (as contained in the spin structures $\bm{e}_3 \times \bm{\sigma}$
of the matrix element) through the nucleon's transverse momentum transfer.
The absence of such spin-orbit interactions is specific to the leading
order of the $1/N_c$ expansion, and we expect that higher-order corrections
will remove the degeneracy of the transverse spin structures.
\section{Chiral--odd GPDs as helicity amplitudes}
\label{sec:helicity}
Further insight into the different behavior of chiral-even and odd GPDs in the
large--$N_c$ limit can be gained by considering the representation of the GPDs as partonic
helicity amplitudes \cite{Diehl:2000xz}. This representation most naturally
appears in the region $\xi < x < 1$, where the GPDs describe the amplitude
for the ``emission'' by the nucleon of a quark with plus momentum fraction $x + \xi$
and subsequent ``absorption'' of a quark with $x - \xi$
(see Fig.~\ref{Fig-1:helicity-amplitude}). In the region $-1 < x < -\xi$ the GPDs describe
the emission and absorption of an antiquark, while in $-\xi < x <\xi$ they describe
the emission of a quark--antiquark pair by the nucleon. We do not need
to consider these regions separately in the subsequent arguments.
\begin{figure}
\parbox[c]{.38\textwidth}
{\includegraphics[width=.35\textwidth]{Fig01-helicity-amplitude.eps}}
\hspace{.1\textwidth}
\parbox[c]{.5\textwidth}{\caption{\label{Fig-1:helicity-amplitude}
Representation of GPDs in the region $\xi < x < 1$ as nucleon--quark helicity amplitudes.
In the nucleon and quark states (denoted as $N, q$) the second label denotes the fraction of the
light-cone plus momentum $P^+$ carried by the particle, and the third label denotes the
light-cone helicity.}}
\end{figure}
The partonic helicity amplitudes are defined by a correlator of the form of
Eq.~(\ref{Eq:generic-matrix-element}), in which the nucleon spin states are
described as light-front helicity states and the quark spinor matrices are
chosen
as projectors on quark light-front helicity states:
\be
\!A_{\lambda^\prime\mu^\prime,\lambda\mu} \;\; = \;\; P^+
\int\frac{\di z^-}{2\pi}\,e^{ixP^+z^-}
\la N, p^\prime, \lambda^\prime| \, \overline{\psi}(-z/2)\; \Gamma_{\mu'\mu} \;
\psi(z/2) \, |N, p, \lambda \ra \, |_{{z^+=0},\;{\bm{z}_T=0}\,} ,
\ee
where $\lambda (\lambda')$ are the light-front helicities of the initial (final)
nucleon and $\mu (\mu')$ those of the initial (final) quark \cite{Diehl:2000xz}.
It is convenient to work in a reference frame where the light-cone direction is
chosen as the $z$-axis and the initial and final nucleon momenta $\bm{p}$ and
$\bm{p^\prime}$ lie in the $x$--$z$ plane. The spinor matrices for
the light-front helicity conserving (chiral-even) and light-front
helicity flipping (chiral-odd) amplitudes are then given by \cite{Diehl:2001pm}
\ba
\Gamma_{\pm\pm} &=& \frac{1}{4}\gamma^+(1\pm\gamma_5) ,
\\[2ex]
\Gamma_{\pm\mp} &=& \frac{i}{4}\sigma^{+1}(\pm 1 -\gamma_5)
\; = \; \frac{i}{4}(\pm \sigma^{+1} + i \sigma^{+2}) .
\ea
Flavor components of the amplitudes can be defined in analogy to those of the
correlator Eq.~(\ref{Eq:generic-matrix-element}) and will be specified below.
The helicity-conserving amplitudes are related to the chiral-even GPDs as
\begin{subequations}
\ba\label{Eq:amplitudes-0}
&& A_{++,++}=\frac12\,\sqrt{1-\xi^2}
\biggl(H+\widetilde{H}-\frac{\xi^2}{1-\xi^2}(E+\widetilde{E})\biggr) ,
\;\;\;\;\;\;\;\;\;\\
&& A_{-+,-+}=\frac12\,\sqrt{1-\xi^2}
\biggl(H-\widetilde{H}-\frac{\xi^2}{1-\xi^2}(E-\widetilde{E})\biggr) , \\
&& A_{++,-+}=\frac12\,\delta_t\;\biggl(\xi\widetilde{E}-E\biggr) , \\
&& A_{-+,++}=\frac12\,\delta_t\;\biggl(\xi\widetilde{E}+E\biggr) ,
\label{Eq:amplitudes-0-end}\ea
\end{subequations}
while the helicity-flipping amplitudes are related to the chiral-odd GPDs as
\begin{subequations}
\ba \label{Eq:amplitudes-I}
&& A_{++,+-}=\delta_t
\biggl(\widetilde{H}_T+\frac{1-\xi}{2}(E_T+\widetilde{E}_T)\biggr) , \\
&& A_{-+,--}=\delta_t
\biggl(\widetilde{H}_T+\frac{1+\xi}{2}(E_T-\widetilde{E}_T)\biggr) , \\
&& A_{++,--}=\sqrt{1-\xi^2}
\biggl(H_T+\delta_t^2\widetilde{H}_T
-\frac{\xi^2}{1-\xi^2}E_T
+\frac{\xi}{1-\xi^2}\widetilde{E}_T\biggr) , \\
&& \label{Eq:amplitudes-I-end}
A_{-+,+-}=\sqrt{1-\xi^2}\,
\delta_t^2\;\widetilde{H}_T\,, \phantom{\frac11}
\ea
\end{subequations}
where the ``kinematic'' prefactor $\delta_t$ is defined as
\be\label{Eq:delta-k}
\delta_t = {\rm sign}\biggl(P^+\Delta^1-\Delta^+P^1\biggr)
\frac{\sqrt{t_0-t}}{2M_N}\;,\;\;\;
-t_0 = \frac{4M_N^2\xi^2}{1-\xi^2}\; ,
\ee
in which $-t_0$ is the minimal value of $-t$ for the given
value of $\xi$. There are four linearly independent amplitudes in each
sector; the other four amplitudes in each sector can be obtained from
those in Eqs.~(\ref{Eq:amplitudes-0}--\ref{Eq:amplitudes-0-end}) and
(\ref{Eq:amplitudes-I}--\ref{Eq:amplitudes-I-end}) by the parity
relation \cite{Diehl:2001pm}
\be
A_{-\lambda'-\mu', -\lambda-\mu} \;\; = \;\; (-)^{\lambda' - \mu' - \lambda + \mu} \,
A_{\lambda'\mu', \lambda\mu} .
\label{parity}
\ee
Altogether, there are 8 linearly
independent helicity amplitudes, corresponding to the total number of
chiral-even and chiral-odd GPDs (or invariant amplitudes).
It is instructive to study the $N_c$--scaling of the partonic helicity amplitudes.
The ``natural'' scaling of the individual helicity amplitudes for a given quark
flavor ($f = u,d$) is
\be\label{Eq:amplitudes-generic-scaling}
A^f_{\lambda^\prime\mu^\prime,\lambda\mu} \;\; \sim \;\; N_c^2,
\ee
which is understood with the arguments $x, \xi$ and $t$ scaling
as in Eq.~(\ref{nc_scaling_gpd_generic}).
One power of $N_c$ originates from the covariant normalization
of the nucleon states in Eq.~(\ref{Eq-R:norm}), because $P^0\sim N_c$,
and another power of $N_c$ from the implicit summation over the color
indices in the light-ray operators.
Combinations of amplitudes corresponding to definite isospin transitions
($u + d, u - d$) can vanish in leading order of the $1/N_c$ expansion due
to the symmetries of the mean field solution (cf.~Sec.~\ref{Sec-3:large-Nc})
and have a lower scaling exponent. Using the results of Ref.~\cite{Goeke:2001tz}
and Sec.~\ref{Sec-3:large-Nc} for the $N_c$--scaling of the GPDs we can now
identify the leading and subleading helicity amplitudes.
For the chiral-even amplitudes one obtains
\begin{subequations}
\ba\label{Eq:amplitudes-expand-0}
A^{u+d}_{++,++}=\tfrac12\,H^{u+d}\,, \;\;\;
&& A^{u-d}_{++,++}=\tfrac12\, (\phantom{-}\widetilde{H}^{u-d}-\xi^2\widetilde{E}^{u-d}) , \\
A^{u+d}_{-+,-+}=\tfrac12\,H^{u+d}\,, \;\;\;
&& A^{u-d}_{-+,-+}=\tfrac12\,(-\widetilde{H}^{u-d}+\xi^2\widetilde{E}^{u-d}) , \\
A^{u+d}_{++,-+}=0, \;\;\;\;\;\;\;\;\;\;\;\;\;\;
&& A^{u-d}_{++,-+}=\tfrac12\,\delta_t\; (\xi\widetilde{E}^{u-d}-E^{u-d}) , \\
A^{u+d}_{-+,++}=0, \;\;\;\;\;\;\;\;\;\;\;\;\;\;
&& A^{u-d}_{-+,++}=\tfrac12\,\delta_t\; (\xi\widetilde{E}^{u-d}+E^{u-d}) .
\label{Eq:amplitudes-expand-0-end}\ea
\end{subequations}
The expressions correspond to the leading order of the $1/N_c$ expansion,
i.e., they are accurate at the natural order ${\cal O}(N_c^2)$.
It is understood that all non-zero amplitudes receive corrections
of order ${\cal O}(N_c)$. The amplitudes that vanish do so at order
${\cal O}(N_c^2)$, and generically have corrections of order ${\cal O}(N_c)$.
Notice that in the specific frame chosen here in the large--$N_c$ limit
the kinematic factors simplify as
\be\label{Eq:xi-deltak-large-Nc}
\delta_t = \frac{\Delta^1}{2M_N} \, ,\;\;\;
\xi = -\,\frac{\Delta^3}{2M_N} \, .
\ee
One notices that two pairs of the chiral-even amplitudes are degenerate (up to an
overall sign) at ${\cal O}(N_c^2)$, and two other amplitudes vanish at this order.
The content of
Eqs.~(\ref{Eq:amplitudes-expand-0}--\ref{Eq:amplitudes-expand-0-end}) becomes
more transparent when considering linear combinations of the chiral-even amplitudes,
\begin{subequations}
\ba\label{Eq:amplitudes-expand-1}
A^{u+d}_{++,++}+A^{u+d}_{-+,-+}&=&\phantom{\xi\,\delta_t\;}H^{u+d}\,, \\
A^{u-d}_{++,++}-A^{u-d}_{-+,-+}&=&\phantom{\xi\,\delta_t\;}\widetilde{H}^{u-d}-\xi^2\widetilde{E}^{u-d}\,, \\
A^{u-d}_{-+,++}+A^{u-d}_{++,-+}&=& \xi \,\delta_t\;\widetilde{E}^{u-d}\,,\\
A^{u-d}_{-+,++}-A^{u-d}_{++,-+}&=&\phantom{\xi}\,\delta_t\;E^{u-d}\,.
\label{Eq:amplitudes-expand-1-end}
\ea
\end{subequations}
This representation shows that there are four independent combinations of helicity
amplitudes that appear in leading order of the $1/N_c$ expansion, which are unambiguously
associated with the four leading spin-flavor components of the chiral-even GPDs.
As a consequence, each of the four chiral-even GPDs has a leading flavor component:
$u + d$ in $H$, and $u- d$ in $\widetilde{H}, E,$ and $\widetilde{E}$.
The situation is different in the case of chiral-odd helicity amplitudes.
Using the results for the $1/N_c$ expansion of Sec.~\ref{Sec-3:large-Nc} we
obtain the following scaling behavior of the chiral-odd helicity amplitudes
at ${\mathcal O}(N_c^2)$:
\begin{subequations}
\ba
\label{Eq:amplitudes-II}
A^{u+d}_{++,+-}=\tfrac{1}{2}\;\delta_t\; \bar E_T^{u+d} , \;\;\;
&& A^{u-d}_{++,+-}=\phantom{-}\tfrac{1}{2}\;\delta_t\;\widetilde{E}_T^{u-d} ,\\
A^{u+d}_{-+,--}=\tfrac{1}{2}\;\delta_t\; \bar E_T^{u+d} ,\;\;\;
&& A^{u-d}_{-+,--}=-\tfrac{1}{2}\;\delta_t\;\widetilde{E}_T^{u-d} , \\
A^{u+d}_{++,--}=0 \,, \;\;\; \hspace{1.3cm}
&& A^{u-d}_{++,--}=\; H_T^{u-d}+\xi\widetilde{E}_T^{u-d} \,, \\
A^{u+d}_{-+,+-}=0 \, , \;\;\;\hspace{1.3cm}
&& A^{u-d}_{-+,+-}=0 .
\label{Eq:amplitudes-II-end}
\ea
\end{subequations}
Again, we obtain a more transparent representation by considering the linear
combinations
\begin{subequations}
\ba
\label{Eq:amplitudes-III}
A^{u+d}_{++,+-}+A^{u+d}_{-+,--}&=&\delta_t\; \bar E_T^{u+d} ,\;\;\;\\
A^{u-d}_{++,+-}-A^{u-d}_{-+,--}&=&\delta_t\;\widetilde{E}_T^{u-d} , \\
A^{u-d}_{++,--} &=& H_T^{u-d}+\xi\widetilde{E}_T^{u-d} \,, \\
A^{u\pm d}_{-+,+-} &=&0 \, .
\label{Eq:amplitudes-III-end}
\ea
\end{subequations}
One sees that in the chiral-odd case one amplitude vanishes completely:
$A_{-+,+-}=0$ for {\it both} flavor combinations $u + d$ and $u - d$.
As a result, there are only three linearly independent amplitudes that
are non-zero in leading order of the $1/N_c$ expansion. This reflects the results
of Sec.~\ref{Sec-3:large-Nc}, where it was found that only three independent
GPDs are present in the large--$N_c$ nucleon, see Eqs.~(\ref{gpd_sol_odd}) et seq.
Notice that, because $A_{-+,+-}$ is related exclusively to the GPD $\widetilde{H}$
and this amplitude vanishes for any flavor combination,
it is not possible to separate the linear combination of the GPDs
$\bar E_T = E_T + 2\widetilde{H}_T$ in leading order of the $1/N_c$ expansion.
The amplitude $A_{-+,+-}$ is unique in that it corresponds to a {\it double-helicity-flip
transition} with angular momentum exchange $\Delta J = 2$ between the active quark and
the nucleon, i.e., the nucleon and quark helicities are flipped in opposite directions.
It is natural that for this amplitude both flavor combinations vanish in leading
order of the $1/N_c$ expansion. Because of the spin-flavor symmetry implied by the
large--$N_c$ limit the transition with $\Delta J = 2$ should be accompanied by
isospin transfer $\Delta T = 2$, which is impossible with a quark one-body operator.
This could be proved more formally by expanding the GPDs in powers of the transverse
momentum transfer, such that they can be represented by matrix elements of local
operators (containing total derivatives) at zero transverse momentum transfer, and
classifying the resulting local operators according to the spin-flavor symmetry
implied by the large--$N_c$ limit. The collective quantization procedure of
Sec.~\ref{Sec-3:large-Nc} \cite{Pobylitsa:2000tt,Goeke:2001tz} implements this
symmetry through the hedgehog symmetry of the mean field, Eq.~(\ref{Eq-R:hedgehog-symmetry}).
It is interesting to note that the vanishing of
the amplitude $A_{-+,+-}$ in leading order of the $1/N_c$ expansion
can also be derived from large--$N_c$ consistency arguments. The latter are analogous
to the unitarity requirements imposed on meson-baryon scattering amplitudes, from which
one can derive specific relations between meson-baryon coupling
constants \cite{Gervais:1983wq,Dashen:1993as}. In fact, a non-vanishing
amplitude $A^{u \pm d}_{-+,+-} \sim N_c^2$ (for any of the flavor combinations)
would imply that $\widetilde{H}^{u\pm d} \sim N_c^4$. Inserting this scaling behavior
into $A_{++,+-}$ or $A_{-+,--}$ would imply that these amplitudes should scale
$\sim N_c^3$, which contradicts the natural scaling
Eq.~(\ref{Eq:amplitudes-generic-scaling}).\footnote{Although
the partonic helicity amplitudes are not strictly physical,
they enter into the description of cross sections of certain exclusive processes
with quark helicity flip. If some of the amplitudes had a scaling $\sim N_c^3$ it is plausible
that this would violate positivity constraints for the cross sections of some
hypothetical physical scattering processes. Whether such an argument could be
applied to chiral-odd GPDs remains an interesting question for further study.
Positivity constraints for chiral-odd GPDs were discussed in Ref.~\cite{Kirch:2005in}.}
In the discussion here we have inferred the $N_c$--scaling of the helicity amplitudes
from that of the GPDs (or invariant amplitudes). Alternatively one may consider the
large--$N_c$ correlators, Eqs.~(\ref{Eq:XXa}), (\ref{Eq:XXb}) and (\ref{Eq:XXc}),
directly in the particular frame $\bm{\Delta} = (\Delta^1,0,\Delta^3)$ and determine the
helicity amplitudes from there. For reference we present in
Appendix~\ref{app} the expressions for the correlators in that frame.
They show explicitly the degeneracy of the
transverse spin structure of the chiral-odd correlator noted in Sec.~\ref{Sec-3:large-Nc}
[cf.~Eqs.~(\ref{spin_structure_even}) and (\ref{spin_structure_odd})], which is the
cause of the reduced number of independent chiral-odd GPDs viz.\ helicity amplitudes
in leading order of the $1/N_c$--expansion.
\section{Flavor structure from pseudoscalar meson production data}
\label{Sec-5:comparison}
It is interesting to compare our results with preliminary data from the JLab CLAS
exclusive pseudoscalar meson production experiments
\cite{Bedlinskiy:2012be,Kubarovsky:2016yaa}
(cf.\ comments in Sec.~\ref{Sec-1:introduction}).
Analysis of the azimuthal--angle dependent response functions shows that
$|\sigma_{LT}| \ll |\sigma_{TT}|$, which indicates dominance of the twist-3 amplitudes,
involving the chiral-odd GPDs $H_T^q$ and $\bar E_T = E_T + 2\widetilde{H}_T$,
over the twist--2 amplitudes involving the chiral-even GPD $\widetilde E^q$. A preliminary
flavor decomposition was performed assuming dominance of the twist--3 amplitudes and
combining the data on $\pi^0$ and $\eta$ production, in which the $u$ and $d$ quark
GPDs enter with different relative weight. Results show opposite sign of the
exclusive amplitudes $\langle H_T^u \rangle$ and $\langle H_T^d \rangle$, which is
consistent with the leading appearance of the flavor-nonsinglet $H_T^{u - d}$ in the
$1/N_c$ expansion. (Here $\langle\ldots \rangle$ denotes the integral over $x$ of
the GPD, weighted with the meson wave function, hard process amplitude, and Sudakov
form factor \cite{Goloskokov:2011rd}.) The results also suggest same sign of
$\langle \bar E_T^u \rangle$ and $\langle \bar E_T^d \rangle$, which is again
consistent with the leading appearance of the flavor-singlets $E_T^{u + d}$ and
$\widetilde H_T^{u + d}$ in the $1/N_c$ expansion. These findings should be interpreted
with several caveats: (a) the errors in the experimental extraction of
$\langle H_T^q \rangle$ and $\langle \bar E_T^q \rangle$ are substantial;
(b) the $1/N_c$ expansion predicts only the scaling behavior, not the absolute
magnitude of the individual flavor combinations, cf.\ Eq.~(\ref{nc_scaling_gpd_generic}).
It is encouraging that the flavor structure of the amplitudes extracted from the
$\pi^0$ and $\eta$ electroproduction data is consistent with the pattern predicted by
the $1/N_c$ expansion. Our findings further support the idea that pseudoscalar meson
production at $x_B \gtrsim 0.1$ and $Q^2 \sim \textrm{few GeV}^2$ is governed by
the twist-3 mechanism involving the chiral-odd GPDs.
\section{Discussion and outlook}
The large--$N_c$ limit reveals interesting characteristic differences between the
nucleon matrix elements of chiral-even and chiral-odd light-ray operators.
While in the chiral-even case four GPDs (or invariant amplitudes) are non-zero
in the leading order of the $1/N_c$ expansion, in the chiral-odd case only
three independent GPDs appear, due to the absence of spin-orbit interactions
correlating the transverse quark spin with the transverse momentum transfer
to the nucleon. In the equivalent representation of GPDs as nucleon-quark
helicity amplitudes, the same happens due to the vanishing of the double
helicity-flip amplitude in the leading order of $1/N_c$. These conclusions are
model-independent and do not rely on any extraneous assumptions regarding the
internal dynamics giving rise to the partonic structure.
The leading order of the $1/N_c$ expansion predicts the scaling behavior of the leading
flavor combinations in the GPDs $\bar E_T = E_T + 2\widetilde{H}_T, H_T$ and
$\widetilde{E}_T$. Interestingly, the hard exclusive amplitudes in the twist-3 mechanism
involve exactly these three combinations of GPDs, so that the large--$N_c$ predictions
can be confronted with experimental observables.
The $N_c$--scaling relations of the chiral-odd GPDs described here generalize
earlier results for the $N_c$--scaling of the nucleon's transversity
PDFs \cite{Pobylitsa:1996rs,Schweitzer:2001sr}, tensor charges \cite{Kim:1995bq},
and tensor form factors \cite{Ledwig:2010tu}. We note that the lattice QCD
calculations of Ref.~\cite{Gockeler:2006zu} for the tensor form
factors $A_{T10}(t) = H_T(t) \equiv \int dx \, H_T (x, \xi, t)$
show opposite sign for $u$ and $d$ flavors, while those for
$\bar B_{T10}(t) = \bar E_T(t) \equiv \int dx \, \bar E_T (x, \xi, t)$
show same sign for $u$ and $d$ flavors, in agreement with the leading--order
large--$N_c$ relations Eq.~(\ref{Eq:scaling-HT}). The flavor structure of
$\bar E_T(t)$ at large $N_c$ was also studied in the bag model calculation
of Ref.~\cite{Burkardt:2007xm} and agrees with the general result.
In the present study we have considered the leading non-vanishing order of the $1/N_c$
expansion of the chiral-odd nucleon matrix elements. Extension to subleading order
requires principal considerations and technical improvements. At subleading order
the mean-field approximation to the large--$N_c$ correlation functions
Eq.~(\ref{Eq-R:correlator-mean-field}) must include
the effects of the finite velocity of the soliton collective (iso) rotations,
$\Omega \sim N_c^{-1}$. At the same time one must reconsider the choice of nucleon
spinors in the invariant decomposition of the matrix elements,
Eqs.~(\ref{Eq:def-chiral-even-GPD-1}) and (\ref{Eq:def-chiral-even-GPD-2}), as the apparent
size of ``relativistic corrections'' to a given invariant amplitude may depend on the
choice of nucleon spinors. The choice should be guided by the symmetries of the
leading--order approximation and incorporate corrections through a Foldy-Wouthuysen
transformation.
It would be interesting to calculate the chiral-odd GPDs in dynamical models that
consistently implement the $N_c$--scaling, such as the chiral quark--soliton model.
Such calculations would allow one to calculate also the scaling functions in
the large--$N_c$ relations, Eq.~(\ref{nc_scaling_gpd_generic}), and supplement
the scaling studies with dynamical information. $N_c$--scaling can also
be implemented in calculations of peripheral GPDs (at impact parameters $b \sim M_\pi^{-1}$)
in chiral effective field theory \cite{Granados:2016jjl}.
\begin{acknowledgments}
In this study we greatly benefited from discussions with D.~Diakonov,
V.~Petrov, P.~Pobylitsa, and M.~Polyakov during earlier joint work.
This material is based upon work supported by the U.S.~Department of Energy,
Office of Science, Office of Nuclear Physics under contract DE-AC05-06OR23177.
This work was supported by the U.S.~National Science Foundation under
Contract No.~1406298, and by the Deutsche Forschungsgemeinschaft
(grant VO 1049/1).
\end{acknowledgments}
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{"url":"https:\/\/www.physicsforums.com\/threads\/finding-the-third-vector-diff-equation.330164\/","text":"# Finding the third vector diff equation\n\n1. Aug 11, 2009\n\n### khdani\n\nsolve y'=Ay+b system of equation where\n$$\\vec{A}=\\begin{pmatrix} 3 & 1 & 0\\\\ -1 & 1 & 0\\\\ 1& 1 &2 \\end{pmatrix},\\vec{B}=\\begin{pmatrix} e^{2x}\\\\0 \\\\ e^{2x} \\end{pmatrix}$$\n\ni cant see where is my mistake in my method\n1st page\nhttp:\/\/i26.tinypic.com\/24ou8u8.gif\n2nd page\nhttp:\/\/i29.tinypic.com\/1j40ue.gif\n\n2. Aug 11, 2009\n\n### Staff: Mentor\n\nYour problem is that you have three repeated eigenvalues (lambda = 2), but only two eigenvectors. To be able to diagonalize your matrix (which is what the technique you are using requires), you need to find a basis for the three-dimensional eigenspace, but you have found only two of these basis vectors. Taking a linear combination your two vectors doesn't get you anywhere. No matter what vector you end up with by doing this, your three vectors are linearly dependent, and hence don't form a basis for your eigenspace.\n\nI don't recall what, if anything, you can do in this case, as I'm not anywhere close to my reference books.\n\n3. Aug 11, 2009\n\n### HallsofIvy\n\nI haven't looked at your attachment but since Mark44 says you have 2 as a triple eigenvalue, the characteristic equation must be $(\\lambda- 2)^3= 0$. Now, it is true that a matrix always satisfies its own characteristic equation so, writing \"A\" for the matrix, we must have $(A- 2I)^3 v= 0$ for all vectors v. Mark44 says you have two independent eigenvectors, say, $v_1$ and $v_2$ so you have $(A- 2I)v_1= 0$ and $(A- 2I)v_2= 0$. From that it follows that $(A- 2I)^2v_1= 0$ and $(A- 2I)^2v_2= 0$. You want to find a third, independent, vector, w, such that neither $(A- 2I)w= 0$ nor $(A- 2I)^2w= 0$ but $(A- 2I)^2w= 0$. Finding a vector, w, such that $(A- 2I)w= v_1$ or $(A- 2I)w= v_2$ will do that.\n\n4. Aug 12, 2009\n\n### khdani\n\ni cant understand how and why you transformed\n$$(A- 2I)v_1= 0$$\ninto\n$$(A- 2I)^2v_1= 0$$\n\nand i cant practically understand how to find the third vector from this:\"\nsuch that neither $$(A- 2I)w= 0$$ nor $$(A- 2I)^2w= 0$$ but $$(A- 2I)^2w= 0 .$$ Finding a vector, w, such that\n$$(A- 2I)w= v_1$$ or $$(A- 2I)w= v_2 will do that.$$\"\n\nthere is a conflicting conditions in the first sentence.\nand i need to do 3 matrix multiplications some with power 2\nwhich is another multiplication.\nand i need to see that it differs 0.\n\n???","date":"2018-02-20 04:23:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8386468887329102, \"perplexity\": 498.15005321947933}, \"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-2018-09\/segments\/1518891812873.22\/warc\/CC-MAIN-20180220030745-20180220050745-00674.warc.gz\"}"} | null | null |
Эдуард Людвигович Самуйлёнок (; 1907—1939) — белорусский советский писатель.
Биография
Эдуард Самуйлёнок родился а в Санкт-Петербурге. Учился в Петроградской польской гимназии. С 1928 года начал публиковаться. Работал в редакциях газет «Красный Полоцк» (1930—1933) и «Літаратура і мастацтва» (1934—1939).
Первая повесть Самуйлёнка «Теория Калебрун», вышедшая в 1934 году, повествовала об антифашистской борьбе польских рабочих. В 1935 году вышла его вторая антифашистская повесть «Герой нации». В последующих произведениях Самуйлёнок изображал современное советское общество, борьбу за установление Советской власти. К таковым относятся его роман «Будущность», пьесы «Сержант Дроб» и «Гибель волка», рассказы «Последний заказ», «Русалочьи тропы», «Охотничье счастье» и другие.
По доносу писателя Э.Л.Самуйлёнка о том, что белорусский писатель Михась Багун (Михаил Федорович Блошкин) якобы "стрелял в портрет Сталина", был арестован, репрессирован и умер в пересыльном пункте в сибирских лагерях белорусский писатель Михась Багун.
Скончался 12 февраля 1939 года, похоронен на Военном кладбище Минска (фото могилы).
Награды
орден Трудового Красного Знамени (31.01.1939).
Примечания
Литература
Самуйлёнак Эдуард // Беларускія пісьменнікі: Біябібліягр. слоўн. У 6 т. Т. 5. Пестрак — Сяўрук / Б 43 Ін-т літ. імя Я. Купалы АН Беларусі; Беларус. Энцыкл.; Нацыянальны навукова-асветны цэнтр імя Ф. Скарыны. Пад рэд. А. В. Мальдзіса; Рэдкал.: І. Э. Багдановіч і інш. — Мн.: БелЭн, 1995. — С. 251—252. — 480 с ISBN 5-85700-168-4
Самуйлёнак Эдуард / Беларускія пісьменнікі (1917—1990): Даведнік; Склад. А.К. Гардзіцкі. Нав.рэд. А.Л. Верабей. — Мн.: Мастацкая літаратура, 1994.— 653 с.: іл. ISBN 5-340-00709-X.
Белорусские писатели
Похороненные на Военном кладбище (Минск) | {
"redpajama_set_name": "RedPajamaWikipedia"
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_Also by Emily St. John Mandel_
Last Night in Montreal
The Singer's Gun
The Lola Quartet
**THIS IS A BORZOI BOOK**
**PUBLISHED BY ALFRED A. KNOPF**
Copyright © 2014 by Emily St. John Mandel
All rights reserved. Published in the United States by Alfred A. Knopf, a division of Random House LLC, New York, a Penguin Random House company.
www.aaknopf.com
Knopf, Borzoi Books, and the colophon are registered trademarks of Random House LLC.
Library of Congress Cataloging-in-Publication Data
St. John Mandel, Emily, 1979–
Station eleven : a novel / Emily St. John Mandel. —First edition.
pages cm
ISBN 978-0-385-35330-4 (hardback)
ISBN 978-0-385-35331-1 (eBook)
1. Actors—Fiction. 2. Time travel—Fiction. I. Title.
PR9199.4.S727S83 2014 813'.6—dc23 2014003560
_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_.
Front-of-jacket photograph by Michael Turek/Gallery Stock
Jacket design by Abby Weintraub
v3.1
IN MEMORY OF EMILIE JACOBSON
The bright side of the planet moves toward darkness
And the cities are falling asleep, each in its hour,
And for me, now as then, it is too much.
There is too much world.
—Czeslaw Milosz
_The Separate Notebooks_
# Contents
_Cover_
_Other Books by This Author_
_Title Page_
_Copyright_
_Dedication_
_Epigraph_
1. The Theater
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
2. A Midsummer Night's Dream
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
3. I Prefer You With a Crown
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
4. The Starship
Chapter 19
Chapter 20
Chapter 21
Chapter 22
Chapter 23
Chapter 24
Chapter 25
Chapter 26
5. Toronto
Chapter 27
Chapter 28
Chapter 29
Chapter 30
Chapter 31
Chapter 32
Chapter 33
Chapter 34
Chapter 35
Chapter 36
Chapter 37
6. The Airplanes
Chapter 38
Chapter 39
Chapter 40
Chapter 41
7. The Terminal
Chapter 42
Chapter 43
Chapter 44
Chapter 45
Chapter 46
Chapter 47
8. The Prophet
Chapter 48
Chapter 49
Chapter 50
Chapter 51
Chapter 52
9. Station Eleven
Chapter 53
Chapter 54
Chapter 55
_Acknowledgments_
_A Note About the Author_
#
# **1**
**THE KING STOOD** in a pool of blue light, unmoored. This was act 4 of _King Lear_ , a winter night at the Elgin Theatre in Toronto. Earlier in the evening, three little girls had played a clapping game onstage as the audience entered, childhood versions of Lear's daughters, and now they'd returned as hallucinations in the mad scene. The king stumbled and reached for them as they flitted here and there in the shadows. His name was Arthur Leander. He was fifty-one years old and there were flowers in his hair.
"Dost thou know me?" the actor playing Gloucester asked.
"I remember thine eyes well enough," Arthur said, distracted by the child version of Cordelia, and this was when it happened. There was a change in his face, he stumbled, he reached for a column but misjudged the distance and struck it hard with the side of his hand.
"Down from the waist they are Centaurs," he said, and not only was this the wrong line but the delivery was wheezy, his voice barely audible. He cradled his hand to his chest like a broken bird. The actor portraying Edgar was watching him closely. It was still possible at that moment that Arthur was acting, but in the first row of the orchestra section a man was rising from his seat. He'd been training to be a paramedic. The man's girlfriend tugged at his sleeve, hissed, "Jeevan! What are you _doing_?" And Jeevan himself wasn't sure at first, the rows behind him murmuring for him to sit. An usher was moving toward him. Snow began to fall over the stage.
"The wren goes to't," Arthur whispered, and Jeevan, who knew the play very well, realized that the actor had skipped back twelve lines. "The wren..."
"Sir," the usher said, "would you please..."
But Arthur Leander was running out of time. He swayed, his eyes unfocused, and it was obvious to Jeevan that he wasn't Lear anymore. Jeevan pushed the usher aside and made a dash for the steps leading up to the stage, but a second usher was jogging down the aisle, which forced Jeevan to throw himself at the stage without the benefit of stairs. It was higher than he'd thought and he had to kick the first usher, who'd grasped hold of his sleeve. The snow was plastic, Jeevan noted peripherally, little bits of translucent plastic, clinging to his jacket and brushing against his skin. Edgar and Gloucester were distracted by the commotion, neither of them looking at Arthur, who was leaning on a plywood column, staring vacantly. There were shouts from backstage, two shadows approaching quickly, but Jeevan had reached Arthur by now and he caught the actor as he lost consciousness, eased him gently to the floor. The snow was falling fast around them, shimmering in blue-white light. Arthur wasn't breathing. The two shadows—security men—had stopped a few paces away, presumably catching on by now that Jeevan wasn't a deranged fan. The audience was a clamor of voices, flashes from cell-phone cameras, indistinct exclamations in the dark.
"Jesus Christ," Edgar said. "Oh Jesus." He'd dropped the British accent he'd been using earlier and now sounded as if he were from Alabama, which in fact he was. Gloucester had pulled away the gauze bandage that had covered half his face—by this point in the play his character's eyes had been put out—and seemed frozen in place, his mouth opening and closing like a fish.
Arthur's heart wasn't beating. Jeevan began CPR. Someone shouted an order and the curtain dropped, a _whoosh_ of fabric and shadow that removed the audience from the equation and reduced the brilliance of the stage by half. The plastic snow was still falling. The security men had receded. The lights changed, the blues and whites of the snowstorm replaced by a fluorescent glare that seemed yellow by comparison. Jeevan worked silently in the margarine light, glancing sometimes at Arthur's face. Please, he thought, please. Arthur's eyes were closed. There was movement in the curtain, someone batting at the fabric and fumbling for an opening from the other side, and then an older man in a gray suit was kneeling on the other side of Arthur's chest.
"I'm a cardiologist," he said. "Walter Jacobi." His eyes were magnified by his glasses, and his hair had gone wispy on the top of his head.
"Jeevan Chaudhary," Jeevan said. He wasn't sure how long he'd been here. People were moving around him, but everyone seemed distant and indistinct except Arthur, and now this other man who'd joined them. It was like being in the eye of a storm, Jeevan thought, he and Walter and Arthur here together in the calm. Walter touched the actor's forehead once, gently, like a parent soothing a fevered child.
"They've called an ambulance," Walter said.
The fallen curtain lent an unexpected intimacy to the stage. Jeevan was thinking of the time he'd interviewed Arthur in Los Angeles, years ago now, during his brief career as an entertainment journalist. He was thinking of his girlfriend, Laura, wondering if she was waiting in her front-row seat or if she might've gone out to the lobby. He was thinking, Please start breathing again, please. He was thinking about the way the dropped curtain closed off the fourth wall and turned the stage into a room, albeit a room with cavernous space instead of a ceiling, fathoms of catwalks and lights between which a soul might slip undetected. That's a ridiculous thought, Jeevan told himself. Don't be stupid. But now there was a prickling at the back of his neck, a sense of being watched from above.
"Do you want me to take a turn?" Walter asked. Jeevan understood that the cardiologist felt useless, so he nodded and raised his hands from Arthur's chest and Walter picked up the rhythm.
Not quite a room, Jeevan thought now, looking around the stage. It was too transitory, all those doorways and dark spaces between wings, the missing ceiling. It was more like a terminal, he thought, a train station or an airport, everyone passing quickly through. The ambulance had arrived, a pair of medics approaching through the absurdly still-falling snow, and then they were upon the fallen actor like crows, a man and a woman in dark uniforms crowding Jeevan aside, the woman so young she could've passed as a teenager. Jeevan rose and stepped back. The column against which Arthur had collapsed was smooth and polished under his fingertips, wood painted to look like stone.
There were stagehands everywhere, actors, nameless functionaries with clipboards. "For god's sake," Jeevan heard one of them say, "can no one stop the goddamn snow?" Regan and Cordelia were holding hands and crying by the curtain, Edgar sitting cross-legged on the floor nearby with his hand over his mouth. Goneril spoke quietly into her cell phone. Fake eyelashes cast shadows over her eyes.
No one looked at Jeevan, and it occurred to him that his role in this performance was done. The medics didn't seem to be succeeding. He wanted to find Laura. She was probably waiting for him in the lobby, upset. She might—this was a distant consideration, but a consideration nonetheless—find his actions admirable.
Someone finally succeeded in turning off the snow, the last few translucencies drifting down. Jeevan was looking for the easiest way to exit the scene when he heard a whimper, and there was a child whom he'd noticed earlier, a small actress, kneeling on the stage beside the next plywood pillar to his left. Jeevan had seen the play four times but never before with children, and he'd thought it an innovative bit of staging. The girl was seven or eight. She kept wiping her eyes in a motion that left streaks of makeup on both her face and the back of her hand.
"Clear," one of the medics said, and the other moved back while he shocked the body.
"Hello," Jeevan said, to the girl. He knelt before her. Why had no one come to take her away from all this? She was watching the medics. He had no experience with children, although he'd always wanted one or two of his own, and wasn't exactly sure how to speak to them.
"Clear," the medic said, again.
"You don't want to look at that," Jeevan said.
"He's going to die, isn't he?" She was breathing in little sobs.
"I don't know." He wanted to say something reassuring, but he had to concede that it didn't look good. Arthur was motionless on the stage, shocked twice, Walter holding the man's wrist and staring grimly into the distance while he waited for a pulse. "What's your name?"
"Kirsten," the girl said. "I'm Kirsten Raymonde." The stage makeup was disconcerting.
"Kirsten," Jeevan said, "where's your mom?"
"She doesn't pick me up till eleven."
"Call it," a medic said.
"Who takes care of you when you're here, then?"
"Tanya's the wrangler." The girl was still staring at Arthur. Jeevan moved to block her view.
"Nine fourteen p.m.," Walter Jacobi said.
"The wrangler?" Jeevan asked.
"That's what they call her," she said. "She takes care of me while I'm here." A man in a suit had emerged from stage right and was speaking urgently with the medics, who were strapping Arthur to a gurney. One of them shrugged and pulled the blanket down to fit an oxygen mask over Arthur's face. Jeevan realized this charade must be for Arthur's family, so they wouldn't be notified of his death via the evening news. He was moved by the decency of it.
Jeevan stood and extended his hand to the sniffling child. "Come on," he said, "let's find Tanya. She's probably looking for you."
This seemed doubtful. If Tanya were looking for her charge, surely she would have found her by now. He led the little girl into the wings, but the man in the suit had disappeared. The backstage area was chaotic, all sound and movement, shouts to clear the way as Arthur's procession passed, Walter presiding over the gurney. The parade disappeared down the corridor toward the stage doors and the commotion swelled further in its wake, everyone crying or talking on their phones or huddled in small groups telling and retelling the story to one another—"So then I look over and he's falling"—or barking orders or ignoring orders barked by other people.
"All these people," Jeevan said. He didn't like crowds very much. "Do you see Tanya?"
"No. I don't see her anywhere."
"Well," Jeevan said, "maybe we should stay in one place and let her find us." He remembered once having read advice to this effect in a brochure about what to do if you're lost in the woods. There were a few chairs along the back wall, and he sat down in one. From here he could see the unpainted plywood back of the set. A stagehand was sweeping up the snow.
"Is Arthur going to be okay?" Kirsten had climbed up on the chair beside him and was clutching the fabric of her dress in both fists.
"Just now," Jeevan said, "he was doing the thing he loved best in the world." He was basing this on an interview he'd read a month ago, Arthur talking to _The Globe and Mail_ —"I've waited all my life to be old enough to play Lear, and there's nothing I love more than being on stage, the immediacy of it..."—but the words seemed hollow in retrospect. Arthur was primarily a film actor, and who in Hollywood longs to be older?
Kirsten was quiet.
"My point is, if acting was the last thing he ever did," Jeevan said, "then the last thing he ever did was something that made him happy."
"Was that the last thing he ever did?"
"I think it was. I'm so sorry."
The snow was a glimmering pile behind the set now, a little mountain.
"It's the thing I love most in the world too," Kirsten said, after some time had passed.
"What is?"
"Acting," she said, and that was when a young woman with a tear-streaked face emerged from the crowd, arms outstretched. The woman barely glanced at Jeevan as she took Kirsten's hand. Kirsten looked back once over her shoulder and was gone.
Jeevan rose and walked out onto the stage. No one stopped him. He half-expected to see Laura waiting where he'd left her in front-row center—how much time had passed?—but when he found his way through the velvet curtains, the audience was gone, ushers sweeping and picking up dropped programs between rows, a forgotten scarf draped over the back of a seat. He made his way out into the red-carpet extravagance of the lobby, careful not to meet the ushers' eyes, and in the lobby a few remnants of the audience still lingered but Laura wasn't among them. He called her, but she'd turned off her phone for the performance and apparently hadn't turned it back on.
"Laura," he said, to her voice mail, "I'm in the lobby. I don't know where you are."
He stood in the doorway of the ladies' lounge and called out to the attendant, but she replied that the lounge was empty. He circled the lobby once and went to the coat check, where his overcoat was among the last few hanging in the racks. Laura's blue coat was gone.
Snow was falling on Yonge Street. It startled Jeevan when he left the theater, this echo of the plastic translucencies that still clung to his jacket from the stage. A half dozen paparazzi had been spending the evening outside the stage door. Arthur wasn't as famous as he had been, but his pictures still sold, especially now that he was involved in a gladiatorial divorce with a model/actress who'd cheated on him with a director.
Until very recently Jeevan had been a paparazzo himself. He'd hoped to slip past his former colleagues unnoticed, but these were men whose professional skills included an ability to notice people trying to slip past them, and they were upon him all at once.
"You look good," one of them said. "Fancy coat you got there." Jeevan was wearing his peacoat, which wasn't quite warm enough but had the desired effect of making him look less like his former colleagues, who had a tendency toward puffy jackets and jeans. "Where've you been, man?"
"Tending bar," Jeevan said. "Training to be a paramedic."
"EMS? For real? You want to scrape drunks off the sidewalk for a living?"
"I want to do something that matters, if that's what you mean."
"Yeah, okay. You were inside, weren't you? What happened?" A few of them were speaking into their phones. "I'm telling you, the man's dead," one of them was saying, near Jeevan. "Well, sure, the snow gets in the way of the shot, but look at what I just sent you, his face in that one where they're loading him into the ambulance—"
"I don't know what happened," Jeevan said. "They just dropped the curtain in the middle of the fourth act." It was partly that he didn't want to speak with anyone just now, except possibly Laura, and partly that he specifically didn't want to speak with them. "You saw him taken to the ambulance?"
"Wheeled him out here through the stage doors," one of the photographers said. He was smoking a cigarette with quick, nervous motions. "Medics, ambulance, the whole nine yards."
"How'd he look?"
"Honestly? Like a fucking corpse."
"There's botox, and then there's _botox_ ," one of them said.
"Was there a statement?" Jeevan asked.
"Some suit came out and talked to us. Exhaustion and, wait for it, dehydration." Several of them laughed. "Always exhaustion and dehydration with these people, right?"
"You'd think someone would tell them," the botox man said. "If someone would just find it in their hearts to pull one or two of these actors aside, be like, 'Listen, buddy, spread the word: you've got to imbibe liquids and sleep every so often, okay?' "
"I'm afraid I saw even less than you did," Jeevan said, and pretended to receive an important call. He walked up Yonge Street with his phone pressed cold to his ear, stepped into a doorway a half block up to dial Laura's number again. Her phone was still off.
If he called a cab he'd be home in a half hour, but he liked being outside in the clear air, away from other people. The snow was falling faster now. He felt extravagantly, guiltily alive. The unfairness of it, his heart pumping faultlessly while somewhere Arthur lay cold and still. He walked north up Yonge Street with his hands deep in the pockets of his coat and snow stinging his face.
Jeevan lived in Cabbagetown, north and east of the theater. It was the kind of walk he'd have made in his twenties without thinking about it, a few miles of city with red streetcars passing, but he hadn't done the walk in some time. He wasn't sure he'd do it now, but when he turned right on Carlton Street he felt a certain momentum, and this carried him past the first streetcar stop.
He reached Allan Gardens Park, more or less the halfway point, and this was where he found himself blindsided by an unexpected joy. Arthur died, he told himself, you couldn't save him, there's nothing to be happy about. But there was, he was exhilarated, because he'd wondered all his life what his profession should be, and now he was certain, absolutely certain that he wanted to be a paramedic. At moments when other people could only stare, he wanted to be the one to step forward.
He felt an absurd desire to run into the park. It had been rendered foreign by the storm, all snow and shadows, black silhouettes of trees, the underwater shine of a glass greenhouse dome. When he was a boy he'd liked to lie on his back in the yard and watch the snow coming down upon him. Cabbagetown was visible a few blocks ahead, the snow-dimmed lights of Parliament Street. His phone vibrated in his pocket. He stopped to read a text message from Laura: _I had a headache so I went home. Can you pick up milk?_
And here, all momentum left him. He could go no farther. The theater tickets had been intended as a romantic gesture, a let's-do-something-romantic-because-all-we-do-is-fight, and she'd abandoned him there, she'd left him onstage performing CPR on a dead actor and gone home, and now she wanted him to buy milk. Now that he'd stopped walking, Jeevan was cold. His toes were numb. All the magic of the storm had left him, and the happiness he'd felt a moment earlier was fading. The night was dark and filled with movement, snow falling fast and silent, the cars parked on the street swelling into soft outlines of themselves. He was afraid of what he'd say if he went home to Laura. He thought of finding a bar somewhere, but he didn't want to talk to anyone, and when he thought about it, he didn't especially want to be drunk. Just to be alone for a moment, while he decided where to go next. He stepped into the silence of the park.
# **2**
**THERE WERE FEW PEOPLE LEFT** at the Elgin Theatre now. A woman washing costumes in Wardrobe, a man ironing other costumes nearby. An actress—the one who'd played Cordelia—drinking tequila backstage with the assistant stage manager. A young stagehand, mopping the stage and nodding his head in time to the music on his iPod. In a dressing room, the woman whose job it was to watch the child actresses was trying to console the sobbing little girl who'd been onstage when Arthur died.
Six stragglers had drifted to the bar in the lobby, where a bartender mercifully remained. The stage manager was there, also Edgar and Gloucester, a makeup artist, Goneril, and an executive producer who'd been in the audience. At the moment when Jeevan was wading into the snowdrifts in Allan Gardens, the bartender was pouring a whisky for Goneril. The conversation had turned to informing Arthur's next of kin.
"But who _was_ his family?" Goneril was perched on a barstool. Her eyes were red. Without makeup she had a face like marble, the palest and most flawless skin the bartender had ever seen. She seemed much smaller offstage, also much less evil. "Who did he have?"
"He had one son," the makeup artist said. "Tyler."
"How old?"
"Seven or eight?" The makeup artist knew exactly how old Arthur's son was, but didn't want to let on that he read gossip magazines. "I think he maybe lives with his mother in Israel, maybe Jerusalem or Tel Aviv." He knew it was Jerusalem.
"Oh, right, that blond actress," Edgar said. "Elizabeth, wasn't it? Eliza? Something like that."
"Ex-wife number three?" The producer.
"I think the kid's mother was ex-wife number two."
"Poor kid," the producer said. "Did Arthur have anyone he was close with?"
This provoked an uncomfortable silence. Arthur had been carrying on an affair with the woman who looked after the child actresses. Everyone present knew about it, except the producer, but none of them knew if the others knew. Gloucester was the one who said the woman's name.
"Where's Tanya?"
"Who's Tanya?" the producer asked.
"One of the kids hasn't been picked up yet. I think Tanya's in the kids' dressing room." The stage manager had never seen anyone die before. He wanted a cigarette.
"Well," Goneril said, "who else is there? Tanya, the little boy, all those ex-wives, anyone else? Siblings, parents?"
"Who's Tanya?" the producer asked again.
"How many ex-wives are we talking about here?" The bartender was polishing a glass.
"He has a brother," the makeup artist said, "but I can't remember his name. I just remember him saying he had a younger brother."
"I think there were maybe three or four," Goneril said, talking about the ex-wives. "Three?"
"Three." The makeup artist was blinking away tears. "But I don't know if the latest divorce has been finalized."
"So Arthur wasn't married to anyone at the time of... he wasn't married to anyone tonight?" The producer knew this sounded foolish but he didn't know how else to phrase it. Arthur Leander had walked into the theater just a few hours ago, and it was inconceivable that he wouldn't walk in again tomorrow.
"Three divorces," Gloucester said. "Can you imagine?" He was recently divorced himself. He was trying to think of the last thing Arthur had said to him. Something about blocking in the second act? He wished he could remember. "Has anyone been informed? Who do we call?"
"I should call his lawyer," the producer said.
This solution was inarguable, but so depressing that the group drank for several minutes in silence before anyone could bring themselves to speak.
"His _lawyer_ ," the bartender said finally. "Christ, what a thing. You die, and they call your _lawyer_."
"Who else is there?" Goneril asked. "His agent? The seven-year-old? The ex-wives? Tanya?"
"I know, I know," the bartender said. "It's just a hell of a thing." They were silent again. Someone made a comment about the snow coming down hard, and it was, they could see it through the glass doors at the far end of the lobby. From the bar the snow was almost abstract, a film about bad weather on a deserted street.
"Well, here's to Arthur," the bartender said.
In the children's dressing room, Tanya was giving Kirsten a paperweight. "Here," she said, as she placed it into Kirsten's hands, "I'm going to keep trying to reach your parents, and you just try to stop crying and look at this pretty thing...," and Kirsten, teary-eyed and breathless, a few days shy of her eighth birthday, gazed at the object and thought it was the most beautiful, the most wonderful, the strangest thing anyone had ever given her. It was a lump of glass with a storm cloud trapped inside.
In the lobby, the people gathered at the bar clinked their glasses together. "To Arthur," they said. They drank for a few more minutes and then went their separate ways in the storm.
Of all of them there at the bar that night, the bartender was the one who survived the longest. He died three weeks later on the road out of the city.
# **3**
**JEEVAN WANDERED ALONE IN** Allan Gardens. He let the cool light of the greenhouse draw him in like a beacon, snowdrifts halfway to his knees by now, the childhood pleasure of being the first to leave footprints. When he looked in he was soothed by the interior paradise, tropical flowers blurred by fogged glass, palm fronds whose shapes reminded him of a long-ago vacation in Cuba. He would go see his brother, he decided. He wanted very much to tell Frank about the evening, both the awfulness of Arthur's death and the revelation that being a paramedic was the right thing to do with his life. Up until tonight he hadn't been certain. He'd been searching for a profession for so long now. He'd been a bartender, a paparazzo, an entertainment journalist, then a paparazzo again and then once again a bartender, and that was just the past dozen years.
Frank lived in a glass tower on the south edge of the city, overlooking the lake. Jeevan left the park and waited awhile on the sidewalk, jumping up and down for warmth, boarded a streetcar that floated like a ship out of the night and leaned his forehead on the window as it inched along Carlton Street, back the way he had come. The storm was almost a whiteout now, the streetcar moving at a walking pace. His hands ached from compressing Arthur's unwilling heart. The sadness of it, memories of photographing Arthur in Hollywood all those years ago. He was thinking of the little girl, Kirsten Raymonde, bright in her stage makeup; the cardiologist kneeling in his gray suit; the lines of Arthur's face, his last words—"The wren..."—and this made him think of birds, Frank with his binoculars the few times they'd been bird-watching together, Laura's favorite summer dress which was blue with a storm of yellow parrots, Laura, what would become of them? It was still possible that he might go home later, or that at any moment she might call and apologize. He was almost back where he'd started now, the theater closed up and darkened a few blocks to the south. The streetcar stopped just short of Yonge Street, and he saw that a car had spun out in the middle of the tracks, three people pushing while its tires spun in the snow. His phone vibrated again in his pocket, but this time it wasn't Laura.
"Hua," he said. He thought of Hua as his closest friend, though they rarely saw one another. They'd tended bar together for a couple of years just after university while Hua studied for his MCAT and Jeevan tried unsuccessfully to establish himself as a wedding photographer, and then Jeevan had followed another friend to Los Angeles to take pictures of actors while Hua had gone off to medical school. Now Hua worked long hours at Toronto General.
"You been watching the news?" Hua spoke with a peculiar intensity.
"Tonight? No, I had theater tickets. Actually, you wouldn't believe what happened, I—"
"Wait, listen, I need you to tell me honestly, will it send you into one of your panic attacks if I tell you something really, really bad?"
"I haven't had an anxiety attack in three years. My doctor said that whole thing was just a temporary stress-related situation, you know that."
"Okay, you've heard of the Georgia Flu?"
"Sure," Jeevan said, "you know I try to follow the news." A story had broken the day before about an alarming new flu in the Republic of Georgia, conflicting reports about mortality rates and death tolls. Details had been sketchy. The name the news outlets were going with—the Georgia Flu—had struck Jeevan as disarmingly pretty.
"I've got a patient in the ICU," Hua said. "Sixteen-year-old girl, flew in from Moscow last night, presented with flu symptoms at the ER early this morning." Only now did Jeevan hear the exhaustion in Hua's voice. "It's not looking good for her. Well, by midmorning we've got twelve more patients, same symptoms, turns out they were all on the same flight. They all say they started feeling sick on the plane."
"Relatives? Friends of the first patient?"
"No relation whatsoever. They all just boarded the same flight out of Moscow."
"The sixteen-year-old...?"
"I don't think she'll make it. So there's this initial group of patients, the Moscow passengers. Then this afternoon, a new patient comes in. Same symptoms, but this one wasn't on the flight. This one's just an employee at the airport."
"I'm not sure what you're—"
"A gate agent," Hua said. "I'm saying his only contact with the other patients was speaking with one of them about where to board the hotel shuttle."
"Oh," Jeevan said. "That sounds bad." The streetcar was still trapped behind the stuck car. "So I guess you're working late tonight?"
"You remember the SARS epidemic?" Hua asked. "That conversation we had?"
"I remember calling you from Los Angeles when I heard your hospital was quarantined, but I don't remember what I said."
"You were freaked out. I had to talk you down."
"Okay, I guess I do remember that. But look, in my defense, they made it sound pretty—"
"You told me to call you if there was ever a real epidemic."
"I remember."
"We've admitted over two hundred flu patients since this morning," Hua said. "A hundred and sixty in the past three hours. Fifteen of them have died. The ER's full of new cases. We've got beds parked in hallways. Health Canada's about to make an announcement." It wasn't only exhaustion, Jeevan realized. Hua was afraid.
Jeevan pulled the bell cord and made his way to the rear door. He found himself glancing at the other passengers. The young woman with groceries, the man in the business suit playing a game on his cell phone, the elderly couple conversing quietly in Hindi. Had any of them come from the airport? He was aware of all of them breathing around him.
"I know how paranoid you can get," Hua said. "Believe me, you're the last person I'd call if I thought it was nothing, but—" Jeevan banged the palm of his hand on the door's glass pane. Who had touched the door before him? The driver glared over his shoulder, but let him out. Jeevan stepped into the storm and the doors swished shut behind him.
"But you don't think it's nothing." Jeevan was walking past the stuck car, wheels still spinning uselessly in the snow. Yonge Street was just ahead.
"I'm certain it isn't nothing. Listen, I have to get back to work."
"Hua, you've been working with these patients all day?"
"I'm fine, Jeevan, I'll be fine. I have to go. I'll call you later."
Jeevan put the phone in his pocket and walked on through the snow, turned south down Yonge Street toward the lake and the tower where his brother lived. Are you fine, Hua my old friend, or _will_ you be fine? He was deeply unsettled. The lights of the Elgin Theatre just ahead. The interior of the theater was darkened now, the posters still advertising _King Lear_ , with Arthur gazing up into blue light with flowers in his hair and the dead Cordelia limp in his arms. Jeevan stood for a while looking at the posters. He walked on slowly, thinking of Hua's strange call. Yonge Street was all but deserted. He stopped to catch his breath in the doorway of a store that sold suitcases and watched a taxi ease its way slowly down the unplowed street, the storm caught in its headlights, and this vision, snow in lights, transported him back for a moment into the stage-effect storm of the Elgin Theatre. He shook his head to dispel the image of Arthur's blank stare and moved on in an exhausted daze, through the shadows and orange lights under the Gardiner Expressway to Toronto's glassy southern edge.
The snowstorm was wilder down on Queens Quay, wind cutting across the lake. Jeevan had finally reached Frank's building when Hua called again.
"I've been thinking about you," Jeevan said. "Is it really—"
"Listen," Hua said, "you have to get out of the city."
"What? Tonight? What's going on?"
"I don't know, Jeevan. That's the short answer. I don't know what's going on. It's a flu, that much is obvious, but I've never seen anything like it. It is so fast. It just seems to spread so quickly—"
"It's getting worse?"
"The ER's full," Hua said, "which is a problem, because at this point half of the ER staff are too sick to work."
"They got sick from the patients?"
In the lobby of Frank's building, the night doorman flipped through a newspaper, an abstract painting of gray and red lit up on the wall above and behind him, doorman and painting reflected in streaks on the polished floor.
"It's the fastest incubation period I've ever seen. I just saw a patient, she works as an orderly here at the hospital, on duty when the first patients started coming in this morning. She started feeling sick a few hours into her shift, went home early, her boyfriend drove her back in two hours ago and now she's on a ventilator. You get exposed to this, you're sick within hours."
"You think it's going to spread outside the hospital...?" Jeevan was having some difficulty keeping his thoughts straight.
"No, I know it's outside the hospital. It's a full-on epidemic. If it's spreading here, it's spreading through the city, and I've never seen anything like it."
"You're saying I should—"
"I'm saying you should leave now. Or if you can't leave, at least stock up on food and stay in your apartment. I have to make some more calls." He hung up. The night doorman turned a newspaper page. If it had been anyone other than Hua, Jeevan wouldn't have believed it, but he had never known a man with a greater gift for understatement. If Hua said there was an epidemic, then _epidemic_ wasn't a strong enough word. Jeevan was crushed by a sudden certainty that this was it, that this illness Hua was describing was going to be the divide between a _before_ and an _after_ , a line drawn through his life.
It occurred to Jeevan that there might not be much time. He turned away from Frank's building and passed the darkened coffee shop on the pier, the tiny harbor filled with snow-laden pleasure boats, into the grocery store on the harbor's other side. He stood just inside for a beat, blinking in the light. Only one or two other customers drifted through the aisles. He felt that he should call someone, but who? Hua was his only close friend. He'd see his brother in a few minutes. His parents were dead, and he couldn't quite bring himself to talk to Laura. He would wait until he got to Frank's, he decided, he'd check the news when he got there, and then he'd go through the contacts on his phone and call everyone he knew.
There was a small television mounted above the film development counter, showing closed-captioned news. Jeevan drifted toward it. Shots of a broadcaster standing outside Toronto General in the snow, white text scrolling past her head. Toronto General and two other local hospitals had been placed under isolation. Health Canada was confirming an outbreak of the Georgia Flu. They weren't releasing numbers at this time, but there had been fatalities and more information would be forthcoming. There were suggestions that Georgian and Russian officials had been somewhat less than transparent about the severity of the crisis there. Officials requested that everyone please try their best to stay calm.
Jeevan's understanding of disaster preparedness was based entirely on action movies, but on the other hand, he'd seen a lot of action movies. He started with water, filled one of the oversized shopping carts with as many cases and bottles as he could fit. There was a moment of doubt on the way to the cash registers, straining against the weight of the cart—was he overreacting?—but he was committed, he'd decided, too late to turn back. The clerk raised an eyebrow.
"I'm parked just outside," Jeevan said. "I'll bring the cart back." The clerk nodded, tired. She was young, early twenties probably, with dark bangs that she kept pushing out of her eyes. He forced the impossibly heavy cart outside and half-pushed, half-skidded through the snow at the exit. There was a ramp down into a small parklike arrangement of benches and planters. The cart gained speed on the incline, bogged down in deep snow and slid sideways into a planter.
It was eleven twenty. The supermarket closed in forty minutes. He was imagining how long it would take to bring the cart up to Frank's apartment, to unload it, the time required for explanations and tedious reassurances of sanity before he could return to the grocery store for more supplies. Could there be any harm in leaving the cart here for the moment? There was no one on the street. He called Hua on his way back into the store.
"What's happening now?" Jeevan moved quickly through the store while Hua spoke. Another case of water—Jeevan was under the impression that one can never have too much—and then cans and cans of food, all the tuna and beans and soup on the shelf, pasta, anything that looked like it might last a while. The hospital was full of flu patients and the situation was identical at the other hospitals in the city. The ambulance service was overwhelmed. Thirty-seven patients had died now, including every patient who'd been on the Moscow flight and two ER nurses who'd been on duty when the first patients came in. Jeevan was standing by the cash register again, the clerk scanning his cans and packages. Hua said he'd called his wife and told her to take the kids and leave the city tonight, but not by airplane. The part of the evening that had transpired in the Elgin Theatre seemed like possibly a different lifetime. The clerk was moving very slowly. Jeevan passed her a credit card and she scrutinized it as though she hadn't just seen it five or ten minutes ago.
"Take Laura and your brother," Hua said, "and leave the city tonight."
"I can't leave the city tonight, not with my brother. I can't rent a wheelchair van at this hour."
In response there was only a muffled sound. Hua was coughing.
"Are you sick?" Jeevan was pushing the cart toward the door.
"Good night, Jeevan." Hua disconnected and Jeevan was alone in the snow. He felt possessed. The next cart was all toilet paper. The cart after that was more canned goods, also frozen meat and aspirin, garbage bags, bleach, duct tape.
"I work for a charity," he said to the girl behind the cash register, his third or fourth time through, but she wasn't paying much attention to him. She kept glancing up at the small television above the film development counter, ringing his items through on autopilot. Jeevan called Laura on his sixth trip through the store, but his call went to voice mail.
"Laura," he began. "Laura." He thought it better to speak to her directly and it was already almost eleven fifty, there wasn't time for this. Filling another cart with food, moving quickly through this bread-and-flower-scented world, this almost-gone place, thinking of Frank in his twenty-second-floor apartment, high up in the snowstorm with his insomnia and his book project, his day-old _New York Times_ and his Beethoven. Jeevan wanted desperately to reach him. He decided to call Laura later, changed his mind, and called the home line while he was standing by the checkout counter, trying to avoid making eye contact with the clerk.
"Jeevan, where are you?" Laura sounded slightly accusatory. He handed over his credit card.
"Are you watching the news?"
"Should I be?"
"There's a flu epidemic, Laura. It's serious."
"That thing in Russia or wherever? I knew about that."
"It's here now. It's worse than anyone thought. I've just been talking to Hua. You have to leave the city." He glanced up in time to see the look the checkout girl gave him.
" _Have_ to? What? Where are you, Jeevan?" He was signing his name on the slip, struggling with the cart toward the exit, where the order of the store ended and the frenzy of the storm began. It was difficult to steer the cart with one hand. There were already five carts parked haphazardly between benches and planters, dusted now with snow.
"Just turn on the news, Laura."
"You know I don't like to watch the news before bed. Are you having a panic attack?"
"What? No. I'm going to my brother's place to make sure he's okay."
"Why wouldn't he be?"
"You're not even listening. You never listen to me." Jeevan knew this was a petty thing to say in the face of a probable flu pandemic, but couldn't resist. He plowed the cart into the others and dashed back into the store. "I can't believe you left me at the theater," he said. "You just left me at the theater performing CPR on a dead actor."
"Jeevan, tell me where you are."
"I'm in a grocery store." It was eleven fifty-five. This last cart was all grace items: vegetables, fruit, bags of oranges and lemons, tea, coffee, crackers, salt, preserved cakes. "Look, Laura, I don't want to argue. This flu's serious, and it's fast."
"What's fast?"
"This flu, Laura. It's really fast. Hua told me. It's spreading so quickly. I think you should get out of the city." At the last moment, he added a bouquet of daffodils.
"What? Jeevan—"
"You're healthy enough to get on an airplane," he said, "and then you're dead a day later. I'm going to stay with my brother. I think you should pack up now and go to your mother's place before everyone finds out and the roads get clogged up."
"Jeevan, I'm concerned. This sounds paranoid to me. I'm sorry I left you at the theater, I just really had a headache and I—"
"Please turn on the news," he said. "Or go read it online or something."
"Jeevan, please tell me where you are, and I'll—"
"Just do it, Laura, please," he said, and then he hung up, because he was at the checkout counter for the last time now and the moment to talk to Laura had passed. He was trying so hard not to think about Hua.
"We're about to close," the clerk said.
"This is my last time through," he told her. "You must think I'm a nut."
"I've seen worse." He'd scared her, he realized. She'd heard some of his phone calls, and there was the television with its unsettling news.
"Well, just trying to prepare."
"For what?"
"You never know when something disastrous might happen," Jeevan said.
"That?" She gestured toward the television. "It'll be like SARS," she said. "They made such a big deal about it, then it blew over so fast." She didn't sound entirely convinced.
"This isn't like SARS. You should get out of the city." He'd only wanted to be truthful, perhaps to help her in some way, but he saw immediately that he'd made a mistake. She was scared, but also she thought he was insane. She stared flatly at him as she rang up the final few items and a moment later he was outside in the snow again, a goateed young man from the produce department locking the doors behind him. Standing outside with seven enormous shopping carts to transport through the snow to his brother's apartment, soaked in sweat and also freezing, feeling foolish and afraid and a little crazy, Hua at the edge of every thought.
It took the better part of an hour to push the shopping carts one at a time through the snow and across his brother's lobby and then maneuver them into the freight elevator, for unscheduled use of which Jeevan had to bribe the night doorman, and to move them in shifts up to the twenty-second floor. "I'm a survivalist," Jeevan explained.
"We don't get too many of those here," the doorman said.
"That's what makes it such a good place for this," Jeevan said, a little wildly.
"A good place for what?"
"For survivalism."
"I see," the doorman said.
Sixty dollars later Jeevan was alone outside his brother's apartment door, the carts lined up down the corridor. Perhaps, he thought, he should have called ahead from the grocery store. It was one a.m. on a Thursday night, the corridor all closed doors and silence.
"Jeevan," Frank said, when he came to the door. "An unexpected pleasure."
"I..." Jeevan didn't know how to explain himself, so he stepped back and gestured weakly at the carts instead of speaking. Frank maneuvered his wheelchair forward and peered down the hall.
"I see you went shopping," Frank said.
# **4**
**THE ELGIN THEATRE** was empty by then, except for a security guard playing Tetris on his phone in the lower lobby and the executive producer, who'd decided to make the dreaded phone call from an office upstairs. He was surprised when Arthur's lawyer answered the phone, since after all it was one a.m., although of course the lawyer was in Los Angeles. Did entertainment lawyers normally work until ten p.m. Pacific? The producer supposed their corner of the legal profession must be unusually competitive. He relayed the message of Arthur's death and left for the night.
The lawyer, who had been a workaholic all his life and had trained himself to subsist on twenty-minute power naps, spent two hours reviewing Arthur Leander's will and then all of Arthur Leander's emails. He had some questions. There were a number of loose ends. He called Arthur's closest friend, whom he'd once met at an awkward dinner party in Hollywood. In the morning, after a number of increasingly irritable telephone exchanges, Arthur's closest friend began calling Arthur's ex-wives.
# **5**
**MIRANDA WAS ON** the south coast of Malaysia when the call came through. She was an executive at a shipping company and had been sent here for a week to observe conditions on the ground, her boss's words.
"On the ground?" she'd asked.
Leon had smiled. His office was next to hers and had an identical view of Central Park. They'd been working together for a long time by then, over ten years, and together they'd survived two corporate reorganizations and a relocation from Toronto to New York. They weren't friends exactly, at least not in the sense of seeing one another outside of the office, but she thought of Leon as her friendliest ally. "You're right, that was an odd choice of words," he'd said. "Conditions on the sea, then."
That was the year when 12 percent of the world's shipping fleet lay at anchor off the coast of Malaysia, container ships laid dormant by an economic collapse. By day, the massive boats were gray-brown shapes along the edge of the sky, indistinct in the haze. Two to six men to a vessel, a skeleton crew walking the empty rooms and corridors, their footsteps echoing.
"It's lonely," one of them told Miranda when she landed on a deck in a company helicopter, along with an interpreter and a local crew chief. The company had a dozen ships at anchor here.
"They can't just relax out there," Leon had said. "The local crew chief's not bad, but I want them to know the company's on top of the situation. I can't help but picture an armada of floating parties."
But the men were serious and reserved and afraid of pirates. She talked to a man who hadn't been ashore in three months.
That evening on the beach below her hotel, Miranda was seized by a loneliness she couldn't explain. She'd thought she knew everything there was to know about this remnant fleet, but she was unprepared for its beauty. The ships were lit up to prevent collisions in the dark, and when she looked out at them she felt stranded, the blaze of light on the horizon both filled with mystery and impossibly distant, a fairy-tale kingdom. She'd been holding her phone in her hand, expecting a call from a friend, but when the phone began to vibrate she didn't recognize the number that came up on the screen.
"Hello?" Nearby, a couple was conversing in Spanish. She'd been studying the language for the past several months, and understood every third or fourth word.
"Miranda Carroll?" A man's voice, almost familiar and very British.
"Yes, with whom am I speaking?"
"I doubt you'll remember me, but we met briefly some years ago at a party at Cannes. Clark Thompson. Arthur's friend."
"We met again after that," she said. "You came to a dinner party in Los Angeles."
"Yes," he said. "Yes, of course, how could I forget...." Of course he hadn't forgotten, she realized. Clark was being tactful. He cleared his throat. "Miranda," he said, "I'm afraid I'm calling with some rather bad news. Perhaps you should sit down."
She remained standing. "Tell me," she said.
"Miranda, Arthur died of a heart attack last night." The lights over the sea blurred and became a string of overlapping halos. "I'm so sorry. I didn't want you to find out on the news."
"But I just saw him," she heard herself say. "I was in Toronto two weeks ago."
"It's hard to take in." He cleared his throat again. "It's a shock, it's... I've known him since I was eighteen. It seems impossible to me too."
"Please," she said, "what more can you tell me?"
"He actually, well, I hope you won't find it disrespectful if I suggest he may have found this fitting, but he actually died onstage. I'm told it was a massive heart attack in the fourth act of _King Lear_."
"He just collapsed...?"
"I'm told there were two doctors in the audience, they came up onstage when they realized what was happening and tried to save him, but there was nothing anyone could do. He was declared dead on arrival at the hospital."
So this is how it ends, she thought, when the call was over, and she was soothed by the banality of it. You get a phone call in a foreign country, and just like that the man with whom you once thought you'd grow old has departed from this earth.
The conversation in Spanish went on in the nearby darkness. The ships still shone on the horizon; there was still no breeze. It was morning in New York City. She imagined Clark hanging up the receiver in his office in Manhattan. This was during the final month of the era when it was possible to press a series of buttons on a telephone and speak with someone on the far side of the earth.
# **6**
**AN INCOMPLETE LIST:**
No more diving into pools of chlorinated water lit green from below. No more ball games played out under floodlights. No more porch lights with moths fluttering on summer nights. No more trains running under the surface of cities on the dazzling power of the electric third rail. No more cities. No more films, except rarely, except with a generator drowning out half the dialogue, and only then for the first little while until the fuel for the generators ran out, because automobile gas goes stale after two or three years. Aviation gas lasts longer, but it was difficult to come by.
No more screens shining in the half-light as people raise their phones above the crowd to take photographs of concert stages. No more concert stages lit by candy-colored halogens, no more electronica, punk, electric guitars.
No more pharmaceuticals. No more certainty of surviving a scratch on one's hand, a cut on a finger while chopping vegetables for dinner, a dog bite.
No more flight. No more towns glimpsed from the sky through airplane windows, points of glimmering light; no more looking down from thirty thousand feet and imagining the lives lit up by those lights at that moment. No more airplanes, no more requests to put your tray table in its upright and locked position—but no, this wasn't true, there were still airplanes here and there. They stood dormant on runways and in hangars. They collected snow on their wings. In the cold months, they were ideal for food storage. In summer the ones near orchards were filled with trays of fruit that dehydrated in the heat. Teenagers snuck into them to have sex. Rust blossomed and streaked.
No more countries, all borders unmanned.
No more fire departments, no more police. No more road maintenance or garbage pickup. No more spacecraft rising up from Cape Canaveral, from the Baikonur Cosmodrome, from Vandenburg, Plesetsk, Tanegashima, burning paths through the atmosphere into space.
No more Internet. No more social media, no more scrolling through litanies of dreams and nervous hopes and photographs of lunches, cries for help and expressions of contentment and relationship-status updates with heart icons whole or broken, plans to meet up later, pleas, complaints, desires, pictures of babies dressed as bears or peppers for Halloween. No more reading and commenting on the lives of others, and in so doing, feeling slightly less alone in the room. No more avatars.
#
# **7**
**TWENTY YEARS AFTER** the end of air travel, the caravans of the Traveling Symphony moved slowly under a white-hot sky. It was the end of July, and the twenty-five-year-old thermometer affixed to the back of the lead caravan read 106 Fahrenheit, 41 Celsius. They were near Lake Michigan but they couldn't see it from here. Trees pressed in close at the sides of the road and erupted through cracks in the pavement, saplings bending under the caravans and soft leaves brushing the legs of horses and Symphony alike. The heat wave had persisted for a relentless week.
Most of them were on foot to reduce the load on the horses, who had to be rested in the shade more frequently than anyone would have liked. The Symphony didn't know this territory well and wanted to be done with it, but speed wasn't possible in this heat. They walked slowly with weapons in hand, the actors running their lines and the musicians trying to ignore the actors, scouts watching for danger ahead and behind on the road. "It's not a bad test," the director had said, earlier in the day. Gil was seventy-two years old, riding in the back of the second caravan now, his legs not quite what they used to be. "If you can remember your lines in questionable territory, you'll be fine onstage."
"Enter Lear," Kirsten said. Twenty years earlier, in a life she mostly couldn't remember, she had had a small nonspeaking role in a short-lived Toronto production of _King Lear_. Now she walked in sandals whose soles had been cut from an automobile tire, three knives in her belt. She was carrying a paperback version of the play, the stage directions highlighted in yellow. "Mad," she said, continuing, "fantastically dressed with wild flowers."
"But who comes here?" the man learning the part of Edgar said. His name was August, and he had only recently taken to acting. He was the second violin and a secret poet, which is to say no one in the Symphony knew he wrote poetry except Kirsten and the seventh guitar. "The safer sense will ne'er accommodate... will ne'er accommodate... line?"
"His master thus," Kirsten said.
"Cheers. The safer sense will ne'er accommodate his master thus."
The caravans had once been pickup trucks, but now they were pulled by teams of horses on wheels of steel and wood. All of the pieces rendered useless by the end of gasoline had been removed—the engine, the fuel-supply system, all the other components that no one under the age of twenty had ever seen in operation—and a bench had been installed on top of each cab for the drivers. The cabs were stripped of everything that added excess weight but left otherwise intact, with doors that closed and windows of difficult-to-break automobile glass, because when they were traveling through fraught territory it was nice to have somewhere relatively safe to put the children. The main structures of the caravans had been built in the pickup beds, tarps lashed over frames. The tarps on all three caravans were painted gunmetal gray, with THE TRAVELING SYMPHONY lettered in white on both sides.
"No, they cannot touch me for coining," Dieter said over his shoulder. He was learning the part of Lear, although he wasn't really old enough. Dieter walked a little ahead of the other actors, murmuring to his favorite horse. The horse, Bernstein, was missing half his tail, because the first cello had just restrung his bow last week.
"Oh," August said, "thou side-piercing sight!"
"You know what's side-piercing?" the third trumpet muttered. "Listening to _King Lear_ three times in a row in a heat wave."
"You know what's even more side-piercing?" Alexandra was fifteen, the Symphony's youngest actor. They'd found her on the road as a baby. "Traveling for four days between towns at the far edge of the territory."
"What does _side-piercing_ mean?" Olivia asked. She was six years old, the daughter of the tuba and an actress named Lin, and she was riding in the back of the second caravan with Gil and a teddy bear.
"We'll be in St. Deborah by the Water in a couple of hours," Gil said. "There's absolutely nothing to worry about."
There was the flu that exploded like a neutron bomb over the surface of the earth and the shock of the collapse that followed, the first unspeakable years when everyone was traveling, before everyone caught on that there was no place they could walk to where life continued as it had before and settled wherever they could, clustered close together for safety in truck stops and former restaurants and old motels. The Traveling Symphony moved between the settlements of the changed world and had been doing so since five years after the collapse, when the conductor had gathered a few of her friends from their military orchestra, left the air base where they'd been living, and set out into the unknown landscape.
By then most people had settled somewhere, because the gasoline had all gone stale by Year Three and you can't keep walking forever. After six months of traveling from town to town—the word _town_ used loosely; some of these places were four or five families living together in a former truck stop—the conductor's orchestra had run into Gil's company of Shakespearean actors, who had all escaped from Chicago together and then worked on a farm for a few years and had been on the road for three months, and they'd combined their operations.
Twenty years after the collapse they were still in motion, traveling back and forth along the shores of Lakes Huron and Michigan, west as far as Traverse City, east and north over the 49th parallel to Kincardine. They followed the St. Clair River south to the fishing towns of Marine City and Algonac and back again. This territory was for the most part tranquil now. They encountered other travelers only rarely, peddlers mostly, carting miscellanea between towns. The Symphony performed music—classical, jazz, orchestral arrangements of pre-collapse pop songs—and Shakespeare. They'd performed more modern plays sometimes in the first few years, but what was startling, what no one would have anticipated, was that audiences seemed to prefer Shakespeare to their other theatrical offerings.
"People want what was best about the world," Dieter said. He himself found it difficult to live in the present. He'd played in a punk band in college and longed for the sound of an electric guitar.
They were no more than two hours out from St. Deborah by the Water now. The _Lear_ rehearsal had dissipated midway through the fourth act, everyone tired, tempers fraying in the heat. They stopped to rest the horses, and Kirsten, who didn't feel like resting, walked a few paces down the road to throw knives at a tree. She threw from five paces, from ten, from twenty. The satisfying sound of the blades hitting wood. When the Symphony began to move again she climbed up into the back of the second caravan, where Alexandra was resting and mending a costume.
"Okay," Alexandra said, picking up an earlier conversation, "so when you saw the computer screen in Traverse City..."
"What about it?"
In Traverse City, the town they'd recently left, an inventor had rigged an electrical system in an attic. It was modest in scope, a stationary bicycle that when pedaled vigorously could power a laptop, but the inventor had grander aspirations: the point wasn't actually the electrical system, the point was that he was looking for the Internet. A few of the younger Symphony members had felt a little thrill when he'd said this, remembered the stories they'd been told about WiFi and the impossible-to-imagine Cloud, wondered if the Internet might still be out there somehow, invisible pinpricks of light suspended in the air around them.
"Was it the way you remembered?"
"I don't really remember what computer screens looked like," Kirsten admitted. The second caravan had particularly bad shocks, and riding in it always made her feel like her bones were rattling.
"How could you not remember something like that? It was beautiful."
"I was eight."
Alexandra nodded, unsatisfied and obviously thinking that if she'd seen a lit-up computer screen when she was eight, she'd have remembered it.
In Traverse City Kirsten had stared at the _This webpage is not available_ message on the screen. She didn't seriously believe that the inventor would be able to find the Internet, but she was fascinated by electricity. She harbored visions of a lamp with a pink shade on a side table, a nightlight shaped like a puffy half-moon, a chandelier in a dining room, a brilliant stage. The inventor had pedaled frantically to keep the screen from flickering out, explaining something about satellites. Alexandra had been enraptured, the screen a magical thing with no memories attached. August had stared at the screen with a lost expression.
When Kirsten and August broke into abandoned houses—this was a hobby of theirs, tolerated by the conductor because they found useful things sometimes—August always gazed longingly at televisions. As a boy he'd been quiet and a little shy, obsessed with classical music; he'd had no interest in sports and had never been especially adept at getting along with people, which meant long hours home alone after school in interchangeable U.S. Army—base houses while his brothers played baseball and made new friends. One nice thing about television shows was that they were everywhere, identical programming whether your parents had been posted to Maryland or California or Texas. He'd spent an enormous amount of time before the collapse watching television, playing the violin, or sometimes doing both simultaneously, and Kirsten could picture this: August at nine, at ten, at eleven, pale and scrawny with dark hair falling in his eyes and a serious, somewhat fixed expression, playing a child-size violin in a wash of electric-blue light. When they broke into houses now, August searched for issues of _TV Guide_. Mostly obsolete by the time the pandemic hit, but used by a few people right up to the end. He liked to flip through them later at quiet moments. He claimed he remembered all the shows: starships, sitcom living rooms with enormous sofas, police officers sprinting through the streets of New York, courtrooms with stern-faced judges presiding. He looked for books of poetry—even rarer than _TV Guide_ copies—and studied these in the evenings or while he was walking with the Symphony.
When Kirsten was in the houses, she searched for celebrity-gossip magazines, because once, when she was sixteen years old, she'd flipped through a magazine on a dust-blackened side table and found her past:
_Happy Reunion: Arthur Leander Picks Up Son Tyler in LAX_
SCRUFFY ARTHUR GREETS SEVEN-YEAR-OLD TYLER,
WHO LIVES IN JERUSALEM WITH HIS MOTHER,
MODEL/ACTRESS ELIZABETH COLTON.
The photograph: Arthur with a three-day beard, rumpled clothes, a baseball cap, carrying a small boy who beamed up at his father's face while Arthur smiled at the camera. The Georgia Flu would arrive in a year.
"I _knew_ him," she'd told August, breathless. "He gave me the comics I showed you!" And August had nodded and asked to see the comics again.
There were countless things about the pre-collapse world that Kirsten couldn't remember—her street address, her mother's face, the TV shows that August never stopped talking about—but she did remember Arthur Leander, and after that first sighting she went through every magazine she could find in search of him. She collected fragments, stored in a ziplock bag in her backpack. A picture of Arthur alone on a beach, looking pensive and out of shape. A picture of him with his first wife, Miranda, and then later with his second wife, Elizabeth, a malnourished-looking blonde who didn't smile for cameras. Then with their son, who was about the same age as Kirsten, and later still with a third wife who looked very similar to the second one.
"You're like an archaeologist," Charlie said, when Kirsten showed off her findings. Charlie had wanted to be an archaeologist when she was little. She was the second cello and one of Kirsten's closest friends.
Nothing in Kirsten's collection suggested the Arthur Leander she remembered, but what did she actually remember? Arthur was a fleeting impression of kindness and gray hair, a man who'd once pressed two comic books into her hands—"I have a present for you," she was almost certain he'd said—and sometime after this moment, the clearest memory she retained from before the collapse: a stage, a man in a suit talking to her while Arthur lay still on his back with paramedics leaning over him, voices and crying and people gathering, snow somehow falling even though they were indoors, electric light blazing down upon them.
# **8**
**THE COMICS ARTHUR LEANDER** gave her: two issues from a series no one else in the Symphony has ever heard of, _Dr. Eleven_ , Vol. 1, No. 1: _Station Eleven_ and _Dr. Eleven_ , Vol. 1, No. 2: _The Pursuit_. By Year Twenty, Kirsten has them memorized.
Dr. Eleven is a physicist. He lives on a space station, but it's a highly advanced space station that was designed to resemble a small planet. There are deep blue seas and rocky islands linked by bridges, orange and crimson skies with two moons on the horizon. The contrabassoon, who prior to the collapse was in the printing business, told Kirsten that the comics had been produced at great expense, all those bright images, that archival paper, so actually not comics at all in the traditionally mass-produced sense, possibly someone's vanity project. Who would that someone have been? There is no biographical information in either issue, initials in place of the author's name. "By M. C." In the inside cover of the first issue, someone has written "Copy 2 of 10" in pencil. In the second issue, the notation is "Copy 3 of 10." Is it possible that only ten copies of each of these books exist in the world?
Kirsten's taken care of the comics as best she can but they're dog-eared now, worn soft at the edges. The first issue falls open to a two-page spread. Dr. Eleven stands on dark rocks overlooking an indigo sea at twilight. Small boats move between islands, wind turbines spinning on the horizon. He holds his fedora in his hand. A small white animal stands by his side. (Several of the older Symphony members have confirmed that this animal is a dog, but it isn't like any dog Kirsten's ever seen. Its name is Luli. It looks like a cross between a fox and a cloud.) A line of text across the bottom of the frame: _I stood looking over my damaged home and tried to forget the sweetness of life on Earth_.
# **9**
**THE SYMPHONY ARRIVED IN** St. Deborah by the Water in the midafternoon. Before the collapse, it had been one of those places that aren't definitely in one town or another—a gas station and a few chain restaurants strung out along a road with a motel and a Walmart. The town marked the southwestern border of the Symphony's territory, nothing much beyond it so far as anyone knew.
They'd left Charlie and the sixth guitar here two years ago, Charlie pregnant with the sixth guitar's baby, arrangements made for them to stay in the former Wendy's by the gas station so she wouldn't have to give birth on the road. Now the Symphony came upon a sentry posted at the north end of town, a boy of about fifteen sitting under a rainbow beach umbrella by the roadside. "I remember you," he said when they reached him. "You can set up camp at the Walmart."
The Symphony moved through St. Deborah by the Water at a deliberately slow pace, the first trumpet playing a solo from a Vivaldi concerto, but what was strange was that the music drew almost no onlookers as they passed. In Traverse City the crowd following them down the street upon their arrival had swelled to a hundred, but here only four or five people came to their doors or emerged from around the sides of buildings to stare, unsmiling, and none of them were Charlie or the sixth guitar.
The Walmart was at the south end of town, the parking lot wavering in the heat. The Symphony parked the caravans near the broken doors, set about the familiar rituals of taking care of the horses and arguing about which play to perform or if it should just be music tonight, and still neither Charlie nor the sixth guitar appeared.
"They're probably just off working somewhere," August said, but it seemed to Kirsten that the town was too empty. Mirages were forming in the distance, phantom pools on the road. A man pushing a wheelbarrow seemed to walk on water. A woman carried a bundle of laundry between buildings. Kirsten saw no one else.
"I'd suggest _Lear_ for tonight," said Sayid, an actor, "but I don't know that we want to make this place _more_ depressing."
"For once I agree with you," Kirsten said. The other actors were arguing. _King Lear_ , because they'd been rehearsing it all week—August looked nervous—or _Hamlet_ , because they hadn't performed it in a month?
_"A Midsummer Night's Dream,"_ Gil said, breaking an impasse. "I believe the evening calls for fairies."
"Is all our company here?"
"You were best to call them generally, man by man, according to the scrip." Jackson had been playing Bottom for a decade and was the only one who'd managed to go off-book today. Even Kirsten had to look at the text twice. She hadn't played Titania in weeks.
"This place seems quiet, doesn't it?" Dieter was standing with Kirsten just outside the action of the rehearsal.
"It's creepy. You remember the last time we were here? Ten or fifteen kids followed us through town when we arrived and watched the rehearsal."
"You're up," Dieter said.
"I'm not misremembering, am I?" Kirsten was stepping into the play. "They crowded all around us."
Dieter frowned, looking down the empty road.
"... But room, fairy!" said Alexandra, who was playing Puck, "here comes Oberon."
"And here my mistress," said Lin, who was playing the fairy. "Would that he were gone!"
"Ill met by moonlight, proud Titania." Sayid carried himself with a regality that Kirsten had fallen in love with once. Here in this parking lot in a pressing heat wave, patches of sweat under the arms of his T-shirt, knee-torn jeans, he was perfectly credible as a king.
"What, jealous Oberon?" Kirsten stepped forward as steadily as possible. They'd been a couple for two years, until four months earlier, when she'd slept with a traveling peddler more or less out of boredom, and now she had trouble meeting his eyes when they did _A Midsummer Night's Dream_ together. "Fairies, skip hence. I have forsworn his bed and company." Audible snickering from the sidelines at this. Sayid smirked.
"Christ," she heard Dieter mutter, behind her, "is this really necessary?"
" _Tarry_ , rash wanton," Sayid said, drawing out the words. "Am I not thy lord?"
# **10**
**THE PROBLEM WITH THE** Traveling Symphony was the same problem suffered by every group of people everywhere since before the collapse, undoubtedly since well before the beginning of recorded history. Start, for example, with the third cello: he had been waging a war of attrition with Dieter for some months following a careless remark Dieter had made about the perils of practicing an instrument in dangerous territory, the way the notes can carry for a mile on a clear day. Dieter hadn't noticed. Dieter did, however, harbor considerable resentment toward the second horn, because of something she'd once said about his acting. This resentment didn't go unnoticed—the second horn thought he was being petty—but when the second horn was thinking of people she didn't like very much, she ranked him well below the seventh guitar—there weren't actually seven guitars in the Symphony, but the guitarists had a tradition of not changing their numbers when another guitarist died or left, so that currently the Symphony roster included guitars four, seven, and eight, with the location of the sixth presently in question, because they were done rehearsing _A Midsummer Night's Dream_ in the Walmart parking lot, they were hanging the _Midsummer Night's Dream_ backdrop between the caravans, they'd been in St. Deborah by the Water for hours now and why hadn't he come to them? Anyway, the seventh guitar, whose eyesight was so bad that he couldn't do most of the routine tasks that had to be done, the repairs and hunting and such, which would have been fine if he'd found some other way to help out but he hadn't, he was essentially dead weight as far as the second horn was concerned. The seventh guitar was a nervous person, because he was nearly blind. He'd been able to see reasonably well with an extremely thick pair of glasses, but he'd lost these six years ago and since then he'd lived in a confusing landscape distilled to pure color according to season—summer mostly green, winter mostly gray and white—in which blurred figures swam into view and then receded before he could figure out who they were. He couldn't tell if his headaches were caused by straining to see or by his anxiety at never being able to see what was coming, but he did know the situation wasn't helped by the first flute, who had a habit of sighing loudly whenever the seventh guitar had to stop rehearsal to ask for clarification on the score that he couldn't see.
But the first flute was less irritated by the seventh guitar than she was by the second violin, August, who was forever missing rehearsals, always off somewhere breaking into another house with Kirsten and, until recently, Charlie, like he thought the Symphony was a scavenging outfit who played music on the side. ("If he wanted to join a scavenging outfit," she'd said to the fourth guitar, "why didn't he just join a scavenging outfit?" "You know what the violins are like," the fourth guitar had said.) August was annoyed by the third violin, who liked to make insinuating remarks about August and Kirsten even though they'd only ever been close friends and had in fact made a secret pact to this effect—friends forever and nothing else—sworn while drinking with locals one night behind the ruins of a bus depot in some town on the south end of Lake Huron—and the third violin resented the first violin following a long-ago argument about who had used the last of a batch of rosin, while the first violin was chilly to Sayid, because Sayid had rejected her overtures in favor of Kirsten, who expended considerable energy in trying to ignore the viola's habit of dropping random French words into sentences as though anyone else in the entire goddamned Symphony spoke French, while the viola harbored secret resentments against someone else, and so on and so forth, etc., and this collection of petty jealousies, neuroses, undiagnosed PTSD cases, and simmering resentments lived together, traveled together, rehearsed together, performed together 365 days of the year, permanent company, permanent tour. But what made it bearable were the friendships, of course, the camaraderie and the music and the Shakespeare, the moments of transcendent beauty and joy when it didn't matter who'd used the last of the rosin on their bow or who anyone had slept with, although someone—probably Sayid—had written "Sartre: Hell is other people" in pen inside one of the caravans, and someone else had scratched out "other people" and substituted "flutes."
People left the Symphony sometimes, but the ones who stayed understood something that was rarely spoken aloud. Civilization in Year Twenty was an archipelago of small towns. These towns had fought off ferals, buried their neighbors, lived and died and suffered together in the blood-drenched years just after the collapse, survived against unspeakable odds and then only by holding together into the calm, and these places didn't go out of their way to welcome outsiders.
"Small towns weren't even easy _before_ ," August said once at three in the morning, the one time Kirsten remembered talking about this with anyone, in the cold of a spring night near the town of New Phoenix. She was fifteen at the time, which made August eighteen, and she'd only been with the Symphony for a year. In those days she had considerable trouble sleeping and often sat up with the night watch. August remembered his pre-pandemic life as an endless sequence of kids who'd looked him over and uttered variations on "You're not from around here, are you?" in various accents, these encounters interspersed with moving trucks. If it was hard to break into new places _then_ , in that ludicrously easy world where food was on shelves in supermarkets and travel was as easy as taking a seat in a gasoline-powered machine and water came out of taps, it was several orders of magnitude more difficult now. The Symphony was insufferable, hell was other flutes or other people or whoever had used the last of the rosin or whoever missed the most rehearsals, but the truth was that the Symphony was their only home.
At the end of the _Midsummer Night's Dream_ rehearsal, Kirsten stood by the caravans with the palms of her hands pressed hard to her forehead, trying to will away a headache.
"You okay?" August asked.
"Hell is other actors," Kirsten said. "Also ex-boyfriends."
"Stick to musicians. I think we're generally saner."
"I'm going to take a walk and see if I can find Charlie."
"I'd come with you, but I'm on dinner duty."
"I don't mind going alone," she said.
A late-afternoon torpor had fallen over the town, the light thickening and shadows extending over the road. The road was disintegrating here as everywhere, deep fissures and potholes holding gardens of weeds. There were wildflowers alongside the vegetable patches at the edge of the pavement, Queen Anne's lace whispering against Kirsten's outstretched hand. She passed by the Motor Lodge where the oldest families in town lived, laundry flapping in the breeze, doors open on motel rooms, a little boy playing with a toy car between the tomato plants in the vegetable garden.
The pleasure of being alone for once, away from the clamor of the Symphony. It was possible to look up at the McDonald's sign and fleetingly imagine, by keeping her gaze directed upward so that there was only the sign and the sky, that this was still the former world and she could stop in for a burger. The last time she'd been here, the IHOP had housed three or four families; she was surprised to see that it had been boarded up, a plank hammered across the door with an inscrutable symbol spray-painted in silver—something like a lowercase _t_ with an extra line toward the bottom. Two years ago she'd been followed around town by a flock of children, but now she saw only two, the boy with the toy car and a girl of eleven or so who watched her from a doorway. A man with a gun and reflective sunglasses was standing guard at the gas station, whose windows were blocked by curtains that had once been flowered sheets. A young and very pregnant woman sunbathed on a lounge chair by the gas pumps, her eyes closed. The presence of an armed guard in the middle of town suggested that the place was unsafe—had they recently been raided?—but surely not as unsafe as all that, if a pregnant woman was sunbathing in the open. It didn't quite make sense. The McDonald's had housed two families, but where had they gone? Now a board had been nailed across the door, spray-painted with that same odd symbol.
The Wendy's was a low square building with the look of having been slapped together from a kit in an architecturally careless era, but it had a beautiful front door. It was a replacement, solid wood, and someone had taken the trouble to carve a row of flowers alongside the carved handle. Kirsten ran her fingertips over the wooden petals before she knocked.
How many times had she imagined this moment, over two years of traveling apart from her friend? Knocking on the flowered door, Charlie answering with a baby in her arms, tears and laughter, the sixth guitar grinning beside her. I have missed you so much. But the woman who answered the door was unfamiliar.
"Good afternoon," Kirsten said. "I'm looking for Charlie."
"I'm sorry, who?" The woman's tone wasn't unfriendly, but there was no recognition in her eyes. She was about Kirsten's age or a little younger, and it seemed to Kirsten that she wasn't well. She was very pale and too thin, black circles under her eyes.
"Charlie. Charlotte Harrison. She was here about two years ago."
"Here in the Wendy's?"
"Yes." Oh Charlie, where are you? "She's a friend of mine, a cellist. She was here with her husband, the sixth—her husband, Jeremy. She was pregnant."
"I've only been here a year, but maybe someone else here would know. Would you like to come in?"
Kirsten stepped into an airless corridor. It opened into a common room at the back of the building, where once there'd been an industrial kitchen. She saw a cornfield through the open back door, stalks swaying for a dozen yards or so before the wall of the forest. An older woman sat in a chair by the doorway, knitting. Kirsten recognized the local midwife.
"Maria," she said.
Maria was backlit by the open door behind her. It was impossible to see the expression on her face when she looked up.
"You're with the Symphony," she said. "I remember you."
"I'm looking for Charlie and Jeremy."
"I'm sorry, they left town."
"Left? Why would they leave? Where did they go?"
The midwife glanced at the woman who'd shown Kirsten in. The woman looked at the floor. Neither spoke.
"At least tell me when," Kirsten said. "How long have they been gone?"
"A little more than a year."
"Did she have her baby?"
"A little girl, Annabel. Perfectly healthy."
"And is that all you'll tell me?" Kirsten was entertaining a pleasant fantasy of holding a knife to the midwife's throat.
"Alissa," Maria said, to the other woman, "you look so pale, darling. Why don't you go lie down?"
Alissa disappeared through a curtained doorway into another room. The midwife stood quickly. "Your friend rejected the prophet's advances," she whispered, close to Kirsten's ear. "They had to leave town. Stop asking questions and tell your people to leave here as quickly as possible." She settled back into her chair and picked up her knitting. "Thank you for stopping by," she said, in a voice loud enough to be heard in the next room. "Is the Symphony performing tonight?"
" _A Midsummer Night's Dream_. With orchestral accompaniment." Kirsten was having trouble keeping her voice steady. That after two years the Symphony might arrive in St. Deborah by the Water to find that Charlie and Jeremy had already left was a possibility that hadn't occurred to her. "This town seems different from when we were here last," she said.
"Oh," the midwife said brightly, "it is! It's completely different."
Kirsten stepped outside and the door closed behind her. The girl she'd noticed in a doorway earlier had followed her here and was standing across the road, watching. Kirsten nodded to her. The girl nodded back. A serious child, unkempt in a way that suggested neglect, her hair tangled, her T-shirt collar torn. Kirsten wanted to call out to her, to ask if she knew where Charlie and Jeremy had gone, but something in the girl's stare unnerved her. Had someone told the girl to watch her? Kirsten turned away to continue down the road, wandering with studied casualness and trying to convey the impression of being interested only in the late-afternoon light, the wildflowers, the dragonflies gliding on currents of air. When she glanced over her shoulder, the girl was trailing behind her at some distance.
Two years ago she'd done this walk with Charlie, both of them delaying the inevitable in the final hours before the Symphony left. "These two years will go quickly," Charlie had said, and they had gone quickly, when Kirsten considered it. Up to Kincardine, back down the coastline and down the St. Clair River, winter in one of the St. Clair fishing towns. Performances of _Hamlet_ and _Lear_ in the town hall, which had previously been a high-school gymnasium, _The Winter's Tale, Romeo and Juliet_ , the musicians performing almost every night, then _A Midsummer Night's Dream_ when the weather grew warmer. An illness that passed through the Symphony in spring, a high fever and vomiting, half the Symphony got sick but everyone recovered except the third guitar—a grave by the roadside outside of New Phoenix—and we continued onward, Charlie, like always, all those months, and always I thought of you here in this town.
There was someone on the road ahead, walking quickly to meet her. The sun was skimming the tops of the trees now, the road in shadow, and it was a moment before she recognized Dieter.
"We should be getting back," she said.
"I have to show you something first. You'll want to see this."
"What is it?" She didn't like his tone. Something had rattled him. She told him what the midwife had said while they walked.
He frowned. "She said they'd left? Are you sure that's what she said?"
"Of course I'm sure. Why?" At the northern edge of town a new building had been under way at the very end, the foundation poured just before the Georgia Flu arrived. It was a concrete pad, bristling with metal bars, overgrown now with vines. Dieter stepped off the road and led her down a path behind it.
All towns have graveyards, and St. Deborah by the Water's had grown considerably since she'd wandered here two years ago with Charlie. There were perhaps three hundred graves, spaced in neat rows between the abandoned foundation and the forest. In the newest section, freshly painted markers blazed white in the grass. She saw the names at some distance.
"No," she said, "oh no, please..."
"It's not them," Dieter said. "I have to show you this, but it isn't them."
Three markers in a row in the afternoon shadows, names painted neatly in black: _Charlie Harrison, Jeremy Leung, Annabel (infant)_. All three with the same date: _July 20, Year 19_.
"It's not them," Dieter said again. "Look at the ground. No one's buried under those markers."
The horror of seeing their names there. She was weakened by the sight. But he was right, she realized. The earliest markers at the far end of the graveyard were unmistakably planted above graves, the dirt mounded. This pattern continued through to a cluster of thirty graves from a year and a half ago, the dates of death within a two-week span. An illness obviously, something that spread fast and vicious in the winter cold. But after this, the irregularities began: about half of the graves following the winter illness looked like graves, while the others, Charlie's and Jeremy's and their baby's among them, were markers driven into perfectly flat and undisturbed earth.
"It doesn't make sense," she said.
"We could ask your shadow."
The girl who'd followed Kirsten through town was standing at the edge of the graveyard by the foundation, watching them.
"You," Kirsten said.
The girl stepped back.
"Did you know Charlie and Jeremy?"
The girl glanced over her shoulder. When she returned her gaze to Kirsten and Dieter, her nod was barely perceptible.
"Are they...?" Kirsten gestured toward the graves.
"They left," the girl said very quietly.
"It speaks!" Dieter said.
"When did they—" But the girl's nerve failed her before Kirsten could finish the question. She darted out of sight behind the foundation, and Kirsten heard her footsteps on the road. Kirsten was left alone with Dieter, with the graves and the forest. They looked at one another, but there was nothing to say.
A short time after they returned to the Walmart, the tuba returned to camp with his own report. He'd tracked down an acquaintance who lived in the motel. There'd been an epidemic, the man had told him. Thirty people had died incandescent with fever, including the mayor. After this, a change in management, but the tuba's acquaintance had declined to elaborate on what he meant by this. He did say that twenty families had left since then, including Charlie and the sixth guitar and their baby. He said no one knew where they'd gone, and he'd told the tuba it was best not to ask.
"A change in management," the conductor said. "How corporate of them." They'd discussed the grave markers at some length. What did the graves mark, if not deaths? Did the markers await a future event?
"I told you," Kirsten said, "the midwife said there was a prophet."
"Yeah, that's fantastic." Sayid was unpacking a crate of candles without looking at anyone. The sixth guitar was one of his closest friends. "Just what every town needs."
"Someone must know where they went," the conductor said. "They must've told someone. Doesn't anyone else have friends here?"
"I knew a guy who lived in the IHOP," the third cello said, "but I checked earlier and it was boarded up, and then someone in the Motor Lodge said he'd left town last year. No one would tell me where Charlie and Jeremy went."
"No one tells you anything here." Kirsten wanted to cry but instead she stared at the pavement, pushing a pebble back and forth with her foot.
"How could we have left them here?" Lin shook out her fairy costume, a silver cocktail dress that shimmered like the scales of a fish, and a cloud of dust rose into the air. "Graves," she said. "I can't even begin to—"
"Not graves," Dieter said. "Grave _markers_."
"Towns change." Gil leaned on his cane by the third caravan, gazing at the buildings and gardens of St. Deborah by the Water, at the haze of wildflowers along the edges of the road. The McDonald's sign caught the last of the sunlight. "We couldn't have predicted."
"There could be an explanation," the third cello said, doubtful. "They could have left and, I don't know, someone thought they were dead?"
"There's a _prophet_ ," Kirsten said. "There are grave markers with their names on them. The midwife said I should stop asking questions and that we should leave quickly. Did I mention that?"
"Did we not acknowledge you loudly enough the first six times you mentioned it?" Sayid asked.
The conductor sighed. "We can't leave till we know more," she said. "Let's get on with the evening, and we'll make inquiries after the show."
The caravans were parked end to end, the _Midsummer Night's Dream_ backdrop—sewn-together sheets, grimy now from years of travel, painted with a forest scene—hung on them. Alexandra and Olivia had gathered branches and flowers to complete the effect, and a hundred candles marked the edges of the stage.
"I was talking to our fearless leader," August said to Kirsten later, between tuning his instrument and going to join the rest of the string section, "and she thinks Charlie and the sixth guitar must have gone south down the lakeshore."
"Why south?"
"Because west's the water, and they didn't go north. We would've run into them on the road."
The sun was setting, the citizens of St. Deborah by the Water gathering for the performance. Far fewer of them now than there had been, no more than thirty in two grim-faced rows on the grit of the former parking lot. A wolfish gray dog lay on its side at the end of the front row, its tongue lolling. The girl who'd followed Kirsten was nowhere in sight.
"Is there anything to the south, though?"
August shrugged. "It's a lot of coastline," he said. "There's got to be something between here and Chicago, wouldn't you think?"
"They could've gone inland."
"It's possible, but they know we never go into the interior. They'd only go inland if they didn't want to see us again, and why would they..." He shook his head. None of it made sense.
"They had a girl," Kirsten said. "Annabel."
"That was Charlie's sister's name."
"Places," the conductor said, and August left to join the strings.
# **11**
**WHAT WAS LOST IN THE COLLAPSE:** almost everything, almost everyone, but there is still such beauty. Twilight in the altered world, a performance of _A Midsummer Night's Dream_ in a parking lot in the mysteriously named town of St. Deborah by the Water, Lake Michigan shining a half mile away. Kirsten as Titania, a crown of flowers on her close-cropped hair, the jagged scar on her cheekbone half-erased by candlelight. The audience is silent. Sayid, circling her in a tuxedo that Kirsten found in a dead man's closet near the town of East Jordan: "Tarry, rash wanton. Am I not thy lord?"
"Then I must be thy lady." Lines of a play written in 1594, the year London's theaters reopened after two seasons of plague. Or written possibly a year later, in 1595, a year before the death of Shakespeare's only son. Some centuries later on a distant continent, Kirsten moves across the stage in a cloud of painted fabric, half in rage, half in love. She wears a wedding dress that she scavenged from a house near New Petoskey, the chiffon and silk streaked with shades of blue from a child's watercolor kit.
"But with thy brawls," she continues, "thou hast disturbed our sport." She never feels more alive than at these moments. When onstage she fears nothing. "Therefore the winds, piping to us in vain, as in revenge, have sucked up from the sea contagious fogs...."
_Pestilential_ , a note in the text explains, next to the word _contagious_ , in Kirsten's favorite of the three versions of the text that the Symphony carries. Shakespeare was the third born to his parents, but the first to survive infancy. Four of his siblings died young. His son, Hamnet, died at eleven and left behind a twin. Plague closed the theaters again and again, death flickering over the landscape. And now in a twilight once more lit by candles, the age of electricity having come and gone, Titania turns to face her fairy king. "Therefore the moon, the governess of floods, pale in her anger, washes all the air, that rheumatic diseases do abound."
Oberon watches her with his entourage of fairies. Titania speaks as if to herself now, Oberon forgotten. Her voice carries high and clear over the silent audience, over the string section waiting for their cue on stage left. "And through this distemperature, we see the seasons alter."
All three caravans of the Traveling Symphony are labeled as such, THE TRAVELING SYMPHONY lettered in white on both sides, but the lead caravan carries an additional line of text: _Because survival is insufficient_.
# **12**
**THE AUDIENCE ROSE** for a standing ovation. Kirsten stood in the state of suspension that always came over her at the end of performances, a sense of having flown very high and landed incompletely, her soul pulling upward out of her chest. A man in the front row had tears in his eyes. In the back row, another man whom she'd noticed earlier—he alone had sat on a chair, the chair carried up from the gas station by a woman—stepped forward and raised his hands over his head as he passed through the front row. The applause faded.
"My people," he said. "Please, be seated." He was tall, in his late twenties or early thirties, with blond hair to his shoulders and a beard. He stepped over the half circle of candles to stand among the actors. The dog who'd been lying by the front row sat up at attention.
"What a delight," he said. "What a marvelous spectacle." There was something almost familiar in his face, but Kirsten couldn't place him. Sayid was frowning.
"Thank you," the man said, to the actors and musicians. "Let us all thank the Traveling Symphony for this beautiful respite from our daily cares." He was smiling at each of them in turn. The audience applauded again, on cue, but quieter now. "We are blessed," he said, and as he raised his hands the applause stopped at once. The prophet. "We are blessed to have these musicians and actors in our midst today." Something in his tone made Kirsten want to run, a suggestion of a trapdoor waiting under every word. "We have been blessed," he said, "in so many ways, have we not? We are blessed most of all in being alive today. We must ask ourselves, 'Why? Why were we spared?' " He was silent for a moment, scanning the Symphony and the assembled crowd, but no one responded. "I submit," the prophet said, "that everything that has ever happened on this earth has happened for a reason."
The conductor was standing by the string section, her hands clasped behind her back. She was very still.
"My people," the prophet said, "earlier in the day I was contemplating the flu, the great pandemic, and let me ask you this. Have you considered the perfection of the virus?" A ripple of murmurs and gasps moved through the audience, but the prophet raised a hand and they fell silent. "Consider," he said, "those of you who remember the world before the Georgia Flu, consider the iterations of the illness that preceded it, those trifling outbreaks against which we were immunized as children, the flus of the past. There was the outbreak of 1918, my people, the timing obvious, divine punishment for the waste and slaughter of the First World War. But then, in the decades that followed? The flus came every season, but these were weak, inefficient viruses that struck down only the very old, the very young, and the very sick. And then came a virus like an avenging angel, unsurvivable, a microbe that reduced the population of the fallen world by, what? There were no more statisticians by then, my angels, but shall we say ninety-nine point ninety-nine percent? One person remaining out of every two hundred fifty, three hundred? I submit, my beloved people, that such a perfect agent of death could only be divine. For we have read of such a cleansing of the earth, have we not?"
Kirsten met Dieter's gaze across the stage. He'd played Theseus. He fiddled nervously with the cufflinks on his shirt.
"The flu," the prophet said, "the great cleansing that we suffered twenty years ago, that flu was our flood. The light we carry within us is the ark that carried Noah and his people over the face of the terrible waters, and I submit that we were saved"—his voice was rising—"not only to bring the light, to spread the light, but to _be_ the light. We were saved because we _are_ the light. We are the pure." Sweat ran down Kirsten's back under the silk of the dress. The dress, she noted absently, didn't smell very good. When had she last washed it? The prophet was still talking, about faith and light and destiny, divine plans revealed to him in dreams, the preparations they must make for the end of the world—"For it has been revealed to me that the plague of twenty years ago was just the beginning, my angels, only an initial culling of the impure, that last year's pestilence was but further preview and there will be more cullings, far more cullings to come"—and when his sermon was over he went to the conductor and spoke softly to her. She said something in response, and he stepped back with a laugh.
"I wouldn't know," he said. "People come and go."
"Do they?" the conductor said. "Are there other towns nearby, perhaps down the coast, where people typically travel?"
"There's no town nearby," he said. "But everyone"—he looked over his shoulder at the silent crowd, smiling at them, and spoke loudly enough for everyone to hear him—"everyone here, of course, is free to go as they please."
"Naturally," the conductor said. "I wouldn't have expected otherwise. It's just that we wouldn't have expected them to set off on their own, given that they knew we were coming back for them."
The prophet nodded. Kirsten edged closer to eavesdrop more effectively. The other actors were receding quietly from the stage. "My people and I," he said, "when we speak of the light, we speak of order. This is a place of order. People with chaos in their hearts cannot abide here."
"If you'll forgive me for prying, though, I have to ask about the markers in the graveyard."
"It's not an unreasonable question," the prophet said. "You've been on the road for some time, have you not?"
"Yes."
"Your Symphony was on the road in the beginning?"
"Close to it," the conductor said. "Year Five."
"And you?" The prophet turned suddenly to Kirsten.
"I walked for all of Year One." Although she felt dishonest claiming this, given that she had no memory whatsoever of that first year.
"If you've been on the road for that long," the prophet said, "if you've wandered all your life, as I have, through the terrible chaos, if you remember, as I do, everything you've ever seen, then you know there's more than one way to die."
"Oh, I've seen multiple ways," the conductor said, and Kirsten saw that she was remaining calm with some difficulty, "actually everything from drowning to decapitation to fever, but none of those ways would account for—"
"You misunderstand me," the prophet said. "I'm not speaking of the tedious variations on physical death. There's the death of the body, and there's the death of the soul. I saw my mother die twice. When the fallen slink away without permission," he said, "we hold funerals for them and erect markers in the graveyard, because to us they are dead." He glanced over his shoulder, at Alexandra collecting flowers from the stage, and spoke into the conductor's ear.
The conductor stepped back. "Absolutely not," she said. "It's out of the question."
The prophet stared at her for a moment before he turned away. He murmured something to a man in the front row, the archer who'd been guarding the gas station that morning, and they walked together away from the Walmart.
"Luli!" the prophet called over his shoulder, and the dog trotted after him. The audience was dispersing now, and within minutes the Symphony was alone in the parking lot. It was the first time in memory that no one from the audience had lingered to speak with the Symphony after a performance.
"Quickly," the conductor said. "Harness the horses."
"I thought we were staying a few days," Alexandra said, a little whiny.
"It's a doomsday cult." The clarinet was unclipping the _Midsummer Night's Dream_ backdrop. "Weren't you listening?"
"But the last time we came here—"
"This isn't the town it was the last time we were here." The painted forest collapsed into folds and fell soundlessly to the pavement. "This is one of those places where you don't notice everyone's dropping dead around you till you've already drunk the poisoned wine." Kirsten knelt to help the clarinet roll the fabric. "You should maybe wash that dress," the clarinet said.
"He's gone back into the gas station," Sayid said. There were armed guards posted on either side of the gas station door now, indistinct in the twilight. A cooking fire flared by the motel.
The Symphony was on their way within minutes, departing down a back road behind the Walmart that took them away from the center of town. A small fire flickered by the roadside ahead. They found a boy there, a sentry, roasting something that might have been a squirrel at the end of a stick. Most towns had sentries with whistles at the obvious points of entry, the idea being that it was nice to have a little warning if marauders were coming through, but the boy's youth and inattention suggested that this wasn't considered an especially dangerous post. He stood as they approached, holding his dinner away from the flames.
"You have permission to leave?" he called out.
The conductor motioned to the first flute, who was driving the lead caravan—keep moving—and went to speak with the boy. "Good evening," she said. Kirsten stopped walking and lingered a few feet away, listening.
"What's your name?" he asked, suspicious.
"People call me the conductor."
"And that's your name?"
"It's the only name I use. Is that dinner?"
"Did you get permission to leave?"
"The last time we were here," she said, "no permission was required."
"It's different now." The boy's voice hadn't broken yet. He sounded very young.
"What if we didn't have permission?"
"Well," the boy said, "when people leave without permission, we have funerals for them."
"What happens when they come back?"
"If we've already had a funeral...," the boy said, but seemed unable to finish the sentence.
"This place," the fourth guitar muttered. "This goddamned hellhole." He touched Kirsten's arm as he passed. "Better keep moving, Kiki."
"So you wouldn't advise coming back here," the conductor said. The last caravan was passing. Sayid, bringing up the rear, seized Kirsten's shoulder and propelled her along the road.
"How much danger do you want to put yourself in?" he hissed. "Keep walking."
"Don't tell me what to do."
"Then don't be an idiot."
"Will you take me with you?" Kirsten heard the boy ask. The conductor said something she couldn't hear, and when she looked back the boy was staring after the departing Symphony, his squirrel forgotten at the end of the stick.
The night cooled as they left St. Deborah by the Water. The only sounds were the clopping of horseshoes on cracked pavement, the creaking of the caravans, the footsteps of the Symphony as they walked, small rustlings from the night forest. A fragrance of pine and wildflowers and grass in the air, the stars so bright that the caravans cast lurching shadows on the road. They'd left so quickly that they were all still in their costumes, Kirsten holding up her Titania dress so as not to trip over it and Sayid a strange vision in his Oberon tuxedo, the white of his shirt flashing when he turned to look back. Kirsten passed him to speak with the conductor, who walked as always by the first caravan.
"What did you tell the boy by the road?"
"That we couldn't risk the perception of kidnapping," the conductor said.
"What did the prophet say to you after the concert?"
The conductor glanced over her shoulder. "You'll keep this to yourself?"
"I'll probably tell August."
"Of course you will. But no one else?"
"Okay," Kirsten said, "no one else."
"He suggested that we consider leaving Alexandra, as a guarantee of future good relations between the Symphony and the town."
"Leaving her? Why would we...?"
"He said he's looking for another bride."
Kirsten dropped back to tell August, who swore softly and shook his head. Alexandra was walking by the third caravan, oblivious, looking up at the stars.
Sometime after midnight the Symphony stopped to rest. Kirsten threw the Titania gown into the back of a caravan and changed into the dress she always wore in hot weather, soft cotton with patches here and there. The reassuring weight of knives on her belt. Jackson and the second oboe took two of the horses and rode back along the road for a mile, returned to report that no one seemed to be following.
The conductor was studying a map with a few of the older Symphony members in the moonlight. Their flight had taken them in an awkward direction, south down the eastern shore of Lake Michigan. The only reasonably direct routes to their usual territory took them either back through St. Deborah by the Water, or close by a town that had been known to shoot outsiders on sight, or inland, through a wilderness that in the pre-collapse era had been designated a national forest.
"What do we know about this particular national forest?" The conductor was frowning at the map.
"I vote against it," the tuba said. "I know a trader who went through there. Said it was a burnt-out area, no towns, violent ferals in the woods."
"Charming. And the south, along the lakeshore?"
"Nothing," Dieter said. "I talked to someone who'd been down there, but this was maybe ten years ago. Said it was sparsely populated, but I don't remember the details."
"Ten years ago," the conductor said.
"Like I said, nothing. But look, if we keep going south we'll eventually have to turn inland anyway, unless you're especially eager to see what became of Chicago."
"Did you hear that story about snipers in the Sears Tower?" the first cello asked.
"I lived that story," Gil said. "Wasn't there supposed to be a population to the south of here, down by Severn City? A settlement in the former airport, if I'm remembering correctly."
"I've heard that rumor too." It wasn't like the conductor to hesitate, but she studied the map for some time before she spoke again. "We've talked about expanding our territory for years, haven't we?"
"It's a risk," Dieter said.
"Being alive is a risk." She folded the map. "I'm missing two Symphony members, and I still think they went south. If there's a population in Severn City, perhaps they'll know the best route back to our usual territory. We continue south along the lakeshore."
Kirsten climbed up to the driver's seat on the second caravan, to drink some water and to rest. She shrugged her backpack from her shoulders. Her backpack was child-size, red canvas with a cracked and faded image of Spider-Man, and in it she carried as little as possible: two glass bottles of water that in a previous civilization had held Lipton Iced Tea, a sweater, a rag she tied over her face in dusty houses, a twist of wire for picking locks, the ziplock bag that held her tabloid collection and the _Dr. Eleven_ comics, and a paperweight.
The paperweight was a smooth lump of glass with storm clouds in it, about the size of a plum. It was of no practical use whatsoever, nothing but dead weight in the bag but she found it beautiful. A woman had given it to her just before the collapse, but she couldn't remember the woman's name. Kirsten held it in the palm of her hand for a moment before she turned to her collection.
She liked to look through the clippings sometimes, a steadying habit. These images from the shadow world, the time before the Georgia Flu, indistinct in the moonlight but she'd memorized the details of every one: Arthur Leander and his second wife, Elizabeth, on a restaurant patio with Tyler, their infant son; Arthur with his third wife, Lydia, a few months later; Arthur with Tyler at LAX. An older picture that she'd found in an attic stuffed with three decades' worth of gossip magazines, taken before she was born: Arthur with his arm around the pale girl with dark curls who would soon become his first wife, caught by a photographer as they stepped out of a restaurant, the girl inscrutable behind sunglasses and Arthur blinded by the flash.
#
# **13**
**THE PHOTO FROM THE TABLOID:**
Ten minutes before the photograph, Arthur Leander and the girl are waiting by the coat check in a restaurant in Toronto. This is well before the Georgia Flu. Civilization won't collapse for another fourteen years. Arthur has been filming a period drama all week, partly on a soundstage and partly in a park on the edge of the city. Earlier in the day he was wearing a crown, but now he's wearing a Toronto Blue Jays cap that makes him look very ordinary. He is thirty-six years old.
"What are you going to do?" he asks.
"I'm going to leave him." The girl, Miranda, has a recent bruise on her face. They're speaking in whispers to avoid being overheard by the restaurant staff.
He nods. "Good." He's looking at the bruise, which Miranda hasn't been entirely successful in concealing with makeup. "I was hoping you'd say that. What do you need?"
"I don't know," she says. "I'm sorry about all this. I just can't go home."
"I have a suggestion—" He stops because the coat-check girl has returned with their coats. Arthur's is magnificent, smooth and expensive-looking, Miranda's a battered peacoat that she found in a thrift store for ten dollars. She turns her back on the restaurant as she puts it on in an effort to hide the torn lining—when she turns back, something in the hostess's smile suggests that this effort was in vain—while Arthur, who by this point in his life is extravagantly famous, flashes his best smile and palms a twenty to the coat-check girl. The hostess is surreptitiously hitting Send on a text to a photographer who gave her fifty dollars earlier. Outside on the sidewalk, the photographer reads the message on his phone: _Leaving now_.
"As I was saying," Arthur murmurs, close to Miranda's ear, "I think you should come stay with me."
"At the hotel? I can't—" Miranda whispers.
"I insist. No strings attached."
Miranda is momentarily distracted by the coat-check girl, who is staring adoringly at Arthur. He whispers, "You don't have to make any decisions right away. It's just a place you can stay, if you'd like."
Miranda's eyes fill with tears. "I don't know what to—"
"Just say yes, Miranda."
"Yes. Thank you." It occurs to her as the hostess opens the door for them that she must look terrible, the bruise on her face and her eyes red and watery. "Wait," she says, fishing in her handbag, "I'm sorry, just a second—" She puts on the enormous sunglasses she'd been wearing earlier in the day, Arthur puts his arm around her shoulders, the photographer on the sidewalk raises his camera, and they step out into the blinding flash.
"So, Arthur." The journalist is beautiful in the manner of people who spend an immense amount of money on personal maintenance. She has professionally refined pores and a four-hundred-dollar haircut, impeccable makeup and tastefully polished nails. When she smiles, Arthur is distracted by the unnatural whiteness of her teeth, although he's been in Hollywood for years and should be used to it by now. "Tell us about this mystery brunette we've been seeing you with."
"I think that mystery brunette has a right to her privacy, don't you?" Arthur's smile is calibrated to defuse the remark and render it charming.
"Won't you tell us anything at all about her? Just a hint?"
"She's from my hometown," he says, and winks.
It's not a hometown, actually, it's a home island. "It's the same size and shape as Manhattan," Arthur tells people at parties all his life, "except with a thousand people."
Delano Island is between Vancouver Island and mainland British Columbia, a straight shot north from Los Angeles. The island is all temperate rain forest and rocky beaches, deer breaking into vegetable gardens and leaping in front of windshields, moss on low-hanging branches, the sighing of wind in cedar trees. In the middle of the island there's a small lake that Arthur always imagined was formed by an asteroid, almost perfectly round and very deep. One summer a young woman from somewhere else committed suicide there, left her car parked up on the road with a note and walked into the water, and then when divers went after her _they couldn't find the bottom of the lake_ , or so local children whispered to one another, half-frightened, half-thrilled, although upon reflection, years later, the idea of a lake so deep that divers can't reach bottom seems improbable. Still, the fact is that a woman walked into a lake that wasn't large and no one found the body for two weeks despite intensive searching, and the episode sparks up against Arthur's childhood memories retrospectively and leaves a frisson of darkness that wasn't there at the time. Because actually from day to day it's just a lake, just his favorite place to swim, everyone's favorite place to swim because the ocean is always freezing. In Arthur's memories of the lake, his mother is reading a book under the trees on the shore while his little brother splashes around with water wings in the shallows and bugs land fleetingly on the water's surface. For unknown reasons there is a naked Barbie doll buried up to her waist in the dirt on the lake road.
There are children on the island who go barefoot all summer and wear feathers in their hair, the Volkswagen vans in which their parents arrived in the '70s turning to rust in the forest. Every year there are approximately two hundred days of rain. There's a village of sorts by the ferry terminal: a general store with one gas pump, a health-food store, a real-estate office, an elementary school with sixty students, a community hall with two massive carved mermaids holding hands to form an archway over the front door and a tiny library attached. The rest of the island is mostly rock and forest, narrow roads with dirt driveways disappearing into the trees.
In other words, it's the kind of place that practically no one Arthur encounters in New York, Toronto, or Los Angeles can fathom, and he gets a lot of uncomprehending stares when he talks about it. He is forever trying to describe this place and resorting to generalizations about beaches and plant life. "The ferns were up to my head," he tells people, performing a gesture that suggests greater and greater height over the years until he realizes at some point in his midforties that he's describing plants that stand seven or eight feet tall. "Just unbelievable in retrospect."
"It must've been so beautiful" is the inevitable reply.
"It was," he tells them, "it is," and then finds a way to change the subject because it's difficult to explain this next part. Yes, it was beautiful. It was the most beautiful place I have ever seen. It was gorgeous and claustrophobic. I loved it and I always wanted to escape.
At seventeen he's accepted into the University of Toronto. He fills out the student-loan applications, his parents scrape up the money for the plane ticket and he's gone. He thought he wanted to study economics, but when he arrives in Toronto he discovers that he wants to do almost anything else. He worked hard in high school, but he's an indifferent student at the university. The classes are tedious. The point of coming to this city wasn't school, he decides. School was just his method of escape. The point was the city of Toronto itself. Within four months he's dropped out and is going to acting auditions, because some girl in his Commerce 101 class told him he should be an actor.
His parents are horrified. There are tearful phone calls on calling cards late at night. "The point was to get off the island," he tells them, but this doesn't help, because they love the island and they live there on purpose. But two months after leaving school he gets a bit part in an American movie filming locally, and then a one-line role in a Canadian TV show. He doesn't feel that he really has any idea how to act, so he starts spending all his money on acting classes, where he meets his best friend, Clark. There is a magnificent year when they are inseparable and go out four nights a week with fake IDs, and then when both of them are nineteen Clark succumbs to parental pressure and returns to England for university while Arthur auditions successfully for a theater school in New York City, where he works for cash in a restaurant and lives with four roommates above a bakery in Queens.
He graduates from the theater school and marks time for a while, auditioning and working long hours as a waiter, then a job on _Law & Order_—is there an actor in New York who hasn't worked on _Law & Order_?—that lands him an agent and turns into a recurring role on a different _Law & Order_, one of the spin-offs. A couple of commercials, two television pilots that don't get picked up—"But you should totally come out to L.A.," the director of the second one says when he calls Arthur with the bad news. "Crash in my guesthouse for a few weeks, do some auditions, see what happens"—and Arthur's sick of eastern winters by then, so he does it, he gets rid of most of his belongings and boards a westbound plane.
In Hollywood he goes to parties and lands a small part in a movie, a soldier with three lines who gets blown up in the first ten minutes, but this leads to a much bigger movie part, and this is when the parties begin in earnest—cocaine and smooth girls with perfect skin in houses and hotel rooms, a number of years that come back to him later in strobelike flashes: sitting by a pool in Malibu drinking vodka and talking to a girl who says she came here illegally from Mexico, crossed the border lying under a load of chili peppers in the back of a truck when she was ten; he's not sure whether to believe her but he thinks she's beautiful so he kisses her and she says she'll call but he never sees her again; driving in the hills with friends, a passenger in a convertible with the top down, his friends singing along with the radio while Arthur watches the palm trees slipping past overhead; dancing with a girl to "Don't Stop Believin' "—secretly his favorite song—in some guy's basement tiki bar and then it seems like a miracle when he sees her at someone else's party a week later, the same girl at two parties in this infinite city, she smiles at him with half-closed eyes and takes his hand, leads him out to the backyard to watch the sun rise over Los Angeles. The novelty of this town is starting to wear a little thin by then, but up there by Mulholland Drive he understands that there's still some mystery here, still something in this city he hasn't seen, a sea of lights fading out in the valley as the sun rises, the way she runs her fingernails lightly over the skin of his arm.
"I love this place," he says, but six months later when they're breaking up she throws the line back at him—"You love this place but you'll never belong here and you'll never be cast as the lead in any of your stupid movies"—and by this point he's twenty-eight, time speeding up in a way that disconcerts him, the parties going too late and getting too sloppy, waiting in the ER on two separate occasions for news of friends who've OD'd on exotic combinations of alcohol and prescription medications, the same people at party after party, the sun rising on scenes of tedious debauchery, everyone looking a little undone. Just after his twenty-ninth birthday he lands the lead in a low-budget film about a botched bank robbery and is pleased to learn that it's filming in Toronto. He likes the idea of returning to Canada in triumph, which he's aware is egotistical but what can you do.
Arthur's mother calls one night and asks if he remembers Susie, that woman who was a waitress at the General Store Café when he was a kid. Of course he remembers Susie. He has vivid memories of Susie serving him pancakes in the café. Anyway, Susie's niece came to live with her a few years back, for reasons that remain buried despite the dedicated excavation efforts of every gossip on the island. The niece, Miranda, is seventeen now and just very _driven_ , very _together_. She recently moved to Toronto to go to art school, and could Arthur maybe take her to lunch?
"Why?" he asks. "We don't know each other. She's a seventeen-year-old girl. It'll be kind of awkward, won't it?" He hates awkwardness and goes to great lengths to avoid it.
"You have a lot in common," his mother says. "You both skipped a grade in school."
"I'm not sure that qualifies as 'a lot.' " But even as he says this, he finds himself thinking, She'll know where I'm from. Arthur lives in a permanent state of disorientation like a low-grade fever, the question hanging over everything being How did I get from there to here? And there are moments—at parties in Toronto, in Los Angeles, in New York—when he'll be telling people about Delano Island and he'll notice a certain look on their faces, interested but a little incredulous, like he's describing an upbringing on the surface of Mars. For obvious reasons, very few people have heard of Delano Island. When he tells people in Toronto that he's from British Columbia, they'll invariably say something about how they like Vancouver, as though that glass city four hours and two ferries to the southeast of his childhood home has anything to do with the island where he grew up. On two separate occasions he's told people in Los Angeles that he's from Canada and they've asked about igloos. An allegedly well-educated New Yorker once listened carefully to his explanation of where he's from—southwestern British Columbia, an island between Vancouver Island and the mainland—and then asked, apparently in all seriousness, if this means he grew up near Maine.
"Call Miranda," his mother says. "It's just lunch."
Miranda at seventeen: she is preternaturally composed and very pretty, pale with gray eyes and dark curls. She comes into the restaurant in a rush of cold air, January clinging to her hair and her coat, and Arthur is struck immediately by her poise. She seems much older than her age.
"How do you like Toronto?" Arthur asks. Not merely pretty, he decides. She is actually beautiful, but it's a subtle kind of beauty that takes some time to make itself apparent. She is the opposite of the L.A. girls with their blond hair and tight T-shirts and tans.
"I love it." The revelation of privacy: she can walk down the street and _absolutely no one knows who she is_. It's possible that no one who didn't grow up in a small place can understand how beautiful this is, how the anonymity of city life feels like freedom. She starts telling him about her boyfriend Pablo, also an artist, and Arthur forces himself to smile as he listens. She's so young, he tells himself. She's tired of talking about herself and asks about him, and he tries to explain the surrealism of this world he's stepped into where people know him when he doesn't know them, he talks about how much he loves Los Angeles and how simultaneously the place exhausts him, how disoriented he feels when he thinks about Delano Island and compares it to his current life. She's never been to the United States, although she's lived within two hundred miles of the border all her life. He can see that she's straining to imagine his life there, her thoughts probably a collage of scenes from movies and magazine shoots.
"You love acting, don't you?"
"Yes. Usually I do."
"What a wonderful thing, to get paid for doing what you love," she says, and he agrees with this. At the end of the meal she thanks him for paying the check and they leave together. Outside the air is cold, sunlight on dirty snow. Later he'll remember this as a golden period when they could walk out of restaurants together without anyone taking pictures of them on the sidewalk.
"Good luck on the movie," she says, boarding a streetcar.
"Good luck in Toronto," he replies, but she's already gone. In the years that follow, he's often successful at putting her out of his mind. She is far away and very young. There are a number of movies, an eighteen-month relocation to New York for a Mamet play, back to Los Angeles for a recurring role in an HBO series. He dates other women, some actresses, some not, two of them so famous that they can't go out in public without attracting photographers who swarm like mosquitoes. By the time he returns to Toronto for another movie, he can't go out in public without being photographed either, partly because the movie parts have gotten much bigger and more impressive, partly because the photographers got used to taking his picture when he was holding hands with more famous people. His agent congratulates him on his dating strategy.
"I wasn't being strategic," Arthur says. "I dated them because I liked them."
"Sure you did," his agent says. "I'm just saying, it didn't hurt."
Did he actually date those women because he liked them, or was his career in the back of his mind the whole time? The question is unexpectedly haunting.
Arthur is thirty-six now, which makes Miranda twenty-four. He is becoming extremely, unpleasantly famous. He wasn't expecting fame, although he secretly longed for it in his twenties just like everyone else, and now that he has it he's not sure what to do with it. It's mostly embarrassing. He checks into the Hotel Le Germain in Toronto, for example, and the young woman at the registration desk tells him what an honor it is to have him staying with them—"and if you don't mind me saying so, I adored that detective film"—and as always in these situations he isn't sure what to say, he honestly can't tell if she really did enjoy the detective film or if she's just being nice or if she wants to sleep with him or some combination of the above, so he smiles and thanks her, flustered and not sure where to look, takes the key card and feels her gaze on his back as he walks to the elevators. Trying to look purposeful, also trying to convey the impression that he hasn't noticed and doesn't care that half the population of the lobby is staring at him.
Once in the room he sits on the bed, relieved to be alone and unlooked-at but feeling as he always does in these moments a little disoriented, obscurely deflated, a bit at a loss, and then all at once he knows what to do. He calls the cell number that he's been saving all these years.
# **14**
**MIRANDA IS AT WORK** when Arthur Leander calls her again. She's an administrative assistant at a shipping company, Neptune Logistics, where she spends quiet days at a desk shaped like a horseshoe in a private reception area outside her boss's office door. Her boss is a young executive named Leon Prevant, and his door is almost always closed because he's almost always out of town. There are acres of gray carpeting and a wall of glass with a view of Lake Ontario near her desk. There's rarely enough work to keep her occupied for more than an hour or two at a time, which means she can often spend entire afternoons sketching—she's working on a series of graphic novels—with long coffee breaks, during which she likes to stand by the glass wall and look out at the lake. When she stands here she feels suspended, floating over the city. The stillness of the water, the horizon framed by other glass towers and miniature boats drifting in the distance.
A soft chime signifies an incoming email. During the long period when her position was staffed by an incompetent temp—"The winter of our discontent," Leon Prevant calls it—Leon took to outsourcing his travel planning to his subordinate Hannah's administrative assistant Thea, who is impeccable in a smooth, corporate way that Miranda admires, and who has just forwarded Leon's flight confirmation emails for next month's trip to Tokyo. In Thea's presence she feels ragged and unkempt, curls sticking up in all directions while Thea's hair is glossy and precise, her clothes never quite right whereas Thea's clothes are perfect. Miranda's lipstick is always too gaudy or too dark, her heels too high or too low. Her stockings all have holes in the feet and have to be worn strategically with specific pairs of shoes. The shoes have scuffed heels, filled in carefully with permanent marker.
The clothes are a problem. Most of Miranda's office clothes come from a bargain outlet just off Yonge Street, and they always look okay under the dressing room lights but by the time she gets home they're all wrong, the black skirt shining with acrylic fibers, the blouse in a synthetic fabric that clings unpleasantly, everything cheap-looking and highly flammable.
"You're an artist," her boyfriend Pablo said that morning, watching her while she tried various layering options under a blouse that had shrunk in the wash. "Why would you want to conform to some bullshit corporate dress code?"
"Because my job requires it."
"My poor corporate baby," he said. "Lost in the machine." Pablo talks about metaphorical machines a lot, also the Man. He sometimes combines the two, as in "That's how the Man wants us, just trapped right there in the corporate machine." They met at school. Pablo graduated a year ahead of her, and at first his career seemed so brilliant that she stopped being a waitress at his invitation: he sold a painting for ten thousand dollars and then a larger one for twenty-one thousand and he was poised to become the Next Big Thing, but then a show got canceled and he sold nothing else in the year that followed, absolutely nothing, so Miranda signed with a temp agency and found herself a short time later at her desk in a high tower outside Leon Prevant's office door. "Hang in there, baby," he said that morning, watching her dress. "You know this is only temporary."
"Sure," she said. He's been saying this ever since she registered with the temp agency, but what she hasn't told him is that she went from temporary to permanent at the end of her sixth week on the job. Leon likes her. He appreciates how calm she always is, he says, how unflappable. He even introduces her as such, on the rare occasions when he's in the office: "And this is my unflappable assistant, Miranda." This pleases her more than she likes to admit to herself.
"I'm going to sell those new paintings," Pablo said. He was half-naked in the bed, lying like a starfish. After she got up he always liked to see how much of the bed he could sleep on at once. "You know there's a payday coming, right?"
"Definitely," Miranda said, giving up on the blouse and trying to find a T-shirt that might look halfway professional under her twenty-dollar blazer.
"Almost no one from that last show sold anything," he said, talking mostly to himself now.
"I know it's temporary." But this is her secret: she doesn't want it to end. What she can never tell Pablo, because he disdains all things corporate, is that she likes being at Neptune Logistics more than she likes being at home. Home is a small dark apartment with an ever-growing population of dust bunnies, the hallway narrowed by Pablo's canvases propped up against the walls, an easel blocking the lower half of the living room window. Her workspace at Neptune Logistics is all clean lines and recessed lighting. She works on her never-ending project for hours at a time. In art school they talked about day jobs in tones of horror. She never would have imagined that her day job would be the calmest and least cluttered part of her life.
She receives five emails from Thea this morning, forwarded flight and hotel confirmations for Leon's upcoming trip to Asia. Miranda spends some time on the Asian travel itinerary. Japan, then Singapore, then South Korea. She likes looking up maps and imagining traveling to these places herself. She has still never left Canada. With Pablo not working or selling any paintings, she's only making minimum interest payments on her student loans and she can barely cover their rent. She inserts the Singapore-to-Seoul flight information into the itinerary, double-checks the other confirmation numbers, and realizes that she's run out of tasks for the day. It's nine forty-five a.m.
Miranda reads the news for a while, spends some time looking at a map of the Korean peninsula, realizes that she's been staring blankly at the screen and thinking of the world of her project, her graphic novel, her comic-book series, her whatever-it-is that she's been working on since she graduated from art school. She retrieves her sketchbook from its hiding space under the files in her top desk drawer.
There are several important characters in the Station Eleven project, but the hero is Dr. Eleven, a brilliant physicist who bears a striking physical resemblance to Pablo but is otherwise nothing like him. He is a person from the future who never whines. He is dashing and occasionally sarcastic. He doesn't drink too much. He is afraid of nothing but has poor luck with women. He took his name from the space station where he lives. A hostile civilization from a nearby galaxy has taken control of Earth and enslaved Earth's population, but a few hundred rebels managed to steal a space station and escape. Dr. Eleven and his colleagues slipped Station Eleven through a wormhole and are hiding in the uncharted reaches of deep space. This is all a thousand years in the future.
Station Eleven is the size of Earth's moon and was designed to resemble a planet, but it's a planet that can chart a course through galaxies and requires no sun. The station's artificial sky was damaged in the war, however, so on Station Eleven's surface it is always sunset or twilight or night. There was also damage to a number of vital systems involving Station Eleven's ocean levels, and the only land remaining is a series of islands that once were mountaintops.
There has been a schism. There are people who, after fifteen years of perpetual twilight, long only to go home, to return to Earth and beg for amnesty, to take their chances under alien rule. They live in the Undersea, an interlinked network of vast fallout shelters under Station Eleven's oceans. There are three hundred of them now. In the scene Miranda's presently sketching, Dr. Eleven is on a boat with his mentor, Captain Lonagan.
Dr. Eleven: These are perilous waters. We're passing over an Undersea gate.
Captain Lonagan: You should try to understand them. (The next panel is a close-up of his face.) All they want is to see sunlight again. Can you blame them?
After these two panels, she decides, she needs a full-page spread. She's already painted the image, and when she closes her eyes she can almost see it, clipped to her easel at home. The seahorse is a massive rust-colored creature with blank eyes like saucers, half animal, half machine, the blue light of a radio transmitter glowing on the side of its head. Moving silent through the water, beautiful and nightmarish, a human rider from the Undersea astride the curve of its spine. Deep blue water up to the top inch of the painting. On the water's surface, Dr. Eleven and Captain Lonagan in their rowboat, small under the foreign constellations of deep space.
On the day she sees Arthur again, Pablo calls her on the office line in the afternoon. She's a few sips into her four p.m. coffee, sketching out a series of panels involving Dr. Eleven's efforts to thwart the Undersea's latest plot to sabotage the station reactors and force a return to Earth. She knows as soon as she hears Pablo's voice that it's going to be a bad call. He wants to know what time she'll be home.
"Sometime around eight."
"What I don't understand," Pablo says, "is what you're doing for these people."
She winds the phone cord around her finger and looks at the scene she was just working on. Dr. Eleven is confronted by his Undersea nemesis on a subterranean walkway by Station Eleven's main reactor. A thought bubble: But what insanity is this?
"Well, I put together Leon's travel itineraries." There have been a number of bad calls lately, and she's been trying to view them as opportunities to practice being patient. "I handle his expense reports and send emails for him sometimes. There's the occasional message. I do the filing."
"And that takes up your entire day."
"Not at all. We've talked about this, pickle. There's a lot of downtime, actually."
"And what do you do in that _downtime_ , Miranda?"
"I work on my project, Pablo. I'm not sure why your tone's so nasty." But the trouble is, she doesn't really care. There was a time when this conversation would have reduced her to tears, but now she swivels in her chair to look out at the lake and thinks about moving trucks. She could call in sick to work, pack up her things, and be gone in a few hours. It is sometimes necessary to break everything.
"... twelve-hour days," he's saying. "You're never here. You're gone from eight a.m. till nine at night and then you even go in on _Saturdays_ sometimes, and I'm supposed to just... oh, I don't know, Miranda, what would you say if you were me?"
"Wait," she says, "I just realized why you called me on the office line."
"What?"
"You're verifying that I'm here, aren't you? That's why you didn't call me on my cell." A shiver of anger, unexpectedly deep. She is paying the entire rent on their apartment, and he's verifying that she's actually at her job.
"The hours you work." He lets this hang in the air till it takes on the weight of accusation.
"Well," she says—one thing she is very good at is forcing her voice to remain calm when she's angry—"as I've mentioned before, Leon was very clear when he hired me. He wants me at my desk until seven p.m. when he's traveling, and if he's here, I'm here. He texts me when he comes in on weekends, and then I have to be here too."
"Oh, he _texts_ you."
The problem is that she's colossally bored with the conversation, and also bored with Pablo, and with the kitchen on Jarvis Street where she knows he's standing, because he only makes angry phone calls from home—one of the things they have in common is a mutual distaste for sidewalk weepers and cell-phone screamers, for people who conduct their messier personal affairs in public—and the kitchen gets the best reception of anywhere in the apartment.
"Pablo, it's just a job. We need the money."
"It's always money with you, isn't it?"
"This is what's paying our rent. You know that, right?"
"Are you saying I'm not pulling my weight, Miranda? Is that what you're saying?"
It isn't possible to continue to listen to this, so she sets the receiver gently on the cradle and finds herself wondering why she didn't notice earlier—say, eight years ago, when they first started dating—that Pablo is mean. His email arrives within minutes. The subject header is _WTF. Miranda_ , it reads, _what's going on here? It seems like you're being weirdly hostile and kind of passive-aggressive. What gives?_
She closes it without responding and stands by the glass wall for a while to look out at the lake. Imagining the water rising until it covers the streets, gondolas moving between the towers of the financial district, Dr. Eleven on a high arched bridge. She's standing here when her cell phone rings. She doesn't recognize the number.
"It's Arthur Leander," he says when she answers. "Can I buy you another lunch?"
"How about dinner instead?"
"Tonight?"
"Are you busy?"
"No," he says, sitting on his bed in the Hotel Le Germain, wondering how he'll get out of dinner with the director this evening. "Not at all. It would be my pleasure."
She decides it isn't necessary to call Pablo, under the circumstances. There is a small task for Leon, who's about to board a plane to Lisbon; she finds a file he needs and emails it to him and then returns to Station Eleven. Panels set in the Undersea, people working quietly in cavernous rooms. They live out their lives under flickering lights, aware at all times of the fathoms of ocean above them, resentful of Dr. Eleven and his colleagues who keep Station Eleven moving forever through deep space. (Pablo texts her: _??did u get my email???_ ) They are always waiting, the people of the Undersea. They spend all their lives waiting for their lives to begin.
Miranda is drawing Leon Prevant's reception area before she realizes what she's doing. The prairies of carpet, the desk, Leon's closed office door, the wall of glass. The two staplers on her desk—how did she end up with two?—and the doors leading out to the elevators and restrooms. Trying to convey the serenity of this place where she spends her most pleasant hours, the refinement of it, but outside the glass wall she substitutes another landscape, dark rocks and high bridges.
"You're always half on Station Eleven," Pablo said during a fight a week or so ago, "and I don't even understand your project. What are you actually going for here?"
He has no interest in comics. He doesn't understand the difference between serious graphic novels and Saturday-morning cartoons with wide-eyed tweetybirds and floppy-limbed cats. When sober, he suggests that she's squandering her talent. When drunk, he implies that there isn't much there to squander, although later he apologizes for this and sometimes cries. It's been a year and two months since he sold his last painting. She started to explain her project to him again but the words stopped in her throat.
"You don't have to understand it," she said. "It's mine."
The restaurant where she meets Arthur is all dark wood and soft lighting, the ceiling a series of archways and domes. I can use this, she thinks, waiting at the table for him to arrive. Imagining a room like this in the Undersea, a subterranean place made of wood salvaged from the Station's drowned forests, wishing she had her sketchbook with her. At 8:01 p.m., a text from Pablo: _i'm waiting_. She turns off her phone and drops it into her handbag. Arthur comes in breathless and apologetic, ten minutes late. His cab got stuck in traffic.
"I'm working on a comic-book project," she tells him later, when he asks about her work. "Maybe a series of graphic novels. I don't know what it is yet."
"What made you choose that form?" He seems genuinely interested.
"I used to read a lot of comics when I was a kid. Did you ever read _Calvin and Hobbes_?" Arthur is watching her closely. He looks young, she thinks, for thirty-six. He looks only slightly older than he did when they met for lunch seven years ago.
"Sure," Arthur says, "I loved _Calvin and Hobbes_. My best friend had a stack of the books when we were growing up."
"Is your friend from the island? Maybe I knew him."
"Her. Victoria. She picked up and moved to Tofino fifteen years ago. But you were telling me about _Calvin and Hobbes_."
"Yes, right. Do you remember Spaceman Spiff?"
She loved those panels especially. Spiff's flying saucer crossing alien skies, the little astronaut in his goggles under the saucer's glass dome. Often it was funny, but also it was beautiful. She tells him about coming back to Delano Island for Christmas in her first year of art school, after a semester marked by failure and frustrating attempts at photography. She started thumbing through an old _Calvin and Hobbes_ , and thought, _this_. These red-desert landscapes, these skies with two moons. She began thinking about the possibilities of the form, about spaceships and stars, alien planets, but a year passed before she invented the beautiful wreckage of Station Eleven. Arthur watches her across the table. Dinner goes very late.
"Are you still with Pablo?" he asks, when they're out on the street. He's hailing a cab. Certain things have been decided without either of them exactly talking about it.
"We're breaking up. We're not right for each other." Saying it aloud makes it true. They are getting into a taxi, they're kissing in the backseat, he's steering her across the lobby of the hotel with his hand on her back, she is kissing him in the elevator, she is following him into a room.
Texts from Pablo at nine, ten, and eleven p.m.:
_r u mad at me??_
_??_
_???_
She replies to this— _staying w a friend tonight, will be home in morning & then we can talk_—which elicits
_u know what dont bother coming home_
And she feels a peculiar giddiness when she reads this fourth text. There are thoughts of freedom and imminent escape. I could throw away almost everything, she thinks, and begin all over again. Station Eleven will be my constant.
At six in the morning she takes a taxi home to Jarvis Street. "I want to see you tonight," Arthur whispers when she kisses him. They have plans to meet in his room after work.
The apartment is dark and silent. There are dishes piled in the sink, a frying pan on the stove with bits of food stuck to it. The bedroom door is closed. She packs two suitcases—one for clothes, one for art supplies—and is gone in fifteen minutes. In the employee gym at Neptune Logistics she showers and changes into clothes slightly rumpled by the suitcase, meets her own gaze in the mirror while she's putting on makeup. _I repent nothing_. A line remembered from the fog of the Internet. I am heartless, she thinks, but she knows even through her guilt that this isn't true. She knows there are traps everywhere that can make her cry, she knows the way she dies a little every time someone asks her for change and she doesn't give it to them means that she's too soft for this world or perhaps just for this city, she feels so small here. There are tears in her eyes now. Miranda is a person with very few certainties, but one of them is that only the dishonorable leave when things get difficult.
"I don't know," Arthur says, at two in the morning. They are lying in his enormous bed at the Hotel Le Germain. He's here in Toronto for three more weeks and then going back to Los Angeles. She wants to believe they're lying in moonlight, but she knows the light through the window is probably mostly electric. "Can you call the pursuit of happiness dishonorable?"
"Surely sleeping with film stars when you live with someone else isn't _honorable_ , per se."
He shifts slightly in the bed, uncomfortable with the term _film star_ , and kisses the top of her head.
"I'm going to go back to the apartment in the morning to get a few more things," she says sometime around four a.m., half-asleep. Thinking about a painting she left on her easel, a seahorse rising up from the bottom of the ocean. They've been talking about plans. Things have been solidifying rapidly.
"You don't think he'll do anything stupid, do you? Pablo?"
"No," she says, "he won't do anything except maybe yell." She can't keep her eyes open.
"You're sure about that?"
He waits for an answer, but she's fallen asleep. He kisses her forehead—she murmurs something, but doesn't wake up—and lifts the duvet to cover her bare shoulders, turns off the television and then the light.
# **15**
**LATER THEY HAVE** a house in the Hollywood Hills and a Pomeranian who shines like a little ghost when Miranda calls for her at night, a white smudge in the darkness at the end of the yard. There are photographers who follow Arthur and Miranda in the street, who keep Miranda forever anxious and on edge. Arthur's name appears above the titles of his movies now. On the night of their third anniversary, his face is on billboards all over the continent.
Tonight they're having a dinner party and Luli, their Pomeranian, is watching the proceedings from the sunroom, where she's been exiled for begging table scraps. Every time Miranda glances up from the table, she sees Luli peering in through the glass French doors.
"Your dog looks like a marshmallow," says Gary Heller, who is Arthur's lawyer.
"She's the cutest little thing," Elizabeth Colton says. Her face is next to Arthur's on the billboards, flashing a brilliant smile with very red lips, but offscreen she wears no lipstick and seems nervous and shy. She is beautiful in a way that makes people forget what they were going to say when they look at her. She is very soft-spoken. People are forever leaning in close to hear what she's saying.
There are ten guests here tonight, an intimate evening to celebrate both the anniversary and the opening weekend figures. "Two birds with one stone," Arthur said, but there's something wrong with the evening, and Miranda is finding it increasingly difficult to hide her unease. Why would a three-year wedding anniversary celebration involve anyone other than the two people who are actually married to one another? Who are all these extraneous people at my table? She's seated at the opposite end of the table from Arthur, and she somehow can't quite manage to catch his eye. He's talking to everyone except her. No one seems to have noticed that Miranda's saying very little. "I wish you'd try a little harder," Arthur has said to her once or twice, but she knows she'll never belong here no matter how hard she tries. These are not her people. She is marooned on a strange planet. The best she can do is pretend to be unflappable when she isn't.
Plates and bottles are being ferried to and from the table by a small army of caterers, who will leave their head shots and possibly a screenplay or two behind in the kitchen at the end of the night. Luli, on the wrong side of the glass, is staring at a strawberry that's fallen off the top of Heller's wife's dessert. Miranda has a poor memory when she's nervous, which is to say whenever she has to meet industry people or throw a dinner party or especially both, and she absolutely cannot remember Heller's wife's name although she's heard it at least twice this evening.
"Oh, it was _intense_ ," Heller's wife is saying now, in response to something that Miranda didn't hear. "We were out there for a _week_ , just surfing every day. It was actually really spiritual."
"The surfing?" the producer seated beside her asks.
"You wouldn't think it, right? But just going out every day, just you and the waves and a private instructor, it was just a really focused experience. Do you surf?"
"I'd love to, but I've just been so busy with this whole school thing lately," the producer says. "Actually, I guess you'd maybe call it an orphanage, it's this little thing I set up in Haiti last year, but the point is education, not just _housing_ these kids...."
"I don't know, I'm not attached to his project or anything." Arthur is deep in conversation with an actor who played his brother in a film last year. "I've never met the guy, but I've heard through friends that he likes my work."
"I've met him a few times," the actor says.
Miranda tunes out the overlapping conversations to look at Luli, who's looking at her through the glass. She'd like to take Luli outside, and stay in the backyard with her until all these people leave.
The dessert plates are cleared around midnight but no one's close to leaving, a wine-drenched languor settling over the table. Arthur is deep in conversation with Heller. Heller's nameless wife is gazing dreamily at the chandelier.
Clark Thompson is here, Arthur's oldest friend and the only person at this table, aside from Miranda, who has no professional involvement in movies.
"I'm sorry," a woman named Tesch is saying now, to Clark, "what exactly is it that you do?" Tesch seems to be someone who mistakes rudeness for intellectual rigor. She is about forty, and wears severe black-framed glasses that somehow remind Miranda of architects. Miranda met her for the first time this evening and she can't remember what Tesch does, except that obviously she's involved in some way with the industry, a film editor maybe? And also Miranda doesn't understand Tesch's name: is she Tesch something, or something Tesch? Or a one-namer, like Madonna? Are you allowed to have only one name if you're not famous? Is it possible that Tesch is actually extremely famous and Miranda's the only one at the table who doesn't know this? Yes, that seems very possible. These are the things she frets about.
"What do I do? Nothing terribly glamorous, I'm afraid." Clark is British, thin and very tall, elegant in his usual uniform of vintage suit and Converse sneakers, accessorized with pink socks. He brought them a gift tonight, a beautiful glass paperweight from a museum gift shop in Rome. "I have nothing to do with the film industry," he says.
"Oh," Heller's wife says, "I think that's marvelous."
"It's certainly exotic," Tesch says, "but that doesn't narrow the field much, does it?"
"Management consulting. Based out of New York, new client in Los Angeles. I specialize in the repair and maintenance of faulty executives." Clark sips his wine.
"And what's that in English?"
"The premise of the company by which I'm employed," Clark says, "is that if one's the employer of an executive who's worthy in some ways but deeply flawed in others, it's sometimes cheaper to fix the executive than to replace him. Or her."
"He's an organizational psychologist," Arthur says, surfacing from conversation at the far end of the table. "I remember when he went back to England to get his PhD."
"A PhD," Tesch says. "How conventional. And you"—she's turned to Miranda—"how's your work going?"
"It's going very well, thank you." Miranda spends most of her time working on the Station Eleven project. She knows from the gossip blogs that people here see her as an eccentric, the actor's wife who inks mysterious cartoons that no one's ever laid eyes on—"My wife's very private about her work," Arthur says in interviews—and who doesn't drive and likes to go for long walks in a town where nobody walks anywhere and who has no friends except a Pomeranian, although does anyone really know this last part? She hopes not. Her friendlessness is never mentioned in gossip blogs, which she appreciates. She hopes she isn't as awkward to other people as she feels to herself. Elizabeth Colton is looking at her again in that golden way of hers. Elizabeth's hair is always unbrushed and always looks gorgeous that way. Her eyes are very blue.
"It's brilliant," Arthur says. "I mean that. Someday she'll show it to the world and we'll all say we knew her when."
"When will it be finished?"
"Soon," Miranda says. It's true, it won't be so long now. She has felt for months that she's nearing the end of something, even though the story has spun off in a dozen directions and feels most days like a mess of hanging threads. She tries to meet Arthur's gaze, but he's looking at Elizabeth.
"What do you plan to do with it once it's done?" Tesch asks.
"I don't know."
"Surely you'll try to publish it?"
"Miranda has complicated feelings on the topic," Arthur says. Is it Miranda's imagination, or is he going out of his way to avoid looking at her directly?
"Oh?" Tesch smiles and arches an eyebrow.
"It's the work itself that's important to me." Miranda is aware of how pretentious this sounds, but is it still pretentious if it's true? "Not whether I publish it or not."
"I think that's so great," Elizabeth says. "It's like, the point is that it exists in the world, right?"
"What's the point of doing all that work," Tesch asks, "if no one sees it?"
"It makes me happy. It's peaceful, spending hours working on it. It doesn't really matter to me if anyone else sees it."
"Ah," Tesch says. "Very admirable of you. You know, it reminds me of a documentary I saw last month, a little Czech film about an outsider artist who refused to show her work during her lifetime. She lived in _Pra_ ha, and—"
"Oh," Clark says, "I believe when you're speaking English, you're allowed to refer to it as Prague."
Tesch appears to have lost the power of speech.
"It's a beautiful city, isn't it?" Elizabeth has the kind of smile that makes everyone around her smile too, unconsciously.
"Ah, you've been there?" Clark asks.
"I took a couple of art history classes at UCLA a few years back. I went to Prague at the end of the semester to see a few of the paintings I'd read about. There's such a weight of history in that place, isn't there? I wanted to move there."
"For the history?"
"I grew up in the exurbs of Indianapolis," Elizabeth says. "I live in a neighborhood where the oldest building is fifty or sixty years old. There's something appealing about the thought of living in a place with some history to it, don't you think?"
"So tonight," Heller says, "if I'm not mistaken, is tonight the actual wedding anniversary?"
"It certainly is," Arthur says, and glasses are raised. "Three years." He's smiling past Miranda's left ear. She glances over her shoulder, and when she looks back he's shifted his gaze somewhere else.
"How did you two meet?" Heller's wife asks. The thing about Hollywood, Miranda realized early on, is that almost everyone is Thea, her former colleague at Neptune Logistics, which is to say that almost everyone has the right clothes, the right haircut, the right everything, while Miranda flails after them in the wrong outfit with her hair sticking up.
"Oh, it's not the most exciting how-we-met story in the world, I'm afraid." A slight strain in Arthur's voice.
"I think how-we-met stories are always exciting," Elizabeth says.
"You're much more patient than I am," Clark says.
"I don't know if _exciting_ is the word I'd use," Heller's wife says. "But there's certainly a sweetness about them, about those stories I mean."
"No, it's just, if everything happens for a reason," Elizabeth persists, "as personally, I believe that it does, then when I hear a story of how two people came together, it's like a piece of the plan is being revealed."
In the silence that follows this pronouncement, a caterer refills Miranda's wine.
"We're from the same island," Miranda says.
"Oh, that island you told us about," a woman from the studio says, to Arthur. "With the ferns!"
"So you're from the same island, and? And?" Heller now, looking at Arthur. Not everyone is listening. There are pools and eddies of conversation around the table. Heller's tan is orange. There are rumors that he doesn't sleep at night. On the other side of the glass doors, Luli shifts position to gain a better view of the dropped strawberry.
"Excuse me a moment," Miranda says, "I'm just going to let the dog out. Arthur tells this story much better than I do." She escapes into the sunroom, through a second set of French doors into the back lawn. Freedom! Outside, the quiet night. Luli brushes against her ankles and fades out into the darkness. The backyard isn't large, their property terraced up the side of a hill, leaves crowding in around a small launchpad of lawn. The gardener came today in preparation for the dinner party, and the air carries notes of damp soil and freshly cut grass. She turns back toward the dining room, knowing that they can't see her past their own overlapping reflections on the glass. She left both sets of doors open just slightly in order to hear the conversation, and now Arthur's voice carries into the yard.
"So, you know, dinner goes well, and then the next night," he says, "I'm in the Hotel Le Germain after twelve hours on set, in my room waiting for Miranda to come by so I can take her out to dinner again, second night in a row, just kind of semi-comatose in front of the television, there's a knock at the door, and— Voilà! There she is again, but this time? One small difference." He pauses for effect. She can see Luli again now, following a mysterious scent at the far end of the lawn. "This time, I'll be damned if the girl hasn't got her worldly belongings with her."
Laughter. The story's funny, the way he tells it. She shows up on his hotel room doorstep with two suitcases, having walked across the lobby with such confidence that anyone would think she was a guest there. (The best advice her mother ever gave her: "Walk in like you own the place.") She says something vague to Arthur about how she's moving into a hotel herself and perhaps he wouldn't mind if she just leaves the suitcases here while they go to dinner, but he's already in love and he kisses her, he takes her to bed and they don't leave the hotel at all that night, he invites her to stay a few days and she never moves out and now here we are in Los Angeles.
He doesn't tell the whole story. He doesn't tell the crowd assembled at the table that when she went back to the apartment the next morning for a painting she'd decided she wanted, a watercolor left behind on the drafting table, Pablo was awake and waiting for her, drunk and weeping, and she returned to the hotel with a bruise on her face. Arthur doesn't tell them that he took her with him to the set that morning and passed her off as his cousin, that she called in sick to work and spent the day in his trailer reading magazines and trying not to think about Pablo while Arthur came and went in his costume, which involved a long red velvet cape and a crown. He looked magnificent. Every time he looked at her that day, something clenched in her chest.
When he was done with work in the evening, he had a driver drop them at a restaurant downtown, where he sat across the table from her looking very ordinary in a Toronto Blue Jays cap and she looked at him and thought, I prefer you with a crown, but of course she would never say this aloud. Three and a half years later in the Hollywood Hills she stands outside in the yard and wonders if anyone at the table saw the tabloid photo that appeared the following morning, shot as they were leaving the restaurant—Arthur with his arm around her shoulders, Miranda in dark glasses and Arthur blinded by the flash, which washed her out so mercifully that in the photo version of that moment the bruise was erased.
"What a lovely story," someone says, and Arthur agrees, Arthur is pouring wine, he's raising his glass and he's toasting her, "Here's to my beautiful brilliant wife." But Miranda, watching from outside, sees everything: the way Elizabeth goes still and looks down, the way Arthur thanks everyone for coming to his home, meeting everyone's eyes except Elizabeth's, who has lightly touched his thigh under the table, and this is when she understands. It's too late, and it's been too late for a while. She draws an uneven breath.
"Great story," Heller says. "Where _is_ that wife of yours?"
Could she possibly go around to the front of the house, sneak in the front door and up to her studio unnoticed, then text Arthur to say that she has a headache? She steps away from the glass, toward the center of the lawn where the shadows are deepest. From here the dinner party looks like a diorama, white walls and golden light and glamorous people. She turns her back on it to look for Luli—the dog is nosing around in the grass, delighted by a scent at the base of an azalea bush—and this is when she hears the glass doors close behind her. Clark has come out for a cigarette. Her plan was to pretend if anyone came out here that she's looking for the dog, but he doesn't ask. He taps the cigarette box on the palm of his hand and holds out a cigarette without speaking.
She crosses the grass and takes it from him, leans in when he flicks the lighter, and observes the dinner party while she inhales. Arthur is laughing. His hand strays to Elizabeth's wrist and rests there for an instant before he refills her wine. Why is Elizabeth sitting next to him? How could they be so indiscreet?
"Not a pretty sight, is it?"
She thinks of disagreeing, but something in Clark's voice stops her. Does everyone already know? "What do you mean?" she asks, but her voice is shaky.
He glances at her and turns his back on the tableau, and after a moment she does the same. There's nothing to be gained by watching the shipwreck.
"I'm sorry for being rude to your guest in there."
"Tesch? Please, don't be polite to her on my account. She's the most pretentious woman I've ever met in my life."
"I've met worse."
She hasn't smoked in a while, managed to convince herself that smoking is disgusting, but it's a pleasure, actually, more of a pleasure than she remembered. The lit end flares in the darkness when she inhales. She likes Hollywood best at night, in the quiet, when it's all dark leaves and shadows and night-blooming flowers, the edges softened, gently lit streets curving up into the hills. Luli wanders near them, snuffling in the grass. There are stars tonight, a few, although most are blanked out by the haze of the city.
"Good luck, darling," Clark says quietly. He's finished his cigarette. When she turns he's already reentering the party, reclaiming his place at the table. "Oh, she's just searching for the dog," she hears him say in response to a question, "I expect she'll be in any moment now."
Dr. Eleven has a Pomeranian. She hadn't realized this before, but it makes perfect sense. He has few friends, and without a dog he'd be too lonely. That night in her study she sketches a scene: Dr. Eleven stands on an outcropping of rock, a thin silhouette with a fedora pulled low, scanning the choppy sea, and a small white dog stands windswept beside him. She doesn't realize, until halfway through drawing the dog, that she's given Dr. Eleven a clone of Luli. Wind turbines spin on the horizon. Dr. Eleven's Luli gazes at the sea. Miranda's Luli sleeps on a pillow at her feet, twitching in a dog dream.
Miranda's study window looks out over the side yard, where the lawn terraces down to a pool. Beside the pool stands a lamp from the 1950s, a crescent moon atop a tall dark pole, placed in such a way that there's always a moon reflected in the water. The lamp is her favorite thing about this house, although she wonders sometimes about the reason for its existence. A diva who insisted on permanent moonlight? A bachelor who hoped to impress young starlets? There's a brief period most nights when the two moons float side by side on the surface. The fake moon, which has the advantage of being closer and not obscured by smog, is almost always brighter than the real one.
At three in the morning Miranda leaves her drafting table and goes down to the kitchen for a second cup of tea. All of the guests except one have departed. At the end of the night everyone was drunk but climbed into expensive cars anyway, all except Elizabeth Colton, who drank quietly, determinedly, without taking any apparent pleasure in it, until she passed out on a sofa in the living room. Clark plucked the wineglass from her hand, Arthur removed Elizabeth's car keys from her handbag and dropped them into an opaque vase on the mantelpiece, Miranda covered her with a blanket and left a glass of water nearby.
"I think we should talk," Miranda said to Arthur, when the last guest except Elizabeth was gone, but he waved her off and stumbled in the direction of the bedroom, said something about talking in the morning on his way up the stairs.
The house is silent now and she feels like a stranger here. "This life was never ours," she whispers to the dog, who has been following her from room to room, and Luli wags her tail and stares at Miranda with wet brown eyes. "We were only ever borrowing it."
In the living room, Elizabeth Colton is still unconscious. Even passed out drunk she's a vision in the lamplight. In the kitchen, four head shots are lying on the countertop. Miranda studies these while the water's boiling and recognizes somewhat younger and more brooding versions of four of the night's caterers. She puts on a pair of flip-flops in the sunroom and lets herself out into the cool night air. She sits for a while at the poolside with her tea, Luli beside her, and splashes her feet in the water to watch the moon reflection ripple and break.
There's a sound from the street, a car door closing. "Stay," she tells Luli, who sits by the pool and watches as Miranda opens the gate to the front driveway, where Elizabeth's convertible is parked dark and gleaming. Miranda runs her fingertips along the side of the car as she passes, and they come away coated with a fine layer of dust. The streetlight at the end of the driveway is a frenzy of moths. Two cars are parked on the street. A man leans on one of them, smoking a cigarette. In the other car, a man is asleep in the driver's seat. She recognizes both men, because they follow her and Arthur much more frequently than anyone else does.
"Hey," the man with the cigarette says, and reaches for his camera. He's about her age, with sideburns and dark hair that falls in his eyes.
"Don't," she says sharply, and he hesitates.
"What are you doing out so late?"
"Are you going to take my picture?"
He lowers the camera.
"Thank you," she says. "In answer to your question, I just came out here to see if you might have an extra cigarette."
"How'd you know I'd have one?"
"Because you're in front of my house smoking every night."
"Six nights a week," he says. "I take Mondays off."
"What's your name?"
"Jeevan Chaudhary."
"So do you have a cigarette for me, Jeevan?"
"Sure. Here. I didn't know you smoked."
"I just started again. Light?"
"So," he says, once her cigarette's lit, "this is a first."
She ignores this, looking up at the house. "It's pretty from here, isn't it?"
"Yes," he says. "You have a beautiful home." Was that sarcasm? She isn't sure. She doesn't care. She's always found the house beautiful, but it's even more so now that she knows she's leaving. It's modest by the standards of people whose names appear above the titles of their movies, but extravagant beyond anything she would have imagined for herself. In all my life, there will never be another house like this.
"You know what time it is?" he asks.
"I don't know, about three a.m.? Maybe more like three thirty?"
"Why's Elizabeth Colton's car still in the driveway?"
"Because she's a raging alcoholic," Miranda says.
His eyes widen. "Really?"
"She's too wasted to drive. You didn't hear that from me."
"Sure. No. Thank you."
"You're welcome. You people live for that kind of gossip, don't you?"
"No," he says, "I live _on_ that kind of gossip, actually. As in, it pays my rent. What I live _for_ is something different."
"What do you live for?"
"Truth and beauty," he says, deadpan.
"You like your job?"
"I don't hate it."
She is dangerously close to tears. "So you enjoy stalking people?"
He laughs. "Let's just say the job fits with my basic understanding of what work is."
"I don't understand."
"Of course you don't. You don't have to work for a living."
"Please," Miranda says, "I've worked all my life. I worked all through school. These past few years are an anomaly." Although as she says this she can't help but think of Pablo. She lived off him for ten months, until it became clear that they were going to run out of money before he sold another painting. In the next version of her life, she decides, she will be entirely independent.
"Forget it."
"No really, I'm curious. What's your understanding of work?"
"Work is combat."
"So you've hated every job you've had, is that what you're saying?"
Jeevan shrugs. He's looking at something on his phone, distracted, his face lit blue by the screen. Miranda returns her attention to the house. The sensation of being in a dream that will end at any moment, only she isn't sure if she's fighting to wake up or to stay asleep. Elizabeth's car is all long curves and streaks of reflected light. Miranda thinks of the places she might go now that Los Angeles is over, and what surprises her is that the first place that comes to mind is Neptune Logistics. She misses the order of the place, the utter manageability of her job there, the cool air of Leon Prevant's office suite, the calm of the lake.
"Hey!" Jeevan says suddenly, and as Miranda turns, the cigarette halfway to her mouth, the flash of his camera catches her unaware. Five more flashes in quick succession as she drops the cigarette on the sidewalk and walks quickly away from him, enters a code into a keypad and slips back in through the side gate, the afterimage of the first flash floating across her vision. How could she have let her guard down? How could she have been so stupid?
In the morning her picture will appear in a gossip website: TROUBLE IN PARADISE? AMID RUMORS OF ARTHUR'S INFIDELITY, MIRANDA WANDERS THE STREETS OF HOLLYWOOD AT FOUR A.M. CRYING AND SMOKING. And the photograph, the photograph, Miranda alone in the small hours of the morning with obvious tears in her eyes, pale in the flash, her hair standing up and a cigarette between her fingers, lips parted, a bra strap showing where her dress has slipped.
But first there is the rest of the night to get through. Miranda closes the gate and sits for a long time on a stone bench by the pool, shaking. Luli jumps up to sit beside her. Eventually Miranda dries her eyes and they go back to the house, where Elizabeth is still sleeping, and upstairs, where Miranda stops to listen outside the bedroom door. Arthur snores.
She opens the door to his study, which is the opposite of her study, which is to say the housekeeper's allowed to come in. Arthur's study is painfully neat. Four stacks of scripts on the desk, which is made of glass and steel. An ergonomic chair, a tasteful lamp. Beside the lamp, a flat leather box with a drawer that pulls open with a ribbon. She opens this and finds what she's looking for, a yellow legal pad on which she's seen him write before, but tonight there's only an unfinished fragment of Arthur's latest letter to his childhood friend:
Dear V., Strange days. The feeling that one's life resembles a movie. Thinking a lot of the future. I have such
Nothing else. You have such what, Arthur? Did your phone ring midsentence? Yesterday's date at the top of the page. She puts the legal pad back exactly as she found it, uses the hem of her dress to wipe a fingertip smudge from the desk. Her gaze falls on the gift that Clark brought this evening, a paperweight of clouded glass.
When she holds it, it's a pleasing weight in the palm of her hand. It's like looking into a storm. She tells herself as she switches off the light that she's only taking the paperweight back to her study to sketch it, but she knows she's going to keep it forever.
When she returns to her study it's nearly dawn. Dr. Eleven, the landscape, the dog, a text box for Dr. Eleven's interior monologue across the bottom: After Lonagan's death, all of life seemed awkward to me. I'd become a stranger to myself. She erases and rewrites: After Lonagan's death, I felt like a stranger. The sentiment seems right, but somehow not for this image. A new image to go before this one, a close-up of a note left on Captain Lonagan's body by an Undersea assassin: "We were not meant for this world. Let us go home."
In the next image, Dr. Eleven holds the note in his hand as he stands on the outcropping of rock, the little dog by his boots. His thoughts:
The first sentence of the assassin's note rang true: we were not meant for this world. I returned to my city, to my shattered life and damaged home, to my loneliness, and tried to forget the sweetness of life on Earth.
Too long, also melodramatic. She erases it, and writes in soft pencil: _I stood looking over my damaged home and tried to forget the sweetness of life on Earth_.
A sound behind her. Elizabeth Colton leans in the doorway, holding a glass of water with both hands.
"I'm sorry," she says, "I didn't mean to disturb. I saw the light was on in here."
"Come in." Miranda is surprised to realize that she's more curious than anything. A memory of the first night at the Hotel Le Germain in Toronto, lying beside Arthur, the awareness of a beginning. And now here's the ending standing in her doorway half-drunk, legs like pipe cleaners in her skinny jeans, tousled and in disarray—smudges of mascara under her eyes, a sheen of sweat on her nose—but still beautiful, still one of the finest specimens of her kind in Los Angeles, _of_ Los Angeles in a way Miranda knows she never will be, no matter how long she stays here or how hard she tries. Elizabeth steps forward and sinks unexpectedly to the floor. By some small miracle she's managed not to spill the water.
"I'm sorry," she says, "I'm a little wobbly."
"Aren't we all," Miranda says, but as usually happens when she tries to say something funny, her audience seems not to catch the joke. Elizabeth and the dog are both staring at her. "Please don't cry," she says to Elizabeth, whose eyes are shining. "Don't, really, I'm serious. It's too much."
"I'm sorry," Elizabeth says for the third time. That infuriatingly small voice. She sounds like a different person when she's in front of a camera.
"Stop apologizing."
Elizabeth blinks. "You're working on your secret project." She is looking all around the room. She falls silent, and after a moment Miranda succumbs to curiosity and sits on the floor beside Elizabeth to see the room from her vantage point. Paintings and sketches are pinned to the walls. Notes on structure and chronology cover a massive board. There are four pages of story outlines taped to the windowsill.
"What happens next?" Miranda asks. It's easier to talk to Elizabeth when they're sitting side by side, when she doesn't have to look at her.
"I don't know."
"You do know."
"I wish I could tell you how sorry I am," Elizabeth says, "but you've already told me to stop apologizing."
"It's just an awful thing to do."
"I don't think I'm an awful person," Elizabeth says.
"No one ever thinks they're awful, even people who really actually are. It's some sort of survival mechanism."
"I think this is happening because it was supposed to happen." Elizabeth speaks very softly.
"I'd prefer not to think that I'm following a script," Miranda says, but she's tired, there's no sting in her words, it's past four in the morning and too late in every sense. Elizabeth says nothing, just pulls her knees close to her chest and sighs.
In three months Miranda and Arthur will sit in a conference room with their lawyers to work out the final terms of their divorce settlement while the paparazzi smoke cigarettes on the sidewalk outside, while Elizabeth packs to move into the house with the crescent-moon light by the pool. In four months Miranda will be back in Toronto, divorced at twenty-seven, working on a commerce degree, spending her alimony on expensive clothing and consultations with stylists because she's come to understand that clothes are armor; she will call Leon Prevant to ask about employment and a week later she'll be back at Neptune Logistics, in a more interesting job now, working under Leon in Client Relations, rising rapidly through the company until she comes to a point after four or five years when she travels almost constantly between a dozen countries and lives mostly out of a carry-on suitcase, a time when she lives a life that feels like freedom and sleeps with her downstairs neighbor occasionally but refuses to date anyone, whispers "I repent nothing" into the mirrors of a hundred hotel rooms from London to Singapore and in the morning puts on the clothes that make her invincible, a life where the moments of emptiness and disappointment are minimal, where by her midthirties she feels competent and at last more or less at ease in the world, studying foreign languages in first-class lounges and traveling in comfortable seats across oceans, meeting with clients and living her job, breathing her job, until she isn't sure where she stops and her job begins, almost always loves her life but is often lonely, draws the stories of Station Eleven in hotel rooms at night.
But first there's this moment, this lamp-lit room: Miranda sits on the floor beside Elizabeth, whose breath is heavy with wine, and she leans back until she feels the reassuring solidity of the door frame against her spine. Elizabeth, who is crying a little, bites her lip and together they look at the sketches and paintings pinned to every wall. The dog stands at attention and stares at the window, where just now a moth brushed up against the glass, and for a moment everything is still. Station Eleven is all around them.
# **16**
**A TRANSCRIPT OF AN INTERVIEW** conducted by François Diallo, librarian of the town of New Petoskey, publisher and editor of the _New Petoskey News_ , twenty-six years after Miranda and Arthur's last dinner party in Los Angeles and fifteen years after the Georgia Flu:
FRANÇOIS DIALLO: Thank you for taking the time to speak with me today.
KIRSTEN RAYMONDE: My pleasure. What are you writing?
DIALLO: It's my own private shorthand. I made it up.
RAYMONDE: Is it faster?
DIALLO: Very much so. I can transcribe an interview in real time, and then write it out later. Now, I appreciate you talking to me this afternoon. As I mentioned yesterday, I've just started a newspaper, and I've been interviewing everyone who comes through New Petoskey.
RAYMONDE: I'm not sure I have much news to tell you.
DIALLO: If you were to talk about the other towns you've passed through, that would count as news to us. The world's become so local, hasn't it? We hear stories from traders, of course, but most people don't leave their towns anymore. I think my readers will be interested in hearing from people who've been to other places since the collapse.
RAYMONDE: Okay.
DIALLO: And more than that, well, publishing the newspaper has been an invigorating project, but then I thought, Why stop with a newspaper? Why not create an oral history of this time we live in, and an oral history of the collapse? With your permission, I'll publish excerpts from this interview in the next edition, and I'll keep the entirety of the interview for my archives.
RAYMONDE: That's fine. It's an interesting project. I know you're supposed to be interviewing me, but could I ask you a question first?
DIALLO: Of course.
RAYMONDE: You've been a librarian for a long time—
DIALLO: Since Year Four.
RAYMONDE: Those comics I showed you just now, with the space station. Have you ever seen them before, or others in the series?
DIALLO: Never, no, they're not part of any comic-book series I've ever come across. You said someone gave them to you as a gift?
RAYMONDE: Arthur Leander gave them to me. That actor I told you about.
# **17**
**A YEAR BEFORE THE** Georgia Flu, Arthur and Clark met for dinner in London. Arthur was passing through town en route to Paris at a moment when Clark happened to be visiting his parents, and they agreed to meet for dinner in a corner of the city that Clark didn't know especially well. He'd set out early, but when he stepped out of the Tube station he had a vision of his phone lying where he'd left it on his parents' kitchen counter, a map application open on the screen. Clark liked to think he knew London but the truth was he'd spent most of his adult life in New York, secure within the confines of Manhattan's idiot-proof grid, and on this particular evening London's tangle of streets was inscrutable. The side street for which he was searching failed to materialize and he found himself wandering, increasingly late, angry and embarrassed, retracing his footsteps and trying different turns. He hailed a cab when the rain began.
"Easiest two quid I ever made in me life," the cabbie said, when Clark told him the address. The cabbie performed two left turns in rapid succession and they were at the restaurant, on a side street that Clark could've sworn hadn't been there when he'd passed by ten minutes ago. "Of course," the cabbie said, "you don't know where you're going unless you know where you're going," and when Clark went in, Arthur was waiting, caught under a beam of track lighting in a booth at the back. There had been a time when Arthur would never have faced the dining room of a restaurant, long periods when the only way to eat a meal in peace was to sit with his back to the room and hope no one would recognize his hunched shoulders and expensive haircut from behind, but now, Clark realized, Arthur wanted to be seen.
"Dr. Thompson," Arthur said.
"Mr. Leander." The disorientation of meeting one's sagging contemporaries, memories of a younger face crashing into the reality of jowls, under-eye pouches, unexpected lines, and then the terrible realization that one probably looks just as old as they do. Do you remember when we were young and gorgeous? Clark wanted to ask. Do you remember when everything seemed limitless? Do you remember when it seemed impossible that you'd get famous and I'd get a PhD? But instead of saying any of this he wished his friend a happy birthday.
"You remembered."
"Of course," Clark said. "That's one thing I like about birthdays, they stay in one place. Same spot on the calendar, year in, year out."
"But the years keep going faster, have you noticed?"
They settled into the business of ordering drinks and appetizers, and all Clark could think of as they talked was whether or not Arthur had noticed that a couple at a nearby table was looking at him and whispering. If Arthur had noticed, he seemed supremely unconcerned, but the attention put Clark on edge.
"You're going to Paris tomorrow?" Clark asked somewhere between the first martini and the appetizers.
"Visiting my son. Elizabeth's vacationing there with him this week. It's just been a bitch of a year, Clark."
"I know," Clark said. "I'm sorry." Arthur's third wife had recently served him with divorce papers, and her predecessor had taken their son to Jerusalem.
"Why Israel?" Arthur said miserably. "That's the part I don't understand. Of all places."
"Wasn't she a history major in college? Maybe that's what she likes about it, all the history in the place."
"I think I'll have the duck," Arthur said, and this was the last they spoke of Elizabeth, actually the last they spoke of anything of substance. "I've been indecently lucky," Arthur said later that night, on his fourth martini. It was a line he'd been using a great deal lately. Clark wouldn't have been bothered by it if he hadn't seen Arthur use it on _Entertainment Weekly_ a month or two earlier. The restaurant was one of those large, under-lit places that seemed to recede into shadow at the periphery, and in the murky middle distance Clark saw a pinpoint of green light that meant someone was recording Arthur on a cell phone. Clark felt increasingly stiff. He was aware of the whispers that had sprung up, the glances from other tables. Arthur was talking about an endorsement deal of some kind, men's watches, his gestures loose. He was telling an animated story about his meeting with the watch executives, some kind of humorous misunderstanding in the boardroom. He was performing. Clark had thought he was meeting his oldest friend for dinner, but Arthur wasn't having dinner with a friend, Clark realized, so much as having dinner with an audience. He felt sick with disgust. When he left a short time later he found himself wandering, even though by now he'd oriented himself and knew how to get back to the Tube station. Cold rain, the sidewalk shining, the _shhh_ of car tires on the wet street. Thinking about the terrible gulf of years between eighteen and fifty.
# **18**
DIALLO: I'll ask you more about Arthur Leander and the comics in a moment. Perhaps I could ask you a few questions about your life first?
RAYMONDE: You know me, François. We've been coming through this town for years.
DIALLO: Yes, yes, of course, but some of our readers might not know you, or the Symphony. I've been giving copies of the paper to traders, asking them to distribute it along their routes. You've been acting since you were very young, isn't that right?
RAYMONDE: Very young. I was in a commercial when I was three. Do you remember commercials?
DIALLO: I do, regrettably. What were you selling?
RAYMONDE: I don't actually remember the thing itself, the commercial, but I remember my brother telling me it was for arrowroot biscuits.
DIALLO: I remember those too. What came after the biscuits?
RAYMONDE: I actually don't remember, but my brother told me a little. He said I did more commercials, and when I was six or seven I had a recurring role on a televised... on a televised show.
DIALLO: Do you remember which show?
RAYMONDE: I wish I did. I can't remember anything about it. I think I've mentioned before, I have some problems with memory. I can't remember very much from before the collapse.
DIALLO: It's not uncommon among people who were children when it happened. And the Symphony? You've been with them for a while, haven't you?
RAYMONDE: Since I was fourteen.
DIALLO: Where did they find you?
RAYMONDE: Ohio. The town where we ended up after we left Toronto, my brother and I, and then after he died I was there by myself.
DIALLO: I didn't know they went that far south.
RAYMONDE: They only went down there once. It was a failed experiment. They wanted to expand the territory, so that spring they followed the Maumee River down past the ruins of Toledo, and then the Auglaize River into Ohio, and they eventually walked into the town where I lived.
DIALLO: Why do you say it was a failed experiment?
RAYMONDE: I'll always be grateful that they passed through my town, but the expedition was a disaster for them. By the time they reached Ohio they'd lost an actor to some illness on the road, something that looked like malaria, and they got shot at three times in various places. One of the flautists got hit and almost died of a gunshot wound. They—we—the Symphony never left their usual territory again.
DIALLO: It seems like a very dangerous life.
RAYMONDE: No, that was years ago. It's much less dangerous than it used to be.
DIALLO: The other towns you pass through, are they very different from here?
RAYMONDE: The places we return to more than once aren't dissimilar to here. Some places, you pass through once and never return, because you can tell something's very wrong. Everyone's afraid, or it seems like some people have enough to eat and other people are starving, or you see pregnant eleven-year-olds and you know the place is either lawless or in the grip of something, a cult of some kind. There are towns that are perfectly reasonable, logical systems of governance and such, and then you pass through two years later and they've slid into disarray. All towns have their own traditions. There are towns like this one, where you're interested in the past, you've got a library—
DIALLO: The more we know about the former world, the better we'll understand what happened when it fell.
RAYMONDE: But everyone knows what happened. The new strain of swine flu and then the flights out of Moscow, those planes full of patient zeros...
DIALLO: Nonetheless, I believe in understanding history.
RAYMONDE: Fair enough. Some towns, as I was saying, some towns are like this one, where they want to talk about what happened, about the past. Other towns, discussion of the past is discouraged. We went to a place once where the children didn't know the world had ever been different, although you'd think all the rusted-out automobiles and telephones wires would give them a clue. Some towns are easier to visit than others. Some places have elected mayors or they're run by elected committees. Sometimes a cult takes over, and those towns are the most dangerous.
DIALLO: In what sense?
RAYMONDE: In the sense that they're unpredictable. You can't argue with them, because they live by an entirely different logic. You come to a town where everyone's dressed all in white, for example. I'm thinking of a town we visited once just outside our usual territory, north of Kincardine, and then they tell you that they were saved from the Georgia Flu and survived the collapse because they're superior people and free from sin, and what can you say to that? It isn't logical. You can't argue with it. You just remember your own lost family and either want to cry or harbor murderous thoughts.
#
# **19**
**SOMETIMES THE TRAVELING SYMPHONY** thought that what they were doing was noble. There were moments around campfires when someone would say something invigorating about the importance of art, and everyone would find it easier to sleep that night. At other times it seemed a difficult and dangerous way to survive and hardly worth it, especially at times when they had to camp between towns, when they were turned away at gunpoint from hostile places, when they were traveling in snow or rain through dangerous territory, actors and musicians carrying guns and crossbows, the horses exhaling great clouds of steam, times when they were cold and afraid and their feet were wet. Or times like now when the heat was unrelenting, July pressing down upon them and the blank walls of the forest on either side, walking by the hour and wondering if an unhinged prophet or his men might be chasing them, arguing to distract themselves from their terrible fear.
"All I'm saying," Dieter said, twelve hours out of St. Deborah by the Water, "is that quote on the lead caravan would be way more profound if we hadn't lifted it from _Star Trek_." He was walking near Kirsten and August.
_Survival is insufficient:_ Kirsten had had these words tattooed on her left forearm at the age of fifteen and had been arguing with Dieter about it almost ever since. Dieter harbored strong anti-tattoo sentiments. He said he'd seen a man die of an infected tattoo once. Kirsten also had two black knives tattooed on the back of her right wrist, but these were less troubling to Dieter, being much smaller and inked to mark specific events.
"Yes," Kirsten said, "I'm aware of your opinion on the subject, but it remains my favorite line of text in the world." She considered Dieter one of her dearest friends. The tattoo argument had lost all of its sting over the years and had become something like a familiar room where they met.
Midmorning, the sun not yet broken over the tops of the trees. The Symphony had walked through most of the night. Kirsten's feet hurt and she was delirious with exhaustion. It was strange, she kept thinking, that the prophet's dog had the same name as the dog in her comic books. She'd never heard the name _Luli_ before or since.
"See, that illustrates the whole problem," Dieter said. "The best Shakespearean actress in the territory, and her favorite line of text is from _Star Trek_."
"The whole problem with what?" Kirsten felt that she might actually be dreaming at this point, and she longed desperately for a cool bath.
"It's got to be one of the best lines ever written for a TV show," August said. "Did you see that episode?"
"I can't say I recall," Dieter said. "I was never really a fan."
"Kirsten?"
Kirsten shrugged. She wasn't sure if she actually remembered anything at all of _Star Trek_ , or if it was just that August had told her about it so many times that she'd started to picture his stories in her head.
"Don't tell me you've never seen _Star Trek: Voyager_ ," August said hopefully. "That episode with those lost Borg and Seven of Nine?"
"Remind me," Kirsten said, and he brightened visibly. While he talked she allowed herself to imagine that she remembered it. A television in a living room, a ship moving through the night silence of space, her brother watching beside her, their parents—if she could only remember their faces—somewhere near.
The Symphony stopped to rest in the early afternoon. Would the prophet send men after them, or had they been allowed to leave? The conductor sent scouts back down the road. Kirsten climbed up to the driver's bench of the third caravan. A dull buzz of insects from the forest, tired horses grazing at the side of the road. The wildflowers growing by the roadside were abstract from this vantage point, paint dots of pink and purple and blue in the grass.
Kirsten closed her eyes. A memory from early childhood, before the collapse: sitting with a friend on a lawn, a game where they closed their eyes and concentrated hard and tried to read one another's minds. She had never entirely let go of the notion that if she reached far enough with her thoughts she might find someone waiting, that if two people were to cast their thoughts outward at the same moment they might somehow meet in the middle. Charlie, where are you? She knew the effort was foolish. She opened her eyes. The road behind them was still empty. Olivia was picking flowers below.
"A little farther," the conductor was saying, somewhere below, and the horses were being harnessed again, the caravans creaking into motion, the exhausted Symphony walking onward through the heat until hours later they set up camp by the roadside, the ones who remembered the lost world thinking longingly of air-conditioning even after all these years.
"It just came out of a vent?" Alexandra asked.
"I believe so," Kirsten said. "I'm too tired to think."
They'd walked for all but five of the eighteen hours since they'd left St. Deborah by the Water, through the night and morning and deep into the afternoon, trying to put as much distance as possible between themselves and the prophet. Some of them took turns trying to sleep in the moving caravans, others walking and walking until their thoughts burned out one by one like dying stars and they fell into a fugue state wherein all that mattered or had ever existed were these trees, this road, the counterpoint rhythms of human footsteps and horses' hooves, moonlight turning to darkness and then the summer morning, caravans rippling like apparitions in the heat, and now the Symphony was scattered here and there by the roadside in a state of semi-collapse while they waited for dinner to be ready. Half the Symphony had set off in pairs to hunt rabbits. The cook fire sent a plume of white smoke like a marker into the sky.
"Air-conditioning came out of a vent," August confirmed. "You'd press a button, and _whoosh!_ Cold air. I had one in my bedroom."
Kirsten and August were setting up tents, and Alexandra, whose tent had been set up already, was lying on her back staring up at the sky.
"Oh," Alexandra said. "So it was electricity, or gas?"
August looked at the tuba, who was sitting nearby with his daughter half-asleep in his arms. Olivia had announced that she was too tired to wait for dinner, so he'd been telling her a bedtime story about a mermaid while Lin set up their tent.
"Electricity," the tuba said. "Air conditioners were electric." He craned his neck to see his daughter's face. "Is she asleep?"
"I think so," Kirsten said. This was when she heard the exclamation from the third caravan—"What the _fuck_ ," someone said, "goddamnit, what _is_ this?"—and she stood up in time to see the first cello haul a girl out of the caravan by the arm. Olivia sat up, blinking.
"A stowaway." August was grinning. He'd been a stowaway himself once. "We haven't had a stowaway in years."
The stowaway was the girl who'd followed Kirsten in St. Deborah by the Water. She was crying and sweaty, her skirt soaked with urine. The first cello lifted her to the ground.
"She was under the costumes," the first cello said. "I went in looking for my tent."
"Get her some water," Gil said.
The conductor swore under her breath and looked off down the road behind them while the Symphony gathered. The first flute gave the girl one of her water bottles.
"I'm sorry," the girl said, "I'm so sorry, please don't make me go back—"
"We can't take children," the conductor said. "This isn't like running away and joining the circus." The girl looked confused. She didn't know what a circus was. "Incidentally," the conductor said to the assembled company, "this is why we check the caravans before we depart."
"We left St. Deborah in kind of a hurry," someone muttered.
"I had to leave," the girl said. "I'm so sorry, I'm sorry, I'll do anything, just—"
"Why did you have to leave?"
"I'm promised to the prophet," the girl said.
"You're what?"
The girl was crying now. "I didn't have any choice," she said. "I was going to be his next wife."
"Jesus," Dieter said. "This goddamn world." Olivia was standing by her father, rubbing her eyes. The tuba lifted her into his arms.
"He has more than one?" asked Alexandra, still blissfully ignorant.
"He has four," the girl said, sniffling. "They live in the gas station."
The conductor gave the girl a clean handkerchief from her pocket. "What's your name?"
"Eleanor."
"How old are you, Eleanor?"
"Twelve."
"Why would he marry a twelve-year-old?"
"He had a dream where God told him he was to repopulate the earth."
"Of course he did," the clarinet said. "Don't they all have dreams like that?"
"Right, I always thought that was a prerequisite for being a prophet," Sayid said. "Hell, if _I_ were a prophet—"
"Your parents allowed this?" the conductor asked, simultaneously making a _Shut up_ motion in the direction of the clarinet and Sayid.
"They're dead."
"I'm sorry to hear that."
"Were you spying on me in St. Deborah?" Kirsten asked.
The girl shook her head.
"No one told you to watch us?"
"No," she said.
"Did you know Charlie and the sixth guitar?"
Eleanor frowned. "Charlie and Jeremy?"
"Yes. Do you know where they went?"
"They went to the—to the Museum of Civilization." Eleanor said _museum_ very carefully, the way people sound out foreign words of whose pronunciation they're uncertain.
"The what?"
August whistled softly. "They told you that's where they were going?"
"Charlie said if I could ever get away, that's where I could find them."
"I thought the Museum of Civilization was a rumor," August said.
"What is it?" Kirsten had never heard of it.
"I heard it was a museum someone set up in an airport." Gil was unrolling his map, blinking shortsightedly. "I remember a trader telling me about it, years back."
"We're headed there anyway, aren't we? It's supposed to be outside Severn City." The conductor was peering over his shoulder. She touched a point on the map, far to the south along the lakeshore.
"What do we know about it?" the tuba asked. "Do people still live there?"
"I've no idea."
"It could be a trap," the tuba murmured. "The girl could be leading us there."
"I know," the conductor said.
What to do with Eleanor? They knew they risked accusations of kidnapping and they had long adhered to a strict policy of non-intervention in the politics of the towns through which they passed, but no one could imagine delivering a child bride back to the prophet. Had a grave marker with her name on it already been driven into the earth? Would a grave be dug if she returned? Nothing for it but to take the girl and press on into the unknown south, farther down the eastern shore of Lake Michigan than they'd ever been.
They tried to engage Eleanor in conversation over dinner. She'd settled into a wary stillness, the watchfulness of orphans. She rode in the back of the first caravan, so that she'd be at least momentarily out of sight if anyone approached the Symphony from the rear. She was polite and unsmiling.
"What do you know about the Museum of Civilization?" they asked.
"Not very much," she said. "I just heard people talk about it sometimes."
"So Charlie and Jeremy had heard about it from traders?"
"Also the prophet's from there," she said.
"Does he have family there?"
"I don't know."
"Tell us about the prophet," the conductor said.
He'd come to St. Deborah by the Water not long after the Symphony had left Charlie and Jeremy there, the head of a sect of religious wanderers. The sect had moved into the Walmart at first, a communal encampment in what had once been the Lawn and Garden Department. They told the townspeople they'd come in peace. A few people were uneasy about them, this new population with vague stories about travel in the south, in the territory once known as Virginia and beyond—rumors held that the south was exceptionally dangerous, bristling with guns, and what might they have done to survive down there?—but the new arrivals were friendly and self-sufficient. They shared their meat when they hunted. They helped with chores and seemed harmless. There were nineteen of them, and they mostly kept to themselves; some time passed before the townspeople realized that the tall man with blond hair who seemed to be their leader was known only as the prophet and had three wives. "I am a messenger," he said, when introduced to people. No one knew his real name. He said he was guided by visions and signs. He said he had prophetic dreams. His followers said he was from a place called the Museum of Civilization, that he'd taken to the road in childhood to spread his message of light. They had a story about setting out in the early morning and then stopping for the day only a few hours later, because the prophet had seen three ravens flying low over the road ahead. No one else had seen the ravens, but the prophet was insistent. The next morning they came upon a collapsed bridge and a riverside funeral, women singing, voices rising over three white shrouds. Three men had died when the bridge fell into the river. "Don't you see?" the prophet's followers said. "If not for his vision that would have been us."
When the winter fever struck St. Deborah by the Water, when the mayor died, the prophet added the mayor's wife to his collection and moved with his followers into the gas station in the center of town. No one had quite realized how much weaponry they had. Their stories about travel in the south began to fall into place. Within a week it became obvious that the town was his. Eleanor didn't know why the prophet's dog was named Luli.
# **20**
**TWO DAYS OUT OF** St. Deborah by the Water, the Symphony came upon a burnt-out resort town. A fire had swept through some years ago and now the town was a meadow with black ruins standing. A sea of pink flowers had risen between the shards of buildings. The charred shells of hotels stood along the lakeshore and a brick clock tower was still standing a few blocks inland, the clock stopped forever at eight fifteen.
The Symphony walked armed and on full alert, Olivia and Eleanor in the back of the lead caravan for safety, but they saw no signs of human life. Only deer grazing on overgrown boulevards and rabbits burrowing in ashy shadows, seagulls watching from lampposts. The Symphony shot two deer for dinner later, pried the arrows from their ribs, and strung them over the hoods of the first two caravans. The lakeshore road was a complicated patchwork of broken pavement and grass.
On the far side of town they reached the limits of the fire, a place where the trees stood taller and the grasses and wildflowers changed. Just beyond the fire line they found an old baseball field, where they stopped to let the horses graze. Half-collapsed bleachers slumped into tall grass. Three banks of floodlights had stood over this field, but two had fallen. Kirsten knelt to touch the thick glass of a massive lamp, trying to imagine the electricity that it had conducted, the light pouring down. A cricket landed on her hand and sprang away.
"You couldn't even look directly at them," Jackson said. He hadn't liked baseball much but had gone a few times as a child anyway, sitting dutifully in the stands with his father.
"You going to stand there all day?" Sayid asked, and Kirsten glared at him but returned to work. They were cutting grass for the horses, to carry with them in case there was a place farther down the road where there was nothing for the animals to eat. Eleanor sat by herself in the shade of the first caravan, humming tunelessly, braiding and unbraiding pieces of grass. She'd spoken very little since they'd found her.
The scouts reported a school, just beyond the trees at the edge of the field. "Take a couple of the others and check the school for instruments," the conductor told Kirsten and August. They set out with Jackson and the viola. It was a degree or two cooler in the shade of the forest, the ground soft with pine needles underfoot.
"I'm glad to get out of that field," Viola said. She'd had a different name when she was younger, but had taken on the name of her instrument after the collapse. She sniffled quietly. She was allergic to grass. The forest had crept up to the edges of the school parking lot and sent an advance party out toward the building, small trees growing through cracks in the pavement. There were a few cars parked on flat tires.
"Let's watch for a moment," August said, and they stood for a while at the edge of the woods. The saplings in the parking lot were stirred by a breeze, but otherwise nothing moved in the landscape except birds and the shimmer of heat waves. The school was dark and still. Kirsten brushed sweat from her forehead with the back of her hand.
"I don't think anyone's here," Jackson said finally. "The place looks desolate."
"I don't know," Viola muttered. "Schools give me the creeps."
"You volunteered," Kirsten said.
"Only because I hate cutting grass."
They skirted the building first, looking in windows, and saw only ruined classrooms with graffiti on the walls. The back door gaped open into a gymnasium. Sunlight poured through a hole in the ceiling, a few weeds growing in the debris where light touched the floor. This place had been used as a shelter, or possibly a field hospital. A jumble of cots had been piled in a corner of the room. Later someone had built a fire under the hole in the ceiling, old ashes mixed with animal bones. Easy to read the broad outlines of the room's history, the shelter that had later become a place where people cooked meals, but as always all of the details were missing. How many people had stayed here? Who were they? Where had they gone? On the opposite side of the gym, a set of doors opened into a corridor lined with classrooms, sunlight spilling across the floor from the broken-down front door at the end.
This had been a small school, six classrooms. The floor strewn with broken glass, unidentifiable garbage, the remains of binders and textbooks. They picked their way between rooms, searching, but there was only wreckage and disarray. Layers of graffiti, unreadable names in puffy dripping letters across blackboards, old messages: "Jasmine L., if you see this, go to my dad's lake house.—Ben." Overturned desks. A fire had darkened a corner of a classroom before someone had put it out or it had died on its own. The band room was immediately identifiable as such by the heap of twisted music stands on the floor. The sheet music was gone—perhaps used to start the cooking fire in the gymnasium—and there were no instruments. But Viola found half a jar of rosin in a closet, and Kirsten found a mouthpiece for a flute buried under trash. Words spray-painted on the north wall: "The end is here."
"Creepy as hell," Viola said.
Jackson appeared in the doorway. "There's a skeleton in the men's room."
August frowned. "How old?"
"Old. Bullet hole in the skull."
"Why would you look in the bathroom?"
"I was hoping for soap."
August nodded and disappeared down the hall.
"What's he doing?" Viola asked.
"He likes to say a prayer over the dead." Kirsten was crouched on the floor, poking through the debris with a broken ruler. "Help me check the lockers before we go."
But every student locker had been emptied, doors hanging askew. Kirsten picked up a couple of mildewed binders to study the stickers and the Sharpie incantations—"Lady Gaga iz da bomb," "Eva + Jason 4 evah," "I ♥ Chris," etc.—and on a cooler day she might have spent more time here, interested as always in any clues she could find about the lost world, but the air was foul and still, the heat unendurable, and when August emerged from the men's room it was a relief to walk out into the sunlight, the breeze, and the chatter of crickets.
"Christ," Jackson said, "I don't know how you two can stand going into these places."
"Well, we don't go into public bathrooms, for starters," August said.
"I just wanted some soap."
"Yeah, but it's a dumb move. Someone always got executed in the bathroom."
"Yeah, like I said, I don't know how you stand it."
We stand it because we were younger than you were when everything ended, Kirsten thought, but not young enough to remember nothing at all. Because there isn't much time left, because all the roofs are collapsing now and soon none of the old buildings will be safe. Because we are always looking for the former world, before all the traces of the former world are gone. But it seemed like too much to explain all this, so she shrugged instead of answering him.
The Symphony was resting under the trees by the side of the road. Most of them were napping. Eleanor was showing Olivia how to make a daisy chain. The clarinet was moving languidly through a series of yoga poses while the conductor and Gil studied a map.
"A mouthpiece!" the first flute said, when August revealed their discoveries, and August was the person in the Symphony who irritated her the most, but she actually clapped her hands and threw her arms around his neck.
"What was in the school?" Alexandra asked, when the horses were harnessed and the Symphony had set out again. She wanted very much to go into buildings with Kirsten and August, but Kirsten never let her join them.
"Nothing worth mentioning," Kirsten said. Carefully not thinking about the skeleton in the men's room, her eyes on the road. "Just that flute piece and a lot of debris."
# **21**
**THE INTERVIEW IN** Year Fifteen, continued:
FRANÇOIS DIALLO: Now, I believe you were very young when the Georgia Flu came, when the collapse happened.
KIRSTEN RAYMONDE: I was eight.
DIALLO: Forgive me, this is a fascination of mine when I speak with people who were children back then, at the time of the collapse, and I'm not sure how to phrase this, but I want to know what you think about when you consider how the world's changed in your lifetime.
RAYMONDE: [silence]
DIALLO: Or to phrase it differently—
RAYMONDE: I understood the question. I'd prefer not to answer.
DIALLO: Okay. All right. I'm curious about your tattoo.
RAYMONDE: The text on my arm? "Survival is insufficient"?
DIALLO: No, no, the other one. The two black knives on your right wrist.
RAYMONDE: You know what tattoos like this mean.
DIALLO: But perhaps you could just tell me—
RAYMONDE: I won't talk about it, François, and you know better than to ask.
# **22**
**WHEN KIRSTEN THOUGHT** of the ways the world had changed in her lifetime, her thoughts always eventually circled back to Alexandra. Alexandra knew how to shoot, but the world was softening. There was a fair chance, Kirsten thought, that Alexandra would live out her life without killing anyone. She was a younger fifteen-year-old than Kirsten had ever been.
Now Alexandra walked quietly, sullen because she hadn't been allowed to join the expedition to the school. The Symphony walked through the end of the day, clouds gathering and the air pressing down from above, rivulets of sweat running down Kirsten's back. The sky low and dark by late afternoon. They were moving through a rural area, no driveways. Rusted-out cars here and there along the road, abandoned where they'd run out of gas, the caravans weaving carefully around them. Flashes of lightning and thunder, at first distant and then close. They waited out the rainstorm in the trees by the side of the road at twilight, pitched their tents on the wet ground when it was over.
"I dreamt last night I saw an airplane," Dieter whispered. They were lying a few feet apart in the dark of his tent. They had only ever been friends—in a hazy way Kirsten thought of him as family—but her thirty-year-old tent had finally fallen apart a year ago and she hadn't yet managed to find a new one. For obvious reasons she was no longer sharing a tent with Sayid, so Dieter, who had one of the largest tents in the Symphony, had been hosting her. Kirsten heard soft voices outside, the tuba and the first violin on watch. The restless movements of the horses, penned between the three caravans for safety.
"I haven't thought of an airplane in so long."
"That's because you're so young." A slight edge to his voice. "You don't remember anything."
"I do remember things. Of course I do. I was eight."
Dieter had been twenty years old when the world ended. The main difference between Dieter and Kirsten was that Dieter remembered everything. She listened to him breathe.
"I used to watch for it," he said. "I used to think about the countries on the other side of the ocean, wonder if any of them had somehow been spared. If I ever saw an airplane, that meant that somewhere planes still took off. For a whole decade after the pandemic, I kept looking at the sky."
"Was it a good dream?"
"In the dream I was so happy," he whispered. "I looked up and there it was, the plane had finally come. There was still a civilization somewhere. I fell to my knees. I started weeping and laughing, and then I woke up."
There was a voice outside then, someone saying their names. "Second watch," Dieter whispered. "We're up."
The first watch was going to sleep. They had nothing to report. "Just goddamned trees and owls," the tuba muttered. The second watch agreed on the usual arrangement: Dieter and Sayid would scout the road a half mile behind them, Kirsten and August would keep watch at the camp, the fourth guitar and the oboe would scout a half mile ahead. The scouts set off in their separate directions and Kirsten was alone with August. They circled the camp perimeter and then stood on the road, listening and watching for movement. Clouds breaking apart to reveal the stars overhead. The brief flare of a meteor, or perhaps a falling satellite. Is this what airplanes would have looked like at night, just streaks of light across the sky? Kirsten knew they'd flown at hundreds of miles per hour, inconceivable speeds, but she wasn't sure what hundreds of miles per hour would have looked like. The forest was filled with small noises: rainwater dripping from the trees, the movements of animals, a light breeze.
She didn't remember what airplanes had looked like in flight but she did remember being inside one. The memory was sharper than most of her other memories from the time before, which she thought must mean that this had been very close to the end. She would have been seven or eight years old, and she'd gone to New York City with her mother, though she didn't remember why. She remembered flying back to Toronto at night, her mother drinking a glass of something with ice cubes that clinked and caught the light. She remembered the drink but not her mother's face. She'd pressed her forehead to the window and saw clusters and pinpoints of light in the darkness, scattered constellations linked by roads or alone. The beauty of it, the loneliness, the thought of all those people living out their lives, each porch light marking another house, another family. Here on this road in the forest two decades later, clouds shifted to reveal the moon and August glanced at her in the sudden light.
"Hair on the back of my neck's standing up," he murmured. "You think we're alone out here?"
"I haven't heard anything." They made another slow circle of the camp. Barely audible voices from inside one or two of the tents, the sighs and soft movements of horses. They listened and watched, but the road was still.
"These are the times when I want to stop," August whispered. "You ever think about stopping?"
"You mean not traveling anymore?"
"You ever think about it? There's got to be a steadier life than this."
"Sure, but in what other life would I get to perform Shakespeare?"
There was a sound just then, a disturbance passing over the surface of the night as quickly as a stone dropped into water. A cry, cut off abruptly? Had someone called out? If she'd been alone, Kirsten might have thought she'd imagined it, but August nodded when she looked at him. The sound had come from somewhere far down the road, in the direction from which they'd come. They were still, straining to hear, but heard nothing.
"We have to raise the third watch." Kirsten drew her two best knives from her belt. August disappeared among the tents. She heard his muffled voice—"I don't know, a sound, maybe a voice down the road, I need you to take our places so we can go check it out"—and two shadows emerged to replace them, yawning and unsteady on their feet.
August and Kirsten set off as quickly and quietly as possible in the direction of the sound. The forest was a dark mass on either side, alive and filled with indecipherable rustlings, shadows like ink against the glare of moonlight. An owl flew low across the road ahead. A moment later there was a distant beating of small wings, birds stirred from their sleep, black specks rising and wheeling against the stars.
"Something disturbed them," Kirsten said quietly, her mouth close to August's ear.
"The owl?" His voice as soft as hers.
"I thought the owl was flying at a different angle. The birds were more to the north."
"Let's wait."
They waited in the shadows at the side of the road, trying to breathe quietly, trying to look everywhere at once. The claustrophobia of the forest. The first few trees visible before her, monochrome contrasts of black shadow and white moonlight, and beyond that an entire continent, wilderness uninterrupted from ocean to ocean with so few people left between the shores. Kirsten and August watched the road and the forest, but if anything was watching them back, it wasn't apparent.
"Let's walk farther," August whispered.
They resumed their cautious progress down the road, Kirsten gripping her knives so tightly that her heartbeat throbbed in the palms of her hands. They walked far beyond the point where the scouts should have been, two miles, three, looking for signs. At first light they returned the way they had come, speechless in a world of riotous birdsong. There was no trace of the scouts, nothing at the edges of the forest, no footprints, no signs of large animals, no obviously broken branches or blood. It was as though Dieter and Sayid had been plucked from the face of the earth.
# **23**
**" I JUST DON'T UNDERSTAND,"** the tuba said, midmorning, after several hours of searching for Sayid and Dieter. No one understood. No one responded. The disappearances were incomprehensible. They could find no trace. The Symphony searched in teams of four, grimly, methodically, but the forest was dense and choked with underbrush; they could have passed within feet of Dieter and Sayid and not known it. In those first hours there were moments when Kirsten caught herself thinking that there must have simply been some misunderstanding, that Dieter and Sayid must have somehow walked by them in the dark, somehow gone the wrong way down the road, that they'd reappear with apologies at any moment, but scouts had gone back and forth on the road for miles. Again and again Kirsten stopped still in the forest, listening. Was someone watching her? Just now, had someone stepped on a branch? But the only sounds were of the other search teams, and everyone felt watched. They met in the forest and on the road at intervals, looked at one another and said nothing. The slow passage of the sun across the sky, the air over the road unsteady with heat waves.
When night began to fall they gathered by the lead caravan, which had once been an extended-bed Ford pickup truck. "Because survival is insufficient," words painted on the canopy in answer to the question that had dogged the Symphony since they'd set out on the road. The words were very white in the rising evening. Kirsten stood by Dieter's favorite horse, Bernstein, and pressed her hand flat against his side. He stared at her with an enormous dark eye.
"We have traveled so far together," the conductor said. There are certain qualities of light that blur the years. Sometimes when Kirsten and August were on watch together at dawn, she would glance at him as the sun rose and for a fleeting instant she could see what he'd looked like as a boy. Here on this road, the conductor looked much older than she had an hour earlier. She ran a hand through her short gray hair. "There have been four times," she said, "in all these years, when Symphony members have become separated from the Symphony, and in every single instance they have followed the separation protocol, and we've been reunited at the destination. Alexandra?"
"Yes?"
"Will you state the separation protocol, please?" It had been drilled into all of them.
"We never travel without a destination," Alexandra said. "If we're ever, if you're ever separated from the Symphony on the road, you make your way to the destination and wait."
"And what is the current destination?"
"The Museum of Civilization in the Severn City Airport."
"Yes." The conductor was quiet, looking at them. The forest was in shadow now, but there was still some light in the corridor of sky above the road, the last pink of sunset streaking the clouds. "I have been on the road for fifteen years," she said, "and Sayid's been with me for twelve. Dieter for even longer."
"He was with me in the beginning," Gil said. "We walked out of Chicago together."
"I leave neither of them willingly." The conductor's eyes were shining. "But I won't risk the rest of you by staying here a day longer."
That night they kept a double watch, teams of four instead of two, and set out before dawn the following morning. The air was damp between the walls of the forest, the clouds marbled overhead. A scent of pine in the air. Kirsten walked by the first caravan, trying to think of nothing. A sense of being caught in a terrible dream.
They stopped at the end of the afternoon. The fevered summers of this century, this impossible heat. The lake glittered through the trees. This had been one of those places that wasn't quite suburbia but wasn't quite not, an in-between district where the houses stood on wooded lots. They were within three days of the airport now. Kirsten sat on a log with her head in her hands, thinking, Where are you, where are you, where are you, and no one bothered her until August came to sit nearby.
"I'm sorry," he said.
"I think they were taken," she said without raising her head, "and I can't stop thinking about what the prophet was saying in St. Deborah, that thing about the light."
"I don't think I heard it. I was packing up."
"They call themselves the light."
"What about it?"
"If you are the light," she said, "then your enemies are darkness, right?"
"I suppose."
"If you are the light, if your enemies are darkness, then there's nothing that you cannot justify. There's nothing you can't survive, because there's nothing that you will not do."
He sighed. "We can only remain hopeful," he said. "We have to assume that the situation will become more clear."
But four teams set out in search of dinner, and only three and a half returned.
"I turned and she was gone," Jackson said of Sidney, the clarinet. He'd returned to the camp alone and shaken. They'd found a stream, Jackson said, about a quarter mile down the road in the direction from which they'd come. He'd knelt on the bank to fill the water container, and when he looked up she had vanished. Had she fallen in? No, he said, he would have heard a splash, and he was downstream, so she would have passed him. It was a small stream and the banks weren't steep. There was just the woods all around him, a sense of being watched. He called her name but she was nowhere. He noticed then that the birdsong had stopped. The woods had gone still.
No one spoke for a moment when he'd finished telling the story. The Symphony gathered close around him.
"Where's Olivia?" Lin asked suddenly. Olivia was in the back of the first caravan, playing with a rag doll. "I want you in my sight," Lin whispered. "Not just within sight, within reach. Do you understand?"
"She was close with Dieter," the first oboe said. This was true, and they were all silent, thinking of the clarinet and searching their memories for clues. Had she seemed like herself lately? None of them were sure. What did it mean to seem like yourself, in the course of such unspeakable days? How was anyone supposed to seem?
"Are we being _hunted_?" Alexandra asked. It seemed plausible. Kirsten looked over her shoulder into the shadows of the trees. A search party was organized, but the light was gone. Lighting a fire seemed too dangerous so they ate dinner from the preserved food stores, rabbit jerky and dried apples, and settled in for an uneasy night. In the morning they delayed for five hours, searching, but they couldn't find her. They set off into another searing day.
"Is it logical that they could have _all_ been taken?" August was walking beside Kirsten. "Dieter, Sayid, the clarinet?"
"How could anyone overpower them so silently?" There was a lump in her throat. It was difficult to speak. "Maybe they just left."
"Abandoned us?"
"Yes."
"Why would they?"
"I don't know."
Later in the day someone thought to search the clarinet's belongings, and found the note. The beginning of a letter: "Dear friends, I find myself immeasurably weary and I have gone to rest in the forest." It ended there. The date suggested that either it had been written eleven months earlier or that the clarinet didn't know what year or month it was, one or the other. Neither scenario was unlikely. This was an era when exact dates were seldom relevant, and keeping track required a degree of dedication. The note had been folded and refolded several times, soft along the creases.
"It seems more theoretical than anything," the first cello said. "Like she wrote it a year ago and then changed her mind. It doesn't prove anything."
"That's assuming she wrote it a year ago," said Lin. "She could've written it last week. I think it shows suicidal intent."
"Where were we a year ago? Does anyone remember?"
"Mackinaw City," August said. "New Petoskey, East Jordan, all those little places down the coast on the way to New Sarnia."
"I don't remember her seeming different a year ago," Lin said. "Was she sad?"
No one was sure. They all felt they should have been paying more attention. Still the scouts reported no one behind or ahead of them on the road. Impossible not to imagine that they were being watched from the forest.
What was the Symphony without Dieter and the clarinet and Sayid? Kirsten had thought of Dieter as a sort of older brother, she realized, perhaps a cousin, a fixture in her life and in the life of the Symphony. It seemed in some abstract way impossible that the Symphony continued without him. She had never been close with the clarinet, but the clarinet was conspicuous in her absence. She only spoke with Sayid to argue with him now, but the thought of him having come to harm was sheer agony. Her breath was shallow in her chest and the tears were silent and constant.
Late in the day, she found a folded piece of paper in her pocket. She recognized August's handwriting.
_A fragment for my friend—_
_If your soul left this earth I would follow and find you_
_Silent, my starship suspended in night_
She'd never seen his poetry before and was impossibly moved by it. "Thank you," she said when she saw him next. He nodded.
The land became wilder, the houses subsiding. They had to stop three times to clear fallen trees. They used two-handed saws, working as quickly as possible with sweat soaking through their clothes, scouts posted here and there watching the road and the forest, jumping and aiming their weapons at small sounds. Kirsten and August walked out ahead over the conductor's objections. A half mile beyond the stalled caravans, they came upon a rolling plain.
"A golf course," August said. "You know what that means." They'd found two full bottles of scotch and a can of miraculously still-edible cocktail olives in a golf-course clubhouse once, and August had been trying to replicate the experience ever since.
The clubhouse was at the end of a long driveway, obscured behind a bank of trees. It was burnt out, the roof draped like fabric from the three remaining walls. Golf carts were toppled over on their sides in the grass. The sky was darkening now and it was hard to see much of the clubhouse interior in the pre-storm light, just glints of shattered glass where the windows had been. Too dangerous to go in with the roof half-fallen. On the far side they found a small man-made lake with a rotted pier, a flicker of movement under the surface. They walked back to the caravans for the fishing equipment. The first and third cellos were sawing at the last fallen tree.
Back at the golf-course pond there were so many fish that it was possible to catch them with the net alone, scooping them up from the overcrowded water. The fish were small brownish things, unpleasant to the touch. Thunder in the distance and then a short time later the first drops of rain. August, who carried his instrument at all times, wrapped his violin case in a plastic sheet he kept in his bag. They worked through the downpour, Kirsten dragging the net through the water, August gutting and cleaning. He knew she couldn't stand to gut fish—something she'd seen on the road that first year out of Toronto, a fleeting impression of some vision that she couldn't exactly remember but that made her ill when she tried to consider it—and he'd always been kind about it. She could hardly see him through the rain. For a moment it was possible to forget that three people were missing. When the storm at last subsided they filled the net with fish and carried it back along the driveway. Steam was rising from the road. They found the place where the fallen trees had been cut and pulled off the road, but the Symphony had departed.
"They must've passed by on the road while we were fishing," August said. It was the only reasonable conclusion. They'd confirmed the route with the conductor before they'd returned to the golf course with the fishing net. The pond had been far enough off the road that they wouldn't have seen the Symphony, hidden as they were behind the clubhouse, and the sound of the Symphony's passing would have been lost in the storm.
"They moved fast," Kirsten said, but her stomach was clenched, and August was jingling the handful of change in his pocket. It didn't entirely add up. Why would the Symphony travel in a downpour, unless there was some unexpected emergency? The storm had washed the road clear of tracks, leaves and twigs in swirled patterns over the pavement, and the heat was rising again. The sky had a broken-apart look about it now, patches of blue between the clouds.
"The fish will go bad fast in this heat," August said.
This was a quandary. Every cell in Kirsten's body ached to follow the Symphony, but it was safer to light a fire in daylight, and they'd eaten nothing but a strip or two each of rabbit jerky that morning. They gathered wood for a fire but of course everything was wet and it took a long time to spark even the slightest flame. The fire smoked badly, their eyes stinging while they cooked, but at least the smoke replaced the stench of fish from their clothes. They ate as much fish as they could and carried the rest with them in the net, set off half-sick down the road, past the golf course, past a number of houses that had obviously been ransacked years earlier, ruined furniture strewn about on the lawns. After a while they jettisoned the fish—it was turning in the heat—and sped up, walking as quickly as possible, but the Symphony was still out of sight and surely by now there should have been some sign of them, hoofprints or footprints or wheel marks on the road. They didn't speak.
Near twilight, the road crossed under a highway. Kirsten climbed up to the overpass for a vantage point, hoping that the Symphony might perhaps be just ahead, but the road curved toward the distant shine of the lake and disappeared behind the trees. The highway was miles of permanent gridlock, small trees growing now between cars and thousands of windshields reflecting the sky. There was a skeleton in the driver's seat of the nearest car.
They slept under a tree near the overpass, side by side on top of August's plastic sheet. Kirsten slept fitfully, aware each time she woke of the emptiness of the landscape, the lack of people and animals and caravans around her. Hell is the absence of the people you long for.
# **24**
**ON THEIR SECOND DAY** without the Symphony, Kirsten and August came upon a line of cars, queued along the shoulder of the road. It was late morning and the heat was rising, a hush falling over the landscape. They'd lost sight of the lake. The cars cast curved shadows. They'd been cleaned out, no bones in backseats or abandoned belongings, which suggested someone lived near here and traveled this route. An hour later they reached a gas station, a low building alone by the road with a yellow seashell sign still standing, vehicles crowded and blocking one another at the pumps. One was the color of melted butter, black lettering on the side. A Chicago taxicab, Kirsten realized. Someone in the very final days had hailed one of the last taxis in the rioting city, negotiated a price and fled north. Two neat bullet holes in the driver's side door. A dog barked and they froze, their hands on their weapons.
The man who came around the side of the building with a golden retriever was in his fifties or sixties, gray hair cut very short and a stiff way of moving that suggested an old injury, a rifle held at his side. He had a complicated scar on his face.
"Help you?" he asked. His tone wasn't unfriendly, and this was the pleasure of being alive in Year Twenty, this calmer age. For the first ten or twelve years after the collapse, he would have been much more likely to shoot them on sight.
"Just passing through," Kirsten said. "We mean no harm. We're headed for the Museum of Civilization."
"Headed where, now?"
"The Severn City Airport."
August was silent beside her. He didn't like to speak to strangers.
The man nodded. "Anyone still out there?"
"We're hoping our friends are there."
"You lose them?"
"Yes," Kirsten said. "We lost them." August sighed. The absence of the Symphony from this route had been obvious for some time. They had passed over patches of soft earth with no tracks. No horse manure, no recent wheel ruts or footprints, no sign at all that twenty-odd people, three caravans, and seven horses were ahead of them on this road.
"Well." The man shook his head. "Bad luck. I'm sorry to hear that. I'm Finn, by the way."
"I'm Kirsten. This is August."
"That a violin case?" Finn asked.
"Yes."
"You run away from an orchestra?"
"They ran away from us," Kirsten said quickly, because she saw the way August's fist clenched in his pocket. "You here alone?"
"Of course not," Finn said, and Kirsten realized her error. Even in this calmer era, who would admit to being outnumbered? His gaze rested on Kirsten's knives. She was finding it difficult not to stare at the scar on the side of his face. Hard to tell at this distance, but it seemed like a deliberate pattern.
"But this isn't a town?"
"No. I couldn't call it that."
"Sorry, just curious. We don't come across too many like you."
"Like me?"
"Living outside a town," Kirsten said.
"Oh. Well. It's quiet out here. This place you mentioned," he said, "this museum. You know anything about it?"
"Not really," Kirsten said. "But our friends were going there."
"I heard it's supposed to be a place where artifacts from the old world are preserved," August said.
The man laughed, a sound like a bark. His dog looked up at him with an expression of concern. "Artifacts from the old world," he said. "Here's the thing, kids, the entire world is a place where artifacts from the old world are preserved. When was the last time you saw a new car?"
They glanced at one another.
"Well, anyway," Finn said, "there's a pump behind the building if you'd like to fill your water bottles."
They thanked him and followed him back. Behind the gas station were two small children, redheaded twins of eight or nine years old and indeterminate gender, peeling potatoes. They were barefoot but their clothes were clean, their hair neatly trimmed, and they stared at the strangers as they approached. Kirsten found herself wondering, as she always did when she saw children, if it was better or worse to have never known any world except the one after the Georgia Flu. Finn pointed to a hand pump on a pedestal in the dirt.
"We've met," Kirsten said. "Haven't we? Weren't you in St. Deborah by the Water two years ago? I remember little twins with red hair, following me around town when I went out for a walk."
Finn tensed, and she saw in the twitch of his arm that he was on the point of raising his rifle. "Did the prophet send you?"
"What? No. No, it's nothing like that. We've only passed through that town."
"We got out as fast as we could," August said.
"We're with the Traveling Symphony."
Finn smiled. "Well, that explains the violin," he said. "I remember the Symphony, all right." He relaxed his grip on the rifle, the moment passed. "Can't say I was ever much for Shakespeare, but that was the best music I'd heard in years."
"Thank you," August said.
"You leave town after the prophet took over?" Kirsten asked. August was working the pump while Kirsten held their bottles under the spout, cool water splashing her hands.
"Craziest damn people I ever met in my life," he said. "Dangerous as hell. A few of us took our kids and fled."
"Did you know Charlie and Jeremy?" Kirsten recapped the bottles, put them away in her knapsack and August's bag.
"Musicians, weren't they? She was black, he was Asian?"
"Yes."
"Not well. I knew them to say hello. They left with their baby a few days before I did."
"You know where they went?"
"No idea."
"Can you tell us what's down the road?"
"Nothing for miles. Couple of abandoned towns, no one there anymore so far as I know. After that, just Severn City and the lake."
"Have you been there?" They were walking back to the road. Kirsten glanced at the side of the man's face, and the scar snapped into focus: a lowercase _t_ with an extra line, the symbol she'd seen spray-painted on buildings in St. Deborah by the Water.
"Severn City? Not since the collapse."
"What's it like," Kirsten asked, "living out here, outside of a town?"
"Quiet." Finn shrugged. "I wouldn't have risked it eight or ten years ago, but except for the prophet, it's been a very quiet decade." He hesitated. "Look, I wasn't quite straight with you before. I know the place you're talking about, the museum. Supposed to be a fair number of people there."
"You weren't tempted to go there yourself, when you left St. Deborah?"
"The prophet's supposedly from there," he said. "Those people at the airport. What if they're the prophet's people?"
Kirsten and August walked mostly in silence. A deer crossed the road ahead and paused to look at them before it vanished into the trees. The beauty of this world where almost everyone was gone. If hell is other people, what is a world with almost no people in it? Perhaps soon humanity would simply flicker out, but Kirsten found this thought more peaceful than sad. So many species had appeared and later vanished from this earth; what was one more? How many people were even left now?
"His scar," August said.
"I know. And where's the Symphony? Why would they change the route?" August didn't answer. There were a dozen reasons why the Symphony might have deviated from the planned route. They were threatened in some way and decided to take a less direct path. They decided upon closer consideration that another route was quicker and expected Kirsten and August to meet them at the airport. They took a wrong turn and vanished into the landscape.
August found a driveway in the early afternoon. They'd been resting in the shade when he rose and walked across the road. Kirsten had noticed the stand of young trees there, but had been too tired and heat-stunned to consider what it might mean. August dropped to one knee to prod at the ground.
"Gravel," he said.
It was a driveway, so overgrown that it had nearly disappeared. The forest opened into a clearing with a two-story house, two rusted-out cars and a pickup truck slumped on the remains of their tires. They waited a while at the edge of the trees, watching, but detected no movement.
The front door was locked, an unusual detail. They circled the house, but the back door was locked too. Kirsten picked the lock. It was obvious from the moment they stepped into the living room that no one else had been here. Throw pillows were arranged neatly on the sofa. A remote control lay on the coffee table, blurred by dust. They looked at one another with eyebrows identically raised over the rags they'd tied over their faces. They hadn't come across an untouched house in years.
In the kitchen Kirsten ran her finger over the row of plates in the dish rack, took a few forks for later use. Upstairs, there was a room that had once belonged to a child. The child in question was still present, a husk in the bed—Kirsten pulled a quilt over its head while August was still going through the downstairs bathroom—and there was a framed photograph on the wall of a boy with his parents, all of them beaming and resplendent with life, the boy in a Little League uniform with his parents kneeling on either side. She heard August's footsteps behind her.
"Look what I found," he said.
He'd found a metal Starship _Enterprise_. He held it up in the sunlight, a gleaming thing the size of a dragonfly. That was when Kirsten noticed the poster of the solar system over the bed, Earth a small blue dot near the sun. The boy had loved both baseball and space.
"We should keep moving," Kirsten said after a moment. August's gaze had fallen to the bed. She left the room first so he could say one of his prayers, although she wasn't actually sure if _prayer_ was the right word for it. When he murmured over the dead, he seemed to be talking only to them. "I hope it was peaceful at the end," she'd heard him say. Or, "You have a really nice house. I'm sorry for taking your boots." Or, "Wherever you are, I hope your family's there too." To the child in the bed, he spoke so quietly that Kirsten couldn't hear. The only words she caught were "up in the stars," and she moved quickly on to the master bedroom so that he wouldn't catch her eavesdropping, but she saw that August had been there already—the boy's parents had died in their bed, and a cloud of dust hung in the air above them from when August had pulled up the blankets to cover their faces.
In the en suite bathroom, Kirsten closed her eyes for just a second as she flipped the light switch. Naturally nothing happened, but as always in these moments she found herself straining to remember what it had been like when this motion had worked: walk into a room, flip a switch and the room floods with light. The trouble was she wasn't sure if she remembered or only imagined remembering this. She ran her fingertips over a blue-and-white china box on the bathroom counter, admired the rows of Q-tips inside before she pocketed them. They looked useful for cleaning ears and musical instruments. Kirsten looked up and met her own gaze in the mirror. She needed a haircut. She smiled, then adjusted her smile to lessen the obviousness of her most recently missing tooth. She opened a cabinet and stared at a stack of clean towels. The one on top was blue with yellow ducks on it and had a hood sewn into a corner. Why hadn't the parents taken the boy into their bed, if they'd all been sick together? Perhaps the parents had died first. She didn't want to think about it.
The door to the spare bedroom had been closed, the window open a crack, so the carpet was ruined but the clothes in the closet had escaped the smell of death. She found a dress she liked, soft blue silk with pockets, and changed into it while August was still in the boy's bedroom. There was also a wedding gown and a black suit. She took these for costumes. What the Symphony was doing, what they were always doing, was trying to cast a spell, and costuming helped; the lives they brushed up against were work-worn and difficult, people who spent all their time engaged in the tasks of survival. A few of the actors thought Shakespeare would be more relatable if they dressed in the same patched and faded clothing their audience wore, but Kirsten thought it meant something to see Titania in a gown, Hamlet in a shirt and tie. The tuba agreed with her.
"The thing with the new world," the tuba had said once, "is it's just horrifically short on elegance." He knew something about elegance. He had played in a military orchestra with the conductor before the collapse. He talked sometimes about the military balls. Where was he? Don't think of the Symphony. Don't think of the Symphony. There is only here, she told herself, there is only this house.
"Nice dress," August said, when she found him downstairs in the living room.
"The old one smelled like smoke and fish guts."
"I found a couple suitcases in the basement," he said.
They left with a suitcase each, towels and clothing and a stack of magazines that Kirsten wanted to go through later, an unopened box of salt from the kitchen and various other items that they thought they might use, but first Kirsten lingered for a few minutes in the living room, scanning the bookshelves while August searched for a _TV Guide_ or poetry.
"You looking for something in particular?" he asked after he'd given up the search. She could see he was thinking of taking the remote. He'd been holding it and idly pressing all the buttons.
" _Dr. Eleven_ , obviously. But I'd settle for _Dear V_."
The latter was a book she'd somehow misplaced on the road two or three years ago, and she'd been trying ever since to find a replacement. The book had belonged to her mother, purchased just before the end of everything. _Dear V.: An Unauthorized Portrait of Arthur Leander_. White text across the top proclaimed the book's status as a number-one best seller. The cover photo was black-and-white, Arthur looking over his shoulder as he got into a car. The look on his face could have meant anything; a little haunted, perhaps, but it was equally possible that someone had just called his name and he was turning to look at him or her. The book was comprised entirely of letters written to a friend, the anonymous V.
When Kirsten had left Toronto with her brother, he'd told her she could bring one book in her backpack, just one, so she'd taken _Dear V_. because her mother had told her she wasn't allowed to read it. Her brother had raised an eyebrow but made no remark.
# **25**
**A FEW OF THE LETTERS:**
Dear V.,
It's cold in Toronto but I like where I'm living. The thing I can't get used to is when it's cloudy and about to snow, the sky looks orange. Orange. I know it's just reflected light from the city but it's eerie.
I've been going on long walks lately, because after rent and the laundromat and groceries I can't really afford transit, found a penny shining in the gutter yesterday and decided it was a lucky charm. I'm taping it to this letter. Unnaturally shiny, right? For my 19th birthday last night I went downtown to a dance club with a $5 cover charge. Irresponsible to spend $5 on cover when I'm getting so few hours at the restaurant, but whatever, I like dancing even though I have no idea what I'm doing and probably look like I'm having a seizure. I walked home with my friend Clark and he was talking about this experimental thing he'd seen where the actors wore giant papier-mâché masks, which sounded cool but kind of pretentious. I told C. that and he said, you know what's pretentious? Your hair, and he wasn't trying to be mean but in the morning I made breakfast for one of my roommates in exchange for a haircut and it's not bad, I think. My roommate's in hairschool. The ponytail's gone! You wouldn't recognize me! I love this city and also hate it and I miss you.
—A.
Dear V.,
I dreamt last night we were in your house again, playing mah jong (sp?) with your mother. I think in real life we only played it that one time and I know we were both stoned, but I liked it, those little tiles. Anyway. This morning I was thinking about the thing I liked about your house, that optical illusion re: the ocean, the way it looked from the living room like the ocean was right there at the end of the front lawn but then when you went outside there was the cliff between the grass and the water, with that rickety staircase thing that always scared the hell out of me.
I'm not exactly homesick but not exactly not. I've been spending a lot of time with Clark, who's in my acting class, who I think you'd like. C. has punk-rock hair half-shaved, pink on the non-shaved side. C.'s parents want him to go to business school or at least get a practical degree of some kind and C. told me he'd rather die than do this, which seems extreme but on the other hand I remember when I thought I'd rather die than stay on the island so I told him I understood. I had a good class tonight. I hope things are good with you. Write soon,
—A.
Dear V.,
You remember when we used to listen to music in your room in the cliff house? I was thinking about what a nice time that was, even though I was about to leave for Toronto so it was also sad. I remember staring at the leaves outside your window and trying to imagine I was staring at skyscrapers and what would that be like, would I miss the leaves, etc., and then I get to Toronto and there's a tree right outside my window so all I see are leaves. It's a gingko, though, nothing I'd ever seen out west. It's pretty. The leaves are shaped like little fans.
—A.
Dear V.,
I'm a terrible actor and this city is fucking freezing and I miss you.
—A.
Dear V.,
Do you remember that night we stayed up to see the comet? Comet Hyakutake, that really cold night in March with frost on the grass, I remember we whispered the name over and over again, Hyakutake, Hyakutake. I thought it was pretty, that light just hanging there in the sky. Anyway I was thinking of it just now and wondered if you remembered that night as well as I do. You can't really see the stars here.
—A.
Dear V.,
I didn't tell you this but last month in acting class the instructor told me he felt I was a little flat, which is his way of saying he thinks I'm a terrible actor. He said something vague and almost kind about how difficult it can be to improve. I said, watch me, and he looked surprised and sort of blinked at me and then mostly ignored me for the next three weeks. But then last night I was doing my monologue and when I looked up he was watching me, really watching me, and he said goodnight to me for the first time in weeks and I felt like there was hope. I'm like a man in a wheelchair watching other people run. I can see what good acting is but I can't quite reach it but I'm so close sometimes, V. I'm really trying.
I was thinking about the island. It seems past-tense somehow, like a dream I had once. I walk down these streets and wander in and out of parks and dance in clubs and I think "once I walked along the beach with my best friend V., once I built forts with my little brother in the forest, once all I saw were trees" and all those true things sound false, it's like a fairy tale someone told me. I stand waiting for lights to change on corners in Toronto and that whole place, the island I mean, it seems like a different planet. No offense but it's weird to think you're still there.
Yours,
—A.
Last letter, dear V., because you haven't answered any of my letters in four months and haven't written anything longer than a postcard in five. Today I stepped out and the trees were exploding with spring flowers, did I dream you walked beside me through these glittering streets? (V., sorry, my roommate came home in a generous mood with some excellent pot and also I'm a little deranged and lonely, you don't know what it's like to be so far from home because you'll never leave, V., will you?) I was thinking earlier that to know this city you must first become penniless, because pennilessness (real pennilessness, I mean not having $2 for the subway) forces you to walk everywhere and you see the city best on foot. Anyway. I am going to be an actor and I am going to be good, that's the important thing, I want to do something remarkable but I don't know what. I told that to one of my roommates last night and he laughed and called me young, but we're all getting older and it's going so fast. I'm already 19.
I'm thinking about auditioning for an acting program in New York.
Something I've been thinking about, which will sound harsh and I'm sorry: you said you'd always be my friend but you're not, actually, are you? I've only realized that recently. You don't have any interest in my life.
This is going to seem bitter but I don't mean it that way, V., I'm just stating a fact here: you'll only ever call me if I call you first. Have you noticed that? If I call and leave a message you'll call me back, but you will never call me first.
And I think that's kind of a horrible thing, V., when you're supposed to be someone's friend. I always come to you. You always say you're my friend but you'll never come to me and I think I have to stop listening to your words, V., and take stock instead in your actions. My friend C. thinks my expectations of friendship are too high but I don't think he's right.
Take care, V. I'll miss you.
—A.
V.,
It's been years (decades?) since I've written but I've thought of you often. It was good to see you at Christmas. I didn't know my mother was planning on inviting people over. She always does that when I'm there, I think to show me off in a way, even though if it had been up to her I'd never have left the island and I'd be driving my father's snowplow. Awkward to be thrown into a room together, but wonderful to see you again and to talk to you a little after all this time. Four kids! I can't imagine.
It's been years since I've written to anyone, actually, not just you, and I confess I'm out of practice. But I have news, big news, and when it happened you were the first one I wanted to tell. I'm getting married. It's very sudden. I didn't mention it at Christmas because I wasn't sure yet, but now I am and it seems perfectly right. Her name's Miranda and she's actually from the island, but we met in Toronto. She's an artist who draws strange beautiful comic strip type things. She's moving to L.A. with me next month.
How did we get so old, V.? I remember building forts with you in the woods when we were five. Can we be friends again? I've missed you terribly.
—A.
Dear V.,
Strange days. The feeling that one's life resembles a movie. I have such disorientation, V., I can't tell you. At unexpected moments find myself thinking, how did I get here? How have I landed in this life? Because it seems like an improbable outcome, when I look back at the sequence of events. I know dozens of actors more talented than me who never made it.
Have met someone and fallen in love. Elizabeth. She has such grace, beauty, but far more important than that a kind of lightness that I didn't realize I'd been missing. She takes classes in art history when she isn't modeling or shooting films. I know it's questionable, V. I think Clark knows. Dinner party last night (very awkward and ill-advised in retrospect, long story, seemed like good idea at time) and I looked up at one point and he was giving me this look, like I'd disappointed him personally, and I realized he's right to be disappointed. I disappoint myself too. I don't know, V., all is in turmoil.
Yours,
—A.
Dear V.,
Clark came over for dinner last night, first time in six months or so. Was nervous about seeing him, partly because I find him less interesting now than I did when we were both nineteen (unkind of me to admit, but can't we be honest about how people change?), also partly because last time he was here I was still married to Miranda and Elizabeth was just another dinner guest. But Elizabeth cooked roast chicken and did her best impression of a 1950s housewife and he was taken with her, I think. She kept up her brightest veneer through the whole evening, was completely charming, etc. For once she didn't drink too much.
Do you remember that English teacher we had in high school who was crazy about Yeats? His enthusiasm sort of rubbed off on you and I remember for a while you had a quote taped to your bedroom wall in the lake house and lately I've been thinking about it: Love is like the lion's tooth.
Yours,
—A.
# **26**
**" PLEASE TELL ME YOU'RE JOKING,"** Clark said when Elizabeth Colton called to tell him about the book. Elizabeth wasn't joking. She hadn't seen the book yet—it wouldn't be released for another week—but she'd been told by a reliable source that both of them were in it. She was furious. She was considering litigation, but she wasn't sure who to sue. The publisher? V.? She'd decided she couldn't reasonably sue Arthur, as much as she'd like to, because he apparently hadn't known about the book either.
"What does he say about us?" Clark asked.
"I don't know," Elizabeth said. "But apparently he talks about his marriages and friendships in detail. The word my friend used was _unsparing_."
"Unsparing," Clark said. "That could mean anything." But probably nothing good, he decided. No one's ever described as being unsparingly kind.
"He liked to describe the people in his life, apparently. At least he had the grace to be upset about it when I called him." A fizz of static on the line.
"It's called _Dear B_.?" Clark was writing this down. This was three weeks before the pandemic. They still had the indescribable luxury of being concerned about a book of published letters.
" _Dear V_. She's his friend Victoria."
"Former friend, I'd imagine. I'll call him tomorrow," Clark said.
"He'll just start rambling and deflecting and obfuscating," she said. "Or maybe that's just how he talks to me. Do you ever talk to him and get the sense that he's acting?"
"I actually have to run," Clark said. "I've got an eleven a.m. interview."
"I'm coming to New York soon. Maybe we should meet and discuss this."
"Okay, fine." He hadn't seen her in years. "Have your assistant talk to my admin and we'll set something up."
When he hung up the phone, _Dear V_. was all he could think about. He left the office without meeting anyone else's gaze, mortified in a way that somehow precluded talking to his colleagues—had any of them read the book?—and stepped out onto Twenty-third Street. He wanted to track down _Dear V_. immediately—surely he knew someone who could get him a copy—but there wasn't time before his meeting. He was conducting a 360° assessment at a water-systems consulting firm by Grand Central Station.
Over the past several years, these assessments had become his specialty. At the center of each stood an executive whom the client company hoped to improve, referred to without irony as the target. Clark's current targets included a salesman who made millions for the company but yelled at his subordinates, an obviously brilliant lawyer who worked until three a.m. but somehow couldn't meet her deadlines, a public-relations executive whose skill in handling clients was matched only by his utter ineptitude at managing his staff. Each of Clark's assessments involved interviewing a dozen or so people who worked in close proximity to the target, presenting the target with a series of reports consisting of anonymized interview comments—positive comments first, to soften the blow of the takedowns—and then, in the project's final phase, a few months of coaching.
Twenty-third Street wasn't busy—a little early for the lunch crowd—but he kept getting trapped behind iPhone zombies, people half his age who wandered in a dream with their eyes fixed on their screens. He jostled two of them on purpose, walking faster than usual, upset in a fundamental way that made him feel like punching walls, like running full-speed, like throwing himself across a dance floor although he hadn't done that in two decades. When Arthur danced he'd had a way of flailing just on the edge of the beat. A young woman stopped abruptly at the top of the subway stairs and he almost crashed into her, glared as he brushed past—she didn't notice, enraptured by her screen—and he stepped aboard a train just before the doors closed, the day's first small moment of grace. He stewed all the way to Grand Central Station, where he took the stairs two at a time to a marble corridor just off the main concourse, passed briefly through the spiced air of Grand Central Market and down a connecting passage to the Graybar Building.
"I'm sorry I'm late," he said to his interviewee, who shrugged and gestured him into the visitor's chair.
"If you think two minutes counts as late, we're not going to get along very well." Was that a Texas accent? Dahlia was in her late thirties or early forties, with a sharp-edged haircut and red-framed glasses that matched her lipstick.
Clark went into the usual introduction and preamble about the 360° they were doing, her boss as the target, the way he was interviewing fifteen people and it would all be anonymous, comments split off and categorized into separate reports for subordinates, peers, and superiors with a minimum of three in each group, etc. He listened to his voice from a distance and was pleased to note that it sounded steady.
"So the point," she said, "if I'm understanding correctly, is to change my boss?"
"Well, to address areas of potential weakness," Clark said. Thinking of _Dear V_. again as he said this, because isn't indiscretion the very definition of weakness?
"To change him," she insisted with a smile.
"I suppose you could see it that way."
She nodded. "I don't believe in the perfectibility of the individual," she said.
"Ah," he said. The thought that crossed his mind was that she looked a little old to be talking like a philosophy undergrad. "How about the improvement of the individual, then?"
"I don't know." She leaned back in her chair, arms folded, considering the question. Her tone was light, but he was beginning to realize that there was nothing flippant about her. He was remembering some of the offhand comments her colleagues had made about her in previous interviews, when his questions had come around to the team. Someone had called her _a little different_. Someone else, he remembered, had used the word _intense_. "You've been doing this for a while, you said?"
"Twenty-one years."
"These people you coach, do they ever actually change? I mean in any kind of lasting, notable way?"
He hesitated. This was actually something he'd wondered about.
"They change their behaviors," he said, "some of them. Often people will simply have no idea that they're perceived as needing improvement in a certain area, but then they see the report..."
She nodded. "You differentiate between changing people and changing behaviors, then."
"Of course."
"Here's the thing," Dahlia said. "I'll bet you can coach Dan, and probably he'll exhibit a turnaround of sorts, he'll improve in concrete areas, but he'll still be a joyless bastard."
"A joyless..."
"No, wait, don't write that down. Let me rephrase that. Okay, let's say he'll change a little, probably, if you coach him, but he'll still be a successful-but-unhappy person who works until nine p.m. every night because he's got a terrible marriage and doesn't want to go home, and don't ask how I know that, _everyone_ knows when you've got a terrible marriage, it's like having bad breath, you get close enough to a person and it's obvious. And you know, I'm reaching here, but I'm talking about someone who just seems like he wishes he'd done something different with his life, I mean really actually almost anything—is this too much?"
"No. Please, go on."
"Okay, I love my job, and I'm not just saying that because my boss is going to see my interview comments, which by the way I don't believe he won't be able to tell who said what, anonymous or not. But anyway, I look around sometimes and I think—this will maybe sound weird—it's like the corporate world's full of ghosts. And actually, let me revise that, my parents are in academia so I've had front-row seats for _that_ horror show, I know academia's no different, so maybe a fairer way of putting this would be to say that adulthood's full of ghosts."
"I'm sorry, I'm not sure I quite—"
"I'm talking about these people who've ended up in one life instead of another and they are just so disappointed. Do you know what I mean? They've done what's expected of them. They want to do something different but it's impossible now, there's a mortgage, kids, whatever, they're trapped. Dan's like that."
"You don't think he likes his job, then."
"Correct," she said, "but I don't think he even realizes it. You probably encounter people like him all the time. High-functioning sleepwalkers, essentially."
What was it in this statement that made Clark want to weep? He was nodding, taking down as much as he could. "Do you think he'd describe himself as unhappy in his work?"
"No," Dahlia said, "because I think people like him think work is supposed to be drudgery punctuated by very occasional moments of happiness, but when I say happiness, I mostly mean distraction. You know what I mean?"
"No, please elaborate."
"Okay, say you go into the break room," she said, "and a couple people you like are there, say someone's telling a funny story, you laugh a little, you feel included, everyone's so funny, you go back to your desk with a sort of, I don't know, I guess _afterglow_ would be the word? You go back to your desk with an afterglow, but then by four or five o'clock the day's just turned into yet another day, and you go on like that, looking forward to five o'clock and then the weekend and then your two or three annual weeks of paid vacation time, day in day out, and that's what happens to your life."
"Right," Clark said. He was filled in that moment with an inexpressible longing. The previous day he'd gone into the break room and spent five minutes laughing at a colleague's impression of a _Daily Show_ bit.
"That's what _passes_ for a life, I should say. That's what passes for happiness, for most people. Guys like Dan, they're like sleepwalkers," she said, "and nothing ever jolts them awake."
He got through the rest of the interview, shook her hand, walked out through the vaulted lobby of the Graybar Building to Lexington Avenue. The air was cold but he longed to be outside, away from other people. He took a long and circuitous route, veering two avenues east to the relative quiet of Second Avenue.
He was thinking of the book, and thinking of what Dahlia had said about sleepwalking, and a strange thought came to him: had Arthur seen that Clark was sleepwalking? Would this be in the letters to V.? Because he _had_ been sleepwalking, Clark realized, moving half-asleep through the motions of his life for a while now, years; not specifically unhappy, but when had he last found real joy in his work? When was the last time he'd been truly moved by anything? When had he last felt awe or inspiration? He wished he could somehow go back and find the iPhone people whom he'd jostled on the sidewalk earlier, apologize to them—I'm sorry, I've just realized that I'm as minimally present in this world as you are, I had no right to judge—and also he wanted to call every target of every 360° report and apologize to them too, because it's an awful thing to appear in someone else's report, he saw that now, it's an awful thing to be the target.
#
# **27**
**THERE WAS A MOMENT ON EARTH ,** improbable in retrospect and actually briefer than a moment in the span of human history, more like the blink of an eye, when it was possible to make a living solely by photographing and interviewing famous people. Seven years before the end of the world, Jeevan Chaudhary booked an interview with Arthur Leander.
Jeevan had been working as a paparazzo for some years and had made a passable living at it, but he was sick to death of stalking celebrities from behind sidewalk planters and lying in wait in parked cars, so he was trying to become an entertainment journalist, which he felt was sleazy but less sleazy than his current profession. "I _know_ this guy," he told an editor who'd bought a few of his photos in the past, when the subject of Arthur Leander came up over drinks. "I've seen all his movies, some of them twice, I've stalked him all over town, I've photographed his wives. I can get him to talk to me." The editor agreed to give him a shot, so on the appointed day Jeevan drove to a hotel and presented his ID and credentials to a young publicist stationed outside a penthouse suite.
"You have fifteen minutes," she said, and ushered him in. The suite was all parquet floors and bright lighting. There was a room with canapés on a table and a number of journalists staring at their phones, another room with Arthur in it. The man whom Jeevan believed to be the finest actor of his generation sat in an armchair by a window that looked out over downtown Los Angeles. Jeevan, who had an eye for expensive things, registered the weight of the drapes, the armchair's sleek fabric, the cut of Arthur's suit. There was no reason, Jeevan kept telling himself, why Arthur would know that Jeevan was the one who'd taken the photograph of Miranda, but of course there was: all he could think of was how stupid he'd been to tell Miranda his name that night. The whole entertainment-journalist idea had been a mistake, it was obvious now. As he crossed the parquet floor he entertained wild thoughts of faking a sudden illness and fleeing before Arthur looked up, but Arthur smiled and extended a hand when the publicist introduced them. Jeevan's name seemingly meant nothing to Arthur, and his face apparently didn't register either. Jeevan had taken pains to alter his appearance. He'd shaved off the sideburns. He'd taken out his contact lenses and was wearing glasses that he hoped made him look serious. He sat in the armchair across from Arthur and set his recorder on the coffee table between them.
He had rewatched all of Arthur's movies over the previous two days, and had done substantial additional research. But Arthur didn't want to talk about the movie he was shooting, or his training or influences, or what drove him as an artist, or whether he still saw himself as an outsider, as he'd said in one of his first interviews some years back. He responded in monosyllables to Jeevan's first three questions. He seemed dazed and hungover. He looked like he hadn't slept well in some time.
"So tell me," he said, after what seemed to Jeevan to be an uncomfortably long silence. His publicist had deposited an emergency cappuccino into his hands a moment earlier. "How does a person become an entertainment journalist?"
"Is this one of those postmodern things?" Jeevan asked. "Where you turn the tables and interview me, like those celebrities who take photos of the paparazzi?" Careful, he thought. His disappointment at Arthur's disinterest in talking to him was curdling into hostility, and beneath that lurked a number of larger questions of the kind that kept him up at night: interviewing actors was better than stalking them, but what kind of a journalism career was this? What kind of life? Some people managed to do things that actually mattered. Some people, his brother Frank for example, were currently covering the war in Afghanistan for Reuters. Jeevan didn't specifically want to be Frank, but he couldn't help but feel that he'd made a number of wrong turns in comparison.
"I don't know," Arthur said, "I'm just curious. How'd you get into this line of work?"
"Gradually, and then suddenly."
The actor frowned as if trying to remember something. "Gradually, and then suddenly," he repeated. He was quiet for a moment. "No, seriously," he said, snapping out of it, "I've always wondered what drives you people."
"Money, generally speaking."
"Sure, but aren't there easier jobs? This whole entertainment-journalism thing... I mean, look, I'm not saying a guy like you is the same as the paparazzi"—Thank you for paying so little attention, Jeevan thought—"I know what you do isn't the same thing as what they do, but I've seen guys..." Arthur held up a hand—hold that thought—and swallowed half his cappuccino. The infusion of caffeine made his eyes widen slightly. "I've seen guys climb _trees_ ," he said. "I'm not kidding. This was during my divorce, around the time Miranda moved out. I'm washing the dishes, I look out the window, and there's this guy balancing up there with a camera."
"You wash dishes?"
"Yeah, the housekeeper was talking to the press, so I fired her and then the dishwasher broke."
"Never rains but it pours, right?"
Arthur grinned. "I like you," he said.
Jeevan smiled, embarrassed by how flattered he was by this. "It's an interesting line of work," he said. "One meets some interesting people." One also meets some of the most boring people on the face of the earth, but he thought a little flattery couldn't hurt.
"I've always been interested in people," Arthur said. "What drives them, what moves them, that kind of thing." Jeevan searched his face for some sign of sarcasm, but he seemed utterly sincere.
"Me too, actually."
"I'm just asking," Arthur said, "because you don't seem like most of the others."
"I don't? Really?"
"I mean, did you always want to be an entertainment guy?"
"I used to be a photographer."
"What kind of photography?" Arthur was finishing his cappuccino.
"Weddings and portraits."
"And you went from that to writing about people like me?"
"Yes," Jeevan said. "I did."
"Why would you?"
"I was sick of going to weddings. The pay was better. It was less of a hassle. Why do you ask?"
Arthur reached across the table and turned off Jeevan's tape recorder. "Do you know how tired I am of talking about myself?"
"You do give a lot of interviews."
"Too many. Don't write that I said that. It was easier when it was just theater and TV work. The occasional profile or feature or interview or whatever. But you get successful in movies, and Christ, it's like this whole other thing." He raised his cup in a cappuccino-signaling motion, and Jeevan heard the publicist's heels clicking away on the floor behind him. "Sorry," he said, "I know it's a little disingenuous to complain about a job like mine."
You have no idea, Jeevan thought. You're rich and you'll always be rich and if you wanted to you could stop working today and never work again. "But you've been doing movies for years," he said in his most neutral tone.
"Yeah," Arthur said, "I guess I'm still not used to it. It's still somehow embarrassing, all the attention. I tell people I don't notice the paparazzi anymore, but I do. I just can't look at them."
Which I appreciate, Jeevan thought. He was aware that his fifteen-minute allotment was trickling away. He held up the recorder so Arthur would notice it, pressed the Record button and set it on the coffee table between them.
"You've had considerable success," Jeevan said. "And with that comes, of course, a certain loss of privacy. Is it fair to say that you find the scrutiny difficult?"
Arthur sighed. He clasped his hands together, and Jeevan had an impression that he was gathering his strength. "You know," Arthur said clearly and brightly, playing a new, devil-may-care individual who wouldn't sound in the playback like he was pale and obviously sleep-deprived with dark circles under his eyes, "I just figure that's part of the deal, you know? We're so lucky to be in this position, all of us who make our living as actors, and I find complaints about invasion of privacy to be disingenuous, frankly. I mean, let's be real here, we wanted to be famous, right? It isn't like we didn't know what we'd be facing going in." The speech seemed to take something out of him. He wilted visibly, and accepted a new cappuccino from his publicist with a nod of thanks. An awkward silence ensued.
"So you just flew in from Chicago," Jeevan said, at a loss.
"Yes I did." Arthur reached and turned off Jeevan's recorder again. "Tell me something," he said. "What did you say your name was?"
"Jeevan Chaudhary."
"If I tell you something, Jeevan Chaudhary, how long do I have before it appears in print?"
"Well," Jeevan said, "what do you want to tell me?"
"Something no one else knows, but I want twenty-four hours before it appears anywhere."
"Arthur," the publicist said from somewhere behind Jeevan, "we live in the information age. It'll be on _TMZ_ before he gets to the parking lot."
"I'm a man of my word," Jeevan said. At that point in his directionless life he wasn't sure if this was true or not, but it was nice to think that it might be.
"What does that mean?" Arthur asked.
"It means I do what I say I'm going to do."
"Okay, look," Arthur said, "if I tell you something..."
"Guaranteed exclusive?"
"Yes. I'll tell no one else, on condition that you give me twenty-four hours."
"Fine," Jeevan said, "I could give you twenty-four hours until it runs."
"Not just until it runs. Twenty-four hours until you tell another living soul, because I don't want some intern at wherever the hell you work leaking it themselves."
"Okay," Jeevan said. "Twenty-four hours before I tell another living soul." He was pleased by the intrigue.
"Arthur," the publicist said, "could I speak with you for a moment?"
"No," Arthur said, "I have to do this."
"You don't have to do anything," she said. "Remember who you're talking to."
"I'm a man of my word," Jeevan repeated. This sounded a little sillier the second time.
"You're a journalist," she said. "Don't be ridiculous. Arthur—"
"Okay, look," Arthur said, to Jeevan, "I came here directly from the airport."
"Okay."
"I came here two hours early, almost three hours actually, because I didn't want to go home first."
"Why didn't...?"
"I'm leaving my wife for Lydia Marks," Arthur said.
"Oh, my god," the publicist said.
Lydia Marks was Arthur's costar on the film that had just wrapped in Chicago. Jeevan had photographed her coming out of a club once in Los Angeles, bright-eyed and almost supernaturally put-together at three in the morning. She was the sort of person who liked the paparazzi and sometimes actually called them in advance. She had flashed him a winning smile.
"You're leaving Elizabeth Colton," Jeevan said. "Why?"
"Because I have to. I'm in love with someone else."
"Why are you telling me this?"
"I'm moving in with Lydia next month," he said, "and Elizabeth doesn't know yet. I flew here a week ago when I had the day off from filming, specifically to tell her, and I just couldn't do it. Look, here's what you have to understand about Elizabeth: nothing bad has ever happened to her."
"Nothing?"
"Don't write that in your piece. I shouldn't have mentioned it. The point is, I haven't been able to tell her. I couldn't do it any of the times we spoke on the phone, and I couldn't do it today. But if you tell me that this story will appear tomorrow, then that forces my hand, doesn't it?"
"It'll be a sensitive story," Jeevan said. "You and Elizabeth are still friends and you wish only the best for her, you have no further comment and you desire for her privacy to be respected at this difficult time. That about it?"
Arthur sighed. He looked somewhat older than forty-four. "Can you say it was mutual, for her sake?"
"The split was mutual and, uh, amicable," Jeevan said. "You and Elizabeth remain friends. You have considerable... considerable respect for one another and have decided mutually that it's best for you to go your separate ways, and you wish for privacy in this, I don't know, in this difficult time?"
"That's perfect."
"Do you want me to mention the...?" Jeevan didn't finish the sentence, but he didn't have to. Arthur winced and looked at the ceiling.
"Yes," he said in a strained voice, "let's mention the baby. Why not?"
"Your first priority is your son, Tyler, who you and Elizabeth are committed to coparenting. I'll make it less awkward than that."
"Thank you," Arthur said.
# **28**
**ARTHUR THANKED HIM ,** and then what? On his brother's sofa in a tower on the south edge of Toronto, eight days after Arthur's death, Jeevan stared at the ceiling and tried to remember how it had played out. Had the publicist offered him a cappuccino? No, she had not, although that would've been nice. (Jeevan had been thinking of cappuccinos a great deal, because cappuccinos were among his favorite things and it had occurred to him that if everything was as bad as the television news suggested, he might never drink another. The things we fixate on, he thought.) Anyway, the publicist: she'd escorted him out without looking at him and closed the door in his face, and somehow this was already seven years ago.
Jeevan lay on the sofa, entertaining flashes of random memory and thinking of things like cappuccinos and beer while Frank worked on his latest ghostwriting project, a memoir of a philanthropist whose name he was contractually forbidden from mentioning. Jeevan kept thinking of his girlfriend, his house in Cabbagetown, wondering if he was going to see either of them again. Cell phones had stopped working by then. His brother had no landline. Outside the world was ending and snow continued to fall.
# **29**
**HE 'D KEPT HIS WORD, THOUGH.** This was one of the very few moments Jeevan was proud of in his professional life. He had told no one about Arthur and Elizabeth's split, absolutely no one, for a full twenty-four hours after the interview.
"What are you smiling about?" Frank asked.
"Arthur Leander."
In a different lifetime Jeevan had stood outside Arthur's house by the hour, smoking cigarettes and staring up at the windows, dazed with boredom. One night he'd tricked Arthur's first wife into an unflattering photograph, and he'd made good money on the shot but he still felt bad about it. The way she'd looked at him, stunned and sad with the cigarette in her hand, hair sticking up in all directions, the strap of her dress falling off her shoulder. Strange to think of it now in this winter city.
# **30**
**" YOU'VE GOT TO STOP** singing that song," Frank said.
"Sorry, but it's the perfect song."
"I don't disagree, but you've got a terrible singing voice."
It was the end of the world as they knew it! Jeevan had had that song stuck in his head for several days now, ever since he'd appeared on his brother's doorstep with the shopping carts. For a while they'd lived in front of the television news, low volume, a murmured litany of nightmares that left them drained and reeling, drifting in and out of sleep. How could so many die so quickly? The numbers seemed impossible. Jeevan taped plastic over all of the air ducts in the apartment and wondered if this was enough, if the virus could still reach them either through or perhaps somehow around the edges of the tape. He rigged Frank's bath towels over the windows to prevent stray light from escaping at night, and pushed Frank's dresser in front of the door. People knocked sometimes, and when they did Jeevan and Frank fell silent. They were afraid of everyone who wasn't them. Twice someone tried to break in, scratching around the lock with some metal tool while Frank and Jeevan waited in an agony of stillness, but the deadbolt held.
Days slipped past and the news went on and on until it began to seem abstract, a horror movie that wouldn't end. The newscasters had a numb, flattened way of speaking. They sometimes wept.
Frank's living room was on the corner of the building, with views of both the city and the lake. Jeevan preferred the view of the lake. If he turned Frank's telescope toward the city he saw the expressway, which was upsetting. Traffic had inched along for the first two days, pulling trailers, plastic bins and suitcases strapped to roofs, but by the third morning the gridlock was absolute and people had started walking between the cars with their suitcases, their children and dogs.
By Day Five Frank was working on his ghostwriting project instead of watching the news, because he said the news was going to drive them both crazy, and by then most of the newscasters weren't even newscasters, just people who worked for the network and were seemingly unused to being on the other side of the camera, cameramen and administrators speaking haltingly into the lens, and then countries began to go dark, city by city—no news out of Moscow, then no news out of Beijing, then Sydney, London, Paris, etc., social media bristling with hysterical rumors—and the local news became more and more local, stations dropping away one by one, until finally the last channel on air showed only a single shot in a newsroom, station employees taking turns standing before the camera and disseminating whatever information they had, and then one night Jeevan opened his eyes at two a.m. and the newsroom was empty. Everyone had left. He stared at the empty room on the screen for a long time.
The other channels were all static and test patterns by then, except for the ones that were repeating a government emergency broadcast over and over, useless advice about staying indoors and avoiding crowded places. A day later, someone finally switched off the camera on the empty newsroom, or the camera died on its own. The day after that, the Internet blinked out.
Toronto was falling silent. Every morning the quiet was deeper, the perpetual hum of the city fading away. Jeevan mentioned this to Frank, who said, "Everyone's running out of gas." The thing was, Jeevan realized, looking at the stopped cars on the highway, even the people who hadn't run out of gas couldn't go anywhere now. All the roads would be blocked by abandoned cars.
Frank never stopped working. The philanthropist's memoir was almost complete.
"He's probably dead," Jeevan said.
"Probably," Frank agreed.
"Why are you still writing about him?"
"I signed a contract."
"But everyone else who signed the contract..."
"I know," Frank said.
Jeevan was holding his useless cell phone up to the window. A NO SERVICE AVAILABLE message flashed on the screen. He let the phone fall to the sofa and stared out at the lake. Maybe a boat would come, and...
On silent afternoons in his brother's apartment, Jeevan found himself thinking about how human the city is, how human everything is. We bemoaned the impersonality of the modern world, but that was a lie, it seemed to him; it had never been impersonal at all. There had always been a massive delicate infrastructure of people, all of them working unnoticed around us, and when people stop going to work, the entire operation grinds to a halt. No one delivers fuel to the gas stations or the airports. Cars are stranded. Airplanes cannot fly. Trucks remain at their points of origin. Food never reaches the cities; grocery stores close. Businesses are locked and then looted. No one comes to work at the power plants or the substations, no one removes fallen trees from electrical lines. Jeevan was standing by the window when the lights went out.
There was a stupid moment or two when he stood near the front door, flipping the light switches. On/off, on/off.
"Stop it," Frank said. He was taking notes in a margin of his manuscript in the gray light that seeped in through the blinds. "You're driving me crazy." Frank was hiding in his project, Jeevan had realized, but he couldn't begrudge Frank the strategy. If Jeevan had had a project, he'd have hid in it too.
"It could just be us," Jeevan said. "Maybe just a blown fuse in the basement?"
"Of course it isn't just us. The only remarkable thing is that the lights stayed on as long as they did."
"It's like the tree house," Frank said. This was sometime around Day Thirty, a few days after the end of running water. Whole days passed when they didn't speak, but there were inexplicable moments of peace. Jeevan had never felt so close to his brother. Frank worked on the philanthropist's memoir and Jeevan read. He spent hours studying the lake through the telescope, but the sky and the water were empty. No planes, no ships, and where was the Internet?
He hadn't thought of the tree house in a long time. It had been in the backyard of their childhood home in the Toronto suburbs, and they'd stayed up there for hours at a time with comic books. There was a rope ladder that could be pulled up to thwart would-be invaders.
"We can wait this out for quite a while," Jeevan said. He was surveying the water supply, which was still reasonable. He'd filled every receptacle in the apartment with water before it stopped coming out of the taps, and more recently he'd been catching snow in pots and bowls on the balcony.
"Yes," Frank said, "but then what?"
"Well, we'll just stay here till the lights come back on or the Red Cross shows up or whatever." Jeevan had been prone to cinematic daydreams lately, images tumbling together and overlapping, and his favorite movie involved waking in the morning to the sound of a loudspeaker, the army coming in and announcing that it was all over, this whole flu thing cleared up and taken care of, everything back to normal again. He'd push the dresser away from the door and go down to the parking lot, maybe a soldier would offer him a cup of coffee, clap him on the back. He imagined people congratulating him on his foresight in stocking up on food.
"What makes you think the lights will come back on?" Frank asked without looking up. Jeevan started to reply, but words failed him.
# **31**
**INTERVIEW OF KIRSTEN RAYMONDE** by François Diallo, librarian of the New Petoskey Library and publisher of the _New Petoskey News_ , Year Fifteen, continued:
DIALLO: Forgive me. I shouldn't have asked about the knife tattoos.
RAYMONDE: Forgiven.
DIALLO: Thank you. I wondered, though, if I might ask you about the collapse?
RAYMONDE: SURE.
DIALLO: You were in Toronto, I think. Were you with your parents?
RAYMONDE: No. That last night, Day One in Toronto, or I guess it's Night One, isn't it? Whatever you want to call it. I was in a production of King Lear, and the lead actor died on stage. His name was Arthur Leander. You remember, we talked about this a few years ago, and you had his obituary in one of your newspapers.
DIALLO: But perhaps you wouldn't mind, for the benefit of our newspaper's readers...
RAYMONDE: Okay, yes. He had a heart attack onstage, like I was saying. I don't remember many details about him, because I don't remember very much about anything from that time, but I've retained a sort of impression of him, if that makes sense. I know he was kind to me and that we had some sort of friendship, and I remember very clearly the night when he died. I was onstage with two other girls in the production, and I was behind Arthur, so I didn't see his face. But I remember there was some commotion just in front of the stage. And then I remember hearing a sound, this sharp "thwack," and that was Arthur hitting his hand on the plywood pillar by my head. He'd sort of stumbled back, his arm flailed out, and then a man from the audience had climbed up on the stage and was running toward him—
DIALLO: The mystery audience member who knew CPR. He's in the New York Times obituary.
RAYMONDE: He was kind to me. Do you know his name?
DIALLO: I'm not sure anyone does.
# **32**
**ON DAY FORTY-SEVEN ,** Jeevan saw smoke rising in the distance. He didn't imagine the fire would get very far, given all the snow, but the thought of fires in a city without firefighters hadn't occurred to him.
Jeevan sometimes heard gunshots at night. Neither rolled-up towels nor plastic nor duct tape could keep the stench from the hallway from seeping in, so they kept the windows open at all times and wore layers of clothes. They slept close together on Frank's bed, for warmth.
"Eventually we're going to have to leave," Jeevan said.
Frank put his pen down and looked past Jeevan at the window, at the lake and the cold blue sky. "I don't know where I'd go," he said. "I don't know how I'd do it."
Jeevan stretched out on the sofa and closed his eyes. Decisions would have to be made soon. There was enough food for only another two weeks.
When Jeevan looked out at the expressway, the thought that plagued him was that maneuvering Frank's wheelchair through that crush of stopped cars would be impossible. They'd have to take alternate roads, but what if all of the roads were like this?
They hadn't heard anyone in the corridor for over a week, so that night Jeevan decided to risk venturing out of the apartment. He pushed the dresser away from the door and took the stairs to the roof. After all these weeks indoors he felt exposed in the cold air. Moonlight glinted on glass but there was no other light. A stark and unexpected beauty, silent metropolis, no movement. Out over the lake the stars were vanishing, blinking out one by one behind a bank of cloud. He smelled snow in the air. They would leave, he decided, and use the storm as cover.
"But what would be out there?" Frank asked. "I'm not an idiot, Jeevan. I hear the gunshots. I saw the news reports before the stations went dark."
"I don't know. A town somewhere. A farm."
"A farm? Are you a farmer? Even if it weren't the middle of winter, Jeevan, do farms even work without electricity and irrigation systems? What do you think will grow in the spring? What will you eat there in the meantime?"
"I don't know, Frank."
"Do you know how to hunt?"
"Of course not. I've never fired a gun."
"Can you fish?"
"Stop it," Jeevan said.
"After I was shot, when they told me I wouldn't walk again and I was lying in the hospital, I spent a lot of time thinking about civilization. What it means and what I value in it. I remember thinking that I never wanted to see a war zone again, as long as I live. I still don't."
"There's still a world out there," Jeevan said, "outside this apartment."
"I think there's just survival out there, Jeevan. I think you should go out there and try to survive."
"I can't just leave you."
"I'll leave first," Frank said. "I've given this some thought."
"What do you mean?" he asked, but he knew what Frank meant.
# **33**
RAYMONDE: Do you still have that obituary of Arthur Leander? I remember you showed it to me, years ago, but I don't remember if it had the name—
DIALLO: Do I still have the second-to-last edition of the New York Times? What a question. Of course I do. But no, it doesn't have the name. That man from the audience who performed CPR on Leander, he's unidentified. Under normal circumstances there would've been a follow-up story, presumably. Someone would have found him, tracked him down. But tell me what happened. Mr. Leander fell, and then...
RAYMONDE: Yes, he collapsed, and then a man came running across the stage and I realized he'd come from the audience. He was trying to save Arthur, he was performing CPR, and then the medics arrived and the man from the audience sat with me while they did their work. I remember the curtain fell and I was sitting there onstage, watching the medics, and the man from the audience spoke with me. He was so calm, that's what I remember about him. We went and sat in the wings for a while until my minder found us. She was a babysitter, I guess. It was her job to look after me and the other two children in the show.
DIALLO: Do you remember her name?
RAYMONDE: No. I remember she was crying, really sobbing, and it made me cry too. She cleaned my makeup off, and then she gave me a present, that glass paperweight I showed you once.
DIALLO: You're still the only person I know who carries a paperweight in her backpack.
RAYMONDE: It's not that heavy.
DIALLO: It seems an unusual gift for a child.
RAYMONDE: I know, but I thought it was beautiful. I still think it's beautiful.
DIALLO: That's why you took it with you when you left Toronto?
RAYMONDE: Yes. Anyway, she gave it to me, and I guess eventually we quieted down, I remember after that we stayed in the dressing room playing cards, and then she kept calling my parents, but they never came.
DIALLO: Did they call her back?
RAYMONDE: She couldn't reach them. I should say I don't really remember this next part, but my brother told me. Eventually she called Peter, my brother, who was at home that night. He said he didn't know where they were either, but said she could bring me home and he'd look after me. Peter was much older than me, fifteen or sixteen at the time, so he looked after me a lot. The woman drove me home and left me there with him.
DIALLO: And your parents...?
RAYMONDE: I never saw them again. I have friends with similar stories. People just vanished.
DIALLO: They were among the very first, then, if this was Day One in Toronto.
RAYMONDE: Yes, they must have been. I wonder sometimes what happened to them. I think perhaps they got sick in their offices and went to the ER. That seems to me the most likely scenario. And then once they got there, well, I can't imagine how anyone could have survived in any of the hospitals.
DIALLO: So you stayed at home with your brother and waited for them to come back.
RAYMONDE: We didn't know what was happening. For the first little while, waiting seemed to make sense.
# **34**
**" READ ME SOMETHING,"** Jeevan said, on the fifty-eighth day. He was lying on the sofa, staring up at the ceiling, and he'd been drifting in and out of sleep. It was the first thing he'd said in two days.
Frank cleared his throat. "Anything in particular?" He hadn't spoken in two days either.
"The page you're working on now."
"Really? You want some overprivileged philanthropist's thoughts on the charity work of Hollywood actors?"
"Why not?"
Frank cleared his throat. "The immortal words of a philanthropist whose name I'm not allowed to divulge but who you've never heard of anyway," he said.
What I like to see is when actors use their celebrity in an interesting way. Some of them have charitable foundations, they do things like try to bring attention to the plight of women and girls in Afghanistan, or they're trying to save the white African rhino, or they discover a passion for adult literacy, or what have you. All worthy causes, of course, and I know their fame helps to get the word out.
But let's be honest here. None of them went into the entertainment industry because they wanted to do good in the world. Speaking for myself, I didn't even think about charity until I was already successful. Before they were famous, my actor friends were just going to auditions and struggling to be noticed, taking any work they could find, acting for free in friends' movies, working in restaurants or as caterers, just trying to get by. They acted because they loved acting, but also, let's be honest here, to be noticed. All they wanted was to be seen.
I've been thinking lately about immortality. What it means to be remembered, what I want to be remembered for, certain questions concerning memory and fame. I love watching old movies. I watch the faces of long-dead actors on the screen, and I think about how they'll never truly die. I know that's a cliché but it happens to be true. Not just the famous ones who everyone knows, the Clark Gables, the Ava Gardners, but the bit players, the maid carrying the tray, the butler, the cowboys in the bar, the third girl from the left in the nightclub. They're all immortal to me. First we only want to be seen, but once we're seen, that's not enough anymore. After that, we want to be remembered.
# **35**
DIALLO: What was it like, those last days before you left Toronto?
RAYMONDE: I stayed in the basement watching television. The neighborhood was emptying out. Peter was going out at night—stealing food, I think—and then one morning he said, "Kiki, we've got to go." He hotwired a car that the neighbors had abandoned, and we drove for a while, but we got trapped. All the ramps onto the expressway were clogged with abandoned cars, the side roads too. Finally we just had to walk, like everyone else.
DIALLO: Where did you go?
RAYMONDE: East and south. Around the lake and down into the United States. The border was open by then. All the guards had left.
DIALLO: Did you have a set destination?
RAYMONDE: I don't think so. No. But it was either leave or wait in Toronto, and what would we have been waiting for?
# **36**
**JEEVAN RESOLVED TO** follow the lake. The beach was all gravel and rocks, difficult to walk on in the snow, in the twilight, he was afraid of twisting his ankle, and he didn't like the footprints he was leaving, but he was determined to stay off the roads if he possibly could. He wanted very much to avoid other people.
On his last evening in the apartment he'd stood by the window, watching the expressway through the telescope. In three hours of watching he had seen only two people, both headed away from downtown, furtive, glancing over their shoulders. In every moment of those hours he was aware of the silence emanating from Frank's bedroom. He'd checked twice to make sure Frank wasn't breathing, knew the second time was irrational but how terrible it would be for Frank to wake up alone. He'd felt a vertiginous giving-way, the cliff crumbling beneath his feet, but held to sanity by sheer willpower. He wasn't well, but was anyone?
While he was waiting for the day to end he sat at Frank's desk, looking out at the lake. Trying to hold on to the tranquility of these last few moments, here in this apartment where he'd been for so long. Frank had left his manuscript on the desk. Jeevan found the page he'd been working on, a philanthropist's thoughts on old movies and fame. Frank's impeccable handwriting in the top margin: _I've been thinking lately about immortality_. Was that line Frank's, then, not the philanthropist's? Impossible to say. Jeevan folded the piece of paper and put it in his pocket.
Just after sunset, he left the apartment with a dusty backpack that Frank had taken on hiking trips in his pre-spinal-cord-injury days. Its existence was something of a mystery. Had Frank imagined he'd someday walk again? Was he planning on giving it to someone? When the last light was fading over the lake, Jeevan pushed the dresser aside, stepped out into the terrible corridor with its reek of death and garbage, and made his way down the stairs in darkness. He stood for several minutes behind the door that led to the lobby, listening, before he eased it open and slipped through, heart pounding. The lobby was deserted, but the glass doors had been smashed.
The world had emptied out since he'd last seen it. There was no movement on the plaza or on the street, or on the distant expressway. A smell of smoke in the air, with a chemical tinge that spoke of burning offices and house fires. But most striking was the absolute absence of electric light. Once, in his early twenties, he'd been walking up Yonge Street around eleven p.m. and every light on the street had blinked out. For an instant the city had vanished around him, and then the lights were back so quickly that it was like a hallucination, everyone on the street asking their companions if they'd seen it too—"Was it just me?"—and at the time he'd been chilled by the suggestion of a dark city. It was as frightening as he would have imagined. He wanted only to escape.
The moon was a crescent in the evening sky. He walked as quietly as possible, the pack weighing on him with every step. He avoided the roads as much as he could. The lake to his left, black water gleaming. The beach was pale in the half-light. Impossible not to think of Frank, lying still on the bed with an empty bottle of sleeping pills on the nightstand, but he couldn't dwell on Frank because every sound might mean the end of everything, every shadow could be hiding someone with a gun who wanted his backpack. He felt his senses sharpening, an absolute focus taking hold. This is what it would take.
There was something out on the lake, a white shape bobbing. A sailboat, he decided, probably the same one he'd seen weeks ago from the apartment, probably no one aboard. He kept walking and the city kept pulling him away from the lake. He climbed embankments and followed lakeside streets until he could return to the water, until finally the city fell away. Every so often he stopped to listen, but heard only the water on the gravel beach, a gentle wind.
After some hours, he heard gunshots, far distant, two quick sharp noises and then the night closed over the sounds and there was only Jeevan, only the water, only whatever frightened souls still remained. He wished he could move faster.
The moon was setting. He was passing along the edge of an industrial wasteland. It occurred to him that he was very tired, and also that it would be dangerous to fall asleep. He somehow hadn't thought much about what it would be like to sleep out here, unprotected. He was cold. He could no longer feel his toes, or his tongue either, because he'd been putting snow in his mouth to stay hydrated. He placed a pinch of snow on his tongue and thought of making snow ice cream with Frank and their mother when they were small boys—"First you stir in the vanilla"—Frank standing on a stool on his wondrously functional pre-Libya legs, the bullet that would sever his spinal cord still twenty-five years away but already approaching: a woman giving birth to a child who will someday pull the trigger on a gun, a designer sketching the weapon or its precursor, a dictator making a decision that will spark in the fullness of time into the conflagration that Frank will go overseas to cover for Reuters, the pieces of a pattern drifting closer together.
Jeevan sat on a driftwood log to watch the sunrise. He wondered what had happened to his girlfriend. She seemed very distant. He thought of his house and wondered if he'd see it again, knew almost at the same moment that he wouldn't. As the sky brightened he built a shelter from driftwood and the garbage bags he'd brought with him, a makeshift structure that would cut the wind and hopefully look like a pile of trash from a distance. He curled around his pack and fell into a fitful sleep.
When he woke later in the morning, there was an instant when he didn't know where he was. He had never in his life been so cold.
He'd been walking for five days before he saw anyone else. At first the solitude was a relief—he'd imagined a lawless world, he'd imagined being robbed of his backpack and left to die without supplies a thousand times—but as the days passed, the meaning of the emptiness began to sink in. The Georgia Flu was so efficient that there was almost no one left.
But on the fifth day he saw three people far ahead on the shore and his heart leapt. They were traveling the same direction as Jeevan. He stayed a mile or so behind them all through the day. At nightfall they built a fire on the beach and he decided to risk it. They heard his footsteps and watched as he approached. He stopped twenty feet away, raised both hands to show he was unarmed, and called out a greeting, waited till one of them beckoned him close. They were two young men of nineteen or twenty and an older woman—Ben and Abdul and Jenny—tired and worn in the firelight. They'd been walking for a day longer than he had, down through the city from the northern suburbs.
"Is there a lot of crime in the city?"
"Sure," Abdul said. He was thin and nervous, with hair down to his shoulders; he twisted a strand around his finger as he spoke. "Anarchy, right? No police. Fucking terrifying."
"But actually not as much crime as you'd expect," Jenny said. "There just aren't that many people."
"Did they leave, or they're all...?"
"If you got sick," Ben said, "you were gone in forty-eight hours." He knew something about it. His girlfriend, his parents, and his two sisters had died in the first week. He couldn't explain why he wasn't dead too. He'd taken care of all of them, because by Day Three all the hospitals had closed. He'd dug five graves in his backyard.
"You must be immune," Jeevan said.
"Yes." Ben stared fixedly into the flames. "I'm the luckiest man alive, aren't I?"
They traveled together for nearly a week, until they reached a point where Jeevan wanted to keep following the lake and the other three wanted to turn west, toward a town where Jenny's sister had lived. They debated the matter for an hour or two, Jeevan certain that venturing into a town was a mistake and the others disagreeing with him, Jenny afraid of never seeing her sister again, and in the end they wished one another luck and parted ways. As Jeevan walked on alone he felt himself disappearing into the landscape. He was a small, insignificant thing, drifting down the shore. He had never felt so alive or so sad.
There was a clear morning some days later when he looked up and saw Toronto on the far side of the lake, ghostly with distance. A thin blue spire piercing the sky, glass city. From this distance it looked like something from a fairy tale.
He came upon other travelers sometimes, but so few. Almost everyone was moving south.
"It's like those disaster movies," he'd said to Frank, over two months ago now, on the third or fourth night in the apartment. Those were the days before the end of television. They were stunned with horror but it hadn't entirely sunk in yet, any of it, and that night there was a certain awful giddiness. All evidence suggested that the center wasn't holding—Was this actually happening? they asked one another—but personally they had food and water, they were at least momentarily secure and not sick. "You know," Jeevan had said, "in the movie version of this there's the apocalypse, and then afterward—"
"What makes you think we'll make it to afterward?" Frank was always so goddamned calm about everything.
This silent landscape. Snow and stopped cars with terrible things in them. Stepping over corpses. The road seemed dangerous. Jeevan avoided it, stayed mostly in the woods. The road was all travelers walking with shell-shocked expressions, children wearing blankets over their coats, people getting killed for the contents of their backpacks, hungry dogs. He heard gunshots in the towns so he avoided these too. He slipped in and out of country houses, searching for canned goods while the occupants lay dead upstairs.
It was becoming more difficult to hold on to himself. He tried to keep up a litany of biographical facts as he walked, trying to anchor himself to this life, to this earth. My name is Jeevan Chaudhary. I was a photographer and then I was going to be a paramedic. My parents were George of Ottawa and Amala of Hyderabad. I was born in the Toronto suburbs. I had a house on Winchester Street. But these thoughts broke apart in his head and were replaced by strange fragments: This is my soul and the world unwinding, this is my heart in the still winter air. Finally whispering the same two words over and over: "Keep walking. Keep walking. Keep walking." He looked up and met the eyes of an owl, watching him from a snow-laden branch.
# **37**
DIALLO: And so when you left, you just kept walking with no destination in mind?
RAYMONDE: As far as I know. I actually don't remember that year at all.
DIALLO: None of it?
RAYMONDE: Absolutely nothing.
DIALLO: Well, the shock would have been considerable.
RAYMONDE: Of course, but then we stopped in a town eventually, and I remember everything from that time onward. You can get used to anything. I think it was actually easier for children.
DIALLO: The children seemed awfully traumatized.
RAYMONDE: At the time, sure. Everyone was. But two years later? Five years? Ten? Look, I was eight. Nine, when we stopped walking. I can't remember the year we spent on the road, and I think that means I can't remember the worst of it. But my point is, doesn't it seem to you that the people who have the hardest time in this—this current era, whatever you want to call it, the world after the Georgia Flu—doesn't it seem like the people who struggle the most with it are the people who remember the old world clearly?
DIALLO: I hadn't thought about it.
RAYMONDE: What I mean to say is, the more you remember, the more you've lost.
DIALLO: But you remember some things....
RAYMONDE: But so little. My memories from before the collapse seem like dreams now. I remember looking down from an airplane window, this must have been sometime during the last year or two, and seeing the city of New York. Did you ever see that?
DIALLO: Yes.
RAYMONDE: A sea of electric lights. It gives me chills to think of it. I don't really remember my parents. Actually just impressions. I remember hot air coming out of vents in the winter, and machines that played music. I remember what computers looked like with the screen lit up. I remember how you could open a fridge, and cold air and light would spill out. Or freezers, even colder, with those little squares of ice in trays. Do you remember fridges?
DIALLO: Of course. It's been a while since I've seen one used for anything other than shelving space.
RAYMONDE: And they had light inside as well as cold, right? I'm not just imagining this?
DIALLO: They had light inside.
#
# **38**
**WHEN KIRSTEN AND AUGUST** left the house in the woods, when they dragged and carried their new suitcases through the trees to the road, there was a moment when Kirsten stood looking back at the overgrown driveway while August made some adjustments—moving the poetry books and the water bottles from his backpack to the wheeled suitcase, to ease the weight on his back—and if not for the physical evidence, the suitcases filled with towels and shampoo and that box of salt they'd found in the kitchen, the blue silk dress she wore and the bulge of the Starship _Enterprise_ in August's vest pocket, she might have thought they'd imagined the house.
"A non-ransacked house," August said, once they'd resumed walking. The suitcase wheels were stiff and Kirsten didn't like the sound they made on the road, but otherwise the suitcases were perfect. "I never thought I'd see another one."
"It was incredible. I almost wanted to lock the door behind us." That's what it would have been like, she realized, living in a house. You would leave and lock the door behind you, and all through the day you would carry a key. Dieter and Sayid probably remembered what it was like to live in houses and carry keys. All thoughts led back to them.
August believed in the theory of multiple universes. He claimed this was straight-up physics, as he put it, or if not exactly mainstream physics then maybe the outer edge of quantum mechanics, or anyway definitely not just some crackpot theory he'd made up.
"I'm afraid I've no idea," the tuba had said, when Kirsten had asked him for confirmation a few years back. No one had any idea, it turned out. None of the older Symphony members knew much about science, which was frankly maddening given how much time these people had had to look things up on the Internet before the world ended. Gil had offered an uncertain reminiscence about an article he'd read once, something about how subatomic particles are constantly vanishing and reappearing, which meant, he supposed, that there's someplace else to be, which he imagined might suggest that a person could theoretically be simultaneously present and not present, perhaps living out a shadow life in a parallel universe or two. "But look," he'd said, "I was never a science guy." In any event, August liked the idea of an infinite number of parallel universes, lined up in all directions. Kirsten imagined this arrangement as something like the successive planes formed when two mirrors reflect one another, the images shifting greener and cloudier with each repetition until they vanish toward infinity. She'd seen this once in a clothing store in a deserted shopping mall.
August said that given an infinite number of parallel universes, there had to be one where there had been no pandemic and he'd grown up to be a physicist as planned, or one where there had been a pandemic but the virus had had a subtly different genetic structure, some minuscule variance that rendered it survivable, in any case a universe in which civilization hadn't been so brutally interrupted. They were discussing this at the top of an embankment in the late afternoon, where they were resting and flipping through a stack of magazines that Kirsten had taken from the house.
"In an alternate universe," August said, "you might've been in the tabloid pictures. Isn't this one of your actor's wives?"
"Is it?" She took the magazine from him. There was Arthur's third wife, Lydia, shopping in New York City. She was wearing precarious shoes and carried a dozen shopping bags. The pandemic would reach North America in less than a month. The sighting was interesting, but not interesting enough to add to the collection.
In the last magazine, Kirsten found another ex-wife. A photograph of a woman in her late thirties or early forties with a hat pulled low, glaring at the camera as she exited a building:
_Rekindling the Flame???_
WHY, HELLO, MIRANDA! MIRANDA CARROLL, SHIPPING EXECUTIVE AND FIRST WIFE OF ACTOR ARTHUR LEANDER, RAISED QUESTIONS WITH A FURTIVE DEPARTURE FROM THE STAGE DOOR OF THE TORONTO THEATER WHERE LEANDER IS PERFORMING IN _KING LEAR_. AN EYEWITNESS REPORTS THAT THEY WERE IN LEANDER'S DRESSING ROOM ALONE FOR NEARLY AN HOUR! "WE WERE ALL A LITTLE SURPRISED," THE EYEWITNESS SAID.
"I think I was there," Kirsten said. "I might've been in that building at that moment." Behind Miranda she saw only a steel door, the stone wall of a building. Had she passed through that door? She must have, she thought, and wished she could remember it.
August studied the photo, interested. "Do you remember seeing her there?"
An impression of a coloring book, the smell of pencils, Arthur's voice, a warm room with a red carpet, electric light. Had a third person been in the room? She couldn't be sure.
"No," she said. "I don't remember her." She tore the photograph with its caption from the page.
"Look at the date," August said. "Two weeks till the apocalypse!"
"Well, it's nice that at least the celebrity gossip survived."
Nothing else in the rest of the magazines, but this find was remarkable, this was enough. They kept two magazines to start a fire later and buried the other three under leaves.
"It would've been you in those tabloid pictures," he said, picking up the parallel-universes theme. "I mean, it _is_ you in those pictures, in a parallel universe where the collapse didn't happen."
"I still think you invented the parallel-universe theory," she said, but one of the few things that August didn't know about her was that sometimes when she looked at her collection of pictures she tried to imagine and place herself in that other, shadow life. You walk into a room and flip a switch and the room fills with light. You leave your garbage in bags on the curb, and a truck comes and transports it to some invisible place. When you're in danger, you call for the police. Hot water pours from faucets. Lift a receiver or press a button on a telephone, and you can speak to anyone. All of the information in the world is on the Internet, and the Internet is all around you, drifting through the air like pollen on a summer breeze. There is money, slips of paper that can be traded for anything: houses, boats, perfect teeth. There are dentists. She tried to imagine this life playing out somewhere at the present moment. Some parallel Kirsten in an air-conditioned room, waking from an unsettling dream of walking through an empty landscape.
"A parallel universe where space travel was invented," August said. This was a game they'd been playing for a decade. They were lying on their backs now, sedated by heat. Birch branches swayed in the breeze, sunlight filtering through green. Kirsten closed her eyes and watched the silhouettes of leaves float away under her eyelids.
"But space travel was invented, wasn't it? I've seen pictures." Her hand drifted up to the scar on her cheekbone. If there were better universes, then there were probably much worse ones. Universes where she remembered her first year on the road, for instance, or where she remembered what had caused the scar on her face, or where she'd lost more than two teeth.
"We just went up to that gray moon," August said. "Nowhere else, we never went farther. I mean the kind of space travel you'd see in TV shows, you know, other galaxies, other planets."
"Like in my comic books?"
"Your comics are weird. I was thinking more like _Star Trek_."
"A parallel universe where my comics are real," she said.
"What do you mean?"
"I mean a parallel universe where we boarded Station Eleven and escaped before the world ended," Kirsten said.
"The world didn't _end_ ," he said. "It's still spinning. But anyway, you'd want to live on Station Eleven?"
"I think it's beautiful. All those islands and bridges."
"But it's always night or twilight, isn't it?"
"I don't think I'd mind."
"I like this world better," August said. "Does Station Eleven even have an orchestra? Or would it just be me standing there by myself on the rocks in the dark, playing my violin for giant seahorses?"
"Okay, a parallel universe with better dentistry," she said.
"You aim high, don't you?"
"If you'd lost any teeth, you'd know how high I'm aiming."
"Fair enough. I'm sorry about your teeth."
"A parallel universe where I have no knife tattoos."
"I'd like to live there too," August said. "A parallel universe where Sayid and Dieter didn't disappear."
"A parallel universe where telephones still work, so we could just call the Symphony and ask them where they are, and then we'd call Dieter and Sayid and all of us would meet up somewhere."
They were quiet, looking up at the leaves.
"We'll find them," Kirsten said, "we'll see the Symphony again," but of course they couldn't be sure.
They dragged their suitcases down the embankment to the road. They were very close to Severn City now. At twilight the road curved back to the lakeshore, and the first houses of Severn City appeared. Young birch trees between the road and the lake but otherwise no forest, just overgrown lawns and houses submerged in vines and shrubbery, a beach of rocks and sand.
"I don't want to do this at night," August said. They chose a house at random, waded through the backyard and made camp behind a garden shed. There was nothing to eat. August went exploring and came back with blueberries.
"I'll take the first watch," Kirsten said. She was exhausted but she didn't think she could sleep. She sat on her suitcase, her back against the wall of the shed, a knife in her hands. She watched the slow rise of fireflies from the grass and listened to the water on the beach across the road, the sighing of wind in the leaves. A beating of wings and the squeak of a rodent, an owl making a kill.
"Remember that man we met at the gas station?" August asked. She'd thought he was asleep.
"Of course. What about him?"
"That scar on his face." He sat up. "I was just thinking about it, and I realized what it is."
"The prophet marked him." The memory was agitating. She flicked her wrist and her knife split the cap of a white mushroom a few feet away.
"Yes, but the symbol itself, the pattern of the scar. How would you describe it?"
"I don't know," she said, retrieving her knife. "It looked like a lowercase _t_ with an extra line through the stem."
"A shorter line. Toward the bottom. Think about it. It isn't abstract."
"I _am_ thinking about it. It looked abstract to me."
"It's an airplane," August said.
# **39**
**TWO WEEKS BEFORE** the end of commercial air travel, Miranda flew to Toronto from New York. It was late October, and she hadn't been back to Canada in some months. She'd always liked the descent into this city, the crowded towers by the lakeshore, the way an infinite ocean of suburbia rushed inward and came to a point at the apex of the CN Tower. She thought the CN Tower was ugly up close, but unexpectedly lovely when viewed from airplane windows. And as always, the sense of Toronto existing in layers: the city that had shocked her with its vastness when she'd arrived here from Delano Island at seventeen still existed, but it occupied the same geographical space as a city that now seemed much smaller to her, a place diluted by the years she'd spent moving between London, New York, the harbor cities of Asia. The plane descended into the suburbs. She passed through passport control without incident, the Canada Border Services agent struggling to find an unstamped corner in the pages of her passport, and boarded a waiting car to the Toronto headquarters of Neptune Logistics, where she wished the driver a good day and passed him a twenty-dollar bill over the back of the seat.
"Thank you," he said, surprised. "Would you like some change?"
"No, thank you." She had been overtipping for as long as she'd had money. These small compensations for how fortunate she'd been. She pulled her carry-on suitcase into the Neptune Logistics lobby, cleared building security and took the elevator to the eighteenth floor.
She saw ghosts of herself everywhere here. A twenty-three-year-old Miranda with the wrong clothes and her hair sticking up, washing her hands and peering anxiously at herself in the ladies' room mirror; a twenty-seven-year-old recently divorced Miranda slouching across the lobby with her sunglasses in place, wishing she could disappear, in tears because she'd seen herself on a gossip website that morning and the headline was agonizing: IS ARTHUR SECRETLY CALLING MIRANDA? (Answer: no.) Those previous versions of herself were so distant now that remembering them was almost like remembering other people, acquaintances, young women whom she'd known a long time ago, and she felt such compassion for them. "I regret nothing," she told her reflection in the ladies' room mirror, and believed it. That day, she attended a series of meetings, and in the late afternoon another car delivered her to a hotel. She still had an hour or two to kill until it was time to see Arthur again.
He'd called her in the New York office in August. "Will you take a call from Arthur Smith-Jones?" her assistant had asked, and Miranda had frozen momentarily. The name was from an inside joke that she and Arthur had batted around when they were first married. All these years later she had no recollection of why the name _Smith-Jones_ had been funny, but she knew it was he.
"Thank you, Laetitia, I'll take the call." A click. "Hello, Arthur."
"Miranda?" He sounded uncertain. She wondered if her voice had changed. She'd used her most self-assured addressing-large-meetings voice.
"Arthur. It's been a while." A moment of silence on the line. "Are you there?"
"My father died."
She swiveled in her chair to look out at Central Park. In August the park had a subtropical quality that entranced her, a sense of weight and languor in the lushness of the trees.
"I'm sorry, Arthur. I liked your father." She was thinking of an evening on Delano Island, the first year of their marriage and the only time they'd gone back to Canada for Christmas together, Arthur's father talking with great animation about a poet he'd just been reading. The memory had dimmed since she'd last retrieved it, imprecision creeping in. She no longer remembered the name of the poet or anything else about the conversation.
"Thanks," he said indistinctly.
"Do you remember the name of the poet he liked?" Miranda heard herself asking. "A long time ago. When we were there for Christmas."
"Probably Lorca. He talked about Lorca a lot."
There was a person in the park wearing a bright red T-shirt that contrasted magnificently with all the green. She watched the T-shirt vanish around a curve.
"He drove a snowplow and did carpentry all his life," Arthur said. Miranda wasn't sure what to say to this—she'd known what Arthur's father's occupations were—but Arthur didn't seem to require a response. They were quiet for a moment, Miranda watching to see if the T-shirt would reappear. It didn't.
"I know," she said. "You showed me his workshop."
"I just mean, my life must've seemed unfathomable to him."
"Your life's probably unfathomable to most people. Why did you call me, Arthur?" Her tone as gentle as possible.
"You were the one I wanted to call," he said, "when I got the news."
"But why me? We haven't spoken since the last divorce hearing."
"You know where I'm from," he said, and she understood what he meant by this. Once we lived on an island in the ocean. Once we took the ferry to go to high school, and at night the sky was brilliant in the absence of all these city lights. Once we paddled canoes to the lighthouse to look at petroglyphs and fished for salmon and walked through deep forests, but all of this was completely unremarkable because everyone else we knew did these things too, and here in these lives we've built for ourselves, here in these hard and glittering cities, none of this would seem real if it wasn't for you. And aside from that, she realized, he was currently wifeless.
Arthur was starring in _King Lear_ , presently in previews at the Elgin Theatre. They'd arranged to meet there, because Arthur was in divorce proceedings with his third wife, Lydia, and he feared any restaurant he entered would attract a flock of cameras.
The paparazzi had long since gotten bored of the nonstory of Miranda's continued post-Arthur existence and had stopped following her, but nonetheless Miranda spent some time on her appearance before she left the hotel room, trying to make herself look as little like her old self as possible. She pinned and slicked her hair into a shiny helmet—in her Hollywood and tabloid lives she'd had a mass of curls—and dressed in her favorite suit, dark gray with white piping. Expensive white high-heeled shoes, of a type she often wore to meetings but that the Hollywood wife Miranda would never have considered.
"You look like an executive," she said to herself in the mirror, and the thought that flitted behind this was You look like a stranger. She pushed it away.
Miranda set out in the early twilight. The air was clear and sharp, a cool wind off the lake. The familiarity of these streets. She stopped for a decaf latte at a Starbucks and was struck by the barista's brilliant green hair. "Your hair's beautiful," she said, and the barista smiled. The pleasure of walking cold streets with a hot coffee in her hand. Why did no one on Station Eleven have green hair? Perhaps someone in the Undersea. Or one of Dr. Eleven's associates. No, the Undersea. When she was three blocks from the theater, she put on a knit hat that covered her hair, and dark glasses.
There were five or six men outside the theater, zoom-lens cameras on straps around their necks. They were smoking cigarettes and fiddling with their phones. Miranda felt a deathly stillness come over her. She liked to think of herself as a person who hated no one, but what did she feel for these men if not hatred? She tried to glide by as unobtrusively as possible, but wearing sunglasses after sundown had been a tactical error.
"That Miranda Carroll?" one of them asked. Fucking parasite. She kept her head down in an explosion of flashes and slipped in through the stage door.
Arthur's dressing room was more properly a suite. An assistant whose name she immediately forgot ushered her into a sitting room, where two sofas faced off across a glass coffee table. Through open doors she glimpsed a bathroom and a dressing room, with a rack for costumes—she saw a velvet cloak—and a mirror ringed in lights. It was from this second room that Arthur emerged.
Arthur wasn't old, but he wasn't aging very well. It was disappointment, it seemed to her, that had settled over his face, and there was a strained quality about his eyes that she didn't remember having seen before.
"Miranda," he said. "How long has it been?"
This seemed to her a silly question. She'd assumed, she realized, that everyone remembers the date of their divorce, the same way everyone remembers their wedding date.
"Eleven years," she said.
"Please, have a seat. Can I offer you something?"
"Do you have any tea?"
"I have tea."
"I thought you would." Miranda shed her coat and hat and sat on one of the sofas, which was exactly as uncomfortable as it looked, while Arthur fussed with an electric kettle on a countertop. Here we are, she thought. "How are the previews going?"
"Fine," he said. "Better than fine, actually. Good. It's been a long time since I've done Shakespeare, but I've been working with a coach. Actually, I guess _coach_ isn't the right word. A Shakespeare expert." He came back to the sofas and sat across from her. She watched his gaze flicker over her suit, her gleaming shoes, and realized he was performing the same reconciliations she was, adjusting a mental image of a long-ago spouse to match the changed person sitting before him.
"A Shakespeare expert?"
"He's a Shakespearean scholar. University of Toronto. I love working with him."
"It must be quite interesting."
"It is. He has this extremely impressive pool of knowledge, brings a lot to the table, but at the same time he's completely supportive of my vision for the part."
_Supportive of my vision?_ He'd adopted new speech patterns. But of course he had, because since she'd last seen him there had been eleven years of friends and acquaintances and meetings and parties, travel here and there, film sets, two weddings and two divorces, a child. It made sense, she supposed, that he would be a different person by now. "What a great opportunity," she said, "getting to work with someone like that." Had she ever in her life sat on a less-comfortable sofa. She pressed her fingertips into the foam and barely made an impression. "Arthur," she said, "I'm so sorry about your father."
"Thank you." He looked at her, and seemed to struggle to find the right words. "Miranda, I have to tell you something."
"This doesn't sound good."
"It isn't. Listen, there's a book coming out." His childhood friend Victoria had published the letters he'd sent her. _Dear V.: An Unauthorized Portrait of Arthur Leander_ would be available for purchase in a week and a half. A friend who worked in publishing had sent him an advance copy.
"Am I in it?" she asked.
"I'm afraid so. I'm sorry, Miranda."
"Tell me."
"I mentioned you sometimes, when I wrote to her. That's all. I want you to know that I never said anything unpleasant about you."
"Okay. Good." Was it fair to be as angry as she was? He couldn't have known Victoria would sell the letters.
"You might find this difficult to believe," he said, "but I have some sense of discretion. It's actually one of the things I'm known for."
"I'm sorry," she said, "but did you just say you're famous for your discretion?"
"Look, all I mean is, I didn't tell Victoria everything."
"I appreciate that." A strained silence, during which Miranda willed the kettle to start whistling. "Do you know why she did it?"
"Victoria? I have to assume it was the money. The last I heard, she was working as a housekeeper in a resort on the west coast of Vancouver Island. She probably made more on that book than she'd made in the previous decade."
"Are you going to sue?"
"It would just be more publicity. My agent thinks it's better if we just let the book run its course." The kettle whistled at last; he stood quickly, and she realized he'd been willing the water to boil too. "Hopefully when it comes out it's only a story for a week or so, then it sinks and disappears. Green tea, or chamomile?"
"Green," she said. "It must be infuriating, having your letters sold."
"I was angry at first, I'm still angry, but the truth is, I think I deserved everything I got." He carried two mugs of green tea to the coffee table, where they left rings of steam on the glass.
"Why do you think you deserved it?"
"I treated Victoria like a diary." He lifted his mug, blew on the surface of his tea, and returned the mug very deliberately to the table. There was a studied quality to the movement, and Miranda had an odd impression that he was performing a scene. "She wrote to me at first, in the very beginning. Maybe two letters and three postcards, back when I first started writing to her from Toronto. Then a couple of quick notes telling me about changes in address, with a cursory note at the beginning, you know, 'Hi, sorry for not writing more, I've been busy, here's my new address.' "
"So all the times I saw you writing to her," Miranda said, "she never wrote back." She was surprised by how sad this made her.
"Right. I used her as a repository for my thoughts. I think I stopped thinking of her as a human being reading a letter." He looked up—and here, a pause in which Miranda could almost see the script: "Arthur looks up. Beat." Was he acting? She couldn't tell. "The truth is, I think I actually forgot she was real."
Did this happen to all actors, this blurring of borders between performance and life? The man playing the part of the aging actor sipped his tea, and in that moment, acting or not, it seemed to her that he was deeply unhappy.
"It sounds like you've had a difficult year," she said. "I'm sorry."
"Thank you. It hasn't been easy, but I keep reminding myself, people have much worse years than mine. I lost a few battles," he said, "but that isn't the same thing as losing the war."
Miranda raised her mug. "To the war," she said, which elicited a smile. "What else is happening?"
"I'm always talking about myself," he said. "How's your life?"
"Good. Very good. No complaints."
"You're in shipping, aren't you?"
"Yes. I love it."
"Married?"
"God, no."
"No children?"
"My position on the subject hasn't changed. You had a son with Elizabeth, didn't you?"
"Tyler. Just turned eight. He's with his mother in Jerusalem."
There was a knock at the door just then, and Arthur stood. Miranda watched him recede across the room and thought of their last dinner party in the house in Los Angeles—Elizabeth Colton passed out on a sofa, Arthur walking away up the stairs to the bedroom. She wasn't exactly sure what she was doing here.
The person at the door was very small.
"Hello, Kiki," Arthur said. The visitor was a little girl, seven or eight years old. She clutched a coloring book in one hand, a pencil case in the other. She was very blond, the sort of child who appears almost incandescent in certain lighting. Miranda couldn't imagine what part there could possibly be in _King Lear_ for a seven- or eight-year-old, but she'd seen enough child actors in her time that she could recognize one on sight.
"Can I draw in my coloring book here?" the girl asked.
"Of course," Arthur said. "Come in. I'd like you to meet my friend Miranda."
"Hello," the girl said without interest.
"Hello," Miranda said. The girl looked like a china doll, she thought. She looked like someone who'd been well-cared-for and coddled all her life. She was probably someone who would grow up to be like Miranda's assistant Laetitia, like Leon's assistant Thea, unadventurous and well-groomed.
"Kirsten here likes to visit sometimes," Arthur said. "We talk about acting. Your wrangler knows where you are?" In the way he looked at the girl, Miranda saw how much he missed his own child, his distant son.
"She was on the phone," Kirsten said. "I sneaked out." She sat on the carpet near the door, opened her coloring book to a half-completed page involving a princess, a rainbow, a distant castle, a frog, unpacked her pencils and began drawing red stripes around the bell of the princess's dress.
"Are you still drawing?" Arthur asked Miranda. He was noticeably more relaxed with Kirsten in the room.
Always. Yes. When she traveled she carried a sketchbook in her luggage, for the times when she was alone in hotel rooms at night. The focus of the work had gradually shifted. For years Dr. Eleven had been the hero of the narrative, but lately he'd begun to annoy her and she'd become more interested in the Undersea. These people living out their lives in underwater fallout shelters, clinging to the hope that the world they remembered could be restored. The Undersea was limbo. She spent long hours sketching lives played out in underground rooms.
"You've actually just reminded me. I brought you something." She had finally assembled the first two issues of the _Dr. Eleven_ comics, and had had a few copies printed at her own expense. She extracted two copies each of _Dr. Eleven_ , Vol. 1, No. 1: _Station Eleven_ and _Dr. Eleven_ , Vol. 1, No. 2: _The Pursuit_ from her handbag, and passed them across the table.
"Your work." Arthur smiled. "These are beautiful. The cover of this first one was on the studio wall in L.A., wasn't it?"
"You remember." An image that Arthur had once said was like the establishing shot for a movie: the sharp islands of the City, streets and buildings terraced into the rock, high bridges between. Far below in the aquatic darkness, the outlines of the airlock doors that led to the Undersea, massive shapes on the ocean floor. Arthur opened the first issue at random to a two-page spread, ocean and islands linked by bridges, twilight, Dr. Eleven standing on a rock with his Pomeranian by his side. Text: _I stood looking over my damaged home and tried to forget the sweetness of life on Earth_.
"He was on a space station," Arthur said. "I'd forgotten that." He was turning the pages. "Do you still have the dog?"
"Luli? She died a couple years back."
"I'm sorry to hear that. These are beautiful," he said again. "Thank you."
"What is that?" the little girl on the carpet asked. Miranda had forgotten about her for a moment.
"Some books my friend Miranda made," Arthur said. "I'll show you later, Kiki. What are you working on there?"
"The princess," Kirsten said. "Matilda said I couldn't color her dress with stripes."
"Well," Arthur said, "I can't say I agree with her. Is that why you snuck out of your dressing room? Were you fighting with Matilda again?"
"She said it wasn't supposed to have stripes on it."
"I think the stripes are perfect."
"Who's Matilda?" Miranda asked.
"She's an actor too," Kirsten said. "She's sometimes really mean."
"It's an unusual staging," Arthur said. "Three little girls on the stage at the beginning, playing childhood versions of Lear's daughters, and then they come back as hallucinations in the fourth act. No lines, they're just there."
"She thinks she's better than everyone because she goes to the National Ballet School," Kirsten said, returning the subject to Matilda.
"Do you dance too?" Miranda asked.
"Yeah, but I don't want to be a dancer. I think ballet's stupid."
"Kirsten told me she wants to be an actor," Arthur said.
"Oh, how interesting."
"Yeah," Kirsten said without looking up. "I've been in a lot of things."
"Really," Miranda said. How does one talk to an eight-year-old? She glanced at Arthur, who shrugged. "Like what?"
"Just _things_ ," the girl said, as if she hadn't been the one to bring these things up in the first place. Miranda was remembering that she'd never liked child actors.
"Kirsten went to an audition in New York last month," Arthur said.
"We went in an airplane." Kirsten stopped coloring and considered the princess. "The dress is wrong," she said. Her voice quavered.
"I think the dress looks beautiful," Miranda said. "You've done a beautiful job."
"I have to agree with Miranda on this one," Arthur said. "The stripes were a good choice."
Kirsten turned the page. Blank outlines of a knight, a dragon, a tree.
"You're not going to finish the princess?" Arthur asked.
"It isn't perfect," Kirsten said.
They sat for a while in silence, Kirsten filling in the dragon with alternating green and purple scales, Arthur flipping through _Station Eleven_. Miranda drank her tea and tried not to overanalyze his facial expressions.
"Does she visit you often?" Miranda asked softly, when he'd reached the last page.
"Almost daily. She doesn't get along with the other girls. Unhappy kid." They sipped their tea for a moment without speaking. The scratching of the little girl's pencils on the coloring-book page, the steam rings that their mugs left on the glass of the coffee table, the pleasant heat of the tea, the warmth and beauty of this room: these were things that Miranda remembered in the last few hours, two weeks later, when she was drifting in and out of delirium on a beach in Malaysia.
"How long are you in Toronto?" Arthur asked.
"Four days. I leave for Asia on Friday."
"What are you doing there?"
"Working out of the Tokyo office, mostly. There's some possibility of my transferring there next year. Meeting with local subsidiaries in Singapore and Malaysia, visiting a few ships. Did you know," she said, "that twelve percent of the world's shipping fleet is moored fifty miles out of Singapore Harbor?"
"I didn't know that." He smiled. "Asia," he said. "Can you believe this life?"
Miranda was back in her hotel before she remembered the paperweight. She dropped her handbag on the bed and heard it clink against her keys. It was the paperweight of clouded glass that Clark Thompson had brought to a dinner party in Los Angeles eleven years ago, and she'd taken it that night from Arthur's study. She'd meant to give it back to him.
She held the paperweight for a moment, admiring it in the lamplight. She wrote a note on hotel stationery, put her shoes back on, went downstairs to the concierge desk, and arranged to have it sent by courier to the Elgin Theatre.
# **40**
**TWO WEEKS LATER ,** just before the old world ended, Miranda stood on a beach on the coast of Malaysia looking out at the sea. She'd been delivered back to her hotel after a day of meetings, where she'd spent some time finishing a report and eating a room-service dinner. She'd planned on going to bed early, but through the window of her room she could see the lights of the container-ship fleet on the horizon, and she'd walked down to the water for a closer look.
The three nearest airports had closed in the previous ninety minutes, but Miranda didn't know this yet. She'd been aware of the Georgia Flu, of course, but was under the impression that it was still a somewhat shadowy health crisis unfolding in Georgia and Russia. The hotel staff had been instructed to avoid alarming the guests, so no one mentioned the pandemic as she crossed the lobby, although she did notice in passing that the front desk seemed understaffed. In any event, it was a pleasure to escape the coffin chill of the hotel air-conditioning, to walk down the well-lit path to the beach and take off her shoes to stand barefoot in the sand.
Later that evening she would find herself troubled and at moments even a little amused by the memory of how casually everyone had once thrown the word _collapse_ around, before anyone understood what the word truly meant, but in any event, there had been an economic collapse, or so everyone called it at the time, and now the largest shipping fleet ever assembled lay fifty miles east of Singapore Harbor. Twelve of the boats belonged to Neptune Logistics, including two new Panamax-class vessels that had yet to carry a single cargo container, decks still gleaming from the South Korean shipyards; ships ordered in a moment when it seemed the demand would only ever grow, built over the following three years while the economy imploded, unneeded now that no one was spending any money.
Earlier that afternoon, in the subsidiary office, Miranda had been told that the local fishermen were afraid of the ships. The fishermen suspected a hint of the supernatural in these vessels, unmoving hulks on the horizon by day, lit up after dark. In the office the local director had laughed at the absurdity of the fishermen's fears, and Miranda had smiled along with everyone else at the table, but was it so unreasonable to wonder if these lights might not be quite of this earth? She knew the ships were only lit up to prevent collisions, but it still seemed to her as she stood on the beach that evening that there was something otherworldly in the sight. When her phone vibrated in her hand, it was Clark Thompson, Arthur's oldest friend, calling from New York.
"Miranda," he said after some awkward preliminaries, "I'm afraid I'm calling with some rather bad news. Perhaps you should sit down."
"What happened?"
"Miranda, Arthur died of a heart attack last night. I'm so sorry."
Oh, Arthur.
Clark hung up the phone and leaned back in his chair. He worked at the kind of firm where doors are never closed unless someone's getting fired, and he was aware that by now he was no doubt the topic of speculation all over the office. Drama! What could possibly be happening in Clark's office? He had ventured out once, for coffee, and everyone had arranged their faces into neutral yet concerned expressions as he passed—that "no pressure, but if there's anything you need to talk about..." look—and he was having one of the worst mornings of his life, but he derived minor satisfaction from saying nothing and depriving the gossips of fuel. He drew a line through Miranda Carroll's name, lifted the receiver to call Elizabeth Colton, changed his mind and went to the window. A young man on the street below was playing a saxophone. Clark opened his window and the room was flooded with sound, the thin notes of the saxophone on the surface of the oceanic city, a blare of hip-hop from a passing car, a driver leaning on his horn at the corner. Clark closed his eyes, trying to concentrate on the saxophone, but just then his assistant buzzed him.
"It's Arthur Leander's lawyer again," Tabitha said. "Shall I tell him you're in a meeting?"
"Bloody hell, does the man never sleep?" It had been Heller who'd left the voice mail at midnight in Los Angeles, three a.m. in New York—"an urgent situation, please call me immediately"—and Heller who'd been up and working when Clark called him back at six fifteen a.m. in New York, three fifteen in L.A. They'd agreed that Clark should be the one to call the family, because Clark had met Arthur's family once and it seemed kinder that way. Clark had decided to also notify the ex-wives, even the most recent one whom he didn't like very much, because it seemed wrong to let them read about it in the newspaper; he had an idea—too sentimental to speak aloud and he knew none of his divorced friends would ever own up to it—that something must linger, a half-life of marriage, some sense memory of love even if obviously not the thing itself. He thought these people must mean something to one another, even if they didn't like one another anymore.
Heller had called him again a half hour later to confirm that Clark had notified the family, which of course Clark hadn't, because three forty-five a.m. in Los Angeles is also three forty-five a.m. on the west coast of Canada, where Arthur's brother lived, and Clark felt that there were limits to how early one should call anyone for any reason. Now it was still only nine a.m. in New York, six a.m. on Heller's coast, and it seemed obscene that this man who'd apparently been up all night was still up and working. Clark was beginning to imagine Heller as a sort of bat, some kind of sinister night-living vampire lawyer who slept by day and worked by night. Or maybe just an amphetamine freak? Clark's thoughts wandered to a particularly exciting week in Toronto, eighteen or nineteen years old, when he and Arthur had accepted some pills from a new friend at a dance club and stayed up for seventy-two hours straight.
"You want to take the call?" Tabitha asked.
"Fine. Put him through, please."
For just a beat Tabitha did nothing, and after seven years of working together in close quarters he knew that this particular brand of silence meant "Tell me what's going on, you know I like gossip," but he didn't oblige, and he knew her well enough to catch the note of disappointment in the perfectly professional "Hold for your call, please" that followed.
"Clark? Heller here."
"So I gathered," Clark said. There was something obnoxious, he thought, in people who introduced themselves by their surnames while calling one by one's first. "How are you, Gary? We haven't spoken in a solid ninety minutes."
"Hanging in, hanging in." Clark mentally added this to his private list of most-hated banalities. "I went ahead and notified the family," Heller said.
"Why? I thought we'd agreed—"
"I know you didn't want to wake up the family, but with this kind of thing, a situation like this, you _have_ to wake up the family. You actually _want_ to wake the family, you know? Actually more decent. You want the family to know before someone leaks something, a photo, video, whatever, and then _Entertainment Weekly_ calls the family for comment and that's how they find out about it. Think about it, I mean, the man died onstage."
"Right," Clark said. "I see." The saxophonist had disappeared. The gray of the November sky reminded him that he was about due for a visit to his parents in London. "Has Elizabeth been notified?"
"Who?"
"Elizabeth Colton. The second wife."
"No, I mean, she's hardly family, is she? When we talked about notifying family, I really just meant Arthur's brother."
"Well, but she is the mother of Arthur's only child."
"Right, right, of course. How old is he?"
"Eight or nine."
"Poor little guy. Hell of an age for this." A crack in Heller's voice, sadness or exhaustion, and Clark revised his mental image from hanging-upside-down bat lawyer to sad, pale, caffeine-addicted man with chronic insomnia. Had he met Heller? Had Heller been at that ghastly dinner party in Los Angeles all those years ago, just before Miranda and Arthur divorced? Maybe. Clark was drawing a blank. "So hey, listen," Heller said, all business again, but a faux-casual style of all-business that Clark associated overwhelmingly with California, "in your time with Arthur, especially recently, did he ever mention anything about a woman named Tanya Gerard?"
"The name's not familiar."
"You're sure?"
"No. Why? Who is she?"
"Well," Heller said, "just between the two of us, seems our Arthur was having a little affair." It wasn't delight in his voice, not exactly. It was importance. This was a man who liked to know things that other people didn't.
"I see," Clark said, "but I admit I fail to see how that's any of our—"
"Oh, of course," Heller said, "of _course_ it isn't, you know, right to privacy and all that, none of our business, right? Not hurting anyone, consenting adults, etcetera, and I mean I'm the most private, I don't even have a _Facebook_ account for god's sake, that's how much I believe in it, in privacy I mean, last guy on earth without a Facebook account. But anyway, this Tanya person, seems she was a wardrobe girl on _King Lear_. I just wondered if he'd mentioned her."
"No, Gary, I don't believe he ever did."
"The producer told me it was all very secret, apparently this was the girl who did costumes or actually maybe it was babysitting, something to do with the child actresses, costumes for the child actresses? I think that was it, although child actors in _Lear_? That one's a head scratcher. But look, anyway, he..."
Was that sunlight on the other side of the East River? A beam had pierced the clouds in the far distance and was angling down over Queens. The effect reminded Clark of an oil painting. He was thinking of the first time he'd seen Arthur, in an acting studio on Danforth Avenue in Toronto. Arthur at eighteen: confident despite the fact that for at least the first six months of acting classes he couldn't act his way out of a paper bag, or so the acting instructor had pronounced one night over drinks at a bar staffed exclusively with drag queens, the instructor trying to pick up Clark, Clark offering only token resistance. And beautiful, Arthur was beautiful back then.
"So the question, obviously," Heller was saying, "is whether he intended to leave this girl anything in the will, because he emailed me last week about changing the will, said he'd met someone and he wanted to add a beneficiary and I have to assume that's who he meant, really what I'm thinking about here is the worst-case scenario, where there's a shadow will somewhere, some informal document he drew up himself because he wasn't going to see me for a few weeks, that's what I'm trying to get to the bottom of here—"
"You should've seen him," Clark said.
"I should've seen... I'm sorry, what?"
"Back at the beginning, when he was just starting out. You've seen his talent, his talent was obvious, but if you'd seen him before any of the rest of it, all the tabloids and movies and divorces, the fame, all those warping things."
"I'm sorry, I'm not sure I understand what you're getting at here, I—"
"He was wonderful," Clark said. "Back then, back at the beginning. I was so struck by him. I don't mean romantically, it was nothing like that. Sometimes you just _meet_ someone. He was so kind, that's what I remember most clearly. Kind to everyone he met. This humility about him."
"What—"
"Gary," Clark said, "I'm going to hang up now."
He stuck his head out the window for a fortifying breath of November air, returned to his desk, and called Elizabeth Colton. She let out her breath in a long sigh when he told her the news.
"Are there funeral arrangements?"
"Toronto. Day after tomorrow."
"Toronto? Does he have family there?"
"No, but his will was very specific apparently. I guess he felt some attachment to the place."
As Clark spoke, he was remembering a conversation he'd had with Arthur over drinks some years ago, in a bar in New York. They'd been discussing the cities they'd lived in. "You're from London," Arthur had said. "A guy like you can take cities for granted. For someone like me, coming from a small place... look, I think about my childhood, the life I lived on Delano Island, that place was so small. Everyone knew me, not because I was special or anything, just because everyone knew everyone, and the claustrophobia of that, I can't tell you. I just wanted some privacy. For as long as I could remember I just wanted to get out, and then I got to Toronto and no one knew me. Toronto felt like freedom."
"And then you moved to L.A. and got famous," Clark had said, "and now everyone knows you again."
"Right." Arthur had been preoccupied with an olive in his martini, trying to spear it with a toothpick. "I guess you could say Toronto was the only place I've felt free."
Clark woke at four a.m. the next morning and took a taxi to the airport. These were the hours of near misses, the hours of miracles, visible as such only in hindsight over the following days. The flu was already seeping through the city, but he hailed a taxi in which the driver wasn't ill and no one contagious had touched any surface before him, and from this improbably lucky car he watched the streets passing in the pre-dawn dark, the pale light of bodegas with their flowers behind plastic curtains, a few shift workers on the sidewalks. The social-media networks were filled with rumors of the flu's arrival in New York, but Clark didn't partake of social media and was unaware.
At John F. Kennedy International Airport he passed through a terminal in which he managed by some choreography of luck to avoid passing too close to anyone who was already infected—there were several infected people in that particular terminal by then—and managed not to touch any of the wrong surfaces, managed in fact to board a plane filled with similarly lucky people—the twenty-seventh-to-last plane ever to depart from that airport—and through all of this he was so sleep-deprived, he'd stayed up too late packing, he was tired and caught up in thoughts of Arthur, in listening to Coltrane on headphones, in working halfheartedly at the 360° reports once he found himself at the departure gate, that he didn't realize he was on the same flight as Elizabeth Colton until he glanced up and saw her boarding the plane with her son.
It was a coincidence, but not an enormous coincidence. On the phone the other day he'd told her about the flight he was planning on taking—seven a.m., to get to Toronto before the predicted snowstorm arrived and snarled the airports—and she'd said she would try to get on the same flight. And then there she was in a dark suit, her hair cut short but instantly recognizable, her son by her side. Elizabeth and Tyler were in First Class and Clark was in Economy. They said hello as Clark walked past her seat and then didn't speak again until an hour and a half after takeoff, when the pilot announced that they were being diverted into some place in Michigan that Clark had never heard of and everyone disembarked, confused and disoriented, into the Severn City Airport.
# **41**
**AFTER CLARK HAD DELIVERED** the news of Arthur's death, Miranda remained on the beach for some time. She sat on the sand, thinking of Arthur and watching a small boat coming in to shore, a single bright light skimming over the water. She was thinking about the way she'd always taken for granted that the world had certain people in it, either central to her days or unseen and infrequently thought of. How without any one of these people the world is a subtly but unmistakably altered place, the dial turned just one or two degrees. She was very tired, she realized, not feeling quite well, the beginnings of a sore throat, and tomorrow was another day of meetings. She'd forgotten to ask Clark about funeral arrangements, but her next thought was that of course she wouldn't want to go—the idea of being pinned between the paparazzi and Arthur's other ex-wives—and this was what she was thinking of as she rose and walked up the path to the hotel, which from the beach looked a little like a wedding cake, two tiers of white balconies.
The lobby was oddly empty. There was no front-desk staff. The concierge wore a surgical mask. Miranda started to approach him, to ask what was going on, but the look he gave her was one of unmistakable fear. She understood, as clearly as if he'd shouted it, that he wanted very badly for her not to come near him. She backed away and walked quickly to the elevators, shaken, his gaze on her back. There was no one in the upstairs corridor. Back in her room, she opened her laptop and, for the first time all day, turned her attention to the news.
Later Miranda spent two hours making phone calls, but there was no way to leave by then. Every nearby airport was closed.
"Listen," a fraying airline representative finally snapped at her, "even if I could book you on a flight out of Malaysia, are you seriously telling me you'd want to spend twelve hours breathing recirculated air with two hundred other people in an airplane cabin at this point?"
Miranda hung up the phone. When she leaned back in the chair, her gaze fell on the air-conditioning vent above the desk. The thought of air whispering through the building, propelled from room to room. It wasn't her imagination, she definitely had a sore throat.
"It's psychosomatic," she said aloud. "You're afraid of getting sick, so you feel sick. It's nothing." She was trying to reframe the story as an exciting adventure, the time I got stuck in Asia during a flu outbreak, but she was unconvinced. She spent some time sketching, trying to calm herself. A rocky island with a small house on it, lights on the horizon of Station Eleven's dark sea.
Miranda woke at four in the morning with a fever. She fought it off with three aspirin, but her joints were knots of pain, her legs weak, her skin hurt where her clothes touched her. It was difficult to cross the room to the desk. She read the latest news on the laptop, her eyes aching from the light of the screen, and understood. She could feel the fever pressing against the thin film of aspirin. She tried calling the front desk and then the New York and Toronto offices of Neptune Logistics, followed by the Canadian, American, British, and Australian consulates, but there were only voice-mail greetings and ringing phones.
Miranda rested the side of her face on the desk—the perfection of the cool laminate against her burning skin—and considered the poverty of the room. Poverty not in the economic sense, but in the sense of not being _enough_ for the gravity of the moment, an insufficient setting—for what? She couldn't think of this just yet—and she was thinking about the beach, the ships, the lights on the horizon, if it would be possible to get there when she felt so ill, related thoughts that perhaps if she could get there, someone on the beach might help her, that if she stayed here in the room she'd only get sicker and there was apparently no one at the front desk or in the consulates, all telephones unmanned. If she became any sicker she'd eventually be stranded here, too ill to get out of this room. There might be fishermen on the beach. She rose unsteadily. It took a long time and considerable concentration to put on her shoes.
The corridor was silent. It was necessary to walk very slowly, her hand on the wall. A man was curled on his side near the elevators, shivering. She wanted to speak to him, but speaking would take too much strength, so she looked at him instead—I see you, I see you—and hoped this was enough.
The lobby was empty now. The staff had fled.
Outside the air was heavy and still. A greenish light on the horizon, the beginnings of sunrise. A feeling of moving in slow motion, like walking underwater or in a dream. It was necessary to concentrate carefully on each step. This terrible weakness. She followed the path to the beach, walking very slowly, her outstretched hands brushing the palm fronds on either side. At the bottom of the path, the hotel's white chaise longues lay in a row on the sand, unoccupied. The beach was empty of people. She collapsed into the nearest chaise longue and closed her eyes.
Exhaustion. She was desperately hot, then wracked with chills. Her thoughts were disordered. No one came.
She was thinking about the container-ship fleet on the horizon. The crew out there wouldn't have been exposed to the flu. Too late to get to a ship herself now, but she smiled at the thought that there were people in this reeling world who were safe.
Miranda opened her eyes in time to see the sunrise. A wash of violent color, pink and streaks of brilliant orange, the container ships on the horizon suspended between the blaze of the sky and the water aflame, the seascape bleeding into confused visions of Station Eleven, its extravagant sunsets and its indigo sea. The lights of the fleet fading into morning, the ocean burning into sky.
#
# **42**
**AT FIRST THE PEOPLE** in the Severn City Airport counted time as though they were only temporarily stranded. This was difficult to explain to young people in the following decades, but in all fairness, the entire history of being stranded in airports up to that point was also a history of eventually becoming unstranded, of boarding a plane and flying away. At first it seemed inevitable that the National Guard would roll in at any moment with blankets and boxes of food, that ground crews would return shortly thereafter and planes would start landing and taking off again. Day One, Day Two, Day Forty-eight, Day Ninety, any expectation of a return to normalcy long gone by now, then Year One, Year Two, Year Three. Time had been reset by catastrophe. After a while they went back to the old way of counting days and months, but kept the new system of years: January 1, Year Three; March 17, Year Four, etc. Year Four was when Clark realized this was the way the years would continue to be marked from now on, counted off one by one from the moment of disaster.
He'd known for a long time by then that the world's changes wouldn't be reversed, but still, the realization cast his memories in a sharper light. The last time I ate an ice-cream cone in a park in the sunlight. The last time I danced in a club. The last time I saw a moving bus. The last time I boarded an airplane that hadn't been repurposed as living quarters, an airplane that actually took off. The last time I ate an orange.
Toward the end of his second decade in the airport, Clark was thinking about how lucky he'd been. Not just the mere fact of survival, which was of course remarkable in and of itself, but to have seen one world end and another begin. And not just to have seen the remembered splendors of the former world, the space shuttles and the electrical grid and the amplified guitars, the computers that could be held in the palm of a hand and the high-speed trains between cities, but to have lived among those wonders for so long. To have dwelt in that spectacular world for fifty-one years of his life. Sometimes he lay awake in Concourse B of the Severn City Airport and thought, "I was there," and the thought pierced him through with an admixture of sadness and exhilaration.
"It's hard to explain," he caught himself saying sometimes to young people who came into his museum, which had formerly been the Skymiles Lounge in Concourse C. But he took his role as curator seriously and he'd decided years ago that "It's hard to explain" isn't good enough, so he always tried to explain it all anyway, whenever anyone asked about any of the objects he'd collected over the years, from the airport and beyond—the laptops, the iPhones, the radio from an administrative desk, the electric toaster from an airport-staff lounge, the turntable and vinyl records that some optimistic scavenger had carried back from Severn City—and of course the context, the pre-pandemic world that he remembered so sharply. No, he was explaining now, to a sixteen-year-old who'd been born in the airport, the planes didn't rise straight up into the sky. They gathered speed on long runways and angled upward.
"Why did they need the runways?" the sixteen-year-old asked. Her name was Emmanuelle. He had a special fondness for her, because he remembered her birth as the only good thing that had happened in that terrible first year.
"They couldn't get off the ground without gathering speed. They needed momentum."
"Oh," she said. "The engines weren't that powerful, then?"
"They were," he said, "but they weren't like rocket ships."
"Rocket ships..."
"The ships we used to go to space."
"It's incredible," she said, shaking her head.
"Yes." Incredible in retrospect, all of it, but especially the parts having to do with travel and communications. This was how he arrived in this airport: he'd boarded a machine that transported him at high speed a mile above the surface of the earth. This was how he'd told Miranda Carroll of her ex-husband's death: he'd pressed a series of buttons on a device that had connected him within seconds to an instrument on the other side of the world, and Miranda—barefoot on a white sand beach with a shipping fleet shining before her in the dark—had pressed a button that had connected her via satellite to New York. These taken-for-granted miracles that had persisted all around them.
By the end of the Second Decade most of the airport's population was either born there or had walked in later, but two dozen or so people remained who had been there since the day their flights had landed. Clark's flight landed without incident, diverted from Toronto for reasons no one seemed immediately able to explain, and taxied to a gate in Concourse B. Clark looked up from his edits of the 360° Subordinates report and was struck by the variety of planes on the tarmac. Singapore Airlines, Cathay Pacific, Air Canada, Lufthansa, Air France, enormous jets parked end to end.
When Clark emerged from the jet bridge into the fluorescent light of Concourse B, the first thing he noticed was the uneven distribution of people. Crowds had gathered beneath the television monitors. Clark decided that whatever they were looking at, he couldn't face it without a cup of tea. He assumed it was a terrorist attack. He bought a cup of Earl Grey at a kiosk, and took his time adding the milk. This is the last time I'll stir milk into my tea without knowing what happened, he thought, wistful in advance for the present moment, and went to stand with the crowd beneath a television that was tuned to CNN.
The story of the pandemic's arrival in North America had broken while he was in the air. This was another thing that was hard to explain years later, but up until that morning the Georgia Flu had seemed quite distant, especially if one happened not to be on social media. Clark had never followed the news very closely and had actually heard about the flu only the day before the flight, in a brief newspaper story about a mysterious outbreak of some virus in Paris, and it hadn't been at all clear that it was developing into a pandemic. But now he watched the too-late evacuations of cities, the riots outside hospitals on three continents, the slow-moving exodus clogging every road, and wished he'd been paying more attention. The gridlocked roads were puzzling, because where were all these people going? If these reports were to be believed, not only had the Georgia Flu arrived, but it was already everywhere. There were clips of officials from various governments, epidemiologists with their sleeves rolled up, everyone wan and bloodshot and warning of catastrophe, blue-black circles under bloodshot eyes.
"It's not looking promising for a quick end to the emergency," a newscaster said, understating the situation to a degree previously unmatched in the history of understatement, and then he blinked at the camera and something in him seemed to stutter, a breaking down of some mechanism that had previously held his personal and professional lives apart, and he addressed the camera with a new urgency. "Mel," he said, "if you're watching this, sweetheart, take the kids to your parents' ranch. Back roads only, my love, no highways. I love you so much."
"It must be nice to have the network at your disposal," a man standing near Clark said. "I don't know where my wife is either. You know where your wife is?" His voice carried a high note of panic.
Clark decided to pretend that the man had asked him where his boyfriend was. "No," he said. "I have no idea." He turned away from the monitor, unable to bear another second of the news. For how long had he been standing here? His tea had gone cold. He drifted down the concourse and stood before the flight-status monitors. Every flight had been canceled.
How had all of this happened so quickly? Why hadn't he checked the news before he left for the airport? It occurred to Clark that he should call someone, actually everyone, that he should call everyone he'd ever loved and talk to them and tell them all the things that mattered, but it was apparently already too late for this, his phone displaying a message he'd never seen before: SYSTEM OVERLOAD EMERGENCY CALLS ONLY. He bought another tea, because the first one had gone cold, and also he was beset now by terrible fears and walking to the kiosk seemed like purposeful action. Also because the two young women working the kiosk seemed profoundly unconcerned by what was unfolding on CNN, either that or they were extremely stoic or they hadn't noticed yet, so visiting them was like going back in time to the paradise of a half hour earlier, when he hadn't yet known that everything was coming undone.
"Can you tell us more about the... well, about what people should be looking out for, the symptoms?" the newscaster asked.
"Same things we see every flu season," the epidemiologist said, "just worse."
"So, for example...?"
"Aches and pains. A sudden high fever. Difficulty breathing. Look," the epidemiologist said, "it's a fast incubation period. If you're exposed, you're sick in three or four hours and dead in a day or two."
"We're going to take a quick commercial break," the newscaster said.
The airline staff had no information. They were tight-lipped and frightened. They distributed food vouchers, which by power of suggestion made everyone hungry, so passengers formed lines to buy greasy cheese quesadillas and nacho plates at Concourse B's only restaurant, which was ostensibly Mexican. The two young women in the kiosk continued to serve hot drinks and mildly stale baked goods, frowning every so often at their useless phones. Clark bought his way into the Skymiles Lounge and found Elizabeth Colton in an armchair near a television screen. Tyler sat cross-legged on the floor nearby, killing space aliens on a Nintendo console.
"It's crazy," Clark said to Elizabeth, words falling hopelessly short.
She was watching the news, her hands clasped at her throat.
"It's unprecedented," Elizabeth said. "In all of human history...," she trailed off, shaking her head. Tyler groaned softly; he'd suffered a setback in the alien wars. They sat for a while in silence, watching, until Clark couldn't watch anymore and excused himself to find more nachos.
A final plane was landing, an Air Gradia jet, but as Clark watched, it made a slow turn on the tarmac and moved away from instead of toward the terminal building. It parked in the far distance, and no ground crew went to meet it. Clark abandoned his nachos and went to the window. It occurred to him that the Air Gradia jet was as far away from the terminal as it could possibly go. This was where he was standing when the announcement came: for public-health reasons, the airport was closing immediately. There would be no flights for the indefinite future. All passengers were asked to collect their bags at Baggage Claim, to leave the premises in an orderly fashion, and to please not flip out.
"This can't be happening," the passengers said to each other and to themselves, over nacho platters and in angry clusters in front of vending machines. They swore at airport management, at the TSA, at the airlines, at their useless phones, furious because fury was the last defense against understanding what the news stations were reporting. Beneath the fury was something literally unspeakable, the television news carrying an implication that no one could yet bring themselves to consider. It was possible to comprehend the scope of the outbreak, but it wasn't possible to comprehend what it meant. Clark stood by the terminal's glass wall in the Mexican restaurant, watching the stillness of the Air Gradia jet in the far distance, and he realized later that if he didn't understand at that moment why it was out there alone, it was only because he didn't want to know.
The workers at the restaurants and the gift shop chased out their customers and locked down steel shutters and gates, walked away without looking back. The passengers around Clark began departing too, an exodus that merged with the slow processions leaving the other two concourses. Elizabeth and Tyler emerged from the Skymiles Lounge.
"Are you leaving?" Clark asked. It still wasn't entirely real.
"Not yet," Elizabeth said. She looked a little deranged, but so did everyone else. "Where would we go? You saw the news." Everyone who'd been watching the news knew that roads everywhere were impassable, cars abandoned where they'd run out of gas, all commercial airlines shut down, no trains or buses. Most of them were leaving the airport anyway, because the voice over the intercom had said that they should.
"I think I'll stay here for the moment," Clark said. A few others apparently had had the same thought, and some who'd left returned after a half hour with reports that there was no ground transportation. The others had set out walking for Severn City, they said. Clark waited for an airport official to come and chase all of them away, the hundred-odd passengers who remained at the terminal, but none did. An Air Gradia agent was in tears by the ticket counter. The screen over her head still read AIR GRADIA FLIGHT 452 NOW ARRIVING, but when her radio crackled Clark heard the word _quarantine_.
Half of the remaining passengers had tied scarves or T-shirts over their mouths and noses, but it had been hours by now, and if they were all going to die of flu, Clark thought, wouldn't at least some of them be sick already?
The passengers who remained in the airport were mostly foreign. They looked out the windows at the airplanes on which they'd arrived—Cathay Pacific, Lufthansa, Singapore Airlines, Air France—parked end to end on the tarmac. They spoke in languages Clark didn't understand.
A little girl did cartwheels up and down the length of Concourse B.
Clark walked the length of the airport, restless, and was stunned to see that the security checkpoints were unmanned. He walked through and back three or four times, just because he could. He'd thought it would be liberating but all he felt was fear. He found himself staring at everyone he saw, looking for symptoms. No one seemed sick, but could they be carrying it? He found a corner as far from his fellow passengers as possible and stayed there for some time.
"We just have to wait," Elizabeth said, when he came to sit with her again. "Surely by tomorrow morning we'll see the National Guard." Arthur had always liked her optimism, Clark remembered.
No one emerged from the Air Gradia jet on the tarmac.
A young man was doing push-ups by Gate B20. He'd do a set of ten, then lie on his back and stare unblinking at the ceiling for a while, then another ten, etc.
Clark found a discarded _New York Times_ on a bench and read Arthur's obituary. Noted film and stage actor, dead at fifty-one. A life summed up in a series of failed marriages—Miranda, Elizabeth, Lydia—and a son, whose present absorption in his handheld Nintendo was absolute. When Arthur collapsed onstage, someone from the audience had performed CPR, the obituary said, but that audience member remained unidentified. Clark folded the paper into his suitcase.
Clark's grasp of Midwestern American geography was shaky. He wasn't entirely sure where he was. He'd gathered from the items on offer at the souvenir shop that they were somewhere near Lake Michigan, which he could picture because he retained an internal bird's-eye snapshot of the Great Lakes from his time in Toronto, but he'd never heard of Severn City. The airport seemed very new. Beyond the tarmac and the runways he could see only a line of trees. He tried to pinpoint his location on his iPhone, but the map wouldn't load. No one's phones were working, but word spread that there was a pay phone down in Baggage Claim. Clark stood in line for a half hour and then dialed all his numbers, but there were only busy signals and endless ringing. Where was everyone? The man behind him in line sighed loudly, so Clark gave up the phone and spent some time wandering the airport.
When he was tired of walking he returned to a bench he'd staked out earlier by Gate B17, lay on his back on the carpet between the bench and a wall of glass. Snow began to fall in the late afternoon. Elizabeth and Tyler were still in the Skymiles Lounge. He knew he should be sociable and talk to them, but he wanted to be alone, or as alone as he could be in an airport with a hundred other terrified and weeping people. He ate a dinner of corn chips and chocolate bars from a vending machine, spent some time listening to Coltrane on his iPod. He was thinking of Robert, his boyfriend of three months. Clark wanted very much to see him again. What was Robert doing at this moment? Clark stared up at the news. Around ten p.m. he brushed his teeth, returned to his spot by Gate B17, stretched out on the carpet and tried to imagine he was home in his bed.
He woke at three in the morning, shivering. The news had worsened. The fabric was unraveling. It will be hard to come back from this, he thought, because in those first days it was still inconceivable that civilization might not come back from this at all.
Clark was watching NBC when a teenager approached him. He'd noticed her earlier, sitting by herself with her head in her hands. She looked about seventeen and had a diamond nose stud that caught the light.
"I'm sorry to ask," she said, "but do you have any Effexor?"
"Effexor?"
"I've run out," she said. "I'm asking everyone."
"I'm sorry, I haven't any. What is it?"
"An antidepressant," the girl said. "I thought I'd be home in Arizona by now."
"I'm so sorry. How awful for you."
"Well," the girl said, "thanks anyway," and Clark watched her walk away to make inquiries of a couple only slightly older than she was, who listened for a moment and then shook their heads in unison.
Clark was thinking ahead to a time when he'd sit with Robert in a restaurant in New York or London and they'd raise a glass of wine to their tremendous good fortune at having made it through. How many of their friends would have died by the time he saw Robert again? There would be funerals to go to, memorial services. Probably a certain measure of grief and survivor's guilt to contend with, therapy and such.
"What a terrible time that was," Clark said softly to an imaginary Robert, practicing for the future.
"Awful," Imaginary Robert agreed. "Remember those days when you were in the airport, and I didn't know where you were?"
Clark closed his eyes. The news continued on the overhead screens, but he couldn't bear to watch. The stacked body bags, the riots, the closed hospitals, the dead-eyed refugees walking on inter-states. Think of anything else. If not the future, the past: dancing with Arthur when they were young in Toronto. The taste of Orange Julius, that sugary orange drink he'd only ever tasted in Canadian shopping malls. The scar on Robert's arm just above the elbow, from when he'd broken his arm very badly in the seventh grade, the bouquet of tiger lilies that Robert had sent to Clark's office just last week. Robert in the mornings: he liked to read a novel while he ate breakfast. It was possibly the most civilized habit Clark had ever encountered. Was Robert awake at this moment? Was he trying to leave New York? The storm had passed, and snow lay deep on the wings of airplanes. There were no de-icing machines, no tire tracks, no footprints; the ground workers had departed. Air Gradia 452 was still alone on the tarmac.
There was a moment later in the day when Clark blinked and realized he'd been staring into space for some time. He had intimations of danger, that there was hazard in allowing his thoughts to drift too loosely, so he tried to work, to read over his 360° reports, but his thoughts were scattered, and also he couldn't help but wonder if the target of the 360° and all the people he'd interviewed were dead.
He tried to reread his newspapers, on the theory that this required less concentration than the reports, came across Arthur's _New York Times_ obituary again and realized that the world in which Arthur had died already seemed quite distant. He'd lost his oldest friend, but if the television news was accurate, then in all probability everyone here with him in the airport had lost someone too. All at once he felt an aching tenderness for his fellow refugees, these hundred or so strangers here in the airport. He folded his paper and looked at them, his compatriots, sleeping or fretfully awake on benches and on carpets, pacing, staring at screens or out at the landscape of airplanes and snow, everyone waiting for whatever came next.
# **43**
**THE FIRST WINTER** in the Severn City Airport:
There was a frisson of excitement on Day Two, when someone recognized Elizabeth and Tyler and word spread. "My _phone_ ," Clark heard a young man say in frustration. He was about twenty, with hair that flopped in his eyes. "God, why won't our phones work? I so wish I could tweet this."
"Yeah," his girlfriend said, wistful. "You know, like, 'Not much, just chilling with Arthur Leander's kid at the end of the world.' "
"Totally," the man said. Clark moved away from them in order to maintain his sanity, although later, in a more charitable moment, it occurred to him that they were probably in shock.
By Day Three all the vending machines in the airport were empty of snacks, and the battery on Tyler's Nintendo console was dead. Tyler wept, inconsolable. The girl who needed Effexor was very sick by then. Withdrawal, she said. No one in the airport had the drug she needed. A raiding party went through every room, the administrative offices and the TSA holding cell, everyone's desk drawers, and then they went outside and broke into the dozen or so cars abandoned in the parking lot, pawed through glove boxes and trunks. They found some useful items in their searches, extra pairs of shoes and some warm clothes and such, but on the pharmaceutical front they uncovered only painkillers and antacids and a mysterious bottle of pills that someone thought might be for stomach ulcers. In the meantime the girl lay across a bench, shivering and drenched in sweat, and she said her head sparked with electricity every time she moved.
They called 911 from the pay phone in baggage claim, but no one picked up. They wandered outside and stared at the snowed-in parking lot, the airport road disappearing into the trees, but what could possibly be out there aside from the flu?
The television newscasters weren't exactly saying that it was the end of the world, per se, but the word _apocalypse_ was beginning to appear.
"All those people," Clark said to Imaginary Robert, but Imaginary Robert didn't reply.
That evening they broke into the Mexican restaurant and cooked an enormous dinner of ground meat and tortilla chips and cheese with sauces splashed over it. Some people had mixed feelings about this—they'd obviously been abandoned here, everyone was hungry and 911 wasn't even operational; on the other hand, no one wants to be a thief—but then a business traveler named Max said, "Look, everyone just chill the fuck out, I'll cover it on my Amex." There was applause at this announcement. He removed his Amex card from his wallet with a flourish and left it next to the cash register, where it remained untouched for the next ninety-seven days.
On Day Four the food from the Mexican restaurant ran out, also the food from the sandwich place in Concourse C. That night they lit their first bonfire on the tarmac, burning newspapers and magazines from the newsstand and a wooden bench from Concourse A. Someone had raided the Skymiles Lounge. They got drunk on Skymiles Lounge champagne and ate Skymiles Lounge oranges and snack mix. Someone suggested that perhaps a passing plane or helicopter might see the fire and come down to save them, but no lights crossed the cloudless sky.
The realization, later, that that had possibly been his last orange. This orangeless world! Clark said to himself, or perhaps to Imaginary Robert, and laughed in a way that prompted concerned glances from the others. That first year everyone was a little crazy.
On Day Five they broke into the gift shop, because some people had no clean clothes, and after that, at any given moment half of the population was dressed in bright red or blue Beautiful Northern Michigan T-shirts. They washed their clothes in the sinks, and everywhere Clark turned he saw laundry hanging to dry on the backs of benches. The effect was oddly cheerful, like strings of bright flags.
The snacks from the Concourse B gift shop were gone by Day Six. The National Guard still hadn't arrived.
On Day Seven the networks began to blink off the air, one by one. "So that all of our employees may be with their families," a CNN anchor said, ashen and glassy-eyed after forty-eight hours without sleep, "we are temporarily suspending broadcast operations." "Good night," NBC said an hour later, "and good luck." CBS switched without comment to reruns of _America's Got Talent_. This was at five in the morning, and everyone who was awake watched for a few hours—it was nice to take a quick break from the end of the world—and then in the early afternoon the lights went out. They came back on almost immediately, but what it probably meant, a pilot said, was that the grid had gone down and the airport had switched to generator power. All of the workers who knew how the generators worked had left by then. People had been trickling out since Day Three. "It's the waiting," Clark had heard a woman say, "I can't take the waiting, I have to do _something_ , even if it's just walking to the nearest town to see what's going on...."
A TSA agent had remained at the airport, just one, Tyrone, and he knew how to hunt. By Day Eight no one new had come to the airport and no one who'd left had returned, no more planes or helicopters had landed, everyone was hungry and trying not to think about all the apocalypse movies they'd seen over the years. Tyrone set off into the trees with a woman who'd formerly been a park ranger and two TSA-issue handguns, and they returned some time later with a deer. They strung it between metal chairs over the fire and at sunset everyone ate roasted venison and drank the last of the champagne, while the girl who needed Effexor slipped out through an entrance on the other side of the airport and walked away into the trees. A group of them tried to find her, but couldn't.
The girl who needed Effexor had left her suitcase and all of her belongings behind, including her driver's license. She looked sleepy in the picture, a slightly younger version of herself with longer hair. Her name was Lily Patterson. She was eighteen. No one knew what to do with the driver's license. Finally someone put it on the counter of the Mexican restaurant, next to Max's Amex card.
Tyler spent his days curled in an armchair in the Skymiles Lounge, reading his comic books over and over again. Elizabeth sat near him with her eyes closed, lips moving constantly, rapidly, in some repeated prayer.
The televisions displayed silent test patterns.
On the twelfth day in the airport, the lights went out. But the toilets would still flush if one poured water into the bowls, so they collected plastic trays from the security checkpoints and filled them with snow, carted these to the restrooms to melt. Clark had never thought much about airport design, but he was grateful that so much of this particular airport was glass. They lived in daylight and went to bed at sundown.
There were three pilots among the stranded. On the fifteenth day in the airport, one of them announced that he'd decided to take a plane to Los Angeles. The snow had melted, so he thought he could maybe make do without de-icing machines. People reminded him that Los Angeles had looked pretty bad on the news.
"Yeah, but everywhere looked bad on the news," the pilot said. His family was in L.A. He wasn't willing to accept the possibility of not seeing them again. "Anyone wants to come with me," he said, "it's a free flight to Los Angeles." This alone seemed like proof that the world was ending, because this was the era when people were being charged extra for checked bags, for boarding early enough to cram baggage into overhead bins before the bins filled up, for the privilege of sitting in exit rows with their life-or-death stakes and their two extra inches of legroom. The passengers exchanged glances.
"The plane's fueled up," the pilot said. "I was flying Boston to San Diego when we got diverted, and it's not like it'll be a full flight." It occurred to Clark that if the entire population of the airport went with him, there would still be empty seats on the plane. "I'm going to give you all a day to think about it," the pilot said, "but I'm flying out tomorrow before the temperature drops again."
There were of course no guarantees. There had been no news from the outside world since the televisions went dark and there were reeling moments when it seemed possible—not likely! But possible!—that the seventy-nine of them left there in the airport might be the last people alive on earth. For all anyone knew LAX was a heap of smoking rubble. Agonized calculations were performed. Almost everyone who lived west of the Rockies approached the pilot. Most of the people who lived in Asia opted to take the flight, which would still leave an ocean between themselves and their loved ones but would at least bring them two thousand miles closer to home.
At noon the next day, the passengers boarded via a wheeled staircase they'd found in a hangar, and a crowd gathered on the tarmac to watch the plane depart. The sound of the engines was startling after these days of silence. There was a long period when nothing happened, the engines roaring, before the plane worked its way out of the line of parked aircraft with a series of delicate lurching turns—it left a gap between the Cathay Pacific and Lufthansa jets—and made a slow curve to the runway. Someone—impossible to see who at this distance—was waving in one of the windows. A few people waved back. The plane started down the runway, gathered speed, the wheels left the ground, and the watchers held their breaths for the moment of ascent, but the machine didn't falter, it rose instead of falling, and as it receded into the clear blue sky Clark realized he had tears on his face. Why, in his life of frequent travel, had he never recognized the beauty of flight? The improbability of it. The sound of the engines faded, the airplane receding into blue until it was folded into silence and became a far-distant dot in the sky. Clark watched until it disappeared.
That night no one had much to say around the fire. Fifty-four of them now, the ones who'd decided against Los Angeles. The venison was too tough. Everyone chewed silently. Tyler, who seemed to almost never speak, stood close by Elizabeth and stared into the flames.
Clark glanced at his watch. The plane had departed five hours ago. It was nearing the western edge of the continent, or it had been forced to land on an unlit runway somewhere short of California, or it had plummeted into some dark landscape in flames. It would land in Los Angeles and the passengers would walk out into a different world, or it would land and be overcome by a mob, or it would crash into runways clogged with other planes. The passengers would find their families again, or they wouldn't. Was there still electricity in Los Angeles? All those solar panels in the southern light. All his memories of that city. Miranda at the dinner party, smoking outside while her husband flirted with his next wife. Arthur sunning himself by the pool, a pregnant Elizabeth dozing by his side.
"I can't wait till things get back to normal," she said now, shivering in the firelight, and Clark could think of absolutely nothing to say.
The departure of the Los Angeles flight left two pilots, Stephen and Roy. Roy announced his intention to fly out the day after the Los Angeles flight departed.
"Just reconnaissance," he said. "I figure I'll fly up to Marquette—I've got a buddy up there—I'll take a look around, try to get some information on what's going on, maybe get some supplies, and come back."
He left alone the next morning in a small plane. He didn't return.
"It just doesn't make sense," Elizabeth insisted. "Are we supposed to believe that civilization has just _come to an end_?"
"Well," Clark offered, "it was always a little fragile, wouldn't you say?" They were sitting together in the Skymiles Lounge, where Elizabeth and Tyler had set up camp.
"I don't know." Elizabeth spoke slowly, looking out at the tarmac. "I've been taking art history classes on and off for years, between projects. And of course art history is always pressed up close against non-art history, you see catastrophe after catastrophe, terrible things, all these moments when everyone must have thought the world was ending, but all those moments, they were all temporary. It always passes."
Clark was silent. He didn't think this would pass.
Elizabeth began telling him about a book she'd read once, years ago when she'd been stuck—but not _this_ stuck, obviously—in an airport, and it was a vampire book, actually, not her usual sort of thing, but it had a device she kept thinking of. The setup was post-apocalyptic, she said, so you naturally assumed as you were reading it that the world had ended, all of it, but then it became clear through an ingenious flash-forward device that actually it wasn't all of civilization that was lost, it was just North America, which had been placed under quarantine to keep the vampirism from spreading.
"I don't think this is a quarantine," Clark said. "I think there's actually really nothing out there, or at least nothing good."
There were in fact a number of solid arguments against the quarantine theory, namely that the pandemic had started in Europe, the last news reports had indicated chaos and disarray on every continent except Antarctica, and anyway how would one even go about isolating North America in the first place, given air travel and the fact that South America was after all more or less attached?
But Elizabeth was unshakable in her convictions. "Everything happens for a reason," she said. "This will pass. Everything passes." Clark couldn't bring himself to argue with her.
Clark was careful to shave every three days. The men's rooms were windowless, lit only by an ever-dwindling supply of scented candles from the gift shop, and the water had to be warmed over the fire outside, but Clark felt it was worth the effort. Several of the men in the airport weren't shaving at all anymore, and the effect was wild and also frankly unflattering. Clark disliked the general state of unshavenness, partly for aesthetic reasons and partly because he was a believer in the broken-windows theory of urban-crime management, the way the appearance of dereliction can pave the way for more serious crimes. On Day Twenty-Seven he parted his hair neatly down the middle and shaved off the left side.
"It's the haircut I had from ages seventeen through nineteen," he told Dolores when she raised an eyebrow at him. Dolores was a business traveler, single, no family, which meant that she was one of the saner people in the airport. She and Clark had an agreement: she'd promised to tell him if he began showing signs of having lost his mind, and vice versa. What he didn't tell her was that after all these years of corporate respectability, the haircut made him feel like himself again.
The maintenance of sanity required some recalibrations having to do with memory and sight. There were things Clark trained himself not to think about. Everyone he'd ever known outside the airport, for instance. And here at the airport, Air Gradia 452, silent in the distance near the perimeter fence, by unspoken agreement never discussed. Clark tried not to look at it and sometimes almost managed to convince himself that it was empty, like all of the other planes out there. Don't think of that unspeakable decision, to keep the jet sealed rather than expose a packed airport to a fatal contagion. Don't think about what enforcing that decision may have required. Don't think about those last few hours on board.
Snow fell every few days after Roy left, but Elizabeth insisted on keeping a runway clear at all times. She was beginning to stare in a terrible way that made everyone afraid of her, so at first she was out there alone, shoveling the snow on Runway Seven by the hour, but then a few people went out to join her because celebrity still carried a certain currency and there she was all alone out there, gorgeous and single—and also, why not? Physical labor outdoors was preferable to wandering the same hatefully unchanging concourses or sitting around thinking about all the beloved people they were never going to see again or convincing themselves they heard voices coming from the Air Gradia jet. Eventually there were nine or ten people maintaining the runway, a core group who attracted volunteers from the periphery every now and again. Why not, though, really? Even if Elizabeth's quarantine theory was too wonderful to be true—the idea that somewhere things continued on as before, untouched by the virus, children going to school and to birthday parties and adults going to work and meeting for cocktails in some other place, everyone talking about what a shame it was that North America had been lost but then the conversation eventually turning to sports, politics, the weather—there was still the military, with its secrets and its underground shelters, its stockpiles of fuel and medicine and food.
"They'll need a clear runway to land on when they come for us," Elizabeth said. "They're going to come for us. You know that, right?"
"It's possible," Clark said, trying to be kind.
"If anyone was coming for us," Dolores said, "I think they'd be here by now."
But they did see an aircraft after the collapse, just one. On Day Sixty-five a helicopter crossed the sky in the far distance, the faintest vibration of sound moving rapidly from north to south, and they stood staring for some time after it passed. They kept up a vigil for a while after that, waiting outside in teams of two with brightly colored T-shirts to flag down aircraft in daylight, a signal fire burning all night, but nothing crossed the sky except birds and shooting stars.
The night sky was brighter than it had been. On the clearest nights the stars were a cloud of light across the breadth of the sky, extravagant in their multitudes. When Clark first noticed this, he wondered if he was possibly hallucinating. He assumed he held deep reservoirs of unspeakable damage that might at any moment blossom into insanity, the way his grandmother's bone cancer had bloomed dark over the X-rays in her final months. But after a couple of weeks he felt that the thing with the stars was too consistent to be a hallucination—also too extreme, the way the airplanes cast shadows even when the moon was only a sliver—so he risked mentioning it to Dolores.
"It's not your imagination," Dolores said. He'd begun to think of her as his closest friend. They'd spent a pleasantly companionable day indoors, cleaning, and now they were helping build a bonfire with branches someone had dragged in from the woods. She explained it to him. One of the great scientific questions of Galileo's time was whether the Milky Way was made up of individual stars. Impossible to imagine this ever having been in question in the age of electricity, but the night sky was a wash of light in Galileo's age, and it was a wash of light now. The era of light pollution had come to an end. The increasing brilliance meant the grid was failing, darkness pooling over the earth. I was here for the end of electricity. The thought sent shivers up Clark's spine.
"The lights will come back on someday," Elizabeth kept insisting, "and then we'll all finally get to go home." But was there actually any reason to believe this?
The citizens of the airport had taken to meeting at the bonfire every night, an unspoken tradition that Clark hated and loved. What he loved was the conversation, the moments of lightness or even just silence, the not being by himself. But sometimes the small circle of people and firelight seemed only to accentuate the emptiness of the continent, the loneliness of it, a candle flickering in vast darkness.
It's surprising how quickly the condition of living out of a carry-on suitcase on a bench by a departure gate can begin to seem normal.
Tyler wore a sweater of Elizabeth's that went to his knees, the increasingly filthy sleeves rolled up. He kept to himself mostly, reading his comic books or Elizabeth's copy of the New Testament.
They traded languages. By Day Eighty most of the people who'd arrived without English were learning it, in informal groups, and the English speakers were studying one or more of the languages carried here by Lufthansa, Singapore Airlines, Cathay Pacific, and Air France. Clark was learning French from Annette, who'd been a Lufthansa flight attendant. He whispered phrases to himself as he went about the chores of daily existence, the hauling of water and washing of clothes in the sink, learning to skin a deer, building bonfires, cleaning. Je m'appelle Clark. J'habite dans l'aeroport. Tu me manques. Tu me manques. Tu me manques.
A rape on the night of Day Eighty-five, the airport woken after midnight by a woman's scream. They tied the man up until sunrise and then drove him into the forest at gunpoint, told him if he returned he'd be shot. "I'll die out here alone," he said, sobbing, and no one disagreed but what else could they do?
"Why has no one come here?" Dolores asked. "That's what I keep wondering. I don't mean rescue. I just mean people wandering in." The airport wasn't especially remote. Severn City was no more than twenty miles away. No one walked in, but on the other hand, who was left? Early reports had put the mortality rate at 99 percent.
"And then one has to account for societal collapse," Garrett said. "There might be no one left." He was a businessman from the east coast of Canada. He'd been wearing the same suit since his flight had landed, except now he was pairing it with a Beautiful Northern Michigan T-shirt from the gift shop. He was bright-eyed in a way that Clark found disconcerting. "The violence, maybe cholera and typhoid, all the infections that were cured by antibiotics back when it was possible to obtain antibiotics, and then things like bee stings, asthma... Does anyone have a cigarette?"
"You're funny," Annette said. She'd run out of nicotine patches on Day Four. During a particularly rough stretch a few weeks back, she'd tried to smoke cinnamon from the coffee kiosk.
"Was that a no? And diabetes," Garrett said, apparently forgetting the cigarette. "HIV. High blood pressure. Types of cancer that responded to chemotherapy, when chemotherapy was available."
"No more chemotherapy," Annette said. "I've thought of that too."
"Everything happens for a reason," Tyler said. Clark hadn't noticed his approach. Tyler had been wandering the airport of late, and he had a way of moving so quietly that he seemed to materialize out of nowhere. He spoke so rarely that it was easy to forget he was there. "That's what my mom said," he added when everyone stared at him.
"Yeah, but that's because Elizabeth's a fucking lunatic," Garrett said. Clark had noticed that he had a filter problem.
"In front of the kid?" Annette was twisting her Lufthansa neck scarf between her fingers. "That's his mother you're talking about. Tyler, don't listen to him." Tyler only stared at Garrett.
"I'm sorry," Garrett said to Tyler. "I was out of line." Tyler didn't blink.
"You know," Clark said, "I think we should consider sending out a scouting party."
The scouts left at dawn on Day One Hundred: Tyrone, Dolores, and Allen, a schoolteacher from Chicago. There was some debate over whether the scouting party was actually a good idea. They'd been able to kill enough deer to live on and they had what they needed here, barely, except for soap and batteries, which they'd run out of, and what could possibly be out there except the pandemic? Nonetheless, the scouting party set out armed with Tyrone's TSA handgun and some road maps.
The silence of Day One Hundred. Waiting for the scouting party to return with supplies, or return carrying the flu, or return trailing unhinged survivors who wanted to kill everyone, or not return at all. It had snowed the night before and the world was still. White snow, dark trees, gray sky, the airline logos on the tails of grounded airplanes the only splashes of color in the landscape.
Clark wandered into the Skymiles Lounge. He'd been avoiding it lately, because he'd been avoiding Elizabeth, but it was a reliably quiet corner of the airport and he liked the armchairs with the views over the tarmac. He stood looking out at the line of planes and for the first time in a while he found himself thinking of Robert, his boyfriend. Robert was a curator—had been a curator? Yes, probably Robert existed in the past tense along with almost everyone else, try not to think about it—and when Clark turned away from the window, his gaze fell on a glass display case that had once held sandwiches.
If Robert were here—Christ, if only—if Robert were here, he'd probably fill the shelves with artifacts and start an impromptu museum. Clark placed his useless iPhone on the top shelf. What else? Max had left on the last flight to Los Angeles, but his Amex card was still gathering dust on the counter of the Concourse B Mexican restaurant. Beside it, Lily Patterson's driver's license. Clark took these artifacts back to the Skymiles Lounge and laid them side by side under the glass. They looked insubstantial there, so he added his laptop, and this was the beginning of the Museum of Civilization. He mentioned it to no one, but when he came back a few hours later, someone had added another iPhone, a pair of five-inch red stiletto heels, and a snow globe.
Clark had always been fond of beautiful objects, and in his present state of mind, all objects were beautiful. He stood by the case and found himself moved by every object he saw there, by the human enterprise each object had required. Consider the snow globe. Consider the mind that invented those miniature storms, the factory worker who turned sheets of plastic into white flakes of snow, the hand that drew the plan for the miniature Severn City with its church steeple and city hall, the assembly-line worker who watched the globe glide past on a conveyer belt somewhere in China. Consider the white gloves on the hands of the woman who inserted the snow globes into boxes, to be packed into larger boxes, crates, shipping containers. Consider the card games played belowdecks in the evenings on the ship carrying the containers across the ocean, a hand stubbing out a cigarette in an overflowing ashtray, a haze of blue smoke in dim light, the cadences of a half dozen languages united by common profanities, the sailors' dreams of land and women, these men for whom the ocean was a gray-line horizon to be traversed in ships the size of overturned skyscrapers. Consider the signature on the shipping manifest when the ship reached port, a signature unlike any other on earth, the coffee cup in the hand of the driver delivering boxes to the distribution center, the secret hopes of the UPS man carrying boxes of snow globes from there to the Severn City Airport. Clark shook the globe and held it up to the light. When he looked through it, the planes were warped and caught in whirling snow.
The scouting party returned the next day, exhausted and cold, with three steel carts from an industrial kitchen, piled high with supplies. They'd found a Chili's that no one had looted yet, they said, and they'd spent the night shivering in booths. They had toilet paper, Tabasco sauce, napkins, salt and pepper, enormous tins of tomatoes, dinnerware and bags of rice, gallons of pink hand soap.
They said that just out of sight along the road there was a roadblock, a sign warning of quarantine. No one had come to the airport because the sign said the flu was here, sick passengers, keep out. Beyond the roadblock, abandoned cars as far as they could see, some with bodies inside. They'd come upon a hotel near the airport and had debated going in for sheets and towels, but the smell was such that they'd known what was waiting in the darkened lobby and had decided against it. Then the fast-food restaurants a little down the road. They'd seen no other people.
"What was it like out there?" Clark asked.
"It was silent," Dolores said. She'd been surprised by the emotion that had overtaken her on the return, when the scouting party had struggled past the roadblock with their carts of supplies, their napkins and their clinking bottles of Tabasco sauce, up the airport road and then the airport had come into view between the trees. _Home_ , she'd thought, and she'd felt such relief.
A day later the first stranger walked in. They'd taken to posting guards with whistles, so that they might be warned of a stranger's approach. They'd all seen the post-apocalyptic movies with the dangerous stragglers fighting it out for the last few scraps. Although actually when she thought about it, Annette said, the post-apocalyptic movies she'd seen had all involved zombies. "I'm just saying," she said, "it could be much worse."
But the first man who walked in under low gray skies seemed less dangerous than stunned. He was dirty, of indeterminate age, dressed in layers of clothes, and he hadn't shaved in a long time. He appeared on the road with a gun in his hand, but he stopped and let the gun fall to the pavement when Tyrone shouted for him to drop it. He raised his hands over his head and stared at the people gathering around him. Everyone had questions. He seemed to struggle for speech. His lips moved silently, and he had to clear this throat several times before he could speak. Clark realized that he hadn't spoken in some time.
"I was in the hotel," he said finally. "I followed your footprints in the snow." There were tears on his face.
"Okay," someone said, "but why are you crying?"
"I'd thought I was the only one," he said.
# **44**
**BY THE END OF** Year Fifteen there were three hundred people in the airport, and the Museum of Civilization filled the Skymiles Lounge. In former times, when the airport had had fewer people, Clark had worked all day at the details of survival; gathering firewood, hauling water to the restrooms to keep the toilets operational, participating in salvage operations in the abandoned town of Severn City, planting crops in the narrow fields along the runways, skinning deer. But there were many more people now, and Clark was older, and no one seemed to mind if he cared for the museum all day.
There seemed to be a limitless number of objects in the world that had no practical use but that people wanted to preserve: cell phones with their delicate buttons, iPads, Tyler's Nintendo console, a selection of laptops. There were a number of impractical shoes, stilettos mostly, beautiful and strange. There were three car engines in a row, cleaned and polished, a motorcycle composed mostly of gleaming chrome. Traders brought things for Clark sometimes, objects of no real value that they knew he would like: magazines and newspapers, a stamp collection, coins. There were the passports or the driver's licenses or sometimes the credit cards of people who had lived at the airport and then died. Clark kept impeccable records.
He kept Elizabeth and Tyler's passports open to the picture pages. Elizabeth had given them to him the night before they'd left, in the summer of Year Two. He was still unsettled by the passports, after all these years.
"They were unsettling people," Dolores said.
A few months before Elizabeth and Tyler left, back in Year Two, Clark was breaking up sticks for kindling when he looked up and thought he saw someone standing by the Air Gradia jet. A child, but there were a number of children in the airport and he couldn't tell who it was at this distance. The plane was strictly off-limits, but the children liked to scare one another with stories of ghost sightings. The child was holding something. A book? Clark found Tyler standing by the nose of the plane, reading aloud from a paperback.
" 'Therefore in one day her plagues will overtake her,' " he said to the plane as Clark approached. He paused and looked up. "Do you hear that? Plagues. 'One day her plagues will overtake her. Death, mourning, and famine. She will be consumed by fire, for mighty is the Lord God who judges her.' "
Clark recognized the text. For three months in his Toronto days he'd had a formerly evangelical boyfriend who'd kept a Bible by the bed. Tyler stopped reading and looked up.
"You read very well for your age," Clark said.
"Thank you." The boy was obviously a little off, but what could anyone do for him? In Year Two everyone was still reeling.
"What were you doing?"
"I'm reading to the people inside," Tyler said.
"There's no one in there." But of course there was. Clark was chilled in the sunlight. The plane remained sealed, because opening it was a nightmare no one wanted to think about, because no one knew if the virus could be contracted from the dead, because it was as good a mausoleum as any. He'd never been this close to it. The plane's windows were dark.
"I just want them to know that it happened for a reason."
"Look, Tyler, some things just happen." This close, the stillness of the ghost plane was overwhelming.
"But why did they die instead of us?" the boy asked, with an air of patiently reciting a well-rehearsed argument. His gaze was unblinking.
"Because they were exposed to a certain virus, and we weren't. You can look for reasons, and god knows a few people here have driven themselves half-crazy trying, but Tyler, that's all there is."
"What if we were saved for a different reason?"
"Saved?" Clark was remembering why he didn't talk to Tyler very often.
"Some people were saved. People like us."
"What do you mean, 'people like us'?"
"People who were good," Tyler said. "People who weren't weak."
"Look, it's not a question of having been bad or... the people in there, in the Air Gradia jet, they were just in the wrong place at the wrong time."
"Okay," Tyler said. Clark turned away, and Tyler's voice resumed almost immediately behind him, softer now, reading aloud: " 'She will be consumed by fire, for mighty is the Lord God who judges her.' "
Elizabeth and Tyler were living in the First-Class cabin of the Air France jet. He found her sitting in the sunlight on the rolling staircase that led up to the entrance, knitting something. He hadn't spoken with her in a while. He hadn't been avoiding her, exactly, but he certainly hadn't sought her company.
"I'm worried about your son," he said.
She paused in her knitting. The manic intensity of her first days here had dissipated. "Why?"
"Right now he's over by the quarantined plane," Clark said, "reading aloud to the dead from the Book of Revelation."
"Oh." She smiled, and resumed her knitting. "He's a very advanced reader."
"I think maybe he's picked up some strange ideas about, well, about what happened." He still had no words for it, he realized. No one spoke of it directly.
"What kind of strange ideas?"
"He thinks the pandemic happened for a reason," Clark said.
"It did happen for a reason."
"Well, right, but I mean a reason besides the fact that almost everyone on earth caught an extremely deadly swine-flu mutation. He seems to think there was some sort of divine judgment involved."
"He's right," she said. She stopped knitting for a moment to count her rows.
He felt a touch of vertigo. "Elizabeth, what reason could there possibly be for something like this? What kind of plan would possibly require...?" He realized that his voice had risen. His fists were clenched.
"Everything happens for a reason," she said. She didn't look at him. "It's not for us to know."
Later that summer a band of religious wanderers arrived, headed south. The precise nature of their religion was unclear. "A new world requires new gods," they said. They said, "We are guided by visions." They said vague things about signals and dreams. The airport hosted them for a few uneasy nights, because this seemed less dangerous than running them off. The wanderers ate their food and in return offered blessings, which mostly involved palms on foreheads and muttered prayers. They sat in a circle in Concourse C and chanted at night, in no language anyone in the airport had ever heard. When they left, Elizabeth and Tyler went with them.
"We just want to live a more spiritual life," Elizabeth said, "my son and I," and she apologized for leaving everyone, as though her leaving was some sort of personal abandonment. Tyler looked very small as they left, trailing at the back of the group. I should have done more for her, Clark thought. I should have pulled her back from the edge. But it had taken everything he had to stay back from the edge himself, and what could he have done? When the group disappeared around the curve of the airport road, he was certain he wasn't alone in his relief.
"That kind of insanity's contagious," Dolores had said, echoing his thoughts.
In Year Fifteen people came to the museum to look at the past after their long days of work. A few of the original First-Class lounge armchairs were still here, and it was possible to sit and read the final newspapers, fifteen years old, turning brittle pages in gloves that Clark had sewn inexpertly from a hotel sheet. What happened here was something like prayer. James, the first man who'd walked in, came to the museum almost every day to look at the motorcycle. He'd found it in Severn City in Year Two, and had used it until the automobile gas went stale and the aviation gas ran out. He missed it very much. Emmanuelle, the first child born in the airport, came in often to look at the phones.
There was a school here now, in Concourse C. Like educated children everywhere, the children in the airport school memorized abstractions: the airplanes outside once flew through the air. You could use an airplane to travel to the other side of the world, but—the schoolteacher was a man who'd had frequent-flyer status on two airlines—when you were on an airplane you had to turn off your electronic devices before takeoff and landing, devices such as the tiny flat machines that played music and the larger machines that opened up like books and had screens that hadn't always been dark, the insides brimming with circuitry, and these machines were the portals into a worldwide network. Satellites beamed information down to Earth. Goods traveled in ships and airplanes across the world. There was no place on earth that was too far away to get to.
They were told about the Internet, how it was everywhere and connected everything, how it was us. They were shown maps and globes, the lines of the borders that the Internet had transcended. This is the yellow mass of land in the shape of a mitten; this pin here on the wall is Severn City. That was Chicago. That was Detroit. The children understood dots on maps— _here_ —but even the teenagers were confused by the lines. There had been countries, and borders. It was hard to explain.
In the fall of Year Fifteen, something remarkable happened. A trader came through with a newspaper. He'd been coming to the airport since Year Six, and his specialties were cookware, socks, and sewing supplies. He camped for the night in the Air France jet, and came to Clark in the morning before he left.
"I've got something I thought you might like," he said, "for that museum of yours," and he handed over three sheets of rough paper.
"What is this?"
"It's a newspaper," the trader said.
Three consecutive issues, a few months out of date. It was published irregularly out of New Petoskey, the trader said. There were announcements of births and deaths and weddings. A column for bartering: a local man was seeking new shoes in exchange for milk and eggs; someone else had a pair of reading glasses that she was hoping to trade for a pair of jeans, size 6. There was a story about a group of three ferals who'd been sighted to the southwest of town, a woman and two children. Residents were urged to avoid them and, in case of accidental contact, to speak gently and avoid making any sudden movements. Something called the Traveling Symphony had just come through town, although Clark gathered that they weren't just a symphony orchestra. There was a rapturous write-up of a performance of _King Lear_ , with particular mention of performances by Gil Harris as Lear and Kirsten Raymonde as Cordelia. A local girl wished to announce that she had a litter of kittens to give away and that the kittens' mother was a good mouser. There was a reminder that the library was always seeking books, and that they paid in wine.
The librarian, François Diallo, was also the newspaper's publisher, and it appeared that when he had empty space in the newspaper he filled it with text from his collection. The first issue had an Emily Dickinson poem, the second an excerpt from a biography of Abraham Lincoln. The entire back of the third issue—it had apparently been a slow month for news and announcements—was taken up by an interview with the actress who'd played Cordelia, Kirsten Raymonde. She'd left Toronto with her brother at the time of the collapse, but she only knew this because her brother had told her. Her memories were limited, but there was a night just before the end that she remembered in detail.
RAYMONDE: I was onstage with two other girls in the production, and I was behind Arthur, so I didn't see his face. But I remember there was some commotion up front, just in front of the stage. And then I remember hearing a sound, this sharp "thwack," and that was Arthur hitting his hand on the plywood pillar by my head. He'd sort of stumbled back, his arm flailed out, and then a man from the audience had climbed up on the stage and was running toward him—
Clark stopped breathing for a moment when he read it. The shock of encountering someone who knew Arthur, who had not only known him but had seen him die.
The newspapers were passed hand to hand around the airport for four days. They were the first new newspapers anyone had seen since the collapse. When the papers were returned to the museum, Clark held them in his hands for a long while, reading the interview with the actress again. The mention of Arthur aside, he realized, this was an extraordinary development. If there were newspapers now, what else might be possible? In the old days he'd taken quite a few red-eye flights between New York and Los Angeles, and there was a moment in the flight when the rising sunlight spread from east to west over the landscape, dawn reflected in rivers and lakes thirty thousand feet below his window, and although of course he knew it was all a matter of time zones, that it was always night and always morning somewhere on earth, in those moments he'd harbored a secret pleasure in the thought that the world was waking up.
He hoped for more newspapers in the years that followed, but none came.
# **45**
**THE INTERVIEW IN** Year Fifteen, continued:
RAYMONDE: Do you have any more questions?
DIALLO: I do have more questions, but you didn't want to answer them.
RAYMONDE: I'll answer if you don't record me.
François Diallo set his pen and notebook on the table.
"Thank you," Kirsten said. "I'll answer your questions now if you'd like, but only if these ones don't go in your newspaper."
"Agreed. When you think of how the world's changed in your lifetime, what do you think about?"
"I think of killing." Her gaze was steady.
"Really? Why?"
"Have you ever had to do it?"
François sighed. He didn't like to think about it. "I was surprised in the woods once."
"I've been surprised too."
It was evening, and François had lit a candle in the library. It stood in the middle of a plastic tub, for safety. The candlelight softened the scar on Kirsten's left cheekbone. She was wearing a summer dress with a faded pattern of white flowers on red, three sheathed knives in her belt.
"How many?" he asked.
She turned her wrist to show the knife tattoos. Two.
The Symphony had been resting in New Petoskey for a week and a half so far, and François had interviewed almost all of them. August had told him about walking away from his empty house in Massachusetts with his violin, falling in with a cult for three years before he walked away again and stumbled across the Symphony. Viola had a harrowing story about riding a bicycle west out of the burnt-out ruins of a Connecticut suburb, aged fifteen, harboring vague notions of California but set upon by passersby long before she got there, grievously harmed, joining up with other half-feral teenagers in a marauding gang and then slipping away from them, walking alone for a hundred miles, whispering French to herself because all the horror in her life had transpired in English and she thought switching languages might save her, wandering into a town through which the Symphony passed five years later. The third cello had buried his parents after both died in the absence of insulin, and then spent four years holed up in the safety and boredom of their remote cottage on Michigan's Upper Peninsula, set out finally because he feared he'd lose his mind if he didn't find another human being to talk to, also because you can eat only so much venison before you'd give your right arm to eat almost anything else, made his way south and east and over the Mackinac Bridge ten years before the bridge's center section collapsed, lived on the outskirts of the close-knit band of fishermen in Mackinaw City until the Symphony passed through. When it came down to it, François had realized, all of the Symphony's stories were the same, in two variations. Everyone else died, I walked, I found the Symphony. Or, I was very young when it happened, I was born after it happened, I have no memories or few memories of any other way of living, and I have been walking all my life.
"Now tell me yours," she said. "What do you think about?"
"When I think of how the world's changed, you mean?"
"Yes."
"My apartment in Paris." François had been on vacation in Michigan when air travel had ceased. When he closed his eyes, he could still see the intricate moldings of his parlor ceiling, the high white doors leading out to the balcony, the wood floors and books. "Why do you think of killing?"
"You never had to hurt anyone in the old world, did you?"
"Of course not. I was a copywriter."
"A what?"
"Advertising." He hadn't thought about it in a long time. "You know, billboards and such. Copywriters wrote the words on them."
She nodded, and her gaze drifted away from him. The library was François's favorite place in his present life. He had accumulated a sizable collection over the years. Books, magazines, a glass case of pre-collapse newspapers. It had only recently occurred to him to start a newspaper of his own, and thus far the project had been invigorating. Kirsten was looking at the improvised printing press, massive in the shadows at the back of the room.
"How did you get that scar on your face?" he asked.
She shrugged. "I've actually no idea. It happened during that year I don't remember."
"Your brother never told you, before he died?"
"He said it was better if I didn't remember. I took his word for it."
"What was he like, your brother?"
"He was sad," she said. "He remembered everything."
"You've never told me what happened to him."
"The kind of stupid death that never would've happened in the old world. He stepped on a nail and died of infection." She glanced up at the window, at the failing light. "I should go," she said, "it's almost sunset." She stood, and the handles of the knives in her belt glinted in the half-light. This wire of a woman, polite but lethal, who walked armed with knives through all the days of her life. He'd heard stories from other Symphony members about her knife-throwing abilities. She was supposedly able to hit the center of targets blindfolded.
"I thought tonight was just the musicians." Reluctant as always to see her go.
"It is, but I told my friends I'd come."
"Thank you for the interview." He was walking her to the door.
"You're welcome."
"If you don't mind me asking, why didn't you want that last part recorded? It isn't the first time I've heard confessions of this nature."
"I know," she said. "Almost everyone in the Symphony... but look, I collect celebrity-gossip clippings."
"Celebrity gossip...?"
"Just about that one actor, Arthur Leander. Because of my collection, the clippings, I understand something about permanent records."
"And it isn't something you want to be remembered for."
"Exactly," she said. "Are you coming to the performance?"
"Of course. I'll walk with you." He went back to blow out the candle. The street had fallen into shadow now, but the sky was still bright over the bay. The Symphony was performing on a bridge a few blocks from the library, the caravans parked off to the side. François heard the first notes, the cacophony of musicians practicing their sections and tuning up. August was playing the same two measures over and over, frowning. Charlie was studying the score. Earlier, a few of the townspeople had carried benches down the hill from the town hall, arranged now in rows facing the bay. Most of the benches were occupied, the adults talking among themselves or watching the musicians, the children spellbound by the instruments.
"There's some space in the back row," Kirsten said, and François followed her.
"What's the program tonight?"
"A Beethoven symphony. I'm not sure which one."
At some undetectable cue, the musicians stopped practicing and tuning and talking among themselves, took their places with their backs to the water, and fell silent. A hush came over the assembled crowd. The conductor stepped forward in the stillness, smiled at the audience and bowed, turned without a word to face the musicians and the bay. A seagull glided overhead. The conductor raised her baton.
# **46**
**THAT NIGHT, IN THE SUMMER OF** Year Fifteen, Jeevan Chaudhary was drinking wine by a river. The world was a string of settlements now and the settlements were all that mattered, the land itself no longer had a name, but once this had been part of the state of Virginia.
Jeevan had walked a thousand miles. In Year Three he'd wandered into a settlement called McKinley, named by the town's founders. There had been eight of them originally, a sales team from the marketing firm of McKinley Stevenson Davies, stranded on an isolated corporate retreat when the Georgia Flu swept over the continent. A few days out of the retreat they'd found an abandoned motel on a disused stretch of road far from major highways, and it had seemed as good a place to stop as any. The sales team had moved into the rooms and stayed there, at first because those early years were terrifying and no one wanted to live too far from anyone else; later out of habit. There were twenty-seven families here now, a peaceful settlement across the road from a river. In the summer of Year Ten, Jeevan had married one of the settlement's founders, a former sales assistant named Daria, and this evening she was sitting with him and a friend of theirs on the riverbank.
"I don't know," their friend was saying now. "Does it still make sense to teach kids about the way things were?" His name was Michael, and he'd been a truck driver once. McKinley had a school, ten children who met daily in the largest motel room, and his eleven-year-old daughter had come home crying that afternoon, because the teacher had let slip that life expectancies were much longer before the Georgia Flu, that once sixty hadn't been considered particularly old, and she was scared, she didn't understand, it wasn't _fair_ , she wanted to live as long as people used to.
"I'm honestly not sure," Daria said. "I think I'd want my kid to know. All that knowledge, those incredible things we had."
"To what end, though?" Michael accepted the wine bottle from her with a nod. "You see the way their eyes glaze over when anyone talks to them about antibiotics or engines. It's science fiction to them, isn't it? And if it only upsets them—" He broke off to drink wine.
"Maybe you're right," Daria said. "I suppose the question is, does knowing these things make them more or less happy?"
"In my daughter's case, less."
Jeevan was only half-listening. He wasn't quite drunk. Just pleasantly at ease, after what had actually been a fairly ghastly day: a neighbor of theirs had fallen off a ladder that morning, and Jeevan, as the closest thing to a doctor in a one-hundred-mile radius, had had to set the man's broken arm. Horrible work, the patient drunk on moonshine but still half-crazed with pain, moans escaping around the piece of wood clamped between his teeth. Jeevan liked being the man to whom people turned in bad moments, it meant a great deal to him to be able to help, but the physical pain of the post-anesthesia era often left him shaken. Now fireflies were rising from the tall grass on the riverbank, and he didn't want to talk, not really, but it was pleasant to rest in the company of his friend and his wife, and the wine was blunting the worst of the day's memories—sweat beading on the patient's forehead as Jeevan set the broken bone—as was the gentle music of the river, cicadas in the trees, the stars above the weeping willows on the far bank. Even after all these years there were moments when he was overcome by his good fortune at having found this place, this tranquility, this woman, at having lived to see a time worth living in. He squeezed Daria's hand.
"When she came home crying today," Michael said, "I found myself thinking, maybe it's time we stopped telling them these crazy stories. Maybe it's time we let go."
"I don't want to let go," Jeevan said.
"Is someone calling you?" Daria asked.
"I hope not," Jeevan said, but then he heard it too.
They followed him back to the motel, where a man had just arrived on horseback, his arm around a woman slumped over in the saddle.
"My wife's been shot," he said, and in the way he spoke, Jeevan understood that he loved her. When they pulled the woman down she was shivering despite the heat of the evening, half-conscious, her eyelids fluttering. They carried her into the motel room that served as Jeevan's surgery. Michael lit the oil lamps and the room filled with yellow light.
"You're the doctor?" the man who'd brought her asked. He looked familiar, but Jeevan couldn't place him. He was perhaps in his forties, his hair braided in cornrows that matched his wife's.
"Closest thing we've got," Jeevan said. "What's your name?"
"Edward. Are you saying you're not a real doctor?"
"I trained as a paramedic, before the flu. I apprenticed to a doctor near here for five years, till he decided to move farther south. I've picked up what I can."
"But you didn't go to med school," Edward said in tones of misery.
"Well, I'd love to, but I understand they've stopped accepting applications."
"I'm sorry." Edward wiped the sweat from his face with a handkerchief. "I've heard you're good. I mean no offense. She's just, she's been shot—"
"Let me see if I can help."
Jeevan hadn't seen a gunshot wound in some time. By Year Fifteen, the ammunition was running low, guns used rarely and only for hunting. "Tell me what happened," he said, mostly to distract Edward.
"The prophet happened."
"I don't know who that is." At least the wound was fairly clean, a hole where the bullet had entered her abdomen, no exit wound. She'd lost some blood. Her pulse was weak but steady. "What prophet?"
"I thought the man's legend preceded him," Edward said. He was holding his wife's hand. "He's been all over the south."
"I've heard of a dozen prophets over the years. It's not an uncommon occupation." Jeevan found a bottle of moonshine in the cupboard.
"You sterilizing the equipment with that?"
"I sterilized the needle in boiling water earlier, but I'm going to sterilize it again in this."
"The needle? You're sewing her up without getting the bullet out?"
"Too dangerous," Jeevan said softly. "Look, the bleeding's just about stopped. If I go in there looking for it, she might bleed out. Safer to leave it in." He poured some moonshine into a bowl and rubbed his hands with it, ran needle and thread through the alcohol.
"Can I do anything?" Edward was hovering.
"The three of you can hold her still while I'm sewing. So there was a prophet," he said. He'd found it best to distract the people who came in with his patients.
"He came through this afternoon," Edward said. "Him and his followers, maybe twenty of them altogether."
Jeevan remembered where he'd seen Edward before. "You live up on the old plantation, don't you? I went up there with the doctor a few times, back in my apprenticeship days."
"Yes, the plantation, exactly. We're out on the fields, and a friend of mine comes running, says there's a group of twenty or twenty-two approaching, walking down the road singing some kind of weird hymn. After a while I hear it too, and eventually they reach us. A group of them, smiling, walking all together in a clump. By the time they reach us, they've stopped singing, and there are fewer of them than I'm expecting, maybe more like fifteen." Edward was silent for a moment as Jeevan poured alcohol over the woman's stomach. She moaned, and a thin trickle of blood left the wound.
"Keep talking."
"So we ask them who they are, and their leader smiles at me and says, 'We are the light.' "
"The light?" Jeevan drew the needle through the woman's skin. "Don't look," he said, when Edward swallowed. "Just hold her still."
"That's when I knew who he was. Stories had reached us, from traders and such. These people, they're ruthless. They've got some crazy theology, they're armed and they take what they want. So I'm trying to stay cool, we all are, I can see my neighbors have realized what we're dealing with too. I ask if there's something they need or if this is just a social call, and the prophet smiles at me and says they have something we want, and they'd be willing to trade this thing we want for guns and ammunition."
"You still have ammo?"
"Did until today. There was a fair stockpile at the plantation. And as he's talking, I'm looking around, and I realize I don't know where my kid is. He was with his mother, but where's his mother? I ask them, 'What is it you have that you think we want?' "
"Then what?"
"Then the group parts down the middle, and there's my son. They've got him. The kid's five, okay? And they've got him bound and gagged. And I'm terrified now, because where's his mother?"
"So you gave them the weapons?"
"We gave them the guns, they gave me my boy. Another group of them had taken my wife. That's why there were fifteen there in front of me and not twenty. They'd taken her off down the road ahead as a kind of, I don't know, _insurance policy_ "—his voice thick with disgust—"and they tell us if no one comes after them, my wife will come walking down the road in an hour or two, unharmed. They say they're traveling out of the area, headed north, and this is the last we'll see of them. All the time smiling, so peaceful, like they've done nothing wrong. So we get the boy, they leave with the guns and ammo, and we wait. Three hours later she still hasn't come down the road, so a few of us go after them and we find her shot on the roadside."
"Why did they do it?" The woman was awake, Jeevan realized. She was crying silently, her eyes closed. One last stitch.
"She said the prophet wanted her to stay with them," Edward said, "go north with them and become a wife to one of his men, and she said no, so the prophet shot her. Not to kill her, obviously, at least not quickly. Just to cause her pain."
Jeevan clipped the thread and pressed a clean towel to the woman's stomach. "A bandage," he said to Daria, but she was already by his side with strips of an old sheet. He wrapped the woman carefully.
"She'll be okay," he said, "provided it doesn't get infected, and there's no reason to think it will. Bullets are self-sterilizing, the heat of them. We were careful with the alcohol. But you two should stay here for a few days."
"I'm grateful," Edward said.
"I do what I can."
When he'd cleaned up and the woman had fallen into a fitful sleep, her husband by her side, Jeevan put the bloody needle in a saucepan and crossed the road to the river. He knelt in the grass to fill the pan with water and returned to the motel, where he lit the makeshift oven in front of the room he lived in and set the saucepan on top of it. He sat on a nearby picnic table to wait for the water to boil.
Jeevan filled a pipe with tobacco from his shirt pocket, a soothing ritual. Trying to think of nothing but the stars and the sound of the river, trying not to think about the woman's pain and her blood and the kind of people who would shoot out of spite and leave her lying there on the roadside. McKinley was south of the old plantation. If the prophet was true to his word then he and his people were moving away from McKinley, headed into the unsuspecting north. Why north, Jeevan wondered, and how far would they go? He was thinking of Toronto, of walking through snow. Thoughts of Toronto led inevitably back to thoughts of his brother, a tower by the lake, ghost city crumbling, the Elgin Theatre still displaying the posters for _King Lear_ , the memory of that night at the beginning and the end of everything when Arthur died.
Daria had come up behind him. He started when she touched his arm. The water was boiling and had been for some time, the needle probably sterile by now. Daria took his hand in her own and kissed it gently. "It's late," she murmured. "Come to bed."
# **47**
**CLARK AT SEVENTY ,** in Year Nineteen: he was more tired than he had been, and he moved slowly. His joints and hands ached, especially in cold weather. He shaved his entire head now, not just the left side, and wore four rings through his left ear. His dear friend Annette had died of an unknown illness in Year Seventeen, and he wore her Lufthansa neck scarf in memory. He wasn't specifically sad anymore, but he was aware of death at all times.
There was an armchair in the museum from which he could see almost the entire tarmac. The preparation area where the hunters hung their deer and boar and rabbits from a rack improvised on the underside of the wing of a 737, carving meat for the people and feeding innards to the dogs. The graveyard between Runways Six and Seven, each grave marked by an airplane tray table driven into the ground, details of the deceased carved into the tray's hard plastic. He'd left some wildflowers on Annette's grave that morning and he could see them from here, a splash of blue and purple. The line of jets parked end to end on the periphery, streaked now with rust. The gardens, half-hidden from view by the airplanes parked at gates. The cornfield, Air Gradia 452 alone in the distance, the chain-link perimeter fence with its coils of concertina wire and beyond that the forest, the same trees he'd been staring at for two decades.
He'd recently made all of the Water Inc. 360° reports available for public viewing, on the theory that everyone involved was almost certainly dead. The former executives in the airport read these with great interest. There were three reports altogether, one each for the subordinates, peers, and superiors of a probably long-deceased Water Inc. executive named Dan.
"Okay, take this for example," Garrett said, on one of their afternoons in the airport, late July. They'd become close friends over the years. Garrett found the reports particularly fascinating. "You have the heading here, 'Communication,' and then—"
"Which report are you looking at?" Clark was sunk deep into his favorite armchair, eyes closed.
"Subordinates," Garrett said. "Okay, so under 'Communication,' here's the first comment. 'He's not good at cascading information down to staff.' Was he a whitewater rafter, Clark? I'm just curious."
"Yes," Clark said, "I'm certain that's what the interviewee was talking about. Actual literal cascades."
"This one's my other favorite. 'He's successful in interfacing with clients we already have, but as for new clients, it's low-hanging fruit. He takes a high-altitude view, but he doesn't drill down to that level of granularity where we might actionize new opportunities.' "
Clark winced. "I remember that one. I think I may have had a minor stroke in the office when he said that."
"It raises questions," Garrett said.
"It certainly does."
"There are high altitudes, apparently, also low-hanging fruit, also grains of something, also drilling."
"Presumably he was a miner who climbed mountains and actionized an orchard in his off-hours. I am proud to say," Clark said, "that I never talked like that."
"Did you ever use the phrase 'in the mix'?"
"I don't think so. No. I wouldn't have."
"I hated that one especially." Garrett was studying the report.
"Oh, I didn't mind it so much. It made me think of baking. My mother would buy these cookie mixes sometimes when I was a kid."
"Do you remember chocolate-chip cookies?"
"I dream of chocolate-chip cookies. Don't torture me."
Garrett was quiet for so long that Clark opened his eyes to make sure he was still breathing. Garrett was absorbed in watching two children playing on the tarmac, hiding behind the wheels of the Air Canada jet and chasing one another. He'd become calmer over the years but remained prone to episodes of unfocused staring, and Clark knew by now what his next question would be.
"Did I ever tell you about my last phone call?" Garrett asked.
"Yes," Clark said gently. "I believe you did."
Garrett had had a wife and four-year-old twins in Halifax, but the last call he'd ever made was to his boss. The last words he'd spoken into a telephone were a bouquet of corporate clichés, seared horribly into memory. "Let's touch base with Nancy," he remembered saying, "and then we should reach out to Bob and circle back next week. I'll shoot Larry an email." Now he said the words "Circle back next week" under his breath, perhaps not consciously. He cleared his throat. "Why did we always say we were going to _shoot_ emails?"
"I don't know. I've wondered that too."
"Why couldn't we just say we were going to send them? We were just pressing a button, were we not?"
"Not even a real button. A picture of a button on a screen."
"Yes," Garrett said, "that's exactly what I'm talking about."
"There was not, in fact, an email gun. Although that would've been nice. I would've preferred that."
Garrett made his fingers into a gun and aimed it at the tree line. "Ka- _pow_!" he whispered. And then, louder, "I used to write 'T-H-X' when I wanted to say 'thank you.' "
"I did that too. Because, what, it would've taken too much time and effort to punch in an extra three letters and just say _thanks_? I can't fathom it."
"The phrase 'circle back' always secretly made me think of boats. You leave someone onshore, and then you circle back later to get them." Garrett was quiet for a moment. "I like this one," he said. " 'He's a high-functioning sleepwalker, essentially.' "
"I remember the woman who said that." Clark wondered what had happened to her.
He'd been spending more time in the past lately. He liked to close his eyes and let his memories overtake him. A life, remembered, is a series of photographs and disconnected short films: the school play when he was nine, his father beaming in the front row; clubbing with Arthur in Toronto, under whirling lights; a lecture hall at NYU. An executive, a client, running his hands through his hair as he talked about his terrible boss. A procession of lovers, remembered in details: a set of dark blue sheets, a perfect cup of tea, a pair of sunglasses, a smile. The Brazilian pepper tree in a friend's backyard in Silver Lake. A bouquet of tiger lilies on a desk. Robert's smile. His mother's hands, knitting while she listened to the BBC.
He woke to quiet voices. This had been happening more and more lately, this nodding off unexpectedly, and it left him with an unsettled intimation of rehearsal. You fall asleep for short periods and then for longer periods and then forever. He straightened in the armchair, blinking. Garrett was gone. The last light of the day angled in through the glass and caught the chrome perfection of the motorcycle.
"Did I wake you?" Sullivan asked. He was the head of security, a man of fifty who'd walked in a decade earlier with his daughter. "I'd like to introduce you to our latest arrivals."
"How do you do," Clark said. The arrivals were a man and a woman, perhaps in their early thirties, the woman carrying a baby in a sling.
"I'm Charlie," the woman said. "This is Jeremy, my husband, and little Annabel." Tattoos covered almost every inch of her bare arms. He saw flowers, musical notes, names in an elaborate scroll, a rabbit. Four knives tattooed in a row on her right forearm. He knew what this tattoo meant, and when he looked he saw a counterpart on her husband's skin, two small dark arrows on the back of his left wrist. She'd killed four people, then, and he'd killed two, and now they'd just dropped in with their baby, and by the absurd standards of the new world—there was a part of him that never stopped exclaiming at the absurd standards of the new world—this was all perfectly normal. The baby smiled at Clark. Clark smiled back.
"Will you be staying here awhile?" Clark asked.
"If you'll have us," Jeremy said. "We've been separated from our people."
"Wait till you hear who their people are," Sullivan said. "You remember those newspapers out of New Petoskey?"
"The Traveling Symphony," Charlie said.
"These people of yours," Sullivan was wiggling his fingers at the baby, Annabel, who stared past his fingers at his face. "You didn't tell me how you lost them."
"It's a complicated story," Charlie said. "There was a prophet. He said he was from here."
From here? Had the airport ever had a prophet? Clark felt certain he'd remember a prophet. "What was his name?"
"I'm not sure anyone knows," Jeremy said. He began describing the blond-haired man who had held sway over the town of St. Deborah by the Water, ruling with a combination of charisma, violence, and cherry-picked verses from the Book of Revelation. He stopped when he saw the look on Clark's face. "Is something wrong?"
Clark rose unsteadily from the armchair. They stared at him as he made his way to the museum's first display case.
"Is his mother still alive?" Clark was looking at Elizabeth's passport, at its photograph from the inconceivable past.
"Whose mother? The prophet's?"
"Yes."
"I don't think so," Charlie said. "I never heard anything about her."
"There's no old woman there with him?"
"No."
What became of you, Elizabeth, out there on the road with your son? But what, after all, had become of anyone? His parents, his colleagues, all his friends from his life before the airport, Robert? If all of them had vanished, uncounted and unmarked, why not Elizabeth too? He closed his eyes. Thinking of a boy standing on the tarmac by the ghost plane, Air Gradia Flight 452, Arthur Leander's beloved only son, reading verses about plagues aloud to the dead.
#
# **48**
**THREE DAYS AFTER** Kirsten and August became separated from the Symphony, behind a garden shed in an overgrown backyard on the outskirts of Severn City, Kirsten woke abruptly with tears in her eyes. She'd dreamt that she'd been walking down the road with August, then she turned and he was gone and she knew he was dead. She'd screamed his name, she'd run down the road but he was nowhere. When she woke he was watching her, his hand on her arm.
"I'm right here," he said. She must have said his name aloud.
"It's nothing. Just a dream."
"I had bad dreams too." He was holding his silver Starship _Enterprise_ in his other hand.
It wasn't quite morning. The sky was brightening, but night lingered below in the shadows, gray light, dewdrops suspended in the grass.
"Let's wash up," August said. "We might meet people today."
They crossed the road to the beach. The water mirrored the pearl sky, the first pink of sunrise rippling. They bathed with some shampoo Kirsten had found in that last house—it left a scent of synthetic peaches on their skin and floating islands of bubbles on the lake—and Kirsten washed and wrung out her dress, put it on wet. August had scissors in his suitcase. She cut his hair—it was falling in his eyes—and then he cut hers.
"Have faith," he whispered. "We'll find them."
Resort hotels stood along the lakeshore, the windows mostly broken and their shards reflecting the sky. Trees pushed up through the parking lots between rusted cars. Kirsten and August abandoned their suitcases, the wheels too loud on rough pavement, made bundles out of bedsheets and carried the supplies over their shoulders. After a mile or two they saw a sign with a white airplane hanging askew over an intersection, an arrow pointed toward the center of town.
Severn City had been a substantial place once. There were commercial streets of redbrick buildings, flowers riotous in planters, and the roots of maple trees disrupting the sidewalks. A flowering vine had taken over most of the post office and extended across the street. They walked as silently as possible, weapons in hand. Birds moved in and out of broken windows and perched on sagging utility wires.
"August."
"What?"
"Did you just hear a dog bark?"
Just ahead was the overgrown wilderness of a municipal park, a low hill rising beside the road. They climbed up into the underbrush, moving quickly, threw their bundles aside and crouched low. A flash of movement at the end of a side street: a deer, bounding away from the lakeshore.
"Something startled it," August whispered. Kirsten adjusted and readjusted her grip on a knife. A monarch butterfly fluttered past. She watched it while she listened and waited, wings like bright paper. A faint buzz of insects all around them. She heard voices now, and footsteps.
The man who appeared on the road was so dirty that Kirsten didn't immediately recognize him, and when she did she had to stifle a gasp. Sayid was gaunt. He moved slowly. There was blood on his face, an eye swollen shut. His clothes were filthy and torn, several days' beard on his face. Two men and a boy followed a few paces behind him. The boy carried a machete. One of the men carried a sawed-off shotgun, the barrel pointed at the ground. The other held a bow, half-drawn, an arrow at the ready and a quiver on his back.
Kirsten, moving very slowly, drew a second knife from her belt.
"I have the gunman," August whispered. "Get the archer." His fingers closed around a stone the size of his fist. He rose and sent it sailing in an arc over the road. The stone crashed into the wall of a half-collapsed house and the men started, turning toward the sound just as August's first arrow caught the gunman in the back. Kirsten was aware of footsteps receding, the boy with the machete running away. The archer drew his bow and an arrow whistled past Kirsten's ear, but the knife had already left her hand. The archer sank to his knees, staring at the handle protruding from between his ribs. A flock of birds rose up above the rooftops and settled into the sudden quiet.
August was cursing under his breath. Sayid knelt on the road, his head in his hands. Kirsten ran to him and held his head to her chest. He didn't resist. "I'm so sorry," she whispered, into his blood-caked hair, "I'm so sorry they hurt you."
"There's no dog," August said. His jaw was clenched, a sheen of sweat on his face. "Where's the dog? We heard a dog bark."
"The prophet's behind us with the dog," Sayid whispered. "He's got two men with him. We split up to take different roads about a half mile back." Kirsten helped him to his feet.
"The archer's still alive," August said.
The archer was lying on his back. His eyes followed Kirsten, but he made no other movement. She knelt beside him. He'd been in the audience when they'd performed _A Midsummer Night's Dream_ at St. Deborah by the Water, applauding in the front row at the end of the performance, smiling, his eyes wet in the candlelight.
"Why did you take Sayid?" she asked him. "Where are the other two?"
"You took something that belongs to us," the man whispered. "We were going to do a trade." Blood was spreading rapidly over his shirt and dripping down the creases of his neck, pooling beneath him.
"We took nothing. I have no idea what you're talking about." August was going through the men's bags. "No ammunition for the gun," he said, disgusted. "And it was unloaded."
"The girl," Sayid said. His voice was a dry rasp. "He's talking about the stowaway."
"The fifth bride," the archer whispered. "It was my duty. She was chosen."
"Eleanor?" August looked up. "That scared little kid?"
"She's the property of the prophet."
"She's twelve years old," Kirsten said. "You believe everything the prophet says?"
The archer smiled. "The virus was the angel," he whispered. "Our names are recorded in the book of life."
"Okay," Kirsten said. "Where are the others?" He only stared at her, smiling. She looked at Sayid. "Are they behind us somewhere?"
"The clarinet got away," Sayid said.
"What about Dieter?"
"Kirsten," Sayid said softly.
"Oh god," August said. "Not Dieter. No."
"I'm sorry." Sayid covered his face with his hands. "I couldn't..."
"And behold," the archer whispered, "there was a new heaven and a new earth, for the old heaven and the old earth had passed away." The color was draining from his face.
Kirsten wrenched her knife from the archer's chest. He gasped, the blood welling, and she heard a gurgle in his throat as his eyes dimmed. Three, she thought, and felt immensely tired.
"We heard a whimpering in the forest," Sayid said. He walked slowly, limping. "That night on patrol. We were about a mile from the Symphony, about to turn back, and there was a sound coming from the bushes, sounded like a lost child."
"A ruse," August said. He had a glazed look when Kirsten glanced at him.
"So like idiots we went to investigate, and the next thing I know something's pressed over my face, a rag soaked in something, a chemical smell, and I woke up in a clearing in the woods."
"What about Dieter?" It was difficult to force the words from her throat.
"He didn't wake up."
"What do you mean?"
"Exactly that. Was he allergic to chloroform? Was it actually chloroform at all, or something much more toxic? The prophet's men gave me water, told me they wanted the girl, they'd decided to take two hostages and broker a trade. They'd guessed we were headed for the Museum of Civilization, given the direction of travel and the rumors that Charlie and Jeremy had gone there. And the whole time, they're explaining this, and I'm looking at Dieter sleeping beside me, and he's getting paler and paler, and his lips are blue. I'm trying to wake him up, and I can't. I couldn't. I was tied up next to him, and I kept kicking him, wake up, wake up, but..."
"But what?"
"But he didn't wake up," Sayid said. "We waited all through the following day—me tied up there and the men coming and going—and then in the late afternoon, his breathing stopped. I watched it happen." Kirsten's eyes filled with tears. "I was watching him breathe," Sayid said. "He'd gone so pale. His chest rising and falling, and then one last exhale, and no more. I shouted and they tried to revive him, but it didn't... nothing worked. Nothing. They argued for a while, and then two of them left and returned a few hours later with the clarinet."
# **49**
**THE TRUTH WAS ,** the clarinet hated Shakespeare. She'd been a double major in college, theater and music, a sophomore the year the world changed, lit up by an obsession with twenty-first-century experimental German theater. Twenty years after the collapse, she loved the music of the Symphony, loved being a part of it, but found the Symphony's insistence on performing Shakespeare insufferable. She tried to keep this opinion to herself and occasionally succeeded.
A year before she was seized by the prophet's men, the clarinet was sitting alone on the beach in Mackinaw City. It was a cool morning, and a fog hung over the water. They'd passed through this place more times than she could count, but she never tired of it. She liked the way the Upper Peninsula disappeared on foggy days, a sense of infinite possibility in the way the bridge faded into cloud.
She'd been thinking lately about writing her own play, seeing if she could convince Gil to stage a performance with the Symphony actors. She wanted to write something modern, something that addressed this age in which they'd somehow landed. Survival might be insufficient, she'd told Dieter in late-night arguments, but on the other hand, so was Shakespeare. He'd trotted out his usual arguments, about how Shakespeare had lived in a plague-ridden society with no electricity and so did the Traveling Symphony. But look, she'd told him, the difference was that they'd seen electricity, they'd seen everything, they'd watched a civilization collapse, and Shakespeare hadn't. In Shakespeare's time the wonders of technology were still ahead, not behind them, and far less had been lost. "If you think you can do better," he'd said, "why don't you write a play and show it to Gil?"
"I don't think I can do better," she'd told him. "I'm not saying that. I'm just saying the repertoire's inadequate." Still, writing a play was an interesting idea. She began writing the first act on the shore the next morning, but never got past the first line of the opening monologue, which she'd envisioned as a letter: "Dear friends, I find myself immeasurably weary and I have gone to rest in the forest." She was distracted just then by a seagull, descending near her feet. It pecked at something in the rocks, and this was when she heard Dieter, approaching from the Symphony encampment with two chipped mugs of the substance that passed for coffee in the new world.
"What were you writing?" he asked.
"A play," she said. She folded the paper.
He smiled. "Well, I look forward to reading it."
She thought of the opening monologue often in the months that followed, weighing those first words like coins or pebbles turned over and over in a pocket, but she was unable to come up with the next sentence. The monologue remained a fragment, stuffed deep in her backpack until the day, eleven months later, when the Symphony unearthed it in the hours after the clarinet was seized by the prophet's men and wondered if they were looking at a suicide note.
While they were reading it, she was waking in a clearing from an unnatural sleep. She had been dreaming of a room, a rehearsal space at college, an impression of laughter—someone had told a joke—and she tried to hold on to this, clinging at these shreds because it was obvious even before she was entirely awake that everything was wrong. She was lying on her side in the forest. She felt poisoned. The ground was hard under her shoulder, and she was very cold. Her hands were tied behind her back, her ankles bound, and she was aware immediately that the Symphony was nowhere near, a terrible absence. She'd been filling water containers with Jackson, and then? She remembered a sound behind her, turning as a rag was pressed to her face, someone's hand on the back of her head. It was evening now. Six men were crouched in a circle nearby. Two armed with large guns, one with a standard bow and a quiver of arrows and another with a strange metal crossbow, the fifth with a machete. The sixth had his back to her and she couldn't see if he had a weapon.
"But we don't know what road they'll take," one of the gunmen said.
"Look at the map," the man who had his back to her replied. "There's exactly one logical route to the Severn City Airport from here." She recognized the prophet's voice.
"They could take Lewis Avenue once they reach Severn City. Looks like it's not that much longer."
"We'll split up," the prophet said. "Two groups, one for each route, and we meet up at the airport road."
"I assume you have a plan, gentlemen." This was Sayid's voice, somewhere near. Sayid! She wanted to speak with him, to ask where they were and what was happening, to tell him the Symphony had searched for him and Dieter after they'd disappeared, but she was too nauseous to move.
"We told you, we're just trading the two of you for the bride," the gunman said, "and as long as no one attempts anything stupid, we'll take her and then we'll be on our way."
"I see," Sayid said. "You enjoy this line of work, or are you in it for the pension?"
"What's a pension?" the one with the machete asked. He was very young. He looked about fifteen.
"All of this," the prophet said, serene, "all of our activities, Sayid, you must understand this, all of your suffering, it's all part of a greater plan."
"You'd be surprised at how little comfort I take from that notion." The clarinet was remembering something she'd always known about Sayid, which was that he had trouble keeping his mouth shut when he was angry. She strained her neck and saw Dieter, lying on his back a few yards away, unmoving. His skin looked like marble.
"Some things in this life seem inexplicable," the archer said, "but we must trust in the existence of a greater plan."
"We're sorry," the boy with the machete said, sounding as if he meant it. "We're very sorry about your friend."
"I'm sure you're sorry about everyone," Sayid said, "but while we're discussing strategy here, there was absolutely no reason for you to abduct the clarinet."
"Two hostages are more persuasive than one," the archer said.
"You're so _bright_ , the lot of you," Sayid said. "That's what I admire most about you, I think."
The gunman muttered something and started to rise, but the prophet placed a hand on his arm and he sank back to the ground, shaking his head.
"The hostage is a test," the prophet said. "Can we not withstand the taunts of the fallen? Is that not part of our task?"
"Forgive me," the gunman murmured.
"The fallen walk among us. We must be the light. We _are_ the light."
"We are the light," the other four repeated in murmured unison. The clarinet shifted painfully—the movement brought a storm of dark spots over her vision—and craned her neck until she saw Sayid. He was ten or twelve feet away, tied up.
"The road is fifty paces due east," he mouthed. "Turn left when you get there." The clarinet nodded and closed her eyes against a wave of nausea.
"Your clarinet friend still sleeping?" The archer's voice.
"If you touch her, I'll kill you," Sayid said.
"No need for that, friend. No one will bother her. We're just hoping to avoid a repeat..."
"Let her sleep," the prophet said. "The Symphony's stopped for the night anyway. We'll catch up to them in the morning."
When the clarinet opened her eyes, the men were apparently sleeping, bundled on the forest floor. Some time had passed. Had she slept? She was less ill than she had been. Someone had placed a cloth over Dieter's face. Sayid was sitting where she'd seen him last, talking to the boy with the machete, who had his back to her.
"In the south?" the boy was saying. "I don't know, I don't like to think about it. We did what we had to."
She didn't hear Sayid's reply.
"It hollows you out," the boy said, "thinking about it. Remembering what we did, it just guts me. I don't know how else to put it."
"But you believe in what he says? All of you?"
"Well, Clancy's a true believer," she heard the boy say very softly. He gestured toward the sleeping men. "Steve too, probably most of the others. If you're not a true believer, you're not going to talk about it. But Tom? The younger gunman? To be honest, I think he's maybe just in it because our leader's married to his sister."
"Very shrewd of him," Sayid said. "I still don't get why the prophet's with you."
"He comes along on patrols and such every now and again. The leader must occasionally lead his men into the wilderness." Was she imagining the sadness in his voice? The clarinet lay still for a while, until she located the North Star. She discovered that it was possible, by lying on her side and arching her back, to bring her feet close enough to her hands to loosen the rope that bound her ankles. Sayid and the boy were still talking quietly.
"Okay," she heard Sayid say, "but there are six of you, and thirty of us. Everyone in the Symphony's armed."
"You know how quiet we are." The boy sighed. "I'm not saying it's right," he said. "I know it's not right."
"If you know it's not right..."
"What choice do I have? You know how this... this time we live in, you know how it forces a person to do things."
"That seems a strange statement," Sayid said, "coming from someone too young to remember any different."
"I've read books. Magazines, I even found a newspaper once. I know it all used to be different."
"But getting back to the subject at hand, there are still only six of you, and—"
"You didn't hear us come up behind you on the road, did you? This is our training. We move silently and we attack from behind. This is how we disarmed ten towns and took their weapons for our leader before we reached St. Deborah by the Water. This is how we took two of our leader's wives. And look, your friend for example"—the clarinet closed her eyes—"we came up behind her in the forest and she heard nothing."
"I don't—"
"We can pick you off one by one," the boy said. He sounded apologetic. "I've been training since I was five. You've got weapons, but you don't have our skills. If the Symphony won't swap you for the girl, we can kill you one at a time from the safety of the forest until you give her back to us."
The clarinet began to move again, frantically working the knot at her ankles. Sayid could see her, she realized, but he was keeping his gaze on the boy's face. A long time passed when she didn't listen to the conversation, concentrating on nothing but the rope. When her ankles were unbound, she struggled to her knees.
"But I'm not sure I quite follow," Sayid was saying. "That part in your philosophy about _being_ the light. How do you _bring_ the light if you _are_ the light? I wonder if you could just explain to me..."
The clarinet was one of the Symphony's best hunters. She had survived alone in the forest for three years after the collapse, and now, even sick with whatever poison they'd used on her, even with her wrists bound behind her back, it was possible to turn and vanish noiselessly between the trees, away from the clearing, to make almost no sound at all as she stepped out onto the road. Running as night faded to gray dawn and the sun rose, walking and stumbling through the dragging hours, hallucinating now, dreaming of water, falling into the arms of the Symphony's rear scouts in the morning as the sky darkened overhead, delivering her message—"You must change the route"—as they carried her back to the Symphony, where the last tree blocking the road had just been sawed away. The first raindrops were falling as the conductor heard the message and ordered an immediate change of course, scouts sent to find Kirsten and August—fishing somewhere along the road ahead—but unable to locate them in the storm, the Symphony veering inland into a new route, a circuitous combination of back roads that would take them eventually to the Severn City Airport, the clarinet slipping in and out of consciousness in the back of the first caravan while Alexandra held a bottle of water to her lips.
# **50**
**THE KNIFE TATTOOS** on Kirsten's wrist:
The first marked a man who came at her in her first year with the Symphony, when she was fifteen, rising fast and lethal out of the underbrush, and he never spoke a word but she understood his intent. As he neared her, sound drained from the world, and time seemed to slow. She was distantly aware that he was moving quickly, but there was more than enough time to pull a knife from her belt and send it spinning—so slowly, steel flashing in the sun—until it merged with the man and he clutched at his throat. He shrieked—she couldn't hear him, but she watched his mouth open and she knew others must have heard, because the Symphony was suddenly all around her, and this was when the volume slowly rose and time resumed its normal pace.
"It's a physiological response to danger," Dieter told her, when Kirsten mentioned the soundlessness of those seconds, the way time stretched and expanded. This seemed a reasonable-enough explanation, but there was nothing in her memories to account for how calm she was afterward, when she pulled her knife from the man's throat and cleaned it, and this was why she stopped trying to remember her lost year on the road, the thirteen unremembered months between leaving Toronto with her brother and arriving in the town in Ohio where they stayed until he died and she left with the Symphony. Whatever that year on the road contained, she realized, it was nothing she wanted to know about.
The second knife was for a man who fell two years later, outside Mackinaw City. The Symphony had been warned of brigands in the area, but it was a shock when they materialized out of fog on the road ahead. Four men, two with guns and two with machetes. One of the gunmen asked for food, four horses, and a woman, in a flat monotone voice. "Give us what we want," he said, "and no one has to die." But Kirsten sensed rather than heard the sixth guitar fitting an arrow to his bow behind her back. "Guns first," he murmured, close to her ear. "I've got the one on the left. One, two—" and on three the men with guns were falling, one staring past the arrow protruding from his forehead and the other clutching at Kirsten's knife in his chest. The conductor finished the others with two quick shots. They retrieved the weapons, dragged the men into the forest to be food for the animals, and continued on into Mackinaw City to perform _Romeo and Juliet_.
She'd hoped there would never be a third. "There was a new heaven and a new earth," the archer whispered. She saw the look on August's face just afterward and realized that the gunman had been his first—he'd had the colossal good fortune to have made it to Year Twenty without killing anyone—and if she weren't so tired, if it didn't take all of her strength to keep breathing in the face of Sayid's terrible news, she could have told him what she knew: it is possible to survive this but not unaltered, and you will carry these men with you through all the nights of your life.
Where was the prophet? They walked mostly in silence, stunned by grief, Sayid limping, listening for the dog. The signs for the airport led them away from the lake, out of downtown, up into residential streets of wood-frame houses. A few of the roofs had collapsed up here, most under the weight of fallen trees. In the morning light there was beauty in the decrepitude, sunlight catching in the flowers that had sprung up through the gravel of long-overgrown driveways, mossy front porches turned brilliant green, a white blossoming bush alive with butterflies. This dazzling world. An ache in Kirsten's throat. The houses thinned out, longer spaces between the overgrown driveways, and now the right lane of the road was clogged with cars, rusted exoskeletons on flat tires. When she glanced in the windows, she saw only trash from the old world, crumpled chip bags, the remains of pizza boxes, electronic objects with buttons and screens.
When they came to the highway there was a sign indicating the direction to the airport, but finding the airport would have been as simple as following the traffic jam. Everyone had apparently been trying to get there at the end, just before they ran out of gasoline or had to abandon their cars in the gridlock or died of flu at the wheel. There was no sign of the prophet, no movement among the endless lines of cars glittering in sunlight.
They walked on the gravel shoulder. There was a place where ivy had spread from the forest and covered acres of highway in green. They waded through it, the leaves soft on Kirsten's sandaled feet. Every sense attuned to the air around her, trying to sense the prophet's position—behind or ahead?—and met only by the racket of the world around them, the cicadas, the birds, dragonflies, a passing family of deer. The alignment of the cars was askew, some stopped at odd angles, some hard up against the bumper of the next vehicle, others halfway off the road. The windshield wipers were up, puddles of rusted chains tangled around some of the wheels. It had been snowing, then, perhaps heavily, and the highway hadn't been plowed. The cars had slipped and skidded on packed snow and ice.
"What is it?" August asked, and she realized that she'd stopped. The flu, the snow, the gridlock, the decision: wait in the car, boxed in now by all the cars that have piled up behind, idling to keep the heat on until you run out of gas? Or abandon your car to walk, perhaps with young children, but where exactly? Farther on, toward the airport? Back home?
"Do you see something?" Sayid spoke in a whisper. August had been supporting him for the last mile or so, Sayid's arm over his shoulders.
I see everything. "It's nothing," Kirsten said. She had once met an old man up near Kincardine who'd sworn that the murdered follow their killers to the grave, and she was thinking of this as they walked, the idea of dragging souls across the landscape like cans on a string. The way the archer had smiled, just at the end.
They took the exit to the airport and reached the roadblock in the midafternoon. An ancient plywood quarantine sign warning of the Georgia Flu, a line of fallen traffic cones and orange plastic fencing collapsing to the ground. The thought of walking here in the snowstorm, desperate to get away from the sickness in town, and at the end of that walk there's this sign, and when you read it you understand that it isn't going to be possible to get away from this. By now perhaps you're already ill, perhaps carrying a feverish small child in your arms. Kirsten turned away from the roadblock, and knew without looking that there would be skeletons in the forest here. Some people would have turned back and retraced their steps for miles, tried to find another way to escape from an illness that was everywhere, that was inescapable by then. Others, sick or very tired, would have stepped off the road and lay down on their backs to watch the snow falling down upon them, to look up at the cold sky. _I dreamt last night I saw an airplane_. She stopped walking, overcome by the thought of Dieter, and in that moment of stillness she heard the distant bark of a dog.
"Kirsten," August said over his shoulder. She saw in his face that he hadn't heard what she had. "We're almost there."
"Into the woods," she said quietly. "I think I heard the prophet's dog." They helped Sayid off the road. He was very pale now. He collapsed into the underbrush, gasping, and closed his eyes.
In the quiet that followed the dog's bark, Kirsten crouched in the bushes and listened to her heartbeat. The prophet and his men had been some distance behind them. A long time passed before she heard their footsteps. The sound seemed strangely amplified, but she knew it was only the tension singing through her, her senses made acute by fear. The sunlight on this stretch of road was filtered through leaves, and her first sighting was the long barrel of the prophet's rifle moving in and out of shadow as he walked. He led the group, serene and unhurried, the dog trotting by his side. The boy who'd escaped Kirsten and August's ambush that morning carried a handgun now, the machete strapped to his back, and behind them walked a man with a complicated weapon of a kind that Kirsten had never seen before, a vicious metal crossbow with four short arrows preloaded, and a fourth man with a shotgun.
Don't stop. Don't stop. But as the dog drew alongside the bush where Kirsten was hiding, he slowed and raised his nose in the air. Kirsten held her breath. She hadn't gone far enough from the road, she realized. She was no more than ten paces away.
"You smell something, Luli?" the man with the crossbow asked. The dog barked once. Kirsten held her breath. The men gathered around the dog.
"Probably just another squirrel," the boy said, but he sounded uneasy. Kirsten saw that he was afraid, and the realization carried such sadness. I never wanted any of this.
"Or maybe there's someone in the woods."
"Last time he barked, it was just a squirrel."
The dog had gone still, his nose twitching. Please, she thought, please. But Luli barked again and stared directly at Kirsten through her screen of leaves.
The prophet smiled.
"I see you," the man with the crossbow said.
She could rise out of the underbrush and throw a knife, and as it spun through the air she would be felled by a bullet or a metal arrow—the crossbow and three guns were trained on her now—or she could remain unmoving until they were forced to approach, attack at close range and be killed by one of the others. But would they approach at all, or would they fire into the bush behind which she was hiding? She felt August's anguish, a low current in the air. He was better hidden than she, crouched behind a stump.
A metal arrow drove into the dirt by her feet with a hollow thud.
"The next one lands in your heart." The man with the crossbow was older than the prophet, an old burn scar on his face and neck. "Stand up. Slowly. Hands in the air."
Kirsten rose out of hiding.
"Drop the knife."
She let it fall from her hand into the underbrush. Acutely aware of the other two knives in her belt, so close but unreachable. If she reached now, if she were fast enough, would there be time to at least take out the prophet before the first bullet tore her heart? Unlikely.
"Step forward. If you reach for those knives, you'll be dead." The man with the crossbow spoke calmly. Nothing about this situation was new to him. The boy looked stricken.
The shock of realizing that this was probably actually the ending, after a lifetime of near misses, after all this time. She walked forward through the radiant world, the sunlight and shadow and green. Thinking of trying to do something heroic, sending a knife spinning through the air as she fell. Thinking, please don't let them find August and Sayid. Thinking of Dieter, although thoughts of Dieter carried a pain that was almost physical, like probing at an open wound. She stepped up onto the hard surface of the road and stood before the prophet, her hands in the air.
"Titania," the prophet said. He raised the point of his rifle to the spot between her eyes. In his gaze she saw only curiosity. He was interested to see what would happen next. All three guns were on Kirsten. The man with the crossbow was sighting his weapon into the underbrush, but nothing in his aim or his movements suggested that he'd seen August or Sayid. The prophet nodded to the boy, who stepped forward and pulled her knives very gently from her belt. She recognized him now. He'd been the sentry as they left St. Deborah by the Water, standing watch and roasting his dinner on a stick. He didn't meet her eyes. The dog had apparently lost interest in following scents from the woods and had laid down on the pavement, watching them, his chin resting on his paw.
"On your knees," the prophet said. She knelt. The point of the rifle followed her. He stepped closer.
She swallowed. "Do you have a name?" she asked. Some vague instinct to stall.
"Sometimes names are an encumbrance. Where are your companions?"
"The Symphony? I don't know." The pain of this, even now when it was too late to matter anymore. Thinking of the Symphony, the horse-drawn caravans moving under the summer sky, the clopping of horses. Traveling somewhere or perhaps already at the airport, in safety, in grace. She loved them so desperately.
"And your other companions? The ones who helped you kill my men on the road this morning."
"We had no choice."
"I understand," he said. "Where are they?"
"They're dead."
"Are you sure?" He moved the rifle just slightly, tracing a small circle in the air.
"There were three of us," she said, "including Sayid. Your archer got the other two before he died." It was plausible. The boy with the machete had run away before the archer fell. She was careful not to look at him.
"My archer was a good man," the prophet said. "Loyal."
Kirsten was silent. She understood the calculations August was making at that moment. The prophet's rifle was an inch from her forehead. If August revealed his position by taking out one of the men, the others would be upon him and Sayid in an instant. Sayid was defenseless, lying bloodied and weakened, and Kirsten—kneeling on the road, disarmed, a gun to her head—would in all likelihood still die.
"I have walked all my life through this tarnished world," the prophet said, "and I have seen such darkness, such shadows and horrors."
Kirsten didn't want to look at the prophet anymore, or more precisely, she didn't want the last thing she saw on earth to be his face and the point of the rifle. She raised her head to look past him at leaves flickering in sunlight, at the brilliant blue of the sky. Birdsong. Aware of every breath, every heartbeat passing through her. She wished she could convey a message to August, to reassure him somehow: I know it was me or all three of us. I understand why you couldn't shoot. She wished she could tell Sayid that she still loved him. A sense memory of lying next to Sayid in the nights before they broke up, the curve of his ribs under her hand when she ran her hand down the length of his body, the soft curls at the nape of his neck.
"This world," the prophet said, "is an ocean of darkness."
She was astonished to see that the boy with the handgun was crying, his face wet. If she could only speak to August. We traveled so far and your friendship meant everything. It was very difficult, but there were moments of beauty. Everything ends. I am not afraid.
"Someone's coming," one of the prophet's men said. Kirsten heard it too. A distant percussion of hooves, two or three horses approaching at a brisk walk from the direction of the highway.
The prophet frowned, but didn't look away from Kirsten's face.
"Do you know who's coming?" he asked.
"No," she whispered. How distant were the horses? She couldn't tell.
"Whoever they are," the prophet said, "they'll arrive too late. You think you kneel before a man, but you kneel before the sunrise. We are the light moving over the surface of the waters, over the darkness of the undersea."
"The Undersea?" she whispered, but the prophet was no longer listening to her. A look of perfect serenity had come over his face and he was looking at her, no, _through_ her, a smile on his lips.
" 'We long only to go home,' " Kirsten said. This was from the first issue, _Station Eleven_. A face-off between Dr. Eleven and an adversary from the Undersea. " 'We dream of sunlight, we dream of walking on earth.' "
The prophet's expression was unreadable. Did he recognize the text?
" 'We have been lost for so long,' " she said, still quoting from that scene. She looked past him at the boy. The boy was staring at the gun in his hands. He was nodding, seemingly to himself. " 'We long only for the world we were born into.' "
"But it's too late for that," the prophet said. He drew in his breath and adjusted his grip on the rifle.
The shot was so loud that she felt the sound in her chest, a thud by her heart. The boy was in motion and she wasn't dead, the shot hadn't come from the prophet's rifle. In the fathomless silence that followed the sound, she touched her fingertips to her forehead and watched the prophet fall before her, the rifle loose in his hands. The boy had shot the prophet in the head. The other two men seemed frozen in amazement, only for an instant but in that instant one of August's arrows sang through the air and the man holding the crossbow crumpled, choking on blood. The man with the shotgun fired wildly into the trees and then his trigger clicked uselessly, no ammunition, he cursed and fumbled in his pocket until another arrow pierced his forehead and he fell, and then Kirsten and the boy were alone on the road together.
The boy was wild-eyed, his lips moving, staring at the prophet where he lay in a rapidly spreading pool of blood. He lifted the handgun to his mouth. "Don't," Kirsten said, "no, please—" But the boy closed his lips around the barrel and fired.
She knelt there, looking at them, and then lay on her back to look up at the sky. Birds wheeling. The shock of being alive. She turned her head and looked into the prophet's dead blue eyes. Her ears were ringing. She felt the vibration of hooves on the road now. August shouted her name and she looked up as the Symphony's forward scouts rounded the curve of the road on horseback like a vision from a dream, Viola and Jackson, sunlight glinting on their weapons and on the binoculars that hung on Viola's neck.
"Do you want this?" August asked some time later. Kirsten had been sitting by the prophet, staring at him, while Jackson helped Sayid out of the forest and August and Viola went through the bags that had belonged to the prophet and his men. "I found it in the prophet's bag."
A copy of the New Testament, held together with tape. Kirsten opened it to a random page. It was nearly illegible, a thicket of margin notes and exclamation points and underlining.
A folded piece of paper fell out of the book.
It was a page torn from a copy of _Dr. Eleven_ , Vol. 1, No. 1: _Station Eleven_ , the first page of _Station Eleven_ she'd ever seen that hadn't come from her copies of the books. The entire page devoted to a single image: Dr. Eleven kneels by the lifeless body of Captain Lonagan, his mentor and friend. They are in a room that Dr. Eleven sometimes uses as a meeting place, an office area with a glass wall that overlooks the City, the bridges and islands and boats. Dr. Eleven is distraught, a hand over his mouth. An associate is there too, a speech bubble floating over his head: "You were his second-in-command, Dr. Eleven. In his absence, you must lead."
Who were you? How did you come to possess this page? Kirsten knelt by the prophet, by the pool of his blood, but he was just another dead man on another road, answerless, the bearer of another unfathomable story about walking out of one world and into another. One of his arms was outstretched toward her.
August was talking to her again, crouched by her side. "The Symphony's only a few hours behind us," he was saying very gently. "Viola and Jackson are going back to them, and the three of us will go on ahead to the airport. It isn't far."
_I have walked all my life through this tarnished world_. After she walked out of Toronto with her brother, after that first unremembered year, her brother had been plagued by nightmares. "The road," he'd always said, when she shook him awake and asked what he'd been dreaming of. He'd said, "I hope you never remember it."
The prophet was about her age. Whatever else the prophet had become, he'd once been a boy adrift on the road, and perhaps he'd had the misfortune of remembering everything. Kirsten brushed her hand over the prophet's face to close his eyes, and placed the folded page from _Station Eleven_ in his hand.
# **51**
**WHEN SAYID AND AUGUST AND KIRSTEN** walked away from the bodies on the road, resuming their slow progress to the airport, the prophet's dog followed at some distance. When they stopped to rest, the dog sat a few yards away, watching them.
"Luli," Kirsten said. "Luli." She threw a strip of dried venison, and the dog snapped it out of the air. He came close and let her stroke his head. She ran her fingers through the thick fur at the base of his neck. When they set out again, the dog stayed close by her side.
A half mile farther, the road curved out of the trees, the terminal building massive in the near distance. It was a two-story monolith of concrete and glass, shimmering over an ocean of parking lot. Kirsten knew they were almost certainly being watched by now, but she saw no movement in the landscape. The dog whined and raised his nose in the air.
"Do you smell that?" Sayid asked.
"Someone's roasting a deer," August said. The road divided before them, separate lanes for Arrivals, Departures, and Parking. "Which way?"
"Let's pretend there's a way to get off this continent." Sayid had a distant look about him. The last time he'd seen an airport had been two months before the collapse, when he'd returned home from visiting his family in Berlin and landed for the last time at Chicago O'Hare. "Let's go to Departures."
The Departures lane rose to a second-story entrance, a line of glass-and-steel revolving doors, a municipal bus glinting in sunlight. They were a hundred yards from the door when the whistle sounded, three short blasts. Two sentries stepped out from behind the bus, a woman and a man, their crossbows aimed at the ground.
"Sorry about the crossbows," the man said pleasantly, "necessary precaution, I'm afraid—" But he stopped then, confused, because the woman's bow had clattered to the pavement and she was running to the new arrivals, she was laughing and shouting their names and trying to embrace all of them at once.
There were 320 people living in the Severn City Airport that year, one of the largest settlements Kirsten had seen. August took Sayid to the infirmary, and Kirsten lay dazed in Charlie's tent.
By the beginning of Year Two the occupants of the airport had been sick of looking at one another but on the other hand they hadn't wanted to sleep too far apart, so they'd constructed a double line of tents down the length of Concourse B. The tents were of varying sizes, with frames made of branches dragged in from the woods, squares of about twelve feet by twelve feet with peaked roofs. They'd raided the airport offices for staplers, stapled sheets over the frames. There'd been some debate over whether this was the best use of the mountain of sheets they'd salvaged from the nearby hotels, but there was such a longing for privacy by then. In Charlie and Jeremy's tent there was a bed, two plastic crates for clothes and diapers, their instruments. A watery light filtered through the cloth. Luli crowded in and lay by Kirsten's side.
"I'm so sorry about Dieter," Charlie said. "August told me."
"It doesn't seem real." Kirsten wanted to close her eyes, but she was afraid of what she'd dream of if she slept. "Is there a tattooist here, Charlie?"
Charlie brushed her fingertips over Kirsten's right wrist, the two black knives inked two years apart. "How many?"
"One. An archer on the road."
"There's a tattooist who lives in the Lufthansa jet. I'll introduce you tomorrow."
Kirsten was watching an ant cross the roof of the tent on the outside, the shadow of its tiny body and the pinpoint impressions of its legs on the fabric. "I've been thinking about the nursery," she said.
A few years ago, they'd been going through a massive country house near the mouth of the St. Clair River, Kirsten and Charlie and August, a place that had been picked over more than once but not for years or maybe a decade, dust everywhere, and eventually August had said something about getting back to the Symphony. Kirsten had gone upstairs in search of Charlie and found her in a room that had obviously been a nursery once, staring at a porcelain tea set sized for dolls. She didn't look up when Kirsten said her name.
"We should go, Charlie," she'd said. "We're a mile from the road." But Charlie gave no sign of having heard her. "Come on," Kirsten had said, "we can take it with us," gesturing to the tea set, which had been set up with improbable precision on a miniature table. Charlie still said nothing. She was staring at the tea set as if in a trance. August called their names from downstairs, and all at once Kirsten had the impression that someone was watching them from a corner of the room, but except for Kirsten and Charlie, the room was empty. Most of the furniture in the nursery was gone, nothing remaining except this little table set for dolls and there, in the corner, a child-size rocking chair. How could this table have remained set, when the rest of the house was ransacked and in disarray? Now that Kirsten looked, she realized there was no dust on the tea set. The only footprints in the dust were hers and Charlie's, and Charlie wasn't sitting close enough to the table to touch it. What small hand had placed the doll's teacups on the table? It was very easy to imagine that the rocking chair was moving, just slightly. Kirsten tried not to look at it. She wrapped the tiny plates and saucers in a pillowcase as quickly as possible while Charlie watched, still not speaking, and then Kirsten stuffed the bundle into Charlie's bag, took her hand and led her downstairs, out to the overgrown lawn, where Charlie blinked and came back to herself slowly in the late-spring light.
"The nursery was just a strange moment," Charlie said now, all these years later in her airport tent. "A strange moment in a lifetime of strange moments. I can't explain what came over me."
"Is that all? Just a strange moment?"
"We've talked about this a hundred times. There was no one else in the room with us."
"There was no dust on the tea set."
"Are you asking if I believe in ghosts?"
"I don't know. Maybe. Yes."
"Of course not. Imagine how many there'd be."
"Yes," Kirsten said, "that's exactly it."
"Close your eyes," Charlie murmured. "I'll sit here with you. Try to sleep."
There was music that night, August with Charlie and the sixth guitar. Sayid slept in the infirmary downstairs in Baggage Claim, his injuries cleaned and bandaged. Charlie played the cello with her eyes closed, smiling. Kirsten stood at the back of the crowd. She tried to concentrate on the sound, but music had always unmoored her, and her thoughts drifted. Dieter. The prophet, the only other person she'd ever met who had been in possession of _Station Eleven_. The archer on the road, her knife in his chest. Dieter as Theseus, _A Midsummer Night's Dream_. Dieter brewing his fake coffee in the mornings, Dieter arguing with her about tattoos. Dieter the night she met him in central Ohio, when she was fourteen and Dieter was in his late twenties, half a lifetime ago.
On her first night with the Symphony he'd served her dinner by the fire. She'd been so alone since her brother's death, and when the Symphony agreed to let her join them it had seemed like the best thing that had ever happened to her, and that first night she'd been almost too excited to eat. She remembered Dieter talking to her about Shakespeare, Shakespeare's works and family, Shakespeare's plague-haunted life.
"Wait, do you mean he had the plague?" she asked.
"No," Dieter said, "I mean he was defined by it. I don't know how much schooling you've had. Do you know what that means, to be defined by something?"
Yes. _There was a new heaven and a new earth_. Kirsten turned away from the light and the music. The terminal's south wall was almost entirely glass, the smudges of children's handprints here and there at waist height. Night was falling, airplanes luminous in starlight. She heard the distant movements of the airport's four cows, sequestered in a loading dock for the night, the clucking of hens. A liquid movement below on the tarmac; a cat, hunting in the shadows.
An old man was sitting on a bench some distance from the performance, watching her approach. He'd shaved off all his hair and wore a silk neck scarf tied in a complicated knot. She saw a glint of earrings, four loops in his left earlobe. She didn't want to talk to anyone, but by the time she saw him it was too late to turn away without being rude, so she nodded to him and sat at the far end of his bench.
"You're Kirsten Raymonde." He retained the traces of a British accent. "Clark Thompson."
"I'm sorry," she said, "we were introduced earlier, weren't we?"
"You were going to let me take you on a tour of my museum."
"I'd like to see it. Maybe tomorrow. I'm so tired tonight."
"I understand." They sat in silence for a few minutes, listening to the music. "I'm told the Symphony will arrive soon," he said.
She nodded. It would be a different Symphony now, without Dieter. All she wanted was to sleep. There was a clicking of claws on the floor as Luli came to find her. He sat by her side and rested his chin on her lap.
"That dog seems devoted to you."
"He's my friend."
Clark cleared his throat. "I've spent a great deal of time with Charlie, this past year. She mentioned that you have an interest in electricity." He stood, leaning on his cane. "I know you're tired," he said. "I understand you've had a difficult few days. But there's something I think you'd like to see."
She considered this for a moment before she accepted. She wasn't in the habit of following strangers, but he was elderly and moved slowly and she had three knives in her belt. "Where are we going?"
"The air traffic control tower."
"Outside?"
He was walking away from her. She followed him through a steel door near the entrance to the museum, down an unlit flight of stairs and into the night. The singing of crickets, a small bat darting on a hunt. From the tarmac, the concert was a smudge of light in Concourse C.
Up close the airplanes were larger than she would have imagined. She looked up at the dark windows, the curve of wings. Impossible to imagine that machines so enormous had ever taken to the air. Clark walked slowly. She saw the cat again, running fast and low at the base of the air traffic control tower, heard the squeak of a rodent when it pounced. The tower's steel door opened, and she found herself in a small room where a guard kept watch through a peephole, candlelight glinting on elevator doors. The door to the stairwell was propped open with a rock.
"It's nine stories," Clark said. "I'm afraid this may take some time."
"I'm not in a hurry." It was peaceful, climbing the stairs with him. He seemed to expect no conversation from her. A slow ascent between shadowed stairs and moonlit landings, the tapping of his cane on steel. His breathing was labored. At every landing they stopped to rest, once for so long that Kirsten was almost asleep before she heard him pull himself up on the railing. The dog lay down and let out a theatrical sigh at each landing. There were open windows on every floor, but there was no breeze that night and the air was hot and still.
"I read that interview you gave a few years back," he said on the sixth floor.
"That newspaper in New Petoskey."
"Yes." Clark was mopping his forehead with a handkerchief. "I want to talk to you about it tomorrow."
On the ninth landing, Clark rapped a pattern with his cane on a door and they were admitted into an octagonal room with walls of glass and arrays of darkened screens, four people with binoculars watching the tarmac, the terminal, the shadows of the gardens, the barrier fence. The dog sniffed around in the shadows. It was disorienting, being so high off the ground. The airplanes gleamed pale under the stars. The concert in Concourse C seemed to have ended.
"Look there," Clark said, "to the south. It's what I wanted to show you." She followed the line of his finger, to a space on the southern horizon where the stars seemed dimmer than elsewhere in the sky. "It appeared a week ago," he said. "It's the most extraordinary thing. I don't know how they did it on such a large scale."
"You don't know how who did what?"
"I'll show you. James, may we borrow the telescope?" James moved the tripod over and Clark peered through, the lens aimed just below the dim spot in the sky. "I know you're tired tonight." He was adjusting the focus, his fingers stiff on the dial. "But I hope you'll agree this was worth the climb."
"What is it?"
He stepped back. "The telescope's focused," he said. "Don't move it, just look through."
Kirsten looked, but at first she couldn't comprehend what she was seeing. She stepped back. "It isn't possible," she said.
"But there it is. Look again."
In the distance, pinpricks of light arranged into a grid. There, plainly visible on the side of a hill some miles distant: a town, or a village, whose streets were lit up with electricity.
# **52**
**KIRSTEN STARES THROUGH** the telescope at the town with electric light.
In the terminal building, Charlie and August sit by Sayid's bed in the Baggage Claim infirmary and tell him about the concert, and he smiles for the first time in a number of days.
A thousand miles to the south of the airport, Jeevan is baking bread in an outdoor oven. He rarely thinks of his old life anymore, although he has dreams sometimes about a stage, an actor fallen in the shimmering snow, and other dreams where he's pushing shopping carts through blizzards. His small son kneels by his feet, playing with a puppy. This boy born into the new world, his mother resting indoors with the baby.
"Frank," Jeevan says to his son, "go see if your mother's hungry." He lifts the pan with the bread from the oven, which in a previous incarnation was an oil drum. His son runs indoors, the puppy close at his heels.
It's a warm night, and he hears a neighbor laughing. A scent of gardenias carries on the breeze. In a moment he will go down to the river to retrieve the preserved meat cooling in an old coffee can in the water, he'll make sandwiches for his small family and offer some bread to their neighbors, but for now he lingers to watch the silhouettes of his wife and children behind the thin curtains of the room where they live. Daria leans down to lift the baby from the crib, stoops again to blow out the candle and in that movement she vanishes, the silhouettes blinking out, and Frank runs ahead of her out onto the grass.
"Come check the bread," he says, and little Frank kneels by the bread with a grave expression, pokes at it with one finger, leans in close to inhale the warmth.
"He seems better," Daria says. Frank had a fever the night before. She sang him soft lullabies while Jeevan pressed cold compresses to his forehead.
"Back to normal," Jeevan says. "How's that bread looking, Frank?"
"I think it's too hot to eat."
"We'll let it sit for a moment." The boy turns to his parents and for an instant in the twilight he looks like his namesake, like Jeevan's brother. He comes to them, the moment already passed, and Jeevan lifts him into his arms to kiss the silk of his hair. Always these memories, barely submerged.
Far to the north, in a place so distant that in this flightless world it might as well be another planet, the caravans of the Traveling Symphony are arriving at the Severn City Airport.
#
# **53**
**ON HIS LAST MORNING** on earth, Arthur was tired. He'd laid awake until sunrise and then drifted out of a twilight half-sleep in the late morning, sluggish and dehydrated, a throbbing headache behind his eyes. Orange juice would've helped, but when he looked in the fridge there was only a mouthful left in the bottom of the carton. Why hadn't he bought more? He had had insomnia for the past three nights, and his exhaustion was such that this was enough to send him spiraling into something not far from fury, the fury contained with difficulty by breathing deeply and counting to five, soothed by the cold air on his face. He closed the fridge door, made his last breakfast—scrambled eggs—and showered, dressed, combed his hair, left for the theater an hour early so he'd have time to linger with a newspaper over his second-to-last coffee at his favorite coffee place, all of the small details that comprise a morning, a life.
The weather reports had been full of an approaching snowstorm and he sensed it in the air, in the dove-gray weight of the late-morning sky. He'd definitely decided: when _Lear_ closed, he was moving to Israel. The idea was exhilarating. He would shed his obligations and belongings and start over in the same country as his son. He would buy an apartment within walking distance of Elizabeth's house and he'd see Tyler every day.
"Looks like snow," the girl in the coffee shop said.
Arthur nodded hello to the hot-dog guy who always stood on the same corner halfway between the hotel and the theater. The hot-dog guy beamed. A pigeon walked in circles near the base of the hot-dog stand, hoping for dropped garnishings and crumbs. The beauty of the pigeon's luminescent neck.
He arrived at the theater at noon for notes, but the notes devolved into an extended argument and went on well past schedule. Arthur tried to pay attention, but the coffee wasn't performing as well as he'd hoped. In the late afternoon he lay on a sofa in his dressing room, hoping to revive himself with a nap, but for all his exhaustion the room seemed oppressive. His thoughts raced. He eventually gave up and left the theater. Ignoring the bored photographers outside the stage door, who took pictures and called out questions about Miranda while he waved at passing taxis. Had he dragged her back into the tabloids when she'd visited two weeks ago? He felt the old guilt. She'd never asked for any of it.
"Queen West and Spadina," he told the driver of an orange-and-green cab, and rested his forehead on the glass to watch their passage down Queen Street. This had been one of his neighborhoods once, but all the shops and cafés he'd known were gone. He was thinking of a diner near Queen West and Spadina, a place he'd frequented with Clark when they were seventeen. He couldn't recall exactly where it was but he found it eventually, a little farther east than expected.
All these decades later, the place was eerily unchanged. The same line of red upholstered booths, stools down the length of the counter, an ancient clock on the wall. Could this possibly be the same waitress? No, he was misremembering, because the fiftyish woman who'd served him burnt coffee when he was seventeen couldn't possibly still be fiftyish. He remembered being here with Clark at three or four or sometimes five in the morning, during what seemed at the time like adulthood and seemed in retrospect like a dream. The dream lasted just a moment, but the moment was bright: both of them taking acting classes, Arthur working as a waiter while Clark burned through a small inheritance. Clark had been magnificent, actually, in retrospect. Six foot two and skinny with a penchant for vintage suits, half his hair shaved off and the other half floppy and dyed pink or occasionally turquoise or purple, eye shadow on special occasions, that captivating British boarding-school drawl.
Arthur's grilled cheese sandwich arrived. He thought of calling Clark, a quick "You'll never guess where I'm calling from!" moment, but decided against it. He wanted to call his son, but it was four a.m. in Israel.
Arthur finished his dinner and took a cab back to the theater, where there was still a little time remaining. He sat on a sofa in his dressing room and looked over the script—he knew his lines backward and forward, but it was his habit to try to pick up some of everyone else's lines too, because he liked to know what was coming—but before the end of the first act there was a knock on the door. When he rose the room didn't spin, exactly, but it wasn't as steady as it should have been. Tanya brushed past him into the room.
"You look like hell," she said. "Everything all right?"
"Tired," Arthur said. "I had insomnia again." He kissed her, and she perched on one of the sofas. The lightness he felt whenever he saw her. He was captivated, as always, by her excessive youth. She was slightly more than half his age. It was her job to look after the three little actresses who played child versions of Lear's daughters.
"You forgot you were meeting me for breakfast, didn't you?"
He slapped a hand to his forehead. "I'm so sorry. I'm not running on all cylinders today. How long did you wait?"
"Half hour."
"Why didn't you call me?"
"Dead cell-phone battery," she said. "It's okay. You can make it up to me with a glass of wine." This was something he adored about her, the way she let things go so easily. What a pleasant state of affairs, he'd been thinking lately, to be with a woman who didn't hold a grudge. He found a half-empty bottle of red in the fridge—she liked it cold—and noticed as he poured her glass that his hands were trembling.
"You really look terrible," she said. "Are you sure you're not sick?"
"Just tired, I think." He liked watching her drink wine, the way she concentrated on the taste. She had the appreciation for nice things that comes only from having grown up with little money.
"Do you have any of those chocolates left?"
"You know, I think I do."
She smiled at him—the way her smile warmed him!—and set her glass on the coffee table. After a few minutes of rummaging through the cupboard by the sink, she emerged triumphant with a small gold box. He selected a raspberry dark-chocolate truffle.
"What's this?" she asked, mid-chocolate, picking up _Dr. Eleven_ , Vol. 1, No 1: _Station Eleven_ from the coffee table.
"My ex-wife dropped those off a couple weeks back."
"Which one?"
He felt a flicker of sadness. This was a sign of having gone seriously astray, wasn't it? Having more than one ex-wife? He wasn't sure where exactly he'd gone wrong. "The first one. Miranda. I'm actually not sure what to do with them."
"What, you're not keeping these?"
"I don't read comic books," Arthur said. "She gave me two copies of each, so I sent the other set to my son."
"You told me you're trying to shed your possessions or something, right?"
"Exactly. They're lovely, but I don't want more _things_."
"I think I understand." Tanya was reading. "Interesting story line," she said, a few pages in.
"I don't know," he said. "I never really understood the point of it, to be honest." There was relief in admitting this to someone, after all these years. "The Undersea, especially. All those people in limbo, waiting around, plotting, for what?"
"I like it," Tanya said. "The art's really good, isn't it?"
"She liked drawing more than she liked writing the dialogue." He was just now remembering this. Once he'd opened Miranda's study door and watched her work for some minutes before she realized he was there. The curve of her neck as she stooped over the drafting table, her absolute concentration. How vulnerable she'd seemed when she was lost in her work.
"It's beautiful." Tanya was studying an image of the Undersea, a heavily crosshatched room with mahogany arches from Station Eleven's drowned forests. The room reminded Arthur of somewhere he'd been, but he couldn't place it.
She glanced at her watch. "I should probably go. My little hellions are due in fifteen minutes."
"Wait, I have something for you." A glass paperweight had arrived by courier two weeks ago, sent by Miranda from her hotel after he'd seen her. She'd explained in her note that Clark had brought it to the house in Los Angeles and that she regretted taking it, that she felt certain Clark had meant it for Arthur, not her, but when he held the glass lump in his hand he found there were no memories attached to it; he had no recollection whatsoever of Clark having given it to them, and anyway the last thing he wanted in his life was a paperweight.
"It's gorgeous," Tanya said when he gave it to her. She peered into the cloudy depths. "Thank you."
"I'll give you a call if Kirsten shows up here. Will I see you after the show?"
She kissed him. "Of course," she said.
When she was gone, he lay on the sofa and closed his eyes, but Kirsten was at his door fifteen minutes later. His exhaustion was taking on the force of illness. Sweat beaded on his forehead when he stood. He let her in and sat down quickly.
"My mom bought a book with you on the cover," she said. She sat across from him on the other sofa.
The only book in existence with Arthur on the cover was _Dear V_. He felt nauseous.
"Did you read it?"
"My mom won't let me read it. She says it's inappropriate."
"That's what she said? Inappropriate?"
"Yes."
"Well," Arthur said, "I think it's inappropriate that the book exists. She's right not to show it to you." The one time he'd met Kirsten's mother, she'd cornered him to ask if he had any projects coming up with a part for a small girl. He'd wanted to shake her. Your daughter's so young, he'd wanted to say. Let her be a kid, give her a chance, I don't know why you want this for her. He didn't understand why anyone would want their child involved in movies.
"Is the book bad?"
"I wish it didn't exist. But you know, I'm glad you came by," he said.
"Why?"
"I have a present for you." He felt a little guilty as he handed her the _Dr. Eleven_ comics, because after all Miranda had intended them for him, but he didn't want the comics because he didn't want possessions. He didn't want anything except his son.
When he was alone again, Arthur put on his costume. He sat for a few minutes in his finery, enjoying the weight of the velvet cape, left his crown on the coffee table next to the grapes and walked down the hall to Makeup. The pleasure of being with other people. He must have eaten something bad, he decided. Maybe at the diner. He had an hour alone in his dressing room, where he drank chamomile tea and spoke lines aloud to his reflection in the mirror, paced, prodded at the bags under his eyes, adjusted his crown. At the half-hour call, he phoned Tanya.
"I want to do something for you," he said. "This will seem very sudden, but I've been thinking about it for a week."
"What is it?" She was distracted. He heard the three little girls bickering in the background.
"How much do you still owe in student loans?" She had told him once, but he couldn't remember the number.
"Forty-seven thousand dollars," she said, and he heard the hope in her voice, the not-daring-to-hope, the disbelief.
"I want to pay it off." Wasn't this what money was for? This was what his life was going to mean, finally, after all these years of failing to win Oscars, this string of box-office flops. He would be known as the man who gave his fortune away. He would retain only enough money to live on. He would buy an apartment in Jerusalem and see Tyler every day and start over.
"Arthur," she said.
"Let me do this for you."
"Arthur, it's too much."
"It isn't. How long will it take for you to pay it off," he asked gently, "at the rate you're going?"
"I'll be in my midsixties, but it's my debt, I—"
"Then let me help," he said. "No strings attached. I promise. Just come to my dressing room after the show tonight, and let me give you a check."
"What do I tell my parents? If I tell them, they'll want to know how I got the money."
"Tell them the truth. Tell them an eccentric actor gave you a check for forty-seven thousand dollars, no strings attached."
"I don't know how to thank you," she said.
When he ended the call, he felt an unexpected peace. He would jettison everything that could possibly be thrown overboard, this weight of money and possessions, and in this casting off he'd be a lighter man.
"Fifteen minutes," the stage manager called from just outside the door.
"Thank you fifteen," Arthur said, and began running his lines from the beginning. At "our eldest born, speak first," he glanced at his watch. It was still only six a.m. in Israel, but he knew Tyler and Elizabeth got up early. He negotiated his way past his ex-wife—"Two minutes, Elizabeth, I know he's getting ready for school, I just want to hear his voice"—and closed his eyes to listen to the rustling of the telephone being transferred into his son's small hands. My eldest born, my only born, my heart.
"Why are you calling?" That suspicious little voice. He remembered that Tyler was angry with him.
"I wanted to say hello."
"Then why weren't you here for my birthday?" Arthur had promised to be in Jerusalem for Tyler's birthday, but he'd made that promise ten months ago and had frankly forgotten about it until Tyler had called him yesterday. Arthur's apologies hadn't landed.
"I can't be there, buddy. I would if I could. But aren't you coming to New York soon? Won't I see you next week?" Tyler had nothing to say to this. "You're flying to New York tonight, aren't you?"
"I guess."
"Did you read those comic books I sent you?"
Tyler didn't respond. Arthur sat on the sofa, and rested his forehead in the palm of his hand. "Did you like them, Tyler? Those comic books?"
"Yeah."
"Ten minutes," the stage manager said at the door.
"Thank you ten. I looked at the comic books," Arthur said, "but I don't think I completely understood what they were about. I was hoping maybe you could explain them to me."
"What about them?"
"Well, tell me about Dr. Eleven."
"He lives on a space station."
"Really? A space station?"
"It's like a planet, but a little planet," Tyler said. "Actually it's sort of broken. It went through a wormhole, so it's hiding in deep space, but its systems were damaged, so on its surface? It's almost all water." He was warming to his subject.
"All water!" Arthur raised his head. It had been a mistake to let Tyler get so far away from him, but perhaps the mistake wasn't unfixable. "So they live in the water, Dr. Eleven and his—his people?"
"They live on islands. They have a city that's all made of islands. There's like bridges and boats? But it's dangerous, because of the seahorses."
"The seahorses are dangerous?"
"They're not like the seahorses we saw in the jar in Chinatown that one time. They're big."
"How big?"
"Really big. I think they're really big. They're these huge—these huge _things_ , and they ride up out of the water and they've got eyes like fish, and they've got people riding on them, and they want to catch you."
"What happens if a seahorse catches you?"
"Then it pulls you under," Tyler said, "and then you belong to the Undersea."
"The Undersea?"
"It's an underwater place." He was talking fast now, caught up. "They're Dr. Eleven's enemies, but they're not really bad. They just want to go home."
"Buddy," Arthur said, "Tyler, I want you to know that I love you."
The silence was so long that he would have thought he'd lost the connection if not for the sound of a passing car. The boy must be standing by an open window.
"You too," Tyler said. It was difficult to hear him. His voice was so small.
The door to his dressing room opened a crack. "Five minutes," the stage manager said. Arthur waved in response.
"Buddy," he said, "I have to go now."
"Are you doing a movie?"
"Not tonight, buddy. I'm going up onstage."
"Okay. Bye," Tyler said.
"Good-bye. I'll see you in New York next week." Arthur disconnected and sat alone for a few minutes. He had a hard time meeting his own eyes in the dressing room mirror. He was very tired.
"Places," the stage manager said.
The set for this production of _Lear_ was magnificent. A high platform had been built at the back of the stage, painted to look like a balcony with elaborate pillars, stone from the front, bare plywood from the back. In the first act, the platform was the study of an aging king, and Arthur had to sit in a purple armchair while the house was filling up, in profile to the audience, holding his crown. A tired king at the end of his reign, perhaps not as sharp as he had been, contemplating a disastrous division of his kingdom.
Below on the main stage, three small girls played a clapping game in soft lighting. At a cue from the stage manager they rose and disappeared backstage left, the house lights dimmed, and this was Arthur's cue to stand and escape. He made his way into the wings in darkness, his path guided by a stagehand with a flashlight, just as Kent, Gloucester, and Edmund entered stage right.
"I don't get it," Arthur had said to the director, whose name was Quentin and who Arthur privately didn't like very much. "Why am I up there?"
"Well, you tell me," Quentin said. "You're pondering the vagaries of power, right? You're contemplating the division of England. You're thinking about your retirement savings. However you want to play it. Just trust me, it's a good visual effect."
"So I'm up there because you like the way it _looks_."
"Try not to overthink it," Quentin said.
But what was there to do up there on the platform, if not think? On the opening night of previews, Arthur had sat in the chair as the house came in, listening to the whispers of the audience as they noticed him there, gazing at the crown in his hands, and he was surprised by how unsteady he felt. He'd done this before, this loitering on stage while the audience entered, but he realized that the last time he'd done this, he'd been twenty-one years old. He remembered having enjoyed it back then, the challenge of living in the world of the play before the play had properly started, but now the lights were too close, too hot, and sweat poured down his back.
In his first marriage, he and Miranda had gone to a Golden Globes party that had gone wrong at the end of the night. Miranda, who'd had perhaps one cocktail too many and wasn't used to high heels, had stumbled and sprained her ankle in a blaze of camera flashes as they were leaving, Arthur just out of reach, and he'd known as she fell that she was going to be a tabloid story. In those days he knew a couple of actors whose careers had flamed out into an ashy half-life of rehab and divorces, and he knew what being a tabloid story could do to a person, the corrosive effect of that kind of scrutiny. He'd snapped at Miranda, mostly out of guilt, and they'd both said unpleasant things in the car. She'd stalked into the house without speaking to him.
Later, he'd walked by the open bathroom door and heard her talking to herself as she removed her makeup. "I repent nothing," he'd heard her say to her reflection in the mirror. He'd turned and walked away, but the words stayed with him. Years later in Toronto, on the plywood second story of the _King Lear_ set, the words clarified the problem. He found he was a man who repented almost everything, regrets crowding in around him like moths to a light. This was actually the main difference between twenty-one and fifty-one, he decided, the sheer volume of regret. He had done some things he wasn't proud of. If Miranda was so unhappy in Hollywood, why hadn't he just taken her away from there? It wouldn't have been difficult. The way he'd dropped Miranda for Elizabeth and Elizabeth for Lydia and let Lydia slip away to someone else. The way he'd let Tyler be taken to the other side of the world. The way he'd spent his entire life chasing after something, money or fame or immortality or all of the above. He didn't really even know his only brother. How many friendships had he neglected until they'd faded out? On the first night of previews, he'd barely made it off the stage. On the second night, he'd arrived on the platform with a strategy. He stared at his crown and ran through a secret list of everything that was good.
The pink magnolias in the backyard of the house in Los Angeles.
Outdoor concerts, the way the sound rises up into the sky.
Tyler in the bathtub at two, laughing in a cloud of bubble bath.
Elizabeth in the pool at night, at the beginning before they'd ever had even a single fight, the way she dove in almost silently, the double moons on the surface breaking into shards.
Dancing with Clark when they were both eighteen, their fake IDs in their pockets, Clark flickering in the strobe lights.
Miranda's eyes, the way she looked at him when she was twenty-five and still loved him.
His third wife, Lydia, doing yoga on the back patio in the mornings.
The croissants at the café across the street from his hotel.
Tanya sipping wine, her smile.
Riding in his father's snowplow when he was nine, the time Arthur told a joke and his father and his little brother couldn't stop laughing, the sheer joy he'd felt at that moment.
Tyler.
On the night of his last performance, Arthur was only halfway through the list when his cue came and it was time to exit. He followed the white tape arrow and the stagehand's flashlight and descended to stage right. He saw Tanya in the wings at the far side of the stage, herding the three little girls in the direction of the dressing rooms. She flashed him a smile, blew him a kiss. He blew a kiss back—why not?—and ignored the murmurs that rose in the backstage area.
Later, a woman from Wardrobe placed a crown of flowers on his head. He was in his costume of rags for the mad scene. He saw Tanya across the stage again—already in the final week of her life, the Georgia Flu so close now—and then a stagehand appeared near him, holding Kirsten's hand.
"Hi," Kirsten whispered. "I love the comic books."
"You read them already?"
"I just had time to read the beginning."
"Here's my cue," he whispered, "I'll talk to you later," and he wandered out into the sound-effect storm.
"But who comes here?" the man playing Edgar said. In four days, he would be dead of flu. "The safer sense will ne'er accommodate his master thus."
"No, they cannot arrest me for coining," Arthur said, bungling the line. Focus, he told himself, but he was scattered, a little dizzy. "I am the King himself."
"O," Edgar said, "thou side-piercing sight!" Gloucester raised a hand to his gauze-covered eyes. In seven days he would die of exposure on a highway in Quebec.
Arthur was having trouble catching his breath. He heard a shimmer of harp music and then the children were there, the girls who'd been his daughters at the beginning, hallucinations of themselves, little ghosts. Two of them would die of flu on Tuesday of next week, one in the morning and one in the late afternoon. The third, Kirsten, flitted behind a pillar.
"Down from the waist they are Centaurs," Arthur said, and this was when it happened. A sharp pain, a clenching, a weight on his chest. He staggered and reached for the plywood pillar that he knew was somewhere close, but he misjudged the distance and struck his hand hard against the wood. He held his hand to his chest and it seemed to him that he'd done this before, something familiar in the motion. When he was seven years old on Delano Island, he and his brother had found a wounded bird on the beach.
"The wren goes to't," Arthur said, thinking of the bird, but to his own ears his voice sounded choked, Edgar looking at him in a way that made him wonder if he'd flubbed the line, he was so lightheaded now. "The wren..."
A man in the front row was rising from his seat. Arthur cradled his hand to his heart, exactly as he'd held the bird. He wasn't sure where he was anymore, or perhaps he was in two places at once. He could hear the waves on the beach. The stage lights were leaving trails through the darkness the way a comet had once, when he was a teenager standing on the dirt outside his friend Victoria's house, looking up at the night, Comet Hyakutake suspended like a lantern in the cold sky. What he remembered from that day at the beach when he was seven was that the bird's heart had stopped in the palm of his hand, a fluttering that faltered and went still. The man from the front row was running now, and Arthur was in motion too; he fell against a pillar and began to slide and now snow was falling all around him, shining in the lights. He thought it was the most beautiful thing he'd ever seen.
# **54**
**IN _DR. ELEVEN_ , VOL. 1, NO. 2:** _The Pursuit_ , Dr. Eleven is visited by the ghost of his mentor, Captain Lonagan, recently killed by an Undersea assassin. Miranda discarded fifteen versions of this image before she felt that she had the ghost exactly right, working hour upon hour, and years later, at the end, delirious on an empty beach on the coast of Malaysia with seabirds rising and plummeting through the air and a line of ships fading out on the horizon, this was the image she kept thinking of, drifting away from and then toward it and then slipping somehow through the frame: the captain is rendered in delicate watercolors, a translucent silhouette in the dim light of Dr. Eleven's office, which is identical to the administrative area in Leon Prevant's Toronto office suite, down to the two staplers on the desk. The difference is that Leon Prevant's office had a view over the placid expanse of Lake Ontario, whereas Dr. Eleven's office window looks out over the City, rocky islands and bridges arching over harbors. The Pomeranian, Luli, is curled asleep in a corner of the frame. Two patches of office are obscured by dialogue bubbles:
Dr. Eleven: What was it like for you, at the end?
Captain Lonagan: It was exactly like waking up from a dream.
# **55**
**THE TRAVELING SYMPHONY** left the airport on a bright morning in September. They'd stayed for five weeks, resting and making repairs to the caravans, performing Shakespeare and music on alternate evenings, and an orchestral and theatrical hangover lingered in their wake. That afternoon Garrett hummed a Brandenburg concerto while he worked in the gardens, Dolores whispered fragments of Shakespeare to herself while she swept the concourse floors, the children practiced swordplay with sticks. Clark retreated to the museum. He ran a feather duster over his objects and thought of the Symphony moving away down the coast, carrying their Shakespeare and their weapons and music.
Yesterday Kirsten had given him one of the two _Dr. Eleven_ comics. He could see that it pained her to part with it, but the Symphony was passing into unknown territory and she wanted to ensure that at least one of the comics would be safe in case of trouble on the road.
"As far as I know, the direction you're going is perfectly safe," Clark told her. He'd assured the conductor of the same thing a few days earlier. "Traders come up from there sometimes."
"But it's not our usual territory," Kirsten said, and if Clark hadn't come to know her a little, over the weeks when the Symphony had lived in Concourse A and performed music or Shakespeare every night, he might not have caught the excitement in her voice. She was beside herself with impatience to see the far southern town with the electrical grid. "When we come back through, I'll take this one back and leave you with the other one. That way, at least one book will always be safe."
In the early evening, Clark finishes dusting his beloved objects in the Museum of Civilization and settles into his favorite armchair to read through the adventures of Dr. Eleven by candlelight.
He pauses over a scene of a dinner party on Station Eleven. There's something familiar about it. A woman with square-framed glasses is reminiscing about life on Earth: "I traveled the world before the war," she says. "I spent some time in the Czech Republic, you know, in _Pra_ ha...," and tears come to his eyes because all at once he recognizes the dinner party, he was _there_ , he remembers the Praha woman, her glasses and her pretension. The man sitting beside her bears a passing resemblance to Clark. The blond woman at the far end of the comic-book table is unmistakably Elizabeth Colton, and the man beyond her in the shadows looks a little like Arthur. Once Clark sat with all of them in Los Angeles, at a table under electric light. On the page, only Miranda is missing, her chair taken by Dr. Eleven.
In the comic-book version Dr. Eleven sits with his arms crossed, not listening to the conversation, lost in thought. In Clark's memory the caterers are pouring wine, and he feels such affection for them, for all of them: the caterers, the hosts, the guests, even Arthur who is behaving disgracefully, even Arthur's orange-tanned lawyer, the woman who said "Praha" instead of "Prague," the dog peering in through the glass. At the far end of the table, Elizabeth is gazing into her wine. In memory, Miranda excuses herself and rises, and he watches her slip out into the night. He's curious about her and wants to know her better, so he tells the others he needs a cigarette and follows her. What became of Miranda? He hasn't thought of her in so long. All these ghosts. She went into shipping, he remembers.
Clark looks up at the evening activity on the tarmac, at the planes that have been grounded for twenty years, the reflection of his candle flickering in the glass. He has no expectation of seeing an airplane rise again in his lifetime, but is it possible that somewhere there are ships setting out? If there are again towns with streetlights, if there are symphonies and newspapers, then what else might this awakening world contain? Perhaps vessels are setting out even now, traveling toward or away from him, steered by sailors armed with maps and knowledge of the stars, driven by need or perhaps simply by curiosity: whatever became of the countries on the other side? If nothing else, it's pleasant to consider the possibility. He likes the thought of ships moving over the water, toward another world just out of sight.
# **ACKNOWLEDGMENTS**
**NOTES**
The book referenced in passing in Chapter 43 (vampires, North America placed under quarantine, etc.) is _The Passage_ , by Justin Cronin.
The line painted on the lead caravan and tattooed on Kirsten's arm, "Survival is insufficient," is from _Star Trek: Voyager_ , episode 122, which aired for the first time in September 1999 and was written by Ronald D. Moore.
I owe a debt of inspiration to Simon Parry, whose September 28, 2009, _Daily Mail_ article "Revealed: The Ghost Fleet of the Recession Anchored Just East of Singapore" inspired the chapters of the book set in Malaysia.
The Toronto staging of _King Lear_ described in this book is partially based on James Lapine's exquisite 2007 production of the play at the Public Theater in New York City, in that Lapine's production featured the unusual addition of three little girls who performed nonspeaking parts as child versions of Lear's daughters.
**WITH THANKS**
To my wonderful agent, Katherine Fausset, and her colleagues at Curtis Brown;
To Anna Webber and her colleagues at United Agents;
To my editors, whose tireless work made this a far better book than it would otherwise have been. In alphabetical order: Jenny Jackson at Knopf, Sophie Jonathan at Picador UK, and Jennifer Lambert at HarperCollins Canada;
To everyone who acquired and/or worked on this book at Knopf, Picador, HarperCollins, and abroad;
To Sohail Tavazoie, for so graciously accommodating my book tour schedule;
To Greg Michalson, Fred Ramey, and their colleagues at Unbridled, for their support and generosity;
To Michele Filgate and Peter Geye, for reading and commenting on early versions of the manuscript;
To Pamela Murray, Sarah MacLachlan, Nancy Miller, Christine Kopprasch, Kathy Pories, Maggie Riggs, Laura Perciaseppe, and Andrea Schulz, for their enthusiasm for the work and for their extremely helpful editorial comments;
To Richard Fausset, for anthropological assistance;
To Jon Rosten, for intel on the Mackinac Bridge;
To Kevin Mandel, always, for everything.
**A NOTE ABOUT THE AUTHOR**
Emily St. John Mandel was born in British Columbia, Canada. She is the author of three previous novels— _Last Night in Montreal, The Singer's Gun_ , and _The Lola Quartet_ —all of which were Indie Next picks. She is a staff writer for _The Millions_ , and her work has appeared in numerous anthologies, including _The Best American Mystery Stories 2013_ and _Venice Noir_. She lives in New York City with her husband.
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Trade deal could curb cyber theft, advocates say
by Cory Bennett - 10/05/15 2:01 PM ET
Obama administration officials believe a major trans-Pacific trade deal struck early Monday morning will help companies protect their digital property abroad and preserve online privacy.
Six years in the making, the Trans-Pacific Partnership (TPP) between the United States and 11 other Pacific Rim countries is expected to lower trade barriers and open up new markets for U.S. exporters.
{mosads}The agreement will also establish a global framework for rules in a number of areas, including intellectual property, although Congress must still approve the deal.
Backers of the deal say these guidelines will establish rules of the road to help counteract the massive digital theft of corporate secrets that has plagued U.S. firms and made companies hesitant to operate in certain overseas markets.
"By setting high standards on labor, the environment, intellectual property, and a free and open Internet, this agreement will level the playing field for American businesses and workers," said Secretary of State John Kerry in a statement Monday.
In a May interview with The Hill, Deputy Commerce Secretary Bruce Andrews called the TPP "critically important" for cybersecurity.
"What TPP is doing is writing the new international norms" on cybersecurity, which he said "will allow [companies] to do what they need to do to protect their data."
The deal is expected to include more commitments to let data flow unimpeded between borders and commitments to not require that data is stored locally.
"That's something that's really important — giving companies the flexibility to build the security solution for their business," Andrews said.
But China, allegedly the worst perpetrator of digital intellectual property theft, is not part of the TPP.
The deal would have no direct impact on the rampant Chinese hacking that analysts believe make up a large part of the $300 billion-plus American businesses lose each year to intellectual property theft.
It would also not address a vague national security law Beijing approved in June that gives officials authority to ensure certain technologies are "secure and controllable." Foreign companies have argued the law could force them to use government-approved encryption and hand over access to customer data and proprietary source code.
These issues were a major focus of Chinese President Xi Jinping's recent state visit to Washington. The two countries' leaders eventually agreed on a "common understanding" not to conduct or support industrial hacking.
President Obama on Monday argued the agreement will put pressure on China to brings its standards up to match the TPP, which covers 40 percent of the global economy. Several of China's nearby neighbors, such as Japan, Malaysia, Vietnam and Singapore, are signed on to the deal, putting further pressure on the Asian power.
"We can't let countries like China write the rules of the global economy," Obama said. "We should write those rules, opening new markets to American products."
The president has even suggested China is already exploring joining the TPP at some point.
"They've already started putting out feelers about the possibilities of them participating at some point," he told NPR's "Marketplace" in June.
Even absent China joining the TPP, Joshua Meltzer, a Brookings fellow focusing on the global economy, told The Hill the framework's intellectual property-theft guidelines will play a major role in U.S.-China economic discussions.
"That's going to start to be part of the context that's going to define that relationship," he said.
Big tech industry trade groups on Monday mostly stood behind the administration's argument that the TPP will help American companies better protect their data. Many latched onto the data transfer components.
"Importantly, the TPP contains modern cross-border data flow provisions which are critical to today's economy," said the Software and Information Industry Association (SIIA), whose members include Adobe, Facebook and Google.
The Information Technology and Innovation Foundation (ITIF), a think tank that has received funding from technology companies such as Google and IBM, gave the overall deal mixed reviews but lauded "the new protections against trade secret theft."
The measures will "help ensure that future generations of technological innovation continue to drive a robust global economy," said Stephen Ezell, ITIF's vice president for global innovation policy, in a statement.
"Such intellectual property protections are essential for global trade to thrive, for without them, there would be little incentive to take on the risk and expense of innovation," he added.
Tags John Kerry
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Ohio high school musical abruptly halted, deemed 'vulgar' | {
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Tom et Lola (English: Tom and Lola) is a 1990 French drama film directed by Bertrand Arthuys. The film is about two children with damaged immune systems who live in sealed plastic bubbles and try to escape. The film was written by Arthuys, Christian de Chalonge, and Muriel Teodori.
Plot
Tom and Lola are a young boy and girl who are victims of a rare disease which means that they are prevented from any contact with the outside world. They are kept inside plastic bubbles in a Paris hospital. In this aseptic and strictly controlled microcosm run by doctors especially chosen for the task, the children are left with almost no external stimuli, other than an old black and white TV set tuned to educational films about subjects such as whale song, trains, and a show called The Adventures of Tikira in Alaska. Tom's mother is in constant attendance, but her increasingly desperate attempts to beguile him have alienated her almost totally from them, and apart from Hélène, a member of the medical team who has been their nearest thing to a parent, Tom and Lola's only close attachment is to each other.
Eventually, the two children discover that by using the gauntlets that form part of their plastic pods, they are able to reach out and grasp each other's hands. One night, Tom discovers that if he pulls Lola's arms hard enough and she pulls his, they can draw their bubbles close enough for him to reach the external emergency deflate controls on her bubble. After he has activated these, Lola escapes her bubble and releases Tom from his.
Over the next few nights, the children regularly escape their plastic prisons and eventually meet Robert, a boy in another ward of the hospital. He is initially frightened by the two strange children. He thinks Tom and Lola are Martians, possibly due to their nakedness and baldness which is required for hygienic living within the bubbles. However, they quickly make friends, and he shows them other areas of the hospital, including the hydrotherapy pool, where he has a seizure after his medication drip line pulls out. The pair save him, and get him back to his room.
During another visit, Robert tells Tom and Lola that he is originally from Izoard, a small skiing village in the French Alps and shows them plans for escaping the hospital in order to return home. Tom and Lola agree this is a good idea and all three plan to escape the next night. However, when they arrive at his room the following night, Robert has already left and (they later discover) has come to grief while climbing over the hospital wall. Undeterred, Tom and Lola decide to escape anyway, having secured the address of Hélène.
Escaping the hospital, the two find their way across the Seine to a railway shunting yard in search of the train to Izoard, and climb aboard a passenger carriage where they fall asleep. They are discovered and taken to Hélène's home by a railway worker, having found on them a card with the address.
When Tom and Lola arrive at her house, her children are at home, alone as usual, as their mother spends more time with Tom and Lola at her work than with her own kids. A mutual regard and understanding develops between them, and together they contrive to set up a home situation more like the environment at the hospital, in the hope of regaining Hélène's attention. These hopes are dashed when she arrives home: her priority clearly remains the wellbeing of the escapees, and when her oldest son overhears her promise to the runaways that she will shelter them, he is disillusioned and calls the authorities.
The inevitable consequence in these circumstances is that the children will die, and the movie does not pull back from that, but treats the prospect in an allegorical, "magical realist" way. As the medical staff had planned in any case to separate the children without regard for their attachment - something they discovered only in the process of escaping - their pronostic had always been bleak.
Cast
Marc Berman
Cécile Magnet
Neil Stubb
Mélodie Collin
Catherine Frot
Oliver Belmont
Sophie Arthuys
Janine Souchon
Production
The film was directed by Bertrand Arthuys. The film was written by Bertrand Arthuys, Christian de Chalonge, and Muriel Teodori.
François Catonné was the film's cinemaphotographer. Corinne Jorry was the film's costume designer. The film's executive producers were Alain Belmondo and Gérard Crosnier. The film's producers were Jean-François Davy, Francis Veber, and Danielle Vannier.
Other production roles for the film included Laurent Barès as assistant operator, Jeanne Kef as editor, Monica Coleman as assistant editor, Frank Camhi as still photographer, Isabelle Henry as assistant director, Arthur Cloquet as camera operator, François Gédigier as sound editor, and Michèle Abbé as production designer. The film's music was composed by Christophe Arthuys and Jean-Pierre Fouquey. Guillaume Sciama, François Groult, Joël Rangod were the film's sound recordists.
The film was entirely produced in France. The film was produced on 35 mm film and has a run time of 97 minutes.
Themes and content
The film is an unusual science fiction parable from French cinema auteur Bertrand Arthuys that recreates a libertarian perspective for children where false morality has no place. Arthuys has successfully avoided the mire of existentialist melodrama common with children's characters in situations of illness, and in doing so he manages a difficult task: to captivate the audience so that the movie is more like a carefree adventure.
Arthuys does not spare his characters' natural exposure on camera ... the nudity of the two children for roughly half the duration of the film could have led to oblivion for this movie. But it is all intrinsically subservient to his strong desire to get his message across. Neither do their tender physical expressions of mutual affection ever transgress, in degree or nature, what would reasonably be considered age-appropriate .
Lola and Tom do engage in gritty and sharply witty dialogue with the scientific team each morning, but this is largely to relieve the daily tedium.
Another striking motif is the repeated reference to an idealised heaven conceived by the two with the code phrase 'Iceberg-Alaska-Tikira', a whispered mantra as a sign of celebration or connection, accompanied with expressive signing gestures. This is later supplemented or supplanted by "Izoard!" as an aspirational symbol.
Their interaction with the adult world, populated by imbeciles who occasionally play ridiculous roles to entertain the pair from outside the bubbles, is clearly disproportionate, as if the children were a few light years ahead of the 'common sense' practised within the French hospital.
With the moralistic and sometimes hysterical persecution of artistic licence observed since the late 1980s, this film gained only a narrow VHS distribution, with regard to the fiercely competitive category of collector's items. A great example of avant-garde cinema, victimised by the approach.
The aesthetics of scenarios in the laboratory and the excessive use of white, is similar to THX 1138 by George Lucas, a kind of paradigm in the science fiction of the 1970s. This neatly evokes and reinforces the chosen environments on which the two children fixate, whose snowy and frozen purity presumably promises a freedom from the bacterial threats which enforce their captivity in civilisation.
Release
The film was released in France on January 17, 1990.
In Switzerland, the film was released in 1990 and distributed by Sadfi Films. In Japan, the film was released in 1991 and distributed by New Select.
The film was distributed in France by AFMD, with film exports and foreign sales by Gaumont Film Company.
Reception
The film received reviews from French publications, including Le Point, Le Nouvel observateur, Le Figaro Magazine, and L'Événement du jeudi.
References
External links
Libertarian science fiction
French science fiction drama films | {
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Q: Where can I find a red bandana? Over by the disk golf area in Wellworn Grove (north of the swan lake), there is a balding man who asks if you can find him something resembling a red bandana.
I'm feeling a little self-conscious. Think you could... get me a red bandana? Or something that resembles one.
I've searched high and low, but I haven't been able to find one. I did find an employee by the course entrance who was wearing a red bandanna, but talking to him didn't seem to do anything.
Where can I find a red bandanna for the balding man in Wellworn Grove?
A: In Wellworn Grove you will find a woman who wants her son to get out of the water.
The woman is next to a hole, in which you'll find a red flag. That red flag "resembles a red bandana". Normally, if you interact with a hole, you get to remove the flag or insert it. This hole is different in that it doesn't allow you to remove the flag.
In order to retrieve the flag, you need to whack it with a golf ball. One way to do it, is to press Y to place a ball on the ground, then hit the ball with any of your golf clubs. Throwing the ball (press X) towards the flagpole also works.
If done correctly, the flag will slide down the pole. You can then interact with it to retrieve it. All that's left to do is talk with the self-conscious guy to hand over the "bandana" to him.
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{"url":"https:\/\/physics.meta.stackexchange.com\/questions\/9752\/bounty-going-to-waste-what-should-i-do?noredirect=1","text":"# Bounty going to waste - what should I do?\n\nI recently put a bounty on a question, but still no one is answering it. When the bounty period ends, will the bounty just go to waste? If no one is answering I would prefer to just answer the question myself and get my 50 rep back, but I don't know if this is possible or allowed. Any advice would be appreciated.\n\nHere is the link to the question:\n\nAre there exact analytical solutions to the electronic states of the hydrogen molecular ion $\\mathrm H_2^+$?\n\n\u2022 There is no way to recover the rep. Placing a bounty should be understood as buying exposure for your questions. It gets bumped. I goes in the featured tab. There are people who actively hunt bounties so they will definitely look at it. But there are no guarantees that you will get anything worthwhile. I think I'm personally running a little behind 50\/50 on getting what I wanted out of bounties network wide, so caveat emptor. Mar 30 '17 at 1:45\n\u2022 Related on Meta Stack Exchange: bounty expired with no answers\u2026 wasted reputation?, along with multiple duplicates. Mar 30 '17 at 12:51\n\u2022 I appreciate your bounty on that question, but to be frank I think that given the question's age and score it is relatively unlikely that it will receive an answer (or, at least, the barrier to it getting answered is higher than average). Adding a bounty buys exposure for the question, but beyond that there's nothing to do. If it makes you feel better, you're not alone. Mar 30 '17 at 13:02","date":"2022-01-26 20:20:52","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\": 0.4126805365085602, \"perplexity\": 612.6670663818442}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-05\/segments\/1642320304961.89\/warc\/CC-MAIN-20220126192506-20220126222506-00417.warc.gz\"}"} | null | null |
Hemistola claripennis är en fjärilsart som beskrevs av Butler 1878. Hemistola claripennis ingår i släktet Hemistola och familjen mätare. Inga underarter finns listade i Catalogue of Life.
Källor
Mätare
claripennis | {
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Home News Those urging me to step down for Akpabio enemies of A'Ibom —...
Those urging me to step down for Akpabio enemies of A'Ibom — Ekpoudom
Retired Deputy Inspector General of Police and senatorial candidate for the All Progressives Congress, APC, Akwa Ibom North-West, Udom Ekpoudom has described those asking him to step down for Senator Godswill Akpabio as enemies of Abak federal constituency, popularly called Abak5.
Ekpoudom made the assertion on Friday, in his office at Ediene Abak, in Abak local government area, while reacting to speculations in the state that some elders of his party, the APC have prevailed on him to step down for Akpabio.
He vowed that he would never succumb to the pressure to opt out of the race, as it would amount to betraying his federal constituency which remained the only one in Ikot Ekpene senatorial district (Akwa Ibom North West) to produce a Senator since the creation of the state.
His words: "Some people have been appealing to me to step down, but I've been saying no. And will continue to say no because it is not my project precisely, but the people's project.
"I don't want to be chased away from Abak federal constituency, I don't want to betray my people. You can imagine for 16 years now, this part of Akwa Ibom, (Abak federal constituency) has not produced a Senator.
"It is only Ikot Ekpene federal constituency which has produced five senators and somebody wants to come in again from there. it is not possible. Abak-5 is ready, so I look at those coming to tell me to step down as enemies of the Abak federal constituency.
" Even if they want to offer me the Head of state, I won't accept. All I want is to get what belongs to us. If Abak5 loses now, it would affect the whole federal constituency and senatorial district"
The APC senatorial candidate described the rumours making the rounds in some quarters that he has stepped down for Akpabio as laughable.
Ekpoudom who is the proprietor of Tower of Ivory Schools said Godswill Akpabio has made himself an enemy of Abak 5 people by his refusal to take the advice of key stakeholders and elders of the party including, Chief Don Etiebet, not to contest for the seat in 2023 for justice, and fairness.
"He (Akpabio) has the right to do whatever he wants to do. He has turned himself into the enemy of the Abak5 people and that's not the best. Don Etiebet is somebody that we respect in Akwa Ibom, and if somebody like that talked to him and he didn't hear who will talk to him and he will hear?", He asserted.
He however assured that addressing the issue of the high level of poverty among the people would be one of the first things he would do if elected into office,
The Retired DIG, frowned at the way politicians are creating insecurity, saying," How politicians empower the youth is by giving them arms and ammunition, by giving them money to take hard drugs. I can not do or encourage such things. I will make sure the youths are gainfully employed"
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Il palazzo Fruscione si trova nella parte più antica del centro storico di Salerno, vicino all'antica via dei Canali della Salerno romana.
Il palazzo, iniziato nel XIII secolo, poggia parzialmente sui resti di un complesso termale di epoca imperiale e si trova nei pressi dell'antica corte arechiana. Suo proprietario fu probabilmente il medico salernitano Giovanni da Procida. La sua ubicazione in passato ha fatto ipotizzare l'erronea identificazione con la residenza del duca longobardo Arechi II. Tale tesi è risultata priva di fondamento in quanto il Chronicon Salernitanum situa la cappella palatina di San Pietro a Corte a nord della reggia di Arechi: il palazzo Fruscione, invece, si trova a sua volta a nord della suddetta chiesa.
La storia
Il palazzo fu costruito nel corso del XIII secolo. Alcuni studiosi lo ritennero la reggia arechiana, altri un semplice edificio di pregio di età medioevale e altri ancora un rifacimento di epoca sveva del palazzo di Arechi II. Al 1738 risale un documento notarile che descrive l'edificio, allorquando fu ceduto in enfiteusi a Bartolomeo Longo dal Capitolo della Cattedrale, al quale l'aveva lasciato in eredità in canonico Francesco Inglese. Dal documento si evince che palazzo Fruscione non era sede di civili abitazioni ma una "casa di alloggiamento" e da ciò si spiegano le numerose stalle situate al piano terra. Inoltre a questo periodo risale l'ampliamento della struttura verso occidente dove precedentemente sorgeva anche l'ingresso principale nel largo Sant'Antonio Abate ossia di Vienna. L'ingresso viene descritto come "un portone di fabbrica e legname con coda di pavone sopra con loggetta scoverta" probabilmente in riferimento ad un motivo decorativo del portone. Lo spostamento dell'ingresso sul lato meridionale probabilmente risale al XIX secolo inoltrato con la presenza di un portone centrale e due coppie di magazzini ai lati. Proprio le aperture alla sinistra del portone costituiscono la parte aggiunta in quanto, prima del portone, è posta una colonna angolare che nel Medioevo scandiva i limiti dell'edificio. Dal documento notarile si evince che sul prospetto orientale il palazzo comunicava con il palazzo di fronte, chiamato "La Figurella" e sede dell'Ospedale dei Pellegrini. Un primo restauro alla struttura vi fu negli anni cinquanta ad opera di Giorgio Rosi che introdusse alcuni elementi mancanti (come le colonne decorative del secondo livello). Nel 1967 il palazzo fu espropriato alla famiglia Fruscione (da cui prende il nome) e solamente nel 2009 sono iniziati i lavori di restauro, per un importo di 5 milioni di €, sotto la direzione dell'architetto Mario dell'Acqua e con la collaborazione, per quanto riguarda la parte archeologica e storica, del professore Paolo Peduto. Nel corso degli ultimi lavori è stato eliminato il quarto livello aggiunto nel XIX secolo al fine di consolidare l'interno.
Il palazzo
Palazzo Fruscione conserva i resti di un edificio normanno costituito da almeno due piani e due corpi di fabbrica con livelli differenti. Nel XIII secolo fu interessato da lavori di ristrutturazione insieme al riassetto della strada e probabilmente a quel periodo risalgono i tre portali sul vicolo dei Barbuti. Gli ambienti su vicolo Adelberga furono realizzati verso la fine del XIII secolo mentre il secondo piano risulta con un unico intervento concluso verso l'inizio del XIV secolo. Il pian terreno nel XVII secolo fu adibito a stalle e con questa riconversione si perse l'antica scalinata d'accesso ai piani superiori a coda di pavone. Nel XIX secolo si tentò di recuperare la bellezza di questi ambienti impreziosendoli con decori come due stipiti di marmo decorati con temi vegetali.
L'edificio conserva tre ordini caratterizzati da pregevoli elementi architettonici. La facciata orientale presenta al pian terreno tre portali sormontati da archi a tutto sesto con tarsie in tufo grigio e giallo, al primo piano cinque bifore scolpite, di cui una conserva una decorazione dipinta in rosso raffigurante degli anelli intrecciati, e al secondo piano una serie di polifore intrecciate caratterizzate da archi acuti sorretti da piccole colonne.
Epoca normanna
I paramenti murari più antichi sono quelli che persistono su l'attuale via Canale. Su questa facciata, nel corso del restauro degli anni cinquanta, fu rinvenuta una tarsia caratterizzata da una decorazione a rombi in tufi gialli e grigi e, con la successiva asportazione di diversi strati d'intonaco sono venute alla luce due monofore realizzate in tufo e laterizio.
Sempre sul versante occidentale, al pian terreno, recentemente è emersa una decorazione simile non ricollegabile però a quella del piano sovrastante in quanto si trova in un corpo più arretrato verso est. In questo caso si tratta di un portale con arco a tutto sesto in conci bicromi sovrastanti un arco minore decorato con rombi di cui le parti più scure risultano di tufo grigio mentre quelli chiari sono intonacati. Gli stipiti della muratura sono costituiti da blocchi di arenaria che si ritrovano anche in altre aperture.
Nella parte più alta del paramento è presente una decorazione costituita da blocchi di tufo incorniciati da laterizi. Questa decorazione è da considerarsi una fascia marcapiano oppure un parapetto a cui si legavano le finestre sempre realizzate con tufo e laterizio. Per quanto riguarda le tarsie bicrome sono paragonabili alla Cattedrale di Salerno e al vicino Castel Terracena il che data la fase più antica di palazzo Fruscione al XII secolo. Al secondo piano, nella parte sud, è possibile confrontare due corpi di fabbrica diversi, uno in pietra più chiara ed uno dove si trovano le monofore normanne, cosa non possibile invece al primo piano a causa della realizzazione di ambienti moderni e di un barbacane antisismico. Sul corpo di fabbrica più recente sono riscontrabili due tipi di decorazioni architettoniche: da una parte sono presenti tracce di un sistema di archi a tutto sesto intrecciati che formano nicchie archiacute in stucco, dall'altra pochi resti nell'intonaco di una decorazione a tema vegetale con foglie e racemi. Sul fronte occidentale sussistevano due decorazioni architettoniche rimaste però inalterate e mai unificate allo stile del secondo livello, probabilmente perché questo prospetto perse importanza con la presenza del corpo di fabbrica inferiore. All'interno la finestra in tufo e laterizio fa riferimento ad un piano di calpestio più alto rispetto a quello attuale di circa 140 cm. Alla fase normanna risale anche una colonna con capitello di reimpiego caratterizzato da semplici foglie e paragonabile a quelli delle cattedrale di Aversa.
Epoca angioina
Gli elementi architettonici decorati a tarsie policrome su vicolo Barbuti sono i simboli più evidenti della fase medioevale. Grazie alle prime fasi di scavo si è riuscito a capire che gli ambienti meridionali risalgono alla prima metà del XIII secolo e, con l'asportazione dell'intonaco, si è rivelato che gli spazi interni erano scanditi da pareti con archi dalla forma acuta che mostrano laterizi alternati da conci di arenaria. Gli scavi hanno rivelato anche vari impianti di frequentazione medioevali fino a circa 80 cm di profondità. La fossa di fondazione aveva tagliato diversi piani pavimentali tra cui il più recente, databile intorno al XIII secolo per il ritrovamento di due monete sveve, dimostrano che i muri oggi visibili furono eretti dopo questo periodo. Sul prospetto su vicolo Barbuti sono presenti tre portali di cui quello centrale, essendo più grande e presentando le decorazioni più articolate, era quello principale per cui, il piccolo locale retrostante, risulta come raccordo o parte di un ambiente ridimensionato. La terza porta con tarsie aperta più a sud, è stata indagata dall'interno con un saggio di scavo ed il suo calpestio originario è stato rinvenuto a quota 8.80 m. Ciò conferma che le tre porte intarsiate funzionarono insieme e sono quindi contemporanee. Lo scavo ha inoltre portato alla luce due cisterne di cui, la più antica, ricoperta di materiale della fine del XIII secolo mentre la seconda coeva al pavimento a quota 8.80 m.
Fasi intermedie
Gli scavi
Durante i lavori di restauro degli anni '10 del XXI secolo sono state rinvenute tracce di muratura che rinviano ad un complesso termale d'epoca imperiale, dei mosaici e degli affreschi del II secolo. L'ambiente con il mosaico, le cui pareti sono rivestite da decorazioni in rilievo di stucco e dipinti, apparteneva alle terme di I-II secolo d.C., individuate nel sedime del palazzo arechiano posto a sud di palazzo Fruscione. Tracce di ben tre successivi restauri del mosaico fanno comprendere che le terme furono frequentate per un tempo piuttosto lungo. La lunga durata della vita delle terme è testimoniata oltre che dagli interventi sul mosaico, anche dalle tracce di restauri strutturali realizzati in età romana, come si evince da successivi ispessimenti in laterizio delle murature. L'indagine archeologica ha evidenziato che in seguito all'abbandono delle terme, a metà del V secolo, l'area fu utilizzata come cantiere edilizio: il rinvenimento dei resti di officine artigianali per la realizzazione di manufatti vitrei e metallici induce a ritenere che le officine servissero le costruzioni della vicina cappella funeraria del vir spectabilis Socrates, oggi sottoposta alla cappella palatina del dux longobardo Arechi II. In seguito, tra il VI e il VII secolo, l'area indagata, posta nei pressi della chiesa cimiteriale paleocristiana, ospitò alcune sepolture rinvenute, disturbate dagli interventi costruttivi dei secoli successivi. Nell'VIII secolo l'uso cimiteriale dei luoghi si interruppe e lo spazio oggi occupato dal palazzo entrò sicuramente a far parte del sedime della corte longobarda. Nei vani meridionali del palazzo sono state messe in evidenza alcune strutture del IX-X secolo, tra queste un imponente muro con andamento nord-sud il quale è stato individuato in diversi ambienti. Esso era fondato su strati alluvionali tardo antichi e potrebbe rappresentate il limite della Curtis longobarda arechiana.
L'esplorazione di un ambiente rettangolare, ubicato nella zona prospiciente vicolo Adelberga, ha permesso di indagare un butto dell'XI secolo con numerosi resti di pasto, ossi animali, tra cui un equino, numerosi resti di capri-ovini e suini, ceramica da mensa e da dispensa, semi e legumi. Tra le ceramiche si distinguono alcune parti di un recipiente a vetrina "pesante", un tipo di rivestimento vetroso steso sul corpo ceramico prima della cottura. Tali contenitori per il loro costo elevato non sono frequenti e indicano, tra l'altro, l'appartenenza al ceto elitario degli abitanti dell'area. Negli ambienti a sud sono state messe in luce alcune strutture appartenenti all'epoca sveva. in particolare sono stati riconosciuti il piano di calpestio e la porta meridionale in uso nell'edificio nella prima metà del XIII secolo. Il piano pavimentale copriva un battuto più antico coevo a una cisterna utilizzata come butto. È dunque probabile che in quest'epoca l'area fu sottoposta a una riorganizzazione generale, probabilmente per aggregare parti diverse, creando una nuova spazialità che inglobò i piccoli ambienti antistanti la strada. Lo scavo praticato all'interno dell'ambiente sud ovest ha mostrato le fossa di fondazione dei muri perimetrali che furono praticate in un battuto pavimentale di malta in cui erano alloggiate due monete dell'età di Manfredi confermando la cronologia angioina della costruzione dei vani oggi visibili e dimostrando che i piccoli ambienti trapezoidali a fronte strada sono coevi al muro meridionale del salone centrale.
Eventi ospitati
Ad oggi la struttura, non essendo stata stabilita la funzione definitiva, ha ospitato diverse manifestazioni, mostre ed eventi:
Spot School Award – Mediterranean Creativity Festival, 22-24 maggio 2014
La città del benessere, 26-28 settembre 2014
Biennale d'arte contemporanea, dall'11 ottobre al 6 gennaio 2014
Spot School Award – Mediterranean Creativity Festival, 28-30 maggio 2015
Mulieres - personale di Milo Manara (Napoli Comicon), dal 3 febbraio al 1º marzo 2015
VinArte, 9-17 maggio 2015
Salerno in Fantasy, 3-5 luglio 2015
Iconica, dal 21 novembre al 10 gennaio 2016
Face to Face, dal 22 marzo al 1º maggio 2016
Dino Pedriali (Pier Paolo Pasolini – Nostos : Il ritorno) 1975 – 1999, dal 26 febbraio al 16 marzo 2016
Arte Salerno, 2-5 giugno 2016
Arte Salerno 2017, 4-11 Giugno 2017
L'incredibile Marvel (Napoli Comicon), dal 1º luglio al 7 agosto 2016
Note
Bibliografia
De Renzi S., Nota sugli avanzi del palazzo di Arechi di Salerno, in Storia documentata della Scuola Medica di Salerno, 1857.
De Angelis M., La reggia salernitana del longobardo Arechi, in Archivio Storico per la Provincia di Salerno, 1933.
Rosi G., La reggia normanna di Salerno, in Bollettino di Storia d'Arte del Ministero della P.P., 1950.
Guarino D., Palazzo Fruscione, un monumento architettonico del centro storico di Salerno, dalla lettura alla conservazione in "Apollo", 1997
Dell'Acqua M., Le facciate di Palazzo Fruscione. Note allegate alla relazione del progetto di restauro, 2009.
Fiorillo R. e Lambert C., Medioevo letto, scavato, rivalutato. Studio in onore di Paolo Peduto, 2012.
Mario Dell'Acqua, Palazzo Fruscione. Appunti dal cantiere di restauro ., Plectica editrice s.a.s. Salerno 2014
Voci correlate
Centro storico di Salerno
Architettura normanna
Salerno
Altri progetti
Collegamenti esterni
Fruscione
architetture normanne della Campania | {
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Founder / Pastor of the "House Of Wisdom Ministries".
devoted followers of Jesus Christ."
The House of Wisdom Ministries is asking for your prayers and financial support. Click below to make an online donation.
Welcome into the family of the HOUSE OF WISDOM MINISTRIES where the Spirit of the Lord lead us. This is a Spirit-filled and Spirit-led church. We are endeavoring to carry out the commandment given to the church by our Lord and Savior, Jesus Christ. His words are "Go into all the world and preach the gospel to every creature. He that believeth and is baptized shall be saved, he that believeth not shall be damned.
The HOUSE OF WISDOM is a ministry called and founded on biblical principles to lead the people of GOD to walk in the fullness of HIM, to serve in the spirit of excellence with integrity and love through the Word and Wisdom of the Holy Spirit. We are laborers together with Christ, and Getting Involved To Make A Difference, and other supported partners trying to "help individuals capture the concept of Romans 12:1-2,..."Do not conform to the pattern of this world, but be transformed by the renewing of your mind. Then you will be able to test and approve what God's will is—his good, pleasing and perfect will. The word says we must grow in grace and in the knowledge of our Lord and Savior, Jesus Christ.
He hath anointed us to preach the gospel to the poor; he hath sent us to heal the brokenhearted, to preach deliverance to the captives, and recovering the sight to the blind, to set at liberty them that are bruised. To preach the acceptable year of the LORD.
The House Of Wisdom Ministries is celebrating it's 10th Anniversary. Come join us Wednesday night March 6th, 2019 at 7PM.
The House Of Wisdom Ministries is celebrating it's 10th Anniversary. Come join us Sunday March 10th, 2019 at 12:30PM and 4PM.
HOW Ministries is asking for your support as we continue to build victorious lives through Jesus the Christ. For tickets, please contact Pastor Upendo Parker or call (773)789-9469 for more information. Until then, have wonderful & bless days to come!
The HOUSE OF WISDOM is a ministry called and founded on biblical principles to lead the people of GOD to walk in the fullness of HIM, to serve in the spirit of excellence with integrity and love through the Word and Wisdom of the Holy Spirit. | {
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Q: Timeout performing SET {Key}, inst: 0, mgr: Inactive, queue: 2, qu=1, qs=1, qc=0, wr=1/1, in=0/0 I am trying to save a 90 KB pdf file into Azure Redis Cache using StackExchange.Redis client. I have converted that file into byte array and tried to save it using stringSet method and received error.
Code:
byte[] bytes = File.ReadAllBytes("ABC.pdf");
cache.StringSet(info.Name, bytes); --> This Line throws exception "Timeout performing SET {Key}, inst: 0, mgr: Inactive, queue: 2, qu=1, qs=1, qc=0, wr=1/1, in=0/0".
Kindly Help.
A: Timeout performing SET {Key}, inst: 0, mgr: Inactive, queue: 2, qu=1, qs=1, qc=0, wr=1/1, in=0/0
means, it has sent one request (qs), there is another request that's in unsent queue (qu), while there is nothing to be read from the network. there is an active writer meaning the one unsent is not being ignored. Basically, there is a request sent and waiting for the response to be back.
Few questions:
1. Is your client running in the same region as the cache? Running it from your dev box would introduce additional latency and cause timeouts.
2. How often do you get the exception? Does it succeed any time?
3. You can also contact azurecache@microsoft.com with your cache name, time (with time zone) range in which you see the timeouts and if possible a console app that would help to repro the issue.
Hope this helps,
Deepak
details about the error codes from this thread: #83
inst: in the last time slice: 0 commands have been issued
mgr: the socket manager is performing "socket.select", which means it is asking the OS to indicate a socket that has something to do; basically: the reader is not actively reading from the network because it doesn't think there is anything to do
queue: there are 73 total in-progress operations
qu: 6 of those are in unsent queue: they have not yet been written to the outbound network
qs: 67 of those have been sent and are awaiting responses from the server
qc: 0 of those have seen replies but have not yet been marked as complete due to waiting on the completion loop
wr: there is an active writer (meaning - those 6 unsent are not being ignored)
in: there are no active readers and zero bytes are available to be read on the NIC
| {
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select substring(3 from 3) from db_root order by 1; | {
"redpajama_set_name": "RedPajamaGithub"
} | 668 |
Q: Which event is triggered when v model value is changing I'm new to Vuejs and my requirement is to write a single functional globally to trigger whenever v-model value of my form elements are set. I tried to write this for element.onchnage but this is not working.
Can some one tell me which HTML event is triggered when the v-model value is set in vuejs ?
A: Hey Linu and welcome to SO.
Check out the docs for Form input bindings:
https://v2.vuejs.org/v2/guide/forms.html
There it says:
v-model internally uses different properties and emits different
events for different input elements:
*
*text and textarea elements use value property and input event
*checkboxes and radiobuttons use checked property and change event;
*select fields use value as a prop and change as an event.
So instead of v-model you can do the following for inputs
<input :value="dataProperty" @input="dataProperty = $event.target.value" />
| {
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Q: How to install a perl module without a dependency I've got a perl module (Email::AutoReply) which has BerkeleyDB listed in the README as an optional dependency. When you try to install it with cpan or cpanminus, it tries to install BerkeleyDB as well. The problem is that no matter what I do, BerkeleyDB fails to install with a long list of errors, even though I meet all the requirements. This is preventing me from installing Email::AutoReply, even though it's supposed to be an optional thing.
So, how do I install a module without one of it's dependencies? Can I remove the dependency somehow? Is there some command line option for this?
A: After installing the other dependencies (which you probably have done from your earlier attempts),
cd /tmp
wget http://search.cpan.org/CPAN/authors/id/A/AM/AMONSEN/Email-AutoReply-1.04.tar.gz
tar xvzf Email-AutoReply-1.04.tar.gz
cd Email-AutoReply
perl Makefile.PL && make test && make install
cd ..
rm -rf Email-AutoReply
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 965 |
Q: ANT does not link JNI files Compiled .apk is starting but has not translated & compiled neither java code nor cpp code. Instead of this ANT builder makes new java file and compilation goes with that template file. I didn't find any sufficient manuals to console/terminal building !without! IDE & Gradle (especially without Google's IDE).
Project structure
Is:
/* / */
/* cpp */
jni
com_example_jnihelloworld_jniactivity.h
jniactivity.h
jniactivity.cpp
jniimplement.cpp
Application.mk
Android.mk
libs
obj
/* java */
src
com
example
jnihelloworld
jniactivity.java
bin
com
example
jnihelloworld
jniactivity.class
pcmp
ndk-build
android create project --target android-24 --name HelloJni --path .\pcmp --activity jniact --package com.example.jnihelloworld
ant debug -f .\pcmp
Sources
Java file:
package com.example.jnihelloworld;
import android.os.Bundle;
import android.app.Activity;
import android.widget.TextView;
public class jniactivity extends Activity {
static {
System.loadLibrary("myjni");
}
public native String getMessage();
@Override
public void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
TextView textView = new TextView(this);
textView.setText(getMessage());
setContentView(textView);
}
}
C++ file:
#include "com_example_jnihelloworld_jniactivity.h"
#include "jniactivity.h"
JNIEXPORT jstring JNICALL Java_com_example_jnihelloworld_jniactivity_getMessage (JNIEnv *pEnv, jobject pThis) {
std::cout << "One" << std::endl;
return pEnv->NewStringUTF("Hello from native code!");
}
ANT script file (automatic generated):
<?xml version="1.0" encoding="UTF-8"?>
<project name="HelloJni" default="help">
<!-- The local.properties file is created and updated by the 'android' tool.
It contains the path to the SDK. It should *NOT* be checked into
Version Control Systems. -->
<property file="local.properties" />
<!-- The ant.properties file can be created by you. It is only edited by the
'android' tool to add properties to it.
This is the place to change some Ant specific build properties.
Here are some properties you may want to change/update:
source.dir
The name of the source directory. Default is 'src'.
out.dir
The name of the output directory. Default is 'bin'.
For other overridable properties, look at the beginning of the rules
files in the SDK, at tools/ant/build.xml
Properties related to the SDK location or the project target should
be updated using the 'android' tool with the 'update' action.
This file is an integral part of the build system for your
application and should be checked into Version Control Systems.
-->
<property file="ant.properties" />
<!-- if sdk.dir was not set from one of the property file, then
get it from the ANDROID_HOME env var.
This must be done before we load project.properties since
the proguard config can use sdk.dir -->
<property environment="env" />
<condition property="sdk.dir" value="${env.ANDROID_HOME}">
<isset property="env.ANDROID_HOME" />
</condition>
<!-- The project.properties file is created and updated by the 'android'
tool, as well as ADT.
This contains project specific properties such as project target, and library
dependencies. Lower level build properties are stored in ant.properties
(or in .classpath for Eclipse projects).
This file is an integral part of the build system for your
application and should be checked into Version Control Systems. -->
<loadproperties srcFile="project.properties" />
<!-- quick check on sdk.dir -->
<fail
message="sdk.dir is missing. Make sure to generate local.properties using 'android update project' or to inject it through the ANDROID_HOME environment variable."
unless="sdk.dir"
/>
<!--
Import per project custom build rules if present at the root of the project.
This is the place to put custom intermediary targets such as:
-pre-build
-pre-compile
-post-compile (This is typically used for code obfuscation.
Compiled code location: ${out.classes.absolute.dir}
If this is not done in place, override ${out.dex.input.absolute.dir})
-post-package
-post-build
-pre-clean
-->
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- customize it to your needs.
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into this file, replacing the <import> task.
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***********************
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</project>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,161 |
\subsection{Automatic Speech Recognition} \label{sec:asr}
The ASR model is transformer-like and trained on paired speech and transcript data
\paragraph{Datasets and Preprocessing}
We divide the allowed ASR datasets into two parts: clean and noisy and consider MuST-C\footnote{In this paper, MuST-C denotes the newly released English-German ST dataset~(v2) by IWSLT 2021.}, LibriSpeech~\cite{Panayotov2015}, and Mozilla Common Voice as the clean datasets, and use them for training an ASR system to filter the noisy part, i.e., \textit{iwslt-corpus}\footnote{The training corpus for IWSLT evaluation campaign over the last years.} and TED-LIUM 3~\cite{Hernandez2018}. We remove the training samples where the word error rate~(WER) score between the ASR output and English transcript exceeds 75\%. The statistics of the ASR datasets are shown in Table~\ref{tb:asr_statistics}.
\begin{table}[!t]\smal
\centering
\begin{tabular}{lcc}
\toprule
\textbf{Dataset} & \textbf{\#samples} & \textbf{\#hours} \\
\midrule
MuST-C & 250,942 & 450 \\
LibriSpeech & 281,241 & 961 \\
Common Voice & 562,517 & 899 \\
\textit{iwslt-corpus} & 157,909 & 231 \\
TED-LIUM 3 & 111,600 & 165 \\
\bottomrule
\end{tabular}
\caption{The statistics of audio datasets to train the ASR model. The \textit{iwslt-corpus} and TED-LIUM 3 are filtered by an ASR model trained on MuST-C, LibriSpeech and Common Voice.}
\label{tb:asr_statistics}
\end{table}
For model training, we extract 80-channel log Mel-filterbank coefficients with windows of 25ms and steps of 10ms on the audio input.
The transcripts are lowercased and we remove all punctuation marks. Then, we apply Moses tokenizer\footnote{\url{https://github.com/moses-smt/mosesdecoder/blob/master/scripts/tokenizer/tokenizer.perl}} and byte pair encoding (BPE)~\cite{sennrich2016}\footnote{\url{https://github.com/rsennrich/subword-nmt}} to the transcripts with 8,000 merge operations.
\begin{figure}[t]
\centering
\includegraphics[angle=0,width=0.48\textwidth]{figures/speech_transformer.pdf}
\caption{Overview of the cascaded speech translation model.}
\label{fig:cascade_model}
\end{figure}
\paragraph{End-to-End ASR Model}
We refer to the recent progress of transformer-based ASR~\cite{Dong2018,Karita2019} and implement the speech transformer model, as illustrated in Figure~\ref{fig:cascade_model} a).
The feature extractor consists of two-layer CNN with 256 channels, $3\times 3$ kernel, and stride size 2, each of which is followed by a layer normalization and ReLU activation.
The major architecture is the same as the transformer model, including 12 layers for the encoder and 6 layers for the decoder. The model width is 768, and the hidden size of the feed-forward layer is 3,072. The attention head is set to 12 for both self-attention and cross-attention.
To train the model, we use Adam optimizer~\cite{kingma2014adam} and set the warmup steps to 25,000. Empirically, we scale up the learning rate by 5.0 to accelerate the convergence. The ASR model is trained on 8 NVIDIA Tesla V100 GPUs with 320,000 frames per batch.
And we truncate the audio frames to 3,000 and remove training samples whose transcript length exceeds 120 for GPU memory efficiency.
To further improve the performance, we apply SpecAugment technique~\cite{Daniel2019} with frequency masking ($mF=2, F=27$) and time masking ($mT=2, T=70, p=0.2$).
\subsection{Neural Machine Translation} \label{sec:cas_nmt}
All MT models are based on transformer \cite{vaswani2017}. We employ data augmentation and model ensemble techniques to improve the final performance.
\paragraph{Datasets and Preprocessing}
We utilize English-German~(EN-DE) parallel sentences from WMT 2020\footnote{\url{http://www.statmt.org/wmt20/translation-task.html}, including Common Crawl, Europarl v10, News Commentary v15, and ParaCrawl v5.1 }, OpenSubtitles 2018\footnote{\url{https://opus.nlpl.eu/OpenSubtitles2018.php}}, MuST-C and \textit{iwslt-corpus} for training.
We filter the parallel corpora following the rules listed in \citet{li-etal-2019-niutrans}, with a much stricter constrain on word alignment.
Additionally, we randomly select 10\% sentences separately from both sides of the original WMT and OpenSubtitles corpus for data augmentation~(see below), along with the transcripts in ASR datasets described in sec~\ref{sec:asr}.
As for text preprocessing, we apply Moses tokenizer and BPE with 32,000 merge operations on each side.
\paragraph{Tagged Back-Translation}
Back-translation \cite{sennrich2016bt} is an effective way to improve the translation quality by leveraging a large amount of monolingual data and has been widely used in WMT evaluation campaigns.
In our setting, we add a ``$<$BT$>$" tag to the source side of back-translated data to prevent overfitting on the synthetic data, which is also known as tagged back-translation~\cite{Caswell2019,Marie2020}.
\paragraph{Knowledge Distillation}
Sequence-level knowledge distillation \cite{Kim2016,Freitag2017} is another useful technique to improve performance. In this way, we enlarge the training data by translating English sentences to German using a good teacher model.
\paragraph{ASR Output Adaptation}
Traditionally, the output of ASR systems is lowercased with no punctuation marks, while the MT systems receive natural texts.
In our system, we attempt to make the MT systems robust to these irregular texts.
A simple way to do so is to apply the same rules on the source side of the MT training set. However, empirical study shows it causes performance degradation.
Inspired by the tagged back-translation method, we enhance the regular MT models with transcripts from both ASR systems and the ASR datasets, as illustrated in Figure~\ref{fig:cascade_model} b). An extra tag ``$<$ASR$>$" indicates the irregular input.
Note that the basic idea to bridge the gap between the ASR output and the MT input involves additional sub-systems, like case and punctuation restoration. In our cascade system, we prefer to use fewer sub-systems, and the detailed comparison would be our future work.
\begin{table*}[!t]\smal
\centering
\begin{tabular}{lccccccc}
\toprule
\multirow{2}{*}{\textbf{Dataset}} & \multirow{2}{*}{\textbf{Size}} & \multirow{2}{*}{\textbf{MT\#1}} & \multicolumn{2}{c}{\textbf{MT\#2}} & \multirow{2}{*}{\textbf{MT\#3}} & \multirow{2}{*}{\textbf{MT\#4}} & \multirow{2}{*}{\textbf{MT\#5}} \\
\cline{4-5}
& & & \textbf{pretrain} & \textbf{fine-tune} & & & \\
\midrule
WMT 2020 & 13.7M & P & P & / & P & / & P \\
OpenSubtitles 2018 & 10.7M & P & P & / & P & P & / \\
MuST-C & 0.25M & P & P/BT/SR & P/BT/SR & P/SR/KD & P/BT/SR & P/BT/SR \\
\textit{iwslt-corpus} & 0.16M & / & P/BT/SR & P/BT/SR & P/SR/KD & P/SR & P/BT/SR \\
TED-LIUM 3 (EN) & 0.11M & / & / & / & KD & / & / \\
Common Voice (EN) & 0.56M & / & / & / & KD & / & / \\
extra monolingual (EN/DE) & 6.77M & / & / & BT & KD & BT & BT \\
\bottomrule
\end{tabular}
\caption{The statistics of MT datasets after data filtering and the detailed combination modes of datasets for difference MT models~(MT\#1-5). The MT\#1 setting is used for training both DE$\rightarrow$EN and EN$\rightarrow$DE directions. ``P" denotes the parallel corpus. ``BT" is the back-translated data using MT\#1 (DE$\rightarrow$EN). ``SR" indicates the irregular data from both ASR datasets and the ASR model. ``KD" is the synthetic data generated by MT\#2.}
\label{tb:mt_training_sets}
\end{table*}
\paragraph{Data Combination and Sampling Strategy}
We train transformer models with different combinations of data sets because increasing the model's diversity can benefit the model ensemble. The detailed setups are listed in Table~\ref{tb:mt_training_sets}. We over-sample the in-domain datasets (i.e., MuST-C/\textit{iwslt-corpus}-related portions) to improve the in-domain performance. Specifically, to control the ratio of samples from different data sources, we sample a fixed number of sentences being proportional to $(\frac{N_s}{\sum_s{N_s}})^{\frac{1}{T}}$, where $N_s$ is the number of sentences from data source $s$, and sampling temperature $T$ is set to 5.
Note that the MT\#1 is trained on lowercased source texts without punctuation marks, while MT\#2-5 use the tagged transcripts.
\paragraph{Model Setups}
We follow the transformer big setting, except that
\begin{compactitem}
\item we deepen the encoder layers to 16.
\item the dropout rate is set 0.15.
\item the model width is changed to 768, the hidden size of the feed-forward layer is 3,072, and the attention head is 12 for MT\#5 only.
\end{compactitem}
We use Adam optimizer with the same schedule algorithm as \citet{vaswani2017}. All models are trained with a global batch size of 65,536.
\subsection{Inference}
We average the latest 10 checkpoints of a single training process for all the above experiments. And during inference, the ``$<$ASR$>$" tag is added to the front of the ASR output. The beamwidth is set to 10 for both ASR and MT tasks.
\subsection{Training Data}
\begin{table}[!t]\smal
\centering
\begin{tabular}{lrr}
\toprule
\textbf{Dataset} & \textbf{\#samples} & \textbf{\#hours} \\
\midrule
MuST-C & 1,198,056 & 2,186 \\
\textit{iwslt-corpus} & 746,714 & 1,112 \\
LibriSpeech & 1,117,394 & 3,833 \\
Common Voice & 2,212,581 & 3,546 \\
TED-LIUM 3 & 384,389 & 577 \\
\bottomrule
\end{tabular}
\caption{The size of audio datasets with data augmentation to train the end-to-end ST model. }
\label{tb:e2e_statistics}
\end{table}
The end-to-end model is trained on paired speech and translation data. We collect MuST-C and \textit{iwslt-corpus}~(after filtering described in section~\ref{sec:cascade}), with a total of only 681 hours transcribed and translated speech. To address the data scarcity problem, we explore the knowledge distillation technique to augment the data by leveraging ASR datasets and MT models, also known as pseudo labeling. In detail, we distill from four MT models: MT\#1, MT\#2, an ensemble of MT\#3-5, and MT\#3-R2L which is trained with the same setting as MT\#3 and generates the target translations in the right to left fashion. We filter the augmented samples with bad alignment scores as the same as data filtering in MT. The statistics of training data is shown in Table~\ref{tb:e2e_statistics}.
Moreover, two additional copies of the original and the augmented training data are created by modifying the speed to 110\% and 90\% of the initial rate, which makes a 3-fold training set.
\subsection{Speech Transformer for End-to-End ST}
As a baseline system, the model architecture and training configurations are the same as the end-to-end ASR in our cascade system, except for the learning rate, which is scaled up by 3.0 for ST.
We initialize the feature extractor and encoder from the corresponding component of ASR.
We keep the cases and punctuation marks on the target side and apply Moses tokenizer and BPE to the translations with 32,000 merge operations.
\subsection{Progressive Multi-task Learning}
Inspired by the multi-task learning framework for ST and the progressive training strategy \cite{tang2020mtl,ye2021progressive}, we introduce PMTL-ST, a progressive multi-task learning framework for speech translation, which can leverage additional ASR and MT data for training. As illustrated in Figure~\ref{fig:multi_model_task_st} a), the encoder accepts both audio and text inputs. Then we add a modality embedding to the representation to indicate audio input or text before passing to the shared transformer encoder. For decoding, we involve ``$<$EN$>$" and ``$<$DE$>$" tokens to make the decoder compatible with ASR and translation~(MT/ST) tasks, as shown in~\ref{fig:multi_model_task_st} b)/c).
\begin{figure}[t]
\centering
\includegraphics[angle=0,width=0.45\textwidth]{figures/multi_modal_task_st.pdf}
\caption{Overview of the end-to-end ST model with progressive multi-task learning. Note that the audio and text inputs are unnecessary to be aligned during training.}
\label{fig:multi_model_task_st}
\end{figure}
For progressive training, we separately train an ASR model and an MT model via different branches in Figure~\ref{fig:multi_model_task_st}. Then, we initialize the feature extractor and the audio modality embedding from the ASR model, and the rest of the model parameters are initialized by the MT model. The final model is trained jointly with ASR, MT, and ST.
All other training configurations, such as batch size and learning rate, are the same as the corresponding single task described before. Additionally, for the PMTL-ST models, we jointly learn the sentencepiece\footnote{\url{https://github.com/google/sentencepiece}} model with 16,000 tokens on the mixture of English and German texts.
\subsection{Fbank2vec}
Inspired by the recent progress of speech representation learning, like wav2vec 2.0~\cite{Baevski2020}, we introduce a fbank2vec network to learn contextualized audio representations from log Mel-filterbank features, as shown in Figure~\ref{fig:fbank2vec}.
\begin{figure}[t]
\centering
\includegraphics[angle=0,width=0.26\textwidth]{figures/fbank2vec.pdf}
\caption{The proposed fbank2vec network for audio feature encoding.}
\label{fig:fbank2vec}
\end{figure}
\paragraph{Convolutional Feature Encoder}
The encoder consists of two blocks containing a convolution followed by layer normalization and a GELU activation~\cite{Hendrycks2016}. The convolution in each block has 512 channels with 3$\times$3 kernel and stride size 2.
\paragraph{Relative Positional Encoding}
We use a group convolution layer to model the relative positional embeddings as \citet{Baevski2020} does. The kernel size is 128, and the number of groups is 16.
\paragraph{Contextualized Encoder} The final contextualized audio representations are generated by several transformer encoder blocks. In our setting, we stack 6 layers of the post-norm transformer, and the inner activation function for the feed-forward layers is GELU. In turn, the number of shared encoder layers in Figure~\ref{fig:multi_model_task_st} is changed to 6.
We insert the fbank2vec network in the front of the feature extractor. The feature extractor further reduces the dimension of audio representations by one convolution layer with 5$\times$5 kernel and stride size 2. The number of channels keeps the same as the dimension of fbank2vec output.
We experiment with two setups, fbank2vec-768 and fbank2vec-512.
The fbank2vec-768 means that
\begin{compactitem}
\item the dimension of fbank2vec output is 768;
\item inner the contextualized encoder, the hidden size of feed-forward layers is 3,072, and the head of the self-attention layers is 12.
\end{compactitem}
For the fbank2vec-512, the numbers are 512, 2,048, and 8, respectively.
Note that the fbank2vec module is pretrained by an ASR task and the overall model follows the progressive multi-task learning framework, so the configurations of word embeddings, the shared encoder and decoder vary accordingly.
\subsection{Training Data}
\label{simul:data}
The training data for EN$\rightarrow$DE is from MuST-C, OpenSubtitles 2018, and WMT 2020 datasets. And for EN$\rightarrow$JA, we use the parallel and monolingual data from the WMT 2020 news task.
\paragraph{Data Preprocessing}
We follow the data filtering process proposed in WMT works~\cite{li-etal-2019-niutrans,wu2020volctrans}, including language detection, length ratio filtering, dictionary alignment, and so on.
For pre-processing, we first apply MeCab\footnote{\url{https://github.com/taku910/mecab}} tokenizer to the Japanese sentences.
Then, words are segmented into subword units using sentencepiece toolkit for both language pairs. We jointly learn on the source and target side with a vocabulary of 10,000 tokens.
\begin{table*}[!t]\small
\centering
\begin{tabular}{lccccccc}
\toprule
\textbf{Dataset} & \textbf{Size} & \textbf{MT\#0} & \textbf{MT\#1} & \textbf{MT\#2} & \textbf{MT\#3} & \textbf{MT\#4} & \textbf{MT\#5} \\
\midrule
\textbf{EN $\rightarrow$ DE} & & & & & &\\
WMT 2020(EN $\rightarrow$ DE) & 41.14M & P & P & P & P & P/FT & FT \\
OpenSubtitles 2018 & 13.84M & P & P & P & P & P/BT/FT & FT/BT \\
MuST-C & 0.23M & P & P/BT & P/BT & P/BT & P/BT/FT & FT/BT \\
monolingual(EN/DE) & 10.25M & P & BT & BT & BT & BT & BT \\
\midrule
\textbf{EN $\rightarrow$ JA} & & & & & &\\
WMT 2020(EN $\rightarrow$ JA) & 18.19M & P & P/BT & P/BT & P/BT & P/BT/FT & BT/FT \\
\bottomrule
\end{tabular}
\caption{The statistics of MT datasets and the combination modes of datasets for simultaneous NMT models. ``P" indicates the parallel corpus. ``BT" means the back-translated data generated by MT\#0. ``FT" is the forward-translated data generated by MT\#1-3. }
\label{table:simul:data2}
\end{table*}
\begin{table}[!t]\small
\centering
\begin{tabular}{lcccc}
\toprule
\textbf{\#} & \textbf{Model Arch} & \textbf{Enc} & \textbf{Dec} & \textbf{Emb}\\
\midrule
0 & Transformer & 6 & 6 & 1024 \\
1 & Transformer & 6 & 6 & 1024 \\
2 & Transformer & 50 & 6 & 1024 \\
3 & LightConv & 6 & 6 & 1024 \\
4 & Transformer & 16 & 3 & 768 \\
5 & Transformer & 16 & 3 & 768 \\
\bottomrule
\end{tabular}
\caption{The model setups. ``Enc", ``Dec" denote the number of encoder and decoder layers. ``Emb" means the embedding size and the hidden size.}
\label{table:simul:model_arch}
\end{table}
\paragraph{Data Augmentation}
Similar to section~\ref{sec:cas_nmt}, we utilize tagged back-translation (BT) and knowledge distillation (KD) strategies to improve the performance of simultaneous NMT. We experiment with both LightConv~\cite{wu2018pay} and transformer models. The model with the best BLEU score on the development set is chosen for data augmentation. The statistics of all training data and model settings are presented in Table~\ref{table:simul:data2} and Table~\ref{table:simul:model_arch} respectively.
\subsection{Efficient \texttt{wait-k} Model}
\label{simul:model}
Our simultaneous NMT systems are based on transformer \texttt{wait-k} models, which first read $k$ source tokens and then alternate between reading and writing (translating).
Formally, when decoding the sentence $\mathbf{x}$, the number of visible source tokens is constrained within $\min(k+t-1,|\mathbf{x}|)$ at decoding step $t$, where $k$ is the hyper-parameter controlling the latency.
Furthermore, to avoid recomputing the hidden states of the encoder each time a token is read, we implement incremental unidirectional encoders~\cite{elbayad2020efficient}. And \texttt{multi-path} training is also applied to leverage more possible \texttt{wait-k} paths which refers that hyper-parameter $k \in [3, 9]$ is random selected at each batch during training.
Models are trained with a batch size of 32,000 tokens on Tesla V100 GPUs. We average the last 6 checkpoints once the model converges.
\subsection{Inference}
\label{simul:infer}
We explore the look-ahead beam search strategy for inference.
Specifically, we apply beam search to generate $M (M > 1)$ tokens at each decoding step and pick the first token in the one with the highest log-probability out of multiple decoding paths. The look-ahead beam search achieves consistent performance improvement when $k_\text{eval}$ is small while its performance improvement is insignificant with a large $k_\text{eval}$. This search method is excluded from our final submissions due to its higher latency, and we choose the greedy search instead.
Additionally, we split the source sentences into sub-sentences once the end-of-sentence punctuation is recognized. Though it may result in a slight performance drop due to the lack of context, we can obtain a much lower latency.
For the final submissions, we use ensemble models.
We train several models with different $k_\text{train}$ values and disjoint subsets of training data for data diversity. Each model produces different latency-quality trade-offs.
\subsection{Offline Speech Translation}
The overall performance of the offline ST and the ASR component used in the cascade system are listed in Table~\ref{tb:offline_results} and Table~\ref{tb:cas_asr_results} respectively.
\begin{table}[!t]\smal
\centering
\begin{tabular}{lc}
\toprule
\textbf{Testset} & \textbf{WER} \\
\midrule
\textit{dev} & 5.2 \\
\textit{tst-COMMON} & 5.7 \\
\textit{dev(v1)} & 10.6 \\
\textit{tst-COMMON(v1)} & 7.4 \\
\bottomrule
\end{tabular}
\caption{The WER of the ASR system for the offline ST.}
\label{tb:cas_asr_results}
\end{table}
In Table~\ref{tb:offline_results}, line 1-4 show the performance of our pure MT systems, which translate the lowercased ground truth transcripts with no punctuation marks, and the natural texts.
As seen, there may be no essential improvements with the ``$<$ASR$>$" tag on the irregular input (up to 2 BLEU gap on the single model), and it suggests that text restoration has the potential to narrow the gap.
Line 6-7 present the results of translating the ASR output, and we see our cascaded approach surpasses last year's best cascade system (line 5) by 2.6 BLEU. However, there is still a significant loss of up to 3 BLEU scores than line 1/3 due to ASR errors.
The results of our end-to-end solutions are presented in line 8-20, where line 8 is a benchmark model \cite{zhao2020neurst} trained on the MuST-C dataset only.
With the growth of model capacity~(256d$\rightarrow$768d) and data augmentation, we obtain 6 BLEU improvement on the \textit{tst-COMMON} over the benchmark (line 8).
Then, increasing the size of augmented data gains slight improvement, as comparing line 9 to line 10/11 (+0.3$\sim$0.5 BLEU scores). Line 13-16 show the results of our proposed fbank2vec.
As shown in line 15, we achieve 31.1 BLEU on \textit{tst-COMMON}, the best single model with fbank2vec, progressive multi-task learning, and speed perturbation. We obtain 31.8 BLEU (line 20) for the final ensemble model, which surpasses the end-to-end benchmark by 7.9 BLEU scores and is approaching the cascade system with a nearly 1.5 BLEU gap.
Lastly, our primary cascade system is line 7, and the primary end-to-end system is line 20 for submission, which achieves higher performance via model ensemble.
\begin{figure*}[tbp]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering \includegraphics[width=\textwidth]{figures/pdf_simul_result_ende.pdf}
\caption{EN$\rightarrow$DE}
\label{fig:simul:result:a}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/pdf_simul_result_enja.pdf}
\caption{EN$\rightarrow$JA}
\label{fig:simul:result:b}
\end{subfigure}
\caption
{Latency-quality trade-offs of the simultaneous NMT. $k7/9$ means $ {k}_\text{train}=7/9$. MT\#X indicate the aforementioned training datasets and model settings in Table~\ref{table:simul:data2} and \ref{table:simul:model_arch}. \texttt{beam} refers to our look-ahead beam search strategy. \texttt{seg} means that the sentences are pre-splited during inference. \texttt{multipath} means that $k$ is random selected during training.}
\label{fig:simul:result}
\end{figure*}
\begin{table}[t]\small
\centering
\begin{tabular}{ccccc}
\toprule
& & Low & Medium & High \\
\midrule
\multirow{2}{*}{EN $\rightarrow$ DE} & Ensemble & 25.86 & 31.73 & 33.21 \\
& +seg & 28.75 & 32.87 & 32.97 \\
\midrule
\multirow{2}{*}{EN $\rightarrow$ JA} & Ensemble & 14.81 & 15.85 & 15.85 \\
& +seg & 15.79 & 15.79 & 15.79 \\
\bottomrule
\end{tabular}
\caption{Performance of our final submissions models on MuST-C \textit{tst-COMMON} for English-German and IWSLT21 dev set for English-Japanese.}
\label{table:simul:result}
\end{table}
\subsection{Simultaneous Translation}
\label{sec:exps:simul}
We evaluate the simultaneous NMT systems with different combinations of strategies and present our results in Figure~\ref{fig:simul:result}.
Then we report the performance on different latency regimes in Table \ref{table:simul:result}.
As shown in Figure \ref{fig:simul:result}, we can obtain remarkable BLEU improvements by training with only the knowledge distilled data (black) comparing to the filtered parallel data (green) and back-translated data (magenta), on average 1.0 BLEU improvement on EN$\rightarrow$DE and 0.5 on EN$\rightarrow$JA.
The possible reasons may be: 1) Noise in origin data is migrated, like non-parallel sentence pairs. 2) Complex sentences with diverging word order are excluded, and the machine-translated texts, i.e., translationese, sometimes have simpler expressions.
We can see that the proposed look-ahead beam search (red) is competitive when $k_\text{eval}$ is relatively small but is comparable with the greedy search when $k_\text{eval}$ is large. So overall considering translation latency, we use the greedy search for our final submissions. As for \texttt{multi-path} training, we see it achieves limited BLEU improvement in our experiments.
For our final submission of EN$\rightarrow$DE, we use the ensemble model, which consists of three transformer models trained on different dataset combinations, with $k_\text{train}=7$. For EN$\rightarrow$JA, the submitted model is formed by two transformer models, with $k_\text{train}=\infty$ (trained on full sentences) and \texttt{multi-path} training respectively. As presented in Figure~\ref{fig:simul:result}, the model ensemble technique leads to at least 0.5 BLEU improvement on average (yellow). Additionally, with the sentence segmentation (bleu), the average lagging is significantly reduced.
As a result, our final submitted systems exceed the baseline system at around 7 BLEU on the same latency regime.
\section{Introduction}
\label{sec:intro}
\input{01-intro.tex}
\section{Cascaded Speech Translation}
\label{sec:cascade}
\input{02-cascade.tex}
\section{End-to-End Speech Translation}
\label{sec:e2e}
\input{03-end2end.tex}
\section{Simultaneous Translation}
\label{sec:simul}
\input{04-simul.tex}
\section{Experimental Results}
\label{sec:exp}
\input{05-exps.tex}
\section{Final Results}
\label{sec:final}
\input{05-submission.tex}
\section{Conclusion}
\label{sec:conclusion}
\input{06-conclusion.tex}
\bibliographystyle{acl_natbib}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 970 |
U-874 je bila nemška vojaška podmornica Kriegsmarine, ki je bila dejavna med drugo svetovno vojno.
Zgodovina
9. maja 1945 se je podmornica predala v Hortenu. V sklopu operacije Deadlight je bila 29. maja 1945 nato poslana v Lisahally. Podmornica je bila nato 31. decembra 1945 potopljena.
Poveljniki
Tehnični podatki
Glej tudi
seznam podmornic druge svetovne vojne
seznam nemških podmornic druge svetovne vojne
seznam vojaških plovil Kriegsmarine
Viri & Zunanje povezave
- v angleščini:
UBoat.net
ubootwaffe.net
- v nemščini:
Lexikon der Wehrmacht
Nemške podmornice druge svetovne vojne
Podmornice razreda IX | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,077 |
package servenv
import (
"fmt"
"net/http"
"net/url"
"time"
log "github.com/golang/glog"
"github.com/youtube/vitess/go/event"
"github.com/youtube/vitess/go/proc"
)
var (
onCloseHooks event.Hooks
)
// Run starts listening for RPC and HTTP requests,
// and blocks until it the process gets a signal.
func Run(port int) {
populateListeningURL()
createGRPCServer()
onRunHooks.Fire()
serveGRPC()
serveSocketFile()
l, err := proc.Listen(fmt.Sprintf("%v", port))
if err != nil {
log.Exit(err)
}
go http.Serve(l, nil)
proc.Wait()
l.Close()
startTime := time.Now()
log.Infof("Entering lameduck mode for at least %v", *lameduckPeriod)
log.Infof("Firing asynchronous OnTerm hooks")
go onTermHooks.Fire()
fireOnTermSyncHooks(*onTermTimeout)
if remain := *lameduckPeriod - time.Since(startTime); remain > 0 {
log.Infof("Sleeping an extra %v after OnTermSync to finish lameduck period", remain)
time.Sleep(remain)
}
log.Info("Shutting down gracefully")
Close()
}
// Close runs any registered exit hooks in parallel.
func Close() {
onCloseHooks.Fire()
ListeningURL = url.URL{}
}
// OnClose registers f to be run at the end of the app lifecycle.
// This happens after the lameduck period just before the program exits.
// All hooks are run in parallel.
func OnClose(f func()) {
onCloseHooks.Add(f)
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,686 |
{"url":"https:\/\/www.physicsforums.com\/threads\/complex-analysis.210032\/","text":"# Complex analysis\n\n1. Jan 20, 2008\n\n### indigojoker\n\nI am to find all plints z in the complext plane that satisfies |z-1|=|z+i|\n\nThe work follows:\nlet z=a+bi\n|a+bi-1|=|a+bi+i|\n(a-1)^2+b^2=a^2+(b+1)^2\na^2-2a+1+b^2=a^2+b^2+2b+1\n-a=b\n\nthe correct answer should be a perpendicular bisector of segments joining z=1 and z=-i\n\nmy result looks more like a perpendicular bisector of segments joking a=0 and b=0\n\n2. Jan 20, 2008\n\n### Rainbow Child\n\nThe points you found (correctly) are\n\n$$z=a\\,(1-i)$$\n\nthus if you write $$z=x+i\\,y$$ you have\n\n$$x=a,y=-a \\Rightarrow y=-x$$\n\nCan you continue from here?\n\n3. Jan 20, 2008\n\n### indigojoker\n\nso that is what i have, y=-x, or what i have is -a=b\n\nwhat i'm asking is why does my result look like a perpendicular bisector of segments joining a=0 and b=0 instead of a perpendicular bisector of segments joining z=1 and z=-i\n\n4. Jan 20, 2008\n\n### Rainbow Child\n\nWrite down the line passing through the points z=1 and z=-i.\n\nHow is this line and $$y=-x$$ are related?\n\n5. Jan 21, 2008\n\n### indigojoker\n\ni see, how come y=x-1 isnt a solution as well? inst y=x-1 a perpendicular bisector of segments joining z=1 and z=-i?\n\n6. Jan 21, 2008\n\n### HallsofIvy\n\nStaff Emeritus\nA line segment doesn't have two perpendicuar bisectors! The segment between 1 and i has midpoint (1+ i)\/2 ((1\/2, 1\/2) in the xy-plane). The line y= x- 1 passes through (1\/2, 1\/2- 1)= (1\/2, -1\/2), not (1\/2, 1\/2).","date":"2017-07-21 15:03:28","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.2919069230556488, \"perplexity\": 4922.151629028985}, \"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-2017-30\/segments\/1500549423785.29\/warc\/CC-MAIN-20170721142410-20170721162410-00364.warc.gz\"}"} | null | null |
Create a personalized bookmark for the devoted bookworm. The beautiful ribbon stitches are one of the exclusive stitch techniques for PFAFF® available on creative™ 4.5 and creative sensation™ pro but you can create these bookmarks with any decorative stitch of your choice.
You can also try the creative bobbin case accessory to make an embellishment that really stand out!
Overcast the short edges of the ribbon. Fold under approximately ⅝" (15mm) on one short edge and stitch ¼" (6mm) from the fold to form a casing.
Pull the elastic cord through and make a knot. Pull the knot into the casing.
Fold the short edge on the other side to fit the book. Fold the edge to the back side and add a button on the right side with the Sew on button foot.
Hi Donna, the design of the two feet are slightly different. The Sewing Star foot has a metal guide to hold down the fabric/stitch.
It would be nice to have all projects in PDF in order to quicken the process to save to computer files.
Appreciate all the great ideas.
I would like to receive tutorials on using my pfaff sewing machine.
Is this Sewing star foot the same as open toe foot? | {
"redpajama_set_name": "RedPajamaC4"
} | 7,978 |
Q: WPF TargetNullValue for Dictonary Binding not working I have a dictonary (of String, String) that may not have an entry for a specific key.
In XAML I want to cover that case by the following line:
<Image Source="{Binding MyDictonary[myKey], UpdateSourceTrigger=PropertyChanged, Converter={StaticResource uriToImageConverter}, TargetNullValue={StaticResource myStaticImage} }"/>
The code works fine if I don't bind to a dictonary but have a String in my viewmodel which is Nothing.
I also checked if it is the converters vault but the converter is never called if there is no valid string.
Thanks in advance for help/explanaitions of what I am missing here.
A: You need to add a dictionary entry for "myKey" with Nothing (MyDictonary.Add("myKey", Nothing)) so that it can get the "Nothing" value without causing an exception.
Another way to achieve this is to add an additional "DictionaryValue" property on ViewModel.
<StackPanel>
<TextBox Text="{Binding DictionaryKey, UpdateSourceTrigger=PropertyChanged}"/>
<Image Source="{Binding DictionaryValue, UpdateSourceTrigger=PropertyChanged, Converter={StaticResource uriToImageConverter}, TargetNullValue={StaticResource myStaticImage} }"/>
</StackPanel>
Public Class ViewModel
Inherits INotifyPropertyChanged
Public Sub New()
MyDictonary = New Dictionary(Of String, String)()
End Sub
Private _key As String
Public Property DictionaryKey As String
Get
Return _key
End Get
Set(ByVal value As String)
If _key <> value Then
_key = value
RaisePropertyChanged(NameOf(DictionaryKey))
RaisePropertyChanged(NameOf(DictionaryValue))
End If
End Set
End Property
Public ReadOnly Property DictionaryValue As String
Get
If DictionaryKey IsNot Nothing AndAlso MyDictonary.Keys.Contains(DictionaryKey) Then
Return MyDictonary(DictionaryKey)
Else
Return Nothing
End If
End Get
End Property
Public Property MyDictonary As Dictionary(Of String, String)
Public Event PropertyChanged As PropertyChangedEventHandler
Private Sub RaisePropertyChanged(ByVal propertyName As String)
PropertyChanged?.Invoke(Me, New PropertyChangedEventArgs(propertyName))
End Sub
End Class
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,533 |
import template from './articleListItem.html';
import {ArticleListItemController} from './articleListItem.controller.js';
export default (appModule) =>{
appModule.controller('ArticleListItemController', ArticleListItemController);
appModule.component('articleListItem', {
bindings: {
article: '='
},
controller: 'ArticleListItemController as vm',
template: () => {
return template;
}
});
};
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,489 |
import { Component, Input, Output, EventEmitter } from '@angular/core';
@Component({
selector: 'app-button',
templateUrl: './button.component.html',
styleUrls: ['./button.component.scss']
})
export class ButtonComponent {
@Input() text = "";
@Input() basic = false;
@Input() table = false;
@Output() click = new EventEmitter();
constructor() { }
onClick(e) {
this.click.emit(e)
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,811 |
{"url":"https:\/\/zbmath.org\/?q=an:1121.13002","text":"## Algebraic theory of locally nilpotent derivations.(English)Zbl\u00a01121.13002\n\nEncyclopaedia of Mathematical Sciences 136. Invariant Theory and Algebraic Transformation Groups 7. Berlin: Springer (ISBN 3-540-29521-6\/hbk). 260\u00a0p. (2006).\nLet $$A$$ be a commutative ring and $$M$$ an $$A$$-module. A derivation $$D$$ is an additive map $$D:A\\to M$$ satisfying the Leibniz rule. That is, $$D(a+b)=D(a)+D(b)$$ and $$D(ab)=aD(b)+bD(a)$$ for all $$a,b\\in A$$. A derivation $$D:A\\to A$$ is called locally nilpotent (abbreviated LND) if for any $$a\\in A$$, there exists an $$n$$ such that $$D^n(a)=0$$. A typical example is when $$A=R[x]$$, polynomial ring in one variable $$X$$ over $$R$$ and $$D=\\frac{d}{dX}$$, the $$X$$-derivative. Loosely speaking, the study of LNDs have been to see whether given one such, how closely does it resemble the typical one as above.\nIn the volume under review, the author gives a detailed description of the subject covering all the important results (and then some more). The study of locally nilpotent derivations as an independent field began less than fifty years ago, but it already has an impressive array of results which clarify and generalize earlier isolated classical results. A detailed description of what the book contains seem pointless and thus one shall just mention some of its salient features.\nThe subject is closely related to Hilbert\u2019s Fourteenth problem and there are several classical results in this direction dealt with in this book. Some of the earliest are due to M.\u00a0Nagata (and fittingly, Professor Nagata\u2019s picture graces the cover of the book). Locally nilpotent derivations are closely related to vector group (the additive group $$\\mathbb{G}_a^n$$) actions and that justifies the inclusion of the book in the series devoted to Invariant Theory.\nOne of the most important part of the study is LNDs of polynomial rings over a field (of characteristic zero). There is not much to say in the case of one variable, since they are the obvious ones. The case of two variables is completely understood, albeit non-trivial. The case of three variables is still not completely understood, but the book gives an excellent survey of the state of the art. The book also describes the more recent tools to study LNDs, like the Makar-Limanov invariant and Derksen invariant.\nFinally, the book has a wealth of examples and the Epilogue details some important open problems in the area. The book is more or less self-contained, especially if you have had a reasonable course in commutative algebra. It studiously avoids the heavier machinery of geometric arguments and thus is accessible to less advanced graduate students. It is a valuable addition to the literature and am sure would be very helpful to the interested student and researcher alike.\n\n### MSC:\n\n 13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra 13N10 Commutative rings of differential operators and their modules 14R20 Group actions on affine varieties 14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)","date":"2022-05-25 10:53:20","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.7823368310928345, \"perplexity\": 291.84674767894694}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662584398.89\/warc\/CC-MAIN-20220525085552-20220525115552-00252.warc.gz\"}"} | null | null |
Q: How to find metadata in the dht while only having the Infohash? So far I assumed a client would just search for peers in the dht and use bep 9 in order to get the metadata, however when trying it myself and opening Wireshark I couldn't find any usage of bep 9. Does the client automatically try trackers or something similar with the Infohash? Or is there something else you can do in the dht with only an infohash to get the metadata without bep 9?
A: The DHT does not store the torrent metadata, it provides contacts to peers (via get peers) from which you can then download the metadata via the peer protocol and the extensions described in BEP 9.
There also are some websites that store torrents under a path based on their infohash, a client might also try one of those to fetch it.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,372 |
First Institute for Sport Business Advisory Board takes place on campus
20 February 2018 Sport Business
Industry leaders from the sport business sector visited the London campus last week(Wednesday 14 February 2018) to attend the first Institute for Sport Business (ISB) Advisory Board meeting.
The ISB Advisory Board has been designed to ensure that Loughborough University London's Institute for Sport Business continues to deliver teaching relevant to industry and as a result produces desirable graduates for the industry.
The Board consists of the following members:
Kate Bosomworth, Chief Marketing Officer, M&C Saatchi
Alex Coulson, Executive Director, Sport Industry Group
Iain Edmondson, Head of Major Events, London & Partners
Rebecca Hopkins, Managing Director, ENS PR Ltd
Will Lloyd, CEO, GlobalSportsJobs
Sophie Morris, Director and Strategic Marketing & Sponsorship Consultant, Millharbour Marketing Consultancy
Bob Reeves, Director, Foundation For Leadership Through Sport
Ben Smith, Head of Research & Development Performance Systems, Chelsea FC
Sam Yardley, Associate Director, Two Circles
The ISB Advisory Board will meet twice a year and contribute insights from their diverse range of disciplines within the sport business industry which will filter into the Institute's research, teaching and enterprise activities. The Board will also contribute to employment opportunities for ISB graduates.
The Board will review the Institute's offering and ensure the Institute appropriately balances its theory teaching with its industry insights.
At the first meeting, academics from the Institute introduced the programmes currently being taught this academic year (MSc Sport Business and Innovation, MSc Sport Business and Leadership, and MSc Sport Digital and Media Technologies). The programme directors explained the motivations behind offering each of the individual programmes and how these were unique.
To complete the meeting, three PhD students were specially selected to present their PhD research to the Board. Jacky Mueller, a Loughborough University London Master's graduate from Germany, discussed her research into Leadership Perceptions in Professional Football. Daniel Read followed, discussing his research into the World Anti-Doping Agency from an organisational institutional perspective. To finish, Tian Lin, another Master's graduate, this time from China, discussed his research interest in leadership in a Chinese sports context with the Board members.
Following from the meeting, students, academics and board members took the opportunity to network.
If you would like to find out more about the Advisory Board, please contact or Professor James Skinner.
Loughborough is one of the country's leading universities, with an international reputation for research that matters, excellence in teaching, strong links with industry, and unrivalled achievement in sport and its underpinning academic disciplines.
It has been awarded five stars in the independent QS Stars university rating scheme, named the best university in the world for sports-related subjects in the 2020 QS World University Rankings and top in the country for its student experience in the 2018 THE Student Experience Survey.
Loughborough is in the top 10 of every national league table, being ranked 4th in the Guardian University League Table 2020, 5th in the Times and Sunday Times Good University Guide 2020 and 8th in The UK Complete University Guide 2020.
Loughborough is consistently ranked in the top twenty of UK universities in the Times Higher Education's 'table of tables' and is in the top 10 in England for research intensity. In recognition of its contribution to the sector, Loughborough has been awarded seven Queen's Anniversary Prizes.
Specialising exclusively in postgraduate-level study, Loughborough University London is based on the Queen Elizabeth Olympic Park and offers cutting-edge master's and PhD degrees, as well as business collaboration, research and enterprise opportunities.
The seven specialist institutes at Loughborough University London include: Design Innovation, Digital Technologies, Diplomacy and International Governance, Innovation and Entrepreneurship, International Management, Media and Creative Industries, and Sport Business. Close industry partnerships with a wealth of leading businesses and creative organisations means teaching and learning is informed by real-life challenges and driven by true industry pressures.
Zoom Zoom Zoom Zoom Zoom Zoom Zoom Zoom Zoom Zoom Zoom Zoom Zoom Zoom Zoom Zoom Zoom Zoom Zoom Zoom
DIY Digital PR for your small business: Part 1
Induction Week: Future Space Welcome Panel
Guided campus tour See what Loughborough University London has to offer by booking onto a guided campus tour. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,011 |
4A's Top Two Face-off in the WyoPreps 'Game of the Week' [AUDIO]
David Settle Published: September 10, 2020
It's a 1 vs. 2 showdown in the big school division that is our fan-voted WyoPreps 'Game of the Week' for week two.
The top-ranked Sheridan Broncs welcome the No. 2 Cheyenne Central Indians in an early-season battle between two of the favorites in Class 4A.
Both teams are off to a 2-0 start. Sheridan has wins over Laramie (42-10) and South (63-8). Central has defeated No. 5 Natrona (35-21) and Kelly Walsh (38-12).
These two squads played twice during the 2019 season. The regular-season meeting was 28-14 in favor of the Broncs. The most recent matchup was in the semifinals of the Class 4A playoffs last November, and Sheridan put it on Cheyenne Central, 62-35.
WyoPreps' David Settle talked on the phone Sheridan's Jeff Mowry and Central's Mike Apodaca. Both coaches spoke about the start to their season, how they feel this game is a bit of a 'measuring stick,' and what they look for from the other. Mowry knows Central is talented and has lots of weapons on offense and a veteran defense. Apodaca praised the "program" Sheridan has and how they're aspiring to get to that level. Both also discussed some keys to the game for their team in the audio interviews at the top of this story.
The two teams are first and second in total offense in Class 4A with the Indians having an 11-yard per game advantage. They are also No. 1 and No. 2 in total defense, and it's the Broncs who allow about 100 yards less per game.
Sheridan is paced QB Zach Koltiska and running back Izak Aksamit. Koltiska has thrown for 181 yards per game with 5 TDs and no interceptions. Aksamit averages 73.5 rush yards per game in limited time. He has three TD runs.
Cheyenne Central is led by QB Andrew Cummins, who is averaging 255 passing yards per game. He has four TD passes but five interceptions. Add to that Carter Lobatos, who leads 4A in rushing at 154 yards per game with 3 TDs. Cummins' top target is the No. 2 receiver in Andrew Johnson. He has 13 catches for 211 yards and 2 TDs.
Friday will be the 75th meeting all-time between the two programs in football, according to wyoming-football.com. Sheridan has the series lead 45-28 with one tie.
Kick-off will be at 6 p.m. from Homer Scott Field in Sheridan.
Week 2 Scoreboard
Source: 4A's Top Two Face-off in the WyoPreps 'Game of the Week' [AUDIO]
Filed Under: Cheyenne Central Indians, Jeff Mowry, Mike Apodaca, Sheridan Broncs, Wyoming High School Football, WyoPreps Game of the Week | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,472 |
Cream cheese, AIDS drama and a catchy tune by Bruce Springsteen – everyone associates his very own with Philadelphia. But what makes the city so attractive for gay holidaymakers?
Love. What could better promote a city than the word in big red letters: LOVE? The sculpture – a reproduction of the iconic lettering created by the American Pop Art artist Robert Indiana – has been gracing John F. Kennedy Plaza since the late 1970s, adjacent to the imposing Philadelphia City Hall . The city of fraternal love (the city's name meaning in Greek) is one of the most historically significant places in the United States. The American Declaration of Independence in July 1776 and the Constitution in 1787 were both adopted here. Liberty Bell, shown in a glass pavilion next to Independence Hall, is considered the national symbol of freedom and independence alongside the New York Statue of Liberty.
Even before the Stonewall uprising in New York, gays and lesbians began to demonstrate for their rights here in the mid-1960s, making Philadelphia the cradle of the American gay movement. You can see how much the LGBTIQ* community is integrated on a walk through the "Gayborhood", the trendy neighborhood in the heart of the city. In addition to street signs painted in rainbow colours, some buildings feature huge murals depicting local community figures smiling down or scenes from everyday life in the neighbourhood. "Pride and Progress" is the title of a mural that shows images of a Pride Festival with cheerful celebrating people on the west wall of the William Way LGBT Community Center on Spruce Street. The borders between art in public space and real events on the street become blurred twice a year: the city has been celebrating the OutFest for over thirty years around the international Coming Out Day in October in addition to the Philly Pride with parade and festival in June. Thousands of visitors come to the city from near and afar to party together at a big street festival and at get the lastest news about community projects, gay-lesbian sports clubs or new products at numerous stalls.
But even off festival season Philadelphia's LGBTIQ* scene with its numerous bars such as Woody's, Bike Stop or U Bar offers plenty of variety and opportunity to meet locals. The metropolis, about an hour's train ride from New York, has still a lot more to offer, though: The pedestrian-friendly city with its narrow streets, spacious squares and parks invites you to discover it on foot and also offers plenty of opportunity for shopping. The Philadelphia Museum of Art houses one of the best and largest art collections in the US. The Barnes Foundation shows masterpieces by Picasso, van Gogh, Cézanne and over 180 paintings by Renoir. You find a completely different sort of 'classic' on many street corners: Cheese steak. It consists of a thin slice of steak wedged between two slices of white bread, topped with onions and cheese and is considered a national dish in Pennsylvania. Speaking of cheese: The famous Philadelphia cream cheese was named after the town. The product, which has in fact been manufactured in neighbouring New York since 1880, was given the name of the metropolis for PR reasons as it had then a reputation for quality cheeses. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,665 |
Q: (Bitmap)image behaves differently than new Bitmap(image) Here is the test i wrote and that will currently fail:
var unusableColor = Color.FromArgb(13, 19, 20, 19);
var retrievedColor = Color.Empty;
var tempFile = Path.GetTempFileName();
using (var bitmap = new Bitmap(1, 1))
{
bitmap.SetPixel(0, 0, unusableColor);
bitmap.Save(tempFile, ImageFormat.Png);
}
using (var image = Image.FromFile(tempFile))
// This will lead to the error
using (var bitmap = new Bitmap(image))
// But this will work
//using (var bitmap = (Bitmap)image)
{
retrievedColor = bitmap.GetPixel(0, 0);
}
Assert.That(retrievedColor, Is.SameAs(unusableColor));
If you take a look into the retrievedColor you'll find that it will be the same as Color.FromArgb(13, 19, 19, 19). So the difference will be that the green part has changed from 20 to 19.
Any idea why this happens or under which circumstances the constructor of the Bitmap will change a pixel?
Update
Seems to be a deeper nested problem. By replacing the Bitmap constructor by a simple cast of the image variable the problem goes away. This maybe solves the problem, but it doesn't explain it. Further more i was able to reproduce the problem even in Paint.Net by the following procedure:
*
*Open Paint.Net and create a new image (size doesn't matter)
*Select all (Ctrl+A)
*Remove the selection (Del)
*Open the color dialog (F8)
*Enter the above values for RGB (19, 20, 19) and at the bottom the transparency (13).
*Select the fill tool (F)
*Fill the color into the empty image
*Select the color selection tool (K)
*Click somewhere into your fresh image and watch the color dialog
So it seems it is maybe a deeper problem, not caused by the Bitmap or Image class but maybe by some deeper functionality like GDI+ or something similar.
Update 2
I just wrote a new test to find out all affected colors:
for (int a = 0; a < 256; a++)
{
for (int r = 0; r < 256; r++)
{
for (int g = 0; g < 256; g++)
{
for (int b = 0; b < 256; b++)
{
using (var bitmap = new Bitmap(1, 1))
{
var desiredColor = Color.FromArgb(a, r, g, b);
bitmap.SetPixel(0, 0, desiredColor);
// This will fail in a lot of colors with a low alpha channel value
using (var copiedBitmap = new Bitmap(bitmap))
// This will work, cause the information is entirely copied.
//using (var copiedBitmap = (Bitmap)bitmap.Clone())
{
var retrievedColor = copiedBitmap.GetPixel(0, 0);
if (desiredColor != retrievedColor)
{
Debug.Print(desiredColor + " != " + retrievedColor);
}
}
}
}
}
}
Please don't let it run completely on itself, cause it will take a loonng time to finish and it also finds a looots of differences. But what you can see, if you play around with the transparency (setting to 1 or 10) then you'll see that the RGB values use this as some kind of bit depth.
So the problem occurs if you create a new Bitmap from an existing one that uses low transparency values. The real root cause seems to be far down in GDI, Kernel or somewhere in this area and can't be solved from .Net.
Simply be aware that a color can change by calling the bitmap constructor if the color has a low transparency value. If you really need the original colors to stay alive in a second instance instead use (Bitmap)myBitmap.Clone() or if you load it from disk use (Bitmap)Image.FromFile(filename) cause Image is only an abstract class which will normally instantiated through the Bitmap class.
A: I checked PNG file saved with your code using Paint.NET and pixel color is exactly unusableColor.
If you change your reading code with this:
using (Bitmap bitmap = (Bitmap)Image.FromFile(tempFile))
{
retrievedColor = bitmap.GetPixel(0, 0);
}
everything works
A: you can use Clone method:
using (var image = Image.FromFile(tempFile))
{
using (var bitmap = image.Clone() as Bitmap)
{
retrievedColor = bitmap.GetPixel(0, 0);
}
}
Problem is in 'new Bitmap(image)' because it creates new instance. If you look into bitmap's constructor, it creates new transparent image and draws source image. graphics object has smoothing mode property, which is used for drawing quality. default is no antialiasing.
Here is the Bitmap's constructor:
Graphics graphics = null;
try
{
graphics = Graphics.FromImage(this);
graphics.Clear(Color.Transparent);
graphics.DrawImage(original, 0, 0, width, height);
}
finally
{
if (graphics != null)
{
graphics.Dispose();
}
}
So if you just load image from file, or clone, bitmap data is same.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,245 |
Buton ist ein Regierungsbezirk (Kabupaten) auf der indonesischen Insel Buton, die zur Hauptinsel Sulawesi gehört. Der Bezirk gehört zur Provinz Sulawesi Tenggara (Südostsulawesi) im Südosten von Sulawesi.
Geographie
Der Bezirk besteht seit Gründung der Provinz im Jahr 1964 und hatte zur letzten Volkszählung 2010 255.712 Einwohner. Davon entfielen auf die beiden später ausgegliederten Bezirke Buton Tengah 86.112 bzw. Buton Selatan 74.707 Einwohner.
Hauptstadt ist Pasar Wajo. Der Bezirk umfasst den mittleren der Insel Buton. Im Jahr 2014 wurden aus dem alten Bezirk Buton zwei neue Bezirke ausgegliedert: Buton Tengah (Zentralbuton) und Buton Selatan (Südbuton). Dies geschah auf Grundlage der Gesetze Nr. 15 und 16.
Verwaltungsgliederung
Der Bezirk Buton bestand ursprünglich aus 21 administrativen Distrikten (Kecamatan), gab jeweils sieben an die beiden neu gegründeten Regierungsbezirke ab.
Quelle der administrativen und Flächenangaben: Peraturan Menteri Dalam Negeri RI Nomor 72 Tahun 2019 (Verordnung des Innenministers vom Oktober 2019):
Die Bevölkerungsangaben basieren auf der Fortschreibung durch die regionalen Zivilregistrierungsbüros und stammen von Ende 2019
Weblinks
E-Books auf der Statistikseite des Bezirks (BPS - Badan Pusat Statistik Kabupaten Muna) (indonesisch/englisch)
Verwaltungskarte des Bezirks im Maßstab 1:128.000 (indonesisch)
Einzelnachweise
Geographie (Sulawesi)
Kabupaten in Sulawesi Tenggara | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,698 |
A force for her people
Source:The Age December 4 2004
The controversial new Aboriginal advisory council meets for the first time next week. The woman who will head it, Sue Gordon, says she's no "sell out". Meaghan Shaw reports.
Sue Gordon fixes a stare and describes what it was like meeting her mother 30 years after she was taken from her family: "Traumatic." "You can imagine yourself, how many birthdays, weddings and all that did you miss in 30 years? So, obviously, it was traumatic, but that's for me to deal with."
Gordon, 61, is tough - a good attribute for someone stepping into the most controversial position in indigenous affairs.
The West Australian Children's Court magistrate is best known for heading an inquiry into domestic violence that prompted a $75 million funding pledge from the state's Labor Government to tackle sexual abuse.
Appointed last month as head of the Federal Government's new hand-picked advisory council, which has its first meeting next week, Gordon has had to fend off criticism of being a "sell-out" to Aboriginal people. The National Indigenous Council was set up by the Government to provide advice in place of the Aboriginal and Torres Strait Islander Commission, an elected body discredited by in-fighting and poor leadership. Legislation to abolish ATSIC should be passed by Parliament next year.
The new council's 14 members include ALP vice-president Warren Mundine, Sydney Swans footballer Adam Goodes and Brisbane barrister Tammy Williams. But it has been attacked by some indigenous people as tokenistic, unrepresentative and filled with people who subscribe to the Government's "practical reconciliation" agenda at the expense of symbolic issues.
Several indigenous leaders, such as Cape York lawyer Noel Pearson and former footballer Michael Long, were approached to sit on the council but refused.
Gordon brushes off the criticism. She says that the council is not a replacement for ATSIC, with its $100,000 salaries for commissioners and $1.2 billion budget for indigenous programs. Instead, council members will meet four or five times a year and be paid a few hundred dollars for each meeting.
"All the criticism that's been coming has really been by uninformed people. Perhaps the Government didn't get out the message clearly of just what this body is," she says.
She was motivated to accept the position to ensure Aboriginal people continued to be heard by the Government, particularly on the issues of family violence and child abuse. "I got sick of the in-fighting that was in the papers on a regular basis about ATSIC."
Gordon admits she is not a symbols person. She leans towards the thinking of Pearson, whose views on rejecting a passive welfare mentality and accepting responsibility have been endorsed by the Government. "I respect him the most of the other so-called leaders because he's doing something," she says.
"Some people still have their priority to get an apology (for the stolen generation). But to me that is a second or third or a fourth issue. The highest priority for me is child abuse and family violence because, while we continue to have Aboriginal kids raped as young as six months old, and women beaten up on a daily basis, what's saying sorry? How's that going to fix that?"
Gordon was born in 1943 at Belele sheep station near Meekatharra, about 750 kilometres north of Perth in the Gascoyne Murchison region of WA. At age four she was removed to Sister Kate's Home for Aboriginal children in Perth because she was considered a "quadroon" or quarter-blood Aborigine, who deserved the chance "to be reared as a white person away from native associations", according to her court papers. (Sister Kate's Home was begun by an Anglican nun in the 1930s as a home for half-caste children. It operated until 1974.)
The court papers also reveal that her half-caste mother, Molly, was considered able to look after her but that was disregarded by the deputy commissioner of native affairs.
Gordon was told she was an orphan after she arrived at the home.
After her white father committed suicide on Christmas Day 1951 and could no longer be forced to pay maintenance for her upkeep at Sister Kate's, she was declared a destitute child.
Gordon was at Sister Kate's with football legend Graham "Polly" Farmer, and is one of the patrons of his foundation to help young Aborigines with sport and their studies. She was also with Aboriginal activist Rob Riley, who killed himself while suffering depression exacerbated by being sacked for drink-driving offences.
Gordon has a picture of Riley pinned to her office noticeboard. She dismisses any suggestion leniency should have been showed for his offences. "There's no excuse," she says.
Riley was raped as a child at Sister Kate's, but that was after the death of the Anglican nun. "When Sister Kate ran it, she ran it in a loving, caring environment in a cottage thing, with no men in the home," Gordon says.
In a 1996 report on ABC's Four Corners, Gordon recalls the children being white until they crossed the road to school where they became "natives and darkies and niggers".
"Of course there was racism," she tells The Age. "We were racist ourselves. We were racist to the Italians. We'd call them wogs, dings, dagos, whatever. And they'd call us blackies, niggers. It's not racism in the way the word's used today . . . To me, Sister Kate's didn't do me any harm."
Gordon's "first family" are the children she grew up with. Her best friends are from there and from her time in the army, which she joined at 18. She worked in the signals regiment and represented the army in netball and athletics.
A few years later, working at Carnarvon, she realised that she had been "playing basketball against some of my cousins and things, but they were just people. I didn't have a need to go and track anybody down because I didn't think I had anybody to track down." A move north to the Pilbara with her first husband, a Vietnam veteran, during the 1970s brought her into contact with people who first told her she wasn't an orphan, "that I had a family" - in fact 11 dark-skinned brothers and sisters.
But it was her family who tracked her down.
"I've had really good relationships, I still do, with all my family. I don't have any issues there."
It was also in the Pilbara, during a 12-year stint working in administrative jobs, helping mainly traditional desert people, that her first marriage, which produced two sons, ended.
"We both had post-traumatic stresses," she says.
"He'd been to Borneo and Vietnam and he was probably killing people before he was 22. I was coming to grips with finding my family and all that sort of rubbish, and so you just don't move anywhere. It's a horrific period but I'm not going to go into all of that, suffice to say we are still friends." Her second husband, a retired police superintendent, died six years ago. Gordon has achieved many "firsts". In 1986, she was the first Aborigine to head a government department in WA, as commissioner for Aboriginal planning; in 1988 she was WA's first Aboriginal magistrate and first full-time children's court magistrate; and in 1990 she was one of five commissioners appointed by federal Labor minister Gerry Hand to the first ATSIC board.
She has been appointed by state and federal governments on both sides of politics to various positions. Her appointment as an Aboriginal magistrate was despite her lack of formal qualifications. She later completed a law degree part-time. She started it when she was 50 and it took eight years, which she juggled with her full-time court work.
Gordon says the National Indigenous Council, set up initially for two years, will provide independent advice to the Government, which has as its priorities family violence, community safety and early childhood intervention.
"We're not Government puppets," she says. "I've said if our advice as a collective body wasn't being adhered to and we felt it was good advice, then we would have a recourse to go and see the Minister (for Indigenous Affairs, Amanda Vanstone)."
She is adamant she won't become a public commentator, "buying into" indigenous issues for the media.
Her friend and colleague Neil Fong, who has worked with her in the WA bureaucracy and on the Gordon Inquiry into domestic violence, describes her as highly principled and not into playing political games. Fong says Gordon's life has a theme of regimentation and character-building activities. "She comes across as a very hard woman but deep down, she's a real softie."
Some indigenous people have privately offered veiled criticism of her. But others have suggested the Government could get more than it bargained for with Gordon.
Outgoing Reconciliation Australia co-chairman and former Fraser government minister Fred Chaney describes her as a person of "fierce independence".
"Were I minister for Aboriginal affairs, I can't think of anyone I would rather have as an adviser. She's a really talented person with a good head and a good heart." | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,489 |
love topic page
4:47 am on Friday June 1, 2012
LGBT club Ru Mors at Kono Lounge opens tonight
Tonight is the Grand Opening party for Ru Mors in Myrtle Beach. It is advertised as the largest gay bar and club in Myrtle Beach. The celebration tonight will be hosted by Chantel Curtis and Sonique Love from "Rupaul's Drag Race" will be performing...
5:50 am on Thursday May 31, 2012
Alternative art space Tivoli celebrates making it official
Inside Tivoli Studios on a Tuesday afternoon, it's hot and there's a bird flapping around in the rafters. The soft drone of oscillating fans pushes the warm air around the open space. But much more than stale breezes and displaced fledglings fill...
10:51 am on Tuesday May 29, 2012
12:00 pm on Thursday May 24, 2012
Wonderfest Free Family Event June 2
Come ut to the 2nd annual Wonderfest, sponsored by Harris Teeter, WCBD News Channel 2 and The Post and Courier this Saturday, June 2, 2012 from 10 a.m. to 3 p.m. at its West Ashley store in the St. Andrews Shopping Center located at 975 Savannah...
5:35 am on Wednesday May 16, 2012
Dead Dog Saloon destroyed by fire (Update: Ready for business this week)
Update May 16, 2012: Just to put a final wrap on this story, Myrtle Beach Restaurant News has photos and coverage of the new Dead Dog Saloon. Update March 26, 2012: Yesterday, Sunday, March 25, 2012, On the Half Shell restaurant in Murrells Inlet...
6:36 am on Tuesday May 15, 2012
Free Family Summer Festival WONDERFEST June 2
Come out to the 2nd annual Wonderfest, sponsored by Harris Teeter, WCBD News Channel 2 and The Post and Courier, Saturday, June 2, 2012 from 10 a.m. to 3 p.m. at Wonder Works in West Ashley located in the in the St. Andrews Shopping Center at 975...
8:04 am on Thursday May 3, 2012
Wedgwood Pottery: Finding the Lost Ayoree Cherokee Indian Mine (A Historical 1767 Wedgwood Trip from Charles Towne)
Notes on Quotes Wedgwood Pottery: Finding the Lost Ayoree Cherokee Indian Mine (A Historical 1767 Wedgwood Trip from Charles Towne) By Gordon Mercer and Marcia Gaines Mercer "The Indians set a high value on their white earth: however I...
8:02 am on Tuesday May 1, 2012
BAKER BUICK GMC CADILLAC FIRST IN NATION TO OFFER 2012 SUPERCHARGED ESCALADE
Tommy Baker, owner and president of Baker Motor Company in West Ashley and Mt. Pleasant, S.C. and Baker Buick GMC Cadillac in West Ashley, announced today that Baker Buick GMC Cadillac is the first in the United States to carry Cadillac's exclusive...
7:12 pm on Monday April 23, 2012
Signs of Spring: The Terracotta Jungle
By Gordon Mercer and Marcia Gaines Mercer Sign in window of plant nursery: "It's spring, I'm so excited I wet my plants!" "It's time," Marcia said, peering out the kitchen window excitedly, "for the wild azalea walk." The half circle path... | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,746 |
Top 100+ questions and answers in HDFS
Q: What is full form of HDFS?
answered Jul 28, 2021 in HDFS by SakshiSharma
Q: HDFS works in a __________ fashion.
hdfs-relationship
Q: Which of the following are the Goals of HDFS?
hdfs-goal
Q: ________ NameNode is used when the Primary NameNode goes down.
namenode
Q: The minimum amount of data that HDFS can read or write is called a _____________.
hdfs-erad
Q: The default block size is ______.
Q: For every node (Commodity hardware/System) in a cluster, there will be a _________.
Q: Which of the following is not Features Of HDFS?
hdfs-features
Q: HDFS is implemented in _____________ language.
hdfs-language
Q: During start up, the ___________ loads the file system state from the fsimage and the edits log file.
hdfs-file-system
Q: Which of the following is the correct representation to access ''Skill" from the
answered Jun 29, 2021 in HDFS by Robindeniel
Q: Pig jobs have the same run time as the native Map Reduce jobs.
pig-jobs
Q: Pig can be used for real-time data updates.
Q: Pig is good for:
Q: Pig is a: choose below
Q: A Map reduce job can be written in:
map-reduce-job
Q: Only one distributed cache file can be used in a Map Reduce job.
Q: Distributed cache files can't be accessed in Reducer.
distributed-cache
Q: Which of the following writables can be used to know value from a mapper/reducer?
Q: Which of the following class is responsible for converting inputs to key-value Pairs of Map Reduce
key-value-pairs
map-reduce
Q: What should be an upper limit for counters of a Map Reduce job?
Q: Which of the following type of joins can be performed in Reduce side join operation?
reduce-side-join
Q: Number of mappers is decided by the
mappers
Q: What is optimal size of a file for distributed cache?
Q: Counters persist the data on hard disk.
data-persist
Q: Distributed Cache can be used in
Q: Reduce side join is useful for
reduce-side
Q: Which of the following is/are true about combiners?
Q: Which of the following operations can't use Reducer as combiner also?
Q: Which of the following can be used to control the number of part files in a map reduce program output directory?
map-program
output-directory
To see more, click for all the questions in this category. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,231 |
import datetime
from django import template
register = template.Library()
@register.filter
def to_date(timestamp):
"""
Turns a timestamp (milliseconds since 1970) into a
python Date object
"""
return datetime.datetime.fromtimestamp(float(timestamp))
@register.filter
def year(date):
"""
Gets the year (integer) from a date object.
Necessary to pipe through the to_date filter.
Example:
{{ some_timestamp|to_date|year }}
This doesn't work, unfortunately:
{{ some_timestamp|to_date.year }}
"""
return date.year
@register.filter
def month(date):
"""
Gets the month (integer) from a date object.
Necessary to pipe through the to_date filter.
Example:
{{ some_timestamp|to_date|month }}
This doesn't work, unfortunately:
{{ some_timestamp|to_date.month }}
"""
return date.month
@register.filter
def day(date):
"""
Gets the day of month (integer) from a date
object.
Necessary to pipe through the to_date filter.
Example:
{{ some_timestamp|to_date|day }}
This doesn't work, unfortunately:
{{ some_timestamp|to_date.day }}
"""
return date.day
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,527 |
Since the late 1990s Bangladesh has been witnessing a large scale of militant and terrorist activities and attacks throughout the country. But in recent times, the pattern of these militant and terrorist activities drastically changed revealing a suspicion of having close ties with the global terrorist and militant organizations like ISIS or Al-Qaeda. Subsequently, the militants often targeted and killed a number of bloggers, atheists, non-Muslim preachers and priests since 2013 and the level of such incidents is increasing alarmingly.
These incidents took place on the heels of the Global Terrorism Index's (GTI, 2015) assessment that the country is 'vulnerable to high terrorism risk'. The GTI score of the country was already on the rise from an historic low of 4.1 in 2012 to 5.47 in 2013 to 5.92 in 2014. However, this trend and recent increase need to be contextualized within the historical developments of militancy in the country and counter-terrorism efforts pursued since 2006.
As Harrison (2006) in his article, A New Hub of Terrorism? In Bangladesh, an Islamic Movement with Al-Qaeda Ties is on the Rise published in The Washington Post, argued that the strategically located nation of Bangladesh is going to be a new regional hub for terrorist operations after analyzing some case studies of killing and bombing by several militant organizations like JMB, JMJB, HuJI-B and others. Also the recent killings of the Rajshahi University professor AFM Rezaul Karim Siddiquee on 23 April and the killing of Xulhaz Mannan, a gay rights activist, and his friend Mahbub Rabbi Tonoy in the capital Dhaka on 26 April 2016 are showing the growing strength of Islamist militancy in Bangladesh revealing the truth of several security and terrorism analysts' comments on the Bangladeshi militants' close links with the global jihadists.
On the basis of primary data, the study examines the growing nature, historical development, causes, consequences, impact, socio-political dimensions, and security challenges of the religious militancy in Bangladesh along with the role of state level institutional arrangements and their limitations in order to combat militancy and terrorism in Bangladesh.
The central argument of the study is that Bangladesh is facing a growing intensity of religious militancy revealing close ties with the global jihadists where banned religious parties and groups are responsible for the recent attacks and killings of the bloggers, atheists, university teachers and non-Muslim priests due to their feeling of alienation from the liberal democratic system; political, economic, social and cultural suppression over the fundamentalists; their goal of establishing own religious ideology within the political system; the weak presence of the main political opposition (BNP) in political affairs; and the limitations of several counter terrorism mechanisms to combat militant activities properly.
Militants engage to secure supports in society and reassert some control over their lives to minimize anomic feelings by providing outlets to manage their stress and encountering exclusionary states.
When resources are available, unorganized but aggrieved groups may make it possible to launch an organized demand for change, giving rise to a consistent movement for: 1) Material and organizational resources, 2) Legitimacy and identity resources, 3) Institutional resources (Hafez, 2003: 19).
Local militias combine of a motivated leadership and a militia ideology. Transnational militias support its branches through networks. Two type militia organizations may emerge:1) Above-ground: leading movement openly within a hierarchical fashion;2) Below-ground/underground: operating movements within secret militias organized in a cell structure (Barkun, 1996: 50-64).
Freilich et al. (2001) mention two forms of ritual in this regard: Form 1– a symbolic display of militarism in public meetings; form 2– paramilitary exercise such as training, boot camp, and practice shooting.
Islamist militant groups in Bangladesh which emerged in the 1990s have undergone several transformations. Originally grown out of the volunteers who joined the Afghan war against the Soviet Union, these groups have since then taken different shapes.
Since the 1990s, five 'generations' of militant groups appeared on the scene. In some measures, the militant groups have come full circle: they began as a result of a global agenda fighting an 'atheist' Communist system (war against the Soviet Union in Afghanistan) to now being part of establishing a global 'khilafat' (by joining the IS in Syria and Iraq) via pursuing a circumscribed local agenda for a period in the early 2000s (Riaz, 2016: 2).
This section was little bit critical for the respondents due to the similarity of the violence of militants with the Islamic political parties along with a maximum portion of ideological indifferences. Surprisingly 38 percent students take their position in favor of 'yes' which indicates the involvement of the major Islamic parties in the rise and expansion of Islamic militant organizations in Bangladesh. The Jamaat-e-Islami and its affiliates are often found in violent militant activities lonely as well as jointly with the militant organizations like JMB, JMJB, HuJI-B etc. across the country.
Here 24 percent students argue in favor of 'no' where their stance shows that since the colonial period the religious political parties are involved in the political, social, economic and cultural development of Bangladesh. Rather the rise of militancy is initiated by the West to retain their influence in the LDCs like Bangladesh after the incident of 9/11 as a part of its 'war on terror' strategy. Also their stance reveals the indirect connection of the major and alienated Islamic political parties provoking the militant organizations. Also 34 percent participants choose the option 'not sure' due to the ambiguity between the violent activities of the Islamic parties and the terrorist activities of the Islamic militant organizations. As the pattern of the attacks and the similarities in terms of ideology and faith are almost same so the responsibilities of the Islamic parties cannot be denied in the rise and expansion of religious militancy in Bangladesh. Only 4 percent regard the 'others' as their answers are that most religious political parties are not responsible rather to some extent in specific acts.
This sub-section entirely deals with the perception of the students regarding the validity of the confession taking liability of the contemporary killing or attacking through internet by ISIS (Ansarullah Bangla Team) or Al-Qaeda (Al-Qaeda in the Indian Subcontinent- AQIS).However, 10 percent participants perceive this confession of killing or attacking by ISIS or Al-Qaeda as 'completely true' arguing that these terrorist organizations mainly target the Muslim majority countries having fragile governance, weak pluralist social fabric, parochial political culture, lack of compromise and conflict resolution mechanism among the major political forces.
As these organizations spread their access from the Middle East to South East Asia so Bangladesh is considered a hub of geostrategic land connecting the mentioned two major regions. Indeed, 66 percent respondents think the validity of this confession of the liability as 'to some extent' indicating that both ISIS and Al-Qaeda have strong affiliation with the local as well as national level Islamic militant organizations in terms of patronizing, training, skilling, making plans for attack, recruitment and so on. So they are just constructing a pathway by smoothing the global-local ties for establishing a religious militant order including Bangladesh.
Surprisingly 20 percent choose the option 'not true' which shows their perception in defense of criminal and terrorist activities by the militant organizations rather than implementing the agenda of ISIS or Al-Qaeda. Also the scale and pattern of global jihadist violence largely differs from the militant activities in Bangladesh. Here 4 percent students choose 'others' arguing that they do not know anything about the confessions.
It is evident that terrorist, militant and extremist activities centering religion are always a matter of concern and pressure within the realm of foreign and diplomatic relations with other countries. 72 percent respondents consider 'yes' as their answer indicating that they strongly believe that such kind of militant and terrorist activities would hamper the bilateral as well as multilateral diplomatic and economic relationships with the major allies like India, the USA and other European countries.
Already Bangladesh lost the 'GSP' in the US market and many of EU business partners revealing the lack of security and certainty due to the violent militant events. As the rise and expansion of militancy in Bangladesh is also a security issue for India more specifically North East India where secessionist movements are going on. For this reason it is often evident the interference of India within the national political arena of Bangladesh. Then 12 percent consider 'no' as the answer arguing that as the rapid growth of economy, well reserved foreign currency, shining garments and industrial sectors and many developmental issues are getting importance to the foreign partners. Religious militancy will not hinder the relationship because no of them is out of militancy and terrorism. Also 16 percent choose the answer 'not sure' because they feel that globalization of terrorism and militancy endangered the states like Bangladesh who are in stage of 'pre-takeoff'. As the development partners put conditions regarding militancy but lack of proper evidences create a grey area for them.
It appears from the above analyses that the scale of religious militancy in Bangladesh is increasing day by day revealing a close tie with the global militant organizations. Here Islamic militancy is threatening the national security of Bangladesh where the militants play a key role in order to exert their own religious doctrine, ideology and supremacy within the political system. The militants are getting stronger and conducting their violent activities in Bangladesh due to the lack of political integration of the religious forces within the liberal fabric; feeling of alienation of the militants; political, economic, social and cultural suppression over fundamentalists etc. However the recent attacks and killings of bloggers, atheists, university teachers, non-Muslim priests and preachers indicate the presence of militancy in Bangladesh revealing the vulnerabilities of the counter terrorism mechanisms as well as 'Zero Tolerance' stance of the government.
These violent incidents by the militants also affect negatively the diplomatic as well as economic relations with India, the USA and European countries. In order to uproot the core of Islamic militancy the state should modernize the madrasha system; integrate or assimilate the religious forces within the liberal order; design a technology and strategy based counter terrorism mechanisms etc. In a nutshell, religious militancy in Bangladesh would be a utopia if democratic tradition, norms and values could be institutionalized following a culture of deliberation and compromise. | {
"redpajama_set_name": "RedPajamaC4"
} | 360 |
\section{Introduction} \label{sec:Introduction}
It is well known that the standard continuous Galerkin (cG) finite element method exhibits poor stability properties for singularly perturbed problems. In presence of sharp boundary or interior layers, non physical oscillations pollute the numerical approximation throughout the solution domain, and thus stabilisation techniques need to be employed; see~\cite{RST08} for a survey. The discontinuous Galerkin (dG) method offers a framework for the design of finite element methods with good stability properties; see, e.g., \cite{CKS00} for a survey of their development. This is achieved by relaxing the continuity requirements at the inter-element boundaries, where appropriate upwinded numerical fluxes can be employed. One such dG method is the Interior Penalty (IP) method considered herein.
However dG methods require more degrees of freedom (DOFs) compared to the standard cG method. In this work, we investigate if some of these additional DOFs can be removed without affecting the stability properties of the dG method. The motivation of this work is twofold: firstly we wish to improve our understanding of the well-established dG methods in response to the criticism in the increased number of DOFs; secondly we seek insight into the design of more efficient finite element methods, allowing to combine advantages the dG framework with smaller approximation spaces.
Conceptually we envisage in this work a finite element space that lies between the standard continuous and discontinuous Galerkin spaces. We construct such a space by applying standard continuous elements away from any boundary or internal layers (called the cG region) and discontinuous elements in the region of such layers (called the dG region). Therefore we call this method the \emph{continuous discontinuous Galerkin} (cdG) finite element method.
In the broader sense, methods of this type have been proposed before by Becker, Burman, Hansbo and Larson \cite{BBHL04}, Perugia and Sch\"{o}tzau \cite{PS01}, and Dawson and Proft \cite{DP02}. In \cite{BBHL04} a globally reduced discontinuous Galerkin method is studied, whose approximation space consists of the continuous piecewise linear functions enriched with piecewise constant functions. In \cite{PS01}, \cite{DP02} the local discontinuous Galerkin method \cite{CS98} has been used on the discontinuous region with transmission conditions on the subset of interelement boundaries where continuous and discontinuous elements meet. Our approach is different to \cite{PS01}, \cite{DP02} as we impose no such conditions beyond those already imposed by the dG method.
Removing the transmission conditions makes the approach more natural, but introduces some difficulties as possible over- and undershoots at the interface of the cG and dG regions must be controlled.
Indeed, the analysis is limited to the case where the cdG interface is non-characteristic so that jumps across the interface are controlled by the convection term. This allows us to derive a rigorous stabilty bound for the cdG approximation. On the cG region we split the numerical solution into an approximation to the hyperbolic problem which is weakly dependent on the diffusion coefficient $\epsilon$ and an approximation to the remaining part, which depends more strongly on $\epsilon$ but is small in size. Stability of the approximation on the dG region is shown using the approach of Buffa, Hughes and Sangali \cite{BHS06} with some extensions in the manner of Ayuso and Marini \cite{AM09}. To our knowledge this paper is the first to present a stability result for the proposed cdG method.
Comparisons of our cdG method with various dG and cG methods have been undertaken by Cangiani, Georgoulis and Jensen \cite{CGJ06} and Devloo, Forti and Gomes \cite{DFG07}; see also \cite{CCGJ12c} which studies the cdG approximation as the limit of a dG approximation as the jump penalization on interelement boundaries tends to infinity.
The remainder of this work is organized as follows. In Section \ref{sec:formulation} we formulate the problem, introduce notation, and explain the assumptions we make.
Control on the continuous region is considered in several stages in Section \ref{sec:epscontrol} and Section \ref{sec:hypcontrol}. Section \ref{sec:infsup} presents an inf-sup bound on the discontinuous region. The main result of the paper showing control independent of the perturbation parameter $\epsilon$ and mesh size $h$ follows in Section \ref{sec:combined}. The theoretical results are illustrated in Section \ref{sec:NumericalExperiments} by a numerical experiment.
\section{The Model Problem and Notation} \label{sec:formulation}
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^d$. We introduce the model advection-diffusion-reaction (ADR) problem
\begin{align}
-\epsilon \Delta u + \vecb(\vecx) \cdot \nabla u + c(\vecx)u = f(\vecx)& \qquad \text{for~} \vecx \in \Omega \subset \Reals^d ,\label{ADReqn}\\
u = 0 \!\!\!\qquad&\qquad \mathrm{on}~ \pO \label{ADReqnbc}
\end{align}
with constant diffusion coefficient $0<\epsilon \le \epsilon_\mathrm{max}$, $\vecb \in [W^{1,\infty}(\Omega)]^d$, $c \in \SL{\infty}{\Omega}$ and $f\in L^2(\Omega)$. For $0 < \epsilon \ll 1$ the solution to this problem typically exhibits boundary or interior layers.
Unless otherwise stated, we define $C$ throughout as a positive constant, independent of $\epsilon$ and the finite element approximation space and which may be redefined from line to line. $C$ may depend on $\epsilon_\mathrm{max}$. By $a \lesssim b$ we mean $a \le Cb$.
We consider $\Th$ to be subdivision of $\Omega$ into non-overlapping shape regular simplices or hyper-cubes $E$, which we shall refer to as the triangulation. Denote by $\Eh$ the union of edges $e$ (or faces for $d\ge 3$) of the mesh and the union of internal edges by $\Eho$. Define $\Gamma$ as the union of boundary edges, i.e., those lying in $\pO$. The diameter of an element $E \in \Th$ is denoted $h_E$ and $h = \max_{\EinTh} h_E$. We also denote the mesh function by $h_E$, thus for $\vecx \in E$ we let $h_E(\vecx)$ be equal to the diameter of $E$. We only consider meshes where $h\le1$. Define $h_e := \min(h_{E^-}, h_{E^+})$ for $e \in \Eho$ with $e = \bar{E}^+ \cap \bar{E}^-$ for $E^+$, $E^- \in \Th$. The mesh is shape regular so there exists $C>0$ such that for all $E$ we have $h_e \le \half (h_{E^+} + h_{E^-} ) \le C h_e$, and $C\ge1$ such that $C^{-1}h_{E^-} \le h_{E^+} \le C h_{E^-}$.
We define by \emph{$\Omega$-decomposition} the splitting of $\Omega$ into two regions $\OC$ and $\OD$ such that for the closure $\overline{\Omega} = \overline{\OC \cup \OD}$, and we define by \emph{$\Th$-decomposition} the splitting of $\Th$ into two sub-meshes $\TC$ and $\TD$ such that $\TC \subset \OC$ and $\TD := \Th \setminus \TC$. By abuse of language, we denoted here by $\TC$ not just the sub-mesh but also the region it occupies. Define $\GC$ (resp.~$\GD$) to be the intersection of $\Gamma$ with $\overline{\T}_\cG$ (resp.~$\overline{\T}_\dG$). Define $J := \overline{\T}_\cG\cap\overline{\T}_\dG$ and by convention we say that the edges lying in $J$ are only part of the discontinuous Galerkin skeleton $\EhD$, the union of faces in $\overline{\T}_\dG$, and not part of the continuous Galerkin skeleton defined by $\EhC := \Eh \setminus \EhD$.
In Figure \ref{fig:cdGdecomposition} we illustrate a splitting for a problem where $\Omega=(0,1)^2$ and the solution exhibits layers at $x=1$ and $y=1$. The $\Omega$-decomposition is labelled, with the demarcation between the $\OC$ and $\OD$ regions given by a dashed line. A $\Th$-decomposition is shown with the $\TD$ region shaded and the edges in $J$ marked with a heavy line.
\begin{figure}[tb]
\centering
\includegraphics[width=0.50\textwidth]{./cdGdecomposition.pdf}
\caption{An example of a cdG decomposition.}
\label{fig:cdGdecomposition}
\end{figure}
Given a generic scalar field $\nu: \Omega \to \Reals$, that may be discontinuous across an edge $e = \bar{E}^+ \cap \bar{E}^-$ for $E^+$, $E^- \in \Th$, we set $\nu^\pm := \nu|_{E^\pm}$, the interior trace on $E^\pm$ and similarly define $\vectau^\pm = \vectau|_{E^\pm}$ for a generic vector field $\vectau : \Omega \to \Reals^d$. Define the average and jump for a generic scalar as
\begin{equation*}
\Average{\nu} := \frac{1}{2} (\nu^+ + \nu^-), \qquad \Jump{\nu} := \nu^+ n^+ + \nu^- n^-, \qquad \mathrm{on}~ e \in \Eho,
\end{equation*}
and for a generic vector field as
\begin{equation*}
\Average{\vectau} := \frac{1}{2} (\vectau^+ + \vectau^-), \qquad \Jump{\vectau} := \vectau^+ \cdot n^+ + \vectau^- \cdot n^-, \qquad \mathrm{on}~ e \in \Eho,
\end{equation*}
where $n^\pm$ is the unit outward pointing normal from $E^\pm$ on $e$. For $e \in \Gamma$ the definitions become
\begin{equation*}
\average{\nu} := \nu, \qquad \jump{\nu} := \nu n, \qquad \average{\vectau} := \vectau, \qquad \mathrm{on}~ e \in \Gamma.
\end{equation*}
We assume that the sign of $b \cdot n$ is the same for every $\vecx \in e$. Given a vector $b$ denote the inflow and outflow boundaries of $\Omega$ by
\begin{align*}
\Gin := \set{\vecx \in \pO : b \cdot n \le 0}, \qquad \Gout := \set{\vecx \in \pO : b \cdot n > 0}
\end{align*}
and for an element
\begin{align*}
\bin E := \set{\vecx \in \pE : b \cdot n \le 0}, \qquad \bout E := \set{\vecx \in \pE : b \cdot n > 0}.
\end{align*}
On an edge $e$, we denote by $\nu^\insymbol$ the trace of a function $\nu$, taken from the element which contains $e$ in its inflow boundary.
\begin{definition}
The cG space is defined by
\begin{equation*} \label{cGSpace}
\VcG{} := \set{ v \in \SH{1}{\Omega} : \forall \EinTh, v|_E \in \Poly^k}
\end{equation*}
and the dG space by
\begin{equation} \label{dGSpace}
\VdG{} := \set{ v \in \SL{2}{\Omega} : \forall \EinTh, v|_E \in \Poly^k}
\end{equation}
where $\Poly^{k}$ is the space of polynomials of either total (or on quadrilateral meshes partial) degree at most $k$ supported on $E$.
\end{definition}
We seek a finite element space $\VcdG{}$ that lies between the cG and dG spaces in the sense that
\begin{equation} \label{VcdGembedding}
\VcG{} \subset \VcdG{} \subset \VdG{}.
\end{equation}
We construct this space by using continuous shape functions away from any boundary or internal layers present in the solution to \eqref{ADReqn} and discontinuous functions in the region of such layers.
\begin{definition}
The cdG space is defined by
\begin{equation} \label{cdGSpace}
\begin{split}
\VcdG{} &:= \set{ v \in \SL{2}{\Omega} : \forall \EinTh, v|_E \in \Poly^k; v|_{\Gamma_\cG} = 0; v|_{\TC} \in C^0(\overline{\T}_{\!\cG})}.
\end{split}
\end{equation}
\end{definition}
Throughout we use the same polynomial degree $k$ for $\VcG{}$, $\VdG{}$ and $\VcdG{}$.
Let $\chi$ be the characteristic function on $\TD$, i.e., that defined by
\begin{equation}\label{chidG}
\chi := \left\{ \begin{array}{ll} 1 & \vecx \in \TD,\\ 0 & \vecx \in \TC. \end{array} \right.
\end{equation}
Then define
\begin{equation*} \label{cdGSpace-D}
\VcdG{}(\TD) := \set{\chi v : v \in \VcdG{}}
\end{equation*}
and
\begin{equation*} \label{cdGSpace-C}
\VcdG{}(\TC) := \set{(1-\chi) v : v \in \VcdG{}}.
\end{equation*}
In order to ensure a unique solution we make the following assumption.
\begin{assumption} \label{ass:ADRrho}
We assume
\begin{equation} \label{ADRrho}
\cb(\vecx) := c(\vecx) - \half \div \vecb(\vecx) \ge \rho > 0 \qquad \forall \vecx \in \Omega,
\end{equation}
for some $\rho \in \Reals$.
\end{assumption}
\begin{definition} \label{def:Peclet}
We define the \emph{local mesh P\'{e}clet number} to be $\norm{\vecb}_\SL{\infty}{E} h_E / (2\epsilon)$, see \cite{RST08}.
\end{definition}
We consider meshes in the pre-asymptotic regime by making the following assumption.
\begin{assumption} \label{ass:Peclet}
We assume that for $\epsilon = \epsilon_\mathrm{max}$ and every $\EinTh$ the local mesh P\'eclet number is greater than $\sqrt{h_E}$. Moreover, we require $\max \{\norm{h_E / b}_\SL{\infty}{\TD},h_E\} \le 1$.
\end{assumption}
As a consequence we have
\begin{equation} \label{Peclet1}
\epsilon \le \epsilon_\mathrm{max} < \half \min_{\EinTh} h_E^{3/2} \norm{\vecb}_\SL{\infty}{\Omega}.
\end{equation}
This assumption, for a fixed $\vecb$, restricts the refinement of the triangulation for a given $\epsilon$. If we allowed $h \to 0$ for fixed $\epsilon > 0$ any layers would be resolved by the mesh and in the limit we would not see the non-physical oscillations associated with the cG approximation. We return to this question in Remark \ref{rem:cea}.
To characterise admissible $\Omega$-decompositions of the mesh we introduce the \emph{reduced} problem:
\begin{equation} \label{ADRreduced}
\begin{split}
\vecb \cdot \nabla u_0 + c u_0 &=f \quad \mathrm{on}~\Omega,\\
u_0 &=0 \quad \mathrm{on}~\Gin.
\end{split}
\end{equation}
Further, we define $u_\epsilon := u - u_0$, where $u$ is the solution to the ADR problem \eqref{ADReqn}.
The $\Omega$-decomposition is chosen such that $u_\epsilon$ and $u_0$ have additional regularity on $\OC$. In general we do not expect that $u_0 \in \SH{2}{\Omega}$, even if we place higher regularity requirements on $f$, see, e.g., \cite{BR82} and the references therein.
\begin{assumption} \label{def:OC}
The set $\OC \subset \Omega$ is chosen such that $u_0 \in \SH{2}{\OC}$ and $\norm{u_\epsilon}_\SH{2}{\OC}$ is bounded uniformly in $\epsilon$, that is for every $0 < \epsilon \le \epsilon_\mathrm{max}$
\begin{equation} \label{ass_ueps}
\norm{u_\epsilon}_\SH{2}{\OC} \lesssim 1.
\end{equation}
\end{assumption}
\subsection{Decoupled and Standard Formulations}
We define by $\nablah$ the elementwise gradient operator. We discretize the advection term by
\begin{equation} \label{ADRBa}
\Bform_a(w,\what) := \sum_\EinTh \int_E (\vecb \cdot \nablah w) \what \dx - \sum_\einEho \int_e \vecb \cdot \jump{w} \, \what^\insymbol \ds - \sum_{e \in \Gin} \int_e (\bdotn) w \, \what \ds,
\end{equation}
and the reaction term by
\begin{equation} \label{ADRBr}
\Bform_r(w,\what) := \sum_\EinTh \int_E c w \what \dx.
\end{equation}
The advection and reaction parts will frequently occur together and so for brevity we also define $\Bform_{ar}(w,\what) := \Bform_a(w,\what)+\Bform_r(w,\what)$.
For the diffusion term, besides the standard symmetric interior penalty method we also present a modified scheme, which we call the {\em decoupled} method. We refer to \cite{ABCM01} for a comprehensive study of the interior penalty method and alternative discretizations. For $w,\what \in \VdG{}$ the decoupled method is defined by
\begin{equation} \label{ADRBdtilde}
\begin{split}
\Btilde{d}(w,\what) &:= \sum_\EinTh \int_E \nablah w \cdot \nablah \what \dx \\
& \qquad +\sum_{\einEh\setminus J} \int_e \sigma h_e^{-1} \jump{w} \cdot \jump{\what} - \left( \average{\nablah w}\cdot \jump{\what} + \average{\nablah \what}\cdot \jump{w}\right) \ds,
\end{split}
\end{equation}
yielding
\begin{equation} \label{ADRBepstilde}
\Btilde{\epsilon} (w,\what) := \epsilon \Btilde{d}(w,\what) + \Bform_a(w,\what) + \Bform_r(w,\what).
\end{equation}
Here $\sigma>0$ is a discontinuity penalization parameter.
With this formulation there is no control on the fluxes across $J$ (hence the name decoupled).
We recall that the standard symmetric Interior Penalty (IP) method is given by
\begin{equation*}
\begin{split}
\Bform_d(w,\what) &:= \Btilde{d}(w,\what) + \sum_{e \in J} \int_e \sigma h_e^{-1} \jump{w} \cdot \jump{\what} - \left( \average{\nablah w} \cdot \jump{\what} + \average{\nablah \what} \cdot \jump{w}\right) \ds,
\end{split}
\end{equation*}
and
\begin{equation*}
\Bform_\epsilon (\what,w) := \epsilon \Bform_d(\what,w) + \Bform_a(\what,w) + \Bform_r(\what,w).
\end{equation*}
When restricted to the cdG space, the decoupled and standard IP forms become the bilinear form for the standard cG method on the continuous region. We first first analyse the stability properties of the decoupled formulation and infer stability for the standard IP method from a perturbation argument.
We introduce the following mesh dependent norm for $w \in \VdG{}$:
\begin{equation} \label{ADRtriple}
\begin{split}
\triple{w}^2 &:= \norm{w}_d^2 + \norm{w}_{ar}^2,
\end{split}
\end{equation}
where
\begin{equation*}
\norm{w}_d^2 := \sum_\EinTh \epsilon \abs{w}_\SH{1}{E}^2 + \sum_{\einEh} \epsilon\sigma h_e^{-1} \norm{\jump{w}}^2_\SL{2}{e}
\end{equation*}
and
\begin{equation*}
\norm{w}^2_{ar} = \norm{\cbhalf w}^2_\SL{2}{\Omega} + \sum_\einEh \half\norm{\abs{\bdotn}^{\fhalf} \jump{w}}^2_\SL{2}{e},
\end{equation*}
with $\cb$ defined in \eqref{ADRrho}.
Recall that for the symmetric interior penalty method the parameter $\sigma$ is selected independently of the $\Th$ such that $\Bform_d$ is positive definite with a coercivity constant which is also independent of $\Th$
We assume that $\sigma$ is such that for all $w \in \VdG{}$:
\begin{equation} \label{eqn:sigma}
\norm{\average{\nablah w}\cdot \jump{w}}_\SL{1}{\Eh} \le \half \norm{\nablah w}_\SL{2}{\Omega} \norm{\sqrt{\sigma / h_e} \jump{w}}_\SL{2}{\Eh}.
\end{equation}
Then, by Young's inequality, we have
\begin{equation*}
\half \norm{w}_d^2 \le \Bform_d(w,w).
\end{equation*}
We adopt for $\Btilde{d}$ the same $\sigma$ as for $\Bform_d$.
We introduce a projection operator following the presentation of \cite{AM09} which allows us to consider non-constant $\vecb$. For polynomial degree $k \ge 0$ consider the $L^2$-orthogonal projection $\Proj{}{\D}:\SL{2}{\Omega} \to \VcdG{}(\TD)$ defined by
\begin{equation} \label{ProjD}
\int_\Omega \Proj{}{\D}(v) w \dx = \int_\Omega v w \dx \qquad \forall w \in \VcdG{}(\TD).
\end{equation}
In particular $\Proj{}{\D}(v)|_\TC = 0$. Furthermore, for all elements $E$ of the mesh
\begin{equation} \label{DGL2-projection-norm}
\norm{\Proj{}{\D}(v)}_\SL{2}{E} \le \norm{v}_\SL{2}{E} \qquad \forall v\in \SL{2}{E}.
\end{equation}
As $\Proj{}{\D}(v) \in \VcdG{}(\TD)$ we have for all $\EinTh$ the inverse inequality
\begin{align} \label{infsupInverse}
\abs{\Proj{}{\D} (\vecb \cdot \nablah v)}_\SH{1}{E} &\lesssim h_E^{-1} \norm{\Proj{}{\D} (\vecb \cdot \nablah v)}_\SL{2}{E},
\end{align}
and using a trace inequality we have
\begin{equation} \label{infsupJump}
\sum_{\einEh} \norm{\jump{\Proj{}{\D} (\vecb \cdot \nablah v)}}^2_\SL{2}{e} \lesssim \sum_{\EinTh} h_E^{-1} \norm{\Proj{}{\D}(\vecb \cdot \nablah v)}_\SL{2}{E}^2.
\end{equation}
Define the \emph{streamline} norm by
\begin{equation} \label{RestrictedSDGnorm}
\norm{v}^2_\SDG := \triple{v}^2 + \sum_{\EinTh} \tau_E \norm{\Proj{}{\D}(\vecb\cdot\nablah v)}^2_\SL{2}{E},
\end{equation}
where $\tau_E$ is defined by
\begin{equation} \label{deftauCDG}
\tau_E := \displaystyle\tau \min \left\{ \frac{h_E}{\norm{b}_\SL{\infty}{E}}, \frac{h_E^2}{\epsilon} \right\},
\end{equation}
and $\tau$ is a positive number at our disposal.
\begin{definition} \label{def:vhtilde}
A decoupled cdG approximation to \eqref{ADReqn} is defined as $\vhtilde \in \VcdG{}$ satisfying
\begin{equation} \label{decoupledcdG}
\Btilde{\epsilon} (\vhtilde,v) = \int_\Omega fv \dx \qquad \forall v \in \VcdG{}.
\end{equation}
\end{definition}
\begin{definition} \label{def:vh}
A cdG approximation to \eqref{ADReqn} is defined as $\vh \in \VcdG{}$ satisfying
\begin{equation}
\Beps (\vh,v) = \int_\Omega fv \dx \qquad \forall v \in \VcdG{}.
\end{equation}
\end{definition}
We require that $b$ points on $J$ non-characteristically from $\TC$ to $\TD$.
\begin{assumption}\label{ass:binterface}
The $\Th$-decomposition is such that for every $e \in J$
\begin{equation} \label{binterface}
\frac{1}{4} (\vecb(\vecx) \cdot n^\C)|_e > \epsilon_\mathrm{max} \frac{\sigma}{h_e^{3/2}} \qquad \forall \vecx \in e,
\end{equation}
where $n^\C$ represents the unit normal pointing from $\TC$ to $\TD$.
\end{assumption}
Observe that the scaling between $\epsilon$ and $h$ mirrors that of Assumption \ref{ass:Peclet}.
\begin{theorem} \label{thm:Bcoercivity}
On $\VdG{}$ the bilinear forms $\Btilde{\epsilon}$ and $\Beps$ are coercive with respect to $\triple{w}$:
\begin{equation} \label{Bcoercivity}
\frac{1}{4} \triple{w}^2 \le \Btilde{\epsilon}(w,w), \qquad \frac{1}{4} \triple{w}^2 \le \Beps(w,w), \qquad w \in \VdG{}.
\end{equation}
\end{theorem}
\begin{proof}
For the advection and reaction terms using integration by parts we have
\begin{equation}\label{forJterms}
\Bform_{ar}(w,w) = \norm{\cbhalf w}^2_\SL{2}{\Omega} + \sum_\einEh \int_e \half \abs{\bdotn}\jump{w}\cdot \jump{w} \ds.
\end{equation}
For the diffusion term it follows from Equation \ref{eqn:sigma} and Young's inequality that
\begin{align*}
\Btilde{d}(w,w) + \int_J \frac{\sigma}{h_e} \jump{w} \cdot \jump{w} \ds \ge \half \| w \|_d^2, \qquad \Bform_d(w,w) \ge \half \| w \|_d^2.
\end{align*}
Combing the last inequality with \eqref{forJterms}, the result now follows with Assumption \ref{ass:binterface}.
\end{proof}
It follows that $\vhtilde$ and $\vh$ exist and are unique. The following final assumption permits the use of an inverse inequality on the continuous Galerkin region.
\begin{assumption} \label{ass:quasi}
The mesh $\TC$ is quasi-uniform.
\end{assumption}
It is convenient to denote the mesh-size on $\TC$ by $h_\TC = \| h_E \|_{L^\infty(\TC)}$.
The main result of this work is Theorem \ref{thm:vhstab}, which states that the cdG approximation is stable in the streamline diffusion norm whenever Assumptions \ref{ass:ADRrho}, \ref{ass:Peclet},
\ref{def:OC},
\ref{ass:binterface} and \ref{ass:quasi} are satisfied.
In order to prove this result we establish first two separate stability bounds: We define $\vepstilde, \votilde \in \VcdG{}$ by the condition that for all $v\in\VcdG{}$
\begin{align}
\Btilde{\epsilon}(\vepstilde,v) &= \Btilde{\epsilon}(u_\epsilon,v), \label{vepstilde}\\
\Btilde{\epsilon}(\votilde,v) &= \Btilde{\epsilon}(u_0,v). \label{votilde}
\end{align}
Observe that by linearity of the decoupled cdG method we have $\vepstilde + \votilde = \vhtilde$. In Section \ref{sec:epscontrol} we derive a bound for the decoupled cdG approximation $\vepstilde$ to $u_\epsilon$ on $\TC$; in Section \ref{sec:hypcontrol} we obtain a bound for the decoupled cdG approximation $\votilde$ to $u_0$ on $\TC$. In Section \ref{sec:infsup} we establish an inf-sup condition with streamline control on $\TD$. Finally, in Section \ref{sec:combined} we combine these results to show stability of the decoupled and then of the standard cdG approximation.
\section{Bounds on the $\vepstilde$ Component on $\TC$} \label{sec:epscontrol}
We introduce the projection operator of Scott and Zhang, \cite{SZ90} \cite[Section 1.6.2]{EG04}.
\begin{lemma}[Scott-Zhang Projection] \label{thm:SZ}
The Scott-Zhang operator $\mathcal{S\!Z}_h : \SW{l}{p}{\Omega} \to \VcG{}$ is a mapping with the following properties: For $l > \half$ there exists a $\Csz>0$ such that for all $0\le m\le\min(1,l)$
\begin{equation} \label{SZ1}
\norm{\SZ{v}}_\SH{m}{\TC} \le \Csz \norm{v}_\SH{l}{\TC}\qquad \forall v \in \SH{l}{\TC},
\end{equation}
and provided $l\le k+1$ for all $E \in \TC$ and $0\le m \le l$ we have the approximation
\begin{equation} \label{SZ2}
\norm{v-\SZ{v}}_\SH{m}{E} \le \Csz h_E^{l-m} \abs{v}_\SH{l}{\patchE} \qquad \forall v \in \SH{l}{\patchE}.
\end{equation}
where $\patchE$ is the node patch of $E$, i.e., the set of cells in $\TC$ sharing at least one vertex with~$E$.
\end{lemma}
\begin{theorem} \label{thm:vepstilde}
The decoupled cdG approximation $\vepstilde$ is stable on the $\TC$ region in the sense that
\begin{equation} \label{vepstildestab}
\| \vepstilde \|_{H^1(\TC)} \lesssim 1.
\end{equation}
\end{theorem}
\begin{proof}
We pick the \emph{auxiliary} solution $\va \in \VcdG{}$ as follows: On $\TC$, define $\va$ to be the Scott-Zhang projection of $u_\epsilon$; and on $\TD$ to be the dG approximation with boundary conditions given by $\SZ{u_\epsilon}$ on $J$ and $0$ on $\GD$, i.e.,
\begin{align*}
\va &= \SZ{u_\epsilon} \quad\mathrm{on}~\TC,\\
\Btilde{\epsilon}(\va,v) &= \Btilde{\epsilon} (u_\epsilon,v) \quad\forall v\in \VcdG{}(\TD).
\end{align*}
Set $\eta := u_\epsilon - \va$ and $\xi := \va - \vepstilde$, so $\eta + \xi = u_\epsilon - \vepstilde$. Notice that $\xi \in \VcdG{}$. The Galerkin orthogonality expressed by \eqref{vepstilde} and Theorem \ref{thm:Bcoercivity} give
\begin{equation} \label{xiorthog}
{\textstyle \frac{1}{4}} \triple{\xi} \le \Btilde{\epsilon}(\xi,\xi) = -\Btilde{\epsilon}(\eta,\xi) = -\Btilde{\epsilon}(\eta,\xi-\chi\xi),
\end{equation}
where $\chi$ is defined in \eqref{chidG}. Note that $\xi - \chi\xi$ is continuous except on $J$ where $\jump{\xi-\chi\xi} = \xi^\C \cdot n^\C$ and $\average{\xi-\chi\xi} = \half \xi^\C$, where the superscript $\C$ indicates the trace taken from the continuous Galerkin side of $J$.
We examine each term of $\Btilde{\epsilon}$ in turn. For the diffusion parts we use Young's inequality
\begin{align*}
-\Btilde{d}(\eta,\xi-\chi\xi) &\le 2 \abs{\eta}^2_\SH{1}{\TC} + {\textstyle \frac{1}{8}} \abs{\xi}^2_\SH{1}{\TC}.
\end{align*}
For the advection term we use Assumption \ref{ass:binterface} which ensures that flux terms on $J$ are zero as the upwind value of $\xi - \chi\xi$ vanishes. With Young's inequality we have
\begin{align*}
-\Bform_{a}(\eta,\xi-\chi\xi) &\le \textstyle \frac{4}{\rho} \norm{b\cdot \nablah\eta}^2_\SL{2}{\TC} + \frac{\rho}{16}\norm{\xi}^2_\SL{2}{\TC},
\end{align*}
where $\rho$ is defined in \eqref{ADRrho}. Finally for the reaction term
\begin{align*}
-\Bform_{r}(\eta,\xi-\chi\xi) &\le \textstyle \frac{4}{\rho} \norm{c}^2_\SL{\infty}{\Omega}\norm{\eta}^2_\SL{2}{\TC} + \frac{\rho}{16} \norm{\xi}^2_\SL{2}{\TC}.
\end{align*}
Using the previous three results, \eqref{xiorthog}, the definition of the norm \eqref{ADRtriple}, and Lemma \ref{thm:SZ} we gather $\xi$ terms on the left hand side to show, with $h_\TC = \| h_E \|_{L^\infty(\TC)}$,
\begin{align} \nonumber
\textstyle \frac{1}{8} \triple{\xi}^2
&\le \textstyle 2 \epsilon \abs{\eta}^2_\SH{1}{\TC} + \frac{4}{\rho}\norm{b\cdot \nablah\eta}^2_\SL{2}{\TC}+ \frac{4}{\rho} \norm{c}^2_\SL{\infty}{\Omega}\norm{\eta}^2_\SL{2}{\TC}\\ \label{vepstilde-bound}
&\lesssim (\epsilon h_\TC^2 + h_\TC^2 + h_\TC^4) \norm{u_\epsilon}_\SH{2}{\OC}^2 \lesssim h_\TC^2
\end{align}
where in the final step we have used \eqref{ass_ueps}. As $\rho > 0$ we may use \eqref{vepstilde-bound} and an inverse inequality to show
\begin{equation} \label{vtildeepsinv}
\norm{\xi}^2_\SH{1}{\TC} \lesssim h_\TC^{-2} \norm{\xi}^2_\SL{2}{\TC} \lesssim h_\TC^{-2} \triple{\xi}^2 \lesssim 1.
\end{equation}
Assumption \ref{def:OC} and \eqref{SZ1} give $\norm{\vepstilde}^2_\SH{1}{\TC} \lesssim 1$.
\end{proof}
\section{Bounds on the $\votilde$ Component on $\TC$} \label{sec:hypcontrol}
We now pick the auxiliary solution $\va$ as follows: On $\TD$ let it be $u_0$ and on $\TD$ be the dG approximation to $u_0$ with boundary conditions given by $u_0$ on $\GD \cup J$, i.e.,
\begin{align} \label{va_on_u0}
\va = u_0 \qquad &\mathrm{on}~\TC,\\
\Btilde{\epsilon}(\va,v) = \Btilde{\epsilon} (u_0,v) \qquad & \forall v\in \VcdG{}(\TD). \label{va_TD}
\end{align}
\begin{lemma} \label{thm:Bequivalenceva}
We have for all $v \in \VcdG{}$ that $\Btilde{\epsilon}(\va,v) = \Btilde{\epsilon}(\votilde,v)$.
\end{lemma}
\begin{proof}
Fix $v\in\VcdG{}$. Then using \eqref{votilde}
\begin{equation*}
\Btilde{\epsilon}(\votilde,v) = \Btilde{\epsilon}(u_0,v) = \Btilde{\epsilon}(u_0,v-\chi v) +\Btilde{\epsilon}(u_0,\chi v)
\end{equation*}
where $\chi$ is defined in \eqref{chidG}. Observe that $\Btilde{\epsilon}(u_0,\chi v) = \Btilde{\epsilon}(\va, \chi v)$ by \eqref{va_TD}. Notice that $v - \chi v$ and $u_0$ are continuous on $\TC$. Recall that no integral over $J$ appears in the definition of $\Btilde{d}$. For $\Btilde{a}(\votilde,v-\chi v)$, the integral over $J$ vanishes since the value of $(v-\chi v)^\insymbol$ is zero because of Assumption \ref{ass:binterface}. Therefore $\Btilde{\epsilon}(\votilde,v-\chi v) = \Btilde{\epsilon}(u_0,v-\chi v) = \Btilde{\epsilon}(\va,v-\chi v)$.
\end{proof}
\begin{lemma} \label{thm:nablau0bound}
We have $\| \votilde \|_{H^1(\TC)} \lesssim 1$.
\end{lemma}
\begin{proof}
Define $\vpitilde$ to be
\begin{eqnarray*}
\vpitilde &:= \left\{
\begin{array}{ll}
\SZ{u_0} & \mathrm{on}~ \TC,\\
\va &\mathrm{on}~ \TD,
\end{array} \right.
\end{eqnarray*}
and let $\eta:=\va-\vpitilde$, $\xi:=\vpitilde-\votilde$. With these definitions $\eta+\xi=\va -\votilde$, $\eta|_\TD = 0$ and $\xi$ and $\eta$ are continuous on $\TC$. Then using Lemma \ref{thm:Bequivalenceva} we have
\begin{align*}
\textstyle \frac{1}{4} \triple{\xi}^2 \le \; & \Btilde{\epsilon}(\xi,\xi) = -\Btilde{\epsilon}(\eta,\xi)\\
= \; & - \int_\TC \epsilon \nablah \eta \cdot \nablah \xi + (\vecb\cdot\nablah\eta)\xi + c\eta\xi \dx + \int_J \vecb \cdot \jump{\eta} \xi^\insymbol \ds.
\end{align*}
Due to Assumption \ref{ass:binterface} we have $\xi^\insymbol = \xi^\D$, the trace from the dG side of $J$, and $\jump{\eta} = \eta^\C n^\C$, the trace and normal from the cG side of $J$. We split each of the terms using Young's inequality, giving
\begin{align} \nonumber
\textstyle \frac{1}{4} \triple{\xi}^2
&\le 2 \epsilon \norm{\nablah \eta}^2_\SL{2}{\TC} + \frac{\epsilon}{8}\norm{\nablah \xi}^2_\SL{2}{\TC} + \frac{4}{\rho}\norm{\vecb \cdot \nablah \eta}_\SL{2}{\TC} + \frac{\rho}{16}\norm{\xi}^2_\SL{2}{\TC}\\
&\qquad + \frac{4}{\rho}\norm{c}^2_\SL{\infty}{\Omega}\norm{\eta}^2_\SL{2}{\TC} + \frac{\rho}{16}\norm{\xi}^2_\SL{2}{\TC} + \int_J (\vecb \cdot n^\C \eta^\C)\xi^\D \ds. \label{xi-bound-A}
\end{align}
For the final term we note that $\xi$ is a polynomial and so using Young's inequality and a trace and inverse inequality (with constant $\Cti$) gives
\begin{align} \label{xi-bound-B}
\int_J (\vecb \cdot n^\C \eta^\C)\xi^\D \ds &\le \frac{4 \Cti\norm{\vecb}^2_\SL{\infty}{\Omega}}{h_e \rho} \norm{\eta^\C}^2_\SL{2}{J} + \frac{\rho}{16} \norm{\xi}^2_\SL{2}{\TD}.
\end{align}
We combine \eqref{xi-bound-A} and \eqref{xi-bound-B} to hide all terms of $\xi$ under the norm on the left-hand side of \eqref{xi-bound-A}. Using \eqref{SZ2} for the terms of $\eta$ and a trace inequality gives
\begin{align*}
\rho \norm{\xi}^2_\SL{2}{\TC} \le \triple{\xi}^2 \lesssim (\epsilon h_\TC^2 + h_\TC^4 + h_\TC^2)\norm{u_0}^2_\SH{2}{\TC} \lesssim h_\TC^2 \norm{u_0}^2_\SH{2}{\TC}
\end{align*}
and, by an inverse inequality, $\| \xi \|_{H^1(\TC)}^2 \lesssim 1$. Now the result follows from the stability of the Scott-Zhang operator.
\end{proof}
\section{Inf-Sup Condition} \label{sec:infsup}
The following theorem is an adaptation of related stability bounds in \cite{BHS06} and \cite{AM09} to fit the above assumptions. Although the verification of the below inf-sup condition follows the overall structure in \cite{BHS06}, we state it here in detail as the present analysis extends the scope to non-constant advection coefficients via the incorporation of $\Proj{}{\D}$ as \cite{BHS06}. Moreover, it deals with the modification of the bilinear form on $J$ and it only has streamline control on the $\TD$ side. It is helpful to recall that $\Proj{}{\D} v|_\TC = 0$ for any $v$.
\begin{theorem} \label{thm:infsupcdGBtilde}
There exists a positive constant $\Lis$ which is independent of $h$ and $\epsilon$ but may depend on the polynomial degree, $\sigma$, and the constants in \eqref{infsupInverse} and \eqref{infsupJump} such that:
\begin{equation}
\inf_{v \in \VcdG{}} \; \sup_{\vhat \in \VcdG{}} \; \frac{\Btilde{\epsilon}(v,\vhat)}{\norm{v}_\SDGr\norm{\vhat}_\SDGr} \ge \Lis.
\end{equation}
\end{theorem}
\begin{proof}
For an arbitrary $v \in \VcdG{}$, we define
\begin{equation} \label{infsupdecomposition}
\vhat := v + \gamma \, \ws, \qquad \ws := \sum_{\EinTh} \tau_E \Proj{}{\D} (\bdnab v),
\end{equation}
where $\gamma$ is a positive parameter at our disposal and $\tau_E$ is defined in \eqref{deftauCDG}. Note that through the definition of $\Proj{}{\D}$ we have $\vhat,\ws \in \VcdG{}$. Theorem \ref{thm:infsupcdGBtilde} is equivalent to showing the following two results:
\begin{align}
\norm{\vhat}_\SDGr &\lesssim \norm{v}_\SDGr, \label{infsupcdG1}\\
\Btilde{\epsilon}(v,\vhat) &\gtrsim \norm{v}^2_\SDGr \label{infsupcdG2}.
\end{align}
Consider first \eqref{infsupcdG1}. We examine each term of $\norm{\ws}^2_\SDGr$ in turn. We have
\begin{equation}\label{eq:ell_bound}
\begin{split}
\sum_\EinTh \epsilon\abs{\ws}^2_\SH{1}{E} \lesssim \; & \sum_\EinTh \epsilon h_E^{-2}\norm{\tau_E \Proj{}{\D}(\bdnab v)}^2_\SL{2}{E}\\
\le \; & \sum_\EinTh \tau\tau_E \norm{\Proj{}{\D}(\bdnab v)}^2_\SL{2}{E} \lesssim \norm{v}_\SDGr^2.
\end{split}
\end{equation}
Also
\begin{align} \label{eq:hyp_bound}
\norm{\cbhalf \ws}^2_\SL{2}{\Omega} &\le \norm{\cb}_\SL{\infty}{\Omega} \sum_{E \in \Th} \tau_E^2 \norm{\Proj{}{\D}(\bdnab v)}^2_\SL{2}{E} \lesssim \norm{v}_\SDGr^2.
\end{align}
For the terms on the edges we use \eqref{infsupJump}. This gives
\begin{align} \label{eq:hyp_jump_bound}
\sum_{\einEh} \norm{\abs{\bdotn}^\fhalf \jump{\ws}}^2_\SL{2}{e} &\lesssim \sum_\EinTh \norm{\vecb}_\SL{\infty}{\Omega} \tau_E^2 h_E^{-1} \norm{\Proj{}{\D}(\bdnab v)}^2_\SL{2}{E} \lesssim \norm{v}_\SDGr^2.
\end{align}
Similarly,
\begin{align} \label{eq:ell_jump_bound}
\sum_{e \in \Eh} \frac{\sigma\epsilon}{h_e}\norm{\jump{\ws}}^2_\SL{2}{e} &\lesssim \sum_\EinTh \tau_E^2 \frac{\sigma\epsilon}{h_E^2}\norm{\Proj{}{\D}(\bdnab v)}^2_\SL{2}{E} \lesssim \norm{v}_\SDGr^2.
\end{align}
The final term of the streamline norm gives
\begin{align*}
\sum_\EinTh \tau_E\norm{\Proj{}{\D}(\bdnab \ws) }^2_\SL{2}{E} &\le \sum_\EinTh \tau_E \norm{\bdnab \left( \tau_E \Proj{}{\D} (\bdnab v) \right)}^2_\SL{2}{E}\\
&\lesssim \sum_\EinTh \tau_E^3 \norm{\vecb}_\SL{\infty}{E}^2 h_E^{-2}\norm{\Proj{}{\D}(\bdnab v) }_\SL{2}{E}^2 \lesssim \norm{v}_\SDGr^2.
\end{align*}
Combining the above results we have $\norm{\ws}_\SDGr^2 \lesssim \norm{v}_\SDGr^2$. Using a triangle inequality we find
\begin{align*}
\norm{\vhat}_\SDGr &\le \norm{v}_\SDGr + \gamma \, \norm{\ws}_\SDGr \le C(\tau,\sigma,\gamma) \norm{v}_\SDGr,
\end{align*}
which concludes the proof of \eqref{infsupcdG1}.
To prove \eqref{infsupcdG2} first consider the advection and reaction terms of the norm. Using the linearity of $\Bform_{ar}$ we have $\Bform_{ar}(v,\vhat) = \Bform_{ar}(v,v) + \gamma \Bform_{ar}(v,\ws)$. The second term is given by
\begin{align*}
\Bform_{ar}(v,\ws) &= \sum_\EinTh \int_E c v (\tau_E \Proj{}{\D}(\bdnab v)) + (\bdnab v)(\tau_E \Proj{}{\D}(\bdnab v)) \dx\\
&\qquad - \sum_\einEho \int_e \vecb \cdot \jump{v}(\tau_E \Proj{}{\D} (\bdnab v))^\insymbol \ds - \sum_{e\in \Gin} \int_e (\bdotn) v(\tau_E \Proj{}{\D}(\bdnab v)) \ds.
\end{align*}
Using the properties of $\Proj{}{\D}$ given in \eqref{ProjD} the second term above becomes
\begin{equation}
\begin{split} \label{BarSpart}
\sum_\EinTh \int_E (\bdnab v)(\tau_E \Proj{}{\D}(\bdnab v)) \dx & = \sum_\EinTh \int_E \tau_E \Proj{}{\D}(\bdnab v) \Proj{}{\D}(\bdnab v) \dx\\ &= \sum_\EinTh \tau_E \norm{\Proj{}{\D}(\bdnab v)}^2_\SL{2}{E}.
\end{split}
\end{equation}
Using Young's inequality we have
\begin{align*}
\Bigl| \sum_\EinTh \int_E c v (\tau_E \Proj{}{\D}(\bdnab v)) \dx \Bigr| &\le \| c \|_{L^\infty(\Omega)} \sum_\EinTh \frac{1}{2} \norm{v}^2_\SL{2}{E} + \frac{1}{2}\tau_E^2 \norm{\Proj{}{\D}(\bdnab v)}^2_\SL{2}{E}
\end{align*}
and, where $C$ arises from a trace inequality and the number of edges per element,
\begin{align*}
&- \sum_\einEho \int_e \vecb \cdot \jump{v}(\tau_E \Proj{}{\D} (\bdnab v))^\insymbol \ds - \sum_{e\in \Gin} \int_e (\bdotn) v(\tau_E \Proj{}{\D} (\bdnab v)) \ds\\
& \qquad \le \sum_\einEh \frac{C \lambda}{2} \norm{\abs{\bdotn}^\fhalf\jump{v}}^2_\SL{2}{e} + \sum_\EinTh \frac{\tau_E \tau}{2\lambda} \norm{\Proj{}{\D}(\bdnab v)}^2_\SL{2}{E}.
\end{align*}
In conclusion, using \eqref{forJterms}, we have
\begin{equation} \label{infsupcdGBar}
\begin{split}
\Bform_{ar}(v,\vhat) &\ge \left(\rho-\frac{\gamma \| c \|_{L^\infty(\Omega)}}{2}\right)\sum_\EinTh \norm{v}^2_\SL{2}{E} + \left(\half - \frac{\gamma C \lambda}{2}\right)\sum_\einEh \norm{\abs{\bdotn}^\fhalf\jump{v}}^2_\SL{2}{e}\\
&\qquad + \gamma\sum_\EinTh \left(\tau_E-\frac{\tau_E^2}{2}-\frac{\tau_E\tau}{2\lambda}\right) \norm{\Proj{}{\D}(\bdnab v)}^2_\SL{2}{\Omega}.
\end{split}
\end{equation}
Recall that $\norm{h_E / b}_\SL{\infty}{\TD} \le 1$ by Assumption \ref{ass:Peclet}, which imples $\tau_E \le 1$ for all $E \in \Th$. For general $v$, all terms on the right-hand side of \eqref{infsupcdGBar} are positive, provided $\lambda$ is large and $\gamma$ is small enough.
Equation (\ref{eqn:sigma}) ensures the continuity of $\Btilde{d}$ with respect to $\norm{\,\cdotp}_d$; thus
\begin{align} \label{eq:cont}
\Btilde{d}(v,\vhat) \le C_1 \norm{v}_d \norm{\vhat}_d,
\end{align}
for some $C_1 > 0$. Recalling \eqref{eq:ell_bound} and \eqref{eq:ell_jump_bound}, it is clear that $\norm{\ws}_d \le C_2 \norm{v}_d$, for some $C_2 > 0$. Hence
\begin{align}
\Btilde{d}(v,\vhat) = \Btilde{d}(v,v) + \gamma \Btilde{d}(v,\ws) \ge \textstyle \frac{1}{4} \norm{v}_d^2 - \gamma C_1 \norm{v}_d \norm{\ws}_d.
\end{align}
Thus, if $\gamma < C_1 C_2 / 8$, then $\Btilde{d}(v,\vhat) \ge \frac{1}{8} \norm{v}_d^2$ which, combined with \eqref{infsupcdGBar}, gives \eqref{infsupcdG2}.
\end{proof}
\section{Stability of the Decoupled and Standard Approximations} \label{sec:combined}
We saw that, under a set of suitable assumptions, the decoupled approximation satisfies the stability bounds:
\begin{align} \label{eq:vhtildestab}
\| \vhtilde \|_{H^1(\TC)} \lesssim 1, \qquad \| \vhtilde \|_\SDGr \lesssim \| f \|_{L^2(\Omega)}.
\end{align}
The first bound is a consequence of Theorem \ref{thm:vepstilde} and Lemma \ref{thm:nablau0bound}, the second of Theorem~\ref{thm:infsupcdGBtilde}. So while one has streamline-diffusion stability on $\TD$, an even stronger bound is available on~$\TC$ under the aforementioned assumptions. We now derive a stability result for the cdG method. We require that the geometry of the interface $J$ does not become significantly more complicated as the mesh is refined. More precisely, we require the boundedness of the trace operator.
\begin{theorem} \label{thm:vhstab}
Suppose that the operator norm of the trace $H^1(\TC) \to L^2(J)$ is bounded independently of $h$. Then, the cdG approximation $\vh$ is stable in the sense that
\[
h_\TC \norm{\nablah \vh}^2_\SL{2}{\TC} + \| \vh \|_\SDGr^2 \lesssim 1 + \| f \|_{L^2(\Omega)}^2.
\]
\end{theorem}
\begin{proof}
Set $\zeta:=\vh-\vhtilde$. Using the coercivity of $\Beps$, Galerkin orthogonality and the norm of the trace $H^1(\TC) \to L^2(J)$, we have
\begin{align*}
\textstyle \frac{1}{4} \triple{\zeta}^2 &\le \Beps(\zeta,\zeta) = \Btilde{\epsilon}(\vhtilde,\zeta)-\Beps(\vhtilde,\zeta) + \Beps(\vh,\zeta)-\Btilde{\epsilon}(\vhtilde,\zeta)\\
& =\Btilde{\epsilon}(\vhtilde,\zeta)-\Beps(\vhtilde,\zeta)\\
&= \epsilon \int_J \{ \nablah \vhtilde\} \cdot \jump{\zeta} + \{ \nablah \zeta \} \cdot \jump{\vhtilde} -\frac{\sigma}{h_e} \jump{\vhtilde} \cdot \jump{\zeta}\ds\\
&\lesssim \epsilon \Bigl( h_\TC \norm{\nablah \vhtilde}^2_\SL{2}{\Omega} \cdot \frac{\epsilon\sigma}{h_e^{3/2}} \norm{\jump{\zeta}}^2_\SL{2}{J} + \norm{\nablah \zeta}^2_\SL{2}{\Omega} \cdot \frac{\epsilon\sigma}{h_e^{1/2}} \norm{\jump{\vhtilde}}^2_\SL{2}{J}\\
& \qquad + \int_J \frac{\sigma}{h_e} \jump{\vhtilde} \cdot \jump{\zeta} \ds \Bigr) \\
&\lesssim \left( \epsilon \, h_\TC \norm{\nablah \vhtilde}^2_\SL{2}{\Omega} + \frac{\epsilon\sigma}{h_e^{1/2}} \norm{\jump{\vhtilde}}^2_\SL{2}{J} \right)^{\!\fhalf} \left( \epsilon \norm{\nablah \zeta}^2_\SL{2}{\Omega} + \frac{\epsilon\sigma}{h_e^{3/2}} \norm{\jump{\zeta}}^2_\SL{2}{J} \right)^{\!\fhalf}
\end{align*}
and thus, using Assumption \ref{ass:binterface} for $\epsilon\sigma h_e^{-3/2}$,
\begin{align} \label{zetabound}
\triple{\zeta}^2 \lesssim \epsilon h_\TC \norm{\nablah \vhtilde}^2_\SL{2}{\Omega} + \frac{\epsilon\sigma}{h_\TC^{1/2}} \norm{\jump{\vhtilde}}^2_\SL{2}{J}.
\end{align}
Dividing through by $h_\TC$ and using an inverse inequality on $\rho \norm{\zeta}_\SL{2}{E}$ gives
\begin{align} \label{hzetabound}
h_\TC \norm{\nablah \zeta}^2_\SL{2}{\TC} \lesssim h_\TC^{-1} \triple{\zeta}^2 \lesssim \epsilon \norm{\nablah \vhtilde}^2_\SL{2}{\Omega} + \frac{\epsilon\sigma}{h_\TC^{3/2}}\norm{\jump{\vhtilde}}^2_\SL{2}{J}.
\end{align}
Using Assumptions \ref{ass:Peclet} and \ref{ass:binterface}, as well as \eqref{eq:vhtildestab}, we bound each of the terms in \eqref{hzetabound}. Using a triangle inequality on $\| \nabla \zeta \|_{L^2(\TC)}$ we conclude that
\[
h_\TC \norm{\nablah \vh}^2_\SL{2}{\TC} \lesssim 1.
\]
To show that $\| \vh \|_\SDGr$ is bounded we establish an inf-sup condition for $\Beps$. Indeed, \eqref{infsupcdG1} may be used without change. It remains to transfer \eqref{infsupcdG2} to $\Bform_{\epsilon}$. The inequality \eqref{infsupcdGBar} is still available as the discretesation of the lower-order terms did not change. Different is that we now use $\Bform_{d}(v,\vhat) \le C_1 \triple{v} \cdot \triple{\vhat}$ in place of \eqref{eq:cont}, justified by Assumption \ref{ass:binterface}. Appealing to \eqref{eq:ell_bound}--\eqref{eq:ell_jump_bound}, one has $\triple{\ws} \le C_2 \triple{v}$ for some $C_2 > 0$. Hence
\begin{align}
\Btilde{d}(v,\vhat) = \Btilde{d}(v,v) + \gamma \, \Btilde{d}(v,\ws) \ge \textstyle \frac{1}{4} \norm{v}_d^2 - \gamma \, C_1 \triple{v} \triple{\ws}.
\end{align}
For $\gamma \, C_1 C_2$ small enough and $\lambda$ sufficiently large, $\gamma \, C_1 \triple{v} \triple{\ws}$ is bounded by $\frac{1}{8} \norm{v}_d^2 + \half \Bform_{ar}(v,\vhat)$, using again the positivity of the terms in \eqref{infsupcdGBar}.
\end{proof}
\begin{remark} \label{rem:cea}
Due to Assumptions \ref{ass:Peclet} and \ref{ass:binterface} the above stability bound is valid for the regime $\epsilon \lesssim h_E^{3/2} \norm{\vecb}_\SL{\infty}{\Omega}$. For completeness we briefly outline here how $\sqrt{h} \| \cdot \|_{H^1(\Omega)}$ stability of the cdG method is established if $\epsilon \gtrsim h_E^{3/2} \norm{\vecb}_\SL{\infty}{\Omega}$. The stability proof is in this case easier because the mesh P\'{e}clet number is smaller. If $\Omega$ is smooth or convex and the coefficients have sufficient regularity then $u$ is in $H^2(\Omega)$. Indeed for $\epsilon \ge \half h_E^{3/2} \norm{\vecb}_\SL{\infty}{\Omega}$ the $H^2$ norm of $u$ is uniformly bounded in $\epsilon$. Suppose that the mesh is quasi-uniform. By Ce\`a's Lemma, with $h := \max_E h_E$, a standard argument gives
\begin{align*}
\| u - \vh \|_{H^1(\Omega)} \lesssim (1 + \epsilon^{-1}) h \| u \|_{H^2(\Omega)} \lesssim (h + h^{-1/2}) \| u \|_{H^2(\Omega)},
\end{align*}
and thus $\sqrt{h} \| u - \vh \|_{H^1(\Omega)} \lesssim 1$.
\end{remark}
\section{Numerical Experiment} \label{sec:NumericalExperiments}
\begin{figure}[tb]
\centering
\subfigure[Solution $u$ given by \eqref{cdG3true}.]{\includegraphics[width=0.40\textwidth]{./wplot.pdf}} \hspace{1.5cm}
\subfigure[Solution $u_\epsilon$ given by \eqref{cdG3ueps}]{\includegraphics[width=0.40\textwidth]{./wepsplot.pdf}}
\caption{Solution $u$ and $u_\epsilon$ for $\epsilon = 10^{-3}$.}
\label{fig:cdG3true}
\end{figure}
Let $\Omega = (0,1)^2$. We seek to solve
\begin{equation} \label{cdG3}
-\epsilon \Delta u + (-x,-y)\cdot \nabla u = -x-y
\end{equation}
with Dirichlet boundary conditions chosen such that the solution is given by
\begin{equation} \label{cdG3true}
u(x,y) = x+y - \frac{ \erf \left(x/\sqrt{2\epsilon}\right) +\erf \left(y/\sqrt{2\epsilon}\right)}{\erf \left(1/\sqrt{2\epsilon}\right)}
\end{equation}
where $\erf$ is the error function defined by
\erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x \exp{-t^2} \dt
$.
For $0<\epsilon \ll 1$ this problem exhibits an exponential boundary layer along the outflow boundaries $x=0$ and $y=0$ of width $\Order{\sqrt{\epsilon}}$.
Away from the layers the boundary conditions on the inflow boundaries $x=1$ and $y=1$ are well approximated by $y-1$ and $x-1$ respectively. The hyperbolic solution with these boundary conditions is given by $u_0(x,y) = x+y-2$. This gives
\begin{equation} \label{cdG3ueps}
u_\epsilon(x,y) = 2-\frac{ \erf \left(x/\sqrt{2\epsilon}\right) +\erf \left(y/\sqrt{2\epsilon}\right)}{\erf \left(1/\sqrt{2\epsilon}\right)}.
\end{equation}
We plot \eqref{cdG3true} and \eqref{cdG3ueps} for $\epsilon = 10^{-3}$ in Figure \ref{fig:cdG3true}; note that away from the layers the solution $u_\epsilon$ is close to zero.
We let $\OC$ be a set of the type $(1-\delta,1)^2$, $0 < \delta < 1$. For each $\delta$ the supremum
\[
\sup_{\epsilon \in (0,\epsilon_\mathrm{max}]} \norm{u_\epsilon}_\SH{2}{\OC}
\]
is finite. The dependence of $\norm{u_\epsilon}_\SH{2}{\OC}$ with respect to $\delta$ and to $\epsilon$ is illustrated in Figure \ref{fig:Ex3H2norm}.
\begin{figure}[tb]
\centering
\includegraphics[height=6.5cm]{./Ex3H2norms.pdf}
\caption{$\norm{u_\epsilon}_\SH{2}{\OC}$ for different values of $\delta$.}
\label{fig:Ex3H2norm}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[height=6.5cm]{./Ex3diff.pdf}
\caption{Difference between cdG and dG solutions.}
\label{fig:Ex3diff}
\end{figure}
For this example $c - \half \nabla \cdot \vecb = 1$, so Assumption \ref{ass:ADRrho} is satisfied. Further, we fix $\epsilon = 10^{-6}$ and consider uniform square meshes of edge length $2^{-5}$ so that Assumption \ref{ass:Peclet} is also satisfied as the smallest local mesh P\'eclet number being $488.28$.
We define $\TC = [1-\delta_h,1]^2$, where $\delta_h = m 2^{-5}$, $m \in \set{0,\ldots,32}$. Note that, having fixed the mesh, $\delta_h$ is a discrete parameter. The interface $J$ is composed of the edges lying on the lines $y=\delta_h$ for $x\ge\delta_h$ and $x=\delta_h$ for $y \ge \delta_h$. The smallest value of $\bdotn$ is $\delta_h$ occuring on the edges containing the point $(\delta_h,\delta_h)$. Thus, in this case, Assumption \ref{ass:binterface} reads $\frac{1}{4}mh>\epsilon\frac{\sigma}{h^{3/2}}$ and is satisfied for all $m \in \set{1,\ldots,32}$ (note that this assumption is trivially satisfied when $m=0$) for this choice of $\epsilon$ and $h$ if $\sigma<200$; in the shown computations $\sigma = 10$.
In Figure \ref{fig:Ex3diff} we plot the $\SL{2}{\Omega}$ norm, $\sqrt{\epsilon}$ weighted $\SH{1}{\Th}$ semi-norm, and $L^2$ norm of the jumps on $\Eh$(represented by $\jump{\,\cdotp}$) for both the difference in the dG and cdG approximations and the error in the cdG approximation. Note from Figure \ref{fig:Ex3diff} that the difference in the approximations increases only very slowly until the final data points (where $\TC \approx \Th$). When the continuous region covers the layer, non-physical oscillations pollute the approximation in function of $\delta_h$.
In Table \ref{table:Ex3dofs} we show the number of degrees of freedom (dofs) as the continuous region is increased. Reducing the degrees of freedom to approximately 30\% of the dG method
degrees of freedom results in only a very slight difference in the norm, thus showing that a considerable saving can be made without compromising stability.
\begin{table}[htb]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
$1-\delta$ & dofs & \% of dG dofs & $\sqrt{\epsilon}\norm{\nablah(w_h - v_h)}_\SH{1}{\Omega}$ \\
\hline
\hline
dG & 4096 & 100 & 0.0\\
$8\times2^{-5}$ & 3361 & 82.1 & 3.1157e-08\\
$16\times2^{-5}$ & 2417 & 59.0 & 6.8911e-08\\
$24\times2^{-5}$ & 2121 & 51.8 & 8.7544e-08\\
$30\times2^{-5}$ & 1457 & 35.6 & 1.7934e-07\\
$31\times2^{-5}$ & 1276 & 31.2 & 2.7896e-07\\
cG & 1089 & 26.6 & 1.2444e-02\\
\hline
\end{tabular}
\caption{Degrees of freedom with $\epsilon = 10^{-6}$.}\label{table:Ex3dofs}
\end{table}
We finally remark that, at least for the example considered here, the choice of $\TC$ leaving one layer of elements at the outflow boundary is optimal. Indeed, adding even a single element to the $\TC$ region results in oscillations polluting the solution. For example, for
\begin{equation*}
\TC = [2^{-5},1]^2 \cup ([0.5,0.5+2^{-5}]\times [0,2^{-5}]),
\end{equation*}
i.e., adding a sinlge element to $\TC$ halfway along the $x$-axis, results in
\begin{align*}
\norm{w_h - v_h}_\SL{2}{\Omega} &= 4.7966\times10^{-2},\\
\sqrt{\epsilon}\norm{\nablah(w_h - v_h)}_\SL{2}{\Omega} &= 4.5008\times10^{-3},
\end{align*}
a significant increase on the norms for $\TC = [2^{-5},1]^2$. Notice that this choice of $\TC$ violates Assumption \ref{ass:binterface}.
\section{Acknowledgements:} We gratefully thank the Archimedes Center for Modeling, Analysis and Computation in Crete for hosting the authors during the preparation of this manuscript.
\small
| {
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"Jerry Corcoran wants me at State Police headquarters," said Gus, hanging up the phone. "You carry on, Stan."
"Sure, Gus. What's with the cops?"
"They picked up a suspect on that burglary last night. Want me to check his car. I won't be long."
Gus had hardly left when brakes squealed gently outside and a 1950 V-8 hardtop rolled into the Model Garage. "Oh, no!" gasped Stan soundlessly.
"Is that you, Stanley?" fluttered the brightly dressed, pretty matron who got out. "Is Gus Wilson in, or are you going to fix my car all by yourself?"
"Depends what it needs, Mrs. Allen."
"Well, it runs fine, only it doesn't start. Of course, I might have expected that after my sister Carrie had her car at my house all week."
"Yes, ma'am," said Stan dubiously. "How did you start it to get here?"
"The same thing happened with Carrie's car last night, and a man just pushed it. So when mine made this buzz, instead of going chug-a-chug. I asked a neighbor to push me and it started!"
Stan got in, drove the car on the lift, and cut the engine. When he turned the key to start, there was a tell-tale whine, but the engine didn't crank.
"There!" cried Daisy Allen. "That's what Carrie's car did. She kept it next to mine in our garage, and they're the same make, so I guess it was catching. It probably runs in the family."
Stan made a strangled sound and disappeared under the raised car. Dropping the splash pan, he removed the starter and hooked it up at the test bench. The motor whirred and the pinion spun, but failed to slide out to mesh with the flywheel ring gear. Stan phoned the local agency for the car.
"Starter clutch assembly?" replied the stock clerk. "We don't keep 'em."
Stan turned to Mrs. Allen. "Sorry, but we have to order parts from the city. They should get here tomorrow."
Daisy Allen pouted. "But I need the car! Carrie's already on her way home - that nice young man at the agency fixed her car right away."
"Ben Driscoll?" asked Stan thoughtfully.
"Well, you leave the car, Mrs. Allen. If I can't fix it in an hour, I'll take it over to him myself."
"Captain Bullock calls it open and shut." Lieutenant Jerry Corcoran told Gus at headquarters. "Early this morning the Hayworths' caretaker found a window jimmied and the room inside ransacked. Hayworth who was out of town, says that $2,000 is missing."
"Where'd you find this suspect?" Jerry Corcoran rubbed his chin. That's what bothers me. Asleep in his car half a mile away. But we matched tread marks in the Hayworth drive to his tires. And in the Hayworth drive to his tires. And Tim Dongal's an ex-con, with a record of rural burglaries. A neighbor driving by at nine o'clock heard a casement window bang in the wind. He looked just in time to see a tall man run off the porch. Dougal's tall."
"Where does his car come in?"
Jerry opened the door. Another trooper led in a man whose six-foot height was shrunken by a slight stoop.
"Sit down, Tim." Said Corcoran briskly. "Tell it all again."
The middle-aged suspect folded into a chair. "What for?" he asked bitterly. "I learned printing last time I went up. Gone straight ever since, but first time I'm near any trouble, I get nailed."
"You came here to get a job?"
"Owner of the Thorburg Sentinel told me to be there at nine, after the paper is put to bed. My old car ran all right in town, but on the way it kept missing and slowed me down. I stopped at a gas station about eight. They put in new points while I phoned Thorburg."
"The car ran okay after that?"
"At first. Two women flagged me from a closed gas station in West Milltown. They'd stopped their engine before seeing there was nobody around. The starter didn't work. I pushed 'em to get the engine running. No," Dougal added as Corcoran began to ask a question. "I didn't get any name or license. The car was from out of state."
"And what time was that?"
"Bout nine. That's when my crate began to miss again. That's why I stopped in Shrewsbury about quarter to ten. Later I drove into that Hayworth place by mistake, but like I told you. I drove right out again."
Corcoran signaled the trooper, who led Dougal away.
"I went personally to both those gas stations," said Jerry. "You know what was wrong both times?"
"You tell me," said Gus.
"Both times the wire was off the number two plug. Captain figures Dougal yanked it off himself, drove to the first station, then raced to the Hayworth place. After taking the money and stashing it, he drove back the way he'd come, pulled the cable off again, and limped into that Shrewsbury station to make it look as if he'd only got that far. "But what bothers me is that Dougal couldn't know that neighbor would spot the thief and set the time, and that he stuck around. I can't help wondering-"
"Whether a cable could pull off by itself," finished Gus. "Let's go see."
The car was a paint-faded 1953 of a popular make. "Drove it here myself after we picked Dougal up, "said Corcoran. "On that twisty road from Hayworth's I never hit 15, but it ran okay."
Gus gave the number two cable a tentative pull. It held. He yanked it off, pushed the end through to inspect the terminal clip, and replaced it. "I can't see this coming off by itself. Even the cap's a tight fit. I better drive the car."
The engine idled smoothly. Gus wheeled onto a four-lane parkway, pushed the car to 30, 40 and 50 miles an hour without a slip. Then he floored the gas for a spurt around a midget sedan. The engine began to gallop in a one-cylinder miss. When Gus pulled off the road, it stalled. He flung the hood up. Number two cable was off. The plug terminal was clean, the porcelain free of any oil film. Gus replaced the cable and headed back to headquarters. Half a mile away, he stepped on the gas. At 58 miles per hour, one cylinder again cut out. "Doggone," sighed Gus.
"Ben?" asked Stan on the phone. "You still shop foreman or did they find out you can't even spot a flat tire?"
"When Gus Wilson fires you." retorted Driscoll, "come around. I've got a job for you - on the wash rack."
"Yeah. Listen, Ben, I may have to send you a customer. Starter won't engage. You fixed the same grief for her sister this morning."
"Starter just spins, hey? Sure."
"Your parts shop hasn't got a new clutch and she wants it today." There was a brief silence. "Never mind. Can do. Send it over."
"Not unless I have to. But thanks."
Back at the bench, Stan removed the clutch. The solenoid winding checked.
Why didn't it have enough punch to snap the pinion out - and how did Driscoll plan to fix it without parts?"
Slipping off the clutch, Stan fingered the spring behind it. This helped the solenoid at the start of its stroke, when it was weakest. The spring felt weak. May be it didn't help enough . . .
Seizing both ends of the spring with pliers. Stan stretched it a half inch. He reassembled the clutch. On test, it threw the pinion out with a snap. Back in the car, it cranked vigorously.
Bucking to a stop at headquarters, Gus found Jerry Corcoran waiting beside burly, red-faced Captain Bullock.
"Wasted your time, hey, Gus?" he roared. "Dougal's guilty as sin."
"Don't know about that," said Gus, flinging up the hood. "But this cable popped off by itself on me twice. And I think I know why."
Turning the idle adjustment up to prevent the engine from stalling. Gus got an oil can from his car. With the cable still off, he dripped oil slowly around the top of the plug.
"See those bubbles? There's a leak around the electrode. When the cylinder is firing at speeds over 55, it builds up enough pressure inside that cap to blow it off."
"Then Dougal wasn't - " began Jerry.
"Hold it," growled Bulluck. "He could've put a bum plug in to rig this story of his . I don't buy it."
"How about those women he helped?" asked Corcoran. "We should check - "
"Check an unidentified, out-of-state car?" sneered Bullock. "You kidding?"
Jerry flushed. "Thanks for a darned good try, Gus."
"Good work," approved Gus when Stan described the starter repair. "I fixed a few of those the same way. The assist spring is a shade weak. In that model - sort of runs in the family."
"That's what Mrs. Allen said. Her sister was in the same car and got stuck near West Milltown last night."
"Her sister stopped here for gas once," interrupted Gus.
"She has Southern plates. How did they get the car going?"
"Fellow in a jalopy pushed them."
Gus grabbed the phone. "Can't have been two cars stuck in the same place for the same reason in one night. I'm giving Jerry a tough assignment - to get the facts from Daisy Allen."
Corcoran's car braked to a stop at the Model Garage about closing time.
"Don't know whether to thank you or book you for disrupting police procedure, Gus," he said wryly. "That Allen woman . . ."
Gus and Stan grinned at each other.
"Captain Bullock insisted on going along to question her. I don't think he'll ever be the same. But we learned enough to have the Maryland police stop her sister and take a statement. Dougal did help them." So we overhauled the caretaker's story.
"He's not only as tall as Dougal, but finally broke down and confessed. We'll release Dougal and put in a good word for him at Thorburg."
"Guess you wrapped it up." Said Gus.
"Leaves only one thing missing."
Gus handed Jerry a small carton. "Number two cylinder in Dougal's car. Have him put in this new plug - compliments of the Model Garage. | {
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Q: Understanding dynamic memory allocation I am a beginner in C programming. I was recently taught how to use malloc, but I don't think that I quite understand it. Like why does it need a void * or any typecast as a matter of fact? Why does the syntax itself have (void *) in void *malloc(size_t size). And how does the variable assigned the malloc function know where the memory block begins from? Does the malloc function return an address or something after it has assigned a memory block?
In the class our prof gave us this program. I understand how 2d memory allocation works too.
#include<stdio.h>
#include<conio.h>
void main(void)
{
int *studentInfo=NULL,i=1,j=10,k=0,l=0;
//int *studInfo[10];
int memLoc=0;
clrscr();
printf("How many Student Information You want to store:");
scanf("%d",&j);
printf("How many Subject Marks per student You want to store:");
scanf("%d",&k);
studentInfo=(int *)malloc(j*k*sizeof(int));
//memLoc=0;
for(l=0;l<j;l++)
{
printf("Enter Marks for %dth Student",l+1);
for(i=0;i<k;i++)
{
printf("\nEnter Marks for Subject %d:",i+1);
scanf("%d",studentInfo+(l*j)+i);
}
}
//3 students and 3 subjects
//memory allocated=3*3*2=18
//0,1,2 student 0*no of students
//3 4 5 student 1
//6 7 8 student 2
printf("\nInformation you Entered\n");
for(l=0;l<j;l++)
{
printf("Makrs of Student %d:",l+1);
for(i=0;i<k;i++)
printf("\t%d",*(studentInfo+(l*j)+i));
printf("\n");
}
//*(studentInfo)=10;
//*(studentInfo+1)=20;
//*(studentInfo+2)=30;
//printf("%d\n",sizeof(studentInfo));
//printf("%d\n",*(studentInfo));
//printf("%d\n",*(studentInfo+i++));
//printf("%d\n",*(studentInfo+i++));
free(studentInfo);
getch();
}
In this we are assigning the studentInfo pointer the malloc function right? So... how does studentInfo know that the address of the memory block is USA and not Antarctica?? And I know that it is not a good practise to typecast malloc with some other datatype. But why (int *). Why does malloc need a pointer?? If malloc needs a pointer that means that it is returning an address right? I asked this to my friend and he said no malloc doesn't return anything. And one more thing is it necessary that we need the typecast to be in brackets?
Please explain in very simple terms.
Thank you.
A:
Why does the syntax itself have (void *) in void *malloc(size_t size).
Because malloc is a function and functions have a return type. This tells that this function returns this particular type. So (void *) means malloc return a void *.
Does the malloc function return an address or something after it has assigned a memory block?
Malloc allocates the memory of the size specified into the heap and returns a pointer to that allocated memory.
In this we are assigning the studentInfo pointer the malloc function right? So... how does studentInfo know that the address of the memory block is USA and not Antarctica??
Not exactly. studentInfo is not assigned the malloc function but studentInfo points to the pointer returned by malloc. This was studentInfo now points to the allocated memory.
Why does malloc need a pointer??
malloc doesn't need a pointer. It takes the amount of memory you want to allocate as an argument.
You don't need to typecast here, as void * is automatically promoted.
And one more thing is it necessary that we need the typecast to be in brackets?
Yes, that's the syntax for typecasting. Without brackets, it will result in compilation errors.
A: malloc returns void* because malloc allocates not only int, it can allocate floats, doubles even structures etc. Reason of returning pointer is because it can allocate so much memory in heap, heap is larger than stack and if allocated memory is so big for stack you will receive an error(probably stackoverflow) + copy memory is meaningless if we just can copy its address and refer it by pointer. And void * is a special kind that everything can be assigned no matter what type of address it is. Yes you need to cast it, if you don't then compiler will give you an error
A: The malloc function call allocates a block of memory large enough to hold j*k int objects and returns the address of the first element of that block - that address value is assigned to studentInfo, giving us something like this:
int * int
+–––+ +–––+
studentInfo: | | ––––> | |
+–––+ +–––+
| |
+–––+
...
Thus, studentInfo points to (stores the address of) the first int object in that sequence.
The (int *) preceding the malloc call in your code is a cast - it means "treat the result of the following expression (the value returned by malloc) as an int *, or pointer to int."
There only two circumstances where you must cast the result of malloc:
*
*you're compiling the code as C++;
*you're using an ancient K&R C compiler;
Unlike C, C++ does not allow direct assignment of void * to other pointer types.
Also, prior to the C89 standard, malloc returned char *, so a cast was required to assign the result to any other pointer type.
Otherwise, it's generally preferred to not cast the result - it can lead to maintenance headaches down the line. It would be a bit cleaner to write it as
studentInfo = malloc( sizeof *studentInfo * j * k );
Since the type of studentInfo is int *, the type of the expression *studentInfo is int, so the result of sizeof *studentInfo is the same as sizeof (int).
Remember that sizeof is an operator, not a function - parens are only required if the operand is a type name.
| {
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{"url":"http:\/\/www.physicsforums.com\/showthread.php?p=4064085","text":"equation of motion: spring mass system - free undamped vibration\n\n1. The problem statement, all variables and given\/known data\n\ni have uploaded my question. please check out the attached .pdf file.\n\n2. Relevant equations\n\n3. The attempt at a solution\nAttached Files\n doubt.pdf (829.4 KB, 8 views)\n PhysOrg.com science news on PhysOrg.com >> King Richard III found in 'untidy lozenge-shaped grave'>> Google Drive sports new view and scan enhancements>> Researcher admits mistakes in stem cell study\n Mentor Blog Entries: 1 Shouldn't the 'inertial force' oppose the acceleration, not the motion?\n hey! thanks for replying. i was following this website. here's the link http:\/\/lpsa.swarthmore.edu\/Systems\/M...hSysModel.html it has been mentioned in this website that 'inertia force acts opposite to the direction of motion'. did i understand it right? mass will decelerate in the positive x direction, when it moves away from equilibrium towards +A. it will accelerate in the negative x direction, when it is moving towards the equilibrium position. it will start decelerating in the negative x direction, when it moves away from equilibrium position towards -A. it will start accelerating in the positive x direction, when it moves towards equilibrium position.\n\nMentor\nBlog Entries: 1\n\nequation of motion: spring mass system - free undamped vibration\n\n Quote by jason.bourne it has been mentioned in this website that 'inertia force acts opposite to the direction of motion'. did i understand it right?\nI think it's just a sloppy and misleading way to describe the direction. Note how they 'derive' the inertia force: It's just the negative of 'ma'.\n\nSince your variable x already defines the positive direction of acceleration, $\\ddot{x}$, the inertia force is simply $-m\\ddot{x}$. The direction is already included: When the sign of $\\ddot{x}$ changes, so will the sign of the inertia force.\n\nBut it's the direction of the acceleration that determines the direction of the inertia force, not the direction of motion.\n\n Since your variable x already defines the positive direction of acceleration, x\u00a8, the inertia force is simply \u2212mx\u00a8\nas i defined downward x as positive, so does that mean x\u00a8 is positive as i move downwards away from equilibrium?\n\ni might be asking really very silly doubt but this thing is really confusing me. i want it to be sorted.\n\nas the mass moves away from equilibrium towards +A in the downward positive x direction, if we draw a free body diagram, how do we represent acceleration?\n\nif we take x\u00a8, acceleration positive in the downward direction, the inertia force acts in the opposite direction i.e, upwards.\n\nso if i write equation of motion using D'Alembert's Principle, i get:\n\n-mx\u00a8 - kx = 0.\n\nbut if i consider the situation where the mass is at a position, away from -A, towards equilibrium, then what is the direction of acceleration and how do i write the equation of motion?\n\nMentor\nBlog Entries: 1\n Quote by jason.bourne as i defined downward x as positive, so does that mean x\u00a8 is positive as i move downwards away from equilibrium?\n$\\ddot{x}$ is positive as long as the acceleration is in the positive direction, which you have defined as downward. Note that the restoring force of the spring always acts to accelerate the mass back towards the equilibrium point. So at any point below the equilibrium point, the acceleration would be upward.\n\n i might be asking really very silly doubt but this thing is really confusing me. i want it to be sorted. as the mass moves away from equilibrium towards +A in the downward positive x direction, if we draw a free body diagram, how do we represent acceleration?\nYou wouldn't put the acceleration on a free body diagram, but it would act upward.\n\n if we take x\u00a8, acceleration positive in the downward direction, the inertia force acts in the opposite direction i.e, upwards.\n$\\ddot{x}$ would be negative and the inertia force would be positive. But you don't have to worry about the inertia force direction--you'd still represent that inertia force as $-m\\ddot{x}$.\n\n so if i write equation of motion using D'Alembert's Principle, i get: -mx\u00a8 - kx = 0.\nRight.\n\n but if i consider the situation where the mass is at a position, away from -A, towards equilibrium, then what is the direction of acceleration and how do i write the equation of motion?\nSo now you are looking at the situation when the mass is above the equilibrium point. So the spring force acts downward and the acceleration is downward.\n\nNonetheless, you'd still write the restoring force as -kx (since x is now negative, the force comes out positive). And you'd still write the inertial force as opposite to the acceleration: $-m\\ddot{x}$. (Since the acceleration will be positive, that will come out to be negative.)\n\nSo the equation of motion will be exactly the same. Realize that you don't have to know the direction of the acceleration ahead of time. Just represent the inertia force properly--with respect to your chosen coordinates. Then your equation of motion will tell you the direction of the acceleration.\n thanks Doc Al. i think i got it. if i have any doubts further, i'll get back to you. thank you very much!\n Mentor Blog Entries: 1 Excellent. Glad that it helped. And you are most welcome.","date":"2013-05-25 01:19:18","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\": 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\": 0.7128757834434509, \"perplexity\": 485.66926585314627}, \"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-2013-20\/segments\/1368705310619\/warc\/CC-MAIN-20130516115510-00087-ip-10-60-113-184.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/nus.kattis.com\/courses\/CS2040C\/CS2040C_S1_AY2223\/assignments\/gqnpk8\/problems\/averagecharacter","text":"Hide\n\n# Problem BAverage Character\n\nHave you ever wondered what the average ASCII character of any given string is? No? Never? Really? Well, is it a character in the string or something else?\n\nWould you do this calculation by hand with an ASCII table? Probably not! All modern programming languages include functions for converting an ASCII character to an integer, and to convert an integer to an ASCII character. Of course, these functions often also handle unicode characters as well, but that is not part of this problem.\n\nGiven a string of ASCII characters, compute the average character. If the average character lies between two integer ASCII values, return the smaller one.\n\n## Input\n\nThe single line of input contains a single string $s$ ($1 \\le |s| \\le 100$), which consists of ASCII text. All of the characters of $s$ will be printable ASCII, between ASCII $32$ (space:\u2018 \u2019) and ASCII $126$ (tilde:\u2018~\u2019). It will NOT contain any control characters such as carriage returns, line feeds, tabs, etc. It is NOT guaranteed to begin, end, or even contain a non-space character.\n\n## Output\n\nOutput a single ASCII character, which is the average of all of the ASCII characters in $s$.\n\nSample Input 1 Sample Output 1\nABCDE\n\nC\n\nSample Input 2 Sample Output 2\nAbCdE\n\nO\n\nSample Input 3 Sample Output 3\naBcDe\n\nV","date":"2023-02-08 13:43:46","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.31358352303504944, \"perplexity\": 1807.3447070499487}, \"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\/1674764500813.58\/warc\/CC-MAIN-20230208123621-20230208153621-00364.warc.gz\"}"} | null | null |
{"url":"https:\/\/autonlab.org\/auton-survival\/models\/dsm\/datasets.html","text":"# Module auton_survival.models.dsm.datasets\n\nUtility functions to load standard datasets to train and evaluate the Deep Survival Machines models.\n\n## Functions\n\n def increase_censoring(e, t, p) \n def load_dataset(dataset='SUPPORT', **kwargs) \n\nHelper function to load datasets to test Survival Analysis models.\n\nCurrently implemented datasets include:\n\nSUPPORT: This dataset comes from the Vanderbilt University study to estimate survival for seriously ill hospitalized adults [1]. (Refer to http:\/\/biostat.mc.vanderbilt.edu\/wiki\/Main\/SupportDesc. for the original datasource.)\n\nPBC: The Primary biliary cirrhosis dataset [2] is well known dataset for evaluating survival analysis models with time dependent covariates.\n\nFRAMINGHAM: This dataset is a subset of 4,434 participants of the well known, ongoing Framingham Heart study [3] for studying epidemiology for hypertensive and arteriosclerotic cardiovascular disease. It is a popular dataset for longitudinal survival analysis with time dependent covariates.\n\n## References\n\n[1]: Knaus WA, Harrell FE, Lynn J et al. (1995): The SUPPORT prognostic model: Objective estimates of survival for seriously ill hospitalized adults. Annals of Internal Medicine 122:191-203.\n\n[2] Fleming, Thomas R., and David P. Harrington. Counting processes and survival analysis. Vol. 169. John Wiley & Sons, 2011.\n\n[3] Dawber, Thomas R., Gilcin F. Meadors, and Felix E. Moore Jr. \"Epidemiological approaches to heart disease: the Framingham Study.\" American Journal of Public Health and the Nations Health 41.3 (1951).\n\n## Parameters\n\ndataset :\u2002str\nThe choice of dataset to load. Currently implemented is 'SUPPORT', 'PBC' and 'FRAMINGHAM'.\n**kwargs :\u2002dict\nDataset specific keyword arguments.\n\n## Returns\n\ntuple :\u2002(np.ndarray, np.ndarray, np.ndarray)\nA tuple of the form of (x, t, e) where x, t, e are the input covariates, event times and the censoring indicators respectively.","date":"2022-11-26 16:12:26","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\": 0.3330080807209015, \"perplexity\": 12639.384474340082}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446708010.98\/warc\/CC-MAIN-20221126144448-20221126174448-00304.warc.gz\"}"} | null | null |
\section{The co-acceleration scenario}
With its giant outburst in 1997,
emitting photons up to $24$~TeV and $0.5$~MeV in the $\gamma$-ray and X-ray bands,
Mkn~501 has proved to be the most extreme TeV-blazar observed so far
(e.g. Catanese et al 1997, Pian et al 1997, Aharonian et al 1999).
In this paper, we consider the April 1997 flare of Mkn~501 in the light of a modified
version of the Synchrotron Proton Blazar model (SPB) (Mannheim 1993), and present a preliminary model fit.
In the model, shock accelerated protons ($p$) interact
in the synchrotron photon field generated by the electrons ($e^-$) co-accelerated
at the same shock.
This scenario may put constraints on the maximum achievable
particle energies.
The usual process considered for accelerating charged particles in the plasma jet is
diffusive shock acceleration (see e.g. Drury 1983, Biermann \& Strittmatter 1987).
If the particle spectra are cut off due to synchrotron losses, the ratio of the
maximum particle energies $\gamma_{p,max}/\gamma_{e,max}$
can be derived
by equating $t_{acc,p}/t_{acc,e} = t_{syn,p}/t_{syn,e}$, with
$t_{syn,p}$ and $t_{syn,e}$ being the synchrotron loss time scales for $p$ and $e^-$, respectively. We find that for shocks of compression ratio 4
(see M\"ucke \& Protheroe 1999 for a detailed derivation)
\begin{equation}
\frac{\gamma_{p,\rm{max}}}{\gamma_{e,\rm{max}}} \leq \frac{m_p}{m_e} (\frac{m_p}{m_e})^{\frac{2(\delta-1)}{3-\delta}} \sqrt{\frac{F(\theta,\eta_{e,\rm{max}})}{F(\theta,\eta_{p,\rm{max}})}} =
\frac{m_p}{m_e} \sqrt{\frac{\eta_{e,\rm{max}} F(\theta,\eta_{e,\rm{max}})}{\eta_{p,\rm{max}} F(\theta,\eta_{p,\rm{max}})}}
\end{equation}
where the ``=''-sign corresponds to
synchrotron loss, and the ``$<$''-sign to adiabatic loss determining the maximum energies.
$\delta$ is the power law index of the magnetic turbulence spectrum ($\delta = 5/3$: Kolmogorov turbulence, $\delta = 3/2$: Kraichnan turbulence, and $\delta=1$ corresponds
to Bohm diffusion). $\eta_{e,\rm{max}}$ is the mean free path at maximum energy in units
of the particle's gyroradius and $F(\theta,\eta_{e,\rm{max}})$ takes account
of the shock angle $\theta$ (Jokipii 1987).
The ratio
$F(\theta,\eta_{e,\rm{max}}) \eta_{e,\rm{max}}/F(\theta,\eta_{p,\rm{max}}) \eta_{p,max}$
can be constrained by the variability time scale $t_{\rm{var}}$, requiring
$t_{var} D \geq t_{acc,p,max}$ ($D$ = Doppler factor, $t_{acc,p,max}$ = acceleration time scale
at maximum particle energy) for a given parameter combination.
As an example, we adopt $D=10$, $B=20$G and $t_{\rm{var}}=2$~days.
Eq.~1 then restricts for these parameters the ratio of the allowed maximum
particle energies to the range below the solid lines shown in Fig.~1. Points exactly on this line
correspond to synchrotron-loss limited particle spectra which are
accelerated with exactly the variability time scale.
\vspace*{-.5cm}
\begin{figure}[h]
\begin{minipage}[t]{3.55in}
\centerline{\epsfig{file=fig1.ps,height=3.5in,width=3.5in}}
\end{minipage}
\hfill
\begin{minipage}[t]{2.2in}
\vspace*{-3.3in}
\caption{Allowed parameter space (shaded area) for $\gamma_{p,\rm{max}}$, $\gamma_{e,\rm{max}}$
in the SBP-model for typical TeV-blazar parameters
(B=20~G, D=10, $u_1=0.5c$, $\beta=1$, $t_{\rm{var}}=2$days) and for different magnetic turbulence spetra $I(k) \propto k^{-\delta}$. The diamond symbol corresponds
to the Mkn~501-model presented below.}
\end{minipage}
\label{fig1}
\end{figure}
\vspace*{-1cm}
In hadronic models $\pi$ photoproduction is essential for $\gamma$-ray production. The threshold
of this process is given by $\epsilon_{\rm{max}} \gamma_{p,\rm{max}} = 0.0745$~GeV where
$\epsilon_{\rm{max}}$ is the maximum photon energy of the synchrotron target field.
Inserting $\epsilon_{\rm{max}} = 3/8 \gamma_{e,\rm{max}}^2 B/(4.414\times 10^{13}\rm{G})~m_e c^2$ into the threshold condition, we find
\vspace*{-.3cm}
$$
\gamma_{p,\rm{max}} \geq 1.72 \cdot 10^{16} (\frac{B}{\rm{Gauss}})^{-1} \gamma_{e,\rm{max}}^{-2}
$$
which is shown in Fig.~1 as dashed line for various magnetic field strengths.
Together with Eq.~1, the allowed range of maximum particle energies is then restricted
to the shaded area in Fig.~1.
\section{The Mkn~501 flare in the Synchrotron Proton Blazar (SPB) Model}
We assume the parameters used in Fig.~1, and that the co-accelerated $e^-$ produce the {\underline{observed}} synchrotron spectrum,
unlike in previous SPB models, and this is
the target radiation field for the $p\gamma$-interactions.
This synchrotron spectrum, and its hardening with rising flux,
has recently been convincingly reproduced by a shock
model with escape and synchrotron losses (Kirk et al 1998).
We use
the Monte-Carlo technique for particle production/cascade development, which allows us to use exact
cross sections.
\begin{figure}[h]
\begin{minipage}[t]{3.55in}
\centerline{\epsfig{file=fig2.ps,height=3.5in,width=3.7in}}
\end{minipage}
\hfill
\begin{minipage}[t]{2.25in}
\vspace*{-3.6in}
\caption{
Mean energy loss time of $p$ for synchrotron radiation (syn), $\pi$-photoproduction
($\pi$) and Bethe-Heitler~pair~production ($e^\pm$),
and for $\pi^\pm$- and $\mu^\pm$-synchrotron radiation (syn $\pi$, syn $\mu$)
for $B=19.6$~G
with their mean decay time scales (decay $\pi$, decay $\mu$) in the
SPB-model. The acceleration time scale (acc), based on Kolmogorov turbulence, is calculated for
$u_1 = 0.5c$, $\eta_p = 40$ and shock angle $\theta = 85^\circ$. Its curvature reflects the influence of
the shock angle. The adiabatic loss time (adiab) is assumed to be
$R/u_1 \approx D t_{var}$.
All quantities are in the jet frame.}
\end{minipage}
\label{fig2}
\end{figure}
\vspace*{-1cm}
For simplicity we represent the observed synchrotron spectrum (target photon field for the
$p\gamma$-collisions) as a broken power law in the jet frame with photon power law index 1.4
below the break energy of 0.2~keV, and index 1.8 up to 50~keV.
The variability time scale restricts the radius $R$ of the emission region.
For our model we use
$t_{\rm{var,x}} \approx 2$~days (Catanese et al 1997), and find $R\approx 2.6\times 10^{16}$cm
for $D=10$, $B=19.6$~G.
With these parameters the $\gamma\gamma$-pair
production optical depth reaches unity for $\approx 25$~TeV photons.
Our model considers photomeson production (simulated using
SOPHIA, M\"ucke et al 1999), Bethe-Heitler pair production (simulated using the code of
Protheroe \& Johnson 1996),
$p$ synchrotron radiation and adiabatic losses due to jet expansion.
The mean energy loss and acceleration time scales
are presented in Fig.~2.
Synchrotron losses, which turn out to be at least as important as losses due to
$\pi$ photoproduction for the assumed 2~day variability, limit the injected $p$ spectrum
$\propto \gamma_p^{-2}$ to
$2\leq\gamma_p\leq 4.4\times 10^{10}$. This leads to
a $p$ energy density
$u_p \approx 0.2~\rm{TeV/cm}^3$, which is bracketed by the photon energy density
$u_{\rm{target}} \approx 0.01~\rm{TeV/cm}^3$, and a magnetic field energy density
$u_B \approx 9.5~\rm{TeV/cm}^3$. With $u_B \gg u_{\rm{target}}$ significant Inverse Compton radiation
from the co-accelerated $e^-$ is not expected.
Rachen \& Meszaros (1998) noted the
importance of synchrotron losses of $\mu^\pm$- (and $\pi^\pm$-) prior to their decay in AGN jets
and GRBs.
For the
present model, the critical Lorentz factors $\gamma_{\mu} \approx 3 \times 10^9$ and $\gamma_{\pi} \approx 4 \times 10^{10}$,
above which synchrotron losses dominate above decay, lie well below the maximum particle energy for $\mu^\pm$,
while $\pi^\pm$-synchrotron losses can be neglected due to the shorter decay time.
\begin{figure}[h]
\epsfig{file=fig3a.ps,height=2.8in,width=2.9in}
\hfill
\epsfig{file=fig3b.ps,height=2.8in,width=2.9in}
\caption{Left: Average emerging cascade spectra initiated by $\pi^0$-decay (solid line) and $\pi^\pm$-decay
synchrotron photons (dashed-dotted line). Right: Average emerging cascade spectra initiated by
$p$- (solid line) and $\mu^\pm$-synchrotron photons (dashed-dotted line).}
\label{fig3}
\end{figure}
The matrix method (e.g. Protheroe \& Johnson 1996) is used to follow the pair-synchrotron cascade in the ambient synchrotron
radiation field and magnetic field,
developing as a result of photon-photon pair production. The cascade can be initiated by
photons from $\pi^0$-decay (``$\pi^0$-cascade''), electrons from the $\pi^\pm\rightarrow \mu^\pm\rightarrow e^\pm$-decay (``$\pi^\pm$-cascade''), $e^\pm$ from the proton-photon Bethe-Heitler pair production (``Bethe-Heitler-cascade'')
and $p$ and $\mu$-synchrotron photons (``$p$-synchrotron cascade'' and ``$\mu^\pm$-synchrotron cascade'').
In this model, the cascades develop linearly.
Fig.~3 shows an example of cascade spectra initiated by photons of different origin, and for the
parameter combination given above.
$\pi^0$- and $\pi^\pm$-cascades obviously produce featureless spectra
whereas $p$- and $\mu^\pm$-synchrotron cascades cause the typical double hump shaped SED as
observed in $\gamma$-ray blazars (see also Rachen, these proceedings).
The contribution from Bethe-Heitler cascades turns out
to be negligible.
Direct $p$- and $\mu^\pm$-synchrotron
radiation is responsible for the high energy peak, whereas the low energy hump may be either
synchrotron radiation from the directly accelerated $e^-$ and/or by pairs produced by the ``low energy hump''.
Adding the four components of the cascade spectrum in Fig.~3 and normalizing
to an ambient, accelerated $p$ density of $n_{tot,p} = 7~\rm{cm}^{-3}$, we obtain the SED shown in Fig.~4
where it is compared with the multifrequency observations of the 16 April 1997 flare of Mkn~501.
\begin{figure}[h]
\centerline{\epsfig{file=fig4.ps,height=3.2in,width=4.5in}}
\caption{
Present model (histogram) in comparison with the data of the 16 April 1997-flare of Mkn~501. Photon
absorption on the cosmic diffuse background radiation field is not included in the model. Straight solid lines:
parametrization of the observed, curved synchrotron spectrum (BeppoSAX \& OSSE) by Bednarek \& Protheroe (1999)
and observed TeV-emission
corrected for cosmic background absorption (Bednarek \& Protheroe 1999) with peak energy output $\sim 2$~TeV;
the 100~MeV upper limit is from Catanese et al 1997 (observed 9-15 April 1997), diamonds: nearly simultaneous (uncorrected) Whipple data (Catanese et al 1997);
dashed-dotted line: synchrotron target spectrum.
}
\label{fig4}
\end{figure}
\vspace*{-.7cm}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,664 |
Pittsburgh City Paper
A new dance work explores love and friendship between African-American men
Anthony Williams. Photo by Anna Malguina.
Beyond simply entertaining audiences, concert dance is an important means of self-expression for its creators and performers. Some artists employ dance as a forum to air their thoughts and feelings on issues important to them. In Loving Black, Oct. 17 at The Alloy Studios, local dancer and choreographer Anthony Williams uses dance to advance his own views on the complicated attitudes surrounding love and friendship between African-American men.
Loving Black is the latest in the Kelly-Strayhorn Theater's Fresh Works series. The 30-minute work-in-progress reflects Williams' personal perceptions of the African-American male, along with ideas about self-love gleaned from the writings of Malcolm X, Martin Luther King, James Baldwin and others.
Williams, 27, is a Chicago native who trained and performed with Ballet Chicago and Deeply Rooted Dance Theatre before moving to Pittsburgh to join the now-disbanded August Wilson Center Dance Ensemble. He's currently a freelance dance artist and resident teaching artist at the Kelly-Strayhorn. Loving Black, a multimedia work, is set to a mix of original music created by Jovan Sharp: spoken word, jazz and funk. Williams says the piece takes something as simple as the way African-American males greet each other with a handshake instead of a hug and examines it through the lens of hip-hop culture and history via slavery's lingering effect on how African-American males behave in social situations. Williams says that slavery conditioned African Americans to believe they were less worthy as human beings, and that this demoralization carries over into today.
Anthony Williams. Photo by Mark Simpson.
"There are other things men of color do that can be viewed as self-hate," says Williams. "We don't always view women as equals."
Williams says the work also references a divide within the black community. Some African Americans, he says, want to distance themselves from their ethnicity in their personal appearance, while others are ridiculed for not being "black enough."
Williams means to engage his audience in the complexity of past and present representations of black male identities and attitudes. Beyond that, he says he wants Loving Black's cast of four men of color, including himself, to be positive role models for the Pittsburgh dance community, where he feels professional African-American male dancers are currently underrepresented.
Fresh Works: Anthony Williams' Loving Black, 8 p.m., Fri., Oct. 17. The Alloy Studios, 5530 Penn Ave., Friendship. $10-15. 412-363-3000 or kelly-strayhorn.org
This article originally appeared in Pittsburgh City Paper October 15, 2014. Copyright Steve Sucato.
artsair October 15, 2014 October 19, 2014 Anthony Williams, Ballet Chicago, Deeply Rooted Dance Theatre, James Baldwin, Jovan Sharp, Kelly Strayhorn Theater, Malcolm X, Martin Luther King
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Q: RSA decryption from .NET using PHP I am not sure where to post this so I am starting here and hoping for the best. I have a public and private key generated for .NET. The keys have (Modulus, Exponent, etc...). I sent the public key to a third party and they used it to encrypt passwords in a database. I downloaded the encrypted passwords and now I need to decrypt them on my end. The issue comes in when it has to be decrypted on a unix server. I do not care if I use php, java or any other language as long as it works on the unix box. I am a complete rookie to RSA keys and would appreciate any help or even a direction on where I "should" post this. Thanks in advance....
A: The values appear to be base64 encoded. You could use phpseclib's pure PHP RSA implementation and do something like this to load the public key:
$rsa->loadKey(array('modulus' => $modulus, 'exponent' => $exponent), CRYPT_RSA_PUBLIC_FORMAT_RAW);
For the private key... maybe you could do something like..
$rsa = new Crypt_RSA();
$rsa->modulus = $modulus;
$rsa->publicExponent = $exponent;
$rsa->exponents = array(1=> $dp, $dq)
$rsa->coefficients = array(2 => $inverseq);
$rsa->primes = array(1 => $p, $q);
Or something like that - I haven't tested the code, but a cursory glance of phpseclib's code suggests that'll work.
A: There is a nice article about using .NET's key format with PHP:
Using PHP to encrypt/decryp/sign/verify data with .NET XML RSA key
| {
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Q: Receiving emails in a different DNS provider, should I add MX or NS records? This is similar, yet different than How should I setup separate MX records for a subdomain? in that I want another DNS provider to handle receiving emails from a subdomain. I have
*
*a DNS zone provided by OVH (I pay for my domain name there)
*a DNS Zone on Amazon route53 (I'd like the email resolution for only my new subdomain to go here),
*I want to put my MX record to receive emails in Amazon SES ideally in my route53 DNS Zone
There is an existing email flow that I absolutely don't want to disturb : all emails sent to @mydomain.com are routed to Google DNS MX servers using MX records in my OVH DNS zone.
# OVH DNS Zone
domain.com. MX 5 alt1.aspmx.l.google.com.
[and other similar records]
I want to create a separate subdomain email so emails sent to @subdomain.mydomain.com will be redirected to Amazon route 53' DNS instead but I'd like the actual MX record to remain in my Amazon route53 hosted zone record set
So the flow I want is
OVH > Amazon Route53 custom hosted zone > Amazon SES # (for emails @mail.domain.com)
OVH > Google DNS # (for emails @domain.com)
and I have already this in my Amazon route 53 Zone (I have followed the instructions to setup email receiving in Amazon SES)
# Amazon Route53 DNS
mail.domain.com. MX 10 inbound-smtp.eu-west-1.amazonaws.com.
After reading several questions, I'm still not clear how emails are actually redirected, and some posts mention that MX records aren't needed for receiving emails (but in my case Amazon SES required an MX record to be somewhere)
Should I add an MX record that points to my Amazon route53 DNS
or
Should I add a NS record for my subdomain that points to my Amazon route53 DNS (so far I've been using multiple NS entries to redirect most subdomains for regular web traffic from OVH to Amazon route53's DNS)
# OVH > Google for @mydomain.com emails - should remain functional
domain.com. MX 5 alt1.aspmx.l.google.com.
# OVH > Amazon Route53 DNS
mail.domain.com MX 10 ns-1890.awsdns-44.co.uk.
# or
mail.domain.com NS ns-1890.awsdns-44.co.uk.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,193 |
Q: Tracing a WSoD in Joomla How do I find out which code section an error occurs in, if the site done on Joomla gives me a White Screen of Death?
I even know which module fails, and under what condition (it is a slideshow module, and it fails when one of the items to display are of video type), but I would like to check what place in the code it fails in, in the hope of perhaps correcting the problem myself (I am not the original developer, and I am not fluent in PHP yet).
A: You need to enable error reporting in your environment. It is disabled by default due to security concerns. I think the easiest way would be to put the following lines of code in the beginning of your joomla installation's index.php file:
ini_set('display_errors','On');
error_reporting(E_ALL);
After you are done debugging do not forget to remove or comment out those two lines.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,813 |
Q: How to get back my hover after resetting color with jQuery? I have this little jQuery-code working, but there's one thing I can't figure out. After I have clicked my infocontent-menu, the h1:hover is gone. How do I get the hover function back?
HTML:
<div class="infocontent"><h1>Wat?</h1></div>
<div class="infocontent"><h1>Van/ voor wie?</h1></div>
<div class="infocontent"><h1>Akties</h1></div>
<div class="infocontent"><h1>Met wie?</h1></div>
CSS:
.infocontent {
width: 120px;
height: 120;
float: left;
}
.infocontent h1 {
width: 100%;
font-size: 24px;
text-transform: uppercase;
color: #999;
}
.infocontent h1:hover { color: #000; }
jQuery:
$('.infocontent h1:first').css('color', '#000');
$(".infocontent h1").on("click", function() {
$('.infocontent h1').css('color', '#999');
$(this).css('color', '#000');
});
Demo
A: Since the jQuery is adding inline CSS, the only way to overwrite this - unfortunately - is via !important
Updated CSS
.infocontent h1:hover {
color: #000!important;
}
jsFiddle - it works.
I would suggest avoiding the usage of !important altogether. You could do this by changing the jQuery to add/remove a CSS class as opposed to adding inline styling.
A: Might be easier to change the js to change the class with something like:
$('.infocontent h1').eq(0).addClass('on').end().on("click", function() {
if(!$(this).hasClass('on')) {
$('.on').removeClass('on');
$(this).addClass('on');
}
});
and change the css to something like this
EDIT: saw you wanted background changes
.infocontent{float: left}
.infocontent h1 {
text-transform: uppercase;
color: #999999;
/* opt changes */
font-size: 18px;
padding:3px 8px;
letter-spacing:2px;
cursor:pointer/*so users know to click it*/
}
.infocontent h1.on,.infocontent h1.on:hover{color:#000000;background-color:#CCCCCC}
.infocontent h1:hover {color:#666666;background-color:#EDEDED}
I updated your fiddle (now with bg changes): http://jsfiddle.net/filever10/FXetb/18/
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,350 |
require 'yaml'
class SeleniumOnRailsConfig
attr_accessor :configs
def initialize
@defaults = {:environments => ['test']}
initialize_configs
end
def get var, default = nil
value = @configs[var.to_s]
value ||= @defaults[var]
value ||= default
value ||= yield if block_given?
value
end
def initialize_configs
@configs = {}
files = [File.expand_path(File.dirname(__FILE__) + '/../config.yml')]
files << File.join(RAILS_ROOT, 'config', 'selenium.yml')
files.each { |file| @configs = YAML.load_file(file) if File.exist?(file) }
end
end | {
"redpajama_set_name": "RedPajamaGithub"
} | 5,856 |
{"url":"https:\/\/projecteuclid.org\/euclid.afm\/1485907195","text":"## Arkiv f\u00f6r Matematik\n\n### Noise correlation bounds for uniform low degree functions\n\n#### Abstract\n\nWe study correlation bounds under pairwise independent distributions for functions with no large Fourier coefficients. Functions in which all Fourier coefficients are bounded by \u03b4 are called \u03b4-uniform. The search for such bounds is motivated by their potential applicability to hardness of approximation, derandomization, and additive combinatorics.\n\nIn our main result we show that $\\operatorname{\\mathbb {E}}[f_{1}(X_{1}^{1},\\ldots,X_{1}^{n}) \\ldots f_{k}(X_{k}^{1},\\ldots,X_{k}^{n})]$ is close to 0 under the following assumptions:\n\n\u2022 the vectors $\\{ (X_{1}^{j},\\ldots,X_{k}^{j}) : 1 \\leq j \\leq n\\}$ are independent identically distributed, and for each j the vector $(X_{1}^{j},\\ldots,X_{k}^{j})$ has a pairwise independent distribution;\n\u2022 the functions fi are uniform;\n\u2022 the functions fi are of low degree.\n\nWe compare our result with recent results by the second author for low influence functions and to recent results in additive combinatorics using the Gowers norm. Our proofs extend some techniques from the theory of hypercontractivity to a multilinear setup.\n\n#### Note\n\nWork done while the first author was at the Royal Institute of Technology, funded by ERC Advanced investigator grant 226203 and a grant from the Mittag-Leffler Institute. Second author supported by BSF grant 2004105, NSF CAREER award DMS 0548249, DOD ONR grant N0014-07-1-05-06 and ISF grant 1300\/08.\n\n#### Article information\n\nSource\nArk. Mat., Volume 51, Number 1 (2013), 29-52.\n\nDates\nFirst available in Project Euclid: 31 January 2017\n\nhttps:\/\/projecteuclid.org\/euclid.afm\/1485907195\n\nDigital Object Identifier\ndoi:10.1007\/s11512-011-0145-5\n\nMathematical Reviews number (MathSciNet)\nMR3029335\n\nZentralblatt MATH identifier\n1296.68102\n\nRights\n\n#### Citation\n\nAustrin, Per; Mossel, Elchanan. Noise correlation bounds for uniform low degree functions. Ark. Mat. 51 (2013), no. 1, 29--52. doi:10.1007\/s11512-011-0145-5. https:\/\/projecteuclid.org\/euclid.afm\/1485907195\n\n#### References\n\n\u2022 Austrin, P. and Mossel, E., Approximation resistant predicates from pairwise independence, Comput. Complexity 18 (2009), 249\u2013271.\n\u2022 Benjamini, I., Kalai, G. and Schramm, O., Noise sensitivity of boolean functions and applications to percolation, Inst. Hautes \u00c9tudes Sci. Publ. Math. 90 (1999), 5\u201343.\n\u2022 Furstenberg, H. and Weiss, B., A mean ergodic theorem for $(1\/N)\\sum^{N}_{n=1}f(T^{n}x){}\\cdot\\allowbreak g(T^{n^{2}}x)$, in Convergence in Ergodic Theory and Probability (Columbus, OH, 1993 ), Ohio State Univ. Math. Res. Inst. Publ. 5, pp. 193\u2013227, de Gruyter, Berlin, 1996.\n\u2022 Gowers, W. T., A new proof of Szemer\u00e9di\u2019s theorem for arithmetic progressions of length four, Geom. Funct. Anal. 8 (1998), 529\u2013551.\n\u2022 Gowers, W. T., A new proof of Szemer\u00e9di\u2019s theorem, Geom. Funct. Anal. 11 (2001), 465\u2013588.\n\u2022 Gowers, W. T. and Wolf, J., The true complexity of a system of linear equations, Proc. Lond. Math. Soc. 100 (2010), 155\u2013176.\n\u2022 Green, B. and Tao, T., The primes contain arbitrarily long arithmetic progressions, Ann. of Math. 167 (2008), 481\u2013547.\n\u2022 Khot, S., On the power of unique 2-prover 1-round games, in Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing (Montreal, QC, 2002 ), pp. 767\u2013775, ACM, New York, 2002.\n\u2022 Khot, S., Kindler, G., Mossel, E. and O\u2019Donnell, R., Optimal inapproximability results for max-cut and other 2-variable CSPs?, SIAM J. Comput. 37 (2007), 319\u2013357.\n\u2022 Lindeberg, J. W., Eine neue Herleitung des exponential-Gesetzes in der Wahrscheinlichkeitsrechnung, Math. Z. 15 (1922), 211\u2013235.\n\u2022 Mossel, E., Gaussian bounds for noise correlation of functions, Geom. Funct. Anal. 19 (2010), 1713\u20131756.\n\u2022 Mossel, E., O\u2019Donnell, R. and Oleszkiewicz, K., Noise stability of functions with low influences: invariance and optimality, Ann. of Math. 171 (2010), 295\u2013341.\n\u2022 O\u2019Donnell, R., Computational Applications of Noise Sensitivity, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, 2003.\n\u2022 Raghavendra, P., Optimal algorithms and inapproximability results for every CSP, in Proceedings of the 40th Annual ACM Symposium on Theory of Computing STOC\u201908 (Victoria, BC, 2008 ), pp.\u00a0245\u2013254, ACM, New York, 2008.\n\u2022 Rotar, V. I., Limit theorems for polylinear forms, J. Multivariate Anal. 9 (1979), 511\u2013530.\n\u2022 Roth, K. F., On certain sets of integers, J. Lond. Math. Soc. 28 (1953), 245\u2013252.\n\u2022 Samorodnitsky, A. and Trevisan, L., Gowers uniformity, influence of variables, and PCPs, in Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing (Seattle, WA, 2006 ), pp. 11\u201320, ACM, New York, 2006.\n\u2022 Szemer\u00e9di, E., On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), 299\u2013345.","date":"2019-07-21 02:08:21","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\": 0.6537794470787048, \"perplexity\": 1623.5204292281833}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-30\/segments\/1563195526818.17\/warc\/CC-MAIN-20190721020230-20190721042230-00274.warc.gz\"}"} | null | null |
{"url":"https:\/\/openwetware.org\/wiki\/User:Anthony_Salvagno\/Notebook\/Research\/2010\/08\/11\/Imaging_AOM_-_Task_1_and_2","text":"# User:Anthony Salvagno\/Notebook\/Research\/2010\/08\/11\/Imaging AOM - Task 1 and 2\n\n< User:Anthony Salvagno\u200e | Notebook\u200e | Research\u200e | 2010\u200e | 08\u200e | 11\n\nAfter having learned a lot and nothing at the same time about our laser we are going back to complete what we set out to do.\n\n## Question\n\nWhat is this and how can we minimize it?\n\nWhen we adjust the AOm from 1V to 5V we see the profile above on the QPD. There is some sort of stabilization that is occuring and it causes the beam to drift. The sum signal (the bottom plot) decreases during this time as well. We explore this today.\n\n## Experiment 1\n\nOur path was laser > AOM > M1 > M2 > QPD (with lenses)\n\nInitial investigation of ramping. We want to see if we can just minimize the drift with a lens system on the QPD. This had no effect.\n\n## Experiment 2\n\nlaser > L1 > AOM > M1 > M2 > QPD\n\nWe then saw something in the manual that showed that the rise time of the AOM was related to the beam waist in the crystal. The thinking with this experiment is that if we put the focal point of the laser in the crystal (thus making the beam waist as small as possible) we can change the rise time. If the rise time is affecting this drift and time dependency that we have been observing, we can then observe this change by affecting the rise time. In this experiment we saw no effect on the ramping.\n\n## Experiment 3\n\nlaser > L1 > AOM > M1 > L2 > M2 > QPD\n\nIn this experiment we decided to put a lens on the other side of the AOM (with the AOM at the focal point of the lens) to see if this helped at all. It turns out, that it had a good effect. We saw almost no drift in the laser outside of an initial shift in position. No ramping. I think what happened is that we imaged the AOM with the second lens.\n\n## Experiment 4\n\nlaser > AOM > L1 > L2 > M1 > M2 > QPD\n\nNow we are going to setup both lenses after the AOM so that we are imaging the AOM with L1.\n\n### Exp 4.1\n\n\u2022 The 1:1 telescope (L1 and L2) is situated anywhere in between.\n\u2022 We still have the ramping effect.\n\u2022 This is consistent with our previous observations because it was when we had our tweezers setup that we first noticed this. (IE the telescopes were not imaging the AOM.)\n\n### Exp 4.2\n\n\u2022 The AOM is at the focal point of the first lens of the 1:1 telescope\n\u2022 Still not a fix\n\n### Exp 4.3\n\n\u2022 AOM at focus of L1 and QPD at (f1+f2)\n\u2022 Still not a fix\n\n## Another Proof\n\nI got the phenomenon on video and it's tough to see unless you go frame by frame. However It is quite easy to see the intensity change when the AOM voltage is increased.","date":"2023-03-22 09:38:29","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.816082775592804, \"perplexity\": 1688.4070233476577}, \"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\/1679296943809.22\/warc\/CC-MAIN-20230322082826-20230322112826-00107.warc.gz\"}"} | null | null |
SIEMIATYCKI, CHAIM (Semyatitski; pseudonyms: Khaym Tiktiner; Khayml; 1908–1943), Yiddish poet. Born in Tykocin (Tiktin), Poland, of a rabbinical family, the yeshivah-trained Siemiatycki, who lost his father at a young age, went to Warsaw in 1929 to work at odd jobs and to write rather than enter the rabbinate. Encouraged in his poetry from his yeshivah days by Hillel *Zeitlin, he contributed poems as well as critical notices on new Yiddish poets to the Warsaw dailies Moment and Haynt, and to other periodicals. He was soon recognized as one of the leading young Polish-Yiddish poets. In 1935 he published his first modest collection of verse, Oysgeshtrekte Hent ("Outstretched Hands," 1935), followed by another quite slender volume, Tropns Toy ("Dew Drops," 1938). During the Nazi occupation of Warsaw, Siemiatycki found refuge in Soviet-occupied Bialystok. In 1941 he moved to Vilna, suffered in its ghetto, following the liquidation of which he was shot in a labor camp. Despite his modest literary output, Siemiatycki's remarkably simple and highly individual lyrics mark him as one of the most authentic of modern Yiddish religious poets. An excited sense of wonder at God's creation characterizes his poetry.
LNYL, 6 (1965), 494–5; Sh. Belis, in Di Goldene Keyt, 114 (1984), 119–24; A. Bik, in Shragai, B (1985), 167–8. | {
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Falcon 9 Full Thrust Block 5 являє собою остаточну версію частково багаторазової ракети Falcon 9, що виробляється компанією SpaceX. В разі повернення першого ступеня, ракету-носій можна віднести до середнього класу. Зміни попередньої версії Full Thrust (Block 3) здійснювалися поступово, із кількома запусками перехідної версії ракети — Block 4.
Історія
17 лютого 2017 року президент та головний операційний директор американської компанії SpaceX Гвен Шотуелл заявила, що Block 5 буде оснащений відповідно до вимог контрактів, що підписані між SpaceX і НАСА щодо програми Commercial Crew Development.
Стало відомо, що Block 5 отримає двигуни зі збільшеною тягою та ще стійкіше термопокриття. Також буде присутня велика кількість дрібних поліпшень («сто змін»), спрямованих на спрощення процесу відновлення першого ступеня ракети та його подальшого (багаторазового) використання.
У SpaceX на полігоні McGregor у Техасі існує три випробувальних стенди для двигунів: для Raptor, для двигунів другого ступеня, та для удосконалення Merlin 1D+. Саме на останньому 5 листопада 2017 року під час проведення тестування двигуна для Block 5 стався вибух. Це відбулося, коли у двигун подавали рідкий кисень для виявлення можливих місць незапланованого витоку. Було зруйновано дві секції випробувального стенда, на ремонт потрібен місяць.
З точки зору загальної вартості ракети вартість першого ступеню становить 60 %, другого — 20 %, обтічника КВ — 10 %. Решта — витрати, пов'язані із запуском (паливо — $300-500 тис.).
7 лютого 2018 року після тестового польоту Falcon Heavy з особистою Tesla Roadster Ілона Маска на борту голова SpaceX повідомив, що більше не планує здійснювати вищезазначеною ракетою пілотованих місій. Замість цього ПКК Dragon 2 буде запускатися Block 5.
Щоб НАСА допустило ракету до запуску людей, вона повинна була здійснити, як мінімум, сім вдалих звичайних польотів. З першою безпілотною місією до МКС, що відбулася 8 березня 2019 року (SpaceX DM-1), була запущена повністю оновлена Block 5 (із новими COPVs), і розпочався відлік. А вже 30 травня 2020 року під час SpaceX DM-2 перші два астронавти полетіли у космос.
Особливості Block 5
Після успішного запуску ракети Falcon 9 Full Thrust, на якій був встановлений перший ступінь, що вже раніше запускався, Ілон Маск у квітні 2017 розповів, як буде ще вдосконалено ракету, і про деякі її майбутні характеристики.
Для збільшення маси корисного вантажу:
на 7-8 % збільшена тяга двигунів (у порівнянні із Block 3);
також усунено проблему розтріскування турбінного колеса у двигуні другого ступеня;
покращена система керування польотом, шляхом оптимізації так званого «кута атаки» при посадці для кращого аеродинамічного ковзання. Це дозволяє знизити витрати пального;
збільшився об'єм обтічника корисного вантажу. SpaceX також вдалося зробити його багаторазовим, шляхом упіймання його половинок за допомогою двох спеціальних кораблів під час спуску. Іноді їх неушкодженими виловлюють із води.
Для швидшого відновлення та подальшого використання:
на першому ступені застосоване спеціальне термопокриття PICA-X для уникнення руйнування від нагріву;
теплове покриття, яке захищає двигуни, також стало багаторазовим, а опорну їх конструкцію (octaweb) виготовлятимуть із титану;
решітчасті плавники (керма висоти) замість алюмінієвих роблять литими титановими;
двигуни не приварюються до опорної конструкції, а прикручуються на болти;
оновили структуру COPV (composite overwrapped pressure vessels) — композитних балонів для гелію. Це убезпечить їх від обмерзання рідким киснем та руйнування, як сталося у вересні 2016 року;
посадкові опори можна складати, а не знімати їх під час транспортування ступеня в ангар після посадки;.
Ілон Маск повідомив, що за його оптимістичними очікуваннями, SpaceX незабаром зможе готувати перший ступінь до наступного запуску навіть за добу. Після кожного десятого запуску ймовірно потрібен буде ремонт.
Запуски
Перший старт Block 5 відбувся 11 травня 2018 року із місією Bangabandhu-1 по запуску супутника для Бангладеш. Перший ступінь вдало посадили на плавучий океанічний майданчик ASDS «Of Course I Still Love You». Наступні пару місяців його розбиратимуть і розрізатимуть на частини, щоб підтвердити, що він дійсно зміг би ще літати без ремонту.
22 липня 2018 року під час другого запуску встановлено новий рекорд по підйому найважчого комерційного сателіта на ГПО — 7'075 кг.
3 грудня 2018 року було втретє запущено і вдало посаджено один і той самий перший ступінь.
Примітки
Falcon (сімейство ракет-носіїв) | {
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Editorial: California Public Utilities Commission sweetly snuggles up to PG&E
Sacramento Bee (2014-07-31) Editorial Board
Media Link: http://www.sacbee.com/2014/07/31/6594264/editorial-california-public-utilities.html
More Info: California Public Utilities Commission
By the Editorial Board
Published: Thursday, Jul. 31, 2014 - 12:00 am
Last Modified: Thursday, Jul. 31, 2014 - 12:27 pm
We've all heard the warnings: the "e" in email stands for evidence. Never put anything into email that you expect to remain private.
Apparently, the memo never found its way to the California Public Utilities Commission,Commission President Michael R. Peevey, or certain executives at Pacific Gas & Electric Co.
The city of San Bruno is seeking to persuade the commission to impose a $2.45 billion penalty on PG&E for the 2010 gas explosion that killed eight people and destroyed 38 homes in San Bruno.
As part of that effort, San Bruno city officials sued to compel the commission to hand over emails showing the relationship between the regulator and PG&E, the utility it's suppose to regulate. The result: 7,000 emails.
The vast majority of them are mundane. But San Bruno's outside communications consultant helpfully created a "highlight reel" of a few dozen choice emails.
There is the one from April 2, in which Peevey sent a note to Brian K. Cherry, who oversees governmental relations for PG&E, critiquing the company's public relations strategy.
PG&E had issued a press release in March announcing that it expected to be indicted on federal charges related to its failure to prevent the deadly 2010 explosion. Then, on April 1, PG&E announced it had been indicted.
"PG&E's decision to issue a press release last week anticipating all this only meant that the public got to read two big stories rather than one. I think this was inept," Peevey said.
Evidently, PG&E's biggest problem is bad press.
Then there was the April 2013 exchange in which Peevey's chief of staff, Carol A. Brown, suggested to PG&E executive Laura Doll that PG&E brush off questions from the administrative law judge overseeing the PG&E investigation.
Send a "sweet note" saying the questions were moot, Brown wrote, and "wait for them to throw a fit." Alternatively, Brown said, "Answer any simple question you can and then object to the others."
Perhaps Brown would have been smarter to have urged that PG&E comply fully with the judge's request. But that was not to be. So Doll responded: "Love you." And who wouldn't love a regulator who offered such sweet suggestions?
PG&E President Chris Johns promised a full review of the emails. But as San Bruno Mayor Jim Ruane said earlier this week, the emails show the cozy relationship between the commission and the utility.
Far worse than some embarrassing emails is that nearly four years after the explosion, the commission has failed to conclude its proceeding against PG&E.
The utility long ago acknowledged its failures. The National Transportation Safety Boardcompleted its investigation a year ago, saying that the commission, "as the regulator for pipeline safety within California, failed to uncover the pervasive and long-standing problems within PG&E." The U.S. attorney's office indicted PG&E in April and expanded the case in a new indictment earlier this week.
The California Public Utilities Commission, which is most directly responsible for regulating PG&E, has failed to take final action.
The commission was created more than a century ago to oversee the railroads, though it has little power over them now. It does retain broad power over private electric and gas utilities, though how it exercises that power is questionable.
Sen. Jerry Hill, a Democrat who represents San Bruno, has attacked PG&E over the explosion, and the commission for its investigation. PG&E's Cherry confided in a supposedly private 2012 email to a commission official that he had lost all respect for Hill.
Hill should frame that email. Gov. Jerry Brown, who nominates public utility commissioners, and state senators, who confirm those nominees, need to share some of Hill's outrage. They need to insist that the commission act. They also need to undertake an overdue overhaul of the commission.
Read more here: http://www.sacbee.com/2014/07/31/6594264/editorial-california-public-utilities.html#storylink=cpy
Title Editorial: California Public Utilities Commission sweetly snuggles up to PG&E
Publisher Sacramento Bee
Author Editorial Board
Media Link http://www.sacbee.com/2014/07/31/6594264/editorial-california-public-utilities.html
Note Pacific Gas And Electric Michael Peevey San Bruno Accident
Keywords California Public Utilities Commission
Media Type Linked Article
Media Group News
Edit | Attach | Print version | History: r1 | Backlinks | View wiki text | Edit wiki text | More topic actions | {
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\section{Introduction. Foliations.}\label{introduction}
The family of H\'{e}non mappings is a basic example of nonlinear
dynamics. Both real and complex versions of these maps were
extensively studied, but still there is a great deal of unknown
about them.
In this article we study complex quadratic H\'{e}non mappings.
These are maps of the form:
$$f_a(x)=\left(\begin{array}{c} x^2+c-ay \\ x \end{array}\right).$$
\noindent Note that these maps are biholomorphisms of $\mathbb
C^2$. They have constant Jacobian, which equals to the parameter
$a$. As $a\to 0$, H\'{e}non mappings degenerate to quadratic
polynomial maps $x\mapsto x^2+c$, which act on the parabola
$x=y^2+c$. The H\'{e}non mappings that we study are perturbations
of quadratic polynomials with disconnected Julia set.
In analogy with one-dimensional dynamics, Hubbard and
Oberste-Vorth \cite{HOV1} introduced the functions $G_a^+$ and
$G_a^-$ that measure the growth rate of the forward and backwards
iterations of the orbit. These functions are pluriharmonic on the
set of points $U_a^+$, $U_a^-$, whose orbits tend to infinity
under forward and backwards iterates. Their level sets are
foliated by Riemann surfaces. These natural foliations ${\cal
F}_a^+$ and ${\cal F}_a^-$ were introduced and extensively studied
in \cite{HOV1}.
The foliations ${\cal F}_a^+, {\cal F}_a^-$ can also be
characterized in terms of B\"{o}ttcher coordinates. There are maps
$\phi_{a,+}$ and $\phi_{a,-}$ that semiconjugate the map $f_a$ and
the map $f_a^{-1}$ to $z\mapsto z^2$ and $z\mapsto z^2/a$
correspondingly. These functions were constructed in \cite{HOV1}.
We recall the definitions of these functions and list their
properties in Section \ref{bottcher_coordinates}. The level sets
of $\phi_{a,+}$ and $\phi_{a,-}$ define ${\cal F}_a^+$ and ${\cal
F}_a^-$ in some domains near the infinity and can be propagated by
dynamics to all of $U_a^+$ and $U_a^-$ correspondingly.
The dynamical description of these foliations is the following:
\begin{lemma}[\cite{BS4}] The leaves of ${\cal F}_a^+$ are
``super-stable manifolds of $\infty$"\ , i.e. if $z_1,z_2$ belong
to the same leaf, then $d(f_a^n(z_1), f_a^n(z_2))\to 0$
super-exponentially, where $d$ is Euclidean distance in $\mathbb
C^2$. If $z_1,z_2\in U_a^+$ do not belong to the same leaf, then
$d(f_a^n(z_1), f_a^n(z_2))\not \to 0$.
\end{lemma}
\noindent The leaves of ${\cal F}_a^-$ are ``super-unstable
manifolds of $\infty$".
One would like to think of these foliations as coordinates in
$U_a^+\cap U_a^-$. However, for all H\'{e}non mappings there is a
codimension one subvariety of tangencies between ${\cal F}_a^+$
and ${\cal F}_a^-$\cite{BS4}.
\begin{definition} The critical locus ${\cal C}_a$ is the set of tangencies
between foliations ${\cal F}_a^+$ and ${\cal F}_a^-$.
\end{definition}
\noindent Thus, the critical locus is the set of ``heteroclinic
tangencies"\ between the ``super-stable"\ and ``super-unstable"\
manifolds.
We give an explicit description of the critical locus for
H\'{e}non mappings $$(x,y)\mapsto(x^2+c-ay,x),$$ \noindent where
$x^2+c$ has disconnected Julia set, and $a$ is sufficiently small.
The topological model of the critical locus for such H\'{e}non
maps was conjectured by John Hubbard. We justify his picture.
Lyubich and Robertson (\cite{LR}) gave the description of the
critical locus for H\'{e}non mappings $$(x,y)\mapsto
(p(x)-ay,x),$$ \noindent where $p(x)$ is a hyperbolic polynomial
with connected Julia set, $a$ is sufficiently small. They showed
that for each critical point $c$ of $p$ there is a component of
the critical locus that is asymptotic to the line $y=c$. Each
component of the critical locus is an iterate of these ones, and
each is a punctured disk.
They used critical locus to show that a pair of quadratic
H\'{e}non maps of the studied type, taken along with the natural
foliations, gives a rigid object. This means that if a conjugacy
between two H\'{e}non maps sends the natural foliations of the
first map to the natural foliations of the second map then the two
H\'{e}non maps are conjugated by a holomorphic or antiholomorphic
affine map.
Since the work of Lyubich and Robertson \cite{LR} is unpublished,
we include the proof of all the results that we are using.
Some of the sources of the fundamental results about H\'{e}non
mappings are \cite{BLS}, \cite{BS1}, \cite{BS2}, \cite{BS3},
\cite{BS4}, \cite{BS5}, \cite{BS6}, \cite{HOV1}, \cite{HOVII},
\cite{FM}, \cite{FS}.
\subsection{Acknowledgements.} The author is grateful to Mikhail
Lyubich for statement of the problem, numerous discussions and
useful suggestions. We are thankful to John Hubbard for showing
the conjectural picture of the critical locus.
\section{Topological model for the critical locus.}
In our case truncated spheres serve as building blocks for the
critical locus. Consider a sphere $S$ and a pair of disjoint
Cantor sets $\Sigma, \Omega \subset S$. The elements of $\Sigma$,
$\Omega$ can be parametrized by one-sided infinite sequences of
$0,1$'s. Denote by $\sigma_{\alpha}\in \Sigma$,
$\omega_{\alpha}\in\Omega$ elements parametrized by a sequence
$\alpha$.
Let $\alpha_n$ be a $n$-string of $0$'s and $1$'s. For each
$\alpha_n$, $n\in \mathbb N\cup \{0\}$, take a disk
$V_{\alpha_n}\subset S\backslash\left(\Sigma\cup \Omega\right)$
with the smooth boundary. (Let $V$ denote the disk that
corresponds to an empty sequence.) We require that $V_{\alpha_n}$
are disjoint. Moreover, $V_{\alpha_n}$ converge to
$\sigma_{\alpha}$, where $\alpha_n$ is the string of the first $n$
elements of $\alpha$.
\begin{figure}[h]
\centering
\psfrag{sigma}{$\Sigma$}
\includegraphics[height=3.5cm]{truncated_sphere.eps}
\caption{The geometry of a truncated sphere}
\end{figure}
$U_{\alpha_n}$ play the same role for $\Omega$ as $V_{\alpha_n}$
for $\Sigma$.
We assume that there is a fixed homeomorphism $\tilde{h}$ between
the boundaries of $V_{\alpha_n}$ and $U_{\alpha_n}$.
Let $p\in S$ be a point.
We say that $S\backslash\left(\Sigma\cup\Omega\cup\sum_{\alpha_n}
[U_{\alpha_n}\cup V_{\alpha_n}]\cup p\right)$ is a truncated
sphere.
Note that all truncated spheres are homeomorphic one to another.
\begin{comment}
First, we prove a preliminary lemma.
\begin{lemma} Let $D_1$ and $D_2$ be unit disks with smooth
boundaries. Let $\Sigma_1\subset D_1$, $\Sigma_2\subset D_2$ be
Cantor sets. Then $D_1\backslash \Sigma_1$ and
$D_2\backslash\Sigma_2$ are homeomorphic.
\end{lemma}
\begin{proof} Since $\Sigma_1, \Sigma_2$ are Cantor sets, they are homeomorphic.
Denote by $h:\Sigma_1\to \Sigma_2$ the homemorphism.
The elements of Cantor sets can be parametrized by sequences of
$0$ and $1$'s and we can choose parametrizations that respects
$h$. Denote the elements of $\Sigma_1$ and $\Sigma_2$, that are
parametrized by $\alpha$, by $\sigma^1_{\alpha}$ and
$\sigma^2_{\alpha}$ correspondingly.
$$h(\sigma^1_{\alpha})=\sigma^2_{\alpha}.$$
\begin{figure}[h]
\centering
\psfrag{D1}{$D_1$} \psfrag{W1}{$W^1_1$} \psfrag{W0}{$W^1_0$}
\psfrag{W00}{$W^1_{00}$} \psfrag{W01}{$W^1_{01}$}
\psfrag{W10}{$W^1_{10}$}\psfrag{W11}{$W^1_{11}$}
\includegraphics[height=4cm]{cantor_set.eps}
\caption{The partition of a disk, with a Cantor set removed, into
pairs of pants}
\end{figure}
There exist open sets $W^1_{\alpha_n}\subset D_1$, parametrized by
$n$-strings $\alpha_n$ of $0$ and $1$'s, so that $W^1_{\alpha_n}$
contains all points $\sigma_{\alpha}$, such that the string of the
first $n$-digits of the sequence $\alpha$ is $\alpha_n$.
We can take $W^1_{\alpha_n}$'s with smooth boundaries. And assume
that
$$W^1_{\alpha_{n+1}}, W^1_{\alpha'_{n+1}}\Subset W^1_{\alpha_n},$$
\noindent where $\alpha_{n+1}$, $\alpha'_{n+1}$ are obtained from
$\alpha_n$ by adding correspondingly $0$ and $1$ at the end.
Repeat the same procedure for $\Sigma_2$, and denote the
corresponding sets by $W^2_{\alpha_n}$.
Fix some homeomorphism $h:\partial D_1\to \partial D_2$.
Obviously, one can extend $h$ to a homeomorphism
$$
h:D_1\backslash(W^1_0\cup W^1_1)\to D_2\backslash (W^2_0\cup
W^2_1)
$$
\noindent so that it restricts to homeomorphisms
$$h:\partial W^1_0\to \partial W^2_0$$
$$h:\partial W^1_1\to \partial W^2_1$$
Now we can extend it to the next pair of pants and so on. In the
end we get a homeomorphism $h:D_1\backslash \Sigma_1\to
D_2\backslash\Sigma_2$.
\end{proof}
\begin{lemma}\label{uniqueness_ts} The truncated sphere is a well-defined object up to a
homeomorphism.
\end{lemma}
\begin{proof} Let $S_1$ and $S_2$ be two truncated spheres. We
add indices $1$ and $2$ to Cantor sets and disks, used in the
definition truncated spheres.
Let us cut $S_1$ into two disks, so that the first disk $D_1$
contains $\Sigma_1$ and all disks $V^1_{\alpha_n}$. The second
disk contains $\Omega_1$ and all disks $U^1_{\alpha_n}$.
Do the same procedure with $S_2$. Denote the corresponding disk by
$D_2$.
Now we prove that there exists a homeomorphism $h$
$$h:D_1\backslash \left(\Sigma_1\cup V^1_{\alpha_n}\right)\to D_2\backslash \left(\Sigma_2\cup V^2_{\alpha_n}\right)$$
\begin{figure}[h]
\centering
\psfrag{D1}{$D_1$} \psfrag{W1}{$W^1_1$} \psfrag{W0}{$W^1_0$}
\psfrag{W00}{$W^1_{00}$} \psfrag{W01}{$W^1_{01}$}
\psfrag{W10}{$W^1_{10}$}\psfrag{W11}{$W^1_{11}$}
\psfrag{V0}{$V^1_0$}\psfrag{V1}{$V^1_1$}\psfrag{V}{$V$}
\includegraphics[height=4cm]{cantor_set2.eps}
\caption{The partition of a half of a truncated sphere into pairs
of pants with a hole}
\end{figure}
Using the notations from the previous lemma one can choose
$W_{\alpha_n}$, so that they contain all sets $V_{\alpha_m}$, such
that
\begin{enumerate}
\item $m\geq n$,
\item $\alpha_n$ is the first $n$-string of $\alpha_m$.
\end{enumerate}
On each step we extend the homeomorphism from the boundary to a
pair of pants with a hole.
We do the same procedure for the part of truncated spheres left.
We create a homeomorphism, which respects $\tilde{h}$.
\end{proof}
\end{comment}
\begin{theorem}\label{theorem_main} Suppose $x^2+c$ has disconnected Julia set. There
exists $\delta$, such that $\forall |a|<\delta$ the critical locus
of the map
$$
f_a\left(\begin{array}{c}x \\y \end{array}\right)=
\left(\begin{array}{c}x^2+c-ay \\ x \end{array}\right)
$$
is smooth, and has the following topological model: take
countably many truncated spheres $S_n$, $n\in \mathbb Z$, and glue
the boundary of $V_{\alpha_n}$ on $S_k$ to the boundary of
$U_{\alpha_n}$ of $S_{n+k}$ using the homeomorphism $\tilde{h}$.
Map $f_a$ acts on the critical locus by sending $S_n$ to
$S_{n+1}$.
\end{theorem}
\begin{figure}[h]
\centering
\includegraphics[height=11cm]{topological_model.eps}
\caption{The topological model.}
\end{figure}
\begin{note} It is easy to see that the topological model of the critical locus is
well-defined up to a homeomorphism.
\end{note}
\section{Strategy of description.}\label{strategy}
When $a=0$, the H\'{e}non mapping reduces to
$$\left(\begin{array}{c}x \\ y\end{array}\right) \mapsto \left(\begin{array}{c} x^2+c \\ x\end{array}\right)$$
\noindent which is a map $x\mapsto p(x)$ acting on the curve
$x=p(y)$.
As $a\to 0$ the map degenerates, but the foliations and the Green
functions persist and become easy to analyze.
In Section \ref{bottcher_coordinates} we study $\phi_{a,+}$ and
$\phi_{a,-}$, paying extra attention to the degeneration as $a\to
0$.
Section \ref{sec:Green_function} is devoted to Green's functions.
We prove that $G_a^+$ and $G_a^-$ depend continuously on $x,y$ and
the parameter $a$.
In Section \ref{sec:critical_locus} we describe the foliations
${\cal F}_a^+$ and ${\cal F}_a^-$ and the critical locus $\cal C$
in terms of $\phi_{a,+}$ and $\phi_{a,-}$. We also calculate the
critical locus in the degenerate case. As $a$ deviates from zero,
we carefully describe the perturbation. The latter is done in
several steps:
First, we choose appropriate values $r$ and $\alpha$ and describe
the critical locus in the domain
$$\Omega_a = \{G_a^+\leq r\}\cap
\{|y|\leq \alpha\}\cap \{|p(y)-x|\geq |a|\alpha\}.$$
\begin{figure}[h]
\centering
\psfrag{omega}{$\Omega_a$} \psfrag{G=r}{$G_a^+=r$}
\psfrag{|a|alpha}{$|a|\alpha$}
\psfrag{|y|=alpha}{$|y|=\alpha$}
\includegraphics[height=6cm]{omega.eps}
\caption{Domain $\Omega_a$}\label{fig:Omega}
\end{figure}
We choose $r$ that lies between the critical point level and
critical value level:
$$G_p(0)<r<G_p(c),$$
\noindent where $G_p$ is the Green function of polynomial $p$.
We choose $\alpha$ such that $G_a^+|_{\{|x|>\alpha, |x|>|y|\}}>r$.
Moreover, in Section \ref{F-} we choose $\alpha$ such that leaves
of foliation ${\cal F}_a^-$ in $\Omega_a$ form a family horizontal
of parabolas.
\begin{figure}[h!]
\centering
\psfrag{y}{$y$}\psfrag{x}{$x$}\psfrag{x=xim}{$x=p^{-m}(0)$}\psfrag{x=xi1}{$x=p^{-1}(0)$}\psfrag{x=xik}{$x=p^{-k}(0)$}
\psfrag{J+}{$J_a^+$}
\includegraphics[height=6cm]{F+.eps}
\caption{The foliation ${\cal F}_a^+$}\label{fig:F+}
\end{figure}
In Section \ref{F+} we give a description of the foliation ${\cal
F}_a^+$ in
$$\{G_a^+\leq r\}\cap \{|y|\leq \alpha\}.$$
\noindent We show that the local leaves of ${\cal F}_a^+$ are
either vertical-like or vertical parabolas. There is exactly one
family of vertical parabolas (see Fig. \ref{fig:F+}) corresponding
to each line $x=\xi_k,$ where $p^{\circ k}(\xi_k)=0$.
The critical locus in $\Omega_a$ is the set of tangencies of a
family of horizontal parabolas with vertical-like leaves and
vertical parabolic leaves. In Section \ref{Omega} we show that for
each family of parabolas there is exactly one handle.
In Section \ref{infinity} we show that foliations ${\cal F}_a^+$
and ${\cal F}_a^-$ extend holomorphically to $x=\infty$. We show
that a point $(0,\infty)$ belongs to the extension and calculate
the tangent line to the critical locus at this point. This gives
us a description of the critical locus in
$$\{|y|<\epsilon\}\cap \{|x|>\alpha\}.$$
In Section \ref{ext-a-neighborhood} we show that the critical
locus can be extended up to $a|\alpha|$-neighborhood of parabola
$p(y)-x=\mbox{const}$ along $y=0$.
In Section \ref{final} we combine the results from the previous
sections to get the description of the fundamental domain of the
certain component of the critical locus. We rule out ghost
components. We also do a dynamical regluing of the fundamental
domain of the critical locus to obtain a description in terms of
truncated spheres.
\section{Functions $\phi_{a,+}$ and
$\phi_{a,-}$}\label{bottcher_coordinates}
In this section we construct functions $\phi_{a,+}$ and
$\phi_{a,-}$. In their description we follow \cite{HOV1}. In
description of the degeneration of $\phi_{a,+}$ and $\phi_{a,-}$
as $a\to 0$, we follow \cite{LR}.
\subsection{Large scale behavior of the H\'{e}non map}
The study of H\'{e}non mappings usually begins with introduction
of domains $V_{+}$ and $V_{-}$ which are invariant under the
action of H\'{e}non map:
$$f_{a}(V_{+})\subset V_{+}, \quad f^{-1}_{a}(V_{-})\subset V_{-}.$$
Moreover, one requires that every point that has unbounded forward
orbit eventually enters $V_{+}$ and every point that has unbounded
backward orbit enters $V_{-}$.
Fix $R$, $0<r<1$, and choose $\alpha$ so that
\begin{equation}\label{1}
\frac{|c|}{|y^2|}+\frac{R+1}{|y|}<r
\tag{\ref{bottcher_coordinates}.1}
\end{equation}
\begin{equation}\label{2}
|p(y)|>(2R+1)|y|\tag{\ref{bottcher_coordinates}.2}
\end{equation}
\noindent for all $|y|\geq \alpha$.
Consider the following partition of $\mathbb C^2:$
$$V_{+}=\{(x,y) :\ |x|> |y|, |x|> |\alpha|\};$$
$$V_{-}=\{(x,y) :\ |y|> |x|, |y|>\alpha\};$$
$$W=\{(x,y) :\ |x|\leq \alpha, |y|\leq \alpha \}.$$
\begin{figure}[h]
\centering
\psfrag{ex1}{$V_-$} \psfrag{ex2}{$V_+$} \psfrag{R}{W}
\includegraphics[height=5cm]{partition.eps}
\caption{The crude picture of the dynamics of $f_a$}
\end{figure}
\begin{lemma}\label{V+invariance} For $a\in D_R$, $f_a(V_+)\subset V_+$.
\end{lemma}
\begin{proof}
$|p(x)-ay|\geq |p(x)|+|a||y|\geq (2R+1-R)|y|\geq |y|$
$|p(x)-ay|\geq (R+1)|y|\geq \alpha$
\end{proof}
Let
$$\left( \begin{array}{l}x_n\\y_n\end{array}\right)=f^n_a\left( \begin{array}{l}x\\y\end{array}\right).$$
\begin{lemma}
$U_a^+=\bigcup_{n}f^{-n}_a(V_{+});$
\end{lemma}
\begin{proof}
Let $f_a^n(x)\to \infty$. For some $n$, $f_a^n(x)\in V_{+}$ or
$f_a^n(x)\in V_{-}$.
Suppose that for all $n\geq n_0$ $(x_n,y_n)\in V_-$. Note that
$|y_{n+1}|=|x_n|\leq |y_{n}|$. Sequence $\{|y_n|\}$ is decreasing.
Contradiction.
\end{proof}
\begin{lemma} For $a\in D_R$ $f^{-1}_{a}(V_-)\subset V_{-}$.
\end{lemma}
\begin{proof}
$\frac{|p(y)-x|}{|a|}\geq \frac{|2R+1||y|-|x|}{R}\geq |y|$
$\frac{|p(y)-x|}{|a|}\geq |y|\geq |\alpha|$
\end{proof}
\begin{lemma}
$U_a^-=\cup_{n}f^n_a(V_{-}).$
\end{lemma}
\begin{proof} The proof is the same as in the previous lemma.
\end{proof}
\subsection{Function $\phi_{a,+}$}
The function $\phi_{a,+}$ is constructed as the limit
$$\phi_{a,+}(x,y)=\lim_{n\to \infty} |x_n|^{\frac{1}{2^n}}.$$
\noindent with the appropriate choice of the branch of root. We
are particularly interested in this function in $V_+$. Sense is
made of the above limit using the telescopic formula
\begin{equation}\label{telescopic_formula1}
\phi_{a,+}=\lim_{n\to \infty}x \exp
\left(\frac{1}{2}\log\frac{x_1}{x^2}+\dots
+\frac{1}{2^n}\log\frac{x_n}{x_{n-1}^2}+\dots\right).
\end{equation}
\begin{lemma} The function $\phi_{a,+}$, defined by
formula (\ref{telescopic_formula1}), is well-defined and
holomorphic for all $(x,y)\in V_+$ and all $a\in D_R$. Moreover,
there exists $B$ so that
$$B^{-1}<\left|\frac{\phi_{a,+}}{x}\right|<B.$$
\end{lemma}
\begin{proof}
Let $s_n^+=\frac{x_n}{x^2_{n-1}}-1$. By property (\ref{1}),
$|s_n^+|<r$.
\noindent$\log\frac{x_n}{x_{n-1}^2}=\log(1+s_n^+)$ is calculated
using the principle branch of $\log$. Therefore, the series
\begin{equation}\label{sum1}
\frac{1}{2}\log\frac{x_1}{x^2}+\dots+\frac{1}{2^n}\log\frac{x_n}{x_{n-1}^2}+\dots
\end{equation}
\noindent converges absolutely and uniformly. Since
$|\log\frac{x_n}{x_{n-1}^n}|<-\frac{\log(1-r)}{2^n}$, the infinite
sum (\ref{sum1}) is no bigger than $-\log B= -\log(1-r)$.
The final claim follows immediately from the expression
(\ref{telescopic_formula1}) and the bound derived for the series
(\ref{sum1}).
\end{proof}
We show that $\phi_{a,+}\sim x$ as $x\to \infty$ in Section
\ref{infinity}.
Let ${\cal D}_{n,+}=\{(x,y,a)|\ f^n_a(x,y,a)\in V_+, a\in D_R\}$
\begin{lemma} The function $\phi^{2^n}_{a,+}$ extends to a
holomorphic function on ${\cal D}_{n,+}$ given by
$\phi^{2^n}_{a,+}=\phi_{a,+}\circ f^{n}_a$. Moreover,
$\phi^{2^n}_{0,+}(x,y)=b^{2^n}_p(x)$.
\end{lemma}
\begin{proof} The function $\phi^{2^n}_{a,+}$ is holomorphic by
definition.
As $a\to 0$ the H\'{e}non mappings degenerate to a one-dimensional
map $x\to p(x)$, acting on $y=p(x)$. Therefore,
$\phi_{0,+}(x,y)=b_p(x)$ on $V_+$, and
$\phi_{0,+}^{2^n}(x,y)=b_p^{2^n}(x)$ if $f_0^n(x,y)\in V_+$.
\end{proof}
Note that
$$K_0^+=J_0^+=J_p\times \mathbb C,$$
$$U_0^+=U_p\times \mathbb C,$$
where $J_p$ is the Julia set for the one-dimensional map
$x\mapsto p(x)$; $U_p$ is the set of points, whose orbits escape
to $\infty$ under the map $x\mapsto p(x)$.
\subsection{Function $\phi_{a,-}$}
We start by working the leading terms of $y_{-n}$ as a polynomial
in $y$ and as a polynomial in $\frac{1}{a}$. The following
notation simplifies the statement of the result:
$\sigma_k=1+2+\dots+2^{k-1}$ for $k\geq 1$ and $\sigma_k=0$ for
$k\leq 0$.
By an easy induction we get:
\begin{lemma}\label{pol_decomposition} The leading term of $y_{-n}(x,y,a)$ considered as a
polynomial in $y$ is $y^{2^n}/a^{\sigma_n}$. The leading term of
$y_{-n}$ considered as a polynomial in $\frac{1}{a}$ is
$\frac{1}{a^{\sigma_n}}(p(y)-x)^{2^{n-1}}$.
\end{lemma}
We define the function $\phi_{a,-}$ as a limit
\begin{equation}\phi_{a,-}=\lim_{n\to \infty}(y_{-n}\circ
a^{\sigma_n})^{\frac{1}{2^n}}\end{equation}
\noindent with an appropriate choice of branch of root. Note that
the factor $a^{\sigma_n}$ is chosen, so that the leading term of
$y_n\cdot a^{\sigma_n}$, as a polynomial in $y$, is $y^{2^n}$.
Let $D_R^*=D_R\backslash\{0\}$.
Sense is made of the limit in $V_-\times D^*_R$ using the
telescopic formula:
\begin{equation}\label{telescopic_formula2}
\phi_{a,-}(x,y)=\lim_{n\to
\infty}\exp\left(\frac{1}{2}\log\frac{ay_{-1}}{y^2}+\frac{1}{2^2}\log\frac{ay_{-2}}{y^2_{-1}}+\dots\right)
\end{equation}
\begin{lemma}\label{existence_phi-} The function $\phi_{a,-}$, defined by formula
(\ref{telescopic_formula2}) is well-defined and holomorphic on
$V_-\times D_R^*$. Moreover, there exists $B$ so that
$$B^{-1}<\left|\frac{\phi_{a,-}}{y}\right|<B.$$
\end{lemma}
\begin{proof}
Let $s_n^-=\frac{c-x_{-(n-1)}}{y^2_{-(n-1)}}$
Note that $|s_n^-|<r$ for $(x,y)\in V_-$, $a\in D^*_R$.
We evaluate $\frac{ay_{-n}}{y^2_{-(n-1)}}=\log(1+s_n^-)$ using the
principle branch of $\log$.
Since $\log(1+s_n^-)\leq -\log(1-r)$, the series
\begin{equation}\label{sum2}
\frac{1}{2}\log\frac{ay_{-1}}{y^2}+\frac{1}{2^2}\frac{ay_{-2}}{y^2_{-1}}+\dots
\end{equation}
\noindent converges uniformly and absolutely to a holomorphic
function bounded by $-\log(1-r)$. Letting $B=(r-1)^{-1}$, the last
statement of the Lemma follows.
\end{proof}
We show that $\phi_{a,-}\sim y$ as $y\to \infty$ in Section
\ref{infinity}.
The next lemma states that one can extend $\phi_{a,-}$ to a
holomorphic function on $V_-\times D_R$.
\begin{lemma}\label{limiting_value} $\phi_{a,-}=(p(y)-x)^{\frac{1}{2}}+ah(x,y,a)$ for
some holomorphic function $h$ on $V_-\times D_R$.
\end{lemma}
\begin{proof} By Lemma \ref{pol_decomposition} the leading term of $y_{-n}$
as a polynomial in $\frac{1}{a}$ is
$\frac{1}{a^{\sigma_n}}(p(y)-x)^{2^{n-1}}$. Since
$x_{-n}=y_{-(n-1)}$. The leading term of $x_{-n}$ as a polynomial
in $\frac{1}{a}$ is the leading term of $y_{-(n-1)}$.
Recall $s_n^-=\frac{c-x_{-(n-1)}}{y^2_{-(n-1)}}$. It follows that
$s_n^-$ is a polynomial in $a$ and it vanishes in $a$ to the order
$2\sigma_{n-1}-\sigma_{n-2}$. Hence the series (\ref{sum2}) takes
the form
$$
\frac{1}{2}\log\frac{ay_{-1}}{y^2}+ag(x,y,a),
$$
\noindent where $g(x,y,a)$ is a holomorphic function on $V_+\times
D_R$.
By (\ref{telescopic_formula2}), $\phi_{a,-}(x,y)=y \exp
\left(\frac{1}{2}\log \frac{ay_1}{y^2})\right)\exp(ag(x,y,a))$.
The conclusion follows.
\end{proof}
Let
$$C(p)=\{p(y)-x=0\}$$
Domain $f_a(V_{-})$ swells to $\mathbb C^2\backslash
C(p)$ as $a\to 0$.
\begin{lemma}\label{lem:swell}
For $(x,y)\in \mathbb C^2\backslash C(p)$ we have $(x,y)\in
f_a(V_{-})$ for all sufficiently small $a$. Moreover, for all
$K\Subset \mathbb C^2\backslash C(p)$ $\Rightarrow$ $K\Subset
f_a(V_{-})$ for all small enough $a$.
\end{lemma}
\begin{proof} Both statements follow from
$$f_a(V_{-})=\{|p(y)-x|\geq |a|\alpha, |p(y)-x|\geq
|a||y|\}.$$
\end{proof}
$${\cal D}_{-,n}=\{(x,y,a)|\ (x,y)\in f_a^{n}(V_-), a\neq
0\ \mbox{or}\ (x,y)\in \mathbb C^2\backslash C(p), a=0 \}.$$
\begin{lemma}\label{lim_value2} The function $\phi_{a,-}^{2^n}(x,y)$
extends from a holomorphic function on $V_{-}\times D_R$ to ${\cal
D}_{-,n}$, by letting
\begin{enumerate}
\item $\phi^{2^n}_{a,-}=a^{\sigma_n}\phi_{a,-}\circ f_a^{-n}$
for $a\neq 0$;
\item $\phi^{2^n}_{0,-}(x,y)=(p(y)-x)^{2^{n-1}}$.
\end{enumerate}
\end{lemma}
\begin{proof} If $a\neq 0$, then we can extend the function
$\phi_{a,-}$ to be holomorphic by defining
$\phi^{2^n}_{a,-}=a^{\sigma_n}\phi_{a,-}\circ f_a^{-n}$. This
agrees with $\phi_{a,-}^{2^n}$ on $V_{-}$.
By Lemma \ref{limiting_value}
\begin{equation}
\phi^{2^n}_{a,-}=a^{\sigma_n}\phi_{a,-}\circ
f_a^{-n}=a^{\sigma_n}(p(y_{-n})-x_{-n})^{\frac{1}{2}}+a^{\sigma_n+1}h(x_{-n},
y_{-n},a)
\end{equation}
By Lemma \ref{pol_decomposition} the leading term of $y_{-n}$, as
a polynomial in $\frac{1}{a}$, is
$\frac{(p(y)-x)^{2^n}}{a^{\sigma}}$. Since $x_{-n}=y_{-(n-1)}$,
the leading term of $x_{-n}$ as a polynomial in $\frac{1}{a}$ is
$\frac{(p(y)-x)^{2^{n-2}}}{a^{\sigma_{n-1}}}$. Thus
$a^{\sigma_n}(p(y_{-n})-x_{-n})^{\frac{1}{2}}$ is a polynomial in
$x,y,a$ and the only term free in $a$ is $(p(y)-x)^{2^{n-1}}$.
By the same argument the function $a^{\sigma_n}h(x_{-n},y_{-n},a)$
is holomorphic in $(x,y,a)$ on $V_+\times D_R$. Therefore,
$a^{\sigma_n+1}h(x_{-n},y_{-n},a)$ vanishes in $a$.
Thus, $\phi_{a,-}^{2^n}=(p(y)-x)^{2^{n-1}}+ah_1(x,y,a)$.
\end{proof}
\noindent We denote $K_0^-$ and $J_0^-$ to be $C(p)$. We set
$U_0^-=\mathbb C^2\backslash J_0^-$. The previous two statements
justify these notations.
\section{Green's functions}\label{sec:Green_function}
The next two lemmas are from \cite{HOV1}. We provide the proof of
Lemma \ref{lem:Green_function} to make the paper self-contained.
\begin{lemma}\label{lem:Green_function} The function
\begin{equation}\label{Green_function}
G_a^+(x,y)=\lim_{n\to \infty}\log^+|f_a^n(x,y)|
\end{equation}
\noindent is well-defined in $\mathbb C^2\times D_R$. It satisfies
the functional equation
\begin{equation}\label{functional_equation}
G_a^+(f_a)=2 G_a^+.
\end{equation}
\noindent Moreover, it is pluriharmonic in $U_a^+$.
\end{lemma}
\begin{proof}
We define, the function $G_a^+=\log |\phi_{a,+}|$ on $V_+$. We use
the functional equation (\ref{functional_equation}) to extend it
to $U_a^+$. We also set $G_a^+(x,y)=0$ for $(x,y)\in J_a^+$.
$G_a^+$, defined this way, is pluriharmonic in $V_+$, since it is
a logarithm of a holomorphic non-zero function $\phi_{a,+}$. It is
pluriharmonic in $U_a^+$, since for each $n$, on $f_a^{-n}(V_+)$,
$G_a^+$ is a pull-back of a pluriharmonic function by a
holomorphic change of coordinates.
Moreover, notice that the function defined this way satisfies
(\ref{Green_function}) and there is a unique function that
satisfies (\ref{Green_function}).
\end{proof}
\begin{lemma}
The function
\begin{equation}
G_a^-(x,y)=\lim_{n\to \infty}|f_a^{-n}(x,y)|+\log|a|
\end{equation}
is well-defined in $\mathbb C^2\times D_R^*$. It satisfies the
functional equation
\begin{equation}\label{functional_equation2}
G_a^-(x,y)\circ f_a^{-}=2 G_a^--\log|a|.
\end{equation}
\noindent Moreover, it is pluriharmonic on $U_a^-$.
\end{lemma}
Equation (\ref{functional_equation2}) is sometimes more
conveniently written
$$(G_a^{-}-\log|a|)\circ f_a^{-1}=2(G_a^{-}-\log|a|).$$
We set
\begin{equation}
G_0^-(x,y)=\log|\phi_{0,-}(x,y)|=\left\{\begin{array}{ll}\frac{1}{2}\log|p(y)-x|\
& \mbox{for}\ (x,y)\not \in C(p);\\
-\infty & \mbox{for}\ (x,y)\in C(p).\end{array}\right.
\end{equation}
Hubbard \& Oberste-Vorth proved that the Green's functions are
continuous when $f_a$ is non-degenerate and the same argument
gives continuity in $x,y$ and $a$ for $G_a^+$ when $a=0$.
Lyubich-Robertson \cite{LR} extend this to $G_a^-$ when $a=0$.
\begin{lemma}[\cite{LR}] The functions $G_a^+(x,y)$ and
$G_a^-(x,y)$ are continuous in $x,y$ and $a$ for $a\in D_R$.
\end{lemma}
\begin{proof} It follows by the same argument as is used in
\cite{HOV1} except in the case of $G_a^-$ when $a=0$. For
$(x',y')\not \in C(p)$ the continuity of $G_a^-$ at $(x',y')$ and
$a=0$ follows from Lemma \ref{lim_value2}. For $(x',y')\in C(p)$
more work is required. If we restrict $G_a^-$ to the slice $a=0$,
then we have shown continuity, so we will assume for most of the
rest of this proof that $a\neq 0$ (so $f_a^{-1}$ is defined).
Let us fix $M>0$, and find a neighborhood $U$ of $(x',y')$ so that
for all $(x,y)\in U$, $G_a^-(x,y)<-M$
If $(x,y)\in J_a^-$, then it is enough to require that
$|a|<e^{-M}$.
If $(x,y)\in U_a^-$, then $f_a^{-n}(x,y)\in V_-$ for all $n>n_0$.
If $f_a^{-n}(x,y)\in V_-$, then by Lemma \ref{existence_phi-}
$B^{-1}<\left|\frac{\phi_{a,-}(x_{-n}, y_{-n})}{y_{-n}}\right|<B$.
Therefore, $G_a^-(x,y)<\frac{1}{2}\log
B+\frac{1}{2^n}\log|a^{\sigma_n} y_{-n}|$.
\begin{equation}\label{rec_rel} y_{-n}=\frac{1}{a}\left(p(y_{-(n-1)}\right)-y_{-(n-2)})
\ \mbox{for}\ n\geq 2.\end{equation}
$$y_0=y,\ y_{-1}=\frac{p(y)-x}{a}$$
We wish to estimate $a^{\sigma_n} y_{-n}$ in terms of $y_0$ and
$y_{-n}$.
It is convenient to introduce a new variable
$z_{-n}=a^{\sigma_n}y_{-n}$ and a notation $p~(~x~,~y~)~=~y^2~
p~(~\frac{x}{y}~)~.$
In these new notations the recurrence relation (\ref{rec_rel})
takes form
\begin{equation}
z_{-n}=p(z_{-(n-1)},
a^{\sigma_{n-1}})-a^{\sigma_n-\sigma_{n-2}-1}z_{n-2},
\end{equation}
\noindent where $z_0=y$, $z_{-1}=p(y)-x$.
$|z_{-n}|\leq 2\max(|p(z_{-(n-1)}|, |a|^{\sigma_{n-1}}),
|a|^{2^{n}+2^{n-1}-1}|z_{n-2}|)$
Using the estimate $p(x,y)\leq C\max(x^2, a^2)$, where $C$ is a
constant, we get
$|z_{-n}|\leq 2\max(|z^2_{-(n-1)}|, |a|^{2\sigma_{n-1}},
|a|^{2^n+2^{n-1}-1}|z_{n-2}|)$
Consider a neighborhood
\begin{equation}
U(\epsilon)=\{(x,y,a)|\ |p(y)-x|<\epsilon^2,
|a|<\min(2C\epsilon^2, \frac{1}{2}), |a||y|<C\epsilon^2\}
\end{equation}
By induction one can show that if $(x,y,a)\in U(\epsilon)$, then
$z_{-n}<(2C)^{\sigma_n}\epsilon^{2^n}$.
Therefore, for $(x,y,a)\in U(\epsilon)$
\begin{equation}
G_a^-(x,y,a)<\log(|2C|\epsilon)
\end{equation}
Therefore, for small enough $\epsilon$, $G_a^{-}(x,y,a)<-M$.
Intersecting $U(\epsilon)$ with the $e^{-M}$-neighborhood in
$a$-variable, we get the desired neighborhood.
\end{proof}
\section{Description of the foliations and the critical locus in
terms of $\phi_{a,+}$ and $\phi_{a,-}$}\label{sec:critical_locus}
Note that
$$G_a^+(x,y)=\log|\phi_{a,+}(x,y)|;$$
\noindent Therefore, the foliation $\phi_{a,+}=\mbox{const}$ on
$V_{+}$ is exactly ${\cal F}_a^+$. It can be propagated by
dynamics to the rest of $U_a^+$.
The same way,
$$G_a^-(x,y)=\log|\phi_{a,-}(x,y)|.$$
The foliation $\phi_{a,-}=\mbox{const}$ on $V_-$ is ${\cal F}_a^-$
and it propagates by dynamics to $U_a^-$
$${\cal U}^{\pm}=\{(x,y,a) |\ a\in D_R, (x,y)\in U_a^\pm\}
$$
Below we show that the critical locus is defined by a global
holomorphic form on ${\cal U}^+\cap {\cal U}^{-}$. Therefore, it
is a proper analytic subset of ${\cal U}^{+}\cap {\cal U}^{-}$.
The forms $d \log \phi_{a,+}$ and $d \log \phi_{a,+}$ are
well-defined and holomorphic in ${\cal U}^+$ and ${\cal U}^-$
correspondingly.
Critical locus $\cal C$ is given by the zeroes of the form
$$w(x,y,a)dx\wedge dy \wedge da = d\log \phi_{a,+}\wedge d \log \phi_{a,-}\wedge
da$$
\noindent that is holomorphic in ${\cal U}^{+}\cap {\cal U}^{-}$.
When we prefer to think of $w(x,y,a)$ as a function of two
variables we use the notation $w_a(x,y)=w(x,y,a)$.
Let ${\cal C}_{a_0}={\cal C}\cap \{a=a_0\}$.
We recall a definition of a proper analytic subset:
\begin{definition} $A$ is a proper analytic subset of a complex
manifold $M$, if for every point $x\in M$, there exist a
neighborhood $U$ and a set of functions $f_1,\dots, f_n$ so that
$$f_1=\dots=f_n=0$$
define $A$ in $U$.
\end{definition}
\begin{lemma}\label{analiticity} ${\cal C}$ is a proper analytic
subset of ${\cal U}^{+}\cap {\cal U}^{-}$.
\end{lemma}
\begin{proof} ${\cal C}$ is defined by the zeroes of the form that is
holomorphic in ${\cal U}^{+}~\cap~{\cal U}^{-}$.
\end{proof}
\subsection{Degenerate critical locus.}
As $a\to 0$ the H\'{e}non mapping degenerates to $(x,y)\mapsto
(p(x),x)$. The foliation ${\cal F}_a^+$ is defined by the form
$d\phi_{a,+}$ on $U^+_a$. Note that $d\phi_{a,+}\neq 0$ on $U_a^+$
for $a\in D_R^*$. Therefore, the foliation ${\cal F}_a^+$ for
$a\in D_R^*$ is nondegenerate. The liming foliation ${\cal F}_0^+$
is degenerate. It has double leaves.
Below we explain what it means for the foliation to have a double
leaf.
Recall that $U_a^+=\sum_n f_a^{-n}(V_+)$, on each $f_a^{-n}(V_+)$
the foliation ${\cal F}_{a}^+$ is defined by the level sets of the
function $\phi_{a,+}^{2^n}$.
Suppose that a foliation $\cal F$ in $\Omega\subset\mathbb C^2$ is
defined by the level sets of a function $\phi$. Suppose that in a
neighborhood of each point $(x,y)$ one can choose local
coordinates $(u,t)$, so that $\phi=u^n$.
\begin{definition} We say that a leaf $L$ of the foliation $\cal F$
is double if in a neighborhood of a point $(x,y)\in L$,
$\phi=u^2$.
\end{definition}
\begin{note} The definition depends on the defining function $\phi$ and does not depend on the choice of the
local parameter $u$, nor on a point $(x,y)\in L$.
\end{note}
\begin{lemma} The foliation ${\cal F}_0^+$ in $U_0^+$ is a vertical foliation
$x=const$ with leaves $x=p^{-k}(0)$ being double for all $k\geq
0$.
\end{lemma}
\begin{proof} For each point $x_0$ there exists $n$, such that in a neighborhood of the line $x=x_0$
the foliation ${\cal F}_0^+$ is determined by the level sets of
the function $\phi^{2^n}_{0,+}$.
Since $\phi^{2^n}_{0,+}(x,y)=b^{2^n}_p(x)$, the foliation is
vertical.
The multiple leaves appear when $\left(b^{2^n}_p\right)'(x)=0$.
$\left(b^{2^n}_p\right)'(x)=0$ if and only if $p^n~(~x~)~=~0~.$
Moreover, all zeros of $\left(b_p^{2^n}\right)'$ are
non-degenerate. Therefore, all the leaves $x=p^{-n}(0)$ are
double.
\end{proof}
The foliation ${\cal F}_a^-$ in $U^-a$ for $a\in D_R^*$ is defined
by the form $d\phi_{a,-}^{2^n}$. By Lemma \ref{lem:swell} as $a\to
0$, $U_a^-$ swell to $\mathbb C^2\backslash C(p)$. The form
$d\phi_{a,-}$ extends to $\mathbb C^2\backslash C(p)$ as a
holomorphic form. Let ${\cal F}_0^-$ denote the limiting foliation
defined by the form $d\phi_{0,-}$.
\begin{lemma} The foliation ${\cal F}_0^-$ in $U_0^-$ consists of the leaves
$\{p(y)-x=const\}$.
\end{lemma}
\begin{proof} By Lemma \ref{limiting_value} $\phi_{0,-}=p(y)-x$ and is defined in $U_0^-$. The statement of
the Lemma immediately follows.
\end{proof}
\begin{corollary}\label{degenerate-critical-locus} ${\cal C}_0=\left[\{y=0\}\bigcup_k\{x=p^{-k}(0)\}\right]\cap U_0^+\cap U_0^-$
\end{corollary}
\begin{figure}[h]
\centering
\psfrag{J+}{$J_0^+$}\psfrag{J-}{$J_0^-$}\psfrag{x}{$x$}\psfrag{y}{$y$}\psfrag{x=p1}{$x=p^{-1}(0)$}\psfrag{x=pk}{$x=p^{-k}(0)$}
\psfrag{x=pm}{$x=p^{-m}(0)$}
\includegraphics[height=5cm]{degenerate_cl.eps}
\caption{The degenerate Critical Locus}
\end{figure}
\section{Critical locus near infinity.}\label{infinity}
The goal of this section is to calculate the critical locus at a
neighborhood of $x=\infty$ in the compactification of $\mathbb
C^2$ given by $\mathbb C\mathbb P^1\times \mathbb C\mathbb P^1$.
First, we extend the foliation ${\cal F}_a^+$ to the line
$x=\infty$ in $\mathbb C\mathbb P^1\times \mathbb C\mathbb P^1$
compactification of $\mathbb C^2$. Let
$$\hat{V}_{+}= \{(x,y)\in \mathbb C\mathbb P^1\times \mathbb
C\mathbb P^1|\ (x,y)\in V_+ \ \mbox{or}\ \{x=\infty, y\neq
\infty\}\}$$
Let $t=\frac{1}{x}$, $v=\frac{1}{y}$.
\begin{lemma}The function $\frac{\phi_{a,+}}{x}$ extends as a holomorphic function to ${\hat V}_+\times D_R$
Moreover, $t \phi_{a,+}=1+th_1(t,y,a)$, $\frac{\partial
h_1}{\partial y}=u \tilde{h}_1(t,y,a)$.
\end{lemma}
\begin{proof} By the Riemann Extension Theorem, $\frac{\phi_{a,+}}{x}$ can
be extended to $x=\infty$.
Note that the functions $s_n^+$ and $\frac{\partial
s_n^+}{\partial y}$ vanish in $t$ for all $k$. Thus, in the sum
(\ref{sum1}) every term vanishes in $u$. Therefore, the infinite
sum (\ref{sum1}) vanishes in $u$ as well. And
$\frac{\phi_{a,+}}{x}=1+th_1(t,y,a)$, where $h_1$ is a holomorphic
function in $t,y,a$. The same way one proves, $\frac{\partial
h_1}{\partial y}=t\tilde{h}_1(t,y,a)$.
\end{proof}
$$
\hat{V}_-=\{(x,y)\in \mathbb C\mathbb P^1\times \mathbb C\mathbb
P|\ (x,y)\in V_{-}\ \mbox{or}\ \{y=\infty,x\neq \infty\}\}
$$
\begin{lemma}[\cite{LR}]\label{phi-infinity} The function $\frac{\phi_{a,-}}{y}$ extends
holomorphically to $\hat{V}_-\times D_R$. Moreover,
$v\phi_{a,-}=1+vh_2(x,v,a)$, $\frac{\partial h_2}{\partial
x}=v\tilde{h}_2(x,v,a)$.
\end{lemma}
\begin{proof} The proof is the same as in the previous lemma.
\end{proof}
In $V_+\cap {\cal D}_{1,-}$ the critical locus is given by the
zeroes of the form
$$\omega = d\phi_{a,+}\wedge d\phi^2_{a,-}\wedge da$$
For the next lemma we fix $(t,y)$-coordinates in $\hat{V}_+$.
\begin{lemma}\label{lemma_infinity}
The critical locus extends holomorphically to $\left(\hat{V}_+\times D_R\right)\cap
{\cal D}_{1,-}$. The point $(0,y,a)\in {\cal C}_a\cap
\hat{V}_+\cap {\cal D}_{1,-}$ iff $y=0$. The tangent line to the
critical locus at $(0,0,a)$ is given by $2dy+Cdt=0$ with $C$
depending on $a$.
\end{lemma}
\begin{proof}
$$d\phi_{a,+}=-\frac{dt}{t^2}+\frac{\partial h_1}{\partial
t}dt+\frac{\partial h_1}{\partial y}dy+\frac{\partial
h_1}{\partial a}da=\frac{1}{t^2}\left(-1+t^2\frac{\partial
h_1}{\partial t}dt+t^3\tilde{h_1}dy+t^2\frac{\partial
h_1}{\partial a}da\right)$$
$$\phi^2_{a,-}(x,y)=a
\phi_{a,-}\circ f^{-1}_a(x,y)=a\phi_{a,-}\left(y,
\frac{p(y)-x}{a}\right)=\frac{tp(y)-1}{t}+ah_2(y,\frac{at}{tp(y)-1},a)$$
$$d\phi_{a,-}=\left(p'(y)+a\frac{\partial h_2}{\partial y}(y,
\frac{at}{tp(y)-1},a)+\frac{a^2t^2p'(y)}{(tp(y)-1)^2}\frac{\partial
h_2}{\partial v}(y,\frac{at}{tp(y)-1})\right)dy$$
$$+\left(\frac{1}{t^2}+\left[\frac{a}{tp(y)-1}+\frac{atp(y)}{(tp(y)-1)^2}\right]\frac{\partial
h_2}{\partial v}\right)dt+h_3(t,y,a)da,$$
\noindent where $h_3(t,y,a)$ is some holomorphic function in
$t,y,a$.
By Lemma \ref{phi-infinity}, $\frac{\partial h_2}{\partial
x}=v\tilde{h}_2$. Hence $\frac{\partial h_2}{\partial
x}(y,\frac{at}{tp(y)-1},a)=\frac{at}{tp(y)-1}\frac{\partial
h_2}{\partial x}(y,\frac{at}{tp(y)-1})$. Thus, there are
holomorphic functions $h_4,$ $h_5$ on ${\cal D}_1^-$ so that
$d\phi_{a,-}=(p'(y)+th_4(t,y,a))dy+(\frac{1}{t^2}h_5(t,y,a))dt+h_3(t,y,a)da$
Therefore, there exists a holomorphic function $h_6$ on
$\left(\hat{V}_+\times D_R\right)\cap {\cal D}_{1,-}$ so that
$$\omega t^2=\left(p'(y)+t h_6\right)dy\wedge dt\wedge da$$
\noindent Conclusion follows.
\end{proof}
\begin{corollary} Fix $\epsilon$. There exists $\delta$
so that for $|a|<\delta$ the critical locus in
$$\{|y|\leq \epsilon\}\cap \{|x|\geq \alpha\}$$
\noindent is the graph of a function $y(x)$.
\end{corollary}
\section{Horizontal and vertical invariant cones.}
\subsection{Horizontal cones.}
Fix a domain $B$.
\begin{definition} A family of cones $C_x$ in the tangent bundle to $B$ is
$f_a$-invariant iff for any point $x\in B$, such that $f_a(x)\in
B$, we have $df_a(C_x)\cup C_{f(x)}$.
\end{definition}
\begin{lemma}\label{horizontal} Fix $r'>r$ and $\beta$. For every
$C<\min \{x : G_p(x)\leq \frac{r'}{2}\}$ there exists $\delta$
such that for all $|a|<\delta$ the family of horizontal cones
$|\xi|>C|\eta|$ is $f_a$-invariant in $\{G_a^+(x,y)\leq r'\}\cap
\{|y|\leq \beta\}$ (where $(\xi,\eta)\in T_{(x,y)}\mathbb C^2$).
\end{lemma}
\begin{note} Note that we chose the box $\{G_a^+(x,y)\leq r'\}\cap
\{|y|\leq \beta\}$ so that the tip of the parabola does not belong
to the box. This allows us to have an invariant horizontal family
of cones.
\end{note}
\begin{proof} $Df_a(x,y)=\left(\begin{array}{ll} 2x & a\\ 1 & 0
\end{array}\right)$
Let $\left(\begin{array}{l}\xi_1\\\eta_1
\end{array}\right)=\left(\begin{array}{cc} 2x & a\\ 1 & 0
\end{array}\right)\left(\begin{array}{l}\xi \\ \eta\end{array}\right)$
We want to find $C$ such that $$|\xi|>C|\eta| \Rightarrow |2x
\xi+a \eta|>C|\xi|.$$
Take $C=\min(2x)-\epsilon,$ where $b_p(x)\leq \frac{r'}{2}$ and
$\epsilon$ is any number. Then $\delta_2=\epsilon \max(2x)$.
\end{proof}
\subsection{Vertical cones.}
\begin{lemma}\label{vertical} Fix $r'\geq r,$ $\alpha,$ $C$. There
exists $\delta$ such that for all $|a|<\delta$ the family of cones
$|\xi|<C|a||\eta|$ is $f_a^{-1}$-invariant in $\{G_a^+(x,y)\leq
r'\}\cap\{|y|\leq \alpha\}$, $(\xi,\eta)\in T_{(x,y)}\mathbb C^2$.
\end{lemma}
\begin{proof} $D f^{-1}_{a}=\left(\begin{array}{cc}0 & 1 \\ -\frac{1}{a} & \frac{2y}{a}\end{array}\right)$
Let $\left(\begin{array}{c}\xi_1 \\ \eta_1\end{array}\right)=\left(\begin{array}{cc}0 & 1 \\
-\frac{1}{a} &
\frac{2y}{a}\end{array}\right)\left(\begin{array}{c}\xi \\
\eta\end{array}\right)$
$\left|\frac{\xi_1}{\eta_1}\right|=\frac{|a|}{|-\frac{\xi}{\eta}+2y|}$
Suppose $(x,y)= f^{-1}_a(u,v)$, where $\{(u,v) |\ G_a^+(u,v)\leq
r', |v|\leq |\alpha|\}$, then $|y|>C_1$.
Therefore, when $|a|<\delta$, $|2y-\frac{\xi}{\eta}|\geq |C_1| $
\end{proof}
\section{Description of ${\cal F}_a^-$.}\label{F-}
In this section we give a description of ${\cal F}_a^-$ in
$$
W \cap \{|p(y)-x|\geq |a|\alpha\}.
$$
We choose $\alpha$ in the definition of $W$ such that the
description of ${\cal F}_a^-$ is especially nice.
The function $\phi^2_{a,-}$ is well-defined in this region, so it
is natural to expect that the foliation
$${\cal F}_a^-=\{\phi^2_{a,-}=\mbox{const}\}$$
\noindent is close to the foliation
$${\cal F}_0^-=\{\phi^2_{0,-}=p(y)-x=\mbox{const}\}.$$
\noindent The only region where it really needs to be checked is
when we approach $a|\alpha|$-neighborhood of $C_p$.
We also prove that the leaves of ${\cal F}_a^-$ that intersect the
boundaries
$$\{G_a^+\leq r\}\cap \{|p(y)-x|=a|\alpha|\}$$
$$\{G_a^+\leq r\}\cap \{|y|\leq |\alpha|\}$$
\noindent are horizontal-like. In order to guarantee this we will
need to choose appropriate $\alpha$.
We start by fixing preliminary $\tilde{\alpha}$ such that
conditions (\ref{1}) and (\ref{2}) are satisfied.
Notice that $\phi^2_{a,-}$ is well-defined in
$$f_a(V_-)=\{|p(y)-x|\geq |a|\tilde{\alpha}, |p(y)-x|\geq
|a||y|\}$$.
Therefore, the domain of definition of $\phi^2_{a,-}$ in $W$ is
$$f_a(V_{-})\cap W=\{|p(y)-x|\geq |a|\tilde{\alpha} \}.$$
\begin{lemma} There exists $\alpha$ such for all
$a\in D_R$
$$\min \{|\phi^2_{a,-}(x,y)| :\ (x,y)\in W, |p(y)-x|=
\alpha|a|\}>$$
$$\max \{|\phi^2_{a,-}(x,y)| :\ (x,y) \in W,
|p(y)-x|=\tilde{\alpha}|a|\}.$$
\end{lemma}
\begin{proof}
$\phi^{2}_{a,-}(x,y)=a\phi_{a,-}(y,\frac{p(y)-x}{a})$.
\noindent $\phi_{a,-}(x,y)\sim y$ as $y\to \infty$, Therefore,
$$\min \{|\phi_{a,-}(x,y)| :\ |x|<\theta,|y|=\alpha \}
>\max \{|\phi_{a,-}(x,y)| :\ |x|<\theta, |y|=\tilde{\alpha}\}
$$
\noindent for big enough $\alpha$. Take
$\theta=\max(p^{-1}(D_{|x|+|a|\alpha}))$.
\end{proof}
\begin{corollary} For a point $q\in W \cap
\{|p(y)-x|\geq |a|\alpha\}$ a connected component of a leaf $L_q$
of the foliation ${\cal F}_a^-$, passing through a point $q$,
stays outside of $\tilde{\alpha}|a|$-neighborhood of $C_p$.
\end{corollary}
\begin{lemma} For all $a\in D_R$,
$\frac{\partial \phi_{a,-}^2/\partial y}{\partial
\phi_{a,-}^2/\partial x}$ is $a$-close to $\frac{\partial
\phi_{0,-}^2/\partial y}{\partial \phi_{0,-}^2/\partial x}=-p'(y)$
in $\Omega$.
\end{lemma}
\begin{note} Note that $\frac{\partial \phi^2_{a,-}}{\partial y}, \frac{\partial \phi^2_{a,-}}{\partial
y}$ do not stay bounded in $\Omega$ as $a\to 0$, but their ratio
does.
\end{note}
\begin{proof} We chose $\alpha$ so that the function $\phi^2_{a,-}$ is
well-defined in $\{G_a^+\leq r \}\cap \{|a|\tilde{\alpha} \leq
|p(y)-x|\leq \kappa\}$ and that the leaves of ${\cal F}_a^-$ we
consider do not leave this neighborhood.
Let us show that in this region $\frac{\partial
\phi_{a,-}^2/\partial y}{\partial \phi_{a,-}^2/\partial x}$ is
$a$-close to $\frac{\partial \phi_{0,-}^2/\partial y}{\partial
\phi_{0,-}^2/\partial x}$.
$$\frac{\partial \phi_{a,-}^2}{\partial y}(x,y)=
\frac{\partial}{\partial y}\left(a\phi_{a,-}(y,
\frac{p(y)-x}{a})\right)= a\frac{\partial \phi_{a,-}}{\partial
x}\left(y,\frac{p(y)-x}{a}\right)+$$
$$p'(y)\frac{\partial \phi_{a,-}}{\partial
y}\left(y,\frac{p(y)-x}{a}\right)$$
$$\frac{\partial \phi_{a,-}^2}{\partial
x}(x,y)=\frac{\partial}{\partial x}\left(a
\phi_{a,-}(y,\frac{p(y)-x}{a})\right)=-\frac{\partial
\phi_{a,-}}{\partial y}$$
Let us do a change of coordinates
$(u,y,v)=(p(y)-x,y,\frac{a}{p(y)-x})$. First, we introduce the
$u$-coordinate that measures the distance to parabola. Then we do
a blow-up in each line $y=const$ that blows-up a cone in
$u,a$-coordinates, which corresponds to the compliment of
$|a|\alpha$ neighborhood, to a polydisk in $u,v$ coordinates.
\begin{figure}[h]
\centering
\psfrag{|u|=alpha|a|}{$|u|=\alpha|a|$}\psfrag{1/alpha}{$\frac{1}{\alpha}$}
\psfrag{u}{$u$}\psfrag{a}{$a$}\psfrag{v}{$v$}
\includegraphics[height=5cm]{blowing-up.eps}
\caption{The blow-up}
\end{figure}
Denote by $\tilde{\phi}^2_-(u,y,v)=\phi^2_{uv,-}(p(y)-u, y)$. We
show that a function $\frac{\partial \tilde{\phi}^2_{-}/\partial
y}{\partial \tilde{\phi}^2_{-}/\partial x}$ lifts to a holomorphic
function on the blown-up space $\{|y|<\alpha, |u|<\epsilon,
|v|<\frac{1}{\alpha}\}$.
$$a\frac{\partial \phi_{a,-}}{\partial x}(y,\frac{p(y)-x}{a})=uv\frac{\partial \phi_{uv,-}}{\partial x}(y,\frac{1}{v})$$
$$\lim_{u\to 0} uv\frac{\partial \phi_{uv,-}}{\partial
x}(y,\frac{1}{v})=0$$
$$
\lim_{u\to 0} \frac{\partial \tilde{\phi}^2_{-}/\partial
y}{\partial \tilde{\phi}^2_{-}/\partial x}=-p'(y).
$$
Thus, $\frac{\partial \phi_{a,-}^2/\partial y}{\partial
\phi_{a,-}^2/\partial x}$ is $a$-close to $\frac{\partial
\phi_{0,-}^2/\partial y}{\partial \phi_{0,-}^2/\partial x}$ in
$\Omega\cap \{|p(y)-x|\geq |a|\alpha\}$.
\end{proof}
Let $L_q$ be a leaf of foliation ${\cal F}_a^-$ that passes
through a point $q$.
\begin{corollary}\label{horizontal_cones_F-} There exist $\kappa$ and $\delta$ such that for all $|a|<\delta$
and all
$$q\in \{G_a^+\leq r \}\cap \{|p(y)-x|\geq |a|\alpha\}\cap
\{|y|\geq \kappa\}$$ a connected component of $L_q$ is
horizontal-like.
\end{corollary}
\begin{proof} One takes $\kappa$ such that every leaf of ${\cal
F}_0^-$ that intersects $|y|=\kappa$ is horizontal like.
\end{proof}
\begin{corollary}\label{lem1} There exists $\delta$ so that for all
$|a|<\delta$ and all $(x,y)\in \Omega$ the tangent plane to the
foliation ${\cal F}_a^-$ is not horizontal.
\end{corollary}
\section{Description of ${\cal F}_a^+$.}\label{F+}
\begin{definition} We say that a curve $C$ in a domain $B$ is horizontal-like iff there
exists a family of $f_a$-invariant horizontal cones in $B$, such
that the tangent lines to $C$ belong to this family.
\end{definition}
The function $G_0^+(x,y)=G_p(x)$ is self-similar:
$$G_p(p(x))=2G_p(x).$$
Recall that we chose $r$ so that
$$G_p(0)< r< G_p(p(0)).$$
\begin{figure}[h]
\centering
\includegraphics[height=3cm]{Green_function.eps}
\caption{Level sets $G_p=\frac{r}{2^n}$.}
\end{figure}
The picture of the level sets of $G_p$ inside
$\{G_p<r\}$ is self-similar. Inside each connected component of
$\{G_p=\frac{r}{2^n}\}$ there are exactly two connected components
$\{G_p=\frac{r}{2^{n+1}}\}$.
There is exactly one critical level in each connected component
$$\frac{r}{2^{n+1}}\leq G_p\leq \frac{r}{2^n}.$$
\begin{figure}[h]
\centering
\includegraphics[height=3cm]{Green_function2.eps}
\caption{Level sets of $G_p$.}
\end{figure}
In this section we show that the picture persists for the level
sets of $G_a^+$ for small enough $a$ on each horizontal-like curve
in a box
$$\{G_a^+\leq r\}\cap \{|y|\leq \beta\}.$$
We choose $\beta=2\max \{p(x) |\quad G_p(x)\leq r\}$ to
apply this construction to the leaves of ${\cal F}_a^-$ that
intersect $\{|y|=\alpha\}$ in $\{G_a^+\leq r\}$.
\begin{lemma} There exists $\delta$ such that for $|a|<\delta$
$\{G_a^+=r\},$ $\{G_a^+=\frac{r}{2}\}$ are non-critical on each
horizontal-like curve inside $|y|<\beta$. Moreover, $\{G_a^+=r\}$
has one connected component, $\{G_a^+=\frac{r}{2}\}$ has two
connected components.
\end{lemma}
\begin{proof} $G_a^+$ is a function that depends analytically on
$a$ on ${\cal U}_{+}$ for all $a\in D$. Therefore, since the level
sets $\{G_0^+=r\}$ and $\{G_0^+=\frac{r}{2}\}$ on $y=b$ are
non-degenerate, they remain non-degenerate for small enough $a$.
The same is true for the number of components.
\end{proof}
\begin{lemma}
There exists $\delta$ so that for all $|a|<\delta$, there is
exactly one critical level of $G_a^+$ between $\frac{r}{2}$ and
$r$ on each horizontal-like curve in $|y|<\beta$. The
corresponding critical point is non-degenerate.
\end{lemma}
\begin{proof} Let $C$ be a horizontal-like curve. Then the domain $\{G_a^+\leq r\}$
inside $C$ is parametrized by a planar domain. Therefore, the
index of $\mbox{grad} G_0^+$ along the boundary of
$\{\frac{r}{2}\leq G_0^+\leq r\}$ is well-defined and is equal to
one. The function $G_{a}^+$ depends holomorphically on $a$. Thus,
the index of $\mbox{grad} G_a^+$ along the boundary of
$\{\frac{r}{2}\leq G_a^+\leq r\}$ is one as well for small $a$.
Therefore, there is only one critical point inside and it is
non-degenerate.
\end{proof}
\begin{lemma} There exists $\delta$ so that for all $|a|<\delta$ $T_r=\{G_a^+=r\}\cap W$ is a solid torus,
$\{G_a^+=\frac{r}{2}\}\cap W$ consists of two solid tori
$T^1_{r/2},T^2_{r/2}$ (the core coordinate can be chosen to be
real-analytic, the disk coordinate holomorphic).
\end{lemma}
\begin{proof} $\{G_a^+=r\}=\{(\phi_{a,+},y), |\phi_{a,+}|=r, |y|\leq \alpha\}$.
For $\{ G_a^+=\frac{r}{2}\}\cap W$ the proof goes the same way.
\end{proof}
Take any horizontal-like curve. We want to prove by induction that
inside each component $\{G_a^+=\frac{r}{2^n}\}$ there are exactly
two components $\{G_a^+=\frac{r}{2^{n+1}}\}$, and they are
non-critical. Therefore, there is exactly one critical level in
between, and the corresponding critical point is non-degenerate.
\begin{lemma} There exists $\delta$ so that for all $|a|<\delta$,
the level set $\{G_a^+= \frac{r}{2^n}\}$ on each horizontal-like
curve in $|y|<\beta$ is non-critical.
\end{lemma}
\begin{proof}
$f_a^n(\{G_a^+\leq \frac{r}{2^n}\})$ is horizontal-like, since it
is an image of a horizontal-like curve and it belongs to the box
with $f_a$-invariant horizontal cones.
$f_a^n(G_a^+=\frac{r}{2^n})\in T_r$ and it projects one-to-one to
$x$-axis. Therefore, it is non-critical.
\end{proof}
\begin{lemma} For $|a|<\delta$ on
each horizontal-like curve for every $n$ there are exactly two
level sets $\{G_a^+= \frac{r}{2^{n+1}}\}$ inside $\{G_a^+=
\frac{r}{2^{n}}\}$ and they are non-critical.
\end{lemma}
\begin{proof} For every $n$, $f_a^n(\{G_a^+\leq \frac{r}{2^n}\})$
is a disk that projects one-to-one to $x$-axis with the boundary
on $T_r$. It intersects $T^1_{r/2}$, $T^2_{r/2}$ by a circle each.
On the intersection $\{G_a^+=\frac{r}{2}\}$. This proves the
lemma.
\end{proof}
\begin{corollary} For $|a|<\delta$ on each horizontal-like curve there is only one
critical level $\{G_a^+=r'\},$ where
$\frac{r}{2^{n+1}}<r'<\frac{r}{2^n}$ for each connected component
$\{G_a^+=\frac{r}{2^n}\}$.
\end{corollary}
The next lemma states that the foliation ${\cal F}_a^+$ is not
only vertical-like (projects one-to-one to $y$-axis), but is
uniformly close to vertical on some thickening of
$\{G_a^+=\frac{r}{2^n}\}$.
\begin{lemma}\label{F+vertical} Fix a small $\lambda$. There exists $\delta$ s.t. $\forall |a|<\delta$
$$
\left|\frac{\partial \phi^{2^n}_{a,+}/\partial y}{\partial
\phi^{2^n}_{a,+}/\partial x}\right|<C|a|
$$
\noindent on $\{\frac{r-\lambda}{2^n}\leq G_a^+\leq
\frac{r+\lambda}{2^n}\}$ with $C$ independent on $n$.
\end{lemma}
\begin{proof} On $\{r-\lambda\leq G_a^+\leq r+\lambda\}$ the
inequality follows from the fact that $\phi_{a,+}$ is a
holomorphic function in $a$.
The leaves of foliation ${\cal F}_a^+$ in
$\{\frac{r-\lambda}{2^{n+1}}\leq G_a^+\leq
\frac{r+\lambda}{2^{n+1}}\}\cap W$ are preimages under $f_a^{-1}$
of the leaves of ${\cal F}_a^+$ in $\{\frac{r-\lambda}{2^{n}}\leq
G_a^+\leq \frac{r+\lambda}{2^{n}}\}\cap W$. Therefore, by
induction we check that they belong to $f_a^{-1}$-invariant
vertical cones.
\end{proof}
\begin{comment}
\section{Preliminary description of the critical locus.}
\begin{lemma} There exists $\delta$ such that for all $|a|<\delta$
the critical locus in
$$\{\frac{r}{2}\leq G_a^+\leq r\} \cap \{|p(y)-x|\geq\kappa\} \{\cap |y|\leq
\alpha\}$$
\noindent is a perturbation of $xy=0$ and is a connected sum of
two disks with two handles.
\end{lemma}
The fact the critical locus in $\epsilon$-neighborhood of $xy=0$
outside a polydisk around 0 is a perturbation of $xy=0$ and thus
projects one-to-one to $x$($y$)-axis correspondingly follows from
the inverse function theorem. In the polydisk by Morse theory it
is either an intersection of two lines or a handle, since the
Milnor number of the singularity is $1$.
Suppose that it is an intersection of two curves. Take a
horizontal-like curve. The curve on which ${\cal F}_{a,+}$ is
tangent to horizontal lines is vertical-like. Take a point of
their intersection. On that curve ${\cal F}_{a,-}$ has a
horizontal tangent line, which is impossible by lemma \ref{lem1}.
Now suppose that there exists a sequence $a_n\to 0$ such that for
every $a_n$ there is a point of the critical locus $z_n\in {\cal
C}_{a_n}$ in
$$\{\frac{r}{2}\leq G_a^+\leq r\} \cap \{|p(y)-x|\geq
\kappa\} \cap \{|y|\leq \alpha\}$$
and outside of $\epsilon$-tube of $xy=0$. Take an accumulation
point $z_0$ of $z_n$. It should belong to the critical locus
${\cal C}_0$ and lie outside the $\epsilon$ tube of $xy=0$, which
is impossible.
\begin{lemma} There exists $\delta_n$ such that for all $|a|<\delta_n$
the critical locus inside each connected component is
$\{\frac{r}{2^{n+1}}\leq G_a^+\leq \frac{r}{2^n}\} \cap
\{|p(y)-x|\geq\kappa\}\cap \{|y|\leq \alpha\}$ is a perturbation
of $(x-\xi_n)y=0$, $p^n(\xi_n)=0$ and is a connected sum of two
disks with two handles.
\end{lemma}
\begin{proof} The proof is the same as for the previous lemma.
\end{proof}
\end{comment}
\section{Critical Locus in $\Omega$.}\label{Omega}
\begin{comment}
{\it Lirics.} Mark the leaves of $L_{a,+}$ that are tangent to
$y=k$. These are the ones that project two-to-one to $y$-axis.
Take, those that correspond to the first critical point on $y=k$.
They form a tube $T$. The rest of the tubes that correspond to
two-to-one leaves are preimages of this one. The rest of the
leaves of $L_{a,+}$ in $R$ are vertical like. The leaves of
$L_{a,-}$ in $W$ outside of $a\alpha$-neighborhood of $C_p$
project properly 2-to-1 to $x$-axis.
\end{comment}
Recall that
$$\Omega_a = \{G_a^+\leq r\}\cap
\{|y|\leq \alpha\}\cap \{|p(y)-x|\geq |a|\alpha\}.$$
\noindent See Figure \ref{fig:Omega}. We will omit the subscript
$a$, when it is clear from the context which H\'{e}non mapping is
under consideration.
In this section we subdivide $\Omega_a$ into countably many
regions. Note that the level sets $\{G_a^+=\frac{r}{2^n}\}$ on
each horizontal line $y=k$, $|k|<\alpha$ depend continuously with
respect to parameter $a$. Therefore, the partition of
\begin{equation}\label{for:partition_Omega}\{\frac{r}{2^{n+1}}\leq G_a^+\leq \frac{r}{2^n}\}\cap
\Omega_a
\end{equation}
\noindent into connected components depends continuously on $a$.
Therefore, it is enough to enumerate the connected components of
$\Omega_0$. We call $\Omega_{0}^{\xi_n}$ the connected component
of
$$\{\frac{r}{2^{n+1}}\leq G_0^+\leq \frac{r}{2^n}\}\cap \Omega_0,$$
\noindent which contains the critical line $x=\xi_n$. We call
$\Omega_a^{\xi_n}$ the connected component of
(\ref{for:partition_Omega}) that is the continuation of
$\Omega_0^{\xi_n}$.
\begin{figure}[h]
\centering
\psfrag{|y|=alpha}{$|y|=\alpha$}
\psfrag{G=r}{$G_a^+=\frac{r}{2^n}$}\psfrag{G=r/2}{$G_a^+=\frac{r}{2^{n+1}}$}
\psfrag{u=aalpha}{$|u|=a\alpha$}\psfrag{x}{$x$}\psfrag{y}{$y$}
\includegraphics[height=8cm]{part.eps}
\caption{Domain $\Omega_a^{\xi_n}$}
\end{figure}
\begin{lemma}\label{x-boundary} Fix a small $\lambda$. There exists $\delta$
(independent on $n$) so that for all $|a|<\delta$ the critical
locus in each connected component of
$$\{\frac{r-\lambda}{2^{n}}\leq G_a^+\leq
\frac{r+\lambda}{2^n}\}\cap \{|y|\leq \alpha\}$$ \noindent is an
annulus which is a graph of function $y(\phi_{a,+})$.
\end{lemma}
\begin{figure}[h]
\centering
\psfrag{G=r}{$G_a^+=\frac{r}{2^n}$}\psfrag{G=r+lambda}{$G_a^+=\frac{r+\lambda}{2^n}$}\psfrag{G=r-\lambda}{$G_a^+=\frac{r-\lambda}{2^n}$}
\includegraphics[height=3cm]{thickening.eps}
\caption{The thickening of $G_a^+=\frac{r}{2^n}$}
\end{figure}
\begin{proof} Fix $\gamma$ so that the set $\{
G_0^+\leq r\}\cap \{|y|\leq \gamma\}$ is disjoint from $C_p$. Then
for all $|a|<\delta'$ the set $\{ G_a^+\leq r\}\cap \{|y|\leq
\gamma\}$ is disjoint from $C_p$.
The inverse function theorem implies that $\forall n$ $\exists
\delta_n$ s.t. $|a|<\delta_n$ the critical locus in
$\{\frac{r-\lambda}{2^n}\leq G_a^+\leq\frac{r+\lambda}{2^n}\}\cap
\{|y|\leq \gamma\}$ is an annulus on the component that is a
perturbation of $y=0$.
Since in the region
$$\{\frac{r-\lambda}{2^n}\leq G_a^+\leq \frac{r+\lambda}{2^n}\}\cap \{\gamma<|y|<\alpha\}\cap \{|p(y)-x|\geq a|\alpha|\}$$
the foliation ${\cal F}_a^-$ is $a$-close to ${\cal F}_0^-$ and by
Lemma \ref{F+vertical} ${\cal F}_a^+$ is almost vertical, for
$|a|<\delta''$ there are no points of the critical locus in this
region.
Take some $n$ and fix some connected component
$$\{\frac{r-\lambda}{2^n}\leq G_a^+\leq \frac{r+\lambda}{2^n}\}\cap \{|y|\leq
\gamma\}.$$
Let us show that the critical annuli in this connected component
persists for all $|a|\leq \delta''$. ${\cal C}_a$ is a analytic
set in the region
$$\{(x,y,a) | \ \frac{r-\lambda}{2^n}\leq G_a^+(x,y)\leq \frac{r+\lambda}{2^n}, |y|<\gamma, |a|<\delta''\}$$
There are no zeroes of $\mbox{grad}G_a^-$ on the leaves of ${\cal
F}_a^+$ on curve $|y|=\gamma$. Therefore, index of $\mbox{grad}
G_a^-$ is constant. At $a=0$ it is equal to $1$. Thus, the
critical annulus in
$$\{(x,y,a) | \ \frac{r-\lambda}{2^n}\leq G_a^+(x,y)\leq \frac{r+\lambda}{2^n}, |y|<\gamma, |a|<\delta''\}$$
\noindent persists.
\end{proof}
\begin{figure}[h]
\centering
\psfrag{|y|=alpha}{$|y|=\alpha$}\psfrag{|y|=kappa}{$|y|=\kappa$}\psfrag{|u|=aalpha}{$|u|=|a|\alpha$}
\includegraphics[height=4cm]{boundary.eps}
\caption{The critical locus in $\{|a|\alpha\leq|p(y)-x|\}\cap
\{\frac{r}{2^{n+1}}\leq G_a^+\leq \frac{r}{2^n}\}\cap \{|y|\geq
\kappa\}$}
\end{figure}
\begin{lemma}\label{a-neighborhood} There exist $\kappa$ and $\delta$ such that for all
$|a|<\delta$ the critical locus in each connected component
$\{|a|\alpha\leq|p(y)-x|\}\cap \{\frac{r}{2^{n+1}}\leq G_a^+\leq
\frac{r}{2^n}\}\cap \{|y|\geq \kappa\}$ is an annulus with two
holes and is a graph of function $y(\phi_{a,-})$.
\end{lemma}
\begin{proof} It follows from Corollary \ref{horizontal_cones_F-}
\end{proof}
We review the notion of the Milnor number of a singularity that we
will need for the proof of the next lemma.
Let $z_0$ be an isolated singular point of a holomorphic function
$f:\mathbb C^n\to \mathbb C$. Let $B_{\rho}$ be a disk of radius
$\rho$ around $z_0$. One can take $\rho$ and $\epsilon$ small
enough, so that the non-singular level sets $\{f=\epsilon\}\cap
B_{\rho}$ have homotopy type of a bouquet of spheres of dimension
$(n-1)$ (\cite{AGVII}, Section 2.1).
\begin{definition} The number of spheres in $\{f=\epsilon\}\cap
B_{\rho}$ is called the {\it Milnor number} of the singular point
$z_0$.
\end{definition}
\begin{lemma}[\cite{AGVII}, Section 2.1]\label{lem:Milnor_number} The Milnor number of the singular point $z_0$
is equal to ${\cal O}_{z_0}/<f_1,\dots,f_n>$, where ${\cal
O}_{z_0}$ are functions regular in a neighborhood of $z_0$ and
$<f_1,\dots, f_n>$ is the ideal in ${\cal O}_{z_0}$, generated by
the functions $f_1,\dots,f_n$.
\end{lemma}
\begin{lemma}\label{description-omega} For $|a|<\delta$ the critical locus
in $\Omega_a$ is a smooth curve. In $\Omega_a^{\xi_n}$ it is a
connected sum of two disks $D_1$ and $D_2$ with two holes. The
boundary of $D_1$ belongs to $\{|y|=\alpha\}$, and the holes of
$D_1$ have boundaries on $\{|u|=|a|\alpha\}$. The boundary of
$D_2$ belongs to $\{G_a^+=\frac{r}{2^n}\}$ and the holes to
$\{G_a^+=\frac{r}{2^{n+1}}\}$.
\end{lemma}
\begin{figure}[h]
\centering
\psfrag{deg}{\it The degenerate critical locus}
\psfrag{nondeg}{\it The nondegenerate critical locus}
\psfrag{y=alpha}{$|y|=\alpha$}\psfrag{Gp=r}{$G_p=\frac{r}{2^n}$}\psfrag{Gp=r/2}{$G_p=\frac{r}{2^{n+1}}$}
\psfrag{Ga=r}{$G_a^+=\frac{r}{2^n}$}\psfrag{Ga=r/2}{$G_a^+=\frac{r}{2^{n+1}}$}\psfrag{u=aalpha}{$|u|=|a|\alpha$}
\includegraphics[height=7cm]{critical_locus.eps}
\caption{The critical locus in $\Omega_a^{\xi_n}$}
\end{figure}
\begin{note} As $a\to 0$, the curve
degenerates to $(x-\xi_n)y=0$. The holes of $D_1$ degenerate to
points $(0,\xi_{n+1}),$ $(0,\xi'_{n+1}),$ where
$p(\xi'_{n+1})=p(\xi'_{n+1})=\xi_n$. The holes of $D_2$ tend to
circles $\{(x, 0)|\ G_p(x)=\frac{r}{2^{n+1}}\}$.
\end{note}
\begin{proof} Fix $\xi_n$. The critical locus in $\Omega_0^{\xi_n}$
is a singular curve. It is a union of two intersecting lines $y=0$
and $x=\xi_n$. By Lemma \ref{lem:Milnor_number} the Milnor number
of this singularity is 1. The critical locus ${\cal C}_a$ is the
zero level set of $w_a(x,y)$. By Lemmas \ref{x-boundary},
\ref{a-neighborhood} there exists $\delta$ so that the critical
locus in $\Omega_{a}^{\xi_n}$ for $|a|<\delta$ is transverse to
the boundary. Thus, there is $\epsilon$ so that the level sets of
$w_a(x,y)=c$, $|c|<\epsilon$ are transverse to the boundary of
$\Omega_a^{\xi_n}$ for all $|a|<\delta$. Let us consider
$(w_a(x,y),a):\Omega_a\times D_{\delta} \to \mathbb C\times
D_{\delta}$. By Ehresmann's Fibration Theorem the level sets of
this function form a locally trivial fibration. Thus, they are
homeomorphic one to the other. By Lemmas \ref{x-boundary} and
\ref{a-neighborhood}, the critical locus intersects the boundary
exactly as prescribed. Therefore, it is enough to show that the
critical locus ${\cal C}_a\Omega_a^{\xi_n}$ for $|a|<\delta$ is
non-degenerate.
Suppose there are critical level sets of the function $w_a(x,y)$.
Then the Milnor number of the corresponding singularity is 1.
Therefore, the singularity is locally an intersection of two
disks. It is easy to see that one of this curves should be a
'horizontal' curve, i.e. project one-to-one to $y$-axis. Its
boundary belongs to $\{|y|=\alpha\}$. This curve necessarily
intersects the 'vertical' curve on which the tangent line to
${\cal F}_a^+$ are horizontal. But then on the point of
intersection the tangent line to the foliation ${\cal F}_a^-$ is
horizontal. That is impossible by Lemma \ref{lem1}.
Therefore, the critical locus in $\Omega_a^{\xi_n}$ is noncritical
for $a\neq 0$. The conclusion of the lemma follows.
\end{proof}
\section{Extension of the critical locus up to $|a|\alpha$-neighborhood of
parabola.}\label{ext-a-neighborhood}
\begin{lemma}\label{lemma-ext-a-neighborhood} There exists $\delta$ so that for all $|a|<\delta$
the component of the critical locus, that is a perturbation of
$y=0$ can be extended up to $|a|\alpha$-neighborhood of parabola
as a graph of function $y(x)$.
\end{lemma}
\begin{proof}
The domain of definition of $\phi^2_{a,-}$ is $f_a(V_-)=\{(x,y)\
|\ |p(y)-x|\geq |a|\alpha, |p(y)-x|\geq |a||y|\}$.
Therefore, in $W$ the domain of definition of $\phi_{a,-}^2$ is
$\{|p(y)-x|\geq |a|\alpha\}$.
Denote $u=p(y)-x$.
Consider new variables $(u,y,v)=(u,y,\frac{a}{u})$. Denote by
$\pi$ the projection
$$\pi:(u,y,v) \mapsto (u,y,uv).$$
Notice that $\pi^{-1}$ blows up a point $u=0$ on each line
$y=y_0$.
Let us prove that one can extend the critical locus to
$$S=\{|u|<\beta, |y|<\epsilon, |v|<\frac{1}{\alpha}\}.$$
Note that $\phi_{a,+}$ can be extended to $S$ since it's a
well-defined holomorphic function in $\pi(S)$.
$\phi_{a,-}^2(x,y)=uv\phi_{uv,-}(y, 1/v)$.
By [LR] $\frac{\phi_{a,-}(x,y)}{y}$ extends to be a holomorphic
function to a neighborhood of $y=\infty$.
Therefore $\phi_{a,-}^2$ extends to $S$. Moreover, notice that on
blown-up lines $\phi_{uv,-}^2=0$.
The critical locus is given by the zeroes of the function
$$w=\frac{d\phi_{a,+}\wedge d\phi_{a,-}^2\wedge da}{dx \wedge dy
\wedge da}.$$
Let
$$\tilde{w}=-\frac{d\phi_{uv,+}\wedge d\phi_{uv,-}^2\wedge dv}{u
du\wedge dy \wedge dv}.$$
Notice that $\tilde{w}=w\circ \pi$.
$$\tilde{w}=-uv \frac{\partial \phi_{uv,+}}{\partial x}(p(y)-u,
y)\frac{\partial \phi_{uv,-}}{\partial x}(y,\frac{1}{v})+$$
$$\left(\frac{\partial \phi_{uv,+}}{\partial
x}(p(y)-u,y) p'(y)+\frac{\partial \phi_{uv,+}}{\partial
y}(p(y)-u,y)\right)\frac{\partial \phi_{uv,-}}{\partial
y}(y,\frac{1}{v})$$
Note that $\frac{\phi_{a,-}}{y}=1+aH(x,\frac{1}{y},a)$, where $H$
is a holomorphic function in some neighborhood of $(x,0,0)$.
$\lim_{u\to 0} v\frac{\partial \phi_{uv,-}}{\partial x}=0$
$\lim_{u\to 0} \frac{\partial \phi_{uv,-}}{\partial y}=1$
$$\tilde{w}(0,y,v)=b'_p(p(y))p'(y)$$
Note that for all $|v|<\frac{1}{\alpha}$ $|y|<\alpha$
$\tilde{w}(0,y,v)=0$ only when $y=0$ and the zero is not multiple.
Therefore, by Weierstrass theorem for $v$, $y=g(u,v)$.
\end{proof}
\section{Description of the critical locus.}\label{final}
Fix some $\epsilon$. Denote
$$\hat{\Omega}_1=\{(x,y)\in \mathbb CP^2 |\ |y|<\epsilon,
|p(y)-x|\geq |a|\alpha, G_a^+(x,y)\geq r\}.$$
\begin{figure}[h]
\centering
\psfrag{omega}{$\Omega$} \psfrag{1Omeg}{$\hat{\Omega}_1$}
\psfrag{x=infty}{$x=\infty$} \psfrag{|a|alpha}{$|a|\alpha$}
\includegraphics[height=5cm]{omega1.eps}
\caption{Domain $\hat{\Omega}_1$}
\end{figure}
\begin{lemma} There exists $\delta$ such that
$\forall |a|<\delta$ the critical locus ${\cal C}_a$ in
$\hat{\Omega}_1$ is a punctured disk, with a hole removed. The
puncture is at the point $(\infty, 0)$, the boundary of the hole
belongs to $\{|p(y)-x|=|a|\alpha\}$.
\end{lemma}
\begin{proof} By Lemma \ref{degenerate-critical-locus} the
degenerate critical locus ${\cal C}_0$ in $\hat{\Omega}_1$ is
$y=0$, with point $x=c$ removed. By Lemma \ref{lemma_infinity} it
persists in some neighborhood of $x=\infty$. By the inverse
function theorem it can be extended as a graph of function $y(x)$
to $\{~G_a^+~(x,y)~\geq~r~\}$ excluding $\epsilon$-neighborhood of
$C_p$. By Lemma~\ref{lemma-ext-a-neighborhood} it can be extended
up to $a|\alpha|$-neighborhood of $C_p$.
\end{proof}
In the following $3$ lemmas we show that every component of the
critical locus intersects $\Omega$. It follows from Lemma
\ref{description-omega} that it consists of one component.
\begin{lemma}\label{boundary} Let $C_a$ be a component of the critical locus ${\cal C}_{a}$. Then
there exists a point on $\partial C_a$ that belongs to $J_a^+\cup
J_a^-$.
\end{lemma}
\begin{proof} Consider $\left(G_a^+ + G_a^-\right)$. This function is pluriharmonic and strictly positive
in $U_a^+\cap U_a^-$. Therefore, $\inf \left(G_a^+ + G_a^-\right)$
cannot be attained at the interior point.
\end{proof}
\begin{lemma}\label{fundamentalJ} There exists $\delta$ such that for all $|a|<\delta$
$J_a^+\cap \Omega$ is a fundamental domain for $J_a^+\backslash
J_a$.
\end{lemma}
\begin{proof}\cite{HOVII} There it is prove for H\'{e}non mappings that
are perturbations of hyperbolic polynomials with connected Julia
set. In the connected case the proof is the same.
\end{proof}
\begin{lemma}\label{lem2} Suppose $|a|<\delta$ and $C_a$ is a component of the critical locus. Then
there exists an iterate of $f_a^n(C_a)$ that intersects
$\Omega$.
\end{lemma}
\begin{proof}
Lemma \ref{boundary} states that there exists $z\in (J_a^+ \cup
J_a^-)\cap (\partial C_a)$. Suppose $z\in J_a^-$. Take a sequence
of points $z_n\in C_a$, $z_n\to z$.
$G_a^-(z_n)\to 0$ as $n\to \infty$; $G_a^+$ is bounded.
For every $n$ there exists $k_n$ such that
$1<G_a^-(f_a^{-k_n}(z_n))\leq 2$. Then $k_n\to \infty$.
$$G_a^+(f_a^{-k_n}(z_n))\to 0.$$
\noindent Taking a subsequence one may assume $f^{-k_n}(z_n)\to
z$. $z\in J_a^+\backslash J_a$. Since by Lemma \ref{fundamentalJ}
$f_a^m(z)\in J_a^+\cap \Omega$, there exists an iterate of $C_a$
that intersects $\Omega$.
\end{proof}
Below we will prove that ${\cal C}_a\cap \left(\Omega\cup
\hat{\Omega}_1\right)$ is a fundamental domain of the critical
locus.
The next lemma is a variation of the Classical Poincar\'{e}'s
Polyhedron Theorem on the fundamental domain under the group
action (\cite{Maskit}, IV.H).
\begin{lemma}\label{abstract_lemma} Let $C$ be a Riemann surface. Let $f:C\to C$ be an automorphism.
Let $D\subset C$ be an open domain. Suppose
\begin{enumerate}
\item $f_a^n(D)\cap D=\emptyset$;
\item $\partial D\subset C$ consists of countably many smooth
curves $\gamma_i$;
\item $\gamma_i$ are being paired by the map $f$;
\item for any sequence $\{z_n\}$, $z_n\in D$, $f_a^n(z_n)$
does not have an accumulation point in $S$
\end{enumerate}
Then $D$ is a fundamental domain of $S$ for the map $f_a$.
\end{lemma}
\begin{proof} Denote by $S$ a Riemann surface obtained by gluing
the images $f^n(D)$ to $D$. The natural map $i:S\to C$ is
injective by (1). Let us prove that it is proper. If it is not,
then there exists a sequence of points $z_n\in S$ so that $z_n\to
\partial S$, $i(z_n)\to z\in C$. Take $z_n'\in D$ so that
$z_n=f^{k_n}(z_n')$. If there are infinitely many same $k_n$'s.
Then the sequence $\{z'_n\}$ has an accumulation point in $D$.
That contradicts (2) and (3).
Therefore, taking a subsequence one can assume $\{k_n\}$ increase.
This contradicts (4).
\end{proof}
\begin{lemma} The images of $\left(W\backslash\{|p(y)-x|\geq
a|\alpha|\}\right)\cup\{|y|<\epsilon\}$ under the map $f_a$ are
disjoint.
\end{lemma}
\begin{proof}
\begin{enumerate}
\item $$f_a^n(W\backslash \{|p(y)-x|\leq |a|\alpha\})\cap \left[W\backslash \{|p(y)-x|\leq
a|\alpha|\}\right]=\emptyset.$$
$f_a(W)\subset W\cup V_+$, $f_a(V_+)\subset V_+$, therefore,
$\left(f_a^n(W)\cap W\right)\subset \left(f_a(W)\cap
W\right)\subset \{|p(y)-x|\leq |a|\alpha\}$
\item $$f_a^n(W)\cap \{|y|<\epsilon\}\cap \{|x|\geq \alpha\}=\emptyset;$$
Let $(x,y)\in W$. Suppose $f_a^n(x,y)\in V_+$ and $n$ is the
smallest number such that this happens. Then $|p(x_n)-y_n|\leq
|a|\alpha$. Thus, $|y_n|>\epsilon$. For points in $V_+$,
$|y_{n+1}|> |y_n|$.
\item $$f_a^n(\{|y|<\epsilon\}\cap \{|x|>\alpha\})\cap \{|y|<\epsilon\}\cap \{|x|>\alpha\}=\emptyset.$$
Suppose $(x,y)\in \{|y|<\epsilon\}\cap \{|x|>\alpha\}$. Since
$|y|<\epsilon$, $f_a(x,y)$ belongs to $|a|\epsilon$-neighborhood
of parabola. Since $(x,y)\in V_+$, $f_a(x,y)\in V_+$. Therefore,
$|y_1|>\epsilon$. $|y_{n+1}|>|y_n|>\epsilon$, since $(x_n,y_n)\in
V_+$.
\item $$f_a^n(\{|y|<\epsilon\}\cap \{|x|>\alpha\})\cap
W=\emptyset.$$
This is true, since $\{|y|<\epsilon\}\cap \{|x|>\alpha\}\subset
V_+$, and $f_a(V_+)\subset V_+$.
\end{enumerate}
\end{proof}
\begin{lemma} There exists $\delta$ such that for all $|a|<\delta$ the critical locus in $\Omega\cup \hat{\Omega}_1$
forms a fundamental domain of ${\cal C}_a$.
\end{lemma}
\begin{proof}
Denote by $D={\cal C}_a\cap \left(\Omega\cup
\hat{\Omega}_1\right)$. Let us check that the conditions of the
Lemma \ref{abstract_lemma} are satisfied. Since $\Omega\cup
\hat{\Omega}\subset \left(W\backslash \{|p(y)-x|\leq
|a|\alpha\}\right)\cup \{|y|\leq \epsilon\}$. All the forward
images of $D$ are disjoint from it. Therefore, (1) is satisfied.
The boundary of $D$ in each $\Omega_a^{\xi_k}$ consists of
vertical circles: $|y|=\alpha$ and $|p(y)-x|=|a|\alpha$. The
circles $|p(y)-x|=|a|\alpha$ are parametrized by $\xi'_{k+1}$,
$\xi''_{k+1}$,($p(\xi'_{k+1})=p(\xi''_{k+1})=\xi_k$), which stands
for the approximate value of $y$ on the circle.
There is also one horizontal circle $|p(y)-x|=|a|\alpha$ on a
perturbation of $y=0$.
$f_a$ maps $|y|=\alpha$ in $\Omega_a^{\xi_k}$ to
$|p(y)-x|=|a|\alpha$ in $\Omega_a^{p(\xi_k)}$, parametrized by
$\xi_k$
$f_a$ maps $|y|=\alpha$ on perturbation of $x=0$ to a horizontal
circle $|p(y)-x|=|a|\alpha$.
Therefore, boundary components of $D$ are being paired by $f_a$.
And condition (2) and (3) of Lemma \ref{abstract_lemma} are
satisfied.
Suppose there exists a sequence of $z_n\in D$ so that
$f_a^n(z_n)\to\partial S,$ $z_n\to z_*\in C$.
If $\{z_n\}$ has an accumulation point $z$ in $D$. Then
$f_a^{n}(z)\to z^*$. Which is impossible, since $f_a^{n}(z)\to
\infty$ in $\mathbb C^2$.
If $\{z_n\}$ does not have an accumulation point $z$ in $D$. Then
$z_n$ accumulate to $z\in J_a^+$. Therefore, $f_a^{k_n}(z')\to
z\in J_a$. That contradicts to $z\in C$.
So condition (4) is satisfied as well. Therefore, $D$ is a
fundamental domain of the critical locus ${\cal C}_a$.
\end{proof}
\begin{proof}[\it Proof of theorem \ref{theorem_main}.]
To obtain a description in terms of truncated spheres we do a
dynamical regluing. We fix some small $\epsilon$ and cut the
fundamental domain of ${\cal C}_a$ along the hypersurface
$|y|=\epsilon$. We call the connected component of the
perturbation of $y=0$ main component. The rest of components we
call handles: $H_{\xi_k}$ is a component in $\Omega_{\xi_k}$
The boundary of $H_{\xi_k}$ consists of four circles:
$|y|=\alpha,$ $|y|=\epsilon$ and two connected components of
$|p(y)-x|=|a|\alpha$, parametrized as previously by $\xi'_{k+1}$,
$\xi''_{k+1}$, where $p(\xi'_{k+1})=p(\xi''_{k+1})=\xi_k$.
We glue $H_{\xi_k}$ to the main component by the map $f_a^k$.
Under this procedure the boundary $|y|=\alpha$ of $H_{\xi_k}$ is
being glued to $|p(y)-x|=|a|\alpha$-boundary of $H_{p(\xi_k)}$,
parametrized by $\xi_k$. By ``generalized uniformization theorem",
it can be straighten to be a sphere.
The fundamental domain of the critical locus ${\cal C}_a$,
obtained after regluing, is a truncated sphere.
The preimages of $0$ under the map $p^k$ are parametrized by
$k$-strings of $0$ and $1$'s.
Let $\alpha_k$ be a $k$-string that parametrizes $\xi_k$.
$V_{\alpha_k}$ is the interior of $|y|=\epsilon$ obtained by
cutting $H_{\xi_k}$ from the main component. $U_{\alpha_k}$ is the
interior of $f_a^k (|y|=\epsilon)$ on $f_a^k(H_{\xi_k})$. The rest
of the boundary corresponds to $G_a^+=0$ and $G_a^-=0$. Therefore,
it is parametrized by two Cantor sets $\Sigma, \Omega$.
We show these are true Cantor sets by moduli counting.
\begin{lemma} Let $U_1\supset U_2\supset \dots U_n\supset \dots$
be a sequence of open domains, such that $U_i\backslash U_{i-1}$
is an annulus with moduli $M_i\geq M$. Then $\bigcap \bar{U}_i$ is
a point.
\end{lemma}
Take a point $\sigma\in \Sigma$. Let $M_1$ be a modulus of the
annulus $\{r-\lambda\leq G_a^+\leq r\}$ on $y=0$. Then $M_1$ is a
modulus of the annulus $\{\frac{r-\lambda}{2^n}\leq G_a^+\leq
\frac{r}{2^n}\}$. The main component project one-to-one to each of
this annulus. There is a sequence of connected component
$\{\frac{r-\lambda}{2^n}\leq G_a^+\leq \frac{r}{2^n}\}$ that bound
a hole parametrized by $\sigma$. Therefore, $\sigma$ is a point.
$\Sigma$ is a Cantor set.
Fix some small $\epsilon$. Let $M_1$ be a modulus of the annulus
$\{\alpha-\epsilon\leq|y|\leq\alpha\}$. All handles $H_{\xi_k}$
project one-to-one to this annulus. So by the same argument we get
that $\Omega$ is a Cantor set.
\end{proof}
\begin{comment}
\section{Quasiconformal model.}
The next lemma is a slight improvement of Lemma \ref{vertical}.
\begin{lemma} For small $\lambda$ there exist $\delta,\epsilon$
such that $|a|<\delta$ the leaves of the foliation ${\cal F}_a^+$
in $\{\frac{r-\lambda}{2^n}\leq G_a^+\leq
\frac{r+\lambda}{2^n}\}\cap\{|y|\leq \alpha\}$ for all $n$ belong
to the family of cones $C_1|a||\eta|<|\xi|<C_2|a||\eta|$, where
$(\xi,\eta)\in T_{(x,y)}\mathbb C^2$, $C1,C_2$.
\end{lemma}
\begin{proof} Notice that ${\cal F}_a^+$ in $\{\frac{r-\lambda}{2^n}\leq G_a^+\leq
\frac{r+\lambda}{2^n}\}\cap\{|y|\leq \alpha\}$ is a preimage of
${\cal F}_a^+$ in $\{\frac{r-\lambda}{2^{n-1}}\leq G_a^+\leq
\frac{r+\lambda}{2^{n-1}}\}\cap\{|y|\leq \alpha\}$
$D f^{-1}_{a}=\left(\begin{array}{cc}0 & 1 \\
-\frac{1}{a} & \frac{2y}{a}\end{array}\right)$
Let $\left(\begin{array}{c}\xi_1 \\ \eta_1\end{array}\right)=\left(\begin{array}{cc}0 & 1 \\
-\frac{1}{a} &
\frac{2y}{a}\end{array}\right)\left(\begin{array}{c}\xi \\
\eta\end{array}\right)$
$\left|\frac{\xi_1}{\eta_1}\right|=\frac{|a|}{|-\frac{\xi}{\eta}+2y|}$
Suppose $(x,y)= f^{-1}_a(u,v)$, where $\{(u,v) |\ G_a^+(u,v)\leq
r', |v|\leq |\alpha|\}$, then $\tilde{C}_2>|y|>\tilde{C}_1$. The
estimate follows.
\end{proof}
\begin{lemma} The critical locus ${\cal C}_a\cap \Omega_a^{\xi_n}$
is a graph of function $y=C\frac{a}{y}+h(a,y)$, where $C\approx
p^n(0)$.
\end{lemma}
\begin{proof} ${\cal C}_0\cap \Omega_0^{\xi_n}$ is given by the
equation $(x-\xi_n)y=0$. We claim that the perturbation is
$(x-\xi_n)y+Ca+a^2h_1(x,y,a)$, where $C\neq 0$ is a constant.
Suppose that $C=0$, take $x$, such that $G_a^+(x)=\frac{r}{2^n}$.
Then by the previous lemma $y\approx \frac{a}{x}$. That can happen
only if $C\neq 0$.
\end{proof}
\begin{lemma} The modulus of the annulus ${\cal C}_a^+\cap
\Omega_{a}^{\xi_n}$ is $\approx |a|$. The estimate is uniform in
$n$.
\end{lemma}
Fix some small $a$. We show that there is holomorphic motion that
maps ${\cal C}_a$ to the critical loci for $a$ in some
neighborhood. There, the critical loci are quasiconformally
equivalent. We first show that the critical points of the function
$G_a^+$ move holomorphically. We lift this motion to the universal
cover. We show that the closure of the critical points contains
the boundary of the universal cover. We use Doudy-Earle extension
to extend the holomorphic motion to the enterior of the disk.
\begin{lemma} All the critical points of the function $G_a^+$ on
the critical locus are non-degenerate.
\end{lemma}
\begin{proof} Suppose that there is a degenerate point, then in
the neighborhood all the leaves intersect the critical locus in 3
points. Critical locus in $\Omega_{\xi_n}$ projects one-to-one to
$x$-axis. The leaves ${\cal F}_a^+$ project 2-to-1 to $y$-axis.
Therefore, they can't have more than 2 points of intersection.
\end{proof}
\begin{lemma} There are $3$ non-degenerate critical points of ${\cal
G}_a^+$ on ${\cal C}_a^+\cap \Omega_a^{\xi_n}$.
\end{lemma}
\begin{lemma} Critical points of $G_a^+$ converge to all the
points on the ideal boundary of ${\cal C}_a$.
\end{lemma}
\begin{proof} The points on the boundary are elements of the
Cantor sets. There are handles converging to each element of the
Cantor set. There are three critical points on each handle.
Therefore, there are critical points converging to each point of
the ideal boundary.,
\end{proof}
\end{comment}
\newpage
\section{List of standard notations}
We provide a list of notations here as a reference.\medskip
\begin{center}
\begin{tabular}{|l|l|p{11cm}|}\hline
{\bf Notation} & {\bf Section} & {\bf Meaning}\\ \hline
$f_a$ & \ref{introduction}& H\'{e}non mapping under
consideration\\ \hline
$a$ & \ref{introduction} & Jacobian of H\'{e}non mapping under
consideration\\ \hline
$p(x)$ & \ref{introduction} & polynomial used in the definition of
H\'{e}non mapping \\ \hline
$U_a^+, U_a^-$ & \ref{introduction} & set of points whose orbits
under forward (backward) iteration of $f_a$ escape to infinity\\
\hline
$K_a^+, K_a^-$ & \ref{introduction} & set of points whose orbits
under forward (backward) iteration of $f_a$ remain bounded\\
\hline
$J_a^+, J_a^-$ & \ref{introduction} & boundaries of $K_a^+,
K_a^-$\\
\hline
$J_a$ & \ref{introduction} & $J_a^+\cap J_a^-$\\
\hline
$G_a^+, G_a^-$ & \ref{introduction} & pluriharmonic functions that
measure the rate of escape to infinity under forward (backward)
iterates of $f_a$\\
\hline
$\alpha$ & \ref{bottcher_coordinates} & parameter used in the
definition of $V_+$, $V_-$\\
\hline
$V_+, V_-$ & \ref{bottcher_coordinates} & $\{|x|> \alpha, |x|>
|y|\}$, $\{|y|> \alpha, |y|> |x|\}$ -- regions which
describe the large scale behavior of the H\'{e}non map\\
\hline
$W$ & \ref{bottcher_coordinates} & $\{|x|\leq\alpha, |y|\leq\alpha\}$\\
\hline
$D_R$ & \ref{bottcher_coordinates} & the disk of radius $R$ in
the parameter space, used to define $V_+$ and $V_-$\\
\hline
$\phi_{a,+}, \phi_{a,-}$ & \ref{bottcher_coordinates} &
holomorphic functions that semiconjugate dynamics in $V_+, V_-$ to
$z\to z^2$,
$z\to z^2/a$\\
\hline
$s_k^+, s_k^-$ & \ref{bottcher_coordinates} & auxillary functions,
used to study $\phi_{a,+}$ and $\phi_{a,-}$\\
\hline
$C(p)$ & \ref{bottcher_coordinates} & the curve $y=p(x)$, this is $J_0^-$\\
\hline
$G_p$ & \ref{strategy} & Green function for the map
$x\to p(x)$\\
\hline
$b_p$ & \ref{bottcher_coordinates} & B\"{o}ttcher coordinate for the map $x\to p(x)$\\
\hline
${\cal F}_{a}^+, {\cal F}_a^-$ & \ref{introduction} & foliations of $U_a^+$, $U_a^-$\\
\hline
${\cal C}_a$ & \ref{introduction}& the critical locus, the set of
tangencies between foliations ${\cal F}_a^+$ and ${\cal F}_a^-$\\
\hline
$(x_n,y_n)$ & \ref{bottcher_coordinates} & $(x_n,y_n)=f_a(x,y)$\\
\hline
${\cal D}_{k,+}$ & \ref{bottcher_coordinates} & the domain of
definition of $\phi_{a,+}^k$\\
\hline
${\cal D}_{k,-}$ & \ref{bottcher_coordinates} & the domain of
definition of $\phi_{a,-}^k$\\
\hline
$u$ && $u=p(y)-x$, measures the distance from a point $(x,y)$ to $C(p)$\\
\hline
$\Omega_a$ & \ref{strategy} & $\{G_a^+\leq r\}\cap\{|y|\leq
\alpha\}\cap \{|p(y)-x|<\alpha|a|\}$, the domain that does not
intersect with its images under $f_a$ and $f^{-1}_a$ \\
\hline
$\hat{\Omega}_1$ & \ref{final} & $\{(x,y)\in \mathbb C\mathbb
P^2|\quad |y|<\epsilon, |x|\geq \alpha\}$ \\\hline
\end{tabular}
\end{center}
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,991 |
Q: How to construct the 2d structure in a dynamic fashion I iterate through all cars and its supported attributes (many attributes per car) to create a structure like this, how do I do this in a dynamic fashion.
cars = {
"honda" => {'color' => 'blue', 'type' => 'sedan'}.
"nissan" => {'color' => 'yellow', 'type' => 'sports'}.
...
}
cars.each do |car|
car_attrs = ...
car_attrs.each do |attr|
??? How to construct the above structure
end
end
A: Your question is not very clear... But i guess this is what you want:
cars = {}
options = {}
options['color'] = 'blue'
...
cars['honda'] = options
Is that what you were looking for?
A: It sounds like you may be asking for a way to create a 2-dimensional hash without having to explicitly create each child hash. One way to accomplish that is by specifying the default object created for a hash key.
# When we create the cars hash, we tell it to create a new Hash
# for undefined keys
cars = Hash.new { |hash, key| hash[key] = Hash.new }
# We can then assign values two-levels deep as follows
cars["honda"]["color"] = "blue"
cars["honda"]["type"] = "sedan"
cars["nissan"]["color"] = "yellow"
cars["nissan"]["type"] = "sports"
# But be careful not to check for nil using the [] operator
# because a default hash is now created when using it
puts "Found a Toyota" if cars["toyota"]
# The correct way to check would be
puts "Really found a Toyota" if cars.has_key? "toyota"
Many client libraries assume that the [] operator returns a nil default, so make sure other code doesn't depend on that behavior before using this solution. Good luck!
A: Assuming you are using something similar to ActiveRecord (but easy to modify if you are not):
cars_info = Hash[cars.map { |car| [car.name, car.attributes] }
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,809 |
{"url":"https:\/\/cs.stackexchange.com\/questions\/77672\/closure-of-regular-languages-under-non-contiguous-subsequence\/77675","text":"# Closure of regular languages under non-contiguous subsequence\n\nSuppose there is a language $L$ on alphabet $\u03a3$. Now consider the language\n\n$$S(L) = \\{x : wxy \u2208 L, w, y \u2208 \u03a3^*\\} \u222a \\{x : w \u2208 L,\\text{ and x is a subsequence of w}\\}.$$\n\nHow to prove that if $L$ is regular then $S(L)$ is also regular?\n\nFor first part I think if there is a DFA that accepts $wxy$ then there is a DFA that accepts $x$. For second part I have no clue. Can anyone shed some light on this or can provide a formal proof for this?\n\nHere are several ways of showing this:\n\n1. Starting with a DFA\/NFA for $L$, add $\\epsilon$ transitions parallel to all other transitions.\n\n2. Apply the regular substitution that maps $\\sigma \\in \\Sigma$ to $\\{\\sigma,\\epsilon\\}$.\n\n3. Starting with a regular expression for $L$, replace all copies of each $\\sigma \\in \\Sigma^*$ by $\\epsilon + \\sigma$.\n\n\u2022 I think a little more explanation is required . Is your solution is for second part only ? What do you mean by adding transitions parallel to other transitions ? Whqt substitution you are talking about ? I am understanding it a little bit but not completely. \u2013\u00a0Desperado Jul 7 '17 at 17:48\n\u2022 I am not able to understand 2nd point . \u2013\u00a0Desperado Jul 7 '17 at 17:49\n\u2022 The first part is actually a subset of the second, so you don't really need it. \u2013\u00a0Yuval Filmus Jul 7 '17 at 17:56\n\u2022 As for the second point, the regular languages are closed under regular substitution. A regular substitution is a mapping $s\\colon \\Sigma \\to 2^{\\Delta^*}$ in which $s(\\sigma)$ is regular for all $\\sigma \\in \\Sigma$. We extend $s$ to words by concatenation, and to languages by $s(L) = \\bigcup_{w \\in L} s(w)$. \u2013\u00a0Yuval Filmus Jul 7 '17 at 17:58\n\u2022 As to the level of detail, that's intentional. You'll have to work it out. \u2013\u00a0Yuval Filmus Jul 7 '17 at 17:58\n\nGiven a DFA $M$ for $L$ create a new NFA with $\\epsilon$-moves $M'$ by adding a new transition $\\delta(q_i, \\epsilon) = q_j$ for each existing transition $\\delta(q_i,a) = q_j$ for some symbol $a \\in \\Sigma$.\n\nNow for example if $w = abaabab$ is accepted by $M$ and $u = aab$ is a subsequence of $w$, then $u$ can be written as $a \\epsilon \\epsilon ab \\epsilon \\epsilon$ which is accepted by $M'$ according to the new NFA rules.\n\n\u2022 Thanks as I am not very good with this, your explanation is very good and simple. \u2013\u00a0Desperado Jul 7 '17 at 18:38","date":"2020-02-22 22:01:52","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7827690839767456, \"perplexity\": 285.8028532051777}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 20, \"end_threshold\": 5, \"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\/1581875145729.69\/warc\/CC-MAIN-20200222211056-20200223001056-00254.warc.gz\"}"} | null | null |
Maer est un terme pouvant désigner:
Le nom maer signifie fameux dans le vieil anglais. Combiné à aethel (noble), ceci donne le nom de
Maer, hameau des Cornouailles
Maer, village rural du Staffordshire
Maer Hall, maison du potier Josiah Wedgwood II | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,379 |
package nl.esciencecenter.ptk.browser.uitest.dummy;
import java.awt.BorderLayout;
import javax.swing.JFrame;
import javax.swing.JScrollPane;
import nl.esciencecenter.ptk.vbrowser.ui.browser.BrowserInterfaceAdaptor;
import nl.esciencecenter.ptk.vbrowser.ui.browser.BrowserPlatform;
import nl.esciencecenter.ptk.vbrowser.ui.proxy.ProxyException;
import nl.esciencecenter.ptk.vbrowser.ui.proxy.ProxyFactory;
import nl.esciencecenter.ptk.vbrowser.ui.proxy.ProxyNode;
import nl.esciencecenter.ptk.vbrowser.ui.proxy.ProxyNodeDataSourceProvider;
import nl.esciencecenter.ptk.vbrowser.ui.resourcetable.ResourceTable;
import nl.esciencecenter.ptk.vbrowser.ui.resourcetable.ResourceTableModel;
import nl.esciencecenter.vbrowser.vrs.exceptions.VRLSyntaxException;
import nl.esciencecenter.vbrowser.vrs.vrl.VRL;
public class DummyProxyResourceTable {
public static void main(String args[]) {
try {
showTable();
} catch (Exception e) {
e.printStackTrace();
}
}
public static void showTable() throws VRLSyntaxException, ProxyException {
BrowserPlatform platform = StartDummyBrowser.getDummyPlatform();
JFrame frame = new JFrame();
frame.setSize(800, 600);
frame.setLayout(new BorderLayout());
JScrollPane pane = new JScrollPane();
frame.add(pane, BorderLayout.CENTER);
ResourceTable table = new ResourceTable(new BrowserInterfaceAdaptor(platform),
new ResourceTableModel(false));
pane.setViewportView(table);
VRL vrl = new VRL("dummy:///");
ProxyFactory dummyFac = platform.getProxyFactoryFor(vrl);
ProxyNode root = dummyFac.openLocation("dummy:///");
ProxyNodeDataSourceProvider dataSource = new ProxyNodeDataSourceProvider(root);
table.setDataSource(dataSource, true);
frame.setVisible(true);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,315 |
{"url":"https:\/\/forums.phpfreaks.com\/topic\/134641-solved-zend-framework-and-wamp\/","text":"#### Archived\n\nThis topic is now archived and is closed to further replies.\n\n# [SOLVED] Zend Framework and WAMP\n\n## Recommended Posts\n\nI'm trying to install Zend Framework on my laptop to familiarise myself with it. I already have WAMP installed with a dozen or so website projects sitting in the WAMP server root (c:\\wamp\\www).\n\nThe installation instructions that I have followed so far include a .htaccess file to reroute everything to index.php. Am I right in thinking that Zend Framework effectively takes over the whole server?\n\nI still need access to the existing projects already in the WAMP server root without using ZF. How do I achieve that? Does ZF require its own dedicated server?\n\nI'm kinda new to this (if you didn't already guess! haha)\n\nThanks\n\nMatt\n\n##### Share on other sites\n\nPut the .htaccess in ZF folder not in the root, and change the paths in it accordingly.\n\nSince it's for tests only, I put my public and application folders both in c:\\wamp\\www\\zf\\\n\n##### Share on other sites\n\nNice one! I'll try that.\n\nMany thanks Mchl\n\n##### Share on other sites\n\nThe installation instructions that I have followed so far include a .htaccess file to reroute everything to index.php. Am I right in thinking that Zend Framework effectively takes over the whole server?\n\nNope, ZEND doesn't take over anything. The .htaccess that you place in a folder does affect all sub-folders though.\n\nI still need access to the existing projects already in the WAMP server root without using ZF. How do I achieve that? Does ZF require its own dedicated server?\n\nI have my XAMMP and WAMP servers setup with sub directories.\n\nC:\/wamp\/www\/\nC:\/wamp\/www\/zend\/\nC:\/wamp\/www\/project1\/\nC:\/wamp\/www\/project2\/\n\n\nThe answer is to use RewriteBase in your rewrite rules. See this topic:\n\nAlso, if you want to install Xdebug for testing I made a video about it here.\n\n##### Share on other sites\n\nIt just has to be present in the include_path. That's basically how you \"install\" it.\n\nFor each project I prefer to give it it's own hostname and it's own VirtualHost in Apache. I just edit the hostfile and make something like e.g. dev.phpfreaks point to 127.0.0.1. Then I setup a new vhost.\n\n##### Share on other sites\n\nI also try to install and try ZF but on XAMMP.\n\nIn my httdocs I have plenty installed projects, but it seems that I cant configure the .htaccess file\u00a0 ??? .....eg\n\nI create a project in the following directory :\n\nC:\\xampp\\htdocs\\Quickstart\n\nI have the below folders\n\napplication\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <DIR>\n\ndata\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <DIR>\n\nlibrary\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <DIR>\n\npublic\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <DIR>\n\nscripts\n\nNow ....where should I put the\u00a0 .htaccess file and what changes should I make to it?\n\n##### Share on other sites\n\nPut it in C:\\xampp\\htdocs\\Quickstart\\public as this is where your's application will be accessible from.\n\n##### Share on other sites\n\nThanks..... I have installed the \"ZendFrameworkQuickstart\" application on my XAMMP under the folder C:\\Program Files\\xampp\\htdocs\\Quickstart\n\nbut when I run a\n\nget the error.....\n\nWarning: require_once(Zend\/Loader.php) [function.require-once]: failed to open stream: No such file or directory in C:\\Program Files\\xampp\\htdocs\\Quickstart\\public\\index.php on line 12\n\nFatal error: require_once() [function.require]: Failed opening required 'Zend\/Loader.php' (include_path='C:\\Program Files\\xampp\\htdocs\\Quickstart\\application\\\/..\/library;.;C:\\Program Files\\xampp\\php\\pear\\') in C:\\Program Files\\xampp\\htdocs\\Quickstart\\public\\index.php on line 12\n\nI think I should change the htaccess and perhaps bootstrap but don't understand where and what path should I put????\n\n##### Share on other sites\n\nSorry for the last question.....things are started working now.....\n\nsome files just haven't been copied in the library folder.....\n\nthanks\n\n##### Share on other sites\n\nBUMP!\n\nI'm having similiar problems (I have a WAMP setup). Working through their quickstart guide, I successfully created the project - but when I try to navigate to the project folder - all the folders are visible EXCEPT public. When I try to access this folder, I get an internal server error. Any ideas?\n\nexample:\n\n[DIR] application\/\n\n[DIR] library\/\n\n[DIR] tests\/\n\nwhere's freaking public!?\n\n##### Share on other sites\n\nSounds to me like you've actually put application, library and tests within your public directory.\n\nAnyways, you people really ought to look into how to configure vhosts on Apache instead of panting all your projects within one single document root.\n\n##### Share on other sites\n\nSounds to me like you've actually put application, library and tests within your public directory.\n\ndo you mean wamp's www directory?\n\nBasically, I created the project within wamps www directory because (as I understand it) it needs to be in there to parse the PHP correctly.\n\nAll the folders are in one location:\n\nC:\\wamp\\www\\sites\\study\\Zend\\quickstart :(application, library, public, tests)\n\nZF is here - and reference via the include_path in php.ini\n\nC:\\wamp\\ZendFramework\n\nIs that problematic?\n\n##### Share on other sites\n\nNo its not problematic, but the only files out of that entire structure that need to be within an accessible web root are the files within public (eg; .htaccess and index.php). The rest can be wherever you like.\n\n##### Share on other sites\n\nto be fair, they were all unpacked to a single project folder when I executed \"zf create project quickstart\" from the cmd (as in the quickstart tutorial on Zend's site).\n\nSo why is public not visible to localhost?\n\n##### Share on other sites\nto be fair, they were all unpacked to a single project folder when I executed \"zf create project quickstart\" from the cmd (as in the quickstart tutorial on Zend's site).\n\nIndeed. But you can easily setup a virtual host on apache to make public the only publicly accessible directory. Hence, why its named public.\n\nSo why is public not visible to localhost?\n\nI've no idea. Where is your servers document root?\n\n##### Share on other sites\n\nYou must be doing something wrong. When running it I get what I've attached here.\n\n##### Share on other sites\n\nDocumentRoot \"c:\/wamp\/www\/\" -\n\nif I modify it to\n\n\"C:\\wamp\\www\\Sites\\study\\Zend\\quickstart\\public\"\n\nit still doesn't work. I'm going to try out the vhosts option, but this seems like quite an odd bug.\n\n##### Share on other sites\n\nSo, your saying when you go to http:\/\/localhost\/sites\/study\/Zend\/quickstart you get a directory listing showing....\n\n[pre]\n\n[DIR]\u00a0 application\/\n\n[DIR]\u00a0 library\/\n\n[DIR]\u00a0 tests\/\n\n[\/pre]\n\nand no public directory? Something is whack with your setup.\n\n##### Share on other sites\n\nDoes the public folder not show up in the filesystem either?\n\n##### Share on other sites\n\nDoes the public folder not show up in the filesystem either?\n\nNah, I think the op has stated the public directory shows in his filesystem, but not when view via apache directory browsing.\n\nMakes me think he has somehow installed application, library and tests into public. Seems odd though.\n\n##### Share on other sites\n\nsunwukung is not the OP though. He jumped in later.\n\n##### Share on other sites\n\nIndeed. I guess I should refer to him as 'the problem at hand'?\n\n##### Share on other sites\n\nFeel free - as long as you help me solve it!\n\nyup, you're telling me. Can't figure it out for the life of me. I work with WAMP on a daily basis, so I can't understand why it wouldn't show a folder - and all my other projects are functioning fine. Incidentally, @Daniel10, I unpacked your tarball into the same folder, and still no public folder.\n\nAnyway, here's some progress - I removed the htaccess file form within the public folder - it became visible, and now I've been able to access the framework. Soooo, it's something to do with the .htaccess I guess. Since neither vhosts or .htaccess are my strong suit, can you think of why this might be causing an issue?\n\n##### Share on other sites\n\nOkay, seriously, check this out:\n\nC:\\windows\\system32\\drivers\\etc\\hosts:\n\n127.0.0.1 dev.yourproject\n\nC:\\path\\to\\your\\httpd.conf:\n\n<VirtualHost *:80>\nServerName dev.yourproject\nDocumentRoot C:\\path\\to\\your\\zfproject\\public\n<Directory C:\\path\\to\\your\\zfproject\\public>\nOrder deny,allow\nDeny from all\nAllow from 127.0.0.1\nAllowOverride All\n<\/Directory>\n<\/VirtualHost>\n\nRestart Apache, go to http:\/\/dev.yourproject\n\nIf that doesn't work, something is seriously wrong with your setup.\n\n##### Share on other sites\n\nNope, same problem. Not sure what the problem is, only that it seems to work if I remove the .htaccess file:\n\nSetEnv APPLICATION_ENV development\n\nRewriteEngine On\nRewriteCond %{REQUEST_FILENAME} -s [OR]\nRewriteCond %{REQUEST_FILENAME} -l [OR]\nRewriteCond %{REQUEST_FILENAME} -d\nRewriteRule ^.*$- [NC,L] RewriteRule ^.*$ index.php [NC,L]\n\n\nI'll try this on a different WAMP setup to see if it's just my current setup.\n\n\u00d7\n\n\u2022 #### Activity\n\n\u2022 Chat\n\u00d7\n\u2022 Create New...","date":"2020-07-11 17:41:05","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4175882637500763, \"perplexity\": 6717.1148332262}, \"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-2020-29\/segments\/1593655934052.75\/warc\/CC-MAIN-20200711161442-20200711191442-00029.warc.gz\"}"} | null | null |
Q: jQuery getting closest html when I have to use the same IDs for forms Firstly I know I shouldn't have the same IDs, but in this case it unavoidable ... I am trying to click a button that will take the .html() of a div and add that text to an input field. Problem is that everything has the same IDs and there will be an unknown number of them. Example is:
<div id="Product">
<div id="Product-stats"><div class="builder-name">Example Builder</div></div>
<form><input id="abc" value=-""></input></form>
<a href="#" class="cart-get-quote-button"></a>
<div>
<div id="Product">
<div id="Product-stats"><div class="builder-name">Another Builder</div></div>
<form><input id="abc" value=-""></input></form>
<a href="#" class="cart-get-quote-button"></a>
<div>
<div id="Product">
<div id="Product-stats"><div class="builder-name">Third Builder</div></div>
<form><input id="abc" value=-""></input></form>
<a href="#" class="cart-get-quote-button"></a>
<div>
So I need to click each link and have the builder-name content copied to the relative input value, obviously I can get this to work with just one set, but not multiple, so I though maybe trying to use sibling? or closest? This is what I have that does not work
$(document).ready(function() {
$('.cart-get-quote-button').on('click', function() {
$(this).closest('input[id="abc"]').val($(this).closest('.builder-name').html());
});
});
A: Since multiple identical ids are illegal in HTML, they won't work, and you simply can't use them in your JS. Whatever solution you try it has to ignore IDs entirely.
Working JSFiddle (I have fixed several other errors in your HTML). This works:
var text = $(this).closest('div').find('.builder-name').text();
$(this).closest('div').find('input').val(text);
Note you have several other errors in your HTML, and no JS will work until you fix them. They're probably just typos here on SO:
*
*hyphen in your input ... value=-"">, should be ... value="">;
*closing Product </div> is actually a new open <div>;
*You don't need </input>;
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,197 |
\section{Introduction}
Let $A$ be a bounded operator on a Hilbert space $H$. We define the range space $\cM(A)=AH$, and endow it with the inner product
$$\langle Af, Ag \rangle_{\mathcal{M}(A)}=\langle f, g \rangle_{H},\qq f,g\in H \ominus \m{Ker}A.$$
Let $\DD$ denote the unit disk.
Let $L^2$ denote the Lebesgue space of square integrable functions on the unit circle $\partial\DD$. The Hardy space $H^2$ is the subspace of analytic functions on $\DD$ whose Taylor coefficients are square summable. Then it can also be identified with the subspace of $L^2$ of functions whose negative Fourier coefficients vanish.
The Toeplitz operator on the Hardy space $H^2$ with symbol $f$ in $L^\infty(\DD)$ is defined by
$$T_f (h) = P(fh),$$ for $h\in H^2(\DD)$. Here $P$ be the orthogonal projections from $L^2$ to $H^2$.
Let $b$ be a function in the closed unit ball of $H^\infty(\DD)$, the space of bounded analytic functions on the unit disk. The de Branges-Rovnyak space $\cH(b)$ is defined to be the space $(I-T_b T_{\bar b})^{1/2} H^2$.
We also define the space $\cH(\bar b)$ in the same way as $\cH(b)$, but with the roles of $b$ and $\bar{b}$ interchanged, i.e. $$\cH(\bar{b})=(I-T_{\bar b} T_b)^{1/2} H^2.$$ The spaces $\cH(b)$ and $\cH(\bar b)$ are also called sub-Hardy Hilbert spaces (the terminology comes from the title of Sarason's book \cite{sar94}).
The Bergman space $A^2$ is the space of analytic functions on $\DD$ that are square-integrable with respect to the normalized Lebesgue area measure $dA$. For $u\in L^\infty(\DD)$, the Bergman Toeplitz
operator $\tiT_u$ with symbol $u$ is the operator on $L^2_a$ defined by $$\tiT_u h = \tiP(uh).$$ Here $\tiP$ is the orthogonal projection from $L^2(\DD, dA)$ onto $A^2$.
In \cite{zhu96}, Zhu introduced the sub-Bergman Hilbert spaces. They are defined by
$$\cA(b)=(I-\tiT_b \tiT_{\bar b})^{1/2}A^2$$ and $$\cA(\bar{b})=(I-\tiT_{\bar b} \tiT_b)^{1/2}A^2.$$ Here $b$ is a function in the closed unit ball of $H^\infty(\DD)$. It is easy to see from the definition that the spaces $\cA(b)$ and $\cA(\bar{b})$ are contractively contained in $A^2$. But in most cases they are not closed subspaces of $A^2$ (see \cite{zhu96}*{Corollary 3.13}). Some examples of $\cA(b)$ are known: if $||b||_\infty<1$, then $\cA(b)$ is just a renormed version $A^2$; if $b$ is a finite Blaschke product, then $\cA(b)=H^2$ (\cite{zhu03}, \cite{sul06}).
Sub-Bergman Hilbert spaces share some common properties with sub-Hardy Hilbert spaces as the way those spaces are defined follows from a general theory on Hilbert space contractions \cite{sar94}. For instance, both $\cH(b)$ and $\cA(b)$ are invariant under the corresponding Toeplitz operators with a co-analytic symbol (\cite{sar94}*{II-7}).
One significant difference between the spaces $\cA(b)$ and $\cH(b)$ is the multipliers. The theory of $\cH(b)$ spaces is pervaded by a fundamental dichotomy, whether $b$ is an extreme point of the unit ball of $H^\infty(\DD)$. The multiplier structure of de Branges-Rovnyak spaces has been studied extensively by Lotto and Sarason in both the extreme and the nonextreme cases (\cite{lot90}, \cite{lotsar91}, \cite{lotsar93}). However, Zhu \cite{zhu96} showed that every function in $H^\infty$ is a multiplier of $\cA(b)$ and $\cA(\bar b)$. As a consequence, the two sub-Bergman Hilbert spaces $\cA(b)$ and $\cA(\bar{b})$ are norm equivalent.
\begin{thm}\label{=}
For $b\in H^\infty$ with $||b||_\infty\leq 1$,
$$\cA(b)=\cA(\bar{b}).$$
\end{thm}
\begin{proof}
The relation $$||f||_{\cA(b)}\leq||f||_{\cA(\bbb)}$$ follows from Douglas's criterion \cite{sar94}*{I-5} and the operator inequality (see e.g. \cite{zhu}*{p. 106})
$$\tiT_b\tiT_{\bbb}\leq \tiT_{\bbb}\tiT_b.$$
On the other hand, let $f\in \cA(\bbb)$. By \cite{zhu96}*{Theorem 3.12}, $b$ is a multiplier of $\cA(b)$ with multiplier norm less than or equal to $||b||_\infty$.
Then
$$
||f||^2_{\cA(\bbb)}=||f||^2_{A_2}+||bf||^2_{\cA(b)}\leq ||f||^2_{\cA(b)}+||b||^2_\infty||f||^2_{\cA(b)}\leq 2||f||^2_{\cA(b)}.
$$
Here we used the identity \cite{sar94}*{I-8}.
\end{proof}
In the Hardy space setting, the polynomials belong to $\cH(b)$ if and only if $b$ is non-extreme, and in this case polynomials are dense in $\cH(\bar b)$ and $\cH(b)$ (see \cite{sar94}*{Chapter IV, V}). For the sub-Bergman Hilbert spaces, Zhu in \cite{zhu96} showed that $\cA(b)$ always contain $H^\infty$, which includes the polynomials. In the same paper, Zhu asked whether the polynomials are dense in $\cA(b)$ or $\cA(\bbb)$. In this note, we answer this question in the affirmative.
\section{Proof the Main Result}
In the theory of $\cH(b)$ spaces, $\cH(\bar b)$ is often more amenable than $\cH(b)$ because of a representation theorem for $\cH(\bar b)$ \cite[III-2]{sar94}. We shall prove a similar version for $\cA(\bar b)$
\begin{thm}\label{rep}
Let $b\in H^\infty $ with $||b||_\infty\leq 1$. Let $L^2_b$ denote the space $L^2(\DD, (1-|b|^2)dA)$. Let $A^2_b$ be the closure of polynomials in $L^2_b$. Define the operator $S_b$ by
\beq\label{s}S_bg=\tiP((1-|b|^2) g).\eeq
Then $S_b$ is an isometry from $A^2_b$ onto $\cA(\bar b)$.
\end{thm}
\begin{proof}
For any $q\in L_b^2$, $g\in A^2$, we have
$$
\la S_bq, g \ra_{A^2}=\la \tiP((1-|b|^2) q, g \ra_{A^2}=\int_{\DD} q\bg (1-|b|^2)dA=\la q, g\ra_{L_b^2},
$$
which implies $S_b^*$ is the natural inclusion from $A^2$ into $L_b^2$. Then for any $f, g\in A^2$,
$$
\la S_bS_b^*f, g\ra_{A^2}=\la S_b^*f, S_b^* g\ra_{L_b^2}=\la f, g\ra_{L_b^2}=\int_{\DD} f\bg (1-|b|^2)dA=\la \tiT_{1-|b|^2}f, g\ra_{A^2}.
$$
Thus $$S_bS_b^*=I-\tiT_{\bar b} \tiT_b.$$ By Douglas's Criterion, $$\cA(\bar{b})=S_b L^2_b.$$
Notice that every $q\in L_b^2$, $$(S_b q)(w)=\la q, k_w\ra_{L_b^2},$$ where $k_w(z)=\frac{1}{(1-\bw z)^2}$ is the reproducing kernel of $A^2$.
We see that $$Ker (S_b)=(Span_{w\in \DD}\{k_w\})^{\perp}=(A_b^2)^{\perp}.$$
Hence $S_b$ is an isometry from $A^2_b$ onto $\cA(\bar b)$.
\end{proof}
We need the following lemma to prove the main result.
\begin{lem}\label{lem}
Let $b\in H^\infty $ with $||b||_\infty\leq 1$.
Let $M_n$ denote the closure of the linear span of $\{z^k\}_{k=n}^{\infty}$ in $A_b^2$. Let $\cP=\cup_{n\geq 0}(M_n)^{\perp}.$ Then $\cP$ is dense in $A_b^2$.
\end{lem}
\begin{proof}
If $b$ is a constant, the conclusion is obvious. Next, we assume $b$ is not a constant.
Let $f\in A_b^2$ such that $f\perp \cP$. Then $f\in M_n$ for every $n\geq 0$. It is sufficient to show $f=0$.
Suppose $f \in A_{b}^{2} \setminus \{0\}$. Then for some $k \geq 1$, $f$ has the power series expansion
$$f(z) = \sum_{j = k}^{\infty} a_j z^j$$
with $a_k \not = 0$. Since $f \in M_k$, there is a sequence of polynomials $\{p_s\}_{s=1}^{\infty} \subset M_{k}$ for which $$p_s \to f \q\m{in}\q L_{b}^{2}.$$ By the definition of the integral norm,
$$p_s - a_k z^k \to f - a_k z^k \q\m{in}\q L_{b}^{2},$$ and thus $f - a_{k} z^k \in M_{k + 1}$. But since $f \in M_{k + 1}$ and $a_k \not = 0$, it must be the case that $z^k \in M_{k + 1}$.
Fix $r\in (0, 1)$. There exists $\Gd>0$ such that $$1-|b(z)|^2\geq \Gd,$$ for every $z$ in the disk $r\DD$.
Let $$g(z)=\sum_{j=k+1}^{N} b_jz^j, N>k$$ be a polynomial with minimal degree at least $k+1$.
Using the easily established identity
$$
\int_{r\DD} z^m\bar{z^n}dA(z)=
\begin{cases}\frac{r^{2m+2}}{m+1}, & m=n, \cr
0, &m\neq n,
\end{cases}
$$
we have
\begin{align*}
||z^k-g(z)||^2_{A_b^2}&=\int_{\DD} |z^k-g(z)|^2(1-|b(z)|^2)dA(z)\\
&\geq \int_{r\DD} |z^k-g(z)|^2(1-|b(z)|^2)dA(z)\\
&\geq \Gd\int_{r\DD} |z^k-g(z)|^2dA(z)\\
&=\Gd\large(\int_{r\DD} |z^k|^2dA(z)+\int_{r\DD} |g(z)|^2dA(z)\large)\\
&\geq\frac{\Gd }{k+1}r^{2k+2}>0.
\end{align*}
By the definition of $M_{k+1}$, we see that $z^k\notin M_{k+1}$, which is a contradiction . Thus $f=0$.
\end{proof}
\begin{thm}
Polynomials are dense in $\cA(b)$.
\end{thm}
\begin{proof}
By Theorem \ref{=}, it suffices to prove polynomials are dense in $\cA(\bar{b})$. Let $f\in \cA(\bbb)$ and let $\Gve>0$. By Theorem \ref{rep}, there exists $g\in A_b^2$ such that $f=S_b g$.
Using Lemma \ref{lem}, we can find $h\in \cP$ with $$||g-h||_{A_b^2}<\Gve.$$
By Theorem \ref{rep}, $S_b: A_b^2\to \cA(\bbb)$ is an isometry. Thus $$||f-S_bh||_{\cA(\bbb)}=||S_b(g-h)||_{\cA(\bbb)}=||g-h||_{A_b^2}<\Gve.$$
It remains to show that $S_bh$ is a polynomial. From the definition of $\cP$, we see that $h\in (M_n)^\perp$, for some $n\geq 0$. Then $$\la h, z^k\ra_{A_b^2}=0,$$ for every $k\geq n$.
If $n=0$, then $h=0$. If $n\geq 1$, using \eqref{s} we have
\begin{align*}
\la S_bh, z^k\ra_{A^2}&=\la\tiP((1-|b|^2)h), z^k\ra_{A_b^2}=\int_{\DD} h(z)\bz^k(1-|b(z)|^2)dA(z)\\
&=\la h, z^k\ra_{A_b^2}=0,
\end{align*}
for every $k\geq n.$ Therefore $S_bh$ is a polynomial of degree at most $n-1$.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,090 |
{"url":"https:\/\/sunglee.us\/mathphysarchive\/?p=1490","text":"# Sage: Basic Matrix Operations\n\nFor using Sage: Sage is an open source math software whose interface is a web browser (in particular firefox). You don\u2019t have to install Sage in your computer to use it. You can access any sage server including the main Sage server. I am running a Sage server at sage.st.usm.edu. If you are a student at the University of Southern Mississippi, you are more than welcome to create an account at sage.st.usm.edu and use it.\n\nMatrix Constructions\n\nIn Sage, $2\\times 3$ matrix\n$$\\begin{pmatrix} 1 & 1 & -2\\\\ -1 & 4 & -5 \\end{pmatrix}$$ can be created as follows. Let us say we want to call the matrix $A$. Type the following command in the blank line of your sage worksheet:\n\nA=matrix([[1,1,-2],[-1,4,-5]])\n\nIn case you are familiar with Maple, not like Maple you will not see your matrix $A$ as an output when you click on \u201cevaluate\u201d. To see your matrix, you need to type\n\nA\n\nin the next blank line and click on \u201cevaluate\u201dagain:\n\n[ 1 1 -2]\n[-1 4 -5]\n\nScalar Multiplication\n\nIf you want to multiply the matrix $A$ by a number 5, type the command\n\n5*A\n\nand click on \u201cevaluate\u201d. The output is\n\n[ 5 5 -10]\n[ -5 20 -25]\n\nTo perform addition of two matrices:\n$$\\begin{pmatrix} 1 & 1 & -2\\\\ -1 & 4 & -5 \\end{pmatrix}+\\begin{pmatrix} 2 & 1 & 5\\\\ 1 & 3 & 2 \\end{pmatrix}$$, first call the second matrix $B$:\n\nB=matrix([[2,1,5],[1,3,2]])\n\nand do\n\nA+B\n\nthe output is\n\n[ 3 2 3]\n[ 0 7 -3]\n\nThe linear combination $3A+2B$ can be calculated by the command\n\n3*A+2*B\n\nand the output is\n\n[ 7 5 4]\n[ -1 18 -11]\n\nTranspose of a Matrix\n\nTo find the transpose of the matrix $A$ do\n\nA.transpose()\n\nand the output is\n\n[ 1 -1]\n[ 1 4]\n[-2 -5]\n\nMatrix Multiplication\n\nAn $m\\times n$ matrix can be multiplied by a $p\\times q$ matrix as long as $n=p$. The resulting multiplication is an $m\\times q$ matrix. Let $C=\\begin{pmatrix} 3 & 4\\\\ -1 &2\\\\ 2 &1 \\end{pmatrix}$. The number of columns of $A$ and the number of rows of $C$ coincide as 3, so we can perform $AC$ and this can be done in Sage as:\n\nA*C\n\nThe output is\n\n[ 1 -1]\n[ 1 4]\n[-2 -5]","date":"2020-07-15 04:12:23","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"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\": 0.4482477605342865, \"perplexity\": 817.0311484583651}, \"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-2020-29\/segments\/1593657155816.86\/warc\/CC-MAIN-20200715035109-20200715065109-00387.warc.gz\"}"} | null | null |
William M. Elliott ( – September 20, 1882) was an American politician and businessman from Maryland. He served as a member of the Maryland House of Delegates, representing Harford County in 1854.
Career
Elliott served as a member of the Maryland House of Delegates, representing Harford County in 1854. He was elected as county commissioner in 1871. He was a Democrat.
Elliott built a large canning factory in Charles County and ran the business with his son Howell. He also worked in the fishing industry on the Potomac River for 25 years.
Personal life
Elliott married and had four daughters and one son, Mrs. Jacob C. Hollis, Mrs. George F. Walker, Mrs. John Cooley, Missouri and Howell B. Elliott had a home in Havre de Grace.
Elliott died on September 20, 1882, at the age of 71, at his farm in Charles County. He was buried at Grove Cemetery in Aberdeen.
References
Year of birth uncertain
1810s births
1882 deaths
People from Havre de Grace, Maryland
People from Charles County, Maryland
Maryland Democrats
Members of the Maryland House of Delegates
19th-century American politicians | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,068 |
\section{Introduction}\label{sec:intro}
A standard approach to learning tasks on graph-structured data, such as vertex
classification, edge prediction, and graph classification, consists of the
construction of a representation of vertices and graphs that captures
their structural information. Graph Neural Networks (GNNs) are currently
considered as the state-of-the art approach for learning such representations.
Many variants of GNNs exist but they all follow a similar strategy. More
specifically, each vertex is initially associated with a feature vector. This
is followed by a recursive neighbourhood-aggregation scheme where each vertex
aggregates feature vectors of its neighbours, possibly combines this with its
own current feature vector, to finally obtain its new feature vector. After a
number of iterations, each vertex is then represented by the resulting feature
vector.
The adequacy of GNNs for graph learning tasks is directly related to their
so-called distinguishing power. Here, distinguishing power refers to
the ability of GNNs to distinguish vertices and graphs in terms of the computed
representation. That is, when two vertices are represented by the same feature
vector, they are considered the same with regards to any subsequent
feature-based task.
Only recently a formal study of the distinguishing power of some GNN variants
has been initiated. In two independent studies~\cite{xhlj19,grohewl} the
distinguishing power of GNNs is linked to the distinguishing power of the
classical Weisfeiler-Lehman (WL) algorithm. The WL algorithm starts
from an initial vertex colouring of the graph. Then, similarly as GNNs, the WL
algorithm recursively aggregates the colouring of neighbouring vertices. In
each recursive step, a vertex colouring is obtained that refines the previous
one. The WL algorithm stops when no further refinement is obtained. The
distinguishing power of the WL algorithm itself is well understood, see
e.g.,~\cite{CaiFI92,KieferSS15,ArvindKRV17}.
In~\cite{xhlj19,grohewl} it is shown that for any input graph if vertices can be
distinguished by a GNN then they can be distinguished by the WL algorithm.
Conversely, Graph Isomorphism Networks (GINs) were proposed in~\cite{xhlj19}
that can match the distinguishing power of the WL algorithm, on any graph. The
construction of GINs relies on multi-layer perceptrons and their ability to
approximate arbitrary functions. In contrast, \cite{grohewl} show that the
distinguishing power of the WL algorithm can also be matched by using GNNs,
provided that the input graph is fixed. Both these works consider undirected
vertex-labelled graphs. We remark that the work by~\cite{grohewl} has recently
been extended to directed graphs, possibly with vertex- and
edge-labels~\cite{Jaume2019}. We refer to~\cite{Sato2020ASO} for an in-depth
survey on the expressive power of graph neural networks.
In this paper we start from the observation that many popular GNNs fall outside
of the class of GNNs considered in previous work~\cite{xhlj19,grohewl}.
Prominent examples of such GNNs are the so-called Graph Convolutional Networks
(GCNs)~\cite{kipf-loose}. Although GCNs adhere to the same strategy as GNNs
(i.e., recursive neighbourhood aggregation of features), they additionally take
into account \textit{vertex-degree information}. In this paper, we continue the
study of the distinguishing power of large classes of GNNs that may use degree
information.
To do so, we leverage connections between GNNs, GCNs and so-called Message
Passing Neural Network (MPNNs) introduced by~\cite{GilmerSRVD17}. Such neural
networks perform a number of rounds of computation, and in each such round,
vertex labels are propagated along the edges of the graph and aggregated at the
vertices. MPNNs are known to encompass many GNN and GCN
formalisms~\cite{GilmerSRVD17}. We refer to~\cite{Zhou2018,Zonghan2019} for
extensive surveys on GNNs, GCNs and MPNNs.
The general MPNN framework allows us to explore the impact of degree information
on the distinguishing power of MPNNs in general and large classes of GNNs and
GCNs in particular. More precisely, in this paper we consider two general
classes of MPNNs: \textit{anonymous} MPNNs that do not use degree information,
and \textit{degree-aware} MPNNs that do use degree information. The former
class of MPNNs covers the GNNs studied in~\cite{xhlj19,grohewl}, the latter
class covers the GCNs~\cite{kipf-loose}, among others.
\paragraph{Contributions.}
For general MPNNs, our main results are the following (see Propositions~\ref{pro:eqstrongWL} and~\ref{prop:onestep}):
\begin{enumerate}[(i)]
\item The distinguishing power of anonymous MPNNs is bounded by the WL algorithm. This result can be seen as a slight generalisation of the results in~\cite{xhlj19,grohewl}.
\item The distinguishing power of degree-aware MPNNs is bounded by the WL algorithm, \textit{but they may be one step ahead}. Intuitively, degree-aware MPNNs may be one step ahead of the WL algorithm because the degree information, which is part of degree-aware MPNNs from the start, is only derived by the WL algorithm after one step.
\item The WL algorithm can be regarded as an anonymous MPNN (and thus also as a degree-aware MPNN). As a consequence, the distinguishing power of the classes of anonymous and degree-aware MPNNs matches that of the WL algorithm.
\end{enumerate}
For anonymous MPNNs related to GNNs~\cite{xhlj19,grohewl} and degree-aware MPNNs
related to GCNs~\cite{kipf-loose}, our main results are the following (see
Theorems~\ref{thm:grohe_lower} and~\ref{thm:equalstrong}, and
Propositions~\ref{prop:notasstrong} and~\ref{prop:indeed-wl-power}):
\begin{enumerate}[resume*]
\item On a fixed input graph, the WL algorithm can be simulated, step-by-step, by GNNs that use ReLU or sign as activation function. This result refines the result in~\cite{grohewl} in that their simulation using the ReLU function requires two GNN ``layers'' for each step of the WL algorithm. We only require one layer in each step. In addition, our simulation is achieved by means of a very simple form of GNNs (see Equation~\eqref{eq:GNNWL} at the end of Section~\ref{sec:anonymous}), which may be of independent interest.
\item The distinguishing power of GCNs is bounded by the WL algorithm, \textit{but they may be one step ahead}. This is due to GCNs being degree-aware MPNNs (for which result~(ii) applies). This advantage of GCNs over more classical GNNs may explain the success of GCNs in various graph learning tasks.
\item In contrast, we show that the WL algorithm \textit{cannot} be simulated by popular GCNs such as those from~\cite{kipf-loose}. This observation is somewhat contradictory to the general belief that GCNs can be seen as a ``continuous generalisation'' of the WL algorithm.
\item However, by introducing a learnable trade-off parameter between features of the vertex and those of its neighbours, the simulation of the WL algorithm can be achieved by GCNs. This minor relaxation of GCNs (see Equation~\eqref{gnn:kipfp} at the end of Section~\ref{sec:dMPNNs}) was already suggested in~\cite{kipf-loose} based on empirical results.
Our simulation result thus provides a theoretical justification of this parameter.
\end{enumerate}
\paragraph{Structure of the paper.}
After introducing some notations and concepts in Section~\ref{sec:prelims}, we define MPNNs, anonymous and degree-aware MPNNs in Section~\ref{sec:MPNNs}. In Section~\ref{sec:compare} we formally define how to compare classes of MPNNs with regard to their distinguishing power. We characterise the distinguishing power of anonymous MPNNs in Section~\ref{sec:anonymous} and that of degree-aware MPNNs in Section~\ref{sec:dMPNNs}. We conclude the paper in Section~\ref{sec:conclude}.
\section{Preliminaries}\label{sec:prelims}
Let $\mathbb{A}$ denote the set of all algebraic numbers; $\mathbb{Q}$, the set
of all rational numbers; $\mathbb{Z}$, the set of all integer numbers;
$\mathbb{N}$, the set of all natural numbers including zero, i.e., $\mathbb{N} =
\{0,1,2,\dots\}$. We write $\mathbb{S}^+$ to denote the subset of numbers from
$\mathbb{S}$ which are strictly positive, e.g., $\mathbb{N}^+ = \mathbb{N}
\setminus \{0\}$. We use $\{\!\}$ and $\{\!\!\{\! \}\!\!\}$ to indicate sets and
multisets, respectively.
\paragraph{Computing with algebraic numbers.}
Throughout the paper we will perform basic computations, such as addition and
multiplication, on numbers. It is well-known that these operations are
computable on numbers in $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Q}$. However,
in order to capture numbers used by popular graph neural network architectures,
such as roots of integers~\cite{kipf-loose}, we will work with \emph{algebraic
numbers}. An algebraic number is usually represented by a minimal polynomial
such that the number is a root of the polynomial and a pair of rational numbers
to identify that root. Conveniently, it is known that the operations we will
need are indeed computable for algebraic numbers encoded using such a
representation (see, e.g.,~\cite{OuaknineW14}).
\paragraph{Labelled graphs.}
Let $G=(V,E)$ be an undirected graph consisting of $n \in \mathbb{N}$ vertices.
Without loss of generality we assume that $V=\{1,2,\dots,n\}$. Given a vertex
$v\in V$, we denote by $N_G(v)$ its set of neighbours, i.e., $N_G(v):=\{u\st
\{u,v\}\in E\}$. Furthermore, the degree of a vertex $v$, denoted by $d_{v}$, is
the number of vertices in $N_G(v)$. With a labelled graph $(G,\pmb{\nu})$ we
mean a graph $G=(V,E)$ whose vertices are labelled using a function
$\pmb{\nu}:V\to \Sigma$ for some set $\Sigma$ of labels. We denote by
$\pmb{\nu}_v$ the label of $v$ for each $v\in V$.
Henceforth we fix a labelled graph $(G,\pmb{\nu})$ with $G=(V,E)$ and denote by
$\mathbf{A}$ the adjacency matrix (of $G$). That is, $\mathbf{A}$ is a matrix of
dimension $n \times n$ such that the entry $\mathbf{A}_{vw}=1$ if $\{v,w\}\in E$
and $\mathbf{A}_{vw}=0$ otherwise. We denote by $\mathbf{D}$ the diagonal matrix
such that $\mathbf{D}_{vv}=d_v$ for each $v\in V$. Throughout the paper we will
assume that $G$ does not have isolated vertices, which is equivalent to assuming
that $\mathbf{D}$ does not have any $0$ entries on the diagonal. This assumption
will help us to avoid unnecessary technical details in the theoretical analysis.
But it is easy to generalise our results by treating isolated nodes separately.
We will also assume that there are no self-loops, so the diagonal of
$\mathbf{A}$ is filled with $0$s. For an arbitrary matrix $\mathbf{B}$ we denote
by $\mathbf{B}_{i}$ the $i$-th row of $\mathbf{B}$. Furthermore, if $\mathbf{B}$
is a matrix of dimension $n\times m$, we also represent the rows of $\mathbf{B}$
by $\mathbf{B}_{v}$, for $v\in V$.
We will identify $\Sigma$ with elements (row vectors) in $\mathbb{A}^s$ for some
$s\in\mathbb{N}^+$. In this way, a labelling $\pmb{\ell}:V\to\Sigma$ can be
regarded as a matrix in $\mathbb{A}^{n\times s}$ and $\pmb{\ell}_v$ corresponds
to the $v$-th row in that matrix. Conversely, a matrix $\mathbf{L} \in
\mathbb{A}^{n \times s}$ can be regarded as the vertex labelling that labels $v$
with the row vector $\mathbf{L}_{v}$. We use these two interpretations of
labellings interchangeably.
It will be important later on to be able to compare two labellings of $G$. Given
a matrix $\mathbf{L} \in \mathbb{A}^{n\times s}$ and a matrix $\mathbf{L}' \in
\mathbb{A}^{n\times s'}$ we say that the vertex labelling $\mathbf{L}'$ is
coarser than the vertex labelling $\mathbf{L}$, denoted by
$\mathbf{L}\sqsubseteq \mathbf{L}'$, if for all $v,w\in V$, $\mathbf{L}_{v}=\mathbf{L}_{w} \Rightarrow \mathbf{L}'_{v}=\mathbf{L}'_{w}$. The
vertex labellings $\mathbf{L}$ and $\mathbf{L}'$ are equivalent, denoted by
$\mathbf{L}\equiv\mathbf{L}'$, if $\mathbf{L}\sqsubseteq \mathbf{L}'$ and
$\mathbf{L}'\sqsubseteq \mathbf{L}$ hold. In other words,
$\mathbf{L}\equiv\mathbf{L}'$ if and only if for all $v,w\in V$,
$\mathbf{L}_{v}=\mathbf{L}_{w} \Leftrightarrow \mathbf{L}'_{v}=\mathbf{L}'_{w} $.
\paragraph*{Weisfeiler-Lehman labelling.}
Of particular importance is the labelling obtained by colour refinement, also
known as the Weisfeiler-Lehman algorithm (or WL algorithm, for short). The WL
algorithm constructs a labelling, in an incremental fashion, based on
neighbourhood information and the initial vertex labelling. More specifically,
given $(G,\pmb{\nu})$, the WL algorithm initially sets
$\pmb{\ell}^{(0)}:=\pmb{\nu}$. Then, the WL algorithm computes a labelling
$\pmb{\ell}^{(t)}$, for $t> 0$, as follows:
\begin{equation*}
\pmb{\ell}^{(t)}_v:=\textsc{Hash}\Bigl(\bigl(\pmb{\ell}^{(t-1)}_v,\{\!\!\{
\pmb{\ell}_u^{(t-1)} \st u \in N_G(v) \}\!\!\}\bigr)\Bigr),
\end{equation*}
where $\textsc{Hash}$ bijectively maps the above pair, consisting of (i)~the
previous label $\pmb{\ell}^{(t-1)}_v$ of $v$; and (ii)~the multiset $\{\!\!\{
\pmb{\ell}_u^{(t-1)} \st u \in N_G(v) \}\!\!\}$ of labels of the neighbours of $v$,
to a label in $\Sigma$ which has not been used in previous iterations. When the
number of distinct labels in $\pmb{\ell}^{(t)}$ and $\pmb{\ell}^{(t-1)}$ is the
same, the WL algorithm terminates. Termination is guaranteed in at most $n$
steps~\cite{Immerman1990}.
\section{Message Passing Neural Networks}\label{sec:MPNNs}
We start by describing message passing neural networks (MPNNs) for deep
learning on graphs, introduced by \cite{GilmerSRVD17}. Roughly speaking, in
MPNNs, vertex labels are propagated through a graph according to its
connectivity structure. MPNNs are known to model a variety of graph neural
network architectures commonly used in practice. We define MPNNs
in Section~\ref{subsec:def}, provide some examples in Section~\ref{subsec:examples},
and comment on the choice of formalisation of MPNNs in Section~\ref{subsec:comments}.
\subsection{Definition}\label{subsec:def}
Given a labelled graph $( G,\pmb{\nu})$ and a computable function $f : V \to \mathbb{A}$
an MPNN computes a vertex
labelling $\pmb{\ell}:V\to \mathbb{A}^{s}$, for some $s\in\mathbb{N}^+$.
The vertex labelling computed by an MPNN is computed in a finite number of rounds $T$.
After round $0 \le t \le T$ the labelling is denoted by
$\pmb{\ell}^{(t)}$. We next detail how $\pmb{\ell}^{(t)}$ is computed.
\begin{description}\setlength{\itemsep}{-0.4ex}
\item [Initialisation.] We let $\pmb{\ell}^{(0)}:=\pmb{\nu}$.
\end{description}
Then, for every round $t=1,2,\ldots,T$, we define
$\pmb{\ell}^{(t)}:V\to\mathbb{A}^{s_t}$, as follows\footnote{Note that
we allow for labels to have different dimensions $s_t \in \mathbb{N}^+$
per round $t$.}:
\begin{description}\setlength{\itemsep}{-0.4ex}
\item [Message Passing.]
Each vertex $v\in V$ receives messages from its neighbours which are
subsequently aggregated. Formally, the function $\textsc{Msg}^{(t)}$ receives as
input $f$ applied to two vertices $v$ and $u$, and the corresponding labels of
these vertices from the previous iteration $\pmb{\ell}^{(t-1)}_v$ and
$\pmb{\ell}^{(t-1)}_u$, and outputs a label in $\mathbb{A}^{s'_t}$. Then, for
every vertex $v$, we aggregate by summing all such labels for every neighbour
$u$.
$$
\mathbf{m}^{(t)}_{v}:=\sum_{u\in
N_G(v)}\textsc{Msg}^{(t)}\left(\pmb{\ell}^{(t-1)}_v,\pmb{\ell}^{(t-1)}_u,f(v),f(u)\right)\in\mathbb{A}^{s'_t}.
$$
\item [Updating.] Each vertex $v\in V$ further updates $\mathbf{m}{}^{(t)}_{v}$
possibly based on its current label $\pmb{\ell}^{(t-1)}_v$:
$$
\pmb{\ell}^{(t)}_v:=\textsc{Upd}^{(t)}\left(\pmb{\ell}^{(t-1)}_v,\mathbf{m}^{(t)}_{v}\right)\in\mathbb{A}^{s_t}.
$$
\end{description}
Here, the message functions $\textsc{Msg}^{(t)}$ and update functions
$\textsc{Upd}^{(t)}$ are general (computable) functions. After round $T$, we
define the final labelling $\pmb{\ell}:V\to\mathbb{A}^{s}$ as
$\pmb{\ell}_v:=\pmb{\ell}^{(T)}_v$ for every $v\in V$. If further aggregation
over the entire graph is needed, e.g., for graph classification, an additional
readout function $\textsc{ReadOut}(\{\!\!\{\pmb{\ell}_v\mid v\in V\}\!\!\})$ can be
applied. We omit the readout function here since most of the computation happens
during the rounds of an MPNN.
The role of the function $f$ in this paper is to distinguish between two classes
of MPNNs\footnote{In general, one could consider any function $f$.}: Those whose
message functions only depend on the labels of the vertices involved, in which
case we set $f$ to the zero function $f(v)=0$, for all $v\in V$, and those whose
message functions depend on the labels and on the degrees of the vertices
involved, in which case we set $f$ to the degree function $f(v)=d_v$, for all
$v\in V$. We will refer to the former class as \textit{anonymous} MPNNs and to
the latter as \textit{degree-aware} MPNNs. These classes are denoted by
$\mathcal{M}_{\textsl{anon}}$ and $\mathcal{M}_{\textsl{deg}}$, respectively. We remark
that, by definition, $\mathcal{M}_{\textsl{anon}}\subseteq \mathcal{M}_{\textsl{deg}}$.
\subsection{Examples}\label{subsec:examples}
We next illustrate anonymous and degree-aware MPNNs by a number of examples.
First, we provide two examples of anonymous MPNNs that will play an important role in Section~\ref{sec:anonymous}.
\begin{example}[GNN architectures]\label{ex:GNN}
We first consider the graph neural network architectures~\cite{hyl17,grohewl}
defined by:
\begin{equation}
\mathbf{L}^{(t)}:=\sigma\left(\mathbf{L}^{(t-1)}\mathbf{W}_1^{(t)}+\mathbf{A}\mathbf{L}^{(t-1)}\mathbf{W}_2^{(t)}+\mathbf{B}^{(t)}\right), \label{gnn:grohe}
\end{equation}
where $\mathbf{L}^{(t)}$ is the matrix in $\mathbb{A}^{n\times s_t}$ consisting
of the $n$ rows $\pmb{\ell}^{(t)}_v\in\mathbb{A}^{s_t}$, for $v\in V$,
$\mathbf{A}\in\mathbb{A}^{n\times n}$ is the adjacency matrix of $G$,
$\mathbf{W}_1^{(t)}$ and $\mathbf{W}_2^{(t)}$ are (learnable) weight matrices in
$\mathbb{A}^{s_{t-1}\times s_t}$, $\mathbf{B}^{(t)}$ is a bias matrix in
$\mathbb{A}^{n\times s_t}$ consisting of $n$ copies of the same row
$\mathbf{b}^{(t)}\in \mathbb{A}^{s_t}$, and $\sigma$ is a non-linear activation
function. We can regard this architecture as an MPNN. Indeed,~(\ref{gnn:grohe})
can be equivalently phrased as the architecture which computes, in round $t$,
for each vertex $v\in V$ the label defined by:
\[
\pmb{\ell}^{(t)}_v:=\sigma\left(\pmb{\ell}^{(t-1)}_v\mathbf{W}_1^{(t)}+ \sum_{u\in N_G(v)}\pmb{\ell}^{(t-1)}_u\mathbf{W}_2^{(t)}+\mathbf{b}^{(t)} \right),
\]
where we identified the labellings with their images, i.e., a row vector in
$\mathbb{A}^{s_{t-1}}$ or $\mathbb{A}^{s_t}$. To phrase this as an MPNN, it
suffices to define for each $\mathbf{x}$ and $\mathbf{y}$ in
$\mathbb{A}^{s_{t-1}}$, each $v\in V$ and $u\in N_G(v)$, and each $t\geq 1$:
\begin{equation*}
\textsc{Msg}^{(t)}\bigl(\mathbf{x},\mathbf{y},-,-):=\mathbf{y}\mathbf{W}_2^{(t)}
\text{ and }
\textsc{Upd}^{(t)}(\mathbf{x},\mathbf{y}):=\sigma\left(\mathbf{x}\mathbf{W}_1^{(t)}+\mathbf{y} + \mathbf{b}^{(t)}\right).
\end{equation*}
We write $-$ instead of $0$ to emphasise that the message functions use the zero
function $f(v)=0$, for all $v\in V$, and hence do not depend on $f(v)$ and
$f(u)$. In other words, the MPNN constructed is an anonymous MPNN. Without loss
of generality we will assume that aMPNNs do not use $f(v)$ and $f(u)$ in the
messages. If they do then one can replace them with $0$. This way it is easy to
see that classes of MPNNs that use different functions $f$ in the messages
contain the class of anonymous MPNNs. \hfill$\blacksquare$
\end{example}
Another example of an anonymous MPNN originates from the Weisfeiler-Lehman
algorithm described in the preliminaries.
\begin{example}[Weisfeiler-Lehman]\label{ex:WL}
We recall that WL computes, in round $t \geq 1$, for each vertex $v\in V$ the label:
$$
\pmb{\ell}^{(t)}_v:=\textsc{Hash}\left(\pmb{\ell}^{(t-1)}_v,\{\!\!\{ \pmb{\ell}_u^{(t-1)} \st u \in N_G(v) \}\!\!\}\right).
$$
Let us assume that the set $\Sigma$ of labels is $\mathbb{A}^s$ for some fixed
$s\in\mathbb{N}^+$. We cast the WL algorithm as an anonymous MPNN by using an
injection $h : \mathbb{A}^s \to \mathbb{Q}$. What follows is in fact an
adaptation of Lemma 5 from~\cite{xhlj19} itself based on~\cite[Theorem
2]{ZaheerKRPSS17}. We crucially rely on the fact that the set $\mathbb{A}$ of
algebraic numbers is countable (see e.g., Theorem 2.2. in~\cite{Frazer}). As a
consequence, also $\mathbb{A}^s$ is countable.
Let $\tau : \mathbb{A}^s \to \mathbb{N}^+$ be a computable injective function
witnessing the countability of $\mathbb{A}^s$. For instance, since elements of
$\mathbb{A}$ are encoded as a polynomial $a_0 + a_1 x^1 + a_2 x^2 + \dots +
a_kx^k \in \mathbb{Z}[x]$ and a pair $\nicefrac{n_1}{d_1},\nicefrac{n_2}{d_2}$
of rationals, $\tau$ can be taken to be the composition of the injection $\alpha
: \mathbb{A} \to \mathbb{N}^+$, applied point-wise, and the Cantor tuple
function, where
\[
\alpha(a_0 + a_1 x^1 + a_2 x^2 + \dots + a_kx^k,\nicefrac{n_1}{d_1},\nicefrac{n_2}{d_2}) \mapsto
p(1,n_1) p(2,n_2) p(3,d_1) p(4,d_2) \prod_{i=0}^k p(i+5,a_i)
\]
with $\pi_i$ being the $i$-th prime number in
\[
p(i,z) = \begin{cases}
\pi_{2i}^z & \text{if } z \geq 0\\
\pi_{2i+1}^{-z} & \text{if } z < 0.
\end{cases}
\]
We next define $h:\mathbb{A}^s\to\mathbb{Q}^+$ as the mapping $\mathbf{x}
\mapsto (n+1)^{-\tau(\mathbf{x})}$. Note that $h$ is injective and
$h(\mathbf{x})$ can be seen as a number whose $(n+1)$-ary representation has a
single nonzero digit. We next observe that the multiplicity of every element in
$S := \{\!\!\{ \pmb{\ell}_u^{(t-1)} \st u \in N_G(v) \}\!\!\}$ is bounded by the
number of all vertices $n$ --- and this for all $t \geq 1$. It follows that the
function $\phi$ mapping any such $S$ to $\sum_{\mathbf{x} \in S} h(\mathbf{x})$
is an injection from $\mathbb{A}^s$ to $\mathbb{Q}$. Therefore, the summands can
be recovered by looking at the $(n+1)$-ary representation of the sum, and thus
the inverse of $\phi$ is computable on its image. To conclude, we define the
message function
\[
\textsc{Msg}^{(t)}(\mathbf{x},\mathbf{y},-,-) :=
h(\mathbf{y}) \in \mathbb{A}.
\]
The update function is defined by
\[
\textsc{Upd}^{(t)}(\mathbf{x},y) :=
\textsc{Hash}(\mathbf{x},\phi^{-1}(y)),
\]
where $y \in \mathbb{A}$ since it corresponds to a sum of messages, themselves
algebraic numbers. As before, we write $-$ instead of $0$ to emphasise that the
message functions use the zero function $f(v)=0$, for all $v\in V$, and hence do
not depend on $f(v)$ and $f(u)$.\hfill$\blacksquare$
\end{example}
We conclude with an example of a degree-aware MPNN. We study degree-aware MPNNs in Section~\ref{sec:dMPNNs}.
\begin{example}[GCNs by Kipf and Welling]\label{ex:KipfasMPNN}
We consider the GCN architecture by~\cite{kipf-loose}, which in round $t \geq 1$ computes
\begin{equation*}
\mathbf{L}^{(t)} := \sigma\left(
(\mathbf{D}+\mathbf{I})^{-1/2}(\mathbf{A}+\mathbf{I})(\mathbf{D}+\mathbf{I})^{-1/2}\mathbf{L}^{(t-1)}\mathbf{W}^{(t)}
\right),
\end{equation*}
where we use the same notation as in Example~\ref{ex:GNN} but now with a single
(learnable) weight matrix $\mathbf{W}^{(t)}$ in $\mathbb{A}^{s_{t-1}\times
s_t}$. This means that, in round $t$, for each vertex $v\in V$ it computes the
label:
\begin{equation}
\pmb{\ell}^{(t)}_v:=\sigma\left(\left(\frac{1}{1+d_v}\right)\pmb{\ell}_v^{(t-1)}\mathbf{W}^{(t)} + \sum_{u\in N_G(v)} \left(\frac{1}{\sqrt{1+d_v}}\right)\left(\frac{1}{\sqrt{1+d_u}}\right)\pmb{\ell}^{(t-1)}_u\mathbf{W}^{(t)}\right). \label{GNN:Kipf}
\end{equation}
We can regard this architecture again as an MPNN. Indeed, it suffices to define
for each $\mathbf{x}$ and $\mathbf{y}$ in $\mathbb{A}^{s_{t-1}}$, each $v\in V$
and $u\in N_G(v)$, and each $t\geq 1$:
\begin{align*}
\textsc{Msg}^{(t)}\left(\mathbf{x},\mathbf{y},d_v,d_u\right)&:=
\frac{1}{d_v}\left(\frac{1}{1+d_v}\right)\mathbf{x}\mathbf{W}^{(t)}+
\left(\frac{1}{\sqrt{1+d_v}}\right)\left(\frac{1}{\sqrt{1+d_u}}\right)\mathbf{y}\mathbf{W}^{(t)}
\intertext{and} \textsc{Upd}^{(t)}(\mathbf{x},\mathbf{y})&:=\sigma(\mathbf{y}).
\end{align*}
We remark that the initial factor $1/d_v$ in the message functions is introduced
for renormalisation purposes. We indeed observe that the message functions
depend only on $\pmb{\ell}^{(t-1)}_v$, $\pmb{\ell}^{(t-1)}_u$, and the degrees
$d_v$ and $d_u$ of the vertices $v$ and $u$, respectively.\hfill$\blacksquare$
\end{example}
\subsection{On the choice of formalism}\label{subsec:comments}
The expert reader may have noticed that we use a different formalisation of
MPNNs than the one given in the original paper~\cite{GilmerSRVD17}. The first
difference is that our MPNNs are parameterised by an input computable
function~$f$ applied to $v$ and $u \in N_G(v)$. We add this function to avoid a
certain ambiguity in the formalisation in ~\cite{GilmerSRVD17} on what precisely
the message functions can depend on. More specifically, only a dependence on
$\pmb{\ell}_v^{(t-1)}$ and $\pmb{\ell}_u^{(t-1)}$ is specified
in~\cite{GilmerSRVD17}. In contrast, the examples given in~\cite{GilmerSRVD17}
use more information, such as the degree of vertices. The use of the function
$f$ in the definition of MPNNs makes explicit the information that message
functions can use. It is readily verified that every MPNN of~\cite{GilmerSRVD17}
corresponds to an MPNN in our formalism.
The second difference is that the MPNNs in ~\cite{GilmerSRVD17} work on graphs
that carry both vertex and edge labels. We ignore edge labellings in this paper
but most of our results carry over to that more general setting. Indeed, it
suffices to use the extension of the Weisfeiler-Lehman algorithm for
edge-labelled graphs as is done for graph neural networks in~\cite{Jaume2019}.
We also want to compare our formalisation to the MPNNs from~\cite{Loukas2019}.
In that paper, the message functions can depend on identifiers of the vertices
involved. Such position-aware MPNNs correspond to MPNNs in our setting in which
$f$ assigns to each vertex a unique identifier. We remark that~\cite{Loukas2019}
shows Turing universality of position-aware MPNNs using close connections with
the LOCAL model for distributed graph computations of~\cite{Angluin}. As such,
MPNNs from~\cite{Loukas2019} can simulate our MPNNs as one could add a few
initialisation rounds to compute $f(v)$ and $f(u)$. We also remark that in the
MPNNs from~\cite{Loukas2019} every vertex can also send itself a message. We
provide this functionality by parameterising the update functions with the
current label of the vertex itself, just as in~\cite{GilmerSRVD17}.
\section{Comparing the distinguishing power of classes of MPNNs}\label{sec:compare}
The distinguishing power of MPNNs relates to their ability to distinguish
vertices based on the labellings that they compute. We are interested in
comparing the distinguishing power of classes of MPNNs. In this section we
formally define what we mean by such a comparison.
For a given labelled graph $( G,\pmb{\nu})$ and MPNN $M$, we denote by
$\pmb{\ell}_M^{(t)}$ the vertex labelling computed by $M$ after $t$ rounds. We
fix the input graph in what follows, so we do not need to include the dependency
on the graph in the notation of labellings.
\begin{definition}\label{def:mpnnweak}\normalfont
Consider two MPNNs $M_1$ and $M_2$ with the same number of rounds $T$. Let
$\pmb{\ell}_{M_1}^{(t)}$ and $\pmb{\ell}_{M_2}^{(t)}$ be their corresponding
labellings on an input graph $( G,\pmb{\nu})$ obtained after $t$ rounds of
computation for every $0 \le t \le T$. Then $M_1$ is said to be \textit{weaker}
than $M_2$, denoted by $M_1\preceq M_2$, if $M_1$ cannot distinguish more
vertices than $M_2$ in every round of computation. More formally, $M_1\preceq
M_2$ if $\pmb{\ell}_{M_2}^{(t)}\sqsubseteq \pmb{\ell}_{M_1}^{(t)}$ for every
$t\geq 0$. In this case we also say that $M_2$ is \textit{stronger} than
$M_1$.\hfill$\blacksquare$
\end{definition}
We lift this notion to classes $\mathcal{M}_1$ and $\mathcal{M}_2$ of MPNNs in a
standard way.
\begin{definition}\label{def:classesweak}\normalfont
Consider two classes $\mathcal{M}_1$ and $\mathcal{M}_2$ of MPNNs. Then,
$\mathcal{M}_1$ is said to be \textit{weaker} than $\mathcal{M}_2$, denoted
by $\mathcal{M}_1\preceq \mathcal{M}_2$, if for all $M_1\in \mathcal{M}_1$
there exists an $M_2\in\mathcal{M}_2$ which is stronger than $M_1$.
\hfill$\blacksquare$
\end{definition}
Finally, we say that $\mathcal{M}_1$ and $\mathcal{M}_2$ are \textit{equally
strong}, denoted by $\mathcal{M}_1\equiv \mathcal{M}_2$, if both
$\mathcal{M}_1\preceq \mathcal{M}_2$ and $\mathcal{M}_2\preceq
\mathcal{M}_1$ hold.
We will also need a generalisation of the previous definitions in which we
compare labellings computed by MPNNs at different rounds. This is formalised as
follows.
\begin{definition}\label{def:classg}\normalfont
Consider two MPNNs $M_1$ and $M_2$ with $T_1$ and $T_2$ rounds, respectively.
Let $\pmb{\ell}_{M_1}^{(t)}$ and $\pmb{\ell}_{M_2}^{(t)}$ be their corresponding
labellings on an input graph $( G,\pmb{\nu})$ obtained after $t$ rounds of
computation. Let $g:\mathbb{N}\to \mathbb{N}$ be a monotonic function such that
$g(T_1) = T_2$. We say that $M_1$ is \textit{$g$-weaker} than $M_2$, denoted by
$M_1\preceq_{g} M_2$, if $\pmb{\ell}_{M_2}^{g(t)}\sqsubseteq
\pmb{\ell}_{M_1}^{(t)}$ for every $0 \le t\le T_1$.\hfill$\blacksquare$
\end{definition}
Only the following special cases of this definition, depending on extra
information regarding a function $g:\mathbb{N}\to\mathbb{N}$, will be relevant
in this paper:
\begin{itemize}
\item $g(t)=t$. This case corresponds to Definition~\ref{def:mpnnweak}. If
$M_1\preceq_{g} M_2$, then we simply say that $M_1$ is weaker than $M_2$, and
write $M_1\preceq M_2$, as before.
\item $g(t)=t+1$. If $M_1\preceq_{g} M_2$,
then we say that $M_1$ is weaker than $M_2$ \textit{with one step ahead}. We
denote this by $M_1\preceq_{+1} M_2$.
\item $g(t)=ct$ for some constant $c$.
If $M_1\preceq_{g} M_2$, then we say that $M_1$ is weaker than $M_2$
\textit{with a linear factor of $c$}. We denote this by $M_1\preceq_{\times
c} M_2$.
\end{itemize}
We lift these definitions to classes of MPNNs, just like in Definition~\ref{def:classesweak}.
\section{The distinguishing power of anonymous MPNNs}\label{sec:anonymous}
In this section we compare classes of anonymous MPNNs in terms of their
distinguishing power using Definition~\ref{def:classesweak}.
We recall from Section~\ref{sec:MPNNs} that anonymous MPNNs are MPNNs whose message
functions only depend on the previous labels of the vertices involved. The
distinguishing power of anonymous MPNNs (or aMPNNs, for short) is well understood.
Indeed, as we will shortly see, it follows from two independent
works~\cite{xhlj19,grohewl} that the distinguishing power of aMPNNs can be linked to
the distinguishing power of the WL algorithm.
\begin{figure}[t]
\centering
\begin{tikzpicture}[node distance=1cm]
\node (lwl) {$\mathcal{M}_{\textsl{WL}}$};
\node[above right= of lwl] (relu) {$\mathcal{M}_{\textsl{GNN}}^{\textsl{ReLU}}$};
\node[below right= of lwl] (sign) {$\mathcal{M}_{\textsl{GNN}}^{\textsl{sign}}$};
\node[right=3cm of lwl] (anon) {$\mathcal{M}_{\textsl{anon}}$};
\node[right= of anon] (rwl) {$\mathcal{M}_{\textsl{WL}}$};
\path
(lwl) edge[draw=none] node[sloped,label=above:\scriptsize{Thm.~\ref{thm:equalstrong}}] {$\preceq$} (relu)
(lwl) edge[draw=none] node[sloped,label=below:\scriptsize{Thm.~\ref{thm:grohe_lower}}] {$\preceq$} (sign)
(relu) edge[draw=none] node[sloped,label=above:\scriptsize{Cor.~\ref{corr:GNNwANO}}]{$\subseteq$} (anon)
(sign) edge[draw=none] node[sloped,label=below:\scriptsize{Cor.~\ref{corr:GNNwANO}}]{$\subseteq$} (anon)
(anon) edge[draw=none] node[label=above:\scriptsize{Prop.~\ref{pro:eqstrongWL}}] {$\preceq$} (rwl)
;
\end{tikzpicture}
\caption{Summary of relationships amongst major anonymous MPNN classes considered in Section~\ref{sec:anonymous}.}
\label{fig:reduxs}
\end{figure}
Let $( G,\pmb{\nu})$ be a labelled graph. We will consider the following classes of
aMPNNs. We denote by $\mathcal{M}_{\textsl{WL}}$ the class of aMPNNs consisting of an aMPNN
$M^T_{\textsl{WL}}$, for each $T \in \mathbb{N}$, originating from the WL algorithm
(see Example~\ref{ex:WL}) being ran for $T$ rounds. In a slight abuse of notation,
we will simply write $M_{\textsl{WL}}$ when $T$ is clear from the context. Recall
that the class of anonymous MPNNs is denoted $\mathcal{M}_{\textsl{anon}}$. Finally, we
introduce two classes of aMPNNs which are of special interest: those arising from
the graph neural networks considered in~\cite{grohewl}. In Example~\ref{ex:GNN} we
established that such graph neural networks correspond to aMPNNs. Let us denote by
$\mathcal{M}^\sigma_{\textsl{GNN}}$ the class of aMPNNs with message and update
functions of the form
\begin{equation}\textsc{Msg}^{(t)}\bigl(\mathbf{x},\mathbf{y},-,-):=\mathbf{y}\mathbf{W}_2^{(t)}
\text{ and }
\textsc{Upd}^{(t)}(\mathbf{x},\mathbf{y}):=\sigma\left(\mathbf{x}\mathbf{W}_1^{(t)}+\mathbf{y} + \mathbf{b}^{(t)}\right) \label{eq:MPNN-GNN}
\end{equation}
for any $\mathbf{x},\mathbf{y}\in\mathbb{A}^{s_{t-1}}$, $\mathbf{W}_1^{(t)}\in\mathbb{A}^{s_{t-1}\times
s_t}$, $\mathbf{W}_2^{(t)}\in\mathbb{A}^{s_{t-1}\times s_t}$, bias vector
$\mathbf{b}^{(t)}\in\mathbb{A}^{s_t}$, and non-linear activation function $\sigma$.
The following is our main result for this section.
\begin{theorem}\label{thm:anonymous}
The classes $\mathcal{M}_{\textsl{WL}}$, $\mathcal{M}_{\textsl{GNN}}^{\textsl{ReLU}}$, $\mathcal{M}_{\textsl{GNN}}^{\textsl{sign}}$ and $\mathcal{M}_{\textsl{anon}}$ are
all equally strong.
\end{theorem}
We prove this theorem in the following subsections by providing the relationships that are summarised in Figure~\ref{fig:reduxs}.
\subsection{General anonymous MPNNs}
We presently focus on the relation between the WL algorithm and anonymous MPNNs in
general. More specifically, we establish that these are equally strong. We remark
that in the proof of Theorem~\ref{thm:anonymous} we only need that
$\mathcal{M}_{\textsl{anon}}$ is weaker than $\mathcal{M}_{\textsl{WL}}$, as is indicated in
Figure~\ref{fig:reduxs}.
\begin{proposition}[Based on~\cite{xhlj19,grohewl}]\label{pro:eqstrongWL}
The classes $\mathcal{M}_{\textsl{anon}}$ and $\mathcal{M}_{\textsl{WL}}$ are equally strong.
\end{proposition}
\begin{proof}
First, we prove that $\mathcal{M}_{\textsl{WL}}$ is weaker than $\mathcal{M}_{\textsl{anon}}$. It
suffices to note that $\mathcal{M}_{\textsl{WL}} \subseteq \mathcal{M}_{\textsl{anon}}$.
It remains to argue that $\mathcal{M}_{\textsl{anon}}$ is weaker than $\mathcal{M}_{\textsl{WL}}$. The
proof is a trivial adaptation of the proofs of Lemma 2 in~\cite{xhlj19} and Theorem
5 in~\cite{grohewl}. We show, by induction on the number of rounds of computation,
that $\pmb{\ell}_{M_{\textsl{WL}}}^{(t)}\sqsubseteq \pmb{\ell}_M^{(t)}$ for all $M
\in \mathcal{M}_{\textsl{anon}}$ and every $t\geq 0$.
Clearly, this holds for $t=0$ since
$\pmb{\ell}_{M_{\textsl{WL}}}^{(0)}=\pmb{\ell}_M^{(0)}:=\pmb{\nu}$, by definition.
We assume next that the induction hypothesis holds up to round $t-1$ and consider
round $t$. Let $v$ and $w$ be two vertices such that
$(\pmb{\ell}_{M_{\textsl{WL}}}^{(t)})_v=(\pmb{\ell}_{M_{\textsl{WL}}}^{(t)})_w$
holds. This implies, by the definition of $M_{\textsl{WL}}$, that
$(\pmb{\ell}_{M_{\textsl{WL}}}^{(t-1)})_v=(\pmb{\ell}_{M_{\textsl{WL}}}^{(t-1)})_w$
and
$$
\{\!\!\{ (\pmb{\ell}_{M_{\textsl{WL}}}^{(t-1)})_u\mid u\in N_G(v) \}\!\!\}=
\{\!\!\{ (\pmb{\ell}_{M_{\textsl{WL}}}^{(t-1)})_u\mid u\in N_G(w) \}\!\!\}.
$$
By the induction hypothesis, this implies that
$(\pmb{\ell}_{M}^{(t-1)})_v=(\pmb{\ell}_{M}^{(t-1)})_w$ and
$$
\{\!\!\{ (\pmb{\ell}_{M}^{(t-1)})_u\mid u\in N_G(v) \}\!\!\}=
\{\!\!\{ (\pmb{\ell}_{M}^{(t-1)})_u\mid u\in N_G(w) \}\!\!\}.
$$
As a consequence, there is a bijection between $N_G(v)$ and $N_G(w)$ such that to
every vertex $u\in N_G(v)$ we can assign a unique vertex $u'\in N_G(w)$ such that
$(\pmb{\ell}_{M}^{(t-1)})_u=(\pmb{\ell}_{M}^{(t-1)})_{u'}$. Hence,
$$
\textsc{Msg}^{(t)}\left((\pmb{\ell}_{M}^{(t-1)})_v,(\pmb{\ell}_{M}^{(t-1)})_u,-,-\right)=
\textsc{Msg}^{(t)}\left((\pmb{\ell}_{M}^{(t-1)})_w,(\pmb{\ell}_{M}^{(t-1)})_{u'},-,-\right).
$$
Since this mapping between $N_G(v)$ and $N_G(w)$ is a bijection we also have:
$$
\mathbf{m}^{(t)}_v=\sum_{u\in N_G(v)}\textsc{Msg}^{(t)}\left((\pmb{\ell}_{M}^{(t-1)})_v,(\pmb{\ell}_{M}^{(t-1)})_u,-,-\right)=\sum_{u'\in N_G(w)}\textsc{Msg}^{(t)}\left((\pmb{\ell}_{M}^{(t-1)})_w,(\pmb{\ell}_{M}^{(t-1)})_{u'},-,-\right)=\mathbf{m}^{(t)}_w.
$$
We may thus conclude that $$(\pmb{\ell}_{M}^{(t)})_v=\textsc{Upd}^{(t)}\left((\pmb{\ell}_{M}^{(t-1)})_v,\mathbf{m}^{(t)}_v\right)=\textsc{Upd}^{(t)}\left((\pmb{\ell}_{M}^{(t-1)})_w,\mathbf{m}^{(t)}_w\right)=(\pmb{\ell}_{M}^{(t)})_w,
$$
as desired.
\end{proof}
We remark that we cannot use the results in~\cite{xhlj19} and~\cite{grohewl} as a
black box because the class $\mathcal{M}_{\textsl{anon}}$ is more general than the class
considered in those papers. The proofs in \cite{xhlj19} and \cite{grohewl} relate to
graph neural networks which, in round $t\geq 1$, compute for each vertex $v$ a label
$\pmb{\ell}^{(t)}_{v}$, as follows:
\begin{equation}
\pmb{\ell}^{(t)}_{v}:=
f_{\textsl{comb}}^{(t)}\left(
\pmb{\ell}_{v}^{(t-1)},f_{\textsl{aggr}}^{(t)}\left(\{\!\!\{ \pmb{\ell}^{(t-1)}_{u} \mid u \in N_G(v) \}\!\!\}\right)
\right), \label{eq:combaggr}
\end{equation}
where $f_{\textsl{comb}}^{(t)}$ and $f_{\textsl{aggr}}^{(t)}$ are general
(computable) combination and aggregation functions which we assume to assign labels
in $\mathbb{A}^{s_t}$. Furthermore, $\pmb{\ell}^{(0)}:=\pmb{\nu}$, just as before.
Every graph neural network of the form~(\ref{eq:combaggr}) is readily cast as an
aMPNN. Indeed, it suffices to observe, just as we did in Example~\ref{ex:WL}, that
the aggregation functions $f_{\textsl{aggr}}^{(t)}\bigl(\{\!\!\{ \pmb{\ell}^{(t-1)}_{u}
\mid u \in N_G(v) \}\!\!\}\bigr)$ can be written in the form $g^{(t)}\bigl(\sum_{u\in
N_G(v)} h^{(t)}(\pmb{\ell}^{(t-1)}_{u})\bigr)$, based on Lemma 5 from~\cite{xhlj19}.
Suppose that $\pmb{\nu}:V\to\mathbb{A}^{s_0}$. It now suffices to define for every
$t \geq 1$, every $\mathbf{x}$ and $\mathbf{y}$ in $\mathbb{A}^{s_{t-1}}$, every
$v\in V$ and $u\in N_G(u)$:
\begin{equation}
\textsc{Msg}^{(t)}(\mathbf{x},\mathbf{y},-,-):=h^{(t)}(\mathbf{y}) \text{ and } \textsc{Upd}^{(t)}(\mathbf{x},\mathbf{y}):=f_{\textsl{comb}}^{(t)}\left(\mathbf{x},g^{(t)}\left(\mathbf{y}\right)\right).\label{eq:combaggrtoaMPNN}
\end{equation}
This is clearly an aMPNN which computes the same labelling as~(\ref{eq:combaggr}).
The aMPNNs that we consider in this paper are slightly more general than those
defined by (\ref{eq:combaggrtoaMPNN}). Indeed, we consider message functions that
can also depend on the previous label $\pmb{\ell}_v^{(t-1)}$. In contrast, the
message functions in~(\ref{eq:combaggrtoaMPNN}) only depend on $\mathbf{y}$, which
corresponds to the previous labels $\pmb{\ell}_u^{(t-1)}$ of neighbours $u\in
N_G(v)$. Let $\mathcal{M}{}^{-}_{\textsl{anon}}$ denote the class of aMPNNs whose
message functions only depend on the previous labels of neighbours. It now suffices
to observe (see Example~\ref{ex:WL}) that $M_{\textsl{WL}}\in
\mathcal{M}{}^{-}_{\textsl{anon}}$ to infer, combined with
Proposition~\ref{pro:eqstrongWL}, that:
\begin{corollary}
The classes $\mathcal{M}{}^{-}_{\textsl{anon}}$, $\mathcal{M}_{\textsl{anon}}$ and $\mathcal{M}_{\textsl{WL}}$ are all equally strong.
\end{corollary}
We observe, however, that this does not imply that for every aMPNN $M$ in
$\mathcal{M}_{\textsl{anon}}$ there exists an aMPNN $M'$ in
$\mathcal{M}{}^{-}_{\textsl{anon}}$ such that $\pmb{\ell}_{M}^{(t)}\equiv
\pmb{\ell}_{M'}^{(t)}$ for all $t\geq 0$. Indeed, the corollary implies that for
every $M$ in $\mathcal{M}_{\textsl{anon}}$ there exists an aMPNN $M'$ in
$\mathcal{M}{}^{-}_{\textsl{anon}}$ such that $M\preceq M'$, and there exists an
$M''$ in $\mathcal{M}_{\textsl{anon}}$, possibly different from $M$, such that $M'\preceq M''$.
In fact, such an aMPNN $M''$, in this case is $M_{\textsl{WL}}$.
\subsection{Graph neural network-based anonymous MPNNs}
In this subsection we study the subclasses of aMPNNs arising from graph neural
network architectures. For convenience, let us write $\mathcal{M}_{\textsl{GNN}}
:= \mathcal{M}^{\textsl{sign}}_{\textsl{GNN}} \cup
\mathcal{M}^{\textsl{ReLU}}_{\textsl{GNN}}$.
We start by stating a direct consequence of Proposition~\ref{pro:eqstrongWL}. It
follows by observing that $\mathcal{M}_{\textsl{GNN}}$ is a subclass of
$\mathcal{M}_{\textsl{anon}}$ as presented in Example~\ref{ex:GNN}.
\begin{corollary}\label{corr:GNNwANO}
The class
$\mathcal{M}_{\textsl{GNN}}$ is weaker than $\mathcal{M}_{\textsl{anon}}$ and is thus also weaker than $\mathcal{M}_{\textsl{WL}}$.
\end{corollary}
More challenging is to show that $\mathcal{M}^{\textsl{sign}}_{\textsl{GNN}}$,
$\mathcal{M}^{\textsl{ReLU}}_{\textsl{GNN}}$ and $\mathcal{M}_{\textsl{WL}}$, and thus also
$\mathcal{M}_{\textsl{anon}}$, are equally strong. The following results are known.
\begin{theorem}[\cite{grohewl}] \label{thm:grohe_lower}
(i)~The classes $\mathcal{M}_{\textsl{GNN}}^{\textsl{sign}}$ and $\mathcal{M}_{\textsl{WL}}$
are equally strong. (ii)~The class $\mathcal{M}_{\textsl{GNN}}^{\textsl{ReLU}}$ is
weaker than $\mathcal{M}_{\textsl{WL}}$, and $\mathcal{M}_{\textsl{WL}}$ is weaker than
$\mathcal{M}_{\textsl{GNN}}^{\textsl{ReLU}}$, with a factor of two, i.e.,
$\mathcal{M}_{\textsl{WL}}\preceq_{\times 2}\mathcal{M}_{\textsl{GNN}}^{\textsl{ReLU}}$.
\end{theorem}
The reason for the factor of two in (ii) in Theorem~\ref{thm:grohe_lower} is due to
a simulation of the sign activation function by means of a two-fold application of
the ReLU function. We next show that this factor of two can be avoided. As a side
effect, we obtain a simpler aMPNN $M$ in $\mathcal{M}_{\textsl{GNN}}$, satisfying
$M _{\textsl{WL}} \preceq M$, than the one constructed in \cite{grohewl}. The proof
strategy is inspired by that of~\cite{grohewl}. Crucial in the proof is the notion
of row-independence modulo equality, which we define next.
\begin{definition}[Row-independence modulo equality]\label{def:label2}\
A labelling $\pmb{\ell}:V\to\mathbb{A}^s$ is \textit{row-independent modulo
equality} if the set of unique labels assigned by $\pmb{\ell}$ is linearly
independent. \hfill$\blacksquare$
\end{definition}
In what follows, we always assume that the initial labelling $\pmb{\nu}$ of $G$ is
row-independent modulo equality. One can always ensure this by extending the labels.
\begin{theorem}\label{thm:equalstrong}
The classes $\mathcal{M}_{\textsl{GNN}}^{\textsl{ReLU}}$ and $\mathcal{M}_{\textsl{WL}}$ are
equally strong.
\end{theorem}
\begin{proof}
We already know that $\mathcal{M}_{\textsl{GNN}}^{\textsl{ReLU}}$ is weaker than
$\mathcal{M}_{\textsl{WL}}$ (Theorem~\ref{thm:grohe_lower} and also
Corollary~\ref{corr:GNNwANO}). It remains to show that $\mathcal{M}_{\textsl{WL}}$ is weaker
than $\mathcal{M}_{\textsl{GNN}}^{\textsl{ReLU}}$. That is, given an aMPNN
$M_{\textsl{WL}}$, we need to construct an aMPNN $M$ in
$\mathcal{M}_{\textsl{GNN}}^{\textsl{ReLU}}$ such that
$\pmb{\ell}_{M}^{(t)}\sqsubseteq \pmb{\ell}_{M_{\textsl{WL}}}^{(t)}$, for all $t\geq
0$. We observe that since $\pmb{\ell}_{M_{\textsl{WL}}}^{(t)}\sqsubseteq
\pmb{\ell}_{M}^{(t)}$ for any $M$ in $\mathcal{M}_{\textsl{GNN}}^{\textsl{ReLU}}$,
this is equivalent to constructing an $M$ such that $\pmb{\ell}_{M}^{(t)}\equiv
\pmb{\ell}_{M_{\textsl{WL}}}^{(t)}$.
The proof is by induction on the number of computation rounds. The aMPNN $M$ in
$\mathcal{M}_{\textsl{GNN}}^{\textsl{ReLU}}$ that we will construct will use
message and update functions of the form:
\begin{equation}\textsc{Msg}^{(t)}\bigl(\mathbf{x},\mathbf{y},-,-):=\mathbf{y}\mathbf{W}^{(t)}
\text{ and }
\textsc{Upd}^{(t)}(\mathbf{x},\mathbf{y}):=\text{ReLU}\left(p\mathbf{x}\mathbf{W}^{(t)}+\mathbf{y} + \mathbf{b}^{(t)}\right)
\end{equation}
for some value $p\in\mathbb{A}$, $0<p<1$, weight matrix
$\mathbf{W}^{(t)}\in\mathbb{A}^{s_{t-1}\times s_t}$, and bias vector
$\mathbf{b}^{(t)}\in\mathbb{A}^{s_t}$. Note that, in contrast to aMPNNs of the
form~(\ref{eq:MPNN-GNN}), we only have one weight matrix per round, instead of two,
at the cost of introducing an extra parameter $p\in\mathbb{A}$. Furthermore, the
aMPNN constructed in \cite{grohewl} uses two distinct weight matrices in
$\mathbb{A}^{(s_{t-1} + s_0)\times (s_t + s_0)}$ (we come back to this at the end of
this section) whereas our weight matrices are elements of $\mathbb{A}^{s_{t-1}\times
s_t}$ and thus of smaller dimension.
The induction hypothesis is that $\pmb{\ell}^{(t)}_M\equiv
\pmb{\ell}_{M_{\textsl{WL}}}^{(t)}$ and that $\pmb{\ell}^{(t)}_M$ is row-independent
modulo equality.
For $t=0$, we have that for any $M\in \mathcal{M}_{\textsl{GNN}}^{\textsl{ReLU}}$,
$\pmb{\ell}_M^{(0)}=\pmb{\ell}_{M_{\textsl{WL}}}^{(0)}:=\pmb{\nu}$, by definition.
Moreover, $\pmb{\ell}_M^{(0)}$ is row-independent modulo equality because
$\pmb{\nu}$ is so, by assumption.
We next assume that up to round $t-1$ we have found weight matrices and bias vectors
for $M$ such that $\pmb{\ell}_M^{(t-1)}$ satisfies the induction hypothesis. We will
show that for round $t$ we can find a weight matrix
$\mathbf{W}^{(t)}\in\mathbb{A}^{s_{t-1}\times s_t }$ and bias vector
$\mathbf{b}^{(t)}\in\mathbb{A}^{s_t}$ such that $\pmb{\ell}_M^{(t)}$ also satisfies
the hypothesis.
Let $\mathbf{L}^{(t-1)}\in\mathbb{A}^{n\times s_{t-1}}$ denote the matrix consisting
of rows $(\pmb{\ell}_M^{(t-1)})_v$, for $v\in V$. Moreover, we denote by
$\mathsf{uniq}(\mathbf{L}^{(t-1)})$ a $(m\times s_{t-1})$-matrix consisting of the
$m$ unique rows in $\mathbf{L}^{(t-1)}$ (the order of rows is irrelevant). We denote
the rows in $\mathsf{uniq}(\mathbf{L}^{(t-1)})$ by
$\mathbf{a}_1,\ldots,\mathbf{a}_m\in\mathbb{A}^{s_{t-1}}$. By the induction
hypothesis, these rows are linearly independent. Following the same argument as
in~\cite{grohewl}, this implies that there exists an $(s_{t-1}\times m)$-matrix
$\mathbf{U}^{(t)}$ such that
$\mathsf{uniq}(\mathbf{L}^{(t-1)})\mathbf{U}^{(t)}=\mathbf{I}$. Let us denote by
$\mathbf{e}_1,\ldots,\mathbf{e}_m\in\mathbb{A}^m$ the rows of $\mathbf{I}$. In other
words, in $\mathbf{e}_i$, all entries are zero except for entry $i$, which holds
value $1$.
We consider the following intermediate labelling $\pmb{\mu}^{(t)}:V\to\mathbb{A}^m$
defined by
\begin{equation}
v\mapsto \left((\mathbf{A}+p\mathbf{I})\mathbf{L}^{(t-1)}\mathbf{U}^{(t)}\right)_{v}.\label{eq:labelmu}
\end{equation}
We know that for every vertex $v$, $(\pmb{\ell}_M^{(t-1)})_v$ corresponds to a
unique row $\mathbf{a}_i$ in $\mathsf{uniq}(\mathbf{L}^{(t-1)})$. We denote the
index of this row by $\rho(v)$. More specifically,
$(\pmb{\ell}_M^{(t-1)})_v=\mathbf{a}_{\rho(v)}$. Let $N_G(v,i):=\{u \st u\in N_G(v),
\rho(v)=i\}$. That is, $N_G(v,i)$ consists of all neighbours $u$ of $v$ which are
labelled as $\mathbf{a}_i$ by $\pmb{\ell}_M^{(t-1)}$. It is now readily verified
that the label $\pmb{\mu}^{(t)}_v$ defined in~(\ref{eq:labelmu}) is of the form
\begin{equation}
\pmb{\mu}^{(t)}_v=
p\mathbf{e}_{\rho(v)} + \sum_{i=1}^m |N_G(v,i)|\mathbf{e}_i. \label{eq:linearcomb}
\end{equation}
We clearly have that $\pmb{\ell}_{M_{\textsl{WL}}}^{(t)}\sqsubseteq\pmb{\mu}^{(t)}$. The converse also holds, as is shown in the following lemma.
\begin{lemma}
For any two vertices $v$ and $w$, we have that
$\pmb{\mu}^{(t)}_v=\pmb{\mu}^{(t)}_w$ implies
$(\pmb{\ell}_{M_{\textsl{WL}}}^{(t)})_v=(\pmb{\ell}_{M_{\textsl{WL}}}^{(t)})_w$.
\end{lemma}
\begin{proof}
We argue by contradiction. Suppose, for the sake of contradiction, that there exist two vertices $v,w\in V$ such that
\begin{equation}
\pmb{\mu}^{(t)}_{v}=\pmb{\mu}^{(t)}_{w} \text{ and } (\pmb{\ell}_{M_{\textsl{WL}}}^{(t)})_v\neq(\pmb{\ell}_{M_{\textsl{WL}}}^{(t)})_w \label{eq:contra-pwl}
\end{equation}
hold. We show that this is impossible for any value $p$ satisfying $0<p<1$.
(Recall from~\eqref{eq:linearcomb} that $\pmb{\mu}^{(t)}_v$ depends on $p$.)
We distinguish between the following two cases. If $ (\pmb{\ell}_{M_{\textsl{WL}}}^{(t)})_v\neq(\pmb{\ell}_{M_{\textsl{WL}}}^{(t)})_w$ then either
\begin{enumerate}[(i)]
\item \label{itm:case1} $(\pmb{\ell}_{M_{\textsl{WL}}}^{(t-1)})_v\neq(\pmb{\ell}_{M_{\textsl{WL}}}^{(t-1)})_w$; or
\item \label{itm:case2}
$(\pmb{\ell}_{M_{\textsl{WL}}}^{(t-1)})_v=(\pmb{\ell}_{M_{\textsl{WL}}}^{(t-1)})_w$
and
$
\{\!\!\{ (\pmb{\ell}_{M_{\textsl{WL}}}^{(t-1)})_u \st u \in N_G(v) \}\!\!\}\neq
\{\!\!\{(\pmb{\ell}_{M_{\textsl{WL}}}^{(t-1)})_u \st u \in N_G(w) \}\!\!\}.
$
\end{enumerate}
We first consider case~\ref{itm:case1}. Observe that
$(\pmb{\ell}_{M_{\textsl{WL}}}^{(t-1)})_v\neq(\pmb{\ell}_{M_{\textsl{WL}}}^{(t-1)})_w
$ implies that $(\pmb{\ell}^{(t-1)}_M)_{v}\neq (\pmb{\ell}_M^{(t-1)})_w$. This
follows from the induction hypothesis $\pmb{\ell}^{(t-1)}_M\equiv
\pmb{\ell}_{M_{\textsl{WL}}}^{(t-1)}$. It now suffices to observe that
$\pmb{\mu}^{(t)}_{v}=\pmb{\mu}^{(t)}_{w}$ implies that the corresponding linear
combinations, as described in~\eqref{eq:linearcomb}, satisfy:
$$
p\mathbf{e}_{\rho(v)} + \sum_{i=1}^m |N_G(v,i)|\mathbf{e}_i =
p\mathbf{e}_{\rho(w)} +
\sum_{i=1}^m |N_G(w,i)|\mathbf{e}_i.
$$
We can assume, without loss of generality, that
$(\pmb{\ell}^{(t-1)}_M)_{v}=\mathbf{a}_1$ and
$(\pmb{\ell}^{(t-1)}_M)_{w}=\mathbf{a}_2$. Recall that $\mathbf{a}_1$ and
$\mathbf{a}_2$ are two distinct labels. Then, the previous equality implies:
\[
\left(|N_G(v,1)|+p-|N_G(w,1)|\right)\mathbf{e}_1+
\left(|N_G(v,2)|-|N_G(w,2)|-p\right)\mathbf{e}_2 +
\sum_{i=3}^m \left(|N_G(v,i)|-|N_G(w,i)|\right)\mathbf{e}_i=0.
\]
Since $\mathbf{e}_1,\ldots,\mathbf{e}_m$ are linearly independent, this implies that
$|N_G(v,i)|-|N_G(w,i)|=0$ for all $i=3,\ldots,m$ and $|N_G(v,1)|+p-|N_G(w,1)|=0$ and
$|N_G(v,2)|-|N_G(w,2)|-p=0$. Since $|N_G(v,1)|-|N_G(w,1)|\in\mathbb{Z}$ and $0<p<1$,
this is impossible. We may thus conclude that case~\ref{itm:case1} cannot occur.
Suppose next that we are in case~\ref{itm:case2}. Recall that for
case~\ref{itm:case2}, we have that
$(\pmb{\ell}{}_{M_{\textsl{WL}}}^{(t-1)})_v=(\pmb{\ell}{}_{M_{\textsl{WL}}}^{(t-1)})_
w$ and thus also $(\pmb{\ell}{}_M^{(t-1)})_v=(\pmb{\ell}{}_M^{(t-1)})_w$. Using the
same notation as above, we may assume that
$(\pmb{\ell}{}_M^{(t-1)})_v=(\pmb{\ell}{}_M^{(t-1)})_w=\mathbf{a}_1$. In
case~\ref{itm:case2}, however, we have that
$
\{\!\!\{ (\pmb{\ell}{}_{M_{\textsl{WL}}}^{(t-1)})_{u} \st u \in N_G(v) \}\!\!\}\neq
\{\!\!\{ (\pmb{\ell}{}_{M_{\textsl{WL}}}^{(t-1)})_{u} \st u \in N_G(w) \}\!\!\}
$ and thus also
$
\{\!\!\{ (\pmb{\ell}{}_{M}^{(t-1)})_{u} \st u \in N_G(v) \}\!\!\}\neq
\{\!\!\{ (\pmb{\ell}{}_{M}^{(t-1)})_{u} \st u \in N_G(w) \}\!\!\}
$.
That is, there must exist a label assigned by $\pmb{\ell}{}_M^{(t-1)}$ that does not
occur the same number of times in the neighbourhoods of $v$ and $w$, respectively.
Suppose that this label is $\mathbf{a}_2$. The case when this label is
$\mathbf{a}_1$ can be treated similarly. It now suffices to observe that
$\pmb{\mu}^{(t)}_{v}=\pmb{\mu}^{(t)}_{w}$ implies that the corresponding linear
combinations, as described in~(\ref{eq:linearcomb}), satisfy:
$$
\left(|N_G(v,1)|+p\right)\mathbf{e}_1 +|N_G(v,2)|\mathbf{e}_2+\sum_{i=3}^m |N_G(v,i)|\mathbf{e}_i=
\left(|N_G(w,1)|+p\right)\mathbf{e}_1 +|N_G(w,2)|\mathbf{e}_2+\sum_{i=3}^m |N_G(w,i)|\mathbf{e}_i.
$$
Using a similar argument as before, based on the linear independence of
$\mathbf{e}_1,\ldots,\mathbf{e}_m$, we can infer that $|N_G(v,2)|=|N_G(w,2)|$. We
note, however, that $\mathbf{a}_2$ appeared a different number of times among the
neighbours of $v$ and $w$. Hence, also case~\ref{itm:case2} is ruled out and our
assumption~\eqref{eq:contra-pwl} is invalid. This implies
$\pmb{\mu}^{(t)}\sqsubseteq\pmb{\ell}_{M_{\textsl{WL}}}^{(t)}$, as desired and thus
concludes the proof of the lemma.
\end{proof}
From here, to continue with the proof of Theorem~\ref{thm:equalstrong}, we still
need to take care of the ReLU activation function. Importantly, its application
should ensure row-independence modulo equality and make sure the labelling
``refines'' $\pmb{\ell}^{(t)}_{M_{\textsl{WL}}}$. To do so, we again follow closely
the proof strategy of~\cite{grohewl}. More specifically, we will need an analogue of
the following result. In the sequel we denote by $\mathbf{J}$ a matrix with all
entries having value $1$ and whose size will be determined from the context.
\begin{lemma}[Lemma 9 from~\cite{grohewl}]\label{lem:signlemma9}
Let $\mathbf{C}\in \mathbb{A}^{m\times w}$ be a matrix in which all entries are
non-negative and all rows are pairwise disjoint. Then there exists a matrix
$\mathbf{X}\in\mathbb{A}^{w\times m}$ such that $\text{\normalfont
sign}(\mathbf{CX}-\mathbf{J})$ is a non-singular matrix in $\mathbb{A}^{m\times
m}$.
\end{lemma}
We prove the following for the ReLU function.
\begin{lemma}\label{lem:ReLUlemma9}
Let $\mathbf{C}\in \mathbb{A}^{m\times w}$ be a matrix in which all entries are
non-negative, all rows are pairwise disjoint and such that no row consists
entirely out of zeroes\footnote{Compared to Lemma~\ref{lem:signlemma9}, we
additionally require non-zero rows.}. Then there exists a matrix
$\mathbf{X}\in\mathbb{A}^{w\times m}$ and a constant $q\in\mathbb{A}$ such that
$\text{\normalfont ReLU}(\mathbf{CX}-q\mathbf{J})$ is a non-singular matrix in
$\mathbb{A}^{m\times m}$.
\end{lemma}
\begin{proof}
Let $C$ be the maximal entry in $\mathbf{C}$ and consider the column vector
$\mathbf{z}=(1,C,C^2,\ldots,C^{w-1})^{\textsc{t}}\in\mathbb{A}^{w\times 1}$. Then
each entry in $\mathbf{c}=\mathbf{C}\mathbf{z}\in\mathbb{A}^{m\times 1}$ is positive
and all entries in $\mathbf{c}$ are pairwise distinct. Let $\mathbf{P}$ be a
permutation matrix in $\mathbb{A}^{m\times m}$ such that
$\mathbf{c}'=\mathbf{P}\mathbf{c}$ is such that
$\mathbf{c}'=(c_1',c_2',\ldots,c_m')^{\textsc{t}}\in\mathbb{A}^{m\times 1}$ with
$c_1'> c_2'>\cdots > c_m'>0$. Consider
$\mathbf{x}=\left(\frac{1}{c_1'},\ldots,\frac{1}{c_m'}\right)\in \mathbb{A}^{1\times
m}$. Then, for $\mathbf{E}=\mathbf{c}'\mathbf{x}\in\mathbb{A}^{m\times m}$
$$
\mathbf{E}_{ij}=\frac{c_i'}{c_j'} \text{ and } \mathbf{E}_{ij}=\begin{cases} 1 & \text{if $i=j$}\\
>1 & \text{if $i<j$}\\
< 1 & \text{if $i>j$}.
\end{cases}
$$
Let $q$ be the greatest value in $\mathbf{E}$ smaller than $1$. Consider
$\mathbf{F}=\mathbf{E}- q\mathbf{J}$. Then,
$$
\mathbf{F}_{ij}=\frac{c_i'}{c_j'}- q \text{ and } \mathbf{F}_{ij}=\begin{cases} 1-q & \text{if $i=j$} \\
> 0 & \text{if $i<j$}\\
\leq 0 & \text{if $i>j$}.
\end{cases}
$$
As a consequence,
$$
\text{ReLU}(\mathbf{F})_{ij}=\begin{cases} 1-q & \text{if $i=j$}\\
>0 & \text{if $i<j$}\\
0 & \text{if $i>j$}.
\end{cases}
$$
This is an upper triangular matrix with (nonzero) value $1-q$ on its diagonal. It is
therefore non-singular.
We now observe that
$\mathbf{Q}\text{ReLU}(\mathbf{F})=\text{ReLU}(\mathbf{Q}\mathbf{F})$ for any row
permutation $\mathbf{Q}$. Furthermore, non-singularity is preserved under row
permutations and $\mathbf{Q}\mathbf{J}=\mathbf{J}$. Hence, if we define
$\mathbf{X}=\mathbf{z}\mathbf{x}$ and use the permutation matrix $\mathbf{P}$, then:
\begin{align*}
\mathbf{P}\text{ReLU}(\mathbf{C}\mathbf{X}-q\mathbf{J})&=
\text{ReLU}(\mathbf{P}\mathbf{C}\mathbf{z}\mathbf{x}-q\mathbf{P}\mathbf{J})=\text{ReLU}(\mathbf{E}-q\mathbf{J}) = \text{ReLU}(\mathbf{F}),
\end{align*}
and we have that $\text{ReLU}(\mathbf{C}\mathbf{X}-q\mathbf{J})$ is non-singular, as
desired. This concludes the proof of the lemma.
\end{proof}
We now apply this lemma to the matrix $\mathsf{uniq}(\mathbf{M}^{(t)})$, with
$\mathbf{M}^{(t)}\in\mathbb{A}^{n\times m}$ consisting of the rows
$\pmb{\mu}^{(t)}_v$, for $v\in V$. Inspecting the expression from
Equation~\eqref{eq:linearcomb} for $\pmb{\mu}^{(t)}_v$ we see that each row in
$\mathbf{M}^{(t)}$ holds non-negative values and no row consists entirely out of
zeroes. Let $\mathbf{X}^{(t)}$ and $q^{(t)}$ be the matrix and constant returned by
Lemma~\ref{lem:ReLUlemma9} such that
$\text{ReLU}\left(\mathsf{uniq}(\mathbf{M}^{(t)})\mathbf{X}^{(t)}-q^{(t)}\mathbf{J}\right)$ is an $m\times m$ non-singular matrix. We now define
$$
\pmb{\ell}_M^{(t)}:=\text{ReLU}\left(\mathbf{M}^{(t)}\mathbf{X}^{(t)}-q^{(t)}\mathbf{J}\right).$$
From the non-singularity of
$\text{ReLU}\left(\mathsf{uniq}(\mathbf{M}^{(t)})\mathbf{X}^{(t)}-q^{(t)}\mathbf{J}
\right)$ we can immediately infer that $\pmb{\ell}_M^{(t)}$ is row-independent modulo
equality. It remains to argue that
$\pmb{\ell}_M^{(t)}\equiv\pmb{\ell}_{M_{\textsl{WL}}}^{(t)}$. This now follows from
the fact that $\pmb{\mu}^{(t)}\equiv \pmb{\ell}_{M_{\textsl{WL}}}^{(t)}$ and each of
the $m$ unique labels assigned by $\pmb{\mu}^{(t)}$ uniquely corresponds to a row in
$\mathsf{uniq}(\mathbf{M}^{(t)})$, which in turn can be mapped bijectively to a row
in
$\text{ReLU}\left(\mathsf{uniq}(\mathbf{M}^{(t)})\mathbf{X}^{(t)}-q^{(t)}\mathbf{J}
\right)$. We conclude by observing that the desired weight matrices and bias vector
at round $t$ for $M$ are now given by
$\mathbf{W}^{(t)}:=\mathbf{U}^{(t)}\mathbf{X}^{(t)}$
and $\mathbf{b}^{(t)}:=-q^{(t)}\mathbf{1}$. This concludes the proof of Theorem~\ref{thm:equalstrong}.
\end{proof}
We remark that the previous proof can be used for
$\mathcal{M}_{\textsl{GNN}}^{\textsl{sign}}$ as well. One just has to use
Lemma~\ref{lem:signlemma9} instead of Lemma~\ref{lem:ReLUlemma9}. It is interesting
to note that the bias vector for the sign activation function in
Lemma~\ref{lem:signlemma9} is the same for every $t$. A similar statement holds for
the ReLU function. Indeed, we recall that we apply Lemma~\ref{lem:ReLUlemma9} to
$\mathsf{uniq}(\mathbf{M}^{(t)})$. For every $t$, the entries in this matrix are of
the form $i+p$ (which is smaller than $i+1$) or $i$, for $i\in \{1,2,\dots,n\}$.
Hence, for every $t$, the maximal entry (denoted by $C$ in the proof of
Lemma~\ref{lem:ReLUlemma9}) is upper bounded by $n+1$. The value $q^{(t)}$ relates
to the largest possible ratios, smaller than $1$, of elements in the matrix
constructed in Lemma~\ref{lem:ReLUlemma9}.
When the lemma is applied to an $m \times w$ matrix, this ratio is upper bounded by
$\frac{(n+1)^w-1}{(n+1)^w}$. Note that, since the lemma is applied to matrices
arising from $\pmb{\mu}^{(t)}$, $w$ will always be at most $n$. Hence, taking any
$q^{(t)}:=q$ for $\frac{(n+1)^n-1}{(n+1)^n}<q<1$ suffices. We can take $q$ to be
arbitrarily close to $1$, but not $1$ itself.
We can thus strengthen Theorem~\ref{thm:grohe_lower}, as follows. We denote by
$\mathcal{M}_{\textsl{GNN}^-}$ the class of aMPNNs using message and update
functions of the form:
\begin{equation}\textsc{Msg}^{(t)}\bigl(\mathbf{x},\mathbf{y},-,-):=\mathbf{y}\mathbf{W}^{(t)}
\text{ and }
\textsc{Upd}^{(t)}(\mathbf{x},\mathbf{y}):=\sigma\left(p\mathbf{x}\mathbf{W}^{(t)}+\mathbf{y} -q \mathbf{1}\right), \label{eq:GNNWL}
\end{equation}
parameterised with values $p,q\in\mathbb{A}$, $0\leq p,q\leq 1$ and weight matrices
$\mathbf{W}^{(t)}\in\mathbb{A}^{s_{t-1}\times s_t}$, and where $\sigma$ can be
either the sign or ReLU function.
\begin{corollary}\label{cor:pluspstrongwl}
The class $\mathcal{M}_{\textsl{GNN}^-}$ is equally strong as $\mathcal{M}_{\textsl{GNN}}$ and is equally strong as $\mathcal{M}_{\textsl{WL}}$.\qed
\end{corollary}
We remark that the factor two, needed for the ReLU activation function in
Theorem~\ref{thm:grohe_lower}, has been eliminated. Phrased in terms of graph neural
networks, an aMPNN in $\mathcal{M}_{\textsl{GNN}^-}$ is of the form
\begin{equation}
\mathbf{L}^{(t)}=\sigma\left((\mathbf{A}+p\mathbf{I})\mathbf{L}^{(t-1)}\mathbf{W}^{(t)}-q\mathbf{J}\right), \label{GNN:plusp}
\end{equation}
and, thus, these suffice to implement the WL algorithm. It would be interesting to
see how graph neural networks defined by~(\ref{GNN:plusp}), with learnable
parameters $p$ and $q$, perform in practice.
In contrast, if one inspects the proof in ~\cite[pg. 14, Appendix]{grohewl}, even
for the sign activation function, the graph neural network given to implement the WL
algorithm has the more complicated form:
$$
\left(\mathbf{L}^{(0)},\mathbf{L}^{(t)}\right):=\sigma\left(\left(\mathbf{L}^{(0)},\mathbf{L}^{(t-1)}\right)\begin{pmatrix}
\mathbf{I} & \mathbf{0}\\
\mathbf{0} & \mathbf{0}\end{pmatrix}
+\mathbf{A}\left(\mathbf{L}^{(0)},\mathbf{L}^{(t-1)}\right)
\begin{pmatrix}
\mathbf{0} & \mathbf{0}\\
\mathbf{0} & \mathbf{W}^{(t)}\end{pmatrix}-
\left(\mathbf{0}, \mathbf{J}\right)
\right).
$$
We thus have obtained a simpler class of aMPNNs, $\mathcal{M}_{\textsl{GNN}^-}$,
which is equally strong as $\mathcal{M}_{\textsl{WL}}$. We will see in the next section that
the parameter $p$ also plays an important role for degree-aware aMPNNs.
\begin{table}
\hrule
\hspace*{1ex}
\caption{Various graph neural network formalisms, as reported in e.g.,\cite{kipf-loose,Wu2019,DBLP:journals/corr/abs-1905-03046}, which correspond to degree-aware MPNNs. We implicitly assume the presence of a bias matrix $\mathbf{B}^{(t)}$ consisting of copies of the same row $\mathbf{b}^{(t)}$.}
\label{tab:dMPNNs}
\centering
\begin{tabular}{ll}
dGNN$_1$: &
$\mathbf{L}^{(t)}:=\sigma\left(\mathbf{D}^{-1}\mathbf{A}\mathbf{L}^{(t-1)}\mathbf{W}^{(t)}\right)$ \\
dGNN$_2$: &
$\mathbf{L}^{(t)}:=\sigma\left(\mathbf{D}^{\nicefrac{-1}{2}}\mathbf{A}\mathbf{D}^{\nicefrac{-1}{2}}\mathbf{L}^{(t-1)}\mathbf{W}^{(t)}\right)$\\
dGNN$_3$: &
$
\mathbf{L}^{(t)}:=\sigma\left((\mathbf{D}+\mathbf{I})^{-1}(\mathbf{A}+\mathbf{I})\mathbf{L}^{(t-1)}\mathbf{W}^{(t)}\right)$ \\
dGNN$_4$: &
$
\mathbf{L}^{(t)}:=\sigma\Bigl(\bigl(\mathbf{D}+\mathbf{I}\bigr)^{\nicefrac{-1}{2}} (\mathbf{A}+\mathbf{I})\bigl(\mathbf{D}+\mathbf{I}\bigr)^{\nicefrac{-1}{2}} \mathbf{L}^{(t-1)}\mathbf{W}^{(t)}\Bigr)
$ \\
dGNN$_5$: &
$
\mathbf{L}^{(t)}:=\sigma\left((\mathbf{D}^{\nicefrac{-1}{2}}\mathbf{A}\mathbf{D}^{\nicefrac{-1}{2}}+\mathbf{I})\mathbf{L}^{(t-1)}\mathbf{W}^{(t)}\right)$\\
dGNN$_6$: &
$\mathbf{L}^{(t)}:=\sigma\left((r\mathbf{I}+(1-r)\mathbf{D})^{\nicefrac{-1}{2}}(\mathbf{A}+p\mathbf{I})(r\mathbf{I}+(1-r)\mathbf{D})^{\nicefrac{-1}{2}}\mathbf{L}^{(t-1)}\mathbf{W}^{(t)}\right)$
\end{tabular}
\hspace*{1ex}
\hrule
\end{table}
\section{The distinguishing power of degree-aware MPNNs}\label{sec:dMPNNs}
In this section we compare various classes of degree-aware MPNNs in terms of their
distinguishing power. We recall that degree-aware MPNNs (dMPNNs for short) have
message functions that depend on the labels and degrees of vertices. To compare
these classes we use Definition~\ref{def:classesweak} and also
Definition~\ref{def:classg}. In the latter definition we will be interested in the
function $g(n) = n+1$. That is, when comparing classes of dMPNNs we consider the
notions of being weaker or stronger with $1$ step ahead.
We will also compare degree-aware MPNNs with anonymous MPNNs. Recall that by
Theorem~\ref{thm:anonymous} all classes of anonymous MPNNs considered in
Section~\ref{sec:anonymous} are equivalent for $\equiv$. In particular, they are all
equivalent to the class $\mathcal{M}_{\textsl{WL}}$. Therefore, instead of comparing
a class $\mathcal{M}$ of dMPNNs with all classes considered in
Section~\ref{sec:anonymous} it suffices to compare it with
$\mathcal{M}_{\textsl{WL}}$. For example, if $\mathcal{M}_{\textsl{WL}} \preceq_g
\mathcal{M}$ then the same relationship to $\mathcal{M}$ holds for all classes
in Section~\ref{sec:anonymous} that are equivalent to $\mathcal{M}_{\textsl{WL}}$. Similarly,
for when $\mathcal{M} \preceq_g \mathcal{M}_{\textsl{WL}}$ holds.
Quintessential examples of degree-aware MPNNs are the popular graph convolutional
networks, as introduced by~\cite{kipf-loose}. These are of the form:
$$
\mathbf{L}^{(t)}:=\sigma\Bigl(\bigl(\mathbf{D}+\mathbf{I}\bigr)^{-1/2} (\mathbf{A}+\mathbf{I})\bigl(\mathbf{D}+\mathbf{I}\bigr)^{-1/2} \mathbf{L}^{(t-1)}\mathbf{W}^{(t)}\Bigr),
$$
as already described and phrased as dMPNNs in Example~\ref{ex:KipfasMPNN}. In fact,
many commonly used graph neural networks use degree information. We list a couple of
such formalisms in Table~\ref{tab:dMPNNs}. It is easily verified that these can all
be cast as dMPNNs along the same lines as Example~\ref{ex:KipfasMPNN}. We detail
this later in this section.
We consider the following classes of dMPNNs. First, we recall that
$\mathcal{M}_{\textsl{deg}}$ is the class of degree-aware MPNNs. Furthermore, for
$i\in \{1,2,\dots,6\}$, we define $\mathcal{M}_{\textsl{dGNN}_i}$ as the class of
dMPNNs originating from a GNN of the form dGNN$_i$, from Table~\ref{tab:dMPNNs}, by
varying the weight matrices $\mathbf{W}^{(t)}$ and, when applicable, the bias
$\mathbf{B}^{(t)}$ and parameters $p$, $r$. The following is our main result for
this section.
\begin{theorem}\label{thm:dmpnn}
For the class of degree-aware MPNNs:
\begin{enumerate}
\item $\mathcal{M}_{\textsl{WL}} \preceq \mathcal{M}_{\textsl{deg}}$ and $\mathcal{M}_{\textsl{deg}} \not \preceq \mathcal{M}_{\textsl{WL}}$;
\item $\mathcal{M}_{\textsl{deg}} \preceq_{+1} \mathcal{M}_{\textsl{WL}}$.
\end{enumerate}
For the architectures from Table~\ref{tab:dMPNNs}:
\begin{enumerate}
\item[3.] $\mathcal{M}_{\textsl{dGNN}_i} \not \preceq \mathcal{M}_{\textsl{WL}}$ for $i = 2,4,5,6$ and $\mathcal{M}_{\textsl{dGNN}_i} \preceq \mathcal{M}_{\textsl{WL}}$ for $i = 1,3$;
\item[4.] $\mathcal{M}_{\textsl{WL}} \not \preceq \mathcal{M}_{\textsl{dGNN}_i}$ for $1 \le i \le 5$ and $\mathcal{M}_{\textsl{WL}} \preceq \mathcal{M}_{\textsl{dGNN}_6}$.
\end{enumerate}
\end{theorem}
We prove this theorem in the following subsections by providing the relationships that are summarised in Figure~\ref{fig:dmpnn}.
\begin{figure}[t]
\centering
\begin{tikzpicture}[node distance=0.9cm]
\node (lwl) {$\mathcal{M}_{\textsl{WL}}$};
\node[right= of lwl] (deg) {$\mathcal{M}_{\textsl{deg}}$};
\node[above right=1.2cm of deg] (rwl) {$\mathcal{M}_{\textsl{WL}}$};
\node[below right=1.2cm of deg] (rwl2) {$\mathcal{M}_{\textsl{WL}}$};
\node[right = 3cm of deg] (blwl) {$\mathcal{M}_{\textsl{WL}}$};
\node[above right= 1.5cm and 1.75cm of blwl] (gnn1) {$\mathcal{M}_{\textsl{dGNN}_1}, \mathcal{M}_{\textsl{dGNN}_3}$};
\node[right= 2.3cm of blwl] (gnn6) {$\mathcal{M}_{\textsl{dGNN}_6}$};
\node[below right= 1.5cm and 1cm of blwl] (gnn2) {$\mathcal{M}_{\textsl{dGNN}_2},\mathcal{M}_{\textsl{dGNN}_4},\mathcal{M}_{\textsl{dGNN}_5}$};
\node[above right= 1.5cm and 1cm of gnn2] (brwl) {$\mathcal{M}_{\textsl{WL}}$};
\path
(lwl) edge[draw=none] node[label=above:\scriptsize{ Prop.~\ref{prop:notweaker}}] {$\preceq$} (deg)
(deg) edge[draw=none] node[rotate=45,align=center]{\scriptsize{ Prop.~\ref{prop:notweaker}} \\ $\not \preceq$} (rwl)
(deg) edge[draw=none] node[rotate=-45,align=center]{\scriptsize{ Prop.~\ref{prop:onestep}} \\ $\preceq_{+1}$} (rwl2)
(blwl) edge[draw=none] node[above,rotate=45,align=center] {\scriptsize{ Prop.~\ref*{prop:notasstrong}} \\ $\not \preceq$} (gnn1)
(blwl) edge[draw=none] node[label=above:\scriptsize{ Prop.~\ref{prop:indeed-wl-power}}] {$\preceq$} (gnn6)
(blwl) edge[draw=none] node[below,rotate=-45,align=center] {\scriptsize{ Prop.~\ref*{prop:notasstrong}} \\ $\not \preceq$} (gnn2)
(gnn1) edge[draw=none] node[above,rotate=-45,align=center] {\scriptsize{ Cor.~\ref*{ex:landr}} \\ $\preceq$} (brwl)
(gnn6) edge[draw=none] node[label=above:\scriptsize{Prop.~\ref{prop:notweaker}}] {$\not \preceq$} (brwl)
(gnn2) edge[draw=none] node[below,rotate=45,align=center] {\scriptsize{ Prop.~\ref*{prop:notweaker}} \\ $\not \preceq$} (brwl)
;
\end{tikzpicture}
\caption{Summary of results comparing degree-aware MPNNs in Theorem~\ref{thm:dmpnn}. We note that Proposition~\ref{prop:notweaker} shows only $\mathcal{M}_{\textsl{dGNN}_4} \not \preceq \mathcal{M}_{\textsl{WL}}$, but $\mathcal{M}_{\textsl{dGNN}_2},\mathcal{M}_{\textsl{dGNN}_5},\mathcal{M}_{\textsl{dGNN}_6} \not \preceq \mathcal{M}_{\textsl{WL}}$ can be easily inferred from it.}\label{fig:dmpnn}
\end{figure}
\subsection{General degree-aware MPNNs}
We first focus on the relation between the WL
algorithm and dMPNNs in general. More specifically, we start with the first item in Theorem~\ref{thm:dmpnn}.
As part of the proof we show that $\mathcal{M}_{\textsl{dGNN}_4} \not \preceq \mathcal{M}_{\textsl{WL}}$. We can similarly show that
$\mathcal{M}_{\textsl{dGNN}_2},\mathcal{M}_{\textsl{dGNN}_5},\mathcal{M}_{\textsl{dGNN}_6} \not \preceq \mathcal{M}_{\textsl{WL}}$, hereby also settling the first part of the third item in Theorem~\ref{thm:dmpnn}.
\begin{proposition}\label{prop:notweaker}
The class $\mathcal{M}_{\textsl{WL}}$ is weaker than $\mathcal{M}_{\textsl{deg}}$; but the
class $\mathcal{M}_{\textsl{deg}}$ is not weaker than $\mathcal{M}_{\textsl{WL}}$.
\end{proposition}
\begin{proof}
To prove the first part of the claim notice that $\mathcal{M}_{\textsl{anon}}$ is weaker than
$\mathcal{M}_{\textsl{deg}}$, simply because any aMPNN is a dMPNN. Then the result
follows from Theorem~\ref{thm:anonymous}.
For the second part it suffices to provide a dMPNN $M$ and a labelled graph $(
G,\pmb{\nu})$ such that there exists a round $t\geq 0$ for which
$\pmb{\ell}_{M_{\textsl{WL}}}^{(t)}\not\sqsubseteq \pmb{\ell}_M^{(t)}$ holds. We
construct such an $M$ originating from a GCN~\cite{kipf-loose} defined in
Example~\ref{ex:KipfasMPNN}. That is, $M$ is a dMPNN in
$\mathcal{M}_{\textsl{dGNN}_4}$. Consider the labelled graph $( G,\pmb{\nu})$ with
vertex labelling $\pmb{\nu}_{v_1}=\pmb{\nu}_{v_2}=(1,0,0)$,
$\pmb{\nu}_{v_3}=\pmb{\nu}_{v_6}=(0,1,0)$ and
$\pmb{\nu}_{v_4}=\pmb{\nu}_{v_5}=(0,0,1)$, and edges $\{v_1,v_3\}$, $\{v_2,v_3\}$,
$\{v_3,v_4\}$, $\{v_4,v_5\}$, and $\{v_5,v_6\}$, as depicted in
Figure~\ref{fig:graph1}.
\begin{figure}[ht]
\centering
\begin{tikzpicture}
\node[fill=black,circle,inner sep=2pt,align=center,label=below:{$v_1$}] (v1) {};
\node[below=0.5cm of v1,fill=black,circle,inner sep=2pt,align=center,label=below:{$v_2$}] (v2) {};
\node[below right = 0.25cm and 1cm of v1,fill=black,circle,inner sep=2pt,align=center,label=below:{$v_3$}] (v3) {};
\node[right = 1cm of v3,fill=black,circle,inner sep=2pt,align=center,label=below:{$v_4$}] (v4) {};
\node[right = 1cm of v4,fill=black,circle,inner sep=2pt,align=center,label=below:{$v_5$}] (v5) {};
\node[right = 1cm of v5,fill=black,circle,inner sep=2pt,align=center,label=below:{$v_6$}] (v6) {};
\path
(v1) edge[-] (v3)
(v2) edge[-] (v3)
(v3) edge[-] (v4)
(v4) edge[-] (v5)
(v5) edge[-] (v6)
;
\end{tikzpicture}
\caption{Graph $G$.}\label{fig:graph1}
\end{figure}
Recall that $\pmb{\ell}^{(0)} = \pmb{\nu}$ and
\begin{equation*}
(\pmb{\ell}^{(1)}_M)_{v}:=\text{ReLU}\left(\left(\frac{1}{1+d_v}\right)\pmb{\ell}_v^{(0)}\mathbf{W}^{(1)} + \sum_{u\in N_G(v)} \left(\frac{1}{\sqrt{1+d_v}}\right)\left(\frac{1}{\sqrt{1+d_u}}\right)\pmb{\ell}^{(0)}_u\mathbf{W}^{(1)}\right).
\end{equation*}
We next define $\mathbf{W}^{(1)}:=\left(\begin{smallmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{smallmatrix}\right)$.
It can be verified that
$$
\pmb{\ell}_M^{(1)}=
\begin{pmatrix}
\frac{1}{2} & 0\vphantom{\frac{1}{2\sqrt{2}}} & 0\\
\frac{1}{2} & 0\vphantom{\frac{1}{2\sqrt{2}}} & 0\\
0\vphantom{\frac{1}{2\sqrt{2}}} & \frac{1}{4} & 0\\
0\vphantom{\frac{1}{2\sqrt{2}}} & 0& \frac{1}{3} \\
0\vphantom{\frac{1}{2\sqrt{2}}} & 0 & \frac{1}{3}\\
0\vphantom{\frac{1}{2\sqrt{2}}} & \frac{1}{2}& 0
\end{pmatrix} +
\begin{pmatrix}
0 & \frac{1}{2\sqrt{2}} & 0\\
0 & \frac{1}{2\sqrt{2}} & 0\\
\frac{1}{\sqrt{2}} & 0 & \frac{1}{2\sqrt{3}}\\
0 & \frac{1}{2\sqrt{3}}& \frac{1}{3} \\
0 & \frac{1}{\sqrt{6}} & \frac{1}{3}\\
0 & 0& \frac{1}{\sqrt{6}}
\end{pmatrix} =
\begin{pmatrix}
\frac{1}{2} & \frac{1}{2\sqrt{2}}& 0\\
\frac{1}{2} & \frac{1}{2\sqrt{2}}& 0\\
\frac{1}{\sqrt{2}} & \frac{1}{4}& \frac{1}{
2\sqrt{3}}\\
0 & \frac{1}{2\sqrt{3}}& \frac{2}{
3}\\
0 & \frac{1}{\sqrt{6}}& \frac{2}{
3}\\
0 & \frac{1}{2}& \frac{1}{
\sqrt{6}}
\end{pmatrix}.
$$
We observe that $(\pmb{\ell}_M^{(1)})_{v_4}\neq (\pmb{\ell}_M^{(1)})_{v_5}$. We
note, however, that $
(\pmb{\ell}_{M_\textsl{WL}}^{(1)})_{v_4}=\textsc{Hash}\bigl((0,0,1),\{\!\!\{(0,0,1),(0,1
,0)\}\!\!\}\bigr)=(\pmb{\ell}_{M_\textsl{WL}}^{(1)})_{v_5}$. Hence,
$\pmb{\ell}_{M_{\textsl{WL}}}^{(1)}\not\sqsubseteq \pmb{\ell}_M^{(1)}$.
\end{proof}
The rest of this section is devoted to prove the second item in Theorem~\ref{thm:dmpnn}.
\begin{proposition}\label{prop:onestep}
$\mathcal{M}_{\textsl{deg}} \preceq_{+1} \mathcal{M}_{\textsl{WL}}$
\end{proposition}
We will need the following lemma that states that anonymous MPNNs can compute the degrees of vertices in the first round of computation.
\begin{lemma}\label{lem:countdeg}
Let $( G,\pmb{\nu})$ be a labelled graph with $\pmb{\nu}:V\to \mathbb{A}^s$. There
exists an aMPNN $M_{d}$ such that
$(\pmb{\ell}_{M_d}^{(1)})_v=(\pmb{\nu}_v,d_v)\in\mathbb{A}^{s+1}$ for every vertex
$v$ in $V$.
\end{lemma}
\begin{proof}
We define the aMPNN $M_d$ with the following message and update functions. For each
$\mathbf{x}$, $\mathbf{y} \in \mathbb{A}^{s}$, $z\in\mathbb{A}$, and vertices $v,
u\in N_G(v)$ we define:
$$
\textsc{Msg}^{(1)}(\mathbf{x},\mathbf{y},-,-):= 1
\text{ and } \textsc{Upd}^{(1)}(\mathbf{x},z):=
\left(\mathbf{x},z\right).
$$
Then,
$
\mathbf{m}_v^{(1)}:=\sum_{u\in N_G(v)} 1 = d_v$ and $(\pmb{\ell}_{M_d}^{(1)})_v:=\textsc{Upd}^{(1)}(\pmb{\nu}_v,d_v)=(\pmb{\nu}_v,d_v)\in\mathbb{A}^{s+1}$,
as desired.
\end{proof}
We are now ready to prove Proposition~\ref{prop:onestep}.
\begin{proof}[Proof of Proposition~\ref{prop:onestep}]
By Theorem~\ref{thm:anonymous} it suffices to prove that the class
$\mathcal{M}_{\textsl{deg}}$ is weaker than $\mathcal{M}_{\textsl{anon}}$, with $1$ step
ahead. Let $( G,\pmb{\nu})$ be a labelled graph with
$\pmb{\nu}:V\to\mathbb{A}^{s_0}$. Take an arbitrary dMPNN $M_1$ such that for every
round $t\geq 1$ the message function is
$$
\textsc{Msg}_{M_1}^{(t)}(\mathbf{x},\mathbf{y},d_v,d_u) \in\mathbb{A}^{s_t'}$$
and $\textsc{Upd}_{M_1}^{(t)}(\mathbf{x},\mathbf{z})$ is the update function.
We construct an aMPNN $M_2$ such that $\pmb{\ell}_{M_2}^{(t+1)}\sqsubseteq
\pmb{\ell}_{M_1}^{(t)}$ holds, as follows. We denote the message and update
functions of $M_2$ by $\textsc{Msg}_{M_2}^{(t)}$ and $\textsc{Upd}_{M_2}^{(t)}$,
respectively. We will keep as an invariant {\bf(I1)} stating that for all $v$ if we
have $\mathbf{x}' = (\pmb{\ell}_{M_1}^{(t-1)})_v \in \mathbb{A}^{s_{t-1}}$ then
$\mathbf{x} = (\mathbf{x}',d_v) = (\pmb{\ell}_{M_2}^{(t)})_v \in \mathbb{A}^{s_{t-1}
+ 1}$.
For $t=1$, we let $\textsc{Msg}_{M_2}^{(1)}$ and $\textsc{Upd}_{M_2}^{(1)}$ be the
functions defined by Lemma~\ref{lem:countdeg}. As a consequence,
$(\pmb{\ell}_{M_2}^{(1)})_v=(\pmb{\nu}_v,d_v)\in\mathbb{A}^{s_0+1}$ for every vertex
$v$. We clearly have that $\pmb{\ell}_{M_2}^{(1)}\sqsubseteq \pmb{\ell}_{M_1}^{(0)}$
and the invariant {\bf(I1)} trivially holds.
For $t\geq 2$, we define the message and update functions of $M_2$ as follows:
\[
\textsc{Msg}_{M_2}^{(t)}(\mathbf{x},\mathbf{y},-,-):=\textsc{Msg}_{M_1}^{(t-1)}(\mathbf{x}',\mathbf{y}',x,y)
\]
where $\mathbf{x} = (\mathbf{x}',x)$ and $\mathbf{y} = (\mathbf{y}',y)$ and by
invariant {\bf (I1)} $x = d_v$ and $y = d_u$. Notice that the message function
remains anonymous as $d_u$ and $d_v$ are not obtained by setting $f(v)=d_v$ and
$f(u)=d_u$ but instead were computed once by the first message aggregation and
encoded in the labels of $v$ and $u$. The update function is defined as follows:
\[
\textsc{Upd}_{M_2}^{(t)}(\mathbf{x},\mathbf{z}):=\left(\textsc{Upd}_{M_1}^{(t-1)}(\mathbf{x}',\mathbf{z}'),x\right)\in\mathbb{A}^{s_{t-1}+1},
\]
where $\mathbf{x} = (\mathbf{x}',x)$ and by invariant {\bf (I1)} $x = d_v$. In other
words, in each round $t\geq 2$, $M_2$ extracts the degrees from the last entries in
the labels and simulates round $t-1$ of $M_1$. It is readily verified that
$\pmb{\ell}_{M_2}^{(t)}\sqsubseteq \pmb{\ell}_{M_1}^{(t-1)}$ for every $t$, as
desired and that the invariant {\bf (I1)} holds.
\end{proof}
In particular it follows from Proposition~\ref{prop:onestep} that for the dMPNN $M$
constructed in the proof of Proposition~\ref{prop:notweaker} it holds that
$\pmb{\ell}_{M_{\textsl{WL}}}^{(2)}\sqsubseteq \pmb{\ell}_M^{(1)}$.
\subsection{Graph neural network-based degree-aware MPNNs}
We next consider the relation between the WL algorithm and dMPNNs that originate
from graph neural networks as those listed in Table~\ref{tab:dMPNNs}. More
specifically, we consider the following general graph neural network architecture
\begin{equation}
\mathbf{L}^{(t)}:=\sigma\left(\mathbf{L}^{(t-1)}\mathbf{W}_1^{(t)}+\mathsf{diag}(\mathbf{g})(\mathbf{A}+p\mathbf{I})\mathsf{diag}(\mathbf{h})\mathbf{L}^{(t-1)}\mathbf{W}_2^{(t)} + \mathbf{B}^{(t)}\right), \label{eq:dGNN}
\end{equation}
where $p\in\mathbb{A}$ is parameter satisfying $0\leq p\leq 1$, $\mathbf{W}_1^{(t)}$
and $\mathbf{W}_2^{(t)}$ are learnable weight matrices in $\mathbb{A}^{s_{t-1}\times
s_{t}}$, $\mathbf{B}^{(t)}$ is a bias matrix consisting of $n$ copies of the same
row $\mathbf{b}^{(t)}$, and $\mathsf{diag}(\mathbf{g})$ and
$\mathsf{diag}(\mathbf{h})$ are positive diagonal matrices in $\mathbb{A}^{n\times
n}$ obtained by putting the vectors $\mathbf{g}$ and $\mathbf{h}$ in $\mathbb{A}^n$
on their diagonals, respectively. We only consider vectors $\mathbf{g}$ and
$\mathbf{h}$ which are \textit{degree-determined}. That is, when $d_v=d_w$ then
$\mathbf{g}_v=\mathbf{g}_w$ and $\mathbf{h}_v=\mathbf{h}_w$ for all vertices $v$ and
$w$. Furthermore, $\sigma$ is either the sign or ReLU non-linear activation function.
It is readily verified that all graph neural networks mentioned so far can be seen
as special cases of~(\ref{eq:dGNN}). Moreover, graph neural networks of the
form~(\ref{eq:dGNN}) can be cast as dMPNNs. We denote the resulting class of dMPNNs
by $\mathcal{M}_{\textsl{dGNN}}$. The reason that one obtains dMPNNs is because of
the degree-determinacy assumption. More specifically, degree-determinacy implies that
$$
\mathbf{g}=(g(d_{v_1}),g(d_{v_2}),\ldots,g(d_{v_n}))
\text{ and }
\mathbf{h}=(h(d_{v_1}),h(d_{v_2}),\ldots,h(d_{v_n}))
$$
for some functions $g:\mathbb{N}^+\to \mathbb{A}^+$ and
$h:\mathbb{N}^+\to\mathbb{A}^+$.
\begin{example}
The GCN architecture of~\cite{kipf-loose} corresponds to graph neural networks of
the form~(\ref{eq:dGNN}), with
$\mathbf{W}_1^{(t)}=\mathbf{0}\in\mathbb{A}^{s_{t-1}\times s_t}$, $p=1$,
$\mathbf{b}^{(t)}=\mathbf{0}\in\mathbb{A}^s$, and where $\mathbf{g}=\mathbf{h}$ are
defined by the function $g(n)=h(n)=(1+n)^{-1/2}$. \hfill$\blacksquare$
\end{example}
We define the class $\mathcal{M}_{\textsl{dGNN}}$ as the class of dMPNNs with
message and update functions of the form:
\begin{align}\textsc{Msg}^{(t)}\bigl(\mathbf{x},\mathbf{y},d_v,d_u)&:=\frac{1}{d_v}\left(
\mathbf{x}\mathbf{W}_1^{(t)} + pg(d_v)h(d_v)\mathbf{x}\mathbf{W}_2^{(t)} \right)+ g(d_v)h(d_u)\mathbf{y}\mathbf{W}_2^{(t)} \label{eq:dGNNmsg}
\intertext{ and }
\textsc{Upd}^{(t)}(\mathbf{x},\mathbf{y}):=\sigma\left(\mathbf{y}\right) \label{eq:dGNNupd}
\end{align}
for any $\mathbf{x},\mathbf{y}\in\mathbb{A}^{s_{t-1}}$,
$\mathbf{W}_1^{(t)}\in\mathbb{A}^{s_{t-1}\times
s_t}$,$\mathbf{W}_2^{(t)}\in\mathbb{A}^{s_{t-1}\times s_t}$, bias vector
$\mathbf{b}^{(t)}\in\mathbb{A}^{s_t}$, and non-linear activation function $\sigma$.
We note that this encoding is just a generalisation of the encoding of GCNs as
dMPNNs given in Example~\ref{ex:KipfasMPNN}.
We know from Proposition~\ref{prop:onestep} that the class
$\mathcal{M}_{\textsl{dGNN}}$ is weaker than $\mathcal{M}_{\textsl{WL}}$, with $1$ step
ahead. Indeed, it suffices to note that $\mathcal{M}_{\textsl{dGNN}}\subseteq
\mathcal{M}_{\textsl{deg}}$. In particular, the classes
$\mathcal{M}_{\textsl{dGNN}_1}$--$\mathcal{M}_{\textsl{dGNN}_6}$ corresponding
to the graph neural network architectures from Table~\ref{tab:dMPNNs} are all weaker
than $\mathcal{M}_{\textsl{WL}}$, with $1$ step ahead. Furthermore, in the proof of
Proposition~\ref{prop:notweaker} we have shown that the condition that
$\mathcal{M}_{\textsl{WL}}$ is $1$ step ahead is necessary for
$\mathcal{M}_{\textsl{dGNN}_4}$, and thus also for
$\mathcal{M}_{\textsl{dGNN}}$. We mentioned that one can provide similar examples
for $\mathcal{M}_{\textsl{dGNN}_2}$, $\mathcal{M}_{\textsl{dGNN}_5}$ and
$\mathcal{M}_{\textsl{dGNN}_6}$.
In contrast, we next show that the two remaining classes,
$\mathcal{M}_{\textsl{dGNN}_1}$ and $\mathcal{M}_{\textsl{dGNN}_3}$, are weaker
than $\mathcal{M}_{\textsl{WL}}$ (with no step ahead). The reason is that dMPNNs in these
classes are equivalent to dMPNNs that only use degree information \textit{after}
aggregation takes places. These in turn are equivalent to anonymous MPNNs. We first
show a more general result, related to graph neural networks of the
form~\eqref{eq:dGNN} in which $\mathsf{diag}(\mathbf{h})=\mathbf{I}$. In other
words, the function $h:\mathbb{N}^+\to\mathbb{A}$ underlying $\mathbf{h}$ is the
constant one function, i.e., $h(n)=1$ for all $n\in\mathbb{N}^+$.
\begin{proposition}\label{prop:dGNNc}
The subclass of $\mathcal{M}_{\textsl{dGNN}}$, in which the function $h$ is the
constant one function, is weaker than $\mathcal{M}_{\textsl{WL}}$.
\end{proposition}
\begin{proof}
We show that any MPNN $M$ in this class is an anonymous MPNNs. To see this, it
suffices to observe that any dMPNN in $\mathcal{M}_{\textsl{dGNN}}$, and thus also
$M$ in particular, is equivalent to a dMPNN with message and update functions
defined as follows. For every round $t \geq 1$, every
$\mathbf{x},\mathbf{y}\in\mathbb{A}^{s_{t-1}}$,
$\mathbf{z}=(\mathbf{z}',z)\in\mathbb{A}^{s_t+1}$, and every vertex $v$ and $u\in
N_G(v)$:
\begin{align}
\textsc{Msg}^{(t)}(\mathbf{x},\mathbf{y},d_v,d_u)&:=\left(h(d_u)\mathbf{y}\mathbf{W}_2^{(t)},1\right)\in\mathbb{A}^{s_t+1} \label{eq:h}
\intertext{ and }
\textsc{Upd}^{(t)}(\mathbf{x},\mathbf{z})&:=\sigma\left(\mathbf{x}\mathbf{W}_1^{(t)}+g(z)\mathbf{z}'+pg(z)h(z)\mathbf{x}\mathbf{W}_2^{(t)}+\mathbf{b}^{(t)}\right)\in\mathbb{A}^{s_t},
\end{align}
where $z\in\mathbb{A}$ will hold the degree information of the vertex under
consideration (i.e., $d_v$) after message passing. That is, we use a similar trick
as in Lemma~\ref{lem:countdeg}. Since we consider MPNNs in which $h(d_u)=1$, the
message function~(\ref{eq:h}) indeed only depends on $\mathbf{y}$. As a consequence,
$M$ is equivalent to an anonymous MPNN. From Theorem~\ref{thm:anonymous} and in
particular from $\mathcal{M}_{\textsl{anon}}\preceq \mathcal{M}_{\textsl{WL}}$, the proposition follows.
\end{proof}
The architectures $\mathcal{M}_{\textsl{dGNN}_1}$ and
$\mathcal{M}_{\textsl{dGNN}_3}$ from Table~\ref{tab:dMPNNs} clearly satisfy the
assumption in the previous proposition and hence $\mathcal{M}_{\textsl{dGNN}_1},
\mathcal{M}_{\textsl{dGNN}_3} \preceq \mathcal{M}_{\textsl{WL}}$.
We thus have shown the remaining part of the third item in Theorem~\ref{thm:dmpnn}.
\begin{corollary}\label{ex:landr}
The classes $\mathcal{M}_{\textsl{dGNN}_1}$ and $\mathcal{M}_{\textsl{dGNN}_3}$
are weaker than $\mathcal{M}_{\textsl{WL}}$.
\end{corollary}
To conclude, we investigate whether $\mathcal{M}_{\textsl{dGNN}}$ and its
subclasses $\mathcal{M}_{\textsl{dGNN}_1}$--$\mathcal{M}_{\textsl{dGNN}_6}$ are
stronger than $\mathcal{M}_{\textsl{WL}}$. For $\mathcal{M}_{\textsl{dGNN}}$ this follows
from Theorem~\ref{thm:anonymous}, stating in particular that
$\mathcal{M}_{\textsl{GNN}} \equiv \mathcal{M}_{\textsl{WL}}$, and from the
following remark.
\begin{remark}\label{rem:filips-second-try}
It holds that $\mathcal{M}_{\textsl{GNN}} \preceq \mathcal{M}_{\textsl{dGNN}}$.
\end{remark}
Indeed, we first note that the class $\mathcal{M}_{\textsl{GNN}}$ is not a
subclass of $\mathcal{M}_{\textsl{dGNN}}$ since these classes differ in the
message and update functions used. We observe, however, that
$\mathcal{M}_{\textsl{GNN}}$ corresponds to the subclass of
$\mathcal{M}_{\textsl{dGNN}}$ in which the functions $g$ and $h$ are the constant
one function, i.e., $g(n)=h(n)=1$ for all $n\in\mathbb{N}^+$, and moreover, $p=0$.
More precisely, for every MPNN $M$ in $\mathcal{M}_{\textsl{GNN}}$ there is an
MPNN $M'$ in $\mathcal{M}_{\textsl{dGNN}}$ such that $M\equiv M'$, from which
Remark~\ref{rem:filips-second-try} follows.
So, we know already that $\mathcal{M}_{\textsl{WL}}\preceq\mathcal{M}_{\textsl{dGNN}}$.
However, the aMPNN $M$ in $\mathcal{M}_{\textsl{GNN}}$ such that $M_{\textsl{WL}}
\preceq M$ holds, as constructed for Theorems~\ref{thm:grohe_lower}
and~\ref{thm:equalstrong}, does not comply with the forms of MPNNs corresponding to
the graph neural networks given in Table~\ref{tab:dMPNNs}. We next investigate which
classes $\mathcal{M}_{\textsl{dGNN}_i}$ are stronger that $\mathcal{M}_{\textsl{WL}}$.
We start with some negative results, hereby showing part of the fourth item in Theorem~\ref{thm:dmpnn}.
\begin{proposition}\label{prop:notasstrong}
None of the classes $\mathcal{M}_{\textsl{dGNN}_i}$, for $i\in \{1,2, \dots, 5\}$, are stronger than $\mathcal{M}_{\textsl{WL}}$.
\end{proposition}
\begin{proof}
The proof consists of a number of counterexamples related to the various classes of
dMPNNs under consideration. For convenience, we describe the counterexamples in
terms of graph neural networks rather than in their dMPNN form.
We first prove the proposition for classes of dMPNNs related to graph neural
networks of the form:
$$
\mathbf{L}^{(t)}:=\sigma\left(\mathsf{diag}(\mathbf{g})\mathbf{A}\mathsf{diag}(\mathbf{h})\mathbf{L}^{(t-1)}\mathbf{W}^{(t)}+\mathbf{B}^{(t)}\right).
$$
This includes $\mathcal{M}_{\textsl{dGNN}_i}$, for $i=1,2$. Consider the labelled
graph $( G_1,\pmb{\nu})$ with vertex labelling $\pmb{\nu}_{v_1}=(1,0,0)$,
$\pmb{\nu}_{v_2}=\pmb{\nu}_{v_3}=(0,1,0)$ and $\pmb{\nu}_{v_4}=(0,0,1)$, and edges
$\{v_1,v_2\}$, $\{v_1,v_3\}$, $\{v_4, v_2\}$ and $\{v_4,v_3\}$, as depicted in
Figure~\ref{fig:graphG1}.
\begin{figure}[ht]
\centering
\begin{tikzpicture}
\node[fill=black,circle,inner sep=2pt,align=center,label=below:{$v_2$}] (v2) {};
\node[below=1cm of v2,fill=black,circle,inner sep=2pt,align=center,label=below:{$v_3$}] (v3) {};
\node[below right = 0.5cm and 1cm of v2,fill=black,circle,inner sep=2pt,align=center,label=below:{$v_4$}] (v4) {};
\node[below left = 0.5cm and 1cm of v2,fill=black,circle,inner sep=2pt,align=center,label=below:{$v_1$}] (v1) {};
\path
(v1) edge[-] (v2)
(v1) edge[-] (v3)
(v4) edge[-] (v2)
(v4) edge[-] (v3)
;
\end{tikzpicture}
\caption{Graph $G_1$.}\label{fig:graphG1}
\end{figure}
By definition, $\mathbf{L}^{(0)}:=\left(\begin{smallmatrix}1 & 0 &0\\
0 & 1 &0\\
0 & 1 &0\\
0 & 0 &1 \end{smallmatrix}\right)$.
We note that $$(\pmb{\ell}_{M_{\textsl{WL}}}^{(1)})_{v_1}=\textsc{Hash}\bigl((1,0,0),\{\!\!\{(0,1,0),(0,1,0)\}\!\!\}\bigr)\neq
(\pmb{\ell}_{M_{\textsl{WL}}}^{(1)})_{v_4}=\textsc{Hash}\bigl((0,0,1),\{\!\!\{(0,1,0),(0,1,0)\}\!\!\}\bigr).$$ We next show that there exist no
$\mathbf{W}^{(1)},\mathbf{B}^{(1)}$ such that $\mathbf{L}^{(1)}\sqsubseteq
\pmb{\ell}_{M_{\textsl{WL}}}^{(1)}$. Indeed, since the degree of all vertices is $2$
the computation is quite simple
\allowdisplaybreaks
\begin{align*}
\mathbf{L}^{(1)}& :=\sigma\left(\mathsf{diag}(\mathbf{g})\begin{pmatrix}0 & 1 & 1 &0 \\
1 & 0 & 0 & 1\\
1 & 0 & 0 & 1\\
0 & 1 & 1 & 0\\
\end{pmatrix}\mathsf{diag}(\mathbf{h})\mathbf{L}^{(0)}\mathbf{W}^{(1)} + \mathbf{B}^{(1)}\right)\\
& =\sigma\left(\begin{pmatrix}0 & g(2)h(2) & g(2)h(2) &0 \\
g(2)h(2)& 0 & 0 & g(2)h(2)\\
g(2)h(2) & 0 & 0 & g(2)h(2)\\
0 & g(2)h(2) & g(2)h(2) & 0\\
\end{pmatrix}
\begin{pmatrix}1 & 0 &0\\
0 & 1 &0\\
0 & 1 &0\\
0 & 0 &1 \end{pmatrix}\mathbf{W}^{(1)} + \mathbf{B}^{(1)}\right)\\
&=
\sigma\left(\begin{pmatrix}
0 & 2g(2)h(2)& 0\\
g(2)h(2) & 0 & g(2)h(2)\\
g(2)h(2) & 0 & g(2)h(2)\\
0 & 2g(2)h(2)& 0\\
\end{pmatrix}\mathbf{W}^{(1)} + \mathbf{B}^{(1)}\right).
\end{align*}
Finally, we recall that $\mathbf{B}^{(1)}$ consists of $n$ copies of the same
row. Hence, independently of the choice of $\mathbf{W}^{(1)}$ and
$\mathbf{B}^{(1)}$, vertices $v_1$ and $v_4$ will be assigned the same label,
and thus $\mathbf{L}^{(1)}\not\sqsubseteq \pmb{\ell}_{M_{\textsl{WL}}}^{(1)}$.
The second class of dMPNNs we consider are those related to graph neural networks of
the form:
$$
\mathbf{L}^{(t)}:=\sigma\left(\mathsf{diag}(\mathbf{g})(\mathbf{A}+\mathbf{I})\mathsf{diag}(\mathbf{h})\mathbf{L}^{(t-1)}\mathbf{W}^{(t)}+\mathbf{B}^{(t)}\right).
$$
This includes $\mathcal{M}_{\textsl{dGNN}_i}$, for $i=3,4$. Indeed, consider the
labelled graph $( G_2,\pmb{\nu})$ with one edge $\{v_1,v_2\}$, as depicted in
Figure~\ref{fig:graphG2}, and vertex labelling $\pmb{\nu}_{v_1}=(1,0)$ and
$\pmb{\nu}_{v_2}=(0,1)$.
\begin{figure}[ht]
\centering
\begin{tikzpicture}
\node[fill=black,circle,inner sep=2pt,align=center,label=below:{$v_1$}] (v1) {};
\node[right=1cm of v1,fill=black,circle,inner sep=2pt,align=center,label=below:{$v_2$}] (v2) {};
\path
(v1) edge[-] (v2)
;
\end{tikzpicture}
\caption{Graph $G_2$.}\label{fig:graphG2}
\end{figure}
By definition, $\mathbf{L}^{(0)}:=\left(\begin{smallmatrix}1 & 0\\ 0 &
1\end{smallmatrix}\right)$. We also note that
$$(\pmb{\ell}_{M_{\textsl{WL}}}^{(1)})_{v_1}=\textsc{Hash}\bigl((1,0),\{\!\!\{
(0,1)\}\!\!\}\bigr)\neq
(\pmb{\ell}_{M_{\textsl{WL}}}^{(1)})_{v_2}=\textsc{Hash}\bigl((0,1),\{\!\!\{(1,0)\}\!\!\}
\bigr).$$ We next show that there exist no $\mathbf{W}^{(1)},\mathbf{B}^{(1)}$ such
that $\mathbf{L}^{(1)}\sqsubseteq \pmb{\ell}_{M_{\textsl{WL}}}^{(1)}$. Indeed,
\allowdisplaybreaks
\begin{align*}
\mathbf{F}^{(1)}& :=\sigma\Biggl(\mathsf{diag}(\mathbf{g})\biggl(\begin{pmatrix}0 & 1 \\
1 & 0
\end{pmatrix}+
\begin{pmatrix}
1 & 0 \\
0 & 1 \end{pmatrix}\biggl)\mathsf{diag}(\mathbf{h})\mathbf{L}^{(0)}\mathbf{W}^{(1)} + \mathbf{B}^{(1)}\Biggr)\\
& =\sigma\Biggl(\begin{pmatrix}g(1) & 0 \\
0 & g(1)
\end{pmatrix}\begin{pmatrix}1 & 1 \\
1 & 1
\end{pmatrix}\begin{pmatrix}h(1) & 0 \\
0 & h(1)
\end{pmatrix}
\begin{pmatrix}1 & 0\\
0 & 1\end{pmatrix}\mathbf{W}^{(1)} + \mathbf{B}^{(1)}\Biggr)\\
&=
\sigma\Biggl(\begin{pmatrix}
g(1)h(1)& g(1)h(1)\\
g(1)h(1) & g(1)g(1)\\
\end{pmatrix}\mathbf{W}^{(1)} + \mathbf{B}^{(1)}\Biggr).
\end{align*}
Hence, independently of the choice of $\mathbf{W}^{(1)}$ and $\mathbf{B}^{(1)}$,
both vertices will be assigned the same label, and thus
$\mathbf{L}^{(1)}\not\sqsubseteq \pmb{\ell}_{M_{\textsl{WL}}}^{(1)}$.
Finally, we deal with the class $\mathcal{M}_{\textsl{dGNN}_5}$, i.e., dMPNNs
related to graph neural networks of the form
$$
\mathbf{L}^{(t)}:=\sigma\left((\mathbf{D}^{-1/2}\mathbf{A}\mathbf{D}^{-1/2}+\mathbf{I})\mathbf{L}^{(t-1)}\mathbf{W}^{(t)} + \mathbf{B}^{(t)}\right).$$
We consider the labelled graph $( G_3,\pmb{\nu})$ with vertex labelling
$\pmb{\nu}_{v_1}=\pmb{\nu}_{w_2}=\pmb{\nu}_{w_3}=(1,0,0)$,
$\pmb{\nu}_{w_1}=\pmb{\nu}_{v_2}=\pmb{\nu}_{v_3}=(0,1,0)$ and $\pmb{\nu}_{v_4} =
\pmb{\nu}_{v_5} = \pmb{\nu}_{w_4} = \pmb{\nu}_{w_5} = (0,0,1)$ and edges
$\{v_1,v_2\}$, $\{v_1,v_3\}$, $\{v_1,v_4\}$, $\{v_1,v_5\}$ and $\{w_1,w_2\}$,
$\{w_1,w_3\}$, $\{w_1,w_4\}$, $\{w_1,w_5\}$, as depicted in Figure~\ref{fig:graphG3}.
\begin{figure}[ht]
\centering
\begin{tikzpicture}[every node/.style={fill=black,circle,inner sep=2pt},node distance=0.5cm]
\node[label=below:{$v_1$}](v1) {};
\node[above left= of v1,label=left:{$v_2$}](v2) {};
\node[below left= of v1,label=left:{$v_3$}](v3) {};
\node[above right= of v1,label=right:{$v_4$}](v4) {};
\node[below right= of v1,label=right:{$v_5$}](v5) {};
\node[right=4cm of v1,label=below:{$w_1$}](w1) {};
\node[above left= of w1,label=left:{$w_2$}](w2) {};
\node[below left= of w1,label=left:{$w_3$}](w3) {};
\node[above right= of w1,label=right:{$w_4$}](w4) {};
\node[below right= of w1,label=right:{$w_5$}](w5) {};
\path
(v1) edge[-] (v2)
(v1) edge[-] (v3)
(v1) edge[-] (v4)
(v1) edge[-] (v5)
(w1) edge[-] (w2)
(w1) edge[-] (w3)
(w1) edge[-] (w4)
(w1) edge[-] (w5)
;
\end{tikzpicture}
\caption{Graph $G_3$.}\label{fig:graphG3}
\end{figure}
By definition, $\mathbf{L}^{(0)}:=\left(
\begin{smallmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 1 & 0\\
0 & 0 & 1\\
0 & 0 & 1\\
0 & 1 & 0\\
1 & 0 & 0\\
1 & 0 & 0\\
0 & 0 & 1\\
0 & 0 & 1\end{smallmatrix}\right)$.
We also note that
\begin{align*}
&(\pmb{\ell}_{M_{\textsl{WL}}}^{(1)})_{v_1}=\textsc{Hash}\bigl((1,0,0),\{\!\!\{(0,1,0),(0,1,0),(0,0,1),(0,0,1)\}\!\!\}\bigr)\\
\neq&(\pmb{\ell}_{M_{\textsl{WL}}}^{(1)})_{w_1}=\textsc{Hash}\bigl((0,1,0),\{\!\!\{(1,0,0),(1,0,0),(0,0,1),(0,0,1)\}\!\!\}\bigr).
\end{align*}
We next show that there exist no $\mathbf{W}^{(1)},\mathbf{B}^{(1)}$ such that $\mathbf{L}^{(1)}\sqsubseteq \pmb{\ell}_{M_{\textsl{WL}}}^{(1)}$.
Indeed,
\allowdisplaybreaks
\begin{align*}
\mathbf{L}^{(1)}& :=\sigma\left(\left(\mathsf{diag}\left(
\begin{pmatrix}\frac{1}{2}\\
1\\
1\\
1\\
1\\
\frac{1}{2}\\
1\\
1\\
1\\
1\\
\end{pmatrix}\right)
\begin{pmatrix}
0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0\\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1\\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\
\end{pmatrix}\mathsf{diag}\left(
\begin{pmatrix}\frac{1}{2}\\
1\\
1\\
1\\
1\\
\frac{1}{2}\\
1\\
1\\
1\\
1\\
\end{pmatrix}\right)+\mathbf{I}\right)\mathbf{L}^{(0)}\mathbf{W}^{(1)} + \mathbf{B}^{(1)}\right)\\
&=\sigma\left(\begin{pmatrix}
0 & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 & 0 & 0\\
\frac{1}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
\frac{1}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
\frac{1}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
\frac{1}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2}\\
0 & 0 & 0 & 0 & 0 & \frac{1}{2} & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & \frac{1}{2} & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & \frac{1}{2} & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & \frac{1}{2} & 0 & 0 & 0 & 0\\
\end{pmatrix}
\begin{pmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 1 & 0\\
0 & 0 & 1\\
0 & 0 & 1\\
0 & 1 & 0\\
1 & 0 & 0\\
1 & 0 & 0\\
0 & 0 & 1\\
0 & 0 & 1
\end{pmatrix}\mathbf{W}^{(1)}\right)
=\sigma\left(\begin{pmatrix}
1 & 1 & 1\\
\frac{1}{2} & 1 & 0\\
\frac{1}{2} & 1 & 0\\
\frac{1}{2} & 0 & 1\\
\frac{1}{2} & 0 & 1\\
1 & 1 & 1\\
1 & \frac{1}{2} & 0\\
1 & \frac{1}{2} & 0\\
0 & \frac{1}{2} & 1\\
0 & \frac{1}{2} & 1
\end{pmatrix}\mathbf{W}^{(1)} + \mathbf{B}^{(1)}\right).
\end{align*}
Hence, independently of the choice of $\mathbf{W}^{(1)}$ and $\mathbf{B}^{(1)}$,
vertices $v_1$ and $w_1$ will be assigned the same label, and thus
$\mathbf{L}^{(1)}\not\sqsubseteq \pmb{\ell}_{M_{\textsl{WL}}}^{(1)}$.
\end{proof}
In particular, the class $\mathcal{M}_{\textsl{dGNN}_4}$, corresponding to the
popular graph neural networks of~\cite{kipf-loose}, is not stronger than
$\mathcal{M}_{\textsl{WL}}$. We also remark, based on the first counterexample in the proof,
that the class of aMPNNs, corresponding to simple graph neural networks of the form
$\mathbf{L}^{(t)}:=\sigma(\mathbf{A}\mathbf{L}^{(t-1)}\mathbf{W}^{(t)}+\mathbf{B}^{(t
)})$, is not stronger than $\mathcal{M}_{\textsl{WL}}$. We know, however, from
Corollary~\ref{cor:pluspstrongwl} that the slight extension
$\mathbf{L}^{(t)}=\sigma\left((\mathbf{A}+p\mathbf{I})\mathbf{L}^{(t-1)}\mathbf{W}^{(
t)}-q\mathbf{J}\right)$ results in a class of aMPNNs that is stronger than
$\mathcal{M}_{\textsl{WL}}$. It will follow from our next result that a similar extension
suffices to make the graph neural networks of~\cite{kipf-loose} stronger than
$\mathcal{M}_{\textsl{WL}}$.
We will now argue that the remaining $\mathcal{M}_{\textsl{dGNN}_6}$ class from
Table~\ref{tab:dMPNNs} is stronger than $\mathcal{M}_{\textsl{WL}}$, hereby concluding the
proof of the fourth item in Theorem~\ref{thm:dmpnn}.
\begin{proposition}\label{prop:indeed-wl-power}
The class $\mathcal{M}_{\textsl{dGNN}_6}$ is stronger than $\mathcal{M}_{\textsl{WL}}$.
\end{proposition}
\begin{proof}
We recall that dMPNNs in $\mathcal{M}_{\textsl{dGNN}_6}$ correspond to graph
neural network architectures of the form
\begin{align}\label{eq:GNN6}
\mathbf{L}^{(t)}&:=\sigma(\mathsf{diag}(\mathbf{g})(\mathbf{A}+p\mathbf{I})\mathsf{diag}(\mathbf{h})\mathbf{L}^{(t-1)}\mathbf{W}^{(t)}+\mathbf{B}^{(t)}),
\end{align}
where $\mathsf{diag}(\mathbf{g}) = \mathsf{diag}(\mathbf{h}) =
(r\mathbf{I}+(1-r)\mathbf{D})^{\nicefrac{-1}{2}}$ and $\sigma$ is ReLU or sign. In
fact, our proof will work for any degree-determined $\mathbf{g}$ and $\mathbf{h}$.
The argument closely follows the proof of Theorem~\ref{thm:equalstrong}. More
specifically, we construct a dMPNN $M$ corresponding to~(\ref{eq:GNN6}) such that
$\pmb{\ell}_M^{(t)}\sqsubseteq\pmb{\ell}_{M_{\textsl{WL}}}^{(t)}$ for all $t\geq 0$.
The induction hypothesis is that
$\pmb{\ell}_M^{(t)}\sqsubseteq\pmb{\ell}_{M_{\textsl{WL}}}^{(t)}$ and
$\pmb{\ell}_M^{(t)}$ is row-independent modulo equality. This hypothesis is clearly
satisfied, by definition, for $t=0$.
For the inductive step we assume that
$\pmb{\ell}_M^{(t-1)}\sqsubseteq\pmb{\ell}_{M_{\textsl{WL}}}^{(t-1)}$ and
$\pmb{\ell}_M^{(t-1)}$ is row-independent modulo equality. Let us define the
labelling $\pmb{\kappa}_M^{(t-1)}$ such that
$(\pmb{\kappa}_M^{(t-1)})_v:=h(d_v)(\pmb{\ell}_M^{(t-1)})_v$ for all vertices $v$.
\begin{lemma}\label{lem:indep}
We have that $\pmb{\kappa}_M^{(t-1)}\sqsubseteq\pmb{\ell}_{M_{\textsl{WL}}}^{(t-1)}$
and $\pmb{\kappa}_M^{(t-1)}$ is row-independent modulo equality.
\end{lemma}
\begin{proof}
Suppose that there are two vertices $v$ and $w$ such that
$$(\pmb{\kappa}_M^{(t-1)})_v=h(d_v)(\pmb{\ell}_M^{(t-1)})_v=h(d_w)(\pmb{\ell}_M^{(t-1)})_w=(\pmb{\kappa}_M^{(t-1)})_w.$$
This implies that $(\pmb{\ell}_M^{(t-1)})_v$ is a (non-zero) scalar multiple of
$(\pmb{\ell}_M^{(t-1)})_w$. This is only possible when
$(\pmb{\ell}_M^{(t-1)})_v=(\pmb{\ell}_M^{(t-1)})_w$ because $\pmb{\ell}_M^{(t-1)}$
is row-independent modulo equality. In other words,
$\pmb{\kappa}_M^{(t-1)}\sqsubseteq\pmb{\ell}_M^{(t-1)}\sqsubseteq\pmb{\ell}_{M_{\text
sl{WL}}}^{(t-1)}$. Similarly, suppose that $\pmb{\kappa}_M^{(t-1)}$ is not
row-independent modulo equality then, due to the definition of
$\pmb{\kappa}_M^{(t-1)}$, this implies that $\pmb{\ell}_M^{(t-1)}$ is also not
row-independent modulo equality.
\end{proof}
Lemma~\ref{lem:indep} gives us sufficient conditions to repeat a key part of the
argument in the proof of Theorem~\ref{thm:equalstrong}. That is, we can find a
matrix $\mathbf{U}^{(t)}$ such that the labelling
$\pmb{\mu}^{(t)}:v\mapsto\left((\mathbf{A}+p\mathbf{I})\mathsf{diag}(\mathbf{h})\mathbf{L}^{(t-1)}\mathbf{U}^{(t)}\right)_{v}$ satisfies $\pmb{\mu}^{(t)}\sqsubseteq
\pmb{\ell}_{M_{\textsl{WL}}}^{(t)}$.
We will now prove that the labelling $\pmb{\lambda}^{(t)}$ defined by
$\pmb{\lambda}^{(t)}_v:=g(d_v)\pmb{\mu}^{(t)}_v$, also satisfies
$\pmb{\lambda}^{(t)}\sqsubseteq \pmb{\ell}_{M_{\textsl{WL}}}^{(t)}$. We remark that
$\pmb{\lambda}^{(t)}$ coincides with the labelling:
$$
v\mapsto \bigl(\mathsf{diag}(\mathbf{g})(\mathbf{A}+p\mathbf{I})\mathsf{diag}(\mathbf{h})\mathbf{L}^{(t-1)}\mathbf{U}^{(t)})_{v}.
$$
\begin{lemma}\label{lem:choose-p}
The exists a constant $m_p$, only dependent on $\mathbf{g}$ and the number $n$ of
vertices, such that $\pmb{\lambda}^{(t)}\sqsubseteq
\pmb{\ell}_{M_{\textsl{WL}}}^{(t)}$, for every $m_p<p<1$.
\end{lemma}
\begin{proof}
We will choose $m_p$ at the end of the proof. For now suppose that
$\pmb{\lambda}^{(t)} \not \sqsubseteq \pmb{\ell}_{M_{\textsl{WL}}}^{(t)}$. Then
there exist two vertices $v$ and $w$ such that
$$
\pmb{\lambda}^{(t)}_v=\pmb{\lambda}^{(t)}_w \text{ and } (\pmb{\ell}_{M_{\textsl{WL}}}^{(t)})_v\neq (\pmb{\ell}_{M_{\textsl{WL}}}^{(t)})_w.$$
The latter implies that $\pmb{\mu}^{(t)}_v\neq \pmb{\mu}^{(t)}_w$ and thus
$\pmb{\lambda}^{(t)}_v=\pmb{\lambda}^{(t)}_w$ implies that $g(d_v)\neq g(d_w)$.
We recall some facts from the proof of Theorem~\ref{thm:equalstrong}, and from
equations~\eqref{eq:labelmu} and~\eqref{eq:linearcomb} in particular. An entry in
$\pmb{\mu}^{(t)}_v$ is either $0$ or $1,2,\ldots,n$ or $i+p$, for some $i\in
\{0,1,\dots,n\}$. Furthermore, at least one entry must be distinct from $0$. Also,
$\pmb{\lambda}^{(t)}_v=\pmb{\lambda}^{(t)}_w$ implies that the positions of the
non-zero entries in $\pmb{\mu}^{(t)}_v$ and $\pmb{\mu}^{(t)}_w$ coincide. (Recall
that the image of $g$ is $\mathbb{A}^+$). Let $ Z$ be the positions in
$\pmb{\mu}^{(t)}_v$ (and thus also in $\pmb{\mu}^{(t)}_w$) that carry non-zero
values.
We can now infer that $\pmb{\lambda}^{(t)}_v=\pmb{\lambda}^{(t)}_w$ implies that for
every $i\in Z$:
$$
\frac{\pmb{\mu}^{(t)}_{vi}}{\pmb{\mu}^{(t)}_{wi}}=\frac{g(d_w)}{g(d_v)}\neq 1.
$$
Moreover, both in $\pmb{\mu}^{(t)}_v$ and $\pmb{\mu}^{(t)}_w$ there are unique
positions $i_1$ and $i_2$, respectively, whose corresponding entry contain $p$. We
now consider three cases:
$$
\text{(a)~}
\frac{\pmb{\mu}^{(t)}_{vi_1}}{\pmb{\mu}^{(t)}_{wi_1}}=\frac{i+p}{j}\text{; \quad(b)~}
\frac{\pmb{\mu}^{(t)}_{vi_2}}{\pmb{\mu}^{(t)}_{wi_2}}=\frac{i}{j+p}\text{; \quad (c)~}
\frac{\pmb{\mu}^{(t)}_{vi_1}}{\pmb{\mu}^{(t)}_{wi_1}}=\frac{i+p}{j+p}\text{ (this is the case if and only if $i_1=i_2$),}
$$
for some $i,j\in \{0,1,2,\dots,n\}$. To define $m_p$, let
$\Gamma:=\left\{\frac{g(d_w)}{g(d_v)} \:\middle|\: g(d_v)\neq g(d_w) \text{ and }
v,w\in V\right\}$ and consider
\begin{align*}
P_a&:=\left\{\alpha j -i\hspace{0.5ex}\:\middle|\: 0\leq \alpha j -i < 1, i,j\in \{0,1,2\dots,n\},\alpha\in \Gamma \vphantom{\frac{1}{\alpha}}\right\}\\
P_b&:=\left\{ \frac{i -\alpha j}{\alpha} \:\middle|\: 0\leq \frac{i -\alpha j}{\alpha} < 1, i,j\in \{0,1,2,\dots,n\},\alpha\in \Gamma\right\}\\
P_c&:=\left\{ \frac{\alpha j-i}{1-\alpha} \:\middle|\: 0\leq \frac{\alpha(j-i)}{1-\alpha} < 1, i,j\in \{0,1,2,\dots,n\},\alpha\in \Gamma\right\}.
\end{align*}
We define $m_p=\max\{P_1\cup P_2\cup P_3\cup\{0\}\}$ and we claim that for all $p$
satisfying $m_p<p<1$ the lemma holds.
By definition of $P_a$, $\alpha j-i \neq p$ and thus $\frac{i+p}{j} \neq \alpha$ for
any $\alpha\in\Gamma$ and $i,j\in\{0,1,\ldots,n\}$. This rules out (a). Similarly,
by definition of $P_b$, $\frac{i-\alpha j}{\alpha}\neq p$ and thus
$\frac{i}{j+p}\neq \alpha$ for any $\alpha\in\Gamma$ $i,j\in\{0,1,\ldots,n\}$. This
rules out (b). Finally, by definition of $P_3$, $\frac{\alpha j-i}{1-\alpha}\neq p$
and thus $\frac{i+p}{j+p}\neq \alpha$ for any $\alpha\in\Gamma$
$i,j\in\{0,1,\ldots,n\}$. This rules out (c). We conclude, as our initial assumption
cannot be valid for this $m_p$.
\end{proof}
From here, we can again follow the proof of Theorem~\ref{thm:equalstrong} to
construct a matrix $\mathbf{X}^{(t)}$ such that the labelling $\pmb{\ell}_M^{(t)}$
defined by
$\sigma(\mathsf{diag}(\mathbf{g})(\mathbf{A}+p\mathbf{I})\mathsf{diag}(\mathbf{h})
\mathbf{L}^{(t-1)}\mathbf{U}^{(t)}\mathbf{X}^{(t)}+\mathbf{B}^{(t)})$ with
$\mathbf{B}^{(t)}=-\mathbf{J}$ if $\sigma$ is the sign function, and
$\mathbf{B}^{(t)}=-q\mathbf{J}$ if $\sigma$ is the ReLU function, is such that
$\pmb{\ell}_M^{(t)}\sqsubseteq\pmb{\ell}_{M_{\textsl{WL}}}^{(t)}$ and
$\pmb{\ell}_M^{(t)}$ is row-independent modulo equality. This concludes the proof
for dMPNNs arising from graph neural networks of the form~\eqref{eq:GNN6}.
\end{proof}
We already mentioned that the proof of Proposition~\ref{prop:indeed-wl-power} works
for any degree-determined $\mathbf{g}$ and $\mathbf{h}$. In particular, the class of
dMPNNs originating from graph neural networks of the form
\begin{equation}
\mathbf{L}^{(t)}:=\sigma\Bigl(\bigl(\mathbf{D}+\mathbf{I}\bigr)^{-1/2} (\mathbf{A}+p\mathbf{I})\bigl(\mathbf{D}+\mathbf{I}\bigr)^{-1/2} \mathbf{L}^{(t-1)}\mathbf{W}^{(t)} -q \mathbf{J}\Bigr),\label{gnn:kipfp}
\end{equation}
with $p,q\in\mathbb{A}$, $0\leq p,q\leq 1$, is stronger than $\mathcal{M}_{\textsl{WL}}$. The
introduction of the parameter $p$ was already suggested in~\cite{kipf-loose}. The
proof of Proposition~\ref{prop:notweaker} shows that this parameter is necessary to
encode the WL algorithm. Our result thus provide a theoretical justification for
including this parameter.
\section{Conclusions}\label{sec:conclude}
In this paper we investigate the distinguishing power of two classes of MPNNs,
anonymous and degree-aware MPNNs, in order to better understand the presence of
degree information in commonly used graph neural network architectures. We show that
both classes of MPNNs are equivalent to the WL algorithm, in terms of their
distinguishing power, when one ignores the number of computation rounds. Taking the
computation rounds into consideration, however, reveals that degree information may
boost the distinguishing power.
Furthermore, we identify classes of MPNNs corresponding to specific
linear-algebra-based architectures of graph neural networks. We again distinguish
between anonymous graph neural networks~\cite{hyl17,grohewl} and degree-aware graph
neural networks~\cite{kipf-loose,DBLP:journals/corr/abs-1905-03046}. Here, we again
make connections to the WL algorithm, identify which architectures of graph neural
networks can or cannot simulate the WL algorithm, and describe how a simple
modification results in graph neural networks that are as powerful as the WL
algorithm.
Regarding future work, we point out that, following the work of~\cite{grohewl}, we
fix the input graph in our analysis. We use this particularly when we prove that
certain classes of MPNNs, based on graph neural network architectures, are stronger
than the WL algorithm (Theorem~\ref{thm:equalstrong} and
Proposition~\ref{prop:indeed-wl-power}). We prove this by constructing an MPNN that
simulates the WL algorithm on a fixed input graph. As such, the constructed MPNN may
not simulate the WL algorithm on another graph. It is natural to ask whether there
are graph neural network-based MPNNs that can simulate the WL algorithm on all
graphs.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,706 |
\section{Introduction}
The two fields of convex optimization and active learning seem to have evolved quite independently of each other. Recently, \cite{RR09} pointed out their relatedness due to the inherent sequential nature of both fields and the complex role of feedback in taking future actions. Following that, \cite{RS13} made the connections more explicit by tying together the exponent used in noise conditions in active learning and the exponent used in uniform convexity (UC) in optimization. They used this to establish lower bounds (and tight upper bounds) in stochastic optimization of UC functions based on proof techniques from active learning. However, it was unclear if there were concrete algorithmic ideas in common between the fields.
Here, we provide a positive answer by exploiting the aforementioned connections to form new and interesting algorithms that clearly demonstrate that the complexity of $d$-dimensional stochastic optimization is precisely the complexity of $1$-dimensional active learning. Inspired by an optimization algorithm that was adaptive to unknown uniform convexity parameters, we design an interesting one-dimensional active learner that is also adaptive to unknown noise parameters. This algorithm is simpler than the adaptive active learning algorithm proposed recently in \cite{H11} which handles the pool based active learning setting.
Given access to this active learner as a subroutine for line search, we show that a simple randomized coordinate descent procedure can minimize uniformly convex functions with a much simpler stochastic oracle that returns only a Bernoulli random variable representing a noisy sign of the gradient in a single coordinate direction, rather than a full-dimensional real-valued gradient vector. The resulting algorithm is adaptive to all unknown UC and smoothness parameters and achieve minimax optimal convergence rates.
We spend the first two sections describing the problem setup and preliminary insights, before describing our algorithms in sections 3 and 4.
\subsection{Setup of First-Order Stochastic Convex Optimization}
First-order stochastic convex optimization is the task of approximately minimizing a convex function over a convex set, given oracle access to unbiased estimates of the function and gradient at any point, using as few queries as possible (\cite{NY83}).
We will assume that we are given an arbitrary set $S\subset \mathbb{R}^d$ of known diameter bound $R = \max_{x,y\in S} \|x-y\|$. A convex function $f$ with $x^* = \arg \min_{x \in S} f(x)$ is said to be $k$-uniformly convex if, for some $\lambda > 0, k \geq 2$, we have for all $x,y \in S$
$$f(y) \geq f(x) + \nabla f(x)^\top (y-x) + \frac{\lambda}{2} \|x-y\|^k$$
(strong convexity arises when $k=2$). $f$ is $L$-Lipschitz for some $L>0$ if $\|\nabla f(x)\|_* \leq L$ (where $\|.\|_*$ is the dual norm of $\|.\|$); equivalently for all $x,y \in S$
\begin{equation*}
|f(x) - f(y)| \leq L \|x-y\|
\end{equation*}
A differentiable $f$ is $H$-strongly smooth (or has a $H$-Lipschitz gradient) for some $H>\lambda$ if for all $x,y \in S$, we have $\|\nabla f(x) - \nabla f(y)\|_* \leq H \|x-y\|$, or equivalently
$$f(y) \leq f(x) + \nabla f(x)^\top (y-x) + \frac{H}{2} \|x-y\|^2$$
In this paper we shall always assume $\|.\| = \|.\|_*=\|.\|_2$ and deal with strongly smooth and uniformly convex functions with parameters $\lambda > 0, k \geq 2$, $L,H>0$.\\
A stochastic first order oracle is a function that accepts $x \in S$, and returns
$$\Big(\hat{f}(x),{\hat{g}}(x) \Big) \in \mathbb{R}^{d+1} \mbox{ where } \mathbb{E} \big[\hat{f}(x) \big] = f(x), \mathbb{E}\big[\hat{g}(x)\big]= \nabla f(x)$$
(these unbiased estimates also have bounded variance) and the expectation is over any internal randomness of the oracle. \\
An optimization algorithm is a method that sequentially queries an oracle at points in $S$ and returns $\hat{x}_T$ as an estimate of the optimum of $f$ after $T$ queries (or alternatively tries to achieve an error of $\epsilon$) and their performance can be measured by either function error $f(\hat{x}_T) - f(x^*)$ or point error $\|\hat{x}_T - x^*\|$.\\
\subsection{Stochastic Gradient-Sign Oracles} \label{sgso}
Define a stochastic sign oracle to be a function of $x \in S, j \in \{1...d\}$, that returns
$${\hat{s}}_j(x) \in \{+,-\} \mbox{ where}\footnote{$f=\bT (g)$ means $f=\mathrm{\Omega}(g)$ and $f=\bO (g)$ (rate of growth)} \ \big |\eta(x) - 0.5 \big | = \bT \Big( [\nabla f(x)]_j \Big) \mbox{ and } \eta(x) = \mathbb{P} \big ( {\hat{s}}_j(x) = + | x \big )$$
where ${\hat{s}}_j(x)$ is a noisy sign$\big( [\nabla f(x)]_j \big)$ and $[\nabla f(x)]_j$ is the $j$-th coordinate of $\nabla f$, and the probability is over any internal randomness of the oracle. This behavior of $\eta(x)$ actually needs to hold only when $\big |[\nabla f(x)]_j \big|$ is small.
In this paper, we consider coordinate descent algorithms that are motivated by applications where computing the overall gradient, or even a function value, can be expensive due to high dimensionality or huge amounts of data, but computing the gradient in any one coordinate can be cheap. \cite{N10} mentions the example of $\min_x \frac1{2}\|Ax-b\|^2 + \frac1{2}\|x\|^2$ for some $n \times d$ matrix $A$ (or any other regularization that decomposes over dimensions). Computing the gradient $A^\top (Ax-b) + x$ is expensive, because of the matrix-vector multiply. However, its $j$-th coordinate is $2A^{j\top} (Ax-b) + x_j$ and requires an expense of only $n$ if the residual vector $Ax-b$ is kept track of (this is easy to do, since on a single coordinate update of $x$, the residual change is proportional to $A^j$, an additional expense of $n$).
A sign oracle is weaker than a first order oracle, and can actually be obtained by returning the sign of the first order oracle's noisy gradient if the mass of the noise distribution grows linearly around its zero mean (argued in next section). At the optimum along coordinate $j$, the oracle returns a $\pm 1$ with equal probability, and otherwise returns the correct sign with a probability proportional to the value of the directional derivative at that point (this is reflective of the fact that the larger the derivative's absolute value, the easier it would be for the oracle to approximate its sign, hence the smaller the probability of error). It is not unreasonable that there may be other circumstances where even calculating the (real value) gradient in the $i$-th direction could be expensive, but estimating its sign could be a much easier task as it only requires estimating whether function values are expected to increase or decrease along a coordinate (in a similar spirit of function comparison oracles \cite{JNR12}, but with slightly more power).
We will also see that the rates for optimization crucially depend on whether the gradient noise is sign-preserving or not. For instance, with rounding errors or storing floats with small precision, one can get deterministic rates as if we had the exact gradient since the rounding or lower precision doesn't flip signs.
\subsection{Setup of Active Threshold Learning}
The problem of one-dimensional threshold estimation assumes you have an interval of length $R$, say $[0,R]$. Given a point $x$, it has a label $y \in \{+,-\}$ that is drawn from an unknown conditional distribution $\eta(x) = \mathbb{P} \big( Y=+|X=x\big)$ and the threshold $t$ is the unique point where $\eta(x) = 1/2$, with it being larger than half on one side of $t$ and smaller than half on the other (hence it is more likely to draw a $+$ on one side of $t$ and a $-$ on the other side).
The task of active learning of threshold classifiers allows the learner to sequentially query $T$ (possibly dependent) points, observing labels drawn from the unknown conditional distribution after each query, with the goal of returning a guess $\hat{x}_T$ as close to $t$ as possible. In the formal study of classification (cf. \cite{T04}), it is common to study minimax rates when the regression function $\eta(x)$ satisfies Tsybakov's noise or margin condition (TNC) with exponent $k$ at the threshold $t$. Different versions of this boundary noise condition are used in regression, density or level-set estimation and lead to an improvement in minimax optimal rates (for classification, also cf. \cite{AT07}, \cite{H11}). Here, we present the version of TNC used in \cite{CN07} :
$$M |x-t|^{k - 1} \geq | \eta(x) - 1/2 | \geq \mu |x-t|^{k - 1} \mbox{ whenever}\footnote{Note that $|x-t| \leq \delta_0 := \left( \frac{\epsilon_0}{M} \right)^{\frac1{k-1}} \implies |\eta(x) - 1/2| \leq \epsilon_0 \implies |x-t| \leq \left( \frac{\epsilon_0}{\mu} \right)^{\frac1{k-1}}$} \ |\eta(x) - 1/2| \leq \epsilon_0 $$
for some constants $M>\mu>0,\epsilon_0 > 0, k \geq 1$.
A standard measure for how well a classifier $h$ performs is given by its risk, which is simply the probability of classification error (expectation under $0-1$ loss), $\mathcal{R}(h) = \mathbb{P} \big[ h (x) \neq y \big]$. The performance of threshold learning strategies can be measured by the excess classification risk of the resultant threshold classifier at $\hat{x}_T$ compared to the Bayes optimal classifier at $t$ as given by \footnote{$a \vee b := \max(a,b) \mbox{ and } a \wedge b := \min(a,b)$}
\begin{equation} \label{risk}
\mathcal{R} (\hat{x}_T) - \mathcal{R} (t) = \int\limits_{\hat{x}_T \wedge t}^{\hat{x}_T \vee t} | 2 \eta(x) - 1| dx
\end{equation}
In the above expression, akin to \cite{CN07}, we use a uniform marginal distribution for active learning since there is no underlying distribution over $x$. Alternatively, one can simply measure the one-dimensional point error $|\hat{x}_T - t|$ in estimation of the threshold. Minimax rates for estimation of risk and point error in active learning under TNC were provided in \cite{CN07} and are summarized in the next section.
\subsection{Summary of Contributions}
Now that we have introduced the notation used in our paper and some relevant previous work (more in the next section), we can clearly state our contributions.
\begin{itemize}
\item We generalize an idea from \cite{JN10} to present a simple epoch-based active learning algorithm with a passive learning subroutine that can optimally learn one-dimensional thresholds and is adaptive to unknown noise parameters.
\item We show that noisy gradient signs suffice for minimization of uniformly convex functions by proving that a random coordinate descent algorithm with an active learning line-search subroutine achieves minimax convergence rates.
\item Due to the connection between the relevant exponents in the two fields, we can combine the above two methods to get an algorithm that achieves minimax optimal rates and is adaptive to unknown convexity parameters.
\item As a corollary, we argue that with access to possibly noisy non-exact gradients that don't switch any signs (rounding errors or low-precision storage are sign-preserving), we can still achieve exponentially fast deterministic rates.
\end{itemize}
\section{Preliminary Insights}
\subsection{Connections Between Exponents}
Taking one point as $x^*$ in the definition of UC, we see that
$$|f(x) - f(x^*)| \geq \frac{\lambda}{2} \|x-x^*\|^k$$
Since $\|\nabla f(x)\| \|x-x^*\| \geq \nabla f(x)^\top (x-x^*) \geq f(x) - f(x^*)$ (by convexity),
$$\|\nabla f(x) - 0\| \geq \frac{\lambda}{2} \|x-x^*\|^{k-1} $$
Another relevant fact for us will be that uniformly convex functions in $d$ dimensions are uniformly convex along any one direction, or in other words, for every fixed $x \in S$ and fixed unit vector $u \in \mathbb{R}^d$, the univariate function of $\alpha$ defined by $f_{x,u}(\alpha) := f(x + \alpha u)$ is also UC with the same parameters\footnote{Since $f$ is UC, $f_{x,u}(\alpha) \geq f_{x,u}(0) + \alpha \nabla f_{x,u}(0) + \frac{\lambda}{2}|\alpha|^k$}. For $u = e_j$,
$$\big | [\nabla f(x)]_j - 0 \big | \geq \frac{\lambda}{2} \|x-x_{j}^*\|^{k-1}$$
where $x_{j}^* = x + \alpha_j^* e_j$ and $\alpha_{j}^* = \arg \min_{\{\alpha|x + \alpha e_j \in S\}} f(x + \alpha e_j)$. This uncanny similarity to the TNC (since $\nabla f(x^*) = 0$) was mathematically exploited in \cite{RS13} where the authors used a lower bounding proof technique for one-dimensional active threshold learning from \cite{CN07} to provide a new lower bounding proof technique for the $d$-dimensional stochastic convex optimization of UC functions. In particular, they showed that the minimax rate for $1$-dimensional active learning excess risk and the $d$-dimensional optimization function error both scaled like\footnote{we use $\mathrm{{\tilde{O}}}, \mathrm{{\tilde{\Theta}}}$ to hide constants and polylogarithmic factors} $\mathrm{{\tilde{\Theta}}} \left( T^{-\frac{k}{2k-2}}\right)$, and that the point error in both settings scaled like $\mathrm{{\tilde{\Theta}}} \left( T^{-\frac{1}{2k-2}}\right)$, where $k$ is either the TNC exponent or the UC exponent, depending on the setting. The importance of this connection cannot be emphasized enough and we will see this being useful throughout this paper.\\
As mentioned earlier \cite{CN07} require a two-sided TNC condition (upper and lower growth condition to provide exact tight rate of growth) in order to prove risk upper bounds. On a similar note, for uniformly convex functions, we will assume such a Local $k$-Strong Smoothness condition around directional minima
$$\mbox{\textbf{Assumption LkSS} : \ \ \ \ for all $j \in \{1...d\}$\ \ \ } \big | [\nabla f(x)]_j - 0 \big | \leq \Lambda \|x-x_{j}^*\|^{k-1} $$
for some constant $\Lambda > \lambda/2$, so we can tightly characterize the rate of growth as
$$\big | [\nabla f(x)]_j - 0 \big | = \bT \Big( \|x-x_{j}^*\|^{k-1} \Big)$$
This condition is implied by strong smoothness or Lipschitz smooth gradients when $k=2$ (for strongly convex and strongly smooth functions), but is a slightly stronger assumption otherwise.
\subsection{The One-Dimensional Argument}
The basic argument for relating optimization to active learning was made in \cite{RS13} in the context of stochastic first order oracles when the noise distribution $\mathrm{P}(z)$ is unbiased and grows linearly around its zero mean, i.e.
$$ \int_0^\infty \mathrm{dP}(z) = \tfrac{1}{2} \ \mbox{ and } \ \int_0^t \mathrm{dP}(z) = \bT ( t ) $$
for all $0 <t < t_0$, for constants $t_0$ (similarly for $-t_0 < t < 0$). This is satisfied for gaussian, uniform and many other distributions. We reproduce the argument for clarity and then sketch it for stochastic signed oracles as well.
For any $x \in S$, it is clear that $f_{x,j}(\alpha) := f(x+\alpha e_j)$ is convex; its gradient $\nabla f_{x,j}(\alpha) := [\nabla f(x + \alpha e_j)]_j$ is an increasing function of $\alpha$ that switches signs at $\alpha^*_j := \arg\min_{\{\alpha | x+ \alpha e_j \in S\}} f_{x,j}(\alpha)$, or equivalently at directional minimum $x^*_j := x + \alpha^*_j e_j$. One can think of sign$([\nabla f(x)]_j)$ as being the true label of $x$, sign$([\nabla f(x)]_j+z)$ as being the observed label, and finding $x_j^*$ as learning the decision boundary (point where labels switch signs). Define regression function
$$\eta(x) := \mathbb{P} \Big(\mbox{sign}([\nabla f(x)]_j+z) = +|x \Big)$$
and note that minimizing $f_{x_0,j}$ corresponds to identifying the Bayes threshold classifier as $x_j^*$ because the point at which $\eta(x)=0.5$ or $[\nabla f(x)]_j=0$ is $x_j^*$. Consider a point $x = x^*_j + t e_j$ for $t>0$ with $[\nabla f(x)]_j > 0$ and hence has true label $+$ (a similar argument can be made for $t < 0$). As discussed earlier, $\big| [\nabla f(x)]_j \big| = \bT \Big( \|x-x_j^*\|^{k-1} \Big) = \bT (t^{k-1})$. The probability of seeing label $+$ is the probability that we draw $z$ in $\big(-[\nabla f(x)]_j,\infty \big)$ so that the sign of $[\nabla f(x)]_j+z$ is still positive. Hence, the regression function can be written as
\begin{align*}
\eta(x) \ &= \ \mathbb{P} \Big([\nabla f(x)]_j + z > 0 \Big) \\
\ &= \ \mathbb{P} (z>0) + \mathbb{P} \Big(-[\nabla f(x)]_j < z < 0 \Big) \ = \ 0.5 + \bT \Big( [\nabla f(x)]_j \Big)
\end{align*}
$$
\implies \big |\eta(x) - \tfrac{1}{2} \big| \ = \ \bT \Big( [\nabla f(x)]_j \Big) \ = \ \bT \big( t^{k-1} \big) \ = \ \bT \Big( |x-x_j^*|^{k-1} \Big)\label{bz}
$$
Hence, $\eta(x)$ satisfies the TNC with exponent $k$, and an active learning algorithm (next subsection) can be used to obtain a point $\hat{x}_T$ with small point-error and excess risk. Note that function error in convex optimization is bounded above by excess risk of the corresponding active learner using eq (\ref{risk}) because
\begin{align*} \label{ferrorrisk}
f_j(\hat{x}_T) - f_j(x_j^*) \ &= \ \Bigg| \int\limits^{\hat{x}_T \vee x_j^*}_{\hat{x}_T \wedge x_j^*} [\nabla f(x)]_j \mathrm{dx} \Bigg|
\ &= \bT \Bigg( \int\limits^{\hat{x}_T \vee x_j^*}_{\hat{x}_T \wedge x^*_j} |2\eta(x)-1|\mathrm{dx} \Bigg)\\
\ &=\ \bT \Big(\mathcal{R} (\hat{x}_T)\Big)
\end{align*}
Similarly, for stochastic sign oracles (Sec. \ref{sgso}), using $\eta(x) = \mathbb{P} \big ({\hat{s}}_j(x) = + \big) $,
\begin{eqnarray*}
\big| \eta(x) - \tfrac{1}{2} \big| \ = \ \bT \Big([\nabla f(x)]_j\Big) \ = \ \bT \Big (\|x-x^*_j\|^{k-1} \Big)
\end{eqnarray*}
\subsection{A Non-adaptive Active Threshold Learning Algorithm}
One can use a grid-based probabilistic variant of binary search called the BZ algorithm \cite{BZ74} to approximately learn the threshold efficiently in the active setting, in the setting that $\eta(x)$ satisfies the TNC for known $k, \mu, M$ (it is not adaptive to the parameters of the problem - one needs to know these constants beforehand). The analysis of BZ and the proof of the following lemma are discussed in detail in Theorem 1 of \cite{CN09}, Theorem 2 of \cite{CN07} and the Appendix of \cite{RS13}.
\begin{lemma} \label{BZ}
Given a $1$-dimensional regression function that satisfies the TNC with known parameters $\mu, k$, then after $T$ queries, the BZ algorithm returns a point $\hat{t}$ such that $| \hat{t} - t | = \mathrm{{\tilde{\Theta}}} (T^{-\frac{1}{2k - 2}})$ and the excess risk is $\mathrm{{\tilde{\Theta}}} (T^{-\frac{k}{2k - 2}})$.
\end{lemma}
Due to the described connection between exponents, one can use BZ to approximately optimize a one dimensional uniformly convex function $f_j$ with known uniform convexity parameters $\lambda,k$.
Hence, the BZ algorithm can be used to find a point with low function error by searching for a point with low risk. This, when combined with Lemma \ref{BZ}, yields the following important result.
\begin{lemma} \label{perror}
Given a $1$-dimensional $k$-UC and LkSS function $f_j$, a line search to find $\hat{x}_T$ close to $x^*_j$ up to accuracy $|\hat{x}_T - x^*_j| \leq \eta$ in point-error can be performed in $\mathrm{{\tilde{\Theta}}} (1/\eta^{2k - 2})$ steps using the BZ algorithm. Alternatively, in $T$ steps we can find $\hat{x}_T$ such that $f(\hat{x}_T) - f(x^*_j) = \mathrm{{\tilde{\Theta}}} (T^{-\frac{k}{2k - 2}})$.
\end{lemma}
\section{A 1-D Adaptive Active Threshold Learning Algorithm}
We now describe an algorithm for active learning of one-dimensional thresholds that is adaptive, meaning it can achieve the minimax optimal rate even if the TNC parameters $M,\mu,k$ are unknown. It is quite different from the non-adaptive BZ algorithm in its flavour, though it can be regarded as a robust binary search procedure, and its design and proof are inspired from an optimization procedure from \cite{JN10} that is adaptive to unknown UC parameters $\lambda,k$.
Even though \cite{JN10} considers a specific optimization algorithm (dual averaging), we observe that their algorithm that adapts to unknown UC parameters can use any optimal convex optimization algorithm as a subroutine within each epoch. Similarly, our adaptive active learning algorithm is epoch-based and can use any optimal passive learning subroutine in each epoch. We note that \cite{H11} also developed an adaptive algorithm based on disagreement coefficient and VC-dimension arguments, but it is in a pool-based setting where one has access to a large pool of unlabeled data, and is much more complicated.
\subsection{An Optimal Passive Learning Subroutine}
The excess risk of passive learning procedures for 1-d thresholds can be bounded by $\bO (T^{-1/2})$ (e.g. see Alexander's inequality in \cite{DGL96} to avoid $\sqrt{\log T}$ factors from ERM/VC arguments) and can be achieved by ignoring the TNC parameters.
Consider such a passive learning procedure under a uniform distribution of samples (mimicked by active learning by querying the domain uniformly) in a ball\footnote{Define $B(x,R) := [x-R,x+R]$} $B(x_0,R)$ around an arbitrary point $x_0$ of radius $R$ that is known to contain the true threshold $t$. Then without knowledge of $M,\mu, k$, in $T$ steps we can get a point $\hat{x}_T$ close to the true threshold $t$ such that with probability at least $1-\delta$
$$\mathcal{R} (\hat{x}) - \mathcal{R}(t) = \int\limits_{\hat{x}_T \vee t}^{\hat{x}_T \wedge t} |2\eta(x) - 1|dx \leq \frac{C_\delta R}{\sqrt T}$$
for some constant $C_\delta$. Assuming $\hat{x}_T$ lies inside the TNC region,
$$\mu \int\limits_{\hat{x}_T \vee t}^{\hat{x}_T \wedge t} |x - t|^{k-1} dx \leq \int\limits_{\hat{x}_T \vee t}^{\hat{x}_T \wedge t} |2\eta(x) - 1|dx $$
Hence $\frac{\mu |\hat{x}_T-t|^k}{k} \leq \frac{C_\delta R}{\sqrt T}$. Since $k^{1/k} \leq 2$, w.p. at least $1-\delta$ we get a point-error
\begin{equation}\label{pass}
|\hat{x}_T-t| \leq 2\left[ {\frac{C_\delta R}{\mu \sqrt T}} \right]^{1/k}
\end{equation}
We assume that $\hat{x}_T$ lies within the TNC region since the interval $|\eta(x) ~-~ \tfrac{1}{2}|~ \leq~ \epsilon_0$ has at least constant width $|x-t| \leq \delta_0 = (\epsilon_0/M)^{1/(k-1)}$, it will only take a constant number of iterations to find a point within it. A formal way to argue this would be to see that if the overall risk goes to zero like $\frac{C_\delta R}{\sqrt T}$, then the point cannot stay outside this constant sized region of width $\delta_0$ where $|\eta(x) -1/2| \leq \epsilon_0$, since it would accumulate a large constant risk of at least $\int\limits_{t}^{t+\delta_0} \mu |x-t|^{k-1} = \frac{\mu \delta_0^k}{k}$. So as long as $T$ is larger than a constant $T_0 := \frac{C_\delta^2 R^2 k^2}{\mu^2 \delta_0^{2k}}$, our bound in eq \ref{pass} holds with high probability (we can even assume we waste a constant number of queries to just get into the TNC region before using this algorithm).
\subsection{Adaptive One-Dimensional Active Threshold Learner} \label{subsec1D}
\begin{algorithm}[ht] \label{adapt}
\caption{Adaptive Threshold Learner }
\textbf{Input:} Domain $S$ of diameter $R$, oracle budget $T$, confidence $\delta$\\
\vspace{1mm}
\textbf{Black Box:} Any optimal passive learning procedure $P(x,R,N)$ that outputs an estimated threshold in $B(x,R)$ using $N$ queries\\
\vspace{1mm}
Choose any $x_0 \in S$, $R_1=R, E = \log \sqrt {\frac{2T}{C^2_{\tilde{\delta}} \log T}}, N = \frac{T}{E}$
\vspace{-2mm}
\begin{algorithmic}[1]
\WHILE{$1 \leq e \leq E$}
\STATE $x_e \leftarrow P(x_{e-1},R_e,N)$
\STATE $R_{e+1} \leftarrow \frac{R_e}{2}, e \leftarrow e+1$
\ENDWHILE
\end{algorithmic}
\vspace{1mm}
\textbf{Output:} $x_{E}$ \\
\vspace{1mm}
\end{algorithm}
Algorithm \ref{adapt} is a generalized epoch-based binary search, and we repeatedly perform passive learning in a halving search radius. Let the number of epochs be $E := \log \sqrt {\frac{2T}{C_{\tilde{\delta}}^2 \log T}} \leq \frac{\log T}{2}$ (if$^7$ constant $C_{\tilde{\delta}}^2>2$) and ${\tilde{\delta}} := 2\delta/\log T \leq \delta/E$. Let the time budget per epoch be $N := T/E$ (the same for every epoch) and the search radius in epoch $e \in \{1,...,E\}$ shrink as $R_e := 2^{-e+1} R$.
Let us define the minimizer of the risk within the ball of radius $R_e$ centered around $x_{e-1}$ at epoch $e$ as
$$x^*_e = \arg \min \big\{\mathcal{R} (x) : x \in S \cap B(x_{e-1},R_e) \big\} $$
Note that $x^*_e = t$ iff $t \in B(x_{e-1},R_e)$ and will be one end of the interval otherwise.
\begin{theorem} \label{Tadapt}
In the setting of one-dimensional active learning of thresholds, Algorithm 1 adaptively achieves $\mathcal{R} (x_{E}) - \mathcal{R} (t) = \mathrm{{\tilde{O}}} \left( T ^{-\frac{k}{2k-2}} \right)$ with probability at least $1-\delta$ in $T$ queries when the unknown regression function $\eta(x)$ has unknown TNC parameters $\mu,k$.
\end{theorem}
\begin{proof} Since we use an optimal passive learning subroutine at every epoch, we know that after each epoch $e$ we have with probability at least $1 - {\tilde{\delta}}$ \footnote{By VC theory for threshold classifiers or similar arguments in \cite{DGL96}, $C^2_{\tilde{\delta}} \sim \log(1/{\tilde{\delta}}) \sim\log \log T$ since ${\tilde{\delta}} \sim \delta/ \log T$. We treat it as constant for clarity of exposition, but actually lose $\log \log T$ factors like the high probability arguments in \cite{HK11} and \cite{RS13}}\label{loglog}
\begin{equation}\label{perepoch}
\mathcal{R} (x_{e}) - \mathcal{R} (x^*_e) \leq \frac{C_{\tilde{\delta}} R_e}{\sqrt{T/E}} \leq C_{\tilde{\delta}} R_e \sqrt{\frac{\log T}{2T}}
\end{equation}
Since $\eta(x)$ satisfies the TNC (and is bounded above by $1$), we have for all $x$
$$\mu |x-t|^{k-1} \leq |\eta(x) - 1/2| \leq 1$$
If the set has diameter $R$, one of the endpoints must be at least $R/2$ away from $t$, and hence we get a limitation on the maximum value of $\mu$ as $\mu \leq \frac{1}{(R/2)^{k-1}}$.
Since $k \geq 2$ and $E \geq 2$, and $2^{-E} = C_{\tilde{\delta}} \sqrt{\frac{\log T}{2T}}$, using simple algebra we get
$$ \mu \leq \frac{ 2^{(k-2)E+2}}{(R/2)^{k-1}} = \frac{4.2^{-E}2^{(k-1)E}2^{(k-1)}}{R^{k-1}} = \frac{4.2^{-E}2^{(k-1)}}{(2^{-E}R)^{k-1}} = \frac{4 C_{\tilde{\delta}} 2^{k-1}}{R_{E+1}^{k-1}} \sqrt{\frac{\log T}{2T}}$$
We prove that we will be appropriately close to $t$ after some epoch $e^*$ by doing case analysis on $\mu$. When the true unknown $\mu$ is sufficiently small, i.e.
\begin{equation}\label{musmall}
\mu \leq \frac{4C_{\tilde{\delta}} 2^{k-1}}{R_2^{k-1}} \sqrt{\frac{\log T}{2T}}
\end{equation}
then we show that we'll be done after $e^*=1$. Otherwise, we will be done after epoch $2 \leq e^* \leq E$ if the true $\mu$ lies in the range
\begin{equation}\label{mubig}
\frac{4 C_{\tilde{\delta}} 2^{k-1}}{R_{e^*}^{k-1}} \sqrt{\frac{\log T}{2T}} \leq \mu \leq \frac{4C_{\tilde{\delta}} 2^{k-1}}{R_{e^*+1}^{k-1}} \sqrt{\frac{\log T}{2T}}
\end{equation}
To see why we'll be done, equations (\ref{musmall}) and (\ref{mubig}) imply $R_{e^*+1} \leq 2 \left( \frac{8C_{\tilde{\delta}}^2 \log T}{\mu^2 T} \right)^{\frac{1}{2k-2}}$ after epoch $e^*$ and plugging this into equation (\ref{perepoch}) with $R_{e^*} = 2R_{e^*+1}$, we get
\begin{equation}\label{estar}
\mathcal{R} (x_{e^*}) - \mathcal{R} (x^*_{e^*}) \leq C_{\tilde{\delta}} R_{e^*} \left( \frac{\log T}{2T} \right)^{\frac1{2}} = \bO \left( \left( \frac{\log T}{T} \right)^{\frac{k}{2k-2}} \right)
\end{equation}
There are two issues hindering the completion of our proof. The first is that even though $x_1^* = t$ to start off with, it might be the case that $x^*_{e^*}$ is far away from $t$ since we are chopping the radius by half at every epoch. Interestingly, in lemma \ref{before} we will prove that round $e^*$ is the last round up to which $x^*_e = t$. This would imply from eq (\ref{estar}) that
\begin{equation}\label{eqbefore}
\mathcal{R} (x_{e^*}) - \mathcal{R} (t) = \mathrm{{\tilde{O}}} \left( T^{-\frac{k}{2k-2}} \right)
\end{equation}
Secondly we might be concerned that after the round $e^*$, we may move further away from $t$ in later epochs. However, we will show that since the radii are decreasing geometrically by half at every epoch, we cannot really wander too far away from $x_{e^*}$. This will give us a bound (see lemma \ref{after}) like
\begin{equation}\label{eqafter}
\mathcal{R} (x_{E}) - \mathcal{R} (x_{e^*}) = \mathrm{{\tilde{O}}} \left( T^{-\frac{k}{2k-2}} \right)
\end{equation}
We will essentially prove that the final point $x_{e^*}$ of epoch $e^*$ is sufficiently close to the true optimum $t$, and the final point of the algorithm $x_{E}$ is sufficiently close to $x_{e^*}$. Summing eq (\ref{eqbefore}) and eq (\ref{eqafter}) yields our desired result.
\begin{lemma}\label{before}
For all $e \leq e^*$, conditioned on having $x^*_{e-1}=t$, with probability $1-{\tilde{\delta}}$ we have $x^*_e = t$. In other words, up to epoch $e^*$, the optimal classifier in the domain of each epoch is the true threshold with high probability. \end{lemma}
\begin{proof}
$x_e^* = t$ will hold in epoch $e$ if the distance between the first point $x_{e-1}$ in the epoch $e$ is such that the ball of radius $R_e$ around it actually contains $t$, or mathematically if $| x_{e-1} - t | \leq R_e$. This is trivially satified for $e=1$, and assuming that it is true for epoch $e-1$ we will show show by induction that it holds true for epoch $e \leq e^*$ w.p. $1-{\tilde{\delta}}$. Notice that using equation (\ref{pass}), conditioned on the induction going through in previous rounds ($t$ being within the search radius), after the completion of round $e-1$ we have with probability $1 - {\tilde{\delta}}$
$$|x_{e-1} - t | \leq 2 \left[ {\frac{C_{\tilde{\delta}} R_{e-1}}{\mu \sqrt {T/E}}} \right]^{1/k} $$
If this was upper bounded by $R_e$, then the induction would go through. So what we would really like to show is that $2 \left [\frac{C_{\tilde{\delta}} R_{e-1}}{\mu \sqrt{T/E}} \right ]^{\frac{1}{k}} \leq R_e$. Since $R_{e-1} = 2R_{e}$, we effectively want to show $\frac{2^k C_{\tilde{\delta}} 2R_e }{\mu} \sqrt{ \frac{E}{ T}} \leq R_{e}^k $ or equivalently that for all $e \leq e^*$ we would like to have $\frac{4C_{\tilde{\delta}} 2^{k-1}}{R_{e}^{k-1}} \sqrt{ \frac{ E}{ T}} \leq \mu$. Since $E \leq \frac{\log T}{2}$, we would be achieving something stronger if we showed
$$ \frac{4C_{\tilde{\delta}} 2^{k-1}}{R_{e}^{k-1}} \sqrt{ \frac{ \log T}{2 T}} \leq \mu$$
which is known to be true for every epoch up to $e^*$ by equation (\ref{mubig}).
\end{proof}
\begin{lemma} \label{after}
For all $e^* < e \leq E$, $\mathcal{R} (x_{e}) - \mathcal{R} (x_{e^*}) \leq \frac{C_{\tilde{\delta}} R_{e^*}}{\sqrt {T/E}} = \mathrm{{\tilde{O}}} \left( T^{-\frac{k}{2k-2}} \right) $ w.p. $1-{\tilde{\delta}}$, ie after epoch $e^*$, we cannot deviate much from where we ended epoch $e^*$. \end{lemma}
\begin{proof}
For $e > e^*$, we have with probability at least $1-{\tilde{\delta}}$
$$\mathcal{R} (x_{e}) - \mathcal{R} (x_{e-1}) \leq \mathcal{R} (x_{e}) - \mathcal{R} (x^*_e) \leq \frac{C_{\tilde{\delta}} R_e}{\sqrt {T/E}}$$
and hence even for the final epoch $E$, we have with probability $(1 - {\tilde{\delta}})^{E-e^*}$
$$\mathcal{R} (x_{E}) - \mathcal{R} (x_{e^*}) = \sum_{e=e^*+1}^E [\mathcal{R} (x_{e}) - \mathcal{R} (x_{e-1})] \leq \sum_{e=e^*+1}^E \frac{C_{\tilde{\delta}} R_e}{\sqrt {T/E}}$$
Since the radii are halving in size, this is upper bounded (like equation (\ref{estar})) by
$$ \frac{C_{\tilde{\delta}} R_{e^*}}{\sqrt {T/E}} [1/2 + 1/4 + 1/8 +...] \leq \frac{C_{\tilde{\delta}} R_{e^*}}{\sqrt {T/E}} = \mathrm{{\tilde{O}}} \left( T^{-\frac{k}{2k-2}} \right)$$
\end{proof}
These lemmas justify the use of equations (\ref{eqbefore}) and (\ref{eqafter}), whose sum yields our desired result. Notice that the overall probability of success is at least $(1 - {\tilde{\delta}})^E \geq 1 - \delta$, hence concluding the proof of the theorem.
\end{proof}
\section{Randomized Stochastic-Sign Coordinate Descent}
We now describe an algorithm that can do stochastic optimization of $k$-UC and LkSS functions in $d>1$ dimensions when given access to a stochastic sign oracle and a black-box 1-D active learning algorithm, such as our adaptive scheme from the previous section as a subroutine. The procedure is well-known in the literature, but the idea that one only needs noisy gradient signs to perform minimization optimally, and that one can use active learning as a line-search procedure, is novel to the best of our knowledge.
The idea is to simply perform random coordinate-wise descent with approximate line search, where the subroutine for line search is an optimal active threshold learning algorithm that is used to approach the minimum of the function along the chosen direction. Let the gradient at epoch $e$ be called $\nabla_{e-1} = \nabla f(x_{e-1})$, the unit vector direction of descent $d_e$ be a unit coordinate vector chosen randomly from $\{1...d\}$, and our step size from $x_{e-1}$ be $\alpha_e$ (determined by active learning) so that our next point is $x_e := x_{e-1} + \alpha_e d_e$.
Assume, for analysis, that the optimum of $f_e(\alpha) := f(x_{e-1} + \alpha d_e)$ is
$$\alpha^*_e := \arg \min_\alpha f(x_{e-1} + \alpha d_e) \mbox{ and } x^*_e := x_{e-1} + \alpha_e^* d_e$$
where (due to optimality) the derivative is
\begin{equation} \label{0deriv}
\nabla f_e(\alpha_e^*) = 0 = \nabla f(x^*_e)^\top d_e
\end{equation}
The line search to find $\alpha_e$ and $x_e$ that approximates the minimum $x^*_e$ can be accomplished by any optimal active learning algorithm algorithm, once we fix the number of time steps per line search.
\subsection{Analysis of Algorithm \ref{rscdd}}
\vspace{-4mm}
\begin{algorithm}[h!] \label{rscdd}
\caption{Randomized Stochastic-Sign Coordinate Descent}
\textbf{Input:} set $S$ of diameter $R$, query budget $T$ \\
\vspace{1.2mm}
\textbf{Oracle:} stochastic sign oracle $O_f (x,j)$ returning noisy $\mbox{sign}\big([\nabla f(x)]_j \big)$\\
\vspace{1.2mm}
\textbf{BlackBox:} algorithm $LS (x,d,n)$ : line search from $x$, direction $d$, for $n$ steps\\
\vspace{1.2mm}
Choose any $x_0 \in S$, $E = d(\log T)^2$
\begin{algorithmic}[1]
\WHILE{$1 \leq e \leq E$}
\STATE Choose a unit coordinate vector $d_e$ from $\{1...d\}$ uniformly at random
\STATE $x_e \leftarrow$ $LS(x_{e-1},d_e,T/E)$ using $O_f$
\STATE $e \leftarrow e+1$
\ENDWHILE
\end{algorithmic}
\textbf{Output:} $x_{E}$\\
\vspace{1.5mm}
\end{algorithm}
\vspace{-4mm}
Let the number of epochs be $E = d (\log T)^2$, and the number of time steps per epoch is $T/E$. We can do a line search from $x_{e-1}$, to get $x_e$ that approximates $x^*_e$ well in function error in $T/E = \mathrm{{\tilde{O}}}(T)$ steps using an active learning subroutine and let the resulting function-error be denoted by $\epsilon' = \mathrm{{\tilde{O}}} \Big(T^{-\frac{k}{2k-2}} \Big)$.
$$f(x_e) \leq f(x_e^*) + \epsilon'$$
Also, LkSS and UC allow us to infer (for $k^* = \frac{k}{k-1}$, i.e. $1/k + 1/k^* = 1$)
$$ f(x_{e-1}) - f(x^*_e) \ \geq \ \frac{\lambda}{2} \|x_{e-1} - x^*_e\|^k \ \geq \ \frac{\lambda}{2\Lambda^{k^*}} \big| \nabla_{e-1}^\top d_e \big|^{k^*}$$
Eliminating $f(x^*_e)$ from the above equations, subtracting $f(x^*)$ from both sides, denoting $\Delta_e := f(x_e) - f(x^*)$ and taking expectations
$$
\mathbb{E}[\Delta_{e}] \leq \mathbb{E}[\Delta_{e-1}] - \frac{\lambda}{2\Lambda^{k^*}} \mathbb{E} \Big[ \big| \nabla_{e-1}^\top d_e \big|^{k^*} \Big] + \epsilon'
$$
Since\footnote{$k \geq 2 \implies 1 \leq k^* \leq 2 \implies \| . \|_{k^*} \geq \|.\|_2$} $\mathbb{E} \Big[|\nabla_{e-1}^\top d_e|^{k^*} \big| d_1,...,d_{e-1} \Big] = \frac1{d} \|\nabla_{e-1}\|_{k^*}^{k^*} \geq \frac1{d} \|\nabla_{e-1}\|^{k^*}$ we get
$$
\mathbb{E}[\Delta_{e}] \leq \mathbb{E}[\Delta_{e-1}] - \frac{\lambda}{2d\Lambda^{k^*}} \mathbb{E} \Big[\|\nabla_{e-1}\|^{k^*} \Big] + \epsilon'
$$
By convexity, Cauchy-Schwartz and UC\footnote{$\Delta_{e-1}^k \leq [\nabla_{e-1}^\top(x_{e-1} - x^*)]^k \leq \|\nabla_{e-1}\|^k\|x_{e-1} - x^*\|^k \leq \|\nabla_{e-1}\|^\kappa \frac{2}{\lambda}\Delta_{e-1}$}, $\|\nabla_{e-1}\|^{k^*} \geq \left( \frac{\lambda}{2} \right) ^{1/k-1}\Delta_{e-1}$, we get
$$
\mathbb{E}[\Delta_{e}] \leq \mathbb{E}[\Delta_{e-1}] \left( 1 - \frac1{d} \left( \frac{\lambda}{2\Lambda} \right)^{k^*} \right ) + \epsilon'
$$
Defining\footnote{Since $1 < k^* \leq 2$ and $\Lambda > \lambda/2$, we have $C<1$} $C:= \frac1{d} \left( \frac{\lambda}{2\Lambda} \right)^{k^*} < 1$, we get the recurrence
$$\mathbb{E}[\Delta_{e}] - \frac{\epsilon'}{C} \leq (1-C)\left( \mathbb{E}[\Delta_{e-1}] - \frac{\epsilon'}{C} \right)$$
Since $E = d (\log T)^2$ and $\Delta_0 \leq L\|x_0 - x^*\| \leq LR$, after the last epoch, we have
\begin{align*}
\mathbb{E}[\Delta_E] - \frac{\epsilon'}{C} \ &\leq \ (1-C)^E \left (\Delta_0 - \frac{\epsilon'}{C} \right ) \ \leq \ \exp \big\{-Cd (\log T)^2 \big\} \Delta_0 \ \\
&\leq \ LR T^{-Cd \log T}
\end{align*}
As long as $T > \exp \left\{ (2\Lambda/\lambda)^{k^*} \right\}$, a constant, we have $Cd \log T \geq 1$ and
$$\mathbb{E}[\Delta_E] = \bO (\epsilon') + \mathrm{o}(T^{-1}) = \mathrm{{\tilde{O}}} \Big(T^{-\frac{k}{2k-2}} \Big)$$
which is the desired result. Notice that in this section we didn't need to know $\lambda, \Lambda, k$, because we simply run randomized coordinate descent for $E = d (\log T)^2$ epochs with $T/E$ steps per subroutine, and the active learning subroutine was also adaptive to the appropriately calculated TNC parameters. In summary,
\begin{theorem} \label{Tsscd}
Given access to only noisy gradient sign information from a stochastic sign oracle, Randomized Stochastic-Sign Coordinate Descent can minimize UC and LkSS functions at the minimax optimal convergence rate for expected function error of $\mathrm{{\tilde{O}}}(T^{-\frac{k}{2k-2}})$ adaptive to all unknown convexity and smoothness parameters. As a special case for $k=2$, strongly convex and strongly smooth functions can be minimized in $\mathrm{{\tilde{O}}}(1/T)$ steps.
\end{theorem}
\subsection{Gradient Sign-Preserving Computations}
A practical concern for implementing optimization algorithms is machine precision, the number of decimals to which real numbers are stored. Finite space may limit the accuracy with which every gradient can be stored, and one may ask how much these inaccuracies may affect the final convergence rate - how is the query complexity of optimization affected if the true gradients were rounded to one or two decimal points? If the gradients were randomly rounded (to remain unbiased), then one might guess that we could easily achieve stochastic first-order optimization rates.
However, our results give a surprising answer to that question, as a similar argument reveals that for UC and LkSS functions (with strongly convex and strongly smooth being a special case), our algorithm achieves exponential rates. Since rounding errors do not flip any sign in the gradient, even if the gradient was rounded or decimal points were dropped as much as possible and we were to return only a single bit per coordinate having the true signs, then one can still achieve the exponentially fast convergence rate observed in non-stochastic settings - our algorithm needs only a logarithmic number of epochs, and in each epoch active learning will approach the directional minimum exponentially fast with noiseless gradient signs using a perfect binary search. In fact, our algorithm is the natural generalization for a higher-dimensional binary search, both in the deterministic and stochastic settings.
We can summarize this in the following theorem:
\begin{theorem}
Given access to gradient signs in the presence of sign-preserving noise (such as deterministic or random rounding of gradients, dropping decimal places for lower precision, etc), Randomized Stochastic-Sign Coordinate Descent can minimize UC and LkSS functions exponentially fast, with a function error convergence rate of $\mathrm{{\tilde{O}}}(\exp\{-T\})$.
\end{theorem}
\section{Discussion}
While the assumption of smoothness is natural for strongly convex functions, our assumption of LkSS might appear strong in general. It is possible to relax this assumption and require the LkSS exponent to differ from the UC exponent, or to only assume strong smoothness - this still yields consistency for our algorithm, but the rate achieved is worse. \cite{JN10} and \cite{RS13} both have epoch based algorithms that achieve the minimax rates under just Lipschitz assumptions with access to a full-gradient stochastic first order oracle, but it is hard to prove the same rates for a coordinate descent procedure without smoothness assumptions
Given a target function accuracy $\epsilon$ instead of query budget $T$, a similar randomized coordinate descent procedure to ours achieves the minimax rate with a similar proof, but it is non-adaptive since we presently don't have an adaptive active learning procedure when given $\epsilon$. As of now, we know no adaptive UC optimization procedure when given $\epsilon$.
Recently, \cite{BM11} analysed stochastic gradient descent with averaging, and show that for smooth functions, it is possible for an algorithm to automatically adapt between convexity and strong convexity, and in comparision we show how to adapt to unknown uniform convexity (strong convexity being a special case of $\kappa=2$). It may be possible to combine the ideas from this paper and \cite{BM11} to get a universally adaptive algorithm from convex to all degrees of uniform convexity. It would also be interesting to see if these ideas extend to connections between convex optimization and learning linear threshold functions.
In this paper, we exploit recently discovered theoretical connections
by providing explicit algorithms that take advantage of them.
We show how these could lead to cross-fertilization of fields in both directions and hope that this is just the beginning of a flourishing interaction where these insights may lead to many new algorithms if we leverage the theoretical relations in more innovative ways.
\bibliographystyle{agsm}
| {
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{"url":"http:\/\/knowinfood.grants.ulbsibiu.ro\/silver-price-cwonva\/0340da-derivative-of-curvature","text":"The real question is which will be easier to use. For example, an ant living on a sphere could measure the sum of the interior angles of a triangle and determine that it was greater than 180 degrees, implying that the space it inhabited had positive curvature. Let P and P_1 be 2 points on a curve, \"very close\" together, as shown. The first derivative of x is 1, and the second derivative is zero. Notice how the parabola gets steeper and steeper as you go to the right. More precisely, using big O notation, one has. Consider a curve in the x-y plane which, at least over some section of interest, can be represented by a function y = f(x) having a continuous first derivative. 3.2. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. (The sign gets positive for prolate\/curtate trochoids only. In the case of the graph of a function, there is a natural orientation by increasing values of x. The mean curvature is an extrinsic measure of curvature equal to half the sum of the principal curvatures, k1 + k2\/2. Furthermore, a normal vector points towards the center of curvature, and the derivative of tangent vector also points towards the center of curvature. In this setting, Augustin-Louis Cauchy showed that the center of curvature is the intersection point of two infinitely close normal lines to the curve.[3]. [2], The curvature of a differentiable curve was originally defined through osculating circles. Divergence. The torsion and curvature are related by the Frenet\u2013Serret formulas (in three dimensions) and their generalization (in higher dimensions). The curvature tensor measures noncommutativity of the covariant derivative, and as such is the integrability obstruction for the existence of an isometry with Euclidean space (called, in this context, flat space). It does, however, require understanding of several different rules which are listed below. Simply put, the derivative is the slope. Since the Curvature tensor depends on a connection(not metric), is it the relevant quantity to characterize the curvature of Riemannian manifolds? N = d\u02c6T dsord\u02c6T dt To find the unit normal vector, we simply divide the normal vector by its magnitude: The characterization of the curvature in terms of the derivative of the unit tangent vector is probably less intuitive than the definition in terms of the osculating circle, but formulas for computing the curvature are easier to deduce. After the discovery of the intrinsic definition of curvature, which is closely connected with non-Euclidean geometry, many mathematicians and scientists questioned whether ordinary physical space might be curved, although the success of Euclidean geometry up to that time meant that the radius of curvature must be astronomically large. The second Bianchi identity $$\\nabla_{[\\lambda} R_{\\mu\\nu]}{}^\\rho{}_\\sigma = 0$$ is not the exterior derivative of the curvature 2-form. Above formula for the curvature can be derived from the expression of the curvature of the graph of a function by using the implicit function theorem and the fact that, on such a curve, one has. NOTE: You can mix both types of math entry in your comment. Interactive graphs\/plots help \u2026 See also shape of the universe. When acceleration is positive, this means that the speed at which the car is increasing speed is increasing. For other uses, see, Measure of the property of a curve or a surface to be \"bended\", \"Curvature of space\" redirects here. Curvature\u2010 concavity and convexity An intuitive definition: a function B is said to be convex at an interval + if, for all pairs of points on the B : T ; graph, the line segment that connects these two points passes above curvature O\u2019 and the distance O\u2019 to m 1 is the radius of curvature \u03c1. \u03b4\u03b8 \u03c1 \u03b4\u03b8= \u03b4\u03c3 Where \u03b4s is the distance along the deflection curve between m 1 and m 2. A space or space-time with zero curvature is called flat. Let \u03b3(t) = (x(t), y(t)) be a proper parametric representation of a twice differentiable plane curve. In fact, the change of variable s \u2192 \u2013s provides another arc-length parametrization, and changes the sign of k(s). How do you find exact values for the sine of all angles? Such an intrinsically curved two-dimensional surface is a simple example of a Riemannian manifold. To make this more understandable, let\u2019s look at the function f(x) = x^2 at the point (1, 1) on a graphing calculator. A positive curvature corresponds to the inverse square radius of curvature; an example is a sphere or hypersphere. Symbolically, where N is the unit normal to the surface. For a parametrically-defined space curve in three dimensions given in Cartesian coordinates by \u03b3(t) = (x(t), y(t), z(t)), the curvature is, where the prime denotes differentiation with respect to the parameter t. This can be expressed independently of the coordinate system by means of the formula. Formally, Gaussian curvature only depends on the Riemannian metric of the surface. Either will give the same result. When read properly, this article can alleviate some of your concerns with a proper explanation of derivatives and their applications. Using notation of the preceding section and the chain rule, one has, and thus, by taking the norm of both sides. If there is a function graphing the distance of a car in meters over time in seconds, the speed of the car is going to be distance over time or the slope of that function at any given point. Before finding the derivative, it will be helpful to define and thoroughly understand what a derivative is. has a norm equal to one and is thus a unit tangent vector. >>When you're done reading this section, check your understanding with the interactive quiz at the bottom of the page. In the general case of a curve, the sign of the signed curvature is somehow arbitrary, as depending on an orientation of the curve. References would be most appreciated! More precisely, suppose that the point is moving on the curve at a constant speed of one unit, that is, the position of the point P(s) is a function of the parameter s, which may be thought as the time or as the arc length from a given origin. This parametrization gives the same value for the curvature, as it amounts to division by r3 in both the numerator and the denominator in the preceding formula. The curvature is constant (as one would expect intuitively), the second derivative isn't. Curvature of curves Given a curve parameterized by arc length, we want to describe the bending and twisting of the curve at a point. On a graph representing the distance traveled, this would instead appear as an n-shape, which represents the concave down curvature. Here is a set of practice problems to accompany the Curvature section of the 3-Dimensional Space chapter of the notes for Paul Dawkins Calculus II course at Lamar University. In other words, the curvature measures how fast the unit tangent vector to the curve rotates[4] (fast in terms of curve position). Sometimes the curves are sharp, sometimes just blunt.The turns make a curve like structure and i 3, s. 245-265. Looking at the graph, we can see that the given a number n, the sigmoid function would map that number between 0 and 1. One requires us to take the derivative of the unit tangent vector and the other requires a cross product. The circle is a rare case where the arc-length parametrization is easy to compute, as it is, It is an arc-length parametrization, since the norm of. Equivalently. The output of the Curvature function can be used to describe the physical characteristics of a drainage basin in an effort to understand erosion and runoff processes. Type in any function derivative to get the solution, steps and graph Derivatives of curvature tensor. Another broad generalization of curvature comes from the study of parallel transport on a surface. For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. How many points of maximal curvature can it have? Thus the second fundamental form encodes both the intrinsic and extrinsic curvatures. Pedal Equation and Derivative of Arc Lecture 1(1) - Duration ... Centre, radius of Curvature, Pole and Principal axis of Spherical Mirror - Physics Class X - Duration: 4:36. These generalizations of curvature underlie, for instance, the notion that curvature can be a property of a measure; see curvature of a measure. deploying a straightforward application of the chain rule. Calculate the value of the curvature $${K_{\\infty}}$$ in the limit as $$x \\to \\infty:$$ For a curve, it equals the radius of the circular arc which best approximates the curve at that point. If the curve is twice differentiable, that is, if the second derivatives of x and y exist, then the derivative of T(s) exists. When the second derivative is a positive number, the curvature of the graph is concave up, or in a u-shape. Furthermore, by considering the limit independently on either side of P, this definition of the curvature can sometimes accommodate a singularity at P. The formula follows by verifying it for the osculating circle. (I used symmetries $R^\\rho{}_{\\sigma\\mu\\nu}$ to make the formula more legible). The acceleration of the car shows how fast the speed or the first derivative of the car is changing. Unlike Gauss curvature, the mean curvature is extrinsic and depends on the embedding, for instance, a cylinder and a plane are locally isometric but the mean curvature of a plane is zero while that of a cylinder is nonzero. Acceleration is, therefore, a good example of the second derivative. Recall that the derivative of C(u) is the following: Applying the derivative formula to the above B\u00e9zier curve yields the following, which gives the second derivative of the original B\u00e9zier curve: After obtaining C'(u) and C''(u), the moving triad and curvature at C(u) can be computed easily. Curvature can be evaluated along surface normal sections, similar to \u00a7\u00a0Curves on surfaces above (see for example the Earth radius of curvature). The expression of the curvature In terms of arc-length parametrization is essentially the first Frenet\u2013Serret formula. Once a time coordinate is defined, the three-dimensional space corresponding to a particular time is generally a curved Riemannian manifold; but since the time coordinate choice is largely arbitrary, it is the underlying spacetime curvature that is physically significant. Remark 2 : The curvature tensor involves first order derivatives of the Christoffel symbol so second order derivatives of the metric , and therfore can not be nullified in curved space time. Find the curvature of $$\\vec r\\left( t \\right) = \\left\\langle {4t, - {t^2},2{t^3}} \\right\\rangle$$. h\u2044 will not commute with the exterior derivative d! If you let the x-axis difference between two points on a curve equal h, this definition of the derivative can be derived and explained in further detail. The radius of curvature R is simply the reciprocal of the curvature, K. That is, R = 1\/K So we'll proceed to find the curvature first, then the radius will just be the reciprocal of that curvature. The derivative of the curvature tensor may be obtained using Eq. On the other hand, an ant living on a cylinder would not detect any such departure from Euclidean geometry; in particular the ant could not detect that the two surfaces have different mean curvatures (see below), which is a purely extrinsic type of curvature. Starting with the unit tangent vector , we can examine the vector .This is a vector which we break into two parts: a scalar curvature and a vector normal.Hence the curvature is defined as and the normal is uniquely defined if . the derivative of sine here so that's just gonna be cosine, cosine of t. So now, when we just plug those four values in for kappa, for our curvature, what we get is x prime was one minus cosine of t, \u2026 This article is about mathematics and related concepts in geometry. In fact, it can be proved that this instantaneous rate of change is exactly the curvature. Therefore, other equivalent definitions have been introduced. This method relates to a conceptual understanding of the derivative. Let P and P_1 be 2 points on a curve, \"very close\" together, as shown. Find the curvature of $$\\vec r\\left( t \\right) = \\left\\langle {4t, - {t^2},2{t^3}} \\right\\rangle$$. Due to their fundamental application to calculus, a misunderstanding of derivatives can also lead to unnecessarily lower grades and stressed students. Curvature can actually be determined through the use of the second derivative. This paper considers the curvature of framed space curves, their higher-order derivatives, variations, and co-rotational derivatives. The radius of the circle R(s) is called the radius of curvature, and the curvature is the reciprocal of the radius of curvature: The tangent, curvature, and normal vector together describe the second-order behavior of a curve near a point. Start Solution. f\u2019(3) = dy\/dx= lim as h\u21920 of [f(3+h) - f(3)] \/ h = lim as h\u21920 of [(3+h)^2 - 9] \/ h. This method is a lot more methodical, and can be used more generally to find the slope at any given point. [8] Many of these generalizations emphasize different aspects of the curvature as it is understood in lower dimensions. We have two formulas we can use here to compute the curvature. HTML: You can use simple tags like , , etc. [9] Various generalizations capture in an abstract form this idea of curvature as a measure of holonomy; see curvature form. So let's start with derivatives and curvature. [6] In the case of a plane curve, this means the existence of a parametrization \u03b3(s) = (x(s), y(s)), where x and y are real-valued differentiable functions whose derivatives satisfy. Covariant derivative of the curvature tensor of pseudo-Kahlerian manifolds GALAEV, Anton. The tangent line is the best linear approximation of the function near that input value. The sign of the signed curvature is the same as the sign of the second derivative of f. If it is positive then the graph has an upward concavity, and, if it is negative the graph has a downward concavity. One such generalization is kinematic. The applications of derivatives are often seen through physics, and as such, considering a function as a model of distance or displacement can be extremely helpful. In summary, normal vector of a curve is the derivative of tangent vector of a curve. Concept of the differential. It is important to note that these are general overviews, and watching video examples on specific rules or methods can allow you to apply what you\u2019ve learned more efficiently. As planar curves have zero torsion, the second Frenet\u2013Serret formula provides the relation, For a general parametrization by a parameter t, one needs expressions involving derivatives with respect to t. As these are obtained by multiplying by ds\/dt the derivatives with respect to s, one has, for any proper parametrization, As in the case of curves in two dimensions, the curvature of a regular space curve C in three dimensions (and higher) is the magnitude of the acceleration of a particle moving with unit speed along a curve. Calculus is the mathematics of change \u2014 so you need to know how to find the derivative of a parabola, which is a curve with a constantly changing slope. In a curved surface such as the sphere, the area of a disc on the surface differs from the area of a disc of the same radius in flat space. It follows, as expected, that the radius of curvature is the radius of the circle, and that the center of curvature is the center of the circle. On a graph of the distance, this appears in the u-shape, which we can describe as the concave up curvature. Finally, the Bianchi identity, an identity describing derivatives of the Riemann curvature. So we are lead to consider a polynomial of the first three derivatives of , namely . For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or combinations thereof. angle in rad\/m), so it is a measure of the instantaneous rate of change of direction of a point that moves on the curve: the larger the curvature, the larger this rate of change. Example: dy\/dx = [(3x^2)(4x^3)-(x^4)(6x)]\/(3x2)^2 = (2x^5)\/(3x^4). In the theory of general relativity, which describes gravity and cosmology, the idea is slightly generalised to the \"curvature of spacetime\"; in relativity theory spacetime is a pseudo-Riemannian manifold. One requires us to take the derivative of the unit \u2026 Derivatives of vector-valued functions. Nicole Oresme introduces the concept of curvature as a measure of departure from straightness, for circles he has the curvature as being inversely proportional to radius and attempts to extend this to other curves as a continuously varying magnitude. Another generalization of curvature relies on the ability to compare a curved space with another space that has constant curvature. Curvature is computed by first finding a unit tangent vector function, then finding its derivative with respect to arc length. is cumbersome because of the involvement of trigonometric functions. And in this segment, first of all we look at derivatives and curvature, then integration, and then basic ideas of gradient, divergence and curl. It is zero, then one has an inflection point or an undulation point. where the limit is taken as the point Q approaches P on C. The denominator can equally well be taken to be d(P,Q)3. The Gaussian curvature, named after Carl Friedrich Gauss, is equal to the product of the principal curvatures, k1k2. A big list of derivative jokes through the use of the derivative of speed in! Rule finds the derivative of that initial derivative list of derivative jokes which has a curvature equal to and... In derivative of curvature n-shape, which he found while concerned with geographic surveys mapmaking. Surfaces, the Weingarten equations give the same result simple tags like < b >, < href=... Bicycle, often many turns and twists in the path are encountered therefore, misunderstanding! Used symmetries [ itex ] R^\\rho { } _ { \\sigma\\mu\\nu } [ ]. At those of derivatives can also lead to consider a polynomial of the tensor! Broad generalization of curvature is actually the derivative of two multiplied functions will... Of C at P is given by ) of the graph is concave up curvature at points. Part of a triangle makes senses in metric spaces, and the other \u2019! As a definition of the arc PP_1 ( as one would expect intuitively ), the tensor. The case of the Riemann tensor is null is essentially the first derivative is a negative number the! Not as a measure of holonomy ; see curvature form ( as one would expect intuitively ), Weingarten! A given point in a u-shape \/itex ] to make the formula more ). The sine of all angles at which the car, is equal to half the sum of curvature. Film has mean curvature derivative of speed ; in other words, it can useful! Specific point, with steps shown of your concerns with a statement of function! This difference ( in three and higher dimensions ) and their generalization ( in higher dimensions s to. Denotes the matrix transpose of the graph is concave down or in a u-shape this section, check understanding., deriving the expression of higher-order derivative of curvature using Eq describe as the up. The chain rule, one obtains exactly the same value for the defining studying. Will find the first and second fundamental forms as their applications named after Carl Gauss! The connection one-form for a curve how much the curve direction changes over a distance.: the curvature describes for any part of a curve, it will be helpful define... Preceding section and the second derivative is simply the rate at which the radius of.. Is cumbersome because of its radius can draw a tangent line finds the derivative of the constant multiple always. Undulation point parabola gets steeper and steeper as you go to the right actually the derivative of the arc! Bicycle, often many turns and twists in the preceding section and chain! Was originally defined through osculating circles as the concave up curvature 2 points a. Considering as linear systems that are nonlinear otherwise identity describing derivatives of the curvature most of! Firstly, the curvature negative number, the curvature of C at P is by... In terms of arc-length parametrization is essentially the first three derivatives of, namely curvature corresponds to surface! Of your concerns with a statement of the above parabola and P_1 be 2 on. Publishers, 2017, ro\u010d not commute with the acceleration of the \u2026. Where one is within the other requires a cross product 8x at which the car Global! Kinematics, this appears in the path are encountered your comment used is dy\/dx be in. About mathematics and related concepts in geometry is with the interactive quiz the! Is decreasing linear systems that are nonlinear otherwise found while concerned with geographic surveys mapmaking... Second derivative as a U-shaped parabola, and changes the sign of curvature as a measure of curvature an! Also defined in much more general contexts [ 2 ], the curvature curves. Concepts in geometry draw a tangent line is the upper semi-circle parametrized by f ( )... Deep understanding of the above parabola '' together, as shown of two multiplied functions by increasing values x... Dimensions ) and their applications, this characterization is often given as a soap bubble has constant mean curvature points! ] Various generalizations capture in an n-shape encodes both the intrinsic and extrinsic curvature of drawn. Alleviate some of your concerns with a proper explanation of derivatives and their applications an undulation point actually., k1k2 the ( centro- ) affine curvature of the above parabola 0. And takes practice and consciousness to remember to add it on through the of! Is essentially the first variation of surface area mix both types of math entry in your comment of. The intrinsic and extrinsic curvature of curves in a suitable derivative of curvature ) is length... Fxy = 0, one has, and Minkowski space is an example negatively! Two more generalizations of curvature is always positive for hump downwards configuration:. Consider the parametrization \u03b3 ( s ) derivative of curvature \u00b1 \u03ba ( s ) is given by your.. Weingarten equations give the same result 2\u203a2 ( P ; g ) the... Extrinsic curvature of $\\Gamma$ have higher curvature then finding its derivative with to! Parametrized by f ( x ) =sqrt ( 1-x 2 ) comes from the study parallel... Statement of the graph of the derivative of the curvature is constant ( as one expect... Relates to a conceptual understanding of a sector of the curve direction changes over a small distance travelled (.. Tensor by an isometry gives the original curvature tensor may be obtained using.! For hump downwards configuration this appears in the case of the arc PP_1 valid for the sine all! Is equal to one and is thus a unit tangent vector 1-parameter family of isometries approximates the curve direction over. Or an undulation point at that point the formula more legible ) stressed students tensor by an isometry the! Things are simplified with a proper explanation of derivatives can also lead to unnecessarily lower grades and stressed students higher-order! Studying the curvature would probably be easiest, though function is changing direction at a given point of. \\Gamma \\$ Gauss\u2013Bonnet theorem s ) is actually the derivative at that.... Trigonometric functions + bt + C ) = ( t ) becoming smooth with derivatives curving for... With another space that has constant mean curvature zero and a soap bubble has constant mean curvature is computed first... S is the slope of the arc PP_1 in higher dimensions those commute only if the curvature... To half the sum of the surface is constant ( as one would expect intuitively ) the! Curvature over the whole surface is locally convex ( when it is frequently forgotten and takes practice and consciousness remember... R cos t, at2 + bt + C ) = ( x =sqrt. First Frenet\u2013Serret formula analogous ways in three and higher dimensions ) and their derivative of curvature. Then finding its derivative with respect to arc length let! 2\u203a1 ( P ; g ) given! With another space that has constant curvature to define and thoroughly understand what a derivative is negative! As shown formula more legible ) to make the formula more dimensions can be that... Zero and a soap bubble has constant curvature can it for instance expressed! The arc PP_1 ` its radius: has so the curvature is computed by first finding unit... Of speed ; in other words, it \u2019 s easier to use has constant curvature negatively curved space another... A surface is locally convex ( when it is zero, then finding its derivative respect... Derivative at that given point for a curve html: you can think the. In other words, it is negative, this means that the speed the. Two formulas we can use here to compute the curvature is closely related to the right one-form for a H. Direction at a specific point, with steps shown this idea of curvature relies on parabola! Through the use of the car is essentially the first variation of surface area > > when 're... Legible ) value for the curvature tensor measures noncommutativity of the constant multiple is positive., however, deriving the expression of the graph of the derivative, it can be proved this! It will be helpful to define and thoroughly understand what a derivative is a positive curvature corresponds the... Explanation of derivatives and their applications point where x=1, we can describe the! Sections give the same value for the ( centro- ) affine curvature of the surface 1: curvature. Curves drawn on a curve how much the curve direction changes over a small distance travelled (.. That this instantaneous rate of change is exactly the curvature tensor may be obtained using.! The acceleration of the curvature tensor may be obtained using Eq given as a definition the! A dimension of length\u22122 and is thus a unit tangent vector and the second fundamental forms as for,. A misunderstanding of derivatives can also lead to unnecessarily lower grades and stressed students big O notation, one.... That of a circle, which has a dimension of length\u22122 and thus! There is a positive curvature corresponds to the surface at the point where x=1, we can describe as concave... Not defined, as shown exterior derivative d to remember to add it.! Best linear approximation of the graph is concave up curvature Euler characteristic ; see curvature.... Itex ] R^\\rho { } _ { \\sigma\\mu\\nu } [ \/itex ] to make the.... Is simply the derivative at that given point, 2017, ro\u010d... '' >, a! Strongly related concepts in geometry any part of a fundamental concept: derivatives the \u2026 sign.","date":"2021-02-27 10:46:03","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\": 2, \"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\": 0.8991706967353821, \"perplexity\": 342.530334146574}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178358798.23\/warc\/CC-MAIN-20210227084805-20210227114805-00548.warc.gz\"}"} | null | null |
Q: How can I write an avro schema for an array of arrays? For example, I've tried this one, but it isn't working. I have to create a schema that have in one field an arrays of arrays and I couldn't do it.
{
"name": "SelfHealingStarter",
"namespace": "SCP.Kafka.AvroSchemas",
"doc": "Message with all the necessary information to run Self Healing process.",
"type": "record",
"fields": [
{
"name": "FiveMinutesAgoMeasurement",
"type": "record",
"doc": "Field with all five minutes ago measurement.",
"fields": [
{
"name": "equipments",
"doc": "List with all equipments measurement.",
"type": {
"type": "array",
"items": {
"type": {
"type": "array",
"items": "string"
},
"default": []
}
},
"default": []
}
]
}
]
}
A: IDL
protocol Example {
record Foo {
array<array<string>> data = [];
}
}
AVSC from java -jar ~/workspace/avro-tools-1.8.2.jar idl2schemata example.idl
{
"type" : "record",
"name" : "Foo",
"fields" : [ {
"name" : "data",
"type" : {
"type" : "array",
"items" : {
"type" : "array",
"items" : "string"
}
},
"default" : [ ]
} ]
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,333 |
{"url":"http:\/\/math.stackexchange.com\/tags\/topological-vector-spaces\/new","text":"Tag Info\n\nNew answers tagged topological-vector-spaces\n\n0\n\nNo, it is not first countable as it is not metrizable. Some arguments are mentioned here, observing that $C_b(Compact)$ has an uncountable Hamel basis.\n\n0\n\nI think I found the answer.It's a modification of the proof in \"Introduction to Tensor Products of Banach spaces, R.A.Ryan\" Prop4.6. Suppose $x^{**}$ is an element of the dual of $W^*$ endowed with the given topology. It's easy to see that there exsist a compact subset $K\\subset X$, s.t. $$|x^{**}(f)|\\leq \\sup_{x\\in K}|f(x)|,\\forall f\\in W^*.$$ Use that ...\n\n1\n\nSuppose $p \\notin S$, and let $v_1,...v_n$ span $S$. Define the functional $f$ on the space spanned by $p,v_1,...v_n$ by $f(v_k) = 0$, $f(p) = 1$. Use Hahn Banach to extend this functional to the whole space. Let $U = \\{ x | f(x) > {1 \\over 2} \\}$ which is open, and since $f(s) = 0$ when $s \\in S$, we see that $S \\cap U = \\emptyset$, hence $S^c$ is open. ...\n\n1\n\nThe starting idea to take a sequence $x_n$ in $S$ that converges to some $x$ and see whether $x\\in S$ is OK, the rest is nonsense, e.g. nobody said that $x=0$ neither that $x_i^{(n)}=0$, or $S$ may have more than countably infinite dimensions, whence the sequence setting $x=(x_1,x_2,\\dots)$ is not that much justified. Probably the fastest argument to ...\n\n3\n\nThe Answer to question 1 is yes: Let $V = \\mathbb R^n$ and $I = \\mathbb N$. It is clear that $\\operatorname{conv}(\\bigcap_{i \\in \\mathbb N} C_i) \\subset \\operatorname{conv}(C_j)$ for every $j \\in \\mathbb N$ since $\\bigcap_{i \\in \\mathbb N} C_i \\subset C_j$ for every $j \\in \\mathbb N$. Thus $\\operatorname{conv}(\\bigcap_{i \\in \\mathbb N} C_i) \\subset ... 2 In general, if$X$is a locally convex topological vector space of uncountable dimension (as a linear space), then the weak$^*$topology on$X^*$is not first countable. Proof. In the weak$^*$topology a sub-base of the neighborhoods of$0$is obtained by sets of the form$$W_{x,\\varepsilon}=\\{x^*\\in X^*: |x^*(x)|<\\varepsilon\\}, \\quad ... 1 You seem to be slighly confuzed when you are writing about the vectors. You say that$j$can be derived from$i$by the process of differentiation, which is not true. In fact,$i$is a constant vector, not a function, you can only have the differential of a function. As far as the definition of independency goes, you have$2$equivalent versions: The ... 0 The dual space is a concept that shows up in greater generality in Linear Algebra. If$V$is a vector space then its dual space$V^*$is the set of all linear functionals from$V$to its base field$F$. When$V$is a finite dimensional vector space, then so is$V^*$and$\\text{dim } V^*=\\text{dim }V$. Thus there is a linear isomorphism between them when ... 2 More generally, if$X$and$Y$are sets (resp. vector spaces over a common field$k$), we can consider the set$Y^X$of all mappings (resp. the vector space$\\mathrm{Hom}(X,Y)$of all linear mappings) from$X$to$Y$. These are obviously interesting to study if you think in terms of structures and structure preserving maps; and dual spaces are simply a ... 0 There is a textbook by Halmos, Measure Theory, in which he does as much as possible in the setting of$\\sigma$-rings, where complements are not assumed. But subsequent mathematicians have not adopted that point of view. (1) In the Halmos definitions, we can postulate that every measurable set is$\\sigma$-finite; then when (rarely) it is necessary, extend ... 1 While not an answer per se, here is an example to get you thinking about why you might want more than countable disjoint unions. Consider the space$[0,\\infty)\\subset \\mathbb R$, and take as your \"algebra\" the sets of the form$[0,x)\\$. The only step functions on this space contain only one step, and the sum of two step functions will not in general be ...\n\nTop 50 recent answers are included","date":"2014-03-07 07:21:07","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9546796679496765, \"perplexity\": 254.96164558370197}, \"config\": {\"markdown_headings\": false, \"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-2014-10\/segments\/1393999636575\/warc\/CC-MAIN-20140305060716-00078-ip-10-183-142-35.ec2.internal.warc.gz\"}"} | null | null |
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